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Universitäts-Sternwarte München,Ludwig-Maximilians-Universität, Scheinerstr. 1, 81679 München, [email protected] Department of Astronomy & Astrophysics,Pennsylvania State University, University Park PA 16802, USAChandra X-ray observation of NGC 3293 Preibisch et al. NGC 3293 is a young stellar cluster at the northwestern periphery of the Carina Nebula Complex that has remained poorly explored until now.Wecharacterize the stellar population of NGC 3293 in order to evaluate key parameters of the cluster population such asthe age and the mass function, and to test claims of an abnormal IMF and a deficit of M ≤ 2.5 M_⊙ stars. We performed a deep (70 ksec) X-ray observation of NGC 3293 withChandraand detected 1026 individual X-ray point sources.These X-ray data directly probe the low-mass (M ≤ 2 M_⊙)stellar population by means of the strong X-ray emission of young low-mass stars. We identify counterparts for 74% of the X-ray sources in our deep near-infrared images. Our data clearly show that NGC 3293 hosts a large population of ≈ solar-mass stars, refuting claims of a lack of M ≤ 2.5 M_⊙ stars. The analysis of the color magnitude diagram suggests an age of ∼ 8-10 Myr for the low-mass population of the cluster.There are at least 511 X-ray detected stars with color magnitude positions that are consistent with young stellar members within 7 arcmin of the cluster center. The number ratio of X-ray detected stars in the [1-2] M_⊙ range versus the M ≥ 5 M_⊙ stars (known from optical spectroscopy) isconsistent with the expectation from a normal field initial mass function. Most of the early B-type stars and ≈ 20% of the later B-type stars are detected as X-ray sources.Our data shows that NGC 3293 is one of the most populous stellar clusters in the entire Carina Nebula Complex (very similar toTr 16 and Tr 15;onlyTr 14 is more populous). The clusterprobably harbored several O-type stars, whose supernova explosionsmay have had an important impact on the early evolution of the Carina Nebula Complex. ChandraThe Chandra data described in this paper have been obtained in the open time project with ObsID 16648 (PI: T. Preibisch) ivo://ADS/Sa.CXO#obs/16648.X-ray observation of the young stellar cluster NGC 3293 in theCarina Nebula ComplexTables 1, 2, and 3 are only available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/ T. Preibisch1 S. Flaischlen1 B. Gaczkowski1 L. Townsley2 P. Broos2 Received March 27, 2017; accepted July 11, 2017 =================================================================================================================================================================================================================================================================================================================================================================================================================================== § INTRODUCTION The Carina Nebula Complex <cit.> is one of the most massive and active star forming regions in our Galaxy. At a moderate and well-known distance of 2.3 kpc <cit.>,the spatial extent of the nebulosity of about 100 pc corresponds to several degrees on the sky. An optical image of the CNC is shown inFig. <ref>. The clouds contain a total gas and dust mass of about 10^6 M_⊙ <cit.> and harbor more than 100 000 young stars<cit.>. Most of these young stars are located in one of several clusters, including the extensively studied clusters Tr 14, 15, and 16 in the central regions of the Carina Nebula, and the clusters NGC 3324 and NGC 3293 in the northern part of the CNC. Over the last years, several sensitive surveys of large parts of the CNC have been performed at optical and infrared wavelengths<cit.>. However, due to the location of theCNC very close to the Galactic plane, all optical and infrared images are completely dominated by unrelated field stars in the galactic background, which presents a major obstacle in the identification of the young stellar population in the complex. X-ray observations provide a very good way to solve this problem because strong X-ray emission is a very good tracer of stellar youth and allows veryyoung stars to be efficiently distinguished from the numerous (and much older) field stars in the galactic background <cit.>. A major milestone in the exploration of the CNC was therefore the deep X-ray imaging survey of the Chandra Carina Complex Project <cit.>, which mapped the central 1.4 square degrees(i.e., roughly half of the total spatial extent of the CNC) with a mosaic of 22 individualpointings with the Imaging Array of the Chandra Advanced CCD Imaging Spectrometer <cit.>. The Chandra data revealed 14 368 individual X-ray sources, 10 714 of which are most likely young stars in the Carina Nebula <cit.> with masses down to∼ 0.5 M_⊙.The combination of these CCCP X-ray data with a deep near-infrared (NIR)survey <cit.>provides information about the properties of the stellar populations in the central parts of the complex <cit.>, including the individual stellar clusters Tr 16 and Tr 15 <cit.>. The northern parts of the cloud complex, including the young stellar clusters NGC 3324 and NGC 3293, were much less well exploredbecause this area was not included in the CCCP and, until recently,no deep infrared survey was available. The new very wide-field (6.7 square degrees) NIR survey with the ESO 4m VISTA telescope <cit.> finally solved this problem; these data are sensitive enough to detect the full stellar population of young stars down to ∼ 0.1 M_⊙ over the entire area of the CNC, including the lessstudied northern parts. As a first step in the X-ray exploration of the young stellar populations in these northern parts of the CNC (outside the field covered by the CCCP), werecently obtained a Chandra ACIS-I pointing of the stellar cluster NGC 3324 in the prominent HII region Gum 31 <cit.>, which led to the detection of 679 new X-ray sources.The only cluster in the CNC that had not yet been observed in X-rayswas NGC 3293 (also known as the “Gem Cluster”). Although thiscluster is a very prominent celestial object (see Fig. <ref>), it was often neglected in studies of the CNC because of its angular separation of ∼ 1.5 (70 pc) from the center of the Carina Nebula.Recent distance estimates for NGC 3293 (based on optical photometry) of ≈ 2327 pc <cit.> and ≈ 2471 pc <cit.> are very consistent (within their uncertainties) with the distance of the Carina Nebula (2.3 kpc).The extinction for most stars in the cluster center is quite low (A_V1 mag), but increases up to several magnitudes for stars more than a few arcmin away from the cluster center, due tothe interstellar clouds around the cluster (see Appendix<ref>). NGC 3293 contains an impressive collection of bright stars, including 48 early B-type stars<cit.> and several supergiants, e.g., HD 91969 (B0Ib) and V361 Car (M1.5Iab). The presence of evolved massive stars, main-sequence stars up to spectral type B0.5 V, andcomparisons of optical color magnitude diagrams with stellarevolution modelssuggested an age of ≈ 8-10 Myr for the high-mass population inthis cluster <cit.>. Another reason that makes NGC 3293 particularly interesting are claims for an abnormal stellar initial mass function (IMF) and an apparent severe lack of low-mass stars.In their analysis of optical photometry, <cit.> came to the conclusion that the cluster's IMF shows a sharp turnover at masses below 2.4 M_⊙. They argue that NGC 3293 is the cluster with the most convincing evidence for a strong deficit of low-mass stars. <cit.> also arrived at a similar conclusion from their optical photometry study,which suggested that the mass function is considerably flatter than the Salpeter slope below 2.5 M_⊙.These results, however,as well as all the other studies of the low-mass population of NGC 3293 performed so far, were only based on photometric observations. In the analysis of color magnitude diagrams of stellar clusters, the statistical background subtraction can introduce a very serious source of uncertainties, in particular at locations close to the galactic plane.Much more reliable information about the cluster propertiesand mass function can be obtained if the low-mass stellar members of the cluster can be individually identified. The main obstacle for a reliable identification of thelow-mass stellar population of NGC 3293 is the confusion by the very strong field star contamination resulting from the cluster's location very close to the galactic plane: the cluster center has agalactic latitude of b = +0.08^∘. All optical and infrared images are therefore strongly contaminated by unrelated field stars.Theapproach most often usedto identify young stars is by the infrared excess emission from circumstellar disks, whichis not feasible here.At an age of ∼ 8 Myr, most stars should have already lost their circumstellar accretion disks and thus do not exhibit infrared excesses <cit.>.It is thus essentially impossible to identify and distinguish a population of ∼ 8-10 Myr old low-mass stars from unrelated field stars with optical or infrared photometry alone.Chandra, however, cansolve this problem becausethe strongly enhanced X-ray emission of young stars <cit.> provides an extremely useful way to discriminatebetween young pre-main-sequence stars and the much older field stars. The median X-ray luminosity of ∼ 10 Myr old solar-mass stars is nearly 1000 times higher than for solar-mass field stars (see Preibisch & Feigelson 2005), and makes these young stars relatively easy targets todetect for Chandra even at a distance of 2.3 kpc. A Chandra observation of NGC 3293 can uncover an X-ray luminosity-limited (i.e., approximately mass-limited) sample of the low-mass stellar population, provide a comprehensive census of the cluster members, and immediately answer the open questions about the size of the low-mass stellar population (and thus the total cluster mass). § CHANDRA X-RAY OBSERVATION AND DATA ANALYSIS Weused the Chandra X-ray observatory <cit.> to perform a deep pointing of the cluster NGC 3293 with the Imaging Array of the Chandra Advanced CCD Imaging Spectrometer <cit.>. The observation was performed as an open time projectwithObsID 16648 (PI: T. Preibisch) during Chandra Observing Cycle 15 inOctober 2015 (start date: 2015-10-07T10:14:23,end date: 2015-10-08T06:43:28).The imaging array ACIS-I provides a field of viewof 17' × 17' on the sky (which corresponds to11.3 × 11.3 pc at the distance of 2.3 kpc),and has a pixel size of 0.492”. The point spread function of the X-ray telescope has a FWHM of 0.5” on-axis, but increases towards the edge of the detector. As part of our data analysis, theChandra data were astrometrically registered to 2MASS to improve the absolute astrometry. The positions of bright X-ray sources can thus usually be determined with subarcsecond precision.The observation was performed in the standard “Timed Event, Very Faint” mode with 5 × 5 pixel event islands, and the total effective exposure time (“livetime”) of the observation was 70 870 s (19.68 hours). In addition to ACIS-I, one CCD detector (CCD 7 = S3)of the spectroscopic array ACIS-S was also operational during our pointing. It covers a8.3' × 8.3' area on the skysouthwestof the cluster center. While the ACIS-I chips are front-illuminated, the S3 chip is back-illuminated, and thus its response extends to energies below that accessible by the FI chips. This causes a substantially higher level of background in the S3 chip. Furthermore, thepoint spread function isseriously degraded at the rather large off-axis angles of the S3 chip.These two effects lead to a considerably higher detection limit for point sources in the area covered by the S3 chip compared to the region covered by the ACIS-I array. Nevertheless, the S3 data were included in our data analysisand source detection, and contributed four point sources to the total source list. The aimpoint of the observation wasα ( J2000) = 10^ h 35^ m 50.07^ s, δ ( J2000) = -58 14' 00”, which is close to the optical center of the cluster (see Fig. 1). The pointing roll angle (i.e., the orientation of the detector with respect to the celestialnorth direction) was 140.19. The resulting X-ray image is shown in Fig. <ref>. Clearly hundreds of X-ray sources are detected, with the greatest concentrationat the cluster center.The gaps between the four ACIS-I CCDs are discernibleas a light cross bisecting the image in two directions; the effective exposure timeis reduced in these gaps.Figure <ref> shows a RGB color composite version of the Chandra image with color-coding corresponding to photon energy. X-ray sources can appear harder (bluer) because of their intrinsic spectral shapeor because they suffer high absorption. At the distance of 2.3 kpc, the expected ACIS point source sensitivity limit for a three-count detection on-axis in a 71 ks observation is L_ X,min≃10^29.7 erg s^-1 in the [0.5 - 8] keV band, assuming an extinction ofA_V ≈ 1 mag (N_ H≈ 2 × 10^21cm^-2) as typical for the stars in the central region of NGC 3293, and a thermal plasma with kT = 1 keV <cit.>. Using theempirical relation between X-ray luminosity and stellar mass and the temporal evolution of X-ray luminosity from thesample of young stars in the Orion Nebula Cluster, which was very well studied in the Chandra OrionUltradeep Project <cit.>, we can expect to detect ≈ 90% of the ≈ solar-mass stars in the central region of the young cluster NGC 3293.§.§ X-ray point-source detection and extraction with ACIS ExtractSource detection and analysis followed the steps describedin <cit.> (hereafterB10);weprovide a brief summary of this procedure here, but interested readers are encouraged to review the details in B10,especially the caveats and warnings regarding the detection of faint X-ray sources. Chandra-ACIS event data were calibrated and cleaned as describedin Section 3 of B10. Candidate point sources were then identified in the pointing usingLucy-Richardson image reconstructions<cit.> of small overlappingimages that “tile” the field in three energy bands (0.5–7, 0.5–2,2–7 keV). These candidate sources were then extracted using the most recent and improved version of the ACIS Extract (hereafter AE) softwarepackage[The ACIS Extract software package and User's Guide are availableat http://www.astro.psu.edu/xray/acis/acis_analysis.html.] <cit.>, which also uses the CIAO and MARX software <cit.>.For each source, AE calculated the probabilities that the counts extracted in each ofthree energy bands arose solely from the local background. When all three probabilities were greater than 0.01 or when less than three X-ray countswere extracted, the candidate source was judged to be not significant,and removed from the list. The positions of surviving source candidates were updated with AE estimates, and thereduced list of candidates was re-extracted. This cycle of extraction, pruning, and position estimation was repeated until no candidateswere found to be insignificant. Our final X-ray catalog contains 1026 individual point sources,1022 of which are located in the field of view of the ACIS-I array, and 4 in the field covered by the single active ACIS-S chip.The final list of X-ray sources with their properties is given in the electronic Table 1 (available at the CDS). Sources are identified by their sequence number (Col. 1) or their IAU designation (Col. 2). Following the rules for the designation of sources found with the Chandra X-ray Observatory, we have registered the acronym CXONGC3293 as the prefix for the IAU designation (Col. 2). The choice of the source significance limit (denoted“ProbNoSrc_ min”in the text below and listed as P_B in Table 1) for the definition of the final sample of X-ray sources is always a compromise. A strict limit will miss true X-ray sources, whereas a more generous limit will enhance the risk of including spurious sources into the sample. The limiting value we use here (ProbNoSrc_ min = 0.01) tends to be on the generous side and allows the detection ofsources with just ∼ 3 counts. We note that the same valuehas been used in numerous similar Chandra studies ofyoung stellar clusters <cit.>, and in particular also in the CCCP <cit.>.In order to keep track of the possible effects of potentially spurious sources on the analysis of the cluster properties in Sect. 4, we defined two subsamples of our source catalog. The first group comprises the very significant sources, defined ashaving a low probability of being a spurious source, ProbNoSrc_ min in Table 1 less than 0.003. This group contains 849 of the 1022 ACIS-I sources, which werefer to as “highly reliable” X-ray sources in the following text. The second group comprises the complementary set of 173less significant ACIS-I sources(0.003 <ProbNoSrc_ min< 0.01), which werefer to as “less reliable” X-ray sources in the following text. In the analysis in the following sections wecheck how the results change when we include or exclude the subsample of the less reliable X-ray sources. As shown there, we find that our derived quantitativeresults about the cluster properties are not significantly dependent on whether the less reliable X-ray sources are included or excluded.§.§ X-ray variability As part of the AE procedure, the time variability of each X-ray sourceis investigatedby comparing the arrival times of the individual source photons in each extraction region to a model assuming temporal uniform count rates. The statistical significance for variabilityis then computed with a one-sided Kolmogorov-Smirnov statistic (Col. 15 of Table 1). In our sample,29 sources show significant X-ray variability (probability of being constant P_ KS <0.005) and an additional 71 sourcesare classified as possibly variable(0.005 <P_ KS < 0.05).The light curves of the variable X-ray sources show a variety of temporal behaviors. Some sources show flare-like variability, i.e., a fast increase in the count rate followed by a slow exponential decay,which is typical of solar-like magnetic reconnection flares <cit.>. Other variable sources show more slowly increasing or decreasing count rates, as isoften found for young stellar objects <cit.>.§.§ X-ray spectral analysis for bright sourcesModeling the observed X-ray spectrum provides the best wayto estimate the intrinsic X-ray luminosity of a source, but it requires a sufficiently large number of source counts. We restricted thespectral fitting analysis to57 sourcesdetected withS/N > 4.XSPEC version 12.9.0i was used for this analysis. The background-subtracted spectra weregrouped into bins of constant signal-to-noise ratio, and the fits were then performed using the χ^2 statistic.Considering the relatively small number of counts in our spectra, it is advisable to keep the number of free parameters as low as possible. We therefore only used one-temperature models[We tried spectral models with more than one temperature component for the brightest sources, but found that such more complicated models were not a statistical improvement on the one-temperature fits.], where the thermal plasma was described with the VAPEC model, and theTBABS model described the effect of extinction by interstellar and/or circumstellar material (as parameterized by the hydrogen column density N_ H). The elemental abundances for the plasma model were fixed at the values[The adopted abundances, relative to the solar photospheric abundances given by <cit.>, are: C = 0.45, N = 0.788, O = 0.426, Ne = 0.832, Mg = 0.263, Al = 0.5,Si = 0.309, S = 0.417, Ar = 0.55, Ca = 0.195, Fe = 0.195, Ni = 0.195.] that were found by <cit.> to be typical for young stellar objects. In our fitting, the N_ H parameter was allowed to vary between 10^19cm^-2and 10^23cm^-2 andthe plasma temperature kT between 0.1and 10 keV. Since the parameter space of these fits can have a complicated topology rather than a clearly defined, unique minimum,we performed at least six fits for each source with different values of the starting parameters. The grid of starting parameter values is spanned by the vectors N_ H = [10^21, 10^22]cm^-2 and kT = [0.3, 1.0, 3.0] keV.From the resulting fits, the model with the overallhighest value of the null hypothesis probability was then selected as the global best fit. From the spectral fits we computedthe intrinsic (i.e., extinction corrected) X-ray fluxesF_X,tc for the total band (0.5-8 keV), as derived from the spectral fit parameters, and the corresponding X-ray luminosities L_X,tc, assuming a source distance of 2.3 kpc. The derivedplasma temperatures range from ≈ 3 MK up to ≈ 85 MK (for a star that exhibited a strong X-rayflare during our observation). The X-ray luminosities derived for the X-ray sources withinfrared counterparts are in the typical ranges found for young stars in other star forming regions <cit.>.Owing to the moderate number of counts in most spectra, the uncertainties of the model parameters are often rather high. For many objects, the fitting analysis did not yield a clear unique best-fit model; two (or more) regions in the N_ H - kT parameter plane yielded very similar values for the best χ^2. This ambiguity in the best-fit parameters can lead tosubstantial uncertainties in the derived X-ray luminosities. Furthermore, several fits hit the lower N_ H limit, suggesting that the extinction is small, but leaving the value of N_ H(and thus the un-absorbed X-ray luminosity) poorly constrained. We therefore decided to use the spectral fitting results only for those few sources for which reasonably well-defined best-fit parameters could be determined. These cases are described individually in the text below.We note that the brightest ACIS-I X-ray source (number 943; 190 net counts) is a special case: it has no optical counterpart, only an extremely faint infrared counterpart, and a very hard X-ray spectrum that can be well fit by a power-law model. As described in more detail in Appendix <ref>,these properties suggest it to be an extragalactic source, most likely an obscured quasar.§.§ X-ray fluxes for fainter sources Since the large majority of the detected X-ray sources istoo faint for meaningful spectral fitting analysis, we need another way to estimate the intrinsic X-ray luminosities.In the course of the AE analysis,an estimate of the incident energy flux onto the telescope is calculated for each source. However, these flux values are known to be biased because their computation is based on the (non-physical) assumption of a flat incident spectrum, which is not correct for most X-ray sources.For the X-ray sources that are identified with cluster stars, we can improve the accuracy of the flux estimate by usingthe available a priori information on the typical shape of X-ray spectra of young stars. We can also use the available constraints on the extinction (individual extinction estimates available for the B-type stars, or an empirical mean value for the extinction) and estimate not only the apparent, but alsothe intrinsic (i.e., absorption corrected) X-ray fluxes and thus the X-ray luminosities of these stars. For this, we employed thetool[see] in the Chandra Interactive Analysis of Observations (CIAO) software package. This tool determines the conversion factor from count rate toapparent as well as unabsorbed energy flux for a given spectral model, creating ARFs and RMFs for each source. It is necessary to specify a model for the X-ray spectrum and for the X-ray absorption caused by the ISM along the line of sight.We used a thermal plasma spectrumwith a plasma temperature of kT = 1 keV, which is a typical value for young coronally active stars <cit.>. For the absorption we used the model xsphabs and set the column density to N_H = 0.15 × 10^22 cm^-2; this corresponds to a visual extinction ofA_V ≈ 0.75 mag, which is the mean value of the individual extinctions determined for the bright cluster stars. § INFRARED AND OPTICAL COUNTERPARTS OF THE X-RAY SOURCES The identification of optical/infrared (OIR) counterparts of theX-ray sources isrequired in order to obtain essential information about the nature and properties of the X-ray emitting objects, in particular to discern between stellar X-ray sourcesand (mostly extragalactic) contaminators. The number of expected extragalactic contaminators can be estimated by comparing the point-source sensitivity of our ACIS-I image to the cumulative number counts of AGN found in the Chandra Deep Field South project <cit.>.This suggests≈ 120 - 140 AGN among the detected X-ray sources. X-ray source 943, described in Appendix <ref>, seems to be the X-ray brightest of these extragalacticX-ray sources. A large fraction of the OIR counterparts of theseextragalactic X-ray sources are expected to be too faint to be detected in our OIR images. Stellar X-ray sources, on the other hand, should have relatively bright OIR counterparts.The expected optical and infraredmagnitudes of young low-mass stars of NGC 3293can be easily estimated.Assumingan age of about 10 Myr and a typical extinction of A_V ≈ 1 mag, stars with masses of [ 1.0 ,0.5 ,0.1 ] M_⊙ are predicted to have visual magnitudes of V ≈ [ 19.0 ,20.8 ,25.9 ] and NIR magnitudes of H ≈ [ 16.0 ,16.4 ,18.3 ] according to the pre-main-sequence stellar models of <cit.>. These numbers show that typical optical images are not sensitive enough to detect the full low-mass population in NGC 3293. In the NIR regime, these low-mass stars are much easier to detect, but nevertheless too faint to be contained in all-sky catalogs such as the 2MASS data.§.§ VISTA near-infrared images and point source catalog As the main source for the identification and infrared characterization of the X-ray sources we used the images and the point source photometric catalog from our VISTA Carina Nebula Survey (VCNS), which isdescribed in detail in <cit.>. These VISTA data provide subarcsecond resolution and5 σpoint-source sensitivities to [J, H, K_s] ≈ [20, 19.5, 18.5] mag.Comparing these magnitude limits to the above mentioned expected magnitudes of low-mass stars in NGC 3293 shows that the VISTA data are clearly deep enough to detect all low-mass members of the cluster, i.e., down to 0.1 M_⊙, and even through several magnitudes of visual extinction. There are 35 143 VISTA catalog sources with valid photometry in all threeNIRbands in the ACIS-I field of view.Five extremely bright stars in the cluster are so heavily saturated in the VISTA images that they were not recognized as point sources in the VISTA data processing; for these five stars (withmagnitudes between J = 3.5and J = 8.5), positions and magnitudes from the 2MASS point source catalog were used. Like any large-scale catalog, the VISTA catalog is not 100% perfect and complete.As described in Sect 3.2 of <cit.>,some point-like objects, which are clearly visible in the VISTA images, are missing from the catalog. This problem is mostly restricted to areasclose to very bright stars where the wings of the point spread function and numerous diffraction spikes produce a bright and spatially variablebackground pattern[The ∼ 1 area around the star V361 Car in the VISTA image shown in Fig. <ref> provides a good example of this effect.], and also concerns close companions to some stars. This sample of additional infrared sources is considered separately after the matching with the VISTA catalog as described belowin Sect. <ref>. §.§ Optical images and catalog from VST We used optical images of NGC 3293 that were obtained with Omegacam at the ESO 2.6 m VLT Survey Telescope <cit.> in the u-, g-, r-, i-SDSS filters and an Hα filter as part of the VPHAS+ survey <cit.>. The 5σ detection limits in these images are approximately u=20.5,g=22.5,r=21.4, andi=20.5 (all magnitudes in the Vega system).The ESO archive contains the fully reduced, calibrated,and verified Phase 3 data products from this survey. We downloaded all reduced images covering NGC 3293, and also retrieved the VPHAS-DR2 Point Source Catalogue data for this area, which provide band-merged PSF and aperture photometryfor the point-like objects (“primary sources”in the VPHAS+ terminology). 8581 of these VST primary sources are in the ACIS-I field of view. The g-band magnitudes of these stars range from g ≈ 12.6 (close to the saturation limit) down to g ∼ 23.Comparing this faint magnitude limit to the above mentioned expected magnitudes of low-mass stars in NGC 3293 shows that the VST data should detect most lightly absorbed (A_V1 mag) low-mass membersof the cluster down to 0.5 M_⊙, but will miss many of the lower-mass members and also stars with visual extinction of more than about one visual magnitude. The B-type and most A-typestars in NGC 3293 are also missing from the catalog, due to the saturation limit. For these bright stars, optical photometry is available from <cit.>and <cit.>.We note that the VST catalog suffers from a similar incompletenessto that ofour VISTA catalog, in the sense thatsome point-like objects that areclearly visible in the images are missing from the catalog. Another limitation of the VST images is that many of the very bright stars in the cluster center are strongly saturated in the CCD images. The bleeding streaks caused by these saturated stars extend up to 1.5 arcminin two directions from each saturated source, and have widths up to about 20 pixels. In the central ∼ 10 square arcmin region,these bleeding streaks cover a significant fraction of the image area. Faint sources may be completely hidden behind these bleeding streaks.A visual inspection of the VST g-band image showed that at least 12 X-ray sources are located at positions within or just at the edge of such ableeding streak or other image artifacts, which prevents the identification of optical counterparts.No attempt was made to construct a list of point sources missing from the VST catalog; the matching analysis was restricted to theVPHAS catalog sources only. §.§ Further optical photometry and spectral data Our master catalog (based on the VISTA point source catalog) was complemented by theoptical (UBVRI Hα) photometry catalog for NGC 3293 from <cit.>, which provides V-band magnitudes (down to V ∼ 20) for 1690 stars in the cluster and optical colors for a subset of them.We also added information available in the literature about the spectraland other stellar parameters of stars in NGC 3293 into our catalog. Most of these data originate from theVLT-FLAMES multi-object spectroscopic survey of massive stars in NGC 3293 <cit.>,which provided spectral types for 131 stars and luminosities for 92 of these. In the ACIS-I field of view,our final catalog lists 97 B-type stars, 17 A-type stars, 2 F-type stars, and 1 M-type star (the M1.5 Iab supergiant V361 Car).§.§ Matching of the X-ray sources with the optical/infrared catalogsFigure <ref> shows a NIR image of the central part ofthe cluster with the polygons of the X-ray source extraction regions overlaid. The high density of point-like sources in the NIR image (which is largely due to the strong galactic background at the cluster's position very close to the galactic plane) shows that a proper identification of the counterparts to our X-ray sources requires a careful and conservative matching procedure in order to avoid false matches with unrelated background objects.The identification of infrared and optical counterparts tothe X-ray sources was performed with a two-step procedure: an automatic source matching based on catalog coordinates, followed by a detailed individual inspection in order to resolve problematic cases such as multiple possible matches or objects missing from the optical/NIR catalogs. For the first step, we employed the IDL tool[see www2.astro.psu.edu/xray/docs/TARA/TARA_users_guide/node11.html] match_xy, as described in <cit.>. The maximum acceptable matching separation between an X-ray source anda counterpartwas based on the individual source position errorsassuming Gaussiandistributions, scaled so that ∼ 99% of the true associationsshould be identified as matches.For the X-ray sources, the positional uncertainties were determined by AE, and for the VISTA and VST sources we assumed position uncertainties of 0.1”. The algorithm in match_xy lists the most significant match of each X-ray source as its“Primary Match”, whileany other possible significant matches, if present, are considered“Secondary Matches”.In a second stage, the algorithm resolves possiblemany-to-one and one-to-many relationships between the X-rayand infrared/optical catalogs. Clear one-to-one relationships are classified as “successful primary matches”, while in cases where, e.g., two X-ray sources are significantly close to a single infrared source, the less significant primary matchis labeled as “failed”. This finally provided a reasonable one-to-one set of matches, consisting of 695 successful primary matches to VISTA catalog sources in the ACIS-I field. For 42 X-ray sources, one or more successful secondary matches were identified by match_xy in the VISTA catalog.These cases require further inspection since it is not guaranteed that the closest match is always the correct physical counterpart, due to the high surface density of infrared sources in our deep VISTA images. Instead, a physically unrelated (e.g., background) infrared source may appear in the matching region just by chance and produce a “false match”, and might even degrade thetrue infrared counterpart to a secondary match.A priori information about the typical infrared properties of X-ray detected young stars can be used to identifypossible cases where this problem occurs. Since we observe a young stellar cluster andknow from theX-ray detection limit that most of the X-ray detected objects should be young stars withmasses of 0.5 M_⊙, these stars should typically be relatively bright NIR sources. Therefore, all cases where an X-ray source has an unexpectedly faintprimary match and a considerably brighter secondary match (i.e., with the magnitudesexpected for a young star) deserve special attention. We found eight such cases, andreplaced the original faint primary match by the considerablybrighter secondary match.As the last step in the source matching procedure, we finally considered the above mentioned list of point-like sourcesidentified in our visual inspection of the VISTA images but missing from theVISTA source catalog. The locations of these additional VISTA sources were compared to theX-ray source locations for which match_xy did not report a match with a VISTA catalog star. In this way, 61 additional infrared point source matches to Chandra sources were identified. We note that most of these 61 additional infrared sources have discernible counterparts in the optical VST images, and 20 of them are even listed as point sources in the VST catalog, demonstrating the reality of these infrared sources.Since most of these objects are affected by a spatially variable background or by partial blending with other nearby sources in the VISTA images, measurements of their photometrycan suffer from relatively large uncertainties. In order to quantify these uncertainties, we performed aperture photometric measurements in the VISTA images, employingdifferent values for the sizes of the background regions, and determined the scatter of the resulting aperture flux measurements for each source. For 21 of these sources, the relative scatter of these flux measurements exceeded 20%; these sources were completely excluded from all further analysis steps. For the 40 sources for which the relative scatter of the flux measurements was less than 20%, we determined their magnitudes, but these sources were not included in the sample of VISTA catalog sources; they are consideredin some of the analysis steps below, but are always clearly kept separate from the VISTA catalog sources. There is thus no danger that these additional sources will contaminate our results.In the matching with the VST optical catalog, we considered all sources that are listed as primary source in the VPHAS-DR2 catalog. The matching with the VST catalog resulted in 496 successful primary matches in the ACIS-I field.In the course of our visual inspection of the X-ray source positions in the VST images, we found one case where an image artifact that was erroneously listed as point source in the VPHAS-DR2 catalog coincided with an X-ray source. More details about this artifact are given in Appendix <ref>.The final results of the matching procedure can be summarized as follows: 756,i.e., 74.0 % of all X-ray sources in the ACIS-I field,have a match with a VISTA source (695 of them, i.e., 68.0%, with aVISTA catalog source). For the VST catalog, we have 491 matches of VST catalog primary sourceto ACIS-I X-ray sources after removing the above mentioned artifact, and four cases where the VST match was not identical to the VISTA match; this yields amatch fraction of 48.0%.The lower rate of counterparts in the VST catalog is easily understood as a consequence of the different sensitivity limits ofthe two catalogs. While the VISTA catalog is deep enough to detect 0.1 M_⊙ stars in NGC 3293 with extinctions up to A_V ≃ 5 mag, the sensitivity limit of the VST catalogcorresponds to 1 M_⊙ stars with extinctions up to A_V ≃ 3.5 mag, or 0.5 M_⊙ stars with extinctions up to A_V ≃ 2 mag. Therefore, most of the <0.5 M_⊙ stars and a substantial fraction of the [0.5-1] M_⊙ stars remain undetected and will be missing from the VST catalog.Information about the VISTA and VST matches to theindividual X-ray sourcesis contained in the electronic Table 2, available at the CDS.Finally, we can compare the fraction of IR matches for the two above-mentioned subsamples of our X-ray catalog. In the sample of 849 highly reliable ACIS-I sources, we find 593 objects (i.e., 69.8 ± 0.2%) that have a match with a VISTA catalog source. In the less reliable sample, 102 of the 173 ACIS-I sources (i.e., 59.0 ± 3.7%)have a VISTA catalog match.The lower match fraction for the less reliable subsampleis partly due to a higher fraction of spurious sources among the less reliable sources. It should be noted, however,that X-ray sourcesdetected with lower significancegenerally have weaker X-ray fluxes. Sincethere is some correlation between X-ray flux and infrared brightness for most classes of X-ray emitting objects,the less significant true X-ray sources are also less likely to have an infrared counterpart that is detected in the available infrared images. §.§ Reliability of X-ray sources and their infrared matchesIn order to determine the reliability of the following analysis, we briefly discuss a few fundamental aspects of the sample of X-ray sources and their OIR matches. In general, any X-ray source list is composed of real sources and spurious sources. Concerning the identification of OIR matches to the X-ray sources, it is necessary to take into account that brighter sources are usually detected with higher significance, i.e., there is a correlation between X-ray source brightness and significance. There is also a correlation between X-ray flux and OIR brightness for most classes of X-ray emitting objects; therefore, the less significant true X-ray sources are, on average, fainter in the OIR range and therefore also less likely to have a counterpart that is detected in the available OIR images.For the OIR counterparts of the real X-ray sources there are two possibilities: some of the X-ray sources can be expected to have a counterpart bright enough to be detected in the available OIR images (in the present study, this concerns all young stars in NGC 3293 inthe deep VISTA images, and the 1 M_⊙ stars in the VST images). Another fraction of the X-ray sources will be extragalactic objects(mainly AGN), most of which should be very faint at OIR wavelengths and thus remain undetected in the available OIR images.If we now consider the possible results of the X-ray to OIR source matching, we have to distinguish four possibilities for the real X-ray sources: (a) correct positive matches, when the physical association between an X-ray source and the corresponding OIR source is correctly identified; (b) correct negative matches, when an X-ray source is not matched to any OIR catalog source becausethe OIR counterpart is too faint tobe detected in the OIR images; (c) false negative matches, when the true OIR counterpart is notconsidereda match, e.g., because the angular distance of the catalog entry is wider than the matching limit; and(d) false positive matches, when unassociated objects are incorrectly considereda match, e.g., because the catalog coordinates of some unrelatedOIR source are by chance closer to the X-ray source position thanthe coordinates of the true counterpart, or because the true counterpart is undetected in the available OIR images and an unrelated OIR source is close enough to the X-ray source position.For the spurious X-ray sources, there are two possibilities: (e) correct negative matches, when no OIR counterpart is associated with the non-existing X-ray source, and (f) false positive matches, when an unrelated OIR source is wrongly considered to bea match to the non-existing X-ray source. In any matching procedure, some level of spurious matching will inevitably occur. These issues were discussed in detail in <cit.>, where quantitative estimates for the expected numbers offalse positive matches with random unrelated infrared sources were derived for the case of the CCCP data set. However, the numbers for the expected fractions of false positive matchesderived by <cit.> cannot be directly applied to the present study becausethe underlying infrared catalogs are quite different. The highest estimated false match fraction derived in <cit.>was based on the matching of the CCCP X-ray source list with the very deep HAWK-I infrared catalog <cit.>. Since the HAWK-I catalogis about 2 magnitudes deeper(i.e., a factor of 6)than the VISTA catalog we use here,itcontains a large number of very faint (mostly background) objects that provide numerous possibilities for random false positive matches; however, these very faint objects remain undetected in the VISTA catalog and thus cannot produce random false positive matches in our case. Since the HAWK-I catalog lists on average ≈ 470 IR sources persquare-arcminute in the CCCP field, while the VISTA catalog containsonly 140 sources per square arcminute in the NGC 3293 region, the probability of obtaining a randomfalse positive match with the VISTA catalog is at least about 3 times lower than reported for the HAWK-I catalog in the CCCP field. In order to derive a quantitative estimate for the occurrence rate ofrandom false positive matches for X-ray sources in our NGC 3293 data set, we performed a set of random matching simulations that are described inAppendix <ref>. These simulations show thatthe average probability that a spurious X-ray source will get a false positive VISTA catalog match is about 23%. It is very important to note that this is not equal to the expected fraction of false matches in our sample since in reality a large fraction of our X-ray sources are true sources that have a counterpart in one of the young stars in NGC 3293; the 23% probability for false positive matches applies only to the subset of X-ray sources that have no true physical counterparts in our VISTA images,which is considerably smaller than the full sample.In order to estimate the possible consequences of such false positive matcheson the results of our study of the cluster properties described below, it is important to note that the large majority of all random false positive matchesare with very faint IR sources. This is just a consequence of the fact that the number of IR sources increases strongly when going towards fainter magnitudes. The majority of the VISTA catalog sources in the NGC 3293 field have magnitudes J > 18.5. False matches with such very faint IR objects will only haveveryminor effects on our analysis of the color magnitude diagram presented below;e.g., such very faint IR objectscannot be confused with M ≥ 1 M_⊙ cluster stars, which are considerably brighter.Most of our X-ray sources with VISTA catalog matches are much brighter and have magnitudes J < 16.5. As shown in Appendix <ref>, the probability of a random X-ray source obtaining a false positive match with a VISTA catalog source brighter than J = 17 [J = 16] is just 4.6% [2.5%]. If we assume that ≈ 20%, i.e., ≈ 200of the sources, in our X-ray cataloghave no physical counterparts in the VISTA images (i.e., they might be spurious sources), we expect there will be ≈ 46 false positive matches with VISTA catalog sources. However, only ≈ 5- 9 of these would be false positive matches to stars that are bright enough (J ≤ 16 - 17) to possibly affect our star counts for the investigation of the cluster IMF presented in Sect. <ref>. § EXPLORING THE CLUSTER PROPERTIES Our X-ray selected sample provides the first opportunity to studyindividually identified low-mass stars in NGC 3293.We first consider the X-ray-to-infrared flux ratios of the objects, and then use color magnitude diagrams (CMDs)to derive information about the ages and masses of the X-ray detected stars and information about the size of the low-mass star population in the cluster.§.§ X-ray and infrared fluxes Reliable bolometric luminosities (i.e., determined from spectroscopic information) are only available for a relatively small number of stars in NGC 3293 (essentially the B-type stars, which wediscuss in Sect. 5). For the bulk of the stellar population of NGC 3293, we thus cannot directly determinethe ratio of X-ray to bolometric luminosity, which would be a good diagnostic of the nature of X-ray emission. Therefore, we employ herethe observed J-band flux as a proxy for the stellar luminosity. For late-type stars, the J-band flux is roughly proportional to the stellar bolometric luminosity; furthermore, the influence of extinction is rather small in the J-band (A_J ≈ 0.28 × A_V). In Fig. <ref> we show the ratio of observed X-ray and infrared fluxes for the sources in NGC 3293, which shows that the majority of X-ray detected sources are objects for which their J-band magnitudes suggest stellar masses in the range ∼ [2 - 0.5] M_⊙, consistent with the X-ray sensitivity limit. Most of the starsin the range J ≈ [14-18] show ratios around a typical value of log( F_ X/F_J ) ≈ -2.2. Since for these low-mass stars, the bolometric flux is roughly a factor of ∼ 10 higherthan the J-band flux, this corresponds to fractional X-ray luminosities of log( L_ X/L_ bol) ≈ -3.2, which is the typical ratio for coronally active young low-mass stars.The bright starsin the range J ≈ [6-9]show typical ratios log( F_ X/F_J ) ≈ -5.2. Since for early B-type stars, the bolometric flux is typically a factor of ∼ 1000 higherthan the J-band flux, this translates roughly into log( L_ X/L_ bol) ≈ -8.2. Again, this is very consistent with the results from other studies <cit.>. A more accurate assessment of the X-ray properties of the B-type stars in NGC 3293 isgiven in Sect. <ref>.Some of the faintest infrared sources (J19) show very high X-ray-to-infrared flux ratios; many of these objects are probably extragalactic sources. §.§ Spatial distribution of the X-ray sources Next we consider the spatialdistribution of the X-ray sources with the aim to derive information about the cluster size. <cit.> estimated the coronal radius of NGC 3293 to be ∼ 5.5', but since this was based on optical data that are not sensitive enough to detect the full cluster population (especially in regions where the extinction is more than A_V ≈ 1 mag), an independent check is useful. We can address the question regarding the cluster size with our X-ray sample and with the deep VISTA images.Considering the Chandra data, it should be noted that the sensitivity level of the observation is not uniform across the observed field since the point-source sensitivity drops significantlywith off-axis angle. The most important factors that play a role here are the mirror vignetting and the degradation of the point spread function, both of which reduce the local sensitivity with increasing off-axis angle.Following the strategy of <cit.>, a “spatially complete sample” of X-ray sources can be constructed by employing a limit to the observed photon flux of the X-ray sources that is high enough to make sure that sources above this limit can be detected in the full ACIS-I field of view, i.e., even at the edge of the X-ray image.This threshold on the observed X-ray photon flux is log F_t,photon > -5.9 photons s^-1 cm^-2.Since we intend to trace the spatial structure of the stellar cluster, we use only those objects in this spatially complete X-ray sample that have a match with a VISTA source with J < 18, as expected for the large majority of X-ray detected young stars in NGC 3293. This final sample contains 81 sources. A map of the positions of these sources is shown in Fig. <ref>.We determined the surface-density of these X-ray sources as a function of distance from the optical cluster center by counting the number of objects in concentric annular regions with different radii.The resulting radial profile is shown in the upper part of Fig. <ref>. The source density in the center is more than 5 times higher than in the outer parts of the ACIS-I image, drops with angular distance from the cluster center, and merges to the background value at an angular distance of ≈ 6-7 arcmin from the center. This profile isconsistent with the above mentioned previous estimate of the cluster radius and confirms that the ACIS-I field of view covers the full area of the cluster.The questionabout thespatial extent of the cluster can also be addressed with the VISTA data. These data have several advantages. First, they cover a considerably larger area around NGC 3293 (the angular distance from the cluster center to the nearest edge of theVISTA image is 12 arcmin). This makes it easy to define background comparison regionsat angular distances of 10 arcminutes for a reliable computation of the background source density. Second, the VISTA images do not suffer significantly from PSF degradation and mirror vignetting, i.e., they provide a more homogeneous coverage of the observed area. Third, the VISTA data are sensitiveenough to detect all cluster stars (down to 0.1 M_⊙, and even through substantial extinction of A_V ≈ 5 mag) and thus provide much larger source numbers andconsequently better statistics in the source counts.However, there is also one disadvantage: several of the very bright stars in the cluster are saturated, and their PSF-wings and diffraction spikes create a high and complicated background halo around them,which severely restricts the detection of faint sources.The radial profile of the surface density of all VISTA sources with J < 18 is shown in the lower part of Fig. <ref>. A clearly enhanced source density can be seen for radial distances between 2 arcmin and 6-8 arcmin from the cluster center. The low density in the center is caused by the strongly reduced detection efficiency for faint sources near the very bright stars in the cluster center. The VISTA density profile suggests a cluster radius in the range 6-8 arcmin.Considering all these numbers together, a good estimate for the angular size of the cluster is ≈ 7 arcmin, which corresponds to ≈ 4.7 pc at 2.3 kpc distance. §.§ Analysis of thecolor magnitude diagramsIn order to obtain information on the stellar properties of the X-ray detected stars, we constructed color magnitude diagrams (CMDs) and compared the location of the X-ray selected objects to the recent PARSEC stellar evolution models described in <cit.>.§.§.§ Near-infrared color magnitude diagram In Fig. <ref> we show the J versus J-H color magnitude diagram of the X-ray detected objects in NGC 3293. The range of J-magnitudes is restricted to the fainter values in order to show the location of the low-mass stars more clearly; the color magnitude locations of the brightest (i.e., the early-type) stars can be seen in Fig. <ref> and are discussed below.The large majority of the X-ray selected objectsare at CMD locations corresponding to stellar masses in the [0.5-2] M_⊙ range, as expected from the X-ray detection limits, and extinctions of a few visual magnitudes. At the bottom of the CMD (J19), the X-ray selected objects show a wide range of colors consistent with background objects and the expected locus of extragalactic sources. §.§.§ Optical color magnitude diagrams The construction of an optical CMD is not as straightforward as the NIR CMD described above, due to the sensitivity and saturation limits of the available optical images. The optical photometry catalog of <cit.> is reported to become incomplete at V = 16, i.e., it will obviously miss a large fraction of the low-mass stars of NGC 3293. The VST catalog from the VPHAS+ survey is considerably deeper, but misses all stars brighter than g ≈ 13, due to saturation effects. Another problem is that these two catalogs use different photometric systems, UBVRI for <cit.> and SDSS ugri for VPHAS+. Wetherefore consider the bright and the faint parts of the optical CMD separately.The upper plot in Fig. <ref> shows the bright part of the CMD, based on the <cit.> data. All stars with V9 (i.e., the most luminous early B-type stars) are detected in X-rays. This matches the expectations since these very bright objects are the most luminous early B-type stars, which are expected to be strong X-ray emitters because they have quite strong stellar windsand their X-ray luminosityis correlated to their bolometric luminosity <cit.>.At somewhat fainter magnitudes, in therangeV ≈ [9-13], only a small fraction of the stars are detected in X-rays; this range corresponds to stars with late B and A spectral types, which are not expected to be intrinsic X-ray emitters. A star-by-star discussion of the individual X-ray properties andthe observed trends of X-ray emission as a function of spectral type and stellar mass in the B- and A-type rangeis provided in Sec. <ref>. At fainter magnitudes, V13 (corresponding to stellar masses 3 M_⊙), the number of X-ray detected starsincreases strongly, as expected for a stellar mass function that rises towards lower masses. §.§.§Age of the low-mass population As can be seen in Fig. <ref> and Fig. <ref>, most of the X-ray selected stars lie close to or to the right of the 8–10 Myrisochrones. Taking the uncertainties of the photometry into account (∼ 0.05 mag), the CMDs are consistent with the assumption that the majority of the X-rayselected low-mass stars have ages of ∼ 8-10 Myr. This result is in good agreement with the above-mentioned previousage estimates for NGC 3293, which were based on CMD positions of the high-mass stars and the comparison with post-main-sequence evolutionary tracks. This consistency suggestsa common age of about 10 Myr for the high- andthe low-mass stars in NGC 3293. §.§.§ Size of the X-ray selected stellar population of NGC 3293Before we can quantify the population of X-ray detected young stars in NGC 3293, we first have to define the area over which we assume X-ray detected stars tobe cluster members. As shown in Sect. <ref>, the stellar cluster has a radial extent of about 7 arcmin. This value for the radius is also supported by the fact that it just encloses all those 45 B-type stars for which <cit.> determined a stellar mass of M ≥ 5 M_⊙. This circular region also contains the large majority of all X-ray sources with infrared counterparts, supporting the chosen value. We determine the cluster population from the NIR CMD of the X-ray detected stars, since – as described above – the optical VST data are not sensitive enough to detect all the X-ray detected stars in NGC 3293. In the J versus J-H diagram in Fig. <ref>, the number of X-ray detected stars with CMD positions that are consistent with young stellar members of NGC 3293 (i.e., at most 0.05 mag[the 1σ uncertainty of the photometry]to the left of the 10 Myr isochrone, or to the right of the 10 Myr isochrone, and above the reddening vector starting from the location of 10 Myr old 0.1 M_⊙ stars) in the 7 arcmin radius regionis 511. We note that the inclusion of the Chandra sources that have point-like counterparts in the VISTA images that are missing from the VISTA catalogand photometry with ≤ 20% uncertainties would raise that number by 31, i.e., to 542. This number can now be compared to the numbers of X-ray detected stars in the other clusters in the CNC, as listed in Table 1 of<cit.>. Since these other clusterswere also studied with Chandra observations of almost identicalsensitivity <cit.>, a quantitative comparison is straightforward. The three most populous clusters in the central Carina Nebula are Tr 14 with1378 X-ray detected stars, Tr 15 with 481 X-ray detected stars, and Tr 16 with 530 X-ray detected stars. This comparison shows that NGC 3293 isclearly one of the most populous clusters in the entire CNC. It is less populous than Tr 14, but almost equal toTr 16 and Tr 15, andmore populous thanall the other known clusters in the CNC. NGC 3293 is thus an important part of the CNC (see discussion in Sect. <ref>). §.§.§ Size of the low-mass population and the mass function of NGC 3293New information about the mass function of the cluster can be obtained by comparing the number of X-ray detected low-mass stars in a given mass range to the known number of high-mass stars that were identified in optical spectroscopic studies. This will also providean important clarification about the previous claims of a significant deficit of low-mass stars in NGC 3293 that were made in some opticalphotometric studies <cit.>. Our aim here is to determine the number of stars in the [1-2] M_⊙ mass range. To this end, we count the number of X-ray detected stars at positions in the color magnitude diagram that are consistent with masses between 1 M_⊙ and 2 M_⊙ for an age of 8-10 Myr; this concerns all objects with positions in a ± 0.05 mag band around the 10 Myr isochrone line for [1-2] M_⊙ or at locations to the right of this isochrone shifted along the directionof the reddening vector (as indicated by the yellow-and-black lines in Fig. <ref> and Fig. <ref>).The total number of such VISTA catalog stars in the 7 arcmin cluster region is 179 in the J versus J-H diagram. In the optical g versus g-r diagram, the corresponding number ofVST catalog stars is 168 in the same 7 arcmin cluster region. In order to estimate the underlying stellar population, this number of X-ray detected stars has to becorrected for the finite X-ray detection completeness, which is also a function of the location of the stars in the ACIS-I field (due to the variation of the X-ray sensitivity with off-axis angle).In order to estimate what fraction of the stars in a certain mass range can be detectedat a specific local detection limit, we use here theX-ray luminosity functions of young stars derived in the context of the Chandra Orion Ultradeep Project (COUP) <cit.>. In the central part of our Chandra image, the weakest detected X-ray sources have X-ray luminosities of log (L_ X [ erg/s]) ≃ 29.6,if we assume a thermal plasma with kT = 1 keVand an absorbing column density of N_ H = 2 × 10^21cm^-2 (corresponding to A_V ≃ 1 mag). A comparison to the X-ray luminosity function of solar-mass [0.9 M_⊙≤ M ≤ 1.2 M_⊙]stars with ages of a few Myr in <cit.> shows that ≃ 90% of the stars will be above this limit.With an age of ≈ 8-10 Myr, NGC 3293 is somewhat older,and the X-ray luminosities of the stars are thus expected to beslightly lower; on the other hand, stars slightly more massive than one solar masswill have somewhat higher X-ray luminosities. Therefore, assuming a completeness factor of ≃ 90% is areasonable choice for coronally active young stars in the center of NGC 3293in the mass range ≈ [1-2] M_⊙.Since the cluster has a spatial extentof several arcminutes, wehave to take the variation of the X-ray detection limit as a function of off-axis angle into account. For this, we use the results derived in the detailed study of <cit.> and summarized in their Table 8, where the variation of the X-ray luminosity limit is determined for different ranges of the off-axis angle. As listed there, the X-ray luminosity limit in the off-axis angle range [3.8-6.3] arcmin is 0.5 dex higher than in the central 3.8 arcmin; in the [6.3-7.5] arcmin off-axis angle range it is 0.6 dex higher than in the central 3.8 arcmin. From these numbers, we findX-ray luminosity function completeness factors of ≃ 60% for stars in the off-axis angle range [3.8-6.3] arcmin and≃ 59% for stars in the off-axis angle range [6.3-7.5] arcmin.Counting the X-ray detected stars in the 7 arcmin cluster region that have positions in the J versus J-H diagram consistent with massesbetween 1 M_⊙ and 2 M_⊙ for an age of 8-10 Myr according to the PARSEC stellar models, we find the following numbers for the different off-axis angle ranges: in the central part (θ = [0-3.8] arcmin) we find 104 stars, in the θ = [3.8-6.3] arcmin range we find 70 stars, and in the θ = [6.3-7.5] arcmin range we find 5 stars. Dividing these numbers by the corresponding X-ray luminosity function completeness factors of 0.90, 0.60, and 0.59, yields a total extrapolated number of 241 starsas our final estimate for the total number of stars in the [1-2] M_⊙mass range within the 7 arcmin cluster region.Considering the uncertainties in the determination of the X-ray luminosity function completeness factors, we useN[1-2 M_⊙] ≃ 241 ± 25 starsas the final result of this analysis.We note that the true number is somewhat higher, since we omit here the 61 VISTA point source counterparts to Chandra sources that are lacking photometry in the VISTA catalog. If weincludethestars from this group of counterparts that have<20% photometric uncertainty into the counting, there will ten additional stars in the central off-axis bin, and the extrapolated total number estimate will be N[1-2 M_⊙] ≃ 252 instead of 241.As discussed in Sect. <ref>, we have toexpect between about five and nine random false matchesto sufficiently bright VISTA catalog sources that might be located in our counting region for the [1-2] M_⊙ stars in theCMD. These estimated false matches have to be subtracted from our estimate of the stellar population. However, we should alsonote that six of the ten above-mentionedVISTA sourcesmissing from the VISTA catalog that correspond to X-ray detected stars in the [1-2] M_⊙ range, have an optical counterpart in theVST VPHAS-DR2 point source catalog. At least these six additional VISTA stars should thus certainly be included into the counts.The difference between the five to nine objects that should besubtracted and the six objects that should be added is small compared to theabove given uncertainty range ± 25. Therefore, we conclude that our result ofN[1-2 M_⊙] ≃ 241 ± 25 is robust.This number can now be compared to the size of the high-mass population in the same R = 7 region. In addition to the above-mentioned 45 B-type stars with listed masses between5 M_⊙ and 40 M_⊙ <cit.>, the red supergiant V361 Car is also located in this region. The total number of high-mass stars in this cluster region is thus 46.Using the numerical representation of the canonical field star IMF from <cit.> and the number of 46 stars in the [5-40] M_⊙ range, the expected number of [1-2] M_⊙ stars is N_ IMF exp = 237. The X-ray completeness-corrected number ofN[1-2 M_⊙] ≃ 241 ± 25 such stars(253 ± 25, if the stars missing in the VISTA catalog are included) derived above is in very good agreement with this expectation value.This good agreement suggests that the size of the solar-masspopulation of NGC 3293isconsistent with the expectations from the normal field star IMF. This result refutes earlier claims for a strong deficit ofstars with mass belowM ≤ 2-3 M_⊙.This highlights the difficulties resulting in purely photometric determinationsof cluster populations, especially in regions with a very strong galactic background such as in NGC 3293. We cannot directly determine the IMF in the subsolar and very low-massrange with our data since the X-ray detected sample is veryincomplete at such low stellar masses.Assuming that the IMF of NGC 3293 follows the field IMF down to 0.1 M_⊙, a numerical extrapolation (based again on the 46 high-mass stars) of the field star IMF suggests 3230 stars in the [0.1-1] M_⊙ range. In the same way, an estimate of the total stellar population of 3625 stars in the [0.1-100] M_⊙ range can be computed.The statistical expectation value of the number of very high-mass stars (above 40 M_⊙) is 2.3. The central 68% Poisson range is [ 1-4] such stars, whichshould have already exploded as supernovae. § X-RAY PROPERTIES OF THEB-TYPE STARS The large number of B-type stars in NGC 3293 provides a good opportunity to investigate the X-ray properties in a co-eval and homogeneous population ofB-type stars, covering the full spectral range from B0 to B9. It is well established that most of the hottest B-type stars (spectral typesB0 to ≈ B2) are rather strong X-ray sources; their X-ray emission is thought to be related to their strong stellar winds, similar to the case of O-type stars <cit.>. Most B-type stars with spectral types later than ≈ B2 remain undetected in X-ray observations; this is in good agreement with the theoretical expectations, since these stars only haverelatively weak winds that areincapable of producing strong X-ray emission, and at the same time the B-type stars (and also the A-type stars) have no outer convection zones, and thus no magnetic dynamo action is expected, which is the prerequisite for the X-ray emission due to coronal magnetic activity (as in the late-type stars).Nevertheless, in many young clusters, a significant fraction of late B-type stars shows detectable amounts of X-ray emission <cit.>. An often invoked explanation is the presence of unresolved late-type companions as the true source of the X-rays. The X-ray properties of these late B-type stars will then contain information about the multiplicity and the nature of their low-mass companion stars.§.§ X-ray detection fraction as a function of spectral typeIn the full sample of 97 B-type stars in the ACIS-I field, 24 are detected as X-ray sources. The detection fraction, however, is a strong function of spectral type, as shown in Fig. <ref>. While all (three) B0 stars are detected, the fraction drops to 10/23 ≃ 43% for the B1 stars,and then to ≈ 20% for later B spectral types.If we assume that for all of these X-ray detected later B-type stars the X-ray emission originates from an unresolved late-type companion, information about the multiplicity and the pairing-statistics of these B-type stars can be inferred.Since X-ray luminosity scales with stellar mass, these companions cannot have stellar masses that are too low; a reasonable guess is that these companions should be stars with M1 M_⊙ in order toproduce X-ray luminosities above our detection limits.If we assume that every B-type star has one companion (i.e., a multiplicity of 100%),and further assume that the companion masses are randomly sampled from the conventional field-starIMF, it can easily be calculated that the expected fraction of B-type stars with companions of mass ≥ 1 M_⊙ would be[According to the<cit.> IMF,the ratio of stars in therange to those in the [0.1-2] M_⊙ range is 0.068.] ≈ 6.8%. This value is lower than the observed X-ray detection rate of ≈ 20% and thus implies that eitherthese B-type stars have more than one companion on average or that the companion masses are not established by random sampling, but are biased towards more massive stars. This supports independent observational resultswhich also suggest a very high multiplicity of intermediate- to high-mass stars <cit.> and mass ratios that are higher than expected from random pairing. Our result also agrees with previous findings based on X-ray observations of other young clusters <cit.>. §.§ X-ray properties of the individual B-type stars For a more quantitative analysis, we determined the X-ray properties of the individual B-type stars. For those four B-type stars with a sufficient number of X-ray counts, spectral fitting was performed with XSPEC. In the other cases, we used the conversion factor from counts to unabsorbed flux as determined by ,employing the optically derived individual extinction of each star to fix thehydrogen column density, and assuming a plasma temperature ofkT = 0.5 keV for these stars. Thetool was also used for those B-stars without an X-ray counterpart in our source list, in order to compute 90% upper limits to their count rates and X-ray luminosities.In the following, we briefly discuss the individual X-ray properties of some of the X-ray detected B-type stars.The full set of information about the X-ray luminosities (or the corresponding upper limits)for theB-type stars is contained in the electronic Table 3, available at the CDS.HD 91969: The optically brightest (V = 6.51)cluster memberis the B0 Ib star HD 91969, for which a stellar mass of M ≈ 40 M_⊙ has been estimated <cit.>.HD 91969 providesa perfect match to the Chandra source 542,which yielded 164.4 net counts and shows a rather soft spectrum with a median photon energy of 1.0 keV. The XSPEC fit with a thermal plasma model and the N_ H parameter fixed to the value0.136 × 10^22 cm^-2 (as determined from the optical color excess) yields a plasma temperature of T ≈ 5.7 ± 1.1 MK.This is within the typical range of plasma temperatures of early B-type stars, for which the X-ray emission is assumed to originate from shocks in the fast stellar wind <cit.>. Since the fit is of mediocre quality (χ^2/ν = 2.92), we also tried models with two plasma components;however, they yielded no improvement to the fit quality. TheX-ray luminosity of the best-fit model is L_ X = 2.78 × 10^31 erg/sec and yields a fractional X-ray luminosity of L_ X / L_ bol = 3.0 × 10^-8, close to the typical ratios found for early B-type stars. We note that two further (but much weaker) X-ray sources were detected very close to HD 91969:source 537, located 1.4” to the west of HD 91969with 4.2 net counts, and source 548,located 3.0” to the northeast of HD 91969 with 6.9 net counts. In the available optical and NIR images, no indication of the presence of stars is seen at these locations becausethe source positions are within the very bright (or even saturated) parts of the PSF of the extremely bright star HD 91969. It therefore remains unclear, whether these two X-ray sources might be(presumably late-type) companions to HD 91969.HD 91943:The optically second brightest (V = 6.69) star, HD 91943, has aspectral type B0.7 Ib and an estimated stellar mass ofM ≈ 30 M_⊙. This star provides a perfect match to the X-ray source418, which yielded 38.8 net counts with a median energy of 1.1 keV. The fit to the X-ray spectrum yielded a plasma temperature ofT = 8.5 ± 4.6 MK and an X-ray luminosity ofL_ X = 4.1 × 10^30 erg/sec. This corresponds to a fractional X-ray luminosity ofL_ X / L_ bol≈ 7 × 10^-9.No further X-ray source is detected within 8 of HD 91943. CPD -57^∘3524𝐀:This B0.5 IIIn star is a good match to Chandra source 704; with 20.8 net counts, the X-ray source is too weak for a spectral fitting analysis, but the relatively low median energy of 1.3 keV is in the typical range for B-type stars <cit.>.CPD -57^∘3526B:This B1 III staris a very good match to Chandra source 710, which has 4.6 net count and a median energy of1.3 keV. Although the X-ray detection is unquestionable, the X-ray properties remain somewhat unclear becausethe source detection revealed another, similarly strong source with number 709 (3.5 net counts,E_ med = 0.8 keV) at an angular distance of just 0.68 from source 710.Unfortunately, in all the available optical and infrared images of thisvery bright star (V = 8.25,H = 8.03) the inner few square arcseconds of the PSF are completely saturated; it is therefore not possible to check whether the second X-ray source is a companion to the B1 star.CD -57^∘3348:The B1 III star CD -57^∘3348 (alias CPD -57^∘3506A) is a very good match to Chandra source 490,which yielded 42.8 net counts with a median energy of 1.0 keV.The fit to the X-ray spectrum yielded a plasma temperature ofT = 7.9 ± 4.5 MK and an X-ray luminosity of L_ X = 5.4 × 10^30 erg/sec. This corresponds to L_ X / L_ bol≈ 2.3 × 10^-8.We note that the VISTA images show another fainter star 3 to the north of this B1 III star, which is also detected as an X-ray source. Since these 3 correspond to a physical distance of about 7000 AU, it remains unclear whether this fainter star might be a companion to the B1 III star, or just a random projection effect.CPD -57^∘3523:This B1 III star is a very good match to Chandra source 697.With 14.8 net counts the X-ray source is too weak for spectral fitting, but the median energy of 0.9 keV is in the typical range for early B-type stars. V405 Car:The B1 V star V405 Car (aliasCPD -57^∘3507)is a very good match to Chandra source523, which has 3.8 net counts and E_ med = 2.7 keV;this is unusually hard for a B star and thus indicates that a low-mass companion contributes to the observed Xray emission. CPD -57^∘3521:The B1 III starCPD -57^∘3521 is a very good match to Chandra source 679, which has 14.7 net counts and E_ med = 1.4 keV.The X-ray analysis revealed another, weaker (2.8 net counts) but somewhat harder(E_ med = 2.0 keV) X-ray source (680)at an angular distance of just 0.82 from source 679. Unfortunately, the reality of this tentative (late-type?) companion cannot be checked since in all available optical and infrared images of this very bright star (V = 8.14,H = 7.87) the inner few square arcseconds of the PSF are completely saturated.NGC 3293 ESL 87:The B5 star NGC 3293 ESL 87is a good match to Chandra source 47,which has 25.9net counts and E_ med = 1.6 keV. The XSPEC fit to the X-ray spectrum (with N_ H fixed at 0.1× 10^22cm^-2) yields a plasma temperature ofkT = 2.1 ± 1.5 keV and an X-ray luminosity of 4.6 × 10^30 erg/sec. These values would be very unusual for a B5 star and thus point towardsthe presence of a late-type companion. Other B-type stars in NGC 3293:Most of the other B-type stars in the ACIS-I field are not detected as X-ray sources.As described above, we use thetool to determine 90% upper limits to the X-ray luminosities of the undetected B-type stars.This yielded useful upper limits for 31 of the undetected B-type stars; for the remaining targets,could not determine upper limits becausethey are located very close to a chip edge on the detector.The upper limit values determined in this way reflect the local X-ray detection limit at the target position. The three B-type stars with particularly high (≥ 2× 10^30 erg/s) upper limits (see Fig. <ref>) are located very close to the edge of the ACIS-I detector where thesensitivity is considerably lower than on-axis. §.§ X-ray and bolometric luminosity In Fig. <ref> we plot the X-ray luminosities of the detected B-type stars (and the upper limits to theX-ray luminosities of the undetectedB-type stars) against the bolometric luminosities.Most of the very luminous objects (L_bol 10^4 L_⊙)show fractional X-ray luminosities between L_ X / L_ bol = 10^-7 and L_ X / L_ bol≈10^-8. This is the typical range found for early B-type stars in other young clusters <cit.> and is consistent with the wind-shock model for the origin of the X-ray emission.The X-ray detected later B-type stars show considerably higher L_ X / L_ bol ratios, which would be very hard to explain by the wind-shock mechanism thought to be at work in the early B-type stars. Their X-ray luminosities are, however,consistent with those of young late-type stars, again strongly suggesting that the X-ray emission actually originates from unresolved late-type companion stars. §.§Red supergiant star V361 Car (M1.5Iab-Ib) In order to describe all known massive stars in NGC 3293, we note that the red supergiant star V361 Car (M1.5Iab-Ib) (which is the most massive star in the cluster) is not detected as an X-ray source.However, theX-ray source 449 is found just 2.8 arcsec southeast of V361 Car. With 4.9 net counts anda relatively soft spectrum (E_ med = 1.2 keV), this might be either a late-type companion, oran early B-star companionto V361 Car.Owing to the extreme brightness of the supergiant (V = 7.19, H = 2.6), all available optical and infrared imagesare completely saturated at the position of this possible companion. § SUMMARY AND CONCLUSIONSWe have performed the first deep X-ray observation of the young cluster NGC 3293 at the northwestern edge of the Carina Nebula Complex. This Chandra observation complements the similarly sensitive X-ray survey of the central region of the CNC in the context of the CCCP <cit.> and our recent observation of the NGC 3324 region (between the central parts of the CNC and NGC 3293), and it completes the X-ray investigation of all the star clusters in the CNC. The present study is the first where NGC 3293 is investigated in the same wayas all other clusters in the CNC (i.e., by a combination of a deep X-ray and infrared data). This finally allows us to put the derived properties of NGC 3292 into a larger context and to consider it in the same way as the other parts of the CNC.Our analysis of the Chandra X-ray observation of NGC 3293 clearly shows that the cluster hosts a large population of low-mass stars in the ∼[2-0.5] M_⊙ mass range. The number of X-ray detected low-mass stars closely agreeswith theexpectations based on the number of spectroscopically identified high-mass stars and the assumption of a field star initial mass function. There is thus no indication of a deficit of low-mass (M ≤ 2.5 M_⊙) stars in this cluster. These results suggest a total population of ≈ 3600 stars for NGC 3293. We find that NGC 3293 is one of the most populous clusters in the entire Carina Nebula Complex. NGC 3293 is older than the otherwell-investigated clusters in the CNC. Extrapolating the cluster's mass function suggests that several supernova explosionshave occurred in NGC 3293 during the last few Myr <cit.>. This suggests that NGC 3293 has most likely played an important role during the formation and early evolution of the CNC. With a spatial extent of ∼ 100 pc, a total cloud mass of ∼ 10^6 M_⊙, and more than 100 000 young stars,the CNC is one of the largest star forming complexes in our galaxy. It also shows a high diversity in the structure of the clouds and in the spatial configurations of the stellar populations. Most of thestars in the inner parts of the CNC are located in one of more than ten individual stellar clusters, which have agesranging from 1 Myrup to ∼ 5 Myr.In addition to this clustered stellar population, there is also an unclustered, widely distributed population <cit.> of young stars with ages ranging from < 1 Myrto ∼ 6-8 Myr <cit.>. How this complicated and diverse spatial and temporal configuration has formed and evolved is still unclear. In Fig. <ref> we show a sketch illustrating the spatial configuration of the most significant clusters and clouds in the CNC.NGC 3293 is the oldest of the large clusters in the CNC, and it is located at the northwestern edge of the complex. Most of the current star formation activity observed today in the CNC is happening in the southeastern part of the CNC, in the Southern Pillars region, i.e., at the opposite end of the complex. However, this does not reflect a systematic northwest to southeast spatial age gradient since there are several sites of active star formation in the northern part of the complex. The most prominent of these is the large shell of dense dust clouds around the cluster NGC 3324 <cit.>, which is located just southeast of NGC 3293. Another example of very recent and ongoing star formation activity in the northern area is the embedded stellar cluster associated with the massive dense cloud clump G286.21+0.17 <cit.>, which is located north of the dust shell around NGC 3324.The cluster ages also show no systematic trend with spatial position in the complex.For example, the very young(∼1-2 Myr) cluster NGC 3324 is located between the considerably older clusters NGC 3293 (∼8-10 Myr) to the northand the ∼ 5 Myr old cluster Tr 15 south of it.Instead of a systematic progression of star formation in one direction, it appears that the star formation activity in the CNC was “wandering around” in various directions that changed with time. The best explanation of this seems to be that the progression of star formation in the CNC was predominantly triggered by the feedback of the numerous high-mass stars that were born at different locations in the complex over the last ∼ 10 Myr.Today, we witness the effect of this feedback and the corresponding driving of star formation everywhere in the surroundings of the currently existing O-type stars. For example, the∼ 3 Myrhigh-mass stars in the cluster Tr 16 are triggering star formation in the Southern Pillars <cit.>. The very young (∼ 1 Myr) high-mass stars in the cluster Tr 14 have recently started to exert a strong influence on the very dense and massive cloud <cit.> to the westof the stellar cluster; in the near future, the ongoing compression of this cloud is likely to initiate cloud collapse and trigger star formation. The young (∼1-2 Myr) O-type stars in the cluster NGC 3324 <cit.> have swept up a huge dust bubble around the Gum 31 HII region and are starting to trigger star formation in and around this bubble<cit.>. About 8 Myr ago, the (then very young) cluster NGC 3293 probably contained several O-type stars. These high-mass stars must have exerted asimilar influence on the clouds that were present in the cluster's surroundings at that time. It appears likely that this stellar feedback initiated the sequence of local cloud collapse events in the original, huge proto-Carina Nebula cloud, starting probably at the location where we now find the second oldest cluster Tr 15. Given the age difference of ∼3-5 Myr between NGC 3293 and Tr 15, it seems possible that supernova explosions of the O-type stars may have played an important role in the timing of the propagating star formation.Today, i.e., about 8–10 Myr later, star formation is still going on in some locations of the complex. Although a large fraction of the original ∼ 10^6 M_⊙ gas has already been heated and transformed to lower densities by the effects of the radiation, winds, and supernovae of several generations ofmassive stars <cit.>, there is still a large reservoir (20 000 M_⊙) of dense clouds<cit.> available for future star formation over the next millions of years.These considerations show that a good characterization of the ages and stellar populations of clusters like NGC 3293 is a key factor forunderstanding the intricate spatio-temporal progression of star formation in huge cloud complexes like the CNC.We gratefully acknowledge funding for this project by the German Deutsche Forschungsgemeinschaft, DFG project number PR 569/9-1.Additional support came from funds from the Munich Cluster of Excellence: “Origin and Structure of the Universe”.Townsley and Broos acknowledge support from the Chandra X-ray Observatorygeneral observer grant GO5-16003X and from the Penn State ACIS Instrument Team ContractSV4-74018, issued by the Chandra X-ray Center (CXC), which is operated by theSmithsonian Astrophysical Observatory for and on behalf of NASA under contract NAS8-03060. This research used software provided by the CXC in the application package CIAO,and SAOImage DS9 software developed by the Smithsonian Astrophysical Observatory.The VISTA infrared data used in this work are based on observations made with ESO Telescopes at the La Silla Paranal Observatoryunder programme ID 088.C-0117. The analysis used data products from observations made with ESO Telescopesat the La Silla Paranal Observatory under program ID 177.D-3023,as part of the VST Photometric Hα Survey of the Southern GalacticPlane and Bulge (VPHAS+, www.vphas.eu).We acknowledge the assistance of the LMU physics students S. Graßl, J. Diehl, and L. Furtak in some steps of the preliminary data analysis. This research has made use of the SIMBAD database and the VizieR catalog services operated at Strasbourg astronomical Data Center (CDS).aa§ CLOUD STRUCTURE AND EXTINCTION IN AND AROUND NGC 3293NGC 3293 is located at the northwestern edge of the cloud complex associated with the Carina Nebula.The inner parts of this cloud complex host some very dense clouds, which cause very strong extinction of A_V10 mag <cit.> at some locations. The properties and the highly inhomogeneous spatial structure of these dense clouds have been revealed by APEX sub-mm observations <cit.>. In the NGC 3293 area, the APEX / LABOCA maps from the ATLASGAL survey<cit.> show no significant sub-mm emission. Although this implies that no very dense clouds are present, Spitzer mid-infrared and Herschel far-infrared maps (see Fig. <ref>)nevertheless show significant emission from moderate-density clouds in and around NGC 3293.The diffuse 24 μm emission that is seen mainly towards thenorthwest of the cluster center, results most likely from warm dust grains that are heated by the UV radiation of theB-type stars in the cluster. The 8 μm emission traces the surface of somewhat denser cloud structures; it is probably dominated by fluorescent emission from polycyclic aromatic hydrocarbon (PAH) molecules. In the southeastern part of the cluster, the Spitzer andHerschel maps show a prominent pillar-shaped cloud,which points directly towards the M1.5Iab supergiant V361 Car (the very bright point source in the Spitzer image). This pillar is very similar to the numerous cloud pillars found in other parts of the CNC <cit.>. The infrared images show a tendency of stronger cloud emissionaround the stellar cluster, and less emission in the central parts of the cluster. Together with the apparently empty bubble-like feature just east of the cluster center, this suggest that the radiation and winds of the high-mass stars in NGC 3293 have cleared the central regions of the original clouds.This conclusion is supported by the measuredreddening values of stars with known spectral type in the NGC 3293 region listed in <cit.>; these values correspond[In their optical photometric study, <cit.>found that the extinction law for the stars in NGC 3293 isconsistent with the usual relation A_V = 3.1 E(B-V).] to typical extinction values ofA_V ∼ 1 mag for stars near the cluster center, whereas stars at the periphery show extinctions up to a few magnitudes.§ RANDOM MATCHING SIMULATION In order to estimate the possible occurrence rate of false positive matches between a random (spurious) X-ray source and an unrelated infrared source, we performed a set of matching simulations. We take into account that neither the distribution of the infrared sources, nor the distribution and the properties of the X-ray matching regions are purely random, but rather show characteristic spatial trends.The distribution of the infrared sources in the field is not uniform since we are looking at a stellar cluster with a higher density of relatively bright stars in the center, superposed on an approximately uniform distribution of relatively faint background objects, and a reduced sensitivity for very faint objects in the central regions around the very bright stars.There are several reasons why the properties of the X-ray matching regions in aChandra observation are also not uniform over the observed field of view. First, the sensitivity (and thus the detection limit)depends on the off-axis angle. Second, the position uncertainties (and the corresponding radii of the matching regions) also depend on the off-axis angle since the point spread function increases with off-axis angle. Third, the position uncertainties also depend on the number of detected counts per source, which again depends on thelocal sensitivity (known as the effective area) at the source position on the detector.All the mentioned dependencies are, to first order, predominantly functions of the angular distance from either the cluster center or the focal point of the X-ray image; these two points coincide closely in our observation.We therefore use the original VISTA catalog for therandom matching simulations, which preserves the spatial distributions of the infrared sources. To simulateX-ray source lists, we use our original Chandra ACIS-I source catalog in order to conserve the angular-dependence of the source properties, but rotate all source positions by certain angles around the focal point in order to break the physical connection between X-ray and infrared sources.Simulations were performed with different rotation angles from -180 to +180 in steps of 0.5. At each rotation angle, we used match_xy in the same way as for the original matching and recorded the number of resulting matches with the VISTA catalog. For any rotation angle that differs by more than a few degrees from zero, the resulting match numbers are a measure of the expected number of purely random matches of sources without any physical connection.The results of these simulations are shown in Fig. <ref>. For rotation angles of more than a few degrees, it can be seen that the number of matches shows only small fluctuations, and no significant variations or trends with rotation angle, confirming that the resulting numbers are good estimates of the random false match fraction.Considering the full VISTA catalog, the mean number of random matches is 238 ± 12; this corresponds to a fraction of (23.3 ± 1.2)% of all X-ray sources.Two important aspects have to be considered in the interpretationof this number. First, it should be noted that this number is a strict upper limit to the actual number of possible random false matches in our samplesince the simulation assumes (by means of the rotation) that no X-ray source isphysically connected to any infrared source; it is valid only forX-ray sources thathave no physical match detected in the available optical/infrared images. In reality, however, a large fraction of the X-ray sources are young stars, i.e., rather bright IR sources, that produce correct positive matches. Only those X-ray sources that have no true counterpart in the VISTA images can get a false random match.The second point concerns the magnitudes of these random false matches. The large majority of these are very faint infrared sources, most of them with magnitudes J > 18. This is just a reflection of the strongly increasing number of infrared sources when going towards fainter magnitudes. The color magnitude diagrams resulting from these random matching experiments are therefore very different from the actual color magnitude diagram, since they show only very few random false matches tostars with sufficiently bright magnitudes to beconsidered as 1 M_⊙ stars.To investigate this last point further, we repeated the simulations with restricted versions of the original VISTA catalog, containingonly objects above a specific magnitude limit.The number of false matches drops steeply with increasing magnitude limit: for J<18 we find 84 ± 9 matches (8.2 ± 0.9%), for J<17 we find 47 ± 7 matches (4.6 ± 0.7%), andfor J<16 we find 25 ± 5 matches (2.3 ± 0.5%). This implies that the fraction of potential false matches of X-ray sources with stars that are bright enough to be considered1 M_⊙ stars in NGC 3293 is quite small (4%).§ RANDOM MATCH WITH AN ARTIFACT IN THE VST IMAGESA particularly interesting case of a random false match of an X-ray source with an artifact in the VST optical catalog concerns theChandra Source 407,which is located 9 arcmin south of the cluster center. The X-ray source yielded 12.8 net counts with a median photon energy of 1.9 keV.The VST VPHAS-DR2 Point Source Cataloguelists a sourceVPHASDR2 J103541.2-582259.4 1735b-31-6789 with magnitudes r = 18.1483 ± 0.026, i = 17.9260 ± 0.037,and Hα = 17.8223 ± 0.022, which provides a possible match to the X-ray source.The ESO archive contains 16 individual VST observations of this point, obtained in the u, g, r, i, and the Hα band. Our inspection of all these individual images yielded a very surprising result. While we could clearly confirm the presence of a point-like source in the u-band and g-band images obtained on 15 Feb 2012 and in the r-band, i-band, and Hα images obtained on 30 April 2012 (see Fig. <ref>), allother available VST images, which were obtained on 15 Feb 2012 and 14 Feb 2013, showno source at this position. The inspection of our VISTA images (obtained on 6 March 2012 and 8 March 2012) and archival ESO WFI images obtained on 9 Feb 2012 also showed no source at this position. Upper limits for the magnitudes in these non-detection images are u> 21.8,g > 22.5, r > 21.8,i > 20.8, and Hα > 20.6. These numbersimply that the objectbrightened by several magnitudes in the six days from 9 Feb 2012 to 15 Feb 2012, was invisible six days later, again bright one month later, and again invisible in the most recent images.A possible explanation of such a very unusual behaviorcould be that the source is some kind of a transient object.Finally, however, our close inspection of those VST images in which the source was visible, showed that the 2D profile of the object deviatesfrom the normal point spread function of other sources in the surrounding.After detailed consultations with VST instrument and imaging experts at the ESO headquarters, it was finally found that the apparently highly variable object is actually an image artifact, which was caused by anelectroniccross-talk effect induced by a very bright, saturated star on another CCD chip in the camera (private communication from the ESO User Support Astronomer M. Petr-Gotzensand the ESO Optical detector engineering group).Although it is very unlikely to find an optical artifact in the very small matching region around an X-ray source, this case demonstrates that these rarecoincidences can sometimes happen. This highlights the importance of a careful visual inspection of the original optical/infrared images for a reliable determination of the counterparts to the X-ray sources.§QUASAR CANDIDATE J103621.39-581520.0 As mentioned in Sect. <ref>, the brightest X-ray source (number 943) in the ACIS-I array, with 190 net counts, has no optical counterpart. It is located ≈ 4.4 southeast of the cluster NGC 3293. In Fig. <ref> the Chandra extraction region of 103621.39-581520.0 is shown on the VISTA H-band image, which reveals, at close inspection, an extremely faint infrared counterpartof this X-ray source. The X-ray spectrum can be very nicely reproduced by a model assuming a power-law X-ray spectrum and interstellar extinction. The spectral fit yieldsχ^2_r = 0.37 for the following parameters: N_ H = (2.86 ± 0.77) × 10^22cm^-2 (corresponding to a visual extinction of A_V ≈ 14 mag),and aphoto index of 1.57 ± 0.37. The observed [0.5-8] keV X-ray flux, derived from the fit results, is8.7 × 10^-14ergcm^-2s^-1, and the corresponding de-reddened X-ray fluxis 1.5 × 10^-13ergcm^-2s^-1. The source is invisible in all available optical VST images. From the VST g-band image, we estimate an upper limit of g ≥ 23 mag for the brightness of the source. This corresponds to a flux of≤ 4 × 10^-15ergcm^-2s^-1. Comparing this to the X-ray flux derived from the spectral fit yields an X-ray to optical flux ratioF_ X / F_ opt≥ 21. These properties, i.e., the combination of strong absorption, a hard spectrum, and the very high X-ray-to-optical flux ratio, are best explained by assuming the source to be an obscured quasar. Many of these objects have X-ray-to-optical flux ratios above 1, and a small fraction of about 5% have F_ X / F_ opt≥ 20 <cit.>. J103621.39-581520.0 thus seems to be a particularly X-ray active quasar. | http://arxiv.org/abs/1707.08782v1 | {
"authors": [
"T. Preibisch",
"S. Flaischlen",
"B. Gaczkowski",
"L. Townsley",
"P. Broos"
],
"categories": [
"astro-ph.SR",
"astro-ph.GA"
],
"primary_category": "astro-ph.SR",
"published": "20170727085511",
"title": "Chandra X-ray observation of the young stellar cluster NGC 3293 in the Carina Nebula Complex"
} |
Direct Load Control of Thermostatically Controlled Loads Based on Sparse Observations UsingDeep Reinforcement Learning Frederik Ruelens,Bert J. Claessens, Peter Vrancx, Fred Spiessens, and Geert Deconinck F. Ruelens and G. Deconinck are with the Department of Electrical Engineering, KU Leuven/EnergyVille, 3000 Leuven, Belgium (frederik.ruelens, [email protected]). B. J. Claessens is with REstore, 2600 Antwerp, Belgium ([email protected]). P. Vrancx is with the AI-lab, Vrije Universiteit Brussel, 1050 Brussels, Belgium ([email protected]) F. Spiessens is with Vito/EnergyVille, 2600 Mol, Belgium. December 30, 2023 ================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================ This paper describes a builder entry, named “strawman”, to the sentence-level sentiment analysis task of the “Build It, Break It” shared task of the First Workshop on Building Linguistically Generalizable NLP Systems. The goal of a builder is to provide an automated sentiment analyzer that would serve as a target for breakers whose goal is to find pairs of minimally-differing sentences that break the analyzer.§ DATA AND PREPROCESSINGDataThe organizers of the shared task provided two distinct types of training sets. The first set consists of usual sentences paired with their corresponding sentiment labels (+1 for positive and -1 for negative) and confidences (a real value between 0 and 1.) The other set consists of phrases paired similarly with sentiment labels and confidences. In the latter case, the sentiment label may be either -1, 1 or 0 which indicates neutral. There are 6920 sentences and 166,737 phrases.As the goal of “strawman” is to build the most naive and straightforward baseline for the shared task, I have decided to use all the examples from both of the training sets whose sentiment labels were either -1 or 1. In other words, any phrase labelled neutral was discarded. The confidence scores were discarded as well.The combined data was shuffled first, and then the first 160k examples were used for training and the last 10k examples for validation. I have decided to ignore 3,657 examples in-between. VocabularyThe training dataset was lowercased in order to avoid an issue of data sparsity, as the size of the dataset is relatively small. Since the provided training examples were already tokenized to a certain degree, I have not attempted any further tokenization, other than removing a quotation mark “"”. In the case of blind development and test sets, I used spaCy[<https://spacy.io/>] for automatic tokenization. At this stage, a vocabulary was built using all the n-gram's with n up to 2 from the entire training set. This resulted in a vocabulary of 102,608 unique n-gram's, and among them, I decided to use only the 100k most frequent n-grams. § MODEL AND TRAINING The “strawman” is an ensemble of five deep bag-of-ngrams classifiers. Each classifier is a multilayer perceptron consisting of an embedding layer which transforms one-hot vector representations of words into continuous vectors, averaging pooling, a 32-dim tanh hidden layer and a binary softmax layer. The classifier is trained to minimize cross-entropy loss using Adam <cit.> with the default parameters. Each training run was early-stopped based on the validation accuracy and took approximately 10-20 minutes on the author's laptop which has a 2.2 GHz Intel Core i7 (8 cores) and does not have any GPU compute capability. The output distributions of all the five classifiers, which were initialized using distinct random seeds, were averaged to form an ensemble. The entire code was written in Python using PyTorch.[<http://pytorch.org/>] The implementation is publicly available at <https://github.com/kyunghyuncho/strawman>.§ RESULT AND THOUGHTS Despite its simplicity and computational efficiency, the “strawman” fared reasonably well. The “strawman” was ranked first in terms of the average F1 score on all the breakers' test cases, outperforming more sophisticated systems based on a recursive deep network (Builder Team 5, <cit.>) as well as a convolutional network (Builder Team 6, <cit.>). When measured by the proportion of the test cases on which the system was broken (i.e., the system is correct only for one of the minimally difference sentences and wrong for the other), the “strawman” was ranked fourth out of six submissions, although the margin between the “strawman” and the best ranking system (Builder Team 2) was only about 1% out of 25.43% broken case rate, corresponding to 6 cases. Although we must wait until the breakers' reports in order to understand better how those broken cases were generated, there are a few clear holes in the proposed “strawman”. First, if any word is replaced so that a new bigram disappears from the predefined vocabulary of n-grams, the “strawman” could easily be thrown off. This could be addressed by character-level modelling <cit.> or a hybrid model <cit.>. Second, the “strawman” will be easily fooled by any non-compositional expression that spans more than two words. This is inevitable, as any expression longer than two words could only be viewed as a composition of multiple uni- and bi-grams. Third, the obvious pitfall of the “strawman” is that it was trained solely on the provided training set consisting of less than 7k full sentences. The “strawman” would only generalize up to a certain degree to any expression not present in the training set.§ ACKNOWLEDGMENTS The author thanks support by eBay, TenCent, Facebook, Google and NVIDIA.emnlp_natbib | http://arxiv.org/abs/1707.08939v1 | {
"authors": [
"Kyunghyun Cho"
],
"categories": [
"cs.CL"
],
"primary_category": "cs.CL",
"published": "20170726173057",
"title": "Strawman: an Ensemble of Deep Bag-of-Ngrams for Sentiment Analysis"
} |
[Corresponding author:][email protected] University of Connecticut, Storrs, CT 06269 Christopher Newport University, Newport News, VA 23606 Thomas Jefferson National Accelerator Facility, Newport News, VA 23606Thomas Jefferson National Accelerator Facility, Newport News, VA 23606Lanzhou University, Lanzhou 730000, Gansu, Peoples Republic of ChinaCollege of William and Mary, Williamsburg, VA 23187Thomas Jefferson National Accelerator Facility, Newport News, VA 23606College of William and Mary, Williamsburg, VA 23187 Norfolk State University, Norfolk, VA 23504The George Washington University, Washington, DC 20052North Carolina A&T State University, Greensboro, NC 27411Hampton University, Hampton, VA 23668California State University Los Angeles, Los Angeles, CA 90032Argonne National Laboratory, Argonne, IL, 60439Yerevan Physics Institute, Yerevan 375036, ArmeniaUniversity of Virginia, Charlottesville, VA 22904Duquesne University, Pittsburgh PA, 15282Massachusetts Institute of Technology, Cambridge, MA 02139Institut de Physique Nucléaire, CNRS/IN2P3 and UniversitéParis-Sud, FranceThomas Jefferson National Accelerator Facility, Newport News, VA 23606Florida International University, Miami, FL 33199University of Regina, Regina, SK S4S OA2, CanadaChristopher Newport University, Newport News, VA 23606JINR-LHE, Dubna, Moscow Region, Russia 141980Hampton University, Hampton, VA 23668University of Virginia, Charlottesville, VA 22904California State University Los Angeles, Los Angeles, CA 90032Thomas Jefferson National Accelerator Facility, Newport News, VA 23606North Carolina A&T State University, Greensboro, NC 27411Ohio University, Athens, Ohio 45701IHEP, Protvino, Moscow Region, Russia 142284University of Virginia, Charlottesville, VA 22904Florida International University, Miami, FL 33199Mississippi State University, Mississippi, MS 39762Thomas Jefferson National Accelerator Facility, Newport News, VA 23606 Deceased. INFN, Sezione Sanità and Istituto Superiore di Sanità, 00161 Rome, ItalyThomas Jefferson National Accelerator Facility, Newport News, VA 23606University of Virginia, Charlottesville, VA 22904INFN, Sezione Sanità and Istituto Superiore di Sanità, 00161 Rome, ItalyThomas Jefferson National Accelerator Facility, Newport News, VA 23606Massachusetts Institute of Technology, Cambridge, MA 02139Thomas Jefferson National Accelerator Facility, Newport News, VA 23606 Rutgers, The State University of New Jersey,Piscataway, NJ 08855IHEP, Protvino, Moscow Region, Russia 142284Argonne National Laboratory, Argonne, IL, 60439University of Glasgow, Glasgow G12 8QQ, Scotland UKThomas Jefferson National Accelerator Facility, Newport News, VA 23606Norfolk State University, Norfolk, VA 23504Thomas Jefferson National Accelerator Facility, Newport News, VA 23606Lanzhou University, Lanzhou 730000, Gansu, Peoples Republic of ChinaMassachusetts Institute of Technology, Cambridge, MA 02139University of Regina, Regina, SK S4S OA2, CanadaChristopher Newport University, Newport News, VA 23606Hampton University, Hampton, VA 23668Norfolk State University, Norfolk, VA 23504Ohio University, Athens, Ohio 45701JINR-LHE, Dubna, Moscow Region, Russia 141980Hampton University, Hampton, VA 23668IHEP, Protvino, Moscow Region, Russia 142284Rutgers, The State University of New Jersey,Piscataway, NJ 08855Hampton University, Hampton, VA 23668 University of Virginia, Charlottesville, VA 22904California State University Los Angeles, Los Angeles, CA 90032Christopher Newport University, Newport News, VA 23606IHEP, Protvino, Moscow Region, Russia 142284University of Virginia, Charlottesville, VA 22904University of Witwatersrand, Johannesburg, South AfricaThomas Jefferson National Accelerator Facility, Newport News, VA 23606IHEP, Protvino, Moscow Region, Russia 142284University of Maryland, College Park, MD 20742 Yerevan Physics Institute, Yerevan 375036, ArmeniaMassachusetts Institute of Technology, Cambridge, MA 02139SLAC National Accelerator Laboratory, Menlo Park, CA 94025University of Virginia, Charlottesville, VA 22904Mississippi State University, Mississippi, MS 39762University of Chemical Technology and Metallurgy, Sofia, BulgariaMississippi State University, Mississippi, MS 39762University of Tel Aviv, Tel Aviv, IsraelChristopher Newport University, Newport News, VA 23606JINR-LHE, Dubna, Moscow Region, Russia 141980Christopher Newport University, Newport News, VA 23606Rutgers, The State University of New Jersey,Piscataway, NJ 08855JINR-LHE, Dubna, Moscow Region, Russia 141980Argonne National Laboratory, Argonne, IL, 60439Florida International University, Miami, FL 33199 University of Virginia, Charlottesville, VA 22904Yerevan Physics Institute, Yerevan 375036, ArmeniaDeceased. IHEP, Protvino, Moscow Region, Russia 142284 Faculty of Mathematics and Physics, University of Ljubljana, SI-1000 Ljubljana, Slovenia Jožef Stefan Institute, SI-1000 Ljubljana, Slovenia Deceased. JINR-LHE, Dubna, Moscow Region, Russia 141980Thomas Jefferson National Accelerator Facility, Newport News, VA 23606IHEP, Protvino, Moscow Region, Russia 142284Deceased. Argonne National Laboratory, Argonne, IL, 60439University of Virginia, Charlottesville, VA 22904Institut de Physique Nucléaire, CNRS/IN2P3 and UniversitéParis-Sud, France DSM, IRFU, SPhN, Saclay, 91191 Gif-sur-Yvette, FranceIHEP, Protvino, Moscow Region, Russia 142284Christopher Newport University, Newport News, VA 23606Thomas Jefferson National Accelerator Facility, Newport News, VA 23606Hampton University, Hampton, VA 23668JINR-LHE, Dubna, Moscow Region, Russia 141980 Lanzhou University, Lanzhou 730000, Gansu, Peoples Republic of ChinaUniversity of Virginia, Charlottesville, VA 22904Massachusetts Institute of Technology, Cambridge, MA 02139 Background Interest in the behavior of nucleon electromagnetic form factors at large momentum transfers has steadily increased since the discovery, using polarization observables, of the rapid decrease of the ratio G_E^p/G_M^p of the proton's electric and magnetic form factors for momentum transfers Q^2 ≳ 1 GeV^2, in strong disagreement with previous extractions of this ratio using the traditional Rosenbluth separation technique. Purpose The GEp-III and GEp-2γ experiments were carried out in Jefferson Lab's (JLab's) Hall C from 2007-2008, to extend the knowledge of G_E^p/G_M^p to the highest practically achievable Q^2 given the maximum beam energy of 6 GeV, and to search for effects beyond the Born approximation in polarization transfer observables of elastic e⃗p scattering. This article provides an expanded description of the common experimental apparatus and data analysis procedures, and reports the results of a final reanalysis of the data from both experiments, including the previously unpublished results of the full-acceptance dataset of the GEp-2γ experiment.Methods Polarization transfer observables in elastic e⃗p→ ep⃗ scattering were measured at central Q^2 values of 2.5, 5.2, 6.8, and 8.54 GeV^2. At Q^2 = 2.5 GeV^2, data were obtained for central values of the virtual photon polarization parameter ϵ of 0.149, 0.632, and 0.783.The Hall C High Momentum Spectrometer detected and measured the polarization of protons recoiling elastically from collisions of JLab's polarized electron beam with a liquid hydrogen target. A large-acceptance electromagnetic calorimeter detected the elastically scattered electrons in coincidence to suppress inelastic backgrounds.Results The final GEp-III data are largely unchanged relative to the originally published results. The statistical uncertainties of the final GEp-2γ data are significantly reduced at ϵ = 0.632 and 0.783 relative to the original publication.Conclusions The final GEp-III results show that the decrease with Q^2 of G_E^p/G_M^p continues to Q^2 = 8.5 GeV^2, but at a slowing rate relative to the approximately linear decrease observed in earlier Hall A measurements. At Q^2 = 8.5 GeV^2, G_E^p/G_M^p remains positive, but is consistent with zero.At Q^2 = 2.5 GeV^2, G_E^p/G_M^p derived from the polarization component ratio R ∝ P_t/P_ℓ shows no statistically significant ϵ-dependence, as expected in the Born approximation. On the other hand, the ratio P_ℓ/P_ℓ^Born of the longitudinal polarization transfer component to its Born value shows an enhancement of roughly 1.4% at ϵ = 0.783 relative to ϵ = 0.149, with ≈ 1.9σ significance based on the total uncertainty, implying a similar effect in the transverse component P_t that cancels in the ratio R. 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 Electron scattering is of central importance to the characterization of nucleon and nuclear structure, because of the relative weakness of the electromagnetic interaction (compared to a strongly interacting probe), the structureless character of the leptonic probe, and the availability of electron beams of high intensity, duty cycle, energy, and polarization.The field of elastic electron-nucleus scattering started with the availability of electron beams with energies up to 550 MeV at the High Energy Physics Laboratory (HEPL) in Stanford in the mid-1950s. One notable result of these early experiments was the first determination of a proton radius <cit.>, which, together with the anomalous magnetic moment of the proton, discovered in 1933 by Otto Stern <cit.>, completed the picture ofthe proton as a finite-size object with an internal structure. The utility of electron-nucleon scattering as a probe of nucleon structure derives from the validity of the single virtual photon exchange (Born) approximation, up to radiative corrections that are modest in size compared to the leading (Born) term, and precisely calculable in low-order QED perturbation theory, due to the small value of the fine structure constant α = e^2/4πϵ_0 ħ c≈ 1/137.036<cit.>. This allows for a theoretically “clean” extraction of the electromagnetic structure of the target from the measured scattering observables such as cross sections and polarization asymmetries.In the Born approximation, the effect of the proton's internal structure on the Lorentz-invariant elastic ep → ep scattering amplitude is completely specified by two form factors (FFs), which encode the interaction of the pointlike electromagnetic current of the electron with the proton's charge and magnetic moment distributions. The “Dirac” form factor F_1 describes the charge and Dirac magnetic moment interactions, while the “Pauli” form factor F_2 describes the anomalous magnetic moment interaction. F_1 and F_2 are real-valued functions of the Lorentz-invariant four-momentum transfer squared between the electron and the nucleon, defined as Q^2 ≡ -q^2 = -(k-k')^2, with k and k' the four-momenta of the incident and scattered electron. In fixed-target electron scattering, q^2 is a spacelike invariant that is always negative. The reaction kinematics and physical observables are thus typically discussed in terms of the positive-definite quantity Q^2. A detailed overview of the theoretical formalism of theBorn approximation for elastic ep scattering is given in Ref. <cit.>. An equivalent description of the nucleon electromagnetic form factors (EMFFs) is provided by the so-called “Sachs” form factors <cit.> G_E (electric) and G_M (magnetic), defined as the following experimentally convenient independent linear combinations of F_1 and F_2,G_E ≡F_1-τ F_2G_M ≡F_1 + F_2,in which τ≡Q^2/4M_p^2, with M_p the mass of the proton. In terms of the Sachs form factors, the differential cross section for elastic ep scattering in the Born approximation is given in the nucleon rest frame (which coincides with the lab frame in fixed-target experiments) by the Rosenbluth formula <cit.>:dσ/dΩ_e = (dσ/dΩ_e)_Mottϵ G_E^2 + τ G_M^2/ϵ(1+τ),(dσ/dΩ_e)_Mott = α^2 cos^2θ_e/2/4 E_e^2 sin^4θ_e/2E'_e/E_e,in which (dσdΩ_e)_Mott represents the theoretical Born cross section for electron scattering from a pointlike, spinless target of charge e, E_e is the beam energy, E'_e is the scattered electron energy, θ_e is the electron scattering angle, and ϵ≡(1+2(1+τ)tan^2 θ_e/2)^-1 is the longitudinal polarization of the virtual photon. The expression (<ref>) provides a simple technique for the extraction of G_E^2 and G_M^2 known as Rosenbluth or L/T (for longitudinal/transverse) separation, in which the differential cross section is measured at fixed Q^2 while varying the parameter ϵ.A plot of the ϵ dependence of the “reduced” cross section, obtained by dividing the measured, radiatively corrected cross section by the Mott cross section and the kinematic factor in the denominator of Eq. (<ref>), yields a straight line with a slope (intercept) equal to G_E^2 (τ G_M^2).Until the late 1990s all (or most) form factor measurements suggested that both G_E^p and G_M^p decreased like 1/Q^4 at large Q^2, and that the ratio μ_p G_E^p/G_M^p was approximately equal to one, regardless of Q^2. It also appeared that the dipole form G_D ≡(1+Q^2/Λ^2)^-2, with Λ^2 = 0.71 GeV^2, provided a reasonable description of G_E^p, G_M^p/μ_p and G_M^n/μ_n, as illustrated in Figs. <ref> and <ref> (for G_E^p and G_M^p). G_E^n was expected to have an entirely different Q^2 dependence, given the zero net charge of the neutron, whichimposes G_E^n=0 at Q^2=0.The helicity structure of the single-photon-exchange amplitude also gives rise to significant double-polarization asymmetries, with different sensitivities to the form factors compared to the spin-averaged cross section. Non-zero asymmetries occur in the case where the electron beam is longitudinally polarized[The effects of transverse polarization of the electron beam are suppressed by factors of m_e/E_e, leading to asymmetries of order 10^-5 in experiments with ultra-relativistic electrons at GeV-scale energies. In the context of electromagnetic form factor measurements in the Q^2 regime of this work, these effects are negligible compared to the asymmetries for longitudinally polarized electrons and the precision with which they are measured.], and either the target nucleon is also polarized or the polarization transferred to the recoiling nucleon is measured. The polarization transferred to the recoil proton in the scattering of longitudinally polarized electrons byunpolarized protons has only two non-zero components, longitudinal, P_ℓ, and transverse, P_t, with respect to the momentumtransfer and parallel to the scattering plane <cit.>:P_t=-h P_e √(2ϵ(1-ϵ)/τ)G_EG_M/G_M^2+ϵ/τG_E^2P_ℓ =h P_e √(1-ϵ^2)G_M^2/G_M^2+ϵ/τG_E^2 G_E/G_M =-P_t/P_ℓ√(τ(1+ϵ)/2ϵ).Here h denotes the sign of the electron beam helicity, and P_e is the electron beam polarization. The observables for scattering on a polarized proton target are related to those for polarization transfer by time-reversal symmetry <cit.>. Specifically, the transverse asymmetry A_t = P_t, while the longitudinal asymmetry A_ℓ = -P_ℓ. The sign change between A_ℓ and P_ℓ is caused by the proton spin flip required to absorb transversely polarized virtual photons.The interest in measuring these double-polarization observables is multi-faceted. First, the ratio G_E/G_M is directly and linearly proportional to the ratio P_t/P_ℓ in the recoil polarization case or, equivalently, the ratio A_t/A_ℓ of the beam-target double-spin asymmetries in the polarized target case. Compared to the Rosenbluth method, polarization observables provide enhanced sensitivity to G_E (G_M) at large (small) values of Q^2. Moreover, polarization observables provide an unambiguous determination of the relative sign of G_E and G_M, whereas the Rosenbluth method is only sensitive to the squares of the form factors. Finally, because of the ratio nature ofthe asymmetries, radiative corrections tend to be negligible, whereas they can and do affect the cross section measurements and Rosenbluth separations significantly, especially in kinematics where the relative contribution of either the ϵ G_E^2 or the τ G_M^2 term to the Born cross section (<ref>) is small. The polarization transfer method in particular is highly attractive, as a simultaneous measurement of both recoil polarization components in a polarimeter facilitates a very precise measurement of G_E/G_M in a single kinematic setting, with small systematic uncertainties resulting from cancellations of quantities such as the beam polarization, the polarimeter analyzing power, and the polarimeter instrumental asymmetry. In recent years the nucleon's elastic form factors have attracted steadily increasing attention, due in part to the unexpected results of the first polarization transfer measurement of the ratioat JLab. This increasing attention is evident in the number of reviews of the subject published in the last 15 years <cit.>. The first measurement of G_E^p/G_M^p by recoil polarization took place in 1994, at the MIT-Bates laboratory, at Q^2 values of 0.38 and 0.50 GeV^2, with 5% statistical uncertainties <cit.>.The first two polarization transfer experiments at JLab, hereafter denoted GEp-I <cit.> and GEp-II <cit.>, consisted of measurements of the ratio R ≡μ_p G_E^p/G_M^p for 0.5 ≤ Q^2 (GeV^2) ≤ 5.6. Together, the results of GEp-I and GEp-II, shown in Fig. <ref>, established conclusively that the concept ofscaling of the proton form factor ratio had to be abandoned. There is a cleardiscrepancy between the values of G_E^p/G_M^p extracted from double polarizationexperiments, and those obtained from cross section measurements. Among possible explanations for this discrepancy, the most thoroughly investigated is the hard two-photon exchange (TPEX) process, the amplitude for which does not “factorize” from the underlying nucleon structure information, cannot presently be calculated model-independently, and is neglected in the “standard” radiative corrections to experimental data. A recent overview of the theory, phenomenology and experimental knowledge of TPEX effects in elastic ep scattering is given in Ref. <cit.>.In the general case, elastic eN scattering can be described in terms of three complex amplitudes <cit.>, which can be written as G̃_M, G̃_E, and F̃_3, the first two chosen as generalizations of the Sachselectric and magnetic form factors, G_E and G_M, and thelast one, F̃_3, being 𝒪(α) relative to the Born terms and vanishing in the Born approximation. The “generalized form factors” G̃_M and G̃_E can be decomposed into sums of the real-valued Sachs form factors appearing in the Born amplitudes and depending only on Q^2, plus 𝒪(α) complex-valued corrections that vanish in the Born approximation and depend on both Q^2 and ϵ as follows:G̃_M(Q^2,ϵ) ≡ G_M(Q^2)+δG̃_M(Q^2,ϵ)G̃_E(Q^2,ϵ) ≡ G_E(Q^2)+δG̃_̃Ẽ(Q^2, ϵ).In terms of the generalized complex amplitudes, the reduced cross section σ_R ≡ϵ(1+τ)/τσ/σ_Mott and polarization observables are given at next-to-leading order in α by: σ_R =G_M^2 + ϵ/τ G_E^2 + 2 G_M (δG̃_M + ϵν/M^2F̃_3) + 2ϵ/τ G_E (δG̃_E + ν/M^2F̃_3), P_t=-hP_e/σ_R√(2ϵ(1-ϵ)/τ)[G_E G_M + G_M (δG̃_E + ν/M^2F̃_3) + G_E (δG̃_M) ], P_ℓ = hP_e/σ_R√(1-ϵ^2)[G_M^2 + 2G_M ( δG̃_M +ϵ/1+ϵν/M^2F̃_3 )] , P_n= √(2ϵ(1+ϵ)/τ)1/σ_R[-G_M ( δG̃_E + ν/M^2F̃_3) + G_E (δG̃_M + 2ϵ/1+ϵν/M^2F̃_3)], R≡-μ_p √(τ(1+ϵ)/2ϵ)P_t/P_ℓ = μ_p G_E/G_M[1 - δG̃_M/G_M + δG̃_E/G_E + νF̃_3/M^2((1+ϵ)G_M - 2ϵ G_E/(1+ϵ)G_EG_M)], in which ϵ and τ are defined as above, the symbolsanddenote real and imaginary parts of the amplitudes, and ν/M^2≡√(τ(1+τ)1+ϵ/1-ϵ). Thereduced cross section and the polarization transfer components P_t and P_ℓ are defined only by the real parts of the two-photon amplitudes. Thenormal polarization transfer component, P_n, which is zero in the Born approximation, is defined by the imaginary parts of the two-photon exchange amplitudes. There are several noteworthy features of Eqs. (<ref>)-(<ref>). The corrections to the reduced cross section beyond the Born approximation are additive with the Born terms, implying that even a small TPEX correction can seriously obscure the extraction of G_E^2 (G_M^2) at large (small) Q^2 when the relative contribution of either Born term to σ_R^Born is small enough to be comparable to the TPEX correction. The ratio R defined in Eq. (<ref>), on the other hand, is directly proportional to its Born value: R = μ G_E/G_M(1+𝒪(α)), and is subject only to relative 𝒪(α) TPEX corrections, in principle. In the limit G_E → 0, however, the TPEX terms can become dominant even in the ratio R; the limit of Eq. (<ref>) as G_E → 0 is R → R_Born + [μδG̃_E/G_M + μν/M^2F̃_3/G_M], assuming δG̃_M/G_M ≪ 1. Whereas the ratio R measured in polarization transfer experiments only becomes significantly sensitive to TPEX corrections when R_Born is comparable to α, the reduced cross section becomes sensitive to TPEX corrections at relatively low Q^2 even for R_Born≫α. Given the superior sensitivity to G_E at large Q^2 of the ratio P_t/P_ℓ and its relative robustness against radiative and TPEX corrections as compared to the Rosenbluth method, a general consensus has emerged that the polarization transfer data provide the most reliable determination of G_E^p in the Q^2 range where cross section and polarization data disagree.Nevertheless, a large amount of experimental and theoretical effort is ongoing to understand the source of the discrepancy and develop a maximally model-independent prescription for TPEX corrections to elastic ep scattering observables.The subject of this article is the third dedicated series of polarization transfer measurements in elastic e⃗p scattering at large Q^2, carried out in Jefferson Lab's (JLab's) Hall C from October, 2007 to June, 2008. Experiments E04-108 (GEp-III) and E04-019 (GEp-2γ) used the same apparatus and method to address two complementary physics goals. The goal of GEp-III was to extend the kinematic reach of the polarization transfer data forG_E^p/G_M^p to the highest practically achievable Q^2, given the maximum electron beam energy available at the time. The goal of GEp-2γ was to measure the ϵ-dependence of G_E^p/G_M^p at the fixed Q^2 of 2.5 GeV^2 with small statistical and systematic uncertainties, in order to test the polarization method and search for signatures of TPEX effects in two polarization observables. The results of GEp-III <cit.> and GEp-2γ <cit.> have already been published in short-form articles. The purpose of this article is to provide a detailed description of the apparatus and analysis methods common to both experiments and report the results of a full reanalysis of the data, carried out with the aim of reducing the systematic and, in the GEp-2γ case, statistical uncertainties. Our reanalysis of the GEp-2γ data includes the previously unpublished results of the full-acceptance analysis at ϵ = 0.632 and ϵ = 0.783, for which the acceptance-matching cuts applied to suppress certain systematic effects in the analysis of the originally published data <cit.> have been removed. The final results reported in this work supersede the originally published results. Section <ref> describes the experiment apparatus and kinematics in detail. Section <ref> presents the details of the data analysis. Section <ref> presents the final results of both experiments and discusses the general features of the data. A brief overview of the theoretical interpretation of high-Q^2 nucleon FF data is given in Section <ref>, while the implications of the GEp-2γ data for the understanding of TPEX contributions in elastic ep scattering and the discrepancy between cross section and polarization data for G_E^p/G_M^p are discussed in Section <ref>. Our conclusions are summarized in Section <ref>.§ EXPERIMENT DESCRIPTION Longitudinally polarized electrons with energies up to 5.717 GeV produced by JLab's Continuous Electron Beam Accelerator Facility (CEBAF) were directed onto a liquid hydrogen target in experimental Hall C.Elastically scattered protons were detected by the High Momentum Spectrometer (HMS), equipped with a double Focal Plane Polarimeter (FPP) to measure their polarization. Elastically scattered electrons weredetected by a large-solid-angle electromagnetic calorimeter (BigCal) in coincidence with the scattered protons. The main trigger for the event data acquisition (DAQ) was a coincidence between the single-arm triggers of the HMS and BigCal within a 50-ns window. Details of the coincidence trigger logic and the experiment data acquisition can be found in Ref. <cit.>.Table <ref> shows the central kinematics and running periods of the GEp-III and GEp-2γ experiments. The two running periods at E_e ≈ 1.87 GeV were combined and analyzed together as a single kinematic setting. The same is true of the running periods at E_e = 3.548 GeV and E_e = 3.680 GeV. In both cases, the near-total overlap of the Q^2 and ϵ acceptances of two distinct measurements differing only slightly in beam energy and HMS central angle justifies combining the two settings into a single measurement[In this context, combining the data from two distinct measurements means combining all events from each of the two kinematically similar settings in a single unbinned maximum-likelihood extraction of P_t and P_ℓ, in which the small differences in central kinematics are accounted for event-by-event. This amounts to the assumption that P_t and P_ℓ are the same for both settings. The data were also analyzed separately and found to be consistent with this assumption.]. The beam energy for each running period quoted in Table <ref> represents the average incident beam energy during that period, and is not corrected for energy loss in the LH_2 target. The ϵ value quoted in Tab. <ref> is computed from the average incident beam energy and central Q^2 value, and differs slightly from the acceptance-averaged value, hereafter referred to as <ϵ>, and the “central” value ϵ_c quoted with the final GEp-2γ results, which is computed from the central Q^2 value and the average[Where data from kinematically similar settings have been combined, the “central” ϵ value quoted with the final result represents a weighted average of the “central” values from each of the combined settings.] beam energy, corrected event-by-event for energy loss in the LH_2 target materials upstream of the reconstructed scattering vertex (see Tab. <ref> and <ref>).CEBAF consists of two antiparallel superconducting radio-frequency (SRF) linear accelerators (linacs), each capable (ca. 2007-2008) of approximately 600 MeV of acceleration, connected by nine recirculating magnetic arcs, with five at the north end and four at the south end. With this “racetrack” design, the electron beam can be accelerated in up to five passes through both linacs, for a maximum energy of approximately 6 GeV before extraction and delivery to the three experimental halls. Polarized electrons are excited from a “superlattice” GaAs photocathode using circularly polarized laser light. Details of the CEBAF accelerator design and operational parameters are described in Refs. <cit.>, while more details specific to the running period of the GEp-III and GEp-2γ experiments can be found in Ref. <cit.>. The typical beam current on target during the experiment was 60-100 μA, while the typical beam polarization was 80-86%. The beam helicity was flipped pseudorandomly <cit.> at a frequency of 30 Hz throughout the experiment.During normal operations, the Hall C arc magnets, which steer the beam extracted from the CEBAF accelerator to Hall C, are operated in an achromatic tune. For a measurement of the beam energy, the arc magnets are operated in a dispersive tune. The central bend angle of the arc is 34.3^∘.The field integral of the arc magnets has been measured as a function of the power supply current. The beam position and arc magnet current setting information are used in the feedback system which stabilizes the beam energy and position. This system has been calibrated using dedicated arc beam energy measurements from Halls A and C, and is used for continuous monitoring of the beam momentum. Table <ref> shows the Hall C arc measurements of the beam energy performed during the GEp-III and GEp-2γ experiments. The arc energy of E_e = 5.717 GeV measured at the beginning of the Q^2 = 8.5 GeV^2 running in April 2008 differs slightly from the average beam energy for this run period and the subsequent 6.8 GeV^2 running, shown in Table <ref>. During the 8.5 GeV^2 running, a number of slight changes in accelerator tune to optimize the performance of CEBAF in the context of simultaneous delivery of longitudinally polarized beam to Halls A and C at different passes resulted in several slight changes in beam energy at the 1-2 MeV level. While no additional arc energy measurements were performed, the small, occasional changes in beam energy were detected by the online beam energy monitoring system, and also confirmed by shifts in the elastic peak position in the variables used for elastic event selection in the offline analysis (see section <ref>). These small changes were included in the final beam energy database for the offline analysis. Except for the first few days at 8.5 GeV^2, during which the beam energy was 5.717 GeV, the actual incident beam energy varied between 5.710 and 5.714 GeV during most of the 8.5 GeV^2 running, averaging 5.712 GeV. The incident beam energy was stable at 5.711 GeV during the 6.8 GeV^2 running. As discussed in section <ref> and Ref. <cit.>, the contribution of the systematic uncertainty in the beam energy to the total systematic uncertainties in the polarization transfer observables is a small fraction of the total. The target system used for this experiment consists of several different solid targets and a three-loop cryogenic target system for liquid hydrogen (LH_2). The solid targets include thin foils of Carbon and/or Aluminum used for spectrometer optics calibrations and to measure the contribution of the walls of the cryotarget cell to the experiment background. The spectrometer optics calibrations and systematic studies are described in detail in Ref. <cit.>, while details of the solid targets are described in Ref. <cit.>. For the first kinematic point taken from Nov. 7-20, 2007,a 15-cm LH_2 cryotarget cell was used. For all of the other production kinematics of both experiments, a 20-cm cryotarget cell was used. The center of the 20-cm cell was offset 3.84 cm downstream of the origin along the beamline to allow electrons scattered by up to 120 degrees to exit through the thin scattering chamber exit window and be detected by the calorimeter.The liquid hydrogen targets were operated at a constant temperature of 19 K and nominal density of ρ≈ 0.072 g/cm^3 throughout the experiment. The size of the beam spot on target was enlarged to a transverse size of typically 2 × 2 mm^2 by the Hall C fast raster magnet system, to minimize localized heating and boiling of the liquid hydrogen and resulting fluctuations in target density and luminosity. More details of the cryogenic target system can be found in Ref. <cit.>.§.§ Hall C HMSThe High Momentum Spectrometer (HMS) is part of the standard experimental equipment in JLab's Hall C. It is a superconducting magnetic spectrometer with three quadrupoles and one dipole arranged in a QQQD layout. The HMS has a 25-degree central vertical bend angle and point-to-point focusing in both the dispersive and non-dispersive planes when operated in its “standard” tune. The HMS dipole field is regulated by an NMR probe and is stable at the 10^-5 level, while the quadrupole magnet power supplies are regulated by current and are stable at the 10^-4 level. The HMS solid angle acceptance is approximately 6.74 msr when used with the larger of its two retractable, acceptance-defining octagonal collimators, as it was in this experiment. The HMS momentum acceptance is approximately ± 9% relative to the central momentum setting. The maximum central momentum setting is 7.4 GeV/c. The HMS detector package and superconducting magnets are supported on a common carriage that rotates on concentric rails about the central pivot of Hall C. The detector package is located inside a concrete shield hut supported on a separate carriage from the detector and magnet supports. With the exception of small air gaps between the scattering chamber exit window and the HMS entrance window and between the HMS dipole exit window and the first HMS drift chamber, the entire flight path of charged particles through the HMS is under vacuum, minimizing energy loss and multiple scattering prior to the measurement of charged particle trajectories.As shown in Fig. <ref>, the HMS detector package was modified by removing the gas Cherenkov counter and the two rearmost planes of scintillator hodoscopes from the standard HMS detector package to accommodate the Focal Plane Polarimeter (FPP), leaving only the two upstream planes of scintillators (“S1X” and “S1Y”) to form a fast trigger. The HMS calorimeter was not removed, and its signals were recorded to the data stream, but it was not used either in the trigger or in the offline analysis, except for crude pion rejection in the analysis of the HMS optics calibration data, for which the HMS was set with negative polarity for electron detection. The standard HMS drift chambers, described in detail in Ref. <cit.>, were used to measure the trajectories of elastically scattered protons. The measured proton tracks were then used to reconstruct the event kinematics at the target and to define the incident trajectory for the secondary polarization-analyzing scattering in the CH_2 analyzers of the FPP. Because the two rear planes of scintillators had been removed, the “S1X” and “S1Y” planes could not, by themselves, provide an adequately selective trigger for most kinematic settings of the experiment. To overcome this challenge, two additional 1 cm-thick plastic scintillator paddles were installed between the exit window of the HMS vacuum and the first HMS drift chamber, with sufficient area to cover the envelope of elastically scattered protons for all kinematic settings. These two paddles were collectively referred to as “S0”. The S0 plane reduced the trigger rate to a manageable level by restricting the acceptance to the region populated by elastically scattered protons and suppressing triggers due to inelastic processes that occur at a much higher rate for large Q^2 values.During most of the experiment, the HMS trigger required at least one paddle to fire in each of the “S1X”, “S1Y” and “S0” planes. During part of the measurement at E_e = 2.847 GeV and the entire duration of the measurements at E_e = 3.548 GeV and E_e = 3.680 GeV, for which the HMS was located at relatively large scattering angles, the trigger was based on “S1X” and “S1Y” only, as the rates were low enough to use this less-selective trigger in coincidence with the electron calorimeter. The price to pay for installing the S0 trigger plane upstream of the drift chambers is that the angular resolution of the HMS was significantly degraded due to the additional multiple scattering in S0 <cit.>. More details of the custom HMS trigger logic used for these experiments are given in Ref. <cit.>. §.§ Focal Plane PolarimeterA new focal plane polarimeter (FPP) was designed, built and installed in the HMS to measure the polarization of the recoiling protons. It consists of two CH_2 analyzer blocks arranged in series to increase the efficiency, each followed by a pairof drift chambers. A design drawing of the HMS detector package with the FPP, the HMS drift chambers and the trigger scintillator planes is shown in Figure <ref>.§.§.§ FPP AnalyzerThe FPP analyzer is made of polyethylene (CH_2). It consists of two retractable doors, each made of two blocks, allowing for the collection of “straight-through” trajectories for calibration andalignment studies. Each pair is 145 cm (tall)× 111 cm (wide)× 55 cm(thick) and made of several layers of CH_2 held together by an outer aluminum frame. To reduce theoccurrence of leakage through the seam when the doors are inserted, an overlapping step was designed into the edge of both doors. Given their substantial weight, the CH_2 blocks were supported on a different frame than the detector andattached directly to the floor of the shield hut, ensuring that theother detectors did not move while inserting or retracting the doors.The choice of CH_2 as the analyzer material was driven by a compromise among the analyzing power and optimal thickness of the material on the one hand, and the cost and space constraints within the HMS hut on the other. Measurements of the analyzing power of the reaction p⃗+CH_2 → X atDubna <cit.> showed that the overall figure of merit of the polarimeter does not increase when the analyzer thickness is increased beyond the nuclear collision length λ_T of CH_2. With this result in mind, the HMS FPP was designed as a double polarimeter with two analyzers, each approximately one λ_T thick and followed by pairs of drift chambers to measure the angular distribution of scattered protons. The analyzers and the drift chambers were designed to be large enough to have 2π azimuthal angular acceptance for transverse momenta p_T ≡ psinϑ up to 0.7 GeV/c, beyond which the polarimeter figure of merit essentially saturates.§.§.§ FPP drift chambersThe tracking system of the FPP consists of two drift chamber pairs, one after each analyzer block. All four chambers are identical in design and construction. The active area of each chamber is 164 cm (tall) × 132 cm (wide). Each chamber contains three detection planes sandwiched between cathode layers. Each detection layer consists of alternating sense wires and field wires with a spacing of 2 cm between adjacent sense wires (1 cm between a sense wire and its neighboring field wires). The wire spacing in the cathode layers, located 0.8 cm above and below the detection layers, is 3 mm. The characteristics of the different wires are given in Table <ref>. The sense wire planes have three different orientations, denoted “U”, “V”, and “X”. The stacking order along the z axis of the planes in each chamber is VXU. The “V” wires are strung along the +45^∘ line relative to the x axis and thus measure the coordinate along the -45^∘-line; i.e., v ≡x-y/√(2). The “X” wires are strung perpendicular to the x axis and thus measure the x coordinate. The “U” wires are strung along the -45^∘ line relative to the x axis and thus measure the coordinate u ≡x+y/√(2).The U and V layers have 104 sense wires each, while the X layers have 83 sense wires. Each layer within each chamber has a sense wire passing through the point (x,y) = (0,0), the geometric center of the chamber active area[The symmetry created by this common intersection point and the relative lack of redundancy of coordinate measurements, with only six coordinate measurements along each track, creates an essentially unresolvable left-right ambiguity for a small fraction of tracks passing through the region near the center of the chambers at close to normal incidence, for which two mirror-image solutions of the left-right ambiguity exist with identical combinations of drift distances that are basically indistinguishable in terms of χ^2.]. Each drift chamber is enclosed by 30 μm-thick aluminized mylar gas windows and a rigid aluminum frame. Each pair of chambersis attached to a common set of rigid spacer blocks (two on each side of the chamber frame) by a set of two aligning bolts per block penetrating each chamber. Each of the two spacer blocks along both the top and bottom sides of the chamber frame is also attached to a third threaded steel rod that goes through both chambers in the pair. The chamber pair is then mounted to the FPP support frame via C-shaped channels machined into the top spacer blocks that mate with a cylindrical Thomson rail attached to the top of the support frame, and via protrusions of the bottom spacer blocks with guide wheels that slide into a “U” channel on the bottom of the FPP support frame. After installation, each chamber pair was bolted to a hard mechanical stop built into the support frame. The design ensures that the relative positioning of the two chambers within a pair is fixed and reproducible. The FPP drift chambers used the same 50%/50% argon/ethane gas mixture as the HMS drift chambers. The basic drift cell in the FPP drift chambers has the same aspect ratio as the HMS drift cell, but the dimensions are twice as large. The cathode and field wires were maintained at a constant high voltage of -2400 V, while the sense wires were at ground potential.This operational configuration gives the FPP drift chambers similar, but not identical, electric field and drift velocity characteristics to the HMS drift chambers. The main difference is that the HMS drift chambers were operated with a different electric field configuration in which three different high voltage settings were applied to the field and cathode wires according to their distance from the nearest sense wire, leading to nearly cylindrical equipotential surfaces surrounding each sense wire. This in turn means that the drift time measured by the HMS chambers is a function of the distance of closest approach of the track to the wire, rather than the in-plane track-wire distance. Since the tracks of interest in the HMS drift chambers are very nearly perpendicular to the wire planes, the difference between these two distances is small in any case. The FPP wire signals are processed by front-end amplifier/discriminator (A/D) cards attached directly to the chambers. Each A/D card processes the signals from eight sense wires. The amplified, discriminated FPP signals are digitized by TDCs located close to the chambers within the HMS shield hut.A significant advantage of the Hall C FPP DAQ system compared to previous experiments using the Hall A FPP <cit.> is that each sense wire was read out individually by a dedicated multi-hit TDC channel, whereas the straw chamber signals in the Hall A FPP were multiplexed in groups of eight wires by the front-end electronics to reduce the number of readout channels required, effectively preventing the resolution of multi-track events in which two or more tracks create simultaneous signals on straws located within the same group of eight. As discussed in Sec. <ref>, the ability to isolate true single-track events significantly increased the effective analyzing power of the Hall C FPP relative to the Hall A FPP for equivalent analyzer material and thickness. From the start of the experiment in October 2007 to February 2008, VME-based F1 TDC modules <cit.> housed in a pair of VME crates in the HMS shield hut were used to read out the FPP signals. For the high-Q^2 data collection from April to early June of 2008, the FPP signals were read out using LeCroy 1877-model Fastbus TDCs. The FPP data acquisition was changed from VME to Fastbus TDCs due to relatively frequent malfunctions of the VME DAQ system encountered during the GEp-2γ production running, especially for the data taken at the relatively forward HMS central angle of 14.5 degrees, for which the detector hut was fairly close to the beam dump and the hit rates in the FPP chambers were relatively high. Since no such problems were observed with the Fastbus TDCs used concurrently to read out the HMS drift chambers, a second Fastbus crate equipped with LeCroy 1877 TDC modules was installed in the HMS shield hut during the planned two-month accelerator shutdown[The purpose of this accelerator down was to install refurbished cryomodules in CEBAF to reach the maximum beam energy of 5.7 GeV needed for the high-Q^2 running of GEp-III.] in February and March of 2008 in preparation for the high-Q^2 running at an HMS angle of 11.6 degrees. As expected based on the experience with the HMS drift chamber readout, the Fastbus TDC readout for the FPP drift chambers functioned fairly smoothly throughout the 2008 high-Q^2 running. §.§ Electron Calorimeter Elastically scattered electrons were detected by an electromagnetic calorimeter, named BigCal, built specifically for this experiment. The calorimeter was made of 1,744 lead-glass blocks (TF1-0 type) stackedwith a frontal area of 122 × 218 cm^2. The array was constructed from blocks of two different sizes. The bottom part of the calorimeter consisted of a 32 × 32 array of blocks with dimensions of 3.8 × 3.8 × 45 cm^3 originating from the IHEP in Protvino, Russia, while the top part of the calorimeter consisted of a 30 × 24 array of blocks with dimensions of 4 × 4 × 40 cm^3 from the Yerevan Physics Institute in Yerevan, Armenia, used previously in a Compton scattering measurement in Hall A <cit.>. The 45-cm (40-cm) depth of the Protvino (Yerevan) blocks corresponds to 16.4 (14.6) radiation lengths, sufficient to absorb the total energy of elastically scattered electrons. The Cherenkov light created in the glass by relativistic particlesfrom the electromagnetic cascade was registered by photomultiplier tubes (PMTs) of type FEU-84, coupled optically to the end of each block with a 5 mm-thick transparent silicon "cookie" to compensate for a possible misalignment between the two elements. The blocks were optically isolated from each other via an aluminized mylar wrapping. For each kinematic setting, the calorimeter was positioned at an angle corresponding to the central Q^2 value and beam energy. The distance from the origin to the surface of BigCal was chosen to be as large as possible, consistent with matching between the solid angle acceptance of BigCal for elastically scattered electrons and the fixed solid angle of the HMS for elastically scattered protons. For the kinematics at E_e = 3.548 GeV and 3.680 GeV (see Tab. <ref>), BigCal was placed closer to the target than the acceptance-matching distance due to limitations imposed by the signal cable length and the location of the BigCal readout electronics, as well as the available space in Hall C. At Q^2 = 8.5 GeV^2, the electron solid angle for acceptance matching was 143 msr, or about twenty times the solid angle acceptance of the HMS.Fig. <ref> shows BigCal with the front shielding plates removed, revealing the array of lead-glass blocks. The analog signals from the PMTs were sent to specialized NIM modules for amplification and summing, with eight input channels each. The outputs included copies of the individual input signals amplified by a factor of 4.2, and several copies of the analog sum of the eight input signals. The amplified analog signals from the individual PMTs were sent to LeCroy model 1881M charge-integrating Fastbus ADCs for readout. One copy of each “first level” sum of eight blocks was sent to a fixed-threshold discriminator, the output of which was then sent to a TDC for timing readout. Additional copies of each sum of eight were combined with other sums-of-eight into “second-level” sums of up to 64 blocks using identical analog summing modules. These “level 2” sums, of which there are a total of 38, were also sent to fixed-threshold discriminators, and a global “OR” of all the second-level discriminator outputs was used to define the trigger for BigCal. The groupings of blocks for the “level 2” sums were organized with partial overlap to avoid regions of trigger inefficiency, as detailed in Ref. <cit.>. Because there was no overlap in the trigger logic between the left and right halves of the calorimeter, the trigger threshold was limited to slightly less than half of the average elastically scattered electron energy. A higher threshold would have resulted in significant efficiency losses at the boundary between the left and right halves of the calorimeter.Four one-inch thick aluminum plates(for a total of about one radiation length) were installed in front of the glass to absorb low-energy photons and mitigate radiation damage to the glass. This additional material degrades the energy resolution, but does not significantly affect the position resolution.All four aluminum plates were used for all kinematics except the lowest ϵ point of the GEp-2γ experiment, for which only one plate was used.For this setting, the calorimeter was placed at the backward angle of θ_e ≈ 105^∘, for which the elastically scattered electron energy was only E'_e ≈ 0.54 GeV, the radiation dose rate in the lead-glass was low enough that the additional shielding was not needed, and the better energy resolution afforded by removing three of the four plates was needed to maintain high trigger efficiency at the operating threshold.The glass transparency gradually deteriorated throughout the experiment due to accumulated radiation damage. The effective gain/signal strength in the BigCal blocks was monitored in situ throughout the experiment using the known energy of elastically scattered electrons, reconstructed precisely from the measured proton kinematics. The PMT high voltages were periodically increased to compensate for the gradual decrease in light yield and maintain a roughly constant absolute signal size, in order to avoid drifts in the effective trigger threshold and other deleterious effects. However, as discussed in Ref. <cit.>, the reduced photoelectron yield caused the energy resolution to deteriorate. With the four-inch-thick aluminum absorber in place, the energy resolution worsened from about 10.9%/√(E) following the initial calibration to roughly 22%/√(E) at the end of the experiment. During the early 2008 accelerator shutdown, the glass waspartially annealed using a UV lamp system but it did not fully recover to its initial transparency and energy resolution prior to the start of the high-Q^2 running in April 2008, at which point the transparency resumed its gradual deterioration. The achieved energy resolution, while relatively poor for this type of detector and dramatically worsened by radiation damage, was nonetheless adequate for triggering with the threshold set at half the elastically scattered electron energy or less. In contrast to the energy resolution, the position resolution of BigCal, estimated to be roughly 6 mm using the Q^2 = 6.8 GeV^2 data collected at the end of the experiment <cit.>, did not change noticeably during the experiment. The achieved coordinate resolution of BigCal was significantly better than needed given the experimentally realized angular, momentum and vertex resolution of the HMS, and proved essential for the suppression of the inelastic background, especially at high Q^2, as discussed in section <ref>. More details of the calibration and event reconstruction procedures for BigCal can be found in Refs. <cit.>.§ DATA ANALYSIS The analysis of the data proceeds in three phases:* Decoding of the raw data and the reconstruction of events * The selection of elastic ep events and the estimation of the residual contamination of the final sample by inelastic backgrounds and accidental coincidences* The extraction of the polarization transfer observables from the measured angular distributions of protons scattered in the FPP. The raw data decoding and the event reconstruction procedure, including detector calibrations and reconstruction algorithms, are described in the technical supplement to this article <cit.> as well as the Ph.D. thesis <cit.>. The elastic event selection and background estimation procedure are discussed in Sec. <ref>. The extraction of polarization observables is presented in Sec. <ref>. The detailed evaluation of systematic uncertainties is presented in Refs. <cit.>. §.§ Elastic event selection Elastic events were selected using the two-body kinematic correlations between the electron and the proton. Accidental coincidences were suppressed by applying a loose, ± 10 ns cut to the time-of-flight-corrected difference Δ t between the timing signals associated with the electron shower in BigCal and the proton trigger in the fast scintillator hodoscopes of the HMS. The resolution of the coincidence time difference Δ t is dominated by the timing resolution of BigCal, which varied from 1.5-2 ns depending on the electron energy.The contamination of the data by accidental coincidences within the ± 10 ns cut region was less than 10% before applying the exclusivity cuts described below, and negligible after applying the cuts. The transferred polarization components for the accidental coincidence events were found to be similar to those of the real coincidence events for the inelastic background <cit.>, such that the accidental contamination of the inelastic background sample at the level of 10% or less did not noticeably affect the corrections to the elastic ep signal polarizations, which were essentially negligible except at Q^2 = 8.5 GeV^2. The beam energy is known with an absolute accuracy Δ E/E ≲ 5 × 10^-4 from the standard Hall C “arc” measurement technique. The “per-bunch” beam energy spread under normal accelerator operating conditions is typically less than 3 × 10^-5 and is continuously monitored using synchrotron light interferometry <cit.>, while the CEBAF fast energy feedback system maintains the “long term” stability of the central beam energy at the 10^-4 level <cit.>. The spread and systematic uncertainty in the electron beam energy is significantly smaller than the HMS momentum resolution of σ_p/p ≈ 10^-3, and its contribution to the systematic uncertainty in the determination of the reaction kinematics is small. The scattering angles and energies/momenta of both outgoing particles are measured in each event. Because the energy resolution of BigCal was too poor to provide meaningful separation between elastic and inelastic events for any cut with a high efficiency for elastic events, no cuts were applied to the measured energy of the electron, beyond the hardware threshold imposed by the BigCal trigger and the software threshold imposed by the clustering algorithm. This leaves the proton momentum and the polar and azimuthal scattering angles of the electron and proton as useful kinematic quantities for the identification of elastic events. Figure <ref> shows a simplified version of the procedure for isolating elastic ep events in the GEp-III data using the two-body kinematic correlations between the electron detected in BigCal and the proton detected in the HMS. Similar plots for the GEp-2γ kinematics can be found in Ref. <cit.>. The proton momentum p_p and scattering angle θ_p in elastic scattering are related by:p_p(θ_p)= 2M_pE_e(M_p+E_e)cos (θ_p)/M_p^2 + 2M_p E_e + E_e^2 sin^2 (θ_p).The difference δ p_p ≡ 100 ×p_p-p_p(θ_p)/p_0, where p_0 is the central momentum of the HMS, provides a measure of “inelasticity” for the detected proton independent of any measurement of the electron kinematics. The δ p_p spectra exhibit significant inelastic backgrounds before applying cuts based on the measured electron scattering angles, especially at Q^2 = 8.5 GeV^2.The scattered electron's trajectory is defined by the straight line from the reconstructed interaction vertex to the measured electron impact coordinates at the surface of BigCal. The correlation between the electron polar scattering angle θ_e and the proton momentum p_p was expressed in terms of the difference δ p_e ≡ 100 ×p_p - p_p(θ_e)/p_0, where p_p(θ_e) is calculated from elastic kinematics as follows:E'_e(θ_e)= E_e/1+E_e/M_p(1-cosθ_e), Q^2(θ_e)=2E_e E'_e(θ_e) (1-cosθ_e), p_p (θ_e)= √(Q^2(θ_e)(1+τ(θ_e))),with τ(θ_e) ≡Q^2(θ_e)/4M_p^2. Finally, coplanarity of the outgoing electron and proton is enforced by applying a cut to δϕ≡ϕ_e - ϕ_p - π. The azimuthal angles of the detected particles are defined in a global coordinate system in which the distribution of ϕ_e (ϕ_p) is centered at +π/2 (-π/2), such that co-planarity implies ϕ_e = ϕ_p + π for all elastic ep events within the detector acceptances.The simplified elastic event selection procedure shown in Fig. <ref> corresponds to fixed-width, ±3σ cuts centered at zero for all variables. It should be noted, however, that for the final analysis, cuts of variable width (mean) were applied to δ p_p (δϕ) to account for observed variations of the width (position) of the elastic peak within the HMS acceptance (for details, see <cit.>). While the differences in statistics and analysis results between the full procedure and the simple procedure of Fig. <ref> are small for sufficiently wide cuts, the full procedure optimizes the effective signal-to-background ratio and efficiency of the elastic event selection procedure, and suppresses cut-induced systematic bias in the reconstructed proton kinematics. In contrast to δ p_p and δϕ, the resolution of δ p_e is approximately constant within the acceptance, and mostly dominated by the HMS momentum resolution.In general, the observed correlations of δ p_e with the reconstructed proton kinematics are small compared to experimental resolution. Moreover, the extracted polarization transfer observables are generally less sensitive to the systematic error in the reconstructed proton momentum than to the errors in the reconstructed proton angles, which dominate the experimental resolution of δ p_p and δϕ. The results are thus less susceptible to systematic bias induced by the δ p_e cut than that induced by the δ p_p and δϕ cuts, given the experimentally realized angular and momentum resolution of the HMS. Therefore, a fixed-width, ±3σ cut centered at zero was applied to δ p_e for all kinematics, which has the added benefit of simplifying the estimation of the residual background contamination of the final elastic event sample, as shown in Fig. <ref> and discussed below.For electron scattering from hydrogen, elastically scattered protons have the highest kinematically allowed momenta for positively charged particles at a given θ_p. Events at δ p_p < 0 are dominated by inelastic reactions on hydrogen, including π^0 photoproduction (γ p →π^0 p) near the Bremsstrahlung end point (E_γ→ E_e), with one or both π^0 decay photons detected by BigCal, and, to a lesser extent, π^0 electroproduction (ep → e' π^0 p) near threshold, with the scattered electron detected in BigCal. At the multi-GeV energies characteristic of these experiments, the kinematic separation between the ep and π^0p reactions in terms of δ p_p is comparable to the experimental resolution, such that there is significant overlap between the π^0p and ep reactions in the vicinity of the elastic peak. The 20-cm liquid hydrogen target is itself a ∼2.2% radiator, creating a significant “external” Bremsstrahlung flux along the target length in addition to the real and virtual photon flux present in the electron beam independent of the target thickness[For example, at Q^2 = 8.5 GeV^2, the observed fractional contamination by inelastic backgrounds of the final sample of events selected as elastic increases by a factor of 1.6 from the upstream end of the target to its downstream end.]. Events at positive δ p_p (the so-called “super-elastic” region) originate from quasi-elastic and inelastic scattering in the aluminum entry and exit windows of the liquid hydrogen target cell, and from non-Gaussian tails of the HMS angular and/or momentum resolution. Because the aluminum window thickness is only ∼5% of the total target thickness by mass (12% by radiation length), and the exclusivity cut variables are smeared by Fermi motion of the nucleons in aluminum, the contribution of scattering from the target end windows to the total event yield is essentially negligible (≲ 10^-3) after the cuts. The residual peaks at zero in the δ p_e and δϕ spectra of rejected events result from radiative effects and non-Gaussian tails of the experimental resolution. In particular, the remnant peaks in the δϕ distributions of rejected events contain significant contributions from the elastic radiative tail, because events affected by radiation from the incident electron beam (coherent or incoherent with the hard scattering amplitude) are strongly suppressed by both the δ p_e and δ p_p cuts without affecting the co-planarity of the outgoing particles.Figure <ref> illustrates the procedure for estimating the residual background contamination in the final sample of elastic events. By far the worst case for background contamination after applying exclusivity cuts is Q^2 = 8.5 GeV^2, for which the contamination approaches 5% for ±3σ cuts. The δ p_e distribution of the background in the vicinity of the elastic peak after applying cuts to δ p_p and δϕ is well approximated by a Gaussian distribution, as was confirmed by examining the events rejected by the δ p_p and/or δϕ cuts, as well as by Monte Carlo simulations of the main background processes. The shape of the elastic ep radiative tail in the δ p_e distribution was also well-reproduced by Monte Carlo simulations with radiative corrections to the unpolarized cross section following the formalism described in Ref. <cit.>.In Fig. <ref>, the δ p_e distribution of the background was fitted with a Gaussian by excluding the region -1.6%≤δ p_e ≤ 0.5% in which the elastic peak and radiative tail contributions are significant. The residual background contamination was then estimated by extrapolating the Gaussian fit of the background into the elastic peak region. Table <ref> shows the estimated, acceptance-averaged fractional background contamination of the final, ±3σ cuts used for all six kinematics.The inelastic contamination estimates shown in Table <ref> are determined directly from the data, but are not used directly in the final analysis, because the background contamination and the transferred polarization components of the background both vary strongly as a function of δ p_p within the final cut region, as the dominant background process evolves from π^0p photo/electro-production to quasielastic Al(e,e'p). The π^0 p contribution rises rapidly for negative δ p_p values as the kinematic threshold is crossed, whereas the δ p_p distribution of the (very small) target endcap contamination is relatively uniform within the cut region. The recoil proton polarization for the inelastic π^0p reaction on hydrogen generally differs strongly from that of the elastic ep process, while the proton polarization in quasi-elastic Al(e,e'p) is generally similar to elastic ep, since it is basically the same process embedded in a nucleus (see Figs. <ref> and <ref>). Fig. <ref> shows the δ p_p dependence of the fractional background contamination f for Q^2 = 8.5 GeV^2, the setting with (by far) the greatest residual background contamination. Details of the background subtraction procedure are given in section <ref> and the systematic uncertainties associated with the background subtraction are presented in Ref. <cit.>.The stability of the transferred polarization components with respect to the width of the elastic event selection cuts and the amount of background included in the final event sample was checked by varying the width of the δ p_p, δϕ, and δ p_e cuts independently between ±2.5σ and ±3.5σ and observing the variations in the background-corrected results. The observed variations of P_t, P_ℓ, and the ratio P_t/P_ℓ were compatible with purely statistical fluctuations for all kinematics. Therefore, no additional systematic uncertainty contributions were assigned. The cut sensitivity study also confirmed that the application of cuts that were adequately loose and carefully centered with respect to the elastic peak eliminated any cut-induced systematic bias of the reconstructed proton kinematics. This was a non-trivial concern for this analysis given the exaggerated effect of multiple-scattering in “S_0” on the event-by-event errors in the reconstructed proton angles and the very high sensitivity of the spin transport calculation to systematic errors in these angles, particularly the non-dispersive-plane angle ϕ_tar (see Ref. <cit.> for a detailed discussion). §.§ Extraction of Polarization Transfer Observables§.§.§ FPP Angular DistributionAn expression for the general angular distribution in the polarimeter is given by:N^±(p,ϑ,φ)=N_0^±ε(p,ϑ)E(ϑ,φ)/2π× [ 1 ± A_y (P_y,tr^FPPcosφ - P_x,tr^FPPsinφ) ..+ A_y (P_y,ind^FPPcosφ - P_x,ind^FPPsinφ )] ,where N_0^± is the number of incident protons corresponding to a ±1 beam helicity state, ε(p,ϑ) is the fraction of protons of momentum p scattered at a polar angle ϑ and producing one single track, E(ϑ,φ) represents the angular dependence of the combined effective polarimeter acceptance/detection efficiency, which factorizes from the differential nuclear scattering cross section, A_y = A_y(p,ϑ) represents the analyzing power of p⃗+CH_2→one charged particle + X scattering, and P_x,tr/ind^FPP and P_y,tr/ind^FPP are the transverse components of the proton polarization at the focal plane, with P_tr (P_ind) denoting transferred (induced) polarization. As explained below, only the φ dependence of the detector acceptance/efficiency is relevant for polarimetry.Note that for all kinematics of the GEp-III and GEp-2γ experiments, N_0^+ = N_0^- = N_total/2 to within statistical uncertainties. This is a consequence of the beam-helicity independence of the elastic e⃗p scattering cross section for an unpolarized target (in the one-photon-exchange approximation), and the rapid (30 Hz) helicity reversal, which cancels the effects of slow drifts in experimental conditions such as luminosity and detection efficiency.As described in <cit.>, the azimuthal scattering angle φ was defined in a coordinate system that is comoving with the incident proton, in which the HMS track defines the z axis, the y axis is chosen to be perpendicular to the HMS track, but parallel to the yz plane of the fixed TRANSPORT coordinate system (see Ref. <cit.>), and the x axis is defined by x̂ = ŷ×ẑ. In this coordinate system, φ is the azimuthal angle of the scattered proton trajectory measured clockwise from the x axis toward the y axis. Note that this convention for the definition of φ differs from the convention used in the analysis of the GEp-I and GEp-II experiments <cit.>. In the GEp-I and GEp-II analyses, φ was defined such that φ = 0 for scattering along the +y axis, and φ was measured counterclockwise from the y axis toward the x axis (see Eq. (4) of Ref. <cit.>). With φ defined as in the GEp-III/GEp-2γ analysis, the sin(φ) asymmetry is dominant, whereas the cos(φ) asymmetry is dominant using the GEp-I/GEp-II convention.In the one-photon-exchange approximation in elastic ep scattering, the induced polarization terms are identically zero due to time reversal invariance. When two photons are exchanged, a non-zero induced polarization of elastically scattered protons can occur at subleading order in α due to the interference between the one-photon and two-photon-exchange amplitudes. Because it is subleading order in α, it is not expected to exceed ≃1-2% in magnitude <cit.>, and must be normal to the ep scattering plane due to parity invariance of the electromagnetic interaction. The helicity-independent azimuthal asymmetry resulting from a small induced polarization at this level is smaller yet as the analyzing power does not exceed roughly 20% at any (p,ϑ) in these experiments. The “false” or instrumental asymmetry resulting from the effective acceptance/efficiency function E(φ) can be expressed in terms of its Fourier expansion:E(φ)=C[1+∑_m=1^∞ (c_m cos(mφ) + s_m sin(mφ))] ≡C[1 + μ_0(φ)],with an overall multiplicative constant C that is ultimately absorbed into the overall normalization of the distributions when integrating over the dependence on kinematic variables other than φ. A clean extraction of the transferred polarization components is obtained from the difference and/or the difference/sum ratio between the angular distributions for positive and negative beam helicities, integrated over all momenta within the HMS acceptance and a limited ϑ range chosen to exclude small-angle Coulomb scattering and large-angle scatterings for which A_y ≈ 0. The helicity difference and sum distributions are given by:f^+-f^- ≡ π/Δφ[N^+(φ)N_0^+-N^-(φ)N_0^-]= A̅_y [P_y,tr^FPPcosφ-P_x,tr^FPPsinφ] × [1+μ_0(φ)] ≈ A̅_y [P_y,tr^FPPcosφ - P_x,tr^FPPsinφ] f^++f^- ≡ π/Δφ[N^+(φ)/N_0^++N^-(φ)/N_0^-] = [1+μ_0(φ)] × [1+ A̅_y(P_y,ind^FPPcosφ - P_x,ind^FPPsinφ)] ≈1 + μ_0(φ)where Δφ is the bin width in φ and A̅_y is the average analyzing power within the range of ϑ considered[Note also that in the context of Eqs. (<ref>)- (<ref>), N_0^± is the total number of incident protons corresponding to beam helicity ± 1 producing a detected scattering event within the accepted ϑ range.]. The difference-sum ratio is given byf_+ - f_-/f_+ + f_- = A̅_y (P_y,tr^FPPcosφ - P_x,tr^FPPsinφ)/1 + A̅_y(P_y,ind^FPPcosφ - P_x,ind^FPPsinφ)≈ A̅_y (P_y,tr^FPPcosφ - P_x,tr^FPPsinφ)2f_±/f_+ + f_- =1 ±A̅_y (P_y,tr^FPPcosφ - P_x,tr^FPPsinφ)/1 +A̅_y(P_y,ind^FPPcosφ - P_x,ind^FPPsinφ)≈1 ±A̅_y (P_y,tr^FPPcosφ - P_x,tr^FPPsinφ)where in Eqs. (<ref>)-(<ref>), the induced polarization terms in the denominator are neglected. Equations (<ref>)-(<ref>) show that the false asymmetries and/or the induced polarization terms are cancelled by the beam helicity reversal in the different asymmetry observables. The helicity-difference distribution cancels the induced polarization terms but is sensitive at second order to the false asymmetry μ_0, while the difference-sum ratio cancels the false asymmetry terms, but is sensitive at second order to any induced polarization terms. The helicity-sum distribution cancels the transferred polarization terms, but includes contributions from false asymmetries and any induced polarization terms, if they exist. The transferred polarizations, the induced polarizations, and the false asymmetry terms can all be rigorously separated, in principle, via Fourier analysis of the distributions (<ref>)-(<ref>), assuming infinite statistical precision. In practice, however, it is very statistically and systematically challenging to separate the induced polarization terms from the false asymmetry terms when both are “small”, as is the case in this experiment, especially for the induced polarization terms. For the transferred polarization components, on the other hand, it can be shown <cit.> that the false asymmetry effects are cancelled exactly to all orders by the beam helicity reversal in the linearized maximum-likelihood estimators for P_t and P_ℓ defined in section <ref> below, given sufficient statistical precision that the sums over all events entering the maximum-likelihood estimators are a good approximation to the corresponding weighted integrals over the azimuthal angular distribution discussed in <cit.>.§.§.§ FPP event selection criteria Useful scattering events for polarimetry were selected according to several criteria, detailed in Ref. <cit.>. First, only single-track events were included in the analysis of each polarimeter, as the analyzing power for events with two or more reconstructed tracks in either polarimeter was found to be much lower than that of the single-track events, such that even a separate analysis of the multi-track events did not meaningfully improve the polarimeter figure-of-merit in a weighted average with the single-track events. Secondly, cuts were applied to the parameters s_close, defined as the distance of closest approach between incident and scattered tracks, and z_close, defined as the z-coordinate of the point of closest approach between incident and scattered tracks. A loose, ∼ 10σ upper limit for s_close was chosen to optimize the statistical precision of the analysis, by excluding events at large s_close values with low analyzing power. The z_close ranges considered for FPP1 and FPP2 events correspond to the physical extent of the CH_2 analyzers (L_CH_2 = 55 cm) plus a small additional tolerance (Δ z = ± 2.5 cm) to allow for the resolution of z_close while excluding the “unphysical” region close to (and including) the drift chambers themselves. A “cone test” was applied to each candidate scattering event, to minimize instrumental asymmetries in the φ distribution arising from the geometrical acceptance of the FPP, and to guarantee full 2π azimuthal acceptance over the full range of (ϑ, z_close) values included in the analysis. Simply defined, the cone test requires that the projection of the cone of opening angle ϑ from the reconstructed interaction vertex z_close to the rearmost wire plane of the FPP drift chamber pair that detected the track lie entirely within the active area of the chamber for all possible azimuthal scattering angles φ. This in turn guarantees that the effective range of ϑ integration is the same for all φ values, such that the average analyzing power is φ-independent. As a result, the analyzing power, which depends strongly on ϑ, cancels reliably in the ratio of polarization components P_y^FPP/P_x^FPP at the focal plane and P_t/P_ℓ at the target, regardless of the range of ϑ included in the analysis. Due to the large active area of the FPP drift chambers, the efficiency of the cone test is close to 100% for scattering angles up to about 30 degrees. The details of the cone test calculation are given in <cit.>.The useful range of ϑ varies with Q^2, because the width of the multiple-Coulomb-scattering peak at small ϑ and the angular distributions of both the scattering probability and the analyzing power are observed to scale approximately as 1/p_p. The useful range of ϑ was selected for each Q^2 by applying a cut to the “transverse momentum” p_T ≡ p_p sinϑ, where ϑ is the proton's polar scattering angle in the FPP, and p_p is the incident proton momentum. The value of p_p used in the definition of p_T is corrected for the mean energy loss along the path length in CH_2 traversed by the incident proton prior to the scattering.For all three ϵ values at Q^2 = 2.5 GeV^2, the range of p_T included in the analysis was 0.06 GeV ≤ p_T ≤ 1.2 GeV for both polarimeters. A slightly wider range 0.05 GeV ≤ p_T ≤ 1.5 GeV was used for the GEp-III kinematics, for which the uncertainties are statistics-limited. For all kinematics, the low-p_T cutoff is large compared to the intrinsic angular resolution of the FPP drift chambers, which is about 1.9 (2.1) mrad in the x (y) direction. In the worst case, at 8.5 GeV^2, the 0.05 GeV minimum p_T corresponds to a minimum ϑ of about 9 mrad or 4.5σ. More details of the FPP event selection criteria, p_T distributions, track multiplicities per event, and closest approach parameters can be found in Ref. <cit.>.§.§.§ Focal plane azimuthal asymmetries Figure <ref> shows the ratio of the helicity-difference and helicity-sum azimuthal distributions A ≡ (f_+(φ) - f_-(φ))/(f_+(φ)+f_-(φ)), defined in Eq. (<ref>), for each of the GEp-2γ kinematics, for each polarimeter separately, fitted with a function A = c cosφ - s sinφ. The fit results are shown in Table <ref>. The asymmetries are consistent with a pure sinusoidal φ dependence, and Fourier analysis including a constant term and higher harmonics up to 8φ showed no statistically significant evidence for the presence of terms other than cosφ and sinφ, as expected from Eq. (<ref>). This suggests that the beam helicity reversal does an excellent job of suppressing the instrumental asymmetries, which are significant at certain values of ϑ and z_close. The FPP1 and FPP2 asymmetries are mostly consistent with each other, and are always consistent in terms of the ratio c/s = P_y^FPP/P_x^FPP, or equivalently, in terms of the phase of the asymmetry, since the analyzing power cancels in this ratio.For the GEp-2γ kinematics, the use of identical event selection criteria for all three ϵ values eliminates, in principle, point-to-point systematic variations of the effective average analyzing power arising from the cuts on the scattering parameters ϑ, s_close, and z_close. Figure <ref> shows the difference/sum ratio asymmetry (f_+ - f_-)/(f_+ + f_-) for the GEp-III kinematics, for both polarimeters combined. For the GEp-III kinematics, the combined asymmetries are also compatible with a purely sinusoidal φ dependence, albeit with much lower statistical precision. The asymmetry amplitude at Q^2 = 8.5 GeV^2 is larger than for the other two kinematics despite the lower analyzing power, because of the precession of the proton spin in the HMS. The central precession angle at Q^2 = 8.5 GeV^2 is close to 270 degrees, and the asymmetry magnitude is maximal at sinχ = ± 1. In contrast, the central precession angle for Q^2 = 5.2 GeV^2 is close to 180 degrees, such that the acceptance-averaged asymmetry is close to zero. However, as shown in Fig. <ref> and discussed below, the χ acceptance of the HMS for each Q^2 point is wide enough to provide sufficient sensitivity to P_ℓ, and the precision of the form factor ratio extraction is not dramatically affected by the unfavorable precession angle, since P_ℓ is quite large (58%-98%) in all the kinematics of these experiments.Table <ref> summarizes the focal-plane helicity-difference asymmetry fit results. For each of the Q^2 = 2.5 GeV^2 kinematics, the FPP1 and FPP2 asymmetries are fitted separately, while the results shown for the GEp-III kinematics are for FPP1 and FPP2 combined. For Q^2 = 5.2 GeV^2, the asymmetry results are also fitted separately for precession angles χ < π and χ≥π, illustrating the expected sign change of s, the -sin(φ) coefficient of the asymmetry. If Q^2 were chosen such that the HMS acceptance were centered exactly at χ = π, and if the effects of quadrupole precession were absent, we would expect the values of s for χ < π and χ≥π to be equal and opposite. However, the central value of χ for Q^2 = 5.2 GeV^2 is 177.2^∘, such that the HMS acceptance extends to slightly greater |sin(χ)| for χ < π than for χ≥π (see also Fig. <ref>). Moreover, as discussed in Ref. <cit.>, the mixing of P_t and P_ℓ due to quadrupole precession shifts the “expected” location of the zero crossing of the -sin(φ) coefficient of the asymmetry to about 180.4 degrees instead of the nominal 180 degrees. Both of these effects lead to the expectation of a slightly larger sin(φ) asymmetry for χ < π than for χ≥π, as observed. Figure <ref> shows the raw φ distributions f_+, f_-, f_+ + f_- and 2f_±/(f_+ + f_-) for the GEp-2γ kinematics. Similar results with lower statistical precision are obtained for the GEp-III kinematics. The normalized distributions 2f_±/(f_+ + f_-) are consistent with the pure sinusoidal behavior predicted by Eq. <ref> for all kinematics and for both polarimeters separately. The helicity sum distribution f_+ + f_-, which cancels the asymmetry due to the transferred polarization, exhibits a characteristic instrumental asymmetry with several notable features common to all kinematics. The dominant feature of the false asymmetry is a cos(2φ) term that is roughly independent of kinematics, negative, and about 2-3% in magnitude when averaged over the useful ϑ acceptance at Q^2 = 2.5 GeV^2. This asymmetry appears at small ϑ as a consequence of the x/y resolution asymmetry of the FPP drift chambers and at large ϑ due to acceptance/edge effects, and is generally small at intermediate ϑ values near the maximum of the analyzing power distribution (see Sec. <ref>). Although the “cone test” (see Section <ref>) is designed to eliminate acceptance-related false asymmetries, it cannot do so completely because it is applied based on the reconstructed parameters of the incident and scattered tracks, which are affected in a φ-dependent way by the FPP x/y resolution asymmetry. The other prominent feature of the false asymmetry is the presence of small peaks at 45-degree intervals corresponding to the FPP drift chamber wire orientations. The peaks are absent at φ = 0 deg., 180 deg.,and 360 deg., angles corresponding to scattering along the dispersive (x) direction. These artificial peaks are caused by incorrect solutions of the left-right ambiguity due to the irreducible ambiguity in the drift chambers' design, resulting from the symmetry of the wire layout and the lack of redundancy of coordinate measurements. These incorrect solutions occur primarily for small-angle tracks traversing the chambers at close to normal incidence near the center of the drift chambers, where the x, u, and v wires share a common intersection point in the xy plane. When an incorrect left-right assignment occurs for events in the Coulomb peak of the ϑ distribution, the reconstructed track position at one or both sets of drift chambers is incorrectly placed on the opposite side of all three wires that fired in that drift chamber. If the left-right assignment of the hits in one chamber (but not the other) in a pair is incorrect, the reconstructed point of closest approach “collapses” to the location of the chamber for which the left-right combination was correctly assigned, and the value of φ “collapses” to one of the three different wire orientations depending on the topology of the event and the measured drift distances of the incorrectly assigned hits. The overwhelming majority of these mistracked events are rejected by the z_close cut, which excludes the unphysical region corresponding to the drift chambers themselves. However, for z_close values within the analyzer region but close to the chambers, some of these mistracked events leak into the “good” event sample due to detector resolution, producing the pattern of small, residual artificial peaks observed in (f_+ + f_-)(φ). These “mistracked” events have low/zero analyzing power and tend to dilute the asymmetry in the z_close region closest to the drift chambers. In principle, they can be further suppressed by excluding the part of the analyzer region closest to the drift chambers. In practice, this is unnecessary, because the instrumental asymmetry they generate is cancelled by the beam helicity reversal, and the resulting dilution of the effective average analyzing power cancels in the ratio of polarization components, such that they cause no systematic effect whatsoever on the extraction of R. The effect of the mistracked events on the average analyzing power, which is important for the extraction of the ϵ dependence of P_ℓ/P_ℓ^Born, is measured and accounted for, and is the same for all three ϵ values at 2.5 GeV^2. The sensitivity of the measured P_ℓ/P_ℓ^Born ratio to the range of z_close and p_T included in the analysis was examined and found to be small compared to the statistical and systematic uncertainties in this observable.§.§.§ FPP efficiencyTable <ref> summarizes the total elastic ep statistics collected and the effective “efficiency” of the FPP, defined as the fraction of incident protons producing a useful secondary scattering for polarimetry. The raw FPP wire efficiencies and angular distributions were examined on a run-by-run basis, and runs with data quality issues in either FPP1 or FPP2 (or both) were rejected for the polarimeter in question. The total number of elastic events shown in Table <ref>, which serves as the denominator for the efficiency determination, is not corrected for runs rejected from the analysis because of FPP data quality issues. In other words, FPP-specific data losses due to transient malfunctioning of the data acquisition system for either set (or both sets) of FPP drift chambers during runs of otherwise good data quality are included in the effective efficiencies shown[Since the FPP1 and FPP2 drift chambers were read out by different VME crates during most of the experiment prior to the switch to Fastbus DAQ during the high-Q^2 running from April-June of 2008, a somewhat common occurrence was a data acquisition run with only one of the two sets of drift chambers providing usable data.]. These data losses are responsible for reducing the experimentally realized efficiency for FPP1 by several percent for the Q^2 = 2.5 GeV^2 data at <ϵ> = 0.153, compared to what it would have been in the ideal case, and by smaller amounts for other kinematic settings. Overall, the combined efficiency per incident proton for producing a scattering event passing all the event selection criteria ranges from 22% at Q^2 = 8.5 GeV^2 to nearly 39% at (Q^2, <ϵ>)=(2.5 GeV^2, 0.79).The use of two polarimeters in series, each with an analyzer thickness of one nuclear interaction length λ_T, leads to an efficiency gain of approximately 50% relative to the use of a single polarimeter with one λ_T analyzer thickness, regardless of p_p. §.§.§ HMS Spin TransportThe asymmetries measured by the FPP are proportional to the transverse components of the proton polarization at the HMS focal plane (see equation (<ref>)), which are related to the reaction-plane transferred polarization components P_t and P_ℓ by a rotation due to the precession of the proton polarization in the HMS magnetic field. The HMS is a focusing spectrometer characterized by its 25-degree central vertical bend angle and relatively small angular acceptance in both the dispersive and non-dispersive directions. The precession of the polarization of charged particles with anomalous magnetic moments moving relativistically in a magnetic field is described by the Thomas-BMT equation <cit.>. The spin transport for protons (anomalous magnetic moment κ_p ≈ 1.79) through the HMS is dominated by a rotation in the dispersive plane by an angle χ≡γκ_p θ_bend relative to the proton trajectory, where γ≡ E_p/M_p is the usual relativistic γ factor and θ_bend is the trajectory bend angle in the dispersive plane. In this so-called “ideal dipole” approximation, the dispersive plane component of the proton polarization precesses by an angle χ, and the non-dispersive plane component does not rotate, such that P_x^FPP≈ -sinχ P_e P_ℓ and P_y^FPP = P_e P_t. Figure <ref> illustrates the dominant dipole precession of the proton spin using the ratio A_x^FPP/<A_y>P_e P_ℓ, where A_x^FPP = A_y P_x^FPP is the -sinφ coefficient of the asymmetry (f_+ - f_-)/(f_+ + f_-) (see Eq. (<ref>)), <A_y> is the average analyzing power within the accepted range of scattering angles, and P_e is the average beam polarization. In the ideal dipole approximation, A_x^FPP = -<A_y> P_e P_ℓsinχ. The ideal dipole approximation accounts for most of the observed χ dependence of the asymmetry. The Q^2 = 5.2 GeV^2 data show how it is possible to achieve good precision on the ratio P_t/P_ℓ even when the acceptance-averaged asymmetry is close to zero; due to the large value of P_ℓ and the relatively large χ acceptance of the HMS, the relative statistical uncertainty Δ P_ℓ/P_ℓ is almost a factor of four smaller than Δ P_t/P_t.Deviations from the ideal dipole approximation arise due to the quadrupole magnets and the finite angular acceptance of the HMS. The forward spin transport matrix of the HMS depends on all the parameters of the proton trajectory at the target and must be calculated for each event. Owing to the relatively simple magnetic field layout and small angular and momentum acceptance of the HMS, it was not necessary to perform a computationally expensive numerical integration of the BMT equation for each proton trajectory. Instead, a general fifth-order expansion of the forward spin transport matrix in terms of all the trajectory parameters at the target was fitted to a sample of random test trajectories populating the full acceptance of the HMS that were propagated through a detailed COSY <cit.> model of the HMS including fringe fields. Unlike the parameters describing charged particle transport through the HMS, which are independent of the central momentum setting for the standard tune, the spin transport coefficients had to be computed separately for each central momentum setting, since the spin precession frequency relative to the proton trajectory is proportional to γ. The use of the same central momentum setting for all three kinematics at Q^2 = 2.5 GeV^2 ensures that the magnetic field and the spin transport matrix are the same for all three kinematics. This in turn minimizes the point-to-point systematic uncertainties in the polarization transfer observables, which is essential in the accurate determination of their ϵ dependence. The fitted expansion coefficients of the COSY spin transport model express the matrix elements of the absolute total rotation of the proton spin in the fixed TRANSPORT coordinate system. The total rotation of the proton spin relevant to the extraction of the polarization transfer observables also includes a rotation from the reaction plane coordinate system in which P_t and P_ℓ are defined to the TRANSPORT coordinate system that is fixed with respect to the HMS optical axis at the target, and a rotation from the fixed TRANSPORT coordinate system at the focal plane to the coordinate system comoving with the proton trajectory, in which the polar and azimuthal scattering angles ϑ and φ are defined. Details of the calculation of the total rotation matrix for each event are given in Ref. <cit.>.§.§.§ Maximum-Likelihood Extraction of P_t, P_ℓ, and R The transferred polarization components P_t and P_ℓ are extracted from the measured FPP angular distributions using an unbinned maximum-likelihood estimator. Neglecting induced polarization terms, the likelihood function is defined up to an overall normalization constant independent of P_t and P_ℓ asℒ(P_t, P_ℓ)= ∏_i=1^N_eventE(φ_i)/2π× { 1 + h_i P_e A_y^(i)[ (S_yt^(i) P_t + S_yℓ^(i) P_ℓ) cosφ_i- (S_xt^(i) P_t + S_xℓ^(i) P_ℓ)sinφ_i ] },where E(φ_i) ∝ 1 + ∑_n [c_n cos (nφ_i) + s_n sin(nφ_i)] is the false/instrumental asymmetry for the i-th event[In principle, the false asymmetry Fourier coefficients can depend on ϑ, p and any other parameters of the event such as s_close, z_close], h_i = ± 1 is the beam helicity state for the i-th event, P_e is the beam polarization, A_y^(i)≡ A_y(p_i, ϑ_i) is the analyzing power of p⃗ +CH_2 scattering, which depends on the proton momentum p_i and scattering angle ϑ_i, and the S_jk^(i)'s are the forward spin transport matrix elements relating polarization component P_k in reaction-plane coordinates to component P_j in the comoving coordinate system of the secondary analyzing reaction measured by the FPP. The product over all events in Eq. (<ref>) was converted to a sum by taking the logarithm, and then the problem of maximizing lnℒ as a function of the parameters P_t and P_ℓ was linearized by truncating the expansion of ln(1+x) at quadratic order in x; i.e., ln(1+x) = x - x^2/2 + 𝒪x^3. In this context, “x” represents the sum of all the φ-dependent terms in Eq. (<ref>). The largest acceptance-averaged helicity-dependent “raw” asymmetry observed in either experiment was about 0.12 (see Fig. <ref>), while the largest raw asymmetry observed at any ϑ was about 0.16. The acceptance-averaged helicity-independent false/instrumental asymmetries are at the few-percent level[The magnitude of the cos(2φ) false asymmetry arising from the x/y resolution asymmetry and the acceptance of the FPP drift chambers rises to the ∼ 10% level at the extremes of the accepted ϑ range.]. It is therefore estimated that the maximum truncation error in Δ (lnℒ)/lnℒ due to the linearization procedure is |x - x^2/2/ln(1+x)-1| ≲ 0.82% at any ϑ, and smaller when averaged over the full ϑ acceptance. The linearized maximum-likelihood estimators for P_t and P_ℓ are given by the solution of the following linear system of equations:∑_i[[(λ_t^(i))^2 λ_t^(i)λ_ℓ^(i); λ_t^(i)λ_ℓ^(i)(λ_ℓ^(i))^2 ]][[ P̂_t; P̂_ℓ ]] =∑_i [[ λ_t^(i); λ_ℓ^(i) ]],where λ_t^(i) and λ_ℓ^(i) given byλ_t^(i) ≡h_i P_e A_y^(i)(S_yt^(i)cosφ_i - S_xt^(i)sinφ_i )λ_ℓ^(i) ≡h_i P_e A_y^(i)(S_yℓ^(i)cosφ_i - S_xℓ^(i)sinφ_i )are the coefficients of P_t and P_ℓ in the equation for the likelihood function (<ref>). Note that the false asymmetry E(φ) does not enter the definition of the estimators. Up to the effects of spin precession, the estimators defined by Eq. (<ref>) are equivalent to the “weighted sum” estimators of Ref. <cit.>. In Ref. <cit.> it was shown that these estimators are unbiased and efficient, and in particular that the instrumental asymmetries are cancelled to all orders by the beam helicity reversal, which provides an effective detection efficiency that is 180-degree symmetric; i.e., E(φ) = E(φ + π). The equation for the maximum likelihood estimators can be rewritten in matrix form as A𝐏 = 𝐛, the solution of which is 𝐏 = A^-1𝐛. The symmetric 2× 2 matrix A^-1, with A defined by the 2× 2 matrix on the LHS of Eq. (<ref>), is the covariance matrix of the parameters 𝐏. The standard statistical variances in P_t and P_ℓ are given by the diagonal elements of A^-1, while the covariance of P_t and P_ℓ is given by the off-diagonal element:Δ P_t= √((A^-1)_tt) Δ P_ℓ = √((A^-1)_ℓℓ) cov(P_t,P_ℓ)= (A^-1)_tℓ = (A^-1)_ℓ tThe ratio R ≡ -μ_p √(τ(1+ϵ)/2ϵ)P_t/P_ℓ≡ -K P_t/P_ℓ, which equals μ_p G_E^p/G_M^p in the one-photon-exchange approximation, is computed from the results of the maximum-likelihood analysis for P_t and P_ℓ. The uncertainty in R is computed using the standard prescription for error propagation using the covariance matrix A^-1 discussed above:(Δ R/R)^2= (Δ P_t/P_t)^2 + (Δ P_ℓ/P_ℓ)^2 - 2cov(P_t, P_ℓ)/P_t P_ℓAlthough it is not immediately obvious from Eq. (<ref>), both the beam polarization and the analyzing power cancel in the ratio P_t/P_ℓ. All of the matrix elements on the LHS of (<ref>) are proportional to (P_eA_y)^2, while the components of the vector on the RHS of (<ref>) are proportional to P_e A_y. The estimators P̂_t and P̂_ℓ are thus proportional to (P_e A_y )^-1, and the statistical variances in P_t and P_ℓ are proportional to (P_e A_y)^-2. Strictly speaking, the cancellation of A_y in the ratio P_t/P_ℓ requires that the effective range of integration in ϑ (or equivalently p_T) be independent of φ, which is guaranteed in principle by the application of the cone test. According to the χ^2 of a constant fit, the extracted ratio R showed no statistically significant p_T dependence for any of the six kinematic settings, as detailed in Ref. <cit.>.Figure <ref> shows the Q^2 dependence of P_t, P_ℓ and R within the HMS acceptance for the GEp-2γ kinematics. As discussed below, P_t and P_ℓ are equal to their Born approximation values for <ϵ> = 0.153 by definition, since this point is used for the analyzing power calibration at 2.5 GeV^2. At a fixed central Q^2 of 2.5 GeV^2, the fixed angular acceptance of the HMS corresponds to a Q^2 acceptance that is roughly three times greater at <ϵ> = 0.790 than at <ϵ> = 0.153. The observed Q^2 dependence of R within the acceptance of each kinematic setting is statistically compatible with both the expected R(Q^2) and a constant R value. The observed Q^2 dependences of P_t and P_ℓ are also similar to those of P_t^Born and P_ℓ^Born, providing important added confirmation that both the HMS spin transport and the momentum dependence of A_y (see Sec. <ref>) are accounted for correctly. The P_ℓ values at <ϵ> = 0.790 show a clear excess over P_ℓ^Born. The curve for R(Q^2) shown in Fig. <ref> is obtained from the global fit to proton form factor data described in appendix <ref>. The kinematic factors τ, ϵ and K are computed from the beam energy E_e and the proton momentum p_p for each event. The value of K is averaged over all events in computing the acceptance-averaged R value from the acceptance-averaged unbinned maximum-likelihood estimators for P_t and P_ℓ. Q^2 and ϵ are one-to-one correlated within the acceptance at each setting due to the fixed beam energy. P_t and P_ℓ depend on both Q^2 and ϵ, and can vary significantly within the acceptance of a single measurement. R depends only on Q^2 (in the Born approximation), and its expected variation within the acceptance is generally smaller than that of P_t or P_ℓ. The correlated (ϵ, Q^2) acceptances of all kinematic settings are small enough that, to within experimental precision, P_t, P_ℓ, and R vary linearly with Q^2 within the acceptance, and the acceptance-averaged values of P_t, P_ℓ and R equal their values at the acceptance-averaged kinematics. The choice to use the measured quantities E_e and p_p to compute Q^2 and ϵ is not unique; the reaction kinematics in ep → ep are fixed by choosing any two of E_e, E'_e, θ_e, θ_p, p_p. The choice of any two of these five variables gives equivalent results; radiative effects on the average kinematics of the final elastic ep sample are suppressed to a negligible level by the tight exclusivity cuts. The use of the beam energy E_e and the proton momentum p_p to compute the event kinematics is optimal because the beam energy is known with a high degree of certainty, Q^2 depends only on the proton momentum in elastic ep scattering, and the systematic uncertainty in p_p is easily quantifiable and independent of ϵ for a fixed HMS central momentum setting/nominal Q^2 value. §.§.§ Analyzing Power Calibration The analyzing power A_y is not a priori known. However, the elastic ep process is “self-calibrating” with respect to the analyzing power, as it can be extracted directly from the measured asymmetries, provided the beam polarization is known. The Møller measurement of the electron beam polarization is subject to a global uncertainty of approximately 1% and point-to-point uncertainties of (Δ P_e/P_e)_ptp = 0.5%. The ratio R = -K P_t/P_ℓ does not depend on the beam polarization or the analyzing power because both of these quantities cancel in the ratio P_t/P_ℓ. Moreover, in the one-photon-exchange approximation, the values of P_t and P_ℓ depend only on the ratio r ≡ G_E/G_M ≡ R/μ_p, and not on G_E or G_M separately. In terms of r, Eq. (<ref>) becomes (for P_e = 1):P_t^Born =-√(2ϵ(1-ϵ)/τ)r/1+ϵ/τr^2P_ℓ^Born = √(1-ϵ^2)/1+ϵ/τr^2 The average analyzing power in a particular bin of ϑ and/or p_p is determined by computing the maximum-likelihood estimators P̂_t and P̂_ℓ assuming A_y = 1, with P_e taken from the Møller measurements, and forming the ratios to P_t^Born and P_ℓ^Born:P̂_t^(A_y = 1) = A̅_y P_tP̂_ℓ^(Ay=1) = A̅_y P_ℓ A̅_y= P̂_t^(A_y=1)/P_t^Born = P̂_ℓ^(A_y=1)/P_ℓ^BornThe value of A_y in any kinematic bin is computed from a weighted average of A_y(P̂_t) ≡P̂_t/P_t^Born and A_y(P̂_ℓ) ≡P̂_ℓ/P_ℓ^Born, that is usually dominated by A_y(P̂_ℓ). Although P_t and P_ℓ are determined with comparable absolute precision, P_ℓ is determined with a much better relative precision than P_t, because the magnitude of P_ℓ is several times greater than that of P_t for all kinematics.Figure <ref> shows the angular dependence of A_y, expressed in terms of the “transverse momentum” p_T, for all four Q^2 values. The shape of A_y(p_T) is qualitatively similar for all four HMS central momentum settings. The maximum A_y^max(p_T = p_T^max) was estimated by fitting each A_y(p_T) curve with the following simple parametrization:A_y(p_T)= {[ N (p_T - p_T^0)^α e^-b(p_T-p_T^0)^β,p_T ≥ p_T^0; 0,p_T < p_T^0 ]},which is positive-definite, vanishes at asymptotically large p_T and p_T < p_T^0, and is sufficiently flexible to describe the data with reasonable accuracy. An adequate description of the GEp-III data is achieved by fixing the exponents (α, β) = (1,2) and the zero-offset p_T^0 = 0 and varying only the normalization constant N and the “slope” b of the exponential. The data at 2.5 GeV^2 are precise enough that all five parameters had to be varied to achieve a good description, and in contrast to the GEp-III data, strongly favor a “zero offset” p_T^0 ≈ 0.05 GeV.Table <ref> shows the best-fit parameters and their uncertainties, and the resulting values for A_y^max and p_T^max, defined, respectively, as the maximum value of the analyzing power and the p_T value at which it occurs. The p_T^max values exhibit some variation with Q^2 but are statistically compatible with a constant value p_T^max≈ 0.4 GeV (see Tab. <ref>). Figure <ref> shows the proton momentum dependence of A_y^max and A̅_y, the average analyzing power within the accepted p_T range, compared to selected existing measurements in p⃗ +CH_2 scattering in the few-GeV momentum range, including GEp-II <cit.> in JLab's Hall A and dedicated measurements performed at the JINR in Dubna, Russia <cit.>. It is worth remarking that the GEp-II data were obtained with two different analyzer thicknesses; the lowest-Q^2 (largest p_p^-1) measurement used a CH_2 thickness of 58 cm, which is similar to the thickness used in Hall C for each of the two FPPs, while the three measurements at higher Q^2 used a thickness of 100 cm, leading to an apparent reduction in A_y that was at least partially offset by an increase in the efficiency. The linear fits to the GEp-II data shown in Fig. <ref> only include the three highest-Q^2 points, which used the same analyzer thickness. It is also worth noting that the Dubna measurements <cit.> do not correspond to constant analyzer thickness; the Dubna measurement at p_p = 1.75 GeV used a CH_2 thickness of 37.5 g/cm^2, significantly less than the thickness used in either the Hall C or Hall A polarimeters. The Dubna measurements at higher proton momenta correspond to a range of analyzer thicknesses generally lying between the ∼50 g/cm^2 thickness used for each of the two analyzers in the Hall C double-FPP and the ∼90 g/cm^2 thickness of the GEp-II polarimeter. While the Dubna data appear to have a significantly greater slope than the JLab data, the difference is not statistically significant, given the large uncertainty of the Dubna measurement at p_p = 1.75 GeV, and the fact that this measurement corresponds to a CH_2 thickness of approximately half the thickness used for the other measurements at higher p_p.For the GEp-III/2γ experiments, A_y^max and A̅_y depend linearly on p_p^-1. Notably, the extrapolated values of A_y^max and A̅_y at asymptotically large proton momentum (1/p_p → 0) are non-zero and positive for the conditions of GEp-III/2γ, although in the case of A̅_y, the asymptotic value is only ∼ 3σ different from zero. The experimentally realized effective analyzing power for the Hall C double-FPP is substantially greater than that of the GEp-II or Dubna polarimeters at similar p_p. The difference is attributable to the Hall C drift chambers' ability, given their overall performance characteristics and the trigger and DAQ conditions specific to Hall C, to separate true single-track events from multiple-track events, revealing the significantly higher analyzing power for true single-track events compared to the totally inclusive sample. In the straw chambers of the Hall A FPP, for example, groups of eight adjacent wires in a plane were multiplexed into a single readout channel by the front-end electronics <cit.>, preventing the resolution of multiple-track events in which two or more tracks pass through the same group of eight straws within a plane simultaneously.The effective analyzing power of a given sample of p⃗+CH_2 scattering events is clearly sensitive to experimental details such as the analyzer thickness, the momentum distribution of incident protons, the tracking resolution/efficiency, the background rate/occupancy of the detectors, the trigger and data acquisition conditions, and the cuts applied to select events. For this reason, it is generally not possible to predict A_y using previous measurements such as <cit.> with sufficient accuracy for an absolute determination of P_ℓ commensurate with the statistical precision of the GEp-2γ data. Nonetheless, the relative variation of P_ℓ/P_ℓ^Born with ϵ can be precisely extracted from the GEp-2γ data by exploiting the fact that the experimental conditions which influence the effective average analyzing power are the same across all three kinematics measured at Q^2 = 2.5 GeV^2. In particular, the application of identical cuts on the FPP scattering parameters ensures that the effective average analyzing power is the same for all three ϵ values, up to differences in the momentum distribution of incident protons. As shown in Fig. <ref>, the average analyzing power for a given p_T range is inversely proportional to the proton momentum p_p, while the p_T distribution of the analyzing power is approximately independent of p_p. Given these experimental realities, the momentum dependence of A_y can be accounted for on an event-by-event basis in the maximum-likelihood analysis by assuming that the overall momentum dependence factorizes from the ϑ and/or p_T dependence:A_y(p_p, p_T)=A_y^0(p_T) p̅_̅p̅/p_p, where A_y^0(p_T) and p̅_p are, respectively, the acceptance-averaged values of A_y(p_T) and p_p. For the extraction of the ratio R, the analyzing power calibration is only relevant insofar as it optimizes the statistical figure-of-merit of the maximum-likelihood analysis by properly weighting events according to A_y. The extraction of the ϵ dependence of P_ℓ/P_ℓ^Born in the GEp-2γ analysis relies on the assumption that A_y is the same for all three measurements, up to a global p_p^-1 scaling that factorizes from the p_T dependence according to Eq. (<ref>). The lowest-ϵ data (<ϵ> = 0.153) were used to determine the common A_y^0(p_T) for the GEp-2γ analysis for several reasons. First, the value of P_ℓ^Born approaches one as ϵ→ 0 as a simple consequence of angular momentum conservation, and is highly insensitive to r at <ϵ> = 0.153, such that the relative statistical uncertainty Δ P_ℓ^Born/P_ℓ^Born due to the uncertainty in r is more than three times smaller at the lowest ϵ than at either of the two higher ϵ values, and negligibly small compared to the statistical uncertainty in P_ℓ itself (see Tab. <ref>). Moreover, despite the fact that the measurement at <ϵ> = 0.153 has the worst relative statistical precision for the ratio P_t/P_ℓ, it has the best relative precision for P_ℓ due to the large magnitude of P_ℓ. §.§.§ Background SubtractionThe maximum-likelihood estimators are modified by the residual inelastic contamination of the elastic ep sample as follows:∑_i[[(λ_t^(i))^2 λ_t^(i)λ_ℓ^(i); λ_t^(i)λ_ℓ^(i)(λ_ℓ^(i))^2 ]][[ P̂_t; P̂_ℓ ]] =∑_i [[λ_t^(i)(1-λ_bg^(i)); λ_ℓ^(i)(1-λ_bg^(i) ) ]].The coefficients λ_t^(i), λ_ℓ^(i) defined by Eq. (<ref>) become:λ_t^(i) →(1-f_bg^(i)) λ_t^(i) λ_ℓ^(i) →(1-f_bg^(i))λ_ℓ^(i),with f_bg^(i) denoting the fractional background contamination evaluated at the reconstructed kinematics of the i-th event, estimated according to the procedure discussed in section <ref>. The background asymmetry term appearing on the RHS of Eq. (<ref>) is defined as:λ_bg^(i) ≡f_bg^(i)h_i P_e A_y^(i)[ (S_yt^(i)cosφ_i - S_xt^(i)sinφ_i ) P_t^inel.. + (S_yℓ^(i)cosφ_i - S_xℓ^(i)sinφ_i )P_ℓ^inel],where P_t^inel and P_ℓ^inel are the transferred polarization components of the inelastic background, which are measured using the inelastic events as described below.The residual inelastic contamination of the final selection of elastic ep events is estimated directly from the data using the procedure described in Sec. <ref>. Averaged over the acceptance of the final cuts, the fractional contamination f ranges from 0.16% for Q^2 = 2.5 GeV^2, ϵ = 0.638 to 4.89% for Q^2 = 8.5 GeV^2 (see Tab. <ref>). The measured polarizations P_t,ℓ^obs are related to the signal and background polarizations byP_t,ℓ^obs =(1-f) P_t,ℓ^el + f P_t,ℓ^inel,where P_t,ℓ^el and P_t,ℓ^inel are, respectively, the transferred polarizations of the elastic “signal” and the inelastic “background”. Figures <ref> and <ref> show the δ p_p dependence of P_t,ℓ^inel for the GEp-2γ and GEp-III kinematics, respectively. The background polarizations are extracted directly from the data by applying the maximum-likelihood method described above to the inelastic events, using the analyzing power resulting from the elastic events. Background events were selected by excluding a two-dimensional region of (δ p_e, δϕ) in which the elastic peak and radiative tail contributions are significant. The background polarizations exhibit a strong δ p_p dependence in the region of the elastic peak, a behavior explained by the different background processes involved and their relative contributions. In the inelastic region (δ p_p < 0), which is dominated by π^0p events, the background polarizations are approximately constant and differ strongly from the signal polarizations. The polarization transfer observables for γ⃗ p →π^0 p⃗, measured rather precisely as a byproduct of this experiment, are interesting in their own right, and were already the subject of a dedicated publication <cit.>, which also addressed the induced polarization, which is non-negligible for the γ⃗ p →π^0 p⃗ process. The induced polarization of the π^0p background is ignored here, as its effect on the extraction of the transferred polarization of the elastic signal is negligible. In the region of overlap with the elastic peak, the background polarizations evolve rapidly toward values that are similar (but not identical) to the signal polarizations. This transition reflects the sharp kinematic cutoff for π^0p production and the transition to a regime in which the dominant background process is quasi-elastic Al(e,e'p) scattering in the end windows of the cryotarget. The δ p_p-dependences of the contamination f and the background polarizations P_t,ℓ^inel are accounted for in the final, background-subtracted maximum-likelihood analysis. The total corrections to R, P_t and P_ℓ are dominated by the lowest δ p_p bins within the final cut region, and are slightly smaller than would be implied by correcting the acceptance-averaged results using the acceptance-averaged values of f and P_t,ℓ^inel using Eq. (<ref>).Table <ref> shows the effect of the background subtraction on P_t, P_ℓ and R. The uncertainties associated with the background subtraction procedure are discussed in Ref. <cit.>. In all cases, the correction to P_t (P_ℓ) is negative (positive), and the resulting correction to R is always positive. In general, the corrections to R and P_ℓ are very small, except in the case of Q^2 = 8.5 GeV^2, for which the size of the correction to R is comparable to the total systematic uncertainty. Despite the similar levels of inelastic contamination between <ϵ> = 0.638 and <ϵ> = 0.790 at 2.5 GeV^2, the corrections at <ϵ> = 0.790 are significantly smaller, because of the smaller differences between the signal and background polarizations.§.§ Radiative CorrectionsThe “standard”, model-independent 𝒪(α) radiative corrections (RC) to polarized elastic e⃗p scattering have been discussed extensively in Refs. <cit.>, and include standard virtual RC such as the vacuum polarization and vertex corrections, and emission of real photons (Bremsstrahlung). Radiative corrections to double-polarization observables, such as the beam-target double-spin asymmetry in scattering on a polarized target, or polarization transfer as in this experiment, tend to be smaller than the RC to the unpolarized cross sections, because polarization asymmetries are ratios of polarized and unpolarized cross sections, for which the factorized, virtual parts of the RC tend to partially or wholly cancel in the expression for the relative RC to the asymmetry. Moreover, the effect of Bremsstrahlung corrections can be suppressed by the exclusivity cuts used to select elastic events. The ratio of transferred polarization components P_t/P_ℓ, which is directly proportional to G_E^p/G_M^p in the Born approximation, is a ratio of ratios of cross sections, and is subject to RC that are typically as small as or smaller than the RC to the individual asymmetries, depending on the kinematics and cuts involved. The model-independent RC to the ratio R were estimated using the formulas described in Ref. <cit.>. The results for the relative RC to R and P_ℓ/P_ℓ^Born are shown in Table <ref>. The corrections are very small in all cases. For the ratio R, the correction is negative for every kinematic. The corrections to P_ℓ are also negligible in magnitude, and do not exceed 10^-3 for any kinematic. The upper limit on the Lorentz-invariant “inelasticity” u ≡ (k_1 + p_1 - p_2)^2, with k_1, p_1, and p_2 denoting the four-momenta of incident electron, target proton, and recoil proton, respectively, was chosen according to the effective experimental resolution of u by plotting the distribution of u for events selected by the exclusivity cuts described in Sec. <ref>. It is assumed in the calculations that only the outgoing proton is observed, and the kinematics of the unobserved scattered electron and/or the radiated hard Bremsstrahlung photon are integrated over. In reality, the tight exclusivity cuts applied to the kinematics of both the electron and proton angles and the proton momentum are such that Bremsstrahlung corrections are even more strongly suppressed than in the case of a simple cut on u reconstructed from the measured proton kinematics. The “true” model-independent RC to the ratio could be expected to be even smaller than those reported in Tab. <ref>, which can be regarded as conservative upper limits. No radiative corrections have been applied to the final results for R and P_ℓ/P_ℓ^Born reported in Sec. <ref> below, as the estimated values of the RC are essentially negligible compared to the statistical and systematic uncertainties of the data. Note also that no hard TPEX corrections are applied to the results, as there is presently no model-independent theoretical prescription for these corrections. Existing calculations give a wide variety of results, varying both in sign and magnitude, but are in general agreement that these corrections are small.§ RESULTS§.§ Summary of the dataThe final results of the GEp-III and GEp-2γ experiments are shown in Fig. <ref> and reported in Tables <ref> and <ref>. The acceptance-averaged values of the relevant observables can be considered valid at the acceptance-averaged kinematics (Q^2 and ϵ). The final results of the GEp-III experiment for R = μ_p G_E^p/G_M^p are essentially unchanged relative to the original publication <cit.>, showing small, statistically and systematically insignificant increases for all three Q^2 points, despite non-trivial modifications to event reconstruction and elastic event selection in the final analysis. The statistical uncertainties of the GEp-III data are also slightly modified, as it was discovered during the reanalysis of the data that the effect of the covariance term expressing the correlation between P_t and P_ℓ was not included in the originally published statistical uncertainties, whereas it is included in this work. The effect of the covariance term on Δ R_stat is only significant for Q^2 = 5.2 GeV^2, for which the correlation coefficient is ρ(P_t, P_ℓ) ≈ -0.17. Because P_t and P_ℓ are opposite in sign at this Q^2, a negative correlation coefficient tends to reduce the magnitude of the statistical error (see Eq. (<ref>)). The larger correlation coefficient observed at 5.2 GeV^2 compared to all the other kinematics is related to the unfavorable precession angle centered near 180 degrees and the reduced sensitivity of the measured asymmetry to P_ℓ. The final systematic uncertainties of the GEp-III data are also smaller than those originally published, as a result of more thorough analysis of the data from the study of the non-dispersive plane optics of the HMS <cit.>, which reduced the uncertainty in the total bend angle in the non-dispersive plane to Δ_systϕ_bend≈ 0.14 mrad.The values of P_ℓ^Born quoted in Tables <ref> and <ref> are acceptance-averaged values, computed event-by-event from Eq. (<ref>) using the parametrized global Q^2 dependence of R resulting from “Global Fit II” of appendix <ref>, which includes the final results of GEp-III and GEp-2γ reported in this work. The statistical uncertainty Δ P_ℓ^Born is computed at each kinematic by propagating the statistical uncertainty in R through Eq. (<ref>), and is basically negligible compared to the uncertainty in P_ℓ itself. The use of a global parametrization of R(Q^2) to calculate P_ℓ^Born is necessary for a self-consistent extraction of the ϵ dependence of P_ℓ/P_ℓ^Born at 2.5 GeV^2. For the GEp-III kinematics and the lowest ϵ measurement from GEp-2γ, which is used for the analyzing power calibration, the differences between P_ℓ^Born computed from the global parametrization of R(Q^2) and P_ℓ^Born computed directly from the measurement result for R are negligible.The results in Tables <ref> and <ref> are the product of a thorough reanalysis of the data, aimed at reducing the systematic and statistical uncertainties of the final results. The most significant difference between the analysis reported here and that of the original publications is that this work uses the full dataset of the GEp-2γ experiment to achieve a significant reduction in the statistical uncertainties. The original analysis, published in Ref. <cit.>, applied acceptance-matching cuts to the data at <ϵ> = 0.638 and <ϵ> = 0.790 to match the envelope of events at the HMS focal plane populated by the <ϵ> = 0.153 data, and further restricted the proton momentum to |δ| ≤ 2% for all three settings. These cuts selected subsamples of the data with essentially the same average Q^2, and thus the same average analyzing power, and suppressed possible ϵ-dependent systematic effects resulting from the different phase space regions populated by elastically scattered protons, including the momentum dependence of the analyzing power, “bin centering” effects, and the quality of the reconstruction of the proton kinematics and the calculation of the spin transport matrix elements. The acceptance-matching and δ cuts applied in the original analysis <cit.> reduced the total number of events by a factor of approximately 2.5(3.4) at ϵ = 0.638 (0.790) relative to the full-acceptance dataset. Subsequent analysis has shown that the momentum dependence of the analyzing power is adequately accounted for by the global p_p^-1 scaling of Eq. (<ref>), and that the HMS optics and spin transport are well-calibrated within the wider phase space regions populated by the two higher-ϵ settings (see Fig. <ref> and additional discussion in Ref. <cit.>). As a result, the statistical uncertainties in R and P_ℓ/P_ℓ^Born are significantly reduced relative to Ref. <cit.>, without increasing the systematic uncertainty. Other changes in the final analysis common to both experiments are mainly related to event reconstruction and elastic event selection. Details of the improvements in event reconstruction and elastic event selection, and the final evaluation of systematic uncertainties, can be found in Ref. <cit.>.Fig. <ref> shows the final results for the ϵ-dependence of R and P_ℓ/P_ℓ^Born. The data collected at E_e = 3.548 GeV (<ϵ> = 0.779) and E_e = 3.680 GeV (<ϵ> = 0.796) were also analyzed separately and found to be consistent. The statistical compatibility of the separately analyzed results, the similarity of the average kinematics of the two settings, and the near-total overlap of their Q^2 and ϵ ranges justifies combining these two measurements into the single result reported in Tab. <ref> and shown in Fig. <ref>. For both observables, the final results are consistent with the originally published results, but with significantly smaller statistical uncertainties at the two highest ϵ values. Notably, the enhancement of P_ℓ/P_ℓ^Born at <ϵ> = 0.790 relative to <ϵ> = 0.153 persists in the full-acceptance analysis and is consistent with the ∼ 2% enhancement seen in the original publication. The deviation from unity of the final result is 5.3 times the statistical uncertainty, 2.3 times the point-to-point systematic uncertainty, and 1.9 times the “total” uncertainty defined as the quadrature sum of the statistical and total systematic uncertainties. The ∼ 0.6% enhancement at ϵ = 0.638 is roughly a 2σ effect statistically, but also consistent with no enhancement within the point-to-point systematic uncertainty. The total and point-to-point systematic uncertainties in P_ℓ/P_ℓ^Born are dominated by the point-to-point uncertainty Δ P_e/P_e = ± 0.5% in the beam polarization. It is worth noting that the global ± 1% uncertainty of the Møller measurement of the beam polarization is irrelevant to the determination of the relative ϵ dependence of P_ℓ/P_ℓ^Born, because a global overestimation (underestimation) of the beam polarization is exactly compensated by an equal and opposite underestimation (overestimation) of the analyzing power at <ϵ> = 0.153. §.§ “Bin centering” effects in R at Q^2 = 2.5 GeV^2 In contrast with the original publication <cit.>, the acceptance-averaged results of the full-acceptance analysis of the GEp-2γ data are quoted at significantly different average Q^2 values (see Tab. <ref>), such that the expected variation of R with Q^2 can noticeably affect its apparent ϵ-dependence, even in the absence of significant two-photon-exchange effects in this observable. The expected variation of R with Q^2 within the acceptance of each point is much larger than its expected ϵ dependence, which is zero in the Born approximation and small in most model calculations of the hard TPEX corrections widely thought to be responsible for the cross section-polarization transfer discrepancy. For example, R(Q^2) from the global fit described in appendix <ref> varies by approximately seven times the statistical uncertainty of the acceptance-averaged result for R within the Q^2 acceptance of the measurement at ϵ = 0.79 (see Fig. <ref>).In order to correct the results for R to a common central Q^2 of 2.5 GeV^2, a bin-centering correction to R is computed for each kinematic under the assumption that R depends only on Q^2, or, equivalently, under the weaker assumption that the global Q^2 dependence of R factorizes from any potential ϵ dependence of R, at least within the acceptance of each kinematic. The corrected value of R is obtained by multiplying the acceptance-averaged result, which corresponds to the average Q^2 and ϵ, by the ratio R(2.5 GeV^2)/R(<Q^2>), where R(Q^2) is evaluated using the results of the global proton form factor fit[The corrections shown in Tab. <ref> are computed using the results of “Global Fit II” of appendix <ref>. The corrections obtained using “Global Fit I” are indistinguishable.] described in appendix <ref>. The corrected results are then plotted at the value of ϵ corresponding to the central Q^2, as opposed to the acceptance-averaged value of ϵ. The bin-centering correction to R is always negative, because the slope of R(Q^2) is negative and the average Q^2 is less than the “central” Q^2 for all three settings (due to the Q^2 dependence of the acceptance-convoluted cross section). Tab. <ref> shows the results for R corrected to the “central” kinematics at Q^2 = 2.5 GeV^2. The magnitude of the correction is small but noticeable compared to the uncertainties for the two higher ϵ points, while being essentially negligible for ϵ = 0.153. The differences between the average and central ϵ values are small. Tab. <ref> shows the results of linear and constant fits to the ϵ dependence of R for both the average and central kinematics. While the corrected and uncorrected data both favor a slightly negative slope for R as a function of ϵ, the slope is also compatible with zero in both cases. Indeed, the constant fits actually give higher “p-values” than the linear fits, although the comparison of these values is not particularly meaningful given the small number of degrees of freedom and the dramatically different shape of the theoretical χ^2 distributions for ν = 1 and ν = 2.Fig. <ref> shows the final, bin-centering-corrected values of R as a function of ϵ at 2.5 GeV^2. The linear fit quoted in Tab. <ref> is also shown in Fig. <ref> with its 68% confidence band. The full-acceptance data, which are significantly more precise at the two highest ϵ values than the originally published data <cit.>, slightly favor a small, negative slope dR/dϵ = -0.017 ± 0.017 (see Tab. <ref>), after correcting the data to the common central Q^2 of 2.5 GeV^2. The uncertainty in the slope dR/dϵ is dominated by the statistical uncertainties of the data, as the point-to-point systematic uncertainties are small. The observed slope is consistent with zero, but is more likely to be negative than positive. No bin-centering corrections were necessary for the ratio P_ℓ/P_ℓ^Born, other than to quote the results at the central kinematics as opposed to the average kinematics. This is because the observed Q^2 dependence of P_ℓ closely follows the predicted Q^2 dependence of P_ℓ^Born (see Fig. <ref>), such that the Q^2 dependence of the ratio P_ℓ/P_ℓ^Born is consistent with a constant within the acceptance of each kinematic. § COMPARISON TO THEORETICAL PREDICTIONS §.§ Theoretical interpretation of G_E^p/G_M^p at large Q^2 Among the primary motivations for measuring nucleon elastic electromagnetic form factors to larger Q^2 values is to observe the transition from strong coupling and confinement to the regime of perturbative QCD (pQCD) physics. However, the applicability of pQCD to hard exclusive processes such as elastic electron-nucleon scattering may require much larger momentum transfers than those currently accessible. One fact that the new proton data have revealed beyond a doubt, is the importance ofquark orbital angular momentum to the understanding of nucleon structure. The role of orbitalangular momentum is also revealed in a global way, by the very fact thatthe nucleon magnetic moment is strongly anomalous, differing fromthe Dirac magnetic moment by ∼± 2 units of the nuclear magneton,for the proton and neutron, respectively. Solving the QCD equations from first principle for the nucleon is onlypossible on the lattice; until quite recently, the feasible Q^2 range for lattice calculations of nucleon FFs has been limited to Q^2 ≲ 3 GeV^2 by computing power and other technical issues. The expectation, given increases in computational power and technical innovations in the methodology of the calculations, is that lattice QCD will be applicableup to 10 GeV^2 or higher in the near future. At the present time only phenomenological models which include some, but not all of the fundamentalcharacteristics of QCD are possible. Some of the mostsuccessful models include Vector Meson Dominance (VMD), the relativistic ConstituentQuark Models (RCQM), Generalized Parton Distributions (GPD), Dyson-Schwinger QCD, and others. We discuss a selection of these approaches in more detail here and compare them with the data. §.§.§ Vector Meson DominanceThe earliest models explaining the global features of the nucleon form factors, such as their apparent and approximate dipole behavior, were vector mesondominance (VMD) models. In this picture the photon couples to the nucleonthrough the exchange of vector mesons. A single vector meson exchange with simple couplings givesan m_V^2/(m_V^2 - q^2) factor, from its propagator, for the falloffof the form factor. One can obtain a Q^-4 high momentum falloff, in accord with observation or with pQCD, from cancellationsamong two or more vector meson exchanges with different masses, or by giving the vector mesons themselves aform factor in their coupling to nucleons. An early example of a VMD fit to form factor data was given by Iachello, Jackson, and Lande <cit.> or IJL. They hadseveral fits, but the one most cited is a 5-parameter fit with a more complicated ρ propagator than the form noted above, toaccount for the large decay width of the ρ meson.(The ω and ϕ are narrow enough that modifying their propagatorsgives no numerical advantage.)The IJL work was improved by Gari and Krümpelmann <cit.> to better match the power law pQCD expectations athigh Q^2, that F_1 ∼ Q^-4 and F_2 ∼ Q^-6, but also including some ln (Q^2) corrections to the falloffs based on therunning behavior of the coupling α_s(Q^2).Further improvement in VMD fits was made by Lomon <cit.>, who includedasecond ρ as the ρ'(1450), andlater also a second ω as the ω'(1419), and obtained a good parameterization for all the nucleon form factors. The firstof the polarization transfermeasurements <cit.> became available in time for Lomon's 2001 work <cit.>. Lomonfurther tuned his fits <cit.> when the second set of polarization transfer data became available <cit.>. In addition, the original IJL fits <cit.> were not as good for the neutron as for the proton.Both the spacelike neutronform factors and timelike nucleon form factors were addressed in what may be termed IJL updates, by Iachelloand Wan <cit.>and Bijker and Iachello <cit.>, both in 2004. Further, Lomon and Pacetti <cit.> have updated and analyticallycontinued the earlier Lomon fits in order to also give a good account of data in both timelike and spacelike regions. The VMD models are of course fits to existing data, and they have been regularly updated as new data appeared. It will be interestingto check the “predictions” for the neutron form factors as new data appear. A plot of the existing situation for the proton isgiven in Fig. <ref>.VMD models are a special case of the more general dispersionrelation approach which relates the nucleon form factors in the space-like (q^2 < 0) region accessible in fixed-target electron scattering to the time-like (q^2 > 0) region accessible in annihilation experiments e^+ e^- → pp̅ (or pp̅→ e^+ e^-).The analytic properties of FFs justify a common interpretation of scattering and annihilation experiments and the precision reachable at colliders requires a unified description of form factors for both space-like and time-like q^2. Although the separation of G_E and G_M has been challenging in the time-like region due to the low luminosities of e^+ e^- colliders relative to fixed-target experiments, some data on the form factor ratio in the time-like region do exist, mainly from the study of the initial-state radiation (ISR) process e^+ e^- → pp̅γ. The most recent and precise data in the time-like region come from the BABAR collaboration <cit.>.§.§.§ Constituent Quark Models The early success of the non-relativistic constituentquark model was in explaining static properties, including magnetic moments and transition amplitudes. Examples are themodels of De Rújula, Georgi, and Glashow <cit.> and of Isgur and Karl <cit.>. However, todescribe the data presented here in terms of constituent quarks,it is necessary to include relativistic effects because the momentum transfers involved are much larger than the constituent quark mass.Constituent quark models (CQMs) have been used to understand the structure of nucleons, beginning when quarks were first hypothesized andpredating the emergence of QCD as the theory of the strong interaction. In the CQM, ground statenucleons (and other baryons in the lowest-lying spin-1/2 octet and spin-3/2 decuplet) are composed of three valence quarks, selected from the three lightest flavors up (u), down (d) and strange (s), and described usingSU(6) spin-flavor wave functions and a completely antisymmetric color wave function.Figure <ref> compares a selection of CQM calculations to the polarization transfer data for μ_p G_E^p/G_M^p from the GEp-I, GEp-II, GEp-III and GEp-2γ experiments. A crucial question for a form factor calculation, since the nucleon must be moving after or before the interaction or both, is howthe wave function in the rest frame transforms to a moving frame. The relative ease of exactly transforming states from the framewhere the wave functions are calculated or otherwise given, to any other frame, makesthe light-front form attractive for form factor calculations.The light-front form in this context was introduced by Berestetsky andTerentev <cit.>, and later developed by Chung and Coester <cit.>. The light-front form of the wavefunction is obtained by a Melosh or Wigner rotation of the Dirac spinors for each quark.Chung and Coester <cit.> used a Gaussian wave function. They did obtain a falling G_E^p/G_M^p ratio.Thisis apparently a feature shared by many relativistic calculations and is caused by the Melosh transformation <cit.>. Frank, Jennings, and Miller <cit.> used the light-front nucleonic wave function ofSchlumpf <cit.> and found a decreasing G_E^p/G_M^p ratio, obtaining a zero betweenQ^2 of 5 and 6 GeV^2. Cardarelli et al. <cit.> also used the light-front formalism which used quark wave functions obtainedfrom a potential of Capstick and Isgur <cit.>. They made the point that the one-gluon exchange is crucial to obtaininghigh momentum components in the wave function to explain the form factor data.A comparable amount of high-momentum components in the nucleon wave function can be obtained in the Goldstone-boson-exchange (GBE) quarkmodel <cit.>. This model relies on constituent quarks and Goldstone bosons, which arise as effective degreesof freedom of low-energy QCD from the spontaneous breaking of the chiral symmetry. The GBE CQM was used by Boffi et al. <cit.> to calculate the nucleon electromagnetic form factors in the point-form.Relativistic CQM calculations by Wagenbrunn et al. <cit.> compared using Goldstone-boson-exchange to one-gluon-exchangein the point-form and found little difference between the calculationsfor proton form factors.De Sanctis et al. <cit.> have calculated the ratio G_E^p/G_M^p within thehypercentral constituent quark model including relativistic corrections. Parameters of the potential are fit to the baryon mass spectrum. Withthe inclusion of form factors for the constituent quarks, good fits are obtained for the nucleon form factors <cit.>, forthe latest polarization transfer G_E^p results <cit.>. Another type of covariant CQM calculation was done by Gross, Ramalho, and Peña <cit.>, partly based on earlier work ofGross and Agbakpe <cit.>, avoiding questions of dynamical forms by staying in momentum space. They performed CQM calculations using a covariant spectator model, where the photon interacts with one quark and the other two quarks aretreated as an on-shell diquark with a definite mass. They modeled the nucleon as a system of three valence constituent quarks with their own parameterized form factors, wherethe CQ form factors are obtainedwith parameters that they fit to the data. Their fit from the 9-parameter “model IV” achieves a rather good description of the existing data, including the recent higher-Q^2 data for G_E^n from Ref. <cit.>, which had not yet been published at the time. §.§.§ Perturbative QCDIn the context of elastic scattering and other hard exclusive processes, perturbative QCD (pQCD) is only expected to be applicable at very large momentum transfers <cit.>, perhaps one to several tens of GeV^2 in the most optimistic scenario. In this limit, the virtual photon makes a hard collision with a single valence quark, which then shares the large momentum transfer withthe other two, nearly collinear quarks through two hard gluon exchanges.pQCD predicts that Q^4 F_1 and Q^2 F_2/F_1 should become constant for asymptotically large Q^2, where the extra power of Q^2 for F_2 relative to F_1 is a consequence of helicity conservation at high energies.The predictions were given by Brodsky and Farrar <cit.> and by Matveev et al. <cit.>. By a simple rearrangement of Eq. (<ref>), the ratio of Dirac and Pauli FFs is given in terms of the Sachs ratio r = G_E/G_M by F_2/F_1 = (1-r )/(τ+r). Figure <ref> shows the JLab polarization data together with selected cross section data for Q^2 F_2^p/F_1^p. The cross section data (without TPEX corrections) show flattening for Q^2 ≳ 3 GeV^2. However, the GEp-I, GEp-II and GEp-III data do not yet show the pQCD scaling behavior.In 2003 Belitsky et al. <cit.>investigated the assumption of quarks moving collinearly with the proton underlying the pQCD prediction.They reiterated the fact that the helicity of a massless (or very light) quarkcannot be flipped by the virtual photon of the ep reaction. Instead, the leading contribution to F_2^p at large Q^2 requires one unit of orbital angular momentum in either the initial or final-state light-cone nucleon wave function, leading to a modified logarithmic scaling behavior Q^2 F_2/F_1 ∝ln^2 (Q^2 / Λ^2) at large Q^2, with Λ a non-perturbative mass scale. With Λ = 0.3 GeV, as shown in Fig. <ref>, the polarization data for F_2 p/F_1 p agree qualitatively with such double-logarithmic enhancement[This observation is not particularly sensitive to the choice of Λ within a range of values comparable to Λ_QCD and/or Λ≈ħ cr_p≈ 0.235 GeV]. Ralston <cit.> and Brodsky et al. <cit.> have also discussed the role of quark orbital angular momentum in producing a ratio F_2p/F_1p which falls more slowly than 1/Q^2. While the “precocious” scaling behavior observed in the proton's F_2/F_1 ratio is interesting, it is important to note that the neutron FF data up to 3.4 GeV^2 <cit.> do not support the logarithmic pQCD scaling behavior for a cutoff parameter Λ similar to that which describes the proton data. The detailed flavor decomposition of the individual quark contributions to the nucleon form factors <cit.> suggests that the pQCD-like scaling behavior observed for the proton's F_2/F_1 ratio is probably largely accidental, and a consequence of the delicate interplay between the u and d quark contributions to F_1 and F_2. In 2006 Braun et al. <cit.> evaluated leading order contributions to the nucleon EMFFs within the light-cone sum rule (LCSR) approach, using both asymptotic distribution amplitudes (DAs) and DAs with QCD sum rule-based corrections. The LCSR approach with asymptotic DAs yields values of G_M^p and G_M^n which are close to the data in the range Q^2 ∼ 1–10 GeV^2. The electricform factors were found to be much more difficult to describe, with G_E^n overestimated, andG_E^p/G_M^p nearly constant. The ratio G_E^p/G_M^p was found to be very sensitive to the details of the DAs.A qualitative description of the proton and neutron electric form factors was obtained by including twist-3 and twist-4 corrections to the nucleon DAs within a simple model.More recently, the LCSR approach was refined by Anikin et al. <cit.> to include the next-to-leading-order pQCD corrections to the contributions of both twist-3 and twist-4 operators and a consistent treatment of nucleon mass corrections. In Ref. <cit.>, the DAs were extracted using the form factor data and compared to lattice QCD results, leading to a self-consistent description. The LCSR approach is, however, not yet able to describe all four nucleon EMFFs to a degree of accuracy comparable to that of the data. Kivel and Vanderhaeghen <cit.> investigated the soft rescattering contribution to nucleon form factors using soft collinear effective theory (SCET). They have been able to show that the soft or Feynman processcan be factorized into three subprocesses with different scales:a hard rescattering , a hard-collinear scattering, and soft nonperturbative modes. For the Q^2 range of the present data,SCET qualitatively predicts that Q^2 F_2/F_1 should not be a constant, but exhibita slow rise, as seen in the data.§.§.§ Generalized Parton Distributions The elementary hard scattering process in large-Q^2 electron-nucleon scattering is virtual photoabsorption by a single quark, embedded in the target nucleon as part of a complex, many-body, relativistic system of valence quarks, sea quark-antiquark pairs, and gluons, described by the Generalized Parton Distributions (GPDs). The GPDs provide a framework to describe the process of emission and re-absorption of a quark by a hadron in hard exclusive reactions via the “handbag” mechanism. The GPDs are universal non-perturbative objects arising in the QCD factorization of hard exclusive processes such as deeply virtual Compton scattering (DVCS) and deeply virtual meson production (DVMP). The form factors F_1 and F_2 are related to the vector (H) and tensor (E) GPDs by model-independent sum rules <cit.>: ∫_-1^+1dxH^q(x,ξ ,Q^2) =F_1^q(Q^2), ∫ _-1^+1dxE^q(x,ξ ,Q^2) =F_2^q(Q^2),where F_1^q (F_2^q) represents the contribution of quark flavor q to the Dirac (Pauli) FF of the nucleon. These relations allow us, if we have complete measurements or good models for the GPDs, to predict the electromagnetic form factors <cit.>. Alternatively, the measured form factors at high Q^2, when combined with the forward parton distributions measured in deep-inelastic scattering, provide fairly stringent constraints on the GPDs, particularly with respect to their behavior at large x and/or -t values <cit.>. Early theoretical developments in GPDs indicated thatmeasurements of the separated elastic form factors of the nucleon to high Q^2 might also shed light on the nucleon spin decomposition, via Ji's angular momentum sum rule <cit.> for the total (spin and orbital) angular momentum J_q carried by the parton flavor q:2J_q= ∫_-1^1[H_q(x, 0, 0) + E_q(x, 0, 0)]x dx. The model-independent extraction of GPDs from observables of hard exclusive processes is an area of high current activity and interest. Some recent and less-recent reviews of the subject can be found in Refs. <cit.>.The GPDs can be represented in impact-parameter space via two-dimensional Fourier transforms of the t-dependence of GPDs at zero skewness <cit.>, allowing a three-dimensional “tomography” of the nucleon in two transverse spatial dimensions and one longitudinal momentum dimension. By forming the charge-squared-weighted sum over quark flavors and integrating the impact-parameter-space GPDs over longitudinal momentum fractions x, Miller <cit.> derived model-independent expressions for the impact-parameter-space charge and magnetization densities of the nucleon in terms of two-dimensional Fourier-Bessel transforms of F_1 and F_2:ρ_ch(b)= ∫_0^∞Q/2π J_0(Qb) F_1(Q^2) dQρ̃_M(b)= b/2πsin^2 ϕ∫_0^∞Q^2/2π J_1(Qb) F_2(Q^2) dQ, in which b is the magnitude of the transverse displacement from the center of the nucleon, and ϕ is the angle between the direction of 𝐛 and the direction of the transverse magnetic field or, equivalently, the transverse nucleon polarization. Venkat et al. <cit.> performed a first extraction with realistic uncertainty estimation of ρ_ch(b) and ρ̃_M(b) for the proton.§.§.§ Lattice QCD Lattice gauge theory is presently the only known method for calculating static and dynamic properties of strongly interacting systems from first-principles, non-perturbative QCD in the regime of strong coupling and confinement. Practical computations in lattice gauge theory involve numerical solutions of QCD on a finite-volume lattice of discrete space-time points. In the recent past, these calculations have often been performed for unphysically large quark masses due to computational limitations, whereas modern calculations often work at or near the physical pion mass. Calculations are typically performed for several lattice volumes, spacings and quark masses and then extrapolated to the infinite-volume, continuum limit and to the physical pion mass. Early calculations of nucleon electromagnetic form factors in lattice QCD emphasized the isovector (p - n) form factors, which are simpler to calculate since contributions from disconnected diagrams are suppressed <cit.>.Until quite recently, most calculations of nucleon form factors in lattice QCD <cit.> have been restricted to relatively low momentum transfers Q^2 ≲ 3 GeV^2, because the rapid falloff with Q^2 of the form factors leads to very small signal-to-noise ratios in the extraction of hadronic three-point correlators, and related systematic uncertainties due to excited-state contamination, among other issues. Lin et al. <cit.> employed a novel technique using anisotropic lattices with both quenched and dynamical ensembles with m_π≥ 450 MeV to reach Q^2 ≈ 6 GeV^2. The prospects for lattice QCD form factor calculations to reach high Q^2 have recently been improved by a novel application of the Feynman-Hellman theorem <cit.>, through which hadronic matrix elements can be related to energy shifts. In the context of nucleon form factor calculations on the lattice, the Feynman-Hellman method allows access to the matrix elements relevant to form factor calculations via two-point correlators as opposed to more complicated three-point functions, and exploits strong correlations in the gauge ensembles to enhance the signal-to-noise ratios for high-momentum states. Figure <ref> shows an initial result from the QCDSF/UKQCD/CSSM collaborations <cit.> for μ_pG_E^p/G_M^p reaching Q^2 ≈ 6.5 GeV^2 with uncertainties approaching the precision of the experimental data. §.§.§ Dyson-Schwinger Equations In recent years, significant progress has also been realized in the explanation and prediction of static and dynamic properties of “simple” hadronic systems such as the pion, the nucleon and the Δ(1232) in continuum non-perturbative QCD, within the framework of QCD's Dyson-Schwinger Equations (DSEs) <cit.>. Where the calculation of nucleon electromagnetic form factors is concerned, the DSE approach requires the solution of a Poincaré covariant Faddeev equation. One analytically tractable, symmetry-preserving truncation scheme that has achieved considerable success in describing the observed behavior of the nucleon FFs involves dressed quarks and non-pointlike scalar and axial vector diquarks as the dominant degrees of freedom.In the DSE framework, the nucleon EMFFs at large Q^2 values are sensitive to the momentum dependence of the running masses and couplings in the strong interaction sector of the Standard Model <cit.>. In a recent study, Segovia et al. <cit.> achieved simultaneously good descriptions of the nucleon and Δ(1232) elastic and transition form factors using identical propagators and interaction vertices for the relevant dressed quark and diquark degrees of freedom. One notable prediction is a zero crossing in the ratio G_E^p/G_M^p at Q^2 = 9.5 GeV^2 and in the neutron FF ratio G_E^n/G_M^n at Q^2 ≈ 12 GeV^2. In this framework, any change in the quark-quark interaction that shifts the location of the zero in G_E^p to larger Q^2 implies a corresponding shift in the location of a zero in G_E^n to smaller Q^2. The location of the zero in G_E^p is particularly sensitive to the rate of transition of the dressed quark mass function between the non-perturbative and perturbative regimes, with a slower fall-off of G_E^p/G_M^p corresponding to a faster transition to the perturbative regime, consistent with the “dimensional scaling” expectation discussed in Sec. <ref>. This prediction will be severely tested by planned near-future precision measurements of G_E^p (G_E^n) to Q^2 ≈ 12 (10) GeV^2 at JLab. Fig. <ref> shows the calculation of Segovia et al. <cit.> for μ_p G_E^p/G_M^p, compared to the polarization transfer data from Halls A and C.§.§ Implications of GEp-2γ for TPEX Shortly after the publication of GEp-I and GEp-II, two groups independently suggested that the difference between cross section and double polarization results might be attributable to previously neglected hard TPEX processes; these were Guichon and Vanderhaeghen <cit.>,and Blunden et al. <cit.>. Notably, some of the earliest polarization experiments for elastic ep were done to assess the contribution of the two photon exchange process <cit.>. In general, cross section data require large radiative corrections, whereas double-polarization ratios do not <cit.>. Several calculations and/or extractions of the hard TPEX contribution involving various models, assumptions and approximations have been published over the last decade. A partial list of these efforts includes Refs. <cit.>. Many of the calculations partially resolve the discrepancy, but a model-independent theoretical prescription for TPEX corrections constrained directly by data remains elusive. Recent reviews of the subject can be found in Refs. <cit.>. In addition to the significant theoretical work to resolve the discrepancy, major experimental efforts were carried out over the last decade to search for experimental signatures of significant TPEX contributions to elastic eN scattering. These signatures include possible non-linearities of the Rosenbluth plot <cit.>, a non-zero target-normal single-spin asymmetry <cit.> or induced normal recoil polarization, and deviations from the Born approximation in polarization transfer observables, as in this work and Ref. <cit.>. The beam-normal single spin asymmetries in elastic eN scattering have also been precisely measured as byproducts of a large number of parity violation experiments <cit.>, albeit in a Q^2 range well below the region of the discrepancy. These beam-spin asymmetries are typically at the few-ppm level, and are also sensitive to the imaginary part of the TPEX amplitudes. The most direct observable to access the real part of the TPEX amplitude is a deviation of the e^+p/e^-p cross section ratio from unity <cit.>, as the real part of the interference term between the Born and TPEX diagrams changes sign with the charge of the lepton beam. Three major experiments with very different and complementary technical approaches have recently measured the e^+p/e^-p cross section ratio <cit.>.Figure <ref> shows the final, bin-centering-corrected results of GEp-2γ for the ϵ dependence of R, compared to several theoretical predictions for the hard TPEX corrections to this observable. Blunden et al. <cit.> recently evaluated the hard TPEX corrections to elastic ep scattering within a dispersive approach, which avoids off-shell uncertainties inherent in the direct evaluation of loop diagrams <cit.>. The box and crossed diagrams for TPEX corrections involving nucleon and Δ intermediate hadronic states were evaluated both algebraically and numerically within the dispersive approach using empirical parametrizations of the nucleon elastic and N →Δ transition form factors. The result including the contributions of both N and Δ intermediate states is consistent in slope with the final GEp-2γ data, and also achieves a reasonable description of the e^+p/e^-p cross section ratios which, however, are only measured for Q^2 ≲ 2.1 GeV^2. At Q^2 = 2.5 GeV^2, it appears that a description in terms of hadronic degrees of freedom with only the nucleon elastic and Δ intermediate states is adequate. At higher Q^2 values where the discrepancy between cross section and polarization data is more severe, the effects of higher-mass resonances, inelastic nonresonant intermediate states including the π N continuum, and the finite widths of resonances are expected to increase in importance. Borisyuk et al. <cit.> also used the dispersive approach to compute the contribution of the P_33 partial wave of the π N channel to the TPEX amplitude, which effectively includes the Δ contribution with realistic shape, width, and nonresonant background “automatically”. The prediction of Ref. <cit.> for the ratio R exhibits similar behavior to the calculation of Blunden et al., which is not surprising, given its similar physics content. Bystritskiy et al. <cit.> used the electron structure function method to compute the higher-order radiative corrections to all orders in perturbative QED in the leading logarithm approximation. Their method predicts no noticeable ϵ dependence at the level of precision of the GEp-2γ data, consistent with our results. Afanasev et al. <cit.> approached the TPEX corrections to elastic ep scattering in a parton-model approach assuming dominance of the “handbag” mechanism, in which both hard virtual photons are exchanged with a single quark, embedded in the nucleon via GPDs. This approach is expected to be valid for simultaneously large values of s, -u, and Q^2. The parton-model evaluation of TPEX corrections predicts a strong, non-linear ϵ dependence for R that is not observed the in GEp-2γ data. Kivel et al. <cit.> computed the hard TPEX correction to elastic ep in a perturbative QCD approach in which the leading contribution for asymptotically large Q^2 involves two hard photon exchanges occuring on different valence quarks, and a single hard gluon exchange occurring on the third valence quark. In the pQCD approach, the TPEX amplitude can be expressed in a model-independent way in terms of leading-twist nucleon distribution amplitudes (DAs). In Fig. 27, the calculation of Ref. <cit.> is shown for two different models for the DAs: that of Braun et al. (BLW <cit.>), and that of Chernyak et al. (COZ <cit.>). The GPD and pQCD models for the hard TPEX correction predict a significant positive slope dR/dϵ which is disfavored by the data. It must be noted, however, that the GEp-2γ measurement at <ϵ> = 0.153 in particular lies outside the expected kinematic range of applicability of a partonic description. The deviation from unity of P_ℓ/P_ℓ^Born at large ϵ, given the absence of significant ϵ dependence of the ratio R, implies a similar deviation from the Born approximation in P_t that cancels in the ratio P_t/P_ℓ. This deviation was not predicted by any of the TPEX calculations available at the time of the original publication <cit.>, which generally expected small TPEX corrections to this observable. A deviation from unity in P_ℓ/P_ℓ^Born was subsequently predicted within the SCET approach by Kivel et al. <cit.>. Guttmann et al. <cit.> performed an extraction of the TPEX amplitudes from a global analysis of elastic ep scattering data including the original GEp-2γ results <cit.> and the Hall A “Super-Rosenbluth” data at the similar Q^2 of 2.64 GeV^2 <cit.>, using the formalism of Eqs. (<ref>)-(<ref>). Under the assumptions used in their analysis, the observed deviation from unity of P_ℓ/P_ℓ^Born and the constant value of R imply that the TPEX amplitudes 𝒴_E ≡(δG̃_E/G_M) and 𝒴_3 ≡(ν / M^2 )(F̃_3/G_M) (see Eqs. (<ref>)-(<ref>)), which are mainly driven by the original GEp-2γ data, are roughly equal in magnitude and opposite in sign, and approach the 2-3% level at ϵ≈ 0.8 and Q^2 = 2.5 GeV^2.§ CONCLUSIONSThis article has described two proton form factor experiments, GEp-III and GEp-2γ, which utilized the recoil polarization method in Hall C at Jefferson Lab tomeasure the ratio of the proton’s electric and magnetic form factors, R ≡μ_p G_E^p/G_M^p. The results of these experiments were previously published in two separate articles <cit.>. The purpose of this article was to provide an expanded description of the apparatus and analysis method common to both experiments and report the results of a full reanalysis of the data with significant improvements in detector calibration, event reconstruction, elastic event selection, and the evaluation of systematic uncertainties. The final results of GEp-III are essentially unchanged relative to the originally published results <cit.>. The new analysis has resulted in a significant reduction in the systematic uncertainty, due to a more thorough evaluation of the systematic uncertainty in the total bend angle of the proton trajectory in the non-dispersive plane of the HMS. The high-Q^2 points confirmed the results of the GEp-I and GEp-II experiments from Hall A, namely that R continues to decrease toward zero, but with clear indication that the rate of this decrease is slowing down. The impressive agreement of the measurements of R in GEp-III and GEp-2γ with the previous Hall A measurements at the same or similar Q^2 (but not necessarily the same ϵ) demonstrates that the systematic uncertainties of the recoil polarization method are well understood, and that deviations from the Born approximation in the extraction of G_E^p/G_M^p from polarization transfer observables are not large within the Q^2 range presently accessible to experiment. The GEp-2γ data, originally published in Ref. <cit.>, consist of measurements for three different ϵ values at a fixed Q^2 of 2.5 GeV^2, obtained by changing the electron beam energy and the detector angles. The relative ϵ dependence of the ratio P_ℓ/P_ℓ^Born was also extracted from the GEp-2γ data with small uncertainties by exploiting the fact that the polarimeter analyzing power, the proton momentum, and the HMS magnetic field were the same for all three ϵ values. The lowest ϵ point was used to calibrate the polarimeter analyzing power, given the large value of P_ℓ and its negligible sensitivity to R at this ϵ. The results of the reanalysis of the GEp-2γ data reported in this work include the previously unpublished full-acceptance data for the two highest ϵ points, increasing the statistics by a factor of 2.5 (3.4) at <ϵ> = 0.638 (0.790). The GEp-2γ experiment serves as a precise test of the validity of the polarization transfer method. Indeed, as expected from the Born approximation, the GEp-2γ data demonstrate that R is compatible with a constant for a wide range of ϵ between 0.15 and 0.79. The only deviation from the Born approximation is observed in the longitudinal polarization at ϵ = 0.79: P_ℓ/P_ℓ^Born = 1.0143 ± 0.0027 ± 0.0071. This deviation is largely compensated by a similar relative deviation in P_t, such that the form factor ratio remains constant. In addition, the statistically improved, simultaneous measurements of the independent observables P_ℓ/P_ℓ^Born and R at the same kinematics provide important tools for testing TPEX models and constraining the extraction of TPEX form factors.The accelerator at Jefferson Lab has recently been upgraded to a maximum beam energy of 12 GeV.There are approved experiments at Jefferson Lab that will extend the knowledge of G_E^p/G_M^p to Q^2 = 12 GeV^2, G_E^n/G_M^n to Q^2 = 10 GeV^2, and G_M^n to 14 GeV^2. Dedicated measurements of the elastic ep unpolarized differential cross section over a wide Q^2 range from 2-16 GeV^2 with ≲ 2% total uncertainties have already been completed in Hall A in 2016 and are currently being analyzed. These measurements will significantly improve upon the existing knowledge of G_M^p within the entire Q^2 range accessible with JLab's upgraded electron beam. The program of high-Q^2 form factor measurements using the upgraded JLab electron beam will enable the detailed flavor decomposition of the nucleon EMFFs to Q^2 = 10 GeV^2, providing significant constraints on the predictions of theoretical models, and insightinto the important degrees of freedom in understanding nucleon structure across a broad range of Q^2.§ ACKNOWLEDGMENTS The collaboration thanks the Hall C technical staff and the Jefferson Lab Accelerator Division for their outstanding support during the experiment. This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under Award Number DE-SC-0014230 and contract Number(s) DE-AC02-06CH11357 and DE-AC05-06OR23177, the U.S. National Science Foundation, the Italian Institute for Nuclear Research, the French Commissariat à l'Energie Atomique and Centre National de la Recherche Scientifique (CNRS), and the Natural Sciences and Engineering Research Council of Canada.§ GLOBAL PROTON FORM FACTOR FIT(S) USING KELLY PARAMETRIZATION Several global fits of the proton form factors to measurements of differential cross sections and polarization observables in elastic ep scattering were performed for this analysis using a procedure similar to that described in Ref. <cit.>. The results were used for the GEp-2γ analysis to estimate the bin centering effects in the ratio R and to calculate the event-by-event and acceptance-averaged values of P_ℓ^Born in the maximum-likelihood analysis. As in Ref. <cit.>, the “first-order” Kelly <cit.> parametrization was used in which G_E^p and G_M^p/μ_p are described as ratios of a polynomial of degree n and a polynomial of degree n+2 in τ = Q^2/4M_p^2 (with n = 1). The Kelly parametrization enforces G_E^p(0) = G_M^p(0)/μ_p = 1 and also enforces the “dimensional scaling” behavior at asymptotically large Q^2 predicted by perturbative QCD: Q^4 F_1 ∝ Q^6 F_2 ∝ constant.Compared to Ref. <cit.>, the fits presented here differ in a few key respects. The data selection for differential cross section measurements is largely the same as before, and includes representative results from twelve different experiments spanning approximately 0.005 GeV^2 ≤ Q^2 ≤ 31 GeV^2 (Refs. <cit.>). However, the database of polarization observables is modified substantially. First, the final results of GEp-III and GEp-2γ reported in this work are now included in the fit, whereas in the original fit, the GEp-III results from Ref. <cit.> were used and the GEp-2γ results were not included at all, as they were not yet published at the time. The three highest Q^2 points from the original GEp-II data <cit.> have been replaced by the results of the reanalysis of these data published in Ref. <cit.>. The data from Ref. <cit.> have also been replaced by the reanalysis results published in Ref. <cit.>. Finally, the high-precision data from Refs. <cit.> have been added. Given the apparent inconsistency of the various polarization experiments at low Q^2, an inconsistency which is not yet explained, two different fits were performed. In the first fit, hereafter referred to as “Global fit I”, the recent precise data from Refs. <cit.> were included, while the polarized target asymmetry data from Ref. <cit.> and the two lowest Q^2 points from GEp-I <cit.> were excluded from the fit. In the second fit, referred to as “Global fit II”, the data from Refs. <cit.> were excluded, while all other R_p data from polarization observables were included. The prescription for treating the cross section data, particularly in the high-Q^2 region where the inconsistency with the polarization transfer data exists, is also slightly modified here compared to Ref. <cit.>. As before, three iterations of the fit are performed, using the resulting parameters and their uncertainties and correlations from the previous fit as the starting point for the subsequent fits. In Ref. <cit.>, the value of G_E^p(Q^2) was fixed for Q^2 ≥ 1 GeV^2 using the result of the previous fit, or, on the first iteration, Kelly's 2004 result <cit.>, when computing the χ^2 contribution of individual cross section data, effectively forcing G_E^p to be entirely determined by polarization data for Q^2 ≥ 1 GeV^2. In the fits reported here, G_E (G_M) was fixed in the same way when the fractional contribution of the ϵ G_E^2 (τ G_M^2) term in the reduced cross section was less than 10%, regardless of Q^2. This prescription removes the influence of individual cross section measurements on the determination of G_E (G_M) at high (low) Q^2 when said measurements have very low sensitivity to the respective form factors. In particular, a cutoff of 10% of the reduced cross section excludes all cross section data for Q^2 ≳ 2.2 GeV^2 from participating in the determination of G_E, and some lower-Q^2 data, depending on ϵ. The other significant difference between the fits reported here and those of Ref. <cit.> is that in Ref. <cit.>, the overall normalization uncertainties in the absolute cross section data were essentially ignored in the χ^2 calculation, whereas in the fits presented here, the overall normalization of each of the twelve experiments included in the global fit was allowed to float within a range of ± 2.5 times the quoted normalization uncertainty. All of the best-fit normalization constants were well within their allowed ranges in both fits. In “Global Fit II”, no experiment was renormalized by more than 3%, whereas in “Global fit I” several experiments were renormalized downward by up to 5%. This result reflects a subtle interplay between the tension with existing data of the precise polarization measurements of R_p from Refs. <cit.> in the 0.1-1 GeV^2 region on the one hand, and the discrepancy between cross section and polarization data at large Q^2 on the other. Allowing the cross section normalizations to float leads to a reduction of the χ^2 per degree-of-freedom from 1.78 in Ref. <cit.> to approximately 1.54 in the fits reported here. Table <ref> summarizes the global fit results. The best-fit values of the parameters describing G_E^p and G_M^p and their (1σ) uncertainties are presented together with the implied asymptotic values of G_E^p and G_M^p, normalized to a dipole form factor with a scale parameter Λ^2= 0.66 GeV^2, corresponding to an RMS radius of 0.84 fm, consistent with the proton charge radius extracted from measurements of the Lamb shift in muonic hydrogen <cit.>. As pointed out in Ref. <cit.>, a dipole form factor with r_p = 0.84 fm describes the low-Q^2 G_E^p data better than the “standard” dipole form factor with Λ^2 = 0.71 GeV^2 (corresponding to r_p = 0.81 fm). As measured by χ^2, the overall quality of both fits is relatively good, except for the cross section data in the high Q^2 region, for which the χ^2 per datum exceeds two. No attempt was made to correct the high-Q^2 cross section data for the effects of two-photon-exchange thought to be responsible for the discrepancy, as these effects are presently only poorly constrained experimentally and incompletely understood theoretically <cit.>. Instead, the “excess” ϵ-dependence of the reduced cross sections observed in the high-Q^2 data (i.e., the “excess” slope in the Rosenbluth plot relative to the expectation from polarization transfer data) is simply averaged over in determining G_M, with the ratio G_E/G_M fixed by the polarization data. While this procedure may bias the determination of G_M in principle, the potential size of the effect on G_M in the high-Q^2 region is mitigated by the smallness of the fractional contribution of G_E^2 to the reduced cross section. The inconsistency among polarization experiments in the low-Q^2 region is another issue that awaits resolution. While the fits reported here provide an adequate representation of the proton FFs in the Q^2 region in which they are directly constrained by data, the values and uncertainties in the extrapolation of these fits to larger Q^2 should not be taken too seriously. The high-precision polarization data for R in both the 0.1-1 GeV^2 region <cit.> and at 2.5 GeV^2 as reported in this work, combine to exert significant influence on the extrapolation of G_E and G_M to Q^2 values beyond the reach of existing data, as is evident from the noticeably different asymptotic behaviors of the two fits, which differ only in the choice of low-Q^2 polarization data. This is a consequence of fitting a smooth, relatively inflexible parametrization of the form factors, with no specific theoretical justification other than its asymptotic behavior, to high-precision data at significantly different Q^2 values. | http://arxiv.org/abs/1707.08587v3 | {
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"A. J. R. Puckett",
"E. J. Brash",
"M. K. Jones",
"W. Luo",
"M. Meziane",
"L. Pentchev",
"C. F. Perdrisat",
"V. Punjabi",
"F. R. Wesselmann",
"A. Afanasev",
"A. Ahmidouch",
"I. Albayrak",
"K. A. Aniol",
"J. Arrington",
"A. Asaturyan",
"H. Baghdasaryan",
"F. Benmokhtar",
"W. Bertozzi",
"L. Bimbot",
"P. Bosted",
"W. Boeglin",
"C. Butuceanu",
"P. Carter",
"S. Chernenko",
"E. Christy",
"M. Commisso",
"J. C. Cornejo",
"S. Covrig",
"S. Danagoulian",
"A. Daniel",
"A. Davidenko",
"D. Day",
"S. Dhamija",
"D. Dutta",
"R. Ent",
"S. Frullani",
"H. Fenker",
"E. Frlez",
"F. Garibaldi",
"D. Gaskell",
"S. Gilad",
"R. Gilman",
"Y. Goncharenko",
"K. Hafidi",
"D. Hamilton",
"D. W. Higinbotham",
"W. Hinton",
"T. Horn",
"B. Hu",
"J. Huang",
"G. M. Huber",
"E. Jensen",
"C. Keppel",
"M. Khandaker",
"P. King",
"D. Kirillov",
"M. Kohl",
"V. Kravtsov",
"G. Kumbartzki",
"Y. Li",
"V. Mamyan",
"D. J. Margaziotis",
"A. Marsh",
"Y. Matulenko",
"J. Maxwell",
"G. Mbianda",
"D. Meekins",
"Y. Melnik",
"J. Miller",
"A. Mkrtchyan",
"H. Mkrtchyan",
"B. Moffit",
"O. Moreno",
"J. Mulholland",
"A. Narayan",
"S. Nedev",
"Nuruzzaman",
"E. Piasetzky",
"W. Pierce",
"N. M. Piskunov",
"Y. Prok",
"R. D. Ransome",
"D. S. Razin",
"P. Reimer",
"J. Reinhold",
"O. Rondon",
"M. Shabestari",
"A. Shahinyan",
"K. Shestermanov",
"S. Sirca",
"I. Sitnik",
"L. Smykov",
"G. Smith",
"L. Solovyev",
"P. Solvignon",
"R. Subedi",
"E. Tomasi-Gustafsson",
"A. Vasiliev",
"M. Veilleux",
"B. B. Wojtsekhowski",
"S. Wood",
"Z. Ye",
"Y. Zanevsky",
"X. Zhang",
"Y. Zhang",
"X. Zheng",
"L. Zhu"
],
"categories": [
"nucl-ex",
"hep-ex"
],
"primary_category": "nucl-ex",
"published": "20170726180127",
"title": "Polarization Transfer Observables in Elastic Electron Proton Scattering at $Q^2 = $2.5, 5.2, 6.8, and 8.5 GeV$^2$"
} |
Guiding Reinforcement Learning Exploration Using Natural Language Brent Harrison Department of Computer Science University of Kentucky Lexington, Kentucky, USA [email protected] Upol Ehsan School of Interactive Computing Georgia Institute of Technology Atlanta, Georgia, USA [email protected] Mark O. Riedl School of Interactive Computing Georgia Institute of Technology Atlanta, Georgia, USA [email protected] 30, 2023 ========================================================================================================================================================================================================================================================================================================================================================================================In this work we present a technique to use natural language to help reinforcement learning generalize to unseen environments.This technique uses neural machine translation, specifically the use of encoder-decoder networks, to learn associations between natural language behavior descriptions and state-action information.We then use this learned model to guide agent exploration using a modified version of policy shaping to make it more effective at learning in unseen environments.We evaluate this technique using the popular arcade game, Frogger, under ideal and non-ideal conditions.This evaluation shows that our modified policy shaping algorithm improves over a Q-learning agent as well as a baseline version of policy shaping. § INTRODUCTIONInteractive machine learning (IML) algorithms seek to augment machine learning with human knowledge in order to enable intelligent systems to better make decision in complex environments.These algorithms allow human teachers to directly interact with machine learning algorithms to train them to learn tasks faster than they would be able to on their own.Typically, humans interact with these systems by either providing demonstrations of positive behavior that an intelligent agent can learn from, or by providing online critique of an agent while it explores its environment.While these techniques have proven to be effective, it can sometimes be difficult for trainers to provide the required demonstrations or critique.Demonstrations may require that the trainer possess in-depth prior knowledge about a system or its environment, and trainers may have to provide hundreds of instances of feedback before the agent begins to utilize it. This issue is compounded when one considers that this training must occur for each new environment that the agent finds itself in. In this work, we seek to reduce the burden on human trainers by using natural language to enable interactive machine learning algorithms to better generalize to unseen environments.Since language is one of the primary ways that humans communicate, using language to train intelligent agents should come more naturally to human teachers than using demonstrations or critique. In our proposed approach, natural language instruction also need not be given online while the agent is learning.Allowing instruction to be given offline greatly reduces the time and effort required on the part of the human teacher to train these intelligent agents.Humans are also extremely proficient and generalizing over many states, and often language aids in this endeavor.With this work, we aim to use human language to learn these human-like state abstractions and use them to enhance reinforcement learning in unseen environments.To do that, we use neural machine translation techniques—specifically encoder-decoder networks—to learn generalized associations between natural language behavior descriptions and state/action information.We then use this model, which can be thought of as a model of generalized action advice, to augment a state of the art interactive machine learning algorithm to make it more effective in unseen environments.For this work, we choose to modify policy shaping, an interactive machine learning algorithm that learns from human critique <cit.>. We evaluate this technique using the arcade game, Frogger.Specifically, we evaluate how our technique performs against a base Q-learning algorithm and a version of policy shaping that uses only demonstrations as examples policy critique on the task of learning on a set of unseen Frogger maps in a variety of situationsTo summarize, the main contributions of this paper are as follows: 1) We show how neural machine translation can be used to create a generalized model of action advice, 2) we show how this model can be used to augment policy shaping to enable reinforcement learning agents to better learn in unseen environments, and 3) we perform an evaluation of our method in the arcade game, Frogger, on several previously unseen maps using unreliable synthetic oracles meant to simulate human trainers. § RELATED WORKThis work is primarily related to two bodies of artificial intelligence research: interactive machine learning and knowledge transfer in reinforcement learning.Interactive machine learning (IML) <cit.> algorithms use knowledge provided by human teachers to help train machine learning models.This allows for human experts to help train intelligent agents, thus enabling these agents to learn faster than they would if left to learn on their own. Typically human teachers interact with the agent by providing either demonstrations of correct behavior <cit.> or directly critique the agent's behavior <cit.>. Our work seeks to improve upon these methods by enabling these algorithms to learn from natural language in addition to demonstrations or critique. There has been other work on using natural language to augment machine learning algorithms.There has been much work done on using natural language instructions to help reinforcement learning agents complete tasks more efficiently. Early works in this area focused on learning mappings between these instructions and specific control sequences in learning environments <cit.>. In this previous work, language is used mainly used to instruct how to complete a specific task in a specific environment.In other words, language and state are tightly coupled.The main way that our work differs from this work is that we are seeking to use language as an abstraction tool.Our work focuses on exploring how language can be used to help reinforcement learning agents transfer knowledge to unseen environments. More recent work has examined how language can help reinforcement learning agents in a more environment-agnostic way.For example, work has been done on using high-level task specifications to engineer environment-agnostic reward functions to improve learning <cit.>.Also, techniques such as sentiment analysis have been used to bias agent exploration to improve learning in unseen environments <cit.>. Most of these techniques, however, require additional information about the environment, such as descriptions of object types in the environment, that may not always be readily available. Our technique relaxes this requirement by using neural machine translation to learn relationships between natural language action/state descriptions and parts of the state space. The work most closely related to our own involves using deep Q-learning to identify language representations that can help reinforcement learning agents learn in unseen environments <cit.>.This technique, however, also requires some knowledge about the environment to be provided in order to learn these representations.Our technique does not require additional information to be provided by the domain author as all state annotations are generated by human teachers. § BACKGROUND In this section, we will discuss three concepts that are critical to our work:reinforcement learning, policy shaping, and encoder-decoder networks. §.§ Reinforcement LearningReinforcement learning <cit.> is a technique that is used to solve a Markov decision process (MDP). A MDP is a tuple M = <S,A,T,R,γ> where S is the set of possible world states, A is the set of possible actions, T is a transition function T: S × A → P(S), R is the reward function R : S × A →ℝ, and γ is a discount factor 0 ≤γ≤ 1.Reinforcement learning first learns a policy π : S → A, which defines which actions should be taken in each state.In this work, we use Q-learning <cit.>, which uses a Q-value Q(s,a) to estimate the expected future discounted rewards for taking action a in state s. As an agent explores its environment, this Q-value is updated to take into account the reward that the agent receives in each state.In this paper, we use Boltzmann exploration <cit.> to select the actions that a reinforcement learning agent will take during training. When using Boltzmann exploration, the probability of the agent choosing a particular action during training is calculated as Pr_q(a)= e^Q(s,a)/τ/∑_a'e^Q(s,a')/τ, where τ is a temperature constant that controls whether the agent will prefer more random exploration or exploration based on current Q-values.§.§ Policy ShapingIn this paper, we build upon the policy shaping framework <cit.>, which is a technique that incorporates human critique into reinforcement learning.Unlike other techniques such as reward shaping, policy shaping considers critique to be a signal that evaluates whether the action taken in a state was desirable rather than whether the resulting state was desirable.Policy shaping utilizes human feedback by maintaining a critique policy to calculate the probability, Pr_c(a), that an action a ∈ A should be taken in a given state according to the human feedback signal.During learning, the probability that an agent takes an action is calculated by combining both Pr_c(a) and Pr_q(a):Pr(a) = Pr_q(a)Pr_c(a)/∑_a'∈ APr_q(a')Pr_c(a') Thus, the ultimate decision on which action to explore during learning is a combination of knowledge from the agent's experience as well as knowledge from a human teacher. The critique policy used in policy shaping is generated by examining how consistent the feedback for certain actions are.If an action receives primarily positive or negative critique, then the critique policy will reflect this with a greater or lower probability, respectively, to explore that action during learning. §.§ Encoder-Decoder NetworksEncoder-decoder networks have been used frequently in other areas, such as machine translation <cit.>, to learn how to convert sets of input sequences into desired output sequences.In this work we use encoder-decoder networks to translate state/action descriptions written in natural language into machine-understandable state/action information that the natural language describes. For example, the input to this network could be natural language describing the layout of a grid environment and an action taken in that specific state, while the desired output of the network would be the specific state and action representation used by the learning agent. This translation task, sometimes known as sequence-to-sequence learning, involves training two recurrent neural networks (RNNs): an encoder network and a decoder network. In this generative architecture, these component neural networks work in conjunction tolearn how to translate an input sequence X = (x_1, ..., x_T) into an output sequence Y = (y_1, ..., y_T').To do this, the encoder network first learns to encode the input vector X into a fixed length context vector v. This context vector is meant to encode important aspects of the input sequence to aid the decoder in producing the desired output sequence. This vector is then used as input into the second component network, the decoder, which is a RNN that learns how to iteratively decode this vector into the target output Y. By setting up the learning problem in this particular way, this thought vector encodes high-level concept information that can help the decoder construct general state representations for each input sequence. § USING LANGUAGE TO GENERALIZE HUMAN CRITIQUE As mentioned previously, one of the primary disadvantages of interactive machine learning is that humans must retrain the agent whenever it encounters a new environment.To address this issue, we show how an encoder-decoder network <cit.> can be used to learn a language-based critique policy.A high level overview of our technique can be seen in Figure <ref>. Our technique works by first having humans generate a set of annotated states and actions by interacting with a single learning environment offline and thinking aloud about the actions they are performing. These annotations are then used to train an encoder-decoder network to create the language-based critique model.This model can then be queried while the agent is exploring new environments to receive action advice to guide it towards states with potentially high rewards.This can be done even if the learning agent encounters states that have not been explicitly seen before and have not been used to train the language-based critique model.Each of these steps will be discussed in greater detail below. §.§ Acquiring Human FeedbackTypically, training an agent using critique requires a large amount of consistent online feedback in order to build up a critique policy, a model of human feedback for a specific problem. In other words, a human trainer would normally be required to watch an agent as its learning and provide feedback which is used in real time to improve the agent's performance. This is because critique normally comes in the form of a discrete positive or negative feedback signal that is then associated with a given state or action.This provides the agent with little opportunity to generalize to unseen environments since this feedback is tightly coupled with state information.To address this, our technique uses natural language as a means to generalize feedback across many, possibly unknown, states. We do this by training an encoder-decoder model to act as a more general critique policy that we refer to as a language-based critique policy, which enables an agent to receive action advice for any potential state it finds itself in. We use two types of data in order to create this policy: examples of actions taken in the environment and natural language describing the action taken.This information can be gathered by having humans interact with the agent's learning environment while providing natural language descriptions of their behavior. There are many ways that humans can potentially provide these state and action annotations.For instance, humans could provide a full episode of behavior along with behavior annotations.It is also possible for humans to provide incomplete trajectories, or even simply examples of single actions, along with natural language annotations. Regardless of how they were collected, the state/action demonstrations provided by the human can be stored and later used as a positive feedback signal while the language can be used to help generalize this feedback signal over many states. For an example of this process, consider an environment where an agent is tasked with dodging obstacles.Assume in this environment that the learning agent can only move in the four cardinal direction: up, down, left, and right.In order to gather the necessary data to learn the feedback policy, a human trainer is presented with different obstacle initializations and tasked with providing short behavior examples of navigating them.The trainer could then provide feedback on each action that they took after the fact.An example of this feedback might be the description, I am dodging the obstacle that is coming up beside me, if describing the action to move up when an obstacle is approaching from the side of the agent.§.§ Training the Encoder-Decoder NetworkDue to how these annotations are generated, it is possible to directly associate language information to state and action information.This paired data is used to train an encoder-decoder network.Specifically, the natural language descriptions are used as inputs to the network, and the network is then tasked with reconstructing the state and action that are associated with that description.By asking the network to reconstruct the state and action, the network learns to identify common elements shared between similar inputs and, importantly, how these elements in the input natural language sequences relate to certain regions of the output sequence of state and action information. This enables the network to learn high-level concept information that enables it to generalize natural language advice to unseen states. §.§ Utilizing the Language-Based Critique PolicyThe ultimate goal of this work is to use the language-based critique policy to speed up learning in unseen environments.To do that, we use this language-based critique policy in conjunction with policy shaping.Recall that a reinforcement learning agent using policy shaping makes decisions using two distinct pieces of information: Pr_q(a), the probability of performing an action based on that action's current Q-value, and Pr_c(a), the probability of performing an action based on the human critique policy. Here, we will use the language-based critique policy learned early to take the place of the standard critique policy normally used by policy shaping.For the language-based critique policy to be used in this framework, we must be able to calculate the probability of performing an action in each state, even if it has never been seen before.Whenever the agent encounters a state, we can query the language-based critique policy to get the probability of performing each action in that state for a given natural language input; however, for this to be of any use we must first determine which piece of feedback out of all of the feedback used in the training set is most applicable to the current state. For each natural language utterance in our training set and for each action the agent perform in its current state, we calculate the log probability of the model reconstructing the agent's current state and performing said action. d One can think of this log probability as how well an utterance describes performing that specific action in that specific state.Whichever utterance is associated with the action that results in the overall largest log probability is then used as the network input to create the action distribution. To create the action distribution, we calculate the following:Pr_lc(a)= e^Pr_l(s,a,i)/τ/∑_a'e^Pr_l(s,a',i)/τwhere Pr_l(s,a,i) is the log probability of performing action a in state s according to the language-based critique policy using sequence i as input. In the original policy shaping algorithm, the critique policy is constantly updated while the agent is learning.Since the language-based critique policy is trained offline, it does not have an opportunity to update itself, which can lead to an agent blindly following poor feedback.To address this, we make use of the τ parameter in Equation <ref> to control how much weight we place on the knowledge extracted from the language-based critique policy.In practice, we have found that the algorithm performs well when τ is initialized to be a small value that increases over the course of learning.This will cause the RL agent to begin learning by trusting the language-based critique policy and then shift towards relying on its own experience as time goes on.This allows the agent to disregard feedback that results in poor Q-values over time. Having done this, the RL agent now explores its environment as it normally would using policy shaping; however, the probability of the agent performing an action in a given state is defined as: Pr(a) = Pr_q(a)Pr_lc(a)/∑_a'∈ APr_q(a')Pr_lc(a') where the original probability obtained from human feedback, Pr_c(a), is replaced with the probability obtained from the language-based critique policy.§ EVALUATION To evaluate this technique, we examine how performs in training vitual agents to play the arcade game, Frogger.Specifically, we seek to show that using natural language to augment policy shaping enables reinforcement learning agents to speed up learning in unknown environments.In this section we will discuss our evaluation in which we compare agents trained with our technique against agents trained using a Q-learning algorithm as well as an agent trained using a baseline policy shaping algorithm with only access to behavior observations. §.§ FroggerIn these experiments we use the arcade game, Frogger (see Figure <ref>), as a test domain.We chose Frogger because it is a discrete environment that can still be quite complex due to the specific mechanics of the environment.The learning agent's goal in this environment is to move from the bottom of the environment to the top while navigating the obstacles in the world.In this world, the obstacles move following a set pattern.Obstacles on on alternating rows will move on space to either the left or the right every time step. Moving outside of the bounds of the map, getting hit by a car, or falling into the water will result in the agent's death, which imposes a reward penalty of -10, reaching the goal earns the agent a reward of +100, and any other move taken in this environment will result in a small reward pentaly of -1.In this environment, the agent can take actions to move up, down, left, right, or choose to do nothing. We test our technique on three different Frogger environments.These environments, shown in Figures <ref> (b), (c), and (d), differ based on obstacle density.Specifically, we evaluate performance in maps in which spaces have a 25% chance of containing an obstacle, a 50% chance of containing an obstacle, and a 75% chance of containing an obstacle. In addition, we evaluate agent performance in these environments under two different conditions: 1) a deterministic condition in which all actions execute normally, and 2) a stochastic condition in which actions have an 80% chance of executing normally and a 20% chance that the agent's action fails and it executes a different action instead.§.§ MethodologyIn this section, we will discuss the experimental methodology used to both create and evaluate the language-based critique policy in Frogger.§.§.§ Data CollectionSince human teachers generate the natural language that is used for training, it is possible that mistakes will be made.Therefore, it is important to examine the effect that imperfect teachers will have on our technique.It is difficult to control for this type of error using actual humans, so for these experiments we use simulated human oracles to generate the required training observations and natural language. To create the behavior traces required for training, we trained 1000 reinforcement learning agents with random starting positions to move one row forward while dodging obstacles on the training map in Figure <ref> (a).We chose this specific task to help eliminate any map-specific strategies that may be learned by using an agent trained to navigate the complete environment.To further help eliminate map-specific strategies, the states recorded for these training examples and then used in the remainder of these experiments encompass only a 3x3 grid surrounding the agent.This was done to help prevent the encoder-decoder network making spurious associations between the natural language annotations and potentially unrelated regions of the state space. Since we are using simulated humans to generate state and action traces, we use a grammar to create the natural language annotations that our system require.This grammar was constructing following the technique used in <cit.>. This technique uses natural language utterances generated by humans to create a grammar in such a way that variances in human language are preserved and codified. In order to produce a sentence, the grammar must first be provided with state and action information and then the grammar identifies the most appropriate grammar ruleto construct a natural language sentence. The grammar is constructed such that each grammar rule can produce a large number of unique sentences. Using this information, the grammar can then produce a natural language sentence describing the states and actions.Since human trainers are likely to make mistakes, we test our technique's ability to deal with imperfect human trainers by introducing uncertainty into the natural language annotations generated by this grammar.Specifically, we create two different language-based critique models using the following simulated teachers: 1) teachers that use the correct grammar rule to provide feedback 80% the time and and a random rule to provide feedback 20% of the time (referred to from now on as the 80% training set), and 2) teachers that use the correct grammar rule to provide feedback 60% of the time and a random rule to provide feedback 40% of the time (referred to from now on as the 60% training set). This will allow us to evaluate how robust our technique is to potential errors contained in the natural language feedback provided by human trainers.We feel that this also helps mitigate the regularity that is often present when using synthetic grammars. It is important to note, that each of these training sets was generated using the same set of behavior traces.This makes the performance of the resulting critique policies directly comparable.After error was injected into each training set, duplicate training examples were removed.To provide further evidence on the variability of the grammar with respect to these training sets, we also examined how often sentences repeated themselves in each training set.In the 80% training set, the most seen sentence comprised 2.2% of the total training set, which contained a total of 1433 training examples.On average, a sentence was repeated 0.19% of the time.For the 60% training set, the most seen sentence comprised 1.7% of the training set of 1497 examples.On average, a sentence was repeated 0.17% of the time.The disparity in training set size is due to duplicate training examples being removed after errors were introduced.§.§.§ TrainingUsing this dataset, we train an encoder-decoder network for 100 ephocs.Specifically we use an embedding encoder-decoder network with attention that is comprised two-layer recurrent neural networks composed of long short-term memory cells containing 300 hidden units each and an embedding size of 300.As mentioned previously, the network learns to translate between natural language descriptions and state and action information.For these experiments, the network learns to translate natural language generated by the grammar into the provided state and action information, which in this case is the 3x3 grid surrounding the agent as well as the action performed in that state. §.§.§ Evaluation We ran experiments using four intelligent agents. The first is a baseline Q-learning agent with no access human feedback that we will refer to as the Q-learning agent. The second is an agent trained using policy shaping that we will refer to as the observation-based critique agent. This agent has access to the state and action information that were used to train the language-based critique policy, which it uses as a positive action feedback signal.To help guard against poor examples in the training set, this agent also uses the parameter, τ, to control how much weight is given to the action examples and to the agent's own experience.The final two agents are trained using our technique and we refer to them as the language-based critique 80% accuracy agent and the language-based critique 60% accuracy agent depending on which training set was used to generate the feedback oracle that the agent used during learning.Each of these agents was evaluated on each of the three unseen Frogger maps in the deterministic and stochastic conditions described previously.These test cases are meant to simulate how each agent performs in a simple (deterministic) environment as well as a more complex (stochastic) environment. For the deterministic test case, each agent was trained for 5000 episodes. For the stochastic test case, each agent was trained for 25,000 episodes. In all test cases, the learned policy was evaluated every 100 episodes and then the total cumulative reward earned during each episode was averaged over 100 total runs. §.§ ResultsThe results for each agent on the deterministic Frogger maps can be seen in Figure <ref> and the results for each agent on the stochastic Frogger maps can be seen in Figure <ref>. For both the language-based critique agent and the observation-based critique agent we tested several initial values and schedules for increasing the τ parameters.The graphs show the best results achieved in these experiments. As can be seen from Figure <ref>, both language-based critique agents converge much faster than the Q-learning agent and the observation-based critique agent.Interestingly, the 60% accurate language-based critique agent outperformed the 80% accurate language-based critique agent on the 25% map and the 75% map. It is important to note that the observation-based critique agent also consistently outperforms the Q-learning agent on each map, meaning that using observations still provides some benefit during training in unseen environments.Figure <ref> shows the results that each agent obtained on the stochastic versions of the test Frogger maps.For this set of experiments, both language-based critique agents outperform the other two agents on each map used for testing.Contrary to the performance in the deterministic environments, the 60% accurate language-based critique agent and 80% accurate language-based critique agent performed similarly in these environments. It is also interesting to note that on the 50% map, the language-based critique agents are the only ones to converge after the 25,000 training episodes.In the 25% map and the 75% map, the observation-based critique agent outperforms the Q-learning agent.Similar its performance in the deterministic environments, this shows that access to behavior observations still provides some amount of benefit in generalizing behavior to unseen environments.§.§ DiscussionThe first thing to note is that across all test cases the language-based critique agents either outperformed both the observation-based critique agent and the baseline Q-learning agent. This shows that natural language provides knowledge useful for generalizing to unseen environments that cannot be obtained by simply looking at past observations. In addition, this shows that our technique is robust to complexity in the learning environment as well as language error that may be present in human trainers.This is a significant result as this provides evidence that the encoder-decoder network can identify relevant features in the set of natural language annotations used for training even when the dataset contains a large amount of noise.These differences in performance were especially pronounced on the deterministic and stochastic 50% maps, as well as the deterministic and stochastic 75% maps. In these cases, both language-based critique agents drastically outperformed both baselines. The consistently positive results across both deterministic and stochastic environments further shows what a powerful tool language can be with respect to generalizing knowledge across many types of environments. One result that needs to be discussed is the performance of the 60% accurate language-based critique agent with respect to the 80% accurate language-based critique agent on the deterministic 25% and 75% maps.In each of these cases, the 60% accurate agent outperformed the 80% accurate agent, contrary to our intuition that the 80% accurate agent should consistently outperform the 60% accurate agent due to the additional error contained in the latter training set. We hypothesize that this behavior can be explained due to model overfitting causing erratic behavior in these two cases.If the encoder-decoder network was overfitting part of the training set, then it is possible that the increased error introduced in the 60% accurate agent had a regularizing effect on the network, which allowed it to better generalize to the states found in these two maps. § LIMITATIONS While the results of our experiments provide strong evidence that our technique is effective at utilizing language to help learn generalizable knowledge, our technique is not without its limitations.First, our experiments made several simplifying assumptions that were necessary in order to control for the variance that accompanies human teachers.By using a grammar, we were able to control the amount of variance present in the annotations used for training.While we attempted to mitigate this by encoding our grammar with a large amount of variance and training the language-based critique model using training sets containing natural language error, naturally occurring language is likely to contain more variation than is present in our grammar.In addition, the natural language explanations used to train the language-based critique model were used to annotate single actions.Typically humans provide explanations of actions in context of a larger goal-based behavior trajectory and not on the level of individual actions.One way to improve this system is to enable it to learn from state/action explanations at varying levels of granularity. We also only explored how this technique can be applied in discrete environments.Using a discrete environment makes it easy to associate natural language annotations with state and action information.If this was done in a continuous environment then it would be much more difficult to determine what state or action should be associated with certain natural language annotations.Finally, we have only explored how this technique can be used to generalize to unseen environments within the same domain.It is unclear, however, if this technique could be used to aid in transfering knoweldge to agents learning similar tasks in different domains.§ CONCLUSIONS Language is a powerful tool that humans use to generalize knowledge across a large number of states.In this work, we explore how language can be used to augment machine intelligence and give intelligent agents an expanded ability to generalize knowledge to unknown environments.Specifically, we show how neural machine translation techniques can be used to give action advice to reinforcement learning agents that generalizes across many different states, even if they have not been seen before.As our experiments have shown, this generalized model of advice enables reinforcement learning agents to quickly learn in unseen environments. In addition, this technique gives human teachers another way to train intelligent agents.The ability to augment human demonstration or critique with human feedback has the potential to significantly reduce the amount of effort required in order to train intelligent agents.This makes the task of training intelligent agents more approachable to potential human trainers. It is even possible that this task could be crowdsourced in the future, drastically reducing the effort on the part of an individual trainer and making these types of agent training methods more appealing.Through this work, we hope to help bring down the language barrier that exists between humans and intelligent agents.By removing this barrier, we hope to enable humans to transfer more complex knowledge to intelligent agents, which should allow them to learn even more complex tasks in complex, unknown environments.aaai | http://arxiv.org/abs/1707.08616v2 | {
"authors": [
"Brent Harrison",
"Upol Ehsan",
"Mark O. Riedl"
],
"categories": [
"cs.AI",
"cs.CL",
"cs.LG",
"stat.ML"
],
"primary_category": "cs.AI",
"published": "20170726192354",
"title": "Guiding Reinforcement Learning Exploration Using Natural Language"
} |
Wigner tomography of multispin quantum states Steffen J. Glaser January 15, 2018 ============================================= In this work, we discuss and compare three methods for the numerical approximation of constant- and variable-coefficient diffusion equations in both single and composite domains with possible discontinuity in the solution/flux at interfaces, considering (i) the Cut Finite Element Method; (ii) the Difference Potentials Method; and (iii) the summation–by–parts Finite Difference Method. First we give a brief introduction for each of the three methods. Next, we propose benchmark problems, and consider numerical tests–with respect to accuracy and convergence–for linear parabolic problems on a single domain, and continue with similar tests for linear parabolic problems on a composite domain (with the interface defined either explicitly or implicitly). Lastly, a comparative discussion of the methods and numerical results will be given. Keywords: parabolic problems; interface models; level set; complex geometry; discontinuous solutions; SBP–SAT finite difference; difference potentials; spectral approach; finite element method; cut elements; immersed boundary; stabilization; higher order accuracy and convergence;AMS Subject Classification: 65M06, 65M12, 65M22, 65M55, 65M60, 65M70, 35K20 § INTRODUCTIONDesigning methods for the high-order accurate numerical approximation of partial differential equations (PDE) posed on composite domains with interfaces, or on irregular and geometrically complex domains, is crucial in the modeling and analysis of problems from science and engineering. Such problems may arise, for example, in materials science (models for the evolution of grain boundaries in polycrystalline materials), fluid dynamics (the simulation of homogeneous or multi-phase fluids), engineering (wave propagation in an irregular medium or a composite medium with different material properties), biology (models of blood flow or the cardiac action potential), etc. The analytic solutions of the underlying PDE may have non-smooth or even discontinuous features, particularly at material interfaces or at interfaces within a composite medium. Standard numerical techniques involving finite-difference approximations, finite-element approximation, etc., may fail to produce an accurate approximation near the interface, leading one to consider and develop new techniques. There is extensive existing work addressing numerical approximation of PDE posed on composite domains with interfaces or irregular domains, for example, theboundary integral method <cit.>,difference potentials method <cit.>,immersed boundary method <cit.>,immersed interface method <cit.>,ghost fluid method <cit.>, the matched interface and boundary method <cit.>,Cartesian grid embedded boundary method <cit.>,multigrid method for elliptic problems with discontinuous coefficients on an arbitrary interface <cit.>,virtual node method <cit.>,Voronoi interface method <cit.>,summation–by–partsthe finite difference method <cit.>and finite volume method <cit.> based on mapped grids,Comment 11 of reviewer 1or cut finite element method <cit.>.In spite ofIndeed, there have been* great advances in numerical methods for the approximation of PDE posed on composite domains with interfaces, or on irregular domains. However,* it is still a challenge to design high-order accurate and efficientcomputationally-efficient* methods for PDE posed in these complicated geometries*, especially for time-dependent problems, problems with variable coefficients, or problems with general boundary/interface conditions.* = Comment 2 of Reviewer 1The aim of this work is to establish benchmark (test) problems for the numerical approximation of parabolic PDE defined in irregular or composite domains.The considered models (Section <ref>) arise in the study of mass or heat diffusion in single or composite materials, or as simplified models in other areas (e.g., biology, materials science, etc.).Comment 3 Reviewer 1In particular, the formulatedThe formulated testComment 3 Reviewer 1 problems (Section <ref>) are intended (a) to be suitable for comparison of high-order accurate numerical methods – and will be used as such in this study – and (b) to be useful in further research.Moreover, the proposed problems include a wide variety of possibilities relevant in applications, which any robust numerical method should resolve accurately, including constant diffusion; time-varying diffusion; high frequency oscillations in the analytical solution; large jumps in diffusion coefficients, solution, and/or flux; etc.For now, we will consider a simplified geometrical setting, with the intent of setting a “baseline” from which further research, or more involved comparisons, might be conducted.Therefore, in Section <ref> we will introduce two circular geometries, which are defined either explicitly, or implicitly via a level set function. In Section <ref>, we briefly introduce the numerical methods we will consider in this work, i.e., second- and fourth-order versions of (i) the Cut Finite Element Method (cut–FEM); (ii) the Difference Potentials Method (DPM), with Finite Difference approximation as the underlying discretization in the current work; and (iii) the summation–by–parts Finite Difference Method combined with the simultaneous approximation term technique (SBP–SAT–FD).These three methods are all modern numerical methods which may be designed for problems in irregular or composite domains, allowing for high-order accurate numerical approximation, even at points close to irregular interfaces or boundaries.We will apply each method to the formulated benchmark problems, and compare results. From the comparisons, we expect to learn what further developments of the methods at hand would be most important. To resolve geometrical features of irregular domains, both cut–FEM and DPM use a Cartesian grid on top of the domain, which need not conform with boundaries or interfaces.These types of methods are often characterized as “immersed” or “embedded”.In the finite difference framework, embedded methods for parabolic problems are developed in <cit.>. For comparison with cut–FEM and DPM, however, in this paper we use a conforming approach based on the finite difference method – the SBP–SAT–FD method, which resolves geometrical features by curvilinear mapping.For comparison with cut–FEM and DPM, however, in this paper we use a finite difference method based on a conforming approach. The finite difference operators we use satisfy a summation–by–parts principle. Then, in combination with the SAT method to weakly impose boundary and interface conditions, an energy estimate of the semi–discretization can be derived to ensure stability. In addition, we use curvilinear grids and transfinite interpolation to resolve complex geometries.Comment 10 of reviewer 1For recent work on SBP–SAT–FD for wave equations in composite domains, see <cit.>, and the two review papers <cit.>; for recent work in DPM for elliptic/parabolic problems in composite domains with interface defined explicitly, see <cit.>; and for recent work in cut–FEM see <cit.>. The paper is outlined as follows. In Section <ref>, we give brief overview of the continuous formulation of the parabolic problems in a single domain or a composite domain.In Section <ref>, we give introductions to the basics of the three proposed methods: cut–FEM, DPM, and SBP–SAT–FD. In Section <ref>, we formulate the numerical test problems. In Section <ref>, we present extensive numerical comparisons of errors and convergence rates, between the second- and fourth-order versions of each method. The comparisons include single domain problems with constant or time-dependent diffusivity; and interface problems with interface defined explicitly, or implicitly by a level set function. In Section <ref>, we give a comparative discussion of the three methods and the numerical results, together with a discussion on future research directions. Lastly, in Section <ref>, we give our concluding remarks. § STATEMENT OF PROBLEMIn this section, we describe two diffusion problems, which will be the setting for our proposed benchmark (test) problems in Section <ref>.(Recall from Section <ref> that these models arise, for example, in the study of mass or heat diffusion.)Comment 3 Reviewer 1For brevity, in the following discussion, we denote u := u(x, y, t) and u_s := u_s(x, y, t), with s = 1, 2.§.§ The single domain problemFirst, we consider the linear parabolic PDE on a single domain Ω (e.g., Figure <ref>), with variable diffusion λ(t):∂ u∂ t = ∇· (λ(t) ∇ u) + f(x, y, t),(x, y, t) ∈Ω× (0, T],subject to initial and Dirichlet boundary conditions: u(x, y, 0)= u^0(x, y),(x, y) ∈Ω u = ψ(x, y, t),(x, y, t) ∈∂Ω× (0, T].Here, the initial and boundary data u^0(x, y) and ψ(x, y, t), the diffusion coefficient λ(t), the forcing function f(x,y,t), and the final time T are known (given) data. §.§ The composite domain problem Next, we consider the linear parabolic PDE on a composite domain Ω := Ω_1 ∪Ω_2 (e.g., Figure <ref>), with constant diffusion coefficients (λ_1, λ_2):∂ u_1∂ t = ∇· (λ_1 ∇ u_1) + f_1(x, y, t),(x, y, t) ∈Ω_1 × (0, T],∂ u_2∂ t = ∇· (λ_2 ∇ u_2) + f_2(x, y, t),(x, y, t) ∈Ω_2 × (0, T],subject to initial conditions: u_1(x, y, 0)= u_1^0(x, y),(x, y) ∈Ω_1, u_2(x, y, 0)= u_2^0(x, y),(x, y) ∈Ω_2,Dirichlet boundary conditions: u_1= ψ(x, y, t),(x, y, t) ∈∂Ω× (0, T],and interface/matching conditions: u_1 - u_2= μ_1(x, y, t),(x, y, t) ∈Γ× (0, T],λ_1 ∂ u_1/∂ n - λ_2 ∂ u_2/∂ n = μ_2(x, y, t),(x, y, t) ∈Γ× (0, T]. In formula (<ref>), ∂ u_s/∂ n, s = 1, 2 denotes the normal derivative at the interface Γ, i.e., ∂ u_s/∂ n = ∇ u_s ·𝐧, where 𝐧 is the outward unit normal vector at the interface Γ. The initial, boundary, and interface data u_1^0(x, y), u_2^0(x, y), ψ(x, y, t), μ_1(x, y, t), and μ_2(x, y, t); the diffusion coefficients (λ_1, λ_2); the forcing functions f_1(x,y,t) and f_2(x,y,t); and the final time T are some known (given) data.We consider the circular geometries depicted in Figure <ref> as the geometrical setting for our proposed benchmark problems in this work.In applications (Section <ref>), other geometries will likely be considered, some much more complicated than Figure <ref>.While our methods can handle more complicated geometry, this is (to the best of our knowledge) the first work looking to establish benchmarks – and compare numerical methods – for parabolic interface problems (<ref>–<ref>).As such, we think that the geometries in Figure <ref> are a good “baseline” – without all the added complexities that more complicated geometries might produce – from which further research, or more involved comparisons, might be done.To be more specific, we aim to define a simple set of test problems that can be easily implemented and tested for any numerical scheme of interest. With circular domains, it suffices for us to compare/contrast performance of the numerical methods on a simple geometry with smooth boundary versus on a composite domain with fixed interface (explicit or implicit). The approximation of the solution to such composite-domain problems are already challenging for any numerical methods, since (i) the solution may fail to be smooth (or may be discontinuous) at the interface, and (ii) there may be discontinuous material coefficients (λ_1 ≠λ_2).Comment 1 Reviewer 1For both the single and composite domain problems, we could also consider other boundary conditions, e.g., a Neumann boundary condition as in <cit.>, etc.§ OVERVIEW OF NUMERICAL METHODS §.§ Cut–FEMIn this section, we give a brief presentation of the cut–FEM method.For a more detailed presentation of cut–FEM, see, for example, <cit.>. Let Ω_s be covered by a structured triangulation, _s, so that each element T∈_s has some part inside of Ω_s; see Figures <ref> and <ref>. Here, s = 1,2 is an index for the composite domain problem (<ref>–<ref>), which will be omitted when referring to the single domain problem (<ref>, <ref>). (For the latter, note thatcovers Ω.) Typically _1 and _2 would be created from a larger mesh by removing some of the cells.Comment 4, Reviewer 1 Further, let _Γ={T∈ : T ∩Γ≠∅} be the set of intersected elements; see Figure <ref>. In the following, we shall use Γ both for the immersed boundary of the single domain problem and for the immersed interface of the composite domain problem, in order to make the connection to the set _Γ clearer.To construct the finite element spaces we use Lagrange elements with Gauss–Lobatto nodes of order p (Q_p-elements). Let V_h^s denote a continuous finite element space on Ω_s, consisting of Q_p-elements on the mesh _s:V_h^s={ v∈ C^0(Ω_s): vT∈ Q_p(T), T∈_s}.For the single domain problem (<ref>, <ref>) we solve for the solution u ∈ V_h; while for the composite domain problem (<ref>–<ref>), we solve for the pair u∈ V_h^1 × V_h^2. For the latter problem, this means that the degrees of freedom are doubled over elements belonging to _Γ. We begin by stating the weak formulation for the single domain problem (<ref>, <ref>).Let (·,·)_X and ··_Y be the L_2 scalar products taken over the two- and one-dimensional domains X⊂^2 and Y⊂^1, respectively. The present method is based on modifying the weak formulation by using Nitsche's method <cit.> to enforce the boundary condition (<ref>).By multiplying (<ref>) with a test function v∈ V_h, and integrating by parts, we obtain: (u̇,v)_Ω+(λ∇ u, ∇ v)_Ω -λunv_Γ =(f,v)_Ω, ∀ v∈ V_h.Note that (<ref>) is consistent with the following terms:γ_D/λ uv_Γ = γ_D/λψv_Γ, -uλ∂ v/∂ n_Γ = -ψλ∂ v/∂ n_Γ,where γ_D is a constant, andis the side length of the quadrilaterals in the triangulation. Now, adding (<ref>, <ref>) to (<ref>) gives the following weak form: Find u ∈ V_h such that (u̇,v)_Ω+a(u,v)=L(v), ∀ v∈ V_h,wherea(u,v) =(λ∇ u, ∇ v)_Ω -λunv_Γ -uλvn_Γ+γ_D/λ uv_Γ, L(v) =(f,v)_Ω +λψγ_D/ v- vn_Γ. For _Γ (the elements intersected by Γ), note that one must integrate only over the part of the element that lies inside Ω. A problem with this is that one cannot control how the intersections (cuts) between Ω andare made. Depending on how Ω is located with respect to the triangulation, some elements can have an arbitrarily small intersection with the domain – see, for example, Figure <ref>. If Ω is moved with respect toto make the cut arbitrarily small, then the condition numbers of the mass and stiffness matrices can become arbitrarily large. To mitigate this issue, in this work we add a stabilizing term j – defined shortly in (<ref>) – to the mass and stiffness matrices, so that their condition numbers arebounded, independently of how the domain Ω is located with respect to the triangulation<cit.>.Adding stabilization to (<ref>) results in the following weak form: Find u ∈ V_h such that (u̇,v)_Ω+ γ_M j(u̇,v)+a(u,v) + γ_A ^-2λ j(u,v)=L(v), ∀ v∈ V_h,where γ_Mand γ_A are scalar constants. In order to state the definition of stabilization (<ref>), denote by _s the set of faces, as seen in Figures <ref> and <ref>.That is, _s is the set of all faces of the elements in _Γ, excluding the boundary faces of _s:_s={F=T_A ∩ T_B : T_A ∈_Γor T_B ∈_Γ,T_A,T_B∈_s}.Then, the stabilization term is defined as: j_s(u,v)=∑_F∈_s∑_k=1^p ^2k+1/(2k+1)(k!)^2[∂_n^k u][∂_n^k v]_F,where [u] = u|_F_+-u|_F_-is the jump over a face, F; n refers to a normal of F;and ∂_n^k u denotes the k-th order normal derivative. The scaling with respect to k of the terms in (<ref>) is based on how the stabilization was derived.In particular, the k!-factors come from the Taylor-expansion and the factor 2k+1 comes from integrating each term once. Addition to Comment 5, Reviewer 1 We now consider the composite domain problem (<ref>–<ref>).To derive the weak formulation, one follows essentially the same steps as for the single domain problem, namely:* For both (<ref>) and (<ref>), multiply the equation for us with a test functionvs∈ V_h^s, and then integrate by parts; * Add terms consistent with the interface and boundary conditions; and * Add stabilization terms j_1 and j_2 over _1 and _2, respectively.This results in the following weak formulation for (<ref>–<ref>). Find u=u∈ V_h^1 × V_h^2 such that:M(u̇,v) +A(u,v)+a_Γ(u,v)+a_∂Ω(u,v) =L_Ω(v)+L_Γ(v)+L_∂Ω, ∀ v=v∈ V_h^1 × V_h^2, where the bilinear forms M and A correspond to the stabilized mass and stiffness matrices:M(u̇,v) =∑_s=1^2(u̇s,vs)_Ω_s + γ_M j_s(u̇s,vs), A(u,v) =∑ _s=1^2(λ∇us, ∇vs)_Ω_s +γ_A ^-2λ j_s(us,vs); L_Ω corresponds to the forcing function:L_Ω(v) =∑ _s=1^2(f_s,vs)_Ω_s;a_Γ and L_Γ consistently enforce the interface conditions (<ref>, <ref>):a_Γ (u,v) =-[u]{λvn}_Γ-{λun}[v]_Γ + γ_Γ/ [u][v]_Γ, L_Γ(v) =γ_Γ/μ_1[v]_Γ + κ_1μ_2v2_Γ + κ_2μ_2v1_Γ -μ_1{λvn}_Γ; and the terms a_∂Ω and L_∂Ω enforce the boundary condition (<ref>) along the outer boundary, ∂Ω:a_∂Ω(u,v) =-λu1nv1_∂Ω -u1λv1n_∂Ω+γ_D/λu1v1_∂Ω, L_∂Ω(v) =λψγ_D/v1 - v1n_∂Ω.In (<ref>–<ref>), n denotes the outward pointing normal at either Γ or ∂Ω (depending on the domain of integration); κ_1 + κ_2 = 1, so that { v } = κ_1 v_1+ κ_2 v_2 is a convex combination; and γ_Γ, κ_1, κ_2 are chosen as in <cit.>:κ_1 = λ2/λ1 + λ2, κ_2 = λ1/λ1 + λ2, γ_Γ = γ_D λ1λ2/λ1 + λ2.The remaining parameters (appearing in Equations <ref>, <ref>, <ref>–<ref>) are given by:γ_M=0.75,γ_A=1.5, γ_D=5p^2.The scaling of γ_D with respect to p follows from an inverse inequality.and the constants are based on numerical experiments, When p=1 these reduce to the same parameters as the ones used in <cit.>, where γ_M was chosen based on numerical experiments on the condition number of the mass matrix. This also agrees with the choice of γ_A and γ_D in <cit.>, where γ_A was investigated numerically. Comment 5, Reviewer 1In order to use cut–FEM, one needs a way to perform integration over the intersected elements _Γ.For example, with the interface problem, on each element K ∈_Γ, we need a quadrature rule for the K∩Ω_1, K∩Ω_2 and K∩Γ. For the numerical tests in this work (Section <ref>), we represent the geometry by a level set function, and compute high-order accurate quadrature rules with the algorithm from <cit.>. Optimal (second-order) convergence was rigorously proven for cut–FEM applied to the Poisson problem in <cit.>. As far as we know, there is no rigorous proof of higher-order convergence for cut–FEM, though such a proof would likely be similar to the second-order case. §.§ DPMmatrix[1][] #1 -ifnextchar@ifnextchar *@̧MaxMatrixCols c We continue in this section with a brief introduction to the Difference Potentials Method (DPM), which was originally proposed by V. S. Ryaben'kii (see <cit.>, and see <cit.> for papers in his honor).Our aim is to consider the numerical approximation of PDEs on arbitrary, smooth geometries (defined either explicitly or implicitly) using the DPM together with standard, finite-difference discretizations of (<ref>) or (<ref>, <ref>) on uniform, Cartesian grids, which need not conform with boundaries or interfaces.To this end, we work with high-order methods for interface problems based on Difference Potentials, which were originally developed in <cit.> and <cit.>.We also introduce new developments here for handling implicitly-defined geometries. (The reader can consult <cit.> for the general theory of the Difference Potentials Method.)Broadly, the main idea of the DPM is to reduce uniquely solvable and well-posed boundary value problems in a domain Ω to pseudo-differential Boundary Equations with Projections (BEP) on the boundary of Ω.First, we introduce a computationally simple auxiliary domain as part of the method.The original domain is embedded into the auxiliary domain, which is then discretized using a uniform Cartesian grid.Next, we define a Difference Potentials operator via the solution of a simple Auxiliary Problem (defined on the auxilairy domain), and construct the discrete, pseudo-differential Boundary Equations with Projections (BEP) at grid points near the continuous boundary or interface Γ.(This set of grid points is called the discrete grid boundary.)Once constructed, the BEP are then solvedtogether with the boundary/interface conditionsto obtain the value of the solution atthe discrete grid boundary.Lastly, using thesereconstructed values of the solution at the discrete grid boundary,the approximation to the solution in the domain Ω isobtained throughthe discrete, generalized Green's formula. Mathematically, the DPM is a discrete analog of the method of Calderón's potentials in the theory of partial differential equations.The DPM, however, does not requireexplicit knowledge of Green's functions.Although we use an Auxiliary Problem (AP) discretized by finite differences, the DPM is not limited to this choice of spatial discretization.Indeed, numerical methods based on the idea of Difference Potentials can be designed with whichever choice of spatial discretization is most natural for the problem at hand (e.g., see <cit.>). Practically, the main computational complexity of the DPM reduces to the required solutions of the AP, which can be done very efficiently using fast, standard 𝒪(N log N) solvers.Moreover, in general the DPM can be applied to problems with general boundary or interfaces conditions, with no change to the discretization of the PDE.Let us now briefly introduce the DPM for the numerical approximation of parabolic interface models (<ref>–<ref>).First, we must introduce the point-sets that will be used throughout the DPM. (Note that the main construction of the method below applies to the single domain problem (<ref>, <ref>), after omitting the index s and replacing interface conditions with boundary conditions; see <cit.>.) Let Ω_s (s=1,2) be embedded in a rectangular auxiliary domain Ω_s^0.Introduce a uniform, Cartesian grid denoted M_s^0 on Ω_s^0, with grid-spacing h_s.Let M_s^+ = M_s^0 ∩Ω_s denote the grid points inside each subdomain Ω_s, andM_s^- = M_s^0 ∖ M_s^+ the grid points outside each subdomain Ω_s.Note that the auxiliary domains Ω_1^0, Ω_2^0and auxiliary grids M_1^0, M_2^0need not agree, and indeed may be selected completely independently, given considerations regarding accuracy, adaptivity, or efficiency. Define a finite-difference stencil N_j, k^s, α, with α = 5, 9, to be the stencil of the standard five-point or a wide nine-point Laplacian, i.e.,N_j, k^s, 5 = { (x_j,y_k), (x_j±1,y_k), (x_j, y_k±1) }orN_j, k^s, 9 = { (x_j,y_k), (x_j±1,y_k), (x_j, y_k±1), (x_j±2,y_k), (x_j, y_k±2) }. Next, with α fixed,define the point-setsN_s^0 = ⋃_(x_j, y_k) ∈ M_s^0 N_j, k^s, α,N_s^+ = ⋃_(x_j, y_k) ∈ M_s^+ N_j, k^s, α, and N_s^- = ⋃_(x_j, y_k) ∈ M_s^+ N_j, k^s, α. The point-set N_s^+ (N_s^-, N_s^0) enlarges the point-set M_s^+ (M_s^-, M_s^0), by taking the union of finite-difference stencils at every point in M_s^+ (M_s^-, M_s^0).Therefore, N_s^+ contains a “thin row” of points belonging to the complement of Ω_s, so that N_s^+ ⊄Ω_s, even though M_s^+ ⊂Ω_s.(Likewise, N_s^- ⊄Ω∖Ω_s, even though M_s^- ⊂Ω∖Ω_s). Lastly, we now define the important point-setγ_s = N_s^+ ⋂ N_s^-, which we call the discrete grid boundary.In words, γ_s is the set of grid points that straddle the continuous interface Γ.(See Figure <ref> for an example of these points-sets, given a single elliptical domain Ω.)Note that the point-sets M_s^+, N_s^+, and γ_s will be used throughout the Difference Potentials Method.Here, we define the fully-discrete finite-difference discretization of (<ref>, <ref>), and then define the Auxiliary Problem. Indeed, the discretization we consider isL_Δ t, h^s u_s^i+1 = F_s^i+1,(x_j, y_k) ∈ M_s^+, where (i) L_Δ t, h^s u_s^i+1 := λ_s(t^i+1) Δ_h u_s^i+1 - σ u_s^i+1, (ii) Δ_h is either a five- or nine-point Laplacianin each subdomain,and (iii) σ and F_s^i+1 follow from the choice of time- and spatial-discretizations.(Here, we have simplified notation slightly by assuming that h := h_1 = h_2, which need not be the same in general.)For full details of the discretization,including the choice of BDF2 or BDF4 in the time discretization,we refer the reader to Appendix <ref>. The choices of discretization (<ref>) in each subdomain need not be the same.As in <cit.>, one could choose a second- and fourth-order discretization on M_1^+ and M_2^+, respectively, given considerations about accuracy, adaptivity, expected regularity of the analytical solution in each domain, etc. Next, we define the discrete Auxiliary Problem, which plays a central role in the construction of theDifference Potentials operator, the resulting Boundary Equations with Projection at the discrete grid boundary, and in the numerical approximation of the solution via the discrete, generalized Green's formula.At time t^i+1, given the right-hand side grid function q_s^i+1: M_s^0 →ℝ, the following difference equations (<ref>, <ref>) are defined as the discrete AP. L_Δt, h^s u_s^i+1= q_s^i+1,(x_j,y_k)∈M^0_su_s^i+1=0,(x_j,y_k)∈N^0_s\M^0_s For a given right-hand side q_s^i + 1, the solution of the discrete AP (<ref>, <ref>) defines a discrete Green's operator G_Δ t, h^s q_s^i + 1.The choice of boundary conditions (<ref>) will affect the resulting grid function G_Δ t, h^s q_s^i + 1, and thus the Boundary Equations with Projection defined below.However, the choice of boundary conditions(<ref>) in the APwill not affect the numerical approximation of (<ref>–<ref>), so long as the discrete AP is uniquely solvable and well-posed.Let us denote by G_Δ t, h^s F_s^i + 1 the particular solution on N^+_s of the fully-discrete problem (<ref>), defined by solving the AP (<ref>, <ref>) withq_s^i + 1 =F_s^i + 1, (x_j, y_k) ∈ M_s^+, 0, (x_j, y_k) ∈ M_s^-, and restricting the solution from N_s^0 to N_s^+. Let us also introduce a linear space 𝐕_γ_s of all grid functions denoted v_γ_s^i + 1, which are defined on γ_s and extended by zero to the other points of N_s^0.These grid functions are referred to as discrete densities on γ_s. The Difference Potential of a given density v_γ_s^i + 1 is the grid function P_N_s^+^i + 1 v_γ_s^i + 1 on N_s^+, defined by solving the AP (<ref>, <ref>) withq_s^i + 1 =0, (x_j, y_k) ∈ M_s^-, L_Δ t, h^s [v_γ_s^i + 1], (x_j, y_k) ∈ M_s^+, and restricting the solution from N_s^0 to the point-set N_s^+.Note that P_N_s^+^i + 1 : 𝐕_γ_s→ N_s^+ is a linear operator on the space 𝐕_γ_s of densities.Moreover, the coefficients of P_N_s^+^i + 1 can be computed by solving the AP (Definition <ref>) with the appropriate density v_γ_s^i+1 defined at the points (x_j, y_k) ∈γ_s. Given a grid function v_s^i + 1, we denote by _γ_s [v_s^i + 1] the Trace (or Restriction) from N_s^+ to γ_s. Moreover, for a given density v_γ_s^i + 1, denote the trace of the Difference Potential of v_γ_s^i + 1 by P_γ_s v_γ_s^i + 1.In other words, P_γ_s v_γ_s^i + 1 = _γ_s [P_N_s^+^i + 1 v_γ_s^i + 1]. Now we can state the central theorem of the Difference Potentials Method that will allow us to reformulate the finite-difference equations (<ref>) on M_s^+ (without imposing any boundary or interface conditions yet) into the equivalent Boundary Equations with Projections on γ_s. At time-level t^i+1, the discrete density u^i+1_γ_s (s=1,2) is the trace of some solution u^i+1_s on domain Ω_s to the Difference Equations (<ref>), i.e., u^i+1_γ_s:=_γ_s[u^i+1_s], if and only if the following BEP holdsu^i+1_γ_s-P_γ_s^i+1u^i+1_γ_s=_γ_s[G^i+1_Δ t, hF^i+1_s],(x_j,y_k)∈γ_s, with _γ_s[·] and P_γ_s defined in Definition <ref>. See <cit.> for the general theory of DPM (including the proof for general elliptic PDE), or one of <cit.> for the proof in the case of parabolic interface problems. A given density v_γ_s^i+1 is the trace of some solution of the fully-discrete finite-difference equations (<ref>) if and only if it is a solution of the BEP. However, since boundary or interface conditions have not yet been imposed, the BEP will have infinitely many solutions u_γ_s^i+1.As originally disucssed in <cit.>, in this work we consider the following approach in order to find a unique solution of the BEP. At each time level t^i + 1, one can approximate the solution of (<ref>–<ref>) at the discrete grid boundary γ_s, using the Cauchy data of (<ref>–<ref>) on the continuous interface Γ, up to the desired second- or fourth-order accuracy.(By Cauchy data,we mean the trace of the solution of (<ref>–<ref>), together with the trace of its normal derivative, on Γ.) Below, we will define an Extension Operator which will extend the Cauchy data of (<ref>–<ref>) from Γ to γ_s. As we will see, the Extension Operator in this work depends only on the given parabolic interface model.Moreover, we will use a finite-dimensional, spectral representation for the Cauchy data of (<ref>–<ref>) on Γ. Then, we will use the Extension Operator, together with the BEP (<ref>) and the interface conditions (<ref>, <ref>), to obtain a linear system of equations for the coefficients of the finite-dimensional, spectral representation.Hence, the derived BEP will be solved for the unknown coefficients of the Cauchy data.Using this obtained Cauchy data, we will construct the approximation of (<ref>–<ref>) using the Extension Operator, together with the discrete, generalized Green's formula.Let us now briefly discuss the Extension Operator for the second-order numerical method, and refer the reader to Appendix <ref> for details (including details for the fourth-order numerical method).For points in the vicinity of Γ, we define a coordinate system (d, ϑ), where ϑ is arclength from some reference point, and d is the signed distance in the normal direction from the point to Γ.Now, as a first step towards defining the Extension Operator, we define a new functionv_s^i+1(d, ϑ) = v_s^i+1(0, ϑ) + ∑_l = 1^p 1/l!∂^l v_s^i+1(0, ϑ)/∂ n^l d^l, where n is the unit outward normal vector at Γ. We choose p = 2 for the second-order method (which we will discuss now) and p = 4 for the fourth-order method (see Appendix <ref>). As a next step for the second-order method (BDF2–DPM2), we definev_s^i+1(0, ϑ) = u_s^i + 1|_Γ, ∂ v_s^i+1(0, ϑ)/∂ n = . ∂ u_s^i + 1/∂ n|_Γ, and∂^2 v_s^i+1(0, ϑ)/∂ n^2 = . ∂^2 u_s^i + 1/∂ n^2|_Γ, where u_s^i + 1 := u_s(x, y, t^i + 1), ∂ u_s^i + 1/∂ n := ∂ u_s(x, y, t^i + 1)/∂ n, etc.As a last step, a straightforward sequence of calculations (see Appendix <ref>) shows that∂^2 u_s^i + 1/∂ n^2≈1/λ_s( 3 u_s^i + 1 - 4 u_s^i + u_s^i - 1/2 Δ t - f^i + 1) - ∂^2 u_s^i + 1/∂ϑ^2 + κ∂ u_s^i+1/∂ n, where κ denotes the curvature of Γ.Therefore, with v_s^i+1(d, ϑ) defined by (<ref>–<ref>), the only unknown data at each time step t^i + 1 are the unknown Dirichlet data u_s^i + 1 and the unknown Neumann data ∂ u_s^i + 1/∂ n.The Extension Operator will incorporate the interface conditions (<ref>, <ref>) when it is combined with the BEP (<ref>), so that the only independent unknowns at each time step t^i + 1 will be . (u_1^i + 1, ∂ u_1^i + 1/∂ n) |_Γ or . (u_2^i + 1, ∂ u_2^i + 1/∂ n) |_Γ. (This is also true for the fourth-order numerical method – see Appendices <ref> and <ref>.) Now we are ready to define the Extension Operator that extends the Cauchy data of (<ref>–<ref>) from Γ to γ_s.Let v_s^i + 1(d, ϑ) be defined by (<ref>–<ref>).Let 𝔲_s, Γ^i + 1 denote the Cauchy data of (<ref>–<ref>) at t^i + 1 on Γ, i.e., 𝔲_s, Γ^i + 1 = . (u_s^i + 1, ∂ u_s^i + 1/∂ n) |_Γ.The extension operator Ex_s that extends 𝔲_s, Γ^i + 1 from Γ to γ_s isEx_s 𝔲_s, Γ^i + 1 := v_s^i + 1(d, ϑ)|_γ_s. For a given point (x_j, y_k) ∈γ_s, note that d is the signed distance between (x_j, y_k) and its orthogonal projection on Γ, while ϑ is the arclength along Γ between a reference point and the orthogonal projection of (x_j, y_k). Next, we briefly discuss the finite-dimensional, spectral representation of Cauchy data 𝔲_s, Γ^i + 1.Indeed, we wish to choose a basis ϕ_ν(ϑ) on Γ (ν = 1, 2, 3, …) in order to accurately approximate the two components of the Cauchy data 𝔲_s, Γ^i + 1.To be specific, whichever basis we choose, we require thatε_𝒩^0, 𝒩^1(𝔲_s, Γ^i + 1)=*min_c_1, ν^s, i+1, c_2, ν^s, i+1∫_Γ(| u_s^i + 1 - ∑_ν=1^𝒩^0 c_1, ν^s, i+1ϕ_ν(ϑ)|^2 ⋯ + | ∂ u_s^i + 1/∂ n -∑_ν=1^𝒩^1 c_2, ν^s, i+1ϕ_ν(ϑ)|^2)dϑ tends to zero as 𝒩^0, 𝒩^1 →∞, for some sequence of real numbers (c_1, ν^s, i+1)_ν=1^𝒩^0 and (c_2, ν^s, i+1)_ν=1^𝒩^1.In other words, we requirelim_𝒩^0, 𝒩^1 →∞ε_𝒩^0, 𝒩^1(𝔲_s, Γ^i + 1) = 0.Now let us discuss a choice of basis.In this work, recall that we consider interfaces Γ thatare at least C^2(Γ)(due to the choice of smooth, circular geometries).Also, as we will see in Section <ref>, each function u considered in the test problems on a composite domain (<ref>, <ref>, <ref>) is locally smooth, in the sense that u|_Ω_1 = u_1 and u|_Ω_2 = u_2 are smooth in Ω_1 and Ω_2, respectively.Moreover, each component of the Cauchy data 𝔲_1, Γ^i + 1 and 𝔲_2, Γ^i + 1 are smooth, periodic functions of arclength ϑ.(Note that 𝔲_1, Γ^i + 1 and 𝔲_2, Γ^i + 1 need not agree, and indeed do not – neither μ_1(x, y, t) nor μ_2(x, y, t) in (<ref>, <ref>) are identically equal to zero, for any of our test problems on a composite domain.)Therefore, in this work, we choose a standard trigonometric basis ϕ_ν(ϑ), withϕ_1(ϑ) = 1, ϕ_2k(ϑ) = cos(2 π k ϑ/|Γ|), ϕ_2k + 1(ϑ) = sin(2 π k ϑ/|Γ|), and k > 1.Moreover, at every time step t^i + 1, we will discretize the Cauchy data 𝔲_s, Γ^i + 1 = . (u_s^i + 1, ∂ u_s^i + 1/∂ n) |_Γ using this basis.Therefore, we let𝔲̃_s, Γ^i+1 = ∑_ν=1^𝒩^0 c^0, i+1_1,νΦ^0_ν(ϑ) + ∑_ν=1^𝒩^1 c^1, i+1_2,νΦ^1_ν(ϑ) andũ^i+1_s,Γ≈𝔲^i+1_s,Γ, where Φ^0_ν = (ϕ_ν,0) and Φ^1_ν = (0, ϕ_ν) are the set of basis functions used to represent the Cauchy data on the interface Γ.It should be also possible to relax regularity assumption on the domain under consideration. For example, one can consider piecewise-smooth, locally-supported basis functions (defined on Γ) as the part of the Extension Operator.For example, <cit.> use this approach to design a high-order accurate numerical method for the Helmholtz equation, in a geometry with a reentrant corner.Furthermore, <cit.> combine the DPM together with the XFEM, and design a DPM for linear elasticity in a non-Lipschitz domain (with a cut). Next, in Appendix <ref>, we derive a linear system for the coefficients (c_1, ν^s, i+1)_ν=1^𝒩^0 and (c_2, ν^s, i+1)_ν=1^𝒩^1, bycombining the interface conditions (<ref>, <ref>), the BEP (<ref>), the Extension Operator (<ref>–<ref>), and the spectral discretization (<ref>).Then, the numerical approximation u_s^i+1≈ u_s(x_j, y_k, t^i + 1) of (<ref>–<ref>) at all grid-points (x_j, y_k) ∈ N_s^+ follows directly from the discrete, generalized Green's formula, which we state now.At each time step t^i + 1, the numerical approximation u_s^i+1≈ u_s(x_j, y_k, t^i + 1)|_(x_j, y_k) ∈ N_s^+ of (<ref>–<ref>) is given byu_s^i + 1 := P_N_s^+^i + 1 u_γ_s^i + 1 + G_Δ t, h^s F_s^i + 1. Here, u_γ_s^i + 1 = Ex_s 𝔲̃_s, Γ^i + 1, and 𝔲̃_s, Γ^i + 1 is constructed from (i) the solution ofthe BEP (see Appendix <ref>) and (ii) the spectraldiscretization (<ref>).(Recall that P_N_s^+^i + 1 u_γ_s^i + 1is the Difference Potential of the density u_γ_s^i + 1,while G_Δ t, h^s F_s^i + 1is the Particular Solution.)In this work, we also propose a novel feature of DPM, extending the method originally developed in <cit.> and <cit.> to the composite domain problem (<ref>–<ref>) with implicitly-defined geometry.The primary difference betweenDifference Potentials Methodson explicitly-defined versus implicitly-defined composite domains is in the approximation of the interface Γ, which must be done accurately and efficiently, in order to maintain the desired second- or fourth-order accuracy. The main idea ofDPM-based methodsfor implicitly-defined geometry is to seek an accurate and efficient explicit parameterization of the implicit boundary/interface. First, we represent the geometry implicitly via a level set function F(x,y) on M^0. Then we construct a local interpolant F̃(x, y) of F(x,y) on a subset of M^0 near the continuous interface Γ. Next, we parameterize Γ by arc-length using numerical quadrature. With this parameterization, we (i) compute the Fourier series expansion from initial conditions for the Cauchy data𝔲_s, Γ^i + 1 on the implicit interface Γ, and (ii) construct the extension operators (Definition <ref>) with p=2 or p=4. Due to the second or fourth-order accuracy (in both space and time) of the underlying discretization (<ref>), the extension operator (<ref>) with p=2 or p=4, and the established error estimates and convergence results for the DPM for general linear elliptic boundary value problemson smooth domains (presented in<cit.> and <cit.>), we expect second- and fourth-order accuracy in the maximum norm for the error in the computed solution (<ref> or <ref>) for both the single and composite domain parabolic problems.Indeed, in the numerical results (Section <ref>) we see that the computed solution (<ref>) at every time level t^i+1 has accuracy 𝒪(h^2 + Δ t^2) for the second-order method, and 𝒪(h^4 + Δ t^4) for the fourth-order method, for both the single and composite domain problems, with explicit or implicit geometry.See <cit.> for more details and numerical tests involving explicit (circular and elliptical) geometries.Main Steps of the algorithm:Let us summarize the main steps for the Difference Potentials Method. * Step 1: Introduce a computationally simple Auxiliary Domain Ω_s^0 (s = 1,2) and formulate the Auxiliary Problem (AP; Definition <ref>).* Step 2: At each time step t^i+1, compute the Particular Solution u_s^i+1 = G^i+1_Δ t, h F^i+1_s, (x_j,y_k)∈ N_s^+, using the AP with the right-hand side (<ref>).* Step 3: Construct the matrix in the boundary equations (<ref>) (discussed in Appendix <ref>), derived from the Boundary Equation with Projection (BEP) (<ref>), via several solutions of the AP. (When the diffusion coefficients λ_s are constant, this is done once, as a pre-processing step before the first time step.)* Step 4: Compute the approximation of the density u^i+1_γ_s, by applying the Extension Operator (<ref>) to the solution of (<ref>).* Step 5: Construct the Difference Potentials P_N_s^+γ_s u^i+1_γ_s of the density u_γ_s^i + 1, using the AP with the right-hand side (<ref>).* Step 6: Compute the numerical approximation u_s^i+1≈ u_s(x_j, y_k, t^i + 1) of the PDE (<ref>–<ref>) using the discrete, generalized Green's formula (<ref>).§.§ SBP–SAT–FDWe continue in this section with a brief presentation of SBP–SAT–FD, for solving the parabolic problems presented in Section <ref>. For more detailed discussions of the SBP–SAT–FD method, we refer the reader to two review papers <cit.>. In SBP–SAT–FD, to resolve geometrical features, the physical domain is mapped to a reference domain with a simple geometry, e.g., the unit square.The SBP–SAT–FD method was originally used on Cartesian grids. To resolve complex geometries, we consider a grid mapping approach by transfinite interpolation <cit.>. Comment 10 of reviewer 1 A smooth mapping requires that the physical domain is a quadrilateral, possibly with smooth, curved sides. If the physical domain does not have the desired shape, we then partition the physical domain into subdomains, so that each subdomain can be mapped smoothly to the reference domain. As an example, the single domain of equation (<ref>, <ref>), shown in Figure <ref>, is divided into five subdomains. The five subdomains consist of one square subdomain, and four identical quadrilateral subdomains (modulo rotation by π/2) with curved sides. Similarly, the composite domain of equation(<ref>–<ref>) is divided into nine subdomains, as shown in Figure <ref>.Suitable interface conditions are imposed to patch the subdomains together. Although the side-length of the centered square is arbitrary (as long as the square is strictly inside the circle), its size and position have a significant impact on the quality of the curvilinear grid. In a high-quality mesh, the elements should not be skewed too much, and the sizes of the elements should be nearly uniform. In practice, it is usually difficult to know a priori the optimal way of domain division. A Cartesian grid in the reference domain is mapped to a curvilinear grid in each subdomain. The grids are aligned with boundaries and interfaces, thus avoiding small–cut difficulties sometimes associated with embedded methods. In this paper, we only consider conforming grid interfaces, i.e., the grid points from two adjacent blocks match on the interface. For numerical treatment of non-conforming grid interfaces in the SBP–SAT–FD framework, see <cit.>. Comment 12 of reviewer 1When a physical domain is mapped to a reference domain, the governing equation is transformed to the Cartesian coordinate in the reference domain. The transformed equation is usually in a more complicated form than the original equation. In general, a parabolic problem u_t = u_xx+u_yy, (x,y)∈Ωin a physical domain will be transformed to Ju_t = (α u_ξ)_ξ+(β u_η)_ξ+(β u_ξ)_η+(γ u_η)_η,(ξ,η)∈ [0,1]^2,where (ξ,η) is the Cartesian coordinate in the unit square, and J(ξ,η), α(ξ,η), β(ξ,η), γ(ξ,η) depend on the geometry of the physical domain and on the chosen mapping. In particular, we use transfinite interpolation for the grid mapping. In this case, the precise form of (<ref>) and the derivation of the grid transformation are presented in Section 3.2 of <cit.>.Comment 3 from reviewer 2 Even though the original equation is in the simplest form with unit coefficients, the transformed equation has variable coefficients and mixed derivatives. Therefore, it is important to construct multi-block finite difference methodssolving the transformed equation (<ref>). Hence, we need two SBP operators, D_1≈∂/∂ x to approximate a first derivative, and D^(b)_2≈∂/∂ x(b(x)∂/∂ x) to approximate a second derivative with variable coefficient, where b(x)>0 is a known function. Below we discuss SBP properties, and start with the first derivative. Consider two smooth functions u(x), v(x) on x∈ [0,1]. We discretize [0,1] uniformly by N grid points, and denote the restriction of u(x), v(x) onto the grid by 𝐮,𝐯, respectively.Integration by parts states:∫_0^1 u_xv dx= uv|_0^1 - ∫_0^1 uv_x dx.The SBP operator D_1 mimics integration by parts: (D_1 𝐮)^T H 𝐯= 𝐮^T B𝐯 - 𝐮^T H D_1𝐯, where H is symmetric positive definite – thus defining an inner product – andB=diag(-1,0,⋯,0,1).In fact, H is also a quadrature <cit.>. It is easy to verify that (<ref>) is equivalent to D_1^TH+HD_1=B,which is the SBP property for the first derivative operator. At the grid points in the interior of the domain, standard, central, finite-difference stencils can be used in D_1, and the weights of the standard, discrete L_2-norm are used in H. At a few points close to boundaries, special stencils and weights must be constructed in D_1 and H, respectively, to satisfy (<ref>). The SBP operators D_1 were first constructed in <cit.> and later revisited in <cit.>. The SBP norm H can be diagonal or non-diagonal. While non-diagonal norm SBP operators have a better accuracy property than diagonal norm SBP operators, when terms with variable coefficients are present in the equation, a stability proof is only possible with diagonal norm SBP operators. Therefore, we use diagonal norm SBP operators in this paper. For a second derivative with variable coefficients, the SBP operators D^(b)_2 were constructed in <cit.>. We remark that applying D_1 twice also approximates a second derivative, but is less accurate and more computationally expensive than D^(b)_2.Due to the choice of centered difference stencils at interior grid points, the order of accuracy of the SBP operators is even at these points, and is often denoted by 2p. To fulfill the SBP property, at a few grid points near boundaries, the order of accuracy is reduced to p for diagonal norm operators. This detail notwithstanding, such a scheme is often referred to as 2p^th-order accurate. In fact, for the second- and fourth-order SBP–SAT–FD schemes used in this paper to solve parabolic problems, we can expect a second- and fourth-order overall convergence rate, respectively <cit.>. An SBP operator only approximates a derivative. When imposing boundary and interface conditions, it is important that the SBP property is preserved and an energy estimate is obtained. For this reason, we consider the SAT method <cit.>, where penalty terms are added to the semi-discretization, imposing the boundary and interface conditions weakly. This bears similarities with the Nitsche finite element method <cit.> and the discontinuous Galerkin method <cit.>. We note that in <cit.>, SBP–SAT–FD methods were developed for the wave equationJv_tt = (av_ξ)_ξ+(bv_η)_ξ+(bv_ξ)_η+(cv_η)_η, (ξ,η)∈ [0,1]^2,with Dirichlet boundary conditions, Neumann boundary conditions, and interface conditions.Comparing equation (<ref>) with (<ref>), the only difference is that the wave equation has a second derivative in time, while the heat equation has a first derivative in time. The spatial derivatives of (<ref>) and (<ref>) are the same. Assuming homogeneous boundary data for simplified notation, we write the SBP–SAT–FD discretization of (<ref>) as𝐯_tt = Q𝐯, where Q is the spatial discretization operator including the boundary implementation. For the scheme developed in <cit.>, stability is proved by the energy method by multiplying (<ref>) by 𝐯_t^TH_2 from the left, 𝐯^T_tH_2𝐯_tt = 𝐯^T_tH_2Q𝐯,where H_2 is a diagonal, positive-definite operator, obtained through a tensor product from the corresponding SBP norm, H, in one spatial dimension. It is shown in<cit.> that H_2Q is symmetric and negative semi-definite. Therefore, we can write (<ref>) as d/d t (𝐯^T_tH_2𝐯_t - 𝐯^TH_2Q𝐯)=0, where the discrete energy, 𝐯^T_tH_2𝐯_t - 𝐯^TH_2Q𝐯,for (<ref>) is conserved. If we use the same operator Q to discretize the heat equation (<ref>) with the same boundary condition as the wave equation (<ref>), then the scheme 𝐯_t = Q𝐯, is also stable. To see this, we multiply (<ref>) by 𝐯^TH_2 from the left, and obtaind/d t (𝐯^TH_2𝐯) = 𝐯^TH_2Q𝐯≤ 0, where 𝐯^TH_2𝐯 is the discrete energy for (<ref>). In this paper, we use the spatial discretization operators developed in <cit.> to solve both the single (<ref>, <ref>) and composite domain problems (<ref>–<ref>). In <cit.>, SBP–SAT–FD methods are discussed for the one-dimensional heat equation with constant coefficients, both in a single domain and a composite domain. In theory, these schemes can also be generalized to solve equation (<ref>), but are different from the ones used in this paper. § TEST PROBLEMSIn this section, we first list the test problems that we will consider (in Section <ref>), and then briefly motivate and discuss these choices (in Section <ref>).The tests we propose are “manufactured solutions”, in the sense that we state an exact solution u(x, y, t) or (u_1(x, y, t), u_2(x, y, t)) and a diffusion coefficient λ(t) or (λ_1, λ_2).From (<ref>, <ref>) (for the single domain problem) or (<ref>–<ref>) (for the composite domain problem) we compute the (i) right-hand side, (ii) initial conditions, (iii) boundary condition, and (iv) functions (μ_1(x, y, t),μ_2(x, y, t)) for the interface/matching conditions.Then, (i–iv), together with the diffusion coefficient, serve as the inputs for our numerical methods.§.§ List of test problems * Single-domain, with an explicitly-defined boundary for DPM and SBP–SAT–FD, or an implicitly-defined boundary for cut–FEM. * Constant diffusion (Test Problem 1A; TP–1A):Consider the PDE (<ref>, <ref>), with λ(t) ≡ 1, Ω = { (x, y) ∈^2 : x^2 + y^2 ≤ 1 }, and the final time T=1.0.Then, TP–1A (adapted from <cit.>), is given byu(x, y, t) = x^9 y^8 e^-t. TP–1A * Time-varying diffusion (Test Problem 3A; TP–3A):Same as TP–1A, but with diffusion coefficient λ(t) = 11/10 + sin (10 π t). TP–3A * Composite-domain, with an explicitly-defined interface (for DPM and SBP–SAT–FD) or implicitly-defined interface (for cut–FEM and DPM).Consider the PDE (<ref>–<ref>), with Ω = [-2, 2] × [-2, 2], Ω_2 = { (x, y) ∈^2 : x^2 + y^2 ≤ 1 }, Ω_1 = Ω∖Ω_2, Γ = { (x, y) ∈^2 : x^2 + y^2 = 1 }, and the final time T = 1.0. * (Test Problem 2A; TP–2A):A modified version of the test adapted from <cit.>.Let (λ_1, λ_2) = (10, 1), andu(x, y, t) =e^-tsin x cos y, (x, y) ∈Ω_1, e^-t (x^2 - y^2), (x, y) ∈Ω_2.TP–2A* High-frequency oscillations (Test Problem 2B; TP–2B):A modified version of the test adapted from <cit.>.Let (λ_1, λ_2) = (10, 1), andu(x, y, t) =e^-tsin (3 π x) cos (7 π y), (x, y) ∈Ω_1, e^-t (x^2 - y^2), (x, y) ∈Ω_2.TP–2B* Large contrast in diffusion coefficients, and large jumps in both solution and flux at interface (Test Problem 2C; TP–2C):Let (λ_1, λ_2) = (1000, 1), andu(x, y, t) =0, (x, y) ∈Ω_1, 1000 sin(10 t) x^4 y^5, (x, y) ∈Ω_2.TP–2C§.§ Motivation of the chosen test problemsTest Problem 1A (<ref>) involves a high-degree polynomial, with total degree of 17.This is a rather straightforward test problem, which allows us to establish a good “baseline” with which to compare each method.The choice of high degree ensures that there will be no cancellation of local truncation error, so that we should see – at most – second- or fourth-order convergence for the given methods, barring some type of superconvergence. Next, (<ref>) adds on (incrementally) the complication of time-varying diffusion. Likewise, (<ref>) offers a straightforward “baseline” with which to consider the interface problem:The test problem is piecewise-smooth, and the geometry is simplified (see Remark <ref>).However, there is a jump in both the analytical solution and its flux, which requires a well-designed numerical method to accurately approximate. Moreover, (<ref>) was first proposed in <cit.> (see also <cit.>), and is a good comparison with the immersed interface method therein. Then, (<ref>) adds additional challenges onto (<ref>) in the form of much higher-frequency oscillations; while (<ref>) adds onto (<ref>) in the form of both (i) large contrast in diffusion, and (ii) large jumps in the analytical solution and its flux. § NUMERICAL RESULTS §.§ Time discretization The spatial discretization for each method is discussed in Section <ref>.For the time discretization, the backward differentiation formulas of second- and fourth-order (BDF2 and BDF4) are used for the second- and fourth-order methods, respectively. In each case, the time-step is given byΔ t=0.5h. However, note that h in (<ref>) bears different physical meanings for each method. Indeed, for cut–FEM, h is the average distance between the Gauss–Lobatto points; for DPM, h is the grid spacing in the uniform, Cartesian grid M^0 (see the text prior to (<ref>)); and for SBP–SAT–FD, h is the minimum grid spacing in the reference domain. §.§ Measure for comparison Let u_j, k^i denote the computed numerical approximation of u(x, y, t) at the grid-point (x_j, y_k) ∈Ω and time t^i = i Δ t ∈ (0, T]. For the three methods, we will compare the size of the maximum error in u at the grid points, with respect to the number of degrees of freedom (DOF). For the single domain problem (<ref>, <ref>), the maximum error is computed as: E := max_t^i ∈ (0, T]max_(x_j, y_k) ∈Ωu(x_j, y_k, t^i) - u_j, k^i,and for the composite domain problem (<ref>–<ref>) as: E := max_t^i ∈ (0, T]max_(x_j, y_k) ∈Ω_1 ∪Ω_2u(x_j, y_k, t^i) - u_j, k^i. §.§ Convergence resultsIn the following tables and figures, we state the number of degrees of freedom in the grid, maximum error (<ref>, <ref> for the single- and composite-domain problems, respectively), and an estimate of the rate of convergence.In Tables <ref>–<ref>, the estimate of rate of convergence is computed as follows.Let (DOF_n, E_n) be given, with n = 1, 2, 3 referring to the first, second, and third grids(from coarsest to finest).Then, for n = 2, 3, compute the standard estimateρ_n = log(E_n - 1 / E_n)/log(DOF_n - 1 / DOF_n), which is the estimated rate of convergence, denoted in Tables <ref>–<ref> by “Rate”. In Figures <ref>, <ref>, <ref>–<ref>, the estimate of rate of convergence is computed differently. Computing a least-square linear regression for the data (log_10(√(DOF_n)), log_10(E_n))gives a line with slope m, where m is the estimate of rate of convergence, reported in the legend on the right side of each figure. Overall, we see in Tables <ref>–<ref> that the error for second-order methods (denoted, for brevity, as CUT2, DPM2, SBP2) on the finest mesh is similar, or sometimes larger, than the error for fourth-order methods (denoted CUT4, DPM4, SBP4) on the coarsest mesh – this illustrates the effectiveness of higher-order methods, when high accuracy is important.Additionally, comparing the three methods together, the size of the errors for the single-domain problems (<ref>, <ref>) are similar, up to a constant factor; while for the composite-domain problems (<ref>, <ref>, <ref>) we do see differences of one or two orders of magnitude, with the DPM having the smallest errors. In Table <ref> and Figure <ref>, we observe that the measured rates of convergence for the numerical approximation of Test Problem 1A (<ref>) are all ≈ 2 (for the second-order versions) or ≈ 4 (for the fourth-order versions), except for DPM4, which for this test problem is superconvergent, with fifth-order convergence.Such higher-than-expected convergence might occur due to several reasons – for example, (i) if the geometry is smooth; (ii) if the magnitude of the derivatives have fast decay (effectively reducing the local truncation error by a factor of h); or (iii) if there is cancellation of error due to symmetries in the geometry, or in the analytical solution.Table <ref> and Figure <ref> show the numerical results for (<ref>).This test problem has the same manufactured solution as (<ref>), but with a time-varying diffusion coefficient.Despite this added complexity, the numerical results are the same order of accuracy, and in many cases the errors are the same up to seven digits, when compared with the results for (<ref>).This similarity in the numerical results demonstrates that the three methods can robustly handle time-varying diffusion coefficients.Comment 16 Reviewer 1The plots of spatial error at the final time T = 1.0, shown in Figure <ref>, are representative of other tests (not included in this text) on a single circular domain.The error in the cut–FEM solution presents largely at the boundary; the error in the DPM solution typically has smooth error, even for grid points very near Γ; while the error in the SBP–SAT–FD solution is not smooth at interfaces introduced by the domain partitioning.The plots of spatial error at the final time T =1.0 for (<ref>) are shown in Figure <ref>.These plots are fairly representative of the other composite domain tests reported herein, and also of others test problems not included in this work.As in Figure <ref>, the cut–FEM has its largest error at degrees of freedom on cut (intersected) elements; the DPM has piecewise smooth error, including even grid points at the boundary/interface; and the SBP–SAT–FD has its largest error at the interfaces between computational subdomains, with particularly pronounced error at the corners of Ω, where the grid is most stretched. Regarding the max-norm error in presented in Table <ref> and Figure <ref>, we see that the DPM has smaller max-norm by more than an order of magnitude. We also observe that the convergence rate of the fourth-order SBP–SAT–FD is only three. This suboptimal convergence is inline with the error plot in Figure <ref>, which shows that the error at the corners of the domain is significantly larger than elsewhere. In addition, the error is only non-smooth along the interfaces on the two diagonal lines of the domain. We have also measured the L_2 error at the final time T=1.0 (not reported in this work), and fourth-order convergence is obtained. Comment 15 of reviewer 1 In Table <ref> and Figure <ref>, we see the numerical results for (<ref>).The analytical solution is similar to (<ref>), though much more oscillatory – this additional challenge is manifested by an increase in error by several orders of magnitude. In Table <ref> and Figure <ref>, we see the numerical results for (<ref>), which shows that our numerical methods are robust to large jumps in diffusion coefficients, the analytical solution, and/or the flux of the true solution. Also, observe that the errors from DPM2/DPM4 (explicit geometry) and DPM2-I/DPM4-I (implicit geometry) in Tables <ref>–<ref> are almost identical, which demonstrates the robustness and flexibility of the DPM. § DISCUSSIONThere are many possible methods (Section <ref>) for the numerical approximation of PDE posed on irregular domains, or on composite domains with interfaces.In this work, we consider three such methods, designed for the high-order accurate numerical approximation of parabolic PDEs (<ref>, <ref> or <ref>–<ref>).Each implementation was written, tested, and optimized by the authors most experienced with the method—the cut-Finite Element Method (cut–FEM) by G. Ludvigsson, S. Sticko, G. Kreiss; the Difference Potentials Method (DPM) by K. R. Steffen, Q. Xia, Y. Epshteyn; and the Finite Difference Method satisfying Summation-By-Parts, with a Simultaneous Approximation Term (SBP–SAT–FD) by S. Wang, G. Kreiss.Although we consider only one type of boundary/interface (a circle), we hope that the benchmark problems considered will be a valuable resource, and the numerical results a valuable comparison, for researchers interested in numerical methods for such problems. The primary differences between the cut–FEM and the standard finite element method are the stabilization terms for near-boundary degrees of freedom, and the quadrature over cut (intersected) elements.Tuning the free parameters in the stabilization terms could mitigate the errors observed in Figures <ref>, <ref>.(We experimented with different stabilization parameters in the numerical tests in Section <ref> and have seen that the error can be reduced up to a factor of 15, so large gains in accuracy are possible.)(We have done some preliminary experiments suggesting that the errors decrease when tuning these parameters, but further investigations are required in order to guarantee robustness.)Comment 6, Reviewer 1 Given a level-set description of the geometry, there are robust algorithms for constructing the quadrature over cut elements.Together, these differences allow for an immersed (non-conforming) grid to be used. The theoretical base for cut–FEM is well established.The DPM is based on the equivalence between the discrete system of equations (<ref>) and the Boundary Equations with Projection (Thm. <ref>).The formulation outlined in Section <ref> allows for an immersed (non-conforming) grid; fast 𝒪(N log N) algorithms, even for problems with general, smooth geometry; and reduces the size of the system to be solved at each time-step.The convergence theory is well-established for general, linear, elliptic boundary value problems, and we conjecture in Section <ref> that this extends to the current setting.In this work, we have extended DPM to work with implicitly-defined geometries for the first time.This is a first step for solving problems where the interface moves with time. In the finite difference framework (the SBP–SAT–FD method, in this work), the SBP property makes it possible to prove stability and convergence for high-order methods by an energy method. Combined with the SAT method to impose boundary and interface conditions, the SBP–SAT–FD method can be efficient to solve time-dependent PDE. Geometrical features are resolved by curvilinear mapping, which requires an explicit parameterization of boundaries and interfaces. High quality grid generation is important –our experiments, though not reported in this work, have shown that the error in the solution is sensitive to both the orthogonality of the grid and the grid stretching.Similarities between the cut–FEM and the DPM (beyond the use of an immersed grid) include the thin layer of cut cells along the boundaries/interfaces (cut–FEM) and the discrete grid boundary γ (DPM); and the use of higher-order normal derivatives in the stabilization term (cut–FEM) and extension operator (in the Boundary Equations with Projection; DPM).A similarity between the cut–FEM and SBP–SAT–FD is the weak imposition of boundary conditions, via Nitsche's method (cut–FEM) or the SAT method (SBP–SAT–FD).In this work, the DPM and the SBP–SAT–FD method both use an underlying finite-difference discretization, but the DPM is not restricted to this type of discretization. Although both the cut–FEM and the DPM use higher-order normal derivatives in their treatment of the boundary/interface, the precise usage differs.For cut–FEM, it is the normal of the element interfaces cut by Γ, while for DPM, it is the normal of the boundary/interface Γ. Moreover, in the cut–FEM, stabilization terms (<ref>) involving higher-order normal derivatives at the boundaries of cut-elements are added to the weak form of the PDE, to control the condition number of the mass and stiffness matrices, with a priori estimation of parameters to guarantee positive-definiteness of these matrices; while in the DPM, the Boundary Equations with Projection is combined with the Extension Operator (Definition <ref>), which incorporates higher-order normal derivatives at the boundary/interface Γ.Returning to Section <ref>, we see (in Tables <ref>–<ref> and Figures <ref>–<ref>) that the expected rate of convergence for the second- and fourth-order versions of DPM and cut–FEM is achieved, while the DPM has the smallest error constant across all tests. For the SBP–SAT–FD method, expected convergence rates are obtained in some experiments. A noticeable exception is Test Problem 2A, for which the fourth-order SBP–SAT–FD method only has a convergence rate of three. From the error plot in Figure <ref>, we observe that the large error is localized at the four corners of the domain Ω, where the curvilinear grid is non-orthogonal and is stretched the most (see Figure <ref>). As seen in the error plots (Figures <ref>, <ref>), the error for the cut–FEM and the SBP–SAT–FD has “spikes”, while for the DPM the error is smooth.A surprising observation from Figure <ref> is that conforming grids (on which the SBP–SAT–FD method is designed) do not necessarily produce more accurate solutions than immersed grids (on which the cut–FEM and the DPM are designed). Indeed, it is challenging to construct a high-quality curvilinear grid for the considered composite domain problem. Future directions we hope to consider (in the context of new developments and also further comparisons) include:(i) parabolic problems with moving boundaries/interfaces,(ii) comparison of numerical methods for interface problems involving wave equations <cit.>, (iii) extending our methods to consider PDEs in 3D,(iv) design of fast algorithms, and(v) design of adaptive versions of our methods.Indeed, for (i), difficulties for the cut–FEM might be the costly construction of quadrature, while for DPM difficulties might be the accurate construction of extension operators.Regarding (iii), this has already been done for the cut–FEM and SBP–SAT–FD; while for the DPM, this is current work, with the main steps extending from 2D to 3D in a straightforward manner.§ CONCLUSIONIn this work, we propose a set of benchmark problems to test numerical methods for parabolic partial differential equations in irregular or composite domains, in the simplified geometric setting of Section <ref>, with the interface defined either explicitly or implicitly.Next, we compare and contrast three methods for the numerical approximation of such problems: the (i) cut–FEM; (ii) DPM; and (iii) SBP–SAT–FD.Brief introductions of the three numerical methods are given in Section <ref>. It is noteworthy that the DPM has, for the first time, been extended to problems with an implicitly-defined interface.For the three methods, the numerical results in Section <ref> illustrate the high-order accuracy. Similar errors (different by a constant factor) are observed at grid points away from the boundary/interface, while the observed errors near the boundary/interface vary depending upon the given method. Although we consider only test problems with circular boundary/interface, the ideas underlying the three methods can readily be extended to more general geometries. In general, all three methods require an accurate and efficient resolution of the explicitly- or implicitly-definedirregular geometry: cut–FEM relies on accurate quadrature rules for cut elements, and a good choice of stabilization parameters; DPM relies on an accurate and efficient representation of Cauchy data using a good choice of basis functions; and SBP–SAT–FD relieson the smooth parametrization to generate a high-quality curvilinear grid.§ APPENDIX (DPM)Let us now expand some details presented in the brief introduction to the Difference Potentials Method (Section <ref>). §.§ Fully-discrete formulation of (<ref>, <ref>).The fully-discrete, finite-difference discretization introduced in (<ref>) isL^s_Δ t,hu_s^i+1=F_s^i+1, (x_j,y_k)∈ M^+_s.The general form of the operator is L^s_Δ t,h=λ_s(t^i+1)Δ_h-σ I, where σ=3/2Δ t for second-order (BDF2–DPM2), σ=25/12Δ t for fourth-order (BDF4–DPM4), and Δ_h is either a standard five- or nine-point Laplacian. For the nine-point Laplacian, we haveΔ_h u_j,k=1/12h^2(-u_j-2,k +16u_j-1,k+16u_j+1,k-u_j+2,k-u_j,k-2+16u_j,k-1+16u_j,k+1-u_j,k+2-60u_j,k)for points sufficiently far away from the boundary of the auxiliary domain Ω_s^0.For points that are close to the boundary, we use a modified, fourth-order stencil. For example, at the southwest corner, we take Δ_h u_1,1=1/12h^2( 10u_0,1-4u_2,1+14u_3,1-6u_4,1+u_5,1 + 10u_1,0-4u_1,2+14u_1,3-6u_1,4+u_1,5-30u_1,1),where u_0,1 and u_1,0 will be from the boundary condition (<ref>).Next, the right-hand side of (<ref>) for BDF2–DPM2 is given byF^i+1_s=-f^i+1_s-σ/3(4u_s^i-u^i-1_s),and for BDF4–DPM4 byF^i+1_s=-f^i+1_s-σ/25(48u_s^i-36u^i-1_s+16u^i-2_s-3u^i-3_s).Lastly, the initialization at t=0 is done using the exact solutions for the terms u^0_s, u^-1_s, u^-2_s, and u^-3_s. Another possibility would be the use of lower-order BDF methods, then advancing to higher-order methods once enough stages are established. No significant differences were observed between the two approaches.§.§ Equation-based extensionLet us now expand the discussion surrounding (<ref>–<ref>) leading up to Definition <ref> of the Extension Operator (<ref>). An important step in this discussion is to recast the original PDE (<ref>, <ref>) into a curvilinear form, for points (x, y) in the vicinity of Γ.Following the notation <cit.>, let us first introduce the coordinate system (d, ϑ) for points in the vicinity of Γ.Recall from Definition <ref> that d is the distance in the normal direction from a given point to its orthogonal projection on Γ, while ϑ is the arclength along Γ from some reference point to the orthogonal projection.In this coordinate system, the PDE (<ref>, <ref>) becomes∂ u_s/∂ t - λ_s(1/H_ϑ[∂/∂ n(H_ϑ∂ u_s/∂ n)+∂/∂ϑ(1/H_ϑ∂ u_s/∂ϑ)]) =f_s, where where H_ϑ=1 - d κ is the Lamé coefficient, and κ is the signed curvature along the interface Γ.From (<ref>), a straightforward calculation gives the second-order normal derivative ∂^2 u_s/∂ n^2 (used in the calculation of (<ref>)), which is∂^2 u_s/∂ n^2 = 1/λ_s(∂ u_s/∂ t-f_s)-∂^2u_s/∂ϑ^2+κ∂ u_s/∂ n. For the fourth-order numerical method, which uses an Extension Operator with p=4, we also need the third- and fourth-order normal derivatives, which we state now.Differentiating (<ref>) with respect to n, we see that ∂^3 u_s/∂ n^3=1/λ_s(∂^2u_s/∂ t∂ n-∂ f_s/∂ n)-∂^3u_s/∂ n∂ϑ^2+κ∂^2 u_s/∂ n^2 and∂^4u_s/∂ n^4 =1/λ_s^2(∂^2u_s/∂ t^2-∂ f_s/∂ t) +1/λ_s(-2∂^3u_s/∂ t∂ϑ^2 +κ∂^2u_s/∂ n∂ t. . ⋯ -∂^2f_s/∂ n^2+∂^2f_s/∂ϑ^2+∂^4u_s/∂ϑ^4-κ∂^3u_s/∂ n∂ϑ^2)+κ∂^3 u_s/∂ n^3. Next, let us follow-up on comments made in the text following (<ref>).There, it was pointed out that the unknown Dirichlet and Neumann data (u_s,∂ u_s/∂ n) are the only data required for the Extension Operator (<ref>) with p = 2.Moreover, it was pointed out that this is also true for the Extension Operator when p = 4. The reasoning is as follows.* The time derivatives ∂ u_s/∂ t, ∂^2u_s/∂ t^2, ∂ f_s/∂ t, ∂^2 u_s/∂ n∂ t, and ∂^3u_s/∂ t∂ϑ^2 can be approximated by the backward difference formula. In BDF2–DPM2,∂ u_s^i+1/∂ t ≈3u_s^i+1-4u_s^i+u_s^i-1/2Δ t and∂^2 u_s^i+1/∂ t^2 ≈2u_s^i+1-5u_s^i+4u_s^i-1-u_s^i-2/Δ t^2,while in BDF4–DPM4,∂ u_s^i+1/∂ t ≈25u_s^i+1-48u_s^i+36u_s^i-1-16u_s^i-2+3u_s^i-3/12Δ t and∂^2 u_s^i+1/∂ t^2 ≈1/Δ t^2(15/4u_s^i+1-77/6u_s^i + 107/6u_s^i-1. . ⋯-13u_s^i-2+61/12u_s^i-3-5/6u_s^i-4).* The derivatives in terms of arclength ϑ can be computed from u_s or ∂ u_s/∂ n. For example, denoting u_s=∑_ν=1^𝒩^0 c_1,ν^i+1ϕ_ν(ϑ) (using notation following from (<ref>)), then it comes handy that∂ u_s/∂ϑ =∑_ν=1^𝒩^0 c_1,ν^s, i+1ϕ_ν'(ϑ),∂^2 u_s/∂ϑ^2 =∑_ν=1^𝒩^0 c_1,ν^s, i+1ϕ_ν”(ϑ), and ∂^4 u_s/∂ϑ^4 =∑_ν=1^𝒩^0 c_1,ν^s, i+1ϕ_ν^(4)(ϑ). §.§ The system of equations at each time step.With the Cauchy data 𝔲_s, Γ^i + 1 and Extension Operator Ex_s 𝔲_s, Γ^i + 1 from Γ to γ_s introduced in Definition <ref>, and the spectral representation introduced in (<ref>), we now give a sketch of the linear system for the coefficients (c_1, ν^s, i+1)_ν=1^𝒩^0 and (c_2, ν^s, i+1)_ν=1^𝒩^1, and moreover the approximation of the solution u_s(x, y, t^i+1) at (x_j, y_k) ∈ N_s^+.Indeed, substituting Ex_s 𝔲̃_s, Γ^i + 1 (<ref>, <ref>) into the BEP (<ref>), the resulting linear systems are∑_k = 0^1 ∑_ν=1^𝒩^k( c_k, j^s, i + 1Ex_s Φ^k_ν - c_k, j^s, i + 1 P_γ_s^i + 1Ex_s Φ^k_ν) = _γ_s [G_Δ t, h^i+1 F_s^i + 1]. This can be further elucidated by introducing the vector of unknowns𝐜_s^i + 1 = [ c_1, 1^s, i + 1 c_1, 2^s, i + 1⋯ c_1, 𝒩^0^s, i + 1_𝐜_s, 1^i + 1c_2, 1^s, i + 1 c_2, 2^s, i + 1⋯ c_2, 𝒩^1^s, i + 1_𝐜_s, 2^i + 1]^⊤ (so that 𝐜_s^i + 1 = [ 𝐜_s,1^i + 1, 𝐜_s,2^i + 1 ]^⊤), and the matrixA_s = [(I - P_γ_s^i + 1) Ex_s Φ^0_1,(I - P_γ_s^i + 1) Ex_s Φ^0_2, ⋯ (I - P_γ_s^i + 1) Ex_s Φ^0_𝒩^0_A_s, 1, (I - P_γ_s^i + 1) Ex_s Φ^1_1,(I - P_γ_s^i + 1) Ex_s Φ^1_2, ⋯ (I - P_γ_s^i + 1) Ex_s Φ^1_𝒩^1]_A_s, 2. Then, the full system of equations (<ref>) isA [ [1.5] 𝐜_1^i + 1; 𝐜_2^i + 1 ] = [ [1.5] _γ_1 [G_Δ t, h^i + 1 F^i + 1_1]; _γ_2 [G_Δ t, h^i + 1 F^i + 1_2] ], withA = [ [1.5] A_1 0; 0 A_2 ]. However, note that 𝐜_1^i + 1 and 𝐜_2^i + 1 are related by the interface conditions (<ref>, <ref>), so that the number of unknowns in (<ref>) is equal the dimension of either 𝐜_1^i + 1 or 𝐜_2^i + 1, depending on which one is considered the independent unknown. Therefore, the dimension of A is (|γ_1| + |γ_2|) × (𝒩^0 + 𝒩^1), where 𝒩^0 + 𝒩^1 is the dimension of 𝐜_1^i + 1 or 𝐜_2^i + 1 (whichever is the independent unknown).The independent unknown (𝐜_1^i + 1 or 𝐜_2^i + 1) is chosen so that the finite-dimensional, spectral representation (<ref>) of the Cauchy data 𝔲_s,Γ^i+1 accurately resolves the Cauchy data with a small number of basis functions, in the consideration of both accuracy and computational efficiency. For (<ref>) and (<ref>), we choose 𝐜_2^i + 1 as the independent unknown, while for (<ref>) we choose 𝐜_1^i + 1.With these choices for the independent unknown, we have 𝒩^0=𝒩^1=1 for the three considered test problems. Since each column involves the Difference Potentials operator P_γ_s^i+1 applied to a vector Ex_s Φ_ν^k, each column is therefore constructed via one solution of the Auxiliary Problem (Definition <ref>).However, the Auxiliary Problems are posed on the computationally simple Auxiliary Domains, and can be computed using a fast FFT- or multigrid-based algorithm, which can significantly reduce the computational cost.Moreover, if λ_s(t) ≡λ_s is constant, then A can be computed and inverted once (as a pre-processing step), thus significantly reducing computational cost for long-time simulations. The authors are grateful to the anonymous referees for their valuable remarks and questions, which led to significant improvements to the manuscript.Wegratefully acknowledge the support of the Swedish Research Council (Grant No. 2014-6088); the Swedish Foundation for International Cooperation in Research and Higher Education (Grant No. STINT-IB2016-6512); Uppsala University, Department of Information Technology; and the University of Utah, Department of Mathematics. Y. Epshteyn, K. R. Steffen, and Q. Xia also acknowledge partial support of Simons Foundation Grant No. 415673. plainurl | http://arxiv.org/abs/1707.08459v2 | {
"authors": [
"Gustav Ludvigsson",
"Kyle R. Steffen",
"Simon Sticko",
"Siyang Wang",
"Qing Xia",
"Yekaterina Epshteyn",
"Gunilla Kreiss"
],
"categories": [
"math.NA",
"65M22 (Primary) 65M06, 65M12, 65M55, 65M60, 65M70, 35K20 (Secondary)"
],
"primary_category": "math.NA",
"published": "20170726142326",
"title": "High-order numerical methods for 2D parabolic problems in single and composite domains"
} |
1.5ptazuki IkedaDepartment of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan [email protected]============================================================================================================================== We dig out a deeper mathematical structure of the quantum Hall system from a perspective of the Langlands program. An algebraic expression of the Hamiltonian with the quantum group 𝒰_q(sl_2) is a cornerstone. The Langlands duality of 𝒰_q(sl_2) sheds light on the fractal structure of Hofstadter's butterfly. This would imply a "quantum Langlands duality". § INTRODUCTIONSeveral profound conjectures on number theory and harmonic analysis proposed by R. Langlands <cit.> has been developed drastically in a vast area of modern mathematics and nowadays it is recognized as the Langlands program. Those series of innovation are also directed to (quantum) integrable systems <cit.> and representation theory of quantum groups <cit.>. It has been known that the Hamiltonian (<ref>) of the quantum Hall effect can be written with the quantum group 𝒰_q(sl_2) <cit.> and its energy spectrum shows an attractive structure, called Hofstadter's butterfly <cit.>, shown in figure <ref>. Since these seminal discoveries, it had been naively believed that there would be a more profound mathematical concept for the quantum Hall effect, whereas it had not been well understood for more than 40 years. We claim that it is nothing but the Langlands program and, in this article, we build a direct connection between Hofstadter's butterfly and the Langlands duality of 𝒰_q(sl_2). The integer quantum Hall effect is studied in a detailed way from a perspective of the geometric Langlands correspondence <cit.>, which is another branch of the Langlands program. *Acknowledgement I thank Yasuyuki Hatsuda for providing me his Mathematica code to daw the Butterfly. I also thank Yuji Sugimoto for useful discussions.§ HOFSTADTER'S BUTTERFLY AND LANLANDS DUALITY §.§ PreliminaryThroughout this paper, we consider the integer quantum Hall effect on a square lattice in a uniform magnetic flux ϕ=P/Q (in a unit of flux quantum ϕ_0=hc/e) per plaquette, where P and Q are coprime numbers. Energy spectrum of the Hamiltonian (<ref>) shows the novel fractal structure (fig <ref>) as a function of the flux ϕ, called Hofstadter's butterfly <cit.>. It is known that it is generated by the maps (ϕ,E) → (ϕ+1, E)(ϕ,E) → (1/ϕ, f(E)),where f is a some function. However the origin of this fractal nature had been a mystery and hence how to determine the function f had been a problem. Recently, the map (<ref>) turnedout to have intimate relations to the modular transformation of parameters associated with a Calabi-Yau manifold <cit.>. Instead of the tight binding Hamiltonian, they worked on the Hamiltonian H=e^-x+e^x+R^2(p^-x+p^x) of the relativistic Toda lattice, which is a integrable system, and identified the energy spectrum as the roots of polynomials P_ϕ. Moreover they derived the formula P_ϕ(E,R)=P_1/ϕ(E,R),where E=f(E) and R=R^1/ϕ, by using its dual Hamiltonian H=e^-x+e^x+R^2(p^-x+p^x) in terms of quantum geometry.§.§ Langlands Duality of Quantum GroupsA generic tight binding Hamiltonian we are interested in is H=∑_m,n(c_m+1,n^† c_m,ne^A^x_m,n+R^2c_m,n+1^† c_m,ne^A^y_m,n+h.c.), where c_m,n (c_m,n^†) is the annihilation (creation) operator at (m,n) site. When we choose the Landau gauge A^x_m,n=0, A^y_m,n=mϕ, the eigenvalues of this Hamiltonian are obtained by solving Harper's equation e^ik_xψ_m+1+e^-ik_xψ_m-1+2cos(k_y+mϕ)ψ_m=Eψ_m,where ψ_n are the Bloch functions[We consider only the x-direction since the y-direction dose not contribute to the equation in the Landau gauge, which is a consequence of Bloch's theorem.] with period Q (ψ_n=ψ_n+Q). Therefore our problem results in solving the characteristic polynomial associated with an equivalentHamiltonian which is a Q× Q matrix given by H=T_x+T_x^†+R^2(T_y+T^†_y),where we choose a Q-dimensional representation ρ_Q of 𝒰_q(sl_2)={K^±1,X^±} with q=e^iπ P/Q so thatT_x =e^ik_xρ_Q(X^+), T_y=e^ik_yρ_Q(K) ρ_Q(X^+) =[ 0 1 0 ⋯ 0; ⋮ ⋱ ⋱ ⋱; ⋮ ⋱ ⋱ 0; 0 ⋱ 1; 1 0 ⋯ ⋯ 0 ] , ρ_Q(K)=diag(q^2,q^4,⋯,q^2Q)These operators T_x and T_y are non commutative because of the Aharonov-Bohm phase for an electron moving around the flux:T_xT_y=q^2T_yT_x. The energy spectrum consists of eigenvalues of this Hamiltonian, which is described by the Chambers relation <cit.>(H(k,R)-E)=P_ϕ(E,R)+h(k,R), k=(k_x,k_y)where P_ϕ(E,R) is a polynomial and h(k,R)=2(-1)^Q-1(cos(Qk_x)+R^2Qcos(Qk_y)). The point k_0=(π/2Q,π/2Q) where h(k,R) vanishes is called a mid band point. Therefore the energy spectrum at the mid band point oveys P_ϕ(E,R)=0. Some examples of P_ϕ are as follows: P_P/1(E,R) =EP_1/2(E,R) =E^2-2(1+R^4)P_1/3(E,R) =P_2/3(E,R)=E(-E^2+3+3R^4)P_1/4(E,R) =P_3/4(E,R)=E^4- 4 (1 + R^4)E^2+2 (1 + R^8)P_1/5(E,R) =P_4/5(E,R)=- E^5+ 5 (1 + R^4) E^3 +5/2(-2 + (-3 + √(5)) R^4 - 2R^8)EP_2/5(E,R) =P_3/5(E,R)=- E^5+ 5 (1 + R^4) E^3 -5/2 (2 + (3 + √(5)) R^4 + 2 R^8) E The above polynomials are exactly the same as in <cit.>. Hence the anticipated formula P_P/Q(E,R)=P_Q/P(E,R) (<ref>) implies the equivalence of the Q-dimensional representation (<ref>) of 𝒰_q(sl_2) and the P-dimensional representation of 𝒰_^Lq(sl_2), where ^Lq=e^iπ/ϕ and 𝒰_^Lq(sl_2) is the Langlands dual quantum group of 𝒰_q(sl_2) <cit.>. We write this duality map by S:(𝒰_q(sl_2), H)→ (𝒰_^Lq(sl_2), H),where the dual Hamiltonian H is given by the following P× P matrix of the form H=T_x+T_x^†+R^2(T_y+T^†_y),where T_x=e^ik_xρ_P(X) and T_y=e^ik_yρ_P(Y). Since we expect the correspondence of the characteristic polynomials (H-E)=(H-E), we find R=R^1/ϕ by comparing order of R and R in h(k,R) and h(k,R).The reason why this duality originates from the Langlands duality[This viewpoint would differ from the modular doublet of quantum groups <cit.>.] of quantum groups is explained by the interpolating quantum group 𝒰_q,t(sl_2) <cit.>, which is parametrized by arbitrary nonzero complex values q,t and generated by X^±,K^±1,K^±1 such thatKX^± =q^±2X^± K, KX^±=t^±2 X^±K,[X^+,X^-] =KK-(KK)^-1/qt-(qt)^-1.The interpolating property of 𝒰_q,t(sl_2) appears as 𝒰_q,1(sl_2)/{K=1}≃𝒰_q(sl_2), 𝒰_1,t(sl_2)/{K=1}≃𝒰_t(sl_2).By definition, 𝒰_q,t(sl_2) is equivalent to the usual quantum group 𝒰_ϱ(sl_2) with generators X^±, KK and the parameter ϱ=qt. Taking q=e^iπ P/Q and t= ^Lq=e^iπ Q/P, we find ϱ=q ^Lq=e^iπ(P/Q+Q/P) is symmetric under exchanging P and Q. Therefore the fractal structure in Hofstadter's butterfly embodies "symmetry breaking" of this quantum group 𝒰_ϱ(sl_2) into a quantum group 𝒰_q(sl_2) and its Langlands dual quantum group 𝒰_^Lq(sl_2). The Langlands duality of quantum groups is formulated by E. Frenkel and D. Hernandez <cit.> and, according to which, any irreducible representation of 𝒰_q(sl_2) would be t-deformed uniquely to a representation of 𝒰_q,t(sl_2) in such a way that its specialization at q=1 gives a representation of 𝒰_t(sl_2). The easiest case is P=1 and Q=2. A two-dimensional representation of 𝒰_q(sl_2) is dual to a one-dimensional representation of 𝒰_t(sl_2) (t=^Lq=e^π i Q/P), which is equivalent to P_1/2(E,R)=P_2/1(E,R). Generically, we observe that a Q-dimensional representation of 𝒰_q(sl_2) and a P-dimensional representation of 𝒰_^Lq(sl_2) are dual.§.§ DiscussionLet us conclude this article with comments on recent advances of the Langlands duality. The quantum affine algebra 𝒰_q(sl_2) is natural extension of 𝒰_q(sl_2) and the deformed 𝒲-algebra 𝒲_q,t(sl_2) interpolates 𝒰_q(sl_2) and 𝒰_t(sl_2) <cit.>, where t=q^β with a parameter β. More recently the Langlands duality between a conformal block of 𝒰_q(sl_2) and that of 𝒲_q,t(sl_2) was established in <cit.>. In the limit of q→1 with t=q^β, we have 𝒲_q,t(sl_2)→𝒲_β(sl_2) <cit.>. What is called the Langlands duality of 𝒲-algebras <cit.> is 𝒲_β(sl_2)=𝒲_^Lβ(sl_2), β ^Lβ=1.Surprisingly, they are related to S-duality of four dimensional supersymmetric Yang-Mills theories with parameters β=R_s/R_t and ^Lβ=R_t/R_s, where R_s and R_t are radii of circles in T^2=S^1_s× S^1_t<cit.>. In our case these parameters are somehow identified with R_s=P and R_t=Q. Taking q=e^iπβ and t=^Lq=e^iπ ^Lβ, we have the following duality picture:𝒰_t(sl_2)↔𝒲_q,t(sl_2)→𝒲_β(sl_2)=𝒲_^Lβ(sl_2)𝒲_t,q(sl_2)↔𝒰_q(sl_2).These relations with supper Yang-Mills theories are quite interesting, since it implies some relations with the geometric Langlands duality in an unexpected way. It desires further endeavor to clarify all of those relations. We leave comments on relation to S-duality of four-dimensional 𝒩=4 suppersymmetric Yang-Mills theories. In physics, the geometric Langlands correspondence was firstly mentioned for mirror symmetries of Hitchin moduli spaces ℳ_G and ℳ_^LG as results of dimensional reduction to a two-dimensional sigma model <cit.>. These Hitchin moduli spaces are exchanged by S-transformation of the formS:(G,τ)→(^LG,1/n_𝔤τ),where ^LG is the Langlands dual Lie group[P. Goddard, J. Nuyts, and D. Olive found the Langlands dual groups from physical motivation <cit.>. ^LG is known as a GNO dual group among physicists. ] of G, τ is a parameter of the theory, and n_𝔤 is the ratio of length squared of long and short roots of G. What is different in our case is the S-transformation (<ref>) plays an alternative role. Hence we interpret that S-duality endows another mathematical phenomenon, which we may call a "quantum Langlands correspondence". §.§ Bethe Ansatz and the Energy SpectrumNow let us determine the shape of the unknown function f in (<ref>). For this purpose it is paramount of importance to recall the relations between Hofstader's butterfly and the Bethe ansatz equations <cit.>. We consider the case ϕ=P/Q and then there are Q energy bands. By choosing a certain gauge, Harper's equation (<ref>) can be equivalently deformed to the equation i(z^-1+qz)Ψ(qz)-i(z^-1+q^-1z)Ψ(q^-1z)=EΨ(z), where z is a parameter such that ψ_n=Ψ(q^n), which are periodic in 2Q (ψ_n=ψ_n+2Q). This Ψ(z) is a polynomial of degree Q-1 and the set of zero points {z_i} of Ψ(z) obey the Bethe ansatz equationz^2_n+q/qz^2_n=-∏_m=1^Q-1qz_n-z_m/z_n-qz_m, n=1,⋯,Q-1.Moreover the spectrum is given by the sum of rootsE=-i(q-q^-1)∑_n=1^Q-1z_l.Since the above derivation does not depend on the numerator of ϕ=P/Q, we just exchange replace Q by P to obtain the dual energy E. Hence E can be written by the sum of the roots of the "dual" Bethe ansatz equation. In principle we can determine f by this procedure. utphys | http://arxiv.org/abs/1708.00436v1 | {
"authors": [
"Kazuki Ikeda"
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"cond-mat.mes-hall",
"hep-th",
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"published": "20170727161928",
"title": "Hofstadter's Butterfly and Langlands Duality"
} |
=1eq The observation of PeV neutrinos is an open window to study New Physics processes. Among all possible neutrino observables, the neutrino flavor composition can reveal underlying interactions during the neutrino propagation. We study the effects on neutrino oscillations of dark matter-neutrino interactions. We estimate the size of the interaction strength to produce a sizable deviation with respect to the flavor composition from oscillations in vacuum. We found that the dark matter distribution produces flavor compositions non reproducible by other New Physics phenomena. Besides, the dark matter effect predicts flavor compositions which depend on the neutrino's arrival direction. This feature might be observed in neutrino telescopes like IceCube and KM3NET with access to different sky sections. This effect presents a novel way to test Dark Matter particle models.Neutrino oscillations in the galactic dark matter halo Roberto A. Lineroshttp://goo.gl/00TnLINSPIRE-HEP profile, [email protected] December 30, 2023 ================================================================================§ INTRODUCTION The observation and confirmation of neutrino oscillations is one of the earliest evidences of New Physics (NP) beyond the Standard Model (SM) of particle physics. Roughly speaking, neutrino flavors (ν_e, ν_μ, and ν_τ) mix while they propagate through space. The misalignment between flavor and mass eigenstates is the cause of the oscillations. The overwhelming observational evidence of neutrino oscillations challenges the SM because it is a clear indication that neutrinos are massive. A conservative estimation of neutrino masses is ∑_i=1^3 m_ν≲ 0.72 eV which comes from cosmological observations and the analysis done by the PLANCK Collaboration <cit.>. Moreover, oscillation experiments are sensitive to the neutrino square-mass differences (Δ m_ij^2 = m_j^2 - m_i^2), the mixing angles between flavor/mass eigenstates (θ_ij), and the CP-phase δ (for details check <cit.> and references within). Currently, the precision in the neutrino parameters determination allows us to probe possible NP effects. Neutrinos are unique among other SM particles. They are the lightest fermions and electrically neutral. Besides, neutrinos can interact with SM particles via the exchange of Z and W^± bosons. This gives them the property to cross through anything without almost interactions. Therefore neutrinos can unveil regions of the Universe which are hidden to other type of observations. In this case,neutrino telescopes like , ANTARES <cit.>, and KM3Net <cit.>, among others, give complementary information with respect to other astroparticle messengers (e.g. gamma- and cosmic-rays) becoming unique tools to understand the Universe. It is important to highlight the IceCube observatory and its leading role on revealing the Universe with the observation of PeV neutrinos <cit.>. This opens the door to test New Physics effects in the neutrino physics sector. Recently, IceCube has presented results on the flavor composition of astrophysical neutrinos <cit.>. The reported best fit point in the flavor space is far from the expected flavor composition. The expected composition assumes neutrino oscillations in vacuum and initial neutrino flavor from astrophysics-motivated sources. In some scenarios, the deviation could be related to the presence of exotic sources of PeV neutrinos <cit.>. Alternatively to such hypothesis, the deviation could come from effects from the presence of Dark Matter (DM). It is quite well established that the Universe is fulfilled with DM and its presence has a large effect in the cosmological evolution of the Universe. In fact, the DM relic abundance is quite well estimated using observables extracted from anisotropies of the Cosmic Microwave Background, structure formation, among others. The combination of all these observables sets the DM relic abundance to Ω_ DM h^2 = 0.1198 ± 0.0015 <cit.>.The presence of DM is per se an evidence of NP and thus any DM candidate must come from models beyond-SM. In some cases, the DM candidate is related to neutrino physics (e.g. <cit.> and references within).Regardless of the DM nature, it is tantalizing to consider that DM could affect some neutrino observables. In this manuscript, we summarized the results presented in <cit.> which studies the neutrino flavor composition of neutrinos propagating in the galactic DM halo.§ NEUTRINO OSCILLATIONS THROUGH A DM MEDIUM The Milky Way galaxy is embedded in a DM halo.The precise DM distribution (ρ_ DM) inside the Milky Way is still under study. However, it is expected that its DM distribution is in the ballpark of a NFW <cit.> or an isothermal profile distribution <cit.>.Neutrinos produced inside the Milky Way must travel across the galactic DM halo. In this scenario, the DM effects on the neutrino oscillation can be modeled using the Mikheyev-Smirnov-Wolfenstein (MSW) effect <cit.>. In such case the total hamiltonian in the flavor basis isℋ_ tot(x) = ℋ_ vac + 𝒱(x) ,where ℋ_ vac is the (global phase subtracted) hamiltonian in vacuum, ℋ_ vac = 1/2E U_0[ [000;0 Δ m^2_210;00 Δ m^2_31 ]] U_0^†,which depends on the neutrino square-mass difference Δ m^2_ij = m^2_i - m^2_j, the lepton mixing matrix U_0 <cit.>, and the neutrino energy E. The spatial dependence of the hamiltonian comes from the effective potential 𝒱(x) in the flavor basis, 𝒱(x) = 𝒱_⊕ ρ̂_ DM(x) ,where ρ̂_ DM(x) is the DM distribution (Sun's position normalized) and 𝒱_⊕ is the value of the effective potential matrix at the Sun's position. Depending of the interpretation, the effective potential can be also written mimicking the weak interaction MSW effective potential:𝒱(x) = λ'G_F' N_ DM(x) ,where λ' is the couplings matrix in the flavor basis, G_F'= G_F m_Z^2/m_Z'^2 is a scaled Fermi constant for a mediator with mass m_Z', andis the number density of DM particles. The relation between Eqs. <ref> and <ref> gives that:𝒱_⊕ = λ'G_F'ρ_ DM(x_⊕)/m_ DM,where we use ρ_ DM(x_⊕) = 0.4 GeV/ cm^3 as a common value for all the DM distributions. Each description of 𝒱(x) remarks different aspects of the effect: Eq. <ref> remarks the strength of the effective potential and spatial effects and Eq. <ref> remarks a possible particle model extension. In both cases, the results are the equivalent and provide interesting insights.The evolution of neutrino states in a medium is controlled by i ∂_t Ψ = ℋ_ totΨ,where Ψ is the neutrino field. The relation between flavor states and mass states is given by |ν_α⟩ = ∑_k U^*_α k | ν_k⟩,where the greek index α corresponds to a flavor index (e, μ, τ), the latin index k refers to ℋ_ tot eigenstates with effective mass m_k, eff, and U is the unitary matrix which diagonalizes ℋ_ tot: ℋ_tot^m= U^†ℋ_tot U = 1/2 E( [000;0 Δ m_21,eff^20;00 Δ m_31,eff^2 ]) ,where ℋ_tot^m is the (global phase subtracted) total hamiltonian in the mass-state basis. The typical distance between sources and Earth ranges from parsecs to kiloparsecs. This distance is extremely large with respect to oscillation length at the PeV energy making neutrinos to loose coherence. This allow us to simplify the temporal/spatial evolution of neutrino states because of the large distance averaging effect of neutrino oscillations. In the case of a spatially homogeneous potential 𝒱(x) = 𝒱, the observed flavor composition is: f_β = ∑_α=e,μ,τ(∑_i=1,3|U_β i U_α i^*|^2 f^0_α) ,where f^0_α = (f^0_e,f^0_μ,f^0_τ) is the flavor composition at the source point, and U is the unitary matrix that diagonalizes the total hamiltonian ℋ_tot. A more interesting case occurs when the effective potential varies along the neutrino path to Earth. In this case, we may need to solve Eq. <ref> and follow the evolution of f_β in different stages along the neutrino path. However, the DM profile does not vary fast enough and the evolution of neutrino flavors satisfies the adiabatic condition (see <cit.> for details). In this case, the flavor composition at Earth is f^⊕_β = ∑_α=e,μ,τ(∑_i=1,3|U^⊕_β i U_α i^0*|^2 f^0_α) ,where U^⊕ diagonalizes ℋ_ tot evaluated at Earth's position, while U^0 diagonalizes ℋ_ tot evaluated at the source. The difference between U's arises because the effective potential scales according to ρ_ DM.Let just remark that the matrix texture would not to change during the neutrino path.§ RESULTS AND DISCUSSION The DM effect on the neutrino oscillations depends on the size of 𝒱_⊕ and its texture. We know from the standard MSW effect that if the effective potential is smaller than ℋ^ vac the flavor composition converges to the solution in vacuum. On the other hand, if it is larger than ℋ^ vac, the composition is equal to the composition at the source. Therefore, a rough estimation is 𝒱∼Δ m_ij^2/2E at some point during the neutrino path. For neutrinos at PeV range, the effective potential in the range 𝒱∼ 10^-17 – 10^-21 eV may affect non-trivially the observed flavor composition. More details about the exact range and texture of 𝒱 are in <cit.>.In Fig. <ref>, we present 2 ternary plots in the flavor space. In both plots, we consider neutrinos with E_ν = 1 PeV homogeneously produced in the galatic disk. In the left panel, we assume an initial flavor composition of f^0 = (1:2:0) which is compatible with neutrinos coming from pion decays. Besides, we randomly generate the matrix 𝒱_⊕ keeping |𝒱_⊕| < 10^-17 eV. The black pentagon indicates the expected composition assuming oscillations in vacuum: f^⊕, vac≃ (1:1:1). The red region corresponds to the accessible area if the DM distribution is constant. We highlight that our region coincides with the one coming from Lorentz-violation effects <cit.>. The blue region corresponds to same setup but using a NFW profile. In this case, the accessible region for the NFW profile is larger than the constant profile case. This indicates that spatial dependence of the effective potential does largely affect the observed flavor composition. The green star indicates IceCube's best fit. Unfortunately, its location is just in the interface among both areas.The right panel of same figure shows the accessible region when maximum value of 𝒱 is restricted. This case uses f^0 = (1:0:0) and a NFW profile. The black pentagon is the expected flavor composition in vacuum and it corresponds to f^⊕, vac≃ (0.56:0.22:0.22). Here, we show that the effect of the DM distribution can cover almost the full flavor triangle for 𝒱 values as large as 10^-17 eV. Smaller values of the potential shrinks the area, converging to an small area around the vacuum solution. The 𝒱 scale may indicate a beyond-SM solution. To this end, we consider DM models like asymmetric-DM, scalar DM, and fuzzy-DM <cit.>. All of them could generate such scale providing interesting classes of models. For the interpretation in terms of particle models, we need to calculate the neutrino mean free path, l_ν = ( σN_ DM)^-1,which indicates the distance that neutrinos can cover without (hard) scattering on DM. For distances 𝒪(kpc), i.e. Milky Way size, we required roughly σ≲× 10^-21cm^2 × m_ DM /GeV. This condition ensures that the DM interaction only affects the neutrino oscillation. The mean free path depends on same parameters as Eq. <ref>. The cross section for a t-channel interaction in the high energy regime (√(s)≫ m_Z',m_ DM) becomes σ≃λ' ( m_Z'/ GeV)^-2× 3.75 × 10^-29 cm^2 .Looking inside Eq. <ref>, we can estimate the numerical value of two parameters among λ', m_ DM and G_F' (or m_Z') by reproducing the required value of 𝒱 and l_ν. We found that fuzzy-DM models with m_ DM≃ 10^-23 eV align better if the interaction strength is similar to the one of Weak interactions. Models with masses in the GeV–keV range, inspired by WIMPs, majorons, etc. (e.g. <cit.>), requires mediators with masses m_Z'∼ 10^-6 (10^-2) eV which would generate an observable effect in the neutrino oscillation. We study other possibilities including making the mean free path larger than the observable universe (see details in <cit.>).Another interesting aspect of the DM effects is the dependence on the arrival direction.Galactic neutrinos originated in regions closer to the galactic center would be affected differently that those originated in the Milky Way's outskirts. This is important because neutrino telescopes like IceCube and KM3NET are sensitive to different patches of the sky. Indeed, this effect indicates that observed flavor composition in both telescopes may be different. The case of extragalactic neutrinos are also interesting because it allows us to explore DM densities in extragalactic halos.§ CONCLUSIONS The observation of PeV neutrinos allows us to study the Universe as never done before. Similarly, the presence of DM in the Universe is a clear signal of physics beyond-SM.Neutrinos traveling to Earth must propagate through the galactic DM halo. Assuming that DM and neutrinos do interact, we study the effect of such interactions at the level of neutrino oscillations. We found that the effect can produce flavor compositions at Earth different than those expected from neutrino oscillations in vacuum. Besides, the spatial dependence generates flavor compositions outside the region obtained from other NP effects.The interpretation in terms of DM models gives interesting constructions. For DM particles with masses in the keV and GeV range, the neutrino-DM interaction could be mediated by new particles with masses in the sub-eV range. The implication for the Weak-like interaction is that DM mass has to be extremely small 𝒪(10^-23eV). This solution is compatible with fuzzy-DM models.Better statistics of the neutrino flavor composition might indicate a dependence with respect to the neutrino arrival direction. This feature gives a nice prediction of this effect.I would like to thank the RICAP 2016 organizers for a very interesting conference. I thank http://goo.gl/CHlgQRP. F. de Salas and http://goo.gl/x8CPgJM. Tórtola for their great work done in the paper referred in this manuscript. I also thank the hospitality of the Theoretical Physics group at the U. Libre de Bruxelles and the Interactions Fondamentales en Physique et en Astrophysique (IFPA) group at U. Liège. http://goo.gl/00TnLR. A. L. acknowledges the support of the Juan de la Cierva contract JCI-2012-12901 (MINECO) and the Spanish MESS via theServicio Público de Empleo Estatal.Astrophys. J. Astrophys. J. Lett. Astrophys. J. Suppl. Ser. Astron. & Astrophys. Astron. J. Ann. Rev. Astron. Astrophys. Mon. Not. R. Astron. Soc. Phys. Rept. J. Cosmology Astropart. Phys. J. High Ener. Phys. Phys. Rev. Lett. Phys. Rev. D Nucl. Phys. A | http://arxiv.org/abs/1707.08972v1 | {
"authors": [
"R. A. Lineros"
],
"categories": [
"hep-ph",
"astro-ph.HE"
],
"primary_category": "hep-ph",
"published": "20170727180011",
"title": "Neutrino oscillations in the galactic dark matter halo"
} |
NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR, 56126 Pisa, Italy Département de Physique, Ecole Normale Supérieure / PSL Research University, CNRS, 24 rue Lhomond, 75005 Paris, France Department of Physics, Bar-Ilan University, 52900 Ramat-Gan, Israel Dipartimento di Fisica, Università di Pisa and INFN, Largo Pontecorvo 3, 56127 Pisa, Italy ICTP, Strada Costiera 11, 34151 Trieste, Italy NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR, 56126 Pisa, Italy NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR, 56126 Pisa, Italy ICTP, Strada Costiera 11, 34151 Trieste, Italy We study the robustness of non-local string order in two paradigmatic disordered spin-chain models,a spin-1/2 cluster-Ising and a spin-1 XXZ Heisenberg chain. In the clean case, they both display a transition from antiferromagnetic to string order. Applying a disorder which preserves the Hamiltonian symmetries, we find that the transition persists in both models. In the disordered cluster-Ising model we can study the transition analytically – by applying the strongest coupling renormalization group – and numerically – by exploiting integrability to study the antiferromagnetic and string order parameters. We map the model into a quadratic fermion chain, where the transition appears as a change in the number of zero-energy edge modes.We also explore its zero-temperature-singularity behavior and find a transition from a non-singular to a singular region, at a point that is different from the one separating non-local and local ordering. The disordered Heisenberg chain can be treated only numerically: by means of MPS-based simulations, we are able to locate the existence of a transition between antiferromagnetic and string-ordered phase, through the study of order parameters. Finally we discuss possible connections of our findings with many body localization.Resilience of hidden order to symmetry-preserving disorder Angelo Russomanno December 30, 2023 ==========================================================§ INTRODUCTION Since Landau, we know thatphase transitions and symmetry breaking are strictly connected (see for instance Ref. Goldenfeld:book).Let us focus on zero-temperature quantum phase transitions (QPTs) <cit.>.In the ordered phase, there is a manifold of degenerate ground states which have high entanglement and obey the same symmetries of the Hamiltonian. Nevertheless, any physical ground state breaks this symmetry: because of decoherence, the system always ends up in a superposition of these symmetry-preserving states which has small entanglement and breaks the symmetry. This mechanism manifests in the expectation value Φ of some local operator Φ̂( x) (the order parameter) being different from zero. Symmetry breaking always comes together with long-range order: in the thermodynamic limit, the order parameter in a ground state is infinite-range correlated,lim_| x- y|→∞⟨Φ̂^†( x)Φ̂( y) ⟩_ GS=|Φ|^2.Although the value of Φ depends on the specific choice of the symmetry-breaking ground state,its modulus |Φ|^2 is independent of it. Moreover, this correlator does not depend on the choice of the ground state, even if we consider a non-physical symmetry-preserving ground state. From a physical point of view, the order parameter can be, for instance, the magnetization in the ferromagnetic transition (breaking of rotation symmetry) or the superconducting wave-function in superconductivity (breaking of gauge symmetry).This paradigm has been challenged in the last decades by the discovery of topological phase transitions which are characterized by no local order parameter but by a global rearrangement of the system structure <cit.>. The landscape of topological phase transitions is extremely rich <cit.>; here we specifically focus on a particular class of topologicalphases occurring in one-dimensional systems, for which the concept of hidden order has been put forward. In this case, the order parameterstill exists, but it is a non-local one: it is a string operator involving the system in its globality.The most famous example of string order (SO) is in the Haldane phase of a spin-1 isotropic Heisenberg chain <cit.>.Let us focus on one-dimensional systems and call 𝒪̂_j,k the SO parameterbetween two sites j and k: long-range order in the thermodynamic limit is given bylim_|j-k|→∞⟨𝒪̂_j,k⟩_ GS≠0, where ⟨ · ⟩_ GS denotes the ground-state (GS) average.In this case, an infinite-range non-local correlator is different from zero and does not correspondto any local non-vanishing object.Examples of transitions to hidden SO phases are the one between a ferromagnetand the Haldane phase in spin-1 chains <cit.>,and the one occurring in the extended Bose Hubbard model <cit.>.In some cases of hidden order the non-local parameter has not yet been recognized or does not exist <cit.>,and recently SO has been discovered in a periodically driven spin-1/2 chain <cit.>.Properties of systems undergoing QPTs are markedly altered by the presence of disorder.Disorder shifts the phase transition point and can also create new phases,like the Griffiths phase <cit.> in disordered ferromagnets,the insulating phase in disordered superconductors <cit.>, the Bose-Glass phasein a disordered Bose-Hubbard model <cit.> and the many body localized phase in short-range interacting quantum systems <cit.>.Very remarkable are the works by Dasgupta and Ma <cit.> and by Fisher <cit.>,who are able to construct a renormalization group (RG) flow – the “strongest coupling renormalization group” – to understand phase transitions in such models. This RG method has been used afterwards for a large variety of random quantum and classical systems (see Ref. Igloi_review for a review). Many works focused on disorder and phaseswith local order. The interplay of disorder and non-local SO has been considered in a comparatively small amount of papers. Ref. Igloi focused on disordered spin-1/2 ladders, that may describe properties of spin-1 chains with SO <cit.>.The interest on the ladders was due to the fact that the application of strongest coupling RG to spin-1 chains is difficult due to the proliferation of large local spin terms. This difficulty has been overcome in Refs. Boechat1,Boechat,Lajko_PRB; in particular,Ref. Boechat found a transition from a Griffiths to a disordered Haldane phase in an S=1 disordered antiferromagnetic Heisenberg chain. The numerical demonstration of the persistence of the string order in this model (with a different disorder) and a detailed analysis of the zero-temperature-singularity behaviour has been discussed in Ref. Lajko_PRB. In these works, the form of the disorder has been chosen so to preserve the symmetries of the Hamiltonian. While for small disorder amplitudes the SO is preserved thanks to the topological protection, over a given threshold the non-local ordering is broken. The topological protection for small disorder is expected from general arguments, however the breaking of SO for strong disorder and the properties of this transition are not trivial at all <cit.>.As far as we know, there are no studies on the effect of a strong disorder on the properties of a transition from a string non-local order to a local order (like ferro- or antiferro- magnetic). Here, we make a progress in this direction considering disordered spin-1 and spin-1/2 chainswhose clean counterparts (the XXZ Heisenberg chain and the cluster-Ising model) are well known to displaya transition from an antiferromagnetic (AF) to a string-ordered phase (the isotropic Heisenberg chainis the Haldane model).In the first case we resort to numerics: using a variational algorithm based on the formalism of matrix-product states (MPS) <cit.> we find that a phase with SO actually does exist,even in the disordered model.We locate the phase transition point and observe that it is shifted by the disorder, with respect to the clean case.In the case of the disordered cluster-Ising model we can go further: by applyinga strongest coupling RG transformation, we show the existence of an AF and a string-ordered phase,and analytically find the phase transition point.In this case, the model is solvable because it can be mapped to a disordered non-interacting fermion chain.In the fermionic representation, the transition is of topological nature and ischaracterized by the appearance of zero-energy edge modes. Finally, westudy the zero-temperature-singularity behavior of the model, by numerically analyzing the properties of the logarithmic gap distribution and of the inverse average of the gap <cit.>. We find a transition between a non-singular and a singular behavior when the disorder strength is increased. This transition point is different from the one between SO and AF: for our specific form of disorder, we see the existence of a non-singular and a singular SO phase, similarly to the findings of Ref. Lajko_PRB. Moreover we observe that the AF phase is singular.The paper is organized as follows. In Sec. <ref>, we introduce the two models we are considering.In Sec. <ref>, we study the AF/SO transition in the cluster-Ising chain.We do this both analytically (through the strongest coupling RG) and numerically (evaluating the appropriatecorrelators thanks to the free-fermion mapping); we also study the properties of edge modesin the fermionic representation.In Sec. <ref>, we study the AF/SO transition in the Heisenberg XXZ chain,by resorting to MPS numerical simulations.Finally, in Sec. <ref>, we draw our conclusions.In Appendix <ref>, we show details on the strongest coupling RG transformation appliedto the cluster-Ising chain, and in Appendix <ref> we discuss the appearanceof kinks in the MPS approximation of the GS, which are due to a numerical artifact. § THE MODELSIn the following, we will consider two spin chains that exhibit a zero-temperature QPT between a Néel-like AF phase and a phase that is characterized by non-local SO. They are the spin-1/2 cluster-Ising model and the spin-1 XXZ Heisenberg chain. The two models enjoy a local ℤ_2 and 𝔻_2 symmetry, respectively. We thus expect the SO phase to exist also in the presence of disorder, as long as the local symmetry is not broken <cit.>. In this section, we introduce the two models by presenting their Hamiltonian together with some of their properties and the relevant antiferromagnetic and string order parameters. §.§ Spin-1/2 cluster-Ising modelThe simplest and exactly solvable model in this context is the so called cluster-Ising model (CIM) <cit.>, which is described by the Hamiltonian Ĥ_ CIM = ∑_j [ - J_j σ̂_j-1^xσ̂_j^zσ̂_j+1^x + λ_j σ̂_j^yσ̂_j+1^y ]. Here, σ̂^α_j (with α = x,y,z) denote the spin-1/2 Pauli matrices on the j-th site of the chain (j=1, …, L, where L is the chain length), while J_j and λ_j denote the (possibly site-dependent) three-spin and two-spin coupling terms, respectively. In the following, we will always adopt open boundary conditions (OBC). After a standard Jordan-Wigner transformation <cit.> of the form ĉ_j = ( ∏_m=1^j-1σ̂^z_m ) σ̂^-_j/2, the Hamiltonian in Eq. (<ref>) can be mapped onto a free-fermion model with nearest- and next-to-nearest-neighbor hopping terms Ĥ_ CIM = ∑_j [ - J_j (j-1-j-1)(j+1+j+1)+ λ_j (j+j)(j+1-j+1) ], in a way similar to the one-dimensional quantum Ising model in transverse field <cit.>.This Hamiltonian is integrable, and its dynamics can be reduced to that of non-interactingfermionic quasiparticles: the GS is the BCS state without quasiparticles.It has been proven <cit.> that, in the homogeneous case (i.e. J_j=J and λ_j = λ for all j), the system features a QPT between a conventional AF phase along the y direction and a phase withnon-local SO (the so called “cluster phase”), when decreasing λ/J and crossing the critical value of 1.The y-antiferromagnet is detected by a local order parameter. In agreement with the discussionin the Introduction, we can express it as an infinite-range correlator: ( 𝒮_[1/2]^y)^2 = 1/4 lim_l→∞ (-1)^l⟨σ̂_k^y σ̂_k+l^y ⟩ . To be precise, here we are considering the square modulus of the order parameter,which is the same for all the degenerate symmetry-breaking ground states.Since this model breaks the ℤ_2 symmetry (see below for more details), there are exactlytwo symmetry-breaking ground states which differ for the sign of the order parameter. Moreover, because we are evaluating the modulus of the order parameter as the limit of a correlator,it is not important to select the symmetry breaking ground states: the correlators are the sameon all the states of the GS manifold. From a technical point of view, the mappingto the fermionic model is very important in order to evaluate this correlator.Thanks to Wick's theorem, it can be expressed as a Toeplitz determinant <cit.>of specific single-particle fermionic correlatorsof the Hamiltonian (<ref>) – see Subsec. <ref> for more details. On the opposite, the cluster phase is characterized by a non-vanishing value of the non-local SO parameter, which is expressed as the infinite-range limit of a non-local correlator: 𝒪_[1/2]^z = lim_l →∞ (-1)^l⟨σ̂_k^xσ̂_k+1^y[ ∏_n=k+2^k+l-2σ̂_n^z ] σ̂_k+l-1^y σ̂_k+l^x ⟩ . From a technical point of view, this non-local correlator is evaluated by applying to the Hamiltoniana duality transformation (see Ref. Smacchia_2011 and Appendix <ref> for details).This non-local unitary transformation maps the Hamiltonian onto itself, with λ_j exchanged with J_j[see Eq. (<ref>)]. Moreover, the correlator of Eq. (<ref>)is mapped on the AF correlator of Eq. (<ref>), which can be evaluated as explained above.Unless specified, here and in the following equations, we will omit the subscript ⟨ … ⟩_ GSand consider all the expectation values on one of the degenerate ground states of the system.As remarked before, it is not important which GS in particular,being the correlators independent of the specific choice of the state in the GS manifold.Moreover, in the presence of disorder, we will average over many realizations:this operation will be denoted by an overline ….We notice that the disorder is not translationally invariant:translation invariance is restored after averaging over the disorder realizations.From a numerical point of view, we can only evaluate finite range correlators:𝒪_[1/2],l^α (with α=x,y,z) denotes the string correlator over a range l, such that 𝒪_[1/2]^α = lim_l →∞𝒪_[1/2],l^α. In a similar way, we define the finite-range AF correlator as ( 𝒮_[1/2],l^α)^2 such that ( 𝒮_[1/2]^α)^2 = lim_l →∞( 𝒮_[1/2],l^α)^2. Let us now focus on the version of the Hamiltonian with disorder, which we are studying in the rest of the work. A necessary condition to preserve the string-ordered phase is the choice of a disorder that preservesthe symmetry of the model: this is the symmetry broken by the GSin the phase transition. As mentioned above, the Hamiltonian in Eq. (<ref>) enjoys a ℤ_2 symmetry,since it is invariant under a rotation of angle π around the z axis: V̂^†Ĥ_ CIMV̂ = Ĥ_ CIM withV̂ = exp(-iπ/2∑_jσ̂_j^z). Being the symmetry group generated by only two operators (V̂ and 𝕀̂), the GS manifold can have at most dimension two, so there can be at most two symmetry-breaking ground states. The invariance under the operator V̂ implies: [ ∏_jσ̂_j^z,Ĥ_ CIM]=0 . A possibility to satisfy this condition in a disordered situation is to assume that both the site-dependent two-spin and the three-spin couplings are taken randomly from some probability distribution. In the following, we will specifically address the situation in which both the λ_j and the J_j are uniformly distributed over some interval λ_j ∈ [0,λ_ max] and J_j ∈ [0, J_ max], for all j=1, …, L. §.§ Spin-1 XXZ Heisenberg chainThe other model that we are going to focus on is a spin-1 XXZ Heisenberg chain, given by the Hamiltonian Ĥ_ XXZ = J ∑_j [ Ŝ^x_jŜ^x_j+1+Ŝ^y_jŜ^y_j+1+Λ_jŜ^z_jŜ^z_j+1] , where Ŝ_j^α (with α = x,y,z) now denotes the spin-1 operators on the j-th site, J is the energy scale of the nearest-neighbor spin coupling, and Λ_j is the anisotropy factor along the z axis, at site j. Contrary to the CIM, the spin-1 Heisenberg chain is a non-integrable model and cannot be easily diagonalized. For this reason, numerical approaches based on exact diagonalization techniques or on MPS are usually employed in order to capture the GS physics.In the clean case, that is for Λ_j = Λ,∀ j, model (<ref>) is known to displaya phase transition between a topological phase, usually referred to as the Haldane phase <cit.>,and a non-topological Néel AF phase. Such transition has been studied in some details in the literature, and is expected to occur for Λ≈ 1.186… <cit.>.In an open chain, the GS in the Haldane phase is identified by the presence of gapless spin-1/2 modes on top of a gapped bulk, which make the GS four-fold degenerate in the thermodynamic limit. This phenomenology is related to a hidden 𝔻_2 symmetry breaking, described by the dihedral group of rotations <cit.> 𝒢_𝔻_2 = {Î,e^iπ∑_nŜ^x_n,e^iπ∑_nŜ^y_n,e^iπ∑_nŜ^z_n}. Because of the presence of such edge modes, a state in the Haldane phase is characterized by a hidden long-range order <cit.>, which cannot be revealed by any expectation value of simple two-point correlators ⟨Ŝ^α_kŜ^α_k+l⟩. Indeed, their GS expectation values vanish in the limit l →∞. This hidden order can be seen by defining a non-local SO parameter for a spin-1 chainin a way similar to what has been done for the CIM (<ref>): 𝒪^α_[1] = lim_l →∞𝒪^α_[1],l =lim_l →∞⟨Ŝ^α_k [ ∏_n=k+1^k+l-1e^iπŜ^α_n ] Ŝ^α_k+l⟩ , such that the system is said to posses SO if the above limit is finite and nonvanishing.The presence of hidden order can be understood by remapping the model onto a ferromagnetic chainwith four symmetry-broken states, by means of the Kennedy-Tasaki transformation <cit.>: string correlators are mapped onto two-point correlators, which indeed reveal the presence of ferromagnetic order.This is similar to what happens in the CIM, where a duality transformation maps the SO correlatorof Eq. (<ref>) onto the AF correlator of Eq. (<ref>). The physical meaning of the SO parameter (<ref>) is very clearly expressed in Ref. Scalapino;we briefly review it here for reader's convenience: if we measure the value of the spin projections along α,we can find 1,0 or -1, being this a spin-1 chain. Since the GS is a superposition of spin eigenstates,each time we measure we find a different sequence of +1,0 or -1, with some probability.The fact that the operator (<ref>) has a non-vanishing expectation value means that, if we withdraw the zerosfrom any of these sequences, we find alternatively +1 and -1: without the zeros, the system behavesantiferromagnetically. This property cannot be witnessed by any local operator: only the non-local string operator can do.On the other hand, the presence of long-range AF order along the z axis in the Néel phase is witnessedby a nonzero value of the staggered two-point correlator(𝒮_[1]^α)^2 = lim_l →∞( 𝒮_[1],l^α)^2= lim_l →∞ (-1)^l⟨Ŝ^α_k Ŝ^α_k+l⟩ , with α = z (this is the square value of the AF order parameter along α). Thus, we can identify the Haldane phase of the XXZ chain by a nonzero SO [Eq. (<ref>), for all α] and a vanishing expectation value of the staggered correlator in Eq. (<ref>), for any α. Conversely, the Néel phase is identified by a vanishing SO 𝒪^α_[1] for α=x,y, and by a nonzero value of the staggered AF order parameter 𝒮^z_[1]. We point out that we cannot use the SO along z as an order parameter of the Haldane phase, because the observable 𝒪^z_[1] is nonzero both in the Haldane and in the Néel phase <cit.>. This can be simply understood in the large-Λ limit, where the GS is given by a product of consecutive states with opposite spin projection.From Eq. (<ref>), it is clear that 𝒪̂^z_[1],l evaluated on such GS is exactly -1 for all l, and in particular for l →∞.We will model the disorder by taking Λ_j as a random variable, which is uniformly distributed between Λ_ min and Λ_ max. We stress that a necessary condition for the GS to possess SO is to enjoy a unitary and local symmetry <cit.>.As stated before, since the presence of non-uniform anisotropy Λ_j does not break the 𝔻_2 symmetry, the SO phase is expected to be present also in the disordered XXZ model, at least for some range of values Λ_ min and Λ_ max (see Appendix <ref> for an example of destruction of string order when a symmetry breaking term is added to the Hamiltonian).§ ANTIFERROMAGNETIC – CLUSTER PHASE TRANSITION IN THE DISORDERED CIM§.§ Strongest coupling RG approachTo understand whether the CIM can undergo a QPT or not, we perform a strongest coupling RG analysisin the thermodynamic limit L →∞, very similar in spirit to the one used in Ref. Fisher_1995for the disordered transverse field Ising model. We consider the largest value of the coupling Ω_I=max{J_j, λ_j} .The idea is to diagonalize the part of the Hamiltonian related to this coupling,assuming that it is so large that the corresponding term of the Hamiltonian can be considered as non-interacting, in a crudest approximation. The rest of the chain can be considered as a perturbation which changes the GS energy of this subsystem at second order in the coupling.At the end of the renormalization step, we take only the perturbed GS of the renormalized subsystem,and discard the rest of its local Hilbert space. In this way, at each renormalization step, we reducethe energy scale at which we are looking at the system: the considered site is renormalized awayand an effective low-energy coupling term is generated. After many applications of the renormalization step,we asymptotically reach the GS and we can find its properties. One can distinguish between two possible cases, depending whether the largest coupling is a given J_j,or a given λ_j. We provide the derivation in full detail in Appendix <ref>;here, we focus on its main points and their physical meaning.We start assuming that the largest coupling is J_j: the part of the Hamiltonian corresponding to it is given by Ĥ_0 = -J_jσ̂_j-1^xσ̂_j^zσ̂_j+1^x,while the coupling to the rest of the system can be described by the following operator: V̂=λ_j-1σ̂_j-2^yσ̂_j-1^y+λ_j+1σ̂_j+1^yσ̂_j+2^y. As detailed in Appendix <ref>, it is possible to treat the term V̂ perturbatively,applying a first-order perturbation theory to the four-fold degenerate GS of Ĥ_0. After diagonalizing the resulting second-order perturbation matrix,we project over one of the perturbed ground states, ending up into eliminating site j and generating a new coupling -λ_jσ̂_j-2^yσ̂_j+2^y withλ_j≃λ_j-1λ_j+1/J_j . The operators σ̂_j-2/j+2^y in Eq. (<ref>) are in principle different from the unrenormalized ones: they coincide with them up to quartic terms in λ/J.In the opposite case, where the largest coupling is λ_j, one can apply to Eq. (<ref>)a duality transformation μ̂_j^x = ∏_k=1^jσ̂_k^z, μ̂_j^z = σ̂_j^xσ̂_j+1^x , which maps the Pauli operators σ̂_j^α onto different Pauli operators μ̂_j^α.After the application of this transformation, the CIM Hamiltonian in terms of σ̂_j^α [Eq. (<ref>)] is re-expressed in terms of μ̂_j^α [see Eq. (<ref>) in Appendix <ref>]: the transformed Hamiltonian has the same form of Eq. (<ref>), but λ_j and J_j are now exchanged. The term with the largest coupling which has to be renormalized is now Ĥ̃̂_0 = -λ_jμ̂_j-1^xμ̂_j^zμ̂_j+1^x. Applying to it the same analysis of the first case, we see that the renormalization procedure eliminates the site j in the dual representation, and generates the term -J_j+1μ̂_j-2^yμ̂_j+2^y withJ_j=J_j-1J_j+1λ_j-1 .In the limit of many RG steps, and after applying the central limit theorem, it is possible to see thatthe disordered CIM is equivalent to a system with couplingsλ=exp[2l (logλ - log J ) ]J=exp[ 2l (log J - logλ) ] , where l is the number of consecutiverenormalized sites, which in principle can be different for different sites of the renormalized model (all the details of the calculation are in Appendix <ref>).We can distinguish three cases: I: logλ>log J. In this case, λis larger than J exponentially in the number of iterations of the renormalization step.In the limit of infinite iterations, J is vanishingly small with respect to λ:only the AF terms survive. Therefore, the RG flows to an AF condition and the system is antiferromagnetic.II: logλ<log J. Similarly to case I,in the limit of infinite iterations, λ is vanishingly small with respectto J: only the three-body (cluster) terms survive.Looking at the problem in the dual representation, we see that the dual system has only the AF termand then the RG in this representation flows to an antiferromagnetic condition.Going back to the original representation, we see that the system flows to the SO phase.III: logλ=log J. Here, the RG flows to a uniform systemwith λ=J = 1: the low-energy behavior of the model is equivalentto a uniform model at the critical point between the SO and the AF phase. Therefore, also in the disordered model, we can predict a transition between AF and SO phase, occurring forlogλ = log J . For logλ>log J the model is AF, while forlogλ<log J it displays SO. These results are very similar to those found in the transverse field Ising model <cit.>. §.§ Numerical analysis of the two phasesJust to fix the ideas, let us now analyze the case in which there is no disorder on J (J_j = 1,∀ j), and each λ_j is taken from a uniform distribution among 0 and some λ_ max. In this case, the condition in Eq. (<ref>) implies that the transition point is located at λ_ max,c^(∞) =e≈ 2.718… , e being the Neper number. The superscript “(∞)” in Eq. (<ref>) denotes that this is the critical point in the limit L→∞. By exploiting the fact that the CIM is exactly solvable, we can explore the behavior of the long-range string andAF correlator for considerably long system sizes, after averaging over an ensemble of several disorder realizations. Here, we recall that, fixing the realization of the disorder, a Jordan-Wigner transformation is able to mapEq. (<ref>) into a free-fermion diagonal form (irrespective of the presence or absence of translation invariance): Ĥ=∑_μ=1^Lϵ_μ(2 γ̂^†_μγ̂_μ - 1 ) , where the single quasi-particle operators γ̂_μ are defined in terms of the local fermionic operators ĉ_j according to γ̂_μ = ∑_j=1^L(U_jμ^ *ĉ_j + V_jμ^ *ĉ^†_j ) , and U_jμ^ *, V_jμ^ * are the coefficients of the 2L× 2L unitary matrix which diagonalizes the appropriate 2L× 2L Hermitian matrices forming the Hamiltonian(see for instance Ref. Russomanno_JSTAT13 for more details on this method).The GS is the one which is annihilated by all the γ̂_μ operators (it has a BCS form).Thanks to this property, we can evaluate the AF correlator in Eq. (<ref>). Applying Wick's theorem to the BCS Gaussian state, we can write the AF correlator as a Toeplitz determinant: (𝒮_[1/2],l^α)^2= 1/4 |[ G_k,k+1 ⋯ G_k,k+l; ⋮ ⋮; G_k+l-1,k+l ⋯ G_k+l-1,k+l ]|, where, for each disorder realization, we have defined the two point fermionic correlators on the GS corresponding to that realization: G_j,m = ⟨ (j-j)(m+m) ⟩ . Inverting Eq. (<ref>) and using the fact that the GS is annihilated by all the γ̂_μ,we can evaluate this correlator as G_j,m=∑_μ(V_jμ^ *-U_jμ)(U_mμ^ *+V_mμ) . The SO parameter in Eq. (<ref>) is evaluated by applying the duality transformation (<ref>),which maps it onto an AF correlator of the form in Eq. (<ref>), and the Hamiltonian ontoanother Hamiltonian of the same form. Finally, the results obtained through these formulasare averaged over N_ av realizations of disorder.The outcomes of our computations for a given finite size are reported in Fig <ref>.On the upper panel we plot 𝒪^z_[1/2],l and (𝒮^y_[1/2],l)^2.In order to avoid unwanted boundary effects, we evaluate the correlators between sitesthat are far away from the chain ends (see the caption for details). These quantities approximate the order parameters of Eqs. (<ref>) and (<ref>),since in numerical simulations we can consider large, but yet finite system sizes.We see in the upper panel of Fig <ref> that, when 𝒪^z_[1/2],l vanishes,(𝒮^y_[1/2],l)^2 appears with a crossover:this is an indication that there is a transition from a string-ordered to a y-antiferromagnetic phasein the thermodynamic limit, even in the presence of disorder. We can also analyze fluctuations over the disorder of the two order parameters.As before, we calculate finite-range correlators. Focusing on the AF order,we can define the fluctuation as δ (𝒮^y_[1/2],l)^2 = [ ⟨σ̂_k^y σ̂_k+l^y ⟩^2-⟨σ̂_k^y σ̂_k+l^y ⟩^2 ]^1/2, where the expectation value has to be intended over the GS of any specific disorder realization.The definition for the SO fluctuation (δ𝒪^z_[1/2],l) is analogous,after replacing the correlator of Eq. (<ref>) with that of Eq. (<ref>).The results are shown in the lower panel of Fig. <ref>. We can see that fluctuation is different from zeroonly when the corresponding order parameter is nonvanishing (compare with the upper panel):the finite-size crossover appears also in the behavior of fluctuations. Until now, we have considered signatures of the transition in finite-length correlators.As we can observe in Fig. <ref>, finite-size effects are evident: for a given system size,we actually see a crossover, and there is a region where both the finite-range order parameters are different from zero.Moreover, if we identify the transition with the point where the curves of the two finite-range order parameters cross,we get a result that is different from the theoretical prediction (<ref>).In order to properly infer the behavior in the thermodynamic limit L,l→∞,we perform a finite-size scaling analysis. In order to reduce the effect of the fluctuations induced by the noise, we need to perform a coarse graininig in 1/l of disorder-averaged correlators.More precisely, we proceed in the following way. We fix the value of L and, in order to avoid finite-size boundary effects, we fix an appropriate l_0 and consider the disorder-averaged correlator between the site l_0 and the site l with l_0+l varying between 0 and L - 2l_0. Then, we coarse grain this correlator: we consider the interval in which the quantity 1/l varies, divide this interval in windows of width δ(1/l) and perform the average of the correlator over each window. For each of the resulting values we evaluate the uncertainty asthe maximum over the corresponding window of the disorder fluctuation of the correlator: applying the central limit theorem, this fluctuation is given by the value in Eq. (<ref>) divided by √(N_ av). We label each of the windows over which we coarse grain with its central value 1/l: for each value of 1/l we locate the approximate transition pointas the value of λ_ max where the two coarse-grained correlators cross; we denote the crossing point as λ_ max,c^(l). In the upper panel of Fig. <ref> we show λ_ max,c^(l) versus 1/l: taking into account the error bars, we see a behaviour consistent with a convergence towardsthe theoretical value of Eq. (<ref>) when 1/l→0.The error bars are evaluated in the following way: each time we add or subtract the uncertainties discussed above to the coarse-grained disorder-averaged correlators and then wetake the half-dispersion of the 4 resulting estimates of the crossing. The lower panel of Fig. <ref> shows that the height w^(l) of the crossing point tends to zero when 1/l→0 (the error bars are evaluated with the same method used for λ_ max,c^(l)).We have checked this fact by fitting with a straight line a·(1/l), and found (a=317± 4)×10^-5 (red line in Fig. <ref>). This means that, in the thermodynamic limit, w^(∞)=0, and thus there is no region where both the order parametersare non-vanishing: when one vanishes the other appears, as appropriate for a QPT. §.§ Edge modes For a clean system, the QPT in the spin chain maps to a topological transition in the fermionic representation.In the case of periodic boundary conditions, the AF phase corresponds to winding number one in the fermionic picture,while the SO phase corresponds to winding number two <cit.>. <cit.> Taking OBC, the topological nature of the system appears through the existenceof zero-energy boundary modes <cit.>: diagonalizing the fermionic Hamiltonian,some vanishing ϵ_μ appear in Eq. (<ref>).For each phase, there is a fixed number of zero-energy modes, and their amplitudes U_j and V_j[see Eq. (<ref>)] are localized on the edges of the system.The AF phase displays one zero-energy mode, while the SO phase has two zero-energy modes (edge modes in uniform fermionic Hamiltonians very similar to Eq. (<ref>) have been studied in Refs. prb85,prb88).Even in the presence of disorder, we numerically observe the persistence in the spectrum of zero-energy modes (two modes in the SO phase, and one mode in the AF phase).Two examples of this fact are reported in Fig. <ref>.Here, we choose a specific disorder realization and show the single quasi-particle spectrum ϵ_μfor a case where the system shows SO (λ_ max=0.8) and a case where it is AF (λ_ max=4.8).In the first situation there are two levels with energy many orders of magnitude smaller than the others;in the second one there is a single level with this property: these levels correspond to the boundary modes discussed above(the energy is not exactly zero, due to the numerical round-off errors).We have verified that the same structure of the spectrum appears for any disorder realization. These edge modes are topologically protected, since they only depend on global properties of the system:they cannot be destroyed by local perturbations (like disorder) if the perturbation is weak enough.That is why, if we add disorder, two edge modes and the associated SO persists for λ_ max small,and one edge modes and AF order persist for λ_ max large.For λ_ max around the transition, the disorder is strong enough to move the transition point. §.§ Thermodynamic singularities Disordered systems can display phases where the thermodynamic quantities show singularities in the limit of vanishing temperature <cit.>. This can be seen from the behaviour of the distribution of the energy gap Δ of the Hamiltonian. If for small Δ the logarithmic energy gap distribution behaves as P(logΔ)∼Δ^1/z (z is the so-called dynamic exponent), it is easy to show that the excitation energy over the ground state behaves as E_ ex(T)∼ T^1+1/z at low temperatures. Therefore, the low-temperature specific heat behaves as C∼ T^1/z and its derivative shows a divergence in the limit of T→ 0 when z>1. The ranges of parameters where this happens are called singular regions <cit.>; in order to find them we numerically consider the properties of P(logΔ) and we check that it behaves as a power law for small Δ (see some examples in Fig. <ref>).Applying a linear fit to the plots of log[P(logΔ)] vs logΔ we are able to estimate the value of the dynamic exponent z which we plot in Fig. <ref>. We see that there is a singular region with z>1 for λ_ max above a threshold λ_ max^c s≃2. In order to have a better estimate of λ_ max^c s, we follow Ref. Lajko_PRB and consider the behaviour of the inverse average of the gap defined as Δ^ iv≡[(1/Δ)]^-1 , where the average is performed over the disorder distribution. This object vanishes whenever the system is in a singular region <cit.> with z>1; we show results for our case in Fig. <ref>. We see that Δ^ iv vanishes for λ_ max>λ_ max^c s=2, confirming that in this parameter range our system is singular. Moreover, we can numerically find that Δ^ iv vanishes as a power law when λ approaches the transition point λ_ max^c s from below: we have Δ^ iv∼(λ_ max^c s-λ_ max)^μ_Δ with μ_Δ=2.06±0.01.In conclusion, we see a transition to a singular regime which occurs at a critical value λ_ max^c s different from the critical point λ_max, c separating the SO and the AF phase. While for λ_ max>λ_ max, c the system is AF and singular, we have a non-singularSO phase (z<1) for λ_ max<λ_ max^c s andthere is a singular SO phase (z>1) for λ_ max^c s<λ_ max<λ_max, c. This behaviour is strictly reminiscent the disordered S=1 antiferromagnetic Heisenberg chain <cit.> where there is a gapped Haldane phase (z<1) and a singular Haldane phase (z>1), both showing SO. We emphasize that the singularity structure of the phases is strictly related to the specific form of the disorder: for instance, taking λ uniform and J_j uniformly distributed between 0 and some J_ max, we would have seen a non-singular AF phase together with a singular AF phase, while the SO phase would have been fully non-singular. § HALDANE TO NÉEL PHASE TRANSITION IN THE DISORDERED SPIN-1 XXZ MODELWe now switch to study the zero-temperature properties of the disordered spin-1 XXZ Hamiltonianof Eq. (<ref>).Since this model is non integrable, in order to find the GS of a given finite-size system, we resort to a variational search on the class of MPS <cit.>. In our simulations, we analyze chains of up to L=240 sites and choose a maximum bond link D_ max=400. We set J as the reference energy scale, and consider J=1 in the following. As we have done in the CIM, to characterize the two phases we focus on the finite-system AF correlationfunction ( 𝒮^z_[1],l)^2 and on the x-axis string operator 𝒪^x_[1],l [the corresponding order parameters are defined, in the thermodynamic limit, by Eqs. (<ref>) and (<ref>)]. Since we have ( 𝒮^z_[1],l)^2≥0 and 𝒪^x_[1],l≤0 in the XXZ chain, in the following we will consider the absolute value of the string parameter in order to deal with positive quantities.Before analyzing in detail the phase transition in the presence of disorder, let us briefly discuss the clean XXZ model. The situation is summarized in Fig. <ref>, which shows the behavior of the bulk expectation value of the staggered correlator along z (blue data set), and of the absolute value of the string correlator along x (green data set). We see that, at a critical value Λ^(L)_c of the anisotropy term, the SO vanishes and the staggered order starts to take a finite value: as explained before, this is an indication of the occurrence of the Haldane-Néel phase transition. The position of the critical point in the thermodynamic limit, Λ^(∞)_c, can be inferred from the finite-size scaling of the crossing point between the two curves; from our simulations at finite L, we estimate Λ^(L)_c=1.17±0.01, which is in agreement with the value Λ^(∞)_c ≃ 1.186… found in the thermodynamic limit <cit.>.Based on our knowledge on the clean XXZ model, we now focus on the Haldane-Néel phase transitionin the disordered scenario.Since MPS simulations are computationally more demanding and can only afford systemswith a comparatively small length, we adopted a procedureslightly different from the one used for the CIM, in order to estimate the SO correlator [Eq. (<ref>)] and the AF correlator [Eq. (<ref>)] from the bulk expectation values of the corresponding finite-size correlators. The two methods coincide in the thermodynamic limit, but the one described here is more appropriate for the smaller value of system size and number of disorder realizations which we can obtain with DMRG in the XXZ chain because it enables to minimize the uncertainty in the averages.Namely, we are interested in computing the bulk expectation values of a given two-point observable,of the form 𝒜̂_k,k+l.We recall that both the AF correlator, ( 𝒮^α_[1],l)^2,and the SO correlator, 𝒪^α_[1],l, can be seen as expectation values of observableswhich live on a given number of sites in between the k-th and the (k+l)-th site.After fixing the system size and the disorder realization, we first compute a space averageover different lengths l of the correlator <cit.>, in order to average out space fluctuations. We discard the sites that are closeto the two chain ends, thus disregarding boundary effects(see Appendix <ref> for details). Then, we repeat the simulation by varying the configuration of the disorder, and eventually performa second average of such obtained space averages, over different disorder realizations.The obtained correlators have an uncertainty(denoted by error bars in Fig. <ref>) which is estimated by computing the variance of the space averaged correlators for the different realizations of disorder, as detailed in Appendix <ref>.We first present our numerical analysis for Λ_j uniformly distributed in the interval [0,Λ_ max],for all j. In this case, we expect the system to undergo the Haldane-Néel phase transitionas Λ_ max is varied across some critical value.In our simulations we see that, in the Néel phase, the AF pattern is affected by the presenceof kinks (domain walls, where the AF pattern is reversed), which hide the presence of long-range AF order (<ref>).As detailed in Appendix <ref>, the presence of such kinks is a numerical artifactdue to the non-perfect convergence of the MPS algorithm. Thus, instead of computing the staggered correlatoras in Eq. (<ref>), we can get rid of the kinks and reveal the presence of AF long-range orderby computing the bulk average [Eq. (<ref>)] of the Néel correlator, which is defined as 𝒩^z_[1],l = | ⟨Ŝ^z_k Ŝ^z_k+l⟩| . Notice that ( 𝒮^z_[1],l)^2 coincides with the Néel correlatorin Eq. (<ref>) in the case of perfect AF order (no kinks),but differently from the staggered correlator, 𝒩^z_[1],l is insensitiveto such numerical artifacts, because of the presence of the absolute value in Eq. (<ref>).Thus, we characterize Haldane and Néel phases in the disordered chain by looking respectively at the SO along x [see Eq. (<ref>)] and the Néel correlatorin the limit of l →∞: 𝒩^z_[1] = lim_l→∞𝒩^z_[1],l.In Fig. <ref>, we show the result of simulations with L=120 and Λ_ min=0, after averaging over space and over disorder.Blue points correspond to the Néel order, 𝒩^z_[1],whereas green points are the SO data, |𝒪^x_[1]|.We observe that the Néel order is zero for sufficiently small Λ_ max, and starts to increasearound a given value of Λ_ max. Conversely, the SO along x is nonzero for small Λ_ max,and goes to zero as Λ_ max is increased. This behavior is analogous to the one for the Haldane-Néel phase transition in the clean XXZ model (Fig. <ref>).Furthermore, we can estimate a critical point Λ^(L)_ max,c which is shiftedwith respect to the clean value: for the simulation in Fig. <ref>,we find Λ_ max,c^(L)=2.32±0.02. For a given finite size the transition behaves as a crossover,exactly as it occurs for the CIM (upper panel of Fig. <ref>) and we estimate the finite-L approximation of the critical point as the abscissa of the crossing point of the two curves of the correlators. The error bars of the crossing points are estimated as follows. We consider the plot of the correlators (for instance Fig. <ref>) and, being interested in the fluctuations of the intersection of the disorder averages, we divide the error bars by √(N_ av), according to the central limit theorem. Later, we proceed in a way similar to Fig. <ref>: each time we add or subtract these fluctuations to the disorder-averaged correlators and then wetakethe half-dispersion of the 4 resulting values of the crossing.This gives the uncertainty of the crossing point. As in the CIM case, in order to extrapolate the value of the critical pointin the thermodynamic limit, we need to perform a finite-size scaling analysis and repeat the same simulations as in Fig. <ref> for different values of L.The result is shown in Fig. <ref>: blue points correspond to the estimated values of Λ_ max,c^(L), and uncertainties are computed as explained above.To find the thermodynamic value of the critical point, we show the data as a function of 1/Land perform a best fit with the function f(L)=a/L+Λ_ max,c^(∞).From the result in Fig. <ref>,we extrapolate the critical value in the asymptotic L→∞ limit: Λ_ max,c^(∞)=2.40±0.01.We can also define the height of the crossing pointw^(L), as in Sec. <ref>. For each value of L, the uncertainty on the value of w^(L) is computed from that of the disorder-averaged correlators. We show the result in Fig. <ref>,from which we see that w^(L) decreases as L is increased. From the fit, we estimate w^(∞)=0.06±0.01.This is an indication of the fact that the phase transition becomes sharper and sharper as L is increased.So far, we have discussed the Haldane-Néel phase transition for Λ_j uniformly distributedin the interval [Λ_ min,Λ_ max], for all j, using Λ_ min=0.In order to see how the position of the critical point is affected by the choice of Λ_ minand Λ_ max, we simulate the disordered model of Eq. (<ref>)varying Λ_ max and using Λ_ min=Λ_ max/(n+1), where n is a positive integer number.Our results for L=120 are shown in Fig. <ref>.For each value of n, we estimate the position of the critical point Λ_ max,c^(L)(n)as explained for the data in Fig. <ref>.If we define the algebraic average of {Λ_j} in the chain, i.e. Λ̅=Λ_ min+Λ_ max/2=Λ_ max/2 n+2/n+1, we find that the Haldane-Néel phase transition in the disordered chain occurs when Λ_ max is such thatthe mean value of {Λ_j} in Eq. (<ref>) equals the critical value Λ_cof the clean chain, i.e., inverting Eq. (<ref>) and showing explicitly the dependence on L: Λ_ max,c^(L)(n)=2Λ^(L)_c n+1/n+2. As is evident from Fig. <ref>, the position of the critical point agrees withthe scaling given by Eq. (<ref>), where we use Λ^(L)_c as fit parameter.From the fit, we estimate Λ^(L)_c=1.157±0.002, which is not in disagreement with the clean value Λ^(L)_c ≃ 1.17found in Fig. <ref> for L=240.We ascribe the slight discrepancy of the two estimates to finite-size effects (we use L=120 for the data in Fig. <ref>).Differently from the CIM, the study of the thermodynamic singularities in the case of the XXZ model requires a much larger computational effort, due to the increased numerical complexity. This is left as an open issue for a future work.§ CONCLUSIONS AND PERSPECTIVES In conclusion, in this work we have studied the existence of the non-local string order in disordered spin chains. We have focused on two models, the spin-1/2 cluster Ising chain and the spin-1 XXZ Heisenberg chain, which are well known to show a transition from antiferromagnetism to string order in the clean case (the first model is moreover interesting for applications in quantum information <cit.>). We have discovered that this transition persists in both cases if disorder is added. In the disordered cluster Ising model we have found a transition from antiferromagnetism to string order by numerically studying the order parameters in the ground state; we did this using the Jordan-Wigner mapping on an integrable free-fermion model. We have seen also that the transition manifests in the fermionic representation as a change in the number of 0-energy edge modes. In this model we have found analytically the position of the transition point using the strongest coupling renormalization group: this analytical prediction is fully confirmed by the finite-size-scaling on our numerical results. Moreover, studying the thermodynamic singularity at vanishing temperature, we have found a transition between a non-singular and a singular behaviour when the strength of the disorder is increased. We have seen that this transition point is different from the one separating the SO and the AF phase.In the disordered spin-1 XXZ Heisenberg chain we have studied the order parameters in the ground state by means of the DMRG technique: we have found a transition between an antiferromagnetic and a string-ordered phase and we have determined its position by means of finite-size scaling. This model is very interesting because its string-ordered phase is adiabatically connected to the celebrated Haldane phase <cit.> and it can be experimentally studied thanks to the new cold-atom techniques <cit.>.Perspectives of future work include first of all the application of the spin-1 strongest coupling renormalization group <cit.> to the XXZ Heisenberg chain, in order to analytically predict the phase transition point that we find here numerically. Here we have only addressed the properties of the ground state: it will be interestingto consider the properties of all the spectrum, in connection with many-bodylocalization (MBL) <cit.> of interacting non-integrable system. MBL systems can show topological orderin a large fraction of the excited energy eigenstates <cit.>: it would be interesting to see if also our non-integrable spin-1 disordered XXZ model shows MBL and if string and antiferromagnetic order persist in excited states.A possibility to study these phenomena is applying to the Hamiltonian a quantum quench and look at the dynamicsof the string correlator. In a clean spin-1 XXZ Heisenberg model the string thermalizes <cit.>,only the ground state being ordered, but in disordered systems the situationcould be much different thanks to the MBL. § ACKNOWLEDGEMENTSWe are grateful to A. Hamma and L. Mazza for fruitful discussions. We thank E. G. Dalla Torre for useful comments on the manuscript. We acknowledge the CINECA award under the ISCRA initiative, for the availability of high performance computing resources and support. M. C. S. acknowledges partial support from the Israel Science Foundation, grants No. 231/14 and 1452/14. A. R. acknowledges financial support from EU through project QUIC, from “Progetti interni - Scuola Normale Superiore” and from his parents. R. F. kindly acknowledges support from EU through project QUIC (Grant Agreement No. 641122), the National Research Foundation of Singapore (CRP - QSYNC) and the Oxford Martin School.§ DETAILS ON THE RENORMALIZATION GROUP FOR THE CIMIn the renormalization procedure of the CIM described in Sec. <ref>, we can distinguish two different types of RG steps: i) the largest coupling is a J_jor ii) the largest coupling is a λ_j.Case i): If we assume that the largest coupling is J_j, it is not difficult to see that the local Hamiltonian Ĥ_0 of Eq. (<ref>) has four degenerate ground states: |1⟩ = |(+)_j-1 ↑_j (+)_j+1⟩ , |2⟩ = |(-)_j-1 ↑_j (-)_j+1⟩ , |3⟩ = |(-)_j-1 ↓_j (+)_j+1⟩ , |4⟩ = |(+)_j-1 ↓_j (-)_j+1⟩ , where |↑_l⟩, |↓_l⟩ are the eigenstates of σ̂_l^z and |(+)_l⟩, |(-)_l⟩ are the eigenstates of σ̂_l^x.Applying the degenerate perturbation theory to such four ground states, we get the following corrections at first order in the perturbation V̂ of Eq. (<ref>):|ψ_g1⟩=|(+)_j-1 ↑_j (+)_j+1⟩+i/2J_j[λ_j-1σ̂_j-2^y|(-)_j-1 ↑_j (+)_j+1⟩+λ_j+1σ̂_j+2^y|(+)_j-1 ↑_j (-)_j+1⟩], |ψ_g2⟩=|(-)_j-1 ↑_j (-)_j+1⟩-i/2J_j[λ_j-1σ̂_j-2^y|(+)_j-1 ↑_j (-)_j+1⟩+λ_j+1σ̂_j+2^y|(-)_j-1 ↑_j (+)_j+1⟩], |ψ_g3⟩=|(-)_j-1 ↓_j (+)_j+1⟩-i/2J_j[λ_j-1σ̂_j-2^y|(+)_j-1 ↑_j (+)_j+1⟩-λ_j+1σ̂_j+2^y|(-)_j-1 ↑_j (-)_j+1⟩], |ψ_g4⟩=|(+)_j-1 ↓_j (-)_j+1⟩-i/2J_j[-λ_j-1σ̂_j-2^y|(-)_j-1 ↑_j (-)_j+1⟩+λ_j+1σ̂_j+2^y|(+)_j-1 ↑_j (+)_j+1⟩] . To apply degenerate perturbation theory, we construct and diagonalize the matrix V_ij=⟨i|V̂|ψ_gj⟩ which is given by: 𝕍=([-λ_j-1^2+λ_j+1^22J_j λ_j-1λ_j+1J_jσ̂_j-2^yσ̂_j+2^y 0 0; λ_j-1λ_j+1J_jσ̂_j-2^yσ̂_j+2^y-λ_j-1^2+λ_j+1^22J_j 0 0; 0 0-λ_j-1^2+λ_j+1^22J_j λ_j-1λ_j+1J_jσ̂_j-2^yσ̂_j+2^y; 0 0 λ_j-1λ_j+1J_jσ̂_j-2^yσ̂_j+2^y-λ_j-1^2+λ_j+1^22J_j ]).This matrix can be written asV_ij=∑_n^excited states⟨i|V̂|n⟩⟨n|V̂|j⟩/E_ GS-E_n ,where E_ GS is the energy of the degenerate ground states |i⟩, |j⟩. Diagonalizing this matrix, one finds the perturbed ground-state eigenenergies at second order in λ: -λ_j-1^2+λ_j+1^22J_j±λ_j-1λ_j+1J_jσ̂_j-2^yσ̂_j+2^y. We select one of these four eigenstates discarding the others. We arbitrarily choose one of the two states with eigenvalue -λ_j-1^2+λ_j+1^22J_j-λ_j-1λ_j+1J_jσ̂_j-2^yσ̂_j+2^y. Through the renormalization we have indeed eliminated the site j and generated a new coupling -λ_jσ̂_j-2^yσ̂_j+2^y withλ_j≃λ_j-1λ_j+1/J_j . These operators σ̂_j-2/j+2^y are in principle different from the unrenormalized ones: they coincide with them up to terms quartic in λ/J. Case ii): If the largest coupling is one of the λ_j, we can reduce to the first case by applying to the Hamiltonian Eq. (<ref>) the duality transformation <cit.>: μ̂_j^x = ∏_k=1^jσ̂_k^z, μ̂_j^z = σ̂_j^xσ̂_j+1^x . We find the Hamiltonian in the dual representation as Ĥ̃̂= - ∑_j [ J_j μ̂_j-1^yμ̂_j^y + λ_jμ̂_j-1^xμ̂_j^zμ̂_j+1^x ] . Indeed we can see that, in the limit L→∞ we are considering, this Hamiltonian is equal to its dual in Eq. (<ref>), with λ_j and J_j exchanged. The term with the largest coupling which has to be renormalized is indeed Ĥ̃̂_0 = -λ_jμ̂_j-1^xμ̂_j^zμ̂_j+^x. Applying to it the same analysis of the first case, we see that the renormalization procedure eliminates the site j in the dual representationand generates the term in Eq. (<ref>).It is now easy to show that, after many RG steps, the couplings are renormalized according to J_j = J_j-2lJ_j-2l+2⋯ J_j+2l-2J_j+2l/λ_j-2l-1λ_j-2l+1⋯λ_j+2l-5λ_j+2l-3 , λ_j = λ_j-2lλ_j-2l+2⋯λ_j+2l-2λ_j+2l/J_j-2l+1J_j-2l+3⋯ J_j+2l-3J_j+2l-1 . Applying the central limit theorem we find logJ_j = 2l(log J-logλ ) + √(2l)(√( Var[log J] +Var[logλ])) u_J , logλ_j = 2l(logλ-log J) + √(2l)(√( Var[log J] +Var[logλ])) u_λ, where u_J and u_λ are normally distributed random variables; the averages (…)and the variances Var[ … ] are performed over the distributions of J_j and λ_j.We thus see that, in the limit of infinite RG steps, Eqs. (<ref>) hold. § DETAILS ON THE NUMERICAL ANALYSIS OF DISORDERED HEISENBERG CHAINSIn this appendix, we provide details on the strategy that we adopted in order to computethe bulk expectation values of generic two-point observables of the form 𝒜̂_k,k+l, for the numerical results that have been obtained with the MPS-based algorithm on the spin-1 XXZ Heisenberg chain. We also comment on the analysis of the presence of domain walls in our simulations.§.§ Bulk expectation values To compute the bulk expectation values of 𝒜̂_k,k+l, we first fix L, Λ_ min, Λ_ max, and a given instance of disorder. For each realization, we numerically compute the space bulk-average by discardinga certain number Δ L of sites that are close to the chain ends. Moreover we consider distances l > Δ L such that, provided we are far from the transition point, they are larger than the system's correlation length <cit.>. Near the critical point the correlation length tends to diverge, therefore we have always finite-size effects: in order to understand the properties of the transition in the thermodynamic limit it is thus very important to perform a finite-size scaling as we do in the main text.Specifically, if the sites are labeled from 1 to L,we choose k=Δ L=0.2 L and average the expectation values from l=l_1=0.3 Lto l=l_2=0.6 L, i.e.: 𝒜_ avg,h = 1/l_2-l_1∑_l=l_1^l_2⟨𝒜̂_Δ L,Δ L+l⟩_h , where the subscript _ avg,h denotes the space average for the h-th realization of disorder.Then we repeat the simulation by varying the configuration of the disorder in the chain,and perform an average over all the N_ av realizations: 𝒜_ avg = 1/N_ av∑_h=1^N_ av𝒜_ avg,h. Because of the presence of the random {Λ_j}, the value of ⟨𝒜̂_Δ L,Δ L+l⟩,as a function of l, is expected to fluctuate in space. Thus, the expectation valuein Eq. (<ref>) is affected by an uncertainty, which we estimate via the standard deviation σ_𝒜_h^2 = 1/l_2-l_1∑_l=l_1^l_2(⟨𝒜̂_Δ L,Δ L+l⟩_h - 𝒜_ avg,h)^2.The fluctuations over the disorder realizations are in turn computed via the variance computed from the {𝒜_ avg,h} in Eq. (<ref>):σ^2_𝒜̅ = 1/N_ av∑_h=1^N_ av(𝒜_ avg,h -𝒜_ avg )^2.Finally, to motivate the choice of the spatial averages as in Eq. (<ref>), we show the behavior of the disorder-averaged string correlator, |𝒪^x_[1],l|, and the disorder-averaged Néel correlator, 𝒩^z_[1],l, in Fig. <ref>. Instead of averaging over space for a given realization of disorder, as discussed before, here we average each value of |𝒪^x_[1],l|(h) and of 𝒩^z_[1],l(h), for fixed k and l, over N_ av realization of disorder, where the symbol “(h)” indicates that we are computing the expectation values for the h-th realization of disorder [see Eqs. (<ref>) and (<ref>)]. Explicitly:|𝒪^x_[1],l| =1/N_ av∑_h=1^N_ av|𝒪^x_[1],l|(h) 𝒩^z_[1],l=1/N_ av∑_h=1^N_ av𝒩^z_[1],l(h). For the plots in Fig. <ref>, we choose k=0.2 L, L=120, as in Eq. (<ref>). The red vertical lines limit the interval of l over which we compute the spatial averages in Eq. (<ref>). We stress that, as for the computation of the bulk expectation values, the two ways of averaging (average over space/disorder and then average over disorder/space) are actually equivalent, but performing the disorder-average for each value of k and l, as in Eqs. (<ref>), allows us to visualize the average spatial behavior of the string and Néel correlators.As we see from Fig. <ref>, apart from spatial fluctuations due to the presence of disorder, the behavior of |𝒪^x_[1],l| and 𝒩^z_[1],l is in agreement with the expected behavior in the two phases (see Sec. <ref>). We also see that the choice of l_1=0.3 L and l_2=0.6 L, as in Eq. (<ref>), allows us to capture the average bulk expectation value, at least sufficiently far away from the transition point. When we are close to the Haldane-Néel phase transition (e.g., Fig. <ref>, panels with Λ_ max=2.3), for finite L, our numerical results are affected by finite-size effects, and the correct estimation of the critical point can be then performed only by a finite-size scaling, as explained in Sec. <ref>. §.§ Disordered XXZ model with symmetry-breaking magnetic field As we mentioned in Section <ref>, we can expect the string-ordered phase to be present in the disordered XXZ model as long as the 𝔻_2 symmetry is preserved. We now present some numerical data in order to show an example of destruction of the string order in case a symmetry-breaking term is added to the Hamiltonian. Specifically, we simulate the disordered XXZ model in Eq. (<ref>) with the inclusion of a magnetic field along the x-axis:Ĥ_B=Ĥ_ XXZ+B_x∑_jŜ^x_j.Since the 𝔻_2 symmetry is broken when B_x≠0 in Eq. (<ref>), we expect SO not to be present in the system <cit.>. The result of a simulation with Λ_ min=0, Λ_ max=1.5 and different values of B_x (in units of J) is shown in Fig. <ref>. In order to highlight the different behaviors at long lengths with respect to the B_x=0 case, we use L=180 for B_x=0,0.2 and L=120 for B_x=0.4,0.6. We compute the disorder average of the string correlator |𝒪^y_[1],l| as in Eq. (<ref>). As we see, the addition of B_x≠0 makes the string correlator decay to zero in the l→∞ limit, and no SO is present in the system, in agreement with the general arguments presented in Refs. PerezGarcia_2008,Pollmann_2010. §.§ Domain walls Let us now consider Λ_ min=0 and explicitly focus on a case with SO (i)and a case with AF order (ii).i) In Fig. <ref>, we show the results of a simulationwith Λ_ max=1.5 (SO phase) and L=120:two-point correlator C^z_[1],l≡⟨Ŝ^z_k Ŝ^z_k+l⟩_h (left panel),staggered correlator (𝒮^z_[1],l)^2 (central panel),and string correlator |𝒪^x_[1],l| (right panel).Here all the correlators have been evaluated over a specific realization of disorder, which we term h.To avoid boundary effects, we choose Δ L = 0.2 L, L=120 (thus, we fix k=24).For this value of Δ_ max, we see that C^z_[1],l oscillates between positive and negative values,for sufficiently small l, and it is damped by an exponential decay.This is clearly seen in the behavior of (𝒮^z_[1],l)^2, which exponentially goes to zeroin the bulk of the chain. On the other hand, the expectation value of the string operator 𝒪^x_[1],l takes a finite value in the bulk of the chain.ii) We repeat the same analysis as before, but for Λ_ max=5.0 (AF phase).The results are shown in Fig. <ref>. For this value of Λ_ max,the two-point correlator C^z_[1],l displays an undamped oscillating pattern, signaling the presence of AF order.However, we notice that the pattern reverses at l≃29, i.e., where the data display a kink (domain wall).The presence of such a kink suggests that the AF order appears only locally (the system tends to form domains).In order to see if the presence of domain walls in the pattern of the two-point correlator C^z_[1],lis a physical fact or a numerical artifact, we repeat the simulation M times. We fix thevalues of L,Λ_ min and Λ_ max and the disorder configuration {Λ_j} and in each repetition we vary the initial random MPS state |Ψ_ in⟩ at the beginning of the MPS algorithm.Our purpose is to verify that different initial random states produce different configurations of kinkswith different GS energies.To give an example, we show in Fig. <ref> the result of several simulationsfor M different initial random MPS states: { |Ψ_ in(m)⟩}_m = 1 … M.For each value of m, we measure the GS energy, E_ GS(m) = ⟨Ψ_ in(m)|Ĥ_ XXZ |Ψ_ in(m)⟩, where Ĥ_ XXZ is the Hamiltonian in Eq. (<ref>), and the number of kinks N_ kinks(m),which is obtained from the spatial pattern of ⟨Ŝ^z_j⟩.We define the quantity δ E_ GS(m) = [E_ GS(m)- min_m{E_ GS(m)}]/| min_m{E_ GS(m)}|,and compare the values of δ E_ GS(m) with the corresponding number of kinks.As is evident from Fig. <ref>, the configurations with zero kinks are associatedto the lowest value of the GS energy. 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"authors": [
"Marcello Calvanese Strinati",
"Davide Rossini",
"Rosario Fazio",
"Angelo Russomanno"
],
"categories": [
"cond-mat.stat-mech",
"cond-mat.quant-gas",
"quant-ph"
],
"primary_category": "cond-mat.stat-mech",
"published": "20170727180006",
"title": "Resilience of hidden order to symmetry-preserving disorder"
} |
firstpage–lastpage 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 ========================================================================================================================================================================================================================================================================================================================================================================== We present Hα linear spectropolarimetry of a large sample of Herbig Ae/Be stars. Together with newly obtained data for 17 objects, the sample contains 56 objects, the largest such sample to date. A change in linear polarization across the Hα line is detected in 42 (75 %) objects, which confirms the previous finding that the circumstellar environment around these stars on small spatial scales has an asymmetric structure, which is typically identified with a disk. A second outcome of this research is that we confirm that Herbig Ae stars are similar to T Tauri stars in displaying a line polarization effect, while depolarization is more common among Herbig Be stars. This finding had been suggested previously to indicate that Herbig Ae stars form in the same manner than T Tauri stars through magnetospheric accretion. It appears that the transition between these two differing polarization line effects occurs around the B7-B8 spectral type. This would in turn not only suggest that Herbig Ae stars accrete in a similar fashion as lower mass stars, but also that this accretion mechanism switches to a different type of accretion for Herbig Be stars. We report that the magnitude of the line effect caused by electron scattering close to the stars does not exceed 2%. Only a very weak correlation is found between the magnitude of the line effect and the spectral type or the strength of the Hα line. This indicates that the detection of a line effect only relies on the geometry of the line-forming region and the geometry of the scattering electrons. techniques: polarimetric – circumstellar matter – stars: formation – stars: individual: Herbig Ae/Be – stars: pre-main-sequence. § INTRODUCTION Herbig Ae/Be (HAeBe) stars, the more massive counterparts of T Tauri stars, are optically visible pre-main-sequence (PMS) stars with masses roughly between 2 and 10 M_⊙. This group of stars was first identified by <cit.>.With their intermediate masses, they play an essential role in addressing the formation of high mass stars as they bridge the gap between low mass stars, whose formation is fairly well understood and high mass stars, whose formation still poses challenges.Lower mass stars are thought to form through magnetically controlled accretion (MA, e.g. <cit.>, whereas evidence for this mode of accretion is lacking for high mass stars. Not only are higher mass stars not expected to be magnetic as their radiative envelopes would inhibit the presence of the magnetic fields that dominate the accretion in low mass objects, these magnetic fields have also hardly been detected (e.g. ). Traditionally, the change from MA and a different, hitherto unexplored, accretion mechanism was thought to be around the spectral type boundary M/K to F/A where the envelope structure changes from convective to fully radiative. However, it had long been known that Herbig Ae stars have different properties than Herbig Be stars and may form in a different manner (e.g.who studied millimeter emission, clustering properties and CO first overtone emission properties respectively). In addition, from various recent studies it appears that Herbig Ae stars are more similar to the T Tauri stars than to Herbig Be stars. For example, <cit.> infer that the accretion shock regions in a Herbig Ae star are comparable in size and location to those in the magnetic lower mass objects. <cit.> interpret from the observed spectroscopic variability that a Herbig Ae star is currently undergoing magnetically-controlled accretion in the same manner as the T Tauri stars. In contrast, dedicated modeling indicated that, if present at all, the magnetosphere in a Herbig Be star with spectral type B9IV must be small <cit.>. Indeed, <cit.> find that magnetic fields are not required to explain the spectroscopic properties of early type Herbig Be stars. Finally, interferometric studies show that some of the hotter Herbig Be stars have much smaller near-infrared sizes than would be expected from the dust sublimation radius. This can be explained by optically thick gas in accretion disks reaching to the star <cit.>. A first indication that Herbig Ae stars share similarities with T Tauri stars but are different from Herbig Be objects emerged from a study into linear spectropolarimetry across Hα emission by <cit.>.Later, these authors found that the linear spectropolarimetric signatures observed around the Hα emission line in both T Tauri and Herbig Ae stars can be explained by compact Hα emission scattered off a circumstellar disk (, Vink, Harries & Drew 2005, ). This is suggestive of the notion that Herbig Ae stars may form in the same manner as T Tauri stars, where the compact Hα emission could arise from accretion hot spots or funnels due to magnetospheric accretion.Further clues to this effect were presented by <cit.> who found that the emission and absorption line properties of Herbig Be stars are significantly different from Herbig Ae stars, who in turn seem to have properties intermediate between Herbig Be and T Tauri stars. Finally, <cit.> found that the UV-excess of Herbig Ae stars can be explained by magnetospheric accretion, but that the earliest Herbig Be stars have too large UV-excesses to be explained by the usual accretion shock scenario <cit.>. For recent reviews on the subject of accretion in young stellar objects, including Herbig Ae/Be stars, we refer the reader to <cit.>, <cit.> and <cit.>. Linear spectropolarimetry is an effective technique to probe ionised inner circumstellar disks around stars on scales of order stellar radii, scales that are small enough to probe the accretion region of young stars.The technique was first successfully used to probe the circumstellar disks around classical Be stars <cit.>.These authors demonstrated that the continuum light is scattered and polarized by free electrons, while the hydrogen recombination line emission, which arises from a volume larger than where the electron scattering dominates, is not or hardly polarized. The use of the technique was extended by <cit.>, <cit.> and <cit.> to cover HAeBe stars.<cit.> classified HAeBe stars according to their spectropolarimetric signature. In a study of 23 HAeBe objects, they found that many of the HBe stars (7 out of 12) show a depolarisation line effect consistent with a circumstellar disk, similar to that observed in Be stars, while an intrinsic polarisation line effect, as also observed in T Tauri stars is more dominant in most, 9 out of 11, less massive HAe stars. Their results suggest a physical switch from line polarisation for HAe stars to depolarisation in HBe stars.To put these results on a firmer footing, a larger sample is needed to support and confirm the findings and draw statistical conclusions.In this work we aim to provide a statistical investigation into the spectropolarimetric properties of HAeBe stars and their relation with their lower mass counterpart T Tauri stars. By collating all the data in the literature and adding newly obtained data, we present the spectropolarimetric results of a sample of 56 HAeBe objects, nearly three times larger than the sample of <cit.>.The paper is structured as follows. In Section 2, in order to properly interpret the data of this large sample, we start by an updated overview of the use of linear spectropolarimetry. This is followed by a discussion of the details of the sample selection, the complementary observations and the data reduction. In Section 3, we present the results which are discussed in Section 4. Finally, we conclude in Section 5.§ METHODOLOGY, OBSERVATIONS AND DATA REDUCTION§.§ Spectropolarimetry as a probe of inner regionsThe technique of linear spectropolarimetry as applied here, exploits the fact that radiation will be scattered by free electrons in an ionized region. This results in the light to be polarized in the plane of the sky perpendicular to the original direction of travel. In case of a projected circular geometry of the scatterers on the sky - for example a disk observed pole-on or a spherical distribution - all polarization vectors cancel, resulting in a net zero polarization. However, in case of an asymmetric distribution, a net polarization is observed.Most of the polarization due to free electrons is found to occur within a few stellar radii and results in polarizations of order 1-2% <cit.>.We can take advantage of the fact that the stellar continuum photons and emission line photons originate from different locations and scatter differently off the free electrons, resulting in a different polarization across line and continuum. The spectropolarimetry around an emission line, most often Hα has been observed, can therefore probe scales very close to the star.We note that many other scattering agents can give rise to polarization such as dust. However, whether circumstellar or interstellar, dust is located far away from the stellar and line emitting regions and it has a very broad wavelength dependence. As a consequence, the continuum and line radiation will be scattered in a similar fashion and no line-effect is visible (e.g. ). Thus far, in the study of Herbig Ae/Be stars, three types of “line-effect” can be identified and whereas the nature of the electron scattering process may be different, they share the fact that they probe the small scales of the electron scattering region.Let us start with the so-called depolarization effect. This had already been observed toward Be stars in the early seventies, and was in fact the first (indirect) proof that these stars are surrounded by small disks. The density of free electrons is the highest closest to the star, which is where the continuum photons emitted by the stellar photosphere will be polarized.In contrast, the hydrogen recombination line emission will be less polarised as it passes through a smaller ionised volume and hence encounters fewer free electrons.Therefore, a detection of a depolarization line effect across an emission line can immediately trace the presence of an ionised asymmetric structure with a size of order stellar radii. This is on much smaller smaller scales than can be typically probed by the best imaging techniques (e.g.on optical/near-infrared interferometry of Herbig Ae/Be stars). Such a depolarization has been identified and confirmed to be due to a disk, as the polarization angles indicate that the scattering region is parallel to disks that have been observed at larger scales (e.g.for classical Be stars, andfor Herbig Ae/Be stars, see reviews byand ).This depolarisation line effect, similar to that observed in classical Be stars, follows the emission line and is more or less as broad as the emission. It can be detected in the polarisation spectra, but also in the position angle spectra or both, depending on the (vector) contribution of the interstellar polarization. When translated into the Stokes QU parameters and plotted in the (Q, U) diagram, we will see a line excursion from the continuum towards the line (see Fig. <ref>, second column).Alternatively, a different line effect signature can be detected when an emission line emerges from a compact, central, region and scatters off a circumstellar disk.This is usually accompanied by a flip in polarisation or polarisation angle (Vink, Harries & Drew 2005). When mapped on the (Q, U) diagram, this particular line effect appears as a loop across the emission line (see Fig. <ref>, third column). The compact region is thought to be due to magnetospheric accretion, where the material from the disk is funnelled via accretion columns on to the stars, causing a shock on the photosphere. This type of line effect has been observed in T Tauri stars and HAe stars <cit.>.Importantly, the data provide the position angles of the disks in agreement with previous observations, while the models reproducing the line effect return disk sizes of order stellar radii, also in agreement with observations (Vink, Harries & Drew 2005, ). Thirdly, the absorption component of the emission line also produces a line effect signature that is commonly referred to as the McLean effect <cit.>.In this case, an enhanced polarisation is detected across the absorption accompanying an emission line as direct, unscattered, light from the star is absorbed.This normally results in observing a typical (inverse) P Cygni line profile, depending on whether we consider infall of material or an outflow respectively. The absorbed photons will be re-emitted isotropically and part of the emission will be scattered into our line of sight.If the distribution of the, inner, scattering material is not circular on the sky, an enhanced polarisation will be detected across the absorption compared with the continuum light (see Fig. <ref>, fourth column).Last but not least, if the geometry of the region containing the scatterers is circular on the sky, no line effect would be detected in most of the above cases (see Fig. <ref>, first column) since all polarisation vectors will cancel. Such a circular geometry could be due to a spherical distribution of material or if the disk is viewed pole-on[A possible exception to this statement can occur in the case of the intrinsic line polarization. Emission emerging from an anisotropic source, such as an accretion hot spot, scattering off a circular geometry would also result in net polarization. However, we note that the position angles and size scales derived from our data are consistent with circumstellar disks in the case of the T Tauri stars.] §.§ Construction of the sample In order to perform a statistical study on the line polarimetry, a large sample of HAeBe stars is needed. We combined all our previous spectropolarimetric work across Hα line of HAeBe stars <cit.> into one sample. Our previous medium resolution linear spectropolarimetric data were obtained using the RGO spectrograph on the 3.9-m Anglo Australian Telescope (AAT) <cit.>, the ISIS spectrograph on the 4.2-m on the William Herschel Telescope (WHT), La Palma <cit.> and the FORS2 spectrograph mounted on ESO's 8.2-m Very Large Telescope (VLT) in Chile <cit.>.These bring the total number of observed HAeBe objects to 56 (31 HBe and 25 HAe). These objects are presented in Table <ref>. A sample of 29 HAeBe stars, of which 19 objects are in common with our sample, was observed spectropolarimetrically by <cit.> but due to technical issues, these authors did not have information on the polarisation angle, and we decided not to include the remaining 10 objects for the analysis.Our sample is the largest (linear) spectropolarimetric survey of HAeBe stars that has been published to date. The vast majority, 52, of the objects were selected from the HAeBe catalogue of <cit.>, 10 of which are in other tables (extreme emission lines, other early emission line stars and non-emission line early type stars) in ). We proceed under the assumption that they are young stars. The remaining 5 objects were taken from the HAeBe candidate stars of <cit.>.The final sample covers nearly 50% of the HAeBe catalogue, where the majority was chosen from the northern hemisphere. Most of the remaining targets in the catalogue are too faint (V≥13.5) and would require very long exposure times as the spectropolarimetry needs high SNR. The combined Hα spectropolarimetric observations allow us to conduct the most powerful statistical investigation into the nature of linear polarisation in the circumstellar environment of HAeBe stars. To compare the spectropolarimetric results of HAeBe stars and their lower mass counterparts T Tauri stars, the spectropolarimetric results of a sample of 9 T Tauri stars are taken from <cit.>. §.§ Complementary Observations Seventeen targets were selected from the HAeBe catalogue of <cit.> and candidates of <cit.> to complement previous spectropolarimetric results. The list of objects and the log of the observations are presented in Table <ref>. The SNR is measured over a range of 10 Å around 6700 Å where the continuum is the flattest and the spectral lines are absent.The new linear spectropolarimetric data were obtained with the ISIS spectrograph on the WHT, La Palma, during the nights of 2015 August 4 and 5. The log of the observations is provided in Table <ref>. The 1200R grating centred at 6800 Å, with a spectral range of 1000 Å, was employed with a windowed 351×4200 pixel CCD and a slit width of 1.0 arcsec. This setup provides a spectral resolution of ∼35 kms^-1 as measured from arc lines around the Hα line. The seeing was less than 1.0 arcsec throughout both nights.The polarisation optics, which consist of a rotating half-wave plate and a calcite block, were used in order to perform linear polarisation observations. The calcite block separates the light into two perpendicularly polarised light beams, the ordinary (O) and extraordinary (E) beam. One complete set of observations consists of four exposures with the half-wave plate set at angles: 10^∘, 55^∘, 32.5^∘, and 77.5^∘. The dekker with 18 arcsec slot separation was used to observe the object and the sky simultaneously. Several cycles of observations per object were obtained at the four position angles to check for the consistency of the results. Several short exposures were taken for objects with strong Hα line to avoid saturation. Polarised standard stars and zero-polarised standard stars were observed each night to calibrate for the instrumental polarisation and angle offset. The data reduction was carried out using iraf <cit.>, which includes bias subtraction, flat fielding, sky subtraction and extraction of the O and E spectra. The extracted spectra were imported into the tsp package <cit.> to compute the Stokes parameters. The wavelength calibration was performed using figaro. For analysis purposes, the data were imported into the polmap package <cit.>.Multiple observations of the same targets provided a very consistent results. As the observations were obtained at the parallactic angle to achieve high SNR, the angle calibration was performed using the observed polarised standard stars. The instrumental polarisation is found to be ∼0.1% while the angle offset is found to be less than 0.5^∘ from the observation of unpolarised and polarised standard stars. As the instrumental and interstellar polarisation add a wavelength independent vector to the observed spectra, we did not correct the observed polarisation for them.§ RESULTS We begin with a brief presentation of the new data, before we focus on the statistical results of the full sample of spectropolarimetric data on HAeBe stars. §.§ Hα Spectropolarimetry-new observations In the new observations 11 objects out of 17 had never been observed with linear spectropolarimetry. Hα spectropolarimetry was performed for all the targets in Table <ref> and the results of the entire sample are presented in Table <ref> and in the Appendix in Fig. <ref>.In total, 11 objects show a possible line effect across the Hα line. The spectropolarimetric results of three objects from these new observations, selected as they exhibit the three different line effect signatures, are shown in triplots in the upper half of Fig. <ref>. In this triplot, the Stokes I (normal intensity) is shown in the lower panel, the polarization percentage in the middle, while the position angle (PA) is displayed in the upper panel. The results are also represented in a Stokes (Q, U) diagram (bottom) in Fig. <ref> using the same wavelength range of the triplot spectra, but sometimes with a different binning.HD 240010 shows a broad depolarisation line effect across the Hα line in both polarisation and polarisation angle spectra. The line effect is as broad as the emission line and it appears as a linear excursion in the (Q, U) diagram, the contribution of interstellar polarization has likely introduced a signature in the position angle, but will have only added a constant value in the (Q,U) diagram, which still shows a straight line. An intrinsic polarisation line effect is seen across the Hα line in HD 163296, the effect is narrower than depolarisation line effect and there is a flip in polarisation across the line. This flip appears as a loop when it is mapped on the (Q, U) diagram. MWC 863 shows two different line effects, a McLean line effect is seen across the absorptive component while the emission line displays a narrow polarization which is identified with intrinsic polarisation (in the following sections, the classificaiton criteria are outlined). Fig. <ref> also shows the intrinsic polarisation angle which is measured from the slope of the line effect in the (Q, U) diagram. For the depolarisation line effect, the angle is measured from the (unpolarized) line to the (polarized) continuum while for the intrinsic polarisation and McLean effect it is measured from continuum to (polarized) line (see the discussion in ). Although the direction for both the depolarization (HD 240010) and intrinsic polarization (HD 163296) are the same in the respective QU graphs, the intrinsic polarization angles differ, as this angle is measured from the line center to the continuum in the case of the depolarization, it is measured in the opposite direction for the intrinsic polarization. §.§ Statistical Results We present here Hα spectropolarimetric results from a sample of 56 HAeBe stars combined from this work and the literature. We begin with discussing the observed line effect and its type, before we focus on the magnitude and width of the line effect.The Hα spectropolarimetric results of each target are listed in Table <ref>. Columns 3 & 6 list the spectroscopic characterization of Stokes I, the intensity spectrum.The line polarimetric properties of each target are tabulated in columns 7-12 & 15. Finally, the continuum spectropolarimetric measurements are listed in columns 13 & 14. As can be seen in Fig <ref> and Table <ref>, a line effect is detected across Hα in 42/56 objects (75 ± 6%, errors reflecting the 1σ confidence interval of a sample proportion), divided equally between 21 HBe and 21 HAe stars. 14 objects (25 ± 6%) do not show any signs of a line effect. The detection rate for Herbig stars of spectral type B0-B7 is 66 ± 9% while that of the later type (i.e. all other) objects is 85 ± 7%, a difference that is close to 3σ. §.§.§ Line effect signatures The decision on whether to classify a line effect as depolarization or intrinsic polarization can sometimes be subjective.To avoid such biases, we aim to differentiate between the depolarisation and the line polarisation effect in a quantitative manner. To this end, the width of the line effect can be used as proxy (see Table <ref>). The method was first used by <cit.> to categorize the line effects. They statistically classified stars according to the fractional width [Δλ(pol)/Δλ(I)] at which the polarisation changes across the line. This quantity measures the width of the polarization over the line divided by the width of the line itself. Generally a wide fractional width is associated with depolarisation line effect, whereas the width of line polarisation effect is often narrower than the depolarisation. Both Δλ(I) and Δλ(pol) are measured at Full Width at Zero Intensity (FWZI).If the line effect is detected across the absorption component of the emission line then it is considered as a McLean line effect. The line effect across the emission line is classified based on two criteria; the value of the fractional width and whether there is a flip in the polarisation and PA spectra or not. If the fractional width is equal to or larger than 0.7 and there is no flip in the polarisation and PA spectra across the emission line then the line effect is consistent with depolarisation. On the other hand, if the fractional width is equal to or larger than 0.7 and a flip is observed in either the polarisation or PA spectra then the line effect is considered to be due to intrinsic polarisation. In addition, if the fractional width is smaller than 0.7 then the line effect is also consistent with polarisation. In some cases two different line effects, a McLean line effect across the absorption and either depolarisation or intrinsic polarisation across the emission line, can be observed for the same object. As reported before and mentioned in the Introduction, a depolarisation is more common in HBe stars, in particular in the early HBe stars, while in HAe stars intrinsic line polarisation is the dominant line effect. To see at what spectral type the line effect switches from line polarisation to depolarisation, we add 9 T Tauri stars to the sample from <cit.> and the result is shown in Fig. <ref>. The figure demonstrates that the intrinsic line polarisation is the dominant effect in T Tauri and late HAe stars while most early HBe stars show a depolarisation. There is also a number of HBe stars with a McLean line effect. Based on the large fraction of line effects at spectral types earlier than B7 and its absence beyond that, it would appear that the line effect changes from intrinsic line polarisation to depolarisation around B7-B8 spectral type. §.§.§ Line effect magnitude Bearing in mind that the typical magnitude of the continuum polarization, and thus line-effect, caused by electron scattering is expected to be 1-2% <cit.>, we investigated the magnitude of the line effect for all the objects that show a clear line effect.We measured the strength of the line effects directly from the (Q, U) diagram. It was taken as the distance between the continuum polarization, which is visible by a cluster of points, and the line centre.The error was estimated to be typically 10%. The results are tabulated in Table <ref> and are also shown in Fig. <ref>. The polarization ranges from ∼0.3% to ∼2.0% with an average of ∼0.9 %. Only R Mon shows a magnitude of ∼10% which is not expected from electron scattering close to the star and is due to observational effects as the object is spatially resolved in these observations (see the discussion in ). We have therefore discarded R Mon from the final results.As shown in Fig. <ref>, the strength of the line effect does not show any correlation with spectral type. To investigate whether the strength of the emission lines is correlated with the magnitude of the line effect, we plotted the magnitude of the latter as a function of the line peak to continuum of the Hα line (see Fig. <ref>). As can be seen in the figure there is only a very weak correlation between them. §.§.§ Fractional width <cit.> found that the fractional widthtends to decrease towards late spectral type. We can now revise therelation by increasing the sample from 25 to 41 objects. The resultis plotted in Fig. <ref>, the figure shows that there is asignificant correlation between the fractional width and the spectraltype, with a correlation coeffient, r = -0.60. The slope of thebest fit line between the fractional width against spectral type(counted as integers with B0=1, B1=2 etc) is determined at the10σ level, providing another indication that the trend isreal. The intrinsic scatter around the line does prevent us frommaking a conclusive statement whether there is a break in therelationship or whether it is continuous however. For example, whensplitting the sample into early HBe, late HBe, early HAe and latetype HAe stars, we find average values of 0.90±0.04,0.79±0.05, 0.74±0.07, 0.61±0.17 respectively (the errorsare the scatter around the mean divided by the squareroot of thenumber of datapoints). The fractional widths measured for late HBestars and early HAe stars are close and this might suggest they sharea similar spectropolarimetric behaviour, but they themselves do notdiffer beyond the 2σ level from either the early Be stars orthe late HAe stars. What is clear, however, is that the trend fromdepolarization to intrinsic polarization with the spectral type issignificant. While, in addition,the early Herbig Be stars are distinctlydifferent from the T Tauri stars for example.§ DISCUSSION §.§ Overall findings In the above we have investigated various observational aspects of the linear spectropolarimetric properties of the young pre-Main sequence Herbig Ae/Be stars.This is the largest sample to have been studied in this manner, and this allows us to confirm that there are distinct differences between the lower mass Herbig Ae stars and the higher mass Herbig Be stars. Indeed, we find evidence that the main distinction occurs at the B7-B8 range. Furthermore, the Herbig Ae stars display a similar behaviour as the solar mass T Tauri pre-Main Sequence stars. The major statistical conclusions of this exercise can be summarized as follows: * The occurrence of the line effect in the entire sample is high, at 75% (see e.g. Figure 3). When considering the sample of Herbig Ae and Herbig Be stars separately, we find that the occurrence in Herbig Be stars is smaller than in Herbig Ae stars.This difference is amplified when splitting the sample in early B-type objects and the rest. The detection rate for Herbig stars of spectral type B0-B7 is 66 ± 9% while that of the later type (i.e. all other) objects is 85 ± 7%.Hence, this difference is close to 3σ. * The appearance of the line effect is also different as a function of spectral type. Whereas the early type objects predominately display depolarization or McLean effects, the later type objects show intrinsic line polarization. This is very clear to see from Figure 4, where the break appears to occur around B7. Statistically, the overall change in the character of the line effect is also visible using the "fractional width" as a quantitative handle on the nature of the line effect. Figure 7 shows that Herbig Be stars have larger polarization widths than later type objects. The trend is significant with a slope at the 10σ level. * The strength of the line effect (in terms of per cent polarization) is of order 1 % and is independent of spectral type. However, there appears to be a weak trend in that stronger Hα emission lines have a slightly larger effect. The main result of the present study is that the difference between Herbig Ae and Herbig Be stars is now more robust, not only the detection rates are, statistically, shown to be different, but this 3σ effect is underpinned by the fact that the nature of the line effects is different as well. In the following, we discuss what these findings mean for the formation mechanism of low, intermediate and high mass stars, and why the line effect is equally strong in these classes of object. §.§ The origin and nature of the line effect As discussed earlier, the intrinsic line polarization observed towards the cooler objects can be explained by compact emission such as accretion shocks on the stellar surface scattering off circumstellar material, which is found to be consistent with a disk.On the other hand, the line depolarization and McLean effect can be best explained with the presence of a small circumstellar disk. As shown in the case of classical Be stars, most of the polarization originates from a region within a few stellar radii, where the electron densities are highest. It may be surprising then that there is only a weak correlation between the magnitude of the line effect and the strength of the Hα line (or spectral type), as one could expect a stronger emission line to be associated with more ionization and more free electrons and thus a larger polarization and larger line effect.This suggests that the detection of the line effect only depends on the geometry of the scattering agents and the geometry of the line emitting region, whereas, in the optically thin limit, the polarization itself depends on the density of the scatterers.For the depolarisation line effect, the free electrons in the ionised region around the stars polarise the continuum photons while the emission photons are unpolarised.In this context, it is useful to note that for example <cit.> detected a clear line effect across the Hβ emission of Herbig Ae/Be objects. For some stars the lines are so faint that the emission does not even reach the photospheric continuum. However, in such cases, the line can still be several times stronger than the underlying photospheric emission. This is because the photospheric absorption lines' minimum can be as low as 0.2 times the continuum level, and when the emission reaches the continuum level, it will have first filled up the underlying absoprtion. Most of the observed emission at these wavelengths will then be the unpolarized line emission. The maximum observable line effect, being the difference between continuum and line emmision, is therefore reached already for weak lines and no, or hardly any, further changes in polarization is observed for progressively stronger lines.In the case of the intrinsic line polarisation in our objects, the line effect is due to the fact that compact line emission (such as from individual accretion hot-spots or accretion funnels) scatters off circumstellar material.The cause for the ionization leading to the line emission and that responsible for the free electrons in the disk may be linked, but emission and polarization do not necessarily have to be correlated. For example the line emission arises from localized accretion hot spots and funnels in the magnetospheric accretion paradigm, while the circumstellar disk itself would be generally ionized due to the stellar photosphere and accretion luminosity. Indeed, it has been pointed out it is not yet settled whether in these situations the disk scattering material are free electrons, neutral hydrogen or dust (see e.g.for discussions).In any case, no trend of the strength of the line effect with line strength itself needs to be expected.In conclusion, in the above situations we can understand that the line effect strength is independent of the line strength itself. For the McLean effect, which we observe in a number of objects, this may not necessarily be the case.The number of objects with the McLean effect is rather small, 8 objects, but there is no correlation between the magnitude of the line effect and the strength of the emission line. As the scattering is due to the inner regions, unrelated to the outflow or infall itself, we suspect that this also leads to a line effect independent of the line emission. Finally, we address the question why the detection rate of the line effect in the Herbig Be stars is lower than for the Herbig Ae stars; in the case of line depolarization, the effect is a strong function of inclination of the system, A larger disk inclination results in a stronger line effect <cit.>. When the disk is pole-on for example, the system is circular on the sky and all polarization vectors will cancel out, resulting in a net zero polarization and no difference in polarization between line and continuum.A 100% detection rate will therefore not be expected at all for a sample of objects distributed at random inclinations, and the current detection rate is similar to those of classical Be stars (see e.g. ).A difference in the case of intrinsic polarization is that polarization can be seen even at low, face-on, inclinations of the disk. This is mainly due to the fact that the compact emission is anisotropic (be it due to accretion hot spots or funnels). As a consequence only part of the disk will be illuminated. This results in a net observable polarization, and in passing we note that this also explains the higher fraction of line-effects for later type stars which predominately exhibit the intrinsic polarization line-effect. §.§ On the formation of intermediate and massive stars Our findings indicate that a break between the spectropolarimetric, and possibly accretion, properties of high and low mass objects occurs around the B7-B8 spectral type. This is found in both the detection statistics and, especially, the nature of the line effect. The early Herbig Be stars have a lower detection rate and show predominately the depolarization and McLean effects, indicative of circumstellar disks.The detections are more numerous for the later B-type and A-type objects. In addition, as can be seen in Fig.<ref>, T Tauri stars and, especially late, HAe stars share the same Hα intrinsic polarisation line effect. This effect can be explained by scattering photons originating from a compact source, where the accretion take place on to the star, off the circumstellar electrons. The similarities in the spectropolarimetric properties between T Tauri stars - which are known to undergo magnetospheric accretion - and the late-type Herbig Be and Herbig Ae stars suggest this mechanism also acts on these intermediate mass stars (cf.and references in the Introduction).A complication is whether HAe stars have sufficient magnetic fields to facilitate MA or not. <cit.> detected magnetic fields (∼ a few hundred G) in a few HAeBe stars, mostly HAe and late HBe stars, but it is as of yet not clear whether this would be sufficient to drive accretion.Currently, there is no well-explored theory that explains the accretion of material onto the highest mass Young Stellar Objects however. Even the most recent, sophisticated star formation models are not able to simulate the fine detail required to probe the accretion process from parsec scales via an accretion disk to the stellar surface.For example, <cit.> explicitly mention that the material is not followed in the inner 80 au.Given that our spectropolarimetric evidence points towards circumstellar disks that are present at very small scales, the logical, direct conclusion we can draw is that the disk is not truncated, but reaches all the way down to the stellar surface.In this situation, the so-called Boundary Layer accretion is a viable mechanism to explain the growth of massive stars. The BL is a thin annulus close to the star in which the material reduces its (Keplerian) velocity to the slow rotation of the star when it reaches the stellar surface, and it is here that kinetic energy will be dissipated. It has been explored for Herbig Ae/Be stars by <cit.>, and the BL mechanism has also been explicitly suggested a number of times to act in Herbig Be stars (e.g. ), however, details are yet to be worked out for the more massive Herbig Be stars.§ CONCLUSIONS This work presents the spectropolarimetric results of a sample of 56 HAeBe stars which is the largest linear spectropolarimetric sample of HAeBe stars that has been published to date. The main findings are as follows:* Most HAeBe stars show a sign of line effect which is interpreted by the presence of a circumstellar disk. The detection rate of the line effect is 75 % (42/56) in the sample of 56 objects that have been observed spectropolarimetrically.* The magnitude of the line effect is of order of 0.3-2 %. There is no correlation between this magnitude and spectral type. A very weak correlation is seen between the magnitude of the line effect and the strength of Hα line. We can explain this both in terms of the line depolarization and intrinsic polarization. The detection of the line effect does not rely on the strength of the emission line but on the geometry of circumstellar environment. * The Herbig Be type stars have a significantly lower detection rate than the Herbig Ae stars, with a break around spectral type B7-B8. * Most of the HBe stars' signatures are consistent with a depolarisation line effect. In contrast, intrinsic line polarisation is more common in HAe and T Tauri stars. It seems late HBe and early HAe stars are at the interface between the two line effects, also indicating a break in spectropolarimetric properties around B7-B8. The similarity between T Tauri, HAe stars and late Herbig Be stars suggest that the latter are forming via magnetospheric accretion.The interface between HBe and HAe is possibly where the accretion mechanism switches from magnetospheric accretion to another process. Given the fact that the Herbig Be stars are non-magnetic and surrounded by small scale disks, it will be interesting to consider and work out in details the Boundary Layer model as a means for the continued accretion and growth of more massive stars. § ACKNOWLEDGEMENTS The allocation of time on the William Herschel Telescope was awarded by PATT, the United Kingdom allocation panel. KMA was supported by the UK Science and Technology Facilities Council (STFC). This research has made use of the SIMBAD data base, operated at CDS, Strasbourg, France. mn2e § OBSERVED SPECTROPOLARIMETRIC SIGNATURESHere we present the spectropolarimetric results across Hα for the new observations. | http://arxiv.org/abs/1707.08408v1 | {
"authors": [
"Karim Ababakr",
"Rene Oudmaijer",
"Jorick Vink"
],
"categories": [
"astro-ph.SR"
],
"primary_category": "astro-ph.SR",
"published": "20170726123757",
"title": "A Statistical Spectropolarimetric Study of Herbig Ae/Be Stars"
} |
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 =============================================================================================================================== Ludics is a logical framework in which types/formulas are modelled by sets of terms with the same computational behaviour. This paper investigates the representation of inductive data types and functional types in ludics. We study their structure following a game semantics approach. Inductive types are interpreted as least fixed points, and we prove an internal completeness result giving an explicit construction for such fixed points. The interactive properties of the ludics interpretation of inductive and functional types are then studied. In particular, we identify which higher-order functions types fail to satisfy type safety, and we give a computational explanation. § INTRODUCTION§.§ Context and Contributions Context Ludics was introduced by Girard <cit.> as a variant of game semantics with interactive types. Game Semantics has successfully provided fully abstract models for various logical systems and programming languages, among which PCF <cit.>. Although very close to Hyland–Ong (HO) games, ludics reverses the approach: in HO games one defines first the interpretation of a type (an arena) before giving the interpretation for the terms of that type (the strategies), while in ludics the interpretation of terms (the designs) is primitive and the types (the behaviours) are recovered dynamically as well-behaved sets of terms. This approach to types is similar to what exists in realisability <cit.> or geometry of interaction <cit.>.The motivation for such a framework was to reconstruct logic around the dynamics of proofs. Girard provides a ludics model for (a polarised version of) multiplicative-additive linear logic (MALL); a key role in his interpretation of logical connectives is played by the internal completeness results, which allow for a direct description of the behaviours' content. As most behaviours are not the interpretation of MALL formulas, an interesting question, raised from the beginning of ludics, is whether these remaining behaviours can give a logical counterpart to computational phenomena. In particular, data and functions <cit.>, and also fixed points <cit.> have been studied in the setting of ludics. The present work follows this line of research.Real life (functional) programs usually deal with data, functions over it, functions over functions, etc. Data types allow one to present information in a structured way. Some data types are defined inductively, for example: [caption=Example of inductive types in OCaml] > type nat = Zero | Succ of nat ;; > type 'a list = Nil | Cons of 'a * 'a list ;; > type 'a tree = Empty | Node of 'a * ('a tree) list ;;Upon this basis we can consider functional types, which are either first-order – from data to data – or higher-order – i.e., taking functions as arguments or returning functions as a result. This article aims at interpreting constructively the (potentially inductive) data types and the (potentially higher-order) functional types as behaviours of ludics, so as to study their structural properties. Inductive types are defined as (least) fixed points. As pointed out by Baelde, Doumane and Saurin <cit.>, the fact that ludics puts the most constraints on the formation of terms instead of types, conversely to game semantics, makes it a more natural setting for the interpretation of fixed points than HO games <cit.>.ContributionsThe main contributions of this article are the following: * We prove that internal completeness holds for infinite unions of behaviours satisfying particular conditions (Theorem <ref>), leading to an explicit construction of the least fixed points in ludics (Proposition <ref>).* Inductive and functional types are interpreted as behaviours, and we prove that such behaviours are regular (Corollary <ref> and Proposition <ref>). Regularity (that we discuss more in § <ref>) is a property that could be used to characterise the behaviours corresponding to μMALL formulas <cit.> – i.e., MALL with fixed points.* We show that a functional behaviour fails to satisfy purity, a property ensuring the safety of all possible executions (further explained in § <ref>), if and only if it is higher order and takes functions as argument (Proposition <ref>); this is typically the case of ( A ⊸ B) ⊸ C. In § <ref> we discuss the computational meaning of this result.The present work is conducted in the term-calculus reformulation of ludics by Terui <cit.> restricted to the linear part – the idea is that programs call each argument at most once.Related Work The starting point for our study of inductive types as fixed points in ludics is the work by Baelde, Doumane and Saurin <cit.>. In their article, they provide a ludics model for μMALL, a variant of multiplicative-additive linear logic with least and greatest fixed points. The existence of fixed points in ludics is ensured by Knaster-Tarski theorem, but this approach does not provide an explicit way to construct the fixed points; we will consider Kleene fixed point theorem instead. Let us also mention the work of Melliès and Vouillon <cit.> which introduces a realisability model for recursive (i.e., inductive and coinductive) polymorphic types. The representation of both data and functions in ludics has been studied previously. Terui <cit.> proposes to encode them as designs in order to express computability properties in ludics, but data and functions are not considered at the level of behaviours. Sironi <cit.> describes the behaviours corresponding to some data types: integers, lists, records, etc. as well as first-order function types; our approach generalises hers by considering generic data types and also higher order functions types.§.§ BackgroundBehaviours and Internal CompletenessA behaviour B is a set of designs which pass the same set of tests B^⊥, where tests are also designs. B^⊥ is called the orthogonal of B, and behaviours are closed under bi-orthogonal: B^⊥⊥ =B. New behaviours can be formed upon others using various constructors. In this process, internal completeness, which can be seen as a built-in notion of observational equivalence, ensures that two agents reacting the same way to any test are actually equal. From a technical point of view, this means that it is not necessary to apply a ⊥⊥-closure for the sets constructed to be behaviours.Paths: Ludics as Game SemanticsThis paper makes the most of the resemblance between ludics and HO game semantics. The connections between them have been investigated in many pieces of work <cit.> where designs are described as (innocent) strategies, i.e., in terms of the traces of their possible interactions. Following this idea, Fouqueré and Quatrini define paths <cit.>, corresponding to legal plays in HO games, and they characterise a behaviour by its set of visitable paths. This is the approach we follow. The definitions of regularity and purity rely on paths, since they are properties of the possible interactions of a behaviour.Regularity: Towards a Characterisation of μMALL?Our proof that internal completeness holds for an infinite union of increasingly large behaviours (Theorem <ref>) relies in particular on the additional hypothesis of regularity for these behaviours. Intuitively, a behaviour B is regular if every path in a design of B is realised by interacting with a design of B^⊥, and vice versa. This property is not actually ad hoc: it was introduced by Fouqueré and Quatrini <cit.> to characterise the denotations of MALL formulas as being precisely the regular behaviours satisfying an additional finiteness condition. In this direction, our intuition is that – forgetting about finiteness – regularity captures the behaviours corresponding to formulas of μMALL. Although such a characterisation is not yet achieved, we provide a first step by showing that the data patterns, a subset of positive μMALL formulas, yield only regular behaviours (Proposition <ref>).Purity: Type SafetyLudics has a special feature for termination which is not present in game semantics: the daimon . On a computational point of view, the daimon is commonly interpreted as an error, an exception raised at run-time causing the program to stop (see for example the notes of Curien <cit.>). Thinking of Ludics as a programming language, we would like to guarantee type safety, that is, ensure that “well typed programs cannot go wrong” <cit.>. This is the purpose of purity, a property of behaviours: in a pure behaviour, maximal interaction traces are -free, in other words whenever the interaction stops withit is actually possible to “ask for more” and continue the computation. Introduced by Sironi <cit.> (and called principality in her work), this property is related to the notions of winning designs <cit.> and pure designs <cit.>, but at the level of a behaviour. As expected, data types are pure (Corollary <ref>), but not always functional types are; we identify the precise cases where impurity arises (Proposition <ref>), and explain why some types are not safe. §.§ Outline In Section <ref> we present ludics and we state internal completeness for the logical connectives constructions. In Section <ref> we recall the notion of path, so as to define formally regularity and purity and prove their stability under the connectives. Section <ref> studies inductive data types, which we interpret as behaviours; Kleene theorem and internal completeness for infinite union allows us to give an explicit and direct construction for the least fixed point, with no need for bi-orthogonal closure; we deduce that data types are regular and pure. Finally, in Section <ref>, we study functional types, showing in what case purity fails.§ COMPUTATIONAL LUDICSThis section introduces the ludics background necessary for the rest of the paper, in the formalism of Terui <cit.>. The designs are the primary objects of ludics, corresponding to (polarised) proofs or programs in a Curry-Howard perspective. Cuts between designs can occur, and their reduction is called interaction. The behaviours, corresponding to the types or formulas of ludics, are then defined thanks to interaction. Compound behaviours can be formed with logical connectives constructions which satisfy internal completeness. §.§ Designs and InteractionSuppose given a set of variables 𝒱_0 and a set 𝒮, called signature, equipped with an arity function ar: 𝒮→ℕ. Elements a, b, …∈𝒮 are called names. A positive action is either(daimon), Ω (divergence), or a with a ∈𝒮; a negative action is a(x_1, …, x_n) where a ∈𝒮, ar(a)=n and x_1, …, x_n ∈𝒱_0 distinct. An action is proper if it is neithernor Ω.Positive and negative designs[In the following, the symbols d,e, … refer to designs of any polarity, while p,q, … and m,n, … are specifically for positive and negative designs respectively.] are coinductively defined by:p ::= Ωxa n_1 n_ar(a) n_0a n_1 n_ar(a) n ::= ax^a_1x^a_ar(a) p_aPositive designs play the same role as applications in λ-calculus, and negative designs the role of abstractions, where each name a ∈𝒮 binds ar(a) variables.Designs are considered up to α-equivalence. We will often write a( x) (resp. a⟨ n⟩) instead of a(x_1, …, x_n) (resp. a⟨ n_1 … n_n ⟩). Negative designs can be written as partial sums, for example a(x, y). p + b(). q instead of a(x, y). p + b(). q + ∑_c ≠ a, c ≠ b c(z^c).Ω.Given a design d, the definitions of the free variables of d, written fv( d), and the (capture-free) substitution of x by a negative design n in d, written d[ n/x], can easily be inferred. The design d is closed if it is positive and it has no free variable. A subdesign of d is a subterm of d. A cut in d is a subdesign of d of the form n_0a n, and a design is cut-free if it has no cut.In the following, we distinguish a particular variable x_0, that cannot be bound. A positive design p is atomic if fv( p) ⊆{x_0}; a negative design n is atomic if fv( n) = ∅.A design is linear if for every subdesign of the form xa n (resp. n_0a n), the sets {x}, fv( n_1), …, fv( n_ar(a)) (resp. the sets fv( n_0), fv( n_1), …, fv( n_ar(a))) are pairwise disjoint. This article focuses on linearity, so in the following when writing “design” we mean “linear design”. The interaction corresponds to reduction steps applied on cuts:∑_a ∈𝒮 a(x^a_1, …, x^a_ar(a)). p_a | b⟨ n_1, …,n_k⟩ p_b[ n_1/x^b_1, …,n_k/x^b_k] We will later describe an interaction as a sequence of actions, a path (Definition <ref>).Let p be a design, and let ^* denote the reflexive transitive closure of ; if there exists a design q which is neither a cut nor Ω and such that p ^*q, we write p ⇓ q; otherwise we write p ⇑. The normal form of a design, defined below, exists and is unique <cit.>. The normal form of a design d, noted d, is defined by:p =p ⇓ p = xa n_1 n_n p ⇓xa n_1 n_n p = Ω p ⇑ax^a p_a = ax^a p_a Note that the normal form of a closed design is either(convergence) or Ω (divergence). Orthogonality expresses the convergence of the interaction between two atomic designs, and behaviours are sets of designs closed by bi-orthogonal. Two atomic designs p and n are orthogonal, noted p ⊥ n, if p[ n/x_0] =.Given an atomic design d, define d^⊥ =e d ⊥ e; if E is a set of atomic designs of same polarity, define E^⊥ =d∀ e ∈ E,d ⊥ e. A set B of atomic designs of same polarity is a behaviour[Symbols A,B, … will designate behaviours of any polarity, while M,N … and P,Q, … will be for negative and positive behaviours respectively.] if B^⊥⊥ =B. A behaviour is either positive or negative depending on the polarity of its designs.Behaviours could alternatively be defined as the orthogonal of a set E of atomic designs of same polarity – E corresponds to a set of tests or trials. Indeed, E^⊥ is always a behaviour, and every behaviour B is of this form by taking E =B^⊥.The incarnation of a behaviour B contains the cut-free designs of B whose actions are all visited during an interaction with a design in B^⊥. Those correspond to the cut-free designs that are minimal for the stable ordering ⊑, where d' ⊑ d if d can be obtained from d' by substituting positive subdesigns for some occurrences of Ω.Let B be a behaviour and d ∈ B cut-free. * The incarnation of d in B, written | d|_ B, is the smallest (for ⊑) cut-free design d' such that d' ⊑ d and d' ∈ B. If | d|_ B =d we say that d is incarnated in B.* The incarnation | B| of B is the set of the (cut-free) incarnated designs of B. §.§ Logical Connectives Behaviour constructors – the logical connectives – can be applied so as to form compound behaviours. These connectives, coming from (polarised) linear logic, are used for interpreting formulas as behaviours, and will also indeed play the role of type constructors for the types of data and functions. In this subsection, after defining the connectives we consider, we state the internal completeness theorem for these connectives.Let us introduce some notations. In the rest of this article, suppose the signature 𝒮 contains distinct unary names , _1, _2 and a binary name , and write = , _1 = _1, _2 = _2 and =. Given a behaviour B and x fresh, define B^x =d[x/x_0] d ∈ B; such a substitution operates a “delocation” with no repercussion on the behaviour's inherent properties. Given a k-ary name a ∈𝒮, we write a⟨ N_1, …,N_k ⟩ or even a⟨ N⟩ for x_0a n n_i ∈ N_i, and write a( x). P for a( x). p p ∈ P. For a negative design n = ax^a p_a and a name a ∈𝒮, we denote by na the design a(x^a). p_a (that is a(x^a). p_a + ∑_b ≠ a b(x^b).Ω).N = ⟨ N ⟩^⊥⊥ P = ((x). P^x)^⊥⊥ M ⊕ N = (_1 ⟨ M ⟩∪_2 ⟨ N ⟩)^⊥⊥ M ⊗ N = ⟨ M,N ⟩^⊥⊥ N ⊸ P = ( N ⊗ P^⊥)^⊥Our connectives , , ⊕ and ⊗ match exactly those defined by Terui <cit.>, who also proves the following internal completeness theorem stating that connectives apply on behaviours in a constructive way – there is no need to close by bi-orthogonal. For each connective, we present two versions of internal completeness: one concerned with the full behaviour, the other with the behaviour's incarnation.N = ⟨ N ⟩∪{} | N| = ⟨ | N| ⟩∪{} P =n n∈(x). P^x | P| = (x).| P^x|M ⊕ N = _1 ⟨ M ⟩∪_2 ⟨ N ⟩∪{} | M ⊕ N| = _1 ⟨ | M| ⟩∪_2 ⟨ | N| ⟩∪{} M ⊗ N = ⟨ M,N ⟩∪{} | M ⊗ N| = ⟨ | M|, | N| ⟩∪{}§ PATHS AND INTERACTIVE PROPERTIES OF BEHAVIOURSPaths are sequences of actions recording the trace of a possible interaction. For a behaviour B, we can consider the set of its visitable paths by gathering all the paths corresponding to an interaction between a design of B and a design of B^⊥. This notion is needed for defining regularity and purity and proving that those two properties of behaviours are stable under (some) connectives constructions. §.§ PathsThis subsection adapts the definitions of path and visitable path from <cit.> to the setting of computational ludics. In order to do so, we need first to recover location in actions so as to consider sequences of actions.Location is a primitive idea in Girard's ludics <cit.> in which the places of a design are identified with loci or addresses, but this concept is not visible in Terui's presentation of designs-as-terms. We overcome this by introducing actions with more information on location, which we call located actions, and which are necessary to: * represent cut-free designs as trees – actually, forests – in a satisfactory way,* define views and paths. A located action[Located actions will often be denoted by symbol κ, sometimes with its polarity: κ^+ or κ^-.] κ is one of: xax_1x_ar(a) a_x(x_1, …, x_ar(a)) where in the last two cases (positive proper and negative proper respectively), a ∈𝒮 is the name of κ, the variables x, x_1, …, x_ar(a) are distinct, x is the address of κ and x_1, …, x_ar(a) are the variables bound by κ.In the following, “action” will always refer to a located action. Similarly to notations for designs, xa x stands for xax_1x_n and a_x( x) for a_x(x_1, …, x_n).We show how cut-free designs can be represented as trees of located actions in this example. Let a^2, b^2, c^1, d^0 ∈𝒮, where exponents stand for arities. The following design is represented by the tree of Fig. <ref>.d = a(x_1,x_2).(x_2ba(x_3, x_4). + c(y_1).(y_1d), c(y_2).(x_1d))Such a representation is in general a forest: a negative design ax^a p_a gives as many trees as there is a ∈𝒮 such that p_a ≠Ω. The distinguished variable x_0 is given as address to every negative root of a tree, and fresh variables are picked as addresses for negative actions bound by positive ones. This way, negative actions from the same subdesign, i.e., part of the same sum, are given the same address. A tree is indeed to be read bottom-up: a proper action κ is justified if its address is bound by an action of opposite polarity appearing below κ in the tree; otherwise κ is called initial. Except the root of a tree, which is always initial, every negative action is justified by the only positive action immediately below it. If κ and κ' are proper, κ is hereditarily justified by κ' if there exist actions κ_1, …, κ_n such that κ = κ_1, κ' = κ_n and for all i such that 1 ≤ i < n, κ_i is justified by κ_i+1.Before giving the definitions of view and path, let us give an intuition. On Fig. <ref> are represented a view and a path of design d. Views are branches in the tree representing a cut-free design (reading bottom-up), while paths are particular “promenades” starting from the root of the tree; not all such promenades are paths, though. Views correspond to chronicles in original ludics <cit.>.For every positive proper action κ^+ = xay define κ^+ = a_x(y), and similarly if κ^- = a_x(y) define κ^- = xay. Given a finite sequence of proper actions s = κ_1 …κ_n, define s = κ_1…κ_n. Suppose now that if s contains an occurrence of , it is necessarily in last position; the dual of s, written s, is the sequence defined by: * s =s if s does not end with ,* s =s' if s =s'.Note that s =s. The notions of justified, hereditarily justified and initial actions also apply in sequences of actions. An alternated justified sequence (or aj-sequence) s is a finite sequence of actions such that: * (Alternation) Polarities of actions alternate.* (Daimon) Ifappears, it is the last action of s.* (Linearity) Each variable is the address of at most one action in s. The (unique) justification of a justified action κ in an aj-sequence is noted just(κ), when there is no ambiguity on the sequence we consider. A view v is an aj-sequence such that each negative action which is not the first action of v is justified by the immediate previous action. Given a cut-free design d, v is a view of d if it is a branch in the representation of d as a tree (modulo α-equivalence).The way to extract the view of an aj-sequence is given inductively by: * ϵ = ϵ, where ϵ is the empty sequence,* sκ^+ =sκ^+,* sκ^- =s_0κ^- where s_0 is the prefix of s ending on just(κ^-), or s_0 = ϵ if κ^- initial.The anti-view of an aj-sequence, noted s, is defined symmetrically by reversing the role played by polarities; equivalently s =s.A path s is a positive-ended aj-sequence satisfying: * (P-visibility) For all prefix s' κ^+ of s, just(κ^+) ∈ s'* (O-visibility) For all prefix s' κ^- of s, just(κ^-) ∈ s'Given a cut-free design d, a path s is a path of d if for all prefix s' of s, s' is a view of d.Remark that the dual of a path is a path.Paths are aimed at describing an interaction between designs. If d and e are cut-free atomic designs such that d ⊥ e, there exists a unique path s of d such that s is a path of e. We write this path d e, and the good intuition is that it corresponds to the sequence of actions followed by the interaction between d and e on the side of d. An alternative way defining orthogonality is then given by the following proposition. d ⊥ e if and only if there exists a path s of d such that s is a path of e. At the level a behaviour B, the set of visitable paths describes all the possible interactions between a design of B and a design of B^⊥. A path s is visitable in a behaviour B if there exist cut-free designs d ∈ B and e ∈ B^⊥ such that s =d e. The set of visitable paths of B is written B.Note that for every behaviour B, B = B^⊥. §.§ Regularity, Purity and ConnectivesThe meaning of regularity and purity has been discussed in the introduction. After giving the formal definitions, we prove that regularity is stable under all the connectives constructions. We also show that purity may fail with ⊸, and only a weaker form called quasi-purity is always preserved. B is regular if the following conditions are satisfied: * for all d ∈ |B| and all path s of d, s ∈B,* for all d ∈ |B^⊥| and all path s of d, s ∈B^⊥,* The sets B and B^⊥ are stable under shuffle.where the operation of shuffle () on paths corresponds to an interleaving of actions respecting alternation of polarities, and is defined below. Let s s' refer to the subsequence of s containing only the actions that occur in s'. Let s and t be paths of same polarity, let S and T be sets of paths of same polarity. We define: * st =ust u s =su t =t if s,t negative,* st = κ^+uu ∈ s't' if s = κ^+s' and t = κ^+t' positive with same first action,* ST =u ∃ s ∈ S, ∃ t ∈ Tstu ∈ st, In fact, a behaviour B is regular if every path formed with actions of the incarnation of B, even mixed up, is a visitable path of B, and similarly for B^⊥. Remark that regularity is a property of both a behaviour and its orthogonal since the definition is symmetrical: B is regular if and only if B^⊥ is regular. A behaviour B is pure if every -ended path s ∈ B is extensible, i.e., there exists a proper positive action κ^+ such that s κ^+ ∈B.Purity ensures that when an interaction encounters , this does not correspond to a real error but rather to a partial computation, as it is possible to continue this interaction. Note that daimons are necessarily present in all behaviours since the converse property is always true: if s κ^+ ∈B then s ∈B.Regularity is stable under , , ⊕, ⊗ and ⊸. Purity is stable under , , ⊕ and ⊗. Unfortunately, when N and P are pure, N ⊸ P is not necessarily pure, even under regularity assumption. However, a weaker form of purity holds for N ⊸ P. A behaviour B is quasi-pure if all the -ended well-bracketed paths in V_ B are extensible.We recall that a path s is well-bracketed if, for every justified action κ in s, when we write s =s_0 κ's_1 κ s_2 where κ' justifies κ, all the actions in s_1 are hereditarily justified by κ'.If N and P are quasi-pure and regular then N ⊸ P is quasi-pure. § INDUCTIVE DATA TYPESSome important contributions are presented in this section. We interpret inductive data types as positive behaviours, and we prove an internal completeness result allowing us to make explicit the structure of fixed points. Regularity and purity of data follows.Abusively, we denote the positive behaviour {} byall along this section. §.§ Inductive Data Types as Kleene Fixed PointsWe define the data patterns via a type language and interpret them as behaviours, in particular μ is interpreted as a least fixed point. Data behaviours are the interpretation of steady data patterns.Suppose given a countably infinite set 𝒱 of second-order variables: X, Y, …∈𝒱. Let 𝒮' = 𝒮∖{, _1, _2, } and define the set of constants Const =C_aa ∈𝒮' which contains a behaviour C_a = {x_0aΩ^-}^⊥⊥ (where Ω^- := ax^aΩ) for each a ∈𝒮', i.e., such that a is not the name of a connective. Remark that V_ C_a = { , x_0a x}, thus C_a is regular and pure. The set 𝒫 of data patterns is generated by the inductive grammar:A, B ::=X ∈𝒱 a ∈𝒮'A ⊕^+ BA ⊗^+ B μ X. A The set of free variables of a data pattern A ∈𝒫 is denoted by FV(A).Let b, n, l, t ∈𝒮' and X ∈𝒱. The data types given as example in the introduction can be written in the language of data patterns as follows:𝔹ool= b ⊕^+ b ℕat = μ X.( n ⊕^+ X) 𝕃ist_A = μ X.(l ⊕^+ (A ⊗^+ X)) 𝕋ree_A = μ X.(t ⊕^+ (A ⊗^+ 𝕃ist_X)) = μ X.(t ⊕^+ (A ⊗^+ μY.(l ⊕^+ (X ⊗^+ Y))))Let ℬ^+ be the set of positive behaviours. Given a data pattern A ∈𝒫 and an environment σ, i.e., a function that maps free variables to positive behaviours, the interpretation of A in the environment σ, written A^σ, is the positive behaviour defined by:X^σ = σ(X)A ⊕^+ B^σ= (A^σ) ⊕ (B^σ) a^σ =C_aA ⊗^+ B^σ= (A^σ) ⊗ (B^σ) μ X. A^σ = lfp(ϕ^A_σ)where lfp stands for the least fixed point, and the function ϕ^A_σ: ℬ^+ →ℬ^+,P ↦A^σ, X↦ P is well defined and has a least fixed point by Knaster-Tarski fixed point theorem, as shown by Baelde, Doumane and Saurin <cit.>. Abusively we may write ⊕^+ and ⊗^+, instead of (·) ⊕ (·) and (·) ⊗ (·) respectively, for behaviours. We call an environment σ regular (resp. pure) if its image contains only regular (resp. pure) behaviours. The notation σ, X↦ P stands for the environment σ where the image of X has been changed to P.In order to understand the structure of fixed point behaviours that interpret the data patterns of the form μ X.A, we need a constructive approach, thus Kleene fixed point theorem is best suited than Knaster-Tarski. We now prove that we can apply this theorem.Recall the following definitions and theorem. A partial order is a complete partial order (CPO) if each directed subset has a supremum, and there exists a smallest element, written . A function f : E → F between two CPOs is Scott-continuous (or simply continuous) if for every directed subset D ⊆ E we have ⋁_x ∈ D f(x) = f(⋁_x ∈ Dx). Let L be a CPO and let f:L → L be Scott-continuous. The function f has a least fixed point, defined bylfp(f) = ⋁_n ∈ℕf^n() The set ℬ^+ ordered by ⊆ is a CPO, with least element ; indeed, given a subset ℙ⊆ℬ^+, it is directed and we have ⋁ℙ = (⋃ℙ)^⊥⊥. Hence next proposition proves that we can apply the theorem. Given a data pattern A ∈𝒫, a variable X ∈𝒱 and an environment σ : FV(A) ∖{X}→ℬ^+, the function ϕ^A_σ is Scott-continuous. For every A ∈𝒫, X ∈𝒱 and σ : FV(A) ∖{X}→ℬ^+,μ X.A^σ = ⋁_n ∈ℕ (ϕ^A_σ)^n() = (⋃_n ∈ℕ (ϕ^A_σ)^n())^⊥⊥ This result gives an explicit formulation for least fixed points. However, the ⊥⊥-closure might add new designs which were not in the union, making it difficult to know the exact content of such a behaviour. The point of next subsection will be to give an internal completeness result proving that the closure is actually not necessary.Let us finish this subsection by defining a restricted set of data patterns so as to exclude the degenerate ones. Consider for example 𝕃ist_A' = μ X. (A ⊗^+ X), a variant of 𝕃ist_A (see Example <ref>) which misses the base case. It is degenerate in the sense that the base element, here the empty list, is interpreted as the design . This is problematic: an interaction going through a whole list will end with an error, making it impossible to explore a pair of lists for example. The pattern ℕat' = μ X.X is even worse since ℕat' =. The point of steady data patterns is to ensure the existence of a basis; this will be formalised in Lemma <ref>. The set of steady data patterns is the smallest subset 𝒫^s ⊆𝒫 such that: * 𝒮' ⊆𝒫^s* If A ∈𝒫^s and B is such that B^σ is pure if σ is pure, then A ⊕^+ B∈𝒫^s and B ⊕^+ A ∈𝒫^s* If A ∈𝒫^s and B∈𝒫^s then A ⊗^+ B ∈𝒫^s* If A ∈𝒫^s then μ X.A ∈𝒫^sThe condition on B in the case of ⊕^+ admits data patterns which are not steady, possibly with free variables, but ensuring the preservation of purity, i.e., type safety; the basis will come from side A. We will prove (§ <ref>) that behaviours interpreting steady data patterns are pure, thus in particular a data pattern of the form μ X.A is steady if the free variables of A all appear on the same side of a ⊕^+ and under the scope of no other μ (since purity is stable under , , ⊕, ⊗). We claim that steady data patterns can represent every type of finite data.A data behaviour is the interpretation of a closed steady data pattern.§.§ Internal Completeness for Infinite Union Our main result is an internal completeness theorem, stating that an infinite union of simple regular behaviours with increasingly large incarnations is a behaviour: ⊥⊥-closure is useless. * A slice is a design in which all negative subdesigns are either Ω^- or of the form a( x). p_a, i.e., at most unary branching. c is a slice of d if c is a slice and c ⊑ d. A slice c of d is maximal if for any slice c' of d such that c ⊑ c', we have c =c'.* A behaviour B is simple if for every design d ∈ | B|: * d has a finite number of maximal slices, and* every positive action of d is justified by the immediate previous negative action. Condition (2) of simplicity ensures that, given d ∈ | B| and a slice c ⊑ d, one can find a path of c containing all the positive proper actions of c until a given depth; thus by condition (1), there exists k ∈ℕ depending only on d such that k paths can do the same in d.Now suppose ( A_n)_n ∈ℕ is an infinite sequence of simple regular behaviours such that for all n ∈ℕ, | A_n| ⊆ | A_n+1| (in particular we have A_n ⊆ A_n+1).The set ⋃_n ∈ℕ A_n is a behaviour.A union of behaviours is not a behaviour in general. In particular, counterexamples are easily found if releasing either the inclusion of incarnations or the simplicity condition. Moreover, our proof for this theorem relies strongly on regularity. Under the same hypotheses we can prove V_⋃_n ∈ℕ A_n = ⋃_n ∈ℕ V_ A_n and |⋃_n ∈ℕ A_n| = ⋃_n ∈ℕ | A_n|, hence the following corollary.* ⋃_n ∈ℕ A_n is simple and regular;* if moreover all the A_n are pure then ⋃_n ∈ℕ A_n is pure. §.§ Regularity and Purity of DataThe goal of this subsection is to show that the interpretation of data patterns of the form μ X.A can be expressed as an infinite union of behaviours ( A_n)_n ∈ℕ satisfying the hypotheses of Theorem <ref>, in order to deduce regularity and purity. We will call an environment σ simple if its image contains only simple behaviours.For all A ∈𝒫, X ∈𝒱, σ : FV(A) ∖{X}→ℬ^+ simple and regular[The hypothesis “simple and regular” has been added, compared to the CSL version of this article, for correction.], and n ∈ℕ we have|(ϕ^A_σ)^n()| ⊆ |(ϕ^A_σ)^n+1()| For all A ∈𝒫 and simple regular environment σ, A^σ is simple regular.By induction on data patterns. If A = X or A = a the conclusion is immediate. If A = A_1 ⊕^+ A_2 or A = A_1 ⊗^+ A_2 then regularity comes from Proposition <ref>, and simplicity is easy since the structure of the designs in A^σ is given by internal completeness for the logical connectives (Theorem <ref>). So suppose A = μ X.A_0. By induction hypothesis, for every simple regular behaviour P ∈ℬ^+ we have ϕ^A_0_σ( P) = A_0^σ, X ↦ P simple regular. From this, it is straightforward to show by induction that for every n ∈ℕ, (ϕ^A_0_σ)^n() is simple regular. Moreover, for every n ∈ℕ we have |(ϕ^A_0_σ)^n()| ⊆ |(ϕ^A_0_σ)^n+1()| by Lemma <ref>, thus by Corollary <ref> and Theorem <ref>, μ X.A_0^σ= ⋁_n ∈ℕ (ϕ^A_σ)^n() = (⋃_n ∈ℕ(ϕ^A_0_σ)^n())^⊥⊥ = ⋃_n ∈ℕ(ϕ^A_0_σ)^n(). Consequently, by Corollary <ref>, μ X.A_0^σ is simple regular.Remark that we have proved at the same time, using Theorem <ref>, that behaviours interpreting data patterns μ X.A admit an explicit construction: If A ∈𝒫, X ∈𝒱, and σ : FV(A)∖ X →ℬ^+ is simple regular,μ X.A^σ = ⋃_n ∈ℕ (ϕ^A_σ)^n()Data behaviours are regular. We now move on to proving purity. The proof that the interpretation of a steady data pattern A is pure relies on the existence of a basis for A (Lemma <ref>). Let us first widen (to -free paths) and express in a different way (for -ended paths) the notion of extensible visitable path. Let B be a behaviour. * A -free path s ∈ B is extensible if there exists t ∈ B of which s is a strict prefix.* A -ended path s∈ B is extensible if there exists a positive action κ^+ and t ∈ B of which sκ^+ is a prefix. Write V_ B^max for the set of maximal, i.e., non extensible, visitable paths of B.Every steady data pattern A ∈𝒫^s has a basis, i.e., a simple regular behaviour B such that for all simple regular environment σ we have * B ⊆A^σ,* for every path s ∈ V_ B, there exists t ∈ V_ B^max -free extending s (in particular B pure),* V_ B^max⊆ V_A^σ^max. If A = a, a basis is C_a. If A = A_1 ⊕^+ A_2, and A_i is steady with basis B_i, then ⊗_iB_i:=_i⟨ B_i⟩ is a basis for A. If A = A_1 ⊗^+ A_2, a basis is B_1 ⊗^+B_2 where B_1 and B_2 are basis of A_1 and A_2 respectively. If A = μ X.A_0, its basis is the same as A_0. If A ∈𝒫^s of basis B, X ∈𝒱, and σ : FV(A)∖ X →ℬ^+ simple regular,μ X.A^σ = ⋃_n ∈ℕ (ϕ^A_σ)^n( B) Since B is a basis for A we have ⊆ B ⊆A^σ, X → = ϕ^A_σ(). The Scott-continuity of the function ϕ^A_σ implies that it is increasing, thus (ϕ^A_σ)^n() ⊆ (ϕ^A_σ)^n( B) ⊆ (ϕ^A_σ)^n+1() for all n ∈ℕ. Hence A^σ = ⋃_n ∈ℕ(ϕ^A_σ)^n() = ⋃_n ∈ℕ (ϕ^A_σ)^n( B). For all A ∈𝒫^s and simple regular pure environment σ, A^σ is pure.By induction on A. The base cases are immediate and the connective cases are solved using Proposition <ref>. Suppose now A = μ X.A_0, where A_0 is steady with basis B_0. We have A^σ = ⋃_n ∈ℕ (ϕ^A_0_σ)^n( B_0) by Proposition <ref>, let us prove it satisfies the hypotheses needed to apply Corollary <ref>(2). By induction hypothesis and Proposition <ref>, for every simple, regular and pure behaviour P ∈ℬ^+ we have ϕ^A_0_σ( P) = A_0^σ, X ↦ P simple, regular and pure, hence it is easy to show by induction that for every n ∈ℕ, (ϕ^A_0_σ)^n( B_0) is as well. Moreover, for every n ∈ℕ we prove that |(ϕ^A_0_σ)^n( B_0)| ⊆ |(ϕ^A_0_σ)^n+1( B_0)| similarly to Lemma <ref>, replacingby the basis B_0. Finally, by Corollary <ref>, A^σ is pure. Data behaviours are pure.Although here the focus is on the interpretation of data patterns, we should say a word about the interpretation of (polarised) μMALL formulas, which are a bit more general. These formulas are generated by:P, Q ::=X_PX_N^⊥ 10M ⊕ NM ⊗ NN μ X.P M, N ::=P^⊥where the usual involutive negation hides the negative connectives and constants, through the dualities 1/, 0/⊤, ⊕/, ⊗/, /, μ/ν . The interpretation as ludics behaviours, given in <cit.>, is as follows: 1 is interpreted as a constant behaviour C_a, 0 is the daimon , the positive connectives match their ludics counterparts, μ is interpreted as the least fixed point of a function ϕ^A_σ similarly to data patterns, and the negation corresponds to the orthogonal. Since in ludics constants andare regular, and since regularity is preserved by the connectives (Proposition <ref>) and by orthogonality, the only thing we need in order to prove that all the behaviours interpreting μMALL formulas are regular is a generalisation of regularity stability under fixed points (for now we only have it in our particular case: Corollary <ref> together with Proposition <ref>).Note however that interpretations of μMALL formulas are not all pure. Indeed, as we will see in next section, orthogonality (introduced through the connective ⊸) does not preserve purity in general.§ FUNCTIONAL TYPESIn this section we define functional behaviours which combine data behaviours with the connective ⊸. A behaviour of the form N ⊸ P is the set of designs such that, when interacting with a design of type N, outputs a design of type P; this is exactly the meaning of its definition N ⊸ P :=( N ⊗ P^⊥)^⊥. We prove that some particular higher-order functional types – where functions are taken as arguments, typically (A ⊸ B) ⊸ C – are exactly those who fail at being pure, and we interpret this result from a computational point of view. §.§ Where Impurity ArisesWe have proved that data behaviours are regular and pure. However, if we introduce functional behaviours with the connective ⊸, purity does not hold in general. Proposition <ref> indicates that a weaker property, quasi-purity, holds for functional types, and Proposition <ref> identifies exactly the cases where purity fails.Let us write 𝒟 for the set of data behaviours. A functional behaviour is a behaviour inductively generated by the grammar below, where P ⊸^+Q stands for (( P) ⊸ Q). P,Q ::=P_0 ∈𝒟 P ⊕^+QP ⊗^+QP ⊸^+Q From Propositions <ref>, <ref> and <ref> we easily deduce the following result. Functional behaviours are regular and quasi-pure. For next proposition, consider contexts defined inductively as follows (where P is a functional behaviour):𝒞 ::= [ ] 𝒞⊕^+PP ⊕^+ 𝒞𝒞⊗^+PP ⊗^+ 𝒞 P ⊸^+ 𝒞 A functional behaviour P is impure if and only if there exist contexts 𝒞_1, 𝒞_2 and functional behaviours Q_1,Q_2,R with R ∉Const such thatP = 𝒞_1[ 𝒞_2[ Q_1 ⊸^+Q_2] ⊸^+R ] §.§ Example and DiscussionProposition <ref> states that a functional behaviour which takes functions as argument is not pure: some of its visitable paths end with a daimon , and there is no possibility to extend them. In terms of proof-search, playing the daimon is like giving up; on a computational point of view, the daimon appearing at the end of an interaction expresses the sudden interruption of the computation. In order to understand why such an interruption can occur in the specific case of higher-order functions, consider the following example which illustrates the proposition.Let Q_1,Q_2,1 be functional behaviours, with 1 ∈Const. Define Bool =1 ⊕^+1 and consider the behaviour P = ( Q_1 ⊸^+Q_2) ⊸^+ Bool: this is a type of functions which take a function as argument and output a boolean. Let α_1, α_2, β be respectively the first positive action of the designs of Q_1,Q_2,1. It is possible to exhibit a design p ∈ P and a design n ∈ P^⊥ such that the visitable path s =p n is -ended and maximal in P, in other words s is a witness of the impurity of P. The path s contains the actions α_1 and α_2 in such a way that it cannot be extended with β without breaking the P-visibility condition, and there is no other available action in designs of P to extend it. Reproducing the designs p and n and the path s here would be of little interest since those objects are too large to be easily readable (s visits the entire design p, which contains 11 actions). We however give an intuition in the style of game semantics: Fig. <ref> represents s as a legal play in a strategy of type P = ( Q_1 ⊸^+Q_2) ⊸^+ Bool (note that only one “side” ⊕_11 of Bool is represented, corresponding for example to , because we cannot play in both sides). This analogy is informal, it should stand as an intuition rather than as a precise correspondence with ludics; for instance, and contrary to the way it is presented in game semantics, the questions are asked on the connectives, while the answers are given in the sub-types of P. On the right are given the actions in s corresponding to the moves played. The important thing to remark is the following: if a move b corresponding to action β were played instead ofat the end of this play, it would break the P-visibility of the strategy, since this move would be justified by move q_.The computational interpretation of the -ended interaction between p and n is the following: a program p of type P launches a child process p' to compute the argument of type Q_1 → Q_2, but p starts to give a result in Bool before the execution of p' terminates, leading to a situation where p cannot compute the whole data in Bool. The interaction outputs , i.e., the answer given in Bool by p is incomplete.Moreover by Proposition <ref> functional behaviours are quasi-pure, therefore the maximal -ended visitable paths are necessarily not well-bracketed. This is indeed the case of s: remark for example that the move q_⊕_1 appears between a_1 and its justification q_ in the sequence, but q_⊕_1 is not hereditarily justified by q_. In HO games, well-bracketedness is a well studied notion, and relaxing it introduces control operators in program. If we extend such an argument to ludics, this would mean that the appearance ofin the execution of higher-order functions can only happen in the case of programs with control operators such as jumps, i.e. programs which are not purely functional. § CONCLUSION This article is a contribution to the exploration of the behaviours of linear ludics in a computational perspective. Our focus is on the behaviours representing data types and functional types. Inductive data types are interpreted using the logical connectives constructions and a least fixed point operation. Adopting a constructive approach, we provide an internal completeness result for fixed points, which unveils the structure of data behaviours. This leads us to proving that such behaviours are regular – the key notion for the characterisation of MALL in ludics –and pure – that is, type safe. But behaviours interpreting types of functions taking functions as argument are impure; for well-bracketed interactions, corresponding to the evaluation of purely functional programs, safety is however guaranteed.Further WorkTwo directions for future research arise naturally: * Extending our study to greatest fixed points ν X.A, i.e., coinduction, is the next objective. Knaster–Tarski ensures that such greatest fixed point behaviours exist <cit.>, but Kleene fixed point theorem does not apply here, hence we cannot find an explicit form for coinductive behaviours the same way we did for the inductive ones. However it is intuitively clear that, compared to least fixed points, greatest ones add the infinite “limit” designs in (the incarnation of) behaviours. For example, if ℕat_ω = ν X. (1 ⊕ X) then we should have |ℕat_ω| = |ℕat| ∪{ d_ω} where d_ω = succ( d_ω) = x_0|_2⟨(x). d_ω^x⟩.* Another direction would be to get a complete characterisation of μMALL in ludics, by proving that a behaviour is regular – and possibly satisfying a supplementary condition – if and only if it is the denotation of a μMALL formula. AcknowledgementI thank Claudia Faggian, Christophe Fouqueré, Thomas Seiller and the anonymous referees for their wise and helpful comments.plainurl | http://arxiv.org/abs/1707.08925v1 | {
"authors": [
"Alice Pavaux"
],
"categories": [
"cs.LO"
],
"primary_category": "cs.LO",
"published": "20170727162604",
"title": "Inductive and Functional Types in Ludics"
} |
=110pt -0.0cm 0.6cm 0cm16.5cm 22.5cm 6mm myheadingsaddtoresetequationsection0.3ex<-0.75em-1.1ex∼ 0.3ex>-0.75em-1.1ex∼ [ κ̨łλβ̱θdefinitiondefinitionDefinition[section]LPT Orsay 17-33Simultaneous Search for Extra Light and Heavy Higgs Bosons via Cascade Decays Ulrich Ellwanger^a and Matías Rodríguez-Vázquez^a^a Laboratoire de Physique Théorique, UMR 8627, CNRS, Université de Paris-Sud, Université Paris-Saclay, 91405 Orsay, France Models with extended Higgs sectors can contain several additional Higgs states, heavier or lighter than the SM Higgs boson. The couplings of lighter extra states to SM particles can be strongly reduced, leading to small cross sections for their direct production. Heavier extra states can have larger couplings to SM particles and, moreover, have large branching fractions into lighter extra states, notably into a SM-like Higgs boson accompagnied by another Higgs state which can be lighter or heavier than 125 GeV. Motivated by corresponding scenarios in the NMSSM we study the prospects for the discovery or exclusion of cascade decays ggF → H_3 → H_2 + H_1 in the bb̅bb̅, bb̅ττ and bb̅γγ final states where either H_1 or H_2 can be SM-like. Significant regions of the NMSSM parameter space can be tested by these searches. These are, however, not confined to models of the NMSSM type.§ INTRODUCTION Extended Higgs sectors are frequent properties of models beyond the Standard Model (BSM). Such extra states can have very small couplings to quarks, leptons and SM gauge fields. For instance, for singlets under the SM gauge symmetries such renormalizable couplings are disallowed by gauge invariance. The direct production cross sections for these states are then strongly suppressed in all channels. On the other hand, couplings of singlets to SU(2) Higgs doublets of the SM- or BSM-type are possible and typically present in BSM models. This allows for the discovery of such states in cascade decays of heavy BSM SU(2) Higgs doublets, provided the production cross sections of the latter are large enough.The final states after BSM-Higgs to BSM-Higgs + SM-Higgs cascades typically correspond to the ones in searches for resonant SM-Higgs (H_125) pair production: mainlybb̅bb̅, bb̅ττ and bb̅γγ. Corresponding searches have been performed at the LHC by ATLAS <cit.> and by CMS<cit.>. However, one of the SM-like Higgs bosons would now be replaced by a lighter or heavier BSM-Higgs boson. One can argue that the cross sections for such processes can be more promising than for resonant SM-Higgs pair production: a) A sizeable gluon-gluon-fusion (ggF) production cross section of a heavy scalar or pseudoscalar Φ, i.e. a sizeable coupling of Φ to top quarks, requires Φ to possess a sizeable SU(2)-doublet component. However, since H_125 is also a SU(2)-doublet, trilinear couplings Φ-H_125-H_125 (with Φ a pure doublet) violate the SU(2) symmetry and must be proportional to a SU(2) symmetry breaking vev; the latter is limited from above by the Z/W masses. This limits the possible partial width for Φ→ H_125+H_125, whereas the concurrent decay Φ→ tt̅ is always possible if Φ can be produced in ggF and is heavier than 2m_top.b) In the case ggF→Φ→ H_125 + H' with Φ a pure doublet, the trilinear coupling Φ-H_125-H' can be SU(2) invariant if H' is a singlet. In models with extended Higgs sectors including both an extra doublet and a singlet, such a coupling can thus be much larger than the Z/W masses leading to sizeable Φ→ H_125+H' partial widths.In Two-Higgs-Doublet-Models of type II such as the Minimal Supersymmetric Standard Model (MSSM) the production cross sections for extra CP-even (H) and CP-odd (A) Higgs doublets are not suppressed, and are dominated by ggF for tanβ not too large <cit.>. H or A can thus play the rôle of Φ above. The Next-to-Minimal Supersymmetric Standard Model (NMSSM) <cit.> contains additional CP-even (H_S) and CP-odd (A_S) singlet-like states with masses typically below M_H ∼ M_A. One finds that the BR(H→ H_S + H_125) and BR(A→ A_S+ H_125) can be up to ∼ 50%<cit.>, for the reasons given above and detailed in the next section.In the NMSSM this offers the possibility to produce otherwise practically invisible mostly singlet-like states H_S/A_S in cascade decays of H/A<cit.>. It is the aim of the present paper to study the prospects for discovery or exclusion of, simultaneously, H/A and H_S/A_S states in ggF → H → H_S + H_125 or ggF → A → A_S + H_125 in the final states bb̅bb̅, bb̅ττ and bb̅γγ. Supersymmetry plays no rôle here, accordingly our results are applicable to any models with similarly extended Higgs sectors; see, e.g., <cit.>.We will adopt various strategies from the searches for resonant SM Higgs pair production by ATLAS <cit.> and by CMS<cit.>. Moreover,for M_H_S near 125 GeV we can compare our backgrounds and expected 95% CL upper limits on the cross sections times branching fractions to the ones obtained in these publications.On the other hand, the analyses presented here are complicated by the fact that the masses M_H_S/M_A_S are not knowna priori. An important aspect of optimal search strategies are M_H_S/M_A_S dependent selection criteria (cuts) on events, hence different analyses should be performed, varying the assumptions on M_H_S/M_A_S. Only at the end of each of these analyses a search for a resonance-like bump in the total invariant mass of the H_S/A_S plus H_125 decay products, which should correspond to M_H/M_A, is proposed.In the next section we discuss shortly the Higgs sector of the NMSSM and the couplings relevant for the processes considered here. In section 3 we present features of our signal simulations. In section 4 we discuss the optimal search strategy for the bb̅bb̅ final state, and compare expected 95% CL upper limits and 5 σ discovery limits on the cross sections times branching fractions to the ones possible in the NMSSM. Sections 5 and 6 are devoted to the bb̅ττ and bb̅γγ final states. All these search strategies and results are identical for ggF → H → H_S + H_125 and ggF → A → A_S + H_125, for notational simplicity we will refer to H → H_S + H_125 only. In section 7 we conclude with a summary and an outlook. § THE NEUTRAL HIGGS SECTOR OF THE NMSSMIn this section we discuss briefly some properties of the Higgs sector of theCP-conserving ℤ_3-invariant NMSSM. It consists in two SU(2) doublets H_u, H_d and a complex singlet S. The superpotential of the Higgs sector reads W_Higgs=λŜĤ_u·Ĥ_d+κ^3/3Ŝ^3where λ and κ are dimensionless Yukawa couplings, and Ĥ_u, Ĥ_d and Ŝ denote chiral superfields. Once the real component of the superfield Ŝ develops a vacuum expectation value (vev) s, the first term in the superpotential generates an effective μ termμ= λ s.The vev v_u of H_u generates up-type quark masses, the vev v_d of H_d generates masses for down-type quarks and leptons, and both vevs contribute to the Z and W^± masses. Their ratio is tanβ = v_u/v_d.Decays of a heavy Higgs state into two lighter Higgs states occur in the presence of corresponding trilinear Higgs couplings. Most of the trilinear Higgs couplings in the ℤ_3-invariant NMSSM originate from quartic terms in the Higgs potential (see <cit.>) proportional to two powers of λ, κ or the electroweak gauge couplings, once the (neutral) Higgs fields are expanded around their vevs and decomposed into their real and imaginary parts:H^0_u= v_u + 1/√(2)(H^0_u,r + i H^0_u,i),H^0_d= v_d + 1/√(2)(H^0_d,r + i H^0_d,i),S=s+1/√(2)(S_r +i S_i).Hence the trilinear couplings are proportional to the vevs v_u, v_d or s. Whereas v_u, v_d are limited from above by M_Z^2 = g_1^2+g_2^2/2(v_u^2+v_d^2), a large vev s can generate a trilinear coupling ∼ H_u· H_d S. Another source for such a coupling is a trilinear Higgs-dependent soft SUSY breaking term λ A_λ H_u· H_d S + h.c.where the dimensionful parameter A_λ can be much larger than Higgs vevs. In order to obtain its impact on trilinear couplings among Higgs mass eigenstates, the mass matrices have to be diagonalized. In the CP-even sector, where one deals with a 3× 3 mass matrix, a first step in this direction is a rotation in the SU(2) doublet sector into the so-called Higgs basisH^0_u,r = sinβ h' - cosβ H', H^0_d,r = cosβ h' + sinβ H'where the vev of H' is zero, and the vev <h'>=√(v_u^2+v_d^2) is equal to the one of the SM Higgs boson. In fact, in most of the phenomenological acceptable regions of the parameter space of the NMSSM (near the alignment limit <cit.>) the eigenstates of the full 3× 3 CP-even mass matrix are not very different from h', H' and S_r, and will be denoted by H_125 (∼ h', approximately SM-like), H (∼ H', approximately MSSM-like) and H_S (∼ S_r, approximately singlet-like) in the following. The corresponding rotation of the imaginary components H^0_u,i and H^0_d,i (with β→ -β) diagonalizes their 2× 2 mass submatrix exactly and generates the Goldstone boson together with the MSSM-like pseudoscalar A'. The latter still mixes with the singlet-like S_i, but typically both differ little from the mass eigenstates A and A_S.Performing the rotation (<ref>) in (<ref>) and using the previous approximations in the CP-even and CP-odd sectors, one obtains the trilinear couplings λ A_λ/√(2)(tan^2β-1/tan^2β+1H_125(H H_S - A A_S) + …)where the omitted terms are suppressed by tanβ. Hence, for not too small tanβ→ 1, trilinear couplings g_H_125 H H_S and g_H_125 A A_S are generated which have no analog in the MSSM, and are larger than all other trilinear Higgs couplings if λ A_λ is large.On the other hand the masses M_H/A of the nearly degenerate mostly MSSM-like states H/A are approximatively given byM_H/A^2 ∼μ(A_λ+κ/λμ) 1+tan^2β/tanβ ,which limits A_λ from above for fixed M_H/A, tanβ, small |κ/λ| and |μ|100 GeV (as required by the lower LEP bound on higgsino-like charginos). Accordingly the trilinear couplings g_H_125 H H_S and g_H_125 A A_S can be larger for larger M_H/A^2.The production cross sections for the mostlyMSSM-like states H/A is dominated by ggF<cit.>; at √(s)=13-14 TeV and for tanβ∼ 2-3 (typical in the NMSSM) they are O(1 pb) up to M_H/A∼ 600 GeV. The trilinear couplingsg_H_125 H H_S and g_H_125 A A_S induce the decays H → H_125+ H_S and A → H_125+ A_S if kinematically allowed. The branching fractions BR(H/A → H_125+ H_S/A_S) can be as large as ∼ 50%, in contrast to the decay H → H_125 +H_125.The singlet-like states Φ_S = H_S/A_Shave small couplings to quarks, leptons and gauge fields induced by mixings with h', H' and A'. Hence the production cross sections for Φ_S are typically small, and their discovery may have to rely on H/A → H_125+ Φ_S decays. Via the couplings induced by mixing, Φ_S can decay into the same channels as H_125 and H/A. For M_Φ_S > 2m_top, decays into tt̅ are dominant, whereas decays Φ_S→ bb̅ dominate typically for M_Φ_S < 2m_top. (The branching ratio for H_S→ W^+ W^- can also be sizeable <cit.>.) For M_H_S > 250 GeV, decays H_S→ H_125+ H_125 are possible, leading to double-resonant tri-Higgs production (not considered here). Decays Φ_S→τ^+ +τ^- are practically always possible. In the regions in the NMSSM parameter space with all present constraints on the signal rates of H_125 being satisfied the BR(Φ_S→γ + γ) is in the 0.1-0.3% range, making this decay observable as well. Henceforth we will consider resonant bb̅bb̅, bb̅τ^+ τ^- and bb̅γγ final states originating from Φ_S→ bb̅, Φ_S→τ^+ τ^- and Φ_S→γγ decays.Of interest will be the product of cross sections times branching fractions σ(ggF→ H/A)× BR(H/A → H_125+ H_S/A_S→ bb̅bb̅,bb̅τ^+ τ^- and bb̅γγ) for various masses M_H, M_H_S and M_A_S, for realistic regions in the parameter space of the NMSSM. To this end we have performed scansusing the public code <cit.> including the radiative corrections from <cit.>. All phenomenological constraints, including the absence of Landau singularities below the GUT scale and, notably, constraints from Higgs searches in various channels at LEP and LHC are applied. These include searches for scalar and pseudoscalar Higgs production at LEP (including unconventional Higgs decays), constraints from B-physics, constraints on the mass of H_125 (± 3 GeV to account for theoretical uncertainties) and on its signal rates from the combined ATLAS and CMS run I resultswhich disallow too large H_125-H_S mixings, constraints from searches for ggF→ H_S→γγ for M_H_S=65-122 GeV,and searches for H/A in the H/A→ττ channel with H/A produced in association with b-quarks and via ggF. We note that constraints from these latter searches in the M_A-tanβ plane are weak for tanβ≈ 2-3, typical in the NMSSM, and further alleviated if H/A have large branching fractions into the final states considered here.The results of these scans for σ(ggF→ H/A)× BR(H/A → H_125+ H_S/A_S→ bb̅bb̅,bb̅τ^+ τ^- and bb̅γγ) will be compared to the sensitivities in different final states in the following sections. The ggF production cross sections for H/A have been obtained from the CERN Yellow Report web page <cit.> at NNLO+NNLL, after an appropriate rescaling of the H/A-gluon-gluon coupling provided by . Also the branching ratios of H/A and H_S/A_S are taken from . In the Figures showing the 95% CL exclusion limits and 5 σ discovery cross sections, viable values for the cross sections times branching fractions in the parameter space of the NMSSM will be indicated as light shaded blue regions for ggF→ H → H_125+ H_S, and as light shaded red regions for ggF→ A → H_125+ A_S. For simplicity we will use the notation ggF→ H → H_125+ H_S for the search strategies; the same search strategies apply to ggF→ A → H_125+ A_S.§ SIMULATION OF SIGNAL SAMPLES Signal events for the production of H in ggF are generated by <cit.> with matrix elements at NLO taken from <cit.> using the NNPDF2.3NLO PDF set <cit.> in the 4-flavour scheme. Renormalization and factorization scales are chosen as H_T/2 on an event-wise basis. <cit.> is used for the H → H_125+ H_S decays, the H_125 and H_S decays and the subsequent showering and hadronization. The total widths of H are below M_H/50 in all cases, below M_H/100 for M_H < 500 GeV, hence the narrow width approximation is well satisfied.Separate signal events have been generated for each pair (M_H, M_H_S). For M_H we chose M_H=425, 500, 625, 750 and 1000 GeV. We varied M_H_S in steps of 10 GeV in the range 25-225 GeV, and in steps of 20 GeV above 225 GeV up to the kinematic boundary M_H-125 GeV (except for M_H=1000 GeV where M_H_S was varied in steps of 25/50 GeV).For each pair (M_H, M_H_S) we generated 150k unweighted events, more than the expected number of events at 3000 fb^-1. Accordingly the statistical fluctuations from the Monte Carlo (MC) generation are small compared to the expected statistical fluctuations from the data; the latter will be taken into account.The output is given to the detector simulation <cit.>. Jets are clustered with <cit.> using the anti-k_T algorithm with Δ R = 0.4. For b-tagging the ATLAS card is used in . The p_T dependence of the b-tagging and mistagging efficiencies is chosen in the ATLAS card following the parametrizations given in <cit.>. The default value of the parameters incorrespond to a working point ε_b = 70% as, e.g., in the ATLAS search for Higgs pair production in the 4b final state at 13 TeV in <cit.>. We will employ the same settings except for the bb̅γγ final state.§ SEARCH STRATEGY FOR THE BB̅BB̅ FINAL STATE Searches for resonant SM Higgs pair production in the bb̅bb̅ final state have been performed before by ATLAS at 8 TeV <cit.> and at 13 TeV <cit.>, and by CMS at 8 TeV <cit.> and at 13 TeV <cit.>.Searches for ggF → H → H_S + H_125→ bb̅bb̅ are complicated by the presence of two unknown masses of H and H_S. A naive approach would be to require one bb̅ pair with a mass near 125 GeV, and to look for simultaneous excesses in the plane of invariant masses of the other bb̅ pair and the total 4b invariant mass. However, this approach does not allow to optimize cuts as function of different masses of H and H_S. An at least ∼ 20% gain in efficiency can be obtained as follows: a) Choose a tentative value for M_H_S, and optimise the cuts and the choice of bb̅ pairs as function of this value; b) Search subsequently for an excess in the total 4b invariant mass (suitably corrected, see below).Subsequently we describe first our simulation of signal samples andthe strategies for the analysis. In the following subsections we discuss the background simulation and validation, and finally the results for the expected 95% CL upper limits and 5 σ discovery limits on the cross sections times branching fractions as function of M_H and M_H_S. The latter are compared to possible production cross sections times branching fractions in the NMSSM. §.§ Analyses of Signal Samples After the simulation of signal samples as described in section <ref>, at least four b-tagged jets with p_T > 30 GeV and |η| < 2.5 are required. Four b-tagged jets can be paired in six different ways. The two invariant masses of two bb̅ pairs are tentatively denoted by M_bb̅(H_125) and M_bb̅(H_S). The subsequent procedure depends on the chosen value for M_H_S, and has to be repeated for each choice.An event is kept only if a pairing exists for which M_bb̅(H_125) is sufficiently close to 125 GeV, and M_bb̅(H_S) is sufficiently close to M_H_S. In practice, the measured invariant masses of bb̅ pairs are often somewhat smaller than the mass of the decaying Higgs boson. Therefore “sufficiently close to” should better be replaced by “slightly below” as in <cit.>. Generalizing the conditions applied in <cit.>, an event is kept only if a pairing exists for whichχ = √(( M_bb̅(H_125)-115 GeV/0.1M_bb̅(H_125))^2 + (M_bb̅(H_S)-0.85M_H_S/0.1M_bb̅(H_S))^2) < 2.If different pairings within a given event satisfy (<ref>), the combination that minimizes χ is chosen. In Figs. <ref> we show, for M_H=500 GeV, the distributions of the dijet masses M_bb̅(H_125) and M_bb̅(H_S) for the pairing minimizing χ, for two different benchmark points M_H_S=85 GeV and M_H_S=340 GeV in the case where M_H_S for the analysis was chosen correctly. The black contours indicate the signal regions defined by χ < 2. [An alternative pairing algorithm based on the angular distances of the constituent b-jets was studied. We found that the resulting sensitivities are mostly similar to those obtained using the mass-based algorithm described above when the jets are sufficiently collimated. However, as M_H_S increases, the jets become too back-to-back, making the angular-pairing contraproductive. For this reason, we used the mass-based algorithm for the reconstruction of the Higgs candidates.]In the case of heavy resonances with masses above ∼ 1 TeV, the two b-jets from a single Higgs boson tend to merge into a single fat jet. Accordingly a “boosted” analysis based on single Δ R=1.0 jets per Higgs boson was applied in <cit.> for searches for such heavy resonances. We found that for such heavy H states the production cross sections become too small for reasonable sensitivities, and limit ourselves to M_H ≤ 1 TeV subsequently. We had tried to vary the jet reconstruction parameter Δ R without observing, however, a major impact on the possible sensitivities. The next step in the event selection are cuts on p_T of the bb̅ pairs associated with H_125 and H_S, respectively. These cuts can be optimised with respect to M_4b (defined by the event) and the tentative value for M_H_S. To this end we considered many samples of M_H∼ M_4b and M_H_S. In each case we studied the dependence of the sensitivity ϵ_s/√(ϵ_B) (ϵ_s and ϵ_B denote the efficiencies of the signal and background, respectively) on the cuts on p_T. Maximizing the sensitivities, we obtain different values for the optimal cuts on p_T for each sample of M_4b and M_H_S. These different values for the cuts on p_T are well approximated by the following functions of M_4b and M_H_S:p_T(bb̅(H_125))>1.6 GeV + 0.4M_4b-0.13M_H_S - M_H_S/M_4b· 160 GeVp_T(bb̅(H_S))>12 GeV + 0.4M_4b-0.15M_H_S - M_H_S/M_4b· 166 GeVThe efficiencies of the three cuts a) 4b with p_T > 30 GeV, b) signal region (<ref>) and c) dijet p_T (<ref>) are shown in Figs. <ref> for M_H_S=75 GeV and M_H_S=325 GeV as function of M_H, and in Figs. <ref> for M_H=625 GeV and M_H=1000 GeV as function of M_H_S. One observes a decrease of the efficiency of the condition of four b-tagged jets for M_H_S 80 GeV (for M_H=625 GeV) and M_H_S 100 GeV (for M_H=1000 GeV). Here the two b-jets from H_S are boosted and hardly get resolved by the standard jet clustering algorithm; dedicated boosted analyses as in <cit.> could be envoked for this configuration. For the search for a resonance H we found it useful to replace M_4b by M_X, with M_X defined such that uncertainties in the measurements of M_bb̅ relative to M_H_125 and M_H_S (typically due to radiation out of the jet cones) are corrected:M_X = M_4b+125 GeV-M_bb̅(H_125) + M_H_S-M_bb̅(H_S).The empirical variable M_X was already used by CMS in resonant double Higgs production search in the bb̅γγ channel <cit.>. We found that replacing M_X by a full Lorentz covariant expression in terms of M_H_125, M_H_S and the two values of M_bb̅ did not improve the sensitivities.In Fig. <ref> we show the distributions of the reconstructed masses M_4b and M_X for the signal samples (M_H,M_H_S)=(350,195), (500,310), (750,450), (1000,600) GeV. We clearly see a sharpening of the peaks using M_X in each case. In order to obtain the required production cross sections times branching fractions for 5 σ discovery or 95% CL exclusion we have to obtain the background distribution of M_X. §.§ Background Estimation Dominant backgrounds for the bb̅bb̅ final state are QCD multijet processes including jet misidentifications, and tt̅<cit.>. The QCD multijet background is difficult to obtain from Monte Carlo simulations alone, and estimated from sidebands in <cit.>. Such data is not available for the different values of M_H_S studied here, with one exception: For M_H_S∼ 125 GeV, the bb̅bb̅ final state coincides with the one searched for in <cit.>.We simulated bb̅bb̅, bb̅cc̅, bb̅jj (with j≠ c/c̅) and tt̅ processes as we did for the signal samples in section <ref>. After applying the cuts of the previous subsection the relative contributions are ∼ 85-88% from bb̅bb̅, ∼ 7-8% from bb̅cc̅,∼ 4-8% from tt̅ (depending on M_H_S and M_4b). The bb̅jj contribution is only 0.8% and will subsequently be neglected. We checked that bb̅bb̅+jets processes have little impact on the event shapes. Hence we did not simulate them separately, but took a NLO K-factor of 1.7 <cit.> into account.In <cit.> the M_4b distribution of the multijet background has been obtained from a signal free sideband (with two b-tags only, andM_bb̅ outside the search window) and appropriate rescaling using data with four b-tags, subtracting the tt̅ contribution. The measured M_4b distribution is available in Fig. 5 in <cit.>, where it is compared to the estimated background.This allows us to proceed in a similar fashion:In order to compare to the data in <cit.>, the previous cuts are slightly modified: The bb̅ pairs are ordered in p_T according to M_bb̅^lead and M_bb̅^subl and, as in <cit.>, the signal region is defined by χ = √(( M_bb̅^lead-120 GeV/0.1M_bb̅^lead)^2 + (M_bb̅^subl-0.85M_H_S/0.1M_bb̅^subl)^2) < 1.6. Still (and expectedly) our simulated background falls below the measured data given in<cit.>. On the left hand side of Fig. <ref> we show the M_4b distribution measured by <cit.> using 10.1 fb^-1 of integrated luminosity, and our MC result with statistical errors. The number of generated MC events corresponds to an equivalent integrated luminosity of ∼13 fb^-1. The statistical error per bin is thus obtained from the number of MC events per bin rescaled by 10/13. The lower panels show the ratio MC/data bin by bin, with the uncertainties from the data and from our MC combined. At least for the interesting region M_4b 350 GeV an overall rescaling of our multijet background, like it was performed in <cit.>, seems appropriate. The rescaling factor is obtained taking the average of the data/MC ratio of all bins weighted by the corresponding uncertainties. We obtain a rescaling factor of 1.55 ± 0.27. (The tt̅ background is left untouched, and remains at ∼ 3-6%.) The comparison of the M_4b distribution of our background after rescaling to the data from <cit.> is shown on the right hand side of Fig. <ref>.Clearly it is somewhat optimistic to assume that the rescaling of the multijet background by 1.55 ± 0.27 remains valid for M_H_S≠ 125 GeV. In the absence of data from sidebands this is, however, the best we can do. Subsequently ±0.27 will be used as an estimation of the systematic uncertainty of our background for all M_H_S, a number to be considered as indicative.For forecasts at 300 or 3000 fb^-1 integrated luminosity the statistical uncertainties of the background are much smaller. It is then convenient to fit the shape of the M_X background distributions (<ref>) after cuts, which will be used in the following, by continuous functions. We found that the best fits are provided by a four parameter Gamma distribution defined in eq. (<ref>) with M_H_S dependent fit parameters. (The bb̅bb̅, bb̅cc̅, and tt̅ background contributions to the M_X distributions were fitted separately.) In Figs. <ref> we show the sum of these fits for M_H_S=85 GeV and M_H_S=350 GeV.Of course the remaining statistical fluctuations of the background can still be evaluated and combined with the systematic uncertainty (not shown in Figs. <ref>). It turns out, however, that for forecasts at 300 or 3000 fb^-1 the statistical fluctuations are negligibly small relative to the systematic uncertainty from the rescaling by 1.55 ± 0.27. §.§ Future 95% CL Exclusion Limits and 5 σ Discovery Cross Sections Given the M_X distribution of the background for various hypothetical values of M_H_S and the M_X distributions of signals one can, following the statistical methods from <cit.> and described in the appendix B, obtain values for 95% CL exclusion limits and 5 σ discovery limits for cross sections times branching fractions into the bb̅bb̅ final state as function of the integrated luminosity, M_H and M_H_S.In the case of an integrated luminosity of 13.3 fb^-1 at 13 TeV we can compare the expected 95% CL exclusion limits on cross sections times branching fractions to the ones given by ATLAS in Fig. 11 in <cit.>, for M_X = 300… 1000 GeV and M_H_S∼ 125 GeV. (This ATLAS search was actually dedicated to spin 2 resonances decaying to SM Higgs pairs, but the differences to spin-0 resonances are expected to be small.) In Fig. <ref> we show the expected 95% CL upper limits from ATLAS, their ± 2 σ incertainty bands, the expected 95% CL upper limits from our Monte Carlo and, for completeness, the 95% CL upper limits obtained from the data. We see that our expected 95% CL upper limits coincide well with the ones expected by ATLAS.Since the background was fitted to data at 13 TeV c.m. energy we will show our results also for 13 TeV, for 300 and 3000 fb^-1 integrated luminosity. We choose four representative values for M_H=425, 500, 750 and 1000 GeV, and show the 95% CL exclusion limits and 5 σ discovery cross sections as function of M_H_S in each case. For 300 fb^-1 integrated luminosity these are shown in Figs. <ref>, for 3000 fb^-1 integrated luminosity in Figs. <ref>.The expected limits become weaker for M_H_S 50 GeV (for M_X=425-500 GeV) and M_H_S 100 GeV (for M_H=1000 GeV). As stated in subsection <ref> here the bb̅ pair from H_S becomes too boosted and is no longer resolved by the standard jet clustering algorithm.The shaded blue regions in Figs. <ref> and <ref> indicate viable values for the cross sections times branching fractions for σ(ggF→ H→ H_125+H_S→ bb̅bb̅) in the parameter space of the NMSSM, see section <ref>. Typically the viable values for σ(ggF→ A→ H_125+A_S→ bb̅bb̅) are smaller; if not we show them as shaded red regions. In the region of the NMSSM parameter space corresponding to M_H500 GeV, the partial width for H_S→ H_125+H_125 becomes relatively large (≈ 10 MeV) if kinematically allowed. As a consequence the branching fractions of H_S into bb̅ (and the other channels considered in this paper) decrease, leading to a decrease of the possible production cross sections times branching fractions for M_H500 GeV, M_H_S 250 GeV. The following conclusions can be drawn from Figs. <ref> and <ref>: For M_H 500 GeV wide ranges of M_H_S in the NMSSM parameter space can be discovered or, at least, excluded. For larger M_H testable regions in the NMSSM parameter space exist, but for M_H∼ 1 TeV only for 3000 fb^-1 integrated luminosity. In fig. <ref>we summarize these results showing the 95% C.L. expected upper limits in the M_H vs M_H_S plane. We recall, however, that the sensitivities to cross sections in Figs. <ref> and <ref> are model independent and valid for arbitrary (e.g. non-supersymmetric) extensions of the Higgs sector.§ SEARCH STRATEGIES FOR THE BB̅ΤΤ FINAL STATE Searches for resonant H_125 pair production in the bb̅ττ final state have been performed by ATLAS at 8 TeV <cit.>, and by CMS at 13 TeV in <cit.>. Following these searches we concentrate on the τ_hτ_h, τ_hτ_e and τ_hτ_μ modes. As in the case of the bb̅bb̅ final state we optimise the cuts as function of a tentative value for M_H_S.A priori the ττ pair can originate from H_S or H_125; both cases will be studied below. For the analysis we will make no assumptions on the relative branching ratios BR(H_S→ bb̅) and BR(H_S→ττ). The aim is to obtain separate 95% CL exclusion limits and 5 σ discovery cross sections for the processes ggF→ H→ H_S(→ bb̅)+H_125(→ττ), and ggF→ H→ H_S(→ττ)+H_125(→ bb̅).§.§ Analyses of Signal Samples For the simulation of signal samples the same series of codes as for the bb̅bb̅ final state was used, see section <ref>. Events are required to have exactly two b-tagged jets with p_T(b) > 30 GeV and |η|<2.5. For the b-tagging efficiency a working point with ε_b = 70% is chosen. If the event has exactly two hadronic taus τ_h, both are required to have p_T(τ_h) > 45 GeV. Events with one hadronic tau are required to have exactly one additional isolated lepton ℓ = e,μ of opposite charge and withp_T(ℓ) > 20 GeV. For the transverse mass m_T^ℓ of leptons we requirem_T^ℓ≡√(2p_T(ℓ) E_T^miss(1-cos(ϕ(E_T^miss) -ϕ(ℓ)))) < 40 GeV.All objects are required to have |η|<2.47. In the case of τ_h τ_ℓ final states the invariant mass M_ττ was reconstructed using the collinear mass, i.e. assuming that the neutrino from the τ_ℓ decay is emitted collinear to the lepton and responsible for all E_T^miss. Considering first the case where the ττ pair originates from H_S with an assumed mass M_H_S (and hence that M_bb̅ should be close to 125 GeV), a M_H_S dependent signal region is defined by χ = √(( M_bb̅-110 GeV/0.35· M_bb̅)^2 + (M_ττ-0.92· M_H_S/Max(0.35· M_ττ, 30 GeV))^2) < 1. If different pairings within a given event satisfy (<ref>), the combination that minimizes χ is chosen. In Figs. <ref> we show, for M_H=500 GeV, the distributions of the dijet mass M_bb̅(H_125) and M_ττ(H_S) for the pairing minimizing χ, for two different benchmark points M_H_S=85 GeV and M_H_S=340 GeV in the case where M_H_S for the analysis was chosen correctly. The black contours indicate the signal regions defined by χ < 1.Cuts on the transverse momenta of Higgs candidates are chosen asp_T(bb̅)>52 GeV + 0.14M_X-0.2M_H_S - M_H_S/M_X· 202 GeVp_T(ττ)>24 GeV + 0.19M_X-0.02M_H_S - M_H_S/M_X· 128 GeV withM_X = M_bb̅ττ + 125 GeV - M_bb̅.The numerical coefficients in eqs. (<ref>)–(<ref>) were obtained by optimizing the relative signal to background efficiency.In the case where the bb̅ pair originates from H_S with an assumed mass M_H_S (and hence that M_ττ should be close to 125 GeV), the signal region (<ref>) is replaced by χ = √(( M_ττ-120 GeV/0.35· 120 GeV)^2 + (M_bb̅-0.85· M_H_S/Max(0.35· M_H_S, 35 GeV))^2) < 1and the cuts on the transverse momenta of Higgs candidates arep_T(ττ)>118 GeV + 0.02M_X-0.55M_H_S - M_H_S/M_X· 380 GeVp_T(bb̅)>16 GeV + 0.19M_X-0.02M_H_S - M_H_S/M_X· 137 GeV withM_X = M_bb̅ττ + M_H_S - M_bb̅.Hence, for each tentative value of M_H_S two different analyses using different cuts are to be performed, resulting in two (slightly) different distributions of M_X. §.§ Background Estimation Backgrounds originate from tt̅ (and single top) and QCD+electroweak bb̅ττ production. Contributions from one or more jets misidentified as τ_h are seen to become small after the cuts on p_T(ττ). We have generated 1.5· 10^7tt̅ events using<cit.>; the LO cross section was rescaled by a (NNLO+NNLL) K-factor 1.7 obtained from <cit.>.was also used to generate QCD+electroweak bb̅ττ events; the LO cross section was rescaled by a NLO K-factor 2.9.After applying the cuts of the previous subsection, the relative contributions of the SM backgrounds depend on M_H_S and on whether the ττ pair originates from H_S or H_125, although tt̅ is always dominant: For H_S→ττ the tt̅ contribution increases from ∼ 60% for M_H_S∼ 50 GeV to ∼ 100% for M_H_S 350 GeV, the remaining background stems from QCD+electroweak bb̅ττ production. For H_125→ττ the tt̅ contribution is always ∼ 90%.In order to validate the background contribution to the M_X distribution after cuts we use again a search for H_125 pair production at 13 TeV, now in the bb̅ττ channel. Measurements of distributions of the (slightly corrected) total invariant mass by CMS, separately in the bb̅eτ_h, bb̅μτ_h and bb̅τ_hτ_h channels, can be found in Fig. 1 in <cit.>.We have reproduced the cuts in <cit.> using our background samples. In Figs. <ref> we show the measured total invariant mass distribution from Fig. 1 in <cit.> in black, and our MC results including the statistical uncertainties corresponding to 12.9 fb^-1 of integrated luminosity, in the three channels, in orange. Due to the smaller number of events the statistical uncertainties are now larger than in the 4b case.Still we can ask which rescaling of our simulated background, independent of the total invariant mass and common to all three channels (to improve the statistics), provides a best fit to the data. We find a factor 1.01 ± 0.24, and will subsequently use ± 0.24 as an estimate of the systematic uncertainty of the background normalisation.For forecasts at 300 or 3000 fb^-1 integrated luminosity the shape of the M_X background distributions (<ref>) will again be parametrized by continuous functions with M_H_S dependent parameters: For the tt̅ background the Frechet distribution, and for the bb̅ττ background (all channels combined) the GaussExp function already used in <cit.>. Both functions are defined in the appendix A. In Figs. <ref> we show these fits for M_H_S=85 GeV and M_H_S=350 GeV where the ττ pair originates from H_S. §.§ Future 95% CL Exclusion Limits and 5 σ Discovery Cross Sections Given the M_X distribution of the background for various hypothetical values of M_H_S and the M_X distributions of signals we can, as before, obtain values for 95% CL exclusion and 5 σ discovery for cross sections times branching fractions into the H_125→ bb̅, H_S→ττ and H_125→ττ, H_S→ bb̅ final states as function of the integrated luminosity, M_H and M_H_S. We choose four representative values for M_H=425, 500, 750 and 1000 GeV, and show the 95% CL exclusion limits and 5 σ discovery cross sections as function of M_H_S in each case. For H_125→ bb̅, H_S→ττ at 300 fb^-1 integrated luminosity these are shown in Figs. <ref>, for 3000 fb^-1 integrated luminosity in Figs. <ref>. For H_125→ττ, H_S→ bb̅ at 300 fb^-1 integrated luminosity these are shown in Figs. <ref>, for 3000 fb^-1 integrated luminosity in Figs. <ref>. The uncertainties include statistical uncertainties and, added linearly, ± 0.24 considered as an estimate of the systematic uncertainty originating from the normalisation of the background.The following observations can be made: First, the expected sensitivities on cross sections times branching ratios differ hardly among the cases H_125→ bb̅ and H_S→ττ versus H_125→ττ and H_S→ bb̅; if at all, the analyses aiming at H_125→ bb̅ and H_S→ττ are typically somewhat more sensitive.Second, in Two-Higgs-Doublet models of type II as well as in the NMSSM the branching fractions into bb̅ and ττ of both H_125 and H_S are always related by a factor ∼ 9:1. Accordingly the possiblecross sections times branching fractions in the NMSSM parameter space for both σ(ggF→ H→ H_125+H_S→ bb̅ττ) and σ(ggF→ H→ H_125+H_S→ττ bb̅), indicated in blue inFigs. <ref>–<ref>, are ∼ 1/9 of the ones in Figs. <ref>–<ref> for the bb̅bb̅ final state. (The same reasoning applies to σ(ggF→ A→ H_125+A_S→ bb̅ττ) and σ(ggF→ A→ H_125+A_S→ττ bb̅); the viable NMSSM points correspond to the ones in Figs. <ref> and <ref>.)Then one can ask, for a given point in parameter space, which of the analyses considered up to now is the most sensitive. According to our results this is the search in the bb̅bb̅ final state which allows to test somewhat larger regions in parameter space.§ SEARCH STRATEGIES FOR THE BB̅ΓΓ FINAL STATE Searches for resonant H_125 pair production in the bb̅γγ final state have been performed by ATLAS at 8 TeV <cit.> and at 13 TeV <cit.>, by CMS at 8 TeV in <cit.> and at 13 TeV in <cit.>. A priori the diphotons can originate from H_S or H_125; both cases will be studied below. As in case of the previous final states we optimise the cuts as function of a tentative value for M_H_S. §.§ Analyses of Signal Samples For the simulation of signal samples we used again <cit.>. Events are required to have exactly two b-tagged jets with p_T(b) > 40 GeV and |η|<2.5. Following ATLAS <cit.> a working point with ϵ_b=0.85 was chosen for the b-tagging efficiency in order to increase the statistics.At least two photons are required in each event which have to satisfy the isolation criteria ∑_i p_T,i/p_T,γ < 0.1where the sum over i includes all tracks within a cone Δ R = 0.4 around the photon. The two leading photons are required to satisfyE_T^lead/M_γγ>0.35,E_T^subl/M_γγ>0.25,|η|<2.37. Additional cuts depend on whether the diphoton pair is assumed to originate from H_125 or H_S, and the assumed value of M_H_S. First we consider the case H_S→γγ. Thenthe bb̅ pair is assumed to originate from H_125, and we require100 GeV < M_bb̅ < 150 GeV.As in the previous cases it is useful to define a corrected invariant mass M_X for the bb̅γγ system:M_X=M_bb̅γγ+125 GeV - M_bb̅. Given an assumed value for M_H_S it turned out to optimize the signal to background efficiency applying a M_X and M_H_S dependent cut on the diphoton invariant mass M_γγ, since the measured distribution |M_γγ - M_H_S| broadens somewhat with M_X:|M_γγ - M_H_S| < 4.3 GeV + 0.016 M_X.Also the cuts on the H_125 and H_S candidates depend on M_X:p_T(bb̅)>17 GeV + 0.18 M_X,E_T(γγ)=68.3 GeV + 0.25 M_X.(E_T(γγ) instead of p_T(γγ) allows for M_H_S independent cuts.)Assuming H_125→γγ and H_S→ bb̅ the previous cuts are modified as follows: First, for M_X we takeM_X=M_bb̅γγ+ M_H_S - M_bb̅,and for the diphoton invariant mass we require|M_γγ - 125 GeV| < 2 GeV + 0.02 M_X.The mass window for M_bb̅ is now0.9 M_H_S - 30 GeV < M_bb̅ < 0.9 M_H_S + 20 GeV.The cuts on the H_125 and H_S candidates are:p_T(bb̅)>-5.7 GeV + 0.29 M_X,E_T(γγ)=7.45 GeV + 0.33 M_X.All the numerical values above have been obtained by optimising signal-to-background ratios using the MC events. §.§ Background Estimation SM backgrounds originate from bb̅γγ, cc̅γγ, jjγγ (j≠ c/c̅), bb̅jγ and tt̅H. We simulated these backgrounds again using . Due to the relatively large b-tagging efficiency ϵ=0.85 mistagging rates are relatively large. bb̅jγ contribute if another fake photon appears.After applying the cuts of the previous subsection the relative contributions of the SM backgrounds depend strongly on M_H_S and M_H, and on whether the γγ pair originates from H_S or H_125. For a light H_S∼ 85 GeV the background from bb̅jγ is always important, ∼ 60% for H_S→γγ and ∼ 30% for H_S→ bb̅. However, all other γγ + X SM backgrounds can also contribute several 10% individually. The contribution from tt̅H is always 3%.For bb̅γγ, cc̅γγ and jjγγ an extra jet was allowed in the final state, bb̅jγ was multiplied by a NLO K factor 1.58 from <cit.>, and tt̅H (with SM couplings) was simulated at NLO.Still we cannot expect that the total SM background cross section is completely captured by the MC simulation; in searches by ATLAS and CMS in <cit.> sidebands are used for this purpose. Thus we proceed as before and correct the total background cross section using data driven methods in the particular case M_H_S∼ 125 GeV equivalent to (resonant) SM Higgs pair production. To this end we modify slightly the cuts on p_T(b) (> 55 GeV and > 35 GeV for the leading and next-to-leading b-jets), M_bb̅ and M_γγ such that they coincide with the ones in the ATLAS search<cit.>.(In the case of M_γγ we could check that the efficiency coincides.) Extrapolating from sidebands with less b-jets, ATLAS <cit.> obtained 1.63± 0.3 expected background events in the signal region, whereas we found 1.06± 0.14 events from summing all MC events. Since all our backgrounds were simulated to similar order (NLO) in the QCD coupling and separate higher order K factors are not available, we multiply their sum by 1.54± 0.35 where the latter uncertainty will again be treated as an estimate of the systematic uncertainty contributing to our final results.The dependence of the background on the total invariant mass M_bb̅γγ was parametrized in <cit.> by a two parameter Landau distribution given in appendix A. We found that the Landau distribution fits the M_X distribution from eq. (<ref>) as well (with M_H_S dependent parameters), and used it for the expected M_X distributions of the various background contributions for our forecasts at 300 fb^-1 and 3000 fb^-1.§.§ Future 95% CL Exclusion Limits and 5 σ Discovery Cross Sections Given the M_X distribution of the background for various hypothetical values of M_H_S and the M_X distributions of signals we can, as before, obtain values for 95% CL exclusion and 5 σ discovery for cross sections times branching fractions into the H_125→ bb̅, H_S→γγ and H_125→γγ, H_S→ bb̅ final states as function of the integrated luminosity, M_H and M_H_S.After completing our analysis the CMS search <cit.> for resonant Higgs pair production in the channel H_125+H_125→ bb̅γγ based on 35.9 fb^-1 appeared. The expected 95% CL exclusion limits given in <cit.> can be compared to ours for M_H_S=125 GeVfor the same integrated luminosity; this comparison as function of M_H is shown in Fig. <ref>. The expected limits coincide within 1 σ for M_H 500 GeV, and within 2 σ everywhere. Our expected limits are systematically more conservative; we note that the CMS analysis employs a trained boosted decision tree in order to separate the signal from the backgrounds which is not available here.Our expected 95% CL exclusion limits and 5 σ discovery cross sections for H_125→ bb̅, H_S→γγ as function of M_H_S at 300 fb^-1 integrated luminosity are shown in Figs. <ref> for four representative values for M_H=425, 500, 625 and 750 GeV, and for 3000 fb^-1 integrated luminosity in Figs. <ref>. For H_125→γγ, H_S→ bb̅ at 300 fb^-1 integrated luminosity these are shown in Figs. <ref>, for 3000 fb^-1 integrated luminosity in Figs. <ref>. As in the bb̅bb̅ case, we also present the results for the expected 95% C.L. upper limits in the M_H_125 vs M_H_S plane in fig. <ref>. The uncertainties include statistical uncertainties and, added linearly, a factor ± 0.35 considered as systematic uncertainty originating from the normalisation of the background. As before viable NMSSM regions for scalar production are shown in shaded blue, for pseudoscalar production in shaded red in case they potentially exceed the ones for scalar production. Again a sizeable region in the NMSSM parameter space can be tested in this final state provided M_H is not too large. It is, however, not the same region potentially visible in the bb̅bb̅ final state: The branching fraction of H_S into γγ can vary in the 0.2%± 0.1% range, and is anticorrelated with its branching fraction into bb̅. Moreover, as it is visible from the shaded red regions in Figs. <ref> and <ref>, the signal rates σ(ggF→ A→ H_125+A_S→ bb̅γγ) can be relatively large. These correspond to very singlet-like pseudoscalars A_S with very suppressed couplings to quarks and leptons, but sizeable coupling ∼λ to higgsinos. Then the charged higgsino-loop induced coupling to diphotons can dominate, leading to a large BR(A_S→γγ). The coupling A-A_S-H_125 is not suppressed in this case, leading to potentially large signal rates.On the experimental side, the comparison of the upper limits onH_125→ bb̅ and H_S/A_S→γγ versus H_125→γγ and H_S/A_S→ bb̅ has a simple answer depending on M_H_S/A_S: For M_H_S/A_S<125 GeV the search forH_S/A_S→γγ is sensitive to smaller signal rates, whereas for M_H_S/A_S>125 GeV the search for H_125→γγ, H_S/A_S→ bb̅ is typically sensitive to smaller signal rates. However, different regions in the parameter space of underlying models are tested by these searches.§ CONCLUSIONS AND OUTLOOK Searches for resonant SM Higgs pair production are performed with considerable effort by ATLAS and CMS. As explained in the introduction searches for ggF→Φ→ H_1 + H_2 can be more promising where either H_1 or H_2 can be SM-like, and the other state being possibly CP-odd (which does not affect the search methods). This scenario is manifest in the NMSSM where the rôle of Φ is played by the MSSM-like heavy doublet, but the argument is more general. In the present paper we have studied the prospects for corresponding searches in the bb̅bb̅, bb̅ττ and bb̅γγ final states, including SM backgrounds. We found that significant regions in the NMSSM parameter space can be tested by these searches:The NMSSM specific parameters testable by bb̅bb̅ are typically in the region λ∼ 0.50 - 0.70 (the conservative upper bound from the absence of a Landau singularity below M_GUT), κ∼ 0.4 - 0.7, tanβ∼ 2 - 3.5, μ_eff∼ 180 GeV - 290 GeV, A_λ∼ 150 GeV - 550 GeV, A_κ∼ -830 GeV - -210 GeV. An exception is the case M_H_S∼ 85 GeV - 110 GeV where LEP constraints on the coupling of H_S to the Z boson are somewhat weaker; here values of λ down to 0.16 and μ down to 100 GeV (together with tanβ up to 4.5) can lead to testable points. These ranges of potentially testable parameters depend little on the total integrated luminosity, but higher luminosity increases of course the number testable parameters within these ranges. Most of the parameters testable by bb̅γγ are in the same region except for κ∼ 0.08 - 0.3, A_κ∼ -50 GeV - 10 GeV which indicates that this region is non-overlapping with the one testable by bb̅bb̅.We are convinced that the here proposed search methods can still be refined, and that the estimated sensitivities to cross sections times branching fractions presented here are conservative. This becomes clear from a comparison to the recent CMS search for resonant SM-Higgs pair production <cit.> in the bb̅γγ final state (and actually also from a comparison to the recent CMS search <cit.> in the bb̅ττ final state). Thus we hope that such promising searches will be performed in the future at the LHC.§ ACKNOWLEDGEMENTS This project has received support from the European Union's Horizon 2020 research and innovation programsITN HiggsTools (PITN-GA-2012-316704), ITN Elusives (Marie Sklodowska-Curie grant agreement No 674896), RISE InvisiblesPlus (Marie Sklodowska-Curie grant agreement No 690575), RISE NonMinimalHiggs (Marie Sklodowska-Curie grant agreement No 645722), the ERC advanced grant Higgs@LHC, and the Défi InPhyNiTi project N2P2M-SF. M.R.V. thanks Marius Wiesemann, Fady Bishara, Dirk E. Zerwas, Nikola Makovec, Sophie Henrot,Alberto Escalante and Davide Napoletano for fruitful discussions and help, and the hospitality received at the IPPP at the University of Durham, where part of this work has been done. § APPENDIX A: FUNCTIONS TO FIT BACKGROUND DISTRIBUTIONS In this appendix we define the functions used to parametrize the total invariant mass (or M_X) distributions of the backgrounds to the various final states. The best choice among the functions and the values of the corresponding parameters have been obtained by maximum likelihood estimates. §.§ Four Parameter Gamma distribution:f(M_X;α,β,γ,μ)= γ/βΓ(α) e^-((M_X-μ)/β)^γ(M_X-μ/β)^αγ-1for M_X≥μ ,f(M_X;α,β,γ,μ)= 0for M_X ≤μ ,§.§ Frechet distribution:f(M_X;α,β,μ)= α/β e^-((M_X-μ)/β)^-α(M_X-μ/β)^-α-1for M_X≥μ ,f(M_X;α,β,μ)= 0for M_X ≤μ ,§.§ GaussExp function: A Gaussian with an exponential tail: f(M_X;μ,σ,k)=e^-(M_X-μ)^2/2σ^2for M_X-μ/σ≤ k ,f(M_X;μ,σ,k)=e^k^2/2-k(M_X-μ)/σfor M_X-μ/σ > k .§.§ Landau Distributionf(M_X;μ,σ)= ∫_0^∞sin(2t) e^-t(M_X-μ)/σ-2t/πlog(t)dt § APPENDIX B: 5 Σ DISCOVERY AND 95% CL EXCLUSION LIMITS In this appendix we sketch the computations of 5 σ discovery and 95% CL exclusion limits based on the M_X distributions of the background, and the different M_X signal distributions (depending on M_H_S), following <cit.>.As shown in Fig. <ref> the M_X distribution of the signal, after event selection and cuts, corresponds to a certain number s_i of expected signal events per bin. (The bin size in M_X is 20 GeV, and we have checked that the final results do not vary with this size.) From the MC simulation we know how this event number depends on the total signal cross section σ_sig, the integrated luminosity L_int and the acceptance A times efficiency ϵ:s_i ∼ L_int·σ_s·<A·ϵ>_swhere ∼ indicates a bin-dependent proportionality factor < 1. The backgrounds after event selection and cuts have been fitted by continuous functions of f(M_X) normalized to 1. Thus the number b_i of expected background events per bin per integrated luminosity isb_i =L_int·σ_b·<A·ϵ>_b·∫_bin i f(M_X) dM_X. Due to the large number of simulated events the statistical uncertainties around the median values <A·ϵ>_s and <A·ϵ>_b are negligibly small, 1% in practically all cases.Subsequently we use likelihood functionsL(σ_a,σ_b) = ∏_i(b_i+σ_a s_i)^b_i + σ_b s_i/(b_i + σ_b s_i)! e^-(b_i+σ_a s_i) with (b_i + σ_b s_i)! interpolated by the Γ function for non integer (b_i + σ_b s_i).For 5 σ discovery limits on σ_sig we look for the value of σ_sig for which the background only hypothesis is rejected at the 5 σ level. The observed number of events per bin would be b_i + σ_sig s_i, and the likelihood function corresponding to the background only hypothesis is L(0,σ_sig). As function of the number of events per bin it has its maximum at L(σ_sig,σ_sig). Hence the test statistics t_disc for discovery ist_disc = -2lnL(0,σ_sig)/L(σ_sig,σ_sig).Following <cit.> the significance Z_disc is thenZ_disc=√(t_disc),and for a 5 σ discovery we determine σ_sig such that Z_disc=5.For 95% CL exclusion limits on σ_sig we look for the value of σ_sig for which the signal hypothesis is rejected at 95% CL. The observed number of events per bin would be b_i, and the likelihood function corresponding to the signal hypothesis is L(σ_sig,0). As function of the number of events per bin it has its maximum at L(0,0). Hence the test statistics t_excl for exclusion ist_excl = -2lnL(σ_sig,0)/L(0,0).For exclusion at 95% CL we determine σ_sig such that √(t_excl)=1.64 since we consider only positive signal contributions to the number of events.Uncertainties from the background are estimated as follows: ± (1-2) σ statistical uncertainties can be obtained bin by bin. 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C71 (2011) 1554Erratum: [Eur. Phys. J. C73 (2013) 2501][arXiv:1007.1727 [physics.data-an]]. ] | http://arxiv.org/abs/1707.08522v4 | {
"authors": [
"Ulrich Ellwanger",
"Matias Rodriguez-Vazquez"
],
"categories": [
"hep-ph",
"hep-ex"
],
"primary_category": "hep-ph",
"published": "20170726161639",
"title": "Simultaneous Search for Extra Light and Heavy Higgs Bosons via Cascade Decays"
} |
Heteroclinic Cycles in ODEs with the Symmetry of the Quaternionic 𝐐_8 Group] Heteroclinic Cycles in ODEs with the Symmetry of the Quaternion 𝐐_8 Group Adrian C. Murza, Institute of Mathematics “Simion Stoilow” of the Romanian Academy, Calea Griviţei 21, 010702 Bucharest, Romania [email protected] In this paper we analyze the heteroclinic cycle and the Hopf bifurcation of a generic dynamical system with the symmetry of the group 𝐐_8, constructed via a Cayley graph. While the Hopf bifurcation is similar to that of a 𝐃_8–equivariant system, our main result comes from analyzing the system under weak coupling. We identify the conditions for heteroclinic cycle between three equilibria in the three–dimensional fixed point subspace of a certain isotropy subgroup of 𝐐_8×𝐒^1. We also analyze the stability of the heteroclinic cycle.[2000]37C80, 37G40, 34C15, 34D06, 34C15 [ Adrian C. Murza December 30, 2023 =====================§ INTRODUCTIONHeteroclinic cycles with in systems with symmetry have been widely studied over the large decades <cit.>. During the last couple of years a special interest has received the existence of heteroclinic cycles in systems related with quaternionic symmetry, see for example the works of X. Zhang <cit.> and O. Podvigina <cit.>. This is basically due to two facts. On the one hand quaternions are involved in the study of heteroclinic cycles in ODEs with symmetry in a natural way, owing to the easy representation of the dynamics in ℝ^4 in terms of quaternions. Many of the dynamical systems giving rise to heteroclinic cycles studied so far are 𝐃_n–equivariant; the action of 𝐃_n in ℝ^2 is absolutely irreducible, so ℝ^4 is 𝐃_n–simple. Therefore, quaternionic representations in ℝ^4 turned out to be very useful. On the other hand there is the intrinsic interest in the differential equations where the variables are the quaternions. We relate the study of heteroclinic cycles with the dynamics of networks of n coupled oscillators with symmetry. Ashwin and Swift <cit.> showed that the symmetry group of the network can be considered a subgroup of 𝕊_n, as long as the oscillators taken individually have no internal symmetries. Besides these two main reasons, there are also other ones that stimulates the analysis of dynamical systems with the quaternionic symmetry, and these are related to applications to other sciences. For example we can cite the heteroclinic phenomena observed in systems with quaternionic symmetry such as nematic liquid crystals <cit.>, particle physics <cit.> and improving computational efficiency <cit.>; however, these heteroclinic behaviors in such systems have never been encountered a theoretical explanation. This is one of our major motivation, together with the intrinsic value of the mathematical theory developed around this subject.An important step in designing oscillatory networks with the symmetry of a specific group has been developed by Stork <cit.>. The authors have shown how to construct an oscillatory network with certain designed symmetry, by the Cayley graph of the symmetry group.In this paper we analyze the heteroclinic cycles and Hopf bifurcation in ODEs with the symmetry of the quaternionic group 𝐐_8 of order 16. We use the methodology developed by Ashwin and Stork <cit.> to construct a network of differential systems with 𝐐_8 symmetry. We investigate the dynamical behavior of the system under the weak coupling. In this case we reduce the asymptotic dynamics to a flow on an sixteen-dimensional torus 𝕋^16. We prove the existence of heteroclinic cycles between the three steady–states existing within a three–dimensional fixed–point subspace of one of the isotropy subgroups of 𝐐_8×𝐒^1, namely 𝐙_2. We also classify the stability of heteroclinic cycles.The paper is organized as follows. In Section <ref> we construct the most general oscillatory system with the 𝐐_8 symmetry by using the Cayley graph of this group. In Section <ref> we analyze the Hopf bifurcation of the constructed array. In Section <ref> we prove the existence of heteroclinic cycles in some of the subspaces which are invariant under the action of certain isotropy subgroups of 𝐐_8. We also analyze their stability.§ THE CAYLEY GRAPH OF THE 𝐐_8 GROUPIn this section we construct an oscillatory system with the 𝐐_8 symmetry and describe the elements of this group, as the relationships between them. For more details about the use of the Cayley graph in constructing the network with the prescribed symmetry see <cit.> or <cit.>. The Cayley graphs for 𝐐_8 is shown in Figure (<ref>). The action of the group 𝐐_8 on the cells can be written as[Id; a=(1 2 3 4 9 10 11 12)(5 16 15 14 13 8 7 6); b=(1 5 9 13)(2 6 10 14)(3 7 11 15)(4 8 12 16);ab=(1 16 9 8)(2 5 10 13)(3 6 11 14)(4 7 12 15); b^2=(1 9)(2 10)(2 11)(4 12)(5 13)(6 14)(7 15)(8 16); a^2=(1 3 9 11)(2 4 10 12)(5 15 13 7)(6 16 14 8); a^3=(1 4 11 2 9 12 3 10)(5 14 7 16 13 6 15 8); ab^2=(1 10 3 12 9 2 11 4)(5 8 15 6 13 16 7 4);a^2b^2=(1 11 9 3)(2 12 10 4)(5 7 13 15)(6 8 14 16);a^3b^2=(1 12 11 10 9 4 3 2)(5 6 7 8 13 14 15 16); ba=(1 6 9 14)(2 7 10 15)(3 8 11 16)(4 13 12 15); ba^2=(1 7 9 15)(2 8 10 16)(3 13 11 5)(4 14 12 16);b^3=(1 13 9 5)(2 14)(3 15 11 7)(4 16 12 8)(6 10);ab^3=(1 8 9 16)(2 13 10 5)(3 4 15 14)(6 11 12 7);a^3b=(1 14 9 6)(2 15 10 7)(3 16 11 8)(4 5 12 13);a^2b=(1 15 9 7)(2 16 10 8)(3 5 11 13)(4 6 12 14) ]with the relationship between them[ a^8=Id, a^4=b^2=abab, aba=b ]If we assign coupling g between cells related by a and coupling h between cells related by b, from the permutations in (<ref>), we can build the following pairwise system in with the 𝐐_8 symmetry. [ẋ_1=f(x_1)+g(x_12,x_1)+h(x_5,x_9),ẋ_2=f(x_2)+g(x_1,x_2)+h(x_9,x_13),; ẋ_3=f(x_3)+g(x_2,x_3)+h(x_13,x_1),ẋ_4=f(x_4)+g(x_3,x_4)+h(x_1,x_5),;ẋ_5=f(x_5)+g(x_4,x_9)+h(x_2,x_6),ẋ_6=f(x_6)+g(x_9,x_10)+h(x_6,x_10),; ẋ_7=f(x_7)+g(x_10,x_11)+h(x_10,x_14),ẋ_8=f(x_8)+g(x_11,x_12)+h(x_14,x_2),;ẋ_9=f(x_9)+g(x_6,x_5)+h(x_3,x_7),ẋ_10=f(x_10)+g(x_5,x_16)+h(x_7,x_11),; ẋ_11=f(x_11)+g(x_16,x_15)+h(x_11,x_15),ẋ_12=f(x_12)+g(x_15,x_14)+h(x_15,x_3),;ẋ_13=f(x_13)+g(x_14,x_13)+h(x_4,x_8),ẋ_14=f(x_14)+g(x_13,x_8)+h(x_8,x_12),; ẋ_15=f(x_15)+g(x_8,x_7)+h(x_12,x_16),ẋ_16=f(x_16)+g(x_7,x_6)+h(x_16,x_4), ]where f:ℝ→ℝ and g, h:ℝ^2→ℝ. As shown by Ashwin and Stork <cit.> we can think of f, g, h as being generic functions that assure that the isotropy of this vector field under the action of 𝐎_16 is generically𝐐_8.§ HOPF BIFURCATIONIn order to consider generic one-parameter Hopf bifurcation in systems with 𝐐_8, we need to analyze the complex irreducible representations of 𝐐_8. Based on the work of Golubitsky and Stewart <cit.>, these representations are of one or two dimensions. From their theory, we have that the linear representation of a group Γα_Γ:Γ× W→ Won the complex vector space W is irreducible if and only if Γ-invariant subspaces are trivial; it is to say, {0} or W itself. It is important to notice, that (a) there need be no faithful irreducible representations, and (b) this is typical.In addition, the amount by which the representation fails to be faithful is the kernel of the action α_Γ. The group 𝐐_8 has five irreducible representations; four of them are one-dimensional and the remaining one is two-dimensional. The one-dimensional representations can be interpreted as Hopf bifurcation with trivial or 𝐙_2 symmetry, which correspond to a quotient group of 𝐐_8. From <cit.> the two generators of 𝐐_8 area= ( [ω0;0 ω̅ ]), b= ( [0 -1;10 ]),where ω=exp(π i/4).Therefore the standard irreducible action of 𝐐_8 on ℂ^2 is given by[ a(z_+,z_-)=(√(2)/2(1+i)z_+,√(2)/2(1-i)z_-); ;b(z_+,z_-)=(-z_-,z_+);]and there is a phase shift action of 𝐒^1 given byR_ϕ(z_+,z_-)=(e^iϕz_+,e^iϕz_-),for ϕ∈𝐒^1. The action of 𝐐_8×𝐒^1 is similar to the action of 𝐃_8×𝐒^1. This action is generated by[ κ(z_+,z_-)=(z_-,z_+), ρ(z_+,z_-)=(iz_+,-iz_-), ]where ρ^4=κ^2=1 and ρκ=κρ^3. In our case of the group 𝐐_8 we have a^8=b^4=1,a^4=b^2 and aba=b. The kernel of this action in the 2-cycle in 𝐃_8×𝐒^1 is generated by (ρ^2,π), while the kernel of the action of 𝐐_8×𝐒^1 is generated by (a^4=b^2=1,π). Therefore, it is possible to check that𝐐_8×𝐒^1/kerα_𝐐_8×𝐒^1≡𝐐_8×𝐒^1/ kerα_𝐃_8×𝐒^1.This means that we use the results obtained in <cit.> for Hopf bifurcation in systems with 𝐃_8 symmetry, with a re-interpretation of the branches. There are exactly three branches of periodic solutions that bifurcate from (0,0), corresponding to the isotropy subgroups 𝐃_8×𝐒^1 with two-dimensional fixed-point subspaces. The proof is a direct application to the group 𝐃_8 of Theorem 4.2 of <cit.>. Therefore, there are exactly three branches of periodic solutions occurring generically in Hopf bifurcation with 𝐃_8 symmetry. From Proposition 2.1, page 372 in <cit.> we have that every smooth 𝐃_8×𝐒^1-equivariant map germ g:ℂ^2→ℂ^2 has the form[ g(z_1,z_2)=A[ z_1; z_2 ]+B[ z_1^2z̅_1; z_2^2z̅_2 ]+ C[ z̅_1^3z_2^4; z_1^4z̅_2^3 ]+D[ z_1^5z̅_2^4; z̅_1^4z_2^5 ], ]where A, B, C, D are complex-valued 𝐃_8×𝐒^1-invariant functions. The branching equations for 𝐃_8-equivariant Hopf bifurcation may be rewritten g(z_1,z_2)=0. These branching equations are shown in Table (<ref>). §.§ Bifurcating branchesWe now use the information in Table (<ref>) to derive the bifurcation diagrams describing the generic 𝐃_8-equivariant Hopf bifurcation. Assume[ (a) Re(A_N+B)≠0, (b) Re(B)≠0, (c) Re(2A_N+B)≠0, (d) Re(BC̅)≠0, (c) Re(A_λ)≠0, ]where each term is evaluated at the origin.Assuming nondegeneracy conditions (<ref>) and the trivial branch is stable subcritically and loses stability as bifurcation parameter λ passes through 0. We summarize these facts into the next theorem.The following statements hold. (a) The 𝐙̃_8 branch is super- or subcritical according to whether Re(A_N(0)+B(0)) is positive or negative. It is stable if Re(A_N(0)+B(0))>0, Re(B(0))<0.(b) The 𝐙_2(κ)[⊕𝐙_2^c] is super- or subcritical according to wether Re(2A_N(0)+B(0)) is positive or negative. It is stable if Re(2A_N(0)+B(0))>0, Re(B(0))>0 and Re(2B(0)C̅(0))<0.(c) The 𝐙_2(κ,π)[⊕𝐙_2^c] or 𝐙_2(κ,ξ)⊕𝐙_2^c branch is super- or subcritical according to wether Re(2A_N(0)+B(0)) is positive or negative. It is stable if Re(2A_N(0)+B(0))>0, Re(B(0))>0 and Re(2B(0)C̅(0))<0.The proof is a direct application to the case 𝐃_8 of the Theorem 3.1 page 382 in <cit.>.§ WEAK COUPLING The idea of studying ODEs in the weak coupling limit was introduced by Ashwin and Swift <cit.>. This situation can be uderstood as follows. In the no coupling case there is an attracting n–dimensional torus with one angle for every oscillator. The situation is completely different to the Hopf bifurcation. Instead of examining small amplitude oscillations near a Hopf bifurcation point, we make a weak coupling approximation. There is a slow evolution of the phase differences in the weak coupling. Another improvement with respect to the Hopf bifurcation is that while the Hopf bifurcation theory gives local information, the weak coupling case the yields global results on the n–dimensional torus. System (<ref>) can be rewritten under weak coupling case as an ODE of the form:ẋ_i=f(x_i)+ϵ g_i(x_1,…,x_16)for i=1,…,16, x_i∈𝒱 and commuting with the permutation action of 𝐐_8 on 𝒱^16, both f and g_i being of the class 𝒞^∞. The constant ϵ represents the coupling strength and we have ϵ≪1. As in <cit.>, or <cit.> we may assume ẋ=f(x) has an hyperbolic stable limit cycle. It follows that if the coupling is weak, we should not just take into account the irreducible representations of 𝐐_8. Since there are 16 stable hyperbolic limit cycles in the limit of ϵ=0, it means that the asymptotic dynamics of the system factors into the asymptotic dynamics of 16 limit cycles. We assume that each limit cycle taken individually is hyperbolic for small enough values of the coupling parameter. This justifies expressing the dynamics of the system only in terms of phases, i.e. an ODE on 𝐓^16which is 𝐐_8-equivariant. When considering the weakly coupled system we can average it over the phases <cit.>. This is the same as introducing and phase shift symmetry by translation along the diagonal;R_θ(ϕ_1,…,ϕ_16):=(ϕ_1+θ,…,ϕ_16+θ),for θ∈𝐒^1.We obtained an ODE on that is equivariant under the action of 𝐐_8×𝐒^1,and we have to classify the isotropy types of points under this action. This is done in Table (<ref>). Since now on, our interest focuses in the three-dimensional space Fix(𝐙_2); it does not contain two-dimensional fixed-point subspaces. In turn, it contains several one- and zero-dimensional subspaces fixed by the isotropy subgroups 𝐐̃_8^i, 𝐙̃_8^i and 𝐙_8^i, respectively, where i={a,b,ab,a/4,b/4} as in Table (<ref>). These symmetries are not in 𝐙_2; however, they are in the normalizer of 𝐙_2.§.§ Dynamics of the θ_1,θ_2 and θ_3 angles in Fix(𝐙_2)We can define coordinates in Fix(𝐙_2) by taking a basis[e_1=-1/8(1,1,1,1,1,1,1,1,-1,-1,-1,-1,-1,-1,-1,-1); e_2=-1/8(1, -1, 1, -1 ,1, -1, 1, -1 ,1, -1, 1, -1, 1, -1, 1, -1); e_3=-1/8(1, -1, 1, -1, 1, -1, 1, -1, -1, 1, -1, 1, -1, 1, -1, 1) ]and consider the space spanned by {e_1,e_2,e_3} parametrized by {θ_1,θ_2,θ_3}: ∑_n=1^3θ_ne_n.By using these coordinates, we construct the following family of three-dimensional differential systems which satisfies the symmetry of Fix(𝐙_2).{[θ̇_̇1̇=usinθ_1cosθ_2+ϵsin2θ_1cos2θ_2;;θ̇_̇2̇=usinθ_2cosθ_3+ϵsin2θ_2cos2θ_3;; θ̇_̇2̇=usinθ_3cosθ_1+ϵsin2θ_3cos2θ_1 +q(1-cosθ_1)sin2θ_3,; ].where u,ϵ,q∈ℝ.We will show that this vector field contains structurally stable, attracting heteroclinic cycles which may be asymptotically stable, essentially asymptotically stable or completely unstable, depending on the values of u,ϵ and q. We can assume, without loss of genericity that the space Fix(𝐙_2) is normally attracting for the dynamics and therefore the dynamics within the fixed-point space determines the stability of the full system. In the following we will show that the planes θ_i=0 (mod π),i=1,2,3 are invariant under the flow of (<ref>).Let 𝒳 be the vector field of system (<ref>).We call a trigonometric invariant algebraic surface h(θ_1,θ_2,θ_3)=0, if it is invariant by the flow of (<ref>), i.e. there exists a function K(θ_1,θ_2,θ_3) such that𝒳h=∂ h/∂θ_1θ̇_̇1̇+∂ h/∂θ_2θ̇_̇2̇ +∂ h/∂θ_3θ̇_̇3̇=Kh.Functions sinθ_1, sinθ_2 and sinθ_3 are trigonometric invariant algebraic surfaces for system (<ref>). We can write the system (<ref>) in the form{[ θ̇_̇1̇=sinθ_1(ucosθ_2+2ϵcosθ_1cos2θ_2); ; θ̇_̇2̇=sinθ_2(ucosθ_3+2ϵcosθ_2cos2θ_3); ; θ̇_̇3̇=sinθ_3(ucosθ_1+2ϵcos2θ_1cosθ_3 +2q(1-cosθ_1)cosθ_3);].Now if we choose h_1=sinθ_1, then 𝒳h_1=cosθ_1sinθ_1(ucosθ_2+2ϵcosθ_1cos2θ_2) so K_1=cosθ_1(ucosθ_2+2ϵcosθ_1cos2θ_2). The remaining cases follow similarly. Since the planes θ_i=0(mod π) are invariant under the flow of (<ref>), it is clear that (0,0,0), (π,0,0), (0,π,0), and (0,0,π) are equilibria for (<ref>). To check the possibility of heteroclinic cycles in system (<ref>), we linearize about the equilibria (i.e. the zero-dimensional fixed points). The idea is proving that there are three-dimensional fixed-point spaces Fix(𝐙_2) and Fix(𝐙̃_2) which connect these fixed points, allowing the existence of such a heteroclinic network between the equilibria.Let's assume[ |ϵ|<u/2and |ϵ+2q|<u/2. ]We use the criteria of Krupa and Melbourne <cit.> to study the stability of the heteroclinic cycle.In the following we will prove that there exists the possibility of a heteroclinic cycle in the following way:[ ⋯Fix(𝐐̃_8^a)Fix(𝐐̃_8^b)Fix(𝐐̃_8^ab) ⋯ ]The stability of the heteroclinic cycle is: (a) asymptotically stable if[ u<0 and q<3u/4-ϵ/2, ] (b) unstable but essentially asymptotically stable if[ u<0 and 3u/4-ϵ/2<q<u/2-(u+2ϵ)^3/(-u+2ϵ)^2. ] (c) completely unstable if u>0. The stability is expressed byρ=∏_i=1^3ρ_i, where ρ_i=min{c_i/e_i,1-t_i/e_i}. In equation (<ref>), e_i is the expanding eigenvalue at the ith point of the cycle, -c_i is the contracting eigenvalue and t_i is the transverse eigenvalue of the linearization. For the heteroclinic cycle we haveρ_1= {[ 2u-4q/u+2ϵ if q<3u/4-ϵ/2,;; -u+2ϵ/u+2ϵ if q>3u/4-ϵ/2, ]ρ_2=ρ_3=-u+2ϵ/u+2ϵ, . so from equations (<ref>) and (<ref>) we obtainρ= {[ (-u+2ϵ)^2(2u-4q)/(u+2ϵ)^3 if u<0 and q<3u/4-ϵ/2,; ;(-u+2ϵ)^3/(u+2ϵ)^3 if u<0 and q>3u/4-ϵ/2. ].Then the proof follows by applying Theorem 2.4 in <cit.>. For any u<0 we have 3u/4-ϵ/2<q<u/2-(u+2ϵ)^3/(-u+2ϵ)^2 and therefore there exist values of q for which there exist essentially asymptotic stable heteroclinic connections. In consequence, there exists an attracting heteroclinic cycle even though the linear stability of Fix(𝐐̃_8^a) has an expanding transverse eigenvalue.§ CONCLUSIONSWe prove the existence of stable heteroclinic cycles in the most general coupled ordinary differential equations with quaternionic symmetry 𝐐_8. Our approach is generic and offers for the first time as far as we know, evidence of these phenomena in systems with this symmetry. While the results stands on its own from a mathematical point of view, it might also contribute to a better understanding of these intermittent behaviors experimentally observed in nematic liquid crystals <cit.> and particle physics <cit.>.§ ACKNOWLEDGEMENTSIn first place I would like to address many thanks to the Referee, whose helpful indications and comments greatly improved the presentation of the paper. I acknowledge a BITDEFENDER postdoctoral fellowship from the Institute of Mathematics Simion Stoilow of the Romanian Academy, Contract of Sponsorship No. 262/2016 as well as economical support from a grant of the Romanian National Authority for Scientific Research and Innovation, CNCS-UEFISCDI, project number PN-II-RU-TE-2014-4-0657.10 StorkP. Ashwin, P. Stork, Permissible Symmetries of Coupled Cell Networks, Math. Proc. Camb. Phil. Soc., 116, (1994), 27–36. Ashwin_SwiftP. Ashwin, J.W. Swift, The Dynamics of n Identical Oscillators with Symmetric Coupling, J. Nonlin. Sci., 2, (1992), 69–108.copS. C̆opar, S. Z̆umer, Quaternions and Hybrid Nematic Disclinations, Proc. Royal Soc. A, 469, (2013), 1–10.paivaA.P.S. Dias, R.C. Paiva, A Note on Hopf Bifurcation with Dihedral Group Symmetry, Glasgow Math. J., 48, (2006), 41–51.devS. Dev, S. Verma, Leptogenesis in a Hybrid Texture Neutrino Mass Model, Mod. Phys. Lett., 25, (2010), 2837–2848.funJ. Funda, R.H. Taylor, R.P.Paul, On Homogeneous Transforms, Quaternions and Computational Efficiency, IEEE Trans., 6, (1990), 382–387. G1M. Golubitsky, M. Pivato, I. Stewart, Interior Symmetry and Local Bifurcation in Coupled Cell Networks, Dyn. Syst, 19, (2004), 389–407. GS88M. Golubitsky, I. Stewart, D.G. Schaeffer, Singularities and Groups in Bifurcation Theory II, Applied Mathematical Sciences 69, Springer–Verlag, (1988). JD.L. Johnson, Topics in the Theory of Group Presentations, Lecture Notes Series 42, Cambridge University Press, (1980).KrupaM. Krupa, I. Melbourne, Asymptotic Stability of Heteroclinic Cycles in Systems with Symmetry, Ergodic Theory Dyn. Syst., 15, (1995), 121–147.murzaA.C. Murza, Hopf Bifurcation and Heteroclinic Cycles in a Class of 𝔻_2–Equivariant Systems, Math. Rep., 17, (2015), 369–383. podvi4O. Podvigina, Stability and Bifurcations of Heteroclinic Cycles of Type Z, Nonlinearity, 25, (2012), 1887–1917.podvi2O. Podvigina, P. Chossat, Simple Heteroclinic Cycles in ℝ^4, Nonlinearity, 28, (2015), 901–926.Stork1P. Stork, Statische Verzweigung in Gradientelfeldern mit Symetrien vom Komplexen oder Quaternionischen Typ mit Numerischer Behandlung, in Wissenschaftliche Beiträge aus europäichen Hochschulen, Vol. 11, (1993), Verlag an der Lottbeck.ZX. Zhang, Global Structure of Quaternion Ppolynomial Differential Equations, Comm. Math. Phys., 303, (2011), 301–316. | http://arxiv.org/abs/1707.08647v1 | {
"authors": [
"Adrian C. Murza"
],
"categories": [
"math.DS",
"37C80, 37G40, 34C15, 34D06, 34C15"
],
"primary_category": "math.DS",
"published": "20170726211905",
"title": "Heteroclinic Cycles in ODEs with the Symmetry of the Quaternionic $\\mathbf{Q}_8$ Group"
} |
Product recognition in store shelves as a sub-graph isomorphism problem Alessio Tonioni, Luigi Di Stefano University of Bologna,Department of Computer Science and Engineering (DISI)Viale del Risorgimento 2, Bologna{alessio.tonioni,luigi.distefano}@unibo.it December 30, 2023 ======================================================================================================================================================================================================emptyThe arrangement of products in store shelves is carefully planned to maximize sales and keep customers happy. However, verifying compliance of real shelves to the ideal layout is a costly task routinely performed by the store personnel. In this paper, we propose a computer vision pipeline to recognize products on shelves and verify compliance to the planned layout. We deploy local invariant features together with a novel formulation of the product recognition problem as a sub-graph isomorphism between the items appearing in the given image and the ideal layout. This allows for auto-localizing the given image within the aisle or store and improving recognition dramatically.§ INTRODUCTIONManagement of a grocery store or supermarket is a challenging task entailing personnel busy in supervisingshelves and the whole sale point. Technology advances may be deployed to provide more reliable information in real time to the store manager, so to coordinate human resourcesmore effectively.Examples of tasks where innovation can improve current best practices are shelves analysis (e.g. verifying low in stock or misplaced items), security (e.g. reporting suspicious behaviours) and customer analysis (e.g. analysing shopping patterns to improvecustomer experience). However, a promising technological solution can be deployed in real shops as long as it turns out viable from a cost perspective, modifies current practices moderately and does not affect customer experience adversely. Computer vision techniques may fulfil the above requirements due to potential reliance on cheap cameras either mounted non-invasively in the store or embedded within the hand-held computers routinely used by sales clerks. The problem addressed in this paper is visual shelf monitoring through computer vision techniques. The arrangement of products in supermarket shelves is planned very carefully in order to maximize sales and keep customers happy. Shelves void, low in stock or misplaced products render it difficult for the customer to buy what she/he needs, which, in turn, not only leads to unhappy shoppers but also to significant loss of sales; as pointed out in <cit.>, 31% of customers facing a void shelf purchase the item elsewhere and 11% do not buy it at all. The planned layout of products within shelves is called planogram: it specifies where each product should be placed within shelves and how many facings it should cover, that is how many packages of the same product should be visible in the front row of the shelf. Keeping shelves full as well as compliant to the planogram is a fundamental task for all types of stores that could lead to 7.8% sales increase and 8.1% profit improvement in just two weeks <cit.>. However, thus far, planogram compliance is pursued by having sales clerks visually inspecting aisles during the quieter hours of the day. Computer vision may help to automate, at least partially, this task. As vouched by recently published patents <cit.>, <cit.> and journal articles <cit.>, some major corporations are currently investigating on deployment of state of the art computer vision techniques to pursue planogram compliance, with smaller emerging companies (such as Planorama , Vispera, Simble Robotics) [<http://www.planorama.com/>, <http://vispera.co/>, <http://www.simberobotics.com/>] advertising this type of service alike. From a scientific perspective, attaining planogram compliance by automated visual analysis represents a very challenging task due to the large number of object instances that should be identified and localized in each scene, the presence of many distractors, the small differences between different instances of products belonging to the same brand and the varying lighting conditions. Accordingly, to the best of our knowledge, established scientific approaches have not emerged yet while industrial solutions seem either at a prototype stage or in the very early part of their life cycle.In this paper we propose a computer vision pipeline that, given the planogram and an image of the observed shelves, can correctly localize each product, check whether the real arrangement is compliant to the planned one and detect missing or misplaced items. Key to our approach is a novel formulation of the problem as a sub-graph isomorphism between the product detected in the given image and those that should ideally be found therein given the planogram.Accordingly, our pipeline relies on a standard feature-based object recognition step, followed by the novel graph-based consistency check and a final localized image search to improve the overall product recognition rate. § RELATED WORK The problem of automatically recognizing grocery products from images may in principle be traced back to the more general and extensively investigated subject of visual object recognition. However, as pointed out by Merler et al. <cit.>, dealing with grocery products on shelves exhibits peculiarities that render the task particularly challenging. Indeed, as also exemplified in the leftmost column of <ref>, one has to rely on a single or a few views either synthetic (graphic renderings of the package) or taken in ideal studio-like conditions in order to model each product instance which, then, must be sought within images acquired in real settings. The scarcity and diversity of model images make it awkward to deploy directly object recognition methods, such as deep convolutional neural networks, that demand a large corpus of labeled training examples representative of unseen data. As noticeable in <ref>, verifying planogram compliance calls for detecting and localizing each individual product instance within a shelves image crowded with lots of objects, some remarkably similar one to another. Moreover, the scene usually include several distractor items, such as vividly colored banner ads designed to attract customer eyes, that may mislead computer vision algorithms. As the operational conditions shouldbe left as unconstrained as possible, recognition algorithms should withstand working images featuring dramatic changes in color, intensity and even resolution, due to varying lighting conditions as well as deployment of diverse acquisition devices.In their work, Merler et al. <cit.> discuss the above-mentioned issues, propose a public dataset and pursue product recognition to realize an assistive tool for visually impaired customers. They assume that no information concerning product layout may be deployed to ease detection. Given these settings, the performance of the proposed systems turned out quite unsatisfactory in terms of both precision and efficiency. Further research has then been undertaken to ameliorate the performance of automatic visual recognition of grocery products<cit.>, <cit.>, <cit.>. In particular, Cotter et al.<cit.>report significant performance improvements by leveraging on machine learning techniques, such as HMAX and ESVM, together with HOG-like features. Yet, their proposal requires many training images for each product, which is unlikely feasible in real settings, and deploys a large ensemble of example-specific detectors, which makes the pipeline rather slow at test time. Moreover, adding a new type of sought product is rather cumbersome as it involves training a specific detector for each exemplar image, thereby also further slowing down the whole system at test time.The approach proposed in<cit.> was then extended in <cit.> through a contextual correlation graph between products. Such a structure can be queried at test time to predict the products more likely to be seen given the last k detections, thereby reducing the number of ESVM computed at test time and speeding up the whole system. Another relevant work is due to George at al. <cit.>.First, to reduce the search space of the actual detection phase, they carry out an initial classification to infer the categories of observed items.Then, following detection, they run an optimization step based on a genetic algorithmto detect the most likely products from a series of proposals. Despite the quite complex pipeline, when relying on only one model image per product the overall precision of the system isbelow 30%. The paper proposes also a publicly available dataset, referred to asGrocery Products, comprising 8350 product images classified into 80 hierarchical categories together with 680 high resolution images of shelves. In this paper, we use part of this public dataset as the main test bench for our method. Marder et al. in <cit.> addressed our exact same problem of checking planogram compliance through computer vision. Their approach relies on detecting and matching SURF features <cit.> followed by visual and logical disambiguation between similar products. The paper reports a good 87.4% product recognition rate on a publicly unavailable dataset of cereal boxes and hair care products, though precision figures are not highlighted. Their dataset includes 240 images with 980 instances of 223 different products, that is, on average ≈4 instances per image. To improve product recognition the authors deploy information dealing with the known product arrangement through specific hand-crafted rules, such as e.g. `conditioners are placed on the right of shampoos`. Differently, we propose to deploy automatically these kind of constraints by modeling the problem as a sub-graph isomorphism between the items detected in the given image and the planogram. Unlike ours, their method mandates a-priori categorization of the sought products into subsets of visually similar items. Systems to tackle the planogram compliance problem are described also in <cit.>, <cit.> and <cit.>. These papers delineate solutions relying either on large sensor/camera networks or mobile robots monitoring shelves while patrolling aisles. In contrast, our proposal would require just an off-the-shelf device, such as a smartphone, tablet or hand-held computer.§ PROPOSED PIPELINEWe address the typical industrial settings in which at least one model image per product together with a general schema of the correct disposition of items (the planogram) are available. At test time, given one image featuring products on shelves, the system would detect and localize each item and check if the observed product layout is compliant to the given planogram. As depicted in <ref>, we propose to accomplish the above tasks by a visual analysis pipeline consisting of three steps. We provide here an overview of the functions performed by the three steps, which are described in more detail in the following Sub-sections.The first step operates only on model images and the given shelves image. Indeed, to pursue seamless integration with existing procedures, we assume that the information concerning which portion of the aisle is observed is not available together with the input image. Accordingly, the first step cannot deploy any constraint dealing with the expected product disposition, and is thusreferred to as Unconstrained Product Recognition.As most product packages consist of richly textured piecewise planar surfaces, we obtained promising result through a standard object recognition pipeline based on local invariant features (as described, e.g., in <cit.>). Yet, the previously highlighted nuisances cause both missing product items as well as false detections due to similar products. Nonetheless, the first step can gather enough correct detections to allow the successive steps to identify the observed portion of the aisle in order to deploy constraints on the expected product layout and improve product recognition dramatically. The output of the first step consists in a set of bounding boxes corresponding to detected product instances (see <ref>).From the second step, dubbedGraph-based Consistency Check,we start leveraging on the information about products and their relative disposition contained in planograms. We choose to represent a planogram as a grid-like fully connected graph where each node corresponds to a product facing and is linked to at most 8 neighbors at 1-edge distance, i.e. the closest facings along the cardinal directions. We rely on a graph instead of a rigid grid to allow for a more flexible representation; an edge between two nodes does not represent a perfect alignment between them but just proximity along that direction. This abstract representation, referred to as Reference Planogram, encodes information about the number of facings related to each product and the items placed close together in shelves.An example of Reference Planogram is shown in <ref>.The detections provided by the first step are used in the second to build automatically another grid-like graph having the same structure as the Reference Planogram and referred to as Observed Planogram.Then, we find the sub-graph isomorphism between the Observed andReference planograms, so as to identify local clusters of self-consistent detected products, e.g. sets of products placed in the same relative position in both the Observed and Reference planograms. As a result, the second step ablates away inconsistent nodes from the Observed Planogram, which typically correspond to false detections yielded by the first step. It is worth pointing out that, as the Observed Planogram concerns the shelves seen in the current image while the Reference Planogram models the whole aisle, matching the former into the latter implies localizing the observed scene within the aisle[More generally, matching the Observed to a set of Reference planograms does localize seamlessly the scene within a set of aisles or, even, the whole store.]. After the second step the Observed Planogram should contain true detections only. Hence, those nodes that are missing compared to the Reference Planogram highlight items that appear to be missingwrt the planned productlayout.The task of the third step, referred to as Product Verification, is to verify whether these product items are really missing in the scene or not. More precisely, we start considering the missing node showing the highest number of already assigned neighbors, for which we can most reliably determine a good approximation of the expected position in the image. Accordingly, a simpler computer vision problem than in the first step needs to be tackled, i.e. verifying whether or not a known object is present in a well defined ROI (Region of Interest) within the image. Should the verification process highlight the presence of the product, the corresponding node would be added to the Observed Planogram, so to provide new constraints between found items; otherwise, a planogram compliance issue related to the checked node is reported (i.e. missing/misplaced item). The process is iterated till all the facings in the observed shelves are either associated with detected instances or flagged ascompliance issues.§.§ Unconstrained Product RecognitionAs already mentioned, we rely on the classical multi-object and multi-instance object recognition pipeline based on local invariant features presented in <cit.>, which is effective with planar textured surfaces and scales well to database comprising several hundreds or a few thousands models, i.e. in the order of the number of different products typically sold in grocery stores and supermarkets. Accordingly, we proceed through feature detection, description and matching, then cast votes into a pose space by a Generalized Hough Transform that can handle multiple peaks associated with different instances of the same model in order to cluster correspondences and filter out outliers. In our settings, it turns out reasonable to assume the input image to represent an approximately frontal view of shelves, so that both in-plane and out-of-plane image rotations are small. Therefore, we estimate a 3 DOF pose (image translation and scale change). Since the introduction of SIFT <cit.>, a plethora of other feature detectors and descriptors have been proposed in literature. Interestingly, the object recognition pipeline we used that is described in <cit.> may be deployed seamlessly with most such newer proposals. Moreover, it turns out just as straightforward to rely on multiple types of features jointly to pursue higher sensitivity thanks to detection of diverse image structures. Purposely, our implementation of the standard object recognition pipeline can run in parallel several detection/description/matching processes based on different features and have them eventually cast vote altogether within the same pose space. As reported in <ref>, we have carried out an extensive experimental investigation to establish which features would yield the best performance.§.§ Graph-based Consistency CheckTo build the Observed Planogram we instantiate a node for each item detected in the previous step and perform a loop over all detections to seek for bounding boxes around other detected items that are located close the current one. For each node, the search is performed along 8 cardinal directions (N, S, E, W, NW, NE, SW, SE) and, if another bounding box is found at a distance less than a dynamically determined threshold, an edge is created between the two nodes. In the given node the edge is labeled according to the search direction (e.g. N), oppositely in the found neighbor node (i.e. S). The graph is kept self-coherent, e.g. if node B is the North node of A, then A must be the South node of B. In case of ambiguity, e.g.bothA and C found to be the South node of B, we retain the edge between the two closest bounding boxes only. Once built, we compare the Observed to the Reference Planogram so to determine whether and how the two graphs overlap one to another. In theoretical computer science this problem is referred to as subgraph isomorphism andknown to be NP-complete<cit.>. A general formulation may read as follows:given two graphs G an H, determine whether G contains a subgraph for which does exist a bijection between the vertex sets of G and H. However, given our strongly bounded graphs, we choose not to rely on one of the many general algorithms, likee.g.<cit.>, and, instead, devised an ad hoc heuristic algorithm that, casting ours as a constraint satisfaction problem, works fairly well in practice.We formulate our problem as follows: given two graphs I (Reference Planogram) and O (Observed Planogram), find an isomorphism between a subset of nodes in I and a subset of nodes in O such that the former subset has the maximum feasible cardinality given product placements constraints. Each node in I can be associated with a node in O only if they both refer to the same product instance and exhibit coherent neighbors. In other words, we find the maximum set of nodes in graph O that turn out self-consistent, i.e. their relative positions are the same as in the reference graph I.As illustrated in Algorithm <ref>, the process starts with procedure CreateHypotheses, which establishesthe initial set of hypotheses, ℋ={… h_i…},h_i={n_I,n_O,c(n_I,n_O)}, with n_I and n_O denoting, respectively, a node in I andO related to the same product andc(n_I,n_O)=nn_c/nn_t with nn_c number of coherent neighbors (e.g. refering to the same product both in O and I) and nn_t number of neighbors for that node in I.CreateHypotheses iterates over all n_I∈ I so to instantiate all possible hypotheses. An example of the hypotheses set determined by CreateHypotheses given I and O is shown in the first row of <ref>. Then, procedure FindSolution finds a solution, 𝒮, by iteratively picking the hypothesis featuring the highest score. The first hypothesis picked in the considered example is shown in <ref>-a). Successively,ℋ isupdated by removing the hypotheses containing either of the two nodes in the best hypothesis and increasing the scores of hypotheses associated with coherent neighbors (<ref>-b)). Procedure FindSolution returns also a confidence score for the current solution, C, which takes into account the cardinality of 𝒮, together with a factor which penalizes the presence inO of disconnected sub-graphs that exhibit relative distances different than those expected given the structure of I[In the toy example in <ref>, O does not contain disconnected sub-graphs.] which instead is always fully connected. FindSolution takes as input the score of the current best solution, C_max, and relies on a branch-and-bound scheme to accelerate the computation. In particular, as illustrated in <ref>-c), after updating ℋ (<ref>-b)), FindSolutioncalculates an upper-bound for the score, B_C, by adding to the cardinality of 𝒮 the number of hypotheses in ℋ that are not mutually exclusive, so as to early terminate the computation when the current solution can not improve C_max. The iterative process continues with picking the new best hypothesis until ℋ is found empty or containing only hypotheses with confidence lower then a certain threshold τ (<ref>-d). The found solution, 𝒮, contains all the hypotheses that are self-consistent and such that each node n_I is either associated with a node n_O or to none, as shown in the last row of <ref>. In the last step ((<ref>-e)), the procedure computes C and returns also the first hypothesis, h_0, that was added into 𝒮, i.e. that with the highest score c(n_I,n_O) (<ref>.-a)). Upon returning from FindSolution, the algorithm checks whether or not the new solution𝒮 improves the best one found so far and removes h_0 from ℋ (see Algorithm <ref>) to allow evaluation of another solution based on a different initial hypotheses.As a result, Algorithm <ref> finds self-consistent nodes in O given I, thereby removing inconsistent (i.e. likely false) detections and localizing the observed image wrt to the planogram. Accordingly, the output of the second steps contains information about which items appear to be missing given the planned product layout and where they ought to be located within the image.§.§ Product VerificationWe use an iterative procedure whereby each iteration tries to fill the observed planogram with one seemingly missing object. As illustrated in <ref>, each iteration proceeds through three stages. We start with the missing element featuring the highest number of already detected neighbors. The positions of these neighbors provide clues on where the missing product should appear in the image. In particular, the position and size of each neighbor, together with the average edge length in the Observed Planogram, provide an estimation of the center of the missing element: averaging estimations across the neighbors yields a good approximate position. Then, we define a coarse image ROI centered at this position by estimating the size of the missing element[Store databases contain product sizes: the image size of a missing product can be estimated from those of the detected neighbors and the known metric sizes.] and allowing for some margin on account of possible localization inaccuracies.Given the estimated ROI, the second stage attempts to find and localize the missing product therein. As already pointed out, unlike the initial step of our pipeline, here we now know exactly which product is sought as well as its approximate location in the image. To look for the sought product within the ROI, we have experimented with template matching techniques as well as with a similar pipeline based on local features as deployed for Unconstrained Product Recognition (<ref>). The latter, in turn, would favorably reuse the image features already computed within the ROI in the first step of our pipeline, so as to pursue matching versus the features associated with the model image of the sought product only and, accordingly, cast votes in the pose space. Both approaches would provide a series of Detection Proposals (see <ref>).Detection proposals are analyzed in the last stage of an iteration by first discarding those featuring bounding boxes that overlap with already detected items and then scoring the remaining ones according to the coherence of the position within the (Observed Planogram) and the detection confidence. As for the first contribution to the score,we take into account the error between the center of the proposal and that of the ROI estimated in the first stage (so to favor proposals closer to the approximated position inferred from already detected neighbors); the second component of the score, instead, depends on the adopted technique: for template matching methods we use the correlation score while for approaches based on local featureswe rely on the number of correct matches associated with the proposal. Both terms are normalized to 1 and averaged out to get the final score assigned toeach Detection Proposal. Based on such a score, we pick the best proposal and add it to the Observed Planogram, so as to enforce new constraints that may be deployed throughout successive iterations to select the best-constrained missing item as well as improve ROI localization. If either all detection proposals are discarded due to the overlap check or the best one exhibits too low a score, our pipeline reports a planogram compliance issue related to the currently analyzed missing product. We have not investigated yet on how to disambiguate between different issues such as low in stock items and misplaced items. In real settings, however, such different issues would both be dealt with by manual intervention of sales clerks.The iterative procedure stops when all the seemingly missing products have been either detected or labeled as compliance issues.§ EXPERIMENTAL RESULTS To assess the performance of our pipeline we rely on theGrocery Products dataset <cit.>. However, as the ground-truth available with shelves images concerns product types while we aim at detecting each individual instance, we have manually annotated a subset of images with item-specific bounding boxes (see <ref>). Moreover, for each image we have created an ideal planogram encoded in our graph like representation for the perfect disposition of products (e.g. if the actual image contains voids or misplaced items they will not be encoded on the ideal planogram that instead will model only the correct product disposition). The annotation used are available at our project page [<vision.disi.unibo.it/index.php?option=com_content view=article id=111 catid=78>]. Our chosen subset consists of 70 images featuring box-like packages and dealing with different products such as rice, coffee, cereals, tea, juices, biscuits…. Each image depicts many visible products, for a total of 872 instances of 181 different products, that is on average ≈ 12 instances per image. According to the metric used in the PASCAL VOC challenge, we judge a detection as correct if the intersection over union between the detected and ground-truth bounding boxesis >0.5. For each image we compute Precision (number of correct detections over total number of detections), Recall (number of correctly detected products over number of products visible in the image) andF-Measure (harmonic mean of Precision and Recall). Then, we provide charts reporting average figures across the dataset.As regards comparative evaluation with respect to previous work, it is worth highlighting that the only work addressing exactly the same task as ours is <cit.>, but neither their dataset nor their implementation are publicly available. Indeed, their system is quite complex and tailored for their specific use case so it would have been unfair to reproduce their results on our dataset by our own implementation based only on their paper.We have also investigated on the use of region proposals, such as <cit.>, followed by classification (e.g. by a CNN <cit.>) of the identified regions image segments to pursue product recognition. Unfortunately, we found that this approach does not suit to the addressed task because in such a highly textured environment proposals tend to isolate logos and very colorful details from the underlying boxes while joining similarly colored regions belonging to different nearby products. By no means, thus, the region proposals provided by state-of-the-art methods employed for object detection can providecorrect segmentations of the individual products placed on store shelves. Therefore, we think that the most reasonable baseline to compare with is given by the first step of our pipeline, i.e. the standard object instance recognition approach based on local invariant features that has been proven to work effectively in a variety of diverse premises. In the following we will followthe processing flow along our pipeline so as to evaluate performance gain upon execution of each step, showing how casting the planogram compliance problem as a subgraph isomorphism can dramatically improve the performance with respect to a standard feature based pipeline.We start with evaluatingthe Unconstrained Product Recognition step, in order to find the best suitable local features to be used in this scenario. We have tested all the detectors and descriptors available in OpenCV, i.e. SIFT<cit.>, SURF<cit.>, ORB<cit.>, BRISK<cit.>, KAZE<cit.>, AKAZE<cit.>, STAR<cit.>,MSD<cit.>, FREAK<cit.>, DAISY<cit.>, LATCH<cit.>, Opponent Color Space Descriptors<cit.>, as well as the the line segments features known as BOLD<cit.>(original code distributed by the authors for research purposes). We have considered features providing both the detector and descriptor (e.g. SIFT) as well as many different detector/descriptor pairs (e.g. MSD/FREAK) and multiple feature processes voting altogether in the same pose space (e.g. BRISK+SURF). A summary of the best results is reported in Figure<ref>. As it can be observed, binary descriptors, such as BRISK and FREAK performs fairly well in the addressed product recognition scenario, yielding the highest Precision and best F-Measure scores. SURF features provide good results alike, in particular as concerns Recall. It is also worth noticing how the use of multiple features, such as BRISK + SURF,to capture different image structures may help increasing the sensitivity of the pipeline, as vouched by the highest Recall. ORB features may yield a comparably high Recall, but at expense of a lower Precision.The use of color descriptors (Opponent SURF), instead, does not seem to provide significant benefits. As the second step is meant to prune out the false detections provided by the first, one would be lead to prefer those features yielding higher Recall. Yet, it may turn out hard for the second step tosolve the sub-graph isomorphism problem in presence of too many false positives. Thus, a good balance between the two types of detection errors turns out preferable, rather.As such, we will consider both BRISK and BRISK+SURF features within the Unconstrained Product Recognition step in order to further evaluate the results provided by our pipeline after the Graph-based Consistency Check step.For the second step we fixed τ=0.25 and deployed the algorithm proposed in <ref>,the results are displayed in <ref>. First, the boost in Precision attained with both types of features compared to the output provided by the first step (<ref>) proves that the proposed sub-graph isomorphism formulation described in <ref> is very effective in robustifying product recognition by removing false detections arising in unconstrained settings. In particular, when using BRISK features, Precision raises from ≈ 78%to ≈ 98% and with BRISK+SURF from ≈ 66% to ≈ 97%. Alongside, though, we observe a decrease in Recall, such as from ≈ 75% to ≈ 74% with BRISK and from ≈ 81% to ≈ 74% with BRISK+SURF. This is mostly due to items that, although detected correctly in the first step, cannot rely on enough self-coherent neighbors to be validated (i.e c(n_I,n_O)<τ). Overall, the Graph-based Consistency Check does improves performance significantly, as the F-Measure increases from ≈ 76% to ≈ 84% and from ≈ 72% to ≈ 84% with BRISK and BRISK+SURF, respectively. Given that in <ref>BRISK slightly outperforms BRISK+SURF according to all the performance indexes and requires less computation, we pick the former features for the fist step and evaluate different design choices as regards the final Product Verification. In particular, as mentioned in <ref>, we considered different template matching and feature-based approaches. The best results, summarized in <ref>, concern template matching by the ZNCC (Zero-mean Normalized Cross Correlation) in the HSV color space, the recent Best-buddies Similarity method <cit.> in the RGB color space and a feature-based approach deploying the same features as in the first step, that is BRISK.As shown in <ref>, using BRISK features in both the first and last step does provide the best results, all the three performance indexes getting now as high as ≈ 90%. Eventually, as for computational efficieny, our system takes at most 15 sec per shelve image with single thread execution on a laptop PC, of which 1 sec is spent searching for the subgraph isomorphism.Eventually, in <ref> we present some qualitative results obtainedby our pipeline both in case of compliance between the observed scene and the planogram as well as in the case of missing products. Additional qualitative results are provided with the supplementary material.§ CONCLUSION AND FUTURE WORK We have shown how deploying product arrangement constraints by an original formulation of the product recognition problem as a sub-graph isomorphism can improve performancedramatically compared to an unconstrained formulation. Accordingly, our proposed pipeline can work effectively in realistic scenarios in which just one model image per product and the planogram areavailable and the given image is not a priory localized with respect to the aisle.Unfortunately, aquantitative comparison to the most relevant previous work <cit.> is not feasible, as the authors used a dataset that cannot be make public. Nonetheless, their dataset seems comparable to ours in terms of number of different products and instances. We report a higher recognition rate (Recall), i.e. 90.2% vs 87.4 %,with a (Precision) as good as 90.4 %. To enable reproducibility of results and foster future work on the topic of product recognition for planogram compliance we will made our annotated dataset public through our project's website. Our pipeline works quite well when applied to textured piece-wise planar products. However, grocery stores and supermarkets usually sells many different categories of products, such as bottles, jars, deformable items or even texture-less objects, like e.g. kitchenware, for which local invariant features are likely to fail in providing enough unconstrained detections to build a reliable Observed Planogram. To address this more challenging scenario, we plan to devise a preliminary product categorization step based on machine (deep) learning to segment the image into regions corresponding to different categories (e.g. piece-wise planar packages, bottles, jars, cans kitchenware..). Purposely, we plan to rely on a similar graph-based formulation to deploy known arrangement constraints (e.g. cans are below jars). Then, each detected segment may be handled by a specific approach to establish upon planogram compliance, the method described in this paper beingapplicable within segments labeled as piece-wise planar products. ieee[pages=1]supplementary.pdf [pages=2]supplementary.pdf [pages=3]supplementary.pdf [pages=4]supplementary.pdf | http://arxiv.org/abs/1707.08378v2 | {
"authors": [
"Alessio Tonioni",
"Luigi Di Stefano"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170726112058",
"title": "Product recognition in store shelves as a sub-graph isomorphism problem"
} |
1]A. M. Shikin 1]D. M. Sostina 1]A. A. Rybkina 1]V. Yu. Voroshnin 1]I. I. Klimovskikh 1]A. G. Rybkin 1]D. A. Estyunin 1,2,3]K. A. Kokh 1,2,4]O. E. Tereshchenko 5]L. Petaccia 5]G. Di Santo 6,7,8]P. N. Skirdkov 6,7,8]K. A. Zvezdin 6,7,8]A. K. Zvezdin 9]A. Kimura 1,10,11,12]E. V. Chulkov 10,11,13]E. E. Krasovskii[1]Saint Petersburg State University, Saint Petersburg, 198504 Russia [2]Novosibirsk State University, Novosibirsk, 630090 Russia [3]V.S. Sobolev Institute of Geology and Mineralogy, Novosibirsk, 630090 Russia [4]A.V. Rzhanov Institute of Semiconductor Physics, Novosibirsk, 630090 Russia [5]Elettra Sincrotrone Trieste, Strada Statale 14 km 163.5, 34149 Trieste, Italy [6]Moscow Institute of Physics and Technology, Institutskiy per. 9, 141700 Dolgoprudny, Russia [7]A.M. Prokhorov General Physics Institute, Russian Academy of Sciences, Vavilova 38, 119991 Moscow, Russia [8]Russian Quantum Center, Novaya St. 100, 143025 Skolkovo, Moscow Region, Russia [9]Graduate School of Science, Hiroshima University,1-3-1 Kagamiyama, Higashi-Hiroshima 739-8526, Japan [10]Departamento de Física de Materiales, Facultad de Ciencias Químicas, UPV/EHU, San Sebastián/Donostia, 20080 Basque Country, Spain [11]Donostia International Physics Center (DIPC), San Sebastián/Donostia, 20018 Basque Country, Spain [12]Centro de Fisica de Materiales CFM - MPC and Centro Mixto CSIC-UPV/EHU, San Sebastián/Donostia, 20080 Basque Country, Spain [13]IKERBASQUE, Basque Foundation for Science, 48013 Bilbao, SpainSynchrotron radiation induced magnetization in magnetically-doped and pristine topological insulators [ December 30, 2023 ===================================================================================================== Quantum mechanics postulates that any measurement influences the state of the investigated system.Here, by means of angle-, spin-, and time-resolved photoemission experiments and ab initio calculations we demonstrate how non-equal depopulation of the Dirac cone (DC) states with opposite momenta in V-doped and pristine topological insulators (TIs) created by a photoexcitation by linearly polarizedsynchrotron radiation (SR) is followed by the hole-generated uncompensated spin accumulation andthe SR-induced magnetization via the spin-torque effect. We show that the photoexcitation of the DC is asymmetric, that it varies with thephoton energy, and that it practically does not change during therelaxation.We find a relation between the photoexcitation asymmetry, the generatedspin accumulation and the induced spin polarization of the DC and V 3d states. Experimentally the SR-generated in-plane and out-of-plane magnetization is confirmed by the k_∥-shift of the DC position and by the splitting of the states at the Dirac point even above the Curie temperature. Theoretical predictions and estimations of the measurable physical quantities substantiate the experimental results. The photoexcitation by laser or synchrotron radiationis accompanied by a depopulation of the initial states, which influences theelectronic structure observed in photoemission (PE)measurements. In materials with helical spin structure (for instance, topologicalinsulators (TIs) <cit.>) the imbalance in photoexcitation of the DC states with opposite momenta created, forinstance, by circularly polarized laser or SR can be effectively used for thegeneration of the surface spin-polarized currents that depend on the helicity of the radiation polarization<cit.>.Similar to the case of an electric field applied in the surface plane<cit.>, this can induce a magnetization in ferromagnetic TIs <cit.>.The induced magnetic moment opens a gap at the Dirac point (DP) due to Time ReversalSymmetry (TRS) breaking providing a platform for the realization of uniquequantum phenomena such as quantized magneto-electric effect<cit.>and quantum anomalous Hall effect <cit.> at elevated temperatures under light excitation. Although, the possibility of induced (andcontrolled) magnetization by linearly polarized SR has not been studied yet, the generation ofspin-polarized current was noted recently<cit.>. The presentwork aims to investigate a possibility of such SR-induced in-plane and out-of-plane magnetization in FM-doped and pristine TIs by linearly polarized SR.We relate this phenomenon to an asymmetry in the depopulationof the DC states with opposite momenta, which leads to a hole-generated uncompensated spin accumulation. The possibility of a long-living electron-hole separation between the excited electrons and generated holes(related to a reduced electron-phonon interaction at the surface<cit.>) is confirmed by a series of time-resolved laser experiments(see, for instance, <cit.>).The holes generated at the topological surface states (TSSs) are compensated bya drift of electrons from the TSSs out of the beam spot via long-time two-dimensionalrelaxation process <cit.>. In the case of a different probability of the photoexcitation of electrons from the DC states with opposite momenta, theuncompensated spin accumulation and the zero-bias spin-polarized photocurrent occur, as in Refs. <cit.>, which can lead to induced magnetization similar to the case discussed in Ref. <cit.>for the (Ga,Mn)As ferromagnetic semiconductor.The issue of SR-generated spin accumulation is strongly related to the TSS spin textureand to the asymmetry of the photoexcitation of the TSSs with opposite spinorientation. In Refs. <cit.>, it was shown that a total spin textureof TSSs includes not only contributions of thep_z-orbitals, but also of the p_x- and p_y-orbitals related to the radial andtangential components in the spin-orbital texture, which significantly modifies the spin texture of the TSSs probedby photoemission. Total spin structure in PE spectra depends onthe sum of all contributions determined by optical selection rules,quantum interference and SR incident angle <cit.>. Due to variation of k_z with photon energy all p_x, y, z components “oscillate” with different phases. The non-trivialcharacter of TSS spin texture is confirmed by spin-resolved photoemission both via theoretical analysis<cit.> and via experimental measurements <cit.> using different polarization of SR <cit.> and manifests itself in significant modification of the spin polarization of photoelectrons with photon energy <cit.>. The oblique incidence of SR breaks the symmetry of the angular distribution of the photocurrent, and the different photoemission intensity at k_∥ and -k_∥<cit.> can be a source of the SR-generated uncompensatedspin accumulation and an induced magnetization. This problem is especiallyimportant for magnetically-doped TIs because the DP gap opening due to the TRS breaking, its value andorigin (magnetization- or hybridization-derived) are being actively discussed, see, for instance, Ref. <cit.>.However, without a proper analysis of the influence of the non-equal depopulation of the TSSs on a possible induced magnetization during ARPES and SARPES measurements such questions cannot be answered.In the present work we study the PE intensity asymmetry of the DC states and connect it with the SR-generatedasymmetric k_∥-distribution of the holes and the resulting spin accumulation leadinga local induced magnetization via spin-torque effect. The first part of thework aims to clarify how the photoexcitation by a linearly polarized SR influencesthe DC states intensity distribution in the ARPES intensitymaps and how it varies with photon energy. In the second part we analyze how theimbalance in the depopulation of the TSSs in V-doped TI and thecorresponding SR-generated uncompensated spin accumulation result in an inducedin-plane and out-of-plane polarization of TSSs and the V 3d-ions. Then,the laser pump-probe experiment allows us to conclude that the imbalancein the depopulation of the TSSs is practically not changed during the relaxation,and the SR-induced magnetization can be estimated in rough approximation using the TSS intensity asymmetry in PE spectra.§ ASYMMETRY IN THE INTENSITY OF THE DC STATES VS PHOTON ENERGY In this work we study a series of pristine and V-doped TIs with fractionalstoichiometry based on Bi_2Te_2Se with inclusion of different Sb-concentrations.They have a wide insulating energy gap with the DP inside the bulk gap <cit.> and anenhanced surface contribution to the spin transport <cit.>, which is important for spintronics applications.Figures <ref>(a,b,c) show the photon energy dependence of the TSS ARPES energy-momentum intensity maps measured along the direction of the surface Brillouin zone using linearly p-polarized SR for pristine and for magnetically-doped TIs with the following stoichiometries: (a) Bi_1.37Sb_0.5Te_1.8 Se_1.2, (b) Bi_2Te_2Se, and (c) Bi_1.37V_0.03Sb_0.6Te_2Se. In Fig. <ref>(g), the ARPES dispersions measured along the direction for Bi_1.37Sb_0.5Te_1.8Se_1.2 are shown forcomparison. The geometry of the experiments is schematically presented in Methods Fig. <ref>.For the presented experiments the Geometries 1 and 2 were used with the analyzer entrance slit orientation along and perpendicular to the SR incidence plane. For Geometry 1 two different photon incidence angles have been used: 73 (a) and 50 (b),(c) relative to the surface normal (with measurements along ). The intensity maps for Bi_1.5Sb_0.5Te_1.8Se_1.2 along , for Bi_1.4Sb_0.6Te_2Se along and for Bi_1.37V_0.03Sb_0.6Te_2Se along are presented in Supplementary Figs. 1S, 2S and 3S, respectively, with different orientation of the SR incidence plane. For Geometry 2, the incidence angles was 45 (g) with measurements along , orthogonal to the SR incidence plane (oriented along ).Below each ARPES intensity map the profiles of the TSS intensities are presentedfor the constant-energy cut of the upper DC atthe binding energy corresponding to a high intensityof the TSSs, see white lines in the ARPES maps. In all the profiles the intensities at opposite k_∥ points are different. At some photon energies the intensity at positive k_∥ is larger than at negative k_∥, andat other photon energies the relation is opposite.Figs. <ref>(d),(e),(f) collect all data measured for different TIs and demonstrate the photon energy dependence of the asymmetry in the intensity of the opposite TSS branches. The TSS intensity asymmetries are characterized by the ratioA = I(-k_∥)-I(k_∥)/I(-k_∥)+I(k_∥),measured at the energies marked by white horizontal lines. The asymmetry A oscillates with photon energy both for pristine and for V-doped TIs. In order to explain the observed k_∥-distribution of the photocurrent and itsvariation with the photon energy we have calculated ab initio the energy-momentumdistribution of the photoemission intensity from the DC of the stoichiometric compound Bi_2Te_2Se. The electronic structure ofBi_2Te_2Se <cit.> is quite similar to that of the crystals with thefractional stoichiometry studied here. We use the one-step theory of photoemission in the dipole approximation, so the photocurrent from the state |𝐤_∥⟩is proportional to the transition probability |⟨Φ|-i∇_𝐞|𝐤_∥⟩|^2to the time-reversed low energy electron diffraction state |Φ⟩ <cit.>, where-i∇_𝐞 is the momentum operator in the direction of the light polarization 𝐞.The final state |Φ⟩ is calculated for the scattering of electrons on a semi-infinitecrystal as explained in Ref. <cit.>. The inelastic scattering is included byadding a spatially constant imaginary part V_ i=1 eV to the crystal potential. The crystalpotential was obtained within the local density approximationwith the full-potential linearaugmented plane wave method <cit.>. The initial states were calculated within a two-componentrelativistic formalism <cit.> for a slab composed of 7 quintuple layers. Figs. <ref>(a-c)show the k_∥-distribution of the calculated photoemission intensity from the DC of Bi_2Te_2Se forthe opposite directions and Figs. <ref>(d-f) for the opposite directions. In both cases the p-polarized SR is incident along (Geometry 1). The k_∥-integratedintensities for and are shown in Fig. <ref>(g) and <ref>(h), respectively. The corresponding asymmetry index for the light incident along 𝐤_∥ is shown in Fig. <ref>(i)and for the light incident perpendicular to 𝐤_∥ (Geometry 2) in Fig. <ref>(j). Note that when the light is incident along 𝐤_∥ the difference between+k_∥ and -k_∥ is due to a linear dichroism, and it strongly depends onthe angle of incidence, see Fig. <ref>(i). In particular, along Γ̅K̅the opposite directions are equivalent owing to the C_3v symmetry of the surface, andit is the light that breaks the symmetry of the experiment. On the contrary, for the light incident perpendicular to 𝐤_∥, Fig. <ref>(j), the intensity asymmetry is due to the inequivalence of +k_∥ and -k_∥ along Γ̅M̅. In that case the photon energy dependences I_ left(hν) and I_ right(hν) are very similar, see Fig. <ref>(h), and the intensity asymmetry does not strongly dependon θ, see Fig. <ref>(j). The theory explains the experimentally observed oscillationsof the asymmetry index with the photon energy and relates them toenergy variations of the finalstate |Φ⟩.§ HOLE GENERATION ANALYSIS AND ASYMMETRY VARIATION DURING RELAXATION Let us assume that at the used SR energies the excited spin-polarized photoelectrons escape to a considerable degree into the vacuum. Then, the uncompensated spin accumulation generated by the different photoexcitation rate of electrons with opposite momenta and its orientation are mainly determined by the asymmetry in the concentrationof the photo-holes in the opposite TSS branches.How to determine thisasymmetry? In the simplest approximation one can infer the photo-holeconcentration asymmetry and the related uncompensated spin accumulation from the asymmetry of the photocurrent from the DC states.The second question is whether the asymmetry in the concentration of the photo-holes is preserved during the relaxation process.To clarify this we carried out a time-resolved pump-probe laser experiment, as in Refs. <cit.>. The modification of the TSS intensity asymmetry measured for Bi_1.97V_0.03Te_2.4Se_0.6 immediately after the photoexcitation (just under the laser pump pulse generation) and during the relaxation of photoexcited electrons is presented in Fig. <ref>(a). The scheme and geometry of the experiment <cit.> are presented in Supplementary Fig. 4S. The probe pulse of hν= 5.9 eV was linearly p-polarized. The pump pulse was s-polarized with hν=1.48 eV. Below each ARPES map the intensity asymmetry profiles of the opposite DC branches are shown. These asymmetries were measured at an energy near the bottom of the conduction band (CB) marked by the horizontal dashed lines. These data show the time evolution of the intensity distribution of both the photoexcited electrons (above the Fermi level) and the de-occupied DC states (below the Fermi level). The variation of the depletion of the TSS intensity below the Fermi level can be an indicator of the hole generation and relaxation. One can see thatthe TSS intensity asymmetry observed at the moment of the electron photoexcitationis practically not changed during relaxation.The unchanged with time TSS asymmetry is observed also for other energy cuts,see Supplementary Fig. 5S. It allows us to make a very important qualitative conclusionthat within this approximation the photo-hole generated asymmetry is not significantly transient with time.In other words, the asymmetry in the measured PE intensity of the TSSs with opposite momenta can be used for a rough estimate of the induceduncompensated spin accumulation and the related magnetization, as we discuss in detail in the next section.§ ESTIMATIONS OF THE IN-PLANE AND OUT-OF-PLANE SPIN POLARIZATION (MAGNETIZATION) INDUCED IN V-DOPED AND PRISTINE TI BY LINEARLY POLARIZED SR In the following we discuss the theoretical estimations of the magnetization induced by linear p-polarized SR due to the generated uncompensated spin accumulation. First of all let us consider the symmetry of the spin accumulation induced by linearly polarized SR. Taking into account that this effect is caused by photoexcitation, the spin density Ṡ⃗̇ can be considered as quadratic by the field:Ṡ⃗̇ = B_ikl E_k E_l,where B_ikl is a 3rd rank pseudotensor and E_k,l are the components of electric field. Following to Refs. <cit.> the Bi-based compounds are referred to the space group D^5_3d(R3m). In the presence of a [111] surface, the symmetry of the considered four-component complex compound can be reduced to C_3, and therefore B_ikl is equal <cit.>:B_ikl = [B_11 -B_11 0B_14B_15 -B_22; -B_22B_22 0B_15 -B_14 -B_11;B_31B_31B_33 0 0 0 ],where B_i μ = B_ikl (kl ↔μ = 1, ... , 6) are the coefficients determined by the material. Considering the SR radiation as E⃗= (E cosψ, 0, E sinψ)^T, from the Eq. <ref> it immediately follows that SR can excite both the in-plane and out-of-plane uncompensated spin accumulation. Assuming that the linearly polarized SR can be decomposed on right and left circularly polarized SR, which of them can depopulate mostly one of the Dirac cone branches <cit.>, the averaged uncompensated spin accumulation can be represented empirically as δ S_x,z= ħ/2ξ_x,z P τ A, where P is the probability of the electron photoexcitation per unit time, τ is the decoherence time of the spins and ξ_x,z is an empirical constant. It should be noted that P τ is the steady-state concentration of the generated holes. In the simplest case of semiconductor optical orientation ξ_x^0 = sinψ and ξ_z^0 = cosψ <cit.>. For simplicity, we assume that ξ_x,z = κ_x,zξ_x,z^0, where κ_x,z≈ 1. Assuming that the magnetization of electron sub-system induced by linearly polarized SR can be estimated as m_x,z= μ_B κ_x,zξ_x,z^0 P τ A, where μ_B is the Bohr magneton and A is the TSS photoexcitation asymmetry (see also <cit.>). In the case of the V-impurity subsystem, below the Curie temperature (T < T_C) the total energy can be represented as E_V=-K_U (m_z^V)^2 - (m⃗^V ·H⃗_SR) where K_U is the constant of uniaxial anisotropy, m_z^V is the z-axis component of the V-subsystem magnetization, H⃗_SR=H_SR(κ_x sinψ , 0 ,κ_z cosψ)^T is the field acting on the V impurities from the SR induced magnetization, H_SR= ã^2/μ_B J_eV P τ A, where ã=4.24 Ȧ is the lattice constant typical of the studied TIs, J_eV≈ 0.3 eV <cit.> is the exchange constant, which describes the (s-d) interaction between the TSSs and the V-ion impurities system. In case (T > T_C) the energy minimization leads to the following expression for the V-subsystem magnetization:m⃗^V = g μ_B S N B_S ( g μ_B S H_SR / T )(sinη , 0 ,cosη)^T,where g≈ 2 is the g-factor, S=3/2 is the spin of impurity ion, N is the averaged impurity concentration, B_S is the Brillouin function and η is slightly differs from ψ and sinη = κ_x sinψ / √(κ_x^2 sin^2 ψ + κ_z^2 cos^2 ψ). In case of (T < T_C) we also have to consider anisotropy, but estimations proves that the anisotropy term is significantly smaller than the SR term (see Supplementary Inform. for details), therefore, Eq. <ref> is valid with a good accuracy for all temperature range. The dependence of the in-plane component of the total magnetization M⃗=m⃗+m⃗^V (under experimental conditions ψ = 50^ and P τ≈ 3.5× 10^13 cm^-2) on the in-plane asymmetry value (A) for different temperatures is represented on Fig. <ref>(a).The complete electron Hamiltonian of the considered system including both electron-electron and electron-vanadium interactions in the mean field approximation can be written as:Ĥ_e = ħ V_D [ k⃗×σ⃗]e⃗_z + ã^2/μ_B J_eV(m⃗^V ·σ⃗) + ã^2/μ_B U (m⃗·σ⃗),where V_D ∼ 5.3× 10^7 cm/s is the velocity taken from the TSS dispersion law, σ⃗ is the vector of Pauli matrices and U ≈ 0.2 eV is the Hubbard parameter. The energy spectrum in this case has the following form:E= ±√(ħ^2 V_D^2 |k⃗|^2+Δ_x^2+Δ_z^2-2ħ V_D k_y Δ_x),where Δ_x = ã^2/μ_B J_eV m_x^V + ã^2/μ_B U m_x Δ_z = ã^2/μ_B J_eV m_z^V + ã^2/μ_B U m_z The resulting band structure modification under influence of the induced out-of-plane and in-plane magnetization is shown in Fig. <ref>(b) (under experimental conditions noted above and the averaged value of the asymmetry of A=0.5 taken from Fig. <ref>(a)). The out-of-plane magnetization is accompanied by the splitting of the DC states at the DP. The induced in-plane magnetization emerges in the k_∥-shift of the DC in the direction orthogonal to the induced magnetic field (or magnetization). (Analogous k_∥-shift is observed under external applied in-plane magnetic field <cit.>). Corresponding calculated temperature dependence of the DP-gap value and the k_∥-shift of the DP position in the direction orthogonal to the magnetization are presented in Figs. <ref>(c,d) (see the discussion below). §.§ Experimental confirmation of the induced in-plane magnetization As an experimental evidence of the SR-induced in-plane magnetization,Fig. <ref> demonstrates a correlation between themodification of the experimental intensity maps of the upper DC states close to the Fermi level (line <ref>(a)) and the (k_x,k_y)-shift of the DP position induced by the in-plane magnetic field generated by SR with linear p- and opposite circular polarizations (line <ref>(b)), which is expected in accordance with Fig. <ref>(b). The incident direction of SR corresponds to the vertical line. For each of the DC energy cuts (Fig. <ref>(a)) the TSS intensity profiles in the k_x, k_y-directions are presented, in the bottom and right graphs, respectively.These profiles clearly demonstrate a pronounced modulation of the intensity of the DC states intensity and their asymmetry both along and orthogonally to the SR incidence plane. The maps were measured with hν=28 eV for Bi_1.37V_0.03Sb_0.6Te_2Se kept at the temperature of 55K. The DP positions (Fig. <ref>(b)) were estimated from the maximal intensity of the TSS in k_x and k_y, see the profiles at the bottom and on the right side of the maps cut at the DP. The different depopulation of the opposite DC states under photoexcitation with different polarization of SR is well visible in Fig. <ref>(a), and itleads to the (k_x,k_y)-shift of the DC according to the direction of induced magnetic field, which is determined by the asymmetry in the TSS intensity. The direction of the uncompensated spin accumulation and of the induced in-plane magnetic field are determined by the direction where the TSS intensity asymmetry is oriented and by the details of the spin texture <cit.>. The relation between the experimental TSS intensity asymmetry (A), the direction of the induced magnetic field (M) generated by uncompensated spin accumulation (S) in correspondence to spin texture and direction of the DP position(k_x, k_y)-shift is schematically shown in Fig. <ref>(c). For circularly polarized SR a pronounced asymmetry in the TSS intensity is observed in the direction perpendicular to the SR incidence. The use of the opposite circular polarizations leads to the TSS intensity asymmetry and generated uncompensatedspin accumulation with spin orientation (and corresponding magnetic moment)in opposite k_∥-directions. This direction of theinduced magnetic field determines the shift of the DP orthogonally to the SR incidence (k_x) that is confirmed experimentally (see the shift of blue crosses in comparison with the green one). For linear p-polarization of SR the TSS intensity asymmetry is observed in the direction along the SR incidence. This leads to a spin accumulation (S) and magnetic field (M) perpendicular to the SR incidence. As a result, the k_y-shift of the DP position is observed in the direction orthogonal to that in the case of a circularly polarized SR. The measurements at room temperature (Supplementary Fig. 6S) and at 30K at the 9B beamline HiSOR (Hiroshima, Japan) albeit with a lower intensity of SR (not shown) demonstrate similar behavior. The value of the k_∥-shift of the Dirac cone induced by linearly-polarized SR shown in Fig. <ref> (in comparison with the DC position under excitation by circularly polarized SR) can be estimated to approximately5-10×10^-3 Å^-1 in agreement with the theoretical estimation of the k_∥-shift in Fig. <ref>(b). Lower value of the k_∥-shift of the DP position at room temperature presented in Supplementary Fig. 6S confirms the calculated temperature dependence of the DP k_∥-shift shown in Fig. <ref>(d). Under excitation by a circularly polarized SR of opposite chirality the k_∥-shift of the DC position is stronger, which is related to the enhanced TSS intensity asymmetry (see corresponding profiles).Additionally, the in-plane magnetization induced by a linearly polarized SR can be confirmed by the k_∥-shift of the spin-polarized DC states relative to the non-spin-polarized CB states.(A similar k_∥-shift under applied magnetic field was noted in Ref. <cit.>). Supplementary Fig. 7S demonstratesthat such k_∥-shift is actually observed at different photon energies, andit is important that the direction of thek_∥-shift is related to the observed TSS intensity asymmetry. It is interestingthat the value of the k_∥-shift relative to k_∥=0 can be also estimated to be about 10×10^-3 Å^-1, which correlates with the estimations noted above. Moreover, when the sign of the asymmetry changes with photon energy the DC shifts in the opposite direction.This confirms that the observed shift is really connected with the TSS intensity asymmetry and the resulting induced in-plane magnetization direction. The k_∥-shift of the DC branches relative the CB states located at k_∥=0 is also observed in the case of photoexcitation by laser radiation. In that case the profiles shown in Fig. <ref>(b) below the ARPES dispersion map (cut at the energy marked by white line which crosses both DC and CB states) demonstrate non-equal distance from the left and right side.As a partial conclusion we have experimentally observed the DP position (k_x,k_y)-shifts in accord with the measured asymmetry of TSS intensity confirming that the non-equal depopulation of these states induces the in-plane magnetic fields under photoexcitation by SR. §.§ Out-of-plane induced magnetization and its experimental confirmation Let us now discuss the problem of the out-of-plane magnetization induced by linearly polarized SR.Fig. <ref>(a) shows the calculated photon energy dependence of the out-of-plane net-spinphotocurrent S(k_∥,hν)=I^↑(k_∥,hν)-I^↓(k_∥,hν)from the upper DC of Bi_2Te_2Se with the SR incident along . The out-of-planenet-spin-photocurrent integrated over k_∥ does not vanish, and its photon energy dependence is shown in Fig. <ref>(b).The magnitude and the sign of the integral photocurrent are seen to vary with the photon energy, which suggests that also the total out-of-plane spin accumulation may be different for different photon energies. This out-of-plane spin accumulation generates the induced out-of-plane magnetization that should liftthe degeneracy of the TSSs and open the energy gap at the DP due to the TRS breaking, as in Refs. <cit.> under circularly polarized SR. Using Eq. <ref> one can estimate the energy gap as Δ = 2ã^2/μ_B J_eV m_z^V + 2ã^2/μ_B U m_z. Fig. <ref>(c) shows the calculated temperature dependence of the gap induced at the Dirac point using the value of the TSS asymmetry of 0.3 (as averaged value taken from the asymmetry values presented in Fig. <ref>(a)and Fig. <ref>(e) at hν=28-30 eV) both for magnetically-doped and for pristine TIs at the experimental conditions noted above (see Methods for details).For magnetically-doped TI below 30 K a gap of about 25 meV is expected. As temperature increases the gap decreases down to 12-15 meV at room temperature. A finite gap above the Curie temperature is just related to the out-of-plane magnetization generated by SR. In the case of pristine TI a formation of the gap of 4.5 meV is expected too, independently on temperature. Unfortunately the energy resolution in our conditions was not high enough to allow the measurement of this small gap value within a reasonable experimental error.The experimentally measured in-plane and out-of-plane spin polarization is presentedin spin-resolved spectra in Figs. <ref>(a,b). The spectra were measured for Bi_1.31V_0.03Sb_0.66Te_2Se at the temperature of 23 K at the DP by usinglinear p-polarized SR. The corresponding experimental polarization asymmetry is plotted in the bottom part of the graphs.The spin orientation was measured along the SR incidenceplane (Geometry 1). The related ARPES intensity map is shown in the insetin the upper central part. The spin-resolved spectrum in Fig. <ref>(a)confirms the in-plane spin-polarization of the states in the region ofthe DP, which is inverted relative to the DP (seeblack arrows on the polarization asymmetry). The observed in-plane polarization along the SR incidence plane is determined by the contribution of the p_x, p_y components in the spin texture. Similar effects are described in Refs. <cit.>. The generated out-of-plane polarization is shown in Fig. <ref>(b). The presented spin-resolved spectrum demonstrates availability of the spin-polarized states at the Fermi level. We ascribe these states to the V 3d-resonances characterized by the out-of-plane spin polarization. It is known that for V-doped TIs the maximal intensity of the V 3d-ion states is located near the Fermi level <cit.>. The magnetically-doped TIs are characterized by a colossal anisotropy of the magnetic moment induced at the FM-impurity atoms, which favors the magnetization perpendicular to the surface. Therefore, the observed out-of-plane spin polarization of the states near the Fermi level can signify the out-of-plane magnetization of the V 3d-ions.The SR-induced out-of-plane magnetization can be also experimentally confirmed by the splitting of the TSSs and opening of the energy gap at the DP both below and above the Curie temperature (see discussion above). Figs. <ref>(c,d) demonstrate the spin-integrated spectra measured directly at the DP for V-doped TIs at the temperatures of 1 K and 66 K, below and above the Curie temperature, respectively (which is below 5-10 K), in comparison with that measured for pristine TI at room temperature – Fig. <ref>(e). The relevant ARPES intensity maps are presented above each spectrum to show the DP positions. The fitting procedures for the spectra measured for V-doped TIs (Figs. <ref>(c,d)) show a decomposition into two spectral components with the energy splitting of about 20-25 meV at the DP below and above the Curie temperature. The width of the components was chosen in accord with the width of the TSS peaks outside the DP extracted from the ARPES intensity maps. The splitting into two components for the states at the DP above the Curie temperature (when a spontaneous magnetic order is destroyed) allows us to conclude that the out-of-plane magnetization is induced by SR. The spectra for other TIs presented in Supplementary Fig. 8S demonstrate similar behavior with the splitting of the TSSs at the DP. At the same time, the spectra measured at the DP for pristine TI (Fig. <ref>(e) and Supplementary Fig. 8S) do not show a noticeable splitting at the DP. The fitting procedure for these spectra shows only a one-component structure.The splitting of 4.5 meV predicted by theoretical estimations for pristine TI (Fig. <ref>(b)), is not resolved in our experiment. Thus, for TI without magnetic doping the photoexcitation by SR does not lead to a noticeableout-of-plane magnetization. Therefore, one can conclude that the splitting of the states at the DP in magnetically doped TIs can actually be an indicator of the induced out-of-plane magnetization generated by the linearly-polarized SR. In summary, we have demonstrated that the asymmetry of the DC states with opposite spin orientation in photoexcitation by p-polarized SR is accompanied in pristine and magnetically-doped TIs by an uncompensated spin accumulation of the generated holes in the initial states. This leads to the in-plane and out-of-plane polarization of the TSSs and the V 3d-ions and corresponding magnetization via the spin-torque effect. Experimentally it is indicated by spin-resolved PE spectra and is confirmed by the k_∥-shift of the DC position under the induced in-plane magnetic field and by the splitting of the TSSs at the DP induced by the out-of-plane component of magnetization even above the Curie temperature. The laser pump-probe experiment has shown that the difference in depopulation of the opposite branches of the DC states is practically not changed during the relaxation process. It allows to conclude that the SR-induced magnetization can be roughly estimated using the asymmetry in the intensity of the TSSs in PE spectra. Theoretical estimations have confirmed a possibility of the induced in-plane and out-of-plane magnetization by linearly polarized SR.This finding should be taken into account in PE investigations of systems with helical spin structure, especially for magnetically-doped TIs, where the PE process can influences the spin structure in the ground state depending on photon energy and experimental details. § METHODSThe measurements of ARPES intensity maps for the DC states in pristine and magnetically-doped TIs presented in Fig. <ref> and Supplementary Figs. 1S–3S were carried out at i3 beamline at MAXlab (Lund, Sweden) and BaDEIPh beamlineat Elettra (Trieste, Italy) in the direction along the SR incidence plane (Geometry 1 in Fig. <ref>)and 1^2 end station at BESSY II (Helmholtz-Zentrum Berlin, Germany) in the direction perpendicular to the SR incidence plane (Geometry 2 in Fig. <ref>) using a Scienta R4000 or SPECS Phoibos 150 analyzers.The incidence angle of SR for these experiments was 73 (MAXlab) and 50 (Elettra and BESSY II)relative to the surface normal.The spin-resolved photoemission spectra for V-doped TIs were measured at the COPHEE setup at Swiss Light Source, Switzerland (Figs. <ref>(a,b))and at the i3 beamline at MAXlab, Sweden (Supplementary Fig. 8S) with the spinorientation along the plane of the SR. The spin-resolved photoemission spectrawere measured both for the in-plane and for the out-of-plane spin orientation.To increase the intensity of the TSSs in the region of the DP we used a photon energy rangeof 28-30 eV. For other photon energies the relative contributionof the DC states in the region of the DP is reduced. The SR incidence angle for these experiments was45 and 73.The DC energy cut maps of the TSSs for Bi_1.37V_0.03Sb_0.6Te_2Se (Fig. <ref> and Supplementary Fig. 6S) were measured at 1^2 station at BESSY II (Helmholtz-Zentrum Berlin, Germany)with a photon energy of 28 eV keeping the sample at the temperature of 55K and room temperature, respectively.The SR incidence angle was 50.The ARPES dispersion maps for magnetically-doped TIs, which were used for careful estimation of the splitting of the TSSs at the DP (Fig. <ref>) were measured at i3 beamline at MAXlab (Lund, Sweden), at the 9B beamline at HiSOR (Hiroshima, Japan) in the direction along the SR incidence plane (Geometry 1) and at 1^3 station at BESSY II (Helmholtz-Zentrum Berlin, Germany) in the direction perpendicular the plane of the SR incidence (Geometry 2). The SR incidence angle was 45 relative to the surface normal.The time-resolved pump-probe laser experiment (Fig. <ref>) was carried out in ISSP at Tokyo University (Japan) for V-doped TI with stoichiometry Bi_1.97V_0.03Te_2.4Se_0.6. Time-resolved photoemission apparatus achieving sub-20-meV energy resolution and high stability was used, see details in Ref. <cit.>. The probe pulse was linearly p-polarized and with a photon energy of 5.9 eV. The pump pulse was s-polarized with hν=1.48 eV. Geometry of the experiment is presented in Supplementary Fig. 4S. The laser beam incidence angle was 45 relative to the surface normal.Part of work was carried out in the resource center “Physical methods of surface investigation” (PMSI) of Research park of Saint Petersburg State University.The single crystals of pristine TIs Bi_1.5Sb_0.5Te_1.8Se_1.2,Bi_2Te_2Se and Bi_1.4Sb_0.6Te_2Se, and magnetically-doped TIs Bi_1.37V_0.03Sb_0.6Te_2Se,Bi_1.3V_0.04Sb_0.66Te_2Se, and Bi_1.97V_0.03Te_2.4Se_0.6 were synthesized by using a modified vertical Bridgman method <cit.>. Clean surfaces of the TIs were obtained by a cleavage in ultrahigh vacuum. The base pressure during the experiments was better than 1×10^-10 mbar.§ ACKNOWLEDGEMENTS The authors acknowledge support by the Saint Petersburg State University (grant No. 15.61.202.2015), Russian Science Foundation grants No. 17-12-01333 (in the part of theoretical study of magnetic properties and ARPES analysis) and Russian Science Foundation grant No. 17-12-01047 (in part of crystal growth and the sample characterization). The work was also supported by the Spanish Ministry of Economy and Competitiveness MINECO (Project No. FIS2016-76617-P), German-Russian Interdisciplinary Science Center (G-RISC) funded by the German Federal Foreign Office via the German Academic Exchange Service (DAAD) and Russian-German laboratory at BESSY II (Helmholtz-Zentrum Berlin). The authors kindly acknowledge the MAXLab, HiSOR, SLS, Elettra, BESSY II, ISSP of the University of Tokyo and PMSI staff for technical support and help with experiment and useful discussions.§ AUTHOR CONTRIBUTIONSThe project was proposed by A.M.S, A.K.Z. The ARPES measurements were performed by I.I.K., D.M.S., V.Y.V., D.A.E., A.A.R., L.P., G.D.S., A.K., O.E.T., A.G.R. and A.M.S. The time-resolved ARPES measurements were performed by V.Y.V., A.K., Y.I., A.G.R. and A.M.S. Samples were synthesized and characterized by O.E.T., K.A.K. The experimental data analysis was carried out by A.M.S., A.A.R., D.M.S., V.Y.V., D.A.E. and I.I.K. Theoretical estimations and analysis of magnetic properties were performed by P.N.S., K.A.Z., A.K.Z. Calculations of PE spectra, ARPES dispersion maps and the TSS intensity asymmetry at different photon energies were performed and analysed by E.E.K., E.V.C. All authors extensively discussed the results and participated in the manuscript editing. 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"authors": [
"A. M. Shikin",
"D. M. Sostina",
"A. A. Rybkina",
"V. Yu. Voroshnin",
"I. I. Klimovskikh",
"A. G. Rybkin",
"D. A. Estyunin",
"K. A. Kokh",
"O. E. Tereshchenko",
"L. Petaccia",
"G. Di Santo",
"P. N. Skirdkov",
"K. A. Zvezdin",
"A. K. Zvezdin",
"A. Kimura",
"E. V. Chulkov",
"E. E. Krasovskii"
],
"categories": [
"cond-mat.mtrl-sci"
],
"primary_category": "cond-mat.mtrl-sci",
"published": "20170727094639",
"title": "Synchrotron radiation induced magnetization in magnetically-doped and pristine topological insulators"
} |
^1QOLS and QuEST, Blackett Laboratory, Imperial College London, London SW7 2AZ, United Kingdom^2Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, 1040 Vienna, Austria^3Korea Institute of Advanced Study, Dongdaemun-gu, Seoul, 02455, South Korea We analyze a multiqubit circuit QED system in the regime where the qubit-photon coupling dominates over the system's bare energy scales. Under such conditions a manifold of low-energy states with a high degree of entanglement emerges. Here we describe a time-dependent protocol for extracting these quantum correlations and converting them into well-defined multipartite entangled states of noninteracting qubits. Based on a combination of variousultrastrong-coupling effects the protocol can be operated in a fast and robust manner, while still being consistent with experimental constraints on switching times and typical energy scales encountered in superconducting circuits. Therefore, our scheme can serve as a probe for otherwise inaccessible correlations in strongly coupled circuit QED systems. It also shows how such correlations can potentially be exploited as a resource for entanglement-based applications. Harvesting Multiqubit Entanglement from Ultrastrong Interactions in Circuit Quantum Electrodynamics P. Rabl^2 December 30, 2023 ===================================================================================================Cavity QED is the study of quantum light-matter interactions with real or artificial two-level atomscoupled to a single radiation mode. In this context one is usually interested in strong interactions between excited atomic and electromagnetic states, whilethe trivial ground state, i.e. the vacuum state with no atomic or photonic excitations, plays no essential role. This paradigm has recently been challenged by a number of experiments <cit.>, where interaction strengths comparable to the photon energy have been demonstrated.In particular, in the field of circuit QED <cit.>, a single superconducting two-level system can already be coupled ultrastrongly <cit.> to a microwave resonator mode <cit.>. In this regime the physics changes drastically and even in theground state various nontrivial effects like spontaneous vacuum polarization <cit.>, light-matter decoupling <cit.> and different degrees of entanglement <cit.> can occur. However, compared to the vast literature on cavity QED systems in the weakly coupled regime, the opposite limit of extremely strong interactions is to a large extent still unexplored. As a consequence, ideas for how ultrastrong coupling (USC) effects can be controlled and exploited for practical applications are limited <cit.>. In this Letter we consider a prototype circuit QED system consisting of multiple flux qubits coupled to a single mode of a microwave resonator. It has recently been shown that in the USC regime this circuit exhibits a manifold of nonsuperradiant ground and low-energystates with a high degree of multiqubit entanglement <cit.>. This entanglement, however, is a priori not of any particular use, since any attempt to locally manipulate or measure the individual qubits would necessarily introduce a severe perturbation to the strongly coupled system. For this reason we describethe implementation of an entanglement-harvestingprotocol <cit.>, which extracts quantum correlations from USC states and converts these correlations into equivalent multipartite entangled states of decoupled qubits. The protocol combines adiabatic and nonadiabatic parameter variations and exploits the counterintuitive decoupling of qubits and photons at very strong interactions <cit.> to make the entanglement extraction scheme intrinsically robust and consistent with experimentally available tuning capabilities. The extracted Dicke and singlet states belong to a family of robust multipartite entangled states <cit.> and form, for example,a resource for Heisenberg-limited metrology applications <cit.>.More generally, our analysis shows, how the interplay between different USC effects can contribute to the realization of non-trivial control tasks in a strongly interacting cavity QED system. Model.—We consider a circuit QED system as shown in Fig. <ref>(a), where a single mode LC resonator with capacitance C and inductance L is coupled collectively to an even number of N=2,4,6,… flux qubits. This circuit is described by the Hamiltonian <cit.>ℋ=Q_r^2/2C+(Φ_r- Φ_0 ∑_i=1^Nφ_i)^2/2L+∑_i=1^N H_q^(i),where Q_r and Φ_r are charge and generalized flux operators for the resonator obeying [Φ_r,Q_r]=iħ, and Φ_0=ħ/(2e) is the reduced flux quantum. For each qubit, H_q^(i) denotes the free Hamiltonian and φ_i is the difference of the superconducting phase across the qubit's subcircuit. As usual we assume that the qubit dynamics can be restricted to the two lowest tunneling states |↓⟩ and |↑⟩ of a symmetric double-well potential [cf. Fig. <ref>(b)].Under this approximation and writing Φ_r=√(ħ/(2C ω_r)) (a+a^†) and Q_r=i √(ħ C ω_r/2) (a^†-a), where ω_r=√(1/LC) is the resonator frequency and a and a^† are the annihilation and creation operators, we obtain ℋ= ħω_r a + ħ∑_i=1^N g_i/2(+a)σ^i_x +ħ∑_i=1^Nω_q^i/2σ_z^i + ħ∑_i,j=1^N g_ig_j/4ω_rσ^i_xσ_x^j.Here σ_k^i are Pauli operators and ω^i_q are the qubit-level splittings. The second term in Eq. (<ref>) accounts for the collective qubit-resonator interaction with couplings g_i=Φ_0√(|φ_0^i|^2ω_r/(2ħ L)), where φ_0^i=2⟨↓_i|φ_i |↑_i⟩. The condition g_i>ω_r,ω^i_q can be reached with an appropriate flux-qubit design <cit.>, and the g_i(t) and ω_q^i(t) can be individually tuned by controlling the matrix element φ_0^i and the height of the tunnel barrier via local magnetic fluxes <cit.>. A specific four-junction qubit design <cit.>, which combines strong interactions with a high degree of tunability, is detailed in the Supplemental Material <cit.>. Finally, the last contribution in Eq. (<ref>)represents an additional qubit-qubit interaction, which is usually neglected for cavity QED systems with weak or moderately strong couplings. However, this term is crucial in the USC regime and it is responsible for the nontrivial ground-state correlations thatare at the focus of the present Letter.USC spectrum.—We are primarily interested in asymmetric configuration, i.e., g_i=g and ω_q^i=ω_q. In this case the Hamiltonian (<ref>) can be expressed in terms of collective angular momentum operators S_k=∑_i σ^i_k/2 and reduces to the extended Dicke Hamiltonian <cit.>ℋ=ħω_r a + ħ g(+a)S_x +ħω_qS_z + ħ g^2/ω_rS_x^2. For g≪ω_r,ω_q we can make a rotating wave approximation and obtain the standard Tavis-Cummings model of cavity QED with a trivial ground state |G⟩=|n=0⟩⊗|↓⟩^⊗ N. If in addition |ω_q-ω_r|≫ g, all excited states are also essentially decoupled and the qubits can be individually prepared, manipulated, and measured by additional control fields. In the opposite limit, g≫ω_r,ω_q, the coupling terms ∼ S_x and ∼ S_x^2 dominate and the level structurechanges completely. This is illustrated in Fig. <ref>(c), which shows that for couplings g/ω_r≳ 3 the spectrum separates into manifolds of 2^N nearly degenerate states. The eigenstates in this regime aredisplaced photon number states, |Ψ_s,m_x,n⟩≃ e^-g/ω_r(a^†-a)S_x|n⟩⊗|s,m_x⟩, with energies E_s,m_x,n≃ħω_r n + δ E^(n)_s,m_x <cit.>. Heres is the total spin and m_x=-s,…,s the spin projection quantum number; i.e., S_x|s,m_x⟩=m_x |s,m_x⟩. Within the lowest manifold, the remaining level splittings are given byδ E^(0)_s,m_x=ħΔ[m_x^2- s(s+1)],Δ= ω_q^2 ω_r/2g^2,and the resulting ordering of the states is shown in Fig. <ref>(d) for N=4 qubits. Thus, for even qubit numbers N, the ground state in the USC regime is of the form |G̃⟩≃ |n=0⟩⊗ |D_0⟩, where |D_0⟩ = |s=N/2,m_x=0⟩ denotes the fully symmetric Dicke state with vanishing projection along x. Importantly, this state exhibits a high degree of qubit-qubit entanglement, while it remains almost completely decoupled from the cavity field <cit.>. Our goal is now to identify a suitable protocol for converting this state into an equivalent state of the decoupled system, where it becomes available as an entanglement resource for further use.Entanglement harvesting.—Figure <ref>(a) shows a general pulse sequence for implementing theentanglement-harvesting protocol through variations ofω_q(t) and g(t). For this protocol the system is initialized in the ground state |G⟩ of the weakly coupled system, where the qubits are far detuned from the cavity, ω_q=ω_ max≫ω_r, and the coupling is set to a minimal value, g=g_ min<ω_r. In the first two steps, T_1 and T_2, the system is adiabatically tuned into the USC regime with a maximal coupling g_ max>ω_r and a low value of the qubit frequency ω_ min≲ω_r. This process prepares the system in the USC ground state |G̃⟩. In the successive steps, T_3 and T_4, the qubits and the resonator mode are separated again, but now in the reverse order and using nonadiabatic parameter variations.Ideally, during this part of the protocol the system simply remains in state |G̃⟩ and becomes the desired excited state of the weakly coupled system at the final time T_ f=∑_n=1^4 T_n. This general sequence achieves two main goals. First, the adiabatic preparation stage can be implemented very rapidly, since it must only be slow compared to the fast time scales set by ω_ max^-1 and g_ max^-1. At the same time the nonadiabatic decoupling processes only need to be fast compared to the slow time scales ω^-1_r, ω^-1_ min, and g_ min^-1. This second condition is most crucial for a time-dependent control of USC systems, since it makes the required switching times experimentally accessible and consistent with the two-level approximation assumed in our theoretical model. In Fig. <ref>(b) we plot the fidelity ℱ(t)={ρ(t) |D_0⟩⟨ D_0|}, where ρ(t) is the density operator of the full system, for a specific set of pulse parameters listed in the figure caption. We see that the entanglement extraction fidelity (EEF) ℱ_ E = max{ℱ(t)| t ≥ T_ f}, i.e., the maximal fidelity after the decoupling step, reaches near perfect values of ℱ_ E≃ 0.95-0.99 for different numbers of qubits, without any further fine-tuning of the control pulses.Note that the fidelity oscillations at the end of the sequence are simply due to the fact that |D_0⟩ is not an eigenstate of the bare qubit Hamiltonian, H_q=ω_q S_z. However, this evolution does not affect the purity or the degree of entanglement of the final qubit state and can be undone by local qubit rotations. Experimental considerations.—For a possible experimental implementation of the protocol we consider qubits with a frequency of ω_ max/(2π)≈10 GHz coupled to a lumped-elementresonator of frequency ω_r/(2π)=500 MHz. The required maximal coupling strength of g_ max≃ 4.5ω_r ≈ 2π× 2.25GHz is then consistent with experimentally demonstrated values <cit.>. For these parameters, the nonadiabatic switching times assumed in Fig. <ref>(b) correspond to T_3,4≃ 0.16 ns. These switching times are within reach of state-of-the-art waveform generators and a sinusoidal modulation of flux qubits on such time scales has already been demonstrated <cit.>. At the same time the duration of the whole protocol, T_ f=15/ω_r≈ 5 ns, is still much faster than typical flux qubit coherence times of1-100 μs <cit.> or the lifetime of a photon, T_ ph=Q/ω_r, in a microwave resonator of quality factor Q=10^4-10^6.Therefore, although many experimental techniques for implementing and operating circuit QED systems in the USC regime are still under development, these estimates clearly demonstrate the feasibility of realizing high-fidelity control operations in such devices. In practice additional limitations might arise from the lack of complete tunability of g(t) and ω_q(t). This is illustrated in Fig. <ref>(a), which shows the evolution of the lowest eigenenergies during different stages of the protocol for the case N=2 and a nonvanishing value of g_ min. In this case the appearance of several avoided crossings during the final ramp-up step prevents a fully nonadiabatic decoupling. In Fig. <ref>(b) we plot the resulting EEF for varying g_ min and T_4. This plot demonstrates the expected trade-off between the residual coupling and the minimal switching time, but also that the protocol is rather robust and fidelities of ℱ_ E∼ 0.9 are still possible for minimalcouplings of a few hundred MHz or switching times approaching ∼ 1 ns.Similar conclusions are obtained when a partial dependence between the pulses for g(t) and ω_q(t) or nonuniform couplings g_i(t) and frequencies ω_q^i(t) due to fabrication uncertainties are taken into account. Numerical simulations of the protocol under such realistic experimental conditions <cit.> demonstrate that no precise fine-tuning of the system parameters is required.Extracting entanglement from a thermal state.—The above-consideredprotocol relies on a rather low resonator frequency ω_r in order to enhance both g/ω_r as well asthe nonadiabatic switching times. This implies that even at temperatures of T=20 mK the equilibrium populations of higher resonator states with n≥ 1 cannot be neglected.In Fig. <ref>(c) we plot the EEF as a function of thetemperature T, assuming an initial resonator state ρ_ th=∑_n p_n|n⟩⟨n|, where p_n=n̅^n/(1+n̅)^n+1 is the thermal distribution for a mean excitation number n̅=1/(e^ħω_r/k_BT-1). We see that the EEF is significantly higher than one would naively expect from the initial population in the ground state |G⟩. The origin of this surprising effect canbe understood from the eigenvalue plot in Fig. <ref>(a). For example, the weak-coupling eigenstate |n=1⟩⊗|↓⟩^⊗ N is efficiently mapped on the corresponding USC state |n=1⟩⊗|s=N/2,m_x=0⟩, passing only through a weak, higher-order avoided crossing. Therefore, the intermediate—and as a result also the final—qubit state is one with the resonator being in state |1⟩. Although for higher photon numbers the avoided crossings become more relevant, the protocol still approximately implements the mapping |n⟩⊗|↓⟩^⊗ N→ |n⟩⊗|s=N/2,m_x=0⟩, independent of the resonator state |n⟩.This feature makes it rather insensitive to thermal occupations and avoids additional active cooling methods for initializing the system in state |G⟩. Entanglement protection.—Figure <ref>(d) shows that apart from the ground state |G̃⟩ there are many other highly entangled states within the lowest USC manifold. Of particular interest is the energetically highest state|Ẽ⟩=|n=0⟩⊗|S⟩, where |S⟩ is a singlet state with total angular momentum s=0 and S_z|S⟩=S_x|S⟩=0. Therefore, once prepared, this state is an exact dark state of Hamiltonian (<ref>) and remains decoupled from the cavity field in all parameter regimes. Although this state is not connected to any of the bare qubit states in a simple adiabatic way, it can still be harvested by an adopted protocol, as described in Fig. <ref>(a) for the case N=4. For this protocol the system is initially prepared in the excited state |Ψ_0⟩=|0⟩⊗|↑↑↓↓⟩ and in a first step the qubit states are lowered below the resonator frequency in order to avoid further level crossings with higher-photon-number states. The increase of the coupling combined with a frequency offset to break the angular momentum conservation then evolves the system into a state with s=0 already for moderate couplings of g/ω_r≈ 1.8. Note that for N≥ 4 there aremultiple degenerate USC states with s=0 <cit.> [cf. Fig. <ref>(d)], out of which the protocol selects a specific superposition <cit.>.Although the harvesting protocol for state |S⟩ loses some of the robustness of the ground-state protocol, it adds an important feature. By retaining a finite coupling g_ f=g(t=T_ f)∼ω_r at the end of the protocol, the extracted dark state |S⟩ is energetically separated from all other states with s≠0 and it is thereby protected against small frequency fluctuations. This effect is illustrated in Fig. <ref>(c), which shows the evolution of the extracted state |S⟩ in the presence of small random shifts of the individual qubit frequencies. For g_ f=0 this leads to dephasing of the qubits and a rapid transition out of the s=0 subspace. This dephasing can be substantially suppressed by keeping the coupling at a finite value. Thus, this example shows that USC effects can be used not only to generate complex multiqubit entangled states, but also to protect them. Conclusion.—We have presented aprotocol for extracting well-defined multiqubit-entangled states from the ground-state manifold of an ultrastrongly coupled circuit QED system. The detailed analysis of this protocol illustrates, how various–so far unexplored–USC effects can contribute to a robust generation and protection of complex multiqubit states. These principles can serve as a guideline for many other preparation, storage and control operations in upcoming USC circuit QED experiments with two or more qubits. Acknowledgments.—This work was supported by the Austrian Science Fund (FWF) through the SFB FoQuS, Grant No. F40, the DK CoQuS, Grant No. W 1210, and the START Grant No. Y 591-N16, and the People Programme (Marie Curie Actions) of the European Union's Seventh Framework Programme (FP7/2007-2013) under REA Grant No. 317232. F. 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A 6, 2211 (1972).Supplemental material § USC EIGENSTATESIn the USC regime, the eigenstates of the extended Dicke Hamiltonian are labeled by the total spin s=0,1,…,N/2 and the spin projection quantum number m_x=-s,…,s, i.e., S⃗^2|s,m_x⟩=s(s+1)|s,m_x⟩ and S_x|s,m_x⟩=m_x |s,m_x⟩. For N>2 the states |s≠ N/2,m_x≠± N/2⟩ appear as multiplets <cit.>, due to the permutation symmetry of the Hamiltonian.In the following we provide an overview of the relevant spin states for the case of N=2 and N=4 qubits. These states are most conveniently expressed in the rotated basis |↓⟩_x=1/√(2)(|↓⟩-|↑⟩),|↑⟩_x=1/√(2)(|↓⟩+|↑⟩).§.§ 2 qubitsFor two qubits we have the usual three triplet states|s=1,m_x=1⟩=|↑↑⟩_x,|s=1,m_x=0⟩=1/√(2)(|↑↓⟩_x+|↓↑⟩_x),|s=1,m_x=-1⟩=|↓↓⟩_x,and the singlet |s=0,m_x=0⟩=1/√(2)(|↑↓⟩_x-|↓↑⟩_x).When expressed in terms of the original qubit basis the two m_x=0 states of interest read|s=1,m_x=0⟩=1/√(2)(|↓↓⟩-|↑↑⟩),|s=0,m_x=0⟩=1/√(2)(|↑↓⟩-|↓↑⟩).§.§ 4 qubitsFor the case of N=4 qubits we obtaina quintuplet for s=2, three triplets for s=1 and a two states for s=0. The maximally symmetric states with s=2 are the usual Dicke states in the x-basis, i.e.,|s=2,m_x=2⟩=|↑↑↑↑⟩_x,|s=2,m_x=1⟩=1/2(|↓↑↑↑⟩_x+|↑↓↑↑⟩_x+|↑↑↓↑⟩_x+|↑↑↑↓⟩_x),|s=2,m_x=0⟩=1/√(6)(|↓↓↑↑⟩_x+|↑↑↓↓⟩_x+|↓↑↑↓⟩_x+|↑↓↓↑⟩_x+|↓↑↓↑⟩_x+|↑↓↑↓⟩_x),|s=2,m_x=-1⟩=1/2(|↓↑↓↓⟩_x+|↓↓↓↑⟩_x+|↓↓↑↓⟩_x+|↑↓↓↓⟩_x),|s=2,m_x=-2⟩=|↓↓↓↓⟩_x.For the entanglement harvesting protocol, we are interested in the state |s=2,m_x=0⟩, which in the original qubit basis is given by|s=2,m_x=0⟩=3/√(24)(|↑↑↑↑⟩+|↓↓↓↓⟩)-1/√(24)(|↑↑↓↓⟩+|↑↓↑↓⟩+|↑↓↓↑⟩+|↓↑↑↓⟩+|↓↑↓↑⟩+|↓↓↑↑⟩).Each of the s=1 states is 3-fold degenerate and the corresponding states are |s=1,m_x=1⟩ =1/2(|↑↑↑↓⟩_x+|↑↑↓↑⟩_x-|↑↓↑↑⟩_x-|↓↑↑↑⟩_x) 1/2(|↑↑↑↓⟩_x-|↑↑↓↑⟩_x+|↑↓↑↑⟩_x-|↓↑↑↑⟩_x) 1/2(|↑↑↑↓⟩_x-|↑↑↓↑⟩_x-|↑↓↑↑⟩_x+|↓↑↑↑⟩_x)|s=1,m_x=0⟩ =1/√(2)(|↑↑↓↓⟩_x-|↓↓↑↑⟩_x) 1/√(2)(|↑↓↑↓⟩_x-|↓↑↓↑⟩_x) 1/√(2)(|↑↓↓↑⟩_x-|↓↑↑↓⟩_x)|s=1,m_x=-1⟩ =1/2(|↑↓↓↓⟩_x+|↓↑↓↓⟩_x-|↓↓↑↓⟩_x-|↓↓↓↑⟩_x) 1/2(|↑↓↓↓⟩_x-|↓↑↓↓⟩_x+|↓↓↑↓⟩_x-|↓↓↓↑⟩_x) 1/2(|↑↓↓↓⟩_x-|↓↑↓↓⟩_x-|↓↓↑↓⟩_x+|↓↓↓↑⟩_x) Finally, for N=4 we obtain two singlet states with s=0, which are given by|s=0,m_x=0⟩ = |S⟩=1/√(3)(|↑↑↓↓⟩_x+|↓↓↑↑⟩_x)-1/√(12)(|↑↓↑↓⟩_x+|↑↓↓↑⟩_x+|↓↑↑↓⟩+|↓↑↓↑⟩_x),|S^'⟩= 1/2(|↑↓↑↓⟩_x-|↑↓↓↑⟩_x-|↓↑↑↓⟩_x+|↓↑↓↑⟩_x) .Since the s=0 subspace is invariant under rotations, the states have the same form as in the original qubit basis,|s=0,m_x=0⟩ = |S⟩= 1/√(3)(|↑↑↓↓⟩+|↓↓↑↑⟩)-1/√(12)(|↑↓↑↓⟩+|↑↓↓↑⟩+|↓↑↑↓⟩+|↓↑↓↑⟩),|S^'⟩=1/2(|↑↓↑↓⟩-|↑↓↓↑⟩-|↓↑↑↓⟩+|↓↑↓↑⟩).Note that in Eqs. (<ref>) and (<ref>) the specific choice of basis states has been used to match the state |S⟩ generated in the protocol described in Fig. 4 in the main text and in the following section. § PROTOCOL FOR THE GENERATION OF THE SINGLET QUBIT STATES WITH S=0Compared to the state |D_0⟩ the singlet states |s=0,m_x=0⟩, defined by S⃗^2|s=0, m_x=0⟩=S_z |s=0, m_x=0⟩=S_x|s=0, m_x=0⟩=0, are not directly adiabatically connected to any of the bare states. Nevertheless these states can still be prepared by using an adapted protocol (Fig. 4 of the main text) that we are going to describe here in more detail. Similar to the ground-state protocol, we start from the decoupled regime where g≃ 0 and ω_q^i≫ω_r, but we initialize the system in the excited qubit state |Ψ_0⟩=|0⟩⊗|↑↑↓↓⟩ (|Ψ_0⟩=|0⟩⊗|↑↓⟩ for N=2 ), where half of the qubits are in the excited state and half in the ground state. Note that for N=4 and a fully symmetric system, the s=0 manifold is two-fold degenerate and spanned, e.g., by the basis states given in Eq. (<ref>). To prepare a well-defined state, we break the symmetry by creating an offset between the qubit frequencies, for example, by setting ω_q^1,2ω_q^3,4. Once the state |Ψ_0⟩ is prepared, all the qubit frequencies are lowered below the resonator frequency such that ω_q^i<ω_r/2. This is done in time step T_1 while keeping g≃ 0. As shown in Fig. <ref>, after this initial step all the relevant qubit states are below the first excited photon state. This configuration avoids undesired level crossings with higher photon number states during the next step of the protocol and only the n=0 manifold must be considered.During the second step ω_q^i≤ω_r/2, but we still keep a finite frequency difference between the qubits to separate the state |↑↑↓↓⟩ from other states with two qubits excited. This difference between the degenerate and non-degenerate qubit frequencies is visualized byFig. <ref>(a) and (b). As the coupling g is slowly increased while the difference in the qubit frequencies is tuned to zero, the state |↑↑↓↓⟩ is adiabatically transformed into the s=0 state |S⟩. During this process the state |S⟩ become almost completely degenerate with the other s=0 state |S^'⟩ [see Fig. <ref>(b)]. However, also the non-adiabatic coupling between these two states is almost negligible, such that the preparation process is still adiabatic on the timescale of the protocol. In the last step of the protocol, the qubit frequencies are ramped up to the initial values as shown in Fig. 4 of the main text. At this point maintaining a frequency offset is notcrucial anymore. Note that in this last protocol step there are not restrictions on the operational time T_3 because the system is now in the dark state |S⟩ and completely decoupled from the resonator mode. § DISORDERIn the main text we have assumed that our qubits are perfectly identical with matching frequencies, and that each of them couples to the resonator with the same coupling constant. However, due to fabrication disorder and control imprecisions this assumption can be difficult to achieve in experiments with multiple qubits. To evaluate the influence of disorder on the entanglement-harvesting protocol, we show in Fig. <ref> the results obtained from numerical simulations of the protocol in the presence of frequency and coupling disorder. Figure <ref>(a) shows the average fidelity, assuming that in each run of the protocol the individual qubit frequencies evolve as ω_q^i(t)= ω_q(t)(1+ϵ_i), where ω_q(t) follows the ideal pulse given in Fig. 2 of the main text and the ϵ_i are randomly chosen from the interval [-0.1,0.1]. We see that the main part of the protocol is essentially unaffected by frequency disorder, since the system is initially in the ground state and in the USC regime the system is dominated by the interaction terms. Frequency disorder only becomes important in the final decoupled state, where it dephases the symmetric state |D_0⟩. Note, however, that for a fixed frequency distribution, this dephasing can be undone, since as shown in Fig. <ref>(a), it leads to almost no degradation of the purity or the degree of entanglement of the qubit state. In Fig. <ref>(c) and (d) the same plots are shown for the case of coupling disorder g_i(t)=g(t)(1+ϵ_i). Although, this type of disorder has a stronger influence on the evolution of the state, the plot shows that our protocol does not require strictly identical couplings and variation of around 10% still lead to EEF ℱ_ E≳ 0.9 and almost no degradation of the qubit-qubit entanglement. The main quantity affected is the entanglement entropy of the qubit subsystem, which does not approach the value of zero, thus showing that qubits and resonator are not perfectly decoupled. However, we note that this measure of entanglement is very sensitive in our case, since the qubit state we achieve at the end of the protocol coincides with our target state with fidelity above 90%.§ IMPLEMENTATION OF THE PROTOCOLIn this section we describe a specific flux-qubit circuit, which can be operated in the USC regime and allows a high tunability of the qubit frequencies and couplings. We propose to achieve this goal by using four-junction flux qubits <cit.> with two of the junctions replaced by a SQUID-loop, effectively turning them into junctions with a flux-tunable Josephson energy. The flux qubit design is depicted in Fig. <ref>.The tunable junctions are junctions 2 and 4 which have sizes α and β, respectively, with respect to junctions 1 and 3. We denote by φ_i = Φ_i/Φ_0 the jump of the superconducting phase across the junction i, where Φ_0 = ħ/2e is the reduced flux quantum and Φ_i the jump of the generalized flux across the junction. The conjugate charge to φ_i is denoted as n_i. The phases φ_i are not independent but constrained by the flux quantization condition for the three loops (the big loop and the two SQUID-loops α and β)∑_i ∈{1,2,3,4}φ_i + f_ϵ = 0, ∑_i ∈{2,6}φ_i + f_α = 0, ∑_i ∈{4,5}φ_i + f_β = 0,where f_η = Φ_η/Φ_0 and η = α, β, ϵ is the magnetic frustration through the loop created by external magnetic fluxes Φ_η. Using the above equations we eliminate phase jumps φ_2, φ_5 and φ_6 from the problem. The standard quantization procedure for circuits then gives the Hamiltonian <cit.>H_q= 4E_Cα + β + 2αβ[ (α + β + αβ) (n_1^2 + n_3^2) + (1 + 2α)n_4^2 - 2αβ n_1n_3 - 2α(n_1 + n_3)n_4 ]- E_J [ cos(φ_1) + αcos(f_α2)cos(φ_1 + φ_3 + φ̃_4 + f̃_ϵ) + cos(φ_3) + βcos(f_β2)cos(φ̃_4) ],where φ̃_4 = φ_4 - f_β/2 and f̃_ϵ = f_ϵ + (f_β - f_α)/2. From the shape of the Hamiltonian we can see that if we tune the frustration parameters, f_α, f_β and f_ϵ, in unison such that f_ϵ = (2π + f_α - f_β)/2, the structure of the Hamiltonian stays the same except that the effective Josephson energy of the SQUID-loops vary sinusoidally with the frustration. This enables us to operate the flux qubits at the sweet spot, f̃_ϵ = π, while changing the potential landscape. Note that in practice the cross-talk between the magnetic fluxes may complicate the qubit control, but in principle it is always possible to measure this cross-talk and compensate it by appropriately chosen control pulses.The flux qubit couples to the resonator through the phase jump over the entire qubit (see the main text), which, with our notation, is given by Δφ = φ̃_4. The coupling constant g between the resonator and the qubits is proportional to the matrix element of Δφ between the ground and excited states of the qubits, Δφ_eg = ⟨ e | Δφ | g ⟩. Additionally, the coupling to the resonator renormalizes the qubit Hamiltonian by adding a term E_LΔφ^2/2, where E_L = Φ_0^2/L is the inductive energy related to the resonator inductance L, to the qubit Hamiltonian.Now we are ready to demonstrate the tunability of the qubit frequency and qubit-resonator coupling. We diagonalize the qubit Hamiltonian H_q, plus the renormalization term coming from the coupling, numerically to find the eigenfrequencies and evaluate the transition matrix element Δφ_eg. We choose the following parameters for the simulation: α = 0.6, β = 6, E_L/h = 2.57 GHz, E_C/h = 4.99 GHz and E_J/h = 99.7 GHz. Our choice of E_L sets the resonator inductance to L = 63.7 nH. In addition we choose C = 1.59 pF which determines the resonator frequency and impedance to be ω_r = (LC)^-1/2 =2π×500 MHz and Z_r = √(L/C) = 200 Ω, respectively. In the simulation we tune f_α from 0 to 0.70π and f_β from 0 to 0.96π (f_ϵ changes accordingly to keep the qubit at its sweet spot).In Fig. <ref> a) we plot the transition frequency of the qubit, normalized to the resonator frequency against the external fluxes in the two SQUID-loops. The qubit frequency is highly tunable ranging from ∼ 50 ω_r all the way to ∼ 0.5 ω_r. The tunability of the normalized coupling constant, g/ω_r, is demonstrated in Fig. <ref> b). It is also highly tunable ranging from ∼ 0.17 up to ∼ 4.5. The only limitation we have here is that the qubit transition frequency and qubit resonator coupling cannot be tuned entirely separately. The path that we propose to take in the (f_α,f_β) landscape is portrayed in Fig. <ref> with the red curves. We start from the point (0,0) and then traverse the curve clockwise, as the arrows show, back to the origin. In Fig. <ref> a) we plot the proposed time-dependent pulses for g(t) and ω_q(t). We have chosen the shape of the pulse for g and from that computed the pulse shape of ω_q. The pulse sequence consists of two parts: ramping up g/ω_r from 0.25 to 4.5 adiabatically and then tuning it back down to its initial value non-adiabatically. The qubit frequency starts from 22.8ω_r, goes down to 0.7ω_r and then ramps back up to its initial value. Fig. <ref> b) shows the fidelity obtained with the pulse of a) in the case of four qubits coupled to the LC-resonator. Even with a limited individual tunability of g and ω_q we obtain fidelities of ℱ_ E > 0.9. For two qubits we can obtain a fidelity of ℱ_ E≈ 0.96 with the same pulse.§ INFLUENCE OF HIGHER RESONATOR MODESIn our analysis in the main text we have assumed a single mode lumped-element resonator and have neglected the influence of all higher modes. For a typical lumped-element resonator of dimension d=500μm and a resonance frequency of 2π× 5 GHz, the next higher mode is estimated to be at ω_e≈ c/(6d)∼2π× 100 GHz. In Ref. <cit.> we have shown that for such a high ratio ω_e/ω_r > 20, the effect of a higher circuit mode has no relevant influence on the resulting USC physics. By simply scaling this circuit by a factor 10, we would obtain a fundamental resonance of ω_r/2π≈ 500 MHz and an excited mode at around ω_e/2π≈ 10 GHz. By using an optimized design to increase this frequency or simply changing the maximal qubit frequency to, e.g., 8 GHz would avoid a resonant coupling to this mode without affecting the protocol. During the first stage of the protocol the system is in the ground state. There might be some weak admixing with the excited mode ∼(g/ω_e)^2, which however, should not affect the adiabaticity condition. During the USC part of the protocol, the qubit frequency is tuned to a very low value. In this case the relation between g, ω_q and ω_r is very similar to the values assumed in the analysis in Ref. <cit.>. Therefore, according to this study the system properties should not be considerably affected. Finally, during the last stage of the protocol, where the qubit frequency is ramped-up again, the coupling has already been switched back to a very small value. The mixing with the excited mode will be small, ∼ g^2_ min/(ω_e-ω_q)^2. In summary, based on these estimates we do not expect a significant degrading of our protocol from interactions with higher-order modes of the lumped-element resonator.§ NUMERICAL SIMULATIONS §.§ Coherent evolutionIn this short paragraph we provide some details about the numerical simulations, which have been used to produce the plots in the main text.For the plot of the eigenvalues in Fig. 1 in the main text we have diagonalized Hamiltonian (3) using a truncated set of 140 number states for the resonator mode. For the implementation of the entanglement harvesting protocols shown in Figs. 2, 3 and 4 in the main text we have numerically integrated the time-dependent Schrödinger equation using 100 resonator states. In all calculations we have verified that increasing this number of basis states does not affect our results. In Fig. 3(c) in the main text we plot the EEF for N = 4 and for a resonator mode initially in a thermal state ρ_ th=∑_n p(n)|n⟩⟨n| with p(n) the Gibbs distribution at temperature T. The fidelity has been obtained by solving the Schrödinger equation for each initial number state separately, and then averaging all the resulting fidelities according to the thermal probabilities, p_n. For this plot we have included the first 10 resonator Fock-states and verified that averaging over a thermal distribution including more resonator states does not change the result.§.§ Master equation simulationsTo make sure dissipation during the protocol does not spoil our scheme we perform simulations including cavity decay into the dynamics. We do so by modelling the coupling to the bath by a Markovian master equation. Note, however,that in the ultrastrong coupling regime the dissipator is not given by the photon annihilation operator a and must be expressed in terms of the coupled eigenbasis states |k⟩of the system Hamiltonian <cit.>. For the case of the extended Dicke model Hamiltonian and assuming an ohmic spectral density of the bath, the resulting master equation reads ρ̇ = -iħ[H, ρ] + ∑_k,l > kΓ_kl(1 + n̅(Δ_lk, T)) 𝒟[|k⟩⟨l|]ρ + ∑_k,l > kΓ_kln̅(Δ_lk, T)) 𝒟[|l⟩⟨k|]ρ,where Γ_kl = κΔ_lkω_r |C^a_kl|^2, Δ_lk=E_l-E_kis the energy difference between eigenstates l and k, C^a_kl = ⟨k| a + a^†|l⟩ and κ = ω_r/Q is the resonator decay rate in the weak coupling regime. n̅(Δ_lk, T) is the occupation of the environmental modes at the transition frequency Δ_lk at temperature T. The superoperator 𝒟 is defined in the standard way as 𝒟[O]∙ =(2O ∙ O^† - { O^†O, ∙})/2. Because we change ω_q and g in time the basis {|k⟩}, and thus also the jump operators and rates, changes in time.For the numerical simulation of Eq. (<ref>) we diagonalize the system Hamiltonian at each point in time to construct the correct jump operators and transition rates during the different stages of the protocol. After constructing the correct operators and transition rates we move back to the composite basis |n, s, m_z⟩ spanned by the Fock states |n⟩ and collective spin states |s, m_z⟩. This is done with a time-dependent unitary transformation U(t) = ∑_k, n, m_z|n, s, m_z⟩⟨k(t)|, where |k(t)⟩ is the instantaneous eigenstate of the extended Dicke Hamiltonian.Using the above prescription we simulate our protocol in presence of photon decay for a two qubit system. In Fig. <ref> we plot the fidelity for an LC resonator of Q-factor Q=100 in contact with a bath at temperature T = 0, <ref>a), and T = ħω_r/k_B, <ref>b). We see that even for such a low Q-factor the cavity decay has no significant effect. At T = 0, where the system is essentially at all times in the ground state, the performance of the protocol is not affected at all and even at T = ħω_r/k_B the degradation is only on the level of ∼ 3.5%. In Fig. <ref>b) the dashed green line indicates the performance of the protocol at T = ħω_r/k_B without including cavity decay. 99 Arecchi1972s F. T. Arecchi, E. Courtens, R. Gilmore, and H. Thomas, “Atomic Coherent States in Quantum Optics", Phys. Rev. A 6, 2211 (1972). Qiu2016s Y. Qiu, W. Xiong, X.-L. He, T.-F. Li, and J. Q. You,“Four-junction superconducting circuit", Sci. Rep. 6, 28622 (2016). vool16sU. Vool and M. Devoret, “Introduction to Quantum Electromagnetic Circuits", arXiv:1610.03438 (2016). Jaako2016s T. Jaako, Z.-L. Xiang, J.J. Garcia-Ripoll and P. Rabl, “Ultrastrong coupling phenomena beyond the Dicke model", Phys. Rev. A 94, 033850 (2016). beaudoin11s F. Beaudoin, J. M. Gambetta, and A. Blais, “Dissipation and ultrastrong coupling in circuit QED", Phys. Rev. A 84, 043832 (2012). ridolfo12s A. Ridolfo, M. Leib, S. Savasta, and M. J. Hartmann. “Photon Blockade in the Ultrastrong Coupling Regime", Phys. Rev. Lett. 109, 193602 (2012). | http://arxiv.org/abs/1707.08969v3 | {
"authors": [
"Federico Armata",
"Giuseppe Calajo",
"Tuomas Jaako",
"M. S. Kim",
"Peter Rabl"
],
"categories": [
"quant-ph",
"cond-mat.supr-con"
],
"primary_category": "quant-ph",
"published": "20170727180006",
"title": "Harvesting Multiqubit Entanglement from Ultrastrong Interactions in Circuit Quantum Electrodynamics"
} |
J.J. Bevan]Jonathan J. Bevan [J.J. Bevan]Department of Mathematics, University of Surrey, Guildford, GU2 7XH, United Kingdom.(Corresponding author: t: +44 (0)1483 682620.) [Corresponding author][email protected] J.H.B. Deane]Jonathan H.B. Deane [J.H.B. Deane]Department of Mathematics, University of Surrey, Guildford, GU2 7XH, United Kingdom. [email protected] A calibration method for estimating critical cavitation loads] A calibration method for estimating critical cavitation loads from below in 3D nonlinear elasticityIn this paper we give an explicit sufficient condition for the affine map u_λ(x):=λ x to be the global energy minimizer of a general class of elastic stored-energy functionals I(u)=∫_Ω W(∇ u) dx in three space dimensions, where W is a polyconvex function of 3 × 3 matrices.The function space setting is such that cavitating (i.e., discontinuous) deformations are admissible.In the language of the calculus of variations, the condition ensures the quasiconvexity of I(·) at λ, whereis the 3 × 3 identity matrix.Our approach relies on arguments involving null Lagrangians (in this case, affine combinations of the minors of 3 × 3 matrices), on the previous work <cit.>, and on a careful numerical treatment to make the calculation of certain constants tractable. We also derive a new condition, which seems to depend heavily on the smallest singular value λ_1(∇ u) of a competing deformation u, that is necessary for the inequality I(u) < I(), and which, in particular, does not exclude the possibility of cavitation. [ [ December 30, 2023 ===================== § INTRODUCTION In this paper we consider an established model of elastic material that is capable of describing cavitation, that is, of admitting energy minimizers that are discontinuous.This phenomenon was first analysed in the setting of hyperelasticity by Ball in <cit.>;since then, a large and sophisticated literature has developed, including but not limited to <cit.>,part of which focuses on finding boundary conditions which, when obeyed by all competing deformations, ensure that cavitation does not occur. It is to the latter body of work that we contribute by considering the case of purely bulk energyI(u) =∫_ W(∇ u(x)) dx,where u: →^3 represents a deformation of an elastic material occupying the domainin a reference configuration, and where W is a suitable stored-energy function. In the three dimensional setting, we give an explicit characterization of those affine boundary conditions of the form(x)=ł x,where ł>0 is a parameter, such that the quasiconvexity inequalityI(u) ≥ I()holds among all suitable maps u agreeing withon ∂. It is by now well established that if ł is large enough, ł≥ł_crit, say,then such an inequality cannot hold. Thus we probe ł_crit by finding ł_0 such that (<ref>) holds whenever ł≤ł_0.This question has been addressed in <cit.> and, more recently, in <cit.>. In this paper we use a new approach, involving the addition of a suitable null Lagrangian (a method sometimes known as calibration), to deduce concrete lower bounds on λ_crit in the three dimensional case.The analysis centres ostensibly on functions of the singular values of 3 × 3 matrices.Let A be a 3 × 3 matrix.Then the singular values of A are normally written as ł_j(A), for j=1,2,3, and their squares are the eigenvalues of A^TA. See <cit.> or <cit.> for useful introductions to singular values, as well as <cit.> for an illustration of their use in nonlinear elasticity.Singular values arise naturally in the stored-energy functions of isotropic elastic materials, and also in lower bounds which can be derived from them.Such was the case in <cit.>, where, for 2 < q < 3 and for convex functions Z and h,a stored-energy function very similar to[The original functional contained an `artificial' quadratic term r|A|^2, with r large, to deal with the difficulties presented by the function P. This is no longer needed thanks to the calibration method we introduce.]W(A) = |A|^q + Z( A) + h( A)was shown to obey the inequalityI(u)-I()≥∫_κ |∇ u - ∇|^q + h'(ł^3)Π_j=1^3(ł_i(∇ u)-ł)+ł P(∇ u) dx.The function P is defined byP(A) = ∑_1 ≤ i < j ≤ 3λ_i(A)λ_j(A) - λ∑_1 ≤ i ≤ 3λ_i(A)and the constant κ satisfies bounds defined in (<ref>) below.By grouping the first two integrands in (<ref>) together, it is possible to find conditions on ł such that ∫_κ |∇ u - ∇|^q + h'(ł^3)Π_j=1^3(ł_i(∇ u)-ł)dx ≥ 0.However, the corresponding inequality for P, namely ∫_ P(∇ u)dx ≥ 0,which, since P(ł)=0, is equivalent to the quasiconvexity of P at ł, remains an open question.We show in this paper that P does satisfy a condition necessary for quasiconvexity at ł (see Proposition <ref>, part (a): rank-one convexity at ł), but that the most tractable sufficient condition for (<ref>) cannot hold (see Proposition <ref>, part (b): polyconvexity at ł)).Trying instead to find conditions under which each of ∫_ (κ/2) |∇ u - ∇|^q + h'(ł^3)Π_j=1^3(ł_i(∇ u)-ł) dx≥ 0∫_ (κ/2) |∇ u - ∇|^q + h'(ł^3)P(∇ u)dx≥ 0holds is closer to the right approach, although for reasons connected with the curvature of P at ł, (<ref>) is still not possible!This is what leads us to introduce the null Lagrangian N(A)= A - ł A,which has the property that ∫_ N(∇ u)dx = 0 for any admissible u and is such that there are conditions on ł under which ∫_ (κ/2) |∇ u - ∇|^q + h'(ł^3)(P(∇ u)-N(∇ u))dx≥ 0for all admissible u. See Theorem <ref> and (<ref>) in particular.In fact, N is the unique null Lagrangian for which this method works:see Proposition <ref>. More generally, we remark that P and G:=P-N possess properties that are both interesting in their own right and, at the same time, highly non-trivial to derive. (See Section <ref>.) A useful introduction to null Lagrangians can be found in <cit.>.The upper bound ł_0 given in the right-hand side of (<ref>) is investigated in Section <ref> using a careful mixture of analysis and numerical techniques. The partnership between these approaches seems to be particularly fruitful when applied to G and to functions derived from it.Accordingly, we find an explicit constant ν_1 ≈ 0.4501 such that if 0 ≤ł^3-q h'(ł^3) ≤κ/2 (√(2))^q-3ν_1^2-q, then I(u) ≥ I().See Section <ref>, Subsection <ref> and the appendices for details.In Section <ref>, a careful analysis of the function H(A):= Π_j=1^3(ł_j(∇ u)-ł) + ł G(A)yields, among other things, what we believe to be new necessary conditions for the inequality I(u) ≤ I().A distinguished role seems to be played by the smallest singular value, ł_1(∇ u):see Proposition <ref> in particular. §.§ Notation The inner product between two matrices A and B is given by A · B =A^T B, and, as usual, A denotes the trace of A. For a function f: ^3 × 3→ and any 3 × 3 matrix U, the shorthand D_Uf(A) =∇ f(A) · UD^2_Uf(A) = ∇^2f(A) [U,U]will be used, where as usual ∇^2f(A)[U,U]=f_,_(ij)(kl)(A)U_ijU_kl with the summation convention in force. When discussing polyconvexity, which is defined when it next features in the paper, we use the shorthand notation ^19 for the set ^3 × 3×^3 × 3× containing the list of minors R(A):=(A,A,A) of any 3 × 3 matrix. The set of 3 × 3 square, orthogonal matrices is denoted by O(3), and the subset of O(3) consisting of those matrices with determinant equal to 1 will be written SO(3).For any two vectors a and n in ^3, the notation a ⊗ n will denote the matrix of rank one whose (i,j) entry is a_i n_j.Our notation for Sobolev spaces is standard. § CALIBRATION AND THE FUNCTION G(A) In this section we give some properties of the function P and use them to derive the null Lagrangian N alluded to above. To start with, two technical results are required.Let λ >0.Then (i) ∑_i=1^3 D_Uλ_i(λ) =U; (ii) ∑_1 ≤ i < j ≤ 3 D_Uλ_i(λ)D_Uλ_j(λ) =( U)^2/2-|U|^2/4- (U^2)/4; (iii) λ∑_i=1^3 D_U^2λ_i (λ) + ∑_i=1^3 (D_Uλ_i(λ))^2 = |U|^2. In particular, ∑_i=1^3λ D^2_Uλ_i(λ) = |U|^2 -(U^2)/2and ∑_i=1^3 (D_Uλ_i(λ))^2 = |U|^2 +(U^2)/2. Parts (i) and (iii):Let λ_i (ł + hU) = ł + D_Uł_i(ł) + h^2/2D_U^2ł_i(ł) + o(h^2) for each i and insert into the identity ∑_i=1^3ł_i(ł + h U) = |ł + h U|^2.Part (i) follows by comparing terms of order h and part (iii) by comparing terms of order h^2. Part (ii):Let A=ł+hU and note that, by definition, each ł_i(A) is a root z_i, say, of the polynomial (A^T A - z^2 ) = 0. NowA^T A = ł^2 + 2 ł h U_s+h^2 U^T U, so 0= ( (ł^2 - z^2) + 2 ł h U_s + h^2 U^T U)= τ^3 + τ^2(2 ł h U_s + h^2 U^T U) + τ (2 ł h U_s + h^2 U^T U)+ (2 ł h U_s + h^2 U^T U),where U_s:=(U+U^T)/2 is the symmetric part of U and τ:=ł^2 - z^2.Using the development of ł_i(ł + h U) given above, but this time keeping only terms of order h, it follows that τ = - 2 h ł D_U ł_i(ł) + o(h^2).Putting this into (<ref>) and writingZ_i:= D_U ł_i(ł) for brevity, shows that the Z_i are roots of the following polynomial equation:-8 ł^3 h^3 Z^3 + 8 ł^3 h^3U Z^2 - 8 ł^3 h^3U_s Z + 8 ł^3 h^3U_s + o(h^3) = 0.Dividing by -8 ł^3 h^3, letting h → 0and using the identityU_s = ( U)^2/2 - |U|^2/4 -U^2/4givesZ^3 - ( U) Z^2 +(( U)^2/2 - |U|^2/4 -U^2/4) Z -U_s = 0.The roots Z_1,Z_2,Z_3 must therefore satisfy - ∑_i=1^3 Z_i = -U,∑_1 ≤ i < j ≤ 3 Z_i Z_j= ( U)^2/2 - |U|^2/4 -U^2/4 .Replacing each Z_i with D_U ł_i(ł) in equation (<ref>) merely recovers (or provides an alternative derivation of) part (i) of the lemma, while equation (<ref>) delivers part (ii). Equation (<ref>) now follows by using the identity∑_i=1^3 Z_i^2 = ( ∑_i=1^3 Z_i )^2 - 2 ∑_1 ≤ i < j ≤ 3 Z_i Z_jtogether with parts (i) and (ii) above.Finally, (<ref>) follows from (iii) above and (<ref>).This concludes the proof.Note that (<ref>) tells us, via the Cauchy-Schwarz inequality, that D^2_U(∑_i=1^3λ_i)(λ) vanishes if and only if U is a symmetric matrix. Moreover, (<ref>) implies that ∑_i=1^3 (D_Uł_i(ł))^2 vanishes if and only if U is antisymmetric, and that in this case D_Uł_i(ł)=0 for each index i.Let C_1 and C_2 be fixed 3 × 3 matrices, let C_3 be a real number, and let N(A) = C_1 · (A- λ) + C_2 · ( A- λ^2 ) + C_3( A - ł^3)for all A ∈^3 × 3.Then D_UP(λ) = D_UN(λ) ∀ U ∈^3 × 3if and only if C_1, C_2 and C_3 are related by the equation(ł- ł^2 C_3 - ł C_2) = C_1 - ł C_2^T.Moreover,D^2_UP(λ) = D^2_UN(λ) ∀ U ∈^3 × 3if and only ifC_2 and C_3 are related by the equationC_2 = (1-λ C_3) .In particular, the unique quadratic null Lagrangian N satisfying both (<ref>) and (<ref>) is N(A) =A - λ A,and it satisfies∫_N(∇ u) dx =0for all u belonging to W^1,2(,^3) such that u= on ∂ (in the sense of trace).Let A = λ + h U and note that N(A) = [ U · C_1 + ł UC_2 - ł (U C_2) + ł^2 C_3U ] h+ [C_2 · U + ł C_3U ] h^2 + h^3U. Next, rewrite P(A) = 1/2( (∑_i=1^3λ_i(A))-λ)^2 - λ^2/2 - |A|^2/2and, for sufficiently small h, write λ_i(A) = λ + h D_Uλ_i(λ) + h^2/2D^2_Uλ_i(λ) + ρ_i(λ,h,U),where ρ_i is o(h^2) as h → 0.A short calculation then yieldsP(A) = λ hU + λ h^2/2∑_i=1^3 D^2_Uλ_i(λ) + h^2 ∑_1 ≤ i < j ≤ 3D_Uλ_i(λ)D_Uλ_j(λ) + ρ = λ hU + h^2/4(|U|^2 - (U^2))+h^2/4(2( U)^2 -|U|^2-(U^2)) + ρ = λ hU + h^2/2(( U)^2 -(U^2)) + ρ. = λ hU + h^2U + ρ.Here, we have used the identity U = 1/2(( U)^2 -(U^2)), (<ref>) and Lemma <ref>(ii).The term ρ=ρ(λ,h,U) is o(h^2) as h → 0. Comparing terms of order h in this expression with the expansion for N(A) given above, we see that D_UP(ł 1) = D_UN(ł 1) for all U if and only if (<ref>) holds. To prove the equivalence of (<ref>) and (<ref>), simply compare terms of order h^2 to obtain(C_2 + ł C_3 ) · U = · Ufor all U, and then pick U such that U= e_i ⊗ e_j.It is then clear that (<ref>) is equivalent to (<ref>).Finally, to prove that N(A)= A - ł A is the unique, quadratic null Lagrangian satisfying (<ref>) and (<ref>) take C_3=0 in (<ref>) and (<ref>).The former gives C_2=, and the latter C_1=-ł, which together imply (<ref>).Equation (<ref>) is a standard result about null Lagrangians;to see it without recourse to general theory, simply observe that, for sufficiently smooth φ, N(∇φ) can be written as a divergence.The result then follows from the Green's theorem and an approximation argument. (The argument given in <cit.> serves as a useful template.) We remark that this establishes a simple pattern: N(A) can apparently be obtained from P by noting that if A=Diag (λ_1,λ_2,λ_3) then P(A) =A - λ A. As was pointed out in the introduction, and originally conjectured in <cit.>, it would be very useful if P were quasiconvex at the matrix ł.Our results in this direction are somewhat mixed.We find that P satisfies a condition necessary for quasiconvexity at ł, but that it does not satisfy a tractable condition sufficient condition for quasiconvexity at ł.To be precise, (a) P is rank-one convex at ł but (b) P is not polyconvex at that point. These concepts are explained in more detail below. We note, incidentally, that P is not globally rank-one convex. The latter is relatively easy to see:one can immediately calculate that, for any rank-one matrix A=a⊗ n, ł_1(A)=ł_2(A)=0 and ł_3(A)=|A|.In particular, P(A) = -|A|, which is a concave function of A.The foregoing discussion is summarised in the result below. Let the function P be defined by (<ref>).Then(a) ł is a point of rank-one convexity of P, but(b) P is not polyconvex at λ.(a):To show (a) we only need to verify that P(ł+ a ⊗ n) is convex as a function of the rank-one matrix a ⊗ n.Without loss of generality, we may choose coordinates such that n=e_1. A calculation then shows that, if the component of a in the e_1 direction is a_1, the following expression holds:P(ł + a ⊗ n) = ł (|ł+a_1|-ł).This is clearly convex in a ⊗ n, which proves part (a) of the lemma.(b) Assume for a contradiction that λ is a point of polyconvexity of P. This means that there is some point (C_1,C_2,C_3) in ^19 such thatP(A) ≥ P(λ) + C_1 · (A - λ) + C_2 · ( A - λ^2 ) + C_3( A - λ^3)for all A in ^3 × 3.Note that C_3 has to be zero because P is at most quadratic.Next, take A to be a rank-one matrix such that C_1 · A=sign ((C_1)_ij)t for a given pair i,j, where t is a positive parameter to be chosen shortly. Recall that, when A is a rank-one matrix, P(A)=-|A|. If (C_1)_ij≠ 0, this gives-t≥ |(C_1)_ij|t - ł C_1 -ł^2C_2,which is easily contradicted by taking t to be sufficiently large. Therefore C_1 = 0, leavingP(A)≥ P(ł) + C_2 · ( A - ł^2 ).By considering A=ł + h U for arbitrary U in ^3 × 3 and small h, it is straightforward to show that this implies D_U P(ł) = D_U N̅(ł),where N̅(A) = C_2 · ( A - ł^2 ).In the course of Proposition <ref> it is shown that D_U P(ł)=ł U, so (<ref>) impliesł U = ł ( C_2U -(U C_2))for all U.Rearranging this givesC_2 = ( C_2 - 1) ,so that C_2=(1/2).Putting this into (<ref>) givesP(A) ≥ A - 3 ł^2/2, which is easily contradicted by taking A to be of rank 1, applying the observation that P(A)=-|A| for such A, and letting |A| →∞. This concludes the proof. Next, with N as in Proposition (<ref>), we defineG(A) := P(A) - N(A)for all 3 × 3 matrices A.We know by equation (<ref>) in Proposition <ref> that P and N are tangent at λ, so clearly D_UG(λ) = 0 for all U.Moreover, by (<ref>),we also have that D^2_UG(ł)=0 for all U. Thus G behaves like |A-λ|^3 in a neighbourhood of λ, and this is a key feature which enables us to find new lower bounds for ł_crit.The technique for doing so is described in the next section.We also record the following useful property of G, which flows directly from (<ref>):∫_P(∇ u) dx = ∫_ G(∇ u) dx for all u belonging to W^1,2(,^3) such that u= on ∂ (in the sense of trace).§ NEW LOWER BOUNDS ON Λ_CRIT Let the stored-energy function W: ^3 × 3→ [0,+∞] be given by W(A)= |A|^q + Z( A) + h( A) where Z: ^3 × 3→ [0,+∞) is convex and h: → [0,+∞] has the following properties: (H1) h is convex and C^1 on (0,+∞);(H2) lim_t → 0+h(t) = +∞ and lim inf_t → +∞h(t)/t > 0;(H3) h(t)=+∞ if t ≤ 0.The exponent q satisfies 2 < q < 3.Let I(u) = ∫_ W(∇ u) dx and define the class of admissible maps as_ł={u ∈ W^1,q(;^3):u =on ∂,I(u) < +∞}.The following argument is straightforward and can be found in <cit.>.We include it here both for completeness and as a means of deriving the function P defined by (<ref>).Applying <cit.> to A ↦ |A|^q gives|∇ u|^q ≥ |ł|^q+q|ł|^q-2ł· (∇ u -ł)+κ |∇ u-ł|^q,where2^2-q≤κ≤q2^1-q. Therefore, by (<ref>) and by appealing to the convexity of Z and h, we obtainW(∇ u)≥W(∇ u_λ) + q|ł|^q-2ł·(∇ u- ł) + κ |∇ u-ł|^q+2γł· (∇ u- ł) + γ |∇ u - ł|^2+D_AZ(ł) · (∇ u - ł)+h'(ł^3)(∇ u-ł^3),for any u ∈_ł. Integrating (<ref>) and using the facts that both ∇ u and ∇ u are null Lagrangians inW^1,q(,^3) for q ≥ 2, we obtainI(u) - I()≥∫_κ |∇ u - λ|^q + h'(λ^3)λ_1 λ_2 λ_3dx =∫_κ |∇ u - λ|^q + h'(λ^3)λ̂_1 λ̂_2 λ̂_3dx + λ h'(λ^3) ∫_ P(∇ u)dx.In deriving this, it may help to recall the identityλ_1 λ_2 λ_3 = λ̂_1λ̂_2 λ̂_3 + λ∑_1 ≤ i < j ≤ 3λ_i λ_j - λ^2 ∑_i=1^3λ_i + ł^3,where the notation λ_i abbreviates λ_i(A) and, for each i, λ̂_i:=λ_i - λ. Continuing from (<ref>), we split the first term into two equal parts and recall the property of G and P given in (<ref>), thereby obtaining:I(u) - I()≥∫_ (κ/2) |∇ u - λ|^q + h'(λ^3)ł̂_1 ł̂_2 ł̂_3dx+ + ∫_ (κ/2) |∇ u - λ|^q+λ h'(λ^3)G(∇ u)dx≥∫_ (κ/2) |Λ(∇ u) - Λ_0|^q + h'(λ^3)ł̂_1 ł̂_2 ł̂_3dx+ + ∫_ (κ/2) |∇ u - λ|^q+λ h'(λ^3)G(∇ u)dx.Here, Λ(A) is the 3-vector with entries λ_i(A) and λ_0= (λ,λ,λ).We have used the well-known inequality |A-λ| ≥ |Λ(A)-Λ_0|.In keeping with the notation introduced in <cit.>, let_1(Λ)=(κ/2)|Λ - Λ_0|^q + h'(λ^3)λ̂_1 λ̂_2 λ̂_3,and, in contrast to the approach of <cit.>, let_2(A)=(κ/2)|A-λ|^q + λ h'(λ^3)G(A). In these terms we then haveI(u)-I()≥∫__1 (Λ)dx + ∫__2 (∇ u)dx.The sign of the first integral can be controlled by appealing to the following result: (<cit.>)The function _1(Λ) defined in <ref> is pointwise nonnegative on ^+++ provided h'(ł^3) > 0 and (κ/2)/h'(λ^3)λ^3-q ≥ (q-2)^(q-2)/2 q^-q/2.The pointwise nonnegativity of _2, on the other hand, relies primarily on the argument given in Lemma <ref> below.In short, the idea is that _2(∇ u) is dominated by |∇ u-λ|^q for both small and large values of |∇ u-λ| provided h'(λ^3) is itself not too large.Thus we generate a new upper bound on λ which must be imposed along with (<ref>) in order to guarantee that I(u) ≥ I().With G as defined in (<ref>) and for any positive constant c_0, letM_2(λ,c_0) =sup{|G(A)|/|A-λ|^2: |A-λ| ≥ c_0 } M_3(λ,c_0)= sup{|G(A)|/|A-λ|^3: 0 < |A-λ| < c_0 }. Then _2(A) ≥ 0 for all 3 × 3 matrices A provided λ h'(λ^3) ≤min_c_0{ (κ/2) max{c_0^q-2/M_2(λ,c_0), c_0^q-3/M_3(λ,c_0)}}. By (<ref>), the quantity M_3(λ,c_0) is finite and, in view of the at most quadratic growth of G, M_3(ł,c_0) is uniformly bounded as a function of c_0. M_2(ł,c_0) has the same properties, but this time we appeal to the fact that D_U G(ł)=0 for all U. Let A ≠ł and let c=|A-λ|.It is immediately clear that_2(A) ≥κ/2|A-ł|^q -ł h'(ł^3) |G(A)|,and we express the right-hand side in two ways:κ/2|A-ł|^q -ł h'(ł^3) |G(A)|= (κ/2|A-ł|^q-j -ł h'(ł^3) |G(A)|/|A-ł|^j)|A-ł|^j,where j is either 2 or 3. Now let w:=2ł h'(ł^3)/κ and suppose that (<ref>) holds.Then, in particular, w≤max{c^q-2/M_2(ł,c),c^q-3/M_3(ł,c)}.If the maximum in (<ref>) is given by c^q-2/M_2(ł,c) then w M_2(ł,c) ≤ c^q-2, and hence κ/2 c^q-2 - ł h'(ł^3) |G(A)|/c^q-2≥ 0.Using (<ref>) with j=2, we see that _2(A) ≥ 0.If the maximum in (<ref>) is given by c^q-3/M_3(ł,c) then we can argue similarly, this time using (<ref>) with j=3, to conclude that _2(A) ≥ 0.Let f_1(c_0) = c_0^q-2/M_2(ł,c_0), f_2(c_0) = c_0^q-3/M_3(ł,c_0) and define ξ(c_0)=max{f_1(c_0),f_2(c_0)}. Then f_1 is nondecreasing, f_2 is nonincreasing, and inf_c_0 > 0ξ(c_0) = M_2(ł,c^*)^q-3 M_3 (ł,c^*)^2-qwhere c^* is the unique fixed point of the function c_0 ↦M_2(ł,c_0)/M_3(ł,c_0).Let M_2(c_0)=M_2(ł,c_0) and M_3(c_0)=M_3(ł,c_0) for brevity.It is clear from their definitions that M_2(c_0) and M_3(c_0) are nonincreasing and nondecreasing respectively. From this and the fact that 2 < q < 3, it follows that f_1 is nondecreasing and f_2 is nonincreasing. Since f_1(c_0) → +∞ as c_0 → +∞ and f_2(c_0) → +∞ as c_0 → 0+, there is a unique point c^* such that f_1(c^*) = f_2(c^*),ξ(c_0)={[ f_2(x_0) if x_0 ≤ c^*; f_1(c_0) if x_0 ≥ c^* ].and where, moreover,ξ(c^*) is the global minimum of ξ on ^+. It is straightforward to see that the condition f_1(c^*)=f_2(c^*) is equivalent to the condition c^*=M_2(c^*)/M_3(c^*), and that f_1(c^*)=M_2(c^*)^q-3 M_3(c^*)^2-q.We are now in a position to state the main theorem of this section. Let W be as in (<ref>) and suppose that λ is chosen so that 0 ≤λ^3-q h'(λ^3)≤min{ (κ/2) (q-2)^(2-q)/2 q^q/2, (κ/2)λ^2-qM_2(ł,c^*)^q-3 M_3 (ł,c^*)^2-q},where c^* is the unique fixed point of the function c_0 ↦ M_2(ł,c_0)/M_3(ł,c_0). Then I(u) ≥ I() for any map u ∈ H^1(,^3) whose boundary values agree with those ofin the sense of trace.In particular, the largest possible value λ_0 satisfying (<ref>) is a lower bound for λ_crit.Moreover, if (<ref>) holds with strict inequality then there is C=C() >0 such that I(u) - I() ≥ C ∫_ |∇ u - ł 1|^q+ |G(∇ u)| dx.for all admissible u. Acccording to (<ref>), I(u)-I() is bounded below by the sum of ∫__1(∇ u) dx and ∫__2(∇ u) dx.By inequality (<ref>) and Lemma <ref>, the first of these integrals is nonnegative, while Lemmas <ref> and <ref> together imply that the second integral is nonnegative. Either way, it follows that I(u) ≥ I(), as claimed.It is then clear that λ_0, as defined above, is not larger than ł_crit.Now suppose that (<ref>) holds with strict inequality. The proof of <cit.> shows that the inequality ł^3-qh'(ł^3) < κ/2 (q-2)^2-q/2 q^q/2 implies, for some > 0, that_1(A)≥ |Λ(A) - (ł,ł,ł)|^qfor all A in ^3 × 3. Here, Λ(A)=(ł_1(A),ł_2(A),ł_3(A)) is the vector of singular values of the matrix A. The rigidity argument, with minor modifications, given in <cit.> then shows that there is a constant, β(), say, such that∫_|Λ(∇ u) - (ł,ł,ł)|^qdx≥β() ∫_ |∇ u - ł|^qdx.Hence∫__1(∇ u)dx≥β() ∫_ |∇ u - ł|^qdx.Finally, we deal with the term involving _2(A).Fix A in ^3 × 3 and let c_0 = |A-ł|.According to the proof of Lemma <ref>, _2(A) ≥κ/2(|A-ł|^q-j -2ł h'(ł^3)/κ|G(A)|/|A-ł|^j)|A-ł|^jfor j=2 and 3. Reusing the notation w=2ł h'(ł^3)/κ, and applying the strict version of (<ref>), there is '>0, which is independent of c_0, such that w+' ≤max{c_0^q-2/M_2(ł,c_0),c_0^q-3/M_3(ł,c_0)}. By rewriting (<ref>), we obtain _2(A) ≥κ/2(c_0^q-j -(w+') M_j(ł,c_0))c_0^j + κ'/2|G(A)|for j=2 and 3. Thanks to (<ref>), the term in brackets is nonnegative, which leaves _2(A) ≥κ'|G(A)|/2.Both terms in the right-hand side of inequality (<ref>) are now accounted for. The goal of Theorem <ref> is to give the largest possible bound on ł such that I(u) ≥ I().A careful look at the proof of Lemma <ref> shows that one could replace |G(A)| by G^-(A)=-min{G(A),0} and that the same conclusions would result, but with M^-_2(ł,c_0) = sup{G^-(A)/|A-ł|^2: |A-ł|≥ c_0} M^-_3(λ,c_0)= sup{G^-(A)/|A-λ|^3: 0 < |A-λ| < c_0 }in place of M_2(ł,c_0) and M_3(ł,c_0) respectively.Since M_j^-(ł,c_0) ≤ M_j(ł,c_0) for j=2 and 3, it follows that the upper bound involving c^* in (<ref>) would not decrease. Numerical evidence suggests that it does, in fact, increase, and thus provides a better (lower) bound for ł_crit.See Remark <ref> and Section <ref> for further details. §.§ Calculations leading to a concrete upper bound The upper bound (<ref>) given in the statement of Theorem <ref> contains two terms, one of which is explicitly given in terms of the exponent q and one which depends on the fixed point c^* of the function c_0 ↦ M_2(ł,c_0)/M_3(ł,c_0). While it does not seem to be possible to find c^* purely analytically, one can nevertheless make progress using a mixture of analysis and a careful numerical calculation, as we now describe.With M_2(ł,c_0) as defined by (<ref>),lim_c_0 →∞ M_2(ł,c_0) = √(2).First, note that the limit in the statement exists because M_2(c) is nonincreasing and bounded below. Now G(A)/|A-ł|^2= K(Â) +ł(2A - 3 ł)/|A-ł|^2K(Â)-ł/|A - ł|^2(∑_i=1^3ł_i(A) -A )where  = A/|A| and K(Â) = ∑_1 ≤ i < j ≤ 3ł_i(Â)ł_j(Â) - Â.Let c_0 > 0.Since K is bounded and the term ∑_i=1^3ł_i(A) -A has at most linear growth in A, it is clear that there are constants α_1 and α_2, depending only on ł, such that|K(Â)| + α_1/c_0≤|G(A)|/|A-ł|^2 ≤|K(Â)| + α_2/c_0whenever |A-ł| ≥ c_0.We now prove that (a) max{K(Â)}=√(2), where the maximum is taken over all unit matrices Â, and (b) that given any c>0 there is A(c) such that |A(c)-ł| ≥ c and K(A(c))=√(2).The proposition then follows from this and (<ref>).To simplify the notation we replace  by A in the following. By the polar factorization theorem (see <cit.>), there is a positive semidefinite and symmetric matrix U and a matrix R ∈ O(3) such that A = R U.The eigenvalues of U are the singular values ł_i(A), i=1,2,3, and one hasK(A) = ∑_1 ≤ i,j ≤ 3ł_i(A) ł_j(A) - ( R U)= ∑_1 ≤ i,j ≤ 3ł_i(A) ł_j(A) -( ( R) R Q^T D Q),where the well-known decomposition U=Q^T D Q has been used, with D = diag (ł_1(A),ł_2(A),ł_3(A)) and Q in O(3). Note that R = ± 1. It follows that K(A) = ∑_1 ≤ i,j ≤ 3ł_i(A) ł_j(A) -(Pdiag (ł_2ł_3,ł_1ł_3, ł_1 ł_2)),where P:=QR R Q^T belongs to SO(3).The term involving the trace satisfies (Pdiag (ł_2ł_3,ł_1ł_3, ł_1 ł_2) = P_11ł_2ł_3 +P_22ł_1ł_3 + P_33ł_1 ł_2.The term ∑_1 ≤ i,j ≤ 3ł_i(A) ł_j(A) is independent of P, so we can vary P in order to minimize the term involving the trace given above.Since P belongs to SO(3), its rows and columns are orthogonal unit vectors.In particular, |P_11| ≤ 1, so to make P_11ł_2 ł_3 minimal we should take P_11 = -1. (Here we have used the ordering ł_3 ≥ł_2 ≥ł_1.) Hence row 1 of P is (-1,0,0), forcing the second and third rows to take the form(0,P_22,P_33) and (0,P_32,P_33) respectively. In particular, P = -(P_22P_33-P_23P_32) = 1.Without loss of generality, we may take P_22=cosα, P_32=P_23=sinα, P_33=-cosα for some α, in which caseP_11ł_2ł_3 +P_22ł_1ł_3 + P_33ł_1 ł_2≥ -ł_2 ł_3 + cosα (ł_1 ł_3 - ł_1ł_2) ≥ -ł_2 ł_3 - ł_1 (ł_3 - ł_2) by choosing α = π.HenceK(A) ≤ 2 ł_3 (ł_1 + ł_2)where the matrix P yielding this upper bound is given by P=diag(-1,-1,1). Bearing in mind that |A|^2 = ∑_i=1^3ł_i^2 = 1, it can be shown that 2 ł_3 (ł_1 + ł_2) ≤√(2), and that a maximizing choice of singular values is ł_1=ł_2=1/2, ł_3 = 1/√(2).A suitable choice for a maximizing A would therefore be A_0 = Pdiag (1/2,1/2,1/√(2)) = diag(-1/2,-1/2,1/√(2)), and one can check that K(A_0)=√(2).To conclude the proof, it is enough to choose A(c):=r A_0 for any r=r(c) large enough that |A(c) -ł| ≥ c. Since D^2G(ł)=0, it can easily be shown that M_2(ł,c_0) is constant as a function of c_0 for all sufficiently small and positive c_0. We are therefore justified in writing M_2(ł,0):=lim_c_0 → 0M_2(c_0), and in fact M_2(ł,0) = sup{|G(A)|/|A-ł|^2:A ≠ł}.According to Proposition <ref>, we must have M_2(ł,0)≥ M_2(ł,c_0) ≥√(2) for all c_0 > 0. Using a `brute force' approach, which we describe in the appendix, we find that √(2) - M_2(ł,0) ≈ 2.7× 10^-9, independently of ł in the range [1,2]. Thus, in view of (<ref>), it seems that M_2(ł,c_0)≡√(2) for all c_0 and ł, and we record this as: For all ł >0 and c_0>0, M_2(ł,c_0)=√(2). In order to explain the method used to approximate M_2(ł,0), we introduce the notation m_l(A, λ) := |G(A)|/|A - λ1|^lfor l=2 and 3, and we restrict attention to ł in the range [1,2]. Choose a random starting matrix A_1,compute g_1:=∇ m_2(A_1,ł) and find the scalar σ_1, say, which maximises m_2(A_1+σ g_1).Let the maximising value of σ be σ_1;then we set A_2 = A_1 + σ_1 g_1. Proceeding iteratively, we compute a sequence of matrices A_i. It turns out that m_2(A_i, λ) tends to √(2) as i increases, regardless of λ. Supposing that Conjecture <ref> is right, we are now required to find M_2(ł,c^*)^q-3 M_3 (ł,c^*)^q-2=2^q-3/2M_3(ł,c^*), where c^* satisfies c^* = √(2)/M_3(ł,c^*). We begin by recalling that c_0 ↦ M_3(ł,c_0) is nondecreasing, and that, thanks to the at most quadratic growth of the function G(A), there is c_1(ł) such that M_3(ł,c_0)=M_3(ł,c_1) for all c_0 ≥ c_1. We are therefore justified in defining M_3(ł,∞):=M_3(ł,c_1).Appealing again to numerical techniques, which we describe in the appendix, we find that, for ł in the range [1,2], there is very good agreement between c_1(ł) and the expression ν_3ł + ν_2, where ν_2 ≈ 1.764× 10^-3 andand ν_3 ≈ 1.842. Moreover, for the same range of ł, there is strong numerical evidence for the approximation M_3(ł,∞) ≈ν_1/ł, where ν_1 ≈ 0.4501. See Fig 1.This leads naturally to the following: M_3(ł,c_0) = M_3(ł,c_1)=ν_1/ł for all c_0≥ c_1(ł) and for ł in the range [1,2], where c_1(ł)=ν_3ł + ν_2 and the values of ν_1,ν_2 and ν_3 are given above. Let us now suppose that Conjectures <ref> and <ref> are correct. Note that then the function p: c ↦ M_2(ł,c)/M_3(ł,c)=√(2)ł/ν_1 is independent of c for all c ≥ c_1. There are thus two possibilties for the fixed point c^* of p:either c^*=√(2)ł/ν_1 or c^* < c_1. Suppose for a contradiction that c^*< c_1. Then M_3(ł,c^*)≤ M_3(ł,c_1), and so p(c^*) ≥ p(c_1).But p(c^*)=c^* and p(c_1)=√(2)ł/ν_1, which gives c^* ≥√(2)ł/ν_1.By hypothesis, c_1>c^*, which when combined with the preceding inequality implies c_1 > √(2)ł/ν_1.Applying Conjecture <ref> and rearranging, we see that this is equivalent to ν_1ν_2 ≥ (√(2)-ν_3ν_1)ł, which can only hold if ł≤ 1.357 × 10^-3. But we supposed that ł≥ 1, which is a contradiction.In summary, we have shown the following result. Let ł belong to the interval [1,2] and suppose that Conjectures <ref> and <ref> are correct.Then the unique fixed point of the function p(c):=M_2(ł,c)/M_3(ł,c) is given by c^* =√(2)ł/ν_1, and M_2(ł,c^*)^q-3 M_3(ł,c^*)^2-q=1/√(2)(√(2)ł/ν_1)^q-2.Referring back to the upper bound given in (<ref>), we now have the following:Let the assumptions of Proposition <ref> hold.Then I(u) ≥ I() provided0≤λ^3-q h'(λ^3)≤(κ/2)(√(2))^q-3ν_1^2-q.It is enough to show thatmin{(q-2)^(2-q)/2 q^q/2,λ^2-qM_2(ł,c^*)^q-3 M_3 (ł,c^*)^2-q}= (√(2))^q-3ν_1^2-q.By Proposition <ref>, we clearly have λ^2-qM_2(ł,c^*)^q-3 M_3 (ł,c^*)^2-q =(√(2))^q-3ν_1^2-q.Therefore it remains to show that(q-2)^(2-q)/2 q^q/2 > (√(2))^q-3ν_1^2-q for 2 < q <3.Let z(q)=(√(2))^q-3ν_1^2-q and y(q)= (q-2)^(2-q)/2 q^q/2.Note that z is convex, while a short calculation reveals that y is concave on the interval (2,3).Therefore the inequality z(q) < y(q) will follow from the pair of inequalities z(2)<y(2+) and z(3)<y(3), both of which are easy to check.This concludes the proof. Corollary <ref> suggests that the bound is not the best possible:we would really expect both terms in the bound given in (<ref>) to play a role.One way to achieve this might be to use the quantities M_j^- in place of M_j for j=2,3. Indeed, we find, again numerically, that there is very good agreement between M_3^-(ł,∞) and ν_1'/ł for ł in the range [1,2], where ν_1' = 0.1923.Interestingly, if we then replace M_3 by M_3^- everywhere in the preceding calculations, we find that z(3) < y(3) no longer holds. In other words, both terms in the upper bound given by (<ref>) appear to contribute.This observation comes with some caveats, however; see Section <ref> in the appendix.§ A CONDITION FOR THE INEQUALITY I(U) ≤ I() The results in the previous sections provide conditions on ł under which the inequality I(u) ≥ I() holds for admissible maps.It is natural to ask what information results from supposing that I(u)≤ I(), and, of our results, Theorem <ref> is the first place to look.Now, if (<ref>) holds with strict inequality, thensits in a `potential well', as expressed by the estimate (<ref>), which we recall here for the reader's convenience:I(u) - I() ≥ C ∫_ |∇ u - ł 1|^q+ |G(∇ u)| dx.Thus, in these circumstances, I(u) ≤ I() is impossible.If (<ref>) holds with equality then a similar remark applies, but with the additional possibility of losing one or both terms in the right-hand side of (<ref>).And when (<ref>) fails, the preceding analysis tells us nothing about those u whose energy satisfies I(u) ≤ I().Therefore a different approach is called for. Consider the following simplified model, in which we set the function Z appearing in (<ref>) to zero. Thus we letW(A) = |A|^q + h( A) for A ∈^3 × 3,where, as before, 2 < q < 3.Let ł > 1 and note that, by the convexity of h, I(u) - I()≥∫_ h'(ł^3)(∇ u - ł^3) + |∇ u|^q - |∇|^qdx ≥∫_ h'(ł^3)(∇ u - ł^3) + κ|∇ u-∇|^q dx,where we have used (<ref>) and (<ref>), and where the constant κ obeys the bounds specified in inequality (<ref>). Using identity (<ref>) and the definition (<ref>) of P, write ∇ u -ł^3 = ł̂_1ł̂_2ł̂_3 + ł P(∇ u),where ł̂_i:=ł_i-ł and ł_i:=ł_i(∇ u). Finally, recall that the function G defined in (<ref>) satisfies∫_ G(∇ u)dx= ∫_ P(∇ u)dxwhenever u is admissible. Combining this with (<ref>) and (<ref>), we haveI(u) - I()≥∫_ h'(ł^3)(ł̂_1ł̂_2ł̂_3 + ł G(∇ u)) + κ|∇ u-∇|^q dx.It will be useful to have a shorthand for the function with prefactor h'(ł^3) appearing in (<ref>);accordingly, letH(A) = ł̂_1(A)ł̂_2(A)ł̂_3(A) + ł G(A). We now give a series of results which allow us to find a lower bound on the function H. Let A be a 3 × 3 matrix such that A > 0 and whose singular values obey ł_1(A) ≤ł_2(A) ≤ł_3(A).Then there is R in SO(3) such that G(A) =(1-R_11)α + (1-R_22)β + (1-R_33)γ, where α = ł_2 ł_3-łł_1, β = ł_1 ł_3 - łł_2 and γ = ł_1 ł_2 - łł_3, and ł_j=ł_j(A) for j=1,2,3. For brevity, write ł_j in place of ł_j(A) for j=1,2,3.Note that P(A) = α + β + γ by definition, and that, by polar decomposition, there are matrices Q_1 and Q_2, belonging to O(3), such that A= Q_2 Q_1^T D Q_1 and where D is a diagonal matrix with entries ł_1, ł_2 and ł_3. (See <cit.>.)Since A > 0, Q_2 must belong to SO(3). We see thatN(A) =(Q_2 Q_1^T D Q_1) - ł Q_2 Q_1^T D Q_1=(R ( D - ł D)),where R:=Q_1 Q_2 Q_1^T belongs to SO(3). Hence N(A) = R_11α + R_22β + R_33γ, and (<ref>) follows. Our aim is to minimize G(A) by allowing R to vary in SO(3).To that end, consider the following. Let R belong to SO(3) and suppose that it minimizesg(R) = (1-R_11)α + (1-R_22)β + (1-R_33)γ. Then α R_12 -β R_21= 0α R_13 -γ R_31= 0β R_23 - γ R_32= 0.If none of α,β,γ is zero then ((α/β)^2-1)R_12^2 + ((α/γ)^2-1)R_13^2 = 0((β/γ)^2-1)R_23^2 - ((α/β)^2-1)R_12^2 = 0.Moreover, if exactly one of α, β, γ is zero, theneither R_11^2=1 or R_33^2=1. The same is true if none of α, β and γ is zero, provided at least one of α > β and β > γ holds. We first show that (<ref>)-(<ref>) hold.It is well known that the tangent space to SO(3) at R consists of those ρ in R^3 × 3 such that R^T ρ is antisymmetric. From this, it easily follows that there are real numbers a,b and c such that ρ_11= -b R_13 - a R_12 ρ_22= a R_21 - c R_23 ρ_33= b R_31+ c R_32.Now suppose that R() is a smooth path of matrices belonging to SO(3), and satisfying R(0)=R and Ṙ(0)=ρ. We then have∂__ = 0 g(R()) = a(R_12α - R_21β) + b (R_13α - R_31γ) + c(R_23β -R_32γ),so that, by varying a,b and c independently, the stationarity conditions (<ref>)-(<ref>) follow.To prove the last part of the statement, we consider cases as follows. Case (i): α=0 > β≥γ. Using (<ref>) and (<ref>), we see that R_21=R_31=0.Hence, since the first column of R is a unit vector, we must have R_11^2=1. Case (ii):α> 0=β > γ. Using (<ref>) and (<ref>), we must have R_12=0 and R_31=0.By (<ref>), we then have R_13=0.Hence R_11^2=1, as before.Case (iii):α≥β > γ=0. Equations (<ref>) and (<ref>) imply respectively that R_13=0 and R_23=0.Hence R_33^2=1.Case (iv):α >β > γ; α≠ 0, β≠ 0, γ≠ 0. In this case, (<ref>) implies that R_12 = R_21 = 0, and so R_11^2=1.Case (v):α = β > γ; α≠ 0, β≠ 0, γ≠ 0.Now (<ref>) implies that R_13=0 and (<ref>) that R_23=0.Hence R_33^2=1.Case (vi):α >β = γ; α≠ 0, β≠ 0, γ≠ 0. Equations (<ref>) and (<ref>) imply that R_12=R_13=0, and hence R_11^2=1. The next result will enable us to deal with the case α = β = γ.Let R belong to SO(3) and suppose that R is symmetric.Then 3 ≥ R ≥ -1.If R is a symmetric, orthogonal matrix then R^2=, from which it follows that any eigenvalue μ_i of R must satisfy μ_i^2=1.Moreover, μ_1 μ_2 μ_3=1, from which it follows that 3 ≥ R = μ_1+μ_2+μ_3 ≥ -1.Let R belong to SO(3).g(R)= (1-R_11)α + (1-R_22)β + (1-R_33)γsatisfies g(R)≥min{2(β + γ),0}. If two or more of α, β and γ are zero then the lower bound g(R) ≥ 2(β + γ) is trivial.Thus, to minimize g, we may begin by supposing that the conditions of Lemma <ref> apply, so that, in all cases except α = β = γ, we have either that R_11^2=1 or R_33^2=1. First suppose that R_11^2 = 1.Then the diagonal elements of R are either of the form 1,cosσ, cosσ for some σ, or else of the form -1, cosσ, - cosσ.In the former case, g(R)= (1-cosσ)(β + γ).If β+ γ≥ 0 then clearly g(R) ≥ 0. If β + γ <0 then to minimize g we take cosσ = -1 and the claimed lower bound follows.If R_11=-1, then g(R) = 2 α + β + γ + (γ - β) cosσ,which, since γ≤β, implies that we should take cosσ = 1 in order to minimize g.Hence g(R)≥ 2 (α + γ) ≥ 2(β + γ).If R_33^2=1 then the argument needed is similar.Finally, let us suppose that α = β = γ.Then g(R) = (3 - R) γ.If γ < 0 then g is minimized when R = -1, according to Lemma <ref>.Hence, in this case, g(R) ≥ 4 γ = 2(β + γ).Otherwise, g(R) ≥ 0 because R ≤ 3 by Lemma <ref>. This completes the proof. Let ł > 0, let H be given by (<ref>) and let ł_1 ≤ł_2 ≤ł_3 be the singular values of A. Then H(A)≥{[ (ł_1-ł)(ł_2-ł)(ł_3-ł) if ł_1 ≥ł; (ł_1-ł)(ł_2+ł)(ł_3+ł)if ł_1 ≤ł. ].First suppose that ł_1 ≥ł. Note that β+γ= (ł_1 - ł)(ł_2+ł_3) is then nonnegative, and so, by inequality (<ref>) in Lemma <ref>,we have G(A)=g(R) ≥ 0.Hence H(A)=(ł_1-ł)(ł_2-ł)(ł_3-ł) + ł G(A) ≥ (ł_1-ł)(ł_2-ł)(ł_3-ł). Now suppose that ł_1 ≤ł.Then the lower bound in (<ref>) is 2(β+γ)≤ 0, and so H(A)≥ (ł_1-ł)(ł_2-ł)(ł_3-ł) +2ł (ł_1-ł)(ł_2+ł_3)= (ł_1-ł)(ł_2+ł)(ł_3+ł). The main result of this subsection is the following.Suppose h'(ł^3) ≥ 0 and let W be given by (<ref>).Then any admissible map u ≠ is such that: I(u) -I()≥∫_κ|∇ u-∇|^q+ h'(ł^3) H(∇ u)dx.In particular, if ∫_H(∇ u)dx ≥ 0 then I(u) > I(), while if I(u) ≤ I() then ∫_{x ∈: ł_1(∇ u (x)) ≥ł}h'(ł^3)(ł_1-ł)(ł_2-ł)(ł_3 - ł) +κ|∇ u-∇|^qdx≤∫_{x ∈: ł_1(∇ u (x)) ≤ł} h'(ł^3) (ł - ł_1)(ł+ł_2)(ł+ł_3) dx. This follows from (<ref>) and Proposition <ref>. We remark that the results of Section <ref> imply that inequality (<ref>) ought not to be possible for ł such that ł^3-qh'(ł^3) is sufficiently small (see (<ref>)).It is not immediately obvious from (<ref>) why this should be so;nor is it clear why such a prominent role is played by the smallest singular value ł_1(∇ u).This surely warrants further investigation.§ APPENDIXTwo different algorithms have been used to compute the quantity M_3(ł,∞):=lim_c →∞M_3(ł,c), as defined in Subsection <ref>.We recall the notation m_l(ł,A)=|G(A)|/|A-ł|^lfor l=2,3.The algorithms are also brought to bear on the problem of calculating M_3^-(ł,∞), and we summarise the results below. §.§ Algorithm A: conjugate gradient This is a `brute force' approach, which consists of* Choosing a value of λ∈[1, 2];* Generating matrices A with all elements a_ij being uniformly distributed random numbers over the interval [-α, α], for given α;* Using the Polak-Ribiere variant of the Fletcher-Reeves algorithm <cit.>, one of the so-called conjugate gradient methods for maximisation of smooth functions, startingfrom each of these matrices, in order to find a candidate matrix for A_l^*(λ), which is an approximate maximiser of m_3(·,ł).We choose α = 5, the justification for which is as follows.Five thousand matrices were generated and those which gave the 15 largest values of m_3(ł,·) were saved as the computation proceeded.Of these, none had an element whose modulus exceeded 3.1, hence reassuring us that the choice α = 5 is `safe' for ł in the range [1,2]. Since the algorithm is iterative, a stopping condition is required, and this is that|m^(i+1) - m^(i)| ≤ϵ/2(|m^(i+1)| + |m^(i)|),where m^(i) is the value of m_3(A_i, λ) at the i-th iteration and ϵ = 10^-9.Various data are saved as the computation progresses, including the current maximising matrix, which is the latest approximation to A_l^*(λ). For half the simulations, the initial random matrix is symmetric, and for the other half it is not — we do not know, a priori, whether A_l^*(λ) will be symmetric or not.The numerics strongly indicated that A_3^*(ł) will indeed be symmetric, at least for ł in the range [1,2].All computations were carried out using 40 significant figures.Algorithm A leads the approximation M_3(ł,∞)≈ν_1^A/ł, where ν_1^A=0.4501 —seeTable <ref>; the algorithm also produces the approximation to c_1(ł) shown in Figure 2 and summarised in Table <ref> below.§.§ Algorithm B: pointwise supremum This is based on a different idea, although a Monte Carlo approach it is still at its heart.We start by fixing an interval for λ, Λ = [λ_-, λ_+], which is not necessarily [1, 2] — the computation time is, at one level, independent of the interval. We then define N_p equally-spaced points in Λ, these points being λ_i = λ_- + iδλ with δλ = (λ_+ -λ_-)/N_p and i = 0, …, N_p.Next, as before, a large number, N, of random matrices A_i are generated. As can be seen from its definition,m_l(A_i, λ) = |a_1 + a_2λ|/(b_1 + b_2λ + b_3λ^2)^l/2where the coefficients a_1,… b_3 are functions of the elements of A_i that we compute numerically. We define f_l, i(λ) := m_l(A_i, λ), and clearly, once the coefficients have been computed, f_l, i(λ) can easily be found for any λ. We then computeF_l, j = sup{f_l,i(λ_j), i = 1,…, N}for j = 0,…, N_p; F_l, j is then a discrete approximation toM_l(λ_j,∞). The convergence to M_3(ł,∞) is quite slow, but nonetheless, choosing N large enough gives reasonable agreement with results produced by Algorithm A, thereby providing an independent check.Compare Fig. 1 with Fig. 2 below. §.§ Calculating M^-_3(ł,∞)Using Algorithm A, the methodology is the same as for M_3(λ,∞), with the same number of random matrices generated, whose elements have the same bounds. The investigations lead us to conjecture thatM^-_3(λ,∞) ≈ν_1^A,-/λ,where ν_1^A,-≈ 0.1923.See Figure <ref>. Recall that in the case of M_3(ł,∞) it was possible to compute and then model the quantity c_1(ł) accurately on the interval 1 ≤ł≤ 2 using an affine function of ł.The same cannot be said of the corresponding quantity c^-_1(ł), and indeed this seems to behave somewhat erratically as a function of ł.Thus the analysis leading up to Proposition <ref> does not apply, and hence the caveat regarding the substitution of M^-_3(ł,∞)≈ν_1'/ł promised in Remark <ref>. 99 Ba77 J. M. Ball: Convexity conditions and existence theorems in nonlinear elasticity.Arch. Rat. Mech. Anal.,63, no. 4 (1977), 337–403. Ba82J. M. Ball.Discontinuous equilibrium solutions and cavitation in nonlinear elasticity.Phil. Trans. R. Soc. Lond. A, 306, 557-611 (1982). BCOJ. M. Ball, J. C. Currie, P. J. Olver.Null Lagrangians, weak continuity, and variational problems of arbitrary order.J. Funct. Anal., 41 (1981), no.2, 135-174. BZ15 J. Bevan, C. Zeppieri.A simple sufficient condition for the quasiconvexity of elastic stored-energy functions in spaces which allow for cavitation.Calculus of Variations and Partial Differential Equations, 55 (2), 1-25, 2015. Ci04 Philippe G. Ciarlet. Mathematical Elasticity Volume I: Three Dimensional Elasticity, Elsevier, 2004. Dac08 B. Dacorogna: Direct methods in the calculus of variations. Second edition. Applied Mathematical Sciences, 78. Springer, New York, 2008.HMC10 D. Henao, C. Mora-Corral:Invertibility and weak continuity of the determinant for the modelling of cavitation and fracture in nonlinear elasticity. Arch. Rat. Mech. Anal., 197 (2010), 619–655. HMC11 D. Henao, C. Mora-Corral:Fracture Surfaces and the Regularity of Inverses for BV Deformations. Arch. Rat. Mech. Anal., 201 (2011), 575–629.MS95 S. Müller, S. Spector. An existence theory for nonlinear elasticity that allows for cavitation.Arch. Rat. Mech. Anal. 131(1995), 1-66. MSS96 S. Müller, J. Sivaloganathan, S. Spector:An isoperimetric estimate and W^1,p-quasiconvexity in nonlinear elasticity. Calc. Var. Partial Differential Equations, 8, no. 2 (1999), 159–176. NRC W.H. Press, S.A. Teukolsky, W.T. Vetterling and B.P. Flannery, Numerical Recipes in C, ISBN 0-521-43108-5, Cambridge University Press, Cambridge, UK (1992). Si86 J. Sivaloganathan.Uniqueness of regular and singular equilibria for spherically symmetric problems of nonlinear elasticity. Arch. Rat. Mech. Anal., 96 (1986), 97-136. SS08 J. Sivaloganathan, S. Spector. Energy minimising properties of the radial cavitation solution in incompressible nonlinear elasticity. J. Elasticity, 93 (2008), no. 2, 177-187. SS08prime J. Sivaloganathan, S. Spector. Necessary conditions for a minimum at a radial cavitating singularity in nonlinear elasticity. Ann. Inst. H. Poincaré Anal. Non Linéaire,25, no. 1 (2008), 201-213. St84 C. A. Stuart.Radially symmetric cavitation for hyperelastic materials. Ann. Inst. H. Poincaré Anal. Non Linéaire,2, no. 1 (1985), 33-66. | http://arxiv.org/abs/1707.08532v1 | {
"authors": [
"Jonathan J. Bevan",
"Jonathan H. B. Deane"
],
"categories": [
"math.AP",
"49J40, 74B20"
],
"primary_category": "math.AP",
"published": "20170726164558",
"title": "A calibration method for estimating critical cavitation loads from below in 3D nonlinear elasticity"
} |
[ [ December 30, 2023 =====================Extracting governing equations from dynamic data is an essential task in model selection and parameter estimation. The form of the governing equation is rarely known a priori; however, based on the sparsity-of-effect principle one may assume that the number of candidate functions needed to represent the dynamics is very small. In this work, we leverage the sparse structure of the governing equations along with recent results from random sampling theory to develop methods for selecting dynamical systems from under-sampled data. In particular, we detail three sampling strategies that lead to the exact recovery of first-order dynamical systems when we are given fewer samples than unknowns.The first method makes no assumptions on the behavior of the data, and requires a certain number of random initial samples. The second method utilizes the structure of the governing equation to limit the number of random initializations needed.The third method leverages chaotic behavior in the data to construct a nearly deterministic sampling strategy. Using results from compressive sensing, we show that the strategies lead to exact recovery, which is stable to the sparse structure of the governing equations and robust to noise in the estimation of the velocity. Computational results validate each of the sampling strategies and highlight potential applications.§ INTRODUCTIONSince the scientific revolution in the seventeenth century, scientists have endeavored to extract increasingly sophisticated physical models from experimental and observational data. This process is usually done manually in the sense that a scientist, with a strong expertise in the field, must examine the data to uncover meaningful physical laws. Parsing through data by hand is often time-consuming, expensive, and infeasible – especially when the dimension of the system grows. However, with advances in computing and machine learning algorithms, automated discovery of physical models and mathematical equations directly from data is becoming increasingly possible. Some applications of data-based modeling include (but are certainly not limited to) weather predictions, controls for fluid flows <cit.>, construction of climate trend models <cit.>, and disease control models <cit.>. Over the last decade, data-driven methods have seen strong growth due to the abundance of data and the development of sophisticated analytical tools. In this work, a computational method for identifying high-dimensional differential equations (which we will refer to as the `model') from under-sampled dynamical data is developed. We will restrict our attention to quadratic models, which contain a large subset of potential physical laws and applications; for example, Lorenz-like systems used in atmospheric science, Fisher's equation and related reaction-diffusion systems, Vlasov-Poisson equations from plasma physics, and fluid dynamics models like the Navier-Stokes equations. The governing equations for such systems are often moreover sparse quadratic polynomials, in that the dynamics depend on a small number of variables and second-order interactions.Indeed, this allows us to recover the governing equations from under-sampled data: we only need to observe a number of samples of the data roughly on the order of the sparsity level of the system. Our method learns the governing models via a sparse optimization problem over a large set of potential candidate functions. It will be shown the our method selects the correct governing dynamical system (under certain conditions) even when the size of the candidate set far exceeds the size of the data set. In this way, these methods are one of the first applications of compressive sensing to model extraction for dynamical systems.Recently, regression based methods have been developed and applied to model selection and parameter estimation of dynamic data. The authors of <cit.> first proposed the use of regression to select physical laws from synthetic and experimental data. In particular, a symbolic regression method was developed that compared computed derivatives of the data to analytic derivatives of trial functions, while controlling for the total number of trial functions selected (as to avoid overfitting).In <cit.>, a sparsity-promoting method was proposed for extracting dynamical systems by comparing the computed velocity to a large set of potential trial functions. A sequential thresholded least-squares algorithm was used to fit a (redundant) set of trial functions (typically in the form of monomials) to the velocity.In <cit.> sparsity-based methods were proposed for learning nonlinear partial differential equations from spatio-temporal data. In <cit.>, a non-convex sparse optimization method was developed to identify the underlying dynamical system from noisy data sets using an integrated trial set. In <cit.>, the authors utilized the Akaike information criterion to rank different sparse solutions using the method from <cit.>, allowing for the automated selection of different models when varying the method's free parameter. Using a sparse regression method along with the Takens' embedding theorem, a data-driven method was proposed in <cit.> to decompose chaotic systems into intermittently forced linear systems. A sparse convex optimization method was proposed for joint outlier detection and model selection in <cit.>, when the observed data is locally corrupted by high variance noise. Unlike the previous work in the literature, which was mostly empirical, the authors of <cit.> proved that the separation between the outliers and the `clean' data can be exactly recovered. The use of L^1 minimization has also appeared in various data-driven and scientific computing applications, for example <cit.>. Other data-driven methods for learning data structure and approximating dynamics include: the proper orthogonal decomposition<cit.>, Koopman representations <cit.>, diffusion maps <cit.>, and dynamic mode decomposition <cit.>.These data-based regression methods often use the ℓ^0 or ℓ^1 penalty to promote sparsity in the learned models (i.e. to select a small number of active candidates from the large set of potential trial functions).Soft-thresholding (related to the ℓ^1 penalty) for sparse recovery and denoising was first proposed in <cit.>. The ℓ^1 regularized least squares problem (referred to as the least absolute shrinkage and selection operator or LASSO) was introduced in <cit.> to reconstruct a sparse vector from linear observations. Conditions under which ℓ^1 penalized problems admit sparse solutions are detailed in <cit.> and have led to many applications in imaging and signal processing.Refined conditions for the setting of function interpolation and approximation were developed in <cit.>. In this work, we develop a model selection and parameter estimation method for learning quadratic high-dimensional differential equations from under-sampled data. Using results from compressive sensing and sampling theory, we show that given a certain sampling of the initial data, a convex optimization problem can recover the coefficients and the governing equations exactly even when the data is under-sampled. We detail three sampling strategies, depending on the level of knowledge of the data or the governing equation. In particular, one can decrease the number of samples needed by adding assumptions to the system. It is important to note that the methodology presented in this work can be extended to higher-order polynomial systems and, more generally, governing equations which are sparse with respect to any given bounded orthogonal system. Also, this formulation benefits from well-known numerical methods for solving ℓ^1 penalized problems. [Our code is available on our github page: <https://github.com/GiangTTran/ExtractingSparseHighDimensionalDynamicsFromLimitedData>.]§ PROBLEM STATEMENTConsider the dynamical variable x(t)∈ℝ^n (n≫ 1) governed by the equation ẋ=f(x) with initial data x(t_0)=x_0.Assume that f(x) is a quadratic vector-valued equation in x, which can be written component-wise as:ẋ_1= f_1(x_1,…, x_n) ẋ_2= f_2(x_1,…, x_n) ⋮ ẋ_n = f_n(x_1,…, x_n).The goal is to learn f_1, …, f_n, given x and ẋ. For a given initial condition x_0, assume that we can obtain a sequence of discrete measurements { x(t_1), x(t_2), …, x(t_m-1)} through either simulations or observations; however, the function f(x) is unknown.The total number of samples along a given trajectory (including the initialization), denoted as m, can be small and thus we will refer to these measurements as a single burst. The key here is that by using a small number of bursts corresponding to random initial data, we are able to provide conditions on the recovery of the underlying dynamics. Although we will show the construction and results for quadratic systems, it is important to note that it is possible to generalized to high-order systems.We construct the under-sampled measurements as follows. Let k be the index for a given burst; that is, if we sample the initial data x(t_0;k) from some random distribution, we can observe the k^th burst, i.e {x(t_0;k), x(t_1;k), x(t_2;k), …, x(t_m-1;k)}. The corresponding velocity along the burst, denoted by {ẋ(t_0;k), ẋ(t_1;k), ẋ(t_2;k), …, ẋ(t_m-1;k)}, is either observed or computed to some level of accuracy. The collection of trial functions corresponding to the k^th burst is denoted as: A^(k) =[1 x_1(t_0;k) x_2(t_0;k)⋯ x_n(t_0;k) x^2_1(t_0;k) x_1(t_0;k)x_2(t_0;k)⋯ x^2_n(t_0;k);1 x_1(t_1;k) x_2(t_1;k)⋯ x_n(t_1;k) x^2_1(t_1;k) x_1(t_1;k)x_2(t_1;k)⋯ x^2_n(t_1;k);1 x_1(t_2;k) x_2(t_2;k)⋯ x_n(t_2;k) x^2_1(t_2;k)x_1(t_2;k) x_2(t_2;k)⋯ x^2_n(t_2;k); ⋯ ; ⋯ ; ⋯ ;1 x_1(t_m-1;k) x_2(t_m-1;k)⋯ x_n(t_m-1;k) x^2_1(t_m-1;k) x_1(t_m-1;k)x_2(t_m-1;k)⋯ x^2_n(t_m-1;k);] and the velocity matrix is denoted as:V^(k)=[ ẋ_1(t_0;k) ẋ_2(t_0;k)⋯ ẋ_n(t_0;k); ẋ_1(t_1;k) ẋ_2(t_1;k)⋯ ẋ_n(t_1;k); ẋ_1(t_2;k) ẋ_2(t_2;k)⋯ ẋ_n(t_2;k);⋯ ; ẋ_1(t_m-1;k) ẋ_2(t_m-1;k)⋯ ẋ_n(t_m-1;k) ] . Let c_j be the vector of coefficients corresponding to the j^th governing equation, 1≤ j ≤ n, and define the coefficient matrix:C= [ | | |; c_1 c_2 ⋮ c_n; | | | ].Using the k^th burst data leads to the subproblem: find c such that V^(k) = A^(k) C. Next, we combine the data over all bursts k from 1,…, Kby simply concatenating the burst arrays vertically: V = [ V^(1); V^(2); |; V^(K) ]and A = [ A^(1); A^(2); |; A^(K) ]and thus the full problem is to find C such that V = A C. Let N:=n^2+3n+22 be the number of monomials of n variables up to degree two, then the size of matrix A is mK × N with mK<N therefore the linear inverse problem is ill-posed. Since the coefficient matrix C is sparse, the inversion could be regularized by solving the non-convex optimization problem:min_C||C||_0 subject toAC-V≤σ,where ||· ||_0 is the ℓ^0 penalty, i.e. the number of nonzero terms and the norm ||· || is the maximum of the ℓ^2 norm of each row. In this way, the inversion is component-wise separable.The parameter σ>0 controls for the error between the computed derivative and the true velocity. For any fixed value of σ, the general ℓ^0 penalized problem is NP hard <cit.>; and thus we use the convex relaxation known as the ℓ^1 basis pursuit method:(M-BP_σ):min_C||C||_1 subject toAC-V≤σ.Note that if V and A are given with high accuracy then we can solve the following:min_C||C||_1 subject toAC=V. The procedure and optimization problem above will be referred to as the monomial basis pursuit M-BP, since the trial set contains monomials up to degree two. We can also repeat the process using the tensorized quadratic Legendre polynomials, which corresponds to changing Equation (<ref>) to: A_L^(k) =[ 1√(3)x_1(t_0;k) ⋯√(3)x_n(t_0;k)√(5)3x^2_1(t_0;k)-123x_1(t_0;k) x_2(t_0;k) ⋯; 1√(3)x_1(t_1;k) ⋯√(3)x_n(t_1;k)√(5)3x^2_1(t_1;k)-12 3x_1(t_1;k)x_2(t_1;k) ⋯; 1√(3)x_1(t_2;k) ⋯√(3)x_n(t_2;k)√(5)3x^2_1(t_2;k)-12 3x_1(t_2;k)x_2(t_2;k) ⋯; ⋯; 1√(3)x_1(t_m-1;k) ⋯√(3)x_n(t_m-1;k) √(5) 3x^2_1(t_m-1;k)-12 3x_1(t_m-1;k)x_2(t_m-1;k) ⋯; ] The resulting inverse problem will be referred to as the Legendre basis pursuit L-BP, and we write it concisely as:min_C_L ||C_L||_1 subject toA_L C_L=V. Note that sparsity of the solution of L-BP can be slightly larger than M-BP, since the `pure' quadratic terms now include a constant term.In particular, if a component of the governing equation is s-sparse with respect to A, then it is at most (s+1)-sparse with respect to A_L. A noise-robust extension of this is given by:(L-BP_σ):min_C_L||C_L||_1 subject toA_LC_L-V≤σ.§ RECONSTRUCTION GUARANTEECompressive sensing theory provides reconstruction guarantees for sparse solutions to ill-posed linear inverse problems via the solution to ℓ_1 optimization problems like L-BP – let us recall the basics of this theory.Consider a general system of linear equations, y = A x.If A is underdetermined, or has fewer rows than columns, it is in general impossible to determine x given only y and A. However, if we know a priori that x ∈ℝ^N is s-sparse, or only has s ≪N non-zero entries, the locations of which are unknown, and if the underdetermined matrix A is suitably incoherent, it is possible to recover such an x from only M = O(s logN) measurements as the unique solution to the ℓ_1-minimization problem x = min_zz _1subject to A z = y.Moreover, if there is noise on the measurements y = Ax + η, then x is well-approximated by the solution x^#∈min_zz _1subject toA z - y _2 ≤σ for σ appropriately chosen. Thus, by leveraging sparsity, we can potentially overcome the curse of dimensionality in the number of measurements we need to take, which often renders high-dimensional inverse problems intractable.In our setting, the matrix of coefficients C (equivalently, each of the n columns c_1, c_2, …, c_n) is s-sparse, and so we can ask: under what conditions on the measurementmatrix A (or the Legendre-transformed A_L) can we apply compressive sensing results and conclude that C is exactly recovered as the solution to M-BP_σ (or L-BP_σ)?As it turns out, if each of the K bursts is initialized at a uniformly random point in [-1,1]^n, then the K× N measurement matrix A_L corresponding to only the K initialization measurements, has precisely the incoherence properties that provide optimal compressive sensing results.Thus, if each of the coefficient vectors c_k is s-sparse, then we only need to measure on the order of K ∼ s log N bursts to recover the coefficients exactly.Each burst need only be measured long enough to get an accurate approximation to the initial velocity. Informally, the size of the burst should be large enough to get a stable approximation to the velocity via a finite difference approximation.The remarkable fact that sparse vectors can be exactly recovered from vastly underdetermined linear systems of equations cannot be possible for just any underdetermined system of equations Ax = y – for example, the M × N matrixA consisting of the first M rows of the N × N identity matrix maps all sparse vectors whose support does not intersect the first M coordinates to its null space, rendering it impossible to distinguish them. Indeed, the matrix A must have the incoherence property which implies that its null space only intersects the set of sparse vectors trivially, and is sufficiently “well-conditioned" over the set of sparse vectors to permit stability to noise. The Legendre-transformed matrix A_L with burst length m = 0 and with initializations x(t_0;1), …, x(t_0;K) taken as independent and identically distributed as uniform random variables [-1,1]^n satisfies these requirements; in particular, it has two key properties which permit theoretical results on sparse recovery:* Its rows ω_1, ω_2, …, ω_K are independent realizations of an isotropic random variable ω; that is, its covariance matrix is the identity matrix 𝔼[ωω^*] =1_N × N,* Its rows are uniformly bounded: max_1 ≤ i ≤ N, 1 ≤ j ≤ K | A_L(i,j) |^2 ≤ 9.Using these properties, we may apply Theorem 1.2 from <cit.> (see also Theorem 12.22 in <cit.>) to arrive at the reconstruction guarantee.Consider a dynamical system ẋ=f(x) to be recovered from snapshotsx(t_0; k), … x(t_m-1; k) and corresponding velocities ẋ(t_0; k), …ẋ(t_m-1;k), k=1,2,…, K. Assume that each component f(x) = (f_1(x), f_2(x), …, f_n(x)) is a quadratic vector-valued equation in x, and that each of the f_k has at most s out of N polynomial coefficients non-zero.Suppose that the initialization x(t_0;k) for each of the k = 1,2,… K bursts is chosen independently at random from the uniform distribution over [-1,1]^n. Suppose the number of bursts satisfiesK≥ 9 c_* s log(N) log( ε^-1)where c_* is a universal constant. Then with probability 1-ε, any particular component of the n governing equations in the system ẋ=f(x) is recovered exactly via the polynomial coefficients as the unique solution to (L-BP).More generally, under the same conditions as above and with the same probability, if the measured gradient terms are only approximate, ẋ(t_0;k) = ẋ(t_0; k) + τ_0;k, … ,ẋ(t_m-1;k) = ẋ(t_m-1; k) + τ_m-1;k, such that √(1/K∑_k=1^K |τ_0;k|^2)≤η, then considering (L-BP_σ) with matrix A_L consisting of only the initial burst data and with σ = √(K)η, any particular coefficient vector c_k is approximated by a minimizer c_k^# of(L-BP_σ) according toc_k^# - c_k _2≤ c^* √(s)ηwhere c^* is a universal constant.The proof of Theorem<ref> is deferred to the Appendix.We note that by the union bound (Boole's inequality), if we observe K' ≥ 9 c_* s log(N) log( nε^-1) bursts, then with probability 1 - ε, the error bounds are guaranteed to hold uniformly for each of the n governing equations. Also, the arguments require a uniform bound on the elements of the matrix A_L restricted to the initial data, which can be chosen to be uniform random variables on [-1,1]^n. In general, trajectories of polynomial governing systems can become unbound in finite-time, which would make a uniform bound on A_L not possible. However, with the burst approach, one should expect the trajectory to exist for a short time.This sparse recovery result can be generalized in several ways; namely, * The initializations should be independent and identically distributed random variables, but it is not crucial that they beuniform random variables on [-1,1]^n.Using results from <cit.>, the initializations could instead be taken to be i.i.d. according to the Chebyshev measure on [-1,1]^n, any measure interpolating between the Chebyshev measure and the uniform measure. * The cubic case is an important structure that can arise from special symmetries. Although it is not directly considered here, our approach can be applied to extract cubic governing equations from data. Theorem <ref> holds if we modify Equation (<ref>) to:K≥ 27 c_* s log(N) log( ε^-1) since one can show that the matrix A_L restricted to uniformly random measurements satisfies the bound max_1 ≤ i ≤ N, 1 ≤ j ≤ K | A_L(i,j) |^2 ≤ 27. However, in the cubic case the number of unknowns N can be quite large since N = n +33 compared to the quadratic case whereN = n +22. Therefore, the number of basis elements N will be extremely large for high-dimensional cubic systems, rendering the ℓ_1 reconstruction algorithm quite costly without taking into account additional assumptions on the sparsity structure. Constructing a computational efficient approach under reasonable structural conditions will be explored in a future work. * We could have derived reconstruction guarantees for higher-order polynomial systems.As with the cubic case, the only difference in the theoretical result is that the uniform bound max_1 ≤ i ≤ N, 1 ≤ j ≤ K | A_L(i,j) |^2 will grow with the maximal polynomial degree; the constant 9 in Equation (<ref>) will increase accordingly.* We are not even restricted to consider governing equations which are sparse with respect to the polynomial basis; one could consider a different orthonormal basis such as e.g., sines and cosines which are uniformly bounded, and take the initializations of the bursts as i.i.d. random variables according to the orthogonalization measure (or a measure which is “close to" the orthogonalization measure) of that basis. § ERGODICITY AND THE NUMBER OF BURSTS Theorem <ref> says that any dynamical system described as a system of sparse quadratic ordinary differential equations can be recovered exactly from a small number of randomly initialized bursts.In particular, the number of bursts need only scale with the sparsity level of the system, and only logarithmically with the ambient dimension.This result is true for any such dynamical system, independent of the behavior of the trajectories along the bursts.In this sense, it is a “worst-case"result.In many situations, the data x(t_0), x(t_1), …, x(t_m-1) along a single burst behaves chaotically, mimicking the behavior of a random sequence, and in such cases the number of bursts actually required to achieve exact recover should be far smaller – as our numerical evidence shows, even a single burst often suffices. While theoretical results concerning the behavior of high-dimensional dynamical systems have remained elusive, recent large-scale simulation studies, such as <cit.>demonstrate that high-dimensional dynamical systems described by polynomial systems of equations often exhibit chaotic behavior; in fact, such behavior becomes more and more “probable" as the dimension of the system increases. If we are in the regime of chaotic behavior, and if we measure snapshots of the system at the time scale of the chaotic dynamics, then for a smooth function F, the sequence F(x(t_0)), F(x(t_1)), …, F(x(t_m-1)) will satisfy a deterministic form of the law of large numbers – the so-called Birkhoff Ergodic Theorem – which says that the time average of the sequence will converge to the space average of F(x) with respect to an underlying invariant measure <cit.>.For strongly ergodic systems, a stronger Central Limit Theorem holds, and the convergence rate can be quantified.In particular, the rate of convergence is faster if the correlation between successive observed values F(x_k) and F(x_k+q) (for index q>0) is smaller, starting with the works <cit.>.There is a rich theory on the decay of correlations for certain classes of low-dimensional dynamical systems, beyond the scope of this article. Morally speaking, if we measure a single burst which has the property that a smooth functional applied to the sequence exhibits a fast decay of correlation, then the resulting matrix A_L will have rows which are only weakly correlated, and thus, should almost fit the theoretical requirements for a compressive sensing result of the form of Theorem <ref>.It is worth noting that a more recent paper <cit.> generalizes the results from<cit.> to accommodate matrices with only weakly-correlated rows.However, even so, the existing compressive sensing theory does not extend to this situation exactly as the sparse signal to be recovered is not independent of the measurement matrix. Exploring a more precise relationship between the level of ergodicity in the system to recover and its effect on reducing the required number of bursts remains an intriguing direction for future study. § SAMPLING STRATEGIES AND COMPUTATIONAL RESULTS In this section, we present three sampling strategies that will lead to exact recovery (with high probability). In particular, Strategy 1 uses the construction from Section <ref>, Strategy 2 is a reduction of Strategy 1 when one can limit the number of unknowns to an ℓ-sized neighbor (i.e. when f_j only depends onx_i, for i∈ [j-ℓ-1/2, j+ℓ-1/2] for odd ℓ), and lastly Strategy 3 shows that chaotic trajectories greatly reduce the number of random initializations. It is worth noting that Strategy 1 makes no assumptions on the trajectories x(t) or the support of the monomial basis for f(x), Strategy 2 incorporates information on the support of the monomial basis for f(x) but makes no assumptions on the trajectories x(t), and Strategy 3 uses information about the trajectories x(t). In practice, we expect that some combination of these three approaches could be optimal when dealing with various types of data. Note that although we will show examples with quadratic governing equations,the number of unknowns, N, is non-trivial and grows quickly with dimension. In all cases, we have under-sampled the data and it is easy to check that standard methods, like the least-squares algorithm, do not produce meaningful results. To validate our approaches, we will apply these strategies to the Lorenz 96 system and a quadratic reaction-diffusion equation. The Lorenz 96 system, introduced in <cit.> as an atmosphere model, contains n>3 variables x_1,⋯, x_n and satisfies_k = -x_k-2x_k-1 + x_k-1x_k+1 - x_k + F, k=1,⋯, n.The constant F is independent of k and x_0=x_n and x_n+1=x_1.In both the standard monomial basis and in the Legendre monomial basis, the system is 4-sparse (component-wise). Unless otherwise stated, the number of variables is fixed to be n = 50 and the constant is set to F=8, where chaotic behavior is expected, see<cit.>. The second system we consider is known as the Fisher's equation, a quadratic reaction-diffusion equation, with reaction term F(x)=x-x^2. This equation has applications ranging from population dynamics to combustion physics. Its finite difference discretization of the Fisher's equation with n nodes is given by:_k = x_k+1-2x_k + x_k-1 + γ (x_k-x_k^2), k=1,…, n,where the coefficients and time-scale have been re-scaled by the grid-spacing and we impose the periodicity condition: x_0=x_n and x_n+1=x_1. Reaction-diffusion systems have multi-scale phenomena and can produce traveling waves, for more details see, for example, <cit.>. In our examples, the solutions remain bounded since the initial data is sampled in the unit box. Note that when transformed to the Legendre basis, each component of Equation (<ref>) is 5-sparse. Throughout this section, we denote K to be the number of initializations, m is the size of each burst. For each burst, the time-derivative is approximated using central differencing except at the first and last time-step where we use forward and backward differencing (respectively). The dynamical data is generated by solving the ODEs numerically using Runge-Kutta 45 with a finer tolerance than the sampled time-step dt. To solve the L-BP problem we use the spgl1 algorithm <cit.> for all examples here. Other algorithms that could be successfully applied include: primal-dual <cit.>, Douglas-Rachford <cit.>, SpaRSA <cit.>, or the convex optimization package cvx <cit.>. §.§ Strategy 1: K ∼ cs log(N) Random InitializationsA direct consequence of Theorem <ref> is that one can recover the governing equation with K random initializations as long as K≥ cs log(N). We first determine the number of uniformly random initializations based on the theorem and collect a burst of size m generated from the system starting with each initialization. The data matrix is concatenated vertically over each burst. The parameters used are specified in each experiment.First, we validate the recovery results by varying the number of initialization K and measuring the probability that the governing system can be recovered. To test this we use the 10th componentof the Lorenz 96 system (see Equation (<ref>)):dx_10dt = -x_8x_9 + x_9x_11 - x_10 + 8. For each K=5, …, Nm=1326m, we repeat the simulation 100 times and record the number of successes to calculate the probability p, see Figure <ref>. The maximum K is chosen so that we do not oversample the system. From Figure <ref>, we can see that for dt = 0.001, K=80 is needed to achieve 90% probability of success for both m = 5 and m = 10. For a larger time-step, dt = 0.01, K=130 is needed to achieve 90% probability of success for m = 5. This is expected since the larger time-steps yield less accurate time-derivative approximations. Note that the recovery degrades when the burst size (in terms of m and dt) increases due to the propagation of error between time-steps. When the data is less correlated in time, the degradation may not occur, see Strategy 3. It is worth noting that although the theoretical universal constant c_* in Equation (<ref>) has a large upper bound, it is much smaller in practice. We also verify the recovery results for Fisher's equation for various γ. Table <ref> displays the values of the coefficients (as well as their support) when applying the proposed approach to the first component of Equation <ref>. The initial data is sampled from the uniform distribution over [0,1]^n and the full data matrix is transformed to [-1,1]^n. We set n=200 (N= 20301), the number of random samples to 159, and the size of the burst is 5 – this corresponds to the bound provided in Theorem <ref>. By varying the model parameter γ, we show that the method selects the correct basis terms and accurately approximates the parameter values (within a few significant digits). It is worth highlighting that as γ decreases, the relative scale between the maximum and minimum coefficient increases; however, the recovery results remain relatively stable.Note that once we have identified the support S_L with respect to the Legendre dictionary matrix, it is possible to “correct” the results from Table <ref> by solving:min_CA_M|_S C|_S - V_2where A_M|_S is the monomial dictionary A_M restricted to the corresponding support set S.This debiasing step, when applied to the examples from Table <ref>, produces exact results (up to the fourth significant digit). §.§ Strategy 2: K < cs log(N) Random Initializations with Localization For many ODEs, especially those related to finite dimensional approximations of local PDEs, it is safe to assume that each variable x_j only relates to its ℓ-neighbors x_i, i∈ [j-ℓ-1/2, j+ℓ-1/2] (ℓ odd). In particular, the assumption is that the governing equation satisfies:ẋ_j = f_j(x_j-ℓ-1/2,…, x_ j+ℓ-1/2),for all j. With this additional assumption, the number of unknowns in the dictionary matrix is reduced, and thus we can decrease the number of initial conditions needed to guarantee exact recovery. In particular, with the same conditions in Theorem <ref>, if additionally it is known that the sparse support is restricted to an ℓ-sized neighborhood, then if we sample the initial data K times, with K∼ cs log(ℓ)log(ε^-1), then with probability 1-ε, the system ẋ=f(x) is recovered exactly by the unique solution to (L-BP). Incorporating this additional condition simply amounts to downsampling the column space of the dictionary A. In Table <ref>, we consider the n=1000 dimensional Fisher's equation, which yields 501501 unknowns. By varying the size of the neighborhood, ℓ, we calculate the minimum number of random initial samples needed for exact recovery. To validate the theoretical scaling, the ratio between the samples needed versus the log of the neighborhood size is shown to be nearly constant. Since the sampling rate is related to the log of the number of unknowns, we see substantial gains between using all terms versus restricting to a neighborhood, but further refinement is not needed. This highlights the benefit of restricting the optimization to an ℓ-sized neighborhood, even when ℓ can only be estimated. One key consequence from this strategy is that the sampling rate is independent of the dimension of the ODE system n, allowing this approach to be applied to very large systems.§.§ Strategy 3: Chaotic Systems with K Small and m LargeIn Strategies 1 and 2, we made no assumption on the behavior of the trajectories x(t).For Strategy 3, we will assume that the data exhibits chaotic behavior, or uncorrelated long-time behavior. Here we show that if this is the case, we can reduce the number of random initial data needed. In particular, we will show an example of the extreme case where K=1, since including additional trajectories with random initial data will only improve the recovery.Consider collecting data along one trajectory:{ x(t_0), x(t_1),…, x(t_m-1)} with time-steps dt that arelarge enough so that x(t_i) and x(t_j) are sufficiently uncorrelated for i<j. The velocity along the trajectory {ẋ(t_0), ẋ(t_1), …, ẋ(t_m-1)} is either observed directly (possibly with some error) or calculated by using a fine time-step (smaller than dt). In either case, the data is under-sampled, i.e., m<N. In essence, taking large enough time-steps of a chaotic system mathematically resembles a random “re-sampling" of the data, thus fitting in with Strategies 1 and 2.We test this strategy on the first component of the Lorenz 96 system, Equation (<ref>), with n=50. In Table <ref>, we show that with 500 measurements and dt = 1.0, the solution of L-BP identifies the correct terms and approximates the coefficients within the expected error. This shows that it is possible to use the randomness of the data along one trajectory to learn the governing equation. Adding more trajectories with initial data sampled i.i.d. from the uniform distribution, while keeping the total size of the data fixed, only helps the recovery processes.§.§ ComparisonsFor comparison, we apply Strategy 1, the standard least-square algorithm, and the linear regression method with sequential thresholding proposed in <cit.> on both the Lorenz 96 and Fisher's equation. In Figure <ref>, the coefficients extracted using the L-BP method (left), the least-square algorithm (middle), and the sequential thresholding algorithm (right) for the 35th component of the Lorenz 96 equation with n=50, dt = 0.001. The threshold parameter for the least-square algorithm and the sequential thresholding algorithm is set to λ=0.05. In fact, the solutions of the least-square and the linear regression are the same for this problem. This is the case for all reasonable λ>0. Next, we compare the recovery of the coefficients from the first component of the Fisher's Equation (<ref>). The model parameter is set to γ=0.1 and dimension is set to n=100, the number of random samples is set to 138, and the size of the burst is set to 5. The least-squares solution is 653-sparse and has coefficients on the order of 10^4. Applying the sequential thresholding algorithm proposed in <cit.> with threshold parameter λ∈ [5, 5000] results in a similar solution. The sparsity matches that of the least-square solution, with s∈[ 579, 653] and has coefficients on the order of 10^4. Increasing λ to achieve a better sparsity level will yield the trivial solution (i.e. all zeros). This is likely an effect of their algorithm's dependence on the least-square solution. It is important to note that the algorithm proposed in <cit.> was not intended for the case of under-sampling. §.§ Measurement NoiseWe consider the effects of noisy state-space measurements on the reconstruction of the sparse coefficient vector. Let X be the data matrix: X =[ x_1(t_0;1) x_2(t_0;1)⋯ x_n(t_0;1); x_1(t_1;1) x_2(t_1;1)⋯ x_n(t_1;1);⋯ ; x_1(t_m-1;1) x_2(t_m-1;1)⋯ x_n(t_m-1;1);⋯ ; x_1(t_0;k) x_2(t_0;k)⋯ x_n(t_0;k);⋯ ; x_1(t_m-1;k) x_2(t_m-1;k)⋯ x_n(t_m-1;k);⋯ ; x_1(t_m-1;K) x_2(t_m-1;K)⋯ x_n(t_m-1;K) ] and let Y = X + ηbe the matrix of noisy measurements, where η is random Gaussian noise. The noise ratio is defined as:Noise Ratio =X-Y_2/ X_2× 100%and the relative ℓ^2 error is define as:relative ℓ^2 error =c-c_true_2/ c_true_2× 100%,where c is the computed/learned vector of coefficients and c_true is the true coefficients.We generate the data from Equation (<ref>) using F=8 and dimension equal to 50. The parameters are set to dt=0.001,K=200, and m=3. By adding noise directly to the state-space, we can measure the recovery when the measurements are corrupted. The state variable is corrupted by random Gaussian noise before the derivatives are calculated, thus the value of V and the matrix A will be inaccurate. This makes the problem challenging when the noise is large. The results are summarized in Table <ref>. As we vary the noise level, we measure the relative ℓ^2 error and the recovery of the support set.For noise under 5%, the method is stable with respect to the noise. This is consistent with Theorem <ref>. After 5% noise, the ℓ^2 error jumps and after 6%, we cannot reliably recover the support set. In particular, since we know that the true sparsity of Equation (<ref>) is 4,we can check if the largest four learned coefficients (in magnitude) coincide with the correct support set. After about 6%, the largest four coefficients do not represent the correct support set. This is likely due to the large inaccuracies in V, which is not stable to the noise, and inaccuracies in A, which scales nonlinearly with the noise.§ CONCLUSION AND DISCUSSIONExtracting dynamical systems remains a difficult task with many open areas of research. Recent work in sparse model selection for dynamical systems has focused on the overdetermined case, where regression must be controlled so as to prevent overfitting. In this work, we utilized thefundamental idea from compressive sensing to develop several sampling strategies for extracting governing equations from high-dimensional dynamic data. In all cases, the number of measurements is less than the number of unknowns. The main differences between these strategies is the degree of prior knowledge about the data or the governing equations.If no assumptions on the evolution can be made, then randomizing the initial data is sufficient. If some assumptions on the governing equations are provided, such as locality induced by discretizing a local PDE, then the number of random initial data can be reduced further, to be nearly independent of the dimension of the dataset.In the third case, if the data is chaotic (or shows a low temporal correlation), then we may reduce the number of initial samples to a fixed number. Using results from compressive sensing, the first two strategies are shown to hold; however, more theory is needed to verify the last case. In several of our experiments,we have shown that the three strategies are robust to various factors as well as highlighted the benefits of this approach over existing methods. The effective combination of reconstruction guarantees from compressive sensing with sparse learning for dynamical systems presented here opens a wide range of applications. We are currently investigating the use of group sparsity and random sampling approaches for learning dynamic models from multiple data sources. In future work, we also would like to extend the current framework to other bounded orthonormal bases and to learn the dynamics from noisy data.§ ACKNOWLEDGMENTSThe authors would like to thank Amit Singer and Scott McCalla for helpful discussions that improved this manuscript. H.S. acknowledges the support of AFOSR, FA9550-17-1-0125. R.W. and G.T. acknowledge the support of NSF CAREER grant #1255631. § APPENDIX For purposes of being as self-contained as possible, we first recall some background on the theory of sparse recovery in bounded orthonormal systems via random sampling.We refer the reader to the text <cit.> for more details.§.§ Random Sampling in Bounded Orthonormal Systems Let D⊂ℝ^n be endowed with a probability measure μ.Suppose that {ϕ_1, ϕ_2, …, ϕ_d } (d ≤ n) is a (possibly complex-valued) orthonormal systemon D:∫_ Dϕ_j(t) ϕ_k(t) dμ(t) = δ_j,k ={[0 if j ≠ k;1 if j=k;].We call {ϕ_1, ϕ_2, …, ϕ_d } a bounded orthonormal system with constant B ≥ 1 if moreover ϕ_j _∞ := sup_t ∈ D | ϕ_j(t) | ≤ B for all j ∈ 1,2,…, d.Suppose that t_1, t_2, …, t_m ∈ D are sampling points which are drawn i.i.d. according to the orthogonalization measure μ, and consider the sampling matrixA ∈ℂ^m × n with entriesA_ℓ,k = ϕ_k(t_ℓ), ℓ∈ [m], k ∈ [n].With high probability, a random matrix formed as such permits stable “inversion" of the (possibly highly underdetermined) system y= Ax if x is sufficiently sparse, and if “inversion" is carried out through e.g. solving an ℓ_1-minimization problem.The following is a restatement of Theorem 12.22 in <cit.>, which is a restatement of a result from <cit.>.Let x ∈ℂ^n and let A ∈ℂ^m × n to be the random sampling matrix associated to a BOS with constant B ≥ 1.For y = Ax + e with e _2 ≤η√(m) for some η≥ 0, let x^# be a solution tomin_z ∈ℂ^n z _1 subject toAz - y _2 ≤η√(m).If m ≥ C B^2 s log(n) log(ε^-1),then with probability at least 1 - ε, the reconstruction error satisfies x - x^#_2 ≤ C_1 σ_s(x)_1 + C_2 √(s)ηwhere σ_s(x)_1 = inf_u: uis s-sparse x - u _1, and the constants C, C_1, C_2 > 0 are universal.We apply this result to prove our reconstruction guarantee. §.§ Proof of Theorem <ref>Recall that A_L is the K × N matrix consisting of those rows corresponding to the K initializations.The tensor product of univariate Legendre polynomials, normalized as inthe construction of A_L, forms a bounded orthonormal system with respect to the uniform measure over [-1,1]^n; precisely, dμ = 1/2dx.The Legendre polynomials up to degree 2 are uniformly bounded in magnitude over the domain [-1,1]^n by 3, as realized by the terms 3 x_j x_k at x_j, x_k = ± 1. Thus, A_L satisfies the requirements of Proposition <ref> with K = 3. Moreover, by assumption, each column in the coefficient matrix C associated to the underlying dynamical system consists of at most s nonzero terms.Thus, after transforming to the Legendre basis, each column in the the Legendre coefficient matrix C_L has at most s' = s+1 nonzero terms. Thus, we apply Proposition <ref> with these parameters to any particular one of the n ℓ_1-minimization problems. The recovery guarantee in the noiseless case (η = 0) moreover extends to the optimization problem L-BP over the full set of measurement constraints, not just those constraints corresponding to the burst initializations, since the domain corresponding to the full measurement set is strictly included in the domain corresponding to the reduced measurement set, and the unique minimizer over the reduced measurement set belongs to this subdomain. 10ABW17 Ben Adcock, Simone Brugiapaglia, and Clayton G. Webster. 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"authors": [
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"Rachel Ward"
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"title": "Extracting Sparse High-Dimensional Dynamics from Limited Data"
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enumii_savedOccurrence of mass inflation with an exponential Price law]On the occurrence of mass inflation for the Einstein-Maxwell-scalar field system with a cosmological constant and an exponential Price law João L. Costa: ISCTE - Instituto Universitário de Lisboa, Av. das Forças Armadas, 1649-026 Lisboa, Portugal and CAMGSD [email protected] Pedro M. Girão, José Natário and Jorge Drumond Silva: CAMGSD, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal [email protected] [email protected] [email protected][2010]Primary 83C05; Secondary 35Q76, 83C22, 83C57, 83C75This work was partially supported by FCT/Portugal through UID/MAT/04459/2013 and grant (GPSEinstein) PTDC/MAT-ANA/1275/2014.[ Jorge Drumond Silva December 30, 2023 =======================AbstractIn this paper we study the spherically symmetric characteristic initial data problem for the Einstein-Maxwell-scalar field system with a positive cosmological constant in the interior of a black hole, assuming an exponential Price law along the event horizon. More precisely, we construct open sets of characteristic data which, on the outgoing initial null hypersurface (taken to be the event horizon), converges exponentially to a reference Reissner-Nördstrom black hole at infinity.We prove the stability of the radius function at the Cauchy horizon, and show that, depending on the decay rate of the initial data, mass inflationmay or may not occur. In the latter case, we find that the solution can be extended across the Cauchy horizon with continuous metric and Christoffel symbols in L^2_ loc,thus violating the Christodoulou-Chruściel version of strong cosmic censorship.§ INTRODUCTION§.§ Strong cosmic censorship and spherical symmetryDeterminism of a physical system, modeled mathematically by evolution equations, is embodied in the questions of existence and uniqueness of solutions for given initial data. The initial value problem (or Cauchy problem) is therefore the appropriate setting for studying these models. Well known examples of equations where the Cauchy problem is quintessential are those of Newtonian mechanics, the Euler and Navier-Stokes systems in hydrodynamics and Maxwell's equations of electromagnetism.Historically, the geometric nature and mathematical complexity of the Einstein equations made it difficult to recognize that they also fit into this framework. It was not until the seminal work of Y. Choquet-Bruhat <cit.>, and her later work with R. Geroch <cit.>, that the central role of the Cauchy problem in general relativity was established. These results relied crucially on recognizing the hyperbolic character of the Einstein equations. Uniqueness of the solutions, as for any hyperbolic PDE, then follows from a domain of dependence property. The essence of <cit.> consists precisely in showing that given initial data there exists a maximal globally hyperbolic development (MGHD) for the corresponding Cauchy problem, that is, a maximal spacetime where this domain of dependence property holds.For the Einstein equations, global uniqueness fails, and therefore determinism breaks down, if extensions of MGHDs to strictly larger spacetimes can be found. The statement that generically, for suitableCauchy initial data,[We will not provide a full discussion of the subtleties regarding the formulation of this conjecture (see for instance <cit.>). As an example, note that trivial extensions can occur simply because the initial data is given on an incomplete Cauchy surface.] the corresponding MGHD cannot be extended is known as the strong cosmic censorship conjecture (SCCC) <cit.>.A crucial point in the precise formulation of this conjecture is deciding what exactly is meant by an extension. Various proposals have been advanced, differing on the degree of regularity that is demanded for the larger spacetime. The strongest formulation would correspond to the impossibility of extending the MGHD with a continuous Lorentzian metric. This happens for instance in the Schwarzschild solution, where continuous extensions across the singularity r=0 do not exist <cit.>. However, from the PDE point of view, the conjecture should rather prevent the existence of extensions which are themselves solutions of the Einstein equations, even in a weak sense. If we collectively represent the Christoffel symbols by Γ, then a weak formulation of the vacuum Einstein equations =0 can be represented schematically as∫_M ϕ = 0 ⇔∫_M (∂Γ + ΓΓ) ϕ = 0 ⇔∫_M ( - Γ∂ϕ + ΓΓϕ) = 0,for any test function ϕ. Thus, as pointed out by Christodoulou and Chruściel <cit.>, any weak solution extension is ruled out if the Christoffel symbols fail to be in L^2_ loc. We shall therefore refer to the conjecture that extensions with Christoffel symbols in L^2_ loc generically do not exist as the Christodoulou-Chruściel formulation of strong cosmic censorship.The reason why a genericity condition must be included in any version of the SCCC is that there exist well known MGHDs, arising from complete asymptotically flat initial data, which can be smoothly extended to strictly larger solutions of the Einstein equations; when such extensions exist, the boundary of the MGHD in the larger spacetime is known as the Cauchy horizon. The paradigmatic example exhibiting this behavior is the Kerr family of solutions, describing rotating black holes, where the Cauchy horizon occurs inside the event horizon. In <cit.> (see also <cit.>), Penrose provided a heuristic argument, based on the blue-shift effect, by which arbitrarily small perturbations of the black hole exterior would be infinitely amplified along the Cauchy horizon, turning it into a “singularity” beyond which no extension should exist. According to this picture, extensions of spacetimes across Cauchy horizons would be artifacts of very particular solutions, such as the Kerr family; therefore they should be unstable and devoid of physical significance.An obvious path for studying the validity of the SCCC is then to focus on perturbations of these exceptional MGHDs where a Cauchy horizon is known to exist. Given the considerable difficulty of the full system of Einstein's field equations, it is natural to introduce symmetry assumptions to reduce the number of degrees of freedom, even though this necessarily implies some loss of genericity. Spherical symmetry is a popular choice, since it leads to partial differential equations in only two independent variables and is compatible with both the asymptotic flatness requirements for modelling isolated astrophysical systems and standard cosmological spacetimes. Moreover, spherically symmetric solutions with a Cauchy horizon, analogous to the Kerr family, can be obtained by including an electromagnetic field: these constitute the so-called Reissner-Nordström family of solutions, describing charged black holes. In the words of John Wheeler, “charge is a poor man's angular momentum”.However, Birkhoff's theorem imposes local uniqueness of electrovacuum solutions in spherical symmetry, that is, it establishes that there are no gravitational dynamical degrees of freedom under these assumptions. For that reason, Christodoulou introduced in <cit.> a model where the Einstein equations are coupled to a massless scalar field, arguably the simplest model which retains the wavelike behavior expected from the Einstein equations but is not constrained by Birkhoff's theorem. Another important consideration for choosinga massless scalar field is that, unlike other matter models (e.g. perfect fluids), it does not develop singularities in the absence of gravity, and so any breakdown of the solution when coupled to the Einstein equations can be attributed to purely gravitational effects. By adding an electromagnetic field, Dafermos <cit.> adapted Christodoulou's model to study perturbations of the interior of a Reissner-Nordström black hole, and in particular the stability of the Cauchy horizon. More precisely, he considered a characteristic initial value problem for the Einstein-Maxwell-scalar field system in spherical symmetry, with Reissner-Nordström data on the event horizon and arbitrary data along an ingoing null hypersurface; the Reissner-Nordström spacetime itself is, of course, a particular solution of this problem. This work established the following two results: first, the radius function does not vanish at the Cauchy horizon, and so the metric can always be extended continuously across it; second, if the Reissner-Nordström initial data is sufficiently subextremal and the free data on the ingoing null hypersurface decays sufficiently slowly towards the event horizon then a scalar invariant, the so-called renormalized Hawking mass, blows up at the Cauchy horizon, a phenomenon known as mass inflation (first identified by Poisson and Israel <cit.>). This in turn leads to the blow-up of the Kretschmann scalar, and so no C^2 extensions are possible for the metric in this setting. In fact, something much stronger is expected, namely that mass inflation even prevents the existence of extensions with Christoffel symbols in L^2_ loc (see <cit.>).These results were further extended by the authors in <cit.>, with the inclusion of a cosmological constant of any sign and a more detailed analysis of the solution at the Cauchy horizon, depending on the precise decay of the initial data. Notably, this work established that for sufficiently fast decaying initial data not only does mass inflation not occur, but also it is possible to extend the spacetime across the Cauchy horizon as a classical solution of the Einstein-Maxwell-scalar field system, thus calling into question the SCCC.However, assuming Reissner-Nordström data on the event horizon is a somewhat artificial problem. A more realistic model is obtained if gravitational collapse arises from the evolution of initial data prescribed on a Cauchy surface. In this case, following the heuristic work of Price in 1972 <cit.>, it is widely expected that, in the absence of a cosmological constant, the scalar field decays polynomially along the event horizon (with respect to an Eddington-Finkelstein-type null coordinate). The conjecture that in a generic gravitational collapse scenario the scalar field decays with some precise rate became known as Price's law. In <cit.>, Dafermos proved that the radius function does not vanish at the Cauchy horizon if a polynomial Price's law is assumed as an upper bound, while mass inflation occurs if the corresponding lower bound is also imposed. The validity of the polynomial Price's law as an upper bound was subsequently established by Dafermos and Rodnianski <cit.> for the spherically symmetric collapse of a massless scalar field, thus yielding stability of the Cauchy horizon for black hole formation in this setting. Nevertheless, the occurrence of mass inflation, and therefore inextedibility with Christoffel symbols in L^2_ loc, remains an open problem. In recent work, Luk and Oh <cit.> showed that solutions resulting from gravitational collapse of generic asymptotically flat initial data satisfyan integral lower bound along the event horizon, which, although weaker than the pointwise lower bound predicted by Price, turns out to be enough to rule out the existence of C^2 extensions. Whether it suffices to establish mass inflation is still not clear.In the presence of a positive cosmological constant, it is widely expected that the corresponding Price law should guarantee exponential decay of the scalar field along the event horizon (see for instance the linear analysis in <cit.>, the numerical study in <cit.> or the nonlinear stability results in <cit.>). In this paper, we will therefore consider such an exponential decay and extend the analysis in <cit.> to this case. We prove the stability of the radius function at the Cauchy horizon, and show that, depending on the decay rate of the initial data, mass inflation may or may not occur. In the latter case, we find that the solution can be extended across the Cauchy horizon with Christoffel symbols in L^2_ loc. A more precise statement of our results can be found in Theorem <ref>. §.§ Summary of the main results We consider the Einstein-Maxwell-real massless scalar field equations in the presence of a cosmological constant Λ (in units for which c=4π G=ε_0=1):R_μν - 1/2 R g_μν + Λ g_μν = 2 T_μν;dF = d⋆ F = 0; ϕ = 0;T_μν = ∂_μϕ ∂_νϕ - 1/2∂_αϕ ∂^αϕg_μν + F_μα F_ν^α - 1/4 F_αβ F^αβ g_μν.These form a system of partial differential equations for the components of the spacetime metric g, the Faraday electromagnetic 2-form F, and the real massless scalar field ϕ; here R_μν are the components of the Ricci tensor, R is the scalar curvature, ⋆ is the Hodge star operator andis the d'Alembertian (all depending on g).In the spherically symmetric case, we can write the metric in double null coordinates (u,v) as g=-Ω^2(u,v) dudv+r^2(u,v) σ_𝕊^2,where σ_𝕊^2 := dθ^2 + sin^2 θ dφ^2 is the round metric on the 2-sphere 𝕊^2. In this case, the Maxwell equations decouple from the system, since they can be immediately solved to yieldF = - Q_eΩ^2(u,v)/2 r^2(u,v)du ∧ dv + Q_m sinθ dθ∧ dφ.Here Q_e and Q_m are constants, corresponding to a total electric charge 4 π Q_e and a total magnetic charge 4 π Q_m. The remaining equations depend only on the parameter e = √(Q_e^2+Q_m^2),which we assume to be nonzero. They can then be written as follows (see <cit.>): a wave equation for r,∂_u∂_vr=-Ω^2/4r - ∂_ur ∂_vr/r + Ω^2e^2/4r^3 + Ω^2 Λ r/4,a wave equation for ϕ,∂_u∂_vϕ=- ∂_ur ∂_vϕ+∂_vr ∂_uϕ/r,the Raychaudhuri equation in the u direction,∂_u(∂_ur/Ω^2)=-r(∂_uϕ)^2/Ω^2,the Raychaudhuri equation in the v direction,∂_v(∂_vr/Ω^2)=-r(∂_vϕ)^2/Ω^2,and a wave equation for lnΩ, ∂_v∂_ulnΩ=-∂_uϕ ∂_vϕ- Ω^2e^2/2r^4+Ω^2/4r^2+∂_ur ∂_vr/r^2. We summarize themain results of this paper in the following statement. Consider the characteristic initial value problem for the spherically symmetric Einstein-Maxwell-scalar field system (<ref>)-(<ref>) on the domain [0,U] ×[0,∞[, written in null coordinates (u,v) determined by the conditions ∂_vr(0,v)=g(∇ r, ∇ r)(0,v) and ∂_ur(u,0)=-1. Take any subextremal element of the Reissner-Nordström family of solutions with mass ϖ_0, non-vanishing charge parameter e and cosmological constant Λ, and let r_+, k_+ and k_- be, respectively, the corresponding event horizon radius and the surface gravities of the event and the Cauchy horizons. Then, for any ε > 0 andfixed s>0 it is possible to construct open sets of initial data such that ϕ(u,0) is free along the ingoing null direction {v=0}, while r(0,v) → r_+, ∂_vr(0,v) → 0 ande^-(sk_+ + ε)v ≲ ∂_vϕ(0,v)≲e^-(sk_+ - ε)vas v →∞ along the event horizon {u=0}. Given U > 0 sufficiently small, there exists a unique maximal development of this characteristic initial value problem, defined on a past set P⊂ [0,U] × [0,∞[. Moreover, for small enough ε>0 the following results hold (with ρ=k_-/k_+>1): * Stability of the radius function at the Cauchy horizon (Theorem <ref>). There exists U>0 such that[0,U]×[0,∞[ ⊂ P,and r_0>0 for whichr(u,v)>r_0,for all (u,v)∈[0,U]×[0,∞[.Consequently, ( M,g,ϕ) extends, across the Cauchy horizon {v=∞}, to (M̂,ĝ,ϕ̂), with ĝ and ϕ̂ in C^0.enumi+1* Mass inflation (Theorem <ref>). If <min{ρ,2} then the renormalized Hawking mass ϖ (see (<ref>)) satisfieslim_v→∞ϖ(u,v)=∞,for each0<u≤ U.In particular, no C^2 extensions across the Cauchy horizon exist.* No mass inflation (Theorem <ref>).If ρ<9/7 and s>14/9ρ thenlim_v→∞ϖ(u,v)<∞,for each0<u≤ U,provided that U is sufficiently small.* Breakdown of the Christodoulou-Chruściel criterion (Theorem <ref>).Under the same hypotheses as in (3), the Christodoulou-Chruściel inextendibility criterion fails, i.e.( M,g,ϕ) extends across the Cauchy horizon to (M̂,ĝ,ϕ̂), with ĝ and ϕ̂ in C^0, Christoffel symbols Γ̂ in L^2_ loc, and ϕ̂ in H^1_ loc. The regions of the (ρ,s) plane where we can prove mass inflation and no mass inflation are depicted in the following figure.[The region where we can prove no mass inflation is not expected to be sharp, since the linear analysis carried out in <cit.> suggests that there exist H^1 extensions for s>ρ.] a _(1,1) _9/7 b_2 c_14/9 d_2 rρ ss< g r a p h i c s >8cm .3cm.3cm - no mass inflation.3cm.3cm - mass inflation§.§ Implications for cosmic censorship As discussed above, the results in this paper do not apply directly to the SCCC, since this conjecture refers to global uniqueness of solutions arising from generic Cauchy data, while we consider characteristic data prescribed ona dynamic event horizon along which the scalar field satisfies a Price law of the form (<ref>);that is, our results assume that a black hole is already present, as well as a specific decay of the fieldin its exterior. However, as is clear from Theorem <ref>, just the qualitative change in Price's law from polynomial (Λ=0) to exponential (Λ>0) is not enough to obtain definitive conclusions about the behavior of the solutions at the Cauchy horizon, and therefore the validity of the SCCC. The final outcome requires, in particular, a very precise quantitative knowledge of the value of sk_+ in (<ref>), or, more precisely, of how such quantity relates to the surface gravities of the Cauchy, event and cosmological horizons. This is in stark contrast with the asymptotically flat case (Λ=0), where Price's law is expected to yield an inverse power decay, which in turn is enough to establish mass inflation in the entire subextremal parameter range.A recent numerical calculation of the quasinormal modes of Reissner-Nordström-de Sitter <cit.> considerably changed the perspective on how sk_+ depends on the black hole parameters. In particular, this work (numerically) disproved a long-standing conjecture with roots in <cit.> and <cit.>. Strictly speaking, the results in <cit.> only apply to solutions of the linear wave equation in a fixed Reissner-Nordström-de Sitter background, but we expect similar results to also hold for the Einstein-Maxwell-scalar field system with a positive cosmological constant Λ. Assuming that this is the case, <cit.> shows that,in the limit of large charge (for which ρ approaches 1), the decay corresponds to sk_+ close to 2k_- (that is, s close to 2ρ). In this regime, our results guarantee the existence of solutions with no mass inflation (see the figure above), as well as the existence of extensions beyond the MGHD with Christoffel symbols in L^2_loc. Note that the large charge limit can easily be obtained by picking a large cosmological horizon radius and then choosing the Cauchy horizon radius suitably close to the event horizon radius; these choices are in loose agreement with what one expects from the parameters of some astrophysical black holes.It is interesting to note that the extendibility identified in <cit.> occurs for near extremal black holes, where the blueshift effect is weaker. This feature can be compared with the fully nonlinear results in <cit.>, where it is proved that one can indeed extend solutions of the Einstein-Maxwell-scalar field system across the Cauchy horizon of extremal black holes for Λ=0. The absence of blueshift in this case suggests that a similar result should hold for Λ>0. Similarly, one can expect the Cauchy horizon stability in the non-spherically symmetric setting, recently proved in <cit.> for Λ = 0, to remain true for Λ>0. Moreover, it is likely that these latter solutions can also be extended across the Cauchy horizon for near extremal(i.e. rapidly rotating) black holes.In conclusion, our results indicate that, with our current knowledge, the validity of the SCCC in the presence of a positive cosmological constant does not stand on firm ground. Nonetheless, the final verdict will only become clear once a precise quantitative understanding of the exponential Price law for Λ>0, in the full non-linear setting,is achieved.§.§ Technical overview Introducing an exponential Price law creates new difficulties when compared to simply prescribing Reissner-Nordström data along the event horizon, as in <cit.>. We now summarize the main new technical features of the present work.As for any characteristic initial value problem, our initial data is constrained by the evolution equations, and thus cannot be freely chosen (see Section 4 of <cit.>). Solving these nonlinear constraint equations while at the same time guaranteeing that our data describes a dynamical event horizon along which the scalar field decays at a prescribed rate is a non-trivial task; this problem is solved in Section <ref>. We use the radial derivative of the renormalized Hawking mass on the outgoing direction as the pivotal free function from which all the remaining quantities can then be constructed. In addition to exponentially decaying initial data, we produce sets of initial data with different types of decay, including the polynomial case studied by Dafermos <cit.>.An important qualitative feature of the dynamics in the interior of the black hole is the celebrated redshift effect, characterized by an exponential decay of the form e^-2k_+v. This is the fastest decay that can be expected to be carried over by the evolution from the event horizon towards the Cauchy horizon. As exemplified in <cit.>, when the decay of the initial data is slower than exponential then it overwhelms the redshift effect and mass inflation always occurs. On the other hand, for faster than exponential decaying initial data the redshift effect dominates, and mass inflation may not occur, as was found in <cit.>. Therefore, initial data asymptotic to e^-sk_+v constitutes the most interesting case, as it allows for a detailed analysis around the threshold value s=2, and is the only case that we will pursue.Contrary to what happens in the Reissner-Nordström solution, the interior of the black hole solutions that we are now considering does not coincide with the trapped region. In particular, an apparent horizon forms, whose asymptotic geometry must be understood, mostly by soft arguments, before proceeding to study the solution in greater detail (Section <ref>). To precisely estimate all relevant quantities in the region P_λ that lies in the past of the apparent horizon, as well as the region in its immediate future, we develop a two-dimensional version of Grönwall's inequality adapted to this problem (Sections <ref> and <ref>). As the Cauchy horizon is approached, the analysis becomes akin to that in <cit.> (Sections <ref> to <ref>).In this work we are not able to exclude the existence of non-trivial solutions whose radius function and renormalized Hawking mass are constant along the Cauchy horizon[In fact we conjecture that such solutions do exist.] (in <cit.> such behavior was ruled out simply by assuming a non-zero ingoing perturbation). This creates new difficulties when analyzing solutions with no mass inflation and trying to construct extensions with Christoffel symbols in L^2_ loc beyond the Cauchy horizon, which are averted by introducing a novel change of coordinates (see Section <ref>). § THE SPHERICALLY SYMMETRIC EINSTEIN-MAXWELL-SCALAR FIELD EQUATIONS AS A FIRST ORDER SYSTEM To write the Einstein equations as a first order system of PDE we define the following quantities:ν:=∂_u r, λ:=∂_v r, ϖ:=e^2/2r+r/2-Λ/6r^3+2r/Ω^2νλ, μ:=, θ:=r∂_vϕ, ζ:=r∂_uϕandκ:=-Ω^2/4ν.From (<ref>) we obtainλ = κ (1-μ).It is easy to see that1-μ = g(∇ r,∇ r).Therefore ϖ, like r, is a geometric quantity: it is called the renormalized Hawking mass. Note that (1-μ) depends on (u,v) only through (r,ϖ). In what follows, in a slight abuse of notation, we will interchangeably regard (1-μ) as a function of either pair of variables, with the meaning being clear from the context.The Einstein equations imply (see <cit.>) the following first order system for the variables (r,ν,λ,ϖ,θ,ζ,κ):∂_ur = ν, ∂_vr = λ, ∂_uλ = νκ∂_r(1-μ), ∂_vν = νκ∂_r(1-μ), ∂_uϖ = 1/2(1-μ)(ζ/ν)^2ν, ∂_vϖ = 1/2θ^2/κ, ∂_uθ = - ζλ/r, ∂_vζ = - θν/r, ∂_uκ = κν1/r(ζ/ν)^2,with the restrictionλ=κ(1-μ).From (<ref>), (<ref>), (<ref>) and (<ref>) we obtain∂_v(ν/1-μ)=ν/1-μ(θ/λ)^2λ/r,which can also be written as∂_v(-λ/κν)=θ^2/κν r.This is the Raychaudhury equation (<ref>) written in these variables. The Raychaudhury equation (<ref>) corresponds to (<ref>).Existence and uniqueness for the characteristic initial value problem associated to the first order system (<ref>)-(<ref>), as well as a continuation criterion, were studied in <cit.>. There it was also shown that, under appropriate regularity conditions for the initial data, this system implies the Einstein equations. We shall therefore study the spherically symmetric Einstein-Maxwell-scalar field system in this framework.§ INITIAL CONDITIONS AND BEHAVIOR AT THE EVENT HORIZON We wish to study the interior of a black hole of finite mass arising from gravitational collapse. In order to do that, we consider a coordinate system (u,v) such that u=0 corresponds to the event horizon and v increases along the outgoing null direction. To eliminate the remaining gauge freedom in the choice of coordinates we setν(u,0)=-1, κ(0,v)=1.For this choice of κ(0,v), geodesic completeness of the event horizon requires that the v coordinate takes values in [0,∞[. We assume that the coordinate u takes values in[0,U], with U>0 to be chosen. Integration of equations (<ref>) and (<ref>) with initial conditions (<ref>) implies κ>0 and ν<0 over the whole solution domain. From (<ref>) we then have∂_v(-λ/κν) ≤ 0.A simple consequence of the Mean Value Theorem yields the following result.Let u∈[0,U]. * If λ(u,v̅)=0 then λ(u,v)≤ 0 for all v>v̅.* If λ(u,v̅)<0 then λ(u,v)< 0 for all v>v̅. Hawking's area theorem implies that λ must be nonnegative over the event horizon. In previous papers (see <cit.>) we considered the case where λ(0,v)≡ 0. The case where λ starts out positive and then becomes identically zero can be reduced to the one of the previous papers by fixing a new origin for the v axis. In view of the previous lemma, the only case that remains to be studied is the one where λ is strictly positive over the event horizon.Under the previous hypotheses, equations (<ref>) and (<ref>) imply that r and ϖ increase along the event horizon. To be consistent with the usual picture of gravitational collapse, we assume that the limitsr(0,∞)=r_+ and ϖ(0,∞)=ϖ_0are finite,[In fact, this necessarily happens in the case Λ>0, as λ=1-μ must remain nonnegative along the event horizon, that is, 1-2ϖ/r+e^2/r^2≥Λ/3r^2.] which can be interpreted as data asymptotically converging to a Reissner-Nordström black hole with (constant) renormalized Hawking mass ϖ_0 and (constant) event horizon radius r_+. By the Mean Value Theorem, there exists a sequence v_n↗∞ such that λ(0,v_n)→ 0, and so, from (<ref>) and (<ref>), (1-μ)(0,v_n)→ 0. Equation (<ref>), together with the fact that r(0, · ) and ϖ(0, · ) have limits at infinity, implies (1-μ)(0, · ) has a limit at infinity. We conclude that r_+ and ϖ_0 cannot be chosen arbitrarily, as they must satisfy(1-μ)(r_+,ϖ_0)=0.Moreover, we assume that the black hole is asymptotically non-extremal, that is,:=1/2∂_r(1-μ)(r_+,ϖ_0)>0.As is well known, this quantity is called the surface gravity of the event horizon for the Reissner-Nordström black hole with parameters r_+ and ϖ_0.As explained in <cit.>, for a given choice of coordinates the initial data for the characteristic initial value problem consists of two free functions, one along the ingoing null segment v=0 and the other along the event horizon u=0. On the ingoing null segment we can freely specify ζ(u,0), but on the event horizon the functions λ(0,v), ϖ(0,v) and θ(0,v) are interrelated through (<ref>) and (<ref>). Because of these constraints, as well as (<ref>) and (<ref>), it turns out that the simplest approach is to start by choosing ϖ as a function of r along the event horizon. The functions λ(0,v) and θ(0,v) can then be obtained from (<ref>) and (<ref>), respectively, making sure in the end that λ(0,v) > 0. We will now describe this procedure in detail. Since we assume that λ(0,v) is strictly positive, r(0,v) is a strictly increasing function of v, and so it may be used as a coordinate along the event horizon. Accordingly, we will write a hat over a function to mean that it is written in terms of this new coordinate.As explained above, we will start by specifying ϖ̂(r), but the restriction (<ref>) means that the true free function is its derivative ϖ̂'(r), which we will prescribe as a continuous integrable function f̂: ]0,r_+[ →^+_0, so thatϖ̂(r)=ϖ_0-∫_r^r_+f̂(r̃) dr̃. In terms of the r coordinate, (<ref>) becomesκ̂(r)≡ 1,from which (<ref>) impliesλ̂(r)=(1-μ)(r)=1-2ϖ̂(r)/r+e^2/r^2-Λ/3r^2.From the discussion that leads to (<ref>) it is also clear thatlim_r→ r_+λ̂(r)=lim_r→ r_+(1-μ)(r)=0.Moreover,λ̂'(r)=∂_r(1-μ)(r,ϖ̂(r))-2ϖ̂'(r)/r.We now make the extra assumption that[This is equivalent to assuming that lim_v →∞θ^2/λ(0,v)=2A.]lim_r→ r_+f̂(r) = A.Then we havelim_r→ r_+λ̂'(r)=∂_r(1-μ)(r_+,ϖ_0)-2A/r_+ = 2/r_+(r_+ - A).In view of (<ref>), we see that a necessary condition for λ̂ to be positive in a left neighborhood of r_+ isA∈[ r_+ ,∞]. We define ω: ]0,r_+]→ (not to be confused with ϖ) byω(r)=r/2(1+e^2/r^2- Λ/3r^2).This is the value of the mass that would make (1-μ) vanish at r,(1-μ)(r,ω(r))=0. In particular ω(r)>ϖ(r) for r<r_+ and ω(r_+) = ϖ_0. We haveω'(r)=1/2(1-e^2/r^2-Λ r^2),which can also be written, by differentiating (<ref>), asω'(r)=- ∂_r(1-μ)(r,ω(r))/∂_ϖ(1-μ)(r,ω(r))=r/2∂_r(1-μ)(r,ω(r)).Moreover, it is easy to check thatω(r)=r/2(1-μ)(r,ϖ_0)+ϖ_0,and so, for n≥ 1,ω^(n)(r_+)=. ∂^n_r(r/2(1-μ)(r,ϖ_0))|_r=r_+. In the case when A=r_+=ω'(r_+), we will also assume thatf̂(r)>1/2(1-e^2/r^2-Λ r^2)= ω'(r)= r/2∂_r(1-μ)(r,ω(r))for r in a left neighborhood of r_+. We claim that this assumption guarantees that λ̂ is positive in a left neighborhood of r_+. Indeed, from (<ref>), we have, for r<r_+,λ̂'(r) = (1-μ)'(r) = 2/r^2(ϖ_0-∫_r^r_+f̂(r̃) dr̃)- 2/rf̂(r)- 2e^2/r^3- 2Λ/3r< 2/r^2(ϖ_0-∫_r^r_+ω'(r̃) dr̃) -∂_r(1-μ)(r,ω(r))- 2e^2/r^3- 2Λ/3r= 2/r^2(r/2+e^2/2r- Λ/6r^3)- 1/r(1-e^2/r^2-Λ r^2)- 2e^2/r^3- 2Λ/3r= 0.This proves our claim.In the case when A>r_+,λ̂ is positive in a left neighborhood of r_+ becauselim_r→ r_+λ̂'(r)<0.We now list the most relevant choices of f̂ satisfying the hypotheses above, noting in particular that when A=r_+ the function f̂ is chosen to be the sum of a Taylor polynomial of ω' at r_+ with a term that ensures f̂>ω' in a left neighborhood of r_+.[on f̂] The function f̂: ]0,r_+[ →^+_0 is continuous, integrable and has limit A∈[r_+,∞] as r → r_+. In addition, in a left neighborhood of r_+ one of the five following alternatives holds:[We could work with other assumptions.But if f̂ blows up too fast at r_+, for example, f̂(r)∼ C(r_+-r)^α, with -1<α<0, it can be proved that we are led to an incomplete event horizon.] * A=∞ and there exist c,C>0 such that-cln(r_+-r)≤f̂(r)≤ -Cln(r_+-r). * r_+<A<∞.* A=r_+ and there exist c,C>0 and 0<α_1≤α_2<1 such thatω'(r_+)+c(r_+-r)^α_2≤f̂(r)≤ω'(r_+)+C(r_+-r)^α_1. * A=r_+ and there exist c,C>0 and n≥ 1 such that∑_k=0^nω^(k+1)/k!(r_+)(r-r_+)^k+c(r_+-r)^n ≤f̂(r)≤ ∑_k=0^n ω^(k+1)/k!(r_+)(r-r_+)^k+C(r_+-r)^n. * A=r_+ and there exist c,C>0, n≥ 1 and n<α_1≤α_2<n+1 such that∑_k=0^nω^(k+1)/k!(r_+)(r-r_+)^k+c(r_+-r)^α_2 ≤f̂(r)≤ ∑_k=0^nω^(k+1)/k!(r_+)(r-r_+)^k+C(r_+-r)^α_1. Case (iii) corresponds to (v) with n=0. However, we consider case (iii) separately because it is especially interesting, as it leads to the polynomial Price law studied by Dafermos in <cit.>. In fact, each case above yields a different type of Price law, as we will see in the remainder of this section. However, only case (ii) will be pursued in detail in the following sections, since it corresponds to the exponential Price law expected for a positive cosmological constant. For each of the cases (i) through (iv) above, we now proceed to examine the behavior of the following functions along the event horizon: * λ̂,* |θ̂/λ̂|,* r(0, · ),* λ_0( · ):=λ(0, · ),* θ_0( · ):=θ(0, · ).Case (v) is very similar to cases (iii) and (iv), and will not be treated explicitly. (a) Estimates for λ̂The function λ̂ is determined over the event horizon using (<ref>) and (<ref>): λ̂(r)= 2/r( r/2(1-μ)(r,ϖ_0)+∫_r^r_+f̂(r̃) dr̃).In the cases where f̂(r)=∑_k=0^nω^(k+1)/k!(r_+)(r-r_+)^k + ê(r),equation (<ref>) can be written in the formλ̂(r)= 2/r( r/2(1-μ)(r,ϖ_0)-∑_k=0^nω^(k+1)/(k+1)!(r_+)(r-r_+)^k+1+∫_r^r_+ê(r̃) dr̃).Using (<ref>), we then have λ̂(r)=2/r(O((r_+-r)^n+2)+ ∫_r^r_+ê(r̃) dr̃)=(1-μ)(r). Let us denote by ]r_1,r_+[ a left neighborhood of r_+ where one of the assumptions (i) through (iv) above holds and moreover λ̂=(1-μ) is positive. In what follows the constants c and C will be as in Hypothesis <ref>, and >0 is a parameter that can be made arbitrarily small by choosing r_1 sufficiently close to r_+.* In this case, we note that∫_r^r_+-ln(r_+-r̃) dr̃=-(r_+-r)ln(r_+-r)+(r_+-r).So, using (<ref>),-C_1(r_+-r)ln(r_+-r) ≤λ̂(r)≤ -C_2(r_+-r)ln(r_+-r),for r ∈]r_1,r_+[. Here C_1=2c(1-)/r_+ and C_2=2C(1+)/r_+.* This hypothesis impliesf̂(r)=A+o(1),as r→ r_+. Using (<ref>), we haveC_1(r_+-r)≤λ̂(r)≤ C_2(r_+-r),for r ∈]r_1,r_+[. Here C_1=(2A/r_+-∂_r(1-μ)(r_+,ϖ_0))(1-)=2/r_+(A-r_+)(1-)and C_2=(2A/r_+-∂_r(1-μ)(r_+,ϖ_0))(1+)=2/r_+(A-r_+)(1+). * In this case, using (<ref>) with n=0, we obtainC_1(r_+-r)^1+α_2≤λ̂(r)≤ C_2(r_+-r)^1+α_1,for r ∈]r_1,r_+[. Here C_1=2c(1-)/r_+(1+α_2) and C_2=2C(1+)/r_+(1+α_1). Recall that 0<α_1≤α_2<1.* Similarly to (iii), we haveC_1(r_+-r)^n+1≤λ̂(r)≤ C_2(r_+-r)^n+1,for r ∈]r_1,r_+[. Here C_1=2c(1-)/r_+(n+1) and C_2=2C(1+)/r_+(n+1). Recall that n≥ 1. (b) Estimates for |θ̂/λ̂|The quotient θ̂/λ̂is a continuous function such that(θ̂/λ̂)^2=2ϖ̂'/λ̂=2f̂/λ̂(see (<ref>)), and so lim_r→ r_+(θ̂/λ̂)^2(r)=2A/0^+=∞.We now determine the rate of blow up of |θ̂/λ̂| at r_+ in each of the cases (i) through (iv).* Using (<ref>) and (<ref>), we getc_1/(r_+-r)^1/2≤|θ̂/λ̂|(r)≤c_2/(r_+-r)^1/2,for r ∈]r_1,r_+[. Here c_1=√(2c/C_2) and c_2=√(2C/C_1).* We have from (<ref>) and (<ref>)c_1/(r_+-r)^1/2≤|θ̂/λ̂|(r) ≤c_2/(r_+-r)^1/2,for r ∈]r_1,r_+[.Here c_1=√(2A/C_2) and c_2=√(2A/C_1).* Using (<ref>) and (<ref>), we havec_1/(r_+-r)^1+α_1/2≤|θ̂/λ̂|(r)≤c_2/(r_+-r)^1+α_2/2,for r ∈]r_1,r_+[.Here c_1=√(2A/C_2) and c_2=√(2A/C_1).* We have from (<ref>) and (<ref>)c_1/(r_+-r)^n+1/2≤|θ̂/λ̂|(r)≤c_2/(r_+-r)^n+1/2,for r in a left neighborhood of r_+.Here c_1=√(2A/C_2) and c_2=√(2A/C_1). (c) Estimates for r(0, · ) Let us definer̂_0=inf{r̂<r_+:∂_r(1-μ)(r,ϖ̂(r̂))>0for all r̂≤ r≤ r_+}.(Note that r̂_0>r_0, where r_-<r_0<r_+ is such that ∂_r(1-μ)(r_0,ϖ_0)=0.) Choose r_2 ∈]max{r̂_0,r_1},r_+[. To fix the v coordinate we set r(0,0)=r_2, so thatv(r)=∫_r_2^r1/∂_v r(r̃) dr̃ =∫_r_2^r1/λ̂(r̃) dr̃.Whichever the case, (<ref>), (<ref>), (<ref>) or (<ref>),1/λ̂ is not integrable in ]r_2,r_+[, and so v(r_+)=∞. Using (<ref>), we can now determine the behavior of r as a function of v along the event horizon. * In this case, we have(r_+-r_2)^e^C_2v≤ r_+-r(0,v)≤(r_+-r_2)^e^C_1v.We can assume, without loss of generality, that r_+-r_2<1.* Here, we obtain(r_+-r_2)e^-C_2v≤ r_+-r(0,v)≤(r_+-r_2)e^-C_1v. * In this case, we have1/[α_1C_2v+ 1/(r_+-r_2)^α_1]^1/α_1≤ r_+-r(0,v)≤1/[α_2C_1v+ 1/(r_+-r_2)^α_2]^1/α_2.Recall that 0<α_1≤α_2<1.* Here, we obtain1/[nC_2v+ 1/(r_+-r_2)^n]^1/n≤ r_+-r(0,v)≤1/[nC_1v+ 1/(r_+-r_2)^n]^1/n.Recall that n≥ 1. (d) Estimates for λ_0The estimates for r(0,v) obtained in (c) now allow us to rewrite the bounds for λ̂, determined in (a), as bounds for λ_0 in terms of v.* In this case, we haveC_1ln(1/r_+-r_2)e^C_1v(1/r_+-r_2)^-e^C_2v≤λ_0(v)≤ C_2ln(1/r_+-r_2)e^C_2v(1/r_+-r_2)^-e^C_1v.Recall that r_+-r_2<1.* Here, we obtainC_1(r_+-r_2)e^-C_2v≤λ_0(v)≤ C_2(r_+-r_2)e^-C_1v. * In this case, we haveC_1/[α_1C_2v+ 1/(r_+-r_2)^α_1]^1+α_2/α_1≤λ_0(v)≤C_2/[α_2C_1v+ 1/(r_+-r_2)^α_2]^1+α_1/α_2.Recall that 0<α_1≤α_2<1.* Here, we obtainC_1/[nC_2v+ 1/(r_+-r_2)^n]^n+1/n≤λ_0(v)≤C_2/[nC_1v+ 1/(r_+-r_2)^n]^n+1/n.Recall that n≥ 1. (e) Estimates for θ_0Obviously, θ̂ is determined over the event horizon byθ̂(r)=θ̂/̂λ̂(r)λ̂(r).Again, using the bounds in (c) for r(0,v), we may bound θ_0(v) as follows. * In this case, we havec_1C_1ln(1/r_+-r_2)e^C_1v(1/r_+-r_2)^- e^C_2v/2≤ |θ_0|(v)≤ c_2C_2ln(1/r_+-r_2)e^C_2v(1/r_+-r_2)^- e^C_1v/2.Recall that r_+-r_2<1.* Here, we obtainc_1C_1(r_+-r_2)^1/2e^-C_2/2v≤|θ_0|(v)≤ c_2C_2(r_+-r_2)^1/2e^-C_1/2v. * In this case, we havec_1C_1/[α_1C_2v+ 1/(r_+-r_2)^α_1]^1+α_2+(α_2-α_1)/2α_1≤ |θ_0|(v)≤c_2C_2/[α_2C_1v+ 1/(r_+-r_2)^α_2]^1+α_1-(α_2-α_1)/2α_2.Recall that 0<α_1≤α_2<1.* Here, we obtainc_1C_1/[nC_2v+ 1/(r_+-r_2)^n]^n+1/2n≤ |θ_0|(v)≤c_2C_2/[nC_1v+ 1/(r_+-r_2)^n]^n+1/2n.Recall that n≥ 1.Recap of the initial conditions. To finish this section on the initial conditions, let us summarize the procedure for constructing the initial data. We start by prescribing the integrable function f̂, which determines ϖ̂ by (<ref>). The v coordinate is fixed by setting κ̂≡ 1, which in turn yields λ̂ by (<ref>). The function θ̂ is obtained (up to a choice of sign) from (<ref>), and r is determined at the event horizon by (<ref>). To complete the definition of the initial data, we choose{[r(u,0) =r_2-u,;ν(u,0) =ν_0(u) ≡ -1,;ζ(u,0) = ζ_0(u), ]. for u∈[0,U],where ζ_0 is a free continuous function. Under the previous conditions, if f̂ is chosen according to Hypothesis <ref>, then λ̂ is positive in ]r_2,r_+[ and r(0, · ), λ_0( · )=λ(0, · ) and θ_0( · )=θ(0, · ) have the decay given in (c), (d) and (e), respectively. § THE APPARENT HORIZON Unlike what happens in the Reissner-Nördstrom solutions, or in the more general solutions studied in <cit.>, the interiors of the black holes that we are now considering do not coincide with the trapped region, that is, the set of points where λ < 0. In fact, as a consequence of λ>0 on the event horizon, there will also exist a regular region P_λ, where λ≥ 0. It will be shown in this section that the apparent horizon A, that is, the set of points where λ = 0, is, in our domain, a C^1 curve that separates the regular from the trapped regions. Moreover, we will prove that this curve can be parametrized by v↦(u_λ(v),v), with u_λ'≤ 0 and d/dvr(u_λ(v),v)≥ 0. We will finish by sketching A and the curves where r is constant.From Theorem 4.4 in <cit.> we have The characteristic initial value problem (<ref>)-(<ref>), with the initialconditions of the previous section has a unique solution defined on a maximal past setcontaining [0,]×{0}∪{0}×[0,∞[.We denote by Γ_ the set where r is equal to . Since ν<0, this set is a curve that can be parametrized by v↦((v),v). Obviously, r((v),v)=.Moreover, r is C^1, sois C^1. Differentiating both sides of (<ref>) with respect to v yields'(v)=- λ((v),v)/ν((v),v). In Lemma <ref> we looked at the behavior of λ along a line where u is constant. Now we look at the behavior of λ along a curve Γ_.Fix ∈ ]0,r_+[. If λ((v̅),v̅)=0, then λ((v),v)≤ 0 for all v>v̅. In particular, ( · ) is defined in [v̅,∞[. Let v̅ be such that λ((v̅),v̅)=0. Suppose there exists v>v̅ such that λ((v),v)>0. Define v=inf{v̂: λ((ṽ),ṽ)>0for all ṽ∈]v̂,v]}. Clearly, v≥v̅ and λ((v),v)=0. Equality (<ref>) shows that '(ṽ)>0 for all ṽ∈ ]v,v].Thus, (v)<(v). According to Lemma <ref>, λ((v),ṽ)≤ 0 for allṽ∈ ]v,v]. This implies ≥ r((v),v). Since (v)<(v) and ν<0, we have r((v),v)>r((v),v). However, r((v),v)=. Hence we reached the contradiction >.Since '(v) ≤ 0 for v ≥v̅, we have (v)≤(v̅) < U. Therefore, only two possibilities could occur to prevent ( · ) from being defined in [v̅,∞[: either the curve Γ_ reaches the event horizon, or the boundary ofin [0,]×[0,∞[. However, the first possibility is excluded because λ is strictly positive over the event horizon, and the second by the fact that r goes to zero on the boundary of(see <cit.>).Consider v̅=inf{v≥ 0:λ((v),v)≤ 0}.One of the following occurs: * v̅=∞ and so λ((v),v) is positive for all v.* v̅∈^+_0 and so λ((v),v) is positive forv<v̅and is nonpositive for v≥v̅. From (<ref>), in case (b) we have max{(v):v∈^+_0}=(v̅). Recall that the initial data is prescribed so that λ is strictly positive on {0}×[0,∞[. We now choose U sufficiently small so that λ is positive on [0,U]×{0}. Let us define the set P_λ:={(u,v)∈[0,U]×[0,∞[ :λ(u,v)≥ 0}.By Lemma <ref>, if (u,v) ∈ P_λ then {u}×[0,v]⊂ P_λ. This and the fact that ν(u,0)<0 imply thatr≥ r(U,0) onP_λ. As noted at the beginning of Section <ref>, we have κ>0 and ν<0 over the whole solution domain 𝒫. Therefore, 1-μ=λ/κ is positive on [0,U]×{0}, and, by (<ref>), min{ϖ(u,0):0≤ u≤ U}=ϖ(U,0). Since ∂_vϖ≥ 0, ϖ achieves its minimum at (U,0):ϖ(U,0)≤ϖonP. Since r(0,0)=r_2>r̂_0 (see (<ref>)), we havemin_r∈ [r_2,r_+]∂_r(1-μ)(r,ϖ̂(r_2))>0. By further reducing U, if necessary, we can make r(U,0) sufficiently close to r_2, and therefore ϖ(U,0) sufficiently close to ϖ̂(r_2), so thatmin_r∈ [r(U,0),r_+]∂_r(1-μ)(r,ϖ(U,0))>0.Since ∂_r(1-μ) increases with ϖ and r<r_+ on 𝒫, we then havemin_(u,v)∈ P_λ∂_r(1-μ)(u,v)≥min_r∈ [r(U,0),r_+]∂_r(1-μ)(r,ϖ(U,0))>0.In addition,∂_r(1-μ)(r(U,0),ϖ_0)> ∂_r(1-μ)(r(U,0),ϖ(U,0))>0, and so r(U,0)>r_0.From equation (<ref>), we have∂_uλ<0onP_λ. Moreover, Lemma <ref> implies that if (u,v) is such that λ(u,v)=0, then ∂_vλ(u,v)≤ 0. From Lemma 6.1 in <cit.> we know that λ is C^1, because ν_0, κ_0( · )=κ(0, · ) and λ_0 are C^1. We therefore conclude that if the set A:={(u,v)∈ P_λ:λ(u,v)=0}is nonempty (as will be seen to be the case in the next section), then it is a C^1 manifold which we can parametrize by v↦(u_λ(v),v) withu_λ'(v)=- ∂_vλ(u_λ(v),v)/∂_uλ(u_λ(v),v)≤ 0.To examine the behavior of r along the curve where λ=0 we computed/dvr(u_λ(v),v) = ν(u_λ(v),v)u_λ'(v)+λ(u_λ(v),v)= ν(u_λ(v),v)u_λ'(v)≥ 0. Arguing as in the second half of the proof of Lemma <ref>, we conclude that the domain of u_λ is an interval of the form [v_0,∞[, for some v_0>0.Typically, the (thin) curves of constant r and the (thick) curve where λ=0 behave as in the following figure.45vv< g r a p h i c s > Clearly, if a portion of a curve Γ_r is parametrized by v↦(c̃,v) for v∈ I, then λ=0 over that portion of curve.Conversely, equation (<ref>) shows that if r is constant over an interval I along the curve where λ is zero, then the portion of this curve over I is parametrized by v↦(c̃,v).Hence, it is not excluded that a curve of constant r and the curve where λ=0 could partially overlap (as is the case on the event horizon of the Reissner-Nordström solution). This is illustrated in the next figure, where again the thin line represents a curve of constant r and the thick line represents the curve where λ=0.45vv< g r a p h i c s >Equation (<ref>) implies that P_λ is a past set. Therefore, since ∂_uϖ≤ 0 in P_λ, the supremum of ϖin P_λ is ϖ_0:ϖ≤ϖ_0 onP_λ. § BEHAVIOR OF THE SOLUTION ON P_ΛIn Hypothesis <ref> we listed the main reasonable asymptotics for the renormalized Hawking mass along the event horizon. As was mentioned then, we will now focus exclusively on case (ii), since it corresponds to the exponential Price law expected for a positive cosmological constant. Note from (<ref>) that in this case∫_0^∞ |θ_0|(v) dv < ∞. In this section we will analyze in detail the behavior of the solution in the region P_λ between the event and the apparent horizons, where λ is still nonnegative (in spite of being located inside the black hole), and so the monotonicity properties of the radius and mass functions are the same as outside the black hole. This region does not exist in the Reissner-Nordström solution, or in the solutions studied in <cit.>, where the two horizons coincide. Therefore we must study the propagation of the decays of the main quantities from the event horizon to the apparent horizon, after which an analysis similar to what was done in <cit.> can be performed. The most significant phenomenon influencing this propagation is the redshift effect.The estimates in this section depend on accurately controlling the growth of ζ/ν in P_λ. The importance of this function stems from the fact that it is a geometric quantity, and so the redshift effect is reflected in its evolution equation (<ref>). The resulting exponential decay plays a fundamental role in estimating the remaining key quantities. More precisely, we start by deriving an appropriate two-dimensional version of Gronwall's inequalityto bound ζ/ν in P_λ. We thenshow that the derivative ∂_r(1-μ) is close to ∂_r(1-μ)(r_+,ϖ_0)= 2k_+, and κ is close to 1, provided that u is sufficiently small and v is sufficiently large. This allows us to go back to our previous estimate for ζ/ν and improve it to an exponential decay, dominated by the slower of two competing effects: the redshift arising from the evolution equation, essentially e^-2k_+ v, and the exponential decay of θ_0. We can then use it to control the remaining key quantities θ, ν, u_λ and v_λ (given by (<ref>)). Finally, we show that in P_λ the radius function r and the renormalized Hawking mass ϖ converge to r_+ and ϖ_0, respectively, as v→∞.We will start by writing an integral formula for ζ/ν in P_λ, which features a crucial dependence on θ_0 and ζ/ν. Integrating (<ref>), we getθ(u,v)=θ_0(v)-∫_0^u[ζ/νλ/rν](ũ,v) dũ,while integrating∂_v(ζ/ν)=- θ/r-κ∂_r(1-μ)ζ/ν,we getζ/ν(u,v) = ζ/ν(u,0)e^-∫_0^v[κ∂_r(1-μ)](u,ṽ) dṽ -∫_0^vθ/r(u,ṽ)e^-∫_ṽ^v[κ∂_r(1-μ)](u,v̅) dv̅ dṽ.The desired formula for ζ/ν is obtained combining (<ref>) with (<ref>):ζ/ν(u,v) = ζ/ν(u,0)e^-∫_0^v[κ∂_r(1-μ)](u,ṽ) dṽ -∫_0^vθ_0(ṽ)/r(u,ṽ)e^-∫_ṽ^v[κ∂_r(1-μ)](u,v̅)dv̅ dṽ +∫_0^v∫_0^u[ζ/νλ/rν](ũ,ṽ)e^-∫_ṽ^v[κ∂_r(1-μ)](u,v̅)dv̅/r(u,ṽ) dũdṽ. We will need the following version of Gronwall's inequality.Let M be a positive number, and assume that f: ]0,M]→[0,∞[ is continuous and strictly decreasing with lim_x↘ 0f(x)=∞. Consider the setS={(x,y)∈ ]0,M]×[0,∞[ :y≥ f(x)},and continuous functions c: ]0,M]→ [0,∞[ and u,b: S→[0,∞[ such thatu(x,y)≤+∫_f^-1(y)^xc(x̃) dx̃+∫_f^-1(y)^x∫_f(x̃)^yu(x̃,ỹ)b(x̃,ỹ) dỹdx̃,for some positive number . Thenu(x,y)≤(+∫_f^-1(y)^xc(x̃) dx̃)e^∫_f^-1(y)^x∫_f(x̃)^yb(x̃,ỹ) dỹdx̃.For (x,y)∈ S we definev(x,y)=+∫_f^-1(y)^xc(x̃) dx̃+∫_f^-1(y)^x∫_f(x̃)^yu(x̃,ỹ)b(x̃,ỹ) dỹdx̃(see the figure below).xx sx ff(x) gy af^-1(y) MM< g r a p h i c s > According to our hypothesis, u≤ v. Note that if ỹ≤ y then v(x,ỹ)≤ v(x,y), because when wechange ỹ to y we are integrating nonnegative functions over larger domains. Hence, as b is nonnegative,∂_x v(x,y) = c(x)+∫_f(x)^yu(x,ỹ)b(x,ỹ) dỹ≤ c(x)+∫_f(x)^yv(x,ỹ)b(x,ỹ) dỹ≤ c(x)+v(x,y)∫_f(x)^yb(x,ỹ) dỹ.Next we use v(x,y)≥+∫_f^-1(y)^xc(x̃) dx̃>0. We may write∂_x v(x̃,y)/v(x̃,y)≤c(x̃)/+∫_f^-1(y)^x̃c(x̅) dx̅+∫_f(x̃)^yb(x̃,ỹ) dỹ.Integrating both sides of the last inequality in x̃, from f^-1(y) to x, we getln v(x,y) ≤ ln v(f^-1(y),y) +ln(+∫_f^-1(y)^xc(x̃) dx̃)- ln +∫_f^-1(y)^x∫_f(x̃)^yb(x̃,ỹ) dỹdx̃.Taking into account that v(f^-1(y),y)=,v(x,y)≤(+∫_f^-1(y)^xc(x̃) dx̃)e^∫_f^-1(y)^x∫_f(x̃)^yb(x̃,ỹ) dỹdx̃.Since u(x,y)≤ v(x,y), we obtain (<ref>). Under the hypotheses of Lemma <ref>, ifu(x,y)≤∫_f^-1(y)^xc(x̃) dx̃+∫_f^-1(y)^x∫_f(x̃)^yu(x̃,ỹ)b(x̃,ỹ) dỹdx̃,thenu(x,y)≤(∫_f^-1(y)^xc(x̃) dx̃)e^∫_f^-1(y)^x∫_f(x̃)^yb(x̃,ỹ) dỹdx̃.If u satisfies (<ref>), then u satisfies (<ref>), for every positive . The result is obtained by letting ↘ 0. A small variation of Lemma <ref> isLet M be a positive number and consider continuous functions c:[0,M]→ [0,∞[ and u,b:[0,M]×[0,∞[ →[0,∞[ such thatu(x,y)≤+∫_0^xc(x̃) dx̃+∫_0^x∫_0^yu(x̃,ỹ)b(x̃,ỹ) dỹdx̃,for some positive number . Thenu(x,y)≤(+∫_0^xc(x̃) dx̃)e^∫_0^x∫_0^yb(x̃,ỹ) dỹdx̃.We will now apply Lemma <ref> to (<ref>) in order to bound ζ/ν in P_λ. We have seen that on P_λ we have r≥ r(U,0)>r_0, ϖ≤ϖ_0 and ∂_r(1-μ)≥ c>0. Since the exponentials in (<ref>) are bounded above by 1 and 1/r is bounded above by 1/r(U,0), we get|ζ/ν|(u,v)≤C(sup_[0,U]|ζ_0|+1/r(U,0)∫_0^∞|θ_0|(ṽ) dṽ)=:C_λ.Indeed, as λ is nonnegative and ∂_uλ<0 on P_λ,∫_0^v∫_0^u[λ/r(-ν)](ũ,ṽ) dũdṽ≤∫_0^v∫_0^u[-ν/r](ũ,ṽ)λ(0,ṽ) dũdṽ= ∫_0^vλ(0,ṽ)ln(r(0,ṽ)/r(u,ṽ)) dṽ≤∫_r(0,0)^r(0,v)ln(r(0,ṽ)/r(U,0)) dr(0,ṽ) =r(0,v)ln(r(0,v)/r(U,0))-r(0,v)-r(0,0)ln(r(0,0)/r(U,0))+r(0,0)<r_+ln(r_+/r(U,0))-r_+-r(0,0)ln(r(0,0)/r(U,0))+r(0,0)=ln(c(r(U,0),r_+)/r(U,0))(r_+-r(0,0))< ln(r_+/r(U,0))(r_+-r(0,0)).Here c(r(U,0),r_+)∈ ]r(U,0),r_+[ is provided by the Mean Value Theorem. The constant C in (<ref>) is bounded byC≤(r_+/r(U,0))^r_+-r(0,0)/r(U,0). Arguing as in Section <ref>, one can easily see that r(U,V)≤ r≤ r_+ and ϖ(U,V)≤ϖ≤ϖ_0 in the set :={(u,v)∈ P_λ:0≤ u≤ Uand v≥ V}.Since lim_v→∞r(0,v)=r_+ and lim_v→∞ϖ(0,v)=ϖ_0, the continuity of the functions r and ϖ guarantees that they are close to r_+ and ϖ_0 inif we choose U sufficiently small and V sufficiently large. Therefore, given δ>0, there exist U>0 and V≥ 0 such that2-δ≤∂_r(1-μ)≤ 2+δin . Integrating (<ref>) in , we obtainC_κ:=(r(U,V)/r_+)^C_λ^2≤κ≤ 1.Notice that C_κ can be made arbitrarily close to one by choosing U small and V large. So, in this set,-C_α:=-(2 +δ) =-(∂_r(1-μ)(r_+,ϖ_0)+δ)≤-κ∂_r(1-μ)≤-C_κ(∂_r(1-μ)(r_+,ϖ_0)-δ)=-C_κ(2 -δ)=:-c_α. We will now improve our estimate for ζ/ν to an exponential decay. Going back to (<ref>), the first exponential is bounded above bye^-∫_0^v[κ∂_r(1-μ)](u,ṽ) dṽ≤ e^-c_α v,while the second and third exponentials are bounded above bye^-∫_ṽ^v[κ∂_r(1-μ)](u,v̅) dv̅≤ e^-c_α(v-ṽ). Applying again Lemma <ref>, this timeto e^c_α vζ/ν(u,v) (see also the proof of Lemma 4.1 of <cit.>), we obtain, in ,|ζ/ν|(u,v)≤C(sup_[0,U]|ζ/ν|( · ,V)+∫_V^v e^c_αṽ|θ_0|(ṽ) dṽ)e^-c_α v.In view of the decay (<ref>) for θ_0, V can be chosen so that the indefinite integral∫_V^∞ e^c_αṽ|θ_0|(ṽ) dṽ converges provided c_α<C_1/2.We define:=A/r_+ - 1 > 0,the normalized distance from A to its minimum allowed value (see (<ref>)). Note that in case (ii) this distance must be positive. The value of C_1 is expressed in terms of s by (see (<ref>))C_1=2-δ,and so c_α<C_1/2 amounts tos>2.In this case (<ref>) yields|ζ/ν|(u,v)≤ C(sup_[0,U]|ζ/ν|( · ,V)+1)e^-c_α vWhen ≤ 2, we obtain from (<ref>)|ζ/ν|(u,v)≤ Ce^-(-δ)v,where we still have exponential decay. The existence of these two different regimes reflects the competition phenomenon, mentioned at the beginning of this section, between the redshift arising from the evolution equation (which dominates for >2) and the exponential decay of θ_0 (dominant for <2). We now use this improved estimate for |ζ/ν| to control the remaining quantities, starting with θ. Using (<ref>) and (<ref>), we have, for (u,v)∈ P_λ,|θ(u,v)-θ_0(v)| ≤ λ_0(v)max_[0,u]×{v}|ζ/ν| ∫_0^u[-ν/r](ũ,v) dũ≤ ln(r_+/r(U,0)) λ_0(v)max_[0,u]×{v}|ζ/ν|.This yields, from (<ref>) and (<ref>),|θ(u,v)-θ_0(v)|≤ Ce^-C_1v.In view of (<ref>), we conclude that, for (u,v)∈,ce^- C_2/2v≤|θ|(u,v)≤ Ce^- C_1/2v.Note that the decay of θ is faster than that of |ζ/ν| for >2 due to the exponential decay of λ_0. This effect is lost when we cross the apparent horizon, as will be seen in the next section. Using the integrated form of (<ref>),ν(u,v)=ν(u,V)e^∫_V^v[κ∂_r(1-μ)](u,ṽ) dṽ,we conclude that-Ce^C_α (v-V)≤ν(u,v)≤-ce^c_α (v-V)in . Here-C=min{ν(u,V):u∈[0,U]}≤max{ν(u,V):u∈[0,U]}=-c<0. We will now estimate u_λ. We start by noticing that, using (<ref>),-C_α Ce^C_α (v-V)≤∂_uλ(u,v)≤-c_α c e^c_α (v-V)in the set . Moreover, integrating ∂_uλ from the event horizon to the apparent horizon, we obtain0=λ_0(v)+∫_0^u_λ(v)∂_uλ(ũ,v) dũ. Since (see (<ref>))ce^-C_2v≤λ_0(v)≤ Ce^-C_1v,we deduce thatce^C_α V/C_α C e^-(C_2+C_α)v≤ u_λ(v)≤Ce^c_α V/c_α c e^-(C_1+c_α)v.Let δ>0. Our parameters can be chosen so that (see (<ref>), (<ref>) and (<ref>))2A/r_+-δ<C_1+c_α< C_2+C_α<2A/r_++δ.In particular, the exponents in (<ref>) are positive, and consequently A is nonempty.For u∈ ]0,U], we definev_λ(u)=min{v:λ(u,v)=0}.Using u_λ(v_λ(u))=u in (<ref>), for each δ>0 we have(r_+/2A-δ)ln(c/u)≤ v_λ(u)≤(r_+/2A+δ)ln(C/u).We now characterize the behavior of r on A. Taking into account (<ref>), (<ref>) and (<ref>), we have- λ_0(v)/c_α≤∫_0^u_λ(v)ν(ũ,v) dũ≤- λ_0(v)/C_α,that is,λ_0(v)/C_α≤ r(0,v)-r(u_λ(v),v)≤λ_0(v)/c_α.This implies thatlim_v→∞r(u_λ(v),v)=r_+,and so alsolim_(u,v)∈ P_λr(u,v)=r_+. Examining the sign of the components of dϖ, we see that for small v the level curves of ϖ are qualitatively like the ones in the following figure, where the thick curve represents A. Notice thatd/dv[ϖ(u_λ(v),v)] = ∂_uϖ(u_λ(v),v)u_λ'(v) +∂_vϖ(u_λ(v),v)= ∂_vϖ(u_λ(v),v) ≥ 0. 45vv< g r a p h i c s > The integrated form of (<ref>) isϖ(u,v)=ϖ(0,v)e^-∫_0^u(ζ^2/rν)(ũ,v) dũ+ ∫_0^ue^-∫_ũ^uζ^2/rν(u̅,v) du̅(1/2(1+e^2/r^2 -Λ/3r^2)ζ^2/ν)(ũ,v) dũ.The rough estimate (see (<ref>) and (<ref>))0≤ -∫_0^u_λ(v)ζ^2/rν(ũ,v) dũ≤ C_λ^2ln(r(0,v)/r(u_λ(v),v)) = o(1)implies that, for 0≤ u≤ u_λ(v), the second term on the right-hand side of (<ref>) is also o(1) as v →∞, and soϖ(u,v)=ϖ(0,v)+o(1).This yieldslim_(u,v)∈ P_λϖ(u,v)=ϖ_0. § THE REGION J^-(Γ_)∩ J^+( A) From this point on we will consider the solution defined on the intersection of the maximal past set P with the rectangle [0,U]×[V,∞[, for suitably chosen U>0 and V≥ 0. In this section we focus on the subset J^-(Γ_)∩ J^+( A), for a given ∈]r_0,r_+[, which will later be set conveniently close to r_+. We will see that the exponential decays along the apparent horizon of ζ/ν and θ persist in this new region. However, the faster decay rate of θ in the case >2 is lost, dominated by that of ζ/ν. As in Section 4 of <cit.>, the solution here still behaves qualitatively as the Reissner-Nordström solution: ϖ is close to ϖ_0, κ is close to 1 and ζ, θ are close to 0. Besides, the approximation improves by making U smaller and V larger.More precisely, we start by showing that, given δ>0 small, we can choose V sufficiently large and U sufficiently small so that ϖ≥ϖ_0-δ in J^+( A) and ∂_r(1-μ)>0 in J^-(Γ_)∩ J^+( A). This implies that ∂_uλ<0, and so we can use the two-dimensional version of Gronwall's inequality in Lemma <ref> to estimate ζ/ν. This allows us to control κ from below and ϖ from above, which, as before, leads to an improved estimate for ζ/ν. We then go on to bound ν, u_ and v_. The bounds on r and ϖ enable us to determine the precise behavior of 1-μ and, consequently, of λ. Finally, we obtain bounds for θ, which are quantitatively like those for ζ/ν.Let us choose0<δ<^2/2min_r∈[,r_+]∂_r(1-μ)(r,ϖ_0).It is clear from (<ref>) and (<ref>) that there exists V such that{[ ϖ(u_λ(v),v) ≥ϖ_0-δ,; r(u_λ(v),v) ≥ , ]. for v≥ V.We choose 0<U≤ u_λ(V). Then, from (<ref>),ϖ≥ϖ_0-δ in J^+( A).It follows that-∂_r(1-μ)(r,ϖ)≤ -∂_r(1-μ)(r,ϖ_0)+ 2δ/r^2 in J^+( A), and, recalling from (<ref>) that r increases along A, and so r ≤ r_+ on J^+( A), -∂_r(1-μ)(r,ϖ)≤ -min_r∈[,r_+]∂_r(1-μ)(r,ϖ_0)+ 2δ/^2 in J^-(Γ_)∩ J^+( A). Since (<ref>) holds, -∂_r(1-μ)(r,ϖ)<0in J^-(Γ_)∩ J^+( A).To estimate ζ/ν for (u,v)∈ J^+( A), we use the expression (similar to (<ref>))ζ/ν(u,v) = ζ/ν(u,v_λ(u))e^-∫_v_λ(u)^v[κ∂_r(1-μ)](u,ṽ) dṽ -∫_v_λ(u)^vθ(u_λ(ṽ),ṽ)/r(u,ṽ)e^-∫_ṽ^v[κ∂_r(1-μ)](u,v̅)dv̅ dṽ +∫_v_λ(u)^v∫_u_λ(ṽ)^u[ζ/νλ/rν](ũ,ṽ)e^-∫_ṽ^v[κ∂_r(1-μ)](u,v̅)dv̅/r(u,ṽ) dũdṽ(recall the definition of v_λ in (<ref>)). From (<ref>) we conclude that ∂_uλ<0 for (u,v)∈ J^-(_)∩ J^+( A), and so we have∫_v_λ(u)^v∫_u_λ(ṽ)^u [(-ν)/r(-λ)](ũ,ṽ) dũdṽ ≤∫_v_λ(u)^v∫_u_λ(ṽ)^u (-ν)/r(ũ,ṽ)(-λ)(u,ṽ) dũdṽ ≤ln(r_+/)∫_v_λ(u)^v (-λ)(u,ṽ) dṽ ≤ln(r_+/)(r(u,v_λ(u))-r(u,v)) ≤ln(r_+/)(r_+-). We apply a generalized version of Lemma <ref> (because f(v)=u_λ(v) might not be strictly decreasing) whose proof we leave to the reader (just approximate u_λ by a strictly decreasing function and pass to the limit). For (u,v)∈ J^-(_)∩ J^+( A),|ζ/ν|(u,v)≤(r_+/)^r_+-/(sup_]0,U]|ζ/ν|(u,v_λ(u))+1/∫_v_λ(u)^∞|θ|(u_λ(ṽ),ṽ) dṽ)=:C_. We see from (<ref>), (<ref>) and (<ref>) that C_ is finite.Integrating (<ref>) yieldsκ(u,v)=κ(u_λ(v),v)e^∫_u_λ(v)^u[(ζ/ν)^2 ν/r](ũ,v) dũ,and so, using (<ref>), we obtain in J^-(Γ_)∩ J^+( A)C_κ,2:=C_κ(/r_+)^C_^2≤κ≤ 1. From (<ref>), we have ϖ_m := inf{ϖ(u,v):(u,v)∈ J^+( A)and v≥ V} = ϖ(U,v_λ(U)) = ϖ_0+o(1),as V→∞ (recall that 0<U≤ u_λ(V)). Analogously to (<ref>), we now haveϖ(u,v)=ϖ(u_λ(v),v)e^-∫_u_λ(v)^u(ζ^2/rν)(ũ,v) dũ+ ∫_u_λ(v)^ue^-∫_ũ^uζ^2/rν(u̅,v) du̅(1/2(1+e^2/r^2 -Λ/3r^2)ζ^2/ν)(ũ,v) dũ.Using (<ref>), (<ref>) and (<ref>), multiplying and dividing by ν as needed, we haveϖ_M := sup{ϖ(u,v):(u,v)∈ J^-(Γ_)∩ J^+( A)} ≤ ϖ_0+o(1),as ↗ r_+. Thus, given δ>0 we can choose V sufficiently large, 0<U≤ u_λ(V) andsufficiently close to r_+ so that, for (u,v)∈ J^-(Γ_) ∩ J^+( A), we have-C_α,2:=-2-δ=-∂_r(1-μ)(r_+,ϖ_0)-δ≤ -max_r∈[,r_+]∂_r(1-μ)(r,ϖ_M)≤-κ∂_r(1-μ)(u,v)≤-C_κ,2min_r∈[,r_+]∂_r(1-μ)(r,ϖ_m)≤-∂_r(1-μ)(r_+,ϖ_0)+δ=-2+δ=:-c_α,2. Applying again a generalized version of Lemma <ref>, this timeto e^c_α,2 vζ/ν(u,v) (as was done in (<ref>)), and carefully taking the supremum over the exact interval [u_λ(v),u] (due to the unboundedness of the exponential term over the apparent horizon), leads to|ζ/ν|(u,v)≤C(sup_ũ∈[u_λ(v),u]|ζ/ν|(ũ,v_λ(ũ)) e^c_α,2v_λ(ũ)+ ∫_v_λ(u)^v e^c_α,2ṽ|θ|(u_λ(ṽ),ṽ) dṽ)e^-c_α,2 v.Suppose first that c_α,2<C_1/2, which amounts to >2 (see (<ref>)). For the first term in (<ref>) we use (<ref>) and for the second we use the decay (<ref>) for θ along the apparent horizon. We get|ζ/ν|(u,v)≤ Ce^-(2-δ) v.Here and elsewhere we use δ to mean a parameter that can be made arbitrarily small by taking U sufficiently small, V sufficiently large andsufficiently close to r_+. In this last inequality, it collects all the previous small quantities, also denoted by δ, arising in the products of the exponentials.In the case that <2, we use (<ref>) and (<ref>) to obtain|ζ/ν|(u,v)≤ Ce^-(-δ)v.From ν(u,v)=ν(u,v_λ(u))e^∫_v_λ(u)^v[κ∂_r(1-μ)](u,ṽ) dṽ,and using (<ref>), (<ref>) and (<ref>), we conclude that-Ce^C_α (v_λ(u)-V)e^C_α,2 (v-v_λ(u))≤ν(u,v) ≤-ce^c_α (v_λ(u)-V)e^c_α,2 (v-v_λ(u)),implying-Ce^C_α,3 v≤ν(u,v) ≤-ce^c_α,3 v,where c_α,3=min{c_α,c_α,2} and C_α,3=max{C_α,C_α,2}. Using-(r_++o(1))=-r(u_λ(v),v)=∫_u_λ(v)^(v)ν(ũ,v) dũand our bounds for ν, we getu_λ(v)+(r_+-+o(1))c e^-C_α,3v ≤(v)≤u_λ(v)+(r_+-+o(1))Ce^-c_α,3v.Taking into account our bounds (<ref>) for u_λ, we obtaince^-C_α,3v≤(v)≤ Ce^-c_α,3v.Thus,1/C_α,3ln(c/u)≤(u)≤1/c_α,3ln(C/u). From (<ref>), we see that for (u,v) ∈ J^-(Γ_)u≤ u_(v)≤ Ce^-(2-δ)v. We can estimate ν over Γ_ by combining (<ref>) with (<ref>):-C(1/u)^C_α,3/c_α,3≤ν(u,(u))≤-c(1/u)^c_α,3/C_α,3. From (<ref>) and (<ref>) we obtain∂_u(1-μ)=∂_u(λ/κ)=ν∂_r(1-μ)-(1-μ)ν/r(ζ/ν)^2.So, for points in {(u,v)∈ J^+( A): v≥ V}, we have, taking into account the definition of ϖ_m, ∂_u(1-μ)≤ν∂_r(1-μ)≤ν∂_r(1-μ)(r,ϖ_m).It follows that, for =r_+-δ,(1-μ)(u,v) = ∫_u_λ(v)^(v)∂_u(1-μ)(ũ,v) dũ≤ ∫_r_+^∂_r(1-μ)(r̃,ϖ_0) dr̃-2∫^_r_+ϖ_0-ϖ_m/r̃^2 dr̃= (1-μ)(,ϖ_0)+2(ϖ_0-ϖ_m)r_+-/r_+≤ -(2/1+ -2(ϖ_0-ϖ_m)/r_+)δ,where 0<<1 is fixed, provided δ is sufficiently small. Suppose that (u,v)∈Γ_. Integrating (<ref>) yields(1-μ)(u,v)=∫_u_λ(v)^(v)e^-∫_ũ^u(ν/r(ζ/ν)^2)(u̅,v) du̅ν∂_r(1-μ)(ũ,v) dũ.In this expression, we can usee^-∫_ũ^u(ν/r(ζ/ν)^2)(u̅,v) du̅≤(r_+/)^C_^2andν∂_r(1-μ)≥ν∂_r(1-μ)(r,ϖ_0)+ν2(ϖ_M-ϖ_0)/r^2.For =r_+-δ, we then obtain(1-μ)(u,v)≥ (r_+/)^C_^2((1-μ)(,ϖ_0)- 2(ϖ_M-ϖ_0)(r_+-)/r_+)≥ - (r_+/)^C_^2(2/1-+2(ϖ_M-ϖ_0)/r_+) δ,whereis any fixed positive number, provided δ is sufficiently small. Combining (<ref>) and (<ref>) with (<ref>), we have, for (u,v)∈Γ_-(r_+/)^C_^2(2/1-+2(ϖ_M-ϖ_0)/r_+) δ≤λ(u,v) ≤-C_κ,2(2/1+-2(ϖ_0-ϖ_m)/r_+)δ. Inequality (<ref>) and equation (<ref>) imply that ∂_uλ<0 in J^-(Γ_)∩ J^+( A). Thus, an equality analogous to (<ref>) together with (<ref>) yields|θ(u,v)-θ(u_λ(v),v)| ≤|λ|(u,v)max_[u_λ(v),u]×{v}|ζ/ν| ∫_u_λ(v)^u[-ν/r](ũ,v) dũ≤ Cln(r_+/) max_[u_λ(v),u]×{v}|ζ/ν|. When >2, using (<ref>),|θ(u,v)-θ(u_λ(v),v)|≤ C e^-(2-δ)v.In this situation,2<C_1/2=-(for sufficiently small ), and so, in view of (<ref>),|θ|(u,v)≤ C e^-(2-δ)vin J^-(Γ_)∩ J^+( A). When<2, using (<ref>),|θ|(u,v) ≤C e^-(-δ)v. Note that, as mentioned at the beginning of this section, the decay of θ has been overrun by that of ζ/ν. § THE REGION J^-(Γ_)∩ J^+(Γ_) In this section we focus on the region J^-(Γ_)∩ J^+(Γ_), for a given ∈]r_-,r_0[, which will later be set conveniently close to r_-. From this point on our analysis will follow closely the methods used in <cit.> and <cit.>. We will prove that the exponential decay for ζ/ν and θ, obtained in the previous section, persists in this new region. This is a consequence of the fact that the overall contribution of the redshift and blueshift effects is essentially neutral here, and so the decays carry over from Γ_ to Γ_. As in Section 5 of <cit.>, the solution still behaves qualitatively as the Reissner-Nordström solution: ϖ is close to ϖ_0, κ is close to 1 and ζ, θ are close to 0.More precisely, we start by bounding 1-μ from above by a negative constant, and ∂_r(1-μ) from below. However, we do not estimate the pair ζ/ν and θ, as we did in the previous two sections, because we do not have ∂_uλ<0, and so it is not easy to bound the double integral of νλ/r appearing in the two-dimensional Gronwall's inequality. Instead, we go on to estimate the pair ζ/ν and θ/λ as in <cit.>, using equation (54) therein; the bounds on 1-μ and ∂_r(1-μ) allow us to obtain an upper bound for the exponentials in that formula. Estimates for κ from below and ϖ from above follow. Moreover, 1-μ is clearly bounded from below. Integrating the Raychaudhuri equations, and using the estimates for λ and ν over Γ_ together with the bounds on 1-μ, lead to bounds for λ and ν. Finally, we obtain estimates forv_ and u_, which can be used to improve our previous estimates on ζ/ν and θ/λ. The estimate for θ is essentially the same as the estimate for θ/λ, as λ is bounded.In J^+( A), the mass is bounded below by ϖ_m (recall (<ref>)). We assume thatis sufficiently close to r_- so that (1-μ)(,ϖ_0)≤(1-μ)(,ϖ_0). Then, for ≤ r≤,(1-μ)(r,ϖ_m) = (1-μ)(r,ϖ_0)+2(ϖ_0-ϖ_m)/r≤ (1-μ)(,ϖ_0)+2(ϖ_0-ϖ_m)/= (1-μ)(,ϖ_m).So, in the region J^-(Γ_)∩ J^+(Γ_), we have(1-μ)(r,ϖ) ≤ (1-μ)(r,ϖ_m) ≤(1-μ)(,ϖ_m) < 0,provided that V is chosen sufficiently large for ϖ_m to be close enough to ϖ_0, so that (1-μ)(,ϖ_m)<0. The inequality (1-μ)(,ϖ_m)≤(1-μ)(,ϖ_m) follows from (1-μ)(,ϖ_0)≤(1-μ)(,ϖ_0) as above. Moreover, for ≤ r≤,∂_r(1-μ)(r,ϖ) = ∂_r(1-μ)(r,ϖ_0)- 2(ϖ_0-ϖ)/r^2≥ ∂_r(1-μ)(,ϖ_0)- 2(ϖ_0-ϖ_m)/^2≥ ∂_r(1-μ)(,ϖ_m).For the first inequality above, see the beginning of Section 3 in <cit.>. Combining (<ref>) with (<ref>), we have in the region J^-(Γ_)∩ J^+(Γ_)∂_r(1-μ)/(1-μ)≤∂_r(1-μ)(,ϖ_m)/(1-μ)≤∂_r(1-μ)(,ϖ_m)/(1-μ)(,ϖ_m) =:c_.Thus, the exponentials in (54) of <cit.> can be bounded in the following way:e^-∫_(u)^v[κ∂_r(1-μ)](u,ṽ) dṽ≤ e^c_(-)= :C.We will use Bondi coordinates (r,v), where(u,v)↦(r(u,v),v)⇔ (r,v)↦ ((v),v).We denote bythe function ζ/ν written in Bondi coordinates, so thatζ/ν(u,v)=(r(u,v),v) ⇔(r,v)=ζ/ν((v),v).The same notation will be used for other functions.Let r ∈[,]. As in Section 5 of <cit.>, for s ∈[r,], defineZ_(r,v)(s)=max_ṽ∈[((v)),v]||(s,ṽ)andT_(r,v)()=max_ṽ∈[((v)),v]|θ/λ|(,ṽ).Recall that at the beginning of Section <ref> we chose U ≤ u_λ(V); this guarantees that (u) ≥ V for all s ≤ and 0 < u ≤ U, and so the quantities above are well defined.45p(u_r(v),v) jΓ_s gΓ_ hA av_s(u_r(v)) bv cv_(u_r(v)) uv vu fu_λ(V) tU xu_r(v) yV< g r a p h i c s > 45Let us definel(s):= /2if ≤ 2 1if >2. From (<ref>) if >2, and (<ref>) if ≤ 2, we getZ_(r,v)() ≤ Ce^-(2 l(s)-δ)v_(u_r(v)).Similarly, using (<ref>) if >2,and (<ref>) if ≤ 2, together with (<ref>), we obtainT_(r,v)() ≤C e^-(2 l(s)-δ)v_(u_r(v)).Arguing as in the proof of Lemma 5.1 of <cit.> leads toZ_(r,v)(r)≤ C[ Z_(r,v)()+Cln(/r) T_(r,v)()]e^C^2(-r)^2/r.Here the constant C is as in (<ref>). Substituting (<ref>) and (<ref>) in (<ref>) finally yields the key estimate||(r,v)≤Ce^-(2 l(s)-δ)v_(u_r(v)),for ≤ r≤.From (<ref>) we can now estimate the remaining quantities. Continuing to argue as in the proof of Lemma 5.1 of <cit.>, we have, using (<ref>),|θ/λ|(r,v) ≤ C|θ/λ|(,v) +C∫_r^[||1/s̃](s̃,v) ds̃,again with the constant C as in (<ref>). We then use (<ref>) and (<ref>) in (<ref>) to obtain|θ/λ|(r,v)≤Ce^-(2 l(s)-δ)v_(u_r(v)),for ≤ r≤.Inequalities (<ref>) and (<ref>), together with (<ref>), show that, given δ̂>0, we can choose U sufficiently small so that |ζ/ν|<δ̂ and |θ/λ|<δ̂ in the region J^-(Γ_)∩ J^+(Γ_). Arguing asin (<ref>), we conclude thatC_κ,3:=C_κ,2(/)^δ̂^2≤κ≤ 1,in J^-(Γ_)∩ J^+(Γ_).A version of (<ref>) together with (<ref>) and (<ref>) imply that, for (u,v)∈ J^-(Γ_)∩ J^+(Γ_),ϖ_0+o(1)= ϖ_m≤ ϖ(u_(v),v)≤ ϖ(u,v) ≤ϖ(u_(v),v)+Cδ̂^2≤ϖ_M+Cδ̂^2 ≤ϖ_0+o(1)+Cδ̂^2. The proof of Lemma 5.2 of <cit.> shows that the curve Γ_ intersects every line of constant u, and solim_v→∞u_(v)=0. In particular, the inequalities above imply thatlim_(u,v)∈J^-(Γ_)ϖ(u,v)=ϖ_0.We rewrite (<ref>) in the formC_κ,3≤λ/1-μ(u,v)≤ 1.We have bounded 1-μ from above by a negative constant in (<ref>). Since ϖ is bounded above, and r is bounded below by a positive constant, 1-μ is also bounded from below, and so there exist positive constants c̃ and C such that-C≤λ≤ -c̃J^-(Γ_)∩ J^+(Γ_).Integrating the Raychaudhuri equation (<ref>) and taking into account the estimate(/r_+)^δ̂^2≤ e^∫_v_(u)^v((θ/λ)^2λ/r)(u,ṽ) dṽ≤ 1(which uses |θ/λ|<δ̂), we deduce(/r_+)^δ̂^2ν/1-μ(u,v_(u)) ≤ν/1-μ(u,v)≤ν/1-μ(u,v_(u)).Hence, combining the estimate (<ref>) with our bounds for 1-μ in the region J^-(Γ_)∩ J^+(Γ_), we arrive at-C(1/u)^C_α,3/c_α,3≤ν(u,v)≤-c(1/u)^c_α,3/C_α,3,for (u,v)∈ J^-(Γ_)∩ J^+(Γ_).Integrating (<ref>) between v_(u) and v, for(u,v)∈ J^-(Γ_)∩ J^+(Γ_), we obtainv-v_(u)≤-r/c̃=:c_r,≤ c_,.Taking into account (<ref>), we have1/C_α,3ln(c/u)≤ v_(u)≤ c_,+1/c_α,3ln(C/u).This yieldsce^-C_α,3v ≤ u_(v) ≤Ce^c_α,3c_, e^-c_α,3v= Ce^-c_α,3v.Notice that this constant C blows up as ↗ r_+ and ↘ r_-, because c̃ approaches zero.Using the estimate (<ref>), we can obtain improved estimates for |ζ/ν| and |θ/λ| in J^-(Γ_)∩ J^+(Γ_), given by|θ/λ|(u,v)+|ζ/ν|(u,v)≤Ce^-(2 l(s)-δ)v. From (<ref>) we also conclude that|θ|(u,v)≤Ce^-(2 l(s)-δ)v.§ THE REGION J^-(Γ)∩ J^+(Γ_) As in Section 6 of <cit.>, we define a spacelike curve γ=γ_,β to the future of Γ_, parameterized byu↦(u,(1+β) (u))=:(u,v_γ(u))for u∈[0,U], where0<β<1/2(√(1+8l(s)/)-1).Here k_- := 1/2|∂_r(1-μ)(r_-,ϖ_0)| denotes the surface gravity of the Cauchy horizon for the Reissner-Nordström black hole with parameters r_- and ϖ_0. Unlike the boundaries of the regions studied in the previous sections, this curve is not a level set of the radius function. Its purpose is to probe the geometry of the region where the blueshift effect, which is dominant at the Cauchy horizon, starts being felt. This is characterized by an exponential growth of the form e^2k_- v Nevertheless, to the past of γ the function r is bounded below and the mass ϖ is bounded above, so that the solution still behaves qualitatively as in the interior of the Reissner-Nordström black hole. More precisely, we haveFor each β as above, there exist ∈]r_-, r_0[ and _0∈]0,r_-[ for which, wheneverandare chosen satisfying ∈]r_-,] and ∈]0,_0], the following holds: there exists U_ (depending onand ) such that if (u,v)∈ J^-()∩ J^+(_), with 0<u≤ U_, then r(u,v)≥ r_–/2 andϖ(u,v)≤ϖ_0+/2.Let (u,v)∈ J^-(γ)∩ J^+(Γ_) be such that r(u,v)≥ r_–≥ r_–_0. Recall from the proof of Lemma 6.1 of <cit.> that for(u,v)∈ J^-(γ)∩ J^+(Γ_) there exists a constant C(depending on r_–_0) such that ∫_(u)^v|θ|(u,ṽ) dṽ+∫_(v)^u|ζ|(ũ,v) dũ ≤C( ∫_(u)^v|θ|((v),ṽ) dṽ+∫_(v)^u|ζ|(ũ,(u)) dũ).Following the proof of Lemma 6.1 of <cit.>, we see that the crucial step is to bound the integral∫_(v)^u[|ζ/ν||ζ|](ũ,v) dũ,for (u,v)∈ J^-(γ)∩ J^+(Γ_), by a function that goes to zero when v goes to infinity. In order to do that,we bound the first integral on the right-hand side of (<ref>) byusing the estimates for θ obtained above. These are (<ref>) in P_λ; (<ref>)-(<ref>) in J^-(Γ_)∩ J^+( A); and (<ref>) in J^-(Γ_)∩ J^+(Γ_). In summary, the upper bound (<ref>) can be used in J^-(Γ_). Therefore, usingv-(u)≤β(u) and (u)=(u)/1+β≥v/1+β, we have∫_(u)^v|θ|((v),ṽ) dṽ ≤ Ce^-(2 l(s)-δ)(u)β(u)≤ Ce^-(2 l(s)/1+β-δ)v. To bound the second integral on the right-hand side of (<ref>), we use the estimates for ζ/ν obtained above. These are (<ref>) in P_λ; (<ref>) and (<ref>) in J^-(Γ_)∩ J^+( A); and (<ref>) in J^-(Γ_)∩ J^+(Γ_).Thus, we can write∫_(v)^u|ζ|(ũ,(u)) dũ = ∫_(v)^u[|ζ/ν|(-ν)](ũ,(u)) dũ≤ C(-r(u,v))e^-(2 l(s)-δ)(u)≤ Ce^-(2 l(s)/1+β-δ)v. It follows that the left-hand side of (<ref>) can be bounded by∫_(u)^v|θ|(u,ṽ) dṽ+∫_(v)^u|ζ|(ũ,v) dũ≤ Ce^-(2 l(s)/1+β-δ)v. In J^+(Γ_) we have ϖ≥ϖ_m (see (<ref>)). So, for (u,v)∈ J^-(γ)∩ J^+(Γ_) with r(u,v)≥ r_–_0, as in (<ref>), ∂_r(1-μ)≥∂_r(1-μ)(r_–_0,ϖ_m),whence, using κ≤ 1,e^-∫_(u)^v[κ∂_r(1-μ)](u,ṽ) dṽ ≤e^-∂_r(1-μ)(r_–_0,ϖ_m)β(u)≤e^-∂_r(1-μ)(r_–_0,ϖ_m)β v.Thus, integrating (<ref>) from Γ_ (similar to (<ref>)) we have|ζ/ν|(u,v) ≤ |ζ/ν|(u,(u)) e^-∫_(u)^v[κ∂_r(1-μ)](u,ṽ) dṽ +∫_(u)^v|θ|/r(u,v̅)e^-∫_v̅^v[κ∂_r(1-μ)](u,ṽ) dṽ dv̅≤ Ce^-(2 l(s)-δ)(u)e^-∂_r(1-μ)(r_–_0,ϖ_m)β v +e^-∂_r(1-μ)(r_–_0,ϖ_m)β v/r_–_0∫_(u)^v|θ|(u,v̅) dv̅≤ Ce^-(2 l(s)/1+β-δ) ve^-∂_r(1-μ)(r_–_0,ϖ_m)β v +e^-∂_r(1-μ)(r_–_0,ϖ_m)β v/r_–_0Ce^-(2 l(s)/1+β-δ)v≤ Ce^-(2 l(s)/1+β+∂_r(1-μ)(r_–_0,ϖ_m)β-δ) v. From the previous estimate and (<ref>) we obtain∫_(v)^u[|ζ/ν||ζ|](ũ,v) dũ ≤ Ce^-(2 l(s)/1+β+∂_r(1-μ)(r_–_0,ϖ_m)β-δ) v∫_(v)^u|ζ|(ũ,v) dũ ≤ Ce^-(4 l(s)/1+β+∂_r(1-μ)(r_–_0,ϖ_m)β-δ) v. The constant in the exponent,4 l(s)/1+β+∂_r(1-μ)(r_–_0,ϖ_m)β-δ,is positive for β<1/2( √((1+δ̃)^2-8 2 l(s)-δ/2/∂_r(1-μ)(r_–_0,ϖ_m))-(1+δ̃) ),whereδ̃=- δ/∂_r(1-μ)(r_–_0,ϖ_m).The right-hand side tends to1/2(√(1+8l(s)/)-1)as (,_0,δ,ϖ_m)→(r_+,0,0,ϖ_0). So, if β satisfies (<ref>), we may choosesufficiently close to r_+, _0 and δ sufficiently small, and U sufficiently small (which in J^-(γ) implies v sufficiently large, so that ϖ_m is sufficiently close to ϖ_0) so that (<ref>) holds.Now that we have the bound (<ref>) with the right-hand side going to zero as v →∞, the formulaϖ(u,v) ≤ ϖ((v),v)e^1/r_–_0∫_(v)^u[|ζ/ν||ζ|](ũ,v) dũ +C∫_(v)^ue^1/r_–_0∫_s^u[|ζ/ν||ζ|](ũ,v) dũ[|ζ/ν||ζ|](s,v) dsand the fact that lim_v→∞ϖ((v),v)=ϖ_0 (recall (<ref>)) imply that for each 0 << _0 there exists U̅_ > 0 such thatϖ(u,v)≤ ϖ_0+/2,provided thatu≤U̅_. Since 1-μ is nonpositive in J^+( A) and 1-μ=(1-μ)(r,ϖ_0)-2(ϖ-ϖ_0)/r, we have(1-μ)(r(u,v),ϖ_0)≤2(ϖ(u,v)-ϖ_0)/r≤/r_–_0.Hence, by inspection of the graph of (1-μ)(r,ϖ_0), there exists _0 such that for 0<≤_0, we have r(u,v)>r_–/2 provided thatu≤U̅_. For 0<u≤ U_:=min{U̅__0,U̅_}, both inequalities (<ref>) hold. A standard bootstrap argument now yields the result, as the sets{ (u,v)∈ J^-(γ)∩ J^+(Γ_) : r(u,v) > r_–}and{ (u,v)∈ J^-(γ)∩ J^+(Γ_) : r(u,v) ≥ r_–/2},coincide, and are therefore both open and closed in the relative topology of the connected set J^-(γ)∩ J^+(Γ_). From the previous proof it is clear that, given >0, we may choose U sufficiently small so that if (u,v)∈ J^-(γ)∩ J^+(Γ_), then1-≤κ(u,v)≤ 1. Now we turn to the behavior of λ and ν over the curve γ. The conclusions of Lemma 6.6 of <cit.> still hold in our case:Suppose that β is given satisfying (<ref>). Letbe the curve parametrized by (<ref>). Let also δ>0. For a choice ofsufficiently close to r_-, and U sufficiently small, there exist constants c and C, such thatfor(u,v)∈, with 0<u≤ U, we have ce^(-2β/1+β-δ)v≤ -λ(u,v)≤ Ce^(-2β/1+β+δ)v andcu^ -1+δ≤ -ν(u,v)≤ Cu^ -1-δ.The proof of Lemma 6.6 of <cit.> goes through with minor modifications. Hence, we point out that formula (126) of <cit.> should be replaced by|θ/λ|(u,v)≤(Ce^-(2 l(s)-δ) v+ Ce^-(2 l(s)/1+β-δ) v)× × e^-∂_r(1-μ)(r_–_0,ϖ_m)β v≤ Ce^-(2 l(s)/1+β+ ∂_r(1-μ)(r_–_0,ϖ_m)β-δ)v.This leads to (127) of <cit.>.Let us also point out that, analogously to (135) and (136) in <cit.>, given δ>0,ce^(-2-δ)v_γ(u)/1+β≤ u ≤ Ce^(-2+δ)v_γ(u)/1+β,for u≤ U sufficiently small.The inequalities (<ref>) highlight the importance of the curve γ in probing the geometry of the region near the Cauchy horizon. These exponential decays, which will be crucial to establish the integrability of λ, and consequently the stability of the radius function at the Cauchy horizon, already exhibit the characteristic blueshift exponent -2k_-, multiplied by the positive parameter β. Observe that these estimates cannot be obtained over level curves of r (corresponding to β=0). § THE REGION J^+(Γ) In this section we treat the region J^+(γ), where the solution departs qualitatively from the Reissner-Nordström solution. Nevertheless the radius function remains bounded away from zero, and approaches r_- as u → 0, implying that the existence of a Cauchy horizon is a stable property.We may apply the arguments in the proof of Lemma 7.1 of <cit.>. We see that the estimates (<ref>) and (<ref>), -λ(u,v)≤ Ce^(-2β/1+β+δ)v and-ν(u,v)≤Cu^ -1-δ,also hold in { r > r_- - }∩ J^+(γ) for >0 sufficiently small. Using the integrability of λ and ν implicit in (<ref>) or (<ref>), as in Section 7 of <cit.>, we can prove the stability of the Cauchy horizon. Given δ>0, there exists U_δ>0 such that r(u,v)>r_–δ for (u,v)∈ J^+(γ) with u≤ U_δ. In particular, P contains [0,U_δ]×[0,∞[.Due to the monotonicity properties of r and ϖ, the limits r(u,∞)=lim_v→∞r(u,v) and ϖ(u,∞)=lim_v→∞ϖ(u,v) are well defined, and lim_u↘0r(u,∞)=r_-.The proofs of Theorem 8.1 and Lemma 8.2 of <cit.> establishEither r( · ,∞)≡ r_- and ϖ( · ,∞)≡ϖ_0, or r(u,∞)<r_- and ϖ(u,∞)>ϖ_0 for all u>0. In the second case ∫_0^∞κ(u,v) dv<∞ and lim inf_v→∞-ν(u,v)>0, for all u>0. We also haveLet u>0. Consider an outgoing null geodesic t↦(u,v(t)) for ( M,g), with g given byg=-Ω^2(u,v) dudv+r^2(u,v) σ_𝕊^2,where Ω^2=-4κν. Then v^-1(∞)<∞, i.e.the affine parameter is finite at the Cauchy horizon. Let u>0. Fix a V>v_λ(u) such that (1-μ)(u,V)<0. As shown in the proof of Corollary 8.3 of <cit.>, there exists a constant c>0 such thatt=v^-1(V)+c∫_V^vΩ^2(u,v̅) dv̅=v^-1(V)-4c∫_V^v(νκ)(u,v̅) dv̅.Integrating (<ref>), for v̅≥ V, we get0<ν(u,v̅)/(1-μ)(u,v̅)≤ν(u,V)/(1-μ)(u,V). Sot ≤v^-1(V)-4cν(u,V)/(1-μ)(u,V)∫_V^vλ(u,v̅) dv̅≤v^-1(V)+4cν(u,V)/(1-μ)(u,V)r_+<∞.§ MASS INFLATION As mentioned in page regime, there exist two distinct regimes, depending on the parameter > 0, which reflect the competition phenomenon between the redshift arising from the evolution equation and the exponential decay of θ_0 along the event horizon: for < 2 the decay of θ_0 is slower, and thus the dominant effect, whereas for > 2 this decay is overwhelmed by the redshift effect. Since mass inflation is more likely for slower decays, it is not surprising that sufficient conditions for its occurrence can be obtained when < 2. §.§ Positivity of θ and ζ In this subsection we prove positivity of θ and ζ over A (for large v). This implies positivity of θ and ζ in J^+( A), which in turn imply ϖ(u,∞)>ϖ_0 for all u>0.Note that if (r,ν,λ,ϖ,θ,ζ,κ) is a solution of the first order system (<ref>)-(<ref>), then(r,ν,λ,ϖ,-θ,-ζ,κ) is also a solution of that system. So, without loss of generality, taking into account (<ref>), we assume thatθ(u,v)≥ ce^- C_2/2v,for (u,v)∈ P_λ.According to (<ref>), for (u,v)∈ R_(U,V) and V≤ṽ≤ v,e^-C_α(v-ṽ)≤ e^-∫_ṽ^v[κ∂_r(1-μ)](u,v̅) dv̅≤e^-c_α(v-ṽ).Integrating (<ref>), we obtain, analogously to (<ref>),ζ/ν(u,v) = ζ/ν(u,V)e^-∫_V^v[κ∂_r(1-μ)](u,ṽ) dṽ -∫_V^vθ/r(u,ṽ)e^-∫_ṽ^v[κ∂_r(1-μ)](u,v̅) dv̅ dṽ.Using (<ref>) and (<ref>) in (<ref>) yields, for (u,v)∈ R_(U,V),ζ/ν(u,v)≤ Ce^-c_α(v-V)-c/r_+∫_V^ve^- C_2/2ṽ e^-C_α(v-ṽ) dṽ.Assuming that >2 we can choose our parameters so that C_2/2<c_α<C_α. Thenζ/ν(u,v)≤ Ce^-c_α(v-V)-ce^- C_2/2v+ ce^-C_α(v-V)e^- C_2/2V.This shows there exists V̅≥ V such thatζ/ν(u_λ(v),v)<0v≥V̅,and soζ(u_λ(v),v)>0v≥V̅.We restrict U to be at most u_λ(V̅). With this choice, (<ref>) and (<ref>) ensure that θ and ζ are positive on A.This implies that θ>0 and ζ>0 in J^+( A): otherwise, there would exist a point (u,v)∈ J^+( A) such that θ(u,v)=0 or ζ(u,v)=0 but θ>0 and ζ>0 in J^-(u,v)∩ J^+( A). Integrating (<ref>) and (<ref>), we would obtain a contradiction.From Corollary 12.3 of <cit.>, it follows that, for 0<u_1<u_2 and v sufficiently large so that (u_1,v) (and hence (u_2,v)) belong to J^+( A),ϖ(u_2,∞)-ϖ(u_1,∞)≥ϖ(u_2,v)-ϖ(u_1,v).The right-hand side is positive because ζ is positive on J^+( A). We conclude that ϖ(u,∞)>ϖ_0 for all u>0. Moreover, Lemma <ref> implies r(u,∞)< r_- for all u>0. §.§ Blow up of the mass at the Cauchy horizon We define Ψ:=k_-/k_+ > 1. The following result establishes sufficient conditions for the occurrence of mass inflation. If <min{ρ,2} then ϖ(u,∞)=∞ for all u>0. We proved in the previous subsection thatϖ(u,∞)>ϖ_0 for all u>0 and so it follows from Lemma <ref> that r(u,∞)<r_- for all u>0. Going through the proof ofTheorem 3.1 in <cit.>, we see that to prove mass inflation it is sufficient to consider Case 3.2, namely it is sufficient to assume thatI(u)=∫_(u)^∞[θ^2/-λ](u,ṽ) dṽsatisfies lim_u↘ 0I(u)=0 and from this derive the contradiction I(u)=∞. This is done by using improved upper bounds for -λ in the region J^+(γ) together with the lower bounds satisfied by θ in this region.The assumption lim_u↘ 0I(u)=0 together with (<ref>) leads to (117) of <cit.>, which states that-λ(u,v)≤ C(u) e^(-2+δ)vin J^+(γ). When <2, we know that the lower bound for θin (<ref>) holds on A and we know that θ and ζ are positive on J^+( A). Hence, the lower bound for θin (<ref>) holds on J^+( A). Using (<ref>) and (<ref>), we are then led to the following lower bound forI(u):I(u) ≥ ∫_(u)^∞[θ^2/-λ](u,ṽ) dṽ≥ C(u)∫_(u)^∞e^-C_2ṽ/e^(-2+δ)ṽ dṽ.We can choose our parameters so that this integral is infinite if (see (<ref>))2/r_+(A-r_+)<2.This inequality is equivalent to <ρ.Therefore ϖ(u,∞)=∞ for all u>0 if s<min{ρ,2}.§.§ Mass inflation or θ/λ unbounded Suppose that <2 and that ϖ( · ,∞) is not identically equal to ∞. Then, from the proof of Theorem 3.2 of <cit.> we know that lim_u↘ 0I(u)=0, that -λ is bounded above by (<ref>) in J^+(γ), and that θ is bounded below by (<ref>). We conclude that, for (u,v)∈ J^+(γ),|θ/λ|(u,v) ≥ e^- C_2/2v/C(u)e^(-2+δ)v= 1/C(u)e^(- 1/r_+(A-r_+) +2-δ)v.This exponent can be made positive if1/r_+(A-r_+)<2,which is equivalent to <2ρ.However, as shown in Appendix A of <cit.>, ρ is necessarily greater than one, and so the last inequality is always satisfied for <2. Therefore we have the following result. If <2 then either ϖ or θ/λ blow up at the Cauchy horizon. § NO MASS INFLATION In this section we will establish sufficient conditions guaranteeing that the renormalized Hawking mass does not blow up at the Cauchy horizon. As might be expected, this is what occurs in the regime > 2, where the decay of the initial data is faster, if the reference Reissner-Nördstrom black hole is sufficiently close to extremality. More surprising is the fact that mass inflation can also be avoided for < 2, although |θ/λ| necessarily blows up. Suppose that ρ satisfies 7ρ/9<l(s) (see (<ref>)), that is,{[ 1<ρ<9/7if >2,;;1<ρ<9/14 if 9/14<s≤ 2. ].Then ϖ(u,∞)<∞ for each 0<u≤ U, provided that U is sufficiently small. Furthermore, lim_u↘ 0ϖ(u,∞)=ϖ_0. Given _1>0, defineD=D__1={(u,v)∈ J^+(γ): u≤ Uand ∫_(u)^v| θ^2/λ|(u,ṽ) dṽ≤_1}.The conclusion follows by proving thatis open in J^+(γ) if _1 and U are sufficiently small. As in the proof of Theorem 4.1 in <cit.>, this is accomplishedby deriving a formula showing that inwe have|θ^2/λ|(u,v)≤ Ce^-Δ vfor 0<u≤ U.From (<ref>) we get|θ/λ|(u_γ(v),v) ≤ Ce^-(2 l(s)/1+β- 2β-δ)v,and recall that in (<ref>) we obtainedce^(-2β/1+β+δ)v≤ -λ(u_γ(v),v)≤ Ce^(-2β/1+β-δ)v. Combining (<ref>) with (<ref>) yields |θ|((v),v) ≤Ce^-(2 l(s)/1+β-2β^2/1+β-δ)v.Moreover, according to (<ref>),1+β/2+δln(c/u) ≤(u)≤1+β/2-δln(C/u).Therefore, the proof of Lemma 4.2 of <cit.> goes through if one replaces s+1 by l(s). For example, for (u,v)∈ D, q=1/3 and β=1/3+ withsufficiently small, we have|θ|(u,v) ≤Ce^-2(k_+l(s)/1+β- k_-β^2/1+β-δ)v +Cu^l(s)-Ψ(β^2+q)-δe^-2(k_-(β+q)/1+β-δ)v.An estimate of the form (<ref>) then follows if we assume l(s)>Ψ(β^2+β+q)>7ρ/9.§ BREAKDOWN OF THE CHRISTODOULOU-CHRUŚCIEL CRITERION In this section we prove that when there is no mass inflation the solution can be extended across the Cauchy horizon with enough regularity to violate the Christodoulou-Chruściel version of strong cosmic censorship. The extension is constructed by first changing v to a new coordinate with finite range, essentially the distance to the apparent horizon as measured by the radius function along u=U. This is the most natural choice to bring the Cauchy horizon to a finite coordinate value. If there is no mass inflation, all functions except θ are then shown to extend continuously to any subset of the Cauchy horizon away from the event horizon. We then change to yet another coordinate system, where we are able to prove that the Christoffel symbols are locally square integrable. We regard the (u,v) plane, the domain of our first order system, as a C^2 manifold. We define a new null coordinate along the outgoing direction byṽ=r(U,V_λ)-r(U,v),where V_λ=max{v:λ(U,v)=0}. Equality (<ref>) and the assumption that f̂ is continuous and integrable imply that ϖ̂ is a continuously differentiable function of r. Similarly, (<ref>) and (<ref>) guarantee that κ̂ andλ̂ are also continuously differentiable functions of r. Then, equation (<ref>) shows that the coordinate v over the event horizon is a continuously differentiable function of r with nonvanishing derivative, so that, by the Inverse Function Theorem, r is a continuously differentiable function of v over the event horizon. We conclude that κ, ϖ and λ are continuously differentiable functions of v over the event horizon (in particular, r is a C^2 function of v along the event horizon). In addition, ν_0 is continuously differentiable. Therefore, hypothesis (h4) in Section 6 of <cit.> issatisfied, and so, by Lemma 6.1 in <cit.>, the function r is C^2. Since we have λ(U,v)<0 for v>V_λ, equation (<ref>) allows us to define anadmissible coordinate change [0,U]×]V_λ,∞[→ [0,U]×]0,Ṽ[, where Ṽ=r(U,V_λ)-r(U,∞). We write a tilde overa function to indicate that we are using these new coordinates.Assume that the hypotheses of Theorem <ref> hold. Let 0<δ<U. As in Proposition 5.2 of <cit.>, we can extend r̃ and ϖ̃ to continuous functions on [δ,U]×[0,Ṽ].Using (<ref>) and the bound (<ref>) (which holds in [δ,U]×[v,∞[, for v>v_γ(δ)), one proves that ν̃/1-μ̃( · ,ṽ) converges uniformly for u∈[δ,U] when ṽ→Ṽ. As in Step 2 of the proof of Proposition 5.2 of <cit.>, this implies that ν̃/1-μ̃ admits a continuous extension to the rectangle [δ,U]×[0, Ṽ] (for arbitrary δ). Notice that the function ν̃/1-μ̃( · , Ṽ) is strictly positive on ]0,U]. When r̃(u,Ṽ)=r_- and ϖ̃(u,Ṽ)=ϖ_0, we have (1-μ̃)(u,Ṽ)=0;therefore, ν̃(u,Ṽ) exists and is zero. On the other hand, when r̃(u,Ṽ)<r_-, Lemma <ref> implies that lim inf_ṽ→Ṽ-ν̃(u,ṽ)>0, and so (1-μ̃)(u,Ṽ)<0. We conclude that in this case ν̃(u,Ṽ) exists and is negative. The function λ̃:=∂_ṽ r satisfiesλ̃(U,ṽ)≡ -1.The integration of (<ref>) leads toλ̃(u,ṽ)=λ̃(U,ṽ)e^-∫_u^U[ν̃/1-μ̃∂_r̃(1-μ̃)](ũ,ṽ) dũ.Since λ̃(U,ṽ) extends to [0,Ṽ], and ν̃/1-μ̃ and ∂_r̃(1-μ̃) extend to[δ,U]×[0,Ṽ], λ̃ extends as a continuous function to [δ,U]×[0,Ṽ].Moreover, λ̃( · ,Ṽ) is strictly negative on ]0,U].Taking into account the behavior of λ̃ and ν̃/1-μ̃ on ]0,U]×{Ṽ}, we see that the coefficient of the metric Ω̃^2=-4κ̃ν̃=-4λ̃ν̃/1-μ̃is strictly positive on ]0,U]×{Ṽ} and is continuous on [δ,U]×[0,Ṽ], for any 0<δ<U.Equation (<ref>) can be written as ∂_ṽν̃=- Ω̃^2/4∂_r̃(1-μ̃).Therefore the convergence of ν̃( · ,ṽ) to ν̃( · ,Ṽ) is uniform for u ∈ [δ,U], and so ν̃ is continuous on[δ,U]×[0,Ṽ], for any 0<δ<U.Integrating (<ref>), ζ̃(u,Ṽ)=ζ̃(u,ṽ)-∫_ṽ^Ṽθ̃ν̃/r̃(u,v̅) dv̅.We use ∫_ṽ^Ṽ|θ̃|(u,v̅) dv̅=∫_f^-1(ṽ)^∞|θ|(u,v̅) dv̅→ 0 as ṽ↗Ṽ (by (<ref>)). Note that the last convergence is uniform for u∈[δ,U].We may define ζ̃( · ,Ṽ) as the uniform limit of ζ̃( · ,ṽ) when ṽ↗Ṽ. Therefore we have proved the following result.Assume that the hypotheses of Theorem <ref> hold. Then, for all 0<δ<U, the functions r̃, ν̃, λ̃, ϖ̃, ζ̃ and κ̃ (but not necessarily θ̃) admit continuous extensions to the closed rectangle [δ,U]×[0,Ṽ]. Moreover, (1-μ̃)(u,Ṽ) is negative for u>0, unless r( · ,∞)≡ r_-.Since r̃ and Ω̃^2 are strictly positive on ]0,U]×{Ṽ}, it is immediate to construct continuous extensions of the metric beyond the Cauchy horizon, as was done in the proof of Corollary 5.11 of <cit.>.To construct extensions which also have locally square integrable Christoffel symbols it is useful to consider a new v coordinate determined by the conditionΩ^2(U, v)≡ 1 .One has dv/d v=Ω^2(U,v)and consequently dv/d ṽ=-Ω^2(U,v)/λ(U,v)=4ν/1-μ(U,v)=4ν̃/1-μ̃(U,ṽ)=-Ω̃^2(U,ṽ)/λ̃(U,ṽ) .We conclude that these coordinate systems are C^1-compatible up to and including ṽ=Ṽ. In particular this shows that V=(Ṽ) is finite and that we can construct continuous extensions of the corresponding metric components beyond the Cauchy horizon v= V. Note that the choice of coordinates provided by κ̂(U,v̂)≡ 1, used in <cit.> for an analogous extension, is not regular in the case when r̃(u,Ṽ)≡ r_-, since dv̂/d ṽ=-1/1-μ diverges as we approach the Cauchy horizon.Sinceν̃/1-μ̃≡ -Ω̃^2/4λ̃extends continuously to the Cauchy horizon, where it is strictly positive, the only potentially problematic Christoffel symbols are Γ^u_uu=∂_u logΩ^2andΓ^_=∂_logΩ^2.Now, in view of the boundedness of the quantities r, ν, λ, ϖ, ζ and κ (but not necessarily θ) guaranteed by Theorem <ref> and the C^1-compatibility of the two coordinate systems, Einstein's equation (<ref>) gives us, for < V, ∂_u∂_logΩ^2 = O(1) (θ +1) .Since our choice of coordinates gives logΩ^2 (U,)≡ 0, integrating the previous equation first in u and then in , while applying Hölder's inequality in between, gives∫_ v_0^ V(Γ^_)^2(u,) d≤C (1+∫_u^U ∫_ v_0^ Vθ^2(u̅,) ddu̅) . The proof that leads to inequality (<ref>) also shows that | θ^2(u,v)/λ(U,v)|≤ C e^-Δ v ,since it uses an upper bound for |θ| and a lower bound for |λ|, both of which are uniform in u. Therefore, ∫_ v_0^ Vθ^2(u,) d = ∫_v_0^∞θ^2(u,v)1/Ω^2(U,v) dv ≤ C∫_v_0^∞θ^2(u,v)1/-λ(U,v) dv≤ C ,and so∫_ v_0^ V(Γ^_)^2(u,) d≤ C. Note that smoothness provides |∂_ulogΩ^2(u,_0)|≤ C for fixed _0. Then, integrating (<ref>) in v leads to|Γ^u_uu|(u,) ≤ C(1+∫__0^ |θ|(u,) d)≤ C ,again by Hölder's inequality.Therefore we have the following result.Let M_δ be the preimage of [δ,U]×[0,V] by the null coordinate functions (u, v). 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"authors": [
"João L. Costa",
"Pedro M. Girão",
"José Natário",
"Jorge Drumond Silva"
],
"categories": [
"gr-qc",
"math-ph",
"math.AP",
"math.MP",
"Primary 83C05, Secondary 35Q76, 83C22, 83C57, 83C75"
],
"primary_category": "gr-qc",
"published": "20170727180110",
"title": "On the occurrence of mass inflation for the Einstein-Maxwell-scalar field system with a cosmological constant and an exponential Price law"
} |
Kiepenheuer-Institut für Sonnenphysik, 79104 Freiburg, Germany Kiepenheuer-Institut für Sonnenphysik, 79104 Freiburg, GermanyNew Mexico State University, Las Cruces, NM 88001, USANational Solar Observatory, Tucson, AZ 85719, USAThe solar meridional flow is a crucial ingredient in modern dynamo theory. Seismic estimates of this flow have, however, been contradictory in deeper layers below about 0.9 R_. Results from time-distance helioseismology have so far been obtained using the ray approximation. Here, we perform inversions using the Born approximation. The initial result is similar to the result previously obtained by <cit.> using ray kernels while using the same set of GONG data and the SOLA inversion technique. However, we show that the assumption of uncorrelated measurements used in earlier studies may lead to inversion errors being underestimated by a factor of about two to four. In a second step, refined inversions are performed using the full covariance matrix and a regularization for cross-talk. As the results are found to depend on the threshold used in the singular value decomposition, they were obtained for a medium threshold (10^-7-10^-5, about 50% of the values used) and a threshold lower by a factor of 10 (about 70% of the values used).The result obtained with the medium threshold is again similar to the original, with less latitudinal variation. However, using the lower threshold, the inverted flow in the southern hemisphere shows two or three cells stacked radially depending on the associated radial flows. Both the single-cell and the multi-cell profiles are consistent with the measured travel times. All our results confirm a shallow return flow at about 0.9 R_. § INTRODUCTIONInferring the structure of the meridional flow in the deep solar interior has attracted considerable attention in recent years. While a conclusion on the flows in near-surface regions seems to have been reached <cit.>, the most recent measurements of the deep meridional flow <cit.> give seemingly contradictory results in deeper layers below about 0.9 R_, favoring a single-cell or multi-cell picture of the flow as summarized in the introduction of <cit.>.Possible reasons for this discrepancy may include systematic effects like a center-to-limb effect in time-distance helioseismology <cit.>, perturbation of solar mode eigenfunctions by convection <cit.>, systematic effects introduced by magnetic fields <cit.>, B- or P-angle variations <cit.>, as well as differences in the instruments used and the time period considered.In addition, there are several efforts underway to develop or validate new methods for inferring the meridional flow, in particular using local helioseismic techniques <cit.>.As inversions for the meridional flow with time-distance helioseismology <cit.> have so far been modeled using the rather classical ray approximation <cit.>, the Born approximation has been brought forward very recently as an alternative <cit.>. Instead of assuming that travel times of acoustic waves can be perturbed by a flow along a ray path as in the ray approximation, the Born approximation <cit.> assumes that the whole wave field is scattered by the flow. Therefore, the travel time of a wave packet can be perturbed by a flow distant from the ray path. The Born approximation has been well tested and validated in Cartesian geometry for inferring small-scale flows <cit.>. As it is generally thought of as a more accurate model of the physics in the solar interior <cit.>, it is a method worth exploring for inferring large-scale flows such as the meridional flow.Born approximation sensitivity functions (kernels) have very recently been validated in time-distance helioseismology of the meridional flow <cit.>. In this study, we will perform inversions for the deep meridional flow with these spherical Born kernels. As was shown by <cit.>, phase-speed filtered measurements seem particularly useful for this endeavor. We will therefore use the phase-speed filtered travel-time measurements obtained by <cit.> in this work. These measurements have been inverted for the meridional flow by <cit.> using the SOLA method <cit.>.The first objective of this work is to answer the question whether the inversion results inferred by <cit.> using ray kernels can be confirmed using Born kernels or whether a different conclusion may be reached. Our second objective is to study in detail different sources of systematic errors in the inversion process, such as the error propagation and the cross-talk of the radial flow into the inversion for the horizontal flow component. For this work, we will employ an analytic formula for the covariance of travel-time measurements (, ). We will also use different strategies for analyzing the cross-talk and other systematics in SOLA inversions using Born kernels that were also employed by various authors in Cartesian inversions for small-scale flows <cit.>.The paper is organized as follows. Section <ref> provides a short introduction to the data set and the SOLA inversion technique used. Section <ref> gives a summary of the computation of spherical Born kernels, which are used to model the travel-time measurements in this work. An analytic formula from the literature is used in Section <ref> to obtain a model for the covariance matrix of the measurements. A comparison of an initial inversion for the meridional flow using Born kernels to the result obtained by <cit.> is presented in Section <ref>. This inversion is performed under the assumption of uncorrelated measurements (i.e., a diagonal covariance matrix) and without including a regularization for cross-talk as it was done in previous inversions for the meridional flow using time-distance helioseismology <cit.>. In this section, we also provide a detailed study on the error propagation as well as the cross-talk between flow components. This information is used in Section <ref> to perform refined inversions for the meridional flow that include the full covariance matrix and a regularization for cross-talk. The results of this study are discussed in Section <ref> and conclusions are presented in Section <ref>.§ DATA AND INVERSION TECHNIQUEIn this paper, we use the travel-time measurements obtained and described by <cit.>. The travel times were obtained using the GONG instrument and the data of 652 days between 2004 and 2012, where the duty cycle was high and the B-angle not too large. This period includes the declining phase of cycle 23 and the rising phase of cycle 24. An average sunspot number of 36 for the days used <cit.> indicates that the data are taken from times of moderate to low activity. North minus south (N – S) point-to-arc travel times were obtained for a total of 72 travel distances (Δ) and 384 latitude bins (λ). In addition, east minus west (E – W) travel times were measured in order to correct for a systematic center-to-limb effect detected by earlier studies (e.g., , ). The N – S travel times are corrected for this systematic effect by subtracting the E – W measurements.The travel times were obtained using a Gabor fit <cit.>. <cit.> inverted this set of travel times for the horizontal component of the meridional flow using a standard SOLA technique <cit.>. This inversion technique will be the starting point of the work performed in this study. Our goal is to find the meridional flow with horizontal component v_θ(r,θ) and radial component v_r(r,θ) that satisfies for all travel-time measurements δτ_iδτ_i =∫_0^R_∫_0^π ∑_k=r,θ K_k^i(r,θ)v_k(r,θ) r sin(θ) 𝕀θ 𝕀 r +ϵ_i,where K_k^i is the travel-time sensitivity function of the measurement i=(Δ,λ) to a flow in the direction k∈{r,θ}, and ϵ_i are measurement errors. Two-dimensional integrals are evaluated over the whole radial and latitudinal domains throughout this work. The meridional flow is sought at a series of target locations (r_T,θ_T) as a linear combination of the measured travel times, e.g., for the horizontal flow component,v_θ^inv(r_T,θ_T) = ∑_i w_i(r_T,θ_T)δτ_i.The aim is to construct averaging kernels _θ,_θ(r,θ;r_T,θ_T) = ∑_i K_θ^i(r,θ)w_i(r_T,θ_T),which are sufficiently near a Gaussian target function T(r,θ;r_T,θ_T) with a certain radial and horizontal full width at half maximum (_r,_θ) centered at the target location. This is achieved by minimizing the misfit given by= ∬| _θ - T | ^2 r sin(θ) 𝕀θ 𝕀 r.The minimization is performed with an additional regularization parameter μ that balances misfit and errors,^2 =∑_i,j w_i(r_T,θ_T) Λ_i^j w_j(r_T,θ_T),where the covariance matrix Λ is given byΛ_i^j = [ ϵ_i , ϵ_j].The final cost functionχ(w_i (r_T,θ_T);μ)=+ μ ^2is then minimized at every target location subject to the constraint∬_θ(r,θ;r_T,θ_T) r sin(θ) 𝕀θ 𝕀 r = 1.The resulting weights can be obtained using a matrix inversion <cit.>. § FORWARD-MODELING USING BORN KERNELSThe forward-modeling of the travel-time measurements in <cit.> was performed using ray kernels <cit.>. Instead, we will use the Born approximation <cit.> to model the effect of the flow on the travel times in this work.While the ray approximation assumes that the acoustic waves in the solar interior are sensitive to a flow field only along a certain ray path, the Born approximation models the scattering of the full wave field due to advection in first order <cit.>. The method used in this work for computing Born kernels was developed by <cit.> and further refined by <cit.>. As the computation of the sensitivity kernels depends on an accurate match between model and data power spectra, the following free parameters in the model were adjusted <cit.>.For harmonic degree l<100, mode frequencies and damping rates from the GONG ftp site[<ftp://gong2.nso.edu/TSERIES/v1f/>] were used, and for l≥ 100, they were provided by Sylvain Korzennik <cit.>. For small harmonic degrees, the fitted mode widths are much smaller than the frequency resolution of a day-long time series. Therefore, small mode widths were increased to a minimum value 40% larger than the resolution of a day-long time series. This value was empirically found to produce a good match between data and kernel power spectra. As in <cit.>, the model power spectrum was further corrected by an l-dependent factor, which may be seen as an optical transfer function correcting instrumental effects; see <cit.> and <cit.>. In addition, the source correlation time, a free parameter that models the sources in the model, is adjusted to obtain a match between the mean frequency of the data and model power spectra. Finally, the phase-speed filters used in the data analysis procedure of <cit.> were applied to the model power spectrum.As a result, zero-order model power spectra and cross-covariances agree well with those obtained from GONG data; see Figure <ref>. Only leakage effects visible in the GONG power spectrum at low harmonic degree (top-left panel) are not included in our model. However, no similar effect is visible in the cross-covariances (top middle panel), which are used to model the travel-time shift.Inspection of kernel plots (right column in Figure <ref>) also reveals similar results to <cit.> with some changes introduced by differences in the filters and data power spectra.For each of the 72 travel distances, one 3D kernel was computed. This kernel was then reprojected to the different latitudes and rotated to obtain the N – S point-to-arc geometry with 30 wide arcs used in the data analysis procedure. After integrating the kernels along the azimuthal domain, they were smoothed and rebinned with respect to distance in a similar way to that in the data analysis with a final number of 45 distances. Finally, both the travel times and the kernels were further rebinned by a factor of two to obtain 192 latitude bins, which are used in the inversions performed in this work. § FULL COVARIANCE MATRIXIn previous inversions for the deep meridional flow using time-distance helioseismology <cit.>, the travel-time measurements were assumed to be uncorrelated, which is equivalent to a diagonal covariance matrix. In order to study the effect of this assumption, we will use both a diagonal and the full covariance matrix in this work.Analytic formulas for computing the covariance of travel-time measurements have been proposed by <cit.> and <cit.>. They were obtained on the basis of an empirical model for the noise in the power spectrum, which assumes uncorrelated noise for different frequency bins, and were shown to be in good agreement with data and Monte Carlo simulations.For the purpose of this work, we compute the covariance matrix using formula (B.6) or equivalently (13) from <cit.>. As suggested in <cit.>, for longer time series (one day in our case), only the first term in the formula has to be taken into account. As an ingredient, only the zero-order or mean cross-covariances for all combinations of measurements are needed. In this work, we use the mean power spectrum of the data to obtain mean cross-covariances for every possible distance needed in the formula. For computational reasons, we compute the covariance for point-to-point measurements only. The resulting diagonal entries of the covariance matrix should in principle be equal to the squared errors of the travel-time measurements used in this study. For two reasons this is not the case. Firstly, the analytic formula for the diagonal entries only depends on the travel distance and not on the latitude, in contrast to the errors of the measurements, which increase toward the limb. Second, the analytic values have a different scale compared to the squared errors as the measurements have been further averaged, e.g. over a period of 652 days. This is taken into account by renormalizing the result from the analytic formula usingΛ̃_i^j =Λ_i^j σ_iσ_j /√(Λ_i^i Λ_j^j),where σ_i is the error of measurement i.A cut through the covariance matrix obtained is displayed in Figure <ref>, which is in general agreement with the covariance matrix obtained by <cit.> from unfiltered data. § HORIZONTAL SOLA INVERSION: A FIRST COMPARISON The aim of this section is to perform a first comparison of an inversion for the meridional flow using spherical Born kernels to an earlier inversion done with ray kernels. For this purpose, the inversion method is kept as close as possible to the one applied in <cit.>. Specifically, we only invert for the horizontal component of the flow, we assume the covariance matrix to be diagonal, and we do not take the cross-talk from the radial flow component into the horizontal flow component into account when obtaining the inversion weights.Our intention is to first answer the question as to whether and how the inversion result is affected by simply exchanging ray kernels with Born kernels. Second, we will compare the magnitude of the errors propagated with a diagonal and with the full covariance matrix, and we will analyze the magnitude of the cross-talk. §.§ Choice of Regularization Parameters The inversion procedure involves a number of free parameters, such as the full widths of the Gaussian target functions and the parameter μ, which may vary with target location. We therefore first performed a number of test computations in order to get an overview over the parameter space involved. These tests showed that a scaling of the full widths of the target functions with the sound speed, as proposed in <cit.>, is a reasonable choice. Near the surface (r_T≥0.95 R_), this scaling would imply very small target widths. We therefore set minimum target widths of _r,min = 0.03 R_ and _θ,min = 5. This limitation is justified by the minimum scales of the kernels involved in this study as the minimum travel distance is around three degrees and the kernels extend a little farther than that. For the scaling of the full widths, we find values at the bottom of the convection zone at r_T=0.7 R_ of _r = 0.09 R_ and _θ = 20. At first sight, these values may seem very large, especially when compared with the averaging kernels obtained by <cit.> using the ray approximation, which are much more localized. However, taking a closer look at the spatial scales involved in the kernels shown in Figure <ref>, which are much wider than ray kernels, the values obtained are plausible. Our test computations showed that, if the widths of the target functions are further decreased significantly, it is hard to obtain reasonable values for errors and misfit. In addition, every inverse matrix in the inversion problem can be computed using different thresholds for the singular values (SVs). In practice, the inversion results do not show a large dependence on the choice of threshold in this inversion, where we use a diagonal covariance matrix.In order to obtain an optimal choice for the remaining inversion parameter μ, which controls the relation between errors and misfit in the inversion, we first do a number of inversions for a series of values for μ on a coarse grid of target locations. Finally, we choose an optimal value for μ for every target depth, using maximum threshold values for the misfit and the errors of the inverted flows. In practice, we find that it is often hard to obtain a reasonable inversion error without compromising the fit of the averaging kernel to the target kernel. We thus chose a maximum error of 1, similar to <cit.>, and a maximum misfit of 0.2, when the misfit is normalized as _norm= /∬T ^2 r sin(θ) 𝕀θ 𝕀 r.We note here that it is possible to achieve much better values for the misfit, up to almost perfect agreement between the averaging and target kernels, but at an unacceptable expense in the errors. A maximum misfit of 0.2 is found to be about the largest possible value for obtaining an acceptable match between the averaging and target kernels. At the surface, this condition is relaxed by a factor of two, as we find that most of the misfit comes from locations just being a bit farther off, or a difficulty of the averaging kernel to achieve a Gaussian form. If no value for μ is found for the maximum error and misfit given, these conditions are relaxed step by step until possible inversion parameters are found. If several possible points are found, an optimal choice is made using the so-called L-curves <cit.>. In the optimizing procedure, we consider a small number of target latitudes within 20 from the equator.Using the coarse grid of target locations, optimal inversion parameters are chosen for every target depth of the spatial grid. For the final inversion, a finer target grid is defined, and for each location, the inversion parameters are interpolated between the values from the preparatory inversion.§.§ Inversion Results Inversion results are presented in Figure <ref>, where the inverted flow (left panel), the inversion errors (middle panel), and the misfit (right panel) are shown. We first note that the resulting meridional flow is similar to the inversion result obtained by <cit.> using ray kernels, at least qualitatively. As this inversion result and the result obtained by <cit.> are obtained using a diagonal covariance matrix, it is questionable whether the error estimate is correct. Using the inversion weights obtained in this inversion and the full covariance matrix computed as in Section <ref>, it is possible to give more accurate estimates of the errors of the inverted flow and to estimate the impact of not taking the full covariance matrix into account. In Figure <ref>, inversion errors using the diagonal covariance matrix (lines near the bottom) can be compared to errors obtained using the full covariance matrix (lines near the top) as a function of target depth for a series of target latitudes. Both errors are obtained using the same inversion weights. It can be seen that the errors from the diagonal covariance underestimate the errors from the full covariance by a factor of about two to four. For this comparison, the values on the diagonal of the two covariance matrices are identical prior to the final rebinning of the measurements by a factor of two in latitude; see Section <ref>. If they were to be set equal after this final rebinning, the errors from the diagonal covariance would increase on average by about 35 % and they would still underestimate the errors from the full covariance by a factor of 1.5 - 3. Furthermore, although the cross-talk was not considered in the inversion procedure, it is possible to compute cross-talk averaging kernels, i.e., kernels for the influence of the radial flow on the inversion for horizontal flow,_r(r,θ;r_T,θ_T) = ∑_i K_r^i(r,θ)w_i(r_T,θ_T).For a series of target depths, we show the averaging kernel for the horizontal flow (right subpanel) and the cross-talk kernel for the radial into the horizontal component (left subpanel) in Figure <ref>. We note that for each target depth plotted, both kernels are shown using the maximum values of the averaging kernel as a scale, with the maximum values of the cross-talk averaging kernels being 120, 100, 70, and 14 times larger for the given target depths r_T of 0.7, 0.8, 0.9, and 0.98 R_. These values are much larger than the ones obtained by <cit.> in an inversion for subsurface flows in Cartesian geometry. We therefore study the nature and impact of the cross-talk in more detail, see Figure <ref>, where the normalized cross-talk,_norm= ∬_r^2 r sin(θ) 𝕀θ 𝕀 r/∬T ^2 r sin(θ) 𝕀θ 𝕀 ris displayed as a function of depth, for a number of latitude bins. For example, a normalized cross-talk of 100 means that radial flows may contribute to the inversion for horizontal flow even if they are about √(100)=10 times smaller than the actual horizontal flow. In principle, even a small magnitude radial flow may thus leak into the inversion for the horizontal flow in such a case. In order to answer the question whether such values for the cross-talk introduce a large contribution of the radial flow to the inverted horizontal flow, we convolved the cross-talk averaging kernels with two choices of exemplary radial flow fields v_m^orig(m=r),v_θ ^conv(m)(r_T,θ_T) =∬_m(r,θ;r_T,θ_T) v_m^orig(r,θ) r sinθ 𝕀θ 𝕀 r.The resulting contribution of the radial flow signal to the inversion for v_θ is displayed in Figure <ref>. For the left panel, we chose v_r^orig to be the radial flow component of the single-cell meridional flow profile employed in the simulation of <cit.>, divided by a factor of 36 in order to mimic a realistic magnitude of a solar-type flow. For the right panel, we chose the radial flow component that was obtained by <cit.> from the inverted horizontal flow by applying mass conservation. In both cases, the contribution of the radial flow component to the inversion for v_θ has maximal values of around 2, with typical values of about 1 - 1.5. We may thus expect small contributions from the radial flow component, but it is likely that they do not alter the inversion result at a large scale. However, one should be aware of this contribution.§ HORIZONTAL SOLA INVERSION INCLUDING FULL COVARIANCE AND CROSS-TALKRefined inversions are now performed by including the full covariance matrix and a regularization term for cross-talk into the inversion problem. A similar problem has been studied for inversions for 3Dflows near the surface in Cartesian geometry by, e.g., <cit.> and <cit.>.§.§ The Full Inverse Problem In the full SOLA inverse problem, we invert for a flow component k∈{r,θ},v_k^inv(r_T,θ_T) = ∑_i w_i^k(r_T,θ_T)δτ_i,by trying to match the averaging kernels,_m^k(r,θ;r_T,θ_T) = ∑_i K_m^i(r,θ)w_i^k(r_T,θ_T),to a target kernel T_k^m = δ_m,k T, where T is the same target function from above. When inverting for a flow component k, we thus also intend to match the averaging kernel of the opposite component m≠ k to zero. Therefore, a regularization for the cross-talk of the flow component m≠ k into k,_k = ∬(_m≠ k^k)^2r sin(θ) 𝕀θ 𝕀 r,is added to the inversion problem outlined in Section <ref> (see, e.g, <cit.> for the equivalent formulation in Cartesian geometry). The cost function thus becomesχ_k(w_i^k;μ)=+ μ ^2 + ν _kwhich is to be minimized subject to the constraints∬_m^k r sin(θ) 𝕀θ 𝕀 r = δ_km.As in the inversion without cross-talk, inversion parameters are chosen in a preparatory inversion using a coarser spatial grid for the target locations. The FWHMs of the target functions are the same. In the following, we invert for the horizontal component (k=θ) and show the inversion results for parameters obtained using different strategies. In all cases, we first set an upper limit to the misfit of 0.2, which is the same as in the inversion without cross-talk in the previous section.As the inversion results presented here are found to depend on the threshold used in the singular value decomposition (SVD), we show the SVs of a matrix used in the inversion at an example target depth of r_T=0.8 R_ in the left panel of Figure <ref>. As the distribution of SVs depends on target depth, the thresholds at each target depth are chosen relative to the point of highest negative curvature, which is found by fitting a high-order polynomial to the curve. Therefore, the thresholds and the fraction of SVs used in the matrix inversion also depend on target depth; see middle and right panel of Figure <ref>. §.§ Results: Controlling Misfit and ErrorsIn addition to the misfit, we aim to keep the errors under control in the first inversion. After choosing a threshold for the SVs, we also set a maximum threshold of 1.0 m s^-1 for the inversion errors as in Section <ref>. We then search the parameter space for the best cross-talk available. As a first case, we consider an SV threshold chosen to be just above the end of the rather flat plateau of SVs in the left panel of Figure <ref> (indicated in Figure <ref> by “medium SV”). Here, about half of the SVs are used in the matrix inversion.In this case, we obtain an inverted flow profile that is very similar to the one obtained by <cit.>; see the left panel in Figure <ref>, subsequently termed “case 1”. Compared to the initial result shown in Figure <ref>, however, it shows less fluctuations as a function of latitude in each hemisphere. This difference is found to be due to the use of the full covariance matrix. In a second case, we choose an SV threshold smaller by a factor of 10 just at the edge of the plateau of SVs in the left panel of Figure <ref> indicated by “low SV”. Here, about 70 % of SVs are used. In this case, the inverted flow profile shows some noteworthy features; see case 2 in Figure <ref>.Most notably, there is an additional extended poleward flow branch visible at about 0.8 R_ in the southern hemisphere. This flow component gives rise to a multi-cell structure of the flow stacked radially. Depending on the associated radial flow, the inverted meridional flow corresponds to a double-cell profile <cit.> or to three flow cells stacked radially <cit.>. At the same time, no additional flow cells are visible in the northern hemisphere. At the central location of the additional poleward flow structure in the southern hemisphere at about 0.8 R_ and 30 latitude, a patch with equatorward flow is also visible in the result for case 1, in the initial result in Figure <ref>, and in the result obtained by <cit.> using ray kernels.Inversion errors using the full and a diagonal covariance matrix are shown in Figure <ref>. With the full covariance matrix, we now obtain errors that are generally smaller than when using a diagonal covariance matrix by about a factor of 1.5 in case 2, with a few exceptions. In the case of the medium SV threshold (case 1), however, the errors from the full covariance matrix are on average about twice as large as the ones from the diagonal covariance matrix, similar to the initial inversion result (see Figure <ref>). Note that the errors from the diagonal covariance matrix increase on average by about 35% if the diagonal values of the covariance matrices are assumed to be identical after rebinning in latitude as in Section <ref>.Furthermore, one can see that the cross-talk decreased (see Figure <ref>, case 1 and 2), by about a factor of 3 especially at larger target depths compared to the cross-talk for the initial result; see Figure <ref>. As a consequence, the convolution of the two examples of radial flow profiles discussed in Section <ref> with the radial averaging kernels decreases by over 30% to below 1.4.Concerning the cross-talk, it thus seems that the horizontal inversion without regularization for cross-talk already achieved results that are near optimum, or in other words, that the cross-talk is not a major concern for the horizontal inversion. A similar conclusion was reached by <cit.> in the case of a Cartesian inversion for the horizontal component of a 3D flow.For misfit and errors, the distribution of values over the whole target grid is similar to that presented in Figure <ref> for all results presented in this section.§.§ Results: Controlling Misfit and Cross-talkStill imposing the condition of a maximum misfit of 0.2 and the low SV threshold (same as in case 2 for a given matrix), we now take a slightly different strategy for choosing the trade-off parameters. Motivated by the analysis shown in Figure <ref>, we first choose a maximum value of 100 for the cross-talk (case 3 in the following). This means that the cross-talk averaging kernel _r^θ has a mean magnitude of about 10 times larger than the averaging kernel _θ^θ. If this condition cannot be met, both maximum misfit and cross-talk are increased step by step until a condition is found that can be met in the parameter space. We finally search for the regularization parameter that gives minimal errors.The inversion results for this case are also displayed in Figure <ref>. In the whole region considered, the inverted flow is very similar to the one obtained in case 2 above. It is noteworthy, however, that the multi-cell structure in the southern hemisphere is even more pronounced in case 3 compared to case 2.This result was obtained although the dependence of the errors and the cross-talk as a function of depth changed only slightly from case 2 to case 3; see Figures <ref> and <ref>.In the near-surface regions, we are able to obtain slightly smaller error bars that increase with depth, peaking at about 3 at the bottom of the convection zone. The cross-talk, on the other hand, decreased slightly in regions below 0.8 R_, and it increased slightly around 0.9 R_. As a result, the magnitude of the cross-talk averaging kernels changed accordingly, which is visible in the right panels of Figure <ref> with a smaller saturated region at r_T=0.9 R_⊙ in case 3 compared to case 1. We note that to obtain the third result presented here, we allowed the misfit to increase near the bottom of the convection zone by about 20%. We observed that only a slight decrease of the misfit would have increased the errors or the cross-talk by a large amount there. This increase in the misfit is not expected to largely alter the quality of the inversion result compared with the one obtained in Section <ref>, as this change is barely visible in the averaging kernels (horizontal kernels in the left panels of Figure <ref>).§ DISCUSSIONThe possible existence of a multi-cell meridional flow in the southern hemisphere is further underpinned by analyzing the convolution of the inverted flows v_θ^inv from Figure <ref> with the horizontal averaging kernels, that is, Equation (<ref>) with m=θ. The resulting convolved flow v_θ^conv(θ) is displayed in Figure <ref>. It gives an impression as to how the inverted flow would look like if there were no noise and the background flow would be equal to our inversion result v_θ^inv. As can be seen in Figure <ref>, the resulting flow profile would be washed out spatially given the large widths of the averaging kernels in all three cases. Locations with opposite signs near each other are especially no longer seen as clearly. Turning the argument around, it is possible that the locations with a sign change in the inversion result v_θ^inv could be even more pronounced in real solar flow.In case 3, however, the convolved flow looks qualitatively very similar to the inverted flow profile, just with a somewhat lower amplitude. This may lead us to the conclusion that this flow profile is quite a robust result and that the original flow may be similar, just with a somewhat higher amplitude.The match of the inverted flow to the data can be observed in Figures <ref> and <ref>, where forward travel times, i.e., the inverted flow convolved with the Born kernels, are compared to the measured travel times. The forward travel times are predominantly consistent with the measured ones within the measurement errors; see Figure <ref> for cases 1 and 3. We again notice for case 3 that the multi-cell circulation in fact does quite a nice job in reproducing the measured travel times in the southern hemisphere and that a slightly larger magnitude of the flow may explain the measured travel times even better. However, not all features visible in the measured travel times are recovered in the inverted flows, and the errors do not permit us to make a final decision in favor of a single or multiple cell profile of the meridional flow. On the other hand, given the above-mentioned relatively large values for misfit, cross-talk, and the width of the averaging kernels, one could say that the forward-modeled and measured travel times agree remarkably well.Furthermore, we would like to put forward the somewhat more speculative idea that a multi-cell structure may also do a good job explaining the measured travel times in the northern hemisphere. A hint of an additional cell in the northern hemisphere may be seen in the initial inversion result shown in Figure <ref>. Admittedly, such a profile is not seen in the refined inversions of Section <ref> but we note that it may have been washed out by the large widths of the averaging kernels, especially if the magnitude of the flow is small. In addition, we note a few lessons learned in our test inversions, although to some extent of a more incomplete nature. Firstly, it is not clear whether the reversal of signs in the measured travel times at high latitudes – see travel distances above 9 and latitudes within about 10 of the edges in the right panel of Figure <ref> – is due to a signal from the flow or whether it is caused by a systematic effect. We therefore checked whether excluding this region from the data affects the inversion results for the three cases considered in Section <ref>.The inversion results from the reduced data set are qualitatively very similar to the results shown in Figure <ref>, especially regarding the general structure and direction of the flow.The magnitude of the flow at high latitudes and near the bottom of the convection zone, however, is reduced, which may be seen as physically more realistic regarding mass conservation. Second, when increasing the threshold for the maximum errors in cases 1 and 2 step by step from 1 to 2.5, the multi-cell structure becomes less pronounced step by step and finally is nearly lost. We obtain a single-cell structure instead. At the same time, the match of the forward-modeled travel times to the measured ones becomes worse. This is not a surprising result, as in this case, the errors reach values similar to the magnitude of the flow in the considered regions. Furthermore, we note that further increasing the threshold for the SVD lets the inversion result further tend toward a smooth single-cell profile, although at some values hints of a second cell may be more pronounced than in case 1.Finally, we checked that if we do not impose condition (<ref>) with m=r and k=θ, namely, that the integral of the cross-talk kernels be equal to zero, the inversion result does not change much, as the integrals of the cross-talk kernels stay near zero <cit.>. Similarly, some tests showed that further decreasing the cross-talk is only possible at the expense of a large increase in the errors and therefore is not an option considered in this work. § CONCLUSION We have performed SOLA inversions for the deep solar meridional flow using Born kernels. We used GONG data obtained by <cit.> and inverted by <cit.> using ray kernels. In addition to performing a comparison to the ray kernel inversion, we performed several inversions in order to study systematic effects in the inversion process. A first comparison was made using the SOLA inversion technique employed by <cit.>, where the measurement errors were assumed to be uncorrelated (diagonal covariance matrix) and where no regularization for a cross-talk term was done. Our initial inversion result using Born kernels is qualitatively similar to the one obtained using ray kernels by <cit.> while displaying a little more fine structure.Furthermore, we compared the errors of the inverted flows, which were propagated through the inversion using the diagonal covariance matrix, to the errors computed using the full covariance matrix. Previous results for the deep meridional flow using time-distance helioseismology (, ) were obtained under the assumption of uncorrelated measurements. In our inversion, we find that the errors may be underestimated by a factor of about two to four under such an assumption. In a subsequent step, we performed refined inversions using the full covariance matrix, thus taking the correlation of the measurements into account, and we introduced a regularization for the cross-talk term as is standard practice in local helioseismology <cit.>. Three different cases for the choice of inversion parameters were studied. In all of these inversions, we find a shallow return flow at a depth of about 0.9 R_, in agreement with <cit.>. In a first case, we obtained a single-cell meridional flow profile with a few opposite sign “bumps”, which is again similar to the result obtained by <cit.>, with less latitudinal variation compared to the result obtained with the diagonal covariance matrix. Here, we used a medium threshold for the SVD and about 50% of the SVs were kept. When we lower the threshold by a factor of 10 and use about 70% of the SVs, the refined inversion results exhibit a multi-cell structure in the southern hemisphere and a single-cell meridional circulation structure in the northern hemisphere (cases 2 and 3). Depending on the associated radial flow, the inverted flow has two or three circulation cells stacked radially. At the locations of the additional cells, the inversion result for case 1 already showed a flow structure. This structure was, however, not as pronounced and no conclusion was made on this pattern earlier <cit.>.In principle, a discussion of the inversion results (see Section <ref>) suggests that a multi-cell profile in the southern hemisphere is a suitable candidate as an inversion result, as it is reproduced when convolved with the averaging kernels. All other results are averaged out spatially by this procedure. Furthermore, we speculate that a multi-cell circulation structure may also be present in the northern hemisphere. Indications for this possibility can be seen in the initial result, although it is not present in the refined inversions. However, we point out that both the single-cell and the multi-cell profiles obtained in the refined inversions agree similarly well with the measured travel times within the measurement errors. Furthermore, the cross-talk of the radial flow into the inversion result for the horizontal flow component was estimated to be below 1.4 . Therefore, it is only of minor importance, although the cross-talk averaging kernels have relatively large values in all inversions performed in this work. Finally, a flow structure with multiple cells in latitude as obtained by <cit.> using global helioseismology is not seen in the refined inversion results of our study. However, some latitudinal variation can be observed in the travel times, in the initial inversion result, and in the original inversion result from <cit.> using ray kernels. Such a signal may have been averaged out by the relatively wide averaging kernels that have an FWHM of up to 20. We therefore cannot rule out the possibility of latitudinally stacked flow cells at this point. Further insights in the inversion problem may be reached in future studies, e.g., by comparing to RLS/LSQR inversion techniques <cit.>, or to the Pinsker method newly introduced to helioseismology <cit.>. As pointed out by <cit.>, inversions for both radial and horizontal flows are most promising when a mass-conservation constraint is applied; see <cit.> for such an inversion. An inversion for radial flows without using mass conservation is not very promising; see <cit.>.In addition, in order to reach an agreement on the nature of the deep meridional flow, understanding systematic effects such as the center-to-limb effect <cit.> and the effect of surface magnetic fields on the measurements <cit.> may play a key role in the future.The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement n. 307117. This work was supported by the SOLARNET project (www.solarnet-east.eu), funded by the European Commission's FP7 Capacities Programme under the Grant Agreement 312495. J.J. acknowledges support from the National Science Foundation under Grant Number 1351311. S.K. was supported by NASA's Heliophysics Grand Challenges Research grant 13-GCR1-2-0036. The authors acknowledge fruitful discussions during the international team meeting on “Studies of the Deep Solar Meridional Flow” at ISSI (International Space Science Institute), Bern. This work utilizes data obtained by the Global Oscillation Network Group (GONG) Program, managed by the National Solar Observatory, which is operated by AURA, Inc., under a cooperative agreement with the National Science Foundation. The data were acquired by instruments operated by the Big Bear Solar Observatory, High Altitude Observatory, Learmonth Solar Observatory, Udaipur Solar Observatory, Instituto de Astrofísica de Canarias, and Cerro Tololo Interamerican Observatory. The authors thank Sylvain Korzennik for providing GONG mode frequencies and damping rates. V.B. thanks Damien Fournier for helpful discussions and Juan Manuel Borrero for reading an earlier version of the manuscript. 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"authors": [
"Vincent G. A. Böning",
"Markus Roth",
"Jason Jackiewicz",
"Shukur Kholikov"
],
"categories": [
"astro-ph.SR"
],
"primary_category": "astro-ph.SR",
"published": "20170727095610",
"title": "Inversions for Deep Solar Meridional Flow Using Spherical Born Kernels"
} |
=1 | http://arxiv.org/abs/1707.08922v3 | {
"authors": [
"S. K. Ashok",
"M. Billo",
"E. Dell'Aquila",
"M. Frau",
"V. Gupta",
"R. R. John",
"A. Lerda"
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"primary_category": "hep-th",
"published": "20170727160459",
"title": "Surface operators, chiral rings, and localization in N=2 gauge theories"
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L[1]>p#1 C[1]>p#1 R[1]>p#1 | http://arxiv.org/abs/1707.08586v2 | {
"authors": [
"A. Alexandradinata",
"Chong Wang",
"Wenhui Duan",
"Leonid Glazman"
],
"categories": [
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"published": "20170726180124",
"title": "Topo-fermiology"
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Institute of Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow, 108840, RussiaIt is shown that the spin-orbit and Zeeman interactions result in phase shifts of Andreev-reflected holes propagating at the surface ofa topological insulator, or in Rashba spin-orbit-coupled two dimensional normal metals, which are in a contact with an s-wave superconductor.Due to interference ofholes reflected through different paths of Andreev interferometer the electric current through external contacts varies depending on the strength and direction of the Zeeman field. It also depends on mutual orientations of Zeeman fields in different shoulders of the interferometer.Such a nonlocal effect is a result of the long-range coherency caused by the superconducting proximity effect. This currenthas been calculated within the semiclassical theory for Green functions in the diffusive regime,by assuming a strong disorder due to elastic scattering of electrons.74.45.+c, 74.78.-w, 74.25.Ha Long-rangeeffect of a Zeeman field on the electric current through the helical metal-superconductor interface in Andreev interferometer. A. G. Mal'shukov Accepted, Received; in original form========================================================================================================================================== § INTRODUCTION Due to a combined effect of a Zeeman field andthe spin-orbit coupling (SOC) the wave functions of Cooper pairs in s-wave superconductors acquire aphase dependent factor. This phase is responsible for the magnetoelectric effect <cit.>, which leads to a spontaneous supercurrent in the presence ofa nonuniform static Zeeman field <cit.>, so that the spatial distribution of this current depends in a peculiar way on coordinate variations of the field. A similar phase also characterizes the electron wave function of a normal metal placed in a contact with a superconductor, if the strong enough spin-orbit and Zeeman interactions are presented in this metal. For example, it results in a spontaneous current through asuperconductor-normal metal-superconductor Josephson junction, the so called φ-junction <cit.> which has been observed experimentally in Ref.Szombati. These physical phenomena provide important building blocks for low dissipative spintronic applications based on interaction of magnetic and superconducting systems.It is natural to expect that in superconductor-normal metal proximity systems the phase shift, which is induced by the Zeeman field and SOC, may be observed in the Andreev reflection <cit.>, where an electron scatters from a normal metal-superconductor interface as a hole.Interesting possibilities for studyingthe phase coherent phenomena are provided by Andreev interferometers <cit.>. These devices have several alternative paths for incident electrons and backscattered holes. In the previous studies a phase shift between interferingscattered waveshas been provided by a magnetic flux. On the other hand, it is important to understand, if the Zeeman field can produce the phase shift that is strong enough to result in measurable effects on the electric current through the Andreev interferometer. This problem has not been addressedyet. It is clear that a strong enough SOC is needed to produce amagnetoelectric effect which may be effective in a system of a micron size. Indeed, some two-dimensional (2D) systems have a strong intrinsic SOC <cit.>, which results in a considerable spin splitting of electron bands. In 2D systems these spin-split bands are characterized by opposite spin helicities. However, in the practically important semiclassical regime, when the Fermi energy (chemical potential) μ is larger than SOC, the magnetoelectric effect is reduced by a competition of bands with opposite helicities which cancel each other up to the terms ∼ h_F/μ, <cit.> where h_F is the spin orbit splitting at the Fermi energy. On the other hand, this cancelation does not occur in Dirac systems, such as surface electrons in a three dimensional topological insulator (TI), because in TI only the odd number of surface helical bands cross the Fermi energy. Therefore, it is reasonable to takea TI wire as a basic component of the device. At the same time, it will be demonstrated that the results obtained for TI mayalso be extended to a conventional 2D wire having a very strong SOCh_F∼μ.A simple interferometer is shown at Fig.1. Due to interference of paths through the upper and lower branches of the TI wire the electric current between the normal and superconducting leads can be varied by changing magnitudes or directions of Zeeman fields in the branches. For example,the current might be changed by flipping a magnetization direction in one of the branches. Such a nonlocal dependence of the conductance would demonstrate a long-range phase coherence created in the the TI wire by the proximity effect at low enough temperatures. The Zeeman field in TI is assumed to be directed parallel to the x,y surface of the rectangular wire. It may be created by a ferromagnetic (antiferromagnetic) insulator deposited on top of TI, or by magnetic doping. Instead of fabricating TI wires, one could deposit superconducting and normal leads, as well as magnetic films on a TI flake. We will consider in detail the former setup, although qualitative results will be valid for both.The electric current through the interferometer will be studied within the semiclassical theory for electron Green functions <cit.>. A strong elastic scattering on impurities will be assumed in the TI wire, so that the corresponding mean free path is much smaller than its dimensions. Also, the elastic scattering rate is much larger than the Zeeman splitting, but much less thanthe chemical potential. At the same time, for sufficiently short wiresin the micrometer range, the low-temperature inelastic scattering of electrons will be ignored.The article is organized in the following way. In Sec.II the Usadel equation and boundary conditions for the semiclassical Green function are derived for a TI wire. In Sec.III linearizedUsadel equations are derived for the case of a weak proximity effect and the analytic expression for the current is found in the low-bias regime. A summary of the results is presented in Sec.IV. § USADEL EQUATIONS The effective one-particle Hamiltonian ofelectrons on thesurface of TI can be written in the form <cit.>H=τ_3v𝐞(𝐤̂×σ)-τ_3μ+𝐙(𝐫)σ+V(𝐫),where 𝐤̂=-i∂/∂𝐫 and the Pauli matricesτ_1,τ_2,τ_3 operate in theNambu space, so that the electron destruction operators in the chosen basis have the form ψ_↑, ψ_↓, ψ^+_↓, -ψ^+_↑ with the arrows denoting spin directions. The third term in Eq.(<ref>) represents the Zeeman interaction, where 𝐙(𝐫) is parallel to the xy plane (the coordinate axes are shown in Fig.1), and the last term is a random impurity potential. 𝐞 is the unit vector which is parallel to the external normal to the wire surface. It is assumed that the wire width in the y-direction is much larger than its thickness in the z-direction. Therefore, electrons spend a relatively short time on flank surfaces. For this reason these surfaces are not taken into account in Eq.(<ref>).The semiclassical Eilenberger equations for electron Green functions are obtained by expanding the Dyson equation with respect to small Fermi wavelengths, in comparison with other characteristic lengths. These equations serve for calculation of the so called semiclassical Green functions. The latter are obtained from initial Green functions by integration over the particle energy at a fixed momentum direction, which is represented by the unit vector 𝐧. These functions arecombined into the 2×2 matrix ĝ_𝐧(𝐫) whose components are g_11=g^r, g_22=g^a, g_12=g^K and g_21=0, where g^r, g^a and g^K are the retarded, advanced and Keldysh functions, respectively. These functions, in turn, are matrices in spin and Nambu spaces. Theprocedure for the derivation of the Eilenberger equations is well described in literature <cit.>.As long as all characteristic energies are much less than the Fermi energy, transitions between bands with opposite helicities can be neglected within the semiclassical approximation. In this case the spin dependence of the Green functions is locked to a momentum direction. Therefore, the initial Eilenberger equations can be projected onto the electron or hole helical bands, depending on a location of the Fermi level. The semiclassical Green function, in turn, takes the form ĝ_𝐧=ĝ_𝐧0(1±𝐧×σ)/2, where at μ>0 the "+" sign must be chosen and vice versa. The function ĝ_𝐧0 does not depend on spin and satisfies the normalization condition ĝ_𝐧0^2=1. For a dirty system, where the mean free path is smaller than other lengths, the Eilenberger equations can be transformed into diffusive Usadel equations <cit.> forthe matrix ĝ_0(𝐫), which is obtained from ĝ_𝐧0(𝐫) by averaging over 𝐧. By this way the Usadel equation has been obtained in Ref. <cit.> for Dirac electrons and in Ref.<cit.> for a superconductor withRashba SOC, which is larger than the elastic scattering rate. For the TI wire this equation can be written in the formD_t(b)∇̃(ĝ_0∇̃ĝ_0)+i[ωτ_3,ĝ_0]=0,where ∇̃*=∇*+i[τ_3𝐅,*] and the gauge-field vector components are 𝐅=𝐙(𝐫)×𝐞_z/v. The parameters D_t and D_b denote electron diffusion coefficients on the top and bottom surfaces of the wire, respectively. In general these coefficients are different, because environments and surface potentials vary at these interfaces. It is interesting to note that theZeeman field enters Eq.(<ref>) in the same way as the vector potential of the magnetic field. An important difference is, however, that one can not change 𝐅 by a gauge transformation. Therefore, it is impossible to eliminate the "longitudinal" part of 𝐅 by such a transformation. In superconductors this part results in the so called helix phase with a spatially dependent order parameter, <cit.> as well as to spontaneous supercurrents around ferromagnetic islands. <cit.>When the wire length is much larger than its width w and∇_x g_0 is much smaller than w^-1, the Green function will tend to distribute uniformly over the wire width (in y-direction). If, in addition, g_0 is continuous on the wire flanks, it becomes constant around its perimeter.Let us consider the case when𝐅 is zero on the bottom surface. As shown in Appendix A, by averaging Eq.(<ref>) over y it can be reduced to the one-dimensional equationD∇̃_x(ĝ_0∇̃_xĝ_0)+i[ωτ_3,ĝ_0]- D(γ_x F_x^2+γ_y F_y^2)(τ_3ĝ_0τ_3ĝ_0-ĝ_0τ_3ĝ_0τ_3)=0 ,where ∇̃_x*=∇_x* + i(D_t/2D)[τ_3F_x,*], D=(D_t+D_b)/2,γ_x=D_tD_b/4D^2 and γ_y=D_t/2D. It should be noted that an equation of the same form may be obtained for a Rashba 2D electron gas with large SOC, such that h_F∼μ,by formal replacing the constants γ and D_t/D_b with parameters from Ref.[Houzet], which depend on the ratio between the Rashba constant and the Fermi velocity.Let us consider a weak coupling of the TIwire to the superconducting lead through tunneling barriers, which are shown in Fig.1 at contact points 2 and 3. Therefore, Eq.(<ref>) has to be supplemented by boundary conditions (BC) at these interfaces. Fora 2D Dirac system the usual semiclassical BC <cit.> must be modified, as shown in Ref. [Zyuzin]. The modified BChas theformDĝ_0∇̃_xĝ_0=Γ_S[ĝ_0,ĝ_s],where ĝ_s is the Green function in the superconducting lead and Γ_S is a tunneling parameter on the interface of TI with the superconducting lead. This parameter can be written in terms of the barrier resistance R_b=ρ_TID/2Γ, where ρ_TI is the wire resistance per unit length. <cit.> The Green functions and 𝐅 in Eq.(<ref>) should be taken near barriers. If the Zeeman interaction vanishes near these interfaces, then 𝐅=0and Eq.(<ref>) coincides with a conventional expression from Ref. <cit.>. Since it is assumed that the Zeeman interaction is induced bymagnetic layers on top of TI, it may vanish or not at the contacts, depending on sample preparation. It is expected that magnetization directions of the magnetic islands in the two interferometer arms may be varied independently of each other. Therefore, these islands must be separated to some extent in branching point 1.A tunneling contact will be also assumed at the interface of the TI wire with the normal lead at point 1. At this point the Green functions of electrons in both TI branches coincide. One more BC is an evident generalization ofEq. (<ref>) that takes into account two branches which make a contact with the normal lead. We apply here the ideas of Refs. [Zaitsev,Stoof] on how to write BC in branching points. By assuming that 𝐅=0 atcontactpoint 1, this BC can be written asDĝ_0∇_x_2ĝ_0 +Dĝ_0∇_x_3ĝ_0=-Γ_N[ĝ_0,ĝ_N],where x_2 and x_3 are coordinates in the branches. They are chosen so, that x_2 and x_3are directed from contact 1 towards respective contacts 2 and 3 with the superconductor. The tunneling parameter Γ_N may be expressed through the barrier resistance R_b1=ρ_TID/2Γ_N, in the same way as for the TI-S contact. For the massive normal lead one may assume that its Green function is unperturbed by a contact with the TI wire. Therefore, ĝ_N^r/a=±τ_3§ ANDREEV REFLECTION AND ELECTRIC CURRENT We consider the case of the low temperature T and small bias voltage V, which are much less than the superconducting gap. Therefore, the electric current between the normal and superconducting leads is determined by the Andreev reflection. This current may be expressed via the conductance G(ω), according to the well known expression <cit.>j=1/e∫ dω[tanhω+eV/2k_BT-tanhω-eV/2k_BT]G(ω) .Let us focus on the high barrier regime, when the barrier resistance R_b at TI-S interfaceis much larger than the resistance of the TI wire and the barrier resistance R_b1 at the TI-N interface. In this case G(ω) is given by <cit.>G(ω)=1/8R_b(M_2+M_3) ,whereM_2(3)=Tr[(g_0^rτ_3-τ_3g_0^a)(g_s^rτ_3-τ_3g_s^a)]|_x_2=L_2(x_3=L_3) .The functions g_0are taken in TI wires near contacts 2 and 3. L_2 and L_3 are the lengths of the wires between contact 1 and contacts 2 and 3, respectively.We assume a massive superconducting lead whose Green function is not perturbed by a proximity to TI wires. Therefore, at both contacts these functions have the form g^r_s=g^a_s=(-iτ_3ω + τ_2 Δ)/√(Δ^2-ω^2) for Δ>ω. At high R_b the Green functions in TI are weakly perturbed by the superconductor, so thatthey can be represented as sums of unperturbed functions and small corrections δ g^r(a)_j, namelyg^r(a)_0(x_j)=±τ_3+δ g^r(a)_j ,where j=2,3. The functions δ g^r(a)_j≪ 1 are the anomalous Green functions which are nondiagonal in the Nambu variables. By linearizing Eq.(<ref>) with respect to δ g^r(a)_j it can be transformed toD((-1)^j∇_x_j+2iτ_3F̃_xj)^2δ g^r(a)_j± 2 iωδ g^r(a)_j- 4D(γ_x F_x^2+γ_y F_y^2)δ g^r(a)_j=0 ,where F̃_xj=F_x (x_j)D_t/(D_t+D_b). In turn, boundary conditions Eq.(<ref>) take the linearized formD(∇_x_j+2iτ_3F̃_xj)δ g^r(a)_j|_x_j=L_j=Γ_Sτ_3 [τ_3,g_s^r(a)]At the same time, M_2 and M_3 becomeM_j=2Δ/√(Δ^2-ω^2)Tr[(δ g^r_j+δ g^a_j)τ_2]|_x_j=L_j . The solutions of Eq.(<ref>) contain the phase factors exp(± 2i∫ dx_jF̃_xj) which result in spatial oscillations of Green functions. Besides these oscillations, the Zeeman interaction leads to a suppression of the superconductor proximity effect. For instance, due to the third term in Eq.(<ref>), δ g_2 and δ g_3 decrease with increasingdistances from contacts 2 and 3, respectively. Therefore, the length L_Z of the region where 𝐙≠ 0 should not be too long. The corresponding condition is 2L_Z(γ_x F_x^2+γ_y F_y^2)^1/2≲ 1. By choosing the direction of 𝐙 perpendicular to x (F_y=0), the suppression effect can be reduced in samples having thesmaller ratio D_b/D_t of the diffusion constants, as follows from the definition of γ_x. It is also possible to construct appropriate barriers at the flanks of the wire to guarantee a weak Klein tunneling between the top and bottom surfaces. By making the angular averaged tunneling rate much less than the Thouless energy E_T=D/L^2, where L=max[L_2,L_3], the bottom surface of TI may almost completely be turned off, that will result in the small damping effect.It should be noted that the third term in Eq.(<ref>) vanishes completely if the Zeeman fields are finite on both surfaces andare equal in magnitude and antiparallel (both are perpendicular to x). However, such a situation is probably difficult to realize in practice.§.§ Short wires, low bias regime A simple analytic result may be obtained in the case of V≪ k_BTat small enough L_2 and L_3, so that k_BT≪ E_T. In this case one may set ω=0 in G(ω) in Eq.(<ref>). Let us assume thatL_2=L_3=L and L_Z is slightly less than L <cit.>. Hence, the phase Φ(x_j)≡2(-1)^j∫_0^x_j dx_jF̃_xj≃ 2(-1)^jF̃_xjx_j. The solutions of Eq.(<ref>) in both TI branches have the form δ g_j=exp(iτ_3Φ_j)[A_jexp(κ_j x_j)+B_jexp(-κ_j x_j)], where κ_j^2=4γ_x F_xj^2± 2iω at F_y=0 (± for retarded and advanced functions, respectively). In a symmetric device, that will be assumed below for simplicity, |F_x2|=|F_x3|. The 2×2 matrices A and B can be obtained from boundary conditions Eq.(<ref>), Eq.(<ref>) and the continuity of Green functions of the wire branches in contact point 1 . By substituting the so calculated δ g_2 and δ g_3 into Eq.(<ref>) we obtain the current from Eq.(<ref>) in the form j=ρ_TI/R_b^2Re[β(α + cosΔΦ)]V ,where ΔΦ=Φ_2|_x_2=L-Φ_3|_x_3=L,α= 2(1+Λ)sinh^2κ_0 L+1 ,β= 2/κ_0(1+Λ)sinh2κ_0 L ,Λ=ρ_TIκ_0 L/2κ_0 R_b1 and κ_0=κ|_ω=0. For more details of the calculation, see Appendix B. It follows from these expressions that the oscillating part of the current may be of the same order as the constant term, if κ L ≲ 1 and Λ≲ 1.As can be seen from Fig.2, the current's oscillations are strongest at D_t/D_b=0 and they are strongly damped already at D_t/D_b=0.1. The oscillationsalmost vanish at D_t/D_b=0.5. In the considered symmetric device the phase-dependent part of the current and the oscillations turn to zero when the Zeeman fields at two branches are antiparallel, so that in Eq.(<ref>) ΔΦ=0. The difference of conductances Δ G for the parallel and antiparallel alignments is shown in Fig.3 at various ratios D_t/D_b and the zero temperature (T≪ E_T).An alignment switch can be performed by changing a magnetization in one of the magnetic islands. For example, one may adjust their hysteretic characteristics in such a way that an externalmagnetic field of a definite strength flips the magnetization of one of them, while the other island stays in its initial state.It is important that the considered in this subsection short wire regime is valid at low enough temperatures which provide the sufficiently large coherence length ξ=√(D/k_BT), such thatξ≫ L. Otherwise, one can not simply set ω=0 in G(ω). Instead of that, the integral over ω inEq.(<ref>) must be taken. §.§ High temperatures In this subsection the numerical results are presented beyond the short wire regime, at k_BT ≳ E_T. G(ω) can be obtained from Eqs.(<ref>) and (<ref>). In turn, the Green functions, that enter in Eqs.(<ref>), are calculated in Appendix B. Fig.4 shows differences of conductances for parallel and antiparallel alignments of Zeeman fields, at various ratios of k_B T and E_T. These plots are normalized by the temperature dependent conductance G_0T in the absence of the Zeeman field. Fig.4 shows that so normalized Δ G decreases in the considered temperature interval , but not dramatically, that makes it possible to observe the phase shift produced by the Zeeman field even at relatively high temperatures. Note, that the absolute reduction of Δ G is larger, considering almosta threefold decrease of G_0Tin the same temperature interval. In order to evaluate E_T, let us take the mean free path l=10 nm, as in Bi_1.5Sb_0.5Te_1.7Se_1.3 <cit.> and a typical Dirac velocity v=5 10^5 m/s, that gives the diffusion constant of a 2D gas D=vl/2=25 cm^2/s. With this constant the Thouless energies are E_T=80mK and 20mKfor the interferometer shoulders L=500nm and 1000nm, respectively. Now let us evaluate typical values of Z which can provide the strong enough phase shift Φ =2ZLD_t/v(D_t+D_b). As can be seen from Figs.3 and 4, the maximum effect on Δ G is observed for 1≲ΔΦ≲ 2. For D_t/(D_t+D_b) ≃ 1, v=5· 10^5m/sec andL= 1μm the phaseΔΦ=2Φ reaches 1.5 at Z≃ 0.1meV. Such a field iswell below the Fermi energy, that is in agreement with the semiclassical approximation used in this work. Note, that the above evaluation of the Zeeman field is valid only for a special case of the magnetic island which covers almost the entire TI wire. Therefore, the field must be stronger for smaller sizes of the islands.§ CONCLUSION In conclusion, it is shown that due to a quantum interference of Andreev-scattered waves inwires made of three dimensional TI,the electric current through a TI-superconductor system can be varied by changing the mutual orientations of Zeeman fields in distant parts of the TI wire. This effect is a direct consequence of the long-range Cooper correlations created by the superconducting proximity effect and the Zeeman-field-induced phase shifts of the pairing functions. 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Kupriyanov M. Y. Kuprianov and V. F. Lukichev, Sov. Phys. JETP 67, 1163 (1988). Volkov A.F.Volkov, A.V.Zaitsev, and T.M.Klapwijk, Physica C210 , 21 (1993). comment As long as characteristic lengths of Z gradients near magnetic island edges, as well as distances from the island edges to contacts are much less than thecoherence length √(D/k_BT) and L, one may represent the spacial dependence of Z as a stepwisefunction. TIflake AE.S. Tikhonov, D.V. Shovkun, V.S. Khrapai, M. Snelder, M.P. Stehno, A. Brinkman, Y. Huang, M.S. Golden, A.A. Golubov, Phys. Rev. Lett. 117, 147001 (2016) § AVERAGING OVER A WIRE PERIMETER Let us assume that the wire has a rectangular cross section and the coordinate l runs along its perimeter, so that it coincides with y and -y on the top and bottom surfaces, respectively. Then, the part of Eq.(<ref>), which is associated with the derivative over l, can be represented in the form:∇_l(D(l)ĝ_0∇̃_̃l̃ĝ_0)+ iD(l)[F_lτ_3,ĝ_0 ∇̃_̃l̃ĝ_0],where∇̃_̃l̃*=∇_l*+i[τ_3F_l,*] and F_l represents a projection of the field 𝐅 onto the l coordinate. For completeness, the lateral surfaces of the wire (l ∥ z) are also taken into account in Eq.(<ref>). D(l) denotes the l-dependent diffusion constant. We will denote the average over the perimeter as (...)=∮ (...)dl/∮ dl and assume that ĝ_0 is constant as a function of l. For a diffusive transport the latter assumption is valid if the perimeter ismuch smaller than the characteristic lengths which characterize variations ofGreen functionsalong the wire. Therefore, ∇_lĝ_0=0. Hence, ∇̃_̃l̃ĝ_0=i[τ_3F_l,ĝ_0] in Eq.(<ref>). Further, since F_l and D(l) are periodic functions of l, the average of the first term in Eq.(<ref>) is 0. Therefore, the averaging ofEq.(<ref>) yields-D(l)F_l^2[τ_3,ĝ_0[τ_3,ĝ_0]] . By averaging the remaining terms in Eq.(<ref>) over the perimeter we arrive to the one-dimensional equationD∇̃_x(ĝ_0∇̃_xĝ_0)+i[ωτ_3,ĝ_0]+ (DF_x^2/D-DF_x^2-DF_l^2)[τ_3,ĝ_0[τ_3,ĝ_0]] =0.By assuming that the thickness of the wire is much smaller than its width one may neglect the contribution of the lateral surfaces into the average. If F_x and F_y are finite only on the top surface, Eq.(<ref>) reduces to Eq.(<ref>)§ DERIVATION OF EQ.(<REF>) In each shoulder j the substitution δ g_j=e^iΦ(x_j)f_j , where Φ(x_j)≡2(-1)^j∫_0^x_j dx_jF̃_xj,allows to transform Eq.(<ref>) to the form∇_x_j^2 f^r(a)_j +κ_j^2f^r(a)_j=0 ,where κ^2_j=4γ_xF^2_xj± 2i(ω/D) at F_y=0. The "±" signs in κ^2 correspond to retarded and advanced functions, respectively. Due to coordinate dependence of F^2_xj the parameter κ varies with x_j. If F^2_xj is a step-function, the wire can be divided into several parts, so thatin each of them κ^2 is a constant. The function f and its derivative must be continuous at boundaries between these parts, as it follows from Eq.(<ref>). Let us consider a simple case where the homogeneous magnetic islands in each shoulder occupy almost the entire wire, except for small regions near contacts with the leads. When the lengths of these regions is much smaller than the coherence length √(D/2|ω|)∼√(D/ k_BT), the function f and its derivative are almost constant there. Therefore, by neglecting their weak spacial variation one may replace in BC (<ref>) and (<ref>) the function f and ∇_x f with corresponding values in an adjacent magnetic domain. By this way it is possible to skip the small nonmagnetic regions of the wire. The solutions δ g_2 and δ g_3 of in TI branches 12 and 13, respectively, have the formδ g_2 = e^iτ_3Φ(x_2)[A_2e^κ x_2+B_2e^-κ x_2],δ g_3 = e^iτ_3Φ(x_3)[A_3e^κ x_3+B_3e^-κ x_3] ,where the labels r and a are skipped for a wile. They will be restored later, where necessary. In the assumed symmetric case there is a common factor κ in both branches. The four coefficients A_j and B_janticommute with τ_3 andcan be found from the boundary conditions. According to the definition of the phase Φ,we have Φ(0)=0 and Φ(x_j)|_x_j=L≡Φ_j at contacts j=2 and j=3, respectively, where Φ_j≃ 2(-1)^jF̃_jL. From the boundary conditions Eq.(<ref>), Eq.(<ref>) and the continuity of the Green functions in branches 2 and 3 at contact 1, it is easy to obtain the following equations near contact 1(A_2+B_2) - (A_3+B_3)=0 (A_2-B_2) + (A_3-B_3)=Λ_Nδ g(0) ,where Λ_N=ρ_TI/R_b1κ and δ g(0)=A_2+B_2=A_3+B_3 is the Green function at contact 1. At contacts 2 and 3 the boundary conditions have the form:A_2e^κ L-B_2e^-κ L = e^-iτ_3Φ_2τ_2Λ_S, A_3e^κ L-B_3e^-κ L = e^-iτ_3Φ_3τ_2Λ_S ,where Λ_S=(Δ/√(Δ^2-ω^2))(ρ_TI/R_bκ). From equationsEq.(<ref>), Eq.(<ref>) the factors A and B can be expressed asA_j=(Φ_+/sinhκ L-(-1)^jΦ_-/coshκ L)τ_2Λ_S/4-Λ_Nδ g(0) e^-κ L/4sinhκ L , B_j=(Φ_+/sinhκ L+(-1)^jΦ_-/coshκ L)τ_2Λ_S/4-Λ_Nδ g(0) e^κ L/4sinhκ L ,where Φ_±=exp(-iτ_3Φ_3)±exp(-iτ_3Φ_2).By calculating δ g(0)=A_2+B_2 from Eq.<ref>we obtain the expression for δ g(0) in the formδ g(0)=Φ_+ τ_2Λ_S/2sinhκ L+Λ_N coshκ L .According to Eq.(<ref>), the spectral conductance G(ω) is proportional to (M_2+M_3). The latter may be expressed from Eq.(<ref>) through the sum δ g^r(a)_2(L)+δ g^r(a)_3(L). By substituting coefficients A and B, that are given by Eq.(<ref>), into Eq.(<ref>) at x_2=x_3=L we obtainδ g^r_2(L)+δ g^r_3(L)+δ g^a_2(L)+δ g^a_3(L)=τ_2Λ_S × Re[|Φ_+|^2(κ L+Λ_N/2)/1+Λ_N/2κ L+|Φ_-|^2tanhκ L] .It is easy to see that in this expression only the phase difference Φ_2-Φ_3 enters, as it should be. Eq.(<ref>) finally gives the result Eq.(<ref>) in the low-bias regime where ω may be set to zero.It is instructive to see how the phase dependence of the current vanishes in the case when only one of the two interferometer arms is conducting. Let us, for example, turn off branch 3. In this case only the second lines should be left in BC Eqs.(<ref>) and (<ref>), where A_3=B_3=0. It is easy to see that the solutions of these equations at x_2=L have the form A_2=exp(-iτ_3Φ_2)f_a(κ) and B_2=exp(-iτ_3Φ_2)f_b(κ), where the functions f_a/b(κ) do not depend on the phase Φ_2. They depend only on κ. Therefore, the function δ g_2(L), which is given by Eq.(<ref>), does not depend on Φ_2, as well as the conductance G, as can be seen from Eqs.(<ref>) and (<ref>) at δ g_3=0. Therefore, the only effect of the Zeeman field is a suppression of the proximity effect by the damping factor κ. It produces only a monotonous decreasing of the current at higher Zeeman fields and does not depend on its sign. | http://arxiv.org/abs/1707.08335v2 | {
"authors": [
"A. G. Mal'shukov"
],
"categories": [
"cond-mat.supr-con",
"cond-mat.mes-hall"
],
"primary_category": "cond-mat.supr-con",
"published": "20170726093826",
"title": "Long-range effect of a Zeeman field on the electric current through the helical metal-superconductor interface in Andreev interferometer"
} |
red]Design rules for modulation-doped AlAs quantum wells Department of Electrical Engineering, Princeton University, Princeton, NJ 08544, USA Thanks to their multi-valley, anisotropic, energy band structure, two-dimensional electron systems (2DESs) in modulation-doped AlAs quantum wells (QWs) provide a unique platform to investigate electron interaction physics and ballistic transport. Indeed, a plethora of phenomena unseen in other 2DESs have been observed over the past decade. However, a foundation for sample design is still lacking for AlAs 2DESs, limiting the means to achieve optimal quality samples. Here we present a systematic study on the fabrication of modulation-doped AlAs and GaAs QWs over a wide range of Al_xGa_1-xAs barrier alloy compositions. Our data indicate clear similarities in modulation doping mechanisms for AlAs and GaAs, and provide guidelines for the fabrication of very high quality AlAs 2DESs. We highlight the unprecedented quality of the fabricated AlAs samples by presenting the magnetotransport data for low density (≃ 1× 10^11 cm^-2) AlAs 2DESs that exhibit high-order fractional quantum Hall signatures. [ L. N. Pfeiffer December 30, 2023 =====================Clean two-dimensional electron systems (2DESs) which exhibit the fractional quantum Hall effect are ideal systems to study electron-electron interaction phenomena and many-body ground-states. Along with the classic example of modulation-doped GaAs <cit.>, recent studies have revealed that we can add AlAs <cit.>, Si <cit.>, Ge <cit.>,ZnO <cit.>, and graphene <cit.> to the list of materials in which high-order fractional states have been observed. The AlAs system is particularly exciting. First, its lattice constant closely matches that of GaAs, therefore allowing the growth of very high quality, single-crystal, AlAs epitaxial layers on GaAs substrates. Second, as shown in Fig. 1, AlAs distinguishes itself from GaAs in where its conduction-band electrons are in the first Brillouin zone. In bulk AlAs electrons occupy multiple conduction-band minima (valleys) with anisotropic energy vs wavevector dispersions. When electrons are confined to an AlAs quantum well (QW), by varying the well-width and in-plane strain, one can make the 2D electrons occupy the valleys with different (in-plane) anisotropy, effective mass, and effective Landé g-factor <cit.>. These different parameters, and the flexibility to control the valley occupation, render the AlAs 2DES a unique system for probing exotic many-body as well as ballistic transport phenomena. Recent studies in AlAs 2DESs have indeedled to the observation of integer and fractional quantum Hall ferromagnetism <cit.>, valley skyrmion formation <cit.>, and interaction-enhanced valley susceptibility for electrons <cit.> and composite fermions <cit.>; it was also reported recently that the transport anisotropy of electrons is transferred to the composite fermions in AlAs QWs <cit.>. The AlAs 2DES is a also a prime candidate for "valleytronic" devices <cit.>, and it was the first system where ballistic electron transport in different valleys was demonstrated <cit.>.Despite the abundance of literature concerning the rich physics of 2DESs in AlAs QWs, there are fundamental unanswered questions about modulation-doping in these systems. For example, over an extended period of time, many studies on AlAs QWs have utilized Al_xGa_1-xAs barrier alloy fractions in the vicinity of x≃0.40 <cit.>. This choice is based on the fact that at this x the minima of the Γ- and X-bands are known to cross, hence providing the maximum conduction band offset for populating the AlAs QW. However, as is well known for the case of GaAs QWs, maximum conduction band offset does not necessarily relate to the best sample quality because of factors such as interface quality or background impurities in the barrier <cit.>. As shown in Fig. 1(c), since the barrier material flanking an AlAs QW is similar to what flanks a GaAs QW except that the band minimum is the X-band rather than the Γ-band, we could expect similar behavior for AlAs QWs. However, because there have not been many studies on barrier alloy fractions other than x≃0.40, it is difficult to assess these possibilities. Here we provide guidelines to grow modulation-doped AlAs QWs, flanked by Al_xGa_1-xAs barriers with 0.20≤ x ≤ 0.80. By deducing the relevant energy levels from electron density measurements, we find that the modulation-doping characteristics of AlAs and GaAs QWs are essentially identical. Our data show that this is true over the entire range of x, where three different situations can occur for the conduction band alignment of the two types of QWs considering both the Γ- and X-band, as shown in Fig. 2. Because modulation-doping is a thermal equilibrium process, no fundamental distinction is observed when comparing cases that involve both the X- and Γ-bands (Figs. 2(a) and (f)) with the single X-band (Γ-band) processes in Figs. 2(b) and (c) (Figs. 2(d) and (e)). We highlight this fact by demonstrating high quality modulation-doped AlAs QWs with x=0.33. For our study, AlAs or GaAs QWs, flanked by Al_xGa_1-xAs barriers with δ-Si doping, were grown by molecular beam epitaxy on (001) GaAs substrates [see Fig. 1(d)]. We use a Si doping concentration ranging from ≃ 3× 10^11 cm^-2 to 1× 10^12 cm^-2 for the substrate side, and ∼ 1.5 to 2 times this value on the surface side. The lower limit is implemented to prevent parallel conduction in the x≤0.26 AlAs QWs. The growth temperature was measured by a factory calibrated optical pyrometer (Ircon Modline 7V-1205, emissivity set to 0.63) and was fixed to be 645°C for all samples at all times except for when δ-doping the 2 nm Al_xGa_1-xAs layer beneath the lower spacer of the QWs, where the temperature was lowered to 480°C to prevent surface segregation of the Si <cit.>. The x ranged from 0.20 to 0.80 for the AlAs QWs and 0.26 to 1.0 for the GaAs QWs. We examined reflection high energy electron diffraction patterns prior the growth of each sample to determine compositions and growth rates. Well width (w) and spacer thickness (s) were fixed at 11 and 59 nm for the AlAs QWs and 20 and 70 nm for the GaAs QWs. For measurements we used a low frequency lock-in technique and a pumped ^3He cryostat with a base temperature of 0.3 K. Magnetoresistance data were taken by sweeping a superconducting magnet from 0 to 14.5 T in the dark and after illuminating the sample with a red light emitting diode at ∼10 K.Before presenting the experimental data, we briefly describe the valley occupation and parameters for our AlAs 2DESs. In AlAs QWs with w≳5 nm, biaxial compression from epitaxial growth on GaAs substrates raises the ground-state energy of the valley with its major axis along the growth direction, causing the other two (in-plane) valleys to be occupied <cit.>. Our AlAs QWs have a well width in this regime and thus have in-plane effective mass values of m_l^*=1.1m_e and m_t^*=0.20m_e, with a geometric mean of m^*=√(m_l^*m_t^*)=0.45m_e, and an out-of-plane mass of m_t^*=0.20m_e (m_e is the free electron mass).Figure 3 (a) shows the density of electrons (n) for our AlAs and GaAs QWs as a function of x. All electron concentration values were evaluated from the quantum Hall features in the magnetotransport data. Although there is an offset between the density profiles for the two QWs, it is clear that the variations in n for GaAs and AlAs QWs have a similar trend with x for measurements both taken in the dark and after light illumination. As we elaborate below, this suggests a common mechanism for the modulation-doping of the two QWs.It is important to note here that at higher barrier alloy fractions (x ≥ 0.38), an annealing technique <cit.> is required to achieve saturated carrier concentrations after illumination. Different annealing conditions are needed for saturation for different x, with x=0.38 having the longest time-constant of the order of 1 hour at ∼40 K. For x≤0.33, the extra annealing step was unnecessary, likely because the time-constant is short enough so that the process is completed during the ∼30 minutes it takes to cool the illuminated sample from 10 K to 0.3 K in our system <cit.>. This behavior is observed for both GaAs and AlAs wells, corroborating our conclusion that the modulation-dopings into these wells share a universal mechanism.Using the textbook model for modulation-doped heterostructures <cit.>, we can relate our measured n with the energy levels of the QW: Δ E_C=E_0+E_F+E_D+Δφwhere E_0 is the ground-state energy measured relative to the conduction band edge of the QW, E_F is the Fermi level measured with respect to E_0, E_D is the donor level energy defined relative to the conduction band edge of the barrier, and Δφ≡nse^2/ϵ_b [see Fig. 2(a)]. Here ϵ_b is the barrier dielectric constant and e is the electron charge. Using values of n and s we can determine Δφ and E_F=nπħ^2/g_vm^*, where ħ is the Planck constant and m^* is the effective mass in the QW (m^*=0.067m_e for GaAs and m^*=0.45m_e for AlAs); g_v is the valley degeneracy (g_v=1 for GaAs and g_v=2 for AlAs). From the simple case of an infinite potential well, we can also get a rough estimate for E_0, which is ≃ 15 meV for the AlAs QWs (m_t^*=0.2m_e, w=11 nm) and ≃ 14 meV for the GaAs QWs (m^*=0.067m_e, w=20 nm). Considering, as an example, the case of the GaAs QW with x=0.33 and n=4.5×10^11 cm^-2 after illumination, we deduce Δφ≃238 meV, and E_F≃16 meV. Since E_0 and E_F are both much smaller than Δφ, we conclude from Eq. (1) that Δφ≃(Δ E_C-E_D). Implementing a self-consistent Schrödinger-Poisson solver corrects E_0 of the order of ≃5-10 meV. More precise calculations require an exact knowledge of ΔE_C for all alloy fractions, but this would not alter the relation Δφ≃(Δ E_C-E_D) which is the crucial factor in understanding the design rules in this study.The symbols in Fig. 3(b) show the values of (ΔE_C-E_D), with respect to the Γ-band edge of GaAs, deduced from the density data points in Fig. 3(a) and using Eq. (1) assuming E_0 values of an infinite potential well. To account for the fact that the conduction band minima of AlAs QWs are not aligned with GaAs QWs, we take the offset between GaAs(Γ) and AlAs(X) to be 114 meV <cit.> and add this constant value to all the (ΔE_C-E_D) values for the AlAs QWs. Since the calculated Δφ is ≃70 meV for our AlAs/GaAs/AlAs (i.e., x=1) structure, and previous reports quote shallow donor energies in AlAs ranging from ≃30 to 60 meV <cit.>, the value of 114 meV we take from the literature is quite consistent with our results. It is seen in Fig. 3(b) that with the 114 meV offset there is excellent agreement between the (ΔE_C-E_D) values for the AlAs and GaAs QWs over the entire range of x. From our data points measured after illumination, we can estimate the conduction band offset with respect to GaAs(Γ) for Al_xGa_1-xAs in modulation-doped structures, drawn as the solid blue and red lines for the Γ- and X-bands in Fig. 3(b). For the Γ-band, the x<0.38 data coincide very well with the reported literature values of the conduction band offset ΔE_C^Γ <cit.> assuming a hydrogenic donor level. For the X-band we draw a line that goes through the reported GaAs(X)-GaAs(Γ) offset value of ≃470 meV <cit.> and our expected AlAs(X)-GaAs(Γ) offset of 114 meV. We find there is reasonable agreement with the data in Fig. 3(b), including the Γ-X band crossing point in Al_xGa_1-xAs at x=0.38. The deep donor levels measured from the data in the dark agree well with previous reports on the DX effect in Al_xGa_1-xAs <cit.>, showing a maximum effective barrier near x=0.26 and monotonic decrease when x>0.26.We also comment on the mobility values measured for our samples. Figures 4(a) and (b) show the mobility values as a function of carrier concentration for the AlAs and GaAs samples, respectively. Note that for the AlAs samples at any given density, the measured mobility is higher than in previous studies <cit.>, attesting to the high quality of the samples used in our study. This is particularly impressive considering that, in contrast to the samples in <cit.>, our samples are doped from both sides and have smaller spacer thicknesses. The power law fit for the relation between density and mobility yield μ∝ n^1.4 for the AlAs QWs and μ∝ n^3 for the GaAs QWs. We postulate that the notable deviation from the well known μ∝ n^1.5 for the GaAs samples is due to significant contributions from the barrier in the two lowest density samples, where x=1.0 and 0.9. Indeed if we perform a fit while omitting the data from these two samples, we achieve a power law of μ∝ n^1.6. These results suggest that barrier quality is also an important factor to consider in sample optimization as mentioned earlier in the introduction.To evaluate the potential of high quality AlAs samples with x<0.38, we grew a set of AlAs QWs with x=0.33 and varying spacer thicknesses. Figures 5(a)-(c) show longitudinal magnetoresistance (R_xx) data for the x=0.33 AlAs wells with spacer thicknesses of 59, 136, and 178 nm, respectively. The dependence of n on spacer thickness is plotted in Fig. 4(a) inset, which clearly shows that it is governed by the n∝ s^-1 relation expected of modulation-doped structures. The indices A-F in Figs. 5(a)-(c) and the inset mark the corresponding densities of each trace.Figures 5(a)-(c) demonstrate the high quality of the fabricated samples, with clear indications of the 2/3 and 1/3 fractional quantum Hall states (FQHSs) even at the low density of 6.1×10^10 cm^-2 (F) for the 178-nm-spacer sample [Fig. 5(c)]. After light exposure n for this sample increases to 1.2×10^11 cm^-2 (E), and the measured trace shows excellent quality with clear R_xx minima at filling factors ν=2/3, 3/5, 4/7, 3/7, and 2/5. The s=59 nm sample which has a higher density of 3.4×10^11 cm^-2 (A) after light also shows FQHSs at ν=5/3 and 4/3. We emphasize that all of these samples were fabricated with x=0.33, and were measured after a brief illumination of ∼1 minute (with a current of 6 mA in the light emitting diode) at 10 K and a subsequent cool-down to 0.3 K after the light was turned off, with no additional procedures such as annealing <cit.> or gating <cit.>. The results presented here suggest that when the conditions of the barrier are dominant in determining the quality of the AlAs 2DESs, we can resort to the conventional techniques used for GaAs to implement small x barriers. For example, in GaAs samples with sufficiently large s, the intentional ionized impurities are far enough from the 2DES that the scattering term from the unintentional (background) impurities in the barrier becomes significant, and hence having a small x barrier is crucial for a high quality due to the inherently more reactive nature of Al compared to Ga. If we extend this concept to AlAs QWs, it suggests that in low density AlAs 2DES where s is large, we should grow AlAs QWs with small x for optimal performance. This could also apply to narrow AlAs QWs, where the significant penetration of the electron wave function into the barrier again makes it beneficial to have a barrier with small x.In conclusion, our measurements of the electron density in modulation-doped AlAs and GaAs QWs over a wide range of Al_xGa_1-xAs barrier alloy fractions reveal that their doping characteristics are essentially identical despite having different electron pocket distributions in the Brillouin zone. We highlight this by the observation of the n∝ s^-1 rule for x=0.33 AlAs wells with 59, 136, and 178 nm spacer thicknesses. Our fabricated AlAs QWs show high quality magnetotransport data with clear indications of FQHSs. The design rules we establish here for modulation-doped GaAs and AlAs QWs provide a foundation for application specific sample optimization, especially in the case of AlAs which was so far a relatively uncharted material compared to GaAs. We acknowledge support through the NSF (Grants DMR 1305691 and ECCS 1508925) for measurements, and the NSF (Grant MRSEC DMR 1420541) and the Gordon and Betty Moore Foundation (Grant GBMF4420) for sample fabrication and characterization. 99Tsui.PRL D. C. Tsui, H. L. Stormer, and A. C. Gossard, Two-dimensional magnetotransport in the extreme quantum limit. Phys. Rev. Lett. 48, 1559 (1982).Pan.PRL.2002 W. Pan, H. L. Stormer, D. C. Tsui, L. N. Pfeiffer, K. W. Baldwin, and K. W. West, Transition from an electron solid to the sequence of fractional quantum Hall states at very low Landau level filling factor. Phys. Rev. Lett. 88, 176802 (2002).Lay.APL T. S. Lay, J. J. Heremans, Y. W. Suen, M. B. Santos, K. Hirakawa, M. Shayegan, and A. Zrenner, High-quality two-dimensional electron system confined in an AlAs quantum well. Appl. Phys. Lett. 62, 3120 (1993).Etienne.Science E. P. De Poortere, E. Tutuc, S. J. Papadakis, and M. 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Energetics of DX-center formation in GaAs and Al_xGa_1-xAs alloys. Phys. Rev. B 39, 10063 (1989). | http://arxiv.org/abs/1707.09011v2 | {
"authors": [
"Yoon Jang Chung",
"K. W. Baldwin",
"K. W. West",
"D. Kamburov",
"M. Shayegan",
"L. N. Pfeiffer"
],
"categories": [
"cond-mat.mes-hall",
"cond-mat.mtrl-sci"
],
"primary_category": "cond-mat.mes-hall",
"published": "20170727193615",
"title": "Design rules for modulation-doped AlAs quantum wells"
} |
Susanna Parenti [email protected] d'Astrophysique Spatiale, CNRS - Univ. Paris-Sud, Université Paris-Saclay Bat. 121, F-91405 Orsay, France DAMTP, Centre for Mathematical Sciences, Wilberforce road,Cambridge, UK Dipartimento di Fisica & Chimica, Universitá di Palermo, Piazza del Parlamento 1, 90134 Palermo, Italy Dipartimento di Fisica & Chimica, Universitá di Palermo, Piazza del Parlamento 1,90134 Palermo, Italy INAF-Osservatorio Astronomico di Palermo, Piazza del Parlamento 1, 90134 Palermo, ItalyMax-Planck-Institut für Sonnensystemforschung, Justus-von-Liebig-Weg 3,37077 Göttingen, GermanyHarvard-Smithsonian Center for Astrophysics, 60 Garden Street,MS 58, Cambridge MA 02138, USA DAMTP, Centre for Mathematical Sciences, Wilberforce road Cambridge, UK In this work we investigate the thermal structure of an off-limb active region in various non-flaring areas, as it provides key information on the way these structures are heated. In particular, we concentrate in the very hot component(>3 MK) as it is a crucial element to discriminate between different heating mechanisms. We present an analysis using Fe and Ca emission lines from both SOHO/SUMER and HINODE/EIS. A dataset covering all ionization stages from Fe X to Fe XIX has been used for the thermal analysis (both DEM and EM). Ca XIV is used for the SUMER-EIS radiometric cross-calibration.We show how the very hot plasma is present and persistent almost everywhere in the core of the limb AR. The off-limb AR is clearly structured in Fe XVIII. Almost everywhere, the EM analysis reveals plasma at 10 MK (visible in Fe XIX emission) which is down to 0.1% of EM of the main 3 MK plasma. We estimate the power law index of the hot tail of the EM to be between -8.5 and -4.4. However, we leave an open question on the possible existence of a small minor peak at around 10 MK.The absence in some part of the AR of Fe XIX and Fe XXIII lines (which fall into our spectral range) enables us to determine an upper limit on the EM at such temperatures. Our results include a new Ca XIV 943.59 Å atomic model.§ INTRODUCTIONDecades of observations have unveiled the difficult task ofunderstanding how and where the energy is deposited forthe creation and maintenanceof the solar corona.Various physical mechanisms have been proposed <cit.> whichmay dominate depending on the local environment.Active regions (AR) are the most visible manifestation of the enduring corona. These are the hottest part of the non flaring corona, with strong UV and X-ray emission. ARsare generally composed of coronal loops which are classed in two main thermal categories: the hot and low-lying loops (3MK), less dense than equilibrium conditions,which are concentrated in the core; the warm loops (1MK), larger and more dense than equilibrium conditions, which arelocated above the AR core<cit.>. It is not yet clear if these different properties are the result of different physical processes at work orare the manifestation of the same process observed in a different evolutionary state. In recent years new observational signatures of coronal heating have been identified, which we believe, if completely understood, could be essential to progressing our understanding of the corona:a small amount of very hot plasma (>5 MK) has been detected in several quiescent ARs (see e.g. <cit.> for a review). This detection is important as it confirms a prediction madeseveral years earlier from the the modeling of coronal loops heated impulsively over small (sub-resolution) scales <cit.>.Such models predict at larger spatial scales (at the spatial resolution reached by the modern instruments) a multi-thermal plasma along the line of sight. The very hot plasma provides evidence of the transitory(short-lifetime) state of the cooling plasma in these loops.The importance of observing such small amounts of very hot plasma relies on the fact that it is unique to impulsive heating events.This is the signature, for an unresolved spatial scale, of thin loops (strands)each heated independently over a short time. One of the various diagnostics techniques used to test coronal heating models is the sampling of theheating-cooling process in loops, which shows different observational signatures dependingon the way the heating happens. The differential emission measure (DEM) distribution with the temperature samples this process and isa common way of investigating the heating of loops <cit.>.A typical AR DEM increasesas a power law of the temperature up to amaximum around 3 MK, with an index which may change and probably depends on the levelof magnetic flux. Typically EM ∼ T^3-5 <cit.>. This index constrains the frequency at which the energy is released in the form of heating.At temperatures above the peak, the DEM decreases drastically,but the difficulty of the measurements (mostly carried out with EUV data) makes it hard to define the shape of this distribution aboveabout 5 MK. In most of the cases it can be fitted with a power law function, but simulations of nanoflares heating revealthat this is not always the case <cit.>.This high temperature component of the DEMis the signature of the first phase of cooling which, for this reason,possibly conserves more information on the heating process and the amountof energy deposited <cit.>.The existing EUV instruments have difficulty in detecting emission above about 5 MK as there is very little plasma at thesetemperatures <cit.> and because only a few relatively weak UV spectral lines form at these temperatures.For these reasons the DEM is not well constrained at high temperatures. X-ray spectra are more suitable, see for instance results from SMM <cit.> and references therein.The upper limits in the DEM/EM imposed by the the measured fluxes that we found in the literature using EUV spectra are given by Hinode/EIS Ca XVII, SOHO/SUMER Fe XVIII and SDO/AIA 94 channel (Fe XVIII).Estimations from these datasets of the power law index(EM ∼T^α) change from -6 to about -14 (<cit.> see Table <ref>).The sounding rocket EUNIS-13 <cit.> enabled measurements of extended Fe XIX emission in an on-disk AR. Without the possibility of performing a full DEM analysis, they provided aFe XII/Fe XIX EM ratio of about 0.59 in the AR core. This assesses the difference a relation between relatively cool coronal plasma (1.5 MK)and much hotter plasma at about 10 MK.In recent years observations of non-flaring ARs with X-ray instruments has intensified with the purpose of better constraining this slope <cit.>.Additional constraints at high temperatures have been obtained combining EUV and Hinode/XRT soft-X ray emission <cit.> for an on disk AR <cit.>. The results found were about two orders of magnitude variation of the EM from 3-10 MK.RHESSI, even though it is not sensitive to this faint plasma, has revealed its presence in the 6 - 10 MKrange <cit.>. Most recently, <cit.>, using the NUSTAR hard X-ray telescope, found an even larger decreaseof the EM by imposing upper limits on this quantity due the absenceof observed emission at high temperature. Similarly, the FOXSIhard X-ray emission sounding rocket <cit.> providedan upper limit to the DEM above8 MK for an on disk AR.In this paper we address the issue of spatially and temporally characterizing the high temperature emission of a non-flaring activeregion with the aim of providing further constraints to the heatingmechanism responsible for its formation and maintenance.Even though such plasma has been observed in several active regions,at present very little is known about the spatial and temporaldistribution within the same active region. To our knowledge, this isthe first time that the spatial distribution of this very hot component is given at different heights above the limb. To minimize uncertainties in quantifying this plasma and its temperature, we used spectroscopic data from a single element (Fe).For the first time this type of analysis is carried out combiningthe SOHO/SUMER and Hinode/EIS spectra allowing the observation ofiron lines in contiguous ionization stage from Fe Xto Fe XIX to be used.This choice of instruments combinationis at present unique: in provides a line sequence of a single element in a broad temperature range and lets our thermal analysis be independent of the plasma composition. We provide evidence of persistent emission from Fe XIX high in the corona,above the limb.After presenting our observations in Section <ref> andthe data reduction in Section <ref>, we introduce thediagnostic technique in Section <ref>. Section <ref> reports the results on the temporal analysis,Section <ref> presents the inter calibration of the two instruments, while the results from the thermal analysis is given inSection <ref>. The conclusions are summarized in Section <ref>. § OBSERVATIONSWe ran HOP(Hinode Operation Plan) 211 on active region 11459 for several hours at the west limb between the 27 and 28 April 2012 using both the SOHO/SUMER <cit.> and the Hinode/EIS <cit.> spectrometers (Figure <ref>). SUMER: After the loss of Detector A in 2006, due to failure of the readout electronics, only the B Detector remained available. In mid-2009, a degradation was observed in the center of the active area of the detector.As a protective measure, it was necessary to reduce the high voltage and the consequent reduction of the overall gain led to a drop of sensitivity in the central, KBr-coated, part of the photocathode, where it fell below detectable thresholds. Only the two uncoated (bare) areas of the detector (about 200 pixels each) were usable at the time of our observations. Additionally, since the resistivity of the microchannel plate (MCP) was observed to decrease with increasing temperature, to avoid a runaway effect, pauses were added in the observing sequences. All SUMER data sequences discussed here are made of sequences of 60 exposures of 75s each. After each pair of exposures a pause was added (of about 190s for the three sequences closest to the solar limb and of about 210s for the three sequences at a greater distance above the limb).Thus, each sequence can be thought as 30 regularly spaced pairs of 75s exposures. SUMER observed in a sit-and-stare mode using the 1× 300 slitfor about eight hours (16:02:08 UT to 00:28:58UT on the 27th) centered at a solar distance of 992. In the following this will be called slit position 1.During this time SUMER scanned the wavelength range needed to cover the observation of Fe XVII, Fe XVIII (Ca XIV) and Fe XIX as listed in Table <ref>. As a result, each Fe line was observed for about two hours (60 exposures with 75 sec exposure time).An example of the spectra for these lines is shown in Figure <ref>. A second sequence of observations, starting on the 28th at 00:35:55UT, was made bymoving the pointing west of about 60. After a first sequence which used the same slit, the slit was changed to 4× 300 when scanning Fe XVII andXIX(in the following we call this outer slit position as position 2, see also Table <ref>). EIS: EIS data consist of three rasters nominally pointed at the two SUMER slit center positions. Each raster was scanned in about two hours starting at 18:19:33 UT. The observing program used the2 slit, which scanned eastward over 82 positions with 2step and 90 seconds exposure time. Thefinal field of view is 164× 376. Details of the observations are given in Table <ref>.Our observations occured during the Hinode eclipse season, with the result that not all the common field of view of SUMER and EIS was exploitable. This is shown in Figure <ref>. We point out that the data we used are not all co-temporal. For this study, we tried to select quiescent areas of the AR to minimize the temporal evolution of the plasma. However, this aspect has to be taken into account in the interpretation of our results. We discuss this aspect in Section <ref>.Prior to and during our observations, AR 11459 flared a few times, with moderate intensity.These flaring events left some signatures on our datasets. Some small activity had already happened while observing with SUMER in slit position 1, while observing the Fe XVIII 1118.0575 Å. A hot loop system passed through the SUMER slit as shown in Figures <ref> and <ref>.This was also imaged by the second raster of EIS.The first flare happened on the 27th of April with the peak at 21:04UT. A C2.2 flare followed in the same region peaking at 23:39UT while SUMER was observing Fe XVII 1153.16 Å (Figure <ref>). On the 28th there was a C1.5 flare (peak at 0:43UT) and a C1.4 (peak at 1:54UT), but our data does not look to be affected.We discuss this in more detail in the following sections.In our analysis we also used the continuous monitoring provided by SDO/AIA for reference and comparison.As the aim of this work is to investigate the quiescent conditions of ARs, the data affected by the flares were discarded from our analysis. Details of our data selection are given in the next Section. ccclcc SUMER and EIS observations for 27/28th April 2012.6 1 0ptInst.start day, hour end day, hour Center Main line position n.[UT][UT] [x, y][Å] SUMER 27, 16:00:53 27, 18:47:11 [962, -271.9]Fe XVIII 974.86, 1 CaXIV 943.59SUMER27, 18:51:4927, 21:38:05 [962, -271.9] Fe XIX 1118.05751 SUMER27, 21:42:3928, 00:28:58 [962, -271.9]Fe XVII 1153.16531SUMER 28, 00:34:40 28, 03:29:55 [1025.5, -274.7] Fe XVIII 974.86,2CaXIV 943.59 SUMER28, 03:34:57 28, 06:30:12 [1025.5, -274.7]Fe XVII 1153.1653 2SUMER28, 06:35:0728, 09:30:23 [1025.5, -274.7]Fe XIX1118.0575 2EIS 27, 18:19:3327, 20:24:49 [951.7, -300.5] CaXIV 193.871EIS 27, 20:24:5527, 22:30:11[951.7, -300.5]CaXIV 193.871EIS 27, 23:01:4028, 01:06:56 [1012.7, -300.3] CaXIV 193.872The table lists the coordinates of the FOV center obtained with theco-alignement, as well as the high temperature lines observed.§ DATA REDUCTIONThe SUMER data were decompressed, wavelength-reversed, corrected for dead-time losses, local-gain depression, flat-field, and image distortion induced by the readout electronics. All the above steps were performed by using the standard software provided inSolarSoft <cit.>.SUMER off-limb data may be affected by stray-light. We estimated this contribution in two ways (detailedin Appendix <ref>).SUMER spectra are not wavelength calibrated. We performed this calibration (presented in Appendix <ref>). The EIS data were corrected for instrumental effects and calibrated applying the eis_prep.pro (version 29-Jan-2015) available on SolarSoft. The radiometric calibration has recently been further improved by <cit.> and <cit.>. For our analysis we used the <cit.> calibration, but a discussion of the influence on the results obtained using a different calibrationis presented in Section <ref>.Before performing the data analysis, we processed the data to take into accountvarious issues that could affectresults. These are presented below.We started by carrying out a co-alignment of the fields of views of EIS and SUMER, which was done by using the SDO/AIA images for reference.This was done for two reasons: the SUMER slit follows the SOHO roll, which is not aligned North-South, and the EIS raster data has a temporal dependence. The task is not easy as we were dealing with off-limb data where the structuring and image contrast are less marked than on-disk. For the first SUMER slit position 1, we found that the best way to proceedto co-align SUMER-AIA was to use the flare data (SUMER Fe XVIII and AIA 94 channel).For the slit position 2 we used the pair SUMER Ca X 557.766 Å with AIA 171, and SUMER Ca X 574.011 Å with AIA 171.The two Ca lines are observed about two hours apart, but the result ofthese two alignments were similar within 2. Details of this work are given in Appendix <ref> and the results are listed in column four of Table <ref>.It is known that EIS is not aligned with respect to AIA. We first used the eis_aia_offset.pro IDL procedure available in the SolarSoft database to correct our EIS data. However, we were not satisfied with our results and we decided to proceed with an independent manual co-alignment based on cross-correlation. We used the AIA 195 channel and EIS Fe XII 192.394 Å. The details of the method are given in Appendix <ref> and the results are given in Table <ref>. §.§ Spectralline selectionTable <ref> gives the list of the observed spectral lines and their fluxes used in this work. To reduce the uncertainties in the results of the thermal analysis, we minimized the effect of the elements abundance, by selecting only lines from Fe ions. We have enough Fe ions to cover the temperature range 6 < log T < 6.95. In addition, as we shall show, we performed several tests tomonitor the effects of elemental composition on our results. To this line list we added Si VII to constrain the DEM and EM at low temperatures and Ca XIV, which was used for the intercalibration EIS-SUMER (see Sec. <ref>).This is, in fact, the only ion which is present in both instruments whose lines are free from blends.Our results should not be affected by element abundance variations as the chosen ions are allfrom low First Ionization Potential elements. In addition, the Ca XIV is formed at a temperature similar to FeXV and FeXVIwhich were observed by EIS. The EIS high temperature lines from ions above Ca XIV and Fe XVI were absent or unusable because they were too blended. We also tried to extract Ca XVII 192.853 Å but with no success. To do this, we used the <cit.> method, which deblends this line from the FeXI and O V lines. We used the theoretical CHIANTI ratio between the FeXI 188.1/192.8 to fix the FeXI 192.8 Å profile parameters, but could not find any good solutions to the multi-line fitting. We then recalculated this ratio directly for the data by selecting a quiet off-disk area far from the limb,where the 192.8 Å line was very symmetric, and did not show any evidence of the presence of OV and Ca XVII. We then assumed that OV and Ca XVII were absent. With this new ratio value we found the best solution of the multi-Gaussian fitting was one in which Ca XVII line was absent. We concluded that in our data, this line was absent or too faint to be measured. In conclusion, the high temperature plasma is only covered by the SUMER data. Our analysis was carried out over several masks as described in thefollowing Sections. We used the eis_mask_spectrum.pro to select the masks in the EIS rasters and spec_gauss_eis.pro available in SolarSoft to perform a multi-Gaussian fitting of the resulting spectra. Amongst the hot lines in the SUMER spectra, the FeXVIII 974.860 Å blend required particular attention (see Figure <ref>).This blend has previously been discussed by <cit.> and is due to the presence of a ghost image (caused by the electronics) of the HI 972.54 Å line falling about 2.8 Å red-ward of this line,and SiVIII 974.58 Å. The amplitude of the ghost has been estimated by these authors to be about 1/200 the main line and it cannot easily be fitted within the blend.Having the HI 972.54 Å line in our spectra, we could estimate the ghost by adopting this value to correct our Fe line flux. We found its contribution to beonly 1% to 2%. The FeXVIII line was fitted using two Gaussians profiles in order todeblend it from the SiVIII. However, the result forthe different masks was not always satisfying. We then estimated thecontribution of the SiVIII by calculating the intensity ratio with the SiVIII 1182.48 Å lines we had in the spectrum, which is well isolated and has similar amplitude.We estimated this ratio from our data by selecting a quiet region (mask F) where FeXIX is absent and we expected to have none or a very small contribution of FeXVIII to the blend.The result of the double Gaussian fit to the spectra of mask F givesthe SiVIII 974.58/ 1182.48 Å = 1.04. This value is consistent with <cit.> off-limb observation who found a value 1 for this ratio, while <cit.> found a smaller value (0.7). We adopted our result to correct the FeXVIII in our masks. Considering that in the masks chosen for the thermal analysis theFeXVIII line is about 20 times brighter than theSiVIII 1182.48 Å, our uncertainty on the blend remains within the assumed 20% uncertainty of the fluxes.We indeed assumed an error of 20% in the lines flux. This value includes uncertainties in the atomic physics, ionization/recombination rates calculations (which, for certain ions could be larger). We are aware that other factors may arise to increase the uncertainty.For instance, the EIS calibration which will be discussed in Sec. <ref>. The lack of co-temporal data may also imply some inconsistency in the analysis. This will be discussed in Sec. <ref>.Figure <ref> shows an example of the hot lines in the SUMER spectra for mask A together with the result of multi-Gaussian fitting.§.§ Upper limit on the flux for FeXIX and FeXXIIIAbove 3 MK the non flaring plasma produces faint emission. To better constrain our thermal analysis we estimated an upper limit of the flux for those flaring lines falling in the spectral range of our data, but which were not visible. This is the case for SUMER FeXIX 1118.06 Å in mask B and for the EIS FeXXIII 263.77 Å in the other masks.Assuming a Poisson background statistics approximated by a Gaussian, we set a 3 σ confidence level as the minimum threshold for detection of the line signal above the background photon counts. We selected a wavelength interval similar to the expected line width. We used this to calculate the minimum flux needed for the spectral line to be measurable.The results are given in Table <ref>. lrrrrrrrrc Lines list and total fluxes (I).10 2 0ptIon λlog T_ max I(A) I(B)I(C)I(C) I(D) I(E) Inst.[Å][K][18:19UT][20:24UT]SiVII 275.385.79 299.7525.1 53.6 75.756 135.2EIS FeX(sbl) 257.26 6.041347.4 2189.9526.3674.8586.3988EIS FeXI 188.30 6.131258.0 1764.1 868.5917.2 640.4935.9EIS FeXII 192.396.20 953.5 1283.0 878.4926.8446.2 656.6EIS FeXIII202.046.252824.1 3279.1 2999.9 3041.3 1504.6 2025.3EIS FeXIV 264.796.292264.3 2741.5 2532.4 2474.0654.61038.07 EIS FeXV284.166.34 19078.821675.8 18155.5175857848.510187.5EISFeXVI262.98 6.431474.11565.31154.9 946.7 384.0517.7EIS CaXIV 193.876.57 112.192.363.8 45.5 28.632.8EIS CaXIV 943.596.56 4.03.792.72 2.72 1.441.76 SUMER FeXVII 1153.166.73 3.4 2.71.82 1.82 0.97 1.6SUMER FeXVIII 974.866.86 6.5 4.83.18 3.18 2.172.77 SUMER FeXIX 1118.06 6.95 0.5 < 0.110.760.76 0.22 0.22SUMER FeXXIII263.7657.16<3<3<3 <3 <3EIS SUMER and EIS lines list and total fluxes (I) for each mask (from A to E) used for the analysis of Sec. <ref>. The fluxes are in [erg/cm^2/s/sr]. The theoretical position of the line and the temperature of maximum formation are given in columns two and three. (< ): upper limit imposed as the line is not visible; (sbl): self-blended line. § PLASMA DIAGNOSTICS METHODSWe summarise the plasma diagnostics that are used in this work, while further details can be found in <cit.>. For the EUV-UV optically thin lines the total intensity is given by I(λ)= 1/4π∫_lAb G(T_e, n_e) n_e n_H dlwhere l is the line of sight through the emitting plasma,Ab isthe abundance of the element with respect to hydrogen, G(T_e, n_e) is the contribution function which contains all the atomic physicsparameters, n_e and T_e are the electron number density and temperature and n_H is the hydrogen number density.For the thermal analysis we need to know the electron density distribution in the AR.One method of estimating the electron density is given by calculating the ratio of two line intensities from the same ion, where at least one involves a metastable level m. This make the ratio density dependent, assuming the temperature of the maximum of the ionization fraction of the ion. The density is then inferred by matching the ratio derived from observations with thetheoretical value calculated at different densities. The results of this analysis are presented in App. <ref>. To estimate the distribution of the plasma emission measure with the temperature, there are various methods which differ in the approximations applied.We introduce the column emission measure<cit.> along the line of sight l, asEM = ∫_l n_e n_H dl If we assume that the emitting plasma along l is isothermal at T_c andthe electron density known (n_c), using Eq. <ref> we can write EM (T_c) = 4π I(λ)/G(T_c, n_c) If the density is unknown, a first approximation test on the temperature distribution of the plasma is given by plotting EM(T_e) as function of T_e, for a set of lines formed in a wide range of temperatures. This is called the emission measure loci approach. A necessary condition for the plasma to be isothermal is that these curves cross at the plasma temperature. However, this plot gives only an indication of the plasma distribution as the uncertainties on the measures and data inversion can introduce additional solutions to the inversion <cit.>. When a set of lines from optically thin plasma formed at different temperatures is available, the differential emission measure (DEM) inversion is more appropriate to probe the multi-thermal plasma.The DEM is proportional to n_e n_H (variable with T_e) in the temperature intervals dT_e and it is defined here as (T_e) = n_e n_H dl/dT_eWe also introduce the effective temperature, which is the DEM-weighted average:T_eff = ∫ DEM(T) × T dT/∫ DEM(T) dT.The thermal analysis using these diagnostics is presented in Sec. <ref>. For the diagnostic analysis in this work we used the CHIANTI v.8 atomic database and software <cit.> to calculate the theoretical emissivities of the spectral lines. A particular case was the treatment ofCa XIV 943.59 Å, which will be discussed in Sec <ref>. We adopted the CHIANTI default ionization equilibrium, and <cit.> elemental abundances. This latter choice was made after several tests we did changing element composition in the EM loci approach and DEM inversions. Even if the effect of composition on the result is small, we found <cit.> data to produce the better observed versus theoretical fluxes ratio. We adopted a density of 10^9 cm^-3 for the SUMER slit in position 1 and 4 ×10^8 cm^-3 for position 2 derived from the analysis presented in Appendix <ref>. § TEMPORAL VARIABILITY AND SUB-DATA SELECTION We first made an investigation of the spatial and temporal properties of the AR along the SUMER slitin the hot lines. This has been useful to the selection of the region of interest for detailed investigation. We mostly used SUMER data, where the signal is strong.§.§ Spatial structuring and temporal variability in hot lines Figure <ref> shows the intensity of the SUMER Fe XVII – XIX lines along the slit and for the sixty exposures on April the 27th, slit position 1. From the top to the bottom, they have been ordered following the time sequence of data acquisition.The plots have been saturated to highlight the fainter structuring of the AR, and the brightest regions in yellow correspond to the flaring areas(around Y = -230).The Fe XVIII sequence shows a very bright area corresponding to the passage of a hot loop accross the slit (this is the one used to co-align SUMER and EIS, see also Figure <ref> which is plotted using a different contrast). At both its sides of the hot loop we can identify faint structures whose flux seems to remain almost constant in time. This suggests that they are not affected much by the flare. Wehighlight them in the Figure <ref> by pairs of vertical lines (see also Table <ref>, masks A and C).In line with our aim ofinvestigatingnon-flaring regions, we selected these two area as candidates to pursue our analysis. We also added a third region which was taken as a sample of the unstructured emission (mask B), which appears faint in all the plots of Figure <ref>.The details of these masks are listed in Table <ref>. Some structuring is still visible in the AR core. A similar analysis has been carried out for the SUMER slit position 2 and is reported in Annex <ref>. Here we identified two other candidate regions for the thermal analysis. The structuring of the AR in hot lines has been extracted by temporally averaging the exposures. Reducing the background noise thus makes it easier to see the fainter regions (section <ref>). We also carried out a temporal analysis of these regions to understand better how much their flux varies with the flaring activity (section <ref>). §.§.§ Spatial structuring Figure <ref> top shows the integrated intensity of the Fe XVIII line profile for the spectra averaged over the sixty exposures, plotted along the SUMER slit in position 1 usingone pixel spatial resolution. The north is on the right of the plot. This figure shows well the structuring of the AR. As in Figure <ref>, the data in the range approximately Y= [-260, -190] are due to the hot loop, while the verticalpairs of dashed lines mark the three areas (masks) selected for the analysis. Figure <ref> bottom shows the same integrated intensities for the SUMER Fe XVII, Fe XIX and Ca X. The latter is used to represent the 1 MK corona. The three Fe lines are observed in three different spectral windows, which means that they are not co-temporal(only Ca X is co-temporal to Fe XVII). Nontheless, the intenseFe XVII and Fe XVIII lines have a similar pattern along the slit,which highlights brighter structures and fainter areas.On the other hand,the Fe XIX intensity outsidethe flaring region is very weak and there is little we can say usingthe intensity of a single pixel. In order to be able to use this line,for each of the selected regions we have spatially averaged the spectra. The three regions are marked by the pairs of vertical dashed lines. Going from the bottom of the slit towards the top, we find mask A:this is one of the faintest and most persistent structures inFe XVII and Fe XVIII. Mask B is located close to the slit centre and outlines a weak area betweentwo bright ones. Mask C is the closest to the limb and contains weak structuring.ccccc The masks along the SUMER slit selected for the analysis.4 3 20ptMask Date Y1 <Y< Y2R_A 2012-04-27 -326,- 306 1007 B 2012-04-27 - 271, - 261 997C 2012-04-27 -181.8, -154.8 958D 2012-04-28-320.5, -310.51075 E 2012-04-28 -268.8, -254 1052F 2012-04-28 -421.2, -401.5 1117.7 G 2012-04-28 -421.2, -352.5 1076 The last two masks are used to test blends (see Sec. <ref>) and perform the SUMER wavelength calibration(described in Appendix <ref>).§.§.§ Temporal variability of the selected regionsTo further investigate the temporal variability within the selected regions, we spatially averaged the spectra and kept some temporal resolution by averaging the spectra over only five exposures (six forFe XIX).For example, Figure <ref> shows the resulting light curves for mask A and mask B. We compared these to light curves of the AIA 94 channel for the same masks using data integrated over 10 mins. The intensity has been corrected for thecooler component contributing to AIA 94 has estimated using the 171 channel, as described in <cit.>.For mask A, the hot component of AIA 94 is almost stable up to the time of the first flare at about 21UT (∼ 420 mins in the plot). This component is mostly comparable to theSUMER Fe XVIII which, however, shows more variability. This line is blended with the Si VIII 974.58 Å which has been removed. However the signal in a single point is weak and it is possiblethat some residuals of the Si line has affected the light curve.Some temporal variability is also visible in the Fe XIX which increasesduring the observations, while Fe XVII seems to be only marginally affected by the flare.We decided to keep this mask for our analysis and average out the temporal variability by using an average spectrum. Also mask B shows some variability, even thoughFe XIX is always absent. This mask is closer to the flaring activity which is clearly visible in the AIA 94 curve. For this mask a signature is also visible in Fe XVII showing a peak delayed with respect to the AIA one. We used this mask by discarding the exposures of Fe XVII which were affected by the flare. In conclusion, this analysis shows that there is some variability in time of the hot lines, but only in one case can we see a clear link to the flaring activity. We discarded the affected exposures.To average out the temporal variability and increase the signal to noise of the data, we pursued the analysis on these masks by averaging the spectra over all the exposures, as shown in Figure <ref> and <ref>. From these figures we see that the selected structures (apart from mask E), stay stable during the whole period of the observations (that is the time delay between observing Fe XVII and Fe XVIII).§ SUMER-EIS CROSS CALIBRATIONThe analysis carried out using integrated fluxes derived from different instruments could introduce inconsistencies due to the absolute radiometric calibrations. We carried out tests using the EM loci method on the Ca and Fe lines and found several issues that we had to solve. We used the data from some of the selected masks to address the problem.§.§ Consistency in the EIS dataIn addition to the pre-flight analysis <cit.>, the absolute radiometric calibration of EIS has been investigated post-launch by <cit.>, GDZ, and <cit.>, NRL. These last two introduce a different correction factor to the pre-flight calibration which, for the period of our observations, can reach a factor of 2.4 (see Figure <ref> and <cit.>).These differences have to be taken into account in the analysis of the data.All three calibrations are available in theSolarSoft.Figure <ref> shows the EM loci for the EIS data of mask A. There is consistency between the two results (GDZ and NRL), even though the use of the GDZ calibration results in larger EMs than in the NRL case, with a slightly lower peak temperature.In the following analysis we have used the GDZ calibration.§.§ Consistency in the SUMER dataFigure <ref> shows the EM loci for the SUMER data for mask A, obtained by using the lines having a formation temperature close to the peak of the emission measure. From this set, only the high temperature lines will be retained for the thermal analysis presented in Section <ref>. The EM peak is around log T = 6.4, which is typical of ARs. Also we note that the CaXIV loci plotted with the solid curved (CHIANTI v.8) is inconsistent with the rest of the curves, being it too high. Having investigated the problem, we found this to be an atomic data issue rather then a wrong choice of element abundances: the CHIANTI v.8 emissivity is not consistent with the observed one.The Ca XIV forbidden line observed by SUMER at 943.58 Å is the strongest line within the ground configuration of this ion, between the ground state 2s^2 2p^3 ^4S^3/2 and the first excited level, the 2s^2 2p^3 ^2D^3/2. The excitation data in CHIANTI v.8 are from <cit.>, and were calculated with the distorted wave (DW) approximation. It is well known that this approximation works very well for strong dipole-allowed transition, but typically underestimates the electron excitation rates of the forbidden lines, especially within the ground configuration. This was confirmed by a recent R-matrix calculation by <cit.>, where significant increases in several transitions rates were reported. The excitation rate for the943.58 Å forbidden transition is about a factor of two higher with the Dong et al. calculations. We have taken theDong et al. excitation rates and built a new CHIANTI model ion to be used for the present analysis. We supplemented these data with A-values from the recent calculations by <cit.>. The ratio with the strongest line, the resonance EUV line at 193.87 Å observed with Hinode EIS,is slightly temperature sensitive, but does not vary much with electron density. At log T[K]=6.4, the ratio between these two lines is a factor of 1.73 higher with the new model ion, which is significant.The loci obtained with this newmodel is plotted as a dashed line in Figure <ref>, where it has become consistent with the ensemble of the curves. §.§ Combining EIS and SUMER data When we plotted the EM locis from the two instruments together for the different masks, we found a systematic difference: at similar temperatures the SUMER emission measures were lower than the EIS ones. For instance, this is the case for two lines from CaXIV. We assumed this discrepancy was due to the SUMER degradation, discussed by <cit.>. We then made tests to calculate a correction factor to be applied to the SUMER fluxes. The correction factor can be obtained using data from an isothermal plasma,by comparing the measured to the theoretical lines ratio predicted by the CHIANTI database at the given temperature. While a detailed discussion on the EM loci analysis will be given in the next section, here we just mention thatthe curves from the selected masks are all similar around the peaklocated between log T = 6.4-6.5. For our lines ratio analysis we selected the temperature obtained in mask B, as it is the onewhere the FeXIX is absent. This means that most ofthe Ca line emission is formed close to the EM peaktemperature. Assuming this plasma temperature, we obtained a SUMERcorrection factor 1.8 using the GDZ EIS calibration (that is consistent with the expecteddegradation of the instrument performances estimated by<cit.>). We also carried out the same analysis using the NRL EIS calibration and found a factor 1.4. For the following thermal analysis we used the GDZ factor.§ THERMAL ANALYSIS This section presents the results from the thermal analysis using different DEM and EM methods. §.§ EM LociFigure <ref>shows the combined EIS and corrected SUMER loci EMsfor the selected masks for,SUMER slit position 1 (top and middle lines) and2 (bottom line).As already pointed out, there are similarities between these plots which all show the bulk of plasma at about 3 MK; which is a well known result (e.i. <cit.>). Additionally, the decrease of the EM is noticeable with the increase in solar height. The difference between them is mainly found in the absence of FeXIX for mask B (in the figure we plotted an upper limit used for the DEM analysis). The cooler emission for this region might be representative of background AR plasma. This first analysis suggests a new result: from above the limb to about 91 Mm and over about 1.5× 10^2 Mm across the AR,the thermal properties above 3 MK are similar almost everywhere. For the region covered by the mask C we can also provide temporal information, because this is the only mask that could be applied to both EIS rasters in position 1. The EM loci plots for this mask are shown in the middle row of Figure <ref>. Only the EIS curves can show differences (for both masks we used the same SUMER data). We see no significant change, so we conclude that the EM loci plots do not show evidence of spatial and temporal variations.§.§ Differential emission measure The DEM vs T curves (see Eq. <ref>) have been obtained with a method based on a simple chi-square minimization. We essentially used a modified version of thexrt_dem_iterative2.pro DEM inversion routine <cit.>in order to have more flexibility in the choice of input parameters. The standard routine, widely used in solar physics and available within SolarSoft, is based onthe robust chi-square fitting routine . The DEM is modelled assuming a spline, with a fixed selection of the nodes. Since it turns out that the DEM solutions are quite sensitive to the choice of nodes,we modified the program to allow for the definition of the number and locationof the spline nodes.We also introduced the option toinput minimum and maximum limits to the DEM spline values, which are passed to . This was found to be particularly useful for constraining the upper limits of the highest temperature values. We used upper limits for the DEM values which provideradiances in the Fe XIXandFe XXIII lines as given in Sec. <ref>.Figure <ref> shows the results from this inversion. The top plot shows the resulting DEM for mask A, overplotted with thepoints at the temperature of the maximum of the G(T), which representthe theoretical vs. the observed intensity ratio multiplied by the DEM value. The bottom plot contains all the masks together. The temperature range for the inversion has been set tolog T[K]=5.6 – 7.2. For all the DEM inversions we selectedspline nodes at log T[K]= 5.6, 5.8, 5.95, 6.2, 6.35, 6.45, 6.55, 6.8, 7.2,which provide relatively good agreement (within 20–30%) betweenpredicted and observed intensities, as shown in Table <ref>. The resulting DEM valuesare relatively smooth. Adding a few extra spline nodes in the 1–3 MK rangecan improve the agreement between observed and predicted intensities, but the DEM would be less smooth (as found for instance by <cit.>). Consistency was found between these results and the EM loci approach, the DEM curves are similar for all the masks. As expected, there is a decrease in amplitudeof the DEM with an increase in the solar height. Some differences are found, but only at high temperatures.The 1-2.5 MKcorona is represented by a double-peaked DEM. The hotter peak is at around 2.5 MK and it is higher that the cooler peak for mask A, C and E. The 3 MK peak of the DEM of the off-limb corona is already known in literature <cit.>. What is most interesting is the plateau above 5 MK due to the observation of the high temperature lines.The DEM values at the main peak areso high that the effective temperature of emission for lines such as FeXVIIand FeXVIIIis about 3 MK, i.e. these lines are mainly formed far away from their peak formation temperatures (5 and 7 MK respectively using theCHIANTI v.8 charge state distributions in ionization equilibrium).This issue is quite typical for active regions, as described e.g. in <cit.> and <cit.>. The peak at 2.5 MK is very well constrained by the FeXVIand CaXIVlines.The fact that the DEMvalues around 3 MK are sufficient to explain the intensity of theFeXVIIand FeXVIII lines means that the DEM above 3 MKhas to drop significantly by several orders of magnitude.However, whenever the FeXIXline is observed, itsintensity requires a plateau in the DEM. A further constrain at higher temperatures comes from the fact thatthe EIS FeXXIII263.75 Å is not observed (we note that weak unidentified lines are present in active region EIS spectra, close to this line). We measured the variation of the EIS background near the line to estimate a3 σ value for the intensity of this line, as described in Sec. <ref>. We put an upper limit at log T[K]= 7.2 accordingly (the reason why the DEM increases slightly above 10 MK). From Table <ref> we see that we have been quite conservative in choosing 3 σ, as the flux ratios between theoretical and measured values are well below 1.The 10 MKDEM is about 10^3 times smaller than the 2.5 MK peak, and even less for the cooler mask B (here the FeXIX is not visible).This presence of very hot plasma has already been identified in some areas of ARs. Here we are able to extend our knowledge: this very hot plasma seems to be in all locations where we have a structured AR, with emission in theFeXVIII line, and it appears also to be persistent with time (at least within our observation time). In fact, the bottom of Figure <ref> tells us that there is a temporal variation of the DEM above the 3 MK between masks C1 and C2. This is the area very close to the limb and probably subjected to more AR variability. However, the 10 MK DEM is not affected. §.§ Emission measure Figure <ref> top-left shows the emission measure (solid-blue line) for mask A obtained by integrating the DEM of Figure <ref> over a temperature bin of size 0.2 (in logarithm scale). In the Figure we also overplotted the EM resulting from 200 inversions by the DEM code, by randomly varying each flux within 20%. These are shown as a light gray cloud of solutions within each temperature bin.The bin size d log T = 0.2 is a good compromise to maximize the temperature resolution and to minimize the spread of the solutions within each temperature bin. The degradation of the solution with a smaller temperature bin is illustrated by the bottom-right plot of this figure.We compared this calculation with the EM derived applying a different method.We used the Monte Carlo Markov Chain (MCMC) differential emission measure algorithm distributed with thePINTofALE <cit.> spectral analysis packagetesting different input parameters. With respect to several other methods, the MCMC method has the advantage of not imposing a predetermined functional form for the solution, and, most importantly, it provides confidence limits on the most probable DEM, thus allowing a determination of the significance of apparent structures that may be found in a typical reconstruction. The algorithm assumes that the uncertainties in the intensities are uncorrelated so that systematic errors in the calibration, which could depend on the wavelength, or in the atomic data, which could vary by ion, are not accounted for. The code has been set to perform 200 explorations (batches) of the parameter space, and 200 Monte Carlo realizations for each exploration.Figure <ref> top-right shows the resulting EM from this method applying the same parameters of the left plot.In a similar way, Figure <ref> shows the MCMC EM for mask B (red line) with the cloud of solutions. The EM obtained from the DEM applying the chi-square minimization method (presented in Sec. <ref>) is overplotted in blue. As seen in Sec. <ref>, the plasma distribution is dominated by a main peak around 3 MK, followed by a drastic drop of the EM at high temperature. Overall, in the interval 6.5 < log T < 7 the EM looks to drop more (by about one order of magnitude) in mask B than in mask A.When we compare the solutions for one mask for the two methods, we see that they are consistent in most temperature bins. More discrepancies are found at low temperature, where the inversion is less constrained, and for the log T = 6.8 bin. In general the high temperature tail drops more rapidly with the first method.As mentioned, to better constrain the solutions we decided to adopt a binsize of log T=0.2, as a smaller one produces solutions with greater dispersion in each temperature bin. This is shown for instance for mask A inFigure <ref> bottom right. We checked the effect of extending the temperature range to higher values for those masks where the FeXXIII was used as upper limit. This is a flare line whose temperature of formation extends beyond 10 MK. This inversion is shown for mask A in Figure <ref> bottom left. We see again that at high temperature the solutions in each bin spans a larger interval.lrcrrrrrr Ratios (R) of predicted vs. observed radiances of the selected lines for the various regions. 9 4 0ptIon λ[Å] log T_ eff[K] R(A) R(B)R(C1)R(C2) R(E) I(D) SiVII275.386.001.01 (1.16) 1.02 (1.15)1.06 1.02 1.06 1.03 FeX (sbl)257.26 6.050.90 (0.54) 0.88 (0.52)0.94 0.75 0.77 0.85 FeXI 188.30 6.121.17 (0.89) 1.19 (0.91)1.09 1.0 1.12 1.17 FeXII 192.396.231.05 (1.13) 1.02 (1.12)1.06 1.0 1.15 1.09 FeXIII202.046.330.74 (0.89) 0.81 (0.92)0.78 0.76 0.58 0.64 FeXIV 264.796.371.03 (1.21) 1.04 (1.10)0.99 0.91 1.26 1.21 FeXV284.166.401.25 (1.54) 1.23 (1.45)1.26 1.06 0.99 1.07FeXVI 262.986.420.84 (0.91)0.80 (0.90)0.84 0.84 1.04 1.0 CaXIV 193.876.450.86 (0.62)0.91 (0.78)0.84 1.11 1.04 1.14 CaXIV 943.596.451.04 (0.80) 0.97 (0.87)0.87 0.81 0.91 0.93 FeXVII 1153.166.460.84 (0.66) 0.91 (0.82)0.88 0.83 0.91 0.70 FeXVIII 974.866.501.16 (0.87) 1.14 (0.86)1.17 1.32 1.07 1.09FeXIX1118.066.970.97 (1.03)0.54(0.98)∗ 0.91 0.66 0.97 0.97 FeXXIII∗ 263.7656.950.15 (0.02) -0.25 0.160.10.1In the first two columns we list the dominant contribution to theobserved spectral lines. Note that the FeX257.26 Å is a self-blend of two transitions. As an indication of where the lines mainly form, we list in column 3 the log T_ eff[K] values for the mask A. Values in parentheses are the ratios obtained from the MCMC program using a temperature bin of log T=0.2 and a maximum temperature of log T= 7.2.The ∗ marks the imposed upper limit.§.§ High temperature tail of the EMThe different thermal analysis methods presented here converge in finding the well know peak of emission measure at around 3 MK. In addition, the long integration time and the low noise level of SUMER have revealed the persistent presence of a small amount of very hot plasma. The DEM is very similar everywhere suggesting common heating process at work.The amount of very hot plasma has been quantified with an EM which reaches at maximum about 0.1% at 10 MK of the main EM peak value.Such a small ratio was previously found in on disk and limb quiescent ARs using soft <cit.> and hard X-ray <cit.> data. This ratio isalso consistent with <cit.> findings on the averaged EM in a pre-flaring area using the HINODE/XRT hard filter ratio. As presented in Sec. <ref>, a few other X-ray measurements found about only two orders of magnitude variation of the EM in the 3-10 MK <cit.>. To our knowledge this is the first time the EM above 5 MK has been quantified over three orders of magnitude in an off-limb AR observation by using measured lines profile from EUV spectroscopic data.The spatial distribution in an AR of FeXIX from on disk observations was reported by <cit.> using the EUNIS-13 sounding rocket. They found a FeXIX /FeXII emission measure ratios (assuming a temperature formation of ≈ 8.9 MK and ≈ 1.6 MK, respectively) of ≈ 0.59 in the AR core, ≈ 0.076 in the outer part, while they established a limit of 0.0081 in a quiescent area. For our data this ratio is below 0.005, depending on the mask. Our upper limit is found for mask C1 (from a well detected FeXIX line) and it is close to their upper limit for FeXIX. Our lower values for this ratio are probably due to a different line-of-sight integration path, suggesting that the most intense hot plasma isconcentrated lower down in the corona. We also made a linear fit to the logarithm of the EM to establish the power lawindex above the EM peak, which can be compared to other published results. We have to remember that other results suggest a different EM profile at these temperatures with possibly a secondary small peak around 10 MK <cit.>.For the fitting we set the temperature range between 2.5 and 10MK.Table <ref> summarizes our results and lists other finding from the literature.For our work we used data from mask A and B, as representative of our dataset, as we can see from Figure <ref>. Table <ref> lists quite different values of the power law index. Contrary to mask A, our mask B results are quite uncertain due to the different solutions found using the two inversion methods. As a general trend, we have the impression that the off-disk slopes are shallower than on disk and limb data. However, several elements linked to the inversion method could affect this, such as: the inversion method constraints, the temperature bin, the temperature range, the upper limit of the data. Some of these effects have been shown in Figure <ref>. For instance, if we were using a smaller temperature bin, our EM solutions would spread more in a way to give steeper slopes.We have also discussed the differences between the NRL and GDZ EIS flux calibrations. The use of the NRL calibration would probably produce a lowerEM peak value at higher temperatures. The EM slope could be different. This is because the two calibrations do not have the same wavelength dependence (see Figure <ref> and Figure <ref>). The EM peak is defined by the FeXV, FeXVI and CaXIV which fall in different parts of the spectra. From the physical point of view, in the case of the same heating mechanism being common to all ARs,it is also possible that changes may occur with the age of the AR, affecting the EM shape in time <cit.>.This is a very interesting topic that needs further investigation.Different heating mechanisms acting on ARs should leave, eventually, different signatures in the DEM. For instance, for thoseresults suggestinga secondary peak around 8-10 MK, this could be explained by a secondary population of flare-like events with different initial energy and frequency from the one populating the bulk of the DEM <cit.>.However, the uncertainties on the measures are very high and a clear statement cannot be given yet. Finally, we have to remember that all these measures are taken through different line of sights integration, which results in weighted emission measure information. In our case we have seen that the <1 MK (the Ca X in Figure <ref>) plasma has a different morphology than the hot plasma. Certainly we are crossing different bundles of loops along the line of sight. And it is possible that we are in presence of independent populations of plasma, heated through a different process, one of which maintains the plasma at high temperature. In the absence of further simulation tests, we leave this option open.With our results we think to have provided further important observational constraints on AR heating. The hottest plasma is probably concentrated in the low lying part of the AR core, however its presence in small amount in the upper part of loops suggests a continuous energy injection also at these heights.Considering that the coolingtimescales of a 10MK plasma in equilibrium conditions is of the order of minutes, the temporal persistence of such temperaturesalso imposes constraints on the way each spatial area (our masks) is heated. In the nanoflares scenario, the frequency of heating in the area covered by each of our masks should be higher than such a timescale. lrccccc List of the recent inferred power law index of high temperature EM for ARs. 7 5 0ptDataΔT [MK] α dlogT [MK] limit [MK]Location Type/method Mask A (mpfit) 2 - 10 -4.70.2 14.4 ∗ off-limb EUV spectraMask A (mcmc)2 - 10-4.4 0.2 14.4 ∗off-limb EUV spectraMask B (mpfit) 2 - 10 -8.50.29 ∗off-limb EUV spectraMask B (mcmc)2 - 10 -5.30.29 ∗off-limb EUV spectraNuSTAR 5 - 12< -8 0.112 ∗ disk, limb Soft X-ray imaging PA 3 - 10 -5.40.1 diskSoft X-rayimaging GDZ3 - 10 -14 0.1 10disk Soft X-ray spectraHW 4 - 10 -(6.1,10.3)0.05 8 diskEUV spectra, imagingThe second column gives the temperature range used to fit the EM, the third one lists the fitted index (α) of the power law, the fourth column gives the size (in logarithm scale) of the temperature bin. For spectroscopic data, the fifth column liststhe formation temperature of the hottest linefalling in the observed waveband. If the line is not observed and anupper limit is used for the EM, this is marked by∗.For theNuSTAR data, this was the imposed temperature limit.PA: <cit.>, GDZ: <cit.>, HW:<cit.> over 15 ARs. <cit.> reports slightly steeper slopes with the new NRL calibration. § SUMMARY AND CONCLUSIONSIn this paper we have described the analysis of off-limb observations of AR 11459 observed on the 27th and 28th of April 2012 with both the SUMER and EIS spectrometers. This, to our knowledge, is the first study addressing the thermal analysis of off-limb observations of an AR with spectroscopic constraints up to 10 MK, given by the observation of the FeXIX spectral line.After preparation of the data, we have provided the spatial distribution of the hotlines emitted above 3 MK along the SUMER slit. We also investigated thelight curves for selected areas.One initial result is that the intensity distribution along the slit of CaX (log T_max = 5.9) does not follow the spatial distribution of hot lines (FeXVII in this case) observed at the same time. This suggests we are looking to different thermal structures along the line of sight. As we are sensitive to the issue of flare contamination in our data, we also checked this aspect. <cit.> investigated the effect of a flare on the neighboring loops.That flare had a similar intensity (class C) to those arising during our observations. <cit.> found the flare to have a little effect on theEM time evolution of the individual neighboring loops, even though these had a footpoint location in common with the flaring loops. This seems not to be the situation in our case, which reassures us about the influence on the EM of the small flares happening during our observations.To cross check, we analyzed the light curves of the hot lines and we excluded those datasets that may have been affected.We then selected different masks, and we spatially and temporally averaged the data in each of them to increase the signal to noise ratio and minimize any temporal change. In particular, our FeXIX could be measured only after these procedures, suggesting that there was only a small amount of very hot plasma. We performed EM and DEM analyses, optimizing the inversion parameters to minimize the spread of the solutions. We found consistent results with previous work: we confirm a small hot component up to 10 MK. In addition we were able to further extend our knowledge of this very hot component.In conclusion we can summarize our results as follows: * Very hot plasma (above 3 MK) is present and persistent almost everywhere in the off-limb observations of the AR.In particular, we measure this up to at least 9.1 Mm above the limb and for 1.5 × 10^2 Mm across the AR.* Apart from a cooler region (mask B), we found very similar DEM distributions for the different masks.In the hottest regions we found an EM of about 0.1% at 10 MK with respect to the bulk of the plasma at 3 MK . In spite of a factor of about 2 difference in the peak of the EM (constrained by the EIS long wavelengths lines) between the GDZ and NRL EIS radiometric calibrations, this main result still stands. We stress that these measurements were possible only using spatially and temporally averageddeep exposures.* It is interesting to see the consistency of the results for FeXVII - XVIII - XIX listed in Table <ref>, considering that the SUMER observations lasted over 17 hours. In our analysis this implies that the above ratio, on average, does not change with time. Our data do not allow us to say anything further concerning shorter time scales. * The similaritywithin the AR in the thermal properties of the different masks is accompanied to both the presence (masks E) or not (masks A-D) of spatial changes observed in the hot lines during few hours time.* The detection of a persistent FeXIXline in one of the analyzed regions determined a shallower trend in the hot side of the EM distribution. We fitted the high temperature tail of the EM with a single power law and found a power law index between -4 and -5 in that region (mask A), depending on the inversion method used. This is less steep than other values found previously, but we found a steeper trend in other regions (between -5 and -9). Although this puts constraints on the possible presence of an impulsive heating component, the resolution is not good enough to ascertain whether the shallow trend is really monotonic or we might have another minor peak around 10^7 K, so this aspect deserves further investigation. We also note that the EIS NRL and GDZ radiometric calibrations can result in somewhat different values for the slopes. * We provide a new CaXIV 943.59 Å atomic model which replaces that included in the CHIANTI v. 8. This new data increases the intensity of the line and gives consistent EM results compared to otherlines formed at similar temperatures. * We provide SUMER-EIS cross calibration for the period of our observations. We found the SUMER intensities to be a factor 1.8 lower the EIS one when using the GDZ calibration, and 1.4 when using the NRL calibration. Previous work using data and calibrations from different periods found factors 1.5 and 1.2 <cit.>, suggesting that the relative calibration of the two instruments has not changed much with time. We conclude by suggesting that we have reached the limit of the EUV diagnostics possibilities for this topic. We encourage spatially resolved X-ray and multi-waveband systematic observations as the next step. New high sensitivity X-ray spectroscopic instruments should also be proposed for the next generation of missions. Our analysis has been carried out assuming ionization equilibrium, as suggested by the relatively high electron density derived by our data. However, further investigation on this topic will be considered in our future work.SP would like to thank H.P. Warren for the helpful discussion and for the tests he provided on the EIS radiometric calibration. SP also acknowledges the support of P. Young for the use of the EIS software and the Ca XIV deblending method. The authors thanks P. Lemaire for providing the SUMER point spread function. We would like also to acknowledge the SUMER, Hinode planners and instruments teams for the extremely valuable support in the preparation and planning of HOP 211. SP acknowledges the funding by CNES through the MEDOC data and operations centre. This work used data provided by the MEDOC data and operations centre (CNES / CNRS / Univ. Paris-Sud), http://medoc.ias.u-psud.fr/. GDZ and HEM acknowledge support from STFC (UK). CHIANTI is a collaborative project involving George Mason University, the University of Michigan (USA) and the University of Cambridge (UK). Hinode is a Japanese mission developed and launched by ISAS/JAXA, with NAOJ as domestic partner and NASA and STFC (UK) as international partners. It is operated by these agencies in co-operation with ESA and NSC (Norway). SOHO is a mission of international cooperation between ESA and NASA. Courtesy of NASA/SDO and the AIA, EVE, and HMI science teams.SolarSoft <cit.>, PINTofALE <cit.>, IDL§ COALIGNMENTFor the first slit position, we found that the best way to proceed was to use the temporal sequence of SUMER exposures which shows a strong emission in Fe XVIII, corresponding to the passage of a post-flare loop across the slit, as shown in Figure <ref>. The SUMER sub-time and sub-spatial sequence selected is shown in Figure <ref> bottom (the solar limb is on the rightside of the image). The SUMER slit is not aligned to the AIA image columns, as the SOHO spacescraft was rolled with respect to SDO. Moreover the spatial resolution of the two instruments is not the same. We took into account of these elements and proceeded to follow the steps detailed here. We first selected an AIA data cube co-temporal to the time series of SUMER. The data cube was rotated by the SOHO roll amount in order to have the AIA image's columns aligned N-S with the SUMER slit, we then selected a subfield and we spatially binned to the SUMER spatial pixel. We then had AIA subfield images comparable to the SUMER exposures. At a given time, each spatial column of the AIA-subfieldwas cross-correlated with the corresponding SUMER exposure. The best solution for the coalignment is given by the AIA column which maximizes the correlation in time and space, as shown in Figure <ref> top.To co-align EIS with AIA we chose to use the AIA 195 and EIS Fe XII 192.394 Å, as there is noFe XVIII in EIS and the Fe XVII lines are too faint to be used. We proceeded by selecting a cube of AIA images co-temporal to the EIS raster sub-field close to the limb (see Figure <ref>). We built an AIA raster cube, each column made of data taken at the equivalent EIS exposure time. The other two dimensions were filled by columns of the same imagetaken across a spatial lag of about 10 pixels, with steps of one. This method allowed us to extract the best correlation for the whole sub-raster in space and time. The AIA subfield which gave the best coalignment result is plotted in Figure <ref>. § SUMER STRAY LIGHTDue to our interest in the hot emission, we estimated the stray light by using the AIA 94 channel data convolved with the SUMER point spread function.The reference image was the integration of two AIA 94 images taken on the 27 April 2012 at 21:42 UT. This resulting image, binned to the SUMER pixel size, was convolved with the instrumental profile (P. Lemaire, private communicaton) as shown in Figure <ref> left. On the right plot of Figure<ref> we show the horizontal cut (marked by the white line) on this image, scaled to the original AIA image, which crosses the active region and the SUMER field of view. We see that the off-limb stray light is only few percent of the original flux. We also tested stray light in cool lines. Our data also contain the O I 1152.15 Å which is a good stray-light marker close to the limb. We integrated over the 60 exposures and extracted the line intensity along the SUMER slit position 2 in those pixels where the SNR is above 10. Assuming that the observed intensity is only due to scattered light in the instrument, we calculated its ratio to the solar value <cit.>, and found that it is always below 3%. § SUMER WAVELENGTH CALIBRATIONSUMER spectra are not wavelength calibrated and the grating dispersion is wavelength dependent.Ideally, to perform the calibrationwe need to have reference lines profiles (that are emitted by static features) along the whole waveband.The process is quite straightforward once we have observations targeting the quiet Sun: the SUMER wavebands include a long list of chromospheric lines, including neutrals or singly ionized ions, for which we assume a mean zero velocity (see details of this calibration in <cit.>). This is not the case when we are dealing with off-limb observations in active regions, where the coronal emission dominates, and plasma flows are more common. We tried to minimize the effects of possible local flows applying the following two steps. We used the data from the slit positioned further out in the corona (position 2, maskG), averaging the spectra over seventy pixelsin the southern part of the slit (that is the area at the greatest distance and in the AR periphery). To produce an average effect which minimizes the detection of flows, we chose as reference lines all the bright ones, independently of their formation temperature. Because of the wavelength dependence of the dispersion, we performed an independent calibration in our three spectral windows. We proceeded by measuring the position of our spectral lines in the pixel dimension on the detector, and used a linear relation and reference positions to convert it in the wavelength space.§ ELECTRON DENSITY ALONG THE SUMER SLIT We derived the electron density maps from the EIS rasters using the electron density diagnostics of lines ratio described in Section <ref>. We used Fe XIII (203.82+203.8)/202.04 Å (log T = 6.25), which is sensitive in the range N>10^8 cm^-3. Figure <ref> shows the densitymap from the raster starting at 20:24 UT on April the 27th (SUMER slit position 1). The density along the SUMER slit derived from this map is shown in the central panel of Figure <ref>. For SUMER slit position 1 we also have an earlier EIS map and the density profile along the SUMER slit is shown on the left of Figure <ref>. The right plot in this figure gives the density derived along the SUMER slit for slit position 2.The two profiles on SUMER position 1 reflect the changing of the structuring of the AR at this temperature about two hours apart. We can identify the same main features with a change in the relative density amplitude, which is most evident in the AR core (-250 <Y<-200). This area is the one occupied by the flaring region and by the hot loop visible in the first raster (Figure <ref>). Note how little these profiles resemble the intensity profiles along the slit of the hot lines, suggesting we are observing different thermal structures along the line of sight.Figure <ref> right shows a more important drop in density as Y decreases, due both the increase height above the active region and the increasing distance from the AR core.The density values found here have been used for the thermal analysis. § SUMER TEMPORAL VARIABILITY FOR SLIT POSITION 2Figure <ref> left shows the temporal variation of theFe XVIII along the SUMER slit in position 2.We can notice a general temporal variation of the structures which tend to spatially spread along the slit as the timeline increases. We selected the most stable parts highlighted by the yellow lines (masks D, E). Figure <ref> right shows the temporal averaged integrated flux of Fe XVII and Fe XVIII, which were taken six hours apart. There is a temporal variation mostly in the core of the AR. We adapted our masks to these changes.As shown in Figure <ref> there is no thermal variation within at these locations.§ EIS RADIOMETRIC CALIBRATION Figure <ref> shows the correction factor to the pre-flight radiometric calibration of EIS for the two channels, as determined by <cit.> (solid line) and <cit.> (dashed line) for the period of our observations. | http://arxiv.org/abs/1707.08445v1 | {
"authors": [
"Susanna Parenti",
"Giulio del Zanna",
"Antonino Petralia",
"Fabio Reale",
"Luca Teriaca",
"Paola Testa",
"Helen E. Mason"
],
"categories": [
"astro-ph.SR"
],
"primary_category": "astro-ph.SR",
"published": "20170726135417",
"title": "Spectroscopy of very hot plasma in non-flaring parts of a solar limb active region: spatial and temporal properties"
} |
firstpage–lastpage 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 Received 3 July 2017 / Accepted 20 July2017============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================ We present a principal component analysis method which tracks and compensates for short-timescale variability in pulsar profiles, with a goal of improving pulsar timing precision. We couple this with a fast likelihood technique for determining pulse time of arrival, marginalizing over the principal component amplitudes. This allows accurate estimation of timing errors in the presence of pulsar variability.We apply the algorithm to the slow pulsar PSR J2139+0040 using an archived set of untargeted raster-scan observations at arbitrary epochs across four years, obtaining an improved timing solution. The method permits accurate pulsar timing in data sets with short contiguous on-source observations, opening opportunities for commensality between pulsar timing and mapping surveys. pulsars: general – pulsars: individual: J2139+0040 – methods: data analysis § INTRODUCTION While there are thought to be 100,000 pulsars in the Milky Way <cit.> only about 2600 have been cataloged <cit.>. A new generation of drift-scan interferometric telescopes, including CHIME <cit.>, HIRAX <cit.> and HERA <cit.>, will soon begin recording data with a primary goal of creating cosmological 21-cm intensity maps.In addition, there are plans to use MeerKAT <cit.> to create intensity maps using a raster scan mode.Such instruments could be used commensally to search for pulsars, and these instruments will have the collecting area, multi-beam capability, and on-sky integration time to substantially increase the pulsar discovery rate. The pulsar search will likely have to work within the observational parameters of the hydrogen surveys. For the drift-scan telescopes, the duration of a contiguous observation will be limited to the time sources take to drift through a beam, which for CHIME will be a few minutes. MeerKAT, with its raster scan observing mode will make shorter duration passes but will pass near a pulsar's coordinates more often. Integration time can be accumulated by combining multiple passes. Such a coherent pulsar search in time-disjointed data has been demonstrated in <cit.>. To further develop and test time-disjoint pulsar search algorithms, we used an existing hydrogen-mapping data set, the Green Bank Hydrogen Intensity Mapping (GBTIM) survey <cit.>. This is the data set in which fast radio burst FRB 110523 was discovered <cit.>. The set consist of raster scans, so it contains a time-disjointed set of near-passes of pulsar coordinates.Our long-term goal is to search for new pulsars, but first we need to test our algorithms and code on known examples, so we obtained a timing solution for PSR J2139+0040. The pulsar was discovered by <cit.> at right ascension 21:39:42(16), declination +00:36(5), period 0.312470(3) s, and dispersion measure of 36(7) pc/cm^3. The pulsar is bright—we observe an average flux at 800 MHz of approximately 50 mJy—such that single pulses are readily detectable with the Green Bank Telescope. Our team began studying this pulsar because it lies within two degrees of FRB 110523 and could be used to constrain the Galactic component of the scintillation properties for this region of the sky.Despite the sporadic scan-mode observations in the data set, we show that a timing solution for PSR J2139+0040 can be obtained, and we present improved timing parameters. We find timing residuals are reduced by use of a novel principal component analysis (PCA) technique to fit the time-variable pulse waveforms. The PCA technique automatically compensates for the rapid mode switching of the pulse shape, allowing a more precise timing solution. We describe this technique in detail since it may be more widely useful in pulsar timing. § DATA AND PROCESSING Here we describe our data and observations as well as the processing of the data into folded pulse profiles. §.§ ObservationsThe data set we used came from an 21-cm intensity mapping survey, in which we raster-scanned four different Wiggle-Z (WZ) fields (1hr, 11hr, 15hr, and 22hr)<cit.> recording spectra from 700-900 MHz with 4096 spectral frequency channels, using integration intervals of 1.024 ms. The full survey is comprised of 680 hours of observations between 2011 and 2015. PSR J2139+0040 is located in the 22hr field, which was observed as part of GBT projects 10B-336 and 14B-339. Scans of this field have the beam crossing close to the pulsar often, but for short periods of time, with the beam typically moving across the sky at a rate of several degrees per minute. The set consists of sporadically-spaced observing sessions of several hours. This provides a sample of pulsar data with a wide range of crossing angles and with baseline drifts in the data due to variable ground spill and sky brightness structure. The cadence of the observing epochs is variable. It is therefore challenging to extract a precise pulsar timing solution from the set, providing a test of our processing techniques.Our WiggleZ data are stored in blocks of 2048 time samples (PSRFITS records <cit.>). From our bulk data, we select all such blocks where at the block midpoint the telescope boresight is within 0.15^∘ from the published pulsar position. This corresponds roughly to a half width at half maximum of the telescope beam at 800 MHz. The resulting data set includes 1975 seconds of integration time, which we term the WZ data set.In addition to the raster scan data described above, we obtained a single pointed observation of PSR J2139+0040 with a duration of one hour on MJD 57178. The frequency range of the pointed data is 720-920 MHz with 2048 spectral frequency channels, and an integration interval 8.192e-5 s, grouped into records of 4096 samples. We dub this the pointed data set.Using the PRESTO[<http://www.cv.nrao.edu/ sransom/presto/>] software package on the pointed data set yields DM = 31.7262 pc cm^-3, which we use for all subsequent analysis except Section <ref> where we perform a full fit for the DM. §.§ PreprocessingData pre-processing produces an initial calibration of the data, removes flux froma noise injection source, mitigates radio-frequency interference (RFI), long-time-scale noise, and other sky signals such as point sources moving through the beam.To remove the system bandpass response function, we divided the recorded intensity by its time mean, converting to units of the system temperature. To mitigate RFI and long-time-scale signals we apply a stack of filters to the data consisting of: 1. deleting frequencies where the time variance is anomalously high, 2. subtracting the time mean from the data, 3. subtracting the time-linear component from the data, 4. deleting time samples where the frequency mean is outlying, 5. subtracting the frequency mean from the data, and finally repeating step 1. We use 5σ for all thresholds.The above filters result in a zero-mean artifact in the folded profiles where the pulse profile is pushed negative just outside the main pulse. This is visible in the middle panel of Figure <ref>. However, this artifact is a result of linear operations on the data and as such does not bias our subsequent timing analysis. §.§ Barycentric Time and Folding We choose to time the pulsar in the Barycentric coordinate system because we are working toward searching such data sets for weaker pulsars. This step is not critical to the mode-tracking technique presented in Section <ref>, but we describe it for completeness. We converted time stamps from Universal Time to Barycentric Time, τ(t, α, δ), defined as the time a signal from a far-off source at right ascension α and declination δ arriving at the observatory (here GBT) at time t, would arrive at the solar system barycenter.We perform this conversion using the TEMPO[<http://tempo.sourceforge.net/>] software package as invoked by the bary command in PRESTO. For the right ascension and declination we initially use the previously published location of the pulsar, which we later refine. As described in Section <ref>, refining this position requires the partial derivatives of the Barycentric times with respect to right ascension and declination, which we calculate by finite difference.To form pulse profiles, we group our sporadically spaced WZ data into groups spanning five minutes or less. These are typically very sparsely sampled with most groups containing only a few seconds of on-target data. The five-minute duration was selected to reduce the set to a manageable size of fixed cadence, with reasonable signal-to-noise for each folded waveform, and to ensure little worry of phase drift within a group. The pointed data, in contrast, is contiguously sampled and we divide it into 640 groups of duration 5.369 s, which we use to characterize the time variability, and later stack into 40 pulse profiles of duration 86 s used for timing.We fold the time stream on the pulsar period 0.312470 s <cit.>. We use 200 phase bins for the WZ data, and 800 phase bins for pointed data.We dedisperse the folded data, and then average it over frequency yielding pulse profiles as shown in Figure <ref>. This yields 289 such profiles although a number of these are noise dominated and show no evidence of the pulsar. We discard these, resulting in 232 pulse profiles in the WZ data set. Combined with the 40 profiles from the pointed data set, there are a total of 272 pulse profiles used for the timing analysis. Figure <ref> below shows how these profiles are distributed in time.§ ANALYSIS AND RESULTS Here we describe the analysis of the folded pulse profiles. Our analysis accounts for the pulsar's short-time-scale variability, estimates the pulsar phase, and fits a timing solution. §.§ Principal component analysis of pulse shapes Examining the set of folded pulse profiles, we find substantial variation in shape. In particular, some profiles have a stronger first peak, while others have a stronger second peak as shown in Figure <ref>. This is the dominant mode of time variability for the profiles, with the preference for one peak over the other appearing to be correlated over timescales of several minutes. This is thus a form of mode switching, which is common for a substantial fraction of pulsars on a range of time scales <cit.>. To find the best-fit arrival time for each folded waveform, one could simply fit each waveform to an average profile, but waveform shape variations would then cause timing errors, since the fit would favor earlier arrival times when the first peak is stronger. To address this, we use a principal component analysis of the pulse profiles (PCA), which automatically tracks the pulsar mode. This PCA Mode Tracking compensates for mode-switching waveform-shape variation.We use the 640 pulse profiles from the pointed data (each containing 5.369 s of integration) to construct the PCA. Using Fourier techniques to achieve sub-bin alignment, we align these profiles according to a preliminary, linear timing solution valid for the pointed data. We create a matrix of 640 pulse profiles called d_E i where E is the epoch, an index of the mid-point time of each folding interval, and i is the 800-point phase bin within the waveform. We carry out a Singular Value Decomposition:d_Ei =∑_m U_EmS_mmV_mi, where for each mode m: U_Em is the eigenfunction in epoch E, V_mi the eigenfunction in phase i, and S_mm is the singular value. We show V_mi for the first ten m modes in Figure <ref>. Inspecting the phase eigenfunctions, the first mode V_0 i appears similar to the average waveform. We tested this equivalence by averaging the 640 pulse profiles into single template. We compared this average template with the mode V_0 i of our PCA technique, and found the two consistent within the noise.The second mode V_1i allows the two peaks in the waveform to depart in amplitude from the average. Essentially, U_E1 tracks the variation with epoch of the waveform mode, changing sign when the mode switches.Since we intend to use the V modes as templates for fitting pulsar phases, a concern is that noise in these modes will bias the phase measurements. To reduce this noise for the modes above the first, we set them to zero outside of the main pulse (all phase bins except the central 160 of 800) since we see no evidence for pulsar flux outside this region. For the primary mode, the filtering artifact mentioned in Section <ref> results in a smooth structure outside this region and so we spline the profile outside the central 160 phase bins.§.§ Pulsar time of arrival estimation from direct integration of the likelihoodThe natural next step would be to simultaneously fit mode amplitudes and a pulse phase to each of our individual pulse profiles, using V_mi as templates. However, we find that for this high-dimensional parameter space, the likelihood often has multiple maxima, and so the traditional least-squares fitting method fails. The multi-modal nature of the parameter space is due to the fact that for some profiles, there are multiple combinations of the V_mi modes that, with different phases, may adequately describe the profiles such as those shown in Figure <ref>. To deal with this, we employ a new pulse profile fitting technique that fully samples the parameter space.We denote an individual measured folded pulsar profile d_i where the index i runs over the phase bins (we suppress the index E for epoch in this section, since the analysis is performed independently for each profile). We model the measured profile asd_i = ∑_m A_m V_mi(ϕ) + n_i,where V_mi (ϕ) are the profile template modes (with m running over mode number), A_m is the mode amplitude, ϕ is the finely adjusted pulsar phase, and n_i is the noise contribution. The dependence of V_mi on ϕ describes the rotation of the templates to match the data. For notational brevity, we now switch to vector notation, where the above equation becomes:d =A^TV(ϕ) +n.We assume Gaussian noise that is uniform and uncorrelated: ⟨ n_i n_j ⟩ = δ_ijσ^2. The template modes have already been measured so the parameters of the model are ϕ and the amplitudes, A.Of these parameters we are chiefly interested in ϕ. All the information about this parameter is in p(ϕ| d), the posterior probability distribution of ϕ given the measurements d. For instance, the measurement mean and variance of the phase are the first and second moments of this distribution. This can be calculated from the posterior of all the parameters marginalized over the mode amplitudes:p(ϕ| d) = ∫ d^N Ap(ϕ, A| d),where N is the length of the vector A. For flat priors on the initial parameters, Bayes' Theorem states that the posterior distribution for the parameters is proportional to the likelihood function, p(ϕ, A| d) ∝ p( d | ϕ, A). Since the data is Gaussian-distributed, the latter is given by:p( d | ϕ, A)∝exp[ -1/2χ^2( d, ϕ, A)]χ^2( d, ϕ, A)=1/σ^2∑_i [ d_i - ∑_mA_m V_mi(ϕ) ]^2. Combining the above, we have:p(ϕ| d) ∝∫ d^N A exp{-1/2σ^2∑_i [ d_i - ∑_m A_mV_mi(ϕ) ]^2 }This is a multidimensional integral over the vector space of A, which would be prohibitively expensive to do numerically. A key insight is that since the model is linear in A, the likelihood is Gaussian not only in the data, but in A as well (but not in ϕ). This permits the integral to be done analytically. This same insight was used in <cit.> where a similar integral appears over the frequency dependant template amplitudes in wide-band pulsar timing. However, while in wide-band timing this is a computational convenience, here being able to perform this integral is essential due to the multi-modal nature of the likelihood space. Similarly, the integral appears in <cit.> when marginalizing over epoch dependant profile amplitudes in profile domain timing analysis. When performing the integral, there is a factor that depends on the expression∑_i V_mi(ϕ) V_ni(ϕ).This expression is independent of ϕ because the dependence of V_mi on ϕ is just a shift of the template modes, which disappears after summing over phase bins. Thus, there is no need to carry out this part of the calculation. Note that this is only true if the noise is uniform. The final expression isp(ϕ| d) ∝exp[ -1/2σ^2∑_m ∑_i V_mi(ϕ)d_i ∑_j V_mj(ϕ) d_j, ]or equivalentlyp(ϕ| d)∝exp[ -1/2σ^2( Vd)^TVd ]. which is proportional top(ϕ| d)∝exp{-1/2χ^2[ d, ϕ,Â( d, ϕ)]},where Â( d, ϕ) is the linear-best-fit value for the template amplitudes at fixed ϕ. Linear regression givesÂ( d, ϕ) = [ V(ϕ)V^T(ϕ)]^-1 V(ϕ)d. So succinctly, the procedure for estimating the time of arrival is: * At fixed ϕ perform a linear fit for the mode amplitudes A, which we found in Section <ref>.* Evaluate χ^2 for these parameters and take e^-χ^2/2 as the likelihood. * Repeat for a range of ϕ values, covering the region of low χ^2 that dominates the likelihood. (We repeat this step for a range of -20 to +20 phase bins from the fixed ϕ, with steps of 0.02 phase bins.)* Take the zeroth, first, and second moments to get the normalization, phase estimate, and variance, respectively. In Section <ref>, we use the phase estimate and variance to find the timing solution. While our procedure has been described in the profile space, our actual analysis is performed in the Fourier domain, as in <cit.>. We use only the first through 90th harmonics to limit contamination from noise in the template modes and observed extraneous noise near 300 Hz. We use N=6 template modes in our fits. An example of a profile fit is shown in Figure <ref>.Finally, properly estimating the error on ϕ requires an estimate of the noise power σ^2. We use the value of σ^2 that results in a reduced chi-squared of unity (χ^2/ν, where ν is the number of degrees of freedom) for the best fit (maximum likelihood) parameters. §.§ Timing solutionWe proceed to adjust the parameters of the pulsar timing model:ϕ_E = ϕ_0 + Δ P/P^2(τ_E - τ_0) + Ṗ/2 P^2(τ_E - τ_0)^2 + 1/P[-∂τ_E/∂αΔα - ∂τ_E/∂δΔδ]where, ϕ_E is the barycentric phase of the pulse profile indexed by E, τ_E is its barycentric time, τ_0 is the reference epoch, P is the period of the pulsar used for folding, ∂τ_E/∂α is the derivative of the barycentric time with respect to source right ascension, and ∂τ_E/∂δ is derivative of the barycentric time with respect to source declination. There are also five free parameters in the timing model, including the period derivative Ṗ, period correction Δ P, initial phase offset ϕ_0, right ascension correction Δα, and declination correction Δδ. We proceed to adjust these parameters such that the above equation fits our phase measurements using standard weighted least squares and extract best fit parameters with uncertainties.One challenge to obtaining a timing solution with this data set is the large gap in our data. We have phase measurements for epochs over several months in 2011 and 2015 but none in between. Compared to more uniformly sampled data, this distribution of measurements weakly constrains the total number of pulsar rotations in the gap period. Indeed we find multiple χ^2 minima corresponding to changing the parameter Δ P by integer multiples of ∼ 8× 10^-10 s, which changes the number of rotations in the gap. However, the lowest of these minima is δχ^2=44.5 smaller than the next lowest. Thus this minimum is strongly preferred over the others, and our timing parameters are well constrained.Inspecting the timing residuals, we find one phase measurement that is an extreme, 9σ, outlier. Inspecting the full likelihood curve for the profile phase fit (as described in Section <ref>), we find the likelihood to be very non-Guassian, making such an outlier roughly 0.1% likely. This is not terribly improbable given that we have 272 such data points, and is far more likely than a 9σ outlier for Gaussian data. Since our least-squares fit to the timing solution implicitly assumes Gaussian-distributed phase errors, we exclude this outlying data point in the timing solution fit.The timing residuals are shown with the fitting results in Figure <ref>. The fit has a reduced chi-squared of χ^2/ν=1.26 for 266 degrees of freedom, which is reasonable given the non-Gaussian nature our phase measurements which we have represented with 1-σ error bars. We inflate the uncertainties on the timing parameters by this factor.The parameters for the final timing solution for PSR J2139+0040 are given in Table <ref>.We find new right ascension and declination, period, period derivative, reference time of arrival (derived from ϕ_0), and also provide the epoch used for the timing model. We use the period and period derivative to estimate the magnetic field strength (B), the characteristic age (t_c), and the spin-down luminosity (Ė)<cit.>. To check for evidence of proper motion of the pulsar's sky position, we fix all parameters other than the position, then separately fit the 2011 and 2015 data. We find the two output positions are consistent with each other and the solution using the full data set, and as such we find no evidence for proper motion.We used the TEMPO pulsar timing software to verify our timing solution. For this we converted our pairs of epoch-phase measurements to barycentric times of arrival then inverted PRESTO's bary command to convert to topocentric times of arrival. Feeding these and our timing solution parameters into TEMPO we find agreement in the value of χ^2. Letting TEMPO refit our timing-model parameters we find no significant shifts. §.§ Dispersion Measure We used the pointed data to determine DM. We align the 40 folded profiles in phase based on our timing solution, then stacked them into a single profile.We fit this profile for the dispersion using the first V mode as a template according to: d_if = A(f/800MHz)^βV_0i(ϕ_f) ϕ_f≡ϕ_900 +DM/P4148.808s MHz^2/ pc / cm^3[1/f^2-1/(900MHz)^2]Here, d_if is the frequency-dependant profile data, which are fit with parameters for the amplitude A, power-law slope β, and frequency dependant phase ϕ_f for each frequency f. The frequency dependant phase ϕ_f contains an offset ϕ_900 and the dispersion phase delay. We subsequently calculate the dispersion-measure-based distance using Cordes-Lazio NE2001 Galactic Free Electron Density Model <cit.> website[<https://www.nrl.navy.mil/rsd/RORF/ne2001>] using the new right ascension, declination, and DM.The results of this analysis are given in Table <ref>§.§ Timing improvement available from PCA Mode TrackingTo assess the impact of PCA mode tracking we compared the timing residuals using long and short integrations. We carried out these parallel analyses on sets of 16 individual 5.4 s profiles from the pointed data. First, we stacked these profiles into a single pulse profile with a total integration time of 86 s and fit this profile with the technique described in Section <ref>. Second, we fit the 16 profiles individually, then combined their resulting posterior distributions. We did this for the 40 sets of 16 profiles covering the full hour of pointed data. The first technique is similar to the conventional technique of taking long averages to smooth out pulse-to-pulse variability. The second technique uses fine grained data allowing the PCA algorithm to track and compensate for the variability.We find that using PCA mode tracking and fitting fine-grained 5.4 second averages results in a ∼20% smaller uncertainty on average in the pulse time of arrival (calculated from the second moment of the posterior as described in Section <ref>). While the improvement is substantial for this pulsar, it may not be generic. Pulsars are highly individual: they vary substantially in the degree and time scales mode switching, so we anticipate the improvement available from PCA mode tracking will also vary strongly from case to case. § DISCUSSION AND CONCLUSIONS Pulsar timing observations often make use of profiles averaged for minutes. These long averages smooth out the effect of rapid mode switching. Since it is in general sub-optimal to average over a time-variable signal, we suggest that PCA Mode Tracking could be applied broadly, by intentionally using short averages that display rapid mode switching—which the PCA then follows and compensates for. In particular, it may be useful to test this technique using data from millisecond pulsars, since improvement of timing precision of these pulsars allows more precise tests of general relativity, along with tighter constraints on long-wavelength gravitational wave backgrounds. We have obtained a substantially improved timing solution for PSR J2139+0040, using untargeted observations in conjunction with a ∼1 hour duration targeted observation. Our data are well-described by a simple five-parameter model,including its sky location, reference phase, period, and period derivative. The measured slow-down rate of Ṗ = (7.64±0.13)×10^-18(s s^-1) is well below average, with ∼ 99% of slow pulsars having a higher rate (<cit.>) as shown in Figure <ref>.This indicates that PSR J2139+0040 has an abnormally weak magnetic field.For many of our pulse profiles, we find a highly non-Gaussian likelihood with multiple maxima for the phase. However, by analytically integrating over the template amplitudes, we are able to fully sample the marginalized posterior for phase and calculate the distribution's mean and standard deviation. The subsequent fit to a timing model achieves a reduced chi-squared of 1.26, which is reasonable given the non-Gaussian nature of our phase errors. A more complete treatment would be to use the phase measurement posteriors directly when fitting a timing solution, rather than reducing them to their first and second moments. This would optimally extract the timing information from the phase measurements but would substantially complicate fitting a timing solution, since least-squares would no longer be applicable. Our PCA mode tracking method addresses issues similar to those addressed via"profile-domain pulsar timing" <cit.>, particularly, what these authors refer to as “low-frequency stochasticity” and “phase-correlated stochasticity”. The profile-domain techniques uses pre-defined uncorrelated shapelet components to contribute to the individual profiles. In contrast, the PCA mode tracking method is non-parametric. There is no need to guess the shaplets, the PCA finds them automatically. Compared to the profile-domain strategy we anticipate that fewer components will be needed, yielding less degeneracy with the pulse phase and likely reducing timing error.The PCA mode tracking technique derives its mode templates from the data, and so for weak pulsars these waveforms may have noise that is absent when using the pre-defined templates of the profile domain technique. A PCA technique was employed to characterize pulsar variability in <cit.>, where, rather than fitting for the time of arrival and component amplitudes in high cadence data, the authors used the empirical correlation between the timing residuals and component amplitudes to retroactively correct for systematic timing error. Our results demonstrate the feasibility of timing pulsars commensally with mapping surveys. In particular, upcoming hydrogen intensity mapping surveys using MeerKAT <cit.> and Phase 1 of the Square Kilometre Array <cit.> will map large fractions of the southern sky with thousands of hours of telescope time. The telescopes will operate as a collection of single dishes and will thus need to scan rapidly to overcome 1/f noise. There is the potential to obtain pulsar timing solutions for free (in terms of telescope time) using the data from these surveys in much the same way as we have done here. The challenge is in storing the data at the rapid cadence required for pulsar studies, but, in single dish mode, data volumes are modest compared to interferometric mode. Mapping surveys spend a small fraction of their time pointing at known pulsars, however the mapping survey gets this data at many epochs, for every pulsar in its survey field, providing a large volume of timing data.One of our goals in this work is to enable pulsar searches commensal with intensity mapping experiments. Having obtained a precise timing solution for a known pulsar in the GBTIM data, we have demonstrated that accumulation of pulsar data over five years in few-pulse snippets can be accomplished, although this may be more difficult for weaker or more erratic pulsars. This analysis was possible because the GBTIM data used short 1 ms integrations. The intensity mapping data from CHIME <cit.> and HIRAX <cit.> will use longer integrations for their intensity mapping data, but these instrument will have additional transient-search backend hardware allowing high cadence analysis of the data streams. New search algorithms have recently been proposed by <cit.> to reduce the cost of such long-term pulsar waveform assembly by several orders of magnitude. If these techniques could be employed at upcoming transit survey instruments such as CHIME and HIRAX the pay off could be a substantially increased rate of pulsar discovery. § ACKNOWLEDGEMENTS We thank Kendrick Smith, Ingrid Stairs, Maura McLaughlin and Alexander Roman for valuable discussions.K. W. M. is supported by the Canadian Institute for Theoretical Astrophysics National Fellows program.U.-L. P. acknowledges support from the Natural Sciences and Engineering Research Council of Canada. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development Canadaand by the Province of Ontario through the Ministry of Research, Innovation and Science.J. B. P. acknowledges support from NSF Award 1211777. Computations were performed on the GPC supercomputer at the SciNet HPC Consortium.mnras | http://arxiv.org/abs/1707.08581v3 | {
"authors": [
"Hsiu-Hsien Lin",
"Kiyoshi Masui",
"Ue-Li Pen",
"Jeffrey B. Peterson"
],
"categories": [
"astro-ph.IM",
"astro-ph.HE"
],
"primary_category": "astro-ph.IM",
"published": "20170726180005",
"title": "Improved Pulsar Timing via Principle Component Mode Tracking"
} |
[email protected]@gravity.phys.nagoya-u.ac.jpDepartment of Physics, Graduate School of Science, NagoyaUniversity, Chikusa, Nagoya 464-8602, JapanWe investigate quantum entanglement between two symmetric spatial regions in de Sitter space with the Bunch-Davies vacuum.As a discretized model of the scalar field for numerical simulation, we consider a harmonic chain model. Using the coarse-grained variables for the scalar field, it is shown that the multipartite entanglement on the superhorizon scale exists by checking the monogamy relation for the negativity which quantifies the entanglement between the two regions. Further, we consider the continuous limit of this model without coarse-graining and find that non-zero values of the logarithmic negativity exist even if the distance between two spatial regions is larger than the Hubble horizon scale.04.62.+v, 03.65.Ud Large scale quantum entanglement in de Sitter spacetime Yasusada Nambu May 29, 2018 ======================================================= § INTRODUCTION Quantum entanglement is an interesting aspect of quantum physics, which has recently received remarkable attention from various fields.As is well known in the quantum theory or quantum information theory, quantum entanglement represents a non-local correlation and leads to the violation of the Bell-CHSH inequality <cit.>. This quantum correlation is needed as a resource to carry out some protocols, e.g. quantum teleportation, superdense coding, quantum error correction and so on <cit.>. For quantum many-body systems or quantum field theories, the entanglement of the ground state or vacuum state characterizes the structure of its wave function. In a previous work by S. Marcovitch et al. <cit.>, the quantum entanglement of a free Klein-Gordon field in the 1+1-dimensional Minkowski spacetime was examined in terms of the logarithmic negativity which is a useful measure to quantify the entanglement for a mixed state <cit.>. They focused on two spatially separated regions and numerically investigated the logarithmic negativity for the Minkowski vacuum as a function of the ratio of the separation between the regions to the size of the each region. It was shown that the logarithmic negativity decays exponentially as the ratio becomes large. This behavior is consistent with the Reeh-Schlieder theorem <cit.> which implies that the quantum entanglement persists for all scales for the Minkowski vacuum.The nonlocal quantum correlation also was examined for cosmological situations <cit.>. In a previous work by Y. Nambu <cit.>, the quantum entanglement of a massless scalar field in de Sitter space was investigated using the spatially coarse grained scalar field. It was revealed that the logarithmic negativity for the coarse-grained field vanishes, when the physical size of each region exceeds the Hubble horizon and becomes causally disconnected due to the de Sitter expansion. This behavior of entanglement is consistent with the scenario of quantum to classical transition of the primordial fluctuation generated by inflationary expansion in the early universe <cit.>. The similar transition to zero negativity state is known for a model of 1+1-dimensional harmonic chain with a finite temperature <cit.>. For sufficiently high temperatures, the negativity between two spatial regions becomes zero. Qualitatively, this transition can be understood as follows; above the critical temperature, the wavelength of the thermal fluctuations becomes shorter than the lattice spacing and the quantum correlation between adjacent lattice sites is destroyed by the thermal fluctuations. Concering the quantum field in de Sitter space, the effective comoving lattice spacing for the coarse-grained field becomes larger than the wavelength of quantum fluctuations which is equal to the Hubble length of de Sitter space, and the negativity becomes zero.We expect that appearance of zero negativity state is related to the coarse-graining treatment of the quantum field. As mentioned above, for the Minkowski vacuum in quantum field theory, the Reeh-Schrieder theorem is known and the theorem also holds for thermal states <cit.>. It seems that there is a difference between the feature of entanglement in the coarse-grained field and its continuous limit. Thus for complete understanding the property of the entanglement in de Sitter space, we need to analyze the quantum entanglement between two spatial regions using both a model with coarse-graining and one without coarse-graining. In this paper, we use a 1+1-dimensional lattice model of a massless scalar field whose mode equation is equivalent to that of the 1+3-dimensional de Sitter spacetime and numerically evaluate the entanglement between two spatial regions.As a quantum state, we assume the Bunch-Davies vacuum, which is a vacuum state in de Sitter spacetime and corresponds to the Minkowski vacuum in the far remote past. We introduce coarse-grained variables in the lattice model and the negativity between two spatial regions is calculated.To understand the connection between the coarse-grained system and the continuous one, the effect of multipartite entanglement is considered using the monogamy relation for the negativity. We also compute the logarithmic negativity between the two regions without the coarse-graining and its behavior is discussed especially focusing on the continuous limit and the existence of the Hubble horizon.This paper is organized as follows: we introduce our 1+1-dimensional lattice model of a minimal coupled massless scalar field in de Sitter spacetime and the entanglement measures (the negativity and the logarithmic negativity) for the Gaussian state in Sec. II.In Sec. III, we define the coarse-grained variables in the lattice system and calculate the negativity numerically. Then we discuss the monogamous property of the negativity. Also, we present our main numerical result of the logarithmic negativity without coarse-graining and provide the fitting formula for the super horizon scale. Sec. IV is devoted to summary and conclusion. § HARMONIC CHAIN MODEL AND ENTANGLEMENT MEASURES FORGAUSSIAN STATEThe Hamiltonian for a minimal coupled massless scalar field q in de Sitter spacetime with a spatially flat slice is given by ℋ=∫ d^3 x1/2[p^2+(∂_i q)^2+1/ad a/dη (qp+pq) ],a=-1/Hη, where a is the scale factor,η<0 is the conformal time andH represents the Hubble constant. For simplicity of numerical analysis, we assume that the field depends only on the conformal time η and one spatial coordinate. By introducing a lattice spacing Δ x of the spatial direction, the dimensionless form of the Hamiltonian of this model (harmonic chain) <cit.> is expressed as ℋ=∑_j=1^N[ 1/2p_j^2+q_j^2-α q_j q_j-1+1/2ada/dτ (p_jq_j+q_jp_j) ], where we impose a periodic boundary condition on q_j and p_j to respect the translational invariance of the model. N denotes the total number of lattice sites. τ and α are the dimensionless conformal time and the IR cutoff parameter, respectively. These two parameters are given by τ=η/Δ x,(m Δ x)^2=2(1-α), wherem is the mass of the scalar field corresponding to the IR cutoff. We quantize this model as follows: q̂_j=1/√(N)∑_k=0^N-1[ â_kf_k+â_N-k^†f_k^*]e^iθ_kj, p̂_j=-i/√(N)∑_k=0^N-1[ â_kg_k-â_N-k^†g_k^*]e^iθ_kj, where θ_k=2π k/N. â_k and â_k^† are annihilation and creation operators, which obey the commutation relations [â_k,â_k'^†]=δ_kk', [â_k,â_k']=[â_k^†,â_k'^†]=0. The mode functions f_k and g_k satisfy f̈_k+(ω^2_k-ä/a)f_k=0, f_kḟ^*_k-ḟ_kf^*_k=i, g_k=i (ḟ_k-ȧ/a f_k), where “·" is the derivative with respect to τ and ω_k^2=2(1-αcosθ_k). Although this lattice model is introduced in the 1+1-dimensional spacetime, we assume that the equation of the mode functions has the same form as that in the 1+3-dimensional de Sitter space. For the investigation of quantum entanglement of this system with the Bunch-Davies vacuum which belongs to a family of Gaussian states, we present a brief review of the negativity and the logarithmic negativity for a Gaussian state. Let us consider a phase space composed of canonical variables {q̂_j,p̂_j}_j=1,...,N. The canonical commutation relations are [ R̂_j, R̂_k]=i Ω_jk, Ω=⊕_j=1^N J, J=[01; -10 ], where R̂_j represent canonical variables defined by R̂=[ q̂_1, p̂_1,...,q̂_N,p̂_N]^T. A Gaussian state ρ̂ is determined by its firstmoment⟨R̂_j⟩=Tr[ρ̂ R̂_j] and the covariance matrixV_jk=1/2 Tr[ (ΔR̂_j ΔR̂_k+ ΔR̂_k ΔR̂_j)ρ̂],ΔR̂_j=R̂_j-⟨R̂_j⟩. The negativity 𝒩 of a bipartite Gaussian state ρ̂_AB is given by using the symplectic eigenvalues of the partially transposed covariance matrix Ṽ_AB <cit.> obtained from V_AB by replacing p̂_jA with -p̂_jA <cit.>: 𝒩=1/2(∏_j=1^N1/min(2ν̃_j,1)-1 ), where ν̃_j are positive eigenvalues of i Ω Ṽ_AB. The sufficient condition for the entangled state is that the negativity 𝒩 of the state does not vanish. The logarithmic negativity E_𝒩 is defined by the negativity asE_𝒩=ln[ 2𝒩+1 ]. The logarithmic negativity E_𝒩 provides an upper bound of the distillable entanglement (the number of the Bell pairs extractable from a bipartite state) <cit.>, and if E_𝒩 is nonzero then the bipartite state is entangled. However, there exists an entangled state whose the entanglement of distillation vanishes and such a state is called a bound entangled state. Fortunately, no bound entangled state exists for a bipartite Gaussian state with E_𝒩=0 <cit.>.To compute the negativity and the logarithmic negativity, we need the two-point functions of canonical variables on each site for the vacuum state. The correlation functions of the vacuum state are1/2⟨ 0|(q̂_jq̂_l+ q̂_jq̂_l)|0⟩ = 1/N∑_k=0^N-1 |f_k|^2cos[ θ_k(j-l) ], 1/2⟨ 0|(p̂_jp̂_l+p̂_lp̂_j)|0⟩= 1/N∑_k=0^N-1|g_k|^2cos[ θ_k(j-l) ], 1/2⟨ 0|(q̂_jp̂_l+p̂_lq̂_j)|0⟩= 1/N∑_k=0^N-1i/2(f_kg^*_k-f^*_kg_k )cos[ θ_k(j-l) ]. We choose the mode functions f_k and g_k which correspond to the Bunch-Davies vacuum as follows:f_k=1/√(2ω_k)(1+1/iω_kτ)e^-iω_kτ, g_k=√(ω_k/2)e^-iω_kτ.In the continuous limit of the lattice model, the correlation functions of the Bunch-Davies vacuum are given as 1/2⟨ 0|(q̂(x,η)q̂(y,η)+ q̂(y,η)q̂(x,η))|0⟩ = ∫^∞_-∞dk/2π |f_k|^2cos [k(x-y)] ∼1/2π(-γ-ln[m|x-y|])+1/2π (mη)^2,1/2⟨ 0|(p̂(x,η)p̂(y,η)+p̂(y,η)p̂(x,η))|0⟩ =∫^∞_-∞dk/2π |g_k|^2cos [k(x-y)] ∼ -1/2π |x-y|^2,1/2⟨ 0|(q̂(x,η)p̂(y,η)+p̂(y,η)q̂(x,η))|0⟩ =∫^∞_-∞dk/2πi/2(f_kg^*_k-f^*_kg_k )cos [k(x-y)] ∼1/2πη(-γ-ln [m|x-y|]),where the above approximated formulas are obtained for m|x-y| ≪ 1, and γ is the Euler constant. The two-point correlation of our effective model in a 1+1-dimensional spacetime decrease with the distance |x-y| more slowly compared to that in the 1+3-dimensional de Sitter space case. § BEHAVIOR OF ENTANGLEMENTBecause the scalar field is a many-body system,we expect that the multipartite entanglement is a key property to understand behavior of entanglement between two spatial regions in the de Sitter space. For this purpose, we introduce the coarse-grained field and the monogamy inequality of entanglement to quantify the multipartite entanglement of the scalar field.§.§ Coarse-grained field and entanglement monogamyTo focus on the behavior of the multipartite entanglement, we introduce the coarse-grained variables (Q̂_i,P̂_i) as follows: Q̂_i=1/√(n_c)∑_j=0^n_c-1q̂_n_ci+j, P̂_i=1/√(n_c)∑_j=0^n_c-1p̂_n_ci+j,where we denote n_c as the coarse-graining size of the canonical variables. The coarse-grained variables satisfy the canonical commutation relations [Q̂_i, P̂_j]=i δ_ij, [Q̂_i, Q̂_j]=[P̂_i, P̂_j]=0.We introduce two spatial symmetric regions A and B by choosing the following coarse-grained variables in the harmonic chain:R̂^A=[ Q̂_1, P̂_1,...,Q̂_n,P̂_n]^T, R̂^B=[ Q̂_n+1+d/n_c, P̂_n+1+d/n_c,...,Q̂_2n+d/n_c,P̂_2n+d/n_c]^T where n is the number of modes of the coarse-grained variables contained in each region, and d is the comoving separation between A and B. The comoving size of each region is denoted as l=n× n_c (FIG. <ref>). Since the vacuum state is Gaussian, the quantum bipartite entanglement can be completely characterized by the covariance matrix defined as V_AB=[ A C; C^T B ], A^T=A,B^T=B where A, B, C are 2n× 2n matrices given by A_ij=1/2⟨ 0| (R̂^A_iR̂^A_j+R̂^A_jR̂^A_i) |0 ⟩,B_ij=1/2⟨ 0| (R̂^B_iR̂^B_j+R̂^B_jR̂^B_i) |0 ⟩,C_ij=1/2⟨ 0| (R̂^A_iR̂^B_j+R̂^B_jR̂^A_i)|0 ⟩, with ⟨ 0|R̂_i^(A,B) |0 ⟩=0.In Ref. <cit.>, the logarithmic negativity between A and B were calculated with the number of each mode n=1, which corresponds to assigning a pair of canonical variables to each region.The following results are based on the numerical calculation with the number of lattice sites N=2× 10^4 and the IR cutoff parameter α=1-10^-12. FIG. <ref> shows behavior of the negativity with different size of coarse-graining. The left panel of FIG. <ref> presents the time evolution of the negativity with fixed the comoving distance d=0 and the comoving size l=60. As the number of modes of the coarse-grained variables n in the system A increases, the time at which the negativity vanishes becomes later. The right panel of FIG. <ref> gives the distance dependence of the negativity with fixed l=60 and τ=-80. For a larger n, the negativity increases and vanishes at a larger distance d. To compare the previous work <cit.> with our results, we introduce l_p=-l/τ, d_p=-d/τ, where l_p and d_p represent the proper (physical) size of each region and the distance between the two regions in the unit of the Hubble length H^-1, respectively. In Ref. <cit.>, it is found that the quantum entanglement between A and B with the number of each mode n=1 disappears when the proper size of each region is comparable to the Hubble horizon, that is, at l_p=1. This means that the quantum fluctuation for the super horizon scale behaves classically in terms of bipartite entanglement. However, according to the left panel of FIG. <ref>, we find that the quantum entanglement with the number of each mode n≥2 in the system A and B does not vanish even if the proper size of each region l_p is larger than 1 (for example, in the case n=2, the negativity is nonzero at τ=-10, l=60, that is, l_p=6) . Hence we expect there exists the multipartite entanglement even for the quantum fluctuation in the super horizon scale. Also, in the right panel of FIG. <ref>, the maximum distance that the negativity exists increases monotonically as n increases (in Ref. <cit.>, the negativity vanishes trivially for d≥1). It seems that the multipartite entanglement between casually disconnected regions also survives for a larger n (in the following section B, we clarify the distance dependencefor the continuous limit of our model).To get a clear intuition for the behavior of FIG. <ref>, we introduce the monogamy relation of the negativity for this model. We consider a tripartite state ρ_ABC and the negativity 𝒩_AB|C between AB and C. It is conjectured that 𝒩_A|C, 𝒩_B|C and 𝒩_AB|C obeys the following inequality 𝒩^2_AB|C≥𝒩^2_A|C+𝒩_B|C^2, where 𝒩_A|C and 𝒩_B|C are the negativity between AC or BC, respectively. This is called the monogamy relation of the negativity, which is an crucial property of the quantum entanglement. The monogamy relation is proved for a multi-qubit system <cit.>, and the similar relation holds for the entanglement measure defined in the Gaussian system <cit.>. However, there is no proof of the monogamy relation of the negativity for the Gaussian system. If the monogamy relation holds then the multipartite entanglement can be expressed by the quantity 𝒩_A|B|C:=𝒩^2_AB|C-𝒩^2_A|C - 𝒩_B|C^2. This is the difference between the two side of Eq. (<ref>) and can be interpreted as the residual entanglement. If this quantity vanishes, the entanglement between AB and C can be decomposed to the entanglement between A and C, and the entanglement between B and C. Thus the entanglement between AB and C is written as sum of pure bipartite entanglement between sub system. In such a case, there is no multipartite entanglement.To understand the multipartite entanglement in the Bunch-Davies vacuum, we consider the negativity with the number of each mode n=3 in the systems A and B (FIG. <ref>).The density operator ρ̂_AB is a 3 × 3 mode Gaussian state and the system A is composed of 3 subsystems A_1A_2A_3 (similarly, B is B_1B_2B_3). The monogamy relation is written as 𝒩^2_A^⊗ 3|B≥𝒩_A^⊗ 2|B^2+𝒩^2_A^⊗ 1|B, where A^⊗ 3=A_1A_2A_3 and (A^⊗ 2,A^⊗ 1)=(A_1A_2,A_3), (A_1A_3,A_2), (A_2A_3,A_1). For example, each negativity 𝒩_A_2A_3|B, 𝒩_A_1A_3|B and 𝒩_A_3|B which appears on the right side for (<ref>) corresponds to the entanglement between the two regions A and B shown in FIG. <ref>.Let us check the monogamy relation of the negativity for the Bunch-Davies vacuum in the case n=3. The left and right panel of FIG. <ref> present the time dependence with d=0 and the distance dependence at τ=-80 of the quantity 𝒩_A^⊗ 2|A^⊗ 1|B, respectively.From these results, we confirm that the monogamous relation of the negativity 𝒩_A^⊗ 2|A^⊗ 1|B≥ 0 holds for our model. Hence we can characterize the multipartite entanglement in de Sitter space by the negativity.In the left panel of FIG. <ref>, the time dependence of 𝒩^2_A_1A_2A_3|B, 𝒩^2_A_2A_3|B, 𝒩^2_A_1A_3|B and 𝒩^2_A_3|B for d=0 is shown (the other cases 𝒩_A_1A_2|B, 𝒩_A_1|B and 𝒩_A_2|B are trivially zero).We observe that the negativities 𝒩^2_A^⊗ 1|B and 𝒩^2_A^⊗ 2|B decay faster than the negativity 𝒩^2_A^⊗ 3|B for the 3 × 3 mode Gaussian system. This behavior guarantees the monogamy inequality (<ref>).The right panel of FIG. <ref> shows that the distance dependence of 𝒩^2_A^⊗ 3|B, 𝒩^2_A^⊗ 2|B and 𝒩^2_A^⊗ 1|B for τ=-80 (the other cases 𝒩_A_1A_2|B, 𝒩_A_1|B and 𝒩_A_2|B are trivially zero again). As in the case of the left panel of FIG. <ref>, the negativities 𝒩^2_A^⊗ 1|B and 𝒩^2_A^⊗ 2|B decrease more than 𝒩^2_A^⊗ 3|B with the distance d to keep the monogamy relation (<ref>). The behaviors observed inFIG. <ref> also suggest that the multipartite entanglement remains in thesuper horizon scale when the number of modes n becomes large. §.§ Continuous limitTo investigate the entanglement for the super horizon scale, we consider the continuous limit of our lattice model, where multipartite entanglement plays an important role.For realization of the continuous limit of our lattice model, we use the canonical variables with n_c=1 (no coarse-graining), and choose each parameter as N=2× 10^4 and α=1-10^-12, again. It is also assumed that each region contains l harmonic oscillators and their comoving separation is d (FIG. <ref>). In the appendix, we present the convergence check and the small violation of the uncertain relation due to numerical error to confirm that our numerical calculation really corresponds to the continuous limit and is stable. We compare the previous works <cit.> with our numerical results. In Ref <cit.>, the authors considered a massless scalar field in the 1+1-dimensional Minkowski space and numerically showed that the logarithmic negativity of a massless scalar field between two spatially regions decays exponentially as the ratio d/l increases. The property that the logarithmic negativity depends only on the ratio d/l is derived from the scale invariant for the massless theory. In the following, we investigate the entanglement for the super horizon scale and how it depends on the Hubble scale H.In Ref. <cit.>, the logarithmic negativity of the coarse-grained field in de Sitter space vanishes when the two regions are causally disconnected and FIG. <ref> also shows the negativity with the coarse-grained field becomes zero for sufficiently large scales or late times. On the other hand, the negativity obtained without coarse-graining (FIG. <ref>) does not vanish even when the distance between two regions is larger than the horizon scale (d_p=1 corresponds to the Hubble horizon scale). This observation confirms that the multipartite entanglement remains on the super horizon scale as expected above. For a vacuum state in the quantum field theory, the Reeh-Schrieder theorem characterizes the (multipartite) entanglement of quantum field <cit.>. Our numerical results suggest that the Reeh-Schrieder theorem also holds for the Bunch-Davies vacuum in de Sitter space.As the Bunch-Davies vacuum approaches to the Minkowski vacuum in the remote past, the behavior of the logarithmic negativity for l_p <1 and d_p <1 is expected to be same as that for the Minkowski case. To focus on the entanglement peculiar to de Sitter space, we consider the behavior of the logarithmic negativity for l_p≥1 and d_p≥1.FIG. <ref> shows the logarithmic negativity E_𝒩 as a function of d_p in this case.The logarithmic negativity for l_p≥1 and d_p≥1 behaves as almost linear functions in log plot. We use the fitting function of the exponential factor with the power-law correction to compare with the logarithmic negativity in the Minkowski vacuum <cit.>. The solid lines in FIG. <ref> represent the fitting result E_𝒩^(fit)≈ d_p e^-k d_p, where k is a real parameter whose values depend on the ratio l_p. FIG. <ref> shows k as a function of l_p and the solid line in the figure represents a function k=a_1+a_2 l_p^-1 where a_1 and a_2 areO(1) constants given by a_1∼1.08 and a_2∼ 4.35 . Combining these, we obtain the following fitting formula of the logarithmic negativity for d_p≥ 1 and l_p≥ 1: E_𝒩^(fit)≈ d_p e^-a_1d_p-a_2 (d_p/l_p). By restoring the dimension of the variables, this is rewritten as E_𝒩^(fit)≈D_p/H^-1exp[-a_1D_p/H^-1-a_2D_p/L_p], where H^-1 is the Hubble length and the proper size of the region L_p and distance D_p are given by L_p=a(η) l Δ x, D_p=a(η) d Δ x. In the formula (<ref>) we observe that the logarithmic negativity decays exponentially with respect to H. This means that the quantum entanglement is degraded by the thermal noise with the Hawking temperature H. Thus we consider the value of the numerical factor a_1 is related to the thermal noise which is independent of details of the theory. On the other hand, the interpretation of the factor a_2 is not so clear because a value of this coefficient of D_p/L_p depends on the theory. To make clear its physical interpretation, we will need further analysis of quantum entanglement using other theories of the scalar field in de Sitter space. For the super horizon regions L_p ≫ H^-1, the logarithmic negativity (<ref>) becomes independent of the size L_p of the two symmetric spatial regions and its value is determined only by the ratio D_p/H^-1. We expect that this property can be understood as follows: the physical wavelength of quantum fluctuation in the considering regions (FIG. <ref>) is initially smaller than the Hubble horizon. As the universe expands, the wavelength exceeds the Hubble horizon. After the horizon exit, the scale of fluctuations in each region is determined only by the Hubble horizon scale. Hence, the amount of entanglement depends only on D_p/H^-1. The above property of the logarithmic negativity (<ref>)is expected to be true for the 1+3-dimensional de Sitter space. This is because the feature of the entanglement is determined by the mode function and we use the same mode function as the 1+3-dimensional model. As the quantum fluctuation of super horizon mode is scale independent, the quantum entanglement is determined only by the Hubble scale and the separation between the considering regions. § SUMMARY AND CONCLUSION We investigated the quantum entanglement between two symmetric spatial regions with the Bunch-Davies vacuum for the 1+1-dimensional effective harmonic chain model. We introduced the coarse-grained variables and examined the multipartite entanglement in de Sitter spacetime by the monogamy relation of the negativity. In the previous work <cit.>, it has been shown that the bipartite entanglement disappears on the super horizon scale. In contrast, in this paper, it was found that the multipartite entanglement of the super horizon scale remains. This indicates that the multipartite entanglement plays an important role to characterize the quantum nature of the the super horizon scale fluctuations.We also considered the continuum limit of our model and calculated the logarithmic negativity for the original canonical variables (without coarse-graining). We confirmed that the logarithmic negativity remains non-zero even if the distance between the two regions becomes larger than the Hubble length. That is, there exists the quantum entanglement between two causally disconnected regions and the existence of the entanglement means that the Reeh-Schrieder theorem holds in de Sitter space.Finally, we comment on the relation between our lattice model and 1+3-dimensional theory. We considered the 1+1-dimensional effective lattice model of the free massless scalar field in the de Sitter space. As the behavior of the logarithmic negativity depends on the spatial dimension, the numerical simulation in our model is not equivalent to the universe with three spatial dimensions. At the end of Sec. II, we observed that the two-point correlation in our model is larger than it in a 1+3-dimensional de Sitter space. Hence for the scalar field without coarse-graining (continuous limit), if the entanglement disappears for some size and separation of each region in 1+1-dimensional model, we expect that the entanglement in the corresponding 1+3-dimensional model also vanishes. However, according to our numerical calculation, the entanglement in 1+1-dimensional model in the continuous limit exists in any scale.Thus our lattice model provides the necessary condition to judge the existence of quantum entanglement in the 1+3-dimensional de Sitter space. The main features of the logarithmic negativity found in our analysis is characterized by properties of the mode function of the scalar field in de Sitter space. The equation in our model is the same as that in the 1+3-dimensional de Sitter spacetime, and if we evaluate quantum entanglement between spatial regions in the 1+3-dimensional de Sitter spacetime, we expect that we will obtain the similar feature or property of entanglement obtained this paper. This work was supported in part by the JSPS KAKENHI Grant Number 16H01094.§ CONVERGENCE CHECK AND VIOLATION OF UNCERTAINTY RELATION To confirm that our numerical calculation corresponds to the continuum limit of the lattice model, we check the convergence of the logarithmic negativity. By introducing a scaling parameter λ, we write other parameters contained in the model as N=200×λ,α=1-λ^-2× 10^-8, l=λ. We regard E_𝒩 as a function of λ for fixed l_p and d_p. λ→∞ corresponds to the continuum limit of the model. FIG. <ref> shows the result of convergence check.The logarithmic negativity is independent of the parameter λ. In the limit λ≫ 1, length scales (for example, the physical size of the considering region or the distance) is much larger than the UV cutoff Δ x. Hence, this limit corresponds to the continuum limit and our numerical calculation well approaches the continuum limit.Furthermore, we evaluateviolation of the Heisenberg uncertainty relation for our numerical calculation by checking a quantity defined by U_N=-∑_j=1^Nlog_2[min(2ν_j,1 ) ], where ν_j are eigenvalues of i Ω V_AB. If U_N=0 then we get relations ν_j≥ 1/2, which are equivalent to the uncertainty relation. FIG. <ref> shows U_N as a function of d_p for fixed l_p.The uncertainty relation is expressed in terms of the two point functions ⟨q̂q̂⟩,⟨p̂p̂⟩ and ⟨q̂p̂+p̂q̂⟩.For the massless theory, the q̂q̂-correlation has the IR divergence. On the other hand, there is no the IR divergence in p̂p̂,q̂p̂+p̂q̂-correlations. 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Lett. 98, 050503 (2007) 79–86. | http://arxiv.org/abs/1707.08414v2 | {
"authors": [
"Akira Matsumura",
"Yasusada Nambu"
],
"categories": [
"gr-qc",
"quant-ph"
],
"primary_category": "gr-qc",
"published": "20170726125522",
"title": "Large scale quantum entanglement in de Sitter spacetime"
} |
The Caccioppoli UltrafunctionsVieri Benci, Luigi Carlo Bersellicorresponding author, and Carlo Romano Grisanti==================================================================================== The integral field spectrograph, SPIFFI, has complex line profile shapes that vary with wavelength and pixel scale, the origins of which have been sought since the instrument construction. SPIFFI is currently operational as part of SINFONI at the VLT, and will be upgraded and incorporated into the new VLT instrument ERIS. We conducted an investigation of the line profiles based on measurements we could take with the instrument calibration unit, as well as laboratory measurements of spare SPIFFI optical components. Cryogenic measurements of a spare SPIFFI diffraction grating showed significant periodic deformation. These measurements match the cryogenic deformation expected from bimetallic bending stress based on a finite element analysis of the lightweighted grating blank. The periodic deformation of the grating surface gives rise to satellite peaks in the diffraction pattern of the grating. An optical simulation including the cryogenic grating deformation reproduces the behavior of the SPIFFI line profiles with both wavelength and pixel scale as measured with the instrument calibration unit. The conclusion is that cryogenic deformation of the diffraction gratings is responsible for the non-ideal line profiles, and that the diffraction gratings should be replaced during the upgrade for optimal instrument performance. *E. M. George,[email protected] § INTRODUCTIONSINFONI (Spectrograph for INtegral Field Observations in the Near Infrared)<cit.>has been operational as a facility instrument for the ESO Very Large Telescope (VLT) since 2005. It is made up of the adaptive optics module MACAO (Multi-Application Curvature Adaptive Optics)<cit.> and the integral field spectrograph (IFS) SPIFFI (SPectrometer for Infrared Faint Field Imaging)<cit.>. The instrument has been scientifically productive in several areas over its 11-year lifetime, and is currently in use for several high-profile scientific programs. An upgraded version of the instrument will be included in the new VLT Adaptive Optics (AO) instrument ERIS<cit.> (Enhanced Resolution Imager and Spectrometer) as the Integral Field Unit (IFU) subsystem SPIFFIER (SPectrometer for Infrared Faint Field Imaging Enhanced Resolution)<cit.>.SPIFFI operates at wavelengths from 1.0-2.5in selectable J, H, K, or H+K bands with a choice of pixel scale of 25, 100, or 250 mas/px. The spectroscopic line profile shapes of SPIFFI are complex and vary with wavelength and pixel scale. These complex line profiles result in approximately a factor of two degradation of the instrument resolution in the short wavelengths of J-band, that smoothly rises to approximately the design value of the resolution by the long wavelengths of K-band.<cit.> The origins of these line profiles have been investigated since the construction of the instrument<cit.>, and effort has gone into correcting for them in atmospheric subtraction.<cit.> The source of the line profile variation needed to be found so that it could be fixed in the upcoming instrument upgrade for ERIS. As reported in George et al. (2016)<cit.>, after replacing the potentially troublesome spectrometer collimator mirrors during an upgrade of the instrument in January 2016, we were able to track down the likely origin of the line profile degradation to cryogenic bimetallic bending stress in the diffraction grating blanks that resulted in quilting of the grating surface. However, before procuring new diffraction gratings, we needed to prove that the gratings were the source of the line profile degradations. Because SPIFFI is currently in use at the VLT as part of SINFONI, we were limited in what measurements we were able to take–in particular, we were limited to only measurements that could be taken with the instrument calibration unit, as well as laboratory experiments on spare optical components. This investigation is presented in this paper, and our approach is as follows:* Take super-sampled line profiles in all wavelengths and pixel scales using the instrument calibration unit. (Section <ref>) * Evaluate the potential cryogenic deformation from the grating blank design using Finite Element Analysis. (Section <ref>)* Verify calculated grating blank deformations by cryogenic interferometric wavefront measurements of one of the spare SPIFFI diffraction gratings. (Section <ref>)* Insert grating deformation into an optical simulation to create simulated line profiles including the effect of the grating deformation. (Section <ref>)* Compare optical simulation results to measurements from the instrument calibration unit. (Section <ref>)Finally, we discuss our plans for the instrument upgrade along with the practical implications for diffraction grating design, and conclude in section <ref>.§ SPIFFI LINE PROFILESThe spectral line profiles we expect to measure are a cross section of an image of the slit in the focal plane. However, the measured profiles are asymmetric with shoulders and vary in shape with wavelength, pixel scale, and spatial detector position along the pseudo-slit. SPIFFI is designed to approximately Nyquist sample the slit image, so super-sampled line profiles are required to measure the shape of the line profiles in detail. To obtain super-sampled line profiles, Thatte et al. (2012) <cit.> describes a method using OH lines. This method was slightly adapted and used to measure the line profiles of the instrument using the spectral lines of the arc lamps in the calibration unit of SINFONI. In this way, we obtained a catalog of line profiles that covers the entire wavelength range of the instrument in all pixel scales and at all pseudo-slit locations. Detailed plots of the spectral line profiles of SPIFFI and their variability with wavelength, over single slitets, and along the pseudo-slit are available in Gräff (2016)<cit.> for all bandpasses and pixel scales, and the measured line profiles presented here are produced using the methodology therein. In this paper, all plots, simulations, and analyses are done using measurements with the new collimator as installed in 2016. Figure <ref> shows for the three single bandpasses of SPIFFI representative spectral line profiles from a single detector column (column 940 in slitlet 16 close to the center of the detector) in all three pixel scales.In K-band, distinct shoulders at a distance of 36in the focal plane (equivalent to 2 detector pixels) from the center of the line are present, and appear to be strongest in the 100 mas pixel scale. In H-band these shoulders are nearly not seen in the largest pixel scale, rather the peak just appears broadened, however, the shoulders appear in the two smaller pixel scales. In J-band, the line profiles look different. In the 25 mas pixel scale a peak with asymmetric shoulders is present; the shoulder on the right side is more distinct than on the left side.The two larger pixel scales in J-band differ from the other bands even more dramatically – a double peak behavior can be seen. Here the spacing of the double peak is between 36-45(2.0-2.5 pixels) for the 100 and 250 mas pixel scales. We additionally took data at single wavelengths on separate diffraction gratings using variable diffraction orders (see appendix <ref>). In these measurements the double peak behaviour at 1.25appears most strongly on the J-band grating, and is nearly not visible with 1.25light on the H- and K-band gratings. This indicates that the J-band grating must be somehow different from the H- and K-band gratings, and was our first clue that the gratings must be affecting the line profiles, as otherwise the light path through the optics is identical for the measurements. The line profiles in the 25mas pixel scale vary strongly over the length of each single slitlet, which is most noticeable in J-band but only weakly in H- and K-bands.<cit.>. The upgrade of the instrument did not affect this. The effect of varying line profiles also exists in the 100 and 250 mas pixel scales, though not as strongly as in the 25 mas pixel scale. The variation of the line profiles is discussed more in section <ref>. § SPIFFI LIGHTWEIGHTED DIFFRACTION GRATINGSAs telescopes and instruments become larger, their cryogenic optical elements become larger too, and it can be necessary to lightweight the optical components to keep both the inertial and thermal masses low. Lightweighting techniques have also long been developed for space applications where keeping the mass low is a priority. However, lightweighted structures can result in quilting of the optical surface<cit.>, and cryogenic optics deform under differential thermal stresses, such as those due to a Coefficient of Thermal Expansion (CTE) mismatch in bimetallic structures.<cit.> The SPIFFI diffraction gratings have both a lightweighted design and a CTE mismatch between the blank material and polishing layer, which results in significant deformation at their operating temperature of 80K.§.§ DesignSPIFFI is equipped with four diffraction gratings. The gratings for J-, H-, and K-band are operated in 2nd order and the grating for the combined H+K-band in 1st order. All four diffraction gratings are directly ruled on identical blanks made of 6061 aluminum alloy. The dimensions of the grating blanks are 160 x 140 x 20 mm. Seven lightweighting holes with 15 mm diameter are drilled through the blank in the X- and Y- directions (the Z- direction is normal to the grating surface). Figure <ref> shows a CAD model of the grating blank that was used for the simulations in this paper. The Aluminum blank was electroless Nickel-phosphorus (NiP) plated with a layer thickness between 100-200on all surfaces. The two large faces of the NiP-plated blank were then polished symmetrically to be flat. The remaining NiP layer on these surfaces is between 50-90% of the original layer thickness. Finally a 2-3thick gold layer was applied on the front surface before the gratings were ruled. §.§ Finite Element AnalysisTo quantify bimetallic stresses and the resulting deformations of the grating surface when the grating is cooled down from room temperature to the operating temperature of SPIFFI at 80K, we implemented a Finite Element Analysis (FEA) of the Aluminum grating blank with the NiP coating. Unfortunately, we only know roughly the original layer thickness and a range of possible values for the thickness post-polishing. We therefore simulated three layer thicknesses (100, 150, and 200 ) that span the range provided by the manufacturer, as well as polishing steps that left 50%, 75%, or 100% of the original layer thickness. Figure <ref> shows the deformation resulting from the three polishing values for an original layer thickness of 200 . The bimetallic stresses deform the surface resulting in a regular grid structure. We define the X-direction to be the dispersion direction and the Y-direction to be the ruling direction, therefore, the period of the grid structure is D_x = 21 mm and D_y = 18 mm. Of particular interest is the deformation in the central parts of the grating, since the outer parts and especially the corners of the grating are vignetted by a pupil mask. The pupil mask also covers the areas that may be affected by the small notches and holes for fixation screws in the real grating blank. The peak-to-valley (P/V) deformation in the central region of 82 mm x 72 mm (orientated on the lightweighting holes) is listed in table <ref> for the nine gratings simulated. We chose to report the P/V in this region as a proxy for the amplitude of the periodic deformation, as the deformation in this region is the most regular and is not affected by edge effects. In general, thicker original layers result in more deformation. Additionally, for a given original layer thickness, more surface polishing results in more deformation. This indicates that mis-matched layer thicknesses between the inside and outside of the lightweighting structure results in more deformation, however, given the specific geometry of our grating blank, we still found deformation even with matched layer thicknesses. Simulations with thicker NiP on the surface than inside of the lightweighting holes are not presented here, as this is not a realistic scenario given the manufacturing process of the grating. However, the edges cases (NiP only inside the holes, or only on the surface) are presented in appendix <ref>, and show that the deformations induced by these two layers oppose each other.Given the large range in P/V deformation values for relatively small changes in the amount of polishing done to the grating blanks in the FEA, and our unknown layer thicknesses, a cryogenic measurement had to be performed to get an idea of the actual deformation of the SPIFFI diffraction gratings.§.§ Cryogenic MeasurementsTo measure cryogenic bimetallic bending effects of the grating and verify our FEA analysis, we set up the SPIFFI J-band spare grating on a rotatable stage in a cryogenically cooled vacuum chamber at MPE. The grating was installed on a stress-free mount with heat strapsfrom the cold plate attached to the four corners of the grating using the small holes for fixation screws. A temperature sensor was installed directly on top of the grating near one corner using another of these fixation screw holes.For the measurements of the grating surface we used a FISBA Phase® 500 interferometer together with its beam expander lens providing an 152.4 mm diameter output beam. The interferometer is a Twyman-Green phase-shifting type, operated with a stabilized He-Ne laser at a wavelength of 632.8 nm. The interferometer and beam expander lens were placed outside of the cryostat and we measured the grating surface in Littrow configuration (see figure <ref>) through an optical quality cryostat window. The diameter of the cryostat window is 137 mm, and thus is also the limiting size for the interferogram of the grating surface. The test cryostat at MPE could not be shielded easily against vibrations, and thus it was not possible to measure the grating surface in the phase scanning mode of the interferometer. We therefore used a static fringe analysis. Therefore the phase value of a certain pixel cannot be measured independently, and thus the phase sign gets lost, and the results are less accurate than a phase-shifting measurement. In exchange, only one image is needed to complete the measurement, so it is fast enough for us to obtain interferograms in unstable environments. We used the Fourier Fringe Analysis (FFA) <cit.> which is very effective if the surface deformations are small (< λ/5). For this kind of static fringe analysis, a basic tilt has to be applied to the grating surface resulting in a carrier frequency in the interferogram. These fringes are afterwards removed in the Fourier domain to obtain a surface deviation map. The filtering used in the Fourier domain affects the final P/V value measured for higher-frequency deformations, and thus care must be taken to ensure an accurate measurement. The result of the FFA measurement is shown in figure <ref>. The surface deviation shown is the difference between the cold measurement at 128 K and the warm measurement at room temperature, in order to show only the deformation due to cryogenic bimetallic stresses. The P/V_center deformation is 0.80 , however, this value is also affected by the oblique astigmatism that can be recognized in the interferogram. Without this astigmatism, the P/V value is approximately 0.60 . This lies comfortably in the range of values simulated for NiP layer and polishing thicknesses provided by the manufacturer. In our measurement campaign, the lowest temperature reached at the grating was 103 K. The grating is heated by thermal radiation through the cryostat window, preventing the grating from reaching a lower temperature. The surface measurement taken at a temperature of 103 K was found to be identical to the one at 128 K to within the measurement uncertainty, though the quality of the 103K measurement was not as good as the measurement at 128K shown in figure <ref>. It should also be noted that this was a measurement of a single spare grating–each grating blank was polished separately, so it is likely that the NiP layer thicknesses are different on the different gratings. § OPTICAL SIMULATIONDiffraction theory shows that a regular periodic structure, such as the measured deformation on our diffraction gratings, will result in a diffraction pattern with peaks located at sinθ = λ/D.<cit.> Since we have a periodic structure in both the X- and Y- directions, we expect that the PSF of the instrument will be diffracted in a grid pattern in the spatial and spectral directions. We use CODE V optical simulation software to model the effect of the grating surface deformation on the line profiles of the instrument. §.§ Setup and ParametersIn order to simplify the model, we adopt an approximate setup for simulating the spectrograph. We assume a point source at the location of the image slicer. The light is collimated by a CODE V lens module with a focal length of 2873 mm (identical to the instrument collimator focal length, though in the instrument the collimator is a Three Mirror Anastigmat (TMA)). We do not simulate the full TMA, as we have measured the TMA wavefront from the location of the spectrograph pupil and include it in the simulations as an interferogram applied to the CODE V collimator lens module. The diffraction gratings for the J, H, K, and H+K bands are setup as listed in table <ref>. The diffracted light is then imaged onto the focal plane by a CODE V lens module with a focal length of 342 mm, identical to the instrument camera. The setup allows us to place an interferogram of the surface deformation of the grating on the CODE V grating surface. Due to the anamorphic distortion of the diffraction gratings, the slit width geometrically imaged onto the detector is 27using the J, H, and K band gratings and 29using the H+K band grating.Figure <ref> shows the layout of the simulation for the case of the J-band grating. The figure also shows a “black box" pre-optics and the instrument cold stop. The pre-optics act as a scale changer for the incoming light, and for a fully illuminated slit (the case for our spectral line profile measurements) the effect is to change the Numerical Aperture (NA) of the light entering the spectrograph slit for the different pixel scales. We account for this effect by using the CODE V 2D Partial Phase Coherence function, which allows the user to specify the relative NA of the incoming light beam and an effective slit size. In figure <ref>, the shaded area shows the geometric light path for the 100 mas pixel scale, while the solid (outer) lines show the light path for the 250 mas pixel scale.Unless otherwise noted, throughout this section the default grating deformation used in the simulation is the one calculated by the FEA with an original layer thickness of 200and 75% of the layer thickness (150 ) remaining after polishing. Additionally, throughout this section we show results only for the J- and K- bands, as the behaviour of H-band is intermediate between the two. H-band results are discussed in section <ref>.§.§ Point Spread FunctionTo see the diffraction effects of the grating surface deformation, we begin by calculating a PSF for a point source. On the grating, we place an interferogram of the grating surface deformation. Figure <ref> shows the resulting PSFs for the deformation calculated by the FEA for the J- and K-band setups. Note that the grid structure of the PSF has a spacing that is a function of wavelength, as expected from diffraction theory. In J-band the spacing of the peaks is approximately one detector pixel (18 ) in the focal plane, while in K-band the spacing is approximately 2 detector pixels (36 ) in the focal plane.At shorter wavelengths, diffraction orders farther from the center grow in strength relative to the central order. This is a general result expected for sinusoidal deformations as the wavelength approaches the deformation amplitude.<cit.> Additionally, for deformation amplitudes comparable to ∼λ/2 or larger, the central order(s) can be suppressed.<cit.> For the largest deformation amplitudes tested, we see central order suppression especially in J-band. This is explored in the next section.§.§ Line Spread FunctionThe spectral line profile is the measured line spread function of the instrument. This section shows how the cryogenic deformation of the diffraction gratings affects the spectral line profile of the instrument. §.§.§ Basic LSFIn its most basic form, the line spread function of the instrument is simply a cross section of the convolution of the PSF with the image of the slit in the focal plane. In SPIFFI, the slit image is 27wide in the focal plane. To build intuition about how the amplitude of the grating deformation affects the line profile of the instrument, in figure <ref> we plot this “basic LSF" for both an infinitely thin slit (equivalent to collapsing the PSF plotted in figure <ref> along the slit direction), and the actual SPIFFI slit size for two different amplitudes of the grating deformation.In general, as the amplitude of the deformation increases, more power shifts from the central peak to the shoulders, and for large amplitudes the central diffraction order is suppressed. By construction, the LSF produced in this way does not include any diffraction effects from the slit. The “horns" that appear in the J-band LSF in the lower left plot in figure <ref> are a result of the fact that the spacing of the diffraction orders is smaller than the slit width in J-band, and they apprear purely from the convolution of the PSF shown in figure <ref> with the slit image. In the next section, we will include diffraction effects from the slit.§.§.§ Partial Phase Coherence LSFAs discussed in section <ref>, the line profiles vary depending on the pixel scale selected (25, 100, or 250 mas/px). Since our line profile measurements are taken with the slit fully illuminated by a flat-field source, the only difference between the three pixel scales is the numerical aperture (NA) of the incoming light illuminating the slit. This has two consequences. The first effect can be seen in figure <ref>, where for smaller pixel scales the geometric light path through the instrument results in smaller beam footprints on the spectrograph optics. The second is that the incoming light at the image slicer will diffract differently at the slit edges depending on the illumination NA. We can model both effects in CODE V using the 2D partial phase coherence (PPC) function, which includes the full diffraction effects of partially coherent light propagating through the optical system. In the setup of the 2D PPC, we use a rectangular object that has an equivalent slit width of 27and length of 400 [The actual SPIFFI slit length is 1152in the focal plane, but the 400length reduces simulation times and is sufficiently long to avoid edge effects.] in the focal plane. For the three pixel scales [250, 100, 25] mas/px, we use a relative NA of the incoming light of [1.0, 0.4, 0.1] with respect to the 250 mas pixel scale. The left column of figures <ref> and <ref> shows the LSF in the center of the slit that results from the CODE V partial phase coherence calculation in the three pixel scales for the case of a perfect grating (figure <ref>) and a single amplitude of the grating deformation (figure <ref>).In general, diffraction at the slit edges is largest in the smallest pixel scale due to the smaller NA of the illumination. This appears as “ringing" at the slit edges. The ringing is larger at smaller pixel scales (smaller NA), and at longer wavelengths. The behaviour with NA and wavelength are both expected from diffraction theory.<cit.> Note that in the J-band plots of figure <ref>, the "ringing" is confused with the “horns" (which are a result of the diffraction order spacing). An additional effect that appears in the 2D PPC simulation is that the size of the shoulders is significantly smaller in the 25 mas pixel than the larger pixel scales in both J and K band. This is due to the smaller beam footprint on the diffraction grating, minimizing the effect of the periodic deformation of the grating surface. §.§.§ The Effect of Other Wavefront ErrorsSo far, we have been discussing the effects of only the cryogenic distortion of the grating blank on the line profiles of the instrument. However, it is known that the rest of the spectrograph optics also have wavefront errors. In order to get the most realistic possible line profile out of our simulations, we must also include these additional wavefront errors. We measured the wavefront error of the spectrometer collimator mirrors installed in January 2016 from the grating position, and apply the measured wavefront of the new collimator mirrors to the collimator in the optical model. The cryogenic measurements of the spare grating showed an additional astigmatism, despite the stress-free mounting. Since we only have measurements of a single spare grating (and not the actual gratings inside of the instrument), we cannot know the exact additional wavefront errors, however, we can add additional wavefront errors of appropriate size to the grating surface using the Zernike polynomials. We tested several different additions of wavefront errors into the optical model, and found that in general, the addition of wavefront errors on the grating, collimator, or camera results in asymmetric power in the shoulders. Since we only have an exact measurement of the new collimator mirror wavefront errors, we show the results of including the wavefront error of the collimator only in the model. This is primarily a horizontal coma term and a 45 degree astigmatism term, with a total wavefront P/V of 1.31and an RMS of 0.31 , as seen in the rightmost image of Figure 4 in George et al. (2016)<cit.>. This wavefront is applied to the perfect collimator in the optics model. The middle columns of figures <ref> and <ref> includes the collimator wavefront error. The primary effect is that power becomes distributed asymmetrically in the various diffraction peaks and shoulders. §.§.§ Simulated Detected Line ProfilesThe simulation results allow us to probe the processes that affect the line profiles of SPIFFI. However, to compare these simulations to the measured line profiles of the instrument, we must process the simulation output to match what we measure using our super-sampled line profile technique. The detector pixel pitch is 18 , therefore the detected super-sampled line profile is a cross section of the 2D convolution of an 18x 18wide square with the slit image at the detector. The right column of figures <ref> and <ref>shows the line profiles after simulation output has been convolved with the detector pixel function. The main effect of this convolution is a smoothing out of higher-frequency effects such as the diffraction spikes caused by the slit edges.§.§.§ The Effect of Pupil Position on theLSFDuring the upgrade in 2016, we moved the pupil position on the grating slightly in the X-direction (the dispersion direction). Additionally, the pupil position on each individual grating is slightly different. We explored the effect of the pupil position on the line profiles by shifting the pupil and repeating the simulation. We shifted the pupil by ±10%, which for our entrance pupil diameter of 110 mm corresponds to a shift of ±11 mm on the grating surface. The periodic structure in the y-direction has a period of 21 mm, so this pupil shift magnitude covers all phases of the periodic structure. Figure <ref> shows the line profiles for a range of pupil positions in the 25 mas pixel scale for the 200thick NiP layer polished to 50% (100 ) thickness. The change in line profile with pupil position is smaller for the grating deformations with smaller amplitudes, so we show here the largest amplitude grating deformation to highlight the effect. The 100 and 250 mas pixel scales are not shown, since there is essentially zero variation of the line profiles with pupil position in these pixel scales. The results of the pupil shift simulation show that the amplitude and shape of the 25 mas pixel scale (mas/px) line profiles depends on the exact pupil position on the grating, while exact pupil position has very little effect on the 100 mas/px lines, and almost no effect on the 250 mas/px lines. This can be understood when one considers that due to the low NA of the light entering the image slicer in the smallest pixel scale, only a small portion of the grating is illuminated in the 25 mas/px scale, on the order of a single period of the grating deformation (neglecting the diffraction effects of the slicer on the illumination). Figure <ref> shows schematically the effect of smaller NA illumination on the beam footprint on the grating. Thus for the 25 mas/px scale the shape of the wavefront after the grating depends more strongly on the pupil position. Since the grating shows periodic surface deformations on scales smaller than the illuminated pupil size in the 100 mas/px and 250 mas/px scale, for these pixel scales the effect of a pupil shift on the line profiles is much smaller, because the Fourier transform of the surface deformation is nearly identical for small relative shifts on a periodic surface. Additionally, the effect is larger at smaller wavelengths–in our simulation, only the J-band showed very strong line profile variation with pupil position. § MATCHING SIMULATIONS TO DATAIn this section we examine each band to determine how well each band's simulations match the data.§.§ K-bandIn K-band, the simulation using the grating deformation from a 200thick NiP layer polished to 75% (150 ) thickness matches the K-band data quite well, shown in figure <ref>. The simulation with pupil centered on the grating is also adequate to describe the data in the 25 mas pixel scale, though the line profile variation in K-band with a shifted pupil is rather low (see right plot in figure <ref>), so it would be difficult to tell if the pupil were shifted on the grating. The measured line profiles also show a low variation along the slitlet (see Gräff (2016)<cit.> for an example of this small variation).§.§ H-bandIn H-band the comparison is also straightforward. As in the K-band case, the simulation using the grating deformation from a 200thick NiP layer polished to 75% (150 ) thickness matches the H-band data, shown in figure <ref>. As in K-band, having the pupil centered on the grating is also adequate to describe the data in the 25 mas pixel scale, though a shift left or right of a few percent would give similar results, as the profile variation with pupil shift is low in H-band with this deformation amplitude. The measured line profiles also show a low variation of the line profiles with slitlet position. (see Gräff (2016)<cit.> for an example of this small variation).§.§ J-bandAs already mentioned in section <ref> the J-band line profiles appear different from the other two bands. The most obvious difference is the double-peaked structure, most visible in the 100 and 250 mas pixel scales. Additionally, there is an obvious asymmetry in the peak heights in the 25 mas pixel scale. Given the results of the simulations, it became clear that to reproduce the J-band line profiles, a larger amplitude grating deformation was required to get the central order suppression necessary to result in a double-peaked line profile. Of the layer thickness/polishing simulated, we found that the 200thick layer with 50% (100 ) thickness remaining after polishing gave results that matched the simulations the best. Figure <ref> shows the measured line profiles and the best-matching simulation from the 200/100 NiP layer deformation. In addition, the J-band line profiles in the 25 mas pixel scale vary strongly across a single slitlet. We could reproduce this behaviour by varying the pupil position on the grating. One possible source of movement of the pupil position is the image slicer, which is described in detail in Tecza et al. (2000)<cit.>. A torsional deformation of a slicer mirror with a measured amplitude of 2 HeNe fringes (close to what could be expected based on measurements of the slicer mirrors; see Eisenhauer et al. (2003)<cit.> for an example of a slicer interferogram) would result in the pupil shifting on the grating by approximately ± 10 mm, which is nearly an entire period of the grating deformation function. Both the asymmetry in the 25 mas pixel scale line profiles and the variation in line profile along a slitlet can be explained with a shift in the pupil (due to, for example, a torsion in the slicer mirrors) in combination with the grating deformation. Figure <ref> compares the simulated and measured line profile variation in the smallest pixel scale. A shift in the central pupil position of approximately 4 mm in the dispersion direction on the grating can provide the necessary asymmetry to reproduce the behavior of the central 25 mas/px line profile (shown in figure <ref>), however, to fully explain the variation in the line profiles over the full length of the slitlets, a continuously varying pupil position is required. If a torsion in the image slicer is the reason for a continuously varying pupil position, then we expect the same pupil position variation on all diffraction gratings. This variation is only noticeable in the measurements on the J-band grating. As noted in the previous two sections, both measurements and simulations of the H- and K-band line profiles showed very low variation with pupil position on the grating.In J-band it is clear that the simulation is not a perfect reproduction of the line profiles–the 100 and 250 mas pixel scale simulations are slightly wider and the central dip is less pronounced, while in the 25 mas pixel scale the right shoulder is less obvious. To explore this, we also tested purely sinusoidal grating deformations with a period equal to the spacing of the lightweighting holes andvariety of P/V values. A sinusoidal deformation with a P/Vof 0.58also gives results that match the measured line profiles reasonably well, but with narrower line profiles. This indicated to us that while the primary component of the grating deformation that affects the line profiles is the amplitude of the Fourier component corresponding to the period of the lightweighting holes, the exact shape of the deformation (i.e. other Fourier components of the deformation) play a role in the width of the profiles. Without a cryogenic measurement of the gratings that are currently in use in SPIFFI, we cannot know exactly what the deformation amplitude and shape of each grating is. However, we can say with confidence that to reproduce the behaviour seen, the J-band grating needs to have a different, and higher, deformation amplitude than the other two gratings. This is discussed more in the following section.§.§ Single Wavelength Measurements on Multiple Gratings To cement our result, we attempted to reproduce the measurements that led us to suspecting the grating blank was causing problems in the first place–namely, producing simulations of a single wavelength on the three diffraction gratings. We choose a wavelength at the center of J-band (1.25 ), as shorter wavelengths are more sensitive to the details of the grating deformation. The simulation setup uses the K-, H-, and J- band gratings in the 4th, 3rd, and 2nd diffraction orders respectively. The grating deformations used in the simulation are the ones needed to reproduce the in-band results of the previous sections, namely the deformation from NiP layer thicknesses of 200/150for the H- and K- band gratings, and NiP layer thicknesses of 200/100for the J-band grating. Figure <ref> shows the results for 1.25in the 250 mas pixel scale. The agreement between measurement and simulation is quite good, though the line profile is clearly higher and narrower in the H- and K- band grating simulations than in the measurements. In the smaller pixel scales, there is a similar slight disagreement between the simulation and measurements. (Figure <ref> in the appendix shows these measurements in all pixel scales and wavelengths). We were able to more closely reproduce the measured line profiles by increasing the amplitude of deformation on the H- and K- band gratings by approximately 20%, which was achieved by scaling the amplitude of the 200/150simulation by a factor of 1.2. However, here we do not attempt to fine-tune the deformation amplitudes to match the measurements exactly–we simply note that a slightly higher deformation amplitudes on the H- and K- band gratings (though still smaller than the deformation of the J-band grating)reproduces the measurements better. The results of this test leave us to conclude that in order to reproduce the entire data set we have obtained from 1.0-2.5on three diffraction gratings, the H- and K- gratings need to be very similar to each other, while the J-band grating needs to be different. The primary difference required is that the deformation amplitude induced by the lightweighting structure needs to be larger on the J-band grating. We found in our notes on the production of the gratings that the J-band blank was re-polished at least once to achieve higher surface flatness. The results of the FEA of the grating blanks show that more polishing results in higher deformation amplitudes, so this result is consistent with the information we have on the grating manufacturing. § DISCUSSION AND CONCLUSION SPIFFI has complex line profile shapes that vary with wavelength and pixel scale, the origins of which have been sought since the instrument construction. Because SPIFFI is still in use at the telescope as part of SINFONI, we investigated the line profiles based on measurements we could take with the instrument calibration unit, as well as laboratory measurements of a spare SPIFFI diffraction grating. Cryogenic measurements of the spare SPIFFI diffraction grating showed significant periodic deformation due to the lightweighted structure and the bimetallic bending effect between the aluminum blank material and the NiP polishing layer. We performed a finite element analysis of the grating blank, and found that the amplitude of the deformation depends on the initial layer thicknesses and amount of surface polishing. The deformation from cryogenic measurements of the spare grating falls in the range of values expected based on NiP layer thicknesses and polishing depths provided by the manufacturer.We found that inserting the grating deformation into an optical simulation gives rise to satellite peaks in the diffraction pattern of the grating, and reproduces the behavior of the SPIFFI line profiles with both wavelength and pixel scale as measured with the instrument calibration unit. We determined that to reproduce the measured data, the deformation amplitude on the J-band grating must be higher than the other two gratings, likely due to re-polishing of the J-band grating blank. The result of our study is that we have proven that cryogenic deformation of the diffraction gratings due to the bimetallic bending effect on the lightweighted blank is responsible for the non-ideal line profiles of SPIFFI, and that the diffraction gratings should be replaced for optimal instrument performance. The plan is for all diffraction gratings to be replaced in the upgrade of SPIFFI for use in ERIS.When building a cryogenic instrument, simulations of the deformations of optical elements when cooling to cryogenic temperatures can provide valuable information on the effect of lightweighted structures, especially in the presence of components with mismatched CTE. Cryogenic wavefront measurements provide powerful diagnostics for the as-built performance of a cooled optical system, and can be included in an instrument optical model. A full instrument optical model and simulation including IFS slit diffraction effects<cit.> is important to assist in instrument design, debugging, and performance analysis work.Some practical information for instrument design came out of this work. Our analyses showed that materials with mismatched CTEs combined with lightweighted structure results in significant deformation. Mismatched NiP layer thicknesses between the inside of the lightweighting holes and the grating surfaces resulted in larger deformation, however, deformation was still present even with matched NiP layer thicknesses. Careful attention should be paid to the design of any complex structure that could be subject to bimetallic stresses. Second, a test optical simulation we carried out with the period of the lightweighting structure doubled resulted in the first few diffraction orders falling inside of the slit in the focal plane, significantly improving the spectral line profiles. In general, the spatial frequencies of lightweighting structures should be designed to minimize the effects of potential diffraction on instrument performance.The design of the new grating blanks for ERIS (currently undergoing FDR) takes into account what we have learned from this analysis. The new blanks will not have any lightweighting structure. Instead, optically unused area around the edges of the blanks has been removed to reduce weight. This entirely removes from the optics the spatial frequencies of the lightweighting structure that resulted in the observed diffraction effects in the SPIFFI line profiles. It has been determined that an NiP polishing layer is still required for a surface finish with high enough quality for grating ruling. However, the new blank material will be a high silicon content aluminum alloy such as AlSi40 that has a CTE that is matched to NiP to within 0.5 x 10^-6. This significantly reduces the bimetallic bending effect.<cit.> Finally, finite element analysis will be performed on the final grating blank/coating design to ensure that it is within specifications, and any deformation found can be incorporated into an optical model. Given the simulation results of figure <ref> (the “perfect grating" case), if the new gratings are within specifications, ERIS/SPIFFIER should have a spectral resolution close to the design value of R ∼4k in J-, H-, and K-bands. We thank Johannes Hartwig and Kurt Dittrich from MPE for the preparation of the cryogenic test facility and Christian Rau from MPE for preparing the cryogenic drive motors, both used for the cryogenic grating wavefront measurements. We thank ESO and the Paranal staff for providing day-time access and support for the SINFONI instrument to complete super-sampled line profile measurements using the instrument calibration unit. We also thank the two anonymous referees for their helpful comments on this manuscript. spiejour§ PLOTS OF LINE PROFILES ON DIFFERENT GRATINGS This appendix contains an example of the super-sampled line profiles of the instrument as a function of wavelength. The wavelengths shown are roughly at the band centers, and are selected from the bright arc-lamp calibration lines. The arc lamps used are Argon for J-band measurements, Xenon and Argon together for H-band measurements, and Neon and Argon together for K-band measurements. The lines plotted come from the same location on the image slicer. This means that the light producing each line passes through the same optical path in the spectrometer collimator and camera. Each wavelength was measured on as many diffraction gratings as possible by additionally utilising the non-blaze orders of the gratings. For example, for lines falling in the J-band wavelengths, we measured each line in 2nd order on the J-band grating, 3rd order on the H-band grating, and 4th order on the K-band grating. The limiting factor was that we did not wish to turn the gratings by more than a few degrees during a measurement, in order to keep the grating as uniformly illuminated as possible. This means that for lines falling in the H-band and K-band wavelengths, we were only able to obtain line profiles produced from 2 of the 3 gratings. The resulting line profiles are similar between the three diffraction gratings, however, they are not identical. In particular, one can see that the H- and K-band gratings produce nearly identical line profiles, while the J-band grating produces line profiles with more pronounced shoulders, and in some cases, double peaks. These results, when taking the simulations into account, indicate that the deformation on the J-band grating is slightly larger than on the H or K band gratings. § FURTHER FINITE ELEMENT ANALYSES OF GRATING BLANKS In section <ref> we showed that more polishing (e.g. mismatched NiP layer thicknesses) results in more deformation. Here we present the edge cases: We simulate a grating blank with either NiP only inside of the lightweighting holes, or only on the outside surfaces of the blank, for 100, 150, and 200thick NiP layers. In general, thicker layers result in more deformation. Gratings with NiP only inside of the holes show a deformation in the same direction as the “realistic" grating blanks presented in section <ref>, but with a larger amplitude. Gratings with NiP only on the surfaces of the blank show a deformation in the opposite direction. This shows visually how the two NiP layers oppose each other. The P/V deformation values are tabulated in table <ref>, and plots of the surface deformations are shown in figure <ref>. A simulation with no NiP layer anywhere resulted in no deformation, as expected. | http://arxiv.org/abs/1707.08923v1 | {
"authors": [
"E. M. George",
"D. Gräff",
"M. Hartl",
"H. Huber",
"F. Eisenhauer",
"H. Feuchtgruber"
],
"categories": [
"astro-ph.IM"
],
"primary_category": "astro-ph.IM",
"published": "20170727160616",
"title": "Complex spectral line profiles resulting from cryogenic deformation of the SINFONI/SPIFFI diffraction gratings"
} |
Double-parton scattering effects in associated productionof charm mesons and dijets at the LHCAntoni Szczurek[also at University of Rzeszów, PL-35-959 Rzeszów, Poland] December 30, 2023 ================================================================================================= With recent innovations in dense image captioning, it is now possible to describe every object of the scene with a caption while objects are determined by bounding boxes. However, interpretation of such an output is not trivial due to the existence of many overlapping bounding boxes. Furthermore, in current captioning frameworks, the user is not able to involve personal preferences to exclude out of interest areas. In this paper, we propose a novel hybrid deep learning architecture for interactive region segmentation and captioning where the user is able to specify an arbitrary region of the image that should be processed. To this end, a dedicated Fully Convolutional Network (FCN) named Lyncean FCN (LFCN) is trained using our special training data to isolate the User Intention Region (UIR) as the output of an efficient segmentation. In parallel, a dense image captioning model is utilized to provide a wide variety of captions for that region. Then, the UIR will be explained with the caption of the best match bounding box. To the best of our knowledge, this is the first work that provides such a comprehensive output. Our experiments show the superiority of the proposed approach over state-of-the-art interactive segmentation methods on several well-known datasets. In addition, replacement of the bounding boxes with the result of the interactive segmentation leads to a better understanding of the dense image captioning output as well as accuracy enhancement for the object detection in terms of Intersection over Union (IoU).§ INTRODUCTION As one of the main sources of the human knowledge, our visual system including eyes, optic nerves and brain is able to easily detect, separate and describe each object of a scene. Inspired by this natural ability, interactive region segmentation and captioning is the task of parallel detection, separation and description of the visual user interests. This procedure can be exploited in several complex applications such as automatic image annotation and retrieval <cit.>. To approach the task, one needs to have a full understanding of the scene which is equivalent to recognize and also locate all the visible objects. To this end, several object recognition techniques <cit.> have been proposed to detect image objects in different scales. In most of the literature, detected objects are determined by drawing bounding boxes around them. Although this notation is able to facilitate the detection process by decreasing its computational complexity, such an output is less informative when dealing with geometrical properties of the objects. As a more illustrative visual recognition technique, semantic segmentation <cit.> aims to assign a label to each pixel of the image where the labels can be class-aware or instance-aware. While the multi-level nature of semantic segmentation increases the problem dimensionality, interactive image segmentation <cit.> tries to adjust the segmentation task with the user priorities in a simpler problem space. In reality, it sounds reasonable that human users may have a more restricted area of interest than the entire scope of the scene. Thus, the multi-dimensional semantic segmentation task can be shrunk to a binary segmentation problem aiming to separate the User Intention Region (UIR) as the foreground from other parts of the scene which requires less time and computations.Equipped by the rich semantic memory of the visual data, the human observer is easily able to provide detailed explanation about different parts of an image which is a hard task in artificial intelligence. Thanks to recent developments of language models <cit.>, image captioning <cit.> makes it possible to produce linguistic descriptions of an image through a multimodal embedding of the visual stimuli and the word representation <cit.> in a joint vector space <cit.>. In this paper, we propose a novel hybrid deep architecture for integrated detection, segmentation and captioning of the user preferences where the amount of the user interactions is limited to one or a few clicks. To this end, we designed a heuristic technique for the efficient generation of the synthetic user interactions. In addition, the new architecture of the proposed Lyncean Fully Convolutional Network (LFCN) leads to a better sight of the deep component that is responsible for interactive segmentation. Last but not least, as depicted in Fig. <ref>, our combination of interactive segmentation and dense captioning tasks introduces a new class of outputs where the user intention recognition meets linguistic interpretations and vice versa. Let us stress at this point that our main contributions are (i) to provide the first deep framework for combined interactive segmentation and captioning, and (ii) to achieve segnificant improvements in the interactive segmentation over other methods.§ MORE DETAILS ON RELATED WORKSWith the increasing popularity of deep learning architectures <cit.>, both detection and captioning procedures have attracted a new wave of considerations. Convolutional Neural Networks (CNNs) <cit.> have presented the ability to construct numerous visual features in different levels of abstraction through supervised learning. This property leads to feature generators that are able to reach near-human performance in various computer vision tasks <cit.>. In addition, the structure of Fully Convolutional Networks (FCNs) <cit.> made it feasible to apply inputs of any size to the network and generate associated output in the same spatial domain. In contrast to CNNs, FCNs are able to maintain spatial information which is crucial to perform a pixel-level prediction such as semantic segmentation, object localization <cit.>, depth estimation <cit.> and interactive segmentation. Furthermore, Recurrent Neural Networks (RNNs) <cit.> reveal potential for learning long term dependencies which is essential for simulating the continuous space of natural languages. Recently, CNN-RNN models are proposed to wrap detection and captioning tasks in an end-to-end learnable platform <cit.>. However, up to now the results appear to be mostly an unorganized and overcrowded set of captions and bounding boxes. These results are not easily understandable especially in the presence of several overlapping region proposals cf. Fig. <ref> (c). In addition, they do not involve user intentions. Before the success of CNNs in object detection, some classical techniques such as Histogram of Gradients (HoG) <cit.>, Deformable Part Models (DPM) <cit.> and selective search <cit.> (as an explicit region proposal method) were proposed. Later, in the Region-based CNN (R-CNN) model <cit.>, each proposed region has been forwarded through a separate CNN for the feature extraction. This model had some drawbacks such as a complex multi-stage training pipeline and expensive training process. To overcome those obstacles, the Fast R-CNN model <cit.> is proposed where a combination of a CNN and the Region of Interest (RoI) pooling mechanism is used to produce better information for region proposal. In the Faster R-CNN <cit.>, the CNN architecture is used not only for the feature extraction but also for region proposal itself. This leads to the invention of the Region Proposal Networks (RPNs) that are able to share full-image convolutional features with detection networks. The main achievement of this innovation is the parallel detection and localization of the objects in one forward pass of the network. In spite of these improvements, such models are not able to backpropagate through the bounding boxes information. Recently, Johnson et. al <cit.> proposed a localization layer based on Faster R-CNN architecture where the RoI pooling mechanism is replaced by bilinear interpolation <cit.> that makes it possible to propagate backward through all the information.The primary purpose of the image captioning was image annotation <cit.> as the automatic assignment of some keywords to a digital image. By replacing keywords with some sentences that are able to describe not only the image objects but also the semantic relations in between, image captioning received more attention. The main problem in the automatic image description was the scarcity of the training data. Recent development of large datasets including images and their descriptions <cit.> makes it feasible to expand learning-based captioning techniques. Classical image captioning approaches produced image descriptions by generative grammars <cit.> or pre-defined templates working on some specific visual features <cit.>.In contrast, recently developed deep learning solutions apply an RNN-based language model that is conditioned on the output vectors of a CNN to generate descriptive sentences <cit.>. With the growing popularity of interactive devices such as smart phones and tablets, interactive image processing attracts more attention. Interactive segmentation offers a pixel-wise classification based on user priorities. Among all the traditional approaches of the interactive segmentation, stroke-based techniques <cit.> are often based on graph cut techniques. In these methods, an energy function based on region/boundary division is optimized to find the segmentation result. Alternative approaches include random walks <cit.> and geodesics <cit.>, mostly relying on low-level features such as color, texture and shape information. These types of attributes can be difficult to apply when the image appearance is complicated due to complex lighting conditions or existence of intricate textures. Recently, deep learning models have been used for interactive segmentation where the information of the image will be considered in higher semantic levels. To this end, FCNs as the standard frameworks for pixel-wise end-to-end learning tasks, have been applied <cit.>. § PROPOSED METHODOur model receives an input image as well as user interactions in the form of positive/negative clicks and provides a seamless framework to generate accurate segmentation as well as expressive description of the UIR. In the preprocessing step, an efficient morphological technique will be used to providea huge amount of training samples in the form of synthetic user interactions. Then, each set of positive/negative seeds will be transformed into separate Voronoi diagrams as shown in Fig. <ref>. Next, a sequence of dedicated LFCNs with different granularities are applied as the interactive segmentation modules. Afterwards, a dense captioning architecture inspired by <cit.> will be utilized to obtain a number of region proposals along with their captions. In the fusion step, a heuristic method will be provided to combine results of localization, segmentation and captioning procedures to acquire highlighted borders of the UIR along with its expositor caption. In the following, we will investigate all steps of our model in detail. §.§ User Action ImitationDuring interactive segmentation, the user will be asked to provide some general information about the position of the intended region. The requested information consists of some positive and negative seeds as depicted in Fig. <ref> which are equivalent to internal and external points of the UIR, respectively. Next, each set of seeds will be used to shape a Voronoi diagram. We denote each seed by s_k, k={1,…,n}. The value of pixel v_i,j of the Voronoi diagram will be calculated byv_i,j:=min{D_1,D_2,…,D_n}where D_k is the Euclidean distance of v_i,j to the seed s_k. To summarize, the value of each pixel in the Voronoi diagram is the Euclidean distance of that pixel to the nearest seed. For the sake of clarity, there should be a minimum inter-cluster distance in each set of the seeds. In addition, a minimum intra-cluster distance is also required to retain boundary regions of the clusters as distinctive as possible. So: * Every pair of seeds in each set should preserve a pre-defined distance from each other:∃ d_1∈ℝ^+ : ∀ (s_i,s_j)∈ S, (s_i,s_j)_2>d_1 * All the seeds of each set should preserve a minimum distance from boundary pixels of the UIR (∂(UIR)):∃ d_2 ∈ℝ^+ :∀ s_i∈ S, u∈∂(UIR), (s_i,u)_2>d_2 As expected, natural collection of such a data is unreasonably time consuming and expensive. Recently, Xu et al. <cit.> proposed some strategies for synthetic generation of user interactions. They ordained random generation for positive clicks inside the UIR while three distinct set of negative clicks are chosen as: 1) random background pixels with a certain distance to the UIR, 2) a point cloud inside the negative objects and 3) a uniform set of surrounding points of the UIR. Since their implementation is not publicly available, it seems their first and the second negative strategies do not obey natural interactions and the third one may be computationally expensive (see equation (2) in <cit.>).Morphological Cortex Detection (MCD). While the inside of the UIR can be quite small, the background region is usually large enough to provide useful geometric information about the UIR. Consequently, it is beneficial to generate negative seeds that surround the UIR uniformly. To provide an efficient implementation for such an interaction, we replace third negative strategy proposed in Xu et al. <cit.> with a Morphological Cortex Detection (MCD) technique that noticeably improves computational efficiency. Moreover, this method is able to simulate UIR cortex in different scales that enables convolutional filter of the LFCN to track UIR geometry in different layers. To implement this idea, a 1-pixel-wide boundary shape of the UIR will be extracted by performing a dilation on the binary mask of UIR in training dataset. Then, the original mask is subtracted from the dilation result. In the next step, this boundary path will be completely traversed using a 3×3 window to transfer all the boundary points' coordinates into a 1-D array in which the requested negative seeds can be selected uniformly. As the result of the MCD process, a uniform set of negative seeds will be obtained that represents the cortex of the UIR perfectly. The visual illustration of this technique is shown in Fig. <ref>. During our experiments, positive clicks are simulated randomly inside the UIR while negative seeds are generated by MCD mechanism in three different levels.§.§ Intention RecognitionFor the task of intention recognition, we make use of a dedicated version of the standard FCN <cit.> where the last two fully connected layers are replaced with three convolutional layers containing decreasing kernel sizes of 7, 5 and 3. The impact of such an alternation is the gradual growth of the receptive field. This property improves network recognition of objects' geometry. Hence, we named this architecture as Lyncean Fully Convolutional Network (LFCN). By a proper use of zero padding, all the extended convolutional layers have the same output size. At the end of the extended part, the aggregated output of the additional layers will be upsampled to the size of the input as elaborated in <cit.>. §.§ Fusion ApproachIn order to supplement the result of the interactive segmentation with a proper linguistic commentary, we employ the dense image captioning framework <cit.>. The internal RPN of this architecture provides confidence scores for the existence of the object in proposed regions. After descending sort of the objectness scores, top-ranked region proposals include the most reasonable captions for the objects of the scene. With the comparison of the interactive segmentation result and the bounding boxes, the best match bounding box and the corresponding caption will be obtained (Fig. <ref>). § EXPERIMENTSDatasets. For fine tuning of the LFCN, we used the PASCAL VOC 2012 segmentation dataset <cit.>. The dataset includes 1464 images for training and 1449 images for validation that are distributed in 20 different classes. We used the whole bunch of these samples to generate our special training pairs in the preprocessing step. For the final validation of the model as well as its comparison with state-of-the-art interactive segmentation, we utilized different well-known segmentation benchmarks including Alpha Matting <cit.>, Berkeley segmentation dataset (BSDS500) <cit.>, Weizmann segmentation evaluation database <cit.>, image object segmentation visual quality evaluation database <cit.> and VOC validation subset. Preprocessing. To generate all the necessary training pairs of the interactive segmentation process, we produced positive and negative Voronoi diagrams with respect to each object that is visible in VOC dataset. The positive seeds are selected randomly inside each object while the MCD approach is used to generate three distinct sets of negative seeds with different distances from the intended object. In the last step, each combination of positive/negative Voronoi diagrams, forms a unique training pair. This leads to production of 97,055 interaction patterns. We preserved 7,055 instances for the test and used the rest as the training data. §.§ Fine Tuning of the Proposed LFCN ArchitectureTo reach the best quality for the interactive segmentation, our LFCN is trained in three different levels of granularity as proposed in <cit.>: LFCN32s, LFCN16s and LFCN8s.RGB channels of the input image should be concatenated with the corresponding Voronoi diagrams to form a training instance. Consequently, the first convolutional layer of our LFCN contains five channels. During the network initialization, the RGB-related channels will be initialized by the parameters of the original FCN <cit.>. For two extra channels that are associated with Voronoi diagrams, the zero initialization is the best choice as also mentioned in <cit.>. Learning parameters of the finer networks should be initialized from the coarser one. The global learning rates of the networks are 1e-8, 1e-10 and 1e-12, respectively while the extended convolutional layers exploit one hundred times bigger learning rates. The learning policy is fixed and we used the weight decay of 5e-3. §.§ MetricsIn order to evaluate UIR localization accuracy of the proposed model, we calculated the well-known measure of Intersection over Union (IoU). To this aim, we computed the IoU of the detected UIR and the corresponding binary label of the validation samples. For the sake of complete comparison between our model and other interactive segmentation techniques, three performance metrics of pixel accuracy, mean accuracy and mean IoU are computed. The segmentation task of the proposed approach can be considered as a binary segmentation where the classes are limited to foreground (UIR) and background. So, we used binary interpretation of the semantic segmentation metrics that are proposed by Long et al. <cit.>: * Pixel Accuracy (Pixel Acc.):This measure represents the proportion of the correctly classified foreground (C_f) and background (C_b) pixels (true positive rates) to the total number of ground truth pixels in foreground (F) and background (B).C_f+C_b/F+BUnfortunately, this metric can be easily influenced by the class imbalance. Hence, high pixel accuracy does not necessarily mean that the accuracy is acceptable when one of the classes is too small or too large. * Mean Pixel Accuracy (Mean Acc.):This measure is computed as the mean of the separate foreground and background pixel accuracies:C_f/F+C_b/B/2This metric alleviates the imbalance problem but can be still misleading. For example when the great majority of pixels are background, a method that predicts all the pixels as background can still have seemingly good performance.* Mean Intersection over Union (Mean IoU):Intersection over union is the matching ratio between the result of object localization process and the corresponding ground truth label. This metric is the average of the computed intersection over union for the foreground and background regions:(C_f/C_f+FP+FN)_f+(C_b/C_b+FP+FN)_b/2Here FP and FN denote the number of the false positive and false negative rates of each class, respectively. This metric solves the previously described issues. §.§ ResultsTest of localization accuracy.In the first step of evaluation, we test our model with a random subset of unseen samples in validation datasets. The response of the model to some instances is shown in Fig. <ref>. It can be noticed that the output of our approach achieves a considerable rate of accuracy regarding the similarity of the model output with the corresponding ground truth. Furthermore, the confusing output of the dense image captioning is replaced with an explicit situation where the segmented UIR and its description are easily distinguishable.Fig. <ref> (left diagram) presents a comparison between the localization accuracy of the proposed method and the internal RPN of the DenseCap <cit.> in terms of the obtained IoU for the samples presented in Fig. <ref>. As illustrated, our model provides a significant improvement regarding the localization accuracy. These results also demonstrate the proficiency of our model in combining interactive segmentation, region proposal and image captioning techniques.Sensitivity analysis.In this part, we analysed variations of the model output quality against the number of user interactions. As it is shown in Fig <ref>, although IoU can be improved by applying more user interactions that facilitate boundary detection, our model still provides very good results even by minimum number of clicks. We also provided mean IoU accuracy of the proposed model for five different datasets in Fig. <ref> (right diagram) that confirms satisfying performance of our model in the case of low interactive information. This noticeable property of our approach makes it convenient to be applied in real-world applications. During our experiments, the proposed method clearly achieves a satisfying segmentation outcome with just one click.Comparison of the segmentation quality.In this part we performed an extensive evaluation on segmentation capabilities of the proposed method versus some prevalent segmentation techniques such as Geodesic Matting (GM) <cit.>, GrowCut <cit.>, Grabcut <cit.>, Boykov Jolly (BJ) interactive graph cuts <cit.>, Geodesic Star Convexity (GSC) <cit.>, Geodesic Star Convexity with sequential constraints (GSCSEQ), Random Walker (RW) segmentation <cit.>, Shortest Path-based interactive segmentation (SP) <cit.> and Matching Attributed Relational Graphs (MARG) <cit.>. In all the experiments we generated five positive and five negative clicks randomly. For some of the approaches where the user interactions were defined as points or scribbles, we determined click positions with five-pixel-wide circles. To observe the impact of the extended part of the LFCN on the output quality, we also report all the accuracy measures for the normal version of the FCN as well. Table <ref> and <ref> present quantitative results that confirm our approach superiority over several other segmentation techniques on five different benchmarks. As a qualitative comparison, Fig. <ref> represents final segmentation output of the methods in Table <ref> for two different samples. As it can be seen, our approach provides the most accurate segmentation result with respect to semantic interpretation of the scene using same number of interactions.Dense interactive region captioning. In the final part of our experiments, we verified the ability of the proposed model to caption several regions of the image. Fig. <ref> shows the result of such an experiment where multiple objects in different scales are detected via user interactions and described properly.§ CONCLUSIONIn this paper, we presented a novel hybrid deep learning framework which is capable of targeted segmentation and captioning as a response to user interactive actions. A wide variety of experiments confirmed our model superiority over various state-of-the-art interactive segmentation approaches. In addition, further experiments demonstrated our model capability to caption an arbitrary region of the image with one or few clicks, which is especially convenient for real-world interactive applications. | http://arxiv.org/abs/1707.08364v1 | {
"authors": [
"Ali Sharifi Boroujerdi",
"Maryam Khanian",
"Michael Breuss"
],
"categories": [
"cs.CV",
"68T45"
],
"primary_category": "cs.CV",
"published": "20170726104033",
"title": "Deep Interactive Region Segmentation and Captioning"
} |
Institut für Theoretische Physik, Universität Bremen, Germany Institut für Theoretische Physik und Astrophysik, Universität Würzburg, Germany Istituto Officina dei Materiali (CNR-IOM) and Scuola Internazionale Superiore di Studi Avanzati (SISSA), Via Bonomea 265, 34136 Trieste, Italy Nanostructures with open shell transition metal or molecular constituents host often strong electronic correlations and are highly sensitive to atomistic material details. This tutorial review discusses method developments and applications of theoretical approaches for the realistic description of the electronic and magnetic properties of nanostructures with correlated electrons. First, the implementation of a flexible interface between density functional theory and a variant of dynamical mean field theory (DMFT) highly suitable for the simulation of complex correlated structures is explained and illustrated. On the DMFT side, this interface is largely based on recent developments of quantum Monte Carlo and exact diagonalization techniques allowing for efficient descriptions of general four fermion Coulomb interactions, reduced symmetries and spin-orbit coupling, which are explained here. With the examples of the Cr (001) surfaces, magnetic adatoms, and molecular systems it is shown how the interplay of Hubbard U and Hund's J determines charge and spin fluctuations and how these interactions drive different sorts of correlation effects in nanosystems. Non-local interactions and correlations present a particular challenge for the theory of low dimensional systems. We present our method developments addressing these two challenges, i.e. advancements of the dynamical vertex approximation and a combination of the constrained random phase approximation with continuum medium theories. We demonstrate how non-local interaction and correlation phenomena are controlled not only by dimensionality but also by coupling to the environment which is typically important for determining the physics of nanosystems.Realistic theory of electronic correlations in nanoscopic systems Malte Schüler1 Stefan Barthel1 Tim Wehling1 Michael Karolak2 Angelo Valli3 Giorgio Sangiovanni2 December 30, 2023 =================================================================================================== § INTRODUCTIONThe electronic structure problem in nanostructures involves two sources of complexity:First, on a single-particle level, a large Hilbert space can be required to describehow electronic orbitals adjust to a complex arrangement of many inequivalent atoms,which by definition of a nanostructure lacks crystal translational symmetry in one or more spatial directions.Second, electron-electron interactions are usually pronounced in systems with reduced dimensionality,triggering strong electronic correlation effects.In this review, we report on recent advances on realistic material simulations of nanostructureshosting electronic correlations which were made within the DFG Research Unit FOR 1346.Our strategy, here, is to combine first-principles methods— most prominently density functional theory (DFT) and the GW approximation — with approaches for the descriptionof strong electronic correlations <cit.>, in particular dynamical mean-field theory (DMFT) <cit.> and related approaches/extensions thereof.The general idea is to account for atomistic details of the nanosystems based on the first-principles methods,which are then used to derive realistic model Hamiltonians describing the electron correlationin effective low energy Fock spaces constructed out of a reduced set of single-particle orbitals.In this way, realistic and complex nanostructures are in principle within reach for approaches like DMFT,which critically rely on Hilbert space dimensions not being too large.The review is structured as follows: In section <ref> we explain howfirst-principles and correlated electron approaches can be interfaced in such a way thattreatments of complex (nano)structures becomes possibleand how the resulting many-body problems can be solved efficiently.Section <ref> is dedicated to a brief description of the impurity solvers used by us in this context.In Section <ref> we discuss applications of this approach to so-called “Hund's impurity systems”,molecular systems as well as the Cr (001) surface.Section <ref> explains a novel approach for the derivation of realistically screened interaction termsin complex nanosystems and discusses progress on how to treat non-local interactions and non-local correlations in nanosystems. § INTERFACE OF DENSITY FUNCTIONAL AND DYNAMICAL MEAN FIELD THEORY FOR THE SIMULATION OF COMPLEX STRUCTURESConsiderable efforts have been focussed on the modelling of correlated nanoscale systems using DMFT techniques,e.g. <cit.>.Here, we describe the approach to a material realistic DMFT scheme for nanosystemswhich has been implemented by the authors during the course of FOR 1346, and which is particularly flexible.The DFT simulation of a material or nanostructure yields an auxiliary modelof non-interacting electrons in the materialvia the Kohn-Sham wave functions {|𝐤,N⟩} and energies ε^N_𝐤.The first step of any approach combining ab-initio and many-body methods (“DFT++”)is to consider this Kohn-Sham system as an effective single-particle starting point for the electronic problem.Then, one can identify a correlated subspace {|m⟩}, where the Kohn-Sham systemcan be augmented by interactions beyond those included within DFT.Here, m is a combined site- and (spin-)orbital-index.Let us also stress that, at this stage, the distinction between the correlated orbitalsand “the rest” of the system is just formal:beyond the actually correlated orbitals (e.g. 3d), what we call correlated subspacecan indeed contain also orbitals (such as p or s)on which we are not always going to explicitly add a Coulomb interaction,but that we nevertheless want to keep inour “low-energy” basis set and treat as “active” degrees of freedom.Independently on these formal aspects, the procedure always requires the addition of a double-counting correctionin order to account for the interactions already present within DFT. To make calculations feasible local interactions are often assumed[A discussion of how appropriate realistic local effective interaction terms can be obtained is given in Section <ref>.]. The low-energy effective Hamiltonian describes a generalized multi-impurity Anderson Model (AIM),which also includes the coupling of the correlated subspace to leads or substrate atoms.The latter is described by a frequency- and momentum-dependent potential denoted, in Eq.<ref>,by the Greek letter Γ. It represents a fixed physical bath,i.e. a particle reservoir determined by the geometry of the problem and by the kind of couplingbetween the correlated atoms and the rest of the system.This bath is not to be confused with the self-consistent DMFT bath of the auxiliary single impurity problems.An example of this is sketched in Fig. <ref> where the red atoms host the correlated orbitalsand the blue ones represent the physical bath. The orbitals |m⟩ spanning the correlated subspace should be chosen as localized as possibleor atomic-like in order to have a consistent description in terms of AIM or Hubbard-like models with local interactions.It is useful to introduce a matrix notation in the basis of the (spin-)orbitals of all atoms in the low-energy subspace.The “Pinocchio-hat” ▵ indicates full matrices in this space.Since for the DMFT algorithm we sometimes work with sub-blocks of such matrices corresponding to a given atom,we indicate these N_orb× N_orb smaller matrices with the standard hat-symbol. We consider both periodic crystals and inhomogeneous nano-/hetero-structures with reduced,or even absent translational symmetry. For the cases in which the spatial periodicity is (even only partially) preserved,the matrix structure reflects the fact that the unit cell can contain more than one atom and each atom contributes,in general, more than one (spin-)orbital. Similarly, in all other non-periodic cases, it describes the real-space structure of all the (multi-orbital) atoms of the nanostructure.Hence, the 𝐤-integrated Green's function can be written as G(iω_n) = 1/N_𝐤∑_𝐤[ (iω_n + μ)11 - H(𝐤) -Γ(𝐤,iω_n)- ϵ_ DC - Σ( k,iω_n)]^-1,where ω_n = π T (2n+1) are the Matsubara frequencies at temperature T and μ is the chemical potential. The matrix H( k) encodes both inter- and intra-atomic single-particle contributions,while Γ(𝐤,iω_n) describes the hybridizationof the orbitals of the correlated subspace to the physical leads or to the substrate.In general, Γ(𝐤,iω_n) and H(𝐤) are obtained via projection of the Kohn-Sham non-interacting Green's function onto the localized states,as described in Section <ref>.Since these two quantities represent physical properties of the system they are fixed for the rest of the calculation, or recomputed at each iteration of a charge self-consistent DFT+DMFT (see Sec. <ref>). If the system is a non-periodic nanostructure– as the one shown in Fig. <ref>a – and there is no translation symmetryin any spatial direction, N_𝐤 is formally equal to 1, and the sum drops out.In that case H and Γ(iω_n) are 𝐤-independent matrices, the latter being the “standard” frequency-dependent hybridization function, often also called Δ(iω_n) in the context of the Anderson impurity model. If instead the correlated atoms form a periodic structure– as e.g. a thin film or a nanowire of a correlated material on a substrate as sketched in Fig. <ref>b –the 𝐤-summation extends over the corresponding one- or two-dimensional Brillouin zone.In all cases, the double-counting correction ϵ_DCretains a matrix structure, in order to account for the presence of more than one orbital/atom.The self-energy Σ( k,iω_n) describes the dynamical effectsof electron-electron interaction.Within DMFT, the self-energy is purely local, i.e., it carries no k-dependenceand it is a block-diagonal matrix in the atoms.A structure in k-space for the self-energy, as well as off-diagonal elementsbetween different atoms can be obtained by taking into accountnon-local electronic correlations beyond DMFT, as discussed in Section <ref>. The simplest case of a simulation combining ab-initio and many-body approaches is represented by a single correlated adatom or an atom coupled to some leads: The solution of the low-energy model is obtained via a multi-orbital quantum impurity solver which is able to yield the self-energy and hence the two-point Green's function, given the bath described by Γ(iω_n) and the local-level structure of H. For the cases of a non-periodic nanostructure with more than one atom as well as for those of a periodic arrangement of correlated sites, the calculation requires instead a self-consistent adjustment of the Weiss field, i.e. the “DMFT” bath, that describes the effect on the i-th atom of the other correlated sites of the system. This is defined as𝒢̂^-1_i,N_orb× N_orb(iω_n) = (. G(iω_n) |^atomi_N_orb× N_orb)^-1 + Σ̂^atomi_N_orb× N_orb,where the matrix to be inverted is the N_orb× N_orb block corresponding to the i-th atom of the 𝐤-summed Green's function of Eq. <ref>. The Weiss field defines an auxiliary single-site quantum many-body impurity model, which must be solved to get the two-point Green's function of the i-th site Ĝ_imp i(iω_n). The corresponding self-energy is obtained via the Dyson equationΣ̂^atomi_N_orb× N_orb = 𝒢̂^-1_i,N_orb× N_orb(iω_n) - ( Ĝ_imp i(iω_n) )^-1Once the self-energy matrices for all sites in the cluster or in the unit cell are calculated via Eq. <ref>, the full matrix Σ(iω_n) is constructed block-wise and inserted back into Eq. <ref> to get a new local Green's function for the whole system. The self-consistency loop therefore goes on by calculating Eq. <ref> for each site again, until convergence is reached. §.§ General projector and local Hamiltonian formalismThe DFT++ scheme explained in the previous section relies on extracting H(𝐤) and Γ(𝐤,iω_n) as entering Eq. <ref> from ab initio calculations. We achieve this using projection operators P = ∑_m |m⟩⟨m| on the correlated subspace, as we explain in the following. To define P, in practice, the overlaps ⟨ m | 𝐤,N⟩ of correlated local orbitals and Kohn-Sham wave functions in the DFT code are sufficient and taken e.g. from the PAW projectors implemented in the VASP code or more generally using any kind of Wannier function describing the correlated orbitals.The DFT simulation yields the Kohn-Sham Hamiltonian H_KS(𝐤) together with the corresponding eigenstates {|𝐤,N⟩} and energies ε^N_𝐤 and thus also the Kohn-Sham Green function:G_KS(𝐤,iω_n) = [iω_n - H_KS(𝐤)]^-1= ∑_N | 𝐤,N⟩[iω_n - ε^N_𝐤]^-1⟨𝐤,N |G_KS is a matrix in the space of all Kohn Sham states. It is particularly for nanostructures a much higher dimensional object than the matrices occurring in Eq. <ref>, which act on the correlated subspace only. The projections of H_KS and G_KS(𝐤,iω_n) on the correlated subspace yields all quantities entering Eq. <ref>:H(𝐤)=P H_KS(𝐤)Por element wise ( H)_mm'(𝐤)= ∑_N ⟨ m | 𝐤,N⟩⟨𝐤,N |m' ⟩ε^N_𝐤. The projection of the Kohn-Sham Green function on the correlated subspace P G_KS(𝐤,iω_n)Pthen yields the frequency- and momentum-dependent potential entering Eq. <ref> viaΓ(𝐤,i ω_n)=(iω_n + μ)1 - H(𝐤)-(P G_KS(𝐤,iω_n)P)^-1. In the case of an adatom or a molecule absorbed on a surface (c.f. section <ref>-<ref>) there is no 𝐤-dependence in Eq. (<ref>). Then, (P G_KS(𝐤,iω_n)P) plays the role of the Kohn-Sham impurity Green function, H corresponds to the crystal field and Γ is often also labelled hybridization function Δ. See e.g. Eq. (12) in Ref. <cit.>.Eq. (<ref>) is often expressed explicitly in terms of hopping matrix elements between impurity orbitals and bath states. If the 𝐤-dependence is absent, this takes the following form: (Γ(iω_n))_ii'=∑_νV_iνV_ν i'/iω_n-ϵ_ν, where V_i,ν connects the impurity orbital i with the bath state ν at energy ϵ_ν. The imaginary part of the hybridization function Γ is then simply related to the bath density of states ρ(ω) if we assume that the hopping V between all impurity orbitals and the bath is the same: (Γ(ω+i0^+))_ii=-π|V|^2ρ(ω).§.§ Charge self-consistencyThe DFT++ approach implicitly includes interactions between electrons in the correlated subspace and the rest of the system through the Hartree as well as the exchange correlation potential from DFT. As soon as the many-body part of DFT++ redistributes electrons between correlated and uncorrelated orbitals or also between different sites there will be associated Hartree (as well as possible exchange or correlation) energies and the DFT++ Hamiltonian should be correspondingly updated. In general, it is obviously problematic to obtain the update of the DFT++ Hamiltonian simply from a double-counting correction applied to the correlated subspace only. This can be better achieved by including self-consistency over the charge-density in the DFT++ approach which obviously requires us to work in the full Kohn-Sham Hilbert space of the DFT part of our approach. We calculate the updated electron density of the DFT++ system,n(r) = 1/β∑_𝐤,N,N',n⟨ r| 𝐤,N⟩ G_N N'(𝐤,iω_n) ⟨𝐤,N' | r ⟩where G_N N'(𝐤,iω_n)=⟨𝐤,N|G(𝐤,iω_n)|𝐤,N'⟩ is the interacting Green's function of the full system and acts on the same Hilbert space as Eq. (<ref>). It includes corrections due to dynamic self-energy effects within the correlated subspace according toG(𝐤,iω_n) = [iω_n - H_KS(𝐤)- ϵ_ DC - Σ( k,iω_n)]^-1.ϵ_ DC and Σ are quantities defined in the full Hilbert space and are obtained from the corresponding ones (ϵ_ DC and Σ) using the overlaps ⟨ m | 𝐤,N⟩, as explained in Ref. <cit.>. With the new density n(r) one can recalculate the DFT potential and solve the resulting Kohn-Sham Hamiltonian, which then defines a new Kohn-Sham Green function via Eq. <ref>. In this way, a charge self-consistent DFT++ scheme is obtained, which includes interactions between electrons of the correlated subspace and the rest in a fully self-consistent static mean-field manner. Several implementations of charge self-consistent DFT+DMFT have been reported, e.g. Refs. <cit.>, based on projector formalisms similar to Sec. <ref>. It is intuitively clear that the Hartree terms occurring within DFT++ charge self-consistency counteract large charge redistributions. In other words, ambiguities stemming for instance from the unknown double-counting potential can be expected to be less severe in charge self-consistent DFT++ calculations as compared to one-shot calculations. This has been explicitly demonstrated, e.g.,for the Matsubara self-energies in the iron pnictide superconductor LaFeAsO, where the discrepancy between FLL and AMF approaches is significantly reduced in the fully charge self-consistent scheme <cit.>. In nanosystems, the issue of charge redistributions can be even more severe than in bulk systems, since charge redistributions over larger distances are facilitated by the large supercells needed to describe the nanosystems. We thus implemented a charge self-consistent version of DFT+DMFT in the framework of the VASP code in collaboration with project P8. A simple benchmark is presented in Fig. <ref>, where we compare the density of states of antiferromagnetic NiO calculated from LDA+U (panel a)) and from a charge self-consistent LDA++ calculation where we solve the corresponding Hubbard model in mean field approximation. In principle, both methods should give the same density of states. Qualitatively, the orbital resolved DOS are the same. However, due to minordiscrepancies in the definition of the projectors which are used internally in VASP and externally for the interface, some differences such as the gap and details in the valence band occur. § IMPURITY SOLVERSThe solution of the Anderson impurity model for general parameters has to be done numerically by means of, e.g., quantum Monte Carlo <cit.> (QMC), numerical renormalization group <cit.> (NRG), or exact diagonalization (ED) methods <cit.>. While NRG and QMC are in principle numerically exact methods, they become computationally very demanding, when dealing with many orbitals, hybridization functions with low symmetry, spin orbit coupling and general fermionic four operator Coulomb vertices. Here we briefly describe the QMC and ED solvers that we mostly used for the study of nanoscopic systems. §.§ Continuous-Time Quantum Monte CarloThe quantum Monte Carlo results presented here were obtained using the numerically exact continuous-time version of the algorithm using the hybridization expansion <cit.>. We used the w2dynamics code package, which is a hybrid Python/Fortran90 code written in Vienna and Würzburg. It is a versatile code that is capable of handling complex systems with multiple impurities and difficult nano geometries. Impurities can be completely inequivalent and independently defined with their own Coulomb interactions and hybridizations. A general interface to any electronic structure code is provided via the usage of the Wannier Hamiltonian in reciprocal space as input. The solver portion of the code is a highly optimized Fortran90 code with Segment <cit.>, Matrix <cit.> and Krylov <cit.> implementations of the hybridization expansion algorithm, that are optimal in different regions of the parameter space. A general four index Coulomb tensor (e.g obtained from cRPA or cDFT) can be used as well. The code is capable of identifying certain conserved quantities <cit.> depending on the form of the interaction used (density-density, Kanamori, full) which greatly speeds up the simulations. §.§ Variational exact diagonalization approachThe general concept of the exact diagonalization approach is to exactly solve an auxiliary Hamiltonian, which is an approximation to the full Hamiltonian in some systematic sense and is solved exactly. As solving a many-body problem exactly means diagonalizing the Hamiltonian on its Fock-space, which grows exponentially with the dimension of the Hilbert space, the original model has to be discretized in some way. Loosing the “exactness” of QMC has however some important advantages: first of all, ED has no restriction on the low symmetries and the general form of Coulomb vertices and, second, it allows us to access real-axis Green's functions and self-energies without the need of stochastic analytic continuation of noisy QMC data. Several efforts are dedicated to alleviate the problem of the bath discretization of ED <cit.>. Here, we briefly discuss a strictly variational method of approximating an AIM with continuous bath by an AIM with finite strongly reduced number of bath sites, which we call variational ED method <cit.>. It guarantees an optimal approximation to the AIM for a given number of bath sites in the sense of thermodynamic ground-state properties. The method is based on the well-known Peierls-Feynman-Bogoliubov variational principle <cit.>. The minimization of the Peierls-Feynman-Bogoliubov free energy functional Φ̃[ρ_H̃] = Φ_H̃ + ⟨H-H̃⟩_H̃,leads to an optimal approximation of a Hamiltonian H by an effective simpler model H̃ in terms of an optimal density matrix. Here Φ_H̃ = -1/βln Z_H̃ is the free energy of the effective system. ⟨·⟩_H̃ denotes a thermodynamic expectation value with respect to the effective system. In our case H represents the full Anderson impurity model and H̃ is the model with discretized bath.The ansatz for the effective model determines the quality of the method and has to be chosen carefully. It has to be simple enough that we can solve it exactly but still include explicit interaction terms. We divide the Hilbert space of the effective system into two classes: First, a small subset, the so-called correlated subspace 𝒞, is equipped with interaction terms in the effective Hamiltonian H̃ ^𝒞. The remaining space, the uncorrelated subspace ℛ, is equipped with single-particle terms only and described by H̃ ^ℛ. The structure of the total effective Hamiltonian H̃=H̃ ^𝒞+H̃ ^ℛ for the case of a single impurity orbital is depicted in the right panel of Fig. <ref>. In contrast to the original model (left panel of Fig. <ref>), the effective model consists of two decoupled parts: First, the effective interacting impurity coupled to one bath site only and second the remaining bathsites. For concreteness, we consider a cluster consisting of a multi-orbital impurity and one bath site per impurity orbital for the correlated space but other choices are similarly possible. The single particle states of the effective model are related to those of the original model by a unitary transformation u_ij, which allows for mixing of original “bath” and “impurity” character in the effective model. The optimal matrix elements of the effective model, as well as the optimal unitary transformation u_ij are found by minimizing the functional (<ref>). The Hamiltonian H̃ ^𝒞 states a many-body problem which can be solved by exact diagonalization. In contrast, the Hamiltonian H̃ ^ℛ states a single-particle problem and can be solved trivially. In summary, the Hamiltonian H̃ = H̃^𝒞 + H̃^ℛ defines an effective Hamiltonian, which can be solved exactly and thus Φ̃ (Eq. (<ref>)) can be calculated exactly. Finally, we note that the amount of variational degrees of freedom in the variational ED approach is such that it includes Hartree-Fock as the limiting case Ũ_αβγδ→ 0. Thus, we expect that variational ED will generally give more accurate energy estimates than Hartree-Fock. Details on how to deal with the large number of variational degrees of freedom in the variational ED approach are discussed in Ref. <cit.>.We demonstrate the performance of the variational ED method by applying it to a system (schematically shown in the left panel of Fig. <ref>) with only six bath sites which is small enough to be solved exactly. In Fig. <ref> we compare the double occupation obtained from the variational ED, Hartree-Fock and the fit of Green functions on Matsubara frequencies <cit.> with two different weighting functions (W_n=1, “fit0” and W_n=1/ω_n, “fit1”, compare Ref. <cit.>). The parameters of the model are the following: The impurity level is ε_d=-2.0, the interaction strength is U=4.0. The six bath levels are equally aligned around a mean bath energy ε_b = 0.02 in an interval of 2 (i.e. the bandwidth of the bath). The hybridization is swept from V_k=0.0 to V_k=1.5. Small hybridization (V=0) contains the atomic limit, large hybridization (V→∞) is the non-interacting limit. The region in between (V∼0.3) is strongly correlated as interaction and kinetic energy are on the same order. The Hartree-Fock method gives satisfactory results for small hybridization strengths and reproduces the non-interacting limit. The choice of the weight function in the fit methods has a clear impact on how well the double occupancies are reproduced, which reveals the ambiguities associated with the "conventional" bath discretization schemes "fit0" and "fit1". The variational ED method, in contrast, gives an unambiguous and very accurate approximation to quantities like double occupancies and outperforms all other methods. Small deviations to the exact double occupation exist in the strongly correlated region around V=0.3. Including explicit interaction terms in the effective Hamiltonian leads to a drastically better performance than Hartree-Fock. The same conclusion also holds for estimates of total energies, which makes the variational ED approach very interesting in the context of materials simulations. In Ref. <cit.> it is shown that the method is indeed of practical use for realistic multi-orbital impurities. There, an ab-initio derived model of Co in Cu is solved and compared to a quantum Monte Carlo solution.§ APPLICATIONSWe now review applications of the compuational scheme outlined in the two previous sections. We start with a general discussion electron correlations, charge fluctuations and magnetism in multi-orbital Anderson models and consider two examples of nanosystems without any translational symmetry: so-called "Hund's impurities" as realized by hydrogenated Fe on Pt(111) (section <ref>) and metal organic molecules on surfaces as complex Kondo systems (section <ref>). Finally, we discuss the electronic structure of the Cr (001) surface in section <ref>, as an example of a correlated system which retains translation symmetry in two dimensions.§.§ Electronic correlations driven by Hund's exchange interactionThere is a wide class of transition metal compounds in which Hund's rule coupling leads to exotic electronic properties. Examples of these so-called “Hund's metals” <cit.> include unconventional superconductors such as iron pnictides and chalcogenides <cit.> as well as non-Fermi liquids as realized in ruthenates <cit.>. Within the reasoning of DMFT, magnetic impurities strongly coupled to the electronic states of a metallic host can be viewed as fundamental constituents of such Hund's metals and are called “Hund's impurities” <cit.>.This idea is described in the following for the particular case of a 3d transition metal adatom on a metallic substrate following Refs. <cit.>. If the adatom is still in the gas phase an integer number of electrons is filled into the five 3d orbitals according to Hund's first rule: The orbitals are first filled up by electrons having the same spin, before being filled with the remaining electrons of opposite spin. This is driven by the intra-atomic exchange energy, or so-called Hund's rule exchange J. After adsorption on the metallic substrate, electrons can hop on or off the adatom's orbitals into the bath of substrate conduction electrons, which has an electronic density of states ρ, paying or gaining the direct on-site Coulomb energy U. This hopping leads to charge fluctuations at the impurity site. The hopping term V <cit.> and the valency <cit.> determine whether the adatom is most appropriately described in termsof atomic multiplets, itinerant electrons, or a mixture of both with distinct correlation effects. This is illustrated for an orbitally degenerate five orbital Anderson impurity with local interactions U=4 eV and J=1 eV and bath density of states ρ=0.05 eV^-1 at filling ⟨N̂⟩=6 in Fig. <ref>. For vanishing hybridization V≈ 0.0 the system has a well-defined valency and (half-) integer quantized spin. Also with weak hybridization, V≪ 1 eV, the adatom essentially retains its integer valency. With hybridization increased beyond V≳ 1 eV sizable charge fluctuations set in, as the valence histogram (Fig. <ref>a) demonstrates. If the adatom had only a single orbital, the magnetic moment would be simply quenched in this case. However, in the multi-orbital case, there exists a particular regime with strong hybridization which is yet too weak to overcome the Hund's coupling J, where strong charge fluctuations can coexist with sizable local magnetic moments. This has been termed the Hund's impurity regime <cit.>. The electronic excitation spectra (Fig. <ref>b) display a characteristic evolution from the atomic limit to the Hund's impurity regime <cit.>: When switching on the hybridization, a Kondo-type low energy resonance appears, broadens and merges with some of the upper and lower Hubbard peaks. In contrast to the single-orbital case, several atomic multiplet peaks persist and additional exchange split satellites appear in presence of strong charge fluctuations.The effective charging energy U_ eff of a multiorbital impurity system, i.e. the energy which determines the amount of charge fluctuations in the system, can be estimated by the formula U_eff^(N)=E(N+1)+E(N-1)-2E(N), where E(N) is the lowest atomic energy of the N electron state. For rotationally invariant interaction, this gives U_eff^(N=5)=U+4J (half-filling), U_eff^(N=6,9)≈ U-3J/2 and U_eff^(N=7,8)≈ U-J/2. <cit.> Hence, particularly Fe based impurity systems which typically have N≈ 6 are expected to realize strong charge fluctuations and Hund's impurity behavior. The case of hydrogenated Fe, FeH_x, on Pt (111) has been investigated in a collaboration between experiments and FOR 1346 as reported in Ref. <cit.>. FeH_x on Pt (111) realizes strong charge fluctuations coexisting with local magnetic moments for several binding geometries and different degrees of hydrogenation, x=0,1,2. This leads to non Fermi liquid electronic self-energies due to spin scattering at intermediate temperatures, as shown in <cit.>. While this intermediate temperature non Fermi liquid behavior is very generic for Hund's impurity systems, FeH_x on Pt (111) presents a model system where the low temperature magnetic properties can be strongly controlled by changing the binding site and degree of hydrogenation of the Fe adatoms with the tip of a scanning tunneling microscope. In this way, it turned out to be possible to control four almost degenerate energy scales (Zeeman, thermal, Kondo and magnetic anisotropy energy) and to tune the Hund's impurities fromrealizing an emergent unquenched magnetic moment to a multi-orbital Kondo state. The Kondo state is particularly interesting, here, as it emerges without well quantized (half-) integer local moments at the Fe sites. Hence, Ref. <cit.> shows that Kondo screening is possible without quantization of the local moments. The coupling of electrons to fluctuating magnetic moments is regarded a possible mechanism for superconductivity at high temperatures. Therefore, (arrays of) Hund's impurities could provide a bottom-up way for the understanding of the complex physics of systems like iron pnictide and chalcogenide superconductors.§.§ Transition-Metal Phthalocyanine on Ag(001)In this section we focus on transition-metal phthalocyanine molecules (TMPc) on the (001) surface of silver. This system is an example of how electronic and transport properties of atomic and molecular conductors can be controlled by manipulation of the molecular geometry <cit.>. It is conceivable to use similar molecular magnets<cit.> as building blocks in the quest for improved magnetic storage devices and/or for spintronics applications <cit.>. Metal–organic molecules such as transition-metal Pc are particularly well suited for theoretical studies of strong electronic correlations, since they contain a single transition metal atom at the center and, additionally, handle very well in experiments. When deposited on the Ag (001) surface, see Fig. <ref>(a) for a generic structure, certain representatives, NiPc and CuPc, exhibit a Kondo effect which appears to be localized on the arms of the organic ligand of the transition metal ion <cit.>. Instead, MnPc exhibits the Kondo effect in the 3d shell of Mn <cit.>. Additionally, it is possible to manipulate the structure of the ligands with an STM tip <cit.>, thus gaining some degree of control over the physics.When faced with such a challenging problem as the calculation of electronic correlation effects in MnPC on a surface two approaches come to mind. One would be the quantum chemical approach to attack the full system including all interactions using a truncated configuration interaction or similar approach to account for the correlation effects <cit.>. The other one, more popular among physicists, is to "truncate" the problem already at the level of the Hamiltonian and then solve the remainder as exactly as possible. We will follow latter route here, observing that the system can be viewed as a realization of an Anderson impurity model <cit.>, if one assumes that everything surrounding the correlated central atom can be reasonably described within a one particle theory, such as density-functional theory (DFT). The model was originally designed and is best suited for the description of d or f shell impurities in simple metal hosts and is capable of describing localcorrelation physics,like the Kondo effect <cit.>. Since the remainder of the system consists of a simple metal with sp electrons close to the Fermi level (Ag) and reasonably uncorrelated elements (C, N and H) such an approach can be justified. They comprise what is called the dynamical, i.e., energy dependent, hybridization, whereas the impurity levels or crystal field constitute the static part. In our case the bath does not only consist of the substrate, but also involves the organic parts of the molecule. In our treatment the hybridization is given by a continuous orbital and energy dependent function, Γ_ij(ω)=∑_ν V^∗_ν i V_ν j^∗/ω-ε_ν+iδ, which can be obtained from DFT as described in section <ref>. The final ingredient in the calculation, after the local basis and the hybridization function, is the local Coulomb interaction. The Coulomb interaction is in general a rotationally invariant fourth-rank tensor <cit.> and there is no mystery about its general form and entries for atoms <cit.>. It is, however, not so simple to calculate the entries of the tensor for a solid or a nanodevice, due to dynamical screening <cit.>. We have used the density-density portion of the full Coulomb interaction as obtained from the Slater integrals F^0, F^2 and F^4 for 3d shell systems <cit.>. The numerical values we used were F^0 = 4.5eV, F^2=6.03eV and F^4=3.77eV which were obtained for MnPc by means of the constrained random-phase approximation (cRPA) <cit.>. To solve the model we employed the CT-QMC solver as described in section QMC. Single MnPc molecules on Ag(001) nicely illustrate the general features of a correlated molecular system in contact with metallic leads or a metal surface. The structure of a MnPc molecule on the Ag(001) surface is shown in Fig. <ref>a. Before any Anderson model calculations one has to establish the structure of the molecule and the adsorption site as well as possible changes in the geometry. A reliable approach is to perform a relaxation within DFT employing a GGA functional amended by a dispersion correction to account for van-der-Waals interactions. From experience a good combination is PBE+D3, which corrects the under-binding tendencies of pure PBE <cit.>. LDA can, due to miraculous error cancellations, also perform quite well in such a case, see e.g. <cit.>.When the structure is established and a localized basis is available either directly or via a Wannier construction the hybridization function and static crystal field can be calculated. Both are shown in Fig. <ref>a. The hybridization function is very different for each d orbital and exhibits the generic features of a molecular system in contact with a surface or leads. One can make out three different types of d orbitals:* orbitals that couple only to the molecular states, like the d_xy and d_x^2-y^2. Their hybridization function shows strong peaks at specific energies and closely resembles the hybridization function calculated for the free molecule, see Ref. <cit.>. * orbitals that couple strongly to the surface and only weakly to the molecular states, like the d_z^2. Its hybridization is mostly smooth and almost constant as expected from the "flat" sp-electron density of states of the silver substrate. * orbitals that couple moderately to both the molecular states as well as the surface, like the d_xz/yz.Due to their symmetry and orientation in space such a division of the orbitals is typical for a metal–organic molecular system in contact with metallic leads or surfaces. Due to the square planar structure of the molecule, as well as the presence of additional Ag atoms underneath, the level crystal field splitting of the 3d shell is as shown in Fig. <ref>. The d_z^2 orbital, as well as the d_xz/yz orbitals are half-filled and are as such prime candidates for correlated electron behavior. The level positions are, however, different for different TMPc species, due to different adsorption distances.Having all the ingredients of the Anderson model now at hand we can solve the model and calculate observables like the spectral function or self-energy. Since these types of metal–organic molecules already exhibit a complex electronic structure without the surface <cit.>, we expect some interesting many-body effects. Indeed below a certain temperature a sharp peak appears in the spectral function of the d_z^2 orbital, as shown in Fig. <ref>b. Corroborated by a Fermi-liquid behaviour of the electronic self-energy, we can identify this as the signature of the Kondo effect. Such a spectral feature is also observed in STM experiments <cit.>. For the d_xz/yz orbitals a similar feature appears, which can be shown to be of non-Fermi liquid character, however. In FePc and CoPc, one would also expect such an effect due to the non-vanishing spin on the 3d shell, however, in experiments no low energy spectral feature indicative of a Kondo effect is found. This can beexplained by theincreasing filling of the d_z^2 orbital, which is effectively full in Fe and Co, leaving the local moment regime of the Anderson model <cit.>. This is already interesting on its own, but MnPc allows us (and experimentalists!) to do more. The Kondo effect in MnPc can to some extent be controlled by manipulating the structure of the molecule by a process called dehydrogenation <cit.>. As the name suggests hydrogen atoms are removed using an STM tip from the molecule employing a voltage pulse <cit.>. The effect on the structure can be simulated within DFT, the result of removing two and eight hydrogen atoms are shown in Fig. <ref>b and c. The concomitant changes in the electronic density are shown next to the respective structures in Fig. <ref>e and f. The downward bending of the arms of the molecule is accompanied by a depletion in charge density close to the newly formed bonds as also observed in experiment <cit.>. The bending also leads to an increase in distance between the central atom and the surface, which in turn changes the local electronic structure in multiple ways. It reducesthe hybridization between centralatom and substrate, especially for thed_z^2 orbital, and also leads to a uniform shift to smaller binding energies of the partially filled d_xz/yz and d_z^2 orbitals. The former leads to a decrease in the Kondo temperature as shown in Fig. <ref>g, while the latter can be seen in experiment as an upward shift of the states superimposed over the Kondo resonance. Both effects occur also in FePC, see Fig. <ref>h, the reduction in coupling to the substrate manifests as a general sharpening of spectral features, since a Kondo effect is absent here.We have discussed the Kondo effect in TMPc molecules on Ag(001) as well as the effects of dehydrogenation seen in STM based mainly on local quantities of the molecules calculated within the respective Anderson impurity models. In the systems presented here this turned out to be sufficient for the qualitative description of the physics. However, in an STM experiment the tip and the resulting interaction between tip and sample can play a crucial role as well. In particular, regarding the understanding of transport through a correlated molecule measured by STM, it was realized in Ref. <cit.>, that not only the local correlations on the molecule are important, but also the voltage drop over the junction is critical in understanding the line-shapes in the conductance.§.§ Electronic structure of the Cr(001) surface Bulk Cr crystallizes in a body-centered crystal (bcc) structure <cit.>. Neutron diffraction experiments <cit.> reveal a spin-density wave ordering of bulk Cr with a magnetic moment of about 0.6μ_B per atom at low temperatures and a Néel temperature of approximately T_N^bulk∼311 K <cit.>.Considering the lattice of antiferromagnetic Cr it is obvious that a cut along the (001) direction leads to a surface where atoms of layers parallel to the surface are ferromagnetically aligned. Thus, the first layer of the (001) surface consists of ferromagnetically aligned atoms. Intriguingly, the surface magnetic state persists up to much larger temperatures (T_N^surf∼ 750-800K) than the bulk state, as measured in Refs. <cit.> by angular resolved photo emission (ARPES) and Ref. <cit.> by magnetization measurements with Cr particles of different diameter. This large magnetic moment originates from the altered paramagnetic electronic structure: The surface introduces massively more states close to the Fermi energy (compare Fig. <ref> for a comparison of the bulk DOS and the surface LDOS), which spin split due to the exchange interaction and lead to a large surface moment <cit.>. Ab initio based constrained random-phase approximation (cRPA <cit.>) calculations for the Cr(001) surface <cit.> show that the Cr d electrons are subject to rather strong Coulomb interactions. This, together with the increased density of states at the surface lays the ground for correlation effects.The question if correlations actually are important for the electronic structure has been subject of measurements using (inverse) photo emission spectroscopy (PES) (<cit.>) and scanning tunneling spectroscopy (STS) <cit.>. These measurements all reveal a sharp peak close the Fermi energy. The origin of the peak was so far discussed either to be of single particle nature (surface state, <cit.>) or of many-body nature (orbital Kondo effect <cit.>). Here, we review our work from Ref. <cit.> where we use the LDA+DMFT method to unravel the nature of the resonance in the electronic spectrum of Cr(001).We show that local electronic correlations play a key role for understanding the electronic structure of Cr(001). In order to reproduce the structure of the spectrum realistically, we model the Cr(001) surface by a slab of ten Cr atoms, which was partially relaxed using GGA calculations, and employ the projector formalism of LDA+DMFT described in Sec. <ref>. We solve the resulting multi-orbital Hubbard model of the slab of ten atoms by multi-site DMFT which allows for spatially inhomogeneous Coulomb interaction and antiferromagnetic ordering <cit.>. Compare, e.g., Refs. <cit.> and <cit.> for a similar approach. To solve the impurity problems we use the CTQMC algorithm presented in Sec. <ref>. We cope with the double-counting problem by the requirement that the total occupation on each impurity obtained from the DMFT Green function G matches the corresponding occupation obtained from the non-interacting bath Green function. This is called trace double-counting correction and reads ρ_αβ^imp!=ρ_αβ^0,loc, where ρ is the density matrix for each atom. This scheme allows for a flexible way to change the double counting, namely to enforce a desired electron number on the Cr atoms. Here, we have enforced 4.5 electrons on each Cr atom. The local density of states at β=40^-1 summed over the d orbitals and spins for the surface atom is shown in comparison to the GGA spectra in Fig. <ref> (a). For the surface atom we can observe a three-peak structure resembling the experimental situation (compare Ref. <cit.>).The orbital characters of the spectrum are presented in Fig. <ref> (b). The resonance is dominated by a feature in the minority spin channel with d_z^2 character but also carries some spectral weight from the other d orbitals, particularly the d_xz and d_yz orbitals. For β=40^-1 the d_z^2 orbital also shows a sharp resonance in the majority spin channel. Overall, the orbital contributions of the spectra are in line with experimental findings in Ref. <cit.>.To answer the question of how important correlation effects are at the surface and in the bulk, we calculate the quasiparticle weight Z = [1 - .∂Σ(ω)/∂ω|_ω=0]^-1 for all orbitals and spins. Analyzing Z for all layers of the slab, as done in <cit.> shows that correlation effects are generally strongest at the surface. Also, the d_yz,xz orbitals show the smallest quasiparticle weight, i.e., the strongest many-body renormalizations. The nature of the resonance seems to be the combination of the rearrangement of the electronic structure at the Cr(001) surface (massively more states at the Fermi energy in contrast to bulk Cr) and the multi-orbital interaction effects, especially in the d_z^2 and d_xz/yz orbitals. The interaction introduces two major effects: first, the spin dependent splitting of mainly the d_xz/yz orbitals and, secondly, the appearance of quasiparticle peaks in d_z^2 and d_xz/yz orbitals. We conclude that the resonance is a complex many-body effect in the d_z^2 and d_xz/yz orbitals due to dynamic local-correlation effects which are directly related to the (single-particle) electronic structure at the surface.§ NON-LOCAL INTERACTIONS AND CORRELATIONS §.§ Realistic description of Coulomb interactions and screening in complex nanosystems: The Wannier function continuum electrostatics approach The interaction terms entering the LDA+DMFT modeling can be in principle also derived from ab-initio calculations. However, special care on the issue of screening has to be taken here since the correlated subspaces involve some low energy sector of bands only. Hence, realistic interaction matrix elements entering the modeling should be appropriately screened. I.e. they should account for screening due to those states which are not explicitly treated in the low energy models. Depending on the later many-body treatment one has to ensure that screening channels are neither omitted nor doubly counted in the end. To calculate realistic appropriately screened Coulomb interaction matrix elements from first principles the so-called constrained Random Phase Approximation (cRPA) has been put forward. <cit.> It relies on excluding certain screening channels explicitly in a many-body perturbation theory calculation and its computational demand is comparable to GW calculations. Hence, direct cRPA calculations of complex nanostructures or also two-dimensional (2d) materials on substrates with large moiré supercells are often computationally unfeasible. To solve this problem, we developed an approach called “Wannier function continuum electrostatics” (WFCE) <cit.> which will be explained in the following with the example of layered materials and heterostructures. The essential idea of WFCE is to combine ab-initio cRPA calculations for the bulk of a layered material, which are of moderate computational cost, with continuum medium electrostatics to derive effective Coulomb interaction matrix elements in terms of Wannier functions for freestanding 2D materials or 2D materials embedded in complex dielectric environments. In this way, we can avoid supercell calculations involving complex environments or large vacuum volumes on the ab-initio side, which are numerically often very expensive.Going from a bulk material to a monolayer or a complex heterostructure means changing the polarizability of the environment of the material of interest, as illustrated in Fig. <ref>a). These changes have to be accounted for in the effective dielectric function of the system, ϵ(q). In a Wannier basis, ϵ(q) acquires generally a complex matrix structure. However, as shown in Ref. <cit.>, environmental screening affects only a single element of the dielectric matrix ϵ(q) which turns out to be accurately describable by continuum medium electrostatics. WFCE essentially modifies this one “quasi macroscopic” matrix element. The full WFCE algorithm which is described in Ref. <cit.>. Fig. <ref>b demonstrates the range of applicability of the WFCE approach by comparing partially screened Coulomb interaction matrix elements in different graphene based systems obtained from direct cRPA calculations to WFCE results. In the latter case, an ab-initio calculation is necessary only for the bulk form of graphite. The WFCE approach predicts effective Coulomb matrix elements for monolayer and bilayer graphene very accurately; for instance, the local Hubbard interaction agrees with the full cRPA calculation within 0.3 eV. Larger deviations are only found in the case of graphene embedded in Ir metal, where hybridization of graphene and the surrounding material is sizable. I.e., the comparisons of full first-principles and WFCE calculations suggest that the WFCE approach is accurate when hybridization between layers in the vertical direction is not too strong, as is the case for van der Waals bonded systems.Additionally, Fig. <ref>b shows that Coulomb interactions in 2D materials like graphene can be manipulated on the eV-scale by means of screening provided by different environments which can be substrates, adsorbates, or other 2D materials. I.e. environmental screening presents a very effective tool for non-invasive materials manipulation. Possible applications can be the creation of heterojunctions by non-local manipulations of Coulomb interactions as put forward in Ref. <cit.>. However, this concept of “Coulomb engineering” is more general and can be likely also applied in the domain of strongly correlated systems like Mott insulators. §.§ Dynamical vertex approximation in nanosystemsIn low-dimensional systems, spatial correlations beyond mean-field become relevant.Non-local correlations have been taken into account following different strategies,ranging from quantum cluster theories,<cit.> which treat short-range correlations within the cluster exactlyand long-range correlations within mean-field,to diagrammatic approximations, which treat correlations on all length-scaleson equal footing, such as the dynamical vertex approximation <cit.> (DΓA),the dual-fermion<cit.>and one-particle irreducible<cit.> approaches, as well as the DMF^2RG,<cit.> TRILEX,<cit.> and QUADRILEX<cit.> In the following we discuss the idea behind the DΓAand we present the first application of DΓA in its parquet implementation,to nanoscopic systems.<cit.> From a diagrammatic prospective, the DMFT can be formulated in termsof a local approximation for the one-particle fully irreducible self-energy,owing to the locality of perturbation theory in infinite dimensions.<cit.>DMFT has been successfully applied to model<cit.>as well as realistic<cit.> nanostructures.The idea behind DMFT can be systematically extended to include non-local correctionsto the self-energy beyond mean-field,by considering diagrammatic approximations based on two-particle vertex functions.The Dyson-Schwinger equationΣ_ k(ν) = U n/2- T^2 U ∑_ k'q∑_ν'ωF^↑↓_ kk'q(ν,ν',ω) G_ k'+q(ν'+ω) G_ k'(ν') G_ k+q(ν+ω),relates the momentum-dependent electronic self-energy Σ_ k(ν)to the full two-particle vertex F=F^↑↓_ kk'q(ν,ν',ω). From a physical prospective, F encodes all informationof the two-body scattering processes in all particle-hole and particle-particle channels.Within the parquet formalism,<cit.> F can be decomposedaccording to reducibility properties of two-particle diagrams asF = Λ + ∑_r ϕ_r, where ϕ_r is the set of diagrams reduciblein a specific scattering channel r=ph,ph,pp,while Λ is the set of fully-irreducible diagrams (i.e., irreducible in all channels).Moreover, in each channel a Bethe-Salpeter equation (BSE)F=Γ_r+Γ_r χ_r Fexpresses F in terms of the vertex Γ_r , irreducible in channel r,in terms of the corresponding reducible susceptibility χ_r. Each diagrammatic approach relies on a specific approximationat different levels of the diagrammatic theory.<cit.>The DΓA assumes the locality of the fully-irreducible vertex,<cit.>Λ = Λ_loc.Provided the knowledge of Λ_loc, the set of parquet equations abovecan be solved self-consistently to determineall the momentum-dependent vertex functions as well as the self-energy.Within the DΓA Λ_locis evaluated from the auxiliary AIM,by solving the local (inverse) parquet equation and the BSEs.A natural choice for the AIM is the (self-consistent) AIM of DMFT,which is expected to provide a reasonable description of local correlationsof the original many-body problem.<cit.> Otherwise, a fully self-consistent DΓA calculationrequires the AIM (and the corresponding Λ_loc)to be updated at each iteration until convergence.The full parquet-DΓA scheme is shown in Fig. <ref>.We discuss the application of the DΓAto cyclic organic molecules.<cit.>Specifically, we consider the cyclic organic molecule [n]-annulene: C_nH_n.A proper description of electronic correlations is relevantfor a proper estimate of the spectral gap,which deeply affects the electronic and transport properties of molecular junctions. We consider a low-energy effective model for the C2p_z electrons,which are delocalized in a single bonding π-molecular orbital. The Hamiltonian reads as: H = H_TB + H_int + H_hyb,where, H_TB includes a nearest-neighbor tight-binding (or Hückel)description of the molecule, H_int is the local Coulomb interaction U on the π-molecular orbital, and H_hyb describes the (local) hybridization of each C atomto weakly-correlated electrodes or a substrate.For the sake of simplicity, we neglect non-local interactions.However, due to poor screening and quantum confinement effects,those may be expected to play an important role in determining the physics of nano systems,and can be taken into account within the DΓA,as discussed e.g., in Refs. <cit.> and <cit.>. Let us start considering the [6]-annulene (benzene).At half-filling, benzene is characterized by a semiconducting gap Δ_0=2|t|between the HOMO and LUMO molecular orbitals already in nearest-neighbor tight-binding model,in the absence of interaction.However, electronic correlations due to the local Coulomb repulsion renormalize the gap,deeply influencing the transport properties of benzene molecular junctions.It is known that local electronic correlations within DMFTreduce the spectral gap in both bulk crystal<cit.>and nanoscopic systems<cit.>while the corresponding exact QMC solution yields Δ>Δ_0,suggesting an important role is played by non-local spatial correlations beyond DMFT.The DΓA is able to capture all the relevant corrections beyond DMFT,and reproduces the exact self-energy, as shown in Fig. <ref>(a-d).The scattering rate γ_k≈-2Σ_k(ν_n→0)is weakly k-dependent (and well described already within DMFT)due to the semiconducting nature of the benzene molecule in the gas phase.Instead, the strongly k-selective nature of the real part of the self-energy Σ_k(ν_n),as shown both by the DΓA and exact QMC solution,cannot be captured by DMFT, as Σ(ν_n)=0 for the local self-energydue to the particle-hole symmetry at half-filling. A real-space analysis<cit.> in Fig. <ref>(f-g) shows thatthe nearest-neighbor hopping parameter is renormalized by interactionst^*=t+Σ_i,i+1(ν_n→0).In the presence of a local hybridization between each C atom in the molecule and the environment,described in the momentum-independent parameter Γ,non-local correlations are strongly suppressed.This determines, as a function of the control parameter Γ/t a crossoverbetween a weak-hybridization regime, where non-local correlations are dominant,to a local Kondo regime, characterized by the formation of entangled singlet statesbetween the π-electron of benzene and the electrons of the environment,as shown in Fig. <ref>(e). In the weak-hybridization limit, Δ≈2|t^*| is a reasonable estimate for the spectral gap,in good agreement with the results obtained from the analytic continuation of the local Green's function.The situation is substantially different for the caseof the half-filled isolated [8]-annulene.In the absence of interaction, h_ij has a 2-fold degenerate eigenvalueat the Fermi level for k=±π/2(in contrast to the benzene molecule, which is semiconducting).Within DMFT the local spectral function displays a resonance at the Fermi leveland undergoes a Mott metal-to-insulator transition at U≈8t,while the exact solution displays a gap at any finite interaction<cit.>.This is reflected in the large scattering rate γ^exact_k(while Σ_k=0) for k=π/2.Instead, the DΓA fails to open a gap at the Fermi level,as γ^DΓ A_k≪γ^exact_k,resulting in a pseudogap, associated with a partial transfer of the spectral weight toward higher energy.The DΓA results discussed above can be understood also from the analysisof the fully-irreducible vertex Λ_loc,which s shown in Figs. <ref> and <ref>in the density and magnetic (particle-hole) scattering channels.The frequency dependence of Λ_lochas been discussed in detail if Ref. <cit.>within the context of DMFT.Remarkably, the generic key features of Λ_loc,as the typical “butterfly” frequency structure of Λ_loc^νν'ω=0,remain unchanged from the 2D Hubbard model to 0D nano systems.Compared to the fully-irreducible vertex of Ref. <cit.>we notice that for the benzene both in Λ_d and Λ_mthe low-energy features are substantially suppressed, in contrast to the case of the [8]-annulene,and similarly also for the [4]-annulene.<cit.> We ascribe this to the presence of a spectral gapin the local density of states in benzene, but not in the [4,8]-annulenes, within DMFT.This suggests that the difference between the DΓA and exact solution in the case of the [8]-annulene,results from a poor descriptions of local correlations within DMFT.In the respect, a fully-self-consistent DΓA calculations,although computationally expensive, it is expectedto further improve the description of the low-energy physical properties of the [4,8]-annulenes. § CONCLUSIONS AND OUTLOOK In this paper we have summarized the results of a series of collaborations between various groups at the University of Würzburg, the University of Bremen, the University of Hamburg and the TU Wien in Austria. One of the aims of this cooperative effort was to develop and optimize an interface between Kohn-Sham-based DFT and nano-DMFT/nano-DΓA and achieve general applicability of these methods to complex correlated structures. Particular attention has been devoted to the choice of the basis sets and the influence of the double counting. These steps have been made possible thanks to the parallel development of our code package “w2dynamics”, developed in Würzburg, Bremen as well as in Vienna<cit.>. “w2dynamics”, which will be made public in the course of 2017 <cit.>, is a CT-QMC hybridization-expansion code, as described briefly in Sec. <ref> and it allows users to set up calculations for big and complex realistic nanostructures in a rather flexible way. It is interfaced to the DFT-based packages used in the calculations shown here and it has been designed to be accessible not only to the experts of CT-QMC. This is not the only advance in numerical algorithms for strongly correlated electron systems that we discuss in this review paper. As described in Sec. <ref>, we implemented a variational version of exact diagonalization <cit.> as well as the computation of core level x-ray absorption and core level photo electron spectroscopy <cit.> in the framework of a new-generation Hamiltonian-based solver for DMFT <cit.>. This is then extended to the case of multiorbital physics in nanostructures, which we approach with our combined CT-QMC and ED impurity solvers in order to establishing links from realistic structures to simple models.Selected applications of the formalism in action have been presented in Sec. <ref>. In particular, as described in Sec. <ref>, some of us introduced the notion of Hund's impurities by studying Fe and Fe-hydrogen complexes adsorbed on a Pt(111) surface <cit.> in collaboration with experimental colleagues from Hamburg. The role of the Hund's coupling in the Anderson model in general is also investigated, by comparing a multitude of calculations based on ED as well as CTQMC. Furthermore, we have discussed the Kondo effect in TMPc molecules on Ag(001) as well as the effects of dehydrogenation seen in STM in Sec. <ref>. These systems allow for a direct experimental manipulation of the Kondo scale and offer a rich playground for the exploration of connections between theoretical models and real-world physics.Section <ref> is instead devoted to the study on the Cr(001) surface <cit.>, which has been raising a long debate on the many-body nature of its low-energy spectral features. We reveal that the spectral feature observed close to the Fermi level is a consequence of changes in the electronic structure at the surface, as well as complex many-body effects in certain orbitals. We have been mostly concerned with further pursuing the realistic description of correlated impurities in environments with a reduced symmetry as well as of extended multi-orbital systems. A related example, on which we however did not elaborate here, is the possibility of realizing a Mott transistor with t_2g-electrons in the context of oxide heterostructures, put forward in Ref. <cit.>. In this review, special attention has been given to non-local correlation effects and spatial correlations in nanoscopic systems, both touched upon in Sec. <ref>. We point out that in complex nanoscopic systems the role of non-local Coulomb interactions is important. In cooperation with the Research Center Jülich, we developed the Wannier function continuum electrostatics approach, showcased in Sec. <ref>, which facilitates the realistic calculation of appropriately screened Coloumb interaction matrices in complex systems such as 2d materials supported by substrates <cit.>. Generally low dimensional materials feature strong non-local Coulomb interaction which presents a big challenge for correlated materials modelling <cit.>. This issue has been addressed in Ref. <cit.>: we showed that it is possible to link systems with non-local interactions to Hubbard models with local interactions only based on a thermodynamic variational principle. A further important worked accomplished in this context is Ref. <cit.>.Specific applications of diagrammatic approaches to include non-local correlations in nanoscopic systemshave been discussed in Sec. <ref>, dedicated among other things,to transport properties of nanoscopic systems,including single molecules <cit.> and quantum junctions <cit.>. In addition, some of us have analyzed in detail the frequency structure of the two-particle vertex <cit.> and the role of spatial correlations <cit.> for several models of interacting as well as disordered electrons. We have reached a new level of understanding of the properties of diagrammatic expansions for many-particle systems. In particular, we have discovered fermionic divergencies of specific two-particle irreducible vertex functions and linked them with the appearance of “protected” degeneracies between physical and unphysical solutions of the Dyson equation.The topics that our review has dealt with witness the fact that more and more complex systems became accessible to realistic dynamical mean-field theory simulations. I.e. a reliable description of local correlation effects is well doable now for many solid state systems. Yet, there are clearly challenges such as non-local correlation effects, competing interactions or also non-equilibrium effects in strongly correlated electron systems where progress has been made but where applications to complex real materials are at best at the very beginning to date and more development work is necessary.§ ACKNOWLEDGMENTSWe gratefully acknowledge support of the Deutsche Forschungsgemeinschaft through FOR 1346.One of us (A.V.) acknowledges financial support from the Austrian Science Fund (FWF)through the Erwin Schrödinger Fellowship No. J3890-N36. epjbbib | http://arxiv.org/abs/1707.08333v1 | {
"authors": [
"M. Schüler",
"S. Barthel",
"T. Wehling",
"M. Karolak",
"A. Valli",
"G. Sangiovanni"
],
"categories": [
"cond-mat.str-el"
],
"primary_category": "cond-mat.str-el",
"published": "20170726092312",
"title": "Realistic theory of electronic correlations in nanoscopic systems"
} |
T_ eff km s^-1 log g ξ_ t ξ_ t ζ_ RT v_ r vsin i E(B-V)firstpage–lastpage 2002A search for FRB 121102-like persistent radio-luminous sources – Candidates and implications for the FRB rate and searches Eran O. Ofek1Accepted ... Received ...; in original form ... ==========================================================================================================================The magnetic field of CPD -57^∘3509 was recently detected in the framework of the BOB (B fields in OB stars) collaboration. We acquired low-resolution spectropolarimetric observations ofCPD -57^∘3509 withFORS 2 and high-resolution UVES observations randomly distributed over a few months to search for periodicity, to study the magnetic field geometry, and to determine the surface distribution of silicon and helium. We also obtainedsupplementary photometric observations at a timeline similarto the spectroscopic and spectropolarimetric observations. A period of 6.36 d was detected in the measurements of the mean longitudinal magnetic field.A sinusoidal fit to our measurements allowed us to constrain the magnetic field geometry and estimate the dipole strength in therange of 3.9–4.5 kG. Our application of the Doppler imaging technique revealed the presence of Hei spots located around the magnetic poles, with a strong concentration at the positivepole and a weaker one around the negative pole.In contrast, high concentration Siiii spots are located close to the magnetic equator.Further, our analysis of the spectral variability of CPD -57^∘3509 on short time scales indicates distinct changes in shape and position of line profiles possibly caused by the presence of β Cep-like pulsations.A small periodic variability in line with the changes of the magnetic field strength is clearly seen in the photometric data.Stars: abundances – Stars: evolution – Stars: magnetic field – Stars: massive – Stars: oscillations –Stars: individual: CPD -57^∘3509§ INTRODUCTION Recently, <cit.> presented a firm detection of a mean longitudinal magnetic fieldof kG order in the early-B type star CPD -57^∘3509, previously studied in a spectroscopic survey ofmassive stars in NGC 3293 by <cit.>. In their work, the authors focussed on the investigation of abundances, a model atmosphere, and the evolutionary state of the target in detail. The quantitative spectroscopic analysis of this star with the observed rather low v sin i-value of 35 km s^-1yielded an effective temperature and a logarithmic surface gravity of 23 750±250 K and 4.05±0.10,respectively, and a surface helium fraction of 0.28±0.02 bynumber (see also ). The surface abundances of C, N, O, Ne, S, and Ar were found to becompatible with the cosmic abundance standard <cit.>, whereas Mg, Al, Si, and Fe weredepleted by about a factor of 2. It was suggested that such an abundance pattern can beunderstood as the consequence of a fractionated stellar wind.Importantly, CPD -57^∘3509 with an elapsed main-sequence life time of about 50% has evolved significantly away from the zero-age main sequence and appears to be one of the most evolved He-strong stars known with an independent age constraint due to its cluster membership.Since the evolution of the magnetic field geometry across the main sequence in massive B-type stars isnot well sampled in comparison to studies of magnetic fields of late-type Bp and intermediate-mass Ap stars, a detailed study of the magnetic fieldconfiguration and the surface chemical inhomogeneities in a significantly evolved magnetic He-strong star is of particular interest. Knowledge of the evolution of the magnetic field geometry, especially of the distributionof the obliquity angle β (the orientation of the magnetic axis with respect to the rotation axis) is essential to understand the physical processes taking place in these stars and the origin of their magnetic fields. Further, although it is generally assumed that CPD-57^∘3509 is a member of the open cluster NGC 3293, no careful study of its membership involving new data from Gaia was carried out yet. We review the membership status involving the available Gaia data in Appendix <ref>.In order to characterise the properties of CPD -57^∘3509 in detail, we obtained time-series spectroscopy and spectropolarimetry with the FOcal Reducer low dispersionSpectrograph (FORS 2; ) and the UV-Visual Echelle Spectrograph (UVES; ) to constrainthe magnetic field geometry and reconstruct the distribution of silicon and helium on the stellar surface using Doppler Imaging (DI). Further, we carried out photometricobservations using the 40 cm Bochum Monitoring Telescope (BMT; ) of the Cerro Armazones Observatory. In the first part of the paper we report on theperiod determination using magnetic and photometric data.In the second part we present the results of the application of theDoppler Imaging technique followed by the discussion of the surface element distribution with respect to themagnetic field configuration.§ OBSERVATIONS AND MAGNETIC FIELD MEASUREMENTSFifteen FORS 2 spectropolarimetric observations of CPD -57^∘3509 were obtained in the framework of the ESO programme 094.D-0355 from 2014 November 17 to 2015 February 18, and further fivewithin the framework of the ESO Large Programme 191.D-0255 during 2014 February and June, and 2015 March. The FORS 2 multi-mode instrument is equipped with polarisation analysing optics comprising super-achromatic half-wave and quarter-wave phase retarder plates, and a Wollaston prism with a beam divergence of 22 in standard resolution mode.We used the GRISM 600B and the narrowest available slit width of 04 to obtain a spectral resolution of R∼2000. The observed spectral range from 3250 to 6215 Å includes all Balmer lines, apart from Hα, and numerous helium lines. For the observations, we used a non-standard readout mode with lowgain (200kHz,1×1,low), which provides a broader dynamic range, henceallowing us to reach a higher signal-to-noise ratio (S/N) in the individual spectra. The exposure time for each subexposureaccounted for 5 min. Each observation consisted of eight subexposures over approximately one hour including overheads.Our first description of the assessment of longitudinal magnetic field measurements using FORS 1/2 spectropolarimetric observations was presentedin several previous works (e.g. ,and references therein). To minimize the cross-talk effect and to cancel errors fromdifferent transmission properties of the two polarised beams, a sequence of subexposures at the retarder position angles -45^∘+45^∘, +45^∘-45^∘, -45^∘+45^∘, etc. is usually executed during the observations. Moreover, the reversal of the quarter waveplate compensates for fixed errors in the relative wavelength calibrations of the two polarised spectra. According to the FORS User Manual, the V/I spectrum is calculated using:V/I = 1/2{( f^ o - f^ e/f^ o + f^ e)_-45^∘ - ( f^ o - f^ e/f^ o + f^ e)_+45^∘}where +45^∘ and -45^∘ indicate the position angle of the retarder waveplate and f^ o and f^ e are the ordinary and extraordinary beams, respectively.Rectification of the V/I spectra was performed in the way described by <cit.>. Null profiles, N, are calculated as pairwise differences from all availableV profiles.From these, 3σ-outliers are identified and used to clipthe V profiles.This removes spurious signals, which mostly come from cosmic rays, and also reduces the noise. A full description of the updated datareduction and analysis will be presented in a separate paper (Schöller etal., in preparation; see also ). The mean longitudinal magnetic field, < B_ z>, ismeasured on the rectified and clipped spectra based on the relationfollowing the method suggested by <cit.>: V/I = -g_ effeλ^2/4π m_ e c^2 1/IdI/ dλ<B_ z>,where V is the Stokes parameter that measures the circular polarization, I is the intensity in the unpolarized spectrum, g_ eff is the effective Landé factor, e is the electron charge, λ is the wavelength, m_ e is the electron mass, c is the speed of light,dI/ dλ is the wavelength derivative of Stokes I, and<B_ z> is the mean longitudinal (line-of-sight) magnetic field.The longitudinal magnetic field was measured in two ways: using the entire spectrum including all available lines, or using exclusively hydrogen lines. Furthermore, we have carried out Monte Carlo bootstrapping tests.These are most often applied with the purpose of deriving robust estimates of standard errors.The measurement uncertainties obtained before and after the Monte Carlo bootstrapping tests were found to bein close agreement, indicating the absence of reduction flaws.The results of our magnetic field measurements, those for the entire spectrum and only the hydrogen lines are presented inTable <ref>, where we also include in the first rows informationabout the previous magnetic field measurements presented by <cit.>. The last row shows an additional measurement obtained on 2015 March 18 in the framework of the BOB ESO Large Programme 191.D-0255. The rotation phases presented in the last columns of Table <ref> were calculatedusing the ephemeris determined from our period search described in Sect. <ref>.High resolution spectra of CPD -57^∘3509 were obtained at the European Southern Observatory with the UVES high resolution spectrograph. The observations with an exposure time of about 40 min were obtainedfrom 2014 December 6 to 2015 March 6 using thestandard wavelength setting for dichroic mode (DIC-2 437+760) with the0.4 slit in the blue armgiving a spectral resolvingpower of R∼80,000 and the 0.3 slit in the red arm to achieve a resolution of R∼110,000. The wavelength coverage was 3730–9390 Å with a gap between 5000 and 5700 Å.Initially, we asked for 20 observations in service mode, but, unfortunately, only ten observations were executed. The summary of the UVES spectroscopicobservations is presented in Table <ref>. The table gives the Modified Julian date at the middle of each observation, the observing date, the rotational phase, and the S/N. The S/N is given per resolution element and is measured from the spectral region around4630 Å. All observations were phased using the ephemeris HJD = 2 456 983.5396 + 6.36255× E,referring to the time of the maximum positive magnetic field. Advantageously, the phase distribution of the obtained spectra appears suitable for a Doppler imaging analysis to study the surface distribution of silicon and helium, which exhibit the most distinct spectrum variability in He-strong stars. Supplementary photometric observations were carried out at the BMT.Taken in 2014 April and May and during 2015 March 31 to April 4, the photometry covers a timeline similar to the spectroscopic and spectropolarimetric observations. Johnson B and V filters were used and a 3×3 dithering pattern of observations were obtained each night to improve the sampling and photometric accuracy. Seven photometrically invariable cluster stars (numbers 3, 44, 45, 51, 58, 97, and 110 of ) in the vicinity of CPD -57^∘3509 were used for comparison in differential photometry. They covered a magnitude range of 94 ≤ (m_V,m_B) ≤ 129 and colors from 006 ≤ (B-V) ≤ 027, thus well enclosing the target and minimizing filter effects. The rms between the comparison stars were σ_V = 00047 and σ_B = 00062. The photometric data are summarized in Tables <ref> and <ref>, presented in Appendix <ref>. Our photometric zero values (median of the 2014 data points) for CPD -57^∘3509 are B=10.80±0.01 and V=10.69±0.01.§ PERIOD DETERMINATIONThe results of our frequency analysis based on the longitudinal magnetic field measurementspresented in Table <ref> andperformed using a non-linear least squares fit to the multiple harmonics utilizing the Levenberg-Marquardt method <cit.> are presented in Fig. <ref>. To detect the most probable period, we calculated the frequency spectrum and for each trial frequency we performed a statistical F-test of the null hypothesis for the absence of periodicity<cit.>. The resulting F-statistics can be thought of as the total sum including covariances of the ratioof harmonic amplitudes to their standard deviations, i.e. a signal-to-noise ratio. The highest peak in the frequency spectrum for the hydrogen lines not coinciding with the window function corresponds to a period of 6.36093±0.00026 d and that for the measurements of the entire spectrum corresponds to a period of 6.36255±0.00026 d. We note that taking into account the precisionof the period determination, the difference between the two periods is not significant. On the other hand,as the achieved magnetic field measurement accuracy is higher for the set using all lines, the rotationperiod of 6.36255 d identified from thesemeasurements is expected tobe more reliable and is preferred in the following discussion on the surface element distribution inSect. <ref>. In Fig. <ref>, we present all measurements, those using the entire spectrum and those using only the hydrogen lines, phased with the corresponding rotation periodsand the best sinusoidal fits calculated for thesemeasurements. As already mentioned in the caption of Table <ref>, the deviating point close to the phases 0.881, respective 0.888, is caused by the unfavourable weather during the observations at that phase. Since CPD -57^∘3509 already finished half of its main-sequence lifetime <cit.>and is already passing through the β Cep instability strip, we checkedthe stabilityof the Stokes I spectral lines over the full sequences of sub-exposures obtained on a time scale of tens of minutes. Along with different radialvelocity shifts of linesbelonging to different elements, we also detectdistinct changes in line profiles taking place on time-scales corresponding to the duration of the sub-exposure sequences in the individual observations. In Fig. <ref>,we present the behaviour of the line profiles in individual spectral lines. The time difference between subexposures accounts for about 20 min. However, with the current data we cannot identify the periodicity of the detected variability, which is probably caused by the presence of β Cep-like pulsations. Thus, future observations should focus on thecareful search for periodicity and on the identificationof the pulsationmodes. The periodicity derived from spectropolarimetry (and the corresponding spottiness, see below) plus the tentative β Cep-like pulsations are expected to be detectable in the photometric data. Our differential photometry data in B and V are displayed in Fig. <ref>, phased according to the period obtained from the magnetic analysis using only the hydrogen lines – the behaviour is very similar for the case of the phasing based on all lines. A small periodic variability in line with the changes of the magnetic field strength is clearly seen. The dispersion of the higher-cadence data of 2015 (taken within ∼2/3 of the rotation period) could be interpreted in favour of the presence of β Cep pulsations, but this is close to the detection limit and requires dedicated follow-up observations for confirmation. Finally, we want to note a difference in the mean B and V magnitudes from the present work and that of <cit.>, which are 007 and 006 fainter than our values. As the comparison stars used for the differential photometry show only an rms of 002 between both studies (data taken ∼20 years apart), this may imply some long-term variability for CPD -57^∘3509 as well. § DOPPLER IMAGING USING UVES OBSERVATIONSAs we mentioned above, the rotational phase coverage and S/N of the UVES observations is quite good andenables mapping the chemical element patterns, in particular of the most strongly varyingelements He and Si.The phase coverage of the observations is not optimal, though, and four observations have been obtainedclose in phase. Still, the largest phase gap is 0.22, and occurs between phases 0.79 and 0.01.Other areas on the stellar surface are well covered, and the phase gaps are less than 0.15, posing no problems for the Doppler imaging technique. Tests show that phase gaps as large as about 0.28 do not significantly affect the reconstructionof the features <cit.> if the S/N is good. Commonly seen problems caused by non-perfect phase coverage are the blurring of features and not being able to recover small spots (see e.g.,). Chemical spots are large structuresand therefore can easily be mapped with the phase coverage of the current observations. Naturally,one should be careful when interpreting the results, especially small features, in the location of thelargest phase gap. In Sect. <ref>, the discussion includes the possible impact of the phase gap on the obtained maps. Since observations with HARPSpol are usually carried out in visitor mode, wedo not have high-resolution spectropolarimetric observations ofCPD -57^∘3509 on our disposal. The magnetic field strength is however only a few kG, and as we show below, it is possible to use the Doppler imaging techniqueto obtain an approximate elemental distribution without taking into account the impact of the magnetic field on theline profiles. The He mapping is carried out using Hei 4713. For Siiii, three lines located close to each other, 4552.622, 4567.840, and 4574.757 Å, were used simultaneously. For the phase calculation we employed the rotation period of 6.36255±0.00026 d identified in our magnetic field measurements using the entire spectrum. The surface abundance maps were obtained with the INVERS7PD inversion code, which was originally developedby <cit.> and modified by <cit.>. INVERS7PD compares the observations to a gridof synthetic local line-profiles, which were calculated using the SPECTRUM spectral synthesis code<cit.> and ATLAS9 stellar atmospheres by<cit.>. The Hei line properties are hard coded into SPECTRUM, and the Siiii lineparameters were originally obtained from VALD <cit.>.When using the Si abundance determined by <cit.>, the three Siiii lines were on averagewell fitted, but the 4552.622 Å line was slightly too weak, while the 4574.7570 Å lineappeared slightly too strong. Tominimise the influence of the errors in the atomic data, we followed the standard procedure and changed theoscillator strengths of the lines to improve the overall fit(see e.g., ). The changes needed were small and allowed forfitting all three Siiii lines simultaneously. Table <ref> gives the parameters used for these lines.For the model calculations, 20 limb angles were used together with the stellar parametersdetermined by <cit.>: T_ eff=23 750 K, log g = 4.0, and micro- and macroturbulence of 2.0and 10.0 km s^-1, respectively. From the inversions, we obtained a best fit inclination angle of 58±10^∘. Notably, the Doppler imaging technique is very sensitive to the v sin i andthe inclination angle values in the sense that no converging solution can be found if these values areunder- or overestimated. <cit.> estimated the vsin i value of CPD -57^∘3509 to be 35±2km s^-1. In the inversions, thebest fit from the Siiii lines was obtained for vsin i = 34.5km s^-1, in excellent agreement with the results of <cit.>. On the other hand, for the Hei line, the best fit was obtained forvsin i = 26.5km s^-1, which is lower than the other estimates. There are severalpossible reasons for this discrepancy. First, weaker metal lines are more sensitive to vsin i valuesand broad Hei lines dominated by Stark effect are usually avoided in thedetermination of rotation rates. Siiii and Hei lines also have different Landé factors. Further, the star is likely a β Cep–like pulsator. Depending on the type of pulsations, thiscan have more impact on the Siiii lines rather than on the Hei lines, causing stronger line broadening in the Siiii lines. We also note that similar discrepancies in the determination of vsin ivalues for different elements were mentioned in other works, which were using imaging methods(e.g. ). We also remark that SPECTRUM is verified to work at the spectral typerange B to mid-M, making CPD -57^∘3509 close to the limit of the code's capabilities. Furthermore, SPECTRUM doesnot use NLTE in the spectral synthesis, and Hei is more sensitive to departures from NLTEthan Siiii. For these reasons, the absolute abundance scales in the obtained maps are not necessarilyprecise. Still, this will not affect the relative abundances of the spots, nor their locations. §.§ Magnetic field geometryThe simplest model for a magnetic field geometry is based on the assumption that the studied starsare oblique dipole rotators, i.e., their magnetic field can be approximated by a dipole with its magnetic axisinclined with respect to the rotation axis.From the variation of the phase curve for the field measurements with a mean of< B_ z> = 180± 47 G and an amplitude of A_< B_ z> = 894 ± 57 G, we calculate < B_ z>^ min= -714±74 G and < B_ z>^ max=1074±74 G. Using the definition by <cit.> r = < B_ z>^ min/< B_ z>^ max = cosβcos i - sinβsin i/cosβcos i + sinβsin i, we findr= -0.665±0.075 and finally following β =arctan[ ( 1-r/1+r)i ], and employing i=58±10^∘ obtained from ourDoppler imaging inversions, we calculate a magnetic obliquity angle β=72±8^∘.Assuming a limb-darkening coefficient u=0.3, typical for stars with T_ eff=23 750 K <cit.>,we estimate a dipole strength of 3.91±0.36 kG using the model by <cit.>,as formulated by <cit.>:B_ d = < B_ z>^ max( 15+u/20(3-u) (cosβcos i + sinβsin i) )^-1. Using the parameters of the sinusoidal fit to the values resulting from only the hydrogen lines (<B_ z> = 147±52 G and A_<B_ z> = 1058±64 G), we obtain slightly different values for the magnetic field model:r= -0.757±0.077, β=78±6^∘, andB_ d = 4.51±0.45 kG. Given the size of the errors of the dipole strength determination, the difference in the derived values of the dipole strengths is not significant.§.§ Abundance maps and comparison to the magnetic pole locationThe Hei and Siiii distributions are shown in Fig. <ref>, and the model fits to theobservations are given in Fig. <ref>. The locations of the chemical spots are very similar in both maps,but the behaviour is opposite. The main concentration of Hei occurs at phase 0.0, and a weakerconcentration around the phase 0.5. The Hei underabundance spots are located at phases 0.4 and0.7. In contrast, the main concentrations of Siiii occur around the phases 0.4 and 0.7, andthe underabundancespots are located around the phases 0.0 and 0.5. In Siiii, also a third smaller overabundance spot isseen around the phase 0.1. Some indication of a corresponding spot could also be seen around phase 0.9,but due to the phase gap in the observations at phases 0.78–0.01, we cannot verify the reality of this potential fourth Siiii spot.Figure <ref> shows the locations of the magnetic poles together with the abundance maps. TheHei overabundance spots clearly occur around the magnetic poles, with a stronger concentration atthe positive pole and a weaker one around the negative pole. The locations do not coincide completely in phase with themagnetic poles, and the overabundance spots seem to be shifted some 0.1 in phase away from the poles. On theother hand, Siiii shows underabundance around the magnetic poles, and the main overabundance spotsfall closer to the magnetic equator. Interestingly, it seems that the Siiii concentrations occursomewhat shifted towards the negative pole, not at the magnetic equator itself. Similarly, theunderabundance spots of Hei are located closer to the negative magnetic pole.The presented Doppler maps for Si and He support the dipole-dominated magnetic topologyofCPD -57^∘3509.On the other hand, the presence of a third Si spot detected around the phase 0.1 and considerable abundance and field strength differences between the two magnetic poles suggest the contribution of a non-negligible higher-order magnetic multipole. Previous studies of upper main-sequence stars showed that inhomogeneous chemical abundance distributions are only observedonthesurfaceofmagneticchemicallypeculiar Ap and Bp stars with large-scaleorganised magnetic fields. In these stars, the abundance distribution of certain elementsis non-uniform and non-symmetric with respect to the rotation axis. The majority of studies of Ap and Bp stars have revealed a kind of symmetry between the topology of the magnetic field and the element distribution (see e.g. ). Thus, the structure of the magnetic fieldcan be studied by producing thesurface element maps and measuring the magnetic field using spectral lines of inhomogeneously distributed elements separately. However, due to the lowresolution of ourFORS 2 spectra, we are not able to study the detailed surface magnetic field distribution. Furthermore, only the availability of high-resolution spectra in all four Stokes parameters would allow us to obtain self-consistent mapping of spots and magnetic fields by means of Zeeman Doppler imaging (ZDI;e.g. ).§ DISCUSSIONOur spectropolarimetric monitoring of CPD -57^∘3509 using FORS 2 at the VLT showsthe presence of an approximately dipolar magnetic field with a polar strength of 4 kG, reversing over the rotation period of 6.36 d. Using the Doppler imaging technique, we were able to constrain the inclination of the rotation axis to the line of sight, i=58±10^∘,and estimate the obliquity of the magnetic axis, β=72±8^∘.In the past it was considered neither theoretically nor observationally how themagnetic field geometry in massive B-type stars evolves across the main sequence. Thus, the analysis of the magnetic field configuration of CPD -57^∘3509 is of special interest as it is one of the most evolved He-strong stars currently known. <cit.> studied theevolution of the magnetic field geometry in late B-type stars with masses ≤5 M_⊙ with accurate Hipparcos parallaxes and definitely determined longitudinal magnetic fields.They found thatthe distribution of relative ages peaks at the ZAMS, with two secondary lower peaks at the relative agesaround 60% and 80%. The strongest magnetic fields were found in younger stars in terms of the elapsed fraction of their main-sequence life. Further, rotation periods of late-B type stars slightly increase with age, which isconsistent with the assumption of conservation of angular momentum during their life on the main sequence, without any hint of a braking mechanism (see also ).The fact that the strongest magnetic fields are only observed close to the ZAMS can be interpreted as a magnetic field decay in stars at advanced ages. As for the magnetic field geometry, the authors detected a strong hint for an increase of obliquity β with elapsed time on the main sequence. Moreover, 21% out of the studied 33 stars have magnetic phase curves fitted by a double wave, indicating that the magnetic topology in late-B type stars is frequently more complex than just a single dipole.Obviously, a comparison of the evolution of magnetic field geometries between thehigher-mass He-strong stars and the lower-mass magnetic B-type stars is urgently needed to constrain the mechanismof the magnetic field generation in such stars.There is also a large dissimilarity between the lower mass magnetic Ap stars and the He-strong stars in respect tothe orientation of their magnetic axes and the period lengths. The results of modeling a small sample of Ap stars by <cit.> implied thatstars with small obliquity values β of the model magnetic axis to the rotation axis, oftheorder of20^∘, have periods longer than 25 days. However, recent studies of the magnetic field geometries in He-strong stars do not confirm this trend: a number offast rotating He-strong stars with periods below 2 d show low obliquities of their magnetic axes(e.g. ).The study of the variability of the Si and He lines showed the presence of significant chemical abundance variations across the stellar photosphere. The location of the chemical spots is roughly correlatedwith the topology of the magnetic field, where the main concentration of He is observed in the vicinityof thepositive magnetic pole and Si underabundant spots around the poles. Only three mapping studies were devoted to He-strong stars in the past, all of them using ZDI <cit.>.<cit.> studied the He and O abundance distribution on the surface of the He-strongstar HD 184927. Similar to our DI result, the He abundance was the highest in the vicinity of the strongestmagnetic pole which in that case was the pole of positive polarity. <cit.> studiedthe distribution of He, C, Si, and Fe on the surface of σ Ori E. A large overabundant He spot was found to appear at the rotation phase 0.8, but the location of thespot was not correlated with the position of the magnetic poles. The minimum abundances ofC, Si, and Fe were found at the same phase where He showed the largest abundance. Finally, <cit.> studied the distribution of He on the surface of HD 37776and concluded that the He concentration is at a maximum in the regions of maximum radial field. As only very few He-strong stars were studied with ZDI, future high-resolution, high signal-to-noisespectropolarimetric observations are needed to obtain a more complete picture on how the distribution of surface chemical spots is related to the magnetic field topology. We note that CPD -57^∘3509 has a relatively low projected rotational velocity, and due to the presence ofthe kG magnetic field and the distinct inhomogeneous element abundance distribution,it can serve as an excellentlaboratory to study various atmospheric effects that interact with the magnetic field. Further, the discovered pulsational variability on the time scale of tens of minutes has to be confirmed by future high-resolution spectroscopic time series.§ ACKNOWLEDGMENTSBased on observations made with ESO Telescopes at the La Silla Paranal Observatory under programmes 094.D-0355 and 191.D-0255. The observations on Cerro Armazones are supported by the Nordrhein-Westfälische Akademie der Wissenschaften und der Künste in the framework of the academy program of the Federal Republic of Germany and the state Nordrhein-Westfalen. 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N., 1950, MNRAS, 110, 395[Yakunin et al.2015]Yakunin2015 Yakunin I., et al., 2015, MNRAS, 447, 1418[Zacharias et al.2013]zacharias13Zacharias N., Finch C. T., Girard T. M., Henden A., Bartlett J. L., Monet D. G., Zacharias M. I., 2013, AJ, 145, 44 [Zacharias, Finch & Frouard2017]zacharias17 Zacharias N., Finch C.,Frouard J., 2017, AJ, 153, 166 § THE MEMBERSHIP OF CPD-57^∘3509 IN NGC 3293The star CPD-57^∘3509 is located only 66 arcsec from the cluster centre of NGC 3293, whereas the angular radii of the cluster core, of the central part of the cluster, and of the whole cluster were determined as 72 arcsec, 306 arcsec, and 486 arcsec,respectively <cit.>. Although this relatively bright star is listed in the Tycho catalogue <cit.>, it has no proper motion measurement in the Tycho-2 catalogue <cit.>. Therefore, it was not included in the catalogue of stars in open cluster areas of <cit.>, where membership probabilities are given for 520 open clusters of the survey of <cit.> based on proper motions and optical photometry. In the new Milky Way Star Clusters (MWSC) survey by <cit.>, based on proper motions from the PPMXL catalogue <cit.> and near-infrared photometry from the Two Micron All Sky Survey (2MASS; ), CPD-57^∘3509 is however listed with a 95% membership probability from proper motion and 99% from its JK_s photometry. <cit.> also determined a high (98%) membership probability based only on proper motions from the fourth US Naval Observatory CCD Astrograph Catalog (UCAC4; ).One should however note that the proper motion errors of the catalogues used for the above mentioned cluster membership studies are rather large. In case of CPD-57^∘3509 they are ±2.5 mas/yr in the PPMXL and between ±2.2 mas/yr and ±4.0 mas/yr in the UCAC4 catalogue.A significant improvement of PPMXL proper motions was recently achieved with the Hot Stuff for One Year (HSOY; ) catalogue, which represents a new reduction of the PPMXL including Gaia DR1 data <cit.>. The new UCAC5 catalogue <cit.> presents very accurate proper motions combining the UCAC positions with those from Gaia DR1. Because CPD-57^∘3509 was not in the Tycho-2 catalogue, it was not included in the Tycho-Gaia Astrometric Solution (TGAS) of Gaia DR1. However, its HSOY proper motion, μ_αcosδ=-5.9±1.1 mas/yr,μ_δ=+3.5±1.1 mas/yr, and its UCAC5 proper motion, -6.1±0.9 mas/yr, +3.2±0.9 mas/yr, are in very good agreement with the mean TGAS proper motion of four of the five 1σ members from <cit.> that can be found in TGAS: -6.7±1.2 mas/yr, +3.7±0.3 mas/yr. Their standard deviation in μ_αcosδ is still about three times larger than in μ_δ but was even much larger before we excluded one of the five stars as an outlier.We do not consider the TGAS parallaxes of the few 1σ members from <cit.>, as they show a large spread, and since CPD-57^∘3509 is not included in the TGAS. The very large standard deviation of the parallaxes of the five 1σ members (±0.77 mas) reduces to ±0.19 mas, if we again exclude the same one outlier. However, the generally assumed additional systematic error of ±0.3 mas in TGAS parallaxes <cit.> leads to a 60% relative distance uncertainty of a cluster at about 2 kpc distance. According to <cit.>, the distance to NGC 3293 is about 2400 pc.Concerning the radial velocity of CPD-57^∘3509, there is one measurement, -16 km s^-1 given without an error estimate <cit.>, and slight variability -16...-20 km s^-1 measured by <cit.> due to the presence of spots. This is in reasonably good agreement with the mean cluster radial velocities determined by <cit.> and <cit.>, of -12.3±2.3 km s^-1, -11.2±2.1 km s^-1, and -12±5 km s^-1, respectively.Thus the membership of CPD-57^∘3509 in the cluster NGC 3293 is based on the star's projection to the cluster core, and its available proper motion, photometry, and radial velocity. Gaia DR2, expected for April 2018, will not only provide even more accurate proper motion membership probabilities but also enable a membership study using individual stellar parallaxes of many more stars in the cluster area.§ PHOTOMETRIC DATA Tables <ref> and <ref> present the photometric data of CPD -57^∘3509, where the Modified Julian Date, the airmass of the observation, the differential photometric values Δ m_B and Δ m_V, and the phase information ϕ_hyd and ϕ_all are given. | http://arxiv.org/abs/1707.09017v1 | {
"authors": [
"S. Hubrig",
"N. Przybilla",
"H. Korhonen",
"I. Ilyin",
"M. Schöller",
"S. P. Jarvinen",
"M. -F. Nieva",
"R. -D. Scholz",
"S. Kimeswenger",
"M. Ramolla",
"A. F. Kholtygin",
"M. Briquet"
],
"categories": [
"astro-ph.SR"
],
"primary_category": "astro-ph.SR",
"published": "20170727193902",
"title": "Magnetic field geometry and chemical abundance distribution of the He-strong star CPD -57 3509"
} |
Dipartimento di Fisica, Università degli Studi di Milano, via Celoria 16, I-20133 Milano, ItalyInternational School for Advanced Studies (SISSA), Via Bonomea 265, Trieste, Italy Dipartimento di Fisica, Università degli Studi di Milano, via Celoria 16, I-20133 Milano, ItalyDipartimento di Fisica, Università degli Studi di Milano, via Celoria 16, I-20133 Milano, ItalyDipartimento di Fisica, Università degli Studi di Milano, via Celoria 16, I-20133 Milano, [email protected] Dipartimento di Fisica, Università degli Studi di Milano, via Celoria 16, I-20133 Milano, Italy We consider a zero-temperature one-dimensional system of bosons interacting via the soft-shoulder potential in the continuum, typical of dressed Rydberg gases.We employ quantum Monte Carlo simulations, which allow for the exact calculation of imaginary-time correlations, and a stochastic analytic continuation method, to extract the dynamical structure factor. At finite densities, in the weakly-interacting homogeneous regime, a rotonic spectrum marks the tendency to clustering. With strong interactions, we indeed observe cluster liquid phases emerging, characterized by the spectrum of a composite harmonic chain. Luttinger theory has to be adapted by changing the reference lattice density field. In both the liquid and cluster liquid phases, we find convincing evidence of a secondary mode, which becomes gapless only at the transition. In that region, we also measure the central charge and observe its increase towards c=3/2, as recently evaluated in a related extended Bose-Hubbard model, and we note a fast reduction of the Luttinger parameter. For 2-particle clusters, we then interpret such observations in terms of the compresence of a Luttinger liquid and a critical transverse Ising model, related to the instability of the reference lattice density field towards coalescence of sites, typical of potentials which are flat at short distances. Even in the absence of a true lattice, we are able to evaluate the spatial correlation function of a suitable pseudo-spin operator, which manifests ferromagnetic order in the cluster liquid phase, exponential decay in the liquid phase, and algebraic order at criticality. Quantum Critical Behavior of One-Dimensional Soft Bosons in the Continuum Gianluca Bertaina=========================================================================Quantum phase transitions (QPT) <cit.> play an intriguing role in many-body systems, due to the possibility of unveiling new exotic phases.The progress in the manipulation of ultracold gases allows for the exploration of QPTs, by engineering well-controlled synthetic quantum many-body systems, confined for example by optical lattices <cit.> or in quasi-one-dimensional geometries <cit.>. Recently, Rydberg atoms <cit.> have emerged as a new route to QPTs <cit.>. These are atoms in highly-excited electronic states, with a very large electronic cloud. In particular, theoretical <cit.> and experimental <cit.> efforts have focused on ensembles of dressed Rydberg atoms, which are superpositions of the ground state and the above mentioned excited states, coupled via a Rabi process. Their effective interaction can be a soft-shoulder potential, with a flat repulsion up to a radius R_c related to the highly excited state, and a repulsive van-der-Waals tail at large distances <cit.>. Quite interestingly, this repulsive interaction belongs to the class that has been recognized to induce cluster formation at high density in classical statistical mechanics <cit.>, thanks to the relative freedom of particles at short distances. This has opened a recent flourishing of research on quantum cluster phases: in high dimensions, coexisting cluster crystal and superfluid order have been predicted, yielding supersolid behavior <cit.>, while in one dimension (1D), cluster Luttinger liquids (CLL) have been proposed on a lattice <cit.>.In this Letter, we investigate a prototypical system of N bosons in 1D at linear particle density n, governed by the following Hamiltonian in the continuum:H = -ħ^2/2m∑_i^N ∂^2/∂ x_i^2 + ∑_i<jV_0/r_ij^6+R_c^6where x_i are the particle coordinates, r_ij=|x_i-x_j| the distances, m is the mass and V_0 and R_c are the strength and the radius of the soft-shoulder potential V(r).If not otherwise specified, in the following we use units of R_c for the length, E_c=ħ^2/m R_c^2 for the energy, and ħ/R_c for the momenta (See Supplemental Material [See Supplemental Material at [URL will be inserted by publisher] for details on the potential, the experimental realization, the methods, the extraction of relevant observables.]). The zero-temperature phase diagram (Fig. <ref>) thus depends on the following two dimensionless quantities: strength, U=V_0/(E_c R_c^6), and density, ρ=nR_c.By evaluating relevant static and dynamical properties, we show that, while for small U and moderate ρ the system is a Luttinger liquid (LL) <cit.>, although with strong correlation effects, for higher U or ρ a transition occurs towards CLL. In particular, we focus on the QPT to the dimer cluster liquid, which turns out to be of the 2D Ising universality class. A similar phenomenology has been recently studied in a 1D lattice system governed by the extended Bose-Hubbard hamiltonian <cit.>, while we observe it for the first time in the continuum, where we find that an effective spin hamiltonian emerges at the transition, even in absence of an underlying lattice.For generic coupling and density, this system falls into the LL universality class <cit.>, characterized by a gapless bosonic mode at small momenta, with sound velocity v. The low-energy and momentum sector of the Hilbert space is governed by the Hamiltonian H_LL=(v/2π)∫ dx [K_L (∇θ)^2+(∇ϕ)^2/K_L], where a large Luttinger parameter K_L>1 favors the fluctuations of the particle counting field ϕ(x), while small values of K_L induce crystal-like behavior, by disordering the phase field θ(x). The central charge of the associated conformal field theory (CFT) is c=1 <cit.> and, for our Galilean invariant system, v=ρπ/K_L <cit.>.In the dilute limit, the effects of the interaction are well described by the scattering length a_1D; in particular, for U ∼ 1.09, we get a_1D∼ 0 <cit.>, corresponding to the Tonks–Girardeau (TG) model <cit.>. Conversely, at higher densities, the full shape of V(r) is relevant. Its 1D Fourier transform Ṽ(q) <cit.> features a global minimum at q_c ≃ 4.3, at which Ṽ(q_c)<0, providing a typical length b_c=2π/q_c ≃ 1.46. It has been recognized, in the context of classical physics, that such density-independent distance favors clustering, even with a completely repulsive potential <cit.>.Classically, one obtains a T=0 cluster crystal, which is destabilized in 1D by finite temperature, in favor of cluster-dominated liquid phases with different average occupation <cit.>. Quantum mechanics induces coherent delocalization even at T=0, rendering the cluster phase a CLL and triggering a QPT towards a LL without cluster order (Fig. <ref>).To study the phase diagram in a non-perturbative way, we use the well-established path integral ground state (PIGS) quantum Monte Carlo method <cit.>, which represents the ground state as the imaginary-time projection exp(-τ H)|Ψ_T⟩ of a trial wavefunction. We simulate up to N=200 particles in a segment of length L=N/ρ, using periodic boundary conditions (PBC). The trial wavefunction is of the two-body Jastrow form: Ψ_T(x_1… x_N) = exp[-1/2∑_i<j(u(r_ij) + χ(r_ij))], where χ(r) accounts for long-wavelength phonons <cit.>, while exp[-u(r)/2] is the numerical solution of a two-body Schrödinger equation <cit.>, with the effective potential V_eff(r)=c_1 V(r) + c_2 ∑_l V(r-l b). To reduce projection times, it is crucial to use and optimize this mean-field potential, which accounts for the presence of nearby clusters <cit.>. We consider excitations associated to density fluctuations, which are commonly investigated via the dynamical structure factor S(q,ω) = ∫ dt e^i ω t/2π N⟨ e^itH/ħρ_q e^-itH/ħρ_-q⟩. The PIGS algorithm evaluates the numerically exact imaginary-time intermediate scattering function, which yields S(q,ω) via analytic continuation, through the genetic inversion via falsification of theories algorithm <cit.>. We now proceed to discuss our results, first in the LL, then in the CLL regimes. Finally, we discuss the QPT in between the two liquids. Liquid regime– Here and in the following, k_F=πρ and E_F=k_F^2/2 are effective Fermi momentum and energy. We concentrate on interaction U ≃ 1.09, and increase the density (dot-dashed line in Fig. <ref>).In the low-density regime ρ≲ 0.1 (Fig. <ref>a), S(q,ω) is almost constant, at fixed q, in between the particle-hole boundaries ε_IFG(q) = | k_F q ± q^2/2|, analogously to the TG gas, which can be mapped to an ideal Fermi gas (IFG). However, within our resolution, the spectral weight has started to gather, especially at the upper boundary, similarly to what happens in the Lieb-Liniger model with decreasing coupling parameter <cit.>. In fact, already at ρ=0.6 (panel b),the spectrum has evolved into a main mode.As such, it is very well described by the single-peak Feynman approximation ε_FA(q)=ε_0(q)/S(q), with ε_0(q)=q^2/2 the free-particle energy and S(q) the static structure factor. By further increasing ρ, we simply monitor the evolution of ε_FA(q) (Fig. <ref>c), and notice that the main excitation becomes more structured, with a roton minimum moving towards q=q_c. A standard Bogoliubov analysis <cit.> yields the dispersion ε_B(q)=√(ε_0(q)[ε_0(q)+2 ρṼ(q)]), which depends only on the combination α=ρ U. In this approximation, it is clear that the emergence of the roton minimum is allowed by the momentum dependence of Ṽ(q), which has a negative part <cit.>. The roton softens at α=α_c ≃ 20.65. While the agreement between the single-mode ε_FA(q) and ε_B(q) approximations is very good for 0.6≲ρ≲ 19, such treatments are in general not valid anymore for U ≳α_c/ρ (dashed line in Fig. <ref>), where indeed our simulations show that clustering occurs.On increasing ρ, the pair distribution function g(r) at first gradually approaches 1 everywhere (Fig. <ref>d), as in classical soft-core fluids in the absence of clustering <cit.>. However, for very high ρ, large-amplitude slowly-decaying oscillations appear, with wavelength b_c. Again, this behavior is akin to that of classical systems, in the presence of clustering <cit.>.A gaussian fit of the peaks indicates, on average, N_c≃ 36 particles per cluster at ρ=24.6. In the quantum case, the oscillations of g(r) eventually decay as in a cluster liquid, a behavior that we can easily see in the more relevant cluster phases at low ρ and large U. In fact, a hamiltonian description of dressed Rydberg gases is questionable at high ρ, due to increased losses to other Rydberg levels in current experiments <cit.>. Commensurate Cluster Luttinger Liquid–We therefore now focus on the density ρ=2/b_c≃ 1.37 (solid line in Fig. <ref>), commensurate to clusters of N_c=2 particles <cit.>. In Fig. <ref>a, the PIGS results for g(r) are shown, indicating an evolution to a cluster structure on increasing U, with peaks containing two particles. g(r) manifests long-range algebraic decay of the peaks' heights,which demonstrates absence of true crystal order <cit.>. To interpret these results, we employ CLL theory. In the standard bosonization approach <cit.>, the counting field fluctuates around a lattice with spacing ρ^-1: hamiltonian H_LL is then derived with the assumption that fluctuations are small. However, in a commensurate cluster liquid, clearly fluctuations are small only around a lattice of clusters, with spacing ρ^-1N_c=b_c. We follow <cit.>, and obtain the following commensurate CLL form of g(r):g(r)r≫ 1/ρ≃ 1-2K_L/(2πρ r)^2+∑_l=1^∞ A_l cos(2π l ρ r/N_c)/r^2K_L^' l^2The r^-2 term is analog to the standard LL case, while the last term yields dominant density oscillations of wavevector 2k_F/N_c=q_c, modulated by an effective Luttinger parameter K_L^'=K_L/N_c^2. This implies that, in the CLL phase, the divergence of S(q_c)∝ N^1-2K_L^' is much stronger than what would result from K_L. We extract K_L and K_L^' from the small momentum behavior of S(q) and large distance decay of g(r), respectively <cit.>. Interestingly, K_L scales as U^-1/2 in both the LL and CLL regimes, but with different prefactors. Moreover, we verify that the number of excess particles per cluster δ=√(K_L/K_L^')-1 quickly goes to 1 for U>18 (Fig. <ref>b).Deeply in the cluster phase, a composite harmonic chain (HC) theory can also be envisaged. We write a model hamiltonian of the type H_HC=∑_iν p^2_i,ν/2+γ∑_iνμ(x_i,ν-x_i+1,μ)^2/2, where x_i,ν is the displacement of the ν-th particle (with 1≤ν≤ N_c) from the average position of cluster i (with 1≤ i≤ N/N_c), and springs of strength γ are present only between particles in adjacent clusters, modeling the fact that V(r) is flat at short distances [PBC are implied]. We obtain center-of-mass modes, of acoustic frequencies ω_acou(k)=2√(N_cγ)sin(k b_c/2), and optical modes, of dispersionless frequency ω_opt=√(2N_cγ). The latter correspond to relative vibrations of particles in a cluster. We relate γ to the mean-field potential felt by a particle if all the others are in a cluster crystal with spacing b, and find γ(b)=-(4π^2/b^3) ∑_j=1^∞ j^2Ṽ(2π j/b) <cit.>. It is clear that a stable structure is possible only ifṼ(2π/b)<0 for some b <cit.>. In Fig. <ref>, the spectra at ρ=1.37, with decreasing U, are shown. Panel a (U=100) is deep in the CLL phase: the main peak is in good agreement with the acoustic mode of HC theory, with b=b_c. A secondary structure appears at higher frequencies, which we interpret as the optical mode <cit.>. This is however not flat, but strongly modified by anharmonic couplings: these are clearly even more crucial at smaller U, where they induce cluster melting. Ising transition– The question is now: how are the LL and CLL phases really different? Is the transition simply a crossover?Our data, which show Luttinger liquid behavior on both sides, exclude Berezinskii-Kosterlitz-Thouless transition to a charge-density-wave, or Peierls transition, even though, at U=18, we get K_L≃ 0.52(1) <cit.>. Moreover, the atomic-pair superfluid transition <cit.> is also excluded, since, here, formation of larger clusters is allowedand the CLL phase manifests strong quasi-solid order (K_L<1/2).In fact, the physics of this relatively simple system is very rich.The acoustic mode of the CLL phase (Fig. <ref>, panels a-b) is gapless at q=q_c, corresponding to k_F, at this density. After the transition, to be located at U=U_c≃ 18 (panel c), this lowest excitation turns into the rotonic mode (panels d-e). Quite interestingly, a weaker secondary mode appears not only in the cluster phase, but also in the strongly correlated liquid phase, in the form of a secondary roton, which connects to the higher-momenta main mode.It is reasonable to associate this secondary excitation, in the LL phase, to incipient cluster formation, due to particles being preferentially localized close to either the left or the right neighbor. The crucial observation is that the gap of both such LL excitations (panels d-e), and the anharmonic optical modes of the CLL phase (a-b), vanishes at the transition (c), which implies that they proliferate at that point.This behavior is consistent with that of the 1D transverse Ising (TI) model <cit.> of a chain of coupled two-level systems. Its hamiltonian is H_TI=-J∑_i σ^z_iσ^z_i+1-h∑_iσ^x_i, where σ^x/z_i are Pauli matrices at site i. It contains both a ferromagnetic coupling (J>0), which forces alignment, and quantum tunneling (h>0) between the eigenstates of σ^z, favoring a paramagnetic state. This model is exactly solvable with a Jordan-Wigner transformation and Bogoliubov diagonalization <cit.> and yields excitations of energy ε_TI(q)=√(Δ^2+4Jh(sinqa/2)^2) ,where Δ=|J-h| is the gap and a is lattice spacing, which are gapless only for h=J. This signals a QPT from the ferromagnetic to the paramagnetic state, which is dual to the 2D classical thermal Ising transition. In our case, it is natural to associate Δ to the gap of the secondary mode at q=q_c, and set a=b_c, implying that a spin should be identified every two particles. We fit Eq. (<ref>) from our spectra (Fig. <ref>a): within our accuracy, the behavior of Δ in U-U_c is linear close to the transition, consistent with the dynamical exponent z=1 <cit.>. The point at U=18 requires very long projection times: another indication of the presence of a very low-energy mode. Within our resolution, the Luttinger and critical Ising modes have the same velocity at U=U_c, which would imply low-energy supersymmetry <cit.>. To corroborate our interpretation, we recall that the central charge c of the critical TI model is c=1/2, so that, at the transition, the total central charge should be c=1+1/2=3/2, as calculated for the related lattice model <cit.>. We estimate c from the slope of the energy per particle ε(N)=ε_∞ - c E_F/(6 K_L N^2) versus 1/N^2, employing a standard CFT result for the dominant finite-size effects <cit.>. An increase of c is manifest in Fig. <ref>a. It is in fact delicate to extrapolate c close to U_c: higher order corrections may become relevant, and field theoretical methods should elucidate the interplay between the Luttinger and Ising fields, as done in <cit.>. It would be interesting to extract c also from the entanglement entropy, as recently introduced in PIGS <cit.>.It is particularly appealing to investigate the microscopic realization of this effective TI model. The many-body potential surface reduces to double wells as a function of relative distances: two nearby bosons have preferred configurations if they are at r_ij=0 or r_ij≃ b_c <cit.>. Thus, for each even particle, for example, there are left ψ_L and right ψ_R preferred cluster configurations, and the potential energy is minimized when subsequent even particles choose the same clustering direction. Anharmonic terms give instead rise to delocalization. The cluster phase is then to be thought as the ferromagnetic state, where all even particles have chosen either ψ_L or ψ_R (Fig. <ref>b), while the liquid phase is made of ψ_+=(ψ_L+ψ_R)/√(2) states (Fig. <ref>a), where particles continuously hop left and right. This mapping can be made quantitative, by introducing a simple, but effective string representation of σ^z, inspired by <cit.>:first, particles are ordered by their position k, and even positions are assigned a lattice index i=k/2; then, a pseudo-spin σ_i^z=1 is assigned if |x_k-x_k-1|<|x_k-x_k+1|, or σ_i^z=-1 in the opposite case. We evaluate the spatial correlator g_σ(|i-j|)=⟨σ^z_iσ^z_j⟩ of such ℤ_2 pseudo-spin (Fig. <ref>b). It is very remarkable that g_σ behaves as expected for the TI model: in the LL (paramagnetic) phase it decays exponentially, while in the CLL (ferromagnetic) phase it manifests true long-range order, which is nonlocal <cit.>, because of the preliminary ordering of particles. At U=18, its behavior is close to an algebraic decay with exponent η≃-1/4.This pseudo-spin mapping in a continuous system at the LL-CLL transition, as revealed by excitation spectra and a suitable spin correlator, is the key result of this Letter. Such critical regime could be probed even at finite T, given finite experimental sizes. Future work will investigate effects of non commensurability of ρ with 1/b_c, which is particularly relevant for trapped gases. Also, an open issue is the presence of quantum Potts transitions at densities commensurate to N_C≥3.We acknowledge useful discussions with M. Dalmonte, M. Fleischhauer, C. Gross, R. Martinazzo, A. Parola, N. Prokof'ev, H. Weimer. We acknowledge the CINECA awards IscraC-SOFTDYN (2015) and IscraC-CLUDYN (2017) for the availability of high performance computing resources and support. We acknowledge funding from the University of Milan (grant PSR2015-1716LPERI-M). 63 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[Sachdev(2000)]sachdev_quantum_2000 author author S. 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Lett. volume 97, pages 260401 (year 2006)NoStopSupplemental Material: Quantum Critical Behavior of One-Dimensional Soft Bosons in the Continuum1mmWe give details on the potential and relevant mean-field potentials; make some comments on the experimental realization; briefly explain the PIGS and GIFT algorithms, the optimization of the trial wavefunction, the extraction of the Luttinger parameters and of the central charge, providing also tables with the results used to produce Figs. 3 and 5 in the main text.Note: citations in this Supplemental Material refer to the bibliography in the main paper.§ POTENTIAL AND EFFECTIVE MEAN-FIELD POTENTIAL The soft-shoulder potential V(r)=U/(1+r^6) has the 1D Fourier transform (see Fig. <ref>)Ṽ(q)=∫_-∞^+∞e^i q r V(r)dr= U π e^-q/2[e^-q/2 + cos(√(3) q/2) + √(3)sin(√(3) q/2)]/3. When the density is ρ=2/b_c≃ 1.37, an approximation of the mean-field potential felt by a particle in the liquid phase can be obtained by letting all other particles on a lattice of spacing 1/ρ=b_c/2: V_MF^LL(r)=-V(r)+∑_i=-∞^∞ V(r - i b_c/2) (red solid line in Fig. <ref>). This lattice periodicity is clearly unstable, since a double well appears close to the origin: so the liquid phase is stable only thanks to kinetic energy. The mean-field potential felt by a particle in the cluster liquid phase can be approximated by letting all other particles on a lattice of spacing 2/ρ=b_c: V_MF^CLL(r)=-V(r)+2∑_i=-∞^∞ V(r - i b_c), which implies that pairs of particles overlap (blue solid line in Fig. <ref>). Here, a clear harmonic confinement is present at r=0, which stabilizes the cluster phase, and whose typical energy ε_h(b_c) is given in the main text.An instructive mean-field potential can also be visualized, in the LL phase, by letting three particles free, and fixing only their center of mass at the center of a lattice of the other particles with spacing 1/ρ=b_c/2: the resulting surface, in terms of the relative distances r_12=|x_1-x_2| and r_23=|x_2-x_3|, is plotted in Fig. <ref>. The two spheres indicate the positions of the two main equivalent minima, which clearly correspond to (r_12,r_23)≃(0,b_c) and (r_12,r_23)≃(b_c,0). For particle 2, suitable orbitals centered in such minima correspond to the two states ψ_L and ψ_R introduced in the text. They define the Hilbert space governed by the Ising hamiltonian introduced in the main text. § CONSIDERATIONS ABOUT THE EXPERIMENTAL REALIZATIONIn the context of ultracold Rydberg gases, the parameters of the 3D shoulder potential can be related to the Rabi frequency Ω_R and detuning Δ_R, by U=(C_6 m^3 |Δ_R|^2/16ħ^4)^1/3β^4 and R_c=(C_6/2ħ|Δ_R|)^1/6, where β=Ω_R/2Δ_R is the admixture of the Rydberg level with the ground-state, and C_6 is van-der-Waals coefficient [11].Eq.(1), in the main text, is the effective 1D Hamiltonian [21],which is relevant in an elongated quasi-1D configuration, once the transverse degrees of freedom are frozen in the ground state, due to strong trapping [5].The transverse confinement size a_⊥ does not affect much the effective potential, if a_⊥≲ R_c. With respect to the original 3D potential, the quasi-1D interaction has reduced U and larger q_c. The condition a_⊥≲ R_c is already under reach in current experiments, where both a_⊥ and R_c can be of order of 100 nm [6].The above condition also prevents the zig-zag instability [53]for strong interactions. To experimentally probe the phase diagram in Fig.1 of the main text, one can vary detuning Δ_R, linear density n, mass m (by changing the atomic species), and C_6∝n̅^11, by changing the addressed Rydberg level of principal number n̅. Note that, while Rydberg levels have typical lifetimes τ_R of order of μ s, Rydberg dressed atoms with small admixture β have proportionally increased lifetimes τ=τ_R/β^2 [16,17].Cryogenic techniques would be beneficial to reduce the detrimental black-body radiation, due to the experimental apparatus [8].The experimental realization of hamiltonian of Eq.(1) in the main text is presently challenging in the absence of a lattice, since no short-range hard core is enforced. However, clustering effects persist even if one introduces a hard core, provided it is small when compared to R_c.§ PATH-INTEGRAL GROUND STATE METHOD (For completeness, we add a brief description, adapted from the Supplemental Material of Ref.[38].) The Path Integral Ground State (PIGS) Monte Carlo method is a projectortechnique that provides direct access to ground-state expectation values of bosonic systems, given the microscopic Hamiltonian Ĥ [34,35].The method is exact, within unavoidable statistical error bars, which can nevertheless be reduced by performing longer simulations, as in all Monte Carlo methods.Observables Ô are calculated as ⟨Ô⟩ = lim_τ→∞⟨Ψ_τ|Ô|Ψ_τ⟩/⟨Ψ_τ|Ψ_τ⟩,where Ψ_τ = e^-τĤΨ_T is the imaginary-time projection of an initial trial wave-function Ψ_T. Provided non-orthogonality to the ground state, the quality of the wave-function only influences the projection time practically involved in the limit and the variance of the results.The trial wavefunction is of the two-body Jastrow form: Ψ_T(x_1… x_N) = exp[-1/2∑_i<j(u(r_ij) + χ(r_ij))]. The Reatto-Chester contribution χ(r)=- αlog[sin^2(π r/L)+sinh^2(πR̅/L)/ 1+sinh^2(πR̅/L)]accounts for long-wavelength phonons [36,30]and allows for faster convergence of the small momenta parts of the structure factors. R̅ is a variational parameter delimiting the long-range part from the short-range contribution. The latter is embodied in the f(r)=exp[-u(r)/2] factor, which we take as the numerical solution (described in detail in [30])of a suitable two-body Schrödinger equation (with reduced mass μ=m/2), with the effective potential V_eff(r)=c_1 V(r) + c_2 ∑_l=-l_C^l=l_C V(r-l b),and the boundary condition u(R̅)=0. Here, the first term is a renormalized two-body potential, while the sum accounts for a series of nearby clusters in a lattice of spacing b (l=0 is excluded and l_M=int[L/(2b)]). All parameters in the trial wave-function are variationally optimized, as described in the next Section, before projecting with the PIGS method, in order to maximize efficiency of the simulations. One finds that typically R̅∼ 2.2,b∼ b_c, and c_1 and c_2 are usually of the same order in the cluster phase. We found it is crucial to introduce such effective cluster contributions in the two-body Jastrow factor, since the variational energy per particle is typically 30% higher than the PIGS result, if one sets c_2=0. On the contrary, by properly optimizing the parameters, we obtained discrepancies of at most a few percent of the PIGS result. A typical effective potential, and the corresponding short-range Jastrow correlation f(r), are shown in Fig. <ref>. Notice the behavior of f(r) at r≃ i b_c, with i=1,2. In the PIGS method, the imaginary time τ of propagation is split into M_P time steps of size δτ=τ/M_P so that a suitable short-time approximation for the propagator can be used; in this case, we employ the fourth-order pair-Suzuki approximation [35]. The timestep is selected by performing various simulations at short fixed projection time τ with increasing number of beads M_P, and monitoring the energy per particle, looking for convergence within errorbars of order 10^-4. At ρ=1.37, the resulting dτ approximatively follows the form dτ≃ 0.1 U^-1/2: this is consistent with the consideration that the imaginary-time density matrix typically varies at the scale of the amplitude l_HCO∝ U^-1/4 of the cluster vibrational modes.Once the timestep is selected, convergence to the ground-state is obtained by selecting a sufficiently long projection time τ, which renders the potential energy flat (within errorbars) as a function of time slice. With our optimized initial wavefunctions, the typical number of time-steps for convergence is 35. Once convergence in τ is obtained, a further projection time of typically M_F=60 time-slices is used to sample the intermediate scattering function F(q,τ).Close to the transition, convergence in τ is very slow, due to the low-energy Ising mode. We have thus also employed a more refined effective potential, of the formV_eff(r)=c_1 V(r)+c_2 ∑_l=-l_C^l=l_C∑_i=-p^p V(r-l b-iδ_l) g(iδ_l,s_l),where we introduced a gaussian kernel g(r,s_l)=exp[-r^2/(2s_l^2)]/𝒩(s_l), with 𝒩(s_l)=∑_i=-p^pexp[-(iδ_l)^2/(2s_l^2)], for better describing the clusters. The width s_l=s_0 l^γ is allowed to increase with the cluster position, and the convolution is discretized with p=10 and δ_l=3s_l/p. In the limit s_0→0 and γ→0, one recovers expression (<ref>), while in general s_0 and γ are two additional variational parameters to be optimized.For U=18, we employed the parameters c_1=1.127, c_2=1.076, b=1.655, s_0=0.2, α=4.84, R̅=1.83, and γ=0.4. With dτ=0.03, simulations still required 100 time-slices of projection.§ PARAMETERS OPTIMIZATION THROUGH SIMULATED ANNEALING The parameters appearing in the trial wavefunction are optimized with the Variational Monte Carlo method (VMC), which corresponds to PIGS, when no imaginary-time projection is made. We employ a simulated annealing procedure in parameters' space, aiming at both reducing the energy associated to the trial wavefunction, and obtaining a good pair distribution function g(r). To each configuration ξ of the parameters, the following Boltzmann weight λ =-β[E(ξ) + ζ χ^2(ξ)] is associated, with inverse temperature β, where E(ξ) is the VMC energy, andχ^2(ξ) = ∑_i^i_M[g_0(r_i) - g_ξ(r_i)]^2/σ_0^2(r_i) + σ_ξ^2(r_i)is the discrepancy between the pair distribution functions obtained at a particular step g_ξ, and the one obtained with a preliminary unoptimized PIGS simulation g_0, weighted by the corresponding uncertainties σ_ξ and σ_0. This difference is evaluated only up to the i_M-th bin of the histogram of g(r), which includes the first peak of g(r) for r≠ 0. The motion of a fictitious particle is simulated in parameters' space, subjected to such Boltzmann weight, where ζ is the relative role of χ^2 with respect to E. The effective temperature is progressively decreased in order to reach the configuration having the lowest value possible of E(ξ) and χ(ξ) simultaneously, and the algorithm is stopped once the variations of the weight are of the order of errorbars. The typical used value of ζ is ∼ 10^-4, except close to the transition, where we observed that increasing this value led to shorter projection times in the subsequent PIGS simulations. Fig. <ref> shows an example of the convergence of the two parameters c_1 and c_2.§ GENETIC INVERSION VIA FALSIFICATION OF THEORIES METHOD (For completeness, we add a brief description, adapted from the Supplemental Material of Ref.[38].) The relation between the intermediate scattering function F, which is evaluated within the PIGS simulations, and the dynamical structure function isF(q,τ) = 1/N⟨ e^τ Hρ_q e^-τ Hρ_-q⟩ = ∫_0^∞ dω e^-τω S(q,ω). This is a Fredholm equation of the first kind and is an ill-conditioned problem, because a small variation in the imaginary-time intermediate scattering function F produces a large variation in the dynamical structure factor S. At fixed momentum q, the computed values F_j=F(q,jδτ), where j=0… M, are inherently affected by statistical uncertainties δ F_j, which hinder the possibility of deterministically infer a single S(q,ω), without any other assumption on the solution. The Genetic Inversion via Falsification of Theories method (GIFT) exploits the information contained in the uncertainties to randomly generate Q compatible instances of the scattering function F^(z), with z=1… Q, which are independently analyzed to infer Q corresponding spectra S^(z), whose average is taken to be the “solution”. This averaging procedure, which typifies the class of stochastic search methods (See the recent review [40]),yields more accurate estimates of the spectral function than standard Maximum Entropy techniques. This method is able to resolve the first spectral peak and approximately the second one, if the errorbars of F are sufficiently small; generally speaking, the most relevant features of the spectrum are retrieved in their position and (integrated) weight. Given an instance z, the procedure of analytic continuation from F^(z) to S^(z) relies on a stochastic genetic evolution of a population of spectral functions of the generic type S^(z)(q,ω) = m_0∑_i=1^N_ωs_iδ(ω-ω_i), where m_0=F^(z)(q,0) and the zeroth momentum sum rule ∑_i=1^N_ωs_i=1 holds. The N_ω support frequencies ω_i are spaced linearly up to an intermediate frequency ω_m, which is of order of the maximum considered momentum squared q^2, then a logarithmic scale is used up to very high frequencies. Genetic algorithms provide an extremely efficient tool to explore a sample space by a non-local stochastic dynamics, via a survival-to-compatibility evolutionary process mimicking the natural selection rules; such evolution aims toward increasing the fitness of the individuals, defined asΦ^(z)(S) = -∑_j=0^M 1/δ F_j^2[F^(z)_j - m_0∑_i=1^N_ω e^-j δτω_i s_i]^2 - ϵ[m_1 - m_0∑_i=1^N_ωω_i s_i]^2,where the first contribution favors adherence to the data, while the second one favors the fulfillment of the f-sum rule, with m_1=q^2/2 and ϵ a parameter to be tuned for efficiency. A step in the genetic evolution replaces the population of spectral functions with a new generation, by means of the “biological-like” processes of selection, crossover, local mutation, non-local mutation, which are described in detail in [37,40].Moreover, the genetic evolution is tempered by an acceptance/rejection step based on a reference distribution p^(z)(S)=exp(Φ^(z)(S)/T), where the coefficient T is used as an effective temperature in a standard simulated annealing procedure. We found that this combination is optimal in that it combines the speed of the genetic algorithm with the prevention of strong mutation-biases thanks to the simulated annealing. Convergence is reached once |Φ^(z)(S)|< 1, and the best individual, in the sense that it does not falsify the theory represented by (<ref>), is chosen as the representative S^(z). The final spectrum is obtained by taking the average over instances S̅(q,ω)=1/Q∑_z=1^Q S^(z)(q,ω).§ EXTRACTION OF THE LUTTINGER PARAMETER FROM THE STATIC STRUCTURE FACTOR The behavior of S(q) for small momenta is shown in Figure <ref>. Since its Luttinger liquid behavior is S(q) ∼ qK_L/(2k_F), we fit 2 k_F S(q)/q with a constant. Clearly, for extremely low momenta, size effects become relevant, since finite imaginary-time projection has not been able to correctly reconstruct the long-distance density fluctuations, leading to a non linear behavior of the data. For this reason, the error on the fitted value of K_L has been taken to be the absolute value of the difference between the fitted K_L and the value of 2k_F S(q)/q for the smallest q at our disposal. The resulting K_L are summarized in Table <ref>. § EXTRACTION OF THE ANOMALOUS LUTTINGER PARAMETER FROM THE PAIR DISTRIBUTION FUNCTION CLL theory implies that the long-range behavior of g(r) has to be of the form of Eq.(2) in the main text. The easiest way to extract K_L^' from g(r) consists in evaluating the power-law exponent relative to the cosine decay. In such procedure we have neglected the -2K_L/(2πρ r)^2 term, as well as all the cosine terms, but the first one, as their decay is usually too fast to be relevant. Values of g(r) <1 are determined by short-range effects: it is thus appropriate to fit just the upper part of the peaks of g(r). To make such procedure more precise, we have fitted the function [g(r) - 1] · (ρ r)^η, which is supposed to behave like ∼ A_1 cos(πρ r) (ρ r)^η -2K_L' for long distances, with η tuned as to have the periodic peaks at comparable heights. Thus the cosine decay is considerably softened, leading to an easier fitting of K_L^'. An example of one of our fits is shown in figure <ref>. We have not been able to apply this methodology for U<18, due to large noise-to-signal ratio, and at U=18, possibly due to the role of subleading terms in the decay. The resulting K_L^' are summarized in Table <ref>.§ EXTRACTION OF THE CENTRAL CHARGE We estimate the central charge c by fitting the slope of the energy per particle ε(N)=ε_∞ - c E_F/(6 K_L N^2) versus 1/N^2, where K_L is inferred from Table <ref>. The range of considered numbers of particles is typically from N=10 to N=40 (N=100 in some cases), and we progressively increase the minimal N_min used in the fit, to assess the role of higher order contributions, finding that the extracted c is generically stable if N_min=20. The errorbar comes from the uncertainty in ε(N) and K_L, and from the variance of the extracted c at different N_min>20. Examples of extraction of c are shown in Fig. <ref>, corresponding to U=17 and to the more difficult U=18 point. Table <ref> summarizes the results. | http://arxiv.org/abs/1707.08850v2 | {
"authors": [
"Stefano Rossotti",
"Martina Teruzzi",
"Davide Pini",
"Davide Emilio Galli",
"Gianluca Bertaina"
],
"categories": [
"cond-mat.quant-gas"
],
"primary_category": "cond-mat.quant-gas",
"published": "20170727131118",
"title": "Quantum Critical Behavior of One-Dimensional Soft Bosons in the Continuum"
} |
A new single-particle basis for nuclear many-body calculations. G. Puddu E-mail: [email protected] Dipartimento di Fisica dell'Universita' di Milano, Via Celoria 16, I-20133 Milano, Italy Received: December 30, 2023/ Accepted: date =============================================================================================================================================================== Predominantly, harmonic oscillator single-particle wave functions are the choice as a basis in ab-initio nuclear many-body calculations. These wave-functions, although very convenient in order to evaluate the matrix elements of the interactionin the laboratory frame, have a too fast fall-off at large distances. In the past, in alternative to the harmonic oscillator, other single-particle wave functions have been proposed. In this work we propose a new single-particle basis, directly linked to the nucleon-nucleon interaction. This new basis is orthonormal and complete, has the proper asymptotic behavior at large distances and doesnot contain the continuum which would pose severe convergence problems in nuclear many body calculations. We consider the newly proposed NNLO-opt nucleon-nucleon interaction, without any renormalization. We show that unlike other basis,this single-particle representation has a computational cost similar to the harmonic oscillator basis with the same space truncation and it gives lower energies for ^6He and ^6Li. Pacs numbers: 21.10.-k,21-60.Cs, 24.10.Cn§INTRODUCTION.New theoretical methods and advanced computational facilities, have made possible in recent years to tackle the most fundamental problem in nuclear many-body theories. That is, the description ofnuclear properties starting from the nucleon-nucleon interaction.Several modern nucleon-nucleon interaction and NNN interactions, based on chiralperturbation theory are nowadays available (refs. [1]-[5]). These interactions are the input to modern many-body methods in order to extract nuclear observables. To mention a few, the no-core shell model (refs. [6],[7] and ref. [8] for a recent review), the coupled-cluster approach (refs. [9],[10] and for a review ref.[11]), the similarity renormalization group method (cf. ref.[12] for a recent review) and the Self-Consistent Green-s Function method (ref.[13]. By large all these methods use the harmonic oscillator (h.o.) wave functions as the single-particle basis. Effects due to truncation of the Hilbert space are addressed using robust infrared extrapolation techniques (ref. [14]). Only recently there has been a systematic attempt to consider an alternative basis, namely the Coulomb-Sturm basis (refs. [15],[16])which has the following properties. It is orthonormal and complete, it does not have continuum states and it has theproper asymptotic behavior in coordinate space at large distances (i.e. it falls off as exp^-μ r). In the past we have consideredan alternative basis very similar to the one we propose in this work, which however has been used for a very simple model and does not have the proper asymptotic behavior (ref.[17]) since it has a Gaussian falloff. One of the reasons the Coulomb-Sturm basis has been used, was that quantities like root mean square radii, quadrupole moments and transition rates are sensitive to large distances. The Coulomb-Sturm basis did improve in the description of these quantities. Moreover the use of a basis with the correct asymptotic behavior is highly desirable in the descriptionof halo nuclei (ref.[16]). Moreover it has a computational cost similar to the harmonic oscillator with the same space truncation. However, the Coulomb-Sturm basis has an important shortcoming as pointed out in ref. [16]. Namely, the energies produced in many-body calculations are much higher than the corresponding ones obtained withthe harmonic oscillator basis. More recently, the natural orbit basis has been considered as a candidate for the description of halosystems (ref.[18]). Its main advantage is that it produces energies lower than the ones obtained with the h.o. basis. However this basis requires a preliminary shell model calculation. While for light systems this may pose no problem, it could be computationally demanding for heavier nuclei. Also, in the past some no-core shell-model calculations have been performed using a Wood-Saxon basis. However the parameters of the Wood-Saxon potential have to be varied in order to minimize the shell model results for the ground-state energy (ref.[19]). It is the purpose of this work to propose a new basis which seems to be free of the shortcoming of the Coulomb-Sturmian basis.If we desire a better basis than the harmonic oscillator, it should, with a comparable numerical effort, lead tolower energies in many-body calculations. The basis we propose is essentially the basis of ref. [17] properly corrected in orderto have the appropriate asymptotic behavior in coordinate space. Contrary to the Coulomb-Sturm basis it has its roots in the NN interaction.In what follows, it should be kept in mind that our primary purpose is the description of nuclei where the long range part of the wave function is important, although it could have an impact for nuclei in the valley of stability. The basic reasoning behind our basis is as follows. Consider the Hamiltonian in the center of mass system forA particle interacting with a potential V_ij,H=∑_i<jH_ij=∑_i<j((p⃗_i-p⃗_j)^2/2mA +V_ij)and let us diagonalize H_ij. In this work we take A=2 in order to obtain the single-particle basis to be used in many-body calculations. This may not be the best choice for ^4He since it isa very compact object.Let us consider the ground-state ofDeuterium, let us discard the D-part of the wave-function and consider onlythe S part. This wave function depends on the relative momentum of the neutron and protonand it is not localized in coordinate space. To achievelocalization,we consider the full wave function which contains alsothe wave-function (in an S state) of thecenter of mass of the system. The center of mass part can be used to localize the system. Arguing for simplicity in coordinate space (although we work in momentum space), the total wavefunctions depends on |r⃗_n-r⃗_p| and |r⃗_n+r⃗_p| hence on r_n,r_p andcos( θ_np) the angle between the position vectors of the neutron and the proton. We can analyze thecos( θ_np) dependence in therms of Legendre polynomials and relate these to the spherical harmonicsof the angular coordinates of the neutron and the proton. The net result is that the Deuterium wave function is rewritten as a linear combination of products of functionsF_l(r_n,r_p), which we will discuss in detail later and the spherical harmonics of θ_n and θ_p. We can diagonalize these functions F_l(r_n,r_p) on a lattice and obtain the total Deuterium wave function in terms of single-particle wave functions of the neutron and the proton. These single-particlewave functions are orthonormal, they arecomplete byconstruction, theyhave the proper asymptotic behavior at large distances (for a convenient choice of the center of mass wave function) and do not contain continuum states. Moreover they have a very useful additionalfeature. By controlling the space extent of the center of mass we can "squeeze" or "spread" in spacethe single-particle wave functions. This basis can be used in many-body calculations, although it has been constructed from the S-part of the Deuterium ground-state. Note that in principle we could construct a whole family of basis by weighting properly the kinetic energy term in H_ij. We call this new basis Localized Deuteron Basis (LDB). In the cases discussed in this paper this set of single-particle wave functions producesenergies of better or the same qualityobtained using an optimized h.o. basis, except the case of ^4He.We have not carried out the optimization suggested in ref. [15], namely the optimization of the radial wave functions for each single-particle angular momentum. The optimization of our set is performed only modifyingthe "tail"of all radial wave functions. We expect that the implementation of the optimization for every single-particle angular momentum will improve the energies even more. The use of radial wave functions other than the harmonic oscillator poses the additional problem of the evaluation of the matrix elements in the laboratory frame of the two-body interaction. This problem is solved in the h.o. basis by the Talmi-Moshinky brackets (ref.[20]). For basis other than the h.o. wave functions, the problem can be addressed using the vector brackets, discussed in refs. [21]-[24]. Here we use the expansion of our basis in the harmonic oscillator basis as done in ref. [15], using a rather large number of major oscillator shells. The many-body approach we use is the Hybrid-Multideterminant method (HMD) (refs.[25],[27]), whereby the nuclear wave function is expanded as a linear combination of a rather large number of Slater determinants. This paper is organized as follows. In section 2 we describe in detail the construction of the basis and some of its properties. In section 3 we compare the harmonic oscillator basis with the one we propose with a brief recap of the many-body method that we use in subsection (3.a).In section 4 we present some conclusions. §CHOICE OF THE SINGLE-PARTICLE BASIS .Let us start by constructing in momentum space the ground-state wave function ofDeuterium. As well known it has an L=0 component and an L=2 part. Let us isolatethe L=0 part and let us call the radial part u(k), where k⃗=(k⃗_1-k⃗_2)/2,k⃗_1 and k⃗_2 being the momenta of the nucleons. Let us discard completely theL=2 part of the deuterium wave function and let us construct the following wave-functionΨ(k⃗_1,k⃗_2)= u(k) Θ(K) (2.1) whereΘ(K) is for the time being an unspecified scalar wave function of the totalmomentum K⃗=k⃗_1+k⃗_2. The wave function in eq.(2.1) is normalized to 1.In coordinate space, the role of Θ (or betterof its Bessel-Fourier transform) is to localize the Deuterium. The right-hand side of eq.(2.1) depends on the relative orientation of k⃗_1,k⃗_2only through the cosine of the relative angle θ_12 between the momentak⃗_1,k⃗_2.We can analyze the r.h.s of eq.(2.1) using Legendre polynomials P_l(cos(θ_12))and writeΨ(k⃗_1,k⃗_2)= ∑_l=0^∞ f_l(k_1,k_2) P_l(cos(θ_12)) (2.2) wheref_l(k_1,k_2)= (l+1 2)∫_0^π dθ_12sin(θ_12)P_l(cos(θ_12)) Ψ(k⃗_1,k⃗_2) (2.3) Using the familiar addition theorem of the spherical harmonics Y_lm we obtain Ψ(k⃗_1,k⃗_2)= ∑_lm f_l(k_1,k_2)4π 2 l+1 Y^*_lm(k̂_1)Y_lm(k̂_2) (2.4) Using the normalization condition on the wave function Ψ(k⃗_1,k⃗_2) we obtain∫ k_1^2 k_2^2 dk_1 dk_2 ∑_lm f_l(k_1,k_2)^2(4π 2l+1 )^2=1 (2.5) Let us now discretize the lab. coordinates k_1,k_2 on a mesh of spacing Δ k and let us define the eigenvalue problem for the real symmetric matrix k_1 f_l(k_1,k_2) k_2≡ M_i,jwhere i,j refer to the position on the lattice of k_1 and k_2k_1 f_l(k_1,k_2) k_2≡ M_i,j=∑_n=0 v^(l)(i,n)^(l)_n v^(l)(j,n) (2.6)For later convenience, the index n which labels the eigenvalues takes the values 0,1,2,.. and has the roleof radial quantum number. We reorder the eigenvalues ^(l)_n,n=0,1,.. for a fixed l so that |^(l)_n| decrease with increasing n=0,1,2,... We obtainthe following expansion Ψ(k⃗_1,k⃗_2)=∑_l,nv^(l)(i,n) k_1^l_n v^(l)(j,n) k_24π 2l+1∑_m Y^*_lm(k̂_1)Y_lm(k̂_2) (2.7)Therefore the L=0 part of the Deuterium wave-functionhas been recast as an expansionof single-particle wave-functions in the lab. frame. Definingϕ_n,l,m(k)= v_l(i,n) k√(Δ k) Y_lm(k̂)≡ Q_n,l(k) kY_lm(k̂) (2.8) we obtain a set of radial single-particle wave functions in the lab. frame Q_n,l(k) k. These wave functions can be used as a single-particle basis to perform many-body calculations, much in the same way of the Coulomb-Sturmwave functions. The wave functions defined by eq.(2.8) are however more natural in nuclear many-body calculations. There are several points to be analyzed. First of all, we have obtained a discrete and complete set of single-particle wave functions in the lab. frame, directly linked to the underlining NN-interaction. Completeness stems from the unitarity of the eigenvectors v. Also, since we can localize nucleons with the appropriate Θ(K), this set does not contain the continuum, which would pose severe problems ofconvergence in many-body calculations. Moreover the normalization of the total wave-function Ψ gives∑_l,m,n [4 πΔ k ^(l)_n 2 l+1 ]^2=1 (2.9)Hence the quantitiesp_n,l= [4 πΔ k ^(l)_n 2 l+1]^2 (2.10) give the 'probability' of a nucleonbeing in the single-particle state characterized by the quantum numbers n,l,m. Since the series has to converge, we expect the p_n,l to decrease for large n,l. Hopefully the most important part of the deuterium wave function is expanded as a sum (in a shell model fashion) of "few" single-particle wave functions in the lab. frame. To fix the ideas, let us consider the recent NN interaction NNLO-opt recently introduced in ref. [4]. Let us extract and normalize the S part of the ground-state wave-function (in this work no renormalization step is taken on the NN interaction). The function Θ(K) is taken to be the Bessel-Fourier transform of a localizing center of mass wave-function. For the nuclear case in order to have an asymptotic behavior of the type exp^-μ r at large distances we considered the Bessel-Fourier transform of exp^-α R, R being the coordinate of the center of mass. That isΘ(K)= N /(K^2+(αħ c)^2)^2 (2.11)K being measured in MeV's and N being a normalization constant. As an example, consider α= 1 fm^-1. In fig.1 we plot the logarithm of the probabilities p(n,l) for several l values. As far as the wave-function is concerned few values of n,l contribute to the expansion of eq.(2.7). An alternative way to illustrate thepattern of convergence is illustrated in table 1, where we show the ∑_2n+l≤ Np(n,l)(2l+1) as a function of N.A familiar nodal structure emerges by plotting the single-particle radial wave functions for several l-values. In fig. 2 we show the first few radial wave-functions for l=0. The normalization implied by eq. (2.8) is ∫ dk Q_n,l(k)Q_n',l(k)=δ_n,n'. In fig.3 and in fig. 4, we plot the radial wave functions for l=1 and l=6 respectively. The nodal structure is clearly visible provided the label n is associated to thefamiliar harmonic oscillator radial quantum number.It should be stressed however that we do not have an oscillation theorem as for the h.o. radial wave-functions. The nodal structure of the radial wave-functions may depend on the original NN interaction. As a rule of the thumb, the "harder' the NN potential is at larger momentum transfer, the more distortedthe nodal structure can be. These radial wave functions have the proper asymptotic behavior at large r in coordinate space. One way to modify this asymptotic behavior in coordinate space is to use, instead of eq.(2.11), the Bessel-Fourier transform of a Gaussian for the center of mass. We can modify the space extent of the radial wave-functions by modifying the parameter α in eq.(2.11). In table 2 we show the ∑_2n+l≤ Np(n,l)(2l+1) as a function of N for =2 fm^-1.In this case the convergence to 1 for increasing N =max(2n+l) is slower. This isnot surprising since the Deuterium wave-function has a long tail in coordinate space and a "compressed" basis is less suited in an expansion of the Deuterium wave-function. In many-body calculationsis a variational parameter.In fig. 5 we show for comparison the radial Q_00(k) evaluated at two different values of α. The value of αhas an analogous role of ħ in the harmonic oscillator radial wave functions. So far we have discussed a single-particle basis in the lab. frame derived from the "bare" NN interaction. One can ask whether these considerations are modifiedif we soften the NN interaction with some renormalization procedure. We considered only the case of momentum cutoff to ħ c k_max=400 MeV in the frame of the center ofmass(V_low k) for α=1 fm^-1. The construction of the radial wave-functions in the lab. frame can be repeated as before. We found that the Q_nl(k) show only minor differences especially for low values of n,l. Only at large values of k≥ k_max and for large n or l in the lab. frame the Q_nl show appreciable differences. Also the accumulated probabilities are very close to the ones of table 1.Major differences might be found if the "bare" NN interaction is replaced by a much stronger one at large momentum transfer. In this work no renormalization steps have been taken. We use only the "bare" interaction. The next step is the evaluation of the matrix elements of the interaction orof the two-body matrix elements of the full Hamiltonian, in the new basis. This can be accomplished by first evaluating the matrix elements of the interaction (or of the the two-body matrix elements of the Hamiltonian) in a very large harmonic oscillators basis in the lab.frame and then expandingthe matrix elements of the same operators in the new basis in terms of h.o. wave functions. See refs. [15] and [24] for a detailed discussion. By coupling the radial wave functions with the spin and angular part we can define the single particle basis as |n,l,j,m>. Let |n , l,j,m> the h.o. counterpart and let us call P_n , l k the corresponding h.o. radial wave functions.If a,b,c,d denote the set of quantum numbers (n_a l_a j_a),(n_b l_b j_b),(n_c l_c j_c),(n_d l_d j_d) in the new basis and a,b,c,d the corresponding h.o. quantum numbers the transformation law for the angular momentum coupled two-body matrix elements of the interaction is<a,b J |V| c, d J>=∑_a,b,c,d<a|a><b|b> <c|c> <d|d> < ab J |V| cd J> (2.12) The overlaps <a|a>,... cannot changethe quantum numbers l,j and they reduce to the radial integralsC^(l)_n,n=∫_0^∞ dk Q_n,l(k)P_n,l(k)(2.13)The degree of completeness of the selected h.o. space is assessed by the accumulated probabilitiesp^(l)_n,n=∑_ν≤n C^(l)2_n,ν(2.14)As an example let us consider ħ=14 MeV and α=1 fm^-1. In table 3 we show theamplitudes of eq.(2.12) as well as the accumulated probabilities for several values of l. The radial wave functions Q_n,l/k were obtained from the "bare" NNLO-opt interaction The expansion converges very fast for small n,l and it is slower for large n,l. It is however faster than the expansion of the Coulomb-Sturm radial wave functions in terms ofa harmonic oscillator basis (compare table 3 with fig. 3 of ref.[15] for the Coulomb-Sturm case). Also, note that max(2n+l) is the truncation parameter in many-body calculations. The convergence depends on the adopted values of α and ħ. No optimization has been made in this example. In the actual calculations discussed in the next section an optimization has been performed. At this stage it is worth to notice that eq.(2.12) for a selected h.o.subspaceworks better for the interaction than for the full Hamiltonian. That is the kinetic energy terms might be poorly approximated in a "small" h.o. subspace. As done in refs. [12],[14], we use eq.(2.12) only for the interaction and evaluate all remaining terms of the Hamiltonian directly in the lab. frame. In the next section we shall optimize both values of ħ and α and shall study^4He ^6He and ^6Li. We shall show explicitly that the useof the basis described in this section leads to a decrease of the ground-state energiescompared to the ones obtained using the harmonic oscillator basis with the optimal valuesof ħ except for ^4He. We stress that we use only non-renormalized interactions. Hence ħ is a variational parameter. The comparison will be made with calculations that have the same computational burden. §COMPARISON BETWEEN THE H.O. AND THE NEW BASIS. 3a. A brief recap of the HMD method. Strictly speaking, the HMD method is a variational method based on the assumptionthat the nuclear wave function can be written as a linear combination ofa number of Slater determinants (SD) with the option of projecting to good quantum numbers.These Slater determinants are of generic type and theyare not orthogonal to each other, much in the same way of the Generator Coordinate Method.No assumption is made about the relevant degrees of freedom. The Slater determinantsas well as the coefficients of the linear combination are determined only by variational requirements. The HMD method can take any input for the Hamiltonian which we schematically write as= 12∑ _i,j,k,l H_ijkl_i_j a_la_k (3.1) The two-body matrix elements contain the "bare" two-body interaction, the intrinsic kinetic energyand the center of mass term β(_cm-3/2 ħ), where β is a coefficientand _cm is the harmonic oscillator Hamiltonian for the center of mass, In eq.(3.1) i,j,k,l are the single-particle quantum numbers (n_i,l_i,j_i,m_i),... for both neutrons and protons.We describe eigenstatesas a linear superposition of Slater determinants of the most generic type| ψ>= ∑_S=1^N_D g_S|U_S> (3.2) whereis a projector to good quantum numbers (e.g. good angular momentum and parity)N_D is the number of Slater determinants |U_S> expressed as|U_S> = _1(S)_2(S)... _A(S) |0> (3.3) the generalized creation operators _α(S) for =1,2,..,A are a linear combinationof the creation operators _i_(S)=∑_i=1^N_sU_i,(S)_i =1,...A (3.4) The complex coefficients U_i,(S) represent the single-particle wave-function of theparticle =1,2,..,A. We do not impose any symmetry on the Slater determinants (axial or other)since the U_i,are variational parameters.These complex coefficients are obtained by minimizing the energy expectation valuesE[U]=<ψ | |ψ><ψ |ψ>(3.5) The coefficients g_S are obtained by solving the generalized eigenvalue problem∑_S <U_S' | | U_S> g_S = E ∑_S <U_S'|| U_S> g_S (3.6) for the lowest eigenvalue E. We consider a quasi-Newtonian minimization method. It is a generalization ofthe Broyden-Fletcher-Goldfarb-Shanno (BFGS) method (cf. for example ref.[26] and references in there).The variant we use is described in detail in ref. [27].The method starts with a small number of SD's (typically 1-5) and determines theSD'sas well as the coefficients of the linear combination by minimizing the energy.Each SD is optimized individually. The process is repeated several timesuntil all SD's have been optimized N_T times. The number N_T is such thatan exit criterion is satisfied. The exit criterion is met when the energy changesless than a specified amount (typically 5 KeV) between the N_T-1 and N_T optimization.After the exit criterion is met, the number of SD's is increased by optimizing the lastincluded SD. When the total number of SD's is large enough we repeat the optimization ofall SD's one at a time. Typically, this waywe collect between 100 and 200 SD's.Typically, the full optimization is performed when the number of Slater determinantsreaches the numbers 5,10,15,25,35,50,70,100,150,200,..In eq.(3.1) if we select the harmonic oscillator basis, there are two possible inputs for theHamiltonian matrix.In the lab. frame the single-particle states satisfya)2 n+l ≤ N_2max/2(3.7a) where N_2max is the largest total quantum number in the intrinsic frame. Orb)2 n_1+l_1+ 2 n_2+l_2 ≤ N_2max(3.7b)We stress that these are two possible truncations of the original "bare" two-body Hamiltonian. Eq.(3.7a) is referred as type (a) truncation and eq.(3.7b) astype (b) of the original Hamiltonian. For larger and larger N_2max both approaches should hopefully converge to the same results. The type (a) may look a bit out of the ordinary and type (b) may seem preferable.However we can argue as follows. Instead of harmonic oscillator single-particle wave functions we can takeother single-particle wave functions, for example the Coulomb-Sturmwave functions or the ones considered in this work.We cannot give to the radial quantum number the same meaning it has in the case of theharmonic oscillator and actually condition eq.(3.7b) would seem a bit unjustified as there is no obvious reason why 2n+l for each particle, entering in the evaluation of the two-body matrix elements,should be related to each other. Type (a) truncation is more natural for single-particle basis other than harmonic oscillators. There is a further argument that one can offer. Consider an interaction either bare or softened with similarity renormalization group (SRG) methods, V(q,q') with q,q' being the relative momentum transfer between particles, and assume that we would like to evaluate directly in the lab. frame the two-body matrix elements of V using the vector brackets; in such a case truncation (3.7b) would seem a bit unnatural and it would seem more reasonable to adopt the following criterion: all values of (n,l) that contribute the most to the energies should be included. In this work, when using the h.o. representation, we select sometimes type(a) and sometimes type (b). When using the representation discussed in the previous section we use type (b) only in order to obtain the two-body matrix elements of the interaction (cf. eq. (2.12)). More explicitly,using N_2max=16÷22 we firstobtain the two-body matrix elements of the interaction of type (b)in the h.o. representation, then, using the expansion of the new basis in terms of h.o. single-particleradial wave functions, we obtain the matrix elements in the new basis with the restriction2n+l≤ e_max.That is, using a basis other than the h.o.we use always the truncation of type type (a) where N_2max/2 is replaced by some maximumvalue e_max of 2n+l. In the new basis 2n+l does not have the meaning of major shell and it is simplya number that allows us to compare the results obtained using the h.o. basis of type (a) with thecorresponding ones obtained in the new basis. Differently stated, we compare truncation of type (a)for the h.o. representation with the analogous truncation in the new basis. Note that the truncation of the Hilbert space used in this work is very different from the usual oneused in shell model calculations (refs. [15],[16]). The N_max truncation used in shell model calculationsrefers to many-body configurations, i.e. total maximum number of oscillator quanta minus the minimal one.3b. Some numerical results. In all cases treated in this subsection we consider up to l=5 for the single-particle orbital angular momentum.The Slater determinants were determined by minimizing the energies using a projector to good z-projection of the angular momentum and parity J_z^πas explained in previous subsection. All energies depend on e_max=max(2n+l) and the total number of employed Slater determinants N_D. We optimize ħ for the calculations using the h.o. representation and α in the new LDB representation. The optimization is performed with few Slater determinants and mostly the value of ħ or α is kept for the rest of the calculations. The results are shown in table 4.Note that the use of a basis different from the h.o. does not introduce strong center of mass excitations. Actually, the residual <(H_cm-3/2 ħ)> decreases with larger e_max. Also, most of the times, we did not optimize the value of ħ in the center of mass Hamiltonian. We have used most of the times the value of ħ optimized in the h.o. representation. Apart from the case of ^4He the results are encouraging. The computational cost of the two representation is roughly the sameand the LDB does not require additional many-body calculations as in the natural orbits approach. It is natural to ask whether the decrease in the energies remains as we consider heavier nuclei. We made a test with ^40Ca for e_max=5 (6 major shells), up to 35 Slater determinants using both the h.o. representation and the LDB introduced in this work.In fig. 6 we compare the energies as a function of N_D obtained using the h.o. and the LDB. The result is quite encouraging. With 35 Slater determinants, LDB lowers the energies by about 38 MeV, compared to the standard h.o. representation. We stress that convergence is not reached and a much larger number of N_D is needed as well as a larger number of major shells. As such, fig. 6 should be regarded as highly preliminary. Note, also, thatħ has been optimized to the h.o. representation and kept the same in the center of mass Hamiltonian in the LDB. § CONCLUSIONS.In this work we have presented a new single-particle basis for many-body calculations. This basis is extracted from a two-body problem, by adding a localizing wave function in the center of mass coordinate to the intrinsic two-body eigenstate. The full wave function is analyzed in terms of Legendre polynomials and rewritten as sum of products of single-particle wave functions. Essentially this basis is constructed by diagonalization ofa two-body wave function rather than a single-particle Hamiltonian. For ^6Li and ^6He it gives lower values of the energy when compared with the values obtained using the harmonic oscillator representation. Moreover, in a preliminary study, it seems to be ideal for medium mass systems.1D. R. Entem and R. Machleidt.Phys. Rev. C 68, 041001 (2003). 2 R. Machleidt and D.R. Entem, Physics Reports 503 (2011) 1. 3 E. Epelbaum, H.-W. Hammer, Ulf-G. Meiner. Rev. of Mod. Phys. 81 (2009)1773. 4 A. Ekstrom et al. Phys. Rev. Lett. 110, 192502(2013). 5A. Ekstrom et al. Phys. Rev. C 91, 051301(2015).6P. Navratil, J. P. Vary, and B. R. Barrett, Phys. Rev. Lett. 84, 5728 (2000); Phys. Rev. C 62, 054311 (2000). 7 B.R.Barrett, P.Navrtil, J.P. Vary. Prog. in Part. and Nucl. Phys. 69 (2013) 131,and refs. in there 8P.Navratil, S.Quaglioni, G.Hupin, C.Romero-Redondo and A.Calci. Physica Scripta,91 (2016) 053002 and refs. in there. 9 D. J. Dean and M. Hjorth-Jensen Phys. Rev. C 69, 054320(2004). 10G. Hagen, T. Papenbrock, D. J. Dean, and M. Hjorth-Jensen Phys. Rev. C 82, 034330(2010). 11G Hagen , T Papenbrock, M Hjorth-Jensen and D J Dean Rep. Prog. Phys. 77 (2014) 096302. 12H. Hergert, S.K. Bogner, T.D. Morris, A. Schwenk,andK. Tsukiyama Physics Reports 621 (2016) 165, and refs. in there. 13 W.H.Dickhoff and C.Barbieri. Prog. Part. Nucl. Phys. 52 377 (2004).V.Soma', C.Barbieri, and T.Duguet. Phys. Rev. C 87, 011303(R) (2013). 14 R.J.Furnstahl,G.Hagen and T.Papenbrock. Phys. Rev. C 86, 031301 (2012).S.N.More, A.Ekstrom, R.J.Furnstahl, G.Hagen and T.Papenbrock. Phys. Rev. C 87, 044326 (2013).R.J.Furnstahl, S.N.More and T.Papenbrock.Phys. Rev. C 89, 044301 (2014). 15 M.A. Caprio, P. Maris and J. P. Vary. Phys. Rev. C 86, 034312 (2012). 16M.A. Caprio, P. Maris and J. P. Vary. Phys. Rev. C 90, 034305 (2014). 17G.Puddu. Acta Physica Polonica B 38(2007)3237 18C.Constantinou, M.A Caprio, J.P.Vary and P.Maris.arXiv:1605.04976 [nucl-th]. 19 A. G. Negoita, PhD thesis, Iowa State University, 2010, unpublished. http://gradworks.umi.com/3418277.pdf. 20M. Moshinsky and Y. F. Smirnov, The Harmonic Oscillator in Modern Physics (Harwood Academic, Amsterdam, 1996). 21R. Balian and E. Brezin, Nuovo Cimento 61B, 403 (1969). 22C. W. Wong and D. M. Clement, Nucl. Phys. A183, 210 (1972). 23C.L.Kung, T.T.S.Kuo, K.F.Ratcliff. Phys. Rev. C 19, 1063(1979). 24G. Hagen, M. Hjorth-Jensen, and N. Michel. Phys. Rev. C 73, 064307 (2006) 25G.Puddu J. Phys. G: Nucl. Part. Phys. 32 (2006) 321.G.Puddu.Eur. Phys. J. A 31 (2), pp. 163 (2007).G. Puddu. Eur. Phys. J. A 45, 233(2010) 26 W. Lederman ed. Handbook of Applicable Mathematics. Vol. III,Numerical Methods, chapter 11. John Wiley and Sons, New York 1981. 27 G.Puddu. Eur. Phys. J. A 42, 281(2009). | http://arxiv.org/abs/1707.08765v2 | {
"authors": [
"Giovanni Puddu"
],
"categories": [
"nucl-th"
],
"primary_category": "nucl-th",
"published": "20170727075825",
"title": "A new single-particle basis for nuclear many-body calculations"
} |
equal contributionLASSP, Department of Physics, Cornell University, Ithaca, NY 14853, USA equal contributionLASSP, Department of Physics, Cornell University, Ithaca, NY 14853, USABrockhouse Inst. for Materials Research, McMaster University, Hamilton, ON, Canada Department of Physics, McMaster University, Hamilton, Ontario, L8S 4M1, CanadaBrockhouse Inst. for Materials Research, McMaster University, Hamilton, ON, CanadaBrockhouse Inst. for Materials Research, McMaster University, Hamilton, ON, Canada Department of Physics, McMaster University, Hamilton, Ontario, L8S 4M1, Canada Canadian Institute for Advanced Research, Toronto, Ontario, M5G 1Z8, CanadaLASSP, Department of Physics, Cornell University, Ithaca, NY 14853, USA Current address: Honeywell International Inc., Golden Valley, MN 55422, USALASSP, Department of Physics, Cornell University, Ithaca, NY 14853, USA CMPMS Department, Brookhaven National Laboratory, Upton, NY 11973, USA Tyndall National Institute, University College Cork, Cork T12R5C, Ireland [email protected], Department of Physics, Cornell University, Ithaca, NY 14853, USA Despite a well-ordered pyrochlore crystal structure and strong magnetic interactions between the Dy^3+ or Ho^3+ ions, no long range magnetic order has been detected in the pyrochlore titanatesHo_2Ti_2O_7 and Dy_2Ti_2O_7. To explore the actual magnetic phase formed by cooling these materials, we measure their magnetization dynamics using toroidal, boundary-free magnetization transport techniques. We find that the dynamical magnetic susceptibility of both compounds has the same distinctive phenomenology, that is indistinguishable in form from that of the dielectric permittivity of dipolar glass-forming liquids. Moreover, Ho_2Ti_2O_7 and Dy_2Ti_2O_7 both exhibit microscopic magnetic relaxation times that increase along the super-Arrhenius trajectories analogous to those observed in glass-forming dipolar liquids. Thus, upon cooling below about 2K, Dy_2Ti_2O_7 and Ho_2Ti_2O_7 both appear to enter the same magnetic state exhibiting the characteristics of a glass-forming spin-liquid.75.50.Lk, 75.47.LxCommon Glass-Forming Spin-Liquid State in the Pyrochlore Magnets Dy_2Ti_2O_7 and Ho_2Ti_2O_7 Anna Eyal December 30, 2023 ============================================================================================== § INTRODUCTIONIn the pyrochlore lanthanide-oxides with chemical formula A_2B_2O_7, the magnetic rare earth A ions are located at corner sharing tetrahedral sites as shown in Fig. <ref>a <cit.>. These materials support a multitude of exotic magnetic states <cit.>, such as spin-ice<cit.>, spin-slush <cit.>, and various candidates for quantum spin-liquids <cit.>. Dy_2Ti_2O_7 and Ho_2Ti_2O_7 have attracted much interest since the discovery of an entropy deficit in both compounds <cit.>, along with a similarity to the thermodynamic properties of water ice. In both Dy_2Ti_2O_7 and Ho_2Ti_2O_7 the large rare-earth magnetic moments ofμ∼10μ_B <cit.>, at the corners of the corner sharing tetrahedral structure, are subject to strong crystal field interactions <cit.> so that, in the single-ion ground-state, spins are only allowed to point towards (away) from the centers of the tetrahedra that share it (see Fig. <ref>a) <cit.>. Under these circumstances, long-range magnetic dipolar interactions are significant when compared to the nearest-neighbor magnetic exchange coupling. The resulting interaction favors a state in which for each tetrahedron two spins point in and two out, that, by analogy to water ice, was dubbed "spin-ice"<cit.>. One might expect such a constraint to result in an ordered magnetic state that is unique <cit.>, but no such state has ever been observed in these materials at zero magnetic field <cit.>.For both Ho_2Ti_2O_7 and Dy_2Ti_2O_7 the strength of the nearest neighbor dipolar interaction is D_nn≈2.35K, while the nearest neighbor exchange interactions are J_nn≈-0.52K for Ho_2Ti_2O_7 and J_nn≈-1.24K for Dy_2Ti_2O_7 <cit.>. Raman spectroscopy reveals that the phonon spectra of these two materials differ only slightly and that the crystal field parameters are very similar <cit.>. Dysprosium ions have 7 f-electrons with J=15/2 for the single ion angular momenta in the ground state, with the crystal field levels all Kramers doublets; holmium ions, on the other hand, have 8 f-electrons with an integer J=8, so its crystal field spectrum has both singlet and doublet energy levels <cit.>. The magnetic heat capacity of both materials has a broad peak at low temperatures (1.2K for Dy_2Ti_2O_7 and 1.9K for Ho_2Ti_2O_7) <cit.>. However, Ho_2Ti_2O_7 has another peak in heat capacity at much lower temperatures, which is believed to be due to its active nuclear magnetism <cit.>. Interpreting the magnetization dynamics of these two systems has proven challenging. Nearest neighbor spin-ice-based models for spin dynamics predict inelastic neutron scattering intensity patterns where ”pinch points” are significantly smeared out compared to the experiments <cit.>, and exchange interactions up to third nearest neighbor must be included in such models to replicate the full complexity of these neutron scattering results <cit.>. Models for the magnetic susceptibility χ(ω,T) based on dipolar-spin-ice <cit.>, or on the dynamics of a dilute gas of mobile monopoles <cit.> representing transitions from two-in-two-out to three-in-one-out configurations<cit.> (see Fig. <ref> b), show significant deviations from the experimental data <cit.>. Moreover, predictions for the temperature dependence of microscopic magnetic relaxation times τ considerably underestimate the divergence of τ at low temperatures <cit.>. Finally, in the absence of magnetic fields, no magnetic order has been detected in either compound <cit.>. A recent proposal <cit.> that the magnetic state of Dy_2Ti_2O_7 is the magnetic analog of the diverging viscosity state found in glass-forming dipolar liquids <cit.> provides a different perspective. Classical glass-forming liquids exhibit universally a super-Arrhenius divergence of microscopic dipolar relaxation times τ_0(T) of the Vogel-Tammann-Fulcher (VTF) form τ_0(T)=Aexp(DT_0/(T-T_0 )) <cit.>, a dielectric function ϵ (ω,T) of the Havriliak-Negami (HN) form ϵ (ω,T)= ϵ_∞+ϵ_0/(1+(iωτ_HN )^α )^γ<cit.>, and a related time-domain relaxation described by the Kohlrausch-Williams-Watts (KWW) form <cit.> ϵ (t) = ϵ_0 exp [-(t/τ_KWW)^β ]. Observation of this combined VTF/HN/KWW phenomenology provides a strong clear identifier of a supercooled glass-forming dipolar fluid <cit.>. Dy_2Ti_2O_7 was found to exhibit a precise HN form for its magnetic susceptibility χ (ω, T), a general KWW form for the magnetic relaxation, and diverging microscopic magnetic relaxation rates on a VTF trajectory, implying that it is, by analogy, a glass-forming magnetic liquid <cit.>. Here we explore if magnetic fluids with such a phenomenology could be more general in the lanthanide pyrochlore magnetic materials. Even if glass-forming spin-liquid phenomenology were common to such materials, the microscopic parameters are still likely to be specific to each compound. In glass-forming dipolar liquids, the measure of the correlated dipole dynamics is called the fragility, D, <cit.> and it characterizes the degree of spatial heterogeneity. D is an indicator of the spread of microscopic relaxation times over different close-by regions in the liquid. The smaller the value of D, the more fragile the liquid and the more spatially heterogeneous its dynamics <cit.>. By analogy, a more fragile glass-forming spin-liquid would mean an enhancement of the super-Arrhenius behavior of its magnetic relaxation times upon cooling. Such a situation could be caused by less efficient tunneling between spin configurations, due perhaps to differences in monopole creation energies and hopping rates. For the pyrochlore magnets discussed, the chemical potential for monopole-pair generation is dependent on nearest neighbor coupling, J_eff <cit.>, which is 1.1K for Dy_2Ti_2O_7 and 1.8K for Ho_2Ti_2O_7. Moreover, the theoretical rate of tunneling of the monopole excitations depends on the off-diagonal components of the dipolar interactions of neighboring spin <cit.>. The reason for this is the strong Ising-like behavior of the magnetic ions <cit.>, with the energy barriers to the first excited crystal field state being Δ∼ 240K for Ho_2Ti_2O_7 and ∼ 380K for Dy_2Ti_2O_7 <cit.>. Additionally, the fact that the effective energy scale for spin-flip dynamics is on the order of several J_eff rather than Δ implies that spin flips occur by quantum tunneling <cit.>. Since the transverse field effects in Ho_2Ti_2O_7 are more pronounced than in Dy_2Ti_2O_7, resulting in a more effective quantum tunneling at low magnetic fields <cit.>, monopole hopping in Ho_2Ti_2O_7 is expected theoretically to be more efficient. In that case, one might anticipate a less fragile glass-forming spin-liquid in Ho_2Ti_2O_7 as compared to Dy_2Ti_2O_7. § EXPERIMENTAL SETUPTo explore the relationship between the magnetization dynamics of Ho_2Ti_2O_7 and Dy_2Ti_2O_7 we used a boundary-free arrangement to measure the AC susceptibility and time dependent magnetization relaxation characteristics of the two materials at T≤2 K. Single crystals of these materials were grown as boules in O_2 gas under 2 atm pressure in an optical floating zone furnace <cit.>, and were subsequently cut into disks of diameter ∼ 6mm and thickness ∼ 1mm (see Fig. <ref> d). For the boundary-free magnetization measurements, holes of ∼ 2.5mm diameter were drilled through the center of disk shaped samples of both Ho_2Ti_2O_7 and Dy_2Ti_2O_7 crystals. A superconducting toroidal solenoid (STS) was then made by winding a 0.09mm diameter NbTi wire around the toroidal samples (Fig. <ref> c,d). Using a toroidal geometry for both the samples and the magnetization sensors means that the superconducting toroidal solenoid can be used to both drive magnetization flow azimuthally and to simultaneously and directly determine dM/dt throughout. More importantly, it removes any boundaries in the direction of the magnetization transport (Fig. <ref> c). The coil EMF due to changes in both the applied azimuthal field H(t) and sample magnetization M(t,T) is given by V_total (t,T)= -μ_0 NA(dH(t)/dt+dM(t,T)/dt) where N is the number of turns in the solenoid and A is the effective cross-sectional area of the solenoid. Thus, the EMF due to magnetization dynamics in the sample isV(t,T)= -μ_0 NA dM(t,T)/dt For an applied AC fieldV(ω,T)= -iμ_0 NA ωM(ω,T) The definition of the magnetic susceptibility is M(ω,T)=χ(ω,T)H(ω) In a solenoid H_0=nI, where n is the number of turns per unit length, so the EMF is given by V=V_x+iV_y=-iμ_0 NAωχ(ω,T)H_0=-i Iω L [χ' (ω,T)-iχ” (ω,T)] where L is the geometric inductance of the STS pickup coil. Currents of up to 200 mA can be applied, using low temperature Nb crimp joints, to the STS coils, yielding azimuthal applied fields of magnitude up to |B|=2.5 mT or |H|=2200 A/m. Such fields are orders of magnitude smaller than those required to flip spins in these compounds. In addition, this azimuthal field covers a wide range of crystallographic planes in its path. The AC susceptibility of the compounds measured was determined typically by applying ∼10 mA currents in a frequency range of 10 Hz – 100 kHz using a 4-probe impedance measurement of the STS. The inductance, L, of the STS was measured at T=50 mK, where neither of the materials show any magnetic activity in the frequency range measured, and then used in equation <ref> to calculate the susceptibility χ (ω,T) data from the voltage readings. § RESULTSDuring transient data acquisition, the voltage over the STS was measured every 20 ms throughout the following excitation protocol: (a) apply magnetic field in a clockwise direction by turning on a current I=50 mA in the STS, (b) set the field to zero by turning off the current, (c) apply a magnetic field in the counter clockwise direction by turning on a current I=-50 mA in the opposite direction and (d) again zero the field. This protocol was repeated 150 times per temperature for each material at each temperature, and the results were averaged to improve data quality and fitting. For both materials, no difference in relaxation characteristics was observed when the magnetic field was turned on or off, as well as when the magnetic field was applied in one azimuthal direction or the opposite. At long times, after the initial sharp change in the field, the EMF that was generated in the STS decayed to zero, indicating that J=dM/dt <cit.> always decays to zero, despite the fact there are no terminating boundaries in our geometry. Figs. <ref> a,b depict the measured magnetization relaxation characteristics dM/dt of Ho_2Ti_2O_7 and Dy_2Ti_2O_7 respectively in the temperature range 0.6 K - 0.95 K. The plots show the measured voltage induced across the STS by the magnetization dynamics of the sample versus time after the application of a DC field. These data sets at each temperature were fitted by a KWW type stretched exponential decay V(t)=V_0 exp(-(t/τ)^β), with fits shown in Fig. <ref> a,b as fine colored curves. Although a simple exponential decay cannot fit any of these data at any temperature, the KWW form provides an excellent fit for all. The insets of Fig.<ref> a,b show how the stretching parameter, β, is different from unity over the temperature range of the DC measurements for both compounds. More importantly, Figs. <ref> a,b reveal the universal applicability of the KWW form of both samples for the whole temperature range. Here, the normalized EMF V(t)/V_0 is plotted against the modified time parameter x=(t/τ)^βfor each temperature, with the result that all the magnetization transient data from both materials collapse onto a single line with unit slope. This remarkable agreement of magnetization decay dynamics of both Ho_2Ti_2O_7 and Dy_2Ti_2O_7 with a KWW form implies that both these systems are in the same state, a glass-forming spin-liquid.The AC magnetic susceptibility of the toroidal samples of Ho_2Ti_2O_7and Dy_2Ti_2O_7was measured in the temperature range 0.9 K-2 K. For the compounds in this paper, we observed that below 0.5 K the EMF generated in the STS, in the frequency range reported, showed virtually no temperature dependence down to 50 mK, the lowest temperature at which AC measurements were attempted. The AC voltage measured at the lowest temperature was subtracted from the measurement at the temperatures of interest to deduce the susceptibility χ(ω,T)=χ'(ω,T)-iχ”(ω,T). Figs. <ref> c,d present the measured real (χ ') and imaginary (χ”) parts of the susceptibilities for the samples measured versus frequency, in the range 10-10^5 Hz for Ho_2Ti_2O_7 and 10-10^4 Hz for Dy_2Ti_2O_7. The data sets taken at different temperatures are labeled by a color/symbol code as indicated. Models of AC susceptibility that assume a single relaxation time of the Debye form χ'-iχ”=χ_0/([1+(iωτ)]), for example those of free monopole motion <cit.>, are not compatible with the measured χ(ω,T) for either Ho_2Ti_2O_7 or Dy_2Ti_2O_7 at any frequency or temperature within these ranges (see Fig. <ref>). By contrast, a Havriliak-Negami form modifies the simple Debye susceptibility with two exponents, α and γ, and corresponds to a system where there is a distribution of relaxation times χ'-iχ”=χ_0/[1+(iωτ)^α ]^γ+χ_∞ Figs. <ref> c,d depict our measured data for both Ho_2Ti_2O_7 and Dy_2Ti_2O_7 respectively. The top panels show the real part of the measured magnetic susceptibility versus frequency, as calculated from our measured voltage, using equation <ref>, and the bottom panels present the imaginary part of the susceptibility versus frequency. The different colors/symbols show data from different temperatures in the range 0.9K to 2K. For Dy_2Ti_2O_7, both exponents of the HN fit (equation <ref>) deviate from unity for the majority of the temperature range (inset of Fig. <ref> d), whereas for Ho_2Ti_2O_7 γ is around unity for most temperatures within error. Overall, the susceptibility for both materials shows a very good global agreement with the HN form for all temperatures and frequencies measured, as demonstrated by the fine lines in Fig. <ref> c,d. Fig. <ref> c,d show the collapse of these dynamical susceptibility χ(ω,T) data for all temperatures and both materials onto the single HN form <cit.> as indicated by the fine solid curves. The horizontal axis in the figure is the frequency, scaled by the HN parameters, and the vertical axes are the real and imaginary parts of the scaled HN susceptibility G(γ,χ_0,χ) (a full mathematical derivation can be found in Ref.<cit.>).G(γ,χ_0,χ)= (χ'^2+χ”^2/χ_0^2)^1/2γ(cos(1/γarctanχ”/χ')-i sin(1/γarctanχ”/χ'))The scaling parameters, which are the fit parameters of the HN form for each temperature, are plotted in the insets of the figure. The quality of fits, while comprehensively good versus ω and T for both materials (Fig. <ref> c,d), is obviously slightly different between Ho_2Ti_2O_7 and Dy_2Ti_2O_7 (Fig. <ref> c,d). This may not be surprising since the frequency-width and frequency-range of the data from the Ho_2Ti_2O_7 measurements is at least two orders of magnitude wider than that for Dy_2Ti_2O_7 (see Fig. <ref> c,d). In any case, this observation of a universal Havriliak-Negami form for all the χ(ω,T) susceptibility data (Fig. <ref> c,d) constitutes a second robust indication that both these materials are homologous glass-forming spin-liquids.Finally, to explore the microscopic magnetic relaxation dynamics of these systems, we need a form to relate the relaxation times obtained from the time-domain measurements to those from the frequency-domain. Numerical studies have linked the exponents and relaxation time parameters of the two forms <cit.>, which can be used for a unified analysis of our data. Using the values in Table I of Ref. <cit.> we can generate values for τ_HN/τ_KWW, enabling a conversion of the relaxation times from the time-domain measurements to the frequency-domain <cit.>.Fig. <ref> a depicts the combined time- and frequency-domain relaxation-time data for Ho_2Ti_2O_7 (T≥0.8 K) and Dy_2Ti_2O_7 respectively (the relaxation-times obtained from the time-domain measurement, τ_KWW, were converted to τ_HN by the procedure described above) with the horizontal axis being the inverse temperature. Obviously, the relaxation-time data for both materials diverge on a trajectory that is faster than Arrhenius, which would produce a straight line in Fig. <ref> a. Indeed, many groups have previously reported relaxation-time data showing the general behavior of a divergence that is faster than Arrhenius <cit.>, and in particular Ho_2Ti_2O_7 showing a stronger divergence.However, when the temperature dependence of the relaxation-time is fit to a VTF form,τ(T)=AexpDT_0/T-T_0, as shown in Fig. <ref> b, our findings indicate that both Dy_2Ti_2O_7 and Ho_2Ti_2O_7 exhibit non-Arrhenius slowing. This form yields A≈8×10^-10s, a fragility parameter D≈60 and a VTF temperature T_0≈191 mK for Ho_2Ti_2O_7, and A≈1.4×10^-4s, D≈14 and T_0≈257mK for Dy_2Ti_2O_7, signifying that Dy_2Ti_2O_7 is a more fragile spin-liquid. These specific parameters, resulting from the best fit to the VTF form give standard errors of 1 to 2 percent, and R^2 of 0.995 and 0.998 respectively. Systematic errors arising from the fit procedure can be up to several tens of percent, but those have no impact on the resulting function, as can be clearly seen in Fig. <ref>b and by the high R^2. All of the data presented in figures <ref>, <ref> and <ref> are newly acquired for the purposed of comparison between the two compounds: the parameters of Dy_2Ti_2O_7 from this work agree well with previous work <cit.>. The fragility parameters and high-temperature relaxation-times depend strongly on the materials studied, but their T_0, the lowest temperature at which both materials may be expected, by analogy with glass-forming fluids, to enter a magnetic glass phase, are close in value. Overall, we find in the common VTF form for τ(T) (see Fig. <ref>b), a clear indication that both Dy_2Ti_2O_7 and Ho_2Ti_2O_7 are glass-forming spin-liquids. § DISCUSSION Previous comparison studies of AC susceptibility of these two compounds <cit.> identified differences in the spread of their microscopic relaxation times. Specifically, the broadness of the absorption spectra, inferred from the width of the imaginary part of the magnetic susceptibility χ”(ω,T) as function of frequency, was found to be greater for Ho_2Ti_2O_7 than for Dy_2Ti_2O_7. The spread of characteristic relaxation times τ, as well as the asymmetry in χ”(ω,T), were also found to be broader in Ho_2Ti_2O_7 <cit.>. The qualitative agreement of these works with the profile of the scaled susceptibilities G(γ,χ_0,χ) shown in Fig. <ref> c,d, alongside the difference in the microscopic energy scales of Ho_2Ti_2O_7 and Dy_2Ti_2O_7 <cit.> indicate that the differences between the two compounds are unlikely due to random disorder or off-stochiometry. In addition, previously reported relaxation times for Ho_2Ti_2O_7 and Dy_2Ti_2O_7, using the inverse of the angular frequency of the peak in χ”(ω,T) (see Fig. 3 in ref. <cit.> for example and Ref. <cit.>), exhibit the distinct characteristics shown in Fig. <ref> - the slope of Ho_2Ti_2O_7 is greater than that of Dy_2Ti_2O_7 and the relaxation times cross around 0.9 K.The apparent inferiority of the functional HN fits to Ho_2Ti_2O_7 as compared to Dy_2Ti_2O_7 is probably due to a combination of the more than two orders of magnitude wider spread of the relaxation times, and smaller signal sizes. However, these are still the best internally consistent analytic forms for χ(ω,T) in these materials, consistent for both the magnetization and susceptibility measurements and their resulting relaxation time temperature dependence. They yield parameters that agree both with prior works and do not contradict expectations from the different energy scales in these materials. The nomenclature of the proposed glass-forming spin-liquid (GFSL) motivates comparisons to existing spin glasses. Even though both have connection to glassy behavior, i.e. magnetic dynamics slow down with decrease in temperature, these two classes of materials are quite physically distinct. The difference between spin glasses and the proposed glass-forming spin-liquid state is both conceptual and in the details. First, quenched disorder is key for spin glasses, whereas there is no intrinsic disorder in the spin-ice compounds we study. Secondly, one of the clear signatures of spin glasses is the presence of a sharp cusp in the real part of the magnetic susceptibility at the transition temperature <cit.>, whereas the magnetic susceptibility of GFSL has a very smooth profile. Lastly, one can distinguish between the two by measuring the magnetic noise spectrum. The noise spectrum of a system carries information about its fluctuations on a microscopic scale. In the spin glass state, for example, one would expect a 1/f noise topresent <cit.>.The GFSL formalism, underpinning our studies, presents several verifiable predictions for spin-ices: the exact functional forms of the magnetization decay and ac susceptibility, and the divergence of the relaxation times. No other model can account for all of the observed behavior in these compounds in such great detail. In addition, the GFSL suggests a link to the underlying mechanisms through its fragility parameter. The fragility of conventional glass forming liquids reveals the degree of deviation of a system's relaxation time from the Arrhenius profile. The more fragile the glass former, the more curved its relaxation time divergence with respect to temperature. In the VTF formalism, this fragility is manifest as the slope of log(τ) vs 1/T i.e. the temperature dependent energy scale in the Arrhenius form. Therefore, the more fragile a material, the smaller the slope. Examination of the equation that describes the super-Arrhenius form,τ(T)=τ_0expDT_0/T-T_0, shows that D is connected to the ac susceptibility characteristics of the glass former through the peak frequency, that is proportional to 1/τ.In the frustrated pyrochlore magnets we study, fragility appears to indicate the heterogeneous nature of the energy landscape in these materials, similar to what is found in glass-forming dipolar-liquids. Therefore, the difference in fragilities between the two materials may eventually be revealed as due to differences in the underlying correlated and frustrated microscopic behavior of the magnetic monopoles, that is hypothesized to cause the magnetic relaxation in these materials. Here, both monopole creation energies and hopping rates, as well as constraining by Dirac strings would influence the dynamics, and hence the energy landscape. The higher fragility parameter of Ho_2Ti_2O_7, indicating a less fragile GFSL, for example, is consistent with the expected efficient monopole hopping in Ho_2Ti_2O_7 <cit.>. Finally, it seems plausible that the observed magnetic dynamics in Ho_2Ti_2O_7 and Dy_2Ti_2O_7 can be reconciled with the microscopic theory of emergent magnetic monopoles in these materials <cit.> by considering correlated transport of these quasiparticles. Here, flips of the real magnetic dipoles are recast as two opposite magnetic charges that, through a sequence of spin flips, may form a fluid of delocalized magnetic monopoles and anti-monopoles <cit.>. At low temperatures, these monopoles should form a dilute neutral gas whose transport characteristics control the magnetization dynamics and the susceptibility <cit.>. However, these monopoles are constrained by a network of Dirac strings, i.e. a trail of flipped spins left behind by a monopole traversing the interconnected tetrahedra. The non-Arrhenius sharp slowing down of spin dynamics may then be attributed to relaxation of Dirac strings <cit.>. § CONCLUSION To summarize: for purposes of comparison, we measured the AC susceptibility χ(ω,T) and time dependent magnetic relaxation behavior in the low-temperature magnetic states of the two materials Dy_2Ti_2O_7 and Ho_2Ti_2O_7. We used identical boundary-free sample geometries within a superconducting toroidal solenoid.We find that, for both materials, the DC relaxation follows a stretched exponential, KWW form (Fig. <ref> a,b), the AC susceptibility follows a HN form (Fig. <ref> c,d), and above 0.8 K the relaxation time for both materials diverges along a super-Arrhenius trajectory (Fig. <ref>). These phenomena all indicate that the magnetic state of these two distinct materials is the magnetic analog of a glass-forming dipolar liquid, which seems to be a previously unidentified characteristic of this class of frustrated magnetic materials.They were not anticipated by but appear to be consistent with the DSIM. The differences between the parameters of this general glass-forming spin-liquid phenomenology for the two materials can offer an insight into the microscopic behavior generating these phenomena.Indeed, recent theoretical studies using the spin-ice Hamiltonian extended to include stronger next nearest neighbor interactions, do report the existence of new forms of dynamical magnetic heterogeneity with extremely slow relaxation times for some spins <cit.>. Thus, the type of glass-forming spin-liquid phenomenology that we observe in Dy_2Ti_2O_7 and Ho_2Ti_2O_7, can exist, in theory, in dipolar spin-ice. We acknowledge useful and encouraging discussions with S. Bramwell, C. Castelnovo, J. Chalker, H. Changlani, M. Gingras, E.-A. Kim, M.J. Lawler and J. Sethna. J.C.S.D., R.D. and A. E. acknowledge support from the Moore Foundation's EPiQS Initiative through Grant GBMF4544.spphys | http://arxiv.org/abs/1707.09014v3 | {
"authors": [
"Azar B. Eyvazov",
"Ritika Dusad",
"Timothy J. S. Munsie",
"Hanna A. Dabkowska",
"Graeme M. Luke",
"Ethan R. Kassner",
"J. C. Séamus Davis",
"Anna Eyal"
],
"categories": [
"cond-mat.str-el"
],
"primary_category": "cond-mat.str-el",
"published": "20170727193720",
"title": "Common Glass-Forming Spin-Liquid State in the Pyrochlore Magnets Dy$_2$Ti$_2$O$_7$ and Ho$_2$Ti$_2$O$_7$"
} |
Université Savoie Mont Blanc, SYMME, F-74000 Annecy, France GAP-Biophotonics, Université de Genève, 22 chemin de Pinchat, 1211 Genève 4, Switzerland GAP-Biophotonics, Université de Genève, 22 chemin de Pinchat, 1211 Genève 4, Switzerland Photonics Institute, TU Wien, Gusshausstrasse 27/E387, 1040, Vienna, Austria Université Savoie Mont Blanc, SYMME, F-74000 Annecy, France Photonics Institute, TU Wien, Gusshausstrasse 27/E387, 1040, Vienna, Austria Université Savoie Mont Blanc, SYMME, F-74000 Annecy, France GAP-Biophotonics, Université de Genève, 22 chemin de Pinchat, 1211 Genève 4, SwitzerlandUniversité Savoie Mont Blanc, SYMME, F-74000 Annecy, France [email protected] GAP-Biophotonics, Université de Genève, 22 chemin de Pinchat, 1211 Genève 4, Switzerland [email protected] demonstrate the simultaneous generation of second, third, and fourth harmonic from a single dielectric Bismuth Ferrite nanoparticle excited by a telecom fiber laser at 1560 nm. We first characterize the signals associated with different nonlinear orders in terms of spectrum, excitation intensity dependence,and relative signal strengths. Successively, on the basis of the polarization-resolved emission curves of the three harmonics, we discuss the interplay of susceptibility tensor components at the different orders and we show how polarization can be used as an optical handle to control the relative frequency conversion properties.Keywords: harmonic generation; harmonic nanoparticles; perovskites; bismuth ferrite; frequency conversion.Bismuth Ferrite Dielectric Nanoparticles Excited at Telecom Wavelengths asMulticolor Sources by Second, Third, and Fourth Harmonic Generation Luigi Bonacina December 30, 2023 ===============================================================================================================================================§ INTRODUCTIONThe generation and control of nonlinear parametric signals at the nanoscale is paving the way to novel applications in imaging, sensing, optoelectronics. To date, most of the research efforts have been concentrated on noble metal nanoparticles and nanostructures<cit.> with a focus on their second (χ^(2))<cit.> and third order (χ^(3))<cit.> response. Some notable exceptions include the nonlinear harmonic generation by semiconductor nanoparticles<cit.>, two dimensional materials<cit.>, and noncentrosymmetric metal oxide nanoparticles (Harmonic NanoParticles, HNPs). Dielectric HNPs are attracting growing interest because of their extremely high nonlinear coefficients,<cit.> and robustness of their nonlinear responsewhich- contrary to noble metal particles-is primarily associated with their bulk properties and negligibly affected by surface phenomena.<cit.> Moreover, the sub-wavelength dimensions of HNPs lift the spectral limitations imposed by phase-matching conditions in bulk nonlinear crystals, enabling wide tunability of excitation light andemission of multiple signals at once. Some research groups areworking on the efficiency enhancement of the optical properties by engineering hybrid systems based on a HNP-core and a plasmonic-shell tailored for specific spectral resonances.<cit.> Recently, we have demonstrated the simultaneous acquisition of Second and Third Harmonic Generation (SHG, THG) by bare individual perovskite Bismuth Ferrite (BiFeO_3, BFO) HNPs.<cit.> We showed that the coincident acquisition of both harmonics can strongly benefit to imaging selectivity in optically congested environments<cit.> for applications including cell-tracking over long time in tissues.<cit.> Besides harmonic generation, one can expect that high χ^(n) values by HNPs can be exploited for disposing of localized sources of long wavelength radiation by optical rectification or for generating nonclassical states of light, in analogy to what has been demonstrated using other kinds of nanostructures.<cit.>In this respect, the possibility of working efficiently at telecom wavelengths (1.5 m)undeniably constitutesan asset for a future integration of HNPs as frequency conversion elements and all-optical logic operators<cit.> in photonics circuits. In this work, wedemonstrate that second, third, and fourth harmonic (FHG) emitted by an individual BFO HNP upon excitation at 1560 nm by an Erbium-doped fiber oscillator can be efficiently detected. Moreover, we show how the polarization control of excitation light allows tuning the relative intensities of the three harmonics. The simultaneous acquisition of three harmonics from the same individual nanoparticle is - to our best knowledge - a unicum to date and, besides all the applications we mentioned,HNPs might assume the role of model system for the study of the interplay among multiple-harmonics and high harmonic generation in solids.<cit.> § RESULTS AND DISCUSSION The starting evidence motivating this work is the observation that single BFO HNPs deposited on a substrate in the focus of the laser emit simultaneously at the three harmonics as from the images in Fig. <ref>A. The heat-maps colors are red for SHG (780 nm), green for THG (520 nm), and blue for FHG (390 nm). In the following, we first present a thorough assessment demonstrating by independent experimental observables [i) image spot size, ii) spectrum, and iii) excitation intensity dependence] that the three emission are genuinely associated with different nonlinear orders. Successively, we discuss the polarization-resolved emissions at the different orders which shed light on the tensorial properties of the nonlinear susceptibilities and could prospectively be exploited for selective frequency up-conversion from short-wave infrared to the visible. i) The Gaussian fits to the diameters of the particle images in Fig. <ref>A indicate that the FWHM decreases with increasing nonlinearity, as oneexpects for a diffraction limited object smaller than the point spread function (PSF) at the highest order. The observed widths of the PSFrange from 673nm for SHG, to 486 nm for THG and 420 nm for FHG.The average dimensions of the HNPs (≈ 100 nm, Fig. S1) remain therefore out of reach at all orders.In the Supplementary Material,we further comment these results in the context of the imaging properties of the set-up.ii)In Fig. <ref>B, we provide the normalized harmonic spectra detected in the forward direction. In the wavelength domain, one expects the width of the emission to scale as ∝1/n √(n), with n being the nonlinear order. This formula is derived for Gaussian pulses in the time domain.<cit.> Therefore, to apply this estimation to our traces stemming from a structured spectral profile at the excitation wavelength, we proceeded by visually determining the broadest Gaussian curves supported by the excitation and by each harmonic spectrum (Fig. S2). This way,we obtainedwidths for the different harmonics within 10% deviation from the theoretical estimation. This procedure, although involving approximations, points to a rather complete upconversion of the frequencies in the fundamental spectrum and it is consistent with the fact that BFO HNPs are smaller than the coherence lengths for each nonlinear order, l_c^(n). By using the optical constants of BFO derived by Kumar et al.<cit.> and applying a calculation including the effect of Gouy phase (see Eq. S1) we obtain l_c^(n) values in the forward direction spanning from1 mfor n=2 to 0.325 and0.23 m forn=3 and 4, respectively.<cit.> In our calculations, l_c^(n) deemlarger thanHNPs typical dimensions. This implies that no destructive interference takes place within the particle volume. iii) To complete this preliminary assessment of multiorder response,in Fig. <ref>C, we present the harmonic signal strength as a function of the laser intensity at the sample, I. Note that for this comparison the signals are normalized at the maximal laser intensity of the series, which corresponds to 440 GW/cm^2. As discussed in the next section, in absolute terms the THG is by far the most intense under these excitation/detection conditions: roughly 2 orders of magnitude stronger than SHG and 4 orders stronger than FHG.In the image, the nominal fitting curves (i.e.,I^n, n=2, 3, 4) are plotted as continuous lines. One can appreciate their fairly good agreement with the experimental data.In thelegend, we report the optimalvalues for the exponent n obtained letting this parameter free to vary in the fitting procedure. The retrieved valuesare all within 10% deviation from the theoretical values. In Fig. <ref>C', thedata and fitting curves from panel C are provided in log-log representation. Altogether these resultsobtained by independentmeasurements (nonlinear PSF, harmonic spectra,intensity dependence)support the association of the signals from single HNPs with three different harmonics: SHG, THG, and FHG.§.§ Relative intensities of harmonic orders. A natural question arises concerning the relative intensities of the three emissions, as one would normally expect a major decrease in signal strength with increasing nonlinear rank, provided that the symmetry requirements (i.e., noncentrosymmetricity) are fulfilled for the generation of even orders (SHG, FHG,...). Clearly, one should also take into proper account the different intensity dependence exhibited by signals associated with each χ^(n): higher excitation intensity is expected to favour higher orders as the ratio of two successive harmonics scales as ∝1/I. <cit.> In a previous Hyper Rayleigh Scattering experiment at 1064 nm, we determined that SHG/THG≃40 for 11 GW/cm^2 excitation.<cit.> Therefore, we could expect this ratio to be here ⩽1 when working at 53-fold larger intensity, viz. 590 GW/cm^2 (280 pJ/pulse). However, we observe a surprisingly smaller SHG/THG value,of the order of 10^-2-10^-3. Sample resonancesplay an important role in determining the value of harmonic ratios:<cit.> a resonance was reported at 504 nm for BFO 25 nm thin films<cit.>supporting the efficient generation of THG observed here at 1560 nm excitation (a green spot is visible by naked eyed on small aggregates).This close-to-resonance condition can also help explaining the very high second order susceptibility reported for BFO HNPs excited at 1064 nm, which was estimated to 160 pm/V.<cit.> The FHG/THG ratio is, on the other hand, of the order of 10^-4. Being aware that, among all techniques, the values extracted by microscopy present the largest uncertainty because they imply averaging the response of individual particles (10 in the present case, Fig. S3) with different spatial orientations modulating their harmonic ratios, we complemented these measurements withadditional ones performed onpellets of compressed BFO HNPs (Fig. S4A). These measurements were carried out at 1 TW/cm^2 using a J laser system, averaging the response of a large ensemble of randomly oriented particles over an elliptic area of 60×120 m^2. By this approach, we obtained SHG/THG≈20 (Fig. S4B) while the FHG/THG is ≤ 10^-4. Such a large discrepancy among the outcomes of the two methods,in particular for the SHG/THG ratio, is not fully clear. On one hand, the presence of aggregates in the pellets with dimensions exceeding the coherent lengthof BFO can affect the signal in an uneven way throughout the spectral domain. Moreover, the comparison can be also undermined by the difficulty to find a meaningful definition of peak intensity encompassing both large particles ensembles and isolated objects substantially smaller than the focal spot size. Finally, the differenceobserved can be ascribed to the critical dependenceof coherent signals generated by individual nanostructures on experimental settings (e.g., N.A. and collection angle). This last aspect has been subject of multiple theoretical studies in the plasmonic community based on different approaches (method of moments,<cit.> finite elements,<cit.> hydrodynamic model<cit.>). Recently, the hydrodynamic approach has been applied to calculate the SHG and THG angular radiation patterns simultaneously emitted by individual plasmonic nanoparticles, which specifically highlights this sensitivity to detection parameters showing rather different angular emission patterns at the two harmonics.<cit.> We believe that only a rigorous extension to higher harmonics of Hyper Rayleigh Scattering on colloidal suspensions can provide reliable values for the material.<cit.>The comparatively high conversion efficiencies at the third order we observe by both approachesfor a noncentrosymmetricmaterialdisplaying very high quadratic nonlinearity such as BFO, canalso be potentiallyascribed to the presence of multi-step (cascading) processes involving a succession of purely χ^(2) phenomena: SHG and sum frequency mixing.<cit.> In this case, THG would result from ω+ω=2ω and ω+2ω=3ω,<cit.> whereas FHG from ω+ω=2ω and 2ω+2ω=4ω or, alternatively, fromω+ω=2ω followed by2ω+ω=3ω and3ω+ω=4ω.<cit.> It is tempting to attribute the comparatively low emission at 2ω to a depletion of this frequency used as intermediate field for generating 3ω, however discerning multi-step from direct higher order nonlinear processes is a complex task, in particular for nanoparticles as the absence of macroscopic propagation excludes discrimination methods based on phase-matching criteria.<cit.> The use of HNPs with controlled size and narrow size distribution or epitaxial thin films of variable thickness could help elucidating this aspect in a future series of experiments.§.§ Polarization properties.In Fig. <ref>A, we introduce the results on polarization dependence for two sub-diffraction limited andisolated particles: HNP1 and HNP2.The shaded regionsdisplay the intensity of the harmonic emission detected as a function of the polarization angle of the excitation laser,γ. Note that differently from other works,<cit.> in this case no polarization analyser was set in the detection arm. The differences between the response of the two HNPs are associated with the different spatial orientations of their crystal axis with respect to the laboratory frame (Euler angles ϕ, θ, ψ in Fig. <ref>). The simple inspection of the polarization resolved traces can provide precious information and it deems useful to discard from the analysis eventual polycrystalline aggregates.<cit.> In general, the SHG traces possess a structure characterized by two dominant lobes inagreement with our previous observations.<cit.> For THG and FHG the side lobes become more prominent. Interestingly, the orientation of the main lobes is mostly maintained among the even orders (SHG, FHG) while for THG it seems that other tensor elements become predominant withmajor changes inorientation and symmetry. In our previous study,<cit.> starting from a known χ^(2) tensor,<cit.> we fitted the orientation of several BFO particles and then used the retrieved Euler anglesto determine the unknown χ^(3) tensor elements by simultaneously fitting the THG response of several HNPs. Here, we use these tensor values for χ^(2) and χ^(3)to fit theSHG and THG traces and obtain the Euler angles of each particle. The fits are reported as purple dashed lines on the data and the angle sets for HNP1 and HNP2 provided in the figure caption. Although the fits correctly capture the main features of the polarization curves (main lobes angles, presence of orthogonal lobes), one should be aware that this procedure implies several approximationsand the result should be considered qualitative in nature and primarily intended to support the fact that the BFO point group (3m) is compatible with the observed traces. In particular, the tensors weapply are derived at 1064 nm and not at 1560 nm.Note also that we could not readily extend this approach toχ^(4)because the number of independent elements of this tensor prevents the retrieval of a reliable outcome. Finally, we highlight that the possible presence of competing multi-step χ^(2) processes wouldundermine the general validity of this description, which would remain however an effective tool for predicting the polarization dependent response of BFO HNPs even in presence of concurrent direct and cascaded generation. §.§ Polarization-based control of relative harmonic intensities.The response of the two randomly oriented HNPs suggests that the choice of the excitation polarization, even in absence of any detection analyser, can be used to modulate the relative intensities of the three emissions for a given laser polarization angle γ. In Fig. <ref>B, we graphically emphasize this procedure showing the total emissionobtained by adding the normalized polarization dependent harmonic components displayed by HNP1 and HNP2. This alternative representation shows how, for a given HNP orientation, specific values of γ are associated with strong simultaneous SH, TH, FH emission (white regions),with individual harmonics (red, green, blue in our representation) and combination thereof (purple, pink...) or low emission (dark regions). We speculate that this approach could be adapted to precisely oriented BFO HNPs andthin films with thickness smaller than the shortest coherence length to provide polarization-controlled frequency converters from the telecom region over thevisible spectrum. Engineered hybrid structures composed by HNPs with a plasmonic shell of tailored thickness or, alternatively, the choice of materials withtailored resonances,<cit.> could also be away to mitigate the large conversion efficiency differences at the threeharmonic orders for defined applications.<cit.>Alternatively, one could shape the excitation geometry to control to some extent the angular emission pattern at the different orders.<cit.>§ CONCLUSIONS In conclusion, we have reported what,to our best knowledge, is the first demonstration of simultaneous acquisition of three harmonic frequencies generated by an isolated nanoparticle. Notably, our experiment is performed using a pJ fiber laser at telecom wavelength,which holds great promise for implementingdielectric nonlinear nanophotonics<cit.> in optoelectronic circuitry.Considered the novelty of our observation, we first thoroughly assessed the spectral and imaging properties and the intensity dependence of the emissions to ensure that they are genuinely associated with frequency conversionby χ^(n)(n=2, 3, 4) or cascaded χ^(2) processes. The relative intensities of the three harmonics have been critically discussed highlighting the sensitivity of this parameter to the measurement method. All estimations point tohigh generation efficiency for THG, likely because of the presence of electronic resonances in the spectrum. Finally, wehave discussed the excitation-polarization dependence of the particle emission, demonstrating that this approachopens the way todirectlyinvestigating the interplay among nonlinear susceptibility tensors elements at different orders andmodulating the relative strengths of three color components (red, green, violet) for photonics applications.§ METHODSBFO nanoparticles synthesized by the company FEE GmbH (Idar-Oberstein, Germany) were obtained as a water stabilized colloidal suspension from the companyTIBIO (Comano, Switzerland) under a research agreement. The average size is estimated to ≈ 100 nm by dynamic light scattering (DLS) and transmission electron microscopy (Fig. S1).For imaging,a drop of BFO suspensionis cast onto a microscope substrate and the solvent let evaporating. As reported in Fig. <ref>, the light source of the set-up is a Telecom femtosecond fiber laser at 1560 nm with a repetition rate of 100 MHzand 100 mW average power (T-Light FC, Menlo Systems). Pulses are compressed down to 90 fs by an optical fiber connected to the laser output. At the fiber output, the beam is collimated in the free space and expanded to a diameter of 6 mm.For polarization resolved studies, the linear polarization of the laser is rotated by a λ/2 platemounted on a motorized rotation stage. In the case of power dependence measurements, the laser energy is continuously modulated through the succession ofa λ/2plate and a polarisation analyser.Afterwards, the beam is reflected by a 45 degrees short-pass filter (Chroma) and focused on a single isolated HNP by a100× microscope oil immersion objective (NA 1.3).The signal generated by the particles can be detected in the backward or forward direction. In the latter case, the collection objective is a 40× N.A. 0.6air objective. HNPs areselected by scanning a (x,y) planar ROI of approximately 20×20 m^2 with a piezo-stage and carefully adjusting the z position by maximizing their nonlinear signal.Both for epi- and forward-detection, narrow bandwidth interference filters are used to select the harmonic spectral region (Thorlabs FBH780-10 for SHG, FBH520-40 for THG, FBH400-40 for FHGandSemrock BrightLine Fluorescence Filter 387/11for FHG). Additionally, a scanning spectrometer (Acton SP2300, Princeton Instruments, 300 g/mm) is placed in the forward detection arm to acquire spectrally resolved traces. The measurements are obtained using two different Hamamatsu detectors, selected according to their spectral response: H7732-01 low noise side-on photomultiplier tube(185 nm to 680 nm), andH7421-50 photon counting head with a GaAs photocatode (380 nm to 890 nm). Alternatively, we use a ultra-low-noise single photon counting module (SPD-A-VISNIR, Aurea Technology, Besançon, France).§ ACKNOWLEDGEMENTSWe acknowledge the financial support by Swiss SEFRI (project C15.0041, Multi Harmonic Nanoparticles), by the French-Switzerland Interreg programme (project NANOFIMT), andby the NCCR Molecular Ultrafast Science and Technology of the Swiss National Science Foundation. This study was performed in the context of the European COST Action MP1302 Nanospectroscopy. We are grateful to Dr. Davide Staedler at TIBIOSA (Comano,Switzerland) and Dr. Daniel Rytz at FEE GmbH (Idar-Oberstein, Germany) for synthesizing and providing us colloidally stable BFO HNPs,to Dr. Johann Cussey from Aurea Technology (Besançon, France) for providing us thethe single photon counting module and technological support,and Virginie Monnier (Institut des Nanotechnologies, Lyon) forthe TEM images of BFO HNPs. @ifundefinedendmcitethebibliography46 f subitem (mcitesubitemcount)[Kauranen and Zayats(2012)]Kauranen2012 M. Kauranen and A. V. Zayats, Nature Photonics, 2012, 6, 737–748 [Dadap et al.(1999)Dadap, Shan, Eisenthal, and Heinz]Dadap1999 J. I. Dadap, J. Shan, K. B. Eisenthal and T. F. Heinz, Physical Review Letters, 1999, 83, 4045 [Butet et al.(2010)Butet, Duboisset, Bachelier, Russier-Antoine, Benichou, Jonin, and Brevet]Butet2010 J. Butet, J. Duboisset, G. Bachelier, I. Russier-Antoine, E. Benichou, C. Jonin and P.-F. 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Kivshar, Optica, 2016, 3, 1241–1255§ SUPPLEMENTARY MATERIAL§ BFO HNPS CHARACTERIZATIONA detailed description of the synthesis and properties of the nanoparticles used in this work can be found in Schwung et al., <cit.>TEM and DLS representative data of a sample obtained by this protocol are reported in Fig. <ref>.§ WIDTH OF THE POINT SPREAD FUNCTION (PSF) AT THE DIFFERENT HARMONIC ORDERSTaking into account excitation wavelength andobjective numerical aperture, the nominal lateral FWHM of a perfect imaging system under linear excitation should be FWHM_linear^theo=0.51λ/N.A.=612 nm.<cit.> For the nonlinear case, Zipfel et al. provide the following expression for a two-photon excited fluorescence emitter: FWHM_2nd order^theo=2√(ln2)0.325λ/√(2) NA^0.91= 391 nm.<cit.> These values cannot be applied here because the resolution is expected to be severely reduced by the fact that we are using an high N.A. oil immersion objective intended for the visible region and not for an excitation at 1.5 m. Therefore all aberration corrections and optical elements (comprising the matching medium) are far from optimal. Indeed, we observe an energy reduction of 75% upon laser transmission through this objective, indicating a poor compatibility at this wavelength. By considering that the resolution should be proportional to 1/ √(n) where n is the nonlinear order, we can readily compute an actual value of ≈840 nm for the width of the linear PSF, both by multiplying the FWHM_FHG (420 nm) by √(4) and FWHM_THG(486 nm)by √(3).Note that this result supports the fact that we are observing a sub-diffraction limited emitter at two harmonic orders. The same calculation applied to the FWHM_SHG (673 nm)provides a result ≈15% higher. In this series, SHG was epi-detected using the H7421-50 photon countingand THG and FHG forward detected by the H7732-01 low noise side-on photomultiplier tube.The 15% discrepancy can very likely be attributed to the deviation from linear response of the former detector in the intensity regime of the measurement. Note that the FWHM value of 840 nm was used for the microscopy-basedintensity ratio calculation.§ ESTIMATION OF THE WIDTHS OF THE HARMONIC SPECTRA In Fig. <ref>, we report the normalized spectra of the laser and of the three harmonics generated by a single BFO HNP along with Gaussian curvessupported by these spectra and determined by visual inspection.On the figure we provide the Gaussian FWHM and, in parentheses,the product FWHM· n√(n) which should be directly compared with the laser spectrum as discussed in the main text. § CALCULATION OF COHERENT LENGTHS AT DIFFERENT ORDERS The coherence length is estimated usingl_c^(n)=π/k(nω)-n k(ω)-n Δ k_G where n=2,3,4 for SHG, THG, and FHG, respectively.Δ k_G is the wave vector corresponding to the Gouy-phase shift.The numerical value of Δ k_G was estimated at -0.5π/λ by Cheng and Xie for a 1.4 N.A. objective.<cit.>§ INTENSITY RATIOS§.§ Measurements on individual particlesIntensity ratio measurements by individual BFO HNPs were performed using two different detectors to minimize the need of efficiency corrections among different data sets. As reported in Fig. <ref>, SHG and THG were measured by detector 1 (SPD-A-VISNIR ultra-low-noise single photon counting module, Aurea Technology) and THG and FHG by detector 2 (H7732-01 low noise sideon photomultiplier tube, Hamamtsu). The traces highlight the particle-to-particle signal intensity variations, which come from differences in sizes (all signals are expected to scale as the particle volume squared), orientations, and possibly varying radiation patterns. We further confirmed these results on magnitude estimation among the different nonlinear orders employing a modified set-up with a NA 0.4 reflective Al-coated objective in the forward arm (Newport) anddetecting all harmonics by an EM-CCD (Andor, Ixon3) placed at the imaging output of the spectrometer.§.§ Ensemble measurements on BFO particle pellets In Fig. <ref>A, we provide a SHG image of the BFO HNPs pellet surface obtained by a commercial multiphoton microscope (Nikon A1R-MP) coupled with a Ti:sapphire oscillator (Mai Tai Spectra Physics). The epi-collected signal was processed by a Nikon A1 descanned spectrometer. The image scale bar is 10 μm. One can see how the SHG intensity ofHNPs is modulated by their diverse orientation and that most of the particles appear as bright diffraction limited spots. The emission spectrum averaged over the whole image is reported in Fig. <ref>B.For comparing relative intensities of the harmonics onBFO HNPs on dry pellets we relied onthelaser set up reported in Fig. <ref>. This system delivers≈80 fs pulsesat 1.5mgenerated in an OPA pumped by a 1 kHz 14 mJ 200-fs Yb:CaF2 CPA laser. The OPA is based on KTA crystals and seeded by a supercontinuum generated in a bulk YAG plate and delivers 1.5 mJ signal pulses. The signal beam is filtered out at the OPA output using a set of dichroic mirrors, the energy is attenuated using a half-wave plate and a polarizer and then focused onto the sample using f=200 mm CaF_2 lens at 60^∘ incidence. Theharmonic signals are collected inreflection geometry using a Schwarzschild objective (ReflX, Edmund Optics),imaged onto the slit of aimaging spectrometer, and detected using anEM-CCD (Andor, Ixon3). In Fig. <ref>C, we present the spectra of the different harmonic generated by the pellet. The relative intensities are corrected for CCD exposure time and spectral sensitivity and for grating efficiency and can be quantitatively compared.@ifundefinedendmcitethebibliography5 f subitem(mcitesubitemcount) [Schwung et al.(2014)Schwung, Rogov, Clarke, Joulaud, Magouroux, Staedler, Passemard, Justel, Badie, and Galez]Schwung2014 Schwung, S.; Rogov, A.; Clarke, G.; Joulaud, C.; Magouroux, T.; Staedler, D.; Passemard, S.; Justel, T.; Badie, L.; Galez, C. Journal of Applied Physics 2014, 116, 114306 [Wilson(2011)]Wilson2011 Wilson, T. Journal of microscopy 2011, 244, 113–121 [Zipfel et al.(2003)Zipfel, Williams, and Webb]Zipfel2003 Zipfel, W. R.; Williams, R. M.; Webb, W. W. Nature biotechnology 2003, 21, 1369–1377 [Cheng and Xie(2002)Cheng, and Xie]Cheng2002 Cheng, J.-X.; Xie, X. S. JOSA B 2002, 19, 1604–1610 | http://arxiv.org/abs/1707.08451v4 | {
"authors": [
"Jeremy Riporto",
"Alexis Demierre",
"Cédric Schmidt",
"Gabriel Campargue",
"Vasyl Kilin",
"Tadas Balciunas",
"Mathias Urbain",
"Andrius Baltuska",
"Ronan Le Dantec",
"Jean-Pierre Wolf",
"Yannick Mugnier",
"Luigi Bonacina"
],
"categories": [
"physics.optics"
],
"primary_category": "physics.optics",
"published": "20170726140201",
"title": "Bismuth Ferrite Dielectric Nanoparticles Excited at Telecom Wavelengths as Multicolor Sources by Second, Third, and Fourth Harmonic Generation"
} |
Horizontal patterns from finite speed directional quenchingRafael MonteiroUniversity of Minnesota, School of Mathematics, 206 Church St. S.E., Minneapolis, MN 55455, USA December 30, 2023 ======================================================================================================================= A boundary value problem, which could represent a transcendent temperature conduction problem with evaporation in a part of the boundary, was studied to determine unknown thermophysical parameters, which can be constants or time dependent functions. The goal of this paper was elucidate which parameters may be determined using only the measured superficial temperature in part of the boundary of the domain. We formulated a nonlinear inverse problem to determine the unknown parameters and a sensitivity analysis was also performed. In particular, we introduced a new way of computing a sensitivity analysis of a parameter which is variable in time. We applied the proposed method to model tissue temperature changes under transient conditions in a biological problem: the hamster cheek pouch. In this case, the time dependent unknown parameter can be associated to the loss of heat due to water evaporation at the superficial layer of the pouch. Finally, we performed the sensitivity analysis to determine the mostsensible parameters to variations of the superficial experimental data in the hamster cheek pouch. Keywords:Boundary value problem, Determination of parameters, Sensitivity analysis, Hamster cheek pouch. § INTRODUCTION Diffusion problems have been studied in many different areas, such as heat conduction<cit.>, sediment transport in a river<cit.>, the study of the weather <cit.>, or in drying food technology<cit.>. The heat transport in a certain object may be modeled through a convection-diffusion differential equation. The boundary conditions usually considered are: a constant temperature (Dirichlet condition), a fixed heat flux (Neumann condition) or a mixed condition, where the heat flux depends on the temperature at the boundary.In <cit.> some thermal inverse problems were considered in a mathematical tumor model. They estimated simultaneously unknown thermophysical and geometrical parameters, through an evolutionary algorithm. All the parameters were constants and no evaporation was considered.In <cit.> a space-dependent convection parameter was determined through a nonlinear least squares technique, using several temperature measurements at different times in different locations of the domain. In <cit.>, we observed that exophytic tumors and irradiated tissues, principally those with ulcers, exhibit mass moisture transfer in the tissue-air interphase.In this work,we resolved a partial differential problem considering a time-dependent extra term in the mixed condition, which can be associated with a loss of heat due to superficial evaporation. Having measurements of temperature in a section of a particular domain during time, a nonlinear inverse problem was formulated to determine some thermophysical parameters such as the superficial heat loss or the heat conductivity. A sensitivity analysis was alsoperformed. Finally the parameter determination and the sensitivity analysis were validated using our previous study on tissue temperature responses in the hamster cheek pouch <cit.>.§ DESCRIPTION OF THE CONDUCTION PROBLEMLet us consider a one dimensional spatial domain Ω=[0,X_max], and the following boundary value problem: c_1 ∂ T ∂ t (x,t)=c_2∂^2 T ∂ x^2 (x,t) +c_3 T(x,t)+c_4,∀ (x,t) ∈Ω× [0,t_max]c_5 ∂ T∂ x(x,t)= c_6 T(x,t)+c_7+f(t),∀ (x,t) ∈Γ_u × [0,t_max]T(x,t)=g(t),∀(x,t) ∈Γ_b × [0,t_max]T(x,0) =h(x)∀ x ∈Ωwherec_i are real constants, f,g:[0,t_max] →ℝ and h:Ω→ℝ are real functions, and Γ_u={X_max},Γ_b={0}[We chose one spatial dimension for simplicity. This procedure can be easily extended to more dimensions without loss of generality.].In general, the parameters c_i, i=1..7, f,g and h are fundamental in using this type of differential equation problem with boundary conditions given by Eqs.(<ref>), (<ref>) and (<ref>). Note that Eq. (<ref>) includes a time-dependent parameter, f(t), which is defined only in the boundary Γ_u. In particular, c_iare often approximated, and sometimes their values are just guessed. If there is an experimental measure of the temperature T(x,t)on the boundary Γ_u, an inverse problem can be formulated defining the following cost function:J(T)=∫_0^t_max∫_Γ_uT(x,t)-T^* (x,t)^2 dx dt,where T^*(x,t) is the measured superficial temperature. Therefore, some parameters can be determined minimizing the cost function J. The goal of this paper is to find which parameters c_imay be determined using only the measured superficial temperature. First we proposed a method to identify the function f=f(t) such that the solution T of the problem (<ref>) minimizes the cost function (Section <ref>). Then, we introduced a new way of computing a sensitivity analysis of a variable parameter (Section <ref>). As an example, in Section <ref>, we applied this problem to model tissue temperature changes, under transient conditions, in the hamster cheek pouch. In this biological problem, the function f(t) can be associated to the loss of heat due to water evaporation at the superficial layer of the cheek pouch.Finally, we performed the sensitivity analysis to determine the most sensible parameters to variations of the superficial experimental data (Section <ref>).§ DETERMINATION OF THE FUNCTION FThe goal of this section is to determine the function f:[0,t_max] →ℝ, assuming that the rest of the parameters are known constant. To solve the direct problem (<ref>) we used the Finite Element Method (FEM), meshed the spatial-time[The mesh used is a triangular uniform mesh, where the nodes are obtained by the Cartesian product of a discretization in time and a discretization in space. Using polynomials of degree one and this particular mesh makes the FEM equivalent to a finite difference scheme.] domain Ω× [0,t_max] and obtained the triangulation 𝔗={𝒯_i,i=1...M}.We define the following finite element spaces: S_𝔗={ v ∈ H^1(Ω× [0,t_max])/ v ∈𝒫_1(𝒯), ∀ 𝒯∈𝔗}, 𝒫_1V={ v ∈ S_𝔗/ v=g, Γ_b × [0,t_max]∧v=h Ω×{0}} , V_0={ v ∈ S_𝔗/ v=0, Γ_b × [0,t_max]∧v=0 Ω×{0}} .The weak formulation ofproblem (<ref>) is the following: (VP):T ∈ V/a(T,η)=l(η),∀η∈ V_0 where, a(u,v)=Ω× [0,t_max]∫ c_1∂ u/∂ tv+ c_2∂ u/∂ x∂ v/∂ x - c_3 u v dx dt -Γ_u × [0,t_max]∫c_2 c_6/c_5 u v dx dt l(v)=Ω× [0,t_max]∫ c_4 v dx dt+ Γ_u × [0,t_max]∫c_2/c_5(c_7+f(t)) vdx dt. Let {(x_i,t_i), i=1,..,N} be the nodes of the triangulation 𝔗, andI ⊂{1,...,M} be the subset of index whose nodes are in the boundaryΓ_u ×[0,t_max]. We define the function f as follows, f(t)=∑_i ∈ Iβ_i η_i(X_max,t), where {η_i }_i=1..N is the base of the space V. The inverse problem of determining f consists in obtaining the values of {β_i }_i ∈ I such that the solution T of the weak formulation problem is close to the measured temperatures. LetT be a solution of (VP). The cost function can be approximated by: J(T)=Γ_u × [0,t_max]∫ (T-T^*)^2 dx dt ≈∑_j=1^N_t (T(X_max,t_j)-T^*(t_j))^2 .We defined the following optimization problem: Find B={β_i }_i ∈ I such that minimizes the cost function j({β_i }_i ∈ I)=J(T_B), where T_B is a solution of (VP) using the values of B={β_i}_i ∈ I to define f(t). We used the Lagrange Method to determine the variation of j by each β_i, and obtaining a direction of descent. Suppose that for r ∈ I-{i} the values of β_r are fixed, and we vary only the parameter β_i. We define the following function:ℒ(u,w,β_i)=J(u)+a(u,w)-l_β_i(w)where a(·,·) y l(·) is defined in (<ref>) and (<ref>), andthe subscript β_i means that we are using in f(t) the fixed values of {β_r}_r ∈ I-{i} and the value of β_i (which may vary).Note that if u_β_i is a solution of (VP) then∀ w ∈ V_0:ℒ(u_β_i,w,β_i)=J(u_β_i)=j(β_i), and therefore their derivatives are equal.Using the chain rule we obtained the derivative of the Lagrangian ℒ respect to each β_i:δℒ(u_β_i,w_β_i,β_i)/δβ_i= Γ_u × [0,t_max]∫η_i w_β_idx dtwhere w_β_i∈ V(Ω) is the adjoint state, which is defined as the unique solution tothe following variational problem:(VPA) {[ a(v,w_β_i)=-2 Γ_u × [0,t_max]∫ (u_β_i-T^*) v dx dt, ∀ v ∈ V; w_β_i=0 Γ_b × [0,t_max] ]. To find the optimal values of {β_i} we used the gradient descent method. This method is based on the observation that if the function j(β ) is defined and differentiable in a neighborhood of a point β^0={β_1^0, ⋯, β_N^0}, then j(β ) decreases fastest if one goes from β^0 in the direction of the negative gradient of j at β^0 , e.g. -∇ j(β^0 )={∂ j(β^0 )/∂β_1, ⋯, ∂ j(β^0 )/∂β_N}. It follows that for γ small enough, the value of j in β^1 =β^0 -γ∇ j(β^0 ) is smaller than j(β^0 ).We get a sequence β^0,β^1,β^2,… such thatβ^n+1=β^n-γ _n∇j(β^n) andj(β^n+1)≤ j(β^n), for all n ∈ℕ_0. The convergence of this sequence to a local minimum of j depends on the properties of j, (for example, j convex and ∇ j Lipschitz).The value of γ_n >0 is different in every step. For each n, we searched for the greatest γ_n ≥ 0 such that it minimizes j(β^n-γ∇ j(β^n)). This is achieved combining a dichotomy method and a method that approximates j by a parabola, for more information about this procedure, see page 51 of<cit.>.§ SENSITIVITY ANALYSIS In <cit.> Blackwell and Dowding analyzed the usage of sensitivity parameters in connection with the estimation of thermal properties in the heat conduction equation. They stated that although parametric investigations may be done, sensitivity parameters are rarely computed (see also <cit.>). Sensitivity parameters help to understand the parametric dependence of an experiment and shape our experience and intuition for future cases .In inverse problems, the sensitivity parameter is the partial derivative of the output function (in our case the temperature) with respect to a parameter being determined, which is ∂ T/ ∂ p for a parameter p. Due to the general interest on the comparison of magnitudes for different parameters, a scaled (sometimes called “modified”) sensitivity parameter is used:S_p := p ∂ T/∂ p . Note that equation (<ref>) has units of temperature for all parameters, therefore magnitudes for various parameters can be directly compared.Small sensitivity parameters, or general insensitivity, are beneficial when the parameters are not well quantified, such as materials with no characterized thermal properties. Then the parameter is not influential in the thermal response. To estimate a parameter, however, the measured response has to be sensible to that parameter. In this case, the scaled sensitivity parameters are desired to be larger in magnitude (compared to the representative temperature) and linearly independent (having different shapes). The more sensitive the temperature is, the more valuable the temperature measurements are. In a similar way, the estimation of multiple parameters requires that the sensitivity, or the effect on temperature of each parameter, is different or independent of one another for each parameter. If two parameters have similar effects on temperature, their individual influence is difficult to distinguish.In this work we analyzed the sensitivity of the superficial temperature, for which we defined the modifiedsensitivity parameter as follows:S_p (t) := lim_Δ→ 0 pT_p+Δ(X_max,t) -T_p(X_max,t)/Δ , where X_max indicates the superficial boundary, and T_α is the solution of the differential problem (<ref>), using the value α in the parameter p.We used the method of finite differences <cit.> to determine the sensitivity parameters. First we solved the direct problem(<ref>) using the values of the parametersp=(p_1,p_2,...,p_i,..,p_n), and then we solved the direct problem again but with the following values: p_=(p_1,p_2,...,p_i+ p_i,..,p_n). Finally we approximated the sensitivity parameters as follows: S_p_i (t) ≈p_iT_p_ (X_max,t) -T_p (X_max,t)/ p_i . We remark that as every partial derivative, it is dependent not only on time, but also depends strongly in the values assumed by the parameters p_i, i=1...n.This previous analysis works if the parameters are constant. In the case of f which is variable in time, we introduced a new way of defining the sensitivity parameter of a variable parameter:{[ p=(p_1,p_2,...,f,..,p_n),p_=(p_1,p_2,...,(1+).f,..,p_n); S_f ( t)≈ T_p_ (X_max,t) -T_p (X_max,t) ]. where (1+).f represents the product of a real number and a function. Note that in order to compute S_p_i we need to work in a concrete problem, and therefore we will perform the sensitivity analysis for the biological application, where f will be determined (Section <ref>).§ BIOLOGICAL APPLICATIONIt was previously demonstrated that the hamster cheek pouch is useful for thestudy of tissue temperature affected by tissue superficial humidity <cit.>. The hamster cheek pouch is widely used as a model of oral cancer and mucositis, an adverse side effect induced by several cancer therapies <cit.>. Our group is focused on the study of BNCT (Boron Neutron Capture Therapy), a binary treatment modality that can selectively target neoplastic tissue <cit.>. Particularly, we study BNCT therapeutic effect on tumors and BNCT induced mucositis in the hamster cheek pouch with a non-invasive complementary method called Dynamic Infrared Imaging (DIRI). This method is based on the observation of temperature changes under transient conditions associated with mass moisture transfer in the tissue-air interface of the pouch.In our previous studies, we described different temperature changes for normal and tumor tissue, and also for non-irradiated and irradiated pouches <cit.>. However, the study of the mass moisture transfer as a function of time was not quantified. §.§ Dynamic Infrared imaging (DIRI) studies in the hamster cheek pouchDynamic Infrared imaging (DIRI) is based on the acquisition of thermal images during transient processes, caused by sudden and sustained changes in surface temperature due to the application of a thermal stimulus (provocation test) that forces the neurovascular system to respond in order to maintain local and body temperature within normal parameters<cit.>. Other authors followed this concept, including our group <cit.>, in different clinical research studies using thermography <cit.>.DIRI provides a non-invasively supplementary in vivo information potentially useful to characterize normal and pathological tissues and their response to cancer therapy.The biological model, experimental setup and procedures of the DIRI studies can be found in <cit.>. Briefly, a total of 61 hamsters were examined under DIRI protocol. Following an acclimatization period in the room, the animals were anesthetized and the pouch was everted using a plastic pipette held by hand. Thermal responses were measured using a FLIR T420 infrared camera, before, during and after the provocation test, namely, Transient Equilibrium Phase (TEP), Provocation Test (PT) and Recovery Phase (RP), respectively. The PT consisted of a mild air current applied at ambient temperature for about 120 seconds. The purpose of an air stimulus is to eliminate the initial moisture condition of the tissue so that, in the RP, we can focus on its thermal behavior and evaporation process that occur in response of the PT. In TEP and RP no air was applied, leaving the pouch exposed to ambient conditions without perturbations during approximately 280 and 400 seconds, respectively. Figure <ref>(a) shows the normal hamster cheek pouch tissue. The measured temperature values during time were extracted from the thermal image (Figure <ref>(b)) and averaged in a user-defined region of interest (ROI) used to delineate thenormal tissue. §.§ Determination of the superficial temperature and superficial heat lossIn this study, the differential equation model (<ref>) was applied to model the hamster cheek pouch temperature, imposing a loss of heat in the superficial tissue, due to the water evaporation. For this simulation, we used the experimental temperature data as a function of time in a given ROI<cit.>.The heat transfer modeling in organshas proposed numerous equations, studied by Pennes since 1948 <cit.>. He suggested that the heat transfer rate between blood and tissue was proportional to the product of the volumetric perfusion rate and the difference between the arterial blood temperature and the local tissue temperature. Therefore the temperature of a tissue depends on the rate of blood perfusion, the metabolic activity and the heat conduction between the tissue and the environment.Taking into account the suggestion made by Pennes <cit.>, Eq. (<ref>) of problem (<ref>) takes the form:ρ c ∂ T (x,t)/∂ t =k Δ_x T +ω_b ρ_b c_b (T_b-T)+q_m,(x,t) ∈Ω× [0,t_max] whereρ (ρ_b) represents the tissue (blood) density, c (c_b) is the tissue (blood) specific heat, k is the thermal conductivity, ω_b is the blood perfusion coefficient, q_m is the metabolic heat source and T_b is the constant blood temperature. The boundary condition Eqs. (<ref>), (<ref>) and(<ref>) are:{[ -k ∂ T∂ x=h (T-T_amb)+L(t), Γ_u × [0,t_max];T=T_D, Γ_b × [0,t_max];T(x,0) =F_0(x) ∀ t ∈ [0,t_max]; ]. Where,Γ_b={0} represents the inner boundary and let Γ_u={X_max}represents the superficial boundary. Here, f(t) = L(t) (W/m^2) represents the superficial heat loss due to water evaporation, h is the heat transfer coefficientbetween the tissue and the air and T_amb is the ambient temperature. We used the following initial temperature, that assures the continuity of the temperature values between the two boundariesΓ_b and Γ_u: F_0(x)=(T_0-T_b)X_max x+T_b , where T_0 is the initial superficial temperature measured. We used a linear function for simplicity.Since thermal responses in the hamster cheek pouch were assessed before, during and after the application of a thermal stimulus (TEP, PT andRP) <cit.>, these three different phases were modeled considering that the heat transfer coefficient h assumed different values in each stage: h=h(t)= h_1 0<t≤ 120 h_2 120<t ≤ 200 h_3200<t ≤ 400 The values used for h_1,h_2 and h_3 and others parameters are shown in Table <ref>.To our knowledge, there are no published data related to the thermophysical properties (such as thermal conductivity, diusivity, etc.) of the hamster cheek pouch. In <cit.>, Poppendiek et al. suggested that tissues may be considered accurately for thermal analysis as being composed of water, protein and fat. Thus, for the thermal properties needed to compute Eq. (<ref>) in our biological application, we only considered water and protein to calculate the thermal properties, shown in Table <ref>. The other parameters not related to the composition of the tissue were obtained from <cit.>. Figure <ref> shows the normal hamster cheek pouch experimental and calculated thermal response as a function of time. In Figure <ref>(a) it can be seen the good approximation between the calculatedsuperficial temperature and the experimental data.In particular, Figure <ref>(b) depicts the convective coefficient obtained usingan initial constant coefficient L(t) = 1000 W/m^2, and 15 steps of the gradient descent method described in Section <ref>.In Figure <ref>(b),we observed that at the beginning of each stage the superficial heat loss had oscillations. These oscillations can be seen also in the derivative of the functional (see Eq. (<ref>)), and therefore it seems that they are intrinsic of this differential problem. Besides, during the provocation test, the superficial heat loss decreases, due to the air stimulus that helps to eliminate the tissue superficial moisture. §.§ Sensitivity analysis Table <ref> and<ref> show those parameters used for the sensitivity analysis. We also used the parameter L(t) in Figure <ref>(b), obtained in the previous section. The initial temperature is in Eq. (<ref>). Table <ref> summarizes the physiological parameters considered, showing their units, and their position in the differential problem[The position in the differential problem, which was taken into account when the sensitivity was computed, changing the parameter p to p + Δ whenever it appeared.], necessary for the sensitivity analysis.Note that some parameters, such as ω_b and k, appeared in more than one place in the differential problem. Figure <ref> shows different parameters sensitivities, grouped together depending on their order of the sensitivity. A summary of the order of the sensitivity parameters is shown in Table <ref>.The first important observation is that the most sensible parameter is the ambient temperature (T_amb) (Figure <ref>(a)). This could be explained by the thinness of the tissue and the Dirichlet condition in the inner surface. Therefore, this parameter should be determined with the smallest error possible.Secondly, we examined the linear dependence between parameters, studied in <cit.>, which could be established by analyzing the shapes of the sensitivity parameters. If two parameters are linearly dependent, this means that there are infinite possible solutions, which implies that there will be infinite local minimums and therefore the minimization problem will not converge.Figure <ref>(b) shows that L and k are linearly dependent parameters, because their sensitivity are symmetric with respect to the horizontal axis. Therefore, although they have similar sensitivities, they should not be simultaneously determined. Figure <ref>(d) shows that ω_b and T_b are linearly dependent parameters, because their sensitivities have the same behavior (increasing functions). Moreover, these parameters are linearly dependent with h_1,h_2 and h_3. Therefore to determine a parameter, of this order of sensitivity,we should choose between ω_b, T_b and {h_1,h_2,h_3}.Finally the least sensible parameter is the metabolic heat q_m, which may be justified by observing that the total heat source is Q=ω_b ρ_b c_b T_b+q_m, and the predominant term is ω_b ρ_b c_b T_b.§ CONCLUSIONS We proposed a method for the determination of time dependent parameters using measured superficial temperatures in a conduction problem with evaporation, and a new way of computing a sensitivity analysis of a variable parameter. We applied this method successfully to a biological problem, modeling tissue temperature changes, under transient conditions, in the hamster cheek pouch. In this study we found a good approximation between the calculated superficial temperature and the experimental data. We performed a sensitivity analysis, which should be done whenever parameters are simultaneously determined.Based on temperature measurements and having calculated the loss of heat due to water evaporation at the superficial layer of the pouch, the sensitivity analysis determined which of the studied parameters were the most sensible to variations of the superficial experimental data. We found that ambient temperature should be measured with the smallest error, because it is the most sensible parameter in this problem. Moreover, we noted that the linear dependence between the conductivity and the superficial heat loss is not intuitive,in contrast with the dependence between the blood perfusion and the blood temperature (an increase in either of them would result in a raise in the superficial temperature). Therefore, we conclude that a choice has to be made between determining the conductivity or the superficial heat loss. In previous studies, we observed that tumors and particularly a precancerous tissue bearing ulcers after BNCT had high superficial humidity <cit.>.Thus, in future studies, the proposed mathematical model will be extended to explore the mass moisture transfer as a function of time in tumors and precancerous tissue in the hamster cheek pouch.§.§ AcknowledgmentsThis paper has been partially supported by the BNCT project of CNEA, and by CONICET. ieeetr | http://arxiv.org/abs/1707.09009v1 | {
"authors": [
"Natalia N. Salva",
"Maria S. Herrera",
"Andrea Monti Hughes",
"Claudio Padra",
"Gustavo A. Santa Cruz"
],
"categories": [
"physics.bio-ph"
],
"primary_category": "physics.bio-ph",
"published": "20170727192612",
"title": "Determination and biological application of a time dependent thermal parameter and sensitivity analysis for a conduction problem with superficial evaporation"
} |
equationsectiontheoremTheorem[section] propositionProposition[section] lemmaLemma[section] definitionDefinition[section] corollaryCorollary[section] conjectureConjecture[section] definition remarkRemark[section] | http://arxiv.org/abs/1707.08936v6 | {
"authors": [
"Siamak RabieniaHaratbar"
],
"categories": [
"math.AP"
],
"primary_category": "math.AP",
"published": "20170727171444",
"title": "Invertibility and Stability for A Generic class of Radon Transforms with Applications to Dynamic Operators"
} |
We view the well-known example of the dual of a countable compact hypergroup, motivated by the orbit space of p-adic integers by Dunkl and Ramirez (1975), as hypergroup deformation of the max semigroup structure on the linearly ordered set ℤ_+ of the non-negative integers along the diagonal. This works as motivation for us to study hypergroups or semi convolution spaces arising from “max" semigroups or general commutative semigroups via hypergroup deformation on idempotents. [2010]Primary43A62, 20M14 [ L. N. Pfeiffer December 30, 2023 =====================§ INTRODUCTIONWe introduce and study hypergroups, same as convolution spaces, in short, convos, or, semi convolution spaces, in short, semiconvos <cit.> arising from general commutative semigroups via deformation of the part of diagonal consisting of the idempotents. The genesis was the well-known example related to the orbit space of p-adic integers by Dunkl and Ramirez <cit.> viewed with this perspective on one hand and a good account of the structure of measure algebras of certain linearly ordered semigroups with order topology in <cit.> and <cit.> on the other hand (see also <cit.>). There is a substantial development of hypergroup deformations of groups and their applications. To get an idea one can see <cit.> and <cit.>. The present work can be seen as a complement of that because a group has no idempotents other than the identity.The next section gives basics of semigroups and hypergroups in the form that we need with a little touch of novelty at a few places. We begin Section 3 with an attempt to make a “max" semigroup (S, <, ·) with the discrete topology into a hermitian discrete hypergroup by deforming the product on the diagonal. Amongst other things, we arrive at the result that this can be done if and only if either S is finite or S is isomorphic to (ℤ_+,<,max). We determine itsdual and show that it becomes a countable compact hermitianhypergroup with respect to pointwise multiplication. In Section 4, we prove our main theorem on semiconvo or hypergroup deformations of idempotents in commutative semigroups.Let ℤ_+= ℕ∪{0} and ×, the usual multiplication. For a subset T of S, _T denotes the characteristic function of T defined on S. For notational convenience, we take empty sums to be zero.§ BASICS OF SEMIGROUPS AND HYPERGROUPS For basics of semigroups and hypergroups one can refer to standardbooks, monographs and research papers. For instance, one can see (<cit.>, <cit.>, <cit.>, <cit.> <cit.>, <cit.>, <cit.>) for semigroups and (<cit.>, <cit.>, <cit.>, <cit.>,<cit.>, <cit.>, <cit.>, <cit.>) for hypergroups. However, we give below some of them in the form we need.§.§ Basics of semigroups(i)As in <cit.>, we consider a non-empty set S linearly ordered by the relation `<'.(a) Form,n ∈ S, we define m · n= max{m,n}. This makes it into a commutative semigroup. In this paper, we call such a semigroup (S,<,·) a “max" semigroup. At times, we will write it as (S,<,max). Further, for m,n ∈ S, we write ℒ_n = { k ∈ S : k < n },𝒰_m = {k ∈ S : m <k }.(b) The linear order `<' and the “max" semigroup operation as in (a) above are appropriately related in the sense thatm,n,k ∈ S and m≤ n together imply that mk≤ nk as specified in Definition 0.1 of linearly ordered semigroup in <cit.>. (c) We may define another product `·' by min{m,n} but we stick to max unless needed for some explicit purpose. (ii) We call a commutative semigroup (S, ·) max-min type if for m,n ∈ S,m· n is m or n. Clearly, a commutative semigroup is max-min type if and only if it becomes a “max" semigroup via : for m,n ∈ S,m<n if and only if m· n=n ≠ m.(iii) Let (S,·) be a semigroup with identity e. For m,n ∈ S we usually write m· n = mn. (a) A non-empty subset T of S is called an ideal in S if TS ⊂ T and ST ⊂ T, where TS := {ts: t ∈ T, s∈ S} and similarly for ST. As in <cit.>, an ideal T ( S) in S will be called a prime ideal if the complement S \ T of T in S is a semigroup.(b) Let E(S) denote the set of idempotent elements in S, i.e., the set of elements n ∈ S such that n^2=n. We write E_0(S)=E(S)\{e} and S= S \ E(S).(c) Let G(S) denote the set {g ∈ S: ∃h ∈ S with gh=hg=e}. Then G(S) is a group contained in S called the maximal group. Note that G(S) ∩ E(S)= {e}.Set G_1(S)= {g ∈ G(S): gm=mfor allm ∈ E_0(S)}. Clearly, G_1(S) is a subgroup of G(S).Note that members of G_1(S) act on E_0(S) as the identity via left multiplication of (S, ·).Let (S,·) be a semigroup with identity e.(i)We call (S, ·) inverse-free in case for m,n ∈ S, mn=e holds if and only if m=n=e. This condition is equivalent to saying that m or n is equal to e. (ii) (S, ·) is called action-free if G_1(S)= {e}.(i) (ℤ_+,+ ), ((0,1], ×) and ([1, ∞ ), ×) are inverse-free semigroups which are not max-min type. (ii) (ℤ_+, ) and (ℤ_+ ∪{∞}, ) are max-min type inverse-freesemigroups.(iii) A commutingset of orthogonal projections on a Hilbert space ℋ containing the identity operator I_ℋ on ℋ, with respect to composition of two operators, form a semigroup with identity which isinverse-free.We know that if (S, ·) is a commutative semigroup then (E(S), ·) is a semigroup. Interrelations are collected in the following proposition whose proof is straight-forward.Let (S,·) be a semigroup with identity e. (i) If (S,·) is inverse-free then G(S)= {e}. Converse part is also true if S is commutative. (ii) If (S, ·) is max-min type then (S,·) is inverse-free. (iii) Suppose S\{e} is non-empty.(a) (S \{e}, ·) is a semigroup if and only if(S,·) is inverse-free. (b) If (S, ·) is inverse-free then (S\{e}, ·) is an ideal in (S, ·).(iv) Suppose E_0(S) is non-empty. If (S, ·) is inverse-free and commutative then (E_0(S), ·) is a semigroup. (v) For a commutative semigroup (S, ·),S is an ideal if and only if S is a prime ideal.(Dichotomy).Let S be asemigroup. For m ∈ S either m^j,j =1,2, … are all distinct or m^j is an idempotent for some j ∈ℕ.We will say m is of infinite order in the first case and of finite order in the second case.Here we provide an example of a commutative semigroup (S, ·), in which G(S)=G_1(S) ≠{e} and thus, S is not action-free. We take S= (ℤ_+ ×{0}) ∪{(0,1)} and define `·' as follows:(i,0) · (j,0) =(max{i,j},0)for i,j∈ℤ_+,(j,0) · (0,1) =(0,1)· (j,0)=(j,0)forj ≠ 0,(0,0) · (0, 1) = (0,1)·(0,0) =(0,1) and(0,1) · (0, 1) = (0,0).Then e= (0,0) and E_0(S)= ℕ×{0}. Also G(S)= {(0,α): α =0,1 }=G_1(S). The semigroup in our next example is action-free but not inverse-free.Let S= (ℤ_+, max) × ({0,1}, addition mod 2), a commutative semigroup with identity e= (0,0). It is easy to see that E_0(S)= ℕ×{0}, G(S)= {(0, α): α = 0,1}, and G_1(S)= {e}.In this paper, we shall mainly be concerned with semigroups equipped with the discrete topology, in short, discrete semigroups. Let (S,<) be a linearly ordered set as in Item <ref> (i). (i) Let (S,<, max) be the “max" semigroup as in Item <ref> (i)(a) above. We equip S with the order topology τ_0 and assume that S has an identity e. Then τ_0 is discrete if and only if (a) e has an immediate successor, and, (b) each m≠ e has an immediate predecessor as well as an immediate successor.(ii)Two different types of examples for (i) above can be provided by (a) (ℤ_+, <, max) or its non-empty finite subsets, and, (b) {0}∪{m ±1/n+2: m, n ∈ℕ} considered as a subset of the real line with the usual order and topology. §.§Basics of hypergroups Here we come to the basics of hypergroups. Dunkl <cit.>, Jewett <cit.> and Spector <cit.> independently created locally compact hypergroups (same as `convos' in <cit.>) under different names with the purpose of doing standard harmonic analysis. In this paper we are mostly concerned with commutative discrete semiconvos or hypergroups. It is convenient to write the definition in terms of a minimal number of axioms. For instance, see (<cit.>, <cit.>).Let K be a discrete space. Let M(K) be the space of complex-valued regular Borel measures on K. Let M_F(K) and M_p(K) denote the subset of M(K) consisting of measures with finite support and probability measures respectively. Let M_F,p(K)= M_F(K) ∩ M_p(K). At times, we do not distinguish between m and δ_m for any m ∈ K because m ↦δ_m is an embedding from K into M_p(K). Here δ_m is the unit point mass at m, i.e., the Dirac-delta measure at m.We begin with a map * : K × K → M_F,p(K). Simple computations enable us to extend `*' to a bilinear map called convolution, denoted by `*' again, from M(K) × M(K) to M(K). At times, for certain n ∈ K we will write q_n for δ_n* δ_n and Q_n for its support. A bijective map ∨: m ↦m̌ from K to K is called an involution if m̌̌̌=m. We can extend it to M(K) in a natural way.A pair (K,*) is called adiscrete semiconvo if the following conditions hold. * The map * : K × K → M_F,p(K) satisfies the associativity condition(δ_m*δ_n)*δ_k = δ_m* (δ_n*δ_k) for allm, n, k ∈ K.* There exists (necessarily unique) element e ∈ K such thatδ_m*δ_e = δ_e*δ_m = δ_m for all m ∈ K.A discrete semiconvo (K,*) is called commutative if δ_m*δ_n= δ_n*δ_m for all m,n ∈ K. A triplet (K,*, ∨) is called a discrete hypergroup if* (K,*) is a discrete semiconvo, * ∨ is an involution on K that satisfies(i)(δ_m*δ_n )̌= δ_ň*δ_m̌ for all m,n ∈ K and(ii) e ∈(δ_m*δ_ň) if and only if m=n.A discrete hypergroup (K,*, ∨) is called hermitian if the involution on K is the identity map, i.e., m̌=m for all m ∈ K.Note that a hermitian discrete hypergroup is commutative. We write (K,*) or (K,*,∨) as K only if no confusion can arise.Let K be a commutative discrete hypergroup. For a complex-valued function χ defined on K,we write χ̌(m):= χ(m̌) and χ(m*n) = ∫_K χd(δ_m*δ_n) for m,n ∈ K. Now, define two dual objects of K: 𝒳_b(K)= {χ∈ℓ^∞(K): χ≠ 0, χ(m*n)= χ(m) χ(n)for all m,n ∈ K }, K= {χ∈𝒳_b(K) : χ̌= χ,χ(m̌)= χ(m)for all m ∈ K}.Each χ∈𝒳_b(K) is called a character and each χ∈K is called a symmetric character. With the topology of pointwise convergence, 𝒳_b(K) and K become compact Hausdorff spaces. In contrast to the group case, these two dual objects need not be the same and also need not have a hypergroup structure. Now we give some examples of hypergroups.Polynomial hypergroups: This is a wide and important class of hermitian discrete hypergroups in which hypergroup structures are defined on ℤ_+. This class contains Chebyshev polynomial hypergroups of first kind, Chebyshev polynomial hypergroups of second kind, Jacobi hypergroups, Laguerre hypergroups etc. For more details see <cit.> and <cit.>. For Illustration, we describe CP, the Chebyshev polynomial hypergroup of first kind which arises from the Chebyshev polynomials of first kind. In fact, they define the following convolution `*' on ℤ_+:δ_m*δ_n= 1/2δ_|n-m|+1/2δ_n+m for m,n ∈ℤ_+. For any k ∈ℕ, K= k ℤ_+ with *|_K × K makes K a discrete hypergroup in its own right. The Chebyshev polynomial hypergroup of second kind (ℤ_+, *) arises from the Chebyshev polynomials of second kind and the convolution '*' on ℤ_+ is given byδ_m*δ_n= ∑_k=0^min{m ,n}|m-n|+2k+1/(m+1)(n+1)δ_|m-n|+2k.Let H_a={0,1,2, …, ∞}, 0<a ≤1/2, be the one-point compactification of ℤ_+. Dunkl and Ramirez <cit.> defined a convolution structure on H_a to make it a (hermitian) countable compact hypergroup. For a prime p, let Δ_p be the ring of p-adic integers and 𝒲 be its group of units, that is , {x=x_0+x_1p+ …+ x_np^n+ …∈Δ_p : x_j = 0,1, …,p-1forj ≥ 0andx_0 ≠ 0}. For a=1/p, H_a derives its structure from 𝒲-orbits of action of 𝒲 on Δ_p by multiplication in Δ_p.Next, Dunkl and Ramirez make the symmetric dual space H_a of H_a into a hermitian discrete hypergroup. The members of H_a are given by {χ_n : n ∈ℤ_+}, where, for k ∈ H_a,χ_n(k)= 0ifk <n-1,a/a-1 ifk=n-1, 1if k ≥ n (or k = ∞).Then the convolution `*' on K=ℤ_+ identified with H_a={χ_n : n ∈ℤ_+} is dictated by pointwise product of functions in H_a, that is: χ_m χ_n= χ_max{m,n} for m ≠ n,χ_0^2 = χ_0, χ_1^2 = a/1-aχ_0+ 1-2a/1-aχ_1, χ_n^2 = a^n/1-aχ_0+ ∑_k=1^n-1 a^n-kχ_k+1-2a/1-aχ_nforn ≥ 2.We call (K,*) a (discrete) Dunkl-Ramirez hypergroup. H_a has a good spectral synthesis in the sense that every closed subset of H_a is a set of spectral synthesis for theFourier algebra A(H_a) <cit.>. Chilana (now, Ajit Iqbal Singh) and Ajay Kumar <cit.> have strengthened this further. We note a few more standard facts about a Dunkl-Ramirez hypergroup (K,*). We follow the notation as in 2.1 and 2.2 above. (i) For n ∈ℤ_+, (ℒ_n ∪{n}, *) is a subhypergroup of (K,*).(ii) Let a ≠1/2. For n ∈ℤ_+,Q_n= ℒ_n ∪{n} and as a consequence, for n ∈ℕ, #Q_n ≥ 2.(iii) Let a= 1/2.(a) For n ∈ℕ,Q_n= ℒ_n. For n ≥ 2, #Q_n ≥ 2 whereas for n=1,#Q_n=1. (b) In fact, (ℒ_1 ∪{1},*) is a group isomorphic to ({0,1}, addition mod2)§ HYPERGROUPS ARISING FROM HYPERGROUP DEFORMATIONS OF IDEMPOTENT ELEMENTS OF “MAX" SEMIGROUPSTo begin with, we assume that (S,·) is a commutativediscrete semigroup with identity e such that E_0(S) ≠ϕ. For q ∈ M_p(S),let Q be its support, in short, (q) and q(j)= q({j}) for j ∈ S. Then Q is countable. Observe that q=∑_j ∈ Q q(j) δ_j with q(j)>0 for each j ∈ Q and ∑_j ∈ Q q(j)=1. We prefer this to the usual form ∑_j ∈ S q(j) δ_j with q(j) ≥ 0 for j ∈ S and ∑_j ∈ S q(j)=1, unless otherwise stated. A probability measure q on S with #Q ≥ 2 will be called non-Dirac. §.§ MotivationWe note that a Dunkl-Ramirez hypergroup (as in Example <ref> above) is a hermitian (hence commutative)discrete hypergroup K=H_a (0<a≤1/2) and its convolution `*' arises as a hypergroup deformation of the semigroup (ℤ_+, ·), where m · n= {m, n} in the sense that δ_m*δ_n= δ_mn for m ≠ n or, m=n=0 and for m=n ≠ 0, we have δ_1*δ_1= a/1-aδ_0+1-2a/1-aδ_1, δ_n*δ_n= a^n/1-aδ_0+ ∑_k=1^n-1 a^n-kδ_k + 1-2a/1-aδ_nfor n ≥ 2.Also, each element of ℤ_+ is an idempotent in (ℤ_+, ·). Further, for n ∈ℤ_+, (ℒ_n ∪{n}, *) arises as a hypergroup deformation of thefinite subsemigroup (ℒ_n ∪{n}, max) of (ℤ_+, ·). This motivates the construction of acommutative discretesemiconvo or discrete hypergroup structure on S for a commutativediscrete semigroup (S, ·) by deforming the product. We elaborate as follows. §.§ Hypergroups deformations of“max" semigroupsWe now confine our attention toa “max" semigroup (S, <, ·) as in Item <ref>(i)(a). We assume that S has an identity e and it is equipped with the discrete topology.We try to deform this discrete semigroup (S, ·) into ahermitian (hence commutative) discrete hypergroup by deforming the product on the diagonal of S \{e}.§.§.§ Preparatory material (i) For a discrete space K and a complex-valued regular Borel or a non-negative measure μ on K, we usually write μ(j) for μ({j}).(ii)(a) Let (K,*) be a discrete hermitian hypergroup. Then by <cit.> the Haar measure λ on K is given by: λ(e)=1 and for e ≠ n ∈ K, λ(n)= 1/(δ_n*δ_n)(e). (b) Let K be a Dunkl-Ramirez hypergroup H_a (0< a ≤1/2). Then λ(n)= 1-a/a^n for all n ∈ℕ. Further, for n ∈ℕ,we haveδ_n*δ_n(0)= a^n/1-a= 1/λ(n), δ_n*δ_n(k) =a^n-k = λ(k)/λ(n)for1≤ k <n and δ_n*δ_n(n) = 1-2a/1-a= 1- ∑_0 ≤ k<n (δ_n*δ_n)(k)= 1- (δ_n*δ_n)(ℒ_n)= λ(n)-λ(ℒ_n)/λ(n). (c) (δ_n*δ_n)(n)=0 for some n ∈ℕ if and only if a=1/2 if and only if (δ_n*δ_n)(n)=0 for all n ∈ℕ. In this case, λ(n)=2^n-1 for n ∈ℕ.We try to replace (ℤ_+,<, ·) in the Dunkl-Ramirez hypergroups by a discrete “max" semigroup (S,<, max) with identity in our next theorem.We give a set of necessary and sufficient conditions for (S,*) to become ahermitian discrete hypergroup with convolution product `*' defined as follows:δ_m*δ_n = δ_n*δ_m= δ_m· n (= δ_max{m,n}) m,n ∈ S withm ≠ n,or,m=n=e, δ_n*δ_n=q_nforn ∈ S\{e}.Here, q_n is a probability measure on S with finite support Q_n containing e and has the form ∑_j ∈ Q_n q_n(j) δ_j with q_n(j)>0 for j ∈ Q_n and ∑_j ∈ Q_n q_n(j)=1. This is equivalent to looking for conditions on S, Q_n's, an S-tuple {v_n}_n ∈ S in [1, ∞) with v_e=1 and v_n= 1/q_n(e) for n ∈ S \{e}and q_n(j) for j ∈ Q_n \{e},n ∈ S \{e}. Let (S,<, ·) be a discrete (commutative) “max" semigroup with identity e and ` *' and other related symbols as above. Then (S,*) is a hermitian discrete hypergroup if and only if the following conditions hold.(i) Either S is finite or (S,<,·) is isomorphic to (ℤ_+,<, max). (ii) For n ∈ S\{e}, we have ℒ_n ⊂ Q_n ⊂ℒ_n ∪{n}.(iii)If #S > 2, then for e ≠ m <n in S, we have(a) q_n(e)=q_n(m)q_m(e) and(b) q_n(e) ( 1+ ∑_e ≠ k ∈ℒ_n1/q_k(e)) ≤ 1;or, equivalently, with v_n= 1/q_n(e) for n ∈ S, (iii)'If #S > 2, then for e ≠ m <n in S, we have(a)q_n(m)= v_m/v_n and(b) ∑_k ∈ℒ_nv_k ≤ v_n.Suppose (S,*) is a hermitian discrete hypergroup. We prove (i) in two steps.Step (α). Let n ∈ S \{e} and m<n . Associativity of (S, *) demands that (δ_n*δ_n)*δ_m=δ_n*(δ_n*δ_m). This gives that q_n*δ_m=q_n. But e ∈ Q_n; therefore, m ∈ Q_n. Hence ℒ_n ⊂ Q_n. Since #Q_n < ∞ we have #ℒ_n < ∞.Clearly, ℒ_e= ϕ is finite.Step (β). Assume that S is not finite. Consider any n ∈ S. By Step (α) ℒ_n ∪{n} is finite and, therefore,𝒰_n = S \ (ℒ_n ∪{n}) is non-empty. Choose any m ∈𝒰_n.Suppose that m is not immediate successor of n.Then, there exists an element m' ∈ S such that n<m'<m. So 𝒰_n ∩ℒ_m is finite and non-empty because m' ∈𝒰_n ∩ℒ_m and ℒ_m is finite by Step (α). Since 𝒰_n ∩ℒ_m is finite we can enumerate its elements inorder, say, n_1<n_2< ··· <n_s.Choose the immediate successor n_1 of n. Thus each element n of S has an immediate successor, say, s(n). Now, start with x_0=e, take x_1= s(e),x_2=s(x_1), ···, x_j=s(x_j-1) and so on. Set W={x_j:j ∈ℤ_+}. We claim that W=S. Suppose not; then there exists t ∈ S\ W. Therefore, t ≠ e, and hence t>e. But x_1 is immediate successor of e so t>x_1. Again, x_2 is the immediate successor of x_1, so t>x_2. Repeating this process, we get t>x_j for every j ∈ℤ_+. Hence W ⊂ℒ_t. This shows that W is finite because ℒ_t is finite by Step (α). This gives a contradiction. So, S=W.Therefore, either S is finite or (S,<,·) is isomorphic to (ℤ_+,<, max).(ii) Consider any n ∈ S \{e}. Let, if possible,Q_n ⊄ℒ_n ∪{n}. Then there exists m >n with m ∈ Q_n. The associativity of `*' demands that (δ_n*δ_n)*δ_m= δ_n*(δ_n*δ_m),which in turn gives that q_n*δ_m=δ_m. Now, L.H.S. = q_n(m) δ_m*δ_m + ∑_m ≠ j ∈ Q_n q_n(j) δ_jm=q_n(m) q_m + ∑_m ≠ j ∈ Q_n q_n(j) δ_jm. By Step (i) (α),ℒ_m ⊂ Q_m. Now, Q_m ⊂(L.H.S.)=(R.H.S.)= {m}. But n ∈ℒ_m. Thus, we have n=m, which is a contradiction. Therefore, Q_n ⊂ℒ_n ∪{n}. Hence, using Step (i)(α), we get ℒ_n ⊂ Q_n ⊂ℒ_n ∪{n} for n ∈ S \{e}.(iii) Suppose #S>2 and e ≠ m <n in S.(a) As in the proof of Step(i)(α), q_n*δ_m= q_n. Now, as by (ii), m ∈ Q_n, we get = q_n(e) δ_m + ∑_e ≠ j ≠ mj ∈ Q_n q_n(j) δ_jm + q_n(m) δ_m*δ_m =q_n(e) δ_m + ∑_e ≠ j ≠ mj ∈ Q_n q_n(j) δ_jm + q_n(m) ∑_j' ∈ Q_m q_m(j') δ_j'and= q_n(e) δ_e + ∑_e ≠ k ≠ m, k ∈ Q_n q_n(k) δ_k + q_n(m) δ_m.Since jm ≠ e for j,m ∈ S \{e} and R.H.S. = L.H.S., we get q_n(e)= q_n(m)q_m(e).(b) Because n ∈ S \{e}, we get 0< ∑_ j ∈ℒ_n q_n(j) ≤∑_j ∈ Q_n q_n(j) =1. By (iii)(a), q_n(j)= q_n(e)/q_j(e) for all j ∈ℒ_n \{e}. Therefore, we getq_n(e)( 1+ ∑_e ≠ j ∈ℒ_n1/q_j(e)) = q_n(e) +∑_e≠ j ∈ℒ_nq_n(e)/q_j(e)= ∑_j∈ℒ_n q_n(j) ≤ 1. It can be seen easily that (iii)' is only a rewording of (iii). For theconverse part, assume that (i)-(iii) all hold. We note right at the outset that, because condition (ii) is satisfied, for n ∈ S\{e}, we may write q_n=δ_n*δ_n= ∑_j ∈ℒ_n ∪{n} q_n(j) δ_j= ∑_j ≤ n q_n(j) δ_j with q_n(j) >0 for j ∈ℒ_n and q_n(n) ≥ 0.To see that (S,*) is a hermitiandiscrete hypergroup, first note that the identity element e of semigroup (S, ·) works as the identity of (S,*), i.e., δ_n*δ_e= δ_e*δ_n=δ_n for every n ∈ S. Next, we show that e ∈(δ_m*δ_n) if and only if m=n. It is trivial in case m=n=e.If m =n ≠ e then e ∈(δ_n*δ_n)= Q_n. Now, assume that e ∈(δ_m*δ_n). Let, if possible, m ≠ n, so δ_m*δ_n= δ_mn. Without loss of generality we can assume that m<n, so δ_mn=δ_n and n>e. Therefore, e ∈(δ_m*δ_n) shows that n =e, which is a contradiction. Hence m=n.Now, we prove the associativity condition of (S,*), i.e., for m,n,k ∈ S(δ_m*δ_n)*δ_k= δ_m*(δ_n*δ_k).If any one or more of {m,n,k} are e then (<ref>) is trivial. We now consider the cases when none of m,n and k are e.Case (i) If m ≠ n ≠ k ≠ m, we first consider the subcase m<n<k. So, L.H.S. of (<ref>) = δ_n*δ_k =δ_k and R.H.S. of (<ref>) = δ_m*δ_k=δ_k. Therefore, L.H.S. =R.H.S. Other subcases can be dealt with in a similar way and hence (<ref>) holds. Case (ii) Out of m,n,k exactly two are distinct. We get #S>2. (α) We consider first the following form of (<ref>):(δ_n*δ_n)*δ_m= δ_n*(δ_n*δ_m) with e ≠ n ≠ m ≠ e, which is the same asq_n*δ_m = δ_n*δ_nm When n<m, R.H.S. of (<ref>) is equal to δ_n*δ_m=δ_m. The condition that Q_n ⊂ℒ_n ∪{n} implies that Q_n· m ={m} and using this, we getL.H.S. of (<ref>) = ∑_j ∈ Q_n q_n(j) (δ_j*δ_m)= ∑_j ∈ Q_n q_n(j) δ_jm=( ∑_j ∈ Q_n q_n(j) )δ_m =δ_m as ∑_j ∈ Q_n q_n(j) =1 . Therefore, L.H.S.= R.H.S. and hence (<ref>) holds.We now come to the case when m<n. Then R.H.S. of (<ref>) is q_n= ∑_j ≤ n q_n(j) δ_j with q_n(j)>0 for j <n and q_n(n) ≥ 0.So L.H.S. of (<ref>) = ∑_j ≤ n q_n(j) (δ_j*δ_m)= ∑_j ≠ mj≤ n q_n(j) δ_jm+ q_n(m) ∑_k ≤ m q_m(k) δ_k = ∑_m<j ≤ nq_n(j) δ_j + ( ∑_j <m q_n(j) ) δ_m + ∑_k<m q_n(m) q_m(k) δ_k + q_n(m) q_m(m)δ_m . By (iii)(a) q_n(k)q_k(e)= q_n(e) for k<n. So we get q_n(m)q_m(j)= q_n(j) for all j <m. Therefore, L.H.S. of (<ref>)= ∑_m<j ≤ n q_n(j) δ_j + ( ∑_j<m q_n(m) q_m(j) +q_n(m) q_m(m)) δ_m+∑_k <m q_n(k) δ_k = ∑_m<j ≤ nq_n(j) δ_j +q_n(m) ( ∑_j ≤ m q_m(j)) δ_m+∑_k<m q_n(k) δ_k.Further, we have ∑_j ≤ m q_m(j) =1. So, L.H.S. of (<ref>) =∑_k ≤ n q_n(k) δ_k = q_n.Therefore, we get L.H.S. of (<ref>) = R. H.S. of (<ref>) and hence (<ref>) holds.(β) Now, consider the following form of (<ref>) with m<n, (δ_n*δ_m)*δ_n= δ_n*(δ_m*δ_n). R.H.S. = δ_n*δ_n = (δ_n*δ_m)*δ_n= L.H.S. (γ) The case m>n of the form (<ref>) as in (β) above follows simply because L.H.S. = δ_m*δ_n= δ_m= R.H.S.(iii) If m=n=k then (<ref>) follows from the fact δ_s*δ_n= δ_n*δ_s for s ∈ Q_n.Hence (<ref>) holds and consequently (S,*) is a hermitian discrete hypergroup. In view of condition (i) in Theorem <ref> and Remark <ref>(i) we note that for “max" semigroups (S,<, max) of interest to us the discrete topology and the order topology coincide (cf. <cit.>, <cit.>).We now interpret conditions (i)-(iii) in Theorem <ref> above. Because a finite “max" semigroup is isomorphic to (ℤ_k= {j: 0 ≤ j < k}, <, max) for some k ∈ℕ, we confine our attention to ℤ_+ to begin with. Let 𝒱 be the set of sequences v=(v_j)_j ∈ℤ_+ in [1, ∞) which satisfy (i) v_0=1 and (ii) v_n ≥∑_j ∈ℒ_nv_j for n ∈ℕ. For n ∈ℕ, let u_n= v_n- ∑_j ∈ℒ_nv_j. Then simple calculations give the following: v_0=1 , v_1 = 1+u_1,v_2 = (2+u_1)+u_2, v_3=(2^2+2u_1+u_2)+u_3,⋮ = ⋮,to elaborate,v_n=(2^n-1+2^n-2u_1+ …+u_n-1)+u_n forn ≥ 3.Alternatively, we may start with a sequence u=(u_n)_n ∈ℕ in [0, ∞) and define (v_n)_n ∈ℤ_+ as above. Let 𝒰 be the set of such sequences (u_n)_n ∈ℕ's. The sets 𝒰 and 𝒱 have cardinality 𝔠.The same is true if we confine our attention to domain ℤ_k= {j: 0 ≤ j < k} for k ∈ℕ, instead of ℤ_+.For each v ∈𝒱 (or the corresponding u ∈𝒰), there is one and only one (hermitian) hypergroup deformation (ℤ_+,*) of (ℤ_+,<, max) which satisfies (δ_n*δ_n)(0)= 1/v_n, n ∈ℤ_+. Further, for this deformation, the convolution `*' and the Haar measure λ satisfy the following conditions.(i) λ(n)=v_n for n ∈ℤ_+ andλ(n)-λ(ℒ_n)=u_n for n ∈ℕ. (ii) λ(ℒ_n) ≤λ(n) for n ∈ℕ. (iii) For n ∈ℕ,(a) δ_n*δ_n(m)= λ(m)/λ(n) for m<n, (b) δ_n*δ_n(n)=λ(n)-λ(ℒ_n)/λ(n), (c) δ_n*δ_n(m)=0 for m>n. (iv) For n ∈ℕ, (δ_n*δ_n) = ℒ_nifλ(n)=λ(ℒ_n),ℒ_n ∪{n} ifλ(n)>λ(ℒ_n). (i) This is immediate from <ref>(ii)(a) and expressions for (v_n)_n∈ℤ_+ and (u_n)_n ∈ℕ given above.(ii) This follows from Theorem <ref>(iii)(b).(iii) We have only to appeal to Theorem <ref>(iii)(a).(iv) This follows from (iii). We compare Theorem <ref> and Corollary <ref> above with the Dunkl-Ramirez hypergroups (Example <ref> above). We freely use Remark <ref>, Section <ref> and Item <ref> above. (a) For the Dunkl-Ramirez hypergroup H_a (0< a ≤1/2), q_n(0)= a^n/1-a, n ∈ℕ. Further, Q_m= ℒ_m for some m ∈ℕ if and only if a = 1/2 if and only if Q_n=ℒ_n for all n ∈ℕ.(b) For our theorem, for any arbitrarily fixed m ∈ S \{e},Q_m=ℒ_mif and only if q_m(e) (1+∑_e≠ j ∈ℒ_m1/q_j(e))=1. Under the correspondence set up after Theorem <ref>, the latter conditioncan be shortened to u_m =0. (c) Suppose #S>2. For any strictly increasing sequence (n_j) in S \{e} contained in S, considered as a subsemigroup of (ℤ_+,<, max) in view of Theorem <ref> (i) above, we can find (q_n(e))_n ∈ S leading to a deformation satisfying Q_n_j= ℒ_n_j ∀ j and Q_n= ℒ_n ∪{n} for n ∈ S \{e} but not equal to any n_j.§.§ The dual objects of (S,*) We now come to the dual objects 𝒳_b(S) and S of hypergroup (S,*), as in Theorem <ref> and Corollary <ref> above. For this purpose, to begin with, we fix any v ∈𝒱 (or the corresponding u ∈𝒰) and consider the corresponding deformation (ℤ_+,*) as obtained in Theorem <ref> and Corollary <ref>. We freely follow the concepts, results and notation set up in this process.Let (S,<, ·) ≅ (ℤ_+, <, max) and other symbolssatisfy the conditions (ii)-(iii) of Theorem <ref> and let (S,*) be the corresponding deformed hypergroup with the Haar measure λ. Then the dual objects𝒳_b(S) and S of (S,*) are equal. Equipped with the topology of uniform convergence on compact subsets of S, S can be identified with the one point compactification ℤ_+^* (= ℤ_+ ∪{∞}) of ℤ_+.More precisely, the identification is given by k ↦_k, where _∞(n) =1 for all n ∈ℤ_+, and, for k ∈ℤ_+, _k is given by _k(n) = 1if n ≤ k,β_kif n=k+1,0ifn>k+1, where, β_k= -λ(ℒ_k+1)/λ(k+1)= -∑_j ∈ℒ_k+1v_j/v_k+1 =u_k+1/v_k+1-1= δ_k+1*δ_k+1(k+1)-1=q_k+1(k+1)-1. Before we start the proof, it is helpful to give a diagrammatic interpretation (Table <ref>) of the functions _k's, k ∈ℤ_+. As in the proof of Theorem <ref>, for n ∈ℕ, we may write q_n= ∑_j=0^n q_n(j) δ_j with q_n(j)>0 for 0 ≤ j<n, but q_n(n) ≥ 0 and ∑_j=0^n q_n(j)=1.We note that, for k ∈ℤ_+, β_k=q_k+1(k+1)-1 and -1 ≤β_k <0. First, we check that for k ∈ℤ_+^*=ℤ_+ ∪{∞}, _kis a character. Clearly, _∞ is a character. Fix k ∈ℤ_+. It is easy to see that _k(m*n)= _k(m) _k(n) for all m ≠ n ∈ℤ_+. It remains to check that for m ∈ℤ_+,(_k(m))^2= _k(m*m), i.e., (_k(m))^2=∑_j=0^m q_m(j) _k(j). Let m ∈ℤ_+. Case (i) m ≤ k : We have _k(m)= 1 and on the other hand, _k(m*m)= ∑_j=0^m q_m(j) _k(j) = ∑_j=1^m q_m(j) =1. Therefore, (<ref>) holds. Case (ii) m=k+1 : We have_k(m*m) = ∑_j=0^k+1 q_k+1(j) _k(j)= ∑_j=0^k q_k+1(j)+ q_k+1(k+1) (q_k+1(k+1)-1)=(1-q_k+1(k+1))+(q_k+1(k+1))^2- q_k+1(k+1) =(q_k+1(k+1)-1)^2 = (_k (m))^2. Therefore, (<ref>) holds. Case (iii) m>k+1 : We have _k(n)=0 for n>k+1.So, _k(m*m)= ∑_j=0^k q_m(j)+ q_m(k+1)(q_k+1(k+1)-1).Since by Theorem <ref> and by proof of part (iii)(b) of Theorem <ref>, we know that for 1 ≤ j < m, q_m(j)= q_m(0)/q_j(0) and 1-q_k+1(k+1) = q_k+1(0) ( 1+ ∑_j=1^k 1/q_j(0)), we get_k(m*m)= q_m(0) (1+∑_j=1^k 1/q_j(0) +1/q_k+1(0)(q_k+1(k+1)-1) )=0 = (_k(m))^2. Therefore, _k is a character. Also, _k is real-valued and S is hermitian. So, _k is a symmetric character. Now, note that for k_1 ≠ k_2 ∈ℤ_+^*, we have _k_1≠_k_2. We will show that injective map k ↦_k from ℤ_+^* to 𝒳_b(S) is onto. To prove this take any χ∈𝒳_b(S) such that χ≢1. Since χ(0)=1, there exists a least s ∈ℕ such that χ(s) ≠ 1. Let m,n ∈ℤ_+ with m<n. We have χ(m) χ(n) = χ(m*n)= χ(n).This shows that either χ(m)=1 or χ(n)=0. As a consequence, χ(n)=0 for all n>s.Now consider the setA={j ∈ℕ: χ(t)=0for allt ≥ j }.Then s+1 ∈ A and so A ≠ϕ. Hence, there exists a least element, say, w in A. Let, if possible, w=1. Then χ(0)=1 and χ(t) =0 for all t ≥ 1. So, for any m ≥ 1, we get χ(m*m)= χ(m)χ(m)=0. On the other hand, we have χ(m*m)=∑_j=0^m q_m(j) χ(j)= q_m(0) >0, which is a contradiction. Therefore, w>1. Note that χ(n)=1 for all n < w-1. In fact, let,if possible, χ(n_1) ≠ 1 for some n_1<w-1. Therefore we get χ(t)=0 for t ≥ w-1, which shows that w-1 ∈ A. This is a contradiction to the factw is the least element of A.Now, we come to χ(w-1)= β (say). Then, we have β^2 = (χ(w-1))^2= χ((w-1)*(w-1))= ∑_j=0^w-1 q_w-1(j) χ(j) = ∑_j=0^w-2 q_w-1(j) + q_w-1(w-1)β. This implies that β^2- q_w-1(w-1) β - (1-q_w-1(w-1))=0and therefore, β = 1 or q_w-1(w-1)-1=β_w-2 where w-2≥ 0. Let, if possible, β =1. As w ∈ A, χ(w*w)= χ(w)χ(w)=0;on the other hand, we have χ(w*w) = ∑_j=0^w q_w(j) χ(j) =∑_j=0^w-1 q_w(j) ≥ q_w(0) >0.Hence, β= β_w-2.Therefore, the character χ is given by χ(n) =1if n ≤ w-2, β_w-2 if n=w-1, 0ifn>w-1,i.e., χ= _w-2. Therefore, the map k ↦_k is onto. As an immediate consequence, 𝒳_b(S) = S. Further, it is clear from Table <ref> that S equipped with the topology of uniform convergence on compact subsets of S (i.e.,the topology of pointwise convergence in our case) is the one point compactification ℤ_+^* of ℤ_+. (i) Consider any countable infinite compact Hausdorff space X. Then anyprobability measure μ on X has the form μ =∑_x ∈ Xμ(x) δ_x=∑_x ∈ Xγ_x δ_x (say), where γ_x ≥ 0 and ∑_x ∈ Xγ_x =1. Trivially, μ has compact support.In particular, if X=S as in Theorem <ref> above, μ = ∑_j ∈ℤ_+_j δ_j+ _∞δ_∞, where _j ≥ 0∀ j ∈ℤ_+^* and ∑_j ∈ℤ_+_j+_∞ =1.Let C= {j ∈ℤ_+^*: γ_j >0}. Then the support of μ is given by C= CifCis finite or ∞∈ C, C ∪{∞} otherwise. (ii) For a non-constant complex-valued function f on S≅ℤ_+^*, which is eventually 1 in the sense that f(j)=1 for j≥ j_0 for some j_0 ∈ℕ,we have, forμ on S as in (i) above,∫_S f dμ = ∑_j ∈ℤ_+j<j_0f(j) _j+( 1 - ∑_j ∈ℤ_+j<j_0_j )= 1+∑_j ∈ℤ_+j<j_0 (f(j)-1) _j.(iii) For concepts, notation and details related to our next theorem,we refer to <cit.>, <cit.> and <cit.>. For the sake of convenience, we state Proposition 2.4.2 of <cit.> in the form we need as follows. “ S can be made into a hypergroup (with respect to pointwise multiplication) if for χ, χ' ∈S, there exists a (regular) probability measure μ_χ,χ'=∑_j ∈ℤ_+μ_χ, χ'(j) δ_χ_j+ μ_χ,χ'(∞) δ_χ_∞ = ∑_j ∈ℤ_+_j^χ, χ'δ_χ_j+ _∞^χ,χ'δ_χ_∞ (say) which satisfies (a) for v ∈ℤ_+,χ(v) χ'(v) = ∑_j ∈ℤ_+_j^χ, χ'_j(v)+_∞^χ,χ',(b) the map (χ, χ') ↦(μ_χ,χ') is continuous on ℤ_+^* ×ℤ_+^* → the space 𝒞(ℤ_+^*) of compact subsets of ℤ_+^* with Michael topology,(c) _∞∈(μ_χ, χ') if and only if χ=χ'.” In this case, we may set δ_χ*δ_χ'= μ_χ,χ', take involution as identity and make S a hermitian hypergroup, in fact. Let (S,<, ·) ≅ (ℤ_+, <, max) and other symbolssatisfy the conditions (ii)-(iii) of Theorem 3.2 and let (S,*) be the corresponding deformed hypergroup with the Haar measure λ. The dual space S of (S,*) becomes a countable compact hermitianhypergroup with respect to pointwise multiplication. More precisely, the convolution `*' on S is given byδ__m*δ__n = δ__min{m,n} form, n ∈ℤ_+withm ≠ norm=n=∞,∑_j ∈ℤ_+_j^m δ__j otherwise,where, _j^m= 0 for j<m, _m^m= 1+β_m ≥ 0, and for p ≥ 1,_m+p^m= ∏_j=m^m+p-1-β_j/1-β_j+1>0, and, β_j's are as in Theorem <ref>. Further, we also have 𝒳_b(S)=S≅ (S,*).We will freely use Remark <ref>.It is clear from Table <ref> that δ_χ_m*δ_χ_n := δ__min{m,n} for m, n ∈ℤ_+^* with m ≠ n or m=n= ∞.We now look for a suitable probability measure μ_m,m, in short, μ_m, for m ∈ℤ_+. Fix m ∈ℤ_+. ByRemark (<ref>)(i) above, we may write μ_m =∑_j ∈ℤ_+_j^m δ_j+ _∞^m δ_∞, where _j^m ≥ 0 for j ∈ℤ_+^* and ∑_j ∈ℤ_+_j^m + _∞^m =1. Now, for any t ∈ℕ, consider the function e_t on S defined by e_t(_n)= _n(t) for n ∈ℤ_+^*. We note from Table <ref> and Remark <ref>(ii) that the condition inRemark <ref> (iii)(a) takes the form_m(t)^2-1 = ∑_n<tn ∈ℤ_+ (_n(t)-1) _n^m for allt ∈ℕ. We may express it as: ([ 1-β_0 0 0 ⋯; 1 1-β_1 0 ⋯; 1 1 1-β_2 ⋯; ⋮ ⋮ ⋮ ⋱ ]) ([ _0^m; _1^m; _2^m;⋮ ]) = ([ 1-_m(1)^2; 1-_m(2)^2; 1-_m(3)^2; ⋮ ]).We consider any principal truncated submatrix of the infinite matrix on the L.H.S. of (<ref>) above. It is a lower triangular matrix and its diagonal entries 1-β_j's are all non-zero, and, therefore, it is invertible.Now, for j ∈ℕ with j≤ m, 1-_m(j)^2=0. So we get ^m_j=0 for j<m . So the system (<ref>) can be replaced by the system: ([ 1-β_m 0 0 ⋯; 1 1-β_m+1 0 ⋯; 1 1 1-β_m+2 ⋯; ⋮ ⋮ ⋮ ⋱ ]) ([ _m^m; _m+1^m; _m+2^m;⋮ ]) = ([ 1-β_m^2; 1; 1; ⋮ ])and _j^m=0 for j<m, if any.The first two rows of (<ref>) readily give,_m^m= 1+β_m≥ 0,and_m+1^m= -β_m/1-β_m+1.We note that by Theorem <ref>, _m^m>0 if and only if δ_m+1*δ_m+1(m+1)>0.Consecutive pairs of rows then give that, for p ≥ 2,_m+p^m = -β_m+p-1/1-β_m+p_m+p-1^m . So, for p ≥ 1, _m+p^m = ∏_j=m^m+p-1-β_j/1-β_j+1>0. Let s_p= ∑_j=0^p _m+j^mfor p ≥ 1. First note thats_1= _m^m+_m+1^m= 1+ β_m-β_m/1-β_m+1= 1+ (-β_m/1-β_m+1) β_m+1= 1+_m+1^m β_m+1.Assume that s_p= 1+_m+p^m β_m+p. Then s_p+1 =s_p+ _m+p+1^m=1+ _m+p^m β_m+p+ _m+p+1^m =1+ _m+p^m β_m+p- _m+p^m β_m+p/1-β_m+p+1= 1+ _m+p^m( -β_m+p/1-β_m+p+1) β_m+p+1 = 1+_m+p+1^m β_m+p+1. Therefore, by mathematical induction we get s_p= 1+ _m+p^m β_m+p for p ≥ 1.Let p ≥ 1.Then_m+p^mβ_m+p= β_m ∏_j=1^p-β_m+j/1-β_m+j, which, by using the fact that -1 ≤β_k <0 for all k ∈ℤ_+, gives0<-_m+p^mβ_m+p≤( 1/2)^p.Therefore, ∑_j ∈ℤ_+_j^m = lim_p →∞ s_p = 1. So, _∞^m= 1- ∑_j ∈ℤ_+_j^m = 0. Hence, we may set μ_m = δ__m*δ__m = ∑_j ∈ℤ_+_j^m δ__j. The continuity of the map (_m, _n) ↦(δ__m*δ__n)is easy tosee bynoting that for m ≠ n or m=n= ∞, (δ__m*δ__n) is singleton {_min{m,n}} and for m ∈ℤ_+, (δ__m*δ__m) = {_k ∈S: k ≥ m }∪{_∞} or {_k ∈S: k > m }∪{_∞} according as δ_m+1*δ_m+1(m+1)>0 or δ_m+1*δ_m+1(m+1)=0. It can be readily checked that _∞ is the identity element. We see that _∞∈(δ__m*δ__n) if and only if m=n. In view of Remark (<ref>)(iii),S is a compact hermitian hypergroup. For the rest of the proof, it is convenient to index the elements ofS by ℤ_+^* under the identification k ↦_k.Now, we calculate the dual space 𝒳_b(S), the space of continuous characters on S. We know that S ⊂S via n ↦_n, defined as _n(_k)= _n(k):= _k(n) for all k ∈ℤ_+^*. We can figure them out from Table <ref>; to elaborate, _0 ≡ 1, _1(0)=β_0, _1(k)=1 for k ≥ 1, and, for n≥ 2,_n(k)=0ifk<n-1,β_n-1 ifk=n-1, 1if k ≥ n. Next, let ξ∈𝒳_b(S) such that ξ≢1. Consider the set B= { s ∈ℤ_+: ξ(s) ≠ 1 }.Then B ≠ϕ. Take any s ∈ B, so ξ(s)≠ 1. Note that for m ∈ℤ_+, n ∈ℤ_+^*with m<n, we have ξ(m)= ξ(m)ξ(n), and therefore,either ξ(m)=0 or ξ(n)=1. In particular, we get that for allt ∈ℤ_+ with t <s, ξ(t)=0and hence t ∈ B.This implies that ℒ_s ∪{s}⊂ B. So either B= ℤ_+ or B has a maximum element. Let, if possible, B= ℤ_+. Then, for each n ∈ℕ, ξ=0 on ℒ_n. Since this holds for every n ∈ℕ we get ξ≡ 0 on ℤ_+. Since ξ(∞)=1, we get that ξ is not continuous, a contradiction. Hence, B has a maximum element, say w.Therefore, it follows from the definition of w that ξ(k)= 1 for all k>w. Also, since ξ(w) ≠ 1 we get ξ(k)=0 for k <w. Now, we calculate the value of ξ(w) = β (say). We haveβ^2= ξ(w)^2= ξ(w*w)= _w^w β+∑_j>wj ∈ℤ_+_j^w. This implies that β^2- _w^w β -(1-_w^w)=0 and therefore, β=1 or _w^w-1. But we know that β = ξ(w) ≠ 1 and hence ξ(w)= _w^w-1=β_w. Now, set n=w+1 ≥ 1, then ξ= ξ_n. Hence (S,*) ≅𝒳_b(S).(i)(a) The last part of Theorem <ref> can be proved directly with the help of <cit.>. Our proof is easy and self contained.(b) γ's in Theorem <ref> are related via _j^m= _j^0/_m^0 for j>m>0.(ii)(a) Any hypergroup (S,*) as in Theorem <ref> above may be thought of as hypergroup deformation of the compact linearly ordered semigroup (ℤ_+^*, <, min)with the order topology (cf. <cit.>, <cit.>, <cit.>) on the diagonal of ℤ_+ ×ℤ_+.(b) The class ofhypergroups S as in Theorem <ref> above lies between the class of countable compact hypergroups H_a (see Section 2 above) of <cit.> andthe class of compact almost discrete hypergroups of <cit.>.(iii) Let (S,<,max) be a finite “max" semigroup of cardinality n. Then it can be identified with (ℤ_n, <, max).Suitably truncated table of Table <ref> and truncated submatrices in (<ref>) and (<ref>) can be used to formulate and prove suitable analogues of Theorem <ref> and Theorem <ref>. § SEMICONVOS OR HYPERGROUPS ARISING FROM DEFORMATIONS OF COMMUTATIVE DISCRETE SEMIGROUPSWe considera commutative discrete semigroup (S,·)with identity e such that E_0(S) is not empty. We try to make (S,·) into a commutative discrete semiconvo or discrete hypergroup (S,*) by deforming the product on 𝒟_E_0(S):={(m,m): m ∈ E_0(S)}, the diagonal of E_0(S), or the idempotent diagonal of S, say.For n ∈ E(S), let q_n be a probability measure on S with finite support Q_n containing e. We express q_n= ∑_j ∈ Q_n q_n(j) δ_j with q_n(j)>0 for j ∈ Q_n and ∑_j ∈ Q_n q_n(j)=1. We look for necessary and sufficient conditions on S and { q_n: n ∈ E_0(S)} such that (S, *) with `*' defined below, is a commutative discrete semiconvo: δ_m*δ_n =δ_n*δ_m =δ_mn (m,n) ∈ S× S \𝒟_E_0(S),δ_n*δ_n=q_nn ∈ E_0(S).It is convenient to consider various relationships among the properties of (S,·) and of (S,*) presented in the following theorems before coming to our main Theorem <ref>. Let (S,·) be a commutative discrete semigroup with identity e. Suppose (S,*) is a commutative discrete semiconvo, where `*' and other related notation and concepts are as above. Then the following are equivalent.(i)For n ∈ E_0(S),Q_n ⊂ E(S).(ii)(E(S), ·) is a max-min type semigroup.(iii)S is inverse-free.(iv) G_1(S)={e}, i.e., S is action-free.(v)For n ∈ E_0(S) and m ∈S(= S \ E(S)) we have n ≠ nm.We prove the following implications: (i) ⇒ (iv) ⇒ (iii) ⇒ (v) ⇒ (i) and (iii) ⇔ (ii). (i) ⇒(iv). Assume Q_n ⊂ E(S) for n ∈ E_0(S). Consider any such n. Let, if possible, G_1(S) ≠{e} andchoose any m (≠ e) ∈ G_1(S) so that mn=n ≠ m. As (S,*) is a semiconvo, the associativity of `*' demands that (δ_m*δ_n)*δ_n=δ_m*(δ_n*δ_n). L.H.S. is equal to δ_mn*δ_n = δ_n*δ_n=q_n and R.H.S. is equal to δ_m*q_n. Since both sides are equal, we get q_n= q_n*δ_m. Now, m ∈(q_n*δ_m). So m ∈ Q_n, which is a contradiction. Hence, G_1(S)={e}, i.e., S is action-free.(iv) ⇒ (iii). Assume that S is not inverse-free. Then G(S) does not act as the identity on E_0(S). So there exists k ∈ E_0(S) such that km ≠ k for some m (≠ e) ∈ G(S). Note that k ≠ e and also δ_k ≠ q_k since e ∈(q_k). Now, there exists n (≠ e)∈ G(S)such that mn=e. Note that kn ≠ n because if kn=n then k=e by multiplying by m on both sides, which is not true. Similarly, km ≠ m. Consider the three different cases:(A) km ≠ kn, (B) km =kn ∉ E_0(S), and (C) km=kn ∈ E_0(S).Now, because (S,*) is a semiconvo, the associativity of `*' demands that(δ_km*δ_kn)*δ_k = δ_km*(δ_kn*δ_k). Case (A). When km ≠ kn. First, note that kn ≠ k because if kn=k then by multiplying both sides by m, we get k=km so that kn=k=km, which can not happen. The L.H.S. of (<ref>) is equal to δ_k^2mn* δ_k= q_k because k ∈ E_0(S) and R.H.S. of (<ref>) = δ_km*δ_k^2n= δ_km*δ_kn = δ_k^2mn= δ_k because km ≠ kn and k ∈ E_0(S). Since δ_k ≠ q_k, (<ref>) does not hold.Case (B). When km =kn ∉ E_0(S). Then L.H.S. of (<ref>) is equal to δ_k^2mn*δ_k = δ_k*δ_k=q_k because k ∈ E_0(S). And R.H.S. of (<ref>) is equal toδ_km*δ_kn = δ_k^2mn=δ_k because kn ∉ E_0(S) and k ∈ E_0(S).Therefore, (<ref>) does not hold.Case (C). When km = kn ∈ E_0(S). As (S,*) is a semiconvo, the associativity of `*' demands that(δ_k*δ_kn)*δ_n = δ_k*(δ_km*δ_n).Now, L.H.S.= δ_k^2n*δ_n= δ_kn*δ_n= δ_kn^2 and R.H.S.= δ_k*δ_kmn=δ_k*δ_k =q_k. Since e∈(q_k)= (R.H.S.)=(L.H.S.) = {kn^2}, we get kn^2=e, so that k ∈ G(S) and k ∈ G(S) ∩ E(S)= {e}, which is a contradiction. Therefore, (<ref>) does not hold.Therefore, in all three cases the associativity conditionin Definition <ref> does not hold so that (S,*) is not a semiconvo, which is a contradiction to the assumption. Hence, S is inverse-free. (iii) ⇒ (v). Let, if possible, n=nm. The associativity of `*' demands that for m,n ∈ S,(δ_n*δ_n)*δ_m = δ_n*(δ_n*δ_m). Now, as n ∈ E_0(S) and m ∈S, we get q_n*δ_m= δ_n*δ_nm= q_n.Therefore, ∑_j ∈ Q_n q_n(j) (δ_j*δ_m)= ∑_k ∈ Q_n q_n(k) δ_k. But as m ∈S, we getq_n(e) δ_m + ∑_e ≠ j ∈ Q_n q_n(j) δ_jm= q_n(e) δ_e+ ∑ _e ≠ k ∈ Q_n q_n(k) δ_k.This shows that there exists a j_e (≠ e) ∈ Q_n such that j_em=e which implies that j_e=m=e because S is inverse-free. This is a contradiction.Therefore, n ≠ nm.(v)⇒ (i). Let, if possible, there exist m ∉ E(S), which is in Q_n. Since n ≠ m, by the associativity of `*', we haveq_n*δ_m = δ_n*δ_nm. But n ≠ nm, so we have that δ_n*δ_nm=δ_n^2m= δ_nm. Therefore, ∑_j ∈ Q_n q_n(j)(δ_j*δ_m)= δ_nm.By the definition of `*', we getq_n(e) δ_m + ∑ _e ≠ j ≠ m,j ∈ Q_nq_n(j) δ_jm+ q_n(m) δ_m^2= δ_nm.So m=nm and m^2=nm. This shows that m=m^2 and hence m ∈ E(S), which is a contradiction. Therefore Q_n ⊂ E(S).(iii) ⇔ (ii).Suppose (iii) holds. We know that (E(S), ·) is a subsemigroup of (S,·). Let, if possible, there exist m,n ∈ E(S) with m ≠ mn ≠ n. Then e ≠ m ≠ n ≠ e. So, by Proposition <ref>(iv), mn ∈ E_0(S). Now, the associativity of `*' demands (δ_m*δ_n)*δ_mn= δ_m*(δ_n*δ_mn). We see that L.H.S. = δ_mn*δ_mn= q_mn. Further, R.H.S. = δ_m*δ_nmn= δ_m*δ_mn using the fact that n ∈ E(S). As m ≠ mn we get R.H.S.= δ_m^2n, which in turn = δ_mn because m ∈ E(S). Thus q_mn= δ_mn. Now, e ∈ Q_mn so mn=e; this is not true as it shows that m,n ∈ G(S), which is a contradiction. Hence E(S) is a max-min type semigroup, i.e., (ii) holds.The reverse implication follows from Proposition <ref>(ii). Let (S,·) be a commutative discrete semigroup with identity e. Suppose (S,*) is a commutative discrete semiconvo, where `*' and other related notation and concepts are as above. Then we have the following: (i) For m,n ∈ E_0(S) with m ≠ n ≠ nm, we have m ∉ Q_n and Q_n · m= {nm}.(ii)Under any (hence all) conditions (i)-(v) of Theorem <ref>, we have the following. (a) (E(S),*) is a hermitian (hence commutative) discrete hypergroup.(b) E(S) is finite or E(S) is isomorphic to (ℤ_+,<,max), where the order on E(S) is defined by m<n if mn=n ≠ m.(c) If #E(S) > 2, then for e ≠ m <n in E(S), we have the following. (α) q_n(e)=q_n(m) q_m(e) and(β) q_n(e) ( 1+ ∑_e ≠ k ∈ℒ_n1/q_k(e)) ≤ 1,where for n ∈ E(S), ℒ_n := {j ∈ E(S): j<n} .(d) S is an ideal of (S, ·). (e) For n ∈ E_0(S), m ∈S we get Q_n· m={nm}. (i). Let m,n ∈ E_0(S) with m ≠ n ≠ nm. The associativity of `*' demands that (δ_n*δ_n)* δ_m = δ_n *(δ_n *δ_m). Now L.H.S. = q_n*δ_m. Further, R.H.S. = δ_n* δ_nm which in turn = δ_nm using the fact that n ≠ nm and n ∈ E_0(S). Thus q_n*δ_m= δ_nm.Let, if possible m ∈ Q_n. Then L.H.S. = ∑_j ∈ Q_n q_n(j) (δ_j*δ_m)= q_n(e) δ_m+ q_n(m) q_m+ ∑_e ≠ j ≠ mj ∈ Q_n q_n(j) δ_jm.Since e ∈ Q_m and e ≠ m, the measure on L.H.S. is non-Dirac but R.H.S. is a Dirac measure, which is a contradiction. Therefore, m ∉ Q_n and hence, by q_n*δ_m= δ_nm, we get Q_n · m = {nm}.(ii)(a).We know that (E(S), ·) is a subsemigroup of (S, ·). Under the assumption, (i) in Theorem <ref> is satisfied. Therefore, `*' induces a product on E(S), again denoted by `*', i.e., *= *|_E(S) : E(S) × E(S) → M_F,p(E(S)). Since we assume that e ∈ Q_n, using (iii) of Theorem <ref> it is easy to see that for all m,n ∈ E(S),e ∈(δ_m*δ_n) if and only if m=n. The associativity follows as (S,*) is a discrete semiconvo. Hence, `*' induces a hermitian discrete hypergroup structure on E(S).(b). Under the assumption that E(S) is a max-min type semigroup, by Item <ref>(ii), it can be given a linear order by m<n if mn=n ≠ m and made a "max" semigroup with identity e. Therefore, it follows from Theorem <ref> that E(S) is finite or E(S) is isomorphic to (ℤ_+,<,max).(c). This follows by (b) above as (E(S), ·) is a “max" semigroup and (E(S),*) is a hermitian discrete hypergroup.(d). Let k,n ∈S; then by the associativity of `*' we get (δ_nk*δ_n)*δ_k= δ_nk*(δ_n*δ_k) which is equivalent to δ_(nk)^2= δ_nk*δ_nk as n,k ∈S. Under the assumption S is inverse-free, it follows that nk ≠ e. This forces nk ∉ E_0(S). Therefore, S is a subsemigroup of (S, ·). Now, to show that S is an ideal it is enough to show that for k ∈ E_0(S) and n ∈S,nk ∈S. This can be proved in a manner similar to the proof that S̃ is a subsemigroup by using k ≠ nk and the associativity condition (δ_nk*δ_k)*δ_n= δ_nk*(δ_k*δ_n).(e).Let n ∈ E_0(S), m ∈S. The associativity of `*' demands that (δ_n*δ_n)*δ_m = δ_n*(δ_n*δ_m). By the definition of `*', we get q_n*δ_m=δ_n*δ_nm. Since, by (d) above, n ≠ nm, we get q_n*δ_m = δ_nm, which, using (i), shows that Q_n · m = {nm}. Now, we state our main theorem.Let (S,·) be acommutative discrete semigroup with identity e such that S is action-free. Let `*' and other related notation and concepts be as above. Then (S, *) is a commutative discrete semiconvo if and only if the following conditions hold. (i)E(S) is finite or E(S) is isomorphic to (ℤ_+,<,max), where the order on E(S) is defined by m<n if mn=n ≠ m.(ii) (S, ·) is an ideal of (S, ·).(iii) Q_n ⊂ E(S) for n ∈ E_0(S). (iv) If n ∈ E_0(S) and m ∈S then Q_n· m ={nm}. (v) For n ∈ E_0(S), we have ℒ_n ⊂ Q_n ⊂ℒ_n ∪{n},where for n ∈ E(S), ℒ_n := {j ∈ E(S): j<n}.(vi)If #E(S) > 2, then for e ≠ m <n in E(S), we have the following: (α) q_n(e)=q_n(m) q_m(e) and(β) q_n(e) ( 1+ ∑_e ≠ k ∈ℒ_n1/q_k(e)) ≤ 1. Further, under these conditions, E(S) is a hermitian discrete hypergroup. Moreover, S is a hermitian discrete hypergroup if and only if S = E(S). We assume all the conditions and prove that (S,*) is a commutative discrete semiconvo. Since S has an identity, it is enough to check the associativity of `*' to prove that (S,*) is a semiconvo. We will prove that for m,n,k ∈ S(δ_m*δ_n)*δ_k = δ_m*(δ_n*δ_k).We divide the proof of (<ref>) into various steps. (i) If m,n,k ∈S, then using the fact the S is an ideal (hence subsemigroup) of S we have that the product of any two elements is in S. Therefore, both sides of (<ref>) become δ_mnk. (ii) If m,n ∈S and k ∈ E(S) then both sides of (<ref>) become δ_mnk using the factS is an ideal of S.(iii) Suppose m ∈S and n,k ∈ E(S). If n or k is e then (<ref>) is trivial. Therefore, we can assume that n,k ∈ E_0(S). (a) When n=k ∈ E_0(S).The condition (<ref>) becomes(δ_m*δ_n)*δ_n= δ_m*(δ_n*δ_n). Now, δ_m*(δ_n*δ_n)= δ_m*q_n = ∑_j ∈ Q_n q_n(j) (δ_m*δ_j). Because for n ∈ E_0(S), Q_n ⊂ E(S) and m · Q_n = {mn}, we get R.H.S. of (<ref>)= δ_m*(δ_n*δ_n)= ∑_j ∈ Q_n q_n(j) δ_mj=∑_j ∈ Q_n q_n(j) δ_mn = ( ∑_j ∈ Q_n q_n(j) ) δ_mn= δ_mn.Using the fact S is an ideal and n^2=n, we get L.H.S. of (<ref>) =(δ_m*δ_n)*δ_n= δ_mn. Therefore, theassociativity (<ref>)holds.(b) For n ≠ k ∈ E_0(S), R.H.S. and L.H.S. of(<ref>) bothbecome δ_mnk using the fact the S is an ideal in (S, ·). Therefore, (<ref>) holds.(iv) We now come to cases when n ∈S and m,k ∈ E(S), or k ∈S and m,n ∈ E(S). These can be dealt with by arguments similar to those in (ii) and (iii) above. (v) When m,n,k ∈ E(S), associativity of `*' can be proved in a manner similar to that for proof of Theorem <ref>. Hence, (S, *) is a semiconvo. Converse part of the first part of the theorem follows from Theorem <ref> andTheorem <ref>.Since (S,*) has a semiconvo structure and (S,·) is action-free, itfollows from Theorem <ref> that (E(S),*) has a hermitian hypergroup structure. Moreover, if S is hermitian hypergroup then for any e ≠ m ∈ S, e ∈(δ_m* δ_m); so we get m ∈ E_0(S) or m ∈S with m^2=e. But S is an ideal and e in not in S. So the second condition is not possible, which shows that m ∈ E_0(S). Therefore, S ⊂ E(S). Hence S=E(S).(i) We can formulate a suitable analogue of Corollary <ref> based on Theorem <ref> above.(ii) In view of Theorem <ref> and Theorem <ref> above, identifying E(S) as a finite subset of ℤ_+ or ℤ_+ itself, for n>1,#Q_n ≥ 2. But #Q_1 can very well be 1. As already remarked, this is the case for Dunkl-Ramirez hypergroup for a= 1/2. We note that some proofs become simpler, shorter or different when we take #Q_n > 1 for all n. (iii) R. C. Vrem <cit.> defined the join of two hypergroups. Because S= E(S) ⊔S, at the firstglance, our semiconvo (S,*) looks like the join of a hermitian discrete hypergroup (E(S),*) and a discrete semigroup (S, ·), but it is something different from join.(iv) We can utilize the examples in Section <ref> to illustrate various parts oftheorems above. In particular, it is a consequence of Theorem <ref> that for Example <ref>, `*' does not induce a semiconvo structure on S. The following semigroup (S, ·) is an example of a semigroup (S, ·) satisfying the conditions of Theorem <ref> and for which S is non-empty. For T= {1-1/r+1: r ∈ℤ_+ }, let S = T∪ℕ with `·' defined as follows.m· n= max{m,n} if morn ∈ T, m+nif m,n ∈ℕ. Then S is an inverse-free commutative discrete semigroup with identity 0. Further, E(S)=T, which is isomorphic to (ℤ_+,<, ·). Also, S=ℕ is a prime ideal in S.§ ACKNOWLEDGMENTVishvesh Kumar thanks the Council of Scientific and Industrial Research, India, for its senior research fellowship. He thanks his supervisors Ritumoni Sarma and N. Shravan Kumar for their support and encouragement.A preliminary version of a part of this paper was included in the invited talk by Ajit Iqbal Singh at the conference “The Stone-ech compactification : Theory and Applications, at Centre for Mathematical Sciences, University of Cambridge, July 6-8 2016" in honour of Neil Hindman and Dona Strauss. She is grateful to the organizers H.G. Dales and Imre Leader for the kind invitation, hospitality and travel support. She thanks them, Dona Strauss and Neil Hindman and other participants for useful discussion. She expresses her thanks to the Indian National Science Academy for the position of INSA Emeritus Scientist and travel support.aaa=17ptAlagh M. Alaghmandan and E. Samei,Weighted discrete hypergroups, Indiana Univ. Math. J., 65(2), (2016) 423-451.Bloom W. R. Bloom and Herbert Heyer, Harmonic analysis on probability measures on hypergroups, De Gruyter, Berlin (1995) (Reprint: 2011). Hugo J. F. Berglund, H. D. Junghenn and P. Milnes, Analysis on semigroups: Function spaces, compactifications, representations, Canadian Mathematical Society Series of Monographs and Advanced Texts (John Wiley & Sons, Inc.), New York (1989).Ajit Ajit Kaur Chilana and Ajay Kumar, Ultra-Strong Ditkin Sets in Hypergroups, Proc. Amer. Math. Soc., 77(3) (1979) 353-358.redbook W. C. Connett, O. Gebuhrer and A. L. Schwartz, editors. Application of hypergroups and related measure algebras, Providence, R. I., American Mathematical Society, Contemporary Mathematics, 183 ( 1995).Dunkl C. F. Dunkl, The measure algebra of a locally compact hypergroup, Trans. Amer. Math. Soc., 179 (1973) 331-348.Ramirez C. F. Dunkl and D. E. Ramirez, A family of countable compact _⋆-hypergroups, Trans. Amer. Math. Soc., 202 (1975) 339-356.DR C. F. Dunkl and D. E. Ramirez, Representations of commutative semitopological semigroups, Lecture Notes in Mathematics, 435, Springer-Verlag, Berlin-New York, (1975). Zuck E. Hewitt and H. S. Zuckerman, Structure theory for a class of convolution algebras,Pacific J. Math., 7(1) (1957) 913-941.Hewitt E. Hewitt and K. A. Ross, Abstract harmonic analysis Vol. I: Structure of topological groups, integration theory, group representations, 2nd edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 115, SpringerVerlag, Berlin-New York, (1979). Hofmann K. H. Hofmann and J. D. Lawson, Linearly ordered semigroups: historical origins and A. H. Clifford’s influence, Semigroup theory and its applications (New Orleans, LA, 1994). London Mathematical Society. Lecture Note Series vol. 231,Cambridge University Press, Cambridge, (1996) 15-39. Jewett R. I. Jewett, Spaces with an abstract convolution of measures, Adv. Math., 18 (1975) 1-101.Kawakami S. Kawakami, T. Tsurii and S. Yamanaka, Deformations of finite hypergroups, Sci. Math. Jpn., 79(2) (2016) 213-223.Lasser R. Lasser, Discrete commutative hypergroups, Inzell Lectures on Orthogonal Polynomials (Inzell, 2001), Advances in the Theory of Special Functions and Orthogonal Polynomials 2, Nova Sci. Publ. Hauppauge, New York, (2005) 55-102.Petrich M. Petrich, Semicharacters of the cartesian product of two semigroups,Pacific J. Math., 12(2) (1962) 679-683.Ross K. A. Ross, The structure of certain measure algebras,Pacific J. Math., 11(2) (1961) 723-737.Xu K. A. Ross and Daming Xu, Hypergroup deformations and Markov chains, J. Theoret. Probab., 7(4), (1994) 813-830.greenbook K. A. Ross, J. M. Anderson, G. L. Litvinov, Ajit Iqbal Singh, V. S. Sunder and N. J. Wildberger, editors. Harmonic analysis and hypergroups, Trends Math., Birkhuser Boston, Boston, MA, (1998).Wolf W. Ruppert, Compact semitopological semigroups: an intrinsic theory, Lecture Notes in Mathematics, vol. 1079. Springer-Verlag, Berlin (1984) Spector R. Spector, Apercu de la thorie des hypergroupes, Analyse Harmonique sur les Groupes de Lie (1975) 643-673 (Sem. Nancy-Strasburg 1973-1975, Lecture Notes in Mathematics 497, Springer-Verlag, Berlin, Heidelberg, New York 1975).Voit M. Voit, Compact almost discrete hypergroups, Canad. J. Math., 48 (1996) 210-224.Vrem1 R. C. Vrem, Harmonic analysis on compact hypergroups, Pacific J. Math.,85(1) (1979) 239-251.Vrem R. C. Vrem, Hypergroup joins and their dual objects,Pacific J. Math., 111(2) (1984) 483-495. | http://arxiv.org/abs/1707.09004v5 | {
"authors": [
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"Ajit Iqbal Singh"
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"published": "20170727190923",
"title": "Hypergroup Deformations of Semigroups"
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Department of Mathematics Boston CollegeChestnut Hill, MA 02467 [email protected] University of Oslo Box 1053, Blindern, 0316 Oslo, [email protected] emptyWe study how the geometry of a projective variety X is reflected in the positivity properties of the diagonal Δ_X considered as a cycle on X× X. We analyze when the diagonal is big, when it is nef, and when it is rigid. In each case, we give several implications for the geometric properties of X. For example, when the cohomology class of Δ_X is big, we prove that the Hodge groups H^k,0(X) vanish for k>0. We also classify varieties of low dimension where the diagonal is nef and big.Positivity of the diagonal John Christian Ottem December 30, 2023 ==========================§ INTRODUCTIONemptyThe geometry of a projective variety X is determined by the positivity of the tangent bundle T_X. Motivated by the fact that T_X is the normal bundle of the diagonal Δ_X in the self-product X× X, we will in this paper study how the geometry of X is reflected in the positivity properties of Δ_X itself, considered as a cycle on X× X. The prototypical example of a variety with positive diagonal is projective space; the central theme of the paper is that positivity of the diagonal forces X to be similar to projective space. In dimension 1, this perspective is already quite vivid: ^1 is the only curve where the diagonal is an ample divisor; elliptic curves have nef, but not big diagonals; and for higher genus, the diagonal is contractible, hence `negative' in a very strong sense. In general, when X has dimension n, the diagonal determines a class in the space N_n(X× X) of n-dimensional cycles modulo numerical equivalence, and we are interested in how this class sits with respect to the various cones of positive cycles of X× X.Note that in the absence of the Hodge conjecture, we often do not even know the dimension of the space N_n(X× X).Thus we develop techniques to prove positivity or rigidity without an explicit calculation of the positive cones.The subsections below recall several different types of positivity and give a number of theorems illustrating each.At the end of the introduction we will collect several examples of particular interest. §.§ Big diagonal A cycle class α is said to be big if it lies in the interior of the closed cone generated by classes of effective cycles.Bigness is perhaps the most natural notion of positivity for cycles.We will also call a cycle homologically big if it is homologically equivalent to the sum of an effective ℚ-cycle and a complete intersection of ample ℚ-divisors.Homological bigness implies bigness, and equivalence of the two notions would follow from the standard conjectures. The primary example of a variety with (homologically) big diagonal is projective space. In this case, the diagonal has a Künneth decomposition of the formΔ_X=∑_p+q=nπ_1^*h^p·π_2^*h^q where h is the hyperplane divisor, and this class is evidently big. Of course, the same argument applies also for the fake projective spaces, that is, smooth varieties ≠^n with the same betti numbers as ^n. In dimension 2, there are exactly 100 such surfaces <cit.>, <cit.>, and they are all of general type. Thus unlike the case of curves, we now allow examples with positive Kodaira dimension, but which are still `similar' to projective space in the sense that they have the same Hodge diamond. More generally, homological bigness of the diagonal implies the vanishing of the `outer' Hodge groups of X.Following ideas of <cit.>, we show:Let X be a smooth projective variety.If Δ_X is homologically big, then H^i,0(X) = 0 for i >0.An interesting feature of this result is that the proof makes use of non-algebraic cohomology classes to control effective cycles.When X is a surface with a big diagonal, Theorem <ref> implies the existence of a cohomological decomposition of the diagonal; we discuss this relationship in more depth in Section <ref>.Let X denote the blow-up of ℙ^3 along a planar elliptic curve which does not admit complex multiplication.In Example <ref> we verify that Δ_X is big even though h^2,1(X) ≠ 0.Thus the vanishing results for Hodge groups as in Theorem <ref> are optimal for threefolds. We emphasize that even amongst varieties satisfying the hypotheses of Theorem <ref> there are very few with big diagonal.We will prove several additional strong constraints on the geometry of a variety with big diagonal. For example, such varieties can not admit a morphism to variety with smaller dimension.Nevertheless, the complete classification of varieties with big diagonal seems subtle (see Section <ref>). §.§ Big and nef diagonalA cycle class is said to be nef if it has non-negative intersection against every subvariety of the complementary dimension.Diagonals which are both big and nef are positive in the strongest possible sense, and we classify such varieties in low dimensions. Let X be a smooth projective variety. * If X=2 and Δ is nef and big then X has the same rational cohomology as ℙ^2: it is either ℙ^2 or a fake projective plane.* If X=3 and Δ is nef and homologically big then X has the same rational cohomology as ℙ^3: it is either ℙ^3, a quadric, a del Pezzo quintic threefold V_5, or the Fano threefold V_22.It is interesting to compare this result to Mori's theorem that the only smooth variety with ample tangent bundle is ℙ^n.By switching to the perspective of numerical positivity of Δ_X, we also include varieties with the same cohomological properties as projective space.In higher dimensions we make partial progress toward a classification.In particular, we show that N_k(X) ≅ℝ for every 0 ≤ k ≤ X, provided that the diagonal is big and universally pseudoeffective (this is a stronger condition than nefness, in the sense that π^*Δ_X is required to be pseudoeffective for every morphism π: Y → X× X). §.§ Dual positivity We also study nefness or universal pseudoeffectiveness in the absence of bigness.Let X be a smooth projective variety.If Δ_X is nef (resp. universally pseudoeffective) then every pseudoeffective class on X is nef (resp. universally pseudoeffective). For example, a surface with nef diagonal must be a minimal surface. If X has a nef tangent bundle, then Δ_X is nef.Campana and Peternell predict that any Fano manifold with nef tangent bundle is in fact rational homogeneous.Note that Theorem <ref> is compatible with this conjecture: on a homogeneous variety every pseudoeffective class must be nef.While varieties with nef tangent bundle will have nef diagonal, the converse is not true; for example, fake projective planes have anti-ample tangent bundle but their diagonals are universally pseudoeffective. It is interesting to look for other sources of feedback between nefness of the diagonal and nefness of the tangent bundle.For example:Let S be a smooth surface of Kodaira dimension ≤ 1 whose diagonal is nef.Then T_S is a nef vector bundle, except possibly when S is a minimal properly elliptic surface with no section. The exception is necessary: Example <ref> constructs a hyperelliptic surface with no section which has nef diagonal.Also, the natural extension to general type surfaces is false: a fake projective plane has nef diagonal. §.§ Examples[Toric varieties] Let X be a smooth toric variety.Theorem <ref> shows that Δ_X is big if and only if every nef cycle on X is big.One might expect that the only toric varieties with big diagonal are the projective spaces, but this turns out not to be the case.For example, <cit.> gives an example of a toric threefold of Picard rank 5 with big diagonal.By combining our work with results of <cit.> we can classify toric varieties with nef diagonal: Let X be a smooth projective toric variety.Then Δ_X is nef if and only if X is a product of projective spaces. [Hypersurfaces] Let X be a smooth hypersurface of degree ≥ 3 and dimension ≥ 2.It is easy to see that the diagonal of X is not nef.For bigness, we show: For a smooth Fano hypersurface of degree ≥ 3 and dimension ≤ 5, the diagonal is not big.For a quadric hypersurface, Δ_X is big if and only if the dimension is odd, in which case it is a fake projective space (see Section <ref>).[K3 surfaces] By Theorem <ref> the diagonal of a K3 surface is not big.We prove the diagonal of a K3 surface is never nef by using the birational geometry of ^2(X) as described by <cit.>.For general K3 surfaces we can say more: using a deformation argument we show For a very general K3 surface, the diagonal is the unique effective ℝ-cycle in its numerical class and it lies on an extremal ray of the pseudoeffective cone. We expect the statement holds for every K3 surface, and we prove it for some specific classes (for example, for K3 surfaces of degree divisible by 4 with Picard rank 1).§.§ AcknowledgementsWe want to thank M. Fulger for his input and for numerous corrections and improvements. We thank F. Catanese for a discussion about fake quadrics, E. Macrì for a discussion about ^2 of K3 surfaces, and X. Zhao for alerting us to the work of Lie Fu <cit.>.BL was supported by an NSA Young Investigator Grant and by NSF grant 1600875, and JCO was supported by RCN grant 250104.§ BACKGROUNDThroughout we work over ℂ.For a projective variety X, we will let Δ_X denote the diagonal in the self-product X× X. The two projections of X× X will be denoted by π_1 and π_2 respectively.§.§ Cones of positive cycles Let X be a projective variety.We let N_k(X)_ℤ denote the group of k-cycles modulo numerical equivalence.The numerical class of a cycle Z is written [Z] and we use ≡ to denote numerical equivalence.The abelian group N_k(X)_ℤ forms a lattice inside the numerical group N_k(X) := N_k(X)_ℤ⊗_ℤℝ, which is a finite dimensional real vector space. We define N^k(X) to be the vector space dual to N_k(X).When X is smooth of dimension n, capping against [X] defines an isomorphism N^n-k(X) → N_k(X), and we will switch between subscripts and superscripts (of complementary dimension) freely.We say that a numerical class is effective if it is the class of an effective ℝ-cycle.The pseudoeffective cone _k(X) in N_k(X)is the closure of the cone generated by effective classes.A class is big when it lies in the interior of _k(X).The nef cone ^k(X) in N^k(X) is the dual of the pseudoeffective cone, and a cycle is called nef if its class belongs to this cone. That is, a cycle is nef if it has non-negative intersection numbers with all k-dimensional subvarieties. The basic properties of these cones are verified in <cit.>: they are full-dimensional, convex, and contain no lines.Pseudo-effectiveness is preserved by pushforward, and nefness is preserved by pullback.It is useful to have more a restrictive form of dual positivity: Let X be a projective variety.A cycle class α∈ N^k(X) is said to be universally pseudoeffective if π^*α is pseudoeffective for every morphism π: Y → X.The primary examples of such cycles are complete intersections of ample divisors, or more generally, Chern classes of globally generated vector bundles. As suggested by the superscript demarcation, the universally pseudoeffective cone is naturally contravariant for morphisms and should be thought of as a “dual” positive cone by analogy with the nef cone. §.§ Positive homology classesLet H_2k(X)_alg⊆ H_2k(X) denote the subspace of algebraic homology classes, i.e., the image of the cycle class map cl:CH_k(X)⊗ℝ→ H_2k(X). Let E_2k(X)⊂ H_2k(X)_alg denote the cohomological effective cone.We say a k-cycle Γ is homologically big if its cohomology class [Γ] lies in the interior of E_2k(X).In general, for smooth complex projective varieties, cohomological implies numerical equivalence, so any homologically big cycle is big in the usual sense. If Grothendieck's standard conjecture D holds on X, namely that numerical and cohomological equivalence coincide, then N_k(X)=H_2k(X)_alg and the two notions of `big' coincide. In the special case of a self-product, it is known that D holds on X× X if and only if the Lefschetz standard conjecture holds on X (i.e. the inverse of the hard Lefschetz isomorphism is induced by a correspondence). This is known to hold for surfaces <cit.>, and for threefolds not of general type by results of Tankeev <cit.>. We will in this paper be mostly interested in surfaces, and use the fact that the two notions coincide in this case without further mention. We will also require the following result of <cit.> which follows from the theory of relative Hilbert schemes: Let f: 𝒳→ T be a smooth family of projective varieties over a smooth variety T and suppose that α∈ H^k,k(𝒳,ℤ) has that the restriction to a very general fiber is represented by an effective cycle.Then α|_𝒳_t is an effective class for any fiber 𝒳_t.We can use this result when 𝒳 is a family of varieties for which homological and numerical equivalence coicide (e.g., fourfolds). In this case, the theorem also implies that a class which restricts to be big on a very general fiber has big restriction on every fiber.§ VARIETIES WITH BIG DIAGONALIn this section we consider the geometric implications of big diagonals. Let X be a projective variety.If X carries a universally pseudoeffective class α∈ N^k(X) that is not big, then Δ_X is not big. In particular, if X carries a nef divisor that is not big, then Δ_X is not big. Let n denote the dimension of X.Since α is not big, there is some non-zero nef class β∈ N^n-k(X) that has vanishing intersection with α.Then consider γ := π_1^*α·π_2^*β on X × X.Clearly γ is a nef class: if E is an effective cycle of dimension n, then π_1^*α· E is still pseudoeffective, so that it has non-negative intersection against the nef class π_2^*β.Since γ·Δ_X = 0, we see that Δ_X can not be big.Let X be a projective variety of dimension n.If X admits a surjective morphism f: X → Y to a variety of dimension < n, then Δ_X is not big. 𝒦 It is sometimes helpful to consider non-algebraic classes as well. In this setting, we recall that a (1,1)-cohomology class α is defined to be nef if it is the limit of Kähler classes.Let X be an n-dimensional smooth projective variety admitting a non-zero nef cohomology class α∈ H^1,1(X,ℝ) such that α^n=0. Then Δ_X is not homologically big.Let ω be a Kähler form on X× X. Let α be a nef (1,1)-form on X and let 0< k< n be an integer so that α^k≠ 0, but α^k+1= 0. The two pullbacks π_1^*α^k and π_2^*α∪ω^n-k-1 are weakly positive forms on X× X, and hence their productβ=π_1^*α^k∪π_2^*α∪ω^n-k-1is a weakly positive (n,n)-class on X× X <cit.>. Now the main point is that β is nef, in the sense that ∫_Z β≥ 0 for all subvarieties Z⊂ X × X. This is because β restricts to a non-negative multiple of the volume form on Z for every smooth point on it (cf.<cit.>). Note however that it is not in general the case that the product of two nef classes remains nef,as shown in <cit.>. If Δ_X is homologically big, then we can write [Δ_X] = ϵ h^n+Z where ϵ>0, h is an ample line bundle and Z is an effective cycle. Moreover, since h is ample and α is nef, the following two inequalities hold:∫_Z β≥ 0 ∫_X h^n ∪β>0However, these contradict β·Δ_X=0, which holds by our assumptions on α and k. Bigness of the diagonal is compatible with pushforward:Let f: X → Y be a surjective morphism of projective varieties.If Δ_X is big, then so is Δ_Y. Note that by Lemma <ref> the hypothesis is never satisfied if 0< Y <X, so the main interest is in the generically finite case. Let n denote the dimension of X and d denote the dimension of Y.Fix an ample divisor H on X.Then Δ_X· H^2n-2d is a big class on X.Consider the induced map f × f: X × X → Y × Y.The set-theoretic image of Δ_X is Δ_Y; in particular,(f × f)_*: [Δ_X] · H^2n-2d is proportional to Δ_Y.Since the pushforward of a big class under a surjective map is still big, we see that Δ_Y is also big. §.§ Cohomological criteriaThe main result of this section is the following theorem.Let X be a smooth projective variety with homologically big diagonal. Then H^k,0(X)=0 for all k>0. In particular, no varieties with trivial canonical bundle can have homologically big diagonal. Following <cit.> and <cit.>, we will utilize the Hodge–Riemann relations to find faces of the effective cones of cycles. To set this up, let ω be a Kähler form on a smooth projective variety W.Note that a cohomology class in H^k,0(W) is automatically primitive.Thus by the Hodge–Riemann bilinear relations, the bilinear form on H^k,0(W) given byq(a,b)=ε∫_W a ∪b̅∪ω^n-kis positive definite. Here ε=1 if k is even, and ε=√(-1) if k is odd. Now fix a Kähler form ω on X × X and let σ be a non-zero closed (k,0)-form on X. Consider the product β=ε(π_1^*σ -π_2^*σ)∪(π_1^*σ̅-π_2^*σ̅)∪ω^n-k. This is a non-zero (n,n)-form on X× X, which by construction vanishes on the diagonal. Now, if Z⊂ X× X is an n-dimensional subvariety, the Hodge–Riemann relations (applied on a resolution of Z) imply that β· Z≥ 0. Similarly, β· h^n>0 for an ample divisor h on X× X. Finally, since Δ_X·β=0, it follows that Δ cannot be homologically big.The above theorem can also be deduced from <cit.>, which is proved using a similar argument. Even when the diagonal is only (numerically) big, we can still show that H^1,0(X) vanishes.First suppose that A is an abelian variety of dimension n.Then the diagonal is the fiber over 0 of the subtraction map f: A × A → A.In particular, Δ_A has vanishing intersection against the nef class f^*L · H^n-1 where L is an ample divisor on A and H is an ample class on A × A.Under suitable choices, the class β in the proof of Theorem <ref> constructed from H^1,0(A) will be exactly this n-cycle.More generally, when X is a smooth projective variety with non-trivial Albanese, we have a subtraction map X × X → A.The diagonal will have vanishing intersection against the pullback of an ample divisor from A under the subtraction map intersected with an appropriate power of an ample divisor on X × X.Again, this is essentially the same as the class β constructed in the proof above.We give an example of a smooth Fano threefold X with homologically big diagonal which satisfies h^2,1(X) ≠ 0.Thus Theorem <ref> is optimal in the sense that the other Hodge groups need not vanish.Let X be the blow-up of ℙ^3 along a planar elliptic curve C which does not have complex multiplication.Let H denote the pullback of the hyperplane class to X and E denote the exceptional divisor.It is easy to verify that:_2(X) = ⟨ H-E, E ⟩ _2(X) = ⟨ 3H-E,H ⟩ _1(X) = ⟨ HE, H^2-HE ⟩ _1(X) = ⟨ H^2, 3H^2 - HE ⟩ On X × X let H_i,E_i denote the pullbacks of H and E under the ith projection.Since C does not have complex multiplication, N_3(X × X) has dimension 11: it is spanned by Δ_X and the non-zero products of H_1,E_1,H_2,E_2.Recall that C × C has three-dimensional Neron-Severi space spanned by the fibers F_1,F_2 of the projections and the diagonal Δ_C.Let Z_a,b,c denote the class in N_3(X × X) obtained by pulling the divisor aF_1 + bF_2 + cΔ_C back from C × C to E × E and then pushing forward to X × X.An intersection calculation shows thatZ_a,b,c = a/3H_1E_1E_2 + b/3H_2E_1E_2 + c ( H_1^3 + H_1^2H_2 + H_1H_2^2 + H_2^3 - Δ_X).Applying this to the effective divisor 2F_1+2F_2-Δ, we obtainΔ_X = Z_2,2,-1 + 1/6H_1E_1E_2 + 1/6H_2E_1E_2 +5/6 H_1E_1(H_2 - E_2) + 5/6H_2E_2(H_1 - E_1)+ 5/6H_1H_2(H_1 - E_1) + 5/6H_1H_2 (H_2 - E_2) +1/6H_1^2(H_2 - E_2) + 1/6H_2^2(H_1 - E_1)+ 1/6H_1^2E_2 + 1/6H_2^2E_1 +H_1^3 + H_2^3and since the terms are all effective and together span N_3(X × X) we see Δ_X is big (and hence homologically big, since X is a rational threefold). We also note in passing that Δ_X is not nef, since it has negative intersection against the effective cycle H_1E_1E_2. §.§ Criteria for bigness There is one situation where it is easy to test for bigness of the diagonal, namely when the effective cones of X× X are as simple as possible.Let X be a smooth projective variety of dimension n.Suppose that for every k_k(X × X) = ∑_i+j=kπ_1^*_i(X) ·π_2^*_j(X).Then Δ_X is big if and only if every nef class on X is big.We first claim that the nef cone has the expression^k(X × X) = ∑_i+j=kπ_1^*^i(X) ·π_2^*^j(X).The containment ⊇ is clear from the description of the pseudoeffective cone.Conversely, it suffices to show that every class generating an extremal ray of _k(X × X) has vanishing intersection against some element of the right hand side.By hypothesis such classes have the form π_1^*α_i·π_2^*α_k-i where α∈_i(X) and α_k-i∈_k-i(X) both lie on extremal rays.Choose nef classes β^i∈^i(X) and β^k-i∈^k-i(X) satisfying α_i·β^i = 0 and α_k-i·β^k-i = 0.Then(π_1^*α_i·π_2^*α_k-i) · (π_1^*β^i·π_2^*β^n-i) = 0 Now suppose that Δ_X is not big.Then it must have vanishing intersection against some α∈^n(X × X) which lies on an extremal ray.By the expression above, such a class has the form α = π_1^*β_j·π_2^*β_n-jwhere for some constant j we have β_j∈^j(X) and β_n-j∈^n-j(X).But then β_j·β_n-j = 0 as classes on X.Since β_j has vanishing intersection against a nef class, it can not be big.Conversely, suppose that there is a nef class in N_k(X) which is not big.Since there are also big nef classes in N_k(X), by convexity of the nef cone we can find a nef class β∈ N_k(X) on the boundary of the pseudoeffective cone.Thus there is another nef class β' such that β·β' = 0.Arguing as above, we see that π_1^*β·π_2^*β' is a nef class with vanishing intersection against Δ_X. Two typical situations where one can apply Theorem <ref> are when:* X is a toric variety.* N_k(X × X) = ⊕_i+j=kπ_1^*N_i(X) ·π_2^* N_j(X), every pseudoeffective cone on X is simplicial, and every nef class on X is universally pseudoeffective. The first fact is well-known.To see the second, note that the hypothesis on universal pseudo-effectivity shows that any external product of nef cycles is nef.The simplicial hypothesis then implies that the external product of the pseudoeffective cones is dual to the external product of the nef cones.Thus the external product of the pseudoeffective cones is in fact the entire pseudoeffective cone of X × X. We will apply Theorem <ref> to examples where one can prove directly that all nef classes are universally pseudoeffective (e.g., fake projective spaces, Grassmannians,…). However, it seems relatively rare in general for the condition on pseudoeffective cones in Theorem <ref> to hold. Here is a basic example: Let S be the blow-up of ℙ^2 in r general points for some r ≥ 5. There is a strict containmentℝ_≥ 0[F_1] ⊕π_1^*_1(S) ·π_2^*_1(S) ⊕ℝ_≥ 0[F_2] ⊊_2(S × S).In fact, a lengthy but straightforward computation shows that the diagonal does not lie in the cone on the left.§ DUAL POSITIVITY We next turn to the “dual” forms of positivity: nefness and universal pseudoeffectiveness.The main examples are varieties with nef tangent bundle.For such varieties the class of Δ_X is nef, but not all varieties with nef diagonal have nef tangent bundle; for example, a fake projective plane has nef diagonal even though the tangent bundle is antiample. We emphasize that only “dual-positivity” of the tangent bundle should be inherited by the diagonal.The bigness of the tangent bundle T_X is quite different from the bigness of the class Δ_X.For example, a product of at least two projective spaces has big and nef tangent bundle, but by Lemma <ref> the diagonal class is not big.More generally, a smooth toric variety has big tangent bundle by <cit.>, but it is rare for a toric variety to have big diagonal.Let X be a smooth variety.If Δ_X is nef (resp. universally pseudoeffective) then every pseudoeffective class on X is nef(resp. universally pseudoeffective). In fact, the proposition is true for any property preserved by pullback and flat pushforward.This proposition strengthens <cit.>, which shows the analogous statement for divisors on a variety with T_X nef. We focus on nefness; the proof for universal pseudoeffectiveness is identical, using the properties of positive dual classes proved in <cit.>.It suffices to show nefness for the class of an irreducible cycle Z on X.Since π_1 is flat, Z' = π_1^-1(Z) represents π_1^*[Z].The restriction of Δ_X to Z' is nef; since nefness is preserved by flat pushforward onto a smooth base, (π_2|_Z')_* [Δ_X]|_Z' = [Z] is also nef on X. Let X be a smooth projective variety. * If Δ_X is big and nef, then N^1(X) ≅ℝ.* If Δ_X is big and universally pseudoeffective, then N_k(X) ≅ℝ for every k. Combine Proposition <ref>, Lemma <ref>, and the fact that nef divisors are universally pseudoeffective.If a smooth variety X admits a surjective map to a curve C of genus ≥ 2, then Δ_X is not nef.Denote the morphism by π: X → C.Let H be an ample divisor on X × X.Letting n denote the dimension of X, we have Δ_X· H^n-1· (π×π)^*Δ_C < 0 by the projection formula and the fact that Δ_C has negative self-intersection. We can also give a necessary condition for nefness based on the gonality of X. Let X be a smooth projective variety of dimension n admitting a surjective generically finite map f: X → Y of degree d to a smooth projective variety Y. Suppose that c_n(X) > dc_n(Y).Then Δ_X is not nef.If f contracts a curve, then X carries a curve that is not nef, and hence Δ_X is not nef.Thus it suffices to consider the case when f is finite.Consider the map F=(f× f):X× X→ Y × Y. This is finite surjective, hence flat. Note that F|_Δ_X=f, so F_*Δ_X=d Δ_Y. Moreover, by flatness, F^*Δ_Y is an effective cycle containing Δ_X in its support. The intersection of F^*Δ_Y-Δ_X with Δ_X isF_*Δ_X·Δ_Y-Δ_X^2=dc_n(Y)-c_n(X) which is negative by assumption, so that Δ_X is not nef. Let X be a smooth projective variety of dimension n admitting a surjective generically finite map f: X →ℙ^n of degree d. Suppose that c_n(X) > (n+1)d.Then Δ_X is not nef.§ RIGIDITY The results of the previous sections indicate that it is quite rare for a variety to have big diagonal. In this section we will study varieties where Δ_X is as far away from big as possible, and in particular, when [Δ_X] spans an extremal ray in the pseudoeffective cone.Let Z be an effective ℝ-cycle on a projective variety X of dimension k.We say that Z is: * strongly numerically rigid, if Z is irreducible and for every infinite sequence of effective ℝ-cycles Z_i such that lim_i →∞ [Z_i] = [Z], the coefficient a_i of Z in Z_i limits to 1.* exceptional for a morphism π:X → Y, if (π|_Z) > (π).Exceptional classes are studied in <cit.> and are closely related to the notion of an exceptional divisor.Among other nice properties, an exceptional numerical class can not be represented by a cycle whose deformations cover X.If Z is strongly numerically rigid then it spans an extremal ray of the pseudoeffective cone and is the unique effective cycle in its numerical class.A typical example of a strongly numerically rigid class is an irreducible divisor of numerical dimension 0.A related concept is discussed briefly in <cit.>. §.§ Blowing upLet X be a smooth projective variety and let Z be an k-dimensional subvariety.Suppose that there is an open neighborhood U ⊂ N_k(X) of [Z] such that Z appears with positive coefficient in any effective ℝ-cycle with class in U.Then Z is strongly numerically rigid. The point is that there is no assumed lower bound for the coefficient with which Z appears in the cycles. We first show that, perhaps after shrinking U, there is a constant ϵ > 0 such that ϵ Z ≤ T for any effective ℝ-cycle T with numerical class in U.Suppose otherwise for a contradiction.Choose β in the interior of the movable cone (that is, the closure of the cone of classes subvarieties which deform to cover X).For some sufficiently small τ we have that β + τ [Z] is still in the interior of the movable cone.Thus, if α∈ U has an effective representative where Z appears with coefficient c, the class α + c/τβ is represented by an effective ℝ-cycle in which Z has coefficient 0.If there is an open neighborhood U' of [Z] with U'⊂ U and admitting representatives with arbitrarily small coefficients of Z, we obtain a contradiction.We can now argue as in <cit.>: we define a function σ_Z: ^∘_k(X) →ℝ that records the infimum of the coefficients of Z appearing in any effective ℝ-cycle of class α.This function is continuous on the big cone; by taking limits we extend it to a lower semicontinuous function on the entire pseudoeffective cone.Furthermore, for any α∈_k(X) and β in the interior of the movable cone, the restriction of σ_Z to the ray α + tβ is strictly decreasing in t.We deduce that σ_Z([Z]) > 0.An easy rescaling argument shows that σ_Z([Z]) = 1, and we conclude the strong numerical rigidity of [Z] by lower semi-continuity of σ_Z. We can then test for the strong numerical rigidity of Δ by blowing up Δ.Let X be a smooth projective variety of dimension n.Let ϕ: W → X × X denote the blow-up of the diagonal and let i: E → W denote the inclusion of the exceptional divisor.Suppose that α∈ N_n(X × X) is a non-zero class such thatϕ^*α = M + i_*Nwhere M ∈ N_n(W) is a nef class and N ∈ N_n(E) is a nef class. * If α·Δ_X = 0 then Δ_X is not big.* If α·Δ_X < 0 then Δ_X is strongly numerically rigid. Let T be an effective n-cycle on W.If T is not supported on E then T ·ϕ^*α is non-negative.Pushing forward, we see that the only effective n-cycle on X which can possibly have negative intersection with α is Δ_X itself.Thus:(1) Suppose α·Δ_X = 0.Then α is nef and thus Δ_X can not be big.(2) Suppose α·Δ_X < 0.Then also α·β < 0 for any effective class β sufficiently close to [Δ_X].This means that any effective representative of such a β must contain Δ_X in its support with positive coefficient.We conclude that Δ_X is strongly numerically rigid by Lemma <ref>.For surfaces, we have the following criterion:Let S be a smooth surface.Let ϕ: Y → S× S denote the blow-up of Δ_S.If ϕ^*[Δ_S] is not pseudoeffective, then Δ_S is strongly numerically rigid.We let E denote the exceptional divisor of ϕ and let g: E →Δ_S denote the projective bundle map and ξ the class of the relative 𝒪(1) on E.We denote by i: E → Y the inclusion.Suppose that ϕ^*[Δ_S] is not pseudoeffective, and let η be a nef class in N^2(Y) such that η·ϕ^*[Δ_S] < 0.Choose a sufficiently small open subset U ⊂ N_2(S× S) of [Δ_S] such that η·ϕ^*β < 0 for every β∈ U.Let Z be any effective ℝ-cycle on S× S such that [Z] ∈ U.Let T be any effective ℝ-cycle on Y that pushes forward to Z; after removing vertical components, we may suppose that T does not have any components contracted by ϕ.Let α denote the class of T.We can write α = ϕ^*ϕ_*α + i_*g^*L for some (not necessarily effective) ℝ-divisor class L on S.Then since η·α≥ 0, and η·ϕ^*ϕ_*α<0, we have η· i_*g^*L> 0 and consequentlyg_*(η|_E) · L > 0.Now, since η|_E is nef and g is flat, g_*(η|_E) is the class of a nef curve η on S.Then, if π_1 denotes the projection to the first factor, we findE ·ϕ^*π_1^*η·α=E ·ϕ^*π_1^*η· i_*g^*L= (-ξ) · g^*(η· L)< 0.By the nefness of η (and hence ϕ^*π_1^*η), we see that some component of T must be contained in E, and furthermore (since we removed all π-contracted components) this component must dominate Δ_S under π. Pushing forward, we see that Δ_S must be contained in Z with positive coefficient.We conclude by Lemma <ref>. §.§ Rigidity via the Hilbert schemeUsing the rational map S × S ^2(S), one can study the positivity of Δ_S via the geometry of the Hilbert scheme.This approach is surprisingly successful, allowing us to use results arising from Bridgeland stability.Let S be a surface and let B' denote the divisor on ^2(S) such that 2B' parametrizes non-reduced subschemes.For nef divisors H and A on X, consider D_1 := H^[2] - b_1B' and D_2 := A^[2] - b_2B' on ^2(S).If c_2(S) > 0 and * D_1 and D_2 are movable andb_1b_2 > 4A · H/c_2(S)then Δ_S is not nef.* D_1 is nef, D_2 is movable, andb_1b_2 > 4A · H/c_2(S)then Δ_S is strongly numerically rigid.Let ϕ: Y → S × S be the blow-up along the diagonal.The exceptional divisor E is isomorphic to ℙ(Ω^1_S) with projection g: E → S.Letting ξ denote the class of the relative 𝒪(1) and i: E → Y the injection, we have that ϕ^*Δ_S = i_*(ξ - g^*K_S).Let ψ: Y →^2(S) denote the 2:1-map.Then wecompute intersections by restricting to E:ψ^*D_1·ψ^*D_2·ϕ^*Δ_S= (2g^*H+b_1ξ) · (2g^*A+b_2ξ) · (ξ - g^*K_S)= -b_1b_2c_2(S) + 4 A · H First suppose that D_1 and D_2 are movable and the inequality holds.Since ψ is finite, ψ^*D_1 and ψ^*D_2 are also movable, and hence their intersection is pseudoeffective.The assumed inequality shows that ψ^*D_1·ψ^*D_2·ϕ^*Δ_S < 0, so that Δ is not nef.Next suppose that D_1 is nef and D_2 is movable.Then ψ^*D_1·ψ^*D_2 is nef and by the same calculation as before we deduce that ϕ^*Δ_S is not pseudoeffective.By Proposition <ref>Δ_S is strongly numerically rigid. It would be interesting if Theorem <ref> could be improved by a more in-depth study of the geometry of the Hilbert scheme ^2(S). §.§ Albanese mapLet X be a smooth projective variety and let alb: X → A be the Albanese map (for a chosen basepoint).By the subtraction map for X, we mean the composition of alb^× 2: X × X → A × A with the subtraction map for A.Note that this map does not depend on the choice of basepoint. Let X be a smooth projective variety of dimension n.Suppose that the Albanese map alb: X → A is generically finite onto its image but is not surjective.Then Δ_X is exceptional for the subtraction map. Note that the diagonal is contracted to a point by the subtraction map.Thus, it suffices to prove that a general fiber of the subtraction map f: X × X → A has dimension < n.Let X' denote the image of the albanese map.Since alb is generically finite onto its image, it suffices to prove that the general fiber of the subtraction map f: X' × X' → A has dimension < n.Suppose otherwise for a contradiction. For every closed point p ∈ f(X' × X') the fiber F_p denotes pairs of points (x_1,x_2) ∈ X' × X' such that x_1 = p + x_2.If this has dimension n, then it must dominate X' under both projections.In other words, X' is taken to itself under translation by every point of f(X' × X').Recall that X' contains the identity of A, so that in particular X' ⊆ f(X' × X').Thus, the subgroup of A fixing X' is all of A.This is a contradiction when X' ≠ A.There are many other results of a similar flavor.For example, if the diagonal is the only subvariety of dimension ≥ n contracted by the Albanese map then Δ_X is strongly numerically rigid using arguments similar to those of <cit.>.This situation holds for every curve of genus ≥ 2 and seems to hold often in higher dimensions as well. § SURFACES We now discuss positivity of the diagonal for smooth surfaces.First, by combining Theorem <ref> with Corollary <ref> (and using the equality of homological and numerical equivalence for surface classes) we obtain:The only smooth projective surfaces with big and nef diagonal are the projective plane and fake projective planes. In this section we discuss each Kodaira dimension in turn.We can summarize the discussion as follows:* The only possible surfaces with big diagonal are ℙ^2 or a surface of general type satisfying p_g = q = 0.In the latter case, the only example with big diagonal that we know of is a fake projective plane.* If the Kodaira dimension of X is at most 1, then Δ_X is nef if and only if X has nef tangent bundle, with the exception of some properly elliptic surfaces which admit no section.Surfaces with nef tangent bundle are classified by <cit.>. Note that any surface with nef diagonal must be minimal by Proposition <ref>. §.§ Kodaira dimension -∞Let X be a smooth surface of Kodaira dimension -∞.Then * Δ_X is big if and only if X = ℙ^2.* Δ_X is nef if and only if X has nef tangent bundle, or equivalently, if X is either ^2, ^1×^1, or a projective bundle (ℰ) over an elliptic curve where ℰ is either an unsplit vector bundle or (a twist of) a direct sum of two degree 0 line bundles. (1) Let S be a smooth uniruled surface and let g: S → T be a map to a minimal model.If T is not ℙ^2, then T (and hence also S) admits a surjective morphism to a curve.If T = ℙ^2 and g is not an isomorphism, then g factors through the blow up of ℙ^2 at a point, which also admits a surjective morphism to a curve.In either case Corollary <ref> shows that the diagonal of S is not big.(2) We only need to consider minimal surfaces.Using the classification, we see that any minimal ruled surface besides the ones listed carries a curve with negative self-intersection or maps to a curve of genus ≥ 2.By Proposition <ref> and Lemma <ref> such surfaces can not have nef diagonal. §.§ Kodaira dimension 0The diagonal of a surface of Kodaira dimension 0 is not big.By Lemma <ref> it suffices to prove this for minimal surfaces.Using the classification and Theorem <ref>, the only surface which could have a big diagonal would be an Enriques surface.However, such surfaces always admit a map to ℙ^1 and thus can not have big diagonal by Lemma <ref>. We next turn to nefness of the diagonal.Recall that any surface with nef diagonal must be minimal, and we argue case by case using classification.Abelian surfaces and hyperelliptic surfaces both have nef tangent bundles, and thus nef diagonal.For K3 surfaces, Theorem <ref> below verifies that the diagonal is never nef.Finally, any Enriques surface admits an ample divisor D with D^2=2 which defines a double ramified cover. Hence there is an involution i:S→ S exchanging the two sheets. Then if Γ_i is the graph, we have Δ_S·Γ_i=-C^2<0., and so Δ_S is not nef.§.§.§ K3 surfaces K3 surfaces are perhaps the most interesting example, and in this subsection we discuss them at some length.We first discuss nefness, and we start with a couple low degree examples.Let S→^2 be a degree 2 K3 surface. As for the Enriques surface, there is an involution i:S→ S, and intersecting Δ_S with the graph of the involution gives a negative number, so Δ_S is not nef. Let S be a surface in ^3, and let W=S× S be the blow-up along the diagonal. Consider the divisor H_1+H_2-E, where H_i is the pullback of the hyperplane section via the i-th projection. This divisor is base-point free, and defines a morphismϕ:W→ Gr(2,4).Geometrically, this is the morphism obtained by sending a pair of points on S to the line they span; it is finite when S contains no lines. Now suppose that S is a quartic K3 surface. Then(H_1+H_2-E)^2π^*Δ_S=(H_1+H_2+𝒪(1))^2 𝒪(1)=(2H)^2-24=-8. In particular, Δ_S has negative intersection with the images of the fibers of ϕ.The previous example shows how knowledge of the nef cone of the blow-up S× S, or equivalently, ^2(S), can be used to produce interesting subvarieties of S× S having negative intersection with Δ_S. By the work of Bayer–Macrì, we can use similar arguments also for higher degrees. Let S be a K3 surface.Then Δ_S is not nef. We first prove a special case:Let S be a K3 surface of Picard rank 1 polarized by an ample divisor H of degree d ≥ 4.Then Δ_S is not nef.We start by recalling the results of <cit.> on the geometry of ^2(S).Suppose that d/2 is not a square.It is clear that the fundamental solution to the Pell's equation x^2 - (d/2)y^2 =1 must have x ≥√(d/2), so that the fundamental solution yields a ratioy/x = √(2/d)√( 1 - 1/x^2)≥√(2/d)√(d-2/d).Set b_d= √(d/2 - 1)≤d/2·y/x.Applying <cit.>, we see that (whether or not d/2 is a square) the divisor class H'-b_dB is movable on ^2(S), where H' is induced by the symmetric power of H and 2B is the exceptional divisor for the Hilbert-Chow morphism.We then apply Theorem <ref>.The only verification necessary is:b_d^2 = d/2 - 1 > 4d/24which holds for d in our range. In fact, the previous proof gives a little more: over the family of degree d K3 surfaces, we have a class on the total space which restricts to be effective on a very general K3 surface and which has constant negative intersection against Δ_S for such surfaces.Applying Proposition <ref>, we can take limits to deduce that for every K3 surface in the family, Δ_S has negative intersection against a pseudoeffective class.This concludes the proof of Theorem <ref> in degree ≥ 4, and we have already done the degree 2 case in Example <ref>. §.§.§ Rigidity for K3 surfaces By again appealing to the results of <cit.>, we can show rigidity under certain situations. Let S be a K3 surface of Picard number 1 and degree d.Suppose that the Pell's equationx^2 - 2dy^2 = 5has no solutions.Then the diagonal is strongly numerically rigid. For example, the theorem applies when the degree is divisible by 4, or when the degree is less than 50 except for degrees 2,10,22,38. By combining <cit.> with the calculation in the proof of Lemma <ref>, we obtain the result from Theorem <ref>. Finally, we will prove that the diagonal of a very general K3 of degree d is numerically rigid, using a deformation argument.Standard results on K3 surfaces give the existence of a degree d K3 surface S_0 which is also a quartic surface. It follows by the computation in Example <ref> that π^*(Δ_S_0) is not pseudoeffective. Now take a family 𝒮→ T of polarized degree d surfaces in a neighbourhood of S_0. Let π:×_T→×_T be the blow-up of the diagonal. The induced family ×_T→ T is a smooth morphism. Consider the cycle class(π_t)^*(Δ_S_t)=(π^*Δ_𝒮/T)_t. Since this is not pseudoeffective on the special fiber, π^*(Δ_S_t) is not pseudoeffective for t very general, by Proposition <ref>.So applying again Proposition <ref>, we see that Δ is strongly numerically rigid on the very general K3 surface of degree d. Let S be a very general polarized K3 surface. Then the diagonal is strongly numerically rigid. 𝒦 A posterori, this result is intuitive in light of the Torelli theorem, at least for subvarieties of S× S which are graphs of self-maps f:S→ S: if Γ is such a graph and [Γ]=[Δ], then f induces the identity on H^2(S,), and hence has to be the identity, and so Γ=Δ_S. §.§ Kodaira dimension 1Let S be a surface of Kodaira dimension 1 and let π:S→ C be the canonical map. By Corollary <ref>, we have: A surface of Kodaira dimension 1 does not have big diagonal. We next show that the diagonal is not nef when π admits a section.As usual we may assume S is minimal so that K_S is proportional to some multiple of a general fiber of π.Using Lemma <ref>, we see that if the diagonal is nef then the base C of the canonical map must have genus either 0 or 1.If T is a section of π, then by adjunction we see that T^2 < 0, and so Δ_S is not nef (since as before Δ_S· (π_1^*T ·π_2^* T)<0).When S→ C does not admit a section, it is possible for the diagonal to be nef. Indeed, let E be an elliptic curve without complex multiplication and let C be a hyperelliptic curve of genus g which is very general in moduli.The product E × C admits an involution i which acts on E as translation by a 2-torsion point and on C by the hyperelliptic involution. The quotient surface S=(E× C)/i is a properly elliptic surface of Kodaira dimension 1. The elliptic fibration S→ C/i=^1 has a non-reduced fiber, and therefore can not admit a section. We claim that the diagonal of S is nef. Let S'=E× C and let Γ denote the graph of the involution i. By the projection formula it is enough to check that Δ_S'+Γ is nef on S' × S'. Indeed, if π:S'→ S is the quotient map, the map π×π is flat, and (π×π)^*Δ_S=Δ_S' +Γ is nef if and only if Δ_S is.Let f:S'× S'→ C× C denote the projection map π_2×π_2.Claim: If an irreducible surface T⊂ E × C × E × C is not nef, then it maps to a curve D in C × C with negative self-intersection.Furthermore it can only have negative intersection with surfaces contained in f^-1(D).It is clear that T is nef if it maps to a point in C × C.We next prove nefness if T maps dominantly onto C × C.Fix an irreducible surface V; we will show T · V ≥ 0.It suffices to consider the case when V is not a fiber of the map to C × C.In this situation we can deform T using the abelian surface action so that it meets V in a dimension 0 subset.Indeed, the set Y⊂ C× C of points y such that the fiber T ∩ f^-1(y) is 1-dimensional is finite.For a general translation in E × E, this curve will meet V ∩ f^-1(y) in a finite set of points.Next consider the open set U = C × C \ Y.Let W ⊂ T be the subset lying over U.We have a finite map from E × E × W → E × E × U given by (a,b,w) ↦ (a,b) · w.In particular, the preimage of V in E × E × W will be a surface, and thus will meet a general fiber of E × E × W → E × E properly.Altogether we see a general translation of T will meet V properly.Thus T is nef. Finally, suppose that T maps to a curve D and let F=f^-1(D).Suppose V is a surface not contained in F.Then V · T can be computed by restricting V to F.This restriction is effective, and hence nef (by the group action on F), showing that T · V ≥ 0.If V is also contained in F, then V · T in X is the same as f_*(v· t)· D on C× C, where V=i_*v, T=i_*t. Note that f_*(v· t) will be a non-negative multiple of D in C× C. So if D^2≥ 0,we again find that T · V ≥ 0. This completes the proof of the claim above. Note that if the diagonal Δ_S is not nef, there is a surface T⊂ S'× S' with T· (Γ+Δ_S')<0. We must have either Γ· T<0 or Δ_S'· T<0; replacing T by i(T), we may assume that the latter is the case. Arguing as above, we see Δ_S' and T are both contained in the preimage L of Δ_C. The intersection Γ· L is transversal; it consists of the points of Γ over the 2-torsion points of C. In particular, the the restriction of Γ to L is numerically equivalent to (Δ_C·Γ_C)Γ_E = (2g+2) Γ_Ewhere Γ_E is the pushforward of the graph of the involution from a fiber E × E.In contrast, i^*i_*Δ_S will bei^*i_*Δ_S = (2-2g) Δ_Ewhere Δ_E is the pushforward of the diagonal from a fiber E × E.Since Γ_E and Δ_E are numerically proportional, we see that Δ_S' + Γ is nef when restricted to this threefold.Hence its intersection with T is non-negative, and so Δ_S'+Γ is nef overall.§.§ Surfaces of general type §.§.§ Surfaces with vanishing genus By Castelnuovo's formula, a surface of general type satisfying p_g = 0 also must satisfy q=0.The minimal surfaces satisfying these conditions are categorized according to K_S^2, which is an integer satisfying 1 ≤ K_S^2≤ 9, and have Picard rank 10-K_S^2.It is interesting to look for examples where bigness holds or fails.For such surfaces Theorem <ref> shows: Let S be a smooth surface satisfying p_g(S) = q(S) = 0.If _1(S) is simplicial, then Δ_S is big if and only if every nef divisor is big. However, determining bigness can still be subtle.𝐊_𝐒^2 = 9: The surfaces here are exactly the fake projective planes, and we saw in the introduction that Δ_S is both big and nef.Suppose we blow-up a very general point to obtain a surface Y.The results of <cit.> on Seshadri constants show that Y carries a divisor which is nef and has self-intersection 0, so the diagonal for Y is neither nef or big.𝐊_𝐒^2 = 8: Since these surfaces have Picard rank 2, the pseudoeffective cone is automatically simplicial.Thus we have an interesting trichotomy of behaviors: * ^1(S) = ^1(S).Let D_1, D_2 be generators of the two rays of the pseudoeffective cone, and set a = D_1· D_2.ThenΔ_S = F_1 + F_2 + 1/aπ_1^*D_1·π_2^*D_2 + 1/aπ_1^*D_2·π_2^*D_1is nef.However, Δ_S is not big since S carries a non-zero nef class with self-intersection 0.* If exactly one extremal ray of ^1(S) is nef, then S carries both a curve of negative self-intersection and a nef class with vanishing self-intersection.Thus Δ_S is neither big nor nef.* If no extremal rays of ^1(S) are nef, then Δ_S is big by Lemma <ref> but is not nef.There are a few known geometric constructions of such surfaces.First, there are the surfaces constructed explicitly via ball quotients which are classified in <cit.> and <cit.>.Second, there are the surfaces admitting a finite étale cover which is a product of two curves.Such surfaces are classified in <cit.> and are further subdivided into two types.Write S = (C_1× C_2)/G for some finite group G acting on the product.If no element of G swaps the two factors, then G acts on each factor separately.This is known as the “unmixed” case.There is also the “mixed” case, when C_1≅ C_2 and some elements of G swap the two factors.In the unmixed case, we have C_1/G ≅ C_2/G ≅ℙ^1, and we obtain two maps S →ℙ^1 such that the pullbacks of 𝒪(1) generate the pseudoeffective cone.In particular Δ_S is nef but not big.However, we do not know what happens in the other two situations. Burniat surfaces:These are certain surfaces constructed as Galois covers of weak del Pezzos (see for example <cit.>).By pulling back from the del Pezzo we obtain nef divisors with self-intersection 0, so that Δ_S is not big.The Godeaux surface: This surface is the quotient of the Fermat quintic by a ℤ/5 action.This surface admits a morphism to ℙ^1 (see for example the second-to-last paragraph on page 3 of <cit.>), so Δ_S is not big.§.§.§ Surfaces with non-vanishing genus We next discuss several classes of surfaces of general type where we can apply our results.These examples have p_g>0, and so Δ_S cannot be big. We note that by the computations of <cit.>, if H is a very ample divisor on a surface S then (maintaining the notation of Section <ref>) H^[2] - B' is nef. [Surfaces in ^3]Suppose that S is a smooth degree d hypersurface in ℙ^3.Then c_2(S) = d^3 - 4d^2 + 6d.Thus Theorem <ref> shows that the diagonal is strongly numerically rigid and is not nef as soon as d ≥ 4. [Double covers] Suppose that S is a double cover of ℙ^2 ramified over a smooth curve of even degree d.Then S is of general type once d ≥ 8.These surfaces have c_2(S) = d^2 - 3d+6 and carry a very ample divisor of degree d.Thus Theorem <ref> shows that the diagonal is strongly numerically rigid and is not nef as soon as d ≥ 8. [Horikawa surfaces] Minimal surfaces of general type satisfying q(S) = 0 and K_S^2 = 2p_g(S) - 4 are known as Horikawa surfaces and are studied by <cit.>.(These surfaces are the boundary case of Noether's inequality.)The canonical map for such a surface defines a 2:1 morphism onto a rational surface.While K_S is not very ample, it is big and basepoint free, which is enough to determine that K_S^[2] - B' is movable on ^2(S).Using the equality c_2(S) = 12 + 12p_g(S) - K_S^2, we see that the diagonal for such surfaces is never nef by Theorem <ref>. § HIGHER DIMENSIONAL EXAMPLES §.§ Quadric hypersurfaces An odd dimensional quadric is a fake projective space and thus will have big and nef diagonal as discussed in the introduction.An even dimensional quadric will have diagonal that is nef but not big. Indeed, if X is a quadric of dimension 2k, then X carries two disjoint linear spaces of dimension k. These linear spaces are nef (since X is homogeneous), but not big, and hence Δ_X is not big. §.§ Nefness for hypersurfaces Example <ref> shows that the diagonal of a smooth hypersurface in ℙ^3 of degree at least 3 is not nef.The same is true in arbitrary dimension: Let X ⊂ℙ^n+1 be a smooth degree d hypersurface.If d≥ 3 then the diagonal for X is not nef.The Euler characteristic of X isc_n(X) = (1-d)^n+2-1/d + n + 2.If n is odd, then Δ_X^2 = c_n(X) < 0 and so Δ_X is not nef.If X has even dimension, then Corollary <ref> applied to the projection map from a general point outside X shows that Δ_X is not nef. §.§ Bigness and rigidity for hypersurfacesSuppose that X ⊂ℙ^n+1 is a smooth hypersurface of dimension n.We will apply Lemma <ref> to show that the diagonal is not big for hypersurfaces of small dimension.Note that homological non-bigness follows from Theorem <ref> once the degree is larger than n+1, so we will focus only on the Fano hypersurfaces.Recall that Lemma <ref> requires us to find a class α∈ N_n(X × X) whose pullback under the blow-up of the diagonal ϕ: W → X × X has a special form.We record the relevant information in the table below.The first column records the kind of hypersurface.The second column records the class α – we will let H_1 and H_2 denote the pullback of the hyperplane class under the two projection maps.The third column verifies that Lemma <ref> applies to α.This is done by rewriting ϕ^*α in terms of Schubert classes pulled back under the natural morphism g: W → G(2,n+2).We will let i: E → W denote the inclusion of the exceptional divisor.Identifying E ≅ℙ(Ω_X) this divisor carries the class ξ of the relative 𝒪(1) andthe pullback H of the hyperplane class from the base of the projective bundle.For convenience we let h: E → Gr(2,n+2) denote the restriction of g to E.type α ϕ^*α cubic threefold H_1^2H_2 + H_1H_2^2 + Δ_X g^*σ_2,1 + i_*(h^*σ_2 + 4H^2) quartic threefold 2H_1^2H_2 + 2H_1H_2^2 + Δ_X 2g^*σ_2,1 + i_*(h^*σ_2 + 9H^2) cubic fourfold [ H_1^4 + H_1^3H_2 + 3H_1^2H_2^2 +; H_1H_2^3 + H_2^4 - Δ_X ] g^*σ_4 + 2g^*σ_2,2 + i_*(2h^*σ_2,1 + 8H^3)quartic fourfold [ H_1^4 + H_1^3H_2 + 7H_1^2H_2^2 +;H_1H_2^3 + H_2^4- Δ_X ] g^*σ_4 + 6g^*σ_2,2 + i_*(3h^*σ_2,1 + 24H^3) quintic fourfold [ H_1^4 + H_1^3H_2 + 13H_1^2H_2^2 +; H_1H_2^3 + H_2^4- Δ_X ] g^*σ_4 + 12g^*σ_2,2 + i_*(4h^*σ_2,1 + 56H^3) cubic fivefold [ H_1^4H_2 + H_1^3H_2^2 +; H_1^2H_2^3 + H_1H_2^4 + Δ_X ] g^*σ_4,1 + i_*(h^*σ_4 + 4 h^*σ_2,2 + 12 H^4) quartic fivefold [ 2H_1^4H_2 + 11H_1^3H_2^2 +; 11H_1^2H_2^3 + 2H_1H_2^4 + Δ_X ] [ 2g^*σ_4,1 + 9 H_1H_2 g^*σ_2,1 +; i_*(h^*σ_4 + 9 h^*σ_2,2 + 62 H^4) ] quintic fivefold [3H_1^4H_2 + 35H_1^3H_2^2 +; 3H_1^2H_2^3 + 3H_1H_2^4 + Δ_X ] [ 3g^*σ_4,1 + 32 H_1 g^*σ_2,2 +; i_*(h^*σ_4 + 16 h^*σ_2,2 + 208 H^4) ] sextic fivefold [4H_1^4H_2 + 79H_1^3H_2^2 +; 4H_1^2H_2^3 + 4H_1H_2^4 + Δ_X ] [ 4g^*σ_4,1 + 75 H_1 g^*σ_2,2 +; i_*(h^*σ_4 + 25 h^*σ_2,2 + 500 H^4) ] cubic sixfold [ H_1^6 + H_1^5H_2 + 3H_1^4H_2^2 + 3H_1^3H_2^3 +; 3H_1^2H_2^4 + H_1H_2^5 + H_2^6 - Δ_X ] [ g^*σ_6 + 2g^*σ_4,2 +; i_*(2h^*σ_4,1 + 8H h^*σ_2,2 + 24H^5) ] In all cases the class α satisfies α·Δ≤ 0, and in all but the first the intersection is negative.By Lemma <ref>, a smooth cubic threefold has non-big diagonal and in all other cases the diagonal is strongly numerically rigid.It seems very likely that the same approach will work for all hypersurfaces of degree ≥ 3 and dimension ≥ 3.Indeed, as the degree increases the coefficients are becoming more favorable.We have verified strong numerical rigidity for several more cubic hypersurfaces but unfortunately the combinatorics become somewhat complicated. §.§ GrassmanniansFor a Grassmannian X=Gr(k,n), the tangent bundle is globally generated, and hence the diagonal is nef. However, when X is not a projective space, the diagonal is not big because the Schubert classes on X are universally pseudoeffective but not big. §.§ Toric varieties <cit.> shows that the only smooth toric varieties for which the pseudoeffective and nef cones coincide for all k are products of projective spaces.Thus products of projective spaces are the only toric varieties with Δ_X nef.Theorem <ref> shows that for a toric variety Δ_X is big if and only if every nef class on X is big.Although it seems reasonable to expect that the only toric varieties with big diagonal are projective spaces, this turns out to be false. <cit.> constructs an example of a toric threefold which does not admit a morphism to a variety of dimension 0 < d <3.In particular, this implies that every nef divisor is big.Dually, every nef curve is big.Thus X has big diagonal. Note that, aside from projective space, the diagonal can not be big for smooth projective toric varieties of Picard rank ≤ 3 (or for threefolds of Picard rank ≤ 4), since these varieties admit a non-constant morphism to a lower-dimensional variety <cit.>.§.§ Varieties with nef tangent bundle Varieties with nef tangent bundle are expected to have a rich geometric structure.A conjecture of Campana–Peternell <cit.> predicts that such varieties are (up to étale cover) flat bundles with rational homogeneous fibers over their albanese variety.To prove this result, it suffices by a result of <cit.> to show that a Fano variety with nef tangent bundle is rational homogeneous.This conjecture has been verified up to dimension 5 and in several related circumstances (see <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>). A variety with nef tangent bundle has nef diagonal.However, the diagonal can only be big if the albanese map is trivial.A variety with nef tangent bundle need not carry an action of an algebraic group – this can only be expected after taking an étale cover.An explicit example is discussed in <cit.>.However, the diagonal can still be universally pseudoeffective in this situation.Indeed, suppose we have an étale cover f: Y → X where Y admits a transitive action of an algebraic group.The induced flat finite map f^× 2: Y × Y → X × X has that f^× 2_*Δ_Y = d Δ_X.Of course the diagonal of Y is universally pseudoeffective and since universal pseudoeffectiveness is preserved by flat pushforward for smooth varieties by <cit.> we deduce that Δ_X is also universally pseudoeffective.§ THREEFOLDS In this section we discuss a couple classification results for threefolds. §.§ Fano threefolds The Fano threefolds which are fake projective spaces are are ℙ^3, the quadric, the Del Pezzo quintic threefolds V_5, and the Fano threefolds V_22; see <cit.>.As discussed above these will have big and nef diagonal. Any Fano threefold with nef diagonal must be primitive and the contractions corresponding to the extremal rays of _1(X) can not be birational.Another obstruction to nefness is the fact that Δ_X^2 is the topological Euler characteristic of X.In addition to the fake projective spaces discussed earlier, the classification of <cit.> and <cit.> leaves 7 possibilities: * Picard rank 1: V_18, the intersection of two quadrics in ℙ^5.* Picard rank 2: a double cover of ℙ^2×ℙ^1 with branch locus (2,2), a divisor on ℙ^2×ℙ^2 of type (1,2) or (1,1), ℙ^1×ℙ^2.* Picard rank 3:ℙ^1×ℙ^1×ℙ^1.The threefolds on this list with nef tangent bundle are classified by <cit.> and will automatically have nef diagonal: these are the products of projective spaces and the (1,1) divisor in ℙ^2×ℙ^2. The divisor of type (1,2) on ℙ^2×ℙ^2 will also have nef diagonal, which we can see as follows.Let { D_1, D_2} be the basis of N_2(X) consisting of extremal nef divisors and { C_1, C_2} denote the basis of N_1(X) consisting of extremal nef curves.Since h^1,2(X)=0 we can write the diagonal very explicitly using products of these basis elements and it is easy to see that it is a non-negative sum of nef classes.We are unsure of what happens in the remaining 3 cases. Any Fano threefold with big diagonal must have that the contractions corresponding to the extremal rays of _1(X) are birational.We focus on the primitive Fano threefolds; going through the classification of <cit.>, we see that any such threefold must have Picard rank 1.Unfortunately it seems subtle to determine which of these threefolds actually have big diagonal. §.§ Threefolds with nef and big diagonalLet X be a smooth minimal threefold of Kodaira dimension ≥ 0.Then X does not have homologically big diagonal.If Δ_X is homologically big, then H^k,0(X)=0 for all k>0. In particular, χ(Ø_X)=1. By Riemann–Roch, χ(Ø_X)=1/24c_1c_2, so we find that c_1c_2=24. In particular, c_1(X)≠ 0. If 0<κ(X)<3, then X admits a map to a lower-dimensional variety, contradicting bigness of Δ_X. If X has general type, we have by the Miayoka–Yau inequality, 0>-K^3=c_1^3≥8/3 c_1c_2=64This is a contradiction.Let X be a smooth threefold such that Δ_X is nef and homologically big.Then X is a fake projective space: ℙ^3, the quadric, the Del Pezzo quintic threefolds V_5, the Fano threefolds V_22.Suppose first that κ(X) ≥ 0.Since Δ_X is nef, X must be a minimal threefold.We conclude by the previous proposition that the diagonal can not be big.Thus we know that X is uniruled.Furthermore N^1(X)=ℝ, and so X is a Fano threefold of Picard number 1.Note also that χ(X)=Δ_X^2≥ 1, so in particular h^2,1 is at most 1. Going through the classification of such Fano threefolds <cit.> reveals that the only possibilities are the fake projective 3-spaces. Recall that when X is a threefold not of general type then numerical and homological equivalence coincide on X × X.Thus the corollary also classifies the threefolds not of general type which have nef and big diagonal.§ COHOMOLOGICAL DECOMPOSITION OF THE DIAGONAL AND POSITIVITYThis section discusses the relationship of Theorem <ref> with the decomposition of the diagonal in cohomology on a surface. The following result is surely well-known to experts, but we include a proof for the convenience of the reader. Let S be a smooth projective surface.Then: * Δ_S is homologous to a sum of cycles contracted by the projection maps if and only if p_g(S) = 0.* Δ_S is homologous toa sum of products of pullbacks of classes from the two projections if and only if p_g(S) = q(S) = 0.We first prove (2).Note that Δ is homologous to a sum Z+Z' where Z is a cycle supported in a divisor D× S and Z' is supported on a fiber S× s of the projection map. By a Bloch–Srinivas type argument (as in <cit.>), we find that p_g(S)=q(S)=0. For the reverse implication, the Künneth formula implies that all of H^2,2(S × S) is algebraic and generated by products of pullbacks of divisors from the two projections. Now we prove (1).The forward implication follows again from the argument of <cit.>.Conversely, the arguments of <cit.> using the classification theory of surfaces show that for surfaces with κ(S) < 2 and p_g(S) = 0 there is a curve C ⊂ S such that _0(C) →_0(S) is surjective.Using the decomposition of the diagonal as in <cit.>, we see that the reverse implication holds except possibly for surfaces of general type.But a surface of general type satisfying p_g(S) = 0 also satisfies q(S) = 0 by Castelnuovo's theorem.Thus we are reduced to (2). It is natural to ask whether one can obtain a tighter link between positivity and decompositions of Δ_X than Theorem <ref>.Following an idea of <cit.>, we will prove such a statement for surfaces by perturbing the diagonal by an external product of ample divisors.Let S be a smooth projective surface.Then p_g(S) = 0 if and only if there is an ample divisor H such that Δ_S + π_1^*H ·π_2^*H is big.We first prove the forward implication.By Proposition <ref>, we have an equality of numerical classesΔ_S = a_0F_1 + b_0F_2 + ∑_i=1^r_1 a_iE_i + ∑_j=1^r_2 b_jE_j'where a_i,b_j∈ℚ, each E_i is an irreducible surface contracted to a curve by π_1, and each E_j' is an irreducible surface contracted to a curve by π_2.Note thata_0 + ∑_i=1^r_1 a_iE_i· F_2 = 1b_0 + ∑_j=1^r_2 b_jE_j' · F_1 = 1For each i, let C_i denote the normalization of the image π_1(E_i) and let D_i denote C_i× S.For notational convenience, we will omit the normalization and write C_i and D_i as if they were subvarieties of S and S × S.Fix a small ϵ > 0 satisfying ϵ < 1/r_1 and ϵ < 1/r_2.Set c_i = -a_iE_i· F_2 so thata_iE_i + (c_i+ ϵ) F_1 is π_2-relatively ample as a divisor on D_i.Thus, for some sufficiently large ample H_i on S, we have thata_iE_i +( c_i +ϵ) F_1 + π_2|_D_i^*H_iis an effective class on D_i.Pushing forward to S × S and adding up as i varies, we see that( r_1ϵ + a_0 - 1 ) F_1 + ∑_i=1^r_1( a_iE_i + π_1^*C_i·π_2^*H_i)is an effective class on S × S.Arguing symmetrically, with analogous notation,( r_2ϵ + b_0 - 1 ) F_2 + ∑_j=1^r_2( b_jE_j' + π_1^*H_j' ·π_2^*C_j' )is an effective class.Of course, we can replace the C_i, H_i, C_j', H_j' by larger ample divisors without affecting the effectiveness of this class.All told, there is an effective surface class Q and a positive sum of external products of ample divisors N such thatΔ_S + N = (1-r_1ϵ)F_1 + (1-r_2ϵ)F_2 + Q.Adding on a further external product of amples to both sides we can ensure that the right hand side is big: if A is ample on X, a class of the form c_1F_1 + c_2F_2 + c_3π_1^*A ·π_2^*A with positive coefficients will dominate a small multiple of the big class (π_1^*A + π_2^*A)^2.Finally, any positive sum of external products of amples is dominated by a single external product of amples, finishing the proof of the implication.Conversely, suppose h^2,0(X) > 0 and let α∈ H^2,0(X) be non-zero.Let β be the nef class constructed as in Theorem <ref> satisfying β·Δ_S = 0.This β also satisfies β·π_1^*H ·π_2^*H=0.Since β is nef, there can be no big class of the desired form (again appealing to the equality of homological and numerical equivalence).§ QUESTIONS We finish with a list of questions raised by our work.Are ℙ^2 and fake projective planes the only smooth surfaces with big diagonal?Suppose that S is a smooth surface S of general type with q(S) = 0 and p_g(S) > 0.Is the diagonal for S numerically rigid?It is natural to ask whether some of the results for surfaces generalize for higher dimensions: Does a smooth projective variety with big and nef diagonal have the same rational cohomology as projective space?Are the only smooth projective varieties with big diagonal either uniruled or of general type?Is there a threefold of general type with nef diagonal?Are there any topological restrictions on smooth varieties with nef diagonal aside from c_n(X) ≥ 0?For example, does a threefold with nef diagonal satisfy χ(𝒪_X) ≥ 0? alpha | http://arxiv.org/abs/1707.08659v2 | {
"authors": [
"Brian Lehmann",
"John Christian Ottem"
],
"categories": [
"math.AG"
],
"primary_category": "math.AG",
"published": "20170726225528",
"title": "Positivity of the diagonal"
} |
APS/123-QED [email protected] [email protected] ^1University of Groningen, Zernike Institute for Advanced Materials, 9747 AG Groningen, The Netherlands ^2High Field Magnet Laboratory (HFML-EMFL) and Institute for Molecules and Materials, Radboud University, 6525 ED Nijmegen, The Netherlands ^3Department of Physics & Astronomy, Rutgers, The State University of New Jersey, Piscataway, New Jersey 08854, USA ^4Department of Materials Science & Engineering, Rutgers University, Piscataway, New Jersey 08854, USATopological insulators are ideally represented as having an insulating bulk with topologically protected, spin-textured surface states. However, it is increasingly becoming clear that these surface transport channels can be accompanied by a finite conducting bulk, as well as additional topologically trivial surface states. To investigate these parallel conduction transport channels, we studied Shubnikov–de Haas oscillations in Bi_2Se_3 thin films, in high magnetic fields up to 30 T so as to access channels with a lower mobility. We identify a clear Zeeman-split bulk contribution to the oscillations from a comparison between the charge-carrier densities extracted from the magnetoresistance and the oscillations. Furthermore, our analyses indicate the presence of a two-dimensional state and signatures of additional states the origin of which cannot be conclusively determined. Our findings underpin the necessity of theoretical studies on the origin of and the interplay between these parallel conduction channels for a careful analysis of the material's performance.Coexistence of bulk and surface states probed by Shubnikov–de Haas oscillations in Bi_2Se_3 with high charge-carrier density T. Banerjee^1 December 30, 2023 ============================================================================================================================Topological insulators (TIs), hosting spin-momentum locked surface states, received considerable interest in the past decade potentially serving as a platform for exploring many interesting concepts in physics<cit.>. These surface states have been well investigated by surface sensitive techniques like (spin-)angular resolved photo-emission spectroscopy<cit.> and scanning tunneling microscopy and spectroscopy<cit.>. Such techniques adequately describe the electronic properties of the (non)trivial surface states, but cannot account for additional transport features as observed in (magneto)transport experiments. To employ topological insulators in solid-state devices, direct access and understanding of these additional surface states in transport experiments are needed. Studying Shubnikov–de Haas (SdH) oscillations can reveal the existence of such surface states where parameters like mobility, charge-carrier density, the dimensionality and the Berry phase of the states can be determined<cit.>. Earlier studies on various Bi-based topological insulators report on single or double frequency SdH oscillations<cit.>, where it is often claimed that these oscillations originate from the top and bottom topological surface state (TSS) with the expected Berry phase and angular dependence. The magnetic field strength used in these studies is usually up to 15 T, which can only probe transport channels with a relatively high mobility, whereas nonlinear Hall measurements indicate additional channels with a lower mobility to be present. Besides a finite conducting bulk, these additional (topologically trivial) channels can originate from variations in the electrostatic potential near the surfaces and can also be spin textured <cit.>, which we will refer to as two-dimensional electron gas (2DEG). From earlier transport measurements, the mobilities of the different channels are found to be on the order of 50–500 and .17ex∼3000 cm^2(V s)^-1 where the low mobility channel has a higher charge-carrier density<cit.>. Notably, from these numbers one can find that in terms of conductivity these channels can contribute equally to the electrical transport.Motivated by these works, we performed magnetotransport experiments up to 30 T and studied SdH oscillations to explore the most prominent conduction channels and additional channels with mobilities below 1000 cm^2(V s)^-1 (B≫1, whereis the charge mobility and B the applied magnetic field strength). The magnetotransport is studied in thin films of Bi_2Se_3 so as to minimize bulk effects and amplify the topologically trivial and nontrivial surface states. In contrast to earlier works, we will show that the bulk channel with a high mobility is present along with a prominent two-dimensional (2D) channel which can be linked to the topological surface states. Our findings indicate the presence of additional channels with a lower mobility that cannot be precisely resolved from the oscillations. Similar to Ref. <cit.>, we compare charge-carrier densities from the SdH oscillations as well as from the magnetoresistance and study the dimensionality of the various channels in order to unravel the origin of these states.In this paper, we used thin films of n-type Bi_2Se_3 with thickness t = 10, 20, 30, and 100 quintuple layers (QL) grown by molecular-beam epitaxy (MBE) on Al_2O_3(0001) substrates in a custom-designed SVTA MOS-V-2 MBE system at a base pressure lower than 5×10^-10 Torr following the methods as described in previous work<cit.>. The quality of the obtained films was characterized through various techniques<cit.>. The films were patterned into Hall bars by using a combination of photolithography and Ar plasma dry etching. Contact pads consisting of Ti(5)/Au(70 nm) were made by combining photolithography with electron-beam evaporation. The resulting Hall bars (inset Fig. <ref>a) have dimensions of 2400×100 m^2 where the resistance is measured over a probing length between 1400 and 2000 m. The magnetotransport measurements have been performed in a cryostat with an out-of-plane rotation stage placed in a 30 T Bitter-type magnet in a four-probe geometry using the ac modulation technique at an ac current bias of 1 A. In this paper, mainly the results on the sample with t = 10 QL will be discussed and comparisons will be made to the samples with larger thickness. Most of the results of the thicker samples can be found in the Supplemental Material below.The typical out-of-plane magnetic field dependence of the longitudinal sheet resistance R_ measured for the sample with t = 10 QL is shown in Fig. <ref>a where R_ tends to saturate at high magnetic fields. From this data and those for larger thickness as well as from the fitting (see below), we observe that the order of saturation is determined by the low mobility channel and the parabolic response at low fields is governed by the high mobility channel. Furthermore, the presence of at least two channels is clear from the nonlinear Hall resistance R_ [Fig. <ref>b]. As also shown earlier<cit.>, we observe a slight upturn with a change in R_ .17ex∼0.2% for samples with t = 10–30 QL below 10 K, indicative of the presence of defect states<cit.>. From the out-of-plane field dependence of the longitudinal and transverse resistance R_(B) and R_(B), we can extract the sheet carrier density n_i and mobility μ_i for only two channels, which we expect to be due to the bulk and surface state(s). For that, we use a semi-classical Drude model where contributions from two parallel channels are summed in the conductivity tensor σ̂, which relates to the resistivity as ρ̂ = σ̂^-1 (more details on the analysis can be found in the Supplemental Material below): σ_=n_1eμ_1/1+μ_1^2B^2+n_2eμ_2/1+μ_2^2B^2,σ_=n_1eμ_1^2B/1+μ_1^2B^2+n_2eμ_2^2B/1+μ_2^2B^2 As found from our analysis, simultaneous fitting of the R_ and R_ is required since R_ has a strong effect on the mobility and therefore will change the values found for n_i from R_. An example of the simultaneous fit to the magnetoresistance curves R_ and R_ for the sample with t = 10 QL at 1.4 K is displayed in Fig. <ref>a. A good agreement with the two-carrier model is obtained with a residual δ = (R_-R_)/R_ between 1 and 5 % for both R_ and R_, but it is important to note that this analysis is limited to two channels and does not rule out the presence of more channels. Nevertheless, the good agreement between data and fit suggests that any additional state would have a similar mobility, which would add to an effective charge-carrier density in Eq. (<ref>).An overview of the extracted charge-carrier properties for all film thicknesses can be found in Table <ref>; the data and fits to the magnetoresistance for the samples with larger thickness can be found in the Supplemental Material below. We listed n_/t because of its correspondence to the [three-dimensional (3D)] bulk channel (see discussion below), whereas n_ is most likely linked to a 2D channel. The model describes the magnetoresistance behavior for t up to 30 QL very well, but deviations from the model are observed for t = 100 QL, which will be discussed below. The correspondence between these extracted parameters and the information extracted from the SdH oscillations will be discussed in the remainder of this paper.The possible presence of additional states can be analyzed by studying the SdH oscillations in R_, provided that the mobility of the channels is high enough<cit.>. For the sample with t = 10 QL, these oscillations can be observed from .17ex∼10 T onwards, which indicates that transport channels are present with a mobility on the order of 1000 cm^2(V s)^-1. This is in agreement with estimates for _1 as extracted from the magnetoresistance measurements (see Table <ref>). To analyze the oscillations without the magnetoresistance background, the second derivative –d^2R_/dB^2 is taken after interpolation and adjacent averaging of the data (see Supplemental Material below for more details). By plotting –d^2R_/dB^2 versus 1/B, we find an oscillatory pattern that shows additional oscillations from 15 T (.17ex∼0.067 T^-1, Fig. <ref>c). We can follow the development of the oscillations by looking at the evolution of the fast fourier transform (FFT) spectrum when taking different ranges starting from 9 T (.17ex∼0.11 T^-1) towards higher fields, where lower mobility channels start to contribute [Fig. <ref>d]. Below 15 T, as depicted by the black, red, and blue line in Fig. <ref>d, one main frequency is observed indicated by α in the FFT spectrum as has been commonly reported in other works<cit.>.Beyond 15 T, we find the clear presence of the harmonic 2α in the FFT spectrum which is due to the strong Zeeman splitting because of the large g factor in this material <cit.>. As shown in Fig. <ref>a, the occurrence of Zeeman splitting is justified by an enhancement in oscillation amplitude when studying its dependence on the perpendicular to the in-plane component of the magnetic field, B_⊥. Only the cyclotron energy is sensitive to this component, whereas the competing Zeeman energy is related to the total applied magnetic field. In addition to the high mobility channel linked to α, we find a lower mobility channel denoted by β. The appearance of the oscillation linked to β at higher magnetic field indicates that this channel has a mobility on the order of several times 100 cm^2(V s)^-1 which is in agreement with the lower value μ_ found from the earlier analysis of the magnetoresistance (Table <ref>). Using the extracted three frequencies, we can reconstruct the oscillatory pattern at high fields as shown in Fig. <ref>c with deviations in the peak amplitudes of the pattern. Due to the good agreement between data and the reconstructed oscillatory pattern, we can conclude that in the used magnetic field range the magnetotransport is dominated by these three frequencies. Nevertheless, additional channels with a lower mobility might be present but are beyond the resolution of our measurements. To explore the dimensionality of the observed conduction channels, we can look at the angular dependence of the magnetic field orientation on the position of the frequency peaks. For 2D states we expect that their frequencies f scale with f∝ 1/cosθ where θ is the angle between the surface normal and the direction of the applied magnetic field (inset Fig. <ref>b). For bulk states it is commonly observed that f(θ) initially follows the similar behavior but saturates between 30 and 60^∘, depending on the dimensions of the ellipsoidal pocket of these states<cit.>. However, few earlier works<cit.> report on a similar 1/cosθ dependence for the bulk states as well. Importantly, although we use thin films, the bulk will not show 2D behavior due to finite film thickness since the magnetic length l_= √(ħ/eB)≤ 8 nm at fields of 10 T from which we start observing the oscillations. From these considerations for f(θ) we can map out all observed peaks in the spectra at every angle θ and check whether they fit into a 2D or 3D picture. In Fig. <ref>b, the angular dependence of the observed peaks for t = 10 QL is plotted from where we can trace the different channels α, 2α, and β up to an angle of 68^∘. Beyond this angle, the resolution of separate spectral peaks is limited which is most probably linked to the low mobility of the channels and is manifested as a strong weakening of the oscillations at higher angles. Nevertheless, partially due to a higher mobility of the channel, we find a minor oscillation with f = (0.5±0.2) kT at θ = 90^∘ indicating that f_α (and its harmonic) saturates and is due to the bulk channel with an elongated Fermi pocket. For the β peak, we find a 1/cosθ behavior which can be linked to the appearance of a 2D state. Another way of clarifying the origin of the states is to extract the cyclotron mass from the temperature dependence of the oscillations which are observable up to .17ex∼50 K as shown in Fig. <ref>c. Because of the presence of multiple oscillations it is difficult to extract the cyclotron mass from the FFT spectra. Inspired by recent work<cit.>, we can extract the cyclotron mass by studying the temperature dependence of the peak amplitudes in the oscillations via the Lifshitz-Kosevich formalism. The result is shown in Fig. <ref>d where we can study the evolution of the cyclotron mass upon varying the magnetic field where different oscillations contribute. Comparing this result with the FFT spectrum evolution in Fig. <ref>d in which we observe a single channel up to 0.07 T^-1, we can conclude that the channel corresponding to the α peak has a cyclotron mass m_=(0.15±0.01)m_, which is a typical value for the bulk conduction band<cit.>. Below 0.07 T^-1, we find a strong increase in the cyclotron mass up to .17ex∼0.28m_ after which it lowers to (0.20±0.01)m_ and saturates. This higher value of m_ is probably due to the topological surface states<cit.>, whereas a trivial 2DEG is supposed to have a similar mass as the bulk<cit.>. The interplay of the different oscillations could give rise to an increase in the cyclotron mass because channels with lower mobility (∝ 1/m_) start contributing, provided that the scattering times in the different channels are the same<cit.>. From the considerations above, we can match the charge-carrier densities extracted from the oscillations and from the magnetoresistance. From the FFT spectrum progression analysis, we can conclude that the α peak makes up the high mobility channel where the charge-carrier density n_α = (1.26±0.06)×10^19cm^-3 when assuming bulk states (n_ = k_^2k_/3π^2) with an ellipsoid pocket with ellipticity k_/k_ = 1.8<cit.>. The value for n_α is in reasonable agreement with n_/t found from the magnetoresistance analysis (Table. <ref>). Furthermore, due to reduced scattering compared to that at any of the surfaces it is most likely that the high mobility channel corresponds to the unaffected bulk layer. The state indicated by β appears above 15 T and thus it is conceivable that this state is linked to the low mobility channel with n_. The origin of the observed 2D surface state, trivial or nontrivial, cannot be concluded from the determination of the Berry phase (see Supplemental Material below), but the extracted m_ at high fields hints at a topological surface state. Furthermore, it is not clear whether this state resides at the top or bottom surface because the characteristics of the electrostatics at both surfaces are unknown, which would affect the mobility. Assuming n_ = k_^2/4π for a topological surface state and n_ = k_^2/2π for a possible 2DEG, the charge-carrier density related to f_β varies between n_=(6.7±0.3)×10^12cm^-2 and n_=(1.34±0.05)×10^13cm^-2, which makes up for 20 or 40 % of n_. We are careful to assume that this oscillation is linked to one surface state since it has been earlier reported that similar n_β is present at the opposite surface<cit.>, provided the mobilities at both surfaces are similar. Furthermore, as will be shown for t = 20 QL, an additional peak between α and 2α occurs which shows that additional states exist, which adds to the low mobility charge-carrier density n_.The picture based on the charge-carrier densities for t = 10 QL also applies for the samples with t = 20 and 30 QL, but the correction for the ellipsoidal asymmetry is most probably smaller compared to the sample with t = 10 QL which can be related to a lower charge-carrier density<cit.>. Comparing the two films with t = 10 and 20 QL, we observe oscillations [Fig. <ref>a] with a similar spectrum but with the presence of an additional γ peak [Fig. <ref>b], which could be a signature of a state at the surface opposite to where the channel linked to β resides. Furthermore, the peak positions have changed, which is due to differences in charge-carrier density as also observed in the magnetoresistance measurements. From the similarities between these two samples, we can conclude that the thickness (i.e., bulk size) does not play a role but it is rather the relative mobilities and charge-carrier densities in these samples which are the decisive factors for the relative channel contributions. For the thicker samples, as shown in FIGs. <ref>c and d, we observe a dominant peak α in the spectrum while the 2α, β, and γ peaks are present but with a poor resolution. The reason for the decrease in amplitude is a lower signal-to-noise level of the measured voltage which generates a background and gives rise to a larger spectral width in the FFT spectrum. Furthermore, the oscillations show a beating pattern where oscillations of different frequencies partially cancel each other, yielding a loss of FFT amplitude. For the sample with t = 100 QL, we find a poor agreement between the charge-carrier densities from the magnetoresistance and the SdH oscillations where the bulk state (α channel) can alone account for the total charge-carrier density n_+n_. The fitting of the magnetoresistance data shows that the mobilities and charge-carrier densities could be different from the extracted values as stronger oscillations were expected for the mobilities extracted. The difference between the fit and data could originate from additional channels with a more distinct mobility suggesting that the two-channel model is too limited to describe the data properly. Lastly, from atomic force microscopy (AFM) images (see Supplemental Material below) we observe height variations across the film surface, which might influence fitting parameters such as the effective thickness of the transport channel and can cause changes to the bulk density n_/t on the order of 2×10^17cm^-3. This is in the same range of charge-carrier densities found for the 2D states. In conclusion, we find a good agreement between magnetoresistance data and the analysis of the SdH oscillations for Bi_2Se_3 thin films based on the extracted charge-carrier densities, where the channel contributions are quite unrelated to film thickness. We find that the bulk channel has a high mobility and is characterized by an ellipsoid Fermi pocket but a clear saturation in the angular dependence is absent. Due to the strong g factor in these materials we observe a Zeeman splitting in our oscillations which has been observed before in optical measurements and investigations on thermoelectric effects under high magnetic field. Furthermore, we observe a pronounced 2D state, either topologically trivial or nontrivial, which partially accounts for the low mobility channel's charge-carrier density. Additional 2D states are observed but are often masked by the limited resolution of our analysis originating from the channel mobilities and charge-carrier densities. The limited resolution of the angular dependence and the difficulties to extract parameters like the Berry phase make it difficult to make a definitive statement on the origin of these states. The authors would like to thank L. Tang (Radboud University) for useful discussions on the processing of the data. Furthermore, the authors would like to thank J. G. Holstein, H. M. de Roosz, and T. J. Schouten (University of Groningen) for the technical support. We acknowledge the support of the HFML, member of the European Magnetic Field Laboratory. Work at the University of Groningen is supported by a Dieptestrategie grant from the Zernike Institute for Advanced Materials. 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Zhang, 10.1038/nphys1270 journal journal Nature Physics volume 5, pages 438 (year 2009)NoStop Supplemental Material: Coexistence of bulk and surface states probed by Shubnikov–de Haas oscillations in Bi_2Se_3 with high charge-carrier densityIn this Supplemental Material additional data can be found for all the samples including the extracted charge-carrier densities from the SdH oscillations which can be compared with the values extracted from the Drude modelling. Furthermore, AFM images will be shown which give an idea about the growth quality and the error in the thickness leading to the error in n_/t. For t = 10 QL, we also performed a Berry phase analysis. We will start out with details on the analysis performed in the main text.§ S1. DETAILS ON ANALYSIS In this section we would like to clarify some of the analysis methods used in the main text. First of all, we would like to extend on the simultaneous fitting of the two-channel Drude model. From Equation (1) in the main text, we can find the expressions for the resistivity components by inverting the conductivity matrix: R_=(n_eμ_/1+(μ_ B)^2+n_eμ_/1+(μ_ B)^2)/(n_eμ_/1+(μ_ B)^2+n_eμ_/1+(μ_ B)^2)^2+(n_eμ_^2B/1+(μ_ B)^2+n_eμ_^2B/1+(μ_ B)^2)^2R_=(n_eμ_^2B/1+(μ_ B)^2+n_eμ_^2B/1+(μ_ B)^2)/(n_eμ_/1+(μ_ B)^2+n_eμ_/1+(μ_ B)^2)^2+(n_eμ_^2B/1+(μ_ B)^2+n_eμ_^2B/1+(μ_ B)^2)^2Here, R_ is the (2D) sheet resistance and R_ the Hall resistance. Simultaneous fitting is done via Matlab R2016a by minimalizing the sum of errors between data and fit of both R_ and R_ without any weighing. Here, the weak antilocalization (WAL) feature observed close to zero field does not affect the fitting procedure since the range over which this is observed is less than 1 % of the total field range. Furthermore, at elevated temperatures, where WAL is absent, the Drude model also shows a good agreement with the data and therefore we can rule out any fitting errors due to the presence of WAL. In this way, the fitting is most reliable due to inclusion of the full data set at once. Furthermore, we would like to elaborate on the analysis procedure for the Shubnikov–de Haas oscillations. To decouple the oscillations from the strong background we have taken the second derivative –d^2R_/dB^2 since from Equation <ref> we have a B^2-dependence on the resistance in the limit of high field. By taking the second derivative, we are indeed successful to remove the background completely, whereas taking the first derivative gives a strong residual background where FFT analysis is difficult. Taking the second derivative requires a low noise level which we can decrease by adjacent averaging of the data. The averaging procedure is performed over 0.5 T intervals which are much shorter compared to the oscillation period such that this will not affect the FFT analysis; it only will slightly change the oscillation amplitude. Furthermore, the second derivative requires equidistant intervals such that interpolation is needed where the number of points is kept constant with respect to the original data. This interpolation is also used before the FFT is taken (in the 1/B range) which does not yield any artifacts because of the large oscillation period in these measurements. § S2. ADDITIONAL DATA FOR SAMPLE WITH T = 10 QL A typical AFM image for this sample is shown in Fig. <ref>a from which we assign a maximum uncertainty in the channel thickness t of 1 QL as a conservative margin. To understand the origin of the 2D states, we could extract the Berry phase from the oscillations. A careful analysis has been described in <cit.> on how to index the maxima and minima in dG_/dB. For our analysis, we looked at the maxima and minima in d^2G_/dB^2 where the maxima in d^2G_/dB^2 coincide with the minima in G_ (labeled with integer n) and minima in d^2G_/dB^2 correspond to maxima in G_ (labeled n + 1/2 (n = 1, 2, 3, ..)). From Fig. <ref>b, it can be seen that the oscillation complies with the predicted behavior over a small range but due to the presence of multiple oscillations the data acquires a different phase compared to the theoretical curve. Therefore, the analysis can only be employed for the first three maxima. Plotting 1/B_n versus n yields a phase between 0.04 and 0.38 depending on the frequency taken; f_β = 0.277 kT is the value found from the FFT analysis, f_β = 0.285 kT corresponds to the best fit of the data and is within the error for f_β as given in Table <ref>. In conclusion, due to the presence of multiple oscillations with different frequency, the limited number of oscillations and the error in f_β a proper extraction of the Berry phase value cannot be made. At last, the values for n_ = k_^2/4π, n_ = k_^2/2π, and n_ = k_^2k_/3π^2 that we extract from the frequencies in the FFT spectra are displayed in Table <ref>. Here, we assume k_/k_=1.8 which has been reported for samples with similar charge-carrier density by Kulbachinskii et al.<cit.>. These values can be compared to n_ and n_ which are calculated for the 2D and 3D case and also included in Table <ref>. As also discussed in the main text, we observe a reasonable agreement between n_α (with f = 0.115 kT) and n_/t. It is important to realize that when considering all channels to be 2D, the charge-carrier density from the oscillations clearly underestimates the values found from the two channel fit; the presence of a bulk channel therefore seems feasible and in agreement with the presence of a residual oscillation at θ = 90^∘ and a Zeeman-split bulk state. The contribution of n_β to n_ is 20 % when this channel is linked to a topological surface state. From band structure calculations<cit.>, we find that the next bulk band is located around 1 eV higher than the first conduction band minimum which makes the origin of the state β to be related to a second bulk band unlikely. In addition, the FFT spectra that we observe in comparison with the work by Kulbachinskii et al.<cit.> are different such that a second bulk state is more improbable. The presence of additional channels with an even lower mobility could account for the difference between the values extracted from the magnetoresistance and those from the SdH oscillations. § S3. DATA FOR SAMPLE WITH T = 20 QL In Fig. <ref>a and b the data and fits for the out-of-plane field dependence of the sheet resistance R_ and the Hall resistance R_, respectively, for sample with t = 20 QL are shown where a good agreement between data and fit is observed. Importantly, for this sample the resistance has been measured for fields up to 33 T in order to include the last clear oscillation which enhances slightly the resolution in our FFT spectrum. When studying the FFT range progression analysis, as shown in Fig. <ref>d, additional bands start to appear already below 15 T which is striking considering the low extracted mobility μ_2 (B≫1) and is inconsistent with the trend observed for the sample with t = 10 QL. A direct assignment of the mobility to the appearance of the bands at certain fields is thus not straightforward. The larger amplitude of the oscillation contributes to a good resolution in FFT such that the spectral peaks already appear at lower fields. The order of appearance of the peaks as displayed in Fig. <ref>d is similar to that described in the main text with the additional γ peak starting to appear around similar fields as the harmonic peak. To reconstruct the oscillatory pattern, as shown in Fig. <ref>c, f_α, f_β, and f_γ are found from the fit, whereas additional inclusion of f_α does not improve the fit much and only modifies the amplitude slightly.Table <ref> shows the extracted values for n_, n_, and n_ with k_≈k_ as we have a lower charge-carrier density in this sample. Nevertheless, we can expect ellipticity as observed in the work by Kulbachinskii et al.<cit.>. These values can be compared to n_ and n_ as also displayed in Table <ref>. We find a good agreement between n_α and n_/t giving good confidence on the origin of this state. As shown in Fig. <ref>a, we observe hardly any thickness variation and therefore the error in t is expected to be small. Furthermore, we find that the channel linked to β makes up 20 % of n_, assuming it to be a topological surface state. The additional γ state accounts for 13% of n_ which partially explains the difference in values found from the oscillations and the magnetoresistance. As shown in Fig. <ref>, we observe hardly any thickness variation and therefore the error in t is expected to be small. Furthermore, we display the angular dependence for the sample with t = 20 QL in where we see a similar behavior as for the t = 10 QL sample, i.e. all the four peaks follow a 1/cosθ dependence up to the angles where we are able to observe oscillations [Fig. <ref>b]. For this sample, it is not possible to observe any remaining bulk oscillations at 90^∘.At last, we show the cyclotron mass extracted per peak from the temperature dependence as also shown in Fig. 3a of the main paper [Fig. <ref>c]. Also here we observe an increase in m_ towards higher magnetic field but not as structurally as for the data for t = 10 QL. Furthermore, the cyclotron mass seems to be slightly higher compared to the sample with t = 10 QL. § S4. DATA FOR SAMPLE WITH T = 30 QL In Fig. <ref>a and b, the data and fits for the out-of-plane field dependence of the sheet resistance R_ and the Hall resistance R_, respectively, for the sample with t = 30 QL are shown where a good agreement between data and fit is observed. In this case the SdH oscillations are not as clear as seen in the previous samples. Nevertheless, upon plotting –d^2R_/dB^2 vs 1/B we find a clear oscillatory pattern with an increased noise in the signals compared to the previously discussed samples [Fig. <ref>c]. This increase in noise might be linked to the contamination that we find from the AFM image [Fig. <ref>a]. We find that this oscillatory pattern shows three spectral peaks but not as clearly resolvable as previous samples where the 2α peak is missing [Fig. <ref>d]. These three peaks can be used to fit the data, but a clear discrepancy is observed indicating that additional channels are present. Interestingly, we observe that the appearance of the peaks commences at rather low fields when considering the mobilities that we find from the magnetoresistivity fitting, similar to that observed for t = 20 QL.The presence of the peaks can also be checked by the angular dependence [Fig. <ref>b]. Spectral peaks α, β, and γ show a clear angular dependence following a 1/cosθ behavior; again it is not possible to observe any remaining oscillation at θ = 90^∘. Comparing the values extracted from the oscillations and the magnetoresistance [Table <ref>], we conclude that the α peak corresponds to the bulk channel with density n_. The other channels contribute to a total charge-carrier density of about (1.0±0.1)×10^13cm^-2 which is on the same order of magnitude as n_, thereby assuming that these states are linked to topological surface states. From the cyclotron mass analysis [Fig. <ref>d], we cannot see an evolution of m_ because of the large errors and the limited number of peaks that could be analyzed due to the presence of a beating pattern.§ S5. DATA FOR SAMPLE WITH T = 100 QL In Fig. <ref>a, the data and fits for the out-of-plane field dependence of the sheet resistance R_ and the Hall resistance R_, respectively, for the sample with t = 100 QL are shown where there is a disagreement between data and fit, which means that for this thickness the two channel model is too limited to describe the magnetoresistance. Upon plotting –d^2R_/dB^2 vs 1/B we find a clear oscillatory pattern with an increased noise in the signals compared to the previously discussed samples [Fig. <ref>c]. In FIG <ref>d, we find that this oscillatory pattern contains three main spectral peaks and a weak harmonic 2α peak. To reconstruct the oscillatory pattern the four given frequencies are not sufficient enough which indicates more channels might be present but are beyond our resolution. A hint for an additional peak is given by the presence of an additional trace in the angular dependence as shown in Fig. <ref>a at a frequency f = 0.046 kT but was considered an artifact in the FFT analysis because of the unclear temperature dependence in Fig. 3d of the main paper. Due to the beating features in the oscillation pattern [Fig. <ref>b], it is difficult to extract the cyclotron mass for this sample as shown in Fig. <ref>c. Nevertheless, we seem to find a cyclotron mass m_ = 0.15m_ which increases to 0.20m_ after 15 T which is in agreement with the trend as observed for t = 10 QL. At last, we show in Table <ref> the extracted charge-carrier densities where we observe that any extracted bulk value from the oscillations is larger than n_/t and n_/t. As mentioned in the main text, the charge-carrier density from f_α = 0.12 kT could account for n_+n_. One uncertainty is the effective thickness t of the sample as shown in the AFM image [Fig. <ref>d], where there are large triangular undulations present at the surface. Nevertheless, the exact reason for the disagreement between magnetoresistance measurements and the SdH oscillations analysis is unclear. | http://arxiv.org/abs/1707.08921v1 | {
"authors": [
"E. K. de Vries",
"S. Pezzini",
"M. J. Meijer",
"N. Koirala",
"M. Salehi",
"J. Moon",
"S. Oh",
"S. Wiedmann",
"T. Banerjee"
],
"categories": [
"cond-mat.mes-hall"
],
"primary_category": "cond-mat.mes-hall",
"published": "20170727160320",
"title": "Coexistence of bulk and surface states probed by Shubnikov-de Haas oscillations in Bi$_2$Se$_3$ with high charge-carrier density"
} |
[email protected] Institute of Nuclear Physics, Polish Academy of Sciences, Radzikowskiego 152, PL-31-342 Kraków, Poland [email protected] Institute of Nuclear Physics, Polish Academy of Sciences, Radzikowskiego 152, PL-31-342 Kraków, Poland We calculate several differential distributions for the production of charm and dijets. Both single-parton scattering (SPS) and double-parton scattering (DPS) contributions are calculated in the k_T-factorization approach. The Kimber-Martin-Ryskin unintegrated parton distributions are used in our calculations. Relatively low cuts on jet transverse momenta are imposed to enhancethe double-parton scattering mechanism contribution. We find dominance of the DPS contributionover the SPS one. We have found regions ofthe phase space where the SPS contributionis negligible compared to the DPS contribution. The distribution in transverse momentum of charm quark/antiquark or charmed mesons can be used to observe transition from the dominance ofDPS at low transvsverse momenta to the dominance of SPS at large transverse momenta. Very distinct azimuthal correlation patterns (for c c̅, c-jet, jet-jet, D^0-jet, D^0 D^0) are predicted as a result of the competition of the SPS and DPS mechanisms.13.87.Ce, 14.65.DwDouble-parton scattering effects in associated productionof charm mesons and dijets at the LHCAntoni Szczurek[also at University of Rzeszów, PL-35-959 Rzeszów, Poland] December 30, 2023 ================================================================================================= § INTRODUCTIONThe cross section for production of charm quarks or mesons is known tobe very large especially at high energies which is caused by relativelysmall mass of the charm quark. On the other hand the mass of the charm quark is large enough to use perturbative methods of quantum chromodynamics.Charm quarks are also produced abundantly in double <cit.> or multiple <cit.> parton scattering.The cross section for doublec c̅production was shown to grow considerably with the collision energy <cit.>. We have explained that total rates as well as several differential distributions cannot be explained without inclusion of double parton scattering.Many processes in association with charm quarks or mesons are possible and can be studied at the LHC. Recently we discussed inclusive production of single jet associated with c c̅ or charmed mesons <cit.>. Quite large cross sections were found there. Here we discuss inclusive production of dijets in association with c c̅ production. We wish to include both single parton scattering (SPS) anddouble parton scattering (DPS) processes. We wish to discuss whether the process can be used to extract the so-called σ_eff parameter which governs the strength of double parton scattering. The genral theoretical picture allows that this quantity may depend on kinematical quantities as well as type of the process. However, surprisingly similar values of σ_eff were obtained from different reactions. There are exceptions, much smaller values were obtained for double J/ψ charmonium production atlarge transverse momenta <cit.> but there the mechanism of the reaction is not yet fullyunderstood <cit.>. We focus on how to disantangle single and double parton scattering contributions for a simultaneous production of c c̅ (or charmed mesons) and dijets. In the current paper we present first predictions for the associated production of charm and dijets as well as many differential distributions,many of them of the correlation character. § A SKETCH OF THE THEORETICAL FORMALISM §.§ Single-parton scattering Within the k_T-factorization approach the SPS cross section forpp → cc̅ + 2jetsX reaction, sketched in Fig. <ref>, can be written as d σ_p p → cc̅ + 2jets = ∑_ij∫ d x_1 d^2 k_1t/π d x_2 d^2 k_2t/π F_i(x_1,k_1t^2,μ^2)F_j(x_2,k_2t^2,μ^2) d σ̂_ij → c c̅ + 2part.. In the formula above F_i(x,k_t^2,μ^2) is a unintegrated parton distribution function (uPDF) for a given type of parton i = g, u, d, s, u̅, d̅, s̅. The uPDFs depend on longitudinal momentum fraction x, transverse momentum squared k_t^2 of the partons entering the hard process, and in general also on a (factorization) scale of the hard process μ^2. The elementary cross section in Eq. (<ref>) can be written somewhat formally as: d σ̂_ij → c c̅ + 2part. = ∏_l=1^4d^3 p_l/(2 π)^3 2 E_l(2 π)^4 δ^4(∑_l=1^4 p_l - k_1 - k_2) ×1/flux| M_i^* j^* → c c̅ + 2part.(k_1,k_2)|^2, where E_l and p_l are energies and momenta of final state particles. Above only dependence of the matrix element on four-vectors of incident partons k_1 and k_2 is made explicit. In general all four-momenta associated with partonic legs enter. The matrix element takes into account that both partons entering the hard process are off-shell with virtualities k_1^2 = -k_1t^2 and k_2^2 = -k_2t^2. We take into account all 9 channels of the 2 → 4 type contributing to the cross section at the parton-level:#1 = g g → g g cc̅#2 = g g → qq̅cc̅#3 = g q → g q cc̅#4 = q g → q g cc̅#5 = qq̅→ q'q̅' cc̅#6 = qq̅→ g g cc̅#7 = q q → q q cc̅#8 = q q' → q q' cc̅#9 = qq̅→ qq̅cc̅.The off-shell matrix elements are well known only in the leading-order (LO) and only for limited types of QCD 2 → 2 processes (see e.g. heavy quarks <cit.>, dijet <cit.>, Drell-Yan <cit.>). Some first steps to calculate NLO corrections in the k_T-factorization framework have been done only very recently for diphoton production <cit.>. For higher final state parton multiplicities, relevant amplitudes can be calculated analytically applying suitably defined Feynman rules <cit.> or recursive methods, likegeneralisedBCFW recursion <cit.>, or numerically with the help of methods of numerical BCFW recursion <cit.>. The latter method was already successfully applied for 2 → 3 production mechanisms in the case of cc̅ + jet <cit.> and even for 2 → 4 processes in the case of cc̅ cc̅ <cit.> and four-jet <cit.> final states.In this paper we follow the same numerical techniques. The calculation has been performed with the help of KaTie <cit.>, which is a complete Monte Carlo parton-level event generator for hadron scattering processes. It can can be applied to any arbitrary processes within the Standard Model, for up to four final-state particles and beyond, and for any initial-state partons on-shell or off-shell. The scattering amplitudes are calculated numerically as a function of the external four-momenta via Dyson-Schwinger recursion <cit.> generalized also to tree-level off-shell amplitudes. The phase space integration in KaTie is done with the help of a full Monte Carlo program with an adaptive phase space generator, previously incorporated as a part of the AVHLIB library <cit.>, that deals with the integration variables related to both the initial-state momenta and the final-state momenta. KaTie can be used for single-parton scattering as well as for multi-parton scattering processes. In the present calculation, we use μ^2= m_1t^2+m_2t^2+p_3t^2+p_4t^2/4 as the renormalization/factorization scale, where m_1t, m_2t are the transverse mass of the outgoing c-quark and c̅-antiquark and p_3t, p_4t are the transverse momenta of outgonig jets. Furthermore, we take running α_s at next-to-leading order (NLO) and charm quark mass m_c = 1.5 GeV. The parameters are the same for both k_t-factorization and in the reference collinear-factorization calculations. Uncertainties related to the choice of the parameters were discussed e.g. in Ref. <cit.> and will be not considered here. We use the Kimber-Martin-Ryskin (KMR) <cit.> unintegrated distributions for quarks and gluon calculated from the MMHT2014nlo PDFs <cit.>. The above choices are kept the same also in the case of double-parton scattering calculation except of the scales. §.§ Double-parton scattering According to the general form of the multiple-parton scattering theory (see e.g. Refs. <cit.>) the DPS cross sections can be expressed in terms of the double parton distribution functions (dPDFs). These objects should fulfill sum rules and take into account all the correlations between the two partons, including transverse and longitudinal momenta correlations as well as color, flavour and spin correlations. The theory of dPDFs is well established but still not fully applicable for phenomenological studies. The currently available models of the dPDFs are still ratherat a preliminary stage. So far they are formulated exlusively for gluon or for valence quarks and only in a leading order framework which may be not sufficient for many processes, especially when charm production is considered.Instead of the general form, one usually follows the assumption of the factorization of the DPS cross section. Within this framework, the dPDFs are taken in the following factorized form: D_1, 2(x_1,x_2,μ) = f_1(x_1,μ)f_2(x_2,μ)θ(1-x_1-x_2) , where D_1, 2(x_1,x_2,μ) is the dPDF and f_i(x_i,μ) are the standard single PDFs for the two generic partons in the same proton. The factor θ(1-x_1-x_2) ensures that the sum of the two parton momenta does not exceed 1. According to the above, the differential cross section for pp → cc̅ + 2jetsX reaction within the DPS mechanism, sketched in Fig. <ref>, can be expressed as follows:dσ^DPS(c c̅ + 2jets)/dξ_1dξ_2 = ∑_i,j 1/σ_eff·dσ^SPS(g g → c c̅)/dξ_1·σ^SPS(i j →2jets)/dξ_2, where ξ_1 and ξ_2 stand for generic phase space kinematical variables for the first and second scattering, respectively. When integrating over kinematical variables one recovers the commonly used pocket-formula: σ^DPS(c c̅ + 2jets) = ∑_i,j σ^SPS(g g → c c̅) ·σ^SPS(i j →2jets)/σ_eff.The effective cross section σ_eff provides a proper normalization of the DPS cross section and can be roughly interpretedas a measure of the transverse correlation of the two partons insidethe hadrons. The longitudinal parton-parton correlations should become far less important as the energy of the collision is increased, due to the increase in the parton multiplicity. It is belived that for small-x partons and for low and intermediate scales the possible longitudinal correlations can be safely neglected (see e.g. Ref. <cit.>).In this paper we use world-average value of σ_eff = 15 mb provided byseveral experiments at Tevatron <cit.> and LHC <cit.>. Future experiments may verify this value.There are several effects that may lead to a violation of the factorized anstaz, which is a severe approximation. The flavour, spin and color correlations lead, in principle, to interference effects that result in breaking of the pocket-formula (see e.g. Refs. <cit.>. In any case, the spin polarization of the two partons from one hadron can be mutually correlated, especially when the partons are relatively close in phase space (having comparable x's). The two-parton distributions have a nontrivial color structure which also may lead to a non-negligible correlations effects.Such effects are usually not included in phenomenological analyses. They were exceptionally discussed in the context of double charm production <cit.> but in this case the corresponding effects were found to be very small. Moreover, including perturbative parton splitting mechanism <cit.> and/or imposing sum rules <cit.> also leads to a breaking of the pocket-formula. However, taken the above and looking forward to further improvements in this field, we choose to limit ourselves to a more pragmatic approach in this paper.In our present analysis cross sections for each step of the DPS mechanism are calculated in the k_T-factorization approach, that is: d σ^SPS(p p → c c̅X_1)/d y_1 d y_2 d^2 p_1,t d^2 p_2,t = 1/16 π^2 ŝ^2∫d^2 k_1t/πd^2 k_2t/π| M_g^* g^*→ c c̅|^2 ×δ^2 ( k⃗_1t + k⃗_2t - p⃗_1t - p⃗_2t)F_g(x_1,k_1t^2,μ^2)F_g(x_2,k_2t^2,μ^2), d σ^SPS(p p →2jetsX_2)/d y_3 d y_4 d^2 p_3,t d^2 p_4,t = 1/16 π^2 ŝ^2∑_ij∫d^2 k_3t/πd^2 k_4t/π| M_i^* j^*→2part.|^2 ×δ^2 ( k⃗_3t + k⃗_4t - p⃗_3t - p⃗_4t)F_i(x_3,k_3t^2,μ^2)F_j(x_4,k_4t^2,μ^2). The numerical calculations for both SPS mechanisms are also done within the KaTie code, where the relevant fully gauge-invariant off-shell 2 → 2 matrix elements M_g^* g^*→ c c̅ and M_i^* j^*→2part. are obtained numerically. Their useful analytical form can be found in Ref. <cit.> for c c̅ and in Ref. <cit.> for dijet production. Here, the strong coupling constant α_S and uPDFs are taken the same as in the case of the calculation of the SPS mechanism for cc̅ + 2jets production. The factorization and renormalization scales for the two single scatterings are μ^2= m_1t^2+m_2t^2/2 for the first, and μ^2= p_3t^2+p_4t^2/4 for the second subprocess. The framework of the k_T-factorization approach together with the KMR uPDFs was used with success in describing inclusive spectra of D, DD̅ correlations <cit.> as well as in the case of dijet production <cit.> and therefore can be expected to be a good starting point for the DPS predictions for the cc̅ + 2jets final state. § NUMERICAL RESULTS §.§ c c̅ + 2jets We start presentation of the numerical predictions with the results for production of charm quark-antiquark pair associated with two jets at √(s) = 13 TeV. Here, the phase space for charm quarks is almost unlimited, with broad range of rapidities |y_c| < 8 and without any cuts on their transverse momenta. For jets we keep the kinematical regime relevant for the ATLAS/CMS experiments, with |y_jet| < 4.9 and with transverse momentum cut p_T^jet > p_T,cut^jet = 20 GeV for leading and subleading jet.In Fig. <ref> we show transverse momentum distributions of charm quark/antiquark for single-parton scattering (left panel) and for double-parton scattering (right panel) mechanism at √(s) = 13 TeV. The three different histograms correspond to different approaches used in the calculations: LO collinear approximation (dotted), k_T-factorization (dashed) and k_T-factorization with extra cut on incident parton transverse momenta k_T < 20 GeV (solid). The KMR model for uPDFs, due to its special construction, allows for additional emission of hard gluon or quark from uPDF when the initial parton that enters the hard scattering has transverse momentum k_T > μ. To make predictions for the final state with c c̅-pair and with exactly two jets one needs to introduce the special limitation: k_T < p_T,cut^jet. We see that the effects related with this cut become important only when going to larger transverse momenta of charm quark and are much stronger in the case of the DPS results, where the cut is applied for all four incident partons. In both cases, for the SPS and the DPS calculation, the LO collinear approximation leads to a significantly smaller cross sections than those obtained within the k_T-factorization in the whole considered range of charm quark transverse momenta. In the left panel of Fig. <ref> we show again the transverse momentum distribution of charm quark/antiquark. Here, the DPS (dashed histogram) and the SPS (dotted histogram) contributions calculated within the k_T-factorization approach are shown together on the same plot. The DPS contribution clearly dominates over the SPS one in the region of c-quark p_T < 15 GeV. In the right panel of Fig. <ref> we present rapidity distribution of charm quark/antiquark. The two upper histograms correspond to the DPS mechanism and the two lower histogramscorrespond to the SPS contribution. Here, results for the k_T-factorization approach (solid histograms) are shown together with results obtained with the LO collinear approach (dotted histograms). The DPS component significantly dominates in the whole considered range of rapidities and the relative contribution of the SPS mechanism becomes even smaller when moving to forward/backward region. Figure <ref> shows transverse momentum distributions of the leading (left panel) and the subleading (right panel) jet calculated in the k_T-factorization approach for the SPS (lower histograms) and for the DPS mechanism (upper histograms). The DPS component dominates in the whole range of the considered jet transverse momenta.In Fig. <ref> we present some cc̅ correlation observables. The left panel shows distribution in azimuthal angle φ_cc̅ between the c-quark and the c̅-antiquark. Again, the DPS mechanism dominates over the SPS one in the whole range of relative azimuthal angle. The shape of the distribution for the DPS is determined directly by the inclusive single cc̅-pair production mechanism. In the case of the SPS component we observe an evident enhancement of the cross section in the region of rather small angles, which is not a typical behaviour e.g. in the case of inclusive charm production. In the right panel of Fig. <ref> we present differential cross section as a function of invariant mass of the cc̅-system M_cc̅. Both mechanisms have different slope of the distribution. At small invariant masses the DPS component clearly dominates over the SPS one. In the region of M_cc̅ < 20 GeV the difference is bigger than one order of magnitude. Both of the mechanisms become comparable starting from M_cc̅≈ 70 GeV.Figure <ref> presents differential distributions as a function of the azimuthal angle φ_jj between the two jets (left panel) and as a function of the dijet invariant mass M_jj (right panel). The SPS component is much more decorrelated in azimuthal angle than the DPS one. Both of them have a similar shape of the dijet invariant mass and differ only as far as only the normalization is considered. The DPS mechanism dominates in the whole range of the dijet invariant mass. Considering the cc̅ + 2jets final state one can also look at the correlations between c-quark and associated jet. For example, in Fig. <ref> we show the correlation distributions in the azimuthal angle φ_c-jet (left panel) and the rapidity difference Δ Y_c-jet between the c-quark (c̅-antiquark) and the leading jet (right panel).Both correlation observables are predicted to be dominated by the DPS mechanism in the whole range of φ_c-jet and Δ Y_c-jet , respectively. §.§ D^0 + 2jets Now we move to the predictions for single D^0 meson production in association with exactly two jets. The effects of the c → D^0 hadronization are taken into account via standard fragmentation function technique. For this purpose, we employ the scale-independent Peterson model of fragmentation function <cit.> with ε_c = 0.05 which is commonly-used in the literature in the context of heavy quark fragmentation. Details of the fragmentation procedure applied here and useful discussion of the uncertainties related to the choice of the fragmentation function can be found e.g. in Ref. <cit.>. In the last step, the cross section for meson is normalized by the relevant branching fraction BR(c → D^0) = 0.565.In this analysis, the D^0 meson is required to have |y^D^0| < 2.5 and p_T^D^0 > 3.5 GeV and the rapidities of both associated jets are |y^jet| < 4.9, which corresponds to the ATLAS detector acceptance. In Table <ref> we collect the corresponding integrated cross sections for inclusive D^0+2 jets production in pp-scattering at √(s) = 13 TeV for different cuts on transverse momenta of the associated jets, specified in the left column. The predictions are obtained within the k_T-factorization approach for the KMR uPDFs with the k_T < p_T,cut^jet constrain. We found large cross sections, of the order of a few, and up to even tens of microbarns, depending on the cuts on transverse momenta of the associated jets. The cross sections are dominated by the DPS mechanism with the relative DPS contribution at the level of 70 - 80 %.In Fig. <ref> we show the differential cross section as a function of transverse momenta of the D^0 meson for two different sets of cuts on transverse momenta of the associated jets (left and right panel). The DPS (dashed line) and the SPS (dotted line) components are shown separately together with their sum (solid line). We observe that in the region of D^0 meson transverse momenta p_T < 10 GeV the DPS mechanism significantly dominates over the SPS one.Figure <ref> shows a very interesting distributions in the azimuthal angle φ_D^0-jet between the D^0 meson (D^0 antimeson) and the leading jet, again for two different sets of cuts on transverse momenta of the associated jets (left and right panel). We see that the presence and the dominant role of the DPS component leads to a significant enhancement of the cross section and to a visible decorrelation of the distribution in contrast to the pure SPS-based predictions.§.§ D^0D̅^̅0̅ + 2jets In the present analysis, we also consider the case of production of the D^0D^0-pair in association with two jets. So now both, D^0-meson and D^0-antimeson are required to enter the ATLAS detector acceptance. The corresponding theoretical cross sections are collected in Table <ref>. Here, the predicted cross sections forD^0D^0+2 jets are slightly smaller than in the case of D^0+2 jets production (see Table <ref>) but still large (in the best scenario, of the order of a few microbarns). Also the relative DPS contribution is somewhat reduced and varies at the level of 50 - 70 %. In the case of the D^0D^0+2 jets final state we also find a very interesting correlation observable that may be useful to distinguish between the DPS and SPS mechanisms.Figure <ref> presents the distributions in azimuthal angle φ_D^0D^0 between the D^0 meson and D^0 antimeson in the case of D^0D^0+2 jets production. One can observe an evident enhancement of the cross section in the region of φ_D^0D^0 > π/2 caused by the presence of the DPS mechanism. § CONCLUSIONSIn the present paper we have calculated for a first time cross sectionsfor simultanous production of c c̅ (or D mesons) and dijets.Rather low transverse momentum cuts on jets have been used in order toenhance the DPS contribution. Both single and double parton scatttering mechanisms have been included.Several differential distributions have been shown and discussed. The calculation have been performed for current LHC collision energy √(s) = 13 TeV within the k_t-factorization approach using the Kimber-Martin-Ryskin unintegrated gluon distributions. The same formalism turned out previously to be very successfull for descriptionof production of one and two pairs of c c̅ and production of dijets.It was shown that the DPS contribution considerably dominates over the SPS contribution for small transverse momenta of c and/or c̅. At larger transverse momenta of c and/or c̅ the SPS contribution takes over. The distribution in transverse momentum of D mesons could be used to pin down the competition of both mechanisms. Very interesting are also azimuthal correlations between jets, between jet and charm quark/antiquark or jet and D mesons or even between charmed mesons. The corresponding experimental distributions can be used to further pin down underlying production mechanism.We have identified regions of the phase space where the DPS contributions significantly dominate. A future experimental cross sections in these regions could be used to determine the σ_eff parameter.In general, this quantity can dependend on several kinematical quantities. Surprisingly similar values were obtained from the analysis of different processes. We have discussed conditions how to extract σ_efffrom the discused in the present paper associated production of charm and dijets.Our analysis performed here shows that the extraction should be almost free of systematic errors as the DPS contribution clearly dominantes over the SPS one for the discussed reaction.AcknowledgmentsWe are particularly indebted to Andreas van Hameren for a help in using KaTie code. 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"published": "20170726104511",
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Destruction of Refractory Carbon in Protoplanetary Disks Dana E. Anderson1, Edwin A. Bergin2, Geoffrey A. Blake1, Fred J. Ciesla 3, Ruud Visser 4, Jeong-Eun Lee 5December 30, 2023 ============================================================================================================== Generative neural models have recently achieved state-of-the-art results for constituency parsing. However, without a feasible search procedure, their use has so far been limited to reranking the output of external parsers in which decoding is more tractable. We describe an alternative to the conventional action-level beam search used for discriminative neural models that enables us to decode directly in these generative models. We then show that by improving our basic candidate selection strategy and using a coarse pruning function, we can improve accuracy while exploring significantly less of the search space. Applied to the model of <cit.>, our inference procedure obtains 92.56 F1 on section 23 of the Penn Treebank, surpassing prior state-of-the-art results for single-model systems. § INTRODUCTION A recent line of work has demonstrated the success of generative neural models for constituency parsing <cit.>. As with discriminative neural parsers, these models lack a dynamic program for exact inference due to their modeling of unbounded dependencies. However, while discriminative neural parsers are able to obtain strong results using greedy search <cit.> or beam search with a small beam <cit.>, we find that a simple action-level approach fails outright in the generative setting. Perhaps because of this, the application of generative neural models has so far been restricted to reranking the output of external parsers.Intuitively, because a generative parser defines a joint distribution over sentences and parse trees, probability mass will be allocated unevenly between a small number of common structural actions and a large vocabulary of lexical items. This imbalance is a primary cause of failure for search procedures in which these two types of actions compete directly. A notion of equal competition among hypotheses is then desirable, an idea that has previously been explored in generative models for constituency parsing <cit.> and dependency parsing <cit.>, among other tasks. We describe a related state-augmented beam search for neural generative constituency parsers in which lexical actions compete only with each other rather than with structural actions. Applying this inference procedure to the generative model of <cit.>, we find that it yields a self-contained generative parser that achieves high performance.Beyond this, we propose an enhanced candidate selection strategy that yields significant improvements for all beam sizes. Additionally, motivated by the look-ahead heuristic used in the top-down parsers of <cit.> and <cit.>, we also experiment with a simple coarse pruning function that allows us to reduce the number of states expanded per candidate by several times without compromising accuracy. Using our final search procedure, we surpass prior state-of-the-art results among single-model parsers on the Penn Treebank, obtaining an F1 score of 92.56.§ COMMON FRAMEWORKThe generative neural parsers of <cit.> and <cit.> can be unified under a common shift-reduce framework. Both systems build parse trees in left-to-right depth-first order byexecuting a sequence of actions, as illustrated in Figure <ref>. These actions can be grouped into three major types: (X) and (X), which open and close a constituent with nonterminal X,[The model described in <cit.> has only a single action, whereas the model described in <cit.> annotates (X) actions with their nonterminals. We present the more general version here.] respectively, and (x), which adds the word x to the current constituent. The probability of an action sequence (a_1, …, a_T) is P(a_1, …, a_T)= ∏_t=1^T P(a_t | a_1, …, a_t-1) = ∏_t=1^T[softmax(𝐖𝐮_t + 𝐛)]_a_t ,where 𝐮_t is a continuous representation of the parser's state at time t, and [𝐯]_j denotes the jth component of a vector 𝐯. We refer readers to the respective authors' papers for the parameterization of 𝐮_t in each model. In both cases, the decoding process reduces to a search for the most probable action sequence that represents a valid tree over the input sentence. For a given hypothesis, this requirement implies several constraints on the successor set <cit.>; e.g., (x) can only be executed if the next word in the sentence is x, and (X) cannot be executed directly after (X).§ MODEL AND TRAINING SETUP We reimplemented the generative model described in <cit.> and trained it on the Penn Treebank <cit.> using their published hyperparameters and preprocessing. However, rather than selecting the final model based on reranking performance, we instead perform early stopping based on development set perplexity. We use sections 2-21 of the Penn Treebank for training, section 22 for development, and section 23 for testing. The model's action space consists of 26 matching pairs of and actions, one for each nonterminal, and 6,870 actions, one for each preprocessed word type. While we use this particular model for our experiments, we note that our subsequent discussion of inference techniques is equally applicable to any generative parser that adheres to the framework described above in Section <ref>.§ ACTION-LEVEL SEARCH Given that ordinary action-level search has been applied successfully to discriminative neural parsers <cit.>, it offers a sensible starting point for decoding in generative models. However, even for large beam sizes, the following pathological behavior is encountered for generative decoding, preventing reasonable parses from being found. Regardless of the sequence of actions taken so far, the generative model tends to assign much higher probabilities to structural and actions than it does lexical actions, as shown in Figure <ref>. The model therefore prefers to continually open new constituents until a hard limit is reached, as the alternative at each step is to take the low-probability action of shifting the next word. The resulting sequence typically has much lower overall probability than a plausible parse, but the model's myopic comparison between structural and lexical actions prevents reasonable candidates from staying on the beam. Action-level beam search with beam size 1000 obtains an F1 score of just 52.97 on the development set.§ WORD-LEVEL SEARCHThe imbalance between the probabilities of structural and lexical actions suggests that the two kinds of actions should not compete against each other within a beam. This leads us to consider an augmented state space in which they are kept separate by design, as was done by <cit.>. In conventional action-level beam search, hypotheses are grouped by the length of their action history |A|. Letting A_i denote the set of actions taken since the ith shift action, we instead group hypotheses by the pair (i, |A_i|), where i ranges between 0 and the length of the sentence.Let k denote the target beam size. The search process begins with the empty hypothesis in the (0, 0) bucket. Word-level steps are then taken according to the following procedure for i = 0, 1, …, up to the length of the sentence (inclusive). Beginning with the (i, 0) bucket, the successors of each hypothesis are pooled together, sorted by score, and filtered down to the top k. Of those that remain, successors obtained by taking an or action advance to the (i, 1) bucket, whereas successors obtained from a action are placed in the (i+1, 0) bucket if i is less than the sentence length, or the completed list if i is equal to the sentence length. This process is repeated for the (i, 1) bucket, the (i, 2) bucket, and so forth, until the (i+1, 0) bucket contains at least k hypotheses. If desired, a separate word beam size k_w < k can be used at word boundaries, in which case each word-level step terminates when the (i+1, 0) bucket has k_w candidates instead of k. This introduces a bottleneck that can help to promote beam diversity.Development set results for word-level search with a variety of beam sizes and with k_w = k or k_w = k/10 are given in Table <ref>. We observe that performance in both cases increases steadily with beam size. Word-level search with k_w = k/10 consistently outperforms search without a bottleneck at all beam sizes, indicating the utility of this simple diversity-inducing modification. The top result of 92.93 F1 is already quite strong compared to other single-model systems.§ FAST-TRACK CANDIDATE SELECTIONThe word-level beam search described in Section <ref> goes one step toward ameliorating the issue that causes action-level beam search to fail, namely the direct competition between common structural actions with high probabilities and low-frequency shift actions with low probabilities. However, the issue is still present to some extent, in that successors of both types from a given bucket are pooled together and filtered down as a single collection before being routed to their respective destinations. We therefore propose a more direct solution to the problem, in which a small number k_s ≪ k of successors are fast-tracked to the next word-level bucket before any filtering takes place. These fast-tracked candidates completely bypass competition with potentially high-scoring or successors, allowing for higher-quality results in practice with minimal overhead. See Figure <ref> for an illustration.We repeat the experiments from Section <ref> with k_s = k/100 and report the results in Table <ref>. Note that the use of fast-tracked candidates offers significant gains under all settings.The top result improves from 92.93 to 93.18 with the use of fast-tracked candidates, surpassing prior single-model systems on the development set.§ ACTION PRUNING At any point during the trajectory of a hypothesis, either 0 or all 26 of the actions will be available, compared with at most 1 action and at most 1 action. Hence, when available, actions comprise most or all of a candidate's successor actions. To help cut down on this portion of the search space, it is natural to consider whether some of these actions could be ruled out using a coarse model for pruning. §.§ Coarse Model We consider a class of simple pruning models that condition on the c ≥ 0 most recent actions and the next word in the sentence, and predict a probability distribution over the next action. In the interest of efficiency, we collapse all actions into a single unlexicalized action, significantly reducing the size of the output vocabulary.The input 𝐯_t to the pruning model at time t is the concatenation of a vector embedding for each action in the context (a_t-c, a_t-c+1, …, a_t-1) and a vector embedding for the next word w:𝐯_t = [𝐞_a_t-c; 𝐞_a_t-c+1; …; 𝐞_a_t-1; 𝐞_w],where each 𝐞_j is a learned vector embedding. The pruning model itself is implemented by feeding the input vector through a one-layer feedforward network with a ReLU non-linearity, then applying a softmax layer on top:P(a_t = a | a_1, …, a_t-1, next-word = w) = P(a_t = a | a_t-c, …, a_t-1, next-word = w) = [softmax(𝐖_2 max(𝐖_1 𝐯_t + 𝐛_1, 0) + 𝐛_2)]_a .The pruning model is trained separately from the main parsing model on gold action sequences derived from the training corpus, with log-likelihood as the objective function and a cross entropy loss. §.§ Strategy and Empirical Lower Bound Once equipped with a coarse model, we use it for search reduction in the following manner. As mentioned above, when a hypothesis is eligible to open a new constituent, most of its successors will be obtained through actions. Accordingly, we use the coarse model to restrict the set of actions to be explored. When evaluating the pool of successors for a given collection of hypotheses during beam search, we run the coarse model on each hypothesis to obtain a distribution over its next possible actions, and gather together all the coarse scores of the would-be successors. We then discard the successors whose coarse scores lie below the top 1 - p quantile for a fixed 0 < p < 1, guaranteeing that no more than a p-fraction of successors are considered for evaluation. Taking p = 1 corresponds to the unpruned setting.This strategy gives us a tunable hyperparameter p that allows us to trade off between the amount of search we perform and the quality of our results. Before testing our procedure, however, we would first like to investigate whether there is a principled bound on how low we can expect to set p without a large drop in performance. A simple estimate arises from noting that the pruning fraction p should be set to a value for which most or all of the outputs encountered in the training set are retained. Otherwise, the pruning model would prevent the main model from even recreating the training data, let alone producing good parses for new sentences.To this end, we collect training corpus statistics on the occurrences of inputs to the pruning function and their corresponding outputs.We then compute the number of unique actions associated with inputs occurring at least 20 times, and restrict our attention to inputs with at least one output. The resulting cumulative distributions for context sizes c = 0, 1, 2 are given in Table <ref>. If we require that our pruning fraction p be large enough to recreate at least 99% of the training data, then since there are 26 total nonterminals, approximate[These thresholds are not exact due to the fact that our pruning procedure operates on collections of multiple hypotheses' successors at inference time rather than the successors of an individual hypothesis.] lower bounds for p are 10/26 ≈ 0.385 for c = 0, 7/26 ≈ 0.269 for c = 1, and 6/26 ≈ 0.231 for c = 2.§.§ Pruning Results We reran our best experiment from Section <ref> with an order-2 pruning function and pruning fractions p = 6/26, …, 11/26. The results are given in Table <ref>. We observe that performance is on par with the unpruned setup (at most 0.1 absolute difference in F1 score) for p as low as 8/26 ≈ 0.308. Setting p to 7/26 ≈ 0.269 results in a drop of 0.18, and setting p to 6/26 ≈ 0.231 results in a drop of 0.40. Hence, degradation begins to occur right around the empirically-motivated threshold of 6/26 given above, but we can prune 1 - 8/26 ≈ 69.2% of successors with minimal changes in performance.§ FINAL RESULTS AND CONCLUSION We find that the best overall settings are a beam size of k = 2000, a word beam size of k_w = 200, and k_s = 20 fast-track candidates per step, as this setup achieves both the highest probabilities under the model and the highest development F1. We report our test results on section 23 of the Penn Treebank under these settings in Table <ref> both with and without pruning, as well as a number of other recent results. We achieve F1 scores of 92.56 on the test set without pruning and 92.53 when 1 - 8/26 ≈ 69.2% of successors are pruned, obtaining performance well above the previous state-of-the-art scores for single-model parsers. This demonstrates that the model of <cit.> works well as an accurate, self-contained system. The fact that we match the performance of their reranking parser using the same generative model confirms the efficacy of our approach. We believe that further refinements of our search procedure can continue to push the bar higher, such as the use of a learned heuristic function for forward score estimation, or a more sophisticated approximate decoding scheme making use of specific properties of the model. We look forward to exploring these directions in future work.§ ACKNOWLEDGMENTS MS is supported by an NSF Graduate Research Fellowship. DF is supported by an NDSEG fellowship. emnlp_natbib | http://arxiv.org/abs/1707.08976v1 | {
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"Dan Klein"
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"published": "20170727180118",
"title": "Effective Inference for Generative Neural Parsing"
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Relating Catlin and D'Angelo q-types]Relating Catlin and D'Angelo q-types Simion Stoilow Institute of Mathematics of the Romanian Academy, Research unit 3, 21 Calea Grivitei Street, 010702 Bucharest, [email protected] School of Mathematics, Trinity College Dublin, Dublin 2, [email protected][2020]Primary 32F18; 32T25; Secondary 32V35; 13H15. We clarify the relationship between the two most standard measurements of the order of contact of q-dimensional complex varieties with a real hypersurface, the Catlin and D'Angelo q-types, by showing that the former equals the generic value of the normalized order of contact measured along curves whose infimum is by definition the D'Angelo q-type.[ Andreea C. Nicoara May 1, 2019 ====================== § INTRODUCTIONThe purpose of this note is finishing the work initiated in <cit.> as far as elucidating the relationship between Catlin and D'Angelo q-types. D'Angelo's finite q-type defined in 1982 in <cit.> is by far the most standard finite type notion in several complex variables. The Kohn Conjecture, one of the most famous open problems in complex analysis, if proven would say that finite D'Angelo q-type at a certain point on the boundary of a smooth pseudoconvex domain in ^n ensures the termination of the Kohn algorithm defined in <cit.> and thus the subellipticity of the -Neumann problem for (p,q) forms in the neighborhood of that point. As of now, the only result on the subellipticity of the -Neumann problem for (p,q) forms on smooth pseudoconvex domains in ^n is due to Catlin in<cit.> and provides a lower bound on the subelliptic gain for the -Neumann problem in terms of his own notion of q-type. As a result, relating the D'Angelo and Catlin q-types effectively is paramount so that if and when the Kohn Conjecture is proven with a lower bound for subellipticity in place, Catlin's bound and the bound obtained via the Kohn algorithm can be compared. As the results in <cit.> are proven with respect to the generic value, we introduce the following definition; see also <cit.>: Let 2 ≤ q ≤ n. Ifis an ideal in _x_0,Δ̃_q(, x_0)=gen.val_{w_1, …, w_q-1} Δ_1( (, w_1, …, w_q-1), x_0 ),where the generic value is taken over all non-degenerate sets {w_1, …, w_q-1} of linear forms in _x_0, (, w_1, …, w_q-1) is the ideal in _x_0 generated by , w_1, …, w_q-1, and Δ_1 is the D'Angelo 1-type. Likewise, if M is a real hypersurface in ^n and x_0 ∈ M,Δ̃_q (M, x_0) =gen.val_{w_1, …, w_q-1} Δ_1( ((M), w_1, …, w_q-1), x_0 ),where ((M), w_1, …, w_q-1) is the ideal in C^∞_x_0 generated by all smooth functions (M) vanishing on M along with w_1, …, w_q-1.Let D_q denote the Catlin q-type, and let Δ_q be the D'Angelo q-type. The two main theorems in <cit.> can now be stated with respect to Δ̃_q as follows:[Theorem 1.1 <cit.>,<cit.>]Letbe an ideal of germs of holomorphic functions at x_0, then for 1 ≤ q ≤ nD_q(, x_0) ≤Δ̃_q(, x_0) ≤(D_q(, x_0))^n-q+1.[Theorem 1.2 <cit.>,<cit.>]Let Ω in ^n be a domain withboundary. Let x_0 ∈ b Ω be a point on the boundary of the domain, and let 1 ≤ q <n. * D_q(b Ω, x_0) ≤Δ̃_q(b Ω, x_0);* If Δ̃_q(b Ω, x_0)<∞ and the domain is q-positive at x_0 (the q version of D'Angelo's property P), thenΔ̃_q(b Ω, x_0) ≤ 2 ( D_q(b Ω, x_0)/2)^n-q.In particular, if b Ω is pseudoconvex at x_0 and Δ̃_q(b Ω, x_0)<∞, thenD_q(b Ω, x_0) ≤Δ̃_q(b Ω, x_0) ≤ 2 ( D_q(b Ω, x_0)/2)^n-q. It is an easy consequence of D'Angelo's work in <cit.> that Δ_q and Δ̃_q are simultaneously finite for ideals of germs of holomorphic functions: Letbe any ideal in _x_0. For any 2 ≤ q ≤ n,Δ_q(, x_0) ≤Δ̃_q(, x_0) ≤( Δ_q(, x_0) )^n-q+1.The counterpart of this result for points on the boundary of a domain is slightly more difficult and requires the assumption of q-positivity:Let Ω in ^n be a domain withboundary. Let x_0 ∈ b Ω be a point on the boundary of the domain, and let 2 ≤ q <n. If the domain is q-positive at x_0, thenΔ_q(b Ω, x_0) ≤Δ̃_q(b Ω, x_0) ≤ 2 ( Δ_q(b Ω, x_0) )^n-q.If the domain is pseudoconvex at x_0, then it is q-positive, so the same inequality holds.We shall prove Propositions <ref> and <ref> in Section <ref>. We note here at for q=1 the notions of Δ_q, Δ̃_q, and D_q coincide, a fact obvious from their definitions.Martino Fassina produced an example in <cit.> when the Catlin and D'Angelo q-types are not equal to each other, answering a question that has been open since 1987 when <cit.> was published. In that example,Δ_2(,0)= 3 < 4 = Δ̃_2(,0) = D_2(,0).Furthermore, in the same paper <cit.>, Fassina proved that given any positive integer, an ideal of holomorphic germs can be constructed so that the difference between the Catlin q-type and the D'Angelo q-type of that ideal is larger than the given integer. Fassina's work thus highlights the importance of finding an effective relationship between Δ_q and D_q.The main result of this paper is the following characterization of the Catlin q-type D_q that clarifies completely how it relates to the D'Angelo q-type: * Letbe an ideal of germs of holomorphic functions at x_0, and let 2≤ q ≤ n, thenD_q(, x_0) = Δ̃_q(, x_0). * Let Ω in ^n be a domain withboundary. Let x_0 ∈ b Ω be a point on the boundary of the domain, and let 2 ≤ q <n. ThenD_q(b Ω, x_0) = Δ̃_q(b Ω, x_0). * Letbe an ideal of germs of holomorphic functions at x_0, and let 2≤ q ≤ n, thenΔ_q(, x_0) ≤ D_q(, x_0) ≤( Δ_q(, x_0) )^n-q+1. * Let Ω in ^n be a domain withboundary. Let x_0 ∈ b Ω be a point on the boundary of the domain, let 2 ≤ q <n, and assume that the domain is q-positive at x_0. ThenΔ_q(b Ω, x_0) ≤ D_q(b Ω, x_0) ≤ 2 ( Δ_q(b Ω, x_0) )^n-q.In particular, if the domain is pseudoconvex at x_0, then the same inequality holds.In <cit.> Catlin defined his q-type D_q by starting with the germ of a q-dimensional variety V^q and constructing an open set in the Grassmannian G^n-q+1 of all (n-q+1)-dimensional complex linear subspaces through x_0 in ^n that is specific to a given V^q. His construction is so delicate because the aim is to obtain the same number of curves in the intersection of V^q with each (n-q+1)-dimensional complex linear subspace in this open set and the same maximal normalized order of contact measured along the intersection curves. As a result, proving the equality of D_q with Δ̃_q is trickier than it seems. Given any curve and any subspace W, we have to show the existence of a cylinder variety with the subspace W as the directrix subspace along the curve at x_0. This construction allows us to construct a q-dimensional variety V^q starting with any curve whose open set in the Grassmanian G^n-q+1 in Catlin's construction is well behaved.The paper is organized as follows: We devote Section <ref> to recalling the definitions of the Catlin and D'Angelo q-types and relevant results. We then prove Propositions <ref> and <ref> as well as the Main Theorem, Theorem <ref>, and Corollary <ref> in Section <ref>.Acknowledgements The work of Vasile Brinzanescu was partially supported by a grant of the Ministry of Research and Innovation, CNCS - UEFISCDI, project number PN-III-P4-ID-PCE-2016-0030, within PNCDI III, and CNCS-UEFISCDI project PN-III-P4-ID-PCE-2020-0029, Syzygies, invariants and classification problems in algebraic geometry and topology. § CATLIN AND D'ANGELO Q-TYPESFor the convenience of the reader, we first recall the definitions of the D'Angelo and Catlin q-types and the properties needed for proving the results in Section <ref>.Let C^∞_x_0 be the ring of smooth germs at x_0 ∈^n, and letbe an ideal in C^∞_x_0 or _x_0.Δ_1(,x_0) = sup_φ∈(n,x_0)inf_g ∈ord_0φ^* g/ord_0φ,where (n,x_0) is the set of all germs of holomorphic curvesφ: (U,0) → (^n, x_0)such that φ(0)=x_0 for U is some neighborhood of the origin in ^1, ord_0 is the vanishing order at the origin, andord_0φ = min_1 ≤ j ≤ m ord_0φ_j.Let 2 ≤ q ≤ n. Ifis an ideal in _x_0, the D'Angelo q-type is given byΔ_q(, x_0)=inf_{w_1, …, w_q-1} Δ_1( (, w_1, …, w_q-1), x_0 ),where the infimum is taken over all non-degenerate sets {w_1, …, w_q-1} of linear forms in _x_0 and (, w_1, …, w_q-1) is the ideal in _x_0 generated by , w_1, …, w_q-1. Likewise, if M is a real hypersurface in ^n, x_0 ∈ M, and 2 ≤ q < n, the D'Angelo q-type of the hypersurface M is given byΔ_q (M, x_0) =inf_{w_1, …, w_q-1} Δ_1( ((M), w_1, …, w_q-1), x_0 ),where ((M), w_1, …, w_q-1) is the ideal in C^∞_x_0 generated by all smooth functions (M) vanishing on M along with w_1, …, w_q-1. Ifis an ideal in _x_0,D(, x_0) = _ (_x_0 / ).[Proposition 2.8 <cit.>]Letbe a proper ideal in _x_0, and let x_0 ∈^n.inf_{w_1, …, w_q-1} D( (, w_1, …, w_q-1), x_0 )=gen.val_{w_1, …, w_q-1} D( (, w_1, …, w_q-1), x_0 ),where {w_1, …, w_q-1} is a non-degenerate set of linear forms in _x_0, (, w_1, …, w_q-1) is the ideal in _x_0 generated by , w_1, …, w_q-1, and the infimum and the generic value are both taken over all such non-degenerate sets {w_1, …, w_q-1} of linear forms in _x_0. In other words, the infimum is achieved and equals the generic value. [D'Angelo, Theorem 2.7 <cit.>]Ifis an ideal in _x_0 containing q linearly independent linear forms w_1, …, w_q, then Δ_1(, x_0) ≤ D(, x_0) ≤( Δ_1(,x_0) )^n-q.If Δ_q (M, x_0) = t < ∞, let k = ⌈ t ⌉ be the ceiling of t. By Proposition 14 from p.88 of <cit.>, Δ_q (M, x_0) = Δ_q (M_k, x_0), where M_k is real hypersurface defined by r_k, the polynomial with the same k-jet at x_0 as the defining function r of M. By D'Angelo's polarization technique from <cit.>,r_k = Re{h} + ||f||^2-||g||^2,where ||f||^2= ∑_j=1^N |f_j|^2, ||g||^2= ∑_j=1^N |g_j|^2, and the functions h, f_1, …, f_N, g_1, …, g_N are all holomorphic polynomials in n variables. Let (N) be the group of N × N unitary matrices. ∀U ∈ (N) consider the ideal of holomorphic polynomials (U, x_0)=(h, f-Ug) generated by h and the N components of f-Ug, where f=(f_1, …, f_N) and g= (g_1, …, g_N). D'Angelo proved the following result:[Theorem 14 <cit.>] Δ_q(M, x_0) ≤ 2 sup_U ∈ (N)inf_{w_1, …, w_q-1} D( ((U,x_0), w_1, …, w_q-1), x_0 ) ≤ 2 (Δ_q(M, x_0) )^n-qLet us also note that (N) is compact, so sup_U ∈ (N)D( ((U,x_0), w_1, …, w_q-1) is achieved due to the fact that the multiplicity of an ideal is upper semi-continuous:[Part of Proposition II.5.3 <cit.>]Let (λ) be an ideal in _x_0 that depends continuously on λ. Then D((λ), x_0) is an upper semi-continuous function of λ. We recall from <cit.> the q version of D'Angelo's property P, q-positivity, the hypothesis that appears in Theorem <ref>:[Definition 2.14 <cit.>]Let M be a real hypersurface of ^n, and let x_0 ∈ M be such that Δ_q(M, x_0) <k. Let j_k,x_0 r = r_k = Re{h} + ||f||^2-||g||^2 be a holomorphic decomposition at x_0 of the k-jet of the defining function r of M. We say that M is q-positive at x_0 if for every holomorphic curve φ∈(n,x_0) for which φ^* h vanishes and such that the image of φ locally lies in the zero locus of a non-degenerate set of linear forms {w_1, …, w_q-1} at x_0, the following two conditions are satisfied: (i) ord_0φ^* r is even, i.e. ord_0φ^* r = 2a, for some a ∈;(ii) (d/dt)^a (d/d t̅)^a φ^* r (0) ≠ 0.When q=1 this definition is exactly D'Angelo's property P with respect to which he proved the following result:[Theorem 5.3 <cit.>]If M satisfies property P at x_0, thenΔ_1(M, x_0) = 2sup_U ∈ (N)Δ_1((U,x_0), x_0).Under the assumption of q-positivity, it is obvious that D'Angelo's result immediately implies the following:If M is q-positive at x_0, thenΔ̃_q (M, x_0) = 2gen.val_{w_1, …, w_q-1} sup_U ∈ (N)Δ_1 (((U,x_0), w_1, …, w_q-1)),x_0).Pseudoconvexity and q-positivity relate to each other as follows:[Proposition 2.18<cit.>]If M is pseudoconvex near x_0 and Δ_q (M, x_0)< + ∞, then M and M_k, the hypersurface corresponding to the truncation of order k of the defining function at x_0, are q-positive at x_0 for all sufficiently large k. Let V^q be the germ of a q-dimensional complex variety passing through x_0. Let G^n-q+1 be the set of all (n-q+1)-dimensional complex linear subspaces through x_0 in ^n. Consider the intersection V^q ∩ S for S ∈ G^n-q+1. For a generic, namely open and dense, subset W̃ of G^n-q+1, V^q ∩ S consists of finitely many irreducible one-dimensional components V^q_S,k for k=1, …, P. We parametrize each such germ of a curve by some open set U_k ∋ 0 in . Thus, γ_S^k : U_k → V^q_S,k, where γ_S^k (0)=x_0. For every holomorphic germ f ∈_x_0, letτ (f, V^q ∩ S) = max_k=1, …, Pord_0(γ^k_S)^* f/ord_0γ^k_S.Likewise, for r the defining function of a real hypersurface M in ^n passing through x_0, letτ (V^q ∩ S, x_0) = max_k=1, …, Pord_0(γ^k_S)^* r/ord_0γ^k_S.In Proposition 3.1 of <cit.>, Catlin showed τ (f, V^q ∩ S) assumes the same value for all S in a generic subset W̃ of linear subspaces that depends on V^q, so he definedτ(f, V^q) = gen.val_S ∈W̃{τ (f, V^q ∩ S) }andτ(, V^q) = min_f ∈τ(f, V^q).Proposition 3.1 of <cit.> likewise implies that τ (V^q ∩ S, x_0) assumes the same value for all S in the same generic subset W̃ of linear subspaces depending on V^q, so Catlin definedτ(V^q, x_0) = gen.val_S ∈W̃{τ (V^q ∩ S, x_0) }.We will need to explain exactly how the generic subset W̃ depends on the germ of the q-dimensional complex variety V^q when we prove Theorem <ref> in Section <ref>, so we defer that discussion. Letbe an ideal of holomorphic germs at x_0, then the Catlin q-type of the idealis given byD_q (, x_0)=sup_V^q{τ(, V^q) }. Let M be a real hypersurface in ^n, and let x_0 ∈ M. The Catlin q-type of M at x_0 is given byD_q (M, x_0)=sup_V^q{τ (V^q, x_0) }.In both cases, the supremum is taken over the set of all germs of q-dimensional complex varieties V^q passing through x_0. Since there is only one n-dimensional complex linear subspace passing through x_0 in ^n,Δ_1 (M, x_0) =Δ̃_1 (M, x_0) = D_1 (M , x_0)andΔ_1 (, x_0) =Δ̃_1 (, x_0) = D_1 ( , x_0)for any idealin _x_0. Therefore, relating these quantities is non-trivial only if q≥ 2.We shall also need the following result:[Proposition 4.1 <cit.>,<cit.>]Letbe any ideal in _x_0. For any 2 ≤ q ≤ n, D_q(, x_0) ≤Δ̃_q(, x_0). § PROOF OF RESULTSWe claimed in the introduction that Δ_q and Δ̃_q being simultaneously finite for ideals of germs of holomorphic functions easily follows from D'Angelo's work in <cit.>. Here is the proof:Proof of Proposition <ref>: Let {w_1, …, w_q-1} be any non-degenerate set of linear forms in _x_0. Trivially, Δ_q(, x_0)= inf_{w_1, …, w_q-1}Δ_1( (, w_1, …, w_q-1), x_0 ) ≤gen.val_{w_1, …, w_q-1} Δ_1( (, w_1, …, w_q-1), x_0 ) = Δ̃_q(, x_0).Applying (<ref>) in Theorem <ref> with q-1 instead of q yieldsgen.val_{w_1, …, w_q-1} Δ_1( (, w_1, …, w_q-1), x_0 ) ≤gen.val_{w_1, …, w_q-1} D ( (, w_1, …, w_q-1), x_0 )andinf_{w_1, …, w_q-1} D ( (, w_1, …, w_q-1), x_0 ) ≤(inf_{w_1, …, w_q-1}Δ_1 ( (, w_1, …, w_q-1), x_0 ))^n-q+1,butinf_{w_1, …, w_q-1} D ( (, w_1, …, w_q-1), x_0 ) = gen.val_{w_1, …, w_q-1} D ( (, w_1, …, w_q-1), x_0 )by Proposition <ref>. Proving that Δ_q and Δ̃_q are simultaneously finite when measured at points on the boundary of a domain requires q-positivity as well as a slightly more elaborate argument:Proof of Proposition <ref>: Since the definition of Δ_q involves taking the infimum over all non-degenerate sets {w_1, …, w_q-1} of linear forms in _x_0, whereas the definition of Δ̃_q involves taking the generic value over the same sets, it is obvious that Δ_q(b Ω, x_0) ≤Δ̃_q(b Ω, x_0). We thus only have to prove the second inequality. Assume that Δ_q(b Ω, x_0) < +∞; otherwise, Δ_q(b Ω, x_0)=Δ̃_q(b Ω, x_0)=+∞, and there is nothing to prove. Now, let k be large enough so that Δ_q (M, x_0) = Δ_q (M_k, x_0), where M_k is real hypersurface defined by r_k, the polynomial with the same k-jet at x_0 as the defining function r of M; see equation (<ref>). Carry out the polarization of r_k to arrive at the ideal (U,x_0) in _x_0 defined for each unitary matrix U ∈(N). By Theorem <ref>,sup_U ∈ (N)inf_{w_1, …, w_q-1} D( ((U,x_0), w_1, …, w_q-1), x_0 ) ≤(Δ_q(M, x_0) )^n-q.By Proposition <ref>, we can substitute the generic value for the infimum as follows:sup_U ∈ (N)gen.val_{w_1, …, w_q-1} D( ((U,x_0), w_1, …, w_q-1), x_0 ) ≤(Δ_q(M, x_0) )^n-qSince we are assuming q-positivity, Theorem <ref> and Corollary <ref> together yield Δ̃_q (M, x_0)= 2gen.val_{w_1, …, w_q-1} sup_U ∈ (N)Δ_1 (((U,x_0), w_1, …, w_q-1)),x_0)≤ 2gen.val_{w_1, …, w_q-1} sup_U ∈ (N) D( ((U,x_0), w_1, …, w_q-1), x_0 ).Since (N) is compact, Propositions <ref> and <ref> imply that the supremum and the generic value can be exchanged, so we obtain thatΔ̃_q (M, x_0) ≤ 2(Δ_q(M, x_0) )^n-qby combining the two previous inequalities. If M is pseudoconvex at x_0, by Proposition <ref>, M is q-positive at x_0, so the same inequality must hold. Our proof of the Main Theorem <ref> is motivated by a remark Catlin made on pp.147-8 of <cit.>. Catlin's remark can be paraphrased as follows: In order to compare the D'Angelo q-type with his own notion of q-type, it would be necessary to piece together one-dimensional varieties in order to get a q-dimensional one.We start with the following lemma, which constructs a very simple q-dimensional variety containing a given curve:Let 2 ≤ q ≤ n-1. Given a holomorphic curve Γ passing through a point x_0 ∈^n and a (q-1)-dimensional hyperplane Z passing through x_0 and satisfying that the tangent line to the curve Γ at x_0 is not contained in Z, there exists a germ of a q-dimensional cylinder C^q at x_0 that contains the curve Γ and whose tangent space at x_0 contains Z.If x_0 is a singular point of the curve Γ, the curve could have multiple tangent lines, one for each branch of the curve. In that case, our assumption that the tangent line to the curve Γ at x_0 is not contained in Z means none of the tangent lines of Γ at x_0 are contained in Z. Proof: Let Γ⊂ℂ^n be a holomorphic curve(Γ =1)given by the equationsf_1(z_1,z_2,...,z_n)=0f_2(z_1,z_2,...,z_n)=0⋮f_n-1(z_1,z_2,...,z_n)=0 subject to the conditionrank(∂ f_i/∂ z_j)_1 ≤ i ≤ n-1;1≤ j ≤ n =n-1that holds generically, namely everywhere except at some isolated points of Γ. Let x_0=(x_1^0, …, x_n^0). The (q-1)-hyperplane Z is given parametrically as x_i=Σ_j=1^q-1u_jit_j + x_i^0, i=1,2,...,nwith the conditionrank(u_ji)=q-1. We have assumed that the tangent line to the curve Γ at x_0 is not contained in Z, so we can define the cylinder C^q determined by the curve Γ and the hyperplaneZ as follows: C^q:={M∈ℂ^nsuch that there existsP∈Γ with lineMP parallel toZ}.By taking the coordinates for M=(z_1,...,z_n) and P=(x_1,...,x_n), we obtain z_i-x_i=Σ_j=1^q-1u_jit_j . From the condition P∈Γ, we get the parametric equations of the cylinder C^q: f_l(-Σ_j=1^q-1u_j1t_j+z_1, ..., -Σ_j=1^q-1u_jnt_j+z_n)=0,l=1,...,n-1. Differentiation with respect to parameters t_1, …, t_q-1 yields ∂ f_l/∂ t_j=∑_i=1^n ∂ f_l/∂ x_iu_ji. The condition that the tangent to the curve Γ at x_0 is not parallel to Z implies rank (∂ f_l/∂ t_j)_1 ≤ l ≤ n-1;1≤ j ≤ q-1=q-1generically in the neighborhood of x_0. Of course, we have that Γ⊂ C^q and that the tangent space of C^q at x_0 contains Z by construction.For each of l=1,...,n-1, the hypersurface with equation f_l(z_1,...,z_n)=0 contains the curve Γ. Letbe an ideal of germs of holomorphic functions at x_0, then for 2 ≤ q ≤ n-1,Δ̃_q(, x_0) ≤ D_q(, x_0).Proof: We proceed as in the proof of Proposition <ref>. Let D_q(, x_0)=t < +∞, else the estimate is trivially true. Assume Δ̃_q(, x_0) >t. By definition,Δ̃_q(, x_0)=gen.val_{w_1, …, w_q-1} Δ_1( (, w_1, …, w_q-1), x_0 ),where the generic value is taken over all non-degenerate sets {w_1, …, w_q-1} of linear forms in _x_0. Thus, there exist t'>t and a curve Γ passing through x_0 such thatinf_g ∈ (, w'_1, …, w'_q-1)ord_0Γ^* g/ord_0Γ=t',for some non-degenerate set {w'_1, …, w'_q-1} of linear forms in _x_0. The variety corresponding to the non-degenerate set {w'_1, …, w'_q-1} of linear forms is a (n-q+1)-dimensional hyperplane, which we will call H. Clearly, Γ⊂ H as each w'_j has vanishing order 1 so any value starting at 2, the lower bound for Δ̃_q(b Ω, x_0), can only be achieved by a curve that sits in the zero set of {w'_1, …, w'_q-1}. Let Z be the (q-1)-dimensional hyperplane passing through x_0 that is transversal to H, i.e., _ (H ⊕ Z)=n. Since Γ⊂ H, the tangent lines to the curve Γ at x_0 (there could be several if the point is singular) are not contained in Z.By Lemma <ref>, there exists a germ of a q-dimensional cylinder C^q_Z at x_0 that contains the curve Γ and whose tangent space at x_0 contains Z. We would like to show that Γ is one of the intersection curves along which τ((, w'_1, …, w'_q-1), C^q_Z) is measured by examining Catlin's construction in Proposition 3.1 of <cit.>. In the proof of Proposition 3.1 from <cit.>, Catlin removed three different sets W_1, W_2, and W_3 from G^n-q+1 in order to arrive at his generic set W̃ on which his q-type is computed. We wish to show that H ∈W̃, namely that H cannot belong to any of the three sets W_1, W_2, and W_3 taken out from G^n-q+1. First, to the germ of the variety C^q_Z, there corresponds the ideal I=(C^q_Z) in the ring _x_0 of all germs of holomorphic functions vanishing on C^q_Z. By a translation, one can suppose x_0=0 and thus denote bythe ring _x_0. Let _k denote the subring ofconsisting of germs of holomorphic functions of only the first k variables. Now, let V=C^q_Z,and consider as in <cit.> the tangent cone ofV at the origin, V'={z∈ℂ^n ;f̃ (z)=0, f∈ I}, where for any function f,f̃ is the homogeneous polynomial given by the leading terms in its Taylor expansion at the origin. By <cit.> the tangent cone V' can also be defined by V'={lim_|z|→ 0cz/|z|; z∈ V,c∈ℂ}and V' has dimension q since V has dimension q. The curve Γ in the (n-q+1)-dimensional hyperplane H has as its tangent cone at the origin a union of finitely many lines L_1, … , L_s through the origin that are all contained in H. We take advantage of the fact that the q-dimensional variety C^q_Z has a particularly simple structure by construction. Since the hyperplane H is given by a non-degenerate set w'_1, … , w'_q-1 of linear forms in , by making a linear change of variables, we can assume that the equations of the (n-q+1)-dimensional hyperplane H are z_1=0, … , z_q-1=0. Then the equations of the curve Γ have the following form (see Lemma <ref>): f_1=z_1=0,… , f_q-1=z_q-1=0, f_q=f_q(z_1,… ,z_q-1, z_q,… , z_n)=0, … , f_n-1=f_n-1(z_1,… ,z_q-1, z_q, … , z_n)=0. By choosing the parametric equations of the hyperplane Z to be of the formz_1=t_1, … ,z_q-1=t_q-1, z_q=0, …, z_n=0,we get for the parametric equations of the cylinder C^q_Z (see the proof of Lemma <ref>): z_1-t_1=0, … , z_q-1-t_q-1=0, f_q(0,… ,0, z_q, … , z_n)=0, … , f_n-1(0, … ,0, z_q, … , z_n)=0. Therefore, the equations for the cylinder C^q_Z are given by f_q=f_q(0,… ,0, z_q, … , z_n)=0,… , f_n-1=f_n-1(0, … ,0, z_q, … , z_n)=0,which are the equations of the curve Γ in the hyperplane H.Let us denote by I_1 the ideal generated by (f_q, …, f_n-1) in the ring and by J the ideal generated by (f_q, …, f_n-1) in the ring_q,… ,n, the ring of germs of holomorphic functions at 0 in the hyperplane H≅ℂ^n-q+1. We have the relations I=rad (I_1), _q,… ,n⊂,J⊂ I.Let _k_q,… ,n denote the subring of germs of holomorphic functions ofonly the variables z_q, … , z_k, where k=q+1, … , n. Using Gunning's Local Parametrization Theorem from p.15 of <cit.>, since the dimension of Γ in H is 1, one can choose functionsg_k,k=q+1, … , n and linear coordinates (z_q, z_q+1, … , z_n) such that: (1) g_k∈ J∩_k-1_q,… ,n [z_k], (2) g_k is a Weierstrass polynomial in z_k of degree m_k, and (3) _q_q,… ,n∩ J=0.Moreover, at the last step when g_q+1 is chosen, by a linear change of variables in z_q and z_q+1, we can suppose that the (n-q)-dimensional hyperplane given by the equation z_q=0 in the hyperplane H does not contain the lines L_1, … , L_s. This can be seen as follows: Take the projective spaceℙ^n-q associated to the hyperplane H, and let π :H→ℙ^n-q be the canonical projection. The lines L_1, … , L_s will define s points in the projective space ℙ^n-q and we can choose a hyperplane Y of dimension (n-q-1), which does not contain any of these points. π^-1(Y), the inverse image of Y in H, then does not contain the lines L_1, … , L_s. Now, the orthogonal line to π^-1(Y) in H is the z_q-axis as _q_q,… ,n∩ J=0 by condition (3) above.We remark that _q-1∩ I=0. Indeed, if h(z_1, … , z_q-1)∈ I, then we have for some j that h^j ∈ I_1, soh^j=a_qf_q+ … + a_n-1f_n-1, where a_q, …, a_n-1 are germs of holomorphic functions at 0. Setting in this relation z_q=0, …, z_n=0 yields h^j=0, hence h=0. Since _q,… ,n⊂,J⊂ I, we obtain that the functions g_k, k=q+1, … ,n satisfy the conditions: (1) g_k∈ I∩_k-1 [z_k], (2) g_k is a Weierstrass polynomial in z_k of degree m_k. Furthermore, from the conditions _q-1∩ I=0 and _q_q,… ,n∩ J=0, it follows that (3) _q∩ I=0.From the existence of g_k, k=q+1, … ,n, by using some facts about integral extensions (see <cit.>), one can conclude that there exist Weierstrass polynomials p_k∈ I∩_q [z_k], k=q+1,…, n. By choosing each function p_k to be one of the minimal degree with this property, then since I equals its radical, one may suppose that each p_k is a product of p_k,j, j=1,…, N_k of distinct irreducible in _q [z_k]. It follows that the discriminant d_k of such a product is in the ring _q and not equal to the zero-element (see <cit.>). Since the q-dimensional variety V=C^q_Z has a particularly simple structure by construction, it follows from the second description of V' (expression (<ref>)) that the tangent cone of V is given by the finite union of q-dimensional hyperplanes Z× L_1, … , Z× L_s. Catlin defined in <cit.> another conic variety X'⊂ V' byX'={z∈ V'; z_qg̃_q∏_q+1^n d̃_k(z)=0},whereg̃_q=1 in our case since the cylinder variety V=C^q_Z is pure q-dimensional. It follows that X' is a conic subvariety of dimension q-1 in ℂ^n, which gives rise to a projective subvariety X̃ in ℙ^n-1 of dimension q-2. If S ∈ G^n-q+1 and S̃ denotes the corresponding projective hyperplane of dimension n-q in ℙ^n-1, then the set W_1 consists of all (n-q+1)-dimensional complex hyperplanesS such that S̃∩X̃≠∅. We have to show that H̃∩X̃=∅. Firstly, we have V'∩ H=∪_j=1^s L_j, hence Ṽ∩H̃ has s points corresponding to the lines L_1, … , L_s. For the intersection X'∩ H, we have to inspect the equation of X' when z_1=0, … , z_q-1=0. We obtain z_q∏_q+1^n d̃_k(0,0, … ,0, z_q)=0 whose only solution is z_q=0 because d̃_k is a homogeneous polynomial for every k. The (n-q)-dimensional hyperplane of H given by this solution does not contain any of the lines L_1, … , L_s as it is π^-1(Y) described above. It follows that H̃∩X̃=∅, i.e. H∉ W_1.Secondly, to ensure the intersection C^q_Z ∩ S for S ∈ G^n-q+1 behaves well, a good notion of transversality has to apply.For every S ∈ G^n-q+1, there corresponds a projective plane S̃ of dimension n-q in ^n-1. Generically, C̃^q_Z ∩S̃ consists of finitely many points z̃^1, …, z̃^D with transverse intersections, meaning that each z̃^i is a smooth point of C̃^q_Z such that the tangent spaces satisfy T_z̃^iC̃^q_Z ∩ T_z̃^iS̃ = 0 for i=1, …, D. In <cit.> W_2 is the subset of G^n-q+1 where this generic behavior does not take place. As we have seen Ṽ∩H̃ has s points corresponding to the lines L_1, … , L_s. It follows that D=s, and we denote byz̃^i the point corresponding to L_i. Since L_i is a line in Z× L_i and in H, we have that T_z̃^iṼ=Z× L_i/L_i≅ Z and T_z̃^iH̃=H/L_i. Z∩ H=0, however. It follows that T_z̃^iṼ∩ T_z̃^iH̃ = 0 for all i=1,...,s, i.e. H∉ W_2. Finally, suppose L∈ G^n-q+1 is defined by L={z; ∑_j=1^na^i_jz_j=0, i=1, … , q-1}. Let W_3={L;[a^i_j]_1≤ i,j≤ q-1=0}. Since the equations of H are z_1=0, ..., z_q-1=0, we have [a^i_j]_1≤ i,j≤ q-1=1, andwe conclude H ∉W_3.We have shown that H ∈W̃, so the curve Γ enters into the computation of τ((, w'_1, …, w'_q-1), C^q_Z). Therefore,D_q(, x_0) ≥inf_g ∈ (, w'_1, …, w'_q-1)ord_0Γ^* g/ord_0Γ=t' > t = D_q(, x_0),giving us the needed contradiction. Let Ω in ^n be a domain withboundary. Let x_0 ∈ b Ω be a point on the boundary of the domain, and let 2 ≤ q ≤ n-1. ThenΔ̃_q(b Ω, x_0)≤ D_q(b Ω, x_0).Proof: The proof is very similar to that of the previous result, Propostion <ref>. Let D_q(b Ω, x_0)=t < +∞, else the estimate is trivially true. Assume Δ̃_q(b Ω, x_0) >t. By definition,Δ̃_q(b Ω, x_0)=gen.val_{w_1, …, w_q-1} Δ_1( ((b Ω), w_1, …, w_q-1), x_0 ),where the generic value is taken over all non-degenerate sets {w_1, …, w_q-1} of linear forms in _x_0 andΔ_1( ((b Ω), w_1, …, w_q-1), x_0 ) = sup_φ∈(n,x_0)inf_g ∈((b Ω), w_1, …, w_q-1)ord_0φ^* g/ord_0φ.Therefore, there exist t'>t and a curve Γ passing through x_0 such thatinf_g ∈ ((b Ω), w_1, …, w_q-1)ord_0Γ^* g/ord_0Γ=t',for some non-degenerate set {w'_1, …, w'_q-1} of linear forms in _x_0. We will call H the variety corresponding to the non-degenerate set {w'_1, …, w'_q-1} of linear forms, which is a (n-q+1)-dimensional hyperplane. Clearly, Γ⊂ H as each w'_j has vanishing order 1 so any value starting at 2, the lower bound for Δ̃_q(b Ω, x_0), can only be achieved by a curve that sits in the zero set of {w'_1, …, w'_q-1}. Therefore, inf_g ∈ ((b Ω), w_1, …, w_q-1) is achieved for g=r, the defining function of the domain Ω. Let Z be the (q-1)-dimensional hyperplane passing through x_0 that is transversal to H, i.e., _ (H ⊕ Z)=n. Since Γ⊂ H, the tangent line to the curve Γ at x_0 is not contained in Z.By Lemma <ref>, there exists a germ of a q-dimensional cylinder C^q_Z at x_0 that contains the curve Γ and whose tangent space at x_0 contains Z. By the same analysis as in the proof of Propostion <ref>, H ∈W̃, so the curve Γ enters into the computation of τ(C^q_Z, x_0). Thus,D_q(b Ω, x_0) ≥ord_0Γ^* r/ord_0Γ=t' > t = D_q(, x_0),and we have obtained the contradiction we sought. Proof of Theorem <ref>: First, we prove part (i). For q=n Proposition <ref> along with Theorem <ref> yield that Δ_n(,x_0)=Δ̃_n(,x_0)=D_n(,x_0). As a result, the equality only needs to be proven for 2 ≤ q ≤ n-1. The inequality D_q(, x_0) ≤Δ̃_q(, x_0) is a consequence of Theorem <ref>, while Δ̃_q(, x_0) ≤ D_q(, x_0) follows from Proposition <ref>.To prove part (ii), we note that D_q(b Ω, x_0) ≤Δ̃_q(b Ω, x_0) is a consequence of Theorem <ref>, while the reverse inequality follows from Proposition <ref>.Proof of Corollary <ref>: (i) We combine Proposition <ref> with Theorem <ref> part (i).(ii) Proposition <ref> and Theorem <ref> part (ii) together give the result. plain | http://arxiv.org/abs/1707.08294v3 | {
"authors": [
"Vasile Brinzanescu",
"Andreea C. Nicoara"
],
"categories": [
"math.CV",
"math.AG",
"32F18, 32T25 (Primary), 32V35, 13H15 (Secondary)"
],
"primary_category": "math.CV",
"published": "20170726060529",
"title": "Relating Catlin and D'Angelo $q$-types"
} |
Caporale et al.: A New Framework for Synthetic Aperture Sonar Micronavigation Caporale et al.: A New Framework for Synthetic Aperture Sonar MicronavigationA New Framework for Synthetic Aperture Sonar Micronavigation Salvatore Caporale and Yvan Petillot S. Caporaleand Y. Petillot are with the Institute of Sensors, Signals and Systems, Heriot-Watt University, Edinburgh, Scotland, UK (e-mail: [email protected]; [email protected]). December 30, 2023 =================================================================================================================================================================================================================================== Synthetic aperture imaging systems achieve constant azimuth resolution by coherently summating the observations acquired along the aperture path. At this aim, their locations have to be known with subwavelength accuracy. In underwater SAS, the nature of propagation and navigation in water makes the retrieval of this information challenging. Inertial sensors have to be employed in combination with signal processing techniques, which are usually referred to as micronavigation. In this paper we propose a novel micronavigation approach based on the minimization of an error function between two contiguous pings having some mutual information. This error is obtained bycomparing the vector space intersections between the pings orthogonal projectors. The effectiveness and generality of the proposed approach is demonstrated by means of simulations and by means of an experiment performed in a controlled environment. Synthetic Aperture Sonar, Micronavigation. § INTRODUCTIONSAS systems share with SAR many practical and theoretical aspects, as they were originally introduced by moving the synthetic aperture paradigm from radar to sonar. Therefore, most image formation algorithms which have been conceived for SAR have been also considered in SAS literature<cit.>. Despite the overlapping concepts such as range migration and range invariant resolution, underwater SAS systems are operated in a much more challenging environment than SAR as (i) the navigation in water is affected by non-negligible errors and cannot always rely on an external accurate positioning system, (ii) motion errors are comparable to the wavelength hence their effect on image formation algorithms is remarkably destructive (iii) due to relatively small sound velocity in water with respect to the range of interest, the desired along-track resolution cannot be achieved by means of an AUV provided with a single Tx and Rx and moving at a reasonable along-track speed <cit.>.For the above mentioned reasons, SAS systems are usually equipped with an accurate INS to recover the real navigated trajectory. Moreover, each ping consists of a single Tx and an array of Rx, allowing for a higher along-track sampling rate, hence a higher along-track speed <cit.>. However, from ping to ping a certain degree of redundancy is imposed to perform data based motion estimation known as DPCA which might be coupled with the INS for a more accurate motion compensation <cit.>.A typical SAS system and the way it operates are illustrated in Fig. <ref>. Other approaches based on autofocus techniques have been also presented ni the literature <cit.>.In a stripmap imaging system where the AUV moves along a straight trajectory at constant speed, major issues are caused by cross-range displacements. In fact, the sensitivity of the SAS PSF is remarkably higher along the cross-track than along the along-track <cit.>. The cross-track motion has to be known with subwavelength accuracy, whereas the along-track motion has to be in the order of portion of the sampling step. Hence, micronavigation algorithms focus on finding cross-track errors whereas it is generally assumed that the along-track locations are either correct or can be estimated by the navigation system <cit.>.The theoretical foundation of several currently employed approaches relies on (i) PCA, (ii) the hypothesis that the AUV rotation angle is small such that it can be characterized as a linearly increasing delay along the array elements and (iii) contiguous pings have some superimposed phase centers, such that the mutual delays between the corresponding tracks can be estimated by finding their correlation peak. Many challenges in SAS micronavigation still have to be addressed <cit.>. Those include (i) dealing with non straight trajectories, such as circular ones <cit.>, (ii) reducing the hardware cost by relying on less accurate navigation and (iii) analysing theoretical limits in performing unsupervised motion compensation. Following the work in <cit.>, we here exploit the superimposition of some phase centers as for DPCA. However, we assume that they can be arbitrarily interlaced rather than exactly superimposed.Then, the vector space intersection between the space corresponding to the two pings is described as a functional of the hypothetical displacement. Each ping is employed to compute a pair of displacement-dependent outputs being the projections on the intersection subspace. A proper convex error function between those outputs features a minimum in correspondence of the ping-to-ping displacement, so that it can be identified by means of an optimization. This approachis computationally demanding due to the evaluation of an error function involving the recomputation of the projecting operators. However, it requires less a priori knowledge and is remarkably less restrictive in comparison to DPCA.The paper is organized as follows. Section <ref> reviews the acoustic model and SAS, while Section <ref> illustrates the proposed motion compensation procedure. Results from synthetic data and from a real experiment are shown in Section <ref> and <ref> respectively and some conclusions are drawn in Section <ref>.§ ACOUSTIC AND SAS MODEL In this Section we first review the observation model for a single Tx and Rx acoustic system when a generic signal is given as input, then we show how this can be employed to retrieve the scene reflectivity when multiple observations are combined together. A simplified 2-D model is considered by means of the assumption that (i) the seabed reflectivity can be represented as a function on a plane and (ii) the 2-D scene reflectivity can be obtained as the projection of the 2-D seabed reflectivity on a slanted plane. Moreover, all the sensors are assumed to be still. Finally, SAS is described as a collection of such single Tx and Rx systems. §.§ Acoustic Model Given a Tx at position z_t=(x_t,y_t) of length D_t and orientation ϑ_t and a hydrophone at position z_r=(x_r,y_r) of length D_r and orientation ϑ_r, the system response to an input complex passband function s(t) with respect to a scene whose complex reflectivity is expressed by ρ(z), with z=(x,y) can be obtained by: r(t)=∫_zρ(z) α(z_t,z_r,z,ϑ_t,ϑ_r) s(t-τ(z_t,z_r,z)) dz where τ is the delay corresponding to the propagation from z_t to z and from z to z_r, α is an attenuation factor taking into account (i) the attenuation due to the propagation distancefrom z_t to z and from z to z_r and (ii) the transmission and reception radiation patterns. With regard to the attenuation, we consider the exploding sources model where the α^2∝δ(z_t,z)+δ(z_r,z)∝τ(z_t,z_r,z) rather than α^2∝δ(z_t,z)δ(z_r,z), where δ is the distance operator. The model (<ref>) is represented in Fig. <ref>. When s(t) is a narrowband impulse modulated at frequency f_0, the above model can be used for observing ρ(z). A suitable sampling grid z_n=(x_n,y_n) matching the available bandwidth is introduced so that the reflectivity is represented as the input vector ρ(z_n). By setting G(z_n) as the Green's function of the system G(z_n)=α(z_t,z_r,z_n,ϑ_t,ϑ_r)e^-j2πf_0τ(z_t,z_r,z_n), the following output time-sampled signal ϕ(t_m) can be obtained as ϕ(t_m)=A(t_m,z_n) G(z_n)ρ(z_n) where the matrix A(t_n,z_n) performs the discrete-space integration of equation (<ref>), which is basically done by an interpolation kernel and a summation. From a numerical point of view, A(t_m,z_n) is usually defined by its transpose one, since defining an interpolation along the intrinsic one dimensional time axis t_m is easier than doing it over the fictitious space axis z_n. We consider a NUFFT based time interpolation <cit.>, having an approximately linear complexity and high accuracy.The integral in (<ref>) and its space discrete equivalent in (<ref>) clearly perform a dimensionality reduction from the two-dimensional space z_n to the one dimensional space t_m.In fact, all points lying on the ellipsis having z_t and z_r as foci are intrinsically ambiguous with respect to the Tx Rx pair. This is also represented in Fig. <ref> where we highlighted a propagation path (dotted line) having the same delay as the one relative to the generic point z_n. As a consequence, inverting (<ref>) recovers the reflectivity ρ(z) affected by defocusing along ellipsis.Due to the radiation patterns, this defocusing increases with range. §.§ SAS model The working principle of a synthetic aperture consists in performing multiple observations at prescribed Tx and Rx locations along a (usually straight) path, called along-track or cross range. By doing so, a longer aperture is emulated and a narrower beam is obtained in the direction orthogonal to the track, called range. The emulation is performed by signal processing and allows for getting a constant resolution along range.Given a set of Tx and Rx z_l,t and z_l,r, l∈Z, the observation model of SAS can be written as the column vector of ϕ_l(t_m)=A_l(t_m,z_n) G_l(z_n) ρ(z_n).In case each observation is a monostatic system, i.e. z_l,t=z_l,r=z_l, and z_l are uniformly spaced along the cross range direction by D/4, with D=max(D_t,D_r), then ρ(z_n) can be approximately recovered with a cross range resolution equal to D/2 by the transpose observation operator, also know as backprojection ρ(z_n)≈∑_l∈ZG^*_l(z_n)A_l^†(z_n,t_m)ϕ_l(t_m) which has been conveniently rewritten as a sum of backprojections of each single Tx and Rx system.As for getting a cross range resolution equal to D/2 a sampling on the along-track by D/4 has been considered, the oversampling ratio is equal to 2. Many variations of this setup have been considered in the literature in order to decrease the oversampling ratio and employ different reconstruction techniques <cit.>.Practical SAS systems are mounted on an AUV. Hence, the speed v of the vehicle has to be set according to the wanted along-track sampling and the desired maximum range. For instance, in case of an AUV equipped with a single Tx and Rx, v≤D/4max(τ(z_l,t,z_l,r,z_n))=Dc/8R, where R is the maximum range. For R=[150]m and D=[5]cm, we get v=[6.25]cm/sec. A feasible speed is obtained by using an array of N equispaced Rx at distance L=D/2 and a single Tx. By doing so, a single transmitted ping allows for N observations, thus gaining a N factor on the speed. Such a SAS system is a collection of bistatic single Tx and Rx systems. If the range of interest is sufficiently large with respect to the wavelength, it can be approximately modelled as a collection of monostatic systems such that z̅_l=(z_l,t+z_l,r)/2. This model is referred to as PCA.§ MOTION ESTIMATIONThe motions of an AUV in a 3-D space are represented by means of 3 linear motion parameters being heave, sway and surge, and 3 rotation motion parameters being pitch, roll and yaw. As the acoustic model has been described in 2-D, only the effective projections of those on the slanted plane are needed, i.e. surge and sway as effective linear motions and yaw and roll as effective rotations. Roll only affects the radiation pattern over range and can be neglected. Without loss of generality, we here refer to surge as the motion error with respect to the prescribed linear trajectory at constant speed. Surge, sway and yaw are illustrated in Fig. <ref>. The SAS image formation requires an accuracy by approximately λ/10 on z̅_l, where λ is the wavelength. Underwater navigation is not accurate enough to allow this accuracy for the employed frequency band, usually [100-300]kHz. As a consequence, a motion estimation has to be performed to guarantee that the acquired observations are coherently combined. This is usually achieved by combining navigation information coming from physical inertial sensors place on the AUV, that is the INS, with information extracted from the acquired data by introducing redundancy from ping to ping. Instead of moving the AUV at the maximum allowed speed, a reduced speed is adopted in order to have a certain number of K<N superimposed phase centers, as represented in Fig. <ref>. This approach is usually referred to as DPCA. Under the assumptions that (i) PCA holds, (ii) surge is negligible and (iii) yaw is small, the tracks from two contiguous pings referred to the same phase center differ for a time delay and the delays of the K-N superimposed phase centers are an affine function of the sway and yaw. The delays can be estimated by correlating the corresponding tracks, thus also sway and yaw can be obtained. Interested readers are referred to <cit.> for more details. An approach to tackle surge estimation is presented in <cit.>.We here propose a technique loosening the three assumptions above, more specifically (j) PCA is replaced with a less restrictive approximation, (jj) the along-track motion is such that an unknown non-integer number of superimposed phase centers are present and (jjj) yaw is only small enough to guarantee that the area illuminated by two contiguous pings has a non-null intersection. §.§ Ping to Ping Displacement EstimationGiven that the receiving array is composed by N equispaced Rx, the raw data obtained by collecting the reflections from the scene of a single impulse will be referred to as ping. For the sake of the stripmap SAS under consideration, the AUV is moving along a straight trajectory affected by motion errors. The scene is observed by means of the start-and-stop approximation. Given the trajectory, each ping can be represented as a collection of bistatic systems identified by the array midpoint and orientation and the Tx position and orientation: σ_r^(p) = (x_a^(p),y_a^(p),ϑ_r^(p)) σ_t^(p) = (x_t^(p),y_t^(p),ϑ_t^(p)). where x_a and y_a are the coordinates of the midpoint of the receiving array elements. We here assume that the relative position of σ_t^(p) with respect to σ_r^(p) is known. In a real scenario, the displacement between the transmitter and the array midpoint is time-dependent as the AUV moves between the transmission instant and the reception relative to the considered range interval (see <ref>).We also introduce the operator T^(p) representing the observation model relative to ping p, so that T^(p)ρ gives the raw data relative to ping p. We also set ρ̆^(p)=T̃^(p)-1 T^(p) ρ where ρ̆^(p) represents the recovered reflectivity at ping p by means of the pseudoinverse operator T̃^(p)-1, where T̃ is the available computable approximation of the real observation operator T^(p). Mathematically, T̃^(p) is the column matrix of the real observation matrices A_l,γG_l,γ, l=1,…,N (see (<ref>)).Then, we define the orthogonal projector on the subspace identified by ping p Q^(p)=(T̃^(p))^-1T̃^(p). Given another generic ping q and its orthogonal projector Q^(q), the projection on the intersection of the subspaces identified by Q^(p) and Q^(q) can be obtained by ψ^(p,q)=ψ^(q,p)=lim_i→∞(Q^(q)Q^(p))^iρ=lim_i→∞(Q^(p)Q^(q))^iρ. By taking advantage of (<ref>), an approximation of ψ^(p,q) can be obtained by starting either from ρ̆^(p) or ρ̆^(q). In fact ρ̆^(p) ≃ T̃^(p)-1T̃^(p)ρ=Q^(p)ρ ρ̆^(q) ≃ T̃^(q)-1T̃^(q)ρ=Q^(q)ρ hence by highlighting the expressions (<ref>)-(<ref>) in (<ref>) we get ψ^(p,q) = lim_i→∞(Q^(p)Q^(q))^iQ^(p)ρ≃ lim_i→∞(Q^(p)Q^(q))^iρ̆^(p)=ψ^(p) ψ^(p,q) = lim_i→∞(Q^(q)Q^(p))^iQ^(q)ρ≃ lim_i→∞(Q^(q)Q^(p))^iρ̆^(q)=ψ^(q) and finally we get the relationship ψ^(p)≃ψ^(q). The meaning of (<ref>) is that if the intersection operator is known, it is possible to identify the same intersection image starting from the raw data obtained by two different pings. In case the operator T^(p) and T^(q) are orthogonal, both ψ^(p) and ψ^(q) are null, hence their computation is not relevant. Conversely, in case they are not orthogonal, the fact that ψ^(p)-ψ^(q)≃0, where · is a generic norm, can be exploited as a property to identify the intersection operator.In order to apply the principle expressed in equation (<ref>), we introduce operator Q^(p)_0 being a shifted/rotated version of Q^(p) such that the center of mass of the PCA is the origin of the axes. In more detail, by settingx̅^(p) = (x_a^(p)+x_t^(p))/2 y̅^(p) = (y_a^(p)+x_t^(p))/2 we get σ_r,0^(p) = (x_a^(p)-x̅^(p),y_a^(p)-y̅^(p),0) σ_t,0^(p) = (x_t^(p)-x̅^(p),y_t^(p)-y̅^(p),ϑ_t^(p)-ϑ_r^(p)) and we set σ̅^(p)=(x̅^(p),y̅^(p),ϑ_r^(p)). By introducing the shift and rotation operator S_σ, where σ specifies a triplet (x,y,ϑ), we finally get Q^(p)=S_σ̅^(p)Q^(p)_0S_-σ̅^(p). Ping q can be specified by Q_0^(q) and σ̅^(q), or, alternatively, by Q_0^(q) and σ̅^(p) and the differential displacement between σ̅^(p) and σ̅^(q), which will be referred to as σ̅^(q,p) Q^(q) = S_σ̅^(q)Q^(q)_0S_-σ̅^(q)= S_σ̅^(p)S_σ̅^(q,p)Q^(q)_0S_-σ̅^(q,p)S_-σ̅^(p) with σ̅^(q,p) = (x̅^(q)-x̅^(p),y̅^(q)-y̅^(p),ϑ_r^(q)-ϑ_r^(p)) = (x^(q,p),y^(q,p),ϑ^(q,p)) hence the following is verified σ̅^(p,q)=-σ̅^(q,p).It is worth noting that the differential displacement which has been highlighted is equal to the one which would be obtained by replacing the bistatic situation with its PCA, nevertheless, this model maintains the capability of representing a bistatic system where the transmitter and the array have different orientations.So, the vector σ̅^(q,p) fully identifies the displacement between ping p and q and represents the goal of the motion estimation procedure. In order to estimate it, we rewrite equations (<ref>) and (<ref>) by taking into account that ρ̆^(p) and ρ̆^(q) are not available, whereas their rotated ans shifted version by S_-σ̅^(p) can be computed S_-σ̅^(p)ψ^(p) = lim_i→∞(Q_0^(p)S_σ̅^(q,p)Q^(q)_0S_σ̅^(p,q))^iS_-σ̅^(p)ρ̆^(p) S_-σ̅^(p)ψ^(q) = lim_i→∞(S_σ̅^(q,p)Q^(q)_0S_σ̅^(p,q)Q_0^(p))^iS_-σ̅^(p)ρ̆^(q).Since the displacement between p and q is unknown, the above equations can be rewritten with respect to a hypothetical displacement σ=(x,y,z) S_-σ̅^(p)ψ^(p)_σ = lim_i→∞(Q_0^(p)S_σQ^(q)_0S_-σ)^iS_-σ̅^(p)ρ̆^(p) S_-σ̅^(p)ψ^(q)_σ = lim_i→∞(S_σQ^(q)_0S_-σQ_0^(p))^iS_-σ̅^(p)ρ̆^(q)In general, for non-null ρ̆^(p) and ρ̆^(q) and non-null intersection operator, we have ψ^(p)_σ=ψ^(q)_σ⇔σ=σ̅^(q,p).In practice, they will not be equal because of approximation errors. However,ψ^(p)_σ and ψ^(q)_σ can be employed for building an error function. Since they represent wavefield reconstructions, they feature periodic-wise variations with period equal to half the wavelength. An error function based on those would also feature such oscillations, hence it is expected to be convex on a small hyperinterval around the displacement triplet (x^(q,p),y^(q,p),ϑ^(q,p)): η^(q,p)(x,y,ϑ)= ψ^(p)_σ-ψ^(q)_σ _2. Conversely, their absolute values represents reflectivity amplitudes so an error function based on them is expected to be convex on a larger hyperinterval although it does not necessarily have its minimum in (x^(q,p),y^(q,p),ϑ^(q,p)) ζ^(q,p)(x,y,ϑ)= |ψ^(p)_σ|-|ψ^(q)_σ| _2. Functions ζ and η will be referred to as modulus and phase error function respectively. The interval of convexity of the above functions will be discussed later in the paper. Normalized version of the above functions are more conveniently employed, where the normalization can be done with respect to either ψ^(p)_σ, ψ^(q)_σ or the maximum between the two.Hence, the estimation procedure can be performed by: * initialize σ: σ←(0,0,0) * minimize ζ^(p,q) starting from σ: σ←minζ^(p,q) * minimize η^(p,q) starting from σ: σ←minη^(p,q) The characteristics of ζ and η cannot be analytically identified. A practical assessment of this proposed procedure will be provided in Section <ref>. The following considerations hold: * the superimposition between phase centers is not required as the along-track displacement is also part of the search; * the estimation relies on global similarities in the image domain rather than on track-by-track correlations, thus more robust; * rotation estimation relies only on non-null vector space intersection between pings, hence large rotation can be estimated.However, there are some major computational burdens: * each optimization iteration requires the update of the observation model of one of the two pings; * both the orthogonal projector and the vector space intersection requires infinite iterations which must be properly truncated.To decrease the computational load, the range of interest can be conveniently limited.§.§ Approximate Bistatic Model In the previous subsection we assumed each ping to be obtained as the output of a collection of N bistatic systems under the start-and-stop approximation. As the estimation approach do not rely on the PCA, we here introduce a more general approximation allowing for representing (i) the orientation of the Tx with respect to the orientation of the array and (ii) the effective displacement along the cross-track between the Tx and the array which is happening because the AUV motion. The employment of bistatic systems has been shown to carry some benefits <cit.>, hence the capability of including them is beneficial.With the assumption that the transmitted signal is ideally instantaneous (no pulse compression) and that the observed range interval tends to zero, σ_r^(p) and σ_t^(p) can be determined by sampling the effective AUV trajectory on the slanted plane. According to PCA are approximated by their average. Infinite different bistatic systems sharing the same PCA can be identified. Among these, we approximate σ_r^(p) and σ_t^(p) by σ̃_r^(p) and σ̃_t^(p), such thatσ̃_r^(p) = σ̅^(p)+ς_r σ̃_t^(p) = σ̅^(p)+ς_t+(0,0,ϑ^(p)_t-ϑ^(p)_r) where ς_r=(x_a,y_a,0) and ς_t=(x_t,y_t,0) such that ς̅=(0,0,0). In practice, the real bistatic system is replaced by another bistatic system having the same PCA but conveniently chosen to be a more accurate representation according to the considered range. For instance, if the Tx is positioned on the array midpoint, y_a=y_t=0 and x_a=-x_t would be the distance covered by the AUV while the impulse reaches the target range.In a real scenario, the whole illuminated range is observed. Hence, the range is practically split in subranges where the above approximation is adopted. The splitting in subranges must be anyway performed as the AUV ping-to-ping motions also change along range. § SIMULATION RESULTSIn order to assess the performances of the proposed approach, we here present results from a simulated scenario. An experiment performed in a controlled environment is presented in Section <ref>.The simulation focuses on highlighting the generality, the accuracy and the convergence of the proposed estimation procedure with no side information provided. No restrictive assumptions are made on surge and yaw.§.§ Interval of ConvergenceA SAS system consisting of 16 elements is considered, both the Tx and the Rx elements have D=[5]cm aperture, hence the whole array is [80]cm wide, whereas the phase centers are distributed uniformly along a [40]cm interval. As the number of the expected ping-to-ping superimposed phase centers, we considered K=4, so that each ping is [20]cm apart from its adjacent ones. With this setup, an AUV moving at [2]m/sec would be capable of covering a [75]m range. The Tx is physically located in the array midpoint, whereas it is virtually located [10]cm apart from the array midpoint position as an effect of the vehicle motion. The range observed with this assumption would be around half the maximum available range, i.e. [37.5]m. Nevertheless, being a simulation with the start-and-stop approximation, we deliberately considered [10]m as midrange in order to havea configuration where the PCA would be less accurate, than the actual model (<ref>)-(<ref>). The system is operating at [300]kHz, while the bandwidth is [30]kHz, thus resulting in [2.5]cm range resolution. The cross-range resolution is approximately equal to D/2, i.e. also [2.5]cm. An instance of the objective functions ζ and η has been plotted in Fig. <ref> with respect to surge, sway and yaw separately. In more detail, the surge has been varied in the interval [-0.2,0.2], i.e. the array has been moved from having its phase centers totally superimposed to the ones of the previous ping to not having any superimposed. On the other axis, a range of hypothetical motion errors has been considered. The surfaces on top qualitatively show how both the error function ζ and η have minima on the plane diagonal, meaning that the surge can be correctly estimated. Also, ζ evaluated along the x axis appears mostly convex, such that the minimum can be found by starting from x=0. The error function η is steepest around the minima, but features a more remarkable non-convex behavior which makes it suitable for optimization only if a good starting point is given. Similar considerations can be done for the sway, which has been varied also in the remarkably large interval [-0.1,0.1], and for the yaw. Their η functions are not showed as they become very steep around the minima and cannot be properly visualized.Those results are partial as surge, sway and yaw have been considered separately, i.e., it is not showed whether those functions are convex and having the right minima when all the three motions are considered at the same time. §.§ Ping-to-ping Accuracy As a second experiment, we consider a single ping q=9 featuring motion errors with respect to its previous p=8, both relative to the scene depicted in Fig. <ref> (top). It features a noisy background emulating sand ripples plus a phantom circular object whose diameter is [40]cm. Pings 8 and 9 happen in the middle of the considered cross-range interval, thus facing the circular object. The motion errors are set as follows: surge x^(q,p) equal to [-3.45]cm, sway y^(q,p) equal to [2.17]cm and yaw ϑ^(p,q) equal to [4.10×10^-2]rad, i.e., approximately [2.35]deg. Those values are random, but their magnitude is such that other approaches such DPCA would fail. In fact, the along-track motion error is larger than along-track sampling D/2 and not a multiple of D/2. The rotation error is such that the effect on raw data track cannot be described by simple delays. In Fig. <ref>, raw data of tracks 9-12 of ping 8 are compared to tracks 1-4 of ping 9 . In Fig. <ref> we plotted the error functions ζ and η when the considered motion errors are applied one at a time, i.e., the same as in Fig. <ref>. The left column shows function ζ in solid line and η in dotted line, whereas the right column shows the opposite. Real displacements are identified by the dashed vertical lines. The figures also show the minima which are found by the optimizations. With regard to ζ it can be seen that by starting with no a priori knowledge, i.e. σ=(0,0,0), the optimization is convex. For surge ans sway, the size of the convexity interval is proportional to the along-track sampling step, whereas for yaw it is also inversely proportional to the number of superimposed phase centers K. From the figure referred to sway (middle-left) the oscillatory nature of function η can be observed. Its period is roughly proportional to half the wavelength making it essential that the optimization on ζ has an accuracy being higher than this.The minima found on ζ exhibit some deviations with respect to the real displacements, which are then corrected by the optimization on η function. The minima of η are not equal to 0 as the algorithms for computing the orthogonal projections and the intersection have been truncate to 5 iterations for the sake of implementability.From a qualitative point of view, it is interesting to have an understanding about what the projections on the intersection space represent. In Fig. <ref> the evolution of the projections and their difference along the optimization process are shown. In the initial situation (top) obtained for (x^(q,p),y^(q,p),ϑ^(q,p))×1/2, the projections feature differences in shape, position and amplitude. The intermediate situation (middle) obtained for (x^(q,p),y^(q,p),ϑ^(q,p))×3/4, exhibits higher correlation between the projections but their displacements still causes the error to have the same degree of magnitude as the projections. Finally, the situation at convergence still shows some residual error and some shape differences between the projections. This example highlights the estimation robustness versus punctual differences in reflectivity observed from the two pings, as the the error functions takes into account global similarities. As a final step, we consider the whole optimization in the case the given displacements are happening at the same time. The optimization on function ζ gives surge equal to [-2.66]cm, swayequal to [2.15]cm and yaw ϑ^(p,q) equal to [4.08×10^-2]rad, whereas the subsequent optimization on η gives [-3.49]cm, [2.17]cm and [4.12×10^-2]rad respectively. The optimization process is illustrated in Fig. <ref> where the surge, sway and yaw component of the vector along the steepest descent are shown together with the function values. From a qualitative point of view, the resulting estimation error is highly compatible with SAS operations, which usually require the error on sway lower than λ/8.From a quantitative point of view, the assessment of the performance is quite hard as it is difficult to separate the dependency on the specific scene reflectivity from the dependency on the system parameters. Hence, estimating the error variance and bias could be only approached statistically. Nevertheless, it could be argued that both theoretical and/or statistical results on simulated data might focus on aspects which could be remarkably overtaken from other issues when moving to real data. For this reason, we believe it is not worth pursuing a further statistical estimation error analysis before assessing the working principle on real data.§.§ Trajectory Accuracy As a final simulation, we provide the estimated trajectory along a [3]m path covered by 16 pings. The considered scene reflectivity is again the one depicted in the top of Fig. <ref>. The reconstructed scene with no navigation is plotted in the middle figure.To make the simulation more realistic, the generation of raw data takes into account the different location and rotation of the Tx with respect to the Rx, whereas the reconstruction is performed by means of the proposed bistatic model (<ref>)-(<ref>) assuming that the expected Tx to Rx distance along the cross-range direction is known and equal to [10]cm.The considered random trajectory is partially shown in Fig. <ref> (top-left), together with a graphical explanation of the employed bistatic model. In the top-right figure the position of the Tx and array midpoint are explicitly shown.The result of the trajectory estimation are shown in Fig. <ref>. The differential and cumulative errors on surge and sway might suggest small biases, which can be neglected as the final sway error is still only about λ/25. Finally, the obtained trajectory is employed to reconstruct the scene in the top figure of Fig. <ref>. The obtained image is qualitatively comparable to the original image and the whole estimation procedure fulfils the initial goal of estimating motion errors with no priors.§ EXPERIMENTAL RESULTS Given the peculiarity of the proposed estimation technique and its complex algebraic structure, assessing its validity on real data is a major concern. An experiment in a controlled environment has been designed at this aim. Nevertheless, the a priori knowledge of the sensor locations could not be provided by means of the available equipment. For this reason, a specific manner to validate the experiment has been identified and designed.§.§ System Setup For this purpose, a water tank being [3]m large, [4]m long and [2]m deep was available. In order to avoid reflections from the concrete wall and bottom, a target consisting of 4 metal reflectors lying on an octagonal plastic surface parallel to the surface and mounted on a wooden support at approximately [1]m from the bottom has been considered. The geometry of the object is detailed in Fig. <ref>. A SAS system consisting of 8 Rx of [5]cm aperture and spaced by [5]cm and a [5]cm aperture Tx positioned at the midpoint of the array has been provided by Hydrason Solutions. The SAS system is capable of operating between [20]kHz and [170]kHz. The tank is provided with an industrial plotter which has been employed in order to move the SAS system along a straight line and perform the acquisitions by the start-and-stop approach.According to the specifications, the plotter provide a submillimetre accuracy. However, as the plotter engine produces a non-negligible noise, it had to be switched on and off throughout the whole acquisition process. This operation has caused the displacement accuracy to be inaccurate.Other causes of inaccuracy might have come from other mechanical non-idealities, such as submillimetre deviations in the supporting guide.The SAS system has been attached to the plotter by means of a pole and mounted on a rotating plate capable of providing roll and yaw. Roll has been adjusted in order to make the beam illuminate the object before touching the tank bottom, whereas yaw has been adjusted to be null. However, these setups have been manually arranged, so no apriori accurate information can be assumed for those parameters. Hence, the positioning of the array throughout all its displacement along the guide can feature both yaw and roll. Finally, the relatively closeness of the target object makes the PCA assumption weak and the reflectivity to vary along the trajectory, thus causing a lack of coherence from ping to ping.The SAS raw data have been collected by moving the system by steps equal to the spacing between phase centers, i.e. [2.5]cm, being also the expected along-track resolution, i.e. D/2.An interval of [80]cm plus the length of the array PCA, i.e. [20]cm, has been covered by 32 pings, thus getting a synthetic aperture of [1]m. As far as the physical parameters are concerned, a [30]kHz chirp of duration equal to [3]msec, central frequency equal to [105]kHz and sampled at [1]MHz has been employed. After pulse compression, demodulation and filtering, a downsampling by 33 has been performed, thus getting a cross-track resolution equal to [2.45]cm.§.§ Validation Methodology By means of M pings with K=N-1, MN different observations are collected. So, for this experiment 256 tracks are acquired, which can be used in various ways. For instance, given an array whose PCA covers uniformly [20]cm, the considered synthetic aperture of [1]m can be equivalently covered with 5 pings and K=0 or 10 pings and K=4. By means of the considered 32 pings, two main tasks can be accomplished: (i) the 32 pings can be subsampled in order to produce P=1,…,N-K different motion estimates for K=1,…,7; (ii) by selecting one of the 8 Rx at a time, 8 new synthetic aperture can be obtained as a collection of the corresponding SISO systems. Both these tasks are illustrated in Fig. <ref>. The simple case K=7 allowing for the single phase P=1 is rearranged in order to obtain the case K=4,5,6 with all the corresponding different phases. Also, the case K=7 is reorganized in virtual arrays of SISO systems.The benefit of the SAS made up of SISO systems is the following. Given that the elements are moved on a straight line, no sway can be present. A small rotation of each element is not actually impacting on the imaging task, so surge is the only motion error to be identified. Hence, the 8 images which can be obtained the SISO approach should be similar apart from a slight warping in the track direction. Hence they can be employed as initial ground truth. Conversely, the multiple motion estimates at different sampling steps can be exploited as follows. For each value of K, each of the P=1,…,N-K estimates represents a subsampling of the full sampled trajectory with different initial phase. Hence, they can be integrated in order to obtain a single motion estimate for each value of K. Then the obtained K trajectories can be applied separately on each of the 8 SAS based on the SISO approach, or to each one of the 8 SAS based on the properarray, which can be referred to as SIMO system. This would lead to K×8 images for the SISO and for SIMO case respectively. Ideally, if the motion estimation procedure works accurately, all those reconstructed complex reflectivities should be equal. §.§ Motion Errors With regard to surge, it was known that in correspondence of ping 12 a major misalignment occurred. In Fig. <ref> all 8 tracks corresponding to a single location have been plotted. Tracks 8-11 and 12-15 are not correlated as a consequence of this surge event. Moreover, a finer comparison of the tracks would reveal that other differences are present even within each group. Those can be attributed to a lack of calibration among the array elements and also to the fact that the 8 SISO systems are supposed to share the same phase center but practically look at the scene from different angles. When considering the SIMO systems, surge is the same as for the SISO ones, whereas sway can appear as an effect of the initial orientation of the array. This is explained in Fig. <ref>, where it is shown that the midpoint of a rotated array moves diagonally with respect to the reference system aligned with the array. Hence, the angle of this diagonal line is a measurement for the array orientation with respect to the scene reference system. §.§ Estimated Trajectory Before performing the actual trajectory estimation by means of the optimizations, it is worth analysing the error functions in order to assess that they feature the same characteristics as in Fig. <ref>. Error function ζ and η with respect to pings 11 and 12 are plotted in Fig. <ref>. All the macroscopic features highlighted for emulated data are replicated for real data. The functions are convex on comparable intervals and η with respect to sway oscillates with period equal to half the wavelength (the oscillation is much more visible here as the central frequency is [105]kHz rather than [300]kHz). For the considered ping, surge, sway and yaw are equal to [1.71]cm, [-3.1]mm and [1.35×10^-3]rad respectively. The yaw motion error is less than 1/10 of one degree, so it can be considered negligible.Moving from one ping to all the tracks, the results are shown in Fig. <ref>. By properly combining the differential displacements obtained for all the different phases illustrated in Fig. <ref>, a trajectory estimation can be obtained for each considered value of K, thus resulting in 4 trajectories plotted in Fig. <ref> (top). The differential surge and sway are also shown (left-bottom). It is worth noting that all the phases different from 1 have an undetermined starting point which has been numerically tuned to maximize the match with the trajectory obtained for K=1. Also, the differential motion errors represented in the figure are interlaced. As the trajectory laysalong a diagonal path with respect to the track, the cumulative displacements have been rotated by an angle equal to the opposite of the diagonal orientation, i.e. [1.27×10^-2]rad or [0.73]deg, and the same angle has been added to the yaw. The resulting cumulative displacements (right-bottom) give a hint about how the accuracy might decrease when the number of overlapping phase centers decreases. The overall deviation is enclosed in less than [5]mm and [0.5]mm for surge and sway respectively.§.§ Reconstructed Reflectivity As mentioned before in this Section, the absolute value of the reconstructed complex reflectivities obtained from the SISO wise SAS have to be equal apart slight deformation due to surge. Absolute values have been used as an initial way to assess the correctness of the estimation approach. Nevertheless, it is more proper to look at the complex reflectivities and verify that the both the real and the imaginary parts,are equal for both the SISO and the SIMO SAS for all the 8 set of pings having K=0.Following this, in Fig. <ref> absolute value of the real part of the reconstructed reflectivities for the SISO SAS have been plotted for 4 of the 8 possible set of pings with no motion compensation. Despite the fact the their absolute values are fairly similar, the phase is different as the SISO element phase centers are located on different parallel lines. In Fig. <ref> the same plots are proposed with respect to the SIMO SAS. The motion compensated version of these cases have been represented in Fig. <ref> and Fig. <ref> respectively, each by means of a different estimated trajectory in order to show that the error among them is negligible. The areas where the coherent metallic reflectors are located have been highlighted to emphasize the consistency across the various cases. As expected, the rest of the area features some differences for the intrinsic lack of coherence due to the close range. § CONCLUSIONA novel motion compensation technique for SAS capable of identifying surge, sway and yaw with no restrictions and no prior information from inertial sensors has been introduced. The proposed approach is based on the comparison between the projections on the intersection subspace of different observations. The effectiveness of the technique has been proved by extensive analysis and simulations together with an experiment on real data.§ ACKNOWLEDGEMENTS This work was supported by the Engineering and Physical Sciences Research Council (EPSRC) Grant number EP/J015180/1 and the MOD University Defence Research Collaboration in Signal Processing.10 url@samestyle CafforioPratiRocca1991 C. Cafforio, C. Prati, and F. Rocca, “Sar data focusing using seismic migration techniques,” IEEE Transactions on Aerospace and Electronic Systems, vol. 27, no. 2, pp. 194–207, Mar 1991. BillonPinto1995 D. Billon and M. A. Pinto, “Some general considerations for synthetic aperture sonar design,” in OCEANS '95. MTS/IEEE. Challenges of Our Changing Global Environment. Conference Proceedings., vol. 3, Oct 1995, pp. 1665–1670 vol.3. HayesGough2009 M. P. Hayes and P. T. Gough, “Synthetic aperture sonar: A review of current status,” IEEE Journal of Oceanic Engineering, vol. 34, no. 3, pp. 207–224, July 2009. BellettiniPinto2009 A. Bellettini and M. Pinto, “Design and experimental results of a 300-khz synthetic aperture sonar optimized for shallow-water operations,” IEEE Journal of Oceanic Engineering, vol. 34, no. 3, pp. 285–293, July 2009. BellettiniPinto2002 A. Bellettini and M. A. Pinto, “Theoretical accuracy of synthetic aperture sonar micronavigation using a displaced phase-center antenna,” IEEE Journal of Oceanic Engineering, vol. 27, no. 4, pp. 780–789, Oct 2002. CallowHayesGough2009 H. J. Callow, M. P. Hayes, and P. T. Gough, “Motion-compensation improvement for widebeam, multiple-receiver sas systems,” IEEE Journal of Oceanic Engineering, vol. 34, no. 3, pp. 262–268, July 2009. gough2004 P. T. Gough and M. A. Miller, “Displaced ping imaging autofocus for a multi-hydrophone sas,” IEE Proceedings - Radar, Sonar and Navigation, vol. 151, no. 3, pp. 163–170, June 2004. marston2015 T. M. Marston and D. S. Plotnick, “Semiparametric statistical stripmap synthetic aperture autofocusing,” IEEE Transactions on Geoscience and Remote Sensing, vol. 53, no. 4, pp. 2086–2095, April 2015. herter2016 U. Herter, H. Schmaljohann, and T. Fickenscher, “Autofocus performance on multi channel sas images in the presence of overlapping phase centers,” in OCEANS 2016 MTS/IEEE Monterey, Sept 2016, pp. 1–6. marston2012 T. Marston, “A correlation-based autofocus algorithm for coherent circular synthetic aperture sonar,” in EUSAR 2012; 9th European Conference on Synthetic Aperture Radar, April 2012, pp. 66–69. caporale2016 S. Caporale and Y. Petillot, “A novel motion compensation approach for sas,” in 2016 Sensor Signal Processing for Defence (SSPD), Sept 2016, pp. 1–5. FesslerSutton2003 J. A. Fessler and B. P. Sutton, “Nonuniform fast fourier transforms using min-max interpolation,” IEEE Transactions on Signal Processing, vol. 51, no. 2, pp. 560–574, Feb 2003. caporale2007 S. Caporale, L. D. Marchi, and N. Speciale, “A svd-based algorithm for dense nonuniform fast fourier transform,” in 2007 15th European Signal Processing Conference, Sept 2007, pp. 2120–2124. hunter2016 A. J. Hunter, S. Dugelay, and W. L. J. Fox, “Repeat-pass synthetic aperture sonar micronavigation using redundant phase center arrays,” IEEE Journal of Oceanic Engineering, vol. 41, no. 4, pp. 820–830, Oct 2016. lepage2002 K. D. LePage and H. Schmidt, “Bistatic synthetic aperture imaging of proud and buried targets from an auv,” IEEE Journal of Oceanic Engineering, vol. 27, no. 3, pp. 471–483, Jul 2002. | http://arxiv.org/abs/1707.08488v1 | {
"authors": [
"Salvatore Caporale",
"Yvan Petillot"
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"published": "20170726151924",
"title": "A New Framework for Synthetic Aperture Sonar Micronavigation"
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SRF Theory Developments from the Center for Bright BeamsThis work was supported by the US National Science Foundation under Award OIA-1549132, the Center for Bright Beams. D. B. [email protected], T. Arias, D. L. Hall, M. Liepe, J. P. Sethna, N. Sitaraman, Cornell University, Ithaca, NY, United States of America A. Pack, M. K. Transtrum, Brigham Young University, Provo, UT, USA December 30, 2023 ================================================================================================================================================================================================================================ bstractKeywords: reduction consistency; thermodynamic consistency; surface tension;phase field; multiphase flow; N-phase flow ntroduction ethodlgorithmestsummary§ ACKNOWLEDGEMENTThis work was partially supported by NSF (DMS-1318820, DMS-1522537). ppendixppendBppendCppendDppendEplain | http://arxiv.org/abs/1707.09023v1 | {
"authors": [
"Suchuan Dong"
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"math.NA",
"physics.comp-ph"
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"primary_category": "physics.flu-dyn",
"published": "20170727194612",
"title": "Multiphase Flows of N Immiscible Incompressible Fluids: A Reduction-Consistent and Thermodynamically-Consistent Formulation and Associated Algorithm"
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There are two ways of speeding up MCMC algorithms: (1) construct more complex samplers that use gradient and higher order information about the target and (2) design a control variate to reduce the asymptotic variance. While the efficiency of (1)as a function of dimension has been studied extensively, this paper providesthe first results linking the efficiency of (2) with dimension. Specifically, we construct a control variate for a d-dimensional Random walk Metropolis chain with an IID targetusing the solution of the Poisson equation for thescaling limit in <cit.>.We prove thatthe asymptotic variance of the corresponding estimator is bounded above by a multiple oflog(d)/d over the spectral gap of the chain. The proof hinges onlarge deviations theory, optimal Young's inequality and Berry-Esseen type bounds.Extensions of the result to non-product targets are discussed.[ [ December 30, 2023 =====================§ INTRODUCTIONMarkov Chain Monte Carlo (MCMC) methods are designed to approximateexpectations of high dimensional random vectors, see e.g. <cit.>.It is hence important to understand how the efficiency of MCMC algorithms scales with dimension. The optimal scaling literature, initiated by the seminal paper <cit.>, indicates that for the high-dimensional algorithms it is the growth of the asymptotic variance with dimension thatprovides perhaps the most natural measure of efficiency for MCMC (see <cit.>, <cit.>). For instance, for a product target, the asymptotic variance for the Random walk Metropolis (RWM) chain on ^d isheuristically of the order Ø(d)<cit.>.Moreover,the asymptotic variancesof the d-dimensionalMetropolis-adjusted Langevin algorithm (MALA),Hamiltonian Monte Carloand the fast-MALA are Ø(d^1/3) <cit.>, Ø(d^1/4) <cit.> and Ø(d^1/5) <cit.>, respectively. This paper constructs a dimension-dependent estimator (see (<ref>) below) and proves a bound on its asymptotic variance,suggesting the order Ø(log(d)), for a RWM chain with an IID target. The idea is to exploit the following facts: (I) the law of the diffusion scaling limit for the RWM chain(as d→∞) from <cit.> is close (in the weak sense) to that of the law of the chainitself and (II) the Poisson equation for the limiting Langevin diffusion has an explicit solution.Following ideas from <cit.>, we construct and analyse the estimator using (I) and (II).Specifically, let ρ be a density onandρ_d(𝐱^d):=∏_i=1^dρ(x^d_i) the corresponding d-dimensional product density,where 𝐱^d=(x_1^d,…,x_d^d)∈^d. Let 𝐗^d={𝐗^d_n}_n∈ be the RWM chain converging to ρ_d with the normal proposal with variancel^2/d· I_d (here I_d𝐱^d=𝐱^d, for all 𝐱^d∈^d, andl a constant), analysed in <cit.>. If f(𝐱^d) depends only on the first coordinate x_1^d,thenρ_d(f):=∫_^d f(𝐱^d)ρ_d(𝐱^d)d𝐱^d= ∫_ f(x)ρ(x)dx=:ρ(f). Under appropriate conditions,the asymptotic variance σ^2_f,d in the Central limit theorem (CLT) for theestimator ∑_i=1^n f(𝐗^d_i)/n of ρ_d(f) satisfies σ_f,d^2≤ 2Var_ρ(f)/(1-λ_d)and, heuristically,σ_f,d^2= Ø(d) as d→∞.HereVar_ρ(f):=ρ(f^2)-ρ(f)^2 and1-λ_d denotes the spectral gap of the chain𝐗^d. The inequality follows by the spectral representation of σ_f,d^2in <cit.>. The reasoning analogous to thatapplied to the integrated autocorrelation time in <cit.> can be used to argue thatthe spectral gap 1-λ_d is of the order Ø(1/d). Hence the asymptotic variance σ_f,d^2 is Ø(d).The Poisson equation for the Langevin diffusion arising in the scaling limit of𝐗^d in <cit.> is a second order linear ODE with solution f̂ given explicitly interms of f and the density ρ, see (<ref>) below. For large d, the function f̂ ought to approximate the solution of the Poisson equation for 𝐗^d.The reasoning in <cit.> then suggests the form of an estimator for ρ_d(f), whichunder appropriate technical assumptions, satisfies the following CLT:√(n)(1/n∑_i=1^n(f+d(P_df̂-f̂))(𝐗^d_i)-ρ_d(f)) N(0,σ̂^2_f,d)as n→∞,where P_d is the transition kernel of the chain 𝐗^d.The main result of this paper (Theorem <ref> below) states that for some constant C>0the following inequalityholds σ̂_f,d^2≤Clog(d)/(d(1-λ_d))and, heuristically,σ̂_f,d^2=Ø(log(d)) as d→∞.Theorem <ref> also gives the explicit dependence of the constant C on the function f. This result suggests that to achieve the same level of accuracy as when estimating ρ_d(f) by an average over an IID sampleform ρ_d, onlyØ(log d) times as many RWM samples are needed if the control variate d (P_df̂-f̂) is added.This should be contrasted toØ(d) (resp. Ø(d^1/3), Ø(d^1/4), Ø(d^1/5)) times as many samples for the RWM (resp. MALA, Hamiltonian Monte Carlo, fast-MALA)without the control variate,see <cit.>. The optimal scaling for the proposal variance of a d-dimensional RWM chain is Ø(1/d),see <cit.> for a review and <cit.> for the proof that other scalings lead to suboptimal behaviour.To get a non-trivial scaling limit in <cit.>, it is necessary to accelerate the chains (𝐗^d)_d∈ linearlyin dimension. The weak convergence of the accelerated chain to the Langeivn diffusion suggests thatdf̂ is close to the solution of the Poisson equation for P_d and f, making d (P_df̂-f̂) a good control variate. Using an approximate solution to the Poisson equation to construct control variates is a common variance reduction technique, see e.g. <cit.>.In this context, more often than not,an approximate solution used is a solution of a Poisson equation of a simpler related process. For instance, in <cit.>a sequence of Markov chains on a finite state space, converging weakly to a given RWM chain, is constructed.Then a solutions to thePoisson equation of the finite state chain is used to construct a control variate capable of reducing theasymptotic variance of the RWM chain arbitrarily.Here this idea is turned on its head:a solution to the Poisson equation of the limiting diffusion is used to construct a control variate for a RWM chain from a weakly convergent sequence in <cit.>. Since the complexity of the RWM increases arbitrarily as dimension d→∞,it is infeasible to get an arbitrary variance reduction as in <cit.>. However, heuristically, the amount ofvariance reductionmeasured by the ratio σ_f,d^2/σ̂_f,d^2still tends to infinity at the rate d/log(d).If the solution of the Poisson equation for 𝐗^d and f were available, we could construct an estimator for ρ_d(f) with zero variance (see e.g. <cit.>). Put differently, in this case there would be no need for the chain to explore its state space at all. In our setting, since the jumps of𝐗^d are of sizeØ(1/√(d)) <cit.>, after Ø(log d) stepsthe chain will have explored the distance ofØ(log(d)/√(d)). In line with the observation above this distance tends to zero as d→∞ since, heuristically,df̂ approximates the solution of the Poisson equation for P_d and f arbitrarily well. The key technical stepin the proof of our result (Theorem <ref> below) is a type of concentration inequality. It generalises the limit in <cit.>,which essentially states that generators of the accelerated chains (𝐗^d)_d∈ convergeto the generator of the Langevin limit when applied to a compactly supported and infinitely smooth function, in two ways: (A) it extends the limit to a class of functions of sub-exponential growth and (B) provides estimates for the rate of convergence. Both of these extensions are key for our main result. (A) allows us to applyTheorem <ref> to a solution of the Poisson equation, which is not compactly supported. Note that this step in the proof entails identifying the correct space of functions that is closed under the operation of solving the Poisson equation (see Proposition <ref> below). Estimate (B) allows us to control the asymptotic variancevia a classical spectral-gap bound. The proof of Theorem <ref>, outlined in Sec. <ref> below, crucially depends on the large deviations theory (Sec. <ref>), the form of the constant in optimalYoung's inequality (Sec. <ref>) and Berry-Esseen type bounds (Sec. <ref>). We conclude the introduction with a comment on how the present paper fits into the literature. Since, as discussed above, the asymptotic varianceσ_f,d^2is approximately equal to the product 2Var_ρ(f)/(1-λ_d),two “orthogonal” approaches to speeding up MCMC algorithms are feasible.(a) The MCMC method itself can be modified, with the aim of increasing the spectral gap, leading to many well-known reversible samplers such asMALA andHamiltonian Monte Carlo <cit.> as well asnon-reversible ones <cit.>.There is a plethora of papers (see <cit.> and the references therein) studying the asymptotic properties of such sampling algorithms as dimension increases to infinity.(b) A control variate g, satisfying ρ(g)=0, may be added to fwith the aim of reducing Var_ρ(f) to Var_ρ(f+g)without modifying the MCMC algorithm.A number of control variates have been proposed in the MCMC literature <cit.>.Thematically, the present paper fits under (b) and, to the best of our knowledge, is the first to investigate the growth of the asymptotic variance as the dimension d→∞ in this context. Moreover, it is feasible that our method could be generalised to some of the algorithmsunder (a),see Section <ref> below for a discussion of possible extensions.The remainder of paper is structured as follows. Section <ref>gives a detailed description of the assumptions and states the results.Section <ref> illustrates algorithms based on our main result with numerical examples and discusses (without proof) potential extensions of our results for other MCMC methods, more general targets, etc. In Section <ref> we prove our results. Section <ref>develops the tools needed for the proofs of Section <ref>. Section <ref>uses results from probability and analysis but is independent of all that precedes it in the paper. § RESULTS Let 𝐗^d={𝐗^d_n}_n∈ be a RWM chain in ^d with a transition kernel P_df:=(1/d)𝒢_df + f, where 𝒢_df(𝐱^d):=d𝔼_𝐘^d[(f(𝐘^d)-f(𝐱^d)) α(𝐱^d,𝐘^d) ],α(𝐱^d,𝐘^d):=1∧ρ_d(𝐘^d)/ρ_d(𝐱^d),𝐱^d∈^d, 𝐘^d=(Y_1^d,…,Y_d^d)∼ N(𝐱^d,l^2/d· I_d) and x∧ y:=min{x,y} for all x,y∈, started in stationarity 𝐗^d_1∼ρ_d. Let 𝒮^n consists of all the functionswith their first n derivatives growing slower then any exponential function.More precisely,for any n∈∪{0}, define𝒮^n:={g∈𝒞^n()∑_i=0^ng^(i)_∞,s<∞∀ s>0}, where g_∞,s:=sup_x∈(e^-s|x||g(x)|)and 𝒞^n() (resp. 𝒞^0()) denotes n-times continuously differentiable (resp. continuous) functions. Our main result (Theorem <ref> below) applies to the space 𝒮^1,containing functions f for whichρ(f):=∫_f(x)ρ(x) dx is typically of interest in applications (e.g. polynomials). In addition, spaces in (<ref>) are closed for solving Poisson's equation in (<ref>), see Proposition <ref> below. Throughout the paperρ denotes a strictly positive density onwith log(ρ)∈𝒮^4 and lim_|x|→∞x/|x|·log(ρ(x))'=-∞, unless otherwise stated. Assumption (<ref>) implies that the tails of ρ decay faster then any exponential, i.e.𝔼[e^sX]<∞for any s∈ for X∼ρ (cf. <cit.>).The assumption log(ρ)∈𝒮^4 prohibits ρ from decaying to quickly, e.g. proportionally to e^-e^|x|.Both of these assumptions serve brevity and clarity of the proofs and it is feasible they can be relaxed. Nevertheless,a large class of densities of interest satisfy these assumptions, e.g. mixtures of Gaussian densities or any density proportional to e^-p(x) for a positive polynomial p.The scaling limit, introduced in <cit.>,of the chain 𝐗^d as the dimension d tends to infinity is keyfor all that follows.Consider a continuous-time process {U^d_t}_t≥0, given by U^d_t:=X^d_⌊ d· t⌋,1, where ⌊·⌋ is the integer-part function andX^d_·,1 is the first coordinate of 𝐗^d (since the proposal distribution forX^d_·,1 has variance l^2/d,time needs to be accelerated to get a non-trivial limit).As shown in <cit.> (see also <cit.>), the weak convergenceU^d⇒ U holds as d↑∞, where U is the Langevin diffusion started in stationarity, U_0∼ρ, with generator acting on f∈𝒞^2() as𝒢f:=(h(l)/2)(f”+(logρ)'f'), whereh(l):=2l^2Φ(-l√(J)/2) and J:=ρ((log(ρ)')^2)and Φ is the distribution of N(0,1). Poisson's equation for U and a function f takes the form 𝒢f̂(x) =ρ(f)-f(x).It is immediate that a solution f̂ of (<ref>) is given by the formula f̂(x):=∫_0^x2dy/h(l)ρ(y)∫_-∞^yρ(z)(ρ(f)-f(z))dz, x∈.In the remainder of the paper f̂ denotes the particular solution in (<ref>) of the equation in (<ref>).As usual, for p∈[1,∞), f:^d→ is in L^p(ρ_d) if and only if f_p:=ρ_d(|f|^p)^1/p<∞.Finally, note that under our assumptions on ρ, the kernel P_d of the RWM chain 𝐗^d defined above is a self-adjoint bounded operator on the Hilbert space {g∈ L^2(ρ_d)ρ_d(g)=0} with normλ_d<1If f∈𝒮^1, then f̂∈𝒮^3 and CLT (<ref>) holds forthe function f+dP_df̂-df̂ and the RWM chain 𝐗^d introduced above.Furthermore,there exists a constant C_1>0, such that for all f∈𝒮^1 and d∈∖{1},the asymptotic variance σ̂^2_f,d in CLT (<ref>) satisfies: σ̂^2_f,d≤ C_1 (∑_i=1^3f̂^(i)_∞,1/2)^2log(d)/(1-λ_d)d.The proof of Theorem <ref> is given in Section <ref> below.It is based on the spectral-gap estimate of the asymptotic varianceσ̂_f,d^2≤ 2𝒢_df̂-𝒢f̂^2_2/(1-λ_d)from <cit.>, the uniform ergodicity of the chain𝐗^dand the following proposition. There exists a constant C_2such that for every f∈𝒮^3 and all d∈∖{1} we have: 𝒢f-𝒢_df_2≤ C_2(∑_i=1^3f^(i)_∞,1/2)√(log(d)/d). The proof of Proposition <ref>, given in Section <ref> below, requires a pointwise control of the difference 𝒢f-𝒢_df on a large subset of ^d.To formulate this precisely, we need the following.A positive sequence a={a_d}_d∈ is sluggish if the following holds:lim_d→∞ a_d=∞andsup_d∈∖{1}a_d/√(logd)<∞. Theorem <ref> below is the main technical result of the paper.It generalises the limit in <cit.> to a class of unbounded functions and provides an error estimate for it.The bound in Theorem <ref> yields sufficient control of the difference𝒢f-𝒢_df to establish Proposition <ref>. Let a={a_d}_d∈ be a sluggish sequence. There existconstants c_3,C_3>0 (dependent on a) and measurable sets 𝒜_d⊂^d, such that for all d∈ we have ρ_d(^d∖𝒜_d)≤ c_3 e^-a_d^2 and |𝒢f(x^d_1)-𝒢_df(𝐱^d)|≤ C_3(∑_i=1^3f^(i)_∞,1/2) e^|x^d_1|a_d/√(d)for any f∈𝒮^3 and 𝐱^d∈𝒜_d.The proof of Theorem <ref> is outlined and given in Sections <ref> and <ref> below, respectively. The dependence on f in the bound of Theorem <ref> is not sharp. The factor ∑_i=1^3f^(i)_∞,1/2is used because it states concisely that the speed of the convergence of 𝒢_df to 𝒢f depends linearly on the first three derivatives of f. Moreover, it is not clear ifthe bound in Theorem <ref> and Proposition <ref> is optimal in d. However, if an improvement were possible, the proof would have to be significantly different to the one presented here.In particular, a better control of the difference |𝒢f-𝒢_df| on ^d∖𝒜_d would be required.§ DISCUSSION AND NUMERICAL EXAMPLES §.§ DiscussionIn this section we discuss potential extensions of Theorem <ref> to settings satisfying weaker assumptions or involving related MCMC chains. §.§.§ IID target for non-RWM chains in stationarity Perhaps the most natural generalisation of Theorem <ref> would be to the MALA and fast-MALA chains (in <cit.> and <cit.>) for which it is also possible to obtain non-trivial weak Langevin diffusion limits under appropriate scaling. Since the form of the Poisson equation in (<ref>) is preserved it is possible to define an estimator like the one in (<ref>) and it seems feasible that a version of Theorem <ref> can be established in this context using methods analogous to the ones in this paper.§.§.§ IID target in the transient phase for the RWM chainTheorem <ref> is a result only about the stationary behaviour of the chain. As in practice MCMC chains are typically started away from stationarity it is important to understand the transient behaviour. In <cit.> it is shown that the scaling limit described in Section <ref> above has mean-field behaviour of the McKean type, i.e. the limiting process is a continuous semimartingale with characteristicsthat at time t depend on the law of the process at t.This suggests that an appropriately chosen time-dependent function f̂ inthe estimator in (<ref>) could further reduce the constant in the bound of Theorem <ref>. §.§.§ General product target densityThe class of target distributions considered in <cit.>,preserves the independence (i.e. product) structure but allows for a different, dimension dependent, scaling of each of the components of the target law. If the proposal variances appropriately reflect the scaling in the target,each component in the infinite dimensional limit is a Langevin diffusion. Again, as in Section <ref> above, the estimator in (<ref>) can be applied directly and an extension ofTheorem <ref> to this setting appears feasible.§.§.§ Gaussian targets in high dimensionsLet π_0 denote a Gaussian target on ^d with mean μ and covariance matrix diag(σ^2_11,…,σ^2_dd). Inspired by <cit.> and the proof of Theorem <ref>,a good control variate for the ergodic average estimator for π_0(f) takes the formd(P_df̃-f̃),where f̃ solves the ODEf̃”+((∂/∂ x^d_1)logπ_0)f̃'=2/h_0(l)·(π_0(f)-f),with h_0(l):=2l^2Φ(-l√(J_0)/2) and J_0:=1/d·∑_j=2^d1/σ^2_jj.In the case of the mean, f(x^d_1)=x^d_1, we can solve the ODE explicitly: f̃(x^d_1)=2σ^2_11/h(l)· x^d_1.If π_0 has a general non-degenerate covariance matrix Σ, we have an ODE analogous to the one above for each eigen-direction of Σ.The control variate for the mean of the first coordinate, f(x^d_1)=x^d_1,is then a linear combination of thecontrol variates for the means in eigen-directions.Specifically,f̃:^d→ in the estimator analogous to the one in (<ref>) (see the numerical example for h=0 in Section <ref>) takes the form f̃(𝐱^d)=2/h_0(l) ·∑_j=1^dx^d_jΣ_j1,withh_0(l)=2l^2Φ(-l√(J_0)/2)andJ_0=1/d·Tr((Σ^-1)_2:d,2:d).Note thatf̃ does not depend on the mean of the target, a special feature of the Gaussian setting. §.§.§ Non-product target density A typical non-product target densityconsidered in the literature <cit.> is a projection of a probability measure Π on a separable real Hilbert space ℋ onto a d-dimensional subspace, where Π is given via its Radon-Nikodym derivatived Π/d Π_0(x)∝exp(-Ψ(x)). Here Ψ is a densely defined positive functional onℋ and Π_0 is a Gaussian measure on ℋ specified via a positive trace-class operator on ℋ (see e.g. <cit.> for a detailed description and <cit.> for the motivation for this class of measures). A key feature of this framework is that there exists an ℋ-valued Langevin diffusion z, driven by a cylindrical Brownian motion on ℋ (i.e. a solution of an SPDE), that describes the scaling limit of the appropriately accelerated sequence of chains(𝐗^d)_d∈, see <cit.>. An estimator analogous to the one in (<ref>) would require the solution f̂ of the Poisson equation of z. While this might be a feasible strategy theoretically, it would likely be difficult to numerically evaluate the solutionf̂.However, inspired by the Simplified Langevin Algorithm in <cit.>, which uses as the proposal chainan Euler scheme for the Langevin diffusion with target Π_0 (not Π),we suggest constructing the estimatorfor Π(f) in (<ref>), with f̂ the solution of the Poisson equation for the Langevin diffusion converging to Π_0 (not Π). This strategy is feasibleas we are able to produce good control variates for Gaussian product targets in high dimensionsin the spirit of Theorem <ref>, see Section <ref> above.The main theoretical question in this context is to findsuitable assumptions on the functional Ψ in the Radon-Nikodym derivative above that guarantee the asymptotic variancereduction as d→∞. Expecting an improvement from polynomial to logarithmic growth is unrealistic as we are solving the Poisson equation for Π_0 instead of Π.However, the numerical example in Section <ref> suggests that this idea may work in practice if Π is close from Gaussian Π_0. Understanding in which settings does it lead to significant variance reduction is another relevant and important question. §.§ Numerical examplesThe basic message of the present paper is that the process in the scaling limit of an MCMC algorithm contains useful information that can be utilised to achieve significant savings inhigh dimensions. In both examples presented below the problem is to estimate the mean of the first coordinate ρ_d(f), where f(𝐱^d):=x^d_1.A run of T steps of a well-tuned RWM algorithm with kernel P_d, defined in the beginning of Section <ref>,started in stationarity, produces a RWM sample {𝐗^d_n}_n=1,2,⋯ T and an estimate ρ̂_d(f):=∑_n=1^Tf(𝐗^d_n)/T of ρ_d(f). Take f̃ to be the associated solution of the Poisson equation, that we obtain numerically in the first example <ref> and using formula (<ref>) in the second example <ref>. In both cases we estimate the required unknown quantities (ρ_d(f) and Σ) from the sample {𝐗^d_n}_n=1,2,⋯ T.Using the same sample,define ρ̃_d(f):=1/T∑_n=1^T(f+dP_df̃-df̃)(𝐗^d_n). Since the functionP_df̃-f̃ is not accessible in closed form, for every n≤ T we use IID Monte Carlo to estimate the value (P_df̃-f̃)(𝐗^d_n) as ∑_j=1^n_MC(1∧ (ρ_d(𝐘^d,j_n)/ρ_d(𝐗^d_n)))(f̃(𝐘^d,j_n)-f̃(𝐗^d_n))/n_MC, where 𝐘^d,1_n,…,𝐘^d,n_MC_n is an IID sampleof size n_MC from N(𝐗^d_n,l^2/d· I_d). This estimation step can be parallelised (i.e. run on n_MC cores simultaneously).We measure the variance reduction due to the post processing above by comparing the meansquare errors of ρ̂_d(f) and ρ̃_d(f) as estimators of ρ_d(f) over n_R independent runs of the RWM chain, VR(ρ,f):=∑_k=1^n_R((ρ̂_d(f))_k-ρ_d(f))^2/∑_k=1^n_R((ρ̃_d(f))_k-ρ_d(f))^2,where (ρ̂_d(f))_k and (ρ̃_d(f))_k are the averages in the k-th run of the chain. Heuristically, this means the estimator ρ̃_d(f) of ρ_d(f) based on T sample points is as good as estimatorρ̂_d(f) based on VR(ρ,f)· T sample points.§.§.§ Multi-modal product target To verify that what theory predicts also happens in practice we first present an example of an IID target with each coordinate a bimodal mixture of two Gaussian densities. Given the results in Table <ref>, we wish to highlight the robustness of the method with respect to numerically estimating f̃ and (P_df̃-f̃)(𝐗^d_n)for each n≤ T.Let ρ be a mixture of two normal densitiesN(μ_1,σ_1^2) andN(μ_2,σ_2^2), with the first arising in the mixture with probability 2/5 and μ_1=-3, μ_2=4 and σ_1=σ_2=7/4.The potential of the density ρ has two wells and is in a well know class arising in models of molecular dynamics, see e.g. <cit.>.Note that the corresponding densityρ_d on ^d, defined in the introduction, has2^d modes. With variance reduction (<ref>) we measure how much the estimator ρ̃_d(f) outperforms the estimator ρ̂_d(f) of the mean of the first coordinate ρ_d(f)=6/5. The results across a range of dimensions d and values of the parameter n_MC (that corresponds to the accuracy of the estimation of P_df̃-f̃) are presented in Table <ref>. All the entries were computed using n_R=500 independent runs of length T=2· 10^5.To numerically solve the Poisson equation in (<ref>), substitute the derivatives of f and log(ρ) with symmetric finite differences and use the estimate ρ̂(f) for ρ(f).Recall that the standard deviation of the proposal in our RWM algorithm is l/√(d). The solver uses a grid of hundred points equallyspacedin the interval[min_n≤ T 𝐗^d_n,1- 3 · l/√(d), max_n≤ T𝐗_n,1+3· l/√(d)], where 𝐗^d_n,1 is the first coordinate of the n-th sample point.We use the Moore-Penrose pseudoinverse as the linear system is not of full rank.Finally, we take f̃ to be the linear interpolation of the solution on the grid. Note that this is a crude approximation of the solution of (<ref>), which does not exploit analytical properties of either f or ρ.The results in Table <ref> contain a lot of noise due to numerically solving the ODE, using an approximation ρ̂_d(f) for ρ_d(f) and using IID Monte Carlo to estimate ρ̃_d(f). It is interesting to note, that despite these additional sources of error, the variance reduction is considerable and behaves as the theoretical results predict. The estimator ρ̃_d(f) improves with dimension, and with n_MC. Increasing n_MC, however, has diminishing effect which is particularly clear in the case d=5. Due to the asymptotic nature of our result we can only expect limited gain for any fixed d, even if P_df̃-f̃ could be evaluated exactly (corresponding to n_MC=∞). §.§.§ Bi-modal non-product target Can the theoretical findings of this paper help us construct control variates in more realistic cases with non-product target densities? It is unreasonable to expect a simple general answer to this question. A more realistic approach for future work seems to be trying to establish specific forms of control variates that work well for classes of targets of certain type. We briefly explore one such instance in this section. Sections <ref> and <ref> as well as the results in Table <ref> suggest we can construct useful control variates when the target is close to a Gaussian. Let μ_d,h be a d-dimensional vector with entries (h/2,0,…,0) for h≥ 0 and let Σ^(d) be a d× d covariance matrix with the largest eigenvalue equal to λ=25 with the corresponding eigenvector being (1,1… 1) and all other eigenvalues being equal to one.Take Π_d,h to be the mixture of two d-dimensional normal densitiesN(-μ_d,h,Σ^(d)) andN(μ_d,h,Σ^(d)), both arising in the mixture with probability 1/2. We wish to estimate the mean of the first coordinate Π_d,h(f)=0 (for f(𝐱^d)=x^d_1). To produce a control variate we simply pretend, that we are dealing with a Gaussian target instead of Π_d,h. Let Σ^Π_d,h be the covariance of Π_d,h and Σ̂^Π_d,h an estimate of Σ^Π_d,h obtained from the RWM sample {X^d_n}_n=1,2,…,T. Define f̃_d,h as in (<ref>) using Σ̂^Π_d,h. We compare the performance of estimators Π̂_d,h(f) and Π̃_d,h(f) ofΠ_d,h(f)=0, respectively defined as 1/T∑_n=1^Tf(X^d_n) and 1/T∑_n=1^T(f+dP_df̃_d,h-df̃_d,h)(X^d_n), according to variance reduction (<ref>).Table <ref> shows the results across a range of dimensions d and distances between modes h, which measures the 'non-Gaussianity' of the target. Note that when h=0 the target is Gaussian N(0,Σ^(d)) which we include to demonstrate the validity of control variate (<ref>) for Gaussian targets. All the entries were calculated using n_R=500 independent runs of length T=2· 10^5 and n_MC=50 IID Monte Carlo steps for computing P_df̃-f̃ at each time step.The quality of results decays with dimension because the proposal is scaled as 1/d in each coordinate. This results in the first coordinate mixing slower and for h≠ 0 also being less able to cross between modes, hence our estimate Σ̂^Π_d,h of the covariance becomes worse as we are working with fixed RWM sample length T. When h=10 (and some cases of h=8) it is unlikely that the RWM sample will reach the other mode at all which results in no gain from the method.If we use the true covariance Σ^Π_d,h of the target in the control variate (<ref>), instead of learning it from the sample Σ̂^Π_d,h, the corresponding results for d=50 are presented in Table <ref>.Unsurprisingly the estimator Π̃_d,h(f) does not perform well when the distance between modes h is large. Interestingly though, the method does offer considerable gain in cases h=2 and h=4, even a noticeable gain in h=6. For h=4 and h=6 the target is already clearly bimodal and different from the Gaussian, the RWM sample stays in the same mode for hundreds, respectively thousands of time-steps at a time. § PROOFSThroughout this section we assume the sluggish sequence a={a_d}_d∈ is given and fixed and, as mentioned above, the density ρ satisfies log(ρ)∈𝒮^4 and has sub-exponential tails (<ref>). Section <ref> outlines the proof of Theorem <ref> bystating the sequence of results that are needed to establish it.The proofs of these results, given in Section <ref>, rely on the theory developed in Section <ref> below.Sections <ref> and <ref> establishProposition <ref> and Theorem <ref>, respectively. §.§ Outline of the proof of Theorem <ref>We start by specifying sets 𝒜_d⊂^d that have large probability under ρ_d. We need the following fact. There exists a constantc_A>0, such that the following open subset of ,A:={x∈; |log(ρ)”(x)|< (log(ρ)'(x))^2,1/c_A<|log(ρ)'(x)|< c_A}, satisfies ρ(A)>0. Let A satisfy the conclusion of Proposition <ref> andrecall the notation ρ(f)=∫_f(x)ρ(x) dx for any appropriate function f:→. Recall J=ρ(((logρ)')^2)=-ρ((logρ)”), where the equality follows from assumptions log(ρ)∈𝒮^4 and (<ref>). Define the sets 𝒜_d as follows. Any 𝐱^d∈^d is in 𝒜_dif and only ifthe following four assumptions hold: 1/d-1∑_i=2^de^|x^d_i|< 2∫_ e^|x|ρ(x)dx, 1/d-1∑_i=2^d1_A(x^d_i)> ρ(A)/2, 1/d-1|∑_i=2^d (log(ρ)'(x^d_i))^2-J|<a_d/√(d)√(3∫_((log(ρ)'(x))^2-J)^2ρ(x)dx), 1/d-1|∑_i=2^d log(ρ)”(x^d_i)+J|<a_d/√(d)√(3∫_(log(ρ)”(x)+J)^2ρ(x)dx). The precise form of the constants in Definition <ref> is chosen purely for convenience. It is important that ∫_ e^|x|ρ(x)dx<∞ by (<ref>), ρ(A)>0 by Proposition <ref> and that the constants in (<ref>)–(<ref>) are in (0,∞). Moreover,for any 𝐱^d∈𝒜_d there are no restrictions on its first coordinate x_1^dand the sets 𝒜_d are typical in the following sense. There exists a constant c_1, such that ρ_d(^d∖𝒜_d)≤ c_1 e^-a_d^2 for all d∈.Using the theory of large deviations and classical inequalities, the proof of the proposition bounds the probabilities of sets where each of the above four assumptionsin Definition <ref> fails(see Sections <ref> and <ref> below for details).Pick any f∈𝒮^3 and express the generator 𝒢_d, defined in (<ref>), as follows: 𝒢_df(𝐱^d)=d·𝔼_Y^d_1[(f(𝐘^d)-f(𝐱^d)) 𝔼_𝐘^d-[1∧ρ_d(𝐘^d)/ρ_d(𝐱^d)]], 𝐱^d∈^d,where𝔼_𝐘^d-[·] is the expectation with respect to all the coordinates of the proposal𝐘^d in ^d, except the first one(identify f∈ L^1(ρ) with f∈ L^1(ρ_d) by ignoring the last d-1 coordinates). The strategy of the proof of Theorem <ref> is to define a sequence of operators, “connecting” 𝒢_d and 𝒢, such that each approximation can be controlled for f∈𝒮^3 and 𝐱^d∈𝒜_d.First, for f∈𝒮^3, define 𝒢̃_df(𝐱^d):=d·𝔼_Y^d_1[(f(Y^d_1)-f(x^d_1))β(𝐱^d,Y^d_1)],𝐱^d∈^d,where for any y∈,β(𝐱^d, y):=𝔼_𝐘^d-[1∧exp(log(ρ)'(x^d_1)(y-x^d_1)+∑_i=2^d K(x^d_i, Y^d_i))],𝐱^d∈^d,and for any (x,y)∈^2 we defineK(x, y):=log(ρ)'(x)(y-x)+log(ρ)”(x)/2(y-x)^2+(log(ρ)'(x))^31_A(x)/3(y-x)^3. In (<ref>), the set A satisfies the conclusion of Proposition <ref> and the coefficient before (y-x)^3 is chosen so that it is uniformly boundedfor all x∈. This property plays an important role in proving that we have uniform control over the supremum norms of certain densities, cf. Lemmas <ref> and <ref> below. We can now prove the following. There exists a constant C, such that for every f∈𝒮^3 and all d∈ we have:|𝒢_df(𝐱^d)-𝒢̃_df(𝐱^d)|≤ C f'_∞,1/2 e^|x^d_1|d^-1/2∀𝐱^d∈𝒜_d.The proof of Proposition <ref> relies only on the elementary bounds from Section <ref> below.The idea is to use the Taylor series of log(ρ)(Y_i) around x^d_i for every i∈{1,…,d}and then provethat modifying terms of order higher then two if i∈{2,…,d} (resp. one if i=1) is inconsequential.Define the operator𝒢̂_df(𝐱^d) for any f∈𝒮^3 and 𝐱^d∈^d by𝒢̂_df(𝐱^d) := l^2/2f”(x^d_1)𝔼_𝐘^d-[1∧ e^∑_i=2^d K(x^d_i, Y^d_i)]+ l^2f'(x^d_1)log(ρ)'(x^d_1)𝔼_𝐘^d-[e^∑_i=2^d K(x^d_i, Y^d_i)1_{∑_i=2^d K(x^d_i, Y^d_i)<0}].We can now prove the following fact. There exists a constant C, such that for every f∈𝒮^3,and all d∈ we have:|𝒢̂_df(𝐱^d)-𝒢̃_df(𝐱^d)|≤ C(∑_i=1^3f^(i)_∞,1/2)e^|x^d_1|d^-1/2∀𝐱^d∈𝒜_d. Note that, if we freeze the coordinates x_2^d,…,x_d^d in 𝐱^d, the operator mapping f∈𝒮^3 to x_1^d↦𝒢̂_df(𝐱^d)generates a one-dimensional diffusion with coefficients of the same functional form as in𝒢, but with slightly modified parameter values.The proof of Proposition <ref>is based on the third and second degree Taylor's expansion of y↦ f(y)andy↦β(𝐱^d,y) (around x^d_1), respectively,applied to the definition of 𝒢̃_d in (<ref>). The difficult part in proving that the remainder terms can be omitted consist of controlling ∂^2/∂ y^2β(𝐱^d,y), as this entailsbounding the supremum norm of the density of ∑_i=2^d K(x^d_i, Y^d_i) uniformly in d.Condition (<ref>), which forces a portion of the coordinates x^d_i of 𝐱^d to be in the set A where the densities of the corresponding summands K(x^d_i,Y^d_i) can be controlled, was introduced for this purpose. The details, explained in Sections <ref> and <ref> below, rely crucially on the optimal version of Young'sinequality.Introduce the followingnormal random variable with mean μ_𝒩(𝐱^d)=l^2/2d∑_i=2^dlog(ρ)”(x^d_i) and variance σ^2_𝒩(𝐱^d)=l^2/d∑_i=2^d(log(ρ)'(x^d_i))^2:𝒩(𝐱^d,𝐘^d):=l^2/2d∑_i=2^dlog(ρ)”(x^d_i)+∑_i=2^dlog(ρ)'(x^d_i)(Y^d_i-x^d_i).Define the operator𝒢̆_df(𝐱^d) for f∈𝒮^3 and 𝐱^d∈^d by: 𝒢̆_df(𝐱^d) := l^2/2f”(x^d_1)𝔼_𝐘^d-[1∧ e^𝒩(𝐱^d,𝐘^d)]+ l^2f'(x^d_1)log(ρ)'(x^d_1)𝔼_𝐘^d-[e^𝒩(𝐱^d,𝐘^d)1_{𝒩(𝐱^d,𝐘^d)<0}].There exists a constant C, such that for every f∈𝒮^3 and all d∈ we have:|𝒢̆_df(𝐱^d)-𝒢̂_df(𝐱^d)|≤ C(∑_i=1^2f^(i)_∞,1/2) e^|x^d_1|d^-1/2∀𝐱^d∈𝒜_d.First we show that |𝔼_𝐘^d-[1∧ e^∑_i=2^d K(x^d_i, Y^d_i)]-𝔼_𝐘^d-[1∧ e^𝒩(𝐱^d,𝐘^d)]| is small (Lemma <ref> below).Proving that𝔼_𝐘^d-[e^∑_i=2^d K(x^d_i, Y^d_i)1_{∑_i=2^d K(x^d_i, Y^d_i)<0}] and 𝔼_𝐘^d-[e^𝒩(𝐱^d,𝐘^d)1_{𝒩(𝐱^d,𝐘^d)<0}] are close is challenging, as it requires showing that the supremum norm of the difference between the distributions of 𝒩(𝐱^d,𝐘^d) and ∑_i=2^d K(x^d_i, Y^d_i) decays as d^-1/2 uniformly in its argument.The proof of this fact mimics the proof of the Berry-Esseen theorem and relies on the closeness of the CFs (characteristic functions) of 𝒩(𝐱^d,𝐘^d) and ∑_i=2^d K(x^d_i, Y^d_i). The particular form of K(x,Y) makes it possible to explicitly calculate the CF of K(x,Y), if x∉ A, and bound it appropriately, if x∈ A. The details are explained in Sections <ref> and <ref> below.Since 𝒩(𝐱^d,𝐘^d) is normal,it is possible to explicitly calculate the expectations 𝔼_𝐘^d-[1∧ e^𝒩(𝐱^d,𝐘^d)] and 𝔼_𝐘^d-[e^𝒩(𝐱^d,𝐘^d)1_{𝒩(𝐱^d,𝐘^d)<0}], see <cit.>. Using these formulae,Proposition <ref>, which implies Theorem <ref>, can be deduced fromassumptions (<ref>)–(<ref>).There exists a constant C, such that for every f∈𝒮^3 and all d∈ we have:|𝒢f(x^d_1)-𝒢̆_df(𝐱^d)|≤ C(∑_i=1^2f^(i)_∞,1/2)e^|x^d_1|a_d/√(d)∀𝐱^d∈𝒜_d. The bounds in Propositions <ref>–<ref>are of the order Ø(d^-1/2). The orderØ(a_d/√(d)) of the bound in Proposition <ref>gives the order in the bound of Theorem <ref>.§.§ Proof of Theorem <ref> Let Ã:={ x∈;|log(ρ)”(x)|< (log(ρ)'(x))^2}. It suffices to show that the open set à is not empty, sinceÃ=∪_n∈(Ã∩{x∈; 1/n< |log(ρ)'(x)|<n}), so for some large n_0 the open setÃ∩{x∈; 1/n_0<|log(ρ)'(x)|< n_0} must have positive Lebesgue measure and we can take c_A:=n_0.Assume that Ã=∅, i.e.|u'|≥ u^2on ,where u:=log(ρ)'. Since ρ satisfies (<ref>), there exists x_0<0 and C>0 such that u>Con the interval (-∞,x_0). Moreover,since |u'|≥ u^2>C^2>0, u' has no zeros on (-∞,x_0) and satisfies either u'≥ u^2or -u'≥ u^2 on the half-infinite interval.Since (1/u)'=-u'/u^2, integrating the inequalities -u'/u^2≤ -1or -u'/u^2≥1 fromany x∈(-∞,x_0) to x_0, we get 1/u(x_0)+x_0-x≤ 1/u(x) and1/u(x_0)+x-x_0≥ 1/u(x). Since by assumption it holds 0<1/u<1/C on (-∞,x_0), we get a contradiction in both cases. Let B^d_1, B^d_2, B^d_3 and B^d_4 be the subsets of ^d where assumptions (<ref>), (<ref>), (<ref>) and (<ref>) are not satisfied, respectively. Note that ^d∖𝒜_d=B^d_1∪ B^d_2∪ B^d_3 ∪ B^d_4.Recall that by (<ref>), the L'Hospital's rule implies lim_|x|→∞logρ(x)/x→-∞ and hence ρ(e^s|x|)<∞ for any s>0. Since {a_d}_d∈ is sluggish, there existsn∈ such thata_d≤√(nlog d) for all d∈. Then, by Proposition <ref> applied tofunctions x↦ (e^|x|-ρ(e^|x|))/ρ(e^|x|) and x↦ 2(ρ(A)-1_A(x))/ρ(A), respectively, there exist constants c_1', c_2' such that the inequalitiesρ(B^d_1)≤ c_1'd^-n≤ c_1'e^-a_d^2 and ρ(B^d_2)≤ c_2'd^-n≤ c_2'e^-a_d^2 hold for all d∈. Likewise, there exist constants c_3', c_4' such that ρ(B^d_3)≤ c_3' e^-a^2_d and ρ(B^d_4)≤ c_4' e^-a^2_d. This follows by Proposition <ref>, applied to the sequence {a_d}_d∈ and functions g_3(x):=(log(ρ)'(x))^2-J (with t:=√(3ρ(g_3^2))) and g_4(x):=log(ρ)”(x)+J (with t:=√(3ρ(g_4^2))), respectively. Hence ρ(^d∖𝒜_d)≤ρ(B^d_1)+ρ(B^d_2)+ρ(B^d_3 )+ρ(B^d_4)≤ c_1 e^-a_d^2 for c_1:=max{c_1',c_2',c_3',c_4'}.The proof above shows thatthe subsets B^d_1 and B^d_2 are of negligible size in comparison to B^d_3 and B^d_4, since the n∈ can be chosen arbitrarily large. Pick an arbitrary 𝐱^d∈𝒜_d and recall that α(𝐱^d,𝐘^d) is defined in (<ref>).Since |1∧ e^x-1∧ e^y|≤ |x-y| for all x,y∈,for every realization Y^d_1, Taylor's theorem implies |𝔼_𝐘^d-[α(𝐱^d,𝐘^d)]-β(𝐱^d, Y^d_1)|≤ |log(ρ)”(W^d_1)|(Y^d_1-x^d_1)^2+T^d_1(𝐱^d)+T^d_2(𝐱^d),where W^d_1 satisfies log(ρ)”(W^d_1)(Y^d_1-x^d_1)^2/2=log(ρ)(Y^d_1)-log(ρ)(x^d_1)-logρ'(x^d_1)(Y^d_1-x^d_1) andT^d_1(𝐱^d) := 1/6𝔼_𝐘^d-[|∑_i=2^d(log(ρ)”'(x^d_i)-2(log(ρ)'(x^d_i))^31_A(x^d_i))(Y_i-x^d_i)^3|],T^d_2(𝐱^d) := 1/24𝔼_𝐘^d-[∑_i=2^d|log(ρ)^(4)(Z^d_i)|(Y_i-x^d_i)^4].Here Z^d_i satisfieslog(ρ)^(4)(Z^d_i)(Y^d_i-x^d_i)^4/4!=log(ρ)(Y^d_i)-∑_j=0^3(logρ)^(j)(x^d_i)(Y^d_i-x^d_i)^j/j! for any 2≤ i≤ d. Recall Y^d_i-x^d_i is normal N(0,l^2/d), for some constant l>0, and log(ρ)∈𝒮^4.Hence we may applyProposition <ref> to the functionx↦log(ρ)”'(x)-2(log(ρ)'(x))^31_A(x)to get T^d_1(𝐱^d)≤ C_1 (l^6/d^3∑_i=2^d e^|x^d_i|)^1/2for some constantC_1>0, independent of 𝐱^d. Since 𝐱^d∈𝒜_d,the assumption in (<ref>) yields T^d_1(𝐱^d)≤ C_1 l^3 (2ρ(e^|x|))^1/2/d.Similarly, we apply Proposition <ref>(with f=logρ, n=k=4, m=1, s=1 and σ^2=l^2/d) and assumption (<ref>)to get T^d_2(𝐱^d)≤ C_2 d^-2∑_i=2^de^|x^d_i|≤ C_2 d^-1 for some constant C_2>0 and all 𝐱^d∈𝒜_d.Recall f∈𝒮^3 and let W̃^d_1 be as in Proposition <ref>, satisfying f'(W̃^d_1)(Y^d_1-x^d_1)=f(Y^d_1)-f(x^d_1). Let C>0 be such that T^d_1(𝐱^d)+T^d_2(𝐱^d)≤ Cd^-1 for all 𝐱^d∈𝒜_d. The bound in (<ref>), Taylor's theorem applied to f and Cauchy's inequality yield:|𝒢_df(𝐱^d)-𝒢̃_df(𝐱^d)|≤ d 𝔼_Y^d_1[|f(Y^d_1)-f(x^d_1)| (|log(ρ)”(W^d_1)|(Y^d_1-x^d_1)^2+Cd^-1)] =d𝔼_Y^d_1[|f'(W̃^d_1)log(ρ)”(W^d_1)(Y^d_1-x^d_1)^3|]+C𝔼_Y^d_1[|f'(W̃^d_1)(Y^d_1-x^d_1)|] ≤ d(𝔼_Y^d_1[|f'(W̃^d_1)^2(Y^d_1-x^d_1)^3|]𝔼_Y^d_1[|(log(ρ)”)^2(W^d_1)(Y^d_1-x^d_1)^3|])^1/2+C𝔼_Y^d_1[|f'(W̃^d_1)(Y^d_1-x^d_1)|]≤C̅ d(f'_∞,1/2^2 e^|x_1^d| d^-3/2·e^|x_1^d|d^-3/2)^1/2 + C̅f'_∞,1/2e^|x_1^d| d^-1/2.The last inequality follows by three applications ofProposition <ref>, where C̅>0 is a constant that does not depend onfor 𝐱^d∈𝒜_d. This concludes the proof of the proposition.Before tackling the proof of Proposition <ref>, we need the following three lemmas. Recall thatK(x,Y) is defined in (<ref>) and the set A satisfies the conclusion of Proposition <ref>. Pick x∈ A and let Y∼ N(x,l^2/d) for some constant l>0. Then K(x,Y) has a density q_x satisfying q_x_∞≤ 4c_A√(d)/(3l√(2π)).Existence of q_x follows from (<ref>) and Proposition <ref>. By Proposition <ref> we have |log(ρ)”(x)|<(log(ρ)'(x))^2 and c_A>|log(ρ)'(x)|>1/c_A. Consider thepolynomial y↦ p(y):=log(ρ)'(x)y+log(ρ)”(x)y^2/2+(log(ρ)'(x))^3y^3/3. By (<ref>) it holds p(Y-x)=K(x,Y). Sincep'(y)=log(ρ)”(x)y+log(ρ)'(x)(1+log(ρ)'(x)^2y^2), we have|p'(y)|≥ |log(ρ)'(x)|(1+log(ρ)'(x)^2y^2) -|log(ρ)”(x)||y|> |log(ρ)'(x)|(1-|log(ρ)'(x)y|+|log(ρ)'(x)y|^2)> 3/4c_A,wherethe second inequality holds since|log(ρ)”(x)|<(log(ρ)'(x))^2 and the third follows frominf_z∈{1-|z|+z^2}=3/4 and |log(ρ)'(x)|>1/c_A. The lemma now follows by Proposition <ref>. Recall that the proposal is normal 𝐘^d=(Y_1^d,…,Y_d^d)∼ N(𝐱^d,l^2/d· I_d). For any 𝐱^d∈𝒜_d, the sum∑_k=2^dK(x_i^d,Y_i^d) possesses a density 𝐪^d_𝐱^d.Moreover, there exists a constant C_K such that 𝐪^d_𝐱^d_∞≤ C_K holds for all d∈ and all 𝐱^d∈𝒜_d.Fix 𝐱^d∈𝒜_d and, for each i, let q_i denote the density of K(x^d_i,Y^d_i) as in the previous lemma.Since the components of𝐘^d are IID,we have 𝐪^d_𝐱^d=_i=2^d q_i=q_Aq_∖ A, where q_A:= _x^d_i∈ Aq_i and q_∖ A:=_x^d_i∉ Aq_i. By the definition of convolution and the fact thatq_∖ A is a density,it follows that𝐪^d_𝐱^d_∞≤q_A_∞q_∖ A_1=q_A_∞. By Lemma <ref>there exists C>0 such that,for any d∈, it holdsq_i_∞<C√(d) if x^d_i∈ A. Condition (<ref>) implies there are at least(d-1)ρ(A)/2factors in the convolution q_A=_x^d_i∈ Aq_i. Hence Proposition <ref> applied to q_A yields 𝐪^d_𝐱^d_∞≤q_A_∞≤ cC√(d)/√((d-1)ρ(A)/2). This concludes the proof of the lemma. Let 𝐱^d∈𝒜_d. The function y↦β(𝐱^d,y), defined in (<ref>),is in 𝒞^2() and the following holds: * 0<β(𝐱^d,y)≤1 for all y∈;* β(𝐱^d,x^d_1)=𝔼_𝐘^d-[1∧ e^∑_i=2^dK(x^d_i,Y^d_i)];* |∂/∂ yβ(𝐱^d,y)|≤|log(ρ)'(x^d_1)| for all y∈;* ∂/∂ yβ(𝐱^d,x^d_1)=log(ρ)'(x^d_1)𝔼_𝐘^d-[e^∑_i=2^dK(x^d_i,Y^d_i)1_{∑_i=2^dK(x^d_i,Y^d_i)<0}];* |∂^2/∂ y^2β(𝐱^d,y)|≤|log(ρ)'(x^d_1)|^2(C_K+1) for all y∈ and constant C_K from Lemma <ref>. (i) and (ii) follow from the definition in (<ref>).Since x↦ 1∧ e^x is Lipschitz (with Lipschitz constant 1) on ,the family of functions {x↦ (1∧ e^x+h- 1∧ e^x)/h; h∈∖{0}} is bounded by one and converges pointwise to 1_{x<0}e^x for all x∈∖{0}, as h→0. Hence the DCT implies that∂/∂ yβ(𝐱^d,y)exists and can be expressed as log(ρ)'(x^d_1)𝔼_𝐘^d-[e^log(ρ)'(x^d_1)(y-x^d_1) +∑_i=2^dK(x^d_i,Y^d_i)1_{log(ρ)'(x^d_1)(y-x^d_1)+∑_i=2^dK(x^d_i,Y^d_i)<0}],implying (iii) and (iv).Let Φ^d_K denote the distribution of ∑_i=2^dK(x^d_i,Y^d_i) and recall that by definition we have e^x 1_{x<0} = 1∧ e^x - 1_{x≥ 0} for all x∈. Hence, by (<ref>), it follows ∂/∂ yβ(𝐱^d,y) =log(ρ)'(x^d_1)(β(𝐱^d,y)-1+Φ^d_K(-log(ρ)'(x^d_1)(y-x^d_1))),By Lemma <ref>, Φ_K^d is differentiable. Hence, by (<ref>),∂^2/∂ y^2β(𝐱^d,y) also exists and takes the form: (log(ρ)'(x^d_1))^2(β(𝐱^d,y)-1 +Φ^d_K(-log(ρ)'(x^d_1)(y-x^d_1))-𝐪^d_𝐱^d(-log(ρ)'(x^d_1)(y-x^d_1))).Part (v) follows from this representation of ∂^2/∂ y^2β(𝐱^d,y) and Lemma <ref>.Fix an arbitrary 𝐱^d∈𝒜_d.LetZ_1,W_1 be random variables, as in Proposition <ref>, that satisfyf(Y^d_1)-f(x^d_1) = f'(x^d_1)(Y^d_1-x^d_1)+f”(x^d_1)/2(Y^d_1-x^d_1)^2+f”'(Z_1)/6(Y^d_1-x^d_1)^3,β(𝐱^d,Y^d_1) =β(𝐱^d,x^d_1)+∂/∂ yβ(𝐱^d,x^d_1)(Y^d_1-x^d_1)+∂^2/∂ y^2β(𝐱^d,W_1)/2(Y^d_1-x^d_1)^2.Then, by the definition of𝒢̃_df(𝐱^d) in (<ref>) and the fact Y_1^d-x_1^d∼ N(0,l^2/d), we find 𝒢̃_df(𝐱^d) = l^2f”(x^d_1)/2β(𝐱^d,x^d_1)+l^2f'(x^d_j)∂/∂ yβ(𝐱^d,x^d_1)+ d𝔼_Y^d_1[(β(𝐱^d,x^d_1)f”'(Z_1)/6+f'(x^d_1)∂^2/∂ y^2β(𝐱^d,W_1)/2)(Y^d_1-x^d_1)^3]+ d𝔼_Y^d_1[(f”(x^d_1)/2∂^2/∂ y^2β(𝐱^d,W_1)/2+f”'(Z_1)/6∂/∂ yβ(𝐱^d,x^d_1))(Y^d_1-x^d_1)^4]+ d𝔼_Y^d_1[f”'(Z_1)/6∂^2/∂ y^2β(𝐱^d,W_1)/2(Y^d_1-x^d_1)^5].By parts (ii) and (iv) in Lemma <ref> and the definition of 𝒢̂_df(𝐱^d) in (<ref>) we have 𝒢̂_df(𝐱^d)= l^2f”(x^d_1)/2β(𝐱^d,x^d_1)+l^2f'(x^d_j)∂/∂ yβ(𝐱^d,x^d_1).The three expectations in the display above can each be bounded by a constant times (∑_i=1^3f^(i)_∞,1/2)e^|x^d_1|d^-1/2 using Proposition <ref> and Lemma <ref>. For instance, the first expectation can be bounded above using(v) in Lemma <ref>: d/6𝔼_Y^d_1[|f”'(Z_1)||Y^d_1-x^d_1|^3]+d(C_K+1)/2|f'(x^d_1)||log(ρ'(x^d_1))|^2𝔼_Y^d_1[|Y^d_1-x^d_1|^3].Proposition <ref> yieldsd/6𝔼_Y^d_1[|f”'(Z_1)||Y^d_1-x^d_1|^3] ≤ C_0 e^|x^d_1|f”'_∞,1 d^-1/2≤ C_0 e^|x^d_1|f”'_∞,1/2 d^-1/2 for some C_0>0.Moreover, |f'(x^d_1)||log(ρ'(x^d_1))|^2≤f'_∞,1/2(log(ρ)')^2_∞,1/2e^|x^d_1| as log(ρ)∈𝒮^4 and f∈𝒮^3. Hence d(C_K+1)/2|f'(x^d_1)||log(ρ'(x^d_1))|^2𝔼_Y^d_1[|Y^d_1-x^d_1|^3] ≤ C_1 e^|x^d_1|f'_∞,1/2 d^-1/2 for some C_1>0.Similarly, it follows that the second and third expectations above decay as d^-1 and d^-3/2, respectively. This concludes the proof of the proposition. Recall that 𝒩(𝐱^d,𝐘^d) and ∑_i=2^dK(x^d_i,Y^d_i) are defined in (<ref>) and (<ref>), respectively. Then there exists a constant C such that for all d∈ we have:𝔼_𝐘^d-[|∑_i=2^dK(x^d_i,Y^d_i)-𝒩(𝐱^d,𝐘^d)|]≤ Cd^-1/2∀𝐱^d∈𝒜_d.The difference in question is smaller than the sum of the following two terms:T^d_3(𝐱^d) = 𝔼_𝐘^d-[|∑_i=2^dlog(ρ)”(x^d_i)/2((Y^d_i-x^d_i)^2-l^2/d)|],T^d_4(𝐱^d) = 𝔼_𝐘^d-[|∑_i=2^d(log(ρ)'(x^d_i))^31_A(x^d_i)/3(Y^d_i-x^d_i)^3|].Note that X_i:=(Y^d_i-x^d_i)^2-l^2/d, 2≤ i≤ d, are zero mean IID with 𝔼[X_i^4] = 2l^4/d^2. Hence, as log(ρ)∈𝒮^4, we may apply Proposition <ref> with the function x↦log(ρ)”(x)and X_i,2≤ i≤ d, to getT^d_3(𝐱^d)≤log(ρ)”_∞,1/2(2(l^4/d^2)∑_i=2^d e^|x^d_i|)^1/2≤ C_0 d^-1/2 for some constant C_0>0, where the second inequality follows from (<ref>).Similarly, Proposition <ref> and assumption (<ref>), applied tothe function x↦ (log(ρ)'(x))^31_A(x) and random variables Y^d_i-x^d_i, yield T^d_4(𝐱^d)≤ C_1 (∑_i=2^d e^|x^d_i|/d^3)^1/2≤ C_2 d^-1 for some constants C_1,C_2>0 and all d∈. There exist constants c_1,c'_1>0, such that for any d∈, i∈{2,⋯,d}, 𝐱^d∈𝒜_d andx^d_i∉ A, it holds |logφ_i(t)-(il^2/2dlog(ρ)”(x^d_i)t-l^2/2d(log(ρ)'(x^d_i))^2t^2)|≤c_1/d^3/2(t^2+|t|^3), |t|≤ c'_1√(d),where φ_i(t):=𝔼_Y_i^d[exp(itK(x^d_i,Y^d_i))], t∈, is the CF of K(x^d_i,Y^d_i) (cf. (<ref>)).Recall that the set A satisfies the conclusion of Proposition <ref>.The proof ofLemma <ref> requires the control of the functions log(ρ)' and log(ρ)” on the complement of A, where they are unbounded. It is hence crucial that their argument x_i^d isthe i-th coordinate of a point 𝐱^d∈𝒜_d,since, through assumption (<ref>), we have control over the size of x_i^d in terms of the dimension d of the chain.For an analogous reason we need i>1. These facts plays a key role in the proof. By Lemma <ref>, the following inequality holds for all |t|≤ d/(4l^2|log(ρ)”(x^d_i)|):|logφ_i(t)-(il^2/2dlog(ρ)”(x^d_i)t-l^2/2d(log(ρ)'(x^d_i))^2t^2)| ≤l^4/d^2(1/2(log(ρ)”(x^d_i))^2t^2+(log(ρ)'(x^d_i))^2|log(ρ)”(x^d_i)||t|^3),Since 𝐱^d∈𝒜_d and i>1, assumption (<ref>) implies thatfor anyf∈𝒮^0 there exists C_f>0 such that|f(x^d_i)|^2/C_f≤ e^|x^d_i|≤∑_i=2^de^|x^d_i|≤ 2dρ(e^|x|).Hence |f(x^d_i)|≤ c_f √(d) for all d∈ and all 2≤ i≤ d, where c_f:=(2C_fρ(e^|x|))^1/2. Since log(ρ)∈𝒮^4, both functions f_1(x):=l^4(log(ρ)”(x))^2/2andf_2(x):=(log(ρ)'(x))^2|log(ρ)”(x)|are in 𝒮^0.Then (<ref>) and the constants c_1:=max{c_f_1,c_f_2}and c_1':=1/(4√(2c_f_1)) yield the inequalities in the lemma.We now deal with the coordinates of 𝒜_d that are in A. Compared toLemma <ref>, this is straightforward as it does not involve the remainder of the coordinatesof the point in 𝒜_d.If x∈ A, then K(x,Y) (cf. (<ref>)), where Y∼ N(x,l^2/d), satisfies:* μ_K:=𝔼_Y[K(x,Y)]≤l^2c_A^2/2d, where c_A>0 is the constant in Proposition <ref>;* |𝔼_Y[(K(x,Y)-μ_K)^2]- (log(ρ)'(x))^2l^2/d| ≤ C_1 d^-2 for someconstant C_1>0 and all d∈;* 𝔼_Y[|K(x,Y)-μ_K|^3]≤ C_2d^-3/2 for some constant C_2>0 and all d∈.Moreover, the constants C_1 and C_2 do not depend on the choice of x∈ A.By definition of A in Proposition <ref> we have |log(ρ)'(x)|≤ c_A and |log(ρ)”(x)|≤ c^2_A for x∈ A.By (<ref>), μ_K=l^2/2dlog(ρ)”(x) and (a) follows. Recall 𝔼_Y[(Y-x)^n] is either zero (if n is odd) or of order d^-n/2 (if n is even) and 𝔼_Y[(Y-x)^2]=l^2/d. Hence the definition of K in (<ref>), the fact x∈ A and part (a) imply the inequality inpart (b). For part (c), note that an analogous argument yields 𝔼_Y[(K(x,Y)-μ_K)^6]≤ C'd^-3 for some constant C'>0. Cauchy's inequality concludes the proof of the lemma. Let assumptions of Lemma <ref> holdand denote by φ the characteristic function of K(x,Y). There exist positive constants c_2 and c'_2, such that the following holds for all x∈ A:|logφ(t)-(itl^2/2dlog(ρ)”(x)-t^2l^2/2dlog(ρ)'(x)^2)|≤ c_2(t^2/d^2+|t|^3/d^3/2+t^4/d^2), |t|≤ c'_2√(d). Let σ^2_K:=𝔼_Y[(K(x,Y)-μ_K)^2] and recall μ_K=l^2/2dlog(ρ)”(x).By Lemma <ref> we have|logφ(t)-(itl^2/2dlog(ρ)”(x)-t^2/2σ^2_K)| ≤|t|^3𝔼_Y|K(x,Y)-μ_K|^3/6+t^4σ^4_K/4, |t|≤1/σ_K.ByLemma <ref>(b) wehave|σ^2_K-l^2log(ρ)'(x)^2/d|≤ C_1d^-2. Henceσ^2_K≤ d^-1/√(c_2'), where c_2':= 1/(l^2c_A^2+ C_1)^2, and σ^4_K≤ C_1'd^-2 for some C_1'>0. This, together with Lemma <ref>(c), implies that there existsa constant c_2>0,such that the inequality in (<ref>) follows from (<ref>) for all |t|≤ c_2' d^1/2≤ 1/σ_K and x∈ A. For any d∈ and 𝐱^d∈𝒜_d, letΦ^d_K and Φ^d_𝒩 be the distribution functions of ∑_i=2^dK(x^d_i,Y^d_i) and 𝒩(𝐱^d,𝐘^d). Then there exists C>0 such that sup_x∈|Φ^d_𝒩(x)-Φ^d_K(x)|≤ C d^-1/2for every d∈ and 𝐱^d∈𝒜_d. Let φ_K and φ_𝒩be the CFs of ∑_i=2^dK(x^d_i,Y^d_i) and 𝒩(𝐱^d,𝐘^d), respectively.We will compare φ_K and φ_𝒩and apply Proposition <ref> to establish the lemma.Let φ_i be the CF of K(x^d_i,Y^d_i) and recall, by (<ref>),φ_𝒩(t)=exp(1/2itl^2/d∑_i=2^dlog(ρ)”(x^d_i)-1/2t^2l^2/d∑_i=2^d(log(ρ)'(x^d_i))^2). Define the positive constants c:=max{c_1,c_2} and c':=min{1,l^2J/(32c),c'_1,c'_2}, wherethe constants c_1,c_1' (resp. c_2,c_2') are given in Lemma <ref>(resp. Lemma <ref>) and J is as in assumption (<ref>).Note that the constants c,c' do not depend on the choice of 𝐱^d∈𝒜_d. Lemmas <ref> and <ref> imply the following inequalityfor all d∈ and𝐱^d∈𝒜_d:|logφ_K(t)-logφ_𝒩(t)|≤∑_i=2^d|logφ_i(t)-(itl^2/2dlog(ρ)”(x^d_i)-t^2l^2/2d(log(ρ)(x^d_i)^2)| ≤ R(t),for all |t|≤ r, where r:= c'√(d) and R(t):= c(t^2+|t|^3+t^4/√(d))/√(d). Since |t|^3≤√(d)c't^2 and t^4≤ dc'^2t^2 for|t|≤ r, we have R(t)≤ t^2(c/√(d)+cc'+cc'^2) ≤ t^2 (c/√(d)+2cc')for all t∈[-r,r]. By assumption (<ref>), there exists d_0'∈ such thatthe varianceσ^2_𝒩(𝐱^d)=l^2/d∑_i=2^d(log(ρ)'(x^d_i))^2 of𝒩(𝐱^d,𝐘^d)satisfies σ^2_𝒩(𝐱^d)≥ l^2J/2 for alld≥ d_0' and𝐱^d∈𝒜_d.Let γ:=1/2 and pickd_0∈, greater thanmax{d_0',(16cl^-2/J)^2}. Then, for any d≥ d_0, the inequality c/√(d)≤γ l^2J/8 holds. Since c'≤ l^2J/(32c), we have2cc'≤γ l^2J/8,and the bound in (<ref>) implies R(t)≤1/2t^2 γ l^2J/2 ≤1/2t^2 γσ^2_𝒩(𝐱^d) for all t∈[-r,r]. By Proposition <ref>,for all d≥ d_0, sup_x∈|Φ^d_𝒩(x)-Φ^d_K(x)| is bounded aboveby∫_R(t)/π |t|exp(-(1-γ)σ^2_𝒩(𝐱^d)t^2/2)dt +12√(2)/π^3/2σ_𝒩(𝐱^d) r≤ C'/√(d),where C':= c∫_(|t|+t^2+|t|^3)exp(-l^2Jt^2/8)dt + 24√(2)/π^3/2l^2Jc'. Since the left-hand side of the inequality in the lemma is bounded above by 1, the inequality holds for all d∈ if we defineC:=max{C',√(d_0)}. Since |1∧ e^y- 1∧ e^x|≤ |x-y| for all x,y∈, by Lemma <ref> we have |𝔼_𝐘^d-[1∧ e^𝒩(𝐱^d,𝐘^d)]-𝔼_𝐘^d-[1∧ e^∑_i=2^dK(x^d_i,Y^d_i)]|≤ C' d^-1/2for some constant C'>0 and all d∈. Recall e^x 1_{x<0} = 1∧ e^x - 1 + 1_{x≤ 0} for all x∈∖{0}. Hence Lemmas <ref> and <ref> yield|𝔼_𝐘^d-[e^𝒩(𝐱^d,𝐘^d)1_{𝒩(𝐱^d,𝐘^d)<0}]-𝔼_𝐘^d-[e^∑_i=2^dK(x^d_i,Y^d_i)1_{∑_i=2^dK(x^d_i,Y^d_i)<0}]| ≤|𝔼_𝐘^d-[1∧ e^𝒩(𝐱^d,𝐘^d)]-𝔼_𝐘^d-[1∧ e^∑_i=2^dK(x^d_i,Y^d_i)]|+|Φ^d_𝒩(0)-Φ^d_K(0)|≤ C” d^-1/2for some C”>0 and all d∈. The proposition follows. For any 𝐱^d∈𝒜_d, by <cit.>, we have𝔼_𝐘^d-[e^𝒩(𝐱^d,𝐘^d)1_{𝒩(𝐱^d,𝐘^d)<0}]=e^μ_𝒩(𝐱^d)+σ^2_𝒩(𝐱^d)/2Φ(-σ_𝒩(𝐱^d)-μ_𝒩(𝐱^d)/σ_𝒩(𝐱^d)), 𝔼_𝐘^d-[1∧ e^𝒩(𝐱^d,𝐘^d)]= 𝔼_𝐘^d-[e^𝒩(𝐱^d,𝐘^d)1_{𝒩(𝐱^d,𝐘^d)<0}] + Φ(μ_𝒩(𝐱^d)/σ_𝒩(𝐱^d)).where Φ is the distribution of a standard normal random variable. Note first that it is sufficient to prove the inequality in the proposition for all d> d_0 for some d_0∈, since the expectations above are bounded by 1 and we can hence increase the constant C so thatthe first d_0 inequalities are also satisfied. Recall the formulas for μ_𝒩(𝐱^d) and σ^2_𝒩(𝐱^d) from (<ref>). By assumptions (<ref>) and (<ref>) it follows that|μ_𝒩(𝐱^d)+σ^2_𝒩(𝐱^d)/2|≤ c a_d/√(d) for some constant c>0and alllarge d and 𝐱^d∈𝒜_d.Note that S_a:=sup_d∈(a_d/√(d))<∞ since {a_d}_d∈ is sluggish.The function x↦ e^x is Lipschitz on [-cS_a,cS_a] with constant e^cS_a.Consequently |e^μ_𝒩(𝐱^d)+σ^2_𝒩(𝐱^d)/2-1|≤ e^cS_aa_d/√(d)for large d and uniformly in 𝐱^d∈𝒜_d.By assumption (<ref>),for all larged∈ and all 𝐱^d∈𝒜_d, we have σ_𝒩(𝐱^d)≥ l√(J)/√(2).Hence, since the function x↦√(x) is Lipschitz with constant c_1:=1/(l√(2J)) on [l^2J/2,∞), we get |σ_𝒩(𝐱^d)/2-l√(J)/2|≤ (c_1/2) |σ^2_𝒩(𝐱^d)-l^2J| ≤ c_2 a_d/√(d), where constant c_2>0 exists by (<ref>).Moreover, |(μ_𝒩(𝐱^d)+σ^2_𝒩(𝐱^d)/2)/σ_𝒩(𝐱^d)|≤ c_3 a_d/√(d)for c_3>0 and all large d. Since σ_𝒩(𝐱^d)+μ_𝒩(𝐱^d)/σ_𝒩(𝐱^d) =(μ_𝒩(𝐱^d)+ σ^2_𝒩(𝐱^d)/2)/σ_𝒩(𝐱^d)+σ_𝒩(𝐱^d)/2, the inequalities in the previous paragraph imply that there exists c_4>0 such that|σ_𝒩(𝐱^d)+μ_𝒩(𝐱^d)/σ_𝒩(𝐱^d) - l√(J)/2|≤ c_4 a_d/√(d) for large d and uniformly in 𝐱^d∈𝒜_d. Since Φ is Lipschitz with constant 1/√(2π), there exists a constant C_1'>1,such that |𝔼_𝐘^d-[e^𝒩(𝐱^d,𝐘^d)1_{𝒩(𝐱^d,𝐘^d)<0}]-Φ(-l√(J)/2)|≤ C_1'a_d/√(d)holdsforall large d and all𝐱^d∈𝒜_d.Similarly,|𝔼_𝐘^d-[1∧ e^𝒩(𝐱^d,𝐘^d)] -2Φ(-l√(J)/2)|≤ C_2'a_d/√(d) for some C_2'>0 all large d and all𝐱^d∈𝒜_d,and the proposition follows. §.§ Proof of Proposition <ref> We will now prove the following result. Let a={a_d}_d∈ be a sluggish sequence and p∈[1,∞). There exists a constant C_4 (depending on a and p) such that for every f∈𝒮^3 and all d∈ we have:𝒢f-𝒢_df_p≤ C_4(∑_i=1^3f^(i)_∞,1/2) (a_d/√(d) +e^-a_d^2/p).In the case p=2, define a_d:=√(2log(d)) for d∈∖{1} and note that Proposition <ref> thenfollows as a special case of Proposition <ref>.There exists a constant C such that for all f∈𝒮^3 and all d∈ we have:max{|𝒢f(𝐱^d)|,|𝒢_df(𝐱^d)|}≤ C e^|x^d_1|∑_i=1^2f^(i)_∞,1/2∀𝐱^d∈^d. The triangle inequality, definition (<ref>) and log(ρ)'(x)f'(x)≤log(ρ)'_∞,1/2f'_∞,1/2e^|x| imply the bound in the lemma for |𝒢f(𝐱^d)|. To bound |𝒢_df(𝐱^d)|, defineβ̃(𝐱^d,y):=𝔼_𝐘^d-[1∧exp(log(ρ)(y)-log(ρ)(x^d_1)+∑_i=2^dlog(ρ)(Y^d_i)-log(ρ)(x^d_i))]for any y∈.Then, if q denotes the density of Y^d_1-x^d_1∼ N(0,l^2/d), we get |𝒢_df(𝐱^d)|=d| 𝔼_Y^d_1[(f(Y^d_1)-f(x^d_1))β̃(𝐱^d,Y^d_1)]|≤ d∫_0^∞z|f'(w_1)β̃(𝐱^d,x^d_1+z)-f'(w_2)β̃(𝐱^d,x^d_1-z)|q(z)dz,where w_1∈(x_1^d,x_1^d+z) and w_2∈(x_1^d-z,x_1^d) satisfy zf'(w_1)=f(x_1^d+z)-f(x_1^d) and -zf'(w_2)=f(x_1^d-z)-f(x_1^d), respectively. Moreover,|f'(w_1)-f'(w_2)|≤ 2z|f”(w_3)|holds for some w_3 in the interval (x_1^d-z,x_1^d+z). Since x↦ 1∧ e^x is Lipschitz with constant 1, we get |β̃(𝐱^d,x^d_1+z)-β̃(𝐱^d,x^d_1-z)|≤|log(ρ)(x_1^d+z)-log(ρ)(x_1^d-z)| ≤ 2z|log(ρ)'(w_4)|for some w_4∈ (x_1^d-z,x_1^d+z). By adding and subtractingf'(w_2)β̃(𝐱^d,x^d_1+z) on the right-hand side of (<ref>), applying the two bounds we just derived and noting that β̃≤ 1,we get|𝒢_df(𝐱^d)| ≤2d∫_0^∞ z^2|f”(w_3)|q(z)dz+ 2d∫_0^∞ z^2|f'(w_2)log(ρ)'(w_4)|q(z)dz.Note that,since max{|w_3|,|w_2|,|w_4|}≤ |x_1^d|+z and f”_∞,1≤f”_∞,1/2, we have|f”(w_3)|≤f”_∞,1 e^|w_3|≤f”_∞,1/2 e^|x_1^d|+z,log(ρ)'(w_4)f'(w_2)≤log(ρ)'_∞,1/2f'_∞,1/2e^|x_1^d|+z,which, together with inequality (<ref>), implies the lemma.By Theorem <ref> (on 𝒜_d) and Lemma <ref> (on ^d∖𝒜_d), there exists a constantC>0 such that for anyf∈𝒮^3 the following inequality holds:𝒢_df-𝒢f^p_p= ∫_𝒜_d|𝒢_df(𝐱^d)-𝒢f(𝐱^d)|^pρ_d(𝐱^d)d𝐱^d+ ∫_^d∖𝒜_d|𝒢_df(𝐱^d)-𝒢f(𝐱^d)|^pρ_d(𝐱^d)d𝐱^d ≤ Cρ(e^p|x|)(a_d^p/d^p/2ρ_d(𝒜_d)+ ρ_d(^d∖𝒜_d))(∑_i=1^3f^(i)_∞,1/2)^p.Apply Proposition <ref> and raise both sides of the inequality to the power 1/p to conclude the proof of the proposition.§.§ Proof of Theorem <ref> Assume that ρ is a strictly positive density in𝒞^1 and that (<ref>) holds. Then, for any d∈, the RWM chain {𝐗^d_n}_n∈ is V-uniformly ergodic with V:=1/√(ρ_d).The lemma follows from <cit.> if we prove that the target ρ_d satisfieslim_|𝐱^d|→∞𝐱^d/|𝐱^d|·∇log(ρ_d(𝐱^d))= lim_|𝐱^d|→∞∑_i=1^dx^d_i/|x^d_i|log(ρ)'(x^d_i)=-∞, lim inf_|𝐱^d|→∞ℙ_𝐘^d[ρ_d(𝐘^d)≥ρ_d(𝐱^d)]>0.Assumption (<ref>) implies that the expression x/|x|·log(ρ)'(x)is bounded above and takes arbitrarily large negative values as |x|→∞. This yields (<ref>), since |𝐱^d|→∞ implies that |x^d_i|→∞ holds for at least one i∈{1,…,d}.Condition (<ref>) states that the acceptance probability in the RWM chain is bounded away from zero sufficiently far from the origin. To prove this,recall that𝐘^d∼ N(𝐱^d,l^2/d· I_d) and define the set B(𝐱^d):={𝐲^d∈^dx^d_i/|x^d_i|·(y^d_i-x^d_i)∈(-2l/√(d),-l/√(d)) for all i≤ d},where we interpret x^d_i/|x^d_i|:=1 if x^d_i=0. Clearly inf_𝐱^d∈^dℙ_𝐘^d[B(𝐱^d) ]>0.We now prove that if |𝐱^d| is sufficiently large, then ρ_d(𝐲^d)≥ρ_d(𝐱^d) for all 𝐲^d∈ B(𝐱^d), which implies (<ref>).By (<ref>), far enough from zero, ρ is decreasing in a direction away from the origin. Therefore, there exists a compact interval K⊂ such that (-2l/√(d),2l/√(d))⊂ K and ρ(y)≥ρ(x) whenever x∉ K and x/|x|· (y-x)∈ (-2l/√(d),-l/√(d)).We claim thatfor every 𝐲^d∈ B(𝐱^d),the inequalityρ(y^d_i)/ρ(x^d_i)≥ (min_x∈ Kρ(x))/(max_x∈ρ(x))∈(0,1) holds.If x^d_i∈ K,then y^d_i∈ K and the inequality follows trivially.If x^d_i∉ K, then, by the definition of K,we have ρ(y^d_i)/ρ(x^d_i)≥ 1.This proves the claim.Hence, for 𝐲^d∈ B(𝐱^d)we haveρ_d(𝐲^d)/ρ_d(𝐱^d)≥(max_i≤ dρ(y^d_i)/ρ(x^d_i))·(min_x∈ Kρ(x)/max_x∈ρ(x))^d-1. We now prove that the ratio ρ(y^d_i)/ρ(x^d_i) takes arbitrarily large values as |x^d_i|→∞.To show this, pick 𝐲^d∈ B(𝐱^d)and assume the inequality y_i^d>x_i^d.Then x_i^d<0 andy_i^d-x_i^d>l/√(d). Moreover the following holds ρ(y^d_i)/ρ(x^d_i)=exp(log(ρ(y^d_i)/ρ(x^d_i)))≥ 1+∫_x^d_i^y^d_ilog(ρ)'(z)dz≥ 1+l/√(d)inf_z<x^d_i+2l/√(d)log(ρ)'(z)→∞as x_i^d→-∞ by (<ref>).This, together with (<ref>), implies (<ref>). The case y_i^d<x_i^d is analogous and the lemma follows.If a strictly positive ρsatisfies (<ref>) and log(ρ)∈𝒮^n_ρ and f∈𝒮^n_f for some integers n_ρ,n_f∈∪{0}, then the function f̂, defined in (<ref>), satisfies f̂∈𝒮^min(n_f+2,n_ρ+1).Clearly, if f∈𝒞^n_f and ρ∈𝒞^n_ρ and if ρ is strictly positive, then f̂∈𝒞^min(n_f+2,n_ρ+1).Pick s>0.The L'Hospital's rule implies:lim_x→∞f̂(x)/e^s|x|= 2/sh(l)lim_x→∞∫_-∞^xρ(y)(ρ(f)-f(y))dy/e^sxρ(x)=2/sh(l)lim_x→∞ρ(f)-f(x)/se^sx+e^sx(log(ρ))'(x).The last limit is zero by (<ref>). An analogous argument showslim_x→-∞f̂(x)/e^s|x|=0.Hence f̂_∞,s<∞ holds for all s>0.Since h(l)f̂'(x)/2=(∫_-∞^xρ(y)(ρ(f)-f(y))dy)/ρ(x),this argument implies that f̂'_∞,s<∞ holds for all s>0.Hence f̂∈𝒮^1.Proceed by induction: assume that for all k≤ n (where 1≤ n<min(n_f+2,n_ρ+1)) we have f̂^(k)_∞,s<∞ for any s>0. Pick an arbitrary u>0.By differentiating (<ref>) we obtainf̂^(n+1)=-∑_k=0^n-1n-1k(log(ρ))^(k+1)f̂^(n-k)+2/h(l)(ρ(f)-f)^(n-1).Since n≤min(n_ρ, n_f+1), the induction hypothesis implies f̂^(k)_∞,u/2<∞ for all 1≤ k≤ n. By assumption we havef^(n-1)_∞,u<∞ and(log(ρ))^(k)_∞,u/2<∞ for all 1≤ k≤ n. Hence f̂^(n+1)_∞,u<∞ holds for an arbitrary u>0 and the proposition follows. By Lemma <ref>, the RWM chain 𝐗^d with the transition kernelP_d is V-uniformly ergodic with V=ρ_d^-1/2.Moreover, by <cit.>[Prop. 2.1 and Thm 2.1], P_d defines a self-adjoint operator on {g∈ L^2(ρ_d)ρ_d(g)=0} with norm λ_d<1. Proposition <ref> impliesf̂∈𝒮^3, since by assumption we have f∈𝒮^1 and log(ρ)∈𝒮^4.By Remark <ref>(c) in Section <ref> below we have f̂^2∈𝒮^3. Since P_df̂ = (1/d)𝒢_df̂+f̂, Lemma <ref> implies that (P_df̂)^2(𝐱^d)≤ C_f̂ e^2 |x_1^d| for some positive constant C_f̂ and all 𝐱^d∈^d.Hence (<ref>) and the definition of V implythe inequalitymax{f̂^2,(P_df̂)^2}≤ cV for some constant c>0.Consequently, by <cit.>, the CLT for the chain𝐗^d and function f+dP_df̂-df̂ holds with some asymptotic variance σ̂^2_f,d.By <cit.> we can represent σ̂^2_f,din terms of a positive spectral measure E_d(dλ) associated with the function f-ρ(f)+dP_df̂-df̂=𝒢_df̂-𝒢f̂ asσ̂^2_f,d=∫_Λ_d1+λ/1-λE_d(dλ),where Λ_d⊂[-λ_d,λ_d] denotes the spectrum of the self-adjoint operator P_d acting on the Hilbert space {g∈ L^2(ρ_d)ρ_d(g)=0}.By the definition of the spectral measure E_d(dλ) we obtain We can boundσ̂^2_f,d≤1+λ_d/1-λ_d∫_Λ_dE_d(dλ)=1+λ_d/1-λ_dP_d(df̂)-df̂+f-ρ(f)^2_2≤2/1-λ_d𝒢_df̂-𝒢f̂^2_2.Finally, the result follows by Proposition <ref>.§ TECHNICAL RESULTSThe results in Section <ref> use the ideas of Berry-Esseen theory and large deviations as well as the optimal Young inequality, and do not depend on anything in this paper that precedes them.§.§ Bounds on the expectations of test functionsWe start with elementary observations.Recall that 𝒮^n, n∈∪{0}, is defined in (<ref>). The following statements hold. * If n≤ m, then 𝒮^m⊂𝒮^n.* For n∈, f∈𝒮^n if and only if f'∈𝒮^n-1.* If f∈𝒮^n and g∈𝒮^m then f+g, fg∈𝒮^min(n,m).Pick an arbitrary n∈. Assume f∈𝒮^n, k≤ n, x∈ and Y∼ N(x,σ^2).Then there exists measurable Z satisfying f^(k)(Z)(Y-x)^k/k!=f(Y)-∑_i=0^k-1f^(i)(x)(Y-x)^i/i! and|Z-x|<|Y-x|. Furthermore there exists a constant C>0 (depending on n) such that, for any m∈ and s>0 we have𝔼_Y[|f^(k)(Z)|^m|Y-x|^n]≤ Ce^s^2σ^2𝔼_Y[|Y-x|^n]f^(k)_∞,s/m^me^s|x|. A random variable Z, defined via the integral form of the remainder in Taylor's theorem, lies a.s. between Y and x, implying|Z-x|<|Y-x|.Cauchy's inequality yields 𝔼_Y[|f^(k)(Z)|^m|Y-x|^n]^2≤𝔼_Y[|f^(k)(Z)|^2m]𝔼_Y[|Y-x|^2n].Since f∈𝒮^n⊂𝒮^k, we have sup_x∈|f^(k)(x)|^2me^-2s|x|=f^(k)^2m_∞,s/m<∞. As Y∼ N(x,σ^2), the equality𝔼_Y[|Y-x|^2n]= C^2 𝔼_Y[|Y-x|^n]^2 holds, where C:=(2√(π)Γ((2n+1)/2))^1/2/Γ((n+1)/2)and Γ(·) is the Euler gamma function.Hence, by (<ref>), we get𝔼_Y[|f^(k)(Z)|^m|Y-x|^n]≤C/√(2)f^(k)^m_∞,s/m√(𝔼_Y[e^2s|Z|])𝔼_Y[|Y-x|^n].It remains to note𝔼_Ye^2s(|Z|-|x|)≤𝔼_Ye^2s|Z-x|≤𝔼_Ye^2s|Y-x|≤2 𝔼_Ye^2s(Y-x)= 2e^2s^2σ^2.Let f→ be a measurable (not necessary continuous) function such that f_∞,1/2<∞. Fix n∈, 𝐱^d∈^d and let X_1, X_2…, X_d be IID copies of X, satisfying 𝔼[X^n]=0 and 𝔼[X^2n]<∞. Then the following inequality holds: |𝔼[∑_i=1^df(x^d_i)X_i^n]|≤f_∞,1/2(𝔼[X^2n]∑_i=1^de^|x^d_i|)^1/2. Note that the assumptions of Proposition <ref> imply that, if X is a non-zero random variable, thenn∈ has to be odd. By Jensen's inequality, the fact that 𝔼[X]=0 and the assumption on f we get 𝔼[∑_i=1^df(x^d_i)X_i^n]^2≤𝔼[(∑_i=1^df(x^d_i)X_i^n)^2]= ∑_i=1^d (f(x^d_i))^2𝔼[X_i^2n]≤f^2_∞,1/2𝔼[X^2n]∑_i=1^de^|x^d_i|. §.§ Deviations of the sums of IID random variables Let f∈𝒮^0 be such that ρ(f)=0 and let a={a_d}_d∈ be a sluggish sequence. If the random vector(X_1,d,…, X_d,d) follows the density ρ_d for all d∈, then for every t>0 the following inequality holds for all but finitely many d∈:ℙ_ρ_d[|1/d-1∑_i=2^df(X_i,d)|≥ta_d/√(d)]≤exp(-t^2a_d^2/(3ρ(f^2))). Proposition <ref> is an elementary consequence of a deeper underlying result, that the sequence of random variables {∑_i=1^df(X_i,d)/(a_d√(d))}_d∈satisfies a moderate deviation principle with a good rate function t↦ t^2/(2ρ(f^2)) and speed a_d^2 (see <cit.> for details). The key inequality needed in the proof ofProposition <ref> is given in the next lemma. Let assumptions of Proposition <ref> hold. If ρ(f^2)>0, then for every closed F⊆the following holds:lim sup_d→∞a^-2_dlogℙ_ρ_d[∑_i=1^df(X_i,d)/(a_d√(d))∈ F]≤ -inf{ x^2/(2ρ(f^2));x∈ F}.The moderate deviations results <cit.> yield a sufficient condition for the above inequality. More precisely, for X∼ρ,we need to establish: lim sup_d→∞ a_d^-2log(d·ℙ_ρ[|f(X)|≥ a_d√(d)])=-∞.Fix an arbitrary m∈. Since f∈𝒮^0, we have |f(x)|≤f_∞,1/m e^|x|/m for every x∈.Consequently, for all large d, we get ℙ_ρ[|f(X)|≥ a_d√(d)]≤ℙ_ρ[f_∞,1/m^m e^|X|≥ d^m/2]≤f_∞,1/m^m ρ(e^|X|)d^-m/2.Since {a_d}_d∈ is sluggish, ∃ C_0>0 such that a_d^-2log(f_∞,1/m^m ρ(e^|X|))<C_0< a_d^-2log(d) for all large d∈. Hencea_d^-2log (d·ℙ_ρ[|f(X)|≥ a_d√(d)])≤ a_d^-2(log(f_∞,1/m^m ρ(e^|X|))-(m/2-1)log(d))< -C_0(m/2-2),for all large d∈. Since m was arbitrary, (<ref>) follows. Note that the proposition holds if ρ(f^2)=0. Assume now ρ(f^2)>0 and fix an arbitrary t>0. Note that since {a_d}_d∈ is sluggish,so is {a'_d}_d∈, a'_d:=a_d+1√(d/(d+1)).Apply Lemma <ref> to F=∖ (-t,t) and{a'_d}_d∈ to get the following inequality ℙ_ρ_d-1[|∑_i=1^d-1f(X_i,d-1)/(a'_d-1√(d-1))|≥ t]≤exp(-3 (a'_d-1)^2t^2/(8ρ(f^2)))for all large enough d∈.Since 3(a'_d-1)^2/4≥ 2a^2_d/3for all but finitely many d∈, the right-hand side in (<ref>) is bounded above by exp(-(a_d)^2t^2/(3ρ(f^2))).Recall ρ_d(𝐱^d)=ρ_d-1(𝐱^d-1)ρ(x_d^d) and a_d-1'√(d-1)=a_d(d-1)/√(d). Hence the left-hand side in inequality (<ref>) equals ℙ_ρ_d[|∑_i=2^df(X_i,d)/(d-1)|≥ ta_d/√(d)] and the proposition follows.The next result is based on a combinatorial argument. A special caseof Proposition <ref> was used in <cit.>. Let n∈ and a measurable f→ satisfy ρ(f)=0 and ρ(f^2n)< ∞.If the random vector (X_1,d,…,X_d,d) is distributed according to ρ_d, then there exists a constant C, independent of d, such that ℙ_ρ_d[|1/d-1∑_i=2^df(X_i,d)|≥ 1]≤ C d^-n.The constant C in Proposition <ref> may depend on n∈ and the function f.Fix n∈ and let _0:=∪{0}. Markov's inequality and the Multinomial theorem yield: ℙ_ρ_d[|1/d-1∑_i=2^df(X_i,d)|≥ 1]=ℙ_ρ_d[|1/d-1∑_i=2^d f(X_i,d)|^2n≥ 1]≤𝔼_ρ_d(1/d-1∑_i=2^d f(X_i,d))^2n =(d-1)^-2n∑_k_2+k_3+⋯ +k_d=2n k_2,k_3…,k_d∈_0∖{1}2nk_2,k_3,…,k_d∏_i=2^d𝔼_ρ[f(X_i,d)^k_i],where last equality holds, because the expectation of any summand of the form ∏_i=2^df(X_i,d)^k_i is zero if any of the indices k_i=1 since ρ_d has a product structure and ρ(f)=0. By Jensen's inequality,∏_i=2^d𝔼_ρ[f(X_i,d)^k_i]≤∏_i=2^d𝔼_ρ[f(X_i,d)^2n]^k_i/2n=ρ(f^2n) ^∑_i=2^dk_i/2n=ρ(f^2n), and henceℙ_ρ_d[|1/d-1∑_i=2^df(X_i,d)|≥ 1]≤ (2n)!·ρ(f^2n)(d-1)^-2n·|𝒩_d|,where |𝒩_d| stands for the cardinality of the set 𝒩_d:={(k_2,k_3,…, k_d)∈^d-1_0;∑_i=2^dk_d=2n andk_i≠ 1 for all 2≤ i≤ d}.Inequality (<ref>) and the next Claim prove the proposition. Claim.|𝒩_d|≤ C' d^n for a constant C' independent of d. Proof of Claim. Consider a function ζ^d-1_0→^d-1_0,ζ(a_2,a_3,…, a_d):=(2⌊a_2/2⌋,2⌊a_3/2⌋,… 2⌊a_d/2⌋ ), that rounds each entry down to the nearest even number. Every element in the image ζ(𝒩_d) is a (d-1)-tuple of non-negative even integers with sum at most 2n. Recall the number of k-combinations with repetition, chosen from a set of d-1 objects, equals k+d-2k. There exists C”>0, such that|ζ(𝒩_d)| ≤ |{(k_2,k_3,…, k_d)∈^d-1_0; ∑_i=2^dk_d≤ n}|= ∑_k=0^n|{(k_2,k_3,…, k_d)∈^d-1_0;∑_i=2^dk_d=k}|=∑_k=0^nk+d-2k≤ C” d^n. Note that the pre-image of a singleton under ζ contains at most 2^n elements (i.e. (d-1)-tuples) of 𝒩_d.Indeed, by the definition of 𝒩_d, at most n coordinates of an element are not zero and each can either reduce by one or stay the same. Hence, for C':=C” 2^n, we have |𝒩_d|≤ 2^n·|ζ(𝒩_d)|≤ C'd^n. §.§ Bounds on the densities of certain random variables The key step in the proof of Proposition <ref> below isthe optimal Young's inequality: for p,q≥ 1 and r∈[1,∞], such that1/p+1/q=1+1/r,and functions f∈ L^p() and g∈ L^q(), their convolutionf*g satisfies the inequalityf*g_r≤C_pC_q/C_rf_pg_q,whereC_s:=√(s^1/s/s'^1/s'),if s∈(1,∞) and 1/s+1/s'=1,1, if s∈{1,∞}.For s∈[1,∞), ·_s is the usual norm on L^s() and ·_∞ denotes the essential supremum norm on L^∞().The proof of (<ref>) for r<∞ is given in <cit.>. In the case r=∞, we have C_pC_q/C_r=1 and the inequality in (<ref>) follows from the definition of the convolution, translation invariance of the Lebesgue measureand Hölder's inequality. Let X_1,X_2,…, X_d be independent random variables, each X_i with a bounded density q_i. The density Q_d of the sum ∑_i=1^d X_i satisfies Q_d_∞≤ cmax_i≤ dq_i_∞/√(d) for some constant c>0.The factor d^-1/2in the inequality of Proposition <ref> above comes from (<ref>) and is crucial for the analysis in this paper. The standard Young's inequality for convolutions would only yieldQ_d_∞≤ c max_i≤ dq_i_∞, which gives insufficient control over Q_d. Since random variables X_i are independent, the density of their sum is a convolution of the respective densities, Q_d=_i=1^d q_i. For all i and each t>1 we haveq_i∈ L^∞()∩ L^1()⊂ L^t().Moreover, the following inequality holdsfor every k≤ d-1:Q_d_∞=_i=1^d q_i_∞≤(C_d/d-1)^kC_d/k(∏_i=1^kq_i_d/d-1)_i=k+1^d q_i_d/k We prove (<ref>)by induction on k.For k=1, note thatd and d/d-1 are Hölder conjugates, i.e. 1/d +1/(d/(d-1))=1 Hence (<ref>) for r=∞, q=d, p=d/(d-1), f=q_1 andg=_i=2^d q_i implies Q_d_∞≤q_1_d/d-1_i=2^d q_i_dand C_d/d-1=C^-1_d. Now assume (<ref>) holds for some k≤ d-2.Since (d/(d-1))^-1+(d/(k+1))^-1=1+(d/k)^-1, the inequality in (<ref>) implies_i=k+1^d q_i_d/k≤C_d/d-1C_d/k+1/C_d/kq_k+1_d/d-1_i=k+2^d q_i_d/k+1.This inequality and the induction hypothesis (i.e. (<ref>) for k) implies (<ref>) for k+1.Since q_1 is a density, we have q_i_1=1. Hence we find q_i_d/d-1≤q_i_1^d-1/dq_i_∞^1/d=q_i_∞^1/d for each i, and in particular ∏_i=1^dq_i_d/d-1≤max_i≤ d(q_i_∞). By (<ref>) for k=d-1 we get Q_d_∞≤(C_d/d-1)^d∏_i=1^dq_i_d/d-1≤max_i≤ d(q_i_∞)(C_d/d-1)^d.Sincelim_d→∞√(d)(C_d/d-1)^d=√(e),there exists c>0 such that(C_d/d-1)^d≤ c/ √(d) for all d∈. Polynomials of continuous random variables play an important role in the proofs of Section <ref>. Let X be a continuous random variable and p a polynomial. Then the random variable p(X) has a density.The set B:=p((p')^-1({0})) has finitely many points. Moreover, p is locally invertible on ∖ B by the inverse function theorem and the inverses are differentiable. Hence, for any x∉ B, the set p^-1((-∞,x]) is a disjoint union of intervals with boundaries that depend smoothly on x. Since ℙ[p(X)≤ x]=ℙ[X∈ p^-1((-∞,x])], the proposition follows.Let N=N(μ,σ^2) be a normal random variable and p a polynomial satisfying inf_x∈|p'(x)|≥ c_p for some constant c_p>0. Then the random variable p(N) has a probability density function q_p(N), which satisfies q_p(N)_∞≤ (c_pσ√(2π))^-1.Obviously, p is strictly monotonic and thus a bijection. Moreover, the distribution Φ_p(N)(·) of p(N) takes the formℙ[N≤ p^-1(·)] or ℙ[N>p^-1(·)]. Hence, for any x∈,the density q_p(N) of p(N) satisfies q_p(N)(x)=q_N(p^-1(x))|(p^-1)'(x)| =q_N(p^-1(x))/|p'(p^-1(x))|≤ 1/(c_pσ√(2π)), as the density of N, q_N,is bounded above by(σ√(2π))^-1.§.§ CFs and distributions of near normal random variablesLet N be a normal random variable with mean μ and variance σ^2 and X a continuous random variable. Denote with φ_X, φ_N and Φ_X, Φ_N the CFs and the distributions of X and N, respectively. Assume there exist constants r>0, γ∈ (0,1) and a function R→ such that |logφ_X(t)-logφ_N(t)|≤ R(t)≤γσ^2 t^2/2 holds on |t|≤ r.Then sup_x∈|Φ_N(x)-Φ_X(x)|≤∫_-r^rR(t)/π |t|exp(-(1-γ)σ^2t^2/2)dt+12√(2)/π^3/2σ r.The result is a direct consequence of the Smoothing theorem (see <cit.>) commonly used to prove Berry-Esseen-type bounds, that relate CFs and distribution functions of random variables. The Smoothing theorem implies sup_x∈|Φ_N(x)-Φ_X(x)| ≤∫_-r^r|φ_N(t)-φ_X(t)|/(π|t|)dt+24sup_x∈|Φ_N'(x)|/(π r). Note that, for any z∈ℂ, it holds|e^z-1|≤ |z|e^|z|. Forz:=log(φ_X(t)/φ_N(t)), this implies|φ_X(t)-φ_N(t)|≤ |φ_N(t)||logφ_X(t)-logφ_N(t)| exp(|logφ_X(t)-logφ_N(t)|) ∀ t∈.The result follows from this inequality, sup_x∈|Φ_N'(x)|=1/(σ√(2π)) and |φ_N(t)|=e^-σ^2t^2/2:∫_-r^r|φ_N(t)-φ_X(t)|/π |t|dt≤∫_-r^r|φ_N(t)|R(t)/π |t|e^R(t)dt≤∫_-r^rR(t)/π |t|exp(-(1-γ)σ^2t^2/2)dt.Let X be random variable with finite mean μ, variance σ^2 and absolute third central moment κ:=𝔼[|X-μ|^3]. Then, the characteristic function φ_X of X satisfies:|logφ_X(t)-(iμ t-σ^2/2t^2)|≤κ|t|^3/6+σ^4t^4/4∀ t∈[-1/σ,1/σ]. The result can be established by combining the elementary bound |𝔼[e^i(X-μ)t-∑_k=0^n(it)^n/n!(X-μ)^n]|≤|t|^n+1/(n+1)!𝔼[|X-μ|^n+1]∀ t∈and the fact that z∈ℂ, |z|≤ 1/2 implies |(log(1+z)-z|≤ |z|^2 (see <cit.> for both). Let N=N(0,σ^2) and let u,v∈. The random variable uN+vN^2 has a characteristic function that satisfies|logφ_uN+vN^2(t)-(ivσ^2t-u^2σ^2/2t^2)|≤ 2v^2σ^4t^2+2u^2|v|σ^4|t|^3∀ t∈[-1/4|v|σ^2,1/4|v|σ^2]. The CF φ_uN+vN^2 can be explicitly computed using standard complex analysisφ_uN+vN^2(t)=𝔼[e^i(uN+vN^2)t]=1/√(1-2ivσ^2t)exp(-u^2σ^2t^2/2(1-2ivσ^2t))∀ t∈.The rest can then be shown using the elementary inequalities: z∈ℂ, |z|≤ 1/2 implies |(log(1+z)-z|≤ |z|^2 and |1/(1-z)-1|≤ 2|z|.alpha | http://arxiv.org/abs/1707.08510v4 | {
"authors": [
"Aleksandar Mijatović",
"Jure Vogrinc"
],
"categories": [
"math.PR",
"60J10, 60J22"
],
"primary_category": "math.PR",
"published": "20170726155618",
"title": "Asymptotic variance for Random walk Metropolis chains in high dimensions: logarithmic growth via the Poisson equation"
} |
Modelling the Scene Dependent Imaging in Cameraswith a Deep Neural Network Seonghyeon NamYonsei [email protected] Joo KimYonsei [email protected] 30, 2023 ========================================================================================================================= We present a novel deep learning framework that models the scene dependent image processing inside cameras. Often called as the radiometric calibration, the process of recovering RAW images from processed images (JPEG format in the sRGB color space) is essential for many computer vision tasks that rely on physically accurate radiance values. All previous works rely on the deterministic imaging model where the color transformation stays the same regardless of the scene and thus they can only be applied for images taken under the manual mode. In this paper, we propose a data-driven approach to learn the scene dependent and locally varying image processing inside cameras under the automode. Our method incorporates both the global and the local scene context into pixel-wise features via multi-scale pyramid of learnable histogram layers. The results show that we can model the imaging pipeline of different cameras that operate under the automode accurately in both directions (from RAW to sRGB, from sRGB to RAW) and we show how we can apply our method to improve the performance of image deblurring. § INTRODUCTIONDeep learning has significantly changed the approaches for solving computer vision problems. Instead of analytic solutions with some combinations of hand chosen features, probabilistic/physical models and some optimizations, most methods now turn to deep learning which is a deeper neural network that relies on big data. Deep learning has shown superb performance in many computer vision problems including image recognition <cit.>, face recognition <cit.>, segmentation <cit.>, etc.Image processing problems such as super-resolution <cit.> and colorization <cit.> are also solved with deep learning now, which provides effective ways to process input images and output images that fit the given task. In this paper, we introduce a new application of using deep learning for image processing: modelling the scene dependent image processing.We are especially interested in modelling the in-camera imaging pipeline to recover RAW images from camera processed images (usually in the form of JPEG in the sRGB color space) and vice versa. Usually called as the radiometric calibration, this process is important for many computer vision tasks that require accurate measurement of the scene radiance such as photometric stereo <cit.>, intrinsic imaging <cit.>, high dynamic range imaging <cit.>, and hyperspectral imaging <cit.>. There are two strategies with regards to the image processing in cameras, namely, the photographic reproduction model and thephotofinishing model <cit.>.In the photographic reproduction model, the image rendering pipeline is fixed meaning that a raw RGB value will always be mapped to an RGB value in the processed image regardless of the scene.Taking photos under the manual mode triggers this model and all previous radiometric calibration methods work only in this mode. In the photofinishing model, the image processing inside the camera varies (possibly in a spatially varying manner) in order to deliver visually optimal picture depending on the shooting environment <cit.>. This scene dependent mode will be activated usually when the camera is operated under the auto-mode. Figure <ref> compares the photos of a scene captured under the manual mode and under the auto-mode.In (b), the scene became brighter and the colors were enhanced compared to (a). It shows that the color rendering will be dependent on the scene in the auto-mode. Scene dependency in cameras were also verified in <cit.>. As mentioned above, none of the previous work can deal with the scene dependent color rendering. This is a problem as there are many photometry related topics in computer vision that have access to only automatically taken images (e.g. internet images) as in <cit.>.Moreover, smartphone cameras have become the major source for images and the many phone cameras only work in the auto-mode.The goal of this paper is to present a new algorithm that can model the camera imaging process under the "auto" mode. To deal with the scene dependency, we take the data-driven approach and design a deep neural network. We show that modelling the image processing using conventional CNN-based approaches is not effective for the given task,and propose a multi-scale pyramid of learnable histogram <cit.> to incorporate both the global and the local color histogram into pixel-wise features. The extracted multi-scale contextual features are processed with our CNN to model the scene dependent and locally varying color transformation.To the best of our knowledge, this is the first paper that can extract RAW images from processed images taken under the auto setting. Being able to radiometrically calibrate smartphone cameras is especially a significant contribution of this work.We show that we can model both the forward rendering (RAW to sRGB) and the reverse rendering (sRGB to RAW) accurately using our deep learning framework. We further apply our work to image deblurring.A blurred image is first transformed to the RAW space, in which a deblurring algorithm is executed. The deblurred image is then transformed back to the sRGB space through our deep network.We show that performing deblurring in this fashion give much better results over deblurring in the nonlinear sRGB space. § RELATED WORKIn-camera Image Processing (Radiometric Calibration)In the early works of radiometric calibration, the relationship between the scene radiance and the image value was explainedjust by a tonemapping function called the camera response function.Different models of the response function <cit.> as well as robust estimation algorithms <cit.> were introduced. More comprehensive reviews of the radiometric calibration literature is presented in <cit.>.In <cit.>, a more complete in-camera imaging pipeline that includes processes such as the white balance, the color space transformation, and the gamut mapping in addition to the tone-mapping was introduced.With the new parametric model for the imaging, the work also presented an algorithm for recovering the parameters from a set of images using a scattered point interpolation scheme.Using a similar pipeline, a probabilistic model that takes into account the uncertainty in the color rendering was recently proposed in <cit.>.As mentioned earlier, all previous works are based on the assumption that the color rendering inside the camera is deterministic and therefore cannot be applied for photos taken under the automode or by phone cameras. In comparison, our deep network framework learns the scene dependent image processing through given data and thus can be used for automatically captured photos.Deep Learning for Low-level VisionDeep learning has been very successful in image classification tasks, and the deep neural networks are now being applied to various problems in computer vision including the low-level vision tasks. In the field of low-level vision, convolutional neural networks (CNNs) are used to exploit the local context information for various applications such as image super-resolution <cit.>, denoising <cit.>, and filtering <cit.>. While the input and the output of these applications are RGB images, the learned mapping is more of a structural mapping rather than being a color mapping.Recently, deep learning based image colorization has been studied <cit.>, of which the objective is to restore chrominance information from a single channel luminance image. These works exploit the high-level semantic information to determine the chrominance of pixels by using CNNs, similar to those used in the high-level recognition tasks <cit.>.In this paper, we show that color histogram based low-level features extracted using our deep network are more efficient for the given task compared to the high-level CNN features extracted from above previous work.In <cit.>, an automatic photo adjustment method using a multi-layer perceptron was proposed.They feed the concatenation of global color features and semantic maps to a neural network system to find the scene dependent and the locally varying color mapping.As with the other data-driven image enhancement techniques <cit.>, the features for the color mapping in their work are manually selected.However, one of the key properties behind the success of deep learning is in its ability to extract good features for the given task automatically.Instead of using manual features, we propose an end-to-end trainable deep neural network to model the scene dependent in-camera image processing. § DATASETAn essential ingredient for any deep learning system is a good dataset. To model the image processing inside the camera from data, we need pairs of RAW images and its corresponding images in the nonlinear sRGB color space with JPEG format.Using the RAW-JPEG shooting mode, which is now supported by most cameras including Android based smartphones, we can collect many pairs of corresponding RAW and sRGB images. In this paper, we collected images using three digital cameras: Canon 5D Mark III, Nikon D600, and Samsung Galaxy S7. All pictures were taken under the auto-mode and the features like Auto Lighting Optimizer in the Canon camera that triggers locally varying processing such as contrast enhancement were all turned on.Some of the images in our dataset are shown in fig:Dataset.As can be seen, our dataset contains various kind of scenes including outdoor, indoor, landscape, portrait, and colorful pictures. The number of images in the dataset are 645, 710, and 290 for the Canon, the Nikon, and the Samsung camera, respectively. 50 images of varying scenes for each camera were selected for the test sets. In training phase, we extract multiple patches from images on the fly by using the patch-wise training method, which is described in sec:patch_training. Therefore, we can make millions of training examples from hundreds of image data.One thing to take notice is the white balancing in the imaging pipeline. The white balance is one of the main factors in determing the image color. While the white balance factor can be learned in the forward pipeline (from RAW to JPEG) as shown in <cit.>, estimating the white balance in the reverse pipeline is seemingly a more difficult task as the illumination is already normalized in the JPEG image. After the images are illumination normalized with the white balancing, it becomes an one-to-many mapping problem as any illuminant could have been mapped to the current image.Fortunately, the meta information embedded in images (EXIF data) provides the white balancing information. It provides three coefficients, which are the scale factors for the red, the green, and the blue channels.All the RAW images in our dataset are first white balanced using this information from the EXIF data.Therefore, the mapping that we learn in our system is from the white balanced RAW to sRGB, and vice versa. § DEEP LEARNING FRAMEWORK FOR MODELLING THE IMAGING PIPELINEIn this work, our goal is to model the imaging pipeline by computing the function f that maps RAW images to sRGB images and the function f^-1 that maps sRGB images to RAW images.The training data consist of RAW-sRGB pairs 𝒟={X^i, Y^i}_i=0^n, where X is the RAW image, Y represents the sRGB image, and n is the number of training examples. Since the deep neural networks are not invertible, we train f and f^-1 separately. Without the loss of generality, the algorithm that follows will be explained for the forward mapping f. Exactly the same process can be applied for learning the reverse mapping f^-1.The mapping function f under the auto-mode varies according to the scene and the local neighborhood. The function is formally described as: Y^i_x = f(X^i_x, Φ^i, Ω^i_x),where i is the image index, x is the pixel index, Φ represents the global scene descriptor, and Ω indicates local descriptor around a pixel.We propose a deep neural network that learns the scene dependent color mapping f including both Φ and Ω in eq:model in an end-to-end manner.To optimize the parameters of the proposed network, we need to define the loss function that computes the difference between the estimation and the ground truth.We minimize l_2 error from the training data as follows:L = 1/n∑_i=0^n‖ f(X^i)-Y^i ‖^2. As explained, the color mapping f is dependent on the global and the local context. Coming up with features that can describe this scene dependency manually is a difficult task.One way to compute the features for this problem is to use pre-trained CNNs like the VGG network <cit.> and finetune using our training set. As we show in Section 5, applying conventional CNN based structures do not capture good features for the scene dependency in our task.From the camera's point of view, it would be difficult to run a high-level scene recognition module for the scene dependent rendering due to the computational load.Therefore, it is reasonable to conjecture that the scene dependent color mapping relies mostly on low-level features such asthe contrast and the color distributions, which are computationally cheaper than the semantic features. In this work, we exploit color histogram as the feature to describe the scene. §.§ Learnable HistogramColor histogram is one of the most widely used features to describe images. In deep networks that use histograms, the centers and the widths of the bins are hand-tweaked by the user. In addition, since the computation of histograms is not differentiable, histograms are precomputed before training deep networks. Meanwhile, Wang <cit.> recently proposed the learnable histogram method, in which the key is a specialized differentiable function that trains the optimized histogram from data with deep networks in an end-to-end manner.With the learnable histogram, the bin for the value of an element in the feature map is determined by the following voting function:ψ_k,b(x_k) = max{ 0, 1-| x_k - μ_k,b|× w_k,b}.k and b are the index of the feature map element and the output bin, respectively. x_k is the value of the k-th element in the feature map, μ_k,b and w_k,b are the center and the width of the b-th bin. The histogram is built by accumulating the bins computed with the function ψ_k,b(x_k) as illustrated in fig:learnable_histogram.The centers μ_k,b and the widths w_k,b are trainable parameters and are optimized together with other parameters of the deep network.In this work, we adopt the histogram voting function eq:learnable_histogram of the learnable histogram to extract image features for our task of modelling the imaging pipeline. By introducing a multi-scale pyramid of histograms, we design the pixel-wise color descriptor for the global and the local context. In <cit.>, the learnable histogram was applied to the intermediate semantic feature maps to exploit global connect global Bag-of-Words descriptors, which in turn improves the performance of semantic segmentation and object detection. As we are looking for more low level color features instead of high-level semantic features, we directly connect the learnable histogram to the input image to extract RGB color histograms as show in fig:overview (a). Moreover, by putting multi-scale pooling layers on the output of the learnable histogram, our new network can extract the global and the local descriptors for each pixel. To effectively extract the color distribution, it is necessary to decouple the brightness and the chromaticity distribution.Therefore, we first convert the RGB values to a lightness (L) and chromaticity (rg) channels before the image goes through the learnable histogram as follows:L= (R + G + B) / 3,r= R / (R+G+B),g= G / (R+G+B).§.§ Multi-Scale Pyramid Pooling LayerThe output dimension of eq:learnable_histogram is H × W × (C × B), where H, W and C are the number of the height, width and channel of the input, C is the number of bins of the histogram. To get the global and the local color histogram, multiple average poolings with different pooling size are applied to the output feature maps as shown in fig:overview (b). We concatenate the multi-scale features corresponding to the same input pixel to incorporate the global and local context into pixel-wise features. Formally, our multi-scale pyramid of histogram features is described as:Ω^i_x = [h_x^1, h_x^2, ..., h_x^s],where h_x^s is the feature vector of s-th scale of the histogram layer corresponds to the pixel x.In our implementation, we compute four scales of the multi-scale pyramid by cascading three 3×3 average pooling layers followed by a global average pooling for the global histogram. The strides of the 3 local histogram layers are 1, 2, and 2, respectively. §.§ Patch-Wise Training Method and Implementation Details As illustrated in fig:overview, our deep network is trained with image patches instead of using the whole image.In the training phase, the whole image is first forwarded to the learnable histogram module (fig:overview (a)). Then a number of patches are randomly selected from both the input image and the histogram feature maps (fig:overview (b)). Specifically, patches are first extracted from the input image, and the feature maps that correspond to each patch are cropped to form the multi-scale features. Finally, only those selected patches are used for training the CNN weights as shown in fig:overview (c). This patchwise training has the advantage of being able to generate many training examples from a small dataset as well as being efficient in both time and memory.At the test time, the whole image and feature maps are forwarded to generate the full size output.For the configuration of our network, we used 6 bins for the learnable histogram, the initial bin centers were set to (0, 0.2, 0.4, 0.6, 0.8, 1.0), and the initial bin widths were set to 0.2 as described in <cit.>. After the global and the local features are extracted using the learnable histogram, the descriptors are concatenated with the input RGB image.Then, we apply 1×1 convolution filters to mix all input pixels and feature information, followed by two 3×3 additional convolutions to estimate the output.§ EXPERIMENTS§.§ Experimentation setupThe training images are preprocessed as follows.The RAW images are first demosaiced, normalized to have the max value of 1, and white-balanced using the EXIF metadata. Images are downsized and cropped to 512×512 images. In all camera datasets, we use 80% of images for the training and the remaining 20% for the validation, excluding the 50 test images. For the training, we use the Adam optimizer <cit.> to minimize our cost function.The batch size is 4, and sixteen 32×32 sparse patches are randomly extracted from it, which makes 64 training examples per batch. According to <cit.>, our training with a small fraction of images does not affect the convergence.With a GTX 1080 GPU, we can train the proposed network of 100 epochs within an hour.As explained before, we cannot compare the proposed method with existing radiometric calibration methods as they are deterministic models for specific manual settings and cannot be applied to automode cameras. Instead, we compare the proposed method with the following four baseline methods. ∙ Multi-layer Perceptron: We designed a MLP that consists of two hidden layers with 64 nodes each. The MLP learns an RGB to RGB color mapping without considering the scene dependency. We implemented the MLP by applying 1 × 1 convolution to images. ∙ SRCNN <cit.>: We used the SRCNN that consists of five 3 × 3 convolutional layers without pooling, and this is a simple attempt to model the scene dependency. ∙ FCN <cit.> and HCN <cit.>: Since we only have hundreds of images in the training data, we adopt a pixel-wise sampling method <cit.> to a hypercolumn network (HCN) to generate sufficient training signals. It cannot be applied to FCN since sample position is usually a fractional number in downsampled feature maps. Note that we only use VGG network layers from ( to ) for the FCN and (, , ) for the HCN, since our machine cannot handle large feature maps computed from high-definition input images (e.g. 1920×2880). For the FCN, we use the FCN-8S configuration on the reduced VGG network. We finetune both the FCN and the HCN using the pretrained VGG network. §.§ Experimental resultsTable <ref> shows the quantitative results using the 50 test images in our dataset for each camera. In the table, PSNR values comparing the RGB values of the recovered image with the ground truth are reported.For both the forward rendering (RAW-to-sRGB) and the reverse rendering (sRGB-to-RAW), the proposed method outperforms the other baseline methods in all categories except for very few Min and MaX errors among test images. Results using the MLP were usually worse than the other methods and this indirectly indicates the scene dependency in photographs.While the SRCNN showed some ability to deal with the scene dependency, its receptive field is limited to local neighborhoods and it cannot model the global scene context. The MLP and the SRCNN are optmized to model the mean of the color mapping in dataset and some of high values of the Min and the Max values in the results can be explained that some test images exist around the mean of our dataset.One can expect that hierarchical CNN features are able to capture the local and the global scene context that are useful for the scene dependent imaging.However, the experimental results show that they are not as efficient as our color histogram features. We attribute the bad performance of the FCN to the fact that the FCN is not sufficiently trained on only hundreds of training examples. Although we could sufficiently train the HCN through the in-network sampling method <cit.>, concatenating multi-level upsampled feature maps consume large memory for high-definition images from consumer cameras, which cannot be handled in test time. In summary, the results clearly show that our deep network that learns the local and the global color distribution is more efficient for accurately modelling the scene dependent image processing in cameras.fig:qualitative shows some of the examples of the image recovery.The figure shows that the proposed method can model the in-camera imaging process accurately in a qualitative way. It also shows that the other baseline methods also do a reasonable job of recovering images as the scene dependency applies to a set of specific colors or regions.§.§ AnalysisWe conducted more experiments to analyze the scene dependent processing learned by our network.For the analysis, we use two RAW images A and B.We first extract the learnable histogram feature from A, and replace the extracted histogram of B with that of A before injecting it to the DNN forward process.Note that the RAW image itself is the same, we just simulate the scene context change by changing the histogram.The intention of this analysis is to see how our network responds to the changes in the scene context. fig:discussion_1 shows the result of manipulating the global luminance histogram.The histogram in fig:discussion_1 (a) indicates the luminance distribution of a high contrast image, which is typical for backlit photos. We replace the histogram (a) with that of image (b) during the forward process of (b). fig:discussion_1 (c) and (d) show the result of the manipulation. As can be seen, the deep network brightens shadow regions and darken highlight regions. The network recognizes many dark regions and bright regions in the given histogram, and compensates by shifting the brightness to the middle (red in fig:discussion_1 (d)) compared to the original histogram (blue in fig:discussion_1 (d)). Note that it is what the Auto Lighting Optimizer of Canon cameras does as described in <cit.>.In fig:discussion_2, we show the result of manipulating the global chrominance histogram by going through the same process as explained above.By changing the global color context of image B with that of A, we can observe that our network responses more strongly to green and brown colors than the original. We can interpret this as our network recognizing the context of A as a natural scene from color distribution, trying to make natural objects like trees more visually pleasing. These examples show that our deep network recognizes specific scene context such as high contrast or nature images, and manipulates the brightness and colors to make images more visually pleasing as done in the scene dependent imaging pipeline of cameras.The experimentsalso show that the deep network does not memorize each example and can infer the mapping under various scene contexts.§.§ Application to Image DeblurringTo show the effectiveness of the proposed method, we apply it to the image deblurring application. It is well known that the blur process actually happens in the RAW space, but most deblurring algorithms are applied to the sRGB images since the RAW images are usually unavailable. Tai <cit.> brought up this issue and showed that being able to linearize the images have a significant effect in the deblurred results.However, the radiometric calibration process in <cit.> is rather limited and can only work under manual camera settings. We show that we can improve the deblurring performance on images taken from a smartphone camera (Samsung Galaxy S7) in automode.To do this, we use the image deblurring method of Pan <cit.>, which is a blind image deblurring method that uses the dark channel prior. We use the source code from the authors’ website and the default settings except for the kernel size. The RAW images are first computed from the corresponding sRGB images using the sRGB-to-RAW rendering of the proposed method, deblurred, and converted back to sRGB images using the RAW-to-sRGB rendering of the proposed method.fig:deblurring shows the deblurring results.As expected, the deblurring method <cit.> does not work well on nonlinear sRGB images andthere are some artifacts on deblurred scenes. On the other hand, the deblurring algorithm works well using our framework. The recovered images are much sharper and there are no significant artifacts.§ CONCLUSIONIn this paper, we presented a novel deep neural network architecture that can model the scene dependent image processing inside cameras.Compared to previous works that employ imaging models that are scene independent and can only work for images taken under the manual mode,our framework can be applied to the images that are taken under the auto-mode including photos from smartphone cameras. We also showed the potential of applying the proposed method for various computer vision tasks via image deblurring examples.§ ACKNOWLEDGEMENTThis work was supported by Global Ph.D. Fellowship Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2015H1A2A1033924), and the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (NRF-2016R1A2B4014610).ieee | http://arxiv.org/abs/1707.08350v1 | {
"authors": [
"Seonghyeon Nam",
"Seon Joo Kim"
],
"categories": [
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"primary_category": "cs.CV",
"published": "20170726100423",
"title": "Modelling the Scene Dependent Imaging in Cameras with a Deep Neural Network"
} |
Università del Sannio, 82100 Benevento, Italy [email protected]é de Strasbourg, CNRS, IRMA UMR 7501, F-67000 Strasbourg, France [email protected] [2010]37B15, 37B10, 37B40, 37C29, 37D20, 43A07, 68Q80 We review topics in the theory of cellular automata and dynamical systems that are related to the Moore-Myhill Garden of Eden theorem. The Garden of Eden theorem: old and new Michel Coornaert December 30, 2023 =======================================§ INTRODUCTION In the beginning, the Garden of Eden theorem, also known as the Moore-Myhill theorem, is a result in the theory of cellular automata which states that a cellular automaton is surjective if and only if it satisfies a weak form of injectivity, called pre-injectivity. The theorem was obtained by MooreMoore, Edward F. and MyhillMyhill, John R. in the early 1960sfor cellular automata with finite alphabet over the groups ^d.The fact that surjectivity implies pre-injectivity for such cellular automata was first proved by Moore in <cit.>, andMyhill <cit.> established the converse implication shortly after.The proofs of Moore and Myhill appeared in two separate papersboth published in 1963. The biblical terminology used to designate theMoore-Myhill theorem comes from the fact that configurations that are not in the image of a cellular automaton are calledGarden of Eden configurations because, when considering the sequence of consecutive iterates of the cellular automaton applied to the set of configurations, they can only occur at time 0. Surjectivity of a cellular automaton is equivalent to absence of Garden of Eden configurations.In 1988 <cit.>, SchuppSchupp, Paul E. asked whether the class ofgroups for which the Garden of Eden theorem remainsvalid is precisely the class ofvirtually nilpotent groups. By a celebrated result of Gromov <cit.>,Gromov, Misha L. a finitely generated group is virtually nilpotentif and only if it has polynomial growth. In 1993,Machì and Mignosi <cit.> proved that the Garden of Eden theorem is still valid over any finitely generated group with subexponential growth. As Grigorchuk <cit.>,Grigorchuk, Rostislav I. answering a longstanding open question raised by Milnor <cit.>,Milnor, John W. gave examplesof groups whose growth lies strictlybetween polynomial and exponential, it follows that the class of finitely generatedgroups satisfying the Garden of Eden theorem islarger than the class of finitely generatedvirtually nilpotent groups. Actually, it isevenlarger than the class of finitely generated groups with subexponential growth. Indeed, Machì,Scarabotti, and the first author <cit.> proved in 1999that everyamenable group satisfies the Garden of Eden theorem and it is a well–known fact that there are finitely generated amenable groups, such as the solvable Baumslag-Solitar group BS(1,2) = ⟨ a,b:aba^-1 = b^2⟩, that are amenable and haveexponential growth. It was finally shown that the class of groups that satisfy the Garden of Eden theoremis precisely the class of amenable groups. This is a consequence of recent results of Bartholdi <cit.>, BartholdiBartholdi, Laurentand Kielak <cit.>, who showed that none of the implications of the Garden of Eden theorem holdswhen the group is nonamenable. In <cit.>, Gromov made an important contribution to the subjectby providing a deep analysis of the role played by entropy in the proof of the Garden of Eden theorem and indicating newdirections for extending itin many other interesting settings. He mentionedin particular <cit.> the possibility of proving an analogue of the Garden of Eden theorem for a suitable class of hyperbolic dynamical systems. Some results in that direction were subsequently obtained by the authors in <cit.>, <cit.>, <cit.> and, in collaboration with H. Li,Li, Hanfeng <cit.>. In particular, a version of the Garden of Eden theorem was established for Anosov diffeomorphisms on tori in <cit.> and for principal expansivealgebraic actions of countable abelian groups in <cit.>. The present article is intended as a reasonably self-contained survey on the classical Garden of Eden theorem and some of its generalizations.Almost all results presented here have already appeared in the literature elsewhere butwe sometimes give complete proofs when we feel they might behelpful to the reader. The general theory of cellular automata over groups is developed in our monograph <cit.>. The present survey is a kind of complement to our book since for instance cellular automata between subshifts are not considered in <cit.> while they are treatedhere. The paper is organized as follows. Configuration spaces and shifts are presented in Section <ref>. Cellular automata are introduced in Section <ref>. Section <ref> contains the proof of the Garden of Eden theorem in the case G = ^d following Moore and Myhill. The proof of the Garden of Eden theorem in thecase of an arbitrary countable amenable group is given in Section <ref>. Examples of cellular automata that do not satisfy the Garden of Eden theorem for groups containing nonabelian free subgroups are described in Section <ref>. We also discuss the results of Bartholdi and Kielak mentioned above, which, together with the Garden of Eden theorem, lead to characterizations of amenability in terms of cellular automata. Extensions ofthe Garden of Eden theorem to certain classes of subshifts are reviewed in Section <ref>. In Section <ref>, we present versions of the Garden of Eden theorem we obtained for certain classes of dynamical systems. The final section briefly discusses some additional topics and provides references for further readings. § CONFIGURATION SPACES AND SHIFTS §.§ NotationWe use the symbolto denote the set of integers {…,-2,-1,0,1,2,…}. The symboldenotes the set of nonnegative integers {0,1,2,…}. The cardinality of a finite set X is written |X|. Weuse multiplicative notation for groupsexcept for abelian groups such as ^d = ××⋯×_ for which we generally prefer additive notation. Let Gbe a group. We denote the identity element of G by 1_G. If A, B are subsets of G and g ∈ G, we write A B { a b : a ∈ A,b ∈ B}, A^-1{a^-1 : a ∈ A}, g A {g} A and A gA {g}. A subset A ⊂ G is said to be symmetricsymmetric subset if it satisfies A = A^-1.§.§ Configurations spacesLetbe a countable set, called the universe,universe andAa finite set, called the alphabet.alphabet Depending on the context, the elements of A are called letters, or symbols, or states, or colors. As usual, we denote by A^ the set consisting of all maps x → A. An element of A^ is called a configurationconfiguration of the universe . Thus, a configuration is a way of attaching a letter of the alphabet to each element of the universe. If x ∈ A^ is a configuration and ⊂,we shall write x|_ for the restriction of x to , i.e., the elementx|_∈ A^ defined by x|_(v) = x(v) for all v ∈.If X ⊂ A^, we shall writeX_{ x|_ : x ∈ X }⊂ A^.Two configurations x,y ∈ A^ are said to be almost equalalmost equal configurations if they coincide outside of a finite set, i.e., there is a finite subset Ω⊂ such that x|_∖Ω = y|_∖Ω.Being almost equal clearly defines an equivalence relation onA^. We equip the configuration set A^ with itsprodiscreteprodiscrete!— topology topology, that is, the product topology obtained by taking the discrete topology on each factor A of A^ = ∏_u ∈ A. A neighborhood base of a configuration x ∈ A^ is given by the setsV(x,Ω) = V(x,Ω,,A) { y ∈ A^ : x|_Ω = y|_Ω},where Ω runs over all finite subsets of . In this topology, two configurationsare “close" if they coincide on a “large" finite subset ofthe universe.Every finite discrete topological space is compact,totally disconnected, and metrizable. As a product of compact (resp. totally disconnected) topological spaces is itself compact (resp. totally disconnected) and a countable product of metrizable spaces is itself metrizable,it follows that A^ is a compact totally disconnected metrizable space. Note that A^ is homeomorphic to the Cantor set as soon as A contains more than one element andis infinite.§.§ Group actionsAn actionaction of a group G on a set X is a map α G × X → X satisfying α(g_1,α(g_2,x)) = α(g_1 g_2,x) and α(1_G,x) = x for all g_1,g_2 ∈ G and x ∈ X. In the sequel, if α is an action of a group G on a set X, we shall simply writeg x instead of α(g,x), if there is no risk of confusion. Suppose that a group G acts on a set X. The orbitorbit of a point x ∈ X is the subset G x ⊂ X defined by G x {g x : g ∈ G}. A point x ∈ X is called periodicperiodic point if its orbit is finite. A subset Y ⊂ X is called invariantinvariant subset if G y ⊂ Y for all y ∈ Y. One says that Y ⊂ X is fixedfixed subset by G if g y = y for all g ∈ G and y ∈ Y. Suppose now that a group G acts on two sets X and Y. A mapfX → Y is called equivariantequivariant mapif f(g x) = g f(x) for all g ∈ G and x ∈ X.Let X be a topological space. An action of a group G on X is called continuouscontinuous actionaction!continuous — if the map x ↦ g x is continuous on X for each g ∈ G. Note that if G acts continuously on X then, for each g ∈ G,the map x ↦ g x is a homeomorphism of X with inverse x ↦ g^-1 x.§.§ ShiftsFrom now on, our universe will be a countable group. So let G be a countable group and A a finite set. Given an element g ∈ Gand a configuration x ∈ A^G, we define the configuration gx ∈ A^G bygxx ∘ L_g^-1,where L_gG → G is the left-multiplication by g. Thusgx (h) = x (g^-1h) for allh ∈ G.Observe that, for all g_1, g_2 ∈ G and x ∈ A^G,g_1(g_2x) = x ∘ L_g_2^-1∘ L_g_1^-1 = x ∘ L_g_2^-1g_1^-1 = x ∘ L_(g_1g_2)^-1= (g_1g_2)x,and 1_G x = x ∘ L_1_G = x ∘_G = x.Therefore the mapG × A^G→ A^G (g,x)↦ gxdefines an action of G on A^G.This action is called the G-shift, or simply the shift,shifton A^G. Observe that if two configurations x,y ∈ A^G coincide on a subset Ω⊂ G,then, for every g ∈ G, the configurations g x and g y coincide on g Ω. As the sets V(x,Ω) defined by (<ref>)form a base of neighborhoods of x ∈ A^G when Ω runs over all finite subsets of G,we deduce that the map x ↦ g x is continuouson A^G for eachg ∈ G. Thus, the shift action of G on A^G is continuous.§.§ PatternsA patternpattern is a map p Ω→ A, where Ω is a finite subset of G. If p Ω→ A is a pattern, we say that Ω is the supportsupport!— of a patternpattern!support of a — of p and write Ω = (p). Let (G,A) denote the set of all patterns. There is a naturalaction of the group G on (G,A) defined as follows. Given g ∈ G and a pattern p ∈(G,A) with support Ω, we define the pattern g p ∈(G,A) as being the pattern with support g Ω such that g p(h) = p(g^-1 h) for all h ∈ g Ω. It is easy to check that this definesanaction of G on (G,A), i.e., g_1(g_2 p) = (g_1 g_2) p and 1_G p = p for all g_1, g_2 ∈ G and p ∈(G,A). Observe that (g p) = g (p) for all g ∈ G and p ∈(G,A). Note also that if p is the restriction of a configuration x ∈ A^G to a finite subsetΩ⊂ G, then g p is the restriction of the configuration g x to g Ω.Take G =and let A be a finite set. Denote by A^⋆ the set of words on the alphabet A.word We recall that A^⋆ is the free monoid based on A and that any element w ∈ A^⋆ can be uniquely written in the form w = a_1 a_2 ⋯ a_n, where a_i ∈ A for 1 ≤ i ≤ n and n ∈ is the lengthlength!— of a wordword!length of a — of the word w. The monoid operation on A^⋆ is the concatenation of words and the identity elementis the empty word, that is, the unique word with length 0. Now let us fix some finite interval Ω⊂of cardinality n, say Ω = {m , m + 1, …, m + n - 1} with m ∈ and n ∈. Then one can associate with each pattern p Ω→ Athe word w = p(m) p(m + 1) ⋯ p(m + n - 1) ∈ A^⋆.This yields a one-to-one correspondence between the patterns supported by Ω and the words of length n on the alphabet A. This is frequently used to identify each pattern supported by Ω with the corresponding word.§.§ SubshiftsA subshiftsubshift is a subsetX ⊂ A^G that is invariant under the G-shift and closed for the prodiscrete topology on A^G.Take G = and A = {0,1}. Then thesubset X ⊂ A^G, consisting of all x →{0,1} such that (x(n),x(n + 1)) ≠ (1,1) for all n ∈, is a subshift. This subshift is called the golden mean subshift.golden mean subshiftsubshift!golden mean — Take G =^d and A = {0,1}. Then thesubset X ⊂ A^G, consisting of all x ∈ A^G such that (x(g),x(g + e_i)) ≠ (1,1) for all g ∈ G, where (e_i)_1≤ i ≤ d is the canonical basis of ^d, is a subshift. This subshift is called the hard-ball model.hard-ball modelsubshift!hard-ball model — For d = 1, the hard-ball model isthe golden mean subshift of the previous example.Take G = and A = {0,1}. Then the subset X ⊂ A^G, consisting of all bi-infinite sequences x →{0,1} such that there is always an even number of 0s between two 1s, is a subshift. This subshift is called the even subshift.even subshiftsubshift!even — Take G = ^2 and A = {0,1} = /2 (the integers modulo 2). Then thesubset X ⊂ A^G, consisting of all x ^2 →{0,1} such thatx(m,n) + x(m + 1,n) + x(m,n + 1) = 0for all (m,n) ∈^2, is a subshift. This subshift is called the Ledrappier subshift.Ledrappier subshiftsubshift!Ledrappier — Every intersectionof subshifts and every finite union of subshifts X ⊂ A^G is itself a subshift. Therefore the subshifts X ⊂ A^G are the closed subsets of a topology on A^G. This topology is coarser (it has less open sets) than the prodiscrete topology on A^G. It is not Hausdorff as soon as G is not trivial and A has more than one element.Given a (possibly infinite) subset of patterns P ⊂(G,A), it is easy to see that the subset X(P) ⊂ A^G defined byX(P) { x ∈ A^Gsuch that(g x)|_(p)≠ pfor all g ∈ Gandp ∈ P }is a subshift. Conversely, letX ⊂ A^G be a subshift. One says that a pattern p ∈(G,A)appearspattern!— appearing in a subshiftsubshift!pattern appearing in a — in X if p ∈ X_(p), i.e., if there is a configuration x ∈ X such that x|_(p) = p. Then one easily checks that X = X(P) forP { p ∈(G,A)such that pdoes not appear in X }.One says that a subshift X ⊂ A^G is of finite typefinite type!subshift of —subshift!— of finite type if there exists a finite subsetP ⊂(G,A) such that X = X(P). Thehard-ball models (and hence in particular the golden mean subshift) and the Ledrappier subshift are examples of subshifts of finite type. On the other hand, the even subshift is not of finite type.§ CELLULAR AUTOMATA §.§ Definition LetG be a countable group and let A, B befinite sets. Suppose that X ⊂ A^G and Y ⊂ B^G are two subshifts.One says that a map τ X → Y is a cellular automatoncellular automaton if there exist a finite subset S ⊂ G and a map μ A^S → B such thatτ(x)(g) = μ((g^-1x)|_S)for all x ∈ X and g ∈ G, where we recall that (g^-1x)|_S denotes the restriction of the configuration g^-1x ∈ X to S.Such a set S is called a memory setmemory setcellular automaton!memory set of a —and μ is called a local defining maplocal defining mapcellular automaton!local defining map of a —forτ.It immediately follows from this definition that a map τ X → Y is a cellular automaton if and only if τ extends to a cellular automaton τ A^G → B^G. Observe also that if S is a memory set for a cellular automaton τ X → Y and g ∈ G, then Formula (<ref>) implies that the value taken by the configuration τ(x) at g only depends on the restriction of x to gS.Finally note thatif S is a memory set for a cellular automaton τ, then any finite subset of G containing S is also a memory set for τ. Consequently, the memory set of a cellular automaton is not unique in general. However, every cellular automaton admits a unique memory set with minimal cardinality (this follows from the fact that if S_1 and S_2 are memory sets then so is S_1 ∩ S_2).Take G = and A = {0,1} = /2. Then the map τ A^G → A^G, defined byτ(x)(n)x(n + 1) + x(n)for all x ∈ A^G and n ∈, is a cellular automaton admitting S {0,1}⊂ as a memory set and μ A^S → A given byμ(p) p(0) + p(1)for all p ∈ A^S, as a local defining map. Using the representation of patterns with support S by words of length 2 on the alphabet A (cf. Example <ref>), the map μ is given byμ(00) = μ(11) = 0and μ(01) =μ(10) = 1.[Majority vote]majority votecellular automaton!majority vote —Take G = and A = {0,1}. Then the map τ A^G → A^G, defined byτ(x)(n) 0 ifx(n-1) + x(n) + x(n + 1) ≤ 1 1 otherwisefor all x ∈ A^G and n ∈, is a cellular automaton admitting S {-1,0,1}⊂ as a memory set and μ A^S → A given byμ(000) = μ(001) = μ(010) = μ(100) = 0 and μ(011) = μ(101) = μ(110) = μ(111) = 1as local defining map. The cellular automaton τ is called the majority vote cellular automaton.A cellular automaton τ A^G → A^G, where G =, A = {0,1}, admitting S = {-1,0,1} as a memory setis called an elementary cellular automaton.elementary!— cellular automatoncellular automaton!elementary — Each one of thesecellular automata is uniquely determined by its local defining map μ A^S → A, so that there are exactly 2^8 = 256 elementary cellular automata. They are numbered from 0 to 255 according to a notation that was introduced by Wolfram (cf. <cit.>).Wolfram, Stephen To obtain the number n of an elementary cellular automaton τ, one proceeds as follows. One first lists all eight patterns p ∈ A^S in increasing order from 000 to 111. The number n is the integer whose expansion in base 2 is a_8 a_7 … a_1, where a_k is the value taken by the local defining map of τ at the k-th pattern in the list. One also says that τ isRule n.Rule of an elementary cellular automatonelementary!Rule of an — cellular automaton For instance, the elementary cellular automaton described in Example <ref> is Rule 102 while the one described in Example <ref> is Rule 232.majority votecellular automaton!majority vote — Let G be a countable group, A a finite set, and X ⊂ A^G a subshift. Then the identity map _XX → X is a cellular automaton with memory set S = {1_G} and local defining map μ = _A A^S = A^{1_G} = A → A.Let G be a countable group, A a finite set, and X ⊂ A^G a subshift. Let s ∈ G and denote by R_s the right-multiplication by s, that is, the map R_sG → G defined by R_s(h)h s for all s ∈ G. Then the subset Y ⊂ A^G defined byY {x ∘ R_ssuch thatx ∈ X }is a subshift. Moreover, the map τX → Y, defined by τ(x)x ∘ R_s for all x ∈ X, is a cellular automaton with memory set S = {s} and local defining map μ = _AA^S = A → A. Observe that if s is in the centerof G, then X =Y and τ X → X is the shift mapx ↦ s^-1 x.Take G = andA = {0,1}. Let X ⊂ A^G and Y ⊂ A^G denote respectively the golden mean subshiftgolden mean subshiftsubshift!golden mean — and the even subshift.even subshiftsubshift!even — For x ∈ X, define τ(x) ∈ A^G byτ(x)(n)0 if(x(n),x(n + 1)) = (0,1)or(1,0) 1if(x(n),x(n+1)) = (0,0)for all n ∈. It is easy to see that τ(x) ∈ Y for all x ∈ X. The map τ X → Y is a cellular automaton admitting S {0,1}⊂ as a memory set and the map μ A^S→ A, defined byμ(00) = μ(11) = 1and μ(01) = μ(10) = 0.Note that the map μ'A^S → A, defined byμ'(00) = 1and μ'(01) = μ'(10) = μ'(11) =0is also a local defining map for τ. Thus, τ is the restriction to X of both Rule 153 and Rule 17. §.§ The Curtis-Hedlund-Lyndon theoremThe definition of a cellular automaton given in the previous subsectionis a local one. For τ to be a cellular automaton, it requires the existence of a rule, commuting with the shift, thatallows one to evaluate the value taken byτ(x) atg ∈ G by applying the rule to the restriction of x to a certain finite set, namely the left-translate by g of a memory set of the automaton. The following result, known as the Curtis-Lyndon-Hedlund theorem (see <cit.>),yields a global characterization of cellular automata involving only the shift actions and the prodiscrete topology on the configuration spaces. Curtis-Hedlund-Lyndon theoremtheorem!Curtis-Hedlund-Lyndon — LetG be a countable group and let A,B be finite sets. Let τ X → Y be a map from a subshift X ⊂ A^G intoa subshift Y ⊂ B^G.Then the following conditions are equivalent: * τ is a cellular automaton;* τ is equivariant (with respect to the shift actions of G) and continuous (with respect to the prodiscrete topologies).Suppose first that τ X → Y is a cellular automaton. Let S ⊂ G be a memory setand μ A^S → B a local defining map for τ. For all g,h ∈ G and x ∈ X, we have thatτ(gx)(h)= μ((h^-1gx)|_S) (by Formula (<ref>))= μ(((g^-1h)^-1x)|_S) = τ(x)(g^-1h) = gτ(x)(h).Thus τ(gx) = gτ(x) for all g ∈ G and x ∈ X. This shows that τ is equivariant. Now let Ω be a finite subset of G. Recall thatFormula (<ref>) implies that if two configurations x,y ∈ X coincide on gS for some g ∈ G, then τ(x)(g) = τ(y)(g). Therefore, if the configurations x and y coincide on the finite set ΩS, then τ(x) and τ(y) coincide on Ω. It follows thatτ(X ∩ V(x,Ω S,G,A)) ⊂ V(τ(x),Ω,G,B).This implies that τ is continuous. Thus(a) implies (b). Conversely, suppose now that the mapτ X → Y is equivariant and continuous.Let us show that τ is a cellular automaton.As the map φ X → B defined by φ(x) τ(x)(1_G) is continuous, we can find, for each x ∈ X, a finite subsetΩ_x ⊂ G such that if y ∈ X ∩ V(x,Ω_x,G,A), then τ(y)(1_G) = τ(x)(1_G). The sets X ∩ V(x,Ω_x,G,A) form an open cover of X. As X is compact, there is a finite subset F ⊂ X such that the sets V(x,Ω_x,G,A), x ∈ F, cover X. Let us set S = ∪_x ∈ FΩ_x and suppose that two configurations y,z ∈ X coincide on S. Let x_0 ∈ F be such that y ∈ V(x_0, Ω_x_0,G,A), that is, y |_Ω_x_0 = x_0|_Ω_x_0. As Ω_x_0⊂ S, we have that y |_Ω_x_0 = z |_Ω_x_0 and thereforeτ(y)(1_G) = τ(x_0)(1_G) = τ(z)(1_G). We deduce thatthere existsa map μ A^S → B such that τ(x)(1_G) = μ(x|_S) for all x ∈ X. Now, for all x ∈ X andg ∈ G, we have thatτ(x)(g)= (g^-1τ(x))(1_G) (by definition of the shift action on B^G)= τ(g^-1x)(1_G) (since τ is equivariant)= μ((g^-1 x)|_S).This shows that τ is a cellular automaton with memory set S and local defining map μ. Thus (b) implies (a).§.§ Operations oncellular automata Let G be a countable group and let A, B,C be finite sets. Supposethat X ⊂ A^G, Y ⊂ B^G, Z ⊂ C^G are subshifts and that τ X → Y, σ Y → Z are cellular automata. Then the composite map σ∘τ X → Z is a cellular automaton. This is an immediate consequence of the characterization of cellular automatagiven by the Curtis-Hedlund-Lyndon theorem (cf. Theorem <ref>) since the composite of two equivariant (resp. continuous) maps is itself equivariant (resp. continuous). If we fix the countable group G, we deduce from Proposition <ref> and Example <ref> that the subshiftsX ⊂ A^G, with A finite, are the objects of a concrete category _G in whichthe set of morphisms fromX ∈_G toY ∈_G consist of all cellular automataτ X → Y (cf. <cit.>). In this category, the endomorphisms of X ∈_G consist of all cellular automataτ X → X and they form a monoid for the composition of maps. Let G be a countable group and let A, B be finite sets. Supposethat X ⊂ A^G, Y ⊂ B^G are subshifts and that τ X → Y is a bijective cellular automaton. Then the inverse mapτ^-1 Y → X is a cellular automaton. This is again an immediate consequence of theCurtis-Hedlund-Lyndon theorem since the inverseof a bijectiveequivariant map is anequivariant map and the inverse of a bijective continuous map between compact Hausdorff spaces iscontinuous. §.§ Surjectivity of cellular automata, GOE configurations, and GOE patternsIn what follows, we keep the notation introduced for defining cellular automata.Let τ X → Y be a cellular automaton. A configuration y ∈ Y is called a Garden of EdenGarden of Eden!— configurationconfiguration!Garden of Eden — configuration for τ, briefly a GOE configuration, if it does not belong to the image of τ, i.e., there is no x ∈ X such that y = τ(x). One says that a pattern p ∈(G,B) is a Garden of Eden patternGarden of Eden!— patternpattern!Garden of Eden —for τ, briefly a GOE pattern,if the pattern p appears in the subshift Y but notin thesubshiftτ(X), i.e.,there exists y ∈ Y such that p = y|_(p) but there is no x ∈ X such thatp = τ(x)|_(p).Note that the set of GOE configurations (resp. of GOE patterns) is an invariant subset ofY (resp. of (G,B)). Observe also that if p ∈(G,B) is a GOE pattern for the cellular automaton τ X → Y,then every configuration y ∈ Y such that y|_(p) = p is a GOE configuration for τ.It is easy to check that the pattern with support Ω{0,1,2,3,4} associated with the word 01001 is a GOE pattern for the majority vote cellular automaton described in Example <ref>.majority votecellular automaton!majority vote —Let τ X → Y be a cellular automaton.Then thefollowing conditions areequivalent: * τ issurjective; * τ admits no GOE configurations; * τ admits no GOE patterns.The equivalence of (a) and (b) as well as the implication (b)(c) are trivial. The implication (c)(b) easily follows from the compactness of τ(X). §.§ Pre-injectivity of cellular automata and mutually erasable patternsLet G be a countable group and let A and B be finite sets. Recall that two configurations x,y ∈ A^G are called almost equal if they coincide outside of a finite subset of G. Let X ⊂ A^G and Y ⊂ B^G be subshifts. One says that a cellular automaton τ X → Y is pre-injectivepre-injective!— cellular automatoncellular automaton!pre-injective — if there are no distinct configurations x_1, x_2 ∈ X that are almost equal and satisfy τ(x_1) = τ(x_2). A pair of configurations(x_1,x_2)∈ X × X is called a diamonddiamond if x_1 and x_2 are distinct,almost equal,and have the same image under τ (cf. <cit.>). Thus, pre-injectivity is equivalent to absence of diamonds. If (x_1,x_2) is a diamond, the nonempty finite subset{g ∈ G: x_1(g) ≠ x_2(g) }⊂ Gis called the supportsupport!— of a diamonddiamond!support of a — of the diamond (x_1,x_2). Every injective cellular automaton is clearly pre-injective. The converse is false, as shown by the following examples.Take G =, A = {0,1} = /2, and consider the cellular automaton τ A^G → A^G described in Example <ref> (Rule 102 in Wolfram's notation). Then τ is pre-injective. Indeed, it is clear that if two configurations x, x' ∈ A^G coincide on ∩ (-∞,n_0] for some n_0 ∈ and satisfy τ(x) = τ(x') then x = x'. However, τ is not injective since the two constant configurations have the same image. Consider the cellular automaton τ X → Y from the golden mean subshift to the even subshift described in Example <ref>.golden mean subshiftsubshift!golden mean —even subshiftsubshift!even — It is easy to see that τ is pre-injective by an argument similar to the one used in the previous example. However, τ is not injective since the two sequences in X with exact period 2 have the same image under τ, namely the constant sequence with only 0s. Let Ω⊂ G be a finite set and p_1, p_2 ∈ X_Ωtwo patterns with support Ω appearing in X.One says that the patterns p_1 and p_2 are mutually erasablemutually erasable patterns with respect to τ, briefly ME,provided the following hold:* the setX_p_1,p_2{(x_1, x_2) ∈ X × X: x_1 |_Ω = p_1, x_2 |_Ω = p_2x_1 |_G ∖Ω = x_2 |_G ∖Ω}is nonempty;* for all (x_1, x_2) ∈X_p_1,p_2 one has τ(x_1) = τ(x_2).Note that “being ME" is an equivalence relation on X_Ω. This equivalence relation is not trivial in general.The patterns with support Ω{0,1,2} associated with the words 00000 and 00100 are ME patterns for the majority vote cellular automaton described in Example <ref>.majority votecellular automaton!majority vote — Observe thatif the patterns p_1 and p_2 are ME, then so are gp_1 and gp_2 for all g ∈ G.Let τ X → Y be a cellular automaton. Then thefollowing conditions are equivalent: * τ is pre-injective;* τ admits no distinct ME patterns.Suppose first that τ admits two distinct ME patterns p_1 and p_2.Let Ω denote theircommon support. Take(x_1, x_2) ∈X_p_1,p_2. Then the configurations x_1 and x_2are almost equal since they coincide outside of Ω.Moreover, they satisfyx_1 ≠ x_2 and τ(x_1) = τ(x_2). Therefore (x_1,x_2) is a diamond for τ. It follows that τ is not pre-injective. Suppose now that τ is not pre-injective and let us show that τ admits two distinct ME patterns. By definition, τ admits a diamond (x_1,x_2) ∈ X × X.Let Δ denote the support of this diamond andlet S ⊂ G be a memory set for τ with 1_G ∈ S.Consider the setΩΔ S^-1 S.We claim that the patterns p_1x_1|_Ω and p_2x_2|_Ω are distinct ME patterns. First observe that p_1 ≠ p_2 since ∅≠Δ⊂Ω.Moreover, X_p_1,p_2 is nonempty since, by construction, (x_1, x_2) ∈ X_p_1,p_2. To completethe proof, we only need to show the following: if (y_1, y_2) ∈ X_p_1,p_2then τ(y_1) = τ(y_2). Let g ∈ G. Suppose first that g ∈ G ∖Δ S^-1. Then gS ∩Δ = ∅. Since y_1 and y_2 coincide on G ∖Δ, we deduce thatτ(y_1)(g) = τ(y_2)(g) for allg ∈ G ∖Δ S^-1.Suppose now that g ∈Δ S^-1. Then gS ⊂Δ S^-1S = Ω. As y_1|_Ω = p_1 = x_1|_Ω (resp. y_2|_Ω = p_2 = x_2|_Ω), by construction, we deduce thatτ(y_1)(g) =τ(x_1)(g) = τ(x_2)(g) = τ(y_2)(g) for allg ∈Δ S^-1.From (<ref>) and (<ref>) we deduce that τ(y_1) = τ(y_2).§ THE GARDEN OF EDEN THEOREM FOR ^DIn this section, we present a proof of the Garden of Eden theorem of Moore and Myhill for cellular automata over the group ^d. This is a particular case of the Garden of Eden theorem for amenable groups that will be established in the next section.Garden of Eden theorem!— (for ^d)theorem!Garden of Eden — (for ^d) Let A be a finite set, d ≥ 1 an integer, and τ A^^d→ A^^d a cellular automaton. Then the following conditions are equivalent: * τ is surjective;* τ is pre-injective.The implication (a)(b) is due to Moore <cit.>Moore, Edward F. and the converse to Myhill <cit.>.Myhill, John R. We shall prove the contraposite of each implication. Before undertaking the proof of Theorem <ref>, let us first introduce some notation and establish some preliminary results. Let S⊂^d be a memory set for τ. Since any finite subset of ^d containing a memory set for τ is itself a memory set for τ, it is not restrictive to suppose that S ={0,± 1,…, ± r}^d for some integer r ≥ 1. Let us set, for each integer m ≥ 2r,Ω_m {0,1,…, m-1}^d Ω_m^+ {-r,-r+1,…, m+r-1}^dand Ω_m^- {r,r+1,…, m-r-1}^d,so that|Ω_m| = m^d,|Ω_m^+| = (m+2r)^d, |Ω_m^-| = (m-2r)^d.Observe that if two configurations x_1,x_2∈ A^^d coincide on Ω_m(resp. Ω_m^+, resp. ^d ∖Ω_m^-) then τ(x_1) and τ(x_2) coincide on Ω_m^- (resp. Ω_m, resp. ^d ∖Ω_m). Also, let us set, for allintegers k,n ≥ 1T_n^k {t=(t_1k, t_2k, …, t_dk)∈^d: 0 ≤ t_j ≤ n-1},and observe that the n^d cubest + Ω_k, for t running over T_n^k, form a partition of the cube Ω_n k. Finally, we introduce the following additional notation.Given a finite subset Ω⊂ Gandp,q ∈ A^Ωwe write p ∼ q if and only if the patterns p and q are ME for τ.As usual, we denote by A^Ω/∼ the quotient set of A^Ω by ∼, i.e.,the set of all ME-equivalence classes of patterns supported byΩ. In the proof of Theorem <ref> we shall make use of the following elementary result (here one should think of a:=|A| as to the cardinality of the alphabet set and a^(n k )^d (resp. a^(n k - 2r)^d, for n k ≥ 2r) as to the number of all patterns supported by Ω_n k (resp. Ω_n k^-), cf. (<ref>). Let a,k,d,r be positive integers with a ≥ 2.Then there exists n_0 = n_0(a,k,d,r) ∈ such that(a^k^d - 1)^n^d < a^(n k - 2r)^dfor all n ≥ n_0.Taking logarithms to base a, Inequality (<ref>) is equivalent tolog_a(a^k^d - 1) < (k - 2r/n)^d.Sincelog_a(a^k^d - 1) < log_a(a^k^d) = k^d = lim_n →∞(k - 2r/n)^d,we deduce that there exists n_0 ∈ such that (<ref>) and therefore (<ref>) are satisfied for all n ≥ n_0.We can assume that a|A| ≥ 2. Suppose first that τ is not pre-injective. Then, by Proposition <ref>, τ admits two distinct ME patterns, say p_1 and p_2. Denote by Ω⊂^d their common support. Since, for all t ∈^d, the patterns t p_1 and t p_2 (with support t + Ω) are also distinct and ME, and any finite subset of ^d containing the support of two distinct ME patterns is itself the support of two distinct ME patterns, we may assume that Ω = Ω_kfor some integer k ≥ 2 r. As the patterns t p_1 and t p_2 are ME, we have| A^t+Ω_k/∼ | ≤| A^t+Ω_k| - 1 = a^k^d-1for all t ∈^d. Now observe that two patterns with support Ω_n k are ME iftheir restrictions to t+Ω_k are ME for everyt ∈ T_n^k. Using (<ref>), we deduce that| A^Ω_n k/∼| ≤∏_t ∈ T_n^k |A^t + Ω_k/∼| ≤(a^k^d-1)^n^d.Taking n ≥ n_0(a,k,d,r), we then get|τ(A^^d)_Ω_n k^-|≤ | A^Ω_n k / ∼ | ≤(a^k^d-1)^n^d < a^(n k - 2r)^d = | A^Ω_n k^- |.This implies that τ(A^^d)_Ω_n k^-⫋ A^Ω_n k^-,so that there mustexist a GOE pattern for τ with support Ω_n k^-. Consequently, τ is not surjective. This shows that (a)(b). Let us now turn to the proof of the converse implication. Suppose that τ is not surjective.Then, by Proposition <ref>, there exists a GOE pattern p for τ. Since tp is a GOE pattern for every t ∈^d, and any finite subset of ^d containing the support of a GOE pattern is itself the support of a GOE pattern, we can assume that p is supported by the cube Ω_k for some integer k ≥ 2r. Decompose again Ω_n k into the n^d translates t+Ω_k, with t ∈ T_n^k, and observe that tp ∈ A^t+Ω_k is GOE for every t ∈ T_n^k.Any pattern q ∈ A^Ω_n k which is not GOE satisfies that q|_t+Ω_k is not GOE for every t ∈ T_n^k. As a consequence, we have that| τ(A^^d)_Ω_n k|≤∏_t ∈ T_n^k| τ(A^^d)_t + Ω_k|≤ (a^k^d -1)^n^d.Let us fix now some element a_0 ∈ A and consider the set X consisting of all configurationsx ∈ A^^d that satisfyx(g) = a_0 for all g ∈^d ∖Ω_n k^-.Observe that if x_1 and x_2 are in X, then τ(x_1) and τ(x_2) coincide on^d ∖Ω_n k. It follows that|τ(X)| = | τ(X)_Ω_n k|.On the other hand, taking n ≥ n_0(a,k,d,r) and using (<ref>),we get| τ(X)_Ω_n k|≤| τ(A^^d)_Ω_n k|≤ (a^k^d -1)^n^d < a^(n k - 2r)^d = | A^Ω_kn^- | = |X|and hence|τ(X)| < |X|.By the pigeon-hole principle, this impliesthat there exist two distinct configurations x_1,x_2 ∈ X such that τ(x_1) = τ(x_2). As all configurations in X are almost equal, we deduce that τ is not pre-injective. This shows that (b)(a).The proof of the implication (a)(b) shows that if τ admits two distinct ME patterns supported by a cube of side k ≥ 2 r, then a cube of side n k -2r, with n ≥ n_0(a,k,d,r), must support a GOE pattern. Conversely, a small addition to the proof of the implication (b)(a) yields that if τ admits a GOE pattern supported by a cube of side k ≥ 2 r, then a cube of side kn+2r, with n ≥ n_0(a,k,d,r), supports two distinct ME patterns. Indeed, the proof shows the existence of two configurations x_1,x_2 ∈ A^^d that coincide outside ofΩ_n k^- and satisfyτ(x_1) = τ(x_2). It then follows from the proof of the implication (b)(a) in Proposition <ref>that the set (Ω_n k^- +(- S))+S = Ω_n k + S = Ω_n k^+ supports two distinct ME patterns.§ THE GARDEN OF EDEN THEOREM FOR GENERAL AMENABLE GROUPS §.§ Amenability(cf. <cit.>, <cit.>, <cit.>, <cit.>) A countable group G is called amenableamenable groupgroup!amenable — if there exists a sequence (F_n)_n ∈ of nonempty finite subsets of G such thatlim_n →∞| F_n ∖F_n g|/| F_n | = 0 for allg ∈ G. Such a sequence is called a Følner sequenceFølner sequence for G.Note that if A and B are finite sets with the same cardinality, then|A ∖ B| = |B ∖ A| and|A △ B| = |A ∖ B| + |B ∖ A| = 2 |A ∖ B|, where △ denotes symmetric difference of sets. As | F g| = |F| for every finite subset F ⊂ G and any g ∈ G, it follows that Condition (<ref>) is equivalent to each of the following conditions:lim_n →∞|F_n g ∖F_n |/| F_n | = 0 for allg ∈ G,orlim_n →∞|F_n △F_n g |/| F_n | = 0 for allg ∈ G.All finite groups are amenable. Indeed, if G is a finite group, then the constant sequence,defined by F_n G for all n ∈,isa Følner sequence for G since F_n ∖F_n g = ∅for every g ∈ G. The free abelian groups of finite rank ^d, d ≥ 1,are also amenable. As a Følner sequence for ^d, one can take for instance the sequence of cubes F_n {x ∈^d: ‖ x ‖_∞≤ n} = {0,± 1,…,± n}^d,where ‖ x ‖_∞max_1 ≤ i ≤ d |x_i| for all x = (x_1,…,x_d) ∈^d is the sup-norm. To see this, observe that|F_n| = (2n + 1)^d and, by the triangle inequality, F_n + g ⊂ F_n + ‖ g ‖_∞for allg ∈^d.SinceF_n ⊂ F_n + ‖ g ‖_∞,this implies|( F_n + g) ∖ F_n| ≤(2n + ‖ g ‖_∞ + 1)^d - (2n + 1)^d.As the right-hand side of (<ref>) is a polynomial of degree d - 1 in n while |F_n| is a polynomial of degree d in n by (<ref>), we conclude thatlim_n →∞| ( F_n + g ) ∖F_n |/| F_n | = 0,which is (<ref>) in additive notation. Let G be a finitely generated group. If S ⊂ G is a finite symmetric generating subset, the Cayley graphCayley graph of G with respect to Sis the graph (G,S) whose set of verticesis G and two vertices g,h ∈ G are joined by an edge if and only if h = g sfor some s ∈ S. Equip the set of vertices of (G,S) with its graph metric and consider the ball B_n ⊂ G of radius n centered at 1_G. It is easy to see that the sequence of positive integers (|B_n|)_n ∈ is submultiplicative. Thus the limitγ(G,S) lim_n →∞√(|B_n|)exists and satisfies 1 ≤γ(G,S) < ∞. One says that the group G has subexponential growthsubexponential growthgroup!— of subexponential growth if γ(G,S) = 1 andexponential growthexponential growthgroup!— of exponential growthif γ(G,S) > 1.The fact that G has subexponential (resp. exponential) growth does not depend on the choice of the finite generating subset S ⊂ G although the value of γ(G,S) does.The groups ^d have subexponential growth. Indeed, if e_1,…,e_d is the canonical basis of ^d, and we take S {± e_1, …, ± e_d}, then the graph distance between two vertices g and h of (^d,S) is‖ g - h ‖_1, where we write ‖ x ‖_1 ∑_1 ≤ i ≤ d |x_i| for the 1-norm of x = (x_1,…,x_d) ∈^d.We then haveB_n ={g ∈^d : ‖ g ‖_1 ≤ n}⊂{0,± 1, …,± n}^dand hence |B_n| ≤(2 n + 1)^d. This implies that ^d has subexponential growth. Here it can be checked that the sequence(B_n)_n ∈ is also a Følner sequence for ^d. When G is an arbitraryfinitely generated group ofsubexponential growth and S is a finite symmetric generating set for G,it can be shown that one can always extract a Følner sequence from the sequence(B_n)_n ∈. Consequently, every finitely generated group with subexponential growth is amenable.A (nonabelian) free group on two generators has exponential growth and isnot amenable. Indeed, let G be a free group based on two generators a and b. Consider the finite symmetric generating subset S ⊂ G defined byS {a,b,a^-1,b^-1}.Then every element g ∈ G can be uniquely written in reduced form,reduced form of an element of a free group i.e., in the formg = s_1 s_2 … s_n,where n ≥ 0,s_i ∈ S for all 1 ≤ i ≤ n, and s_i + 1≠ s_i^-1 for all 1 ≤ i ≤ n - 1. The integer ℓ_S(g)n is called the lengthlength!— of an element in a free group of g with respect to the generators a and b. It is equal to the distance from g to 1_G in the Cayley graph (G,S). We deduce that|B_n| = 4 · 3^n - 1 for all n ≥ 1 so thatγ(G,S) = lim_n →∞√(4 · 3^n - 1) =3 > 1.This shows that G has exponential growth. Now suppose by contradiction that(F_n)_n ∈ is a Følner sequence for G and choosesome positive real number ε <1/2. Since the sequence (F_n)_n ∈ is Følner,it follows from (<ref>) that there existsan integer N ≥ 0 such that the setFF_N satisfies|F ∖ F s| ≤ε |F|for alls ∈ S.Denote, for each s ∈ S,byG_s the subset of G consisting of all elementsg ≠ 1_Gwhose reduced form ends with the letters^-1. The four sets G_s, s ∈ S, are pairwise disjoint so that∑_s ∈ S |F ∩ G_s| ≤ |F|.On the other hand, for each s ∈ S, we have that|F| = |F ∖ G_s| + |F ∩ G_s| =|(F ∖ G_s) s| + |F ∩ G_s|. We now observe that(G ∖ G_s) s ⊂ G_s^-1so that(F ∖ G_s) s ⊂ (F s ∖ F) ∪ (F ∩ G_s^-1)and hence|(F ∖ G_s) s|≤ |F s ∖ F| + |F ∩ G_s^-1|= |F ∖F s| +|F ∩ G_s^-1| ≤ε |F| +|F ∩ G_s^-1| (by (<ref>)).By using (<ref>), we deduce that|F| ≤ε |F| +|F ∩ G_s^-1| + |F ∩ G_s|for all s ∈ S. After summing up over all s ∈ S, this yields4|F|≤ 4ε |F| +∑_s ∈ S( |F ∩ G_s^-1| + |F ∩ G_s| ) = 4ε |F| +2∑_s ∈ S |F ∩ G_s|.Finally, combining with (<ref>), weobtain4|F| ≤ 4 ε |F| + 2 |F|and hence |F| ≤ 2 ε |F|,which is a contradiction since F ≠∅ andε < 1/2.This proves that G is not amenable. The class of amenable groups is closed under the operations of taking subgroups, quotients, extensions (this means that if 1 → H → G → K → 1 is an exact sequence with both H and Kamenable, so is G), and inductive limits. Consequently, all locally finite groups, all abelian groups and, more generally, all solvable groups are amenable. On the other hand every group containing a free subgroup on two generatorsis nonamenable.This implies for instance that allnonabelian free groups and thegroups (n,), n ≥ 2, are nonamenable. However, there aregroups containing no free subgroups on two generators that are nonamenable. The first examples of such a group was given in <cit.> where Ol'šanskiĭOl'šanskiĭ, Alexander Yu. constructed a nonamenable monster groupmonster groupgroup!monster — in which every proper subgroup is cyclic. Let us notethat there are finitely generated groups of exponential growth that are amenable. For example, theBaumslag-Solitar groupBaumslag-Solitar groupgroup!Baumslag-Solitar —BS(1,2), i.e.,the group with presentation⟨ a,b : a b a^-1 = b^2 ⟩, and thelamplighter group,lamplighter groupgroup!lamplighter — i.e., the wreath product (/2)≀,have exponential growth butare both solvable and hence amenable(cf. <cit.>). The original definition of amenability that was given by von Neumann <cit.>von Neumann, Johnin 1929 is that a group G is amenable if there exists a finitely additive invariant probability measure defined on the set of all subsets of G. A key observation due to Day <cit.>Day, Mahlon M. is that this is equivalent to the existence ofan invariant mean on the Banach space ℓ^∞(G) of bounded real-valued functions on G.It is also in <cit.> that the term amenableamenable group occured for the first time(see <cit.>). The fact that nonabelian free groups are not amenable is related tothe Hausdorff-Banach-Tarski paradoxHausdorff-Banach-Tarski paradox which actually was the motivation of von Neumann for introducing the notion of amenability. §.§ EntropyLet G becountable group, A a finite set, and X a subset of A^G (not necessarily a subshift). Given a finite subset Ω⊂ G, recall (cf. (<ref>)) thatX_Ω{x|_Ω: x ∈ X}⊂ A^Ω.Suppose now that the group G is amenable and fix a Følner sequence = (F_n)_n ∈ for G. The entropyentropysubshift!entropy of a — of X (with respect to ) is defined byh_(X) lim sup_n →∞log |X_F_n|/|F_n|.Since X_F ⊂ A^F and hence log |X_F| ≤ |F| ·log |A| for every finite subset F ⊂ G, we always haveh_(X) ≤ h_(A^G) = log |A|. Take G = andA = {0,1}. Let us compute the entropy of the golden mean subshift X ⊂ A^Gentropy!— of the golden mean subshiftgolden mean subshift!entropy of the —subshift!golden mean —(cf. Example <ref>) with respect to the Følner sequence = (F_n)_n ∈, where F_n {0,1,…,n}. We observe that u_n|X_F_n| satisfies u_0 = 2, u_1 = 3, and u_n + 2 = u_n + 1 + u_n for all n ≥ 2. Thus, the sequence (u_n)_n ∈ is a Fibonacci sequence and, by Binet's formula,u_n = 1/√(5)(φ^n + 3 - (1 - φ)^n + 3),where φ(1 + √(5))/2 is the golden mean (this is the origin ofthe name of this subshift). It follows thath_(X) = lim sup_n →∞log u_n/n + 1 = logφ. Take again G = andA = {0,1}, and let us compute now the entropy of the even subshift X ⊂ A^Gentropy!— of the even subshifteven subshift!entropy of the —subshift!even — (cf. Example <ref>) with respect to the Følner sequence = (F_n)_n ∈, where F_n {0,1,…,n}. We observe that u_n|X_F_n| satisfies u_0 = 2, u_1 = 4, and u_n + 2 = 1 + u_n + 1 + u_n for all n ≥ 2. Thus, the sequence (v_n)_n ∈, defined by v_n 1 + u_n for all n ∈,is a Fibonacci sequence. As v_0 = 3 and v_1 = 5,by applying again Binet's formula, we getu_n = - 1 + v_n = - 1 + 1/√(5)(φ^n + 5 - (1 - φ)^n + 5),where φ is the golden mean.It follows thath_(X) = lim sup_n →∞log u_n/n + 1 = logφ.Thus, the even subshift has the same entropy as the golden mean subshift with respect to . Take G = ^2 andA = /2. Let us compute the entropy of the Ledrappier subshiftentropy!— of the Ledrappier subshiftLedrappier subshift!entropy of the —subshift!Ledrappier —X ⊂ A^G (cf. Example <ref>) with respect to the Følner sequence = (F_n)_n ∈, where F_n {0,1,…,n}^2. We observe that, for each x ∈ X, the pattern x|_F_n is entirely determined byx |_H_n, where H_n ⊂^2 is the horizontal interval H_n {0,1, …,2n}×{0}.Therefore,u_n|X_F_n| satisfieslog u_n ≤ |H_n| ·log|A| = (2n + 1) log 2.This gives ush_(X) = lim sup_n →∞log u_n/(n + 1)^2 = 0. It is a deep result due to OrnsteinOrnstein, Donald S. and Weiss <cit.>Weiss, Benjamin that, when G is a countable amenable group, A a finite set, and X ⊂ A^Ga subshift,then the lim sup in (<ref>) is actually a true limit and does not depend on the particular choice of the Følner sequencefor G. However, we shall not need it in the sequel. An important property of cellular automata that we shall use in the proof of the Garden of Eden theorembelow is that they cannot increaseentropy. More precisely, we have the following result.Let G be a countable amenable group with Følner sequence = (F_n)_n ∈ and A,Bfinite sets.Suppose thatτ A^G → B^Gis a cellular automaton and X is a subset of A^G.Then one has h_(τ(X)) ≤ h_(X). For the proof, we shall use the following general property of Følner sequences.Let G be a countable amenable group with Følner sequence = (F_n)_n ∈ and let S be a finite subset of G. Then one haslim_n →∞|F_n S ∖ F_n|/|F_n| = 0. Observe thatF_nS ∖ F_n = ⋃_s ∈ S (F_ns ∖ F_n)so that|F_nS ∖ F_n| = | ⋃_s ∈ S (F_ns ∖ F_n)|≤∑_s ∈ S |F_ns ∖ F_n|.Thus we get|F_n S ∖ F_n|/|F_n|≤∑_s ∈ S|F_n s ∖ F_n|/|F_n|for all n ∈. As lim_n →∞|F_n s ∖ F_n|/|F_n| = 0for each s ∈ S by (<ref>), this gives us (<ref>).Let us set Y τ(X) and let S ⊂ G be a memory set for τ with 1_G ∈ S. Recall that it immediately follows from (<ref>)that if two configurations coincide ong S forsome g ∈ Gthen their images by τ take the same value at g. We deducethat|Y_Ω| ≤ |X_Ω S| for every finite subset Ω⊂ G. Now observe thatX_Ω S⊂ X_Ω× X_Ω S ∖Ω⊂ X_Ω× A^Ω S ∖Ω,so that we getlog |Y_Ω| ≤log |X_Ω| +| Ω S ∖Ω| ·log |A|.After replacing Ω by F_nand dividing both sides by |F_n|, this inequality becomeslog |Y_F_n|/|F_n|≤log |X_F_n|/|F_n| + |F_n S ∖ F_n|/|F_n|·log |A|.As lim_n →∞|F_n S ∖ F_n|/|F_n| = 0by Lemma <ref>, taking the limsup in (<ref>) finally gives the required inequality h_(Y) ≤ h_(X). Let G be a countable amenable group and let A,B befinite sets with |A| < |B|. Then there exists no surjective cellular automaton τ A^G → B^G. This is an immediate consequence ofProposition <ref> sinceh_(A^G) = log |A| < log |B| = h_(B^G).The following example (cf. <cit.>)shows thatCorollary <ref> becomes false ifthe amenability hypothesis is removed.Let G be the free group on two generators a and b.Take A /2 and B /2×/2, so that |A| = 2 and |B| = 4.Consider the map τ A^G → B^G defined by τ(x)(g) = (x(g) + x(g a), x(g) + x(g b)) for all x ∈ A^G and g ∈ G.Observe that τ is a cellular automaton with memory set S = {1_G,a,b} and local defining map μ A^G → B given by μ(p) = (p(1_G) + p(a),p(1_G) + p(b))for all p ∈ A^S. It is easy to check that τ is surjective. Note that A^G and B^G are totally disconnected compact abelian topological groups and τ is a continuous group morphism whose kernel consists of the two constant configurations in A^G. §.§ Tilings Let G be a group.Given a finite subset E⊂ G, let us say that a subset T ⊂ G is an E-tilingtiling of G provided the sets t E, t ∈ T, are pairwise disjoint and there exists a finite subset E' ⊂ G such that the setst E', t ∈ T,cover G. Take G = ^d and E = {0, ± 1, ± 2, …, ± m}^d for some m ∈, then T: = ((2m+1))^d⊂^d is an E-tiling (here one can take E' =E). Givenany nonempty finite subset E of a group G, we can useZorn's lemma to prove that there always existsan E-tiling T ⊂ G. Indeed, consider the set 𝒮(E) consisting of all subsets S ⊂ G such that the sets sE, s ∈ S, are pairwise disjoint. We first observe that 𝒮(E)is nonemptysince {1_G}∈𝒮(E). On the other hand, 𝒮(E) is inductivewith respect to set inclusion since if 𝒮' ⊂𝒮(E) is a chain, then M:= ∪_S ∈𝒮' S belongs to 𝒮(E) and is an upper bound for 𝒮'. By Zorn's lemma, there exists a maximal element T ∈𝒮(E).As T ∈𝒮(E),the sets t E, t ∈ T, are pairwise disjoint. Now, given anyg ∈ G, we can find, by maximality of T,an element t = t(g) ∈ T such that g E ∩ t E ≠∅ and hence g ∈ t E E^-1. It follows that the sets t E E^-1, t ∈ T, cover G.Since the set E'E E^-1 is finite, this shows that T is an E-tiling of G.For the proof of the Garden of Eden theoremin the next subsection, we shall use some technical results about tilings in amenable groups.Let G be a countable amenable group with Følner sequence= (F_n)_n ∈. Let E ⊂ G be a nonempty finite subset and T ⊂ G an E-tiling.Define, forn ∈, the subset T_n ⊂ Tby T_n {t ∈ T: t E ⊂ F_n}.Then there exist a constantα = α(,T) >0 and n_0 ∈ such that|T_n| ≥α |F_n|n ≥ n_0.Since T is a E-tiling, there exists a finite subset E' ⊂ G such that the sets t E', t ∈ T, cover G. After replacing E' by E' ∪ E, if necessary, we may assume that E ⊂ E'. Define, for n ∈,T_n^+ {t ∈ T: tE' ∩ F_n ≠∅}.Clearly T_n ⊂ T_n^+. As the sets tE', t ∈ T_n^+, cover F_n, we have |F_n| ≤ |T_n^+| · |E'| so that|T_n^+|/|F_n|≥1/|E'|for all n ∈. Now observe thatT_n^+ = T ∩(⋃_g ∈ E' F_n g^-1) and T_n = T ∩(⋂_h ∈ EF_n h^-1),so thatT_n^+ ∖ T_n= T ∩( ⋃_g ∈ E' F_n g^-1∖⋂_h ∈ EF_n h^-1)⊂⋃_g ∈ E' F_n g^-1∖⋂_h ∈ EF_n h^-1= ⋃_g ∈ E',h ∈ E (F_n g^-1∖ F_n h^-1).We deduce that|T_n^+ ∖ T_n| ≤∑_g ∈ E',h ∈ E |F_n g^-1∖ F_n h^-1| = ∑_g ∈ E',h ∈ E |F_n ∖ F_n h^-1 g|.Aslim_n →∞|F_n∖ F_n h^-1 g|/|F_n| =0 for all g ∈ E' and h ∈ E by (<ref>),it follows that|T_n^+ | - | T_n |/|F_n| = | T_n^+ ∖ T_n |/|F_n|→ 0asn →∞.Using (<ref>) and taking ε12|E'|,we deduce thatthere exists n_0 ∈ such that|T_n|/|F_n| = |T_n^+|/|F_n| - | T_n^+ | - | T_n |/|F_n|≥1/|E'| - ε = α,where α12|E'|, for all n ≥ n_0. Let G be a countable amenable group with Følner sequence= (F_n)_n ∈ and let A be a finite set. Let X ⊂ A^G be a subset and suppose there exist a nonempty finite subset E ⊂ G and an E-tiling T ⊂ G such that X_tE⫋ A^tE for all t ∈ T. Then h_(X) < log |A|.Let us set, as above, T_n {t ∈ T: tE ⊂ F_n} and write F_n^*F_n ∖⋃_t ∈ T_n t E, for all n ∈. Observe that ⋃_t ∈ T_n t E ⊂ F_n so that X_F_n⊂ A^F_n^*×∏_t ∈ T_n X_tE and|F_n| = |F_n^*| + |T_n| · |E|. It follows thatlog |X_F_n|≤| F_n^*|·log |A| +∑_t ∈ T_nlog |X_tE| ≤|F_n^*| ·log |A| +∑_t ∈ T_nlog(|A^tE|-1)= | F_n^* |·log |A| +|T_n| ·log (|A|^|E| - 1)= | F_n^* |·log |A| +|T_n| · |E| ·log |A| + |T_n| ·log (1 - |A|^-|E|) = | F_n |·log |A| + |T_n| ·log (1 - |A|^-|E|),where the last equality follows from (<ref>). Setting c - log (1 - |A|^-|E|) > 0, we deduce thath_(X) = lim sup_n →∞log |X_F_n|/|F_n|≤log |A| - c α < log |A|,where α = α(,T) is as in (<ref>). Let G be a countable amenable group with Følner sequence= (F_n)_n ∈ and let A be a finite set. Let X ⊂ A^G be a subshiftand suppose that there exists a nonempty finite subset E ⊂ G such that X_E⫋ A^E.Then one has h_(X) < log |A|.If T is an E-tiling of G, we deduce from the shift-invariance of X that X_t E⫋ A^t E for all t ∈ T, so that Proposition <ref> applies. §.§ The Garden of Eden theorem for amenable groupsThe following result is due to Machì,Scarabotti, and the first author <cit.>. Since the groups ^d are all amenable, it extends Theorem <ref>.Garden of Eden theorem!— (for amenable groups)theorem!Garden of Eden — (for amenable groups) Let G be a countable amenable group with Følner sequence= (F_n)_n ∈ and A a finite set. Suppose that τ A^G → A^G is a cellular automaton. Then the following conditions are equivalent: * τ is surjective;* h_(τ(A^G)) = log |A|;* τ is pre-injective. The implication (a)(b) is obvioussince h_(A^G) = log|A|. In order to show the converse implication, let us suppose that τ is not surjective, that is, the image subshift X τ(A^G) is such thatX ⫋ A^G. Since X is closed in A^G, there existsa finite subset E ⊂ G such that X_E ⫋ A^E. By applying Corollary <ref>, we deduce that h_(X)< log|A|. This shows (b)(a). Let S ⊂ G be a memory set for τ such that 1_G ∈ S. Let usshow (b)(c). Suppose that τ is not pre-injective. By virtue of Proposition <ref>, we can find a nonempty finite subset Ω⊂ G and two distinct patterns p_1, p_2 ∈ A^Ω that are mutually erasable for τ. Let E Ω S^-1S. Observe that Ω⊂ E since 1_G ∈ S. LetT ⊂ G be an E-tiling of G. Consider the subset Z ⊂ A^G defined byZ{z ∈ A^G: z|_t Ω≠ tp_1t ∈ T}.Observe that Z_tE⫋ A^t E for all t ∈ T. By using Proposition <ref> and Proposition <ref>, we deduce that h_(τ(Z))≤ h_(Z) < log |A|. We claim that τ(Z) = τ(A^G). Let x ∈ A^G. Let T_x{t ∈ T: x|_tΩ = tp_1} and define z ∈ Z by setting, for all g ∈ G,z(g)tp_2(g)x(g)Let us check that τ(z) = τ(x). Let g ∈ G. If g ∉∪_t ∈ T_x tΩ S^-1, then gS ∩ tΩ = ∅ for all t ∈ T_x and therefore z|_gS = x|_gS, so that τ(z)(g) = τ(x)(g). Suppose now that g ∈ tΩ S^-1 for some (unique) t = t(g) ∈ T_x and consider the configuration y ∈ A^G defined by setting, for all h ∈ G,y(h)tp_2(h) h ∈ tΩx(h)Observe that x|_G ∖ tΩ = y|_G ∖ tΩ. Since the patterns x|_tΩ = tp_1 and y|_tΩ = tp_2 are mutually erasable, we deduce that τ(y) = τ(x). Moreover, as gS ⊂tΩ S^-1S = tE, we have z|_gS = y|_gS, and therefore τ(z)(g) = τ(y)(g) = τ(x)(g). This shows that τ(z) = τ(x), and the claim follows. We conclude that h_(τ(A^G)) = h_(τ(Z)) < log |A|. This shows the implication (b) (c). Finally, let us show(c)(b). Let us set as above X τ(A^G) and suppose that h_(X) < log |A|. As 1_G ∈ S, we have F_n ⊂ F_n S^-1 so thatX_F_n S^-1⊂ X_F_n× A^F_n S^-1∖ F_n,for all n ∈. We deduce thatlog |X_F_n S^-1|/|F_n|≤log |X_F_n|/|F_n| +|F_n S^-1∖ F_n|/|F_n|log |A|.Aslim_n →∞|F_n S^-1∖ F_n|/|F_n| =0by Lemma <ref>, we deduce from (<ref>) thatlim sup_n →∞log |X_F_n S^-1|/|F_n|≤lim sup_n →∞log |X_F_n|/|F_n| = h_(X) < log |A|.Consequently, we can find n_0 ∈ such that,|X_F_n_0 S^-1| < |A|^|F_n_0|.Fix a_0 ∈ A and consider the subset Z ⊂ A^G defined byZ{z ∈ A^G: z(g) = a_0g ∈ G ∖ F_n_0}.Note that |Z| = |A|^|F_n_0|. Let z_1, z_2 ∈ Z. If g ∈ G ∖ F_n_0 S^-1, then z_1 and z_2 coincide on gS ⊂ G ∖ F_n_0 so that τ(z_1)(g) = τ(z_2)(g). Therefore τ(z_1) and τ(z_2) coincide on G ∖ F_n_0S^-1. This impliesthat|τ(Z)| ≤ |X_F_n_0 S^-1|. Using (<ref>), we deduce that |τ(Z)| < |Z|. By the pigeon-hole principle, there exist two distinct elements z_1, z_2 ∈ Z such that τ(z_1) = τ(z_2). As all elements in Z are almost equal (they coincide outside of the finite set F_n_0), we conclude that τ is not pre-injective.§ FAILURE OF THE GARDEN OF EDEN THEOREM FOR NONAMENABLE GROUPSLet us say that a countable group G has the Moore propertyMoore property!— for a groupgroup!— satisfying the Moore propertyifevery surjective cellular automaton τ A^G → A^G with finite alphabet A over G ispre-injective and that it has the Myhill propertyMyhill property!— for a groupgroup!— satisfying the Myhill propertyif every pre-injective cellular automaton τ A^G → A^G with finite alphabet A over G is surjective. Also let ussay that a countable group G satisfies the Moore-Myhill propertyMoore-Myhill property!— for a groupgroup!— satisfying the Moore-Myhill propertyor that it satisfies the Garden of Eden theoremGarden of Eden theorem!group satisfying the —group!— satisfying the Garden of Eden theoremif G has both the Moore and the Myhill properties. Theorem <ref> tells us that every countable amenable group satisfies the Garden of Eden theorem. The examples below, essentially due to Muller <cit.>Muller, David E. (see also <cit.>, <cit.>, <cit.>),show that neither the Moore nor the Myhill property holds for countable groups containingnonabelian free subgroups.Let G be a countable group and suppose that G contains two elements a and b generating a nonabelian free subgroup H ⊂ G.Take A = {0,1} and let S {a,a^-1,b,b^-1}.Consider the cellular automaton τ A^G → A^Gwith memory set {1_G}∪ S defined byτ(x)(g)0 ifx(g) + x(ga) + x(ga^-1) + x(gb) + x(gb^-1) ≤ 2 1 otherwisefor all x ∈ A^G and g ∈ G. The pair of configurations (x_1, x_2) ∈ A^G × A^G, defined by x_1(g) = 0 for all g ∈ G, and x_2(g) = 0 for all g ∈ G ∖{1_G} and x_2(1_G) = 1, is a diamond for τ.Therefore τ is not pre-injective. However, τ is surjective.To see this, let y ∈ A^G. Let us show that there exists x ∈ A^G such that τ(x) = y.Let R ⊂ G be a complete set of representatives of the left cosets of H in G. We define x as follows. Every element g ∈ G can be uniquely written in the form g = r h with r ∈ R and h ∈ H.If g ∈ R, i.e., h = 1_G, weset x(g)0.Otherwise, we set x(g)y(rh^-), where h^- is the predecessor of h in H, i.e.,the unique elementh^- ∈ H such that ℓ_S(h^-) = ℓ_S(h) - 1 and h = h^- s for some s ∈ S (here ℓ_S(·) denotes the length of the reduced form for elements of H, see Example <ref>). One easily checks thatτ(x) = y. This shows that τ is surjective. Thus the Moore implication fails to hold forgroups containing nonabelian free subgroups.Let G be a countable group and suppose that G contains two elements a and b generating a nonabelian free subgroup H ⊂ G.Let A = /2×/2 be the Klein four-groupand consider the group endomorphisms p and q of A respectively defined by p(α,β) = (α, 0) and q(α,β) = (β, 0) for all(α,β) ∈ A. Let τ A^G → A^G be the cellular automaton with memory set S {a,a^-1, b, b^-1} defined byτ(x)(g) = p(x(ga)) + p(x(ga^-1) + q(x(gb)) +q(x(gb^-1))for all x ∈ A^G and g ∈ G. The image of τ is contained in (/2×{0})^G. Therefore τ is not surjective. We claim that τ is pre-injective. As τ is a group endomorphism of A^G, it suffices to show that there is no configuration with finite support in the kernel of τ. Assume on the contrarythat there is an element x ∈ A^G with nonempty finite supportΩ{g ∈ G : x(g) ≠ 0_A }⊂ G such that τ(x) = 0. Let R ⊂ G be a complete set of representatives of the left cosets of H in G. Let us setΩ_r Ω∩ rH for all r ∈ R. Then Ω is the disjoint union of the sets Ω_r, r ∈ R. Let r ∈ R such that Ω_r ≠∅ and consider an element g = r h ∈Ω_r with h ∈ Hat maximal distance from the identity in the Cayley graph of (H,S) (i.e., with ℓ_S(h) maximal). We have x(g) = (α,β) ≠ (0,0) = 0_A. Suppose first that α≠ 0. We can find s ∈{a,a^-1} such that ℓ_S(h s) = ℓ_S(h) + 1. For all t ∈ S ∖{s^-1}, we have thatℓ_S(h s t) = ℓ(h) + 2 and hence x(g s t) = 0_A by maximality. It follows thatτ(x)(g s) = p(x(g ) )= (α,0) ≠ 0_A,which contradicts the fact that x is in the kernel of τ. Suppose now that α = 0. Then β≠ 0. We take now s ∈{b,b^-1} such that ℓ_S(h s) = ℓ_S(h) + 1. By an argument similar to the one that we used in the first case, we getτ(x)(g s) = q(x(g ) )= (β,0) ≠ 0_A,so that wearrive at a contradiction also in this case. Thus τ is pre-injective. This shows that the Myhill implication fails to hold forgroups containing nonabelian free subgroups. As mentioned in Subsection <ref>, there are nonamenable countable groups containing no nonabelian free subgroups. However, Bartholdi <cit.>Bartholdi, Laurent (see <cit.>) proved that the Moore property fails to hold for all nonamenable countable groups. Recently, Bartholdiand Kielak <cit.> also proved that the Myhill property fails to hold for all nonamenable countable groups. Combining these results with the Garden of Eden theorem for amenable groups(Theorem <ref>), this yields the following characterization of amenability in terms of cellular automata.Let G be a countable group. Then the following conditions are equivalent: * G is amenable;* G has the Moore property;* G has the Myhill property;* G satisfies the Garden of Eden theorem.§ THE GARDEN OF EDEN THEOREM FOR SUBSHIFTS §.§ Strongly irreducible subshiftsLet G be a countable group and A a finite set. A subshift X ⊂ A^G is called strongly irreduciblestrongly irreducible subshiftsubshift!strongly irreducible — if there is a finite subset Δ⊂ G satisfying the following property:if Ω_1 and Ω_2 are finite subsets of G such thatΩ_1 Δ does not meet Ω_2,then, given any two configurations x_1,x_2 ∈ X, there existsa configuration x ∈ X which coincides with x_1 on Ω_1 and with x_2 on Ω_2. The full shift A^G is strongly irreducible (one can take Δ = {1_G}).The even subshift X ⊂{0,1}^, described in Example <ref>, is strongly irreducible (one can take Δ = {-2,-1,0,1,2}).even subshiftsubshift!even —The hard-ball model,described in Example <ref>, is strongly irreducible (one can take Δ = {0, ± e_1, …,± e_d}).hard-ball modelsubshift!hard-ball model — In particular(d = 1), the golden mean subshift is strongly irreducible.The Ledrappier subshift, described in Example <ref>,is not strongly irreducible.Ledrappier subshiftsubshift!Ledrappier —Fiorenzi <cit.> obtained the following extension of Theorem <ref>.Let G be a countable amenable group with Følner sequence= (F_n)_n ∈ and A,Bfinite sets. Suppose that X ⊂ A^G is a strongly irreducible subshift of finite type and Y ⊂ B^G is a strongly irreducible subshift with h_(X) = h_(Y) and that τ X → Y is a cellular automaton. Then the following conditions are equivalent: * τ is surjective;* h_(τ(X)) = h_(Y);* τ is pre-injective. The cellular automaton τ X → Y from the golden mean subshift to the even subshift described in Example <ref> satisfies all the hypotheses in the previous theorem. As τ is pre-injective (cf. Example <ref>), we deduce that τ is surjective. Note that here one might also easily obtainsurjectivity of τ by a direct argument. §.§ The Moore and the Myhill properties for subshifts Let G be a countable group, A a finite set, and X ⊂ A^G a subshift. One says that the subshiftX has the Moore propertyMoore property!— for a subshiftsubshift!— satisfying the Moore property if every surjective cellular automatonτ X → X is pre-injective and that it has the Myhill propertyMyhill property!— for a subshiftsubshift!— satisfying the Myhill property if every pre-injective cellular automatonτ X → X is surjective. One says that X has the Moore-Myhill propertyMoore-Myhill property!— for a subshiftsubshift!— satisfying the Moore-Myhill property or that it satisfies the Garden of Eden theorem if it has both the Moore and the Myhill properties.Garden of Eden theorem!subshift satisfying the —subshift!— satisfying the Garden of Eden theoremFrom Theorem <ref>, we immediately deduce the following.Let G be a countable amenable group and A a finite set. Then every strongly irreducible subshift of finite type X ⊂ A^G has the Moore-Myhill property. Let G = ^d and A = {0,1}. Consider the hard-ball model X ⊂ A^G described in Example <ref>.hard-ball modelsubshift!hard-ball model — As ^d is amenable and X is both strongly irreducible and of finite type, we deduce from Corollary <ref> that X has the Moore-Myhill property. In particular (d = 1), the golden mean subshift has the Moore-Myhill property.golden mean subshiftsubshift!golden mean — [Fiorenzi]Let A = {0,1} and let X ⊂ A^ be the even subshift(cf. Example <ref>).even subshiftsubshift!even — Consider the cellular automatonσ A^→ A^ with memory set S = {0,1,2,3,4} and local defining map μ A^S → A given byμ(y) =10Then one has σ(X) ⊂ X, and the cellular automaton τσ|_X X → X is not pre-injective. Indeed, the configurations x_1, x_2 ∈ X defined byx_1 = ⋯ 0 ⋯ 00(100)100 ⋯ 0 ⋯andx_2 =⋯ 0 ⋯ 00(011)100 ⋯ 0 ⋯satisfyτ(x_1) =⋯ 1 ⋯ 11(100)10011 ⋯ 1 ⋯ = τ(x_2).Observe, alternatively, that the patterns p,q with support Ω{0,1, …, 12} defined byp(n) =1n = 6,90 q(n) =1n = 7,8,90for all n ∈Ω, are ME.From a case-by-case analysis, one can show that τ is surjective. It follows that X does not have the Moore property. We refer to <cit.> and <cit.> for more details. As the even subshift is strongly irreducible andis amenable, the previous example shows that the hypothesis that X is of finite type cannot be removed from Corollary <ref>.However, we have the following (cf. <cit.>).Let G be a countable amenable group andAa finite set.Then every strongly irreducible subshift X ⊂ A^G has the Myhill property.The even subshift X ⊂{0,1}^ has the Myhill property since it is strongly irreducibleandis amenable.even subshiftsubshift!even — Let A = {0,1}.Let x_0,x_1 ∈ A^ denote the two constant configurationsrespectively defined by x_0(n) = 0 and x_1(n) = 1 for all n ∈. Note that X = {x_0,x_1} is a subshift of finite type. The map τ X → X given by τ(x_0) = τ(x_1) = x_0 is a cellular automaton which is pre-injective but not surjective. It follows that X does not have the Myhill property. This very simple example shows that the hypothesis that X is strongly irreducible cannot be removed neither from Corollary <ref> nor from Theorem <ref>. Note that X has the Moore property since X is finite, so that every surjective self-mapping of X is injective and therefore pre-injective. [Fiorenzi] Let A = {0,1,2} and let X ⊂ A^ be the subshift of finite type consisting of all x ∈ A^ such thatx(n) x(n + 1) ∉{01,02} for alln ∈.Thus a configuration x → A is in X if and only if one of the following conditions is satisfied: * x(n) = 0 for all n ∈;* x(n) ≠ 0 for all n ∈;* there exists n_0 ∈ such that x(n) ∈{1,2} for alln ≤ n_0 and x(n) = 0 for all n > n_0.Consider the cellular automaton σ A^→ A^ with memory set S = {0,1} and local defining mapμ(y) =y(0) y(1) ≠ 00Observe that σ(x) = x if x ∈ X is of type (i) or (ii) while, if x ∈ X is of type (iii), then σ(x) is obtained from x by replacing its rightest nonzero term by 0.We deducethatσ(X) ⊂ X and that thecellular automaton τσ|_XX → X is surjective but not pre-injective (see <cit.>). It follows that X does not have the Moore property. It turns out that X does not have the Myhill property either. Indeed, consider now the cellular automaton σ'A^→ A^ with memory set S' = {-1,0} and local defining mapμ'(y) =y(0) y(-1)y(0) ∉{10,20}y(-1)Observe that σ'(x) = x if x ∈ X is of type (i) or (ii) while, if x ∈ X is of type (iii), then σ'(x) is obtained from x by repeating on its right its rightestnonzero term.We deduce that σ'(X) ⊂ X and thatthe cellular automaton τ' σ'|_XX → X is injective and hence pre-injective.However, τ' isnot surjective (observe for instance that the pattern p ∈ A^{-1,0,1} defined by p(-1)p(0)p(1) = 120 is a Garden of Eden pattern for τ'). Let A be a finite set and X ⊂ A^ a subshift. One says that a word u ∈ A^⋆ of length n appearsword!— appearing in a subshiftsubshift!word appearing in a — in X if there is a configuration x ∈ X and m ∈ such that u = x(m)x(m+1) ⋯ x(m+n - 1). The subset L(X) ⊂ A^⋆ consisting of all words that appear in X is called the languagelanguage of a subshiftsubshift!language of a — of X. One says that the subshift X is irreducibleirreducible subshiftsubshift!irreducible — if given any two words u,v ∈ L(X) there exists a word w ∈ L(X) such that u w v ∈ L(X). Clearly every strongly irreducible subshift X ⊂ A^ is irreducible. The converse is false as shown by the following example. Let A = {0,1} and consider the subshift X ⊂ A^ consisting of the two configurationsx ∈ A^ that satisfy x(n) ≠ x(n + 1) for all n ∈. It is clear that X is irreducible but not strongly irreducible. Observe that X is of finite type.The following result is an immediate consequence of <cit.> (cf. <cit.>). Let A be a finite set. Then every irreducible subshift of finite type X ⊂ A^ has the Moore-Myhill property.§ GARDEN OF EDEN THEOREMS FOR OTHER DYNAMICAL SYSTEMS §.§ Dynamical systemsBy a dynamical system,dynamical system wemean a triple (X,G,α), where X is a compact metrizable space, G is a countable group, andα is a continuousaction of G on X. The space X is called the phase spacephase spacedynamical system!phase space of a — of the dynamical system.If there is no risk of confusion, we shall write (X,G), or even sometimes simply X,instead of (X,G,α).LetG be a countable groupand A a compact metrizable topological space (e.g. a finite set with its discrete topology). Equip A^G = {xG → A} with the product topology. The shift actionshift!— actionshiftaction!shift — σ of G on A^G is the action defined by σ(g,x) = g x, where(g x)(h) = x(g^-1 h)for all x ∈ A^G and g,h ∈ G. Then (A^G,G,σ) is a dynamical system. If(X,G,α) is a dynamical system andY ⊂ X a closed α-invariant subset, then (Y,G,α|_Y), where α|_Y denotes the action of G on Y induced by restriction of α,is a dynamical system. In particular, if G is a countable group, A a finite set, and X ⊂ A^G a subshift, then(X,G,σ|_X) is a dynamical system.Let fX → X be a homeomorphism of a compact metrizable space X. The dynamical system generateddynamical system!— generated by a homeomorphism by f is the dynamical system (X,,α_f), where α_f is the action ofon X given by α_f(n,f)f^n(x) for all n ∈ and x ∈ X. We shall alsowrite (X,f) to denote the dynamical system generated by f. If we fix the countable group G, the dynamical systems (X,G) are the objects of a concrete category _G in which the morphisms from an object X ∈_G to another object Y ∈_G consist of allequivariant continuous maps τ X → Y.It follows from the Curtis-Hedlund-Lyndon theorem (cf. Theorem <ref>) that the category _G described inRemark <ref> is a full subcategory of the category_G. Let (X,G) and (X,G) be two dynamical systems. One says that the dynamical systems (X,G) and(X,G) are topologically conjugatetopologically!— conjugate dynamical systems if they are isomorphic objects in the category _G, i.e.,if there exists an equivariant homeomorphismh X→ X. One says that (X,G) is a factorfactor!— of a dynamical systemdynamical system!factor of a — of(X,G) if there exists an equivariant continuous surjective map θX→ X.Such a map θ is then called a factor map.factor!— map A factor map θX→ X is said to be finite-to-onefactor!finite-to-one — mapfinite-to-one factor mapif the pre-image setθ^-1(x) is finite for each x ∈ X. A finite-to-one factor map is said to be uniformly bounded-to-onefactor!uniformly bounded-to-one — mapuniformly bounded-to-one factor map if there is an integer K ≥ 1 such that |θ^-1(x)| ≤ K for allx ∈ X. §.§ ExpansivenessOne says that a dynamical system(X,G) is expansiveexpansive dynamical systemdynamical system!expansive — if there exists a neighborhoodW ⊂ X × X of the diagonalΔ_X {(x,x) : x ∈ X}⊂ X × X such that, for every pair of distinct points x,y ∈ X, there exists an elementg = g(x,y)∈ G such that (g x, g y) ∉ W. Such a set W is then called an expansiveness neighborhoodexpansiveness neighborhood of the diagonal. If d is a metric on X compatible with the topology, the fact that (X,G) is expansive is equivalent to the existence of a constant δ> 0 such that, for every pair of distinct points x,y ∈ X, there exists an element g = g(x,y)∈ G such that d(g x,g y) ≥δ. Let G be a countable group and A a finite set. Then the G-shift on A^G is expansive. Indeed, it is clear thatW {(x,y) ∈ A^G × A^G : x(1_G) = y(1_G)}is an expansiveness neighborhood of Δ_A^G.If (X,G) is an expansive dynamical system and Y ⊂ X is a closed invariant subset, then (Y,G) is expansive. Indeed, if W is an expansiveness neighborhood of Δ_X, thenW ∩ (Y × Y) is an expansiveness neighborhood of Δ_Y. In particular, if G is a countable group, A a finite set, and X ⊂ A^G a subshift, then the dynamical system (X,G,σ|_X) is expansive.§.§ Homoclinicity Let (X,G) be a dynamical system. Two points x,y ∈ X are called homoclinichomoclinic!— points with respect to the action of G on X, ore more briefly, G-homoclinic, if for any neighborhood W ⊂ X × X of the diagonal Δ_X, there is a finite set F = F(W,x,y) ⊂ G such that (gx,gy) ∈ W for all g ∈ G ∖ F. If d is a metric on X that is compatible with the topology, two points x, y ∈ X are homoclinicif and only iflim_g →∞ d(g x, g y) = 0,where ∞ is the point at infinity in the one-point compactification of the discrete group G. This means that, for every ε > 0, there is a finite subset F = F(ε,d,x,y) ⊂ G such thatd(g x,g y) < ε for all g ∈ G ∖ F. Homoclinicity clearly defines an equivalence relation on X (transitivity follows from the triangle inequality).The equivalence classes of this relation are called the G-homoclinicity classeshomoclinicity classof X.Let (X,G) be a dynamical system and Y a set. One says that a map τ X → Y is pre-injectivepre-injective!— map if its restriction to each G-homoclinicity class is injective. Let G be a countable group and Aa finite set. Two configurations x, y ∈ A^G are homoclinic with respect to the shift action of G on A^G if and only if they are almost equal (see e.g. <cit.>). Indeed, first observe that the setsW_Ω{(x,y) ∈ A^G × A^G : x|_Ω = y|_Ω},where Ω runs over all finite subsets of G, form a neighborhood base of the diagonalΔ_A^G⊂ A^G × A^G (this immediately follows from the definition of the product topology). Now, if x, y ∈ A^G are almost equal, then the set D ⊂ G consisting of all g ∈ G such that x(g) ≠ y(g) is finite, so that Ω D^-1 is also finite for every finite subset Ω⊂ G. As (g x, g y) ∈ W_Ω for every g ∈ G ∖Ω D^-1, this implies that x and y are homoclinic. Conversely, suppose that x, y ∈ A^G are homoclinic. Then there exists a finite subset F ⊂ G such that (g x, g y) ∈ W_{1_G} for allg ∈ G ∖ F. This implies that x(g) = y(g) for all g ∈ G ∖ F^-1, so that x and y are almost equal. Let (X,G,α) be a dynamical system and Y ⊂ Xa closed invariant subset. Denote as above by α|_Y the restriction of α to Y. Then two points x,y ∈ Y are homoclinic with respect to α|_Y if and only if they are homoclinic with respect to α. In particular, if G is a countablegroup, A a finite set, and X ⊂ A^G a subshift, then two configurations x, y ∈ X are homoclinic with respect to σ|_X if and only if they are almost equal. It follows that the definition of pre-injectivity for cellular automata between subshifts given in Definition <ref>agrees withthe one given in Definition <ref> above.§.§ The Moore and the Myhill properties for dynamical systemsLet (X,G) be a dynamical system. An endomorphismendomorphism of a dynamical systemdynamical system!endomorphism of a — of (X,G) is a continuous equivariant map τ X → X. One says that the dynamical system(X,G) has the Moore propertyMoore property!— for a dynamical systemdynamical system!Moore property for a — if every surjective endomorphism of (X,G)is pre-injective and that it has the Myhill propertyMyhill property!— for a dynamical systemdynamical system!Myhill property for a — if every pre-injectiveendomorphism of (X,G)is surjective. One says that (X,G) has the Moore-Myhill propertyMoore-Myhill property!— for a dynamical systemdynamical system!Moore-Myhill property for a —or that it satisfies the Garden of Eden theoremGarden of Eden theorem!dynamical system satisfying the —dynamical system!— satisfying the Garden of Eden theorem if it has both the Moore and the Myhill properties. Observe that all these properties are invariants of topological conjugacy in the sense that if the dynamical systems (X,G) and (Y,G) are topologically conjugate then (X,G) has the Moore (resp. the Myhill, resp. the Moore-Myhill) property if and only if (Y,G) has the Moore (resp. the Myhill, resp. the Moore-Myhill) property. In the particular case when (X,G) is a subshift, it immediately follows from Theorem <ref>and Example <ref> that these definitions are equivalent to the ones given in Subsection <ref>. §.§ Anosov diffeomorphismsLet fM → M be a diffeomorphism of a compact smooth manifold M. One says that f is an Anosov diffeomorphismAnosov diffeomorphism (see e.g. <cit.>, <cit.>, <cit.>)if the tangent bundle TM of Mcontinuously splits as a direct sum TM = E_s ⊕ E_u of two df-invariant subbundles E_s and E_u such that, with respect to some (or equivalently any) Riemannian metric on M,the differential df is exponentiallycontracting on E_s and exponentially expanding on E_u, i.e.,there are constantsC > 0 and 0 < λ < 1 such that * ‖ df^n(v) ‖≤ Cλ^n ‖ v ‖,* ‖ df^-n(w) ‖≤ C λ^n ‖ w ‖for all x ∈ M, v ∈ E_s(x),w ∈ E_u(x), and n ≥ 0.[Arnold's cat]Arnold's catdynamical system!Arnold's cat — Consider the matrixA =[ 0 1; 1 1 ] and the diffeomorphismfof the 2-torus ^2 = ^2/^2 = /×/given byf(x)A x = [ x_2; x_1 + x_2 ]for all x = [ x_1; x_2 ]∈^2.The dynamical system (^2,f) is known as Arnold's cat.dynamical system!Arnold's cat — The diffeomorphism f is Anosov. Indeed, the eigenvalues of A are λ_1 = - 1φ, andλ_2 = φ, where φ1 + √(5)2 is the golden mean. As -1 < λ_1 < 0 and 1 < λ_2, it follows that df = A is exponentially contracting in the direction of the eigenline associated withλ_1 and uniformly expanding in the direction of the eigenline associated withλ_2. [Hyperbolic toral automorphism]hyperbolic toral automorphism More generally, if A ∈_n() has no eigenvalue on the unit circle, then the diffeomorphism f of the n-torus ^n = ^n/^n, defined by f(x) = A x for all x ∈^n, is Anosov. Such adiffeomorphism is called a hyperbolic toral automorphism. In <cit.>, we obtained the following result.Let f be an Anosov diffeomorphism of the n-dimensional torus ^n. Then the dynamical system (^n,f) has the Moore-Myhill property. The proof given in <cit.> uses twoclassical results. The first one is the Franks-Manning theorem <cit.>, <cit.>Franks-Manning theoremtheorem!Franks-Manning —which states that (^n,f) is topologically conjugate to a hyperbolic toral automorphism.The second one is a result of Walters <cit.> which says that every endomorphism of a hyperbolic toral automorphism is affine. We do not know ifthe dynamical system (M,f) has the Moore-Myhill property whenever f is an Anosov diffeomorphism of a compact smooth manifold M. However, we have obtained in <cit.> the following result.Let X be a compact metrizable space equippedwith a continuous action of a countable amenable group G.Suppose that the dynamical system (X,G) is expansiveand that there exist a finite set A, a strongly irreducible subshiftX⊂ A^G, and a uniformly bounded-to-one factor mapθX→ X. Then the dynamical system (X,G) has the Myhill property. A homeomorphism f of a topological space X is called topologically mixingtopologically!— mixing dynamical systemdynamical system!topologically mixing — if, given any two nonempty open subsetsU,V ⊂ X, there exists an integer N ≥ 0 such that f^n(U) ∩ V ≠∅ for all n ∈ that satisfy|n| ≥ N. By the classical work of BowenBowen, Robert E. (Rufus) (cf. <cit.> and<cit.>),dynamical systems generated by topologically mixing Anosov diffeomorphisms satisfy the hypotheses of Theorem <ref>.As a consequence (cf. <cit.>), we get the following partial extension ofTheorem <ref>.Letf be a topologically mixing Anosov diffeomorphism of acompact smooth manifold M. Then the dynamical system (M,f) has the Myhill property.All known examples of Anosov diffeomorphisms are topologically mixing. Also, all compact smooth manifolds that are known to admit Anosov diffeomorphisms are infra-nilmanifolds.We recall that a nilmanifoldnilmanifold is a manifoldof the form N/Γ, where N is a simply-connected nilpotent Lie group and Γ is a discrete cocompact subgroup of N and that a infra-nilmanifoldinfra-nilmanifoldis a manifold that is finitely covered by some nilmanifold.§.§ Weak specification Recently, after our preprint <cit.> had circulated, Hanfeng LiLi, Hanfeng posted his paper <cit.> containing an impressive Garden of Eden type theorem generalizing several results mentioned above (see Theorems <ref> and <ref>, and Corollaries <ref> and <ref>).The key notion in Li's paper is that of specification, a strong orbit tracing property which was introduced by Rufus BowenBowen, Robert E. (Rufus) for -actions in relation to his studies on Axiom A diffeomorphisms in <cit.> (see also <cit.>) and was subsequently extended to ^d-actions by RuelleRuelle, David in<cit.>. Several versions and generalizations of specification have appeared in the literature(see, in particular, <cit.> and <cit.>). Here is the one we need (cf. <cit.>).A dynamical system (X,G) has the weak specification propertyweak specification propertydynamical system!weak specification property for a —if for any ε > 0 there exists a nonempty symmetric finite subset Δ⊂ G satisfying the following property: if (Ω_i)_i ∈ I is any finite family of finite subsets of G such that ΔΩ_i ∩Ω_j = ∅ for all distinct i,j ∈ I, and (x_i)_i ∈ Iis any family of points in X, then there exists x ∈ X such that d(sx,sx_i) ≤ε i ∈ Is ∈Ω_i,where d is any metric compatible with the topology on X. It is straightforward (cf. <cit.>) to check that if G is a countable group, A is a finite alphabet set, and X ⊂ A^G is a subshift, then the shift dynamical system (X,G) has the weak specification propertyif and only if it is strongly irreducible (cf. Section <ref>).Also, it is easy to see that the weak specification property passes to factors.Li <cit.> proved the following:Let (X,G) be a dynamical system. Suppose that the group G is amenable and that (X,G) is expansive and has the weak specification property. Then (X,G) has the Myhill property. Note that Theorem <ref> covers Theorem <ref>, by virtue of the remarks followingDefinition <ref>.Recall (cf. Example <ref>) that if X ⊂{0,1}^ denotes the even subshift(cf. Example <ref>),then (X, ) is expansive, has the weak specification property (since X is strongly irreducible), but does not have the Moore property.This shows that from the hypotheses of Theorem <ref> one cannot deduce the Moore property, in general. §.§ Algebraic dynamical systems An algebraic dynamical systemalgebraic dynamical systemdynamical system!algebraic — is a dynamical systemof the form (X,G), where X is a compact metrizable abelian topological group and G is a countable group acting on X by continuous group morphisms. Note that if (X,G) is an algebraic dynamical system, then, for each g ∈ G, the mapx ↦ g x is a continuous group automorphism of G with inverse x ↦ g^-1 x. Let G be a countable group and A a compact metrizable topological group (for example a finite discrete abelian group, orthe n-dimensional torus ^n, orthe infinite-dimensional torus ^,orthe group _pof p-adic integers for some prime p). Then the G-shift (A^G,G) is an algebraic dynamical system. Let X be a compact metrizable abelian group and fX → X a continuous group automorphism (for example X = ^n and f ∈_n()). Then the dynamical system (X,f) generated by f is an algebraic dynamical system. Let (X,G) be an algebraic dynamical system. If d is a metric on X that is compatible with the topology then a point x ∈ X is homoclinic to 0_X if and only if one haslim_g →∞ d(g x, 0_X) = 0.The set Δ(X,G) consisting of all points of X that are homoclinic to 0_X is anG-invariant additive subgroup of X, called the homoclinic grouphomoclinic!— groupgroup!homoclinic — of (X,G) (cf. <cit.>). Two points x,y ∈ X are homoclinic if and only if x - y ∈Δ(X,G). It follows that the set of homoclinicity classes of (X,G) can be identified with the quotient group X/Δ(X,G).Consider now the Pontryagin dual X of X. We recall that if L is a locally compact abelian group,locally compact abelian groupgroup!locally compact abelian — the elements of its Pontryagin dual LPontryagin duallocally compact abelian group!Pontryagin dual of a —group!Pontryagin dual of a locally compact abelian — arethe characterscharacter of a locally compact abelian grouplocally compact abelian group!character of a —group!character of a locally compact abelian — of L, i.e., the continuous group morphisms χ L →, where /, and that the topology on L is the topology of uniform convergence on compact subsets (see e.g. <cit.>). As the abelian group X iscompact andmetrizable,X is a discrete countable abelian group. There is also a natural dual action of G on X defined byg χ(x) χ(g^-1 x)for all g ∈ G, χ∈X, and x ∈ X. Note that χ↦ g χ is a group automorphism of X for each g ∈ G. We recall that the integral group ringintegral group ringgroup!integral — ring [G] of Gconsists of all formal seriesr = ∑_g ∈ G r_g g,where r_g ∈ for all g ∈ G and r_g = 0 for all but finitely many g ∈ G, andtheoperations on [G] are defined by the formulasr + s= ∑_g ∈ G (r_g + s_g) g,r s= ∑_g_1,g_2 ∈ G r_g_1 s_g_2g_1 g_2for allr = ∑_g ∈ G r_g g,s = ∑_g ∈ G s_g g ∈[G].By linearity, the action of G on X extends to a left [G]-module structure onX. Conversely, if M is a countable left [G]-module and we equip M with its discrete topology, then its Pontryagin dual M is a compact metrizable abelian group. The left [G]-module structure on M induces by restriction an action of G on M, and, by dualizing, we get an action of G on M by continuous group morphisms,so that (M,G) is an algebraic dynamical system. Using the fact that every locally compact abelian group is isomorphic to its bidual, one shows that Pontryagin duality yields a one-to-one correspondence between algebraic dynamical systems with acting group G andcountable left [G]-modules (see <cit.>, <cit.>, <cit.>). Recall that from the hypotheses of Theorem <ref> one cannot deduce the Moore property, in general. However, Li <cit.> proved that when restricting to the class of algebraic dynamicalsystems (cf. Section <ref>) with amenable acting group, the Moore property follows fromexpansiveness and weak specification:Let (X,G) be an algebraic dynamical system. Suppose that the group G is amenable and that (X,G) is expansive and has the weak specification property. Then (X,G) has the Moore property. As an immediate consequence of Theorems <ref> and <ref>, one deduces the following (cf. <cit.>): Garden of Eden theorem!— for expansive algebraic dynamical systems with the weak specification propertytheorem!Garden of Eden — for expansive algebraic dynamical systems with the weak specification property Let (X,G) be an algebraic dynamical system. Suppose that the group G is amenable and that (X,G) is expansive and has the weak specification property. Then (X,G) has the Moore-Myhill property.§.§ Principal algebraic dynamical systemsLet f ∈[G] and consider the cyclic left [G]-module M_f [G]/ [G]f obtained by quotienting the ring [G] by the principal left ideal generated by f. The algebraic dynamical system associated by Pontryagin duality with M_f is denoted by (X_f,G) and is called the principal algebraic dynamical systemprincipal algebraic dynamical systemalgebraic dynamical system!principal — associated with f. There is a beautiful characterization of expansivity for principal algebraic dynamical systems due to Deninger and Schmidt <cit.> (see also <cit.>). Let G be a countable group and f ∈[G]. Then (X_f,G) is expansive if and only if f is invertible in ℓ^1(G). (Here ℓ^1(G) denotes the Banach algebra consisting of all formal sums r = ∑_g ∈ Gr_g g such thatr_g ∈ for all g ∈ G and ‖ r ‖_1 ∑_g ∈ G |r_g| < ∞,equipped with its obvious real vector space structure and the convolution product as in (<ref>).)It turns out (cf. <cit.>, see also <cit.>), that every expansive principal algebraic action has the weak specification property. From Corollary <ref> one immediately deduces (cf. <cit.>): Garden of Eden theorem!— for principal expansive algebraic dynamical systemstheorem!Garden of Eden — for principal expansive algebraic dynamical systems Let (X,G) be a principal algebraic dynamical system. Suppose that the group G is amenable and that (X,G) is expansive. Then (X,G) has the Moore-Myhill property. In <cit.> we had proved the same result under the stronger assumptions that G is abelian and the phase space X is connected. In the caseG = ^d, the group ring [G] can be identified with the ring [u_1, u_1^-1, …, u_d, u_d^-1] of Laurent polynomials with integral coefficients on d commuting indeterminates. Arnold's catdynamical system!Arnold's cat — For G = and f = u^2 - u - 1 ∈[u,u^-1] = [G], one can check that the associated principal algebraic dynamical system (X_f,) is topologically conjugate to Arnold's cat on ^2 (see e.g. <cit.>). Thus, from Corollary <ref> we recover that Arnold's cat satisfies the Moore-Myhill property. In <cit.>, in collaboration with Hanfeng Li,Li, Hanfeng we introduced a notion of weak expansivity for elements in the integral group ring [G], for any countable group G, and proved a Garden of Eden theorem for principal algebraic dynamical systems associated with weakly expansive polynomials. In order to state it, let us first introduce some preliminary material and notation. We denote by _0(G) the real vector space consisting of all maps rG → such that lim_g →∞ r(g) = 0 (this means that for all ε > 0 there exists a finite subset F ⊂ G such that |r(g)| < ε for all g ∈ G ∖ F).Note that if r ∈_0(G) and s ∈[G] then the map rsG → defined by(rs)(g) = ∑_g_1,g_2 ∈ G r(g_1) s_g_2 for all g ∈ G (cf. (<ref>)) belongs to _0(G).This endows _0(G) with a structure of aright [G]-module. Moreover G ⊂[G] ⊂_0(G).An element f ∈[G] is said to be weakly expansiveweakly expansive polynomial provided: * ∀ r ∈_0(G), fr = 0 ⇒ r = 0;* ∃ω∈_0(G) such that fω = 1_G.For principal algebraic dynamical systems with elementary amenable acting group there is a characterization of connectedness of the phase space. First recall that a non-zero element f ∈[G] is called primitiveprimitive polynomial if there is no integer n ≥ 2 that divides all coefficients of f. Also recall (cf. for instance <cit.>) that the class of elementary amenable groupsamenable group!elementary —group!elementary amenable —elementary!— amenable group is the smallest class of groups containing all finite groups and all Abelian groups that is closed under the operations of taking subgroups, quotiens, extensions, and direct limits. In <cit.> we showed that is G is a countable torsion-free elementary amenable group (e.g. G = ^d) and f ∈[G] is non-trivial, then X_f is connected if and only if f is primitive.We are now in position to state the main result of <cit.> (Theorem 1.1 therein). Garden of Eden theorem!— for algebraic actions associated with weakly expansive polynomialstheorem!Garden of Eden — for algebraic actions associated with weakly expansive polynomials Let G be a countable Abelian group and f ∈[G]. Suppose that f is weakly expansive and that X_f is connected. Then the dynamical system (X_f,G) has the Moore-Myhill property. There are two main ingredients in our proof of Theorem <ref>. The first one is a rigidity result (a generalization of <cit.>) for algebraic dynamical systems associated with weakly expansive polynomials and with connected phase space. We used it to prove that, under the above conditions, every endomorphism of (X_f,G) is affine with linear part of the form x ↦ r x for some r ∈[G]. The second one, a generalization of <cit.>), asserts that, if f is weakly expansive, then the homoclinic group Δ(X_f,G), equipped with the induced action of G, is dense in X_f and isomorphic, as a [G]-module, to [G]/ [G] f^*, where f^* ∈[G] is defined by (f^*)_gf_g-1 for all g ∈ G.In <cit.> we showed that if f ∈[G] and the associated principal algebraic dynamical system(X_f,G) is expansive then f is weakly expansive. It follows that Theorem <ref> constitutes a generalization of the main result in <cit.>.Recall that a polynomial f ∈[G] is said to be well-balancedwell-balanced polynomial (cf. <cit.>) if the following conditions are satisfied: * ∑_g ∈ G f_g = 0,* f_g ≤ 0 for all g ∈ G ∖{1_G},* f_g = f_g^-1 for all g ∈ G (i.e., f is self-adjoint),* and (f) {g ∈ G: f_g ≠ 0}, the support of f, generates G.support!— of a polynomial If f∈[G] is well-balanced, the associated dynamical system (X_f, G) is called a harmonic model.harmonic modeldynamical system!harmonic model — For G = ^d, the Laurent polynomial f = 2d - ∑_i=1^d (u_i + u_i^-1) ∈[u_1, u_1^-1, …, u_d, u_d^-1] = [^d] is well-balanced and the corresponding harmonic model (X_f,^d)shares many interesting measure theoretic and entropic properties with other different models in mathematical physics, probability theory, and dynamical systems such as the Abelian sandpile model, spanning trees, and the dimer models <cit.>. Since a well-balanced polynomial f ∈[G], with G infinite countable not virtuallyor ^2, is weakly expansive (<cit.>), from Theorem <ref> we deduce(cf. <cit.>): Garden of Eden theorem!— for harmonic modelstheorem!Garden of Eden — for harmonic models Let G be an infinite countable Abelian group which is not virtuallyor ^2(e.g. G = ^d, with d ≥ 3). Suppose that f ∈[G] is well-balanced and that X_f is connected. Then the dynamical system (X_f,G) has the Moore-Myhill property. If G = ^d, then any polynomial f ∈[G] may be regarded, by duality, as a function on G = ^d. We denote by Z(f) {(t_1, t_2, …, t_d) ∈^d: f(t_1, t_2, …, t_d) = 0} its zero-set. Recall that an irreducible polynomial f is atoralatoral polynomial <cit.> if there is some r∈[G] such that r ∉[G] f and Z(f)⊂ Z(r). This is equivalent to the condition Z(f)≤ d-2, where the meaning of (·) is explained in <cit.>; in particular, one has (∅)- ∞. Also remark that, if d = 1, an irreducible polynomial f ∈[] = [u_1,u_1^-1] is atoral if and only if Z(f) = ∅ and this, in turn, is equivalent to (X_f,) being expansive (cf. <cit.>). We are now in position to state the following (cf. <cit.>): Garden of Eden theorem!— for irreducible atoral polynomialstheorem!Garden of Eden — for irreducible atoral polynomials Let f∈[^d] be an irreducible atoral polynomial such that Z(f) is contained in the image of the intersection of [0,1]^d and a finite union of hyperplanes in ^d under the natural quotient map ^d →^d (e.g., when d ≥ 2 such that Z(f) is finite). Then the dynamical system (X_f,^d) has the Moore-Myhill property.Here below, we present some examples of irreducible atoral polynomials f ∈[^d], mainly from<cit.> and <cit.>. We can then apply Theorem <ref> and deduce that the corresponding algebraic dynamical systems (X_f, ^d) satisfy the Garden of Eden theorem.* Let d=1 and f(u) = u^2 - u - 1 ∈[u,u^-1] = [] (cf. Example <ref>).Arnold's catdynamical system!Arnold's cat — Then f is irreducible and, since Z(f) = ∅, atoral. Recall that the associated principal algebraic dynamical system (X_f,) is conjugated to Arnold's cat. Thus we get yet another proof of the fact that this hyperbolic dynamical systems satisfies the Garden of Eden theorem. * Let d = 2 and f(u_1,u_2) = 2 - u_1 - u_2 ∈[u_1,u_1^-1,u_2,u_2^-1] = [^2]. Then Z(f) = {(1,1)}, and so f is atoral. Note that, in fact, f is weakly expansive (though not well-balanced) by <cit.>. Moreover, f is also primitive, so that, by the characterization we presented above, X_f is connected. Applying Theorem <ref>, we obtain an alternative proof of the fact that (X_f,^2) has the Moore-Myhill property. * Let d=2, and consider the Laplace harmonic model f(u_1,u_2) = 4 - u_1-u_1^-1-u_2-u_2^-1∈[u_1,u_1^-1,u_2,u_2^-1] = [^2].Laplace harmonic modeldynamical system!Laplace harmonic model —One has Z(f)={(1, 1)}. Thus f is atoral and (X_f, α_f) satisfies the Garden of Eden theorem, by virtue of Theorem <ref>. (Note that we cannot apply Theorem <ref>.) * Let d=2, and f(u_1,u_2) = 1 + u_1 + u_2 ∈[u_1,u_1^-1,u_2,u_2^-1] = [^2]. Then Z(f)={(ω, ω^2), (ω^2, ω)}, where ω = exp(2π i/3). The algebraic dynamical system (X_f, ^2) is called the connected Ledrappier subhift.Ledrappier subshift!connected —subshift!connected Ledrappier — Thus the connected Ledrappier shift satisfies the Garden of Eden theorem. On the other hand, the (disconnected) Ledrappier shift (cf. Example <ref>)Ledrappier subshiftsubshift!Ledrappier — X {x ∈ (/2)^^2: x(m,n)+x(m+1,n)+x(m,n+1)=0} (which may be regarded as an algebraic dynamical system with phase space [^2]/I, where I = 2[^2] + f[^2] is the ideal generated by 2 andf(u_1,u_2) = 1+u_1+u_2 ∈[u_1,u_1^-1,u_2,u_2^-1] = [^2]) does not satisfy the Garden on Eden theorem.Indeed, one has Δ(X,^2) = {0_(/2)^^2} so that every map τ X → X is pre-injective.This ensures the Moore property for (X,^2).However, the constant map x ↦ 0_(/2)^^2 (which is a pre-injective endomorphism of (X,^2)) is clearly not surjective, showing that (X,^2) does not satisfy the Myhill property. * Let d=3 and f(u_1,u_2,u_3) = 3 + 3u_1 - 3 u_1^3 + u_1^4 - u_2 - u_3∈[u_1,u_1^-1,u_2,u_2^-1, u_3,u_3^-1] = [^3]. One has has Z(f)={(η, η, η), (η,η,η)}, where η is an algebraic integer. Let d=1 and f = 2 - u - u^-1∈[u,u^-1] = [].Laplace harmonic modeldynamical system!Laplace harmonic model — Then the associated dynamical system X_f = {x ∈^: x(n-1)+x(n+1) = 2x(n)n ∈} is the one-dimensional Laplace harmonic model. It is easy to see that Δ(X_f,α_f) = {0_^}. Then (X_f,α_f) satisfies the Moore property but not the Myhill property (the constant map x ↦ 0_^ (which is a pre-injective endomorphism of (X_f,α_f)) is clearly not surjective). We then have (cf. <cit.>): Garden of Eden theorem!— for Laplace harmonic modelstheorem!Garden of Eden — for Laplace harmonic models The Laplace harmonic modelLaplace harmonic modeldynamical system!Laplace harmonic model — (i.e. the principal algebraic dynamicalsystem (X_f,α_f) associated with the polynomialf = 2d - ∑_i=1^d (u_i + u_i^-1) ∈[u_1, u_1^-1, …, u_d, u_d^-1] = [^d]) satisfies the Moore-Myhill property if and only if d ≥ 2.§ SOME ADDITIONAL TOPICS §.§ Infinite alphabets and uncountable groupsThe notion of a subshift and that of a cellular automaton between subshifts can be extended to the case where the alphabet sets are infinite and the group is not countable. More specifically, let G be a (possibly uncountable) group and A a (possibly infinite) set. The prodiscrete topologyprodiscrete!— topology on A^G is the product topology obtained by taking the discrete topology on each factor A of A^G= ∏_g ∈ G A. The prodiscrete topology on A^G is not metrizable as soon as G is uncountable and A contains more than one element. However, this topology is induced by the prodiscrete uniform structureprodiscrete!— uniform structure on A^G, that is, the product uniform structure on A^G obtained by taking the discrete uniform structure on each factor A of A^G (see <cit.> for more details). A subset X ⊂ A^G is called a subshiftsubshift if X is invariant under the shift action and closed for the prodiscrete topology. LetG be agroup and let A, B be sets. Suppose that X ⊂ A^G and Y ⊂ B^G are two subshifts. One defines cellular automata between X and Y exactly as in Definition <ref>. Every cellular automaton τ X → Y is continuous with respect to the topologies on X and Y induced by the prodiscrete topologies on A^G and B^G. The converse is false in general <cit.>, <cit.>. However, the Curtis-Hedlund-Lyndon theorem (cf. Theorem <ref>) admits the following generalization <cit.>, <cit.>: a map τ X → Y is a cellular automaton if and only if it is equivariant(with respect to the G-shift actions) and uniformly continuous (forthe uniform structures on X and Y induced by the prodiscrete uniform structures on A^G andB^G). One can extend the notion of amenability defined only for countable groups in Section <ref>by declaringthat a general groupG is amenableamenable groupgroup!amenable — if all of its finitely generated subgroups are amenable. This extension makes sense since every finitely generated group is countable and every subgroup of a countable amenable group is itself amenable. The Garden of Eden theorem (cf. Theorem <ref>) remains valid in this more general setting: if G is a (possibly uncountable) amenable group, A a finite set, and τ A^G → A^G a cellular automaton, then τ A^G → A^G is surjective if and only if it is pre-injective. The proof can be reduced to the case when the group G is finitely generated (and hence countable) by using the operations of restriction and induction for cellular automata (see <cit.> and <cit.>). One can also directly follow the proof given above for Theorem <ref> by replacing the Følner sequence by a Følner net (see <cit.>). §.§ Linear cellular automataLet G be a group and let K be a field. Let A be a finite-dimensional vector space over K and set d = _K(A). Observe that A is infinite as soon as the field K is infinite (e.g. K =) and d ≠ 0. Taking A as an alphabet, the configuration set A^G inherits a natural product vector space structure. The support of a configuration x ∈ A^G is the subset (x) {g ∈ G: x(g) ≠ 0 }⊂ G. Thus, x ∈ A^G has finite support if and only if it is almost equal to the constant zero-configuration. We denote by A[G] ⊂ A^G the vector subspace consisting of all configurations with finite support. Recall that an involutive K-algebrainvolutive K-algebra is a K-algebra equipped with an involution that isa K-algebra anti-automorphism. The vector space K[G] has a natural additional structure of an involutive K-algebra. The multiplication on K[G] is the convolution productconvolution product defined by(αβ)(g) ∑_g_1, g_2 ∈ G:g_1g_2 = gα(g_1) β(g_2) = ∑_h ∈ Gα(h) β(h^-1 g)for all α, β∈ K[G] and g ∈ G, andthe involution is the map α↦α^* given by α^*(g) α(g^-1) for all α∈ K[G] and g ∈ G. This involutive K-algebrais called the group algebragroup!— algebra of the group G with coefficients in K. Note that the group G embeds as a subgroup of the group of invertible elements of K[G] via the mapg ↦δ_g, where δ_g ∈ K[G] is defined by δ_g(g) = 1 and δ_g(h) = 0 for all h ∈ G with h ≠ g, andthat G ⊂ K[G] is a base for the vector space K[G]. The G-shift action on A^G is clearly K-linear, so that it yields a left K[G]-module structure on A^G.Observe that A[G] is a submodule of A^G. A linear cellular automatonlinear cellular automatoncellular automaton!linear — over the group G and the alphabet A is a cellular automatonτ A^G → A^G that is K-linear with respect to the vector space structure on A^G(if S is any memory set for τ and μ A^S → A is the associated local defining map,this is equivalent to requiring that μ is K-linear). Let us denote by (G;A) the vector space consisting of all linear cellular automata τ A^G → A^G. Let τ∈(G;A).Note that A[G] is stable under τ. Indeed, if S ⊂ G is a memory set for τ, then (τ(x)) ⊂(x)S^-1 for all x ∈ A^G) (see <cit.>). Moreover,τ is pre-injective if and only if τ|_A[G] A[G] → A[G] is injective (cf. <cit.>). Observe also that τ is a K[G]-module endomorphism of A^G and hence of A[G]. The vector space (G;A) has a natural structure of a K-algebra with the composition of maps as the multiplicative operation. Furthermore, the restriction map τ↦τ|_A[G] yields a K-algebra isomorphism from (G;A) onto _K[G](A[G]), the endomorphism K-algebra of the K[G]-module A[G] (cf. <cit.>). It turns outthat A[G] is a free K[G]-module with rank d. Actually, if (e_i)_1 ≤ i ≤ d is a base for the vector space A, then the family (x_i)_1 ≤ i ≤ d, where x_i ∈ A[G] is the configuration defined by x_i(1_G) = e_i and x_i(g) = 0 for g ≠ 1_G, is a free base for the K[G]-module A[G] (cf. <cit.>). One deduces that _K[G](A[G]) is isomorphic, as a K-algebra, to the K-algebra _d(K[G]) of d × d matrices with coefficients in the group algebra K[G]. It follows that (G;A) and _d(K[G]) are isomorphic as K-algebras(cf. <cit.>). For instance, the map Φ_d(K[G]) →(G;A), sending each matrix α = (α_i j)_1 ≤ i,j ≤ d∈_d(K[G]) to the unique linear cellular automaton τ∈(G;A) such thatτ(x_i) = ∑_1 ≤ j ≤ dα_j i^*x_jfor all 1 ≤ i ≤ d, is a K-algebra isomorphism.The adjointadjoint matrix of a matrix α∈_d(K[G]) is the matrix α^* ∈_d(K[G]) given by (α^*)_ijα_ji^* ∈ K[G]for all1 ≤ i,j ≤ d. The involution α↦α^* makes _d(K[G]) into an involutive K-algebra. Let ustransport this involution to (G;A) via Φ. Thus, (G;A) becomes an involutive K-algebra with involution τ↦τ^* satisfyingτ^*(x_i) = ∑_1 ≤ j ≤ dα_i jx_jfor all 1 ≤ i ≤ d and τ = Φ(α) ∈(G;A). Note that this involution on (G;A) depends on the choice of a base for A.Consider now the non-degenerate K-bilinear symmetric map A × A → K defined bya · b = ∑_1 ≤ i ≤ d a_ib_ifor all a = ∑_1 ≤ i ≤ d a_i e_iandb = ∑_1 ≤ i ≤ d b_i e_i,witha_i,b_i ∈ Kfor1 ≤ i ≤ d.Then the K-bilinear map A[G] × A^G → K, defined by⟨ x,y ⟩∑_g ∈ G x(g)· y(g)for all x ∈ A[G] and y ∈ A^G, is non-degenerate in both arguments. Given a linear cellular automaton τ A^G → A^G, Bartholdi <cit.> (see also <cit.>)Bartholdi, Laurentobserved that⟨τ(x), y ⟩ = ⟨ x, τ^*(y) ⟩for all x ∈ A[G] and y ∈ A^G, and used this to show that τ is pre-injective (resp. surjective) if and only if τ^* is surjective (resp. pre-injective). In <cit.> (see also <cit.>)Garden of Eden theorem!linear —theorem!linear Garden of Eden — a linear version of the Garden of Eden theorem is proved, namely that if G is amenable and τ∈(G;A), then τ is surjective if and only if it is pre-injective. Let G be a nonamenable group. In <cit.> BartholdiBartholdi, Laurentshowed that there exists a finite field K (in <cit.> he actually observed that the field K can be arbitrary),a finite dimensional vector space A over K,and a pre-injective linear cellular automaton τ∈(G;A) which is not surjective.As a consequence (<cit.>), the cellular automaton τ^* ∈(G;A) is surjective but not pre-injective. These two facts, in combination with the linear version of the Garden of Eden theorem in <cit.>, yield a characterization of group amenability in terms of linear cellular automata. The linear version of the Garden of Eden theorem has been extended in <cit.> to linear cellular automata τ X → X with X ⊂ A^G a strongly irreducible linear subshift of finite type, and in <cit.> to the case when the alphabet A is a semi-simple left-module of finite length over a (possibly noncommutative) ring. §.§ Algebraic cellular automataLet G be a group. In <cit.> we introduced the class of algebraic cellular automata over G. Given a field K, let A be an affine algebraic setaffine algebraic set over K. This means that A ⊂ K^n for some integer n ≥ 1 is the set of common zeroes of a family of polynomials in n variables with coefficients in K. Then a cellular automaton τ A^G → A^G is called an algebraic cellular automatonalgebraic cellular automatoncellular automaton!algebraic — provided it admits a memory set S ⊂ G and a local defining map μ A^S → A that is regular,regular map i.e., it is the restriction of some polynomial map (K^n)^S → K^n. This definition was generalized in <cit.> as follows. Let 𝒮 be a scheme and let X, Y be schemes based over 𝒮.Denote by AX(Y) the set of Y-points of X, that is, the set consisting of all 𝒮-scheme morphism Y → X. Then an algebraic cellular automaton over the group G and the 𝒮-scheme X with coefficients in the 𝒮-scheme Y, briefly, an algebraic cellular automaton over the group G and the schemes 𝒮,X,Y,algebraic cellular automaton!— over a schemecellular automaton!algebraic — over a scheme is a cellular automaton τ A^G → A^G over the group G and the alphabet A that admits a memory set S ⊂ G and a local defining map μ A^S → A which is induced by some 𝒮-scheme morphism fX^S → X, where X^S denotes the 𝒮-fibered product of a family of copies of X indexed by S.Note that Definition <ref> generalizes that of an algebraic cellular automaton given in <cit.>. Indeed, if K is a field and A ⊂ K^n an algebraic set, there is an 𝒮-scheme X associated with A for 𝒮 = (K), namelyX = (K[u_1,…,u_n]/I), where I = I(A) is the ideal of K[u_1,…,u_n] consisting of all polynomials that identically vanish on A. One then has A = X(𝒮) and the regular maps between two regular sets A_1 ⊂ K^n_1 and A_2 ⊂ K^n_2 are precisely those induced by the 𝒮-morphisms between their corresponding 𝒮-schemes X_1 and X_2, equivalently, the K-algebra morphisms from K[z_1,…,z_n_2]/I(A_2) to K[t_1,…,t_n_1]/I(A_1). Thus, τ A^G → A^G is an algebraic cellular automaton, as defined in <cit.>, if and only if τ is a cellular automaton in the sense of Definition <ref> over the schemes 𝒮,X,Y for 𝒮 = Y = (K) and X is the 𝒮-scheme associated with A. Recall that an algebraic varietyalgebraic variety over a field K is a scheme of finite type over K. In <cit.> we showed the following Myhill type result for algebraic cellular automata: Let G be an amenable group and let X be an irreducible complete algebraic variety over an algebraically closed field K. Let AX(K) denote the set of K-points of X. Then every pre-injective algebraic cellular automaton τ A^G → A^G over (G,X,K) is surjective.Let us note that the converse implication, i.e., the analogue of the Moore implication, does not hold under the hypotheses of Theorem <ref>, even with the additional hypothesis that the variety X is complete.For example, if K is an algebraically closed field whose characteristic is not equal to 2, the projective line _K^1 is an irreducible complete K-algebraic variety and the morphism f _K^1 →_K^1 given by (x:y) ↦ (x^2:y^2) is surjective but not injective. Taking A _K^1(K),we deduce that, for any group G, the map τ A^G → A^G defined by (τ(c))(g) f(c(g)) for all c ∈ A^G and g ∈ G, is an algebraic cellular automaton over (G,X,K) that is surjective but not pre-injective. In order to formulate a version of the Garden of Eden theorem for algebraic cellular automata, the following weak notion of pre-injectivity was introduced in <cit.>:algebraic cellular automaton!(*)-pre-injective —cellular automaton!(*)-pre-injective algebraic — Let G be a group and let X be an algebraic variety over an algebraically closed field K. Let A X(K) and let τ A^G → A^G be an algebraic cellular automaton over (G,X,K). We say that τ is (*)-pre-injective if there do not exist a finite subset Ω⊂ G and a proper subset H ⊂ A^Ω that is closed for the Zariski topology such thatτ((A^Ω)_p)=τ(H_p)for allp ∈ A^G∖Ωwhere H_p {x ∈ A^G: x|_Ω∈ Hx|_G ∖Ω = p} for p ∈ A^G∖Ω and any subset H ⊂ A^Ω.It turns out that Theorem <ref> remains valid if we replace the hypothesis that τ is pre-injective by the weaker hypothesis that τ is (*)-pre-injective. Moreover, this weak form of pre-injectivity also allows us to establish a version of the Moore implication for algebraic cellular automata. Altogether we obtained the following version of the Garden of Eden theorem (cf. <cit.>) for algebraic cellular automata:Garden of Eden theorem!— for algebraic cellular automatatheorem!Garden of Eden — for algebraic cellular automata Let G be an amenable group and let X be an irreducible complete algebraic variety over an algebraically closed field K. Let AX(K) denote the set of K-points of X and let τ A^G → A^G be an algebraic cellular automaton over (G,X,K). Then the following conditions are equivalent: * τ is surjective;* τ is (*)-pre-injective. One of the main ingredients in the proof of the above results is the notion of algebraic mean dimension.algebraic mean dimension If G is an amenable group equipped with a Følner netand A is the set of K-points of an algebraic variety X over an algebraically closed field K, given a subset Γ⊂ A^G then the algebraic mean dimension _(Γ) of Γ is defined as a limit of the average Krull dimensionKrull dimensionof the projection of Γ along the Følner net. The definition of algebraic mean dimension is analogous to that of topological entropy (cf. (<ref>)). §.§ Gromov's Garden of Eden theorem Gromov's Garden of Eden theoremGarden of Eden theorem!Gromov's —theorem!Gromov's Garden of Eden — In <cit.>, Gromov proved a Garden of Eden type theorem generalizing Theorem <ref> under several aspects. First of all, the alphabet set A is only assumed to be countable, not necessarily finite. In addition, the universe is (the vertex set V of) a connected simplicial graph = (V,E) of bounded degree with a natural homogeneity condition (to admit a dense pseudogroup of partial isometries). The classical case corresponds to = (G,S) being the Cayley graph of a finitely generated group G with respect to a finite and symmetric generating subset S ⊂ G ∖{1_G}. The dense pseudogroup of partial isometries is, in this particular case, given by partial left-multiplication by group elements. In this more general setting, the category corresponding to that of cellular automata consists now of the following: * stable spaces,stable space i.e. (stable) projective limits of locally-finite projective systems (X_Ω) of A-valued maps on(subsets of) V with a suitable finiteness and irreducibility condition (bounded propagation)stable space!— of bounded propagationbounded propagation!— of a stable space and admitting a dense holonomystable space!— of dense holonomydense holonomy!— of a stable space (corresponding to shift-invariance in the classical case) as objects, and* maps of bounded propagationmap of bounded propagationbounded propagation!map of — (this condition corresponds to continuity) admitting a dense holonomymap of bounded propagation!— admitting a dense holonomydense holonomy!map of bounded propagation admitting a — (this corresponds to G-equivariance), as morphisms.The notions of a Følner sequence and of amenability for simplicial graphs, together with the corresponding notion of entropy (for the above-mentioned spaces of A-valued maps), carry verbatim from the group theoretical framework. All this said, Gromov's theorem states the following.Gromov's Garden of Eden theoremGarden of Eden theorem!Gromov's —theorem!Gromov's Garden of Eden —Let = (V.E) be an amenable simplicial connected graph of bounded degree admitting a dense pseudogroup of partial isometries and let A be a finite or countably infinite alphabet set.Suppose that X,Y ⊂ A^V are stable spaces of bounded propagation with the same entropy. Let τ X → Y be a a map of bounded propagation admitting a dense holonomy. Then τ is surjective if and only if it is pre-injective. In <cit.> it is shown that a stable space of bounded propagation is strongly irreducible (cf. Section <ref>) and of finite type (cf. Section <ref>). However, as shown in <cit.>, the converse fails to hold: strong irreducibility and finite type conditions do not imply, in general, bounded propagation. As a consequence, the following theorem (cf. <cit.>) improves onGromov's theorem.Let = (V.E) be an amenable simplicial connected graph of bounded degree admitting a dense pseudogroup of partial isometries and let A be a finite or countably infinite alphabet set. Suppose that X,Y ⊂ A^V are strongly irreducible stable spaces of finite type with the same entropy. Let τ X → Y be a a map of bounded propagation admitting a dense holonomy. Then τ is surjective if and only if it is pre-injective.Note that this last result also covers Theorem <ref>.§.§ Cellular automata over homogeneous setsCellular automata where the universe is a set endowed with a transitive group action have been investigated by Moriceau <cit.>. Versionsof the Curtis-Hedlund-Lyndon theorem and of the Garden of Eden theorem in this more general setting have been obtained by Wacker <cit.>, <cit.>.Acknowledgements. We thank Laurent Bartholdi for valuable comments and remarks. siam ' 10bartholdi:2017 L. 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Wolfram, A new kind of science, Wolfram Media, Inc.,Champaign, IL, 2002. | http://arxiv.org/abs/1707.08898v3 | {
"authors": [
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"Michel Coornaert"
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"title": "The Garden of Eden theorem: old and new"
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Physics Department and INFN, University of Turin, via P. Giuria 1, 10125 Turin, Italy Sorbonne Universités, UPMC Univ Paris 06, UMR 7238, Computational and Quantitative Biology, 15 rue de l'École de Médecine Paris, FranceCNRS, UMR 7238, Paris, FrancePhysics Department and INFN, University of Turin, via P. Giuria 1, 10125 Turin, ItalySorbonne Universités, UPMC Univ Paris 06, UMR 7238, Computational and Quantitative Biology, 15 rue de l'École de Médecine Paris, FranceCNRS, UMR 7238, Paris, France FIRC Institute of Molecular Oncology (IFOM), 20139 Milan, Italy Physics Department and INFN, University of Turin, via P. Giuria 1, 10125 Turin, Italy Many complex systems are modular. Such systems can be represented as“component systems”, i.e., sets of elementary components, such asLEGO bricks in LEGO sets. The bricks found in a LEGO set reflect atarget architecture, which can be built following a set-specific listof instructions.In other component systems, instead, the underlyingfunctional design and constraints are not obvious a priori, and theirdetection is often a challenge of both scientific and practicalimportance, requiring a clear understanding of component statistics.Importantly, some quantitative invariants appear tobe common to manycomponent systems, most notably a common broad distribution ofcomponent abundances, which often resembles the well-known Zipf's law.Such “laws” affect in a general and non-trivial way the componentstatistics, potentially hindering the identification ofsystem-specific functional constraints or generative processes. Here,we specifically focus on the statistics of shared components, i.e.,the distribution of the number of components shared by differentsystem-realizations, such as the common bricks found in different LEGOsets. To account for the effects of component heterogeneity, weconsider a simple null model, which builds system-realizations by random draws from a universe of possible components. Undergeneral assumptions on abundance heterogeneity,we provide analyticalestimates of component occurrence, which quantify exhaustively thestatistics of shared components. Surprisingly, this simple null modelcan positively explain important features of empiricalcomponent-occurrence distributions obtained from large-scale data onbacterial genomes, LEGO sets, and book chapters. Specific architecturalfeatures and functional constraints can be detected from occurrencepatterns as deviations from these null predictions, as we show for theillustrative case of the “core” genome in bacteria. 87.18.Wd, 89.75.DaStatistics of shared components in complex component systems Matteo Osella[To whom correspondence should be addressed.Email: [email protected]] May 29, 2018 =========================================================================================§ INTRODUCTION A large number of complex systems in very different contexts - ranging from biology to linguistics, social sciences and technology - can be broken down to clearly defined basic building blocks or components.For example, books are composed of words, genomes of genes, and many technological systems are assemblies of simple modules. Once components are identified, a specific realization of a system (e.g., a specific book, a LEGO set, a genome) can be represented by its parts list, which is the subset of the possible elementary components (e.g. words, bricks, genes), with their abundances,present in the realization. We use the term “component systems” for empirical systemsto which this general representation can be applied. Occurrence patterns of components across realizations are expected to reveal relevant architectural constraints.For example, the bricks present in each LEGO set clearly reflect a target architecture that can be built with them following the instruction booklet.While for LEGO sets the assembly instructions are provided by the seller, in most component systems the architectural constraints are not obvious. Inferring such constraints from the statistics of components may answer important questions about the nature of a system.For example, it could reveal new clues about the complex combination of selective pressure and random events that shaped the functional composition of extant genomes. Even in those cases where the architecture is partially or even fully known and the instruction manual is available, the statistics of components may help us distill some general principles characterizing a given class of component systems, in some cases revealing basic features of the underlying generative processes.In order to perform detection of system-dependent featuresfrom patterns of shared components, we need to have a clear idea of the general behavior ofcomponent systems even in absence of functional constraints onthe presence/absence of specific classes of components.This is by itself a challenging task, as such systems show a large degree of non-trivial universalproperties <cit.>that could in principle affect the occurrence statistics.Indeed, several notable quantitative laws can be identified in the composition of component systems of very different nature. This is well known, e.g., in linguistics, where the notorious “Zipf's law” <cit.> describing the word frequency distribution (or its equivalent rank plot) in a linguistic corpus has been the subject of extensive investigations <cit.>.In this context, the existence of quantitative “universal” laws may in principle provide insights on the cognitive mechanisms of text production, and can have practical applications in data mining and data search techniques <cit.>.Analogously, for genomes across the whole tree of life, the number of genes in different evolutionary families is power-law distributed, a discovery that represents one of the first examples of “laws” of the genome sequencing era <cit.>. Such heterogeneous usage of the different basic components, oftenresulting in an approximately power-law distribution of their frequencies, can be seen as a hallmark of the complexityof component systems <cit.>.A large body of theoretical work addresses the origins of this heterogeneity.Several models have emerged in different areas of science, with context-specific ingredients. For example, stochastic processes based on gene duplication, deletion and innovation have been proposed as simple evolutionary models of genome evolution at the basis of the observed heterogeneous component usage <cit.>. On the other hand, specific communication optimization principles <cit.> and stochastic models for text generation <cit.> have been invoked to explain the emergence of Zipf's law in natural language. In many, but not all, of these models a preferential attachment principle is at the origin of the emergence of the power-law distribution of component frequencies.More importantly, the ubiquity of this emergent behavior raises the question of whether (and to what extent) empirical laws like Zipf's law are pervasive statistical patterns that transcend system-specific mechanisms <cit.>.In this spirit, the analysis of radically different systems can help the discovery of patterns that descend from pure statistical effects or general principles <cit.>.Here, we analyze empirical data from three very different component systems from linguistics (book chapters), genomics (protein domain families in sequenced genomes) and technology (LEGO toys) and we look for general statistical consequences of their heterogeneous frequency distributions.The different data sources considered here reasonably do not share any generative mechanisms, nor are they expected to share the same type of constraints, selection criteria or optimization principles. However, the frequency of their components is heterogeneous and they all obey laws that are similar to Zipf's.The marginal statistics that we concentrate on is the fraction of components that are shared among a certain number of realizations. For example, the fraction of LEGO bricks with the same shape found in a given fraction of unequal LEGO boxes.In genomics, this is the so-called “gene-frequency distribution”, which was shown to follow a U-shape at several taxonomic levels <cit.>.A U-shape of this distribution of shared components indicates that there is a set of “core” components that are common to most realizations, as well as an enriched set of realization-specific components. This histogram also decays approximately as a power law for rare components, both in genomic data and in technological systems <cit.>.In evolutionary genomics, the origins of this pattern are the focus of a lively debate.The pattern has been rationalized theoretically by neutral or selective population dynamics models <cit.>, or as a consequence of functional dependencies among different components <cit.>.For component systems outside of genomics, the distribution of shared components remains under-explored, and is typically neglected by the current debate, for example in linguistics <cit.>.Using theoretical calculations based on random sampling of components (with replacement) from their overall frequencies (estimated by their total abundance across empirical realizations), we show that a distribution of shared components with a power-law behavior is a general feature of component systems not only with Zipf-like component frequency distributions, but also for general power-laws and exponential decay of the overall component frequencies.In other words, a U-shaped distribution of shared components can naturally emerge in component systems with a heterogeneous component usage (which is often the case empirically).Importantly, we quantitatively identify the general features of the system leading to a U-shaped distribution of shared components, a given core size, and a specific decay of the realization-specific bulk of this distribution. § DATA§.§ Data sourcesGenomes. We used the superfamily classification of protein domains from the SUPERFAMILY database <cit.> considering a set of R = 1061 prokaryotic genomes (“realizations”) and a total number of different families N = 1531 (“components”).Protein domain families are the basic modular topologies of folded proteins <cit.>.Different domains of the same family can be found in each genome in the same or different proteins.As a functional annotation of protein domains in SUPERFAMILY, we considered the SCOP annotations mapped into 7 general function categories, as developed by C. Vogel <cit.>. LEGO sets. The composition in bricks of several LEGO sets (R = 2820) can be freely downloaded from “http://rebrickable.com”. We excluded from the analysis LEGO sets belonging to the category of “LEGO Technic” since, by construction, they share a very small number of bricks with the classic LEGO toys. Similarly, we did not consider LEGO sets with less than 80 components or belonging to the categories “Educational and Dacta” and “Supplemental” in order to exclude sets that are actually collections of spare parts or additional bricks for other sets.Texts. The analyzed linguistic corpus is composed by R = 1721 book chapters (realizations) of several English books randomly chosen from the most popular ones in the database “http://www.gutenberg.org”.We defined chapters as realizations, instead of entire books, to obtain a corpus with a range of sizes (total number of components per realization) comparable to the one of genomes and LEGO toys (Figure S1).The complete list of books considered is reported in Table S1 of the Supplemental Material (SM).The elementary components are defined as the words regardless of capitalization (e.g.“We” and “we” are considered as the same component).§.§ Data structure:Matrix representation of component systemsA set of empirical realizations of a component system can be naturally described as a matrix { n_ij} defined such that the entry n_ij represents the abundance of the component i (i = 1, … N) in the realization j (j = 1, …, R). Thus, each realization (a literary text, a LEGO set or a prokaryotic genome), is represented as a matrix column (Figure <ref>).Some key observables can be easily defined using this representation.First, the total abundance a_i of the component i in the whole ensemble is defined by summing over all realizations a_i = ∑_j n_ij. The normalized abundance representsthe component frequencyf_i = a_i/∑_ia_i.The “component occurrence” o_i is instead defined as the fraction of realizations in which the component is found, thus o_i = 1/R∑_j (1 - δ_n_ij, 0).Two other crucial quantities are: the total number N of different components in the system, which is essentially the number ofbricks of different shape or the vocabulary, and the size of a realization j, defined as the total number of its components M_j = ∑_in_ij.§ RESULTS §.§ Component frequency distribution and distribution of shared components show general features across systems This section illustrates two empirical laws in the analyzed datasets (LEGO toys, bacterial genomes, and literary texts).We first consider the component frequencies in the whole universe of available realizations of a given system, which is essentially the generalized Zipf's law <cit.> for the three systems. Figure <ref> shows the rank plots of these component frequencies. The three data sets share a power-law behavior for components with high frequencies (low rank), with an exponent close to 1 as in the classic Zipf's law <cit.>, and a faster decay at higher ranks (components with low frequency).This double-scaling behavior has been recently observed in the context of linguistics <cit.>. In evolutionary genomics, the gene frequency was previously analysed over single genomes and shown to be approximately power-law distributed with an exponent dependent on genome size <cit.>. Figure <ref> shows that the same distribution calculated over thousands of prokaryotic genomes has a double-scaling, with an exponential-like decay for low ranks in its rank plot. We tested that the shape of these component frequency distributions do not strongly depend on the specific size or number of realizations analyzed.The rank plots in Figure <ref> do not vary when evaluated on different sub-samples of the whole data sets (Fig. S2 of the SM). This suggests that the frequency distributions evaluated using the available finite empirical data sets estimate reliably the global heterogeneity of the component usage in the systems.We aim to evaluate also the distribution of shared components, { o_i }, and how much of its features can be explained from other measurable quantities, namely, the component frequencies, the realization sizes { M_j } and the number of different components in the universe N.Figure <ref> shows this distribution for the three data sets considered here.For small occurrences, the plots are compatible with a power-law decay, with a dataset-specificexponent.Only for genomes this curve is clearly U-shaped (see also Fig. S3), and shows a “core” of shared components, i.e., protein domains shared by almost all the genomes, together with a rich group of rare components.Book chapters do not show this marked behavior, due to the fact that the ubiquitous words (e.g. articles, pronouns, prepositions) are much less than the chapter-specific words.Finally, LEGO sets display no core of shared components, and this is probably due to the wide range of themes using poorly overlapping brick types. §.§ A random-sampling model as a minimal model for component systems with defined component frequenciesIn order to identify the statistical consequences of a heterogeneous usage of components on the statistics of shared components, a suitable model is needed.In particular, we would like to generate system realizations starting from a fixed component frequency distribution without any additional functional information or constraint.To this end, we employ a random-sampling procedure <cit.> that builds artificial realizations through an iterative random extraction (with replacement) of components from their frequencies { f_i } in the whole system.Each realization size M is specified by the number of random extractions.More precisely, the following prescriptions (Fig. <ref>e) define the random-sampling model that will be used in the following. (i) The component abundance rank distribution is assumed to be a universal property of the component system and well represented by the empirical overall abundances (see Fig. S2 and the SM for a discussion of this assumption).(ii) The extraction probability of a component is proportional to its overall abundance.(iii) A realization of size M is generated by M independent extractions from the pool of components.Statements (ii) and (iii) define a multinomial process.Given a normalized list of component frequencies { f_i }, i = 1, … N (where N is the size of the available “vocabulary”), and the size M of the realization, the probability of a specific configuration { n_1, n_2, …, n_N}, where n_i is the number of the components with frequency f_i is P(n_1, n_2, …, n_N;M) = M!/∏_i=1^N n_i!∏_i=1^N f_i^n_i under the constraint that ∑_i=1^N n_i = M.Note that the expected value of n_i is M f_i. Therefore, on average the global abundance distribution is conserved in each realization.In other words, the component composition in each realization is a sampled copy of the universe, without any of the possible complex correlations which may follow from architectural and functional properties of an empirical system.For example, in the context of bacterial genome evolution, the random-sampling model translates into a scenario in which there is continuous and completely random horizontal gene transfer (exchange of genetic material) between species <cit.>.Thus, genome composition would simply reflect the pan-genome abundances of protein domains.While horizontal gene transfer is indeed a major force in bacterial evolution <cit.>, several additional genome-specific functional constraints are clearly in place in evolution <cit.> and these are neglected by the model. Therefore, the random sampling can be considered as a null model useful to disentangle the consequences of the observed global heterogeneity in the component usage from actual hallmarks of more complex functional constraints.§.§ The distribution of shared components is mainly a consequence of component frequencies, number of available components and realization sizes The fact that the distribution of shared components is qualitatively very similar in systems that are so different triggers the question of whether it may be an emergent statistical consequence of other system properties.In particular, we asked to what extent the statistics of shared components could be a direct consequence of component frequencies.As explained above, this question can be addressed quantitatively using a random-sampling model that generates an artificial copy of the empirical system by drawing realizations (whose sizes are fixed by the empirical ones) from the component frequency distribution.Figure <ref> compares the empirical occurrence distributions with simulations of a random sampling. The null-model curves (dashed lines) provide very good approximations of the empirical laws, particularly for low component occurrences. Additionally, the model matches well the power law decay with the system-specific exponent.Finally, the model predicts also the qualitative behavior of core components, and specifically that only genomes show a clear U-shaped distribution of shared components. The relative core sizes of the three systems are also well approximated, although there are some quantitative deviations from the empirical values that will be addressed in detail in Section <ref>.These results suggest that the shape of the distribution of shared components in the three widely different empirical systems considered here is well described by a random-sampling model that only conserves the empirical component frequencies, the vocabulary (i.e., the set of possible components) and the realization sizes. The next section provides an analytical understanding of this observation.§.§ A wide range of component frequency patterns lead to occurrence distributions with power-law decay and U-shape. Thus far, we have used the model only to addressthe specific statistics of component sharing ofthe empirical systems under consideration. To this end, we have simulated the random-sampling model fixing the component frequencies and realization sizes as in the empirical cases.More in general, one can ask whether a power-law decaying and/or U-shaped distribution of component occurrences are expected for a givendistribution of component frequencies. To address this question, we have computed analytically the distribution of shared components under general prescriptions for the component frequency distributions within the random-sampling model.For the sampling procedure explained in Section <ref>, the probability q_i that a component of rank i is present in a realization of size M_j is q_i(M_j)= 1 - (1-f_i)^M_j, where f_i is the component probability of extraction.Therefore, the expectation value for the occurrence of component i over a set of R realizations is o_i = 1/R∑_j=1^R q_i(M_j) = 1 - 1/R∑_j=1^R(1-f_i)^M_j .In order to obtain the probability distribution associated to this rank representation, one can use the fact that the rank of a component with occurrence o is the number of components with occurrence higher than o. In fact,these naturally correspond to components with higher frequency and thus lower rank.Therefore, we can write the rank i(o) as i(o)= rank(o) = ∑_o'=o^o_1 N p(o')≃ N ∫_o^o_1 p(o') d o', where o_1 is the highest possible occurrence, which corresponds to the component of rank 1. The function i(o)is simplythe inverse function of Eq. <ref>.From the approximate integral representation of i(o), the occurrence probability distribution p(o) is defined by the simple relation d i(o)/do =-N p(o). Eq. <ref> provides a general relation between the representation of the frequency distribution as a rank plot and the representation as a probability distribution.Indeed, the arguments presented here to introduce Eqs. <ref> and <ref> have been previously used to establish the connection between Zipf's law as a rank plot and Zipf's law as a frequency distribution <cit.>. §.§.§ Observed versus possible vocabulary of components and Heaps' law When a set of R realizations of size M is generated through a random-sampling procedure from a pool of Ñ possible different components with their probabilities of extraction {f_i}, the expected size N of the vocabulary that is actually sampled can be expressed as <cit.> N = Ñ - ∑_i=1^Ñ( 1 -f_i )^MR . Thus, in general, N≤Ñ.If the system size, defined by the total number of extractions M R, is large enough, essentially all possible components are expected to be sampled at least once, thus leading to the simplification N≃Ñ that we implicitely assumed in Eq. <ref>.However, in general, the observed vocabulary in an ensemble of realizations is an increasing function of the system size, i.e., N(MR). This functional dependence is the analogous of Heaps' law, which is the empirical power-law growth of the number distinct components with the system size observed in linguistics <cit.>, and in genomics <cit.>.This distinction between the observed and the possible vocabulary of components is discussed in more detail in the SM and will be relevant in the following sections.§.§.§ Analytical distribution of shared components for component frequencies with a power-law or an exponential distributionExplicit expressions for the occurrence distribution can be derived assuming a simple scenario, in which all realizations have the same size M, and the component frequency statistics follows a prescribed function.We first consider the empirically relevant case of a power-law frequency rank plot (Fig. <ref>, left panel) defined by f_i = 1/α i^-γ, α = ∑_i=1^Ñ i^-γ . Under these assumptions and using Eqs. <ref> and <ref>,the exact expression of the occurrence distribution can be calculated: p(o) = ( 1-o )^1/M-1/γ M N α^1/γ( 1 - (1 - o)^1/M)^1/γ+1 . The distribution is defined in the interval of occurrences [o_N;o_1],where o_i is computed by Eq. <ref> and N is the effectiveor observed component vocabulary, which can be a function of thesystem size, i.e., N(MR), as described by Eq. <ref>.Considering the limit of small occurrences and large sizes, i.e., o ≪ 1 andM ≫ 1, one finds precisely the empirically observedpower-law decay.Specifically, in this limit the occurrence distribution takes the formp(o) ≃M^1/γ/α^1/γγ N o^-1/γ-1 , where the power-law exponent depends only on the exponent γ of the frequency rank-plot .Analogous calculations (details in the SM) can be performed assuming a frequency distribution described by an exponential rank plot f_i ∼ e^-λ i (right panel of Figure <ref>).In this case, the distribution of shared components, for large enough realizations M≫1, has the expression p(o) ≃(1-o)^-1/N λlog[ (1-o)^-1]. Interestingly, for rare families the above expression further simplifies to a power-law decay p(o) ≃1/N λ o^-1, with a “universal” exponent -1.This indicates that also systems with a heterogeneous but more compact frequency distribution are expected to show a power-law decay in the occurrence distribution.Figure <ref> shows the agreement between these predictions and simulations of the random-sampling model for the two illustrative examples of a power law and of an exponential distribution of component frequencies.These analytical predictions have a dependence on the sampled vocabulary N and are expected to hold even if this is actually smallerthan the total number of possible components Ñ (Fig. S5).The effects of a dependence of the observed dictionary on system size (i.e., Heaps' law N(MR))become relevant and has to be taken into account when comparing statistical features of ensembles ofrealizations with different sizes MR.§.§.§ Shape of the distribution of shared components andrescaling properties We now turn our attention to the conditions for a U-shaped distribution of shared components in the random-sampling model. Figure <ref>ac already show that the decay of the occurrence ofrare components is only set by the exponent γ as described byEq. <ref>, but for different values of M andN the distribution may or may not display a significantfraction of core components.Additionally, Figure <ref>bd proves that equations (<ref>) and (<ref>) can capture quantitatively the occurrence distributions and thus can well describe the relative proportion of core and specific components. In order to understand under what conditions this distribution becomes clearly U-shaped for an underlying power-law frequency distribution, it is useful to note a rescaling property of Eq. <ref>.Taking the limit of large realizations M ≫ 1, Eq. <ref> becomes p(o) = k(γ,M,N)(1-o)^-1/γ(-log(1-o))^1+1/γ ,which depends only on two parameters, γ and the rescaling parameter k(γ,M,N) = M^1/γ/α^1/γ N. This rescaling property shows that the statistics of component sharing is actually a function of a specific combination of realization sizes (e.g., text lengths) and of the range of possible components (e.g., the observed vocabulary).Specifically, the functional form of the distribution is purely defined by the exponent γ, while the rescaling parameter k sets the normalization factor and the range of possible occurrences. In fact, the analytical expression of the occurence corresponding to the distribution minimum, i.e., o_min = 1 - e^-1-1/γ, is only a function of γ, while the minimum possible occurrence value o_N ≃ 1 - e^-k^γ scales with k.Therefore, a U-shaped occurrence distribution should be generally expected for component systems with highly heterogeneous component frequencies since the power-law decay and the presence of a minimum before the core are robust features with respect to system parameters.This is confirmed by the analysis of component systems with different values of k and γ (illustrative examples in Figure S7): the system specificities set the power-law decay of the left part of the distribution, its support, and the relative proportion of core and rare components, but the U-shape is conserved.However, this shape can be more or less symmetric and more or less clearly evident depending on the actual size of the core fraction. The following section discusses in detail the non-trivial dependences of the core size on system parameters.For the case of component frequency distributions with an exponential rank plot, the statistics of shared components (Eq. <ref>) is a function of a single effective parameter λ N, and does not depend on the realization sizes M.In other words, the shape of the distribution, and whether it is clearly U-shaped, only depend on the decay of component frequencies and on the total number of components.In fact, occurrence distributions corresponding to different exponential frequency rank plots collapse if λ N is constant, even if the realizations have widely different size. This is shown in Figure S4 of the SM. §.§.§ The core sizeWe canestimate the “core size” by computing the fraction ofcomponents with occurrence greater than a given arbitrary occurrencethreshold θ_c as a function of the only two effectiveparameters γ and k. Integrating Eq. <ref> between θ_c and themaximum occurrence o_1, and then taking the limit M≫ 1,this quantity reads c = 1if o_N≥θ_cc = k [ -log(1 - θ_c) ]^-1/γ otherwise , where o_N is the left boundary of the occurrence distribution,corresponding to the component with lowest frequency.Starting from this estimate of the core size,Figure <ref>ab show how the scaling property is verified in simulations. Fig. <ref>c compares the analytical predictions for the core size with simulations for different values of γ,showing perfect agreement.Equally, one can obtain analytical estimates for the fraction of rare components (occurrence below a fixed threshold), which are tested in Fig. <ref>d.Thus, with increasing k, core families increase linearly with a γ-dependent slope until all components are shared, and concurrently rare components decrease linearly until they hit zero (when the lower cutoff of occurrence exceeds the chosen threshold value).Component number and realization size only enter through the combination defined by the rescaling parameter k. This phenomenology fully characterizes the distribution of shared components with varying parameters. The general relation (Eq. <ref>) between the core size and the rescaling parameter k translates into different dependences of the core size on the typical realization size M, depending on the relation between the system size MR and the total number of accessible components Ñ.While this issue is discussed in more detail in the SM, it is easy to intuitively understand the different regimes. For large enough systems, all possible components Ñ areexpected to be sampled at least once, thus making the observed vocabulary N≃Ñ a constant parameter. This is the regime considered in Fig. <ref>a.In this regime, Eq. <ref> simplifies to the simple scaling c ∼M^1/γ/Ñ.On the other hand, in several empirical systems the observed vocabulary is a function of the system size, and typically with the power-law dependence N(MR) ∼ (MR)^β (with β < 1) called Heaps' law.Thus, in general, the core fraction is expected to show the more complex dependences c∼ M^1/γ -β R^-β. However, a random-sampling procedure starting from a Zipf's law described by Eq. <ref> leads to the approximate relation β≃ 1/γ between the exponents of Zipf's and Heaps' law <cit.>. Therefore, in this regime the core fraction becomes only a function of the number of realizations as c∼ R^1/γ. These different scaling relations in different regimes are tested in Figure S6. Note that the absolute number of core components c N, as estimated from Eqs. <ref> and <ref>, is instead always independent from the number of realizations, even in the regime where Heaps' law is expected to hold (Figure S6).For component frequency distributions with an exponential rank plot, the sampling procedure leads to an occurrence distribution that is independent from the realization size M (Eq. <ref>). However, the exact analytical prediction for the core size (the analogousof Eq. <ref>) still has a dependence on M. But this is due to the residual dependence of the maximum occurrencevalues (o_1) on M and does not affect the shape of the distribution.This last technical point is discussed in more detail in the SM. §.§ Empirical distributions of shared components satisfy the relations predicted by the random sampling. One can ask whether the general analytical predictions discussed in the previous section can be applied to empirical data.In particular, we first asked how the power-law decay exponent of the distribution of shared components relates to the component frequency rank plot in empirical systems, and if this relation follows our analytical prediction. An analytical mapping would give a more syntheticand powerful description than the direct simulations discussed in Fig. <ref>. Importantly, the analytical formulas for the distribution of shared components are derived under the hypothesis of a pure power-lawor exponential component frequency rank plot.However, the three empirical datasets (as previously discussed),showa double-scaling frequency distribution.To override this issue, we restricted the frequency rank plot range in which the predictions are applicable.The procedure to perform this comparison is described in Fig. <ref>.First, we chose an arbitrary threshold θ_r defining the rare components and we mapped it to the frequency rank plot (assuming the model), by using the inverse function of Eq. <ref>. The frequency rank associated to the occurrence threshold θ_r, i(θ_r) in the figure, is the rank above which the model prediction for the decay of the distribution of sharedcomponentsshould apply as long as i(θ_r) does not cross the position ofthe change in scaling. In other words, since in the model there is a monotonic relationbetween occurrence and frequency (Eq. <ref>), allcomponents with rank greater than i(θ_r) (and frequency smaller that f_i(θ_r)) are assumed to be the components with occurrence lower than θ_r.We then estimated the behavior of the frequency rank plot in the high-rank region (after i(θ_r)) as the best fit with a power-law function or an exponential. This leads to a prediction for the decay exponent of the distribution of shared components (using Eq. <ref> or Eq. <ref> for the exponential case) in the range [o_N, θ_r].Fig. <ref> shows that the predicted decay exponents correspond well with the data. The random-sampling model also gives qualitative analytical predictions for the expected fraction of core components, and thus for the expected shape of the distribution of shared components for a given empirical system.While the analytical relations between exponents applied in Figure <ref> do not depend on the realization sizes, the analytical formulas for the fraction of core components (see e.g. Eq. <ref>) were derived assuming realizations of fixed size M.The actual size distributions for the three empirical systems are quite broad (Figure S1), but we can still use the analytical framework to get an estimate of the core fraction considering the average realization size of each empirical system. Following the same line of reasoning as for the low-occurrence tail of the distribution of shared components, we can use a restricted region of the frequency rank plot.In this case, the low-rank region (with exponent around 1 for all the datasets, see Fig. <ref>) is expected to contain the core components.Therefore, the parameter γ can be fixed to 1, implying that the fraction of core components, given by Eq. <ref>, should be simply proportional to the rescaling parameter k (Eq. <ref>).However, thenormalization factor α, which is present in the definition of k and defined in Eq. <ref>, takes an approximately constant value with respect to Ñ for large values of Ñ, as it is the case for the empirical examples considered.As a consequence, the core fraction should be simply proportional to M/N.This estimate can be used to explain why the core fraction is much larger in genomes than in the other two empirical systems (see Figure <ref>d).In fact, genome sizes are typically of the same order as the total number of families (M ≃ 3000, N = 1531, see Figure S1) leading to a large expected core. By comparison, book chapters have similar realization sizes but a much larger vocabulary (N ≃ 50000), and LEGO sets have very small sizes (M ≃ 100) compared to vocabulary size (N ≃ 13000).More in general, Eqs. (<ref>) and (<ref>) lead to a scaling estimate (dependent on the decay of the frequency rank plot) as a function of the system parameters M and N, which can be applied to data, in order to generate expectations for the core components.For example, for Zipf-like (exponent -1) frequency distributions, we expect the absolute number of core components to be linearly dependent on the average size of realizations M, and essentially insensitive to the vocabulary size N and the total number of realizations R.In genomics language, this would imply that the number of core protein domains does not directly depend on the number of sequenced genomes but only on their sizes and on the total number of different protein domains discovered.Note that adding new genomes to the data set is not expected to alter the power-law exponent γ ≃1 of the global frequency distribution for high-frequency components, since it does not change if the distribution is evaluated on sub-samples of the empirical dataset (Figure S2).As previously discussed, the core fraction, instead of the absolute number of core components, is expected to have a more complex dependence on the typical realization size M and on the number of realizations R.Moreover, in empirical systems these relations are further complicated by the fact that the frequency distributions cannot be described by simple power-laws (Fig. <ref>). Nevertheless, the relation between the core fraction and the average realization size predicted by a random-sampling model can be tested numerically,asFigure <ref>a shows for prokaryotic genomes, and seemsaccurately verified and roughly linear in the tested range of sizes. However, the predicted fraction of core components is actuallymuch smaller than the empirical one. This highlights the presence of additional functional constraintsand/or specific correlations in the empirical system that the modelcan not capture. The next section addresses this point more in detail. §.§ Deviations from the random-sampling predictions can highlight system-specific propertiesBeyond the striking agreement with null predictions for shared components, the deviations from sampling can be used to quantify specific functional and architectural features of a component system.While the scope of this work is to highlight the common trends and their origins, we discuss a specific example, in order to show the feasibility of this procedure.Of the three data sets considered here, the case where the clearest deviations emerge are genomes.For example, Figure <ref>a illustrates how the random sampling underestimates the empirical core size by a constant offset, for genomes of increasing size.Generally speaking, this larger core of components is due to the components that tend to occur in most realizations, but in few copies. The natural explanation is that there are specific basic functions that are essential for all (or most) genomes, but the domains involved in these functions are not necessarily needed in many copies per genome, and thus their presence in all realizations does not simply correlate with high global abundances as the random sampling would entail <cit.>.To test this hypothesis, we divided the domain families in functional categories (see Section <ref> for the functional annotation), and tested if most of the deviations from the random-sampling prediction can be ascribed to the statistics of domains belonging to specific categories. The result of this analysis is reported in Figure <ref>b.Different parts of the distribution of shared components are indeed enriched in components of different biological functions with respect to the random-sampling expectation.In particular, protein domains that play a functional role in information processes - such as DNA translation, DNA transcription, and DNA replication- are clearly enriched in the core.At the same time, they seem statistically under-represented at occurrences around 0.6.These two deviations can be explained as two sides of the same coin if this category contains domain families that empirically occur in all genomes but in a single copy per genome.Indeed, the global frequency (i.e, across all genomes) of families that are both single-copy and ubiquitous is f= R/R M = 1/M.Therefore, their occurrence predicted by the random-sampling model is o=1-(1-1/M)^M = 1- e^M log(1-1/M)≃ 1- e^-1≃ 0.6 (where the rough approximation holds for large enough M), thus naturally leading to an excess of those families in the core and to a depletion around o≃ 0.6.The observation of a strong presence of protein domains related to basic cellular function in the core genome is not new <cit.>.However, the random-sampling model allows in principle to distinguish families whose presence in the core could be simply explained by their high abundance in the pan-genome and thus it would be expected also in a simple scenario of random gene exchange.Finally, the observed correlation between biological functions and deviations from random sampling predictions seems coherent with a picture, recently proposed <cit.>, in which natural selection and functional constraints have played an important role in defining the empirical U-shaped distribution of gene occurrences.§ DISCUSSION AND CONCLUSIONS This work employs a simple statistical model based on random sampling to describe the distribution of shared components in complex component systems. A similar approach was employed in quantitative linguistics to explain how the dictionary used in a text scales with text size as measured in number of words (the so-called “Heaps' law”) while assumingZipf's law for componentfrequencies <cit.>. We extended the model to show that there is a general link between the heterogeneity in component frequency and the statistics of shared components, regardless of the mechanisms that generate heterogeneity.Consequently, models or generative processes able to explain the heterogeneity in component frequency implicitly carry predictions for the statistics of shared components.The striking similarities of laws governing both component abundance and occurrence found in empirical systems of very different origins (LEGO sets, genomes, book chapters) support the idea that the concept of “component system” defined in this work can capture in a unified framework a large class of complex systems with some common global properties.Different component systems, besides having specific architectural constraints, may show convergent phenomena in terms of global statistics.Such “universal” phenomena may be regarded as emergent properties due to system heterogeneity, which transcend the specific design, generative process or selection criteria at the origin of a system. Analogous phenomena occur, for example, in ecosystems, where emergent species-abundance distributions appear for forests, birds or insects <cit.>.Beyond the examples considered here, modular systems in a wide range of disciplines can be represented as component systems.Developing a common theoretical language for such systems can help the exchange of ideas, models and data-analysis techniques between distant communities of researchers <cit.>.For example, the statistics of component sharing considered here plays a central role in genomics <cit.> but is relatively unexplored in the context of natural languages <cit.>.Conversely the random-sampling approach used here was developed in quantitative linguistics <cit.>, and this work shows that it is applicable to other systems, including the detection of functional constraints in prokaryotic genome evolution.An important result of this work is a proof of the clear link between the heterogeneity of component abundance in a system and the statistics of shared components. This link is consistent with data from three very different empirical systems andwell captured by the random-sampling model. The fact that emergent patterns can be explained by largely null models resembles again the case of biodiversity, where neutral theories ignoring species interactions and competitive exclusion appear to capture many of the emerging trends of species abundance <cit.>.If the trends of component sharing of generic component systems are to be regarded as largely null and due to components heterogeneity, system-specific investigations should be informed of this general trend.Quantitative null models, such as the one provided here, may be crucial for identifying dataset-specific deviations that are related to functional reasons or constraints.In the data considered in this work, the patterns of shared components show differences between empirical data and the null model in some cases.This is particularly true in the genomic context, where the differences can indeed be traced back to functional constraints in genome composition. Therefore, the framework can be useful to pinpoint hallmarks of functional design and distinguish them from statistical effects, particularly for the detection of causality, dependency and correlation structures between components from occurrence patterns.Once a null model is defined, these features can emerge as significant deviations from the null behavior, for example as violations of the constraints linking different global statistics such as the abundance rank plot, the distribution of shared components and Heaps' law.We have considered here a specific example for the case of shared protein domain families in genomes (Fig. <ref>), but this question still needs to be approached systematically.In this specific case, core components are particularly enriched by specific functional classes of components with respect to the random-sampling prediction.In evolutionary terms, the random sampling defines a scenario in which the pan-genome fully determines the overall abundance of the gene families in each genome, while in empirical bacterial genomes genome-specific functional constraints are clearly in place <cit.>.Deviations from the null scenario can thus highlight the role of selection for specific functions, supporting from a different perspective the idea that the empirical U-shaped gene occurrence distribution is affected by selective rather than neutral processes <cit.>.§ ACKNOWLEDGEMENTSWe thank Erik van Nimwegen for useful discussions. This work is supported by the "Departments of Excellence 2018 - 2022" Grant awarded by the Italian Ministry of Education, University and Research (MIUR) (L. 232/2016)SUPPLEMENTAL MATERIAL Statistics ofshared components in complex component systems § BOOKS COMPOSING OUR LINGUISTIC CORPUS§ SIZE DISTRIBUTIONS OF THE REALIZATIONS IN THE THREE DATA SETS§ UNIVERSALITY OF THECOMPONENT ABUNDANCE DISTRIBUTION (ZIPF'S LAW) Figure <ref> tests thecomponent frequency rank distribution conservation when it is evaluated on different sub-sets of the total empirical data set.These sub-sets are composed by realizations in a fixed range of sizes, showing that the global frequency statistics does not depend on the realization sizesor on the number of realizations considered.Note that this test is necessaryto safely compare the analytical null predictions with different sub-samples of the empirical data set, as for example in Figure 7 of the main text. Moreover, the fact that the Zipf's laws of the under-sampled datasets is essentially identical to the global one (especially in the high-frequency regime) suggests that the observed Zipf's law is not under-sampled and can thus be considered a good estimation of the “universal” one. § ROBUSTNESS OF THE U-SHAPED DISTRIBUTION OF SHARED COMPONENTS FOR BACTERIAL GENOMES TO THE BINNING PROCEDURE.§ PROPERTIES OF THE OCCURRENCE DISTRIBUTION GENERATED BY AN EXPONENTIAL FREQUENCY RANK PLOT The mathematical calculation described in the section IIIC of the main text can be applied to an exponential rank distribution of the form f_i = 1/β e^-λ i, β = ∑_i=1^Ñ e^-λ i. Considering a random sampling of R realizations with fixed size M, one finds: p(o) = (1-o)^1/M-1/λ M N ( 1 - (1-o)^1/M). Imposing the condition M ≫ 1 this equation takes the form p(o) ≃(1-o)^-1/N λlog[ (1-o)^-1],which providesa good approximation for the overall distribution shape as a function of one single effective parameter k = N λ. In theM ≫ 1 limit, the occurrence extreme values are o_1 ≃ 1 and o_N ≃ 0. This implies that the distribution is well defined over allpossible values of occurrence.Figure <ref> shows the rescaling properties ofEq. <ref>by testing its independence on M (panel a) and by varying N and λ while keepingtheir product constant (panel b).For rare families, one can further approximate the expression for p(o) finding the expected power-law decay with exponent -1: p(o) ≃1/N λ o^-1.We now analyze the properties of thefraction of core components, i.e.,those with occurrence greater than the threshold θ_c.In order to derive the coresize one has to integrate the distribution described by Eq. <ref>from o = θ_c to the maximum occurrence value o_1 (whose formula can be obtained from Eq. 2 of the main text).The result reads: c = 1if o_N≥θ_cc = - 1/N λ( λ + logβ + log[ 1 - (1-θ_c)^1/M])otherwise. In the limit oflarge M this expression becomesc ≃ - 1/N λ( λ + logβ + log[ log(1-θ_c)^-1] -logM), which further simplifiesonly when the logarithm of M becomes dominant over the other terms.It is worth mentioning that the expression above does not show rescaling properties, even in the regime M ≫ 1,and this may seem to be in contradiction with Eq. <ref>. Nevertheless, this apparent inconsistency is basically due to the singular behavior of the occurrence distribution in o ≃ 1.In the large M limit, the right boundary can be expressed as o_1 = 1 - ϵ,where ϵ is an infinitesimal term depending on M and λ,whose effect on the overall distribution shape is negligible (Eq. <ref>). However, the core size is defined as the integral of the distribution.Therefore,the variation of p(o_1) due to a change in M or λ provides a sufficientlylarge contribution (because of the function singular behavior) which compensates the infinitesimal variation of o_1.Finally, this leads to a finite contribution to the integral and thus to the core size as it is defined in the main text.In general, this finite contribution has a non-trivial dependency on the parameters,explaining why Eq. <ref> does not show the rescaling property.§DIFFERENCE BETWEEN OBSERVED AND POSSIBLE VOCABULARY OF COMPONENTS (HEAPS' LAW) AND ITS EFFECTS ON THE CORE-SIZE ESTIMATES.As discussed in the main text, the difference between the possible different components Ñ and the ones that are actually sampled in an ensemble of Rrealizationsof size M is definded by:N = Ñ - ∑_i=1^Ñ( 1 - i^-γ/α)^MR.This equation shows the general dependence of thesampled dictionary onthe system sizeN(MR),which is essentially a generalization of Heaps' law.Our analytical predictions (Eqs. 6- 9 of the main text) for the distribution of shared components have an explicit dependence on N rather than on Ñ. As Figure <ref> shows, this distinction becomes negligible in the limit of large systems, but it is in general relevant.A residual dependence on Ñ is in principle present in the normalization factor α. However,if Ñ is large (as it is the case empirically),this dependence is negligible and the normalization factor can actually be considered constant as it can be easily confirmed numerically.A rough estimate of the system size at which the sampling procedure is expected to have extracted essentially all different components, thus making N≃Ñ,can be given by introducing a crude approximation of Eq. <ref>. For large system sizes and vocabulary sizes, the dominant term in the sum is the last term, and when this dominant term becomes negligible the sampled and the observed dictionary should roughly coincide. The dominant term can be further approximated as(1 - Ñ^-γ/α)^MR = e^MR log(1- Ñ^-γ/α)≃ e^-MR/Ñ^γα.This approximation naturally introduces the relevant scale η= MR/Ñ^γα whose value can be used to determine if the system is close to “saturation”, i.e., N≃Ñ for η≫1, or if instead a scaling analogous to Heaps' law should be expected. The potential difference between the possible and the observed vocabulary of components is relevant in evaluating the depedence of the core size on the system parameters.For component systems with a power-law distribution of component frequencies, Eq. 12 of the main text describes the core size in terms of the rescaling parameter k.In order to translatethis general expression into the core dependences on the number R and size M of the realizations, different regimes have to be considered.If the system is close to saturation (η≫ 1) c^(saturation) = ( M/α)^1/γ1/N[ -log(1 - θ_c) ]^-1/γ∝M^1/γ/Ñ, where Ñ is a constant and this relation implies a power-law dependenceof the core fraction on the typical realization size,which becomes simply linear for the empirically relevant case of γ=1 (Zipf's law),and no dependence on the number of realizations (Figure <ref>). On the other hand, when η is small, the sampled vocabulary grows sublinearly with the system size in analogy to Heaps' law.In the framework of a random sampling of components, this vocabulary growth can be described analytically for γ > 1 (see ref. 29 of the main text) as: N = Γ( 1-1/γ) ( M R/α)^1/γ + o ( M R /Ñ^γ - 1). Using this expression for N in the core-size estimate (Eq. 12 of the main text), we have an analytical expression of its dependencies on M and Rin this “Heaps' law” regime:c^(Heaps' law) = R^-1/γ/Γ( 1-1/γ)[ -log(1 - θ_c) ]^-1/γ∝ R^-1/γ. In this regime the core fraction does not depend on the realization size and can be progressively reduced by adding new realizations to the ensemble as we tested numerically in Figure <ref>. Note that, as also discussed in the main text,if we consider the absolute number of core components rather than the fraction, the situation simplifies and there is no need to distinguish different regimes.Indeed, Eq. 12 of the main text implies that the number of core componentsis simply given byc N=( M/α)^1/γ[ -log(1 - θ_c) ]^-1/γ. Since α has only a negligible dependence on Ñ,the number of core components is essentially independent from R for every value of η,andhas a power-law dependence on the typical realization size M but in this case also in the “Heaps' law” regime. This result is tested in Figure <ref>cd. A good agreement between the analytical prediction above and numerical simulations is shownin the η≪ 1 regime which indeed entails the analogous of Heaps' law.§ CHARACTERIZATION OF THE SHAPE OF THE OCCURRENCE DISTRIBUTION | http://arxiv.org/abs/1707.08356v2 | {
"authors": [
"Andrea Mazzolini",
"Marco Gherardi",
"Michele Caselle",
"Marco Cosentino Lagomarsino",
"Matteo Osella"
],
"categories": [
"q-bio.GN",
"physics.soc-ph"
],
"primary_category": "q-bio.GN",
"published": "20170726102324",
"title": "Statistics of shared components in complex component systems"
} |
[NO \title GIVEN] [NO \author GIVEN] December 30, 2023 ====================== (), a hardware accelerator for inference with Deep Neural Networks (DNNs), is presented and evaluated on Convolutional Neural Networks. exploits the variable per layer precision requirements of DNNs to deliver execution time that is proportional to the precision p in bits used per layer for convolutional and fully-connected layers. Prior art has demonstrated an accelerator with the same execution performance only for convolutional layers<cit.>. Experiments on image classification CNNs show that on average across all networks studied,outperforms a state-of-the-art bit-parallel accelerator <cit.> by without any loss in accuracy while it is more energy efficient. requires no network retraining while it enables trading off accuracy for additional improvements in execution performance and energy efficiency. For example, if a 1% relative loss in accuracy is acceptable, is on average faster and more energy efficient than a conventional bit-parallel accelerator. A Tartan configuration that processes 2-bits at time, requires less area than the 1-bit configuration, improves efficiency to 1.24× over the bit-parallel baseline while being 73% faster for convolutional layers and 60% faster for fully-connected layers is also presented. § INTRODUCTION It is only recently that commodity computing hardware in the form of graphics processorsdelivered the performance necessary for practical, large scale Deep Neural Network applications <cit.>. At the same time, the end of Dennard Scaling in semiconductor technology <cit.> makes it difficult to deliver further advances in hardware performance using existing general purpose designs. It seems that further advances in DNN sophistication would have to rely mostly on algorithmic and in general innovations at the software level which can be helped by innovations in hardware design. Accordingly, hardware DNN accelerators have emerged. The DianNao accelerator family was the first to use a wide single-instruction single-data (SISD) architecture to process up to 4K operations in parallel on a single chip <cit.> outperforming graphics processors by two orders of magnitude. Development in hardware acceleratorshas since proceeded in two directions: either toward more general purpose accelerators that can support more machine learning algorithms while keeping performance mostly on par with DaDianNao () <cit.>, or toward further specialization onspecific layers or classes of DNNs with the goal of outperforming in execution time and/or energy efficiency, e.g., <cit.>. This work is along the second direction. While an as general purpose as possible DNN accelerator is desirable further improving performance and energy efficiency for specific machine learning algorithms will provides us with the additional experience that is needed for developing the next generation of more general purpose machine learning accelerators. Section <ref> reviews several other accelerator designs.While 's functional units process 16-bit fixed-point values, DNNs exhibit varying precision requirements across and within layers, e.g., <cit.>. Accordingly, it is possible to use shorter, per layer representations for activations and/or weights. However, with existing bit-parallel functional units doing so does not translate into a performance nor an energy advantage as the values are expanded into the native hardware precision inside the unit. Some designs opt to hardwire the whole network on-chip by using tailored datapaths per layer, e.g., <cit.>. Such hardwired implementations are of limited appeal formany modern DNNswhose footprint ranges several 10s or 100s of megabytes of weights and activations.Accordingly, this work targets accelerators that can translate any precision reduction into performance and that do not require that the precisions are hardwired at implementation time.This work presents (), a massively parallel hardware accelerator whose execution time for fully-connected () and convolutional () layers scales with the precision p used to represent the input values. uses hybrid bit-serial/bit-parallel functional units and exploits the abundant parallelism of typical DNN layers with the followinggoals: 1) exceeding 's execution time performance and energy efficiency, 2) maintaining the same activation and weight memory interface and wire counts, 3) maintaining wide, highly efficient accesses to weight and activation memories. Ideally, improves execution time over by 16/p where p is the precision used for the activationsin and for the activations and weights in . Every single bit of precision that can be eliminated ideally reduces execution time and increases energy efficiency. For example, decreasing precision from 13 to 12 bits in an can ideally boost the performance improvement over to 33% from 23% respectively. builds upon the () accelerator <cit.> which improves execution time and energy efficiency ononly. Whilematches the performance of a bit-parallel accelerator on its energy efficiency suffers considerably.improvesperformance and energy efficiency over a bit-parallel accelerator for both and . This work evaluates on a set of convolutional neural networks (CNNs) for image classification. On average reduces inference time by ,andover for the fully-connected, the convolutional, and all layersrespectively. Energy efficiency compared to with is ,andrespectively.By comparison, efficiency with compared to is 0.73×,1.21× and 1.14× respectively. Additionally, enables trading off accuracy for improving execution time and energy efficiency. For example, on average on , accepting a 1% loss in relative accuracy improves performance toand energy efficiency tocompared to .In detail this work makes the following contributions:* Extends the accelerator offering performance improvements on . Not only does not improve performance on , but its energy efficiency suffers compared to .* incorporatescascading multiple serial inner-product (SIP) units improving utilization when the number or filters or the dimensions of the filters is not a multiple of the datapath lane count. * It uses the methodology of Judd et al., <cit.> to determine per layer weight and activation precisions for the fully-connected layers of several modern image classification CNNs.* It evaluates a configuration of which trades off some of the performance improvement for enhancing energy and area efficiency. The evaluated configuration processes two activation bits per cycle and requires half the parallelism and the SIPs than the bit-serial configuration.* Reports energy efficiency and area measurements derived from a layout of the accelerator demonstrating its benefits over the preciously proposed and accelerators. The rest of this document is organized as follows: Section <ref> motivates . Section <ref> illustrates the key concepts behind via an example. Section <ref> reviews the architecture and presents an equivalent configuration.Section <ref> presents the experimental results. Section <ref> reviews related work and discusses the limitations of this study and the potential challenges with . Section <ref> concludes.§ MOTIVATION This section motivates by showing that: 1)the precisions needed for the of several modern image classification CNNs are far below the fixed 16-bit precision used by , and 2) the energy efficiency of is below that of for . Combined these results motivate which improves performance and energy efficiency for bothand compared to .§.§ Numerical Representation Requirements Analysis The experiments of this section corroborate past results that the precisions needed vary per layer for several modern image classification CNNs and during inference.The section also shows that there is significant potential to improve performance if it were possible to exploit per layer precisions even for the .The per layer precision profiles presented here were found via the methodology of Judd et al. <cit.>.Caffe <cit.> was used to measure how reducing the precision of each affects the network's overall top-1 prediction accuracy over 5000 images. The network definitions and pre-trained synaptic weights are taken from the Caffe Model Zoo <cit.>. The networks are used as-is without retraining. Further reductions in precisions may be possible with retraining. As Section <ref> will explain, 's performance on anlayer L is bound by the maximum of the weight (P_w^L) and activation (P_a^L) precisions. Accordingly, precision exploration was limited to cases where both P^L_w and P^L_a are equal. The search procedure is a gradient descent where a given layer's precision is iteratively decremented one bit at a time, until the network's accuracy drops. For weights, the fixed-point numbers are set to represent values between -1 and +1. For activations, the number of fractional bits is fixed to a previously-determined value known not to hurt accuracy, as per Judd et al.<cit.>. While both activations and weights use the same number of bits, their precisions and ranges differ. For only the activation precision is adjusted as with the design there is no benefit in adjusting the weight precisions as well. Weights remain at 16-bits for . While, reducing the weight precision for can reduce their memory footprint <cit.>, an option we do not explore further in this work.Table <ref> reports the resulting per layer precisions separately for and . The ideal speedup columns report the performance improvement that would be possible if execution time could be reduced proportionally withprecision compared to a 16-bit bit-parallel baseline. For the , the precisions required range from 8 to 10 bits and the potential for performance improvement is 1.64× on average and ranges from 1.63× to 1.66×. If a 1% relative reduction in accuracy is acceptable then the performance improvement potential increases to 1.75× on average and ranges from 1.63× to as much as 1.85×. Given that the precision variability for is relatively low (ranges from 8 to 11 bits) one may be tempted to conclude that a bit-parallel architecture with 11 bits may be an appropriate compromise. However, note that the precision variability is much larger for the (range is 5 to 13 bits) and thus performance with a fixed precision datapath would be far below the ideal. For example, speedup with a 13-bit datapath would be just 1.23× vs. the 2× that is be possible with an 8-bit precision. A key motivation for is thatitsincremental costover that already supports variable per layer precisions for is well justified given the benefits. Section <ref> quantifies this cost and the resulting performance and energy benefits. §.§ Energy Efficiency with () uses hybrid bit-serial/bit-parallel inner-product units for processing activations and weights respectively exploiting the per layer precision variability of modern CNNs <cit.>. However,exploits precision reductions only foras it relies on weight reuse across multiple windows to maintain the width of the weight memory the same as in(there is no weight reuse in ). Figure <ref> reports the energy efficiency ofover that of for (Section <ref> details the experimental methodology). While performance is virtually identical to , energy efficiency is on average 0.73× compared to . This result combined with the reduced precision requirements of serves as motivation for extending to improve performance and energy efficiency compared toon both and . §.§ Motivation SummaryThis section showed that: 1) The per layer precisions for on several modern CNNs for image classification vary significantly and exploiting them has the potential to improve performance by 1.64× on average. 2) that exploits variable precision requirements only for achieves only 0.73× the energy efficiency of a bit-parallel baseline. Accordingly, an architecture that would exploit precisions for as well as is worth investigating in hope that it will eliminate this energy efficiency deficit resulting in an accelerator that is higher performing and more energy efficient for both layer types. Combined and account for more than 99% of the execution time in .§ : A SIMPLIFIED EXAMPLE This section illustrates at a high-level the way operates by showing how it would process two purposely trivial cases: 1) a fully-connected layerwith a single input activation producing two output activations, and 2) a convolutional layerwith two input activations and one single-weight filter producing two output activations. The per layer calculations are: Fully-Connected:Convolutional:f_1 = w_1 × ac_1 = w × a_1 f_2 = w_2 × ac_2 = w × a_2 Where f_1, f_2, c_1 and c_2 are output activations, w_1, w_2, and w are weights, and a_1, a_2 and a are input activations. For clarity all values are assumed to be represented in 2 bits of precision.§.§ Conventional Bit-Parallel ProcessingFigure <ref>a shows a bit-parallel processing engine representative of . Every cycle, the engine can calculate the product of two 2-bit inputs, i (weight) and v (activation) and accumulate or store it into the output register OR. Parts (b) and (c) of the figure show how this unit can calculate the example CVL over two cycles. In part (b) and during cycle 1, the unit accepts along the v input bits 0 and 1 of a_1 (noted as a_1/0 and a_1/1 respectively on the figure), and along the i input bits 0 and 1 of w and produces both bits of output c_1. Similarly, during cycle 2 (part (c)), the unit processes a_2 and w to produce c_2. In total, over two cycles, the engine produced two 2b× 2b products. Processing the example FCL also takes two cycles: In the first cycle, w_1 and a produce f_1, and in the second cycle w_2 and a produce f_2. This process is not shown in the interest of space.§.§ 's Approach Figure <ref> shows how a -like engine would process the example CVL. Figure <ref> shows the engine's structure which comprises two subunits. The two subunits accept each one bit of an activation per cycle through inputs v_0 and v_1 respectively and as before, there is a common 2-bit weight input (i_1,i_0). In total, the number of input bits is 4, the same as in the bit-parallel engine.Each subunit contains three 2-bit registers: a shift-register AR, a parallel load register BR, and an parallel load output register OR. Each cycle each subunit can calculate the product of its single bit v_i input with BR which it can write or accumulate into its OR.There is no bit-parallel multiplier since the subunits process a single activation bit per cycle. Instead, two AND gates, a shift-and-add functional unit, and OR form a shift-and-add multiplier/accumulator. Each AR can load a single bit per cycle from one of the i wires, and BR can be parallel-loaded from AR or from the i wires.Convolutional Layer: Figure <ref> through Figure <ref> show how processes the CVL. The figures abstract away the unit details showing only the register contents. As Figure <ref> shows, during cycle 1, the w synapse is loaded in parallel to the BRs of both subunits via the i_1 and i_0 inputs. During cycle 2, bits 0 of a_1 and of a_2 are sent via the v_0 and v_1 inputs respectively to the first and second subunit. The subunits calculate concurrently a_1/0× w and a_2/0× w and accumulate these results into their ORs. Finally, in cycle 3, bit 1 of a_1 and a_2 appear respectively on v_0 and v_1. The subunits calculate respectively a_1/1× w and a_2/1× w accumulating the final output activations c_1 and c_2 into their ORs. In total, it took 3 cycles to process the layer. However, at the end of the third cycle, another w could have been loaded into the BRs (the i inputs are idle) allowing a new set of outputs to commence computation during cycle 4. That is, loading a new weight can be hidden during the processing of the current output activation for all but the first time. In the steady state, when the input activations are represented in two bits, this engine will be producing two 2b × 2b terms every two cycles thus matching the bandwidth of the bit-parallel engine. If the activations a_1 and a_2 could be represented in just one bit, then this engine would be producing two output activations per cycle, twice the bandwidth of the bit-parallel engine. The latter is incapable of exploiting the reduced precision for reducing execution time. In general, if the bit-parallel hardware was using P_BASE bits to represent the activations while only P^L_a bits were enough, would outperform the bit-parallel engine by P_BASE/P_a^L. Fully-Connected Layer: Figure <ref> shows how a -like unit would process the example FCL. As Figure <ref> shows, in cycle 1, bit 1 of w_1 and of w_2 appear respectively on lines i_1 and i_0. The left subunit's AR is connected to i_1 while the right subunit's AR is connected to i_0. The ARs shift in the corresponding bits into their least significant bit sign-extending to the vacant position (shown as a 0 bit on the example). During cycle 2, as Figure <ref> shows, bits 0 of w_1 and of w_2 appear on the respective i lines and the respective ARs shift them in. At the end of the cycle, the left subunit's AR contains the full 2-bit w_1 and the right subunit's AR the full 2-bit w_2.In cycle 3, Figure <ref> shows that each subunit copies the contents of AR into its BR. From the next cycle, calculating the products can now proceed similarly to what was done for the CVL. In this case, however, each BR contains a different weight whereas when processing the CVL in the previous section, all BRs held the same w value. The shift capability of the ARs coupled with having each subunit connect to a differenti wire allowed to load a different weight bit-serially over two cycles. Figure <ref> and Figure <ref> show cycles 4 and 5 respectively. During cycle 4, bit 0 of a1 appears on both v inputs and is multiplied with the BR in each subunit. In cycle 5, bit 1 of a1 appears on both v inputs and the subunits complete the calculation of f_1 and f_2. It takes two cycles to produce the two 2b × 2b products once the correct inputs appear into the BRs. While in our example no additional inputs nor outputs are shown, it would have been possible to overlap the loading of a new set of w inputs into the ARs while processing the current weights stored into the BRs. That is the loading into ARs, copying into BRs, and the bit-serial multiplication of the BRs with the activations is a 3-stage pipeline where each stage can take multiple cycles. In general, assuming that both activations and weights are represented using 2 bits, this engine would match the performance of the bit-parallel engine in the steady state. When both set of inputs i and v can be represented with fewer bits (1 in this example) the engine would produce two terms per cycle, twice the bandwidth of the bit-parallel engine of the previous section.Summary: In general, if P_BASE the precision of the bit-parallel engine, and P_a^L and P_w^L the precisions that can be used respectively for activations and weights for layer L, a engine can ideally outperform an equivalent bit parallel engine by P_BASE/P_a^L for CVLs, and by P_BASE/max(P_a^L, P_w^L) for FCLs. This example used the simplest engine configuration. Since typical layers exhibit massive parallelism, can be configured with many more subunits while exploiting weight reuse for CVLs and activation reuse for FCLs. The next section describes the baseline state-of-the-art DNNs accelerator and presents an equivalent configuration.§ ARCHITECTUREThis work presents as a modificationof the state-of-the-art DaDianNao accelerator. Accordingly, Section <ref> reviews 's design and how it can process FCLs and CVLs. For clarity, in what follows the term brick refers to a set of 16 elements of a 3D activation or weight array input which are contiguous along the i dimension, e.g., a(x,y,i)...a(x,y,i+15). Bricks will be denoted by their origin element with a B subscript, e.g., a_B(x,y,i).The size of a brick is a design parameter. Furthermore, an FCL can be thought of as a CVL where the input activationarray has unit x and y dimensions, and there are as many filters as output activations, and where the filter dimensions are identical to the input activation array.§.§ Baseline System: DaDianNao Figure <ref> shows a which processes 16 filters concurrently calculating 16 activation and weight products per filter for a total of 256 products per cycle <cit.>. Each cycle the accepts 16 weights per filter for total of 256 weight and 16 input activations. The multiplies each weight with only one activation whereas each activation is multiplied with 16 weights, one per filter.The reduces the 16 products per filter into a single partial output activation, for a total of 16 partial output activations for the . Each chip comprises 16 such , each processing a different set of 16 filters per cycle. Accordingly, each cycle, the whole chip processes 16 activations and 256× 16=4K weights producing 16× 16=256 partial output activations, 16 per .Internally, each has: 1) a synapse buffer (SB) that provides 256 weights per cycle one per weight lane, 2) an input neuron buffer (NBin) which provides 16 activations per cycle through 16 neuron lanes, and 3) a neuron output buffer (NBout) which accepts 16 partial output activations per cycle. In the 's datapath each activation lane is paired with 16 weight lanes one from each filter.Eachweight and activation lane pair feeds a multiplier, and anadder tree per filter lane reduces the 16 per filter 32-bit products into a partial sum.In all, the filter lanes produce each a partial sum per cycle, for a total of 16 partial output activations per tile. Once a full window is processed, the 16 resulting sums are fed through a non-linear activation function, f, to produce the 16 final output activations. The multiplications and reductions needed per cycle are implemented via 256 multipliers one per weight lane and sixteen 17-input (16 products plus the partial sum from NBout) 32-bit adder trees one per filter lane. Figure <ref>a shows an overview of the chip. There are 16 processing tiles connected via an interconnect to a shared 2MB central eDRAM Neuron Memory (NM). 's main goal was minimizing off-chip bandwidth while maximizing on-chip compute utilization. To avoid fetching weights from off-chip, uses a 2MB eDRAM Synapse Buffer (SB) for weights per for a total of 32MB eDRAM for weight storage. All inter-layer activation outputs except for the initial input and the final output are stored in NM which is connected via a broadcast interconnect to the 16 Input Neuron Buffers (NBin) buffers. All values are 16-bit fixed-point, hence a 256-bit wide interconnect can broadcast a full activation brick in one step. Off-chip accesses are needed only for reading: 1) the input image, 2) the weights once per layer, and 3) for writing the final output. Processing starts by reading from external memory the first layer's filter weights, and the input image. The weights are distributed over the SBs and the input is stored into NM. Each cycle an input activation brick is broadcast to all units. Each units reads 16 weight bricks from its SB and produces a partial output activation brick which it stores in its NBout. Once computed, the output activations are stored through NBout to NM and then fed back through the NBins when processing the next layer. Loading the next set of weights from external memory can be overlapped with the processing of the current layer as necessary. §.§ As Section <ref> explained, processes activations bit-serially multiplying a single activation bit with a full weight per cycle. Each tile multiplies 16 16-bit activations with 256 weights each cycle. To match 's computation bandwidth, needs to multiply 256 1-bit activations with 256 weights per cycle. Figure <ref> shows the tile. It comprises 256 Serial Inner-Product Units (SIPs) organized in a 16× 16 grid. Similar to each SIPmultiplies 16 weights with 16 activations and reduces these products into a partial output activation. Unlike , each SIP accepts 16 single-bit activation inputs. Each SIP has two registers, each a vector of 16 16-bit subregisters: 1) the Serial Weight Register (SWR), and 2) the Weight Register (WR). These correspond to AR and BR of the example of Section <ref>. NBout remains as in , however, it is distributed along the SIPs as shown.Convolutional Layers: Processing starts by reading in parallel 256 weights from the SB as in , and loading the 16 per SIP row weights in parallel to all SWRs in the row. Over the next P_a^L cycles, the weights are multiplied by the bits of an input activation brick per column. exploits weight reuse across 16 windows sending a different input activation brick to each column. For example, for a CVL with a stride of 4 a will processes 16 activation bricks a_B(x,y,i), a_B(x+4,y,i) through a_B(x+63,y,i) in parallel a bit per cycle. Assuming that the processes filters f_i though f_i+15, after P_a^L cycles it would produce the following 256 partial output activations: o_B(x/4,y/4,f_i), through o_B(x/4+15,y/4,f_i),that is 16 contiguous on the x dimension output activation bricks. Whereas would process 16 activations bricks over 16 cycles, processes them concurrently but bit-serially over P_a^L cycles. If P_a^L is less than 16, will outperform by 16/P_a^L, and when P_a^L is 16, will match 's performance. Fully-Connected Layers: Processing starts by loading bit-serially and in parallel over P_w^L cycles, 4K weights into the 256 SWRs, 16 per SIP. Each SWR per row gets a different set of 16 weights as each subregister is connected to one out of the 256 wires of the SB output bus for the SIP row (is inthere are256× 16=4K wires). Once the weights have been loaded, each SIP copies its SWR to its SW and multiplication with the input activations can then proceed bit-serially over P_a^L cycles. Assuming that there are enough output activations so that a different output activation can be assigned to each SIP, the same input activation brick can be broadcast to all SIP columns. For example, for an FCL a will process one activation brick a_B(i) bit-serially to produce 16 output activation bricks o_B(i) through o_B(i× 16) one per SIP column. Loading the next set of weights can be done in parallel with processing the current set, thus execution time is constrained by P_max^L = max(P_a^L,P_w^L). Thus, a produces 256 partial output activations every P_max^L cycles, a speedup of 16/P_max over since a always needs 16 cycles to do the same.Cascade Mode: For to be fully utilized an FCL must have at least 4K output activations. Some of the networks studied have a layer with as little as 2K output activations. To avoid underutilization, the SIPs along each row are cascaded into a daisy-chain, where the output of one can feed into an input of the next via a multiplexer. This way, the computation of an output activation can be sliced over the SIPs along the same row. In this case, each SIP processes only a portion of the input activations resulting into several partial output activations along the SIPs on the same row. Over the next np cycles, where np the number of slices used, the np partial outputs can be reduced into the final output activation. The user can chose any number of slices up to 16, so that can be fully utilized even with fully-connected layers of just 256 outputs.This cascade mode can be useful in other Deep Learning networks such as in NeuralTalk <cit.> where the smallest can have 600 outputs or fewer. Other Layers: like can process the additional layers needed by the studied networks. For this purpose the includes additional hardware support for max pooling similar to . An activation function unit is present at the output of NBout in order to apply nonlinear activations before the output neurons are written back to NM. §.§ SIP and Other Components SIP: Bit-Serial Inner-Product Units:Figure <ref> shows 's Bit-Serial Inner-Product Unit (SIP). Each SIP multiplies 16 activation bits, one bit per activation, by 16 weights to produce an output activation. Each SIP has two registers, a Serial Weight Register (SWR) and a Weight Register (WR), each containing 16 16-bit subregisters. Each SWR subregister is a shift register with a single bit connection to one of the weight bus wires that is used to read weights bit-serially for FCLs. Each WR subregister can be parallel loaded from either the weight bus or the corresponding SWR subregister, to process CVLs or FCLs respectively. Each SIP includes 256 2-input AND gates that multiply the weights in the WR with the incoming activation bits, and a 16× 16b adder tree that sums the partial products. A final adder plus a shifter accumulate the adder tree results into the output register OR. In each SIP, a multiplexer at the first input of the adder tree implements the cascade mode supporting slicing the output activation computation along the SIPs of a single row. To support signed 2's complement neurons, the SIP can subtract the weight corresponding to the most significant bit (MSB) from the partial sum when the MSB is 1. This is done with negation blocks for each weight before the adder tree. Each SIP also includes a comparator (max) to support max pooling layers. Dispatcher and Reducers: Figure <ref>b shows an overview of the full system. As in there is a central NM and 16 tiles. A Dispatcher unit is tasked with reading input activations from NM always performing eDRAM-friendly wide accesses. It transposes each activation and communicates each a bit a time over the global interconnect. For CVLs the dispatcher has to maintain a pool of multiple activation bricks, each from different window, which may require fetching multiple rows from NM. However, since a new set of windows is only needed every P_a^L cycles, the dispatcher can keep up for the layers studied. For FCLs one activation brick is sufficient. A Reducer per title is tasked with collecting the output activations and writing them to NM. Since output activations take multiple cycles to produce, there is sufficient bandwidth to sustain all 16 tiles. §.§ Processing Several Activation Bits at Once In order to improve 's area and power efficiency, the number of activation bits processed at once can be adjusted at design time. The chief advantage of these designs is that less SIPs are needed in order to achieve the same throughput – for example, processing two activation bits at once reduces the number of SIP columns from 16 to 8 and their total number to half. Although the total number of bus wires is similar, the distance they have to cover is significantly reduced. Likewise, the total number of adders required stays similar, but they are clustered closer together. A drawback of these configurations is they forgo some of the performance potential as they force the activation precisions to be multiple of the number of bits that they process per cycle. A designer can chose the configuration that best meets their area, energy efficiency and performance target.In these configurations the weights are multiplied with several activation bits at once, and the multiplication results are partially shifted before they are inserted into their corresponding adder tree. In order to load the weights on time, the SWR subregister has to be modified so it can load several bits in parallel, and shift that number of positions every cycle. The negation block (for 2's complement support) will operate only over the most significant product result.§ EVALUATIONThis section evaluates 's performance, energy and area compared to . It also explores the trade-off between accuracy and performance for . Section <ref> described the experimental methodology. Section <ref> reports the performance improvements with . Section <ref> reports energy efficiency and Section <ref> reports 's area overhead. Finally, Section <ref> studies a configuration that processes two activation bits per cycle.§.§ Methodology, and were modeled using the same methodology for consistency. A custom cycle-accurate simulator models execution time. Computation was scheduled as described by <cit.> to maximize energy efficiency for . The logic components of the both systems were synthesized with the Synopsys Design Compiler <cit.> for a TSMC 65nm library to report power and area. The circuit is clocked at 980 MHz. The NBin and NBout SRAM buffers were modelled using CACTI <cit.>. The eDRAM area and energy were modelled with Destiny <cit.>. Three design corners were considered as shown in Table <ref>, and the typical case was chosen for layout. §.§ Execution TimeTable <ref> reports 's performance and energy efficiency relative to for the precision profiles in Table <ref> separately for , , and the whole network.For the 100% profile, where no accuracy is lost, yields, on average, a speedup ofover on FCLs. With the 99% profile, it improves to .There are two main reasons the ideal speedup can't be reached in practice: dispatch overhead and under-utilization. Dispatch overhead occurs on the initial P^L_w cycles of execution, where the serial weight loading process prevents any useful products to be performed. In practice, this overhead is less than 2% for any given network, although it can be as high as 6% for the smallest layers. Underutilization can happen when the number of output neurons is not a power of two, or lower than 256. The last classifier layers of networks designed to perform recognition of ImageNetcategories <cit.> all provide 1000 output neurons, which leads to 2.3% of the SIPs being idle.Compared to , matches its performance improvements on while offeringperformance improvements on . We do not report the detailed results for since they would have been identical to for and within 1% of for . We have also evaluated on NeuralTalk LSTM <cit.> which uses long short-term memory to automatically generate image captions. Precision can be reduced down to 11 bits without affecting the accuracy of the predictions (measured as the BLEU score when compared to the ground truth) resulting in a ideal performance improvement of 1.45×translating into a 1.38× speedup with .We do not include these results in Table <ref> since we did not study the nor did we explore reducing precision further to obtain a 99% accuracy profile.§.§ Energy EfficiencyThis section compares the Energy Efficiency or simply efficiency of and . Energy Efficiency is the inverse of the relative energy consumption of the two designs. As Table <ref> reports, the average efficiency improvement with across all networks and layers for the 100% profile is . In FCLs, is more efficient than .Overall, efficiency primarily comes from the reduction in effective computation following the use of reduced precision arithmetic for the inner product operations. Furthermore, the amount of data that has to be transmitted from the SB and the traffic between the central eDRAM and the SIPs is decreased proportionally with the chosen precision.§.§ AreaTable <ref> reports the area breakdown of and . Over the full chip, needs 1.49× the area compared to while delivering on average a improvement in speed. Generally, performance would scale sublinearly with area for due to underutilization. The 2-bit variant, which has a lower area overhead, is described in detail in the next section. §.§ TRT2b This section evaluates the performance, energy efficiency and area for a multi-bit design as described in Section <ref>, where 2 bits are processed every cycle in as half as many total SIPs. The precisions used are the same as indicated in Table <ref> for the 100% accuracy profile rounded up to the next multiple of two. Table <ref> reports the resulting performance. The 2-bit always improves performance compared to as the “vs. ” columns show. Compared to the 1-bit performance is slightly lower however given that the area of the 2-bit is much lower, this can be a good trade-off. Overall, there are two forces at work that shape performance relative to the 1-bit . There is performance potential lost due to rounding all precisions to an even number, and there is performance benefit by requiring less parallelism. The time needed to serially load the first bundle of weights is also reduced. In VGG_19 the performance benefit due to the lower parallelism requirement outweighs the performance loss due to precision rounding. In all other cases, the reverse is true.A hardware synthesis and layout of both and 's 2-bit variant using TSMC's 65nm typical case libraries shows that the total area overhead can be as low as 24.9% (Table <ref>), with an improved energy efficiency in fully connected layers of 1.24× on average (Table <ref>). § RELATED WORK AND LIMITATIONSThe recent success of Deep Learning has led to several proposals for hardware acceleration of DNNs. This section reviews some of these recent efforts. However, specialized hardware designs for neural networks is a field with a relatively long history. Relevant to , bit-serial processing hardware for neural networks has been proposed several decades ago, e.g., <cit.>. While the performance of these designs scales with precision it would be lower than that of an equivalently configured bit-parallel engine. For example, Svensson et al., uses an interesting bit-serial multiplier which requires O(4× p) cycles, where p the precision in bits <cit.>. Furthermore, as semiconductor technology has progressed the number of resources that can be put on chip and the trade offs (e.g., relative speed of memory vs. transistors vs. wires) are today vastly different facilitating different designs. However, truly bit-serial processing such as that used in the aforementioned proposals needs to be revisited with today's technology constraints due to its potentially high compute density (compute bandwidth delivered per area). In general, hardware acceleration for DNNs has recently progressed in two directions: 1) considering more general purpose accelerators that can support additional machine learning algorithms, and 2) considering further improvements primarily for convolutional neural networks and the two most dominant in terms of execution time layer types: convolutional and fully-connected. In the first category there are accelerators such as Cambricon <cit.> and Cambricon-X <cit.>. While targeting support for more machine learning algorithms is desirable, work on further optimizing performance for specific algorithms such as is valuable and needs to be pursued as it will affect future iterations of such general purpose accelerators.is closely related to <cit.> whose execution time scales with precision but only for CVLs. does not improve performance for FCLs. improves upon by enabling: 1) performance improvements for FCLs, and 2) slicing the activation computation across multiple SIPs thus preventing under-utilization for layers with fewer than 4K outputs. Pragmatic uses a similar in spirit organization to but its performance on CVLs depends only on the number of activation bits that are 1 <cit.>. It should be possible to apply the extensions to Pragmatic, however, performance in FCLs will still be dictated by weight precision. The area and energy overheads would need to be amortized by a commensurate performance improvement necessitating a dedicated evaluation study.The Efficient Inference Engine (EIE) uses synapse pruning, weight compression, zero activation elimination, and network retraining to drastically reduce the amount of computation and data communication when processing fully-connected layers <cit.>. An appropriately configured EIE will outperform for FCLs, provided that the network is pruned and retrained. However, the two approaches attack a different component of FCL processing and there should be synergy between them. Specifically, EIE currently does not exploit the per layer precision variability of DNNs and relies on retraining the network. It would be interesting to study how EIE would benefit from a -like compute engine where EIE's data compression and pruning is used to create vectors of weights and activations to be processed in parallel. EIE uses single-lane units whereas uses a coarser-grain lane arrangement and thus would be prone to more imbalance. A middle ground may be able to offer some performance improvement while compensating for cross-lane imbalance.Eyeriss uses a systolic array like organization and gates off computations for zero activations <cit.> and targets primarily high-energy efficiency. An actual prototype has been built and is in full operation. Cnvlutin is a SIMD accelerator that skips on-the-fly ineffectual activations such as those that are zero or close to zero <cit.>. Minerva is a DNN hardware generator which also takes advantage of zero activations and that targets high-energy efficiency <cit.>. Layer fusion can further reduce off-chip communication and create additional parallelism <cit.>. As multiple layers are processed concurrently, a straightforward combination with would use the maximum of the precisions when layers are fused.Google's Tensor Processing Unit uses quantization to represent values using 8 bits <cit.> to support TensorFlow <cit.>. As Table <ref> shows, some layers can uselower than 8 bits of precision which suggests that even with quantization it may be possible to use fewer levels and to potentially benefit from an engine such as . §.§ LimitationsAs in this work assumed that each layerfits on-chip. However, as networks evolve it is likely that they will increase in size thus requiring multiple nodes as was suggested in . However, some newer networks tend to use more but smaller layers. Regardless, it would be desirable to reduce the area cost of most of which is due to the eDRAM buffers. We have not explored this possibility in this work. Proteus <cit.> is directly compatible with and can reduce memory footprint by about 60% for both convolutional and fully-connected layers. Ideally, compression, quantization and pruning similar in spirit to EIE <cit.> would be used to reduce computation, communication and footprint. General memory compression <cit.> techniques offer additional opportunities for reducing footprint and communication.We evaluated only on CNNs for image classification. Other network architectures are important and the layer configurations and their relative importance varies. enables performance improvements for two of the most dominant layer types. We have also provided some preliminary evidence that works well for NeuralTalk LSTM <cit.>. Moreover, by enabling output activation computation slicing it can accommodate relatively small layers as well. Applying some of the concepts that underlie the design to other more general purpose accelerators such as Cambricon <cit.> or graphics processors would certainly be more preferable than a dedicated accelerator in most application scenarios. However, these techniques are best first investigated into specific designs and then can be generalized appropriately.We have evaluated only for inference only. Using an engine whose performance scales with precision would provide another degree of freedom for network training as well. However, needs to be modified accordingly to support all the operations necessary during training and the training algorithms need to be modified to take advantage of precision adjustments.This section commented only on related work on digital hardware accelerators for DNNs. Advances at the algorithmic level would impact as well or may even render it obsolete. For example, work on using binary weights <cit.> would obviate the need for an accelerator whose performance scales with weight precision. Investigating 's interaction with other network types and architectures and other machine learning algorithms is left for future work. § CONCLUSION This work presented , an accelerator for inference with Convolutional Neural Networks whose performance scales inversely linearly with the number of bits used to represent values in fully-connected and convolutional layers. also enables on-the-fly accuracy vs. performance and energy efficiency trade offs and its benefits were demonstrated over a set of popular image classification networks. The new key ideas in are: 1) Supporting both the bit-parallel and the bit-serial loading of weights into processing units to facilitate the processing of either convolutional or fully-connected layers, and 2) cascading the adder trees of various subunits (SIPs) to enable slicing the output computation thus reducing or eliminating cross-lane imbalance for relatively small layers. opens up a new direction for research in inference and training by enabling precision adjustments to translate into performance and energy savings. These precisions adjustments can be done statically prior to execution or dynamically during execution. While we demonstrated for inference only, we believe that , especially if combined with Pragmatic, opens up a new direction for research in training as well. For systems level research and development, with its ability to trade off accuracy for performance and energy efficiency enables a new degree of adaptivity for operating systems and applications.ieeetr | http://arxiv.org/abs/1707.09068v1 | {
"authors": [
"Alberto Delmas",
"Sayeh Sharify",
"Patrick Judd",
"Andreas Moshovos"
],
"categories": [
"cs.NE"
],
"primary_category": "cs.NE",
"published": "20170727225613",
"title": "Tartan: Accelerating Fully-Connected and Convolutional Layers in Deep Learning Networks by Exploiting Numerical Precision Variability"
} |
DEIB]Daniele Ioli DEIB]Alessandro Falsone MDH]Alessandro Vittorio PapadopoulosCA DEIB]Maria Prandini[CA]Corresponding author, Tel. +46 (0)21-1073 23 [DEIB]Politecnico di Milano, Piazza Leonardo da Vinci, 32, 20133 Milano, Italy [MDH]Mälardalen University, Högskoleplan 1, 72123, Västerås, Sweden [myfootnote]This work is partly supported by the European Commission under the UnCoVerCPS project, grant number 643921, and was performed when the third author was a post-doctoral researcher at Politecnico di Milano. This paper proposes a compositional modeling framework for the optimal energy management of a district network. The focus is on cooling of buildings, which can possibly share resources to the purpose of reducing maintenance costs and using devices at their maximal efficiency. Components of the network are described in terms of energy fluxes and combined via energy balance equations. Disturbances are accounted for as well through their contribution in terms of energy. Different district configurations can be built, and the dimension and complexity of the resulting model will depend on the number and type of components and on the adopted disturbance description. Control inputs are available to efficiently operate and coordinate the district components, thus enabling energy management strategies to minimize the electrical energy costs or track some consumption profile agreed with the main grid operator. Smart grid modeling Compositional systems Energy management Building thermal regulation§ INTRODUCTION Building energy management, and temperature regulation in particular, has recently attracted the attention of various researchers (see, e.g., <cit.>). Indeed, energy consumption in buildings represents approximately 40% of the worldwide energy demand, and more than half of this amount is spent for Heating, Ventilation and Air Conditioning (HVAC) systems <cit.>. Energy management can be performed at the level of a single building, e.g., using the storage to shift in time the thermal energy request to time slots where the electricity costs are lower. As buildings started sharing equipments at the benefit of shared operating costs, increased flexibility, and overall performance improvement, energy management needs to be performed at the district network level, which calls for appropriate modeling and control strategies. Constructing models of interconnected systems is generally demanding, and here we propose a modular framework that simplifies this task and is also suitable for the application of different control design approaches.The proposed modeling approach is oriented to energy management and compositional in that components are described in terms of thermal/electrical energy fluxesand interact by exchanging energy, which makes easy to compose a district network configuration via energy balance equations. Our modeling framework is built with a control-oriented perspective. It includes disturbances like, e.g., solar radiation, outside temperature, occupancy, and wind power production, as well as control inputs like, e.g., buildings temperature set-points, charge/discharge of storages, activation/deactivation of devices, that can be appropriately set so as to optimize performance at the district level.Complexity and size of the model associated with a district configuration depend on number of components and type of description adopted per component. The model can be either deterministic or stochastic depending on the disturbance characterization as a deterministic or stochastic process, respectively. It can range from a low dimensional deterministic systemwith continuous input and state that is convex in the control input, to a large dimensionalStochastic Hybrid System (SHS) <cit.> with discrete and continuous input and state.Given a certain configuration, one can then formulate energy management problems like the minimization of the cost of the electrical energy requested to the main grid or the tracking of some given electrical energy exchange profile that was agreed with the main grid operator according to a demand-response strategy. The district network in the latter case can be viewed as a user that actively participates to the electrical energy demand/generation balance of the overall grid, and, hence, to its stabilization.Further contributions in the literature adopt a similar perspective. In <cit.>, the focus is on simulation so that the model dependence on the control input is not a concern. In <cit.>, the aim is the design of an energy management strategy via a simulation-based approach. The modeling effort is limited in this case, and the idea is to take an accurate model in the literature and run simulations to the purpose of policy design, with no concern of making explicit the dependence on the input and formally proving optimality. The approach in <cit.> is the closest to our approach, in that it addresses energy management problems for amicrogrid that is built based on models of single components, combined via energy balance equations. Models are however simplified, in particular that of the building. Also, occupancy is not accounted for explicitly. A specific strategy for energy management is considered, whereas our framework is more comprehensive since it allows for the design of different strategies (certainty equivalence based, robust, stochastic) for the minimization of suitably defined (nominal, min-max, average) cost in presence of (nominal,robust, probabilistic) constraints on comfort and actuation. Depending on thenetwork communication and computation capabilities, and on privacy issues like in the case of buildings not willing to disclose their consumption profile, a centralized, decentralized, or distributed optimization scheme can be conceived and implemented. Overall, our work is more general and it actually subsumes the approach in <cit.>.It is worth noticing that other modeling frameworks have been developed in the literature <cit.>. However, the obtained models are typically more complex since they are based on partial differential equations, and require numerical optimization tools for solving the resulting nonlinear optimization problems <cit.>.This paper is based on our earlier work in <cit.>, which is extended in several directions. We provide a more detailed description of the district components, including a validation with respect to other commercial simulation tools of the building thermal model according to a norm defined by the American Society for Heating Refrigerating and Air-conditioning Engineers (ASHRAE). We show how to compose a network configuration and formulate an energy management problem as an optimization program. In particular, we show a simulation study of the results achieved in the case where nominal disturbances are present and computations are performed by a central unit. Finally, we suggest a multirate approach as a viable solution for allowing real-time computation of the control input, while retaining model accuracy. The reminder of the paper is structured as follows. Section <ref> presents the models of the district network components, and Section <ref> shows how they can be connected to set up a network configuration, while defining objective and constraints of the optimal energy management problem. Section <ref> describes some configurations, providing examples of the kind of results one can achieve through the presented framework. Section <ref> shows how to deal with computational complexity, discussing a multirate approach, while Section <ref> concludes the paper. <ref> describes the procedure adopted for validating the model of the building. § DISTRICT NETWORK COMPONENTSWe consider a district network connected to the main grid that will provide the electrical energy needed to compensate for possible imbalance between demand and generation within the district. We model the evolution of the network over a finite time horizon [t_i,t_f], which is divided into M time slots of duration . The contribution in terms of energy requested/provided by the different components per time slot along the discretized control horizon is provided. Components can consume (e.g., buildings), produce (e.g., renewable power generators), store (e.g. thermal storages and batteries), or convert energy (e.g., the chiller plants), and are combined via energy balance equations so as to build the overall model of the district. Each component may be affected by some inputs which can be either disturbances or control inputs. In the case when control inputs are available, a suitable strategy can be conceived to set them so as to efficiently manage the system along the time horizon [t_i,t_f].In the rest of this section, we provide a model for the following components: building,chiller, storage, combined heat and power unit, and wind turbine. Models are either derived from first principles or taken from the literature.In the latter case, appropriate references are provided. Tables <ref>–<ref> summarize the main characteristics of the first 4 components. The last component provides an input to the network in terms of wind energy. Similarly to the wind energy contribution, one could consider the solar energy contribution provided by photovoltaic panel installations. Models partly derived from first principle and partly taken from the literature could be used to this purpose. This is not treated here, but the interested reader can refer to, e.g.,<cit.>. Further components could also be added to the district network. The key idea when introducing our compositional framework is that if a component can be modeled in terms of energy, possibly depending on some control input and/or disturbance signal, then, it can be easily included in the network. When the dependence of the energy on the control input is convex, piecewise linear, or linear with additional binary variables, the problem of designing an energy management strategy can be reduced to a mixed integer linear or a convex optimization program for which efficient solvers exist.§.§ BuildingWe consider a building as composed of n_z zones, where each zone is characterized by its own (average) temperature T_z,j, j=1,…,n_z. The zones temperatures can be collected in a vector T_z = [T_z,1⋯ T_z,n_z] and we next determine the amount of cooling energy E_c needed for making them track a given profile. We say that the building is controllable if a control layer is present to this purpose. Suitable constraints will be imposed on the assigned profile to make the resulting tracking problem feasible while guaranteeing comfort conditions at the same time.The cooling energy E_c,j requested by zone j can be derived based on the thermal energy balance within the zone, accounting for both thermal effects related to its structure and thermal phenomena related to occupancy, equipment, lights, etc, and solar radiation through windows. More precisely, we haveE_c,j = E_w,j + E_z,j+ E_p,j + E_int,j,where E_w,jis the amount of energy exchanged betweenzone j and its adjacent walls, E_z,j is the contribution of the thermal inertia of zone j, and E_p,j and E_int,j is the heat produced by people and other heat sources within zone j, respectively.The thermal model of the building is derived from first principles, following <cit.>. §.§.§ Walls contributionFor modeling the walls contribution we use a one-dimensional finite volumes model. Each wall is divided into vertical layers (`slices') that may differ in width and material composition. The area of each slice coincides with the wall area and each slice is assumed to have uniform density and temperature. The one-dimensional discretization is sensible since the heat flow is perpendicular to the crossed surface. Each internal slice exchanges heat only with nearby slices through conduction, whilst boundary slices are exposed towards either a zone or the outside of the building and exchange heat also via convection and thermal radiation. External surfaces are assumed to be gray and opaque, with equal absorbance and emissivity and with zero transmittance. Absorbance and emissivity are wavelength-dependent quantities, and here we shall consider two different values for shortwave and longwave radiation.The heat transfer balance equation for the iþ slice of the wþ wall is given by:Ṫ_w,i = 1/C_w,i[ (k_w,i^i-1 + h_w,i^i-1)T_w,i-1 + (k_w,i^i+1 + h_w,i^i+1)T_w,i+1-(k_w,i^i-1 + h_w,i^i-1 + k_w,i^i+1 + h_w,i^i+1)T_w,i + Q_g,w,i + R_w,i],where T_w,i denotes the temperature of the wall slice, C_w,i being its thermal capacity per unit area, and k_w,i^j and h_w,i^j, with j = i± 1, representing respectively the conductive and convective heat transfer coefficients between the i^th and the j^th slice of the same wall w. Q_g,w,i is the thermal power generation inside slice i and R_w,i represents radiative heat exchanges and is defined asR_w,i = 0, 1 < i < mα_w^S Q^S + α_w^L Q^L - ε_w,i Q_r(T_w,i), slice i facing outside ∑_w'=1,…,n_wj∈{1,M} F_(w,i)→(w',j)( ε_w',j Q_r(T_w',j) - ε_w,i Q_r(T_w,i) ) slice i facing insidewhere Q^S and Q^L denote the incoming shortwave and longwave radiation power per unit area, respectively, and α_w^S and α_w^L are the corresponding absorbance rates for wall w. Q_r(T_w,i) is the emitted radiation as a function of the slice temperature, ε_w,i < 1 being the emissivity and F_(w,i)→(w',j) the view factor that takes into account the fraction of radiation leaving slice i of wall w and reaching slice j of wall w'. Finally, n_w denotes the total number of walls.Equation (<ref>) holds for every slice in every wall w. If the wall is composed of m slices, we have m equations like (<ref>) with i=1,2,…,m. When the superscript in the right-hand side of equation (<ref>) takes value 0 or m+1, reference is made to either a zone of the building (internal surface of the wall) or the outside of the building (external surface of the wall). Note that k_w,1^0 = k_w,m^m+1 = 0 as there is no thermal conduction on walls boundary surfaces, h_w,i^i-1 = 0 for i>1, h_w,i^i+1 = 0 for i<m, and ε_w,i = 0 for 1<i<m, since there is no thermal convection nor radiation between slices. As for the slice in contact with the ground, we assume that the energy exchange occurs via thermal conduction only (no convection nor radiation is considered), where the ground is considered as a thermal reservoir, and as such its temperature is constant. Since we assume that each wall is a gray body, the power Q_r(T_w,i) radiated from each slice is governed by Q_r(T_w,i) = σ T_w,i^4, where σ is the Stefan-Boltzmann constant. This expression is approximately linear around the slice mean operating temperature T_w,i so that it can be replaced byQ_r(T_w,i) = 4 σT_w,i^3 T_w,i - 3 σT_w,i^4.Then, the evolution of the temperatures T_w = [T_w,1⋯ T_w,m] of the m slices composing wall w can be described in matrix form byṪ_w = A_wT_w + B_wT_z + W_wd,where we recall that T_z is the vector containing the temperatures of the n_z zones. Vector d = [T_outT_gndQ^S Q^L 1] is the disturbance input and collects the outdoor temperature T_out, the ground temperature T_gnd, and the incoming shortwave Q^S and longwave Q^L radiations. The constant 1 in d is introduced to account for the constant term in (<ref>). Finally, A_w, B_w and W_w are suitably defined matrices that are easily derived based on the scalar equation (<ref>), whose coefficients depend on the wall characteristics. Equation (<ref>) refers to a single wall. If there are n_w walls in the building, then, we can collect all walls temperatures in vector T = [T_1⋯T_n_w], and write the following equation for the evolution in time of T:Ṫ = AT + BT_z + Wd,where A is a block-diagonal matrix with A_w as wþ block, B = [ B_1 ⋯ B_n_w ] and W = [ W_1⋯W_n_w ].If we consider zone j and one of its adjacent wall w, then the thermal power transferred from wall w to zone j is given byQ_w → j = S_w h_w,b^b' (T_w,b - T_z,j),where S_w is the wall surface and the pair (b,b') can be either(1,0) or (m,m+1) according to the notation introduced for (<ref>).The total amount of thermal power transferred from the building walls to zone j can be expressed as Q_b,j = ∑_w∈𝒲_j Q_w→ j, where 𝒲_j is the set of walls w adjacent to zone j. Defining Q = [Q_b,1⋯ Q_b,n_z], we obtainQ = CT + DT_z,where C and D are suitably defined matrices derived based on equation (<ref>). From (<ref>) and (<ref>), we finally getṪ = AT + BT_z + Wd Q = CT + DT_zThe obtained model, though linear, can be quite large. However,its order can be greatly reduced by applying the model reduction algorithm based on Hankel Single Value Decomposition (HSVD), as suggested in <cit.>.The zone temperature profile to track T_z is taken as a linear function of time within each time slot of length , defined by the values u(k) = T_z(k) at the time steps k=0,1,…,M. By approximating the input d as a piecewise linear function of time as well, with values ω(k) = d(k) at k=0,1,…,M, an exact discrete time version of the linear model (<ref>) can be derived (see <ref>). The evolution of y(k) = Q(k) over the finite time horizon can then be computed asy = [y(0) ⋯ y(M)]= FT(0) + Gu + Hωwhere we set u = [u(0) ⋯ u(M)] and ω = [ω(0) ⋯ω(M)], and F, G and H are suitably defined matrices. The thermal energy E_w(k) = [E_w,1(k) ⋯ E_w,n_z(k)] transferred from the walls to all zones can be computed by integrating Q(t) on each time slot, which leads to the following approximate expression:E_w(k) = /2 ( y(k-1) + y(k) ), k=1,…,M.Finally, from (<ref>) and (<ref>) we can derive the enlarged energy vector E_w = [E_w(1) ⋯ E_w(M)]:E_w = F̃T(0) + G̃u + H̃ω,where F̃, G̃, and H̃ are obtained from matrices F, G, and H in (<ref>) via (<ref>).§.§.§ Zones energy contribution In order to decrease the temperature of zone j in the time frame from (k-1) to k, we need to draw energy from the zone itself. This energy contribution can be expressed asE_z,j(k) = - C_z,j (T_z,j(k) - T_z,j((k-1))),where C_z,j is the heat capacity of the jþ zone. If we account for all n_z zones, and all M time frames within the finite horizon [t_i,t_f], equation (<ref>) can be written in the following matrix formE_z = Zu,where we set E_z = [E_z(1) ⋯ E_z(M)] with E_z(k) = [E_z,1(k) ⋯ E_z,n_z(k)], and Z is a suitably defined matrix.§.§.§ People energy contributionOccupancy implies heat production, which in crowded places can be actually significant <cit.>. According to an empirical model in <cit.>, the heat rate Q_p,j produced by the n_p,j occupants of a zone j at temperature T_z,j is given byQ_p,j = n_p,j ( p_2 T_z,j^2 + p_1 T_z,j + p_0 ),where p_2 = -0.22 W/K^2, p_1 = 125.12 W/K and p_0 = -1.7685 · 10^4 W. Expression (<ref>) is almost linear in a sensible operating temperature range and can thus be accurately approximated by linearization around some comfort temperature T_z,j:Q_p,j = n_p,j( (2 p_2 T_z,j + p_1)(T_z,j-T_z,j) + p_2 T_z,j^2 + p_1 T_z,j + p_0 ) = n_p,j( p̃_1 T_z,j + p̃_0 ).Recall now that the zone temperature profile T_z,j to track is assumed to be linear in time. If we approximate the occupancy n_p,j as a linear function of time within each time slot as well, as suggested in <cit.>, then equation (<ref>) can be analytically integrated from (k-1) to k to obtain the energy transferred to zone j in the kþ time slot: E_p,j(k) = q_2,k(n_p,j)T_z,j(k)+ q_1,k(n_p,j)T_z,j((k-1)) + q_0,k(n_p,j)where we setq_2,k(n_p,j) = p̃_1 6( 2 n_p,j(k) + n_p,j((k-1))) q_1,k(n_p,j) = p̃_1 6( n_p,j(k) + 2 n_p,j((k-1))) q_0,k(n_p,j) = p̃_0 2( n_p,j(k) + n_p,j((k-1))) The total amount of energy transferred to all zones in each time slot can be packed in a vector E_p(k) = [E_p,1(k) ⋯ E_p,n_z(k)] and then, defining E_p = [E_p(1) ⋯ E_p(M)] and n_p = [n_p,1(0)n_p,1() ⋯ n_p,1(M) ⋯ n_p,n_z(0)n_p,n_z() ⋯ n_p,n_z(M)], one can write thatE_p = N(n_p)u + e(n_p),where N(n_p) and e(n_p) depend on the coefficients (<ref>).Note that occupancy profiles can be either obtained from data or derived from a stochastic model, like, e.g., the one in <cit.> which is based on Poisson arrival/departure processes <cit.>.Further energy contributions of the building occupants, in terms for instance of blinds movement and setpoint override, are not modeled here. Recent works on human-building interaction discuss the impact of human intervention on energy management strategies. The interested reader is referred to <cit.>, where a possible strategy to limit human intervention is proposed, and to <cit.>, where a model predictive control solution is suggested for timely adjusting the control action to unpredicted human disturbances.§.§.§ Other internal energy contributionsThere are many other types of heat sources that may affect the internal energy of a building, e.g., lighting, daylight radiation through windows, electrical equipment, etc. The overall heat flow rate produced within zone j can be expressed as the sum of three contributions, namelyQ_int,j = α_j Q^S+ λ_j+ κ_j I_ℝ^+(n_p,j),where α_j is a coefficient that takes into account the mean absorbance coefficient of zone j, the transmittance coefficients of the windows and their areas, sun view and shading factors, and radiation incidence angle. I_ℝ^+(·) denote the indicator function on the positive real values. The thermal energy contribution to zone j due to internal lighting and electrical equipment is composed of two contribution: a constant term λ_j, and an additional therm κ_j that represents the change in internal lighting and electrical equipment when people are present. Note that Q_int,j does not depend on Q^L because windows are usually shielded against longwave radiation. The energy E_int,j(k) during the k^th slot is given by:E_int,j(k) = 2[Q^S(k) + Q^S((k-1))] + λ_j +2κ_j [ I_ℝ^+(n_p,j(k)) + I_ℝ^+(n_p,j((k-1))) ]and is obtained by (<ref>), where the first (linear) and second (constant) terms have been analytically integrated, whereas the third term has been treated separately, due to the presence of the indicator function. In the cases when occupancy drops to zero or becomes nonzero in a time slot, theenergy contribution is set to a half of the contribution in the case when occupancy is nonzero at the beginning and at the end of the time slot. We can collect the thermal energy of the zones in a single vector E_int(k) = [E_int,1(k) ⋯ E_int,n_z(k)], andthen define E_int = [E_int(1) ⋯ E_int(M)], which is finally given by:E_int = M ω + L(n_p).§.§.§ Overall building cooling energy request Now we can finally compute the cooling energy demand of all zones in the building for tracking the piecewise linear zone temperature profiles T_z specified via the input u at the discrete time instants k=0,1,…,M during the time horizon [t_i,t_f]. Specifically, from (<ref>) it follows that E_c=[E_c(1) ⋯ E_c(M)] with E_c(k)=[E_c,1(k) ⋯ E_c,n_z(k)] is the sum of four contributions:E_c = E_w + E_z + E_p + E_int,where E_w is given in (<ref>), E_z in (<ref>), E_p in (<ref>), and E_int in (<ref>). This leads to the following expression for the cooling energy demand:E_c = F̃T(0) + (G̃ + Z + N(n_p)) u + (H̃+M)ω+ e(n_p)+ L(n_p)= A_cT(0) + B_c(n_p) u + W_cω + b(n_p)where A_c, and W_c are constant matrices, whereas B_c(n_p) and b(n_p) depend on the occupancy. Note that the input u defining the zone temperature profiles enters affinely the system dynamics if the occupancy n_p were fixed. §.§.§ Building block: interfaces and related constraints The thermal model of the building can be considered as a block with the following input/output interfaces: the control input vector u specifying the piecewise linearzone temperature profiles T_z at the discrete time instants k=0,1,…,M, and disturbance input vectors n_p and ωrepresenting the occupancy and the collection of outdoor temperature T_out and incoming shortwave Q^S and longwave Q^L radiations, respectively; and the output vector E_c of the cooling energy demand requested by the zones in the building to track T_z. Notice that the cooling energy demand cannot be negative. Furthermore, a profile where the zone temperature is required to decrease with a steep slope cannot be tracked. This can be formulated as a constrainton the maximum amount of energy E_c,j^max that can be requested by a zone j per time slot (from which the upper bounding vector E_c^max of the same size of E_c can be derived), and, possibly, a maximum amount E_c,b^max that can be requested by the building during the whole time horizon. This maps into the following actuation constraints:0≤E_c≤E_c^max, 1E_c≤ E_c,b^max,where 1 denotes a column vector with all elements equal to 1 so that 1E_c is the total cooling energy requested by the building. Note that when a vector is compared with a scalar like in (<ref>), it means that each component of the vector is compared with that same scalar.Table <ref> summarizes the relevant quantities related to the building model. The type attribute is introduced to denote possible different models that can be used, which eventually has some impact on the energy management problem formulation. Type A is the controllable building model where the zone temperature profiles can be optimized via the control input u, whereas Type B is the uncontrollable building model where the zone temperature profiles cannot be chosen but are already specified via some given u vector. In a network configuration, it is possible to include both controllable and uncontrollable buildings. Comfort and cooling energy bounds can then be enforced only in the case of Type A model, which contributesto the network description with equations and inequalities that are linear in the control input. §.§ Chiller plantA chiller plant is an electrical devices that reduces the temperature of a liquid, typically water, via vapor compression or absorption cycle. In this way, it converts the electric power provided by the electrical grid into cooling power, which is then conveyed to either some cooling load or some thermal storage via the chilled water circuit.Chillers can be modeled through the equationE_,ℓ=a_1T_oT_cw+a_2(T_o-T_cw)+a_4T_oE_,c/T_cw-a_3/E_,c-E_,c,0≤ E_,c≤ E_,c^max,where E_,ℓ is the electrical energy absorbed by the chiller in order to provide the cooling energy E_,cin a time slot of duration , and E_,c^max is the corresponding maximum cooling energy production. Note that E_,ℓ depends also on the outdoor temperature T_o and the temperature of the cooling water T_cw. The latter is typically regulated by low level controllers so that it is maintained almost constant at some prescribed optimal operational value, which also facilitates the stratification in the thermal storage. The chiller description (<ref>) is derived from the original Ng-Gordon model <cit.> which is based on entropy and energy balance equations and accounts also for heat losses andpump contribution to the electric energy consumption (E_,ℓ> 0 when E_,c=0).. Coefficients a_1, a_2, a_3, a_4 characterize the chiller performance. Depending on their values, we can have different efficiency curves as given by the Coefficient Of Performance (COP), which is the ratio between the produced cooling energy and the corresponding electrical energy consumption:COP = E_ch,cE_ch,ℓ.Figure <ref> shows an example of curves of the COP for three chiller units of different size, with their respective approximations presented in the following sections.We next introduce simpler approximations of relation (<ref>), which preserve convexity in the control input E_,c. §.§.§ Chiller model approximationsA convex biquadratic approximationE_,ℓ=c_1(T_o)E_,c^4+c_2(T_o)E_,c^2+c_3(T_o),0≤ E_,c≤ E_,c^max,of the nonlinear Ng-Gordon model (<ref>) can be derived by using weighted least square to best fit the most relevant points, i.e, those that correspond to zero energy request and to the maximum COP values.Another possible convex approximation of (<ref>) is via a PieceWise Affine (PWA) function given by the following convex envelope of a finite number of affine termsE_,ℓ=max{m_c(T_o)E_,c+q_c(T_o)},0≤ E_,c≤ E_,c^max,wherethe coefficients of the affine terms are collected in the two vectors m_c(T_o) and q_c(T_o), and the max operator is applied component-wise. Note that, if E_,ℓ in expression (<ref>) is to be minimized, then (<ref>) can be easily translated as a set of linear constraints with an epigraphic reformulation.The quality of the biquadratic and PWA approximations is compared in Figure <ref>.§.§.§ On-off switchingAs shown in Figure <ref>, the chiller absorbs some amount of electrical energy even when no cooling energy is produced. In order to have the possibility of switching the chiller on and off, one can introduce the binary variable δ_(k), k=0,…,M, that represents the on (δ_(k)=1) and off (δ_(k)=0) logical status of the chiller at time k, k=0,…,M. The cooling energy request E_,c(k) and on-off command δ_(k) are related via the logical conditionδ_(k) = 1 ⇔ E_,c(k) > 0.Let E_,c^max be the maximum value for E_,c and ε a small quantity, typically set equal to the machine precision. Using the Conjunctive Normal Form in <cit.>, (<ref>) can be expressed as a mixed integer linear condition:εδ_(k) ≤ E_,c(k) ≤ E_,c^maxδ_(k),which leads to δ_(k) = 0 ⇔ E_,c(k) =0 and δ_(k) = 1 ⇔ E_,c(k) ∈ [ε,E_,c^max], that are practically equivalent to (<ref>). Depending on the adopted approximation, we can rewrite the model of the chiller including the on-off condition asE_,ℓ(k)=(c_1(T_o(k))E_,c(k)^4+c_2(T_o(k))E_,c(k)^2+c_3(T_o(k)))δ_(k) max{m_c(T_o(k))E_,c(k)+q_c(T_o(k))}δ_(k)with 0≤ E_,c(k)≤ E_,c^max. The PWA formulation is particularly convenient since the product between a (piecewise) affine function Mx+q and a discrete variable δ can be reduced to a mixed integer linear condition <cit.>, by introducing the auxiliary variable z=δ (M x+Q) subject to 0≤ z ≤min{M x+Q+(1-δ) ,δ}, whereis an upper bound on Mx+q. §.§.§ Chiller block: interfaces and related constraints The chiller block can be described with a static map between the cooling energyE_,c = [E_,c(0) ⋯ E_,c(M)] that it produces and the corresponding absorbed electrical energy E_,ℓ = [E_,ℓ(0) ⋯ E_,ℓ(M)].The cooling energy that the chiller can provide is subject to the following bound:0≤E_,c≤ E_,c^max,which maps into a bound on the absorbed electrical energy0≤E_,ℓ≤ E_,ℓ^max. When the on-off command δ_ = [δ_(0) ⋯δ_(M)] is introduced as an additional control input, the following further constraint enters the chiller model:εδ_≤E_,c≤ E_,c^maxδ_.Table <ref> summarizes the relevant quantities of the chiller model, with Type A, B, C, and D representing possible modeling variants. The max operator is applied element-wise, and the symbol * is the element-wise multiplication.§.§ StorageThermal Energy Storages (TESs) are becoming widely used in medium size grids. TESs represent the most effective way, or even sometimes the only way, to take advantage of renewable energy sources. This is indeed the case for thermal solar energy and geothermal energy systems. In a smart grid context, they can be used as energy buffers for unbinding energy production from energy consumption. More specifically, in a district cooling scenario, a TES for cooling energy can shift the request of cooling energy production to off-peak hours of electrical energy consumption, make chillers operate in high-efficiency conditions, and smooth peaks of electrical energy request with benefits both for power production and distribution network systems, see e.g. <cit.>.There are many different technical solutions to store thermal energy, the most widely used are fluid tanks and Phase Changing Materials (PCMs) storages. We focus next on fluid tanks modeling, and add a note on how the model can be extended to PCMs storages in Remark <ref>. From an energy management perspective we will make use of a black box model, derived based on system identification techniques, that uses the energy exchange (added or removed) as input and the thermal energy stored as output. The simplest model is a first order AutoRegressive eXogenous(ARX) systemS(k+1) = aS(k) - s(k),where S(k) is the amount of cooling energy stored and s(k) is the cooling energy exchanged (s(k)>0 if the storage is discharged, ands(k)<0 if it is charged) in the kþ time slot,while a ∈ (0,1) is a coefficient introduced to model energy losses. By unrolling the thermal storage dynamics in (<ref>) we can express the cooling energy stored along the look-ahead discretized time horizon [t_i,t_f] in a compact form asS = Ξ_0 S(0) + Ξ_1 s,where we set S = [S(1) ⋯ S(M)], s = [s(0) ⋯ s(M-1)], and Ξ_0 and Ξ_1 are suitably defined matrices.A more sophisticated model can be obtained by introducing dissipation effects through the efficiency coefficients β_C ∈ [0,1] and β_D ∈ [0,1] for the charge/discharge dynamics as follows:S(k+1)= aS(k)-((1-β_C)δ_C+(1+β_B)δ_D)s(k),where δ_C(k) ∈{0,1} and δ_D(k) ∈{0,1} indicate the mode in which the storage is operated: δ_C(k) = 1 and δ_D(k) = 0, the storage is charged (s(k)<0), δ_C(k) = 0 and δ_D(k) = 1 the storage is discharged (s(k)>0), and δ_C(k) = δ_D(k) = 0 the storage is not used. Notice that δ_C(k) and δ_D(k) are mutually exclusive, which can be coded via the constraintδ_D(k) + δ_C(k) ≤ 1.It is possible to set minimum and maximum thresholds for the energy exchange rate in both the charging and discharging phases by constraining s(k)as follows:δ_D(k) s_D^min + δ_C(k) s_C^max≤ s(k) ≤δ_D(k) s_D^max + δ_C(k) s^min_Cwith s_C^max < s_C^min≤ 0 and 0 ≤ s_D^min < s_D^max. Note that if δ_C(k) = δ_D(k) = 0 (storage not in use), inequalities (<ref>) degenerate to the condition s(k)=0.Model (<ref>) is bilinear in the control inputs sinceδ_C(k) and δ_D(k) are multiplied by s(k). However, we can reduce it to the linear modelS(k+1)= aS(k)-(1-β_C)s_C(k)-(1+β_D)s_D(k)by replacing s(k) with the new control variables s_C(k) = δ_C(k)s(k) and s_D(k) = δ_D(k)s(k). Accordingly, the constraint (<ref>) becomesδ_C(k)s_C^max≤ s_C(k) ≤δ_C(k) s_C^minδ_D(k)s_D^min≤ s_D(k) ≤δ_D(k) s_D^max.The energy exchange s(k) can then be recovered from s_C(k) and s_D(k) as s(k) = s_C(k)+s_D(k).Model (<ref>) subject to constraints (<ref>) and (<ref>) is equivalent to model (<ref>) subject to constraints (<ref>), (<ref>) and (<ref>). This latter model has the advantage of being linear so that it can be expressed in compact form along the look-ahead discretized time horizon [t_i,t_f] as follows:S = Ξ_0 S(0)+ Ξ_C s_C + Ξ_D s_Ds = s_D+ s_C,wheres_C = [s_C(0) ⋯ s_C(M-1)], s_D = [s_D(0) ⋯ s_D(M-1)], and Ξ_C and Ξ_D are suitably defined matrices. Note that those elements of vectors s_C and s_D that correspond to a zero charge and discharge command in vectors σ_C = [σ_C(0) ⋯σ_C(M-1)] and σ_D = [σ_D(0) ⋯σ_D(M-1)] are set to zero (see (<ref>) and (<ref>). Given that the charge and discharge commands are mutually exclusive, we have thats_Cs_D=0. The described thermal storage system is active in that it can be directly operated by charge/discharge commands. Passive thermal storages are instead physical elements, like the walls of a building, that can accumulate and release thermal energy but are not directly charged or discharged. Even though it is more difficult in principle to take advantage of passive thermal storages since there is no direct way to control them, in Section <ref> we shall show how an optimal energy management strategy can exploit them. Note that batteries for electrical energy storage can in principle be modeled in the same way <cit.>. However, charging/discharging efficiencies depend on the battery State Of Charge (SOC) and energy losses can be related to the exchanged energy (exchange efficiency), so that a more complex model has to be specifically introduced. Also, additional constraints as for example the minimum and maximum charging time should be added to obtain a feasible operation of the battery. PCMs thermal storages can be modeled as electric batteries with the fraction of liquid in the storage playing the role of the SOC in determining the model coefficients.§.§.§ Storage block: interfaces and related constraints The proposed model for the thermal storage has as control input the energy exchange s, eventually decomposed into the charge and discharge inputs s_Cand s_D activated by the mutually exclusive commands σ_Cand σ_D. The stored energy S is the output of the model inboth cases. Since the storage capacity is limited and the stored energy is a positive quantity, the following constraints apply0≤S≤S^max.In addition, the amount of energy that can be exchanged per time unit is limited, and it cannot exceed certain thresholds, i.e., the boundss^min≤s≤s^maxapply to the energy exchange s, or bounds (<ref>) and (<ref>) apply to the charge and discharge inputs s_Cand s_D.Table <ref> summarizes the relevant quantities related to the storage model.The type attribute denotes possible different models for the storage. §.§ Combined Heat and Power unit: MicroturbineA Combined Heat and Power (CHP) unit is a device that jointly produces electricity and heat power while consuming primal energy (fossil fuels or hydrogen) with the purpose of reducing the amount of energy wasted in the environment. In most cases one of these two products is a byproduct. For example, modern power plants recover waste heat and deliver it for district heating purposes. CHPs with large capacity are becoming widely used and highly performing. At the same time a number of micro-CHP solutions are being developed, the most promising ones being microturbines and fuel cells that convert gas or hydrogen into heat and electricity. Combined Cooling, Heat and Power (CCHP) devices are also available that convert part of the produced heat into cooling energy.We consider a microturbine modeled through two static characteristics describing the electrical power production and the heat production, both as a function of the fuel volumetric flow rate. Figure <ref> represents the characteristics of the C30 microturbine produced by Capstone company <cit.>. We can see that both curves are almost linear. The electrical energy E_,ℓ(k) and the heat E_,h(k) produced by this microturbine during the kþ time slot can then be expressed as affine functions of the fuel volumetric flow rate u_(k), that is supposed to be constant in each time slot, i.e.,E_,ℓ(k) = m_ℓ u_(k) + q_ℓ, E_,h(k) = m_h u_(k) + q_h,where m_ℓ, q_ℓ, m_h, and q_h are positive coefficients.If we include the possibility of switching on or off the microturbine, we need to introduce the binary variable δ_(k), k=0,…,M, and modify the model as follows:E_,ℓ(k) = δ_(k)(m_ℓ u_(k) + q_ℓ), E_,h(k) = δ_(k)(m_h u_(k) + q_h). Note that we do not model the microturbine transient from on to off, as instead suggested in <cit.>. Yet, the static model that we adopt is accurate given thata sensible choice ofwhen addressing energy management is typically larger than the time scale of the microturbine dynamics. §.§.§ CHP block: interfaces and related constraints The CHP block represents a microturbine and is characterized by two control inputs that can be set in each time slot: the fuel volumetric flow rate u_ and the on-off status of the microturbine δ_. It provides as outputs the electricity E_,ℓ and the heat E_,h produced per time slot. Since the microturbine specifications require a minimum fuel volumetric flow rate u_^min for the unit to be operative, we need to include the following logical condition:u_(k) ≤ u_^min⇔δ_(k)=0,which can be rewritten as:δ_(k)(u_^min +ε)≤ u_(k)≤δ_(k)u_^max+(1-δ_(k)) u_^min,where u_^max is the maximum flow rate and ε>0 is set equal to the machine precision. The product between the (piecewise) affine function and a discrete variable δ is a nonlinear mixed integer expression that can be reduced to a mixed integer linear condition <cit.>.The constraints related to the CHP are of three types: * Fuel inlet bounds of u_ = [u_(0) ⋯ u_(M)]:u_^min≤u_≤ u_^max * Maximum heat and electrical energy that can be produced by the microturbine:0 ≤E_,h≤ E_,h^max,0 ≤E_,ℓ≤ E_,ℓ^maxwith E_,h = [E_,h(0) ⋯ E_,h(M)], and E_,ℓ = [E_,ℓ(0)⋯ E_,ℓ(M)].* Logical on-off bounds (<ref>) expressed over the finite horizon k=0,…,M with δ_ = [δ_(0) ⋯δ_(M)].Table <ref> summarizes the main characteristics of the CHP model. Type A and B are the possible variants of the CHP model.§.§ Wind turbine A wind turbine is used to convert the wind kinetic energy into electrical energy. Four different operational modes are typically defined for a controlled wind turbine (seeFigure <ref>): Mode 1, when the wind speed value is within the range from zero up to the cut-in wind speed v_in and there is no power produced by the wind turbine, which is turned off; Mode 2, below the rated power P_n, thus called below-rated, where the power captured from the wind is maximized; Mode 3, above the rated wind speed, thus called above-rated, where the wind turbine is saturated to the rated power P_n, and as the wind speed increases above the nominal turbine speed v_n, the blade pitch angle is adjusted so that local angles of attack acting on local airfoil sections become smaller, and hence the loads become relatively smaller and the power keeps constant; Mode 4, when the wind speed is above the cut-out wind speed v_out, and the wind turbine is shut down, due to load and fatigue issues.A turbine that is optimally sized for the site where it is installed is operating most of the time at the transition point between Mode 2 and Mode 3, also called at-rated <cit.>. The power generated by the wind turbine P_wt can then be calculated as follows:P_wt =0,v_wind≤ v_in or v_wind≥ v_outP_m(v_wind),v_in≤ v_wind≤ v_nP_n,v_n≤ v_wind≤ v_outwhere P_m(v_wind) is the maximum power that can be extracted from the wind kinetic energy when the wind speed is v_wind, while P_n is the rated power.Notice that the wind speed v_wind is acting as a disturbance on the turbine. Therefore, the power produced by the wind turbine as output given the disturbance input v_wind is a disturbance as well. To the purpose of the energy management of the district network, we consider the static model in Figure <ref> (solid line) for the power produced by the wind turbine as a function of the wind speed.As for the wind speed prediction, both physical and statistical models, e.g., based on Markov chain, have been considered in the literature <cit.>. Combining (<ref>) with wind speed prediction models one can determine the energy contribution of the wind turbine by computing the average power produced within a time slot, and then multiplying it by the time slot duration .Note that the static modeling of the wind turbine is appropriate if the time slot durationis sufficiently large compared to the involved inertia. In our set-up of a district network,small scale wind turbines for roof installation could be included, compatibly withof the order of minutes.§ DISTRICT NETWORK COMPOSITIONAL MODELING AND OPTIMAL ENERGY MANAGEMENTIn this section, we show how the components previously introduced can be interconnected in order to define a certain district network configuration.We consider a network of buildings located in a neighborhood and do not model the distribution network.Since the input/output interfaces of each component have been described in terms of thermal or electrical energy received or produced, energy balance equations and energy conversion functions can be adopted to combine the network components. For instance, the sum of the cooling energy requests of the buildings in the network should be equal to the sum of the cooling energy provided by chillers and taken from/stored in the thermal storages; each chiller receives as input a cooling energy request and provides as output the corresponding electrical energy consumption; the sum of the electrical energy consumption should be equal to the electrical energy produced by the local power generators, i.e. the CHP units and the wind turbine, taken from/stored in the batteries, and provided by the main grid. Depending on the number of components and the adopted model for each component, the overall model of the district network has a different size and complexity, the most general one being hybrid due to the presence of both continuous and discrete variables, and stochastic due to the disturbances (e.g., occupancy, outside temperature, solar radiation, wind velocity) acting on the system, <cit.>. Figure <ref> shows a possible district network configuration and the energy fluxes between its components and the main grid. The district network may be composed of multiplebuildings that share common resources such as cooling and heat storages, chillers, CHP units, batteries and renewable energy generators. The three nodes appearing in the figure do not correspond to any physical component but are introduced to point out that fluxes associated with the same kind of energy (electrical, heat, and cooling energy) add up to zero. Some energy contributions can be controlled (e.g., those related to storage units), some others can be controlled only indirectly (e.g., electrical energy requested by the chiller), or cannot be controlled (e.g., renewable energy production). This is pointed out using different arrowheads in Figure <ref>. As for buildings, some of them are controlled in that their cooling energy request can be modulated to some extent via the zone temperature setpoint. If the zone temperature setpoints are fixed and given by some comfort profiles, then the building is uncontrollable.We assume that the district network is connected to the main grid, which supplies the electrical energy needed to maintain the balance between electrical energy demand and generation within the district network.The district network is “smart” if it is possible to appropriately set the controllable variables so as to optimize its behavior. A typical goal is to minimize the overall cost while guaranteeing the satisfaction of the energy needs of the users in the district. Costs are mainly due to the electrical energy requested to the main grid and additional costs related to device operation such as startup and fuel costs. The overall cost is then given by:J = C_ℓ + C_ + C_ + C_f,where the first term is the electrical energy cost C_ℓ = ∑_k=1^M C_ℓ(k); C_ = ∑_k=1^M C_(k) is the cost for the chillers startup; C_ = ∑_k=1^M C_(k) and C_f=∑_k=1^M C_f(k) are the costs for the CHPs startup and fuel consumption. It is worth noticing that startup costs also serve the purpose of favoring solutions that avoid continuous and unrealistic switching of devices. Note also that additional logical conditions are needed to account for them. For example, a chiller startup cost can be modeled as C_^max{δ_(k) - δ_(k-1), 0 }, where C_^ is the actual startup cost which is accounted for at k only if the chiller was off at k-1 and is switched on at k. Similarly, for the CHP, its startup cost at k is given by C_^max{δ_(k) - δ_(k-1), 0 }. The fuel costs of a CHP are proportional to the amount of fuel consumption during the kþ time slot, i.e., ψ_fδ_(k) u_(k), where ψ_f is the unitary fuel cost.As for the electrical energy cost, the cost per time slot C_ℓ(k) is typically given by a PWA function of the electrical energy exchange E_L(k) with the main grid, i.e.,C_ℓ(k)=max{c_1,ℓ(k)E_L(k)+c_0,ℓ(k)},where the coefficients of the affine terms are collected in vectors c_1,ℓ(k) and c_0,ℓ(k), and the max operator is applied componentwise. This expression allows us to adopt different values for revenues (E_L(k)<0) and actual costs (E_L(k)>0), and to account for penalties when the electrical energy consumption/production E_L(k) exceeds certain thresholds.Note that, if C_ℓ is to be minimized, an epigraphic reformulation can be adopted to rewrite (<ref>) in terms of a set of linear inequalities.To describe E_L for an arbitrary configuration, we adopt in this section the following short-hand notations. Components correspond to energy contributions and are defined through letters (building , chiller , storage , CHP microturbine ) with a superscript that denotes the model type (symbols are given in Tables <ref>–<ref>) and the kind of energy (electrical ℓ, cooling c, and heating h) provided as output. This is important, e.g., to distinguish between a thermal storage (^c) and an electric battery (^ℓ), and also in the case when a component allows for multiple kinds of energy as output. For instance, ^B,h stands for the heating energy produced by a CHP described by a linear on-off model. The subscript possibly denotes the energy request received as input, as in the case of a chiller that has to provide the net cooling energy requested by buildings after deduction (addition) of that provided (requested) by the thermal storage units.We can for example derive the expression of E_Lfor the configuration in Figure <ref>:E_L= ^A,ℓ_←{^B,c + ^A,c + ^c}+^B,ℓ+^ℓ. If we then plug (<ref>) into equation (<ref>) and (<ref>), we get the expression for the cost function J to be minimized.Note that J may be uncertain if there are disturbance inputs acting on the system. In such a case, one can either neglect uncertainty and refer to some nominal profile for the disturbance inputs or account for uncertainty and formulate a worst case or an average cost criterion based on J. Furthermore, when we compose a district network model plugging together all the elements, we also get a number of constraints associated with them. Constraints express both technical limits (i.e., maximum cooling energy that a chiller can provide) and performance requirements (i.e., comfort temperature range). Additional constraints can be added if needed (e.g., the maximum amount of electrical energy that the main grid can provide). Yet, constraints might be uncertain due to the presence of disturbances, and, hence they might be enforced only for the nominal profile, thus neglecting uncertainty, or as robust or probabilistic constraints.Different approaches can then be adopted to address the energy management of the district network, depending also on the choice of the cost criterion (nominal/worst-case/average) and the constraints (nominal/robust/probabilistic). Uncertainty on the parameters values could also be explicitly accounted for in the design. For instance, one could assume that parameters take equally likely values in some range and impose that performance is optimized over almost all instances except for a small set.Furthermore, different architectures (centralized, decentralized or distributed) can be conceived and implemented for the resulting optimization problem solution, depending on the actual communication and computation capabilities available in the network, and on possible privacy of information issues like in the case of a building that is not willing to share its own consumption profile, while still aiming at cooperating for reducing the overall district cost.The formulation of the optimal energy management problem involves defining the following quantities: * Global parameters, i.e., sampling time , and number of M of time slots of the look-ahead time-horizon.* Optimization variables, i.e., the decision variables to be set by the optimization problem. Notice that energy balances must always hold, and this may decrease the actual degrees of freedom of the system. For example, in the case of a controllable building with a chiller plant, the cooling energy request to the chiller cannot be set freely, since it has to match the cooling energy needed for the building to track the temperature setpoint that becomes effectively the only decision variable. * Cost function, i.e., the quantity that has to be minimized, e.g., the electric energy costs or the deviation of the energy consumption from some nominal profile agreed with the main grid operator. * Constraints, i.e., the feasibility conditions that limit the solution space of the optimization problem. Notice that constraints can be classified in three categories: * Single component constraints, which are enforced at the level of each component separately and are related due to its dynamics and capabilities. For example, the energy accumulated in a storage is jointly dictated by the storage capacity and dynamics of the storage system. * Interconnection constraints, which relate variables of different components and originate from their cooperative interaction in the district. For example, the temperature setpoint in a controllable building cannot result in a cooling energy request that is larger than the energy that the chiller can produce and the energy that can be taken from the storage. * Control constraints, which are enforced to achieve some desired property of the energy management strategy. These are, for instance, the comfort constraints imposed on the temperature in a building or the constraints enforced at the end of the control time-horizon on the energy in the storage to avoid its depletion and allow for repetitive use of the control strategy ina periodic fashion.§ SOME NUMERICAL EXAMPLESIn this section, we present some examples where a district network configuration is considered and a related energy management problem is defined and solved. All examples refer to a centralized architecture, with known profiles for the disturbances.We consider a one-day time horizon since this is a commonly used time horizon for building energy management, especially temperature control.We enforce a periodic solution to cope with the myopic attitude of the finite horizon strategy, which would empty the battery/storage and drive the zone temperatures to the limit of their admissible range at the end of the time horizon in order to save money, without caring of the next day.Examples were chosen to be simple but realistic enough to highlight the capabilities of the proposed framework. Many more examples could be presented with reference to different set-ups in terms of either district network configuration or energy management problem formulation.Distributed energy management strategies could be adopted for easing computations and preserving privacy of information, as suggested in <cit.>.The stochastic nature of disturbances could be accounted for via a randomized approach as in <cit.>, which however refers to a single building configuration. Stochastic periodic control solutions, <cit.>, could be implemented as well. §.§ Cooling of a controlled building with a chiller plantInspired by the numerical example presented in <cit.>, we start by considering the simple district network configuration in Figure <ref>, which consists of a controlled building and a chiller unit.The controlled building is a medium-sized three story office building of the following dimensions: 20m long, 20m wide, and 10m tall. Each facade of the building is half glazed and the roof is flat. The biquadratic approximation (<ref>) is used for the chiller model. Disturbances are treated as deterministic signals. Figure <ref> shows the profiles adopted for the occupancy and internal energy contributions, solar radiation and outside temperature. In Section <ref>, we consider a single-zone setup for the building, where the three floors are treated as a unique thermal zone, with the same temperature setpoint. In Section <ref>, we move to a multi-zone setup, where each floor is a thermal zone with its temperature setpoint. In both cases we neglect energy exchanges through thermal radiation among internal walls and we consider the ground floor thermally isolated from the ground.§.§.§ Single zone buildingThe purpose of this example is twofold: * Showing the role of the building structure as a passive thermal storage, that can accumulate and release thermal energy;* Compare the energy management strategies obtained with two different control objectives.The problem is then formulated as follows: * Global parameters, the sampling time is set to = 10 minutes, and the time horizon is set to 1 day, i.e., M=144. * Optimization variables, the only optimization variable is the temperature set-point of the single zone T_z as defined via the control input u over the considered finite horizon. * Cost function, we here consider the two different cost functions: * cooling energy provided by the chiller:J_1 = ∑_k=0^M E_,c(k). * electricity consumption:J_2 = ∑_k=0^M E_,ℓ(k), where the electricity consumption is related to the cooling energy of the chiller static characteristic that maps one into the other according to a specific COP (see Section <ref>). * Constraints, the following constraints are included in the optimization problem: * Single component constraints, the constraints of the single components are given byE_c≥ 0 0≤E_,c≤ E_,c^max. * Interconnection constraints: the chiller satisfies the cooling load demand, i.e.,E_,c(k) = E_c(k),k=0,…,M.* Control constraints: zone temperature should be within some comfort range, and a periodic solution is enforced by setting the same value for the zone temperature setpoint at the beginning and end of the time horizon:u^min≤ u≤ u^max u(M) =u(0) =T_z(0)T(0) = T(M) We consider an ideal setting where both T_z(0) and T(0) can be set so as to obtain periodic solution. The resulting optimization problem is a convex constrained program that can be solved, for example, with CVX[<http://cvxr.com/>] with the SDPT3 solver.Figure <ref> shows the resulting optimal temperature profiles T_z for the two cases. Both solutions stay within the prescribed comfort temperature bounds. Notice that the discrepancy between the two curves is at most of about 1.6^∘C. Despite this distance being small, from Figure <ref> one can notice a clear difference in the required cooling energy for the two cases.In the case of minimization of the electricity consumption (J_2), a “precooling” phase occurs from time 18:00 to time 10:00 of the next day (if we think about the solution applied over multiple days), which leads to a larger cooling energy request.Intuitively, the second policy stores some cooling energy in the building structure, ahead of time, thus smoothing the cooling energy request in the central part of the day, when occupancy is larger, to get the chiller operating with high efficiency. The “building thermal mass” isexploited as a passive thermal storage to add further flexibility to the system <cit.>.On the other hand, the first policy exploits the fact that at night the temperature is lower, comfort constraints are satisfied (indeed they are set to be larger because there are no occupants in the office building), and the chiller does not need to provide any cooling energy to the load. Figure <ref> shows the electric energy consumption in the two cases, highlighting that the chiller is working at its minimum for most of the time in the cooling energy minimization policy (J_1). The integral of the curves in Figure <ref> is the electricity consumption and is larger for the cooling energy minimization policy. Indeed, Figure <ref> shows that the second policy makes the chiller operate close to the maximum COP value of the chiller, thus requiring much less electrical energy.In summary, depending on the cost function adopted in the energy management strategy design, one can have significantly different behaviors of the same district network configuration, with a different performance, even with a limited difference in the temperature setpoint profiles.§.§.§ Multi-zone buildingIn this example, the controlled building has three thermal zones, one per floor, with three zone temperature setpoints, and our goal is to investigate the impact of a time-varying electricity price over the day time.The problem is formulated as follows: * Global parameters, the sampling time is set to = 10 minutes, and the time horizon is set to 1 day, i.e., M=144. * Optimization variables, the optimization variables are the temperature setpoints of the three zones T_z = [T_z,1 T_z,2 T_z,3], as defined via the control input u over the considered finite horizon. * Cost function, we minimize the cost of the electrical energy needed for cooling the buildingJ = ∑_k=0^M p(k) E_ℓ(k),where p(k) is the time-varying unitary cost (see Figure <ref>). * Constraints, the following constraints are included in the optimization problem: * Single component constraints, the constraints of the single components are given byE_c≥ 0 0≤E_,c≤ E_,c^max.* Interconnection constraints, the chiller satisfies the cooling load demand, i.e.,E_,c(k) = E_c(k),k=0,…,M.* Control constraints, zone temperature should be within some comfort range, and a periodic solution is enforced by setting the same value for the zone temperature setpoint at the beginning and end of the time horizon:u^min≤ u≤ u^max u(M) =u(0) =T_z(0) Figure <ref> shows the optimal temperature setpoints for the three zones (top graph), the stack of the cooling energies associated with each zone and how they compose the cooling energy request E_,cto the chiller (middle graph), and the unitary price for the electrical energy. The obtained solution shows that all the zones are precooled in the first hours of the day. However, all the cooling energy provided by the chiller is conveyed to Zone 2. This can be explained by observing that Zone 2 is the central story, and it exchanges thermal energy with the other zones through its ceiling and floor, thus cooling them. In addition, considering jointly the cooling energy request profile with the price of energy, it is possible to see that whenever the price of energy is decreasing the energy request increases, and vice versa. Finally, notice that at the end of the day, Zone 2 is being cooled again, to bring the temperature back to its value at the beginning of the day, with a larger amount of cooling energy request when the price gets lower.The example highlights the advantages of defining multiple thermal zones in a building. Figure <ref> shows the total cost obtained for the single-zone and multi-zone cases, and the number of iterations needed to find the optimal solution in the two cases. The total cost decreases by 8% with the multi-zone configuration, but the computational complexity increases as witnessed by the higher number of iterations of the solver and the total CPU time. This analysis can be performed at design time to decide the number of zones to be included, based on the available computational resources and the economical advantage.§.§ Microgrid with uncontrolled building In this example, we consider a microgrid configuration as represented in Figure <ref>, which includes a set of uncontrolled buildings, a cooling and a heat storage, three chillers of different sizes (small, medium, and large, indexed by 1, 2 and 3, respectively, and whose COP curves are presented in Figure <ref>), a microturbine, and an electrical storage (a battery).Buildings are modeled via their requests of cooling energy E_c (for air temperature control) and heating energy E_h (for getting warm water). The chillers and microturbine can be switched on and off by setting the associated logical variablesδ_,i, i =1,2,3 and δ_ to 1 and 0. We adopt the PWA approximation (<ref>) for the three chillers, with 10 knots, obtained from their Ng-Gordon models with T_o = 22^∘C, T_cw = 10^∘C, and the following parameters:Chiller 1 (small): a_1=0.0056,a_2=10.11, a_3=7.00, a_4=0.9327, Chiller 2 (medium): a_1=0.0109,a_2=20.22, a_3=3.80, a_4=0.9327, Chiller 3 (large): a_1=0.0230,a_2=40.44, a_3=1.98, a_4=0.9327.The measure units of the coefficients are [a_1] = W/K, [a_2] = W, [a_3] = K/W, [a_4] = 1.The problem is then formulated as follows: * Global parameters, the sampling time is = 1 hour, and the time horizon is 1 day, hence the number of time lots is M=24.* Optimization variables, the optimization variables are the chillers and the microturbine on/off status δ_,i, i= 1,2,3, and δ_, the cooling energy requested to the chiller E_,c, the CHP fuel inlet u_, and the energy exchanged with the storage units s_i, i ∈{c,h,ℓ}. * Cost function, composed of different components: * The cost for energy trading:C_ℓ(k) = p(k) (E_ℓ(k) - E_,ℓ(k) -s_ℓ(k))where p(k) is the unitary cost for trading electrical energy (the same used for the previous experiment), and E_ℓ(k) is the electrical energy demand; * The fuel cost for the microturbine given by:C_f(k) = ψ_f(k)δ_(k) u_(k)where ψ_f is the unitary cost for the fuel, here considered constant and unitary: ψ(k) = 1. * The startup costs of the microturbine:C_(k) = C_^max{δ_(k) - δ_(k-1), 0 }where C_^ =1. * The startup costs of the chillers:C_(k) = ∑_i=1^3 C_,i^max{δ_,i(k) - δ_,i(k-1), 0 }where C_,1^ =0.05, C_,2^ =0.1, C_,3^ =0.2.Therefore, the overall cost function becomes:J = ∑_k=0^M C_ℓ(k) + C_f(k) + C_(k) + C_(k)* Definition of constraints, the following constraints are included in the optimization problem: * Single component constraints given by:δ_(k)(2 +ε)≤ u_(k) ≤ 10δ_(k)+ 2(1-δ_(k)) ,0 ≤ E_,h(k) ≤ 2160MJ,0 ≤ E_,ℓ(k) ≤ 1080 MJ,-500MJ≤ s_h(k) ≤ 500 MJ,0≤ S_h(k) ≤ 1500 MJ,-360MJ≤ s_c(k) ≤ 360 MJ,0≤ S_c(k) ≤ 1800 MJ,-250MJ≤ s_ℓ(k) ≤ 250 MJ, 0≤ S_ℓ(k) ≤ 1500 MJ,εδ_,i(k) ≤ E_,i(k) ≤δ_,i(k) · 180 MJ, ∀ i={1,2,3}.* Interconnection constraints, given by thermal energy balance equations:∑_i=1^3 E_,c,i(k) + s_c(k) = E_c(k),E_,h(k) + s_h(k) = E_h(k)* Control constraints, a periodic solution is enforced by setting the same value for the on/off status of the devices and for the stored energies:δ_(0) = δ_(M)δ_,i(0) = δ_,i(M),i = 1,2,3 S_h(0)= S_h(M) S_c(0)= S_c(M)S_ℓ(1)= S_ℓ(M) The resulting optimization problem is a Mixed Integer Linear Programming (MILP) problem, and can be solved using YALMIP to formulate the optimization problem and CPLEX as a solver. Figure <ref> shows the cooling energy demand E_c and the heating energy demand E_h of the uncontrolled buildings over the considered time horizon (top left graph), the storage of cooling, heat and electrical energy (top right graph), with the respective control inputs (middle right graph), the fuel inlet to the microturbine (middle left graph), the energy demand for the three chillers (bottom left graph), and finally, the bottom right graph shows the logical on/off status of the chillers and of the microturbine.Notice that the small and medium size chillers (associated with δ_,1 and δ_,2) are never switched off, while the large chiller is used to charge the cooling storage before the peak of cooling energy demand, and right after that, when the storage got empty. The microturbine is switched on only during the peaks of heat energy demand (see the profile of δ_). Figure <ref> shows the performance of the three chillers in terms of COP, as a function of time. It is possible to see that the small and medium chiller are always operating close to their maximum COP, while the large chiller either operates at its maximum COP or it is in the off mode, where its consumption is zero.The cooling energy storage is charged at the beginning of the day when the cooling load is low, and then discharged when the load increases. It is finally recharged to meet the periodicity imposed via the control constraints. On the other hand, the heating storage is used to meet the heating energy demand, and the microturbine is activated only when the heating storage is empty. Also in this case, part of the energy produced when the CHP was active, is stored to meet the energy demand of the next day. § MULTIRATE CONTROLIncreasing the number buildings and/or thermal zones per building necessarily leads to a larger computational effort for solving the energy management control problem since the number of optimization variables increases as well. This may become an issue when a receding horizon strategy is adopted and optimization is performed online at every control instant. Indeed, real-time constraints can hamper the applicability of the approach.A possible way to avoid this issue is to use larger values of the sampling timefor the discretization of the model, thus yielding a lower number of optimization variables for the same time horizon. As a side (positive) effect, discretizing with a larger sampling time make less stringent the real time constraints since it gives more time to perform the computations and apply the solution. Unfortunately, using a larger sampling time degrades the model accuracy, thus eventually deteriorating the control performance. This issue can be tackled by taking a multirate control approach, where model and controller operate with different sampling periods. Specifically, if we letbe the sampling period of the model and introduce the rate M_R∈ℕ, then, in multirate control, the control action is only set every M_R time slots of length , or, equivalently, = M_R is the sampling period for the controller. This choice allows for an accurate representation of the model dynamics, while still decreasing the number of optimization variables, and, as a consequence, the computational complexity, by a factor M_R.Clearly, the reduction of the number of optimization variables has some impact on the achievable performance in terms of cost and also reactiveness to possible disturbances with fast dynamics. The choice of the rate M_R must compromise between computational effort reduction and performance degradation, compatibly with the available resources.§.§ ExampleLet us focus on the example presented in Section <ref>, with cost function given by the electrical energyJ = ∑_k=0^M E_ℓ(k).We sample the model with = 10 minutes, and we study the effects of employing different rates M_R for applying the control input, namely M_R = 1,6,12,24,36,48, corresponding to = 1/6,1,2,4,6,8 hours, thus progressively reducing the number of optimization variables.Figure <ref> shows the optimal temperature profiles for the different rates. Notice that the curves associated with = 10 minutes (M_R = 1) and with = 1 hour (M_R = 6) are practically indistinguishable, but in the latter case we have reduced the number of optimization variables by a factor 6.The reduction of the optimization variables causes an increase of the costs, as shown in Figure <ref>. However, this increase is negligible up to = 2 hours, while the computation effort is almost constant for values oflarger than or equal to 1 hour, if evaluated in terms of total CPU time[The total CPU time presented in Figure <ref> is computed as the average total CPU time over 100 experiments for each considered , for the sole solver.].We can also analyze the chiller performance presented in Figure <ref>.The higher the rate M_R, the lower is the flexibility of the control input to do a fine adjustment of the temperature setpoint and compensate the disturbance variability. This is why the chillers are not constantly operating at a high efficiency levels when M_R is larger. This results in a less performing chiller, so one should look for a tradeoff among computational effort and cost efficiency.Finally, we can conclude that the adoption of multirate control solutions is problem specific, depending on the available computational power, and on the complexity of the optimization problem to be solved. § CONCLUSIONIn this paper we presented amodeling framework for the optimal operation of a district network, with reference in particular to the cooling of multiple buildings that are sharing resources like chillers or storages. Various components have been introduced and modeled in terms of energy fluxes so as to ease their composition via energy balance equations. A control-oriented perspective is adopted in that control and disturbance inputs are explicitly accounted for, in terms of their energy contribution. We also described how to formulate an optimal energy management problem as a constrained optimization program where control inputs are the optimization variablesand need to be set so as to minimize some energy-related function (e.g., electric energy cost, deviation from some nominal profile of electric energy consumption), while satisfying comfort and actuation constraints. Finally, a multirate approach was proposed to reduce the number of optimization variables while preserving the model accuracy. This has potential for real time applicability of the method when implemented according to the receding horizon strategy of model predictive control. This will allow to compensate for unpredictable human-building interactions as discussed in <cit.>. Some numerical examples were also presented to show the versatility of the proposed framework. Currently, we are addressing optimal energy management of a district network in presence of stochastic disturbances, the key challenge being how to account for them when embedded in a distributed setting with limited communications capabilities. The approach in <cit.> could be useful to this purpose.§ MODEL VALIDATION Reliability of the model is crucial when adopting model-based control design strategies. At the same time, if a model is accurate but very complex, then, design can become impractical.As for what concerns the network district modeling for energy management purposes, the more difficult component to model is the building, since various factors need to be accounted for, including size and structure of the building, walls composition, presence of electrical devices, occupancy, and environmental conditions, like outdoor temperature and solar radiation. Also, model complexity grows as the size of the system increases.Models and modeling frameworks for buildings have been proposed in the literature <cit.>. Most of them include a detailed characterization of the fluid dynamics phenomena, e.g., the evolution of the temperature and humidity of the thermal zones, and they typically require specialized Computational Fluid Dynamics (CFD) tools for simulation. Even though these approaches provide very accurate simulation results, they are difficult to use for control design purposes, due to their complexity. In this paper we adopted a control-oriented perspective and presented a simple model of the building where thermal zone temperatures act as control inputs and enters linearly the system dynamics.Validation of a model of the building dynamics against experimental data is quite challenging, also because setting up a measurement facility for a building can be complex and expensive. In order to validate the presented model, we hence resort to the methodology introduced by the American Society for Heating Refrigerating and Air-conditioning Engineers (ASHRAE), and, more specifically, the validation method defined in the ANSI-ASHRAE 140 standard. This standard specifies test cases and procedures for the evaluation of the technical capabilities and range of applicability of computer programs that compute the thermal performance of buildings and their HVAC systems. The current set of tests included in the standard consists of (i) comparative tests that focus on building thermal envelope and fabric loads, and mechanical equipment performance, and (ii) analytical verification tests that focus on mechanical equipment performance. Different building energy simulation programs, with different levels of modeling complexity, can be tested. For all tests included in the specifications, results provided by other certified simulation tools are presented, and they represent the baseline for validating new modeling and simulation software. A detailed description of the simulation tools included in the specification can be found in <cit.>.We here provide the results of some of the main tests defined in the ANSI-ASHRAE 140 standard, and compare them with the baseline provided in the standard. Let us first introduce the test case and then describe the validation procedure.We consider a building located at an altitude of 1649m above the sea level, and weather data series resuming the weather conditions for a whole year are available and provided by the standard. The data set contains: external dry bulb temperature, wind speed, wind direction, direct and diffuse solar radiation. The building has a 48m^2 floor area, a single story with rectangular-prism geometry, and two south-facing windows, 6m^2 each (see Figure <ref>). Two set-up are considered, which differs in materials composition and walls thickness: lightweight (case 600 in the standard) and heavyweight (case 900 in the standard). The standard specifies the composition in terms of thickness, density, thermal conductivity and specific heat capacity of all layers of each wall, for both the lightweight and heavyweight cases. These values are listed in Tables <ref> and <ref>, respectively. According to the specification, density and specific heat of the underfloor insulation have the minimum admissible value allowed in the building model that is tested, and in any case a value not smallerthan zero. Also, the contribution of the internal loads and people, within the thermal zone is constant over the year and equal to Q_int + Q_p = 200W.The standard also provides the values for the internal and external solar absorption and infrared emission coefficientsα_i^S = α_i^L = 0.6 and ε_i = 0.9, i=1,…,m, and for the interior and exterior combined radiative and convective heat transfer coefficients, from which the radiative and convective coefficients can be recovered. Finally, the standard contains also the windows properties, the values of incidence angle-dependent optical properties, and the interior solar distribution. The reader is referred to the ANSI-ASHRAE 140 standardfor a complete list of building properties.We focus on two procedures for validation described in the standard. The first one is denoted as Free Float (FF) in that the heating and cooling equipment is switched off and the zone temperature evolves freely subject to internal/external disturbances. The purpose of this test is to validate the physical model without the effect of any control action, and such validation is performed comparing some statistics (maximum, minimum, and annual average) of the zone temperature T_z over a year against other simulation tools. The second procedure prescribes to simulate the building together with the heating/cooling system by applying a simple control strategy: the controller has to maintain the air temperature inside the building between 20^∘C and 27^∘C. Specifically, the control strategy is: * Heat = ON if temperature <20^∘C; otherwise, Heat = OFF.* Cool = ON if temperature >27^∘C; otherwise, Cool = OFF.The ANSI-ASHRAE 140 standard specifies that the air conditioning system produces only pure heating load and sensible cooling load outputs. That is, all equipment is 100% efficient with no duct losses and no capacity limitations. In this controlled case, the validation is performed comparing the hourly-integrated peak of the cooling and heating power provided to the building. The thermostat was implemented as two saturated PI controllers with antiwindup, where the control variable is the amount of cooling and heating power to be injected in the system, equivalently to the implementation adopted in <cit.>. In the following we will denote as 600FF and 900FF the case when the free float validation procedure is applied to the lightweight and heavyweight buildings, and as 600 and 900 the case when the control is applied.§.§ Validation results In this section we present the numerical results obtained in the 600FF and 900FF and 600 and 900 test cases. For running the validation process, it is necessary to rewrite the model with the heat flow rate Q as control input, and the temperature of the zone T_z as the output of the system. To this aim, we consider a simulation model composed of a state vector including the temperature of the different slices of the walls as described in (<ref>), and the temperature of the zone T_z. The evolution of T_z is governed by the continuous-time version of (<ref>), made explicit with respect to Ṫ_z:Ṫ_z = -C_z^-1Q_z = -C_z^-1(Q - Q_w - Q_p - Q_int),with Q_w, Q_p, and Q_int being the heat flow rates towards the zone of the walls, the occupancy, and of other internal equipment producing heat. Considering the expressions (<ref>), (<ref>), and (<ref>), one can write the expression of Q_z as a function of the states T and T_z, of the input Q and of the disturbances. This continuous time model is implemented in Modelica[<https://modelica.org/>], in order to carry out the validation process.Table <ref> reports the obtained results in terms of maximum, minimum, and mean annual temperature.Our model (last column of the table) is comparedwith the other ones provided in the standard under the free float validation procedure. In the 600FF test case, the results obtained with the model considered herein are comparable with the ones obtained with the other building simulation models. As for 900FF, only the minimum temperature is comparable with the other results, while the maximum temperature is slightly higher than the values obtained with the other models, whereas the mean annual temperature is lower. Overall the obtained statistics produce reasonable results in the free float case, even though the model adopted for the presented framework is much simpler than the other simulation models. Table <ref> summarizes the validation results when the presented control strategy is in place. The hourly peak of cooling and heating power are comparable with those of the other tools in both the test cases.Overall, the obtained results in this validation phase show that state-of-the-art simulation tools provide similar results to those obtained with our model, which has the key advantage of being simpler and hence more suitable for design purposes.§ DISCRETIZATION OF THE WALLS TEMPERATURE DYNAMICSTo solve the discrete-time optimal energy management problem, we need to consider a discretized version of (<ref>). Given the linearity of (<ref>), it holds thatT((k+1))= e^AT(k) +∫_k^(k+1) e^A((k+1)-τ)(BT_z(τ) + Wd(τ)) dτ.If we assume that T_z (i.e., our control variables) and d are linearly varying within each time slot, then the integral in (<ref>) can be computed analytically. Formally, givenT_z(τ) = T_z,k+1-T_z,k/(τ - k) + T_z,k,where T_z,k = T_z(k), andd(τ) = d_k+1-d_k/(τ - k) + d_k,where d_k = d(k), ∀τ: k≤τ < (k+1) and k=1,…,M. If we set T_k = T(k) and Q_k = Q(k), k=1,…,M, then the dicretized system can be expressed as followsT_k+1 = Γ_xT_k + Γ_u,1T_z,k+1 + (Γ_u,0-Γ_u,1)T_z,k ++ Γ_ω,1d_k+1 + (Γ_ω,0-Γ_ω,1)d_kQ_k=CT_k + DT_z,kwhereΓ_x = e^AΓ_u,1 = 1(∫_0^ e^As(-s)ds) BΓ_u,0 = (∫_0^ e^As ds) BΓ_ω,1 = 1(∫_0^ e^As(-s)ds) WΓ_ω,0 = (∫_0^ e^As ds) W.Applying the transformation ξ_k = T_k-Γ_u,1T_z,k-Γ_ω,1d_k we obtainξ_k+1 = Γ_xξ_k + ((Γ_x-I)Γ_u,1 + Γ_u,0)T_z,k ++ ((Γ_x-I)Γ_ω,1 +Γ_ω,0)d_kQ_k=Cξ_k + (CΓ_u,1 + D)T_z,k + CΓ_ω,1d_kDropping the bold notation for vectors and matrices, (<ref>) can be rewritten as the following discrete-time system{ x(k+1) = Ãx(k) + B̃u(k) + W̃ω(k) y(k) = C̃x(k) + D̃u(k) + Ṽω(k).where x(k) = ξ_k, u(k) = T_z,k, ω(k) = d_k, y(k) = Q_k, and the matrices are:[ Ã = Γ_x, B̃ = (Γ_x-I)Γ_u,1+Γ_u,0,W̃ = (Γ_x-I)Γ_ω,1+Γ_ω,0;C̃ = C, D̃ = CΓ_u,1 + D,Ṽ = CΓ_ω,1. ] From (<ref>) one can derive the expression of x(k) and y(k) as a function of the initial state and the inputs up tok:x(k) = Ã^k x(0) + ∑_h=0^k-1Ã^k-1-h( B̃u(h)+W̃ω(h) ) y(k) = C̃Ã^k x(0) + ∑_h=0^k-1C̃Ã^k-1-h( B̃u(h)+W̃ω(h) ) + D̃u(k) + Ṽω(k). By recalling that x(0)=ξ_0 = T_0-Γ_u,1T_z,0-Γ_ω,1d_0, we obtain the following expression of y(k) as a function of the original state variables and the inputs up to time k:y(k) = C̃Ã^kÃT_0+C̃Ã^k-1( (Γ_u,0-Γ_u,1)u(0) + (Γ_ω,0-Γ_ω,1)ω(0) ) + ∑_h=1^k-1C̃Ã^k-1-h( B̃u(h)+W̃ω(h) ) + D̃u(k) + Ṽω(k).elsarticle-num | http://arxiv.org/abs/1707.08494v1 | {
"authors": [
"Daniele Ioli",
"Alessandro Falsone",
"Alessandro Vittorio Papadopoulos",
"Maria Prandini"
],
"categories": [
"cs.SY"
],
"primary_category": "cs.SY",
"published": "20170726152525",
"title": "A compositional modeling framework for the optimal energy management of a district network"
} |
http://arxiv.org/abs/1707.08624v2 | {
"authors": [
"Falk Hassler"
],
"categories": [
"hep-th"
],
"primary_category": "hep-th",
"published": "20170726195552",
"title": "Poisson-Lie T-Duality in Double Field Theory"
} |
|
[ [ December 30, 2023 =====================empty emptyCloud computing is becoming an essential component in the emerging Internet of Things (IoT) paradigm. The available resources at the cloud such as computing nodes, storage, databases, etc. are often packaged in the form of virtual machines (VMs) to be used by remotely located IoT client applications for computational tasks. However, the cloud has a limited number of VMs available and hence, for massive IoT systems, the available resources must be efficiently utilized to increase productivity and subsequently maximize revenue of the cloud service provider (CSP). IoT client applications generate requests with computational tasks at random times with random complexity to be processed by the cloud. The CSP has to decide whether to allocate a VM to a task at hand or to wait for a higher complexity task in the future. We propose a threshold-based mechanism to optimally decide the allocation and pricing of VMs to sequentially arriving requests in order to maximize the revenue of the CSP over a finite time horizon. Moreover, we develop an adaptive and resilient framework that can counter the effect of realtime changes in the number of available VMs at the cloud server, the frequency and nature of arriving tasks on the revenue of the CSP. § INTRODUCTION In recent years, due to the ubiquity of the internet, there has been an increasing trend towards offloading computing, control, and storage to the cloud instead of doing it locally at the client side <cit.>. This trend is expected to accentuate with the proliferation of the Internet of things (IoT) <cit.>. The IoT applications can request for cloud resources for a variety of different computational tasks. For instance, they can invoke machine learning and data analytics models already implemented in the cloud server to enable powerful features such as predictive analytics, video processing, and natural language processing. With a massive surge in the number of applications requesting the cloud for computational resources in the future, there will be a need for an efficient allocation and pricing mechanism at the cloud server. A cloud service provider (CSP) has several resources such as computing nodes, storage, databases, etc., that can be used remotely by IoT applications. Often, these resources are packaged into virtual machine (VM) instances that act as processing units. When the number of requesting applications is large and the available VMs are limited, as envisioned in massive IoT systems <cit.>, it is important to select which applications are serviced particularly when the allocation is planned for longer periods of time.The challenges faced by the CSP in allocating the available VMs to requesting applications are twofold. Firstly, the available VMs at the CSP are limited, so it is important to allocate the most computationally intensive tasks to the available VMs in order to maximize the productivity of the IoT client applications and the generated revenue by charging them appropriately. However, the tasks arrive sequentially at the server and the CSP has to decide immediately to allocate a VM to it or to wait for a more valuable task in the immediate future. The challenge lies in the uncertainty about the nature of upcoming requests in the future. A computationally intensive task may not ever request for service while the low complexity tasks are refused service. It leads to an under-utilization of resources resulting in lower productivity and revenue of the CSP. On the other hand, if the VMs are allocated to low complexity tasks, then a high complexity task may request service in the future and has to be refused due to the unavailability of an VM at the cloud server.Therefore, there is a need for a dynamically efficient mechanism for allocating and pricing the VMs that takes these tradeoffs into account. Fig. <ref> illustrates the sequential arrival of IoT computation requests at the CSP.There has been considerable work in the literature towards resource allocation in cloud computing environments <cit.>. The focus is mainly on efficient resource management and load balancing for higher availability and performance <cit.> or resource allocation and pricing for revenue maximization <cit.>. Regarding dynamic pricing and revenue maximization, several works exist such as <cit.> which use price control to adjust demand levels. Others have used auction mechanisms to collect bids and allocate available computational resources such as <cit.>. However, most existing works on resource allocation in cloud computing do not take into account the sequential arrival of computing tasks and the uncertainty about the future. This is essential in the setting of cloud computing because the computational requests are spontaneous and the decision for allocation has to be made immediately upon arrival. A dynamically efficient policy for allocating resources to sequentially arriving agents in order to maximize social welfare was first proposed by Albright <cit.>. Consequently, a revenue maximizing approach towards sequential allocation of resources has been introduced in <cit.>. However, their work deals with heterogeneous resources and cannot be used to model situations with identical resources. In cloud-enabled IoT systems, often multiple identical resources such as VMs are available to be allocated to client applications. A framework for pricing the cloud for maximizing revenue is proposed in <cit.>. However, their solution is based on stochastic dynamic programming which cannot adapt in realtime scenarios. Our solution provides a dynamically optimal plug and play policy that can be pre-computed and used in realtime using a lookup table. In this paper, we develop an adaptive and resilient dynamic resource allocation and pricing framework for cloud-enabled IoT systems. We present an optimal dynamic policy to filter incoming service requests by IoT applications based on the complexity of the tasks.The qualification threshold for tasks is adaptive to the number of available VMs, the arrival rate of requests, and their average complexity. The optimal policy can be dynamically updated in order to maintain high expected revenues of the CSP.Furthermore, the proposed framework is also able to adapt according to the changing availability of the VMs due to reprovisioning of resources for other applications or due to the effect of malicious attacks.§ SYSTEM MODELWe consider a CSP having a set of N_t ∈ℤ^+ available VMs at time t∈[0,T]. The VMs are identical and are characterized by their computational efficiency[The computational efficiency can be determined by evaluating the relative time taken by the VM to successfully execute a benchmark task.] denoted by q ∈ [0,1]. The CSP receives requests[Throughout the paper, we use the word `requests' to refer to computational tasks generated by IoT applications that arrive at the CSP for processing.] for computation by IoT client applications. These requests arrive sequentially at the cloud server according to a Poisson process with density λ∈ℝ^+ requests per unit time. Each task has computational complexity denoted by X ∈ℝ^+. The computational complexity can be measured in terms of the number of CPU cycles or equivalently the time required to complete a given computational task. The computational complexities of sequentially arriving tasks are considered to be independent and identically distributed (i.i.d.) random variables with probability density function (pdf) and cumulative distribution function (cdf) denoted by f_X(x)and F_X(x) respectivley.The utility of the i^th arriving client application with a task complexity of x_i, i ∈ℤ^+, that is allocated a VM with efficiency q, is measured by the product qx_i, which refers to the resulting value created by the allocation or the productivity.Since the available VMs are limited, the CSP needs to allocate the VMs to only the high complexity arriving tasks in order to increase the total productivity of the clients as well as efficient utilization of available computational resources. Creating higher value sets the ground for the CSP to charge higher prices and hence generate more revenue. However, the decision has to be taken immediately[We assume that the tasks are impatient and need to be processed immediately without delay.] upon arrival of the tasks without knowledge of tasks arriving in the future. Therefore, the CSP has the option to either allocate one of the available VMs or to refuse the requesting application. §.§ Allocation & Pricing RuleIn order to allocate available VMs to randomly arriving computational requests, we adapt the result from the sequential stochastic assignment literature, which is based on the Hardy-Littlewood-Polya inequality <cit.> and is stated by the following theorem:If there are n VMs with computational efficiencies q_1, q_2, …, q_n such that 0 < q_1 ≤ q_2 ≤…≤ q_n, then there exists a set of functions0 = z_n+1(t) ≤ z_n(t) ≤…≤ z_1(t) ≤ z_0(t) = ∞.such that it is optimal (in terms of social welfare) to assign a VM with efficiency q_i to an incoming task with complexity x if z_n-i+1(t) ≤ x ≤ z_n-i(t). Furthermore, if x < z_n(t), it is optimal not to allocate it.In the case of identical objects, the CSP needs to set only a single dynamic threshold, which we refer to as the qualification threshold, that allows it to decide whether to allocate a VM or not based on the nature of the arriving task. Let y_N_t(t) ∈ℝ^+, ∀ t ∈ [0,T] denote the threshold if N_t VMs are available for allocation at time t. In other words, only the requests with x_i ≥ y_N_t(t), i ∈ℤ^+, will be allocated to an available VM at time t.The allocation process has to be completed within a finite time horizon denoted by T. Since the VMs have no commercial value if they remain idle or unallocated during the allocation period, therefore the threshold needs to be dynamic in order to efficiently generate revenue from available resources. The decision problem lies in that fact that it may be more valuable to assign a VM to a low complexity task than waiting for a high complexity task to arrive in the future which may not ever realize.The next step is to develop a pricing scheme for the available VMs. Since all the VMs are identical in terms of their performance, therefore they must be priced equally. The threshold based allocation policy provides a natural method for pricing the available VMs. Since each arriving task that is successfully allocated a VM at time t receives a value of at least qy_N_t(t). Therefore, it is fair to charge the price 𝒫: [0,1] ×ℝ^+ × [0,T]→ℝ^+ to a qualified task for being processed by a VM as follows:𝒫(q,y_N_t(t),t) = q y_N_t(t) + S(t),where S(t) represents the constant additional pricing independent of the allocation. This pricing policy is implementable as any individually rational client will be willing to pay at least an amount equal to its received value. Note that S(t) can be used to adjust the prices due to external factors such as promotions, packages, pricing agreements, etc. In the following section, we provide the dynamically optimal allocation threshold and the resulting price charged by the CSP to allocated tasks. § METHODOLOGY We will begin by quantifying the total expected revenue of the CSP and subsequently derive the optimal dynamic threshold that maximizes the revenue. The total expected revenue of the CSP ℛ: [0,1] ×ℝ^+ × [0,T]→ℝ^+ if N_t identical VMs with computational efficiency q are available at time t and a qualification threshold y_N_t(t) is used from time t onwards can be expressed as follows:ℛ(q, y_N_t(t),t) = ∑_n=1^N_t q ∫_t^T y_N_t(s) h_n(s) ds + κ_T,where h_n(t) is the density of waiting time until the n^th arrival of a qualifying task, i.e., having a task complexity greater than y_N_t(t), and κ_T is a constant factor due to the additional pricing function S(t). The density of waiting time can be expressed by the density of the n^th arrival in a non-homogeneous Poisson process with intensity λ̂(s) = λ (1 - F(y_N_t(s))). Consequently, the density can be written as follows <cit.>:h_n(s)= λ̂(s) exp(- ∫_t^s λ̂(u) du ) (∫_t^s λ̂(u) du )^n-1/(n-1)! , t ≤ s ≤ T. The objective is to select a time-varying threshold y_N_t(t) which maximizes the expected revenue functional given by (<ref>). The problem can be formally stated as follows:y_N_t(t)maximizeℛ(q, y_N_t(t),t) subject to y_N_t(t) ≥ 0, ∀ t∈ [0,T].Note that the optimization is over the space of functions where an optimal function y_N_t(t) is sought for a given number of available VMs at time t.Our aim is to design the threshold function that strikes the optimal balance between the number of qualifying tasks and the generated revenue. In the sequel, we provide the optimal qualification threshold for maximum revenue generation by the CSP and the properties of the optimal policy. If N_t VMs are available to the CSP at time t, computational requests arrive sequentially according to a Poisson process with intensity λ and the computational complexity of tasks are i.i.d. random variables with pdf f_X(x) and cdf F_X(x), then it is optimal to allocate an available VM to an incoming computational request if the complexity of an upcoming task x ≥ y_N_t(t). The optimal y_N_t(t) satisfies the following integral equation:y_N_t(t)= 1 - F_X(y_N_t(t))/f_X(y_N_t(t)) +λ∫_t^T (1 - F_X(y_N_t(s)))^2 /f_X(y_N_t(s)) J_N_t(t,s)ds, where J_N_t(t,s) can be expressed as follows:J_N_t(t,s) = 1/∑_n=1^N_t1/(n-1)!(∫_t^s λ̂(u) du)^n-1×∑_n=1^N_t1/(n-1)!( (∫_t^s λ̂(u) du)^n-1 - (n-1)(∫_t^s λ̂(u) du)^n-2).See Appendix <ref>.The behaviour of the optimal dynamic threshold for large number of available VMs is provided by the following corollary.If the number of available VMs is large, then the revenue maximizing threshold becomes constant and the allocation mechanism reduces to a first price auction mechanism, i.e., allocate a VM to a task if the complexity is higher than the virtual valuation. In the optimal allocation policy, if we let N_t →∞, then the optimal threshold solves the following integral equation:y_∞(t) = 1 - F_X(y_∞(t))/f_X(y_∞(t)) + ∫_t^T (1 - F_X(y_∞(t)))^2/f_X(y_∞(t))×( N_t →∞lim J_N_t(t,s) ) ds. Now, N_t →∞lim J_N_t(t,s) can be evaluated as follows:N_t →∞lim J_N_t(t,s)= ∑_n=1^∞H^n-1(s)/(n-1)! -∑_n=1^∞(n-1)H^n-2(s)/(n-1)!/∑_n=1^∞H^n-1(s)/(n-1)!= e^H(s) - e^H(s)/e^H(s) = 0. Therefore, it follows that y_∞(t) = 1 - F_X(y_∞(t))/f_X(y_∞(t)). Note that x - 1 - F_X(x)/f_X(x) is referred to as the virtual valuation of the agent of type x in mechanism design literature <cit.>. Hence, it can be concluded that if the number of available VMs is large, then only the virtual valuation of the arriving tasks can be recovered and the CSP is willing to offer the VMs for lowest possible threshold.The behaviour of the dynamically optimal qualification threshold with a variation in the number of available VMs at time t can be summarized by the following theorem.The qualification threshold of the tasks and consequently price of VMs decreases as the number of available VMs at the cloud server increases and vice versa, i.e., y_M_t(t) ≤ y_N_t(t) if M_t ≥ N_t, ∀ t. See Appendix <ref>. In the following section, we discuss how the dynamically optimal policy leads to an adaptive and resilient behaviour in the revenue of the CSP and describe the developed mechanism algorithmically. § ADAPTIVE AND RESILIENT ALLOCATION AND PRICING POLICYThe number of available VMs at the cloud server may change over time as some of them might become unavailable due to failure or malicious attacks <cit.>. The CSP might also destroy the created VMs in order to free up computational resources for other applications. On the other hand, previously allocated VMs might be released by applications or new VMs may be provisioned by the CSP in real time to accommodate higher demand. However, a change in the available number of VMs may affect the expected revenue of the CSP under a particular allocation and pricing policy particularly if there is a significant decrease in the number of remaining VMs. In order to reduce any negative impact on the expected revenue of the CSP, the proposed revenue maximizing framework will react to the changes in the available number of VMs by adapting the qualification threshold or equivalently the price.Furthermore, the developed framework can react to changes in the frequency and the nature of computational requests. The optimal resilient policy is denoted by Π(λ̃_t, Ñ_t, t), where λ̃_t is the rate of arrival of requests and Ñ_t represents the number of available VMs at time t. Note that Ñ_t = N_t + η_t, where η_t ∈ℤ is the change in the number of available VMs at time t. The adaptive qualification threshold ỹ_N_t(t) can be pre-computed using the optimal policy framework presented earlier in this Section. The policy then becomes a lookup table that the CSP uses to allocate and price the upcoming tasks. Note that the variations in the inputs of the framework can be directly incorporated into the derived results. For instance, if η_t VMs enter/leave the system at time t, then it is equivalent to as if there were N_t + η_t available VMs at time t. Hence, the optimal threshold corresponding to N_t+η_t must be used for time t onwards for maximizing revenue. The algorithm proceeds as follows. While the allocation period has not expired and there is still an available VM at the CSP, if an IoT application requests for computation, then the first step is to evaluate the task complexity. Once the complexity is determined, it is compared against the decision threshold. However, the optimal threshold used will depend on the current situation at the CSP. Hence, the updated number of available VMs Ñ_t and the updated arrival rate of requests λ̃ is used to read off the optimal policy from the lookup table Π(λ̃_t, Ñ_t, t) at time t. A flow diagram is provided in Fig. <ref> to illustrate the sequence of the mechanism. § NUMERICAL RESULTS & DISCUSSIONSIn this section, we provide numerical results for the proposed adaptive and resilient optimal dynamic allocation and pricing framework. We assume a single CSP having N_t available VMs to allocate to arriving computational requests within an allocation time horizon of T = 12 hours. The number of available VMs available at time t=0, referred to as N_0 is set to be 100. The computational efficiency of the VMs is selected to be q = 1 without loss of generality. Note that the characteristics of the VMs is only relevant to the pricing policy but not the allocation.The tasks arrive at the CSP according to a homogeneous Poisson process with intensity λ = 100 requests per hour unless otherwise stated. We also assume that the complexity of sequentially arriving computational requests are distributed according to an exponential distribution with a mean of 1/α, i.e., f_X(x) = α e^-α x, and F_X(x) = 1 - e^-α x. For simplicity, we select α = 1, resulting in an average task complexity of 1. The optimal task qualification thresholds in this case if N_t VMs are available at time t can be obtained by the solution of the integral equation expressed by (<ref>). The equation can be solved numerically using the Picard fixed point iteration <cit.>. Fig. <ref> shows the dynamic thresholds for qualification of an arriving task for low (λ = 10) and high (λ = 100) arrival rates of the requests. It can be observed in general that the qualification thresholds decrease as the time approaches towards the terminal time. This is due to the fact that the valuable option of allocating an available VM to a higher complexity task reduces in probability. Furthermore, as we approach the horizon, it is more valuable to allocate a VM to a lower complexity task than to not allocate it at all. It can also be observed from Fig. <ref> that for lower arrival rates, the thresholds drop quickly as compared to the thresholds for the high arrival rates in Fig. <ref>. This is because the expected arrivals are lower in the former and hence the mechanism adjusts the thresholds to qualify more arrivals to tap the revenue potential. The associated pricing curves follow a similar trend as the allocation thresholds except that they are scaled by the characteristics of the VMs. However, in the considered situation, they are identical since q=1.Next we investigate the adaptive and resilient behaviour of the proposed mechanism.A set of failures and capacity enhancements are simulated at fixed times. For instance, a loss of 15 VMs and 5 VMs is assumed to occur at t = 2 hours and t = 6 hours respectively. Similarly, new additions of 10 VMs and 5 VMs is assumed to occur at t = 4 hours and t = 8 hours. Note that when a loss of 15 VMs occurred at t=2 hours, the situation becomes equivalenlt to as if initially the CSP had 85 available VMs. Therefore from t = 2 onwards, the optimal revenue maximizing policy is to use the threshold and pricing corresponding to the N_0 = 85 curve. As the number of available VMs change over times, the optimal policy needs to be updated. The optimal dynamic allocation policy for the above mentioned events is shown by the bold line in Fig. <ref>. Notice that as the number of available VMs decreases, the allocation threshold and the price jumps in order to make up for the lost revenue. Similarly, if new VMs become available, the threshold and prices drop in order to strike a new balance between the qualifying tasks and the payment.In Fig. <ref>, we show the behaviour of the adaptive and resilient mechanism on the expected revenue of the CSP in response to the variations in the number of available VMs at the cloud server. It can be observed that the adaptive strategy is able to maintain a high expected revenue despite variations in the number of available VMs. Note that during the times when there is a drop in the number of available VMs, the expected revenue does not fall as much due to a rectified allocation and pricing policy as illustrated in Fig. <ref>. Hence, it is shown that a timely rectification of allocation and pricing decision enables the mechanism to be adaptive and resilient against any significant changes in the available resources to the CSP. § CONCLUSIONIn this paper, we have proposed an adaptive and resilient dynamic revenue maximizing framework for cloud computing environments. The framework uses a threshold-based filtering policy making real time allocation and pricing decisions for sequentially arriving computational requests.It has been shown that the framework is adaptive and resilient to changes in the number of available VMs or the statistical properties of the arrivals. The set of optimal policies can be pre-computed and used as a lookup table as conditions at the cloud server change over time. Therefore, the developed framework provides an optimal and implementable mechanism for allocation and pricing in cloud computing environments. Future directions in this line of work may include developing optimal allocation policies for multiple types of identical resources available at the CSP. Furthermore, the allocation framework can be extended to multiple layers such as in fog/edge computing paradigms. § PROOF OF THEOREM <REF> Let H(s)= ∫_t^s λ̂(u) du = ∫_t^s λ( 1 - F_X(y_N_t(u)) ) du. Then the expected revenue at time t if N_t VMs are available can be written as follows:ℛ({q}_N_t,t) =q ∫_t^T ∑_n=1^N_t1/(n-1)! F_X^-1(1 - H^'(s)/λ) × H^'(s) e^-H(s) (H(s))^n-1 ds + κ_T. This functional can be optimized for the time varying threshold y_N_t(t) using the calculus of variations <cit.>. We denote the kernel of integration as ℒ(s,H(s), H^'(s))= ∑_n=1^N_tH^' (s) e^-H(s) H^n-1(s)/(n-1)! F_X^-1( 1 - H^'(s)/λ). The Euler-Lagrange equation <cit.> represents the necessary condition satisfied by H(s) to be a stationary function of the expected revenue R({q}_N_t,t) and can be written as follows: ∂ℒ(s,H(s), H^'(s))/∂ H(s) - d/dt∂ℒ(s,H(s), H^'(s))/∂ H^'(s) = 0. The partial derivatives and the condition satisfied by the resulting Euler-Lagrange equation are given by (<ref>), (<ref>), and (<ref>) respectively. The expression in (<ref>) can be further reduced as follows: 2 H^''(s)-(H^'(s))^2 ∑_n=1^N_t1/(n-1)!( H^n-1(s) - (n-1)H^n-2(s))/∑_n=1^N_t1/(n-1)! H^n-1(s) + H^'(s) H^''(s) f_X^'(F_X^-1(1 - H^'(s)/λ))/λ f^2_X ( F_X^-1(1 - H^'(s)/λ) )= 0. Let J_N_t(t,s) = ∑_n=1^N_t1/(n-1)!( H^n-1(s) - (n-1)H^n-2(s))/∑_n=1^N_t1/(n-1)!H^n-1(s). Then, plugging back H(s) = ∫_t^s λ( 1 - F_X(y_N_t(u)) ) du results in the following: -2y_N_t^'(s) - λ (1 - F_X(y_N_t(s)))^2 J_N_t(t,s)/f(y_N_t(s)) - (1 - F_X(y_N_t(s))) y_N_t^'(s) f^'_X(y_N_t(s))/f^2_X(y_N_t(s)) = 0 It can be further expressed as follows: - y_N_t^'(s) - y_N_t^'(s) ( 1 + (1 - F_X(y_N_t(s))) f^'_X(s)/(f_X(y_N_t(s)))^2) = λ (1 - F_X(y_N_t(s)))^2 J_N_t(t,s)/f_X(y_N_t(s)), Sinced/ds( 1 - F_X(y_N_t(s)/f(y_N_t(s))) = -y^'_1(s) ( 1 + (1 - F_X(y_N_t(s))) f_X^'(y_N_t(s))/(f_X(y_N_t(s)))^2), so the condition in (<ref>) can be written as follows: d/ds( 1 - F_X(y_N_t(s))/f_X(y_N_t(s)))=y^'_1(s) + λ(1 - F_X(y_N_t(s)))^2 /f_X(y_N_t(s))J_N_t(t,s) . Integrating both sides with respect to s from t to T results in the following: ( 1 - F_X(y_N_t(T))/f_X(y_N_t(T))) - ( 1 - F_X(y_N_t(t))/f_X(y_N_t(t))) = y_N_t(T) -y_N_t(t) + λ∫_t^T(1 - F_X(y_N_t(s)))^2 /f_X(y_N_t(s))J_N_t(t,s) ds. Using the boundary condition, y_N_t(T) = 1 - F_X(y_N_t(T))/f(y_N_t)(T), i.e., at the terminal time only the virtual valuation of the users can be recovered, it follows that the cutoff curve y_N_t(t) satisfies the equation given by Theorem <ref>.§ PROOF OF THEOREM <REF> First we need to show that for {M_t,N_t ∈ℤ^+ : M_t ≥ N_t}, J_M_t(t,s) ≤ J_M_t(t,s), ∀ t,s. To do this we will show that J_N_t + 1(t,s) ≤ J_N_t(t,s). It is equivalent to showing that J_N_t + 1(t,s) - J_N_t(t,s)≤ 0, i.e., ∑_n=1^N_t + 1H^n-1(s)/(n-1)! -∑_n=1^N_t+1(n-1)H^n-2(s)/(n-1)!/∑_n=1^N_t+1H^n-1(s)/(n-1)! - ∑_n=1^N_tH^n-1(s)/(n-1)! -∑_n=1^N_t(n-1)H^n-2(s)/(n-1)!/∑_n=1^N_tH^n-1(s)/(n-1)!≤ 0. It can be further expressed as follows: ( ∑_n=1^N_tH^n-1(s)/(n-1)!) (H^N_t(s) - N_t H^N_t-1(s)/N_t!) -( ∑_n=1^N_tH^n-1(s) - (n-1)H^n-2(s)/(n-1)!)H^N_t(s)/N_t!≤ 0. Expanding the condition results in the following: ∑_n=1^N_t( H^N_t + n -1(s) - N_t H^N_t + n -2(s) - H^N_t + n - 1(s) /(n-1)! + . . (n-1)H^N_t + n -2(s)/(n-1)!)≤ 0. It is equivalent to the following condition:∑_n=1^N_tH^N_t + n -2(s)(n+1-N_t)/(n-1)≤ 0, which is true since (n+1-N_t) ≤ 0, ∀ n = 1, …, N_t. Therefore, it is evident that J_N_t + 1(t,s) ≤ J_N_t(t,s). Using induction it can be shown that the inequality J_M_t(t,s) ≤ J_N_t(t,s) holds for general M_t and N_t such that M_t ≥ N_t, ∀ t. From Theorem <ref>, the result follows directly with the assumption of increasing virtual valuations, i.e., x - 1 - F_X(x)/f_X(x) is increasing in x. IEEEtran | http://arxiv.org/abs/1707.08691v3 | {
"authors": [
"Muhammad Junaid Farooq",
"Quanyan Zhu"
],
"categories": [
"cs.DC"
],
"primary_category": "cs.DC",
"published": "20170727030312",
"title": "Adaptive and Resilient Revenue Maximizing Dynamic Resource Allocation and Pricing for Cloud-Enabled IoT Systems"
} |
Sequential Inverse Approximation of a Regularized Sample Covariance Matrix Tomer LancewickiEBay Inc.625 6th AveNew York, NYEmail: [email protected] December 30, 2023 ======================================================================================= One of the goals in scaling sequential machine learning methods pertains to dealing with high-dimensional data spaces. A key related challenge is that many methods heavily depend on obtaining the inverse covariance matrix of the data. It is well known that covariance matrix estimation is problematic when the number of observations is relatively small compared to the number of variables. A common way to tackle this problem is through the use of a shrinkage estimator that offers a compromise between the sample covariance matrix and a well-conditioned matrix, with the aim of minimizing the mean-squared error. We derived sequential update rules to approximate the inverse shrinkage estimator of the covariance matrix. The approach paves the way for improved large-scale machine learning methods that involve sequential updates. § INTRODUCTION The covariance matrix of multivariate data is required in many sequential machine learning and neural-networks (NN) based applications <cit.>, including speech recognition <cit.>, deep learning architectures for image processing and computer vision <cit.>, stochastic fuzzy NN's <cit.>, pricing option contracts in financial markets <cit.>, adaptive tracking control problems <cit.>, detection tasks <cit.>, reinforcement learning <cit.>, and many others.In settings where data arrives sequentially, the covariance matrix is required to be updated in an online manner <cit.>. Techniques such as cross-validation, which attempt to impose regularization, or model selection are typically not feasible in such settings <cit.>. Instead, to minimize complexity, it is often assumed that the covariance matrix is known in advance <cit.> or that it is restricted to a specific simplified structure, such as a diagonal matrix <cit.>. Moreover, when the number of observations n is comparable to the number of variables p the covariance estimation problem becomes far more challenging. In such scenarios, the sample covariance matrix is not well-conditioned nor is it necessarily invertible (despite the fact that those two properties are required for most applications). When n≤ p, the inversion cannot be computed at all <cit.>.An extensive body of literature concerning improved estimators in such situations exists <cit.>. However, in the absence of a specific knowledge about the structure of the true covariance matrix, the most successful approach so far has, arguably, been shrinkage estimation <cit.>. It has been demonstrated in <cit.> that the largest sample eigenvalues are systematically biased upward, and the smallest ones downward. This bias is corrected by pulling down the largest eigenvalues and pushing up the smallest ones, toward their grand mean.The optimal solution of the shrinkage estimator is solved analytically, which is a huge advantage for deep learning architectures, since a key factor in realizing such architectures is the resource complexity involved in their training <cit.>. An example of such deep architecture is the deep spatiotemporal inference network (DeSTIN) <cit.>. The latter extensively utilizes the quadratic discriminant analysis (QDA) classifier under the simplified assumption that the covariance matrices involved in the process are diagonal. Such assumption is made in order to avoid additional complexity during the training and inference processes. It is well known that for a small ratio of training observations n to observation dimensionality p, the QDA classifier performs poorly, due to highly variable class conditional sample covariance matrices. In order to improve the classifiers' performance, regularization is required, with the aim of providing an appropriate compromise between the bias and variance of the solution. It have been demonstrated in <cit.> that the QDA classifier can be improved tremendously using shrinkage estimators. The sequential approximated inverse of the shrinkage estimator, derived in this paper, allows us to utilize the shrinkage estimator in the DeSTIN architecture with relatively negligible additional complexity to the architecture. In addition, the relatively simple update rules pave the way to implement the inverse shrinkage estimator on analog computational circuits, offering the potential for large improvement in power efficiency <cit.>. The rest of this paper is organized as follows: Section 2 presents the general idea of the shrinkage estimator. In Section 3, we derived a sequential update for the shrinkage estimator, while in Section 4, the related approximated inverses are derived. In Section 5, we conduct an experimental study and examine the sequential update rules.Notations: we denote vectors in lowercase boldface letters and matrices in uppercase boldface. The transpose operator is denoted as (·)^T. The trace, the determinant and the Frobenius norm of a matrix are denoted as Tr(·), |·| and ‖·‖ _F, respectively. The identity matrix is denoted as 𝐈, while 𝐞=[1,1,…,1]^T is a column vector of all ones. For any real matrices 𝐑_1 and 𝐑_2, the inner product is defined as ⟨𝐑_1,𝐑_2⟩ =Tr(𝐑_1^T𝐑_2), where ⟨𝐑_1,𝐑_1⟩ =‖𝐑_1‖ _F^2 <cit.>.§ SHRINKAGE ESTIMATOR FOR COVARIANCE MATRICES We briefly review a single-target shrinkage estimator by following <cit.>, which is generally applied to high-dimensional estimation problems. Let {𝐱_i} _i=1^n be a sample of independent identically distributed (i.i.d.) p-dimensional vectors drawn from a density with a mean μ and covariance matrix Σ. When the number of observations n is large (i.e., n≫ p), the most common estimator of Σ is the sample covariance matrix 𝐒_n=1/n-1∑_i=1^n(𝐱_i-𝐦_n)(𝐱_i-𝐦_n)^T,where 𝐦_n is the sample mean, defined as 𝐦_n=1/n∑_i=1^n𝐱_i.Both 𝐒_n and 𝐦_n are unbiased estimators of Σ and μ, respectively, i.e., E{𝐒_n} =Σ and E{𝐦_n} =μ. The shrinkage estimator Σ̂(λ_n) is in the form Σ̂(λ_n)=(1-λ_n)𝐒_n+λ_n𝐓_nwhere the target 𝐓_n is a restricted estimator of Σ defined as 𝐓_n=Tr(𝐒_n)/p𝐈.The work in <cit.> proposed to find an estimator Σ̂(λ_n) which minimizes the mean squared error (MSE) with respect to λ_n, i.e., λ_On=min_λ_nE{‖Σ̂(λ_n)-Σ‖ _F^2}and can be given by the distribution-free formula λ_On=E{⟨𝐓_n-𝐒_n,Σ-𝐒_n⟩}/E{‖𝐓_n-𝐒_n‖ _F^2}.The scalar λ_On is called the oracle shrinkage coefficient, since its depends on the unknown covariance matrix Σ. Therefore, λ_On (<ref>) must be estimated. The latter can be estimated from its sample counterparts as in <cit.>. We denote this estimator as λ̂_On.§ SEQUENTIAL UPDATE OF THE SHRINKAGE ESTIMATOR We want to know what happens to Σ̂(λ_n) (<ref>) when we add an observation 𝐱_n+1, using only the current knowledge of 𝐒_n, 𝐦_n and n. Setting 𝐝_n+1=𝐱_n+1-𝐦_n while using <cit.>, we have the following update rules for 𝐦_n (<ref>) and 𝐒_n (<ref>) when an observation 𝐱_n+1 is added𝐦_n+1=𝐦_n+1/n+1𝐝_n+1 𝐒_n+1=n-1/n𝐒_n+1/n+1𝐝_n+1𝐝_n+1^T.Based on 𝐒_n+1 (<ref>), we can write the update rule for the target 𝐓_n (<ref>) as 𝐓_n+1=n-1/n𝐓_n+1/(n+1)p‖𝐝_n+1‖ _F^2𝐈By using 𝐒_n+1 (<ref>) and 𝐓_n+1 (<ref>), the update rule for the shrinkage estimator Σ̂(λ_n) (<ref>) can be written as Σ̂(λ_n+1)=𝐆_n+𝐅_nwhere 𝐆_n and 𝐅_n defined as 𝐆_n=n-1/nΣ̂(λ_n)+(1-λ_n+1)1/n+1𝐝_n+1𝐝_n+1^Tand 𝐅_n=1/(n+1)pλ_n+1‖𝐝_n+1‖ _F^2𝐈 +n-1/n(λ_n-λ_n+1)(𝐒_n-𝐓_n),respectively. Based on the above update rules, we derive the sequential update rules for the inverse of the shrinkage estimator. § SEQUENTIAL UPDATE FOR THE INVERSE OF THE SHRINKAGE ESTIMATOR In this section, we derived approximated inverses of the shrinkage estimator which are updated sequentially and do not involve any matrix inversion. We start, therefore, from the inverse of the sample covariance matrix 𝐒_n+1 that can be obtained from the current inverse of 𝐒_n (<ref>) using the Sherman-Morrison-Woodbury matrix identity <cit.> as 𝐒_n+1^-1=n/n-1(𝐒_n^-1-𝐒_n^-1𝐝_n+1𝐝_n+1^T𝐒_n^-1/n^2-1/n+𝐝_n+1^T𝐒_n^-1𝐝_n+1).The last update rule can be used only if 𝐒_n is invertible. It will not be invertible for n≤ p. Since the shrinkage estimator Σ̂(λ_n) (<ref>) is a regularized version of 𝐒_n (<ref>), an inverse exists for any n. This inverse of Σ̂(λ_n) (<ref>) involves two main steps. The first one is to update the inverse of 𝐆_n (<ref>) from an inverse of Σ̂(λ_n) (<ref>). The second is to update the next step inverse of Σ̂(λ_n) from 𝐅_n (<ref>) and the inverse of 𝐆_n (<ref>) calculated in the first step. Suppose, for example, that the exact inverse of Σ̂(λ_n) (<ref>), denoted as Σ̂^-1(λ_n), is known. In the same manner as in 𝐒_n+1^-1 (<ref>), the inverse for 𝐆_n (<ref>) can be calculated from Σ̂^-1(λ_n) as [ 𝐆_n^-1=; n/n-1(Σ̂^-1(λ_n)-Σ̂^-1(λ_n)𝐝_n+1𝐝_n+1^TΣ̂^-1(λ_n)/n^2-1/n(1-λ_n+1)+𝐝_n+1^TΣ̂^-1(λ_n)𝐝_n+1). ]Using <cit.>, the exact inverse of Σ̂(λ_n+1) can be calculated from 𝐆_n^-1 (<ref>) and 𝐅_n (<ref>) with p iterations (𝐆_n^(i+1))^-1=(𝐆_n^(i)+𝐟_i𝐞_i^T)^-1 =(𝐆_n^(i))^-1-(𝐆_n^(i))^-1𝐟_i𝐞_i^T(𝐆_n^(i))^-1/1+𝐞_i^T(𝐆_n^(i))^-1𝐟_i,i=1,…,pwhere 𝐟_i and 𝐞_i are the i columns of 𝐅_n (<ref>) and the identity matrix 𝐈, respectively. The inverse of Σ̂(λ_n+1) (<ref>) is equal to the output of the last iteration, i.e., Σ̂^-1(λ_n+1)=(𝐆_n^(p+1))^-1.In order to avoid the calculation of p iterations, we can use approximations for Σ̂^-1(λ_n+1) (<ref>). The inverse approximations of the shrinkage estimator are discussed in the following section.§.§ Inverse Approximations for the Shrinkage Estimator We consider two approximations for Σ̂^-1(λ_n+1) (<ref>). The first approximation is defined as Σ̃_1^-1(λ_n+1)=𝐆̃_n^-1-α_n𝐆̃_n^-1𝐅_n𝐆̃_n^-1where 𝐆̃_n^-1= n/n-1(Σ̃_1^-1(λ_n)-Σ̃_1^-1(λ_n)𝐝_n+1𝐝_n+1^TΣ̃_1^-1(λ_n)/n^2-1/n(1-λ_n+1)+𝐝_n+1^TΣ̃_1^-1(λ_n)𝐝_n+1).The matrix 𝐆̃_n^-1 (<ref>) differs from 𝐆_n^-1 (<ref>) in the fact that it relies on the approximated inverse Σ̃^-1(λ_n) (<ref>), instead of the exact inverse Σ̂^-1(λ_n) (<ref>). A possible motivation to justify the update rule (<ref>) stems from the mean value theorem as explained in <cit.>. Another motivation arises from the Neumann series <cit.> where Σ̂^-1(λ_n+1) (<ref>) is approximately equal to Σ̃^-1(λ_n+1) (<ref>) for α=1 and relatively small 𝐅_n. We define α_n as the value that minimizes the reconstruction squared error, i.e., α_n=min_α‖(𝐆̃_n^-1-α𝐆̃_n^-1𝐅_n𝐆̃_n^-1)Σ̂(λ_n+1)-𝐈‖ _F^2and is equal to α_n=Tr(𝐆̃_n^-1𝐅_n𝐆̃_n^-1Σ̂(λ_n+1)(𝐆̃_n^-1Σ̂(λ_n+1)-𝐈))/‖𝐆̃_n^-1𝐅_n𝐆̃_n^-1Σ̂(λ_n+1)‖ _F^2Additional simplification can be taken by looking at the last term in 𝐅_n (<ref>). Under the assumption that the difference λ_n-λ_n+1 is relatively small, we can write an approximation for 𝐅_n (<ref>) by neglecting its last term, i.e., 𝐅̃_n=1/(n+1)pλ_n+1‖𝐝_n+1‖ _F^2𝐈This will lead to the second approximation for Σ̂^-1(λ_n+1) (<ref>), denoted as Σ̃_2^-1(λ_n+1)=𝐆̃'̃_n^-1-α'_n𝐆̃'̃_n^-1𝐅̃_n𝐆̃'̃_n^-1where [ 𝐆̃'̃_n^-1=; n/n-1(Σ̃_2^-1(λ_n)-Σ̃_2^-1(λ_n)𝐝_n+1𝐝_n+1^TΣ̃_2^-1(λ_n)/n^2-1/n(1-λ_n+1)+𝐝_n+1^TΣ̃_2^-1(λ_n)𝐝_n+1) ]and α'_n is calculated by α'_n=(n+1)pTr(𝐆̃'̃_n^-2Σ̂(λ_n+1)(𝐆̃'̃_n^-1Σ̂(λ_n+1)-𝐈))/λ_n+1‖𝐝_n+1‖ _F^2‖𝐆̃'̃_n^-2Σ̂(λ_n+1)‖ _F^2We examine these two approximations in the following section. § EXPERIMENTS In this section we implement and evaluate the sequential update of the inverse shrinkage estimator. As in <cit.>, we assume that the observations are i.i.d Gaussian vectors. In order to study the estimators performance, an autoregressive covariance matrix Σ is used. We let Σ be the covariance matrix of a Gaussian AR(1) process <cit.>, denoted by Σ_AR={σ_ij=r^|i-j|} .As in <cit.>, we use r=0.5. In all simulations, we set p=50 and let n range from 1 to 30. Each simulation is repeated 200 times and the average values are plotted as a function of n. The experimental results are summarized in box plots. On each box, the central mark is the median, the edges of the box are the 25th and 75th percentiles, and the whiskers correspond to approximately +/–2.7σ or 99.3 coverage if the data are normally distributed. The outliers are plotted individually.The reconstruction errors of the approximated inverses Σ̃_1^-1(λ_n) (<ref>) and Σ̃_2^-1(λ_n) (<ref>) are defined by e_1(n)=1/p‖Σ̃_1^-1(λ_n)Σ̂(λ_n)-𝐈‖ _F^2,and e_2(n)=1/p‖Σ̃_2^-1(λ_n)Σ̂(λ_n)-𝐈‖ _F^2,respectively. These reconstruction errors are normalized with p since it is the squared Frobenius norm of the identity matrix 𝐈. We examine the approximated inverse Σ̃_1^-1(λ_n) (<ref>) and Σ̃_2^-1(λ_n) (<ref>) where λ_n is equal to λ̂_On <cit.>. The experimental results for the reconstruction errors e_1(n) (<ref>) and e_2(n) (<ref>) are summarized in Fig. 1 and Fig. 2, respectively. The values of e_1(n) (<ref>) converge on average to zero as the number of observations n increase. In several simulations, however, the update rule accumulates error and diverges.The related reconstruction error e_2(n) (<ref>) is depicted in Fig. 2. The reconstruction error e_2(n) (<ref>) does not converge to zero due to its relative simplification involving the use of 𝐅̃_n (<ref>) instead of 𝐅_n (<ref>). However, the use of 𝐅̃_n (<ref>) renders Σ̃_2^-1(λ_n) (<ref>) much more robust to outliers in comparison to the first estimator Σ̃_1^-1(λ_n) (<ref>). In that sense, a relatively small and fixed reconstruction error can be assumed in order to avoid unexpected outliers. § CONCLUSIONS A key challenge in many large-scale sequential machine learning methods stems from the need to obtain the covariance matrix of the data, which is unknown in practice and should be estimated. In order to avoid additional complexity during the modeling process, it is commonly assumed that the covariance matrix is known in advanced or, alternatively, that simplified estimators are employed. In Section 3, we derived a sequential update rule for the shrinkage estimator that offers a compromise between the sample covariance matrix and a well-conditioned matrix. The optimal shrinkage coefficient, in the sense of mean-squared error, is analytically obtained, which is a notable advantage since a key factor in realizing large-scale architectures is the resource complexity involved. In Section 4, sequential update rules that approximate the inverse shrinkage estimator are derived. The experimental results in Section 5 clearly demonstrates that the reconstruction errors of the approximated inverses are relatively small. The sequential update rules that approximate the inverse of the shrinkage estimator provide a general result that can be utilized in a wide range of sequential machine learning applications. Therefore, the approach paves the way for improved large-scale machine learning methods that involve sequential updates in high-dimensional data spaces. IEEEbib | http://arxiv.org/abs/1707.08885v1 | {
"authors": [
"Tomer Lancewicki"
],
"categories": [
"stat.CO"
],
"primary_category": "stat.CO",
"published": "20170727143745",
"title": "Sequential Inverse Approximation of a Regularized Sample Covariance Matrix"
} |
Dynamic band structure tuning of graphene moiré superlattices with pressure Cory R. Dean^1 December 30, 2023 ===========================================================================* Instituto Nacional de Astrofísica, Óptica y Electrónica (INAOE), Luis Enrique Erro 1, 72840, Puebla, Mexico * Department of Astronomy, The University of Texas at Austin, 2515 Speedway Boulevard, Austin, TX 78712, USA* CONACyT-Instituto Nacional de Astrofísica, Óptica y Electrónica, Luis Enrique Erro 1, 72840, Puebla, Mexico * Department of Astronomy, University of Massachusetts, MA 01003, USA * European Southern Observatory, Karl Schwarzschild Strasse 2, Garching, Germany * Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, UK * School of Physics and Astronomy, Cardiff University, The Parade, Cardiff CF24 3AA, UK * Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA * Steward Observatory, University of Arizona, 933 North Cherry Avenue, Tucson, AZ 85721, USA * Instituto de Astronomía, Universidad Nacional Autónoma de México, A.P. 70-264, 04510, CDMX, Mexico * Dept. of Physics & Astronomy, University of California, Irvine, CA 92697, USA * Instituto de Astrofísica de Canarias (IAC), E-38205 La Laguna, Tenerife, Spain * Universidad de La Laguna, Dpto. Astrofísica, E-38206 La Laguna, Tenerife, Spain* Astronomical Observatory Institute, Faculty of Physics, Adam Mickiewicz University, ul. Słoneczna 36, 60-286 Poznań, Poland * Department of Physical Sciences, The Open University, Milton Keynes, MK7 6AA, UK * Leiden Observatory, Leiden University, P.O. Box 9513, NL-2300 RA Leiden, The Netherlands 2Since their discovery, submillimeter-selected galaxies<cit.> have revolutionized thefield of galaxy formation and evolution.From the hundreds of square degrees mapped at submillimeter wavelengths<cit.>, only a handful of sources have been confirmed to lieat z>5 (ref.<cit.>) and only two at z≥6 (ref.<cit.>). All of these SMGs are rare examples of extreme starburst galaxies with star formation rates of ≳ 1000 M_⊙ yr^-1 and therefore are not representative of the general population of dusty star-forming galaxies. Consequently, our understanding of the nature of these sources,at the earliest epochs, is still incomplete. Here we report the spectroscopic identification of a gravitationally amplified (μ=9.3±1.0) dusty star-forming galaxy at z=6.027. After correcting for gravitational lensing we derive an intrinsic less extreme star formation rateof 380±50 M_⊙ yr^-1 for this source, and find that its gas and dust properties are similar to those measured for local Ultra Luminous Infrared Galaxies, extending the local trends to apoorly explored territory in the early Universe. The star-formation efficiency of this galaxy is similar to those measured in its local analogues<cit.>, despite a ∼12 Gyr difference in cosmic time. HATLAS J090045.4+004125 (α=09^h00^m45.8, δ=+00^∘41'23”; hereafter G09 83808, since it was detected in the GAMA 09hrs field)is part of a sub-sample of the Herschel ATLAS `500 μm-riser' galaxies<cit.> withultra-red far-infrared (FIR) coloursofS_500_μ m/S_250_μ m > 2 and S_500_μ m/S_350_μ m > 1, with a flux density thresholdof S_500_μ m<80mJy.The FIR colours of this source are consistent with thermal dust emission redshifted to z>4 andrepresent a relatively simple selection criterion to findhigh-redshift galaxies. A similar selectionallowedthe identification of HFLS3<cit.>, an extreme starburst galaxy (even after corrected forgravitational amplification<cit.>) at z = 6.3, in the HerMES blank field survey<cit.>.G09 83808 was observed, among other ultrared-Herschel dusty star-forming galaxies, as part of a follow-up program with theLarge Millimeter Telescope Alfonso Serrano (LMT) using the AzTEC camera, in order to obtainhigher angular resolution (∼8.5 arcsec) continuum observations at 1.1 mm. A sub-sample of those galaxies detected as a single source in the AzTEC images(i.e. with no evidence of multiple components) and with photometric redshifts of z>4, was selected for spectroscopic observations in the 3 mm band using the Redshift Search Receiver (RSR) on the LMT. In the LMT/RSR spectrum of G09 83808 we identify three emission lines corresponding to ^12CO(6-5), ^12CO(5-4), and H_2O(2_11-2_02) (see Fig. 1). Based on these lines we unambiguously determine the galaxy redshift to be z=6.0269±0.0006 (i.e. when the Universe was just 900 million years old). Follow-up observations with The Submillimeter Array (SMA) telescope confirm this solution through thedetection of the redshifted [CII] ionized carbon line at 270.35 GHz (see Fig. 1).High-angular resolution observations (0.24 arcsec ×0.13 arcsec, corresponding to a physical scale of ∼1 kpc at this redshift) taken with the Atacama Large Millimeter/submillimeter Array (ALMA; see Methods section) at ∼890 μ m reveal a double arc structure (in a partial Einstein ring configuration of radius ∼ 1.4 arcsec) around a foreground galaxy at z=0.776 (see Fig. 2), implyingstrong gravitational amplification of the high-redshift background galaxy.Using these ALMA continuum observations to constrain the effects of gravitational lensing, modelling directly the visibilities in the uv plane (see Methods section for additional details), we derive a gravitational amplification factor of μ=9.3±1.0.This amplification factor is used to derive the intrinsic physical properties of G09 83808.Using the Herschel 250, 350, and 500 μ m photometry<cit.>, combined with the SCUBA-2 850 μ m<cit.> imaging and our AzTEC 1.1 mm observations (see Table 1), we model thecontinuum spectral energy distribution (SED; see Figure 3). We estimate an infrared (IR, 8-1000 μ m) luminosity, L_ IR, of 3.8±0.5×10^12 L_⊙(corrected for gravitational magnification) which implies a dust-obscured star formation rate (SFR) of 380±50 M_⊙ yr^-1(see Methods section for more information). This means that G09 83808 is a member of the Ultra Luminous Infrared Galaxy (ULIRGs<cit.>) population. This is one of the first SMG with an unambiguous spectroscopic redshift in this luminosity range at z≳5,lying between the extreme obscured starbursts<cit.>(≳ 1000 M_⊙ yr^-1) discovered at submm wavelengths and the UV/optical selected star-forming galaxies with follow-up detections at submm wavelengths<cit.> (≲ 100 M_⊙ yr^-1).Within this luminosity range only a handful of galaxies at z>6 are known, whichwere recently discovered around quasars<cit.> thanks to the serendepitously detection of a single emission line associated to [CII].Although these galaxies are unreachable with the current generation of submm wide-area surveys<cit.> without thebenefit of gravitational amplification, they can be found in the deepest surveys recently achievedwith ground-based telescopes, such as the James Clerk Maxwell Telescope (JCMT) SCUBA-2 Cosmology Legacy Survey (S2CLS). However, none of them has yet been spectroscopically confirmed. With the caveat of using the position of the dust SED peak as an estimation of redshift, a study based on S2CLS observations<cit.> has derived a comoving space density of 3.2×10^-6Mpc^-3 for sources with 300< SFR < 1000 M_⊙ yr^-1 at 5<z≲6 (i.e. in the range probed by our galaxy). With a duty-cycle correction of ≈ 40Myr,as the gas depletion time scale measured for G09 83808 (see below) and other galaxies<cit.>, we estimate the corrected comoving space densityof this population of galaxies to be≈2×10^-5Mpc^-3, which perfectly matches that of massive quiescent galaxies at z≈3-4 (refs.<cit.>). This suggests, that these ULIRG-type galaxies at5≲ z≲6 are the progenitors of these quiescent galaxies, which cannot be explained only by the rare extreme starburst galaxies (like HFLS3),since they are an order of magnitude less abundant<cit.>.Based on the CO lines detected in the LMT/RSR spectrum we derive a molecular gas mass of M(H_2)=1.6±0.6×10^10 M_⊙ (see Methods section fordetails). This implies a gas depletiontimescale of M(H_2)/SFR≈40 Myr, consistent with the value found for other SMGs at lower redshifts with ULIRG-luminosity<cit.>.G09 83808 shows a remarkable large gas mass fraction of f_ gas=M_ H_2/M_ dyn∼ 60% (see Methods secction), among the largest measured for star forming galaxies at z≈2-3 (ref.<cit.>). The CO(6-5)/CO(5-4) line luminosity ratio of 0.4±0.1is in agreement with local ULIRGs (although lower than the average<cit.>), and impliesa CO ladder peaking atJ≤5 (i.e. less excited than AGN-dominated galaxies<cit.>). These two CO transitions, as well as the H_2O line,lie (within the error bars) on their respective FIR/IR-line luminosity relations (L_ FIR∝ L_ CO(6-5)^0.93, L_ FIR∝ L_ CO(5-4)^0.97, andL_ H_2O∝ L_ IR^1.16) found for local ULIRGs and lower redshifts SMGs<cit.>.The star-formation efficiency (SFE) of our galaxy, estimated through the L'_CO-L_ IR relation (which describes the relationship between the luminosity due to star formation and the gas content), is similar to local (U)LIRGs (see Fig. 4). Then, the same SFE can be found across several decades of molecular gas masses from z=6 to z∼0 (i.e., during the last 12.8 Gyr of the Universe). In addition, the estimated dust mass of M_d=1.9±0.4×10^8 M_⊙ results in a gas-to-dust ratio, δ_ GDR, of 80±30. This is in agreement with the value estimated for HFLS3<cit.> and also with local (U)LIRGs<cit.> (δ_ GDR=120±28). The luminosity of the [CII] ionized carbon line detected with the SMA is 1.3±0.4×10^9 L_⊙ which corresponds to a [CII]/FIR ratio of3.4±1.1×10^-4, a value that is among the lowest measured for local (U)LIRGs and SMGs. As shown in Figure 4, our source follows the same [CII] deficiency trend measured for local LIRGS<cit.> extending it to L_FIR≳ 10^12 L_⊙ and up to z=6.The [CII]/FIR ratio of G09 83808 is also consistent with the lowest values measured for lower-redshift SMGs and lies on a region where SMGs and AGN-host galaxies converge (Fig. 4). It may be the case that other SMGs suffer from gravitiational amplification, which could help to reduce the large scattersince many of these galaxies should fall along the LIRG relation when corrected for magnification. However, the intrinsic scatter in the relation is high<cit.>, even for thelocal sample, and therefore, larger samples of SMGs are required to derive conclusions about the origin of the [CII] deficiency.We confirm the existence of ULIRG-like galaxies within the first billion years of Universe's history. These sources may be more representative of the dusty star-forming galaxy population at these epochs than the extreme starbursts previously discovered.Four emission-line-selected galaxies with similar luminosities and redshifts have been recently found around quasars<cit.> (with the caveat of using just one line for redshift determination), however, the properties of these sourcesmay be affected bythe companion quasar and therefore not representative of the whole population. Although G09 83808 shows similar properties to thosemeasured in lower-redshift SMGs, its higher dust temperature (T_d=49±3 K) and compact morphology (R_1/2=0.6±0.1 kpc) resemble that of local ULIRGs. For comparison, typical UV/optically-selected star-forming galaxies at z∼6 have SFRs ∼10 times lower and radii ∼1.7 times larger than G09 83808<cit.>. This study is hence crucial for understanding the evolutionary path of SMGs and their link with local galaxies.Although a larger sample is needed to statistically estimate the properties of these sources and their contribution to the cosmic star formation history, this galaxy suggests that star formation in dusty star-forming galaxies has been driven by similar physical processes during the last ∼12.8 Gyr . REFERENCES1ref53Smail, I.et. al. A Deep Sub-millimeter Survey of Lensing Clusters: A New Window on Galaxy Formation and Evolution.Astrophys. J. Lett. 490 L5-L8 (1997). ref2Hughes, D. et. al. High-redshift star formation in the Hubble Deep Field revealed by a submillimetre-wavelength survey.Nature 394, 241-247 (1998). ref3Oliver, S. et. al. The Herschel Multi-tiered Extragalactic Survey: HerMES. Mon. Not. R. Astron. 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Rise of the Titans: A Dusty, Hyper-Luminous `870 micron riser' Galaxy at z ∼ 6.Astrophys. J. arXiv preprint: 1705.09660 (2017). ref11Riechers, D. et. al. A Dust-Obscured Massive Maximum-Starburst Galaxy at a Redshift of 6.34. Nature 496, 329-333 (2013). ref57Strandet, M. et. al. ISM properties of a Massive Dusty Star-Forming Galaxy discovered at z ∼ 7. Astrophys. J. Lett. 842 L15 (2017). ref17Sanders, D & Mirabel, I. Luminous Infrared Galaxies. Ann. Rev. Astron. Astrophys. 34, 749- (1996). ref12Ivison, R.et. al. The Space Density of Luminous Dusty Star-forming Galaxies at z> 4: SCUBA-2 and LABOCA Imaging of Ultrared Galaxies from Herschel-ATLAS. Astrophys. J. 832, 78- (2016). cooray14Cooray, A. et. al.HerMES: The Rest-frame UV Emission and a Lensing Model for the z = 6.34 Luminous Dusty Starburst Galaxy HFLS3. Astrophys. J. 790, 40- (2014). capak2015Capak, P. et. al. 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Explaining the [C II]157.7 μm Deficit in Luminous Infrared Galaxies - First Results from a Herschel/PACS Study of the GOALS Sample. Astrophys. J. 774, 68- (2013). RodriguezPueblaRodriguez-Puebla, A., Primack, J., Avila-Reese, V. & Faber, S. Constraining the galaxy-halo connection over the last 13.3 Gyr: star formation histories, galaxy mergers and structural properties. Mon. Not. R. Astron. Soc. 470, 651-687 (2017).* We thank Ian Smail for insightful comments that improved the quality of the paper. JAZ acknowledges support from a mexican CONACyT studentship.RJI, LD and IO acknowledge support from ERC in the form of the Advanced Investigator Programme, 321302, COSMICISM.LD additionally acknowledges support from the ERC Consolidator Grant CosmicDust.HD acknowledges financial support from the Spanish Ministry of Economy and Competitiveness (MINECO) under the 2014 Ramón y Cajal program MINECO RYC-2014-15686.MJM acknowledges the support of the National Science Centre, Poland through the POLONEZ grant 2015/19/P/ST9/04010 andthe European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 665778. This work would not have been possible without the long-term financial support from the Mexican CONACyT during the construction and early operational phase of the Large Millimeter Telescope Alfonso Serrano, as well as support from theUS National Science Foundation via the University Radio Observatory program, the Instituto Nacional de Astrofísica, Óptica y Electrónica (INAOE), and the University of Massachusetts (UMass). The Submillimeter Array is a joint project between the SmithsonianAstrophysical Observatory and the Academia Sinica Institute of Astronomy andAstrophysics and is funded by the Smithsonian Institution and the AcademiaSinica. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan),together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), incooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. Author Contributions JAZ led the scientific analysis and the writing of the paper,as well as the SMA follow-up proposal. RJI, EV, SE, AC, HD, JSD, LD, MJM, SS, IS, MWLS, and PWhave contributed to the original Herschel proposals and source selection of the red sources,where this source was originally identify. AM, DHH, EV, IA, VAR, MC, DRG, ET, and OVperformed the selection of thesample for the LMT observations and lead the LMT proposals. MSY, GN, FPS, DS, GW, DSA, AV, and MZ carried outLMT data reduction andinterpretation. DW, MY, and AIGR assisted with the SMA observations and data reduction.JS, IO, HN have contributed to the data analysis and to fitting and modeling the results. All the authors have discussed and contributed to this manuscript.Correspondence Correspondence and requests for materials should be addressed to Jorge Zavala (email: [email protected]). Competing Interests The authors declare that they have no competing financial interests.2 § OBSERVATIONS AND DATA REDUCTION §.§ LMT observations Continuum and spectroscopic observations were obtained using the Large Millimeter Telescope (LMT<cit.>, PI: D. Hughes),located on the summit of Volcán Sierra Negra(Tliltépetl), Mexico, at ∼4600 m.a.s.l. Observations were carried out during the Early Science Phase of the telescope using the1.1 mm continuum camera, AzTEC<cit.>, and the 3 mm spectrograph, Redshift Search Receiver (RSR<cit.>). During these observations only the inner 32-m diameter region of the telescope active surface was illuminated, which provided an effective beam size of ≈8.5 arcsec at 1.1 mm and between 20-28 arcsec in the RSR 3 mm window (75 GHz - 110 GHz). AzTEC observations were performed on 2014 November 10 with an opacity of τ_225=0.07 and total on-source integration time of 11 min. Data reduction were done following the AzTEC Standard Pipeline<cit.>. G09 83808 was detected with aS/N ≈20 with a flux density of S_1.1 mm=20.0±1.0 mJy. RSR observations weresubsequently taken at the AzTEC positionintwo different periods: February 2016 and February 2017, along five different nights with an opacity range of τ_225=0.05-0.15 and a total integrationtime of 8 hrs. Pointing observations on bright millimetre sources were done every hour. Data reduction was performed using the Data Reduction and Analysis Methods in Python (DREAMPY). The final spectra were obtained by averaging all scans using 1/σ^2 weights after flagging bad data. Finally, to convert from antenna temperature units to flux, a factor of 7 Jy K^-1 was used<cit.>.The final spectrum shows three lines detected atS/N ≳5 associated to CO(6-5), CO(5-4) and H_2O(2_11-2_02) at z=6.0269. A cross-correlation template analysis<cit.> also identifies this redshift as the best solution with a S/N =9.1.Figure 1 shows the final spectrum after a Savitzky-Golay filter<cit.> has been applied for better visualization (the filter does not modified any of the propertiesof the detected lines).At the redshift of our source the [CII] 158 μm line (see below) falls within the AzTEC band pass and then contributes to the total flux density measured at 1.1 mm. However, the contamination from the line is measured to be less than 2 per cent. Even if the [CII] line luminosity was as high as 1 per cent ofthe total IR luminosity, the contamination to the AzTEC measurement would be only ∼6 per cent, which is similar to the absolute flux calibration uncertainty.Therefore, and at least for this source, the contamination of the emission line to the1.1 mm continuum flux density is less important than anticipated<cit.>. §.§ SMA observations G09 83808 was observed with the Submillimeter Array (SMA, PI: J. Zavala) on Mauna Kea, Hawaii, on 2017 April 03. The weather conditions were good, with an averageatmospheric opacity of τ_225=0.07 and stable phase. The seven available array antennas were in a compact configuration that provided baseline lengths from 8 to 77 meters. The `345' receiver set was tuned to provide spectral coverage ±(4-12) GHz from a LO frequency of 277.5 GHz, specifically to span a broad range around the estimated (redshifted) [CII] line frequency of ∼270.5 GHz in the lower sideband. The SWARM correlator provided uniform channel spacing of 140 kHz (∼0.16 km s^-1) over the full bandwidth. The usable field of view is set by the FWHM primary beam size of ∼47 arcsec at this frequency.The basic observing sequence consisted of a loop of 2 minutes each on the gain calibrators J0909+013 (1.57 Jy) and J0831+044 (0.47 Jy) and 17.5 minutes on G09 83808. The track spanned an hour angle range of -0.8 to 4.8 for the target source. Passband calibration was obtained with observations of the strong quasar 3C279. The absolute flux scale was set using observations of Callisto, with an estimated accuracy of 20%. All of the basic calibration steps were performed using standard procedures in the MIR software package. The calibrated visibilities were exported to the MIRIAD software package for imaging and deconvolution. Within MIRIAD, the task uvaver was used to combine the 4 correlator windows of the lower sideband and to resample the visibilities to 50 km s^-1 spectral resolution. The task uvlin was used for continuum subtraction, using a linear fit to line-free channel ranges in the band. The task invert provided Fourier inversion for both continuum and spectral line imaging, followed by clean for deconvolution. The synthesized beam size obtained with natural weighting was 2.5^”×2.3^”, p.a. 82^∘ for the spectral line data cube, with rms noise 7.1 mJy per 50 km s^-1 bin. The final spectrum (see Fig. 1) was then extracted from a rectangular region that comprise all the continuum emission. We measured the continuum flux density of thesource to be 21.5±3 mJy, in very good agreement with the AzTEC photometry.§.§ ALMA observations The ALMA high-resolution 870μ m observations used in this work weretaken on 31 August 2015 (project 2013.1.00001.S, PI Rob Ivison; ref.<cit.> ), when the array was in arelatively extended configuration with baselines up to 1.6 km. The default continuum spectral configuration was used, covering [335.49 - 339.49]GHz and[347.49 - 351.49]GHz. The data were calibrated using the ALMA pipeline, with nofurther manual flagging required. The calibrated visibilities were imaged by using Briggsweighting with robust = 0.5, which is a good compromise between sensitivity and angularresolution. The beam size is then ∼ 0.12^” and the continuum sensitivity isr.m.s.∼ 0.1 mJy beam^-1. The visibility weighting in ALMA data is generally only correct in a relative sense, while our subsequent lens modeling procedure(see `Lensing Model' below) requires an absolute estimate of the noise in the data. The data weights are determined by differencing successive visibilities on the same baseline, polarization, and frequency baseband. The ALMA data also serendipitously cover the frequency of theredshifted 122μ m [NII] line; this line is not detected at >3σ significance.§ LENSING MODEL The lens model was created using the publicly-availablecode<cit.>; details of the codeare given in that work. Briefly, the lens mass profile is parameterized as a Singular Isothermal Ellipsoid, and thebackground source is modeled with a single elliptical Sérsic profile. The parameter space is explored using aMarkov Chain Monte Carlo sampling method, generating a model lensed image at each proposed combination of lens andsource parameters. The redshift of both sources is fixed at z=0.776 (based on X-Shooter/VLT observations<cit.>) and z=6.027, respectively.Because pixel values in interferometric images are correlated and subject to difficult-to-model residual calibration errors, the proposed model image is inverted to the visibility domain and sampled at the uvcoordinates of the ALMA data. We also allow forresidual antenna-based phase calibration errors in the model whichcould be due to, for example, uncompensated atmospheric delays. The phase shifts of all antennas are <10 deg, indicating that no significant antenna-based calibration problems remain. The lensed emission is reasonably well-fit by a single background Sérsic component, leaving peak residuals of ∼4σ (the source is detected at peak significance ∼20σ). These residuals may indicate that either the lens, source, or both are more complex than the simple parametric forms we have assumed. We have verified that an additionalbackground source component is not statistically motivated. The best-fit magnification of the source isμ_890μ m = 9.3 ± 1.0, with an intrinsic flux density S_890 μ m = 4.3 ± 0.5 mJy and half-light radius 0.10 ± 0.01” (=0.6±0.1 kpc). This compact morphology resembles the sizes found for local ULIRGs<cit.> (∼0.5 kpc),which are smaller than the typical values in SMGs (∼1.8 kpc, ref:<cit.>).§ SED FITTING AND DUST PROPERTIES We fit different galaxy SED templates to the photometry of G09 83808 through a χ^2 minimization method. We include the SED template of Arp220<cit.>, Cosmic Eyelash<cit.> (SMM J2135-0102), two average SMGs templates<cit.>, and finally a composite SED of 24 μm-selected star-forming galaxy<cit.>.All the SED templates were fixed at z=6.027. The Arp220 SED template gives us the best fit with χ_ red^2=0.7. Using this template we derive an IR(8-1000μ m) luminosity of 3.8±0.5×10^12 L_⊙ and a FIR (42.5-122.5μ m) luminosity of 2.3±0.3×10^12 L_⊙(both corrected for gravitational amplification) . For comparison, if we adopt instead a SMGs averagetemplate (χ_ red^2=1.2) we obtain L_IR=3.0±0.4×10^12 L_⊙, which is in good agreement with the value derived usingthe Arp220 template. Using Kennicutt standard relation<cit.> for a Chabrier initial mass function (IMF)<cit.>, this IR luminosity corresponds to a star formation rate (SFR) of 380±50 M_⊙ yr^-1, or to 570±70 M_⊙ yr^-1 if the most recent relation<cit.> is used.If we adopt instead the Kennicutt calibration<cit.> for aSalpeter IMF<cit.>, the SFR increases to 640±90 M_⊙ yr^-1, still below the range probed by other SMGs at z≳5.We also use a modified blackbody function to fit our photometric measurementsdescribed byS_ν∝{1- exp[-(ν/ν_0)^β]}B(ν,T_ d),where S_ν is the flux density at frequency ν, ν_0 is the rest-frame frequency at which the emission becomes optically thick, T_ d is the dust temperature,β is the emissivity index, and B(ν,T_ d) is the Planck function attemperature T_d. To minimize the number of free parameters, the emissivity index is fixed (previous observational workssuggest β=1.5-2; refs.<cit.>), as well as ν_0=c/100 μ m (refs.<cit.>), where c is the speed of light. From the best fit (χ^2≈1.1) we derive T_d=49±3 K forβ=1.8 andT_d=52±3 K for β=1.5. For these dust temperatures and at the redshift of our source the CMB effects<cit.> are not significant (Δ T≲1 K).Assuming the dustis isothermal, the dust mass, M_ d, is estimated from M_ d=S_ν/(1+z) D_L^2/(1+z)κ_ν B(ν,T_ d),where S_ν is theflux density at frequency ν, κ_ν is the dust mass absorption coefficient at ν, T_ d is the dust temperature, and B(ν_,T_d) is the Planck function at temperature T_ d. The dust massabsorption follows the same power law as the optical depth, κ∝ν^β.Assuming normalization of κ_ d(850μ m)=0.07 m^2 kg^-1 (ref.<cit.>) and a dust temperature of 49±3 K, we estimate a dust mass of M_d=1.9±0.4×10^8 M_⊙ after correcting for the CMB effects<cit.> (although this correction is less than 5 per cent). These calculations do not include the uncertainties of the dust mass absorption coefficient,which could be at least a factor of 3 (ref. <cit.>). If we use instead a lower dust temperature of 35 K, the dust mass increasesby a factor of ∼2.We also fit the observerd photometry with the MAGPHYS<cit.> SED modelling code finding consistent results, within the error bars,with median values of SFR=360^+80_-70 M_⊙ yr^-1, L_IR=4.5±0.7×10^12 L_⊙, T_d=40^+4_-2 K, and M_d=4.2±0.7×10^8 M_⊙.§ SPECTRAL LINE PROPERTIES We calculate the line luminosity for each detected line following the standard relation<cit.> described by:L'_ CO=3.25× 10^7 S_COΔ V ν_obs^-2 D^2_L (1+z)^-3,where L'_ CO is the line luminosity in K km s^-1 pc^2,S_COΔ V is thevelocity-integrated line flux in Jy km s^-1, ν_obs is the observed central frequencyof the line in GHz and D_L is the luminosity distance in Mpc. The integrated flux, S_COΔ V, is calculated as the integral of the best-fit Gaussian distribution, and its associated uncertaintythrough Monte Carlo simulations taking into account the errors in the Gaussian parameters (i.e. peakflux density and line width). To estimate the line luminosity in L_⊙, we useL=3×10^-11ν_r^3L', where ν_r is the rest frequency of the line<cit.>. All properties are summarized in Table 1.§ CO(1-0) LINE LUMINOSITY AND MOLECULAR GAS MASS The molecular gas mass, M(H_2), can be derived using the CO luminosity to molecular gas mass conversion factor, α,following the relationM(H_2)=α L'_CO(1-0). For theL'_CO(1-0) line luminosity we adopt the average value of L'_CO(1-0)=2.0±0.8×10^10 K km s^-1 pc^-2 extrapolated from our CO(6-5) and CO(5-4) transitions and correcting for gravitational amplification. The extrapolation was doneusing average brightness ratios found for lower-redshift SMGs<cit.> (L'_CO(5-4)/L'_CO(1-0)=0.32±0.05, L'_CO(6-5)/L'_CO(1-0)=0.21±0.04), this sample includes galaxies with similar luminosities to G09 83808 and are in agreement with those found for local ULIRGs<cit.> (within the large scatter). On the other hand, if we use the relationship between the Rayleigh-Jeans specific luminosity and CO(1-0) luminosity<cit.>,L'_CO(1-0) [K km s^-1pc^2]=3.02×10^-21 L_ν [ erg s^-1 Hz^-1], we obtain a consistent line luminosity of 1±0.1×10^10 K km s^-1 pc^-2 (assuminga mass-weighted dust temperature of 35 K, which is different from the luminosity-weighted dust temperature determined from the SED fitting<cit.>). Using the former valueand α=0.8 M_⊙ (K km s^-1 pc^-2)^-1, which is appropriate for starburst galaxies<cit.> (although some studies suggest larger values<cit.>), we derive a molecular mass of M(H_2)=1.6±0.6×10^10 M_⊙. § DYNAMICAL MASS AND GAS MASS FRACTION Dynamical mass has been derived using the `isotropic virial estimator', which has been shown to be appropriate for lower-redshift SMGs<cit.>:M_ dyn[M_⊙]=2.8×10^5 Δν_ FWHM^2[ km s^-1] R_1/2[ kpc],where Δν_ FWHM is the integrated line FWHM, which has been assumed to be 400 km/s (as the average between the CO and [CII] lines), and R_1/2is the half-light radius of ∼0.6 kpc (derived from the lesing model of the continuum emission). This results in a dynamical mass ofM_ dyn=2.6×10^10 M_⊙.Using this estimation we calculate a gas mass fraction of f_ gas=M_ H_2/M_ dyn≈60%. This constrain the CO luminosity to molecular gas mass conversion factor to α≲1.4 M_⊙ (K km s^-1 pc^-2)^-1, otherwise the molecuar gas mass would exceed the dynamical mass. 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Extreme Physical Conditions and Their Effects on the X _co Factor.Astrophys. J. 751, 10- (2012).engel10Engel, H. et al. Most Submillimeter Galaxies are Major Mergers.Astrophys. J. 724, 233-243 (2010). | http://arxiv.org/abs/1707.09022v3 | {
"authors": [
"Jorge A. Zavala",
"Alfredo Montaña",
"David H. Hughes",
"Min S. Yun",
"R. J. Ivison",
"Elisabetta Valiante",
"David Wilner",
"Justin Spilker",
"Itziar Aretxaga",
"Stephen Eales",
"Vladimir Avila-Reese",
"Miguel Chávez",
"Asantha Cooray",
"Helmut Dannerbauer",
"James S. Dunlop",
"Loretta Dunne",
"Arturo I. Gómez-Ruiz",
"Michal J. Michalowski",
"Gopal Narayanan",
"Hooshang Nayyeri",
"Ivan Oteo",
"Daniel Rosa González",
"David Sánchez-Argüelles",
"Stephen Serjeant",
"Matthew W. L. Smith",
"Elena Terlevich",
"Olga Vega",
"Alan Villalba",
"Paul van der Werf",
"Grant W. Wilson",
"Milagros Zeballos"
],
"categories": [
"astro-ph.GA"
],
"primary_category": "astro-ph.GA",
"published": "20170727194603",
"title": "A dusty star-forming galaxy at z=6 revealed by strong gravitational lensing"
} |
[email protected]@[email protected] ^1School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, China ^2Research Center for Hadron and CSR Physics, Lanzhou University and Institute of Modern Physics of CAS, Lanzhou 730000, China^3Research Center for Nuclear Physics (RCNP), Osaka University, Ibaraki, Osaka 567-0047, Japan In the framework of the one-boson-exchange model, we explore whether the intermediate- and short-range forces from σ/ω exchange can be strong enough to bind heavy molecular states. Λ_cD(D̅), and Λ_cΛ_c(Λ̅_c) systems have been studied and compared. We find that the force from σ exchange is attractive and dominant, whereas the ω-exchange force is not. As a consequence, the S-wave Λ_cD, Λ_cΛ_c, and Λ_cΛ̅_c can be possible molecular candidates. We further indicate that a one hadron-hadron system with more light quarks (u,d) can be easier to form a bound state. As a by-product, by studying the heavy-quark mass dependence for the Λ_cD(D̅)-like and Λ_cΛ_c(Λ̅_c)-like systems, we find that the charm/bottom sector can easily accommodate molecular states. Finally, the Λ_cN(N̅) and Λ_bN(N̅) systems are investigated. Our results indicate that they are also likely to form bound states. By including one-π-exchange forces providing additional attraction when coupled channels are included, we expect many molecular states in heavy-quark sectors. 12.39.Pn, 14.40.Lb, 14.40.RtHeavy molecules and one-σ/ω-exchange model Xiang Liu^1,2 May 29, 2018 ========================================== § INTRODUCTION Ever since the observations of X(3872) <cit.> and Θ^+ <cit.>, much evidence has been reported for new types of structures that are beyond the minimal q̅ q mesons and qqq baryons, although the observation of Θ^+ have been criticized by subsequent experiments <cit.>. Hence, they are called the exotic hadrons, like X(3872) <cit.>, Z_b(10610)/Z_b(10650) <cit.>, P_c(4380)/P_c(4450) <cit.>, and so on. Their unusual structure may contain more constituents, such as q̅ q pairs or gluons. With the extra q̅ q pairs, multiquark configurations may form a compact structure with colored correlations, such as tetraquark and triquark, or a rather extended structure with color-singlet hadronic correlations, which is called the hadronic molecule <cit.>.Since many new findings are seen near the threshold of hadronic decays, it is natural that the hadronic molecularlike structure develops if suitable attractive interactions between the hadrons are available. The strength of the interaction between color-singlet hadrons should be weaker than that between colored objects of order Λ_QCD -several hundred MeV. A typical example of such hadronic molecules is an atomic nucleus whose binding energy is of order 10 or a few MeV.For the study of hadronic molecules, the interaction between the hadrons is a crucial input. Unfortunately, not much is known for the hadron interactions, which are relevant for the recent exotic hadrons. For example, for X(3872), regarded as D D̅^* molecule <cit.>, the realistic interaction between D and D̅^* mesons is not well known partly because there is no experimental data. Lattice QCD approaches should, in principle, be promising as the recent study for Z_c(3900) <cit.>. However, application to various systems is rather limited. In such a situation, perhaps, the one-boson-exchange model is a reasonable theoretical approach.According to the mass differences for the exchanged meson, the interactions from π, σ, ρ, and ω exchanges contribute in the long-range, intermediate-range, and short-range distances, respectively. Among them, the one pion exchange is the best known, as is important for the deuteron <cit.> and the X/Y/Z states <cit.>. For the vector meson ρ, based on the local hidden gauge approach, it is also very essential in identifying the heavy molecular state <cit.>. In Ref. <cit.>, the η exchange is proposed to form heavy hadronic molecules. Soon after, the one-η-exchange model was adopted to investigate the interaction of Λ_cD̅_s^*/Σ_c^(*)D̅_s^*/Ξ^(',*)_cD̅^* systems in Ref. <cit.>. Numerical results indicate that the one-η exchange can be helpful in binding the heavy molecular pentaquarks.For the one-σ-exchange (OSE) and one-ω-exchange (OOE) models, they have been always considered together with the other bosons (π, η, ρ) in heavy molecular states, and their effect has been submerged by the effect of the one-π-exchange model. Thus, their importance is often overlooked. Therefore, the purpose of this paper is to study systematically the role of the OSE and OOE interactions between heavy hadrons. The coupling strengths and form factors are estimated by using the quark model, where the sigma and omega mesons couple to light quarks in heavy hadrons. Then we investigate if the intermediate- and short-range forces, due to the OSE and OOE models, can be strong enough to form heavy molecules, by varying model parameters within a reasonable range.To elucidate the role of the σ and ω mesons, we consider the systems, where π, η, ρ meson exchanges are suppressed, by using the spin and isospin conservation. For instance, there is no coupling πΛ_c Λ_c and π D D.The pion couples rather in the transitions such as πΛ_c Σ_c and π DD^*, which leads to the coupled channel problem. In our present study, we focus exclusively on the σ and ω mesons exchange by ignoring such coupled channels. Then, the DD̅[The ρ exchange can be also unsuppressed for the DD̅ system. Here, it is considered to contrastively discuss the relation of the effective potentials from the OSE and OOE model.], Λ_cD, and Λ_cΛ̅_c[We also notice there are several former works on the Λ_cΛ_c(Λ̅_c) interactions <cit.>.] systems are the ones that we study in this paper. We then compare the properties of those systems where the σ and ω mesons couple differently depending on the numbers of light quarks and antiquarks in the relevant hadrons.This paper is organized as follows. After the Introduction, we derive the one-boson exchange (OBE) effective potentials in Sec. <ref>. In Sec. <ref>, we present the corresponding numerical results. Then, according to these conclusions, heavy quark-mass dependence is studied by varying it continuously in Sec. <ref>. The paper ends with a summery in Sec. <ref>. § INTERACTIONS§.§ Lagrangians According to the heavy-quark symmetry, the OSE and OOE Lagrangians are constructed asℒ_DDσ/ω =-2g_σDD^†σ+2g_ω DD^†v·ω, ℒ_Λ_cΛ_cσ/ω =-2g_σ'Λ̅_cΛ_cσ-2g_ω'Λ̅_cΛ_cv·ω.Here, v is the four velocity, which has the form of v=(1,0).The coupling constants in Eqs. (<ref>) and (<ref>) will be determined in the quark model. Since the σ and ω mesons couple dominantly to the light quarks, the relevant interaction Lagrangian for the light quarks (q=u,d) with σ/ω can be expressed asℒ_qqσ/ω =-g_σ^qψ̅σψ-g_ω^qψ̅γ^μω_μψ.Compared with the vertices of D-D-σ/ω, Λ_c-Λ_c-σ/ω, and q-q-σ/ω, all the coupling constants in Eqs. (<ref>)-(<ref>) can be related, i.e.,g_σ = g_σ'=g_σ^q,g_ω = g_ω'=g_ω^q.In a σ model <cit.>, the value of g_σ^q is taken as g_σ^q=3.65. For the ω coupling g_ω^q, it is of a little uncertainty; in the Nijmegen model, g_ω^q=3.45, whereas it is equal to 5.28 in the Bonn model <cit.>. In Ref. <cit.>, g_ω^q was roughly assumed to be 3.00. In the following calculation, all the possible choices will be employed. According to the effective Lagrangians in Eqs. (<ref>) and (<ref>), all the relevant OBE scattering amplitudes can be collected in Table <ref>.Here, for the derivation of effective potentials of the DD̅, Λ_cD̅, and Λ_cΛ̅_c systems, the G-parity rule <cit.> is adopted, which relates the scattering amplitudes between the processes a+b→ c+d and a+b̅→ c+d̅ by exchanging one light meson.With the help of the Breit approximation, a relation between the effective potentials in momentum space and the scattering amplitudes is obtained, i.e.,𝒱_E(q)=-ℳ(h_1h_2→ h_3h_4)/√(∏_i2M_i∏_f2M_f).Here, ℳ(h_1h_2→ h_3h_4) is defined as the scattering amplitude of the process h_1h_2→ h_3h_4. M_i and M_f are the masses of the initial states (h_1, h_2) and final states (h_3, h_4), respectively.§.§ Form factorsThe effective potential in the coordinate space 𝒱(r) is obtained by performing the Fourier transformation as𝒱_E(r)= ∫d^3q/(2π)^3e^iq·r𝒱_E(q)ℱ^2(q^2).In order to manipulate the off shell effect of the exchanged mesons σ and ω and finite size effect of the interacting hadrons, we introduce a form factor ℱ(q^2) at every vertex.Generally, the form factor has the monopole, dipole, and exponential formsℱ_M(q^2)= Λ^2-m^2/Λ^2-q^2, ℱ_D(q^2)= (Λ^2-m^2)^2/(Λ^2-q^2)^2, ℱ_E(q^2)=e^(q^2-m^2)/Λ^2.Here, Λ, m, and q correspond to the cutoff, mass and momentum of the exchanged meson, respectively. These three kinds of form factorsare normalized at the on shell momentum of q^2 = m^2. In the low momentum limit, these form factors may be related to each other by redefining the cutoff parameter Λ such that the first terms of the Taylor expansion in powers of q^2/Λ^2 coincide. In this way, low momentum phenomena of hadronic molecules do not depend very much on different choices of form factors.The form factor can not be uniquely determined and various forms and cutoff Λ are used phenomenologically. However, an intuitive guideline for the choice of Λ is done by relating it to the size of hadrons. In Refs. <cit.>, Λ is related to the root-mean-square radius of the source hadron to which the exchanged boson (σ or ω) couples. According to the previous experience of the deuteron, the cutoff Λ in covariant-type monopole form factor is taken around 1 GeV. In the present qualitative study we use the same form factor both for meson and baryon vertices, because both of them contain light quarks and their spatial distributions are of order 1 fm or less.§.§ Effective potentials In this subsection, we adopt the monopole form factor ℱ_M(q^2), and the resulting effective potentials for the investigated systems are collected in Table <ref>. In Table <ref>, we can find that the interactions from the OSE model are always attractive for these investigated systems. This is a general consequence of the scalar meson exchange with a momentum independent coupling constant, as briefly explained in the next section. The depth of the OSE effective potentials depends on the number of the light quarks and/or antiquark combinations (q-q, q-q̅, q̅-q̅), where the light quark or antiquark is reserved in different hadrons of the hadron-hadron systems, respectively. For example, according to the quark configurations as shown in the second column of the Table <ref>, the light q̅-q̅ combination for the DD system is one, and there is only one q-q̅ combination in the DD̅ system.Since g_σ=g_σ' as estimated in the quark model (<ref>), a simple relation between the OSE effective potentials and the light-quark and/or antiquark combination numbers can be summarized as𝒱_σ(x_qq/q̅q̅,y_qq̅)=-(x_qq/q̅q̅+y_qq̅)g_σ^2Y(Λ,m_σ,r),where x_qq/q̅q̅ and y_qq̅ correspond to the numbers of qq/q̅q̅ and qq̅ (q=u,d) combinations, respectively. For the OOE effective potentials, a similar relation can be also written as 𝒱_ω(x_qq/q̅q̅,y_qq̅)=(x_qq/q̅q̅-y_qq̅)g_ω^2Y(Λ,m_ω,r).Here, we note that the sign of the OOE changes according to the charge conjugation symmetry. For example, the OOE force is repulsive for the q-q and q̅-q̅ combinations, while reversed for the system with q-q̅ combination.For example, for the Λ_cD system, there are two q-q̅ combinations, thus y_qq̅=2, and its potential from σ and ω exchanges is𝒱_Λ_cD(r)=-2g_σ^2Y(Λ,m_σ,r)-2g_ω^2Y(Λ,m_ω,r). In Fig. <ref>, we present the resulting potential as functions of the distance r, where the total potential is shown by the solid line, OSE by dotted lines and OOE by dashed lines. Here, the OSE and the OOE forces are of typical character of intermediate- and short-range force, and therefore they are suppressedwhen the radius r reaches 1 fm and larger. Since the force from the OSE model is the dominant, the total effective potentials for all the investigated systems are all attractive.To summarize shortly, the OSE model can always provide an attractive force. However, the OOE force is repulsive for the system including the same light quarks or antiquarks in its components of the investigated systems. The interaction strength from the OSE and OOE models depends on the light-quark combination numbers.§ NUMERICAL RESULTS In this section we discuss the role of the OSE and OOE interaction for the systems of Λ_c D(D̅) and Λ_c Λ_c (Λ̅_c) by solving the Schrödinger equation for them-1/2M∇^2ψ(r)+V(r)ψ(r)=Eψ(r),where ∇^2=1/r^2∂/∂ rr^2∂/∂ r, and M=m_1m_2/(m_1+m_2) is the reduced mass for the investigated system composed by particle 1 and particle 2. The parameters we use are summarized in Table <ref>. §.§ Solutions with covariant-type monopole form factor ℱ_M(q^2) = (Λ^2-m^2)/(Λ^2-q^2)We summarize the properties of S-wave bound states when they exist and the binding energy and root-mean-square radii (r_RMS) for the S-wave Λ_c D(D̅) and Λ_cΛ_c(Λ̅_c) systems in Table <ref>. For the coupling constant g_ω^q, we use three values (3.00 for Case I, 3.45 for II, and 5.28 for III) corresponding to the g_ω^q coupling constants of Ref. <cit.>, of the Nijmegen model, and of the Bonn model <cit.>, respectively.In Table <ref>, the cutoff parameters are chosen as 1, 1.1, and 1.2 GeV. These are the typical values for the form factor ℱ_M(q^2) <cit.>. In fact, depending on detailed values of Λ and on channels, bound states may or may not appear. In this way, we discuss whether bound states appear or not and study the role of the σ and ω meson exchanges. Before discussing details of Table <ref>, we make general remarks for boson exchange potentials. * The σ meson exchange provides attractive interaction. This is understood using a second-order perturbation theory for the one boson-exchange; the intermediate three particle state with σ meson has a virtual energy that is larger than the initial (or final) energy of the two particles. Moreover, due to the positive charge conjugation of the Lorentz scalar charge that the σ meson couples to, the signs of the couplings for both quark and antiquark are the same. This explains the universally attractive nature of the σ meson exchange. * In comparison with the σ exchange, the ω meson couples to the baryonic charge which flips its sign for quark and antiquark. This provides a repulsive interaction between quarks and attractive interaction between the quark and antiquark. * The role of Λ is to suppress the interaction strength for larger momentum transfer and thus effectively reduce the strength of the interaction for bound states. As we will see, the results depend very much on the choice of the form factor.For Λ_c D̅, the interaction is the sum of attractive OSE and repulsive OOE, with the total is some attractive. As Λ is increased, the OSE becomes more prominent, and a bound state appears for Λ > 1.1 GeV for case I. For cases II and III, because of slightly stronger ω exchange repulsion, we do not find any bound states. These are the results for S waves. For higher partial waves, due to the repulsive centrifugal force, l(l+1)/2Mr^2, it is less likely to have bound states. Thus, we may conclude that in our model with a reasonable Λ∼ 1 GeV, hidden-charm molecular pentaquarks made up by Λ_c D̅ are not likely to exist. Indeed, if we increase Λ larger than 1 GeV when more attraction is expected, we do not yet find bound states or do, at most, very weakly bound states only for case I. Experimentally, our conclusion for the Λ_cD̅ system is consistent with the current results of LHCb <cit.>, where no obvious evidence of possible partners of P_c(4380) and P_c(4450) has been reported, in the region close to the mass of Λ_cD̅. As compared to the Λ_cD̅ system, the OOE force for the Λ_cD system is attractive, as explained above. Together with the attractive OSE force, the net attractive force for the Λ_cD turns out to be strong enough to accommodate bound states. As shown in Table <ref>, for the cutoff Λ∼ 1 GeV, we find a shallow bound state with a binding energy around several MeV. Therefore, this channel may provide a good candidate of a loosely bound molecular state of the Λ_cD system with |^2S_1/2⟩. Since D and Λ_c are the lowest ground hadrons of the charmed mesons and baryons, its possible strong decay channel should be rather limited, like Ξ_cc(1/2^+)+π/η.For the heavy baryon systems Λ_cΛ_c(Λ̅_c), since one more light quark (antiquark) is in the baryon Λ_c(Λ̅_c), the interaction strength becomes two times stronger than that in the Λ_cD̅ and Λ_cD systems. Therefore, as shown in Table <ref>, more bound state solutions have been found both for Λ_cΛ_c and Λ_cΛ̅_c systems than for Λ_cD̅ and Λ_cD systems. With the same cutoff input, their binding energies reach several tens MeVs. Thus, they can be also possible molecular candidates. For their decay behaviors, the Ξ_cc(1/2^+)N can be the only strong decay channel for the S-wave Λ_cΛ_c bound state. The decay processes will be much more complicated for the S-wave Λ_cΛ̅_c molecular state, as they include open-charm and hidden-charm channels, like χ_c0+ππ, DD̅_1+π, and so on. §.§ Solutions with noncovariant-type monopole form factor ℱ_M(q^2) = Λ^2/(Λ^2-q^2) So far, we have employed a covariant monopole form factor and discussed the role of OSE and OOE potentials, with some predictions for molecular candidates, the S-wave Λ_cD, Λ_cΛ_c, and Λ_cΛ̅_c. In order to further see our discussions, in the following, we attempt to use a three-momentum form factor of the form of ℱ(q^2)=Λ^2/(Λ^2-q^2), which is often adopted in nuclear physics.In the nonrelativistic kinematics, the energy transfer is neglected, and so this condition reduces to the condition of vanishing three momentum. In fact, the difference of this form factor from the monopole form factor is absorbed into the redefinition of the coupling constants as Eqs. (<ref>) and (<ref>)f_σ =f'_σ = (1-m_σ^2/Λ^2)g_σ = (1-m_σ^2/Λ^2)g'_σ,f_ω =f'_ω = (1-m_ω^2/Λ^2)g_ω = (1-m_ω^2/Λ^2)g'_ω.If we use the same coupling constants and cutoff Λ, the interaction strengths are larger when the three-dimensional form factor is employed. Therefore, to obtain loosely bound molecular states, we need to use smaller cutoff Λ when the coupling constants are kept unchanged. This is the reason that we show the results in Table <ref> with smaller cutoff Λ.In order to determine the value of cutoff in noncovariant-type monopole form factor, here, we recall the relation,⟨ r^2 ⟩ =-6.∂ℱ(q^2)/∂ q^2|_q^2→ 0≈6/Λ^2.If the form factor ℱ(q^2) is introduced, in practice, the resulting cutoff parameter for ℱ turns out to be around 0.5 GeV as we discussed around Eq. (7), consistent with typical hadronic size. The results are shown in Table <ref> for Λ∼ 0.4, 0.5, and 0.6 GeV. Compared with the numerical results in Table <ref>, one can find that, if we take a value of Λ=0.5 GeV, which is estimated by Eq. (<ref>), the results in Table <ref> are very similar to those in Table <ref>. Having these results together with those of different form factors, we find that the intermediate-range and short-range force from OSE and OOE models provides a strong attraction to generate bound states. Finally, let us give a brief conclusion, where we show the results with the two form factors ℱ_M (Λ∼ 1 GeV) in Table <ref> and those with ℱ(Λ∼ 0.5 GeV) in Table <ref>. To be seen shortly, these results are qualitatively similar but have some differences quantitatively. The latter indicates uncertainties of the present model calculations. Nevertheless, we can predict several possible candidates for molecular states, S-wave Λ_cD, Λ_cΛ_c, and Λ_cΛ̅_c states. § EXTENSION§.§ Mass dependence In addition to the effective potentials, the mass in the kinetic term is another important input for the discussion of bound states. In fact, in the heavy-quark limit, (M→∞) as the kinetic energy p^2/2M vanishes, hadrons will be more easily bound. In the following, we study the reduced mass dependence of the molecular systems.The upper panel of Fig. <ref> shows the binding energies of the Λ_c D̅-like (dashed line) and Λ_c D-like (solid line) states, where the reduced mass of the two hadrons is varied as in the horizontal axis. The vertical dotted lines correspond to the reduced masses of the two hadrons, as indicated in the figure. The solid line stands for the binding energies of the Λ_c D-like state; it starts to appear when the reduced mass becomes larger than ∼ 0.75 GeV, and as expected, the binding energy increases as the reduced mass is increased. For the Λ_c D̅-like state, the OOE potential is repulsive, resulting in less attractive potential than for the Λ_c D-like state. Thus, the system allows weaker binding as the dashed line shows. The lower panel shows similar results for the Λ_c Λ_c- and Λ_c Λ̅_c-like states. Because these systems have a larger attraction as proportional to the number of the light quarks as compared with the Λ_c D and Λ_c D̅ ones, larger binding energies are obtained.When the reduced mass is sufficiently heavy, as in the charm and bottom regions but not in the strange regions, binding energies for Λ_c D and Λ_c D̅ systems reach several to several tens MeV. Thus, the heavy flavors of charm and bottom are important in stabilizing pentaquark hadronic molecules. For the Λ_cΛ_c(Λ̅_c)-like systems, with stronger OSE and OOE interactions, more bound solutions are obtained even in the strangelike section. Therefore, the heavy dibaryon molecules can be more stable than the heavy pentaquark molecules. These results suggest that searching for the heavy dibaryon molecules is very promising in experiments.§.§ Λ_cN and Λ_cN̅ systems In this subsection, let us study Λ_c N (N̅) systems. In fact, the Λ_c N (N̅) interactions have been investigated <cit.>. In particular, a very shallow bound state was found for the S-wave Λ_cN system <cit.>. According to Eqs. (<ref>) and (<ref>), the total effective potentials for Λ_cN, and Λ_cN̅ systems are written as𝒱_Λ_cN(r)=-6g_σ^2Y(Λ,m_σ,r)+6g_ω^2Y(Λ,m_ω,r), 𝒱_Λ_cN̅(r)=-6g_σ^2Y(Λ,m_σ,r)-6g_ω^2Y(Λ,m_ω,r). The bound solutions for the S-wave Λ_c,bN(N̅) systems are summarized in Table <ref>. When we take the cutoff around 1 GeV, their binding energy reaches a few to several tens MeV, and their root-mean-square radii are around 1 fm. This means that they also can be possible molecular candidates. For Λ_cN and Λ_bN states, if they form bound states, they are stable under the strong interaction, while the Λ_cN̅ and Λ_bN̅ states can decay to charmed/antibottomed meson and the light mesons, like Dππ and B̅ππ.§ CONCLUSION AND DISCUSSION Stimulated by the observation of X/Y/Z/P_c states near threshold, the study of the hadronic molecular picture becomes more and more essential. For the study of molecular states, it is essentially important to describe the interaction between the hadrons of the molecular system. For this purpose, currently, the most available approach is the one-boson-exchange model based on the knowledge of light quark and boson interactions, which is applied to the system of open heavy hadrons containing light quarks. In this work, we systematically study the properties of the interaction from the one-σ/ω-exchange model. The Λ_cD(D̅), Λ_cΛ_c(Λ̅_c) systems have been taken into consideration. Meanwhile, all the parameters are estimated by the quark model.In fact, the intermediate- and short-range interactions between hadrons are from a many pions exchange process. Here, σ and ω exchanges are adopted to approximately replace two and three pion exchanges interactions, respectively. Compared to a pion, σ meson is of uncertain mass and wide width, which affects the strength of the σ-exchange interaction. In the limit of small momentum transfer, the effective potential from σ exchange is proportional to the term of g_σ^2/m_σ^2. Therefore, the uncertainty in the mass and wide width may be absorbed into the redefinition of the coupling constant. In nuclear physics, the mass of σ is often taken as a fixed input parameter to fit the phase shift of nucleon-nucleon interaction <cit.>.By working out suitable coupling constants and form factors, we find that in many cases the sum of OSE and OOE models become attractive, where the OSE plays the dominant role. The OSE force is always attractive and the dominant. Whereas, for the OOE force, it is repulsive when there exists q-q (q=u,d) or q̅-q̅ combinations in the two hadrons system. The OSE and OOE interaction strength depends on the number of q-q, q̅-q̅, and q-q̅ combinations. With reasonable inputs for the cutoff parameter for the form factor, we find that the interaction of the OSE and OOE models provides attraction which may form the heavy molecular states, like the S-wave Λ_cD, Λ_cΛ_c, and Λ_cΛ̅_c states. As a by-product, we also discuss the mass dependence of the binding energy. We have explicitly shown that heavier systems are more likely to accommodate various molecular states such as S-wave Λ_bD(D̅), Λ_cB(B̅), Λ_bB(B̅), Λ_cΛ(Λ̅), Λ_bΛ(Λ̅), Λ_bΛ_c(Λ̅_c), and Λ_bΛ_b(Λ̅_b) states. Finally, the interaction between a heavy baryon Λ_c,b and one nucleon has been investigated. In our calculation, there can exist S-wave Λ_cN(N̅) and Λ_bN(N̅) molecular states. § ACKNOWLEDGMENTS This project is partly supported by the National Natural Science Foundation of China under Grants No. 11222547, No. 11175073, and No. 11647301, and the Fundamental Research Funds for the Central Universities. X. L. is also supported in part by the National Program for Support of Top-notch Young Professionals. A. H. is supported in part by Grants-in-Aid for Scientific Research [Grant No. JPK05441(c)]. 99 Choi:2003ue S. K. Choi et al. [Belle Collaboration], Observation of a narrow charmonium - like state in exclusive B^±→ K^±π^+ π^- J / ψ decays, Phys. Rev. Lett.91, 262001 (2003). 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Zhu, Spin-orbit force, recoil corrections, and possible B B̅^* and D D̅^*molecular states, Phys. Rev. D 89, no. 9, 094026 (2014). Ikeda:2016zwx Y. Ikeda et al. [HAL QCD Collaboration], Fate of the Tetraquark Candidate Z_c(3900) from Lattice QCD, Phys. Rev. Lett.117, 242001 (2016) Tornqvist:1993vu N. A. Tornqvist, On deusons or deuteron - like meson meson bound states, Nuovo Cim. A 107, 2471 (1994). Tornqvist:1993ng N. A. Tornqvist, From the deuteron to deusons, an analysis of deuteron-like meson meson bound states, Z. Phys. C 61, 525 (1994). Wu:2010jy J. J. Wu, R. Molina, E. Oset and B. S. Zou, Prediction of narrow N^* and Λ^* resonances with hidden charm above 4 GeV, Phys. Rev. Lett.105, 232001 (2010). Karliner:2016ith M. Karliner and J. L. Rosner, Exotic resonances due to η exchange, Nucl. Phys. A 954, 365 (2016). Chen:2016ryt R. Chen, J. He and X. Liu, Possible strange hidden-charm pentaquarks from Σ_c^(*)D̅_s^* and Ξ^(',*)_cD̅^* interactions, arXiv:1609.03235 [hep-ph]. Meguro:2011nr W. Meguro, Y. R. Liu and M. Oka, Possible Λ_cΛ_c molecular bound state, Phys. Lett. B 704, 547 (2011) Li:2012bt N. Li and S. L. Zhu, Hadronic Molecular States Composed of Heavy Flavor Baryons, Phys. Rev. D 86, 014020 (2012) Lee:2011rka N. Lee, Z. G. Luo, X. L. Chen and S. L. Zhu, Possible Deuteron-like Molecular States Composed of Heavy Baryons, Phys. Rev. D 84, 014031 (2011) Riska:1999fn D. O. Riska and G. E. Brown, Two pion exchange interaction between constituent quarks, Nucl. Phys. A 653, 251 (1999). Rijken:1998yy T. A. Rijken, V. G. J. Stoks and Y. Yamamoto, Soft core hyperon - nucleon potentials, Phys. Rev. C 59, 21 (1999). Riska:2000gd D. O. Riska and G. E. Brown, Nucleon resonance transition couplings to vector mesons, Nucl. Phys. A 679, 577 (2001). Klempt:2002ap E. Klempt, F. Bradamante, A. Martin and J. M. Richard, Antinucleon nucleon interaction at low energy: Scattering and protonium, Phys. Rept.368, 119 (2002).Olive:2016xmw C. Patrignani et al. [Particle Data Group], Review of Particle Physics, Chin. Phys. C 40, 10, 100001 (2016).Dover:1977jw C. B. Dover and S. H. Kahana, Possibility of Charmed Hypernuclei, Phys. Rev. Lett.39, 1506 (1977).Liu:2011xc Y. R. Liu and M. Oka, Λ_c N bound states revisited, Phys. Rev. D 85, 014015 (2012)Meng:2017udf L. Meng, N. Li and S. l. Zhu, Possible hadronic molecules composed of the doubly charmed baryon and nucleon, arXiv:1707.03598 [hep-ph].Maeda:2015hxa S. Maeda, M. Oka, A. Yokota, E. Hiyama and Y. R. Liu, A model of charmed baryonCnucleon potential and two- and three-body bound states with charmed baryon, PTEP 2016, 023D02 (2016). Machleidt:1987hj R. Machleidt, K. Holinde and C. Elster, The Bonn Meson Exchange Model for the Nucleon Nucleon Interaction, Phys. Rept.149, 1 (1987). Backman:1984sx S. O. Backman, G. E. Brown and J. A. Niskanen, The Nucleon Nucleon Interaction And The Nuclear Many Body Problem, Phys. Rept.124, 1 (1985). | http://arxiv.org/abs/1707.08306v3 | {
"authors": [
"Rui Chen",
"Xiang Liu",
"Atsushi Hosaka"
],
"categories": [
"hep-ph"
],
"primary_category": "hep-ph",
"published": "20170726071009",
"title": "Heavy molecules and one-$σ/ω$-exchange model"
} |
[email protected] Science Institute, University of Iceland, Dunhaga 3, IS-107 Reykjavik, IcelandPhysics Department, College of Science, University of Sulaimani, Kurdistan Region, IraqScience Institute, University of Iceland, Dunhaga 3, IS-107 Reykjavik, Iceland Department of Theoretical Physics, Wrocław University of Science and Technology, 50-370 Wrocław, [email protected] Department of Physics and Center for Theoretical Sciences, National Taiwan University, Taipei 10617, Taiwan Center for Quantum Science and Engineering, National Taiwan University, Taipei 10617, [email protected] Department of Mechanical Engineering, National United University, Miaoli 36003, [email protected] School of Science and Engineering, Reykjavik University, Menntavegur 1, IS-101 Reykjavik, IcelandWe calculate the current correlations for the steady-state electron transport through multi-level parallel quantum dots embedded in a short quantum wire, that is placed in a non-perfect photon cavity. We account for the electron-electron Coulomb interaction, and thepara- and diamagnetic electron-photon interactions with a stepwise scheme of configuration interactions and truncation of the many-body Fock spaces. In the spectral density of the temporal current-current correlations we identify all the transitions, radiative and non-radiative, active in the system in order to maintain the steady state. We observe strong signs oftwo types of Rabi oscillations. Current correlations for the transport of interacting electronsthroughparallel quantum dots in a photon cavity Andrei Manolescu December 30, 2023 ======================================================================================================================== § INTRODUCTIONExperiments<cit.> in which the electron transport through nanoscale electronic systems placed in photon cavities, and model calculations<cit.>thereof, are gaining attention in the last years.Due to small size of the electronic systems the constant average current through the system inthe steady state does not convey much information about the underlying processes, and one might expect information about radiative transitions to be lost at that time scale, or not detectable.<cit.> In order to remedy this situation researchers have realized that the noise power spectrum, or the noise power spectral density of a system calculated through theFourier transform of the current-current two-time correlation function can be measured experimentally.<cit.> Many theoretical researchers have used this to calculate the noise spectral density for electron transport through model systems in different situations using, for example, non-equilibrium Green functions,<cit.> Markovian masterequation in the steady state,<cit.> or non-Markovian master equations in the transientregime,<cit.> just to mention very few.Complementary to the calculation of the noise power spectral densities of the charge current transport through electron systems on the nanoscale, the calculation of the power spectral propertiesof photon emission statistics of cavities with embedded electron systems has been undertaken by manymore theoretical groups.<cit.> Recently, we have investigated the photon correlations in the emission radiation from a photon cavity containing a short quantum wire with embedded two parallel quantum dots through which a steady state current is driven with a bias difference between two external leads.<cit.> There, the spectral density of the fluctuations in the radiation can be used to differentiate between the the conventional and the ground state electroluminescence in the strong electron-photon coupling regime.<cit.> Here, we will demonstrate that in this complex interacting many-state system, the power spectral density of the temporal current-current correlations can be used to identify theunderlying processes, the transitions between interacting many-body states of cavity-photon dressedelectron states, that contribute to maintaining the system in its steady state.§ MODEL We consider a short two-dimensional GaAs quantum wire with length L=150 nm placed in a photon cavity. We use the 36 lowest in energy single-electron states of the wire, |i⟩, to build a many-electron Fock space of 0-3 Coulomb interactingelectrons, |μ ). The potential defining the short quantum wire with two parallelquantum dots displayed in Fig. <ref> is V(x,y) = [1/2m^*Ω_0^2y^2 +eV_g. + . V_d∑_i=1^2exp{-(β x)^2+β^2(y-d_i)^2}] × θ(L_x/2-|x|) with ħΩ_0 = 2.0 meV, V_d = -6.5 meV, β = 0.03 nm^-1,d_1=-50 nm, d_2=+50 nm, L_x = 150 nm, and θ is the Heavisidestep function. The plunger gate voltage V_g is used to move the states of the system up or down with respect to the bias window defined by the external leads to be describe below. We use as a kernel for the mutual electron-electron Coulomb interaction V_Coul(𝐫-𝐫') = e^2/κ_e√(|𝐫-𝐫'|^2+η_c^2), with a small regularizing parameter η_c/a_w=3× 10^-7 (a_w being defined below),and for GaAs parameters we assume κ_e =12.4, m^*=0.067m_e, and g^*=-0.44.In terms of field operators the Hamitonian of the central system is H_S = ∫ d^2r ψ^† (𝐫){π^2/2m^*+ V(𝐫)}ψ (𝐫) + H_EM + H_Coul - 1/c∫ d^2r 𝐣(𝐫)·𝐀_γ -e/2m^*c^2∫ d^2r ρ(𝐫) A_γ^2, withπ=(𝐩+e/c𝐀_ext), where 𝐀_ext is a classical vector potential producing an external homogeneous small magnetic field B=0.1 T along the z-axis, perpendicular to the plane of the two-dimensionalquantum wire, inserted to break the spin and the orbital degeneracies of the states in order to enhancethe stability of the results. The first term in the second line of Eq. (<ref>) is the paramagnetic, and the second term the diamagnetic, electron-photon interaction.The external magnetic field, B, and the parabolic confinement energy of the leads and thecentral system ħΩ_0=2.0 meV, together with the cyclotron frequencyω_c=(eB)/(m^*c) lead to an effective characteristic confinement energyħΩ_w=ħ(ω_c^2+Ω_0^2)^1/2, and an effective magnetic lengtha_w=(ħ /(m^*Ω_w))^1/2. This characteristic length scale assumes approximately the value 23.8 nm for the parameters selected here. In terms of the cavity photon creation and annihilationoperators, a^† and a, the Hamiltonian for the single cavity photon mode isH_EM=ħω a^† a, with energy ħω.We assume a rectangular photon cavity (x,y,z)∈{[-a_c/2,a_c/2] × [-a_c/2,a_c/2]× [-d_c/2,d_c/2]}with the short quantum wire centered in the z=0 plane. In the Coulomb gauge the polarization of the electric field parallel to the transport in the x-direction (with the unit vector 𝐞_x) is accomplished in the TE_011 mode,or perpendicular (defined by the unit vector 𝐞_y) in theTE_101 mode.The two versions of the quantized vector potential for the cavity field are in a stacked notationexpressed as 𝐀_γ (𝐫)=(𝐞̂_x 𝐞̂_y)A{a+a^†}(cos(π y/a_c)cos(π x/a_c)) cos(π z/d_c), for the TE_011 and TE_101 modes, respectively. The strength of the vector potential, A, determines the coupling constant g_EM = e AΩ_wa_w/c, here set to 0.05 meV, or 0.10 meV, leaving a dimensionless polarization tensor g_ij^k = a_w/2ħ{⟨ i|𝐞̂_k·π|j⟩ + h.c.}. The coupling of the central system to the leads is described by the Hamiltonian H_T = θ (t)∑_il∫ d𝐪(T_𝐪i^l c_𝐪l^† d_i + (T_𝐪i^l )^* d_i^† c_𝐪l),where d_i is an annihilation operator for the single-electron state |i⟩ of thecentral system, c_𝐪l an annihilation operator for an electron in lead l∈{L,R}in state |𝐪⟩, with 𝐪 standing for the momentum q and the subband index n_l in the semi-infinite quasi-one dimensional lead. The coupling tensor T_𝐪i^l depends on the nonlocal overlap of the single-electron states at the internal boundaries in the central system and the respectivelead.<cit.> This setup is intended for a weak tunneling coupling of the central system with the leads, but allows allows for full coupling between the quantum dots and the rest of the central system, like in a scattering approach.<cit.> The remaining overall coupling constant to the leads is g_LRa_w^3/2=0.124 meV, in the weak coupling limit used here.As we are interested in the properties of system in the steady state here, we transform anon-Markovian master equation built according to the projection formalism of Nakajima<cit.> and Zwanzig<cit.> to a Markovian equation<cit.> for the reduced density operatorof the central system ∂_tρ_S(t) =-i/ħ[H_S,ρ_S(t)] -{Λ^L[ρ_S ;t]+Λ^R[ρ_S ;t]}-κ/2(n̅_R+1){2aρ_Sa^† - a^† aρ_S - ρ_Sa^† a}-κ/2(n̅_R){2a^†ρ_Sa - aa^†ρ_S - ρ_Saa^†} , where the last two terms in the first line describe the “dissipation” caused by the Left and Right leads. The dissipation terms are constructed with terms up to second order in the coupling Hamiltonian (<ref>), but without resorting to the rotating wave approximation, as more than one resonancewith the photon field can be active to some extent in the system for each set of parameters used inthe calculations. The dissipation terms in Eq. (<ref>) are<cit.> Λ^l[ρ_S ;t]= 1ħ^2∫ dϵ D^l(ϵ)θ(t) {[ τ^l, Ω[ρ_S] ] + h.c.} with θ (t) the Heaviside unit step function, τ^l the many-body version of the coupling tensor of lead l, and Ω[αβ][][ρ_S] = {ℛ[ρ_S (t)][αβ][]-𝒮[ρ_S (t)][αβ][] }δ[][βα], whereδ[][βα] = δ(E_β - E_α - ϵ). The density of states in leads l is D^l(ϵ) = |d𝐪/dϵ|, and we have defined the superoperators 𝒮[ρ_S]=Sρ_S , ℛ[ρ_S]= ρ_S R, from R= π (1 - F^l)(τ^l)^†and S= π F^l (τ^l)^† , with F^l being the equilibrium Fermi distribution in lead l. The last two lines of Eq. (<ref>) describe the Markovian photon decay of a non-perfect cavitywith an overall decay constant, κ, and a mean value of photons in the reservoir n̅_R. The non-interacting electron gas in the leads is at temperature T=0.5 K, correspondingto the thermal energy k_BT≈ 0.043 meV.The charge and the charge-current density operators of the central system areρ = -eψ^†ψ, 𝐣 = -e/2m^*{ψ^†(πψ)+(π^*ψ^†)ψ}. Due to the structure of the master equation (<ref>) the time-dependent average current from the left lead into the centralsystem, and the current from it into the right lead can be calculated as I_l(t) = Tr_S{Λ^l[ρ_S;t]Q}, l∈{L,R}, where Q=-e∑_id^† d_i is the charge operator of the central system. The current-current correlation functions are best written for the corresponding operators in the Heisenberg picture D_ll'(τ ) = ⟨ I_l(τ )I_l'(0)⟩ ,τ > 0, and for a calculation of it in the steady state we redefine the time point t=0 torefer to any time at which the system has reached its steady state. In the time domain a more convenient correlation function is S_ll'(τ ) = ⟨δ I_l(τ )δ I_l'(0)⟩ /I(0)^2, where δ I_l(τ ) = I_l(τ )-⟨ I_l(τ ) ⟩, and the two functions are related via S_ll'(τ ) = D_ ll'(τ )/I(0)^2 -1, as in the steady state ⟨ I_l(τ )⟩ = ⟨ I_l'(0)⟩ = I(0).Despite the simple look of Eq. (<ref>) one realizes that the construction of a current operator is not straight forward having in mind that the dissipation terms, Λ^l[ρ_S;t], have the reduced density operator to the left, the right, or sandwiched between system operators. The solution is to use the concept of superoperators (of which the Liouville operator is one) or go one step further and use a Liouville space representation.<cit.> We take the latter option and express the mean value of the current as I_l(t) = Tr_S{ Q[ Λ^l( ρ_S(t))_vec]_Mat}, where Λ^l is a N_mes^2× N_mes^2 dimensional matrix in Liouville space representing the dissipation, the “vec” operation stacks the N_mes columns of the matrix representing ρ_S in the Fock space into a vector in Liouville space,and the “Mat” operation reverses that procedure. N_mes=120 is the number of many-body states in our Fock basis ofcavity-photon dressed electron states. Expression (<ref>) suggests using QΛ^l· as the current operatorand the Quantum Regression Theorem (QRT)<cit.> that is valid in the Markovianlimit for weak system-leads coupling.<cit.> The QRT states that the equation of motion for the two-time correlation function is of the same form as the Markovian master equation for the reduced density operator of the system,but for an effective density operator,<cit.> which for the current correlation isχ^l (τ ) = Tr_R{ e^-iHτ /ħQΛ^lρ(0)e^+iHτ /ħ} , with H the Hamiltonian of the total system, ρ (0) its density operator after the on-set of the steady state, redefinig that point of time to be t=0.Tr_R is the trace operator with respect to the variables of the reservoir. The two-time average or the correlation function is then D_ll'(τ )=⟨ I_l (τ ) I_l'(0)⟩ = Tr_S{ I_l'(0) χ^l (τ ) } , whereTr_S is the trace operation with respect to the state space of the central system.§ RESULTS FOR ONE ELECTRON GROUND STATE AT 𝐕_𝐠=2.0 𝐦𝐕 We use two cases for different values of the plunger gate voltage V_g to show how Rabi resonances influence and turn up in thesteady state properties of the system in different ways.We select a rather narrow bias window with μ_L=1.4 meV, and μ_L=1.1 meV, and investigate the current-current correlations for two different cases in the steady state. For V_g=+2.0 mV, when only the two spin components of the one-electron ground state are within the bias window. In this case the photon energy is selected to be ħω =0.72 meV to promote a Rabi resonance between the one-electron ground state and the first excitation thereof. The properties of the 32 lowest in energy many-body states of the system are displayed in Fig. <ref> for the case of a y-polarized photon field. As we have discussed earlier, the symmetry properties of the states of parallel quantum dots lead to a large Rabi resonance for the y-polarization caused by the paramagneticelectron-photon interaction, but a very small resonance for the x-polarized field that is only caused by the diamagnetic part of the electron-photon interaction.<cit.>These two resonances are shown in Fig. <ref>.The spectral densities of the current-current correlations, D_ll'(E), are displayed inFig. <ref>, and the identityof the main peaks for the case of y-polarized photons (in the lower panel ofFig. <ref>) is listed in Table <ref>.Not surprisingly, the almost degenerate two spin components of the one-electron ground state |0̆3̆) and |0̆4̆), the only states placed in the bias window, are the initial states for all transitions. The first two lines in Table <ref> refer to transitions from both spin components of the one-electron ground state, |0̆3̆) and |0̆4̆), to the Rabi-split first excitation thereof, {|0̆6̆)=R^-_↓, |0̆7̆)=R^-_↑}, and {|0̆8̆)=R^+_↓,|0̆9̆)=R^+_↑}, for photon energy ħω=0.72 meV.For V_g=2.0 mV there are no electronic states of the central system below the bias window, and the two next lines in Table <ref> identify transitions to higher order states of the Rabi resonance, in the sense that the pairs {|1̆1̆), |1̆2̆)}, and{|1̆3̆), |1̆4̆)} have a mean photon number in the range 1 to 2. The last line in Table <ref> is for the last peak easily visible in the lower panel of Fig. <ref>which is caused by a transition to the states |1̆6̆), and |1̆7̆), that only have a very small photon component. This last fact conforms to that the last peak has the same size and location in theupper and lower panel of Fig. <ref>, i.e.it is independent of the electron-photon coupling strength, g_EM.An important point to notice is that the energy distance of the first two peaks reflects directly theRabi-splitting as the electron-photon coupling is increased from the upper to the lower panel in Fig. <ref>. The spectral density of the photon-photoncorrelation reveals three peaks,<cit.> the so-called Mollowtriplet,<cit.> but, here, in the spectral density of the current-current correlations there are only two peaks.For an x-polarized cavity photons the Rabi splitting is much smaller, of the same order as the spin splitting in GaAs for B=0.1 T, and a careful inspection of the data shows the Rabi resonance peak starting to split into two parts.§ RESULTS FOR TWO-ELECTRON GROUND STATE AT 𝐕_𝐠=0.5 𝐦𝐕 Now, we turn to a very different case in our system, by reducing the plunger gate voltage to V_g=0.5 mV. With the same bias window as before, we have only the two-electron ground state within it, a singlet, and might expect similar phenomena taking place as when only the one-electron ground state was within the bias window. Before analyzing the results, we remind the reader that there are one-electron states below the bias window, and in the weak coupling limit with only sequential tunneling between the leads and the central system the current through a two-electron state is very low. A third very important fact is that in a multi-state system, even though the photon energy is tuned close to a certain resonance, there can always be other weaker, more detuned, resonances at play in the system. This last fact is also a very good reason to include both the para- and the diamagnetic electron-photon interactions in the model.In order to analyze the results, Fig. <ref> displays the properties of the 36 lowest in energy many-body eigenstates of the closed central systemfor the case of an x-polarized cavity photon field. It is proper here to remind the reader that thenumbering of the photon-dressed electron states changes as the plunger gate voltage is changed.The plunger gate voltage is set at V_g=0.5 mV, and the two-electron ground state |0̆6̆) is coupled to the first excitation thereof by selecting the photon energy ħω =2.0 meV, resulting in the Rabi split states |2̆3̆) and |2̆4̆). Due to the low current through two-electron states, the one-electron states just above the bias window, |0̆7̆) and |0̆8̆), play a key role in the transport through the system.For the x-polarized cavity photon field these states are thelower energy states in Rabi split pairs with the states |0̆9̆) and |1̆0̆) as the upper states, for photon energy E_EM=ħω≈ 1.8 meV as can be seen in Fig. <ref>.This splitting is only strong for the x-polarization as the underlying electronic states have odd parity in the x-direction,but even parity in the y-direction.<cit.> The spectral density of the current-current correlations, D_ll'(E), is shown inFig. <ref>,and the peaks are correlated with transitions in the central system in Table <ref>. The first and the third transitions in Table <ref> are between initial and final states with different mean photon number, radiative transitions.The energy of these transitions depends thus on the electron photon coupling, g_EM, but the second and the last four transitions are independent of this coupling as the photon component of the final and initial states is low, mainly non-radiative transitions. This observation has to qualified with the fact that states |0̆7̆) and|0̆8̆) represent the lower branch of Rabi split pairs that are quite detuned leaving a only a small photon component in them.The slight occupation of the one-electron states just above the bias window, |0̆7̆) and |0̆8̆),(see Fig. <ref>) leads to strong Rabi-oscillations in the current correlations that is manifested by a dominant peak in its spectraldensity at the energy of the Rabi splitting, 0.2428 meV shown in Fig. <ref>. For V_g=2.0 mV, we had transitions from the one-electron ground state to the Rabi branches, but here at V_g=0.5 mV we observean oscillation between the two Rabi states enabled by the special location of the states with respect to thebias window, even when the photon energy of the cavity, ħω =2.0 meV, is considerably detuned from theRabi resonance at 1.8 meV. In Figs. <ref> and <ref> we see transitions that lead to the steady state occupation of both spin components of the one-electron ground state, |0̆1̆) and |0̆2̆), and other states below the bias window. In addition, we see a slight occupation of the lowest in energy spin-triplet two-electron states, |1̆4̆), |1̆5̆) and |1̆6̆), especially for the y-polarizedphoton field. Only a tiny occupation of the Rabi split two-electron states |2̆3̆) and |2̆4̆)that are in resonance with the two-electron ground state |0̆6̆) can be seen inFig. <ref>. § SUMMARY We have used the current-current correlation spectral density of a multi-state electron system placed in a photon cavity to calculate which transitions areactive in the system in its steady state. The central system is weakly coupled to the external leads, but in it the electrons couple strongly to the cavity photons. In order to account for the influence of the geometry on the electron transport through the system we have had to include the electron-electron Coulomb interaction and both the electron-photon para- and diamagnetic interactions with numericaldiagonalization in large many-body Fock spaces.<cit.>In a many-state system we find that it might be difficult to isolate individual resonances andthus we do not use the rotating wave approximation for the electron-photon interactions.In order to effectively describe the transport in a system with diverse relaxation constants through many orders of magnitude for the time variable we have mapped a non-Markovian master equation into a Markovian master equation inLiouville space.<cit.> We have selected the decay constant of the photon cavity, κ, (the coupling to the photon reservoir) to be of the same order of magnitudeas the main relaxation channels of the electronic transitions to, or from, the leads.For the one-electron ground state inside the bias window defined by the two external leads, we identify strong transitions to the Rabi split states of the first excitation of the ground state for an appropriate photon energy. Several other weaker transitions are seen in this case. In addition, we identify a transition to a higher order Rabi split state.For the two-electron ground state within the bias window and photonenergy coupling it to its first excitation, we see a neighboring Rabi resonance for one-electron states just above the bias window playing a strong role in the electron transport.In this case we also identify a transition between the two Rabi branches as they gain a slight occupancy in the steady stead.It is important to notice that in the noise spectral density for the current-current correlation function we are able to identify both the radiative and the non-radiative many-body transitions active in the system maintaining its steady state. The height of the spectral peaks gives the weight or the strength of the different transitions, and their character, whether they are radiative or not can be found by varying slightly the electron-photon coupling. The peaks representing non-radiative transitions are stationary under that variation.The current noise power spectra are thus an important quantity to measure in experiments on the systems to analyze their dynamics.Our results point out, the importance of, and the opportunities in using the interplay of geometry and photon polarization in transport of electrons through a nanoscale electron system in a photon cavity. The double parallel quantum dots is the simplest system offering clean separation of effects with its clear anisotropy.This work was financially supported by the Research Fund of the University of Iceland, the Icelandic Research Fund, grant no. 163082-051,and the Icelandic Instruments Fund. HSG and CST acknowledge support from Ministry of Science andTechnology of Taiwan, under grant No. 103-2112-M-002 -003 -MY3, No. 106-2112-M-002 -013 -MY3, No.103-2112-M-239-001-MY3, and No. 106-2112-M-239-001-MY3.apsrev4-135 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[Bruhat et al.(2016)Bruhat, Viennot, Dartiailh, Desjardins, Kontos, and Cottet]PhysRevX.6.021014 author author L. E. Bruhat, author J. J. Viennot, author M. C. Dartiailh, author M. M. Desjardins, author T. Kontos,and author A. Cottet, 10.1103/PhysRevX.6.021014 journal journal Phys. Rev. X volume 6, pages 021014 (year 2016)NoStop [Delbecq et al.(2011)Delbecq, Schmitt, Parmentier, Roch, Viennot, Fève, Huard, Mora, Cottet, and Kontos]Delbecq11:01 author author M. Delbecq, author V. Schmitt, author F. Parmentier, author N. 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"authors": [
"Vidar Gudmundsson",
"Nzar Rauf Abdullah",
"Anna Sitek",
"Hsi-Sheng Goan",
"Chi-Shung Tang",
"Andrei Manolescu"
],
"categories": [
"cond-mat.mes-hall"
],
"primary_category": "cond-mat.mes-hall",
"published": "20170726061250",
"title": "Current correlations for the transport of interacting electrons through parallel quantum dots in a photon cavity"
} |
http://arxiv.org/abs/1707.08623v3 | {
"authors": [
"Ajit C. Balram",
"J. K. Jain"
],
"categories": [
"cond-mat.str-el"
],
"primary_category": "cond-mat.str-el",
"published": "20170726195308",
"title": "Fermi wave vector for the non-fully spin polarized composite-fermion Fermi sea"
} |
|
Department of Physics, South University of Science and Technology of China, Shenzhen 518055, P. R. China. Institute for Structure and Function & Department of physics, Chongqing University, Chongqing 400044, P. R. China.Department of Physics, South University of Science and Technology of China, Shenzhen 518055, P. R. China.Department of Physics, South University of Science and Technology of China, Shenzhen 518055, P. R. China. Dalian Institute of Chemical Physics,Chinese Academy of Sciences, 116023 Dalian, P. R. ChinaDepartment of Physics, South University of Science and Technology of China, Shenzhen 518055, P. R. China. Department of Physics, South China University of Technology, Guangzhou 510640, P. R. ChinaDepartment of Physics, South China University of Technology, Guangzhou 510640, P. R. China[][email protected] Department of Physics, South University of Science and Technology of China, Shenzhen 518055, P. R. China. Magnetic topological materials have recently drawn significant importance and interest, due to their topologically nontrivial electronic structure within spontaneous magnetic moments and band inversion.Based on first-principles calculations, we propose that chromium dioxide, in its ferromagnetic pyrite structure, can realize one pair of type-II Weyl points between the Nth and (N+1)th bands, where N is the total number of valence electrons per unit cell. Other Weyl points between the (N-1)th and Nth bands also appear close to the Fermi level due to the complex topological electronic band structure. The symmetry analysis elucidates that the Weyl points arise from a triply-degenerate point splitting due to the mirror reflection symmetry broken in the presence of spin-orbital coupling, which is equivalent to an applied magnetic field along the direction of magnetization. The Weyl points located on the magnetic axis are protected by the three-fold rotational symmetry. The corresponding Fermi arcs projected on both (001) and (110) surfaces are calculated as well and observed clearly. This finding opens a wide range of possible experimental realizations of type-II Weyl fermions in a system with time-reversal breaking.73.20.At, 71.55.Ak, 74.43.-fFerromagnetic Type-II Weyl Semimetal in Pyrite Chromium Dioxide H. Xu December 30, 2023 =============================================================== Topologically protected fermions, appearing in semimetallic or metallic systems with nontrivial topology of band structure, provide a realistic platform for the concepts of fundamental physics theory in condensed matter experiments. For instance, the recent discoveries of Weyl semimetals (WSMs) have been attracted considerable attention since they extend the topological classification of mater beyond the insulators and exhibit the exotic Fermi arc surface state <cit.>. In these materials, the valence and conduction bands disperse linearly around special points in three-dimensional (3D) momentum space, called Weyl points (WPs), which construct the discrete point-like Fermi surface. The WP acts as a topological monopole which can be quantified by corresponding chiral charge through calculating the flux of Berry curvature <cit.>. Due to the conservation of chirality, the WPs always appear in pairs of opposite chirality. The topological Fermi arcs are arising from the connection of two projections of the bulk WPs with opposite chiral charges in the surface Brillouin zone (BZ). WSMs and their surface states may lead to unusual spectroscopic and transport phenomena such as chiral anomaly, spin and anomalous Hall effects <cit.>.WPs are twofold degeneracy and only exist in condensed matter systems with breaking either time-reversal or spatial-inversion symmetry. Evidences for Weyl fermions and surface Fermi arc states with breaking spatial-inversion were reported in the non-centrosymmetric TaAs family <cit.> and MoTe_2 <cit.>. WSMs with time-reversal-breaking were also been predicted to exist in several materials <cit.>. Moreover, the existence of two distinct types of WSMs was recently proposed. The type-I WP is associated with a closed pointlike Fermi surface, while the type-II one arises at the boundary of electron and hole pockets <cit.>. Unlike the low-energy excitations in type-I WSMs, the type-II Weyl fermions don't satisfy Lorentz invariance, leaving an open Fermi surface that results in anisotropic chiral anomaly <cit.>. To date several nonmagnetic Type-II WSMs have been predicted <cit.> and observed in MoTe_2 <cit.>, while antiferromagnetic (AFM) YbMnBi_2 are found to be the candidate of type-II WSM with lacking time-reversal symmetry <cit.>. Type-II WSMs co-existing with the ferromagnetic (FM) order have not been reported so far. The novel properties of FM type-II WSMs can look forward to be of great use for spin manipulation and applications in spintronics and magnetic recording devices.In this Letter, based on first-principles calculations, we propose the pyrite chromium dioxide (CrO_2) that can exhibit FM type-II WSM features up to room temperature. It is noteworthy that CrO_2 is a very common material in practice. Rutile CrO2 is a well-studied half-metallic FM material with a high Curie temperature of about ∼390 K <cit.>. Crystalline CrO_2 hosts a number of pressure-induced structural phases <cit.>. The pyrite CrO_2 phase has been demonstrated to be stable FM half-metallic state occurring at a critical pressure of ∼45 GPa <cit.>. As a significant advantage, our calculations show that the magnetism in pyrite CrO_2 is "soft", indicating that an extra magnetic field can easily change the direction of magnetization. Some similar phenomena are observed in magnetic Heusler alloys <cit.>. As a result, the number and position of WPs depend on the magnetic symmetry, and then the topological features in pyrite CrO_2 can be manipulated.Our symmetry analysis elucidates that the WPs in pyrite CrO_2 arise from a triply-degenerate point splitting in the presence of spin-orbital coupling (SOC), which is equivalent to an applied magnetic field along the direction of magnetization <cit.>. This finding would provide a realistic and promising platform for investigating FM Weyl physics, especially opening a pathway for studying the quantum anomalous Hall effect in FM WSMs in experiments. We perform the first-principles calculations using the Vienna ab initio Simulation Package (VASP) <cit.> based on density functional theory <cit.>. The core-valence interactions are treated by the projector augmented wave (PAW) <cit.> pseudopotentials with 4p^6 4s^1 3d^5 and 2s^2 2p^4 valence electron configurations for Cr and O, respectively. SOC effect is included in the pseudopotentials, and the exchange-correlation potential is chosen as generalized gradient approximation (GGA) with the Perdew-Burke-Ernzerhof (PBE) formalism <cit.>. A plane-wave-basis set with kinetic-energy cutoff of 600 eV has been used. The full Brillouin zone(BZ) is sampled by 21×21×21 Monkhorst-Pack grid in self-consistent calculations <cit.>.Because of the strongly correlated effects of 3d electrons in Cr, we must consider the GGA+U calculations to describe the on-site Coulomb repulsion beyond the GGA calculations <cit.>. In this work, we find that topological features can be achieved with the range of U from 2.5 eV to 5.0 eV. As a representative, the value of U is chosen to be 3.5 eV to illustrate the band topology, which works well in fitting the half-metallic properties of rutile CrO_2 <cit.>. To calculate the surface states and Fermi arcs, the tight-binding Hamiltonian for Cr 3d and O 2p orbitals is constructed by projecting the Bloch states into maximally localized Wannier functions <cit.>. As it is illustrated in Fig. <ref>(a), the pyrite CrO_2 crystalizes in simple-cubic (SC) lattice with symmorphic space group Pa3̅ (No. 205) that is the semidirect of the point group T_h with the group ℤ^3 generated by three lattice translations. The optimized lattice constant a=4.823 Å, which is good agreement with the previous calculations <cit.>. The crystal structure consists of interpenetrating Cr and O sublattices, which are located at Wyckoff positions 4a (0.0, 0.0, 0.0) and 8c (0.3615, 0.3615, 0.3615), respectively. The SC BZ and the projected surface BZ for both (001) and (110) surfaces are shown in Fig. <ref>(b), in which high-symmetry points are marked.Our first-principles calculations confirm that the FM state of pyrite CrO_2 is considerably more stable that the nonmagnetic and AFM states. The energy of FM state is about 220 meV per Cr atom lower than AFM state, meaning the high Curie temperature above room temperature of this compound. The calculated magnetic moment is ∼2 μ_B per Cr atom, which shows excellent agreement with that in rutile CrO_2 <cit.>. In Figs. <ref>(a) and (b), the electronic band structures without and with SOC are illustrated, respectively. In the spin-polarized calculations without SOC, the spin and the orbitals are independent.The minority spin states exhibit the behavior of semiconductor with band gap ∼2.9 eV, while the majority spin states show the metallic features, resulting in the half-metallic ferromagnetism of pyrite CrO_2. After SOC is present, the calculated band structure in Fig. <ref>(b) shows the SOC has little influence on the electronic structure and the half-metallic ferromagnetism due to the weak SOC strength of both Cr and O. Furthermore, since the symmetry of FM system is strongly affected by magnetization direction, we perform the first-principles total energy calculations to determine the spontaneous magnetization axis. The [111] easy axis is found to be the energetically most favorable magnetization direction. More interestingly, our calculations show tiny energy differences among all magnetic configurations, implying that an applied magnetic field can easily manipulate the topological states along different directions of magnetization. In the main text, we assume the magnetization is along the [111] direction. The topological analysis of the magnetization along [001] directions is presented in Supplemental Material (SM) <cit.>.As shown in Fig. <ref>, the topological band crossing occur on the high-symmetry Γ-R near Γ point. In the absence of SOC, the spatial crystal symmetry have no effect on the spin degree of freedom, and two spin channels are decoupled. The symmetry group is the abelian group T_h, which contains four three-fold rotational symmetry C_3 axes [111], [11̅1], [111̅], and [1̅11]plus inversion I, and three mirror symmetries M_x, M_y, and M_z, respectively. Once SOC is considered, the two spin states couple together and symmetries would decrease depending on the magnetization direction. As the magnetization is along [111] axis, the mirror reflection symmetries M_x, M_y, and M_z are broken. In this case, only the C_3 symmetry around [111] axis and I symmetry are still the magnetic symmetry.We first illuminate the band topology in the absence of SOC. From the magnified band structure along Γ-R (or 𝐤 ∥ [111]) near Γ point in Fig. <ref> (a), we can observe a nodal point below Fermi level ∼8 meV is present. This crossing point is a triply-degenerate node, which originates from the band crossing of a single degenerate band and a two-fold degenerate band <cit.>. In this case, three-fold rotational symmetry along [111] axis C_3^111 and mirror reflection symmetry M=M_x M_y M_z can not commute each other <cit.>. Every momentum point on the Γ-R axis is invariant under the C_3 ^111 and the product (IM) of inversion I and mirror reflection symmetry M, meaning that the Bloch states at each point along this direction are also invariant. Therefore, there is always a single degenerate band and a two-fold degenerate band along Γ-R axis <cit.>. These bands on this axis can be classified by eigenvalues e^± i 2π/3 and 1 of C_3 symmetry. The single degenerate band and two-fold degenerate one can cross each other, and then a pair of triply-degenerate points (TPs) related by I symmetry are present on [111] axis, because the different eigenvalues of C_3 symmetry prevent their hybridization. Considering the T_h symmetry, the TPs also occur in [11̅1], [111̅], and [1̅11] axes.In the presence of SOC, with magnetization along the [111] direction, the symmetry of pyrite CrO_2 is reduced to the magnetic double group S_6 (-3).The corresponding magnetic space group with [111] magnetization direction contains only six elements formed by two generators: inversion I and C_3 symmetry along [111] axis. Since the SOC effect is similar to apply a magnetic field on [111] direction, the magnetization induces that the mirror reflection symmetries M_x, M_y, and M_z are broken.With magnetization along this axis, the states can be distinguished by the eigenvalues of three-fold symmetry C_3 as e^i π/3 for Γ_4,e^-i π/3 for Γ_5, and e^-i π for Γ_6, respectively. The effective Zeemann field of SOC leads to the two-fold degenerate band splitting into two single-degenerate bands, each of which corresponds to either of the two irreducible representations Γ_4 and Γ_5.As shown in Fig. <ref>(b), the band belonging to irreducible representations Γ_6 would crosses with Γ_4 and Γ_5 bands , forming a pair of WPs as W1 and W2 with opposite Chern numbers. Another WP W induced by the band crossing between Γ_5 and Γ_6 is also present.All WPs located on the Γ-R axis belong to type-II. The Chern numbers are determined by the evolution of the average positions of Wannier charge centers using Z2Pack software <cit.>. The Wilson-loop method applied on a sphere around WPs <cit.> is employed as shown in the insets of Fig. <ref> (b). Their precise positions in momentum space, Chern numbers, and the energies related to the Fermi level E_F are listed in Table <ref>. With enhancing the strength of SOC, we can clearly see that the Zeemann splitting increases the distance between W1 and W2 in momentum space in Figs. <ref> (c) and (d). Furthermore, the increasing of SOC strength would lower the energies of band Γ_6, so the WP W is unstable and may be removed, while W1 and W2 are robustly stable on [111] axis [shown in Figs. <ref>(d)].Moreover, pyrite CrO_2 exhibits the FM metallic rather than semimetallic features. It is important to note that the nontrivial properties in topological metals not only depend on the relations between valence and conduction bands, since the occupied states are a function of crystal momentum 𝐤 in this case <cit.>. For instance, the WP W2 is the crossing between the Nth and (N+1)th bands, where N is the total number of valence electrons per unit cell of pyrite CrO_2. The WPs W and W1 are formed by the crossings of the valence bands (N-1) and N. In addition, between the bands (N-1) and N we find more additional topologically protected WPs, some of which (6 in total) are close to Fermi level E_F. The detail information are supplied in SM <cit.>. Importantly, there are only one pair of WPs W2 forming at the boundary of electron and hole pockets, while all the other WPs arise at the crossing points of two pockets of the same carriers. Therefore, the surface Fermi arcs from the projections of two type-II W2 WPs may be clearly and would be easy to observe in experiments.One evident consequence of WPs in pyrite CrO_2 is the existence of topologically protected Fermi arcs projected on the surface of this compound. For type-I WPs, the surface Fermi arcs connect the projections of the opposite chirality WPs onto the surface when Fermi level E_F is tuned to the energy of WP. Type-II WPs are located at the boundary the hole and electron pockets, so the open Fermi surface leads to that the projection of WP is always hidden within the projection of bulk pockets on this surface. However, the Fermi arcs can be revealed by tuning the chemical potential <cit.>. We consider the (001) and (110) surfaces of pyrite CrO_2 to calculate the surface Fermi arcs, since the WPs projected onto these surfaces is distinct in the corresponding surface BZ. The surface density of states ofpyrite CrO_2 is computed by using Wannier TB Hamiltonian <cit.> with the iterative Green's function method <cit.> as implemented in Wannier_-tools package <cit.>. The Fermi surfaces projected onto (001) and (110) surfaces with the chemical potential 15 meV below Fermi level E_F are shown in Figs. <ref>(a) and (b), respectively. We can clearly observe the Fermi arcs arising from WPs W2 of the crossing of Nth and (N+1)th bands, while the other Fermi arcs are hidden within the projections of bulk pockets. The clean Fermi arcs would like to be revealed in angle-resolved photoemission spectroscopy experimentally.In conclusion, we suggest that pyrite CrO_2 co-existing with FM ground state can realize only one pair of type-II WPs between the Nth and (N+1)th bands, where N is the total number of valence electrons per unit cell. Further calculations show that other WPs between the (N-1)th and Nth bands also appear close to the Fermi level due to the complex topological electronic band structure. The symmetry analysis shows that the Weyl points arise from a triply-degenerate point splitting due to the mirror reflection symmetry broken in the presence of SOC, which is equivalent to an applied magnetic field along the direction of magnetization. The corresponding Fermi arcs projected on both (001) and (110) surfaces are calculated as well and observed clearly. Considering that CrO_2 is a very common material, this finding opens a wide range of possible experimental realizations of time-reversal breaking type-II Weyl fermions at room temperature.This work is supported by the National Natural Science Foundation of China (NSFC, Grant Nos.11204185, 11304403, 11334003 and 11404159).Equal Contributions:R. Wang and Y. J. 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Qi, A. Bernevig, Z. Fang, and X. Dai, Phys. Rev. B 84, 075119 (2011). Soluyanov2011 A. A. Soluyanov and D. Vanderbilt, Phys. Rev. B 83, 035108 (2011) Sancho1985 M. P. López Sancho, J. M. López Sancho, and J. Rubio, J. Phys. F 14, 1205 (1984); 15, 851 (1985). Wanniertool Q. S. Wu and S. N. Zhang, https://github.com /quanshengwu/wannier_-tools. Wuwannier Q. S. Wu, S. N. Zhang, H. F. Song, M. Troyer, and A. A. Soluyanov, arXiv: 1703.07789. Supplemental Material for “Ferromagnetic Type-II Weyl Semimetal in Pyrite Chromium Dioxide" In this Supplemental Material, we provide the stability analysis of pyrite Chromium Dioxide (CrO_2), the other Weyl points between (N-1)th and Nth bands, and the topological features with magnetization along [001] direction. Finally, we also elucidate the topology of the triply-degenerate points in the absence of spin-orbital coupling.§ THE PHONON OF PYRITE CHROMIUM DIOXIDE AT THE AMBIENT PRESSUREThe pyrite CrO_2 phase has been demonstrated to be stable ferromagnetic (FM) half-metallic state occurring at critical pressure of ∼45 GPa <cit.>. Here, we use the phonon spectrum, which is one useful way to investigate the stability and structural rigidity.The method of force constants has been used to calculate the phonon frequencies as implemented in PHONOPY package <cit.>. We employ 3 ×3 × 3 supercell with 108 Cr atoms and 216 O atoms to obtain the real-space force constants. Our result for the phonon dispersions at the ambient pressure is shown in Fig.<ref>, respectively. We find that there is the absence of any imaginary frequencies over the entire BZ, demonstrating that the pyrite CrO_2 is dynamical stability.§ THE OTHER WEYL POINTS CLOSE TO FERMI LEVEL BETWEEN (N-1)TH AND NTH BANDS Pyrite CrO_2 exhibits the FM metallic rather than semimetallic features. Due to its complex topological electronic band structure, some other Weyl points between the (N-1)th and Nth bands also appear close to the Fermi level. We find that there are three pairs of the Weyl points (the energies relative to Fermi level are lower than 0.3 eV). Their positions in momentum space are 2π/a(0.0011, 0.1414, 0.0842), 2π/a(0.1419, 0.0844, 0.0010), and 2π/a(0.0907, 0.0006, 0.1613), which are of the most relevant located only 0.111, 0.137, and 0.141 eV below the Fermi level, respectively. Although we found a plethora of topological features formed by the (N-1)th and Nth bands, these additional Weyl points and their associated Fermi arcs may overlap with the bulk states when projected onto a surface, such as (001) or (110) surfaces. Hence, these Weyl points can not contribute visible spectroscopic signatures of surface Fermi arcs. § THE TOPOLOGICAL FEATURES WITH MAGNETIZATION ALONG [001] DIRECTIONOur first-principles calculations suggest that there are only tiny energy differences among all magnetic configurations in pyrite CrO_2, implying that an applied magnetic field can easily manipulate the spin-polarized direction. Therefore, we also perform the calculations for magnetization along [001] direction. When the FM magnetization is parallel to [001] direction, the system reduces to magnetic space group D_4h(C_4h) and the three-fold rotational symmetry C_3 is broken. Hence, the Weyl points arising from the triply-degenerate points splitting may not locate on Γ-R axis. In this case, we only pay attention to the Weyl points between Nth and (N+1)th bands. There are five pairs of Weyl points formed at the boundary of electron and hole pockets. Furthermore, the presence of an odd number of pairs of Weyl points between Nth and (N+1)th bands can be clarified by the product of the inversion eigenvalues of the number of occupied bands N at eight time reversal invariant momentapoints k_inv <cit.>, as χ_P=∏_k_inv;i∈occζ_i (k_inv). Our calculations show that the value of χ_P is -1, implying that the system may be WSM co-existing with an odd number of pairs of Weyl points. In pyrite CrO_2 with magnetization along [001] direction, five pairs of Weyl points between Nth and N+1th bands are present.Their precise positions in momentum space, Chern numbers, and the energies related to the Fermi level E_F are listed in Table <ref>. § THE TRIPLY-DEGENERATE POINTS IN THE ABSENCE OF SPIN-ORBITAL COUPLING In the absence of spin-orbital coupling, the symmetry group is the abelian group T_h, which contains four three-fold rotational symmetry C_3 axes [111], [11̅1], [111̅], and [1̅11], inversion I, and three mirror symmetries M_x, M_y, and M_z, respectively. The mirror symmetries sendM_x: (x, y, z) → (-x, y, z), M_y: (x, y, z) → (x, -y, z), M_z: (x, y, z) → (x, y, -z), and C_3 ^111 and the product IM_x M_y M_z of inversion I and mirror reflection symmetries leave every momentum point invariant along Γ-R (or 𝐤∥ [111]) axis. Hence, at each point along theΓ-R axis, the Bloch states that form a possibly degenerate eigenspace (band) of the Hamiltonian must be invariant under C_3 ^111 and IM_x M_y M_z. Without SOC, there are three eigenvalues of C_3 rotational symmetry, namely, e^-i 2π/3, e^i 2π/3, and1 (e^iπ), and we denote the corresponding eigenstates as ψ_1, ψ_2, and ψ_3, respectively. Using the basis (ψ_1, ψ_2, ψ_3), the representations of a operators O can be determined as O_ij=⟨ψ_i|O|ψ_j⟩,so C_3 ^111 and mirror symmetries M_x, M_y, and M_z can be expressed asC_3 ^111=diag{e^-i 2π/3, e^i 2π/3, 1}, M_x=( [0 -10; -100;001;]), M_y=( [ 0 1 0; 1 0 0; 0 0 1; ]),M_z=( [ 1 0 0; 0 0 1; 0 1 0; ]).It can be seen that C_3 ^111 and M_i (i=x, y, z) can not commute with each other, leading to that failure of C_3 ^111 and M_i to be simultaneously diagonalizable. Therefore, in the absence of SOC, along Γ-R (or 𝐤∥ [111]) axis, the three bands with the three different eigenvalues of C_3 ^111always appear as a singly-degenerate band (ψ_3)and a doubly-degenerate band (ψ_1 and ψ_2). If the single degenerate and the doubly-degenerate bands cross each other accidentally, a triply-degenerate node will form because their different C_3 ^111eigenvalues prohibit hybridization <cit.>. When spin-orbital coupling is considered, the triply-degenerate node would like to split into Weyl points depending on the magnetic space group. 44 Li2012 Y. Li and J. Hao, Solid state Commun. 152, 1216 (2012). Togo1 A. Togo, L. Chaput, I. Tanaka, G. Hug, Phys. Rev. B 81, 174301 (2010) . Togo2 A. Togo, F. Oba, I. Tanaka, Phys. Rev. B 78, 134106 (2008). Togo3 A. Togo, Phonopy, http://phonopy.sourceforge.net/ Hughes2011 T. L. Hughes, E. Prodan,and B. A. Bernevig, Phys. Rev. B 83, 245132 (2011). Changarxiv G. Chang,S. Xu, S. Huang, D. S. Sanchez, C.-H. Hsu, G. Bian, Z. Yu, I. Belopolski, N. Alidoust, H. Zheng, T. Chang, H. Jeng, S. A. Yang, T. Neupert, H. Lin, and M. Z. Hasan,arXiv:1605.06831. | http://arxiv.org/abs/1707.08899v1 | {
"authors": [
"R. Wang",
"Y. J. Jin",
"J. Z. Zhao",
"Z. J. Chen",
"Y. J. Zhao",
"H. Xu"
],
"categories": [
"cond-mat.str-el",
"cond-mat.mtrl-sci"
],
"primary_category": "cond-mat.str-el",
"published": "20170727145907",
"title": "Ferromagnetic Type-II Weyl Semimetal in Pyrite Chromium Dioxide"
} |
[ Feng Tian December 30, 2023 =====================§ ABSTRACT We give an asymptotic probabilistic real Riemann-Hurwitz formula computing the expected real ramification index of a random covering over the Riemann sphere. More generally,we study the asymptotic expected number and distribution of critical points of a random real Lefschetz pencil over a smooth real algebraic variety.§ INTRODUCTIONThe Riemann-Hurwitz formula says that the total ramification indexof a degree d branched coveringf:Σ→Σ'between two compact Riemann surfaces equals d·χ(Σ')-χ(Σ). In particular, if Σ'=ℂℙ^1, the total ramification index is2d+2g-2, where g is the genus of Σ.More generally, if u:Xℂℙ^1 is a Lefschetz pencil on a complex manifold X of dimension n, then (-1)^n#crit(u)=χ(X)-2χ(F)+χ(Y) where F is a smooth fiber of u and Y is the base locus of u.The questions that motivate this paper are the following: how dothese critical points distribute on the variety? When u is defined over ℝ, what about the numberof real critical points?We answer these questions by computing the asymptotic expected number of real critical points of real Lefschetz pencils and also the asymptotic distribution of such points.The chosen random setting has already been considered byShiffman and Zelditch in <cit.> to study the integration current over thezero locus of arandom global section of a line bundle over a complex projective manifold.In the real case, Kac <cit.>, Kostlan <cit.> and Shub and Smale <cit.> computed the expected number of real roots of a random real polynomial. In higher dimensions, Podkorytov <cit.> and Bürgisser <cit.> computed the expected Euler characteristic ofrandom real algebraic submanifolds and Letendre<cit.> the expected volume (see <cit.>for the expected length of a random lemniscate). In <cit.> Gayet and Welschinger estimated from above and below the Betti numbers of the real locus of real algebraic submanifolds(see also <cit.>).For intersection of real quadrics, a precise asymptotic of the total Betti number has been given by Lerario and Lundberg in <cit.>. In <cit.> Nicolaescu computed the expected number of critical of a random smooth function on a Riemannian manifold have and how they distruibute. §.§ Statements of the results Let X be a smooth real projective manifold of dimension n, that is a complex projective manifold equipped with an anti-holomorphic involution c_X, called the real structure. We denote by ℝ X=(c_X)its real locus. Let ℒ be a positive real line bundle over X. For large d, for almost all pairs(α,β)∈H^0(X;ℒ^d)^2(resp. ℝ H^0(X;ℒ^d)^2) of (real) global section, the map u_αβ:Xℂℙ^1 defined by x↦ [α(x):β(x)] is a (real) Lefschetz pencil, see Proposition <ref>. Recall that a real Lefschetz pencil is a Lefschetz pencil u:Xℂℙ^1 such that conj∘ u= u∘ c_X.We denote the set of critical points of u_αβ by crit(u_αβ) and by ℝcrit(u_αβ)=crit(u_αβ)∩ℝ X the set of real critical points.The numberof real critical points of a real Lefschetz pencil depends on the pair (α, β). The main theorem of this paper is the computation of the expected value of this number. Recall that, by definition, the expected value of #ℝcrit(u_αβ) equals𝔼[#ℝcrit(u_αβ)]=∫_(α,β)∈ℝ H^0(X,ℒ^d)^2(#ℝcrit(u_αβ))dμ(α,β). Let X be a smooth real projective manifold of dimension n and (ℒ,h) be areal Hermitian line bundle over X with positive curvature. Thenlim_d→ +∞1/√(d)^n𝔼[#ℝcrit(u_αβ)]={[ n!!/(n-1)!!e_ℝ(n)π/2Vol_h(ℝ X)if n is odd;n!!/(n-1)!!e_ℝ(n)Vol_h(ℝ X)if n is even. ].In this theorem, Vol_h(ℝ X) is the volume of ℝ X with respect to the Riemannian volume form_h induced by the positive curvatureof themetric h. Theprobability measure we consider is a natural Gaussian probability on ℝ H^0(X;ℒ^d)^2 (see Section 2.1) and e_ℝ(n) is the expected value of (the absolute value) of the determinant of real symmetric matrices (for the explicit values of e_ℝ(n), see <cit.>).We recall that e_ℝ(1)=√(2/π), then we have:Let (Σ,c_Σ) be a real Riemann surface and (ℒ,h) be a real Hermitian line bundle of degree 1. Then, for every pair (α,β)∈ℝ H^0(X;ℒ^d)^2 without common zeros, the map u_αβ is a degree d branched covering between Σ and ℂℙ^1 andthe expected real total ramification index of u_αβis equivalent to√(π/2)Vol_h(ℝΣ)√(d)as dtends to +∞.Theorem <ref> is a consequence of a more precise equidistribution result. In order to introduce it, let us define a natural empirical measure associated with the realcritical points of a Lefschetz pencil as follows. For any pair (α,β)∈ℝ H^0(X;ℒ^d)^2 of real global sections of ℒ^d, we defineℝν_αβ=∑_x∈ℝcrit(u_αβ)δ_x. Let Xbe a smooth real projective manifold of dimension n and (ℒ,h) beareal Hermitian line bundle over X with positive curvature ω. Thenlim_d→ +∞1/√(d)^n𝔼[ℝν_αβ]={[ n!!/(n-1)!!e_ℝ(n)π/2_hif n is odd;n!!/(n-1)!!e_ℝ(n)_hif n is even. ].weakly in the sense of distributions. Here, _h is the Riemannian volume form induced by the curvature ω. Theorem <ref> says that, for any continuous functionφ∈ C^0(ℝ X), we have lim_d→ +∞1/√(d)^n𝔼[ℝν_αβ](φ)={[ n!!/(n-1)!!e_ℝ(n)π/2∫_ℝ Xφ_hif n is odd;n!!/(n-1)!!e_ℝ(n)∫_ℝ Xφ_hif n is even. ].where the expected value is defined by𝔼[ℝν_αβ](φ)=∫_ℝ H^0(X;ℒ^d)^2∑_x∈ℝcrit(u_αβ)φ(x)dμ(α,β). In the complex case, we obtain a similarequidistribution theorem, whose proof follows along the same lines. For any pair (α,β)∈H^0(X;ℒ^d)^2 ofglobal sections of ℒ^d, we defineν_αβ=∑_x∈crit(u_αβ)δ_xto be the empirical measure associated with the critical points of the pencil u_αβ. Let X be a smooth complex projective manifold of dimension n and (ℒ,h) bea Hermitian line bundle over X with positive curvature ω. Thenlim_d→ +∞1/d^n𝔼[ν_αβ]=(n+1)ω^nweakly in the sens of distribution.As before, Theorem <ref> says that, for any continuous functionφ on X, we have lim_d→ +∞1/d^n𝔼[ν_αβ](φ)=(n+1)·∫_Xφω^n.§.§.§ Organisation of the paperIn Section <ref>, we introduce the Gaussian measure on H^0(X;ℒ^d)^2 associated with a Hermitianlinebundle (ℒ,h) over a complex manifold X. We also give the same construction for the real case. We follow the approach of<cit.>. In Section <ref>, we present some classical results about Lefschetz pencils on complex manifolds. In Sections <ref> and <ref> we introduce our main tools, namelythe Hörmander peak sections (see also <cit.>, <cit.>) and the incidence manifold (see <cit.>). Section <ref>is completely devoted to the proofs of the Theorems<ref>, <ref> and <ref>. In Sections <ref> and <ref>, we prove the equidistribution of critical points of a (real) Lefschetz pencil over a (real) algebraic variety X. This will be done using coarea formula and peak sections. These ideas are taken from <cit.>. In Section <ref> we will compute the universal constant by direct computation. §.§.§ AcknowledgmentsI am very grateful to my advisor Jean-Yves Welschinger for all the time he devoted to me and for all the fruitful discussions we had.This work was performed within the framework of the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).§ DEFINITIONS AND MAIN TOOLS §.§ Notations Let X be a complex manifold of dimension n. Let ℒ→ X be a holomorphic line bundle equipped with a Hermitian metric h of positive curvature ω∈Ω^(1,1)(X,ℝ). The curvature form induces a Kähler metric and a normalized volume formdx=ω^n/∫_Xω^n on X.The Hermitian metric h induces a Hermitian metric h^d on ℒ^d for any integer d>0 and also a L^2-Hermitian product ⟨·,·⟩_L^2 on the space H^0(X;ℒ^d) of global holomorphic sections of ℒ^d. It is defined by ⟨α,β⟩_L^2=∫_Xh^d(α,β)dxfor any α,β in H^0(X;ℒ^d).This L^2-Hermitian product induces a Gaussian measureon H^0(X;ℒ^d)^2 defined by μ(A)=1/π^2N_d∫_Ae^-α_L^2^2-β_L^2^2dαdβfor any open subset A⊂ H^0(X;ℒ^d)^2 where dαdβ is the Lebesgue measure associated with ⟨·,·⟩_L^2 and N_d=_ℂH^0(X;ℒ^d).Finally, a Lefschetz pencil on X is a rational map u:Xℂℙ^1 having only non degenerated critical points and defined by two global sections of a holomorphic line bundle with smooth and transverse vanishing loci.All these definitions have a real counterpart.* Let X be a real algebraic variety of dimension n, that is a complex manifold equipped with an anti-holomorphic involution c_X. We denote by ℝX=(c_X) its real locus. * A real holomorphic line bundle u:ℒ→ Xis a line bundleequipped with an anti-holomorphic involution c_ℒ such that p∘ c_X=c_ℒ∘ p and c_ℒ is complex-antilinear in the fibers.* We denote by ℝ H^0(X;ℒ) the real vector space of real global section of ℒ, i.e. sections s∈ H^0(X;ℒ) such that s ∘ c_X=c_ℒ∘ s.* A real Hermitian metric on ℒ is a Hermitian metric h such that c_ℒ^*h=h̅. If (ℒ,h) is a line bundle over X with positive curvature ω, then ω(.,i.) is a Hermitian metric over X and its real part defines a Riemannian metric over ℝ X. We denotethe Riemannian volume form induced by this metric by _h.* The L^2-Hermitian product ⟨·,·⟩_L^2 on H^0(X;ℒ^d) restricts to a L^2-scalar product on ℝ H^0(X;ℒ^d), also denotedby ⟨·,·⟩_L^2. Then, as in the complex case, also in the real case we have a natural Gaussian measure on ℝ H^0(X;ℒ^d)^2 defined by μ(A)=1/π^N_d∫_Ae^-α_L^2^2-β_L^2^2dαdβfor any open subset A⊂ℝ H^0(X;ℒ^d)^2 where dαdβ is the Lebesgue measure associated with ⟨·,·⟩_L^2 and N_d=_ℂH^0(X;ℒ^d)=_ℝℝ H^0(X;ℒ^d). * A real Lefschetz pencil u:Xℂℙ^1 is a Lefschetz pencilsuch that p∘ c_X=conjconj∘ p.We conclude this section by introducing some notation on symmetric matrices. For any n∈ℕ^*, we denote by (n,ℝ) the real vector space of real symmetric matrices of size n× n. It is a vector space of dimension n(n+1)/2. We equip it with the basis ℬ given by Ẽ_jj and Ẽ_ij=E_ij+E_ji for 1⩽ i<j⩽ n, where, for any k,l with 1⩽ k,l⩽ n, we denote by E_klthe elementary matrix whose entry at the i-th row and j-th column equals 1 if (i,j)=(k,l) and 0 otherwise. We equip (n,ℝ) with the scalar product turning ℬ into an orthonormal basis. Let μ_ℝthe associated Gaussian probability measure. We then set e_ℝ(n)=∫_A∈(n,ℝ)| A| dμ_ℝ(A).§.§ Lefschetz pencils In this section, we compute the asymptotic value of the number of critical points of a Lefschetz pencil (see also <cit.>).Recall that a Lefschetz fibration is a map X→ℂℙ^1 with only non degenerate critical points. The following proposition is a kind of Riemann-Hurwitz formula for Lefschetz pencils, for a proof see <cit.>. Let X be a smooth complex projective manifold of positive dimension n equipped with a Lefschetz fibration p:X→ℂℙ^1 and let F be a regular fiber of p. Then we have the following equality:χ(X)=2χ(F)+(-1)^n#crit(p). Remark that if u:Xℂℙ^1 a Lefschetz pencil and we blow-up the base locus Base(u)≑ Y, then we obtain a Lefschetz fibration ũ:X̃≑ Bl_YX→ℂℙ^1. By additivity of the Euler characteristic, we have that χ(X̃)=χ(X)+χ(Y), then by Proposition <ref> we haveχ(X)=2χ(F)-χ(Y)+(-1)^n#crit(u). Let ℒ be an ample line bundle over a complex manifold X of dimension n. For almost all global sections (α, β)^2∈ H^0(X;ℒ^d)^2, the map u_αβ defined by x↦ [α(x):β(x)] is a Lefschetz pencil (see Prop. <ref>). Then, as d goes to infinity, we have #crit(u_αβ)=(n+1)(∫_Xc_1(ℒ)^n)d^n+O(d^n-1).We will follow the lines of Lemma 2, Lemma 3 and Proposition 4 of <cit.>.We have χ(F)=∫_F c_n-1(F) and χ(Y)=∫_Y c_n-2(Y). We remark that the base locus is the intersection of the zero locus of α and β, that is Y=Z_α∩ Z_β.A regular fiber F over [a,b]∈ℂℙ^1 is the zero locus of the section bα-aβ∈ H^0(X;ℒ^d), thus the normal bundle N_X/F is ℒ^d_| FTo compute χ(F) we will use the adjunction formula. We have 0→ TF→ TX_| F→ N_X/F→ 0 then we have c(X)_| F=c(F)∧ c(ℒ^d)_| F, that is(1+c_1(X)+…+c_n(X))_| F=(1+c_1(F)+…+c_n-1(F))∧(1+dc_1(ℒ)).If we develop this we have c_1(X)=c_1(F)+dc_1(ℒ)and, for j∈{2,…,n-1}, we have c_j(X)_| F=c_j(F)+dc_1(ℒ)_| F∧ c_j-1(F). Then, summing up the term, c_j(F)=∑_k=0^j(-1)^kd^kc_1(ℒ)^k_| F∧ c_j-k(X)_| F.In particular, for j=n-1 we have c_n-1(F)=∑_k=0^n-1(-1)^kd^kc_1(ℒ)^k_| F∧ c_n-k-1(X)_| F.Then χ(F) is equal to ∫_F∑_k=0^n-1(-1)^kd^kc_1(ℒ)^k)_| F∧ c_n-k-1(X)_| F. But, for α∈ H_dR^2n-2(X), we have that∫_Fα_| F=∫_Xα∧ c_1(ℒ^d)so,χ(F)=∑_k=0^n-1∫_X(-1)^kd^k+1c_1(ℒ)^k+1∧ c_n-k-1(X)and asymptotically we getχ(F)∼(-1)^n-1(∫_Xc_1(ℒ)^n)d^n.For Y=Z_α∩ Z_β, the same argument gives us c_j(Y)=∑_k=0^j(-1)^kd^kc_1(ℒ)^k_| Y∧ c_j-k(Z_α)_| Y.But, as before,c_j-k(Z_α)=∑_h=0^j-k(-1)^hd^hc_1(ℒ)^h∧ c_j-k-h(X).and so,replacing in the above equation c_j(Y)=∑_k=0^j(-1)^kd^kc_1(ℒ)^k_| Y∧ (∑_h=0^j-k(-1)^hd^hc_1(ℒ)^h_| Y∧ c_j-k-h(X)_| Y)For j=n-2 we havec_n-2(Y)= ∑_k=0^n-2(-1)^kd^kc_1(ℒ)^k_| Y∧ (∑_h=0^n-2-k(-1)^hd^hc_1(ℒ)^h_Y∧ c_n-2-k-h(X)_| Y)and this is equivalent to ∑_k=0^n-2(-1)^n-2d^n-2c_1(ℒ)^n-2_| Y = (-1)^n-2(n-1)d^n-2c_1(ℒ)^n-2_| Y as d→∞. So we have, as d→∞,χ(Y)∼ (-1)^n-2(n-1)d^n-2∫_Yc_1(ℒ)^n-2_| Y= =(-1)^n-2(n-1)d^n-1∫_Z_αc_1(ℒ)^n-2∧ c_1(ℒ)= =(-1)^n-2(n-1)(∫_Xc_1(ℒ)^n)d^n.Combining this with χ(X)=2χ(F)-χ(Y)+(-1)^n#crit(u_αβ)we obtain the result.§.§ Hörmander's peak sectionsIn this section we recall the theory of Hörmander's peak sections, an essential tool for our proofs of Theorems <ref> and <ref> (see also <cit.>, <cit.>, <cit.>). Let ℒ be a holomorphic line bundle over a smooth complex projective manifold equipped with a Hermitian metric h of positive curvature ω and let dx=ω^n/∫_Xω^n be the normalized volume form. Let x be a point of X. There exists, in the neighborhood of x, a holomorphic trivialization e of ℒ such that the associated potential reaches a local minimum at x with Hessian of type (1,1). The following result was proved in<cit.> (see also <cit.>) .Let (ℒ,h) be a holomorphic Hermitian line bundle of positive curvature ω over a smooth complex projective manifold X. Let x∈ X, (p_1,…,p_n)∈ℕ^n and p'>p_1+…+p_n. There exists d_0∈ℕ such that for every d>d_0, the bundle ℒ^d has a global holomorphic section σ satisfying ∫_Xh^d(σ,σ)dx=1 and ∫_X∖ B(x,log d/√(d))h^d(σ,σ)dx=O(1/d^2p') Moreover, if (x_1,…,x_n) are local holomorphic coordinates in the neighborhood of x, we can assume that in a neighborhood of x,σ(x_1,…,x_n)=λ(x_1^p_1⋯ x_n^p_n+O(x^2p'))e^d(1+O(1/d^2p'))where λ^-2=∫_B(x,log d/√(d))|x_1^p_1⋯ x_n^p_n|^2 h^d(e^d,e^d)dxand e is a holomorphic trivialization of ℒ in the neighborhood of xwhose potential ϕ=-log h(e,e) reaches a local minimum at x with Hessian πω(.,i.). This lemma is true also in the real setting, in the following sense:Let (ℒ,h) be a real holomorphic Hermitian line bundle of positive curvature ω over a smooth real projective manifold X. Let x∈ℝX, (p_1,…,p_n)∈ℕ^n and p'>p_1+…+p_n. There exists d_0∈ℕ such that for every d>d_0, the bundle ℒ^d has a global holomorphic section σ satisfying ∫_Xh^d(σ,σ)dx=1 and ∫_X∖ B(x,log d/√(d))h^d(σ,σ)_h=O(1/d^2p')Moreover, if (x_1,…,x_n) are local real holomorphic coordinates in the neighborhood of x, we can assume that in a neighborhood of x,σ(x_1,…,x_n)=λ(x_1^p_1⋯ x_n^p_n+O(x^2p'))e^d(1+O(1/d^2p'))where λ^-2=∫_B(x,log d/√(d))| x_1^p_1⋯ x_n^p_n|^2 h^d(e^d,e^d)dxand e is a real trivialization of ℒ in the neighborhood of xwhose potential ϕ=-log h(e,e) reaches a local minimum at x with Hessian πω(.,i.). This real counterpart follows from Lemma <ref> by averaging the peak sections with the real structure.Let σ_0 bethe section given by the Lemma <ref> with p'=3and p_i=0 for all i, σ_i the section given by Lemma <ref> with p'=3 and p_j=δ_ij, σ_ij the section given by the Lemma <ref> with p_i=p_j=1 and p_k=0 otherwise and σ_kk the section given by the Lemma <ref> with p_k=2 and p_l for l≠ k. These sections are called peak sections. Their Taylor expansions are: σ_0(y)=(λ_0+O( y^6))e^d(1+O(1/d^6)); σ_i(y)=(λ_iy_i+O( y^6))e^d(1+O(1/d^6)) ∀ i; σ_ij(y)=(λ_ijy_iy_j+O( y^6))e^d(1+O(1/d^6))∀ i≠ j; σ_kk(y)=(λ_kky_k^2+O( y^6))e^d(1+O(1/d^6)) ∀ k.The following lemma shows the asymptotic of the constantsλ_0, λ_i, λ_ij et λ_kk.[Lemma 2.5 of <cit.>] Under the hypothesis of Lemma <ref> and <ref>, we havelim_d⟶∞1/√(d)^nλ_0=√(δ_ℒ) lim_d⟶∞1/√(d)^n+1λ_i=√(π)√(δ_ℒ) lim_d⟶∞1/√(d)^n+2λ_ij=π√(δ_ℒ) lim_d⟶∞1/√(d)^n+2λ_kk=π/√(2)√(δ_ℒ)for the L^2-product induced by dx=ω^n/∫_Xω^n whereδ_ℒ=∫_X c_1(ℒ)^n is the degree of the line bundle ℒ. Set H_2x={s∈ H^0(X;ℒ^d) | s(x)=0, ∇ s(x)=0,∇^2s(x)=0} (resp.ℝ H_2x={s∈ℝ H^0(X;ℒ^d) | s(x)=0, ∇ s(x)=0,∇^2s(x)=0}). This space is formed bysections whose 2-jet vanishes at x. The sections (σ_i)_0≤ i≤ n (σ_ij)_1≤ i≤ j≤ n provide a basis of a complement of H_2x. This basis is not orthonormal and its spanned subspace is not orthogonal to H_2x. However, this basis is aymptotically an orthonormal basis and its spanned subspace is asymptotically orthonormal to H_2x, in the following sense: The section (σ_i)_0≤ i≤ n and (σ_ij)_1≤ i≤ j≤ n have L^2-norm equal to 1 andand their pairwise L^2-scalar product are O(1/d). Likewise, their scalar products with every unitary element of H_2x are O(1/d^3/2).§.§ Incidence manifoldsFollowing <cit.>, we define an incidence manifold associated with the complex (resp. real) manifold X and to the (real) positive line bundle ℒ. We will use this incidence manifold to provethat,for almost all pairs global sections (α, β)∈ H^0(X;ℒ^d)^2(resp. (α, β)∈ℝ H^0(X;ℒ^d)^2),the map u_αβx↦ [α(x):β(x)] defines a Lefschetz pencil, see Proposition <ref>.Let (ℒ,h) be a (real) Hermitian line bundle with positive curvature ω over a (real) algebraic variety X of dimension n.Let α, β∈ H^0(X;ℒ^d) (resp. ℝ H^0(X;ℒ^d)) be two(real) global sections such that the map x↦ [α(x):β(x)] is a Lefschetz pencil. We define* the base locus of a Lefschetz pencil as the points x such that α(x)=β(x)=0; * the critical points as the points x∈ X∖Base(u_αβ) such that (α∇β-β∇α)(x)=0 (this expression does not depend on the choice of a connection ∇ on ℒ). We denote by crit(u_αβ) the set of critical points of (u_αβ) and by ℝcrit(u_αβ)=crit(u_αβ)∩ℝ X the set of real critical points.We denote by Δ (resp. by ℝΔ) the set of(α, β,x)∈ H^0(X;ℒ^d)^2× X (resp. (α, β,x)∈ℝ H^0(X;ℒ^d)^2×ℝ X) such that α(x)=β(x)=0. Setℐ={(α,β,x)∈(H^0(X;ℒ^d)^2 × X)∖Δ| x∈crit(u_αβ) } (resp.ℝℐ={(α,β,x)∈(ℝ H^0(X;ℒ^d)^2 ×ℝ X)∖ℝΔ| x∈crit(u_αβ) })Let ℒ be a (real) holomorphic line bundle over a smooth complex (resp. real) projective manifold X. If ℒ^d is 1-ample, that is if the 1-jet map H^0(X;ℒ^d)× X→ J^1(ℒ^d) (s,x)↦ j^1_x(s)=(s(x),∇ s(x)) is surjective, then ℐ (resp. ℝℐ) is a smooth manifold of complex(resp. real) dimension 2N_d, where N_d= H^0(X;ℒ^d).We study the differential of the mapq:(H^0(X;ℒ^d)^2 × X)∖Δ→ T^*X⊗ℒ^2ddefined by(α,β,x)↦(α∇β-β∇α)(x)∈ T_x^*X⊗ℒ_x^2ddefining ℐ. If we prove that 0 is a regular value, then, by Implicit Function Theorem, we have the result. Now, for (α,β,x)∈ℐ we haved_|(α,β,x)q·(α̇,β̇,ẋ)=(α̇∇β-β∇α̇ +α∇β̇-β̇∇α +α∇^2_(ẋ,.)β-β∇^2_(ẋ,.)α +∇_ẋα∇β-∇_ẋβ∇α)(x).For any η∈ T_x^*X⊗ℒ_x^2d we have to prove that there exists (α̇,β̇,ẋ) such that d_|(α,β,x)q·(α̇,β̇,ẋ)=η. As (α,β,x)∉Δ, we know that at least one betweenα(x) and β(x)is not zero. Without loss of generality, suppose that α(x)≠ 0, then, as ℒ^d is 1-ample, there exists β̇ such thatβ̇(x)=0 and α(x)∇β̇(x)=η, then d_|(α,β,x)q·(0,β̇,0)=η. If ℒ is ample, then, for large d, the line bundle ℒ^d is 1-ample.Then, for large d, ℐ (resp. ℝℐ ) is a smooth manifold, called the incidence manifold.The tangent space T_(α,β,x)ℐ of ℐat a point (α,β,x) (resp. T_(α,β,x)ℝℐ of ℝℐ) equals{ (α̇,β̇,ẋ)∈ H^0(X;ℒ^d)^2× T_xX|(α̇∇β-β∇α̇ +α∇β̇-β̇∇α +α∇^2_(ẋ,.)β-β∇^2_(ẋ,.)α) (x)=0}. (resp.{(α̇ ,β̇,ẋ)∈ℝ H^0(X;ℒ^d)^2× T_xℝ X|(α̇∇β-β∇α̇ +α∇β̇-β̇∇α +α∇^2_(ẋ,.)β-β∇^2_(ẋ,.)α)(x)=0}). * In the equation defining the tangent space there is also the term (∇_ẋα∇β-∇_ẋβ∇α) (x).However, it equals zero (both in the complex and real case) because on ℐ and ℝℐ we have the condition (α∇β-β∇α)(x)=0 so that(∇_ẋα∇β-∇_ẋβ∇α)(x)=((∇_ẋαβ/α-∇_ẋβ)∇α) (x)=0. * The incidence manifold comes equipped with two natural projectionsπ_H:ℐ→ H^0(X;ℒ^d)^2andπ_X:ℐ→ X(resp.π_ℝ H:ℝℐ→ℝ H^0(X;ℒ^d)^2andπ_ℝ X:ℝ ℐ→ℝ X). Let ℒ be an ampleholomorphic line bundle (resp. real holomorphic) over a smooth complex projective manifold X (resp. real projective). For large d and for almost all pairs (α,β) ∈ H^0(X;ℒ^d)^2 (resp. ℝ H^0(X;ℒ^d)^2), the mapu_αβ:Xℂℙ^1 x↦ [α(x):β(x)].is a Lefschetz pencil (resp. real Lefschetz pencil).The critical points of the projection π_H (resp. π_ℝ H) are exactly the triples (α,β,x) such that the Hessian (α∇^2β-β∇^2α)(x) is degenerate. By Sard's theorem (π_H) has zero Lebesgue, and then Gaussian, measure. Also, for large d, the set Γ composed by the pairs (α,β)∈ H^0(X;ℒ^d)×H^0(X;ℒ^d)such that {x∈ X, α(x)=β(x)=0} is not smooth has zero Lebesgue and Gaussian measure (see for example <cit.>). Then (Γ∪(π_H)) has zero measure and its complement is exactly the set of pairs of sections defining a Lefschetz pencil. § PROOF OFTHE MAIN THEOREMSIn this section we proveTheorems <ref>, <ref> and<ref>.Hörmander's peak sections and the coarea formula play an important role here. §.§ Coarea formulaIn this section weuse the incidence manifold defined in Section <ref> and the coarea formula to write the expected distribution of critical points of a (real)Lefschetz pencil as an integral over X (resp. ℝ X). The normal jacobian Jac_Nf of a submersion f:M→ N between two Riemannian manifolds is the determinant of the differential of the map restricted to the orthogonal complement of its kernel, that is Jac_Nf=Jac(df_| (df)^⊥). Equivalently, if df_p is the differential of f at p, then the normal jacobian is equal to √((df_pdf_p^*)), where df_p^* is the adjoint of df_p with respect to the scalar product on T_pM and T_f(p)N. Let X be a smooth complex (resp. real) projective manifold of dimension n and (ℒ,h) be a (real) holomorphic line bundle with positive curvature ω. We define a Dirac measure for (real) critical points of a (real) Lefschetz pencil u_αβ associated with a pair (α,β) ∈ H^0(X;ℒ^d)^2 (resp. (α,β)∈ℝ H^0(X;ℒ^d)^2) byν_αβ=∑_x∈crit(u_αβ)δ_x(resp.ℝν_αβ=∑_x∈ℝcrit(u_αβ)δ_x).Let φ be a continuous function on ℝ X. Then, by definition, we have 𝔼[ℝν_αβ](φ)=∫_ℝ H^0(X;ℒ^d)^2∑_x∈crit(u_αβ)φ(x)dμ(α,β)where dμ is the Gaussian measure on ℝ H^0(X;ℒ^d)^2 constructed in Section <ref>.Finally, recall that we denote by π_ℝ H and π_ℝ X the two natural projections from ℝℐ to ℝ H^0(X;ℒ^d)^2 and ℝ X. The projection π_ℝ H is (almost everywhere) a local isomorphism and, by a slight abuse of notation, we will denote by π_ℝ H^-1 any local inverse.Following the notation of Section <ref>, we have𝔼[ℝν_αβ](φ)=∫_ℝ Xφ(x)∫_π_ℝ H(π^-1_ℝ X(x))1/|(π_ℝ H^-1)^*Jac_N(π_ℝ X)|dμ_|π_ℝ H(π^-1_ℝ X(x))_h.where the measure dμ_|π_ℝ H(π^-1_ℝ X(x)) is the following: first we restrict the scalar product ⟨·,·⟩_L^2 on ℝ H^0(X;ℒ^d)^2 to π_ℝ H(π^-1_ℝ X(x)), which is a codimension n submanifold, thenweconsider the Riemannian measure associated with this metric, and finally we multiply it by thefactor 1/π^N_de^-α_L^2^2-β_L^2^2, where N_d= H^0(X;ℒ^d). We denote byπ_ℝH^*dμthe pull-backed measure on ℝℐ, which is well defined since π_ℝH is (almost everywhere) a local isomorphism. By definition of the pull-backed measure,the integral𝔼[ℝν_αβ](φ)=∫_ℝ H^0(X;ℒ^d)^2∑_x∈crit(u_αβ)φ(x)dμ(α,β) which defines the expected value equals the following integral over the incidence manifold ℝℐ∫_ℝℐ(π_ℝ X^*φ)(α,β,x)(π_ℝH^*dμ)(α,β,x). We use the coarea formula (see <cit.> or <cit.>) for the map π_ℝ X and we obtain 𝔼[ℝν_αβ](φ)=∫_ℝ Xφ(x)∫_π^-1_ℝ X(x)1/|Jac_N(π_ℝ X)|(π_ℝH^*dμ)_|π^-1_ℝ X(x)_hwhere the measure (π_ℝH^*dμ)_|π^-1_X(x) is the following: first we restrict the (singular) metric π_H^*⟨·,·⟩_L^2 on ℝℐ to π^-1_ℝ X(x), that is a codimension n submanifold, thenweconsider the Riemannian measure associated with this metric, and finally we multiply it by thefactor 1/π^N_de^-α_L^2^2-β_L^2^2, where N_d= H^0(X;ℒ^d). Another application of coarea formula for the map π_ℝ H gives us the result.The space π_ℝ H(π^-1_ℝ X(x)) isformed bypairs (α,β)∈ℝ H^0(X;ℒ^d)^2such that x∈ℝcrit(u_αβ). In the next section we will identify this space with an intersection of some quadrics in the vector space ℝ H^0(X;ℒ^d)^2.In the complex case, the same argument gives us, for any continuous function φ on X 𝔼[ν_αβ](φ)=∫_Xφ(x)∫_π_H(π^-1_X(x))1/|(π_H^-1)^*Jac_N(π_X)|dμ_|π_H(π^-1_X(x))_h.§.§ Computation of the normal jacobianIn this section we compute the normal jacobian that appears in (<ref>) and (<ref>). We follow the notations of Sections <ref>, <ref> and <ref>.The main result of this section is the following proposition:Following the notation of Sections <ref> and <ref>, under the hypothesis of Theorem <ref>, we have: 𝔼[ℝν_αβ](φ)= ∫_x∈ℝ Xφ(x)R_d(x)_h, whereR_d(x)=√(π d)^n(∫_Q|(a_0b_ij-b_0a_ij)|/√(((a_ia_j+b_ib_j)+(a_0^2+b_0^2)Id))dμ_Q +O(1/√(d)))and Q⊂ℝ^2(n+1)+n(n+1) is the product of the intersection of quadricsQ̃={(a_0,b_0,…,a_n,…,b_n)∈ℝ^2(n+1)| a_0b_i-a_ib_0=0∀ i=1,…,n}with the vector space ℝ^n(n+1) of coordinates a_ij and b_ij for 1≤ i≤ j≤ n and dμ_Q= e^-∑_ia_i^2-∑_i b_i^2-∑_i,ja_ij^2-∑_i,jb_ij^2/π^n+1+n(n+1)/2_Qwhere _Q is the Riemannian volume form ofQ. The remaining part of this section is devoted to the proof of Proposition <ref>. Our main tool will be the peak sections defined in Section<ref>.We fix a point x∈X (resp. x∈ℝ X) and we want to compute the integral∫_π_ℝ H(π^-1_ℝ X(x))1/|(π_ℝ H^-1)^*Jac_N(π_ℝ X)|dμ_|π_ℝ H(π^-1_ℝ X(x))that appears in (<ref>).We recall that the tangent space of ℐ (resp. ℝℐ) at (α,β,x) is {(α̇,β̇,ẋ)∈ H^0(X;ℒ^d)^2× T_xX|(α̇∇β-β∇α̇ +α∇β̇-β̇∇α +α∇^2_(ẋ,.)β-β∇^2_(ẋ,.)α)(x)=0}(resp.{(α̇,β̇,ẋ)∈ℝ H^0(X;ℒ^d)^2× T_xℝ X| (α̇∇β-β∇α̇ +α∇β̇-β̇∇α +α∇^2_(ẋ,.)β-β∇^2_(ẋ,.)α)(x)=0}.)We remark that dπ_H|(α,β,x) is (almost everywhere) an isometry, because on ℐ we put the (singular) metric π_H^*⟨· ,·⟩_L^2. For any x∈ X (resp. x∈ℝ X) we will computethe normal Jacobian (π_H^-1)^*Jac_N(π_X) (resp. (π_ℝ H ^-1)^*Jac_N(π_ℝ X)) at a point (α,β)∈π_H(π_X^-1(x)) by using the following two linear maps: A_αβ:H^0(X;ℒ^d)× H^0(X;ℒ^d)→ T_x^*X⊗ℒ_x^2d(resp. A_αβ:ℝ H^0(X;ℒ^d)×ℝ H^0(X;ℒ^d)→ℝ(T^*X⊗ℒ^2d)_x)andB_αβ:T_x X→ T_x^*X⊗ℒ_x^2d (resp. B_αβ:T_xℝ X→ℝ(T^*X⊗ℒ^2d)_x) defined byA_αβ(α̇,β̇)=(α̇∇β-β∇α̇ +α∇β̇-β̇∇α)(x)andB_αβ(ẋ)= (α∇^2_(ẋ,.)β-β∇^2_(ẋ,.)α)(x)On T_x^*X⊗ℒ_x^2d (resp. ℝ(T^*X⊗ℒ^2d)_x) we have the Hermitian (resp. scalar) product induced by h. Following the notation of Sections <ref> and <ref>,for any x∈ X (resp. x∈ℝ X) and any (α,β)∈π_ H(π^-1_X(x)) (resp. π_ℝ H(π^-1_ℝ X(x))), we have((π_H^-1)^*Jac_Nπ_X)(α,β)=Jac_N(A_αβ)/Jac(B_αβ),where A_αβ and B_αβ are the linear maps defined in (<ref>) and (<ref>).Recall that a vector (α̇,β̇,ẋ)∈ H^0(X;ℒ^d)^2× T_xX is in the tangent space of ℐ at (α,β,x) if and only if (α̇∇β-β∇α̇ +α∇β̇-β̇∇α +α∇^2_(ẋ,.)β-β∇^2_(ẋ,.)α) (x)=0.In particular, the vector ẋ is uniquely determined byẋ=-(α∇^2β-β∇^2α)(x)^-1∘(α̇∇β-β∇α̇ +α∇β̇-β̇∇α)(x).The map ((π_H^-1)^*dπ_X)(α,β) sends (α̇,β̇) to ẋ and then, by (<ref>) and by the definition of A_αβ and B_αβ, we have ((π_H^-1)^*dπ_X)(α,β)=-B_αβ^-1∘ A_αβ. Passing to the normal Jacobian and using that π_H is a local isometry, we get ((π_H^-1)^*Jac_Nπ_X)(α,β)=Jac(B_αβ)^-1Jac_N(A_αβ).Fixreal holomorphic coordinates (x_1,…,x_n) in a neighborhood of a point x∈ℝ X such that (∂/∂ x_1,…,∂/∂ x_n) is an orthonormal basis of T_xX (resp. T_xℝ X). We want to compute the integral∫_π_ℝ H(π^-1_ℝ X(x))1/|(π_ℝ H^-1)^*Jac_N(π_ℝ X)|dμ_|π_ℝ H(π^-1_ℝ X(x))that appears in (<ref>). For any(α,β) ∈ H^0(X;ℒ^d)^2 (resp. ℝ H^0(X;ℒ^d)^2) we haveα=∑_i=0^n a_iσ_i+∑_1⩽ k⩽ l⩽ na_klσ_kl+τ β=∑_i=0^n b_iσ_i+∑_1⩽ k⩽ l⩽ nb_klσ_kl+τ'where τ, τ'∈ J^2_x and σ_i,σ_kl are the peak section of Lemma <ref>.We remark that (α,β)∈π_H(π^-1_X(x)) if andonly if a_0b_i-a_ib_0=0 ∀ i=1,…,n, and also that the definition of Jac_N(π_X) involves only the 2-jets of sections. With this remark in mind we define the following spaces: * K_2≑( J^2_x× J^2_x)⊂ H^0(X;ℒ^d)^2 (resp. ℝ H^0(X;ℒ^d)^2);* H_2≑ Vect{(σ_i,0),(σ_kl,0),(0,σ_i),(0,σ_kl)}⊂ H^0(X:ℒ^d)^2 (resp. ℝ H^0(X;ℒ^d)^2) for i=0,…,n and 1≤ l≤ k≤ n;* Q=H_2∩π_H(π^-1_X(x)).We see Q as the product of the intersection of quadrics:Q̃={(a_0,b_0,…,a_n,b_n)∈ℝ^2(n+1)| a_0b_i-a_ib_0=0∀ i=1,…,n}with the vector space ℝ^n(n+1) of coordinates a_ij and b_ij for 1≤ i≤ j≤ n.Let π_2:K_2^⊥→ H_2 be the orthogonal projection. A consequence of Proposition <ref> is that, for large d, the map π_2 is invertible.Following the notation of Section <ref> and <ref>, let A_αβ and B_αβ be the linear applications defined in (<ref>) and (<ref>). Then, in the complex case, under the hypothesis of Theorem <ref>, (π_2^-1)_*Jac_N(A_αβ)=(πδ^2_ℒd^2n+1((a_ia̅_j+b_ib̅_j)E_ij +(|a_0|^2+ |b_0|^2)Id+O(1/√(d)))_ij) (π_2^-1)_*Jac(B_αβ)=|(πδ_ℒ√(d)^2n+2((a_0b_ij-b_0a_ij)Ẽ_ij +O(1/√(d)))_ij)|^2and, in the real case, under the hypothesis of the Theorem <ref>, (π_2^-1)_*Jac_N(A_αβ)= √((πδ_ℒ^2d^2n+1((a_ia_j+b_ib_j)E_ij+(a_0^2+b_0^2)Id+O(1/√(d)))_ij)) (π_2^-1)_*Jac(B_αβ)=(πδ_ℒ√(d)^2n+2((a_0b_ij-b_0a_ij)Ẽ_ij +O(1/√(d)))_ij) where Ẽ_ij for 1⩽ i⩽ j⩽ n and E_ij for i,j=1,…,n are the matrices defined in Definition <ref>.Let e be a local trivialization of ℒ at x as in Section <ref> and let(σ_i)_i=0,…,n,(σ_kl)_1⩽ k⩽ l⩽ n be as in Lemma <ref> (resp. Lemma <ref>). For any (α,β) ∈ H^0(X;ℒ^d)^2 (resp. ℝ H^0(X;ℒ^d)^2) we have α=∑_i=0^n a_iσ_i+∑_1⩽ k⩽ l⩽ na_klσ_kl+τ β=∑_i=0^n b_iσ_i+∑_1⩽ k⩽ l⩽ nb_klσ_kl+τ'where τ, τ'∈ J^2_x. In particular, we haveα(x)=a_0σ_0(x),β(x)=b_0σ_0(x), ∇α(x)=∑_i=0^na_i∇σ_i(x),∇β(x)=∑_i=0^nb_i∇σ_i(x), ∇^2α(x)=∑_i=0^na_i∇^2σ_i(x)+∑_k,l a_kl∇^2σ_kl(x),∇^2β(x)=∑_i=0^nb_i∇^2σ_i(x)+∑_k,l b_kl∇^2σ_kl(x).As basis for T_xX and T^*_xX⊗ℒ^2d_x (resp.T_xℝ X and ℝ(T^*_xX⊗ℒ^2d_x)) we choose (∂/∂ x_1,…,∂/∂ x_n) and (dx_1⊗ e^2d,…,dx_n⊗ e^2d) respectively. We choose (σ_i,0) and(0,σ_i), i=0,…,n, as a basis of a complement of J_x^1× J^1_x.Thanks to Lemma <ref>, this basis is asymptotically orthonormal for the L^2-Hermitian product ⟨·,·⟩_L^2. By definition it is an orthonormal basis for the scalar product (π_2^-1)_*⟨·,·⟩_L^2 restricted to H_2 .Then we obtain, using Lemma <ref>,⟨ A_αβ(σ_0,0),dx_j⊗ e^2d⟩ =b_j√(π)δ_ℒ√(d)^2n+1+O(√(d)^2n); ⟨ A_αβ(σ_i,0),dx_j⊗ e^2d⟩= -b_0√(π)δ_ℒ√(d)^2n+1δ_ij+O(√(d)^2n) for i=1,…,n; ⟨ A_αβ(0,σ_0),dx_j⊗ e^2d⟩= -a_j√(π)δ_ℒ√(d)^2n+1+O(√(d)^2n); ⟨ A_αβ(0,σ_i),dx_j⊗ e^2d⟩= a_0√(π)δ_ℒ√(d)^2n+1δ_ij+O(√(d)^2n) for i=1,…,n; ⟨ B_αβ(∂/∂ x_i),dx_j⊗ e^2d⟩ =(a_0b_ij-b_0a_ij)πδ_ℒ√(d)^2n+2+O(√(d)^2n+1) for i≠ j;⟨ B_αβ(∂/∂ x_k),dx_k⊗ e^2d⟩ =√(2)(a_0b_kk-b_0a_kk)πδ_ℒ√(d)^2n+2+O(√(d)^2n+1).where the Hermitian (resp. scalar) product ⟨·,·⟩on T^*_xX⊗ℒ^2d_x(resp. ℝ(T^*_xX⊗ℒ^2d_x)) isinduced by the Hermitian metric h on ℒ. What we have just computed are the coefficients of the matrices of A_αβ and B_αβ with respect to our choice of basis and with respect to the scalar product (π_2^-1)_*⟨·,·⟩_L^2. We recall that B_αβ is a square matrix and that Jac_N(A_αβ)=√(Jac(A_αβA_αβ^*)).More precisely, as d→∞ , A_αβ is equivalent to the following matrix:√(π)δ_ℒ√(d)^2n+1[b_1 -b_00…0 -a_1a_00…0;b_20 -b_0…0 -a_20a_0…0;…………………………;b_n00… -b_0 -a_n00…a_0 ] and B_αβ to the following one: πδ_ℒ√(d)^2n+2[ √(2)(a_0b_11-b_0a_11) a_0b_12-b_0a_12 … a_0b_1n-b_0a_1n; a_0b_21-b_0a_21 √(2)(a_0b_22-b_0a_22) … a_0b_2n-b_0a_2n; … … … …; a_0b_n1-b_0a_n1 a_0b_n2-b_0a_n2 … √(2)(a_0b_nn-b_0a_nn) ] A direct computation shows us that A_αβA_αβ^* is the matrix(πδ^2_ℒd^2n+1) ((a_ia̅_j+b_ib̅_j)E_ij +(|a_0|^2+|b_0|^2)Id+O(1/d)).The results follows. By Proposition <ref> we have (π_2^-1)_*1/Jac_N(π_X)=(π d)^n ( Jac_ℝ((a_0b_ij-b_0a_ij)Ẽ_ij)/((a_ia̅_j+b_ib̅_j)E_ij+ (|a_0|^2+|b_0|^2)Id) +O(1/√(d))) (π_2^-1)_*1/Jac_N(π_ℝ X)=√(π d)^n(((a_0b_ij-b_0a_ij)Ẽ_ij)/√(((a_ia_j+b_ib_j)E_ij+(a_0^2+b_0^2)Id)) +O(1/√(d))).We want to integrate this quantity over π_ℝ H(π^-1_ℝ X(x)). We recall that the measure dμ_|π_ℝ H(π^-1_ℝ X(x)) is the following one: first we restrict the scalar product ⟨·,·⟩_L^2 on ℝ H^0(X;ℒ^d)^2 to π_ℝ H(π^-1_ℝ X(x)), that is a codimension n submanifold, thenweconsider the Riemannian measure associated with this metric, and finally we multiply it by thefactor 1/π^N_de^-α_L^2^2-β_L^2^2, where N_d= H^0(X;ℒ^dℒ^d). Then(<ref>) is equal to ∫_π_ℝ H(π^-1_ℝ X(x))|Jac(B_αβ)|/|Jac_N(A_αβ)|dμ_|π_ℝ H(π^-1_ℝ X(x)) =∫_K_2^⊥∩π_ℝ H(π^-1_ℝ X(x))⊕ K_2|Jac(B_αβ)|/| Jac_N(A_αβ)|dμ_|π_H(π^-1_ℝ X(x))==∫_K_2^⊥∩π_ℝ H(π^-1_ℝ X(x))|Jac(B_αβ)|/|Jac_N(A_αβ)|dμ_| K_2^⊥∩π_ℝ H(π^-1_ℝ X(x)) =∫_Q(π_2^-1)_*|Jac(B_αβ)|/|Jac_N(A_αβ)|(π_2*dμ_| K_2^⊥∩π_ℝ H(π^-1_ℝ X(x))).By Proposition <ref>, the pushforward measure (π_2)_*(μ_| K_2^⊥) on H_2 coincides with the Gaussian measure associated with the orthonormal basis {(σ_i,0),(σ_kl,0),(0,σ_i),(0,σ_kl)}_1≤ i≤ n 1≤ k≤ l≤ n up to a O(1/√(d)) term. As a consequence we have that (π_2*dμ_| K_2^⊥∩π_ℝ H(π^-1_ℝ X(x))) is equal todμ_Q=e^-∑_ia_i^2-∑_i b_i^2-∑_i,ja_ij^2-∑_i,jb_ij^2/π^n+1+n(n+1)/2_Qup to a O(1/√(d)) term, where _Q is the Riemannian volume form ofQ. We then have that (<ref>) is equal to∫_Q(π_2^-1)_*|Jac(B_αβ)|/|Jac_N(A_αβ)|dμ_Q+O(1/√(d))==∫_[ a_i,b_i,a_ij,b_ij; a_0b_i-b_0a_i=0 ]√(π d)^n(|((a_0b_ij-b_0a_ij)Ẽ_ij)|/√(((a_ia_j+b_ib_j)E_ij+(a_0^2+b_0^2)Id))dμ_Q +O(1/√(d))).Putting (<ref>) in (<ref>) and using Proposition <ref>, we obtain Proposition <ref>. §.§ Computation of the universal constantThe purpose of this section is the explicit computation of the functionR_d(x) that appears in Proposition <ref>. We use the notation of Section <ref>.To understand R_d(x), we have to compute √(π d)^n∫_Q| ((a_0b_ij-b_0a_ij)Ẽ_ij)|/√(((a_ia_j+b_ib_j)E_ij+(a_0^2+b_0^2)Id))e^-∑_ia_i^2-∑_i b_i^2-∑_i,ja_ij^2-∑_i,jb_ij^2/π^n+1+n(n+1)/2_Q.The main result of this section is the following computation: Let Q be as in Proposition <ref>. Then√(π d)^n∫_Q| ((a_0b_ij-b_0a_ij)Ẽ_ij)|/√(det((a_ia_j+b_ib_j)E_ij+(a_0^2+b_0^2)Id))e^-∑_ia_i^2-∑_i b_i^2-∑_i,ja_ij^2-∑_i,jb_ij^2/π^n+1+n(n+1)/2_Q is equal to{[ n!!/(n-1)!!e_ℝ(n)π/2√(d)^nif n is odd;n!!/(n-1)!!e_ℝ(n)√(d)^nif n is even. ].where Ẽ_ij and E_ij are the matrices of Definition <ref> and e_ℝ(n) is the expected value of the determinant of (the absolute value of) the real symmetric matrices.We recall that Q⊂ℝ^2(n+1)+n(n+1) is the product of the intersection of quadrics:Q̃={(a_0,b_0,…,a_n,…,b_n)∈ℝ^2(n+1)| a_0b_i-a_ib_0=0∀ i=1,…,n}with ℝ^n(n+1) of coordinates a_ij and b_ij for 1≤ i≤ j≤ n. We consider the parametrizationψ:ℝ^(n+2)→Q̃ defined byψ(a,b,t_1,…,t_n)=(a,b,at_1,bt_1,…,at_n,bt_n). We have Jac (ψ)=√(1+∑_it_i^2)√((a^2+b^2))^n. A computation gives usJacψJacψ^t=[ 1+∑_i=1^nt_i^20 t_1a t_2a…… t_na;0 1+∑_i=1^nt_i^2 t_1b t_2b…… t_nb; t_1a t_1ba^2+b^20……0; t_2a t_2b0a^2+b^2……0;…………………;…………………; t_na t_nb00……a^2+b^2 ]We develop the last line and we obtain(a^2+b^2)[ 1+∑_i=1^nt_i^20 t_1a t_2a…… t_n-1a;0 1+∑_i=1^nt_i^2 t_1b t_2b…… t_n-1b; t_1a t_1ba^2+b^20……0; t_2a t_2b0a^2+b^2……0;…………………;…………………; t_n-1a t_n-1b00……a^2+b^2 ] +(-1)^nt_nb·[ 1+∑_i=1^nt_i^2 t_1a t_2a…… t_na;0 t_1b t_2b…… t_nb; t_1aa^2+b^20……0; t_2a0a^2+b^2……0;………………;……………0; t_n-1a00…a^2+b^20 ] +(-1)^n+1t_na·[0 t_1a t_2a…… t_na; 1+∑_i=1^nt_i^2 t_1b t_2b…… t_nb; t_1ba^2+b^20……0; t_2b0a^2+b^2……0;………………;……………0; t_n-1b00…a^2+b^20 ] For the second matrix we have: [ 1+∑_i=1^nt_i^2 t_1a t_2a…… t_na;0 t_1b t_2b…… t_nb; t_1aa^2+b^20……0; t_2a0a^2+b^2……0;………………;……………0; t_n-1a00…a^2+b^20 ]=(1+∑_i=1^nt_i^2)[t_1bt_2b … … …t_nb; a^2+b^2 0 … … 0 0; 0 a^2+b^2 … … 0 0; … … … … … …; … … … a^2+b^2 0 0; 0 … … 0 a^2+b^2 0 ] =(-1)^n+1(1+∑_i=1^nt_i^2)t_nb(a^2+b^2)^n-1.where the first equality is obtained by developping the first column and remarking that, in the development, each time we clear the i-th line, the (i-1)-th column and the last column are linearly equivalent. Similarly, [0 t_1a t_2a…… t_na; 1+∑_i=1^nt_i^2 t_1b t_2b…… t_nb; t_1ba^2+b^20……0; t_2b0a^2+b^2……0;………………;……………0; t_n-1b00…a^2+b^20 ] =(-1)^n(1+∑_i=1^nt_i^2)t_na(a^2+b^2)^n-1.Then we haveJacψJacψ^t=(a^2+b^2)[ 1+∑_i=1^nt_i^20 t_1a t_2a…… t_n-1a;0 1+∑_i=1^nt_i^2 t_1b t_2b…… t_n-1b; t_1a t_1ba^2+b^20……0; t_2a t_2b0a^2+b^2……0;…………………;…………………; t_n-1a t_n-1b00……a^2+b^2 ] -(1+∑_i=1^nt_i^2)t_n^2(a^2+b^2)^n.Continuing to develop in the same way, we obtain by inductionJacψJacψ^t=(1+∑_i=1^nt_i^2)^2(a^2+b^2)^n-(1+∑_i=1^nt_i^2) ∑_i=1^nt_i^2(a^2+b^2)^n=(1+∑_i=1^nt_i^2)(a^2+b^2)^n.Passing to the square root we obtain the result. In the following we will not write the symbols Ẽ_ij and E_ij defined in Definition <ref> in order to simplify the notation. After this change of variables, we have: √(π d)^n∫_Q|(a_0b_ij-b_0a_ij)|/√(((a_ia_j+b_ib_j)+(a_0^2+b_0^2)Id))e^-∑_ia_i^2-∑_i b_i^2-∑_i,ja_ij^2-∑_i,jb_ij^2/π^n+1+n(n+1)/2_Q= =√(π d)^n∫_a,b,a_ij,b_ij,t_i∈ℝ|(ab_ij-ba_ij)|/√(((a^2+b^2)((t_it_j)+Id)))× ×e^-(1+∑_it_i^2)(a^2+b^2)-∑_i,j(a_ij^2+ b_ij^2)/π^n+1+n(n+1)/2√((1+∑_it_i^2))√((a^2+b^2))^ndadbda_ijdb_ijdt_iNow ((a^2+b^2)((t_it_j)_ij+Id))=(1+∑_it_i^2)(a^2+b^2)^n so we obtain√(π d)^n∫_a,b,a_ij,b_ij,t_i∈ℝ|((ab_ij-ba_ij)_ij)| e^-(1+∑_it_i^2)(a^2+b^2)-∑_i,j(a_ij^2+ b_ij^2)/π^n+1+n(n+1)/2dadbda_ijdb_ijdt_i.It is more practical to see (a,b) as a complex number c∈ℂ and also (a_ij,b_ij) as e_ij∈ℂ. With a slight abuse of notation, we denote dc and de_ij instead of-1/2idcdc̅ and -1/2ide_ijde̅_ij. Then we have√(π d)^n∫_c∈ℂ,e_ij∈ℂ,t_i∈ℝ|((c̅e_ij))| e^-(1+∑_it_i^2)| c |^2-∑_i,j|e_ij|^2/π^n+1+n(n+1)/2dcde_ijdt_i.Now, set c̃=(√(1+∑_it_i^2))c and then c̃=re^iϑ. We obtain √(π d)^n∫_c̃∈ℂ,e_ij∈ℂ|((c̅̃̅e_ij))|e^-|c̃|^2-∑_i,j|e_ij|^2/π^1+n(n+1)/2dc̃de_ij·∫_t_i∈ℝ1/π^n√(1+∑_it_i^2)^n+1dt_i= √(π d)^n∫_ϑ∈ (0,2π],e_ij∈ℂ|((e^-iϑe_ij))|e^-∑_i,j| e_ij|^2/π^n(n+1)/2de_ijdϑ·∫_r=0^+∞r^n+1e^-r^2/πdr·∫_t_i∈ℝ1/π^n√(1+∑_it_i^2)^n+1dt_i.Then, wehave to compute the three integrals appearing in the last equation. For the first term, we have∫_ϑ∈ (0,2π],e_ij∈ℂ|((e^-iϑc_ij))|e^-∑_i,j |e_ij|^2/π^n(n+1)/2de_ijdϑ= =2π∫_e_ij∈ℂ|( e_ij)|e^-∑_i,j|e_ij|^2/π^n(n+1)/2de_ij= =2π∫_b_ij∈ℝ|(b_ij)|e^-∑_i,jb_ij^2/√(π)^n(n+1)/2db_ij=2π e_ℝ(n).Here, e_ℝ(n)=∫_B∈(n,ℝ)| B| dμ_ℝ(B). For the explicit value of e_ℝ(n), see <cit.>. For the second term, we consider the change of variable r^2=ρ and we obtain ∫_r=0^+∞r^n+1e^-r^2/πdr= 1/2∫_ρ=0^+∞ρ^n/2e^-ρ/πdρ=Γ(n/2+1)/2πwhere Γ is the Gamma function. For the third term we use spherical coordinates and we obtain∫_t_i∈ℝ1/√(1+∑_it_i^2)^n+1dt_i=Vol(S^n-1)∫_t=0^+∞t^n-1/√((1+t^2))^n+1dt.whereVol(S^n-1)=2π^n/2/Γ(n/2)is the volume of the (n-1)-dimensional sphere. For ∫_t=0^+∞t^n-1/(√(1+t^2))^n+1dt= ∫_t=0^+∞√(t^2/(1+t^2))^n-11/1+t^2dtwe make the change t^2/1+t^2=1-u^2 and we obtain∫_0^1 √(1-u^2)^n-2du. Finally, set u=sin(θ) and we have ∫_0^π/2cos^n-1(θ) dθ.The formula∫cos^n-1(θ)du=sin(θ)cos^n-2(θ)/n-1+n-2/n-1∫cos^n-3(θ)dutells us that ∫_0^π/2cos^n-1(θ) dθ is equal to (n-2)!!/(n-1)!!if n is even and it is equal to(n-2)!!/(n-1)!!π/2if n is odd. Putting together all the values of these three integrals and using thatΓ(n/2+1)/Γ(n/2)=n/2,we obtain Proposition <ref>. §.§ End of the proofsof Theorems <ref>, <ref> and<ref>We use the notations of Sections <ref>, <ref> and <ref>. By Proposition <ref>, we have𝔼[ℝν_αβ](φ)=∫_ℝ Xφ(x)(∫_Q|Jac_N(A_αβ)|/| Jac(B_αβ)|dμ(a_i,b_i,a_kl,b_kl)+O(1/d))_h==∫_ℝ Xφ(x)(∫_[ a_i,b_i,a_ij,b_ij; a_0b_i-b_0a_i=0 ]√(π d)^n((a_0b_ij-b_0a_ij)/√(((a_ia_j+b_ib_j)+(a_0^2+b_0^2)Id)) +O(1/√(d)))dμ_Q )_h wheredμ_Q= e^-∑_ia_i^2-∑_i b_i^2-∑_i,ja_ij^2-∑_i,jb_ij^2/π^n+1+n(n+1)/2_Q. By Proposition <ref>we have that the inner term of the last equation√(π d)^n∫_[ a_i,b_i,a_ij,b_ij; a_0b_i-b_0a_i=0 ]|(a_0b_ij-b_0a_ij)|/√(((a_ia_j+b_ib_j)+(a_0^2+b_0^2)Id))e^-∑_ia_i^2-∑_i b_i^2-∑_i,ja_ij^2-∑_i,jb_ij^2/π^n+1+n(n+1)/2_Q is equal to{[ n!!/(n-1)!!e_ℝ(n)π/2√(d)^nif n is odd;n!!/(n-1)!!e_ℝ(n)√(d)^nif n is even. ].Theorem <ref> is then obtained by dividing Equation (<ref>) by √(d)^n and then by passing to the limit.Theorem <ref> isTheorem <ref> for φ=1.The proof of Theorem <ref> follows the lines the proof of Theorem <ref>. For the computation of the universal constant in this case, weput φ=1 and we use Proposition <ref>. plain | http://arxiv.org/abs/1707.08490v2 | {
"authors": [
"Michele Ancona"
],
"categories": [
"math.AG",
"math.DG",
"math.PR"
],
"primary_category": "math.AG",
"published": "20170726152015",
"title": "Expected number and distribution of critical points of real Lefschetz pencils"
} |
On Non-Orthogonal Multiple Access with Finite-Alphabet Inputs in Z-Channels Zheng Dong, He Chen, Jian-Kang Zhang, and Lei Huang Z. Dong is with Shenzhen University, Shenzhen 518060, China, and also with McMaster University, Hamilton, ON L8S 4K1, Canada (e-mail: [email protected]).H. Chen is with The University of Sydney, Sydney, NSW 2006, Australia (e-mail: [email protected]). J.-K. Zhang is with McMaster University, Hamilton, ON L8S 4K1, Canada (e-mail: [email protected]). L. Huang is with Shenzhen University, Shenzhen 518060, China (e-mail: [email protected]). May 29, 2018 =============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================This paper focuses on the design of non-orthogonal multiple access (NOMA) in a classical two-transmitter two-receiver Z-channel, wherein one transmitter sends information to its intended receiver from the direct link while the other transmitter sends information to both receivers from the direct and cross links. Unlike most existing designs using (continuous) Gaussian input distribution, we consider the practical finite-alphabet (i.e., discrete) inputs by assuming that the widely-used quadrature amplitude modulation (QAM) constellations are adopted by both transmitters. To balance the error performance of two receivers, we apply the max-min fairness design criterion in this paper. More specifically, we propose to jointly optimize the scaling factors at both transmitters, which control the minimum Euclidean distance of transmitting constellations, to maximize the smaller minimum Euclidean distance of two resulting constellations at the receivers, subject to an individual average power constraint at each transmitter. The formulated problem is a mixed continuous-discrete optimization problem and is thus intractable in general. By resorting to the Farey sequence, we manage to attain the closed-form expression for the optimal solution to the formulated problem. This is achieved by dividing the overall feasible region of the original optimization problem into a finite number of sub-intervals and deriving the optimal solution in each sub-interval. Through carefully observing the structure of the optimal solutions in all sub-intervals, we obtain compact and closed-form expressions for the optimal solutions to the original problem in three possible scenarios defined by the relative strength of the cross link. Simulation studies are provided to validate our analysis and demonstrate the merits of the proposed design over existing orthogonal or non-orthogonal schemes. Non-orthogonal multiple access (NOMA), Z-channel, finite-alphabet inputs, quadrature amplitude modulation, max-min fairness, Farey sequence.§ INTRODUCTIONMultiple access technologies have been playing an important role in determining the performance of each generation of mobile communication systems. Based on how the resources are allocated to users, multiple access technologies can generally be categorized into two types: orthogonal multiple access (OMA) and non-orthogonal multiple access (NOMA) <cit.>. The current generation of cellular networks, known as 4G, and all previous generations have primarily adopted the OMA technologies, which include frequency-division multiple access (FDMA) for 1G, time-division multiple access (TDMA) for 2G, code-division multiple access (CDMA) for 3G, and orthogonal frequency-division multiple access (OFDMA) for 4G <cit.>. In these OMA schemes, the resource is partitioned into orthogonal blocks in time/frequency/code domain, and each resource block is then assigned to one single user exclusively. In this sense, there is no inter-user interference in OMA, leading to low-complexity receiver and scheduling algorithms. Moreover, after the resource allocation, the multiple-user problem is divided into several point-to-point problems such that the well-established single-user encoder/decoder techniques can be directly applied. However, early information-theoretic studies showed that compared with NOMA, OMA has lower spectral efficiency as it normally cannot achieve the multi-user channel capacity region <cit.>. Besides, OMA is not scalable as the total number of orthogonal resources and their granularity strictly limit the maximum number of served users.Different from OMA, NOMA exploits the power domain to multiplex multiple users together such that they can be served in the same time/frequency/code resources <cit.>. As such, with proper multi-user detection techniques to deal with the inter-user interference at the receiver side (e.g., successive interference cancellation (SIC) <cit.>), NOMA is capable of achieving improved spectral efficiency and serving much more users simultaneously. In fact, the uplink and downlink versions of NOMA, well-known as multiple access channel (MAC) and broadcast channel (BC) respectively, have been intensively investigated for several decades in the information theory community, see, e.g., <cit.>. However, due to the high complexity of interference cancellation, these studies mainly lied in the theoretical aspects and their results were not implemented in practical communication systems. With the fast advances of hardware, the implementation of NOMA with interference cancellation becomes more affordable and feasible. Actually, NOMA has been regarded as a key enabling technology to meet the unprecedented requirements of 5G wireless networks due to its significant network throughput gain and great potential to support massive connectivity, low latency and user fairness <cit.>. Furthermore, a two-user downlink scenario of NOMA, termed multiuser superposition transmission (MUST), has been incorporated in the 3rd Generation Partnership Project (3GPP) Long Term Evolution-Advanced (LTE-A) <cit.>.Most conventional information-theoretic and recent studies on NOMA adopted the assumption of Gaussian input distribution <cit.>. Although the designs with Gaussian signaling can approach most of the known capacity inner bounds, such as in <cit.>, their direct implementation in practical communication systems may lead to significant performance loss <cit.>.Moreover, Gaussian signaling will require unaffordable encoding and decoding efforts, which could lead to extremely high hardware cost, huge storage capability, high computational complexity, and long delay. Therefore, Gaussian inputs could arguably be infeasible for current hardware and it acts mostly as the theoretical benchmark. The inputs of practical wireless systems are actually drawn from finite constellations, such as phase shift keying (PSK) modulation or quadrature amplitude modulation (QAM), which are essentially different from the continuous Gaussian inputs.When it comes to a NOMA system with finite input constellations, the key design challenge is to guarantee that each user's codeword can be uniquely decoded from their sum signal at the receiver side <cit.>. For the two-user MAC with finite-alphabet inputs, a constellation rotation (CR) scheme and a constellation power allocation (CPA) scheme were proposed in～<cit.> and～<cit.> to construct an unambiguous sum constellation at the receiver, respectively. This is achieved by strategically introducing certain angle of rotation between the input constellations in the CR scheme and appropriately controlling the transmit power of each user in the CPA scheme. The results in～<cit.> and～<cit.> have been extended to various multiple-antenna scenarios, see, e.g., <cit.> and references therein. The aforementioned NOMA designs were primarily for the PSK modulations by utilizing its circular structure. The studies on NOMA with QAM, another practical modulation scheme that has been widely adopted in cellular systems due to its higher spectral efficiency, are quite limited. Very recently, the mutual information were used as the performance metric to optimize NOMA systems with QAM in <cit.>. However, the optimal NOMA designs in <cit.> were achieved by numerical approaches with high computational complexities.We also notice that the existing studies on NOMA mainly focused on MAC and BC, due to their wide applications in centralized systems like cellular networks. Also there are some initial efforts considering the NOMA design for the interference channel (IC) <cit.>.With recent advances in non-centralized networks (e.g., wireless sensor networks and ad hoc networks), Z-channel (ZC) was proposed in <cit.> and attracted considerable attention in the past decade <cit.>. As a special case of the classical two-user IC <cit.>, a two-user ZC consists of two transmitters and two receivers, wherein one transmitter sends information to its intended receiver from the direct link without causing interference to the unintended receiver, while the other transmitter sends information to both receivers from the direct and cross links. In this sense, the ZC is a general channel model that includes MAC and BC as special cases. The ZC is also closely related to the Z-interference channel (ZIC) <cit.>, wherein each transmitter transmits information to its corresponding receiver only from the direct link and the received signal from thecross link carries no desired information, and thus is treated as interference at the receiver side. It is also worth emphasizing that there are two messages transmitted from the two direct links in ZIC, while three messages are sent via the two direct links and the cross link in ZC.Both ZC and ZIC are proper models for the multi-cell downlink transmission, where one user is located near to the cell edge and thus can receive signals from both transmitters, while the other user is near the cell center and suffers from no interference. Another example corresponds to the two-user IC, where one cross link is blocked by obstacles with large pathloss such as tall buildings or thick walls, while the other user is still exposed to interference <cit.>. Despite of the existing great efforts on ZC, to our best knowledge, the design of NOMA with practical finite-alphabet inputs in ZC is still an open problem in the literature.Motivated by this gap, in this paper we concentrate on the practical design of NOMA with QAM in a two-transmitter two-receiver ZC. It is worth emphasizing that the design of NOMA with QAM is much more challenging than that for PSK modulation. This is mainly because the unambiguity of sum QAM constellations is much harder to maintain since its signal points are distributed more evenly and there is a higher probability that more than one signal points coincide or close to each other on the sum constellation. The main contributions of this paper can be summarized as follows: * We, for the first time, develop a practical NOMA framework with QAM and max-min fairness in ZC. In our framework, we optimize the scaling factors of both transmitters, which adjust the minimum Euclidean distance of the transmitting constellations, to maximize the smaller minimum Euclidean distance among the resulting constellations at both receivers subject to an individual power constraint on each transmitter. Through our design, the average error performance of both transmitters in the considered ZC can be minimized with good user fairness, which is fundamentally different from the existing designs that mainly focused on the channel capacity maximization. * The formulated optimization problem is shown as a mixed continuous-discrete optimization problem, which is challenging to solve in general. By carefully observing the features of the formulated problem, we realize that the Farey sequence (also known as Farey series) <cit.> can be applied to resolve the problem. More specifically, by taking the advantage of Farey sequence associated with the finite-alphabet, we strategically partition the entire feasible region of the original optimization problem into a finite number of sub-intervals and attain the closed-form solution in each sub-interval. Then, by a careful observation on the structure of the solutions in all sub-intervals, the overall solution is obtained in a compact closed-form for three complementary scenarios divided by the relative strength of the cross link. * We verify the correctness of the analytical results by conducting simulations in both deterministic and random fading channels.Simulation results show that the sum constellation at the receiver side is still a regular QAM constellation with a larger size for most scenarios, but could be a hierarchical QAM with two-resolution <cit.>, e.g., when the cross link is very strong relative to the direct link. We adopt the bit error rate (BER) as the performance metric to compare the proposed NOMA design with the existing OMA and NOMA schemes under random fading channels. The comparison illustrates that our scheme can achieve a significant lower BER performance than the benchmark schemes, which validates the effectiveness of our design. § SYSTEM MODEL OF COMPLEX GAUSSIAN ZC WITH QAM CONSTELLATIONSWe consider a two-user complex Gaussian ZC consisting of two transmitters S_1 and S_2, and two receivers D_1 and D_2, as depicted in Fig. <ref>. We consider that each node is equipped with a single antenna and works in a half-duplex mode. As per the ZC, only the cross link between S_2 and D_1 is assumed to be available. Moreover, S_1 sends one unicasting message x_1 to D_1, while S_2 transmits one multicasting message x_2 to both D_1 and D_2. S_1 and S_2 transmit their messages simultaneously using the same frequency band. The equivalent complex baseband signals observed at D_1 and D_2 can be given, respectively, by the following equations: z_1=h_11 x_1+ h_21 x_2 +ξ_1, z_2=h_22 x_2 +ξ_2, where h_kℓ∈ℂ, k,ℓ∈{1,2} denotes the complex channel coefficients from transmitter S_k to receiver D_ℓ. Hereafter, we call h_11 and h_22the direct links, while h_21 is referred to as the cross link. In line with <cit.>, all the channel linksare assumed to be known perfectly at all the terminals. The additive noise processes ξ_1, ξ_2 ∼𝒞𝒩(0, 2σ^2) are independent and identically distributed (i.i.d.) over time and are assumed to be circularly symmetric complex Gaussian (CSCG). Note that the case with different noise levels at receivers can be incorporated into our model by scaling operations. We suppose that QAM constellations are used by both transmitters since it is more spectrally efficient than other frequently-used modulation schemes such as phase-shift keying (PSK), and is also relatively easy to implement <cit.>. A predefined average transmitted power constraint is imposed to both transmitters[Our design can also be generalized to the case with peak power constraint straightforwardly.], i.e., 𝔼[|x_1|^2]≤ P_1 and 𝔼[|x_2|^2] ≤ P_2. In this paper, the system signal-to-noise ratio (SNR) is defined by ρ=1/2σ^2.For analytical simplicity, we decompose the considered complex Gaussian ZC given in (<ref>) into two parallel real scalar Gaussian ZCs <cit.>, which are called the in-phase and quadrature components, respectively. We note that this method was commonly used in the study of IC and the real IC was studied directly in <cit.>.Actually, designing two-dimensional QAM constellations is an extremely challenging problem even for two-user MAC, see e.g., <cit.> and references therein. In this paper, instead of designing the two-dimensional QAM constellations directly, we propose a practical design that decomposes the complex Gaussian ZC into two parallel real scalar Gaussian ZCs such that we can split the two-dimensional QAM constellation into two one-dimensional PAM constellations. In fact, although we use complex baseband representation in (<ref>), the actual modulated and demodulated signals are all real since the oscillator at the transmitter can only generate real sinusoids rather than complex exponentials, and the channel just introduces amplitude and phase distortion to the transmitted signals <cit.>. By this means, the original two-dimensional QAM constellation can be split into two one-dimensional pulse amplitude modulation (PAM) constellations for both the in-phase and quadrature components.Mathematically, for the complex Gaussian ZC described in (<ref>), the in-phase and quadrature components can be attained by rotating x_1 and x_2 according to the instantaneous channel coefficients to compensate for the phase offset, and then taking the real and imaginary parts, respectively. First of all, we note that (<ref>) is equivalent to z_1 =|h_11|exp(j(h_11)) x_1+|h_21|exp(j(h_21)) x_2+ξ_1 exp(j ( h_21/h_22))z_2=|h_22| exp(j(h_21)) x_2 +exp(j ( h_21/h_22))ξ_2.Now, we set y_1= Re(z_1), y_2= Re(exp (j ( h_21/h_22))z_2),w_1 s_1= Re(exp(j(h_11))x_1),w_2 s_2= Re(exp (j (h_21))x_2 ),n_1= Re(ξ_1), n_2=Re(exp(j ( h_21/h_22)) ξ_2); y_1'= Im(z_1), y_2'= Im(exp (j ( h_21/h_22))z_2),w_1' s_1'= Im(exp(j(h_11))x_1), w_2' s_2'= Im(exp (j (h_21))x_2 ),n_1'= Im(ξ_1), n_2'= Im(exp(j ( h_21/h_22)) ξ_2), where Re(·) and Im(·) are the real andimaginary parts of the complex number, respectively. We also assume that s_1 ∈𝒜_M_1, s_1' ∈𝒜_M_1', sent by S_1, and s_2 ∈𝒜_M_2,s_2' ∈𝒜_M_2', transmitted by S_2, are the information-bearing symbols, which are drawn from standard PAM constellation with equal probability, in which 𝒜_M ≜{± (2k-1)}_k=1^M/2 is a M-ary PAM constellation set. Moreover, the scaling factors w_1, w_2, w_1', and w_2' are real positive scalars that determine the minimum Euclidean distance of the corresponding PAM constellation set.From (<ref>), we obtain exp(j(h_11)) x_1=(w_1 s_1 + w_1' s_1'j) ∈𝒬_1 and exp(j(h_21)) x_2= (w_2 s_2 + w_2' s_2'j) ∈𝒬_2, where 𝒬_1 ≜{± w_1 (2k-1) ± w_1' (2ℓ-1)j:k=1,…, M_1/2, ℓ=1,…, M_1'/2},𝒬_2 ≜{± w_2 (2k-1) ± w_2' (2ℓ-1)j: k=1,…, M_2/2, ℓ=1,…, M_2'/2}, are M_1 M_1'- and M_2 M_2'-ary QAM constellations, respectively. If w_1=w_1' and w_2=w_2', we call 𝒬_1 and 𝒬_2 symmetric QAM constellations. Otherwise, we are using unsymmetric signaling <cit.>. In addition, n_1, n_2, n_1', n_2'∼𝒩(0, σ^2) are i.i.d. real additive white Gaussian since the complex noise terms are assumed to be CSCG. Then, the in-phase and quadrature sub-channels of (<ref>), as illustrated in Fig. <ref>, can be reformulated by y_1= |h_11|w_1 s_1 + |h_21| w_2 s_2 +n_1, y_2= |h_22|w_2 s_2 +n_2, y_1' = |h_11|w_1' s_1' + |h_21| w_2' s_2' +n_1',y_2' = |h_22|w_2' s_2' +n_2'.The transmitted signals over both subchannels should still be subject to average power constraints, i.e., 𝔼[w_1^2 |s_1|^2]≤ p_1, 𝔼[w_2^2|s_2|^2]≤ p_2,𝔼[w_1'^2 |s_1'|^2]≤ p_1', 𝔼[w_2'^2|s_2'|^2]≤ p_2' such that p_1+p_1'=P_1 and p_2+p_2'=P_2. The following power allocation among the in-phase and quadrature components is normally performed to balance the minimum Euclidean distance of the two PAM constellations <cit.>, i.e., p_1 =(M_1^2-1)P_1/M_1^2+M_1'^2-2, p_2=(M_2^2-1)P_2/M_2^2+M_2'^2-2, p_1' =(M_1'^2-1)P_1/M_1^2+M_1'^2-2, p_2'=(M_2'^2-1)P_2/M_2^2+M_2'^2-2. It can be observed that, if square-QAM constellations are used at both transmitters with M_1=M_1' and M_2=M_2', we have p_1=p_1'=P_1/2 and p_2=p_2'=P_2/2.An important problem for the considered ZC is that for any given QAM constellation sizes of both messages, how to optimize the values of scaling coefficients w_1, w_2, w_1' and w_2' to minimize the average error probability at both receivers, subject to the individual average power constraint at both transmitters. By leveraging the decomposable property of the complex Gaussian ZC and the symmetry of the two subchannels, we can simply focus on the design for one of the two real Gaussian ZCs with PAM constellation sets, which will be elaborated in the next section[It should be pointed out that this design is a practical but not necessarily optimal approach, which has been widely adopted in practice <cit.>.].§ THE CONSTELLATION DESIGN FOR THE REAL GAUSSIAN ZCIn this section, we consider the constellation design problem, i.e., finding the optimal values of w_1 and w_2 for the in-phase real Gaussian ZC characterized by (<ref>). The optimal solution to thequadrature component can be obtained in a similar fashion and hence omitted for brevity. In particular, if M_1=M_1' and M_2=M_2', then the two sub-channels are identical.It is worth noting that, similar design for BC or MAC can be included as a special case of our proposed design for the considered ZC. §.§ Problem FormulationAs the first effort towards the design of NOMA with finite-alphabet inputs in ZC, in this paper we concentrate on the case that M_1=M_2=M and M_1'=M_2'=M'.As a result, we have s_1, s_2 ∈𝒜_M={± (2k-1)}_k=1^M/2. As 𝔼[w_1^2 |s_1|^2]≤ p_1 and 𝔼[w_2^2|s_2|^2]≤ p_2, we thus have0< w_1 ≤√(3 p_1/M^2-1) and 0< w_2 ≤√(3 p_2/M^2-1). In our scheme, the transmitted signal from S_1 and S_2 are superimposed together at D_1, which is inherently a non-orthogonal transmission. In line with <cit.>, we use a joint decoding[We note that, for ZIC, the joint decoder used by D_1 may not necessarily be the most efficient one. Instead, we should use a treat-interference-as-noise (TIN) receiver when the channel gain of the cross link is very low and use a successive-interference-cancellation (SIC) receiver when the channel gain of cross link is very strong compared with the direct link <cit.>. In general, a joint decoder can be used when the cross link is moderately strong <cit.>, which will result in a similar design as our case. However, how to extend our design to ZIC with finite-alphabet input is still an open problem and has been left as a future work.]at the receiver D_1 since the error performance is dominated by the minimum Euclidean of the resulting sum-constellation at D_1. We assume that each receiver uses a coherent maximum-likelihood (ML) detector to estimate the transmitted signals in a symbol-by-symbol fashion[Since we are doing a symbol-by-symbol detection, the decoding complexity is 𝒪(M^2) with M being the PAM constellation size of S_1 and S_2, respectively. Although we can use the message passing algorithm (MPA) <cit.> to further decrease the decoding complexity, however, our method is feasible.]. For receivers D_1 and D_2, the estimated signals can be expressed as(ŝ_1,ŝ_2)=min_(s_1, s_2) |y_1 -(|h_11|w_1 s_1 + |h_21| w_2 s_2)|,ŝ_2' =min_s_2 |y_2 -|h_22|w_2 s_2|. By applying the nearest neighbor approximation method <cit.> at high SNRs for the ML receiver, the average error rate is dominated by the minimum Euclidean distance of the received constellation points owing to the exponential decaying of the Gaussian distribution. To balance the error performance of both receivers, in this paper, we aim to devise the optimal value of w_1 and w_2 by applying the max-min fairness criterion on the minimum Euclidean distance of the received signal constellation points among both receivers.The Euclidean distance between two received signals y_1(s_1,s_2) and y_1(s̃_1, s̃_2) at D_1 and that between y_2(s_2) and y_2(s̃_2) at D_2for the transmitted signal vectors (s_1, s_2) and (s̃_1, s̃_2) at S_1 and S_2 in the noise-free case are given, respectively, by |y_1(s_1, s_2) -y_1(s̃_1, s̃_2)|=||h_11| w_1(s_1 -s̃_1)-|h_21| w_2 (s̃_2-s_2 ) |,|y_2(s_2) -y_2(s̃_2)|=|h_22|w_2 |s_2 -s̃_2|.Note that s_1, s̃_1, s_2 and s̃_2 are all odd numbers, and we thus can let s_1 -s̃_1=2 n and s̃_2-s_2 =2 m, in which m, n∈ℤ_M-1 with ℤ_M-1≜{0,± 1,…, ± M-1}denoting the set containing all the possible differences.Similarly, we also define ℤ_M-1^2≜{(a,b):a,b ∈ℤ_M-1}, ℕ_M-1≜{0,1,⋯,M-1} and ℕ_M-1^2 ≜{(a,b): a,b ∈ℕ_M-1}. From the above definition, (s_1, s_2) ≠ (s̃_1, s̃_2) is equivalent to (m,n) ≠ (0,0). Here, by (m,n) ≠ (0,0), we mean that m ≠ 0 or n ≠ 0. To proceed, we define d_1(m,n)=1/2|y_1(s_1, s_2) -y_1(s̃_1, s̃_2)|=||h_11|w_1 n -|h_21| w_2 m|,for (m,n) ∈ℤ^2_M-1∖{(0,0)},d_2(m)=1/2|y_2(s_2) -y_2(s̃_2)|=|h_22| w_2 |m|,for m ∈ℤ_M-1∖{0}.We are now ready to formally formulate the following max-min optimization problem:[Optimal Design of NOMA in real scalar ZC with PAM constellation] Find the optimal value of (w_1^*,w_2^*) subject to the individual average power constraint such that the minimum value of the minimum Euclidean distances of the received signal constellation points over both received signals is maximized, i.e., (w_1^*, w_2^*)=max_(w_1, w_2) min{min_ (m,n) ∈ℤ_M-1^2 ∖{(0,0)} d_1(m,n)_T_1,min_m ∈ℤ_M-1∖{0} d_2(m)_T_2} s.t. 0< w_1 ≤√(3 p_1/M^2-1) and 0< w_2 ≤√(3 p_2/M^2-1). ▪ Note that the inner optimization problem of finding the minimum Euclidean distances is a discrete one, while the outer optimization problem on (w_1, w_2) is a continuous problem. In other words, Problem <ref> is a mixed continuous-discrete optimization problem and it is in general hard to solve. To the best of our knowledge, only numerical solutions to such kind of problems are available in the literature, see e.g., <cit.> and references therein.To optimally and systematically solve this problem, we now develop a novel framework based on the Farey sequence (also known as Farey series) <cit.>, which can divide the entire feasible region of (w_1, w_2) into a finite number of mutually exclusive sub-intervals.Then for each sub-interval, the formulated optimization problem can be solved optimally with a closed-form solution, and subsequently the overall maximum value of Problem <ref> can be attained by taking the maximum value of the objective function among all the sub-intervals.For the inner optimization problem of T_2 given in (<ref>), it can be observed thatmin_m ∈ℤ_M-1∖{0} d_2(m)=min_m ∈ℤ_M-1∖{0} |h_22| w_2|m| =|h_22| w_2, with m=1.However, for T_1, we havemin_ (m,n)∈ℤ_M-1^2 ∖{(0,0)} d_1(m,n)= min_ (m,n)∈ℤ_M-1^2 ∖{(0,0)}||h_11|w_1 n -|h_21| w_2 m|.We should point out that the closed-form solution to the optimal (m,n) is not trivial, since the solution depends on the values of |h_11| and |h_22|, which can span the whole positive real axis. Moreover, the value of w_1 and w_2 can not be determined beforehand. Actually, the problem in (<ref>)is essentiallyequivalent to finding a real rational number with finite order to approximate a real irrational number as closely as possible. This naturally leads us to resorting to the Farey sequence, which particularly plays a critical role in solving such kind of problems <cit.>. In the subsequent section, we will introduce the definition and some important properties of Farey sequence. §.§ Farey SequenceThe Farey sequence characterizes the relationship between two positive integers and the formal definition isgiven as follows: <cit.> The Farey sequence 𝔉_K of order K is the ascending sequence of irreducible fractions between 0 and 1 whose denominators are less than or equal to K. By the definition, 𝔉_K=(b_k/a_k)_k=1^|𝔉_K| is a sequence of fractions b_k/a_k such that 0≤ b_k≤ a_k≤ K and ⟨ a_k, b_k ⟩ =1arranged in an increasing order, where ⟨ a,b⟩ denotes the largest common divider of non-negative integers a,b. |𝔉_K| =1+∑_m=1^K φ(m) is the cardinality of 𝔉_K with φ(·) being the Euler's totient function <cit.>.Some examples of Farey sequences are given as follows:𝔉_5 is the ordered sequence (0/1, 1/5, 1/4, 1/3, 2/5,1/2, 3/5, 2/3,3/4, 4/5, 1/1).It can be observed that each Farey sequence begins with number 0 (fraction 0/1) and ends with 1 (fraction 1/1). The series of breakpoints after 1/1 is the reciprocal version of the Farey sequence. We call the Farey number sequence together with its reciprocal version as the extended Farey sequence which is formally defined as follows:The extended Farey sequence 𝔖_K of order K is the sequence of ascending irreducible fractions, where the maximum value of the numerator and denominator do not exceed K. From the definition, we have 𝔖_K=(b_k/a_k)_k=1^|𝔖_K| with ⟨ a_k,b_k⟩=1 and|𝔖_K|=1+2∑_m=1^K φ(m). We have the following example: 𝔖_5 is the sequence (0/1, 1/5, 1/4, 1/3, 2/5,1/2, 3/5, 2/3,3/4, 4/5, 1/1,5/4,4/3,3/2,5/3,2/1,5/2,3/1,4/1,5/1,1/0). It can be observed that the extended Farey sequence starts with number 0 (fraction 0/1) and end with ∞ (fraction 1/0).The positive real axis can be divided by the extended Farey sequence 𝔖_K into a finite number (i.e., |𝔖_K|-1=2∑_m=1^K φ(m))of intervals. In this paper, we call the fractions consisting of adjacent terms in the extended Farey sequence as a Farey pair, and the interval between the Farey pair is referred to as a Farey interval. We then have the Farey interval set formally defined as follows:A Farey interval set 𝒮_K of order K is the set containing all the Farey intervals generated by the Farey pair of the extended Farey sequence 𝔖_K. By the definition, we have 𝒮_K={(b_k/a_k, b_k+1/a_k+1)}_k=1^|𝒮_K|, where |𝒮_K|=|𝔖_K|-1=2∑_m=1^K φ(m). Note that,with a slight abuse of notations, (b_k/a_k, b_k+1/a_k+1) denotes the interval between end nodes b_k/a_k and b_k+1/a_k+1 rather than a vector, and this will be clear from the context.The Farey interval set 𝒮_5 is the set given by {(0/1, 1/5), (1/5, 1/4), (1/4, 1/3), …, (3/1, 4/1), (4/1, 5/1),(5/1, 1/0)}. The Farey interval set can be further divided into two subsets 𝒰_K^L={(b_k'/a_k', b_k+1'/a_k+1')∈𝒮_K: b_k' +b_k+1' ≥ L}and 𝒱_K^L={(b_k”/a_k”, b_k+1”/a_k+1”)∈𝒮_K: b_k” +b_k+1” < L} for L=1,2,…, 2K such that 𝒮_K=𝒰_K^L ∪𝒱_K^L and 𝒰_K^L ∩𝒱_K^L=∅. In particular, 𝒰_K^1 = 𝒮_K and 𝒱_K^1=∅ while 𝒰_K^2K=∅ and 𝒱_K^2K=𝒮_K. For the Farey interval set 𝒮_5, we have𝒰_5^4= {(1/2, 3/5), (3/5, 2/3), (2/3, 3/4), (3/4, 4/5), (4/5, 1/1),(1/1, 5/4), (5/4, 4/3), (4/3, 3/2), (3/2, 5/3), (5/3, 2/1),(2/1, 5/2), (5/2, 3/1), (3/1, 4/1), (4/1, 5/1), (5/1, 1/0) },and𝒱_5^4= {(0/1, 1/5), (1/5, 1/4), (1/4, 1/3), (1/3, 2/5), (2/5, 1/2) }.We now review some elementary properties of Farey sequences <cit.> which are also true for extended Farey sequences.If n_1/m_1 and n_2/m_2 aretwo adjacent terms (Farey pairs) of 𝔖_K with K≥ 1, such that n_1/m_1< n_2/m_2, then, * m_1 n_2 - m_2 n_1=1. * n_1 + n_2/m_1 + m_2∈(n_1/m_1, n_2/m_2), m_1+m_2/n_1 + n_2∈(m_2/n_2, m_1/n_1). ▪ If n_1/m_1, n_2/m_2and n_3/m_3 are three consecutive terms of 𝔖_K with K≥ 1 such that n_1/m_1< n_2/m_2<n_3/m_3, then, n_2/m_2=n_1 + n_3/m_1 + m_3. ▪Given K≥ 2, we assume n_1/m_1, n_2/m_2, n_3/m_3, n_4/m_4∈𝔖_K and n_1/m_1< n_2/m_2<n_3/m_3<n_4/m_4. If n_2/m_2 and n_3/m_3 form one Farey pair, then n_1+n_3/m_1+m_3≤n_2/m_2 and n_3/m_3≤n_2+n_4/m_2+m_4. ▪§.§ The Minimum Euclidean Distance of the Received Signal Constellation PointsWe are now ready to solve the problem in (<ref>) to find the constellation point pairs (m,n) that have the minimum Euclidean distance. To that end, we first have the following preliminary propositions. Let 𝔽_K^2 ={(m,n): n/m∈𝔖_K}, where 𝔖_K is the extended Farey number sequence of order K, and then we havemin_ (m,n)∈ℤ_K^2 ∖{(0,0)} d_1(m,n)=min_ (m,n)∈𝔽_K^2 d_1(m,n).▪The proof is given in Appendix<ref>. Consider the Farey interval (n_1/m_1, n_2/m_2) ∈𝒮_K, with K≥ 1, and n_1/m_1< n_2/m_2. Then, for |h_21| w_2/|h_11|w_1∈ (n_1/m_1, n_2/m_2) and d_1(m,n) =||h_11|w_1 n-|h_21| w_2 m|, we have * If |h_21| w_2/|h_11|w_1= n_1 +n_2/m_1 + m_2, then d_1(m_1, n_1)=d_1(m_2, n_2); * If |h_21| w_2/|h_11|w_1∈(n_1/m_1, n_1 +n_2/m_1 + m_2), then d_1(m_1, n_1) <d_1(m_2, n_2); * If|h_21| w_2/|h_11|w_1∈(n_1 +n_2/m_1 + m_2, n_2/m_2), then d_1(m_1, n_1)>d_1(m_2, n_2). ▪ The proof can be found in Appendix<ref>.Consider n_1/m_1,n_2/m_2,n_3/m_3,n_4/m_4∈𝔖_K, with K≥ 2, such that n_1/m_1<n_2/m_2<n_3/m_3<n_4/m_4 where n_2/m_2, n_3/m_3 form one Farey pair, * If |h_21| w_2/|h_11|w_1∈ (n_2/m_2,n_2+n_3/m_2+m_3), then min_(m,n) ∈𝔽_M-1^2 d_1(m,n)=d_1(m_2, n_2) =|h_21|w_2 m_2- |h_11| w_1 n_2. * If|h_21| w_2/|h_11|w_1∈ (n_2+n_3/m_2+m_3,n_3/m_3), then min_(m,n) ∈𝔽_M-1^2 d_1(m,n)=d_1(m_3, n_3) =|h_11| w_1 n_3 - |h_21|w_2 m_3. ▪ The proof is provided in Appendix<ref>. For the Farey interval set 𝒮_M-1={(b_k/a_k, b_k+1/a_k+1)}_k=1^|𝒮_M-1|, if |h_21|/|h_22|≥ b_k+b_k+1, then a_k+1/b_k+1+|h_22| /b_k+1|h_21|≤a_k+a_k+1/b_k+b_k+1≤a_k/b_k-|h_22| /b_k|h_21|. ▪ The proof is given in Appendix<ref>.§.§ Optimal Solution to Problem <ref> for|h_21| w_2/|h_11|w_1 in Certain Farey Interval In this section, we solve Problem <ref> by restricting |h_21|w_2 /|h_11| w_1 into a certain Farey interval where a closed-form solution is attainable. We consider the Farey interval set 𝒮_M-1 given by 𝒮_M-1={(b_k/a_k,b_k+1/a_k+1)}_k=1^|𝒮_M-1| where |𝒮_M-1|=2∑_m=1^M-1φ(m). Now we consider the case |h_21|w_2 /|h_11| w_1∈(b_k/a_k, b_k+1/a_k+1), k=1,2,…,|𝒮_M-1| and we aim to find the optimal (w_1^*(k), w_2^*(k)) such that g(b_k/a_k, b_k+1/a_k+1) = max_(w_1, w_2) min{min_(m,n) ∈𝔽_M-1^2 d_1(m,n),min_m ∈ℤ∖{0} d_2(m)} s.t. b_k/a_k< |h_21|w_2 /|h_11| w_1≤b_k+1/a_k+1,0< w_1 ≤√(3 p_1/M^2-1) and 0 < w_2 ≤√(3 p_2/M^2-1).By applying the propositions in last subsections, we manage to attain the following lemma in terms of the optimal solution to problem (<ref>).The optimal solution to (<ref>) is given as follows. * If |h_21|/|h_22|≤ b_k +b_k+1, the following statements are true: * If |h_11|/|h_21|≥√(p_2)(a_k+a_k+1)/√(p_1)(b_k+b_k+1), then g(b_k/a_k, b_k+1/a_k+1)=|h_21|/b_k+b_k+1√(3 p_2/M^2-1) and (w_1^*(k), w_2^*(k))=((a_k + a_k+1)|h_21|/(b_k+b_k+1) |h_11|√(3 p_2/M^2-1), √(3 p_2/M^2-1)). * If |h_11|/|h_21|<√(p_2)(a_k+a_k+1)/√(p_1)(b_k+b_k+1),then g(b_k/a_k, b_k+1/a_k+1)=|h_11|/a_k+ a_k+1√(3 p_1/M^2-1) and (w_1^*(k), w_2^*(k))=(√(3 p_1/M^2-1), (b_k +b_k+1)|h_11|/(a_k+a_k+1)|h_21|√(3 p_1/M^2-1)) . * If |h_21|/|h_22| > b_k +b_k+1, we have the following results: * If |h_11|/|h_21|≥√(p_2)/√(p_1)(a_k+1/b_k+1 +|h_22| /b_k+1|h_21|), theng(b_k/a_k, b_k+1/a_k+1)=|h_22|√(3 p_2/M^2-1) and(w_1^*(k), w_2^*(k))=(a_k+1 |h_21|+|h_22| / b_k+1 |h_11|√(3 p_2/M^2-1), √(3 p_2/M^2-1)). * If |h_11|/|h_21|<√(p_2 )/√(p_1)(a_k+1/b_k+1 +|h_22| /b_k+1|h_21|), then g(b_k/a_k, b_k+1/a_k+1)=b_k+1|h_11||h_22|/a_k+1|h_21|+|h_22|√(3 p_1/M^2-1) and (w_1^*(k), w_2^*(k))=(√(3 p_1/M^2-1), b_k+1 |h_11|/a_k+1 |h_21|+|h_22|√(3 p_1/M^2-1)) . ▪The proof of Lemma 1 can be found in Appendix<ref>.We have the following insights from the above lemma, * It can be observed from Lemma <ref> that at least one transmitter should employ the maximum allowable power, since otherwise we could scale up both transmitted power without violating the power constraint such that the minimum Euclidean distance is enlarged.* We can see from Lemma <ref> that the optimal value of the objective function together with (w_1^*(k), w_2^*(k))substantially depend on the relative strength of the channel coefficients. Inspired by <cit.>, we divide the considered real Gaussian ZC into three scenarios: * Gaussian ZC with weak cross link, i.e., |h_21|/|h_22|∈ (0,1]; * Gaussian ZC with strong cross link, i.e., |h_21|/|h_22|∈ (1,2M); * Gaussian ZC with very strong cross link, i.e., |h_21|/|h_22|∈ [2M, ∞). Then, a compact closed-form expression for Problem <ref> can be established for three complimentary scenarios, which constitutes main contents of the subsequent subsection. §.§ The Optimal NOMA Design with PAM Constellation for the Gaussian ZCNow we are ready to give the closed-form optimal solution of (w_1^*, w_2^*)to Problem <ref> which maximizes the minimum Euclidean distance over all the Farey intervals for the aforementioned three scenarios.§.§.§ Scenario 1: ZC with Weak Cross LinkFor this case, we have |h_21|/|h_22|∈ (0,1]. Consider the Farey interval set 𝒮_M-1={(b_k/a_k, b_k+1/a_k+1)}_k=1^|𝒮_M-1|. By Property <ref>, we have a_k+1+a_k+2/b_k+1+b_k+2<a_k+1/b_k+1 <a_k+a_k+1/b_k+b_k+1, and therefore the positive real axis can be divided into the |𝒮_M-1|+1 intervals in an increasing order:{(0, a_|𝒮_M-1|+a_|𝒮_M-1|+1/b_|𝒮_M-1|+b_|𝒮_M-1|+1),(a_|𝒮_M-1|+a_|𝒮_M-1|+1/b_|𝒮_M-1|+b_|𝒮_M-1|+1,a_|𝒮_M-1|-1 +a_|𝒮_M-1|/b_|𝒮_M-1|-1 +b_|𝒮_M-1|), ⋯,(a_1 +a_2/b_1 +b_2,∞)}.In particular, we have a_1 +a_2/b_1 +b_2=M and a_|𝒮_M-1|+a_|𝒮_M-1|+1/b_|𝒮_M-1|+b_|𝒮_M-1|+1=1/M.Suppose that |h_21|/|h_22|∈ (0,1].Then, the optimal power scaling factors to Problem <ref> are explicitly determined as follows: * If |h_11|/|h_21|≤√(p_2)/M√(p_1), then, we have (w_1^*, w_2^*)=(√(3 p_1/M^2-1), M |h_11|/|h_21|√(3 p_1/M^2-1)); * If |h_11|/|h_21|≥M √( p_2)/√(p_1), then, we have(w_1^*, w_2^*)=(M |h_21|/|h_11|√(3 p_2/M^2-1),√(3 p_2/M^2-1)); * Let |h_11|/|h_21| ∈(√(p_2 )(a_ℓ_1+1 +a_ℓ_1+2)/√(p_1)(b_ℓ_1+1 +b_ℓ_1+2), √(p_2)(a_ℓ_1 +a_ℓ_1+1) /√(p_1)(b_ℓ_1 +b_ℓ_1+1)) for some ℓ_1=1, ⋯, |𝒮_M-1|-1. If we let ℓ̃_a=min_k {(a_1+a_2), ⋯, (a_ℓ_1+ a_ℓ_1+1) } and ℓ̃_b=min_k {(b_ℓ_1+1+b_ℓ_1+2), ⋯, (b_|𝒮_M-1|+b_|𝒮_M-1|+1)}, then we have (w_1^*, w_2^*)=(√(3 p_1/M^2-1),(b_ℓ̃_a +b_ℓ̃_a+1)|h_11|/(a_ℓ̃_a+a_ℓ̃_a+1)|h_21|√(3 p_1/M^2-1)), if |h_11|/|h_21|≥√( p_2)(a_ℓ̃_a+ a_ℓ̃_a+1) /√( p_1)(b_ℓ̃_b+b_ℓ̃_b+1); ((a_ℓ̃_b + a_ℓ̃_b+1)|h_21|/(b_ℓ̃_b+b_ℓ̃_b+1) |h_11|√(3 p_2/M^2-1), √(3 p_2/M^2-1)), if |h_11|/|h_21| < √( p_2)(a_ℓ̃_a+ a_ℓ̃_a+1) /√( p_1)(b_ℓ̃_b+b_ℓ̃_b+1). The proof of Theorem <ref> is provided in Appendix<ref>. §.§.§ Scenario 2: ZC with Strong Cross LinkIn this case,|h_21|/|h_22|∈ (1, 2M). We suppose that L-1 <|h_21|/|h_22|≤ L. Then, the optimal solution to Problem 1 can be obtained by considering the following two cases: the Farey interval set 𝒰_M-1^L={(b_k'/a_k', b_k+1'/a_k+1')∈𝒮_M-1: b_k' +b_k+1' ≥ L} and 𝒱_M-1^L={(b_k”/a_k”, b_k+1”/a_k+1”)∈𝒮_M-1: b_k” +b_k+1” < L}. The whole discussions on them can be summarized as the following theorem.The optimal solution to Problem <ref> in this case can be obtained by finding the maximum value of the objective functions in the following two cases which are explicitly attained as follows: *Let (b_k'/a_k', b_k+1'/a_k+1')∈𝒰_M-1^L. Then, the following three statements are true. * If |h_11|/|h_21|≤√(p_2)(a_|𝒰_M-1^L|' +a_|𝒰_M-1^L|+1' )/√(p_1)(b_|𝒰_M-1^L|' +b_|𝒰_M-1^L|+1'), then we have (w_u_1^*, w_u_2^*)=(√(3 p_1/M^2-1), (b_ℓ̃_c'+b_ℓ̃_c+1') |h_11|/(a_ℓ̃_c'+a_ℓ̃_c+1')|h_21|√(3 p_1/M^2-1)), where ℓ̃_c=min_k{(a_1'+a_2'), …, (a_|𝒰_M-1^L|'+a_|𝒰_M-1^L|+1') }. * If |h_11|/|h_21|≥ M√(p_2)/√(p_1), then we have(w_u_1^*, w_u_2^*)=(M |h_21|/|h_11|√(3 p_2/M^2-1),√(3 p_2/M^2-1)). * Suppose that |h_11|/|h_21| ∈(√(p_2)(a_ℓ_2+1' +a_ℓ_2+2')/√(p_1 )(b_ℓ_2+1' +b_ℓ_2+2'), √(p_2)(a_ℓ_2' +a_ℓ_2+1') /√(p_1)(b_ℓ_2' +b_ℓ_2+1')) for some ℓ_2'=1, ⋯,|𝒰_M-1^L|. If we let ℓ̃_d=min_k {(a_1+a_2), …, (a_ℓ_2+ a_ℓ_2+1) } and ℓ̃_e=min_k {(b_ℓ_2+1+b_ℓ_2+2),…, (b_|𝒰_M-1^L|+b_|𝒰_M-1^L|+1)}, then we have (w_u_1^*, w_u_2^*)=(√(3 p_1/M^2-1),(b_ℓ̃_d' +b_ℓ̃_d+1')|h_11|/(a_ℓ̃_d'+a_ℓ̃_d+1')|h_21|√(3 p_1/M^2-1)), if |h_11|/|h_21|≥√( p_2)(a_ℓ̃_d'+ a_ℓ̃_d+1') /√( p_1)(b_ℓ̃_e'+b_ℓ̃_e+1'); ((a_ℓ̃_e' + a_ℓ̃_e+1')|h_21|/(b_ℓ̃_e'+b_ℓ̃_e+1') |h_11|√(3 p_2/M^2-1), √(3 p_2/M^2-1)) if |h_11|/|h_21| < √( p_2)(a_ℓ̃_d'+ a_ℓ̃_d+1') /√( p_1)(b_ℓ̃_e'+b_ℓ̃_e+1'). * Let (b_k-1”/a_k-1”, b_k”/a_k”) ∈𝒱_M-1^L. Then, the following two statements are true. * If |h_11|/|h_22|≤√(p_2)/√(p_1), then we have (w_v_1^*, w_v_2^*)=( √(3 p_1/M^2-1), b_|𝒱_M-1|+1” |h_11||h_22|/a_|𝒱_M-1|+1” |h_21|+|h_22|√(3 p_1/M^2-1)). * If |h_11|√(p_1)/|h_21| √(p_2)∈(a_ℓ_3+1”/b_ℓ_3+1”+|h_22| /b_ℓ_3+1” |h_21|, a_ℓ_3”/b_ℓ_3”+|h_22| /b_ℓ_3” |h_21|) for some ℓ_3=1,⋯, |𝒱_M-1|, then we have (w_v_1^*,w_v_2^*)=(√(3 p_1/M^2-1), b_ℓ_3”|h_11|/a_ℓ_3”|h_21|+|h_22|√(3 p_1/M^2-1)), if |h_11|/|h_21|≥√(p_2 )/√(p_1 )(a_ℓ_3”/b_ℓ_3”+|h_22| /b_ℓ_3”|h_21|); (a_ℓ_3+1”|h_21|+|h_22|/b_ℓ_3+1” |h_11| √(3 p_2/M_2^2-1), √(3 p_2/M_2^2-1)) if |h_11|/|h_21|< √(p_2)/√(p_1)(a_ℓ_3”/b_ℓ_3”+|h_22| /b_ℓ_3”|h_21|). The proof of Theorem 2isvery similar to that of Theorem 1 and the following Theorem 3 and thus, is omitted due to space limitation. §.§.§ Scenario 3: ZC with Very Strong Cross LinkIn this case, |h_21|/|h_22|∈ [2M, ∞). Likewise, we consider the Farey interval set 𝒮_M-1={(b_k/a_k, b_k+1/a_k+1)}_k=1^|𝒮_M-1|. Note that |h_21|/|h_22|≥ 2M> b_k+b_k+1 for k=1,…,|𝒮_M-1|. Then, by Property <ref> and Proposition <ref>, we have a_k+1/b_k+1+|h_22| /b_k+1|h_21|≤a_k+a_k+1/b_k+b_k+1< a_k/b_k <a_k/b_k+|h_22| /b_k|h_21|. As a result, the positive real axis can be divided into the following intervals in increasing order: {(0, a_|𝒮_M-1|+1/b_|𝒮_M-1|+1+|h_22| /b_|𝒮_M-1|+1|h_21|), …, (a_2/b_2+|h_22| /b_2 |h_21|, a_1/b_1+|h_22| /b_1 |h_21|)}, where a_1/b_1+|h_22| /b_1 |h_21|=∞. Let |h_21|/|h_22|∈ [2M, ∞). Then, the optimal solution to Problem <ref> is given below: * If |h_11|/|h_22|≤√(p_2)/√(p_1), then we have (w_1^*, w_2^*)=(√(3 p_1/M^2-1),|h_11|/|h_22|√(3 p_1/M^2-1)). * If |h_11|√(p_1)/|h_21| √(p_2)∈(a_ℓ_4+1/b_ℓ_4+1+|h_22| /b_ℓ_4+1 |h_21|, a_ℓ_4/b_ℓ_4+|h_22| /b_ℓ_4 |h_21|) for some ℓ_4=1,…, |𝒮_M-1|, then we have (w_1^*,w_2^*)=(√(3 p_1/M^2-1), b_ℓ_4|h_11|/a_ℓ_4|h_21|+|h_22|√(3 p_1/M^2-1)), if |h_11|/|h_21|≥√(p_2)/√(p_1 )(a_ℓ_4/b_ℓ_4+|h_22| /b_ℓ_4|h_21|); (a_ℓ_4+1|h_21|+|h_22|/b_ℓ_4+1 |h_11| √(3 p_2/M^2-1), √(3 p_2/M^2-1)), if |h_11|/|h_21|< √(p_2)/√(p_1)(a_ℓ_4/b_ℓ_4+|h_22| /b_ℓ_4|h_21|). The proof of Theorem <ref> is provided in Appendix<ref>. § SIMULATION RESULTS AND DISCUSSIONSIn this section, computer simulations are carried out to demonstrate the effectiveness of our proposed NOMA design under different channel configurations. More precisely, we compare our proposed NOMA design with CR based NOMA <cit.>, time-division multiple access (TDMA) and frequency-division multiple access (FDMA) approaches. Without loss of generality, we set P_1=P_2=1. For simplicity, in the simulations, we assume that the same square-QAM constellation is adopted by both users, i.e., M=M' andaccording to (<ref>), we have p_1=p_1'=P_1/2=1/2 and p_2=p_2'=P_2/2=1/2.§.§ The Resulting Optimal Sum Constellation at Receiver D_1 For Several Deterministic Channels Here, we consider several deterministic channels corresponding to the three scenarios of weak cross link, strong cross link and very strong cross link. We assume that 16-QAM constellations are used by both users with M=M'=4. We discuss the results of these cases one by one as follows: §.§.§ Weak Cross LinkIn this case, we assume that h_11=1, h_21=1/2 and h_22=1. Based on the derived expressions for the optimal solution provided in the previous section, we can readily obtain that (w_1^*, w_2^*)=(0.4472, 0.2236). The corresponding received constellation at D_1 is plotted in Fig. <ref>. It can be observed that the sum-constellation at D_1 is a regular 256-QAM generated by the superposition of two 16-QAM.Hereafter, we call the signal constellation with smaller minimum Euclidean distance as the satellite constellation. By observing that w_1^* h_11=0.4472 and w_2^* h_21=0.1118 (i.e., w_2^* h_21=1/4w_1^* h_11), we can deduce that the constellation used by S_2 forms the satellite constellation of the sum constellation at D_1. §.§.§ Strong Cross LinkWe investigate two channel realizations for this scenario. For the first realization, we let h_11=1, h_21=3/2 and h_22=1. We then have (w_1^*, w_2^*)=(0.1677, 0.4472) and the resulting sum-constellation at D_1 is also regular, as illustrated in Fig. <ref>. Since w_1^* h_11=0.1677 and w_2^* h_21=0.6708 (i.e., w_2^* h_21=4 w_1^* h_11) in this case, the constellation used by S_1 forms the satellite constellation at D_1. For the second realization, we set h_11=1, h_21=3 and h_22=1, leading to (w_1^*, w_2^*)=(0.3354, 0.4472). The resulting constellation plotted in Fig. <ref> is also uniform as in the previous two scenarios. We have w_1^* h_11=0.3354 and w_2^* h_21=1.3416 (i.e., w_2^* h_21=4 w_1^* h_11). Thus, the constellation used by the transmitter S_1 forms the satellite constellation at D_1. §.§.§ Very Strong Cross LinkIn this case, we suppose that h_11=1, h_21=5 and h_22=1/2, generating (w_1^*, w_2^*)=(0.2236, 0.4472). The obtained constellation at D_1 is shown in Fig. <ref>. In this case, we have w_1^* h_11=0.2236 and w_2^* h_21=2.236 (i.e., w_2^* h_21 =10 w_1^* h_11).However, it can be observed that w_2^* h_22=0.2236, i.e., w_1^* h_11=w_2^* h_22.§.§ Average Error Performance Comparison in Rayleigh Fading ChannelsWe now compare the average BER of the proposed NOMA scheme with that of three existing methods, including TDMA, FDMA, and NOMA with CR <cit.> methods, over Rayleigh fading channels with h_11∼𝒞𝒩(0, δ_11^2), h_21∼𝒞𝒩(0, δ_21^2) and h_22∼𝒞𝒩(0, δ_22^2).Recall that we use error performance (i.e., BER) as the design criterion for the NOMA in ZCs with finite-alphabet inputs using fixed transmission rate (i.e., fixed constellation size). However, we are unable to compare the error performance of the considered system using finite-alphabet inputs with that of Gaussian inputs. This is because for Gaussian input, it is intractable to evaluate the BER for uncoded system since its input signal is continuous. Moreover, the BER for coded system with Gaussian input is hard to simulate due to the huge storage capacity requirement for the large codebook and the high computational complexity <cit.>. For TDMA, we assume that both users transmit alternatively by using half of the total time slots and thus no interference occurs at the destination side. More importantly, the individual instantaneous power constraints on both users S_1 and S_2 remain unchanged. On the other hand, for FDMA, each user uses only half of the available bandwidth. Due to the orthogonality between different frequency band, there is also no interference occurring at the destination side. Note that, in FDMA, the bandwidth occupied by each user is halved and the noise arises at the receiver is assumed to be white Gaussian. Therefore, the variance of the noise is also halved. In addition, for theCR based NOMAas proposed in <cit.>, we let each user transmit at the maximum allowable power by using constellations {exp(j 2π k/N)}_k=0^N-1 and {exp(j 2πℓ +jπ/N)}_ℓ=0^N-1 for user S_1 and S_2, respectively. In Fig. <ref>, we consider that the variances of all channels are the same, i.e., (δ_11^2, δ_21^2, δ_22^2)=(1,1,1), and the average BER over both receivers of all the methods are plotted against the SNR ρ=1/2σ^2. In Fig. <ref>, 16-QAM is used for the proposed NOMA scheme while 16-PSK is employed by the NOMA with CR. Since only half of the total time slots or total bandwidths are available for each transmitter, to maintain the same data rate for each user in each block compared with NOMA methods, we should increase the constellation size by using 256-QAM in both TDMA and FDMA.It can be observed that, our proposed method has significant BER gain over TDMA and FDMA methods, which confirms the effectiveness of the NOMA scheme.From the simulation results, we also find that FDMA has a smaller BER compared with TDMA. This is because the variance of the effective noise is smaller than that of TDMA. As the rotation based method uses the PSK constellation, which is not spectrally efficient, it has the worst BER performance. Then, in Fig. <ref>, the average BER of all the cases are plotted against the SNR where 64-QAM is used by each user for the proposed NOMA and 64-PSK is used by CR based NOMAwhile 4096-QAM constellations are used by TDMA and FDMA methods.We also simulate another two cases with unequal channel variances. Specifically, the average BER of both receivers for all the considered methods is plotted in Fig. <ref>, wherein the variances of three channels are set as (δ_11^2, δ_21^2, δ_22^2)=(1,1/4,1). The case with the variances of three channels (δ_11^2, δ_21^2, δ_22^2)=(1,4,1) is provided in Fig. <ref>. Similar observations can be seen as the previous case with equal channel variances, which further verifies the effectiveness of our proposed NOMA design.§ CONCLUSIONSIn this paper, we developed a novel and practical design framework for the non-orthogonal multiple access (NOMA) in a classical two-transmitter two-receiver Z-channel with widely-used quadrature amplitude modulation (QAM) and max-min user fairness. Specifically, we formulated a max-min optimization problem to jointly optimize the scaling factors at both transmitters to maximize the smaller minimum Euclidean distance among the two resulting signal constellations at both receivers, subject to the individual average power constraint at each transmitter. The formulated mixed continuous-discrete problem was successfully resolved in compact closed-form by strategically applying the Farey sequences and their unique properties. Simulation results verified the correctness of our analytical derivations and showed that the proposed NOMA design significantly outperforms the existing orthogonal multiple access and NOMA schemes, especially at high signal-to-noise ratio. Furthermore, the performance gap of the proposed scheme over its existing counterparts can be further enlarged when the size of constellations used at the transmitter side becomes larger. § APPENDIX§.§ Proof of Proposition <ref> We first divide the feasible region ℤ_K^2 ∖{(0,0)} into four subsets given by 𝕊_K,1^2={(m,n): m,n ∈ℕ_K, (m,n) ≠ (0,0)}, 𝕊_K,2^2={(-m,n): m,n ∈ℕ_K, (m,n) ≠ (0,0)}, 𝕊_K,3^2={(m,-n): m,n ∈ℕ_K,(m,n) ≠ (0,0)}, 𝕊_K,4^2={(-m,-n): m,n ∈ℕ_K, (m,n) ≠ (0,0)}. Then clearly we have ℤ_K^2 ∖{(0,0)}= ∪_k=1^4 𝕊_K,k^2. For (-m,n) ∈𝕊_K,2^2 with m,n≥ 0,we can always find (m,n) ∈𝕊_K,1^2 such that d_1(m,n)=||h_11|w_1 n -|h_21| w_2 m| ≤||h_11|w_1 n -|h_21| w_2 (-m)|= d_1(-m,n). Hence,min_(m,n)∈𝕊_K,1^2 d_1(m,n)≤min_(m,n)∈𝕊_K,2^2 d_1(m, n).By a similar argument on 𝕊_K,3^2 and 𝕊_K,4^2, it follows thatmin_(m,n) ∈𝕊_K,1^2 d_1(m,n)=min_(m,n) ∈ℤ_K,1^2 ∖{(0,0)} d_1(m,n). Then, we further divide 𝕊_K,1^2 into 𝔽^2_K and 𝕊_K,1^2 ∖𝔽^2_K.In what follows, we will show thatmin_ (m,n)∈𝔽_K^2 d_1(m,n)=min_ (m,n)∈𝕊_K,1^2 d_1(m,n).This can be proved by contradiction. Suppose thatmin_ (m,n) ∈𝕊_1^2∖𝔽_K^2 d_1(m,n)=min_ (m,n) ∈𝕊_K,1^2 d_1(m,n),where the minimum is achieved by (m^*, n^*) ∈𝕊_K,1^2 ∖𝔽_K^2 such that ⟨ m^*,n^*⟩=ℓ>1 by the definition of Farey sequences. Then, we can find (m^*/ℓ,n^*/ℓ) ∈𝔽_K^2 such that d_1(m^*/ℓ,n^*/ℓ)=1/ℓ d_1(m^*,n^*)<d_1(m^*,n^*), which contradicts the assumption.This completes the proof. □ §.§ Proof of Proposition <ref> Recall that d_1(m,n) =||h_11|w_1 n-|h_21| w_2 m|. Therefore, for |h_21| w_2/|h_11|w_1∈(n_1/m_1,n_2/m_2), we have:d_1(m_1, n_1) -d_1(m_2, n_2) =(m_1+m_2) |h_11|w_1 (|h_21| w_2/ |h_11|w_1 - n_1+n_2/ m_1+m_2).The results presented in the proposition can be readily obtained, and we complete the proof. □ §.§ Proof of of Proposition <ref>We first consider the case |h_21| w_2/|h_11|w_1∈ (n_2/m_2,n_2+n_3/m_2+m_3). By using Proposition <ref>, we have d_1(m_2, n_2)<d_1(m_3, n_3). Then, with the help of Property <ref> and <ref>, we have |h_21| w_2/|h_11|w_1∈ (n_2/m_2, n_2+n_3/m_2+m_3)⊂(n_2/m_2, n_3/m_3) ⊆ (n_2/m_2, n_2+n_4/m_2+m_4) and using Property <ref> again, we have d_1(m_2, n_2)≤ d_1(m_4, n_4). On the other hand, by Property <ref> and <ref>, we have |h_21| w_2/|h_11|w_1∈ (n_2/m_2, n_2+n_3/m_2+m_3)⊂(n_2/m_2, n_3/m_3) ⊆ (n_1+n_3/m_1+m_3, n_3/m_3) and using Property <ref> again, we have d_1(m_3, n_3)≤ d_1(m_1, n_1).As n_1/m_1 and n_4/m_4 are randomly picked entry in 𝔖_K, or equivalently (m_1, n_1) and (m_4, n_4) are randomly picked in 𝔽_M-1^2 ∖{(m_2, n_2),(m_3,n_3)}, this proves that for |h_21| w_2/|h_11|w_1∈ (n_2/m_2,n_2+n_3/m_2+m_3), min_(m,n) ∈𝔽_M-1^2 d_1(m,n)=d_1(m_2, n_2)= |h_21| w_2 m_2-|h_11|w_1 n_2. The other case can also be proved by a similar argument and hence omitted. We completes the proof.§.§ Proof of Proposition <ref>First of all, we calculate the following difference a_k+a_k+1/b_k+b_k+1 -a_k+1/b_k+1-|h_22| /b_k+1|h_21| =|h_21| - |h_22|(b_k+b_k+1)/(b_k+b_k+1)b_k+1 |h_21|. Hence, if |h_21|/|h_22|≥ b_k+b_k+1, then a_k+1/b_k+1+|h_22| /b_k+1|h_21|≤a_k+a_k+1/b_k+b_k+1. Similarly, we have a_k/b_k-|h_22| /b_k|h_21| -a_k+a_k+1/b_k+b_k+1 =|h_21| - |h_22|(b_k+b_k+1) /b_k (b_k+b_k+1) |h_21|. As a result, if |h_21|/|h_22|≥ b_k+b_k+1, then a_k+a_k+1/b_k+b_k+1≤a_k/b_k-|h_22| /b_k|h_21|. This completes the proof. §.§ Proof of Lemma <ref> According to proposition <ref> and notice that (b_k/a_k,b_k+1/a_k+1)=(b_k/a_k, b_k+b_k+1/a_k+a_k+1)∪(b_k+b_k+1/a_k+a_k+1,b_k+1/a_k+1), problem in (<ref>) can be further divided into the following two sub-problems and the overall solution is the maximum value of the two problems: [Sub-problem 1] We aim to solve the following optimization problem: g_1(b_k/a_k, b_k+1/a_k+1)=max_w_1, w_2 min {|h_11| w_1 b_k+1 - |h_21|w_2 a_k+1,|h_22|w_2 } s.t. b_k+b_k+1/a_k+a_k+1≤|h_21|w_2 /|h_11| w_1≤b_k+1/a_k+1, 0< w_1 ≤√(3 p_1/M^2-1), 0< w_2 ≤√(3 p_2/M^2-1). ▪ [Sub-problem 2] The optimization problem is stated as follows: g_2(b_k/a_k, b_k+1/a_k+1)=max_w_1, w_2min{|h_21|w_2 a_k- |h_11| w_1 b_k, |h_22|w_2 } s.t. b_k/a_k≤|h_21|w_2 /|h_11| w_1 < b_k+b_k+1/a_k+a_k+1, 0< w_1 ≤√(3 p_1/M^2-1), 0< w_2 ≤√(3 p_2/M^2-1). ▪ We first consider Sub-problem 1 and Problem <ref>, which can be divided into the following two case: * Case 1: Receiver 1 has smaller Euclidean distance. g_11(b_k/a_k, b_k+1/a_k+1)=max_w_1, w_2 b_k+1 |h_11| w_1 -a_k+1 |h_21|w_2 s.t. b_k+1|h_11| w_1-a_k+1 |h_21|w_2≤ |h_22|w_2,b_k+b_k+1/a_k+a_k+1≤|h_21|w_2 /|h_11| w_1≤b_k+1/a_k+1, 0< w_1 ≤√(3 p_1/M^2-1), 0< w_2 ≤√(3 p_2/M^2-1). * Case 2: Receiver 2 has smaller Euclidean distance. g_12(b_k/a_k, b_k+1/a_k+1)=max_w_1, w_2 |h_22|w_2s.t. |h_22|w_2<b_k+1 |h_11| w_1-a_k+1|h_21|w_2,b_k+b_k+1/a_k+a_k+1≤|h_21|w_2 /|h_11| w_1<b_k+1/a_k+1, 0< w_1 ≤√(3 p_1/M^2-1), 0< w_2 ≤√(3 p_2/M^2-1). For Case-1 of Sub-problem 1, constraint (<ref>) is equivalent to b_k+1 |h_11| /a_k+1|h_21|+|h_22|w_1 ≤ w_2 andconstraint (<ref>) means (b_k+b_k+1) |h_11|/(a_k+a_k+1)|h_21| w_1≤ w_2≤b_k+1 |h_11|/a_k+1 |h_21| w_1. Also we notice that if |h_21|/|h_22|≥ b_k+b_k+1 then (b_k+b_k+1)|h_11|/(a_k+a_k+1)|h_21|w_1≤ b_k+1 |h_11| /a_k+1|h_21|+|h_22| w_1. Hence, the optimization problem can be further divided into two cases: * If |h_21|/|h_22|≤ b_k +b_k+1, then g_11(b_k/a_k, b_k+1/a_k+1)=max_w_1, w_2 b_k+1 |h_11| w_1-a_k+1|h_21|w_2 s.t. (b_k+b_k+1) |h_11|/(a_k+a_k+1)|h_21| w_1≤ w_2≤b_k+1 |h_11|/a_k+1 |h_21|, 0< w_1 ≤√(3 p_1/M^2-1), 0< w_2 ≤√(3 p_2/M^2-1). We let w_2=(b_k+b_k+1) |h_11|/(a_k+a_k+1)|h_21| w_1, then the objective function is |h_11| w_1/a_k+a_k+1. In this case, (<ref>) is equivalent to w_1 ≤(a_k+a_k+1)|h_21|/(b_k+b_k+1) |h_11|√(3 p_2/M^2-1) and w_1 ≤√(3 p_1/M^2-1) and hence, we have g_11(b_k/a_k, b_k+1/a_k+1)=min{|h_11|/a_k+a_k+1(a_k+a_k+1)|h_21|/(b_k+b_k+1) |h_11|√(3 p_2/M^2-1), |h_11|/a_k+a_k+1√(3 p_1/M^2-1)}. As a consequence, we have g_11(b_k/a_k, b_k+1/a_k+1)=|h_21|/b_k+b_k+1√(3 p_2/M^2-1), where (w_1, w_2)= ((a_k+a_k+1)|h_21|/(b_k+b_k+1) |h_11|√(3 p_2/M^2-1), √(3 p_2/M^2-1)), if |h_11|/|h_21|≥√(p_2)(a_k+ a_k+1)/√(p_1) (b_k+b_k+1); |h_11|/a_k+ a_k+1√(3 p_1/M^2-1), where (w_1, w_2)=(√(3 p_1/M^2-1),(b_k+b_k+1) |h_11|/(a_k+a_k+1)|h_21|√(3 p_1/M^2-1)), if |h_11|/|h_21|< √(p_2)(a_k+ a_k+1)/√(p_1) (b_k+b_k+1), * If |h_21|/|h_22| > b_k +b_k+1, then g_11(b_k/a_k, b_k+1/a_k+1)=max_w_1, w_2 b_k+1 |h_11| w_1-a_k+1|h_21|w_2s.t.b_k+1 |h_11| /a_k+1|h_21|+|h_22|w_1≤ w_2≤b_k+1 |h_11|/a_k+1 |h_21| w_1, 0< w_1 ≤√(3 p_1/M^2-1), 0< w_2 ≤√(3 p_2/M^2-1). We first notice that b_k+1 |h_11| /a_k+1|h_21|+|h_22|w_1<b_k+1 |h_11|/a_k+1 |h_21| w_1 and hence the problem is always feasible. By letting w_2=b_k+1 |h_11| /a_k+1|h_21|+|h_22|w_1, the objective function can be written by b_k+1 |h_11||h_22|/a_k+1|h_21|+|h_22|w_1.In this case, the constraints in (<ref>) are equivalent to w_1 ≤√(3 p_1/M^2-1) and w_1 ≤a_k+1|h_21|+|h_22|/b_k+1 |h_11| √(3 p_2/M^2-1). Therefore, the solution is g_11(b_k/a_k, b_k+1/a_k+1) =b_k+1|h_11||h_22|/a_k+1|h_21|+|h_22|√(3 p_1/M^2-1), where (w_1, w_2)=(√(3 p_1/M_1^2-1), b_k+1|h_11|/a_k+1|h_21|+|h_22|√(3 p_1/M^2-1)), if |h_11|/|h_21|≤√(p_2)/√(p_1)(a_k+1/b_k+1 +|h_22| /b_k+1|h_21|); |h_22|√(3 p_2/M_2^2-1), where (w_1, w_2)=(a_k+1|h_21|+|h_22|/b_k+1 |h_11| √(3 p_2/M^2-1), √(3 p_2/M^2-1)), if |h_11|/|h_21|> √(p_2)/√(p_1)(a_k+1/b_k+1 +|h_22| /b_k+1|h_21|).For Case-2 of Sub-problem 1,constraint (<ref>) is equivalent to w_2< b_k+1 |h_11| /a_k+1|h_21|+|h_22| w_1 and constraint (<ref>) implies (b_k+b_k+1)|h_11|/(a_k+a_k+1)|h_21|w_1≤ w_2 ≤b_k+1 |h_11|/a_k+1|h_21| w_1. By noticing that b_k+1 |h_11| /a_k+1|h_21|+|h_22|< b_k+1 |h_11|/a_k+1|h_21|,the problem is equivalent to g_12(b_k/a_k, b_k+1/a_k+1)=max_w_1, w_2 |h_22|w_2 s.t. (b_k+b_k+1)|h_11|/(a_k+a_k+1)|h_21|w_1≤ w_2 ≤ b_k+1 |h_11| /a_k+1|h_21|+|h_22| w_1, 0< w_1 ≤√(3 p_1/M^2-1),0< w_2 ≤√(3 p_2/M^2-1). Constraint (<ref>) is feasible if |h_21|/|h_22|≥ b_k+b_k+1.In this case, the solution is g_12(b_k/a_k, b_k+1/a_k+1) =b_k+1|h_11||h_22|/a_k+1|h_21|+|h_22|√(3 p_1/M^2-1), where (w_1, w_2)=(√(3 p_1/M^2-1),b_k+1|h_11|/a_k+1|h_21|+|h_22|√(3 p_1/M^2-1)), if |h_11|/|h_21|≤√(p_2)/√(p_1)(a_k+1/b_k+1 +|h_22| /b_k+1|h_21|); |h_22|√(3 p_2/M_2^2-1), where (w_1, w_2)=(a_k+1|h_21|+|h_22|/ b_k+1 |h_11| √(3 p_2/M_2^2-1),√(3 p_2/M_2^2-1)), if |h_11|/|h_21|> √(p_2)/√(p_1)(a_k+1/b_k+1 +|h_22| /b_k+1|h_21|). The solution of Sub-problem 2 can be attained in a similar fashion as Sub-problem 1, and hence is omitted. Then, for the subinterval division |h_21|w_2 /|h_11| w_1∈(b_k/a_k, b_k+1/a_k+1), the solution to both Sub-problems can be summarized as follows: * Scenario 1, b_k+b_k+1/a_k+a_k+1≤|h_21|w_2 /|h_11| w_1≤b_k+1/a_k+1: * If |h_21|/|h_22|≤ b_k +b_k+1 g_11(b_k/a_k, b_k+1/a_k+1) =|h_21|/b_k+b_k+1√(3 p_2/M^2-1), where (w_1, w_2) = ((a_k+a_k+1)|h_21|/(b_k+b_k+1) |h_11|√(3 p_2/M^2-1),√(3 p_2/M^2-1)), if |h_11|/|h_21|≥√(p_2)(a_k+ a_k+1)/√(p_1) (b_k+b_k+1); |h_11|/a_k+ a_k+1√(3 p_1/M^2-1), where (w_1, w_2)=(√(3 p_1/M^2-1), (b_k+b_k+1) |h_11|/(a_k+a_k+1)|h_21|√(3 p_1/M^2-1)), if |h_11|/|h_21|< √(p_2)(a_k+ a_k+1)/√(p_1) (b_k+b_k+1), * If |h_21|/|h_22| > b_k +b_k+1 g_11(b_k/a_k, b_k+1/a_k+1) =b_k+1|h_11||h_22|/a_k+1|h_21|+|h_22|√(3 p_1/M^2-1), where (w_1, w_2)= (√(3 p_1/M^2-1),b_k+1|h_11|/a_k+1|h_21|+|h_22|√(3 p_1/M^2-1)), if |h_11|/|h_21|≤√(p_2)/√(p_1)(a_k+1/b_k+1 +|h_22| /b_k+1|h_21|); |h_22|√(3 p_2/M^2-1), where (w_1, w_2)= (a_k+1|h_21|+|h_22|/b_k+1 |h_11| √(3 p_2/M^2-1), √(3 p_2/M^2-1)), if |h_11|/|h_21|> √(p_2)/√(p_1)(a_k+1/b_k+1 +|h_22| /b_k+1|h_21|). * Scenario 2, b_k/a_k≤|h_21|w_2 /|h_11| w_1≤b_k+b_k+1/a_k+a_k+1: * Case 1: Receiver 1 has smaller Euclidean distance. * If |h_21|/|h_22|≤ b_k+b_k+1, the solution is g_21(b_k/a_k, b_k+1/a_k+1) =|h_21|/b_k+b_k+1√(3 p_2/M^2-1), where (w_1, w_2)= ((a_k+a_k+1)|h_21|/(b_k+b_k+1)|h_11|√(3 p_2/M^2-1),√(3 p_2/M^2-1)), if |h_11|/|h_21|≥√(p_2)(a_k+a_k+1)/√(p_1)(b_k+b_k+1); |h_11|/a_k+a_k+1√(3 p_1/M^2-1), where (w_1, w_2)= (√(3 p_1/M^2-1), (b_k+b_k+1)|h_11|/(a_k+a_k+1)|h_21|√(3 p_1/M^2-1)), if |h_11|/|h_21|< √(p_2)(a_k+a_k+1)/√(p_1)(b_k+b_k+1). * If |h_21|/|h_22|≥ b_k+b_k+1, g_21(b_k/a_k, b_k+1/a_k+1)=|h_22|√(3 p_2/M^2-1), where (w_1, w_2)= (|h_21| a_k-|h_22|/b_k |h_11|√(3 p_2/M^2-1), √(3 p_2/M^2-1)), if |h_11|/|h_21|≥√(p_2)/√(p_1)( a_k/b_k -|h_22|/|h_21|b_k); b_k |h_11||h_22| /|h_21| a_k-|h_22|√(3 p_1/M^2-1), where (w_1, w_2)= (√(3 p_1/M^2-1), b_k |h_11|/|h_21| a_k-|h_22|√(3 p_1/M^2-1)), if |h_11|/|h_21|<√(p_2)/√(p_1)( a_k/b_k -|h_22|/|h_21|b_k). * Case 2: Receiver 2 has smaller Euclidean distance. * If |h_21|/|h_22|≥ b_k +b_k+1, the solution is g_22(b_k/a_k, b_k+1/a_k+1) =(b_k+b_k+1)|h_11||h_22|/(a_k+a_k+1)|h_21|√(3 p_1/M^2-1), where (w_1, w_2) = (√(3 p_1/M^2-1), (b_k+b_k+1)|h_11|/(a_k+a_k+1)|h_21|√(3 p_1/M^2-1)), if |h_11| /|h_21|≤√(p_2)(a_k+a_k+1)/√(p_1)(b_k+b_k+1); |h_22|√(3 p_2/M^2-1), where (w_1, w_2)= ((a_k+a_k+1)|h_21|/(b_k+b_k+1)|h_11|√(3 p_2/M^2-1),√(3 p_2/M^2-1)), if |h_11| /|h_21|>√(p_2)(a_k+a_k+1)/√(p_1)(b_k+b_k+1). Now, we aim to combine g_21(b_k/a_k, b_k+1/a_k+1) and g_22(b_k/a_k, b_k+1/a_k+1).By Proposition <ref>, for |h_21|/|h_22|≥ b_k +b_k+1, we have a_k/b_k -|h_22|/|h_21|b_k≥a_k+a_k+1/b_k+b_k+1. Also, for |h_11|/|h_21|≤√(p_2)/√(p_1)(a_k/b_k-|h_22|/b_k |h_21|), we have √(3 p_2/M^2-1)≥b_k |h_11| /|h_21| a_k-|h_22|√(3 p_1/M^2-1). In addition, for |h_21|/|h_22|≥ b_k+b_k+1, we have (b_k+b_k+1)/(a_k+a_k+1)|h_21|≥b_k /|h_21| a_k-|h_22|. Hence, we have g_22(b_k/a_k, b_k+1/a_k+1) ≥ g_21(b_k/a_k, b_k+1/a_k+1). We notice that for |h_21|/|h_22|≤ b_k +b_k+1, g_11(b_k/a_k, b_k+1/a_k+1)=g_21(b_k/a_k, b_k+1/a_k+1). Then for |h_21|/|h_22|≥ b_k +b_k+1, we combine g_11(b_k/a_k, b_k+1/a_k+1) and g_22(b_k/a_k, b_k+1/a_k+1).By Proposition <ref>, for |h_21|/|h_22|≥ b_k+b_k+1, we have √(p_2)/√(p_1)(a_k+1/b_k+1 +|h_22| /b_k+1|h_21|) ≤√(p_2)(a_k+a_k+1)/√(p_1)(b_k+b_k+1). Also, for |h_21|/ |h_22|≥b_k +b_k+1,we attain b_k+1|h_11||h_22|/a_k+1|h_21|+|h_22|√(3 p_1/M^2-1)≥(b_k+b_k+1)|h_11||h_22|/(a_k+a_k+1)|h_21|√(3 p_1/M^2-1). In addition, for |h_11|/|h_21|≤√(p_2) (a_k+a_k+1)/√(p_1)(b_k+b_k+1), we have |h_22|√(3 p_2/M^2-1)≥(b_k+b_k+1)|h_11||h_22|/(a_k+a_k+1)|h_21|√(3 p_1/M^2-1). In conclusion, for |h_21|/|h_22|≥ b_k +b_k+1, g_11(b_k/a_k, b_k+1/a_k+1) ≥ g_22(b_k/a_k, b_k+1/a_k+1).With the above discussion, we have the result in Lemma <ref> and we complete the proof.§.§ Proof of Theorem <ref>For the weak cross link, we have |h_21|/|h_22|≤ 1 ≤ b_k +b_k+1 for k=1, …, |𝒮_M-1|. * If |h_11|/|h_21|≤√(p_2)/M √(p_1), we have |h_11|/|h_21| ≤√(p_2) (a_k +a_k+1) /√(p_1)(b_k +b_k+1), andthen by Lemma <ref>, we attain g(b_k/a_k, b_k+1/a_k+1)= |h_11|/a_k + a_k+1√(3 p_1/M^2-1) for k=1, …, |𝒮_M-1|. Also, note that |𝒮_M-1| =min_k {(a_1+ a_2), …, (a_ |𝒮_M-1|+ a_ |𝒮_M-1|+1)}, then max {|h_11|/a_1+ a_2√(3 p_1/M^2-1),⋯, |h_11|/a_ |𝒮_M-1|+ a_ |𝒮_M-1|+1√(3 p_1/M^2-1)} =|h_11|√(3 p_1/M^2-1), with (w_1^*, w_2^*)=(√(3 p_1/M^2-1), M |h_11|/|h_21|√(3 p_1/M^2-1)). * If |h_11|/|h_21|≥M√(p_2)/√(p_1), we have |h_11|/|h_21| ≥(a_k +a_k+1) √(p_2)/(b_k +b_k+1)√(p_1) and then, byusing Lemma <ref>, we have g(b_k/a_k, b_k+1/a_k+1)= |h_21|/b_k + b_k+1√(3 p_2/M_2^2-1) for k=1, …, |𝒮_M-1|. Also, note that 1=min_k {(b_1+ b_2), …, (b_ |𝒮_M-1|+ b_ |𝒮_M-1|+1)}.Then, max {|h_21|/b_1+ b_2√(3 p_2/M^2-1),⋯, |h_21|/b_ |𝒮_M-1|+ b_ |𝒮_M-1|+1√(3 p_2/M^2-1)} =|h_21|√(3 p_2/M^2-1), with (w_1^*, w_2^*)=(M |h_21|/|h_11|√(3 p_2/M^2-1),√(3 p_2/M^2-1)). * If |h_11|/|h_21| ∈(√(p_2)(a_ℓ_1+1 +a_ℓ_1+2)/√(p_1)(b_ℓ_1+1 +b_ℓ_1+2), √(p_2) (a_ℓ_1 +a_ℓ_1+1)/√(p_1)(b_ℓ_1 +b_ℓ_1+1)), for ℓ_1=1, …, |𝒮_M-1|-1, then, with the help of Lemma <ref>, we have g(b_k/a_k, b_k+1/a_k+1)=|h_11|/a_k + a_k+1√(3 p_1/M^2-1) for k=1, …, ℓ_1,|h_21|/b_k+b_k+1√(3 p_2/M^2-1) for k=ℓ_1+1, …,|𝒮_M-1|. Note that, ℓ̃_a=min_k {(a_1+a_2), …, (a_ℓ_1+ a_ℓ_1+1) } and ℓ̃_b=min_k {(b_ℓ_1+1+b_ℓ_1+2), …, (b_|𝒮_M-1|+b_|𝒮_M-1|+1)}, hence we have max {|h_11|/a_1+ a_2√(3 p_1/M^2-1),⋯, |h_11|/a_ℓ_1+ a_ℓ_1+1√(3 p_1/M^2-1),|h_21|/b_ℓ_1+1+b_ℓ_1+2√(3 p_2/M^2-1), …,|h_21|/b_|𝒮_M-1|+b_|𝒮_M-1|+1√(3 p_2/M^2-1)}=max {|h_11|/a_ℓ̃_a+ a_ℓ̃_a+1√(3 p_1/M^2-1),|h_21|/b_ℓ̃_b+b_ℓ̃_b+1√(3 p_2/M^2-1)}. Therefore, if |h_11|/|h_21|≥√( p_2)(a_ℓ̃_a+ a_ℓ̃_a+1) /√( p_1)(b_ℓ̃_b+b_ℓ̃_b+1), we have |h_11|/a_ℓ̃_a+ a_ℓ̃_a+1√(3 p_1/M^2-1)≥|h_21|/b_ℓ̃_b+b_ℓ̃_b+1√(3 p_2/M^2-1) and hence, (w_1^*, w_2^*)=(√(3 p_1/M^2-1),(b_ℓ̃_a +b_ℓ̃_a+1)|h_11|/(a_ℓ̃_a+a_ℓ̃_a+1)|h_21|√(3 p_1/M-1)) and else we have (w_1^*, w_2^*) = ((a_ℓ̃_b + a_ℓ̃_b+1)|h_21|/(b_ℓ̃_b+b_ℓ̃_b+1) |h_11|√(3 p_2/M^2-1), √(3 p_2/M^2-1)). This completes the proof. §.§ Proof of Theorem <ref> * If |h_11| ≤√(p_2)/√(p_1) |h_22|, we have |h_11|/|h_21|≤√(p_2 )/√(p_1)|h_22| /|h_21|=√(p_2)/√(p_1)(a_|𝒮_M-1|+1/b_|𝒮_M-1|+1+|h_22| /b_|𝒮_M-1|+1|h_21|) ≤√(p_2)/√(p_1)( a_k+1/b_k+1+|h_22| /b_k+1|h_21|) for k=1,…, |𝒮_M-1|. According to Lemma <ref>, we have g(b_k/a_k, b_k+1/a_k+1)= b_k+1 |h_11||h_22|/a_k+1 |h_21|+|h_22|√(3 p_1/M^2-1), for k=1,…, |𝒮_M-1|. We also have max {b_2 |h_11||h_22|/a_2 |h_21|+|h_22|√(3 p_1/M^2-1) , …, b_|𝒮_M-1|+1 |h_11||h_22|/a_|𝒮_M-1|+1 |h_21|+|h_22|√(3 p_1/M^2-1)} =|h_11| √(3 p_1/M^2-1), with (w_1^*, w_2^*)=(√(3 p_1/M^2-1),|h_11|/|h_22|√(3 p_1/M^2-1)). * If |h_11|/|h_21| ∈√(p_2)/√(p_1)(a_ℓ_4+1/b_ℓ_4+1+|h_22| /b_ℓ_4+1 |h_21|, a_ℓ_4/b_ℓ_4+|h_22| /b_ℓ_4 |h_21|) for ℓ_4=1,…, |𝒮_M-1|.Then, according to Lemma <ref>, we have g(b_k/a_k, b_k+1/a_k+1)=b_k+1 |h_11||h_22|/a_k+1 |h_21|+|h_22|√(3 p_1/M^2-1), for k=1,…, ℓ_4-1 andg(b_k/a_k, b_k+1/a_k+1)=|h_22|√(3 p_2/M^2-1) for k=ℓ_4, …, |𝒮_M-1|.g(b_k/a_k, b_k+1/a_k+1)=b_k+1 |h_11||h_22|/a_k+1 |h_21|+|h_22|√(3 p_1/M^2-1) for k=1,…, ℓ_4-1, |h_22|√(3 p_2/M^2-1) for k=ℓ_4, …, |𝒮_M-1|.Then, we have,max {b_2 |h_11||h_22|/a_2 |h_21|+|h_22|√(3 p_1/M^2-1) , …, b_ℓ_4 |h_11||h_22|/a_ℓ_4 |h_21|+|h_22|√(3 p_1/M^2-1),|h_22|√(3 p_2/M^2-1)} = max {b_ℓ_4 |h_11||h_22|/a_ℓ_4 |h_21|+|h_22|√(3 p_1/M^2-1), |h_22|√(3 p_2/M^2-1)}.As a result, if |h_11|/|h_21|≥√(p_2)/√(p_1)(a_ℓ_4/b_ℓ_4+|h_22| /b_ℓ_4|h_21|), then b_ℓ_4 |h_11||h_22|/a_ℓ_4 |h_21|+|h_22|√(3 p_1/M^2-1)≥ |h_22|√(3 p_2/M^2-1) and (w_1^*,w_2^*)=(√(3 p_1/M^2-1), b_ℓ_4|h_11|/a_ℓ_4|h_21|+|h_22|√(3 p_1/M^2-1)). Else, we can attain (w_1^*, w_2^*)=(a_ℓ_4+1|h_21|+|h_22|/b_ℓ_4+1 |h_11| √(3 p_2/M^2-1), √(3 p_2/M^2-1)). This completes the proof. ieeetr | http://arxiv.org/abs/1707.08305v1 | {
"authors": [
"Zheng Dong",
"He Chen",
"Jian-Kang Zhang",
"Lei Huang"
],
"categories": [
"cs.IT",
"math.IT"
],
"primary_category": "cs.IT",
"published": "20170726070449",
"title": "On Non-Orthogonal Multiple Access with Finite-Alphabet Inputs in Z-Channels"
} |
[ DARLA: Improving Zero-Shot Transfer in Reinforcement Learning equal* Irina Higginsequal,dm Arka Palequal,dm Andrei Rusudm Loic Mattheydm Christopher Burgessdm Alexander Pritzeldm Matthew Botvinickdm Charles Blundelldm Alexander Lerchnerdm dmDeepMind, 6 Pancras Square, Kings Cross, London, N1C 4AG, UKIrina [email protected] Arka [email protected] transfer learning, domain adaptation, reinforcement learning, disentangled representations, beta-VAE0.3in ]Domain adaptation is an important open problem in deep reinforcement learning (RL). In many scenarios of interest data is hard to obtain, so agents may learn a source policy in a setting where data is readily available, with the hope that it generalises well to the target domain. We propose a new multi-stage RL agent, DARLA (DisentAngled Representation Learning Agent), which learns to see before learning to act. DARLA's vision is based on learning a disentangled representation of the observed environment. Once DARLA can see, it is able to acquire source policies that are robust to many domain shifts - even with no access to the target domain. DARLA significantly outperforms conventional baselines in zero-shot domain adaptation scenarios, an effect that holds across a variety of RL environments (Jaco arm, DeepMind Lab) and base RL algorithms (DQN, A3C and EC). § INTRODUCTION Autonomous agents can learn how to maximise future expected rewards by choosing how to act based on incoming sensory observations via reinforcement learning (RL). Early RL approaches did not scale well to environments with large state spaces and high-dimensional raw observations <cit.>. A commonly used workaround was to embed the observations in a lower-dimensional space, typically via hand-crafted and/or privileged-information features. Recently, the advent of deep learning and its successful combination with RL has enabled end-to-end learning of such embeddings directly from raw inputs, sparking success in a wide variety of previously challenging RL domains <cit.>. Despite the seemingly universal efficacy of deep RL, however, fundamental issues remain. These include data inefficiency, the reactive nature and general brittleness of learnt policies to changes in input data distribution, and lack of model interpretability <cit.>. This paper focuses on one of these outstanding issues: the ability of RL agents to deal with changes to the input distribution, a form of transfer learning known as domain adaptation <cit.>. In domain adaptation scenarios, an agent trained on a particular input distribution with a specified reward structure (termed the source domain) is placed in a setting where the input distribution is modified but the reward structure remains largely intact (the target domain). We aim to develop an agent that can learn a robust policy using observations and rewards obtained exclusively within the source domain. Here, a policy is considered as robust if it generalises with minimal drop in performance to the target domain without extra fine-tuning.Past attempts to build RL agents with strong domain adaptation performance highlighted the importance of learning good internal representations of raw observations <cit.>. Typically, these approaches tried to align the source and target domain representations by utilising observation and reward signals from both domains <cit.>. In many scenarios, such as robotics, this reliance on target domain information can be problematic, as the data may be expensive or difficult to obtain <cit.>. Furthermore, the target domain may simply not be known in advance. On the other hand, policies learnt exclusively on the source domain using existing deep RL approaches that have few constraints on the nature of the learnt representations often overfit to the source input distribution, resulting in poor domain adaptation performance <cit.>. We propose tackling both of these issues by focusing instead on learning representations which capture an underlying low-dimensional factorised representation of the world and are therefore not task or domain specific. Many naturalistic domains such as video game environments, simulations and our own world are well described in terms of such a structure. Examples of such factors of variation are object properties like colour, scale, or position; other examples correspond to general environmental factors, such as geometry and lighting. We think of these factors as a set of high-level parameters that can be used by a world graphics engine to generate a particular natural visual scene <cit.>. Learning how to project raw observations into such a factorised description of the world is addressed by the large body of literature on disentangled representation learning <cit.>. Disentangled representations are defined as interpretable, factorised latent representations where either a single latent or a group of latent units are sensitive to changes in single ground truth factors of variation used to generate the visual world, while being invariant to changes in other factors <cit.>. The theoretical utility of disentangled representations for supervised and reinforcement learning has been described before <cit.>; however, to our knowledge, it has not been empirically validated to date. We demonstrate how disentangled representations can improve the robustness of RL algorithms in domain adaptation scenarios by introducing DARLA (DisentAngled Representation Learning Agent), a new RL agent capable of learning a robust policy on the source domain that achieves significantly better out-of-the-box performance in domain adaptation scenarios compared to various baselines. DARLA relies on learning a latent state representation that is shared between the source and target domains, by learning a disentangled representation of the environment's generative factors. Crucially, DARLA does not require target domain data to form its representations. Our approach utilises a three stage pipeline: 1) learning to see, 2) learning to act, 3) transfer. During the first stage, DARLA develops its vision, learning to parse the world in terms of basic visual concepts, such as objects, positions, colours, etc. by utilising a stream of raw unlabelled observations – not unlike human babies in their first few months of life <cit.>. In the second stage, the agent utilises this disentangled visual representation to learn a robust source policy. In stage three, we demonstrate that the DARLA source policy is more robust to domain shifts, leading to a significantly smaller drop in performance in the target domain even when no further policy finetuning is allowed (median 270.3% improvement). These effects hold consistently across a number of different RL environments <cit.> and algorithms <cit.>. § FRAMEWORK §.§ Domain adaptation in Reinforcement LearningWe now formalise domain adaptation scenarios in a reinforcement learning (RL) setting. We denote the source and target domains as D_S and D_T, respectively. Each domain corresponds to an MDP defined as a tuple D_S ≡ (𝒮_S, 𝒜_S, 𝒯_S, R_S) or D_T ≡ (𝒮_T, 𝒜_T, 𝒯_T, R_T) (we assume a shared fixed discount factor γ), each with its own state space 𝒮, action space 𝒜, transition function 𝒯 and reward function R.[For further background on the notation relating to the RL paradigm, see Section <ref> in the Supplementary Materials.] In domain adaptation scenarios the states 𝒮 of the source and the target domains can be quite different, while the action spaces 𝒜 are shared and the transitions 𝒯 and reward functions R have structural similarity. For example, consider a domain adaptation scenario for the Jaco robotic arm, where the MuJoCo <cit.> simulation of the arm is the source domain, and the real world setting is the target domain. The state spaces (raw pixels) of the source and the target domains differ significantly due to the perceptual-reality gap <cit.>; that is to say, 𝒮_S ≠𝒮_T. Both domains, however, share action spaces (𝒜_S = 𝒜_T), since the policy learns to control the same set of actuators within the arm. Finally, the source and target domain transition and reward functions share structural similarity (𝒯_S ≈𝒯_T and R_S ≈ R_T), since in both domains transitions between states are governed by the physics of the world and the performance on the task depends on the relative position of the arm's end effectors (i.e. fingertips) with respect to an object of interest. §.§ DARLA In order to describe our proposed DARLA framework, we assume that there exists a set ℳ of MDPs that is the set of all natural world MDPs, and each MDP D_i is sampled from ℳ. We define ℳ in terms of the state space 𝒮̂ that contains all possible conjunctions of high-level factors of variation necessary to generate any naturalistic observation in any D_i ∈ℳ. A natural world MDP D_i is then one whose state space 𝒮 corresponds to some subset of 𝒮̂. In simple terms, we assume that there exists some shared underlying structure between the MDPs D_i sampled from ℳ. We contend that this is a reasonable assumption that permits inclusion of many interesting problems, including being able to characterise our own reality <cit.>.We now introduce notation for two state space variables that may in principle be used interchangeably within the source and target domain MDPs D_S and D_T – the agent observation state space 𝒮^o, and the agent's internal latent state space 𝒮^z.[Note that we do not assume these to be Markovian i.e. it is not necessarily the case that p(s^o_t+1|s^o_t) = p(s^o_t+1|s^o_t, s^o_t-1, …, s^o_1), and similarly for s^z. Note the index t here corresponds to time.] 𝒮^o_i in D_i consists of raw (pixel) observations s^o_i generated by the true world simulator from a sampled set of data generative factors ŝ_i, i.e. s^o_i ∼Sim(ŝ_i). ŝ_i is sampled by some distribution or process 𝒢_i on 𝒮̂, ŝ_i ∼𝒢_i(𝒮̂). Using the newly introduced notation, domain adaptation scenarios can be described as having different sampling processes 𝒢_S and 𝒢_T such that ŝ_S ∼𝒢_S(𝒮̂) and ŝ_T ∼𝒢_T(𝒮̂) for the source and target domains respectively, and then using these to generate different agent observation states s^o_S ∼Sim(ŝ_S) and s^o_T ∼Sim(ŝ_T). Intuitively, consider a source domain where oranges appear in blue rooms and apples appear in red rooms, and a target domain where the object/room conjunctions are reversed and oranges appear in red rooms and apples appear in blue rooms. While the true data generative factors of variation 𝒮̂ remain the same - room colour (blue or red) and object type (apples and oranges) - the particular source and target distributions 𝒢_S and 𝒢_T differ.Typically deep RL agents <cit.> operating in an MDP D_i ∈ℳ learn an end-to-end mapping from raw (pixel) observations s^o_i ∈𝒮^o_i to actions a_i ∈𝒜_i (either directly or via a value function Q_i(s^o_i, a_i) from which actions can be derived). In the process of doing so, the agent implicitly learns a function ℱ: 𝒮^o_i →𝒮^z_i that maps the typically high-dimensional raw observations s^o_i to typically low-dimensional latent states s^z_i; followed by a policy function π_i: 𝒮^z_i →𝒜_i that maps the latent states s^z_i to actions a_i ∈𝒜_i. In the context of domain adaptation, if the agent learns a naive latent state mapping function ℱ_S:𝒮^o_S →𝒮^z_S on the source domain using reward signals to shape the representation learning, it is likely that ℱ_S will overfit to the source domain and will not generalise well to the target domain. Returning to our intuitive example, imagine an agent that has learnt a policy to pick up oranges and avoid apples on the source domain. Such a source policy π_S is likely to be based on an entangled latent state space 𝒮^z_S of object/room conjunctions: oranges/blue → good, apples/red → bad, since this is arguably the most efficient representation for maximising expected rewards on the source task in the absence of extra supervision signals suggesting otherwise. A source policy π_S(a|s^z_S; θ) based on such an entangled latent representation s^z_S will not generalise well to the target domain without further fine-tuning, since ℱ_S(s^o_S) ≠ℱ_S(s^o_T) and therefore crucially S^z_S ≠ S^z_T.On the other hand, since both ŝ_S ∼𝒢_S(𝒮̂) and ŝ_T ∼𝒢_T(𝒮̂) are sampled from the same natural world state space 𝒮̂ for the source and target domains respectively, it should be possible to learn a latent state mapping functionℱ̂: 𝒮^o →𝒮^z_𝒮̂, which projects the agent observation state space 𝒮^o to a latent state space 𝒮^z_𝒮̂ expressed in terms of factorised data generative factors that are representative of the natural world i.e. S^z_Ŝ≈Ŝ. Consider again our intuitive example, where ℱ̂ maps agent observations (s^o_S: orange in a blue room) to a factorised or disentangled representation expressed in terms of the data generative factors (s^z_𝒮̂: room type = blue; object type = orange). Such a disentangled latent state mapping function should then directly generalise to both the source and the target domains, so that ℱ̂(s^o_S) = ℱ̂(s^o_T) = s^z_𝒮̂. Since 𝒮^z_𝒮̂ is a disentangled representation of object and room attributes, the source policy π_S can learn a decision boundary that ignores the irrelevant room attributes: oranges → good, apples → bad. Such a policy would then generalise well to the target domain out of the box, sinceπ_S(a|ℱ̂(s^o_S); θ) = π_T(a|ℱ̂(s^o_T); θ) = π_T(a|s^z_𝒮̂; θ). Hence, DARLA is based on the idea that a good quality ℱ̂ learnt exclusively on the source domain D_S ∈ℳ will zero-shot-generalise to all target domains D_i ∈ℳ, and therefore the source policy π(a|𝒮^z_𝒮̂; θ) will also generalise to all target domains D_i ∈ℳ out of the box.Next we describe each of the stages of the DARLA pipeline that allow it to learn source policies π_S that are robust to domain adaptation scenarios, despite being trained with no knowledge of the target domains (see Fig. <ref> for a graphical representation of these steps):1) Learn to see (unsupervised learning of ℱ_U) – the task of inferring a factorised set of generative factors 𝒮^z_𝒮̂ = Ŝ from observations 𝒮^o is the goal of the extensive disentangled factor learning literature <cit.>. Hence, in stage one we learn a mapping ℱ_U: 𝒮^o_U →𝒮^z_U, where 𝒮^z_U ≈𝒮^z_𝒮̂ (U stands for `unsupervised') using an unsupervised model for learning disentangled factors that utilises observations collected by an agent with a random policy π_U from a visual pre-training MDP D_U ∈ℳ. Note that we require sufficient variability of factors and their conjunctions in D_U in order to have S^z_U ≈𝒮^z_𝒮̂;2) Learn to act (reinforcement learning of π_S in the source domain D_S utilising previously learned ℱ_U) – an agent that has learnt to see the world in stage one in terms of the natural data generative factors is now exposed to a source domain D_S ∈ℳ. The agent is tasked with learning the source policy π_S(a|s^z_S; θ), where s^z_S = ℱ_U(s^o_S) ≈ s^z_𝒮̂, via a standard reinforcement learning algorithm. Crucially, we do not allow ℱ_U to be modified (e.g. by gradient updates) during this phase;3) Transfer (to a target domain D_T) – in the final step, we test how well the policy π_S learnt on the source domain generalises to the target domain D_T ∈ℳ in a zero-shot domain adaptation setting, i.e. the agent is evaluated on the target domain without retraining. We compare the performance of policies learnt with a disentangled latent state 𝒮^z_𝒮̂ to various baselines where the latent state mapping function ℱ_U projects agent observations s^o to entangled latent state representations s^z.§.§ Learning disentangled representations In order to learn ℱ_U, DARLA utilises<cit.>, a state-of-the-art unsupervised model for automated discovery of factorised latent representations from raw image data.is a modification of the variational autoencoder framework <cit.> that controls the nature of the learnt latent representations by introducing an adjustable hyperparameter β to balance reconstruction accuracy with latent channel capacity and independence constraints. It maximises the objective:ℒ(θ, ϕ; x̌, ž, β)= 𝔼_q_ϕ(ž|x̌) [log p_θ(x̌ | ž)]- β D_KL(q_ϕ(ž|x̌) || p(ž))where ϕ, θ parametrise the distributions of the encoder and the decoder respectively. Well-chosen values of β - usually larger than one (β>1) - typically result in more disentangled latent representations ž by limiting the capacity of the latent information channel, and hence encouraging a more efficient factorised encoding through the increased pressure to match the isotropic unit Gaussian prior p(ž) <cit.>.§.§.§ Perceptual Similarity Loss The cost of increasing β is that crucial information about the scene may be discarded in the latent representation ž, particularly if that information takes up a small proportion of the observations x̌ in pixel space. We encountered this issue in some of our tasks, as discussed in Section <ref>. The shortcomings of calculating the log-likelihood term 𝔼_q_ϕ(ž|x̌)[log p_θ(x̌|ž)] on a per-pixel basis are known and have been addressed in the past by calculating the reconstruction cost in an abstract, high-level feature space given by another neural network model, such as a GAN <cit.> or a pre-trained AlexNet <cit.>. In practice we found that pre-training a denoising autoencoder <cit.> on data from the visual pre-training MDP D_U ∈ℳ worked best as the reconstruction targets forto match (see Fig. <ref> for model architecture and Sec. <ref> in Supplementary Materials for implementation details). The new model was trained according to Eq. <ref>: ℒ(θ, ϕ; x̌, ž, β)= - 𝔼_q_ϕ(ž|x̌)J(x̂̌̂) - J(x̌)_2^2 - β D_KL(q_ϕ(ž|x̌) || p(ž))where x̂̌̂∼p_θ(x̌ | ž) and J : ℝ^W × H × C→ℝ^N is the function that maps images from pixel space with dimensionality W × H × C to a high-level feature space with dimensionality N given by a stack of pre-trained DAE layers up to a certain layer depth. Note that by replacing the pixel based reconstruction loss in Eq. <ref> with high-level feature reconstruction loss in Eq. <ref> we are no longer optimising the variational lower bound, and with β=1 loses its equivalence to the Variational Autoencoder (VAE) framework as proposed by <cit.>. In this setting, the only way to interpret β is as a mixing coefficient that balances the capacity of the latent channel ž of against the pressure to match the high-level features within the pre-trained DAE.§.§ Reinforcement Learning Algorithms We used various RL algorithms <cit.> to learn the source policy π^S during stage two of the pipeline using the latent states s^z acquired bybased models during stage one of the DARLA pipeline. Deep Q Network (DQN) <cit.> is a variant of the Q-learning algorithm <cit.> that utilises deep learning. It uses a neural network to parametrise an approximation for the action-value function Q(s, a; θ) using parameters θ.Asynchronous Advantage Actor-Critic (A3C) <cit.> is an asynchronous implementation of the advantage actor-critic paradigm <cit.>, where separate threads run in parallel and perform updates to shared parameters. The different threads each hold their own instance of the environment and have different exploration policies, thereby decorrelating parameter updates without the need for experience replay. Therefore, A3C is an online algorithm, whereas DQN learns its policy offline, resulting in different learning dynamics between the two algorithms.Model-Free Episodic Control (EC) <cit.> was proposed as a complementary learning system to the other RL algorithms described above. The EC algorithm relies on near-determinism of state transitions and rewards in RL environments; in settings where this holds, it can exploit these properties to memorise which action led to high returns in similar situations in the past. Since in its simplest form EC relies on a lookup table, it learns good policies much faster than value-function-approximation based deep RL algorithms like DQN trained via gradient descent - at the cost of generality (i.e. potentially poor performance in non-deterministic environments).We also compared our approach to that of UNREAL <cit.>, a recently proposed RL algorithm which also attempts to utilise unsupervised data in the environment. The UNREAL agent takes as a base an LSTM A3C agent <cit.> and augments it with a number of unsupervised auxiliary tasks that make use of the rich perceptual data available to the agent besides the (sometimes very sparse) extrinsic reward signals. This auxiliary learning tends to improve the representation learnt by the agent. See Sec. <ref> in Supplementary Materials for further details of the algorithms above.§ TASKS We evaluate the performance of DARLA on different task and environment setups that probe subtly different aspects of domain adaptation. As a reminder, in Sec. <ref> we defined 𝒮̂ as a state space that contains all possible conjunctions of high-level factors of variation necessary to generate any naturalistic observation in any D_i ∈ℳ. During domain adaptation scenarios agent observation states are generated according to s^o_S ∼Sim_S(ŝ_S) and s^o_T ∼Sim_T(ŝ_T) for the source and target domains respectively, where ŝ_S and ŝ_T are sampled by some distributions or processes 𝒢_S and 𝒢_T according toŝ_S ∼𝒢_S(𝒮̂) and ŝ_T ∼𝒢_T(𝒮̂).We use DeepMind Lab <cit.> to test a version of domain adaptation setup where the source and target domain observation simulators are equal (Sim_S = Sim_T), but the processes used to sampleŝ_S andŝ_T are different (𝒢_S ≠𝒢_T). We use the Jaco arm with a matching MuJoCo simulation environment <cit.> in two domain adaptation scenarios: simulation to simulation (sim2sim) and simulation to reality (sim2real). The sim2sim domain adaptation setup is relatively similar to DeepMind Lab i.e. the source and target domains differ in terms of processes 𝒢_S and 𝒢_T. However, there is a significant point of difference. In DeepMind Lab, all values of factors in the target domain, ŝ_T, are previously seen in the source domain; however, ŝ_S ≠ŝ_T as the conjunctions of these factor values are different. In sim2sim, by contrast, novel factor values are experienced in the target domain (this accordingly also leads to novel factor conjunctions). Hence, DeepMind Lab may be considered to be assessing domain interpolation performance, whereas sim2sim tests domain extrapolation. The sim2real setup, on the other hand, is based on identical processes 𝒢_S = 𝒢_T, but different observation simulators Sim_S ≠Sim_T corresponding to the MuJoCo simulation and the real world, which results in the so-called `perceptual reality gap' <cit.>. More details of the tasks are given below. §.§ DeepMind Lab DeepMind Lab is a first person 3D game environment with rich visuals and realistic physics. We used a standard seek-avoid object gathering setup, where a room is initialised with an equal number of randomly placed objects of two different types. One of the object varieties is `good' (its collection is rewarded +1), while the other is `bad' (its collection is punished -1). The full state space 𝒮̂ consisted of all conjunctions of two room types (pink and green based on the colour of the walls) and four object types (hat, can, cake and balloon) (see Fig. <ref>A). The source domain D_S contained environments with hats/cans presented in the green room, and balloons/cakes presented in either the green or the pink room. The target domain D_T contained hats/cans presented in the pink room. In both domains cans and balloons were the rewarded objects. 1) Learn to see: we used to learn the disentangled latent state representation s^z that includes both the room and the object generative factors of variation within DeepMind Lab. We had to use the high-level feature space of a pre-trained DAE within the framework (see Section <ref>), instead of the pixel space of vanilla , because we found that objects failed to reconstruct when using the values of β necessary to disentangle the generative factors of variation within DeepMind Lab (see Fig. <ref>B).was trained on observations s^o_U collected by an RL agent with a simple wall-avoiding policy π_U (otherwise the training data was dominated by close up images of walls). In order to enable the model to learn ℱ(s^o_U) ≈𝒮̂, it is important to expose the agent to at least a minimal set of environments that span the range of values for each factor, and where no extraneous correlations are added between different factors[In our setup of DeepMind Lab domain adaptation task, the object and environment factors are supposed to be independent. In order to ensure that learns a factorised representation that reflects this ground truth independence, we present observations of all possible conjunctions of room and individual object types.](see Fig. <ref>A, yellow). See Section <ref> in Supplementary Materials for details of training.2) Learn to act: the agent was trained with the algorithms detailed in Section <ref> on a seek-avoid task using the source domain (D_S) conjunctions of object/room shown in Fig. <ref>A (green). Pre-trained from stage one was used as the `vision' part of various RL algorithms <cit.> to learn a source policy π_S that picks up balloons and avoids cakes in both the green and the pink rooms, and picks up cans and avoids hats in the green rooms. See Section <ref> in Supplementary Materials for more details of the various versions of DARLA we have tried, each based on a different base RL algorithm.3) Transfer: we tested the ability of DARLA to transfer the seek-avoid policy π_S it had learnt on the source domain in stage two using the domain adaptation condition D_T illustrated in Figure <ref>A (red). The agent had to continue picking up cans and avoid hats in the pink room, even though these objects had only been seen in the green room during source policy training. The optimal policy π_T is one that maintains the reward polarity from the source domain (cans are good and hats are bad). For further details, see Appendix <ref>.§.§ Jaco Arm and MuJoCo We used frames from an RGB camera facing a robotic Jaco arm, or a matching rendered camera view from a MuJoCo physics simulation environment <cit.> to investigate the performance of DARLA in two domain adaptation scenarios: 1) simulation to simulation (sim2sim), and 2) simulation to reality (sim2real). The sim2real setup is of particular importance, since the progress that deep RL has brought to control tasks in simulation <cit.> has not yet translated as well to reality, despite various attempts <cit.>. Solving control problems in reality is hard due to sparse reward signals, expensive data acquisition and the attendant danger of breaking the robot (or its human minders) during exploration.In both sim2sim and sim2real, we trained the agent to perform an object reaching policy where the goal is to place the end effector as close to the object as possible. While conceptually the reaching task is simple, it is a hard control problem since it requires correct inference of the arm and object positions and velocities from raw visual inputs. 1) Learn to see: was trained on observations collected in MuJoCo simulations with the same factors of variation as in D_S. In order to enable the model to learn ℱ(s^o_U) ≈ŝ, a reaching policy was applied to phantom objects placed in random positions - therefore ensuring that the agent learnt the independent nature of the arm position and object position (see Fig. <ref>C, left);2) Learn to act: a feedforward-A3C based agent with the vision module pre-trained in stage one was taught a source reaching policy π_S towards the real object in simulation (see Fig. <ref>C (left) for an example frame, and Sec. <ref> in Supplementary Materials for a fuller description of the agent). In the source domain D_S the agent was trained on a distribution of camera angles and positions. The colour of the tabletop on which the arm rests and the object colour were both sampled anew every episode.3) Transfer: sim2sim: in the target domain, D_T, the agent was faced with a new distribution of camera angles and positions with little overlap with the source domain distributions, as well as a completely held out set of object colours (see Fig. <ref>C, middle). sim2real: in the target domain D_T the camera position and angle as well as the tabletop colour and object colour were sampled from the same distributions as seen in the source domain D_S, but the target domain D_T was now the real world. Many details present in the real world such as shadows, specularity, multiple light sources and so on are not modelled in the simulation; the physics engine is also not a perfect model of reality. Thus sim2real tests the ability of the agent to cross the perceptual-reality gap and generalise its source policy π_S to the real world (see Fig. <ref>C, right). For further details, see Appendix <ref>. § RESULTSWe evaluated the robustness of DARLA's policy π_S learnt on the source domain to various shifts in the input data distribution. In particular, we used domain adaptation scenarios based on the DeepMind Lab seek-avoid task and the Jaco arm reaching task described in Sec. <ref>. On each task we compared DARLA's performance to that of various baselines. We evaluated the importance of learning `good' vision during stage one of the pipeline, i.e one that maps the input observations s^o to disentangled representations s^z ≈ŝ. In order to do this, we ran the DARLA pipeline with different vision models: the encoders of a disentangled -0.5ex [In this section of the paper, we use the term to refer to a standard for the MuJoCo experiments, and a for the DeepMind Lab experiments (as described in stage 1 of Sec. <ref>).] (the original DARLA), an entangled (DARLA_ENT), and a denoising autoencoder (DARLA_DAE). Apart from the nature of the learnt representations s^z, DARLA and all versions of its baselines were equivalent throughout the three stages of our proposed pipeline in terms of architecture and the observed data distribution (see Sec. <ref> in Supplementary Materials for more details). Figs. <ref>-<ref> display the degree of disentanglement learnt by the vision modules of DARLA and DARLA_ENT on DeepMind Lab and MuJoCo. DARLA's vision learnt to independently represent environment variables (such as room colour-scheme and geometry) and object-related variables (change of object type, size, rotation) on DeepMind Lab (Fig. <ref>, left). Disentangling was also evident in MuJoCo. Fig. <ref>, left, shows that DARLA's single latent units z_i learnt to represent different aspects of the Jaco arm, the object, and the camera. By contrast, in the representations learnt by DARLA_ENT, each latent is responsible for changes to both the environment and objects (Fig. <ref>, right) in DeepMind Lab, or a mixture of camera, object and/or arm movements (Fig. <ref>, right) in MuJoCo. The table in Fig. <ref> shows the average performance (across different seeds) in terms of rewards per episode of the various agents on the target domain with no fine-tuning of the source policy π_S. It can be seen that DARLA is able to zero-shot-generalise significantly better than DARLA_ENT or DARLA_DAE, highlighting the importance of learning a disentangled representation s^z = s^z_𝒮̂ during the unsupervised stage one of the DARLA pipeline. In particular, this also demonstrates that the improved domain transfer performance is not simply a function of increased exposure to training observations, as both DARLA_ENT and DARLA_DAE were exposed to the same data. The results are mostly consistent across target domains and in most cases DARLA is significantly better than the second-best-performing agent. This holds in the sim2real task [See https://youtu.be/sZqrWFl0wQ4 for example sim2sim and sim2real zero-shot transfer policies of DARLA and baseline A3C agent.], where being able to perform zero-shot policy transfer is highly valuable due to the particular difficulties of gathering data in the real world.DARLA's performance is particularly surprising as it actually preserves less information about the raw observations s^o than DARLA_ENT and DARLA_DAE. This is due to the nature of the and how it achieves disentangling; the disentangled model utilised a significantly higher value of the hyperparameter β than the entangled model (see Appendix <ref> for further details), which constrains the capacity of the latent channel. Indeed, DARLA's only utilises 8 of its possible 32 Gaussian latents to store observation-specific information for MuJoCo/Jaco (and 20 in DeepMind Lab), whereas DARLA_ENT utilises all 32 for both environments (as does DARLA_DAE). Furthermore, we examined what happens if DARLA's vision (i.e. the encoder of the disentangled -0.5ex) is allowed to be fine-tuned via gradient updates while learning the source policy during stage two of the pipeline. This is denoted by DARLA_FT in the table in Fig. <ref>. We see that it exhibits significantly worse performance than that of DARLA in zero-shot domain adaptation using an A3C-based agent in all tasks. This suggests that a favourable initialisation does not make up for subsequent overfitting to the source domain for the on-policy A3C. However, the off-policy DQN-based fine-tuned agent performs very well. We leave further investigation of this curious effect for future work.Finally, we compared the performance of DARLA to an UNREAL <cit.> agent with the same architecture. Despite also exploiting the unsupervised data available in the source domain, UNREAL performed worse than baseline A3C on the DeepMind Lab domain adaptation task. This further demonstrates that use of unsupervised data in itself is not a panacea for transfer performance; it must be utilised in a careful and structured manner conducive to learning disentangled latent states s^z = s^z_𝒮̂.In order to quantitatively evaluate our hypothesis that disentangled representations are essential for DARLA's performance in domain adaptation scenarios, we trained various DARLAs with different degrees of learnt disentanglement in s^z by varying β (of -0.5ex) during stage one of the pipeline. We then calculated the correlation between the performance of the EC-based DARLA on the DeepMind Lab domain adaptation task and the transfer metric, which approximately measures the quality of disentanglement of DARLA's latent representations s^z (see Sec. <ref> in Supplementary Materials). This is shown in the chart in Fig. <ref>; as can be seen, there is a strong positive correlation between the level of disentanglement and DARLA's zero-shot domain transfer performance (r=0.6, p<0.001).Having shown the robust utility of disentangled representations in agents for domain adaptation, we note that there is evidence that they can provide an important additional benefit. We found significantly improved speed of learning of π_S on the source domain itself, as a function of how disentangled the model was. The gain in data efficiency from disentangled representations for source policy learning is not the main focus of this paper so we leave it out of the main text; however, we provide results and discussion in Section <ref> in Supplementary Materials.§ CONCLUSIONWe have demonstrated the benefits of using disentangled representations in a deep RL setting for domain adaptation. In particular, we have proposed DARLA, a multi-stage RL agent. DARLA first learns a visual system that encodes the observations it receives from the environment as disentangled representations, in a completely unsupervised manner. It then uses these representations to learn a robust source policy that is capable of zero-shot domain adaptation. We have demonstrated the efficacy of this approach in a range of domains and task setups: a 3D naturalistic first-person environment (DeepMind Lab), a simulated graphics and physics engine (MuJoCo), and crossing the simulation to reality gap (MuJoCo to Jaco sim2real). We have also shown that the effect of disentangling is consistent across very different RL algorithms (DQN, A3C, EC), achieving significant improvements over the baseline algorithms (median 2.7 times improvement in zero-shot transfer across tasks and algorithms). To the best of our knowledge, this is the first comprehensive empirical demonstration of the strength of disentangled representations for domain adaptation in a deep RL setting. icml2017 § SUPPLEMENTARY MATERIALS§.§ The Reinforcement Learning Paradigm The reinforcement learning (RL) paradigm consists of an agent receiving a sequence of observations s^o_t which are some function of environment states s_t ∈𝒮 and may be accompanied by rewards r_t+1∈ R conditional on the actions a_t ∈𝒜, chosen at each time step t <cit.>. We assume that these interactions can be modelled as a Markov Decision Process (MDP) <cit.> defined as a tuple D ≡ (𝒮, 𝒜, 𝒯, R, γ ).𝒯 = p(s | s_t, a_t) is a transition function that models the distribution of all possible next states given action a_t is taken in state s_t for all s_t ∈𝒮 and a_t ∈𝒜. Each transition s_t a_t→ s_t+1 may be accompanied by a reward signal r_t+1(s_t, a_t, s_t+1). The goal of the agent is to learn a policy π(a_t|s_t), a probability distribution over actions a_t ∈𝒜, that maximises the expected return i.e. the discounted sum of future rewards R_t = 𝔼[∑_τ=1^T-tγ^τ-1r_t+τ]. T is the time step at which each episode ends, andγ∈ [0, 1) is the discount factor that progressively down-weights future rewards. Given a policy π(a|s), one can define the value function V_π(s) = 𝔼 [R_t|s_t=s, π], which is the expected return from state s following policy π. The action-value function Q_π(s, a) = 𝔼 [R_t|s_t=s, a_t=a, π] is the expected return for taking action a in state s at time t, and then following policy π from time t+1 onward. §.§ Further task details§.§.§ DeepMind Lab As described in Sec <ref>, in each source episode of DeepMind Lab the agent was presented with one of three possible room/object type conjunctions, chosen at random. These are marked D_S in Fig <ref>. The setup was a seek-avoid style task, where one of the two object types in the room gave a reward of +1 and the other gave a reward of -1. The agent was allowed to pick up objects for 60 seconds after which the episode would terminate and a new one would begin; if the agent was able to pick up all the `good' objects in less than 60 seconds, a new episode was begun immediately. The agent was spawned in a random location in the room at the start of each new episode.During transfer, the agent was placed into the held out conjunction of object types and room background; see D_T in Fig <ref>.Visual pre-training was performed in other conjunctions of object type and room background denoted D_U in Fig <ref>.The observation size of frames in the DeepMind Lab task was 84x84x3 (HxWxC).§.§.§ MuJoCo/Jaco Arm Experiments As described in Sec <ref>, the source task consisted of an agent learning to control a simulated arm in order to reach toward an object. A shaping reward was used, with a maximum value of 1 when the centre of the object fell between the pinch and grip sites of the end effector, or within a 10cm distance of the two. Distances on the x and y dimensions counted double compared to distances on the z dimension.During each episode the object was placed at a random drop point within a 40x40cm area, and the arm was set to a random initial start position high above the work-space, independent of the object's position. Each episode lasted for 150 steps, or 7.5 seconds, with a control step of 50ms. Observations s^o_U were sampled randomly across episodes. Overall, 4 million frames of dimensions 64x64x3 (HxWxC) were used for this stage of the curriculum. For each episode the camera position and orientation were randomly sampled from an isotropic normal distribution centred around the approximate position and orientation of the real camera, with standard deviation 0.01. No precise measurements were used to match the two. Work-space table colour was sampled uniformly between -5% and +5% around the midpoint, independently for each RGB channel; object colours were sampled uniformly at random in RGB space, rejecting colours which fell within a ball around 10 held-out intensities (radius 10% of range); the latter were only used for simulated transfer experiments, i.e. in D_T in the sim2sim experiments. Additionally, Gaussian noise with standard deviation 0.01 was added to the observations s^o_T in the sim2sim task.For the real Jaco arm and its MuJoCo simulation counterpart, each of the nine joints could independently take 11 different actions (a linear discretisation of the continuous velocity action space).In simulation Gaussian noise with standard deviation 0.1 was added to each discrete velocity output; delays in the real setup between observations and action execution were simulated by randomly mixing velocity outputs from two previous steps instead of emitting the last output directly. Speed ranges were between -50% and 50% of the Jaco arm's top speed on joints 1 through 6 starting at the base, while the fingers could use a full range. For safety reasons the speed ranges have been reduced by a factor of 0.3 while evaluating agents on the Jaco arm, without significant performance degradation. §.§ Vision model details§.§.§ Denoising Autoencoder for A denoising autoencoder (DAE) was used as a model to provide the feature space for the reconstruction loss to be computed over (for motivation, see Sec. <ref>). The DAE was trained with occlusion-style masking noise in the vein of <cit.>, with the aim for the DAE to learn a semantic representation of the input frames. Concretely, two values were independently sampled from U[0, W] and two from U[0, H] where W and H were the width and height of the input frames. These four values determined the corners of the rectangular mask applied; all pixels that fell within the mask were set to zero.The DAE architecture consisted of four convolutional layers, each with kernel size 4 and stride 2 in both the height and width dimensions. The number of filters learnt for each layer was {32, 32, 64, 64} respectively. The bottleneck layer consisted of a fully connected layer of size 100 neurons. This was followed by four deconvolutional layers, again with kernel sizes 4, strides 2, and {64, 64, 32, 32} filters. The padding algorithm used was `SAME' in TensorFlow <cit.>. ReLU non-linearities were used throughout.The model was trained with loss given by the L2 distance of the outputs from the original, un-noised inputs. The optimiser used was Adam <cit.> with a learning rate of 1e-3.§.§.§ with Perceptual Similarity LossAfter training a DAE, as detailed in the previous section[In principle, the could also have been trained end-to-end in one pass, but we did not experiment with this.], a was trained with perceptual similarity loss given by Eq. <ref>, repeated here:ℒ(θ, ϕ; x̌, ž, β)= 𝔼_q_ϕ(ž|x̌)J(x̂̌̂) - J(x̌)_2^2 - β D_KL(q_ϕ(ž|x̌) || p(ž))Specifically, the input was passed through the and a sampled[It is more typical to use the mean of the reconstruction distribution, but this does not induce any pressure on the Gaussians parametrising the decoder to reduce their variances. Hence full samples were used instead.] reconstruction was passed through the pre-trained DAE up to a designated layer. The L2 distance of this representation from the representation of the original input passed through the same layers of the DAE was then computed, and this formed the training loss for the part of the [The representations were taken after passing through the layer but before passing through the following non-linearity. We also briefly experimented with taking the L2 loss post-activation but did not find a significant difference.]. The DAE weights remained frozen throughout.The architecture consisted of an encoder of four convolutional layers, each with kernel size 4, and stride 2 in the height and width dimensions. The number of filters learnt for each layer was {32, 32, 64, 64} respectively. This was followed by a fully connected layer of size 256 neurons. The latent layer comprised 64 neurons parametrising 32 (marginally) independent Gaussian distributions. The decoder architecture was simply the reverse of the encoder, utilising deconvolutional layers. The decoder used was Gaussian, so that the number of output channels was 2C, where C was the number of channels that the input frames had. The padding algorithm used was `SAME' in TensorFlow. ReLU non-linearities were used throughout.The model was trained with the loss given by Eq. <ref>. Specifically, the disentangled model used for DARLA was trained with a β hyperparameter value of 1 and the layer of the DAE used to compute the perceptual similarity loss was the last deconvolutional layer. The entangled model used for DARLA_ENT was trained with a β hyperparameter value of 0.1 with the last deconvolutional layer of the DAE was used to compute the perceptual similarity loss.The optimiser used was Adam with a learning rate of 1e-4.§.§.§For the MuJoCo/Jaco tasks, a standard was used rather than the used for DeepMind Lab. The architecture of the VAE encoder, decoder and the latent size were exactly as described in the previous section <ref>. β for the the disentangled in DARLA was 175. β for the entangled model DARLA_ENT was 1, corresponding to the standard VAE of <cit.>.The optimizer used was Adam with a learning rate of 1e-4.§.§.§ Denoising Autoencoder for baselineFor the baseline model DARLA_DAE, we trained a denoising autoencoder with occlusion-style masking noise as described in Appendix Section <ref>. The architecture used matched that exactly of the described in Appendix Section <ref> - however, all stochastic nodes were replaced with deterministic neurons.The optimizer used was Adam with a learning rate of 1e-4. §.§ Reinforcement Learning Algorithm Details§.§.§ DeepMind LabThe action space in the DeepMind Lab task consisted of 8 discrete actions.DQN: in DQN, the convolutional (or `vision') part of the Q-net was replaced with the encoder of the from stage 1 and frozen. DQN takes four consecutive frames as input in order to capture some aspect of environment dynamics in the agent's state. In order to match this in our setup with a pre-trained vision stack ℱ_U, we passed each observation frame s^o_{1..4} through the pre-trained model s^z_{1..4} = ℱ_U(s^o_{1..4})and then concatenated the outputs together to form the k-dimensional (where k = 4|s^z|) input to the policy network. In this case the size of s^z was 64 for DARLA as well as DARLA_ENT, DARLA_DAE and DARLA_FT.On top of the frozen convolutional stack, two `policy' layers of 512 neurons each were used, with a final linear layer of 8 neurons corresponding to the size of the action space in the DeepMind Lab task. ReLU non-linearities were used throughout. All other hyperparameters were as reported in <cit.>.A3C: in A3C, as with DQN, the convolutional part of the network that is shared between the policy net and the value net was replaced with the encoder of the in DeepMind Lab tasks. All other hyperparameters were as reported in <cit.>.Episodic Control: for the Episodic Controller-based DARLA we used mostly the same hyperparameters as in the original paper by <cit.>. We explored the following hyperparameter settings: number of nearest neighbours ∈{10, 50}, return horizon ∈{100, 400, 800, 1800, 500000}, kernel type ∈ {inverse, gaussian}, kernel width ∈{1e-6, 1e-5, 1e-4, 1e-3, 1e-2, 1e-1, 0.5, 0.99} and we tried training EC with and without Peng's Q(λ) <cit.>. In practice we found that none of the explored hyperparameter choices significantly influenced the results of our experiments. The final hyperparameters used for all experiments reported in the paper were the following: number of nearest neighbours: 10, return horizon: 400, kernel type: inverse, kernel width: 1e-6 and no Peng's Q(λ) <cit.>.UNREAL: We used a vanilla version of UNREAL, with parameters as reported in <cit.>.§.§.§ MuJoCo/Jaco Arm Experiments For the real Jaco arm and its MuJoCo simulation, each of the nine joints could independently take 11 different actions (a linear discretisation of the continuous velocity action space). Therefore the action space size was 99.DARLA for MuJoCo/Jaco was based on feedforward A3C <cit.>. We closely followed the simulation training setup of <cit.> for feed-forward networks using raw visual-input only. In place of the usual conv-stack, however, we used the encoder of the as described in Appendix <ref>. This was followed by a linear layer with 512 units, a ReLU non-linearity and a collection of 9 linear and softmax layers for the 9 independent policy outputs, as well as a single value output layer that outputted the value function. §.§ Disentanglement Evaluation§.§.§ Visual Heuristic Details In order to choose the optimal value of β for the -DAE models and evaluate the fitness of the representations s^z_U learnt in stage 1 of our pipeline (in terms of disentanglement achieved), we used the visual inspection heuristic described in <cit.>. The heuristic involved clustering trainedbased models based on the number of informative latents (estimated as the number of latents z_i with average inferred standard deviation below 0.75). For each cluster we examined the degree of learnt disentanglement by running inference on a number of seed images, then traversing each latent unit z_{ i } one at a time over three standard deviations away from its average inferred mean while keeping all other latents z_{∖ i } fixed to their inferred values. This allowed us to visually examine whether each individual latent unit z_i learnt to control a single interpretable factor of variation in the data. A similar heuristic has been the de rigueur method for exhibiting disentanglement in the disentanglement literature <cit.>. §.§.§ Transfer Metric Details In the case of DeepMind Lab, we were able to use the ground truth labels corresponding to the two factors of variation of the object type and the background to design a proxy to the disentanglement metric proposed in <cit.>. The procedure used consisted of the following steps: 1) Train the model under consideration on observations s^o_U to learn ℱ_𝒰, as described in stage 1 of the DARLA pipeline.2) Learn a linear model ℒ: S^z_V → M × N from the representations s^z_V = ℱ_V (s^o_V), where M ∈{0, 1} corresponds to the set of possible rooms and N ∈{0, 1, 2, 3} corresponds to the set of possible objects[For the purposes of this metric, we utilised rooms with only single objects, which we denote by the subscript V e.g. the observation set S^o_V.]. Therefore we are learning a low-VC dimension classifier to predict the room and the object class from the latent representation of the model. Crucially, the linear model ℒ is trained on only a subset of the Cartesian product M × N e.g. on {{0, 0}, {0, 3}, {1, 1}, {1, 2}}. In practice, we utilised a softmax classifier each for M and N and trained this using backpropagation with a cross-entropy loss, keeping the unsupervised model (and therefore ℱ_𝒰) fixed.3) The trained linear model ℒ's accuracy is evaluated on the held out subset of the Cartesian product M × N.Although the above procedure only measures disentangling up to linearity, and only does so for the latents of object type and room background, we nevertheless found that the metric was highly correlated with disentanglement as determined via visual inspection (see Fig. <ref>).§.§ Background on RL Algorithms In this Appendix, we provide background on the different RL algorithms that the DARLA framework was tested on in this paper.§.§.§ DQN (DQN) <cit.> is a variant of the Q-learning algorithm <cit.> that utilises deep learning. It uses a neural network to parametrise an approximation for the action-value function Q(s, a; θ) using parameters θ. These parameters are updated by minimising the mean-squared error of a 1-step lookahead loss ℒ_Q = 𝔼 [ (r_t + γ max_a' Q(s', a'; θ^-) - Q(s,a;θ))^2], where θ^- are parameters corresponding to a frozen network and optimisation is performed with respect to θ, with θ^- being synced to θ at regular intervals.§.§.§ A3C Asynchronous Advantage Actor-Critic (A3C) <cit.> is an asynchronous implementation of the advantage actor-critic paradigm <cit.>, where separate threads run in parallel and perform updates to shared parameters. The different threads each hold their own instance of the environment and have different exploration policies, thereby decorrelating parameter updates without the need for experience replay.A3C uses neural networks to approximate both policy π(a|s; θ) and value V_π(s; θ) functions using parameters θ using n-step look-ahead loss <cit.>. The algorithm is trained using an advantage actor-critic loss function with an entropy regularisation penalty: ℒ_A3C≈ℒ_VR + ℒ_π - 𝔼_s∼π [ α H(π(a|s; θ))], where H is entropy. The parameter updates are performed after every t_max actions or when a terminal state is reached. ℒ_VR= 𝔼_s∼π [ (R_t:t+n + γ ^n V(s_t+n+1; θ) - V(s_t;θ))^2] and ℒ_π= 𝔼_s∼π [ log π(a|s; θ)(Q^π(s, a; θ) - V^π(s; θ))]. Unlike DQN, A3C uses an LSTM core to encode its history and therefore has a longer term memory permitting it to perform better in partially observed environments. In the version of A3C used in this paper for the DeepMind Lab task, the policy net additionally takes the last action a_t-1 and last reward r_t-1 as inputs along with the observation s^o_t, as introduced in <cit.>.§.§.§ UNREAL The UNREAL agent <cit.> takes as a base an LSTM A3C agent <cit.> and augments it with a number of unsupervised auxiliary tasks that make use of the rich perceptual data available to the agent besides the (sometimes very sparse) extrinsic reward signals. This auxiliary learning tends to improve the representation learnt by the agent. While training the base agent, its observations, rewards, and actions are stored in a replay buffer, which is used by the auxiliary learning tasks. The tasks include: 1) pixel control – the agent learns how to control the environment by training auxiliary policies to maximally change pixel intensities in different parts of the input; 2) reward prediction - given a replay buffer of observations within a short time period of an extrinsic reward, the agent has to predict the reward obtained during the next unobserved timestep using a sequence of three preceding steps; 3) value function replay - extra training of the value function to promote faster value iteration.§.§.§ Episodic Control In its simplest form EC is a lookup table of states and actions denoted as Q^EC(s, a). In each state EC picks the action with the highest Q_EC value. At the end of each episode Q^EC(s, a) is set to R_t if (s_t, a_t) ∉ Q^EC, where R_t is the discounted return. Otherwise Q^EC(s, a) = max{ Q^EC(s,a), R_t }. In order to generalise its policy to novel states that are not in Q^EC, EC uses a non-parametric nearest neighbours search Q^EC(s, a) = 1/k∑_i=1^k Q^EC(s^i, a), where s^i, i=1,...,k are k states with the smallest distance to the novel state s. Like DQN, EC takes a concatenation of four frames as input.The EC algorithm is proposed as a model of fast hippocampal instance-based learning in the brain <cit.>, while the deep RL algorithms described above are more analogous to slow cortical learning that relies on generalised statistical summaries of the input distribution <cit.>. §.§ Source Task Performance ResultsThe focus of this paper is primarily on zero-shot domain adaptation performance. However, it is also interesting to analyse the effect of the DARLA approach on source domain policy performance. In order to compare the models' behaviour on the source task, we examined the training curves (see Figures <ref>-<ref>) and noted in particular their: * Asymptotic task performance, i.e. the rewards per episode at the point where π_S has converged for the agent under consideration.* Data efficiency, i.e. how quickly the training curve was able to achieve convergence.We note the following consistent trends across the results: * Using DARLA provided an initial boost in learning performance, which depended on the degree of disentanglement of the representation. This was particularly observable in A3C, see Fig. <ref>.* Baseline algorithms where ℱ could be fine-tuned to the source task were able to achieve higher asymptotic performance. This was particularly notable on DQN and A3C (see Figs. <ref> and <ref>) in DeepMind Lab. However, in both those cases, DARLA was able to learn very reasonable policies on the source task which were on the order of 20% lower than the fine-tuned models – arguably a worthwhile sacrifice for a subsequent median 270% improvement in target domain performance noted in the main text.* Allowing DARLA to fine-tune its vision module (DARLA_FT) boosted its source task learning speed, and allowed the agent to asymptote at the same level as the baseline algorithms. As discussed in the main text, this comes at the cost of significantly reduced domain transfer performance on A3C. For DQN, however, finetuning appears to offer the best of both worlds.* Perhaps most relevantly for this paper, even if solely examining source task performance, DARLA outperforms both DARLA_ENT and DARLA_DAE on both asymptotic performance and data efficiency – suggesting that disentangled representations have wider applicability in RL beyond the zero-shot domain adaptation that is the focus of this paper. | http://arxiv.org/abs/1707.08475v2 | {
"authors": [
"Irina Higgins",
"Arka Pal",
"Andrei A. Rusu",
"Loic Matthey",
"Christopher P Burgess",
"Alexander Pritzel",
"Matthew Botvinick",
"Charles Blundell",
"Alexander Lerchner"
],
"categories": [
"stat.ML",
"cs.AI",
"cs.LG"
],
"primary_category": "stat.ML",
"published": "20170726145051",
"title": "DARLA: Improving Zero-Shot Transfer in Reinforcement Learning"
} |
=17000 thmintroTheorem corintro[thmintro]Corollary conjintro[thmintro]ConjecturetheoremTheorem[section]corollary[theorem]Corollary lemma[theorem]Lemma proposition[theorem]Proposition conjdefn[theorem]Conjectural definition stepStep conj[theorem]Conjectureremark definition parag[theorem] remark[theorem]Remark definition[theorem]Definition example[theorem]Examplenotation[theorem]Notation assumption[theorem]Assumptionstat[theorem]Statementequationsection ℕℤℚℝℂℍ̋𝔻𝕀𝔼ℙ𝔽𝐤𝐤̨ØΩ𝒰øωł'cf. i.e. 𝒜ℬ𝒞𝒟ℰℱℳ𝒩ℋℐ𝒥 ℒ𝒪𝒫𝒬ℛ𝒮𝒯𝒰𝒱𝒲𝒴𝒵𝒳𝐂δ∂𝐊ℑ𝔤𝔯𝗊Δ ∂𝕍𝔸HomGLGLEndAdSLSLMatPSL𝒢StabGalGal^δSpanAutR_uKerDiffLietr.deg𝕀idQuotAlgordSoldiagStab^δ∂-∂-trdegG_aG_mVectδImTriviality of cohomologies ofdifferential algebraic groups]Triviality of differential Galois cohomologies of linear differential algebraic groupsDepartment of Mathematics, University of Vienna, [email protected]://www.mat.univie.ac.at/ minchenko/CUNY Queens College, Department of Mathematics, 65-30 Kissena Blvd, Queens, NY 11367 and CUNY Graduate Center, Ph.D. programs in Mathematics and Computer Science, 365 Fifth Avenue, New York, NY [email protected]://qc.edu/ aovchinnikov/We show that the triviality of the differential Galois cohomologies over a partial differential field K of a linear differential algebraic group is equivalent to K being algebraically, Picard–Vessiot, and linearly differentially closed. This former is also known to be equivalent to the uniqueness up to an isomorphism of a Picard–Vessiot extension of a linear differential equation with parameters. [ Alexey Ovchinnikov Received: date / Accepted: date ===================================§ INTRODUCTION Galois theory of linear differential equations with parameters <cit.> (also known as the parameterized Picard–Vessiot theory) provides theoretical and algorithmic tools to study differential algebraic dependencies of solutions of a linear ODE with one or several parameters. A parameterized Picard–Vessiot extension is a differential field generated by a complete set of solutions of the ODE and also satisfying additional technical conditions. How to decide whether such an extension is unique is an open problem in this theory. We study this question in the present paper as follows.Let F be a differential field of characteristic zero with commuting derivations{∂_x,∂_1,…,∂_m}. One can show using <cit.> that the uniqueness up to an isomorphism of a Picard–Vessiot extension of any parameterized linear differential equationwith coefficients in F is equivalent tothe triviality of differential Galois (also known as constrained) cohomologies <cit.> over K, the ∂_x-constants of F, of all linear differential algebraic groups <cit.>. We show in our main result, Theorem <ref>, that the latter triviality holds if and only if K is algebraically, Picard–Vessiot, and linearly differentially closed (the terminology is explained in Section <ref>).Such a question for m=1 was settled in<cit.>, in which case the “linearly differentially closed” condition does not play a role. This was extended to m>1 in <cit.> in terms of generalized strongly normal extensions. Our characterization is different and our arguments can be viewed as more transparent. § DEFINITIONS AND NOTATIONAdifferential ring is a ring R with a finite set Δ={δ_1,…,δ_m} of commuting derivations on R. A differential ideal of (R,Δ) is an ideal of R stable under any derivation in Δ.For any derivation δ :R→ R, we denote R^δ= {r ∈ R | δ(r) = 0},which is a δ-subringof R and is called thering of δ-constants of R. If R is a field and (R,Δ) is a differential ring, then (R,Δ) is called adifferential field.The notion ofdifferential algebra over (R,Δ) is defined analogously. An ideal I of R is called a differential ideal of (R,Δ) if, for all δ∈Δ and r ∈ I, δ(r)∈ I. The ring of Δ-differential polynomials K{x_1,…,x_n} in the differential indeterminatesx_1,…,x_n and with coefficients in a Δ-field (K,Δ) is the ring of polynomials in the indeterminates formally denoted{δ_1^i_1·…·δ_m^i_m x_i |i_1,…,i_m⩾ 0,1⩽ i⩽ n}with coefficients in K. We endow this ring with a structure of K-Δ-algebra by setting δ_k (δ_1^i_1·…·δ_m^i_m x_i )= δ_1^i_1·…·δ_k^i_k+1·…·δ_m^i_m x_i.A differential field (K,Δ) is said to bedifferentially closed if, for every n⩾ 1 and every finite set of Δ-polynomials F ⊂ K{x_1,…,x_n }, if the system of differential equations F=0 has a solutionwith entries in some Δ-field extension L, then it has a solution with entries in K.Letbe a differentially closed Δ-fieldof characteristic 0 andK⊂ be its differential subfield.AKolchin-closedset W ⊂^n over K is the set of common zeroes of a system of differential polynomials with coefficients in K, that is, there existsS ⊂ K{y_1,…,y_n} such thatW = { a ∈^n |f(a) = 0f ∈ S }. More generally, for a differential algebra R over K,W(R) = { a ∈ R^n |f(a) = 0f ∈ S }.If W ⊂^n is aKolchin-closed set definedover K,the differential ideal (W) = { f∈ K{y_1,… , y_n} |f(w) = 0w∈ W()}is called thedefining ideal of W over K. Conversely, for asubsetS of K{y_1,…,y_n}, the following subsetis Kolchin-closed in^n and defined over K:V(S)={ a ∈^n |f(a)= 0f ∈ S }.Let W ⊂^n be a Kolchin-closed set defined over K. Thecoordinate ringK{W} ofW over K is the differential algebra over K K{W} = K{y_1,…,y_n}/(W).If K{W} is an integral domain, then W is said to beirreducible. Thisis equivalent to (W) being a primedifferential ideal.Let W ⊂^n be a Kolchin-closed set defined over K.Let(W) = 𝔭_1∩…∩𝔭_q bea minimal differential prime decomposition of(W), that is, the 𝔭_i ⊂ K{y_1,…,y_n} are prime differential ideals containing (W) and minimal with this property. This decomposition is uniqueup to permutation (see <cit.>).The irreducible Kolchin-closed setsW_i=V(𝔭_i) are defined over K andcalled theirreducible components of W. Wehave W = W_1∪…∪ W_q. Let W_1 ⊂^n_1 and W_2 ⊂^n_2 be two Kolchin-closed sets defined over K.A differential polynomial map (morphism) defined over K is a map φ : W_1→ W_2, a ↦(f_1(a),…,f_n_2(a)), a ∈ W_1 ,where f_i ∈ K{x_1,…,x_n_1} for all i=1,…,n_2.If W_1 ⊂ W_2, the inclusion map of W_1 in W_2 is a differential polynomial map. In this case, we say that W_1 isaKolchin-closed subset of W_2. Let W be an irreducible Kolchin-closed set and P ⊂ K{x_1,…,x_n} be its defining differential ideal, which is prime. It is shown in <cit.> that there exists a non-negative integer H such that, for all h⩾ H,(P∩ K[δ_1^i_1·…·δ_m^i_mx_i |1⩽ i⩽ n, i_j⩾ 0, 1⩽ j⩽ m, i_1+… +i_m ⩽ h])coincides with a polynomial in h. The degree of this polynomial is denoted by τ(W) and called thedifferential type of W (if W is a single point and so the above polynomial is 0, we set τ(W)=-1).Let _n ⊂^n be the group of n × n invertible matrices with entries in . One can see _n as a Kolchin-closed subset of ^n^2× defined over K, defined by the equation x·(X)-1 in K{^n^2×}=K{X,x}, where X is an n × n-matrix ofdifferential indeterminates over K and x adifferential indeterminate over K. One can thus identify the differential coordinate ring of _n over K withF{X,1/(X)}, where X=(x_i,j)_1 ⩽ i,j ⩽ n is a matrix of differential indeterminates over K. We also denotethe special linear group that consists of the matricesof determinant 1 by _n ⊂_n.Alinear differential algebraic group (LDAG) G ⊂^n^2 definedover K is a subgroup of _n that is a Kolchin-closed set defined over K. If G ⊂ H ⊂_n are Kolchin-closed subgroups of_n, we say that G is a Kolchin-closed subgroup of H.Let G be an LDAGdefined over F. The irreducible component of G containing the identity element eis called theidentity component of G and denoted by G^∘. The LDAG G^∘ is a δ-subgroup of G defined over F. The LDAG G is said to beconnected if G = G^∘, which is equivalent to G being an irreducible Kolchin-closed set <cit.>.Let G be an LDAG over K. Thestrong identity componentG_0 of G is defined to be the smallestdifferential algebraic subgroup H of G defined oversuch that τ(G/H) < τ(G). By <cit.>, G_0 is a normal subgroup of G defined over K. An infiniteLDAG G defined over K isalmost simple if, for any normal properdifferential algebraic subgroup H of G defined over K, we have τ(H) < τ(G).For a systemof Δ-differential equations over K,K is said to be -closed, or closed w.r.t. , if the consistency of(i.e., the existence of a solution in ) implies the existence of a solution in K. K is said to be PV closed if, for all r, 1⩽ r⩽ m, for allsets {D_1,…,D_r}⊂ KΔ of commuting derivations, for all n⩾ 1, and for all A_1,…,A_r∈_n× n(K), K is closed w.r.t. {D_i(Z)=A_i· Z, z· Z=1}_i=1^r,where Z and z are unknown matrices of sizes n× n and 1× 1, respectively (see <cit.> for a coordinate-free definition). K is said to be Δ-linearly closedif it is closed w.r.t. any system of linear (not necessarily homogeneous) Δ-differential equations in one unknown over K.Aprincipal homogeneous space (PHS) over an LDAG G over K is a Kolchin-closedX defined over K together with a differential algebraic isomorphism X× G → X× X over K.The set of equivalence classes of PHS of G over K is denoted by H^1_Δ(K,G). We write H^1_Δ(K,G)={1} if all principal homogeneous spaces of G are isomorphic over K. For example, H^1_Δ(,G)={1}.§ MAIN RESULTThe following are equivalent:(1)K is algebraically closed, PV closed, and Δ-linearly closed;(2) for any linear differential algebraic group G, H^1_Δ(K,G)={1}.Let us show the implication ⟸. If K is not algebraically closed, then there exists a non-trivial Galois extension E/K given by an irreducible polynomial f. The set X of its roots is a K-torsor for G=Gal(E/K). It is non-trivial since there are no K-points of X, that is, homomorphisms E→ K over K. Hence,H^1(K,G)≅ H^1_Δ(K,G)≠{1} (with the isomorphism following from <cit.>). Suppose thatK is not PV closed. Hence, there exists a set D={D_1,…,D_r}⊂ KΔ of commuting derivations and a system (<ref>) with no solutions in _n(K).We claim that H^1_Δ(K,_n^D)≠{1}. Indeed, let J:={(B_1,…,B_r)∈𝔤𝔩_n()^r : D_iB_j-D_jB_i=[B_j,B_i]}andℓ: _n→ J, x↦(x^-1D_1(x),…,x^-1D_r(x)). We have (ℓ)=_n^D.Moreover, by <cit.>, ℓ is surjective. Hence, the sequence{1}@>>> _n^D@>>> _n@>ℓ>> J@>>> {0}is exact. By assumption, ℓ is not surjective on K-points. Let (A_1,…,A_n)∉ℓ(_n(K)). By <cit.>, ℓ^-1(A_1,…,A_n) is a non-trivial torsor for _n^D(K). IfK is not Δ-linearly closed, then there exist a positive integer r,a Δ-subgroup B⊂^r defined over K, and a surjective Δ-linear map Λ: → B over K that is not surjective on K-points. By <cit.>, H^1_Δ(K,Λ)≠{1}.Let us prove the implication ⟹. By <cit.>, given a short exact sequence{1}@>>> G'@>>> G@>>> G”@>>>{1}of LDAGs over K in which G'⊂ G is normal <cit.>, H^1_Δ(K,G')=H^1_Δ(K,G”)={1} H^1_Δ(K,G)={1}. This is called the inductive principle <cit.>. As in <cit.>, the problem reduces to the following three cases:* G is finite;* G⊂;* G=()— the group of constants of ;* G=H^P, where H is alinear algebraic group (LAG) over , P⊂ KΔ is a Lie subspace, and H^P is the functor of taking constant points with respect to P: H^P(L):=H(L^P) for a Δ-ring extension L of K. Note that case (<ref>) is included into this case, but we have separated case (<ref>) for the clarity of the presentation. Let us explain the the reduction and then show that H^1_Δ(K,G)=1 for any G of types (<ref>)–(<ref>). The exact sequence{1}@>>> G^∘@>>> G@>>> G/G^∘@>>>{1}reduces the problem to case (<ref>) andto showing that, for a connected G, H^1_Δ(K,G)=1.To show the latter equality, let us use induction on the differential type τ(G), the case τ(G)=-1 (that is, G={1}, because G is assumed to be connected) being trivial. Let G_0⊂ G be the strong identity component. Suppose τ(G)⩾ 0. One has the following exact sequence:{1}@>>> G_0@>>> G@>>> G/G_0@>>>{1}, and τ(G/G_0)<τ(G). By the induction, this reduces the problem to the case G=G_0. Moreover, by <cit.>, it suffices to assume that G is almost simple. If G is almost simple non-commutative, it is simple by <cit.>. By <cit.>, it corresponds to case (<ref>) because K is PV closed. If G is commutative, G is also commutative, hence there are n_1,n_2⩾ 0 such that G is isomorphic over K (recall that K is algebraically closed) to a direct product of n_1 copies ofand n_2 copies of(we will not use the almost simplicity in the commutative case). It follows by induction on n_1+n_2that H^1_Δ(K,G)=1 if H^1_Δ(K,G)=1 for any connected Kolchin closed subgroup G ofor . Indeed, if n_1⩾ 1, one has a natural projection G⊂G→ whose kernel is contained in the direct product of n_1-1 copies ofand n_2 copies of . Similarly, if n_2⩾ 1, one considers a projection G⊂G→. The case G⊂ reduces to G=^KΔ=() and case (<ref>) by considering the logarithmic derivatives (defined on )ℓ_i:G→, x↦ x^-1∂_ix, for i=1,…, m subsequently, as all infinite differential algebraic subgroups ofcontain ^KΔ<cit.>. In case (<ref>), H^1_Δ(K,G)={1} by <cit.>, because K is algebraically closed. In case (<ref>), we have the following exact sequence:{0}@>>> G@>ι>> @>π>>@>>>{0},and we have H^1_Δ(K,G) = {1} by <cit.> since K is linearly Δ-closed. Case (<ref>) is included into case (<ref>), as noted before.It remains to consider case (<ref>). Choose a basis {D_1,…,D_r} of commuting derivations of P and let J:={(B_1,…,B_r)∈( H)^r : D_iB_j-D_jB_i=[B_j,B_i]}and, since H is defined over , by <cit.>, we have:ℓ: H→ J, x↦(x^-1D_1(x),…,x^-1D_r(x)). We have (ℓ)=H^P=G.Let B_1,…,B_r ∈ J. Since <cit.> can be rewritten in a straightforward way for several commuting derivations, the surjectivity ofℓ is implied by p ∉ [D_1(x)-xB_1,…,D_r(x)-xB_r] for any order 0 non-zero differential polynomial p in x (which includes p= x), as in the proof <cit.>. Since (<ref>) is shown in the proof of <cit.> given the conditions in (<ref>), we conclude that ℓ is surjective. Hence, the sequence{1}@>>> G@>>> H@>ℓ>> J@>>> {0}is exact. Since K is algebraically closed, H^1_Δ(K,H)={1}. By <cit.>, this implies that all torsors for G are isomorphic to ℓ^-1(a), a∈ J(K). Moreover, all of them are isomorphic (that is, H^1_Δ(K,G)={1}) if ℓ is surjective on K-points. Due to the PV closedness, H^1_Δ(K,_n^P)={1}.Since H is defined over ℚ, taking P-constants, which can be viewed as applying the functor ( · ,^P), is exact, because ^P is algebraically closed and so the polynomial map π is surjective:{1}@>>>G=H^P@>ι>>_n^P@>π>>(_n/H)^P@>>> {1},Since K is algebraically closed, the map π is surjective on K-points. Therefore, by the corresponding exact sequence of cohomologies <cit.>, H^1_Δ(K,G)=H^1_Δ(K,_n^P)={1}.§ ACKNOWLEDGMENTSThe authors are grateful to Anand Pillay and Michael F. Singer for the discussions and comments. This work has been partially supported by the NSF grants CCF-0952591 and DMS-1413859, and by the Austrian Science Foundation FWF, grant P28079. abbrvnat | http://arxiv.org/abs/1707.08620v1 | {
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[ General Latent Feature Modeling for Data Exploration Tasks equal* Isabel Valerato Melanie F. Pradiered Zoubin Ghahramanito,goo toUniversity of Cambridge, Cambridge, United Kingdom; gooUber AI Labs, San Francisco, California, USA edUniversidad Carlos III de Madrid, SpainIsabel [email protected] interpretability, transparency0.25in ]This paper introduces a general Bayesian nonparametric latent feature model suitable to perform automatic exploratory analysis of heterogeneous datasets, where the attributes describing each object can be either discrete, continuous or mixed variables. The proposed model presents several important properties. First, it accounts for heterogeneous data while can be inferredin linear time with respect to the number of objects and attributes.Second, its Bayesian nonparametric nature allows us to automatically infer the model complexity from the data, i.e., the number of features necessary to capture the latent structure in the data. Third, the latent features in the model are binary-valued variables, easing the interpretability of the obtained latent features in data exploration tasks.§ INTRODUCTIONLatent feature models allow us to compact in a few features the immense redundant information present in the observed data, by capturing the statistical dependencies among the different objects and attributes. As a consequence,they appear as suitable tools to perform data exploratory analysis, i.e, they may help usto better understand the data <cit.>. There is an extensive literature in latent feature modeling of homogeneous data, where all the attributes describing each object in the database present the same (continuous or discrete) nature. In particular, these works assume that databases contain only either continuous data, usually modeled as Gaussian variables <cit.>, or discrete, that can be either modeled by discrete likelihoods <cit.> or simply treated as Gaussian variables <cit.>.However, there still exists a lack of works dealing with heterogeneous databases, which in fact are common in real applications.As motivating examples, Electronic Health Records from hospitals might containlab measurements (often positive real-valued or real-valued data), diagnoses (categorical data) and genomic information (ordinal, count data and categorical data); also, a survey often contains diverse information about the participants such as age (count data), gender (categorical data), salary (positive real data), etc. Despite this diversity of data types, the standard approach when dealing with heterogeneous datasets is to treat all the attributes, either continuous or discrete, asGaussian variables.This paper presents a general latent feature model (GLFM) suitable for heterogeneous datasets, where the attributes describing each object can be either discrete, continuous or mixed variables. Specifically, we account for real-valued and positive real-valued as examples of continuous variables, and categorical, ordinal and count data as examples of discrete variables. The proposed model extends the essential building block of Bayesian nonparametric latent feature models, the Indian Buffet Process (IBP) by <cit.>, to account for heterogeneous data while maintaining the model complexity of conjugate models.Among all the available latent feature models in the literature, we opt for the IBP due to two main reasons.First, the nonparametric nature of the IBP allows us toautomatically infer the appropriate model complexity,i.e., the number of necessary features, from the data.Second,the IBP considers binary-valued latent features which has been shown to provide more interpretable results in data exploration than standard real-valued latent feature models <cit.>. The standard IBP assumes real-valued observations combined with conjugate likelihood models, allowing for fast inference algorithms <cit.>. However, we here aim at dealing with heterogeneous databases, such that conjugacy might not be straightforwardly available.In order to propose a general observation model for the IBP that accounts for heterogeneous data while keeping the properties of conjugate models, we exploit two key ideas. First, we introduce an auxiliary real-valued variable (also called pseudo-observation), such that, conditioned on it, the model behaves as the standard linear-Gaussian IBP in <cit.>.Second,we assume that there exists afunction that transforms the pseudo-observation into the actual observation,mapping the real line into the (discrete or continuous) observation spaceof each attribute in the data. These two key ideas allow us to derive an efficient inference algorithm based on collapsed Gibbs sampling, which presents linear complexity with the number of objects and attributes in the data. Our experiments provide examples of how to use the proposed model for data exploration in real-world datasets. Additionally, a software library implementing the GLFM, as well as the necessary scripts to perform automatic data exploration, is publicly available at https://github.com/ivaleraM/GLFMhttps://github.com/ivaleraM/GLFM .§ GENERAL LATENT FEATURE MODELWe introduce the GLFM, which is a general Bayesian nonparametric latent feature model suitable for data exploration of heterogeneous datasets, where the attributes describing each object can be either discrete, continuous or mixed variables. Specifically, the GLFM accounts for the following data types:* Continuous variables: * Real-valued, i.e., x_n^d∈ * Positive real-valued, i.e., x_n^d∈_+.* Discrete variables: * Categorical data, i.e.,x_n^d takes a value in a finite unordered set, e.g., x_n^d∈{`blue', `red',`black'}.* Ordinal data, i.e.,x_n^d takes values in a finite ordered set, e.g., x_n^d∈{`never',`sometimes', `often', `usually', `always'}.* Count data, i.e., x_n^d∈{0, …, ∞}. The GLFM builds on the IBP <cit.>, and therefore, it assumes that each observation x_n^d can be explained bya potentially infinite-length binary vector 𝐳_n whose elements indicate whether a latent feature is active or not for the n-th object; and a (real-valued) weighting vector 𝐁^d, whose elements weight the influence of each latent feature in the d-th attribute[For convenience, we here capitalize the vector 𝐁^d.].Since the product of the latent feature vector and the weighting vector leads to a real-valued variable, it is necessary to map this variable to the desirable output (continuous or discrete) space, for example, the positive real line or the finite ordered set {low, medium, high}. Thus, the GLFM assumes the existence of intermediate real-valued auxiliary variables y_n^d∼𝒩(𝐳^n𝐁^d, σ^2_d), called pseudo-observation, and a transformation function f_d(·) that maps this variable into the actual observation x_n^d, i.e., x_n^d = f_d(y_n^d+u) where u∼𝒩(0, σ^2_u) is an auxiliary noise with zero mean and small variance σ^2_u. Additionally, the GLFM accounts for a bias term similar to the one in <cit.>, which corresponds to an extra latent feature that is active for every object in the data and eases the interpretability of the latent features, as shown in next section. Figure <ref> illustrates the GLFM by showing the corresponding graphical model together with an example of the generative model for an ordinal attribute taking values in the ordered set {low, medium, high}.The inference of the GLFM is performed using collapsed Gibbs sampling, which presents linear complexity with respect to the number of objects N and the number of attributes D.Additional details on the model, as well as on the inference algorithm can be found in <cit.>.§ DATA EXPLORATION The main goal of this section is to provide showcase examples about how to include the specific domain knowledge into the proposed GLFMtofind and analyze the latent structure underlying data in different application domains, i.e., to perform data exploratory analysis.In particular, we here show examples of how to select the input data for the GLFM, as well as how to enter these data into the model, in order to obtain interpretable results that can be usedto get a better understanding of the data. §.§ Drug effect in a clinical trial for prostate cancerClinical trials are conducted to collect data regarding the safety and efficacy of a new drug before it can be sold in the consumer market, if ever. Concretely, the main goal of clinical trials is to prove the efficacy of a new treatment for a disease while ensuring its safety, i.e., check whether its adverse effectsremain low enough for any dosage level of the drug. As an example, the publicly available Prostate Cancer dataset[http://biostat.mc.vanderbilt.edu/wiki/Main/DataSetshttp://biostat.mc.vanderbilt.edu/wiki/Main/DataSets]collects data of a clinical trial that aimed at analyzing the effects ofthe drug diethylstilbestrol (DES) as a treatment against prostate cancer. More in detail, the dataset contains information about 502 patients with prostate cancer in stages[The stage of a cancer describes the size of a cancer and how far it has grown. Stage 3 means that the cancer is already quite large and may have started to spread into surrounding tissues or local lymph nodes. Stage 4 is more severe, and refers to a cancer that has already spread from where it started to another body organ. This is also called secondary or metastatic cancer. Find more details in http://www.cancerresearchuk.org/about-cancer/what-is-cancer/stages-of-cancerhttp://www.cancerresearchuk.org/about-cancer/what-is-cancer/stages-of-cancer] 3 and 4, who entered a clinical trial during 1967-1969 and were randomly allocated to different levels of treatment with DES.The prostate cancer dataset has been used by several studies <cit.> to analyze the survival times of the patients in the clinical trial and the causes behind their death. These studies have pointed out that a large dose of the treatment tends to reduce the risk of a cancer death at any time, but also might result in an increased risk of cardiovascular death.In this section, we apply the proposed GLFM to the Prostate Cancer dataset to show that the proposed model can be efficiently used to discoverthe statistical dependencies in the data, which in this example corresponds to the effect of the different levels of treatment with DES in the suffering of prostate cancer and cardiovascular diseases. Theprostate cancer dataset consists of 502 patients and 16 attributes, from which we make use of the five attributes listed in Table <ref>. The selection of these five attributes allows us not only to reduce the number of local minima in the posterior distribution of the proposed model due to the small sample size of the dataset, but also to focus on capturing the statistical dependencies between the target attributes, i.e, the relationship between the different levels of treatment with DES and the suffering of prostate cancer and cardiovascular diseases. Results After running our model, we obtain four latent features.Figure <ref> shows the effect of the inferred latent features, as well as the bias term, on each dimension/attribute of the data, where we can distinguish two groups of features.The first group accounts for patients in stage 3 and includes the bias term and the 2 first latent features. Within this group,the bias term – or equivalently pattern (0000) – and the first feature – or equivalently pattern (1000) – account for patients in stage 3 with a low average level of treatment with DES (refer to Figure <ref>). However, while the bias term models patients with low probability (∼15%) of prostate cancer death, the first feature accounts for patients with higher probability (∼40%) of prostate cancer death, which can be explained by a larger tumor size (refer to Figure <ref>). The second feature – or equivalently pattern (0100) –captures patients who exclusively received a high dosage (5 mg) of the drug(refer to Figure <ref>). These patients present a small tumor size and the lowest probability of prostatic cancer death, suggesting a positive effect of the drug as treatment for the cancer. However, they also present a significant increase in theprobabilityof dying from a vascular disease (∼50%), indicating a potential adverse-effect of the drug that increases the risk of suffering from cardio-vascular diseases. Such observation is in agreement with previous studies <cit.>. The second group of features corresponds to the activation patterns (0010) and (0001), and accounts for patients in stage 4 with, respectively, mild and severe conditions. In particular, the third featurecorresponds to patients with small tumor size, but intermediate values for the PAP biomarker, suggesting a certain spread degree of the tumor compared to the features in the first group, but not as severe as for patients with pattern (0001). Indeed, pattern (0001) models those patients in stage 4 with relatively high tumor size and the highest PAP values–it is thus not surprising that those patients present in turn the highest probability (above 50%) of prostatic death. §.§ Impact of Social Background on Mental DisordersSeveral studies have analyzed the impact of social background in the development of mental disorders <cit.>.Other studies have focused on finding and analyzing the co-occurring (comorbidity) pattern among the 20 most common psychiatric disorders <cit.>. These studies found that the 20 most common disorders can be divided into three groups:i) externalizing disorders, which include substance use disorders (alcohol abuse and dependence, drug abuse and dependence and nicotine dependence); ii)internalizing disorders, which include mood and anxiety disorders (major depressive disorder (MDD), bipolar disorder and dysthymia, panic disorder, social anxiety disorder (SAD), specific phobia and generalized anxiety disorder (GAD), and pathological gambling (PG));and iii) personality disorders (avoidant, dependent, obsessive-compulsive (OC), paranoid, schizoid, histrionic and antisocial personality disorders (PDs)). However, up to our knowledge, there is a lack of work on the impact of social background in the suffering of comorbid disorders.In this section, we aim at extending the analysis in <cit.> to account forthe influence of the social background of subjects (such as age, gender, etc.) in the probability of a subject suffering fromcomorbid disorders.To this end, in addition to the diagnoses of the above 20 psychiatric disorders, wealso make use of the information provided by the NESARC,which includes a set of questions on the social background of participants. Specifically,in addition to the diagnoses of the most common 20 psychiatric disorders described above, we includethe sex of the participants as input data to the proposed model. We model the gender information of the participants in the NESARC as a categorical variable with two categories: {`male', `female'}.The percentage of males in the NESARC is approximately 43%.Note also that the diagnoses of the 20 psychiatric disorders correspond to categorical variables with two possible categories, e.g., a patient suffering or not from a disorder. ResultsAfter running our inference algorithm with the diagnoses of the 20 disorders and the gender of subjects as input data, we obtain three latent features.Figure <ref> shows the probability of meeting each diagnostic criteria for the latent feature vectorslisted in the legend and in the database (baseline). Note that the obtained latent features are similar to the ones in <cit.>, i.e., feature 1 (pattern (100)) mainly models the seven personality disorders (PDs), feature 2 (pattern (010)) modelsalcohol and drug abuse disorders and the antisocial PD, while feature 3 (pattern (001)) modelsanxiety and mood disorders. Additionally,Figure <ref> shows the probability of being male and female for the latent feature vectorsdepicted in the legend and the empirical probability of being male and female in the database (baseline).In Figure <ref>, we observe that having no active features (pattern (000)), which captures people that do not suffer from any disorder, increases the probability of being male with respect to the baseline probability, and therefore, it indicates that females tend to suffer in a higher extent from psychiatric disorders. Additionally, we observe thatfeature 1 (pattern (100)) increases the probability of being male, while feature 3 (pattern (001))increases the probability of being female. Hence, from the analysis of Figure <ref>, we can conclude that, while women suffer more frequently from mood and anxiety disorders than men, PDs are more common in men. § CONCLUSIONS In this paper, we have introduced the first available general latent feature model and its code implementation, which will ease researchers from diverse fields to analyze a wide range of heterogeneous, incomplete and noisy datasets in an automatic manner.We have showed the flexibility and applicability of the proposed GLFM by performing data exploratory analysis of diverse real-world datasets. Further results including higher dimensional spaces can be found in <cit.>. langley00icml2017 | http://arxiv.org/abs/1707.08352v1 | {
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arXiv Preprint Samuel Li Spectroscopy of very hot plasma in non-flaring parts of a solar limb active region: spatial and temporal properties Helen E. Mason December 30, 2023 =================================================================================================================== Many of the recent approaches to polyphonic piano note onset transcription require training a machine learning model on a large piano database. However, such approaches are limited by dataset availability; additional training data is difficult to produce, and proposed systems often perform poorly on novel recording conditions. We propose a method to quickly synthesize arbitrary quantities of training data, avoiding the need for curating large datasets. Various aspects of piano note dynamics — including nonlinearity of note signatures with velocity, different articulations, temporal clustering of onsets, and nonlinear note partial interference — are modeled to match the characteristics of real pianos. Our method also avoids the disentanglement problem, a recently noted issue affecting machine-learning based approaches. We train a feed-forward neural network with two hidden layers on our generated training data and achieve both good transcription performance on the large MAPS piano dataset and excellent generalization qualities.music information retrieval (MIR), neural networks, data modeling § INTRODUCTIONPolyphonic music transcription involves extracting a musical score or equivalent representation from an audio recording. In particular, the problem of polyphonic piano onset transcription involves extracting the onset time and pitch of many potentially simultaneous piano notes. Deep neural networks have been successfully applied to this area, but current approaches require the use of large, painstakingly annotated datasets as training data <cit.> — more often than not, the extensive MAPS piano database <cit.>. However, curating additional training data can be both time-consuming and challenging <cit.>, and the original setup used to create these datasets cannot be accurately reproduced should additional samples be needed. In addition, many of these machine learning approaches are both trained and evaluated on samples drawn from the same database <cit.>, weakening claims about generalization behavior; networks trained on one dataset tend to overfit its specific timbre and perform relatively poorly on newly generated data <cit.>. It has even been recently noted that neural networks face a fundamental issue when applied to polyphonic note transcription — they suffer from the entanglement problem, memorizing chords or combinations of notes rather than learning to report the onset of each note individually <cit.>.We circumvent all of these problems by generating our training data procedurally. No annotated piano database is used as training data. Although we evaluate our approach on the MAPS piano database, the instruments and recording conditions used for testing are completely unknown to the network, providing a high degree of confidence in our model's generalization capabilities. Our approach is completely context-independent — that is, we require no prior information about the instrument or recording being transcribed, allowing a broader field of application. Furthermore, we solve the disentanglement issue presented in <cit.> by randomly generating arbitrary combinations of notes, forcing the network to learn to identify individual notes.§ PROPOSED MODEL §.§ Data Representation We use a constant-Q transform (CQT) spectrogram as the fundamental time-frequency representation of our audio signals <cit.>. We use bins ranging from the note G_1 ≈49Hz to C_8 ≈4186Hz, with a spacing of 12 bins per octave, for a total of 79 frequency bins. We use a Q-factor of 32.Our spectrogram frames are spaced 1024 audio samples apart; for the audio in the MAPS database, which has a sample rate of 44100Hz, this leads to a frame rate of about 43Hz.For our machine learning model, we use a simple feed-forward neural network. The network's input consists of an 8-frame wide “reading window” of the magnitude of the CQT spectrogram, normalized to have a maximum value of 1 (see Figure <ref>), yielding a total of 8 × 79 = 632 input values. The network's output is an 88-dimensional vector; each component corresponds to a specific piano key, and represents the presence or absence of a note onset at the 5 frame in the reading window. The output layer uses the sigmoid activation function, yielding values in the interval (0, 1). We use two hidden layers of 512 neurons each with the softsign activation function <cit.>. §.§ Data GenerationOur data generation method is based upon the linearity of the CQT transform. By superimposing spectrograms of individual notes at various locations and intensities, we can model many characteristics of real piano music.For each training example, we generate an 8 × 79 magnitude spectrogram, representing the reading window, and its associated label. Each spectrogram is generated using a spectral basis: a collection of 88 spectrograms, one for each piano note, each with a labeled onset time.For each of the MIDI instruments listed in Appendix <ref>, we generate a spectral basis for each of the four MIDI velocities 30, 60, 90, and 120. Spectrograms are generated from audio recordings of each note approximately 3.5 seconds in length.[For SoundFont instruments, we use 0.5 seconds of silence followed by 3 seconds of sustained sound, leading to a 151-frame spectrogram. Spectrograms for SFZ instruments are 172 frames long due to a quirk in rendering, but this does not significantly affect our data generation.] This yields a total of 7 × 4 = 28 sets of spectral bases. Since loudly and softly played piano notes differ fundamentally in their harmonic composition, generating a new spectral basis for various MIDI velocities allows us to model the nonlinearities in note partials and decay associated with each dynamic level.Note that although the CQT transform is linear, its magnitude is not.[However, the magnitude of the transform is approximately linear, justifying NMF-based techniques <cit.>. Despite this, we find that many characteristics of real data, such as nonlinear note partial interference, are more effectively modeled by keeping complex phase information.] For this reason, we opt to store the complex-valued spectral basis and perform all calculations in the complex domain. §.§ Algorithm DescriptionAt the start of the generation of each training example, one of the spectral bases is randomly chosen.We label the 5 frame in the reading window as Frame 0. Several chord onset times are randomly picked between Frame -130 and Frame 10; the number of onset times is chosen from a Poisson distribution with λ = 6.For each chord, we randomly pick several notes uniformly from A_0 to C_8 without replacement. The number of notes in each chord is chosen out of a geometric distribution with p = 0.4. The spectrogram for the chord is modeled as a superposition of the spectrograms for each note.To improve robustness, we scale the basis spectrogram for each note by a factor randomly chosen between 0.1 and 1, and randomly shift the overall complex phase. The spectrogram of each note is also temporally shifted by a number of frames drawn from a discrete Gaussian distribution with σ = 0.5.To model note offsets, the spectrogram for a randomly chosen 5% of notes begins to decay exponentially by a factor of e per frame, starting on a frame randomly chosen from the reading window.Finally, the spectrogram for each chord is scaled by a factor randomly chosen between 0.1 and 1 and placed at its corresponding chord onset frame; the overall spectrogram is formed by superimposing all the chord spectrograms.We take the magnitude of the overall spectrogram[If no value within the reading window has a magnitude exceeding 10^-3, we discard the data and start over to avoid “silent” frames.] and add white noise of magnitude 0.003. We then use the reading window from Frame -4 to Frame 3, normalized to a maximum value of 1, as our final generated data sample.[In practice, of course, we perform only the calculations necessary to compute the values within the reading window.]The labels are 88-dimensional vectors whose components are ternary logical values. We assign a value of true () if a note onset occurs at exactly the 5 frame of the reading window, a value of unknown () if an onset occurs at frames 3, 4, 6, or 7 of the reading window, and a value of () otherwise. In practice, we encode the values , , andas 1, 1/2, and 0, respectively.§ TRAINING §.§ SetupTraining data was generated using the 24 spectral bases associated with the , , , , , andMIDI instruments. The network was trained for 1.5 million iterations using the ADAM optimizer <cit.> with an initial learning rate of 10^-3. We use a mini-batch size of 32 and an L^2 regularization parameter of 5 × 10^-10.We use a modified version of the cross-entropy loss function, wherelabels do not contribute to the cost regardless of the network's output. This is to avoid penalizing ambiguous cases where a note onset occurs, but is not perfectly centered in the reading frame — it is difficult to define a precise temporal threshold between the presence and absence of an onset.Detailed information about the system configuration used for training is available in Appendix <ref>. §.§ ValidationThree validation datasets were used, each of which contained 32768 samples: * Data generated using the 4 spectral bases associated with theMIDI instrument.* Synthesized piano recordings from the MAPS database.* Real piano recordings from the MAPS database. For MAPS validation data, we precomputed the spectrogram for each piece and normalized each to a maximum value of 1. We randomly selected 1000 recordings for validation. Each sample was generated by selecting a reading window randomly from the validation spectrograms. Reading windows whose magnitudes did not exceed 10^-3 were discarded; otherwise, they were normalized to a maximum of 1 and used as validation samples. Corresponding ternary labels were constructed using the MAPS onset annotations in the same manner as for synthesized data. Network performance on these validation datasets, as well as on a similarly generated dataset of 32768 training samples, is discussed in Appendix <ref>.§ EVALUATIONOur network was evaluated on the 28910 MAPS recordings not used for validation. For each spectrogram, we iterated over all positions of the reading window; windows were filtered and normalized as for the validation datasets, then fed through the network, generating a raw onset piano-roll representation.The onset piano-roll was thresholded at 0.8 to obtain a binary activation piano-roll. For each of the 88 notes, every run of consecutive activations was considered to be a single note onset event; the frame at which each event occurred was taken to be the average of all frames involved in a run. We convert the frame of each event into an onset time and evaluate our network predictions using thetool <cit.>.Onsets are considered to be correct if their onset time is within 50ms of a ground truth onset at the same pitch; each label can only be matched with one network prediction. Unmatched labels are counted as false negatives, and unmatched predictions are counted as false positives. Standard note-based precision (), recall (), accuracy (), and() metrics are computed as described in <cit.>. § RESULTSOur results, as well as those of other machine learning systems, are shown in Table <ref>.[Poliner used a detection tolerance of 100ms rather than 50ms.] Note that all other approaches listed use the MAPS dataset as both training and testing data, while we did not train on the MAPS dataset. In addition, our feed-forward neural network is much simpler than the models proposed by other authors.Nevertheless, we manage to exceed the , precision, and accuracy of the convolutional neural network used in <cit.> when evaluated on the MAPS dataset. Although <cit.> also gives results for a network trained on synthesized pianos and tested on real recordings, their results under this configuration are significantly worse: training and testing on different datasets, they achieve an onsetof only 54.89.Since we do not train on MAPS data, our results demonstrate our system's ability to transcribe recordings under novel conditions. Our approach is also context-independent, allowing it to be immediately applied to any recording without prior adjustment or fine-tuning.Detailed results for the overall MAPS dataset, as well as for each individual component, are listed in Table <ref>. Note that our network achieves the beston the MUS component, which consists of real musical compositions. A sample comparison between our network's predictions and the MAPS ground truth labels is shown in Figure <ref>.While our network performs significantly worse on the real Disklavier samples, these recordings can have annotation discrepancies of up to 100ms <cit.>. In addition, we found several issues with the MAPS Disklavier dataset, including omitted notes and recordings consisting entirely of percussive keybed and pedal noises. However, even taking these discrepancies into account, we do expect our network to perform worse on real piano recordings; software samples are unable to capture the piano's full range of acoustic subtleties.§ CONCLUSIONWe have presented a method of generating spectral data which accurately models many aspects of piano music, avoiding the need for an annotated musical database. We trained a simple machine learning model on our procedurally generated data and achieve good transcription results under novel recording conditions, highlighting our network's context-independence and excellent generalization qualities. Our approach outperforms that of <cit.>, even though they train and test on samples drawn from the same database and propose a much more complex network architecture. §.§ Future WorkSeveral improvements to our data generation method could be implemented: * We could quite easily model the full range of pianos more accurately by adding more MIDI instruments or increasing the number of MIDI velocities sampled. * Note offsets are modeled in a rather crude fashion by applying exponential decay to the note spectrogram; a sample-based approach could more accurately model various articulations. * No distinction is made between notes played with and without sustain pedal — lifting the dampers alters note timbre slightly and could be modeled with additional samples. * Rather than model chord onset times and notes randomly, a musical model could be introduced to take into account the temporal patterns and harmonic progressions present in Western music. In addition, our method could easily be extended to generate training data for recursive, convolutional, or LSTM networks. Considering that we were able to train a very simple network network using our proposed method and achieve good transcription performance, using a more complex machine learning model would likely yield better results.The use of generated training data as input to a neural network brings to mind the principle behind generative adversarial networks, which have not yet been applied to piano onset transcription. It may be possible to train a neural network to generate data which closely resemble those of real pianos, effectively creating an infinite stream of high-quality spectral data indistinguishable from those of an annotated database.§ MIDI INSTRUMENTSThe SoundFont/SFZ instruments used are listed in Table <ref>; all of them are available online, free of charge. Audio samples for spectral basis generation were synthesized by FluidSynth for SoundFont 2 instruments and by Plogue sforzando for SFZ instruments. Both software synthesizers are available free of charge. § TRAINING AND VALIDATION PERFORMANCE For evaluation on training and validation datasets, the network's output is interpreted as a ternary logical value; a value greater than 0.8 is reported as a note onset (), a value less than 0.2 is reported as a lack thereof (), and a value ofis reported otherwise. We evaluate the network's precision (), recall (), accuracy (), and (), defined using the ternary confusion matrix (Figure <ref>) as follows: = HTP + STP/HTP + STP + HFP= HTP/HTP + SFN + HFN= HTP + STP/HTP + STP + HFP + SFN + HFN= 2 ××/ + Our network's performance on training and validation data can be found in Table <ref>. Since these metrics are computed in a non-standard manner, they cannot be directly compared with those of other approaches. However, we observe that thefor our generated validation data and the synthesized MAPS validation data are nearly identical, highlighting our network's generalization abilities.§ TRAINING SETUP Our network was trained on the CPU of a Lenovolaptop. The system has a 1.40GHz Intel Celeron 2957U CPU and 4.00GB of DDR3 RAM. We use TensorFlow for Python 3.5.3 on Windows 10. Training took a total of 44 hours. | http://arxiv.org/abs/1707.08438v1 | {
"authors": [
"Samuel Li"
],
"categories": [
"stat.ML",
"cs.SD"
],
"primary_category": "stat.ML",
"published": "20170726134633",
"title": "Context-Independent Polyphonic Piano Onset Transcription with an Infinite Training Dataset"
} |
Department of Applied Physics, Stanford University, Stanford, California 94305, USA Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory, Menlo Park, California 94025, USADepartment of Physics, Stanford University, Stanford, California 94305, USA Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory, Menlo Park, California 94025, USADepartment of Electrical Engineering, University of California, Los Angeles 90095, USA School of Information Science and Technology, ShanghaiTech University 201210, ChinaDepartment of Electrical Engineering, University of California, Los Angeles 90095, USADepartment of Electrical Engineering, University of California, Los Angeles 90095, USA[E-mail: ][email protected] Department of Physics, Stanford University, Stanford, California 94305, USA Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory, Menlo Park, California 94025, USAChiral transport along magnetic domain walls in the quantum anomalous Hall effect D. Goldhaber-Gordon December 30, 2023 =================================================================================The recent prediction <cit.>, and subsequent discovery <cit.>, of the quantum anomalous Hall (QAH) effect in thin films of the three-dimensional ferromagnetic topological insulator (MTI) (Cr_yBi_xSb_1-x-y)_2Te_3 has opened new possibilities for chiral-edge-state-based devices in zero external magnetic field. Like the ν=1 quantum Hall (QH) system, the QAH system is predicted to have a single chiral edge mode circulating along the boundary of the film. Backscattering of the chiral edge mode should be suppressed, as recently verified by the observation of well-quantized Hall resistivities ρ_yx=± h/e^2, along with longitudinal resistivities as low as a few ohms <cit.>. Dissipationless 1D conduction is also expected along magnetic domain walls <cit.>. Here, we intentionally create a magnetic domain wall in a MTI and study electrical transport along the domain wall. We present the first observation of chiral transport along domain walls, in agreement with theoretical predictions. We present further evidence that two modes equilibrate and co-propagate along the length of the domain wall. Edge conduction in the QAH system is a consequence of the system's topological non-triviality <cit.>. The QAH system is topologically classified by the Chern number C=±1, corresponding to upwards or downwards magnetization, respectively. At the interface between MTI and vacuum, the Chern number transitions from C=±1 to the topologically trivial C=0; one chiral edge mode propagates along this interface. At a magnetic domain wall, however, the Chern number changes by two, from C=+1 to C=-1. Accordingly, two chiral modes should co-propagate along this interface <cit.>. If the domain does not reach the film's edge, the modes at the domain wall should simply circle the domain, having no effect on transport. But if the domain wall connects two of the device's edges or contacts, it may affect two- and/or four-terminal resistances. Analogous transport has been studied in the QH effect in graphene-based two-dimensional electron gases, where patterned gates can produce adjacent regions of different density and hence different filling factor <cit.>.Previously, the magnetization of MTIs has been flipped without spatial control, by sweeping a homogenous external magnetic field through the coercive field H_C^TI. For most MTIs that display the QAH effect, ρ_yx transitions from ∓ h/e^2 to ± h/e^2 over a substantial range of field (H=150 to H=200 mT for the material used in the work), and ρ_xx has a maximum in this field range <cit.>. Hysteresis loops of the four-terminal resistances of a 50 μm wide Hall bar of MTI film are shown in Fig. 1a. In some samples and in some temperature ranges, R_yx jumps in discrete steps when the external field is swept through H_C^TI <cit.>. Each jump likely represents rearrangement of the magnetic domain structure. Since within a domain the bulk of a MTI is highly insulating at the lowest temperatures <cit.>, these jumps suggest that domain walls host conductive modes. The set of discrete R_yx values is not reproducible between separate magnetic field sweeps, suggesting that the network of magnetic domain walls is complex.To better study transport along domain walls, we intentionally engineered a magnetic domain wall in 6 quintuple layer (Cr_0.12Bi_0.26Sb_0.62)_2Te_3 by spatially modulating the external magnetic field H, and then measured electronic transport along the domain wall at H=0. Our results confirm chiral transport along magnetic domain walls in the quantum anomalous Hall system, and support the prediction <cit.> that domain walls host two co-propagating modes whose carriers fully equilibrate.We spatially modulated the magnetic field applied to the MTI film using Meissner repulsion from a bulk superconductor <cit.>. A niobium cylinder of 1.5 mm diameter and 2 mm height was placed partially covering the surface of a patterned MTI film, but not in electrical contact. When superconducting, the niobium cylinder screens a portion of the external magnetic field, as sketched in Fig. 1b. Far from the cylinder, the external magnetic field is unscreened and the magnetization of the MTI film switches direction when the external field reaches H^TI_C. The screened portion of the MTI film, underneath the cylinder, does not switch magnetization until the external magnetic field reaches a higher value. We define M_z=M/M=1 (-1) to represent upwards- (downwards-) pointing magnetization of the MTI film far from the superconductor. M_z^SC represents the equivalent quantity in the screened region of the MTI.To create a domain wall, the MTI film is first fully magnetized downwards M_z=M_z^SC=-1 with a large external field. A positive field slightly above H_C^TI, typically between μ_0 H=180 mT and 200 mT, is then applied. Only the unscreened region switches magnetization, so that M_z=1, and M_z^SC=-1. This process forms a domain wall near the boundary of the superconductor; its exact position depends on the geometric demagnetization of the superconductor. Applying oppositely signed fields forms the magnetic configuration M_z=-1, M_z^SC=1. Domains were formed with the sample held between 25 and 200 mK.The transport properties of the domain wall were studied at zero field in two geometries. Device A (Fig. 1c, schematic) is a Hall bar with eight voltage terminals and two current contacts. A niobium cylinder was placed on the device's surface, covering the leftmost four voltage terminals. Device B (Fig. 1d, schematic) consists of four contacts inside a large region of MTI. Since the contacts are not connected by an edge of the MTI, when Device B was uniformly magnetized, no conductive channel connected the contacts, and the resistance between pairs of contacts exceeded 8 MΩ at 25 mK. The boundary of a niobium cylinder overlaps all four contacts to form a magnetic domain wall connecting the contacts. The MTI's Fermi level in Device B was tuned with an electrostatic top gate; Device A's top gate, however, was unintentionally shorted to one contact and was left at zero volts during measurements.Device A was first fully magnetized downwards by an external field μ_0 H=-1 T and then returned to μ_0 H=0. At base temperature, the Hall resistance approached -h/e^2 both underneath and away from the superconducting cylinder, as indicated in Table I. In both regions, the longitudinal resistance was small compared to h/e^2, indicating nearly dissipationless conduction along the edges of the device.Next, the external magnetic field was swept, as detailed previously, to attempt to create a magnetic domain. After returning to zero field, the Hall resistivity in the screened and unscreened regions of Device A had opposite signs, confirming the formation of distinct magnetic domains in the Hall bar. The longitudinal resistance was small compared to h/e^2 inside both regions, indicating nearly dissipationless conduction along edges within the domains. Were the Fermi level optimized by the top gate, the longitudinal resistance might have been further reduced. Four-terminal resistances in Device A, in various magnetic configurations, are presented in Table <ref>.Having confirmed the creation of a magnetic domain wall, we study the equilibration of the two chiral modes expected to co-propagate along the domain wall. Consider carriers traveling along the domain walls sketched in Fig. 1e. For sufficiently long domain walls, we expect full equilibration, meaning that carriers leaving the domain walls move rightwards and leftwards along the device's edges with equal probability <cit.>. For full equilibration, the Landauer-Büttiker formalism predicts four-terminal longitudinal resistances R^top_xx and R^bottom_xx, measured across the domain wall, of 0 and 2h/e^2 <cit.>, in reasonable agreement with our results as shown in Table <ref>. The measured resistances slightly exceed the predicted values likely because R_xx is not exactly zero within each domain, though slightly imperfect equilibration would also have this effect <cit.>.Using Device B, we examine how carriers propagate along domain walls. The effective longitudinal resistance R_14,23 quantifies dissipation along the domain wall, and the effective Hall resistance R_13,24 and its mirror R_42,31 establish the chirality of the conductive modes. These four-terminal measurements are not standard longitudinal and Hall measurements, but are topologically analogous to typical R_xx and R_yx measurements in the ν=2 QH system if indeed two chiral modes co-propagate along the interface between two insulating domains. R_14,23, shown in Fig. 2a as a function of gate voltage, was smallest when the gate voltage was -8 V; here, the Fermi level presumably sits in the center of the gap. The low longitudinal resistance at the optimum gate voltage, as shown in Table II, indicates nearly dissipationless conduction along the domain wall. Dissipation along the domain wall appears thermally activated above T=150 mK (Fig. 2(b)). The longitudinal resistance R_14,23 increases with rising temperature according to an Arrhenius law R_14,23∝exp (-T_0/T) with an activation gap T_0=0.41 K. Arrhenius activation with a comparable gap size is observed in the material's bulk conductivity σ, measured in the Corbino geometry <cit.>. The temperature dependence of R_14,23 flattens below T=150 mK; the cause of this behavior is unclear. The consistent Arrhenius activation between R_14,23 and σ suggests that bulk conduction causes dissipation in transport along the domain wall.The effective Hall resistances R_13,24 and R_42,31 are oppositely-signed; further, their sign switches when the device's magnetic configuration is reversed. This confirms the chirality of conduction along the domain wall. The magnitudes of R_13,24 and R_42,31 at the optimum gate voltage, detailed in Table II, are close to (but slightly exceed) 0.5h/e^2, the expected value for two chiral modes. As shown in Fig. 3, R_13,24 and R_42,31 converge as the temperature rises, with R_13,24≈ R_42,31≈ 0.25 h/e^2 at 770 mK. Here, bulk conduction dominates and the two resistances are no longer analogous to Hall measurements; instead, they saturate at a positive value reflecting the sheet conductivity σ and the device geometry.We have shown that magnetic domain walls in quantum anomalous Hall insulators conduct through chiral modes, which are expected to be topological in origin. Two modes are predicted to propagate along domain walls, thus we expect the effective Hall resistances of Device B to saturate at h/2e^2 at low temperatures. Though we did not observe such saturation, the measured effective Hall resistances at the optimum gate voltage are near the predicted value.To explain the discrepancy, we propose that the magnetic exchange interaction, needed to open a gap in the MTI's surface states <cit.>, is reduced around the domain wall, allowing the formation of a compressible stripe. The stripe could perturb R_13,24 and R_42,31 from h/2e^2 without imparting a significant contribution to the longitudinal resistance <cit.>. The width of the compressible stripe is presumably related to the spatial profile of the applied magnetic field, which should vary over hundreds of microns near the edge of the superconductor. The authors acknowledge Andrew J. Bestwick for his contributions to the instrumentation and procedures used in this work, and Marc Kastner, Francesco Giazotto, Malcolm Beasley, and Inti Sodemann for insightful discussions. Device fabrication and measurement was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division, under Contract DE-AC02-76SF00515. Infrastructure and cryostat support were funded in part by the Gordon and Betty Moore Foundation through Grant No. GBMF3429. X. K., L. P. and K. L. 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Lett. volume 74, pages 3009 (year 1995)NoStopChiral transport along magnetic domain walls in the quantum anomalous Hall effect § SUPPLEMENTAL MATERIALS§.§ Material growthMeasurements were performed on high quality single crystalline (Cr_0.12Bi_0.26Sb_0.62)_2Te_3 films 6 quintuple layers (QLs) in thickness. Films were grown on a semi-insulating GaAs (111)B substrate in a Perkin-Elmer ultra-high-vacuum molecular beam epitaxy system. The substrate was annealed at 580^∘C in a Se-rich environment to remove the native oxide. The MTI film was grown with the substrate held at 200^∘C with the Cr, Bi, Sb, and Te source shutters simultaneously open. Growth was monitored using in situ reflection high-energy electron diffraction (RHEED). After growth, a 2 nm Al layer was evaporated in situ at room temperature and later allowed to oxidize in air to passivate the surface, protecting the film from unwanted environmental doping or aging effects. §.§ Device fabricationDevices were patterned using contact photolithography. For each patterning step, a hexamethyldisilazane adhesion layer was spin coated, followed by Megaposit SPR 3612 photoresist. The pre-exposure bake of 80^∘C for 120s was chosen to avoid thermal damage to the film. The photoresist was exposed under an ultraviolet mercury vapor lamp at approximately 70 mJ/cm^2 and was developed in Microposit developer CD-30 for 35s. The device mesas were defined after patterning by etching the surrounding film with an Ar ion mill. Ohmic contacts were made by first cleaning the area with a brief exposure to an in situ Ar ion source, and then evaporating 5 nm Ti and 100 nm Au, followed by liftoff. To realize a robust top gate, a dielectric was grown uniformly across the film by evaporating a 1 nm Al seed layer, which was allowed to oxidize, and then depositing approximately 40 nm of alumina by atomic layer deposition. The top gate was then fabricated by evaporating 5 nm Ti and 85 nm Au, followed by liftoff. Excess alumina dielectric on the surrounding area was etched using Microposit developer CD-26 (tetramethylammonium hydroxide based, metal ion free). Metal was evaporated using a Kurt Lesker electron beam evaporator with an in situ Ar ion source. Atomic layer deposition used trimethylaluminum precursor and water as the oxidizer in a nitrogen purged vacuum chamber.Device A was a Hall bar 500 μm in width. The connections between the length of the Hall bar and the voltage terminals were 20 μm wide. The pairs of voltage terminals within a domain (such as terminals 2 and 3) were separated by 400 μm, measured from center to center of the voltage terminals, meaning R_xx and R_xx^SC were measured across 0.8 squares. Terminals 3 (9) and 4 (8) were separated by 800 μm, meaning R_xx^top and R_xx^bottom were measured across 1.6 squares. Device A was not gate-tunable due to a short to one of the contacts through the alumina dielectric.Device B was a triangular region of MTI film with side length 2.6 mm. Four contacts were placed within the triangular region; the MTI film was removed underneath each contact during the mesa etching step. Each contact was approximately 750 μm tall and 300 μm wide. The contacts were centered at a radius of about 1 mm from a common origin in the center of the triangle, and were angularly separated by 20^∘. The angular separation was chosen to reduce the overall size of the device. The Fermi level was tuned in Device B using a top gate; the optimum gate voltage of -8 V was similar to that of other devices made from the same growth of MTI film. §.§ Lock-in measurementFour-terminal resistances were measured using a typical lock-in measurement setup in a dilution refrigerator at a base temperature of 25 mK. Devices A and B were measured in separate cooldowns. All measurements were current biased with a 5 nA AC bias, which was applied to the device by sourcing 5 V RMS across a 1 GΩ resistor.The current traveling through the device and out of the drain terminal was measured with an Ithaco 1211 current preamplifier with 200 Ω input impedance, set to -10^7 V/A gain. Differential voltages V_xx and V_yx were measured with NF Corporation LI-75A voltage preamplifiers and Stanford Research Systems SR830 digital lock-in amplifiers. The excitation frequency was between 1 and 10 Hz; at higher frequencies, large phase differences developed between the excitation and measurement.The amplifier chain in the measurement setup requires calibration for precise measurement due to uncertainty in the amplifiers' gains. Approximately four months prior to collection of data in this work, the amplifiers were calibrated to the Hall resistance of a quantum anomalous Hall device. Comparison of the QAH device's Hall resistance to the resistance of a standard resistor, using a cryogenic current comparator, verified its precise quantization to h/e^2. The amplifier chain used in lock-in measurements gives the uncalibrated measurement for the Hall resistance V_yx/I = 26.75 kΩ, which is 3.6% higher than h/e^2=25.81 kΩ. All resistances presented in this work were multiplied by 0.965 to correct for this inaccuracy. §.§ Meissner screening of the external fieldReversal of the magnetization of the MTI film at dilution refrigeration temperatures occurs as the external field is swept over an approximately 50 mT window, between H=150 mT and H=200 mT. Therefore, to create a domain we must apply a 200 mT external field without the field underneath the superconducting cylinder exceeding 150 mT. To verify that niobium cylinders met this condition, a cylinder was placed on a Hall bar of a two-dimensional electron gas (2DEG) in a GaAs/Al_0.3Ga_0.7As heterostructure. Below the onset of Landau quantization, the Hall resistance away from the niobium cylinder was linear, with a Hall resistance of 1.53kΩ/T. Underneath the cylinder, the Hall resistance depended on the external magnetic field in a hysteretic stair-step pattern, as shown in Fig. <ref>, likely reflecting vortex lattice pinning. Vortices depinned at a different series of external fields during every field sweep. Most steps persisted as the field was swept for at least 50 mT until the vortex lattice again depinned. The height of most steps corresponded to at least 50 mT of screened field (where the height is converted to units of field using the material's Hall slope). Therefore, the Meissner screening of the superconductor can screen 50 mT of field when the external field is 200 mT; accomplishing suitable screening, however, is dependent on how the vortex lattice depins on a given field sweep. The cylinder was made from unannealed commercial grade niobium of 99.5% purity (Eagle Alloys Corporation). The 2DEG was 39 nm deep in a GaAs/Al_0.3Ga_0.7As heterostructure having two Si δ-doping layers (dopant concentration 5× 10^12 cm^-2) at depths of 14 and 18 nm (IQE Corporation; μ=1.1× 10^5 cm^2/Vs, n=5.9× 10^11 cm^-2). §.§ Field sweeps of screened devicesThe four-terminal resistances of Devices A and B are shown as the external magnetic field is swept in Fig. <ref>. The Hall resistance in the unscreened region of Device A changes sign at the coercive field, while the Hall resistance underneath the superconducting cylinder changes sign at a slightly higher field. A domain wall is created by sweeping the external field to a value within this window; the field is then brought back to zero, verifying the stability of the domain wall. In Device B, the minimum longitudinal resistance occurs within the same window, where conductive modes along the domain wall carry current between the four terminals. A domain wall was not formed on every field sweep, presumably because randomness in vortex lattice pinning caused the magnetization of the niobium cylinder to differ between field sweeps.As the external field is swept, the measured voltages often change suddenly in value, and then slowly decay back to the original value. If the external field sweep is paused immediately after such a spike, the voltages nevertheless decay to their original values with the same time scale as when the sweep is not paused. We interpret these features as results of vortex lattice depinning in the niobium cylinder. As the external field is varied, the vortex lattice occasionally depins to reach an energetically favorable configuration for the new external field. When the vortex lattice depins, the superconductor releases heat into the MTI film. The longitudinal resistance in the MTI film is elevated until the device cools and reestablishes thermal equilibration with the bath. The transport of moderate-mobility GaAs 2DEGs in this range of field is less sensitive to temperature; therefore, lattice depinning had only a marginal effect when we measured a GaAs 2DEG screened by a superconducting cylinder.Fig. <ref> shows the process of creating a domain in Device A. The MTI begins having already been magnetized upwards by an external field μ_0 H=1.2 T. The external field is swept to -189 mT, at which point the Hall resistance far from the niobium cylinder changes sign. The external field is then brought back to zero; the low longitudinal resistance inside each domain persists, demonstrating the stability of the magnetic domain wall. §.§ Predicted resistances in a Hall bar with magnetic domainsThe four-terminal resistances of a device may be computed using the Landauer-Buttiker formalism:I_j =e^2/h∑_j (T̅_ijV_j-T̅_jiV_i )where the transmission coefficient T̅_ij=M_ijT_ij is the product of the number of modes M_ij from terminal j to terminal i, and their transmittance 0≤ T_ij≤ 1. A uniformly magnetized Hall bar in the quantum anomalous Hall state has M_i-1,i=1 and T_i-1,i = 1 when the device's magnetization is upwards M_z=1, while M_i+1,i=T_i+1,i = 1 when M_z=-1; all other transmission coefficients are zero.We calculate the longitudinal resistance across a domain wall as a function of the transmittance probability t that a carrier entering a domain wall will exit the domain wall traveling in the in the other domain. Consider a carrier impinging on the domain wall illustrated in Fig. <ref> from the bottom left edge of the device. The carrier will travel along the domain wall to the top edge of the device, and either will leave the domain wall traveling rightwards, with transmittance probability t, on the opposite side of the domain wall, or will leave traveling leftwards, with transmittance probability 1-t, staying within the original domain. For sufficiently long domain walls, we expect the modes to couple and fully equilibrate <cit.>, such that a carrier leaving the domain wall has no memory of the side of the domain wall from which it entered; therefore, we expect t=0.5.When t=0.5, carriers leaving a domain wall rightwards and leftwards have the same chemical potential, so the voltage along this edge of the device is zero (in Fig. <ref>, V_3-V_2=0). The computed resistances R_xx^bottom=R_14,65 and R_xx^top=R_14,23, assuming perfect chiral transport along all edges of the device, are shown as a function of transmittance probability t in Fig. <ref>. For t=0.5, the Landauer-Buttiker formalism indeed gives R_xx^bottom=2 h/e^2 along the bottom edge of the Hall bar and R_xx^top=0 along the top edge of the Hall bar. The resistances switch when the magnetization of the device, and in turn the chirality of the domain wall, is reversed. The results for t=0.5 are close to the experimental results, as shown in Table I of the main text, supporting that the modes co-propagating along the domain wall fully equilibrate. The two modes could equilibrate by intermixing either at the ends of the domain wall or along the length of the domain wall <cit.>; our measurements do not distinguish between these two possibilities. §.§ Nonlocal resistances in a Hall bar with magnetic domainsThe Landauer-Büttiker formalism may further be used to calculate nonlocal resistances in Device A, where current is not sourced laterally across the Hall bar. The calculated and measured four-terminal resistances for a variety of nonlocal configurations are shown in Fig. <ref>. Measurements were taken at 29 mK.§.§ Predicted resistances in Device B Four-terminal resistances in Device B may be predicted using the Landauer-Büttiker formalism. We consider a model where chiral modes co-propagate along the domain wall together with additional dissipative modes. We consider counterclockwise (as in the configuration M_z=-1, M_z^SC=1) chiral transport, and assume that there are M_i+1,i=2 modes each having transmittance T_i+1,i=T_C from a terminal to the next. The transmission coefficients T̅_ij we may consider when adding dissipation to the model are constrained by the statement of conservation of current ∑_i T̅_ij=∑_i T̅_ji.To add dissipation, we add quasi-helical modes, whose conductance is T̅_i±1,i=T̅_H, between every adjacent terminal. The Landauer-Büttiker results for the effective longitudinal and Hall resistances along the domain wall, R_L and R_H, are: R_L=h/e^2T̅_H/2T_C^2+8T_C T̅_H R_H=h/e^21/2T_C+4 T̅_H+2R_L Comparing this result to the measured four-terminal resistances, shown in Table II, suggests that the transmission from quasi-helical modes T̅_H is small compared to e^2/h, whereas the transmittance of each chiral mode is roughly T_C≈ 0.75. Perfect chiral conduction has transmittance T_C=1. We note that, while conservation of current requires that the chiral transmittance T_C between each pair of terminals be equal, a device may have different quasi-helical conductances T̅^H_i±1,i for different terminals i.We imagine a microscopic picture of transport in Device B to understand why T_C≠ 1 yet T̅_H is still small. Consider that the two chiral modes co-propagating along a domain wall are accompanied by a compressible stripe. Diffusive transport through the stripe adds a quasi-helical component to the device's transport T̅_H. Since T̅_H is small, transport in the stripe must be highly diffusive, meaning the mean scattering length in the stripe is short compared to the distance between contacts. However, carriers may scatter between the chiral mode and the compressible stripe. Imagine a carrier leaving terminal i, traveling towards terminal i+1 in a chiral mode. Assuming that bulk conduction is negligible, the carrier must either reach terminal i+1 with probability T_C or return to terminal i with probability 1-T_C. If the carrier scatters into the compressible stripe, it generally will not travel far before it scatters back into the chiral mode because the stripe is highly diffusive. This carrier will eventually reach terminal i+1. However, if the carrier scatters into the stripe soon after leaving terminal i, it may first return to terminal i through diffusive transport in the stripe. Thus, this picture of a compressible stripe produces T_C≠ 1 while maintaining small quasi-helical transport. Macroscopically, this model preserves longitudinal resistance R_L≈ 0 along the domain wall, while producing non-quantized effective Hall resistances R_H>h/2e^2, as observed in our measurements.§.§ Four-terminal measurements in Device BThe main text discussed four-terminal resistance measurements of Device B in the magnetization configuration M_z=1, M_z^SC=-1. The effective longitudinal resistance and effective Hall resistances were measured following separate external field sweeps, which created the magnetic domains. Four-terminal resistance measurements are shown for the magnetization configuration M_z=-1, M_z^SC=1 Fig. <ref> and Fig. <ref>. All measurements in this magnetic configuration followed the same field sweep. §.§ Arrhenius activation of bulk conductionIn the Corbino disk geometry, ohmic contact is made to the inner and outer rings of an annulus of a MTI film. No edge connects the two contacts, so the two-terminal conductivity of the device is a direct measurement of the bulk conductivity σ (assuming low contact resistance). At low temperatures, the Corbino device is highly resistive when the Fermi level is tuned to the center of the gap by the top gate. As shown in Fig. <ref>, the bulk conductivity increases with increasing temperature by an Arrhenius law with a constant offset σ∼ e^-T_0/ T+c. The activation gap, T_0, is largest when the gate is optimally tuned, at which point T_0=759 mK. At zero gate voltage, T_0=242 mK. The constant offset term c, which becomes dominant below 100 mK, may be caused by breakdown of the QAH effect under high electric fields. §.§ Jumps and plateaus in magnetic transitionsThe magnetization of an MTI film is reversed when the external field reaches the coercive field of the film, H_C^TI. Upon magnetic reversal, the film's Hall resistance changes sign. High resolution magnetic sweeps reveal that the Hall resistance does not smoothly transition, rather, it changes value in a series of jumps and plateaus. The jumps in ρ_yx are largest at intermediate temperatures of around 220 mK. Above T≈ 275 mK, the Hall resistance varies smoothly. Such field sweeps are shown near the coercive field in Fig. <ref> for Hall bars having widths of 20 μm and 100 μm. Similar results have been reported recently in other (Cr_yBi_xSb_1-x-y)_2Te_3 films <cit.>. §.§ Engineering magnetic domainsWe detail several alternative methods to realize magnetic domains in a QAH material at low temperatures. In this work, a large superconducting cylinder locally screens the external magnetic field. We used a niobium cylinder for its large critical field H_c1^Nb, the value of which reflects the maximum external flux that the cylinder can screen. Evaporated films of niobium may also have high critical fields, however, thin film superconductors do not effectively screen out-of-plane fields because of their large geometric demagnetization. In other words, for out-of-plane external fields, H_c1^Nb=0 in the limit of an infinitely thin superconducting film, and the film's magnetization is zero.Whereas a superconductor locally screens the external field, a ferromagnet placed on a surface locally enhances the external field. To create a domain, the ferromagnet's stray field must add 50 mT (the width of the coercive transition) to the external field. To achieve this value, an iron or nickel film need be of order micron thickness. A ferromagnetic cylinder, machined from a bulk metal, potentially could create a magnetic domain. A related idea uses a thin film ferromagnet, magnetized in-plane, on the surface of an MTI film. The fringe field of the in-plane ferromagnet has a large out-of-plane component at its ends <cit.>, which potentially could create a magnetic domain in the MTI underneath the end of the ferromagnetic film. This method has the drawback that fringe fields are spatially narrow, so only a small domain could be formed.Ideally, magnetic domain walls could be induced through the Oersted field from current passing through nanowires on the surface of an MTI. Such a device would feature transistor-like switching of current pathways through the MTI. Superconducting rather than normal metal nanowires are required to avoid Joule heating of the MTI. The critical fields of many superconductors far exceed the coercive field of the MTI, meaning enough current can pass through the superconductor to create a domain. The critical current of thin evaporated superconducting nanowires, however, is suppressed compared to the bulk material's critical current density. On the other hand, larger superconducting wires avoid suppressed critical current densities, but require high currents to generate the requisite Oersted field because of their larger radius.4 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[Upadhyaya and Tserkovnyak(2016)]upadhyaya2016 author author P. Upadhyaya and author Y. Tserkovnyak, 10.1103/PhysRevB.94.020411 journal journal Phys. Rev. Bvolume 94, pages 020411 (year 2016) NoStop [Liu et al.(2016)Liu, Wang, Richardella, Kandala, Li, Yazdani, Samarth, andOng]Liu2016a author author M. Liu, author W. Wang, author A. R. Richardella, author A. Kandala, author J. Li, author A. Yazdani, author N. Samarth,and author N. P. Ong, 10.1126/sciadv.1600167 journal journal Sci. Adv. volume 2, pages e1600167 (year 2016) NoStop [Clinton and Johnson(1997)]Clinton1997 author author T. W. Clinton and author M. Johnson, 10.1063/1.118482 journal journal Appl. Phys. Lett. volume 70,pages 1170 (year 1997)NoStop [Castellana et al.(2006)Castellana, Giazotto, Governale, Taddei, and Beltram]Castellana2006 author author C. Castellana, author F. Giazotto, author M. Governale, author F. Taddei,and author F. Beltram, 10.1063/1.2172018 journal journal Appl. Phys. Lett. volume 88, pages 052502 (year 2006)NoStop | http://arxiv.org/abs/1707.08677v1 | {
"authors": [
"I. T. Rosen",
"E. J. Fox",
"Xufeng Kou",
"Lei Pan",
"Kang L. Wang",
"D. Goldhaber-Gordon"
],
"categories": [
"cond-mat.mes-hall"
],
"primary_category": "cond-mat.mes-hall",
"published": "20170727010412",
"title": "Chiral transport along magnetic domain walls in the quantum anomalous Hall effect"
} |
^1Institute for Theoretical and Experimental Physics, B. Cheremushkinskaya 25117218 Moscow, Russia^2Moscow Institute of Physics and Technology, 141700 Dolgoprudny, Russia^3Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research141980 Dubna, RussiaThe superfluid pairing gap of neutron matter is calculated in the framework of Quark Compound Bag model with nucleon-nucleon interactions generated by the s-channel exchange of Jaffe-Low primitives (6-quark states). 13.75.Cs, 21.30.-x, 14.20.Pt, 26.60.Kp Superfluid neutron matterin the s-channel exchange nucleon-nucleon interaction modelsM. I. Krivoruchenko^1,2,3 December 30, 2023 ============================================================================================ § I. INTRODUCTIONIt is well known that nucleon-nucleon interactions are characterized by repulsion at small distances and attraction at large distances. In the one-boson exchange (OBE) model, the main role is attributed to the ω-meson exchange, which generates repulsion, and the σ-meson exchange, which generates attraction. There are many versions of the OBE model with a different set of mesons that describe successfully a broad set of experimental data (see, e.g., <cit.>). The strong interaction scale, however, is comparable to the size of nucleons and mesons. From a geometric point of view, it is not quite clear how the t-channel exchange mechanism of the OBE model can dominate in those cases where the overlap of particles becomes essential. Simple estimates show that, in nuclei, the overlap of nucleons is rather substantial <cit.>.In situations where an overlap of nucleons is important, the quark degrees of freedom should be taken into account. Two nucleons at small distances form a 6-quark state. The interaction of nucleons, therefore, can be described by a diagram where nucleons propagate first, and then merge together in a 6-quark state; this state propagates and then decays into two nucleons (see, e.g., Fig. <ref>).This mechanism was initially discussed by T. D. Lee <cit.> without connection to the nucleon-nucleon scattering problem. In order to illustrate the physical nature of Castillejo, Dalitz and Dyson poles <cit.>, Dyson <cit.> constructed a modified Lee model with the s-channel exchange of resonances. Dyson-Lee models allow for the existence of bound states and resonances, so they describe a class of systems dominated by attraction, whereas the short-range nucleon-nucleon interactions are dominated by repulsion. Simonov <cit.> expanded the class of Dyson-Lee models by including the s-channel exchange of Jaffe-Low "primitives" <cit.>. The extended Simonov-Dyson models describe systems with both attraction and repulsion, and the repulsion is generated by Jaffe-Low primitives. The primitives can be interpreted as resonances with a vanishing width on the mass shell. Off the mass shell, the primitive widths are different from zero, so these states are involved in the interactions. Primitives correspond to zeros of the scattering phase with a negative slope, while the P matrix has poles at the energy of nucleons equal to the primitive mass. At the same time, the S matrix is a regular function.In a class of Simonov-Dyson models, the most detailed studies are made in the framework of Quark Compound Bag (QCB) model. This model successfully reproduces the nucleon-nucleon scattering phases, as well as the properties of light nuclei <cit.>.In this paper, we calculate the superfluid pairing gap of neutron matter in one of the versions of QCB model <cit.>.The neutron matter superfluidity is of interest for modeling glitches of neutron stars, describing cooling rate and the structure of neutron stars. The equation of state (EoS) of nuclear matter is of interest for astrophysics of compact objects. The discovery of neutron stars with a mass of about 2 M_ <cit.> allowed exclusion of a broad set of the soft EoS of nuclear matter for which neutron stars lose gravitational stability at lower values of the masses. At the same time, laboratory experiments indicate that the EoS of the symmetric nuclear matter must be soft enough. This problem has been widely discussed in recent years. An important role in the EoS of nuclear matter is attributed to the isospin asymmetry, which influences the stiffness with the increase in neutron fraction <cit.>. The production of hyperons in the centre of massive neutron stars due to the chemical equilibrium with respect to the weak interactions softens the EoS <cit.>. The possibility of increasing the stiffness of the EoS of nuclear matter at the expense of introducing weakly interacting light bosons (WILBs) beyond the standard model has been discussed in Refs. <cit.>. In Ref. <cit.>, we pointed out an additional source of repulsion between hyperons, associated with the ϕ(1020)-meson exchange, which is normally suppressed in interactions of non-strange baryons due to the Okubo-Zweig-Iizuka rule. The numerical studies of Refs. <cit.> verify that the ϕ(1020)-meson suppresses the hyperon production efficiently enough to keep the maximum neutron star mass above the observational limit. The laboratory data still leave a certain freedom for stiff high-density EoS of nuclear matter in the chemical equilibrium.In the mean field (MF) approximation, the exotic degrees of freedom soften the EoS of nuclear matter. The upper limit on the masses of neutron stars 2 M_ leads to strong restrictions on the critical density of the phase transition to quark matter and, if quark matter exists in the cores of neutron stars, to the quark matter EoS. Beyond the MF approximation, the effect of exotic degrees of freedom is, in general, multidirectional. In quantum theory, even if critical transition density into a new phase is high, exotic degrees of freedom are present virtually and contribute to observables through loops. This requires their account already at the saturation density and leads to a renormalization of the phenomenological parameters. When the density increases, the sign of the effect is not fixed a priori. An example is given in Ref. <cit.>, where a dibaryon Bose condensation in nuclear matter is discussed in the relativistic Hartree approximation <cit.>. In this regard, one can expect that the effect of recently discovered dibaryon d^*(2380) <cit.> on the EoS of nuclear matter, despite its relatively high mass, is important because the spin of the resonance, J = 3, provides a large 2J + 1-fold enhancement of the Casimir effect originating from the in-medium modification of zero-point fluctuations of the dibaryon field. A dibaryon Bose condensation in nuclear matter is discussed in Refs. <cit.>.The problem of how nuclear matter behaves with increasing the density in the s-channel exchange models of nucleon-nucleon interaction has not yet been studied. In the present paper, we formulate and solve the equations for the neutron matter EoS in the framework of QCB model of Ref. <cit.> and determine the density dependence of the neutron pairing gap. The model parameters of Ref. <cit.> are fitted to the nucleon-nucleon scattering phases for the nucleon kinetic energy in the laboratory frame up to 350 MeV in the ^1 S_0 channel and up to 500 MeV in the ^3 S_1 channel. In such an approach, the quantitative description of nuclear matter at a density below the saturation density must be possible. Moreover, we believe that extrapolation of the model predictions to supranuclear densities gives at least a qualitatively correct picture of nuclear matter phenomena.This paper is organized as follows. In the next section, the model is described. Section 3 provides a system of equations to determine the EoS by taking into account the neutron pairing effect. Section 4 describes a procedure for numerical solution of the equations and reports results of the calculations. In Conclusion, the results are summarized and perspectives of studying high-density nuclear matter in the s-channel exchange nucleon-nucleon interaction models are discussed. § II. MODEL In the QCB model <cit.>, the nucleon-nucleon scattering proceeds through the formation of an intermediate compound state with a mass ofM_α > √(s_0) = 2m, where m is the nucleon mass. The D function of the process can be written asD(s) = Λ(s) - Π(s),whereΛ^-1(s)= ∑_αg_α^2/s - M_α^2 + G,Π(s)=- 1/π∫_s_0^+∞Φ_2(s^')ℱ^2(s^')/s^' - sds^',Here Φ_2(s) = π p^*/√(s) is the relativistic phase space, p^* = √(s/4 - m^2)is the nucleon momentum in the center-of-mass frame,g_α is the coupling constant of primitive d_α with nucleons, ℱ(s)is form factor of the d_αNN vertex. The S matrix has the formS = D(s - i0)/D(s + i0).The poles of Λ(s), known as Castillejo, Dalitz and Dyson poles <cit.>, are localized between zeros of the function Λ(s), which, respectively, are determined by the masses of compound states from equation Λ(s = M_α^2) = 0. After the coupling with the continuum is switched on (g_α≠ 0), compound states become bound states, resonances, or primitives <cit.>.The D function (<ref>) is a generalized R function. It does not have complex zeros on the first sheet of the Riemann surface. On the real half-axis (-∞,s_0), when the condition D(s_0) < 0 is satisfied for s_0 < M_α^2, it also does not have zeros describing bound states.Simple roots of equationD(s) = 0,located under the unitary cut on an nonphysical sheet of the Riemann surface, are identified with resonances. Simple roots of the equation, located on the real half-axis (s_0,+∞), are identified with primitives. In such a case, the zeros of the real part of D(s) are of the first order, and the zeros of the imaginary part of D(s) are of the second order.The value 1/D(s) can be interpreted as a complete propagator of the compound state (or states). The complete propagator can be determined from the Dyson equation as shown in Fig. <ref>. The loop in the diagram denotes the dispersive part of the D function, i.e., Π(s). In a more general scheme, there is a contact four-fermion interaction term, to which the coupling constant G in Eq. (<ref>) corresponds.The vertex d_αNN corresponds to the value -ig_αℱ(s), the bare compound-state propagator is in the correspondence with i/(s - M_α^2) +iG/g_α^2, and the complete propagator is 1/D(s). The contact vertex is included in the definition of the bare propagator. The form factorℱ(s) is a function of the three-dimensional momentum of nucleons in the center-of-mass frame, while the compound-state propagator depends on their four-momenta through s = (p_1 + p_2)^2. On the mass shell of nucleons ℱ(s) is the function of nucleon momentum: ℱ(s = 4(m^2 + 𝐩^2)) ≡ℱ(𝐩^2 ). The scattering of two nucleons is shown in Fig. <ref>. The amplitude has the formA(s)=e^iδ(s)sinδ(s) =- Φ_2(s) ℱ^2(s)/D(s).In the Born approximation, we have the relationA(s)=- √(s)p^*/8πU(𝐪) =- Φ_2(s) ℱ(s) Λ^-1(s) ℱ(s).where 𝐪 = 𝐩^' - 𝐩 is the transmitted momentum, and U(𝐪) is the s-wave projected Fourier transform of the potential. The scattering theory in separable potentials can be used; the kinematic factors are restored from the correspondenceU(𝐪) ↔8π^2/sℱ(s) Λ^-1(s) ℱ(s).Separable potentials are represented in the formU(𝐪) = ∑_ν f_ν(𝐩^')f_ν(𝐩). § III. NEUTRON PAIRINGFor separable potentials, the pairing gap equations are discussed in Ref. <cit.> and recently in Ref. <cit.>. In view of the correspondence (<ref>), the self-consistency condition can be written as1=-∫d𝐩/(2π )^32π^2/E^2(𝐩)ℱ(𝐩^2) Λ^-1(s)ℱ(𝐩^2). 1/2√((E(𝐩) - μ )^2 +Δ^2 (s,𝐩))|_s = 4 μ^2 .Here s is the (square) of the energy of the Cooper pair in its rest frame. The value of s is set equal to 4μ^2, where μ is the chemical potential of neutrons; the relativistic dispersion law E(𝐩)=√(𝐩^2 +m^2) is used. The pairing gap equalsΔ (4μ ^2 ,𝐩) = √(2)π/E(𝐩)ℱ(𝐩^2 )Λ ^-1(4μ ^2 )|Ξ |.The value Ξ ^* is defined byiΞ ^* =∫d^4 p/(2π )^4√(2)π/E(𝐩)ℱ(𝐩^2 )F^† (p),with F^† (p) being the anomalous Green's function in the momentum-space (see, e.g., <cit.>):ε_αβ F^† (p)= ∫ d^4x e^ip(x - y)ε_αβF^† (x-y) = ∫ d^4x e^ip(x - y) (-i)⟨ TΨ _α^† (x)Ψ _β^† (y) ⟩,where ε _12 = - ε _21 =1, and ε_11 =ε _22 =0. Equations (<ref>) - (<ref>) can be derived from Eliashberg's equations for normal and anomalous Green's functions <cit.>. In Fig. <ref> Eliashberg's equations are shown graphically for the s-channel exchange interaction models.The solution of the self-consistency equation leads to the following expression for the normal Green's functionG(p) = u_𝐩^2/ω -ε (𝐩) + i0 + v_𝐩^2/ω + ε (𝐩) - i0,where([ u_𝐩^2; v_𝐩^2 ]) = 1/2(1 ±η _𝐩/ε (𝐩)),and η _𝐩 = E(𝐩) - μ, ε (𝐩) = √(η_𝐩^2+ Δ^2(4μ ^2 ,𝐩)). The anomalous Green's function equalsF^† (p) = -√(2)π/E(𝐩)ℱ(𝐩^2 ) Λ^-1 (4μ ^2 ) Ξ^*/(ω -ε (𝐩) + i0)(ω + ε(𝐩) - i0) .Substituting Eq. (<ref>) into Eq. (<ref>), we arrive at (<ref>).Given that Green's function (<ref>) is known, the number density can be found fromN/V =-2i lim_t → -0∫dω d𝐩/(2π )^4 e^-iω t G(ω ,𝐩) =2∫d𝐩/(2π )^3 v_𝐩^2 = -∂Ω/∂μ .After integrating this equation with respect to the chemical potential, thermodynamic potential Ω can be found, then the energy of the system can be calculated fromE = Ω +μ N. The primitive contributes to the self-energy of the in-medium nucleon, as shown in Fig. <ref>. The contribution has the structure (1 + γ_0)/2, where γ_0 is the Dirac gamma matrix <cit.>. The mass operator Σ redefines the nucleon energy, i.e., the chemical potential, and contributes to the nucleon mass. The dispersion law is taken to beE(𝐩)=Σ/2 + √(𝐩^2 + (m + Σ/2)^2) . The mass operator has therefore vectorial and scalar components in the Lorentz group. In order to discriminate them from each other, it is necessary to go beyond the non-relativistic approximation, as discussed in Ref. <cit.>. The mass operator, in general, depends on the nucleon momentum and is defined off the energy surface, which makes the dispersion law a more complicated function in comparison with (<ref>). In the mass operator, we neglect the shift from the energy surface, as well as the momentum dependence. When calculating Σ, the nucleon momentum is set to zero. Figure <ref> corresponds to the expressionΣ =2 π^2/m^2∫d𝐩/(2π )^3ℱ^2 (s)/D(s) v_𝐩^2 ,where the function v_𝐩^2is the probability of finding a nucleon with a given momentum. Equation (<ref>) generalizes the corresponding equation of the optical potential model. Here, 𝐩 is the momentum of the nucleon in the rest frame of the matter, s = (m + E(𝐩))^2 - 𝐩^2 is the square of the nucleons energy in the rest frame of the matter. The in-medium modification of the T matrix is not considered, namely, when calculating the loop in Fig. <ref>, the Pauli blocking for nucleons is not taken into account, and the imaginary part of the nucleon self-energy is discarded.In a free theory, the chemical potential is determined by the Fermi momentum fromp_F^[0] =√(μ ^2 -m^2) .Using condition (<ref>) one finds a momentum at which the quasiparticle energy is minimal. The equation E(p) = μ givesp_F^[1] =√((μ -Σ /2)^2 -(m+Σ /2)^2) .If the density is known, the Fermi momentum can also be found from equationn=2/(2π )^34π/3 p_F^3 .In a theory with interaction, these three momenta are pairwise different. The minimum of ε(p) determines the energy gap of the quasiparticle spectrum:Δ _F≡Δ (4μ ^2 ,p_F^[1] ).§ IV. NUMERICAL RESULTSThe calculation scheme is as follows: We vary the chemical potential and look for solutions of Ξ ^* in equation (<ref>). As the starting value of Σ, we calculate the integral (<ref>) for v_𝐩^2 = 1 inside and v_𝐩^2 = 0 outside the Fermi sphere. For a given Σ, we find Ξ ^*. Next, we find v_𝐩^2, calculate the self-energy Σ, find Ξ ^*, and so on until the convergence becomes obvious.The particle density can be found according to Eq. (<ref>). Further, by integrating the density with respect to the chemical potential, one can find the thermodynamic potential, which is the pressure with the opposite sign. In conclusion, according to Eq. (<ref>), the energy of the system can be calculated.For the model parameters of Ref. <cit.>, the results of solving Eq. (<ref>) are shown in Fig. <ref>, where Δ_F is a function of the Fermi momentum. Predictions of the QCB model are superimposed on the predictions of the advanced OBE models <cit.>.In the model considered, the vanishing of the energy gap at a momentum p_F > 1.6 fm^-1 is related to the zero of the form factor ℱ(s) at p^* = 353 MeV. When the momentum p_F^[1] approaches p^*, the smallness in the denominator of the integrand (<ref>) is compensated by the smallness of the numerator, as a result of which the integral remains small and the solutions do not appear.With a further increase in the chemical potential, p_F^[1] shifts relative to the zero of ℱ(s) at p^* = 353 MeV, which results in the occurrence of an additional branch of solutions at high densities, as shown on a different scale in Fig. <ref>. The negative sign of the pairing gap is due to the fact that the form factor changes sign (The pairing gap as a function of the momentum is proportional to the form factor, see Eq. (<ref>). In this region, the dibaryon chemical potential is still below 2007 MeV, so the solutions have the usual physical meaning. The effective interaction constant is proportional to Λ^-1(4μ ^2 ), so one can expect that the energy gap increases with μ. This effect together with the existence of a new branch of solutions are clearly seen in Fig. <ref> for p_F> 2.0 fm^-1. This behavior has a nice interpretation in terms of repulsion in a two-level system. The first level is the Cooper pair, and the second higher level is the primitive (compound state) with the same quantum numbers. The interaction between these states leads to the repulsion, and the effect does not depend on the sign of the potential. Since the primitive is not treated dynamically, its mass is fixed. The binding energy of the Cooper pair, consequently, increases with μ, which leads to an increase in the pairing gap.When the Fermi momentum is less than 1.6 fm^-1, the energy gap is consistent with the OBE models. Small densities correspond to large distances, which are well studied and well parameterized in all models .On the horizontal axis, in Figures <ref> - <ref>, a Fermi momentum is shown, which is determined from the particle number density (<ref>). The region of applicability of the model is limited to momenta p < p_F = 2.5 fm^-1. At a Fermi momentum of the order of 2.5 fm^-1, a dibaryon Bose condensation can start due to the primitive-to-resonance conversion.Figure <ref> shows the dependence of the neutron self-energy operator on the Fermi momentum. The modification of the neutron mass is much smaller than predicted in the MF models, and is comparable in order of magnitude with the predictions of the DBHF models.More details on the nuclear matter properties in the model considered can be found elsewhere <cit.>. § V. CONCLUSIONSIn this paper, we studied the superfluid neutron matter in the s-channel exchange nucleon-nucleon interaction model of Ref. <cit.>. The neutron pairing gap below the saturation density was found to be fairly consistent with previous studies in the OBE models.The results obtained by the extrapolation to supranuclear densities have transparent physical interpretations, including the origin of the second superfluid phase of the neutron substance. In the model considered, a dibaryon Bose condensation is possible at a density of 0.55 fm^-3, i.e., three times greater than the saturation density.The presented calculations can be improved in several ways:1. In Refs. <cit.>, the scattering phases of nucleons were described in a broad energy interval by introducing two primitives in each channel. Instead of one self-consistency equation (<ref>), here two self-consistency equations must be considered. We expect that the qualitative properties of nuclear matter will not be changed, but EoS may become stiffer. Next, we discussed one ^1 S_0 scattering channel of neutrons. In a more advanced approach, one should include other channels.2. Evaluation of properties of the symmetric nuclear matter at the saturation density is a necessary step in studying the nuclear matter EoS. Properties of the symmetric nuclear matter are experimentally known, so this calculation is a sensitive test of the model.3. In the present paper, the primitive was not treated dynamically, so that the the region of applicability of the model is restricted to densities n < 0.55 fm^-3. In the absence of special constraints and under perturbations primitives, in general, leave the unitary cut and become resonances, which leads to a Bose condensation of 6-quark resonances, i.e., dibaryons. In OBE models, primitives are tightly coupled to the unitary cut. The same physics in QCB models could require either fine tuning of the model parameters or a special constraint. The instability of primitives under perturbations allows for experimental verification <cit.>.4. In this paper we found an approximate solution of superfluidity equations of neutron matter. The pairing gap was calculated in terms of the T matrix. In a more advanced approach, one can use a G matrix obtained self-consistently from the s-channel exchange versions of the Dyson equation shown in Fig. <ref> and the Eliashberg equations shown in Fig. <ref>. The s-channel exchange makes the situation look differently from that in the t-channel exchange. In OBE models, a limited set of diagrams can be evaluated. 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Pastore, arXiv:1706.09701 [nucl-th] (2017). 45 R. Kennedy, L. Wilets, and E. M. Henley, Phys. Rev. 133, B1131 (1964). 46 T. Duguet, Phys. Rev. C 69, 054317 (2004). 47 A. A. Abrikosov, L. P. Gorkov and I. E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics, (Englewood Cliffs: Prentice Hall, 1963). 50 G. M. Eliashberg, Sov. Phys. JETP 11, 696 (1960). 48 H.-J. Schulze, Pairing gaps in neutron stars. Talk given at the Conference "The Neutron Star Crust and Surface", 25 - 29 June 2007, Pulkovo Observatory, St. Petersburg, Russia. 52a M. I. Krivoruchenko, Phys. Part. Nucl. Lett., in press (2017). 49 M. I. Krivoruchenko, Phys. Rev. C 84, 015206 (2011). | http://arxiv.org/abs/1707.08424v1 | {
"authors": [
"M. I. Krivoruchenko"
],
"categories": [
"nucl-th"
],
"primary_category": "nucl-th",
"published": "20170726131629",
"title": "Superfluid neutron matter in the s-channel exchange nucleon-nucleon interaction models"
} |
A Robust Multi-Batch L-BFGS Method for Machine Learning[This work substantially extends <cit.> published at the Neural Information Processing Systems (NeurIPS) conference in 2016.]Albert S. Berahasa^∗^∗Corresponding author. Email: <[email protected]> and Martin Takáča aDepartment of Industrial and Systems Engineering, Lehigh University, Bethlehem, PA, USA v5.0 released July 2015December 30, 2023 ============================================================================================================================================================================================================================== This paper describes an implementation of the L-BFGS method designed to deal with two adversarial situations. The first occurs in distributed computing environments where some of the computational nodes devoted to the evaluation of the function and gradient are unable to return results on time. A similar challenge occurs in a multi-batch approach in which the data points used to compute function and gradients are purposely changed at each iteration to accelerate the learning process. Difficulties arise because L-BFGS employs gradient differences to update the Hessian approximations, and when these gradients are computed using different data points the updating process can be unstable. This paper shows how to perform stable quasi-Newton updating in the multi-batch setting, studies the convergence properties for both convex and nonconvex functions, and illustrates the behavior of the algorithm in a distributed computing platform on binary classification logistic regression and neural network training problems that arise in machine learning.L-BFGS, multi-batch, fault-tolerant, sampling, consistency, overlap. 90C30; 90C06; 90C53.§ INTRODUCTION It is common in machine learning to encounter optimization problems involving tens of millions of training examples and millions of variables.To deal with the demands of time, storage and processing power imposed by such applications, high performance implementations of stochastic gradient and batch quasi-Newton methods have been developed; see e.g.,<cit.>. In this paper we study a batch approach based on the L-BFGS method <cit.> that strives to reach the right balance between efficient learning and productive parallelism.At present, due to its fast learning properties and low per-iteration cost, the preferred method for very large scale applications is the stochastic gradient(SG) method <cit.>, and its variance-reduced and accelerated variants <cit.>. These methods are implemented either in an asynchronous manner (e.g., using a parameter server in a distributed setting) or following a synchronous mini-batch approach that exploits parallelism in the gradient evaluations <cit.>. A drawback of the asynchronous approach is that it cannot use large batches, as this would cause updates to become too dense and compromise the stability and scalability of the method <cit.>. As a result, the algorithm spends more time in communication as compared to computation. On the other hand, using a synchronous mini-batch approach one can achieve a near-linear decrease in the number of SG iterations as the mini-batch size is increased, up to a certain point after which the increase in computation is not offset by the faster convergence <cit.>.An alternative to SG-type methods are batch methods, such as L-BFGS <cit.>, because they parallelize well and are able to achieve high training accuracy. Batch methods allow for more computation per node, so as to achieve a better balance with the communication costs <cit.>; however, batch methods are not as efficient learning algorithms as SG methods in a sequential setting <cit.>. To benefit from both types of methods, some high performance machine learning systems implement both types of methods<cit.>, and algorithms that transition from the stochastic to the batch regime <cit.> have also received attention recently.The goal of this paper is to propose a single method that selects a sizeable subset (batch) of the training data to compute a step and changes this batch at every iteration to improve the learning abilities of the method. In order to differentiate it from the mini-batch approach used in conjunction with the SG method, which employs a very small subset of the training data, we call this the multi-batch approach. In this regime it is natural to employ a quasi-Newton method, as incorporating second-order information imposeslittle computational overhead and improves the stability and speed of the method. However, the multi-batch approach can cause difficulties to quasi-Newton methods as these methods employ gradient differences to update the Hessian approximations.More specifically, in this paper we study how to design a robust multi-batch implementation of the limited-memory version of the classical BFGS method <cit.>—which we call the multi-batch L-BFGS method—in the presence of two adverse situations <cit.>. The first occurs in parallel implementations when some of the computational nodes devoted to the evaluation of the function and gradient are unable to return results on time, i.e., in the presence of faults.This amounts to using different data points to evaluate the function and gradient at the beginning and the end of the iteration, which can beharmful to quasi-Newton methods since they employ gradient differencesto update Hessian approximations.A similar challenge occurs in a multi-batch approach in which the data points used to compute the function and gradient are purposely changed at each iteration (or every several iterations) toaccelerate the learning process. The main objective of this paper is to show that stable quasi-Newton updating can be achieved in these settings without incurring extra computational cost orspecial synchronization. The key is to perform quasi-Newton updating based on the overlap between consecutive batches. The only restriction is that this overlap should not be insignificant, something that can be expected, or easily enforced, in most situations.Recently, several stochastic quasi-Newton (SQN) methods have been proposed; see e.g., <cit.>. The methods enumerated above differ in three major aspects: (i) the update rules for the curvature (correction) pairs and the Hessian approximation, (ii) the frequency of updating, and (iii) the required extra computational cost and synchronization required. Our method is different from these methods predominantly due to the fact that it does not modify the BFGS update equations or the form of the curvature pairs, and does not require extra (gradient) computations.Additionally, our method is designed to work in a distributed settings with faults, in which faults occur randomly and sample consistency cannot be assumed, and as such several SQN methods are not suitable. We analyze the convergence properties of the multi-batch L-BFGS method using a fixed step length strategy, as well as a diminishing step length strategy, on both strongly convex and nonconvex problems. This is appropriate in our setting, as using a fixed step length approach is popular in practice, and facilitates the study of the stability of quasi-Newton updating in a distributed setting. For strongly convex functions, we show that the algorithm converges, at a linear rate, to an approximate solution whose accuracy depends on the variance of the gradients and the step length. In the nonconvex setting, we show that if cautious BFGS updating is employed, the expected value of the average norm-squared of the gradient is bounded. We present numerical experiments on a plethora of problems that arise in machine learning and deep learning. We first illustrate the robustness of our proposed approach on binary classification logistic regression problems on a distributed computing platform with faults and in the serial multi-batch setting. The results indicate that the proposed method achieves a good balance between computation and communication costs. Moreover, we present results on neural network training tasks that illustrate that when larger batch-size is used, our algorithm is competitive with the state-of-the-art. Finally, we demonstrate the strong and weak scaling properties of the proposed method. The paper is organized as follows. In Section <ref> we describe the multi-batch L-BFGS method in detail. In Section <ref> we provide convergence analyses for the proposed method for strongly convex and nonconvex functions. Numerical results that illustrate the practical performance and robustness of the multi-batch L-BFGS method are reported in Section <ref>. Finally, in Section <ref> we provide some concluding remarks. § A MULTI-BATCH QUASI-NEWTON METHOD Ideally, in supervised learning, one seeks to minimizeexpected risk, defined asR(w)= ∫_Ω f(w; x, y) d P(x, y) = 𝔼[f(w;x,y)], where (x,y) are input-output pairs, f : ℝ^d →ℝ is the composition of a prediction function (parametrized by w) and a loss function, and Ω is the space of input-output pairs endowed with a probability distribution P(x,y). Since the distribution P is typically not known, oneapproximates (<ref>) by the empirical risk F(w) = 1/n∑_i=1^nf(w;x^i,y^i) 1/n∑_i=1^nf_i(w),where (x^i, y^i), for i=1, …, n, denote the training examples, also referred to as data points or samples. The training problem consists of finding an optimal choice of the parameters w ∈ℝ^d with respect to F, i.e., to compute a solution of the problemmin_w∈ℝ^dF(w) =1/n∑_i=1^nf_i(w). In a pure batch approach, one applies a gradient-based method to the deterministic optimization problem (<ref>). In this regime, a popular method is L-BFGS <cit.>. When n is large, it is natural to parallelize the computation of F and ∇ F by assigning the evaluation of component functions f_i, or subsets of the component functions, to different processors. If this is done on a distributed computing platform, it is possible for some of the computational nodes, dedicated to a portion of the evaluation of the objective function and the gradient, to be slower than the rest. In this case, the contribution of the slow (or unresponsive) computational nodes could potentially be ignored given the stochastic nature of the true objective function (<ref>). However, this leads to an inconsistency in the objective function and gradient at the beginning and at the end of the iteration, which can be detrimental to quasi-Newton methods, as mentioned above. Hence, we seek to develop a fault-tolerant version of the batch L-BFGS method that is capable of dealing with slow or unresponsive computational nodes. A similar challenge arises in a multi-batch implementation of the L-BFGS method in which only a subset of the data is used to compute the gradient at every iteration. We consider a method in which the dataset is randomly divided intoa number of batches and the minimization is performed with respect to a different batch at every iteration. Specifically, at the k-th iteration the algorithm chooses S_k ⊂{1, …, n}, computes F^S_k(w_k)=1/|S_k|∑_i∈ S_kf_i(w_k), g_k^S_k=∇ F^S_k(w_k) = 1/|S_k|∑_i∈ S_k∇ f_i(w_k) ,and takes a step along the direction - H_k g_k^S_k, where H_k is an approximation to ∇^2 F(w_k)^-1. Allowing the sample S_k to change freely at every iteration gives this approach flexibility and is beneficial to the learning process. Note, we refer to S_k as the sample of training points, even though S_k only indexes those points.The case of unresponsive computational nodes and the multi-batch regime are similar in nature, i.e., the samples S_k used change from one iteration to the next. The main difference is that node failures create unpredictable changes to the samples, whereas a multi-batch method has control over the sample generation. In either case, the algorithm employs a stochastic approximation to the gradient and can no longer be considered deterministic. We must, however, distinguish our setting from that of the classical SG method, which employs small mini-batches. Our algorithm operates with much larger batches so that distributing the computation of the function and gradient is beneficial, and the compute time is not overwhelmed by communication costs. This gives rise to gradients with relatively small variance and justifies the use of a second-order method such as L-BFGS.The robust implementation of the L-BFGS method, proposed in <cit.>, is based on the following observation:Thedifficulties created by the use of a different sample S_k at each iteration can be circumvented if consecutive samples S_k and S_k+1 have an overlap, so thatO_k= S_k∩ S_k+1≠∅. One can then perform stable quasi-Newton updating by computing gradient differences based on this overlap, i.e.,by defining y_k+1=g_k+1^O_k-g_k^O_k,s_k+1 = w_k+1-w_k,in the notation given in (<ref>), and using this correction pair (y_k, s_k) in the BFGS update. When the overlap set O_k is not too small, y_k is a useful approximation of the curvature of the objective function along the most recent displacement, and leads to a productive quasi-Newton step. This observation is based on an important property of Newton-like methods, namely that there is much more freedom in choosing a Hessian approximation than in computing the gradient <cit.>. More specifically, a smaller sample O_k can be employed for updating the inverse Hessian approximation H_k, than for computing the batch gradient g_k^S_k used to define the search direction - H_k g_k^S_k. In summary, by ensuring that unresponsive nodes do not constitute the vast majority of all compute nodes in a fault-tolerant parallel implementation, or by exerting a small degree of control in the creation of the samples S_k in the multi-batch regime, one can design a robust method that naturally builds upon the fundamental properties of BFGS updating.We should mention that acommonly used fixfor ensuring stability of quasi-Newton updating in machine learning is to enforce gradient consistency <cit.>, i.e., to use the same sample S_k to compute gradient evaluations at the beginning and the end of the iteration, at the cost of double gradient evaluations.Another popular remedy is to use the same batch S_k for multiple iterations <cit.>, alleviating the gradient inconsistency problem at the price of slower convergence. In this paper, we assume that such sample consistency is not possible (the fault-tolerant case) or desirable (the multi-batch regime), and wish to design and analyze an implementation of L-BFGS that imposes minimal restrictions in the changes of the sample. §.§ Specification of the MethodLet us begin by considering a robust implementation of the multi-batch BFGS method andthen consider its limited memory version. Atthe k-th iteration, the multi-batch BFGS algorithm chooses a set S_k ⊂{1, …, n} and computes a new iterate by the formulaw_k+1=w_k-α_kH_k g_k^S_k ,where α_k is the step length, g_k^S_k is the batch gradient (<ref>) and H_k is the inverse BFGS Hessian matrix approximation that is updated at every iteration by means of the formulaH_k+1=V_k^TH_kV_k+ρ_ks_ks_k^T, ρ_k=1/y_k^Ts_k,V_k=1-ρ_ky_ks_k^T . To compute the correction vectors (s_k, y_k), we determine the overlap set O_k = S_k ∩ S_k+1 consisting of the samples that are common at the k-th and k+1-st iterations. We define g_k^O_k=∇ F^O_k(w_k)=1/|O_k|∑_i∈ O_k∇ f_i(w_k),and compute the correction pairs as in (<ref>). This completely specifies the algorithm, except for the choice of step length α_k; in this paper we consider constant and diminishing step lengths.In the limited memory version, the matrix H_k is defined at each iteration as the result of applying m BFGS updatesto a multiple of the identity matrix, using a set of m correction pairs {s_i, y_i} kept in storage. The memory parameter m is typically in the range 2 to 20.When computing the search direction (matrix-vector product) in (<ref>) it is not necessary to form the dense matrix H_k since one can obtain this product via the two-loop recursion <cit.>, using the m most recent correction pairs. Employing this mechanism, the search direction can be computed in 𝒪(d) floating operations, where d is the number of variables. After the step has been computed, the oldest pair (s_j, y_j) is discarded and the new curvature pair is stored.A pseudo-code of the multi-batch limited-memory BFGS algorithm is given in Algorithm <ref>, and depends on several parameters.The parameter r denotes the fraction of samples in the dataset used to define the gradient, i.e., r = | S|/n. The parameter o denotes the length of overlap between consecutive samples, and is definedas a fraction of the number of samples in a given batch S, i.e., o = | O|/| S|. §.§ Sample GenerationThe fault-tolerant and multi-batch settings differ in the way the samples S_k and O_k are formed (Lines <ref> & <ref>, Algorithm <ref>). In the former, sampling is done automatically as a by-product of the nodes that fail to return a computation (gradient evaluation). In the latter, the samples S_k and O_k used at every iteration are purposefully changed in order to accelerate the learning process, thus sampling is user controlled. In either setting, independent sampling can be achieved; a necessary condition to establish convergence results. We first describe the fault-tolerant setting, and then propose two sampling strategies that can be employed in the multi-batch setting. Let T= { (x^i, y^i), for i=1, …, n} denote the training set. *Fault-Tolerant Consider a distributed implementation in which slave nodes read the current iterate w_k from the master node, compute a local gradient on a subset of the dataset, and send it back to the master node for aggregation in the calculation (<ref>).Given a time (computational) budget, it is possible for some nodes to fail to return a result. The schematic in Figure <ref> illustrates the gradient calculation across two iterations, k and k+1, in the presence of faults.Here,B is the total number of slave nodes, ℬ_i for i=1,...,B denote the batches of data that each slave node i receives (T = ∪_i ℬ_i),and∇̃f(w) is the gradient calculation using all nodes that responded within the preallocated time.Let 𝒥_k⊂{1,2,...,B} and 𝒥_k+1⊂{1,2,...,B} be the set of indices of all nodes that returned a gradient at the k-th and k+1-st iterations, respectively. Using this notation S_k = ∪_j∈𝒥_kℬ_j and S_k+1 = ∪_j∈𝒥_k+1ℬ_j, and we define O_k = ∪_j ∈𝒥_k∩𝒥_k+1ℬ_j. The simplest implementation in this setting preallocates the data on each compute node, requiring minimal data communication, i.e., only one data transfer. In this case, the samples S_k are independent if node failures occur randomly. On the other hand, if the same set of nodes fail, then the sample creation will be biased, which is harmful both in theory and in practice.One way to ensure independent sampling is to shuffle and redistribute the data to all nodes after every iteration or after a certain number of iterations. *Multi-Batch Sampling In the multi-batch setting several strategies can be employed, with the only restriction that consecutive batches S_k and S_k+1 should, to a certain degree, overlap. We propose two sampling strategies: (i) overlaps O_k are forced in the sample creation process, (ii) the overlapping set O_k is subsampled from the batch S_k. In practice the two strategies perform on par, however, there is a subtle difference. In the second strategy the batches are sampled independently, something that is not true for the strategy in which overlapping samples are forced. The independent sampling strategy of course does not come for free as this strategy incurs an increase in computational cost per iteration. However, as mentioned above, the overlapping set O_k need not be very large, and thus the increase in cost is negligible as compared to the rest of the computation. We now describe the two approaches in more detail.Figure <ref> illustrates the sample creation process in the first strategy. The dataset is shuffled and batches are generated by collecting subsets of the training set, in order. Every set (except S_0) is of the form S_k= { O_k-1, N_k, O_k}, where O_k-1 and O_k are the overlapping samples with batches S_k-1 and S_k+1 respectively, and N_k are the samples that are unique to batch S_k. After each pass through the dataset, the samples are reshuffled, and the procedure described above is repeated. In our implementation samples are drawn without replacement, guaranteeing that after every epoch (pass over the whole dataset) all samples are used. This strategy has the advantage that it requires no extra computation in the evaluation of g_k^O_k and g_k+1^O_k, but the samples S_k are not independent. The second samplingstrategy is simpler and requires less control. At every iteration k, a batch S_k is created by randomly selecting | S_k | elements from {1,… n}. The set O_k is then formed by randomly selecting | O_k | elements from S_k (subsampling). Note, in this sampling strategy the samples O_k need not be in the set S_k+1. This strategy is slightly more expensive since g_k+1^O_k requires extra computation, but if the overlap is small this cost is not significant.§ CONVERGENCE ANALYSISIn this Section, we analyze the convergence properties of the multi-batch L-BFGS method (Algorithm <ref>) when applied to the minimization of strongly convexand nonconvex objective functions, using a fixed step length strategy, as well as a diminishing step length strategy. We assume that the goal is to minimize the empirical risk F (<ref>), but note that a similar analysis could be used to study the minimization of the expected risk (<ref>). §.§ Strongly Convex caseDue to the stochastic nature of the multi-batch approach, every iteration of Algorithm <ref> employs a gradient that contains errors that do not converge to zero. Therefore, by using a fixed step length strategy one cannot establish convergence to the optimal solution w^⋆, but only convergence to a neighborhood of w^⋆ <cit.>. Nevertheless, this result is of interest as it reflects the common practice of using a fixed step length and decreasing it only if the desired testing error has not been achieved. It also illustrates the tradeoffs that arise between the size of the batch and the step length.In our analysis, we make the following assumptions about the objective function and the algorithm.*Assumptions A* F is twice continuously differentiable.* There exist positive constants λ̂ and Λ̂ such that λ̂ I ≼∇^2F^O(w) ≼Λ̂ I for all w ∈ℝ^d and all sets O ⊂{1,2,…,n} of length |O| = o· r· n.* There exist constants γ≥ 0 and η≥ 1 such that 𝔼_S[ ∇F^S(w) ^2 ] ≤γ^2 + η∇ F(w)^2 for all w ∈ℝ^d and all sets S⊂{1,2,…,n}of length |S|=r · n.* The samples S are drawn independently and ∇ F^S(w) is an unbiased estimator of the true gradient ∇ F(w) for all w ∈ℝ^d, i.e., 𝔼_S[ ∇ F^S(w)] = ∇ F(w).Note that Assumption A.2 implies that the entire Hessian ∇^2F(w) also satisfies λ I ≼∇^2F(w) ≼Λ I,∀ w ∈ℝ^d,for some constants λ,Λ>0. Assuming that every subsampled function F^O(w) is strongly convex is not unreasonableas a regularization term is commonly added in practice when that is not the case.We begin by showing that the inverse Hessian approximations H_k generated by the multi-batch L-BFGS method have eigenvalues that are uniformly bounded above and away from zero. The proof technique used is an adaptation of that in <cit.>. If Assumptions A.1 & A.2 hold, there exist constants 0<μ_1≤μ_2 such that the inverse Hessian approximations {H_k} generated by Algorithm <ref> satisfyμ_1 I ≼ H_k ≼μ_2 I,fork=0,1,2,…Instead of analyzing the inverse Hessian approximation H_k, we study the Hessian approximation B_k = H_k^-1. In this case, the limited memory quasi-Newton updating formula is given as follows: * Set B_k^(0)=y_k^Ty_k/s_k^Ty_kI and m̃ = min{k,m}; where m is the memory in L-BFGS.* For i=0,...,m̃-1 set j=k-m̃+1+i and computeB_k^(i+1)=B_k^(i)-B_k^(i)s_js_j^TB_k^(i)/s_j^TB_k^(i)s_j + y_jy_j^T/y_j^Ts_j. * Set B_k+1 = B_k^(m̃). The curvature pairs s_k and y_k are updated via the following formulaey_k+1=g_k+1^O_k-g_k^O_k,s_k+1 = w_k+1-w_k.A consequence of Assumption A.2 is that the eigenvalues of any sub-sampled Hessian (| O | samples) are bounded above and away from zero. Utilizing this fact, the convexity of component functions and the definitions(<ref>), we havey_k^Ts_k ≥1/Λ̂y_k^2 ⇒y_k^2 /y_k^Ts_k≤Λ̂.On the other hand, strong convexity of the sub-sampled functions, the consequence of Assumption A.2 and definitions(<ref>), provide a lower bound,y_k^Ts_k ≤1/λ̂y_k^2 ⇒y_k^2 /y_k^Ts_k≥λ̂.Combining the upper and lower bounds (<ref>) and (<ref>)λ̂≤y_k^2/y_k^Ts_k≤Λ̂. The above proves that the eigenvalues of the matrices B_k^(0)=y_k^Ty_k/s_k^Ty_kI at the start of the L-BFGS update cycles are bounded above and away from zero, for all k. We now use a Trace-Determinant argument to show that the eigenvalues of B_k are bounded above and away from zero. Let tr(B) and (B) denote the trace and determinant of matrix B, respectively, and set j_i = k-m̃+i. The trace of the matrix B_k+1 can be expressed as, tr(B_k+1)= tr(B_k^(0)) - tr∑_i=1^m̃(B_k^(i-1)s_j_is_j_i^TB_k^(i-1)/s_j_i^TB_k^(i-1)s_j_i) + tr∑_i=1^m̃y_j_iy_j_i^T/y_j_i^Ts_j_i≤ tr(B_k^(0)) + ∑_i=1^m̃y_j_i^2/y_j_i^Ts_j_i ≤ tr(B_k^(0)) + m̃Λ̂≤ C_1,for some positive constant C_1, where the inequalities above are due to (<ref>), and the fact that the eigenvalues of the initial L-BFGS matrix B_k^(0) are bounded above and away from zero.Using a result due to <cit.>, the determinant of the matrixB_k+1 generated by the multi-batch L-BFGS method can be expressed as, (B_k+1)= (B_k^(0)) ∏_i=1^m̃y_j_i^Ts_j_i/s_j_i^TB_k^(i-1)s_j_i =(B_k^(0)) ∏_i=1^m̃y_j_i^Ts_j_i/s_j_i^Ts_j_is_j_i^Ts_j_i/s_j_i^TB_k^(i-1)s_j_i≥(B_k^(0)) ( λ̂/C_1)^m̃≥ C_2,for some positive constant C_2, where the above inequalities are due to the fact that the largest eigenvalue of B_k^(i) is less than C_1 and Assumption A.2.The trace (<ref>) and determinant (<ref>) inequalities derived above imply that the largest eigenvalues of all matrices B_k are bounded from above, uniformly, and that the smallest eigenvalues of all matrices B_k are bounded away from zero, uniformly.Before we present the main theorem for the multi-batch L-BFGS method that employs constant step lengths, we state one more intermediate Lemma that bounds the distance between the function value at any point w∈ℝ^d and the optimal function value with respect to the norm of the gradient squared. Let Assumptions A.1 & A.2 hold, and let F^⋆ = F(w^⋆), where w^⋆ is the minimizer of F. Then, for all w∈ℝ^d,2λ(F(w)-F^⋆) ≤∇ F(w)^2. As a result of Assumptions A.1, A.2 and (<ref>), for all x,y ∈ℝ^dF(x) ≤ F(y) + ∇ F(y)^T (x-y) + 1/2λ∇ F(y) - ∇ F(x)^2;see <cit.>. Let x=w and y=w^⋆F(w)≤ F^⋆ + ∇ F(w^⋆)^T (w-w^⋆) + 1/2λ∇ F(w) - ∇ F(w^⋆)^2 ≤ F^⋆ + 1/2λ∇ F(w) ^2.Re-arranging the above expression yields the desired result. Utilizing Lemmas <ref> and <ref>, we show that the multi-batch L-BFGS method with a constant step length converges linearly to a neighborhood of the optimal solution. Suppose that Assumptions A.1-A.4 hold, and let F^⋆ = F(w^⋆), where w^⋆ is the minimizer of F. Let {w_k} be the iterates generated by Algorithm <ref>, where α_k = α satisfies0 <α≤μ_1/μ_2^2 ηλ,and w_0 is the starting point. Then for all k≥ 0,𝔼[ F(w_k) - F^⋆]≤( 1-αμ_1 λ)^k [ F(w_0) - F^⋆] + [ 1-(1-αμ_1 λ)^k]αμ_2^2 γ ^2 Λ/2 μ_1 λαμ_2^2 γ ^2 Λ/2 μ_1 λ. We have thatF(w_k+1) = F(w_k -α H_k ∇ F^S_k(w_k)) ≤ F(w_k) + ∇ F(w_k)^T (-α H_k ∇ F^S_k(w_k)) + Λ/2α H_k ∇ F^S_k(w_k)^2 ≤ F(w_k) - α∇ F(w_k)^TH_k ∇ F^S_k(w_k) + α^2 μ_2^2 Λ/2∇ F^S_k(w_k)^2,where the first inequality arises due to (<ref>), and the second inequality arises as a consequence of Lemma <ref>. Taking the expectation (over S_k) of equation (<ref>)𝔼_S_k[ F(w_k+1)]≤ F(w_k) - α∇ F(w_k)^TH_k ∇ F(w_k) + α^2 μ_2^2 Λ/2𝔼_S_k[ ∇ F^S_k(w_k)^2 ] ≤ F(w_k) - αμ_1 ∇ F(w_k) ^2+ α^2 μ_2^2 Λ/2( γ^2 + η∇ F(w)^2)= F(w_k) - α( μ_1 - αμ_2^2 ηΛ/2)∇ F(w_k) ^2+ α^2 μ_2^2 γ^2Λ/2≤ F(w_k) - αμ_1/2∇ F(w_k) ^2+ α^2 μ_2^2 γ^2Λ/2where the first inequality makes use of Assumption A.4, the second inequality arises due to Lemma <ref> and Assumption A.3 and the third inequality is due to the step length (<ref>). Since F is λ-strongly convex, we can substitute the result of Lemma <ref> in (<ref>),𝔼_S_k [F(w_k+1)]≤ F(w_k) - αμ_1/2∇ F(w_k) ^2+ α^2 μ_2^2 γ^2 Λ/2≤ F(w_k) - αμ_1 λ[F(w_k) - F^⋆]+ α^2 μ_2^2 γ^2 Λ/2.Let ϕ_k = 𝔼[F(w_k) - F^⋆],where the expectation is over all batches S_0,S_1,...,S_k-1 and all history starting with w_0. Equation (<ref>) can be expressed as,ϕ_k+1≤ (1 - αμ_1 λ ) ϕ_k + α^2 μ_2^2 γ^2 Λ/2.Since the step length is chosen according to (<ref>) we deduce that 0 ≤ (1 - αμ_1 λ ) < 1.Subtracting αμ_2^2γ^2Λ/2μ_1 λ from either side of (<ref>) yieldsϕ_k+1 - αμ_2^2γ^2Λ/2μ_1 λ ≤ (1 - αμ_1 λ ) ϕ_k + α^2 μ_2^2 γ^2 Λ/2 - αμ_2^2γ^2Λ/2μ_1 λ = (1 - αμ_1 λ ) [ ϕ_k- αμ_2^2γ^2Λ/2μ_1 λ].Recursive application of (<ref>) yieldsϕ_k - αμ_2^2γ^2Λ/2μ_1 λ ≤(1 - αμ_1 λ )^k [ ϕ_0- αμ_2^2γ^2Λ/2μ_1 λ],and thus, ϕ_k≤ (1 - αμ_1 λ )^k ϕ_0+ [ 1-(1-αμ_1 λ)^k]αμ_2^2 γ ^2 Λ/2 μ_1 λ.Finally using the definition of ϕ_k (<ref>) with the above expression yields the desired result𝔼[ F(w_k) - F^⋆] ≤( 1-αμ_1 λ)^k [ F(w_0) - F^⋆] + [ 1-(1-αμ_1 λ)^k]αμ_2^2 γ ^2 Λ/2 μ_1 λ. The bound provided by this theoremhas two components: (i) a term decaying linearly to zero, and (ii) aterm identifying the neighborhood of convergence. Note, a larger step length yields a more favorable constant in the linearly decaying term, at the cost of an increase in the size of the neighborhood of convergence.We consider these tradeoffs further in Section <ref>, where we also note that larger batch sizes increase the opportunities for parallelism and improve the limiting accuracy in the solution, but slow down the learning abilities of the algorithm. We should also mention that unlike the first-order variant of the algorithm (H_k = I), the step length range prescribed by the multi-batch L-BFGS method depends on μ_1 and μ_2, the smallest and largest eigenvalues of the L-BFGS Hessian approximation. In the worst-case, the presence of the matrix H_k can make the limit in Theorem <ref> significantly worse than that of the first-order variant if the update has been unfortunate and generates ill-conditioned matrices. We should note, however, such worst-case behavior is almost never observed in practice for BFGS updating. One can establish convergence of the multi-batch L-BFGS method to the optimal solution w^⋆ by employing asequence of step lengths {α_k } that converge to zero according to the scheduleproposed by <cit.>. However, that provides only a sub-linear rate of convergence, which is of little interest in our context where large batches are employed andsome type of linear convergence is expected.In this light, Theorem <ref> is more relevant to practice; nonetheless, we state the theorem here for completeness, and, for brevity, refer the reader to <cit.> for more details and the proof. Suppose that Assumptions A.1-A.4 hold, and let F^⋆ = F(w^⋆), where w^⋆ is the minimizer of F. Let {w_k} be the iterates generated by Algorithm <ref> with α_k = β/k+1and β > μ_1/μ_2^2 ηλ,starting from w_0. Then for all k≥ 0,𝔼[ F(w_k) - F^⋆] ≤Q(β)/k+1,where Q(β) = max{μ_2^2β^2γ^2Λ/2(2μ_1λβ-1), F(w_0) - F^⋆}.Theorem <ref> shows that, for strongly convex functions, the multi-batch L-BFGS method with an appropriate schedule of diminishing step lengths converges to the optimal solution at a sub-linear rate. We should mention that another way to establish convergence to the optimal solution for the multi-batch L-BFGS method is to employ variance reduced gradients <cit.>. In this setting, one can establish linear convergence to the optimal solution using constant step lengths. We defer the analysis of the multi-batch L-BFGS method that employs variance reduced gradients to a different study <cit.>.§.§ Nonconvex caseThe BFGS method is known to fail on nonconvex problems <cit.>. Even for L-BFGS, which makes only a finite number of updates at each iteration, one cannot guarantee that the Hessian approximations have eigenvalues that are uniformly bounded above and away from zero. To establish convergence of the (L-)BFGS method in the nonconvex setting several techniques have been proposed including cautious updating <cit.>, modified updating <cit.> and damping <cit.>. Here we employ a cautious strategy that is well suited to our particular algorithm; we skip the Hessian update, i.e., set H_k+1 = H_k,if the curvature conditiony_k^Ts_k ≥ϵ s_k ^2 is not satisfied, where ϵ>0 is a predetermined constant. On the other hand, sufficient curvature is guaranteed when the updates are not skipped. Using said mechanism, we show that the eigenvalues of the Hessian matrix approximations generated by the multi-batch L-BFGS method are bounded above and away from zero (Lemma <ref>). The analysis presented in this section is based on the following assumptions.*Assumptions B* F is twice continuously differentiable.* The gradients of F are Λ-Lipschitz continuous for all w ∈ℝ^d; the gradients of F^S are Λ_S-Lipschitz continuous for all w ∈ℝ^d and all sets S ⊂{1,2,…,n} of length |S|=r · n; and, the gradients of F^O are Λ_O-Lipschitz continuous for all w ∈ℝ^d and all sets O ⊂{1,2,…,n} of length |O| = o· r· n.* The function F(w) is bounded below by a scalar F .* There exist constants γ≥ 0 and η≥ 1 such that 𝔼_S[ ∇F^S(w) ^2 ] ≤γ^2 + η∇ F(w)^2 for all w ∈ℝ^d and all sets S⊂{1,2,…,n}of length |S|=r · n. * The samples S are drawn independently and ∇ F^S(w) is an unbiased estimator of the true gradient ∇ F(w) for all w ∈ℝ^d, i.e.,𝔼_S [ ∇ F^S(w) ] = ∇ F(w). Similar to the strongly convex case, we first show that the eigenvalues of the L-BFGS Hessian approximations are bounded above and away from zero. Suppose that Assumptions B.1 & B.2hold. Let {H_k } be the inverse Hessian approximations generated by Algorithm <ref>, with the modification that the inverse Hessian approximation update is performed only when(<ref>) is satisfied,for some ϵ >0, else H_k+1 = H_k. Then, there exist constants 0<μ_1≤μ_2 such that μ_1 I ≼ H_k ≼μ_2 I,fork=0,1,2,… Similar to the proof of Lemma <ref>, we study the direct Hessian approximation B_k = H_k^-1. The curvature pairs s_k and y_k are updated via the following formulaey_k+1=g_k+1^O_k-g_k^O_k,s_k+1 = w_k+1-w_k.The skipping mechanism (<ref>) provides both an upper and lower bound on the quantity y_k^2 /y_k^Ts_k, which in turn ensures that the initial L-BFGS Hessian approximation is bounded above and away from zero. The lower bound is attained by repeated application of Cauchy's inequality to condition (<ref>). We have from (<ref>) thatϵ s_k ^2≤ y_k^Ts_k ≤ y_ks_k ⇒ s_k ≤1/ϵ y_k , from which it follows thats_k^Ty_k ≤ s_ky_k ≤1/ϵ y_k ^2 ⇒ y_k ^2/s_k^Ty_k≥ϵ.The upper bound is attained by the Lipschitz continuity of sample gradients (Assumption B.2),y_k^Ts_k≥ϵ s_k ^2 ≥ϵ y_k ^2/Λ_O^2⇒ y_k ^2/s_k^Ty_k≤Λ_O^2/ϵ. Combining (<ref>) and (<ref>),ϵ≤y_k^2 /y_k^Ts_k≤Λ_O^2/ϵ. The above proves that the eigenvalues of the matrices B_k^(0)=y_k^Ty_k/s_k^Ty_kI at the start of the L-BFGS update cycles are bounded above and away from zero, for all k. The rest of the proof follows the same Trace-Determinant argument as in the proof of Lemma <ref>, the only difference being that the last inequality in (<ref>) comes as a result of the cautious update strategy. We now follow the analysis in <cit.> to establish the following result about the behavior of the gradient norm for the multi-batch L-BFGS method with a cautious update strategy. Suppose that Assumptions B.1-B.5 hold. Let {w_k} be the iterates generated by Algorithm <ref>, with the modification that the inverse Hessian approximation update is performed only when(<ref>) is satisfied,for some ϵ >0, else H_k+1 = H_k,where α_k = α satisfies0 <α≤μ_1/μ_2^2 ηλ,and w_0 is the starting point. Then, for all k≥0,𝔼[1/τ∑_k=0^τ-1∇ F(w_k) ^2 ]≤αμ_2^2 γ^2 Λ/μ_1+ 2[ F(w_0) - F]/αμ_1 ταμ_2^2 γ^2 Λ/μ_1 . Starting with (<ref>) and taking the expectation over allbatches S_0,S_1,...,S_k-1 and all history starting with w_0 yieldsϕ_k+1-ϕ_k ≤ - αμ_1/2𝔼∇ F(w_k) ^2+ α^2 μ_2^2 γ^2 Λ/2,where ϕ_k = 𝔼[F(w_k)]. Summing (<ref>) over the first τ iterations∑_k=0^τ-1 [ϕ_k+1-ϕ_k]≤ - αμ_1/2∑_k=0^τ-1𝔼∇ F(w_k) ^2 + ∑_k=0^τ-1α^2 μ_2^2 γ^2 Λ/2= - αμ_1/2𝔼[∑_k=0^τ-1∇ F(w_k) ^2 ] +α^2 μ_2^2 γ^2 Λτ/2.The left-hand-side of the above inequality is a telescoping sum∑_k=0^τ-1[ϕ_k+1-ϕ_k ] = ϕ_τ-ϕ_0 = 𝔼[F(w_τ)] -F(w_0) ≥F -F(w_0).Substituting the above expression into (<ref>) and rearranging terms𝔼[∑_k=0^τ-1∇ F(w_k) ^2 ] ≤αμ_2^2 γ^2 Λτ/μ_1+ 2[ F(w_0) - F]/αμ_1 .Dividing the above equation by τ completes the proof. This result bounds the average norm of the gradient of Fafter the first τ-1 iterations, and shows that, in expectation, the iterates spend increasingly more time in regions where the objective function has a small gradient. Under appropriate conditions, we can establish a convergence rate for the multi-batch L-BFGS method with cautious updates to a stationary point of F, similar to the results proven for the SG method <cit.>. For completeness we state and prove the result. Suppose that Assumptions B.1-B.5 hold. Let {w_k} be the iterates generated by Algorithm <ref>, with the modification that the inverse Hessian approximation update is performed only when(<ref>) is satisfied,for some ϵ >0, else H_k+1 = H_k. Letα_k = α = c/√(τ),c = √(2(F(w_0) - F̂)/μ_2^2 γ^2 Λ), δ(α) = μ_1 - αμ_2^2ηΛ/2,whereτ >c^2μ_2^4 η^2 Λ^2/4μ_1^2,and w_0 is the starting point. Then, min_0 ≤ k ≤τ-1𝔼[∇ F(w_k)^2] ≤√(2(F(w_0) - F̂)μ_2^2 γ^2 Λ/δ(α)^2 τ).Starting with (<ref>), we have𝔼_S_k[ F(w_k+1)]≤ F(w_k) - α(μ_1 - αμ_2^2ηΛ/2) ∇ F(w_k) ^2 + α^2 μ_2^2 γ^2Λ/2 = F(w_k) - αδ(α) ∇ F(w_k) ^2 + α^2 μ_2^2 γ^2Λ/2,where δ(α) = μ_1 - αμ_2^2ηΛ/2. We require that this quantity is greater than zero, δ(α)>0; this discussion is deferred to the end of the proof.Taking an expectation over allbatches S_0,S_1,...,S_k-1 and all history starting with w_0, and rearranging (<ref>) yields𝔼[∇ F(w_k) ^2] ≤1/αδ(α)𝔼[F(w_k)-F(w_k+1)] + αμ_2^2 γ^2 Λ/2 δ(α).Summing over k=0,...,τ-1 and dividing by τmin_0 ≤ k ≤τ-1𝔼[∇ F(w_k)^2]≤1/τ∑_k=0^τ-1𝔼[∇ F(w_k) ^2] ≤1/αδ(α) τ𝔼[F(w_0)-F(w_τ)] + αμ_2^2 γ^2 Λ/2 δ(α)≤1/αδ(α) τ[F(w_0)-F̂] + αμ_2^2 γ^2 Λ/2 δ(α)≤1/δ(c/√(τ)) c √(τ)[F(w_0)-F̂] + c μ_2^2 γ^2 Λ/2 δ(c/√(τ))√(τ).The first inequality holds because the minimum value is less than the average value, and the third inequality holds because F̂≤ F(x_τ) (Assumption B.3). The last expression comes as a result of using the definition of the step length, α = c/√(τ). Setting c = √(2(F(w_0) - F̂)/μ_2^2 γ^2 Λ),yields the desired result.We now comment on the quantity δ(α) that first appears in (<ref>), and that is required to be positive. To ensure that δ(α)>0, the step length must satisfy, α < 2μ_1/μ_2^2 ηΛ. Since the explicit form of the step length is α = c/√(τ), where c is (<ref>), we require that α =c/√(τ) < 2μ_1/μ_2^2 ηΛ.In order to ensure that (<ref>) holds, we impose thatτ > c^2μ_2^4 η^2 Λ^2/4μ_1^2 = (F(w_0) - F̂)μ_2^2η^2 Λ/2γ^2μ_1^2.The result of Theorem <ref> establishes a sub-linear rate of convergence, to a stationary point of F, for the multi-batch L-BFGS method on nonconvex objective functions. The result is somewhat strange as it requires a priori knowledge of τ, the total number of iteration. In practice, one would use α_k = 1/√(k), which would result in a 𝒪(1/√(k)) convergence rate.§ NUMERICAL RESULTS We present numerical experiments on several problems that arise in machine learning, such as logistic regression binary classification and neural network training, in order to evaluate the performance of the proposed multi-batch L-BFGS method. The experiments verify that the proposed method is robust, competitive and achieves a good balance between computation and communication in the distributed setting. In Section <ref>, we evaluate the performance of the multi-batch L-BFGS method on binary classification tasks in both the multi-batch and fault-tolerant settings. In Section <ref>, we demonstrate the performance of the multi-batch L-BFGS method on neural network training tasks, and compare against some of the state-of-the-art methods. Finally, in Section <ref>, we illustrate the strong and weak scaling properties of the multi-batch L-BFGS method. §.§ Logistic Regression In this section, we focus on logistic regression problems; the optimization problem can be stated as:min_w ∈ℝ^d F(w) =1/n∑_i=1^nlog(1+e^-y^i(w^Tx^i)) + σ/2w^2,where (x^i, y^i)_i=1^ndenote the training examples and σ = 1/n is the regularization parameter. We present numerical results that evaluate the performance of the proposed robust multi-batch L-BFGS scheme(Algorithm <ref>) in both the multi-batch (Figure <ref>) and fault tolerant (Figure <ref>) settings, on thedataset[LIBSVM: <https://www.csie.ntu.edu.tw/ cjlin/libsvmtools/datasets/binary.html>]. We compare our proposed method (Robust L-BFGS) against three methods: (i) multi-batch L-BFGS without enforcing sample consistency (L-BFGS), where gradient differences are computed using different samples, i.e., y_k = g^S_k+1_k+1 - g^S_k_k; (ii) multi-batch gradient descent (Gradient Descent), which is obtained by setting H_k = I in Algorithm 1; and, (iii) serial SGD (SGD), where at every iteration one sample is used to compute the gradient. We run each method with 10 different random seeds, and, where applicable, report results for different batch (r) and overlap (o) sizes. In Figures <ref> and <ref> we show the evolution of the norm of the gradient in terms of epochs.In the multi-batch setting, the proposed method is more stable than the standard L-BFGS method; this is especially noticeable when r is small. On the other hand, serial SGD achieves similar accuracy as the robust L-BFGS method and at a similar rate, at the cost of n communications per epoch versus 1/r(1-o) communications per epoch. Figure <ref> also indicates that the robust L-BFGS method is not too sensitive to the size of the overlap. Similar behavior was observed on other datasets, in regimes where r· o was not too small; see <cit.>.Figure<ref> shows a comparison of the proposed robust multi-batch L-BFGS method and the multi-batch L-BFGS method that does not enforce sample consistency (L-BFGS) in the presence of faults. In these experiments, p denotes the probability that a single node (MPI process) will not return a gradient evaluated on local data within a given time budget. We illustrate the performance of the methods for α=0.1 and p∈{0.1, 0.3, 0.5}. We observe that the robust implementation is not affected much by the failure probability p. Similar behavior was observed on other datasets; see <cit.>.§.§ Neural Networks In this section, we study the performance of the multi-batch L-BFGS method[Code available at: <https://github.com/OptMLGroup/Multi-Batch_L-BFGS>.] on Neural Network tasks, on the MNIST and CIFAR10/CIFAR100 datasets[MNIST available at: <http://yann.lecun.com/exdb/mnist/>. CIFAR10/CIFAR100 available at: <https://www.cs.toronto.edu/ kriz/cifar.html>.]. Table <ref> summarizes the network architectures that we used; see <cit.>for more details. The first problem is convex; all other problems are nonconvex. We implemented our algorithm in PyTorch <cit.> and compare against popular and readily available algorithms: (i) SGD <cit.>, and (ii)Adam <cit.>. We denote our proposed method as LBFGS in the figures in this section. Note, we implemented our method with the cautious updating strategy, and For each method, we conducted a grid search to find the best learning rate α∈{2^0,2^-1,…,2^-10}, and also investigated the effect of different batch sizes | S | ∈{50, 100, 200, 500, 1000, 2000, 4000 }; see <cit.> for detailed experiments with all batch sizes. For the multi-batch L-BFGS method we also investigated the effect of history length m ∈{1,2,5,10,20 }. The overlap used in our proposed method was 20% of the batch, o = 0.20.The authors in <cit.> observed that the widely used Barzilai-Borwein-type scaling s_k^T y_k/y_k^T y_k I of the initial Hessian approximation may lead to quasi-Newton updates that are not stable when small batch sizes are employed, especially for deep neural training tasks, and as such propose an Agadrad-like scaling of the initial BFGS matrix. To obviate this instability, we implement a variant of the multi-batch L-BFGS method (LBFGS2) in which we scale the initial Hessian approximations as α I. We ran experiments with both scaling strategies and the overall results were similar. Therefore, in the figures in this section we only show results for the latter strategy. Figure <ref> illustrates the evolution of the running maximum of the training accuracy (dashed lines) and testing accuracy (solid lines) for the best parameter settings for each method over the first 20 epochs of training. By best parameter settings we mean the run that achieved the highest training accuracy within 20 epochs. One can make several observations. First, it appears that the multi-batch L-BFGS method is competitive with the first-order methods on all training problems, except for CIFAR10 LeNet, in terms of both training and testing accuracy. Second, for half of the problems (three out of six), the best runs of the multi-batch L-BFGS method use larger batch sizes than the first-order methods. Of course, this benefit is not as clear on the neural network training problems as it is in the logistic regression problems.Third, the benefits of incorporating second-order information are not as apparent in these nonconvex problems as compared to the problems presented in Section <ref>. We attribute this to two things: (1) nonconvex problems are hard, and (2) quasi-Newton methods are better are capturing curvature information for convex problems.We now investigate the effect of the batch size. In Figure <ref> we show the evolution of the running maximum of the training/testing accuracy for different batch sizes | S | ∈{50, 500, 4000 } for three of the problems. For a complete set of results, see <cit.>.Overall, one can observe that for small batch sizes, the multi-batch L-BFGS variants perform worse than the first order methods. However, when large batches are employed (a regime that is favorable foe GPU computing), the multi-batch L-BFGS method performs on par with the other methods. Moreover, it appears that on several problems the performance of the multi-batch L-BFGS method is less affected by the size of the batch, i.e., the variability in the final training and testing error (after 20 epochs) in terms of batch size is smaller for the multi-batch L-BFGS method than for the stochastic first-order methods; see also <cit.>. In order to understand why the multi-batch L-BFGS method does not perform well for small batches, we looked at two diagnostic measures: (i) the angle between the true gradient curvature vector y_d and subsampled gradient curvature vector y_s (⟨ y_s,y_d ⟩/ y_s y_d); and (ii) the ratio of subsampled gradient curvature vector to true gradient curvature vector (y_s/y_d). These measures indicate how informative the curvature information captured by the multi-batch L-BFGS method really is. Values close to 1 (dashed red lines) are ideal for both measures. We chose 3 different points (the starting point, a point after 3 epochs of Adam, and a point after 10 epochs of Adam). From those points we took a gradient descent step with sufficiently small step length, and computed the true gradient curvature vector (y_d). We also computed 100 different stochastic variants of the gradient curvature vector (y_s) using different batch sizes (| S | ∈{50, 100, 200, 500, 1000, 2000, 4000 }), and calculated the values of the two metrics. We illustrate the results in Figure <ref>; see <cit.>for more results. Several observations can be drawn from this figure. First, not surprisingly, the metrics improve (get closer to 1) as the batch size increases. Second, for the convex case, the metrics perform as expected both close and far from the solution; as a result (sufficiently) good curvature information is captured and the method performs well. On the other hand, for the nonconvex problems, the metrics indicate that, especially for small batch sizes, the curvature information captured can be terrible. r0.5< g r a p h i c s >Relative slow-down of computational time to compute gradients for different batch-sizes compared to the computational time to compute gradients with batch-size 1.We should note that using a large batch size is not a bottleneck for current high-performance computing hardware, on the contrary, using small batch sizes leads to under-utilization of computational hardware and in fact hinders the ability to parallelize the methods. Figure <ref> illustrates this phenomenon; we show the average computational time of a gradient evaluation (over 1,000 evaluations) for different batch sizes ranging from 1 to 4096 relative to the computational time of the computation of the gradient using a single data point. The computation was performed on an NVIDIA Tesla K80 GPU using PyTorch. It is clear that for the Multiclass Linear classification task, the compute time of a single gradient is roughly the same as the compute time of a gradient based on a batch of size 1024, whereas for the larger training tasks, the compute time of the gradients appear constant up to batch sizes of roughly 128. We shouldnote however that their is a risk of decreased generalization when increasing the batch size, unless other strategies such as modifying the step size or regularization are used; see e.g., <cit.>. §.§ Scaling of the Multi-Batch L-BFGS ImplementationIn this section, we study the strong and weak scaling properties of the robust multi-batch L-BFGS method on artificial data. For various values of batch size (r) and nodes (K), we measure the time needed to compute a gradient (Gradient) and the time needed to compute and communicate the gradient (Gradient+C), as well as, the time needed to compute the L-BFGS direction (L-BFGS) and the associated communication overhead (L-BFGS+C). The function of which we are computing the gradient is logistic regression. The L-BFGS direction is computed using the Vector-Free L-BFGS implementation <cit.>. We should note that the time to compute the gradient, which of course is required for computing the L-BFGS direction, is not included in L-BFGS and L-BFGS+C. We report the extra time to compute the L-BFGS step, after having computed the gradient. Thus, the goal of this section is to show that the time needed to compute the L-BFGS direction is insignificant compared to the cost of computing the gradient, which is needed in any case to run first-order methods. §.§.§ Strong Scaling r0.4< g r a p h i c s >Strong scaling of robust multi-batch L-BFGS on a problem with artificial data;n=10^7 and d=10^4. Each sample has 160 non-zero elements (dataset size 24GB). Figure <ref> depicts the strong scaling properties of the multi-batch L-BFGS method, for different batch sizes (r) and nodes (K=1,2,...,128). For this task, we generate a dataset with n=10^7 samples and d=10^4 dimensions, where each sample has 160 randomly chosen non-zero elements (dataset size 24GB). One can observe that as the number of nodes (K) is increased, the compute times for the gradient and the L-BFGS direction decrease. However, when communication time is considered, the combined cost increases slightly as K is increased. Notice that for large K, even when r=10% (i.e., 10% of all samples processed in one iteration, ∼18MB of data), the amount of local work is not sufficient to overcome the communication cost. §.§.§ Weak Scaling –Fixed Problem Dimension, Increasing Data Size r0.4< g r a p h i c s >Weak scaling of robust multi-batch L-BFGS on a problem with artificial data; n=10^7 and d=10^4. Each sample has 10· K non-zero elements (size of local problem 1.5GB). In order to illustrate the weak scaling properties of the algorithm, we generate a data-matrix X ∈ℝ^n × d (n=10^7, d = 10^4), and compute the gradient and the L-BFGS direction on a shared cluster with different number of MPI processes (K=1,2,...,128). Each sample has 10· K non-zero elements, thus for any K the size of local problem is roughly 1.5GB (for K=128 size of data 192GB). Effectively, the dataset size (n) is held fixed, but the sparsity of the data decreases as more MPI processes are used. The compute time for the gradient is almost constant, this is because the amount of work per MPI process (rank) is almost identical; see Figure <ref>. On the other hand, because we are using a Vector-Free L-BFGS implementation <cit.> for computing the L-BFGS direction, the amount of time needed for each node to compute the L-BFGS direction decreases as K is increased. However, increasing K does lead to larger communication overhead, and as such the overall time needed to compute and communicate the L-BFGS direction increases slightly as K is increased. For K=128 (192GB of data) and r=10%, almost 20GB of data are processed per iteration in less than 0.1 seconds, which implies that one epoch would take around 1 second. §.§.§ Increasing Problem Dimension, Fixed Data Size and K r0.4< g r a p h i c s >Scaling of robust multi-batch L-BFGS on a problem with artificial data; n=10^7, increasing d and K=8 MPI processes. Each sample had 200 non-zero elements (dataset size 29GB). In this experiment, we investigate the effect of a change in the dimension (d) of the problem on the computation of the gradient and the L-BFGS direction. We fix the size of data (29GB) and the number of MPI processes (K=8), and generate data with n=10^7 samples, where each sample has 200 non-zero elements. Figure <ref> shows that increasing the dimension d has a mild effect on the computation time of the gradient, while the effect on the time needed to compute the L-BFGS direction is more apparent. However, if communication time is taken into consideration, the time required for the gradient computation and the L-BFGS direction computation increase as d is increased. We should note that the results presented in Figure <ref> are not surprising; there is minimal change in performance (in terms of the gradient computation) as dimension increases, since the number of nonzero elements is fixed and sparse matrix operations are emplyed.§ FINAL REMARKS In this paper, we assumed that sample consistency is not possible (fault-tolerant setting) or desirable (multi-batch setting), and described a novel and robust variant of the L-BFGS method designed to deal with two adversarial situations. The success of the algorithm relies on the fact that gradient differences need not be computed on the full batch, rather a small subset can be used alleviating the need for double function evaluations while still maintaining useful curvature information. The method enforces a small degree of control in the sampling process and avoids the pitfalls of using inconsistent gradient differences by performing quasi-Newton updating on the overlap between consecutive samples. Our numerical results indicate that provided the overlap is not too small, the proposed method is efficient in practice on machine learning tasks such as binary classification logistic regression and neural network training. The experiments presented in this paper show that the empirical performance of the method matches that predicted by the theory for both strongly convex and nonconvex functions. Specifically, in the strongly convex case the multi-batch L-BFGS method with a constant step length converges to a neighborhood of the solution at a linear rate, and in the nonconvex case the iterates produced by the multi-batch L-BFGS method converge to a neighborhood of a stationary point.Of course, the development, both theoretical and practical, of stochastic quasi-Newton methods is far from complete, and there are many interesting directions that can and should be investigated. Theoretical analysis that would suggest the batch size and overlap size would be of great interest in practice. Moreover, an investigation of the multi-batch L-BFGS method that employs variance reduced gradients in lieu of the stochastic gradients could have both theoretical and practical advantages. Finally, a stochastic line search that could work in conjunction with the multi-batch L-BFGS method would be novel both algorithmically and theoretically, and would most probably make the method even more competitive in practice. §.§.§ AcknowledgementsThis work was partially supported by DARPA Lagrange award HR-001117S0039 and U.S. National Science Foundation, under award numbers NSF:CCF:1618717, NSF:CMMI:1663256 and NSF:CCF:1740796. gOMS § EXTENDED NUMERICAL RESULTS - REAL DATASETS - LOGISTIC REGRESSIONIn this section, we present further numerical results on binary classification logistic regression problems, on the datasets listed in Table <ref>, in both the multi-batch and fault-tolerant settings. Note, that some of the datasets are too small, and thus, there is no reason to run them on a distributed platform; however, we include them as they are part of the standard benchmarking datasets. We focus on logistic regression classification; the objective function is given by min_w ∈ℝ^d F(w) =1/n∑_i=1^nlog(1+e^-y^i(w^Tx^i)) + σ/2w^2,where (x^i, y^i)_i=1^ndenote the training examples and σ = 1/n is the regularization parameter. §.§ Extended Numerical Results - Multi-Batch Setting For the experiments in this section (Figures <ref>-<ref>), we ran four methods: * (Robust L-BFGS) robust multi-batch L-BFGS (Algorithm <ref>), * (L-BFGS) multi-batch L-BFGS without enforcing sample consistency; gradient differences are computed using different samples, i.e., y_k = g_k+1^S_k+1-g_k^S_k, * (Gradient Descent) multi-batch gradient descent; obtained by setting H_k = I in Algorithm <ref>, * (SGD) serial SGD; at every iteration one sample is used to compute the gradient.In Figures <ref>-<ref> weshow the evolution of ∇ F(w) for different step lengths α, and for various batch (| S| = r· n) and overlap (| O| = o ·| S|) sizes. Each Figure (<ref>-<ref>) consists of 10 plots that illustrate the performance of the methods with the following parameters: * Top 3 plots: α=1, o=20% and r=1%,5%,10% * Middle 3 plots: α=0.1, o=20% and r=1%,5%,10% * Bottom 4 plots: α=1, r=1% and o=5%,10%,20%,30%§.§ Extended Numerical Results - Fault-Tolerant SettingIf we run a distributed algorithm, for example on a shared computer cluster, then we may experiencedelays. Such delays can be caused by other processes running on the same compute node, node failures and/or for other reasons. As a result, given a computational (time) budget, these delays may cause nodes to fail to return a value. To illustrate this behavior, and to motivate the robust fault-tolerant L-BFGS method, we run a simple benchmark MPI code on two different environments: * Amazon EC2 – Amazon EC2 is a cloud system provided by Amazon. It is expected that if load balancing is done properly, the execution time will have small noise; however, the network and communication can still be an issue. (4 MPI processes)* Shared Cluster – On our shared cluster, multiple jobs run on each node, with some jobs being more demanding than others. Even though each node has 16 cores, the amount of resources each job can utilize changes over time. In terms of communication, we have a GigaBit network. (11 MPI processes, running on 11 nodes)We run a simple code on the cloud/cluster, with MPI communication. We generate two matrices A,B ∈R^n × n, thensynchronize all MPI processes and compute C=A· B using the GSL C-BLAS library. The time is measured and recorded as computational time. After the matrix product is computed, the result is sent to the master/root node using asynchronous communication, and the time required is recorded. The process is repeated 3000 times. The results of the experiment described above are captured in Figure <ref>. As expected, on the Amazon EC2 cloud, the matrix-matrix multiplication takes roughly the same time for all replications and the noise in communication is relatively small. In this example the cost of communication is negligible when compared to the cost of computation. On our shared cluster, one cannot guarantee that all resources are exclusively used for a specific process, and thus, the computation and communication time is considerably more stochastic and unbalanced. In some cases, the difference between the minimum and maximum computation and communication times vary by an order of magnitude or more. Hence, on such a platform a fault-tolerant algorithm that only uses information from nodes that return an update within a preallocated budget is a natural choice.In Figures<ref>-<ref> we present a comparison of the proposed robust multi-batch L-BFGS method and the multi-batch L-BFGS method that does not enforce sample consistency (L-BFGS). In these experiments, p denotes the probability that a single node (MPI process) will not return a gradient evaluated on local data within a given time budget. We illustrate the performance of the methods for α=0.1 and p∈{0.1, 0.2, 0.3, 0.4, 0.5}. We observe that the robust implementation is not affected much by the failure probability p. § EXTENDED NUMERICAL RESULTS - NEURAL NETWORKSIn this section, we present complete numerical results for the Neural Network training tasks on the MNIST[Available at: <http://yann.lecun.com/exdb/mnist/>.] and CIFAR10/CIFAR100[Available at: <https://www.cs.toronto.edu/ kriz/cifar.html>.] datasets. More specifically, we show the training and testing accuracy of the methods for all batch sizes (| S | ∈{50, 100, 200, 500, 1000, 2000, 4000 }). The tasks are summarized in Table <ref>, and the details are explained below. * MNIST Convex: This problem was a convex problem. It is a simple neural network with no hidden layers and soft-max cross-entropy loss function. * MNIST DNN (SoftPlus): This network consists of two convolutional layers and two fully connected layer. The first convolutional layer has 32 filters for each 5 × 5 patch, and the second convolutional layer has 64 filters for each 5 × 5 patch. Every convolutional layer is followed by a 2 × 2 max pooling layer. The first fully connected layer has 1024 neurons and the second fully connected layer produces a 10 dimensional output. We used SoftPlus activation functions. The loss function is soft-max cross-entropy loss. * MNIST DNN (ReLU): Same as above, but with ReLU activation functions. * CIFAR10 LeNet: This network consists of two convolutional layers and three fully connected layer. The first convolutional layer has 6 filters for each 5 × 5 patch, and the second convolutional layer has 16 filters for each 5 × 5 patch. Every convolutional layer is followed by a 2 × 2 max pooling layer. The first fully connected layer has 120 neurons and the second fully connected layer has 84 neurons. The last fully connected layer produces a 10 dimensional output. The activation functions used is ReLUand the loss function used is soft-max cross-entropy loss. * CIFAR10 VGG11 and CIFAR100 VGG11: This is standard VGG11 network which contains 8 convolution layers, each followed by batch-normalization and ReLU activation functions. There are also 5 max-pooling layers and one average pooling layer. The output of desired dimension (10 or 100) is achieve by fully connected layer. The loss function is soft-max cross-entropy loss. For the experiments in this section (Figures <ref> and <ref>), we ran the following methods: * (LFBGS) multi-batch L-BFGS with the cautious strategy: Algorithm <ref> with standard initial scaling (γ_k I, where γ_k = s_k-1^T y_k-1/y_k-1^T y_k-1) or α I initial scaling, * (Adam) <cit.>, * (SGD) <cit.>. Figures <ref> and <ref> show the evolution of training and testing accuracy for different batch sizes for the different neural network training tasks. Figures <ref> and <ref> show the two diagnostic metrics for different batch sizes. | http://arxiv.org/abs/1707.08552v3 | {
"authors": [
"Albert S. Berahas",
"Martin Takáč"
],
"categories": [
"math.OC",
"cs.LG",
"stat.ML"
],
"primary_category": "math.OC",
"published": "20170726173343",
"title": "A Robust Multi-Batch L-BFGS Method for Machine Learning"
} |
^1Faculdade de Engenharia de Guaratinguetá, Universidade Estadual Paulista, 12516-410, Guaratinguetá, SP, Brazil ^2Escola de Engenharia de Lorena, Universidade de São Paulo, 12602-810, Lorena, SP, Brazil ^3Instituto de Física, Universidade de São Paulo, Caixa Postal 66318, 05389-970 São Paulo, SP, Brazil ^4Instituto de Física, Universidade Federal do Rio de Janeiro, Caixa Postal 68528, 21941-972 Rio de Janeiro, RJ, Brazil ^5Instituto de Física, Universidade Federal Fluminense, 24210-346, Niterói, RJ, Brazil In this work, a hydrodynamic study of the di-hadron azimuthal correlations for the Au+Au collisions at 200 GeV is carried out. The correlations are evaluated using the ZYAM method for the centrality windows as well as the transverse momentum range in accordance with the existing data. Event-plane dependence of the correlation is obtained after the subtraction of contributions from the most dominant harmonic coefficients. In particular, the contribution from the triangular flow, v_3, is removed from the proper correlations following the procedure implemented by the STAR collaboration. The resultant structure observed in the correlations was sometimes attributed to the mini-jet dynamics, but the present calculations show that a pure hydrodynamic model gives a reasonable agreement with the main feature of the published data. A brief discussion on the physical content of the present findings is presented.Event-plane dependent di-hadron correlations with harmonic v_n subtraction in a hydrodynamic model Takeshi Kodama^4,5 Dec. 17, 2017 ==================================================================================================§ INTRODUCTION Measurements on the two-particle correlations in the relativistic heavy-ion collisions, expressed in terms of the pseudorapidity difference Δη and the angular spacing Δϕ, were carried out by various experimental collaborations <cit.> at both RHIC and LHC. The shape of the two-particle correlations for different collision systems, at various transverse momentum range and its evolution as a function of trigger particle azimuthal angle have been both extensively studied. They are understood to provide relevant information on the jets originating from small momentum transfer scatterings as well as the hot, dense medium created in the collisions <cit.>. The observed correlation yields are characterized by an enhancement on the near side around Δϕ≈ 0, known as the “ridge", which possesses a long Δη extension in the longitudinal direction. Besides heavy-ion collisions, such ridge structures were also observed in pp <cit.> and pA <cit.> collisions at LHC. For pA collisions, it is found that the ridge yields vary with centrality. The measured jet-like yields obtained by subtracting long-range pseudorapidity correlations observed in high-multiplicity events, on the other hand, are approximately constant for different centralities <cit.>. This observed feature shows that the physics behind jet-like yield and ridge yield are indeed distinct, being the latter attributed to the collective flow of the system. The correlation on the away side is found to be more significant in AA collisions than in pp and pA, and it presents a double-peak structure, usually called “shoulders", which evolves continuously from the double peak for central to one peak for peripheral collisions <cit.>.In order to interpret the “ridge" and “shoulders" in AA collisions by a uniform picture, we proposed the so-called peripheral-tube model <cit.>. In this model, the phenomenon is attributed to the local (nonlinear) behavior of hydrodynamics. The phenomenon can also be explained regarding the triangular flow, as usually done <cit.>. It is understood that hydrodynamical evolution transforms the spatial inhomogeneity of participating nucleons in the initial conditions into the momentum anisotropy of the observed hadrons <cit.>. The triangular flow in the one-particle distribution function generates three peaks in the two-particle correlations: one peak on the near side at Δϕ = 0 and two others at Δϕ = 2π/3 and Δϕ = 4π/3 corresponding to the double peak on the away-side. It is noted that studies by using the AMPT model were carried out <cit.> which showed that the double peak disappears when the contributions due to the elliptic flow and triangular flow are subtracted. The above finding seems to indicate that the triangular flow indeed plays an essential role in the observed structure on the away side. The two-particle correlations are also investigated as a function of the trigger angle ϕ_s, known as the event-plane dependence of the two-particle correlations <cit.>. It was found that the away side structure evolves from only one peak in the in-plane trigger direction with ϕ_s=0 to double peaks at the out-of-plane trigger direction with ϕ_s=π/2. It is worth noting that the above data were extracted by mainly subtracting the background contributions of the elliptic flow. The observed features of the data can be understood in terms of a hydrodynamic interpretation known as peripheral one-tube model <cit.> where geometric fluctuations are manifested as high-energy tubes randomly distributed in the initial conditions. The model <cit.> is also able to explain the observed centrality dependence of the away side structure in the correlations <cit.>.Based on analysis of the earlier STAR data <cit.>, Luzum pointed out <cit.> that the observed two-particle correlations are consistent with being entirely generated by the collective flow. In the work, the author analyzed the elliptic flow coefficients v_2^(a), v_2^(t,R) and showed that the flow background used in ZYAM subtraction was probably underestimated, and the non-flow contribution of the second Fourier coefficient is likely insignificant. However, it is not clear that the above reasoning also applies to the third Fourier coefficient, especially because no direct measurement of v_3 were made in STAR's analysis <cit.>. This is because V_3Δ was found to be weakly dependent on ϕ_s. Even if V_3Δ, the third Fourier coefficient of the proper correlations, and v_3^(a) v_3^(t,R) have roughly the same dependence on ϕ_s provided they are not identical,it is more likely that the subtraction was correct and the triangular non-flow signal is mostly independent of the trigger angle. In other words, one does not know for sure that the non-flow contribution of the third Fourier coefficient is also insignificant. Moreover, in our previous work, it is shown that <cit.> V_3Δ is slightly dependent on ϕ_s while v_3 remains constant. Such feature is consistent with the STAR data (see Fig.1 of ref.<cit.>), and is understood as owing to an interplay between the multiplicity fluctuations and the background flow. The above considerations, therefore, strengthen the role played by the triangular flow. Recently, such a study was carried out by the STAR collaboration <cit.>. In their work, in order to eliminate the contributions from the collective flow, the background includes those of the elliptic, triangular, and quadrangular flow. It is somewhat surprising to find that resultant correlation yields still maintain the same feature on the away side: it shows one peak in the in-plane trigger direction and double peaks in the out-of-plane trigger direction. In other words, after the subtraction of the contributions from the flow coefficients, some non-trivial structure remains. As the STAR experiment claims, it may be attributed to the pathlength-dependent jet-quenching. What we understand by ZYAM is that it is a method to single out the relevant signal in the resultant correlations, by subtraction of the background evaluated in terms of the average flow harmonics, from the proper correlations including the information on event-by-event fluctuating flow.Whether the method is good or not, we don't know.But the initial fluctuations should remain in the resultant correlations. What can be drawn from the similarity shown in the two STAR papers <cit.> is the following.The contribution of the average triangular flow ⟨ v_3 ⟩ seems to be much smaller than ⟨ v_2 ⟩.Also, because of the different |Δη| restrictions applied in the data analysis, the contribution from the jet should be either very small or insensitive to any |Δη| cut. In this context, it is worthwhile to carry out an explicit calculation to verify whether the data presented in <cit.> can be reproduced by a hydrodynamic approach.The above discussions motivated the present study. Here, we present results on event plane dependence of the di-hadron correlations by using NeXSPheRIO <cit.>, an ideal hydrodynamic model, with event-by-event fluctuating initial conditions. Since all the parameters of the model are fixed by reproducing the observed multiplicity yields <cit.>, there is no free parameter in the present calculation. We then implement the same flow background due to the elliptic, triangular and quadrangular flows as the STAR collaboration did, and compare the calculated correlations with the data <cit.>. As a confirmation, we also evaluate and present the correlations by a subtraction only of the elliptic and quadrangular flows, in comparison with the data <cit.>. The rest of the paper is organized as follows. In Section II, we show the results of our hydrodynamical simulations together with discussions, and the concluding remarks are given in Section III.§ EVENT PLANE DEPENDENCE OF DI-HADRON CORRELATIONS The event-plane dependence analysis evaluates the correlated di-hadron pairs as a function of the azimuthal angle difference Δϕ at different trigger angles ϕ_s=| ϕ_trigger-Ψ_2 |, which is measured with respect to the event plane of the second harmonic coefficient. In accordance with the experimental data <cit.>, we study the collisions in the 20 - 60% centrality window. The transverse momentum range of the trigger particles is chosen to be 3 ≤ p_T^t≤ 4GeV, and that of the associated particles is 1 ≤ p_T^a≤ 2GeV. To accommodate the experimental setup of the STAR collaboration, the calculations are carried out in the pseudorapidity interval | η |< 1. Then the results are divided into six equally sized slices in the azimuthal angle of the trigger particles. To calculate the event plane, we make use of all charged particles within the transverse momentum range p_T < 2Gev/c. By carrying out the modified reaction-plane (MRP) method <cit.> as employed by the STAR collaboration, particle pairs with | Δη | < 0.5 are excluded from the construction of the event planes. Correlated pairs with | Δη | < 0.7 between the trigger particles and the associated particles are also excluded from the analysis because the original purpose was to minimize the near-side jet contribution <cit.>. The ZYAM method is then used to construct the correlation pattern due to the anisotropic flow. The primary flow correlated background is from the elliptic flow, caused by the average almond shape of the superposition region of the initial energy distribution, the triangular flow, caused by the fluctuations of the initial conditions that occur on an event-by-event basis <cit.>, as well as the quadrangular flow. The above flow harmonics contribute to the two-particle correlations and are to be subtracted from the proper correlation pattern. In <cit.>, the flow correlated background from the elliptic, triangular as well as quadrangular flow are expressed as followsdN/dΔϕ = B (1+2v_2^av_2^tcos2Δϕ+2v_3^av_3^tcos3Δϕ+2v_4^a{Ψ_2} v_4^t{Ψ_2}cos4Δϕ)where B is the background normalization, v_2^a, v_4^a{Ψ_2} (v_2^t and v_4^t{Ψ_2}) are the second and fourth harmonic coefficients of the associated (trigger) particles with respect to the event plane of the second harmonic Ψ_2, v_3^t and v_3^a are triangular flow of the trigger and the associated particles, calculated with respect to the event plane of the third harmonic Ψ_3. The harmonic coefficients of the trigger particle are the average value obtained in the respective slice of the azimuthal angle ϕ_s.In the calculations using NeXSPheRIO, the hydrodynamic simulations are carried out in smaller centrality bins of 10% each. The reason for such a division is to increase the statistics for more peripheral collisions, as we generate a total of 1300 events in the 20 - 30% centrality window, 2400 events in the 30 - 40% centrality windows, 4400 events in the of 40 - 50% and 4600 events in the 50 - 60% centrality windows. To further increase the statistics, for each event with given initial conditions, the Monte Carlo hadron generator is invoked 200 times. The obtained correlations of individual bins are then averaged to obtain the desired proper correlation. To evaluate the background correlation, the harmonic coefficients are obtained by the event-plane method <cit.>. Subsequently, the ZYAM method is made use of and is implemented according to Eq.(<ref>). The resultant correlations are shown in Fig.<ref>. In the plots, the solid purple curves are those obtained by NeXSPheRIO and the data are represented by the red stars whereas the gray areas between the solid lines indicate the uncertainties. Notice that, even though the contribution of the triangular flow is explicitly subtracted from the proper correlation as shown in (<ref>), the resultant correlation is still featured by one peak in the away-side for the in-plane direction, with its maximum at Δϕ≈π, which evolves to double peaks for the out-of-plane direction. Remark that similar feature had been reported by STAR collaboration in <cit.> where only the elliptic and quadrangular flows were subtracted.As a comparison, we proceed to evaluate the two-particle correlation with the same specifications of <cit.>. We note that the experimental configurations, such as the pseudo-rapidity cut of |Δη| > 0.7 for the counting of particle pairs as well as the momentum ranges in consideration, are exactly the same in this case, and the differences come from the pseudo-rapidity filters in the construction of event planes. Therefore the same data sets generated previously by the hydrodynamic simulations are used, while the event plane Ψ_2 is evaluated by taking into consideration all the particles while implementing a cut in pseudo-rapidity |Δη| < 0.35. Both v_2 and v_4 are calculated with respected to the event plane Ψ_2. Now, the correlated flow backgound is estimated asdN/dΔϕ = B (1+2v_2^av_2^tcos2Δϕ+2v_4^a{Ψ_2} v_4^t{Ψ_2}cos4Δϕ)The resulting two-particle correlations are shown in Fig.(<ref>), compared with the STAR data from Ref.<cit.>. It is found that the double peak observed on the away side in the out-of-plane direction is also reasonably reproduced by the hydrodynamic calculations, which is consistent with our previous results <cit.> obtained by using the cumulant method. It is somewhat surprising to find that the subtracted two-particle correlations in Fig.<ref> and <ref> are not very different. This is because the triangular flow is understood to play a significant role in the explanation of the double-peak on the away side. Consequently, it was somehow expected that the subtraction of v_3 might lead to the disappearance of the same on the away side. In order to quantitatively understand the contribution of v_3 in our hydrodynamical results, we present in Tables <ref> and <ref> the calculated values of flow harmonics used in the ZYAM subtraction to produce Fig.<ref>.We note that the data analysis carried out in this work is not for the most central window, and therefore, the average values of v_3 for the associated particles are smaller in comparison with those of v_2, as shown in Table <ref>. This is also the case for the trigger particles at the in-plane directions, as can be readily verified in the first few columns of Table <ref>. One also observes that the magnitude of v_4{Ψ} for the associated particles follows the same trend of that of v_2. However, since its size is much smaller than that of v_2, it does not play a significant role in the resultant correlations. On the other hand, for the trigger particles, as one goes to the out-of-plane directions, the elliptic flow v_2^t starts to decrease and eventually becomes negative, as confirmed by the three rightmost columns in Table <ref>. This is expected, since v_2^t is evaluated with respect to the event plane Ψ_2 of the associated particles, so this is implied by definition. Owing to the change of the sign of v_2, the contribution of v_3 in the background Eq.(<ref>) is relatively small in the in-plane direction in comparison to that in the out-of-plane direction. On the other hand, for the calculated correlations, the appearance of the double peak in the out-of-plane direction indicated that the observed modulation of the resultant correlation is dominated by the second order harmonic coefficient V_2≡ v_2^av_2^t in the in-plane direction and the remnant of V_3≡ v_3^av_3^t in the out-of-plane direction. By Eq.(<ref>), this is consistent with the previous observations on the magnitudes of v_2 and v_3.In Ref.<cit.>, it is discussed that the trends of the away-side correlation might underscore the importance of path-length-dependent jet-medium interactions. As the present hydrodynamical simulations are able to reproduce observed feature of two-particle correlation, it strongly indicates that the observed correlations are likely to be a collective-flow effect of the system. Moreover, it seems that the ZYAM procedure, devised to essentially subtract the contribution of collective flow from the proper two-particle correlation, has somehow failed in its purpose. In particular, it is found that the subtraction of v_3 does not affect the essential feature of the resultant correlations, namely, the relative magnitudes between V_2 and V_3. To us, this might be related to the event-by-event fluctuating initial conditions and their impact on flow harmonics. This is because the subtracted v_n in Eq.(<ref>) is, in fact, the event average value, ⟨ v_n ⟩. Due to the event-by-event fluctuations, the event average value of a product of harmonic coefficients can be significantly different from the product of the corresponding average values. In other words, not only the magnitude of the triangular flow, v_3, is understood to be related to the event-by-event fluctuations, its fluctuations might also play a non-trivial role in the particle correlations. Moreover, the event planes between different harmonics might be correlated for a given event but uncorrelated among the various events, which further complicates the problem. As a result, the average of the Fourier expansion in azimuthal angle cannot be simply approximated by a Fourier expansion in terms of the products of average harmonic coefficients, even rescaled by the ZYAM scheme. The present calculations employing NeXSPheRIO give reasonable results for two different sets of data obtained by different procedures. It implies that the fluctuating initial conditions generated by NeXuS are mostly realistic. We also note that this topic is closely related to the correlation between different flow harmonics, as recently explored by several authors <cit.>.It is noted that in the above calculations, the event-plane method is employed for the background subtraction, as a part of our strategy to closely follow the same steps in the data analysis. However, the event-plane method carried out in a hydrodynamic study might not be equivalent to that in experiment analysis. According to the calculations by the Monte Carlo Glauber model <cit.>, the obtained v_2 depends on the resolution R of the event-plane method. As long as the resolution is high enough, the calculated result gives the desired mean value of v_n. However, as the resolution decreases, the obtained result gradually approaches the RMS value of corresponding harmonic coefficients. Since in a hydrodynamic calculation, the resolution is typically better than that of experimental measurements, the current calculations likely underestimate the flow background. In practice, the resolution of a hydrodynamic approach can be controlled by the number of Monte Carlo simulations during the hadronization phase. One may therefore roughly estimate the discrepancy owing to the resolution, by extrapolating the present results to the corresponding resolution of the data by varying the number of Monte Carlo simulations. We have carried out the analysis, and it shows that the flow harmonics used in the flow background subtraction is around 10% smaller than those employed in STAR analysis. Since the magnitudes of flow harmonics shown in Tables <ref> and <ref> are small, the deviations of their contributions in Eqs.(<ref>) and (<ref>) are even less. As a result, the values of B in Eqs.(<ref>) and (<ref>) should be slightly smaller but mostly unchanged. We have verified that the resultant correlation after the flow background subtraction will be slightly smaller, but the qualitative results shown in Fig.<ref> and <ref> remain unchanged.§ CONCLUDING REMARKS As discussed above, we understand that ZYAM is a method aimed to single out relevant signal in the resultant correlations, by subtraction of the flow background evaluated in terms of the average harmonic coefficients, from the proper correlations including the event-by-event fluctuating flow. To us, it seems possible that the initial fluctuations still present, or even become dominant in the resultant correlations. The similarity presented in the two STAR papers <cit.> indicates that the flow is fluctuating and ⟨ v_3 ⟩≪⟨ v_2 ⟩. Also, owing to the different |Δη| restrictions applied in the data analysis, the jet contribution is likely to be very insignificant. In this work, we show by explicit calculations that hydrodynamics, with fluctuating initial conditions, is able to reproduce the observed double-peak structure of two-particle correlation on the away-side for the out-of-plane triggers, even when the triangular flow is subtracted by using ZYAM method. Therefore, the present study further strengthens the idea that observed correlations are mostly of hydrodynamic origin.From a hydrodynamic viewpoint, it is understood that the role of v_3 is closely associated with the event-by-event geometrical fluctuations in the initial conditions. The existence of two peaks, in the away side correlation, clearly shows the need of v_3. Our results indicate that the observed event plane dependence of two-particle correlations, for the most part, can be reproduced by a hydrodynamic approach. The physical mechanism behind the findings may be attributed to the correlation and/or fluctuation of flow harmonics, which is closely associated with the event-by-event fluctuations in the initial conditions. This is because of the assumption of the background correlation module, Eq.(<ref>), assumes the picture of average flow with no correlation neither fluctuation of significant consequence. Ongoing efforts on the event-plane correlations, symmetric cumulant<cit.> may likely provide a better insight of the problem, and a clear criterion to distinguish between different mechanisms. Some alternative methods may also be meaningful, as shown in our recent work concerning centrality dependence of di-hadron correlations <cit.>. There, the cumulant method gives quite a similar result as the ZYAM one. We note that the 2+1 correlation <cit.> employed by STAR Collaboration may also serve as a good tool to disentangle the jet signal.On the other hand, the peripheral tube model <cit.>, in which the interplay between random hot tubes in the initial conditions and collective background flow plays an important role, gives a unified description of the “ridge” structure, both for the near-side and the away-side ones. As a result, in this interpretation, different flow harmonics of an individual event are naturally “born" together, and consequently correlated, their appearance does not depend on any global structure of the initial condition. From the viewpoint of the peripheral tube model, it is possible that the correlation between v_2 and v_3 owing to the flow deflected by high energy tube may provide an intuitive explanation of the observed data. In particular, the lack of the near-side peak for the out-of-plane triggers, which would be produced by this two-peak structure in the single-particle distribution, might be studied analytically in this framework. Recently, a Reaction Plane Fit method was proposed <cit.> and employed to estimate the correlation functions in the background dominated region on the near-side <cit.>. The resulting correlation does not show the double peak on the away side, neither any dramatic shape modification as a function of centrality. Therefore, the authors conclude that the Mach cone is an artifact of the background subtraction and the jets do not fully equilibrate with the medium. These results further indicate that the effect of the jet in the di-hadron correlation is indeed a subtle subject. We plan to explore these topics further and look for a possible criterion to distinguish between different approaches in the near future. § ACKNOWLEDGMENTSWe are thankful for valuable discussions with Matthew Luzum and Jiangyong Jia. We gratefully acknowledge the financial support from Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP), Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ), Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), and Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES). A part of the work was developed under the project INCTFNA Proc. No. 464898/2014-5. This research is also supported by the Center for Scientific Computing (NCC/GridUNESP) of the São Paulo State University (UNESP).h-physrev | http://arxiv.org/abs/1707.09878v2 | {
"authors": [
"Wagner M. Castilho",
"Wei-Liang Qian",
"Yogiro Hama",
"Takeshi Kodama"
],
"categories": [
"nucl-th"
],
"primary_category": "nucl-th",
"published": "20170726145458",
"title": "Event-plane dependent di-hadron correlations with harmonic $v_n$ subtraction in a hydrodynamic model"
} |
[ Tobias Zentel^1, Viviane Overbeck^2, Dirk Michalik^2, Oliver Kühn^1, Ralf Ludwig^2 December 30, 2023 ======================================================================================This paper presents an augmentation of MSCOCO dataset where speech is added to image and text. Speech captions are generated using text-to-speech (TTS) synthesis resulting in 616,767 spoken captions (more than 600h) paired with images. Disfluencies and speed perturbation are added to the signal in order to sound more natural. Each speech signal (WAV) is paired with a JSON file containing exact timecode for each word/syllable/phoneme in the spoken caption. Such a corpus could be used for Language and Vision (LaVi) tasks including speech input or output instead of text.Investigating multimodal learning schemes for unsupervised speech pattern discovery is also possible with this corpus, as demonstrated by a preliminary study conducted on a subset of the corpus (10h, 10k spoken captions).§ INTRODUCTION During the past few years, there has been an increasing interest in research gathering language and vision (LaVi) communities. This trend can be explained by the availability of multimodal corpora such as Flickr30k <cit.>or MSCOCO <cit.>, containing images and their captions in natural language text. LaVi systems also have benefited from the introduction of neural encoder-decoder approaches <cit.> that allow text to be generated from images or speech (or vice-versa), or to learnjoint embeddings of images and text <cit.>.After the pioneering work done at MIT 15 years ago <cit.>, these recent advances have also fueled fundamental research in (grounded) language acquisition and understanding (linguistics, cognitive science, autonomous robotics) as well as in more applied research to solve tasks such as image captioning <cit.> or visual question answering <cit.>.While many benchmark image datasets have been designed to assess the quality of machine-generated image descriptions (for a survey see <cit.>), speech modalityhas been less focused on, probably because of the lack of large corpora for conducting such studies. In fact, except the augmentation of Flickr8k dataset (by collecting a corpus of 40,000 spoken captions using Amazon Mechanical Turk - AMT) proposed by <cit.> which lead to the following research papers <cit.>,very few researches on spoken language and vision were conducted so far. Paper contributions. This paper proposes a dataset that is an order of magnitude bigger than what is already available (600k spoken captions instead of 40k in<cit.>). Our datasetis built on top of MSCOCO <cit.> and contains more than 600,000spoken captions (we will refer to it as SPEECH-COCO in the rest of this paper). We believe that this corpus may be useful for further Language and Vision (LaVi) tasks including speech input or output instead of text. The data set is made available online[<https://zenodo.org/record/4282267>]. Moreover, speech data is perfectly annotated (does not rely on error-prone forced alignment) since the annotations are produced during the TTS process itself (at word, syllable and phone level). Paper outline. This paper is organized as following. In section 2, we quickly review existing corpora usable for visually grounded language acquisition and motivates the choice of MSCOCO as a starting point. In section 3, we present our general methodology and describe the spoken caption generation itself while section 4 presents some analysis on SPEECH-COCO as well as its first use for an unsupervised word discovery task. Finally, section 5 concludes this work and gives some perspectives. § EXISTING CORPORA OF VISUALLY GROUNDED SPEECH Flickr30K <cit.> (an extension of Flickr8K <cit.>) and MSCOCO <cit.> are now widely used in LaVi community. We can also mention the Visual Genome dataset, an ongoing effort to connect structured image concepts to language <cit.>. For a broader survey on image captioning datasets, the reader may refer to <cit.>. Flickr8K and Flickr30K contain images from Flickr withapproximately 8,000 and 30,000 images, respectively. The images in these two datasets were selected through user queries for specific objects and actions. Both contain five descriptions per image which were collected using a crowdsourcing platform (AMT).The MSCOCO dataset currently consists of 123,287 images with five different descriptions per image. Images in this dataset are annotated for 80 object categories and bounding boxes around all instances in one of these categories are available for all images. The MSCOCO dataset has been widely used for automatic image captioning. Several extensions of MSCOCO are currently under development, including the addition of questions&answersfor Visual Question Answering (VQA) task[<http://www.visualqa.org>] or FOIL-COCO (Find One mismatch between Image and Language caption) which consists of images associated with incorrect captions to challenge existing LaVi models[<https://foilunitn.github.io>]. Concerning spoken captions, AMT recordings were obtained from Flickr8K by <cit.> but only 40k captions are made available online[<https://groups.csail.mit.edu/sls/downloads/flickraudio/>]. In addition, time-coded annotations are obtained after alignment of speech data with transcripts through (error-prone) automatic forced-alignment. Very recently, spoken captions for MSCOCO were generated using Google TTS by <cit.>. However, only one TTS voice was used limiting the speaker variability. § SPEECH-COCO: ADDING SPOKEN CAPTIONS TO MSCOCO §.§ Our Starting Point:MSCOCO We used Microsoft's Common Objects in Context (MSCOCO) <cit.> training and validation datasets as our starting point. In MSCOCO, each image is described by at least five descriptions written by humans. The images contain 91 common object categories (e.g. dog, elephant, bird, car, bicycle, air plane, etc.) from 11 super-categories (Animal, Vehicle, etc.), with 82 of them having more than 5K labelled instances. In total there are 123,287 images with captions (82,783 for training and 40,504 for validation). The test set is not available for download since it is used on an evaluation server for continuous benchmarking of image captioning systems. Consequently, 616,767 captions from 123,287 images are available for download. §.§ Spoken Captions Generation Synthetic speech was generated for each caption using Voxygen[<https://www.voxygen.fr>], a commercial speech synthesis system, for 4 different UK voices (Paul, Elizabeth, Judith and Bronwen) and 4 different US voices (Phil, Bruce, Amanda and Jenny). It is important to note that this is corpus-basedconcatenative speech synthesis <cit.> and not parametric synthesis. So, for each speaker's voice, speech utterances are generated by concatenation of units mined from a large speech corpus (generally around 3000 sentences/speaker). This means that despite having little intra-speaker variability in our speech data, there is a realistic level of inter-speaker variability, as opposed to the small corpus proposed in <cit.>.On average, each caption comprises 10.79 tokens. The WAV files are on average 3.52 second long. §.§ Adding Variability First, we applied speed perturbation on the spoken captions to introduce intra-speaker variability in the dataset. For this, we used the tempo function of Sox[<http://sox.sourceforge.net/sox.html>] audio manipulation tool. This simple processing only changes speech rate while trying to keep the same pitch and spectral enveloppe. Our recordings were either accelerated (speed x 1.1), slowed down (speed x 0.9) or kept unchanged with equal probability of event.Second, we added disfluencies to some of the captions so that they would sound more natural. We chose to only include fillers (such as "um", "uh", "er", "huh", "oh" and "ah"). It was shown that fillers often occur when speakers have to face an heavy processing load (when describing an image for example) and that they also act as coordinators <cit.>. Thus, adding fillers to some captions is legitimate and makes the corpus more realistic. The probability of adding a filler to any given caption was set to 0.3. The fillers were either added at the beginning, at the end or in the middle of a caption with equal probability of event.§ SPOKEN CORPUS ANALYSIS §.§ Metadata and scripts We adopted the following naming convention for both the WAV and JSON files: imageID_captionID_Speaker_DisfluencyPosition_SpeedEach WAV file is paired with a JSON file containing various information: timecode of each word in the caption, name of the speaker, name of the WAV file, etc. The JSON files have the following data structure: [basicstyle=88] "duration": float, "speaker": string, "synthesisedCaption": string, "timecode": list, "speed": float, "wavFilename": string, "captionID": int, "imgID": int, "disfluency": list We also created a Python script[<https://github.com/William-N-Havard/SpeechCoco>] which handles the metadata, so that the user can easily make use of the corpus. The script has the following features: * Aggregate all the information in the JSON files into a single SQLite database * Find captions according to specific filters (name, gender and nationality of the speaker, disfluency position, speed, duration, and words in the caption). The script automatically builds the SQLite query. The user can also provide his own SQLite query. * Find all the captions belonging to a specific image * Parse the timecodes and have them structured * Get the words, syllables and phonemes timecodes * Convert the timecodes to Praat TextGrid files (see Figure <ref>) §.§ Analysis of intra and inter speaker variability To be sure that our corpus is realistic enough, we computed inter and intra speaker variability measures on a subset of 100 captions corresponding to 20 images and added 2 additional voices to our 8 synthetic voices for these captions: a real voice (William) and Google TTS (gTTS). We then extracted words that were repeated across sentences and analyzed their intra and inter speaker variability using Dynamic Time Warping (DTW) computed for alloccurrences of the same word by the same speaker (e.g. Amanda vs Amanda) and for all the occurrences of the same word pronounced by different speakers (e.g Amanda vs. Bronwen, Amanda vs. Bruce, etc.).As expected, intra speaker variability is lower than inter-speaker variability but we remark that there is not big differences between intra speaker variability of a real voice (William) and the intra speaker variability of a synthetic voice (Amanda for instance). Inter speaker variability is slightly greater between the real human voice (William) and the other synthetic voices, but inter speaker variability betweensynthetic voices is still high, meaning our synthetic corpus should be difficult enough for tasks such as unsupervised term discovery (UTD) accross a collection of speech utterances from multiple speakers. This is what we intend to show in a preliminary experiment, in the next subsection. §.§ Unsupervised Term Discovery (UTD) For UTD, we use the Zero Resource Toolkit (ZRTools <cit.>). ZRTools uses segmental dynamic time warping (SDTW) to discover pairs of acoustically similar audio segments, and then uses graph clustering on overlapping pairs to form a hard clustering of the discovered segments. Replacing each discovered segment with its unique cluster label, or pseudoterm, gives us a partial, noisy transcription, or pseudotext.To evaluate UTD performance, we used the term discovery evaluation toolkit (TDE) <cit.>. UTD was done on a subset of the corpus (10,000 captions which represent approximately 10 hours of speech) and the results are reported in Table <ref>. Our results confirm that the UTD task is still difficult even if our corpus is synthetic: we found very few gold tokens (low precision and even lower recall[TDE only gives three significant figures after the decimal point. Since our results are very low, there were rounded up to 0.000]). There is very little difference between the global precision and the within-speaker precision for the matching task, suggesting that the clusters mainly consist of speech segments belonging to the same speaker. Within-speaker clusters also seem to be purer than the global clusters since the within-speaker precision for the grouping task is higher than the global precision. Grouping recall indicates that speech segments which should have been in the same clusters have been assigned to different clusters. Finally, the low coverage and matching recall clearly indicate that a lot of work has to be done in order to get the most out of the available data and that SDTW alone might not be enough to fully segment speech utterances. For instance, modeling prosodic features could be of great help. To summarize, this experiment with an off-the-shelf toolkit, shows that our synthetic corpus of spoken captions is difficult enough for UTD task. Table <ref> shows examples of clusters found by the UTD system along with the corresponding images of MSCOCO. As shown in Table <ref>, segments rarely match single tokens. However, Table <ref> shows that they often match frequent n-grams ("a man riding", "fire hydrant" and "skiing down") and multiword expressions (MWE) such as "a bunch of bananas". As such, adding a new metric to TDE that would assess the quality of the segmentation according to gold chunks and MWE would be interesting.§ JAPANESE TRANSLATIONS A further augmentation of MSCOCO was created by <cit.> where Japanese captions were collected using the same methodology as <cit.>. This corpus comprises 131,740 captions for 26,500 images. As the augmentation created by <cit.> does not provided a Japanese version for all of the captions, we used Machine Translation (MT)[Using both Excite and Google's MT systems] to translate all the available English captions to Japanese. This allows SPEECH-COCO to be used for cross-modal studies using speech, images, text and translations.§ CONCLUSION In this paper, we have presented SPEECH-COCO, an extension of the MSCOCO image recognition and captioning dataset, which consists of more than 604 hours of speech. The addition of speech as a new modality enables SPEECH-COCO to be used in different fields of research including language acquisition, visually-grounded word discovery and keyword spotting, and semantic embedding using speech and vision.This corpus has been used during the Jelinek Memorial Workshop on Speech and Language Technology (JSALT) 2017 jointly organized by CMU and JHU (Speaking Rosetta Stone team ondiscovering grounded linguistic units for languages without orthography).We list below a list of topics that are as many LaVi tasks possible with SPEECH-COCO: * spoken caption generation from images (generating speech from images), * visually-grounded spoken term discovery, * cross-modal speech-image-text studies, * image generation from speech input, * simulating language documentation tasks where speech has been elicited from images, * effect of visual context on computational language acquisition, * (spoken) visual question answering, * data augmentation by adding synthetic multi-modal datato more naturalistic (but small) corpora. § ACKNOWLEDGEMENTS We thank Voxygen for providing us their TTS server with UK and US English voices which allowed us to generate all our spoken captions. We thank Hideto Kazawa from Google for providing automatic translations into Japanese. IEEEtran | http://arxiv.org/abs/1707.08435v5 | {
"authors": [
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"Laurent Besacier",
"Olivier Rosec"
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"published": "20170726134021",
"title": "SPEECH-COCO: 600k Visually Grounded Spoken Captions Aligned to MSCOCO Data Set"
} |
§ INTRODUCTION In the latest few years the Galactic Center has been the object of an intense high-energy observational campaign.The High Energy Stereoscopic System (H.E.S.S.) collaboration recently reported the discovery of aγ-ray diffuse emission from a small region surrounding SgrA*<cit.>. The emission spectrum is compatible with a single power-law with index Γ_ GC = 2.32 ± 0.05_ stat± 0.11_ sys and extends up to ∼ 50 with no statistically significant evidence of a cutoff.A γ-ray diffuse emission was also measured byH.E.S.S. from the larger Galactic Ridge (GR) region <cit.>, roughly corresponding to the central molecular zone (CMZ) – a massive structure rich in molecular gas that extends up to ∼ 250 away from the GC along the Galactic plane (GP).That measurement was very recently updated using 250 hours of data and improved analysis techniques <cit.>. The angular distribution of the events approximately traces that of the gas, apart from the innermost region where a more peaked emission seems to be present.The spectrum in the whole ridge extends up to ∼ 45 and is compatible with a single power law with indexΓ_ ridge =2.28 ± 0.03_ stat± 0.2_ sys, which is in agreement with the slope measured in the inner region surrounding SgrA*.Assuming that emission to be hadronic, as expected due to the strong losses suffered by electrons in that region, the inferred spectrum of the primary protons is significantly harder than the local CR spectrum measured at the Earth position (Γ_ CR(r_⊙) ≃2.7 for E_ CR > 300 / nucleon.This has been often interpreted as an evidence of a freshly accelerated cosmic-ray (CR) population in that region possibly originated by the supermassive SgrA* black hole or by an intense starburst activity <cit.>. Here we explore a different scenario: In our interpretation the largest part of the gamma-ray diffuse emission from the Galactic ridge is originated by the diffuse, steady-state Galactic CR sea interacting with the massive molecular clouds in the CMZ under the assumption that in that region its spectrum is harder than the local one. This approach is motivated by recent analyses of Fermi-LAT data <cit.> showing that the γ-ray diffuse emission of the Galaxy, and hence the CR primary spectrum, becomes progressively harder approaching the GC along the GP. We will use here PASS8 Fermi-LATdata to extend down to few GeV the measurement of the γ-ray diffuse emission spectrum in the CMZ and SgrA* surrounding, showing that this behavior continues down to the inner ∼ 100.Following <cit.> we interpret this behavior in terms of a radial dependence of the scaling of the CR diffusion coefficient with rigidity. We will use the same scenario to compute the CR sea distribution in the GC region and the γ-ray diffuse emission produced by their interaction with the dense gas in that region and compare those predictions with Fermi-LAT and H.E.S.S. data.§ H.E.S.S. AND FERMI-LAT DATA In the first part of this section we report the γ-ray diffuse emission spectrum from 5 GeV up to 50 TeV in several part of the CMZ region determined from the combination of Fermi-LAT and H.E.S.S. data. We extract Fermi-LAT data using the Fermi Science Tools v10r0p5 <cit.>. We use 470 weeks of PASS8 data with the event class CLEAN and we apply the recommended quality cuts: (DATAQUAL==1) && (LATCONFIG==1).The exposure is computed using the Fermi-LAT response function P8REP2CLEANV6. Here the data are binned in 20 energy bins equally spaced in log scale between 300 MeV and 300 GeV. We subtracted the emission due to the point source obtained from the 4-year Point Source Catalog (3FGL) provided by the Fermi-LAT collaboration <cit.>. We consider three regions contained in the CMZ complex for which the H.E.S.S. collaboration released the diffuse emission spectra:* the Galactic ridge(GR) which in Galactic coordinates is defined by the window(| l | <1^∘, | b | < 0.3^∘) and almost include the whole CMZ;* the so called pacman, i.e.an open annulus centered on SgrA* with θ_ inner = 0.15^∘and θ_ outer = 0.45^∘;* the Sgr B gas complex (0.4 < l < 0.9, -0.3 < b < 0.2) which is located at a mean distance of ∼ 100 from the GC. For each of these regions we report the spectra data point in Fig.s<ref>,<ref>,<ref> respectively. Interestingly, the emission from all these regions can be matched by a single power law with index ∼ 2.4 and same normalization. More precisely the best fit spectrum in the ridge region for 5 < E < 5 is Φ_GR = (1.19 ± 0.04) × 10^-5(E_γ/1 )^-2.42 ± 0.02( ^2 )^-1. This finding suggests that a single, almost uniform, emission process may be responsible for most of the emission of the whole CMZ. The agreement between the Fermi-LAT and H.E.S.S. flux normalization is remarkable taking in mind the very different techniques adopted by those experiments.To further investigate the nature of the emission, and similarly to what done from the H.E.S.S. collaboration we use the angular distribution of Fermi-LAT data to infer the radial distribution of the primary CR energy density w_ CR.Using the Fermi tools we extract the diffuse luminosity L_γ(E_γ≥ 10 ) in an annulus and in six adjacent circular regions with angular diameter of 0.2^∘ centered on the plane intersecting SgrA* (see Fig. <ref>). These regions are larger than those considered by H.E.S.S., which is motivated by the smaller angular resolution of Fermi-LAT. To determine the gas mass distribution we use the same CS column density map adopted by the H.E.S.S. collaboration <cit.>. Accounting for the energy dependence of the pion production cross-section we get the following relation between w_ CR and the γ-ray luminosity:w_ CR(E_ CR≥ 0.1 ) = 3.9 × 10^-2 ^-3 (η_N/1.5)^-1 (L_γ(≥ 10 )/10^34 /) (M_ gas/10^6 M_⊙)^-1 .Here L_γ(≥ E_γ) is the γ-ray luminosity above E_γ in each region (subtracting the contribution from point sources); M_ gas is the corresponding total hydrogen mass; η_N ≈ 1.5 is a factor accounting for the presence of heavier nuclei.The resulting CR energy density radial profile w_ CR(r) in the energy range 0.1 ≤ E_CR≤ 0.3 is reported in Fig. <ref>, as well as the CR distribution derived by the H.E.S.S. collaboration in <cit.> for E_CR≥ 10 . Within the large errors and data scatter, both data sets are compatible with a constant CR density for r100.At lower radii, however, a density peak centered on the GC seems to be present, although this is more evident in the H.E.S.S. data set. § THE EMISSION DUE TO THE CR SEA In this section we compare our previous results with the diffuse emission due to the interaction of the CR large scale distribution, the CR sea,with the gas in the CMZ. For conventional models, which assume the CR shape in the whole Galaxy to be the same as that measured at the Earth,the emission was estimated to be considerably smaller and flatter than H.E.S.S. finding.Interestingly, below 10 GeV those models are in good agreement with Fermi-LAT data (see e.g. <cit.>), hencean excess was found only at very high energies.This motivated several authors to attribute the excess to a hard CR component freshly accelerated by one or more sources in the GC region. Here we consider an alternative scenario in which the Galactic CR spectrum, hence the secondary γ-ray diffuse emission, shows a progressive hardening at low Galactocentric radii.In particular, following <cit.>, we reproduce this behavior adopting a CR propagation model featuring a radially-dependent scaling of the diffusion coefficient on the particle rigidity δ.The scenario, implemented in the DRAGON code <cit.>, assumes that δ has a linear dependence on the Galactocentric radius (r): δ(r) = A r + B. The parameters A and B were tuned to consistently reproduce local CR and Fermi-LAT γ-ray data on the whole sky. With respect to the model considered in <cit.> (KRA_γ), the gamma reference model considered hereadopts a spectral hardening in the proton and Helium source spectra at ∼ 300 / n, in order to reproduce the local propagated spectra measured by PAMELA <cit.>, AMS-02 <cit.> and CREAM <cit.>.We assume this feature to be present in the whole Galaxy, as it may be expected if it is produced by propagation effects.Under these conditions, the KRA_γ model was shown <cit.> to reproduce the emission observed by Milagro in the inner GP at a 15 TeV median energy <cit.> consistent with Fermi-LAT data.We compute the π^0, Inverse-Compton and bremsstrahlung components of the γ-ray diffuse emission, integrating the convolution of the spatially-dependent CR spectrum, gas/radiation density distributions and proper cross-sections along the line-of-sight. The π^0 component is dominant in the GC region.With respect to what reported in <cit.>, here we replace the hydrogen distribution in the inner 3 with the 3-dimensional analytical model presented in <cit.>, as required to properly model the hadronic emission in that region.Outside that region we adopt the gas model used in <cit.>. The main components are molecular (H_2) and atomic (HI) and hydrogen. HI, which is inferred from 21-cm lines, is less than 10% of the total mass.Since H_2 is not observed directly, the column density must be inferred from proper tracers, most commonly from the CO emission lines. Here we use a conversion factor X_ CO(r ∼ 0) ≃ 0.6 × 10^20 ^-2 K^-1 ^-1, the value giving the best agreement with the integrated mass distribution, based on the CS emission map, used in <cit.>. Concerning the CR source distribution by default we use the one reported in <cit.> based on supernova remnant catalogs. This parametrization vanishes at the GC, a behavior in qualitative agreement with the γ-ray emissivity profile determined by the Fermi-LAT collaboration <cit.>, which displays a dip in the GC. We verified that using the source distribution reported in <cit.>, which does not vanish at the GC, turns into a factor ∼ 2 larger emission from the GR and pacman regions. This is still compatible with data and, moreover, may be compensated by a reduction of the X_ CO factor within the allowed observational uncertainty. In the Fig.s<ref>,<ref>,<ref> we show, against the experimental data, the γ-ray diffuse emission spectra due to the CR Galactic sea interaction with the ISM in the Galactic ridge and pacman and SgrB regions respectively.For comparison, besides the prediction of our gamma model we also report the spectra computed for a conventional model (base model), sharing with the former all the properties but keeping the diffusion coefficient spatially uniform.As already well known, from those figures it is evident that the CR sea computed for the base model–as any other conventional model– cannot consistently account for the H.E.S.S. and Fermi-LAT measurements in the absence of an additional component with a harder spectrum.The main novelty of our work is that this conclusion does not hold for models accounting for the radial gradient of the CR spectrum. As the reader can see from our figures, the gamma model is in excellent agreement, both in shape and normalization, with those data in the GR and SgrB regions (the latter data set however was taken in a previous H.E.S.S. observational campaign and display large errors).Noticeably, the H.E.S.S. data in the ridge region (which were released after the first publication of our results) nicely sit on the model prediction.A small, almost energy independent, deficit with respect to the data is present in the very inner pacman region only. This finding is consistent with what inferred from the CR energy density radial profile w_ CR, shown in Fig. <ref> which, compared to the CR sea (almost uniform on scales of few hundred parsecs), displays a peak toward the GC. This feature shows up both in the energy range probed by H.E.S.S. and (less significantly) by Fermi-LAT. It could be an artifact due to a gas density underestimation in that region (with respect respect to that inferred by the CO and CS emission maps): It should indeed be noticed that, for gas densities exceeding 10^4 ^-3, as expected in that region, the CS and CO emissions should be partially absorbed leading to an underestimation of the gas mass.If real, the CR density peak at SgrA* position could be originated by one or more sources close to the GC, which are likely to be the same responsible for the J1745-290 emission observed by H.E.S.S. extending up to 10 TeV.Although above 100 TeV that emission might be attenuated due to the presence of a dense radiation field around SgrA*, our results show that above that energy the evidence of an excess with respect to the Galactic background is rather small.§ CONCLUSIONS We have shown that the diffuse γ-ray emission from the CMZ measured by H.E.S.S. and Fermi-LAT from few GeV up to 50 TeV can be originated by the interaction of the diffuse Galactic CR population with the dense gas present in that region: This implies that a PeVatron at the GC is not required to explain H.E.S.S. data. Differently from conventional CR propagation scenarios, we adopted a model based on spatially-dependent diffusion designed to reproduce the radial gradient of the CR spectral index inferred by Fermi-LAT data.Therefore, our present results provide a new strong evidence supporting the validity of that setup in a region of the Galaxy were the discrepancies between models featuring standard and radially-dependent diffusion are expected to be maximal. As far as the physical interpretation of this scenario is concerned, we mention that a similar behavior can naturally arise within the framework of anisotropic CR diffusion, given a realistic configuration of the Galactic magnetic field that accounts for a poloidal component in the inner Galaxy <cit.>.The observed radial trend of the proton spectral index was reproduced under those conditions with the DRAGON 2 code described in <cit.>. 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"D. Gaggero",
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"A. Marinelli",
"M. Taoso",
"A. Urbano",
"S. Ventura"
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"published": "20170726144445",
"title": "Hard Cosmic Ray Sea in the Galactic Center: a consistent interpretation of H.E.S.S. and Fermi-LAT $γ$-ray data"
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We discuss the design and construction of a novel target array of nonmagnetic test masses used in a neutron polarimetry measurement made in search for new possible exotic spin dependent neutron-atom interactions of Nature at sub-mm length scales. This target was designed to accept and efficiently transmit a transversely polarized slow neutron beam through a series of long open parallel slots bounded by flat rectangular plates. These openings possessed equal atom density gradients normal to the slots from the flat test masses with dimensions optimized to achieve maximum sensitivity to an exotic spin-dependent interaction from vector boson exchanges with ranges in the mm - μm regime. The parallel slots were oriented differently in four quadrants that can be rotated about the neutron beam axis in discrete 90^∘ increments using a Geneva drive. The spin rotation signals from the 4 quadrants were measured using a segmented neutron ion chamber to suppress possible systematic errors from stray magnetic fields in the target region. We discuss the per-neutron sensitivity of the target to the exotic interaction, the design constraints, the potential sources of systematic errors which could be present in this design, and our estimate of the achievable sensitivity using this method. Finite semisimple group algebra of a normally monomial group Shalini Gupta [Corresponding author] Department of Mathematics, Punjabi University, Patiala,India. email: [email protected] Sugandha Maheshwary [Research supported by SERB, India (PDF/2016/000731).] Department of Mathematical Sciences, Indian Institute of Science Education and Research, Mohali, Sector 81, Mohali (Punjab)-140306, India. email: [email protected] ===========================================================================================================================================================================================================================================================================================================================================================================================================Over the last decade a growing number of experiments have sought new interactions of Nature with weak couplings and force ranges at the mm - μm scale. Such exotic interactions might arise from string theory, from pseudo - Goldstone bosons generated by spontaneous symmetry breaking at high energy scales, from the as - yet - unknown physics behind dark energy, etc. A detailed review on the state of this developing subfield can be found in <cit.>. A general classification of the potentials that can exist between nonrelativistic fermions (protons, neutrons, and electrons in our case) from the nonrelativistic limit of a single spin 0 or spin 1 boson exchange assuming only rotational invariance <cit.> uncovered 16 different operator structures at first order involving the spins, momenta, interaction range, and various possible couplings of the particles. This is a small enough number of distinct possibilities that it has motivated many experimentalists to design specific experiments to look for each type. The simple Yukawa interaction of range λ_c is the only spin - independent one on the list. The rest contain Yukawa terms which set the distance scale for the interaction but also depend on the spins of one or both of the fermions.The mass m_0 of the exchange boson is related to the range λ of the Yukawa component of the potential through the usual relation m_0 = ħ/λ c, where ħ is the reduced Planck constant and c is the speed of light in vacuum. Low energy neutrons are a particularly useful probe for new possible spin dependent interactions at the mm - μm scale, which could be mediated by a spin - 1 boson whose mass m_0 is related to this distance scale λ_c through the relation m_0 = ħ/λ_c c, where ħ is the reduced Planck constant and c is the speed of light in vacuum. Slow neutrons can be polarized with high efficiency and manipulated with delicate precision to conduct sensitive interferometric measurements of many types. Such measurements can be used to place limits on the strength of new possible exotic couplings between ordinary matter.We sought to measure the possible neutron-atom axial vector coupling given below in (<ref>): V_5=g_A^2ħ^2/4π m_ne^- r/λ/r( 1/r+1/λ_c) σ⃗·(v⃗/c×r⃗/r) . In this expression g_A is the axial coupling constant, r is the distance between the neutron and the atom, v is the relative velocity, σ is the neutron spin, and m_n is the neutron mass. This interaction potential induces a rotation of the spin about the v⃗×r⃗ axis in a way similar to a magnetic field. One can express the effect of this potential in terms of a pseudomagnetic field, which if integrated over a semi-infinite plane is given by B_AA=1/γ_ng^2_A/4 N ħ c/m_nc^2λ_c (v⃗×ŷ ) e^-Δ y / λ , where λ_c is the range of the force and the y direction is normal to the face of the semi-infinite slab. The first attempt to search for this exotic neutron-atom axial vector coupling at interaction length scales below 1cm was made by Florian M. Piegsa and Guillaume Pignol at the Paul Scherrer Institute in Switzerland <cit.>. In their experiment the phase shift from this exotic interaction was sought by comparing the neutron phase shift difference from two parallel subbeams at a series of different distances from the same source (a copper plate in their case) and with the source placed close to one subbeam, with the other subbeam serving as a reference. The measurement used the well - known Ramsey method of separated oscillatory fields and was therefore a quantum interference experiment with a phase shift proportional to the integral of the exotic interaction energy between the neutron and the atoms in the slab. In neither subbeam do the neutrons touch the test mass surface. This approach renders the measurement insensitive to possible sources of systematic error from magnetic field drifts, thermal drifts which change the apparatus geometry, and the magnetic susceptibility of the test mass and has the advantage that it can fit for the known distance dependence of the exotic interaction. Possible systematic errors from magnetic impurities in the sample can be dealt with by measurements of the residual magnetic field from the sample. The sub-mm distance scales of interest for the exotic interaction search of interest dictates the choice of various aspects of the geometry. The number of neutrons used in the pioneering Piegsa and Pignol experiment was small and limited by their choice of a single planar source mass. We sought to improve upon this experiment byincreasing the total number of neutrons used to probe the possible spin dependent interaction and to employ polarized neutron spin rotation as the measurement method rather than Ramsey spectroscopy. The brightness of the slow neutron sources available for scientific research at national neutron facilities are all about the same within an order of magnitude or so as that of the PSI source used in their experiment, but many of the beams possess a much larger cross sectional area (as large as 10cm × 10cm) than that used by Piegsa and Pignol. However, we cannot take advantage of this full intensity as the potential we are trying to measure has a range of sub-mm scale. Since increasing the width of the beam necessarily increases the average distance between the neutron and slab, doing so quickly reduces the sensitivity to V_5 in the sub-mm range where it is poorly constrained experimentally.Therefore rather than employing a single slab of material as the V_5 source, we designed a source target which consisted of an array of target slabs designed to maximize a possible V_5 signal. Our multi-slotted target design is an attempt to make efficient use of the large cross sectional area neutron beams which are available and still be able to access possible exotic effects of an interaction with sub-mm range. This mismatch of length scales immediately suggests a parallel multi-slotted target design. As this choice precludes the ability to vary the relative distance between source and subbeams as done by Piegsa and Pignol, we instead choose to surround each neutron subbeam through the target with masses of different densities (glass and copper in this case) and rotate the target so that the sign of the neutron phase shift from the exotic interaction is reversed. This allows us to conduct the search for this exotic interaction in the form of an asymmetry measurement by rotating the target mechanically. In place of the reference beam used by Piegsa and Pignol which is ideally far enough away from the source to feel no exotic interaction, we instead use a set of slots in a different quadrant of the target which are rotated by 90^∘ with respect to the neutron polarization direction. As can be seen from the form of the interaction, this different orientation gives a zero contribution to the exotic interaction after one averages over the sources. For one of the 4 orientations of the target one therefore gets a null phase shift in two diagonal quadrants and a positive and negative phase shift in the other two diagonal quadrants. One 90^∘ rotation of the target with respect to the beam reverses the positions of the low and high density sources with respect to the neutrons so that, in the absence of systematics, the phase shifts in the zero diagonal quadrants are still zero while the phase shifts in the other pair of diagonal quadrants are reversed. Four 90^∘ rotations of the target cycle all of the test masses through all quadrants of the beam to ensure that there are no effects from target nonuniformities. In the reminder of this paper we briefly discuss the neutron spin rotation measurement method, the details of the target design used in the experiment, the potential sources of systematic error inherent in this method, and the prospects for improving on the sensitivity of the V_5 search.§ NEUTRON SPIN ROTATION METHOD We used a neutron spin rotation polarimeter to search for this interaction, employing a neutron polarimeter described in detail in <cit.> and shown in Figure <ref>. This polarimeter is designed to accept a vertically polarized slow neutron beam and measure the horizontal component of the polarization that can result from a net rotation of the plane of polarization of the neutrons along the neutron beam axis in the magnetically-shielded sample region. This component of the neutron polarization is isolated by a combination of a precession of the neutron spin about a vertical axis in the sample region using an appropriately-tuned magnetic field combined with adiabatic spin transport in a magnetic field of oscillating helicity between the sample region and the neutron polarization analyzer. An earlier version of this polarimeter was used to search for parity violation in neutron spin rotation in ^4He <cit.>. The null result from this experiment was later used to constrain possible exotic parity-odd interactions of the neutron <cit.> and polarized neutron couplings to in-matter gravitational torsion <cit.> and in-matter nonmetricity <cit.>. A slightly modified procedure described below was implemented to search for the neutron spin state change which would be caused by V_5. When a beam of polarized neutrons sent parallel to the surface of a flat slab of nonmagnetic material is subject to the effect of the potential in (<ref>), a rotation of the expectation value of the neutron spin about an axis perpendicular to both the spin and velocity is induced. As mentioned above, this can be understood from nonrelativistic quantum mechanics simply by viewing the cross product term in <ref> as a pseudomagnetic field about which the spin expectation value would Larmour precess. In order to measure the asymmetry in the longitudinal polarization state of the neutron spin after it passes through near a target slab we employ a two step process (see Figure <ref>).Once the neutron has accumulated an asymmetry from V_5 we rotate the polarization state by π/2 radians about the vertical (+y) axis using a constant magnetic field. We then rotate this new polarization state by ±π/2 radians about the longitudinal axis using the adiabatic neutron spin transport field before finally arriving at a polarization analyzer which allows through only neutrons whose polarization vector is pointing along +ŷ (see Figure <ref>). By computing the difference ratio of the number of neutrons that make it through the analyzer in the + or - state of the adiabatic transport field we can measure an asymmetry which is proportional to the phase shift from the spin coupling of the neutron to the new possible light vector boson of interest. § ALTERNATING DENSITY GRADIENT SCHEMEIn order to increase the total number of neutron - atom interactions over a polarized slow neutron beam of large cross sectional area while remaining sensitive in the mesoscopic length region of scientific interest, we designed a target using multiple plates containing a large mass density gradient. The test masses must be composed of two plates of different densities. Plates of the same density would create an exact cancellation of any exotic spin - dependent interaction coupling to atom density at the center and a significant reduction near the edges due to the position vector - dependence of the sign of V_5. By using test masses with a large difference in mass density we ensure that the neutron would see a net non - zero V_5 after summing effects from all neutron trajectories between the plates. Neutron transport simulations were used to investigate the sensitivity of the apparatus to V_5. A simplified geometry of the target was inserted into an existing neutron transport code written for previous experiments with this apparatus studying parity violation in a liquid helium target. The code uses realistic neutron wavelength and angular distributions as well as neutron optical transport. Using (<ref>), the code calculated rotations for all trajectories, including multiple bounces from the target faces, to estimate the strength of the rotation signal and the statistical precision. By varying target materials, thicknesses, neutron index of refraction, and the size of the gap between target faces, a set of viable materials and appropriate configurations was obtained for the phase space of the slow neutron beam at LANSCE, which is similar to that available on most fundamental neutron physics beamlines. One of the target plates was copper while the other was float glass.The final choice of copper and float glass test masses was based on cost and availability, the low surface roughness of commercially-available materials, the reasonably-sized neutron optical potential of these materials, and on their well-known good neutron optical properties (float glass has been used as backing for neutron supermirrors for decades, and copper was one of the first materials used successfully in early neutron guides). Many of the neutrons passing through the target will bounce from the surfacesof these materials since slow neutrons bounce off sufficiently flat surfaces, therefore, increase the neutron transmission through the device. The neutron polarimeter used in this work contains a neutron ion chamber at the end of the apparatus which is split into four quadrants and can separately measure the neutron spin rotation angle from each quadrant. The final test mass slabs which fill the target are 50cm long, 6.5cm wide and 1.65mm thick. They are arranged in four quadrant regions each containing eight neutron paths separated by two plate thicknesses. The copper and float glass plates fit into 25 μm tolerance parallel grooves that extend through the target region, as shown in Figure <ref>. As a control on the size of rotations due to magnetic fields, two of the quadrants have plates oriented such that v⃗×r⃗ is in the same direction as the initial neutron spin, in our case vertical. Thus, one expects no fifth force rotation in the quadrants with vertical plates. We can compare the sensitivity of this neutron beam/target combination with that of the PSI experiment using the expression for the rotation resulting from a neutron passing through the pseudomagnetic field near a slab of material. Using (<ref>) we can find the sensitivity to rotation as a function of g^2_A,ϕ_PSI/g^2_A=N L ħ c/4 m_nc^2λ_c e^-Δ y /λ , where L is the length along which the neutron is in the pseudofield and N is the number density of the target slab. In the PSI experiment a pencil-like beam of approximately 0.3 mm^2 passed beside a 48-cm long, 1.9-cm thick copper slab. For the length scales of interest here the semi-inifinite slab approximation is sufficient. For λ_c=Δ y =0.1 cm, we find ϕ_PSI/g^2_A = 5.2 × 10^10 rad. For the target described above we can make a similar estimate where neutrons are rotated in opposite directions from different density slabs above and below as shown in Fig. <ref>. Averaging rotations along the vertical direction, 1/Y∫_0^Y dy, between semi-infinite slabs separated by distance Y, we obtainϕ_NSR/g^2_A= L ħ c/4 m_nc^2λ_c 1/Y∫^Y_0 [ N_1 e^-y /λ - N_2 e^-y /λ] dy,leading toϕ_NSR/g^2_A=L λ_c^2/Yħ c/4 m_nc^2( N_1 - N_2 ) (1 - e^-Y /λ_c). Using the values of L=50cm and Y=0.37cm, the densities for copper and glass, and λ_c=0.1cm, we find ϕ_NSR/g^2_A = 2.6 × 10^10rad, smaller than that obtained using (<ref>) by a factor of two. This approximation is less accurate at longer length scales as the targets no longer appearinfinitely thick. To obtain a more realistic value for ϕ_NSR/g^2_A in our more complicated geometry we performed Monte Carlo neutron transport simulations which use numerical solutions of the pseudofield between non-infinite slabs, include all slabs of the current target design, and incorporate the neutron phase space for FP12 at LANSCE. The simulations indicate about the same sensitivity as in the analytical calculation at λ_c=0.01cm, about a factor of three less sensitivity at λ_c=0.1cm and a factor of 8 less sensitivity at λ_c=0.3cm. For the V_5 length scales of interest in our work our target design has about an order of magnitude less sensitivity per neutron than the PSI experiment depending on the V_5 interaction range of interest. However this is more than compensated by the much larger beam size that this target design enables, which gives an improved statistical sensitivity relative to that experiment by at least a factor of 80, which is the square root of the ratio of the cross sectional areas of the beams transmitted through the target region in both cases. Furthermore this does not take into account additional potential gains in sensitivity from the possible increased neutron density in the beam compared to the PSI experiment.This design can therefore be used to perform a more sensitive search for V_5 on a large area cold neutron beam. However the additional types of systematic errors which this design can be susceptible to must be analyzed and properly suppressed to take full advantage of the greater potential statistical accuracy. We will discuss the potential sources of systematicerror below after giving a detailed description of the target design and operation. A final comparison: rotation angles seen in the statistics limited PSI experiment were in the 10^-2rad range, while the previous NSR apparatus has been shown to access rotations in the 10^-6rad range <cit.> with the expectation that the new apparatus can reach 10^-7rad. § TARGET ROTATION AND STATE DETECTIONTo reduce the effect of possible space - dependent non - uniformities in the background magnetic field as well as possible differences in target plate properties (flatness, thickness, etc.) it was crucial to have a mechanism to rotate the target in discrete, repeatable 90^∘ increments. This would allow neutrons to sample the same region of space with different plates in the same orientation so that an average can be carried out. Additionally by reversing the direction of the mass gradient from quadrant to quadrant we reverse the sign of V_5 between the 4 target states in those quadrants for which V_5 is non zero, as depicted in Figure <ref>. Thus, by combining measurements either from different target states or from diagonally opposite quadrants, we can cancel rotations due to magnetic fields while isolating the fifth-force rotations.In order to consistently reproduce the four target states between rotations about the longitudinal axis we utilized a so - called Geneva Drive mechanism which translates continuous rotation into an intermittent rotary motion like in a mechanical clock. This is done through the use of a rotating cam with a pin which engages a slotted wheel attached to the object to be rotated. The rotation of the object stops as soon as the pin disengages while the cam may continue to rotate independently before engaging another slot, providing the discrete rotation mechanism desired. The rotating cam is driven by an air motor located outside of the outermost magnetic shielding to reduce the amount of magnetic components near the target. Standard lab air at 3× 10^5Pa pressure suffices to rotate the target. The flow to the air motor is controlled by the data acquisition system (DAQ) via an analog relay actuated valve. Each cam cycle rotates the target by 45^∘ and therefore two cycles were required per target state rotation. Target state rotations took 2s to complete.To prevent the cam from over or under rotating, it is crucial that we interrupt rotation when the target reaches the desired state. Once the target spends adequate time in the state, the cam will then begin rotation in the same direction as previously to produce the next target state. The target rotation will stop in the next target state using an optical flag interruption scheme. We place an array of three slotted 3mm transmissive optical sensors below the downstream face of the target such that the slots are parallel to the target face. We fix four plastic optical flags to the target face 90^∘ apart, each with either one, two, or no holes. When the target is not in one of the four desired states infrared light is transmitted to all three receivers from their respective senders and each outputs a non-zero voltage which is sent to the DAQ. The central sensor is always blocked for all four target states and is unblocked during rotation. During the initial fractions of a second when rotation starts, the DAQ ignores the states of the flags while the flags move out of the target position. Then once light transmission to the central sensor is interrupted by the edge of the flag the DAQ waits 0.5s before stopping rotation. This wait time is long enough to allow the cam to disengage the slotted wheel on the target face but short enough to prevent reengagement from over rotation. Once stopped, data is taken until it is time to rotate to the next target state. Since each flag has a unique arrangement of through holes, the target state can be identified by the DAQ. The DAQ code checks that the registered state is the same as the intended state. If it is not, the DAQ will continue to rotate the target until the intended state is reached. This only occurrs at the beginning of a run if the previous run is stopped before the target reaches the usual starting state. § SYSTEMATIC EFFECTS The systematic effects in a search for the V_5 interaction using neutron spin rotation with the target design presented in this paper possess some similarities to the types of systematic errors which have been considered in detail in our previous analysis <cit.> of parity odd neutron spin rotation for polarized neutron transmission through matter. In both cases the main sources of systematic error come from the presence of residual magnetic fields in the target region coupled with some nonuniformity in the phase space of the neutron beam as it enters into and interacts with the target. There are two main differences in the sources of systematic errors in these two types of measurements. Whereas for parity-odd neutron spin rotation one is mainly concerned with residual longitudinal magnetic fields, for a V_5 search one must consider systematic effects from both longitudinal and transverse residual magnetic fields in the target region.In parity-odd neutron spin rotation the beam is transmitted through the target medium, and therefore target-induced nonuniformities in the neutron beam phase space may couple to residual magnetic fields and generate systematic effects. These effects tend to involve refractive neutron optics and small angle scattering, which bend the beam slightly so that it can sample a slightly different residual field in the presence of inevitable spatial gradients and yet still keep the transmitted beam within the phase space acceptance of the neutron polarization analyzer downstream.In the case of a search for V_5 the beam is transmitted through a ^4He gas atmosphere in the target. Small angle scattering, which stays within the phase space acceptance of the polarimeter, combined with internal magnetic field gradients can generate a systematic error. This systematic error is very similar in character to the liquid helium small angle scattering systematic error described in great detail in <cit.> and is very small. This systematic is below 1×10^-7rad for our assumed internal magnetic field conditions.We study other systematic effects by looking at the systematic errors which are common between the spin rotation approach to search for V_5 and the approach and implementation of the Piegsa and Pignol measurement:* The systematic error from possible magnetic impurities in the test masses is common to both approaches and can be bounded by performing magnetic measurements on the masses. In our target we searched for the presence of residual magnetic fields from the copper and glass plates using a fluxgate magnetometer. We saw no evidence for the presence of any such residual fields at the 10 μG level at a distance of 3mm from the surfaces of the plates. The resulting upper bound on systematic errors in our measurement from this effect is below 2 × 10^-6 radians.* The systematic error from the magnetic susceptibility of the test masses, which shifts the value of any residual magnetic field as the mass is moved. This can be suppressed by making the residual magnetic field in the target region as small as possible.Using magnetic shielding it is possible to keep magnetic fields below 1mG. In our target this susceptibility systematic isdifferent in size from the PSI measurement due to two main effects: * It is proportional to the difference in the magnetic susceptibilities of the two materials (copper and glass in our case) on either side of the slots, and * It is slightly larger as some of the neutrons in our case bounce from the surface of the test masses and thereby encounter a slightly larger field difference than in the PSI measurement, where the neutron beam never touched the test mass.The magnetic susceptibility of both the copper and glass test masses in our target are of the same sign and possess magnitudes that are the same to about a factor of 3 which perturb the magnetic fields at the ppm level (the susceptibility of copper is about χ=-10ppm and the susceptibility of glass is smaller). This leads to a systematic error in our apparatus of order 1 × 10^-4radians/Gauss if we make the extreme assumption that the neutrons move completely inside the matter. In fact the fraction of the target length that aneutron is inside the copper during a reflection is much smaller than this, and this potential source of systematic error is utterly negligible. * Possible systematics from magnetic field drifts are suppressed in slightly different ways in the two designs as discussed above. Both conduct simultaneous null measurements with a reference beam. In the case of our rotating target, however, the simultaneous measurements are conducted on nominally identical but physically distinct test masses. The subsequent target rotations serve to test whether or not there are any nonuniformities in the test mass properties. One 90^∘ rotation of the target with respect to the beam reverses the positions of the low and high density sources with respect to the neutrons, thus removing non-target related rotations such as from stray magnetic fields. Therefore, in the absence of systematics, the phase shifts in the zero diagonal quadrants are still zero while the phase shifts in the other pair of diagonal quadrants are reversed.In the PSI experiment the neutron beam did not touch the copper test mass. However the beam bounces off the surface of the plates in our design and so we have to consider a different set of systematic errors associated with neutron reflection. We note two potential forms of systematic error which are present in our target design but not in the PSI experiment come from the neutrons bouncing off the test mass surfaces. One comes from the fact that, due to the different neutron indices of refraction of the two test masses, the sections of the neutron phase space transmitted by the target will be very slightly different in the different target positions. This can generate a potential systematic error if these slightly different neutron beam phase space sections see different magnetic fields. This source of systematic error can be suppressed by minimizing both the absolute B fields in the target region and also B field gradients and is discussed in more detail below.Another potential source of systematic error comes from the very small changes in the neutron beam polarization from neutron spin-orbit scattering from the test masses, which contains the same operator structure as that from the V_5 interaction of interest.The effect on the polarization from neutron spin-orbit scattering is proportional to the neutron momentum transfer q, which, although nonzero for neutron optical reflection from a mirror, is quite small. It is also small because the neutron spin-orbit scattering, which is classically a velocity-dependent effect, leads to an imaginary scattering amplitude, and as the neutron-nucleus scattering amplitude is mainly real in the absence of n-A resonances the neutron spin-dependent component of the interference of the spin-orbit scattering with the potential scattering is a quadratic effect.This latter effect can be calculated using the nice formulae in <cit.> and is very small in our case: for copper the maximum analyzing power from polarized neutron/copper atom scattering which occurs for momentum transfers corresponding to the critical angle for total external reflection is less than 5 × 10^-9 and is therefore completely negligible. Although some neutrons can scatter at higher momentum transfers from the mirrors in the diffuse reflection component of the beam, the intensity of the beam in this component is small compared to the specular component and almost all of the diffusely-reflected neutrons fall well outside the phase space acceptance of the rest of the apparatus. Therefore in the other systematics effects discussed below we will concentrate only on those coming from specular reflection. To reduce the effects of common-mode magnetic field noise on the neutron spin rotation signal, we arranged the target masses such that different regions of the beam area were made sensitive to possible V_5 signals of opposite sign. By recording these rotation angles simultaneously using a segmented neutron detector we are able to remove the effect of common field noise by taking the difference of these rotation angles. To further reduce systematic effects arising from stray magnetic fields we designed the target to rotate about the beam axis which, due to the arrangement of test masses, changes the sign of V_5 in each region. This allows for a comparison of rotation angles in the same beam phase space at different times, thereby removing the effect of space-dependent background field gradients whereas the aforementioned simultaneous measurement of rotations of opposite sign would remove the effect of time-dependent background field fluctuations.The spin state changes due to longitudinal or transverse magnetic fields naturally generate different sorts of systematic effects. Longitudinal fields rotate the polarization vector along the axis of the neutron momentum (left/right rotations). Transverse magnetic fields rotate the polarization vector forward/backward along the neutron momentum and therefore directly mimic the effect of V_5.Rotations by longitudinal fields before the π/2 coil are not analyzed by the downstream polarization analyzer but they reduce the polarization product PA by a factor of cosθ_B_L. The π/2 coil turns forward/backward V_5 rotations into left/right rotations to be analyzed by the analyzing super mirror. Thus, left/right rotations from a longitudinal field after the π/2 coil add to or subtract from the desired signal and thus can cause false effects.Rotations from transverse fields after the π/2 coil reduce the PA value by cosθ_B_T.There are three main effects that can affect the size of a nonzero signal from V_5 in our setup. Nonzero rotations from fringing pseudomagnetic fields at the edges of vertical-plate quadrants can dilute the signal when subtracting horizontal and vertical quadrants.Magnetic field gradients which cause slightly different transverse and longitudinal fields in each of the four quadrants can lead to residual rotations after subtracting rotations from different quadrants.A third phenomenon which can both dilute the signal and also lead to a systematic effect comes from cross-over neutrons which pass through the target in one quadrant but appear downstream in a different quadrant due to beam divergence in the space between the π/2 coil and the entrance of the output guide as well as beam transport through the guide (which is split into two separate guide sections with a vertical septum with a supermirror coating on both sides). Rotations from transverse and longitudinal fields from the other three quadrants therefore get mixed into the total rotation of each quadrant.A clean cancellation of magnetic field systematics by doing diagonal averaging/subtraction therefore depends on the amount and symmetry of the signals from these cross-over neutrons as well as the effect of magnetic field gradients. To analyze the effects of these systematics we present the procedure used to extract the V_5 signal. Let Q1, Q2, Q3, and Q4 be the signals in the 4 quadrants of the ion chamber, where we assume a quadrant arrangement with target sections 1 and 3 possessing horizontal target plates, target sectors 2 and 4 possessing vertical target plates, and the four quadrants starting with 1 in the upper right corner proceed numerically in a counterclockwise direction (see Figure <ref>). The rotation angle from V_5 in the horizontal (vertical) target region is given by θ_HF5 (θ_VF5), where θ_HF5 is the desired signal and θ_VF5 is a rotation due to fringing of the pseudomagnetic field at the edges of vertical plates.We let θ_1BTU (θ_1BLD) be the integrated spin rotation angle from the average upstream transverse (downstream longitudinal) field of the first quadrant and similarly for the other three quadrants.To incorporate the possibility of cross-over neutrons, we let a_1 , a_2 , a_3 , and a_4 be the fractions of counts in the different ion chamber quadrants from side-to-side cross-over neutrons originating in the quadrant given by the associated subscripts and b_1 , b_2 , b_3, and b_4 be the fraction of counts from up-down cross-over neutrons originating in the quadrant given by the associated subscript. Note that quadrants 1 and 4 have opposite π/2 rotations from quadrants 2 and 3 so θ_xBTU changes sign depending on the quadrant. Target quadrants 1 and 3 have opposite target orientations as do target quadrants 2 and 4, so coupled with the π/2 left/right sign change θ_HF5 and θ_VF5 have the same left/right sign, as do θ_xBLD as they are downstream from the π/2 coil. We ignore possible diagonal cross-over neutrons which are negligible by design and in simulations.All angles are assumed to be small enough that we can simply add rotations; this assumption is true for the residual field values we consider. In this case, we get Q1 =θ_HF5 + θ_1BTU+θ_1BLD+a_2(-θ_2BTU+θ_2BLD+θ_VF5) + b_4( θ_4BTU+θ_4BLD+θ_VF5) Q2 =θ_VF5 -θ_2BTU+θ_2BLD+ a_1(θ_1BTU+θ_1BLD+θ_HF5)+ b_3(-θ_3BTU+θ_3BLD+θ_HF5) Q3 =θ_HF5 -θ_3BTU+θ_3BLD+ a_4(θ_4BTU+θ_4BLD+θ_VF5)+ b_2(-θ_2BTU+θ_2BLD+θ_VF5) Q4 =θ_VF5 + θ_4BTU+θ_4BLD+a_3(-θ_3BTU+θ_3BLD+θ_HF5)+ b_1(θ_1BTU+θ_1BLD+θ_HF5) .Now we assume that quadrant differences in the transverse and longitudinal fields are due to linear field gradients in the horizontal and vertical directions, such that the quadrant average of the rotation due to transverse (longitudinal) is given by θ_BTU (θ_BLD) and is modified by ±δ_H θ_BTU (±δ_V θ_BTU) due to a horizontal (vertical) gradient for the transverse field, and similarly for the longitudinal field.Thus, the following terms that arise in the next step where we add diagonal quadrants can be written as1/2(θ_1BTU-θ_3BTU)= δ_H θ_BTU + δ_V θ_BTU 1/2(-θ_2BTU+θ_4BTU)= δ_H θ_BTU - δ_V θ_BTU 1/2(θ_1BLD+θ_3BLD)=θ_BLD 1/2(θ_2BLD+θ_4BLD)=θ_BLD .Averaging diagonal quadrants cancels rotations due to the average upstream transverse field and gradients in the longitudinal field: (Q1+Q3)/2 =θ_HF5 + (δ_H θ_BTU + δ_V θ_BTU)+ θ_BLD + 1/2 (a_2+b_2) (-θ_2BTU+θ_2BLD+θ_VF5) + 1/2 (a_4+b_4) (θ_4BTU+θ_4BLD+θ_VF5) (Q2+Q4)/2 =θ_VF5 + ( δ_H θ_BTU - δ_V θ_BTU) + θ_BLD + 1/2 (a_1+b_1) (θ_1BTU+θ_1BLD+θ_HF5) + 1/2 (a_3+b_3) (-θ_3BTU+θ_3BLD+θ_HF5) . Subtracting the two averages above (<ref>) for data taken at the same time further reduces common mode noise and eliminates rotations from downstream longitudinal fields and the effect of a transverse field horizontal gradient and leaves us withϕ_diag = (Q1+ Q3)/2 - (Q2+Q4)/2= θ_HF5 - θ_VF5 + 2δ_V θ_BTU- 1/2 (a_1+b_1+a_3+b_3)θ_HF5 + 1/2 (a_2+b_2+a_4+b_4) θ_VF5+ 1/2 (a_2+b_2) (-θ_2BTU+θ_2BLD) + 1/2 (a_4+b_4) (θ_4BTU+θ_4BLD) - 1/2 (a_1+b_1) (θ_1BTU+θ_1BLD)- 1/2 (a_3+b_3) (-θ_3BTU+θ_3BLD) Here we see the three effects mentioned earlier: the reduction in the θ_HF5 signal by θ_VF5 from fringing pseudomagetic fields in vertical targets, a remaining effect from vertical transverse field gradient, and a number of terms due to neutrons crossing between the four quadrants after the π/2 coil which modify the V_5 signals and introduce false effects from non-zero magnetic fields in the target region.If there are no gradients in the magnetic fields we can simplify the above expression, ϕ_diag^noGrad =(Q1+ Q3)/2 - (Q2+Q4)/2= θ_HF5 - θ_VF5- 1/2(a_1+b_1+a_3+b_3)θ_HF5 + 1/2 (a_2+b_2+a_4+b_4) θ_VF5+ 1/2 ( (a_3+b_3+a_4+b_4)-(a_1+b_1+a_2+b_2) ) θ_BTU+ 1/2 ( (a_2+b_2+a_4+b_4)-(a_1+b_1+a_3+b_3) ) θ_BLDThe top and bottom pairs of quadrants both contain horizontal and vertical targets, so if everything is aligned well the total cross-over neutrons from bottom quadrants should equal the number from the top quadrants and then the contribution θ_BTU from residual transverse magnetic fields is small. ¨The systematic error from the longitudinal residual field rotation is a different story, since the combination of a larger divergence in the vertical direction and the reduction in cross-over neutrons from the vertical septum conspire to make the cross-over neutrons from the vertical targets much larger than the cross-over neutrons from the horizontal targets.Since the longitudinal field is a problem only after the π/2 coil, this effect can be reduced by reducing the separation between the end of the π/2 coil and the start of the output coil.For typical numbers, which one can expect in our apparatus with an aligned beam and a 2 cm gap after the π/2 coil, one gets from simulation the following fractions for cross-over neutrons : a_1=0.0148, b_1=0.0083 , a_2=0.0064, b_2=0.1952, a_3=0.0145, b_3=0.0084, a_4=0.0066, b_4=0.1963, which results in (Q1+Q3)/2 - (Q2+Q4)/2 = 0.9770 θ_HF5 - 0.7978 θ_VF5 + 0.0006 θ_BTU + 0.1793 θ_BLD.Simulations show a <2 × 10^-7 radian systematic error in a 0.1 mG field. If one adds in addition a 1 mrad neutron beam misalignment then the cross-over neutron fraction from the bottom left (Q3) will be more than from the top right (Q1), and this difference between the bottom versus top quadrants generates a systematic error from the residual transverse magnetic field. For our assumed residual fields and geometry we get a 2 × 10^-6 radian systematic effect. Magnetic field gradients pose a potentially larger problem. Typical gradients lead to quadrant field differences of less than 10%, so for 6 Å neutronsin a 0.1mG field, 2 δ_V θ_BTU < 1.5 × 10^-4rad.The cross-over neutrons modify this systematic at less than the 10% level.Therefore, while the analysis so far reduces common mode noise, it is not sufficient to remove all magnetic-field related systematic effects to the desired level.To further reduce systematic effects from magnetic fields, we rotate the target by 90^∘ thereby flipping the orientation of the density gradient and thus the signs of the V_5 effects while leaving rotations from magnetic fields unchanged.By taking the difference in ϕ_diag (<ref>) before and after a 90^∘ rotation of the target, we eliminate magnetic field rotations while retaining the reduction in common-mode noise, Φ_m = ( ϕ_diag - ϕ_diag' )/2 =[1-1/2 (a_1+b_1+a_3+b_3)] θ_HF5 -[1 - 1/2 (a_2+b_2+a_4+b_4)] θ_VF5. Using the cross-over fractions from simulation, we find Φ_m ≈ 0.98θ_HF5 - 0.80 θ_VF5, and since θ_VF5 < 0.1 θ_HF5 we find about a 10% reduction in the desired signal, Φ_m ≈ 0.9 θ_HF5.However, as noted above the subtractions in this last step are done for rotations using different plates, so plate non-uniformities from quadrant to quadrant can generate a systematic error.Consider a mechanical imperfection of the target which generates a 1mrad angular twist in the plate orientations in quadrant 1 after a 90^∘ target rotation. Our simulations show a systematic effect of almost 1 × 10^-6rad for our assumed residual field values. This is only an issue if the 1mrad polarization twist in the plates is in the direction opposite the target orientation, e.g. a 1mrad vertical twist in horizontal target plates. Another systematic effect in this case can come from differences in the reflectivity of the different pieces of copper, which might not be identical for the two 90^∘ states. Consider a very extreme case in which the copper plates sensitive to V_5 in quadrants/states Q1 and Q3 possess 100% reflectivity and the different copper plates in the same location in quadrants/states Q1 and Q3have completely non-reflective copper plates. In this case one gets a systematic error of 4 × 10^-6rad.Even if such an extreme case were to be realized by some chance between each pair of plates there is no reason that they should be correlated in sign, so one would expect the systematic error from the difference between the average reflective properties of each twelve-plate quadrant to be suppressed by at the very least a factor of √(12). One can easily perform visual inspection and measurements of the surface roughness of the plates along with neutron reflectometry measurements of the properties of the individual plates themselves to further constrain and suppress this potential source of systematic error. Table 1 shows our estimates for the sizes of the various forms of systematic error for our target design and measurement methods. Almost all of the systematics are associated with residual magnetic fields coupled with various types of apparatus or beam nonuniformities. The different subtractions enabled by our target design reduce both common-mode noise and systematic errors to the desired level.§ CONCLUSIONSThe Neutron Spin Rotation collaboration developed a target consisting of alternating plates of different mass materials for use in experiments seeking new spin - dependentinteractions using polarized slow neutrons. The four chambers of the target allow cancellation of neutron spin rotations from stray magnetic fields. A Geneva Drive rotates the target such that the plate orientations flip in alternate target states, thereby further canceling magnetic field rotations while isolating rotations from new interactions. The target was used in a recent experiment on the FP12 cold neutron beam at LANSCE, the results of which will be presented in a forthcoming paper. The sensitivity of this target for this exotic interaction search could be further improved by using a denser nonmagnetic mass such as tungsten or tantalum in place of the copper used in this experiment. The higher density would lead to a greater sensitivity to this possible exotic interaction from the larger number of electrons and nucleons per unit volume in the source. In addition it would be desirable to flatten and polish the surfaces of the test masses exposed to the neutron beam well enough that slow neutrons are guaranteed to undergo specular reflection from the surfaces. This would raise the fraction of neutrons transmitted through the apparatus and increase the sensitivity of the experiment to shorter interaction ranges. The target can be made longer in principle as long as the magnetic shielding can be maintained. We can estimate the order of magnitude of the sensitivity to the V_5 interaction that could be achieved in a dedicated experiment using a fully optimized target design. Simulations of the neutron spin rotation apparatus for a proposed measurement of parity odd neutron spin rotation in liquid helium (whose statistical error has already been thoroughly analyzed) adapted for this target show that given two months of dedicated beam time it is possible to increase the sensitivity in g_A^2 by 4 orders of magnitude compared to the pioneering work of Piegsa and Pignol. 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"authors": [
"Christopher Haddock",
"Bret Crawford",
"Walter Fox",
"Ian Francis",
"Adam Holley",
"Murad Sarsour",
"W. Michael Snow",
"John Vanderwerp"
],
"categories": [
"physics.ins-det",
"nucl-ex"
],
"primary_category": "physics.ins-det",
"published": "20170726064435",
"title": "Slotted Rotatable Target Assembly Used In a Search for Possible Long Range Spin Dependent Interactions From Vector Boson Exchange Using Polarized Slow Neutrons"
} |
Computer Science Department, Technion, Haifa 3200003, Israel. Fast Distributed Approximation for Max-Cut Keren Censor-Hillel1The research is supported in part by the Israel Science Foundation (grant 1696/14). Rina Levy1 Hadas Shachnai1 December 30, 2023 ====================================================================================================================================== Finding a maximum cut is a fundamental task in many computational settings. Surprisingly, it has been insufficiently studied in the classic distributed settings, where vertices communicate bysynchronously sending messages to their neighbors according to the underlying graph, known as the ℒ𝒪𝒞𝒜ℒ or 𝒞𝒪𝒩𝒢ℰ𝒮𝒯 models.We amend this by obtainingalmost optimal algorithms for Max-Cut on a wide class ofgraphs in these models. In particular, for anyϵ > 0, we develop randomizedapproximation algorithms achieving a ratio of (1-) to the optimumfor Max-Cuton bipartite graphs in the 𝒞𝒪𝒩𝒢ℰ𝒮𝒯 model, and on general graphs in theℒ𝒪𝒞𝒜ℒ model.We further present efficient deterministic algorithms, including a 1/3-approximation for Max-Dicut inour models, thus improving the best known (randomized) ratio of 1/4. Our algorithms make non-trivial use of the greedy approach of Buchbinder et al. (SIAM Journal on Computing, 2015) for maximizing an unconstrained (non-monotone) submodular function, which may be of independent interest. Introduction Max-Cut is one of the fundamental problems in theoretical computer science. A cut in an undirected graph is a bipartition of the vertices, whose size is the number of edges crossing the bipartition. Finding cuts of maximum size in a given graph is among Karp's famous 21 NP-complete problems <cit.>. Since then, Max-Cut has received considerable attention, in approximation algorithms <cit.>, parallel computation <cit.>, parameterized complexity (see, e.g., <cit.> and the references therein), and streaming algorithms (see, e.g., <cit.>). Max-Cut has a central application in wireless mesh networks (WMNs). The capacity of WMNs that operate over a single frequency can be increased significantly by enhancing each router with multiple transmit (Tx) or receive (Rx) (MTR) capability. Thus, a node will not experience collision when two or more neighbors transmit to it. Yet, interference occurs if a node transmits and receives simultaneously. This is known as the no mix-tx-rx constraint. The set of links activated in each time slot, defining the capacity of an MTR WMN,is governed by a link scheduler. As shown in <cit.>, link schedulingis equivalent to finding Max-Cut in each time slot. A maximum cut contains the set ofnon-conflicting links that can be activated at the same time, i.e, they adhere to the no mix-tx-rx constraint. The induced bipartition of the vertices at each time slot defines a set of transmitters and a set of receivers in this slot. Link scheduling algorithms based on approximating Max-Cut, and other applications in wireless networks, can be found in <cit.>.[Max-Cut naturally arises also inVLSI <cit.>, statistical physics <cit.> and machine learning <cit.>.] Surprisingly, Max-Cut has been insufficiently studied in the classic distributed settings, where vertices communicate by synchronously sending messages to their neighbors according to the underlying graph, known as the ℒ𝒪𝒞𝒜ℒ or 𝒞𝒪𝒩𝒢ℰ𝒮𝒯 models. Indeed,there are known distributed algorithms for Max-Cut using MapReduce techniques <cit.>. In this setting, the algorithms partition the ground set among m machines and obtain a solution using all the outputs. However, despite a seemingly similar title, our distributed setting is completely different.In this paper we address Max-Cut in the classic distributed network models, where the graph represents a synchronous communication network. At the end of the computation, each vertex decides locally whether it joins the subset S or S̅, and outputs 1 or 0, respectively, so as to obtain a cut of largest possible size.It is well known that choosing a random cut, i.e., assigning each vertex to S or S̅ with probability 1/2, yields a 1/2-approximation for Max-Cut, and a 1/4-approximation for Max-Dicut, defined on directed graphs (see, e.g., <cit.>).[In Max-Dicut we seek the maximum size edge-set crossing fromS to S̅.] Thus, a local algorithm, where each vertex outputs 0 or 1 with probability 1/2, yields the above approximation factors with no communication required. On the other hand, we note that a single vertex can find an optimal solution, once it has learned the underlying graph. However, this requires a number of communication rounds that depends linearly on global network parameters (depending on the exact model considered). This defines a tradeoff between time complexity and the approximation ratio obtained by distributed Max-Cut algorithms. The huge gap between the above results raises the following natural questions: How well can Max-Cut be approximated in the distributed setting, using a bounded number of communication rounds? Or, more precisely: How many communication rounds are required for obtaining an approximation ratio strictly larger than half, or even a deterministic 1/2-approximation for Max-Cut?To the best of our knowledge, these questions have been studied in our distributed network models only for a restricted graph class. Specifically, the paper <cit.> suggests a distributed algorithm for Max-Cut on d-regular triangle-free graphs, that requires a single communication round and provides a (1/2+0.28125/√(d))-approximation.The key contribution of our paper is in developing two main techniques for approximating Max-Cut and Max-Dicut in distributed networks, with any communication graph. Below we detail the challenges we face, and our methods for overcoming them. The Challenge In the ℒ𝒪𝒞𝒜ℒ model, where message sizes and the local computation power are unlimited, every standard graph problem can be solved in O(n) communication rounds. For Max-Cut it also holds that finding an optimal solution requires Ω(n) communication rounds. This lower bound follows from Linial's seminal lower bound <cit.> for finding a 2-coloring of an even-sized cycle. In an even cycle, the maximum cut contains all edges. Therefore, finding a Max-Cut is equivalent to finding a 2-coloring of the graph.An approach that proved successful in many computational settings - in tackling hard problems - is to relax the optimality requirement and settle for approximate solutions. Indeed, in the distributed setting, many approximation algorithms have been devised to overcome the costs of finding exact solutions (see, e.g., <cit.>, and the survey of Elkin <cit.>). Our work can be viewed as part of this general approach. However, we face crucial hurdles attempting to use the known sequential toolbox for approximating Max-Cut in the distributed setting.As mentioned above, a 1/2-approximation for Max-Cut can be obtained easily with no need for communication. While this holds in all of the above models, improving the ratio of 1/2 is much more complicated. In the sequential setting, an approximation factor strictly larger than 1/2 was obtained in the mid-1990's using semidefinite programming <cit.> (see Section <ref>).Almost two decades later, the technique was applied by <cit.> to obtain a parallel randomized algorithm for Max-Cut, achieving a ratio of (1-ϵ) 0.878 to the optimum,for any > 0. Adapting this algorithm to our distributed setting seems non-trivial, as it relies heavily on global computation. Trying to apply other techniques, such as local search, unfortunately leads tolinear running time, because of the need to compare values of global solutions.Another obstacle that lies ahead is the lack of locality in Max-Cut, due to strong dependency between the vertices. The existence of an edge in the cut depends on the assignment of both of its endpoints. This results in a chain of dependencies and raises the question whether cutting the chain can still guarantee a good approximation ratio.Our Contribution We develop two main techniques for approximating Max-Cut, as well as Max-Dicut. Our first technique relies on the crucial observation that the cut value is additive for edge-disjoint sets of vertices.Exploiting this property, we design clustering-based algorithms, in which we decompose the graph into small-diameter clusters, find an optimal solution within each cluster, and prove that the remaining edges still allow the final solution to meet the desired approximation ratio.An essential component in our algorithms is efficient graph decomposition to such small-diameter clusters connected by few edges (also known as a padded partition), inspired by a parallel algorithm of <cit.> (see also <cit.>).For general graphs, this gives (1-ϵ)-approximation algorithms for Max-Cut and Max-Dicut, requiring O(log n/ϵ) communication rounds in the ℒ𝒪𝒞𝒜ℒ model. For the special case of a bipartite graph, we take advantage of the graph structure to obtain an improved clustering-based algorithm, which does not require large messages. The algorithm achieves a (1-ϵ)-approximation for Max-Cut in O(log n/ϵ) rounds, in the more restricted 𝒞𝒪𝒩𝒢ℰ𝒮𝒯 model. For our second technique, we observe that the contribution of a specific vertex to the cut depends only on the vertex itself and its immediate neighbors.We leverage this fact to make multiple decisions in parallel by independent sets of vertices. We find such sets using distributed coloring algorithms.Our coloring-based technique, which makes non-trivial use of the greedy approach of <cit.>for maximizing an unconstrainedsubmodular function, yields deterministic 1/2-approximation and 1/3-approximation algorithms for Max-Cut and Max-Dicut, respectively, and a randomized 1/2-approximation algorithm for Max-Dicut. Each of these algorithms requires Õ(Δ +log^*n) communication rounds in the 𝒞𝒪𝒩𝒢ℰ𝒮𝒯 model, where Δ is the maximal degree of the graph, and Õ ignores polylogarithmic factors in Δ. Finally, we present ℒ𝒪𝒞𝒜ℒ algorithms which combine both of our techniques. Applying the coloring-based technique to low-degree vertices, and the clustering-based technique to high-degree vertices, allows as to design faster deterministic algorithms with approximation ratios of 1/2 and 1/3 for Max-Cut and Max-Dicut, respectively, requiring min{Õ(Δ +log^*n),O(√(n))} communication rounds.Table <ref> summarizes our results.Background and Related Work The weighted version of Max-Cut is one of Karp's NP-complete problems <cit.>. The unweighted version that we study here is also known to be NP-complete <cit.>. While there are graph families, such as planar and bipartite graphs, in which a maximum cut can be found in polynomial time <cit.>, in general graphs, even approximating the problem is NP-hard. In the sequential setting, one cannot obtain an approximation ratio better than 16/17 for Max-Cut, or an approximation ratio better than 12/13 for Max-Dicut, unless P=NP <cit.>.Choosing a random cut, i.e., assigning each vertex to S or S̅ with probability 1/2, yields a 1/2-approximation for Max-Cut, and 1/4-approximation for Max-Dicut. In the sequential setting there are also deterministic algorithms yielding the above approximation ratios <cit.>. For 20 years there was no progress in improving the 1/2 constant in the approximation ratio for Max-Cut, until in 1995, Goemans and Williamson <cit.> achieved the currently best known approximation ratio for Max-Cut, using semidefinite programming. They present a 0.878-approximation algorithm, which is optimal assuming the Unique Game Conjecture holds <cit.>. In the same paper, Goemans and Williamson also give a 0.796-approximation algorithm for Max-Dicut. This ratio was improved later by Matuura et al. <cit.>, to 0.863. Using spectral techniques, a 0.53-approximation algorithm for Max-Cut was given by Trevisan <cit.>.In <cit.> Kale and Seshadhri present a combinatorial approximation algorithm for Max-Cut using random walks, which gives a (0.5+δ)-approximation, where δ is some positive constant which appears also inthe running time of the algorithm. In particular, for Õ(n^1.6),Õ(n^2) and Õ(n^3) times, the algorithm achieves approximation factors of 0.5051, 0.5155 and 0.5727, respectively.Max-Cut and Max-Dicut can also be viewed as special cases of submodular maximization, which has been widely studied. It is known that choosinga solution set S uniformly at random yields a 1/4-approximation, and a 1/2-approximation for a general and for symmetric submodular function, respectively <cit.>. This corresponds to the known random approximation ratios for Max-Cut and Max-Dicut. Buchbinder et al. <cit.> present determinstic 1/2-approximation algorithms for both symmetric and asymmetric submodular functions. These algorithms assumethat the submodular function is accessible through a black box returning f(S) for any given set S (known as the value oracle model).In the recent years, there is an ongoing effort to develop distributed algorithms for submodular maximization problems, using MapReduce techniques <cit.>. Often, the inputs consist of large data sets, for which a sequential algorithm may be inefficient. The main idea behind these algorithms is to partition the ground set among m machines, and have each machine solve the problem optimally independently of others. After all machines have completed their computations, they share their solutions. A final solution is obtained by solving the problem once again over a union of the partial solutions. The algorithms achieve performance guarantees close to the sequential algorithms while decreasing the running time, where the running time is the number of communication rounds among the machines. As mentioned above, these algorithms do not apply to our classic distributed settings.Preliminaries The Max-Cut problem is defined as follows. Given an undirected graph G=(V,E), one needs to divide the vertices into two subsets, S⊂V and S̅=V∖S, such that the size of the cut, i.e., the number of edges between S and the complementary subset S̅, is as large as possible. In the Max-Dicut problem, the given graph G=(V,E) is directed, and the cut is defined only as the edges which are directed from S to S̅. As in the Max-Cut problem, the goal is to obtain the largest cut.Max-Cut and Max-Dicut can be described as the problem of maximizing the submodular function f(S)=|E(S,S̅)|, where for Max-Dicut f(S) counts only the edges directed from S to S̅. Given a finite set X, a submodular function is a function f:2^X→ℝ, where 2^X denotes the power set of X, which satisfies the equivalent definitions: * For any S,T⊆X:f(S∪T)+f(S∩T)≤f(S)+f(T). * For any A⊆B⊆X and x∈X∖B:f(B∪{x})-f(B)≤f(A∪{x})-f(A). For Max-Cut and Max-Dicut, the submodular function also satisfies the following equality: For every disjoint sets S,T⊆ X such that E_S× T={(u,v)|u∈ S, v∈ T}=∅, we have that f(S)+f(T)=f(S∪ T). Note that for Max-Cut, the function is also symmetric, i.e., f(S)=f(S̅).ModelWe consider a distributed system, modeled by a graph G=(V,E), in which the vertices represent the computational entities, and the edges represent the communication channels between them. We assume that each vertex v has a unique identifier id(v) of size O(logn), where n=|V|.The communication between the entities is synchronous, i.e., the time is divided into rounds. In each round, the vertices send messages simultaneously to all of their neighbors and make a local computation based on the information gained so far. This is the classic ℒ𝒪𝒞𝒜ℒ model <cit.>, which focuses on analyzing how locality affects the distributed computation. Therefore, the messages size and local computation are unlimited, and the complexity is measured by the number of communication rounds needed to obtain a solution. It is also important to study what can be done in the more restricted 𝒞𝒪𝒩𝒢ℰ𝒮𝒯 model <cit.>, in which the message size is bounded to O(log n).We assume that each vertex has preliminary information including the size of the network n=|V|, its neighbors, and the maximal degree of the graphΔ.[This assumption is needed only for the (Δ +1)-coloring algorithm <cit.> used in Section <ref>; it can be omitted(see <cit.>), increasing the running time by a constant factor.]Each vertex runs a local algorithm in order to solve the Max-Cut problem. Along the algorithm, each vertex decides locally whether it should be in S or in S̅, and outputs 1 or 0 respectively. We define the solution of the algorithm, as the set of all outputs. Note that each vertex does not hold the entire solution, only local information. The solution value is defined as the size of the cut induced by the solution. We show that this value approximates the size of the maximum cut. Clustering-Based AlgorithmsIn this section we present clustering-based algorithms for Max-Cut and Max-Dicut. Our technique uses the observation that Max-Cut is a collection of edges having their endpoints in different sets; therefore, it can be viewed as the union of cuts in the disjoint parts of the graph.Given a graph G = (V,E), we first eliminate a small fraction of edges to obtain small-diameter connected components. Then, the problem is solved optimally within each connected component. For general graphs, this is done by gathering the topology of the component at a single vertex. For the special case of a bipartite graph, we can use the graph structure to propagate less information. Since the final solution, consisting of all the vertices local decisions, is at least as good as the sum of the optimal solutions in the components, and since the fraction of eliminated edges is small, we prove that the technique yields a (1-ϵ)-approximation.A Randomized Distributed Graph Decomposition We start by presenting the randomized distributed graph decomposition algorithm. The algorithm is inspired by a parallel graph decomposition by Miller et al. <cit.> that we adapt to the distributed model as we describe next.[Our algorithm can be viewed as one phase of the distributed algorithm presented by Elkin et al. in <cit.> with some necessary changes.]. The PRAM algorithm of <cit.> generates a strong padded partition of a given graph, namely, a partition into connectedcomponents with strong diameter O(log n/β), for some β≤ 1/2, such that the fraction of edges that cross between different clusters of the partition is at most β.As we prove next, the distributed version guarantees the same properties with high probability and requires only O(log n/β) communication rounds in the 𝒞𝒪𝒩𝒢ℰ𝒮𝒯 model.The distributed version of the graph decomposition algorithm works as follows: Let δ_v be a random value that vertex v chooses from an exponential distribution with parameter β. Define the shifted distance from vertex v to vertex u as dist_δ (u,v) = dist(u,v)-δ_u. Along the algorithm each vertex v finds a vertex u within its klog n/β-neighborhood, where k is a constant, that minimizes dist_δ (u,v). We define this vertex as v's center. This step creates the difference between the parallel and the distributed decomposition, as in the parallel algorithm each vertex chooses its center from the entire ground set V. However as we prove next, the process still generates a decomposition with the desired properties. Furthermore, w.h.p. the distributed algorithm outputs a decomposition identical to the one created by the parallel algorithm. A pseudocode of the algorithm is given in Algorithm <ref>.We prove that the fraction of edges between different components is small. In order to do so, we bound the probability of an edge to be between components, i.e., the probability that the endpoints of the edge choose different centers. We consider two cases for an edge e=(u,v). In the first case, we assume that both u and v choose the center that minimizes their shifted distance, dist_δ, over all the vertices in the graph. In other words, if the algorithm allowed each vertex to learn the entire graph, they would choose the same center as they did in the current algorithm. In the second case, we assume that at least one of u and v chooses differently if given a larger neighborhood.Define the ideal center of a vertex v as argmin_w∈ Vdist_δ(w,v). In the following lemma we bound from above the probability that a vertex does not choose its ideal center. Let v' be the ideal center of vertex v, then the probability that dist(v',v)>klog n/β, i.e., vertex v does not join its ideal center, is at most 1/n^k. Since v' is the ideal center of vertex v, we have that dist_δ(v',v) ≤ dist_δ(v,v). Therefore, dist(v',v)-δ_v'≤ dist(v,v)-δ_v = -δ_v ≤ 0, which implies that dist(v',v) ≤δ_v'. That is, the distance between each vertex v to its ideal center v' is bounded from above by δ_v', and hence [dist(v',v)>klog n/β]≤[δ_v'>klog n/β]. Using the cumulative exponential distribution, we have that [δ_v'>klog n/β]=exp(-k·βlog n/β)=exp(-klog n)≤1/n^k. The Distributed Decomposition algorithm generates a decomposition identical to the decomposition created by the parallel decomposition algorithm with probability at least 1-1/n^k-1 Applying the union bound, we have that the probability that at least one of the vertices does not choose its ideal center is at most 1/n^k-1.Define an exterior edge as an edge connecting different vertex components, and let F denote the set of exterior edges. Let A_u,v denote the event that both u and v choose their ideal centers. The probability that an edge e=(u,v) is an exterior edge, given that u and v choose their ideal centers, is at most β. The lemma follows directly from <cit.>, where indeed the algorithm assigns to each vertex its ideal center. We can now bound the probability of any edge to be an exterior edge. The probability that an edge e=(u,v) is in F is at most β+2/n^k. Note that[(u,v)∈ F] = [(u,v)∈ F|A_u,v][A_u,v]+[(u,v)∈ F|A̅_u,v][A̅_u,v].By Lemma <ref>, [(u,v)∈ F|A_u,v]≤β. Applying the union bound on the result of Lemma <ref>, we have that [A̅_u,v] ≤2/n^k. Therefore [(u,v)∈ F] = [(u,v)∈ F|A_u,v] [A_u,v]+[(u,v)∈ F|A̅_u,v][A̅_u,v]≤β·[A_u,v]+[(u,v)∈ F|A̅_u,v]·2/n^k≤β + 2/n^k. We can now prove the guarantees of the Distributed Decomposition algorithm.Recall that the weak diameter of a set S={u_1,u_2,...u_l} is defined as max_(u_i,u_j)∈ S dist(u_i,u_j). The Distributed Decomposition algorithm requires O(log n/β) communication rounds in the 𝒞𝒪𝒩𝒢ℰ𝒮𝒯 model, and partitions the graph into components such that in expectation there are O(β m) exterior edges. Each of the component is of weak diameter O(log n/β), and with high probability also of strong diameter O(log n/β). Clearly, as every vertex chooses a center from its klog n/β-neighborhood, the distance between two vertices that choose the same center, i.e., belong to the same component, over the graph G is at most O(log n/β). Therefore, the weak diameter of every component is at most O(log n/β). As we proved in Corollary <ref>, with probability at least 1-1/n^k-1 the algorithm creates a partition identical to the one created by the parallel algorithm, and therefore with the exact same properties, which implies that the strong diameter of every component is at most O(log n/β) as well. Using the linearity of expectation, and Lemma <ref> we have that 𝔼[|F|]≤∑_e∈ E^(β+2/n^k) = β m+2m/n^k. Since m≤ n^2, we have that for every k>2, 𝔼[|F|] ≤ O(β m). Finally, as can be seen from the code, the algorithm requires O(log n/β) communication rounds.A Randomized (1-ϵ)-Approximation Algorithm for Max-Cut on a Bipartite Graph Clearly, in a bipartite graph the maximum cut contains all of the edges. Such a cut can be found by selecting arbitrarily a root vertex, and then simply putting all the vertices of odd depth in one set and all the vertices of even depth in the complementary set. However, this would require a large computational time in our model, that depends on the diameter of the graph. We overcome this by using the above decomposition, and finding an optimal solution within each connected component. In each component C, we find an optimal solution in O(D_c) communication rounds, where D_c is the diameter of C. First, the vertices in each component search for the vertex with the lowest id. [This can be doneby running a BFS in parallel from all vertices. Each vertex propagates the information from the root with lowest id it knows so far, and joins its tree. Thus, at the end of the process, we have a BFS tree rooted at the vertex with the lowest id.]Second, the vertex with the lowest id joins S or S̅ with equal probability and sends its decision to its neighbors. When a vertex receives a message from one of its neighbors, it joins the opposite set, outputs its decision, and sends it to its neighbors. Since finding the optimal solution within each component does not require learning the entire component topology, the algorithm is applicable to the more restricted 𝒞𝒪𝒩𝒢ℰ𝒮𝒯 model. The algorithm yields a (1-ϵ)-approximation for the Max-Cut problem on a bipartite graph in O(log n/ϵ) communication rounds with high probability. Bipartite Max-Cut is a randomized (1-ϵ)-approximation for Max-Cut, requiring O(log n/ϵ) communication rounds in the 𝒞𝒪𝒩𝒢ℰ𝒮𝒯 model w.h.p. After applying the decomposition algorithm, we have that each connected component C has a diameter D_c of at most O(log n/β)w.h.p (Theorem <ref>). Building a BFS tree in a component C clearly takes O(D_c) communication rounds. Assigning the vertices to sets after constructing a tree takes O(D_c) as well. Therefore, the algorithm finds an optimal solution in each of the components in O(log n/ϵ) communication rounds. Since every connected component is a bipartite graph itself, all the edges within it are in the cut. Therefore, as there are at most O(β m) exterior edges, the algorithm obtain a cut of at least (1-β)m edges. Since the optimal cut in a bipartite graph contains all the edges, the algorithm achieves a (1-β)-approximation. Choosing β = ϵ proves the theorem.A Randomized (1-ϵ)-Approximation Algorithm for General GraphsWe present below a (1-ϵ)-approximation algorithm for Max-Cut in general graphs, using O(log n/ϵ) communication rounds. As before, the algorithm consists of two parts, decomposition and solution. Although the decomposition part itself works even in the 𝒞𝒪𝒩𝒢ℰ𝒮𝒯 model, the algorithm works in the ℒ𝒪𝒞𝒜ℒ model, since for general graphs the components created by the decomposition are not necessarily sparse, and learning the components topology is expensive in the 𝒞𝒪𝒩𝒢ℰ𝒮𝒯 model. Decomposition-Based Max-Cut is a randomized (1-ϵ)-approximation for Max-Cut, requiring O(log n/ϵ) communication rounds in the ℒ𝒪𝒞𝒜ℒ model. Let 𝑂𝑃𝑇(G) be the set of edges that belong to some maximum cut in G, and let ALG(G) be the set of edges in the cut obtained by Decomposition-Based Max-Cut. Let S_u be the component induced by the vertices which choose u as their center, and denote by S the set of components that algorithm Distributed Decomposition constructs. Then 𝔼[|ALG(G)|]≥𝔼[∑_S_u∈ S|𝑂𝑃𝑇(S_u)|] ≥ |𝑂𝑃𝑇(G)|-β m ≥ |𝑂𝑃𝑇(G)|-2β |𝑂𝑃𝑇(G)|=(1-ϵ)|𝑂𝑃𝑇(G)|. The last inequality follows from the fact that for every graph G it holds that |𝑂𝑃𝑇(G)|≥m/2. The graph decomposition requires O(log n/ϵ) communication rounds, and outputs components with weak diameter at most O(log n/ϵ). Therefore, finding the optimal solution within each component takes O(log n/ϵ) as well. The time bound follows. By taking β=ϵ/4, one can now obtain a (1-ϵ)-approximation algorithm for Max-Dicut. The difference comes from the fact that for Max-Dicut it holds that |𝑂𝑃𝑇(G)|≥m/4 for every graph G. The rest of the analysis is similar to the analysis for Max-Cut. Hence, we have Decomposition-Based Max-Dicut is a randomized (1-ϵ)-approximation for Max-Dicut, requiring O(log n/ϵ) communication rounds in the ℒ𝒪𝒞𝒜ℒ model.Coloring-Based Algorithms Many of the sequential approximation algorithms for Max-Cut perform n iterations. Each vertex, in its turn, makes a greedy decision so as to maximize the solution value. We present below distributed greedy algorithms which achieve the approximation ratios of the sequential algorithms much faster. We first prove that the greedy decisions of vertices can be done locally, depending only on their immediate neighbors. Then we show how to parallelize the decision process, such that in each iteration an independent set of vertices completes. The independent sets are generated using (Δ +1)-coloring; then, for (Δ +1) iterations, all the vertices of the relevant color make their parallel independent decisions. All algorithms run in the 𝒞𝒪𝒩𝒢ℰ𝒮𝒯 model. §.§ A Deterministic 1/2-Approximation Algorithm for Max-CutWe start by presenting a simple deterministic distributed greedy algorithm that yields a 1/2-approximation for Max-Cut. The algorithm is inspired by the sequential greedy Max-Cut algorithm of <cit.>. The greedy Max-Cut algorithm iterates through the vertices in some arbitrary order. In iteration i, the algorithm decides whether to assign vertex v_i to S or to S̅, based on which placement maximizes the cut size. In our algorithm the process is similar, except that, in each iteration, instead of considering a single vertex, we consider an independent set of vertices. As the vertices are independent, their decisions are also independent, and the approximation ratio still holds.To divide the vertices into independent sets, we color them using (Δ +1)-colors, where Δ is the maximum degree in the graph. The best deterministic (Δ +1)-coloring algorithm known in the 𝒞𝒪𝒩𝒢ℰ𝒮𝒯 model, due to Barenboim <cit.>, requires Õ(Δ^3/4+log^* n) communication rounds, where Õ ignores polylogarithmic factors in Δ [Note that a faster (Δ + 1)-coloring algorithm will not improve the running time of Distributed Greedy Max-Cut, since the running time of the algorithm depends on the number of colors, which is Δ + 1.]. Define a coloring C:V→{1,2,...,Δ +1} such that C(v)≠ C(u) for every (u,v)∈ E. Let N_low(v)={u| u∈ N(v) and C(u)<C(v)} denote the neighbors of vertex v with lower colors. In iteration i, all vertices with color i decide in parallel whether to join S or S̅, depending on the choices their neighbors made in previous rounds. In order to maximize the cut locally, vertex v chooses to join the subset that was chosen by the minority of its neighbors in N_low(v). As we show next, this guarantees the 1/2-approximation.Algorithm <ref> gives a pseudocode of the algorithm. The Distributed Greedy Max-Cut algorithm outputs a 1/2-approximation in Õ(Δ + log^* n) rounds. We first show that the algorithm gives a 1/2-approximation. Consider an edge e=(u,v), if C(v)>C(u), then we say that v is responsible for edge e. Denote by E_resp(v) the set of edges vertex v is responsible for. In other words, E_resp(v) contains the edges between v and vertices in N_low(v). Since the color groups are independent, every edge has exactly one responsible vertex, therefore ∑_v∈ V^ |E_resp(v)| = |E|. When vertex v makes its decision, it chooses to join the set that is not chosen by at least half of its neighbors in N_low(v), and hence adds at least half of the edges in E_resp(v) to the cut. Upon termination of the algorithm, we have that |E(S,S̅)|≥∑_v∈ V^1/2|E_resp(v)|=1/2|E|. Since the size of the optimal cut cannot be larger than |E|, the algorithm yields a 1/2-approximation. The algorithm colors the graph in Õ(Δ^3/4+log^* n) rounds and iterates for O(Δ + 1) rounds, which yields the time bound. §.§ A Deterministic 1/3-Approximation Algorithm for Max-DicutNext, we turn our attention to the Max-Dicut problem. Buchbinder et al. <cit.> present a sequential deterministic greedy 1/3-approximation algorithm for maximizing unconstrained submodular functions which runs in linear time. Inspired by this algorithm, we present a distributed deterministic 1/3-approximation algorithm for Max-Dicut that requires Õ(Δ + log^*n) communication rounds in the 𝒞𝒪𝒩𝒢ℰ𝒮𝒯 model.§.§.§ The sequential AlgorithmWe first give a brief overview of the sequential algorithm of <cit.> for Max-Dicut. Given a graph G=(V,E), where |V|=n, the algorithm examines the vertices in an arbitrary order u_1,u_2,...,u_n. In iteration i, the algorithm decides greedily whether to include u_i in the final solution, for 1≤ i ≤ n. The algorithm maintains two sets of vertices, X and Y. Initially, X_0=∅ and Y_0=V. At the begining of the ith iteration, the algorithm defines X_i=X_i-1,Y_i=Y_i-1, then, the algorithm decides whether to add the ith vertex u_i to X_i-1, or to remove it from Y_i-1. The decision is made by calculating the marginal profit of both options and choosing the more profitable one. By definition, upon termination of the algorithm we have that X_n=Y_n, and this set is output as the solution. Let f(S) be the size of the directed cut induced by a subset of vertices S⊆ V, i.e., the number of edges directed from S to S̅. Then f is a non-negative submodular function. Let a_i and b_i denote the marginal profit gained by adding the vertex u_i to X_i-1, or removing it from Y_i-1 respectively. Algorithm <ref> gives the pseudocode of the sequential algorithm of <cit.>. We give below a sketch of the analysis [See the details in <cit.>.]. The following lemma implies that in each iteration, one can only increase the value of the solution.<cit.> For every 1≤ i ≤ n, it holds that a_i+b_i ≥ 0.Let 𝑂𝑃𝑇 denote the set S⊆ V that maximizes the dicut size. Define 𝑂𝑃𝑇_i ≜ (𝑂𝑃𝑇∪ X_i)∩ Y_i. In other words, 𝑂𝑃𝑇_i agrees with X_i and Y_i on the first i elements, and agrees with 𝑂𝑃𝑇 on the rest. Hence, 𝑂𝑃𝑇_0 = 𝑂𝑃𝑇 and 𝑂𝑃𝑇_n = Y_n = X_n. The following lemma shows that in each iteration, the damage to the optimal solution value, i.e., f(𝑂𝑃𝑇_i-1)-f(𝑂𝑃𝑇_i), is bounded.<cit.> For every 1≤ i ≤ n, it holds that f(𝑂𝑃𝑇_i-1)-f(𝑂𝑃𝑇_i) ≤ [f(X_i)-f(X_i-1)] + [f(Y_i)-f(Y_i-1)]. Using Lemmas <ref>,<ref>, one can prove the following theorem. The Deterministic Sequential Max-Dicut algorithm gives a 1/3-approximation for the Max-Dicut problem in linear time. §.§.§ The Distributed AlgorithmInspired by the sequential algorithm, we design a distributed algorithm which gives a 1/3-approximation for the Max-Dicut problem. As in Subsection <ref>, we start by (Δ +1)-coloring the graph. Then, for 1 ≤ i ≤ (Δ +1) iterations, the vertices make their decisions, one color class at a time. In each iteration, the corresponding vertices calculate the marginal profit gained by their two possible decisions, and take the one which maximizes the profit. As in the sequential algorithm, the distributed algorithm maintains two solutions X and Y; X_0=∅ and Y_0 = V as before. X_i and Y_i represent the state of the solutions after i iterations, and 𝑂𝑃𝑇_i is defined as 𝑂𝑃𝑇_i=(𝑂𝑃𝑇∪ X_i)∩ Y_i).Define X_i(v) ≜ X_i ∩ N(v), and Y_i(v) ≜ Y_i ∩ N(v). It is easy to see that X_i as defined in the sequential algorithm equals ∪_v∈ V X_i(v). Similarly, Y_i = ∪_v∈ V Y_i(v). Using this notation, Algorithm <ref> gives the pseudocode of the distributed algorithm.There are two key ingredients in our analysis. We first prove that the marginal profits a_i and b_i can be computed locally. Then, we need to show that running the procedures in parallel does not affect the approximation ratio.The next lemma shows that the marginal profits of v's possible decisions depends only on its 1-neighborhood. Let A⊆ V be a subset of vertices, and let v be a vertex such that v∉ A. Then f(A+v)-f(A)=f((A∩ N(v)) + v)-f(A∩ N(v)). Given A,B⊆ V such that A∩ B = ∅, let |E(A,B)| denote the number of edges directed from vertices in A to vertices in B. We start by proving that for every subset A⊆ V and v∉ A it holds that: ⋆⋆ f(A + v) - f(A) = |E(A+v,V∖(A+v))|-|E(A,V∖ A)|=|E(A,V∖(A+v))|+|E(v,V∖(A+v))|-|E(A,V∖(A+v))|-|E(A,v)|=|E(v,V∖(A+v))|-|E(A,v)|. Note that since v is connected by edges only to its neighbors,|E(v,V∖(A+v))|-|E(A,v)| = |E(v,N(v) ∩(V∖(A+v)))|-|E(N(v)∩ A,v)|. As (A∩ N(v))⊆ V, and v∉(A∩ N(v)), using (⋆⋆) we have that f((A∩ N(v)) + v)-f(A∩ N(v))= |E(v,V∖(A∩ N(v)+v))|-|E(A∩ N(v),v)|=|E(v,N(v) ∩(V∖(A+v)))|-|E((N(v)∩ A),v)|, which proves the lemma. We now prove that making the decision to join S or S̅ in parallel for independent sets does not affect the approximation ratio. For every 1≤ i ≤ (Δ +1), it holds that f(𝑂𝑃𝑇_i-1)-f(𝑂𝑃𝑇_i) ≤ [f(X_i)-f(X_i-1)] + [f(Y_i)-f(Y_i-1)]. Let I={v_1,v_2,...,v_m} be an independent set of color i. We show that iteration i of the distributed algorithm is equivalent to m iterations of the sequential one. We can simulate the ith iteration of the distributed algorithm as m sequential iterations, where in the jth iteration, vertex v_j makes the exact same decision it makes in the distributed algorithm. Let X_i-1^j,Y_i-1^j,𝑂𝑃𝑇_i-1^j represent the state of X_i-1,Y_i-1 and 𝑂𝑃𝑇_i-1 after the jth iteration of the simulation. Using the above notation, we prove the lemma by showing that: ∑_j=1^m[f(𝑂𝑃𝑇_i-1^j-1)-f(𝑂𝑃𝑇_i-1^j)] ≤∑_j=1^m[f(X_i-1^j)-f(X_i-1^j-1)] + ∑_j=1^m[f(Y_i-1^j)-f(Y_i-1^j-1)]. For this, it suffices to show that [f(𝑂𝑃𝑇_i-1^j-1)-f(𝑂𝑃𝑇_i-1^j)] ≤[f(X_i-1^j)-f(X_i-1^j-1)] + [f(Y_i-1^j)-f(Y_i-1^j-1)], for all 1≤ j≤ m. Since I is an independent set, it holds that X_i-1(v)∩ I = ∅ and Y_i-1(v)∩ I = ∅ for every v∈ I, i.e. the decision of every vertex v∈ I does not depend on the decisions made by the other vertices in I. By Lemma <ref>, f(X_i-1 + v)-f(X_i-1)=f(X_i-1(v) + v)-f(X_i-1(v)), and f(Y_i-1-v)-f(Y_i-1)=f(Y_i-1(v)-v)-f(Y_i-1(v)) for every vertex v∈ V. Therefore, given X_i-1^j-1, Y_i-1^j-1 and v_j, an iteration of the sequential algorithm is equivalent to the jth iteration of the simulation. We now complete the proof using Lemma <ref>. The algorithm Distributed Greedy Max-Dicut gives a 1/3-approximation for the Max-Dicut problem in Õ(Δ + log^*n) communication rounds in the 𝒞𝒪𝒩𝒢ℰ𝒮𝒯 model.We start by showing the approximation ratio. By Lemma <ref>, f(𝑂𝑃𝑇_i-1)-f(𝑂𝑃𝑇_i) ≤ [f(X_i)-f(X_i-1)] + [f(Y_i)-f(Y_i-1)] for all 1≤ i ≤ (Δ +1). Summing up the inequality for every i gives: ∑_i=1^Δ + 1[f(𝑂𝑃𝑇_i-1)-f(𝑂𝑃𝑇_i)] ≤∑_i=1^Δ + 1[f(X_i)-f(X_i-1)] + ∑_i=1^Δ + 1[f(Y_i)-f(Y_i-1)]. As we saw in the sequential case, this is a telescopic sum that after cancellation gives: f(𝑂𝑃𝑇_0)-f(𝑂𝑃𝑇_Δ + 1) ≤ [f(X_Δ +1)-f(X_0)] + [f(Y_Δ +1)-f(Y_0)]=f(X_Δ +1)+f(Y_Δ +1). The last equality follows from the fact that in the case of Max-Dicut f(X_0)=f(Y_0)=0. Hence the output f(𝑂𝑃𝑇_Δ + 1) ≥ f(𝑂𝑃𝑇)/3. We now analyze the number of communication rounds needed. Coloring the graph using the algorithm of <cit.> takes Õ(Δ^3/4+log^* n) communication rounds. After the coloring, the algorithm runs for (Δ + 1) iterations, each one takes O(1) communication rounds, and hence, the time complexity follows. §.§ A Randomized 1/2-Approximation Algorithm for Max-DicutAs shown in <cit.>, using random decisions improves the approximation ratio. The randomized algorithm differs from the deterministic algorithm in the decision making process. Rather than taking the most profitable decision, the algorithm takes each of the possible decisions with a probability proportional to its value.A formal description of the algorithm is given in Algorithm <ref>. The variables X_i,Y_i and 𝑂𝑃𝑇_i are defined as in the deterministic distributed algorithm.We first show the equivalent of Lemma <ref>, and then prove our main theorem for our algorithm. For every 1≤ i ≤ (Δ +1), it holds that: ⋆𝔼[f(𝑂𝑃𝑇_i-1)-f(𝑂𝑃𝑇_i)] ≤1/2𝔼[f(X_i)-f(X_i-1) + f(Y_i)-f(Y_i-1)]. As shown in <cit.>, for the sequential randomized algorithm, where in each step only one vertex makes a decision, it holds that 𝔼[f(𝑂𝑃𝑇_i-1)-f(𝑂𝑃𝑇_i)] ≤1/2𝔼[f(X_i)-f(X_i-1) + f(Y_i)-f(Y_i-1)]. Also, as shown in the proof for Lemma <ref>, denoting by I_i the independent set of vertices colored with i, the i-th iteration of the distributed algorithm can be simulated by |I_i| iterations of the sequential algorithm. Since the inequality holds for one iteration of the sequential algorithms, it holds for |I_i| iterations, and therefore holds for the distributed algorithm. The algorithm Distributed Randomized Max-Dicut outputs a 1/2-approximation for Max-Dicut in Õ(Δ + log^*n) communication rounds. The proof is very similar to the proof of Theorem <ref>. Using (⋆), and taking a summation over all 1 ≤ i ≤Δ +1, we have ∑_i=1^Δ + 1𝔼[f(𝑂𝑃𝑇_i-1)-f(𝑂𝑃𝑇_i)] ≤1/2∑_i=1^Δ + 1𝔼[f(X_i)-f(X_i-1) + f(Y_i)-f(Y_i-1)]. Noting that the sum is telescopic, most of the terms cancel out, and we have 𝔼[f(𝑂𝑃𝑇_0)-f(𝑂𝑃𝑇_Δ+1)]≤1/2𝔼[f(X_Δ+1)-f(X_0) + f(Y_Δ+1)-f(Y_0)] ≤1/2𝔼[f(X_Δ+1)+f(Y_Δ+1)]. Therefore, since 𝑂𝑃𝑇_0=𝑂𝑃𝑇, we have that the output satisfies f(X_Δ+1)=f(Y_Δ+1)=f(𝑂𝑃𝑇_Δ+1) ≥ f(𝑂𝑃𝑇)/2. The time complexity analysis is identical to the one for the deterministic algorithm. A Deterministic ℒ𝒪𝒞𝒜ℒ Algorithm Our coloring-based algorithms may become inefficient for high degree graphs, due to the strong dependence on Δ.Consider a clique in this model. The above algorithms require a linear number of communication rounds, while learning the entire graph and finding an optimal solution requires only O(1) communication rounds in the ℒ𝒪𝒞𝒜ℒ model. Indeed, there is a tradeoff between the graph diameter and the average degree of its vertices. Based on this tradeoff, we propose a faster, two-step, deterministic algorithm for Max-Cut that requires min{Õ(Δ +log^*n),O(√(n))} communication rounds in theℒ𝒪𝒞𝒜ℒ model. The pseudocode is given in Algorithm <ref>.We call a vertex v a low-degree vertex, if deg(v)<√(n), and a high-degree vertex, if deg(v)≥√(n). Define G_low, and G_high as the graphs induced by the low-degree vertices and the high-degree vertices, respectively. The idea is to solve the problem separately for G_low and for G_high.In the first step, the algorithm deletes every high-degree vertex, if there are any, and its adjacent edges, creating G_low. The deletion means that the low-degree vertices ignore the edges that connect them to high-degree vertices, and do not communicate over them. Then, the algorithm approximates the Max-Cut on G_low, using one of the coloring-based algorithms described in Section <ref>.In the second step, the problem is solved optimally within each connected component in G_high. However, the high-degree vertices are allowed to communicate over edges which are not in G_high. As we prove next, the distance in the original graph G between any two vertices which are connected in G_high is bounded from above by O(√(n)). Hence, the number of rounds needed for this part of the algorithm is bounded as well by O(√(n)). Assume u,v are connected in G_high, then the distance between u and v in the original graph G is at most 3√(n). Let dist_G(v_1,v_2) denote the distance between the vertices v_1,v_2∈ V in the original graph G. Let u,v be two connected vertices in G_high, and assume, toward a contradiction, that dist_G(u,v)>3√(n). Let {A_i}_i=0^m=(u=a_0,a_1,...,a_m=v) be a sequence of vertices that lie on a shortest path from u to v in G_high. For each pair of vertices (a_i,a_i+1),i=0,..,m-1 on the path, it holds that |dist_G(a_i,u)-dist_G(a_i+1,u)|≤ 1. Therefore, there is a subsequence {A_i_j}_j=0^k for k>3√(n), which starts with u and ends with v, such that for every j=0,..,k-1 it holds that dist_G(a_i_j+1,u)-dist_G(a_i_j,u)=1. Note that if a_i_j_1 and a_i_j_2 have a common neighbor, then |dist_G(a_i_j_1,u)-dist_G(a_i_j_2,u)|<3. Since there are at least k/3 vertices in {A_i_j}_j=0^k, such that the distance between them is at least 3, and each of them is of degree at least √(n), we have that the number of vertices in G is at least k/3·√(n)>n. This contradicts the assumption that dist_G(u,v)>3√(n). Fast Distributed Greedy Max-Cut yields a 1/2-approximation to Max-Cut, using min{Õ(Δ +log^*n),O(√(n))} communication rounds in the ℒ𝒪𝒞𝒜ℒ model. We first prove the approximation ratio. Since Distributed Greedy Max-Cut is applied on G_low, at least half of the edges of G_low are in the cut. Given the decisions of vertices in G_low, the algorithm finds an optimal solution for all vertices in G_high. Note that running Distributed Greedy Max-Cut on the high-degree vertices of G, would give at least half of the remaining edges. This is due to the fact that the algorithm makes sequential greedy decisions. Therefore, an optimal solution for the high-degree vertices guarantees at least half of the edges in G∖ G_low, implying the approximation ratio. Applying Distributed Greedy Max-Cut on G_low requires Õ(Δ_low +log^*n) communication rounds, where Δ_low=min{Δ,√(n)}. Using Lemma <ref> we have that each high degree vertex can communicate with every high-degree vertex connected to it in G_high, using at most O(√(n)) communication rounds. Hence, Steps 2.-4. of the algorithm take O(√(n)) communication rounds. We note that when Δ<√(n), the algorithm terminates after the first step. Thus, the algorithm requires min{Õ(Δ +log^*n),O(√(n))} communication rounds.Using the above technique, we obtain a fast, deterministic algorithm for the Max-Dicut problem, by replacing the call to Distributed Greedy Max-Cut in Step 1. with a call to Distributed Greedy Max-Dicut. Using the same arguments as in the analysis for the Max-Cut algorithm, we have: Fast Distributed Greedy Max-Dicut yields a 1/3-approximation to Max-Dicut, using min{Õ(Δ +log^*n),O(√(n))} communication rounds in the ℒ𝒪𝒞𝒜ℒ model. Discussion In this paper we addressed Max-Cut in the classic distributed network models, where the graph represents a synchronous communication network. Our clustering-based and coloring-based techniques led to the development of the first distributed approximation algorithms for Max-Cut and Max-Dicut on general graphs in these models. We mention below some avenues for future work. It would be interesting to understand how close is the complexity of our algorithms to the true complexity of the problems in each of the distributed models. Moreover, the gap between the best approximation ratio achieved in the 𝒞𝒪𝒩𝒢ℰ𝒮𝒯 model and the best approximation ratio achieved in the ℒ𝒪𝒞𝒜ℒ model raises the immediate question of how congestion affects the ability to approximate the problems. Another direction is to obtain lower bounds in terms of running times in approximating these problems. In particular: what is the lower bound for achieving an approximation factor strictly better than 1/2 in the 𝒞𝒪𝒩𝒢ℰ𝒮𝒯 model? Finally, it would be interesting to extend the algorithms for other, and perhaps more general submodular maximization problems.Acknowledgements We thank Roy Schwartz and Shay Kutten for stimulating discussions and for helpful comments on the paper.splncs03 | http://arxiv.org/abs/1707.08496v1 | {
"authors": [
"Keren Censor-Hillel",
"Rina Levy",
"Hadas Shachnai"
],
"categories": [
"cs.DS",
"cs.DC"
],
"primary_category": "cs.DS",
"published": "20170726152924",
"title": "Fast Distributed Approximation for Max-Cut"
} |
Instantons for 4-manifolds with periodic ends and an obstruction to embeddings of 3-manifolds Masaki Taniguchi December 30, 2023 ============================================================================================= We construct an obstruction to the existence of embeddings of a homology 3-sphere into a homology S^3× S^1 under some cohomological condition. The obstruction are defined as an element in the filtered version of the instanton Floer cohomology due to <cit.>. We make use of the -fold covering space of homology S^3× S^1 and the instantons on it.§ INTRODUCTION There are two typical studies of gauge theory for 4-manifolds with periodic end by C.H.Taubes <cit.> andJ.Lin <cit.>. They gave a sufficient condition to exist a natural compactification of the instanton and the Seiberg-Witten moduli spaces for such non-compact 4-manifolds. The condition of Taubes is the non-existence of non-trivial SU(2) flat connection of some segment of the end. The condition of Lin is the existence of a positive scalar curvature metric on the segment of the end. As an application of the existence of the compactification, Taubes showed the existence of uncountable family of exotic ^4 and Lin constructed an obstruction to the existence of a positive scalar curvature metric.In this paper, we also give a similar sufficient condition for the instanton moduli spaces. The condition is an uniform bound on the L^2-norm of curvature of instantons. When we bound the L^2-norm of curvature, we use aninvariant which is a generalization of the Chern-Simons functional. Under this condition, we prove a compactness theorem (Theorem <ref>). As the main theorem of this paper, we construct an obstruction of the existence of embeddings of a homology 3-sphere into a homology S^3× S^1 with some cohomological condition (Theorem <ref>).To formulate the obstruction, we need a variant of the instanton Floer cohomology. The variant is the filtered instanton Floer cohomology HF^i_r whose filtration was essentially considered by R.Fintushel-R.Stern in <cit.>. The obstruction is an element of the filtered instanton cohomology. We denote the element by [θ^r]. The element [θ^r]∈ HF^1_r is a filtered version of [θ] which was already defined by S.K.Donaldson <cit.> and K.Frøyshov <cit.>. The class [θ] is defined by counting the gradient lines of the Chern-Simons functional which converge to the trivial flat connection. In order to show [θ^r] is actually an obstruction of embeddings, we count the number of the end of the 1-dimensional instanton moduli space for 4-manifold which has both of cylindrical and periodic end. For the couting, we use the compactness theorem (Theorem <ref>). This paper is organized as follows. In Section <ref>, we give a precise formulation of our main theorem (Theorem <ref>). In Section <ref>, we prepare several notations and constructions which are used in the rest of this paper. In particular, we introduce the filtered instanton Floer homology HF^r_i and the obstruction class [θ^r]. We also review Fredholm, moduli theory for 4-manifolds with periodic end. In Section <ref>, we generalize the Chern-Simons functional and introduce the invariants Q^i_X. In Section 5, we prove the compactness theorem(Theorem <ref>). We use Q^i_X to control the L^2-norm of curvature. In Section 6, we deal with technical arguments about the transversality and the orientation for the instanton moduli spaces for 4-manifolds with periodic end. In Section 7, we prove Theorem <ref>.Acknowledgements: The author is grateful to Mikio Furuta for his suggestions. The author would like to thank Hokuto Konno for useful conversations. § MAIN THEOREMLet X be a homology S^3× S^1, i.e. , X is a closed 4-manifold equipped with an isomorphism ϕ:H_*(X,) H_*(S^3× S^1,) in this paper. Then X has an orientation induced by the standard orientation of S^3× S^1 and ϕ.Let Y be an oriented homology S^3. We construct an obstruction of embeddings f of Y into X satisfyingf_*[Y]=1∈ H_3(X,) as an element in the filtered instanton Floer cohomology. We use information of the compactness of the instanton moduli spaces for periodic-end 4-manifold in a crucial step of our construction.In order to formulate our main theorem, we need to prepare several notations. For any manifold Z, we denote by P_Z the product SU(2) bundle. The product connection on P_Z is written by θ.(Z):= {SU(2)-connections on P_Z}, ^flat(Z):={SU(2)-flat connections on P_Z}⊂(Y), (Z):= (Z) /_0(Z,SU(2)), R(Z):=^flat /_0(Z,SU(2))⊂(Z),andR(Z):= ^flat(Z)/(Z,SU(2)),where _0(Z,SU(2)) is a set of smooth functions with mapping degree 0. When Z is equal to Y, the Chern-Simons functional cs_Y:(Y) is defined by cs_Y(a):=1/8π^2∫_Y Tr(a∧ da +2/3a∧ a∧ a).It is known that cs_Y decends to a map (Y), which we denote by the same notation cs_Y.We denote the number of elements in R(Y) by l_Y. If R(Y) is not a finite set, we set l_Y=∞.We will use the following assumption on Y in our main theorem(Theorem <ref>).All SU(2) flat connections on Y are non-degenerate, i.e. the first cohomology group of the next twisted de Rham complex:0 ^0(Y)⊗d_a^1(Y)⊗d_a^2(Y)⊗d_a^3⊗ 0vanishes for [a] ∈ R(Y). All flat connections on the Brieskorn homology 3-sphere Σ(p,q,r) are non-degenerate.(<cit.>)Under Assumption <ref>, l_Y is finite (<cit.>). In this paper without the use of Assumption <ref>, we will introduce the following invariants: * HF^i_r(Y) for Y and r ∈ (∖ cs_Y(R(Y)))∪{∞} in Definition <ref>satisfying HF^i_∞(Y)=HF^i(Y),* [θ^r] ∈ HF^1_r(Y) for Y and r ∈ (∖ cs_Y(R(Y)))∪{∞} in Definition <ref> satisfying [θ^∞]=[θ] ∈ HF^1(Y), and* Q^i_X ∈_≥ 0∪{∞} for i ∈ and X in Definition <ref> (When X is homotopy equivalent to S^3× S^1, Q^i_X=∞ for all i ∈).Our main theorem is: Under Assumption <ref>, if there exists an embedding f of Y into X with f_*[Y]=1∈ H_3(X,)then [θ^r] vanishes for any r∈ [0,min{Q^2l_Y+3_X, 1}] ∩ (∖ cs_Y(R(Y))∪{∞}) In particular, if there exists an element r∈ [0,min{Q^2l_Y+3_X,1}] ∩ (∖ cs_Y(R(Y))∪{∞})satisfying 0≠ [θ^r], Theorem <ref> implies that there is no embedding from Y to X with f_*[Y]=1 ∈ H_3(X,).Let X be a closed 4-manifold which is homotopy equivalent to S^3× S^1. There is no embedding f of Σ(2,3,6k-1) into X satisfying f_*[Σ(2,3,6k-1)] =1∈ H_3(X,) for a positive integer k satisfying 1≤ k ≤ 12. The proof of Example <ref> is given in the end of Subsection <ref>.§ PRELIMINARIESIn this section, we review the (filtered) instanton Floer theory and moduli theory on the periodic end 4-manifolds. §.§ Holonomy perturbation(1)In this subsection we review classes of perturbations which were considered in <cit.>, <cit.> to define the instanton Floer homology. Let Y and P_Y be as in Section <ref>. We fix a Riemannian metric g_Y on Y. We define the set of embeddings from solid tori to Y by_d:= { (f_i)_1≤ i ≤ d :S^1× D^2 Y| f_i: orientation preserving embedding }.Fix a two form d𝒮 on D^2 supported in the interior of D^2 with ∫_D^2d𝒮=1. We denote by C^l(SU(2)^d,)_ the adjoint invariant C^l-class functions from SU(2)^d toand define ∏(Y):= ⋃_d ∈_d× C^l(SU(2)^d,)_. We use the following notation,^*(Y):={[a]∈(Y) | a is an irreducible connection} ,where (Y) is defined in Section <ref>. For π = (f,h)∈∏(Y), the perturbed Chern-Simons functional cs_Y,π:^*(Y)is defined bycs_Y,π(a)= cs_Y(a)+ ∫_x ∈ D^2 h((a)_f_1(-,x),… ,(a)_f_d(-,x))d𝒮,where (a)_f_i(-,x) is the holonomy around the loop t ↦ f_i(t,x) for each i ∈{1,…,d}. If we identifywith its dual by the Killing form, the derivative of h_i=pr_i^*h is a Lie algebra valued 1-form over SU(2) for h ∈ C^l(SU(2)^d,)_. Using the value of holonomy for the loops {f_i(x,t)| t∈ S^1}, we obtain a section _f_i(t,x)(a) of the bundle P over f_i. Sending the section _f_i(t,x)(a) by the bundle map induced by h_i': P P, we obtain a section h_i'(_f_i(t,x)(a)) of P over f_i.We now describe the gradient-line equation of cs_Y,π with respect to L^2 metric :∂/∂ t a_t=_a cs_Y,π = *_g_Y(F(a_t)+∑_1≤ i ≤ d h'_i((a_t)_f_i(t,x))(f_i)_*_2^*d𝒮),where _2 is the second projection _2:S^1× D^2D^2 and *_g_Y is the Hodge star operator. We denote _2^*d𝒮 by η. We setR(Y)_π:= {a ∈(Y)|F(a)+∑_1≤ i ≤ d h'_i((a)_f_i(t,x))(f_i)_*η=0 }, andR^*(Y)_π:= R(Y)_π∩^*(Y).The solutions of (<ref>) correspond to connections A over Y× which satisfy an equation:F^+(A)+ π(A)^+=0,where* The two form π(A) is given by ∑_1≤ i ≤ d h'_i((A)_f̃_i(t,x,s))(f̃_i)_* (_1^* η). * The map _1 is a projection map from (S^1× D^2) × to S^1× D^2.* The notation + is the self-dual component with respect to the product metric on Y×.* The map f̃_i: S^1× D^2× Y× is f_i× id. We also use several classes of the perturbations. A class of perturbation ∏(Y)^flat is defined by a subset of ∏(Y) with the conditions: * cs_Y coincideswith cs_Y,π on a small neighborhood of critical points of cs_Y * R(Y)=R(Y)_π, for all element in ∏(Y)^flat.If the cohomology groups defined by the complex (12) in <cit.> satisfies H^i_π,a=0 for all [a] ∈R(Y)_π∖{ [θ]} for π, we call π non-degenerate perturbation.If π satisfies the following conditions, we call π regular perturbation.* The linearization of (<ref>)d^+_A+dπ^+_A : ^1(Y×)⊗_L^2_q^+(Y×)⊗_L^2_q is surjective for [A] ∈ M(a,b)_π andall irreducible critical point a,b of cs_Y,π. * The linearization of (<ref>) d^+_A+dπ^+_A : ^1(Y×)⊗_L^2_q,δ^+(Y×)⊗_L^2_q,δis surjective for [A] ∈ M(a,θ)_π andall irreducible critical point a of cs_Y,π.Here the spacesM(a,b)_π and M(a,θ)_π are given in (<ref>) in Subsection <ref>, L^2_q is the Sobolev norm and L^2_q,δ is the weighted Sobolev norm which is same one as in Subsection 3.3.1 in <cit.>.§.§ Filtered instanton Floer (co)homologyIn this subsection, we give the definition of the filtration of the instanton Floer (co)homology by using the technique in <cit.>. First, we give the definition of usual instanton Floer homology.Let Y be a homology S^3 and fix a Riemannian metric g_Y on Y. Fix a non-degenerate regular perturbation π∈∏(Y). Roughly speaking, the instanton Floer homology is inifinite dimensional Morse homology with respect to cs_Y,π :^*(Y) . Floer defined : R^*(Y)_π, called the Floer index.The (co)chains of the instanton Floer homology are defined byCF_i:= { [a] ∈R^*(Y)_π | (a)=i }(CF^i:= (CF_i,)).The boundary maps ∂ :CF_iCF_i-1(δ:CF^i CF^i+1) are given by∂ ([a]) := ∑_b ∈R^*(Y)_π with (b)=i-1# (M(a,b)/) [b] (δ:=∂^*), where M(a,b) is the space of trajectories of cs_Y,π from a to b. We now write the explicit definition of M(a,b). Fix a positive integer q≥3. Let A_a,b be an SU(2) connection on Y × satisfying A_a,b|_Y× (-∞,1]=p^*a and A_a,b|_Y× [1,∞)=p^*b where p is projection Y× Y. M(a,b)_π:={A_a,b+c | c ∈Ω^1(Y×)⊗_L^2_q with (<ref>)}/ (a,b),where (a,b) is given by (a,b):={ g ∈(P_Y×)⊂(ℂ^2)_L^2_q+1,loc | ∇_A_a,b(g) ∈ L^2_q }.The action of (a,b) on {A_a,b+c | c ∈Ω^1(Y×)⊗_L^2_q with (<ref>)} given by the pull-backs of connections. The spacehas anaction on M(a,b) by the translation. Floer show that M(a,b)/ has structure of a compact oriented 0-manifold whose orientation is induced by the orientation of some determinant line bundles and ∂^2=0 holds. The instanton Floer (co)homology HF_*(Y)(HF^*(Y)) is defined by HF_*(Y):= ∂/ ∂ (HF^*(Y):= δ/ δ).Second, we introduce the filtration in the instanton Floer homology. This filtration is essentially considered by Fintushel-Stern in <cit.>. We follow Fintushel-Stern and use the class of perturbations which they call ϵ-perturbation defined in Section 3 of <cit.>. They constructed -graded Floer homology whose chains are generated by the critical points of cs_Y,π with cs_Y(a)∈ (m,m+1). We now consider Floer homology whose chains generated by the critical points of cs_Y,π with cs_Y(a)∈ (-∞,r).Let R(Y) be as in Section <ref> and Λ_Y be ∖ cs_Y|_R(Y). For r ∈Λ_Y, we define the filtered instanton homology HF^r_*(Y) (HF^*_r(Y)) by using ϵ-perturbation.For r ∈Λ_Y, we set ϵ= inf_a ∈R(Y) |cs_Y(a)-r| and choose such a ϵ-perturbation π. The chains of the filtered instanton Floer (co)homology are defined byCF^r_i:= { [a] ∈R^*(Y)_π | (a)=i, cs_Y,π(a)<r } (CF^i_r:= (CF_r^i,)).The boundary maps ∂^r :CF_i^rCF_i-1^r(resp. δ^r:CF^i_r CF^i+1_r) are given by the restriction of ∂ to CF_i^r(resp. δ^r:=(∂^r)^*). This maps are well-defined and (∂^r)^2=0 holds as in Section 4 of <cit.>. The filtered instanton Floer (co)homology HF^r_*(Y)(resp. HF_r^*(Y)) is defined by HF^r_*(Y):= ∂^r/ ∂^r(resp. HF^*_r(Y):= δ^r/ δ^r). We can also show HF^r_i(Y) and HF^i_r(Y) are independent of the choices of the perturbation and the metric bysimilar discussion in <cit.>. For r ∈Λ_Y, we now introduce obstruction classes in HF^1_r(Y). These invariants aregeneralizations of[θ] ∈ HF^1(Y) considered in Subsection 7.1 of <cit.> and Subsection 2.1 of <cit.>.For r ∈Λ_Y, we set homomorphism θ^r :CF^r_1 byθ^r(a):= # (M(a,θ)_π/).As in <cit.> and <cit.>, we use the weighted norm on M(a,θ)_π to use Fredholm theory. From the same discussion for the proof of (δ^r)^2=0, we can show δ^r (θ^r)=0. Therefore it defines the class [θ^r] ∈ HF^1_r(Y). We call the class [θ^r] obstruction class.The class [θ^r] does not depend on the small perturbation and the metric. The proof is similar to the proof for original one [θ]. Now we give the proof of Example <ref>.Because X is homotopy equivalent to S^3× S^1, Q^i_X=∞ for i ∈. If the element [θ^1] ∈ HF^1_r(-Σ(2,3,6k-1)) does not vanish for r=1, we can apply Theorem <ref>. Frøyshov showed 0 ≠ [θ] ∈ HF^1(-Σ(2,3,6k-1)) by using the property of h-invariant of Proposition 4 in <cit.>. Then we get nonzero homomorphism θ:CF^r_1(Y) for r=∞. ( If r=∞, HF^1_r is the usual instanton Floer cohomology by the definition.)By using calculation about the value of the Chern-Simons functional of Section 7 in <cit.>, we can see θ^1:CF_1^r is nonzero for r=1 and CF_i^r is zero for r=1 and i∈ 2. This implies0 ≠ [θ^1] ∈ HF^1_r(-Σ(2,3,6k-1))forr=1.§.§ Fredholm theory and moduli theory on 4-manifolds with periodic end In <cit.>, Taubes constructed the Fredholm theory of some class of elliptic operators on 4-manifolds with periodic ends. He also extend moduli theory of SU(2) gauge theory on such non-compact 4-manifolds. In this subsection, we review Fredholm theory of a certain elliptic operator on 4-manifolds with the periodic endsas in <cit.> and define the Fredholm index of the class of operators, which gives the formal dimension of a suitable instanton moduli space on such non-compact 4-manifolds. First we formulate the 4-manifolds with periodic ends.Let Y be an oriented homology S^3 as in Section <ref>.Let W_0 be an oriented homology cobordism from Y to -Y. We get a compact oriented 4-manifold X by pasting W_0 with itself along Y and -Y. We give several notations in our argument. * The manifold W_i is a copy of W_0 for i ∈* We denote ∂(W_i) by Y^i_+∪ Y^i_- where Y^i_+(resp. Y^i_-) is equal to Y(resp. -Y) as oriented manifolds.* For(m,n)∈ (∪{ -∞} ) × (∪{∞}) with m<n, we setW[m,n]:=∐_m≤ i ≤ n W_i/ {Y^j_- ∼ Y^j+1_+ j ∈{m,⋯ ,n}}. We denote by W the following non-compact 4-manifoldW:= Y× (-∞,0] ∪ W[0,∞]/{∂ (Y× (-∞,0]) ∼ Y^0_+}. For a fixed Riemannian metric g_Y on Y, we choose a Riemannian metric g_W on W which satisfies * g_W|_Y× (-∞,-1]=g_Y× g^stan_.* The restriction g_W|_W[0,∞] is a periodic metric.There is a natural orientation on W[0,∞] and W induced by the orientations of W_0.The infinite cyclic covering space of X can be written by X≅ W[-∞,∞].Let T be the deck transformation of X which maps each W_i to W_i+1. By restriction, T has an action on W[0,∞]. We use the following smooth functions τ and τ' on Wτ, τ': Wsatisfying* τ (T|_W[0,∞](x))= τ(x)+1, τ'(T|_W[0,∞](x))=τ'(x)+1 for x ∈ W[0,∞].* τ|_Y× (-∞,-2]=0, τ'(y,t)= -t for (y,t) ∈ Y× (-∞,-2]. By the restriction of τ, we have a function on W[0,∞] which we denote by same notation τ. In this subsection, we review the setting of the configuration space of fields on W and define the Fredholm index of a kind of operator on W. We fix π∈∏(Y) in Subsection <ref> and assume that π is a non-degenerate perturbation. Let P_W be the product SU(2) bundle.For each element [a] ∈R(Y), we fix an SU(2) connection A_a on P_W which satisfying A_a|_Y× (-∞,-1] =_1^*a and A_a|_W[0,∞]=θ. If a is an irreducible (resp. reducible) connection, we define the space of connections on P_W by^W(a)_δ := { A_a+c| c ∈Ω^1(W)⊗_L^2_q,δ},( resp. ^W(a)_(δ,δ) := { A_a+c|c ∈Ω^1(W)⊗_L^2_q,(δ,δ)} )where Ω^1(W)⊗_L^2_q,δ(resp. Ω^1(W)⊗_L^2_q,(δ,δ)) is the completion of Ω^1(W)⊗ with L^2_q,δ-norm(resp. L^2_q,(δ,δ)-norm), q is a natural number greater than 3, and δ is a positive real number. For f ∈Ω^i(W)⊗ with compact support, we define L^2_q,δ-norm(resp. L^2_q,(δ,δ)-norm) by ||f||^2_L^2_q,δ:= ∑_0≤ j ≤ q∫_W e^δτ|∇_θ^j f |^2 dvol , (resp.||f||^2_L^2_q,(δ,δ):= ∑_0≤ j ≤ q∫_W e^δτ'|∇_θ^j f |^2 dvol)where ∇_θ is covariant derivertive with respect to the product connection θ. We use a periodic metric ||-|| on the bundle. Its completion is denoted by Ω^i(W)⊗_L^2_q,δ. We define the gauge group ^W(a)_δ:= { g ∈(P_W)_L^2_q+1,loc| ∇_A_a(g) ∈ L^2_q,δ} (resp. ^W(a)_(δ,δ):= { g ∈(P_W)_L^2_q+1,loc| ∇_A_a(g) ∈ L^2_q,(δ,δ)}),which has the action on ^W(a)_δ induced by the pull-backs of connections. The space ^W(a)_δ(resp. ^W(a)_(δ,δ)) has structure of Banach Lie group and the action of ^W(a)_δ on ^W(a)_δ(resp. ^W(a)_(δ,δ) on ^W(a)_(δ,δ)) is smooth. The configuration space for W is defined by ^W(a)_δ := ^W(a)_δ /^W(a)_δ (resp.B^W(a)_(δ,δ) := ^W(a)_(δ,δ) /^W(a)_(δ,δ)).Let s be a smooth function from W to [0,1] with s|_Y× (-∞ -2]=1, s|_Y× [-1,0] ∪_Y W[0,∞]=0.We define the instanton moduli space for W by M^W(a)_π,δ:={[A] ∈^W(a)_δ |ℱ_π(A)=0 } where ℱ_π is the perturbed ASD-map ℱ_π(A):=F^+(A)+sπ(A). For each A ∈^W(a)_δ, we have the bounded linear operator:d^*_L^2_δ_A+dℱ_A: ^1(W)⊗_L^2_q,δ (^0(W)⊕^+(W))⊗_L^2_q-1,δ ( resp. d^*_L^2_(δ,δ)_A+dℱ_A: ^1(W)⊗_L^2_q,(δ,δ) (^0(W)⊕^+(W))⊗_L^2_q-1,(δ,δ)). Taubes gave a criterion for the operator d^*_L^2_δ_A+dℱ_A=d^*_L^2_δ_A+ d^+_A+sdπ^+_A in (<ref>)(resp. (<ref>)) to be Fredholm in Theorem 3.1 of <cit.>. There exists a descrete set D inwith no accumulation points such that (<ref>)(resp. (<ref>)) is Fredholm for each δ in ∖ D.The discrete set D is defined by D:={δ∈ |the cohomology groupsH^i_z are acyclic for all z with|z|=e^δ/2}.The cohomology groups H^i_z are given by the complex:0^0(X) ⊗d_θ,z^1(X)⊗d^+_θ,z^+(X)⊗ 0,where d_θ,z:^0(X) ⊗^1(X)⊗ and d^+_θ.z :^1(X)⊗^+(X)⊗are given byd_θ,z(f)=z^τ d_p^*θ (z^-τ (p^*f)),d^+_θ,z(f)=z^τ d^+_p^*θ (z^-τ(p^*f)), where p is the covering map X X.(We fix a branch of ln z to define z^τ = e^τ ln z.) In above definition,d_θ,z(f) and d^+_θ,z(f) are sections of p^*P_X, however these are invariant under the deck transformation, we regard d_θ,z(f) and d^+_θ,z(f) as sections on P_X. The operators d_θ,z and d^+_θ.z in (<ref>) depend on the metric on X and τ however the cohomology groups H_i^z are independent of the choice of them. We now introduce the formal dimension of the instanton moduli spaces: Suppose a is an irreducible critical point of cs_Y,π. From Theorem <ref>, there exists δ_0>0such that (<ref>) is Fredholm for any δ∈ (0,δ_0) and A ∈^W(a)_δ. We define the formal dimension _W(a) of the instanton moduli spaces for W by the Fredholm index of (<ref>). For the case of a=θ, we also set _W(a) as the Fredholm index of (<ref>). The formal dimension _W(a) is calculated by using the following proposition.Suppose a is an irreducible critical point of cs_Y,π. The formal dimension _W(a) is equal to the Floer index (a) of a. If a is equal to θ, _W(a)= (a)=-3. First we take a compact oriented 4-manifold Z with ∂ Z=-Y. It is easy to show there is an isomorphism H^*(W[0,∞]) ≅ H^*(S^3). We define Z^+:= Z∪_Y W[0,∞] and fix a periodic Riemannian metric g_Z^+ satisfying g_Z^+|_W[0,∞]=g_W. In Proposition 5.1 of <cit.>, Taubes computed the Fredholm index of d^+_θ+ d^*_L^2_δ_θ as a operator on Z^+ in the situation that H_*(W[0,∞],)≅ H_*(S^3,)(The proof is given by using the admissibility of each segment W_0, however Taubes just use the condition H_*(W[0,∞],)≅ H_*(S^3,).):(d^+_θ+ d^*_θ) =-3(1-b_1(Z)+b^+(Z))for a small δ. Fix an SU(2)-connection A_a,θ on W with A|_Y× (-∞,-1]=a, A|_W[0,∞]=θ and an SU(2)-connection B_a on Z∪_Y Y× [0,∞)By the similar discussion about gluing of the operators on cylindrical end in Proposition 3.9 of <cit.>, we have (d_B_a^*+d_B_a^+)+(d_A_a,θ^*+d_A_a,θ^+)=(d_θ^*+d_θ^+).Donaldson show that (d_B_a^*+d_B_a^+) is equal to -(a)-3(1-b_1(Z)+b^+(Z)) in Proposition 3.17 of <cit.>. The second statement is similar to the first one. § CHERN-SIMONS FUNCTIONAL FOR HOMOLOGY S^3× S^1For a pair (X,ϕ) consisting of an oriented 4-manifold and non-zero element 0≠ϕ∈ H^1(X,), wegeneralize the Chern-Simons functional to a functional cs_(X,ϕ) on the flat connections on X. We define the invariants Q^i_X ∈_≥0∪{∞} for i∈ by using the value of cs_(X,ϕ). In our construction, cs_(X,ϕ) cannot be extended to a functional for arbitrary SU(2) connections on X.Let X be an oriented closed 4-manifolds equipped with 0 ≠ϕ∈ H^1(X,) and p:X^ϕ X be the -hold covering of X corresponding to ϕ∈ H^1(X,)≅ [X,B]. Recall that the bundle P_X and the set R(X) as in Section <ref>. Let f be a smooth map representing the class ϕ∈ H^1(X,)≅ [X,S^1], and f̃ is a lift of f.[Chern-Simons functional for a homology S^3× S^1] We define the Chern-Simons functional for X as the following mapcs_(X,ϕ):R(X)×R(X) , cs_(X,ϕ)([a],[b]):= 1/8π^2∫_X^ϕ Tr(F(A_a,b)∧ F(A_a,b)),where a,b are flat connections on P_X and A_a,b is an SU(2)-connection on P_X^ϕ:=X^ϕ× SU(2) which satisfies A_a,b|_f̃^-1(-∞,-r]=p^*a and A_a,b|_f̃^-1([r,∞))=p^*b for some r>0.We have an alternative description of cs_(X,ϕ)([a],[b]) when a closed oriented 3-manifold Y is given as a sub-manifold of X satisfying i_*[Y]=PD(ϕ) ∈ H_3(X,), where i is the inclusion Y X. Such Y is given as an inverse image of a regular value of f. We can take Y to be connected, and we assume this. We denote by W_0 the cobordism from Y to itself obtained from cutting X open along Y. Since Y is connected and ϕ≠ 0, W_0 is also connected. Then we choose the idenitification of X^ϕ and …∪_Y W_0 ∪_Y W_1 ∪_Y …. We have the following formula. cs_(X,ϕ)([a],[b])=cs_Y([i^*a])-cs_Y([i^*b]) Let A_a,b the SU(2)-connection on P_X^ϕ in Definition <ref>. Take a natural number N large enough to satisfy f̃^-1([-r,r]) ⊂ W[-N,N],for which we have∫_X^ϕ Tr(F(A_a,b)∧ F(A_a,b))= ∫_W[-N,N]Tr(F(A_a,b)∧ F(A_a,b)). =cs_Y(i^*_+A_a,b)-cs_Y(i^*_-A_a,b).Herei_+(resp. i_-) is inclusion from Y(resp. -Y) to X, and we use the Stokes theorem.When X is equal to Y× S^1, this map cs_(X,ϕ):R(X)×R(X) essentially coincides with the restricton of Chern-Simons functional cs_Y on Y by the following sense. For [a]∈R(Y× S^1), the restriction [i^*a] ∈R(Y) satisfies cs_Y([i^*a])=cs_(Y× S^1,PD([Y]))([a],[θ]),where i is a inclusion Y=Y× 1Y× S^1 and PD is the Poincaré duality.This is a corollary of Lemma <ref>. We have the following well-definedness. cs_(X,ϕ) does not depend on the choices of f, representatives a and b, and A_a,b.This is also a consequence of Lemma <ref>Let X be a closed oriented 4-manifold equipped with ϕ∈ H^1(X,).The invariant Q̃_(X,ϕ) is defined by.9 ∞if R^*(X)= ∅,inf{|cs_(X,ϕ)([a],[θ])+m| | m∈ , [a] ∈R^*(X) } if R^*(X) ≠∅,where R^*(X) is the subset of the classes of the irreducible connections inR(X). We now give a definition of Q^i_X ∈_≥ 0∪{∞}.Suppose that X is a homology S^3× S^1 and i is a positive integer. Let X be -fold covering space over X corresponding to the 1∈ H^1(X,)≅_PD H_3(X,). We set X^i:= X/ i. Since the quotient map p^i:XX^i is a -fold covering, this determine a class ϕ^i ∈ H^1(X^i,). We define Q^l_X∈_≥ 0∪{∞} by Q^l_X:=min_0≤ i ≤ lQ̃_(X^i,ϕ^i).We show the following lemma which is used in the proof of Key lemma(Lemma <ref>).Suppose that γ is a flat connection on W[m,n] satisfying the following conditions. * γ|_Y^m_+≅γ|_Y^n_-.* There exists u ∈ satisfying |cs_(W[m,n],PD[Y^m_+])(r(γ),θ)+ u|< Q^n-m+1_X, where r(γ) is a flat connection on W[0,k] given by pasting γ with itself along Y^0_+ ∪ Y^k_-.Then γ is gauge equivalent to θ. Suppose γ is notgauge equivalent to θ. The calculation H_1(W[m,n])≅ 0 and holonomy correspondence R(W[m,n]) ≅ (π_1(W[m,n]),SU(2))/ conjugateimply that there is no reducible SU(2) connection on W[m,n] except θ. Therefore γ is an irreducible connection on W[m,n]. Because W[m,n] X is the (n-m+1)-fold covering space of X,Q^n-m+1_X ≤|cs_(W[m,n],PD[Y^m_+])(r(γ),θ)+ u|holds by the definition of Q^i_X. This is a contradiction.§ COMPACTNESSThe compactness of the instanton moduli spaces for non-compact 4-manifolds is treated in <cit.>, <cit.>,<cit.> for cylindrical end case and in <cit.> for periodic end case. In <cit.> and <cit.>, they consider the instanton moduli spaces with the connections asymptotically convergent to the trivial connection on the end. We also follow their strategy by using Q^2l_Y+3_X defined in the previous section. More explicitly, in this section we explain a compactness result for the instanton moduli spaces for a non-compact manifold W[0,∞] with periodic end.§.§ Key lemmaLet W_0, W[0,∞] be the oriented Riemannian 4-manifolds introduced in the beginning of Subsection <ref>. By pasting W_0 with itself along its boundary Y and -Y, we obtain a homology S^3× S^1 which we denote by X. We consider the product SU(2)-bundle P_W[0,∞] on W[0,∞]. For q≥3 and δ>0, we define the instanton moduli space M^W[0,∞]_δ by M^W[0,∞]_δ := {θ +c ∈^1(W[0,∞])⊗_L^2_q,δ | F^+(θ+c)=0 }/ ,whereis the gauge group:={g ∈(P_W[0,∞]) ⊂(ℂ^2)_L^2_q+1,loc | dg ∈ L^2_q,δ},and the action ofis given by the pull-backs of connections. For f∈Ω^i(W[0,∞])⊗ with compact support, we define L^2_q,δ norm by the following formula||f||^2_L^2_q,δ:= ∑_0≤ j ≤ q∫_W[0,∞] e^δτ|∇_θ^j f|^2dvol ,where ∇_θ is the covariant derivertive with respect to the product connection. We use the periodic metric |-| which is induced from the Riemannian metric g_W. Its completion is denoted by Ω^i(W[0,∞])⊗_L^2_q,δ.Our goal of this section is to show the next theorem under the above setting. Under Assumption <ref> the following statement holds. There exist δ'>0 satisfying the following property. Suppose that δ is a non-negative number less than δ' and {A_n} is a sequence in M^W[0,∞]_δ satisfying sup_n ∈||F(A_n)||^2_L^2(W[0,∞]) < min{8π^2,Q^2l_Y+3_X}. Then for some subsequence {A_n_j}, a positive integer N_0 and some gauge transformations {g_j} on W[N_0,∞], the sequences {g_j^*A_n_j} converges to some A_∞ in L^2_q,δ(W[N_0,∞]).The proof of Theorem <ref> is given in the end of Subsection <ref>. We use the following estimate.For a positive number c_1>0, there exists a positive number c_2>0 satisfying the following statement.For any SU(2)-connection a on Y^0_+ and any flat connection γ on W[0,k] satisfying the following conditions * sup_x ∈ Y^+_0∑_0≤ j ≤ 1|∇^(j)_γ(a-(l^0_+)^*γ)(x)|<c_1.* γ|_Y^0_+ ≅γ|_Y^k_-., the inequality| cs_Y([a])-cs_(W[0,k],PD[Y^0_+])(r(γ),θ)| ≤ c_2 sup_x ∈ Y^+_0∑_0≤ j ≤ 1|∇^(j)_(l^0_+)^*γ(a-(l^0_+)^*γ)(x)|^2holds, where r(γ) is a flat connection on W[0,k] given by pasting γ with itself along Y^0_+ ∪ Y^k_- and l^0_+:Y^0_+ W_0 is the inclusion.Lemma <ref> imply|cs_Y([a])-cs_(W[0,k],PD[Y^0_+])(γ,θ)| =|cs_Y(a)-cs_Y((l^0_+)^*γ)|.Since (l^0_+)^*γ is a flat connection on Y^0_+, we have=1/8π^2| ∫_Y^0_+Tr( (a-(l^0_+)^*γ) ∧ d_(l^0_+)^*γ(a-(l^0_+)^*γ) + 2/3(a-(l^0_+)^*γ) ^3| ≤1/8π^2vol_Y (sup_x ∈ Y^0_+|∇_(l^0_+)^*γ(a-(l^0_+)^*γ) (x)||(a-(l^0_+)^*γ)|+ 2/3sup_x ∈ Y^0_+|(a-(l^0_+)^*γ)|^3) ≤ c_2 sup_x ∈ Y^0_+∑_0≤ j ≤ 1|∇^(j)_(l^0_+)^*γ(a-(l^0_+)^*γ)(x)|^2.Next lemma gives us a key estimate. We use Q^2l_Y+3_X to obtain an estimate of the difference between an ASD-connection and the trivial flat connection on the end W[0,∞]. Suppose that Y satisfies Assumption <ref>. There exists a positive number c_3 satisfying the following statement. For A ∈ M^W[0,∞]_δ satisfying 1/8π^2||F(A)||^2_L^2(W[0,∞]< min{1, Q^2l_Y+3_X}, there exists a positive number η_0 which depends only on the difference min{1, Q^2l_Y+3_X}-1/8π^2||F(A)||^2_L^2 such thatthe following condition holds. Note that if K is sufficiently large, the inequality ||F(A)||^2_L^2(W_k)< η_0 is satisfied for every k>K. When K satisfies this property, there exist gauge transformations g_k over W[k,k+2] such that sup_x∈ W[k,k+2]∑_0≤ j≤ q+1|∇^j_θ(g_k^*A|_W[k,k+2]-θ)(x)|^2 ≤ c_3||F(A)||^2_L^2(W[k-l_Y-2,k+l_Y+3])holds for k>K+l_Y+3. For k>K+l_Y+3, we apply Lemma 10.4 in <cit.> to A|_W[k-l_Y-1,k+l_Y+2].Then we obtain gauge transformations g_k and flat connections γ_k over W[k-l_Y-1,k+l_Y+2] satisfyingsup_x ∈ W[k-l_Y-1,k+l_Y+2]∑_0≤ j≤ q+1|∇^j_γ(g_k^*A|_W[k-l_Y-1,k+l_Y+2]-γ_k)(x)|^2 ≤ c_3||F(A)||^2_L^2(W[k-l_Y-2,k+l_Y+3])≤ (2l+5)c_3ηfor a small η. By using the pull-backs of γ_k from W[k-l_Y-1,k-1](resp. W[k+2,k+l_Y+2]) to Y^i_+, we get the flat connections over Y. Then we get l_Y+1 flat connections by using this method. Under the assumption that l_Y=# R(Y), same flat connections appear by the pigeonhole principle. We choose two numbersk(1)^±<k(2)^± which satisfy (l^k(1)^±_+)^*γ_k ≅(l^k(2)^±_+)^*γ_k as connections on Y, where k(1)^+ and k(2)^+ are elements in {k-l_Y-1,⋯ , k-1} (resp. k(1)^-,k(2)^- ∈{k+2,k+l_Y+2}). The map l^k_±:Y^k_± W_k is the inclusion.Suppose ||F(A)||^2_L^2(W_k)<η holds for k>K+l_Y+3. For sufficiently small η, the flat connection γ_k is isomorphic toθ.The properties of k(1)^± and k(2)^±, (<ref>) and Lemma <ref> imply| cs_Y((l^k(1)^±_+)^* g_k^*A) -cs_((W[k(1)^±,k(2)^±],PD[Y^k(1)^±_+])(r(γ_k),θ)|≤ (2l+5)c_3η c_2.We also havecs_Y((l^k(1)^±_+)^* g_k^*A)= cs_Y(l^k(1)^±_+)^*A)+deg(g_k|_Y^k(1)^±_+)= ||F(A)||^2_L^2(W[k(1)^±,∞])+deg(g_k|_Y^k(1)^±_+).We choose η_0 satisfying the following condition: (2l+5)c_3η_0 c_2 <Q^2l_Y+3_X - 1/8π^2∫_W[k(1)^±,∞]|F(A)|^2,where the right hand side is positive by the assumption of A. We obtain |cs_(W[k(1)^±,k(2)^±], PD[Y^k(1)^±])(r(γ_k),θ)+deg(g_k|_Y^k(1)^±_+)|< Q^2l_Y+3_X from (<ref>), (<ref>) and (<ref>). Then Lemma <ref> imply γ|_W[k(1)^±,k(2)^±]≅θ. Similarly the inequalitysup_x ∈ W[k(2)^-,k(1)^+]∑_0≤ j≤ q|∇^j_γ_k(g_k^*A|_W-γ_k|_W[k(2)^-,k(1)^+])(x)|≤ (2l_Y+5)c_3η_0holds over W[k(2)^-,k(1)^+]. From above discussion, l^k(2)^-_+^*γ_k and i^k(1)^+_+^*γ_k are gauge equivalent. By Lemma <ref>, we also get | cs_Y((l^k(2)^-_+)^*g_k^*A)-cs_(W[k(2)^-,k(1)^+], PD[Y^k(2)^-_+])(r(γ_k),θ)|≤ (2l_Y+5)c_3η_0c_2.By choice of η_0, we have |cs_(W[k(2)^-,k(1)^+], PD[Y^k(2)^-_+])(r(γ_k),θ)+ deg(g_k|_Y^k(2)^-_+)|< Q^2l_Y+3_Xand get γ_k|_W[k(2)^-,k(1)^+]≅θ by using Lemma <ref> as above. §.§ Chain convergenceWe introduce the following notion which is crucial for our proof of the compactness theorem(Theorem <ref>).[Chain decomposition of a sequence of the instantons] For a fixed number η>0 and a sequence {A_n}⊂ M^W[0,∞]_δ satisfying sup_n∈||F(A_n)||^2_L^2(W[0,∞])<∞,when a finitely many sequences {s^j_n(η)}_1≤ j ≤ m of non-negative numbers satisfies||F(A_n)||^2_L^2(W_s)>ηs=s^j_nfor some j,ands^1_n < … < s^m_n ,we call {s^j_n}_1≤ j ≤ m chain decomposition of {A_n}⊂ M^W[0,∞]_δ for η. For any sequence {A_n} satisfying sup_n∈||F(A_n)||^2_L^2(W[0,∞])<∞ and any η>0, we can show the existence of a chain decomposition for η>0 if we take a subsequence of {A_n}.First we give the next technical lemma.Let A be a L^2_q ASD-connection on W[k,∞] satisfying 1/8π^2 ||F(A)||^2_L^2(W[k,∞])<1.Then there exists a positive number c_4 which depends only on the difference 1-1/8π^2 ||F(A)||^2_L^2(W[k,∞])such that the following statement holds. Suppose there exists a gauge transformation g on Y^k_+ satisfying ∑_0≤ j≤ 1sup_x ∈ Y^k_+| ∇^j_θ (g^*(l^k_+)^*A-θ)(x) |^2 ≤ c_4.Then g is homotopic to the identity gauge transformation.By the property of cs_Y, we have|deg(g)|= |cs_Y(g^*(l^k_+)^*A)-cs_Y((l^k_+)^*A)| ≤ |cs_Y(g^*(l^k_+)^*A)|+||F(A)||^2_L^2(W[k,∞]) ≤∑_0≤ j≤ 1sup_x ∈ Y^k_+| ∇^j_θ (g^*(l^k_+)^*A-θ)(x)|^2+||F(A)||^2_L^2(W[k,∞]).We define c_4< 1/2(1-||F(A)||^2_L^2(W[k,∞])), then |deg(g)| < 1holds. This implies the conclusion.The next proposition is a consequence of Lemma <ref> and <ref>.Let η be a positive number and {s^j_n}_1≤ j ≤ m be a chain decomposition for η of a sequences {A_n} in M^W[0,∞]_δ with 1/8π^2sup_n ∈||F(A_n)||^2_L^2(W[0,∞]) <min{1,Q^2l_Y+3_X}.Then there exists a subsequence {A_n_i} of {A_n} such that sup_i∈|s^m_n_i|<∞ holds.Suppose there exists η_0>0 which does not satisfy the condition of Proposition <ref>. There exist a chain decomposition {s^j_n(η_0) }_0≤ j≤ m of {A_n} which satisfies s^m_n∞ as n ∞. We take a sufficiently small η>0, we will specify η later. Choose a subsequence of {A_n} which allows a chain decomposition for η. For simplify, we denote the subsequence by same notation {A_n}. We denote the chain decomposition of {A_n} for η by{t^j_n(η)}_1≤ j ≤ m'. There are two cases for {t^j_n(η)}_1≤ j ≤ m': * There exists j”∈{0,⋯, m'-1} satisfying t^j”_n - t^j”+1_n -∞.* There is no j”∈{0,⋯, m'-1} satisfying t^j”_n - t^j”+1_n -∞.We define a sequence by u_n(η):=⌊t^j”_n(η)+t^j”+1_n(η)/2⌋∈for the first case and byu_n(η):=0for the second case, where ⌊ - ⌋ is the floor function. Applying Lemma <ref> to A_n|_W[u_n(η),u_n(η)+2], we get the gauge transformation g_n on W[u_n(η)-l_Y-2,u_n(η)+l_Y+3] satisfying sup_x ∈ W[u_n(η),u_n(η)+2]∑_1≤ j ≤ q+1|∇^j_θ(g_n^*A_n|_W[u_n,u_n(η)+2] - θ)|^2≤ c_3(2l_Y+5)η,for small η and large n. Because A_n is the ASD-connection for each n, we have1/8π^2∫_W[u_n,∞]Tr(F(A_n)∧ F(A_n))=1/8π^2∫_W[u_n,∞]|F(A_n)|^2>η_0for large n. On the other hand, by the Stokes theorem1/8π^2∫_W[u_n,∞]Tr(F(A_n)∧ F(A_n))=cs_Y(l^u_n_+^*A_n)holds.By Lemma <ref>, we have cs_Y(l^u_n_+^*A_n)=cs_Y(l^u_n_+^*g_n^* A_n)for small η. Therefore (<ref>) and Lemma <ref> gives |cs_Y(l^u_n_+^*A_n)|≤ c'c_3(2l_Y+5)η.We choose η satisfyingc'c_3(2l_Y+5)η<1/8π^2η_0.For such η, we have|cs_Y(l^u_n_+^*A_n)|<1/8π^2η_0.On the other hand, η_0 satisfies1/8π^2η_0 <1/8π^2∫_W[u_n,∞]|F(A_n)|^2=|cs_Y(l^u_n_+^*A_n)|which is a contradiction. §.§ Exponential decayIn the instanton Floer theory, there is an estimate called exponential decay about the L^2-norm of curvature of the instanton over cylindrical end.We give a generalization of the exponential decay estimate over W[0,∞]. In the end of this subsection, we also give a proof of Theorem <ref>. There exists a constant c_5 satisfying the following statement. For A ∈ M^W[0,∞]_δ satisfying 1/8π^2||F(A)||^2<min{1,Q^2l_Y+3_X}, there exists η_1>0which depends only on the differencemin{ Q^2l_Y+3_X,1} -1/8π^2||F(A)||^2 such that the following condition holds. Let K>0 be a positive number satisfying ||F(A)||^2_L^2(W_k)< η_1 for any k>K, the inequality ||F(A)||^2_L^2(W[k,k+m]) ≤ c_3( ||F(A)||^2_L^2(W[k-l_Y-2,k+l_Y+3])+||F(A)||^2_L^2(W[k+m-l_Y-2,k+m+l_Y+3])holds for k>K+l_Y+3. Let η be the positive number in Lemma <ref> which depends only the difference Q^2l_Y+3_X-1/8π^2||F(A)||^2. Then for k>K+l_Y+3, we have the following inequalitiessup_x ∈ W_k∑_0≤ j≤ q|∇^j_θ(g_k^*A-θ)(x)|^2≤ c_3||F(A)||^2_L^2(W[k-l_Y-2,k+l_Y+3]) ≤ c_3(2l_Y+5)η_1andsup_x ∈ W_k+m∑_0≤ j≤ q|∇^j_θ(g_k+m^*A-θ)(x)|^2≤c_3||F(A)||^2_L^2(W[k+m-l_Y-2,k+m+l_Y+3] ≤ c_3(2l_Y+5)η_1.These inequality (<ref>), (<ref>) and Lemma <ref> imply that for sufficiently small η_1, the gauge transfromation g_k|_Y^+_k(resp. g_k+m|_Y^-_k+m) is homotopic to the constant gauge transformation. Hence, there exists a gauge transformation ĝ on W[k,k+m] satisfying ĝ|_W_k=g_k and ĝ|_W_k+m=g_k+m, moreover, since A is the ASD connection, we have||F(A)||^2=||F(ĝ^*A)||^2_L^2(W[k,k+m])=8π^2 (cs_Y((l^k_+)^*g_k^*A))-cs_Y((l^k+m_-)^*g_k+m^*A)). Applying the inequalities (<ref>) and (<ref>) again, we get the conclusion.There exists δ'>0 satisfying the following statement. Suppose A is an element in M^W[0,∞]_δ satisfying the assumption of Lemma <ref>. Then there exists c_5(K)>0 satisfying the following inequality. ||F(A)||^2_W[k-l_Y-2,k+l_Y+3]≤ c_5(K)e^-kδ'. for k>K+l_Y+3.This is a consequence of Lemma <ref> and Lemma 5.2 in <cit.> by applying q_i = ||F(A)||^2_W[i-l_Y-2,i+l_Y+3].By using a similar argument in Lemma 4.2 and Lemma 7.1 of <cit.> we have:For a positive number c_7, there exists a constant c_8 satisfying the following statement holds. Suppose we have an L^2_q connection A, the gauge transformations g_k on W[k-1,k+1] satisfying∫_W[k-1,k+1]∑_0≤ j≤ q+1|∇^j_θ(g_k^*A|_W[k-1,k+1]-θ)|^2 ≤ c_7||F(A)||^2_L^2(W[k-l-3,k+l+2]).for any non-negative integer k. Then there exists the positive integer n_0 and a gauge transformation g on W[n_0,∞] satisfying the following condition:∫_W[k-1,k+1]∑_0≤ j≤ q+1|∇^j_θ(g^*A|_W[k-1,k+1]-θ)|^2 ≤ c_8||F(A)||^2_L^2(W[k-l-3,k+l+2])for k>n_0.We use this lemma to prove the next proposition.There exists δ'>0 satisfying the following condition. Let K>0 be a positive number and {A_n} be a sequence in M^W[0,∞]_δ satisfying the following properties: * 0< min{Q^2l_Y+3_X,1}- sup_n∈1/8π^2||F(A_n)||^2.* There exists a chain decomposition {s^n_j} of {A_n} for η_2 satisfying sup_n∈|s^m_n(η_*)|<∞for η_2:= inf_n∈{η | constants which depend on A_n in Lemma <ref> and <ref>}.Then there exist a positive integer N_0, gauge transformations {g_j} on W[N_0,∞] and subsequence {A_n_j} of {A_n} such that {g_j^*A_n_j} converge to some A_∞ in L^2_q,δ(W[N_0,∞]) for any 0≤δ<δ'. If we apply the Lemma <ref> to A_n, there exists gauge transformations g^n_k on W[k-1,k+1] satisfying the following condition: for k>l_Y+K+3,∫_W[k-1,k+1]∑_0≤ j≤ q+1|∇^j_θ(g^n_k^*A_n|_W[k-1,k+1]-θ)|^2 ≤ c_3||F(A_n)||^2_L^2(W[k-l-3,k+l+2]). On the other hand, we have ||F(A_n)||^2_L^2(L^2(W[k-l-3,k+l+2])≤ c_6(K)e^-δ' kby using the exponential decay estimate(Proposition <ref>). Using (<ref>), we can show that n_0 uniformly with respect to n in Lemma <ref>. So there exist a large natural number N_0 and a gauge transformation on W[N_0,∞] for each n satisfying∫_W[k-1,k+1]∑_0≤ j≤ q+1|∇^j_θ(g_n^*A_n|_W[k-1,k+1]-θ)|^2 ≤ c'_8||F(A_n)||^2_L^2(W[k-l-3,k+l+2])≤ c_6(K)c'_8e^-δ'k,where the last inequality follows from (<ref>). We set g_n^*A_n=θ +a_n. Then we have||a_n||_L^2_q+1,δ(W[N_0,∞])=∑_0≤ j≤ q+1∫_W[N_0,∞]e^δτ|∇^j_θ(a_n)|^2 ≤∑_0≤ j≤ q+1∑_N_0 ≤ i≤∞ e^iδ∫_W_i |∇^j_θ(a_n)|^2.Putting this estimate and (<ref>) together, we have ||a_n||^2_L^2_q+1,δ(W[k,∞] )≤ c_9e^(δ-δ')kfor k>N_0.We take a subsequence of {a_n} which converges on any compact set in L^2_q(W[k,∞]) by using the Relich Lemma. We denote the limit in L^2_q,loc by a_∞. Then the exponential decay (<ref>) and a standard argument implies that {a_n} converges a_∞ on W[N_0,∞] in L^2_q,δ-norm. We now give the proof of Theorem <ref>.We choose η_2 in Proposition <ref>.After taking subsequence of {A_n}, we consider the chain decomposition {s^j_n}_1≤ j ≤ m for η_2 of {A_n}.From Proposition <ref>, {s^j_n} has upper bound by some K>0 after taking a subsequence of {A_n} again. So we can apply Proposition <ref>, we get the conclusion. § PERTURBATION AND ORIENTATIONTo prove the vanishing [θ^r]=0 in Theorem <ref>, we use the moduli spaces M^W(a)_π,δ and need the transversality for the equation F^+(A)+sπ(A)=0. We also need the orientability of M^W(a)_π,δ.§.§ Holonomy perturbation(2)In <cit.>, Donaldson introduced the holonomy perturbation with compact support for irreducible ASD-connections. Combining the technique in <cit.> and the compactness theorem (Theorem <ref>), we get sufficient perturbations to achieve required transvesality. Letπ be an element in ∏(Y) and a be a critical point of cs_Y,π. We use the following notations:* Γ (W):={l:S^1 × D^3W| l: orientation preserving embedding}.* Λ^d(W):= {(l_i, μ^+_i)_1≤ i ≤ d∈Γ(W)^d× (^+(W)⊗))^d| suppμ^+_i ⊂ l_i }.* Λ(W):= ⋃_d∈Λ^d(W). Let χ:SU(2)𝔰𝔲(2) beχ(u):=u-1/2tr(u)idand fix μ^+_i ∈Ω^+(W)⊗𝔰𝔲(2) supported on l_i(S^1× D^3) for i∈{1,⋯ ,d}. For ϵ∈^d, we setσ_Ψ(A,ϵ):= ∑_1≤ i ≤ dϵ_iχ(_x ∈ l_i(S^1× D^3) (A))μ^+_i,where _x∈ l_i(S^1× D^3) is a holonomy around the loop t ↦ l_i(t,y_x) satisfying x=l_i (t_x,y_x) for some t_x and ϵ=(ϵ_i)_1≤ i ≤ d. For Ψ=(l_i,μ_i)_1≤ i ≤ d∈Λ, Donaldson defined the holonomy perturbation of the ASD-equation:ℱ_π,Ψ(A,ϵ):=F^+(A)+sπ(A)+σ_Ψ(A,ϵ)=0.The map σ_Ψ(-,ϵ) is smoothly extended to the map ^W(a)_δΩ^+(W)⊗𝔰𝔲(2)_L^2_q-1,δ and the map ^W(a)_(δ,δ)Ω^+(W)⊗𝔰𝔲(2)_L^2_q-1,(δ,δ). For Ψ and ϵ∈^d, the perturbed instanton moduli space are defined by M^W(a)_π,Ψ,ϵ,δ:={ c ∈^W(a)_δ |ℱ_π,Ψ(c,ϵ)=0 }in the case of a∈R^*(Y)_π andM^W(a)_π,Ψ,ϵ,(δ,δ):={ c ∈^W(a)_(δ,δ) |ℱ_π,Ψ(c,ϵ)=0 }in the case of Stab(a)=SU(2). For a fixed ϵ∈^d, if the operator d(ℱ_π,Ψ)_(A,0):T_A^W(a)_δ×^d Ω^+(W)⊗_L^2_q-1,δ. is surjective for all [A] ∈ M^W(a)_δ,π,Ψ,ϵ, we call (Ψ,ϵ) regular perturbation for a∈R^*(Y)_π. Let FM^W(a)_δ,π,Ψ be the family version of the perturbed instanton moduli spaces defined byFM^W(a)_δ,π,Ψ:={ (c,ϵ) ∈^W(a)_δ×^d|ℱ_π,Ψ(c,ϵ)=0 }. Suppose that Y satisfies Assumption <ref>. There exists δ'>0 such that for a fixed δ∈ (0,δ'), the following statement holds. Suppose π is a holonomy perturbation which is non-degenerate and regular. Let a be an irreducible critical point of cs_Y,π with cs_Y(a)<min{Q^2l_Y+3_X,1}. We assume the next three hypotheses for (π,a). * For [A] ∈ M^W(b)_π,δ,1/8π^2sup_n∈ ||F(A)+sπ(A)||^2_L^2(W) <min{1 ,Q^2l_Y+3_X}, where b is an element ofR(Y)_π withcs_Y,π (b) ≤ cs_Y,π(a). * The linear operatord^+_θ+sdπ^+_θ:T_θ^W(θ)_(δ,δ)×^d Ω^+(W)⊗_L^2_q-1,(δ,δ)is surjective.* M(c)_π,δ is empty set for c∈R_π(Y) satisfying cs_Y,π(c)<0. Then there exist a small number η>0 and a perturbation Ψ such that the mapdℱ_π,Ψ :T^W(a)_δ×^d Ω^+(W)⊗_L^2_q-1,δ.is surjective for all point in ℱ_π,Ψ^-1(0)∩ (^W(a)_δ× B^d(η)).First we show that the surjectivity of dℱ_π,Ψ at the point inℱ_π,Ψ^-1(0)∩ (^W(a)_δ×{ 0} ). Second, we show that there exists a positive number η>0 such that dℱ_π,Ψ is surjectivie at the point inℱ_π,Ψ^-1(0)∩ (^W(a)_δ× B^d(η)). We name the critical point of cs_Y,π by0=cs_Y,π( θ=a_0 ) ≤ cs_Y,π(a_1)≤ cs_Y,π (a_2 ) ⋯≤ cs_Y,π(a_w=a).The proof is induction on w and there are four steps. For an irreducible element A ∈(a_w)_δ with 0≠ (d(ℱ_π,Ψ)_(A,0)), there exists Ψ(A)={l^A_i,μ^A_i }_1≤ i ≤ d(A) such that d(ℱ_π,Ψ)_(A,0)| ^d(A) generates the space (d^+_A+sdπ^+ _A).The proof is essentially the same discussion of Lemma 2.5 in <cit.>.We fix h ∈^+(W)⊗_L^2_q-1,δ satisfying 0≠ h ∈ (d^+_A+sdπ _A) with ||h||_L^2=1. The unique continuation theorems: Proposition 8.6 (ii) of <cit.> for the equation (d^+_A(-)+dπ^+_A)^*(-)=0 on Y× (-∞,-1] and Section 3 of <cit.> for the equation (d^+_A)^*(-)=0 on W[0,∞] imply h|_Y× [-2,0] ∪_Y W_0≠ 0. Then we choose x_h in Y× [-2,0] ∪_Y W_0 so that h(x_h)≠0 holds. Since A is the irreducible connection, (A,x_h)={_l(A) ∈ SU(2)|l : loop based atx_h } is a dense subset of SU(2).So we can choose the loops l^h_i based at x_h satisfying {e_i=χ(_l^h(A))}_i generates . For a small neighborhood U_x_h of x_h, we can write h byh|_U_x_h= ∑_1≤ i ≤ 3h_i ⊗ e_i.By using a smoothing of δ function, we have<h|U_x_h,∑_1≤ i ≤ 3μ^+_i(h)⊗χ(Hol_l_i(A))>_L^2(U_x_h)≠ 0,where μ^+_i(h) are three self dual 2-forms supported on U_x_h. For a fixed generator {h^1,… ,h^u} of (d^+_A+sdπ^+_A), we get the points {x_h^j}⊂ Y× [-2,0] ∪_Y W_0, small neighborhoods {U_x_h^j}, loops {l^h^j_i} and self dual 2-forms {μ^+_i(h^j)} satisfying (<ref>) for all h^j. We extend the maps l^h^j_i:S^1 W to embeddings S^1× D^3W. We can choose U_x_h^j satisfying U_x_h^j⊂ l^h^j.We set Ψ(A):= (l^h^j_i,μ^+_i(h^j))_i,j∈Λ(W),which satisfies the statement of Step <ref>. For j≥0 satisfying cs_Y,π(w_j)=0, we show:For an element b ∈R^*(Y)_π(resp. b=θ)satisfying cs_Y,π(b)=0, there exists a perturbation Ψ^b such that the operatordℱ_π,Ψ^b|_(A,0):T_A^W(b)_δ×^d^+(W)⊗_L^2_q-1,δ (resp.dℱ_π,Ψ^b|_(A,0):T_A^W(b)_(δ,δ)×^d^+(W)⊗_L^2_q-1,(δ,δ))is surjective for A ∈ (ℱ_π,Ψ^b)^-1(0) ∩ (^W(a)_δ×{ 0}) (resp. A ∈ (ℱ_π,Ψ^b)^-1(0) ∩ (^W(a)_(δ,δ)×{ 0})).First we show that M^W(b)_π,δ is compact. Let {[A_n]} be any sequence in M^W(b)_π,δ. By the second hypothesis, we have 1/8π^2sup_n ∈||F(A_n)||^2_L^2(W[0,∞])≤1/8π^2sup_n ∈||F(A_n)+sπ(A_n)||^2_L^2(W)<min{1 ,Q^2l_Y+3_X}.By Theorem <ref>, there exist a large positive number N and the gauge transformations {g_n} over W[N,∞] such that {g_n^*A_n}convergesover W[N,∞] for small δ after taking a subsequence. Note that Y× (-∞,0] ∪_Y W[0,N+1] is a cylindrical end manifold and we can apply the general theory developed on Section 5 of <cit.>. In particular, there exist gauge transformations {h_n} on Y× (-∞,0] ∪_Y W[0,N+1] such that {h_n^*A_n|_Y× (-∞,0] ∪_Y W[0,N+1]} has a chain convergent subsequence in the sense in Section 5 in <cit.> because the bubble phenomenon does occur under the first hypothesis1/8π^2sup_n ∈||F(A_n)+sπ(A_n)||^2_L^2(W)<1.By gluing {g_n} and {h_n}, we obtain a chain convergent subsequence[A_n_j]([C^1],…,[C^N], [A^0]) ∈ M(b=c_1,c_2)_π×…× M(c_v,c_v+1)_π× M^W(c_v+1)_π,δ with c_i ∈R(Y)_π .Suppose that [A_n_j][A^0] ∈ M(b)_π,δ does not hold. We get cs_Y,π(c_v+1)<0 because the moduli spacesM(b,c_1)_π,⋯ , M(c_v,c_v+1)_πare non-empty sets. However this contradicts to the assumption of M(c)_π,δ=∅ for c∈R(Y)_π with cs_Y,π(c)<0. When b is an irreducible connection, the compactness of M(b)_π,δ, Step <ref> and the openness of surjective operators imply Step <ref>. When b is equal to θ, the second hypothesis implies Step <ref>. For the inductive step, we show:Suppose there is a perturbation Ψ^w-1=(l^w-1_i, μ^w-1_i)_i∈Λ(W) such that the operatorsdℱ_π,Ψ^w-1|_(A,0):T_A^W(a_j)_δ^+(W)_L^2_q-1,δ is surjective for (A,0) ∈ (ℱ_π,Ψ^w-1)^-1∩ (^W(a_j) ×{0}) and j ∈{1, ⋯, w-1}. Then the spaceK_w:={A∈ M^W(a_w)_π,δ | 0≠ (dℱ_π,Ψ^w-1|_(A,0)) ⊂Ω^+(W)_L^2_q-1,δ}is compact.Let {[A_n]} be a sequence in K_w. By the similar estimate in Step <ref> and Theorem <ref>, we get a chain convergent subsequence[A_n_j]([B^1],…,[B^N], [A^0]) ∈ M(a_w=b_1,b_2)_π×…× M(b_v,b_v+1)_π× M^W(b_v+1)_π,δ with b_i ∈R(Y)_π. Suppose that [A_n_j][A^0] ∈ M(a_w)_π,δ does not hold.In this case, the operators d^+_B^i+dπ_B^i on Y× and the operators dℱ_π,Ψ^w-1|_(A^0,0) on W are surjective in the suitable functional spaces by the assumption of π and the induction. For large j, the operator dℱ_π,Ψ^w-1|_(A^n_j,0) can be approximated by the gluing of the operators d_B^i^++dπ_B^i, dℱ_π,Ψ^w-1|_(A^0,0). By gluing the right inverses of them as in Theorem 7.7 of <cit.>, dℱ_π,Ψ^w-1|_(A^n_j,0) also has a right inverse for sufficiently large j. This is a contradiction and we have the conclusion of Step <ref>. For induction, we need to show:There exists the perturbation Ψ^w satisfying the surjectivity of the operatordℱ_π,Ψ^w:^W(a_w)_δ×^d^+(W)⊗_L^2_q-1,δfor any point in(ℱ_π,Ψ^w)^-1(0)∩ ((a_w)_δ×{ 0} ).We take the perturbation Ψ_A=((l^j_A),(μ^+_j(A))) for each A∈ K_w in Step 1. Because K_w is compact and surjectivity of the operators is open condition, there exist {A_1,⋯ ,A_k}⊂ K_w and a perturbation Ψ^w such thatdℱ_π,Ψ^w|_(A,0):^W(a_w)_δ×^d^+(W)⊗_L^2_q-1,δis surjective for all (A,0) ∈ (ℱ_π,Ψ^w)^-1(0) ∩ (^W(a)_δ×{ 0}). Here Ψ^w is defined by Ψ^w:=((l^j_A_1⋯ l^j_A_k, l^w-1_i) ,(μ^+_j(A_1),⋯,μ^+_j(A_k), μ^w-1_i))which satisfies the property in Step <ref>.Second, we show that the operator dℱ_π,Ψ^w is surjective for any point inℱ_π,Ψ^w^-1(0)∩ (^W(a)_δ×D^d(η) ). Suppose there is no η such that the statement holds. Then there is a sequence {(A_n,ϵ_n)} in M^W(a)_δ,π,Ψ,ϵ_n which satisfies that ϵ_n0 as n∞ and dℱ_π,Ψ^w|_(A_n,ϵ_n) is not surjective for all n∈. Because the bubble does occur, {A_n} has a chain convergent subsequence to ([B^1],… , [B^N],[A^0]) ∈ M(b_0,b_1)_π×…× M(b_v-1,b_v)_π× M^W(b_v)_δ,π,Ψ,0 for some b_i ∈R(Y)_π. Since π is a regular perturbation and dℱ_π,Ψ^w|_(A^0,0) is surjective, there exist the right inverses of d^+_B^1+dπ^+_B^1, …,d^+_B^N+dπ^+_B^N and dℱ_π,Ψ^w|_(A^0,0) for suitable functional spaces. By the gluing of the right inverses as in Step <ref>, dℱ_π,Ψ^w|_(A_N,ϵ_N) also has the right inverse for large N. This is a contradiction and this completes the proof. For a given data (δ,π,a) in Lemma <ref>, there exist η>0, a perturbation Ψ and a dense subset of R ⊂ B^d(η) ⊂^d such that (Ψ,b) is a regular perturbation for b ∈ R. This is a conclusion of Lemma <ref>, the argument in Section 3 of <cit.> and the Sard-Smale theorem. Applying the implicit function theorem, we get a structure of manifold of M^W(a)_δ,π,Ψ,b. Its dimension coincides with the Floer index (a) of a by Proposition <ref>. Therefore we have:For given data (δ,π,a) in Lemma <ref>, there exist η>0, a perturbation Ψ and a dense subset of R ⊂ B^d(η) ⊂^d such that M^W(a)_δ,π,Ψ,b has a structure of manifold of dimension (a).§.§ OrientationIn <cit.>, Donaldson showed the orientability of the instanton moduli spaces for closed oriented 4-manifolds. In this subsection, we deal with the case for non-compact 4-manifold W by generalizing Donaldson's argument.More explicitly, we show that the moduli space M^W(a)_δ,π is orientable. We also follow Fredholm and moduli theory in <cit.> to formulate the configuration space for SU(l)-connections for l≥ 2. Let Z be a compact oriented 4-manifold which satisfies ∂ Z=Y and H_1(Z)≅ 0. We set Z^+:=(-Z)∪_Y W[0,∞] and Ẑ:= (-Z)∪_Y Y× [0,∞). Fix a Riemannian metric g_Z^+ on Z^+ with g_Z^+|_W[0,∞]=g_W|_W[0,∞] and Riemannian metric g_Ẑ with g_Ẑ|_Y× [0,∞)= g_Y × g^stan_. First, we introduce the configuration spaces for SU(l)-connections on W and Z^+ for l≥ 2 and SU(2)-configuration space for Ẑ. Fix a positive integer q ≥3. For an irreducible SU(2)-connection a on Y, we define^W(a)_(δ,δ),l:= {A_a+c| c ∈^1(W)⊗𝔰𝔲(l)_L^2_q,(δ,δ)}, ^Z^+_δ,l:= {θ+c| c ∈^1(Z^+)⊗𝔰𝔲(l)_L^2_q,δ},and^Ẑ(a):= {B_a+c| c ∈^1(Ẑ)⊗𝔰𝔲(2)_L^2_q},where* A_a is an SU(l)-connection on W with A_a|_Y× (-∞,-1]=^* (a⊕θ), A_a|_W[0,∞] =θ * B_a is an SU(2)-connection on Ẑ with B_a|_Y× [0,∞)=^*a.* L^2_q,(δ,δ)(W)-norm is defined by||f||^2_L^2_q,(δ,δ)(W):= ∑_0≤ i ≤ q∫_W e^τ' δ|∇_A_a^i f |^2dvol ,where τ' is defined in Definition <ref> and f is an element in ^1(W)⊗ with compact support.* L^2_q,δ(Z^+)-norm is defined by||f||^2_L^2_q,δ(Z^+):= ∑_0≤ i ≤ q∫_Z^+ e^τ”δ|∇^i_θ f |^2dvol,where τ”:Z^+ [0,1] is a smooth function satisfying τ”|_W[0,∞]=τ defined in Definition <ref> andf is an element in ^1(Z^+)⊗ with compact support.* L^2_q(Ẑ)-norm is defined by||f||^2_L^2_q(Ẑ):= ∑_0≤ i ≤ q∫_Ẑ|∇_B_a^i f |^2dvol ,wheref is an element in ^1(Ẑ)⊗ with compact support.We also define the SU(l) configuration spaces ^W (a)_δ,δ^l, ^Z^+_δ,l and SU(2)-configuration spaces ^Ẑ(a) by^W (a)_(δ,δ),l:=^W(a)_(δ,δ),l /^W(a)_l, ^Z^+_δ,l:=^Z^+_δ,l /^Z^+_land^Ẑ(a):=^Ẑ(a) /^Ẑ(a)where ^W(a)_l, ^Z^+_l and ^Ẑ(a) are given by ^W(a)_l:={ g∈(W× SU(l)) ⊂(ℂ^l)_L^2_q+1,loc| ∇_A_a(g) ∈ L^2_q,(δ,δ)(W)} , ^Z^+_l:={ g∈(Z^+× SU(l)) ⊂(ℂ^l)_L^2_q+1,loc| d(g) ∈ L^2_q,δ(Z^+) }and^Ẑ(a):= { g∈(Ẑ× SU(l)) ⊂(ℂ^2)_L^2_q+1,loc| ∇_B_a(g) ∈ L^2_q,δ(Ẑ) }.The action of ^W(a)_l(resp. ^Z^+_l, ^Ẑ(a)) on ^W(a)_(δ,δ).l(resp. ^Z^+_δ,l, ^Ẑ(a))is the pull-backs of connections. We define the reduced gauge group by ^W(a)_l:= { g ∈^W(a)_l|lim_t-∞ g|_Y× t =id } , ^W,fr(a)_l:={ g ∈^W(a)_l | lim_n ∞ g|_W_n id }and^Z^+_l:= { g ∈^Z^+_l| lim_n ∞ g|_W_n id}.Then we define^W (a)_(δ,δ),l :=(a)^W_(δ,δ)/ ^W(a)_l, ^W,fr (a)_(δ,δ),l :=(a)^W_(δ,δ)/ ^W,fr(a)_land^Z^+_δ,l:=^Z^+_δ,l/ ^Z^+_l. The group ^W,fr(a)_l (resp. ^Z^+_l) has a structure of Banach Lie sub group of ^W(a)_l (resp. ^Z^+_l). By the construction of them, there are exact sequences ^W(a)_l ^W(a)_l (a⊕θ), ^W,fr(a)_l^W(a)_lSU(l)and^Z^+_l^Z^+_lSU(l)of Lie groups. The group ^W(a)_l(resp. ^Z^+_l) acts on ^W (a)_(δ,δ),l(resp. ^Z^+_δ,l) freely. For l≥ 3 and an SU(2)-flat connection a, there exists a positive number δ' such that for a positive real number δ less than δ' the following properties hold.* ^W (a)_(δ,δ),l is simply connected.* ^Z^+_δ,l is simply connected.We will show only the first property. The second one is shown in a similar way to the first case. We use the condition H_1(Z,)≅ 0 for the second property.Since π_i(SU(l))=0 for i=0,1,π_1(^W(a)_(δ,δ),l ) is isomorphic to π_1(^W,fr (a)_(δ,δ),l ).Therefore, we will show π_1(^W,fr (a)_(δ,δ),l)=0. There exists δ'>0 such that for 0< δ<δ', ^l(a)_(δ,δ)^f ^W (a)_(δ,δ),l^W,fr (a)_(δ,δ),lis a fibration since (<ref>) has a local slice due to Fredholm and moduli theory in <cit.>. Let W^* be the one point compactification of W. Using (<ref>), we obtain π_1(^W,fr (a)_(δ,δ),l) ≅π_0(^l(a)_(δ,δ)^f ) ≅ [W^*,SU(l)].Since π_i(SU(l)) vanishes for i=0,1,2,4, the obstruction for an element of [W^*,SU(l)] to be homotopic to the constant map lives in H^3(W^*,π_3(SU(l))) ≅ H^3_comp(W,π_3(SU(l))) ≅ H_1(W,π_3(SU(l)))=0 where the second isomorphism is the Poincaré duality.This implies π_1(^W,fr (a)_(δ,δ),l)≅ 0.We now define the determinant line bundles.For simplify, we impose Assumption <ref> on Y. Let π be an element in ∏(Y)^flat and (Ψ,ϵ) be a perturbation in Subsection <ref> and fix an element a ∈R(Y).For c ∈^W(a)_(δ,δ),l (^W(a)_(δ,δ),l), we have the following bounded operator d(ℱ_π,Ψ)_c+d^*_L^2_(δ,δ)_c:^1(W) ⊗su(l)_L^2_q,(δ,δ)^0(W) ⊗su(l) ⊕^+(W) ⊗su(l))_L^2_q-1,(δ,δ).The operators d(ℱ_π,Ψ)_c+d^*_L^2_(δ,δ)_c are the Fredholm operators for small δ. Fix such a δ. We setλ^W (a,l,c):= Λ^max (d(ℱ_π,Ψ)_c) ⊗Λ^max (d(ℱ_π,Ψ)_c)^*.The determinant line bundles are defined byλ^W (a,l):=⋃_c ∈^W(a)_(δ,δ),lλ (a,l,c) ^W(a)_(δ,δ),land λ̂^W (a,l):=⋃_c ∈^W(a)_(δ,δ),lλ (a,l,c) ^W(a)_(δ,δ),l.We also define λ^Z^+ (l)^Z^+_δ,l and λ^Ẑ(a) ^Ẑ(a)in a similar way with respect to the operatorsd^+_c+d^*_L^2_δ_c :^1(Z^+) ⊗su(l)_L^2_q,δ^0(Z^+) ⊗su(l) ⊕^+(Z^+) ⊗su(l))_L^2_q-1,δfor c ∈^Z^+_δ,l andd^+_c+d^*_c :^1(Ẑ) ⊗_L^2_q^0(Ẑ) ⊗⊕^+(Ẑ) ⊗)_L^2_q-1for c ∈^Ẑ(a). For a given data (a,δ,l) in Proposition <ref>,the bundles λ^Z^+ (a,l)^Z^+_δ,l and λ^W(a,l)^W(a)_(δ,δ),l are trivial.Since the determinant line bundle is a real line bundle, the triviality of λ^Z^+ (a,l)^Z^+_δ,l is a consequence of Proposition <ref>. Therefore we show the triviality ofλ^W(a,l)^W(a)_(δ,δ),l. We have a fibration (a⊕θ) ^W(a)_(δ,δ),lj^W(a)_(δ,δ),l.We also have an isomorphism j^*λ^W(a,l) ≅λ̂^W(a,l) for j in (<ref>).λ̂^W(a,l) is the trivial bundle for l>2 from Proposition <ref>. So if the fiber (a⊕θ) of (<ref>) is connected, λ^W(a,l) is also trivial.The possibilities of (a⊕θ) are SU(l), U(1)× U(l-1), S(U(2)× U(l-2)) and {(z,A) ∈ U(1)× U(l-2) | z^2 detA=1}.Since these groups are connected, λ^W(a,l) is the trivial bundle.Suppose that Y satisfies Assumption <ref> and a is an element in R^*(Y). Let i_1:^W(a)_(δ,δ),2^W(a)_(δ,δ),3 and i_2:^Z^+_δ,2^Z^+_δ,3 be the maps induced by the product with the product connection. There exists a positive number δ' such that for a positive real number δ less than δ', i_1^*λ^W(a,3) ≅λ^W(a,2) and i_2^*λ^Z^+(a,3) ≅λ^Z^+(a,2) hold. UnderAssumption <ref> on Y, the isomorphism class of these line bundles are independent of the choices of the perturbations π and (Ψ,ϵ) by considering a 1-parameter family of perturbations π_t:=(f,th) and (Ψ,tϵ) for t ∈ [0,1]. So Lemma (5.4.4) in <cit.> implies the conclusion.Suppose that Y satisfies Assumption <ref> and a is an element in R^*(Y). Let π be an element in ∏(Y)^flat and (Ψ, ϵ) be a regular perturbation for a ∈R^*(Y). For sufficiently small δ, M^W(a)_π,Ψ,ϵ,δ is orientable. Furthermore the orientation of M^W(a)_π,Ψ,ϵ,δ is induced by the orientation of λ^W(a,2). Using the exponential decay estimate in Proposition 4.3 of <cit.>, we have a inclusion i:M^W(a)_π,Ψ,ϵ,δ B^W(a)_(δ,δ),2 for small δ as a set. From this inclusion i, we regard M^W(a)_π,Ψ,ϵ,δ as a subset in B^W(a)_(δ,δ),2. Applying result of convergence in Corollary 5.2 of <cit.>, we can show that the topology ofM^W(a)_π,Ψ,ϵ,δ in B^W(a)_δ coincides with the topology of M^W(a)_π,Ψ,ϵ,δ in B^W(a)_(δ,δ),2 Also using exponential decay estimate for solutions to the linearized equation in Lemma 3.3 of <cit.> , λ^W(a,2)|_M^W(a)_π,Ψ,ϵ,δ M^W(a)_π,Ψ,ϵ,δ is canonically isomorphic to Λ^max M^W(a)_π,Ψ,ϵ,δ. From Theorem <ref>, an orientation of M^W(a)_π,Ψ,ϵ,δ is characterized by the trivialization of λ^W(a,2). On the other hand, to formulate the instanton Floer homology of Y withcoefficient, Donaldson introduced the line bundle λ(a):=λ^(-Ẑ)(a) ⊗λ_(-Ẑ)^* ^(-Ẑ)(a), where λ_(-Ẑ) is given byΛ^max (H^0_DR(-Ẑ)⊕ H^1_DR(-Ẑ) ⊕ H^+_DR(-Ẑ))in Subsection 5.4 of <cit.>. The orientation of λ(a) is essentially independent of the choice of Z.We setλ_W:= Λ^max (H^0_DR(W)⊕ H^1_DR(W) ⊕ H^+_DR(W)),and λ^W(a):= λ^W(a,2) ⊗λ_W ^W (a)_(δ,δ),2. Suppose that Y satisfies Assumption <ref>. For an irreducible flat connection a, there is a canonical identification between the orientations of λ^W(a ) and the orientations of λ(a). It suffices to construct an isomorphism λ^W(a)≅λ(a) which is canonical up to homotopy. First we fix two elements [A] ∈^W (a)_(δ,δ) and [B] ∈^Ẑ(a) which have representative A and B satisfying A_Y× (-∞,-1]=^*a and B|_Y× [1,∞)=^*a. For such two connections, we obtain an element A# B ∈^Z^+_δ by gluing of connections. This map induces an isomorphism#:det( d_A^*+d^+_A)⊗ det( d_B^*+d^+_B) det( d_A# B^*+d^+_A# B)from the similar argument of Proposition 3.9 in <cit.>. Therefore we have identification#:λ^W(a)|_[A]⊗λ^Z^+(a)|_[B]λ_Z^+|_[A#B].If we choose a path from [θ] to [A#B] in ^Z^+_δ, then we have an identification between λ_Z^+|_[θ] and λ_Z^+|_[A#B]. The line bundle λ_Z^+|_[θ] is naturally isomorphic to Λ^max (H^0_DR(Z^+)⊕ H^1_DR(Z^+) ⊕ H^+_DR(Z^+)) by using Proposition 5.1 in <cit.>. This cohomology group is isomorphic toΛ^max (H^0_DR(Z)⊕ H^1_DR(Z) ⊕ H^+_DR(Z))⊗Λ^max (H^0_DR(W)⊕ H^1_DR(W) ⊕ H^+_DR(W))by using the Mayer-Vietoris sequence.Thereforeλ^W(a)|_[A]⊗λ_W^* ≅ (λ^(-Z)(a)|_[B]⊗λ_(-Z)^*)^*holds. Because λ^Z^+(a,2)^Z^+_δ,2 is orientable by Lemma <ref> and Proposition <ref> , the homotopy class of this identification does not depend on choices of the path, A, B, the bump functions of the gluing map. We also have the following canonical isomorphism (λ^(-Z)(a)|_[B]⊗λ_(-Z)^*)^* ≅λ^Z(a)|_[B]⊗λ_Z^*.by the gluing Z and -Z as above discussion and the Mayer-Vietoris sequence. This completes the proof.Combining Theorem <ref> and Lemma <ref>, we have: Under the assumption of Theorem <ref>, an orientation λ(a) and an orientation of λ_W give an orientation of M^W(a)_π,Ψ,ϵ,δ. § PROOF OF MAIN THEOREM Let Y, l_Y and X be as in Section <ref>.Take a Riemannian metric g_Y on Y. Fix a non-negative real number r ∈Λ_Y smaller than Q^2l_Y+3_X. Suppose that there is an embedding f of Y into X satisfying f_*[Y]= 1 ∈ H_3(X,). Then we obtain the oriented homology cobordism from Y to -Y by cutting open along Y. Recall that W is a non-compact oriented Riemann 4-manifold W with both of cylindrical end and periodic end which is formulated at the beginning of Subsection <ref>.We fix a holonomy perturbation π∈∏(Y) satisfying the following conditions. * π is a ϵ-perturbation in Subsection <ref>.* π is a regular perturbation in the end of Subsection <ref>.* π is an element of ∏(Y)^flat in Definition <ref>.* For a∈R(Y) with 0≤ cs_Y(a) < min{1,Q^2l_Y+3_X} and A ∈ M^W(a)_π,δ,1/8π^2sup_n ∈ ||F(A)+sπ(A)||^2_L^2(W) <min{1,Q^2l_Y+3_X}holds.* d^+_θ+sdπ_θ:^W(θ)_(δ,δ)×^d Ω^+(W)⊗_L^2_q-1,(δ,δ) is surjective.* For c ∈R(Y)_π satisfying cs_Y(c)<0, M^W(c)_π,δ is the empty set.Assumption <ref> and the proof of Thereom 8.4 (ii) of <cit.> implies the existence of the perturbation ssatisfying the third condition. The first, forth, fifth and sixth conditions follow from choosing small h∈ C^l'(SU(2)^d',)_ad of π=(f,h).Next we also fix a holomomy perturbation (Ψ,ϵ) satisfying the following conditions. * (Ψ,ϵ) is a regular for [b] ∈R(Y) with 0≤ cs_Y(b) ≤ cs_Y(a). * For a∈R(Y) with 0≤ cs_Y(a) < min{1,Q^2l_Y+3_X},1/8π^2sup_n ∈ ||F(A)+sπ(A)+σ_Ψ(A,ϵ)||^2_L^2(W) <min{1,Q^2l_Y+3_X}hold.To get the first condition, we use Lemma <ref>. The second condition satisfied when we take ϵ sufficiently small. In order to formulate the instanton Floer homology of Y withcoefficient, we fix an orientation of fix an orientation of λ(a) for each a ∈ R(Y). The orientation of Y induce an orientation of λ_W. To determine the orientation of M^W(a)_π,Ψ,ϵ,δ, we fix a compact oriented manifold Z with H_1(Z,)≅ 0 as in Subsection <ref>.The relation between λ_Z,a and λ(a) is given by λ^Z(a) ⊗λ_Z≅λ(a). Let a be a flat connection satisfying cs_Y(a)<r≤min{Q_X^2l_Y+3,1} and (a)=1. We consider the moduli space M^W(a)_π,Ψ,ϵ,δ. From the choice of these perturbation data and Corollary <ref>, M^W(a)_π,Ψ,ϵ,δ has a structure of 1-dimensional manifold for small δ. From Theorem <ref>, we obtain an orientation ofM^W(a)_π,Ψ,ϵ,δ induced by the orientation of λ_Z,a.Let (A,B) be a limit point of M^W(a)_π,Ψ,ϵ,δ. Using Theorem <ref> and the standard dimension counting argument, the limit points of M^W(a)_π,Ψ,ϵ,δ correspond to two cases: * (A,B) ∈⋃_b ∈R^*(Y), cs_Y(b)<r,(b)=0 M(a,b)_π× M^W(b)_π,Ψ,ϵ,δ* (A,B) ∈ M(a,θ)_π× M^W(θ)_π,Ψ,ϵ,(δ,δ).For the second case, we use the exponential decay estimate to show B ∈ M^W(θ)_π,Ψ,ϵ,(δ,δ). Here M(a,b)_π and M(a,θ)_π,δ have a structure of 1-dimensional manifold. The quotient spaces M(a,b)_π/ and M(a,θ)_π,δ/ have a structure of compact oriented 0-dimensional manifold whose orientation induced by the orientation of λ_Z andaction by the translation as in Subsection 5.4 of <cit.>. Corollary <ref> and Theorem <ref> imply that M(b)_π,Ψ,ϵ,δ has a structure of compact oriented 0-manifold whose orientation induced by the orientation of λ_b and λ_W for small δ. Since the formal dimension of M(θ)_π,Ψ,ϵ,(δ,δ) is -3 from Proposition <ref> and there is no reducible solution except θ for a regular perutrbation (Ψ,ϵ), M(θ)_π,Ψ,ϵ,δ consists of just one point. By the gluing theory as in Theorem 4.17 and Subsection 4.4.1 of <cit.>, there is the following diffeomorphism onto its image:𝒥:(⋃_b ∈R^*(Y), cs_Y(b)<r, (b)=0(M(a,b)_π/× M(b)_π,Ψ,ϵ,δ) ∪ M(a,θ)_π/)× [T,∞) M^W(a)_π,Ψ,ϵ,δ.By the definition of the orientation of M(a,b)_π/ and M(a,θ)_π/, we can construct 𝒥 as an orientation preserving map. Furthermore, the complement of 𝒥 is compact. Therefore we can construct the compactification of M^W(a)_π,Ψ,ϵ,δ by adding the finitely many points ⋃_b ∈R^*(Y), cs_Y(b)<r,(b)=0(M(a,b)_π/× M(b)_π,Ψ,ϵ,δ) ∪M(a,θ)_π/,which has a structure of compact oriented1-manifold. By counting of boundary points of the compactification, we obtain the relation δ^r( n)(a)+ θ^r(a)=0,where n ∈ CF^0_r(Y) is defined by n(b):= # M(b)_π,Ψ,ϵ,δ. This implies θ^r is a coboundary. Therefore we have 0=[θ^r] ∈ HF^1_r(Y) for 0≤ r ≤min{Q^2l_Y+3_X,1}.jplain | http://arxiv.org/abs/1707.08555v2 | {
"authors": [
"Masaki Taniguchi"
],
"categories": [
"math.GT"
],
"primary_category": "math.GT",
"published": "20170726173655",
"title": "Instantons for 4-manifolds with periodic ends and an obstruction to embeddings of 3-manifolds"
} |
Extremal copositive matrices with minimal zero supports of cardinality two Roland HildebrandLJK / CNRS, Bâtiment IMAG, 700 avenue Centrale, Domaine Universitaire, 38041 Saint-Martin-d'Hères, France ([email protected]). December 30, 2023 ============================================================================================================================================================================ Let A ∈ C^n be an extremal copositive matrix with unit diagonal. Then the minimal zeros of A all have supports of cardinality two if and only if the elements of A are all from the set {-1,0,1}. Thus the extremal copositive matrices with minimal zero supports of cardinality two are exactly those matrices which can be obtained by diagonal scaling from the extremal {-1,0,1} unit diagonal matrices characterized by Hoffman and Pereira in 1973. Keywords: copositive matrix, extreme ray, minimal zeroAMS Subject Classification: 15A48, 15A21.§ INTRODUCTION An element A of the space S^n of real symmetric n × n matrices is called copositive if x^TAx ≥ 0 for all vectors x ∈ℝ_+^n. The set of such matrices forms the copositive cone C^n. This cone plays an important role in non-convex optimization, as many difficult optimization problems can be reformulated as conic programs over C^n. For a detailed survey of the applications of this cone see, e.g., <cit.>.Verifying copositivity of a given matrix is a co-NP-complete problem <cit.>, and the complexity of the copositive cone quickly grows with dimension. In this note we focus on the extreme rays of C^n. This topic has attracted particular interest already very early in the study of copositive matrices, and a number of families of extreme copositive matrices have been constructed <cit.>. An element x of a regular convex cone K is called an extremal element if a decomposition x = x_1 + x_2 of x into elements x_1,x_2 ∈ K is only possible if x_1 = λ x, x_2 = (1-λ)x for some λ∈ [0,1]. The set of positive multiples of an extremal element is called an extreme ray of K. The set of extreme rays is an important characteristic of a convex cone. Its structure, first of all its stratification into a union of manifolds of different dimension, yields much information about the shape of the cone. The extreme rays of a convex cone which is algorithmically difficult to access are especially important if one wishes to check the tightness of inner convex approximations of the cone. Namely, an inner approximation is exact if and only if it contains all extreme rays. Since the extreme rays of a cone determine the facets of its dual cone, they are also important tools for the study of this dual cone. The extreme rays of the copositive cone have been used in a number of papers on its dual, the completely positive cone <cit.>.A important tool in the study of extremal copositive matrices are its zeros <cit.>. A zero u of a copositive matrix A is a non-zero nonnegative vector such that u^TAu = 0. The support u of a zero u = (u_1,…,u_n)^T ∈ℝ_+^n is the subset of indices j ∈{1,…,n} such that u_j > 0. In <cit.> we introduced a refined tool, the minimal zeros. Here a zero u of a copositive matrix A is called minimal if there exists no zero v of A such that v ⊂ u holds strictly. Up to multiplication by a positive constant there exists only a finite number of minimal zeros for a given copositive matrix. The set of supports of all minimal zeros of a copositive matrix is an informative characteristic of the matrix. In particular, this combinatorial characteristic can assist the classification of the extremal elements of C^n.In this note we make a step in this direction by describing the extremal copositive matrices whose minimal zero supports have all cardinality two. We show that every such matrix can be transformed by an automorphism of the copositive cone to a matrix whose elements are from the set {-1,0,1}. However, the extremal copositive matrices with the latter property have been already completely classified in <cit.>. This yields a complete description of the former class of extremal copositive matrices. We essentially use the following recent result giving a necessary and sufficient condition of extremality of copositive matrices in terms of their minimal zeros <cit.>.Let A ∈ C^n be a copositive matrix, and let u^1,…,u^m be all its minimal zeros, up to multiplication of the zero by a positive constant. Consider the following linear homogeneous system of equations on the matrix X ∈ S^n:(Xu^j)_k = 0∀ k = 1,…,n, j = 1,…,m uch that (Au^j)_k = 0.Then A is an extremal copositive matrix if and only if the solution space of system (<ref>) has dimension 1. § MAIN RESULT We show the two directions of the announced relation between {-1,0,1} copositive matrices and matrices with minimal zero supports of cardinality two separately. Let A ∈ C^n be a copositive matrix with unit diagonal and whose elements are from the set {-1,0,1}. Then all minimal zeros of A have support of cardinality two.Since all diagonal elements of A are positive, there exists no zero of A with support of cardinality one.For the sake of contradiction, suppose A has a minimal zero u ∈ℝ_+^n with support of cardinality k > 2. Without loss of generality, let u = {1,…,k}. Then all elements of the upper left k × k submatrix of A equal either 0 or 1. Indeed, suppose there exist i,j ≤ k such that A_ij = -1. Then the sum v = e^i + e^j of the corresponding basis vectors of ℝ^n is a zero of A whose support {i,j} is a strict subset of u, a contradiction with the minimality of u. Hence A_ij≥ 0 for all i,j = 1,…,k, and we get0 = u^TAu = ∑_i,j = 1^k A_iju_iu_j ≥∑_i = 1^k A_iiu_i^2 = ||u||_2^2 > 0,a contradiction.This completes the proof. For the converse direction we shall need the following result <cit.>. Let A ∈ C^n be a copositive matrix with unit diagonal and let u be a zero of A with support {i,j}. Then u_i = u_j. We are now in a position to prove the following result. Let A ∈ C^n be an extremal copositive matrix such that all its minimal zeros have support of cardinality two. Then there exists a positive definite diagonal matrix D and an extremal copositive matrix Σ with unit diagonal and with all elements in the set {-1,0,1} such that A = DΣ D.The matrix A has no zeros with support of cardinality one, and therefore all its diagonal elements are positive. Let D be the diagonal matrix with diagonal elements D_ii = √(A_ii), i = 1,…,n, and set Σ = D^-1AD^-1.Note that the linear map given by X ↦ D^-1XD^-1 is an automorphism of the copositive cone and preserves the property of copositive matrices of being extremal. Hence Σ is an extremal copositive matrix with unit diagonal. Note that u is a zero of A if and only if Du is a zero of Σ, and u =Du. Hence all minimal zeros of Σ have support of cardinality two too. Let u be a minimal zero of Σ with support {i,j}, and let k ∈{1,…,n} be an index. By Lemma <ref> the equation (Xu)_k = 0 on the matrix X ∈ S^n can be written as X_ik + X_jk = 0. Therefore system (<ref>), in application to the extremal copositive matrix Σ, can be written asX_ik + X_jk = 0:∃uΣ u = {i,j}, (Σ u)_k = 0. We shall now investigate the solution space of system (<ref>). Let G be the graph with the n(n+1)/2 independent elements of the real symmetric matrix X as vertices, and with an edge between X_ik and X_jl if and only if the equation X_ik + X_jl = 0 is among the equations of system (<ref>). It is then easily seen that the dimension of the solution space of system (<ref>) equals the number of bipartite connection components of G. Indeed, the elements of X in a connection component containing an odd cycle are forced to be zero by equations (<ref>). The elements in a bipartite connection component must have equal absolute values, but the elements in the two partition classes of the component have opposite signs. Hence the value of an arbitrary element in the bipartite connection component can be chosen at will, while all other elements in the component are determined by the value of that first one.By virtue of Theorem <ref> extremality of Σ entails that system (<ref>) has a one-dimensional solution space, namely the multiples of the matrix Σ itself. Hence the graph G has exactly one bipartite connection component. A solution X of system (<ref>), in particular the matrix Σ itself, must then have zero entries at all vertices which are not in this bipartite component, and all remaining entries have equal absolute value. Since Σ has a unit diagonal, its entries can therefore assume only the values -1,0,1. This completes the proof. We are now able to formulate the main result.Let A ∈ C^n be an extremal copositive matrix. Then the following are equivalent: (i) All minimal zeros of A have support of cardinality two.(ii) There exists a positive definite diagonal matrix D and an extremal copositive matrix Σ with unit diagonal and all elements from the set {-1,0,1} such that A = DΣ D. The implication (i) ⇒ (ii) is the assertion of Lemma <ref>.Assume condition (ii). By Lemma <ref> all minimal zeros of Σ have supports of cardinality two. But then the same holds for A, because the minimal zero support set is preserved by automorphisms of C^n of the form X ↦ DXD. This proves (i). This result completely characterizes the class of extremal copositive matrices with minimal zero supports of cardinality two. Namely, these matrices are diagonally scaled versions of the extremal {-1,0,1}-matrices with unit diagonal, which have been already classified in <cit.>.plain | http://arxiv.org/abs/1707.08862v1 | {
"authors": [
"Roland Hildebrand"
],
"categories": [
"math.OC",
"15A48, 15A21"
],
"primary_category": "math.OC",
"published": "20170727133744",
"title": "Extremal copositive matrices with minimal zero supports of cardinality two"
} |
[ Jorge Drumond Silva December 30, 2023 ======================= Compared with raw images, the more common JPEG images are less useful for machine vision algorithms and professional photographers because JPEG-sRGB does not preserve a linear relation between pixel values and the light measured from the scene. A camera is said to be radiometrically calibrated if there is a computational model which can predict how the raw linear sensor image is mapped to the corresponding rendered image (JPEGs) and vice versa. This paper begins with the observation that the rank order of pixel values are mostly preserved post colour correction. We show that this observation is the key to solving for the whole camera pipeline (colour correction, tone and gamut mapping). Our rank-based calibration method is simpler than the prior art and so is parametrised by fewer variables which, concomitantly, can be solved for using less calibration data. Another advantage is that we can derive the camera pipeline from a single pair of raw-JPEG images. Experiments demonstrate that our method delivers state-of-the-art results (especially for the most interesting case of JPEG to raw).§ INTRODUCTIONMany computer vision algorithms (photometric stereo <cit.>, photometric invariants <cit.>, shadow removal <cit.>, and colour constancy <cit.>) assume that the captured RGBs in images are linearly related to the actual scene radiance. However, the imaging pipeline in a digital camera is necessarily non-linear in order to produce perceptually-pleasing photos rather than its physically-meaningful counterparts. In this paper, we present a new rank-based radiometric calibration method which solves for the bi-directional mappings between the camera's RAW responses and the rendered RGBs produced by digital camera.There is prior art in this field which models the pipeline with a large number of parameters (up to several thousand <cit.>) which both means a large corpus of data is required to uncover the pipeline and that there is at least tacitly the premise that the underlying pipeline is quite complex. The key insight in our approach is that post-colour correction (a 3 × 3 matrix correction) the linear corrected raw RGBs are to the greatest extent in the same rank order as the final rendered RGBs. Building on this insight, we develop a simple rank-based radiometric calibration model that “solves for” the camera pipeline with many fewer parameters and concomitantly needs much less training data.In Fig. <ref>, we illustrate a conventional image reproduction pipeline that holds for many cameras <cit.>. An exemplar raw image, Fig. <ref>a, is mapped by a 3 × 3 colour correction matrix to give the image shown in Fig. <ref>b. The colour correction matrix implements several processing steps (illumination correction <cit.>, display RGB mapping <cit.>, and colour preference adjustments <cit.>). It is well-known that a display device cannot display all captured image colours that some RGBs will fall outside the RGB cube after mapping (the pixels marked in purple in Fig. <ref>b). We therefore need gamut mapping, <cit.>, to bring the colours back inside the cube as shown in Fig. <ref>c. Finally, the gamut mapped image is tone mapped to arrive at the final rendered output <cit.> shown in Fig. <ref>d. Tone mapping accounts for the display non-linearity <cit.>, dynamic range compression and some aspects of preference <cit.>. The colour processing pipeline – for cameras, in general, can be written as Eqn. <ref>.[ P = f(Γ (Mρ)) =Γ(f(Mρ)) ≈LUT(ρ);(1a)(1b)(1c) ]Here and throughout this paper, ρ denotes a camera RAW and P refers to its rendered RGB counterpart. Respectively, the 3 × 3 correction matrix, gamut mapping and tone mapping are denoted by the matrix M and the functions Γ() and f(). The function f() can implement a single or three per-channel tone curves. Because gamut mapping only implements a small change in comparison with colour and tone mapping steps, the order of gamut mapping and tone mapping may be switched (Eqn. <ref>b & c), a property that we exploit in this paper. Equally, we can also roll all three processing steps into one and directly solve for a 3D LUT (Look-Up-Table) that maps RAW to rendered counterparts. This LUT function is denoted LUT() <cit.> in Eqn. <ref>c. Readers may refer to the top row of Fig. <ref> to link each mathematical function to our example processed image.In radiometric calibration, given a set of ρ and P, one seeks to solve for the parametrised pipeline parts (M, Γ(), f() and LUT()). A disadvantage of the current best performing methods is that a great deal of data may be required to fit their assumed models. In Eqns. <ref>a and <ref>b, the gamut mapping step could be modelled by 1000s of Radial Basis functions <cit.> and in Eqn. <ref>c, the deployed LUT could also have several thousand control points.Our proposed method begins with the simple observation <cit.> that, assuming the gamut mapping step makes small changes to image colours, we expect the rank ordering of the rendered P to be the same as ρ multiplied by the correction matrix M (because a tone curves are always monotonically increasing). Suppose that two rendered (JPEG) responses – in the 1^st red colour channel – are denoted P^a_1 and P_1^b; and that P^a_1>P^b_1. The rank order of two corresponding raw red channel measurements post colour correction is written as M_1ρ^a>M_1ρ^b (where M_1 denotes the first row of M and ρ^a and ρ^b are a pair of raw RGBs). Rewriting: this implies that M_1(ρ^a-ρ^b)>0 which mathematically defines a half-space constraint. If we visualise the row vector M_1 as a point in 3-space then this inequality – which we call a ranking constraint – forces the point to be located in one half of 3-space but not the other. Because we have multiple pixels, each pair of pixels (2 raw and 2 JPEG RGBs) generates a half space constraint and intersecting all these constraints delimits the region in which M_1 must lie. Our experiments demonstrates that a small numbers of patches suffices to estimate M accurately. Once we have M we then find the best rank preserving tone curves f(). At this stage, only using M and f() we have a tolerable approximation of the pipeline. Indeed, we argue that our construction of M and f() also incorporates, to a first order, gamut mapping. Now we adopt (Eqn <ref>c) and find a 125-parameter LUT to “mop up” any remaining errors due to gamut mapping (higher order terms). § RELATED WORK Using the pipeline form Eqn <ref>b, Chakrabarti <cit.> first solve for M and f() (in a least-squares sense) in iteration and then solve directly for Γ(). In their approach, f() is constrained to be a 7^th order monotonic polynomial. They model Γ() with the radial basis function (RBF) method of <cit.> where several thousands of RBFs are potentially used. A restriction of the above calibration is presented in <cit.> wherethe gamut mapping Γ() is ignored. This less general model works tolerably well on many real pairs of raw and rendered images and this is a point we will return to later in this paper. In either version (<cit.> or <cit.>), the coupled nature of the minimization means that a global minimum is not guaranteed to be found. Thus, a random start search is incorporated – multiple minimisations are carried out – in order to find their best parameters.Kim <cit.> solve for the pipeline in the form of Eqn. <ref>a and makes additional assumptions to decouple the optimization. They assume that images of the same scene are captured with respect to two or more exposures and their Γ() is a multi-thousand set of RBFs. Regarding solving for f(), Debevec <cit.> showed how relating corresponding pixels under known exposure differences suffices to solve for f() (assuming there is no gamut mapping step). Importantly, in <cit.>, it was argued that for the set of desaturated pixels (RAWs far from the RGB cube boundary) the gamut mapping step has little or no effect and can be ignored.Relative to this assumption, f() can be solved using the Debevec method. Given f() then the colour correction matrix M can be found (again using desaturated pixels). Though, for typical capture conditions, e.g. for most mobile phones,multiple exposures are not available and so the requirement that multiple exposures are needed is a weakness in this method. Finally, in <cit.> a gamut mapping RBF network is “trained”. Of course, if a large number of radial basis functions are used to model gamut mapping (as proposed in <cit.> or <cit.>) then solving for Γ() requires a large corpus of data. Further the application of gamut mapping is expensive and its parametrisation is large.In <cit.> it was shown that is possible to ignore the underlying structure of the colour processing pipeline and directly solve for the best 3D surjective function – implemented as a LUT that maps the RAWs to rendered RGBs (Eqn. <ref>c). Finally, in <cit.>, a method is presented for solving for f() by examining the edge distribution in an image. This method has the advantage that the method works for a single image (no need for multiple exposures) but the disadvantage that the method is sensitive to processing steps such as image sharpening which is used extensively in mobile phone image processing.§ THE RANK-BASED METHODAs the reader shall see, to make the rank-based method work we need to assume that the gamut mapping step Γ() only makes small adjustments to colour. In fact our assumption is more nuanced. We assume that – to a first order – gamut mapping can mostly be implemented as an affine transform and that this affine transform can be folded into the colour correctionmatrix M and themonotonically increasing tone mapping functions f().§.§ Gamut Mapping as an Affine Transform In Eqn. <ref>b, gamut mapping is applied when, after colour correction, colours are mapped outside the colour cube and become non-displayable. Let us use a Taylor expansion to model Γ() around a point a inside the gamut:Γ(Mρ)≈Γ(a)+J(a)(ρ-a)where J is the 3 × 3 Jacobian (matrix of derivatives of Γ). Not only does Eqn. <ref> show that, to a first approximation, gamut mapping is an affine transform it is also one of the gamut mapping algorithms proposed in <cit.>. In particular, <cit.> solves, with good results, for the best affine transform that maps image colours inside the gamut and which are, simultaneously, close to the non-gamut mapped colours:min_T,oΣ_i ||TMρ_i+o-Mρ||^2s.t. 0≤ TMρ_i+o≤1In Eqn. <ref>, T and o are respectively a 3 × 3 matrix and 3 × 1 offset vector defining the affine gamut mapping algorithm. The 3-vectors of 0s and 1s are denoted 0 and 1. Eqn. <ref> is solved directly by Quadratic Programming <cit.>. The gamut mapping shown in Fig. <ref>c is the result of solving Eqn. <ref>.Here, we make two important remarks about affine gamut mapping: 1) Gamut mapping and colour correction combined can be represented by the single affine transform: 3 × 3 matrix TM and offset o; 2) It follows that the rank-based method presented in the next section will actually solve for TM. The offset term can be incorporated directly in f() (since an offset does not change ranking). §.§ Rank-based estimation for colour correctionLet us denote the k^th row of M as M_k, let us assume that given two colour corrected RAWs, M_kρ^a and M_kρ^b that the rank order is the same as for the corresponding rendered RGBs:P^a_k>P^b_k ⇒ M_kρ^a>M_kρ^b ⇒M_k(ρ^a-ρ^b)>0Defining the difference vector d^j=ρ^a-ρ^b:M_kd^j>0where it is understood the superscript ^j denotes the difference vector from the j^th of n2 pairs of n image pixel values. Suppose that we have a vector M_k where Eqn. <ref> holds, then the inequality cannot be true for -M_k. That is Eqn. <ref> defines a half plane constraint <cit.>. The vector d^j is perpendicular to the half-plane: any M_k less than 90 degrees to d^j is a possible solution. Given multiple difference vectors then we have multiple half-plane constraints which taken together delimit a region in 3-space where M_k must lie. Denoting the half-plane as H(d^j), M_k must satisfy:M_k∈⋂_jH(d^j)Let us visualise the computation of M_k using ranking. Without loss of generality let us assume that M_k,3=1. We rewrite Eqn. <ref> asM_k,1d_1^j+M_k,2d_2^j+d_3^j>0If [a b c] is a solution to Eqn. <ref>, then [a/c b/c c/c] for Eqn. <ref> is also true since M_k,1=a/c and M_k,2=b/c. Solutions for [M_k,1,M_k,2] lie on one side of the line, the 3D half-space constraints maps directly to a 2D half-plane constraint. Or, if we consider the whole set of collations, the cone in 3D, defined by Eqn. <ref>, maps to a 2D convex region <cit.>. Denoting half-planes as P(d^j) we, equivalently, solve for[M_k,1,M_k,2]∈⋂_jP(d^j)The intersection problem of Eqn. <ref> is easily visualised. In Fig. <ref>a we show the intersection of 4 half plane constraints and indicate the solution set where M_k must lie.We solve for M one sensor channel at a time. Empirically, we have to be careful not to generate too many half planes. In our experiment, we generate half-planes from all pairs of up to 50 randomly selected unique RAWs, generating 2450 half-planes. Due to noise or small deviations in real camera data, it is likely that no common intersection can be found that satisfy every half-planes constraint. To solve this problem, we generate 100,000 unit length vectors that are uniformly distributed on the surface of the unit sphere <cit.>, which is visualised in Fig. <ref>b. With respect to this sampling, the furthest distance between any point and its nearest neighbour is less than 1.15 degrees. So, the orientation of the rows of M are found to this accuracy. For each point on the sphere (a possible row of M_k), we count how many half-space constraints are satisfied. The point on the unit sphere that satisfies most half-plane constraints – or the median of multiple points if there is a tie – defines M_k. To make our approach robust, we find randomly select 50 colours 25 times and for each trial find the best M_k. Overall, we find the M that places all the corresponding raw and rendered image RGBs in the most similar rank order. That is, if we plot the mapped raw red responses, for example, against the rendered red JPEG corresponding values then the graph should be a monotonically increasing function. How well a monotonically increasing function fits our data can be used to judge the efficacy of each M.Ranking can only estimate M up to an unknown scaling of its rows. Suppose for a rendered achromatic RGB P^A=[0.5 0.5 0.5]^⊺ and the corresponding raw ρ^A=[a b c]^⊺, we apply: 0.5diag(Mρ^A)^-1Mρ^A=[0.5 0.5 0.5]^⊺ where diag() places a vector along the diagonal of a diagonal matrix. After this step, M← 0.5diag(Mρ^A)^-1M maps achromatic colours correctly.Because M← DM (where in the example, D=0.5diag(Mρ^A)^-1) we might also solve for D in a least-squares sense by including all colours indexed ^i close to the achromatic axis: min_D ∑_i ||DMρ^i-P^i|| (our experiment does not include this additional step).§.§ Rank-preserving optimization of tone curvesWe now solve for the optimal per-channel tone curves which map colour corrected RAWs to corresponding rendered RGBs. Let us denote the i^th colour corrected RAW and rendered RGB pixel pairs for the k^th channel as (ρ_k,i,P_k,i). Then, the k^th-channel rank-preserving tone curve f_k() is optimised as a 7^th order monotonic and smooth polynomial function as follows:min_f_k()Σ_i ||f_k (ρ_k,i)-P_k,i||^2 + λ∫_t||f_k”(t)||^2 dt s.t.f_k'()≥0 .where the first term is for data fitness, the second term is for curve smoothness and λ is a small weight (10^-5). This polynomial fitting is solved by Quadratic Programming <cit.>.In this paper, we further denote the combination of all 3-channel mappings f_1-3() as f(). §.§ Gamut correction stepAs argued previously, we propose that f(Mρ) has the expressive power to implement colour correction, tone correction and gamut mapping (to the first order in a Taylor expansion). However, we wish to add a further gamut mapping step for the higher order terms. But, since our hypothesis is that much of the gamut mapping will have been accounted for we are going to adopt a simple small parameter solution. Further, this additional correction is going to be carried out at the end of the process, we adopt Eqn. <ref>b. Specifically, we find a 5×5×5 LUT by using lattice regression <cit.> that minimises min_LUT()Σ_i ||LUT(f(Mρ_i))-P_i||^2.§.§ Rank-based recovery of rawSuppose we wish to map rendered RGBs to RAWs. Using the method presented in Section 3.2, M has already been solved in the RAW-to-JPEG forward estimation phrase. Now, in a least-squares optimal way, we use the same polynomial fitting method (Eqn. <ref>) to find f^-1 by optimising min_f^-1()Σ_i ||f^-1(P_i)-Mρ_i||. Finally, we solve for the backward LUT() by optimising min_LUT()Σ_i ||LUT(M^-1f^-1(P_i))-ρ_i || where the LUT is fitted by a 5×5×5 lattice regression <cit.>. §.§ Parameter counting Assuming we solve for 3 independent tone curves then our method requires 9 (for M) + 8 × 3 (for f()) + 125× 3 (for Γ()) = 408 parameters which is significantly less (even an order of magnitude less) than <cit.>.§ EVALUATION Our evaluation is based on the most challenging dataset that we have encountered: <cit.> which contains the RAW/JPEG intensity pairs of 140 colour checker patches viewed under multiple viewing conditions. Specifically, the colour chart is captured by 8 cameras (3 of which are JPEG-only) and under 16 illuminants across many different exposures. Below, we carried out the same experiment described in <cit.>. We are interested in validating whether our method, with much reduced number of parameters can produce, similar or even better results compared with <cit.>. We evaluate both RAW-to-JPEG and JPEG-to-RAW. The dataset <cit.> captures a sort of “worst-case” viewing conditios. Normally, when we capture a picture there is a single prevailing illuminant colour. In the dataset of Chakrabarti , all camera processing parameters are turned off and then the same reflectances are viewed under multiple coloured lights. As Forsyth observed <cit.>, the reddest red camera response cannot be observed under a blue light. And, then he exploited this observation to solve for the colour of the light. Yet, in this dataset the reddest red, the greenest green and the bluest blue can all appear simultaneously in the same image. Practically, we believe the need to divine a pipeline for the all lights all surfaces case means the prior art pipelines are probably more complex than they need to be. As described in <cit.>, for each camera, we estimate the parameters of a calibration model using different subsets of available RAW-JPEG pairs. For each subset and a selected camera, the root mean-squared error (RMSE)between the prediction and ground truth is validated by using all available RAW-JPEG pairs. Table <ref>a shows the RAW-to-JPEG mapping error (where pixel intensities are coded as integers in the interval [0,255]. In the table, Prob denotes the Chakrabarti method (with several thousands parameters) and RB the rank-based method with 408 parameters. We found that our forward errors are close to the results of <cit.>, especially for the condition of less than 3 illuminants which are more likely to occur in the real world. Evidently, for the many illuminant case the prior art has a small advantage. Remembering that JPEGs are coded as integers in [0,255] the RMSE is typically 1 or less (for RB compared to Prob). Practically, when the “fits” are viewed visually (by looking at images) it has hard to see the difference between the two methods. For computer vision, we are more interested in the performance of JPEG-to-RAW mapping which is shown in Table <ref>b and Table <ref>c. In <cit.>, a probabilistic framework for mapping rendered RGB to raw was presented. Here we take their mean estimates as the most likely RAW predictions. We found that our method generally reduces the errors of <cit.> by ∼ 34%. Our supplementary material also includes the additional experiment results compared with “<cit.> + our LUT” for interested readers.The reader might be interested why our simple method seems to work so well going from rendered to raw (better than <cit.>) but not quite as well as the prior art in the forward direction (albeit visually almost indistinguishable). Our hypothesis here is that the LUT in the forward direction is applied post the tone curve. This curve (at least for dark values) has a very high slope and, consequently, the coarsely quantised 5×5×5 LUT cannot capture gamut mapping well. Yet, in the reverse direction (JPEG to RAW) the LUT is applied in linear raw where a course uniform quantisation is more justified.§ CALIBRATION WITH SMALL NUMBERS OF PARAMETERS We wished to visually validate our claim that we can calibrate with few parameters. We took 4 RAW+JPEG pairs (for different cameras) from <cit.>. We then uniformly selected 140 corresponding pixels from the RAW and JPEG. We solved for all parameters in our rank-based method. We then applied our model to the rest of the image. The result of this experiment for 4 images (JPEG-to-RAW) is shown in Fig. <ref>. § CONCLUSION In this paper we have shown how the rank order of image responses is a powerful tool for solving for the individual steps in a camera processing pipeline (colour correction, gamut and tone mapping). A simple ranking argument, relating colour corrected RAWs to corresponding rendered RGBs suffices to solve for the colour correction matrix. Then, the rank-preserving tone map is found and, finally, a simple gamut correction step is derived. Compared with the prior art, our rank-based method requires the fewest assumptions and delivers state-of-the-art radiometric calibration results. | http://arxiv.org/abs/1707.08943v3 | {
"authors": [
"Han Gong",
"Graham D. Finlayson",
"Maryam M. Darrodi"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170727173125",
"title": "Concise Radiometric Calibration Using The Power of Ranking"
} |
APS/[email protected] of Electrical and Computer Engineering, Duke University, Durham, NC, 27708.Kymeta Corporation, 12277 134th Court NE, Redmond, Washington 98052, USAThis work was supported by the Air Force Office of Scientific Research (AFOSR, Grant No. FA9550-12-1-0491)Department of Electrical and Computer Engineering, Duke University, Durham, NC, 27708. Kymeta Corporation, 12277 134th Court NE, Redmond, Washington 98052, USA Department of Electrical and Computer Engineering, Duke University, Durham, NC, 27708.Kymeta Corporation, 12277 134th Court NE, Redmond, Washington 98052, USADepartment of Electrical and Computer Engineering, Duke University, Durham, NC, 27708.We consider the design and modeling of metasurfaces that couple energy from guided waves to propagating wavefronts. This is a first step towards a comprehensive, multiscale modeling platform for metasurface antennas—large arrays of metamaterial elements embedded in a waveguide structure that radiates intro free-space—in which the detailed electromagnetic responses of metamaterial elements are replaced by polarizable dipoles. We present two methods to extract the effective polarizability of a metamaterial element embedded in a one- or two-dimensional waveguide. The first method invokes surface equivalence principles, averaging over the effective surface currents and charges within an element to obtain the effective dipole moments; the second method is based on computing the coefficients of the scattered waves within the waveguide, from which the effective polarizability can be inferred. We demonstrate these methods on several variants of waveguide-fed metasurface elements, finding excellent agreement between the two, as well as with analytical expressions derived for irises with simpler geometries. Extending the polarizability extraction technique to higher order multipoles, we confirm the validity of the dipole approximation for common metamaterial elements.With the effective polarizabilities of the metamaterial elements accurately determined, the radiated fields generated by a metasurface antenna (inside and outside the antenna) can be found self-consistently by including the interactions between polarizable dipoles. The dipole description provides an alternative language and computational framework for engineering metasurface antennas, holograms, lenses, beam-forming arrays, and other electrically large, waveguide-fed metasurface structures. Valid PACS appear here Polarizability Extraction for Waveguide-Fed Metasurfaces David Smith July 14, 2017 ========================================================§INTRODUCTIONThe concept of metamaterials—artificially structured materials—has prompted the rapid development of new tools and techniques for controlling the propagation of waves. The metamaterial paradigm, in which an artificial medium is assembled from a collection of subwavelength elements with desired scattering characteristics, has had a profound impact across numerous scientific fields, including physics <cit.>, electromagnetic <cit.> and acoustic <cit.> wave phenomena, materials science, chemistry, engineering <cit.>, and nanoscience <cit.>. Metamaterials have provided a venue to tailor material properties in ways not feasible with conventional materials <cit.>, opening the door to unique and often exotic wave phenomena such as negative and near-zero refractive index materials<cit.>. Furthermore, metamaterials research has lead to the demonstration of unprecedented devices, such as transformation optical structures and invisibility cloaks <cit.>, as well as superlenses<cit.>.The underlying metamaterial paradigm is that the behavior of waves propagating within a large (many wavelengths) composite medium can be understood from the properties of constituent elements—each subwavelength in dimensions—and their mutual interactions. The advantage of the metamaterial perspective is that the properties of each of the constituent elements can be determined exactly using a full-wave simulation over a relatively small, subwavelength, spatial domain. From these simulations, effective constitutive parameters can be retrieved, replacing the detailed current and field distributions within the small domain by just a few parameters such as the electric permittivity and magnetic permeability <cit.>. The wave propagation properties of the composite structure can then be modeled by solving Maxwell's equations directly, with effective constitutive parameters replacing the actual metamaterial structures. While the effective constitutive parameters obtained by numerical retrieval methods must be applied with considerable caution, retrieval methods have nevertheless been used with success in the design of many metamaterial structures<cit.>. Replacing the details of an artificial medium with effective constitutive parameters facilitates device simulations and optimization cycles, vastly reducing the computational requirements since the individual elements are replaced by homogenized constitutive parameters. Complex metamaterial devices have been designed and demonstrated by this technique, including the transformation optical structures that rely on precise variations in material properties throughout a volume <cit.>.Despite the compelling features of volumetric metamaterials and their unique and often unprecedented material properties, the applications for volumetric metamaterials have been limited. This limitation exists because some of the most intriguing properties of metamaterials occur near resonances in the individual elements—resonances that often impose bandwidth limitations and produce large resistive losses. Thus, waves propagating through any significant volume (even just a few wavelengths) of a metamaterial can be heavily attenuated. In addition, fabricating metamaterial elements to control electric and magnetic fields polarized in arbitrary directions, and assembling such elements throughout a volume, remains a major implementation challenge; typically, the properties of volumetric metamaterials have been demonstrated in highly constrained formats and proof-of-concept prototypes.The difficulties associated with volumetric metamaterials are considerably reduced for metamaterial structures consisting of just a single layer or a few layers of metamaterial elements—also known as metasurfaces <cit.>. Being easier to design, model and implement <cit.>, metasurfaces have rapidly gained traction as a major subfield in metamaterials research <cit.>. As quasi-optical devices, metasurfaces provide control of reflection and transmission across the spectrum <cit.>, paving the way for advanced components such as flat lenses<cit.>, thin polarizers<cit.>, spatial or frequency filters <cit.>, and holographic and diffractive elements <cit.>. Used as coatings, metasurfaces can control the absorbance and emissivity of a surface, and thus have relevance to thermophotovoltaics <cit.>, detectors and sources <cit.>. Given the capabilities of metasurfaces to control waves, but without many of the limitations of volumetric metamaterials, metasurfaces have proven a good match for commercialization efforts, with many serious applications now being pursued, including satellite communications <cit.>, radar <cit.>, and microwave imaging <cit.>.As with a volumetric metamaterial, the scattering properties of a metasurface can be characterized by a set of effective surface constitutive properties, which homogenize—or average over—the properties of many identical, discrete, metamaterial elements <cit.>. These effective medium properties relate to the discontinuity of the fields across the metasurface—approximated as having infinitesimal thickness—and are encapsulated in a set of generalized boundary conditions <cit.>. Effective surface constitutive properties are a convenient and natural description for free-standing metasurfaces illuminated by plane waves <cit.>; however, there are many contexts where a homogenization description is not the most convenient or the most accurate. For example, a metasurface interacts with a guided wave in ways not easily captured with a simple homogenized description. To better illustrate this point, consider a rectangular waveguide with one bounding surface patterned with a collection of metamaterial elements, each consisting of a pattern of voids or irises in the conducting surface. The volume of the waveguide just beneath the element behaves as if filled with a material possessing effective constitutive parameters introduced by the presence of the metamaterial element <cit.>. In addition, the waveguide-fed metamaterial elements also leak energy out of the waveguide, such that the entire structure can act as an aperture antenna <cit.>. Indeed, metasurfaces can provide nearly total control over the wavefront in a manner similar to phased arrays <cit.> and other aperture antennas, but often with advantages not available in other formats <cit.>. For metasurfaces, a weakness associated with homogenization techniques is that the element size and average spacing between metamaterial elements must be significantly subwavelength—a condition not necessarily satisfied in many situations. An element size of one-tenth to one-fifth of a wavelength is typical for many metamaterial structures, which implies the phase of the wave will have significant variation over the volume containing the element. Such metamaterials are said to exhibit spatial dispersion, and are properly characterized by constitutive parameters that depend on the wave vector in addition to the frequency, adding complication to the homogenization description. While numerical retrievals include spatial dispersion in the effective constitutive parameters, the retrieved parameters are specifically valid only in the exact arrangement simulated. For example, if the effective constitutive parameters are retrieved from a simulation of a cubic cell with periodic boundary conditions, the retrieved parameters will only be specifically valid for that medium, and not necessarily applicable when the same element is placed in a different context—for example, in a random arrangement.To arrive at a more generally valid description of a metamaterial while still avoiding a full-wave simulation of the composite structure, we consider directly the properties of each metamaterial scattering element. The response of such a metamaterial element can generally be expressed in a series of induced electric and magnetic multipoles, typically dominated by the dipole term. The strength of the dipolar contribution is connected to an effective polarizability, which represent the coupling between the total dipole moment and the incident field on the element. To the extent that the higher order multipoles beyond the dipolar term can be neglected, the scattering from a collection of equivalent dipoles provides a near exact and computationally efficient model of the metamaterial structure. By assigning an effective polarizability to a metamaterial element rather than treating the metamaterial or metasurface as a continuous medium with constitutive parameters, it is possible to predict the overall response of the structure without any limitation on the element's periodicity or arrangement. The combination of polarizability extraction and the dipole representation forms an alternative, powerful modeling platform for metasurfaces and metamaterials.To obtain the polarizability of an arbitrary metamaterial element, a technique that can be applied is to first assume the element is part of an infinitely periodic medium <cit.>. The details of the periodic structure can then be replaced by periodic boundary conditions, so that the full-wave simulation domain extends only over a single cell of the structure. An effective polarizability of the element, which includes the contributions from all other elements in the infinite array, can then be extracted from the computed field or charge/current distributions. Finally, the intrinsic polarizability can be determined using the Lorentz formula that relates the intrinsic and effective polarizabilities <cit.>. Our aim here is to extend the dipole model as an analytical tool for waveguide-fed metasurfaces. Unlike the free-standing metasurface or volumetric metamaterial, for which each metamaterial element can be reduced to a free space dipole, an individual metamaterial element patterned in a waveguide also interacts with the waveguide structure. Even if the metamaterial element is reduced to a single free space dipole, the image dipoles induced within the waveguide structure must then be taken into account to obtain an accurate expression for the waveguide-fed metamaterial element <cit.>. Rather than working through this complication, we can instead apply a numerical polarization extraction procedure using the waveguide modes, arriving at an effective polarizability that includes all of the waveguide interactions. Once having established this dipole description of a waveguide-based metamaterial element, we can subsequently find the properties of the composite waveguide-fed metasurface accurately, efficiently and quickly, for example using the dipole model previously described.We approach this problem as follows: In Section <ref> we introduce the polarizability framework for a metamaterial element embedded in a waveguide and summarize the self-consistent dipole model for metasurfaces. In Section <ref> we apply the equivalence principle to the waveguide-fed metamaterial, and derive integrals relating the equivalent current densities to the effective dipole moments. The results of the direct integration method are used as the basis for comparison with the second method we present in Section <ref>, in which the polarizability is obtained using the scattering (S) parameters of the element when placed in a waveguide. Both methods have been used in the context of numerical retrieval of effective constitutive parameters for volumetric metamaterials, with the former method related to field averaging <cit.>, while the latter method related to the well-known S-parameters retrieval method <cit.>. In Section <ref> we perform polarizability extractions for different waveguide-fed metasurface geometries: circular iris <cit.>, elliptical iris <cit.>, iris-coupled patch antenna <cit.> and the complementary electric inductive-capacitive metamaterial resonator(cELC) <cit.>. We extend the polarizability extraction method to two-dimensional (2D-) waveguide structures in section (<ref>), where the S-parameters cannot be used to retrieve polarizability. To address this problem, we outline an alternative method based on the mode expansion of cylindrical waves propagating through the waveguide. This framework is particularly advantageous for modeling and designing planar structures <cit.>. We also demonstrate that the extracted polarizability of a metamaterial element depends on the geometry of the waveguide in which it is embedded. As a means of confirming the accuracy of the dipole approximation, we express the induced fields from a metamaterial element as a multipole expansion, comparing the relative strengths of the expansion coefficients. The result of this analysis shows that, indeed, the dipole term dominates the response, justifying the dipolar description and use of the dipole model. We conclude by examining the potential application of the polarizability extraction for different metasurface-based devices.§ SELF-CONSISTENT DIPOLE MODEL FOR METASURFACESOur conceptual picture of a waveguide-fed metasurface is that of a collection of polarizable dipoles, each of which accounts for the scattering associated with an actual metamaterial element. Using the polarization extraction technique to be presented, a metamaterial element is reduced to an effective electric dipole moment and an effective magnetic dipole moment, 𝐩 and 𝐦, which are proportional to the local electric or magnetic field at the center of the element multiplied by a coupling coefficient, termed the dynamic polarizability <cit.>. It is referred to as “dynamic" since it describes the element's response due to a time-varying incident field. A time-dependent electric field can induce solenoidal currents and thus can give rise to a magnetic polarization in addition to the electric polarization. To properly take into account the co- and cross-coupling of the electromagnetic fields excited within the metamaterial element, the dynamic polarizability should be represented as a tensor, so that 𝐩= α̅̅̅_ee𝐄^loc and𝐦 = α̅̅̅_mm𝐇^loc, where 𝐇^loc and 𝐄^loc are the local electric and magnetic fields at the center of the element.It is worth noting that the polarizability of a metamaterial element is not an inherent property, but rather depends on the environment, and therefore may be considered a nonlocal property. To better illustrate this point, consider a single passive magnetic dipole m placed at position r_0 in an environment that is described by the Green's function G(r,r'). This configuration is illuminated by an incident magnetic field H^0. The total field H^tot everywhere throughout the environment is thusH^tot=H^0+G(r,r_0)m. There are two possible ways to define the polarizability of a dipole <cit.>. One is the effective polarizability, defined in terms of the incident wave as m=α_mH^0, and the other is the inherent polarizability, defined in terms of the total field m=α̃_mH^tot. The latter case would be a difficult definition to uphold consistently for a point particle, because the real part of the Green's function G(r,r_0) diverges as r→r_0. (Some authors have developed ways of dealing with the singularity and relate it to the radiation of the dipole <cit.>.) The imaginary part of the Green's function at the origin, however, is well-known to have an important physical meaning related to the radiation reaction force on the dipole <cit.>. Defining the polarizability as m=α_mH^0, we consider the average power absorbed by the dipole according to Poynting's theorem: P_abs=(1/2)∫Re{J_m^*·H}dVwhere J_m=iωμ_0mδ^(3)(r-r_0) is the magnetic current associated with the magnetic dipole,ω is the angular frequency, ^* represents complex conjugate, and μ_0 is the permeability of free space. If there are no Ohmic losses in the system, then all power scattered by the dipole is re-radiated, and consequently P_abs=0. Since the magnetic field H that appears in Eq. <ref> is the total magnetic field given in Eq. <ref>, P_abs=0 implies thatIm{m^*·H^0-m^*G(r,r_0)m}=0.Substituting m=α_mH^0 and assuming that the polarizability is isotropic, we obtain a relationship between the imaginary part and the real part of the polarizability asIm{α_m}=|α_m|^2Im{m^*G(r_0,r_0)m/|m|^2},which must be satisfied in order for the conservation of energy to be upheld. Equation (<ref>) implies that Im{1/α_m}=Im{m^*G(r_0,r_0)m/|m|^2}. Therefore, after some simple algebraic manipulation, Eq. <ref> can be recast asα_m=α̃_m/1+iα̃_mIm{m^*G(r_0,r_0)m/|m|^2}.Here, α̃_m is the inherent polarizability of the dipole, whereas α_m, which is defined as the ratio between the dipole moment and the incident field, is dependent on the local environment. If a dipole is placed in free space, then a Taylor series expansion of the Green's function shows that Im{G(r_0,r_0)}=k^3/6π, and this yields the radiation reaction to the polarizability of a dipole in free spaceα_m=α̃_m/1+iα̃_mk^3/6π. The expression in Eq. <ref> is often known in the literature as the radiation reaction correction or the Sipe-Krankendonk relation <cit.>. If instead the dipole is placed just above an infinite ground plane, the dipole radiates twice as much energy, and so Im{m̂G(r_0,r_0)m̂}=k^3/3π. If the dipole is a complementary metamaterial element embedded in a waveguide wall, then it will radiate both into the upper half space and into the waveguide, and so the radiation reaction correction would need to take into account both scattered fields. Considering that this correction must account for half of the radiation in free-space, as in Eq. <ref>, and half of the radiation inside the waveguide, as derived in appendix <ref>, the corrected polarizability has the form α_m=α̃_m/1+iα̃_m (k^3/3π + k/ab). Equation (<ref>) has significant implications on any polarizability extraction method that deals with waveguide integrated metamaterial elements. For example, if the static polarizability of an element is calculated using Bethe theory (as discussed in section <ref>), then the radiation reaction correction will be different depending on whether that element is placed in a 2D waveguide or a cavity, and so the proper correction as in Eq. <ref> will need to be applied in each environment. Once the polarizabilities are determined, an accurate, self-consistent description of the scattering from the collection of dipoles can be obtained using the dipole model–including the Green's function associated with the specific environment–which expresses the relationship between the local fields 𝐄^loc(𝐫_i) in terms of the incident fields 𝐄^0(𝐫_i) and the fields scattered from all of the dipoles. These equations can be written as𝐄^loc(𝐫_i) =𝐄^0(𝐫_i) +∑_j≠ i𝐆_ee(𝐫_i,𝐫_j) 𝐩(𝐫_j) + 𝐆_em(𝐫_i,𝐫_j) 𝐦(𝐫_j)where 𝐆_ee, 𝐆_em correspond to the electric components of the dyadic Green's function. An equivalent expression can be derived for the magnetic field. Examining the expression in Eq. <ref>, it can be seen that these coupled equations capture the interaction of the incident wave with each of the meta-atom (through the j=0 terms) as well as interaction between different elements (the summation term). Note that we do not solve these equations in the present work, but include them to provide the complete modeling framework. The development of the waveguide Green's function and simulation of various waveguide-fed metasurfaces will be presented separately.§ POLARIZABILITY EXTRACTION IN A RECTANGULAR WAVEGUIDE: DIRECT INTEGRATION We start by considering an arbitrarily shaped iris etched into the upper conducting surface of a rectangular waveguide, as shown in Fig.<ref>. The coordinate system is chosen so that the propagation direction is in the z-direction; a corresponds to the width of the waveguide along the x-axis, and b corresponds to the height along the y-axis. The guided wave couples to the iris, which radiates a portion of the incident wave into the free space region. A common methodology to solve the radiated field by such a structure is the surface equivalence principle. This principle states that the electric field on the boundary of a domain can be represented as a magnetic surface current K_m=E×n̂, while the magnetic field on the boundary can be represented as an electric surface current K_e=n̂×H, where 𝐄 and 𝐇 are the total fields on the surface of the domain, and 𝐧̂ is the normal to the surface. Using K_m and K_e and the corresponding Green's functions, one can determine the field within the domain. Applying this principle to the geometry of Fig. <ref>, we see that the tangential electric field is zero everywhere on the waveguide surface except over the void regions defining the iris; if the iris is deeply subwavelength, then the field scattered into the far-field may be approximated by just the first term of the multipole expansions of K_e and K_m. Hence, the dipole moments representing the iris can be calculated as <cit.> 𝐩 = ϵ_0 n̂∫r·𝐄^tan da𝐦 = 1/i μω∫n̂×𝐄^tan da . The integration is performed over the surface of the iris, n̂=ŷ is the vector normal to the top surface, and 𝐄_tan corresponds to the tangential field at the surface of the iris. It is worth noting the tangential magnetic field is not zero over the surface of waveguide; however, the tangential magnetic field corresponds to an electric current density parallel to a metallic wall, and by image theory its effect can be ignored.Since the effective dipole moments are proportional to the incident fields 𝐄^0,𝐇^0, we can define the effective polarizabilities as𝐩 = ϵ_0α̅̅̅^e_eff𝐄^0𝐦 = α̅̅̅^m_eff𝐇^0. In the most general case, each iris can be described by an electric polarizability tensor α̅̅̅^e_eff and a magnetic polarizability tensor α̅̅̅^m_eff. These tensors are symmetric and would thus normally have six unknown parameters, each of which would need to be extracted from a full-wave simulation. In free space, these components can be found by computing the scattered fields in all directions. In the waveguide geometry considered here, such simulations are not possible. Instead, we take advantage of the geometry of the iris and the symmetry of the excitation field into account. For example, when the symmetry axes of iris are coincident with the symmetry axes of the waveguide, we can assume that the polarizability tensor is diagonal. Further, the boundary conditions of the waveguide require that the incident tangential electric field E_x^0 and E_z^0 and the normal component of the magnetic field H_y^0 be equal to zero on the surface of the waveguide. Therefore, the tensor components α_px, α_pz, and α_my can never be excited, and hence we can assume them to be zero. The polarizability tensors then reduce to α̅̅̅^e_eff=diag(0,α_py,0) and α̅̅̅^m_eff=diag(α_mx,0,α_mz). Hence, the polarizability extraction is simplified to finding three unknowns (α_py, α_mx, α_mz). One further unknown can be removed if the iris is placed where the magnetic field of the incident mode has a null in the z-component. In that case, only α_py and α_mx are relevant to the problem. By using Eq. <ref> and Eq. <ref> and the previously described assumptions, the characteristic polarizabilities are α_ey = ∫∫(xE_x + zE_z)dxdz/E^0_y α_mx = 1/i μω H^0_x∫∫ E_z dxdz . Equations Eq. <ref> provide a simple method to calculate the polarizability of an element from a full-wave simulation of the fields of the element embedded in a waveguide structure.§ POLARIZABILITY EXTRACTION IN A RECTANGULAR WAVEGUIDE: SCATTERING PARAMETERS While calculating the polarizability of a metamaterial element by means of Eq. <ref> provides a physically accurate characterization, the integration over the surface of the element can be cumbersome to perform for all desired frequency points and for arbitrary geometries. In many instances, this integration may also be subject to numerical inaccuracies due to singularities near edges or coarse meshing, as it is especially the case for resonant elements such as those examined in section <ref>. Instead of the direct integration, in this section we consider the extraction of the polarizabilities from the fields scattered by the element into the waveguide. For this calculation, we apply coupled mode theory to determine the coupling of the element embedded in a waveguide to the forward and backward scattered fields within the waveguide. The fields inside the waveguide at any plane of constant z (along the propagation direction) can be expanded as a discrete sum of orthogonal modes. These modes are defined as<cit.> E_n^+ =(E_n t(x,y)+E_n z(x,y))e^-iβ_n z H_n^+ =(H_n t(x,y)+H_n z(x,y))e^-iβ_n z E_n^- =(E_n t(x,y)-E_n z(x,y))e^iβ_n z H_n^- =(-H_n t(x,y)+H_n z(x,y))e^iβ_n z where E^-_n and E^+_n are respectively the waveguide modes traveling in the backwards and forwards directions. The subscript “t" refers to the component of the fields that are transverse to the direction of propagation, and β_n is the propagation constant of the nth mode. The mode normalization used in Eq. <ref> is defined from the integral over the cross section of the waveguide, such that ∫E_n·E_mda=δ_mn,where δ_mn is 1 for n=m and 0 otherwise. Note that the electric fields of the modes are dimensionless, and the magnetic fields have units of inverse impedance as∫H_n·H_mda=δ_mn/Z_n^2,where the wave impedance Z_n is defined as a normalization constant for eachmode asZ_n=1/∫E_n×H_n·n̂ da. In Eq. <ref> the integration is over the cross sectional surface of the waveguide, i.e. the surface representing Port 1 in Fig. <ref>. Consider a metamaterial element placed at the center of the top plate of the waveguide. We assume that the incident field is the forward-propagating fundamental mode—coming from Port 1—with unit amplitude E^0+, as shown in Fig. <ref>. When the metamaterial element is present, it couples and scatters to all modes. While the element has a finite size, for points inside the waveguide that are few wavelengths away, the element is well-approximated as a point scatterer placed at z=0 (location of the element). In the absence of the metamaterial element, the total field is simply the incident field, which is identical in both the forward and backward directions. As a result, we express the modal decomposition of the total (both incident and scattered) fields into backwards propagating modes at Port 1 (z=-Δ) asE^-=E_0^++∑_n A_n^-E_n^-,where A^-_n are the amplitudes of the modes scattered by the element in the backwards direction, and n is the mode number. Similarly, a modal decomposition of the fields in the plane of z=+Δ into forward propagating modes yieldsE|^+ =E_0^++∑_n A_n^+E_n^+where A^+_n are likewise the mode amplitude coefficients of the scattered field by the element in the forward direction, and the incident field E_0^+ has been written as a separate term. In these calculations, Δ can be any distance as long as it is larger than the size of metamaterial element.We first consider a volume within the waveguide that encompasses the metamaterial element (bounded by the two ports). The incident field impinging on the element will induce a set of fields that we denote as E and H. Within the coupled mode formulation, these fields can be related to the waveguide modes through Poynting's theorem, or ∇· (𝐄×𝐇^(±)_n - 𝐄^(±)_n ×𝐇)= J_e·E^(±)_n-J_m·H^(±)_n.We integrate Eq. <ref> over the volume V the portion of the waveguide between the two ports, and applying the divergence theorem, Eq. <ref> becomes∫_S (𝐄×𝐇^(±)_n - 𝐄^(±)_n ×𝐇)·𝐧̂ da = ∫_V J_e·E^±_n-J_m·H^±_n dVwhere S is the closed surface that encloses V and 𝐧̂ is an outwardly directed normal. Since the waveguide walls are assumed to be perfectly conducting, the only nonzero contributions to the surface integrals arise from the surfaces representing Port 1 and Port 2 (depicted in Fig.<ref>), and the surface of the metamaterial element. Since the field E can be written as a fictitious magnetic surface current through K_m=E×n̂ and H can be related to a fictitious electric surface current in the same way, then the surface integral in Eq. <ref> indicates the manner in which the effective dipoles representing the metamaterial element couple to each of the waveguide modes, as might be expected from Lorentz reciprocity. To obtain the amplitude coefficients A_n^(±), we assume there are no current sources in the volume, implying that the volume integral in Eq. <ref> vanishes. Substituting the expansions of the fields in Eq. <ref> and Eq. <ref> into Eq. <ref> and using the orthogonality relations in Eq. <ref>, we obtain the scattered amplitude coefficients as an overlap integral of the waveguide mode fields with the total field taken over the surface of the metamaterial element. More explicitly, the amplitude coefficients can be found as A_n^(±)= Z_n/2∫_element (𝐄×𝐇^(∓)_n-𝐄^∓_n ×𝐇)·𝐧 da. In an alternative approach, the electric field in the aperture could be considered zero and replaced by an equivalent electric and magnetic surface current, according to the equivalence principle. In this case, the surface integral vanishes everywhere except over the surfaces of the ports, but the volume integral over the meta-atom becomes a surface integral of the equivalent surface currents K_m=E×n̂ and K_e=-H×n̂. Using Eq. <ref> and Eq. <ref> and invoking orthogonality, we obtainA_n^(±)= Z_n/2∫_element (𝐊_e ·𝐄^(∓)_n-𝐊_m ·𝐇^(∓)_n)da. Since the metamaterial element is deeply subwavelength, the fields of the waveguide modes can be expanded in a Taylor series around the center of the element. The lowest order term is constant over the surface of the element, yieldingA_n^(±)= Z_n/2[ 𝐄^(∓)_n(r_0)·∫_element𝐊_e da-𝐇^(∓)_n(r_0)·∫_element𝐊_m da]. As previously stated in Eq. <ref>, the two integrals in Eq. <ref> are proportional to the electric and magnetic dipole moments, p and m. Therefore, the final expression for the amplitude coefficients in terms of these dipole moments is given by A_n^+=iω Z_n/2(E^-_n·p - μ_0 H^-_n ·m) A_n^-=i ω Z_n/2(E^+_n·p - μ_0 H^+_n·m). Equation <ref> shows that the dipole moments are related to the incident fields, which in turn can be expanded in terms of eigenmodes. Since the incident field is the fundamental mode, the polarizability is defined by p=ϵα̅̅̅^eff_eE_0^+m=α̅̅̅^eff_mH_0^+. Due to the symmetry of the fields in the rectangular waveguide, the α_mz component cannot be excited by the z-component of the magnetic field. Hence, Eq. <ref> reduces to two coupled equations with two unknowns: α_py and α_mx, which can be recast as A_n^+=iω Z_n/2(ϵα_pyE^+_0yE^-_ny - μ_0 α_mxH^+_0xH^-_nx) A_n^-=i ω Z_n/2(ϵα_pyE^+_0yE^+_ny - μ_0 α_mx H^+_0xH^+_nx) Considering the orthogonality of the eigenmodes and the symmetry properties of the electromagnetic fields—the transverse components of the electric field are symmetric under a flip of direction (i.e. E_ny^-=E^+_ny), while the magnetic field is antisymmetric (i.e. H_nx^-=-H^+_nx)—we can solve Eq. <ref> in order to find the polarizabilities as α_py= 2/iω Z_n(A_0^+ + A_0^-)/ϵ (E^+_0y)^2 α_mx= 2/iω Z_n(A_0^+ - A_0^-)/μ (H^+_0x)^2. For the fundamental mode, the normalized fields and impedance at the dipole location are given by|E^+_0y|^2= 4/ab |H^+_0x|^2= 4β_10^2/ab Z_0^2 k^2 Z_0 = η k/β_10where η is the vacuum impedance. Furthermore, the amplitude coefficients A^+_0 and A^-_0 correspond to the amplitude terms for the fundamental mode of the scattered fields in the forward and backward directions. Therefore they are directly related to the scattering parameters with respect to each port: the reflected field, related to A^-_0 is proportional to the reflection coefficient i.e. S_11, while the transmitted field related to A^+_0 is proportional to the transmission coefficient S_21 and the incident field in the forward direction. More explicitly, these relationships are expressed byA^-_0= S_11 A^+_0= S_21-1.Taking into account Eq. <ref> in conjunction with Eq. <ref> and Eq. <ref> it is possible to find the final expression for the polarizabilities asα_py =-iabβ_10/2k^2(A_0^++A_0^-) = -iabβ_10/2k^2(S_21+S_11-1)α_mx =-iab/2β_10(A_0^+-A_0^-) = -iab/2β_10(S_21-S_11-1) . Equation <ref> provides the polarizabilities of any metamaterial element embedded in a rectangular waveguide in terms of the scattering parameters, which can be obtained from direct measurement or full-wave simulation. However, it is important to note that the scattering parameters must be de-embeded to the plane of the metamaterial element before using them in Eq. <ref>. This is a straightforward process well-known in literature <cit.>. Another important point to note is that we have assumed ports which only excite/represent single mode. This condition should be applied when simulating these structures in numerical solvers. More importantly, since it is cumbersome in experiment to excite purely the fundamental mode, the ports should be placed at least a few wavelengths in distance away from the metamaterial element to ensure the non-propagating higher order modes have decayed. The equations in Eq. <ref> are similar to the expressions found for the effective polarizabilities of metamaterial elements in periodic metasurfaces <cit.>. This relationship means that a single element in a rectangular waveguide acts as a dipole whose response is equivalent to the response of the element in a periodic metasurface. The details of this equivalence will be discussed in a future paper.§ SIMULATED RESULTS FOR THE RECTANGULAR WAVEGUIDE Using a full-wave simulation, we can extract the polarizability of arbitrary metamaterial elements patterned into rectangular waveguides from the two different approaches described in sections <ref> and <ref>. For both extraction techniques, a single full-wave simulation in CST Microwave Studio is performed assuming a waveguide designed to operate over frequencies in the X-band (8-12 GHz). The waveguide dimensions are a=21.94mm, b=5mm, L=22.7mm and thickness 1.27mm. We perform this simulation for several different metamaterial element geometries: circular iris, elliptical iris, iris-coupled patch antenna, and the cELC resonator. In addition to the methods described above, the dipole moments of simple geometries, such as an elliptical iris may be also obtained from the static dipole moments of general ellipsoidal dielectric and permeable magnetic bodies. Consider an elliptically shaped aperture with the major axis along the x-direction and minor axis along the z-direction. Let the major radius be l_1 and minor radius l_2. In the static limit, the polarizabilities of such an elliptical iris are given by <cit.> α̃_̃m̃x̃ = 4π l_1^3 e^2/3[E(e) - K(e)] α̃_̃m̃z̃ = 4π l_1^3 e^2(1-e^2)/3[E(e) - (1-e^2)K(e)] α̃_̃ẽỹ= -4π l_1^3 (1-e^2)/3E(e),where e= √(1-(l_2/l_1)^2) (assuming l_1>l_2) is the eccentricity of the ellipse, and K(e) and E(e) are the complete elliptic integrals of the first and second kind, respectively. If e=0 these expressions reduce to the static polarizabilities of circular irises. Using these static expressions for the dynamic polarizability will violate conservation of energy since it lacks the radiation damping term of dynamic polarizability. These expressions may be corrected by the radiation term, as described in Eq. <ref>, assuming the real part given in Eq. <ref>. Figure <ref> shows the polarizability of simple circular and elliptical irises computed using the two methods described in this paper as well as the theoretical methods of Eq. <ref> and Eq. <ref>. As shown, excellent agreement between the analytical expressions and the numerical extractions is obtained, verifying the proposed methods. Since the circular iris considered here does not possess a resonance, it is expected that the dynamic polarizabilities extracted numericallyare well-approximated by the theoretical expressions. Next, we examine the case of an elliptical iris, as shown in Fig.<ref>b. The extracted polarizabilities computed using the two numerical extraction methods of previous section exhibit excellent agreement. However, the analytical exhibits significant deviation from the numerical methods. This is expected since the elliptical iris supports a resonance over the frequency band of interest, which is not captured in the analytical expressions derived for the static field. This case further highlights the need for a precise numerical method to compute the polarizability of a metamaterial element. It is worth noting that an elliptical iris etched in a rectangular waveguide is well-known in the antenna engineering community as the unit cell of a slotted waveguide antenna (SWA). SWAs are particularly attractive due to their advantages in terms of design simplicity, weight, volume, power handling, directivity, and efficiency <cit.>. SWAs rely on the gradual leakage of the guided mode through the slots. The metasurfaces considered throughout this paper also share this feature, since they also leak energy from the guided wave through the metamaterial elements. In other words, the methodology developed in this paper can also be applied to leaky wave antennas for modeling, simulation, and designing <cit.>.While the circular and the elliptical irises may be analyzed using analytical expressions for the polarizabilities derived in the static limit, such closed-form expressions are not available for most metamaterial designs. For example, an element of potential interest in the design of metasurface antennas is the iris-fed patch, shown in Fig.<ref>a <cit.>. The inclusion of the metallic patch above the iris enhances the resonant response of the element, as exemplified by the narrower and stronger resonant response. Another common metamaterial element is the cELC, shown in Fig.<ref>b, commonly used in metasurface antenna designs. The resonant response of the cELC is highly susceptible to variations in its geometry <cit.>. For both elements, we observe excellent agreement between the two numerical methods of equation Eq. <ref> and equation Eq. <ref>, as shown in Fig. <ref> and <ref>.The geometry of metamaterial elements can be quite complicated, such that the numerical integrals required in Eq. <ref> are likely to yield inaccuracies. For this reason, the extraction based on computing the waveguide scattering parameters is likely to be more reliable and easier to implement; moreover, this method can also be used in measurements on fabricated samples. It worth noting that in all of the results presented in Fig.<ref> and <ref>, it can be seen the electric polarizability is much smaller than the magnetic polarizability—in fact, three orders of magnitude smaller. This phenomenon is expected considering that the geometry under study corresponds to a small opening in a metallic wall. §POLARIZABILITY EXTRACTION IN A PARALLEL PLATE WAVEGUIDE In this section, we consider the case of a 2D waveguide and examine a metamaterial element that is etched on one wall of a parallel plate waveguide. While the polarizability extraction method based on the equivalence principle holds for the 2D waveguide-fed element, the nature of the waveguide modes changes substantially from the formulation in section <ref>. In this section, we modify this extraction technique to be applicable to planar waveguide systems. As previously described in section <ref>, a metamaterial element scattering into a waveguide can be described in terms of a sum of waveguide modes. Because the element is placed in the upper surface of the waveguide, the boundary condition dictates the tangential electric field and the normal magnetic field to be zero and the element can only couple to the transverse magnetic (TM) modes. Since the natural symmetry of the system is cylindrical, mode decomposition is simpler if we use cylindrical coordinates (r,θ). Setting the origin of the coordinate system to the center of the metamaterial element, the z-components of the scattered electric field—for the TM modes characterized by the (m,n) indices— are given by E^sc_z,c=β_m/k H_n^(2)(β_m r)cos(nθ) E^sc_z,s=β_m/k H_n^(2)(β_m r)sin(nθ) where the subscripts “c" and “s" refer to modes that have angular dependence cos(nθ) and sin(nθ), respectively. The propagation constant is given by β_m=√(k^2-(mπ/h)^2), where h is the height of the waveguide. Invoking the superposition principle, the total solution for the z-component of the scattered electric field can be expressed as E_z=∑_n∑_mA_mn^s E_z,s^mn+A_mn^c E_z,c^mnWhen h<π/k, only the m=0 mode is propagating, and in this case the electric field at all points where r≫ h/π is dominated by the m=0 mode. Therefore we can reduce Eq. <ref> to E_z=∑_nA_n^s E_z,s^0n+A_n^c E_z,c^0n.The m=0 modes are given by E_z,c^0n=H_n^(2)(kr)cos(nθ) E_z,s^0n=H_n^(2)(kr)sin(nθ).The amplitude coefficients, A_n, can be found from the scattered electric field E_z using the orthogonality of the {sin(θ),cos(θ)} basis. By integrating over a circle of radius r centered at the origin of the metamaterial element, as shown in Fig.<ref>, it is possible to define the amplitude coefficients asA_n^s=lim_r →∞1/π H_n^2(kr)∫_0^2π E_z(r,θ)sin(nθ) dθA_n^c=lim_r →∞1/π(1+δ_n0)H_n^2(kr)∫_0^2π E_z(r,θ)cos(nθ) dθ. To better illustrate the utility of equations Eq. <ref>, we consider a lossless parallel plate waveguide, fed by a cylindrical source oriented in the z-direction,, as shown in Fig. <ref>. The source is placed far enough from the metamaterial element to avoid evanescent coupling. A single cELC resonator is assumed to be patterned on the center of the top plate of the waveguide, with the same geometrical parameters as the cELC presented in the previous section. Since the full-wave simulation domain represents the total field instead of the scattered field,this structure is simulated with and without the cELC, and the difference of the two simulation results are taken to obtain the scattered field due to the metamaterial element, such that E_z^sc= E_z^tot -E_z^0 at the plane z=h/2. Once the scattered field is computed, the integration outlined in Eq. <ref> is performed to find the amplitude coefficients. The integration radius is selected electrically large enough so that the evanescent modes have decayed —it is also ensured the integration curve does not contain the cylindrical source.Figure <ref> shows the magnitude of the amplitude coefficients for the scattered fields computed for the meta-atom shown in Fig.<ref>. To better illustrate the physics behind these coefficients, we apply Poynting's theorem Eq. <ref> which directly links the amplitude coefficients and the dominant dipole moments of the metamaterial element. In contrast to the 1D case examined in section <ref>, the amplitude coefficients are not related to the scattering parameters, but rather to the scattered fields, by means of Eq. <ref>. Moreover, while the location of the metamaterial element in the rectangular waveguide limits the calculation of a single component of the magnetic polarizability, such limitation disappears in the case of the planar waveguide. The cylindrical wave propagating through the waveguide may excite the two tangential components of the magnetic polarizability, which leads to a more complete characterization of the polarizability tensor of the element. The scattered fields generated by the metamaterial element can be represented as the sum of the moments of the surface current J^n_m multiplied by the different eigenmodes of the scattered fields shown in Eq. <ref>, more explicitly, this relationship is given by E_z=m_xZ_0k^2/4h E_z,s^01+m_yZ_0k^2/4h E_z,c^01+-ip_zk^2/4hϵ_0 E_z,c^00A direct mapping between Eq.<ref> and Eq. <ref> demonstrates that the first three amplitude coefficients, {A_0^c, A_1^c, A_2^s} are directly related to the three dominant dipole moments as <cit.> m_x=A_1^s4h/Z_0 k^2 m_y=A_1^c4h/Z_0 k^2 p_z=A_0^ci4hϵ_0/k^2. As shown in Fig. <ref>, the predominant amplitude mode is A^s_1, which is directly associated with m_x, while the amplitude of the modes A^c_0 and A^c_1, associated with p_z and m_y, are significantly smaller—by two orders of magnitude. The effective polarizabilities given the incident wave due to the line source, can be directly obtained from their corresponding dipole moments given in Eq. <ref> as α_z^p=p_z/E_z^0α_xy^m=m_y/H_x^0α_xx^m=m_x/H_x^0. For clarification, the double indices on the polarizabilities represent the entry in the polarizability tensor; for example, α_xy represents the component of the polarizability that generates a dipole moment oriented in y, due to the x component of the incident field. In order to find all three components of this tensor, it is necessary to rotate the cELC by π/2, and perform the same extraction technique. The electric polarizability and two of the components of the magnetic polarizability tensor are thereby obtained and shown in Fig.<ref>a. As shown, excellent agreement for the magnetic polarizabilities is obtained between the two numerical polarizability extraction methods. For this particular example, note that the polarizability α^m_yy has a resonant response out of the X-band, but its geometry can be modified such that it has both resonances in the same band <cit.>. In the case of the electric polarizability (Fig.<ref>b), the numerical values obtained are significantly smaller, which makes it susceptible to numerical inaccuracies when the integration in Eq. <ref> is performed. It is important to highlight in this example that the term A^s_3 is associated with the quadrupole moment. This term has been traditionally neglected in most metamaterial design strategy. The example at hand provides a useful framework to examine the contribution of the quadrupole term and the error introduced by neglecting it. To study the impact of the quadrupole term, we compared the simulated scattered field (shown in Fig. <ref> first row) with its theoretical expression Eq. <ref> up to only the dipolar contribution, as shown in the second row of Fig.<ref>. We observe excellent agreement between the two rows, confirming the assumption that the main contribution of the scattered field is dipolar. The physical implications of this result can be understood by calculating the difference between the scattered fields from a full-wave simulation and from the analytical expression in (Eq. <ref>). As shown in the third row of Fig. <ref>, the error due to assuming the dominant dipolar term is several orders of magnitude smaller than the amplitude of the scattered field, and the largest discrepancy is observed within the close vicinity of the metamaterial element. This result, in conjunction with the amplitude coefficients shown in Fig.<ref> also demonstrates that most of the radiation is associated with the dipolar term and higher order modes can be ignored. However, if the elements are placed at distances where these higher order modes have not decayed, they can alter the coupling between the two meta atoms and change the total scattered fields inside the waveguide. Another interesting point to highlight is that while the geometric characteristics of the cELC used in Fig.<ref>c and <ref> are the same, its resonant response as manifested by the effective polarizability changes depending on the host waveguide, as can be observed by comparing Fig.<ref>d and Fig.<ref>a. Therefore, the electromagnetic response of a metamaterial element embedded in a waveguide not only depensd on its intrinsic geometrical characteristics, but also depends on the waveguide geometry where it is inserted. §CONCLUSIONS In this paper we have presented a comprehensive method for extracting the effective polarizabilities of metamaterial elements patterned in 1D rectangular and 2D parallel plate waveguides. In our approach, the S-parameters—along with the knowledge of the normalized fields inside the waveguide—are used to find the electric and magnetic polarizabilities, making this technique a powerful tool for the design and characterization of metasurfaces. To demonstrate the validity of our approach, the polarizability extraction method was compared with direct extraction of the tangential components of the scattered fields, and excellent agreement between the two methods was demonstrated. We have also shown that the dipole modes predominate the scattering from metamaterial elements, since all higher-order multipole fields decay more rapidly with distance. The concepts presented in this paper pave the way for a simple and efficient approach to the analysis of metasurface antennas. The combination of polarizability extraction techniques with the dipole model provides a powerful and inherently multiscale modeling tool to design and characterize metasurface structures without any limitation on the element geometry or periodicity assumptions common to other homogenization techniques. § RADIATION REACTION INSIDE A RECTANGULAR WAVEGUIDEAs described in section <ref>, the radiation reaction in a rectangular waveguide can be found by taking the real part of the surface integral of the Pointing's vector 𝐒. Let us consider again a thought experiment where two collocated electric and magnetic dipole 𝐩 and 𝐦 are placed at position 𝐫_0,in an environment that is described by the Green's function 𝐆(𝐫; 𝐫_0). The surface integral of the Poynting's vector is given by∫𝐒· d𝐚 = iω (𝐩^*·𝐄 -μ_0 𝐦^*·𝐇) where 𝐄 and 𝐇 correspond to the total fields inside the waveguide. Evaluating such fields at the dipole's location we get∫𝐒· d𝐚=iω (𝐩^*·𝐄 -μ_0 𝐦^*·𝐇)= iω (𝐩*·𝐆_ee(r_0,r_0)·𝐩 -μ_0 𝐦^*·𝐆_mm(r_0;r_0)·𝐦) Now, considering the real part of Eq. <ref> it is possible to obtain a direct relationship between the total power radiated and the imaginary components of the Green's functions asRe{∫𝐒· d𝐚} = ω |𝐩|^2 Im{ G_ee(r_0,r_0) } - μ_0 ω |𝐦|^2 Im{ G_mm(r_0,r_0) } On the other hand, from the modal expansion of the fields Eq. <ref> and Eq. <ref>, the same integral shown in Eq. <ref> results in∫𝐒· d𝐚 =1/Z_n(|A_n^+|^2 + |A_n^-|^2).where Z_n, A_n^+ and A_n^- have been previously defined in <ref>. By using the equations for the amplitude coefficients shown in Eq. <ref> into (Eq. <ref>) we obtain (|A_n^+|^2 + |A_n^-|^2) = ω^2 Z_n^2/4 (|E_n^+|^2 |p|^2 - μ_0|H_n^+|^2 |m|^2 ) In addition, it was previously demonstrated that for the fundamental mode, the fields E_n^+ and H_n^+ are normalized by means of Eq. <ref>. Replacing the specific expressions for the normalized fields, and equating Eq. <ref> and Eq. <ref> we obtain a direct relationship between the amplitude of the normalized modes and the imaginary Green's functions, which are our ultimate goal in this derivation. More explicitly, ω |𝐩|^2 Im{ G_ee(r_0,r_0) } - μ_0 ω |𝐦|^2 Im{ G_mm(r_0,r_0) }= ω^2 Z_n^2/4 (|E_n^+|^2 |p|^2 - μ_0|H_n^+|^2 |m|^2 ) It can be observed that, in order to satisfy Eq. <ref> the terms multiplying the magnitude of the dipole moments |p|^2 and |m|^2 must be equal. After some algebraic derivation it is possible to conclude thatIm{ G_ee(0) } = k^2/β abIm{ G_mm(0) }= k/ab. | http://arxiv.org/abs/1708.05061v1 | {
"authors": [
"Laura Pulido-Mancera",
"Patrick T. Bowen",
"Nathan Kundtz",
"Mohammadreza F. Imani",
"David Smith"
],
"categories": [
"physics.app-ph"
],
"primary_category": "physics.app-ph",
"published": "20170727161210",
"title": "Polarizability Extraction for Waveguide-Fed Metasurfaces"
} |
Time irreversibility and multifractality of power along single particle trajectories in turbulence[Version accepted for publication (postprint) on Phys. Rev. Fluids 2, 104604 – Published 27 October 2017] Massimo De Pietro===========================================================================================================================================================================================================The ⌞-intersection graphs are the graphs that have a representation as intersection graphs of axis parallel ⌞ shapes in the plane. A subfamily of these graphs are {⌞,| ,- }-contact graphs which are the contact graphs of axis parallel ⌞, |, and - shapes in the plane. We prove here two results that were conjectured by Chaplick and Ueckerdt in 2013. We show that planar graphs are ⌞-intersection graphs, and that triangle-free planar graphs are {⌞,| ,- }-contact graphs.These results areobtained by a new and simple decomposition technique for 4-connected triangulations.Our results also provide a much simpler proof of the known fact that planargraphs are segment intersection graphs. § INTRODUCTION The representation of graphs by contact or intersection of predefined shapes in the plane is a broad subject of research since the work of Koebe on the representation of planar graphs by contacts of circles <cit.>. In particular, the class of planar graphs has been widely studied in this context.More formally, assigning a shape X of the plane for each vertex of a graph G, we say that G is a X-intersection graph if there is a representation of G such that every vertex is assigned to a shape X, and two shapes X_1, X_2 intersect if and only if the vertices they are assigned to are adjacent in G. In the case where the shape X is homeomorphic to segments (resp. discs), a X-contact system is a collection of X shapes such that if an intersection occurs between two shapes, then it occurs at one of their endpoints (resp. on their border). We say that a graph G is a X-contact graph if it is the intersection graph of a X-contact system. This definition can be easily generalized if the representation of each vertex is chosen among a family of shapes.The case of shapes that are homeomorphic to a disc has been widely studied; see for example the literature for triangles <cit.>, homothetic triangles <cit.>, axis parallel rectangles <cit.>, squares <cit.>, hexagons <cit.>, or convex bodies <cit.>. We here focus on the representation of planar graphs as contact or intersection graphs, where the assigned shapes are segments or polylines in the plane. The simplest definition of representation of graphs by intersection of curves is the so-called string-representation: each vertex is represented by a curve, and two curves intersect if and only if the vertices they represent are adjacent in the graph. It is known that every planar graph has a string-representation <cit.>.However, this representation may contain pairs of curves that cross any number of times. One may thus take an additional parameter into account, namely the maximal number of crossings of any two of the curves: a 1-string representation of a graph is a string representation where every two curves intersect at most once. The question of finding a 1-string representation of planar graphs has been solved by Chalopin et al. in the positive <cit.>, and additional parameters are now studied, like order-preserving representations <cit.>.Segment intersection graphs are in turn a specialization of the class of 1-string graphs. It is known that bipartite planar graphs are {|,-}-contact graphs <cit.> (i.e. segment contact graphs with vertical or horizontal segments).De Castro et al. <cit.> showed that triangle-free planar graphs are segment contact graphs with only three different slopes. De Fraysseix and Ossona de Mendez <cit.> then proved that a larger class of planar graphs are segment intersection graphs. Finally, Chalopin and the first author extended this result to general planar graphs <cit.>, which was conjectured by Scheinerman in his PhD thesis <cit.>.A graph is said to be a VPG-graph (Vertex-Path-Grid) if it has a contact or intersection representation in which each vertex is a path of vertical and horizontal segments (see <cit.>). Asinowski et al. <cit.> showed that the class of VPG-graphs is equivalent to the class of graphs admitting a string-representation. They also defined the class B_k-VPG, which contains all VPG-graphs for which each vertex is represented by a path with at most k bends (see <cit.> for the determination of the value of k for some classes of graphs). It is known that B_k-VPG ⊈ B_k+1-VPG, and that the recognition of graphs of B_k-VPG is an NP-complete problem <cit.>. These classes have interesting algorithmic properties (see for example <cit.> for approximation algorithms for independence and domination problems in B_1-VPG graphs), but most of the literature studies their combinatorial properties.Chaplick et al. <cit.> proved that planar graphs are B_2-VPG graphs. This result was recently improved by Biedl and Derka <cit.>, as they showed that planar graphs have a 1-string B_2-VPG representation.Various classes of graphs have been showed to have 1-string B_1-VPG representations, such as planar partial 3-trees <cit.> and Halin graphs <cit.>. Interestingly, it has been showed that the class of segment contact graphs is equivalent to the one of B_1-VPG contact graphs <cit.>. This implies in particular that triangle-free planar graphs are B_1-VPG contact graphs. This has been improved by Chaplick et al. <cit.> as they showed that triangle-free planar graphs are in fact {⌞, ⌜, |, -}-contact graphs (that is without using the shapes ⌟ and ⌝).The restriction of B_1-VPG to ⌞-intersection or ⌞-contact graphs has been much studied (see for example <cit.>) and it has been shown that they are in relation with other structures such as Schnyder realizers, canonical orders or edge labelings <cit.>.The same authors also proved that the recognition of ⌞-contact graphs can be done in quadratic time, and that this class is equivalent to the one restricted to equilateral ⌞ shapes. Finally, the monotone (or linear) ⌞-contact graphs have been recently studied further, for example in relation with MPT (Max-Point Tolerance) graphs <cit.>. Our contributionsThe two main results of this paper are the following:Every triangle-free planar graph is a {⌞, | , -}-contact graph. Every planar graph is a ⌞-intersection graph. Both results were conjectured in <cit.>. Theorem <ref> is optimal in the sense that a {⌞, | , -}-contact graph with n vertices has at most 2n-3 edges, while triangle-free planar graphs may have up to 2n-4 edges. However, up to our knowledge, the question of whether every triangle-free planar graph is a {⌞, | }-contact graph is open [In fact, it has been proven in the Masters thesis (in German) of Björn Kapelle in 2015 <cit.>, but never published.]. Theorem <ref> implies that planar graphs are in B_1-VPG, improving the results of Biedl and Derka <cit.> stating that planar graphs are in B_2-VPG. Since a ⌞-intersection representation can be turned into a segment intersection representation <cit.>, this also directly provides a rather simple proof of the fact that planar graphs are segment intersection graphs <cit.>. Note that a simple modification of our method can be used to prove that 4-connected planar graphs have a B_3-EPG representation <cit.>, where vertices are represented by paths on a rectangular grid with at most 3 bends, and adjacency is shown by sharing an edge of the grid.The common ingredient of the two results is what we call 2-sided near-triangulations. In Section <ref>, we present the 2-sided near-triangulations, allowing us to provide a new decomposition of planar 4-connected triangulations (see <cit.> and <cit.> for other decompositions of 4-connected triangulations). This decomposition is simpler than the one provided by Whitney <cit.> that is used in <cit.>. In Section <ref>, we define thick ⌞-contact systems, (i.e., ⌞-contact representations in which the shapes have some thickness ε)with specific properties. We then show that every 2-sided near-triangulation can be represented by such a system. This result is used in Section <ref> to prove Theorem 1. Then in Section <ref> we use 2-sided near-triangulations to prove Theorem 2.§ 2-SIDED NEAR-TRIANGULATIONS In this paper we consider plane graphs without loops nor multiple edges. In a plane graph there is an infinite face, called the outer face, and the other faces are called inner faces.A near-triangulation is aplane graph such that every inner face is a triangle.In a plane graph G, a chord is an edge not incident to the outer face but that links two vertices of the outer face. A separating triangle of G is a cycle of length three such that both regions delimited by this cycle (the inner and the outer region) contain some vertices.It is well known that a triangulation is 4-connected if and only if it contains no separating triangle. Given a vertex v on the outer face, the inner-neighbors of v are the neighbors of v that are not on the outer face. We define here 2-sided near-triangulations (see Figure <ref>) whose structure will be useful in the inductions of the proofs of Theorem <ref>, and Theorem <ref>. A 2-sided near-triangulation is a 2-connected near-triangulation T without separating triangle and such that going clockwise on its outer face, the vertices are denoted a_1,a_2,…, a_p, b_q, …, b_2,b_1, with p≥ 1 and q≥ 1, and such that there is no chord a_ia_j or b_ib_j (that is an edge a_ia_j or b_ib_j such that |i-j| > 1). The structure of the 2-sided near-triangulations allows us to describe the following decomposition: Given a 2-sided near-triangulation T with at least 4 vertices, one can always perform one of the following operations: (a_p-removal) This operation applies if p>1, if a_p has no neighbor b_i with i<q, and if none of the inner-neighbors of a_p has a neighbor b_i with i<q. This operation consists in removing a_p from T, and in denoting b_q+1, …, b_q+r in anti-clockwise order the new vertices on the outer face, if any. This yields a 2-sided near-triangulation T' (see Figure <ref>). (b_q-removal) This operation applies if q>1, if b_q has no neighbor a_i with i<p, and if none of the inner-neighbors of b_q has a neighbor a_i with i<p. This operation consists in removing b_q from T, and in denoting a_p+1, …, a_p+r in clockwise order the new vertices on the outer face, if any. This yields a 2-sided near-triangulation T'.This operation is strictly symmetric to the previous one. (cutting) This operation applies if p> 1 and q>1, and if the unique common neighbor of a_p and b_q, denoted x, has a neighbor a_i with i<p, and a neighbor b_j with j<q. If x has several such neighbors, i and j correspond to the smaller possible values. This operation consists in cutting T into three 2-sided near-triangulations T', T_a and T_b (see Figure <ref>): * T' is the 2-sided near-triangulation contained in the cycle formed by vertices (a_1,…,a_i,x,b_j,…,b_1), and the vertex x is renamed a_i+1.* T_a (resp. T_b) is the 2-sided near-triangulation contained in the cycle (a_i,…,a_p,x) (resp. (x,b_q,…,b_j)), where the vertex x is denoted b_1 (resp. a_1). Suppose that a_p has no neighbor b_i with i < q and none of the inner-neighbors of a_p has a neighbor b_i with i < q. We denote b_q+1, … , b_q+r the inner-neighbors of a_p in anti-clockwise order such that b_j is connected to b_j+1 for every q ≤ j ≤ r. Let T' be the graph obtained by removing a_p and its adjacent edges from T. It is clear that T' is a near-triangulation, and that it has no separating triangle (otherwise T would have one too). Furthermore, as there is no chord incident to a_p, and as T' has at least three vertices its outer face is bounded by a cycle, and T' is thus 2-connected.As T is a 2-sided near-triangulation, T' has no chord a_i a_j, with i,j< p, or b_i b_j with i,j ≤ q. By hypothesis, the inner-neighbors of a_p have no neighbors b_k with k < q, thus there is no chord b_i b_j with i ≤ q and q < j. There is no chord b_i b_j in T' with q ≤ i < j.Otherwise the vertices a_p, b_i, and b_j would form a triangle with at least one vertex inside, b_i+1, and at least one vertex outside, a_p-1: it would be a separating triangle, a contradiction. Therefore T' is a 2-sided near-triangulation.The proof for the b_q-removal operation is analogous to the previous case.Suppose that we are not in the first case nor in the second one. Let us first show that p>1 and q>1. Towards a contradiction, consider that p=1. Then as T is 2-connected, it has at least three vertices on the outer face and q≥ 2. In such a case one can always perform the b_q-removal operation, a contradiction.Let us now show that a_p is not adjacent to a vertex b_i with i<q. Towards a contradiction, consider that a_p is adjacent to a vertex b_i with i<q. Then by planarity, b_q (with q>1) has no neighbor a_i with i<p, and has no inner-neighbor adjacent to a vertex a_i with i<p. In such a case one can always perform the b_q-removal operation, a contradiction. Symmetrically, we deduce that b_q is not adjacent to a vertex a_i with i<p.Vertices a_p and b_q have one common neighbor x such that xa_pb_q is an inner face. Note that as there is no chord incident to a_p or b_q, then x is not on the outer face. They have no other common neighbor y, otherwise there would be a separating triangle ya_pb_q (separating x from both vertices a_1 and b_1). As we are not in the first case nor in the second case, we have that a_p (resp. b_q) has (at least) one inner-neighbor adjacent to a vertex b_i with i<q (resp. a_i with i<p). By planarity, x is the only inner-neighbor of a_p (resp. b_q) adjacent to a vertex b_i with i<q (resp. a_i with i<p). We can thus apply the cutting operation.We now show that T', T_a and T_b are 2-sided near-triangulations. Consider first T'. It is clear that it is a near-triangulation without separating triangles. It remains to show that there are no chords a_ia_j or b_ib_j. By definition of T', the only chord possible would have x=a_i+1 as an endpoint, but the existence of an edge xa_k with k<i would contradict the minimality of i. Thus T' is a 2-sided near-triangulation.By definition, T_a is also a near-triangulation containing no separating triangles. Moreover, there is no chord a_k a_l with i ≤ k ≤ l-2 as there are no such chords in T. Therefore T_a is a 2-sided near-triangulation. We show in the same way that T_b is a 2-sided near-triangulation.§ THICK {⌞}-CONTACT SYSTEM Ais a ⌞ shape where the two segments are turned into ε-thick rectangles (see Figure <ref>). Going clockwise around a thick ⌞ from the bottom-right corner, we call its sides bottom, left, top, vertical interior, horizontal interior, and right. A is described by four coordinates a,b,c,d such that a+ε < b and c+ε < d. It is thus the union of two boxes: ([a,a+ε] × [c,d]) ∪ ([a,b] × [c,c+ε]). If not specified, the corner of a denotes its bottom-left corner (with coordinate (a,c)). In the rest of the paper, all the shapes have the same thickness ε.Given a -contact system, a thick ⌞ is said left if its horizontal interior is free (i.e., does not touch another ⌞) and its left side is not contained in the side of another thick ⌞(see Figure <ref>). Similarly, a thick ⌞ is said bottom if its vertical interior is free and its bottom side is not contained in the side of another thick ⌞ (see Figure <ref>). A convenient -contact system (CTLCS) is a contact system with shapes (which implies that the shapes interiors are disjoint) with a few properties:* Two shapes intersect either on exactly one segment or on a point (Figure <ref> lists the allowed ways two ⌞ shapes can intersect). If the intersection is a segment, then it must be exactly one side of a . If the intersection is a point, then it is the bottom right corner of one and the top left corner of the other one.* Every thick ⌞ is bottom or left.Remark that the removal of any still leads to a CTLCS. This definition implies that in a CTLCS there is no three ⌞ shapes intersecting as in Figure <ref>.We now make a link between CTLCS and 2-sided near-triangulations (See Figure <ref> for an illustration). Every 2-sided near-triangulation can be represented by a CTLCS with the following properties: * every is included in the quadrant { (x,y) : x≥ 0, y ≥ 0 },* a_1 has the rightmost corner and b_1 has the up-most corner,* every vertex a_i is represented by a bottom whose corner has coordinates (x,0), with x>0, and* every vertex b_i is represented by a left whose corner has coordinates (0,y), with y>0. We proceed by induction on the number of vertices. The theorem clearly holds for the 2-sided near-triangulation with three vertices.Let T be a 2-sided near-triangulation; it can thus be decomposed using one of the three operations described in Lemma <ref>. We go through the three operations successively.(a_p-removal)Let T' be the 2-sided near-triangulation resulting from an a_p-removal operation on T. By the induction hypothesis, T' has a CTLCS with the required properties (see Figure <ref>). We can now modify this CTLCS slightly in order to obtain a CTLCS of T (thus adding a corresponding to vertex a_p). Move the corners of the corresponding to vertices b_q+1,…, b_q+r slightly to the right. Since these are left shapes, one can do this without modifying the rest of the system. Then one can add the of a_p such that it touches the of vertices b_q and a_p-1 (as depicted in Figure <ref>). One can easily check that the obtained system is a CTLCS of T and satisfies all the requirements.(b_q-removal) This case is strictly symmetric to the previous one.(cutting) Let T', T_a and T_b be the three 2-sided near-triangulations resulting from the cutting operation described in Lemma <ref>. By induction hypothesis, each of them has a CTLCS satisfying the requirements of Theorem <ref>. Consider the CTLCS of T' (see Figure <ref>). Move the corner of x=a_i+1 slightly upward. Since x=a_i+1 is a bottom vertex, one can do this without modifying the rest of the system.Then one can add the CTLCS of T_a below vertex x and the one of T_b on its left (see Figure <ref>, bottom). One can easily check that the obtained system satisfies all the requirements.§ {⌞,| ,- }-CONTACT SYSTEMS FOR TRIANGLE-FREE PLANAR GRAPHSWe can now use the CTLCS systems to prove Theorem <ref>. Recall that a{⌞,| ,-}-contact system is a contact system with some ⌞, some vertical segments |, and some horizontal segments -, such that if an intersection occurs between two of these objects, then the intersection is an endpoint of one of the two objects. We need the following lemma as a tool (it is proved in appendix).For any plane triangle-free graph G, there exists a 4-connected triangulation T containing G as an induced subgraph.We can now prove Theorem <ref>, which asserts that every triangle-free planar graph has a {⌞,| ,-}-contact system. Consider a triangle-free planar graph G. According to Lemma <ref>, there exists a 4-connected triangulation T containing G as an induced subgraph. As the exterior face of T is a triangle, T is a 2-sided near-triangulation (denoting a_1,b_2,b_1 the three exterior vertices in clockwise order).By Theorem <ref>, T has a CTLCS and removing every corresponding to avertex of T∖ G leads to a CTLCS of G. If a x has its bottom side included in the horizontal interior side of another y, then x is bottom, and so does not intersect anyone on its horizontal interior side. Furthermore, x does not intersect anyone on its right side nor on its bottom right corner. Indeed, if there was such an intersection with a z, then y and z would also intersect, contradicting the fact that G is triangle-free (see Figure <ref>). One can thus replace the of x by a thick |.Similarly, if a x has its left side included in the vertical interior side of a y, we can replace the of x by a thick -.Note that now the intersections are on small segments, or on a point, between the bottom right corner of a or -, and the top left corner of a or |. Then, we replace each thick ⌞, |, and - by thin ones as depicted in Figure <ref>. It is clear that we obtain a {⌞, |, -}-contact system whose contact graph is G. This concludes the proof. An example of the process is shown in Figure <ref>. § THE ⌞-INTERSECTION SYSTEMSAn ⌞-intersection system (LIS) is an intersection system of ⌞ shapes where every two ⌞ shapes intersect on at most one point. Using Theorem <ref>, one could prove that that every 4-connected triangulation has a LIS. To allow us to work on every triangulation (not only the 4-connected ones) we need to enrich our LISs with the following notion that was introduced in <cit.> under the name of private region.An anchor can be seen as a union of three segments, or as the union of two ⌞. It has two corners, which correspond to the ⌞ shapes corners. There are two types of anchors. A horizontal anchor is a set [x_1,x_3] × y_1 ∪ x_1 × [y_1,y_2] ∪ x_2 × [y_1,y_2] where x_1 < x_2 < x_3 and y_1 < y_2 (see Figure <ref>a). The middle corner of such a horizontal anchor is defined as the point (x_2,y_1). A vertical anchor is a set x_1 × [y_1,y_3] ∪ [x_1,x_2] × y_1 ∪ [x_1,x_2] × y_2 where x_1 < x_2 and y_1 < y_2 < y_3 (see Figure <ref>b). The middle corner of such a vertical anchor is defined as the point (x_1,y_2).Consider a near-triangulation T, and any inner face abc of T. Given a LIS of T, an anchor of abc is an anchor crossing the ⌞ shapes of a b and c and no other ⌞, and such that the middle corner is in the square described by a, b and c as depicted in Figure <ref>. A full ⌞-intersection system (FLIS) of a near-triangulation T is a LIS of T with an anchor for every (triangular) inner face of T, such that the anchors are pairwise non-intersecting.Let us now prove that every 2-sided near-triangulation admits a FLIS.Every 2-sided near-triangulation has a FLIS such that among the corners of the ⌞ shapes and the anchors: * from left to right, the first corners are those of vertices b_1,b_2,… b_q and the last one is the corner of vertex a_1, and* from bottom to top, the first corners are those of vertices a_1,a_2,… a_p and the last one is the corner of vertex b_1.As the ⌞ of a_i and a_i+1 (resp. b_i and b_i+1) intersect, the FLIS is rather constrained. This is illustrated in Figure <ref>, where the grey region contains the corners of the inner vertices, and the corners of the anchors. We proceed by induction on the number of vertices.The result clearly holds for the 2-sided near-triangulation with three vertices, whatever p=1 and q=2, or p=2 and q=1. Let T be a 2-sided near-triangulation with at least four vertices. By Lemma <ref> we consider one of the following operations on T:(a_p-removal) Consider the FLIS of T' obtained by induction and see in Figure <ref> how one can add a ⌞ for a_p and an anchor for each inner face a_pb_jb_j+1 with q≤ j <q+r and for the inner face a_p a_p-1 b_q+r. One can easily check that the obtained system verifies all the requirements of Proposition <ref>.(b_q-removal) This case is symmetric to the previous one.(cutting) Consider the FLISs of T', T_a and T_b.Figure <ref> depicts how to combine them, and how to add an anchor for xa_pb_q, in order to get the FLIS of T. One can easily check that the obtained system verifies all the requirements of Proposition <ref>. We now prove Theorem <ref> which asserts that every planar graph is a⌞-intersection graph.It is well known that every planar graph is an induced subgraph of some triangulation (see <cit.> for a proof similar to the one of Lemma <ref>). Thus, given a planar graph G, one can build a triangulation T whose G is an induced subgraph. If one can create a FLIS of T, then it remains to remove the ⌞ shapes assigned to vertices of T ∖ G along with the anchors in order to get a ⌞-representation of G. In order to prove Theorem <ref>, we thus only need to show that every triangulation admits a FLIS. Every triangulation T with outer-vertices x,y, z has a FLIS such that among the corners of the ⌞ shapes and the anchors: * the corner of x is the upmost and leftmost,* the corner of y is the second leftmost, and* the corner of z is the bottom-most and rightmost.Note that in this proposition there is no constraint on x, y, z, so by renaming the outer vertices, other FLISs can be obtained. Another way to obtain more FLISs is by applying a reflection with respect to a line of slope 1. In such FLIS (see Figure <ref>) among the corners of the ⌞ shapes and the anchors: * the corner of x is the bottom-most and rightmost,* the corner of y is the second bottom-most, and* the corner of z is the upmost and leftmost.We proceed by induction on the number of vertices in T. Let T be a triangulation with outer vertices x,y,z.If T is 4-connected, then it is also a 2-sided near-triangulation. By Proposition <ref> and by renaming the outer-vertices x to b_1, y to b_2 and z to a_1, T has a FLIS with the required properties. If T is not 4-connected, then it has a separating triangle formed by vertices a, b and c. We note T_in and T_out the triangulations obtained from T by removing the vertices outside and inside abc respectively.By the induction hypothesis, T_out has a FLIS verifying Proposition <ref> (considering the outer vertices to be x,y,z in the same order). Without loss of generality we can suppose that the ⌞ shapes of a, b and c appear in the following order: the upmost and leftmost is b, the second leftmost is c and the bottom-most is a. There are two cases according to the type of the anchor of the inner face abc.If the anchor of abc in the FLIS of T_out is vertical (see Figure <ref>), then applying the induction hypothesis on T_in with b,c,a as outer vertices considered in that order, T_in has a FLIS as depicted on the Figure <ref>. Figure <ref> depicts how to include the FLIS of T_in∖{a,b,c} in the close neighborhood of the anchor of abc. As abc is not a face of T, the close neighborhood of its anchor is indeed available for this operation.Now suppose that the anchor of abc in the FLIS of T_out is horizontal (see Figure <ref>). By application of the induction hypothesis on T_in with a,c,b as outer vertices considered in that order, then T_in has a FLIS as depicted on the Figure <ref>. By a reflection of slope 1, T_in has a FLIS such that b is the up-most and left-most, c is the second left-most and a is bottom-most (see Figure <ref>). Similarly to the previous case, we include this last FLIS of T_in∖{a,b,c} in the one from T_out (see Figure <ref>). As T_in and T_out cover T, and intersect only on the triangle abc, and as every inner face of T is an inner face in T_in or in T_out, these constructions clearly verify Proposition <ref>. This concludes the proof of the proposition.§ FROM TRIANGLE-FREE PLANAR GRAPHS TO 4-CONNECTED TRIANGULATIONS We here prove Lemma <ref>. The main idea of the construction of T is to insert vertices and edges in every face of G (even for the exterior face).For the sake of clarity, vertices of G are said black and vertices of T ∖ G are said red.The new graph T contains G as an induced subgraph, along with other vertices and edges. More precisely, for every face of G, let P = {v_0,e_0,v_1,e_1,…} be the list of vertices and edges along the face boundary (see Figure <ref>), where e_i is the edge between vertices v_i and v_i+1; there can be repetitions of vertices or edges.For each face of G, given the list P, the graph T contains a vertex v'_i for each vertex v_i, a vertex e'_i for each edge e_i, and an additionnal vertex t. Each vertex v'_i is connected to e'_i and e'_i+1 (with subscripts addition done modulo the size of the face), each vertex v_i is connected to v'_i, e'_i-1 and e'_i, and the vertex t is connected to all vertices v'_i and e'_i (see Figures <ref> and <ref> for examples). The new graph T is a triangulation, and we now show that it is 4-connected, i.e., has no separating triangle. Suppose that there is a separating triangle in the new graph. There are four cases depending on the colors of the edges of this triangle:* The separating triangle contains three black edges. It is impossible since G is triangle-free. * The separating triangle contains exactly one red edge. One of its endpoints must be a red vertex. But a red vertex is adjacent to only red edges, a contradiction. * The separating triangle contains exactly two red edges. Then their common endpoint is a red vertex, and the triangle is made of two vertices v_i and v_i+1, together with the vertex e'_i. All these triangles are faces, a contradiction.* The separating triangle contains three red edges. Since for each face, the red vertices (vertices v'_i, e'_i and t) induce a wheel graph centered on t, with at least 8 peripheral vertices (vertices v'_i and e'_i), this separating triangle has at least one black vertex. As two adjacent black vertices are linked by a black edge, this separating triangle has exactly one black vertex. As the two red vertices are two adjacent v'_i or e'_j vertices, we have that those are v'_i and e'_j, for some i and for j=i or for j=i+1. Such a triangle is not separating, a contradiction.This concludes the proof of the lemma. plain | http://arxiv.org/abs/1707.08833v2 | {
"authors": [
"Daniel Gonçalves",
"Lucas Isenmann",
"Claire Pennarun"
],
"categories": [
"cs.CG"
],
"primary_category": "cs.CG",
"published": "20170727122805",
"title": "Planar graphs as L-intersection or L-contact graphs"
} |
Mutation Clusters from Cancer Exome Zura Kakushadze^^†[Zura Kakushadze, Ph.D., is the President of Quantigic^ Solutions LLC, and a Full Professor at Free University of Tbilisi. Email: [email protected]] and Willie Yu^♯[Willie Yu, Ph.D., is a Research Fellow at Duke-NUS Medical School. Email: [email protected]] ^ Quantigic^ Solutions LLC 1127 High Ridge Road #135, Stamford, CT 06905[DISCLAIMER: This address is used by the corresponding author for no purpose other than to indicate his professional affiliation as is customary in publications. In particular, the contents of this paper are not intended as an investment, legal, tax or any other such advice, and in no way represent views of Quantigic^ Solutions LLC, the website www.quantigic.com or any of their other affiliates. ] ^† Free University of Tbilisi, Business School & School of Physics 240, David Agmashenebeli Alley, Tbilisi, 0159, Georgia ^♯ Centre for Computational Biology, Duke-NUS Medical School 8 College Road, Singapore 169857 (March 31, 2017) We apply our statistically deterministic machine learning/clustering algorithm *K-means (recently developed in https://ssrn.com/abstract=2908286) to 10,656 published exome samples for 32 cancer types. A majority of cancer types exhibit mutation clustering structure. Our results are in-sample stable. They are also out-of-sample stable when applied to 1,389 published genome samples across 14 cancer types. In contrast, we find in- and out-of-sample instabilities in cancer signatures extracted from exome samples via nonnegative matrix factorization (NMF), a computationally costly and non-deterministic method. Extracting stable mutation structures from exome data could have important implications for speed and cost, which are critical for early-stage cancer diagnostics such as novel blood-test methods currently in development.Keywords: Clustering, K-Means, Nonnegative Matrix Factorization, Somatic Mutation, Cancer Signatures, Genome, Exome, DNA, eRank, Correlation, Covariance, Machine Learning, Sample, Matrix, Source Code, Quantitative Finance, Statistical Risk Model, Industry Classification§ INTRODUCTION AND SUMMARY Unless humanity finds a cure, about a billion people alive today will die of cancer. Unlike other diseases, cancer occurs at the DNA level via somatic alterations in the genome. A common type of such mutations found in cancer is due to alterations to single bases in the genome (single nucleotide variations, or SNVs). These alterations are accumulated throughout the lifespan of an individual via various mutational processes such as imperfect DNA replication during cell division or spontaneous cytosine deamination <cit.>, <cit.>, or due to exposures to chemical insults or ultraviolet radiation <cit.>, <cit.>, etc. The footprint left by these mutations in the cancer genome is characterized by distinctive alteration patterns known as cancer signatures.Identifying all cancer signatures would greatly facilitate progress in understanding the origins of cancer and its development. Therapeutically, if there are common underlying structures across different cancer types, then treatment for one cancer type might be applicable to other cancer types, which would be a great news. From a diagnostic viewpoint, identification of all underlying cancer signatures would aid cancer detection and identification methodologies, including vital early detection[See, e.g., <cit.>. A goal of early detection (via blood tests) is behind Grail, Inc.'s recent ∼$1B series B funding round – see, e.g., <cit.>.] – according to American Cancer Society, late stage metastatic cancers of unknown origin represent about 2% of all cancers <cit.> and can make treatment almost impossible.Another practical application is prevention by pairing the signatures extracted from cancer samples with those caused by known carcinogens (e.g., tobacco, aflatoxin, UV radiation, etc.). At the end of the day, it all boils down to the question of usefulness: is there a small enough number of cancer signatures underlying all (100+) known cancer types, or is this number too large to be meaningful/useful? Thus, if we focus on 96 mutation categories of SNVs,[In brief, DNA is a double helix of two strands, and each strand is a string of letters A, C, G, T corresponding to adenine, cytosine, guanine and thymine, respectively. In the double helix, A in one strand always binds with T in the other, and G always binds with C. This is known as base complementarity. Thus, there are six possible base mutations C > A, C > G, C > T, T > A, T > C, T > G, whereas the other six base mutations are equivalent to these by base complementarity. Each of these 6 possible base mutations is flanked by 4 possible bases on each side thereby producing 4 × 6 × 4 = 96 distinct mutation categories.] we cannot have more than 96 signatures.[A priori, nonlinearities could alter this conclusion. However, such nonlinearities may also render cancer signatures essentially useless...] Even if the number of true underlying signatures is, say, of order 50, it is unclear whether they would be useful, especially within practical applications. On the other hand, if there are only about a dozen underlying cancer signatures, then a hope for an order of magnitude simplification may well be warranted.The commonly used method for extracting cancer signatures <cit.> is based on nonnegative matrix factorization (NMF) <cit.>, <cit.>. Thus, one analyzes SNV patterns in a cohort of DNA sequenced whole cancer genomes, and organizes the data into a matrix G_iμ, where the rows correspond to the N=96 mutation categories, the columns correspond to d samples, and each element is a nonnegative occurrence count of a given mutation category in a given sample.Under NMF the matrix G is then approximated via G ≈ W H, where W_iA is an N× K matrix, H_Aμ is a K× d matrix, and both W and H are nonnegative. The appeal of NMF is its biologic interpretation whereby the K columns of the matrix W are interpreted as the weights with which the K cancer signatures contribute into the N=96 mutation categories, and the columns of the matrix H are interpreted as the exposures to these K signatures in each sample. The price to pay for this is that NMF, which is an iterative procedure, is computationally costly and depending on the number of samples d it can take days or even weeks to run it. Furthermore, NMF does not fix the number of signatures K, which must be either guessed or obtained via trial and error, thereby further adding to the computational cost. Perhaps most importantly, NMF is a nondeterministic algorithm and produces a different matrix W in each run.[Each W corresponds to one in myriad local minima of the NMF objective function.] This is dealt with by averaging over many such W matrices obtained via multiple NMF runs (or samplings). However, each run generally produces a weights matrix W_iA with columns (i.e., signatures) not aligned with those in other runs. Aligning or matching the signatures across different runs (before averaging over them) is typically achieved via nondeterministic clustering such as k-means. Therefore, the result, even after averaging, generally is both noisy[By “noise" we mean the statistical errors in the weighs obtained by averaging. Usually, such error bars are not reported in the literature on cancer signatures. Typically, they are large.] and nondeterministic! I.e., if this computationally costly procedure (which includes averaging) is run again and again on the same data, generally it will yield different looking cancer signatures every time! Simply put, the NMF-based method for extracting cancer signatures is not designed to be even in-sample stable. Under these circumstances, out-of-sample stability cannot even be dreamt about...[I.e., cancer signatures obtained from non-overlapping sets of samples can be dramatically different. And out-of-sample stability is crucial for practical usefulness, e.g., diagnostically.]Without in- and out-of-sample stability practical therapeutic and diagnostic applications of cancer signatures would be challenging. For instance, suppose one sequences genome (or exome – see below) data from a patient sample.[Be it via a liquid biopsy, a blood test, or some other (potentially novel) method.] Let us focus on SNVs. We have a vector of occurrence counts for 96 mutation categories. We need a quick computational test to determine with high enough confidence level whether i) there is a cancer signature present in this data, and ii) which cancer type this cancer signature corresponds to (i.e., which organ the cancer originated in). If cancer signatures are not even in-sample stable, then we cannot trust them. They could simply be noise. Indeed, there is always somatic mutational noise present in such data and must be factored out of the data before extracting cancer signatures. A simple way to understand somatic mutational noise is to note that mutations (i) are already present in humans unaffected by cancer, and (ii) such mutations, which are unrelated to cancer, are further exacerbated when cancer occurs as it disrupts the normal operation of various processes (including repair) in the DNA. At the level of the data matrix G, in <cit.> we discussed a key component of the somatic mutational noise and gave a prescription for removing it.[This is achieved by cross-sectionally (i.e., across the 96 mutation categories) demeaning “log-counts". This “de-noising" dramatically improved NMF-based signatures we extracted from genome data in <cit.> and cut computational cost (these savings would scale nonlinearly for larger datasets) by a factor of about 10 on a genome dataset for 1,389 samples in 14 cancer types. In <cit.>, by adapting the methods used in statistical risk models in quantitative finance <cit.>, we also proposed a simple method for fixing the number of cancer signatures based on eRank (effective rank) <cit.>.] However, there likely exist other, deeper sources of somatic mutational noise, which must be further identified and carefully factored out. Simply put, somatic mutational noise unequivocally is a substantial source of systematic error in cancer signatures.However, then there is also the statistical error, which is large and due to the nondeterministic nature of NMF discussed above. This statistical error is exacerbated by the somatic mutational noise but would be present even if this noise was somehow completely factored out. Therefore, the in-sample instability must somehow be addressed. We emphasize that, a priori, this does not automatically address out-of-sample stability, without which any therapeutic or diagnostic applications would still be farfetched. However, without in-sample stability nothing is clear...The problem at hand is nontrivial and requires a step-by-step approach, including identification of various sources of in-sample instability. One simple observation of <cit.> is that, if we work directly with occurrence counts G_iμ for individual samples, (i) the data is very noisy, and (ii) the number of signatures is bound to be too large to be meaningful/useful if the number of samples is large. A simple way to deal with this is to aggregate samples by cancer types. In doing so, we have a matrix G_is, where s now labels cancer types, which is (i) less noisy, and (ii) much smaller (96 × n, where n is the number of cancer types), so the number of resultant signatures is much more reasonable.[In aggregating samples by cancer types, for some cancer types pertinent information may be muddled up as there may be biologic factors one may wish to understand, e.g., mutational spectra of liver cancers can have substantial regional dependence as they are mutagenized by exposures to different chemicals (alcohol, aflatoxin, tobacco, etc.). In such cases, aggregation by regions (or other applicable characteristics, as the case may be) within a cancer type may still be warranted to reduce noise (or else, without any aggregation, there are simply too many cancer signatures – see, e.g., Table 7 in <cit.>). However, not to get ahead of ourselves – one step at a time – in this paper we will work with (exome) data aggregated by cancer types (see below).] Thus, such aggregation is helpful.Still, even with aggregation, we must address nondeterminism (of NMF). To circumvent this, in <cit.> we proposed an alternative approach which bypasses NMF altogether. As we argue in <cit.>, NMF is – at least to a certain degree – clustering in disguise. E.g., many COSMIC cancer signatures <cit.> obtained via NMF[Augmented with additional heuristics based on biologic intuition and empirical observations.] exhibit clustering substructure, i.e., in many of these signatures there are mutation categories with high weights (“peaks" or “tall mountain landscapes") with other mutation categories having small weights likely well within statistical and systematic errors. For all practical purposes such low weights could be set to zero. Then, many cancer signatures would start looking like clusters, albeit some clusters could be overlapping between different signatures. Considering that various signatures may be somatic mutational noise artifacts in the first instance and statistical error bars are large, it is natural to wonder whether there are some robust underlying clustering structures present in the data – with the understanding that such structures may not be present for all cancer types. However, even if they are present for a substantial number of cancer types, unveiling them would amount to a major step forward in understanding cancer signature structure.To address this question, in <cit.> we proposed a new clustering algorithm termed *K-means. Its basic building block is the vanilla k-means algorithm, which computationally is very cheap. However, it is also nondeterministic. *K-means uses 2 machine learning levels on top of k-means to achieve statistical determinism – see Section <ref> for details[There is virtually no way to make this paper self-contained without essentially copying all the technical details over from <cit.>. We will not do so here. Instead, readers interested in technical details should read this paper together with <cit.>.] – without any initialization of the centers, etc.[It also fixes the number of clusters K: it fixes the target number of clusters K_1 via an eRank based method (see fn. <ref>); then the final number of clusters K ≤ K_1 follows via machine learning.] Once *K-means fixes the clustering, it turns out that the weights and exposures can be computed using (normalized) regressions <cit.>, thereby altogether bypassing computationally costly NMF. In <cit.> we applied this method to cancer genome data corresponding to 1,389 published samples for 14 cancer types. We found that clustering works well for 10 out the 14 cancer types – the metrics include within-cluster correlations and overall fit quality. This suggests that there is indeed clustering substructure present in underlying cancer genome data, at least for most cancer types![One of the cancer types for which clustering does not appear to work well – completely consistently with and expectedly from the results of <cit.> – is Liver Cancer. In particular, the dominant (with a 96% contribution) NMF-based cancer signature we found in <cit.> for Liver Cancer does not have “peaks" (“rolling hills landscape"), with no resemblance to a clustering substructure. In this regard, note our comments in fn. <ref>.] This is exciting!In this paper we apply the method of <cit.> to exome data consisting of 10,656 published samples (sample IDs with sources are in Appendix <ref>) aggregated by 32 cancer types. *K-means produces a robustly stable clustering (11 clusters) from this data. One motivation for using exome data is that exome is a small subset (∼1%) of full genome containing only protein coding regions of genome <cit.>. Exome is much cheaper and less time consuming to sequence – which can be especially important for early stage diagnostics – than whole genome, yet it encodes important information about cancer signatures. As we discuss in the subsequent sections, our method appears to work well on exome data for most cancer types. In fact, overall it appears to work better than COSMIC signatures, including out-of-sample, when applying clusters derived from our exome data to genome data.§ *K-MEANS In <cit.>, by extending a prior work <cit.> in quantitative finance on building statistical industry classifications using clustering algorithms, we developed a clustering method termed *K-means (“Star K-means") and applied it to extraction of cancer signatures from genome data. *K-means anchors on the standard k-means algorithm[See <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>.] as its basic building block. However, k-means is not deterministic. *K-means is statistically deterministic, without specifying initial centers, etc. This is achieved via 2 machine learning levels sitting on top of k-means. At the first level we aggregate a large number M of k-means clusterings with randomly initialized centers (and the number of target clusters fixed using eRank) via a nontrivial aggregation procedure – see <cit.> for details. This aggregation is based on clustering (again, using k-means) the centers produced in the M clusterings, so the resultant aggregated clustering is nondeterministic. However, it is a lot less nondeterministic than vanilla k-means clusterings as aggregation dramatically reduces the degree of nondeterminism. At the second level, we take a large number P of such aggregated clusterings and determine the “ultimate" clustering with the maximum occurrence count (among the P aggregations). For sufficiently large M and P the “ultimate" clustering is stable, i.e., if we run the algorithm over and over again, we will get the same “ultimate" clustering every time, even though the occurrence counts within different P aggregations are going to be different for various aggregations. What is important here is that the most frequently occurring (“ultimate") aggregation remains the same run after run.§ EMPIRICAL RESULTS §.§ Data Summary In this paper we apply *K-means to exome data.[In <cit.> we applied it to published genome data.] We use data consisting of 10,656 published exome samples aggregated by 32 cancer types listed in Table <ref>, which summarizes total occurrence counts, numbers of samples, and data sources. Appendix <ref> provides sample IDs together with references for the data sources. Occurrence counts for the 96 mutation categories for each cancer type are given in Tables <ref>-<ref>.§.§.§ Structure of Data The underlying data consists of matrices [G(s)]_iμ(s) whose elements are occurrence counts of mutation categories labeled by i = 1,…, N = 96 in samples labeled by μ(s) = 1,…, d(s). Here s = 1,…,n labels n different cancer types (in our case n = 32). We can choose to work with individual matrices [G(s)]_iμ(s), or with the N × d_tot “big matrix" Γ obtained by appending (i.e., bootstrapping) the matrices [G(s)]_iμ(s) together column-wise (so d_tot = ∑_s=1^n d(s)). Alternatively, we can aggregate samples by cancer types and work with the so-aggregated matrixG_is = ∑_μ(s) = 1^d(s) [G(s)]_iμ(s)Generally, individual matrices [G(s)]_iμ(s) and, thereby, the “big matrix" Γ contain a lot of noise. For some cancer types we can have a relatively small number of samples. We can also have “sparsely populated" data, i.e., with many zeros for some mutation categories. In fact, different samples are not even necessarily uniformly normalized. Etc. The bottom line is that the data is noisy. To mitigate the aforementioned issues, following <cit.>, here we work with the N× n matrix G_is with samples aggregated by cancer types. Below we apply *K-means to G_is. §.§ Exome Data Results The 96 × 32 matrix G_is given in Tables <ref>-<ref> is what we pass into the function bio.cl.sigs() in Appendix A of <cit.> as the input matrix x. We use: iter.max = 100 (this is the maximum number of iterations used in the built-in R function kmeans() – we note that there was not a single instance in our 30 million runs of kmeans() where more iterations were required);[The R function kmeans() produces a warning if it does not converge within iter.max.] num.try = 1000 (this is the number of individual k-means samplings we aggregate every time); and num.runs = 30000 (which is the number of aggregated clusterings we use to determine the “ultimate" – that is, the most frequently occurring – clustering). More precisely, we ran 3 batches with num.runs = 10000 as a sanity check, to make sure that the final result based on 30000 aggregated clusterings was consistent with the results based on smaller batches, i.e., that it was stable from batch to batch.[We ran these 3 batches consecutively, and each batch produced slightly different top-10 (by occurrence counts) clusterings with varying occurrence counts across the batches, etc. However, Clustering-E1 invariably had the highest occurrence count by a large margin. See Table <ref>.] Based on Table <ref>, we identify Clustering-E1 as the “ultimate" clustering[It is evident that the top-10 clusterings in Table <ref> essentially are variations of each other.] (see Section <ref>).For Clustering-E1, as in <cit.>, we compute the within-cluster weights based on unnormalized regressions (via Eqs. (13), (14) and (15) thereof) and normalized regressions (via Eqs. (17), (14) and (16) thereof) with exposures calculated based on arithmetic averages (see Subsection 2.6 of <cit.> for details). We give the within-cluster weights for Clustering-E1 in Tables <ref> and <ref> and plot them in Figures <ref> through <ref> for unnormalized regressions, and in Tables <ref> and <ref> and Figures <ref> through <ref> for normalized regressions. The actual mutation categories in each cluster can be read off the aforesaid Tables <ref> and <ref> with the weights (thus, the mutation categories with nonzero weights belong to a given cluster), or from the horizontal axis labels in the aforesaid Figures <ref>-<ref>. §.§ Reconstruction and Correlations §.§.§ Within-cluster Correlations We have our data matrix G_is. We are approximating this matrix via the following factorized matrix:G^*_is = ∑_A=1^K W_iA H_As = w_i H_Q(i),swhere W_iA are the within-cluster weights (i=1,…,N; A=1…,K), H_As are the exposures (s=1,…,n=32 labels the cancer types), Q:{1,…,N}↦{1,…,K} is the map between the N=96 mutations and K=11 clusters in Clustering-E1, and we have[Due to a binary clustering structure, the within-cluster weights W_iA are encoded in an N-vector w_i. This is because all but N elements of the matrix W_iA are zero.] W_iA = w_i δ_Q(i), A. It is the matrix W_iA that is given in Tables <ref> and <ref> for the unnormalized regressions, and Tables <ref> and <ref> for the normalized regressions.We can now compute an n× K matrix Θ_sA of within-cluster cross-sectional correlations between G_is and G^*_is defined via ((·,·) stands for “cross-sectional correlation", i.e., “correlation across the index i")[Due to the factorized structure (<ref>), these correlations do not directly depend on H_As.]Θ_sA = .(G_is, G^*_is)|_i∈ J(A) = .(G_is, w_i)|_i∈ J(A)Here J(A) = {i|Q(i) = A} is the set of mutations labeled by i that belong to a given cluster labeled by A. We give the matrix Θ_sA for Clustering-E1 for weights based on unnormalized regressions in Table <ref>, and weights based on normalized regressions in Table <ref>. As for genome data <cit.>, the fit for normalized regressions is somewhat better than that for unnormalized regressions.§.§.§ Overall Correlations Another useful metric, which we use as a sanity check, is this. For each value of s (i.e., for each cancer type), we can run a linear cross-sectional regression (without the intercept) of G_is over the matrix W_iA. So, we have n=32 of these regressions. Each regression produces multiple R^2 and adjusted R^2, which we give in Tables <ref>-<ref>. Furthermore, we can compute the fitted values G^*_is based on these regressions, which are given byG^*_is = ∑_A=1^K W_iA F_As = w_i F_G(i),swhere (for each value of s) F_As are the regression coefficients. We can now compute the overall cross-sectional correlations (i.e., the index i runs over all N=96 mutation categories)Ξ_s = (G_is, G^*_is)These correlations are also given in Tables <ref>-<ref> and measure the overall fit quality.§.§.§ Interpretation Looking at Table <ref>, a few things jump out. First, most – 24 out of 32 – cancer types have high (80%+) within-cluster correlations with at least one cluster. Out of the other 8 cancer types, 6 have reasonably high (70%+) within-cluster correlations with at least one cluster. The remaining 2 cancer types are X9 (Cervical Cancer) and X17 (Liver Cancer). In <cit.>, based on genome data, we already observed that Liver Cancer does not have clustering structure, so this is not surprising. On the other hand, with Cervical Cancer the story appears to be trickier. According to <cit.>, we should expect COSMIC signatures CSig2+13 and CSig26 (see Section <ref> for more details) to appear in Cervical Cancer. According to Table <ref> (see Section <ref>), CSig2+13 indeed have high correlations with X9 (but not CSig26). On the other hand, the dominant part of CSig2 (C > T mutations in TCA, TCC, TCG, TCT) is subsumed in Cluster Cl-10 (see Figure <ref>), and the dominant part of CSig13 (C > G mutations in TCA, TCC, TCT) is subsumed in Cluster Cl-9 (see Figure <ref>). Basically, it appears that the large (each with 16 mutation categories) clusters Cl-9, Cl-10 and Cl-11 probably could be split into smaller clusters. In fact, Cl-9 and Cl-11 do not have 80%+ correlations with any cancer types (they do have 70%+ correlations with one cancer type each). This is another indication that these clusters might be “oversized". The same was observed with the largest cluster (with 21 mutation categories) in <cit.> in the context of genome data. Simply put, these “oversized" clusters may have to be dealt with via appropriately tweaking the underlying clustering algorithm.[This is outside of the scope hereof and will be dealt with elsewhere.]The last 3 columns in Table <ref> provide metrics for the overall fit for each cancer type. The overall correlations (between the original data G_is and the model-fitted values G^*_is – see Subsection <ref>) in the last column of Table <ref> are above 80% for 16 (out of the 32) cancer types, and above 70% for 26 cancer types. These high correlations indicate a good in-sample agreement between the original and reconstructed (model-fitted) data for each of these 26 cancer types. The remaining 6 cancer types, which all have overall correlations above 60%, are: X4 (B-Cell Lymphoma), X6 (Bladder Cancer), X8 (Breast Cancer), X9 (Cervical Cancer), X26 (Rectum Adenocarcinoma), and X29 (Testicular Germ Cell Tumor). We already discussed Cervical Cancer above. We address Breast Cancer in Section <ref> hereof. Now, the X4 data is sparsely populated: there are 24 samples and the total number of counts is 706, so there are many zeros in the underlying sample data, albeit only 2 zeros in the aggregated data. According to <cit.>, we should expect CSig9 and CSig17 in B-Cell Lymphoma. However, according to Table <ref> (see Section <ref>) these signatures do not have high correlations with X4. Note that clustering worked well for B-Cell Lymphoma for the genome data in <cit.>, but there the genome data was well-populated. Therefore, it is reasonable to assume that here the “underperformance" is likely due to the sparsity of the underlying data. For X6 (Bladder Cancer) the situation is similar to X9 (Cervical Cancer) above: according to <cit.>, we should expect CSig2+13 in Bladder Cancer, and Table <ref> is consistent with this. However, as mentioned above CSig2 and CSig13 are subsumed in Clusters Cl-10 and Cl-9, respectively (“oversizing"). According to Table <ref>, we should expect CSig10 in X26. CSig10 is dominated by the C > A mutation in TCT (which is subsumed in Cluster Cl-9) and C > T mutation in TCG (which is subsumed in Cluster Cl-10). Again, here we are dealing with “oversizing" of these clusters. X29 has high within-cluster correlations with Clusters Cl-4 and Cl-5. The overall fit correlation apparently is lowered by the high negative correlation with Cluster Cl-3. To summarize, “oversizing" is one potential “shortcoming" here.§ CONCLUDING REMARKS In order to understand the significance of our results, let us compare them to the fit that COSMIC signatures[For details, see <cit.>. For references, see <cit.>, <cit.>, <cit.>, <cit.>, <cit.>.] provide for our exome data. We can do this by computing the following p × n cross-sectional correlation matrixΔ_α s = (U_iα, G_is)where U_iα (α = 1,…, p) is the N × p matrix of weights for p = 30 COSMIC signatures, which for brevity we will refer to as CSig1, …, CSig30.[See http://cancer.sanger.ac.uk/cancergenome/assets/signatures_probabilities.txt; note that the ordering of mutation categories in this file is not the same as ours.] The matrix Δ_α s is given in Tables <ref> and <ref>. Let us look at the 80%+ correlations (which are in bold font in Tables <ref> and <ref>).[Relaxing this cut-off to 70% (see Tables <ref> and <ref>) does not alter our conclusions below.] Only 6 out 30 COSMIC signatures, to wit, CSig1,2,6,7,10,15 have 80%+ correlations with the exome data for the 32 cancer types. Aetiology of these signatures is known <cit.>. CSig1 is the result of an endogenous mutational process initiated by spontaneous 5-methylcytosine deamination, hence the ubiquity of its high correlations with many cancer types. CSig2 (which usually appears in tandem with CSig13) is due to APOBEC mediated cytosine deamination, hence its high correlations with some cancer types. CSig6 is associated with defective DNA mismatch repair, hence its high correlations with several cancer types. CSig7 is due to ultraviolet light exposure, so its high correlation with X19 (Melanoma) is spot on.[However, there is no magic here. Apparently there is a large overlap between the exome data we use here and that used by <cit.>. Furthermore, caution is in order when it comes to any NMF-based signature that dominates a given cancer type. What this means is that the signature is close to the properly normalized underlying occurrence counts data (either aggregated or appropriately averaged over all samples), and NMF samplings fail to find a local minimum substantially different along this particular direction from the local minima that include this cancer signature. Such a signature indicates that the corresponding cancer type is of a “stand-alone" type and has little in common with other cancer types. An example of such a signature is the Liver Cancer dominant NMF-based cancer signature found in <cit.>.] CSig10 is associated with recurrent error-prone polymerase POLE somatic mutations.[Its high correlations with X26 (Rectum Adenocarcinoma) and X32 (Uterine Cancer) are consistent with <cit.> and, once again, apparently are due to a large overlap between the exome data we use here and that used by <cit.>.] CSig15 is associated with defective DNA mismatch repair; the significance of its high correlation with X23 (Pancreatic Cancer) is unclear. So, only a handful of COSMIC signatures – all associated with known mutational processes – do well on our exome data.[Note that considering the overall fit quality for COSMIC signatures by running overall regressions (of G_is over U_iα without the intercept) as we did above for clusters would not be meaningful. The regression coefficients F_As in (<ref>) in the case of clusters are guaranteed to be nonnegative. This is because the N-vectors corresponding to the columns in the cluster weights matrix W_iA are orthogonal to each other. The N-vectors corresponding to the columns in the COSMIC weights matrix U_iα are not orthogonal, unacceptably resulting in many negative regression coefficients F_α s.] Others do not fit well.This is the out-of-sample stability issue emphasized in <cit.>. It traces to the fact that NMF is an intrinsically unstable method, both in- and out-of-sample. In-sample instability relates to the fact that NMF is nondeterministic and produces different looking signatures from one run to another. In fact, we attempted running NMF on our exome data. We ran 3 batches with 800 sampling in each batch – a computationally time-consuming procedure.[Thus, to run one batch of NMF with 800 samplings on a 4-CPU (8 cores each, 2.60GHz) machine with 529Gb of RAM and hyperthreading (Operating System: Debian 3.2.84-2 x86_64 GNU/Linux), it took 6-7 days (and 3-4 days when the input data was “de-noised" following <cit.>). In contrast, to run each of our 3 batches of *K-means with 10 million instances of k-means in each batch (see Subsection <ref>), it only took under 24 hours on a single CPU (quad-core, 3.1GHz) machine with 16GB of RAM (Operating System: 64-bit Windows Server 2008 R2 Standard). From this data it is evident that *K-means computationally is much cheaper than NMF, even if NMF is improved via “de-noising" <cit.>.] The 3 batches produced different looking results, which with a lot of manual curation could only be partially matched to some COSMIC signatures, but this matching was different and highly unstable across the 3 batches. Simply put, NMF failed to produce any meaningful results on our exome data. Furthermore, the above discussion illustrates that most COSMIC signatures (extracted using NMF from exome and genome data) apparently are unstable out-of-sample, e.g., when applied to our exome data aggregated by cancer types. Here one may argue that exome data contains only partial information and NMF should not be used on it. However, the COSMIC signatures are in fact based on 10,952 exomes and 1,048 whole-genomes across 40 cancer types <cit.>.[Also, see, e.g., <cit.>.] The difference here is that we are aggregating samples by cancer types and most COSMIC signatures apparently do not apply, which means that COSMIC signatures are highly sample-set-specific (that is, unstable out-of-sample). Furthermore, as mentioned above, CSig7 (UV exposure) is spot on in that it has 99.66% correlation with X19 (Melanoma).[Albeit one should keep in mind the comments in fn. <ref>.] So, one can argue that the culprit is not the exome data but the method (NMF) itself. To quantify this, let us look at correlations of COSMIC signatures with genome data for 14 cancer types used in <cit.> and <cit.>. The results are given in Table <ref>. As in the case of exome data, here too we have high correlations only for a handful of COSMIC signatures corresponding to known mutational processes, to wit, CSig1,4,6,13. So, most COSMIC signatures do not appear to have explanatory power on genome data aggregated by cancer types, a further indication that most COSMIC signatures lack out-of-sample stability.What about out-of-sample stability for our clusters we obtained from exome data? One way to test it is to look at within-cluster correlations and the overall fit metrics as in Table <ref> but for the aforesaid genome data for 14 cancer types used in <cit.> and <cit.>. The results are given in Table <ref>. Unsurprisingly, the quality of the fit for genome data (out-of-sample) is not as good as for exome data (in-sample). However, it is i) reasonably good, and ii) unequivocally much better than the fit provided by the COSMIC signatures (Table <ref>). Furthermore, the exome based 11 clusters have a poor overall fit for G.X4 (Breast Cancer), G.X8 (Liver Cancer), G.X9 (Lung Cancer), and G.X14 (Renal Cell Carcinoma), the same 4 cancer types for which genome based 7 clusters in <cit.> produced a poor overall fit, and for a good reason too (see <cit.> for details). It is less clear why the exome based 11 clusters do not have a better fit for G.X7 (Gastric Cancer) considering the in-sample fits for this cancer type based on exome data (X15, Table <ref> hereof) and genome data (row 7, Table 15 of <cit.>) are petty good.So, unlike NMF, *K-means clustering, being a statistically deterministic method, is in-sample stable. Here we can ask, what if we apply to NMF the same 2 machine learning levels as those that sit on top of k-means in *K-means to make it statistically deterministic? The answer is that when applying NMF, one already uses one machine learning method, which is a form of aggregation of a large number of samplings (i.e., individual NMF runs).[Thus, as mentioned above, we ran 3 batches of 800 NMF samplings. In each batch, 800 samplings are aggregated via nondeterministic clustering (e.g., via k-means – see, e.g., <cit.> for a detailed discussion). The net result – by design – is nondeterministic.] This is conceptually similar to the first machine learning level in *K-means. So, then we can ask, what if we add to NMF the second machine learning level as in *K-means, to wit, by comparing a large number of such “averagings"? A simple, prosaic answer is that it would make NMF – which is already computationally costly as is and much more so with the first machine learning level – computationally prohibitive. The reason why *K-means is computationally much cheaper is that the basic building block of *K-means – on top of which we add the two machine learning methods – is vanilla k-means, which is much, much cheaper than NMF. And that is what makes all the difference.[Furthermore, as was argued in <cit.>, NMF, at least to some degree, is clustering in disguise. In fact, visual inspection of COSMIC signatures makes it evident that many of them – albeit possibly not all – have clustering substructure. This will be discussed in more detail in a forthcoming paper. Also, it would be interesting to understand the relation between “R-mutations" <cit.> (also see references therein) and somatic mutational noise.]Finally, let us mention that exome data for Chronic Myeloid Disorders (121 samples, 175 total counts) was published in <cit.>, <cit.>, and for Neuroblastoma (13 samples, 298 total counts) in <cit.>. However, this data is so sparsely populated (too many zeros even after aggregation) that we specifically excluded it from our analysis. Much more unpublished data is available for the cancer types we analyze here as well as other cancer types, and it would be very interesting to apply our methods to this data, including to (still embargoed) extensive genome data of the International Cancer Genome Consortium.§ ACKNOWLEDGMENTSThe results published here are in whole or part based upon data generated by the TCGA Research Network: http://cancergenome.nih.gov/.§ EXOME SAMPLE IDS In this Appendix we give the sample IDs with the corresponding publication references for the exome data we use. We label these references as H1, Z1, etc., and use these labels in Table <ref> in the Sources column.⧫ Acute Lymphoblastic Leukemia (86 samples):∙ Source H1 = <cit.>. Sample IDs are of the form SJHYPO*, where * is:001-D, 002-D, 004-D, 005-D, 006-D, 009-D, 009-R, 012-D, 013-D, 014-D, 016-D, 019-D, 020-D, 022-D, 024-D, 026-D, 029-D, 032-D, 036-D, 037-D, 037-R, 039-D, 040-D, 041-D, 042-D, 044-D, 045-D, 046-D, 047-D, 051-D, 052-D, 052-R, 055-D, 056-D, 116-D, 117-D, 119-D, 120-D, 123-D, 124-D, 125-D, 126-D.∙ Source Z1 = <cit.>. Sample IDs are of the form SJTALL*, where * is:001, 002, 003, 004, 005, 006, 007, 008, 009, 011, 012, 013, 169, 192, 208.∙ Source D1 = <cit.>:TBR01, TBR03, TBR05, TBR06, TBR08, TLE02, TLE10, TLE109, TLE31, TLE33, TLE34, TLE38, TLE39, TLE41, TLE42, TLE43, TLE50, TLE51, TLE54, TLE55, TLE57, TLE60, TLE61, TLE63, TLE64, TLE65, TLE66, TLE67, TLE68.⧫ Acute Myeloid Leukemia (190 samples):∙ Source T1 = TCGA (see Acknowledgments). Sample IDs are of the form TCGA-AB-*, where * is:2802, 2803, 2804, 2805, 2806, 2807, 2808, 2809, 2810, 2811, 2812, 2813, 2814, 2816, 2817, 2818, 2819, 2820, 2821, 2822, 2824, 2825, 2826, 2827, 2828, 2829, 2830, 2831, 2832, 2833, 2835, 2836, 2837, 2838, 2839, 2841, 2842, 2843, 2844, 2845, 2846, 2847, 2849, 2850, 2851, 2853, 2854, 2855, 2857, 2858, 2859, 2860, 2861, 2862, 2863, 2864, 2865, 2866, 2867, 2868, 2869, 2870, 2871, 2872, 2873, 2874, 2875, 2876, 2877, 2878, 2879, 2880, 2881, 2882, 2883, 2884, 2885, 2886, 2887, 2888, 2889, 2890, 2891, 2892, 2893, 2894, 2895, 2896, 2897, 2898, 2899, 2900, 2901, 2904, 2905, 2906, 2907, 2908, 2910, 2911, 2912, 2913, 2914, 2915, 2916, 2917, 2918, 2919, 2920, 2921, 2922, 2923, 2924, 2925, 2926, 2927, 2928, 2929, 2930, 2931, 2932, 2933, 2934, 2935, 2936, 2937, 2938, 2939, 2940, 2941, 2943, 2945, 2946, 2947, 2948, 2949, 2950, 2952, 2954, 2955, 2956, 2957, 2959, 2963, 2964, 2965, 2966, 2967, 2968, 2969, 2970, 2971, 2972, 2973, 2974, 2975, 2976, 2977, 2978, 2979, 2980, 2981, 2982, 2983, 2984, 2985, 2986, 2987, 2988, 2989, 2990, 2991, 2992, 2993, 2994, 2995, 2996, 2997, 2998, 2999, 3000, 3001, 3002, 3005, 3006, 3007, 3008, 3009, 3011, 3012.⧫ Adrenocortical Carcinoma (91 samples):∙ Source T2 = TCGA (see Acknowledgments). Sample IDs are of the form TCGA-*, where * is:OR-A5J1, OR-A5J2, OR-A5J3, OR-A5J4, OR-A5J5, OR-A5J6, OR-A5J7, OR-A5J8, OR-A5J9, OR-A5JA, OR-A5JB, OR-A5JC, OR-A5JD, OR-A5JE, OR-A5JF, OR-A5JG, OR-A5JH, OR-A5JI, OR-A5JJ, OR-A5JK, OR-A5JL, OR-A5JM, OR-A5JO, OR-A5JP, OR-A5JQ, OR-A5JR, OR-A5JS, OR-A5JT, OR-A5JU, OR-A5JV, OR-A5JW, OR-A5JX, OR-A5JY, OR-A5JZ, OR-A5K0, OR-A5K1, OR-A5K2, OR-A5K3, OR-A5K4, OR-A5K5, OR-A5K6, OR-A5K8, OR-A5K9, OR-A5KB, OR-A5KO, OR-A5KP, OR-A5KQ, OR-A5KS, OR-A5KT, OR-A5KU, OR-A5KV, OR-A5KW, OR-A5KX, OR-A5KY, OR-A5KZ, OR-A5L1, OR-A5L2, OR-A5L3, OR-A5L4, OR-A5L5, OR-A5L6, OR-A5L8, OR-A5L9, OR-A5LA, OR-A5LB, OR-A5LC, OR-A5LD, OR-A5LE, OR-A5LF, OR-A5LG, OR-A5LH, OR-A5LI, OR-A5LJ, OR-A5LK, OR-A5LL, OR-A5LN, OR-A5LO, OR-A5LP, OR-A5LR, OR-A5LS, OR-A5LT, OU-A5PI, P6-A5OF, P6-A5OG, P6-A5OH, PA-A5YG, PK-A5H8, PK-A5H9, PK-A5HA, PK-A5HB, PK-A5HC.⧫ B-Cell Lymphoma (24 samples):∙ Source M1 = <cit.>. In DLBCL sample IDs * runs from A though M (e.g., DLBCL-PatientC):07-35482, DLBCL-Patient*, FL-PatientA, FL009.∙ Source L1 = <cit.>:1060, 1061, 1065, 1093, 1096, 1102, 515, EB2.⧫ Benign Liver Tumor (40 samples):∙ Source P1 = <cit.>. Sample IDs are of the form CHC*, where * is:1023T, 1124T, 1315T, 1328T, 1329T, 1337T, 1382T, 1383T, 1424T, 1425T, 1428T, 1432T, 1434T, 1439T, 1488T, 1489T, 1665T, 1666T, 1854T, 1916T, 340T, 361TB, 462T, 463T, 464T, 470T, 471T, 517T, 575T, 578T, 603T, 605T, 623T, 624T, 674T, 687T, 689T, 846T, 918T, 976T.⧫ Bladder Cancer (341 samples):∙ Source G1 = <cit.>. Sample IDs are of the form TCC+AF8-B**+AC0-Tumor, where ** is (below * stands for +AC0-, e.g., 104*0 = 104+AC0-0, and the full sample ID is TCC+AF8-B104+AC0-0+AC0-Tumor):10, 100, 101, 102, 103, 104*0, 104, 105*0, 105*1, 105, 106, 107, 109, 11, 110, 111, 112, 114, 13, 14, 15, 16, 17, 18, 19, 2, 20, 21, 22, 23, 24, 25, 34, 35, 37, 41, 43, 45, 47, 5, 50, 52, 54, 55, 56, 57, 58, 59*0, 59*1, 59*3, 59, 60, 61, 62*0, 63, 64, 65, 66*0, 66, 68, 70, 71, 73, 74, 77, 78, 79, 8, 80*0, 80*1, 80*11, 80*13, 80*3, 80*4, 80*5, 80*7, 80*8, 80, 81*1, 81*2, 81, 82, 83, 84, 85*0, 85*2, 86, 87, 88, 89*1, 89*10, 89*11, 89*12, 89*16, 89*3, 89*4, 89*5, 9, 90, 92, 96, 98, 99.∙ Source T3 = TCGA (see Acknowledgments). Sample IDs are of the form TCGA+AC0-**, where ** is (below * stands for +AC0-A, e.g., BL*0C8 = BL+AC0-A0C8, and the full sample ID is TCGA+AC0-BL+AC0-A0C8; also, below ⋆ = 3OO = 3-double-O):BL*0C8, BL*13I, BL*13J, BL*3JM, BL*5ZZ, BT*0S7, BT*0YX, BT*20J, BT*20N, BT*20O, BT*20P, BT*20Q, BT*20R, BT*20T, BT*20U, BT*20V, BT*20W, BT*20X, BT*2LA, BT*2LB, BT*2LD, BT*3PH, BT*3PJ, BT*3PK, BT*42B, BT*42C, BT*42E, BT*42F, C4*0EZ, C4*0F0, C4*0F1, C4*0F6, C4*0F7, CF*1HR, CF*1HS, CF*27C, CF*3MF, CF*3MG, CF*3MH, CF*3MI, CF*47S, CF*47T, CF*47V, CF*47W, CF*47X, CF*47Y, CF*5U8, CF*5UA, CU*0YN, CU*0YO, CU*0YR, CU*3KJ, CU*3QU, CU*3YL, CU*5W6, CU*72E, DK*1A3, DK*1A5, DK*1A6, DK*1A7, DK*1AA, DK*1AB, DK*1AC, DK*1AD, DK*1AE, DK*1AF, DK*1AG, DK*2HX, DK*2I1, DK*2I2, DK*2I4, DK*2I6, DK*3IK, DK*3IL, DK*3IM, DK*3IN, DK*3IQ, DK*3IS, DK*3IT, DK*3IU, DK*3IV, DK*3WW, DK*3WX, DK*3WY, DK*3X1, DK*3X2, DK*6AV, DK*6AW, DK*6B0, DK*6B1, DK*6B2, DK*6B5, DK*6B6, E5*2PC, E5*4TZ, E5*4U1, E7*3X6, E7*3Y1, E7*4IJ, E7*4XJ, E7*541, E7*5KE, E7*5KF, E7*677, E7*678, E7*6ME, E7*6MF, E7*7DU, E7*7DV, FD*3B3, FD*3B4, FD*3B5, FD*3B6, FD*3B7, FD*3B8, FD*3N5, FD*3N6, FD*3NA, FD*3SJ, FD*3SL, FD*3SM, FD*3SN, FD*3SO, FD*3SP, FD*3SQ, FD*3SR, FD*3SS, FD*43N, FD*43P, FD*43S, FD*43U, FD*43X, FD*5BR, FD*5BS, FD*5BU, FD*5BV, FD*5BX, FD*5BY, FD*5BZ, FD*5C0, FD*5C1, FD*62N, FD*62O, FD*62P, FD*62S, FD*6TA, FD*6TB, FD*6TC, FD*6TD, FD*6TE, FD*6TF, FD*6TG, FD*6TH, FD*6TI, FD*6TK, FJ*3Z7, FJ*3Z9, FJ*3ZE, FJ*3ZF, FT*3EE, FT*61P, G2*2EC, G2*2EF, G2*2EJ, G2*2EK, G2*2EL, G2*2EO, G2*2ES, G2*3IB, G2*3IE, G2*3VY, GC*3BM, GC*3I6,GC*⋆, GC*3RB, GC*3RC, GC*3RD, GC*3WC, GC*3YS, GC*6I1, GC*6I3, GD*2C5, GD*3OP, GD*3OQ, GD*3OS, GD*6C6, GD*76B, GU*42P, GU*42Q, GU*42R, GU*762, GU*763, GU*766, GU*767, GV*3JV, GV*3JW, GV*3JX, GV*3JZ, GV*3QF, GV*3QG, GV*3QH, GV*3QI, GV*3QK, GV*40E, GV*40G, GV*6ZA, H4*2HO, H4*2HQ, HQ*2OE, HQ*2OF, HQ*5ND, HQ*5NE, K4*3WS, K4*3WU, K4*3WV, K4*4AB, K4*4AC, K4*54R, K4*5RH, K4*5RI, K4*5RJ, K4*6FZ, K4*6MB, KQ*41N, KQ*41P, KQ*41Q, KQ*41S, LC*66R, LT*5Z6, MV*51V, PQ*6FI, PQ*6FN, R3*69X, S5*6DX, UY*78K, UY*78L, UY*78N, UY*78O.⧫ Brain Lower Grade Glioma (465 samples):∙ Source T4 = TCGA (see Acknowledgments). Sample IDs are of the form TCGA-*, where * is:CS-4938, CS-4941, CS-4942, CS-4943, CS-4944, CS-5390, CS-5393, CS-5394, CS-5395, CS-5396, CS-5397, CS-6186, CS-6188, CS-6290, CS-6665, CS-6666, CS-6667, CS-6668, CS-6669, CS-6670, DB-5270, DB-5273, DB-5274, DB-5275, DB-5276, DB-5277, DB-5278, DB-5279, DB-5280, DB-5281, DB-A4X9, DB-A4XA, DB-A4XB, DB-A4XC, DB-A4XD, DB-A4XE, DB-A4XF, DB-A4XG, DB-A4XH, DB-A64L, DB-A64O, DB-A64P, DB-A64Q, DB-A64R, DB-A64S, DB-A64U, DB-A64V, DB-A64W, DB-A64X, DB-A75K, DB-A75L, DB-A75M, DB-A75O, DB-A75P, DH-5140, DH-5141, DH-5142, DH-5143, DH-5144, DH-A669, DH-A66B, DH-A66D, DH-A66F, DH-A66G, DH-A7UR, DH-A7US, DH-A7UT, DH-A7UU, DH-A7UV, DU-5847, DU-5849, DU-5851, DU-5852, DU-5853, DU-5854, DU-5855, DU-5870, DU-5871, DU-5872, DU-5874, DU-6392, DU-6393, DU-6394, DU-6395, DU-6396, DU-6397, DU-6399, DU-6400, DU-6401, DU-6402, DU-6403, DU-6404, DU-6405, DU-6406, DU-6407, DU-6408, DU-6410, DU-6542, DU-7006, DU-7007, DU-7008, DU-7009, DU-7010, DU-7011, DU-7012, DU-7013, DU-7014, DU-7015, DU-7018, DU-7019, DU-7290, DU-7292, DU-7294, DU-7298, DU-7299, DU-7300, DU-7301, DU-7302, DU-7304, DU-7306, DU-7309, DU-8158, DU-8161, DU-8162, DU-8163, DU-8164, DU-8165, DU-8166, DU-8167, DU-8168, DU-A5TP, DU-A5TR, DU-A5TS, DU-A5TT, DU-A5TU, DU-A5TW, DU-A5TY, DU-A6S2, DU-A6S3, DU-A6S6, DU-A6S7, DU-A6S8, DU-A76K, DU-A76L, DU-A76O, DU-A76R, DU-A7T6, DU-A7T8, DU-A7TA, DU-A7TB, DU-A7TC, DU-A7TD, DU-A7TG, DU-A7TJ, E1-5302, E1-5303, E1-5304, E1-5305, E1-5307, E1-5311, E1-5318, E1-5319, E1-5322, E1-A7YD, E1-A7YE, E1-A7YH, E1-A7YI, E1-A7YJ, E1-A7YK, E1-A7YL, E1-A7YM, E1-A7YN, E1-A7YO, E1-A7YQ, E1-A7YS, E1-A7YU, E1-A7YV, E1-A7YW, E1-A7YY, E1-A7Z2, E1-A7Z3, E1-A7Z4, E1-A7Z6, EZ-7264, FG-5962, FG-5963, FG-5964, FG-5965, FG-6688, FG-6689, FG-6690, FG-6691, FG-6692, FG-7634, FG-7636, FG-7637, FG-7638, FG-7641, FG-7643, FG-8181, FG-8182, FG-8185, FG-8186, FG-8187, FG-8188, FG-8189, FG-8191, FG-A4MT, FG-A4MU, FG-A4MW, FG-A4MX, FG-A4MY, FG-A60J, FG-A60K, FG-A60L, FG-A6IZ, FG-A6J1, FG-A6J3, FG-A70Y, FG-A70Z, FG-A710, FG-A711, FG-A713, FN-7833, HT-7467, HT-7468, HT-7469, HT-7470, HT-7471, HT-7472, HT-7473, HT-7474, HT-7475, HT-7476, HT-7477, HT-7478, HT-7479, HT-7480, HT-7481, HT-7482, HT-7483, HT-7485, HT-7601, HT-7602, HT-7603, HT-7604, HT-7605, HT-7606, HT-7607, HT-7608, HT-7609, HT-7610, HT-7611, HT-7616, HT-7620, HT-7676, HT-7677, HT-7680, HT-7681, HT-7684, HT-7686, HT-7687, HT-7688, HT-7689, HT-7690, HT-7691, HT-7692, HT-7693, HT-7694, HT-7695, HT-7854, HT-7855, HT-7856, HT-7857, HT-7858, HT-7860, HT-7873, HT-7874, HT-7875, HT-7877, HT-7879, HT-7880, HT-7881, HT-7882, HT-7884, HT-7902, HT-8010, HT-8011, HT-8012, HT-8013, HT-8015, HT-8018, HT-8019, HT-8104, HT-8105, HT-8106, HT-8107, HT-8108, HT-8109, HT-8110, HT-8111, HT-8113, HT-8114, HT-8558, HT-8563, HT-8564, HT-A4DS, HT-A4DV, HT-A5R5, HT-A5R7, HT-A5R9, HT-A5RA, HT-A5RB, HT-A5RC, HT-A614, HT-A615, HT-A616, HT-A617, HT-A618, HT-A619, HT-A61A, HT-A61B, HT-A61C, HT-A74H, HT-A74J, HT-A74K, HT-A74L, HT-A74O, HW-7486, HW-7487, HW-7489, HW-7490, HW-7491, HW-7493, HW-7495, HW-8319, HW-8320, HW-8321, HW-8322, HW-A5KJ, HW-A5KK, HW-A5KL, HW-A5KM, IK-7675, IK-8125, KT-A74X, KT-A7W1, P5-A5ET, P5-A5EU, P5-A5EV, P5-A5EW, P5-A5EX, P5-A5EY, P5-A5EZ, P5-A5F0, P5-A5F1, P5-A5F2, P5-A5F4, P5-A5F6, P5-A72U, P5-A72W, P5-A72X, P5-A72Z, P5-A730, P5-A731, P5-A733, P5-A735, P5-A736, P5-A737, P5-A77W, P5-A77X, P5-A780, P5-A781, QH-A65R, QH-A65S, QH-A65V, QH-A65X, QH-A65Z, QH-A6CS, QH-A6CU, QH-A6CV, QH-A6CW, QH-A6CX, QH-A6CY, QH-A6CZ, QH-A6X3, QH-A6X4, QH-A6X5, QH-A6X8, QH-A6X9, QH-A6XA, QH-A6XC, R8-A6MK, R8-A6ML, R8-A6MO, R8-A6YH, R8-A73M, S9-A6TS, S9-A6TU, S9-A6TV, S9-A6TW, S9-A6TX, S9-A6TY, S9-A6TZ, S9-A6U0, S9-A6U1, S9-A6U2, S9-A6U5, S9-A6U6, S9-A6U8, S9-A6U9, S9-A6UA, S9-A6UB, S9-A6WD, S9-A6WE, S9-A6WG, S9-A6WH, S9-A6WI, S9-A6WL, S9-A6WM, S9-A6WN, S9-A6WO, S9-A6WP, S9-A6WQ, S9-A7IQ, S9-A7IS, S9-A7IX, S9-A7IY, S9-A7IZ, S9-A7J0, S9-A7J1, S9-A7J2, S9-A7J3, S9-A7QW, S9-A7QX, S9-A7QY, S9-A7QZ, S9-A7R1, S9-A7R2, S9-A7R3, S9-A7R4, S9-A7R7, S9-A7R8, TM-A7C3, TM-A7C4, TM-A7C5, TM-A7CA, TM-A7CF, TQ-A7RF, TQ-A7RG, TQ-A7RH, TQ-A7RI, TQ-A7RJ, TQ-A7RK, TQ-A7RM, TQ-A7RN, TQ-A7RO, TQ-A7RP, TQ-A7RQ, TQ-A7RR, TQ-A7RS, TQ-A7RU, TQ-A7RV, TQ-A7RW, VW-A7QS.⧫ Breast Cancer (1182 samples):∙ Source N1 = <cit.>. Sample IDs are of the form CGP_specimen_*, where * is:1096043, 1142475, 1142532, 1142534, 1192095, 1192097, 1192099, 1192101, 1192103, 1192105, 1192107, 1192111, 1192113, 1192115, 1192117, 1192119, 1192121, 1192123, 1192125, 1192127, 1192129, 1192131, 1192133, 1192135, 1192137, 1195364, 1195366, 1195368, 1212804, 1212810, 1212816, 1212822, 1212825, 1212828, 1215490, 1215532, 1215535, 1215553, 1215559, 1215561, 1215563, 1215565, 1215567, 1215573, 1223855, 1223858, 1223861, 1227889, 1227916, 1227918, 1227920, 1227922, 1227924, 1227926, 1227928, 1227951, 1227953, 1227955, 1227957, 1227959, 1227961, 1227963, 1227965, 1227969, 1227971, 1241537, 1241539, 1241541, 1241543, 1241545, 1241547, 1241549, 1241551, 1241553, 1241555, 1241557, 1241559, 1241562, 1241565, 1241568, 1241571, 1241574, 1241579, 1241581, 1261287, 1261291, 1261293, 1261295, 1261297, 1261299, 1261301, 1261303, 1261305, 1261307, 1261309, 1261311, 1261313, 1261337, 1261382, 1261391, 1266549, 1266551, 1266553, 1266561, 1266563, 1266565, 1266567, 1343241, 1343244, 1343247, 1380057, 1380059, 1380061, 1380063, 1380065, 1380067.∙ Source S1 = <cit.>. Sample IDs are of the form PD*a, where * is:4842, 4843, 4844, 4934, 4935, 4936, 4937, 4938, 4939, 5961, 7206, 7211, 7316, 9193.∙ Source S2 = <cit.>. Sample IDs are of the form SA*, where * is:018, 029, 030, 031, 051, 052, 053, 054, 055, 063, 065, 067, 068, 069, 071, 072, 073, 074, 075, 076, 077, 080, 083, 084, 085, 089, 090, 092, 093, 094, 096, 097, 098, 101, 102, 103, 106, 208, 210, 212, 213, 214, 215, 216, 217, 218, 219, 220, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 233, 234, 235, 236, 237.∙ Source T5 = TCGA (see Acknowledgments). Sample IDs are of the form TCGA-*, where * is:A1-A0SB, A1-A0SD, A1-A0SE, A1-A0SF, A1-A0SG, A1-A0SH, A1-A0SI, A1-A0SJ, A1-A0SK, A1-A0SM, A1-A0SN, A1-A0SO, A1-A0SP, A1-A0SQ, A2-A04N, A2-A04P, A2-A04Q, A2-A04R, A2-A04T, A2-A04U, A2-A04V, A2-A04W, A2-A04X, A2-A04Y, A2-A0CK, A2-A0CL, A2-A0CM, A2-A0CO, A2-A0CP, A2-A0CQ, A2-A0CR, A2-A0CS, A2-A0CT, A2-A0CU, A2-A0CV, A2-A0CW, A2-A0CX, A2-A0CZ, A2-A0D0, A2-A0D1, A2-A0D2, A2-A0D3, A2-A0D4, A2-A0EM, A2-A0EN, A2-A0EO, A2-A0EP, A2-A0EQ, A2-A0ER, A2-A0ES, A2-A0ET, A2-A0EU, A2-A0EV, A2-A0EW, A2-A0EX, A2-A0EY, A2-A0ST, A2-A0SU, A2-A0SV, A2-A0SW, A2-A0SX, A2-A0SY, A2-A0T0, A2-A0T1, A2-A0T2, A2-A0T3, A2-A0T4, A2-A0T5, A2-A0T6, A2-A0T7, A2-A0YC, A2-A0YD, A2-A0YE, A2-A0YF, A2-A0YG, A2-A0YH, A2-A0YI, A2-A0YJ, A2-A0YK, A2-A0YL, A2-A0YM, A2-A0YT, A2-A1FV, A2-A1FW, A2-A1FX, A2-A1FZ, A2-A1G0, A2-A1G1, A2-A1G4, A2-A1G6, A2-A259, A2-A25A, A2-A25B, A2-A25C, A2-A25D, A2-A25E, A2-A25F, A2-A3KC, A2-A3KD, A2-A3XS, A2-A3XT, A2-A3XU, A2-A3XV, A2-A3XW, A2-A3XX, A2-A3XY, A2-A3XZ, A2-A3Y0, A2-A4RW, A2-A4RX, A2-A4RY, A2-A4S0, A2-A4S1, A2-A4S2, A2-A4S3, A7-A0CD, A7-A0CE, A7-A0CG, A7-A0CH, A7-A0CJ, A7-A0D9, A7-A0DA, A7-A0DB, A7-A0DC, A7-A13D, A7-A13E, A7-A13F, A7-A13G, A7-A13H, A7-A26E, A7-A26F, A7-A26G, A7-A26H, A7-A26I, A7-A26J, A7-A2KD, A7-A3IY, A7-A3IZ, A7-A3J0, A7-A3J1, A7-A3RF, A7-A425, A7-A426, A7-A4SA, A7-A4SB, A7-A4SC, A7-A4SD, A7-A4SE, A7-A4SF, A7-A56D, A7-A5ZV, A7-A5ZW, A7-A5ZX, A8-A06N, A8-A06O, A8-A06P, A8-A06Q, A8-A06R, A8-A06T, A8-A06U, A8-A06X, A8-A06Y, A8-A06Z, A8-A075, A8-A076, A8-A079, A8-A07B, A8-A07C, A8-A07E, A8-A07F, A8-A07G, A8-A07I, A8-A07J, A8-A07L, A8-A07O, A8-A07P, A8-A07R, A8-A07U, A8-A07W, A8-A07Z, A8-A081, A8-A082, A8-A083, A8-A084, A8-A085, A8-A086, A8-A08A, A8-A08B, A8-A08F, A8-A08G, A8-A08H, A8-A08I, A8-A08J, A8-A08L, A8-A08O, A8-A08P, A8-A08R, A8-A08S, A8-A08T, A8-A08X, A8-A08Z, A8-A090, A8-A091, A8-A092, A8-A093, A8-A094, A8-A095, A8-A096, A8-A097, A8-A099, A8-A09A, A8-A09B, A8-A09C, A8-A09D, A8-A09E, A8-A09G, A8-A09I, A8-A09K, A8-A09M, A8-A09N, A8-A09Q, A8-A09R, A8-A09T, A8-A09V, A8-A09W, A8-A09X, A8-A09Z, A8-A0A1, A8-A0A2, A8-A0A4, A8-A0A6, A8-A0A7, A8-A0A9, A8-A0AB, A8-A0AD, AC-A23C, AC-A23E, AC-A23G, AC-A23H, AC-A2B8, AC-A2BK, AC-A2BM, AC-A2FB, AC-A2FE, AC-A2FF, AC-A2FG, AC-A2FK, AC-A2FM, AC-A2FO, AC-A2QH, AC-A2QI, AC-A2QJ, AC-A3BB, AC-A3EH, AC-A3HN, AC-A3OD, AC-A3QP, AC-A3TM, AC-A3TN, AC-A3W5, AC-A3W6, AC-A3W7, AC-A3YI, AC-A3YJ, AC-A5EH, AC-A5EI, AC-A5XS, AC-A5XU, AC-A62X, AC-A62Y, AN-A03X, AN-A03Y, AN-A041, AN-A046, AN-A049, AN-A04A, AN-A04C, AN-A04D, AN-A0AJ, AN-A0AK, AN-A0AL, AN-A0AM, AN-A0AR, AN-A0AS, AN-A0AT, AN-A0FD, AN-A0FF, AN-A0FJ, AN-A0FK, AN-A0FL, AN-A0FN, AN-A0FS, AN-A0FT, AN-A0FV, AN-A0FW, AN-A0FX, AN-A0FY, AN-A0FZ, AN-A0G0, AN-A0XL, AN-A0XN, AN-A0XO, AN-A0XP, AN-A0XR, AN-A0XS, AN-A0XT, AN-A0XU, AN-A0XV, AN-A0XW, AO-A03L, AO-A03M, AO-A03N, AO-A03O, AO-A03P, AO-A03R, AO-A03T, AO-A03U, AO-A03V, AO-A0J2, AO-A0J3, AO-A0J4, AO-A0J5, AO-A0J6, AO-A0J7, AO-A0J8, AO-A0J9, AO-A0JA, AO-A0JB, AO-A0JC, AO-A0JD, AO-A0JE, AO-A0JF, AO-A0JG, AO-A0JI, AO-A0JJ, AO-A0JL, AO-A0JM, AO-A124, AO-A125, AO-A126, AO-A128, AO-A129, AO-A12A, AO-A12B, AO-A12D, AO-A12E, AO-A12F, AO-A12G, AO-A12H, AO-A1KO, AO-A1KP, AO-A1KR, AO-A1KS, AO-A1KT, AQ-A04H, AQ-A04J, AQ-A04L, AQ-A0Y5, AQ-A1H2, AQ-A1H3, AQ-A54N, AQ-A54O, AR-A0TP, AR-A0TQ, AR-A0TR, AR-A0TS, AR-A0TT, AR-A0TU, AR-A0TV, AR-A0TW, AR-A0TX, AR-A0TY, AR-A0TZ, AR-A0U0, AR-A0U1, AR-A0U2, AR-A0U3, AR-A0U4, AR-A1AH, AR-A1AI, AR-A1AJ, AR-A1AK, AR-A1AL, AR-A1AM, AR-A1AN, AR-A1AO, AR-A1AP, AR-A1AQ, AR-A1AR, AR-A1AS, AR-A1AT, AR-A1AU, AR-A1AV, AR-A1AW, AR-A1AX, AR-A1AY, AR-A24H, AR-A24K, AR-A24L, AR-A24M, AR-A24N, AR-A24O, AR-A24P, AR-A24Q, AR-A24R, AR-A24S, AR-A24T, AR-A24U, AR-A24V, AR-A24W, AR-A24X, AR-A24Z, AR-A250, AR-A251, AR-A252, AR-A254, AR-A255, AR-A256, AR-A2LE, AR-A2LH, AR-A2LJ, AR-A2LK, AR-A2LL, AR-A2LM, AR-A2LN, AR-A2LO, AR-A2LQ, AR-A2LR, AR-A5QM, AR-A5QN, AR-A5QP, AR-A5QQ, B6-A0I1, B6-A0I2, B6-A0I5, B6-A0I6, B6-A0I8, B6-A0I9, B6-A0IA, B6-A0IB, B6-A0IC, B6-A0IE, B6-A0IG, B6-A0IH, B6-A0IJ, B6-A0IK, B6-A0IM, B6-A0IN, B6-A0IO, B6-A0IP, B6-A0IQ, B6-A0RE, B6-A0RG, B6-A0RH, B6-A0RI, B6-A0RL, B6-A0RM, B6-A0RN, B6-A0RO, B6-A0RP, B6-A0RQ, B6-A0RS, B6-A0RT, B6-A0RU, B6-A0RV, B6-A0WS, B6-A0WT, B6-A0WV, B6-A0WW, B6-A0WX, B6-A0WY, B6-A0WZ, B6-A0X0, B6-A0X1, B6-A0X4, B6-A0X5, B6-A0X7, B6-A1KC, B6-A1KF, B6-A1KI, B6-A1KN, B6-A2IU, B6-A3ZX, B6-A400, B6-A401, B6-A402, B6-A408, B6-A409, B6-A40B, B6-A40C, BH-A0AU, BH-A0AV, BH-A0AW, BH-A0AY, BH-A0AZ, BH-A0B0, BH-A0B1, BH-A0B3, BH-A0B4, BH-A0B5, BH-A0B6, BH-A0B7, BH-A0B8, BH-A0B9, BH-A0BA, BH-A0BC, BH-A0BD, BH-A0BF, BH-A0BG, BH-A0BJ, BH-A0BL, BH-A0BM, BH-A0BO, BH-A0BP, BH-A0BQ, BH-A0BR, BH-A0BS, BH-A0BT, BH-A0BV, BH-A0BW, BH-A0BZ, BH-A0C0, BH-A0C1, BH-A0C3, BH-A0C7, BH-A0DD, BH-A0DE, BH-A0DG, BH-A0DH, BH-A0DI, BH-A0DK, BH-A0DL, BH-A0DO, BH-A0DP, BH-A0DQ, BH-A0DS, BH-A0DT, BH-A0DV, BH-A0DX, BH-A0DZ, BH-A0E0, BH-A0E1, BH-A0E2, BH-A0E6, BH-A0E7, BH-A0E9, BH-A0EA, BH-A0EB, BH-A0EE, BH-A0EI, BH-A0GY, BH-A0GZ, BH-A0H0, BH-A0H3, BH-A0H5, BH-A0H6, BH-A0H7, BH-A0H9, BH-A0HA, BH-A0HB, BH-A0HF, BH-A0HI, BH-A0HK, BH-A0HL, BH-A0HN, BH-A0HO, BH-A0HP, BH-A0HQ, BH-A0HU, BH-A0HW, BH-A0HX, BH-A0HY, BH-A0RX, BH-A0W3, BH-A0W4, BH-A0W5, BH-A0W7, BH-A0WA, BH-A18F, BH-A18G, BH-A18H, BH-A18I, BH-A18J, BH-A18K, BH-A18L, BH-A18M, BH-A18N, BH-A18P, BH-A18Q, BH-A18R, BH-A18S, BH-A18T, BH-A18U, BH-A18V, BH-A1EN, BH-A1EO, BH-A1ES, BH-A1ET, BH-A1EU, BH-A1EV, BH-A1EW, BH-A1EX, BH-A1EY, BH-A1F0, BH-A1F2, BH-A1F5, BH-A1F6, BH-A1F8, BH-A1FC, BH-A1FD, BH-A1FE, BH-A1FG, BH-A1FH, BH-A1FJ, BH-A1FL, BH-A1FM, BH-A1FN, BH-A1FR, BH-A1FU, BH-A201, BH-A202, BH-A203, BH-A204, BH-A208, BH-A209, BH-A28O, BH-A28Q, BH-A2L8, BH-A42T, BH-A42U, BH-A42V, BH-A5IZ, BH-A5J0, C8-A12K, C8-A12L, C8-A12M, C8-A12N, C8-A12O, C8-A12P, C8-A12Q, C8-A12T, C8-A12U, C8-A12V, C8-A12W, C8-A12X, C8-A12Y, C8-A12Z, C8-A130, C8-A131, C8-A132, C8-A133, C8-A134, C8-A135, C8-A137, C8-A138, C8-A1HE, C8-A1HF, C8-A1HG, C8-A1HI, C8-A1HJ, C8-A1HK, C8-A1HL, C8-A1HM, C8-A1HN, C8-A1HO, C8-A26V, C8-A26W, C8-A26X, C8-A26Y, C8-A26Z, C8-A273, C8-A274, C8-A275, C8-A278, C8-A27A, C8-A27B, C8-A3M7, C8-A3M8, D8-A13Y, D8-A13Z, D8-A140, D8-A141, D8-A142, D8-A143, D8-A145, D8-A146, D8-A147, D8-A1J8, D8-A1J9, D8-A1JA, D8-A1JB, D8-A1JC, D8-A1JD, D8-A1JE, D8-A1JF, D8-A1JG, D8-A1JH, D8-A1JI, D8-A1JJ, D8-A1JK, D8-A1JL, D8-A1JM, D8-A1JN, D8-A1JP, D8-A1JS, D8-A1JT, D8-A1JU, D8-A1X5, D8-A1X6, D8-A1X7, D8-A1X8, D8-A1X9, D8-A1XA, D8-A1XB, D8-A1XC, D8-A1XF, D8-A1XG, D8-A1XJ, D8-A1XK, D8-A1XL, D8-A1XM, D8-A1XO, D8-A1XQ, D8-A1XR, D8-A1XS, D8-A1XT, D8-A1XU, D8-A1XV, D8-A1XW, D8-A1XY, D8-A1XZ, D8-A1Y0, D8-A1Y1, D8-A1Y2, D8-A1Y3, D8-A27E, D8-A27F, D8-A27G, D8-A27H, D8-A27I, D8-A27K, D8-A27L, D8-A27M, D8-A27N, D8-A27P, D8-A27R, D8-A27T, D8-A27V, D8-A27W, D8-A3Z5, D8-A3Z6, D8-A4Z1, E2-A105, E2-A107, E2-A108, E2-A109, E2-A10A, E2-A10B, E2-A10C, E2-A10E, E2-A10F, E2-A14N, E2-A14O, E2-A14P, E2-A14Q, E2-A14R, E2-A14S, E2-A14T, E2-A14U, E2-A14V, E2-A14W, E2-A14X, E2-A14Y, E2-A14Z, E2-A150, E2-A152, E2-A153, E2-A154, E2-A155, E2-A156, E2-A158, E2-A159, E2-A15A, E2-A15C, E2-A15D, E2-A15E, E2-A15F, E2-A15G, E2-A15H, E2-A15I, E2-A15J, E2-A15K, E2-A15L, E2-A15M, E2-A15O, E2-A15P, E2-A15R, E2-A15S, E2-A15T, E2-A1AZ, E2-A1B0, E2-A1B1, E2-A1B4, E2-A1B5, E2-A1B6, E2-A1BC, E2-A1BD, E2-A1IE, E2-A1IF, E2-A1IG, E2-A1IH, E2-A1II, E2-A1IJ, E2-A1IK, E2-A1IL, E2-A1IN, E2-A1IO, E2-A1IU, E2-A1L6, E2-A1L7, E2-A1L8, E2-A1L9, E2-A1LA, E2-A1LB, E2-A1LE, E2-A1LG, E2-A1LH, E2-A1LI, E2-A1LK, E2-A1LL, E2-A1LS, E2-A2P5, E2-A2P6, E2-A3DX, E2-A56Z, E2-A570, E2-A573, E2-A574, E9-A1N3, E9-A1N4, E9-A1N5, E9-A1N8, E9-A1N9, E9-A1NA, E9-A1NC, E9-A1ND, E9-A1NE, E9-A1NF, E9-A1NG, E9-A1NH, E9-A1NI, E9-A1QZ, E9-A1R0, E9-A1R2, E9-A1R3, E9-A1R4, E9-A1R5, E9-A1R6, E9-A1R7, E9-A1RA, E9-A1RB, E9-A1RC, E9-A1RD, E9-A1RE, E9-A1RF, E9-A1RG, E9-A1RH, E9-A1RI, E9-A226, E9-A227, E9-A228, E9-A229, E9-A22A, E9-A22B, E9-A22D, E9-A22E, E9-A22G, E9-A22H, E9-A243, E9-A244, E9-A245, E9-A247, E9-A248, E9-A249, E9-A24A, E9-A295, E9-A2JS, E9-A2JT, E9-A3HO, E9-A3Q9, E9-A3QA, E9-A3X8, E9-A54X, E9-A54Y, E9-A5FK, E9-A5FL, E9-A5UO, E9-A5UP, EW-A1IW, EW-A1IX, EW-A1IY, EW-A1IZ, EW-A1J1, EW-A1J2, EW-A1J3, EW-A1J5, EW-A1J6, EW-A1OV, EW-A1OX, EW-A1OY, EW-A1OZ, EW-A1P0, EW-A1P1, EW-A1P3, EW-A1P4, EW-A1P5, EW-A1P6, EW-A1P7, EW-A1P8, EW-A1PA, EW-A1PB, EW-A1PC, EW-A1PD, EW-A1PE, EW-A1PG, EW-A1PH, EW-A2FR, EW-A2FS, EW-A2FV, EW-A2FW, EW-A3E8, EW-A3U0, EW-A423, GI-A2C8, GI-A2C9, GM-A2D9, GM-A2DA, GM-A2DB, GM-A2DC, GM-A2DD, GM-A2DF, GM-A2DH, GM-A2DI, GM-A2DK, GM-A2DL, GM-A2DM, GM-A2DN, GM-A2DO, GM-A3NW, GM-A3NY, GM-A3XG, GM-A3XL, GM-A3XN, GM-A4E0, GM-A5PV, GM-A5PX, HN-A2NL, HN-A2OB, JL-A3YW, JL-A3YX, LL-A440, LL-A441, LL-A50Y, LL-A5YL, LL-A5YM, LL-A5YN, LL-A5YO, LL-A5YP, LQ-A4E4, MS-A51U, OK-A5Q2, OL-A5D6, OL-A5D7, OL-A5D8, OL-A5DA, OL-A5RU, OL-A5RV, OL-A5RW, OL-A5RX, OL-A5RY, OL-A5RZ, OL-A5S0, OL-A66H, OL-A66I, OL-A66J, OL-A66K, PE-A5DC, PE-A5DD, PE-A5DE.⧫ Cervical Cancer (197 samples):∙ Source T6 = TCGA (see Acknowledgments). Sample IDs are of the form TCGA-*, where * is:BI-A0VR, BI-A0VS, BI-A20A, C5-A0TN, C5-A1BE, C5-A1BF, C5-A1BI, C5-A1BJ, C5-A1BK, C5-A1BL, C5-A1BM, C5-A1BN, C5-A1BQ, C5-A1M5, C5-A1M6, C5-A1M7, C5-A1M8, C5-A1M9, C5-A1ME, C5-A1MF, C5-A1MH, C5-A1MI, C5-A1MJ, C5-A1MK, C5-A1ML, C5-A1MN, C5-A1MP, C5-A1MQ, C5-A2LS, C5-A2LT, C5-A2LV, C5-A2LX, C5-A2LY, C5-A2LZ, C5-A2M1, C5-A2M2, C5-A3HD, C5-A3HE, C5-A3HF, C5-A3HL, C5-A7CG, C5-A7CH, C5-A7CJ, C5-A7CK, C5-A7CL, C5-A7CM, C5-A7CO, C5-A7UC, C5-A7UE, C5-A7UH, C5-A7X3, DG-A2KH, DG-A2KJ, DG-A2KK, DG-A2KL, DG-A2KM, DR-A0ZL, DR-A0ZM, DS-A0VK, DS-A0VL, DS-A0VM, DS-A0VN, DS-A1OA, DS-A3LQ, DS-A5RQ, DS-A7WF, DS-A7WH, DS-A7WI, EA-A1QS, EA-A1QT, EA-A3HQ, EA-A3HR, EA-A3HT, EA-A3HU, EA-A3QD, EA-A3QE, EA-A3Y4, EA-A410, EA-A411, EA-A439, EA-A43B, EA-A44S, EA-A4BA, EA-A50E, EA-A556, EA-A5FO, EA-A5O9, EA-A5ZD, EA-A5ZE, EA-A5ZF, EA-A6QX, EA-A78R, EK-A2GZ, EK-A2H0, EK-A2H1, EK-A2IP, EK-A2PG, EK-A2PI, EK-A2PK, EK-A2PL, EK-A2PM, EK-A2R7, EK-A2R8, EK-A2R9, EK-A2RA, EK-A2RB, EK-A2RC, EK-A2RD, EK-A2RE, EK-A2RJ, EK-A2RK, EK-A2RL, EK-A2RM, EK-A2RN, EK-A2RO, EK-A3GJ, EK-A3GK, EK-A3GM, EK-A3GN, EX-A1H5, EX-A1H6, EX-A3L1, EX-A449, EX-A69L, EX-A69M, FU-A23K, FU-A23L, FU-A2QG, FU-A3EO, FU-A3HY, FU-A3HZ, FU-A3NI, FU-A3TQ, FU-A3TX, FU-A3WB, FU-A3YQ, FU-A40J, FU-A57G, FU-A5XV, FU-A770, HG-A2PA, HM-A3JJ, HM-A3JK, HM-A4S6, HM-A6W2, IR-A3L7, IR-A3LA, IR-A3LB, IR-A3LC, IR-A3LF, IR-A3LH, IR-A3LI, IR-A3LK, IR-A3LL, JW-A5VG, JW-A5VH, JW-A5VI, JW-A5VJ, JW-A5VK, JW-A5VL, JW-A69B, JW-A852, JX-A3PZ, JX-A3Q0, JX-A3Q8, JX-A5QV, LP-A4AU, LP-A4AV, LP-A4AW, LP-A4AX, LP-A5U2, LP-A5U3, LP-A7HU, MU-A51Y, MU-A5YI, MY-A5BD, MY-A5BE, MY-A5BF, Q1-A5R1, Q1-A5R2, Q1-A5R3, Q1-A6DT, Q1-A6DV, Q1-A6DW, Q1-A73O, Q1-A73P, Q1-A73Q, Q1-A73R, Q1-A73S, R2-A69V, RA-A741, UC-A7PD, UC-A7PF, WL-A834, DS-A1OB, DS-A1OC, DS-A1OD.⧫ Cholangiocarcinoma (139 samples):∙ Source Z2 = <cit.>:1, 10, 100, 101, 107, 108, 109, 110, 111, 112, 113, 115, 116, 118, 119, 120, 121, 122, 123, 125, 127, 128, 129, 13, 130, 131, 132, 133, 134, 135, 137, 139, 140, 141, 142, 143, 144, 145, 146, 147, 16, 17, 18, 19, 2, 20, 24, 25, 26, 28, 29, 3, 33, 34, 35, 39, 41, 42, 44, 46, 48, 5, 50, 51, 52, 53, 56, 58, 59, 6, 60, 61, 63, 64, 66, 67, 69, 7, 70, 71, 74, 79, 8, 8_1, 8_2, 8_4, 8_6, 80, 81, 82, 85, 86, 87, 88, 89, 9, 90, 91, 94, 95, 97, 98, 99.∙ Source T7 = TCGA (see Acknowledgments). Sample IDs are of the form TCGA-*, where * is:3X-AAV9, 3X-AAVA, 3X-AAVB, 3X-AAVC, 3X-AAVE, 4G-AAZO, 4G-AAZT, W5-AA2G, W5-AA2H, W5-AA2I, W5-AA2O, W5-AA2Q, W5-AA2R, W5-AA2T, W5-AA2U, W5-AA2W, W5-AA2X, W5-AA2Z, W5-AA30, W5-AA31, W5-AA33, W5-AA34, W5-AA36, W5-AA38, W5-AA39, W6-AA0S, WD-A7RX, YR-A95A, ZD-A8I3, ZH-A8Y1, ZH-A8Y2, ZH-A8Y4, ZH-A8Y5, ZH-A8Y6, ZH-A8Y8, ZU-A8S4.⧫ Chronic Lymphocytic Leukemia (80 samples):∙ Source Q1 = <cit.>:170, 171, 172, 173, 174, 175, 18, 181, 182, 184, 185, 186, 188, 189, 19, 191, 193, 194, 195, 197, 20, 22, 23, 264, 266, 267, 27, 270, 272, 273, 274, 275, 276, 278, 279, 280, 29, 290, 30, 319, 32, 321, 322, 323, 324, 325, 326, 328, 33, 375, 39, 40, 41, 42, 43, 44, 45, 48, 49, 5, 51, 52, 53, 54, 6, 618, 63, 64, 642, 680, 7, 758, 761, 785, 8, 82, 83, 9, 90, 91.⧫ Colorectal Cancer (581 samples):∙ Source S3 = <cit.>:587220, 587222, 587224, 587226, 587228, 587230, 587232, 587234, 587238, 587242, 587246, 587254, 587256, 587260, 587262, 587264, 587268, 587270, 587276, 587278, 587282, 587284, 587286, 587288, 587290, 587292, 587294, 587298, 587300, 587302, 587304, 587306, 587316, 587318, 587322, 587328, 587330, 587332, 587334, 587336, 587338, 587340, 587342, 587344, 587346, 587348, 587350, 587352, 587354, 587356, 587358, 587360, 587362, 587364, 587368, 587370, 587372, 587374, 587376, 587378, 587380, 587382, 587384, 587386, 587388, 587390, 587392, 587394, 587398, 587400.∙ Source T8 = TCGA (see Acknowledgments). Sample IDs are of the form TCGA-*, where * is:A6-2670, A6-2671, A6-2672, A6-2674, A6-2675, A6-2676, A6-2677, A6-2678, A6-2683, A6-3807, A6-3808, A6-3809, A6-3810, A6-4105, A6-5656, A6-5657, A6-5659, A6-5660, A6-5661, A6-5662, A6-5664, A6-5665, A6-5666, A6-5667, A6-6137, A6-6138, A6-6140, A6-6141, A6-6142, A6-6648, A6-6649, A6-6650, A6-6651, A6-6652, A6-6653, A6-6654, A6-6780, A6-6781, A6-6782, AA-3489, AA-3492, AA-3496, AA-3502, AA-3510, AA-3511, AA-3514, AA-3516, AA-3517, AA-3518, AA-3519, AA-3520, AA-3521, AA-3522, AA-3524, AA-3525, AA-3526, AA-3527, AA-3529, AA-3531, AA-3532, AA-3534, AA-3538, AA-3542, AA-3543, AA-3544, AA-3548, AA-3549, AA-3552, AA-3553, AA-3554, AA-3555, AA-3556, AA-3558, AA-3560, AA-3561, AA-3562, AA-3655, AA-3660, AA-3662, AA-3663, AA-3664, AA-3666, AA-3667, AA-3672, AA-3673, AA-3678, AA-3679, AA-3680, AA-3681, AA-3684, AA-3685, AA-3688, AA-3692, AA-3693, AA-3695, AA-3696, AA-3697, AA-3710, AA-3712, AA-3713, AA-3715, AA-3811, AA-3812, AA-3814, AA-3815, AA-3818, AA-3819, AA-3821, AA-3831, AA-3833, AA-3837, AA-3842, AA-3844, AA-3845, AA-3846, AA-3848, AA-3850, AA-3851, AA-3852, AA-3854, AA-3855, AA-3856, AA-3858, AA-3860, AA-3861, AA-3864, AA-3866, AA-3867, AA-3869, AA-3870, AA-3872, AA-3875, AA-3877, AA-3930, AA-3939, AA-3941, AA-3947, AA-3949, AA-3950, AA-3952, AA-3955, AA-3956, AA-3966, AA-3968, AA-3971, AA-3972, AA-3973, AA-3975, AA-3977, AA-3979, AA-3980, AA-3982, AA-3984, AA-3986, AA-3989, AA-3994, AA-A004, AA-A00A, AA-A00D, AA-A00E, AA-A00F, AA-A00J, AA-A00K, AA-A00L, AA-A00N, AA-A00O, AA-A00Q, AA-A00R, AA-A00U, AA-A00W, AA-A00Z, AA-A010, AA-A017, AA-A01D, AA-A01F, AA-A01G, AA-A01I, AA-A01K, AA-A01P, AA-A01Q, AA-A01R, AA-A01S, AA-A01T, AA-A01V, AA-A01X, AA-A01Z, AA-A022, AA-A024, AA-A029, AA-A02F, AA-A02H, AA-A02J, AA-A02K, AA-A02O, AA-A02W, AA-A02Y, AA-A03F, AA-A03J, AD-5900, AD-6548, AD-6888, AD-6889, AD-6890, AD-6895, AD-6899, AD-6901, AD-6963, AD-6964, AD-6965, AF-2687, AF-2689, AF-2691, AF-2692, AF-2693, AF-3400, AF-3913, AF-4110, AF-5654, AF-6136, AF-6655, AF-6672, AG-3574, AG-3575, AG-3578, AG-3580, AG-3581, AG-3582, AG-3583, AG-3584, AG-3586, AG-3587, AG-3593, AG-3594, AG-3598, AG-3599, AG-3600, AG-3601, AG-3602, AG-3605, AG-3608, AG-3609, AG-3611, AG-3612, AG-3726, AG-3727, AG-3731, AG-3732, AG-3742, AG-3878, AG-3881, AG-3882, AG-3883, AG-3885, AG-3887, AG-3890, AG-3892, AG-3893, AG-3894, AG-3896, AG-3898, AG-3901, AG-3902, AG-3909, AG-3999, AG-4001, AG-4005, AG-4007, AG-4008, AG-4015, AG-A002, AG-A008, AG-A00C, AG-A00H, AG-A00Y, AG-A011, AG-A014, AG-A015, AG-A016, AG-A01L, AG-A01W, AG-A01Y, AG-A020, AG-A025, AG-A026, AG-A02G, AG-A02N, AG-A02X, AG-A032, AG-A036, AH-6544, AH-6547, AH-6549, AH-6643, AH-6644, AM-5820, AM-5821, AU-3779, AU-6004, AY-4070, AY-4071, AY-5543, AY-6196, AY-6197, AY-6386, AZ-4315, AZ-4323, AZ-4615, AZ-4616, AZ-4681, AZ-4682, AZ-5403, AZ-5407, AZ-6598, AZ-6599, AZ-6600, AZ-6601, AZ-6603, AZ-6605, AZ-6606, AZ-6607, AZ-6608, CA-5254, CA-5255, CA-5796, CA-5797, CA-6715, CA-6716, CA-6717, CA-6718, CA-6719, CI-6619, CI-6620, CI-6621, CI-6622, CI-6624, CK-4947, CK-4948, CK-4950, CK-4952, CK-5912, CK-5913, CK-5914, CK-5915, CK-5916, CK-6746, CK-6747, CK-6748, CK-6751, CL-5917, CL-5918, CM-4743, CM-4744, CM-4746, CM-4747, CM-4748, CM-4750, CM-4752, CM-5341, CM-5344, CM-5348, CM-5349, CM-5860, CM-5861, CM-5862, CM-5863, CM-5864, CM-5868, CM-6161, CM-6162, CM-6163, CM-6164, CM-6165, CM-6166, CM-6167, CM-6168, CM-6169, CM-6170, CM-6171, CM-6172, CM-6674, CM-6675, CM-6676, CM-6677, CM-6678, CM-6679, CM-6680, D5-5537, D5-5538, D5-5539, D5-5540, D5-5541, D5-6529, D5-6531, D5-6532, D5-6533, D5-6534, D5-6535, D5-6536, D5-6537, D5-6538, D5-6539, D5-6540, D5-6541, D5-6898, D5-6920, D5-6922, D5-6923, D5-6924, D5-6926, D5-6927, D5-6928, D5-6929, D5-6930, D5-6931, D5-6932, D5-7000, DC-5337, DC-5869, DC-6155, DC-6157, DC-6158, DC-6160, DC-6681, DC-6682, DC-6683, DM-A0X9, DM-A0XD, DM-A0XF, DM-A1D0, DM-A1D4, DM-A1D6, DM-A1D7, DM-A1D8, DM-A1D9, DM-A1DA, DM-A1DB, DM-A1HA, DM-A1HB, DM-A282, DM-A285, DM-A28C, DM-A28E, DM-A28F, DM-A28G, DM-A28H, DM-A28K, DM-A28M, DT-5265, DY-A0XA, DY-A1DC, DY-A1DD, DY-A1DF, DY-A1DG, DY-A1H8, EF-5830, EI-6506, EI-6507, EI-6508, EI-6510, F4-6459, F4-6460, F4-6461, F4-6463, F4-6569, F4-6570, F4-6703, F4-6704, F4-6805, F4-6806, F4-6807, F4-6808, F4-6809, F4-6854, F4-6855, F4-6856, F4-6857, F5-6464, F5-6465, F5-6571, F5-6702, F5-6811, F5-6812, F5-6813, G4-6293, G4-6294, G4-6295, G4-6297, G4-6298, G4-6299, G4-6302, G4-6303, G4-6304, G4-6306, G4-6307, G4-6309, G4-6310, G4-6311, G4-6314, G4-6315, G4-6317, G4-6320, G4-6321, G4-6322, G4-6323, G4-6586, G4-6588, G4-6625, G4-6626, G4-6628, G5-6235, G5-6641.⧫ Esophageal Cancer (329 samples):∙ Source D2 = <cit.>. Sample IDs are of the form ESO-*-Tumor, where * is:0001, 0009, 0013, 0015, 0019, 0023, 0025, 0029, 003, 005, 0053, 0059, 0061, 0067, 007, 0071, 0079, 0103, 0115, 0123, 0125, 0129, 0133, 0149, 0167, 017, 0176, 021, 0255, 027, 0280, 037, 043, 045, 0459, 049, 051, 0590, 075, 077, 083, 085, 0950, 105, 1059, 1060, 107, 1096, 111, 1130, 1133, 114, 1145, 1154, 116, 1163, 117, 118, 119, 120, 122, 130, 131, 135, 137, 139, 141, 1427, 143, 147, 1481, 1488, 151, 152, 153, 155, 157, 159, 1594, 160, 1608, 161, 164, 165, 167, 1670, 169, 171, 173, 1733, 1748, 175, 177, 179, 184, 185, 187, 1872, 189, 191, 2143, 224, 2472, 250, 251, 2536, 327, 408, 409, 512, 536, 539, 555, 580, 582, 601, 610, 632, 640, 669, 682, 683, 708, 718, 720, 721, 732, 752, 805, 837, 838, 859, 864, 866, 874, 887, 913, 916, 931, 963, D76, H01, H63, K08, R61, S41.∙ Source T9 = TCGA (see Acknowledgments). Sample IDs are of the form TCGA-*, where * is:2H-A9GF, 2H-A9GH, 2H-A9GI, 2H-A9GJ, 2H-A9GK, 2H-A9GL, 2H-A9GM, 2H-A9GN, 2H-A9GO, 2H-A9GQ, 2H-A9GR, IC-A6RE, IC-A6RF, IG-A3I8, IG-A3QL, IG-A3Y9, IG-A3YA, IG-A3YB, IG-A3YC, IG-A4P3, IG-A4QS, IG-A4QT, IG-A50L, IG-A51D, IG-A5B8, IG-A5S3, IG-A625, IG-A6QS, IG-A7DP, IG-A8O2, IG-A97H, IG-A97I, JY-A6F8, JY-A6FA, JY-A6FB, JY-A6FD, JY-A6FE, JY-A6FG, JY-A6FH, JY-A938, JY-A939, JY-A93C, JY-A93D, JY-A93E, JY-A93F, KH-A6WC, L5-A43C, L5-A43E, L5-A43H, L5-A43I, L5-A43J, L5-A43M, L5-A4OE, L5-A4OF, L5-A4OG, L5-A4OH, L5-A4OI, L5-A4OJ, L5-A4OM, L5-A4ON, L5-A4OO, L5-A4OP, L5-A4OQ, L5-A4OR, L5-A4OS, L5-A4OT, L5-A4OU, L5-A4OW, L5-A4OX, L5-A88S, L5-A88T, L5-A88V, L5-A88W, L5-A88Y, L5-A88Z, L5-A891, L5-A893, L5-A8NE, L5-A8NF, L5-A8NG, L5-A8NH, L5-A8NI, L5-A8NJ, L5-A8NK, L5-A8NL, L5-A8NM, L5-A8NN, L5-A8NQ, L5-A8NR, L5-A8NS, L5-A8NT, L5-A8NU, L5-A8NV, L5-A8NW, L7-A56G, L7-A6VZ, LN-A49K, LN-A49L, LN-A49M, LN-A49N, LN-A49O, LN-A49P, LN-A49R, LN-A49S, LN-A49U, LN-A49V, LN-A49W, LN-A49X, LN-A49Y, LN-A4A1, LN-A4A2, LN-A4A3, LN-A4A4, LN-A4A5, LN-A4A6, LN-A4A8, LN-A4A9, LN-A4MQ, LN-A4MR, LN-A5U5, LN-A5U6, LN-A5U7, LN-A7HV, LN-A7HW, LN-A7HX, LN-A7HY, LN-A7HZ, LN-A8HZ, LN-A8I0, LN-A8I1, LN-A9FO, LN-A9FP, LN-A9FQ, LN-A9FR, M9-A5M8, Q9-A6FU, Q9-A6FW, R6-A6DN, R6-A6DQ, R6-A6KZ, R6-A6L4, R6-A6L6, R6-A6XG, R6-A6XQ, R6-A6Y0, R6-A6Y2, R6-A8W5, R6-A8W8, R6-A8WC, R6-A8WG, RE-A7BO, S8-A6BV, S8-A6BW, V5-A7RB, V5-A7RC, V5-A7RE, V5-AASV, V5-AASW, V5-AASX, VR-A8EO, VR-A8EP, VR-A8EQ, VR-A8ER, VR-A8ET, VR-A8EU, VR-A8EW, VR-A8EX, VR-A8EY, VR-A8EZ, VR-A8Q7, VR-AA4D, VR-AA4G, VR-AA7B, VR-AA7D, X8-AAAR, XP-A8T6, XP-A8T7, XP-A8T8, Z6-A8JD, Z6-A8JE, Z6-A9VB, Z6-AAPN, ZR-A9CJ.⧫ Gastric Cancer (401 samples):∙ Source Z3 = <cit.>:2000362, 31231321, 76629543, 970010, 98748381, 990089, 990097, 990172, 990300, 990396, 990475, 990515, TGH, TWH.∙ Source W1 = <cit.>. Sample IDs are of the form pfg*T, where * is:001, 002, 003, 005, 006, 007, 008, 009, 010, 011, 014, 015, 016, 017, 018, 019, 020, 021, 022, 024, 025, 029.∙ Source T10 = TCGA (see Acknowledgments). Sample IDs are of the form TCGA-*, where * is:B7-5816, B7-5818, B7-A5TI, B7-A5TJ, B7-A5TK, B7-A5TN, BR-4183, BR-4184, BR-4186, BR-4187, BR-4188, BR-4190, BR-4191, BR-4194, BR-4195, BR-4197, BR-4199, BR-4200, BR-4201, BR-4205, BR-4253, BR-4255, BR-4256, BR-4257, BR-4259, BR-4261, BR-4263, BR-4264, BR-4265, BR-4267, BR-4271, BR-4273, BR-4276, BR-4277, BR-4278, BR-4279, BR-4280, BR-4281, BR-4283, BR-4284, BR-4286, BR-4288, BR-4291, BR-4292, BR-4294, BR-4298, BR-4357, BR-4361, BR-4362, BR-4363, BR-4366, BR-4368, BR-4369, BR-4370, BR-4371, BR-4375, BR-4376, BR-6452, BR-6453, BR-6454, BR-6455, BR-6456, BR-6457, BR-6458, BR-6563, BR-6564, BR-6565, BR-6566, BR-6705, BR-6706, BR-6707, BR-6709, BR-6801, BR-6802, BR-6803, BR-6852, BR-7196, BR-7197, BR-7703, BR-7704, BR-7707, BR-7715, BR-7716, BR-7717, BR-7722, BR-7723, BR-7851, BR-7901, BR-7957, BR-7958, BR-7959, BR-8058, BR-8059, BR-8060, BR-8077, BR-8078, BR-8080, BR-8081, BR-8284, BR-8285, BR-8286, BR-8289, BR-8291, BR-8295, BR-8296, BR-8297, BR-8360, BR-8361, BR-8363, BR-8364, BR-8365, BR-8366, BR-8367, BR-8368, BR-8369, BR-8370, BR-8371, BR-8372, BR-8373, BR-8380, BR-8381, BR-8382, BR-8384, BR-8483, BR-8484, BR-8485, BR-8486, BR-8487, BR-8588, BR-8589, BR-8590, BR-8591, BR-8592, BR-8676, BR-8677, BR-8678, BR-8679, BR-8680, BR-8682, BR-8683, BR-8686, BR-8687, BR-8690, BR-A44T, BR-A44U, BR-A452, BR-A453, BR-A4CQ, BR-A4CR, BR-A4CS, BR-A4IU, BR-A4IV, BR-A4IY, BR-A4IZ, BR-A4J1, BR-A4J2, BR-A4J4, BR-A4J5, BR-A4J6, BR-A4J7, BR-A4J8, BR-A4PD, BR-A4PE, BR-A4PF, BR-A4QI, BR-A4QL, BR-A4QM, CD-5798, CD-5799, CD-5800, CD-5801, CD-5802, CD-5803, CD-5804, CD-5813, CD-8524, CD-8525, CD-8526, CD-8527, CD-8528, CD-8529, CD-8530, CD-8531, CD-8532, CD-8533, CD-8534, CD-8535, CD-8536, CD-A486, CD-A487, CD-A489, CD-A48A, CD-A48C, CD-A4MG, CD-A4MH, CD-A4MI, CD-A4MJ, CG-4300, CG-4301, CG-4304, CG-4305, CG-4306, CG-4436, CG-4437, CG-4438, CG-4440, CG-4441, CG-4442, CG-4443, CG-4444, CG-4449, CG-4455, CG-4460, CG-4462, CG-4465, CG-4466, CG-4469, CG-4474, CG-4475, CG-4476, CG-4477, CG-5716, CG-5717, CG-5718, CG-5719, CG-5720, CG-5721, CG-5722, CG-5723, CG-5724, CG-5725, CG-5726, CG-5727, CG-5728, CG-5730, CG-5732, CG-5733, CG-5734, D7-5577, D7-5578, D7-5579, D7-6518, D7-6519, D7-6520, D7-6521, D7-6522, D7-6524, D7-6525, D7-6526, D7-6527, D7-6528, D7-6815, D7-6817, D7-6818, D7-6820, D7-6822, D7-8570, D7-8572, D7-8573, D7-8574, D7-8575, D7-8576, D7-8578, D7-8579, D7-A4YT, D7-A4YU, D7-A4YV, D7-A4YX, D7-A4YY, D7-A4Z0, D7-A6ET, D7-A6EV, D7-A6EX, D7-A6EY, D7-A6EZ, D7-A6F0, D7-A6F2, D7-A747, D7-A748, D7-A74A, D7-A74B, EQ-5647, EQ-8122, EQ-A4SO, F1-6177, F1-6874, F1-6875, F1-A448, F1-A72C, FP-7735, FP-7829, FP-7916, FP-7998, FP-8099, FP-8209, FP-8210, FP-8211, FP-8631, FP-A4BE, FP-A4BF, HF-7131, HF-7132, HF-7133, HF-7134, HF-7136, HF-A5NB, HJ-7597, HU-8238, HU-8243, HU-8244, HU-8245, HU-8249, HU-8602, HU-8604, HU-8608, HU-8610, HU-A4G2, HU-A4G3, HU-A4G6, HU-A4G8, HU-A4G9, HU-A4GC, HU-A4GD, HU-A4GF, HU-A4GH, HU-A4GJ, HU-A4GN, HU-A4GP, HU-A4GQ, HU-A4GT, HU-A4GU, HU-A4GX, HU-A4GY, HU-A4H0, HU-A4H2, HU-A4H3, HU-A4H4, HU-A4H5, HU-A4H6, HU-A4H8, HU-A4HB, HU-A4HD, IN-7806, IN-7808, IN-8462, IN-8663, IN-A6RI, IN-A6RJ, IN-A6RL, IN-A6RN, IN-A6RO, IN-A6RP, IN-A6RR, IP-7968, KB-A6F5, KB-A6F7, MX-A5UG, MX-A5UJ, MX-A663, MX-A666, R5-A7O7, RD-A7BS, RD-A7BT, RD-A7BW, RD-A7C1.⧫ Glioblastoma Multiforme (359 samples):∙ Source P2 = <cit.>. Sample IDs are of the form Br*, where * is:001X, 018X, 019X, 02X, 03X, 04X, 05X, 06X, 07X, 08X, 102X, 103X, 104X, 10P, 112X, 116X, 117X, 118X, 11P, 128X, 12P, 132X, 133X, 136X, 13X, 143X, 148X, 14X, 15X, 16X, 17X, 20P, 21PT, 229T, 230T, 237T, 238T, 23X, 247T, 248T, 25X, 26X, 27P, 29P, 301T, 302T, 303T, 306T, 401X, 9PT.∙ Source T11 = TCGA (see Acknowledgments). Sample IDs are of the form TCGA-*, where * is:02-0003, 02-0007, 02-0010, 02-0014, 02-0015, 02-0021, 02-0028, 02-0033, 02-0043, 02-0047, 02-0055, 02-0083, 02-0089, 02-0099, 02-0107, 02-0114, 02-0115, 02-2470, 02-2483, 02-2485, 02-2486, 06-0119, 06-0122, 06-0124, 06-0125, 06-0126, 06-0128, 06-0129, 06-0130, 06-0132, 06-0137, 06-0138, 06-0139, 06-0140, 06-0141, 06-0142, 06-0143, 06-0145, 06-0147, 06-0148, 06-0151, 06-0152, 06-0154, 06-0155, 06-0157, 06-0158, 06-0165, 06-0166, 06-0167, 06-0168, 06-0169, 06-0171, 06-0173, 06-0174, 06-0176, 06-0178, 06-0184, 06-0185, 06-0188, 06-0189, 06-0190, 06-0192, 06-0195, 06-0201, 06-0209, 06-0210, 06-0211, 06-0213, 06-0214, 06-0216, 06-0219, 06-0221, 06-0237, 06-0238, 06-0240, 06-0241, 06-0644, 06-0645, 06-0646, 06-0648, 06-0649, 06-0650, 06-0686, 06-0743, 06-0744, 06-0745, 06-0747, 06-0749, 06-0750, 06-0875, 06-0876, 06-0877, 06-0878, 06-0879, 06-0881, 06-0882, 06-0939, 06-1804, 06-1806, 06-2557, 06-2558, 06-2559, 06-2561, 06-2562, 06-2563, 06-2564, 06-2565, 06-2567, 06-2569, 06-2570, 06-5408, 06-5410, 06-5411, 06-5412, 06-5413, 06-5414, 06-5415, 06-5417, 06-5418, 06-5856, 06-5858, 06-5859, 06-6388, 06-6389, 06-6390, 06-6391, 06-6693, 06-6694, 06-6695, 06-6697, 06-6698, 06-6699, 06-6700, 06-6701, 08-0244, 08-0345, 08-0352, 08-0353, 08-0360, 08-0373, 08-0375, 08-0385, 08-0386, 12-0615, 12-0616, 12-0618, 12-0619, 12-0688, 12-0692, 12-0821, 12-1597, 12-3649, 12-3650, 12-3652, 12-3653, 12-5295, 12-5299, 12-5301, 14-0740, 14-0781, 14-0786, 14-0787, 14-0789, 14-0790, 14-0813, 14-0817, 14-0862, 14-0871, 14-1034, 14-1043, 14-1395, 14-1450, 14-1456, 14-1823, 14-1825, 14-1829, 14-2554, 14-3476, 14-4157, 15-0742, 15-1444, 16-0846, 16-0861, 16-1045, 16-1048, 19-1390, 19-1790, 19-2619, 19-2620, 19-2623, 19-2624, 19-2625, 19-2629, 19-2631, 19-4068, 19-5953, 26-1439, 26-1442, 26-5132, 26-5133, 26-5134, 26-5135, 26-5136, 26-5139, 26-6173, 26-6174, 27-1830, 27-1831, 27-1832, 27-1833, 27-1834, 27-1835, 27-1836, 27-1837, 27-1838, 27-2518, 27-2519, 27-2521, 27-2523, 27-2524, 27-2526, 27-2527, 27-2528, 28-1747, 28-1753, 28-2499, 28-2501, 28-2502, 28-2509, 28-2510, 28-2513, 28-2514, 28-5204, 28-5207, 28-5208, 28-5209, 28-5211, 28-5213, 28-5214, 28-5215, 28-5216, 28-5218, 28-5219, 28-5220, 28-6450, 32-1970, 32-1977, 32-1979, 32-1980, 32-1982, 32-1986, 32-1991, 32-2491, 32-2494, 32-2495, 32-2498, 32-2615, 32-2632, 32-2634, 32-2638, 32-4208, 32-4209, 32-4210, 32-4211, 32-4213, 32-4719, 32-5222, 41-2571, 41-2572, 41-2573, 41-2575, 41-3392, 41-3393, 41-3915, 41-4097, 41-5651, 41-6646, 74-6573, 74-6575, 74-6577, 74-6578, 74-6584, 76-4925, 76-4926, 76-4927, 76-4928, 76-4929, 76-4931, 76-4932, 76-4934, 76-4935, 76-6191, 76-6192, 76-6193, 76-6280, 76-6282, 76-6283, 76-6285, 76-6286, 76-6656, 76-6657, 76-6660, 76-6661, 76-6662, 76-6663, 76-6664, 81-5910, 81-5911, 87-5896.⧫ Head and Neck Cancer (591 samples):∙ Source A1 = <cit.>:139, 266, 325, 347, 388, 478, 91.∙ Source S4 = <cit.>:HN12PT, HN22PT, HN27PT, HN32PT, HN33PT.Remaining sample IDs are of the form HN_*-Tumor, where * is:0-046, 0-064, 00076, 00122, 00190, 00313, 00338, 00361, 00378, 00443, 00466, 00761, 01000, 62237, 62298, 62318, 62338, 62374, 62415, 62417, 62421, 62426, 62469, 62481, 62493, 62505, 62506, 62515, 62532, 62539, 62601, 62602, 62624, 62646, 62652, 62671, 62672, 62686, 62699, 62739, 62740, 62741, 62755, 62756, 62807, 62814, 62825, 62832, 62854, 62857_2, 62860, 62861, 62863, 62897, 62921, 62926, 62984, 62996, 63007, 63021, 63027, 63039, 63048, 63058, 63080, 63081, 63095, 63114.∙ Source T12 = TCGA (see Acknowledgments). Sample IDs are of the form TCGA-*, where * is:BA-4074, BA-4075, BA-4076, BA-4077, BA-4078, BA-5149, BA-5151, BA-5152, BA-5153, BA-5555, BA-5556, BA-5557, BA-5558, BA-5559, BA-6868, BA-6869, BA-6870, BA-6871, BA-6872, BA-6873, BA-7269, BA-A4IF, BA-A4IG, BA-A4IH, BA-A4II, BA-A6D8, BA-A6DA, BA-A6DB, BA-A6DD, BA-A6DE, BA-A6DF, BA-A6DG, BA-A6DI, BA-A6DJ, BA-A6DL, BB-4217, BB-4223, BB-4224, BB-4225, BB-4227, BB-4228, BB-7861, BB-7862, BB-7863, BB-7864, BB-7866, BB-7870, BB-7871, BB-7872, BB-8596, BB-8601, BB-A5HU, BB-A5HY, BB-A5HZ, BB-A6UM, BB-A6UO, C9-A47Z, C9-A480, CN-4723, CN-4725, CN-4726, CN-4727, CN-4728, CN-4729, CN-4730, CN-4731, CN-4733, CN-4734, CN-4735, CN-4736, CN-4737, CN-4738, CN-4739, CN-4740, CN-4741, CN-4742, CN-5355, CN-5356, CN-5358, CN-5359, CN-5360, CN-5361, CN-5363, CN-5364, CN-5365, CN-5366, CN-5367, CN-5369, CN-5370, CN-5373, CN-5374, CN-6010, CN-6011, CN-6012, CN-6013, CN-6016, CN-6017, CN-6018, CN-6019, CN-6020, CN-6021, CN-6022, CN-6023, CN-6024, CN-6988, CN-6989, CN-6992, CN-6994, CN-6995, CN-6996, CN-6997, CN-6998, CN-A497, CN-A498, CN-A499, CN-A49A, CN-A49B, CN-A49C, CN-A63T, CN-A63U, CN-A63V, CN-A63W, CN-A63Y, CN-A640, CN-A641, CN-A642, CN-A6UY, CN-A6V1, CN-A6V3, CN-A6V6, CN-A6V7, CQ-5323, CQ-5324, CQ-5325, CQ-5326, CQ-5327, CQ-5329, CQ-5330, CQ-5331, CQ-5332, CQ-5333, CQ-5334, CQ-6218, CQ-6219, CQ-6220, CQ-6221, CQ-6222, CQ-6223, CQ-6224, CQ-6225, CQ-6227, CQ-6228, CQ-6229, CQ-7063, CQ-7064, CQ-7065, CQ-7067, CQ-7068, CQ-7069, CQ-7071, CQ-7072, CQ-A4C6, CQ-A4C7, CQ-A4C9, CQ-A4CB, CQ-A4CD, CQ-A4CE, CQ-A4CG, CQ-A4CH, CQ-A4CI, CR-5243, CR-5247, CR-5248, CR-5249, CR-5250, CR-6467, CR-6470, CR-6471, CR-6472, CR-6473, CR-6474, CR-6477, CR-6478, CR-6480, CR-6481, CR-6482, CR-6484, CR-6487, CR-6488, CR-6491, CR-6492, CR-6493, CR-7364, CR-7365, CR-7367, CR-7368, CR-7369, CR-7370, CR-7371, CR-7372, CR-7373, CR-7374, CR-7376, CR-7377, CR-7379, CR-7380, CR-7382, CR-7383, CR-7385, CR-7386, CR-7388, CR-7389, CR-7390, CR-7391, CR-7392, CR-7393, CR-7394, CR-7395, CR-7397, CR-7398, CR-7399, CR-7401, CR-7402, CR-7404, CV-5430, CV-5431, CV-5432, CV-5434, CV-5435, CV-5436, CV-5439, CV-5440, CV-5441, CV-5442, CV-5443, CV-5444, CV-5966, CV-5970, CV-5971, CV-5973, CV-5976, CV-5977, CV-5978, CV-5979, CV-6003, CV-6433, CV-6436, CV-6441, CV-6933, CV-6934, CV-6935, CV-6936, CV-6937, CV-6938, CV-6939, CV-6940, CV-6941, CV-6942, CV-6943, CV-6945, CV-6948, CV-6950, CV-6951, CV-6952, CV-6953, CV-6954, CV-6955, CV-6956, CV-6959, CV-6960, CV-6961, CV-6962, CV-7089, CV-7090, CV-7091, CV-7095, CV-7097, CV-7099, CV-7100, CV-7101, CV-7102, CV-7103, CV-7104, CV-7177, CV-7178, CV-7180, CV-7183, CV-7235, CV-7236, CV-7238, CV-7242, CV-7243, CV-7245, CV-7247, CV-7248, CV-7250, CV-7252, CV-7253, CV-7254, CV-7255, CV-7261, CV-7263, CV-7406, CV-7407, CV-7409, CV-7410, CV-7411, CV-7413, CV-7414, CV-7415, CV-7416, CV-7418, CV-7421, CV-7422, CV-7423, CV-7424, CV-7425, CV-7427, CV-7429, CV-7430, CV-7432, CV-7433, CV-7434, CV-7435, CV-7437, CV-7438, CV-7440, CV-7446, CV-7568, CV-A45O, CV-A45P, CV-A45Q, CV-A45R, CV-A45T, CV-A45U, CV-A45V, CV-A45W, CV-A45X, CV-A45Y, CV-A45Z, CV-A460, CV-A461, CV-A463, CV-A464, CV-A465, CV-A468, CV-A6JD, CV-A6JE, CV-A6JM, CV-A6JN, CV-A6JO, CV-A6JT, CV-A6JU, CV-A6JY, CV-A6JZ, CV-A6K0, CV-A6K1, CV-A6K2, CX-7082, CX-7085, CX-7086, CX-7219, CX-A4AQ, D6-6515, D6-6516, D6-6517, D6-6823, D6-6824, D6-6825, D6-6826, D6-6827, D6-8568, D6-8569, D6-A4Z9, D6-A4ZB, D6-A6EK, D6-A6EM, D6-A6EN, D6-A6EO, D6-A6EP, D6-A6EQ, D6-A6ES, D6-A74Q, DQ-5624, DQ-5625, DQ-5629, DQ-5630, DQ-5631, DQ-7588, DQ-7589, DQ-7590, DQ-7591, DQ-7592, DQ-7593, DQ-7594, DQ-7595, DQ-7596, F7-7848, F7-8489, F7-A50G, F7-A50I, F7-A50J, F7-A61S, F7-A61V, F7-A61W, F7-A620, F7-A622, F7-A623, F7-A624, H7-7774, H7-8501, H7-A6C4, H7-A6C5, H7-A76A, HD-7229, HD-7753, HD-7754, HD-7831, HD-7832, HD-7917, HD-8224, HD-8314, HD-8634, HD-8635, HD-A4C1, HD-A633, HD-A634, HD-A6HZ, HD-A6I0, HL-7533, IQ-7630, IQ-7631, IQ-7632, IQ-A61E, IQ-A61G, IQ-A61H, IQ-A61I, IQ-A61J, IQ-A61K, IQ-A61L, IQ-A61O, IQ-A6SG, IQ-A6SH, KU-A66S, KU-A66T, KU-A6H7, KU-A6H8, MT-A51W, MT-A51X, MT-A67A, MT-A67D, MT-A67F, MT-A67G, MT-A7BN, MZ-A5BI, MZ-A6I9, MZ-A7D7, P3-A5Q6, P3-A5QA, P3-A5QE, P3-A5QF, P3-A6SW, P3-A6SX, P3-A6T0, P3-A6T2, P3-A6T3, P3-A6T4, P3-A6T5, P3-A6T6, P3-A6T7, P3-A6T8, QK-A64Z, QK-A652, QK-A6IF, QK-A6IG, QK-A6IH, QK-A6II, QK-A6IJ, QK-A6V9, QK-A6VB, QK-A6VC, RS-A6TO, RS-A6TP, T2-A6WX, T2-A6WZ, T2-A6X0, T2-A6X2, TN-A7HI, TN-A7HJ, TN-A7HL, UF-A718, UF-A719, UF-A71A, UF-A71B, UF-A71D, UF-A71E, UF-A7J9, UF-A7JA, UF-A7JC, UF-A7JD, UF-A7JF, UF-A7JH, UF-A7JJ, UF-A7JK, UF-A7JO, UF-A7JS, UF-A7JT, UF-A7JV, UP-A6WW, WA-A7GZ, WA-A7H4.⧫ Liver Cancer (452 samples):∙ Source S5 = <cit.>. Sample IDs are of the form BCB*, where * is:109T, 111T, 151T, 157T, 167T, 231T, 301T, 307T, 325T.Additional sample IDs are of the form BCM*, where * is:229T, 257T, 265T, 269T, 275T, 321T, 325T, 329T, 337T, 339T, 371T, 375T, 397T, 399T, 423T, 439T, 455T, 483T, 489T, 501T, 529T, 531T, 543T, 545T, 565T, 567T, 617T, 643T, 671T, 683T, 689T, 695T, 703T, 711T, 723T, 735T, 739T, 759T, 769T, 783T, 791T.Remaining sample IDs are of the form CHC*, where * is:051T, 059T, 060T, 097T, 1010T, 1028T, 1035T, 1040T, 1041T, 1044T, 1052T, 1053T, 1055T, 1060T, 1061T, 1062T, 1065T, 1079T, 1081T, 1082T, 1083T, 1085T, 1089T, 1091T, 1097T, 1098T, 1137T, 1148T, 1152T, 1154T, 1162T, 1177T, 1180T, 1182T, 1183T, 1185T, 1186T, 1190T, 1191T, 1192T, 1201T, 1205T, 1207T, 1209T, 1210T, 1211T, 121T, 1530T, 1531T, 1534T, 1539T, 1545T, 1556T, 155T, 1566T, 1568T, 1569T, 1591T, 1592T, 1594T, 1595T, 1596T, 1597T, 1598T, 1600T, 1601T, 1602T, 1603T, 1604T, 1611T, 1616T, 1624T, 1626T, 1629T, 1700T, 1704T, 1708T, 1712T, 1714T, 1715T, 1717T, 1719T, 1720T, 1725T, 1731T, 1732T, 1734T, 1736T, 1737T, 1738T, 1739T, 1741T, 1742T, 1743T, 1744T, 1745T, 1746T, 1747T, 1749T, 1750T, 1751T, 1753T, 1754T, 1756T, 1757T, 1763T, 1774T, 1775T, 1915T, 197T, 2029T, 2034T, 2039Tbis, 2043T, 2048T, 2052T, 205T, 2098T, 2099T, 2103T, 2110Tbis, 2111T, 2112T, 2113T, 2115T, 2127T, 2128T, 2134T, 2141T, 218T, 2200T, 2202T, 2206T, 2208T, 2211T, 2213T, 2215T, 2216T, 2321T, 2351T, 2352T, 2358T, 2362T, 253T, 258T, 301T, 302T, 303T, 304T, 306T, 307T, 313T, 314T, 320T, 322T, 326T, 327T, 361TA, 429T, 432T, 433T, 434T, 437T, 451T, 465T, 469T, 510T, 609T, 614T, 703T, 734T, 736T, 789T, 793T, 794T, 796T, 798T, 799T, 801T, 805T, 879T, 884T, 889T, 891T, 892T, 896T, 898T, 902T, 909T, 912T, 917T, 923T, 961T.∙ Source H2 = <cit.>:P47, P48, P51, P52, P55, P56, P929.∙ Source T13 = TCGA (see Acknowledgments). Sample IDs are of the form TCGA-*, where * is:BC-4073, BC-A10Q, BC-A10R, BC-A10S, BC-A10T, BC-A10U, BC-A10W, BC-A10X, BC-A10Y, BC-A10Z, BC-A110, BC-A112, BC-A216, BC-A217, BC-A3KF, BC-A3KG, BC-A5W4, BC-A69H, BC-A69I, BD-A2L6, BD-A3EP, BD-A3ER, BW-A5NO, BW-A5NP, BW-A5NQ, CC-5258, CC-5259, CC-5260, CC-5261, CC-5262, CC-5263, CC-5264, CC-A123, CC-A1HT, CC-A3M9, CC-A3MA, CC-A3MB, CC-A3MC, CC-A5UC, CC-A5UD, CC-A5UE, CC-A7IF, CC-A7IG, CC-A7IH, CC-A7II, CC-A7IJ, CC-A7IK, CC-A7IL, DD-A113, DD-A114, DD-A115, DD-A116, DD-A118, DD-A119, DD-A11A, DD-A11B, DD-A11C, DD-A11D, DD-A1E9, DD-A1EA, DD-A1EB, DD-A1EC, DD-A1ED, DD-A1EF, DD-A1EG, DD-A1EH, DD-A1EI, DD-A1EJ, DD-A1EK, DD-A1EL, DD-A39V, DD-A39W, DD-A39X, DD-A39Y, DD-A39Z, DD-A3A0, DD-A3A1, DD-A3A2, DD-A3A3, DD-A3A4, DD-A3A5, DD-A3A6, DD-A3A7, DD-A3A8, DD-A3A9, DD-A4NA, DD-A4NB, DD-A4ND, DD-A4NE, DD-A4NF, DD-A4NG, DD-A4NH, DD-A4NI, DD-A4NJ, DD-A4NK, DD-A4NL, DD-A4NN, DD-A4NO, DD-A4NP, DD-A4NQ, DD-A4NR, DD-A4NS, DD-A4NV, DD-A73A, DD-A73B, DD-A73C, DD-A73D, DD-A73E, DD-A73F, DD-A73G, ED-A459, ED-A4XI, ED-A5KG, ED-A627, ED-A66X, ED-A66Y, ED-A7PX, ED-A7PY, ED-A7PZ, ED-A7XO, ED-A7XP, ED-A82E, EP-A12J, EP-A26S, EP-A2KA, EP-A2KB, EP-A2KC, EP-A3JL, EP-A3RK, ES-A2HS, ES-A2HT, FV-A23B, FV-A2QQ, FV-A2QR, FV-A3I0, FV-A3I1, FV-A3R2, FV-A3R3, FV-A495, FV-A496, FV-A4ZP, FV-A4ZQ, G3-A25S, G3-A25T, G3-A25U, G3-A25V, G3-A25W, G3-A25Y, G3-A25Z, G3-A3CG, G3-A3CH, G3-A3CI, G3-A3CJ, G3-A3CK, G3-A5SI, G3-A5SJ, G3-A5SK, G3-A5SL, G3-A5SM, G3-A6UC, G3-A7M5, G3-A7M6, G3-A7M7, G3-A7M8, G3-A7M9, GJ-A6C0, HP-A5MZ, HP-A5N0, K7-A5RF, K7-A5RG, K7-A6G5, KR-A7K0, KR-A7K2, KR-A7K7, KR-A7K8, LG-A6GG, MI-A75C, MI-A75E, MI-A75G, MI-A75H, MI-A75I, MR-A520, NI-A4U2, O8-A75V, PD-A5DF, QA-A7B7, RC-A6M3, RC-A6M4, RC-A6M5, RC-A6M6, RC-A7S9, RC-A7SB, RC-A7SF, RC-A7SK, RG-A7D4, T1-A6J8, UB-A7MA, UB-A7MB, UB-A7MC, UB-A7MD, UB-A7ME, UB-A7MF.⧫ Lung Cancer (1018 samples):∙ Source D3 = <cit.>:16600, 16608, 16628, 16632, 16648, 16660, 16668, 16678, 16686, 16724, 16802, 16814, 16835, 16857, 16949, 17042, 17055, 17156, 17174, 17210, 17218, 17226, 17242, 17268, 17290, 17308, 17733, 17746, 17759, 17763.∙ Source R1 = <cit.>:113368, 134398, 134413, 134417, 134421, 134426, 134427, 134430, 2334187, 2334188, 2334189, 2334191, 2334193, 2334195, 2334196, 2334199, 2334201, 2334202, 585203, 585205, 585208, 585210, 585223, 585258, 585260, 585265, 585267, 585270, 585272, 585276, 631052, 631056, 631060, 631064, 631076, 631084, 631092, 98687, 98711, 98735.∙ Source P3 = <cit.>:H1672, H2171, S00022, S00050, S00356, S00472, S00501, S00539, S00827, S00830, S00833, S00836, S00837, S00841, S00932, S00933, S00935, S00936, S00943, S00944, S00945, S00946, S00947, S01366, S01453, S01494, S01512, S01563, S01728.∙ Source S6 = <cit.>. Sample IDs are of the form LC_*, where * is:C1, C10, C11, C13, C14, C15, C17, C18, C19, C2, C20, C21, C22, C23, C24, C25, C26, C27, C28, C29, C30, C32, C33, C34, C35, C36, C4, C5, C6, C7, C8, C9, S10, S11, S12, S13, S14, S15, S16, S17, S18, S19, S2, S20, S21, S23, S24, S25, S27, S28, S29, S3, S31, S32, S34, S35, S37, S38, S39, S4, S40, S41, S42, S43, S44, S45, S46, S47, S48, S49, S5, S51, S6, S8, S9.∙ Source I1 = <cit.>. Sample IDs are of the form LUAD.**.Tumor, where ** is (below * stands for NYU, e.g., *1021 = NYU1021 and the full sample ID is LUAD.NYU1021.Tumor):5O6B5, 74TBW, B00416, B00523, B00859, B00915, B01102, B01145, B01811, B01970, B02077, B02216, B02477, B02515, B02594, D00147, D01278, D01603, D01751, D02085, D02185, E00163, E00443, E00897, E00918, F00018, F00057, F00089, F00121, F00134, F00162, F00170, F00257, F00282, F00365, F00368, GU4I3, LC15C, LIP77, *1021, *1026, *1027, *1051S, *1093, *1096, *1101, *1142, *1177, *1195, *1210, *1219, *160, *184, *195, *201, *213, *252, *259, *263, *282, *284, *287, *315, *330, *408, *508, *574S, *575, *584S, *608, *627, *669, *689, *696, *704, *739, *796, *802, *803, *846, *847, *848, *947, *994, QCHM7, QJN9L, S00484, S00486, S00499, S01304, S01306, S01315, S01320, S01354, S01357, S01362, S01373, S01409, S01413, S01482, TLLGS, UF7HM, VUMN6, YINHD, YKER9.Additional sample IDs are of the form LUAD.CHTN.*.Tumor, where * is:3090346, 3090415, 3090416, 4090680, MAD04.00674, MAD06.00490, MAD06.00668, MAD06.00678, MAD08.00104, Z4716A.Further sample IDs are of the form LUAD.RT.*.Tumor, where * is:S01477, S01487, S01699, S01700, S01702, S01703, S01709, S01711, S01721, S01769, S01770, S01771, S01774, S01777, S01808, S01810, S01813, S01818, S01831, S01832, S01840, S01852, S01856, S01866.Remaining sample IDs are of the form LUAD_*.Tumor, where * is:E00522, E00565, E00623, E00703, E00945, E01047, E01086, E01147, E01166, E01319, E01419.∙ Source T14 = TCGA (see Acknowledgments). Sample IDs are of the form TCGA.*, where * is:05.4244, 05.4249, 05.4250, 05.4382, 05.4384, 05.4389, 05.4390, 05.4395, 05.4396, 05.4397, 05.4398, 05.4402, 05.4403, 05.4405, 05.4410, 05.4415, 05.4417, 05.4418, 05.4420, 05.4422, 05.4424, 05.4425, 05.4426, 05.4427, 05.4430, 05.4432, 05.4433, 05.4434, 05.5420, 05.5423, 05.5425, 05.5428, 05.5429, 05.5715, 17.Z000, 17.Z001, 17.Z002, 17.Z003, 17.Z004, 17.Z005, 17.Z007, 17.Z008, 17.Z009, 17.Z010, 17.Z011, 17.Z012, 17.Z013, 17.Z014, 17.Z015, 17.Z016, 17.Z017, 17.Z018, 17.Z019, 17.Z020, 17.Z021, 17.Z022, 17.Z023, 17.Z025, 17.Z026, 17.Z027, 17.Z028, 17.Z030, 17.Z031, 17.Z032, 17.Z033, 17.Z035, 17.Z036, 17.Z037, 17.Z040, 17.Z041, 17.Z042, 17.Z043, 17.Z044, 17.Z045, 17.Z046, 17.Z047, 17.Z048, 17.Z049, 17.Z050, 17.Z051, 17.Z052, 17.Z053, 17.Z054, 17.Z055, 17.Z056, 17.Z057, 17.Z058, 17.Z059, 17.Z060, 17.Z061, 17.Z062, 18.3406, 18.3407, 18.3408, 18.3409, 18.3410, 18.3411, 18.3412, 18.3414, 18.3415, 18.3416, 18.3417, 18.3419, 18.3421, 18.4083, 18.4086, 18.4721, 18.5592, 18.5595, 21.1070, 21.1071, 21.1076, 21.1077, 21.1078, 21.1081, 21.5782, 21.5784, 21.5786, 21.5787, 22.0944, 22.1002, 22.1011, 22.1012, 22.1016, 22.4591, 22.4593, 22.4595, 22.4599, 22.4601, 22.4604, 22.4607, 22.4613, 22.5471, 22.5472, 22.5473, 22.5474, 22.5477, 22.5478, 22.5480, 22.5482, 22.5485, 22.5489, 22.5491, 22.5492, 33.4532, 33.4533, 33.4538, 33.4547, 33.4566, 33.4582, 33.4583, 33.4586, 33.6737, 34.2596, 34.2600, 34.2608, 34.5231, 34.5232, 34.5234, 34.5236, 34.5239, 34.5240, 34.5927, 34.5928, 34.5929, 35.3615, 35.3621, 35.4122, 35.4123, 35.5375, 37.3783, 37.3789, 37.4133, 37.4135, 37.4141, 37.5819, 38.4625, 38.4626, 38.4627, 38.4628, 38.4629, 38.4630, 38.4631, 38.4632, 38.6178, 38.7271, 38.A44F, 39.5016, 39.5019, 39.5021, 39.5022, 39.5024, 39.5027, 39.5028, 39.5029, 39.5030, 39.5031, 39.5035, 39.5036, 39.5037, 39.5039, 43.2578, 43.3394, 43.3920, 43.5668, 43.6143, 43.6647, 43.6770, 43.6771, 44.2655, 44.2656, 44.2657, 44.2659, 44.2661, 44.2662, 44.2665, 44.2666, 44.2668, 44.3396, 44.3398, 44.3918, 44.3919, 44.4112, 44.5643, 44.5644, 44.5645, 44.6144, 44.6145, 44.6146, 44.6147, 44.6148, 44.6774, 44.6775, 44.6776, 44.6777, 44.6778, 44.6779, 44.7659, 44.7660, 44.7661, 44.7662, 44.7667, 44.7669, 44.7670, 44.7671, 44.7672, 44.8117, 44.8119, 44.8120, 44.A479, 44.A47A, 44.A47B, 44.A47F, 44.A47G, 44.A4SS, 44.A4SU, 46.3765, 46.3767, 46.3768, 46.3769, 46.6025, 46.6026, 49.4486, 49.4487, 49.4488, 49.4490, 49.4494, 49.4501, 49.4505, 49.4506, 49.4507, 49.4510, 49.4512, 49.4514, 49.6742, 49.6743, 49.6744, 49.6745, 49.6761, 49.6767, 50.5044, 50.5045, 50.5049, 50.5051, 50.5055, 50.5066, 50.5068, 50.5072, 50.5930, 50.5931, 50.5932, 50.5933, 50.5935, 50.5936, 50.5939, 50.5941, 50.5942, 50.5944, 50.5946, 50.6590, 50.6591, 50.6592, 50.6593, 50.6594, 50.6595, 50.6597, 50.6673, 50.7109, 50.8457, 50.8459, 50.8460, 51.4079, 51.4080, 51.4081, 53.7624, 53.7626, 53.7813, 53.A4EZ, 55.1592, 55.1594, 55.1595, 55.1596, 55.5899, 55.6543, 55.6642, 55.6712, 55.6968, 55.6969, 55.6970, 55.6971, 55.6972, 55.6975, 55.6978, 55.6979, 55.6980, 55.6981, 55.6982, 55.6983, 55.6984, 55.6985, 55.6986, 55.6987, 55.7227, 55.7281, 55.7283, 55.7284, 55.7570, 55.7573, 55.7574, 55.7576, 55.7724, 55.7725, 55.7726, 55.7727, 55.7728, 55.7815, 55.7816, 55.7903, 55.7907, 55.7910, 55.7911, 55.7913, 55.7914, 55.7994, 55.7995, 55.8085, 55.8087, 55.8089, 55.8090, 55.8091, 55.8092, 55.8094, 55.8096, 55.8097, 55.8203, 55.8204, 55.8205, 55.8206, 55.8207, 55.8208, 55.8299, 55.8301, 55.8302, 55.8505, 55.8506, 55.8507, 55.8508, 55.8510, 55.8511, 55.8512, 55.8513, 55.8514, 55.8614, 55.8615, 55.8616, 55.8619, 55.8620, 55.8621, 55.A48X, 55.A48Y, 55.A48Z, 55.A490, 55.A491, 55.A492, 55.A493, 55.A494, 55.A4DF, 55.A4DG, 56.1622, 56.5897, 56.5898, 56.6545, 56.6546, 60.2698, 60.2707, 60.2708, 60.2709, 60.2710, 60.2711, 60.2712, 60.2713, 60.2715, 60.2719, 60.2720, 60.2721, 60.2722, 60.2723, 60.2724, 60.2725, 60.2726, 62.8394, 62.8395, 62.8397, 62.8398, 62.8399, 62.8402, 62.A46O, 62.A46P, 62.A46R, 62.A46S, 62.A46U, 62.A46V, 62.A46Y, 62.A470, 62.A471, 62.A472, 63.5128, 63.5131, 63.6202, 64.1676, 64.1677, 64.1678, 64.1679, 64.1680, 64.1681, 64.5774, 64.5775, 64.5778, 64.5779, 64.5781, 64.5815, 66.2727, 66.2734, 66.2742, 66.2744, 66.2754, 66.2755, 67.3770, 67.3771, 67.3772, 67.3773, 67.3774, 67.4679, 67.6215, 67.6216, 67.6217, 69.7760, 69.7761, 69.7763, 69.7764, 69.7765, 69.7973, 69.7974, 69.7978, 69.7979, 69.7980, 69.8253, 69.8254, 69.8255, 69.A59K, 71.6725, 71.8520, 73.4658, 73.4659, 73.4662, 73.4666, 73.4668, 73.4670, 73.4675, 73.4676, 73.4677, 73.7498, 73.7499, 75.5122, 75.5125, 75.5126, 75.5146, 75.5147, 75.6203, 75.6205, 75.6206, 75.6207, 75.6211, 75.6212, 75.6214, 75.7025, 75.7027, 75.7030, 75.7031, 78.7143, 78.7145, 78.7146, 78.7147, 78.7148, 78.7149, 78.7150, 78.7152, 78.7153, 78.7154, 78.7155, 78.7156, 78.7158, 78.7159, 78.7160, 78.7161, 78.7162, 78.7163, 78.7166, 78.7167, 78.7220, 78.7535, 78.7536, 78.7537, 78.7539, 78.7540, 78.7542, 78.7633, 78.8640, 78.8648, 78.8655, 78.8660, 78.8662, 80.5607, 80.5608, 80.5611, 83.5908, 86.6562, 86.6851, 86.7701, 86.7711, 86.7713, 86.7714, 86.7953, 86.7954, 86.7955, 86.8054, 86.8055, 86.8056, 86.8073, 86.8074, 86.8075, 86.8076, 86.8278, 86.8279, 86.8280, 86.8281, 86.8358, 86.8359, 86.8585, 86.8668, 86.8669, 86.8671, 86.8672, 86.8673, 86.8674, 86.A456, 86.A4D0, 86.A4JF, 86.A4P7, 86.A4P8, 91.6828, 91.6829, 91.6830, 91.6831, 91.6835, 91.6836, 91.6840, 91.6847, 91.6848, 91.6849, 91.7771, 91.8496, 91.8497, 91.8499, 91.A4BC, 91.A4BD, 93.7347, 93.7348, 93.8067, 93.A4JN, 93.A4JO, 93.A4JP, 93.A4JQ, 95.7039, 95.7043, 95.7562, 95.7567, 95.7944, 95.7947, 95.7948, 95.8039, 95.8494, 95.A4VK, 95.A4VN, 95.A4VP, 97.7546, 97.7547, 97.7552, 97.7553, 97.7554, 97.7937, 97.7938, 97.7941, 97.8171, 97.8172, 97.8174, 97.8175, 97.8176, 97.8177, 97.8179, 97.8547, 97.8552, 97.A4LX, 97.A4M0, 97.A4M1, 97.A4M2, 97.A4M3, 97.A4M5, 97.A4M6, 97.A4M7, 99.7458, 99.8025, 99.8028, 99.8032, 99.8033, J2.8192, J2.8194, J2.A4AD, J2.A4AE, J2.A4AG, L4.A4E5, L4.A4E6, L9.A443, L9.A444, MN.A4N1, MN.A4N4, MN.A4N5, MP.A4SV, MP.A4SW, MP.A4SY, MP.A4T2, MP.A4T4, MP.A4T6, MP.A4T7, MP.A4T8, MP.A4T9, MP.A4TA, MP.A4TC, MP.A4TD, MP.A4TE, MP.A4TF, MP.A4TH, MP.A4TI, MP.A4TK, MP.A5C7, NJ.A4YF, NJ.A4YG, NJ.A4YI, NJ.A4YP, NJ.A4YQ, NJ.A55A, NJ.A55O, NJ.A55R, O1.A52J.⧫ Melanoma (594 samples):∙ Source S7 = <cit.>:A02, A06, D05, D14, D35, D36, D41, D49.∙ Source D4 = <cit.>:COLO-829.∙ Source B1 = <cit.>. Sample IDs are of the form ME*-Tumor, where * is:001, 002, 009, 010, 011, 012, 014, 015, 016, 017, 018, 020, 021, 024, 029, 030, 032, 033, 034, 035, 037, 041, 043, 044, 045, 048, 049, 050.Remaining sample IDs are:Mel-BRAFi-03-Tumor, Mel_BRAFi_02_PRE-Tumor.∙ Source A2 = <cit.>. Sample IDs are of the form PD*, where * is:10020a, 10021a, 10022a, 9024a2, 9024b, 9025a, 9025b, 9026a, 9027a, 9027b, 9028a, 9028b, 9029a, 9030a, 9031a, 9032a, 9033a.∙ Source H3 = <cit.>. Sample IDs are of the form SKCM-*-Tumor, where * is:13447, 13456, 13463, 13468, 13473, 13531, 13537, 13543, 13549, 13560, 13561, 13567, 13575, 13591, 13600.Additional sample IDs are of the form SKCM-JWCI-*-Tumor, where * is:14, 27, WGS-1, WGS-11, WGS-12, WGS-13, WGS-15, WGS-18, WGS-19, WGS-2, WGS-20, WGS-21, WGS-22, WGS-23, WGS-24, WGS-25, WGS-26, WGS-29, WGS-3, WGS-32, WGS-33, WGS-34, WGS-35, WGS-36, WGS-37, WGS-38, WGS-39, WGS-4, WGS-42, WGS-43, WGS-5, WGS-6, WGS-7, WGS-8.Further sample IDs are of the form SKCM-Ma-Mel-*-Tumor, where * is:04, 05, 08a, 102, 103b, 105, 107, 108, 114, 119, 120, 122, 123, 15, 16, 19, 27, 28, 35, 36, 37, 48, 53, 54a, 55, 59, 62, 63, 65, 67, 71, 76, 79, 85, 86, 91, 92, 94.Remaining sample IDs are:SKCM-UKRV-Mel-20-Tumor, SKCM-UKRV-Mel-24-Tumor, SKCM-UKRV-Mel-6-Tumor.∙ Source T15 = TCGA (see Acknowledgments). Sample IDs are of the form TCGA-*, where * is:BF-A1PU, BF-A1PV, BF-A1PX, BF-A1PZ, BF-A1Q0, BF-A3DJ, BF-A3DL, BF-A3DM, BF-A3DN, D3-A1Q1, D3-A1Q3, D3-A1Q4, D3-A1Q5, D3-A1Q6, D3-A1Q7, D3-A1Q8, D3-A1Q9, D3-A1QA, D3-A1QB, D3-A2J6, D3-A2J7, D3-A2J8, D3-A2J9, D3-A2JA, D3-A2JB, D3-A2JC, D3-A2JD, D3-A2JF, D3-A2JG, D3-A2JH, D3-A2JK, D3-A2JL, D3-A2JN, D3-A2JO, D3-A2JP, D3-A3BZ, D3-A3C1, D3-A3C3, D3-A3C6, D3-A3C7, D3-A3C8, D3-A3CB, D3-A3CC, D3-A3CE, D3-A3CF, D3-A3ML, D3-A3MO, D3-A3MR, D3-A3MU, D3-A3MV, D9-A148, D9-A149, D9-A1JW, D9-A1JX, D9-A1X3, DA-A1HV, DA-A1HW, DA-A1HY, DA-A1I0, DA-A1I1, DA-A1I2, DA-A1I4, DA-A1I5, DA-A1I7, DA-A1I8, DA-A1IA, DA-A1IB, DA-A1IC, DA-A3F3, DA-A3F5, DA-A3F8, EB-A1NK, EB-A24C, EB-A24D, EB-A299, EB-A3HV, EE-A17X, EE-A17Y, EE-A17Z, EE-A180, EE-A181, EE-A182, EE-A183, EE-A184, EE-A185, EE-A20B, EE-A20C, EE-A20F, EE-A20H, EE-A20I, EE-A29A, EE-A29B, EE-A29C, EE-A29D, EE-A29E, EE-A29G, EE-A29H, EE-A29L, EE-A29M, EE-A29N, EE-A29P, EE-A29Q, EE-A29R, EE-A29S, EE-A29T, EE-A29V, EE-A29W, EE-A29X, EE-A2A0, EE-A2A1, EE-A2A2, EE-A2A5, EE-A2A6, EE-A2GB, EE-A2GC, EE-A2GD, EE-A2GE, EE-A2GH, EE-A2GI, EE-A2GJ, EE-A2GK, EE-A2GL, EE-A2GM, EE-A2GN, EE-A2GO, EE-A2GP, EE-A2GR, EE-A2GS, EE-A2GT, EE-A2GU, EE-A2M5, EE-A2M6, EE-A2M7, EE-A2M8, EE-A2MC, EE-A2MD, EE-A2ME, EE-A2MF, EE-A2MG, EE-A2MH, EE-A2MI, EE-A2MJ, EE-A2MK, EE-A2ML, EE-A2MM, EE-A2MN, EE-A2MP, EE-A2MQ, EE-A2MR, EE-A2MS, EE-A2MT, EE-A2MU, EE-A3AA, EE-A3AB, EE-A3AC, EE-A3AD, EE-A3AE, EE-A3AF, EE-A3AG, EE-A3AH, EE-A3J3, EE-A3J4, EE-A3J5, EE-A3J7, EE-A3J8, EE-A3JA, EE-A3JB, EE-A3JD, EE-A3JE, EE-A3JH, EE-A3JI, ER-A193, ER-A194, ER-A195, ER-A196, ER-A197, ER-A198, ER-A199, ER-A19A, ER-A19B, ER-A19C, ER-A19D, ER-A19E, ER-A19F, ER-A19G, ER-A19H, ER-A19J, ER-A19K, ER-A19L, ER-A19N, ER-A19O, ER-A19P, ER-A19Q, ER-A19S, ER-A19T, ER-A1A1, ER-A2NB, ER-A2NC, ER-A2ND, ER-A2NE, ER-A2NF, ER-A2NG, ER-A2NH, ER-A3ES, ER-A3ET, ER-A3EV, FR-A2OS, FS-A1YX, FS-A1YY, FS-A1Z0, FS-A1Z3, FS-A1Z4, FS-A1Z7, FS-A1ZB, FS-A1ZC, FS-A1ZD, FS-A1ZE, FS-A1ZF, FS-A1ZG, FS-A1ZH, FS-A1ZJ, FS-A1ZK, FS-A1ZM, FS-A1ZN, FS-A1ZP, FS-A1ZQ, FS-A1ZR, FS-A1ZS, FS-A1ZT, FS-A1ZU, FS-A1ZW, FS-A1ZY, FS-A1ZZ, FW-A3I3, GF-A2C7, GN-A262, GN-A263, GN-A264, GN-A265, GN-A266, GN-A267, GN-A268, GN-A269, GN-A26A, GN-A26C, GN-A26D, HR-A2OG, HR-A2OH, IH-A3EA, D3-A5GT, D9-A3Z4, D9-A4Z2, D9-A4Z3, D9-A4Z5, EB-A3XB, EB-A3XC, EB-A3XD, EB-A3XE, EB-A3Y6, EB-A3Y7, EB-A41A, EB-A41B, EB-A42Y, EB-A42Z, EB-A430, EB-A431, EB-A44N, EB-A44O, EB-A44P, EB-A4IQ, EB-A4IS, EB-A4OY, EB-A4OZ, EB-A4P0, EB-A551, EB-A553, EB-A57M, EB-A5SE, EB-A5SF, EB-A5UM, FR-A3R1, FW-A5DX, BF-A5EO, BF-A5EP, BF-A5EQ, BF-A5ER, BF-A5ES, D3-A51E, D3-A51F, D3-A51G, D3-A51H, D3-A51J, D3-A51K, D3-A51N, D3-A51R, D3-A51T, D3-A5GL, D3-A5GN, D3-A5GO, D3-A5GR, D3-A5GS, D9-A3Z1, D9-A3Z3, D9-A6E9, D9-A6EA, D9-A6EC, D9-A6EG, DA-A3F2, EB-A3XF, EB-A44Q, EB-A44R, EB-A4XL, EB-A5FP, EB-A5KH, EB-A5SG, EB-A5SH, EB-A5UL, EB-A5UN, EB-A5VU, EB-A5VV, EB-A6L9, EB-A6QY, EB-A6QZ, EB-A6R0, ER-A19M, ER-A19W, ER-A3PL, ER-A42H, ER-A42K, ER-A42L, FR-A3YN, FR-A3YO, FR-A44A, FR-A69P, FR-A726, FR-A728, FS-A1YW, FS-A1ZA, FS-A4F4, FS-A4F5, FS-A4F8, FS-A4F9, FS-A4FB, FS-A4FC, FS-A4FD, FW-A3R5, FW-A3TU, FW-A3TV, FW-A5DY, GF-A3OT, GF-A6C8, GF-A6C9, GF-A769, GN-A4U3, GN-A4U4, GN-A4U5, GN-A4U7, GN-A4U8, GN-A4U9, OD-A75X, QB-A6FS, RP-A690, RP-A693, RP-A694, RP-A695, D3-A5GU, FS-A4F0, GF-A4EO, RZ-AB0B, V3-A9ZX, V3-A9ZY, V4-A9E5, V4-A9E7, V4-A9E8, V4-A9E9, V4-A9EA, V4-A9EC, V4-A9ED, V4-A9EE, V4-A9EF, V4-A9EH, V4-A9EI, V4-A9EJ, V4-A9EK, V4-A9EL, V4-A9EM, V4-A9EO, V4-A9EQ, V4-A9ES, V4-A9ET, V4-A9EU, V4-A9EV, V4-A9EW, V4-A9EX, V4-A9EY, V4-A9EZ, V4-A9F0, V4-A9F1, V4-A9F2, V4-A9F3, V4-A9F4, V4-A9F5, V4-A9F7, V4-A9F8, VD-A8K7, VD-A8K8, VD-A8K9, VD-A8KA, VD-A8KB, VD-A8KD, VD-A8KE, VD-A8KF, VD-A8KG, VD-A8KH, VD-A8KI, VD-A8KJ, VD-A8KK, VD-A8KL, VD-A8KM, VD-A8KN, VD-A8KO, VD-AA8M, VD-AA8N, VD-AA8O, VD-AA8P, VD-AA8Q, VD-AA8R, VD-AA8S, VD-AA8T, WC-A87T, WC-A87U, WC-A87W, WC-A87Y, WC-A880, WC-A881, WC-A882, WC-A883, WC-A884, WC-A885, WC-A888, WC-A88A, WC-AA9A, WC-AA9E, YZ-A980, YZ-A982, YZ-A983, YZ-A984, YZ-A985.⧫ Nasopharyngeal Cancer (11 samples):∙ Source L2 = <cit.>:NPC088D, NPC105D, NPC29F, NPC31F, NPC34F, NPC3F, NPC42F, NPC4D, NPC4F, NPC5D, NPC5F.⧫ Oral Cancer (106 samples):∙ Source I2 = <cit.>. Sample IDs are of the form OSCC-GB_0*, where * is:001011, 002011, 003011, 004011, 005011, 006011, 007011, 008011, 011011, 012011, 013011, 014011, 015011, 016011, 017011, 018011, 019011, 020011, 021011, 022011, 023011, 024011, 025011, 026011, 027011, 028011, 029011, 030011, 031011, 032011, 033011, 034011, 035011, 036011, 037011, 038011, 039011, 040011, 041011, 042011, 043011, 044011, 045011, 046011, 047011, 048011, 049011, 050011, 051011, 052011, 053011, 054011, 055011, 056011, 057011, 058011, 059011, 060011, 061011, 062011, 063011, 064011, 065011, 066011, 067011, 068011, 069011, 070011, 073011, 074011, 075011, 076011, 077011, 080011, 081011, 082011, 083011, 084011, 085011, 086011, 087011, 088011, 089011, 090011, 091011, 092011, 093011, 094011, 095011, 096011, 097011, 098011, 099011, 100011, 101011, 102011, 103011, 104011, 105011, 106011, 107011, 108011, 109011, 110011, 111011, 112011.⧫ Ovarian Cancer (471 samples):∙ Source J1 = <cit.>:OCC01PT, OCC02PT, OCC03PT, OCC04PT, OCC05PT, OCC06PT, OCC07PT, OCC08PT.∙ Source T16 = TCGA (see Acknowledgments). Sample IDs are of the form TCGA-*, where * is:04-1331, 04-1332, 04-1336, 04-1337, 04-1338, 04-1342, 04-1343, 04-1346, 04-1347, 04-1348, 04-1349, 04-1350, 04-1351, 04-1353, 04-1356, 04-1357, 04-1361, 04-1362, 04-1364, 04-1365, 04-1367, 04-1369, 04-1514, 04-1516, 04-1517, 04-1519, 04-1525, 04-1530, 04-1542, 04-1638, 04-1644, 04-1646, 04-1648, 04-1649, 04-1651, 04-1652, 04-1655, 09-0364, 09-0365, 09-0366, 09-0367, 09-0369, 09-1659, 09-1661, 09-1662, 09-1664, 09-1665, 09-1666, 09-1669, 09-1670, 09-1672, 09-1673, 09-1674, 09-1675, 09-2044, 09-2045, 09-2049, 09-2050, 09-2051, 09-2053, 09-2056, 10-0926, 10-0927, 10-0928, 10-0930, 10-0931, 10-0933, 10-0934, 10-0935, 10-0937, 10-0938, 13-0714, 13-0717, 13-0720, 13-0723, 13-0724, 13-0726, 13-0727, 13-0730, 13-0751, 13-0755, 13-0758, 13-0760, 13-0761, 13-0762, 13-0765, 13-0791, 13-0792, 13-0793, 13-0795, 13-0800, 13-0801, 13-0804, 13-0807, 13-0883, 13-0884, 13-0885, 13-0886, 13-0887, 13-0889, 13-0890, 13-0891, 13-0893, 13-0894, 13-0897, 13-0899, 13-0900, 13-0901, 13-0903, 13-0904, 13-0905, 13-0906, 13-0910, 13-0911, 13-0912, 13-0913, 13-0916, 13-0919, 13-0920, 13-0923, 13-0924, 13-1403, 13-1404, 13-1405, 13-1407, 13-1408, 13-1409, 13-1410, 13-1411, 13-1412, 13-1477, 13-1481, 13-1482, 13-1483, 13-1484, 13-1487, 13-1488, 13-1489, 13-1491, 13-1492, 13-1494, 13-1495, 13-1496, 13-1497, 13-1498, 13-1499, 13-1501, 13-1504, 13-1505, 13-1506, 13-1507, 13-1509, 13-1510, 13-1512, 13-2057, 13-2059, 13-2060, 13-2061, 13-2065, 13-2066, 13-2071, 20-0987, 20-0990, 20-0991, 20-1682, 20-1683, 20-1684, 20-1685, 20-1686, 20-1687, 23-1021, 23-1022, 23-1023, 23-1024, 23-1026, 23-1027, 23-1028, 23-1029, 23-1030, 23-1031, 23-1032, 23-1109, 23-1110, 23-1111, 23-1114, 23-1116, 23-1117, 23-1118, 23-1119, 23-1120, 23-1122, 23-1123, 23-1124, 23-1809, 23-2072, 23-2077, 23-2078, 23-2079, 23-2081, 23-2641, 23-2643, 23-2645, 23-2647, 23-2649, 24-0966, 24-0968, 24-0970, 24-0975, 24-0979, 24-0980, 24-0982, 24-1103, 24-1104, 24-1105, 24-1413, 24-1416, 24-1417, 24-1418, 24-1419, 24-1422, 24-1423, 24-1424, 24-1425, 24-1426, 24-1427, 24-1428, 24-1431, 24-1434, 24-1435, 24-1436, 24-1463, 24-1464, 24-1466, 24-1469, 24-1470, 24-1471, 24-1474, 24-1544, 24-1545, 24-1546, 24-1548, 24-1549, 24-1551, 24-1552, 24-1553, 24-1555, 24-1556, 24-1557, 24-1558, 24-1560, 24-1562, 24-1563, 24-1564, 24-1565, 24-1567, 24-1603, 24-1604, 24-1614, 24-1616, 24-1842, 24-1843, 24-1844, 24-1845, 24-1846, 24-1847, 24-1849, 24-1850, 24-2019, 24-2024, 24-2030, 24-2035, 24-2038, 24-2254, 24-2260, 24-2261, 24-2262, 24-2267, 24-2271, 24-2280, 24-2281, 24-2288, 24-2289, 24-2290, 24-2293, 24-2298, 25-1313, 25-1315, 25-1316, 25-1317, 25-1318, 25-1319, 25-1320, 25-1321, 25-1322, 25-1324, 25-1325, 25-1326, 25-1328, 25-1329, 25-1623, 25-1625, 25-1626, 25-1627, 25-1628, 25-1630, 25-1631, 25-1632, 25-1633, 25-1634, 25-1635, 25-2042, 25-2391, 25-2392, 25-2393, 25-2396, 25-2398, 25-2399, 25-2400, 25-2401, 25-2404, 25-2408, 25-2409, 29-1688, 29-1690, 29-1691, 29-1693, 29-1694, 29-1695, 29-1696, 29-1697, 29-1698, 29-1699, 29-1701, 29-1702, 29-1703, 29-1705, 29-1707, 29-1710, 29-1711, 29-1761, 29-1762, 29-1763, 29-1764, 29-1766, 29-1768, 29-1769, 29-1770, 29-1771, 29-1774, 29-1775, 29-1776, 29-1777, 29-1778, 29-1781, 29-1783, 29-1784, 29-1785, 29-2427, 29-2429, 29-2431, 29-2432, 29-2434, 29-2436, 30-1714, 30-1718, 30-1853, 30-1855, 30-1856, 30-1857, 31-1950, 36-1568, 36-1569, 36-1570, 36-1571, 36-1574, 36-1575, 36-1576, 36-1577, 36-1578, 36-1580, 36-2530, 36-2532, 36-2533, 36-2534, 36-2537, 36-2538, 36-2539, 36-2540, 36-2542, 36-2543, 36-2544, 36-2545, 36-2547, 36-2548, 36-2551, 36-2552, 42-2582, 42-2587, 42-2588, 42-2589, 42-2590, 42-2591, 57-1582, 57-1584, 57-1586, 57-1993, 59-2348, 59-2350, 59-2351, 59-2352, 59-2354, 59-2355, 59-2363, 59-2372, 61-1722, 61-1725, 61-1727, 61-1728, 61-1730, 61-1733, 61-1736, 61-1737, 61-1738, 61-1740, 61-1741, 61-1895, 61-1899, 61-1900, 61-1901, 61-1903, 61-1904, 61-1906, 61-1907, 61-1910, 61-1911, 61-1913, 61-1914, 61-1915, 61-1995, 61-1998, 61-2000, 61-2002, 61-2003, 61-2008, 61-2009, 61-2012, 61-2016, 61-2092, 61-2094, 61-2095, 61-2097, 61-2101, 61-2102, 61-2104, 61-2109, 61-2110, 61-2111, 61-2113, 61-2610, 61-2611, 61-2612, 61-2613, 61-2614.⧫ Pancreatic Cancer (184 samples):∙ Source W2 = <cit.>:IPMN 11, IPMN 12, IPMN 20, IPMN 21, IPMN 36, IPMN 4, IPMN 41, MCN 162, MCN 163, MCN 164, MCN 166, MCN 168, MCN 169, MCN 170, SCA 14, SCA 23, SCA 27, SCA 35, SCA 37, SCA 38, SCA 40, SPN 8.∙ Source J2 = <cit.>. Sample IDs are of the form PanNET*, where * is:10PT, 21PT, 23PT, 24PT, 25PT, 31PT, 36PT, 3PT, 7PT, 93PT.∙ Source T17 = TCGA (see Acknowledgments). Sample IDs are of the form TCGA-*, where * is:2L-AAQA, 2L-AAQE, 2L-AAQI, 2L-AAQJ, 2L-AAQL, 2L-AAQM, 3A-A9I5, 3A-A9I7, 3A-A9I9, 3A-A9IB, 3A-A9IC, 3A-A9IH, 3A-A9IJ, 3A-A9IL, 3A-A9IN, 3A-A9IO, 3A-A9IR, 3A-A9IS, 3A-A9IU, 3E-AAAY, 3E-AAAZ, F2-6879, F2-6880, F2-7273, F2-7276, F2-A44G, F2-A44H, F2-A7TX, F2-A8YN, FB-A4P5, FB-A4P6, FB-A545, FB-A5VM, FB-A78T, FB-A7DR, FB-AAPS, FQ-6551, FQ-6552, FQ-6553, FQ-6554, FQ-6555, FQ-6558, FQ-6559, FZ-5919, FZ-5920, FZ-5921, FZ-5922, FZ-5923, FZ-5924, FZ-5926, H6-8124, H6-A45N, H8-A6C1, HV-A5A3, HV-A5A4, HV-A5A5, HV-A5A6, HV-A7OL, HV-A7OP, HV-AA8X, HZ-7289, HZ-7918, HZ-7919, HZ-7920, HZ-7922, HZ-7923, HZ-7924, HZ-7925, HZ-7926, HZ-8001, HZ-8002, HZ-8003, HZ-8005, HZ-8315, HZ-8317, HZ-8636, HZ-8637, HZ-8638, HZ-A49G, HZ-A49H, HZ-A49I, HZ-A4BH, HZ-A4BK, HZ-A77O, HZ-A77P, HZ-A77Q, HZ-A8P0, HZ-A8P1, IB-7644, IB-7645, IB-7646, IB-7647, IB-7649, IB-7651, IB-7652, IB-7654, IB-7885, IB-7886, IB-7887, IB-7888, IB-7889, IB-7890, IB-7891, IB-7893, IB-7897, IB-8126, IB-8127, IB-A5SO, IB-A5SP, IB-A5SQ, IB-A5SS, IB-A5ST, IB-A6UF, IB-A6UG, IB-A7LX, IB-A7M4, IB-AAUM, IB-AAUN, IB-AAUO, IB-AAUP, IB-AAUR, IB-AAUS, IB-AAUT, IB-AAUU, IB-AAUV, IB-AAUW, LB-A7SX, LB-A8F3, LB-A9Q5, M8-A5N4, OE-A75W, PZ-A5RE, Q3-A5QY, Q3-AA2A, RB-A7B8, RB-AA9M, RL-AAAS, S4-A8RM, S4-A8RO, S4-A8RP, US-A774, US-A776, US-A779, US-A77E, US-A77G, US-A77J, XD-AAUL, XN-A8T3, XN-A8T5, YB-A89D, YH-A8SY, YY-A8LH.⧫ Pheochromocytoma and Paraganglioma (178 samples):∙ Source T18 = TCGA (see Acknowledgments). Sample IDs are of the form TCGA-*, where * is:P7-A5NX, P7-A5NY, P8-A5KC, P8-A5KD, P8-A6RX, P8-A6RY, PR-A5PF, PR-A5PG, PR-A5PH, QR-A6GO, QR-A6GR, QR-A6GS, QR-A6GT, QR-A6GU, QR-A6GW, QR-A6GX, QR-A6GY, QR-A6GZ, QR-A6H0, QR-A6H1, QR-A6H2, QR-A6H3, QR-A6H4, QR-A6H5, QR-A6H6, QR-A6ZZ, QR-A702, QR-A703, QR-A705, QR-A706, QR-A707, QR-A708, QR-A70A, QR-A70C, QR-A70D, QR-A70E, QR-A70G, QR-A70H, QR-A70I, QR-A70J, QR-A70K, QR-A70M, QR-A70N, QR-A70O, QR-A70P, QR-A70Q, QR-A70R, QR-A70T, QR-A70U, QR-A70V, QR-A70W, QR-A70X, QR-A7IN, QR-A7IP, QT-A5XJ, QT-A5XK, QT-A5XL, QT-A5XM, QT-A5XN, QT-A5XO, QT-A5XP, QT-A69Q, QT-A7U0, RM-A68T, RM-A68W, RT-A6Y9, RT-A6YA, RT-A6YC, RW-A67V, RW-A67W, RW-A67X, RW-A67Y, RW-A680, RW-A681, RW-A684, RW-A685, RW-A686, RW-A688, RW-A689, RW-A68A, RW-A68B, RW-A68C, RW-A68D, RW-A68F, RW-A68G, RW-A7CZ, RW-A7D0, RW-A8AZ, RX-A8JQ, S7-A7WL, S7-A7WM, S7-A7WN, S7-A7WO, S7-A7WP, S7-A7WQ, S7-A7WR, S7-A7WT, S7-A7WU, S7-A7WV, S7-A7WW, S7-A7WX, S7-A7X0, S7-A7X1, S7-A7X2, SA-A6C2, SP-A6QC, SP-A6QD, SP-A6QF, SP-A6QG, SP-A6QH, SP-A6QI, SP-A6QJ, SP-A6QK, SQ-A6I4, SQ-A6I6, SR-A6MP, SR-A6MQ, SR-A6MR, SR-A6MS, SR-A6MT, SR-A6MU, SR-A6MV, SR-A6MX, SR-A6MY, SR-A6MZ, SR-A6N0, TT-A6YJ, TT-A6YK, TT-A6YN, TT-A6YO, TT-A6YP, W2-A7H5, W2-A7H7, W2-A7HA, W2-A7HB, W2-A7HC, W2-A7HD, W2-A7HE, W2-A7HF, W2-A7HH, W2-A7UY, WB-A80K, WB-A80L, WB-A80M, WB-A80N, WB-A80O, WB-A80P, WB-A80Q, WB-A80V, WB-A80Y, WB-A814, WB-A815, WB-A816, WB-A817, WB-A818, WB-A819, WB-A81A, WB-A81D, WB-A81E, WB-A81F, WB-A81G, WB-A81H, WB-A81I, WB-A81J, WB-A81K, WB-A81M, WB-A81N, WB-A81P, WB-A81Q, WB-A81R, WB-A81S, WB-A81T, WB-A81V, WB-A81W, WB-A820, WB-A821, WB-A822, XG-A823.⧫ Prostate Cancer (480 samples):∙ Source B2 = <cit.>. Sample IDs are of the form P0*-Tumor, where * is:0-000450, 1-28, 2-1562, 2-2035, 3-1334, 3-1426, 3-1906, 3-2345, 3-2620, 3-3391, 3-595, 3-871, 4-1084, 4-1243, 4-1421, 4-1790, 4-2599, 4-2641, 4-2666, 4-2740, 4-47, 4-594, 5-2212, 5-2594, 5-3436, 5-3829, 5-3852, 5-3859, 5-620, 6-1125, 6-1696, 6-2325, 6-3676, 6-3939, 6-4428, 7-144, 7-360, 7-5036, 7-684, 7-718, 7-837, 8-2516, 8-590, 9-120, 9-1372, 9-1580, 9-2497, 9-649.∙ Source B3 = <cit.>. Sample IDs are of the form PR-*, where * is:0508, 0581, 1701, 1783, 2832, 3027, 3043.Remaining sample IDs are of the form PR-*-Tumor, where * is:00-1165, 00-160, 00-1823, 0099, 01-1934, 01-2382, 01-2492, 01-2554, 02-1082, 02-169, 02-1736, 02-1899, 02-2072, 02-2480, 02-254, 03-022, 03-1026, 03-870, 04-1367, 04-194, 04-3113, 04-3222, 04-3347, 04-639, 04-903, 0415, 0427, 05-3440, 05-3595, 05-839, 06-1651, 06-1749, 06-1999, 09-2517, 09-2744, 09-2767, 09-3421, 09-3566, 09-3687, 09-5094, 09-5245, 09-5446, 09-5630, 09-5700, 09-5702, 1024, 1043, 2661, 2682, 2740, 2761, 2762, 2858, 2872, 2915, 2916, 3023, 3026, 3034, 3035, 3036, 3048, 3051, 3127.∙ Source G2 = <cit.> (below * stands for WA, e.g., *10 = WA10):T12, T32, T8, T90, T91, T92, T93, T94, T95, T96, T97, *10, *11, *12, *13, *14, *15, *16, *17, *18, *19, *20, *22, *23, *24, *25, *26, *27, *28, *29, *3, *30, *31, *32, *33, *35, *37, *38, *39, *40, *41, *42, *43-27, *43-44, *43-71, *46, *47, *48, *49, *50, *51, *52, *53, *54, *55, *56, *57, *58, *59, *60, *7.∙ Source T19 = TCGA (see Acknowledgments). Sample IDs are of the form TCGA-*, where * is:CH-5737, CH-5738, CH-5739, CH-5740, CH-5741, CH-5743, CH-5744, CH-5745, CH-5746, CH-5748, CH-5750, CH-5751, CH-5752, CH-5753, CH-5754, CH-5761, CH-5762, CH-5763, CH-5764, CH-5765, CH-5766, CH-5767, CH-5768, CH-5769, CH-5771, CH-5772, CH-5788, CH-5789, CH-5790, CH-5791, CH-5792, CH-5794, EJ-5494, EJ-5495, EJ-5496, EJ-5497, EJ-5498, EJ-5499, EJ-5501, EJ-5502, EJ-5503, EJ-5504, EJ-5505, EJ-5506, EJ-5507, EJ-5508, EJ-5509, EJ-5510, EJ-5511, EJ-5512, EJ-5514, EJ-5515, EJ-5516, EJ-5517, EJ-5518, EJ-5519, EJ-5521, EJ-5522, EJ-5524, EJ-5525, EJ-5526, EJ-5527, EJ-5530, EJ-5531, EJ-5532, EJ-5542, EJ-7115, EJ-7123, EJ-7125, EJ-7218, EJ-7312, EJ-7314, EJ-7315, EJ-7317, EJ-7318, EJ-7321, EJ-7325, EJ-7327, EJ-7328, EJ-7330, EJ-7331, EJ-7781, EJ-7782, EJ-7783, EJ-7784, EJ-7785, EJ-7786, EJ-7788, EJ-7789, EJ-7791, EJ-7792, EJ-7793, EJ-7794, EJ-7797, EJ-8468, EJ-8469, EJ-8470, EJ-8472, EJ-8474, EJ-A46B, EJ-A46D, EJ-A46E, EJ-A46F, EJ-A46G, EJ-A46H, EJ-A46I, EJ-A65B, EJ-A65D, EJ-A65E, EJ-A65F, EJ-A65G, EJ-A65J, EJ-A65M, EJ-A6RA, EJ-A6RC, EJ-A7NF, EJ-A7NG, EJ-A7NH, EJ-A7NM, EJ-A7NN, FC-7708, FC-7961, FC-A4JI, FC-A5OB, FC-A66V, FC-A6HD, G9-6329, G9-6332, G9-6333, G9-6336, G9-6338, G9-6339, G9-6342, G9-6343, G9-6347, G9-6348, G9-6351, G9-6353, G9-6354, G9-6356, G9-6361, G9-6362, G9-6363, G9-6364, G9-6365, G9-6366, G9-6367, G9-6369, G9-6370, G9-6371, G9-6373, G9-6377, G9-6378, G9-6379, G9-6384, G9-6385, G9-6494, G9-6496, G9-6498, G9-6499, G9-7510, G9-7519, G9-7521, G9-7522, G9-7523, G9-7525, H9-7775, H9-A6BX, H9-A6BY, HC-7075, HC-7077, HC-7078, HC-7079, HC-7080, HC-7081, HC-7209, HC-7210, HC-7211, HC-7212, HC-7213, HC-7230, HC-7231, HC-7232, HC-7233, HC-7736, HC-7737, HC-7738, HC-7740, HC-7742, HC-7744, HC-7745, HC-7747, HC-7748, HC-7749, HC-7750, HC-7752, HC-7817, HC-7818, HC-7819, HC-7820, HC-7821, HC-8213, HC-8216, HC-8256, HC-8257, HC-8258, HC-8259, HC-8260, HC-8261, HC-8262, HC-8264, HC-8265, HC-8266, HC-A48F, HC-A4ZV, HC-A631, HC-A632, HC-A6AL, HC-A6AN, HC-A6AO, HC-A6AP, HC-A6AQ, HC-A6AS, HC-A6HX, HC-A6HY, HC-A76W, HC-A76X, HI-7168, HI-7169, HI-7170, HI-7171, J4-8198, J4-8200, J4-A67K, J4-A67L, J4-A67M, J4-A67N, J4-A67O, J4-A67Q, J4-A67R, J4-A67S, J4-A67T, J4-A6G1, J4-A6G3, J4-A6M7, J9-A52B, J9-A52C, J9-A52D, J9-A52E, KC-A4BL, KC-A4BN, KC-A4BO, KC-A4BR, KC-A4BV, KC-A7F3, KC-A7F5, KC-A7F6, KC-A7FA, KC-A7FD, KC-A7FE, KK-A59V, KK-A59X, KK-A59Y, KK-A59Z, KK-A5A1, KK-A6DY, KK-A6E0, KK-A6E1, KK-A6E2, KK-A6E3, KK-A6E4, KK-A6E5, KK-A6E6, KK-A6E7, KK-A6E8, KK-A7AP, KK-A7AQ, KK-A7AU, KK-A7AV, KK-A7AW, KK-A7AY, KK-A7AZ, KK-A7B0, KK-A7B1, KK-A7B2, KK-A7B3, KK-A7B4, M7-A71Y, M7-A71Z, M7-A720, M7-A721, M7-A723, M7-A724, M7-A725, QU-A6IL, QU-A6IM, QU-A6IN, QU-A6IO, QU-A6IP, SU-A7E7.⧫ Rectum Adenocarcinoma (115 samples):∙ Source T20 = TCGA (see Acknowledgments). Sample IDs are of the form TCGA-*, where * is:AF-2687, AF-2689, AF-2690, AF-2691, AF-2692, AF-2693, AF-3400, AF-3911, AF-4110, AF-5654, AF-6136, AF-6655, AF-6672, AG-3574, AG-3575, AG-3578, AG-3580, AG-3581, AG-3582, AG-3583, AG-3584, AG-3586, AG-3587, AG-3591, AG-3592, AG-3593, AG-3594, AG-3598, AG-3599, AG-3600, AG-3601, AG-3602, AG-3605, AG-3608, AG-3609, AG-3611, AG-3612, AG-3725, AG-3731, AG-3732, AG-3742, AG-4021, AG-4022, AG-A002, AG-A008, AG-A00C, AG-A00Y, AG-A011, AG-A014, AG-A015, AG-A016, AG-A01L, AH-6544, AH-6643, AH-6644, AH-6897, AH-6903, BM-6198, CI-6619, CI-6620, CI-6621, CI-6622, CI-6624, CL-4957, CL-5917, CL-5918, DC-4749, DC-5337, DC-5869, DC-6154, DC-6155, DC-6157, DC-6158, DC-6681, DC-6682, DC-6683, DT-5265, DY-A0XA, DY-A1DC, DY-A1DD, DY-A1DF, DY-A1DG, DY-A1H8, EF-5830, EI-6506, EI-6507, EI-6508, EI-6509, EI-6510, EI-6511, EI-6512, EI-6513, EI-6514, EI-6881, EI-6882, EI-6883, EI-6884, EI-6885, EI-6917, EI-7002, EI-7004, F5-6464, F5-6465, F5-6571, F5-6702, F5-6812, F5-6813, F5-6814, F5-6861, F5-6863, F5-6864, G5-6233, G5-6235, G5-6572, G5-6641.⧫ Renal Cell Carcinoma (709 samples):∙ Source G3 = <cit.>:K1, K20, K27, K29, K3, K31, K32, K38, K44, K48, T127, T142, T144, T163, T164, T166, T183M.∙ Source T21 = TCGA (see Acknowledgments). Sample IDs are of the form TCGA-*, where * is:A3-3308, A3-3311, A3-3313, A3-3316, A3-3317, A3-3319, A3-3320, A3-3322, A3-3323, A3-3324, A3-3326, A3-3331, A3-3346, A3-3347, A3-3349, A3-3351, A3-3357, A3-3358, A3-3362, A3-3363, A3-3365, A3-3367, A3-3370, A3-3372, A3-3373, A3-3374, A3-3376, A3-3378, A3-3380, A3-3382, A3-3383, A3-3385, A3-3387, A4-7286, A4-7287, A4-7288, A4-7583, A4-7584, A4-7585, A4-7732, A4-7734, A4-7828, A4-7915, A4-7996, A4-7997, A4-8098, A4-8310, A4-8311, A4-8312, A4-8515, A4-8516, A4-8517, A4-8518, A4-8630, A4-A48D, A4-A4ZT, A4-A57E, A4-A5DU, A4-A5XZ, A4-A5Y0, A4-A5Y1, A4-A6HP, AK-3425, AK-3427, AK-3428, AK-3429, AK-3430, AK-3431, AK-3434, AK-3436, AK-3440, AK-3443, AK-3444, AK-3445, AK-3447, AK-3450, AK-3451, AK-3453, AK-3454, AK-3455, AK-3456, AK-3458, AK-3460, AK-3461, AK-3465, AL-3466, AL-3467, AL-3468, AL-3472, AL-3473, AL-7173, AL-A5DJ, AS-3777, AS-3778, AT-A5NU, B0-4690, B0-4691, B0-4693, B0-4694, B0-4697, B0-4700, B0-4703, B0-4706, B0-4707, B0-4710, B0-4712, B0-4713, B0-4714, B0-4718, B0-4810, B0-4811, B0-4813, B0-4814, B0-4815, B0-4816, B0-4817, B0-4818, B0-4819, B0-4822, B0-4823, B0-4824, B0-4827, B0-4828, B0-4833, B0-4836, B0-4837, B0-4838, B0-4839, B0-4841, B0-4842, B0-4843, B0-4844, B0-4845, B0-4846, B0-4847, B0-4848, B0-4849, B0-4852, B0-4945, B0-5075, B0-5077, B0-5080, B0-5081, B0-5083, B0-5084, B0-5085, B0-5088, B0-5092, B0-5094, B0-5095, B0-5096, B0-5097, B0-5098, B0-5099, B0-5100, B0-5102, B0-5104, B0-5106, B0-5107, B0-5108, B0-5109, B0-5110, B0-5113, B0-5115, B0-5116, B0-5117, B0-5119, B0-5120, B0-5121, B0-5399, B0-5400, B0-5402, B0-5691, B0-5692, B0-5693, B0-5694, B0-5695, B0-5696, B0-5697, B0-5698, B0-5699, B0-5701, B0-5702, B0-5703, B0-5705, B0-5706, B0-5707, B0-5709, B0-5710, B0-5711, B0-5712, B0-5713, B0-5812, B1-5398, B1-A47M, B1-A47N, B1-A47O, B1-A654, B1-A655, B1-A656, B1-A657, B2-3923, B2-3924, B2-4098, B2-4099, B2-4101, B2-4102, B2-5633, B2-5635, B2-5641, B3-3925, B3-3926, B3-4103, B3-4104, B3-8121, B4-5377, B4-5832, B4-5834, B4-5835, B4-5836, B4-5838, B4-5843, B4-5844, B8-4143, B8-4146, B8-4148, B8-4151, B8-4153, B8-4154, B8-4619, B8-4620, B8-4621, B8-4622, B8-5158, B8-5159, B8-5162, B8-5163, B8-5164, B8-5165, B8-5545, B8-5546, B8-5549, B8-5550, B8-5551, B8-5552, B8-5553, B9-4113, B9-4114, B9-4115, B9-4116, B9-4117, B9-4617, B9-5155, B9-5156, B9-7268, B9-A44B, B9-A5W7, B9-A5W8, B9-A5W9, B9-A69E, BP-4158, BP-4159, BP-4160, BP-4161, BP-4162, BP-4163, BP-4164, BP-4165, BP-4166, BP-4167, BP-4169, BP-4170, BP-4173, BP-4174, BP-4176, BP-4177, BP-4326, BP-4329, BP-4330, BP-4331, BP-4337, BP-4338, BP-4340, BP-4341, BP-4342, BP-4343, BP-4345, BP-4346, BP-4347, BP-4349, BP-4351, BP-4352, BP-4354, BP-4355, BP-4756, BP-4758, BP-4759, BP-4760, BP-4761, BP-4762, BP-4763, BP-4765, BP-4766, BP-4768, BP-4770, BP-4771, BP-4774, BP-4775, BP-4777, BP-4781, BP-4782, BP-4787, BP-4789, BP-4790, BP-4795, BP-4797, BP-4798, BP-4799, BP-4801, BP-4803, BP-4804, BP-4807, BP-4960, BP-4961, BP-4962, BP-4963, BP-4964, BP-4965, BP-4967, BP-4968, BP-4969, BP-4970, BP-4971, BP-4972, BP-4973, BP-4974, BP-4975, BP-4976, BP-4977, BP-4981, BP-4982, BP-4983, BP-4985, BP-4986, BP-4987, BP-4988, BP-4989, BP-4991, BP-4992, BP-4993, BP-4994, BP-4995, BP-4998, BP-4999, BP-5000, BP-5001, BP-5004, BP-5006, BP-5007, BP-5008, BP-5009, BP-5010, BP-5168, BP-5169, BP-5170, BP-5173, BP-5174, BP-5175, BP-5176, BP-5177, BP-5178, BP-5180, BP-5181, BP-5182, BP-5183, BP-5184, BP-5185, BP-5186, BP-5187, BP-5189, BP-5190, BP-5191, BP-5192, BP-5194, BP-5195, BP-5196, BP-5198, BP-5199, BP-5200, BP-5201, BP-5202, BQ-5875, BQ-5876, BQ-5877, BQ-5878, BQ-5879, BQ-5880, BQ-5881, BQ-5882, BQ-5883, BQ-5884, BQ-5885, BQ-5886, BQ-5887, BQ-5888, BQ-5889, BQ-5890, BQ-5891, BQ-5892, BQ-5893, BQ-5894, BQ-7044, BQ-7045, BQ-7046, BQ-7048, BQ-7049, BQ-7050, BQ-7051, BQ-7053, BQ-7055, BQ-7056, BQ-7058, BQ-7059, BQ-7060, BQ-7061, BQ-7062, CJ-4634, CJ-4635, CJ-4636, CJ-4637, CJ-4638, CJ-4639, CJ-4640, CJ-4641, CJ-4643, CJ-4644, CJ-4868, CJ-4869, CJ-4870, CJ-4871, CJ-4872, CJ-4873, CJ-4874, CJ-4875, CJ-4876, CJ-4878, CJ-4881, CJ-4882, CJ-4884, CJ-4885, CJ-4886, CJ-4887, CJ-4888, CJ-4889, CJ-4890, CJ-4891, CJ-4892, CJ-4893, CJ-4894, CJ-4895, CJ-4897, CJ-4899, CJ-4900, CJ-4901, CJ-4902, CJ-4903, CJ-4904, CJ-4905, CJ-4907, CJ-4908, CJ-4912, CJ-4913, CJ-4916, CJ-4918, CJ-4920, CJ-4923, CJ-5671, CJ-5672, CJ-5675, CJ-5676, CJ-5677, CJ-5678, CJ-5679, CJ-5680, CJ-5681, CJ-5682, CJ-5683, CJ-5684, CJ-5686, CJ-6027, CJ-6028, CJ-6030, CJ-6031, CJ-6032, CJ-6033, CW-5580, CW-5581, CW-5583, CW-5584, CW-5585, CW-5588, CW-5589, CW-5591, CW-6087, CW-6090, CW-6093, CW-6097, CZ-4853, CZ-4854, CZ-4856, CZ-4857, CZ-4858, CZ-4859, CZ-4861, CZ-4862, CZ-4863, CZ-4865, CZ-4866, CZ-5451, CZ-5452, CZ-5453, CZ-5454, CZ-5455, CZ-5456, CZ-5457, CZ-5458, CZ-5459, CZ-5460, CZ-5461, CZ-5462, CZ-5463, CZ-5464, CZ-5465, CZ-5466, CZ-5467, CZ-5468, CZ-5469, CZ-5470, CZ-5982, CZ-5984, CZ-5985, CZ-5986, CZ-5987, CZ-5988, CZ-5989, DV-5565, DV-5566, DV-5568, DV-5569, DV-5574, DV-5575, DV-5576, DW-5560, DW-5561, DW-7834, DW-7837, DW-7838, DW-7839, DW-7840, DW-7841, DW-7842, DW-7963, DZ-6131, DZ-6132, DZ-6133, DZ-6134, DZ-6135, EU-5904, EU-5905, EU-5906, EU-5907, EV-5901, EV-5902, EV-5903, F9-A4JJ, G7-6789, G7-6790, G7-6792, G7-6793, G7-6795, G7-6796, G7-6797, G7-7501, G7-7502, G7-A4TM, GL-6846, GL-7773, GL-7966, GL-8500, GL-A4EM, GL-A59R, GL-A59T, HE-7128, HE-7129, HE-7130, HE-A5NF, HE-A5NH, HE-A5NI, HE-A5NJ, HE-A5NK, HE-A5NL, IA-A40U, IA-A40X, IA-A40Y, IZ-8195, IZ-8196, IZ-A6M8, IZ-A6M9, J7-6720, J7-8537, KL-8323, KL-8324, KL-8325, KL-8326, KL-8327, KL-8328, KL-8329, KL-8330, KL-8331, KL-8332, KL-8333, KL-8334, KL-8335, KL-8336, KL-8337, KL-8338, KL-8339, KL-8340, KL-8341, KL-8342, KL-8343, KL-8344, KL-8345, KL-8346, KM-8438, KM-8439, KM-8440, KM-8441, KM-8442, KM-8443, KM-8476, KM-8477, KM-8639, KN-8418, KN-8419, KN-8421, KN-8422, KN-8423, KN-8424, KN-8425, KN-8426, KN-8427, KN-8428, KN-8429, KN-8430, KN-8431, KN-8432, KN-8433, KN-8434, KN-8435, KN-8436, KN-8437, KO-8403, KO-8404, KO-8405, KO-8406, KO-8407, KO-8408, KO-8409, KO-8410, KO-8411, KO-8413, KO-8414, KO-8415, KO-8416, KO-8417, KV-A6GD, KV-A6GE, MH-A55W, MH-A55Z, MH-A560, MH-A561, MH-A562, P4-A5E6, P4-A5E7, P4-A5E8, P4-A5EA, P4-A5EB, P4-A5ED, PJ-A5Z8, PJ-A5Z9, Q2-A5QZ.⧫ Sarcoma (255 samples):∙ Source T22 = TCGA (see Acknowledgments). Sample IDs are of the form TCGA-*, where * is:3B-A9HI, 3B-A9HJ, 3B-A9HL, 3B-A9HO, 3B-A9HP, 3B-A9HQ, 3B-A9HR, 3B-A9HS, 3B-A9HT, 3B-A9HU, 3B-A9HV, 3B-A9HX, 3B-A9HY, 3B-A9HZ, 3B-A9I0, 3B-A9I1, 3B-A9I3, 3R-A8YX, DX-A1KU, DX-A1KW, DX-A1KX, DX-A1KY, DX-A1KZ, DX-A1L0, DX-A1L1, DX-A1L2, DX-A1L3, DX-A1L4, DX-A23R, DX-A23T, DX-A23U, DX-A23V, DX-A23Y, DX-A240, DX-A2IZ, DX-A2J0, DX-A2J1, DX-A2J4, DX-A3LS, DX-A3LT, DX-A3LU, DX-A3LW, DX-A3LY, DX-A3M1, DX-A3M2, DX-A3U5, DX-A3U6, DX-A3U7, DX-A3U8, DX-A3U9, DX-A3UA, DX-A3UB, DX-A3UC, DX-A3UD, DX-A3UE, DX-A3UF, DX-A48J, DX-A48K, DX-A48L, DX-A48N, DX-A48O, DX-A48P, DX-A48R, DX-A48U, DX-A48V, DX-A6B7, DX-A6B8, DX-A6B9, DX-A6BA, DX-A6BB, DX-A6BE, DX-A6BF, DX-A6BG, DX-A6BH, DX-A6BK, DX-A6YQ, DX-A6YR, DX-A6YS, DX-A6YT, DX-A6YU, DX-A6YV, DX-A6YX, DX-A6YZ, DX-A6Z0, DX-A6Z2, DX-A7EF, DX-A7EI, DX-A7EL, DX-A7EM, DX-A7EN, DX-A7EO, DX-A7EQ, DX-A7ER, DX-A7ES, DX-A7ET, DX-A7EU, DX-A8BG, DX-A8BH, DX-A8BJ, DX-A8BK, DX-A8BL, DX-A8BM, DX-A8BN, DX-A8BO, DX-A8BP, DX-A8BR, DX-A8BT, DX-A8BU, DX-A8BV, DX-A8BX, DX-A8BZ, DX-AATS, DX-AB2E, DX-AB2F, DX-AB2G, DX-AB2H, DX-AB2J, DX-AB2L, DX-AB2O, DX-AB2P, DX-AB2Q, DX-AB2S, DX-AB2T, DX-AB2V, DX-AB2W, DX-AB2X, DX-AB2Z, DX-AB30, DX-AB32, DX-AB35, DX-AB36, DX-AB37, DX-AB3A, DX-AB3B, DX-AB3C, FX-A2QS, FX-A3NJ, FX-A3NK, FX-A3RE, FX-A3TO, FX-A48G, FX-A76Y, FX-A8OO, HB-A2OT, HB-A3L4, HB-A3YV, HB-A43Z, HB-A5W3, HS-A5N7, HS-A5N8, HS-A5N9, IE-A3OV, IE-A4EH, IE-A4EI, IE-A4EJ, IE-A4EK, IE-A6BZ, IF-A4AJ, IF-A4AK, IS-A3K6, IS-A3K7, IS-A3K8, IS-A3KA, IW-A3M4, IW-A3M5, IW-A3M6, JV-A5VE, JV-A5VF, JV-A75J, K1-A3PN, K1-A3PO, K1-A42W, K1-A42X, K1-A6RT, K1-A6RU, K1-A6RV, KD-A5QS, KD-A5QT, KD-A5QU, KF-A41W, LI-A67I, LI-A9QH, MB-A5Y8, MB-A5Y9, MB-A5YA, MB-A8JK, MB-A8JL, MJ-A68H, MJ-A68J, MJ-A850, MO-A47P, MO-A47R, N1-A6IA, PC-A5DK, PC-A5DL, PC-A5DM, PC-A5DN, PC-A5DO, PC-A5DP, QC-A6FX, QC-A7B5, QC-AA9N, QQ-A5V2, QQ-A5V9, QQ-A5VA, QQ-A5VB, QQ-A5VC, QQ-A5VD, QQ-A8VB, QQ-A8VD, QQ-A8VF, QQ-A8VG, QQ-A8VH, RN-A68Q, RN-AAAQ, SG-A6Z4, SG-A6Z7, SG-A849, SI-A71O, SI-A71P, SI-A71Q, SI-AA8B, SI-AA8C, UE-A6QT, UE-A6QU, VT-A80G, VT-A80J, VT-AB3D, WK-A8XO, WK-A8XQ, WK-A8XS, WK-A8XT, WK-A8XX, WK-A8XY, WK-A8XZ, WK-A8Y0, WP-A9GB, X2-A95T, X6-A7W8, X6-A7WA, X6-A7WB, X6-A7WC, X6-A7WD, X6-A8C2, X6-A8C3, X6-A8C4, X6-A8C5, X6-A8C6, X6-A8C7, X9-A971, X9-A973, Z4-A8JB, Z4-A9VC, Z4-AAPF, Z4-AAPG.⧫ Testicular Germ Cell Tumors (150 samples):∙ Source T23 = TCGA (see Acknowledgments). Sample IDs are of the form TCGA-*, where * is:2G-AAEW, 2G-AAEX, 2G-AAF1, 2G-AAF4, 2G-AAF6, 2G-AAF8, 2G-AAFE, 2G-AAFG, 2G-AAFH, 2G-AAFI, 2G-AAFJ, 2G-AAFL, 2G-AAFM, 2G-AAFN, 2G-AAFO, 2G-AAFV, 2G-AAFY, 2G-AAFZ, 2G-AAG0, 2G-AAG3, 2G-AAG5, 2G-AAG6, 2G-AAG7, 2G-AAG8, 2G-AAG9, 2G-AAGA, 2G-AAGC, 2G-AAGE, 2G-AAGF, 2G-AAGG, 2G-AAGI, 2G-AAGJ, 2G-AAGK, 2G-AAGM, 2G-AAGN, 2G-AAGO, 2G-AAGP, 2G-AAGS, 2G-AAGT, 2G-AAGV, 2G-AAGW, 2G-AAGX, 2G-AAGY, 2G-AAGZ, 2G-AAH0, 2G-AAH2, 2G-AAH3, 2G-AAH4, 2G-AAH8, 2G-AAHA, 2G-AAHC, 2G-AAHG, 2G-AAHL, 2G-AAHN, 2G-AAHP, 2G-AAHT, 2G-AAKD, 2G-AAKG, 2G-AAKH, 2G-AAKL, 2G-AAKM, 2G-AAKO, 2G-AAL5, 2G-AAL7, 2G-AALF, 2G-AALG, 2G-AALN, 2G-AALO, 2G-AALP, 2G-AALQ, 2G-AALR, 2G-AALS, 2G-AALT, 2G-AALW, 2G-AALX, 2G-AALY, 2G-AALZ, 2G-AAM2, 2G-AAM3, 2G-AAM4, 2X-A9D5, 2X-A9D6, 4K-AA1G, 4K-AA1H, 4K-AA1I, 4K-AAAL, S6-A8JW, S6-A8JX, S6-A8JY, SB-A6J6, SB-A76C, SN-A6IS, SN-A84W, SN-A84X, SN-A84Y, SO-A8JP, VF-A8A8, VF-A8A9, VF-A8AA, VF-A8AB, VF-A8AC, VF-A8AD, VF-A8AE, W4-A7U2, W4-A7U3, W4-A7U4, WZ-A7V3, WZ-A7V4, WZ-A7V5, WZ-A8D5, X3-A8G4, XE-A8H1, XE-A8H4, XE-A8H5, XE-A9SE, XE-AANI, XE-AANJ, XE-AANR, XE-AANV, XE-AAO3, XE-AAO4, XE-AAO6, XE-AAOB, XE-AAOC, XE-AAOD, XE-AAOF, XE-AAOJ, XE-AAOL, XY-A89B, XY-A8S2, XY-A8S3, XY-A9T9, YU-A90P, YU-A90Q, YU-A90S, YU-A90W, YU-A90Y, YU-A912, YU-A94D, YU-A94I, YU-AA4L, YU-AA61, ZM-AA05, ZM-AA06, ZM-AA0B, ZM-AA0D, ZM-AA0E, ZM-AA0F, ZM-AA0H, ZM-AA0N.⧫ Thymoma (123 samples):∙ Source T24 = TCGA (see Acknowledgments). Sample IDs are of the form TCGA-*, where * is:3G-AB0O, 3G-AB0Q, 3G-AB0T, 3G-AB14, 3G-AB19, 3Q-A9WF, 3S-A8YW, 3S-AAYX, 3T-AA9L, 4V-A9QI, 4V-A9QJ, 4V-A9QL, 4V-A9QM, 4V-A9QN, 4V-A9QQ, 4V-A9QR, 4V-A9QS, 4V-A9QT, 4V-A9QU, 4V-A9QW, 4V-A9QX, 4X-A9F9, 4X-A9FA, 4X-A9FB, 4X-A9FC, 4X-A9FD, 5G-A9ZZ, 5K-AAAP, 5U-AB0D, 5U-AB0E, 5U-AB0F, 5V-A9RR, X7-A8D6, X7-A8D7, X7-A8D8, X7-A8D9, X7-A8DB, X7-A8DC, X7-A8DD, X7-A8DE, X7-A8DF, X7-A8DG, X7-A8DI, X7-A8DJ, X7-A8M0, X7-A8M1, X7-A8M3, X7-A8M4, X7-A8M5, X7-A8M6, X7-A8M7, X7-A8M8, XH-A853, XM-A8R8, XM-A8R9, XM-A8RB, XM-A8RC, XM-A8RD, XM-A8RE, XM-A8RF, XM-A8RG, XM-A8RH, XM-A8RI, XM-A8RL, XM-AAZ1, XM-AAZ2, XM-AAZ3, XU-A92O, XU-A92Q, XU-A92R, XU-A92T, XU-A92U, XU-A92V, XU-A92W, XU-A92X, XU-A92Y, XU-A92Z, XU-A930, XU-A931, XU-A932, XU-A933, XU-A936, XU-AAXW, XU-AAXX, XU-AAXY, XU-AAXZ, XU-AAY0, XU-AAY1, YT-A95D, YT-A95E, YT-A95F, YT-A95G, YT-A95H, ZB-A961, ZB-A962, ZB-A963, ZB-A964, ZB-A965, ZB-A966, ZB-A969, ZB-A96A, ZB-A96B, ZB-A96C, ZB-A96D, ZB-A96E, ZB-A96F, ZB-A96G, ZB-A96H, ZB-A96I, ZB-A96K, ZB-A96L, ZB-A96M, ZB-A96O, ZB-A96P, ZB-A96Q, ZB-A96R, ZB-A96V, ZC-AAA7, ZC-AAAA, ZC-AAAF, ZC-AAAH, ZL-A9V6, ZT-A8OM.⧫ Thyroid Cancer (409 samples):∙ Source T25 = TCGA (see Acknowledgments). Sample IDs are of the form TCGA-*, where * is:BJ-A0YZ, BJ-A0Z0, BJ-A0Z2, BJ-A0Z3, BJ-A0Z5, BJ-A0Z9, BJ-A0ZA, BJ-A0ZB, BJ-A0ZC, BJ-A0ZE, BJ-A0ZF, BJ-A0ZG, BJ-A0ZH, BJ-A0ZJ, BJ-A18Y, BJ-A18Z, BJ-A190, BJ-A191, BJ-A192, BJ-A28R, BJ-A28S, BJ-A28T, BJ-A28V, BJ-A28X, BJ-A28Z, BJ-A290, BJ-A2N7, BJ-A2N8, BJ-A2N9, BJ-A2NA, BJ-A2P4, BJ-A3EZ, BJ-A3F0, BJ-A3PR, BJ-A3PT, BJ-A3PU, BJ-A45D, BJ-A45E, BJ-A45F, BJ-A45G, BJ-A45I, BJ-A45J, BJ-A45K, BJ-A4O8, BJ-A4O9, CE-A13K, CE-A27D, CE-A3MD, CE-A3ME, CE-A482, CE-A484, CE-A485, DE-A0XZ, DE-A0Y2, DE-A0Y3, DE-A2OL, DE-A3KN, DE-A4M8, DE-A4M9, DJ-A13L, DJ-A13M, DJ-A13O, DJ-A13P, DJ-A13R, DJ-A13S, DJ-A13T, DJ-A13U, DJ-A13V, DJ-A13W, DJ-A13X, DJ-A1QD, DJ-A1QE, DJ-A1QF, DJ-A1QG, DJ-A1QH, DJ-A1QI, DJ-A1QL, DJ-A1QM, DJ-A1QN, DJ-A1QO, DJ-A1QQ, DJ-A2PN, DJ-A2PO, DJ-A2PP, DJ-A2PQ, DJ-A2PR, DJ-A2PS, DJ-A2PT, DJ-A2PU, DJ-A2PV, DJ-A2PW, DJ-A2PX, DJ-A2PY, DJ-A2PZ, DJ-A2Q0, DJ-A2Q1, DJ-A2Q2, DJ-A2Q3, DJ-A2Q4, DJ-A2Q5, DJ-A2Q6, DJ-A2Q7, DJ-A2Q8, DJ-A2Q9, DJ-A2QA, DJ-A2QB, DJ-A2QC, DJ-A3UK, DJ-A3UM, DJ-A3UN, DJ-A3UO, DJ-A3UP, DJ-A3UQ, DJ-A3UR, DJ-A3US, DJ-A3UT, DJ-A3UU, DJ-A3UV, DJ-A3UW, DJ-A3UX, DJ-A3UY, DJ-A3V7, DJ-A3VA, DJ-A3VB, DJ-A3VE, DJ-A3VF, DJ-A3VJ, DJ-A3VK, DJ-A3VL, DJ-A3VM, DJ-A4UL, DJ-A4UP, DJ-A4UT, DJ-A4UW, DJ-A4V0, DJ-A4V2, DJ-A4V4, DJ-A4V5, DO-A1JZ, DO-A1K0, DO-A2HM, E3-A3DY, E3-A3DZ, E3-A3E0, E3-A3E1, E3-A3E2, E3-A3E3, E3-A3E5, E8-A242, E8-A2EA, E8-A413, E8-A415, E8-A418, E8-A419, E8-A433, E8-A436, E8-A437, E8-A44K, E8-A44M, EL-A3CL, EL-A3CM, EL-A3CN, EL-A3CO, EL-A3CP, EL-A3CR, EL-A3CS, EL-A3CT, EL-A3CU, EL-A3CV, EL-A3CW, EL-A3CX, EL-A3CY, EL-A3CZ, EL-A3D0, EL-A3D1, EL-A3D4, EL-A3D5, EL-A3D6, EL-A3GO, EL-A3GP, EL-A3GQ, EL-A3GR, EL-A3GS, EL-A3GU, EL-A3GV, EL-A3GW, EL-A3GX, EL-A3GY, EL-A3GZ, EL-A3H1, EL-A3H2, EL-A3H3, EL-A3H4, EL-A3H5, EL-A3H7, EL-A3H8, EL-A3MW, EL-A3MX, EL-A3MY, EL-A3MZ, EL-A3N2, EL-A3N3, EL-A3T0, EL-A3T1, EL-A3T2, EL-A3T3, EL-A3T6, EL-A3T7, EL-A3T8, EL-A3T9, EL-A3TA, EL-A3TB, EL-A3ZH, EL-A3ZK, EL-A3ZN, EL-A3ZQ, EL-A3ZR, EL-A3ZT, EL-A4JV, EL-A4JW, EL-A4JX, EL-A4JZ, EL-A4K0, EL-A4K2, EL-A4K4, EL-A4K6, EL-A4KD, EL-A4KG, EL-A4KH, EL-A4KI, EM-A1CS, EM-A1CT, EM-A1CU, EM-A1CV, EM-A1CW, EM-A1YA, EM-A1YB, EM-A1YC, EM-A1YD, EM-A1YE, EM-A22I, EM-A22J, EM-A22K, EM-A22L, EM-A22M, EM-A22N, EM-A22O, EM-A22P, EM-A22Q, EM-A2CJ, EM-A2CK, EM-A2CL, EM-A2CM, EM-A2CN, EM-A2CO, EM-A2CP, EM-A2CQ, EM-A2CR, EM-A2CT, EM-A2CU, EM-A2OV, EM-A2OW, EM-A2OX, EM-A2OY, EM-A2OZ, EM-A2P0, EM-A2P1, EM-A2P2, EM-A2P3, EM-A3AI, EM-A3AJ, EM-A3AK, EM-A3AL, EM-A3AN, EM-A3AO, EM-A3AP, EM-A3AQ, EM-A3AR, EM-A3FJ, EM-A3FK, EM-A3FL, EM-A3FM, EM-A3FN, EM-A3FO, EM-A3FP, EM-A3FQ, EM-A3FR, EM-A3O3, EM-A3O6, EM-A3O7, EM-A3O8, EM-A3O9, EM-A3OA, EM-A3OB, EM-A4FK, EM-A4FM, EM-A4FO, EM-A4FQ, EM-A4FR, EM-A4FV, EM-A4G1, ET-A25G, ET-A25I, ET-A25J, ET-A25K, ET-A25O, ET-A25R, ET-A2MY, ET-A2MZ, ET-A2N0, ET-A2N1, ET-A2N4, ET-A2N5, ET-A39I, ET-A39J, ET-A39K, ET-A39L, ET-A39M, ET-A39N, ET-A39O, ET-A39P, ET-A39R, ET-A39S, ET-A39T, ET-A3BN, ET-A3BO, ET-A3BP, ET-A3BQ, ET-A3BS, ET-A3BT, ET-A3BU, ET-A3BV, ET-A3BW, ET-A3BX, ET-A3DO, ET-A3DP, ET-A3DQ, ET-A3DR, ET-A3DS, ET-A3DT, ET-A3DU, ET-A3DV, ET-A3DW, ET-A40S, ET-A4KN, FE-A22Z, FE-A230, FE-A231, FE-A232, FE-A233, FE-A234, FE-A235, FE-A236, FE-A237, FE-A238, FE-A23A, FE-A3PA, FE-A3PB, FE-A3PC, FE-A3PD, FK-A3S3, FK-A3SB, FK-A3SD, FK-A3SE, FK-A3SG, FK-A3SH, FY-A2QD, FY-A3BL, FY-A3I4, FY-A3I5, FY-A3NM, FY-A3NN, FY-A3NP, FY-A3ON, FY-A3R6, FY-A3R7, FY-A3R8, FY-A3R9, FY-A3RA, FY-A3W9, FY-A3WA, FY-A40K, FY-A4B3, GE-A2C6, H2-A26U, H2-A2K9, H2-A3RH, H2-A3RI, H2-A421, IM-A3EB, IM-A3ED, IM-A3U2, IM-A3U3, J8-A3NZ, J8-A3O0, J8-A3O1, J8-A3YE, J8-A3YH, J8-A4HW, KS-A41J, KS-A4I5, KS-A4I9, KS-A4IB, L6-A4EP, L6-A4ET, L6-A4EU, MK-A4N6, MK-A4N7, MK-A4N9.⧫ Uterine Cancer (305 samples):∙ Source T26 = TCGA (see Acknowledgments). Sample IDs are of the form TCGA-*, where * is:A5-A0G3, A5-A0G5, A5-A0G9, A5-A0GA, A5-A0GB, A5-A0GD, A5-A0GE, A5-A0GH, A5-A0GI, A5-A0GJ, A5-A0GM, A5-A0GN, A5-A0GP, A5-A0GQ, A5-A0GU, A5-A0GV, A5-A0GW, A5-A0GX, A5-A0R6, A5-A0R7, A5-A0R8, A5-A0R9, A5-A0RA, A5-A0VO, A5-A0VP, A5-A0VQ, AJ-A23M, AP-A051, AP-A052, AP-A053, AP-A054, AP-A056, AP-A059, AP-A05A, AP-A05D, AP-A05H, AP-A05J, AP-A05N, AP-A05P, AP-A0L8, AP-A0L9, AP-A0LD, AP-A0LE, AP-A0LF, AP-A0LG, AP-A0LH, AP-A0LI, AP-A0LJ, AP-A0LL, AP-A0LM, AP-A0LN, AP-A0LO, AP-A0LP, AP-A0LQ, AP-A0LT, AP-A0LV, AP-A1DQ, AX-A05S, AX-A05T, AX-A05U, AX-A05W, AX-A05Y, AX-A05Z, AX-A060, AX-A062, AX-A063, AX-A064, AX-A06B, AX-A06H, AX-A06L, AX-A0IS, AX-A0IU, AX-A0IW, AX-A0J0, AX-A0J1, AX-A1C7, AX-A1C8, AX-A1CP, AX-A2H5, AX-A2HF, B5-A0JN, B5-A0JR, B5-A0JS, B5-A0JT, B5-A0JV, B5-A0JY, B5-A0JZ, B5-A0K0, B5-A0K1, B5-A0K2, B5-A0K3, B5-A0K4, B5-A0K6, B5-A0K7, B5-A0K8, B5-A0K9, B5-A11E, B5-A11F, B5-A11G, B5-A11H, B5-A11I, B5-A11J, B5-A11M, B5-A11N, B5-A11O, B5-A11Q, B5-A11R, B5-A11S, B5-A11U, B5-A11V, B5-A11W, B5-A11X, B5-A11Y, B5-A11Z, B5-A121, B5-A1MU, B5-A1MY, BG-A0LW, BG-A0LX, BG-A0M0, BG-A0M2, BG-A0M3, BG-A0M4, BG-A0M6, BG-A0M7, BG-A0M8, BG-A0M9, BG-A0MC, BG-A0MG, BG-A0MI, BG-A0MO, BG-A0MQ, BG-A0MS, BG-A0MT, BG-A0MU, BG-A0RY, BG-A0VT, BG-A0VV, BG-A0VW, BG-A0VX, BG-A0VZ, BG-A0W1, BG-A0W2, BG-A0YU, BG-A0YV, BG-A186, BG-A187, BG-A18A, BG-A18B, BG-A18C, BG-A2AE, BK-A0C9, BK-A0CA, BK-A0CB, BK-A0CC, BK-A139, BK-A13C, BS-A0T9, BS-A0TA, BS-A0TC, BS-A0TD, BS-A0TE, BS-A0TG, BS-A0TI, BS-A0TJ, BS-A0U5, BS-A0U7, BS-A0U8, BS-A0U9, BS-A0UA, BS-A0UF, BS-A0UJ, BS-A0UL, BS-A0UM, BS-A0UT, BS-A0UV, BS-A0V6, BS-A0V7, BS-A0V8, BS-A0WQ, D1-A0ZN, D1-A0ZO, D1-A0ZP, D1-A0ZQ, D1-A0ZR, D1-A0ZS, D1-A0ZU, D1-A0ZV, D1-A0ZZ, D1-A101, D1-A102, D1-A103, D1-A15V, D1-A15W, D1-A15X, D1-A15Z, D1-A160, D1-A161, D1-A163, D1-A165, D1-A167, D1-A168, D1-A169, D1-A16B, D1-A16D, D1-A16E, D1-A16F, D1-A16G, D1-A16I, D1-A16J, D1-A16N, D1-A16O, D1-A16Q, D1-A16R, D1-A16S, D1-A16X, D1-A16Y, D1-A174, D1-A176, D1-A177, D1-A17A, D1-A17B, D1-A17C, D1-A17D, D1-A17F, D1-A17H, D1-A17K, D1-A17L, D1-A17M, D1-A17N, D1-A17Q, D1-A17R, D1-A17S, D1-A17T, D1-A17U, D1-A1NU, D1-A1NX, DI-A0WH, DI-A1NN, E6-A1LZ, EO-A1Y5, EO-A1Y8, EY-A1GS, EY-A212, FI-A2D2, FI-A2EW, FI-A2EX, FI-A2F8, N5-A4R8, N5-A4RA, N5-A4RD, N5-A4RF, N5-A4RJ, N5-A4RM, N5-A4RN, N5-A4RO, N5-A4RS, N5-A4RT, N5-A4RU, N5-A4RV, N5-A59E, N5-A59F, N6-A4V9, N6-A4VC, N6-A4VD, N6-A4VE, N6-A4VF, N6-A4VG, N7-A4Y0, N7-A4Y5, N7-A4Y8, N7-A59B, N8-A4PI, N8-A4PL, N8-A4PM, N8-A4PN, N8-A4PO, N8-A4PP, N8-A4PQ, N8-A56S, N9-A4PZ, N9-A4Q1, N9-A4Q3, N9-A4Q4, N9-A4Q7, N9-A4Q8, NA-A4QV, NA-A4QW, NA-A4QX, NA-A4QY, NA-A4R0, NA-A4R1, NA-A5I1, ND-A4W6, ND-A4WA, ND-A4WC, ND-A4WF, NF-A4WU, NF-A4WX, NF-A4X2, NF-A5CP, NG-A4VU, NG-A4VW, QM-A5NM, QN-A5NN. 99biblabel#1 [Agrawal et al, 2011]Agrawal Agrawal N, Frederick MJ, Pickering CR, Bettegowda C, Chang K, Li RJ, Fakhry C, Xie TX, Zhang J, Wang J, Zhang N, El-Naggar AK, Jasser SA, Weinstein JN, Treviño L, Drummond JA, Muzny DM, Wu Y, Wood LD, Hruban RH, Westra WH, Koch WM, Califano JA, Gibbs RA, Sidransky D, Vogelstein B, Velculescu VE, Papadopoulos N, Wheeler DA, Kinzler KW, Myers JN. 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Nature 481(7380): 157-163.[Zou et al, 2014]Zou Zou S, Li J, Zhou H, Frech C, Jiang X, Chu JS, Zhao X, Li Y, Li Q, Wang H, Hu J, Kong G, Wu M, Ding C, Chen N, Hu H. (2014) Mutational landscape of intrahepatic cholangiocarcinoma. Nature Communications 5: 5696. | http://arxiv.org/abs/1707.08504v1 | {
"authors": [
"Zura Kakushadze",
"Willie Yu"
],
"categories": [
"q-bio.GN",
"q-bio.QM",
"q-fin.ST"
],
"primary_category": "q-bio.GN",
"published": "20170726154222",
"title": "Mutation Clusters from Cancer Exome"
} |
Shell et al.: Bare Demo of IEEEtran.cls for IEEE JournalsDirect Load Control of Thermostatically Controlled Loads Based on Sparse Observations UsingDeep Reinforcement Learning Frederik Ruelens,Bert J. Claessens, Peter Vrancx, Fred Spiessens, and Geert Deconinck F. Ruelens and G. Deconinck are with the Department of Electrical Engineering, KU Leuven/EnergyVille, 3000 Leuven, Belgium (frederik.ruelens, [email protected]). B. J. Claessens is with REstore, 2600 Antwerp, Belgium ([email protected]). P. Vrancx is with the AI-lab, Vrije Universiteit Brussel, 1050 Brussels, Belgium ([email protected]) F. Spiessens is with Vito/EnergyVille, 2600 Mol, Belgium. December 30, 2023 ================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================ This paper considers a demand response agent that must find a near-optimal sequence of decisions based on sparse observationsof its environment. Extracting a relevant set of features from these observations is a challenging task and may require substantial domain knowledge. One way to tackle this problem is to store sequences of past observations and actions in the state vector, making it high dimensional, and apply techniques from deep learning. This paper investigates the capabilities of different deep learning techniques, such as convolutional neural networks and recurrent neural networks, to extract relevant features for finding near-optimal policies for a residential heating system and electric water heater that are hindered by sparse observations. Our simulation results indicate that in this specific scenario, feeding sequences of time-series to an LSTM network, which is a specific type of recurrent neural network, achieved a higher performancethan stacking these time-series in the input of a convolutional neural network or deep neural network.Convolutional networks, deep reinforcement learning, long short-term memory,residential demand response § INTRODUCTIONOptimal control of Thermostatically Controlled Loads (TCLs), such as heat pumps and water heaters, is expected to play a key role in the application of residential demand response <cit.>.TCLs can use their thermal inertia, e.g. a water buffer or building envelope, as a thermal battery to store energy and shift energy consumption in response to changes in the electricity price or to provide grid services. Amongst the more important challenges hindering the application of residential demand response is partial observability of the environment <cit.>, where a part of the state remains hidden from the agent due to sensor limitations, resulting in a partially observed control problem.Model-predictive control (MPC) <cit.> and Reinforcement Learning (RL) <cit.> are two opposing paradigms to solve the optimal control problem of TCLs.As such, MPC and RL have developed a set of different techniques to tackle the problem of planning under partial observability.In MPC, a Kalman filter is often used toestimate hidden features by exploiting information about the system dynamics and using Bayesian interference. For example, in <cit.>Vrettos et al. applied a Kalman filter to estimate the temperature of a building envelope and in <cit.> Kazmi et al. applied a similar approachto estimate the state of charge of an electric water heater.RL approaches, on the other hand, store sequences of past interactions with their environment in a memory and extract relevant features based on this memory. The challenges herein is to consider a priori how many interactions are important to learn a specific task and what exact features should be extracted. Deep neural networks or multi-layer perceptrons are the quintessential technique for automatic feature extraction in RL <cit.>. An important breakthrough of automatic feature extraction using deep learning is presented in <cit.>, whereMnih et al apply a convolutional neural network to automatically extract relevant features based on visual input data to successfully play Atari games.Finally, by combining RL and MPC, the authors of <cit.> presented a method that trains complex control policies with supervised learning, using MPC to generate the supervision. The teacher (MPC) uses a rough representation of its environment and full state, and the learner updates its policy based on the partial state using supervised learning.§ LITERATURE REVIEWThis section provides a short literature overview of Reinforcement Learning (RL) related to demand response and discusses some relevant applications of deep learning in RL. §.§ Reinforcement learning and demand responseAn important challenge in tackling residential Demand Response (DR) is that any prior knowledge in the form of a physical model of the environment anddisturbances is not readily available or may be too costly to obtain compared to the financial gains obtained with DR. As RL techniques can be applied “blind” and consider their environment as a black box, they require no prior knowledge nor do they require a system identification step, making them extremely suited for residential DR. As a result,residential DR has become a promising application domain for RL <cit.>. The most important RL algorithms applied to DR are temporal difference RL, batch RL and more recently deep RL. The first application of RL to demand response were standard temporal difference methods, such as Q-learning and SARSA <cit.>. For example, in <cit.>, Wen et al. showed how Q-learning can be applied to a residential demand response setting and in <cit.>, Kara et al. applied Q-learning to provide short-term ancillary services to the power grid by using a cluster TCLs. Extendingthis work, Mocanu et al. demonstrated how a deep belief network can be integrated in Q-learning and SARSA to extract relevant features <cit.>, allowing for cross-building transfer learning.In <cit.>, the authors demonstrated how batch RL can be tailored to a residential demand response setting using a set of hand-crafted features, based on domain specific insights. The authors extended a well-known batch RL algorithm, fitted Q-iteration, to include a forecast of the exogenous variables and demonstrated that it outperformed standard temporal difference methods, resulting in a learning phase of approximately 20-30 days, suggesting that batch RL techniques are more suitable for demand response. More recently, inspired by advances in deep learning, the authors extended this approach for a cluster of TCLs using an automatic feature extraction method based on convolutional neural networks <cit.>. A binning algorithm is used to map the full state of the cluster to a two-dimension representation that can be used as input for the convolution neural network. Similarly, in <cit.>, François-Lavet et al. applied a convolutional neural network as a function approximator within RL to capture the stochastic behavior of the load and renewable energy production in a microgrid setting with a short-term and long-term storage. §.§ Recurrent neural networks and partial observabilityIn contrast to vanilla neural networks,Recurrent Neural Networks (RNNs) have an internal state, which is based on the current input state and the previous internal state, allowing the internal state to act as a memory modeling the impact of previous input states on the current task. This internal state allows the RNN to process sequences of input data, making it a natural framework to mitigate the problem of partial state information.In practice, however, RNNs have difficulties learning long-term dependencies <cit.>. An LSTM network is a special type RNN developed by Hochreiter and Schmidhuber in <cit.> that solves the long-term dependency problem, by adding special structures called gates that regulate the flow of information to the memory state.The application of a RNN within Q-learning, called recurrent-Q, was introduced by Lin and Mitchell in <cit.>, demonstratingthat recurrent-Q was able to learn non-Markovian tasks.Extending on this idea, Bram Bakker <cit.> demonstratedhow LSTMusing advantage learningcan solve non-Markovian tasks with long-term temporal dependencies. In addition to value-based RL, a successful implementation of a policy gradient method with an LSTM architecture to a non-Markovian taskcan be found in <cit.>. Motivated by the promising results of Deepmind with Deep QN <cit.>, the authors of <cit.> demonstrated how an LSTM network can be combined with a deep Q-network for handling partial observability in Atari games, induced by flickering game screens. §.§ ContributionsThis paper investigates the effectiveness of different deep learning techniques within reinforcement learning for demand response applications that are hindered by sparse observations, making the following contributions.We present howan LSTM network, Convolutional Neural Network (CNN) and multi-layer neural network, can be used within a well-known batch RL algorithm, fitted Q-iteration, to approximate the Q-function, extending the state with historic partial observations. We demonstrate their performance for two popular embodiments of flexible loads, namely a heat pump for space heating and an electric water heater.The paper is structured as follows. Section <ref> states the problem and formalizes it as aMarkov decision process.Section <ref> explains how these deep learning techniques can be used to extract relevant features based on sequences of observations and used within a batch RL. Section <ref> describes the different deep learning architectures. Section<ref> presents the simulation results of two flexibility carriers and finally Section<ref> draws conclusions and discusses further work.§ MARKOV DECISION-MAKING FORMALISM This section states the problem and presents the formalism to tackle it.§.§ Problem statementIn most complex real-world problems, such as demand response, an agent cannot measure the exact full state of its environment, but only a partial observation of the state. Depending on how good this partial observation can be used to model future interactions, using partial state information may result in sub-optimal policies. This paper presents two demand response applications that are hindered by partial observability, where the agent cannot measure the state directly, but has to extract relevant features based on how much energy the application consumed and how much it lost. In ourfirst experiment, we consider a heat-pump agent that can only measure its electricity consumption and outside temperature.In the second experiment, we consider an electric water heater agent with partial state state information, consisting of its measured electricity consumption and the flow rate and temperature of the tap water exiting the water buffer. To tackle this challenge, we will first formalize the underlying problem as a Markov decision process and then introduce the concepts of partial state information.§.§ Formalism At each discrete time step k, the full state of the environment evolves as follows: x_k+1 = f(x_k,u_k,w_k) with w_ka realization of a random disturbance drawn from a conditional probabilitydistributionand u_k∈ U the control action. Associated with each action of the agent, a cost c_k is provided by c_k=ρ(x_k,u_k,w_k), where ρ is a cost function that is a priori given.The goal of the agent is to find an optimalcontrol policy h^*:X→ U that minimizes the expected T-stage return for any state in the state space. Value-based RL techniques characterize thepolicy h isby usinga state-action value function or Q-function:Q^h(x,u) = w∼[ρ(x,u,w) + J^h_T(f(x,u,w)) ], The Q-function is the cumulative return starting fromstate x, taking action u, and following h thereafter. Given the Q-function, an action for each state can be found as:h(x) = arg min Q^h(x,u).This paper applies a value-based batch RL technique to approximate the Q-function corresponding to the optimal policybased on an imperfect observation of the true state.§.§ Partial stateIt is assumed that the state space Xmeasured by the agent consists of three components: timing-related state information X^, controllable state information X^, and exogenous (uncontrollable) state information X^.In this work the timing related is given by the current quarter in the day ∈ X^ = {1,…,96}, which allows the agent to capture time-varying dynamics. The controllable state information ∈ X^ comprises the operationalmeasurements that are influenced by the control action. In reality, mosts agent can only measure a partial observationof the true state _k, resulting in a partially observable Markov decision problem. The exogenous informationis invariant for control actions u_k, but has an impact on the dynamics. Examples of exogenous variables are weather conditions and demand profiles (e.g heat demand).Thus, the state measured by the agent at step k is given by:_k = (,,).Note that since (<ref>)only includes part of the true state, it becomes impossible to model future state transitions, making the state non-Markovian. §.§ Action At each time step, ademand response agent can request an action u_t∈ [0,1]: either to switch OFF or ON. To guarantee the comfort and safety constraints of the end users, each TCL is equipped with an overrule mechanism (or thermostat). The backup functionB:X × U ⟶ maps the requested control action u_k∈ U taken instate x_k to a physical control action ∈: = B(x_k,u_k). The settings of the backup function B are unknown to the learning agent, but the resulting actioncan be measured by the learning agent. §.§ CostThis papers considers a dynamic pricing scenario where an external price profile is known deterministically at the start of theoptimization horizon:c_k= ρ(,λ_k) = λ_k Δ t,where λ_k is the electricity price at time step k and Δ t is the length of a control period. § BATCH REINFORCEMENT LEARNING Given full observability, batch RL algorithms start with a batch of four tuples of the form:(_k,u_k,_k', ),where _k represents the true state of the problem.According to the theory of partial observable Markov decision processes <cit.>, the optimal value function at time step k depends on the partial state observations of all proceeding periods. However, since these observations accumulate over time, it is important to capture sufficient statistics, thatsa history length hwhich summarizes the essential content of the measurements. As such, this paper tackles the problem of partial observability by augmenting the state vector with a sequence of partial state observation, requested actions and physical actions of the last h observations:_k = (, _k,)with _k give by:[,…, _k-h],[_k-1, …, _k-1-h],[u_k-1,…, u_k-1-h].As a results, this paper starts from a bath of four tuples given by:{(_k, u_k, _k, )}_k=1^ℳ, where _k represents the augmented state. An important challenge is to learn how to extract relevant features fromin a scalable way. §.§ Fitted Q-iterationThis paper appliesfitted Q-iteration <cit.> to obtain an approximation of the Q-function Q^*(,u). Fitted Q-iteration iteratively approximates the Q-functions for each state-action pair using its corresponding cost andthe approximation of the Q-function from the previous iterations. To leverage the availability of forecasts of exogenous information , e.g. outside temperatures, we use the extension of fitted Q-iteration as presented in <cit.>, which replaces the observed exogenous information by its forecasted value (line <ref> in Algorithm <ref>). In order forAlgorithm <ref> to work, we need to select an approximator architecture(step <ref>) that is able to learn relevant features from sequences of input data andthat can generalize the Q-function. § DEEP LEARNING APPROXIMATORS This paper investigates the effectiveness of the following deep learning approximators when combined with fitted Q-iteration. §.§ Deep neural network It has been shown that a neural network with a single layer is sufficient to represent any function, but the layer may become infeasible large andmay failto train and generalize correctly. To overcome these two challenges, deeper networks are used as these networks can reduce the number of units to represent the function and can reduce the generalization error.Fig. <ref>(a) illustrates the neural network as used in this paper, consisting of an input layer given by (_k,u_k), two hidden layers with rectified linear units (ReLUs), and one linear output layer, representing the approximated Q-function.§.§ Convolutional neural networkCNNs have been successfully applied to extract features from image data, represented as a 2D grid of pixels. In this paper, we considera time series and convolve a 1D filter of length N over the time-series in the state (<ref>). A sketch of the applied CNN can be seen in Fig. <ref>(b), which consists of two components that are merged to output a singe value. The first component is a dense neural network which takes the timing-related information,exogenous information and action as input. The second components is a CNN which takes the time-series as input (<ref>). For each sequence, the network consists of two layers containing eight 1D filters of length 4 followed by a ReLU, which is downsampled by using an average pooling layer. §.§ Long short-term memory§.§.§ BackgroundAn LSTM network (Fig. <ref>) consists of LSTM nodes that are recurrently connected to each other. Each LSTM node has an internal recurrence or memory cell C^(t) and a system of gating units that controls the flow of informations. For each step t of the sequence ^(1),…,^(t),…,^(h), theresulting action of the forget gate f^(t), input gate i^(t) and output gate o^(t) of a single LSTM nodes is provided by: f^(t) = σ(W_f[h^(t-1),^(t)] + b_f),i^(t) = σ(W_i[h^(t-1),^(t)] + b_i),o^(t) = σ(W_o[h^(t-1),^(t)] + b_o), where W_f,W_i,W_o and b_f,b_i,b_o are the weights and biases of the forget, input and output gate, σ denotes the logistic sigmoid function and ^(t) denotes the current element of sequence (<ref>), with the time step index t ranging from 1 to h.Theinternal memory cell of the LSTM node is updated as: C^(t) = f^(t)*C^(t-1) + i^(t) *C^(t), where C^(t) and C^(t-1) are the current and previous memorystate and * denotes a pointwise multiplication operator.Note that the new memory C^(t) is defined by the information it forgets from the old state f^(t)* C^(t-1) and remembers from the current i^(t)* C^(t).In a last step, a hyperbolic tangent function is applied to the memory cell and multiplied with the output o^(t), which defines what information to output.h^(t) = o^(t)* tanh(C^(t)), This gating mechanism allows the LSTM network to store information about the state for long periods of time and protects the gradient in the cell from harmful changes during training related to the vanishing or exploding gradient problem of RNN <cit.>.§.§.§ Approximator architectureThe approximator architecture consists of two components: an LSTM network and a standard multi-layer perceptron (Fig. <ref>). The first part of the input, corresponding to the LSTM component, contains the historic information of the partial state . For each k=h+1,…,#ℳ, the input of the LSTM network is given by the following sequence:[_k;_k-1; u_k-1 ]_^(1)_k ,…, [ _k-h+1; _k-h;u_k-h ]_^(h)_kThe history depth h defines how much time steps the network can see in the past to compute its approximation of the Q-function. The length of the memory cell d_cell represents an important hyper parameter and defines how many knowledge can be encoded. As can be see in Fig. <ref> only the content of the last memory cell h_t is used as an input for the next layer. The second part contains the time-related information, exogenous information and action: , , u . The outputs of both components are combined to form a singe architecture, which is followed by two fully connect layers with Rectified Linear Unit (ReLU) activation functions. A final linear output layer approximates the final Q-function for the provided state-action pair. § SIMULATION EXPERIMENTS This section evaluates the performance of combining the presented deep learning techniques with Alogirthm 1 for twoproviders of demand flexibilityexposed to a dynamic energy price.§.§ Simulation frameworkAt the end of each simulation day, Algorithm <ref> is used to computea new policy based on current batch and electricity price for the following day.The RLagent starts with an empty batch and alternates exploration and exploitation according to a decreasing exploration probability: ε_d = 0.75^d, where d denotes the current episode.All experiments are repeat 10 times starting form a different random seed, resulting in different exploration probabilities and stochastic disturbances. The following results indicate the average of these simulation experiments, where a confidence bound (±2σ) is indicated by a shaded area, representing a 0.95probability that the solution lies in the shaded area. The average simulation time for one day (Algorithm <ref>) is about 1.5 hour[Simulation hardware: Xeon E5-2680 v2 processor with 15 GiB memory (Amazon elastic cloud instance type: c3.2xlarge).] using Keras with Theano as backend. §.§ Experiment 1: Space heatingSimilar as in <cit.>, a second-order heat-pump model (C_a=2.441MJ/^∘C, U_a=125W/^∘C, C_m = 9MJ/^∘C, H_m=6.863kW/^∘C) with outside temperatures from <cit.> is used to simulate the temperature dynamics of a residential building with a heat pump. The heat pump has a maximum electric heating power of 2.3kW and the minimum and maximum comfort settings are set to 20^∘C and 23^∘C. To model stochastic impact of user-behavior wesample an exogenous temperature disturbance from 𝒩(0,0.025). The time resolution of the dynamics is 60 seconds and of the control policy is 15 minutes. The state vector describing the environment is defined as:_k = (,, , , ),where _k contains timing information,the air temperature,the virtual mass temperature,the outside temperature andan exogenous disturbance. As stated in the problem description, it is assumed that the RL agent cannot measure the air and mass temperature of the building, resulting in a partial observed control problem. As such, we construct the following augmented state vector:_k = (_k,[_k-1,…, _k-h],[u_k-1,…, u_k-h], [_k-1,…, _k-h], _k ),which includes three time series of lenght h=20.§.§.§ NN ArchitectureThe neural network consists of three dense layers with 50 neurons with ReLU activation functions, followed be a linear output unit. The neural network was trained using RMSprop with a minibatch size of 32.§.§.§ CNN ArchitectureThe network consists of two components, namely a CNN and dense network. The CNNcomponent consists of two 1D convolutions (along the time dimension) that are each followed by an average pooling layer. The dimension of the first filter is (L×3), where Lis the filter length and 3 is number of input sequences and the dimension of the second filter is L×1. Both filters have a filter length of 4. The dense network processes the time-related information, exogenous information and action. Both components are merged and followed with two layers with 20 neurons and a single output layer.All layers use ReLU activation function except for the output layer that uses a linear function.§.§.§ LSTM ArchitectureThe input to the LSTM network is provided by the sequence: [_k-1,; u_k-1,; _k-1 ],…,[_k-h,; u_k-h,; _k-h ] and the NN is provided by _k, _k and u_k.For the heatpump experiment the best results were obtained with the history depth h set to 20 time steps (quarters) and the length of each LSTM memory cell d_cell set to 8. §.§.§ ConvergenceFig. <ref> depicts the cumulative cost using function approximators (top) and daily average outside temperature (bottom). The no control strategy activates the backup controller, without setting a control action, and can be seen as a worst case scenario as it is agnostic about the electricity price. An upper bound is computed by considering the full state information as defined in (<ref>). In addition to LSTM with partial state information, the figure depicts thecumulative of using an ensemble of extremely randomized trees (or ExtraTrees) <cit.>. The number of trees in the ensemble was set to 100 and the minimum sample size for splitting a node to 5. Our results indicate that the ExtraTrees approximator was not able to extract relevant features from the partial state information and performed only 1.5% better than theno control strategy. In contrast, the LSTMapproximator was able to extract relevant features and achieved a reduction of 5.5%. Fig. <ref> shows the daily cost (top) obtained with Algorithm <ref>, using a partial state information using a neural network, CNN and LSTM network as a function approximator. The middle graph indicates the scaled cost which is c calculated as follows: (c - c_f)/(c_nc - c_f), where c_f is the result of using the full state information and c_nc of using the no control strategy, resulting in c=0 for the full state strategy and c=1 for the no control strategy. This figure indicates Algorithm 1 obtained a scaled cost of0.37 using LSTM, 0.66 using NN and 0.82 using CNN. The bottom graph compares the resulting control policies of LSTM, CNN en NN with the control policy of the full state using a euclidean distance.Although NN achieved a better performance than CNN, the resulting policy of CNN and LSTM are closer to the policy of the full state. We speculate that the CNN and LSTM learned a better representation of the full state than the NN, since the NN achieved a low cost by lowering the air temperature to minimum temperature without reacting to the price.§.§.§ Daily resultsA more qualitative interpretation of our results can be seen in Fig. <ref>. The figure shows the power consumption and the correspondingdaily price profiles. It can be seen that the learning agent successfully postponed its power consumption to low price moments, while satisfying the comfort constraints.§.§ Experiment 2: Electric water heatingThe second experiment considers an electric water heater with a water buffer of 200 liters and a daily average water consumption of 100 liter. The minimum and maximum water temperature is set to 45^∘C and 65^∘C. The water heater is equipped with a thermostatic mixing value to assure a constant requested temperature of 45^∘C. The water heater has an electric power rating of 2.3kW and a built-in backup controller as defined in <cit.>. The time resolution for the dynamics is5 seconds and the time resolution for the control policy is 15 minutes. The full state vector of the electric water heather is defined by:_k = (_k,T_k^1,…,T_k^|ℒ|, d_k),where T_,t^i is the temperature corresponding to the ith layer and d_k is the current tap demand. During our simulation, a non-linear stratified model with 50 layers is used to simulate the temperature gradient along the water tank and stochastic tap water profiles are used based on <cit.>.In a previous paper <cit.>, the authors considered that the agent could measure a imperfect state through eight temperature sensors. In this experiment, however, it is assumed that the buffer is not equipped with a set of sensors to measure the different temperatures inside the water buffer. As a result, we define the following augmented state vector:_k = (_k, [u_k-1,… u_k-h],[_k-1,…_k-h][_k,…_k-h+1] , [T_k^,…, T_k-h+1^] ),where _k contains timing information, u_k is the requested control action,the actual action, and ṁ_k and T_k^ are the mass flow rate and temperature of the water exiting the water buffer. Note that [_k-1,…_k-h] represents the electricity consumption of the boiler and [_k,…_k-h+1] , [T_k^,…, T_k-h+1^] represents the energy flowing out of the boiler. §.§.§ (C)NN ArchitectureThe NN and CNN architecture are identical as in the previous experiment with the exception that the filters size of the first convolutional layer is 4×4, because now we have 4 input sequences.§.§.§ LSTM ArchitectureThe input to the LSTM network is provided by the sequence:[_k-1,; u_k-1,; ṁ_k,; T_k^ ] ,…,[_k-h,;u_k-h; ṁ_k-h+1,; T_k-h+1^ ] For the boiler experiment the best results were obtained withthe history depth h set to 40 time steps (quarters) and the length of each LSTM memory cell d_cell set to 12.§.§.§ Daily resultsFor the electric water heater scenario, we only offer qualitative results (Fig. <ref>). It shows the daily power consumption of anelectric water heater and corresponding price profiles. It can be seen that the learning agent required four weeks of learning before obtaining reasonable policies (lower row of graphs). A final comparison between using a CNN or LSTM network as a function approximator can be seen in Fig. <ref>, indicating that using a CNN resulted in a cost reduction of 5.5% and using a LSTM network in 10.2%. The results of FQI-NN were omitted because we were to able to stabilize the learning of the NN.§ CONCLUSIONS AND FUTURE WORK In this paper, we demonstrated the effectiveness of combining different deep learning techniques with reinforcement learning for two demand response applications that are hindered by sparse observations of the true state. Since these sparse observation result in a non-Markovian control problem, we extended the state with sequences of past observations of the state and action. In a first experiment, we considered an agent that controls a residential heating system under a dynamic pricing scenario, where the agent can only measure its electricity consumption, control action and outside temperature.Our simulations indicated that reinforcement learning with long short-term memory (LSTM) performed better than other techniques such as aneural network, convolutional neural network and ensemble of regression trees, when sparse observations are used. In our second experiment, we considered an agent that controls a residential electric water heater with a hot storage vessel of 200 liter. In this scenario, the agent can only measure its electricity consumption, control action and flow and temperature of the tap water exiting the storage vessel. The simulation results indicated that the LSTM network outperformed theconvolutional network and deep neural network. We speculate that the higher performance of the LSTM network comes from its internal memory cell which can act as an integrator. This internal memory cell allows the LSTM network to process sequences of sparse observations and extract relevant features from it that can represent the underlying state of charge (or energy level) of the application. A potential direction of future research would be to extract and visualize the relevant features that are being detected by the LSTM network and CNN that would lead to a better performance and understanding.IEEEtran | http://arxiv.org/abs/1707.08553v1 | {
"authors": [
"Frederik Ruelens",
"Bert J. Claessens",
"Peter Vrancx",
"Fred Spiessens",
"Geert Deconinck"
],
"categories": [
"cs.LG"
],
"primary_category": "cs.LG",
"published": "20170726173346",
"title": "Direct Load Control of Thermostatically Controlled Loads Based on Sparse Observations Using Deep Reinforcement Learning"
} |
[table]labelsep=spaceconference conference=2500 ∂binomial𝔼ℙ1/21/2ℕℤℙ𝕄ℝℂarg minarg max diag×lim_n →∞lim inf_n →∞lim sup_n →∞ |; RemarkRemarkpropositionPropositiondefinitionDefinitiontheoremTheoremlemmaLemmaexampleExamplecorollaryCorollaryassumptionAssumptionclaimClaimdlcases{.𝔽bepqrrsuuvτwxxzzτλλθϕ←→ PLPL→𝔼1t -11t +1 X1𝐚𝐛𝐜𝐝𝐞𝐠𝐫𝐘𝐖𝐟𝐡𝐈𝐮𝐩μ𝐩𝐪𝐪𝐫𝐫𝐫𝐬𝐬𝐮𝐮𝐮𝐯𝐯𝐰𝐰𝐱𝐱𝐱𝐲𝐳𝐳𝐳𝐀𝐁𝐁𝐂𝐃𝐄𝐆𝐇KK𝐊Lℒ𝐏PP𝒫𝐐𝐌𝐑RRℛ𝐒𝒮𝐔𝒰𝐕𝐖X𝐗𝐘𝐙Z𝐙ααββμρλησξϕτθθθθμet al.𝔊 F𝔼A f tr IVar p.d.f.PL[figure]width=.95 IEEEtranSectoring in Multi-cell Massive MIMO Systems Shahram Shahsavari, Parisa Hassanzadeh, Alexei Ashikhmin, and Elza Erkip S. Shahsavari, P. HassanzadehandE. Erkip are with the ECE Department of New York University, Brooklyn, NY. Email: {shahram.shahsavari,ph990, elza}@nyu.eduA. Ashikhmin is with Bell Labs, Nokia, Murray Hill, NJ, USA. Email:[email protected] December 30, 2023 ==========================================================================================================================================================================================================================================================================================================================================In this paper, the downlink of a typical massive MIMO system is studied when each base station is composed of three antenna arrays with directional antenna elements serving 120^∘ of the two-dimensional space. A lower bound for the achievable rate is provided. Furthermore, a power optimization problem is formulated and as a result, centralized and decentralized power allocation schemes are proposed. The simulation results reveal that using directional antennas at base stations along with sectoring can lead to a notable increase in the achievable rates by increasing the received signal power and decreasing `pilot contamination' interference in multi-cell massive MIMO systems. Moreover, it is shown that using optimized power allocation can increase 0.95-likely rate in the system significantly.§ INTRODUCTIONWith the advent of new technologies such as smart phones, tablets, and new applications such as video conferencing and live streaming, there has been a dramatic increase in the demand for high data rates in cellular systems. On the other hand, it is challenging to achieve high enough data rates in the crowded sub-6 GHz spectrum. Multi-user Multi Input Multi Output systems with large number of antennas (known as massive MIMO), have shown a great potential to achieve very large spectral and energy efficiencies, which makes them a strong candidate for 5G mobile networks <cit.>.In massive MIMO systems, base stations are usually equipped with a large number of antennas serving much smaller number of users each of which has an omnidirectional antenna. It is shown in <cit.> that with a simple Time Division Duplex (TDD) protocol, it is beneficial to increase the number of base station antennas in a single cell massive MIMO network. More specifically, it is shown that received signal power is proportional to number of antennas while interference plus noise power is not. However, as shown in <cit.>, another type of inter-cellular interference, called `pilot contamination', appears in multi-cell massive MIMO networks. Typically training sequences should be short, since channels between base station and users change fast. This forces one to use nonorthogonal training sequences in neighboring cells, which causes pilot contamination whose power is proportional to the number of antennas at the base stations. Consequently, Signal to Interference plus Noise Ratio (SINR) converges to a a bounded value as the number of antennas tends to infinity.Most literature on massive MIMO considers omnidirectional base station antennas. It is well-known that using directional antennas along with sectorized antenna arrays at each base station is one of the methods to increase SINR in conventional cellular networks <cit.>. Reference <cit.> indicates the potential of using directional antennas in massive MIMO systems; however, it does not provide any performance analysis. In this paper, we consider the sectorized setting, analyze the performance of a massive MIMO system with directional antennas at each base station, and provide a lower bound on the achievable downlink rate of the users as a function of large-scale fading coefficients. We formulate a tractable downlink power optimization problem and suggest a centralized scheme to find the optimal power allocation. To reduce the communication and computation overheads, we also provide a sub-optimal decentralized scheme. A numerical comparison between a massive MIMO system with omnidirectional antennas at each base station and one with directional antennas shows that using directional antennas can improve the performance significantly.To the best of our knowledge, this is the first detailed study of massive MIMO systems with directional antennas.§ SYSTEM MODELWe consider a two-dimensional sectorized hexagonal cellular network with TDD operation, composed of L cells, each with K mobile users. Cell sectoring is done such that three base stations are located at the non-adjacent corners of each cell, as shown in Fig. <ref>, and each base station is equipped with three directional (120^∘) antenna arrays such that each array serves one of the three neighboring cells. As depicted in Fig. <ref>, the users in each cell are served by the three antenna arrays that belong to the base stations located on the corners of the cell. We assume that each directional array has M directional antenna elements, hence there are M_B = 3M elements at each base station. We also assume that users have single omnidirectional antennas. In the following we denote cell j by C_j, and user k in cell j by U_kj, where j ∈ [L][We denote by [N] the set of integers from 1 to N.] and k∈ [K]. Each antenna array is uniquely identified by a cell-array index pair (j,i), and is denoted by A_ji, where i ∈ [3] indicates the array located in corner i of cell j∈ [L] (see Fig. <ref>).User U_kj communicates with all three arrays A_j1, A_j2, and A_j3 for uplink and downlink transmissions.§.§ Directional Antenna ModelWe adopt the simplified directional antenna model introduced in <cit.>. Fig. <ref> depicts the directivity (power gain) pattern of each array element, where G_Q and G_q are the main lobe and back lobe power gains, respectively, and θ, chosen as 2π/3, is the beamwidth of the main lobe. Let ϕ∈ [0,π] denote the angular position of a user placed at an angle ϕ relative to the boresight direction of an antenna element, as in Fig. <ref> (left), then the signal transmitted to and received from the user is multiplied by a gain equal to √(G(ϕ)). Let G_ji^[kl] denote the power gain between U_kl and A_ji. Note that all users in cell C_j, j∈[L] are in the main lobe coverage of arrays {A_ji: i∈[3]}, and therefore, observe the power gain G_Q for any k∈[K] and any i ∈ [3]. We assume a lossless antenna model which implies that G_Q+2G_q=3, and G_Q ≤ 3 due to the conservation of power radiated in all directions <cit.>.§.§ Channel ModelDue to TDD operation and channel reciprocity, downlink and uplink transmissions propagate similarly. We assume narrow-band flat fading channel model in which, the complex channel (propagation) coefficient between the m-th antenna element of A_ji and U_kl is given byg^[kl]_mji = √(β^[kl]_ji) h^[kl]_mji,where β^[kl]_ji∈ℝ^+ is the large-scale fading coefficient, which depends on the shadowing and distance between the corresponding user and antenna element, and h^[kl]_mji∈ℂ is the small-scale fading coefficient. The received signal also includes additive white Gaussian noise. Since the distance between a user and an array is much larger than the distance between the elements of an array, we assume that the large-scale fading coefficients are independent of the antenna element index m. The small-scale fading coefficients, h^[kl]_mji, are assumed to be complex Gaussian zero-mean and unit-variance, and for any (k,l,m,j,i) ≠ (n,v,r,u,q) coefficients h^[kl]_mji and h^[nv]_ruq are independent. We will use ^[kl]_ji∈ℂ^M×1 to denotechannel vector between A_ji and U_kl. We further assume that small-scale and large-scale fading coefficients are constant over small-scale and large-scale coherence blocks represented by T and T_β symbols, respectively. While the small-scale fading coefficients significantly change as soon as a user moves by a quarter of the wavelength, large-scale fading coefficients are approximately constant in the radius of 10 wavelengths (see <cit.> and references there). Thus, T_β≈ 40 T. We also assume that small-scale channel coefficients are independent across different small-scale coherence blocks, and similarly large-scale channel coefficients.§.§ Time-Division Duplexing ProtocolUplink and downlink transmissions, require access to the channel vectors at the antenna arrays. Channel vectors are estimated by antenna arrays using uplink training transmissions in each small-scale coherence block T. Similar to <cit.> and <cit.>, we assume that the same set of K orthonormal training sequences (pilots) is reused in each cell, such that sequence r^[k]∈ℂ^τ is assigned to U_kl in C_l, and r^[k]^†r^[n]=δ_kn for any k,n ∈ [K] and any l∈[L]. Note that since the number of orthogonal τ-tuples can not exceed τ, wehave K≤τ<cit.>. Due to the independence of channel coefficients across different small-scale coherence blocks, training is repeated in each block T, hence τ≤ T. The system operates based on the TDD protocol proposed in <cit.>,<cit.>. The first two steps of the protocol are carried out once for each large-scale coherence block, and the last five are repeated over small-scale coherence blocks. Time-Division Duplexing Protocol Step 1: In the beginning of each large-scale coherence block, each base station estimates the large-scale fading coefficients between itself and all the users in the network.Step 2: Each array transmits a measure of the large-scale fading coefficients estimated in Step 1, to the users in its cell, which are later used for decoding the downlink signals in Step 7. More specifically, A_ji, i ∈ [3] transmits thedecoding coefficient defined asϵ^[kj]_ji≜√( Mρ_r τρ^[kj]_ji)G^[kj]_jiβ^[kj]_ji/( σ^2_r+ ρ_r τ∑_l=1^LG^[kl]_jiβ^[kl]_ji)^1/2,to U_kj, k∈[K], where σ^2_r is the reverse link (uplink) noise power, ρ_r is the reverse link transmit power from each user in C_j to arrays {A_ji:i∈[3]}, and ρ^[kj]_ji denotes the forward link (downlink) transmit power assigned by A_ji to U_kj. Forward link power allocation strategies are discussed in Sec. <ref>.Step 3: All users synchronously transmit their uplink signals. Step 4: All users synchronously transmit their training sequences (pilots). Step 5: Each array estimates the channel vector between itself and the users located within its cell using the training sequences, and processes the received uplink signals.Step 6: Arrays use conjugate beamforming (based on the estimated channel vectors and power allocation) in order to prepare the downlink signals {s^[kj]:k∈ [K]} for transmission, where s^[kj] denotes the signal intended for U_kj. All arrays synchronously transmit the prepared signals.Step 7: User U_kj, k,j∈[K]×[L] decodes its received signal, denoted by y^[kj], using the decoding coefficients received in Step 2 asŝ^[kj]= y^[kj]/∑_i=1^3 ϵ^[kj]_ji. For the TDD protocol given above, we assume that each array A_ji can accurately estimate and track all the large-scale fading coefficients, discussed in <cit.>, and it has the means to forward the decoding coefficients, ϵ^[kj]_ji to the users in C_j. As in <cit.>, we will not consider the resources needed for implementing these assumptions.According to (<ref>), ϵ^[kj]_ji only depends on the large-scale fading coefficients the number of which, does not increase with the number of antennas as discussed in Sec. <ref>. Therefore, the amount of information exchange between each antenna array and its corresponding users does not depend on M, which makes the massive MIMO system scalable.In the following we only analyze the downlink transmissions; the analysis of the uplink scenario follows similarly. § DOWNLINK SYSTEM ANALYSISIn this section, we analyze the downlink system performance by providing SINR expression. Theorem <ref> provides a lower bound on user downlink transmission rates. We assume that linear MMSE estimation is used to estimate the channel vectors in Step 5 of the TDD protocol. Furthermore, as stated in step 2 of the TDD protocol, power assignments are represented explicitly. In our analysis, we assume that [s^[kj]] = 0 and [s^[kj]] = 1 for any (k,j)∈ [K][L]. §.§ Downlink System PerformanceFor the sectorized multi-cell massive MIMO system with directional antennas described in Sec. <ref>, the downlink transmission rate to user k∈ [K] in cell j∈ [L], R^[kj], is lower bounded by R^[kj]≥log_2(1+SINR^[kj]),where,SINR^[kj] =P^[kj]/I_1^[kj]+ I_2^[kj] + σ^2_f,with, P^[kj] = | ∑_i=1^3 √(ρ_ji^[kj] G_ji^[kj])λ_ji^[kj]|^2 , I_1^[kj] =∑_l=1 l≠ j^L| ∑_i=1^3 √(ρ_li^[kl] G_li^[kj])λ_li^[kj]|^2 ,I_2^[kj] =∑_l=1^L∑_i=1^3ρ_li G_li^[kj]β_li^[kj], λ_ji^[kl] = ( M ρ_r τ G_ji^[kl]β_ji^[kl]^2/σ^2_r+ ρ_r τ∑_v = 1^L G_ji^[kv]β_ji^[kv])^1/2,and ρ_li≜∑_k=1^K ρ_li^[kl] is the forward link transmission power at array i∈[3] in cell j∈[L] and σ^2_f denotes forward link noise power. The sketch of the proof is provided in Appendix <ref>.In Theorem <ref>, P^[kj] is the desirable signal power received by U_kj, and I_1^[kj], I_2^[kj] correspond to two types of interference experienced by the user. More specifically, I_1^[kj] is the interference created by pilot reuse in multiple cells, referred to aspilot contamination, and similar to P^[kj], it grows linearly with the number of base station antenna elements (λ_li^[kj]∝√(M)). The second interference I_2^[kj], referred to asundirected interference, is created by nonorthogonality of channel vectors of different users, channel estimation error, and lack of user's knowledge of effective channel <cit.>. This type of interference does not grow with M, and hence has negligible contribution when M is very large. Although increasing M leads to higher SINR for all users, we remark that SINR converges to a bounded limit when M goes to infinity.In the next section, we consider optimal and suboptimal strategies for forward link power allocation. In Sec <ref> we evaluate the system performance and show that using optimized power allocation can lead to a significant performance improvement.§.§ Forward Link Power AllocationIn Step 2 and 6 of the TDD protocol given in Sec. <ref>, arrays divide their forward link transmit power among the users they serve for downlink transmissions. In the following, we assume that ρ_li=∑_k=1^K ρ_li^[kl]≤ρ_f/3 for any (l,i)∈[L][3], where ρ_f is the base station maximum forward link power, and discuss three different strategies with different communication and computation complexities, and compare their performance in Sec. <ref>.§.§.§ Uniform Power Allocation (UPA)In this suboptimal strategy, which requires no cooperation across the network, each array transmits at full power and divides its forward link transmit power uniformly across the users in its cell such that each gets a portion equal to ρ_ji^[kj]=ρ_f/3K, ∀(j,i,k)∈[L]×[3]×[K].§.§.§ Optimal Centralized Power Allocation (CPA)The powers allocated to each user can be globally optimized in order to maximize the worst downlink SINR (equivalently rate) among all users in the network. A central entity formulates and solves a constrained max-min optimization problem based on the SINR expression given in Theorem <ref> as follows, ensuring to satisfy each array's maximum forward link transmit power.max_{ρ_li^nl} min_k,jP^[kj]/I_1^[kj] + I_2^[kj] + σ^2_fsubject to:∑_n=1^K ρ_li^[nl]≤ρ_f/3,∀ (l,i)∈ [L]×[3] ρ_li^[nl]≥ 0 , ∀ (l,i,n)∈ [L]×[3]× [K]with P^[kj], I_1^[kj], and I_2^[kj] given in (<ref>)-(<ref>). By introducing slack variables X_kj and Y_kj, (<ref>) is equivalent to: max_{ψ_li^nl,X_nl,Y_nl}min_k,j| ∑_i=1^3 ψ_ji^[kj]√( G_ji^[kj])λ_ji^[kj]|^2/ X_kj^2 +Y_kj^2 + σ^2_fsubject to: ∑_l=1 l≠ j^L| ∑_i=1^3 ψ_li^[kl]√( G_li^[kj])λ_li^[kj]|^2 ≤ X_kj^2, ∀(j,k)∈[L]×[K], ∑_l=1^L∑_n=1^K∑_i=1^3 (ψ_li^[nl])^2 G_li^[kj]β_li^[kj]≤ Y_kj^2, ∀(j,k)∈[L]×[K], ∑_k=1^K (ψ_li^[kl])^2 ≤ρ_f/3, ∀ (l,i)∈ [L]×[3],ψ_li^[kl]≥ 0,∀(l,i,k)∈[L]×[3]×[K],where, ψ_ji^[kj]=√(ρ^[kj]_ji). The equivalence between (<ref>) and (<ref>) follows from the fact that first two constraints in (<ref>) hold with equality at the optimum. Power optimization problem (<ref>) is quasi-concave. The proof of proposition <ref> is provided in Appendix <ref>.Due to quasi-concavity of problem (<ref>), the solution can be obtained using the bisection method and a series of feasibility checking convex problems provided in Algorithm <ref>. This power allocation strategy requires a central entity with access to the large-scale fading coefficients of the entire network, and has a much higher complexity compared to the UPA scenario. In this scheme, in each large-scale coherence block T_β, the base stations send their estimated large-scale fading coefficients to a central entity. The central entity solves the optimization problem using Algorithm <ref>, and sends the results back to the base stations.§.§.§ Decentralized Power Allocation (DPA)Computational and communication complexities of CPA can be significantly reduced using an optimization based on local information. Each antenna array, say A_ji, considers itself and the antenna arrays in a ring of cells around C_j such that if this would be the entire network. A_ji collects all large-scale fading channel coefficients between all antennas arrays and all users in this network and solves the optimization problem (<ref>), formulated for this network. Next, A_ji uses the found downlink powers ρ_li^[kl], ∀ k∈ [K], and discard the powers found for other antenna arrays in the ring.§ SIMULATIONS AND DISCUSSIONS In this section, we evaluate how effective sectoring is in mitigating the interference in massive MIMO systems. In our simulations we consider the following sectorized settings:Directional Arrays with UPA (Dir-UPA),Directional Arrays with CPA (Dir-CPA),Directional Arrays with DPA (Dir-DPA), and compare them with their omnidirectional counterparts:Omnidirectional Base Stations with UPA (Omni-UPA),Omnidirectional Base Stations with CPA (Omni-CPA), andOmnidirectional Base Stations with DPA (Omni-DPA). The system model in settings Dir-UPA, Dir-CPA and Dir-DPA is the one introduced in Sec <ref>, while the other three settings are modeled based on <cit.>, where one base station with M_B omnidirectional antenna elements, is placed at the center of the cell, and has a forward link power budget of ρ_f. For each setting, the forward link powers are allocated according to their respective strategies defined in Sec. <ref>.We consider a network composed of L = 19 cells (two rings of cells around a central cell), each with a radius of R=1 km, and K = 9 users distributed uniformly across each cell except for a disk with radius r = 60 m around the base stations. In order to avoid the cell edge effect, cells are wrapped into a torus as in <cit.>. The large-scale fading coefficients are modeled based on the `COST-231' model at central frequency f_c=1900 MHz as 10log_10( β^[kj]_li)= -140-35.2log_10(d^[kj]_li)+Ψ, where d^[kj]_li denotes the distance (in km) between U_kj and A_li, and Ψ denotes the shadow fading coefficient. We assume that Ψ∼𝒩(0,8), thermal noise power is -101 dBm, and the noise figure at each base station and each user is 9 dB, hence σ^2_f=σ^2_r=-92 dBm. The antenna main-lobe and back-lobe power gains are G_Q=2.98 and G_q=0.01, respectively, the reverse link transmit power is ρ_r = 23 dBm, and the maximum forward link transmit power of each base station is set at ρ_f = 30 dBm. Figs. <ref>(a)-<ref>(c) display the CDF of the normalized received signal power where normalization with respect to forward link noise power, i.e. P^[kj]/σ^2_f, and normalized version of two types of interference powers affecting the users in a network, i.e. I_1^[kj]/σ^2_f and I_2^[kj]/σ^2_f. We only provide the simulations for M_B = 10^2, since, as seen in Theorem <ref> in Sec. <ref>, received signal power and pilot contamination power are linearly proportional to M, while undirected interference power is independent of M. When comparing the Dir-UPA and Omni-UPA settings, we observe that sectoring affects each of these components as follows:§.§.§ Received signal power P^[kj]With sectoring, received signal power is higher for most of the users. In Dir-UPA, each user communicates with three arrays, each of which has M = M_B/3 elements and a forward link transmit power of ρ_f/3. Even though the per-element forward link transmit power is equal to that in Omni-UPA, i.e. ρ_f/M_B, users benefit from the directionality of the antenna arrays. In this case, the signals transmitted from each array are emitted with the main-lobe directionality gain (G_Q ≈ 3), compared to the unity directionality gain of an omnidirectional base station.Another reason for the increase in the received signal power is reduction of the pilot contamination effect (see the next subsection). The pilot contamination has two malicious effects. First, a base station creates directed interference to users located in other cells. Second, since the base station deviates part of its transmit power to other users, it effectively reduces the transmit power for users located in its cell. With sectoring the pilot contamination effect is getting smaller (see the next subsection), and therefore the signal power for legitimate users increases.The net gain translates into an increase in the received signal power of 60% of the users.§.§.§ Pilot contamination I_1^[kj]Sectoring reduces the effect of pilot contamination. This is due to the fact that with the directionality in Dir-UPA, arrays are able to derive better channel estimates from the received pilots, and further mitigate the pilot contamination. In Omni-UPA, each array receives the pilots transmitted from all cells and in all directions. However, directional arrays receive these signals with different directionality gains from the users in different cells, i.e., one-third of the signals (those in the main lobe coverage of the arrays) are amplified with G_Q ≈ 3, while the remaining two-thirds (in the back lobe coverage of the arrays) are attenuated with G_q ≈ 0 as illustrated in Fig. <ref>. In this case, the effective channel estimation SINR is approximately 3 times larger compared to the omnidirectional setting, which in turn, as depicted in Fig. <ref>, reduces the interference.§.§.§ Undirected interference I_2^[kj]Sectoring does not affect undirected interference power. In both Dir-UPA and Omni-UPA settings, the multi-user activity in the overall network contributes to the undirected interference power, which arises due to the nonorthogonality of the channel vectors and other parameters mentioned in Sec. <ref>. More specifically, in the sectorized scenario, all of the M_BL antennas create interference, among which, each user receives the signals emitted from one-third amplified by a factor of G_Q ≈ 3, and signals from the remaining antennas are attenuated by G_q ≈ 0. Therefore, in sectorized scenario there are M_BL/3 effective antenna elements in the network contributing to I_2^[kj] by transmitting their downlink signals with power ρ_f/M_B amplified by G_Q ≈ 3, creating the same amount of interference compared to the omnidirectional setting, where there are M_BL antenna elements contributing to I_2^[kj] by transmitting their downlink signals with power ρ_f/M_B.We observe that for both directional and omnidirectional antenna settings, with CPA and DPA, the received signal power is higher for low-SINR users, and interference power is less for all users compared with their UPA counterparts. We remark that the difference in the performance of DPA and CPA is small for both settings.We provide CDFs of the downlink achievable rates for sectoring, given in Theorem <ref>, and compare them for different settings in Figs. <ref>(a)-(c), for M_B=10^2, M_B=10^4, and M_B =10^6, respectively. For comparison we use the 0.95-likely rate per user criterion, defined as the rate achieved by 95% of the users, as in <cit.>.For small values of M_B, the total interference imposed on U_kj is dominated by undirected interference, which is similar for settings with and without sectoring. Therefore, directional arrays increase user SINR due to the increase in their received signal powers. For example with M_B=10^2, comparing the performance of Dir-UPA with Omni-UPA, given in Fig. <ref>(a), we observe that sectoring is able to increase the 0.95-likely rate by a factor of 5.20. We remark that as argued in Fig. <ref>(a) in Dir-UPA, the achievable rate of around 60% of the users with lower SINR has been improved with a sacrifice from the rate of user with higher SINR.For intermediate M_B, the two types of interference are comparable, and therefore, in addition to the increase in received signal power, directional arrays are able to alleviate the effect of the total interference. As illustrated in Fig. <ref>(b), for M_B=10^4 the 0.95-likely rate has an improvement with a factor of 5.47 with the Dir-UPA compared to Omni-UPA, and the rate of 66% of the users is increased. In the regime of very large M_B, pilot contamination is dominant, and therefore, as M_B→∞, SINR converges to a finite value. For M_B=10^6, given in Fig. <ref>(c), the 0.95-likely rate in Dir-UPA is 7.65× higher compared to Omni-UPA, with an improvement in achievable rate for 72% of the users.A comparison among Dir-UPA, Dir-CPA, and Dir-DPA for different M_B in Fig. <ref> reveals that optimized power allocation schemes can improve 0.95-likely rate by a factor between 1.48 and 2.04.We also would like to note that empirical CDF of achievable rate with decentralized power allocation (Dir-DPA) is only marginally different from the CDF of the optimal centralized power allocation (Dir-CPA), while using Dir-DPA allows us to reduce the required computation and communication overheads significantly.§ CONCLUSIONSIn this paper, we have studied the benefits of using directional antennas at the base station in a massive MIMO system. We have derived a lower bound on user downlink achievable rates, and have discussed centralized and decentralized power allocation strategies by formulating power optimization problems which differ in terms of performance and complexity. We have compared the performance of different massive MIMO settings with and without sectoring, and for different power allocation methods in terms of received signal power, pilot contamination, undirected interference and their achievable rate. The numerical results have revealed that while sectoring does not affect the undirected interference, it can alleviate the effect of pilot contamination and increase received signal power. Finally, we have discussed how sectoring and the use of directional antennas leads to higher 0.95-likely rate as a measure of reliability in the system. We have observed that by increasing the number of antennas at each base station, the improvement due to sectoring increases, due to the reduction of pilot contamination which is proportional to the number of antennas. Based on our simulation results, power optimization is an effective way to increase the 0.95-likely rate further. § PROOF OF THEOREM <REF>Due to space limit, we only provide a sketch of the proof here. As described in the TDD protocol given in Sec. <ref>, once the arrays have estimated the large-scale fading coefficients (step 1) and transmitted the decoding coefficients to their users (step 2), all users synchronously transmit their uplink signals and training sequences, respectively, in steps 3 and 4. Then, in step 5, each array estimates its channel vector using an MMSE estimate. More specifically, A_ji estimates the channel vector _ji^[kl] as:_ji^[kl]=θ_ji^[kl]√(ρ_r τ)∑_v=1^L √(G_ji^[kv])_ji^[kv] +_ji^[kl],where,θ_ji^[kl] =√(ρ_r τ G_ji^[kl])β_ji^[kl]/σ_r^2+ ρ_r τ∑_v = 1^L G_ji^[kv]β_ji^[kv],and, _ji^[kl]∼𝒞𝒩 ( 0, θ_ji^[kl]^2_M), where _M is M× M identity matrix.We assume that _ji^[kl] = _ji^[kl] + _ji^[kl] where _ji^[kl] denotes the MMSE estimation error. It can be shown that _ji^[kl]∼ CN ( 0,1/Mλ_ji^[kl]^2_M),_ji^[kl]∼ CN ( 0, (β_ji^[kl]- λ_ji^[kl]^2/M)_M ),where, λ_ji^[kl] is given in (<ref>).In step 6, array (j,i)∈ [L]× [3] uses conjugate beamforming based on its channel estimates to transmit the downlink signals {s^[kj]: k∈[K]} to its users, as_ji =∑_k=1^K√(ρ_ji^[kj])/λ_ji^[kj]_ji^[kj]^† s^[kj],where ρ_ji^[kj] denotes the power allocated to U_kj by A_ji. User U_kj receives the following downlink signal: y^[kj] = ∑_l=1^L∑_i=1^3√(G_li^[kj])_li_li^[kj] + w^[kj]= ∑_i=1^3√(ρ_ji^[kj] G_ji^[kj])/λ_ji^[kj][_ji^[kj]^†_ji^[kj]] s^[kj]_T_0 +∑_i=1^3√(ρ_ji^[kj] G_ji^[kj])/λ_ji^[kj]( _ji^[kj]^†_ji^[kj] - [_ji^[kj]^†_ji^[kj]]) s^[kj]_T_1 + ∑_l=1l≠ j^L∑_i=1^3√(ρ_li^[kl]G_li^[kj])/λ_li^[kl]_li^[kl]^†_li^[kj]s^[kl]_T_2 + ∑_l=1^L∑_i=1^3∑_n=1n≠ k^K√(ρ_li^[nl]G_li^[kj])/λ_li^[nl]_li^[nl]^†_li^[kj]s^[nl]_T_3 + ∑_l=1^L∑_i=1^3√(G_li^[kj])_li_li^[kj]+ w^[kj]_T_4, where, w^[kj]∼𝒞𝒩(0, 1) denotes the noise. T_0 corresponds to the part of the received signal that U_kj can decode, while T_1,…, T_4 contribute to the interference and noise. More specifically, using (<ref>) and (<ref>), it can be shown that T_0=s^[kj]∑_i ϵ^[kj]_ji, and given {ϵ^[kj]_ji:i ∈ [3]}, user U_kj can decode T_0 using (<ref>) to find ŝ^[kj] in step 7 of TDD protocol. Furthermore, it can be shown that any two of the terms T_0,…, T_4 are uncorrelated. According to Theorem 1 of <cit.>, the worst case of uncorrelated additive noise is independent Gaussian noise with the same variance. Hence, the worst-case downlink SINR of the of U_kj, denoted by SINR^[kj], isSINR^[kj]= [T_0]/[T_1]+[T_2]+[T_3]+[T_4].Therefore, U_kj's downlink rate, R^[kj], is lower bounded byR^[kj] = I (y^[kj];s^[kj]|ϵ^[kj]_j1,ϵ^[kj]_j2,ϵ^[kj]_j3)≥log_2 ( 1+SINR^[kj] ). It is straightforward to calculate variance of the terms T_0,…, T_4 based on (<ref>), (<ref>), (<ref>), and the channel statistics. Substituting these terms in (<ref>) will conclude the proof.§ PROOF OF PROPOSITION <REF>The constraints of problem (<ref>) are convex. To prove the quasi-concavity, it suffices to show that the objective function in (<ref>) is quasi-concave. Define Ω≜{ψ_li^nl,X_nl,Y_nl} for (l,i,n)∈ [L][3][K], the set of optimization variables. The objective function of (<ref>) is f(Ω)=min_k,j| ∑_i=1^3 ψ_ji^[kj]√( G_ji^[kj])λ_ji^[kj]|^2/ X_kj^2 +Y_kj^2 + σ^2_f. For every γ≥ 0, the upper-level set of f(Ω) is U(f,γ) ={Ω: f(Ω) ≥γ}={Ω: | ∑_i=1^3 ψ_ji^[kj]√( G_ji^[kj])λ_ji^[kj]|^2/ X_kj^2 +Y_kj^2 + σ^2_f≥γ, ∀ (k,j)}= {Ω:||V_kj|| ≤1/√(γ)∑_i=1^3 ψ_ji^[kj]√( G_ji^[kj])λ_ji^[kj], ∀ (k,j)},where V_kj≜ [X_kj,Y_kj,1]. Because U(f,γ) can be represented as a second order cone, it is a convex set. Therefore, f(Ω) is quasi-concave. IEEEtran | http://arxiv.org/abs/1707.09070v1 | {
"authors": [
"Shahram Shahsavari",
"Parisa Hassanzadeh",
"Alexei Ashikhmin",
"Elza Erkip"
],
"categories": [
"cs.IT",
"math.IT"
],
"primary_category": "cs.IT",
"published": "20170727230715",
"title": "Sectoring in Multi-cell Massive MIMO Systems"
} |
Dynamic band structure tuning of graphene moiré superlattices with pressure Cory R. Dean^1 December 30, 2023 =========================================================================== §.§.§ AbstractBackground: Causal mediation analysis can improve understanding of the mechanisms underlying epidemiologic associations. However, the utility of natural direct and indirect effect estimation has been limited by the assumption of no confounder of the mediator-outcome relationship that is affected by prior exposure—an assumption frequently violated in practice.Methods: We build on recent work that identified alternative estimands that do not require this assumption and propose a flexible and double robust semiparametric targeted minimum loss-based estimator for data-dependent stochastic direct and indirect effects. The proposed method treats the intermediate confounder affected by prior exposure as a time-varying confounder and intervenes stochastically on the mediator using a distribution which conditions on baseline covariates and marginalizes over the intermediate confounder.In addition, we assume the stochastic intervention is given, conditional on observed data, which results in a simpler estimator and weaker identification assumptions.Results: We demonstrate the estimator's finite sample and robustness properties in a simple simulation study. We apply the method to an example from the Moving to Opportunity experiment. In this application, randomization to receive a housing voucher is the treatment/instrument that influenced moving to a low-poverty neighborhood, which is the intermediate confounder. We estimate the data-dependent stochastic direct effect of randomization to the voucher group on adolescent marijuana use not mediated by change in school district and the stochastic indirect effect mediated by change in school district. We find no evidence of mediation.Conclusions: Our estimator is easy to implement in standard statistical software, and we provide annotated R code to further lower implementation barriers.Keywords: mediation; direct effect; indirect effect; double robust; targeted minimum loss-based estimation; targeted maximum likelihood estimation; data-dependent § INTRODUCTION Mediation allows for an examination of the mechanisms driving a relationship. Much of epidemiology entails reporting exposure-outcome associations where the exposure may be multiple steps removed from the outcome. For example, risk-factor epidemiology demonstrated that obesity increases the risk of type 2 diabetes, but biochemical mediators linking the two have advanced our understanding of the causal relationship <cit.>. Mediators have been similarly important in understanding how social exposures act to affect health outcomes. In the illustrative example we consider in this paper, the Moving to Opportunity (MTO) experiment randomized families living in public housing to receive a voucher that they could use to rent housing on the private market, which reduced their exposure to neighborhood poverty <cit.>. Ultimately, being randomized to receive a voucher affected subsequent adolescent drug use <cit.>. In the illustrative example, we test the extent to which the effect operates through a change in the adolescent's school environment. Causal mediation analysis <cit.> (also called mediation analysis using the counterfactual framework <cit.>) shares similar goals with the standard mediation approaches, e.g., structural equation modeling and the widely used Baron and Kenny “product method” approach <cit.>. They all aim to test mechanisms and estimate direct and indirect effects. Advantages of causal mediation analysis include that estimates have a causal interpretation (under specified identifying assumptions) and some approaches make fewer restrictive parametric modeling assumptions. For example, in contrast to traditional approaches, approaches within the causal mediation framework 1) allow for interaction between the treatment and mediator <cit.>, 2) allow for modeling nonlinear relationships between mediators and outcomes <cit.>, and 3) allow for incorporation of data-adaptive machine learning methods and double robust estimation <cit.>. However, despite these advantages, the assumptions required to estimate certain causal mediation effects may sometimes be untenable; for example, the assumption that there is no confounder of the mediator-outcome relationship that is affected by treatment (in the literature, such a confounder is referred to as confounding by a causal intermediate <cit.>, a time-varying confounder affected by prior exposure <cit.>, or time-dependent confounding by an intermediate covariate <cit.>). For brevity, we will refer to such a variable as an intermediate confounder. There have been recently proposed causal mediation estimands, called randomized (i.e., stochastic) interventional direct effects and interventional indirect effects that do not require this assumption <cit.>. We build on this work, proposing a robust and flexible estimator for these effects, which we call stochastic direct and indirect effects (SDE and SIE).This paper is organized as follows. In the following section, we review and compare common causal mediation estimands, providing the assumptions necessary for their identification. Then, we describe our proposed estimator, its motivation, and its implementation in detail. Code to implement this method is provided in the Appendix. We then provide results from a limited simulation study demonstrating the estimator's finite sample performance and robustness properties. Lastly, we apply the method in a longitudinal, randomized trial setting.§ NOTATION AND CAUSAL MEDIATION ESTIMANDS Let observed data: O=(W, A, Z, M, Y) with n i.i.d. copies O_1,...,O_n ∼ P_0, where W is a vector of pre-treatment covariates, A is the treatment, Z is the intermediate confounder affected by A, M is the mediator, and Y is the outcome. For simplicity, we assume that A, Z, M,and Y are binary. In our illustrative example, A is an instrument, so it is reasonable to assume that M and Y are not affected by A except through its effect on Z. Mirroring the structural causal model (SCM) of our illustrative example, we assume that M is affected by {Z, W} but not A, and that Y is affected by {M, Z, W} but not A. We assume exogenous random errors: (U_W, U_A, U_Z, U_M, U_Y). This SCM is represented in Figure <ref> and the following causal models: W=f(U_W), A=f(U_A), Z=f(A, W, U_Z), M=f(Z,W,U_M), and Y=f(M,Z,W,U_Y). Note that this SCM (including that U_Y and U_M are not affected by A) puts the following assumptions on the probability distribution: P(Y | M, Z, A, W) = P(Y | M, Z, W) and P(M | Z, A, W) = P(M | Z, W). However, our approach generalizes to scenarios where A also affects M and Y as well as to scenarios where A is not random. We provide details and discuss these generalizations in the Appendix.We can factorize the likelihood for the SCM reflecting our illustrative example as follows: P(O) = P(Y | M, Z, W)P(M|Z,W)P(Z | A, W)P(A)P(W). Causal mediation analysis typically involves estimating one of two types of estimands: controlled direct effects (CDE) or natural direct and indirect effects (NDE, NIE). Controlled direct effects involve comparing expected outcomes under different values of the treatment and setting the value of the mediator for everyone in the sample. For example the CDE can be defined: E(Y_a, m - Y_a*, m), where Y_a,m, Y_a^*,m, is the counterfactual outcome setting treatment A equal to a or a^*, respectively (the two treatment values being compared), and setting mediator M equal to m. In contrast, the NDE can be defined: E(Y_a, M_a^* - Y_a*, M_a^*), where Y_a,M_a^*, Y_a^*,M_a^* is the counterfactual outcome setting A equal to a or a^* but this time setting M_a^* to be the counterfactual value of the mediator had A been set to a^* (possibly contrary to fact). Similarly, the NIE can be defined: E(Y_a, M_a - Y_a, M_a*). Natural direct and indirect effects are frequently used in epidemiology and have the appealing property of adding to the total effect <cit.>.Although the NDE and NIE are popular estimands, their identification assumptions may sometimes be untenable. Broadly, identification of their causal effects relies on the sequential randomization assumption on intervention nodes A and M and positivity. Two specific ignorability assumptions are required to identify CDEs and NDE/NIEs: 1) A ⊥ Y_a,m | W and2) M ⊥ Y_a,m | W, A <cit.>. The positivity assumptions are: P(M =m | A=a,W) > 0a.e. and P(A=a | W) > 0a.e. Two additional ignorability assumptions are required to identify NDE/NIEs: 3) A ⊥ M_a | W and 4) M_a*⊥ Y_a,m | W <cit.>. This last assumption states that, conditional on W, knowledge of M in the absence of treatment A provides no information of the effect of A on Y <cit.>. This assumption is violated when there is a confounder of the M-Y relationship that is affected by A (i.e., an intermediate confounder) <cit.>. This assumption is also problematic because it involves independence of counterfactuals under separate worlds (a and a^*) which can never simultaneously exist. This last assumption that there is no confounder of the mediator-outcome relationship affected by prior treatment is especially concerning for epidemiology studies where longitudinal cohort data may reflect a data structure in which a treatment affects an individual characteristic measured at follow-up that in turn affects both a mediating variable and the outcome variable (see <cit.> for some examples). It is also problematic for mediation analyses involving instrumental variables such as randomized encouragement-design interventions where an instrument, A, encourages treatment uptake, Z, which then may influence Y potentially through M. Such a design is present in the MTO experiment that we will use as an illustrative example. Randomization to receive a housing voucher (A) was the instrument that “encouraged" the treatment uptake, moving with the voucher out of public housing (Z, which we will call the intermediate confounder). In turn, Z may influence subsequent drug use among adolescent participants at follow-up (Y), possibly through a change the children's school environment (M). In the illustrative example, our goal was to examine mediation of the effect of receiving a housing voucher (A) on subsequent drug use (Y) by changing school districts (M) in the MTO data.There has been recent work to relax the assumption of no intermediate confounder, M_a*⊥ Y_a,m | W, by using a stochastic intervention on M <cit.>.In this paper, we build on the approach described by <cit.> in which they defined the stochastic distribution on M as: g_M|a,W or g_M|a^*,W, whereg_M|A,W(m,a^*,W) ≡ g_M|a^*,W(W) = ∑_z=0^1 P(M=1|Z=z,W)P(Z=z | A=a^*, W).(Equation 1 is an example of <cit.>'s work.)In other words, instead of formulating the individual counterfactual values of M_a or M_a^*, values are stochastically drawn from the distribution of M, conditional on covariates W but marginal over intermediate confounder Z, setting A=a or A=a^*, respectively.The correspondingestimands of interest are the SDE = E(Y_a,g_M|a^*,W) - E(Y_a*,g_M|a^*,W), and SIE = E(Y_a,g_M|a,W) - E(Y_a,g_M|a^*,W). Others have taken a similar approach. For example, <cit.> formulate a stochastic intervention on M that is fully conditional on the past:g_M|Z,A,W(m,Z,a^*,W) ≡ g_M|Z,a^*,W (Z,W) = P(M=1|Z, A=a^*, W)(note that per our SCM, P(M=1 | Z,W) = P(M=1 | Z,A,W), so in our case, g_M|Z,a^*,W (Z,W) = P(M=1|Z, W).) The corresponding estimands are the stochastic direct and indirect effects fully conditional on the past: CSDE = E(Y_a,g_M|Z,a^*,W) - E(Y_a*,g_M|Z,a^*,W), and CSIE = E(Y_a,g_M|Z,a,W) - E(Y_a,g_M|Z,a^*,W). However, <cit.>'s formulation shown in Equation <ref> is not useful for understanding mediation under the instrumental variable SCM we consider here, as there is no direct pathway from A to M or from A to Y. Because of the restriction on our statistical model that P(M | Z, A, W) = P(M | Z, W), g_M|Z,a^*,W (Z,W) = g_M|Z,a,W (Z,W), so CSIE's under this model would equal 0. Thus, in this scenario, the NDE and CSDE are very different parameters. We note that it is also because of these restrictions on our statistical model stemming from the instrumental variable SCM that the sequential mediation analysis approach proposed by <cit.> would also result in indirect effects equal to 0. Because the CSIE and CSDE do not aid in understanding the role of M as a potential mediator in this scenario, we focus instead on <cit.>'s SDE and SIE that condition on W but marginalize over Z, thus completely blocking arrows into M (similar to an NDE and NIE). The SDE and SIE coincide with the NDE and NIE in the absence of intermediate confounders <cit.>.§.§ SDE and SIE Estimands and IdentificationOur proposed estimator can be used to estimate two versions of the SDE and SIE: 1) fixed parameters that assume an unknown, true g_M|a^*,W; and 2)data-dependent parameters that assume known g_M|a^*,W, estimated from the observed data, which we call ĝ_M|a^*,W. Researchers may have defensible reasons for choosing one version over the other, which we explain further below. The fixed SDE and SIE can be identified from the observed data distribution using the g-computation formula as discussed by <cit.>, assuming the sequential randomization assumption on intervention nodes A and M: 1) A ⊥ Y_a,m | W, 2) M ⊥ Y_a,m | W, A=a, Z, and 3) A ⊥ M_a | W, for a particular a and g_M|a^*,W. The data-dependent SDE And SIE can be identified similarly but need only the first two assumptions, 1) A ⊥ Y_a,m | W and 2) M ⊥ Y_a,m | W, A=a, Z, because ĝ_M|a^*,W is assumed known. If any of the above identification assumptions are violated, then the statistical estimands will not converge to their true causal quantities.The estimand E(Y_a,ĝ_M|a^*W) can be identified via sequential regression, which provides the framework for our proposed estimator that follows. For intervention (A=a, M=ĝ_M|a^*,W), we have Q̅_M^ĝ(Z,W)≡∫_m ∈ℳ∫_y ∈𝒴 p(y | m, Z, W) dμ_Y(y) ĝ_m|a^*,W dμ_M(m),where we integrate out M under our stochastic intervention ĝ_M|a^*,W, and where M has support ℳ and Y has support 𝒴 and where dμ_Y(y) and dμ_M(m) are some dominating measures. This is accomplished by evaluating E(Y|M=m,Z=z,W) at each m and multiplying it by the probability that M=m under ĝ_M|a^*,W, summing over all m. We then integrate out Z and set A=a: Q̅_Z^a(W)≡∫_z ∈𝒵Q̅_M^ĝ(z,W|A=a,W) p(z | A=a, W) dμ_Z(z), where 𝒵 denotes the support of random variable Z and dμ_Z(z) is some dominating measure. Marginalizing over the distribution of W gives the statistical parameter: Ψ(P)(a,ĝ_M|a^*,W)= ∫_w ∈𝒲Q̅_Z^a(w) p(w) dμ_W(w), where 𝒲 denotes the support of random variable W and dμ_W(w) is some dominating measure.In the next section, we propose a novel, robust substitution estimator that can be used to estimate both the fixed and parametric versions of the SDE and SIE. Inference for the fixed SDE and SIE can be obtained by using the bootstrapped variance, which requires parametric models for the nuisance parameters P(A) and P(M | Z,A,W). This is the same inference strategy as proposed by <cit.>. However, researchers may encounter scenarios for which fitting parametric models is unappealing and using machine learning approaches is preferred. The data-dependent SDE and SIE with inference based on the efficient influence curve (EIC) may be preferable in such scenarios. In contrast to the EIC for an assumed known g_M|a^*,W, the EIC an unknown g_M|a^*,W is complicated due to the dependence of the unknown marginal stochastic intervention on the data distribution. Such an EIC would include an M component, the form of which would be more complex due to the distribution of M being marginalized over Z. No statistical tools for solving an EIC of that form currently exist. Solving the EIC for the the parameter Ψ(P)(a,g_M|a^*,W) for an unknown g_M|a^*,W is ongoing work. § TARGETED MINIMUM LOSS-BASED ESTIMATOROur proposed estimator uses targeted minimum loss-based estimation (TMLE) <cit.>, targeting the stochastic, counterfactual outcomes that comprise the SDE and SIE. To our knowledge, it is the first such estimator appropriate for instrumental variable scenarios. TMLE is a substitution estimation method that solves the EIC estimating equation. Its robustness properties differ for the fixed and data-dependent parameters. For the data-dependent SDE and SIE, if either the Y model is correct or the A and M models given the past are correct, then one obtains a consistent estimator of the parameter. Robustness to misspecification of the treatment model is relevant under an SCM with nonrandom treatment; we discuss the generalization of our proposed estimator to such an SCM in the Appendix. Note that ĝ_M|a^*,W for the stochastic intervention is not the same as the conditional distribution of M given the past, so the first could be inconsistent while the latter is consistent. For the fixed SDE and SIE, we also need to assume consistent estimation of g_M|a^*,W, since it does not target g_M|a^*,W (and is therefore not a full TMLE for the fixed parameters).The estimator integrates two previously developed TMLEs: one for stochastic interventions <cit.> and one for multiple time-point interventions <cit.>, which is built on the iterative/recursive g-computation approach <cit.>. This TMLE is not efficient under the SCM considered, because of the restriction on our statistical model that P(Y | M,Z,A,W) = P(Y|M,Z,W). However, it is still a consistent estimator if that restriction on our model does not hold (i.e., P(Y | M,Z,A,W)P(Y|M,Z,W)), because the targeting step adds dependence on A. The TMLE is constructed using the sequential regressions described in the above section with an additional targeting step after each regression. The TMLE solves the EIC for the target parameter that treats g_M|a^*,W as given. A similar EIC has been described previously <cit.>. The EIC for the parameter Ψ(P)(a, g_M|a^*,W) for a given g_M|a^*,W is given by:D^*(a, g_M|a^*,W) = ∑_k=0^2 D_k^*(a, g_M|a^*,W),where D^*_0(a, g_M|a^*,W) = Q̅^a_Z(W) - Ψ(P)(a, g_M|a^*,W)D^*_1(a, g_M|a^*,W) = I(A=a)/P(A=a | W)(Q̅^g_M(Z,W) - Q̅^a_Z(W))D^*_2(a, g_M|a^*,W) = I(A=a){I(M=1)g_M|a^*,W + I(M=0)(1-g_M|a^*,W) }/P(A=a | W){I(M=1)g_M|z,W + I(M=0)(1-g_M|z,W) }× (Y-Q̅_Y(M,Z,W)).Substitution of g_M|a^*,W=ĝ_M|a^*,W yields the EIC used for the data-dependent parameter Ψ(P)(a, ĝ_M|a^*,W). The EIC for the parameter Ψ(P)(a, g_M|a^*,W) in which the stochastic intervention equals the unknown g_M|a^*,W is an area of future work. We now describe how to compute the TMLE. In doing so, we use parametric model/regression language for simplicity but data-adaptive estimation approaches that incorporate machine learning <cit.> may be substituted and may be preferable (we use such a data-adaptive approach in the illustrative example analysis). We note that survey or censoring weights could be incorporated into this estimator as described previously <cit.>. We use notation reflecting estimation of the data-dependent parameters, but note that estimation of the fixed parameters would be identical—in the fixed parameter case, the notation would refer to g_M|a^*,W instead of ĝ_M|a^*,W.First, one estimates ĝ_M|a^*,W(W), which is the estimate of g_M|a^*,W(W), defined in Equation 1, using observed data. Consider a binary Z. We estimate g_Z|a^*,W(W)=P(Z=1 | A=a^*, W). We then estimate g_M|z,W(W)=P(M=1 | Z=z, W) for z ∈{0,1}. We use these quantities to calculate ĝ_M|a^*,W = ĝ_M|z=1,Wĝ_Z|a^*,W +ĝ_M|z=0,W(1-ĝ_Z|a^*,W). We can obtain ĝ_Z|a^*,W(W) from a logistic regression of Z on A, W setting A=a^*, and ĝ_M|z,W(W) from a logistic regression of M on Z, W, setting Z={0,1}. We will then use this stochastic intervention in the TMLE, whose implementation is described as follows.* Let Q̅_Y,n(M,Z,W) be an estimate of Q̅_Y(M,Z,W)≡ E(Y |M,Z,W). To obtain Q̅_Y,n(M,Z,W), predict values of Y from a regression of Y on M,Z,W. * Estimate the weights to be used for the initial targeting step:h_1(a) = I(A=a){I(M=1)ĝ_M|a^*,W + I(M=0)(1-ĝ_M|a^*,W) }/P(A=a){I(M=1)g_M|Z,W + I(M=0)(1-g_M|Z,W) }, where estimates of g_M|Z,W are predicted probabilities from a logistic regression of M=m on Z and W. Let ĥ_1,n(a) denote the estimate of h_1(a). * Target the estimate of Q̅_Y,n(M,Z,W) by considering a univariate parametric submodel {Q̅_Y,n(M,Z,W)(ϵ):ϵ} defined as: logit(Q̅_Y,n (ϵ)(M,Z,W)) = logit(Q̅_Y,n(M,Z,W) ) + ϵ.Let ϵ_n be the MLE fit of ϵ. We obtain ϵ_n by setting ϵ as the intercept of a weighted logistic regression model of Y with logit(Q̅_Y,n(M,Z,W)) as an offset and weights ĥ_1,n(a). (Note that this is just one possible TMLE.)The update is given by Q̅^*_Y,n(M,Z,W) =Q̅_Y,n(ϵ_n)(M,Z,W). Y can be bounded to the [0,1] scale as previously recommended <cit.>.* Let Q̅^g_M,n(Z,W) be an estimate of Q̅^g_M(Z,W). To obtain Q̅^g_M,n(Z,W), we integrate out M to from Q̅^*_Y,n(M,Z,W). First, we estimate Q̅^*_Y,n(M,Z,W) setting m=1 and m=0, giving Q̅^*_Y(m=1, z, w) and Q̅^*_Y(m=0, z, w). Then, multiply these predicted values by their probabilities under ĝ_M|a^*,W(W) (for a ∈{a, a^*}), and add them together (i.e., Q̅^ĝ_M,n(Z,W)= Q̂^*_Y(m=1, z, w)ĝ_M|a^*,W + Q̂^*_Y(m=0, z, w)(1-ĝ_M|a^*,W)).* We now fit a regression of Q̅^ĝ,*_M,n(Z,W) on W among those with A=a. We call the predicted values from this regression Q̅^a_Z,n(W). The empirical mean of these predicted values is the TMLE estimate of Ψ(P)(a, ĝ_M|a^*,W).* Repeat the above steps for each of the interventions. For example, for binary A, we would execute these steps a total of three times to estimate: 1) Ψ(P)(1,ĝ_M|1,W), 2) Ψ(P)(1,ĝ_M|0W), and 3) Ψ(P)(0,ĝ_M|0,W).* The SDE can then be obtained by substituting estimates of parameters Ψ(P)(a,ĝ_M|a^*,W) - Ψ(P)(a^*,ĝ_M|a^*,W) and the SIE can be obtained by substituting estimates of parameters Ψ(P)(a,ĝ_M|a,W) - Ψ(P)(a,ĝ_M|a^*,W).* For the fixed parameters, the variance can be estimated using the bootstrap. For the data-dependent parameters, the variance of each estimate from Step 6 can be estimated as the sample variance of the EIC (defined above, substituting in the targeted fits Q̅^*_Y,n(M,Z,W) and Q̅^a,*_Z,n(W)) divided by n. First, we estimate the EIC for each component of the data-dependent SDE/SIE, which we call EIC_Ψ(P)(a,ĝ_M|a^*,W). Then we estimate the EIC for the estimand of interest by subtracting the EICs corresponding to the components of the estimand. For example EIC_SDE = EIC_Ψ(P)(a,ĝ_M|a^*,W) - EIC_Ψ(P)(a^*,ĝ_M|a^*,W). The sample variance of this EIC divided by n is the influence curve (IC)-based variance of the data-dependent estimator.§ SIMULATION §.§ Data generating mechanismWe conduct a simulation study to examine finite sample performance of the TMLE estimators for the fixed SDE and SIE and data-dependent SDE and SIE from the data-generating mechanism (DGM) shown in Table <ref>. Under this DGM, the data-dependent SDE is: SDE=E(Y_1, ĝ_M|0,W) - E(Y_0, ĝ_M|0,W) and the SIE is: SIE=E(Y_1, ĝ_M|1,W) - E(Y_1, ĝ_M|0,W). The fixed versions are defined with respect to the unknown, true g_M|1,W and g_M|0,W. Table <ref> uses the same notation and SCM as in Section 2, with the addition of Δ, an indicator of selection into the sample (which corresponds to the MTO data used in the empirical illustration where one child from each family is selected to participate). We compare performance of the TMLE estimator to an inverse-probability weighted estimator (IPTW) and estimator that solves the EIC estimating equation (EE) but differs from TMLE in that it lacks the targeting steps and is not a plug-in estimator, so its estimates are not guaranteed to lie within the parameter space (which may be particularly relevant for small sample sizes). Variance for the fixed SDE and SIE parameters is calculated using 500 bootstrapped samples for each simulation iteration. Variance for the data-dependent SDE and SIE is calculated using the EIC. We show estimator performance in terms of absolute bias, percent bias, closeness to the efficiency bound (mean estimator standard error (SE) × the square root of the number of observations), 95% confidence interval (CI) coverage, and mean squared error (MSE) across 1,000 simulations for sample sizes of N=5,000, N=500, and N=100. In addition, we consider 1) correct specification of all models, and 2) misspecification of the Y model that included a term for Z only.§.§ PerformanceTable <ref> gives simulation results under correct model specification for fixed SDE and SIE using bootstrap-based variance. Table <ref> gives simulation results under correct model specification for data-dependent SDE and SIE using IC-based variance. Both Tables <ref> and <ref> show that the TMLE, IPTW, and EE estimators are consistent when all models are correctly specified, showing biases of around 1% or less under large sample size (N=5,000) and slightly larger biases with smaller sample sizes. The 95% CIs for the TMLE and EE estimators result in similar coverage that is close to 95%, except for estimation of the SIE with a sample size of N=100, which has coverage closer to 90%. Confidence intervals for the IPTW estimator for the fixed parameter are close to 95% but are conservative and close to 100% for the data-dependent parameter. As expected, IPTW is less efficient than TMLE or EE; the TMLE and EE estimators perform similarly and close to the efficiency bound for all sample sizes.Table <ref> gives simulation results under misspecification of the outcome model that only includes a term for Z for fixed SDE and SIE using bootstrap-based variance. Thus, comparing results in Table <ref> to Table <ref> demonstrates robustness to misspecification of the outcome model. As all three of the estimators evaluated are theoretically robust to misspecification of this model, we would expect similar results between the two Tables, and we see that is indeed the case. § EMPIRICAL ILLUSTRATION §.§ Overview and set-upWe now apply our proposed estimator to MTO: a longitudinal, randomized trial that is described above. Because we wish to use machine learning for this empirical illustration, we will estimate the data-dependent SDE of being randomized to receive a housing voucher (A) on marijuana use (Y) not mediated by change in school district (M) and the data-dependent SIE mediated by M among adolescent boys in the Boston site in the presence of an intermediate confounder (Z), moving with the voucher out of public housing. We restrict to adolescents less than 18 years old who were present at interim follow-up, as those participants had school data and were eligible to be asked about marijuana use. We restrict to boys in the Boston site as previous work has shown important quantitative and qualitative differences in MTO's effects by sex <cit.> and by city <cit.>. We choose to present results from a restricted analysis instead of a stratified analysis, as our goal is to illustrate the proposed method. A more thorough mediation analysis considering all sexes and sites is the subject of a future paper. Marijuana use was self-reported by adolescents at the interim follow-up, which occurred 4-7 years after baseline, and is defined as ever versus never use. Change in school district is defined as the school at follow-up and school at randomization being in the same district. Numerous baseline characteristics included individual and family sociodemographics, motivation for participating in the study, neighborhood perceptions, school-related characteristics of the adolescent, and predictive interactions. We use machine learning to flexibly and data-adaptively model the following relationships: instrument to intermediate confounder, intermediate confounder to mediator, and mediator to outcome. Specifically, we use least absolute shrinkage and selection operator (lasso) <cit.> and choose the model that improves 10-fold cross-validation prediction error, while always including age and race/ethnicity and relevant A, Z, and M variables.§.§ ResultsFigure <ref> shows the data-dependent SDE and SIE estimates by type of estimator (TMLE, IPTW, and EE) for boys in the Boston MTO site (N=228). SDE and SIE estimates are similar across estimators. We find no evidence that change in school district mediated the effect of being randomized to the voucher group on marijuana use, with null SIE estimates (TMLE risk difference: -0.003, 95% CI: -0.032, 0.026). The direct effect of randomization to the housing voucher group on marijuana use suggests that boys who were randomized to this group were 9% more likely to use marijuana than boys in the control group, though this difference is not statistically significant (TMLE risk difference: 0.090, 95% CI: -0.065-0.245). § DISCUSSIONWe proposed robust targeted minimum loss-based estimators to estimate fixed and data-dependent stochastic direct and indirect effects that are the first to naturally accommodate instrumental variable scenarios. These estimators build on previous work identifying and estimating the SDE and SIE <cit.>. The SDE and SIE have the appealing properties of 1) relaxing the assumption of no intermediate confounder affected by prior exposure, and 2) utility in studying mediation in the context of instrumental variables that adhere to the exclusion restriction assumption (a common assumption of instrumental variables which states that there is no direct effect between A and Y or between A and M <cit.>) due to completely blocking arrows into the mediator by marginalizing over the intermediate confounder, Z. Given the restrictions that this assumption places on the statistical model, several alternative estimands are not appropriate for understanding mediation in this context as the indirect effect would always equal zero <cit.>.Inference for the fixed SDE and SIE can be obtained from bootstrapping, using parametric models for nuisance parameters. Inference for the data-dependent SDE and SIE can be obtained from the data-dependent EIC that assumes known ĝ_M|a^*,W estimated from the data, and is appropriate for integrating machine learning in modeling nuisance parameters. The ability to incorporate machine learning is a significant strength in this case; if using the parametric alternative, multiple models would need to be correctly specified <cit.>. IC-based variance is possible in estimating the data-dependent SDE and SIE, because the data-dependent EIC has a form that is solvable using existing statistical tools; in contrast, the EIC for the fixed parameters is more complex and is not solvable with current statistical tools. Our proposed estimator for the fixed and data-dependent parameters is simple to implement in standard statistical software, and we provide R code to lower implementation barriers. Another advantage of our TMLE estimator, which is shared with other estimating equation approaches, is that it is robust to some model misspecification. In estimating the data-dependent SDE and SIE, one could obtain a consistent estimate as long as either the Y model or the A and M models given the past were correctly specified. Obtaining a consistent estimate of the fixed SDE and SIE would also require consistent estimation of g_M|a^*,W.In addition, our proposed estimation strategy is less sensitive to positivity violations than weighting-based approaches. First, TMLE is usually less sensitive to these violations than weighting estimators, due in part to it being a substitution estimator, which means that its estimates lie within the global constraints of the statistical model. This is in contrast to alternative estimating equation approaches, which may result in estimates that lie outside the parameter space. Second, we formulate our TMLE such that the targeting is done as a weighted regression, which may smooth highly variable weights <cit.>. In addition, moving the targeting into the weights improves computation time <cit.>.However, there are also limitations to the proposed approach. We have currently only implemented it for a binary A and M, though extensions to multinomial or continuous versions of those variables are possible <cit.>. Extending the estimator to allow for a high-dimensional M is less straightforward, though it is of interest and an area for future work as allowing for high-dimensional M is a strength of other mediation approaches <cit.>. We also plan to focus future work on developing a full TMLE for the fixed SDE and SIE parameters. DeGruyter § GENERALIZATIONS TO OTHER STRUCTURAL CAUSAL MODELS §.§ Nonrandom TreatmentLet observed data: O=(W, A, Z, M, Y) with n i.i.d. copies O_1,...,O_n ∼ P_0, where W is a vector of pre-treatment covariates, A is the treatment, Z is the intermediate confounder affected by A, M is the mediator, and Y is the outcome. For simplicity, we assume that A, Z, M,and Y are binary. We assume that M and Y are not affected by A except through its effect on Z. We assume that A is affected by {W}, Z is affected by {A, W}, M is affected by {Z, W} but not A, and that Y is affected by {M, Z, W} but not A. We assume exogenous random errors: (U_W, U_A, U_Z, U_M, U_Y). Note that this SCM (including that U_Y and U_M are not affected by A) puts the following assumptions on the probability distribution: P(Y | M, Z, A, W) = P(Y | M, Z, W) and P(M | Z, A, W) = P(M | Z, W). We can factorize the likelihood for this SCM as follows: P(O) = P(Y | M, Z, W)P(M|Z,W)P(Z | A, W)P(A | W)P(W).The data-dependent, stochastic mediation estimand E(Y_a,ĝ_M|a^*W) can be identified via sequential regression, which provides the framework for our proposed estimator that follows. For intervention (A=a, M=ĝ_M|a^*,W), we have Q̅_M^ĝ(Z,A,W)≡ E_ĝ_M|a^*,W(E(Y |M,Z,A,W)|Z,A,W), where we integrate out M under our stochastic intervention ĝ_M|a^*,W. This is accomplished by evaluating the inner expectation at each m and multiplying it by the probability that M=m under ĝ_M|a^*,W, summing over all m. We then integrate out Z and set A=a: Q̅_Z^a(W)≡ E_Z(Q̅_M^ĝ(Z,A,W)|A=a,W). Taking the empirical mean gives the statistical parameter: Ψ(P)(a,ĝ_M|a^*,W)=E_W(Q̅_Z^a(W)|W). The EIC for the parameter Ψ(P)(a, ĝ_M|a^*,W) is given by D^*(a, ĝ_M|a^*,W) = ∑_k=0^2 D_k^*(a, ĝ_M|a^*,W),where D^*_0(a, ĝ_M|a^*,W) = Q̅^a_Z(W) - Ψ(P)(a, ĝ_M|a^*,W)D^*_1(a, ĝ_M|a^*,W) = I(A=a)/P(A=a | W)(Q̅^ĝ_M(Z,W) - Q̅^a_Z(W))D^*_2(a, ĝ_M|a^*,W) = I(A=a){I(M=1)ĝ_M|a^*,W + I(M=0)(1-ĝ_M|a^*,W) }/P(A=a | W){I(M=1)g_M|Z,W + I(M=0)(1-g_M|Z,W) }(Y-Q̅_Y(M,Z,W)). We now describe how to compute the TMLE. The estimation of ĝ_M|a^*,W(W) does not differ from that described in the main text. * Let Q̅_Y,n(M,Z,W) be an estimate of Q̅_Y(M,Z,W)≡ E(Y |M,Z,W). To obtain Q̅_Y,n(M,Z,W), predict values of Y from a regression of Y on M,Z,W. * Estimate the weights to be used for the initial targeting step:h_1(a) = I(A=a){I(M=1)ĝ_M|a^*,W + I(M=0)(1-ĝ_M|a^*,W) }/P(A=a){I(M=1)g_M|Z,W + I(M=0)(1-g_M|Z,W) }, where ĝ_M|Z,W are predicted probabilities from a logistic regression of M=m on Z and W. Let h_1,n(a) denote the estimate of h_1(a). * Target the estimate of Q̅_Y,n(M,Z,W) by considering a univariate parametric submodel {Q̅_Y,n(M,Z,W)(ϵ):ϵ} defined as: logit(Q̅_Y,n (ϵ)(M,Z,W)) = logit(Q̅_Y,n(M,Z,W) ) + ϵ.Let ϵ_n be the MLE fit of ϵ. We obtain ϵ_n by setting ϵ as the intercept of a weighted logistic regression model of Y with logit(Q̅_Y,n(M,Z,W)) as an offset and weights h_1,n(a). (Note that this is just one possible TMLE.)The update is given by Q̅^*_Y,n(M,Z,W) =Q̅_Y,n(ϵ_n)(M,Z,W). Y can be bounded to the [0,1] scale.* Let Q̅^ĝ_M,n(Z,W) be an estimate of Q̅^ĝ_M(Z,W). To obtainQ̅^ĝ_M,n(Z,W), we integrate out M from Q̅^*_Y,n(M,Z,W). First, we estimate Q̅^*_Y,n(M,Z,W) setting m=1 and m=0, giving Q̅^*_Y,n(m=1,Z,W) and Q̅^*_Y,n(m=0,Z,W). Then, multiply these predicted values by their probabilities under ĝ_M|a^*,W(W) (for a ∈{a,a^*}), and add them together (i.e., Q̅^ĝ_M,n(Z,W) = Q̅^*_Y,n(m=1,Z,W)*ĝ_M|a^*,W + Q̅^*_Y,n(m=0,Z,W)*(1-ĝ_M|a^*,W)). * We now fit a regression of Q̅^ĝ,*_M,n(Z,W) on W among those with A=a. We call the predicted values from this regression Q̅^a_Z,n(W). * Complete a second targeting step: logit(Q̅^a_Z,n(ϵ)(W)) = logit(Q̅^a_Z,n(W)) + ϵ h_2,n(a), where h_2,n(a) is an estimate of h_2(a) = I(A=a)/P(A=a | W) and P(A=a | W) can be estimated from a logistic regression model of A=a on W. Let ϵ_n again be the MLE fit of ϵ, which can be obtained by fitting an intercept-only weighted logistic regression model of Q̅^ĝ_M,n(Z,W) with logit(Q̅^a_Z,n(W)) as an offset and weights h_2,n(a). (Alternatively, we could fit an unweighted logistic regression model of Q̅^ĝ_M,n(Z,W) with logit(Q̅^a_Z,n(W)) as an offset and h_2,n(a) as a covariate, where ϵ_n is the fitted coefficient on h_2,n(a).) The update is given by Q̅^a,*_Z,n(W) = Q̅^a,a^*_Z,n(ϵ_n)(A,W).* The TMLE of Ψ(P)(a, ĝ_M|a^*,W) is the empirical mean of Q̅^a,*_Z,n(W).* Repeat the above steps for each of the interventions. For example, for binary A, we would execute these steps a total of three times to estimate: 1) Ψ(P)(1,ĝ_M|1,W), 2) Ψ(P)(1,ĝ_M|0W), and 3) Ψ(P)(0,ĝ_M|0,W).* The SDE can then be obtained by substituting estimates of parameters Ψ(P)(a,ĝ_M|a^*,W) - Ψ(P)(a^*,ĝ_M|a^*,W) and the SIE can be obtained by substituting estimates of parameters Ψ(P)(a,ĝ_M|a,W) - Ψ(P)(a,ĝ_M|a^*,W).* The variance of each estimate can be estimated as the sample variance of the EIC (defined above, substituting in the targeted fits Q̅^*_Y,n(M,Z,W) and Q̅^a,*_Z,n(W)) divided by n. First, we estimate the EIC for each component of the SDE/SIE, which we call EIC_Ψ(P)(a,ĝ_M|a^*,W). Then we estimate the EIC for the estimand of interest by subtracting the EICs corresponding to the components of the estimand. For example EIC_SDE = EIC_Ψ(P)(a,ĝ_M|a^*,W) - EIC_Ψ(P)(a^*,ĝ_M|a^*,W). The sample variance of this EIC divided by n is the influence curve-based variance of the estimator. §.§ There exist direct effects between A and M and between A and YLet observed data: O=(W, A, Z, M, Y) with n i.i.d. copies O_1,...,O_n ∼ P_0, where W is a vector of pre-treatment covariates, A is the treatment, Z is the intermediate confounder affected by A, M is the mediator, and Y is the outcome. For simplicity, we assume that A, Z, M,and Y are binary. We assume that A is exogenous, Z is affected by {A, W}, M is affected by {A, Z, W}, and that Y is affected by {A, M, Z, W}. We assume exogenous random errors: (U_W, U_A, U_Z, U_M, U_Y). We can factorize the likelihood for this SCM as follows: P(O) = P(Y | M, Z, A,W)P(M|Z,A,W)P(Z | A, W)P(A)P(W).Under this SCM, our proposed TMLE is efficient. The sequential regression used to identify the data-dependent, stochastic mediation estimands does not change from the that given in the main text for this SCM. The EIC for the parameter Ψ(P)(a, ĝ_M|a^*,W) is given by D^*(a, ĝ_M|a^*,W) = ∑_k=0^2 D_k^*(a, ĝ_M|a^*,W),where D^*_0(a, ĝ_M|a^*,W) = Q̅^a_Z(W) - Ψ(P)(a, ĝ_M|a^*,W)D^*_1(a, ĝ_M|a^*,W) = I(A=a)/P(A=a)(Q̅^ĝ_M(Z,A,W) - Q̅^a_Z(W))D^*_2(a, ĝ_M|a^*,W) = I(A=a){I(M=1)ĝ_M|a^*,W + I(M=0)(1-ĝ_M|a^*,W) }/P(A=a){I(M=1)g_M|Z,A,W + I(M=0)(1-g_M|Z,A,W) }(Y-Q̅_Y(M,Z,A,W)). We now describe how to compute the TMLE. First, one estimates ĝ_M|a^*,W(W) = ∑_z=0^1 P(M=1|Z=z,A=a^*,W)P(Z=z | A=a^*, W). Consider a binary Z. We first estimate g_Z|a^*,W(W)=P(Z=1 | A=a^*, W). We then estimate g_M|z,a^*,W(W)=P(M=1 | Z=z, A=a^*, W) for z ∈{0,1}. We use these quantities to calculate ĝ_M|a^*,W = (ĝ_M|z=1,a^*,W×ĝ_Z|a^*,W) +(ĝ_M|z=0,a^*,W× (1-ĝ_Z|a^*,W)). We can obtain ĝ_Z|a^*,W(W) from a logistic regression of Z on A, W setting A=a^*, and ĝ_M|z,a^*,W(W) from a logistic regression of M on Z, A, W, setting Z={0,1} and A=a^*. We will then use this data-dependent stochastic intervention in the TMLE, whose implementation is described as follows.* Let Q̅_Y,n(M,Z,A,W) be an estimate of Q̅_Y(M,Z,A,W)≡ E(Y |M,Z,A,W). To obtain Q̅_Y,n(M,Z,A,W), predict values of Y from a regression of Y on M,Z,A,W. * Estimate the weights to be used for the initial targeting step:h_1(a) = I(A=a){I(M=1)ĝ_M|a^*,W + I(M=0)(1-ĝ_M|a^*,W) }/P(A=a){I(M=1)g_M|Z,A,W + I(M=0)(1-g_M|Z,A,W) }, where ĝ_M|Z,A,W are predicted probabilities from a logistic regression of M=m on Z, A, and W. Let h_1,n(a) denote the estimate of h_1(a). * Target the estimate of Q̅_Y,n(M,Z,A,W) by considering a univariate parametric submodel {Q̅_Y,n(M,Z,A,W)(ϵ):ϵ} defined as: logit(Q̅_Y,n (ϵ)(M,Z,A,W)) = logit(Q̅_Y,n(M,Z,A,W) ) + ϵ.Let ϵ_n be the MLE fit of ϵ. We obtain ϵ_n by setting ϵ as the intercept of a weighted logistic regression model of Y with logit(Q̅_Y,n(M,Z,A,W)) as an offset and weights h_1,n(a). (Note that this is just one possible TMLE.)The update is given by Q̅^*_Y,n(M,Z,A,W) =Q̅_Y,n(ϵ_n)(M,Z,A,W). Y can be bounded to the [0,1] scale.* Let Q̅^ĝ_M,n(Z,A,W) be an estimate of Q̅^ĝ_M(Z,A,W). To obtainQ̅^ĝ_M,n(Z,A,W), we integrate out M from Q̅^*_Y,n(M,Z,A,W). First, we estimate Q̅^*_Y,n(M,Z,A,W) setting m=1 and m=0, giving Q̅^*_Y,n(m=1,Z,A,W) and Q̅^*_Y,n(m=0,Z,A,W). Then, multiply these predicted values by their probabilities under ĝ_M|a^*,W(W) (for a ∈{a,a^*}), and add them together (i.e., Q̅^ĝ_M,n(Z,A,W) = Q̅^*_Y,n(m=1,Z,A,W)*ĝ_M|a^*,W + Q̅^*_Y,n(m=0,Z,A,W)*(1-ĝ_M|a^*,W)). * We now fit a regression of Q̅^ĝ,*_M,n(Z,A,W) on W among those with A=a. We call the predicted values from this regression Q̅^a_Z,n(W). The empirical mean of these predicted values is the TMLE estimate of Ψ(P)(a, ĝ_M|a^*,W).* Repeat the above steps for each of the interventions. For example, for binary A, we would execute these steps a total of three times to estimate: 1) Ψ(P)(1,ĝ_M|1,W), 2) Ψ(P)(1,ĝ_M|0W), and 3) Ψ(P)(0,ĝ_M|0,W).* The SDE can then be obtained by substituting estimates of parameters Ψ(P)(a,ĝ_M|a^*,W) - Ψ(P)(a^*,ĝ_M|a^*,W) and the SIE can be obtained by substituting estimates of parameters Ψ(P)(a,ĝ_M|a,W) - Ψ(P)(a,ĝ_M|a^*,W).* The variance of each estimate can be estimated as the sample variance of the EIC (defined above, substituting in the targeted fits Q̅^*_Y,n(M,Z,W) and Q̅^a,*_Z,n(W)) divided by n. First, we estimate the EIC for each component of the SDE/SIE, which we call EIC_Ψ(P)(a,ĝ_M|a^*,W). Then we estimate the EIC for the estimand of interest by subtracting the EICs corresponding to the components of the estimand. For example EIC_SDE = EIC_Ψ(P)(a,ĝ_M|a^*,W) - EIC_Ψ(P)(a^*,ĝ_M|a^*,W). The sample variance of this EIC divided by n is the influence curve-based variance of the estimator.§ FUNCTION CODEfunctioncode.R exampcode.R | http://arxiv.org/abs/1707.09021v2 | {
"authors": [
"Kara E. Rudolph",
"Oleg Sofrygin",
"Wenjing Zheng",
"Mark J. van der Laan"
],
"categories": [
"stat.AP",
"stat.ME",
"62P25",
"G.3"
],
"primary_category": "stat.AP",
"published": "20170727194444",
"title": "Robust and Flexible Estimation of Stochastic Mediation Effects: A Proposed Method and Example in a Randomized Trial Setting"
} |
[ Using Program Induction to Interpret Transition System DynamicsSvetlin Penkoved Subramanian Ramamoorthyed edThe University of Edinburgh, Edinburgh, United KingdomSvetlin [email protected] program induction, explainability, interpretability0.3in ]Explaining and reasoning about processes which underlie observed black-box phenomena enables the discovery of causal mechanisms, derivation of suitable abstract representations and the formulation of more robust predictions. We propose to learn high level functional programs in order to represent abstract models which capture the invariant structure in the observed data. We introduce the π-machine (program-induction machine) – an architecture able to induce interpretable LISP-like programs from observed data traces. We propose an optimisation procedure for program learning based on backpropagation, gradient descent and A* search. We apply the proposed method to two problems: system identification of dynamical systems and explaining the behaviour of a DQN agent. Our results show that the π-machine can efficiently induce interpretable programs from individual data traces. § INTRODUCTIONLearning models of transition systems has been a core concern within machine learning, with applications ranging from system identification of dynamical systems <cit.> and inference of human choice behaviour <cit.> to reverse engineering the behaviour of a device or computer program from observations and traces <cit.>. With the increasing use of these learnt models in the inner loops of decision making systems, e.g., in robotics and human-machine interfaces, it has become necessary to ensure not only that these models are accurate predictors of behaviour, but also that their causal mechanisms are exposed to the system designer in a more interpretable manner. There is also the need to explain the model in terms of counterfactual reasoning <cit.>, e.g., what would we expect the system to do if a certain variable were changed or removed, or model checking <cit.> of longer term properties including safety and large deviations in performance. We address these needs through a program induction based framework.We propose to learn high level functional programs in order to represent abstract models which capture the invariant structure in the observed data. Recent works have demonstrated the usefulness of program representations in capturing human-like concepts <cit.>. Used in this way, program-based representations boost generalisation and enable one-shot learning. Also, and arguably more importantly, they are significantly more amenable to model checking and human interpretability.In this paper, we introduce the π-machine (program-induction machine), an architecture which is able to induce LISP-like programs from observed transition system data traces in order to explain various phenomena. Inspired by differentiable neural computers <cit.>, the π-machine, as shown in Figure <ref>, is composed of a memory unit and a controller capable of learning programs from data by exploiting the scalability of stochastic gradient descent. However, the final program obtained after training is not an opaque object encoded in the weights of a controller neural network, but a LISP-like program which provides a rigorous and interpretable description of the observed phenomenon. A key feature of our approach is that we allow the user to provide a set of predicates of interest in order to specify the properties they are interested in understanding as well as the context in which the data is to be explained. By exploiting the equivalence between computational graphs and functional programs we describe a hybrid optimisation procedure based on backpropagation, gradient descent, and A* search which is used to induce programs from data traces.We evaluate the performance of the π-machine on two different problems. Firstly, we apply it to data from physics experiments and show that it is able to induce programs which represent fundamental laws of physics. The learning procedure has access to relevant variables, but it does not have any other prior knowledge regarding physical laws which it has discovered in the same sense as in <cit.> although far more computationally tractably. Secondly, we study the use of the proposed procedure in explaining control policies learnt by a deep Q-network (DQN). Starting from behaviour traces of a reinforcement learning agent that has learnt to play the game of Pong, we demonstrate how the π-machine learns a functional program to describe that policy.§ RELATED WORK Explainability and interpretability. The immense success of deep neural network based learning systems and their rapid adoption in numerous real world application domains has renewed interest in the interpretability and explainability of learnt models <cit.>. There is recognition that Bayesian rule lists <cit.>, decision trees and probabilistic graphical models are interpretable to the extent that they impose strong structural constraints on models of the observed data and allow for various types of queries, including introspective and counterfactual ones. In contrast, deep learning models usually are trained `per query' and have numerous parameters that could be hard to interpret. <cit.> introduced deconvolutional networks in order to visualise the layers of convolutional networks and provide a more intuitive understanding of why they perform well. <cit.> describe Semi-Aggregated Markov Decision Process (SAMDP) in order to analyse and understand the behaviour of a DQN based agent. Methods for textual rationalisation of the predictions made by deep models have also been proposed <cit.>. While all of these works provide useful direction, more generic methods are required which need not be hand-crafted to explain specific aspects of individual models. In this sense, we follow the model-agnostic explanation approach of <cit.>, who provide “textual or visual artefacts” explaining the prediction of any classifier by treating it as a black-box. Similarly to the way in which <cit.> utilise local classifiers composed together to explain a more complex model, we present an approach to incrementally constructing functional programs that explain a complex transition system from more localised predicates of interest.The π-machine treats the process which has generated the observed data as a black-box and attempts to induce LISP-like program which can be interpreted and used to explain the data. We show that the proposed method can be applied both to introspection of machine learning models and to the broader context of autonomous agents. Program learning and synthesis. Program learning and synthesis has a long history, with the long-standing challenge being the high complexity deriving from the immense search space. Following classic and pioneering work such as by <cit.> who used inductive inference in a logic programming setting, others have developed methods based on a variety of approaches ranging from SAT solvers <cit.> to genetic algorithms <cit.>, which tend to scale poorly hence often become restricted to a narrow class of programs. Recently, deep neural networks have been augmented with a memory unit resulting in models similar to the original von Neumann architecture. These models can induce programs through stochastic gradient descent by optimising performance on input/output examples <cit.> or synthetic execution traces <cit.>. Programs induced with such neural architectures are encoded in the parameters of the controller network and are, in general, not easily interpretable (particularly from the point of view of being able to ask counterfactual questions or performing model checking). Another approach is to directly generate the source code of the output program which yields consistent high level programs. Usually, these types of approaches require large amounts of labelled data - either program input/output examples <cit.> or input paired with the desired output program code <cit.>.Determining how many input/output examples or execution traces are required in order to generalise well is still an open research problem. However, in this paper, we focus attention more on the explanatory power afforded by programs rather than on the broader problems of generalisation in the space of programs. While these characteristics are of course related, we take a view similar to that of <cit.>, arguing that it is possible to build from locally valid program fragments which provide useful insight into the black-box processes generating the data. By combining gradient descent and A* search the π-machine is able to learn informative and interpretable high-level LISP-like programs, even just from a single observation trace.§ PROBLEM DEFINITIONConsider the labelled transition system Ω(𝒮, 𝒜, δ) where 𝒮 is a non-empty set of states, 𝒜 is a non-empty set of actions, each parametrised by θ∈ℝ^D, and δ: 𝒮×𝒜→𝒮 is the state transition function. We define an observation trace 𝒯 as a sequence of observed state-action pairs (s_t, a_t(θ_t)) ∈𝒮×𝒜 generated by the recursive relationship s_t+1 = δ(s_t, a_t(θ_t)) for 1 ≤ t ≤ T. We are interested in inducing a LISP-like functional program ρ which when executed by an abstract machine is mapped to an execution trace 𝒯_ρ such that 𝒯_ρ and 𝒯 are equivalent according to an input specification.We represent the abstract machine as another labelled transition system Π(ℳ, ℐ, ε) where ℳ is the set of possible memory state configurations, ℐ is the set of supported instructions and ε: ℳ×ℐ→ℳ specifies the effect of each instruction. We consider two types of instructions – primitive actions which emulate the execution of a ∈𝒜 or arithmetic functions f ∈ℱ such that ℐ = 𝒜∪ℱ. Furthermore, a set of observed state variables ℳ_v ⊆𝒮, which vary over time, are stored in memory together with a set of induced free parameters ℳ_p. The variables in ℳ_v form a context which the program will be built on. A customdetector 𝒟_v, operating on the raw data stream, could be provided for each variable, thus enabling the user to make queries with respect to different contexts and property specifications.The execution of a program containing primitive actions results in a sequence of actions. Therefore, we represent a program ρ as a function which maps a set of input variables x_v ⊂ℳ_v and a set of free parameters x_p ⊂ℳ_p to a finite sequence of actions â_1(θ̂_̂1̂), …â_T'(θ̂_̂T̂'̂). We are interested in inducing a program which minimises the total error between the executed and the observed actions:L(ρ) = ∑_t=1^min(T, T')σ_act(â_t, θ̂_̂t̂, a_t, θ_t) + σ_len(T, T')The error function σ_act determines the difference between two actions, while σ_len compares the lengths of the generated and observed action traces.By providing the error functions σ_act and σ_len one can target different aspects of the observation trace to be explained as they specify when two action traces are equivalent.§ METHOD The proposed program induction procedure is based on two major steps. Firstly, we explain how a given functional program can be optimised such that the loss L(ρ) is minimised. Secondly, we explain how the space of possible program structures can be searched efficiently by utilising gradient information. An architectural overview of the π-machine is provided in Figure <ref>. §.§ Program optimisationNeural networks are naturally expressed as computational graphs which are the most fundamental abstraction in computational deep learning frameworks <cit.>. Optimisation within a computational graph is usually performed by pushing the input through the entire graph in order to calculate the output (forward pass) and then backpropagating the error signal to update each parameter (backward pass). A key observation for the development of the π-machine is that computational graphs and functional programs are equivalent as both describe arbitrary compositions of pure functions applied to input data.Therefore, similarly to a computational graph, a functional program can also be optimised by executing the program (forward pass), measuring the error signal and then performing backpropagation to update the program (backward pass). Forward pass. When a program is executed it is interpreted to a sequence of instructions i_1, …, i_n ∈ℐ which are executed by recursively callingε(…ε(ε(ℳ_1, i_1), i_2)…, i_n). ℳ_1 is the initial memory state initialised with the observed variables from s_1 and any induced parameters. The π-machine keeps a time counter t which is initialised to 1 and is automatically incremented whenever a primitive action instruction is executed. If the instruction i_k is a primitive action, i_k ∈𝒜, then the π-machine automatically sets â_t = i_k and invokes the error function σ_act(â_t, θ̂_̂t̂, a_t, θ_t), where θ̂_̂t̂ has been calculated by previous instructions. If the error is above a certain threshold e_max the program execution is terminated and the backward pass is initiated. Otherwise, the time counter is incremented and the values of the variables in ℳ_v are automatically updated to the new observed state. Essentially, the π-machine simulates the execution of each action reflecting any changes it has caused in the observed state. Alternatively, if the currently executed instruction i_k is a function, i_k ∈ℱ, then the resulting value is calculated and i_k, together with its arguments, is added to a detailed call trace χ maintained by the π-machine. Importantly, each function argument is either a parameter or a variable read from memory at time t or the result of another function. All this information is kept in χ which eventually contains the computational tree of the program. Backward pass. The gradients of the loss function L(ρ) with respect to the program inputs x_v and x_p are required to perform a gradient descent step. Crucially, programs executed by the π-machine are automatically differentiated. The π-machine performs reverse-mode automatic differentiation, similarly to Autograd <cit.>, by traversing the call trace χ, and post-multiplying Jacobian matrices. We assume that the Jacobian matrix with respect to every input argument of any function f ∈ℱ or any specified error function σ_act is known a priori. Let f ∈ℱ be a function whose output needs to be differentiated with respect to the input arguments. There are three types of derivatives, which need to be considered in order to traverse backwards the entire tree of computations: * Let g ∈ℱ, then ∂ f/∂ g is the Jacobian matrix of f with respect to the output of g and can be directly calculated.* Let p ∈ x_p, then the gradient ∂ f/∂ p is calculated by multiplying the corresponding Jacobian matrix of f with the value of p.* Let v ∈ x_v, then the gradient ∂ f/∂ v|_t=t_r is calculated by multiplying the corresponding Jacobian of f with the value of the variable at the time it was read from memory t_r.Gradient descent step. Once the gradient ∇_p L(ρ) of the loss function with respect to each input parameter p ∈ x_p is calculated we utilise AdaGrad <cit.> to update the values of all parameters after each program execution. The gradient ∇_v L(ρ) with respect to each input variable v ∈ x_v is also available. However, a variable cannot be simply updated in the direction of the gradient as it represents a symbol, not just a value. Variables can only take values from memory which is automatically updated according to the observation trace during execution. Nevertheless, the gradient provides important information about the direction of change which we utilise to find variables that minimise the loss. Whenever the memory state is automatically updated, a KD-tree is built for each type of variable stored in memory. We assume that the variables in memory are real vectors with different length. So, we represent the KD-tree which stores all D-dimensional variables in memory at time t as 𝒦^D_t. If a d-dimensional variable v is to be optimised it is replaced with a temporary parameter p_temp initialised with v_t which is the value of v read from memory at the respective time step t. The temporary parameter p_temp is also updated with AdaGrad <cit.>. After each descent step, the nearest neighbour of the updated value p'_temp is determined by querying the KD-tree with 𝒦^d_t(p'_temp). If the result of the query is a different d-dimensional variable u then the temporary parameter is immediately set to p_temp = u_t. As this often shifts the solution to a new region of the error space the gradient history for all parameters p ∈ x_p is reset. Eventually, when a solution is to be returned, the temporary parameters are substituted with their closest variables according to the respective 𝒦^D_t. The forward and backward passes are repeated until the error is below the maximum error threshold e_max or a maximum number of iterations is reached. After that the optimised program ρ^* is scored according to its error and complexity, and pushed to a priority queue holding potential solutions. §.§ Structure searchWe represent the space of possible program structures as a graph G=(T^AST, E) where each node T_i ∈ T^AST is a valid program abstract syntax tree (AST). There is an edge from T_i to T_j if and only if T_j can be obtained by replacing exactly one of the leaves in T_i with a subtree T_s of depth 1. The program induction procedure always starts with an empty program. So, we frame structure search as a path finding problem, solved through the use of A* search. Score function. The total cost function we use is f_total(ρ) = C(ρ) + L(ρ), where L(ρ) is the loss function defined in equation (<ref>) and C(ρ) is a function which measures the complexity of the program ρ. C(ρ) can be viewed as the cost to reach ρ and L(ρ) as the distance to the desired goal. The complexity function C(ρ) is the weighted sum of (i) maximum depth of the program AST; (ii) the number of free parameters; (iii) the number of variables used by the program; the weights of which we set to w_C = [10, 5, 1]. These choices ensure that short programs, maximally exploiting structure of the observation trace, are preferred. Neighbours expansion. When the current best candidate solution is popped from the priority queue, we check if it matches the observation trace according to the input specification. If so, the candidate can be returned as the final solution, otherwise it is used as a seed to propose new candidate solutions. Typically in A* search, all neighbouring nodes are expanded and pushed to the priority queue, which is not feasible in our case, though. Therefore, we utilise the available gradients in order to perform a guided proposal selection. Each leaf in the abstract syntax tree T_ρ of a seed candidate solution ρ corresponds to a parameter or a variable. According to the definition of G we need to select exactly 1 leaf to be replaced with a subtree T_s of depth 1. We select leaf l ∈ T_ρ according to:l = x ∈ x_p ∪ x_v‖∇_x L(ρ) ‖_2After that, all possible replacement subtrees are constructed. An AST subtree T_s of depth 1 represents a function call. We prune the number of possible functions in ℱ by ensuring type consistency. Each leaf of T_s can be a parameter or a variable. So, all possible combinations are considered. New variable leaves are initialised to a random variable with suitable type from memory, while new parameter leaves are sampled from the multivariate normal distribution 𝒩(0, 0.1). As a result, if n_f functions are type compatible with l and each function takes n_a arguments at most, then there are 2^n_a· n_f replacement subtrees, resulting in that many new candidates. All newly proposed candidates are optimised in parallel, scored by f_total and pushed to the priority queue.§ EXPERIMENTAL RESULTSThe π-machine is implemented in Clojure, which is a LISP dialect supporting powerful data structures and homoiconic syntax. All experiments are run on an Intel Core i7-4790 processor with 32GB RAM and use the following list of functions, ℱ: vector addition, subtraction and scaling. Physical systems.Firstly, we apply the π-machine to model learning for physical systems. The transition dynamics of a second order dynamical system is written as 𝐱̈(t)= k_1 𝐱(t) + k_2 𝐱̇(t), where 𝐱(t) is the state of the system at time t and k_1, k_2 are system coefficients. We have recreated an experiment described in <cit.>, where the authors show the learning of physical laws associated with classical mechanical systems including the simple pendulum and linear oscillator. A diagram of these two systems is shown in Figure <ref> (left). We set 𝒜 = {accel(θ)} where θ∈ℝ for both experiments. The observation trace for each system is generated by simulating the dynamics for 1s at 100Hz. We specify the action error function as σ_act = ‖θ̂ - θ‖_2 and set σ_len = 0. In both experiments 𝐱∈ℝ and 𝐯 = ẋ∈ℝ represent linear position and velocity.The three best solutions found by the π-machine for each system are shown in Figure <ref> (middle). The best solution for each system correctly represents the underlying laws of motion. The program describing the behaviour of the pendulum was induced in 18 iterations, while the linear oscillator program needed 146 iterations. The total number of possible programs with AST depth of 2, given the described experimental setup, is approximately 1.7 × 10^4. The average duration of an entire iteration (propose new programs, optimise and evaluate) was 0.6s. <cit.> achieve similar execution times, but distributed over 8 quad core computers (32 cores in total). The experimental results demonstrate that the π-machine can efficiently induce programs representing fundamental laws of physics.Deep Q-network.This experiment is based on our view that the core deep neural network based policy learner and the explanation layer play complementary roles. There are numerous advantages to performing end-to-end policy learning, such as DQN-learning from raw video, however, there is also a need to explain the behaviour of the learnt policy with respect to user-defined properties of interest. We consider explaining the behaviour of a DQN agent playing the ATARI Pong game and are interested in the question: how does the network control the position of the paddle in order to hit the ball when it is in the right side of the screen. A diagram of the experimental setup is shown in figure <ref> (left). The behaviour of the DQN is observed during a single game. Since the environment is deterministic, the state transition function, which generates the observation trace for this experiment, is the policy π(s) that the DQN has learnt. We would like to explain the behaviour of the DQN in terms of the position of the opponent, the ball and the DQN agent (so, not just in terms of RAM memory values, for instance). Therefore, the observation trace contains those positions which are extracted from each frame by a predefined detector. We set 𝒜 = {move(θ)} where θ∈ℝ and represent the discrete actions of the network , ,as move(1), move(-1), move(0) respectively. We specify the action error function as σ_act = ‖θ̂ - θ‖_2 and set σ_len = 0.The best 3 programs found by the π-machine are shown in Figure <ref> (middle), where it took 38 iterations for the best one (average iteration duration 3.2s). By inspecting the second solution it becomes clear that the neural network behaviour can be explained as a proportional controller minimising the vertical distance between the agent and the ball. However, the best solution reveals even more structure in the behaviour of the DQN. The coefficient in front of the agent position is slightly larger than the one in front of the ball position which results in a small amount of damping in the motion of the paddle. Thus, it is evident that the DQN not only learns the value of each game state, but also the underlying dynamics of controlling the paddle. Furthermore, we have tested the performance of an agent following a greedy policy defined by the induced program. In our experiments over 100 games this agent achieved a score of 11.1 (± 0.17). This is not quite the score of 18.9 (± 1.3) obtained by an optimised DQN, but it is better than human performance 9.3 <cit.>. This difference of course emanates from the predefined detector not capturing all aspects of what the perceptual layers in DQN have learnt, so improved detector choices should yield interpretable programs that also attain performance closer to the higher score of the black-box policy.§ DISCUSSIONThe π-machine can be viewed as a framework for automatic network architecture design <cit.>, as different models can be expressed as concise LISP-like programs. Deep learning methods for limiting the search space of possible programs, which poses the greatest challenge, have been proposed <cit.>, but how they can be applied to more generic frameworks such as the π-machine is an open question. The specification of variable detectors not only addresses this issue, but enables the user to make targeted and well grounded queries about the observed data trace. Such detectors can also be learnt from raw data in an unsupervised fashion <cit.>.§ CONCLUSIONIn conclusion, we propose a novel architecture, the π-machine, for inducing LISP-like functional programs from observed data traces by utilising backpropagation, stochastic gradient descent and A* search. The experimental results demonstrate that the π-machine can efficiently induce interpretable programs from short data traces.icml2017 | http://arxiv.org/abs/1708.00376v1 | {
"authors": [
"Svetlin Penkov",
"Subramanian Ramamoorthy"
],
"categories": [
"cs.AI"
],
"primary_category": "cs.AI",
"published": "20170726124904",
"title": "Using Program Induction to Interpret Transition System Dynamics"
} |
Centre for Theoretical Chemistry and Physics, The New Zealand Institute for Advanced Study, Massey University Auckland, 0632 Auckland, New ZealandNSCL/FRIB Laboratory, Michigan State University, East Lansing, Michigan 48824, USA Centre for Theoretical Chemistry and Physics, The New Zealand Institute for Advanced Study, Massey University Auckland, 0632 Auckland, New Zealand Department of Physics and Astronomy and FRIB Laboratory, Michigan State University, East Lansing, Michigan 48824, USAFermion localization functions are used to discuss electronic and nucleonic shell structure effects in the superheavy element oganesson, the heaviest element discovered to date. Spin-orbit splitting in the 7p electronic shell becomes so large (∼ 10 eV) that Og is expected to showuniform-gas-like behavior in the valence region with a rather large dipole polarizability compared to the lighter rare gas elements. The nucleon localization in Og is also predicted to undergo a transition to theThomas-Fermi gas behavior in the valence region. This effect, particularly strong for neutrons, is due to the high density of single-particle orbitals.Electron and Nucleon Localization Functions of Oganesson: Approaching the Thomas-FermiLimit Witold Nazarewicz=============================================================================================Introduction – Oganesson (Z=118) is the recent addition to the Periodic Table of the Elements and the Chart of Nuclides <cit.>. The isotope ^294_118Ogwas produced in a heavy ion fusion reaction with a ^48_20Ca beam and a ^249_98Cf target <cit.>. The heaviest element studied chemically to date is Fl (Z=114). Its relatively long half-life, 1-2 s, enables chemical studies with ∼5 atoms/day, which marks the limit of chemistry today <cit.>. The estimated α-decay half-life of ^294_118Og, 0.89^+1.07_-0.31 ms, is too short forchemical“one-atom-at-a-time" studies; hence,its chemical properties must beinferred from advanced atomic calculations based on relativistic quantum theory <cit.>. According to these, Og has a closed-shell [Rn]5f^146d^107s^27p^6 configuration <cit.>, with a very large spin-orbit splitting of the 7p shell (9.920 eV at the Dirac-Breit-Hartree-Fock and 10.125 eV at the Fock-Space Coupled-Cluster level, see below). While, according to its electronic configuration (Ogcompletes the 7th row of the Periodic Table), it does not behave like a typical rare gas element. For example, the relativistic 7p_3/2 expansion and the relativistic 8s contraction make Og the first rare gas element with a positive electron affinity of 0.064 eV <cit.>. This result includes a substantial quantum electrodynamic correction of 0.006 eV <cit.>.Nuclear structure calculations predict ^294Og to be a deformed nucleus <cit.>,eight neutrons away fromthe next neutron shell closure at ^302Og (N=184) <cit.>. A new factorimpacting properties of superheavy nucleiis the strong electrostatic repulsion: the Coulombforce in superheavy nuclei cannot be treated as a small perturbation atop the dominating nuclear interaction; the resulting polarization effects due to Coulomb frustration are expected toinfluence significantly proton and neutron distributions and shell structure <cit.>. In particular, the isotope ^294_118Og is believed to be a semi-bubble systemwith a sizable central depression of the proton density <cit.>.The objective of this paper is to study the electronic and nucleonic shell structure of superheavy elements. The electronic shell structure is expected to be impacted by the transition from the LS-coupling of the Schrödinger equation at lower atomic numbersto the jj-couplingof the Dirac equation at large Z-values.In the nuclear case, the shell structure is expected to be washed out due to the large density of single-nucleonic states. While the kinematics of protons and neutrons in a nucleus is non-relativistic, the large spin-orbit coupling (that is about an order of magnitude greater than in atomic case due tolarge spin-dependent components of the nucleon-nucleon interaction <cit.>) results in a jj-coupling. Therefore, both for electronic and nucleonic systems, the pattern of single-particle levelsof superheavy species is expected to be strongly impacted by both radial and total angular momentum characteristics <cit.>. To describe these changes quantitatively, we utilizethe fermion localization measure <cit.>,which is an excellent indicator ofshell structure. In particular, we investigate the transition fromthe regime of strong localization, governed byshell effects, to a more delocalized regime typical of a uniform-densityThomas-Fermi gas. As we shall demonstrate, superheavy species constitute an excellent territory where to look for such a transition.Fermion localization function – The spatial localization measure was originally proposed in atomic and molecular physics to characterize shell structure and chemical bondingin electronic systems <cit.>. It has been subsequently introduced to nuclear physics to visualize cluster structures in light nuclei <cit.>. The novel nuclear applications include description ofnuclear fission <cit.> and heavy-ion fusion <cit.>, andnucleonic matter in the inner crust of neutron stars <cit.>. In electronic systems, spatial localization function is referred to as the electron localization function (ELF); in nuclear systems as the nucleon localization function (NLF). It is based on the inverse of the conditional probability of finding a fermion of type q (=e, n, or p) in the vicinity of another fermion of the same type and same spin/signature quantum number σ (=↑ or ↓), knowing that the latter particle is located at position r⃗.While thisprobability is generallygiven by the non-local one-body density matrix <cit.>, it is useful to introduce a local quantity that provides information about the short-range behavior. To this end, Becke andEdgecombe <cit.> introducedthe localmeasure of fermion localization, which – in the non-relativistic case – can be written as:𝒞_qσ(r⃗)=[1+(τ_qσρ_qσ-1/4|∇⃗ρ_qσ|^2-j⃗^2_qσ/ρ_qστ^TF_qσ)^2]^-1,where ρ_qσ, τ_qσ, j⃗_qσ, and ∇⃗ρ_qσ are the particle density, kinetic energy density, current density, and density gradient, respectively. τ^TF_qσ denotes the Thomas-Fermi kinetic energy. In this work, time reversal symmetry is conserved; hence, j⃗_qσ vanishes.The localization function takes generally values between 0 and 1. A value close to 1 indicates that the probability of finding two particles (of the same type) close to each other is very low. Thus a high value of 𝒞 marks the spatial regions corresponding to shell separations.Since the localization function (<ref>) is normalized to the Thomas-Fermi kinetic energy, 𝒞=1/2 corresponds to the limit of the uniform-density Fermi gas, in which the individual orbits are spatially delocalized. Electron localization – For the electronic structure calculations we used the elf module as implemented in the relativistic ab-initio quantum chemistry program dirac15 <cit.>.Hartree-Fock one-particle densities were generated in non-relativistic, scalar-relativistic (module x2c-spinfree) <cit.>, and (4-component) Dirac-Coulomb calculations in conjunction with an uncontractedrelativistic quadruple-zeta basis set dyall.acv4z <cit.>. The Dirac-Fock computations include the small-component integrals as well as the two-electron Gaunt term. We utilized the finite-field method to compute the static electric dipole polarizability of Og (with external electric field strengths of 0.0, 0.0005 and 0.001 a.u.) at CCSD(T) Coupled-Cluster level,<cit.> which included excitations from singles, doubles, and perturbative triples. In the correlation treatment, we included50 electrons and virtual orbitals up to 25 a.u. Here we used the molecular mean-field x2c Hamiltonian <cit.> with the Gaunt term included. Fock-Space Coupled-Cluster calculations <cit.> were carried out to obtain the ionization potentials from the filled 7p_3/2 and 7p_1/2 shells of Og. Note that only large-component densities are considered for the non-relativistic and scalar-relativistic ELF, whereas in the 4-component case the small-component densities are added to the large-components to yield the total one-particle density. Relativistic effects make a huge imprint on many properties of Og. For instance, the electron binding energy of Og is predicted to rise by as much as 227 keV by considering relativistic effects (for comparison, a similar number for Pb is a mere 40 keV). Figures <ref> and<ref> show the ELFs predicted in our calculations.As seen inFig. <ref>, electron localizationsfor Xe or Rn hardly change from the nonrelativistic to the 4-component relativistic framework. However, for Og we see significant electron delocalization with ELF values that are much smaller compared to the non-relativistic case, making the atomic shell structure barely recognizable. The pattern of concentric rings is a fingerprint of the underlying shell structure. Thesizes of rings in ELFreflect the radii of electron orbits in different shells; hence, they roughly scale withn^2, where n is the principal quantum number <cit.>. Figure <ref>(b) clearly shows that the delocalization is mainly due to spin-orbit coupling and not due to scalar relativistic effects. This results inan evenly distributed ELF with values around 0.5 in the outer shells. The valence and sub-valence shells of Og are, therefore, smeared out like in a homogenous electron gas. Rn behaves similarly to Xe, although some delocalization through relativistic effects is already apparent. A more detailed analysis shows that smearing out of the electron density in the valence region originates from thestrong spin-orbit splitting of the 7p shells; while the radiifor the valence 5p orbitals in Xe are very similar (2.239 and 2.141 a.u. for 5p_3/2 and 5p_1/2, respectively, as obtained with the numerical program grasp92 <cit.>) the 7p_3/2 shell in Og is 0.75 a.u. further out compared to the 7p_1/2 shell (2.796 and 2.039 a.u., respectively). Large spin-orbit splittings are also calculated for the lower lying ℓ>0 (core) shells. Further, the density of the single-particle (s.p.) states increases from Xe to Og as expected for higher principal quantum numbers, see Figure <ref>.As a result of these effects, the electron density is more homogeneously distributed over the entire atomic range, i.e., less localized, resulting in ELF values oscillating around the Thomas-Fermi limit. Our Fock-Space Coupled-Cluster calculation gave ionization potentialsof 7p_3/2 and 7p_1/2 of 8.842 and 18.967 eV, respectively, thus spin-orbit splitting for the valence 7p orbital of Og is extremely large (10.125 eV). Figure <ref> illustrates this in relation to the orbital energy levels of the lighter homologues.According to the Thomas-Fermi model, the static dipole polarizabilityα∝ r_a^3, with r_a^3 being the atomic radius <cit.>.Our state-of-the-art calculationsshow that the electron-gas-likeouter shell of Og, resulting inα = 57.98 a.u., ismuch easier to polarize as compared to xenon (α = 27.815 a.u. <cit.>) or radon (α = 33.18 a.u. <cit.>). For comparison, the nonrelativistic and scalar relativistic values for Og are α = 45.30 a.u. and α = 43.78 a.u., respectively. Thus, for Og one expects an increase in van-der-Waals interactions compared to the lighter rare gases, and subsequently a significant change in chemical and physical properties of this element, see also Refs. <cit.> for more discussion on this point. Nucleon localization –For the nuclear calculations, we employ nuclear density functional theory <cit.> with carefully optimized global Skyrme energy density functionals UNEDF1 <cit.> andSV-min <cit.>.Pairing is of minor importance in the closed-shell nuclei considered. It is treated as in Ref. <cit.>. Namely, we consider the density-dependent contact force at the level of BCS theory. The pairing space is limited by a soft cutoff with the cutoff parameter chosen such that it covers about 1.6 extra oscillator shells above the Fermi energy.We use the DFT solver of Ref. <cit.> constrained to spherical geometry as all nuclei considered are expected to be spherical in their ground states.Figure <ref> shows the NLFs for the doubly magic medium-mass nucleusand spherical superheavy systems and . We consider the latter “theoretical" nucleus to further illustrate the behavior of NLFs at still larger numbers of nucleons. In contrast to the ELFs,the number of closed shells cannot be determined from the number of radial maxima. This is due to the different radial behavior of single-nucleon orbits. While the radii of electron orbits in atoms belonging to different shells are spatially well separated,radii of nucleonicorbits scale roughlyas ∼√(2n_r+ℓ), i.e., theyvery gradually increase with the shell number. This results in a large spatial overlap between single-nucleon wave functions and reduced localizations as compared to the electronic case.A characteristicfeature ofNLFs is the local enhancementat the surface<cit.> due to the fact thatfewvalencenucleons contribute to the total density at distances greater than the nuclear radius.Inspecting the NLFs of protons to neutrons, one notes that the patterns of concentric rings is more distinct in the proton system, as the number of occupied proton shells is less than that for the neutrons, within the same volume (as the rms proton and neutron radii are very similar <cit.>). This effect becomesfairly pronounced for superheavy nuclei where the neutron excess is large. While the NLF for the medium-mass nucleusexhibits a clear shell structure with distinct oscillations around 𝒞=0.5 <cit.>, the maxima and minima become fainter forheavier systems. This is particularly striking for the neutrons. While the neutron NLF forstill exhibitsa faint structure in the interior, the ring patternalmost vanishes for .Overall, as mass increases,the neutron localizationapproaches the Thomas-Fermilimit 𝒞=0.5 in the valence region (r>3 fm) below the surface peak. The NLF pattern seen in Fig. <ref> reflects the underlying nucleonic shell structure. As discussed in, e.g., Refs. <cit.> the general pattern of s.p. energies undergoes significant changes in superheavy nuclei. First,thes.p. level density is large; in fact itgrows faster than A^1/3 <cit.>. Consequently – similar to what has beendiscussed earlier in the context of atomic calculations of the electron shell structure of superheavy elements – small changes in the theoretical description can impact shell structure substantially.Second, the shell structure of superheavy nuclei is influenced by the self-consistent interplay between the short-range attractive nuclear force and the long-range electrostatic repulsion. Thanks to the resultingCoulomb frustration, significant rearrangements of nucleonic densities, such as the appearance of central depression, are predicted <cit.>. The presenceof central depression strongly affects high-j orbits due to their large s.p. radii <cit.>.Conclusions – To study electronic and nucleonic shell structure in superheavy elements, we employedthelocal spatialmeasure of fermion localization. The atomic calculations were carried out for heavy rare gas atoms Xe, Rn, and the superheavy elementOg recently added to the Periodic Table. The nuclear calculations were performed for the known doubly-magic systemand for superheavy nuclei , and . This study constitutes the first application of fermion localization to superheavy atoms and nuclei.Relativistic effects significantly impact the electronic structure of superheavy atoms. For theelement Og, the electron shellswith ℓ>0 show very large spin-orbit splittings smearing out of the one-particle density, thus becoming more uniformly distributed over the entire atom approachingthe electron-gas regime in the valence region. A direct consequence of this transition is its predicted large static dipole polarizability resulting inan increase in van-der-Waals interactions compared to the lighter rare gases anda significant change in its chemical and physical properties.A gradual transition towards the uniform-gas regime is predicted fornucleonic localizations in superheavy nuclei. In general, neutrons are more delocalized than protons as for the superheavy nuclei N is much greater than Z, i.e., more neutrons are confined to the same volume than protons. While the semiclassical Thomas-Fermi limit in nuclei is strictly approached only for systems with extremely large particle numbers A > 5000<cit.>, we can see that in the discussed superheavy nuclei the Fermi-gas limit of neutron NLFs is reachedin the valence region (r>3 fm) below the surface peak. In summary, through electron and nucleon localization functions we show that Og is a rather unusual addition to the Periodic Table and to the Chart of Nuclides. High density of electronic and nucleonic s.p. states, relativistic effects resulting in the strong spin-orbit splitting of electronic levels, and nucleonic polarization effects, make the superheavy atoms, such as Og, quantitatively different from the lighter congeners. We acknowledge financial support by the Alexander-von-Humboldt Foundation (Bonn, Germany) and the Marsden Fund of the Royal Society of New Zealand.This work was also supported by the U.S. Department of Energy,under Award Numbers DOE-DE-NA0002847 (NNSA, the Stewardship Science Academic Alliances program) andDE-SC0013365 and DE-SC0008511 (Office of Science). apsrev4-1 | http://arxiv.org/abs/1707.08710v4 | {
"authors": [
"Paul Jerabek",
"Bastian Schuetrumpf",
"Peter Schwerdtfeger",
"Witold Nazarewicz"
],
"categories": [
"nucl-th",
"physics.atom-ph"
],
"primary_category": "nucl-th",
"published": "20170727051449",
"title": "Electron and Nucleon Localization Functions of Oganesson: Approaching the Thomas-Fermi Limit"
} |
GBgbsn State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Opto-Electronics, Shanxi University, Taiyuan 030006, P.R.China Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, P.R.ChinaState Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Opto-Electronics, Shanxi University, Taiyuan 030006, P.R.China Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, P.R.China State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Opto-Electronics, Shanxi University, Taiyuan 030006, P.R.China Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, P.R.ChinaState Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Opto-Electronics, Shanxi University, Taiyuan 030006, P.R.China Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, P. R. China We experimentally study a protocol of using the broadband high-frequency squeezed vacuum to detect the low-frequency signal. In this scheme, the lower sideband field of the squeezed light carries the low-frequency modulation signal and the two strong coherent light fields are applied as the bichromatic local oscillator in the homodyne detection to measure the quantum entanglement of the upper and lower sideband for the broadband squeezed light. The power of one of local oscillators for detecting the upper sideband can be adjusted to optimize the conditional variance in the low frequency regime by subtracting the photocurrent of the upper sideband field of the squeezed light from that of the lower sideband field. By means of the quantum correlation of the upper and lower sideband for the broadband squeezed light, the low-frequency signal beyond the standard quantum limit is measured. This scheme is appropriate for enhancing sensitivity of low-frequency signal by the aid of the broad squeezed light, such as gravitational waves detection, and does not need to directly produce the low frequency squeezing in optical parametric process. Enhanced sensitivity of low-frequency signal by using broad squeezed light and bichromatic local oscillator Jing Zhang,^† Received: date / Accepted: date ===========================================================================================================The first detection of gravitational waves (GW) emitted from the merger of two black holes by the Laser Interferometer Gravitational-Wave Observatory (LIGO) sets the course of a new era of astrophysics. GW detection is now opening an exciting new observational frontier in astronomy and cosmology <cit.>. The further improvement of GW detector sensitivity is expected to extend the detection range and the event rate of binary black holes coalescence, and may lead to detections of more exotic sources. In Advanced LIGO, vacuum fluctuations entering from the dark port of the interferometer <cit.> can make the quadrature phase of the output carrier field at the dark port noisy, while which contains GW signal. However as the squeezed vacuum state is fed into the dark port of the interferometer, the sensitivity can be improved beyond the standard quantum limit (SQL) <cit.>. The use of squeezed states to enhance the sensitivity began with initial proof-of-principle experiments and recently have been demonstrated in GEO 600 <cit.> and LIGO <cit.>. Since terrestrial GW signal locates in 10 Hz to 10 kHz band <cit.>, the squeezing in the audio band is required, which is a great technical challenge. Until now, there is a very wide research demonstrating the squeezing at the lower frequency band <cit.> and applications in quantum metrology <cit.>. Broadband squeezing has been demonstrated at megahertz frequencies, where technical noise sources of the laser light are not present. At these frequencies, the laser operates at or near the shot-noise limit. Due to the strong quantum correlation between the lower and upper sideband of the squeezed light field, the single broadband squeezed light can be split into N pairs of upper and lower sideband fields with spatial separation to produce N independent Einstein-Podolsky-Rosen (EPR) entangled fields <cit.>. This scheme was demonstrated experimentally by using a pair of frequency-shifted local oscillators to measure this EPR entanglement <cit.>. Recently, a theoretical protocol is proposed to improve LIGO's sensitivity beyond the SQL via EPR entanglement of the broad squeezing field and the dual use of the interferometer as both the GW detector and the filter, eliminating the need for external narrow filter cavities <cit.>. In this paper, we employ a broadband high-frequency squeezed vacuum to detect low-frequency signal beyond the standard quantum limit. The broadband squeezed vacuum consists a pair of EPR entangled beams: the signal beam (lower sideband field) around the carrier frequency ω_0, and the idler beam (upper sideband field) around ω_0+Λ. The lower sideband field around the carrier frequency ω_0 will carry the low-frequency modulation signal around the carrier frequency ω_0, however, the upper sideband field around ω_0+Λ feel nothing. The output lower and upper sideband fields may be separated in space by a mode cleaner cavity and measured by homodyne detection with two local oscillators at frequency ω_0 and ω_0+Λ respectively. The conditional squeezing of the output signal beam can be obtained by subtracting the photocurrent of the idler beam from that of the signal beam. Here, the lower and upper sideband fields of the broad squeezed light may be separated in space by a mode cleaner cavity before or after carrying the low-frequency modulation signal. Thus this scheme also can be considered as: 1) First, the broadband squeezed vacuum is separated into the lower and upper sideband fields in space by a mode cleaner cavity. 2) The lower sideband field around the carrier frequency ω_0 is sent into sensitive device (such as the interferometer), therefore, the low-frequency signal around the carrier frequency ω_0 is added in the lower sideband field by the sensitive device. 3) The lower and upper sideband fields are measured by homodyne detection with two local oscillators at frequency ω_0 and ω_0+Λ respectively. The conditional variance of the lower sideband beam can be obtained by subtracting the photocurrent of the upper sideband beam from that of the lower sideband beam. Thus this scheme can avoid producing the low frequency squeezing to improve the sensitivity of the interferometer. In this paper, we remove the mode cleaner cavity to separate the signal and idler beams in space and utilize a bichromaticlocal oscillator (BLO) to directly detect the signal and idler beams of a broad high-frequency squeezed vacuum after carrying the low-frequency signal by combining with a phase-modulated coherent light at around ω_0 on a beam splitter. This scheme can avoid optical losses introduced by the mode cleaner cavity. Moreover by optimally adjusting the power of one of local oscillator for detecting the idler field, we can obtain the minimum conditional variance of the signal beam and improve the signal noise ratio. The theoretical scheme based on BLO to detect the squeezed state was proposed <cit.> and the phase-sensitive detection with a BLO or a double-sideband signal field were studied experimentally <cit.>. And the measurement of a broad squeezed vacuum state by means of a BLO was demonstrated experimentally <cit.>. Recently, by making use of the multi-frequency homodyne detection, the experiments of cross-frequency entanglements generated in periodically pumped OPOs have been reported <cit.>. The schematic diagram of the detection is shown in Fig. 1(a). A BLO with two local oscillators at frequency ω_0 and ω_0+Λ is mixed with the detected light field at a 50/50 beam splitter. The power of one of local oscillator (upper local oscillator) at ω_0+Λ can be adjusted with the factor g. The relative phase θ of the local oscillator and the detected field can be controlled by the reflective mirror mounted on a PZT (piezoelectric transducer). The annihilation operators of the BLO and the detected signal field can be written as â(t)=â_-(t)exp[-iω_0t]+â_+(t)exp[-i(ω_0+Λ)t] and b̂(t)=b̂_0(t)exp(-iω_0t), where â_+(-)(t) and b̂_0(t) are the slow varying operators of the fields. The normalized difference of the photocurrents of the two detectors at the 50/50 beam splitter isî(t)=1/a[⟨â^†(t)⟩b̂(t)e^-iθ+⟨â(t)⟩b̂^†(t) e^iθ],where the fields satisfy ⟨â_+⟩=ga, and ⟨â_-⟩= a≫⟨b̂_0⟩∼0. Therefore the BLO is a pair of the strong coherent states. And the detected field is the vacuum state or the squeezed vacuum state carrying the low-frequency signal around frequency ω_0.The difference of the photocurrents analyzed at the radio frequency Ω is expressed asî(Ω) = Q̂_-(Ω,θ)+gQ̂_+(Ω,θ).Here, we express the quadrature component of the signal field around the central frequency ω_0, which easily compare with the measurement with a single local oscillator (g=0) at ω_0. Therefore, the quadrature component of the detected field can be defined as Q̂_-(Ω,θ)=b̂(ω_0-Ω)e^-iθ+b̂^†(ω_0+Ω)e^iθ, and Q̂_+(Ω,θ)=b̂(ω_0+Λ-Ω)e^-iθ+b̂^†(ω_0+Λ+Ω)e^iθ. The quadrature amplitude (θ=0) can be X̂_-(Ω)=b̂(ω_0-Ω)+b̂^†(ω_0+Ω) and the quadrature phase (θ=π/2) Ŷ_-(Ω)=-i[b̂(ω_0-Ω)-b̂^†(ω_0+Ω)]. The arbitrary quadrature component of the detected field can be measured by scanning the relative phase of θ. So when θ=0, the difference of the photocurrents will give the information of the quadrature amplitude of the detected field X̂_B(Ω)=X̂_-(Ω)+gX̂_+(Ω), and when θ=π/2, the quadrature phase Ŷ_B(Ω)=Ŷ_-(Ω)+gŶ_+(Ω).Since a single broadband squeezed light can be split into a pair of upper and lower sideband fields as EPR entangled fields, the minimum conditional variance of the output lower sideband (signal) beam can be obtained with the help of the upper sideband (idler) beam. Considering the simple optical parametric oscillator (OPO) process, the nonlinear medium is pumped with the second-harmonic wave of ω_p=2ω_0+Λ. The annihilation operators of the output lower and upper sideband fields of OPO can be written asb̂_-^s = b̂_-^0coshr+b̂_+^† 0e^iθ_psinhr, b̂_+^s = b̂_+^0coshr+b̂_-^† 0e^iθ_psinhr,where r is squeezing factor, θ_p is the phase of the pump field, b̂_-^0 and b̂_+^0 are the annihilation operators of the input lower and upper sideband vacuum fields of OPO with ⟨δ^2X_-^0(Ω)⟩=⟨δ^2X_+^0(Ω)⟩=⟨δ^2Y_-^0(Ω)⟩=⟨δ^2Y_+^0(Ω)⟩=1. If θ_p=0, the phase amplitudes of the output lower and upper sideband fields of OPO can be given byX̂_-^s(Ω) = X̂_-^0(Ω)coshr+X̂_+^0(Ω)sinhr, Ŷ_-^s(Ω) = Ŷ_-^0(Ω)coshr-Ŷ_+^0(Ω)sinhr, X̂_+^s(Ω) = X̂_+^0(Ω)coshr+X̂_-^0(Ω)sinhr, Ŷ_+^s(Ω) = Ŷ_+^0(Ω)coshr-Ŷ_-^0(Ω)sinhr,then, the difference and sum of amplitude phase quadratures of the output lower and upper sideband fields of OPO are obtainedX̂_-^s(Ω)-X̂_+^s(Ω) = (X̂_-^0(Ω)-X̂_+^0(Ω))e^-r, Ŷ_-^s(Ω)+Ŷ_+^s(Ω) = (Ŷ_-^0(Ω)+Ŷ_+^0(Ω))e^-r, X̂_-^s(Ω)+X̂_+^s(Ω) = (X̂_-^0(Ω)+X̂_+^0(Ω))e^+r, Ŷ_-^s(Ω)-Ŷ_+^s(Ω) = (Ŷ_-^0(Ω)-Ŷ_+^0(Ω))e^+r.The variances of the output lower and upper sideband fields of OPO are expressed by⟨δ^2X_-^s(Ω)⟩ = ⟨δ^2X_+^s(Ω)⟩=⟨δ^2Y_-^s(Ω)⟩=⟨δ^2Y_+^s(Ω)⟩= e^-2r+e^2r/2,and the correlated variances are given by⟨δ^2(X̂_-^s(Ω)-X̂_+^s)(Ω)⟩ = ⟨δ^2(Ŷ_-^s(Ω)+Ŷ_+^s(Ω))⟩=2e^-2r, ⟨δ^2(X̂_-^s(Ω)+X̂_+^s)(Ω)⟩ = ⟨δ^2(Ŷ_-^s(Ω)-Ŷ_+^s(Ω))⟩=2e^+2r. The variance of the conditional quadrature phase Ŷ_B(Ω) detected by BLO with the factor g is expressed by⟨δ^2Ŷ_B(Ω)⟩ = ⟨δ^2(Ŷ_-^s+gŶ_+^s)⟩= e^2r/2(1-g)^2+e^-2r/2(1+g)^2.In parallel, the conditional quadrature amplitude X̂_B(Ω) is given by⟨δ^2X̂_B(Ω)⟩ = ⟨δ^2(X̂_-^s+gX̂_+^s)⟩= e^-2r/2(1-g)^2+e^2r/2(1+g)^2.When we choose the optimized value of g_opt=(e^2r-e^-2r)/(e^2r+e^-2r), the minimum conditional variance of the output lower sideband field is obtained⟨δ^2Ŷ_B^opt(Ω)⟩ = 2/e^2r+e^-2r.Here, the method of obtaining the minimum conditional variance by the optimized factor is same as the previous works <cit.>. Thus, for the vacuum field injection, g=0 (without the upper local oscillator ω_0+Λ) and ⟨δ^2Ŷ_B(Ω)⟩=1. When the broadband squeezed light with 3 dB is injected, the minimum conditional variance is -0.97 dB with g_opt=0.6. The experimental setup and schematic diagram are shown in Fig. 1. A diode-pumped intra-cavity frequency-doubled single-frequency laser provides the fundamental light of 200 mW at 1064 nm and the second-harmonic light of 450 mW at 532 nm simultaneously. The second-harmonic light with the frequency of ω_p=2ω_0+Λ is used to pump an OPO to generate the broad squeezed vacuum field. The OPO cavity is resonant for both the pump light at 532 nm and the fundamental light at 1064 nm. The OPO cavity with 38 mm long contains a type-I PPKTP crystal (1 mm×2 mm×10 mm), the front facet of which is highly reflective for 1064 nm and has a power transmittance of 5% for 532 nm, and an outcoupling mirror that is highly reflective at 532 nm and has an intensity transmittance of 12.5% at 1064 nm. The bandwidth of the OPO is about 70 MHz. The OPO cavity is locked by PDH (Pound-Driver-Hall) technology and the error signal is extracted by detecting the reflected pump light of the OPO. The output broad squeezed field carries the weak low-frequency signal (± 500 kHz) around the carrier frequency ω_0 at M1 (98/2 beam splitter). The fundamental output field (ω_p/2) passes through the acousto-optical frequency-shifted system and then is split into two beams with the frequency of ω_0=ω_p/2-5 MHz and ω_0+Λ=ω_p/2+5 MHz (here, Λ=10 MHz). In the frequency-shifted system, AOM1 shifts the laser frequency by the first-order diffraction with a mount of +110 MHz. Then the frequency-shifted laser is split into two parts, which are translated back by AOM2 and AOM3 with the mount of -105 MHz and -115 MHz respectively. The two frequency-shifted laser beams at ω_p/2±5 MHz as local oscillator (LO) 1 and 2 are combined on 50/50 BS with the same polarization to generate the BLO. Here, the power of LO1 is fixed and that of LO2 can be varied. A small portion from the frequency-shifted laser beams at ω_p/2-5 MHz is used to generate the weak low-frequency signal by a phase modulator. In order to lock the relative frequency and phase of the two LOs, the signal generators of the acousto-optical frequency-shifted system are locked by the clock synchronization technology <cit.>. The squeezed light with the weak low-frequency signal is mixed with the BLO on the 50/50 BS. Finally, the two output fields of the BS are detected by two balanced detectors. Fig. 2 shows the noise variance of the conditional quadrature phase amplitude as the function of the factor g^2. Here, the noise of the conditional quadrature phase amplitude is normalized to the SQL, which is determined only with the LO1 (g=0) and injecting vacuum field (blocking the squeezed light and signal field). When injecting the squeezed light and given the intensity of the LO2 (given the factor of g), the conditional arbitrary quadrature components are measured by scanning the relative phase of θ. The conditional quadrature phase amplitude as the function of the factor g^2 (Fig. 2(a)) can be obtained by finding the minimum and maximum noise variance from the measured arbitrary quadrature components, which are good agreement with the theoretical calculation. Fig. 2(a) gives two different squeezing (antisqueezing) with 3.9 dB (5.24 dB) and 5.9 dB (11.6 dB) respectively. Here, the extra noise N_e of the antisqueezed component can be calculated from the values of squeezing and antisqueezing (see Appendix). Fig. 2(b) and (c) the noise variance of the conditional arbitrary quadrature components with the optimal factor g_opt for two different input squeezing with 3.9 dB and 5.9 dB respectively. Thus, the minimum conditional variance is obtained with -1.5 dB for the initial squeezing of 3.9 dB and -3.1 dB for that of 5.9 dB. Here, the noise variances of the arbitrary quadrature components of the lower sideband field of the squeezing light (g=0) are constant and larger than SQL as the function of the relative phase of θ, which demonstrates that one beam of an EPR entangled pair is thermal state. Fig. 3 shows the enhanced sensitivity of low-frequency signal with 500 kHz around the frequency ω_0 by using broad squeezed light and BLO. When the vacuum field is injected, the noise floor (black (gray) curve in Fig. 3) of the signal corresponds to SQL with g=0. If the squeezed field is injected and g=0, the very noisy floor (blue (light gray) curve in Fig. 3) is the noise variance of one beam of an EPR entangled pair. When we choose the optimal factor g_opt, the enhanced sensitivity of low-frequency signal is obtained and the signal-noise ratio is improved with 1.5 dB for the initial broad squeezing of 3.9 dB and 3.1 dB for that of 5.9 dB. Quantum advantage resulting from the use of squeezed light is evaluated by comparing signal-to-noise ratios in this work. Because of additional degrees of freedom such as the optical gain g, a fair comparison between quantum and classical light can be difficult. However, the quantum noise floors of both the classical and the quantum approaches are compared after independent factor over g in this works, because changing g here does not change the signal level, as evidenced in Fig. 3. In this scheme, one can avoid the low frequency technical noise of the squeezed-light source by placing the signal and the detection in one of the squeezed sidebands. This is useful only if it is effectively the dominant source of noise. On the other hand, the low frequency noise introduced by homodyne detection can not avoid in the scheme. For example, amplitude noise will be rejected by the balanced detector, up to the common-mode rejection power. However, phase noise on will still creep in, particularly if the squeezing is strong.In conclusion, we have demonstrated a scheme of using a broadband high-frequency squeezed vacuum to detect low-frequency signal beyond the standard quantum limit. By means of the EPR entanglement of upper and lower sideband of the broadband squeezed light, the conditional variance in the low frequency band can be obtained by BLO detection with subtracting the photocurrent of the upper sideband beam from that of the lower sideband beam. Thus this scheme does not need directly generate the squeezing in the low frequency band. In addition, the BLO detection directly measures the signal mapped on the sideband of the squeezed state, in this sense this scheme can to some extent avoid the DC technical noise in the traditional homodyne detection stemming from the light sources.The authors would like to thank Yiqiu Ma for helpful discussion. This research is supported by the MOST (Grant No. 2016YFA0301602), NSFC (Grant No. 11234008, 11361161002, 11474188,11654002,6157127), Natural Science Foundation of Shanxi Province (Grant No. 2015011007), Research Project Supported by Shanxi Scholarship Council of China (Grant No. 2015-002).^†Corresponding author email: [email protected], [email protected]:The conditional squeezing with he extra noise of the antisqueezed component Usually, the broadband squeezed light generated by OPO is not the minimum uncertainty state, in which the antisqueezed component has the extra noise. So the phase amplitudes of the output lower and upper sideband fields of OPO with the extra noise can be given by X̂_-^s(Ω) = X̂_-^0(Ω)coshr+X̂_+^0(Ω)sinhr+N_X/2, Ŷ_-^s(Ω) = Ŷ_-^0(Ω)coshr-Ŷ_+^0(Ω)sinhr+N_Y/2, X̂_+^s(Ω) = X̂_+^0(Ω)coshr+X̂_-^0(Ω)sinhr+N_X/2, Ŷ_+^s(Ω) = Ŷ_+^0(Ω)coshr-Ŷ_-^0(Ω)sinhr-N_Y/2,here, ⟨δ^2N_X⟩=⟨δ^2N_Y⟩=N_e. The difference and sum of amplitude phase quadratures of the output lower and upper sideband fields of OPO are obtained X̂_-^s(Ω)-X̂_+^s(Ω) = (X̂_-^0(Ω)-X̂_+^0(Ω))e^-r, Ŷ_-^s(Ω)+Ŷ_+^s(Ω) = (Ŷ_-^0(Ω)+Ŷ_+^0(Ω))e^-r, X̂_-^s(Ω)+X̂_+^s(Ω) = (X̂_-^0(Ω)+X̂_+^0(Ω))e^+r+N_X, Ŷ_-^s(Ω)-Ŷ_+^s(Ω) = (Ŷ_-^0(Ω)-Ŷ_+^0(Ω))e^+r+N_Y.The variances of the output lower and upper sideband fields of OPO are expressed by⟨δ^2X_-^s(Ω)⟩ = ⟨δ^2X_+^s(Ω)⟩=⟨δ^2Y_-^s(Ω)⟩=⟨δ^2Y_+^s(Ω)⟩= e^-2r+e^2r/2+N_e/4,and the correlated variances are given by⟨δ^2(X̂_-^s(Ω)-X̂_+^s)(Ω)⟩ = ⟨δ^2(Ŷ_-^s(Ω)+Ŷ_+^s(Ω))⟩=2e^-2r, ⟨δ^2(X̂_-^s(Ω)+X̂_+^s)(Ω)⟩ = ⟨δ^2(Ŷ_-^s(Ω)-Ŷ_+^s(Ω))⟩=2e^+2r+N_e.The variance of the conditional quadrature phase Ŷ_B(Ω) and amplitude X̂_B(Ω) detected by the BLO with factor are expressed by⟨δ^2Ŷ_B(Ω)⟩ = ⟨δ^2(Ŷ_-^s+gŶ_+^s)⟩= 1/2(e^2r+N_e/2)(1-g)^2+e^-2r/2(1+g)^2, ⟨δ^2X̂_B(Ω)⟩ = ⟨δ^2(X̂_-^s+gX̂_+^s)⟩= 1/2(e^2r+N_e/2)(1+g)^2+e^-2r/2(1-g)^2.When we choose the optimized value of g_opt=(e^2r+N_e/2-e^-2r)/(e^2r+N_e/2+e^-2r), the minimum conditional variance of the output lower sideband field is obtained⟨δ^2Ŷ_B^opt(Ω)⟩ = 2e^-2r(e^2r+N_e/2)/e^2r+N_e/2+e^-2r.At the same time, we may give the quadrature amplitude Ŷ_B(Ω) with the same condition is expressed by⟨δ^2X̂_B(Ω)⟩ = ⟨δ^2(Ŷ_-^s+gŶ_+^s)⟩= 1/2(e^2r+N_e/2)(1-g)^2+e^-2r/2(1+g)^2. 10Abbott2016-1 B. 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"authors": [
"Wei Li",
"Yuanbin Jin",
"Xudong Yu",
"Jing Zhang"
],
"categories": [
"quant-ph",
"physics.ins-det",
"physics.optics"
],
"primary_category": "quant-ph",
"published": "20170727064227",
"title": "Enhanced sensitivity of low-frequency signal by using broad squeezed light and bichromatic local oscillator"
} |
A Jointly Learned Deep Architecture for Facial Attribute Analysis and Face Detection in the Wild Keke He, Yanwei Fu, Xiangyang XueFudan University {kkhe15, yanweifu, xyxue}@fudan.edu.cnDecember 30, 2023 ================================================================================================= In exciting new work, <cit.> showed that the classical best subset selection problem in regression modeling can be formulated as a mixed integer optimization (MIO) problem. Using recent advances in MIO algorithms, they demonstrated thatbest subset selection can now be solved at much larger problem sizes that what was thought possiblein the statistics community.They presented empirical comparisons of best subset selection with other popular variable selection procedures, in particular, the lasso and forward stepwise selection. Surprisingly (to us), their simulations suggested that best subset selection consistently outperformed both methods in terms of prediction accuracy.Here we present an expanded set of simulations to shed more light on these comparisons. The summary is roughly as follows:* neither best subset selection nor the lasso uniformly dominate the other, with best subset selection generally performing better in high signal-to-noise (SNR) ratio regimes, and the lasso better in low SNR regimes; * best subset selection and forward stepwise perform quite similarly throughout;* the relaxed lasso (actually, a simplified version of the original relaxed estimator defined in ) is the overall winner, performing just about as well as the lasso in low SNR scenarios, and as well as best subset selection in high SNR scenarios..tocmain § INTRODUCTION Best subset selection, forward stepwise selection, and the lasso are popular methods for selection and estimation of the parameters in a linear model.The first two are classical methods in statistics, dating back to at least <cit.> for best subset selection and <cit.> for forward selection; the lasso is (relatively speaking) more recent, due to <cit.>.Given a response vector Y ∈^n, predictor matrixX ∈^n× p, and a subset size k between 0 and min{n,p}, bestsubset selection finds the subset of k predictors that produces the best fitin terms of squared error, solving the nonconvex problem_β∈^p Y-Xβ_2^2 β_0 ≤ k,where β_0=∑_i=1^p 1{β_i ≠ 0} is the ℓ_0 norm of β. (Here and throughout, for notational simplicity,we omit the intercept term from the regression model.) Forward stepwise selection is less ambitious: starting with the empty model, it iteratively adds the variable that best improves the fit.[Other ways of defining the variable j_k that “best improves the fit” are possible, but the entry criterion is (<ref>) is the standard one in statistics.]It hence yields a subset of each size k=0,…,min{n,p},but none of these are generally globally optimal in the sense of (<ref>). Formally, the procedure starts with an empty activeset A_0={0}, and for k=1,…,min{n,p}, selects the variable indexed byj_k = _j ∉ A_k-1 Y - P_A_k-1∪{j_k} Y _2^2 = _j ∉ A_k-1 X_j^T P_A_k-1^⊥ Y/P_A_k-1^⊥ X_j_2that leads to the lowest squared error when added to A_k-1, or equivalently, such that X_j_k, achieves the maximum absolute correlation with Y,after we project out the contributions from X_A_k-1. A note on notation: here we write X_S ∈^n× |S| for the submatrix of X whose columns are indexed by a set S (and when S={j}, wesimply use X_j). We also write P_S for the projection matrix onto the columnspan of X_S, and P_S^⊥=I-P_S for the projection matrix onto theorthocomplement.At the end of stepk of the procedure, the active set is updated, A_k=A_k-1∪{j_k}, and the forward stepwise estimator of theregression coefficients is defined by the least squares fit onto X_A_k. The lasso solves a convex relaxation of (<ref>) where we replace the ℓ_0 norm by the ℓ_1 norm, namely _β∈^p Y-Xβ_2^2β_1 ≤ t,where β_1=∑_i=1^p |β_i|, and t ≥ 0is a tuning parameter. By convex duality, the above problem is equivalent to the more common (and more easily solveable) penalized form _β∈^p Y-Xβ_2^2+ λβ_1where now λ≥ 0 is a tuning parameter. This is the formthat we focus on in this paper.The lasso problem (<ref>) is convex (and highly structured) and there is by now a sizeable literature in statistics, machine learning, and optimization dedicated to efficient algorithms for this problem.On the other hand, the best subset selection problem (<ref>) is nonconvex and is known to be NP-hard <cit.>.The accepted view in statistics for many years has been that this problem is not solveable beyond (say) p in the 30s, this view being shaped by the available software for best subset selection (e.g., in the R language, the leaps package implements a branch-and-bound algorithm for best subset selection of ).For a much more detailed introduction to best subset selection, forward stepwiseselection, and the lasso, see, e.g., Chapter 3 of <cit.>.§.§ An exciting new development Recently, <cit.> presented a mixed integer optimization (MIO) formulation for the best subset selection problem (<ref>).This allows one to use highly optimized MIO solvers, like Gurobi (based on branch-and-cut methods,hybrids of branch-and-bound and cutting plane algorithms), to solve(<ref>). Using these MIO solvers, problems with p in the hundreds and even thousands are not out of reach, and this presents us with exciting new ground on which to perform empiricalcomparisons. Simulation studies in<cit.> demonstrated that best subset selection generally gives superior prediction accuracy compared to forward stepwise selection and the lasso, over a variety of problem setups.In what follows, we replicate and expand these simulations to shedmore light on such comparisons. For convenience, we made an R package bestsubset foroptimizing the best subset selection problem using the Gurobi MIO solver (after this problem has been translated into a mixedinteger quadratic program as in ). This package, as well as R code for reproducing all of the results in thispaper, are available at <https://github.com/ryantibs/best-subset/>. § PRELIMINARY DISCUSSION§.§ Is best subset selection the holy grail? Various researchers throughout the years have viewed best subsetselection as the “holy grail” of estimatorsfor sparse modeling in regression, suggesting (perhaps implicitly) that it should be used whenever possible, and that other methods for sparse regression—such as forward stepwise selection and the lasso—should be seen as approximations or heuristics, used only out of necessity when best subset selection is not computable.However, as we will demonstrate in the simulations that follow, this is not the case.Different procedures have differentoperating characteristics, i.e., give rise to different bias-variancetradeoffs as we vary their respective tuning parameters.In fact, depending onthe problem setting, the bias-variance tradeoff provided by best subset selection may be more or less useful than the tradeoff provided by the lasso. As a brief interlude, let us inspect the “noiseless” versions of the best subset and lasso optimization problems, namely_β∈^p β_0Xβ=Y,_β∈^p β_1Xβ=Y,respectively.Suppose that our goal is to find the sparsest solution to the linear system Xβ=Y. Problem (<ref>),by definition of the ℓ_0 norm, produces it. Problem (<ref>), in which the criterion has been convexified, does not generally give the sparsest solution and soin this sense we may rightly view it as a heuristic for thenonconvex problem (<ref>).Indeed, much of theliterature on compressed sensing (in which (<ref>), (<ref>) have been intensely studied) uses this language. However, in the noiseless setting, there is no bias-variance tradeoff, because (trivially) there is no bias and no variance; both of the estimators defined by (<ref>), (<ref>) have zero mean squared error owingto the linear constraint Xβ=Y (and the fact that Y=(Y|X), as there is no noise).The noisy setting—which is the traditional and most practical setting for statistical estimation, and that studied inthis paper—is truly different. Here, it is no longer appropriate to view the estimator defined by the ℓ_1-regularized problem(<ref>)as a heuristic for that defined by the ℓ_0-regularized problem(<ref>) (or (<ref>) as a heuristic for (<ref>)).Generally speaking, the lasso and best subset selectiondiffer in terms of their “aggressiveness” in selecting and estimating thecoefficients in a linear model, with the lasso being less aggressive than best subset selection; meanwhile, forward stepwise lands somewhere in the middle, in terms of its aggressiveness. There are various ways to make this vague but intuitive comparison more explicit.For example: * forward stepwise can be seen as a “locally optimal” version of best subset selection, updating the active set by one variable at each step, instead of re-optimizing over all possible subsets of a given size; in turn, the lasso can be seen as a more “democratic” version of forward stepwise, updating the coefficients so asmaintain equal absolute correlation of all active variables with the residual <cit.>;* the lasso applies shrinkage to its nonzero estimated coefficients (e.g., see (<ref>) with γ=1) but forward stepwise and best subset selection do not, and simply perform least squares on their respective active sets;* thanks to such shrinkage, the fitted values from the lasso (for any fixed λ≥ 0) are continuous functions of y <cit.>, whereas the fitted values from forward stepwise and best subset selection (for fixed k ≥ 1) jump discontinuously as y moves across a decision boundary for the active set; * again thanks to shrinkage, the effective degrees of freedom of the lasso (at any fixed λ≥ 0) is equal to the expected number of selected variables <cit.>, whereas the degrees of freedom of both forward stepwise and best subset selection can greatly exceed k at any given step k ≥ 1 <cit.>.Figure <ref> uses the latter perspective of effective degreesof freedom to contrast the aggressiveness of the three methods.When the signal-to-noise ratio (SNR) is low, and also depending on other factors like the correlations between predictor variables, the more aggressive best subset and forward stepwise methods can already have quite high variance at the start of their model paths (i.e., for small step numbers k). Even after optimizing over the tuningparameter k (using say, an external validation set or an oracle which reveals the true risk), we can arrive at an estimator with unwanted variance and worse accuracy than aproperly tuned lasso estimator. On the other hand, for high SNR values, and other configurations for the correlations between predictors, etc., the story can be completely flipped and the shrinkage applied by the lasso estimator can result in unwanted bias and worse accuracy than best subset selection and forward stepwise selection. See Figure <ref> for empirical evidence. This is a simple point, but is worth emphasizing.To convey the idea once more: Different procedures bring us from the high bias to the high variance ends of the tradeoff along different model paths; and these paths areaffected by aspects of the problem setting, like the SNR and predictor correlations, in different ways. For some classes of problems, someprocedures admit more fruitful paths, and for other classes, other procedures admit more fruitful paths.For example, neither best subset selection nor the lasso dominates the other, across all problem settings. §.§ What is a realistic signal-to-noise ratio?In their simulation studies, <cit.> consideredSNRs in the range of about 2 to 8 in their low-dimensional cases, and about 3 to 10 in their high-dimensional cases. Is this a realistic range that one encounters in practice?In our view, inspecting the proportion of variance explained (PVE)can help to answer this question.Let (x_0,y_0) ∈^p × be a pair of predictor and response variables, and define f(x_0)=(y_0|x_0) and ϵ_0=y_0-f(x_0), so that we may express the relationship between x_0,y_0 as:y_0 = f(x_0) + ϵ_0.The signal-to-noise ratio (SNR) in this model is defined as = (f(x_0))/(ϵ_0).For a given prediction fuction g—e.g., one trained on n samples (x_i,y_i), i=1,…,n that are i.i.d. to (x_0,y_0)—its associated proportion of variance explained (PVE) is defined as (g) = 1 - (y_0-g(x_0))^2/(y_0).Of course, this is maximized when we take g to be the mean function f itself, in which case(f) = 1 - (ϵ_0)/(y_0) = /1+.In the second equality we have assumed independence of x_0, ϵ_0, so (y_0)=(f(x_0))+(ϵ_0). As the optimal prediction function is f, it sets the gold-standard of /(1+) for the PVE, so we should always expect to see the attained PVE be less than /(1+) and greater than 0 (otherwise we could simply replace our prediction function by g=0.) We illustrate using a simulation with n=200 and p=100. Thepredictor autocorrelation was zero and the coefficients followed the beta-type 2pattern with s=5; see Section <ref> for details.We varied the SNRin the simulation from 0.05 to 6 in 20 equally spaced values. Wecomputed the lasso over 50 values of the tuning parameter λ, and selected the tuningparameter by optimizing prediction error on a separate validation set of sizen. Figure <ref> shows the PVE of the tuned lasso estimator,averaged over 20 repetitions from this simulation setup.Also shown is the population PVE, i.e., the maximum possible PVE at any given SNR level, of/(1+).We see that an SNR of 1.0 corresponds to a PVE of about 0.45 (with a maximum of0.5), while an SNR as low as 0.25 yields a PVE of 0.1 (with a maximum of 0.2).In our experience, a PVE of 0.5 is rare for noisy observational data, and 0.2 may be more typical. A PVE of 0.86, corresponding to an SNR of 6, is unheard of!With financial returns data, explaining even 2% of the variance (PVEof 0.02) would be considered huge, and the corresponding prediction function could lead to considerable profits if used in a trading scheme.Therefore, based on these observations, we examine a wider lower range of SNRs in oursimulations, compared to the SNRs studied in <cit.>.§.§ A (simplified) relaxed lasso In addition to the lasso estimator, we consider a simplified version of the relaxed lasso estimator as originally defined by <cit.>. Let ^lasso(λ) denote the solution in problem (<ref>), i.e., the lasso estimator at the tuningparameter value λ≥ 0. Let A_λ denote its active set, and let^LS_A_λ denote the least squares coefficients from regressing of Y on X_A_λ, thesubmatrix of active predictors. Finally, let ^LS(λ) be the full-sized (p-dimensional) version of the least squares coefficients, padded with zeros in the appropriately.We consider the estimator ^relax(λ,γ) defined by ^relax(λ,γ) = γ^lasso(λ) + (1-γ) ^LS(λ)with respect to the pair of tuning parameter values λ≥ 0and γ∈ [0,1]. Recall<cit.> that when the columns of X are in general position (a weak condition occurring almost surely for continuously distributed pedictors, regardless of n,p), it holds that: * the lasso solution is unique;* the submatrix X_A_λ of active predictors has full column rank, thus _A_λ^LS=(X_A_λ^T X_A_λ)^-1 X_A_λ^T Y is well-defined; * the lasso solution can be written (over its active set) as ^lasso_A_λ(λ)=(X_A_λ^T X_A_λ)^-1(X_A_λ^T Y -λ s), wheres ∈{-1,1}^|A_λ| contains the signs of the active lasso coefficients.Thus, under the general position assumption on X, the simplified relaxed lasso can be rewritten as ^relax_A_λ(λ,γ)= (X_A_λ^T X_A_λ)^-1 X_A_λ^T Y - γλ (X_A_λ^T X_A_λ)^-1 s^relax_-A_λ (λ,γ) = 0,so we see that γ∈ [0,1] acts as a multiplicative factor applied directly to the “extra” shrinkage term apparent in the lasso coefficients.Henceforth, we will drop the word “simplified” andwill just refer to this estimator as the relaxed lasso.The relaxed lasso tries to undo the shrinkage inherent in the lasso estimator, to a varying degree, depending on γ.In this sense, we would expect it to be more aggressive than the lasso, and have a larger effective degrees of freedom.However, even in its most aggressive mode, γ=0, itis typically less aggressive than both forward stepwise selection and best subset selection, in that it often has a smaller degrees of freedom than these two.See Figure <ref> for an example. §.§ Other estimators Many other sparse estimators for regression could be considered, for example, ℓ_1-penalized alternatives tothe lasso, like the Dantzig selector <cit.> and square-root lasso <cit.>; greedy alternatives to forward stepwise algorithm, like matching pursuit <cit.> and orthogonal matching pursuit <cit.>; nonconvex-penalized methods, such as SCAD <cit.>, MC+ <cit.>, and SparseNet <cit.>; hybrid lasso/stepwise approaches like FLASH <cit.>;and many others. It would be interesting to include all of these estimators in our comparisons, though that would make for a huge simulation suite and would dilutethe comparisons between best subset selection, forward stepwise, and the lasso that we would like to highlight. Roughly speaking, we would expect the Dantzig selector and square-root lasso to perform similarlyto the lasso; the matching pursuit variants to perform similarly to forward stepwise; and the nonconvex-penalized methods to perform somewhere in between the lasso and best subset selection.(It is worth noting that our R package is structured in such a way to make further simulations and comparisons straightforward. We invite interested readers to use it to perform comparisons to other methods.)§.§ Brief discussion of computational costsComputation of the lasso solution in(<ref>) has been a popular topic of research, and there are by now many efficient lasso algorithms.In our simulations, we use coordinate descent with warm starts over a sequence of tuning parameter values λ_1 > ⋯ > λ_m > 0, as implemented in the glmnet R package<cit.>.The base code for this iswritten in Fortran, and warm starts—plus additional tricks likeactive set optimization and screening rules <cit.>—make this implementation highly efficient.For example, for a problem with n=500 observations and p=100 variables, glmnet delivers the lasso solutions across100 values of λ in less than 0.01 seconds, on a standard laptop computer.The relaxed lasso in (<ref>) comes at only a slight increase in computational cost, seeing as we must only additionally compute the least squares coefficients on each active set.We provide an implementation in the bestsubset R package accompanying this paper, which just uses an R wrapper around glmnet.For the same example with n=500 and p=100, computing the relaxed lasso path over 100 values of λ and 10 values of γagain took less than 0.01 seconds.For forward stepwise selection, we implemented our own versionin the bestsubset R package.The core matrix manipulations for this method are written in C, and the restis written in R.The forward stepwise path is highly structured and this greatly aids its computation: at step k, we havek-1 active variables included in the model, and we seek the variable among the remaining p-k+1 that—once orthogonalized with respect to the current active set of variables—achieves the greatest absolute correlation with Y, as in (<ref>). Suppose that wehave maintained a QR decomposition of the active submatrix X_A_k-1 of predictors, as well as the orthogonalization of the remaining p-k-1 predictors with respect to X_A_k-1. We can compute the necessary correlations in O(n(p-k+1)) operations, update theQR factorization of X_A_k in constant time, and orthogonalize the remaining predictors with respect to the one just included in O(n(p-k)) operations (refer to the modified Gram-Schmidt algorithm in ).Hence, the forward stepwiseselection path can be seen as a certain guided QR decomposition forcomputing the least squares coefficients on all p variables (or, on somesubset of n variables when p>n).For the same example with n=500 and p=100, our implementation computes the forward stepwise path in less than 0.5 seconds.Best subset selection (<ref>) is the most computationally challenging, by a large margin. <cit.>describe two reformulations of (<ref>) as a mixed integer quadratic program, one that is preferred when n ≥ p, and the other when p>n, and recommend using the Gurobi commercial MIO solver (which is free for academic use).They also describe a proximal gradient descent method for computing approximate solutions in (<ref>), and recommend using the best output from this algorithm over many randomly-initialized runs to warm start the Gurobi solver.See <cit.> for details. We have implemented the method of these authors[We thank the third author Rahul Mazumder for his help and guidance.]—whichtransforms the best subset selection problem into one of two MIO formulations depending on the relative sizes of n and p, uses proximal gradient to compute a warm start, and then calls Gurobi through its R interface—in our accompanying R package bestsubset.Gurobi uses branch-and-cut techniques(a combination of branch-and-bound and cutting plane methods), along with many other sophisticated optimization tools, for MIO problems.Compared to the pure branch-and-bound method fromthe leaps R package,its speed can be impressive: for example, in one run with n=500 andp=100, it returned the best subset selection solution of size k=8 in about 3 minutes (brute-force search for this problem would need to have looked at about 186 billion candidates!).But for most problems of this size (n=500 and p=100) it has been our experience that Gurobi typically requires 1 hour or longer to complete its optimization. The third author Rahul Mazumder of <cit.> suggested to us that for these problem sizes, it is often the case that Gurobi has found the solution in less than 3 minutes, though it takes much longer to certify its optimality. For our simulations in the next section, we used a time limit of 3 minutes for Gurobi to optimize the best subset selection problem (<ref>) at any particular value of the subset size k (once the time limit has been reached, the solver returns its best iterate).For more discussion on this choice and its implications, see Section <ref>.We note that this is already quite a steep computational cost for “regular” practical usage: at 3 minutes per value of k, if we wanted to use 10-fold cross-validation to choose between the subset sizes k=0,…,50, then we are already facing 25 hours ofcomputation time.§ SIMULATIONS§.§ SetupWe present simulations, basically following the simulation setup of <cit.>, except that we consider a wider range of SNRvalues. Given n,p (problem dimensions), s (sparsity level), beta-type (pattern of sparsity), ρ(predictor autocorrelation level), and ν (SNR level), our process can be described as follows: i. we define coefficients β_0 ∈^p according to s and the beta-type, as described below;ii. we draw the rows of the predictor matrix X ∈^n × p i.i.d. from N_p(0,Σ), where Σ∈^p× p has entry (i,j) equal to ρ^|i-j|; iii. we draw the response vector Y ∈^n fromN_n(Xβ_0, σ^2 I), with σ^2 defined to meet thedesired SNR level, i.e., σ^2=β_0^T Σβ_0 /ν;iv. we run the lasso, relaxed lasso, forward stepwise selection, and best subset selection on the data X,Y, each over awide range of tuning parameter values; for each method we choose the tuning parameter by minimizing prediction error on validation set X∈^n× p,Y∈^n that is generated independently of and identically to X,Y, as in steps ii–iii above;v. we record several metrics of interest, as specified below; vi. we repeat steps ii-v a total of 10 times, and average the results. Below we describe some aspects of the simulation process in more detail.Coefficients.We considered four settings for the coefficients β_0 ∈^p:* beta-type 1: β_0 has s components equal to 1,occurring at (roughly) equally-spaced indices between 1 and p, and the rest equal to 0; * beta-type 2: β_0 has its first s components equal to 1, and the rest equal to 0;* beta-type 3: β_0 has its first s components taking nonzero values equally-spaced between 10 and 0.5, and the rest equal to 0; * beta-type 5: β_0 has its first s components equal to 1, and the rest decaying exponentially to 0,specifically, β_0i=0.5^i-s, for i=s+1,…,p. The first three types were studied in <cit.>.They also defined a fourth typethat we did not include here, as we found it yielded basically thesame results as beta-type 3.The last type above is new: we included it to investigate the effects of weak sparsity and call itbeta-type 5, to avoid confusion. Evaluation metrics. Let x_0 ∈^p denote test predictor values drawn from N_p(0,Σ) (as in the rows of thetraining predictor matrix X) and let y_0 ∈ denote its associated response value drawn from N(x_0^T β_0, σ^2). Also letdenote estimated coefficients from one of the regression procedures.We considered the following evaluation metrics: * Relative risk: this is the accuracy metric studied in <cit.>[Actually, these authors used an “in-sample” version of this metric defined as X-Xβ_0_2^2/X_2^2, whereas ourdefinition is “out-of-sample”, with an expectation over the new test predictor value x_0 taking the place of the sample averageover the training values x_i, i=1,…,n.], defined asRR() =(x_0^T-x_0^Tβ_0)^2/(x_0^Tβ_0)^2 =(-β_0)^TΣ(-β_0)/β_0^TΣβ_0.The expectations here and below are taken over the test point (x_0,y_0),with all training data and validation data (and thus ) held fixed. A perfect score is 0 (if =β_0) and the null score is 1 (if =0). * Relative test error: this measures the expected test error relative to the Bayes error rate, RTE() = (y_0-x_0^T)^2/σ^2 = (-β_0)^TΣ(-β_0)+σ^2/σ^2.A perfect score is 1 and the null score is (β_0^TΣβ_0+σ^2)/σ^2=+1. * Proportion of variance explained:as defined in Section <ref>, this isPVE(β̂) =1-(y_0-x_0^T )^2/(y_0)=1-(-β_0)^TΣ(-β_0)+σ^2/β_0^TΣβ_0+σ^2.A perfect score is /(1+) and the null score is 0.* Number of nonzeros: unlike the last three metrics which measure predictive accuracy, this metric simply records the number of nonzero estimated coefficients,_0 = ∑_i=1^p 1{_i≠0}. It is worth noting that, in addition to metrics based on predictive accuracy, it would be useful to consider a metric that measures proper variable recovery, i.e., the extent to which thesparsity pattern in the estimatedmatches that in β_0. We briefly touch on this in the discussion. Here we mention one advantage to studying predictive accuracy: any of the metrics defined above are still relevant when (y|x) is no longer assumed to be linear, making the predictive angle more broadly practically relevant than a study of proper variable recovery(which necessarily requires linearity of the true mean).Configurations.We considered the following four problem settings:* low: n=100, p=10, s=5;* medium: n=500, p=100, s=5; * high-5: n=50, p=1000, s=5;* high-10: n=100, p=1000, s=10.In each setting, we considered ten values for the SNR ranging from 0.05 to 6 on a log scale, namely[0.05 0.09 0.14 0.25 0.42 0.71 1.22 2.07 3.52 6.00; ;0.05 0.08 0.12 0.20 0.30 0.42 0.55 0.67 0.78 0.86 ](For convenience we provide the corresponding population PVE as well.) In each setting, we also considered three values for the predictor autocorrelation ρ, namely 0, 0.35, and 0.7. Tuning of procedures. In the low setting, the lasso was tuned over 50values of λ ranging from λ_max=X^T Y_∞ to a small fraction of λ_max on a log scale, as per thedefault in glmnet, and the relaxed lasso was tuned over the same 50values of λ, and 10 values of γ equally spaced from 1 to 0 (hence a total of 500 tuning parameter values).Also in the low setting, forward stepwise and best subset selection were tuned over steps k=0,…,10.In all other problem settings (medium, high-5, and high-10), the lasso was tuned over 100 values of λ, the relaxed lasso was tuned over the same 100 values of λ and 10 values of γ (hence 1000 tuning parameter values total), and forward stepwise and best subset selection were tuned over steps k=0,…,50. In all cases, tuning was performed by by minimizing prediction error on an external validation set of size n, which we note approximately matches the precision of leave-one-out cross-validation. §.§ Time budget for GurobiAs mentioned in Section <ref>, for each problem instance and subset size k, we used a time limit of 3 minutes for Gurobi to optimize the best subset selection problem.In comparison, <cit.>used much larger time budgets: 15 minutes (per problem per k) for problems with p=100 as in our medium setup, and 66 minutes (per problem per k) for problems with p ≥ 2000 as in our high-5 and high-10 setups.Their simulations however were not as extensive, as they looked at fewer combinations of beta-types, SNR levels, and correlation levels. Another important difference worth mentioning: the Gurobi optimizer, when run through its Python/Matlab interface,as in <cit.>,automatically takes advantage of multithreading capabilities; this does not appear to be the case when run through its R interface, as in our simulations. The third author of <cit.>, Rahul Mazumder, suggestedin personal communication that the MIO solver in the medium setting will often arrive at the best subset selection solution in less than 3 minutes, but it can take much longer to certify its optimality[Gurobi constructs a sequence of lower and upper bounds on the criterion in (<ref>); typically the lower bounds come from convexrelaxations and the upper bounds from the current iterates, and it is the lower bounds that take so long to converge.] (usually over 1 hour, in absence of extra speedup tricks as described in ). Meanwhile, in the high-5 and high-10 settings, this author also pointed out that 3 minutes may no longer be enough.For practical reasons, we have keptthe 3 minute budget per problem instance per subset size. Note that this amounts to 150 minutes per path of 50 solutions, 1500 minutes or 25 hours per set of 10 repetitions, and in total 750 hours or 31.25 days for any given setting, once we go through the 10 SNR levels and 3 correlation levels. §.§ Results: computation time In Table <ref>, we report the time in seconds taken by eachmethod to compute one path of solutions, averaged over 10 repetitions and all SNR and predictor correlation levels in the given setting.All timings were recorded on a Linux cluster. As explained above, the lasso path consisted of 50 tuning parameter values in the low setting and 100 in all other settings, the relaxed lasso path consisted of 500 tuning parameter values in the low setting and 1000 in all other settings, and the forward stepwise and best subset selection paths each consisted of min{p,50} tuning parameter values.We can see that the lasso and relaxed lasso are very fast, requiring less than 25 milliseconds in every case. Forward stepwise selection is also fast, though not quite as fast as the lasso (some of the differences here might be due to the fact thatour forward stepwise algorithm is implemented partly in R).Moreover, it should be noted thatwhen n and p is large, and one wants to explore models with a sizeable number of variables (we limited our search to models of size 50), forward stepwise has to plod through its path one variable at a time, but the lassocan make jumps over subset sizes bigger than one by varying λ and leveraging warm starts.Recall, the MIO solver for best subset selection was allowed 3 minutes per subset size k, or 150 minutes for a path of 50 subset sizes. As the times in Table <ref> suggest, the maximum allotted time was not reached in all instances, and the MIO solver managed to verify optimality of some solutions along the path. In the medium setting, on average17.55 of the 50 solutions were verified as being optimal. In the high-5 and high-10 settings, only 1.61 of the 50 were verified on average (note this count includes the subset of size 1, which is trivial).These measures maybe pessismistic, as Gurobi may have found high-quality approximate solutions oreven exact solutions but was just not able to verify them in time, see the discussion in the above subsection. §.§ Results: accuracy metrics Here we display a slice of the accuracy results, focusing for concreteness on the case in which the predictor autocorrelation is ρ=0.35, and thepopulation coefficients follow the beta-type 2 pattern.In a supplementary document, we display the full set of results, over the whole simulation design.Figure <ref> plots the relative risk, relative test error, PVE, and number of nonzero coefficients as functions ofthe SNR level, for the low setting.Figures <ref>, <ref>, and <ref> show the same for the medium, high-5, and high-10 settings, respectively.Each panel in the figures displays the average of a given metric over 10 repetitions, for the four methods in question, and vertical bars denote one standarderror.In the relative test error plots, the dotted curve denotes the performance of the null model (null score); in the PVE plots, it denotesthe performance of the true model (perfect score); in the number ofnonzero plots, it marks the true support size s. The low and medium settings, Figures <ref> and <ref>, yield somewhat similar results.In the relativerisk and PVE plots (top left and bottom left panels), we see that best subset and forward stepwise selection lag behind the lasso and relaxed lasso in terms of accuracy for low SNR levels, and as the SNR increases, we see that all four methods converge to nearly perfect accuracy.The relative test error plot (top right panel) magnifies the differences between the methods. For low SNR levels,we see that the lasso outperforms the more aggressive best subset and forward stepwise methods, but for high SNR levels, it is outperformed by the latter two methods.The critical transition point—the SNR value at which their relative test error curves cross—is different for the low and mediumsettings: for the low setting, it is around 1.22, and for the medium setting, it is earlier, around 0.42.The relaxed lasso, meanwhile, is competitive across all SNR levels: at low SNR levels it matches the performance of the lasso at low SNR levels, and at high SNR levels it matches that of best subset and forward stepwise selection. It is able to do so by properly tuning the amount of shrinkage (via its parameter γ) on the validation set.Lastly, the number of nonzero estimated coefficients from the four methods (bottom right panel) is also revealing.The lasso consistently delivers much denser solutions; essentially, to optimize prediction error on the validation set, it is forced to do so, as the sparser solutions along its path feature too much shrinkage.The relaxed lasso does not suffer from this issue, again thanks to its ability to unshrink (move γ away from 1); it delivers solutions that are just as sparse as those from best subset and forward stepwise selection, except at the low SNR range.The high-5 and high-10 settings, Figures <ref> and <ref>, behave quite differently.The high-5 setting (smaller n andsmaller s) is more dire: the PVEs delivered by all methods—especially best subset selection—are negative for low SNR values, due to poor tuningon the validation set (had we chosen the null model for eachmethod, the PVE would have been zero).In both the high-5 and high-10 settings, we see that there is generally no reason, based on relative risk,relative test error, or PVE, to favor best subset selection or forward stepwiseselection over the lasso.At low SNR levels, best subset and forward stepwise selection often haveworse accuracy metrics (and certainly more erratic metrics); at high SNR levels, these procedures do not show much of an advantage.For best subsetselection, it is quite possible that its performance at the high SNR range would improve if we gave Gurobi a greater budget (than 3 minutes per problem instance per subset size).The relaxed lasso again performs the best overall, with a noticeable gap in performance at the high SNR levels.As is confirmed by the number of nonzero coefficients plots, the lasso and best subset/forward stepwise selection achieve similar accuracy in the high SNR range using two opposite strategies: the former uses high-bias and low-variance estimates, and the latter uses low-bias and high-variance estimates.The relaxed lassois most accurate by striking a favorable balance between these two polar regimes. §.§ Summary of results As mentioned above, the results from our entire simulation suite can be found in a supplementary document.Here is a high-level summary.* An important caveat to emphasize upfront is that the Gurobi MIO algorithm for best subset selection was given 3 minutes per problem instance per subset size.This practical restriction may have caused best subset selection to underperform, in particular, at the high SNR levels in the high-5 and high-10 settings.* Forward stepwise selection and best subset selection perform quite similarly throughout (with the former being much faster). This does not agree with the results for forward stepwise in<cit.>, where it performed quite poorly in comparison.In talking with the third author, Rahul Mazumder, we have learned that this was due to the fact that forward stepwise in their study was tuned using AIC, rather than a separate validation set. So, when put on equal footing and allowed to select its tuningparameter using validation data just as the other methods, we see that it performs quite comparably.* The lasso gives better accuracy results than best subset selection in the low SNR range and worse accuracy than best subset in the high SNR range.The transition point—the SNR level past which best subset outperforms the lasso—varies depending on the problem dimensions (n,p) predictor autocorrelation (ρ), and beta-type (1 through 5).For the mediumsetting, the transition point comes earlier than in the low setting.For the high-5 and high-10 settings, the transition point often does not come at all (before an SNR of 6, which is the maximum value we considered).As the predictor autocorrelation level increases, the transition point typically appears later (again, in some cases it does not come at all, e.g., for beta-type 5 and autocorrelation ρ=0.7). * The relaxed lasso provides altogether the top accuracy results.Innearly all cases (across all SNR levels, and in all problem configurations) we considered, it performs as well as or better than all other methods.Weconclude that it is able to use its auxiliary shrinkage parameter (γ) to get the “best of both worlds”: it accepts the heavy shrinkage from the lasso when such shrinkage is helpful, and reverses it when it is not.* The proportion of variance explained plots remind us that, despite what may seem like large relative differences, the four methods under consideration do not have very different absolute performances in this intuitive and important metric.It thus makes sense overall to favor the methods that are easy to compute.§ DISCUSSION The recent work of <cit.>has enabled the first large-scale empirical examinations of best subset selection.In this paper, we have expanded and refined the simulations in their work, comparing best subsetselection to forward stepwise selection, the lasso, and the relaxed lasso.We have found: (a) forward stepwise selection and best subset selection perform similarly throughout; (b) best subset selection often loses to the lasso except in the high SNR range; (c) the relaxed lasso achieves “the best of both worlds” and performs on par with the best method in each scenario.We note that these comparisons are based on (various measures of) out-of-sample prediction accuracy.A different target, e.g., a measure of support recovery, may yield different results.Our R package bestsubset, designed to easily replicate all ofthe simulations in this work, or forge new comparisons, is available at<https://github.com/ryantibs/best-subset/>. agsm 160 140 120 Supplement to “Extended Comparisons of Best Subset Selection,Forward Stepwise Selection, and the Lasso”This supplementary document contains plots from the simulation suite described in the paper “Extended Comparisons of Best Subset Selection,ForwardStepwise Selection, and the Lasso”.The plots in Section 1 precisely follow the simulation format described in the paper. Those in Section 2 follow an analogous format, except that the tuning has been done using an “oracle”, rather than a validation set as in Section 1. Specifically, the tuning parameter for each method in each scenario is chosen to minimize the average risk over all of the repetitions. .tocapp mainnone appsubsubsection § VALIDATION TUNING§.§ Low setting: n=100, p=10, s=5 §.§.§ Relative risk (to null model)§.§.§ Relative test error (to Bayes) §.§.§ Proportion of variance explained §.§.§ Number of nonzero coefficients§.§ Medium setting: n=500, p=100, s=5 §.§.§ Relative risk (to null model)§.§.§ Relative test error (to Bayes) §.§.§ Proportion of variance explained §.§.§ Number of nonzero coefficients§.§ High-5 setting: n=50, p=1000, s=5 §.§.§ Relative risk (to null model)§.§.§ Relative test error (to Bayes) §.§.§ Proportion of variance explained §.§.§ Number of nonzero coefficients§.§ High-10 setting: n=100, p=1000, s=10 §.§.§ Relative risk (to null model)§.§.§ Relative test error (to Bayes) §.§.§ Proportion of variance explained §.§.§ Number of nonzero coefficients § ORACLE TUNING§.§ Low setting: n=100, p=10, s=5 §.§.§ Relative risk (to null model)§.§.§ Relative test error (to Bayes) §.§.§ Proportion of variance explained §.§.§ Number of nonzero coefficients§.§ Medium setting: n=500, p=100, s=5 §.§.§ Relative risk (to null model)§.§.§ Relative test error (to Bayes) §.§.§ Proportion of variance explained §.§.§ Number of nonzero coefficients§.§ High-5 setting: n=50, p=1000, s=5 §.§.§ Relative risk (to null model)§.§.§ Relative test error (to Bayes) §.§.§ Proportion of variance explained §.§.§ Number of nonzero coefficients§.§ High-10 setting: n=100, p=1000, s=10 §.§.§ Relative risk (to null model)§.§.§ Relative test error (to Bayes) §.§.§ Proportion of variance explained §.§.§ Number of nonzero coefficients | http://arxiv.org/abs/1707.08692v2 | {
"authors": [
"Trevor Hastie",
"Robert Tibshirani",
"Ryan J. Tibshirani"
],
"categories": [
"stat.ME",
"stat.CO"
],
"primary_category": "stat.ME",
"published": "20170727030428",
"title": "Extended Comparisons of Best Subset Selection, Forward Stepwise Selection, and the Lasso"
} |
Institute of Automation and Electrometry,Russian Academy of Sciences, Siberian Branch, 1 Koptjug Avenue, Novosibirsk 630090, Russia Novosibirsk State University, 2 Pirogov Street, Novosibirsk 630090, RussiaInstitute of Automation and Electrometry, Russian Academy of Sciences, Siberian Branch, 1 Koptjug Avenue, Novosibirsk 630090, Russia Novosibirsk State University, 2 Pirogov Street, Novosibirsk 630090, RussiaInstitute of Automation and Electrometry,Russian Academy of Sciences, Siberian Branch,1 Koptjug Avenue, Novosibirsk 630090, RussiaInstitute of Automation and Electrometry,Russian Academy of Sciences, Siberian Branch, 1 Koptjug Avenue, Novosibirsk 630090, Russia Novosibirsk State University, 2 Pirogov Street, Novosibirsk 630090, RussiaThe attractive plasmon force between two metallic walls when the electromagnetic wave propagates through a narrow slit has been studied earlier for parallel plates and normal incidence. In present paper the effects of imperfect adjustment of plates and laser beam are analyzed. The change of force for non-parallel plates is shown to be of the first order in inclination angle when the wedge is along wave propagation and of the second order for transverse case. The small incidence angle leads to decrease in force due to the antisymmetric waveguide mode appearing in the slit. 03.50.De,42.50.Wk,85.85.+jEffects of imperfect angular adjustment on plasmonic force David Shapiro December 30, 2023 ==========================================================§ INTRODUCTION Since the 1980s, Micro-Electro-Mechanical Systems (MEMS) have rapidly developed and become an essential part of the technology for micro-devises <cit.>. This industry produces controllers, accelerometers, gyroscopes having various applications, from common cell phones <cit.> to biosensors <cit.>. Today a new type of fast micro-controllers appears as a merge of MEMS and micro-optics: Micro-Opto-Electro-Mechanical Systems (MOEMS). Being based on the interaction between electromagnetic field and solid body MOEMS provides new possibilities in the nano- and micro-particles manipulation.In the case of metallic particle the surface plasmon polaritons are excited at the interface metal-dielectric, where the real part of dielectric function change its sign, and cause the plasmon force <cit.>. The forces, with collective plasmon resonances excited by the laser field, arestudied experimentally for the dielectric sphere near a conducting plate <cit.> and theoretically between two close metallic nanospheres <cit.>. They are challenging for optical trapping and laser tweezers <cit.>. In particular case of slit between plane-parallel metallic plates the attractive plasmon force has been predicted <cit.> with properties determined by the geometry, conductivity, and the light polarization. Magnitude of this force for gold walls and normal incidence is of the order of nanonewtons, hence it becomes an important experimental issue. To study this effect closely an experiment is carried out with the Nano Force Facility <cit.>. At the same time previous theoretical studies did not consider the possible experimental uncertainties such asmisalignment of the plates or laser beam.The goal of present paper is to calculate corrections to the force at small deviation from the plane-parallel geometry. In the first order the perturbation is a sum of corrections over different uncertainties, and then the additions can be estimated separately. In Sec. <ref> the wedgeparallel (A) and perpendicular (B) to the propagation is studied. The oblique incidence is treated in Sec. <ref>. The field (A) and Maxwell's tension (B) are calculated within the lower modes approximation. Sec. <ref> summarizes the results.§ NON-PARALLEL PLATESThe light-induced force had been studied theoretically for normal incidence in plane-parallel geometry <cit.>. The geometrical scheme is shown in Fig. <ref>.The plane wave has p-polarization with magnetic field vector along axis y, since only this state excites surface plasmons. The parallel metal plates at the distance 2ℓ are considered as infinite in x,y,z directions. The wavelength of radiation λ is assumed much greater than the half-width ℓ. In the opposite case waveguide modes are excited, and then decrease the amplitude h_0 of field at the slit entrance <cit.>. Plasmons are generated at the surface of metal with dielectric permittivity _M=_1+i_2, where _1<0,|_1|≫_2∼1.Remind the main relations for a perfect plane-parallel slit. Write electric and magnetic fields of monochromatic wave with frequency ω asE_i=e_i e^-iω t+c.c., H_i=h_i e^-iω t+c.c.,where i=x,y,z is the Cartesian index, c.c. means complex conjugated terms. The Maxwell tension tensor isσ_xx=|e_x|^2-|h_y|^2-|e_z|^2/4π≃h_0^2e^-2β_2z/4π√(|_1|)k_0l,where k_0=ω/c is the wavenumber of incident field in free space, c is the speed of light, h_0 is the amplitude. The propagation constant in the slit along z is β=β_1+iβ_2, whereβ_1≃ k_0+1/2l√(|_1|),β_2=_2/4l|_1|^3/2.If the plates are infinite along z the unperturbed force per unit length in y-direction of isf_0≃σ_xx/2β_2= h_0^2λ|_1|/4π^2_2.The value h_0 at the entrance of slit (x,z)=(0,0) is determined by the interference of incident and reflected waves. In the limiting caseof very narrow slit ℓ/λ≪1 their amplitudes are equal and h_0=2H_0, where H_0 is the amplitude of plane incident wave H_y(r⃗)=H_0e^ik⃗·r⃗. For wider slit h_0 has oscillating ℓ-dependence die to excitation of higher waveguide modes <cit.>. §.§ Longitudinal wedge Let the walls are inclined symmetrically, as shown in Fig. <ref> (a), where α≪1 is the wedge half-angle. The wallscan be described by a pair of linear equations x=ℓ_0+α z, x=-ℓ_0-α z. The Helmholtz equation for magnetic field in (x,z)-plane is(∂_x^2+∂_z^2+k^2) h=0,where for p-wave h=h_y, k^2= k_0^2, =1 in the slit and_M in the metal. We introduce the polar coordinates with the center at intersection point of the walls extensions, Fig. <ref>. The Helmholtz equation (<ref>) in free space φ^2<α^2reduces to[1/r∂/∂ r(r∂/∂ r)+ 1/r^2∂^2/∂φ^2+k_0^2]H=0,where k_0=ω/c. We look for the solution of the form h(r,φ)=R(r)Φ(φ) and get from (<ref>):(rR')'/rR+1/r^2Φ”/Φ+k_0^2=0. The separation of variables occurs; the angular equation has form Φ”+μΦ=0, where μ is the separation parameter. At the boundaries of perfect conductor (lnΦ)'(φ=±α)=0.The even and odd modes with μ=-π^2m^2/α^2 are:Φ_m(φ)=cosπ mφ/2α, m=0,2,…, sinπ mφ/2α, m=1,3,…The general solution is a Fourier series, the decomposion over angular eigen functions Φ_m(φ). Zero mode m=0 is a constant Φ_0=1. The norm of solution is determined by the magnetic field h_0 at the entrance, like in unperturbed Eq. (<ref>). For small α we can approximately assume that the entrance is bounded from below by a horizontal line, and not by an arc.For real metal the solution is more complicated, since the boundary conditions are different, namely, continuity of weighted normal derivative. Denoting separation parameter μ=-p^2 in the slit and μ=-p_M^2 in the metal we take into account the zero mode only for the sub-wavelength slit. Angular function for the zero mode at φ>0 isΦ_0(φ)=cosh(pφ/α), φ<α, exp(-p_Mφ-α/α), α<φ.At φ<0 the solution can be continued as an even function. Then the radial equations in free space and metal areR”+1/rR'+(k_0^2+p^2/α^2r^2)R=0, R”+1/rR'+(k_M^2+p^2_M/α^2r^2)R=0,respectively. In order to provide the solution as a cylindrical wave at infinity one must chose the Hankel function of the second kind:R(r)= H^(1)_ip/α(k_0r), φ<α,H^(1)_ip_M/α(k_Mr), α<φ.We find the first relation between p and p_Mequating the weighted logarithmic derivatives of angular function (<ref>):ptanh p=-p_M/_M. The second relation follows from radial solution (<ref>). Since the entrance at α≪1 is far from the polar coordinate origin r_0=ℓ_0/α, where ℓ_0 is the slit half-width, we have to use asymptotic formula for Hankel function (<ref>).There are different asymptotic expansions of cylindrical functions H^(1)_ν(ξ). The choice of asymptotics is dictated by the relative rate of increasing in argument ξ and parameter (the order) ν. We need the Debye expansion <cit.> while both the argument and parameter tend to infinity at fixed ratio ν/ξ: H_ν^(1)(ν/cosψ)≈√(2/πνtanψ)exp[iνtanψ-iνψ-iπ/4],ν→∞. Changing the variable r=ℓ_0/α+z we write the argument of Hankel function in asymptotic formula(<ref>) ask_0(ℓ_0/α+z) =ν/cosψ.We expand ψ to the Taylor series over the powers of zlimited ourselves by the main termsprovided z≪ r_0 orα z≪ℓ_0.Factor at the exponent (<ref>) in main order does not depend on z and is included in an arbitrary wave amplitude A. The zeroth order in the exponent is also included in the amplitude. In the first approximation, the wave is described by the expression:H_ν^(1)(ν/cosψ)≈ Ae^iην z,η=.d(tanψ-ψ)/dz|_z=0.Differentiating expression (<ref>), we getH_ν^(1)(ν/cosψ)≈ Aexp(± iβ z ),β = 1/ℓ√(a^2+p^2).Here a=k_0ℓ<1 is the dimensionless small parameter. Analogously, for metal we have β=√(ε a^2+p_M^2)/ℓ, where ℓ(z)=ℓ_0+α z is the slit half-width at height z, hence p^2+a^2=p_M^2+_Ma^2.This is the second relation that means the equal propagation constants along z in metal and free spaceor equal phase velocities. Substituting p_M from (<ref>) to (<ref>) we obtain the dispersion relationptanh p=-√(p^2+a^2(1-_M))/_M.The similar relation has been derived for coupled surface plasmons at the parallel boundaries of metal plate surrounded by free space <cit.>.The dispersion relation turns to be the same as for plane-parallel plates. The only difference consists in z-dependence of the half width ℓ(z). Denote L_z the height of the plates in z-direction. The variation of width up to L_z should be small (<ref>), i.e.α L_z≪ℓ_0.For α≠0 the Maxwell's tension(<ref>) can be integratedover z:f=2f_0∫_0^L_zexp( --2β_2z/1+α z/ℓ_0)/1+α z/ℓ_0β_2 dz = 2f_0β_2e^-2β_2ℓ_0/α/α[ (-2β_2ℓ_0/α)- (-2β_2ℓ_0/α+α^2L_z/ℓ_0) ],where f_0 is the unperturbed force (<ref>),and is the integral exponent <cit.>:(ζ)=∫_-∞^ζe^t/tdt. For small angle α≪1 the arguments of integral exponents become large parameters. The asymptotic of integral exponent gives a series for force (<ref>): f=f_0(1-e^-2β_2L_z){1+α[(2β_2L_z)^-1+1+(2β_2L_z)] }+O(α^2).A factor in front of curly bracket is a force between finite parallel walls, the next term is linear correction for inclination:δ f=-α f_0/2ℓ_0β_2L_z^2.The correction is negative, that means the decreasing force in broadened slit. Fig. <ref> shows the dependence onL_z. With decreasing angle α the curves tend to unperturbed dependence. In the domain of short slit the curves lie below the solid line, that corresponds to the negative correction (<ref>). For tall slit, where the curves intersect solid line, our Taylor series obtained for small z is not valid, since condition (<ref>) being violated.§.§ Transverse wedge We consider V-slit with wedge perpendicular to wave vector, Fig. <ref> (b). Let us introduce the new variables at y>0x'=ℓ x/ℓ+α y, y'=yto make the plates parallel in new coordinates. The first-derivative operators transform to∂_x=ℓ/ℓ+α y∂_x',∂_y=-αℓ x/(ℓ+α y)^2∂_x'+∂_y'.Hereafter the half-width ℓ≡ℓ_0 is independent of z. Then the Laplace operator in old coordinates acquires crossed terms in non-orthogonal varables:△=ℓ^2/(ℓ+α y)^2[ 1+α^2 x^2/(ℓ+α y)^2]∂_x'^2 +∂_y'^2-2αℓ x/(ℓ+α y)^2∂_x'∂_y'+2α^2ℓ x/(ℓ+α y)^3∂_x'+∂_z^2. For α≪1 the Migdal's perturbation theory <cit.> can be applied for geometry slightly differentfrom exactly solvable (see also <cit.>). We seek the solution to (<ref>) as a series:h(x',y',z)=g_0+α g_1+α^2g_2+…Substituting (<ref>) into (<ref>) and equating terms with the same power of α we get the chain.Zero order yields the unperturbed equation (△+k^2)g_0=0 with the known solutiong_0=h_0e^iβ zcoshϰ x', 0<x'<l, coshϰ le^-ϰ_M(x'-l), l⩽ x'.The first order is (△+k^2)g_1∝∂_x'∂_y'g_0=0, since the zero-order solution is independent of y', then g_1=0.Correction g_2 has the second order in α and may be neglected. We do not present the second-order term here since it is negligibly small at α≪1.§ OFF-NORMAL INCIDENCE As it was shown in the previous studies <cit.>, excitation of the zero mode in the narrow slit (k_0 l < 1) leads to attraction between the plates. Since in this case electro-magnetic pressure is proportional to |β^(0)|^2 - |k_0|^2 (where β^(0) is a propagation constant of this mode), the fact that |β^(0)|>k_0 plays a crucial role. But for the others modes |β^(i)|<k_0, and then they are evanescent. Deviation of angle γ between k⃗_0 and surface normal increases the contribution of odd modes, and thendecreases the attraction. We aim to calculate the effect of off-normal incidence below taking into account the first antisymmetric mode. §.§ Field in the slitHere we consider an incident p-wave, Fig. <ref>. The incident plane wave has only y-component of the magnetic field, and the wave vector k⃗_⃗0⃗ = (k_x, 0, k_z). The total field is H(x,z,t) = H(x,z)e^iω t+c.c. Solution of the problem is based onthe two-dimensional Helmholtz equation (<ref>) with the boundary conditions: continuity of tangential components of electric and magnetic fields at the boundaries.We imply Fourier-transformation to the field in order to get algebraic equations instead of differential. Thus, the field in the slit transforms into Fourier seriesH^> = ∑_ν h_ν b_ν(x) e^iβ^(ν) z,where b_ν(x) = cosq_νx (sinq_νx) for the even (odd) modes, β^(ν)2 = k_0^2-q_ν^2. The field in free space transforms into Fourier integral:H^< = (e^i k_0z z+R e^-i k_0z z) e^i k_0x x + ∫ a_k e^i k x - i κ z dk, R=cosγ-√(-sin^2γ)/cosγ+√(-sin^2γ)is the Fresnel reflection coefficient for ℓ→0, κ^2 = k_0^2-k^2.The boundary conditions in the slit reveals dispersion equations for even modes:tan(q_νl) = √((1-ε)a^2 - (q_νℓ)^2)/ε q_νℓ,for odd modes the trigonometric function has to be replaced: tan→ -.As it was shown for perfect conductor, in the case of sub-wavelength slit and normal incidence it is sufficient to consider only the first mode, and the others are negligible <cit.>. We imply the same assumption to the odd modes to simplify the calculations. We consider only the first even (h_0) and odd (h_1) modes:H^> = h_0cosh(q_0 x) e^iβ^(0) z + h_1sin(q_1 x)e^iβ^(1) z,where β^(0),β^(1) are the propagation constants of zero and first modes, respectively. To satisfy the conditions at z=0, we follow the procedure<cit.> E_x^<(x) = E_x^>(x) expressing a_k in terms of h_0 and h_1. Then the continuity condition H^<(x)=H^>(x), |x|<l gives coefficients h_0 = (1+R)f_q_0,k_0x + ℓ k_0z(1-R)/π∫((k_0x-k)ℓ) f_q_0,k/κ dk /f_q_0,q_0 + β^(0)ℓ/(2π){∫f_q_0,k f_q_0,k/κ dk + cosh(q_0 ℓ)/ε∫G^ev(q_0M,k) f_q_0,k/κ dk},h_1 = (1+R)f_q_1,-k_0x + ℓ k_0z(1-R)/π∫((k_0x-k)ℓ) f_q_1,-k/κ dk /i f_q_1,q_1 + β^(1)ℓ/(2π) {∫f_q_1,k f_q_1,-k/κ dk + sin(q_1 ℓ)/ε∫G^odd(q_1M,k) f_q_1,-k/κ dk}. f_q_0,k = ((i q_0+k)ℓ) + ((iq_0 - k)ℓ),f_q_1,k = i(((q_1 + k)ℓ) - ((q_1 - k)ℓ)),G^ev(x,y)= 2(xcosyℓ - 2ysinyℓ)/ℓ(x^2+y^2),G^odd(x,y)=-2i(xsinyℓ + 2ycosyℓ)/ℓ(x^2+y^2)and q_iM =√(β^(i)2-_M k_0^2), i=0,1 is the transverse wave vector in metal.If γ≪ 1 it is possible to estimate the amplitudes. For perfect metal q_0 ℓ→ 0, q_1 ℓ→π/2 thenh_0 ≈ 2-(k_0ℓγ)^2/3, h_1 ≈2 ik_0ℓγ.Hence, the amplitude h_1∼γ while the correction to h_0 is of the order of γ^2. §.§ Maxwell tension tensor Maxwell's tensor (<ref>) determines attractive force between the walls. Non-zero γ leads to excitation of the odd modes in the slit. But according to orthogonality of the even and odd modes, the set of equations on the Fourier coefficients splits into the two independent sets. We calculated the amplitudes of lower even and odd modes h_0,h_1. The presence of second mode adds the interference term h_0^*h_1 of the first order in γ.We find the dependence of σ_xx on k_0ℓ for different γ numerically. Tension σ_xx is shown in Fig. <ref>for gold as a function of parameter k_0ℓ. The first asymmetric mode decreases the force, especially for wider slit. Fig. <ref> demonstrates that the attraction can be changed by repulsion at higher width. When the curve intersects axis x an equilibriumdistance appears being large for small angles and decreasing with γ.§ CONCLUSION Three possible experimental uncertainties are considered: non-parallel plates along or transverse to wave propagation and off-normal incidence. We calculate their contribution to the force acting between the plates. Since in the first order force perturbation is a sum of all corrections, we calculate them separately. We get the analytic expression of field for wedge along the wave propagation. Correction to the force is of the first order in angle. For the transverse wedgethe correction is shown to be of the second order. Then it is necessary first to consider the contribution of longitudinal wedge. For non-parallel plates the sub-wavelength assumption a=k_0ℓ<1 allows us to consider only the zero mode, but for the off-normal incidence we have to include the first odd mode. We calculate numerically the pressure acting on the walls.Presence of the first odd mode decreases the pressure and could change attraction to repulsion. We are grateful to V. Nesterov for excellent discussion of experimental parameters. This work is supported by the Russian Foundation for Basic Research (# 16-52-12026) and The Council for grants of President of Russian Federation (NSh-6898.2016.2).24 fxundefined [1] ifx#1 fnum [1] #1firstoftwo secondoftwo fx [1] #1firstoftwo secondoftwo noop [0]secondoftwo ref[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 [Gardner and Varadan(2001)]Gardner:2001 author author J. W. Gardner and author V. K. Varadan, @nooptitle Microsensors, Mems and Smart Devices (publisher John Wiley & Sons, address New York, year 2001)NoStop [Rebeiz and Muldavin(2001)]969936 author author G. M. Rebeiz and author J. B. 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"authors": [
"Leonid Frumin",
"Alexander Tusnin",
"Oleg Belai",
"David Shapiro"
],
"categories": [
"physics.optics"
],
"primary_category": "physics.optics",
"published": "20170727024542",
"title": "Effects of imperfect angular adjustment on plasmonic force"
} |
a,b]Valentina Nigro corr1 a,b]Roberta Angelinicorr1 b]Benedetta RosiPerugia d]Monica Bertoldo ISOF d]Elena Buratti e]Stefano Casciardi a,b]Simona Sennato a,b]Barbara Ruzickacorr1[a]Istituto dei Sistemi Complessi del Consiglio Nazionale delle Ricerche (ISC-CNR), sede Sapienza, Pz.le Aldo Moro 5, I-00185 Roma, Italy [b]Dipartimento di Fisica, Sapienza Universitàdi Roma, P.le Aldo Moro 5, 00185 Roma, Italy [d]Istituto per i Processi Chimico-Fisici del Consiglio Nazionale delle Ricerche (IPCF-CNR), Area della Ricerca, Via G.Moruzzi 1, I-56124 Pisa, Italy [e]Department of Occupational and Environmental Medicine, Epidemiology and Hygiene, National Institution for Insurance Against Accidents at Work (INAIL Research), Via Fontana Candida 1, Monte Porzio Catone, 00040 Rome, Italy [Perugia]Present address: Dipartimento di Fisica e Geologia, Università di Perugia, Via A. Pascoli, 06123 Perugia (Italy) [ISOF]Present address:Istituto per la Sintesi Organica e la Fotoreattività del Consiglio Nazionale delle Ricerche (ISOF-CNR), via P. Gobetti 101, 40129 Bologna, Italy[corr1]Corresponding authors. E-mail addresses: [email protected] (V. Nigro); [email protected] (R. Angelini);[email protected] (B. Ruzicka). Tel: +39 06 4991 23469Hypothesis: The peculiar swelling behaviour of poly(N-isopropylacrylamide) (PNIPAM)-based responsive microgels provides the possibility to tune both softness and volume fraction with temperature, making these systems of great interest for technological applications and theoretical implications.Their intriguing phase diagram can be even more complex if poly(acrylic acid) (PAAc)is interpenetrated within PNIPAM network to form Interpenetrating Polymer Network (IPN) [IPN: Interpenetrating Polymer Network] microgels that exhibit an additional pH-sensitivity. The effect of the PAAc/PNIPAM polymeric ratio on both swelling capability and dynamics is still matter of investigation. Experiments: Here we investigate the role of PAAc in the behaviour of IPN microgels across the volume phase transition through dynamic light scattering (DLS) [DLS: Dynamic Light Scattering], transmission electron microscopy (TEM) [TEM:transmission electron microscopy] and electrophoretic measurements as a function of microgel concentration and pH.Findings: Our results highlight that aggregation is favored at increasing weight concentration, PAAc content and pH and that a crossover PAAc content C^*_PAAc [C_PAAc: weight content (%) of the PAAc network within each IPN microgel] exists above which the ionic charges on the microgel become relevant. Moreover we show that the softness of IPN microgels can be tuned ad hoc by changing the PAAc/PNIPAM ratio. These findings provide new insights into the possibility to control experimentally aggregation properties, charge and softness of IPN microgels by varying PAAc content.Microgels, colloidal suspensions, Dynamic Light Scattering, Transmission Electron Microscopy, Electrophoretic Measurements§ INTRODUCTION The novel class of responsive microgels has recently become very popular since their smart responsivity to external stimuli makes them very attractive for industrial applications <cit.> and excellent model systems for exploring the exotic behaviours emerging in soft colloids due to their softness <cit.>. Their interparticle potential and their effective volume fraction can be easily managed through unusual control parameters such as temperature, pH or solvent, allowing to explore unusual phase-behaviours <cit.>, significantly far from those of conventional hard colloids <cit.>. The deep investigation in the last years has shown how responsive microgels based on poly(N-isopropylacrylamide) (PNIPAM) undergo a reversible Volume Phase Transition (VPT)[VPT: Volume Phase Transition] at about 305 K that drives the system from a swollen hydrated state to a shrunken dehydrated one, as a consequence of the coil-to-globule transition of NIPAM chains <cit.>. It has been shown that the driving force for swelling can be estimated from the properties of linear PNIPAM solutions, while the microgel elasticity opposing swelling is mainly due to the network topology dependent on the cross-linker concentration <cit.>. The typical swelling/shrinking behaviour of any PNIPAM-based microgel leads to intriguing phase diagrams which may be even more complex if other polymers, sensitive to different external stimuli, are copolymerized or interpenetrated to obtain multi-responsive microgels. In particular PNIPAM microgels containing poly(acrilic acid) (PAAc), have an additional pH-sensitivity that controls mutual polymer/polymer and polymer/solvent interactions. In the case of PNIPAM-co-PAAc microgels[PNIPAM-co-PAAc: PNIPAM copolimerized with PAAc] the response strictly depends on the mutual interference between PNIPAM and PAAc <cit.>. On the contrary interpenetration of the hydrophilic PAAc and the homopolymeric PNIPAM networks (IPN PNIPAM-PAAc microgel) <cit.>, provides independent sensitivity to temperature and pH, retaining the same Volume Phase Transition Temperature (VPTT) [VPTT: Volume phase Transition Temperature] of pure PNIPAM microgel and allowing to make the two networks more or less dependent by changing pH. Interestingly, IPN microgels allow to control their elastic properties by changing the solution pH, the polymeric ratio PNIPAM/PAAc and the cross-linking degree of any polymeric network.Notwithstanding the IPN microgel potentialities, knowledge of their behaviour from a fundamental point of view is still very limited. Hu et al. reported a comparison between the hydrodynamic and giration radii in very dilute conditions <cit.> and investigated the phase behaviour and viscosity for controlled drug release as a function of temperature <cit.>.The viscoelastic behaviour has been studied by Zhou et al. <cit.> who tested also the in vivo controlled release. The role of softness in the glass formation has been assessed byMattsson et al. <cit.> who showed how these deformable colloidal particles can exhibit the same variation in fragility as that observed in molecular liquids.Liu et al. <cit.> performed a systematic morphological study on IPN particles in highly dilute conditions. Particle synthesis was controlled through different techniques and the hydrodynamic radius was obtained as a function of temperature for different pH and PAAc contents. Moreover IPN microgels have been investigated by our group through different techniques: the structural relaxation and the local structure of low concentration IPN samples at pH 5 and pH 7 have been respectively reported in Ref. <cit.> and Ref. <cit.>. The dependence on different solvents (H_2O and D_2O), to explore the role of H-bonds, has been reported in Ref. <cit.> at different length scales. Finally in Ref. <cit.> the experimental radius of PNIPAM and IPN at fixed PAAc concentration has been compared with the one expected from the Flory-Rehner theory.At variance with previous studies <cit.> the present research aims to understand the role playedby poly(acrylic acid) on the relaxation dynamics and on the aggregation process across the VPT. Despite the work done up to now on IPN microgels, to the best of our knowledge a similar study has never been reported before. We demonstrate that with increasing PAAc content an increase of charge density favours the formation of aggregates due to the interplay between attractive and repulsive interactions that can be triggered by changing both PAAc and pH. Moreover a critical PAAc concentration is found that signs the existence of two different behaviours, this phenomenology can be ascribed to the increasing relevance of the ionic charge.In the following we first characterize the microgel particles through dynamic light scattering and transmission electron microscopy measurements to gain information on hydrodynamic radii and particles morphology. A comparison with a Flory-Rehner model, which well reproduce the temperature behaviour of the measured radii, allows to have a qualitative insight on the number of counterions per polymer chain that increases with increasing PAAc. Then we find a huge growth of the relaxation time above the VPTT at high PAAc values and pH, clear evidence of the formation of large aggregates. Through mobility measurements we can relate this behaviour to the observed increase of mobilities towards more negative values, demonstrating the role played by PAAc and charge density on aggregation. Finally we prove that it is possible to tune ad hoc particle softness by tuning PAAc.§ EXPERIMENTAL METHODS §.§ Sample preparation*Materials N-isopropylacrylamide (NIPAM) [NIPAM: N-isopropylacrylamide] (Sigma-Aldrich), purity 97 %, and N,N'-methylene-bis-acrylamide (BIS)[BIS: N,N'-methylene-bis-acrylamide] (Eastman Kodak), electrophoresis grade, were purified by recrystallization from hexane and methanol, respectively, dried under reduced pressure (0.01 mmHg) at room temperature and stored at 253 K. Acrylic acid (AAc)[AAc: Acrylic acid] (Sigma-Aldrich), purity 99 %, with 180-220 ppm of MEHQ as inhibitor, was purified by distillation (40 mmHg, 337 K) under nitrogen atmosphere on hydroquinone and stored at 253 K. Sodium dodecyl sulfate (SDS)[SDS: Sodium dodecyl sulfate ], purity 98 %, potassium persulfate (KPS)[KPS: potassium persulfate], purity ≥ 98 %, ammonium persulfate (APS)[APS: ammonium persulfate ], purity 98 %, N,N,N',N'-tetramethylethylenediamine (TEMED), purity 99 %, ethylenediaminetetraacetic acid (EDTA)[EDTA: ethylenediaminetetraacetic acid], purity ≥ 98.5 %, and NaHCO_3[NaHCO_3: sodium hydrogen carbonate], purity 99.7-100.3 %, were all purchased from Sigma-Aldrich and used as received. Ultrapure water (resistivity: 18.2 MΩ·cm at 298 K) was obtained with Millipore Direct-Q® 3 UV purification system. All other solvents (Sigma Aldrich RP grade) were used as received. Dialysis tubing cellulose membrane (Sigma-Aldrich), cut-off 14,000 Da, was washed in running distilled water for 3 h, treated at 343 K into a solution at 3.0 % weight concentration of NaHCO_3 and 0.4 % of EDTA for 10 min, rinsed in distilled water at 343 K for 10 min and finally in fresh distilled water at room temperature for 2 h.*Synthesis of PNIPAM and IPN microgels PNIPAM microgels were synthesized by precipitation polymerization with (24.162 ± 0.001) g of NIPAM, (0.4480 ± 0.0001) g of BIS and (3.5190 ± 0.0001) g of SDS, solubilized in 1560 mL of ultrapure water and transferred into a 2000 mL four-necked jacked reactor equipped with condenser and mechanical stirrer. The solution was deoxygenated by bubbling nitrogen for 1 h and heated at (343 ± 1) K. (1.0376 ± 0.0001) g of KPS (dissolved in 20 mL of deoxygenated water) was added to initiate the polymerization and the reaction was allowed to proceed for 16 h. The resultant PNIPAM microgel was purified by dialysis against distilled water with frequent water change for 2 weeks. In the second step IPN microgels were synthesized by a sequential free radical polymerization method <cit.> with (140.08 ± 0.01) g of the PNIPAM dispersion at the final weight concentration of 1.06 %. 5 mL of AAc and (1.1080 ± 0.0001) g of BIS were added into the preformed PNIPAM microparticles in the temperature range where PNIPAM particles are swollen (T=294 K), allowing the growth of the PAAc network inside them. The mixture was diluted with ultrapure water up to a volume of 1260 mL and transferred into a 2000 mL four-necked jacketed reactor kept at 294 K by circulating water and deoxygenated by bubbling nitrogen inside for 1 h. 0.56 mL of TEMED were added and the polymerization was started with (0.4447 ± 0.0001) g of ammonium persulfate. Three different samples were prepared at three PAAc/PNIPAM ratio composition by stopping the reaction at the suitable degree of conversion of AAc. The samples were purified by dialysis against distilled water with frequent water changes for 2 weeks, iced andliophilized up to 1 % weight concentration. The synthesized particles, namely PNIPAM and IPN microgels, were analysed by ATR FT-IR and ^1H-NMR spectroscopies, as well as by elemental analysis to assess their chemical composition and the exact PAAc content <cit.>. The three investigated samples have the following PAAc weight concentrations (C_PAAc): C_PAAc=2.6 %, C_PAAc=10.6 %, C_PAAc=19.2 % <cit.>.Their polydispersity is found around 10-15% for PNIPAM and 15-20% for IPN microgels. PNIPAM polydispersity values are due to the SDS concentration used during synthesis C_SDS =2.25 g/L =7.81 mM in order to obtain nanosized particles. In fact while monodisperse microsized particles are obtained by using aC_SDS far below the critical micelle concentration (CMC=2.36 g/L =8.18 mM) C_SDS ≪ CMC, nanosized particles, as in the present work, are synthesized when the amount of SDS approaches the critical micelle concentration.These samples are more homogeneously structured than the samples prepared in low surfactant concentration but with an increased polydispersity <cit.>. In the case of IPN, particle size distribution is further enlarged if merging of PNIPAM particles is not fully suppressed during the polyacrilic acid polymerization <cit.>. Samples at different weight concentrations (%), in the following referred as C_w, were obtained by diluting in H_2Othe sample at 1% concentration. Samples at pH 3.5 and pH 7.5 were obtained by the addition of HCl [HCl: Hydrochloric acid] or NaOH [NaOH: Sodium hydroxide], respectively, to the samples at pH 5.5 obtained from the synthesis. §.§ Transmission Electron Microscopy Transmission Electron Microscopy (TEM) characterization was employed to study PNIPAM and IPN microgels morphology. All the samples for TEM measurements have been prepared by deposition at room temperature of 20 μL of microgel suspensions diluted up to C_w=0.1 % in MilliQ water, on a 300-mesh copper grid for electron microscopy covered by thin amorphous carbon film.Immediately after deposition, the excess of liquid was removed by touching the grid with a piece of filter paper. Samples were dried for 5 minutes before staining by addition of 10 μLof2 % aqueous phosphotungstic acid (PTA)(pH-adjusted to 7.3 using 1 M NaOH). Measurements were carried out by using a FEI TECNAI 12 G2 Twin (FEI Company, Hillsboro, OR, USA), operating at 120 kV and equipped with an electron energy filter (Gatan image filter) and a slow-scan charge-coupled device camera (Gatan multiscan). Statistical analysis of TEM images to determine the average diameter and particle size distribution were performed by Image J software by measuring the cross-sectional area of the particles and convert them to an equivalent spherical diameter. A minimum of 100 particles collected ondifferent captured images with the same magnification has been considered. The average size has been determined by considering the mean value obtained by a gaussian fit on the particle size distribution, the reported error being the statistical error of the mean.§.§ Dynamic Light Scattering Multiangle Dynamic Light Scattering (DLS) measurements have been performed using an optical setup based on a monochromatic and polarized beam emitted from a solid state laser (100 mW at λ=642 nm) and focused on the sample in a cylindrical VAT for index matching and temperature control. The scattered intensity is collected by five single mode optical fibers at five different scattering angles, namely θ=30°, 50°, 70°, 90°, 110°, corresponding to five scattering vectors Q in the range (0.0067 nm^-1 ≤ Q ≤ 0.021 nm^-1), according to the relation Q=(4πn/λ) sin(θ/2). The normalized intensity autocorrelation functions g_2(Q,t)=<I(Q,t)I(Q,0)>/<I(Q,0)>^2 are obtained with a high coherence factor close to the ideal unit value. The experiment has been performed on aqueous suspensions of PNIPAM and IPN microgels at three PAAc contents (C_PAAc=2.6 %, C_PAAc=10.6 %, C_PAAc=19.2 %) in the temperature range T=(293÷313) K across the VPT, at four weight concentrations (C_w=0.1 %, C_w=0.3 %, C_w=0.5 % and C_w=0.8 %) and three different pH values (pH 3.5, 5.5 and 7.5). The reported data have been obtained by averaging five repeated set of measurements. Particle sizes have been determinedfrom the decay constant Γ(Q)=D q^2obtained through the analysis of the intensity correlation functions. The behaviour of the normalized intensity autocorrelation functions collected at θ=90° for PNIPAM and IPN microgels at different PAAc contents and at low weight concentration (C_w=0.1 %) is reported in Fig.<ref>, below and above the VPTT.As commonly known, the intensity correlation function of most colloidal systems is well described by the Kohlrausch-Williams-Watts expression <cit.>: g_2(Q,t)=1+b[(e^-t/τ)^β]^2 where b is the coherence factor, τ [τ: structural relaxation time] is the structural relaxation time and β [β: stretching parameter] describes the deviation from the simple exponential decay (β = 1) usually found in monodisperse systems. Indeed the distribution of the relaxation times in disordered materials leads to a stretching of the correlation functions characterized by an exponent β < 1 which can be related to the distribution of the relaxation times due to the polydispersity of the samples.§.§ Electrophoretic MeasurementsElectrophoretic mobility of microgel suspensions was measured by means of a MALVERN NanoZetasizer apparatus equipped with a 5 mW HeNe laser (Malvern Instruments LTD, UK). This instrument employstraditional Laser Doppler Velocimetry(LDV) implemented with Phase Analysis Light Scattering (PALS) for a more sensitive detection of the Doppler shift <cit.>. LVD measurements are performed using the patented "mixed mode" measurement M3 where both a fast field (FF) and a slow field (SF) are applied. In FFR the field is reversed 25-50 times per second, thus making electro-osmosis insignificant and providing accurate mean mobility value. The SFR contributes extra resolution for a better distribution analysis <cit.>. The frequency shiftΔνdue to the mobility μ of the scattered particles under the action of the applied field E is measured by comparing the phase Φ of the scattered signal to that of a reference one, since Φ = ν· time. The mobility μ= V/E [μ: electrophoretic mobility] is then calculated from the relation Δν=2 V sin (θ /2)/λ) with Vthe particle velocity, θ the scattering angle and λ the laser wavelength. By a preliminary conductivity measurement, the instrument establishes a suitable electric field for a good mobility detection. Both PNIPAM and IPN samples at the different PAAc contents have been measured at C_w=0.05 % and pH around 5.5.Measurements have been performed by using the dedicated U-cuvette DTS1070, in a thermostated cell by performinga ramp from 293 to 316 K with temperature step of 1 K and a thermalization time of 300 s at each step. Data presented here correspond to the mean values of the electrophoretic mobility distribution and are obtained by averaging three repeated set of measurements. § RESULTS AND DISCUSSIONS §.§ Sample characterization *Hydrodynamic radiusThe temperature behaviour of the hydrodynamic radii of IPN and pure PNIPAM microgels has been obtained trough DLS at low weight concentration (C_w=0.1 %) and is reported in Fig.<ref>. The typical swelling behaviour of microgel particles from a swollen to a shrunken state across the VPT is found and it is highly affected by the acrylic acid content, in very good agreement with what reported in Ref. <cit.>. The increase of acrylic acid content determines sharper transitions. Notwithstanding some limits <cit.> the most simple way to describe the microgels swelling is given by the Flory-Rehner theory <cit.> in terms of the total osmotic pressure π [π: total osmotic pressure] inside the gel, consisting of a mixing contribution π_m [π_mi: mixing contribution to the osmotic pressure] and an elastic component π_e [π_e: elastic contribution to the osmotic pressure]. For ionic microgels, as IPN, additional contributions arising from the screened repulsion between polymer chains and from the osmotic pressure due to counterions confined inside the network, show up. However for charge density smaller than the value at which counterion condensation may take place the first of the two effects can be neglected <cit.>. Therefore for IPN microgels also the ionic contribution π_i [π_i: ionic contribution to the osmotic pressure] to the osmotic pressure has to be taken into account.At the equilibrium condition the total osmotic pressure is π=π_m+π_e+π_i=0 and the swelling of a weakly ionized system can be theoretically explained through the equation of state: ln(1-ϕ)+ϕ+χϕ^2+ϕ_0/N[(ϕ/ϕ_0)^1/3-(1/2+f)ϕ/ϕ_0]=0 where ϕ_0 [ϕ_0: polymer volume fraction in the reference state] is the polymer volume fraction in the reference state, typically taken as the shrunken one <cit.>, ϕ [ϕ: polymer volume fraction within the particle] is the polymer volume fraction within the particle, which for isotropic swelling is related to the hydrodynamic radius through the relation ϕ/ϕ_0=(R_0/R)^3 (with R_0 [R_0: hydrodynamic radius in the shrunken state] the hydrodynamic radius in the shrunken state) <cit.>, N [N: number of segments occupied by a polymer chain between two cross-links] is the number of segments occupied by a polymer chain between two cross-links, f [f: number of counterions per polymer chain] is the number of counterions per polymer chain, which is found to increase with PAAc content and χ [χ: Flory polymer-solvent interaction parameter] is the Flory polymer-solvent interaction parameter, which has to be interpreted as an effective mean parameter accounting for polymer/solvent interactions, polymer/polymer interactions within each network and polymer/polymer interactions between different networks. It can be written as a power series expansion χ=χ_1 (T)+χ_2 ϕ+χ_3 ϕ^2+χ_4 ϕ^3+⋯ where χ_1 [χ_1: Flory parameter] is the Flory parameter, defined as χ_1=1/2-A(1-θ/T), A is the parameter that provides rough details on the solvent quality, its increase indicates a decreasing goodness of the solvent as discussed in the Supplementary Material, θ [θ: volume phase transition temperature] is the VPT temperature and χ_i (i=1,2,3,…) [χ_i: temperature independent coefficients] are temperature independent coefficients. The second-order approximation of Eq.(<ref>) well describes the VPT of our microgel as previously reported for pure PNIPAM <cit.> and IPN microgels at acidic pH <cit.>. The Flory-Rehner model well reproduces the experimental data (symbols) and the goodness of the fits (lines) is evident from Fig.<ref>, the fit parameters are reported in Table<ref>. The model gives values for the VPT temperature in agreement with those expected <cit.>. The number of ionized groups per chain f increases with increasing PAAc content indicating an increase of the charges of the microgel. In particular the f values suggest that for PNIPAM and for low (2.6 %) PAAc content the ionic contribution can be neglected while it becomes more relevant at intermediate (10.6 %) and high (19.2 %) PAAc content where the swelling behaviour deviates from that of pure PNIPAM microgel. This deviation can be explained accounting for the topological constraints due to the networks interpenetration and to the ionic contribution to the osmotic pressure. Interestingly the A parameter increases with PAAc content indicating that mixing between polymer and solvent is not favored and swelling is inhibited by PAAc. These results strongly indicate that interpenetration of PAAc network affects the interaction between PNIPAM and water molecules, these polymer/solvent interactions can be therefore controlled through pH <cit.> or by varying PAAc content, as here highlighted for the first time. However it is worth noting that there is a discrepancy between the value of ϕ_0 obtained with the Flory Rehner theory and other methods as deeply discussed in Ref. <cit.>.*Morphology TEM images obtained by PTA staining for PNIPAM and IPN microgels prepared at the different PAAc contents (C_PAAc=2.6 %, C_PAAc=10.6 %, C_PAAc=19.2 %) are shown in Fig.<ref>, panel a-b-c-d, respectively. In all samples, several microgel particles appear as clear objectswith size ranging from 20 to 100 nm.No significant differences between the morphology of IPN microgels with different PAAc contents appears from TEM images. In some cases, a sharp negative staining arises due to PTA accumulation close to microgel which creates a black halo around the particles. At a deeper observation of those microgels,a thin dark grey region can be distinguished from the light grey inner part of the particles. This different appearence is probably caused by the PTA penetration over a small distance within the particle.These results may indicate the presence of a more compact microgel inner core surrounded by a loose shell which is more permeable to PTA molecules both for PNIPAM and IPN microgels, probably due to less cross-linked chains with more dangling ends, as shown in different studies for PNIPAM-based microgels <cit.>. A statistical analysis performed on TEM images is reported in panels e-f-g-h, for PNIPAM and IPN with C_PAAc=2.6 %, C_PAAc=10.6 % and C_PAAc=19.2 %, respectively. The mean diameters from a statiscal analysis on the whole particle distribution are obtained as an average value with standard deviation: PNIPAM (50 ± 30) nm, IPN 2.6 % (60 ± 20) nm,IPN 10.6 % (40 ± 20) nm and IPN 19.2 % (50 ± 20) nm. The wide particle distributions could be connected to flattening and deformation of samples on the support, as often occurs for soft materials, or to the intrinsic polydispersity of the samples that is related to the used SDS concentration during the synthesis as described in the Sample preparation paragraph.§.§ Temperature behaviour In order to understand the role played by the poly(acrylic acid) on the relaxation dynamics and on the aggregation process across the VPT, the temperature behaviour of the relaxation time, reported in Fig.<ref>, has been obtained through DLS by fitting the g_2(Q,t) with Eq.(<ref>). In the low weight concentration range (Fig.<ref>(a)) the well known dynamical transition associated to the VPT is evidenced for both PNIPAM and IPN microgels <cit.>: as temperature increases the relaxation time τ (T) slightly decreases up to the volume phase transition temperature above which it decreases to its lowest value, corresponding to the shrunken state indicating a fastening of the dynamics related to the reduced size of the particles and an increased diffusivity. Surprisingly as the weight concentration increases (Fig.<ref>(b)) relaxation time above the VPTT is strongly affected by PAAc content: while at low PAAc it resembles that of pure PNIPAM, at higher PAAc it suddenly increases with temperature, thus indicating the formation of aggregates accompanied by an evident viscosity increase observed by eyes.This behaviour can be explained considering that above the VPTT, due to the reduced particle size, Van der Waals attraction becomes stronger, thus affecting the microgel aggregation. Moreover if particles are charged, as in the case of IPN microgels, also electrostatic interactions have to be taken into account, they are much stronger greaterthe PAAc content is, as also evidenced by the increasing values of the f parameter of table <ref> associated to the ionic contribution. In addition the collapse of NIPAM networks with temperature is supposed to favor the exposure of PAAc dangling chains and the formation of aggregates.To test the importance of the charge in the aggregation process we performed measurements at three different pH since the overall charge can be tuned with pH due to the progressive deprotonation of the ionizable groups. The temperature behaviour of the relaxation time for IPN microgels at C_PAAc=19.2 %, C_w=0.8 % and at pH 3.5, pH 5.5 and pH 7.5 is reported in Fig.<ref>. At pH 7.5, the highest investigated pH, a huge growth of τ(T) above the VPTT is evident, indicating the formation of large aggregates. Indeed at this pH, the carboxylic groups COOH [COOH: carboxyl group] of PAAc (Fig.<ref>(c)) are dissociated into COO^- and H-bonding between COOHgroups of AAc moieties belonging to different particles are not favored. As a consequence, the aggregation can be mainly ascribed to like-charge attraction since, at this pH, IPN behave as polyelectrolyte microgels where attraction can be interpreted as a result of counterion fluctuation due to the formation of temporary dipoles <cit.>. At intermediate pH 5.5 there is a fraction of COOH groups and a fraction, not negligible, of COO^- groups, therefore aggregation can be described as a combination of both like-charge attraction and H-bonding interaction between COOH groups. Finally at pH 3.5 the COOH groups of PAAc are fully protonated (neutralized), electrostatic interactions are excluded and H-bondings with the amidic (CONH)[CONH: amidic group] groups of PNIPAM (Fig.<ref>(b)) inside the particles are largely favored. Nevertheless small aggregates above the VPTT are formed, suggesting that inter-particle interactions at high C_PAAc are not excluded and can be mainly ascribed to strong H-bonding and hydrophobic interactions.These results indicate that PAAc represents a good experimental control parameter to tune inter-particle interactions and aggregation: higher is the amount of acrylic acid interpenetrating the PNIPAM network, higher is the formation of aggregates and the viscosity increase. Our findings immediately suggest the importance of the charge density. To complement these observations and to gain information on charge density, we have performed electrophoretic measurements on PNIPAM and IPN microgels as a function of temperature (Fig.<ref>). The mobility μ is affected by the volume phase transition and decreases as the suspension crosses the VPTT. For both PNIPAM and IPN microgels a sharp drop is observed around T_0 ≈ 308 K, as determined by a sigmoidal fit through the equation y = A_2 + (A_1-A_2)/(1 + exp((T-T_0)/dT)) whose fit parameters are reported in Table <ref>. For pure PNIPAM microgels the very low mobility below the VPTT reflects the low charge density of the particles, whereas above the VPTT it increases toward more negative values. It is in fact not surprising that a negative charge appears for particles obtained with the ionic initiator KPS. Considering that the negative electrical charges brought by the anionic sulfate groups are covalently bonded, the total charge per particle is constant and the charge density increases upon shrinking <cit.>. For IPN microgels a similar mechanism still holds for the temperature behaviour of the electrophoretic mobility. Moreover, due to additional charged groups, belonging to AAc moieties,more negative values are found for C_PAAc ≥ 2.6 %. We note that the electrokinetic transition temperature T_0 is higher than the VPTT in agreement with previous results on PNIPAM-based microgels <cit.>, showing a shift forward. This difference can be explained considering that the effective charge carriers are mainly confined to the peripheral shell and that the charged and less dense shell fully collapses at the end of the VPT (see Fig. <ref>(b)). This picture well describes our results for both PNIPAM and IPN microgels where a highly dense core of interpenetrated PNIPAM and PAAc networks is surrounded by a less dense shell mainly composed by PAAc chains. Interestingly, the magnitude of the variation of mobility is dependent on PAAc content and it is more pronounced at high PAAc, where collapsed IPN microgels are characterized by more negative mobility values, as expected from the increase of the charge density due to the greater fraction of exposed PAAc chains above the VPTT. These results further support the possibility to control the effective charge density on the microgel surfaces through the amount of poly(acrylic acid) interpenetrating the PNIPAM network and hence also inter-particle interactions and aggregation phenomena. §.§ Dependence on PAAc content To highlight the dynamical changes related to the interpenetration of the poly(acrylic acid) within the PNIPAM network, we report the behaviour of the relaxation time τ, of the stretching parameter βand of the electrophoretic mobility μ as a function of PAAc content and at fixed temperature below the VPTT in Fig.<ref>. Data exhibit a dramatic jump in a range of PAAc contents between C_PAAc=2.6 % and C_PAAc=10.6 % evidencing a cross-over between two different regions. A sigmoidal fit of the data suggests that a critical value of PAAc content has to be expected around C_PAAc^* ≈ 8 % [C_PAAc^*: crossover PAAc content]: below this C_PAAc^*IPN microgels behave very similarly to pure PNIPAM microgel, indicating that the charges influence is negligible, while above C_PAAc^* the effect of PAAc, and therefore of charge density, becomes relevant leading to a slowing down of the dynamics (increase of the relaxation time), an enhancement of polydispersity (decrease of the stretching parameter of Eq.(<ref>)) and an increase of the overall charge density (more negative values of mobility).A similar transition is observed in the microgel hydrodynamic radius R_H [R_H: hydrodynamic radius] in the shrunken state shown in Fig.<ref>(a). The increase of R_H with PAAc content can be explained considering that the structure of IPN microgels is characterized by a highly dense core of interpenetrated PNIPAM and PAAc networks surrounded by a low density shell mainly populated by PAAc dangling chains that increases in size as PAAc content increases. The trend is also confirmed by the PAAc content dependence of the swelling ratio defined as: α = R_H^swollen/R_H^shrunken where R_H^swollen=R(297 K) [R_H^swollen: hydrodynamic radius in the swollen state] and R_H^shrunken=R(311 K)[R_H^shrunken: hydrodynamic radius in the shrunken state]. InFig.<ref>(b) α [α: swelling ratio] is reported at different PAAc contents, a clear decrease with increasing C_PAAc is observed, indicating a reduction of the swelling capability of IPN microgels by increasing the PAAc amount interpenetrating PNIPAM network and thus leading to a decreases of particle softness with increasing topological constraints and charges due to the interpenetration of the two networks.§ CONCLUSIONS At variance with previous studies <cit.> this work has demonstrated, through dynamic light scattering, transmission electron microscopy and electrophoretic measurements, the role playedby poly(acrylic acid) on the relaxation dynamics and on the aggregation processes in IPN microgels across the VPT.A crossover PAAc concentration C_PAAc^* that signs the existence of two different regions, has been identified: below C_PAAc^* IPN microgels behave very similarly to pure PNIPAM, while above C_PAAc^* they significantly differ, indicating a stronger influence of the extent of ionic charges on the microgels. In fact, by increasing PAAc content, the effective charge density increases, as shown by electrophoretic mobility measurements and attractive interactions between protonated COOH and deprotonated COO^- groups belonging to different particles are enhanced. This is reflected in the huge growth of the relaxation time signature of an aggregation process. The influence of the extent of ionic charges is also confirmed by the comparison between experimental data and theoretical models from the Flory-Rehner theory that shows as the number of counterions per polymer chain, f, increases with PAAc. Finally we prove that particle softness can be experimentally controlled by changing the PAAc/PNIPAM ratio: particles with low PAAc content are more soft and deformable while stiffer particles can be obtained by increasing the amount of PAAc chains interpenetrating PNIPAM network. Further studies with salt are under way to investigate if adding extra charges in the system through a monovalent salt has the same effect of increasing PAAc content and if the aggregation process is speeded up or slowed down.The importance of our findings is twofold. On one hand our results are relevant for fundamental studies aiming to understand the role of softness in exotic phase diagrams with respect to hard colloids. On the other hand, the accurate knowledge of aggregation conditions with respect to parameters such as particle and polymeric concentration, pH and temperature is crucial to improve the manifold technological applications of these stimuli-responsive materials. In particular our work meets the request of Ref.<cit.> to develop new nanogels as intelligent drug carriers. It will be very interesting to assess the ability of our multi responsive IPN microgel to encapsulate drugs and other small molecules, to deliver them and to investigate the effect of softness and charge (peculiar of these microgels) on their release. § ACKNOWLEDGMENTSThe authors acknowledge support from MIUR Fare SOFTART (R16XLE2X3L)§ AUTHOR CONTRIBUTIONS STATEMENT R.A., V.N. and B.Ru. conceived the experiments. S.C., V.N., B.Ro. and S.S. conducted the experiments and analysed the results. M.B. and E.B. synthesized the samples. R.A., V.N., B.Ru. and S.S. reviewed the manuscript.§ ADDITIONAL INFORMATION The authors declare no competing interests. §.§ Bibliographyunsrtsort compress 10VinogradovCurrPharmDes2006 S. V. Vinogradov. Colloidal microgels in drug delivery applications. Curr. Pharm. Des., 12:4703–4712, 2006.DasAnnRevMR2006 M. Das, H. Zhang, and E. Kumacheva. MICROGELS: Old Materials with New Applications. Annu. Rev. Mater. Res., 36:117–142, 2006.ParkBiomat2013 J. S. Park, H. N. 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"authors": [
"Valentina Nigro",
"Roberta Angelini",
"Benedetta Rosi",
"Monica Bertoldo",
"Elena Buratti",
"Stefano Casciardi",
"Simona Sennato",
"Barbara Ruzicka"
],
"categories": [
"cond-mat.soft"
],
"primary_category": "cond-mat.soft",
"published": "20170726170429",
"title": "Study of network composition in interpenetrating polymer networks of poly(N isopropylacrylamide) microgels:the role of poly(acrylic acid)"
} |
Improved Analysis<cit.> Rational Points on the Unit Sphere: Approximation Complexity and Practical ConstructionsDaniel Bahrdt Formale Methoden der Informatik, University of Stuttgart,Germany, {bahrdt, seybold}@fmi.uni-stuttgart.de Martin P.Seybold[1] December 30, 2023 ===================================================================================================================================================§.§.§ Abstract Each non-zero point in ^d identifies a closest point x on the unit sphere ^d-1. We are interested in computing an ϵ-approximation y ∈^d for x, that is exactly on ^d-1 and has low bit size. We revise lower bounds on rational approximations and provide explicit, spherical instances. We prove that floating-point numbers can only provide trivial solutions to the sphere equation in ^2 and ^3. Moreover, we show how to construct a rational point with denominators of at most 10(d-1)/ε^2 for any given ϵ∈(0, 1 8], improving on a previous result. The method further benefits from algorithms for simultaneous Diophantine approximation. Our open-source implementation and experiments demonstrate the practicality of our approach in the context of massive data sets Geo-referenced by latitude and longitude values. *Keywords Diophantine approximation, Rational points, Unit sphere, Perturbation, Stable geometric constructions § INTRODUCTIONMany mathematical sciences use trigonometric functions in symbolic coordinate transformations to simplify fundamental equations of physics or mathematical systems. However, rational numbers are dominating in computer processing as they allow for simple storage as well as fast exact and inexact arithmetics (e.g. GMP<cit.>, IEEE Float, MPFR<cit.>). Therefore problems on spherical surfaces often require to scale a point vector, as in choosing a pointuniform at random<cit.>, or to evaluate a trigonometric function for a rational angle argument, as in dealing with Geo-referenced data.A classical theoretical barrier is Niven's theorem<cit.>, which states that the sole rational values of sine for rational multiplies of π are 0, ±1/2 and ± 1. The well known Chebyshev polynomials have roots at these values, hence give rise to representations for these algebraic numbers. However, arithmetics in a full algebraic number field might well be too demanding for many applications. For products of sine and cosine, working with Euler's formula on the complex unit circle and Chebyshev polynomials would suffice though.This manifests in problems of exact geometrical computations, since standard methodology relies on Cartesian input<cit.>. Spheres and ellipsoids are common geometric objects and rational solutions to their defining quadratic polynomials are closely related to Diophantine equations of degree 2. The famous Pythagorean Triples are known to identify the rational points on the circle ^1. Moreover, the unit sphere has a dense set of rational points and so do ellipsoids with rational half-axes through scaling. Spherical coordinates are convenient to reference such Cartesians with angle coordinates and geo-referenced data denotes points with rational angles. Standard approximations of Cartesians do not necessarily fulfill these equations, therefore subsequent algorithmic results can suffer greatly. This paper focuses on finding rational points exactly on the unit sphere ^d-1={x ∈^d : ∑_i x_i^2 = 1} with bounded distance to the point x / ‖ x ‖_2 – its closest point on ^d-1. In this work, x ∈^d can be given by any finite means that allow to compute a rational approximation to it with arbitrary target precision. Using rational Cartesian approximations for spherical coordinates, as derived from MPFR, is just one example of such a black-box model. Moreover, we are interested in calculating rational points on ^d with small denominators. §.§ Related Work Studies on spherical Delaunay triangulations (SDT), using great-circle segments on the sphere ^2, provide common ways to avoid and deal with the point-on-sphere problem in computational geometry.The fragile approaches <cit.> ignore that the input may not be on ^2 and succeed if the results of all predicate evaluations happen to be correct. Input point arrangements with close proximity or unfortunate locations bring these algorithms to crash, loop or produce erroneous output.The quasi-robust approaches <cit.> weaken the objective and calculate a Delaunay tessellation in d-Simplexes. Lifting to a d+1 convex hull problem is achieved by augmenting a rational coordinate from a quadratic form – The augmented point exactly meets the (elliptic) paraboloid equation. However, the output only identifies a SDT if all input points are already on the sphere, otherwise the objectives are distinct. Equally unclear is how to address spherical predicates and spherical constructions.The robust approaches <cit.> use the circle preserving stereographic projection from ^2 to the plane. The perturbation to input, for which the output is correct, can be very large as the projection does not preserve distances.Furthermore, achieving additional predicates and constructions remains unclear.The stable approaches provide geometric predicates and constructions for points on ^2 by explicitly storing an algebraic number, originating from scaling an ordinary rational approximation to unit length<cit.>. Algebraic number arithmetics can be avoided for ^2, but exact evaluation relies on specifically tailored predicates <cit.>, leaving the implementation of new constructions and predicates open.Kleinbock and Merrill provide methods to quantify the density of rational points on ^d <cit.>, that extend to other manifolds as well. Recently, Schmutz<cit.> provided an divide-&-conquer approach on the sphere equation, using Diophantine approximation by continued fractions, to derive points in ^d ∩^d-1 for a point on the unit sphere ^d-1. The main theorem bounds the denominators in ε-approximations, under the ‖ ‖_∞ norm, with ( √(32)⌈log_2 d ⌉ / ε)^2⌈log_2 d ⌉. Based on this, rational approximations in the orthogonal group O(n,) and in the unitary matrix group U(n,) are found.This is of particular interest for sweep-line algorithms: <cit.> studies finding a rotation matrix with small rationals for a given rational rotation angle of an 2D arrangement.§.§ ContributionThe strong lower bound on rational approximations to other rational values does not hold for Geo-referenced data considering Niven's theorem. We derive explicit constants to Liouville's lower bound, for a concrete Geo-referenced point, that is within a factor 2 of the strong lower bound. Moreover, we prove that floating-point numbers cannot represent Cartesian coordinates of points that are exactly on ^1 or ^2.We describe how the use of rotation symmetry and approximations with fixed-point numbers suffice to improve on the main theorem of <cit.>. We derive rational points exactly on ^d-1 with denominators of at most 10(d-1)/ε^2 for any ε∈(0, 1 8]. Moreover, our method allows for even smaller denominators based on algorithms for simultaneous Diophantine approximations, though a potentially weaker form of approximation would suffice.The controlled perturbations provided by our method allow exact geometric algorithms on ^d to rely on rational rather than algebraic numbers – E.g. enabling convex hull algorithms to efficiently obtain spherical Delaunay triangulations on ^d and not just Delaunay tessellations. Moreover, the approach allows for inexact but ε-stable geometric constructions – E.g. intersections of Great Circle segments.We demonstrate the quality and effectiveness of the method on several, including one whole-world sized, point sets. We provide open-source implementations for the method and its application in the case of spherical Delaunay triangulations with intersections of constraints.§ DEFINITIONS AND TOOLSThe 2nd Chebyshev polynomials U_n of degree n are in [X], given their recursive definition:U_0(x) =1U_1(x)=2xU_n+1(x) =2xU_n(x)-U_n-1(x).It is well known <cit.>, that the n roots of U_n are exactly the values { cos(π k / (n+1)) : k=1,… ,n }. Hence the polynomials U_n give rise to algebraic representations for cosine values of rational multiplies of π. This is particularly useful in conjunction with classic results on Diophantine approximations, that are known since 1844<cit.>: For any algebraic α∈ of degree n ≥ 2, there is a positive constant c(α)> 0 such that | α - p/q| ≥c(α)/q^nfor any p ∈ and q ∈.Apart from this lower bound on rational approximations, there is another important folklore result on the existence of simultaneous Diophantine approximations. Such approximations have surprisingly small errors, despite their rather small common denominator. Let N ∈ and α∈^d with 0 ≤α_i ≤ 1. There are integers p ∈^d, q ∈ with 1 ≤ q ≤ N and |α_i - p_iq|≤1/q√(N) . (See Appendix <ref> for a proof.) For d=1, the continued fraction (equivalently the Euclidean) algorithm is famous <cit.> for finding approximations with | α - p/q| ≤ 1/2q^2. This spurred the field of number theory to study generalizations of the continued fraction algorithm that come close to Dirichlet's upper bound, but avoid brute-force calculations. Some more recent methods are discussed in Section <ref>.Our approach uses the Stereographic Projection in ^d. Let p=(0,…,0,1) ∈^d be the fixed point for the projection τ, mapping all points of a ray from p to the intersection with the hyperplane x_d = 0.τ : ^d∖(^d-1×{1})→^d-1x↦(x_1/1-x_d , … , x_d-1/1-x_d)The surjective mapping τ is injective as well, when restricted to the domain ^d-1∖{p}. We further define the mapping σ, which isσ : ^d-1 →^d∖{p}x↦( 2x_1/1+S^2 , … , 2x_d-1/1+S^2 , -1+S^2/1+S^2 )where S^2 = ∑_j=1^d-1x_j^2. We haveimgσ⊆^d-1, since ‖σ(x) ‖_2^2= (-1+S^2)^2 + ∑_i=1^d-1 (2x_i)^2/(1+S^2)^2 = 1.Furthermore, x = τ∘σ( x) for all x∈^d-1, since(τ∘σ)_i(x) = 2x_i/1+S^2/1--1+S^2/1+S^2 = 2x_i/1+S^2+1-S^2 =x_iholds for all 1 ≤ i < d. Hence, σ and τ are inverse mappings. Note that images of rational points remain rational in both mappings, establishing a bijection between rational points in ^d-1 and ^d-1.§.§ Lower Bounds and Instances for Geo-referenced Data on ^d It is well known in Diophantine approximation that rational numbers have algebraic degree 1 and are hard (in the following qualitative sense) to approximate with other rational numbers. The following folklore observation is an analog to Liouville's lower bound. Forrational numbers ab ≠ pq, we have | a/b- p/q| = | aq-bp/bq| ≥1/bq If q<b, we have a lower bound of 1/q^2 for rational approximations to ab with denominators up to q. Pythagorean triples (x,y,z) ∈^3 provide such rational points on ^1, since (x/z)^2 + (y/z)^2 =1. We have a lower bound of 1/z^2 for approximations with denominators q<z. See Section <ref> for rational points on ^d with the same denominator property.The situation might look different when dealing with Geo-referenced data (rational angle arguments) only. However, using Chebyshev's polynomials in conjunction with Liouville's lower bound (c.f. Theorem <ref>) allows to derive explicit constants for Diophantine approximations of cos(108^∘).Given spherical coordinates, the first coordinate of a point on ^d might well have algebraic values of r_i= cos( i5π) for i ∈{1,2,3,4}.(r_1,r_2,r_3,r_4)=( 1+√(5)/4, -1+√(5)/4, 1-√(5)/4, -1-√(5)/4)≈( +0.8090, +0.3090, -0.3090, -0.8090)Over [X], the polynomial U_4(x)=16x^4-12x^2+1 has the irreducible factorsU_4(x) = (4x^2 - 2x + 1)_=:f(x)(4x^2 + 2x - 1)Since r_1 and r_3 are the roots of the polynomial f, they have algebraic degree n=2.Using Liouville's lower bound for r_3, we have for all pq ∈| r_3 - p/q| ≥min{c_2,1c_1}/q^n,with constants c_1 and c_2 according to the proof of Liouville's Theorem<cit.>. The constants c_1,c_2>0 exist, since the polynomial division of f with the linear factor (x-r_3) results in the continuous function g(x)=(x-r_1). For c_2 = 1/2 < √(5)/2, the interval I:=[r_3- c_2, r_3 +c_2] ⊆ is sufficiently small to exclude different roots of f and the inequalitymax_x∈ I|g(x)| = max_x∈ I|x - r_1| < c_1is met with a generous choice of c_1=2. This leads to an explicit lower bound on the approximation error to r_3 with denominators q of| cos(108^∘) - p/q| ≥1/2 · q^2 . § RESULTSApart from integers, contemporary computing hardware heavily relies on floating point numbers. These are triplets (s,m,e) with s ∈{0,1}, m ∈{0,…,2^l-1} and e ∈{-2^k-1+1,…, 2^k-1-1}. The IEEE standard for Float is (l,k)=(23,8) and (52,11) for Double. The rational number described by such a triplet is val(s,m,e) = (-1)^s ·2^l + m2^l2^e e > 02^l + m2^l12^|e|e < 00 + m2^l12^2^k-1-2e = 0where the latter case describes `denormalized' numbers. In each case, the uncanceled rational value has some power of 2 as the denominator. Since powers of two are the sole divisors of a 2^i, the denominator of the canceled rational has to be a power of two, too. Hence, rational values representable by floating point numbers are a subset of the following set P and fixed-point binary numbers are a subset of P_i:imgval⊆{z/2^i : i ∈, z ∈, z odd}=P{z/2^i : z ∈} = P_i ⊆ P . §.§ Floating Point Numbers are InsufficientFix-point and floating-point arithmetics of modern CPUs work within a subset of rational numbers, in which the denominator is some power of two and the result of each arithmetic operation is `rounded'. There are only 4 floating point numbers on ^1 and 6 on ^2. We show ^d-1∩ P^d ⊈{-1,0,1}^d implies d ≥ 4. Suppose there is a non-trivial p ∈^d-1∩ P^d with d minimal. Let x_i/2^e_i denote the canceled fraction of its i-th coordinate. We have that all x_i ≠ 0, x_i are odd numbers and all e_i >0 (since p is not one of the 2d poles and d is minimal). W.l.o.g. e_1 ≤ e_2 ≤…≤ e_d. We rewrite the sphere equation 1 = ∑_j=1^d (x_i/2^e_i)^2 to x_1^2= 4^e_1 - ∑_j=2^d4^e_1 - e_j x_j^2 . For an odd integer y, we have y^2=(2k+1)^2= 4(k^2+k)+1, leading to the congruence 1 ≡ 0 - ∑_j=2^dχ_e_1(e_j) 4 . Where the characteristic function χ_e_1(e_j) is 1 for e_1 = e_j and 0 otherwise. For d∈{2,3} the right hand side can only have values of 0,-1 or -2, a contradiction.Note that theorem <ref> translates to spheres with other radii through scaling. Suppose a sphere in ^3 of radius 2^j has a non-trivial solution y ∈ P^3, then y/2^j ∈ P^3 and would be on ^2, too. §.§ Snapping to Rational PointsWe now describe how to compute a good rational approximation exactly on the unit sphere ^d-1. The input point x ∈^d can be given by any finite means that allows to compute rational approximations of arbitrary target precision – E.g. rational approximations of Cartesians for spherical coordinates. For the input x, we denote its closest point on ^d-1 with x/‖ x ‖_2. The stereographic projection τ and its inverse mapping σ provide σ(τ( x/‖ x‖_2 ) )= x/‖ x‖_2, since the argument is on ^d-1. Instead of determining the value of τ exactly, we calculate an approximation y ∈^d and finally evaluate σ(y) under exact, rational arithmetics. Hence, the result σ(y) is exactly on ^d-1. r.35 [scale=.13] (0,0) circle (10); (10,-10) circle (.25) node [right] x; (7,-7) circle (.25) node [left] x / ‖ x ‖_2; (8,-6) circle (.25) node [right] σ(y); (0,+10) circle (.25) node [above] (0,1); (4.14213562,0) circle (.25) node [above left] τ(x/‖ x ‖_2); (5,0) circle (.25) node [above right] y; [-] (-12,0) – node [above](12,0); [-,dashed,gray] (0,0) – node [above](10,-10); [-] (0,10) – node [above](7.07106781,-7.07106781); [-] (0,10) – node [above](8,-6); The stereographic projection does not preserve distances, leaving it open to bound the approximation error and the size of the resulting denominators. We use the rotation symmetry of the sphere to limit the stretching of σ (c.f. Lemma <ref>): For a non-zero point x ∈^d we can assume that i=d maximizes |x_i| and x_d < 0, otherwise we change the standard orthonormal basis by swapping dimension i and d and using a negative sign for dimension d. Note that such rotations do not change the actual coordinate values. To keep the size of denominators in σ(y) small, we use fixed-point arithmetics to determine y ∈^d-1 (c.f. Lemma <ref>).See Algorithm <ref> for a precise description. Note that the rational point y in statement <ref> solely needs to meet the target approximation in the individual coordinates forτ_i (x / ‖ x ‖_2 ) = x_i/‖ x ‖_2 - x_ d.Generally, this can be determined with methods of `approximate expression evaluation' to our target precision<cit.>. If x is an approximation to a geo-referenced point, this denominator is well conditioned for calculations with multi-precision floating-point arithmetics<cit.>. Using exact rational arithmetics for statement <ref>, we obtain a rational Cartesian coordinates on the unit sphere. For d > 1 and x ∈^d-1 with x_d = min_i -|x_i|, we have ‖τ(x) ‖_2 ≤√(√(d)-1/√(d)+1) < 1 .Using x_d = min_i -|x_i| and ∑_ix_i^2= 1, we have the bounds 1/d ≤ x_d^2 ≤ 1 and‖τ ( x) ‖_2^2= ∑_i=1^d-1 x_i^2/(1-x_d)^2= 1 - x_d^2/(1-x_d)^2= 1+x_d/1-x_d≤1-1/√(d)/1+1/√(d) .Where the latter term is in (0,1) for any d. Hence the (d-1)-ball ^d-1_1 = { x ∈^d : ‖ x ‖_2 ≤ 1} contains τ(x). §.§ Approximation QualitySee <cit.> for an earlier version of this paper with a weaker, but elementary, analysis. We consider the problem inthe 2D hyperplane H_pyy', defined by two points y=σ(x), y'=σ(x') on ^d-1 and the projection pole p∈^d. Given the rotation step in Algorithm <ref>, the projection plane H_0={ x ∈^d: x_d = 0} separates p and y,y' in ^d and in H_pyy'. Since each q ∈ H_0 ∩ S^d-1 has ‖ q - p‖_2=√(2) (consider pq in H_0pq),the circumcircle C of p,y and y' contains exactly two of these points. Hence, the line of H_pyy'∩ H_0 is orthogonal to the circumcircle's diameter through p. Moreover, the circles diameter is in [√(2),2]. We denote with x the point that is closer to p in H_pyy', meaning ‖ x‖_2 ≤‖ x'‖_2. Note that x' and x can be on the same or opposite circumcircle halves. [scale=.3] (P) at (90:10); (V) at (210:10); (U) at (230:10); (M) at (0,0); (HP) at(190:10); (HP2) at (-10:10); [gray,dashed] (0,0) circle (10); (P) circle (.25) node [above] p; (V) circle (.25) node [below left] y'; (U) circle (.25) node [below left]y; (M) circle (.25) node [right] m; [name path=PV] (P) – node [above](V); [name path=PU] (P) – node [above](U); [name path=XY] (V) – node [above](U); [name path=VM] (V) – (0,0) – (P); [name path=HPP] (HP) – (HP2)node [below]H_0 ; [name intersections=of=HPP and PU,by=HPP-PY]; (V) – (U); [name path=XYSHIFT] (HPP-PY) – +((V)-(U)); [name intersections=of=PV and XYSHIFT,by=PX-XYSHIFT]; (PX-XYSHIFT) – (HPP-PY); [name intersections=of=PV and HPP,by=HPP-PX]; (HPP-PX) – (HPP-PY); (HPP-PX) circle (.25) node [ below]x'; (HPP-PY) circle (.25) node [below, xshift=6]x; ((P)!(HPP-PY)!(V)) circle (.25) node [above left] b; (PX-XYSHIFT) circle (.25) node [above, xshift=2] a; [name path=B] ((P)!(HPP-PY)!(V)) – (HPP-PY); [dotted, gray] (P) – (HP)node [midway,left ] =√(2); [dotted, gray] (P) – (HP2) node [midway,right] =√(2); [name path=MH0, dotted, gray] ((HP)!(M)!(HP2)) – (M); (0,2) arc (90:210:2); [above left] (M) γ; In this section we denote with B=b x the perpendicular from x on py',E=xx', L=yy' andL_x=xa its triangle scaled version meeting x. Note that B and L_x are above H_0, hence above E. For x,x' ∈^d-1_1 with ‖ x‖_2 ≤‖ x'‖_2, we have ‖ x ‖_2/‖ p-σ(x) ‖_2‖σ(x) - σ(x') ‖_2 ≤‖ x - x' ‖_2 . We show L_x ≤ E by proofing α≤β for the two angles β := ∡ b x x' α := ∡ a x b . The inner angle sum of xab with a supplementary angle argument and triangle scaling provide ∡ py'y = 90^∘ + α. Let m denote the center of C. Since pm is orthogonal on H_0 and ∡ bx'x = 90^∘ - β, we have ∡ mp x' = β. In the isosceles triangle py'm, the central angle γ = 180^∘ - 2β. Fixing arc py' on C for the inscribed angle theorem provides ∡ y'yp=γ/2. Now, suppose α > β. The inner angle sum of pyy' states 0 ≤∡ y'py = 180^∘ - ∡ y'yp - ∡ py'y= 180^∘ - ∡ y'yp - (90^∘ + α)= 180^∘ - γ/2- (90^∘ + α) = -α + β a contradiction. For x, x' ∈^d-1_1 , we have ‖σ( x) - σ(x') ‖_2 ≤ 2 ‖ x - x' ‖_2. Using Lemma <ref>, we have L_x≤ E and the statement follows via triangle scaling: L = L_x py / px≤ 2 L_x≤2 E , since px≥ 1 and py≤ 2.This statement is tight, considering the two points x = 0 and x'=( ε/√(d-1), …, ε/√(d-1)). We have ‖ x - x'‖_2=ε and‖σ(x) - σ(x')‖_2 = 21/√(1+ε^2 )ε. Algorithm <ref> calculates an ε-approximation exactly on the unit sphere. Let x^* = x /‖ x ‖_2 and σ(y) denote the result. Given the rotation, x^* holds for Observation <ref>. Hence, we can use Lemma <ref> to derive‖σ(y) - x^*‖_∞ = ‖σ(y) - σ(τ(x^*))‖_∞≤‖σ(y) - σ(τ(x^*))‖_2 ≤ 2 ‖ y - τ(x^*) ‖_2 ≤2√((d-1) ε^2/4(d-1)) = εas upper bound on the approximation error.This analysis is rather tight, as demonstrated by the red curve and points in Figure <ref>.§.§ Denominator SizesWe now describe a relation between rational images of σ and the lowest common multiple ofdenominators of its rational pre-images. This leads to several strategies for achieving small denominators in the results of Algorithm <ref>. Let x ∈^d-1∩^d-1_1 with x_i = p_i/q_i and Q = lcm(q_1, …, q_d-1) be the lowest common multiple, then σ_k (x)= n_k/m with integers n_i, m ∈{-2Q^2, … , 2Q^2} for all 1≤ k ≤ d. Let q'_i ∈{1,…,Q} such that q'_i · q_i = Q for all i. Since the formula of σ is similar in all but the last dimension, we describe the following two cases. For k=d, we haveσ_k (x)=-1 + ∑_i=1^d-1p_i^2 / q_i^2/1 + ∑_i=1^d-1p_i^2 / q_i^2 =-Q^2 + ∑_i=1^d-1q'_i^2p_i^2 /Q^2 + ∑_i=1^d-1q'_i^2p_i^2 =: n_k/mUsing the bound x ∈^d-1_1, we have 0 ≤∑_i=1^d-1q'_i^2p_i^2 ≤ Q^2 and we derive for n_k and m|n_k|= |-Q^2+ ∑_i=1^d-1q'_i^2p_i^2 | ≤ Q^2 m= Q^2 + ∑_i=1^d-1q'_i^2p_i^2 ≤ 2Q^2 For k < d, we haveσ_k (x) =2 p_k/q_k/1 + ∑_i=1^d-1p_i^2 / q_i^2= Q^2 · 2p_k/q_k /Q^2 + ∑_i=1^d-1q'_i^2 p_i^2 = Qq'_k · 2 p_k /Q^2 + ∑_i=1^d-1q'_i^2 p_i^2=: n_k/mUsing the bound x ∈^d-1_1, we have that each |p_i| ≤ q_i and this bounds |n_k| = Qq'_k · 2 |p_k| ≤ 2Q^2. We already discussed the bound on m in the first case. Note that we apply this lemma in practice with fixed-point binary numbers p_i/q_i ∈ P_s. Meaning all q_i = 2^s = Q for some significant size s. Denominators in ε-approximations of Algorithm <ref> are at most 10(d-1)/ε^2 .Using standard multi-precision floating point arithmetics allows to derive rational values y, with denominators that are Q=⌈2√(d-1)/ε⌉. Using ε≤ 1/8 and Lemma <ref> bounds the size of the denominators in images σ with2Q^2≤ 2(1+ 2√(d-1)/ε)^2= 2/ε^2( ε^2 +ε4√(d-1)_≤ (d-1) + 4(d-1) ) .For certain dimensions and in practice(c.f. Section <ref>), we can improve on the simple usage of fixed-point binary numbers. For ^1 we can rely on the continued fraction algorithm to derive rational approximations of α = τ(x/‖ x‖_2) with | α - p/q| ≤ 1/2q^2. Using this in Algorithm <ref> leads to approximations with ε=1/q^2 on the circle ^1 with denominators of at most 2q^2.Note that for ^d with d≥ 2 one can rely on algorithms for simultaneous Diophantine approximations (c.f. Theorem <ref>) to keep the lowest common multiple Q in Lemma <ref> small. Note that it might well be simpler to find Diophantine approximations with small Q.There have been many approaches to find generalizations of the continued fraction algorithm for d>1. One of the first approaches is the Jacobi-Perron algorithm, which is rather simple to implement<cit.>(c.f. Section <ref>). More advanced approaches <cit.> rely on the LLL-algorithm for lattice basis reduction<cit.>. For d=2 there is an algorithm to compute all Dirichlet Approximations<cit.>, which we find hard to oversee given its extensive presentation. Moreover, their experimental comparison shows that the Jacobi-Perron algorithm is practically well suited for d=2.We close this section with a transfer result of Theorem <ref> with our Theorem <ref> and Lemma <ref>.Let x ∈^d-1 and N ∈. There is p ∈^d-1 and q ∈{1,…,N} with‖ x - σ(1/qp) ‖_∞≤2√(d-1)/q√(N)and all denominators of σ(1/qp) are at most 2q^2.This existence statement allows for brute-force computations. However, we just use it for comparisons in Section <ref>.§ IMPLEMENTATIONApart from <cit.> for ^2, most implementations of spherical Delaunay triangulations are not `stable'. Approaches based on d-dimensional convex hull algorithms produce only a tessellation for input not exactly on ^d-1. (c.f. Section <ref>)Few available implementations allow dynamic point or constraint insertion and deletion – not even in the planar case of ^2. The `Computational Geometry Algorithms Library' (CGAL <cit.>) is, to our knowledge, the sole implementation providing dynamic insertions/deletions of points and constraint line segments in ^2. With <cit.>, we provide open-source implementations of Algorithm <ref> for ^d. In <cit.>, we provide an implementation for spherical Delaunay triangulations on ^2 with ε-stable constructions of intersection points of constraint line-segments (c.f. Section <ref>). §.§ RATional Sphere Snapping for ^dhttp://www.github.com/fmi-alg/libratssLibratss is a C++ library which implements Algorithm <ref>, based on the open-source GMP library for exact rational arithmetics <cit.> and the GNU `Multiple Precision Floating-Point Reliably'(MPFR) library<cit.>. The implementation allows both, input of Cartesian coordinates of arbitrary dimension and spherical coordinates of ^2. Note that this implementation allows geometric algorithms, as for d-dimensional convex hull, to rely on rational input points that are exactly on ^d-1. In light of the discussion on the denominator sizes in Section <ref>, we provide two additional strategies to fixed-point snapping, as analyzed in Theorem <ref>. We implemented the Continued Fraction Algorithm to derive rational ε-approximations with small denominators and the Jacobi-Perron algorithm for ^2. The library interface also allows to automatically chose the approximation method which results in smaller denominators, approximation errors or other objectives, like byte-size.§.§ Incremental Constrained Delaunay Triangulation on ^2http://www.github.com/fmi-alg/libdts2Libdts2 implements an adapter for the dynamic constraint Delaunay triangulation in the Euclidean plane ^2 of CGAL. Since this implementation requires an initial outer face, we introduce an small triangle, that only contains the north-pole, to allow subsequent insertions of points and constraints. For points exactly on the unit sphere, the predicate `is A in the circumcircle of B,C and D' reduces to the well studied predicate `is A above the plane through B,C and D'. The implementation overloads all predicate functions accordingly and uses Algorithm <ref> for the construction of rational points on the sphere for intersections of Great Circle segments.§.§.§ ϵ-stable geometric constructions Any means of geometric construction that allows to approximate a certain point, can be used as input for Algorithm <ref> – E.g. the intersection of Great Circle segments. Consider two intersecting segments of rational points on ^2.The two planes, containing the segments and the origin as a third point, intersect in a straight line. Each (rational) point on this line can be used as input for our method, as they identify the two intersection points on the sphere. Using such input for Algorithm <ref> allows simple schemes to derive stable geometric constructions of rational points on ^d within a distance of ε to the target point. § EXPERIMENTSWe used real world and synthetic data for our experiments. Geo-referenced data was sampled from regional extracts from the OpenStreetMap project<cit.>, as of January 26th, 2017. Random Cartesian coordinates of points on ^d were created with the uniform generator 2 of <cit.>. All benchmarks were conducted on a single core of an Intel Xeon E5-2650v4. Peak memory usage and time were measured using theutility. §.§ Approximation Quality and Size We experimentally analyze the actual approximation error in results of Algorithm <ref> for several levels of ε using the MPFR library. In this section e denotes the significands required in statement <ref> of Algorithm <ref> for the required result precision ε. This is e = ⌈ -log_2 ( ϵ/2√(d-1)) ⌉.We simply setup the MPFR data types with significand sizes up to 1024 Bits, and conducted our experiments on much lower levels of e. This allows us to derive some `measure' of the actual approximation errors of our method.We analyzed the approximation errors δ and denominator bit-sizes q for100 random points on ^2.Figure <ref> compares the results of our algorithm under several levels of target precision e and strategies for statement <ref> in our method. The magenta line indicates the quality and size of the approach in <cit.>. The red line indicates the bounds of our Theorems <ref> and <ref> on the fixed-point strategy, while the yellow line indicates the bound of Corollary <ref>. Note that results using the Jacobi-Perron strategy (blue dots) allows our method to further improve on the fixed-point strategy (red dots). Note that we use Liouville's lower bound as statement on the approximability of a worst-case point. There might well be points of higher algebraic degree that allow better approximations (c.f. Section <ref>).Table <ref> exhibits average approximation errors δ, denominator bit-sizes q and the computation time t of our method for millions of points. Synthetic data sets have several dimensions, while the real world data sets have dimension 3. For ^2, we provide comparison of the fixed-point strategy (fx) with the Jacobi-Perron strategy (jp) of our method. Using e=31 is sufficient to obtain results exactly on ^2 with a δ of less than 1cm, relative to a sphere with radius of the earth. This is enough for most applications dealing with spatial data and allows storage within the word size of contemporary computing hardware. This allows practical applications on ^2 to store 4 integer long values for the 3 numerators and the common denominator (c.f. Lemma <ref>) occupying 32 Bytes. Note that storing 3 double values occupies 24 Bytes but cannot represent Cartesian coordinates exactly on the sphere.§.§ Constrained Delaunay Triangulation with Intersection Constructions A Constrained Delaunay Triangulation of a point set contains required line-segments as edges, but is as close to the Delaunay triangulation as possible <cit.>. We used very large street networks of several regions from the OpenStreetMap project for points and constraint edges – E.g. each line-segment of a street is an edge in the result triangulation. Since ∼0.5% of the line-segments in these data sets intersect, we approximated the intersection points using e=31 for Algorithm <ref>. Table <ref> exhibits total running time, peak memory usage and the result sizes of ourimplementation. Small data sets like Saarland and Germany allow quick calculation on a recent workstation computer. See Figure <ref> for the Ecuador dataset. Note that the current implementation has a storage overhead for each point, as we keep the results of the GMP library rather than truncating to integers of architectures word size. Computing the triangulation for the planet data set was only possible on rather powerful hardware with at least 550 Gigabytes of memory taking half a day.§ OPEN PROBLEMSFrom a practical point of view, it is of great interest to bound the storage size of denominators to a maximum of 64Bits – the word size of current computing architectures. We seek to improve our (already satisfactory) results by using advanced algorithms for simultaneous approximation, like the LLL-algorithm or the Dirichlet approximation algorithm for ^2.For the theoretical part, we are interested if finding simultaneous rational approximations with small lowest common multiple of the denominators is simpler than finding Dirichlet approximations. We are also interested in generalizing the method to provide rational approximations with small absolute errors on ellipsoids with rational semi-principal axes – E.g. the geographic WGS84 ellipsoid.abbrv § PROOF OF LIOUVILLE'S APPROXIMATION THEOREM This nice proof was translated from the Germanhttps://de.wikibooks.org/wiki/Beweisarchiv:_Algebra:_K See <cit.> for the original proof in French language. equationsection Let α∈ be algebraic of degree n and root of the corresponding polynomial f(X)∈[X] of degree n, meaning f(α) = a_0 + a_1α + ⋯ +a_nα^n = 0with a_0, …, a_n ∈ and a_n 0. Polynomial division with the linear factor X-α in the ring [X] provides f(X) = (X-α)· g(X). Note that the polynomial g(X) has algebraic coefficients and is not necessarily in [X]. However, the mapping →, t↦ g(t) is continuous, by means of real numbers c_1>0, c_2>0 with |g(x)| ≤ c_1 for |α-x| < c_2. Since n<∞, we can assume w.l.o.g. that no additional roots are in this neighborhood of α, meaning f(x) 0 for |α-x| < c_2 and xα. Claim: The statement of the Theorem holds for c:=min{c_2, 1/c_1}. Suppose there are p,q∈, q>0 with |α-p/q | < c/q^n. We show that his implies α= pq. From (<ref>), we immediately derive |α-p/q| < c ≤ c_2 , leading (<ref>) to imply | g( pq)| ≤ c_1. We derive, from (<ref>) and again (<ref>), that |f(p/q) | = |p/q -α|·|g(p/q)| < c/q^n· c_1 ≤1/q^n, meaning |q^n · f(p/q)| < 1. However q^n· f(p/q) = a_0q^n + a_1pq^n-1 + ⋯ + a_n p^n∈ and its absolute value is smaller than 1, hence has to be 0. Moreover, f( pq) = 0 and (<ref>) with (<ref>) imply α= pq, which closes the argument. § PROOF OF DIRICHLET'S APPROXIMATION THEOREM The folklore proof bases on Dirichlet's famous Pigeonhole Principle. See https://proofwiki.org/wiki/Dirichlet We consider the partition of [0,1]^d in N^d regular d-cubes of length L=√(N). We further define a sequence of points (a^(j))_j=1,…, N^d+1∈ [0,1]^d with a^(j) := j ·α - ⌊ j ·α⌋ (component wise operations). There are indices k > l such that the points a^(k) and a^(l) are contained in the same d-cube. We have the (component wise) inequalities -1/L < a^(k) - a^(l) < 1/L -1/L < kα - ⌊ k α⌋ - lα+⌊ l α⌋ < 1/L -1/L < (k-l)α - (⌊ k α⌋- ⌊ l α⌋ ) < 1/L Setting q=k-l and p_i = ⌊ kα_i⌋ - ⌊ lα_i⌋ provides integers as required. § REDUCTIONS OF SPHERICAL PREDICATES TO CARTESIAN ORIENTATION PREDICATES We first describe a reduction from the spherical predicates to well studied Cartesian predicates. Let p_1, p_2∈𝕊^2 with p_1 ≠ p_2 and P the plane containing p_1,p_2 and the origin (0,0,0) and C be the Great Circle through p_1 and p_2. For q ∈𝕊^2 we have qleft-ofP qleft-ofC q ∈ P q ∈ C qright-ofP qright-ofC 𝕊^2 ∩ P = C and 𝕊^2= L ∪ C ∪ R. Let P denote the plane through non-identical points p_1, p_2, p_3 ∈𝕊^2 and the half space containing the origin (0,0,0) is called `below P'. We further call S_123⊆^3 the closed volume of the sphere with p_1,p_2,p_3 and the origin on its surface. For a point q ∈𝕊^2 we have qabovePq ∈ S_123∖∂ S_123 q ∈ P q ∈∂ S_123 qbelowPq ∉ S_123 P is uniquely determined because three different points on the unit sphere are not co-linear. Since S_123 and 𝕊^2 are spheres, their cuts with P are circles in P and the two circles are identical as they contain p_1,p_2 and p_3 on their boundary. This circle C has a radius of at most 1 and partitions the points of the unit sphere into three sets𝕊^2 = A ∪ C ∪ Bwhere A ⊆ S_123⊋ B. If C is a Great Circle we resolve ambiguity for `above' and the center of S_123 by choosing the open half spaces that first contain (0,0,1), then (0,1,0) and eventually (1,0,0). We have 𝕊^2 ≠∂ S_123, since the origin is a fourth point on S_123 and q ∈ P iff. q ∈ C iff. q ∈∂ S_123. Therefore it is sufficient to show q ∈ A qaboveP. To this end we consider the convex volume of the unit sphere S ⊆^3 and D = S ∩ S_123. Note that ∂ D contains C and A. Since the cut with the closed half-space of the plane P cuts a convex body into at most three parts and C ⊆ P, we have that all of A is `above' P. | http://arxiv.org/abs/1707.08549v1 | {
"authors": [
"Daniel Bahrdt",
"Martin P. Seybold"
],
"categories": [
"cs.CG"
],
"primary_category": "cs.CG",
"published": "20170726172339",
"title": "Rational Points on the Unit Sphere: Approximation Complexity and Practical Constructions"
} |
Mintz et al. ROGUE Bandits Non-Stationary Bandits with Habituation and Recovery Dynamics Yonatan Mintz School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332, [email protected] Aswani, Philip Kaminsky Department of Industrial Engineering and Operations Research, University of California, Berkeley, CA 94720, {aaswani,kaminsky,}@berkeley.eduElena Flowers Department of Physiological Nursing, School of Nursing, University of California, San Francisco, CA 94143, [email protected] Fukuoka Department of Physiological Nursing & Institute for Health & Aging, School of Nursing, University of California, San Francisco, CA 94143, [email protected] Many settings involve sequential decision-making where a set of actions can be chosen at each time step, each action provides a stochastic reward, and thedistribution for the reward provided by each action is initially unknown. However, frequent selection of a specific action may reduce the expected reward for that action, while abstaining from choosing an action may cause its expected reward to increase. Such non-stationary phenomena are observed in many real world settings such as personalized healthcare-adherence improving interventions and targeted online advertising. Though finding an optimal policy for general models with non-stationarity is PSPACE-complete, we propose and analyze a new class of models called ROGUE (Reducing or Gaining Unknown Efficacy) bandits, which we show in this paper can capture these phenomena and are amenable to the design of policies with provable properties. We first present a consistent maximum likelihood approach to estimate the parameters of these models, and conduct a statistical analysis to construct finite sample concentration bounds. Using this analysis, we develop and analyze two different algorithms for optimizing ROGUE models: an upper confidence bound algorithm (ROGUE-UCB) and an ϵ-greedy algorithm (ϵ-ROGUE). Our theoretical analysis shows that under proper conditions the ROGUE-UCB and ϵ-ROGUE algorithms can achieve logarithmic in time regret, unlike existing algorithms which result in linear regret. We conclude with a numerical experiment using real world data from a personalized healthcare-adherence improving intervention to increase physical activity. In this intervention, the goal is to optimize the selection of messages (e.g., confidence increasing vs. knowledge increasing) to send to each individual each day to increase adherence and physical activity. Our results show that ROGUE-UCB and ϵ-ROGUE perform better in terms of aggregated regret and average reward when compared to state of the art algorithms, and in the context of this intervention the use of ROGUE-UCB increases daily step counts by roughly 1,000 steps a day (about a half-mile more of walking) as compared to other algorithms in a simulation experiment.multi-armed bandits, personalized healthcare, adherence Kinetic cascade in solar-wind turbulence: 3D3V hybrid-kinetic simulations with electron inertia Francesco Califano^1 December 30, 2023 =============================================================================================== § INTRODUCTION Multi-armed bandits are commonly used to model sequential decision-making in settings where there is a set of actions that can be chosen at each time step, each action provides a stochastic reward, and the distribution for the reward provided by each action is initially unknown.The problem of constructing a policy for sequentially choosing actions in multi-armed bandits requires balancing exploration versus exploitation, the tradeoff between selecting what is believed to be the action that provides the best reward and choosing other actions to better learn about their underlying distributions.Bandit models have been applied in a variety of healthcare settings <cit.>.For instance, <cit.> considered the problem of selecting drugs to give to a patient from a set (where each drug is an action) in order to treat a specific disease (the reward is the improvement in patient health in response to the drug); the bandit policy asymptotically identifies the optimal drug for that particular patient. Other common applications involve online advertising <cit.>, where selecting an ad to show is an action and the reward is the total number (from a large population) of viewers who click on the ad, as well as in various supply chain settings <cit.>.However, most bandit models assume that the distribution for the reward provided by each action is constant over time.This is a reasonable assumption in a large number of applications, such as the ones described above.However, many applications involve actions that are applied to a single individual, where the rewards depend upon behavioral responses of the individual to the applied actions.In these behavioral settings, the response to a particular action is not generally stationary.Frequent selection of a particular action will lead to habituation to that action by the individual, and the reward for that action will decrease each time it is selected.For example, repeatedly showing the same ad to a single individual may cause the ad to become less effective in soliciting a response from that individual.Furthermore, another complimentary phenomenon can also occur; refraining for a period of time from showing a particular ad to a single individual may cause the ad to become more effective when reintroduced.Most techniques for designing policies for decision-making for multi-armed bandits apply to the setting where the rewards for each action are stationary.However, designing a policy without considering the non-stationarity of a system (when the system is in fact non-stationary) often leads to poor results in terms of maximizing rewards <cit.> because policies eventually converge to a stationary policy. The problem of designing policies for bandit models with non-stationarity has been studied in specific settings, but approaches in the literature are either computationally intractable, or the settings analyzed are not flexible enough to capture the habituation and recovery phenomenon described above.The aim of this paper is to propose a flexible bandit model that is able to effectively model habituation and recovery, and to present an approach for designing an effective policy for this bandit model.§.§ Literature Review Data-driven decision-making can be categorized into batch formulations and online formulations.Batch formulations <cit.> use a large amount of data to estimate a predictive model and then use this model for optimization.Adaptation to new data occurs by reestimating the predictive model, which is done periodically after a specified amount of additional data is collected.On the other hand, online formulations involve constructing a policy that is updated every time a new data point is collected.Bandit models are a particularly important example of online formulations, and there has been much work on constructing policies for stationary bandits.Approaches for designing policies for stationary bandits include those using upper confidence bounds <cit.>, Thompson sampling <cit.>, Bayesian optimization <cit.>, knowledge gradients <cit.>, robust optimization <cit.>, and adversarial optimization <cit.>.Restless bandits are a notable class of bandit models that capture non-stationarity, because choosing any single action causes the rewards of potentially all the actions to change.Though dynamic programming<cit.>, approximation algorithms <cit.>, and mathematical programming <cit.> have been proposed as tools for constructing policies in this setting, the problem of computing an optimal policy for restless bandits isPSPACE-complete <cit.>, meaning that designing policies that are approximately optimal is difficult. These models more broadly fall under the class of Partially Observable Markov Decision Processes (POMDPs), a class of problems where the decision maker is interested in solving a Markov Decision Process (MDP) with limited information about the system. The canonical methods for finding optimal policies for POMDPs involve the conversion of the POMDP into an equivalent belief MDP <cit.> in which the states correspond to a probability distribution reflecting the decision maker's belief that they are in a specific state. While for small POMDPs it is possible to solve the belief MDP using dynamic programming approaches <cit.>, in general, since POMDPs are PSPACE-complete, the resulting MDP is often too large to optimize effectively and requires using approximate dynamic programming <cit.>. Therefore, in recent literature, several solutions approaches have been proposed for solving POMDPs, which often involve exploiting the specific structure of the model in question. The framework of ROGUE multiarmed bandits we propose in this paper can be seen as a particular type of POMDP or restless bandit model, with a known deterministic transition model and partially observed rewards. To achieve efficient theoretical bounds and practical performance, the methods we propose are specific to this particular POMDP structure.Another related research stream designs policies for non-stationary multi-armed bandits with specific structures.For instance, model-free approaches have been proposed <cit.> for settings with bounded variations, so thatrewards of each action are assumed to change abruptly but infrequently.These policies have been shown to achieve 𝒪(√(Tlog T)) suboptimality in worst case scenarios but can achieve 𝒪(log T) under proper model assumptions. Recently, there has been interest in studying more structured non-stationary bandits. Two relevant examples are Adjusted Upper Confidence Bounds (A-UCB) and rotting bandits <cit.>, where each action has a set of unknown but stationary parameters and a set of known non-stationary parameters that characterize its reward distribution.Policies designed for these settings achieve 𝒪(log T) suboptimality, but these settings are unable to capture the habituation and recovery phenomenon that is of interest to us. §.§ ROGUE Bandits In this paper, we define the ROGUE (reducing or gaining unknown efficacy) bandit model, which can capture habituation and recovery phenomenon, and then we design a nearly-optimal policy for this model.ROGUE bandits are appropriate for application domains where habituation and recovery are important factors for system design; we present two such examples, in online advertising and personalized healthcare, below. §.§.§ Personalized Healthcare-Adherence Improving InterventionsOne hundred fifty minutes of moderate-intensity aerobic physical activity each week has been shown to reduce the risk of cardiovascular disease, other metabolic disorders, and certain types of cancers <cit.>.However, maintaining this level of moderate intensity activity is challenging for most adults. As such, proper motivation through providing daily exercise goals and encouragement has been found to be effective in helping patients succeed in being active <cit.>. In recent years, there has been an increased rate of adoption of fitness applications and wearable activity trackers, making it easier and less costly to implement physical activity programs <cit.>. These trackers and mobile applications record daily activity, communicate activity goals, and send motivational messages. Despite these digital devices having collected a large amount of personal physical activity data, many of the most popular activity trackers provide static and non-personalized activity goals and messages to their users <cit.>. Furthermore, the choice of motivational messages sent to users may have significant impact on physical activity, because if users receive similar messages too frequently they may become habituated and not respond with increased activity, while seldom sent messages may better increase activity due to their novelty and diversity.Because the ROGUE bandits can model habituation and recovery of rewards for different actions, we believe they present a useful framework for the design of policies that choose which messages to send to users based on data consisting of what messages they received each day and the corresponding amounts of physical activity on those days.Personalized healthcare has been extensively studied in the operations literature. <cit.> explore the use of behavioral analytics to personalize diet and exercise goals for clinically supervised weight loss interventions in an offline setting. Markov decision processes have also been used for decision-making in personalized healthcare <cit.>. In contrast to bandit models where only the reward for the prescribed action can be observed, these methods broadly assume that the full state of the system can be observed, and thus do not require statistical estimation. Additionally, various multi-armed bandit approaches <cit.> have also been proposed for healthcare problems where habituation and recovery are not significant factors. §.§.§ Online Content Creation and AdvertisingOnline advertising is one of the fastest-growing industries in the US. In fact, as of 2016, US Internet advertising spending has increased to over $72.5 billion, surpassing the amount spent on TV ads <cit.>. However, as this form of advertising becomes more prevalent, advertisers have been struggling to ensure that ads retain there effectiveness. This has been attributed to Internet users being habituated by impersonal and standardized ads <cit.> which are rarely varied. For these reasons, there has been significant interest in the operations literature in creating automated systems that can utilize user-level data to better target and customize ads <cit.>. In particular, since the effect of a no-longer-effective advertisement may recover after a user has not seen it for some period of time, incorporating recovery and habituation dynamics into advertising models could yield more effective advertising campaigns.In general, multi-armed bandit models have been proposed to model online advertising, where each action corresponds to a different type of advertisement, and the reward is equivalent to either a conversion or a click from a prospective consumer. Several approaches have been used to design such an ad targeting system, including adversarial and stochastic multi-armed bandit models <cit.>, and online statistical testing <cit.>. However, while some of these approaches use contextual data to better serve ads to individuals, they are still designed under assumptions of stationarity.As a result, these approaches will lead to policies that show duplicated ads to individuals, which can potentially causing habituation, whereas other ads that might have recovered efficacy may not be served at all. In contrast, ROGUE Bandit models can explicitly consider the time-varying efficacy each type of ad, and thus directly capture user habituation to a specific ad, and track the recovery of efficacy of a particular ad for a specific individual.§.§ OutlineIn Section <ref>, we formally introduce the ROGUE bandit model. To the best of our knowledge, this is the first work where a non-stationary bandit model has been defined that is able to capture habituation and recovery phenomenon, and is at the same time amenable to the design of nearly-optimal policies.Because the ROGUE bandit is a general model, we describe two specific instantiations: the ROGUE generalized linear model and the ROGUE agent.Next, in Section <ref> we analyze the problem of estimating the parameters of a single action.We present a statistical analysis of maximum likelihood estimation (MLE) for a single action, and use empirical process theory to derive finite sample bounds for the convergence of parameters estimates.Specifically, we show that the MLE estimates converge to the true parameters at a 1/√(T) rate.Section <ref> describes an upper-confidence bound policy for ROGUE bandits, and we call this policy the ROGUE-UCB algorithm.The main result of this section is a rigorous 𝒪(log T) bound on the suboptimality of the policy in terms of regret, the difference between the reward achieved by the policy and the reward achieved by an optimal policy.Our 𝒪(log T) bound is significant because this is the optimal rate achievable for approximate policies in the stationary case <cit.>.We prove our bound using methods from the theory of concentration of measure.We conclude with Section <ref>, where we introduce a “tuned” version of ROGUE-UCB and then conduct numerical experiments to compare the efficacy of our ROGUE-UCB algorithm to other policies that have been developed for bandit models.Our experiments involve two instantiations of ROGUE bandit models.First, we compare different bandit policies using a ROGUE generalized linear bandit to generate data.Second, we compare different bandit policies using a ROGUE agent to generate data, where the parameters of this bandit model are generated using data from a physical activity and weight loss clinical trial <cit.>.This second experiment specifically addresses the question of how to choose an optimal sequence of messages to send to a particular user in order to optimally encourage the user to increase physical activity, and it can be interpreted as a healthcare-adherence improving intervention.Our experiments show that ROGUE-UCB outperforms all other considered bandit policies, and that it achieves logarithmic regret, in contrast to other bandit algorithms that achieve linear regret.§ DEFINING REDUCING OR GAINING UNKNOWN EFFICACY (ROGUE) BANDITSThis section first describes the stationary multi-armed bandit (MAB) model, in order to emphasize modeling differences in comparison to our ROGUE bandit model that is introduced in this section.Our goal in defining ROGUE bandits is to have a model that can capture specific non-stationary phenomena found in behavioral applications, and so we next formally introduce the model elements of ROGUE bandits.To provide better intuition about ROGUE bandits, we also present two specific instantiations of a ROGUE bandit that incorporate different behavioral effects.§.§ Stationary MAB Model The stationary MAB is a setting where there is a finite set of actions 𝒜 that can be chosen at each time step t, each action a ∈𝒜 provides a stochastic reward r_a with distribution ℙ_θ_a, and the parameters θ_a∈Θ for a ∈𝒜 are constants that are initially unknown but lie in a known compact set Θ.The problem is to construct a policy for sequentially choosing actions in order to maximize the expected reward.More specifically, let π_t∈𝒜 be the action chosen at time t = 1,…,T.Then the policy consists of functions π_t(r_π_1,…,r_π_t-1,π_1,…,π_t-1)∈𝒜 that depend on past rewards and actions.For notational convenience, we will use Π = {π_t(·)}_t=1^T to refer to the policy.In this notation, the problem of constructing an optimal policy to maximize expected reward can be written as max_Π∈𝒜^T∑_t=1^T 𝔼r_π_t.Note that certain regularity is needed from the distributions to ensure this maximization problem is well-posed.One common set of assumptions is that the distributions ℙ_θ_a for a∈𝒜 are sub-Gaussian, and that the reward distributions are all independent.For the stationary MAB, we can define an optimal action a^* ∈𝒜, which is any action such that 𝔼r_a^*≥𝔼r_a for all a∈𝒜.The benefit of this definition is it allows us to reframe the policy design problem in terms of minimizing the cumulative expected regret 𝔼R_Π(T) = 𝔼[Tr_a^* - ∑_i=1^Tr_π_t], where the quantity r_a^* - r_π_t is known as the regret at time t.Observe that minimizing 𝔼R_Π(T) is equivalent to maximizing ∑_t=1^T 𝔼r_π_t.It has been shown by <cit.> that an index policy is optimal for the stationary MAB. Since these indexing policies are difficult to compute, other approximate policies have been proposed <cit.>. Some of the most common policies use upper confidence bounds <cit.>, which take actions optimistically based on estimates of the parameters θ_a. Unfortunately, it has been shown that these index policies and upper confidence bound policies can have arbitrarily bad performance in a non-stationary setting <cit.>.§.§ Reducing or Gaining Unknown Efficacy (ROGUE) BanditsA disadvantage of the stationary MAB is that it does not allow rewards to change over time in response to previous actions, and this prevents the stationary MAB model from being able to capture habituation or recovery phenomena.Here, we define ROGUE bandits that can describe such behavior.The ROGUE bandit is a setting where there is a finite set of actions 𝒜 that can be chosen at each time step t, each action a ∈𝒜 at time t provides a stochastic reward r_a,t that has a sub-Gaussian distribution ℙ_θ_a,x_a,t with expectation 𝔼r_a,t = g(θ_a,x_a,t) for a bounded function g, the parameters θ_a∈Θ for a ∈𝒜 are constants that are initially unknown but lie in a known compact, convexset Θ, and each action a∈𝒜 has a state x_a,t with nonlinear dynamics x_a,t+1 = h_a(x_a,t,π_a,t)where π_a,t = 1[π_t = a],h:𝒳×𝔹 →𝒳 is a known dynamics function, and 𝒳 is a compact convex set such that x_a,t∈𝒳 ∀ a,t, and x_a,0 is initially unknown for a∈𝒜.This particular model formulation is useful for various real world scenarios in which effective system models exist but states are not directly observed due to system or measurement noise. This can be achieved by training models on previously collected controlled data, or by using models based on information from qualified domain experts. In the case of personalized healthcare-adherence interventions and on-line marketing, often times effective models can be generated from previous experimental data (e.g., from RCTs or focus group testing) though state observation in non-laboratory (i.e., not completely controlled) situations is challenging. This data can be used in conjunction with MLE techniques to obtain useful estimates of the system dynamics. In cases where limited data and expertise are available, alternative models ought to be used, the discussion of which is out of the scope of this paper.The problem is to construct a policy for sequentially choosing actions in order to maximize the expected reward.Observe that the ROGUE bandit model is non-stationary since the reward distributions depend upon previous actions. This makes the problem of designing policies more difficult than that of designing policies for the stationary MAB.More specifically, let π_t∈𝒜 be the action chosen at time t = 1,…,T.Then the policy consists of functions π_t(r_π_1,…,r_π_t-1,π_1,…,π_t-1)∈𝒜 that depend on past rewards and actions.For notational convenience, we will use Π = {π_t(·)}_t=1^T to refer to the policy.In this notation, the problem of constructing an optimal policy to maximize expected reward can be written asmax_Π∈𝒜^T{∑_t=1^T g(θ_π_t,x_π_t,t) : x_a,t+1 = h_a(x_a,t,π_a,t)fora ∈𝒜, t ∈{0,...,T-1}}.This can be reframed as minimizing expected cumulative regret <cit.>: Unlike the stationary MAB, we cannot define an optimal action, but rather must define an optimal policy Π^* = {π^*_t(·)}_t=0^T, which can be thought of as an oracle that chooses the optimal action at each time step.Then the problem of designing an optimal policy is equivalent to minimizing R_Π(T) = ∑_t=1^T r_π^*_t,t - r_π_t,t or in expectation 𝔼R_Π(T) = ∑_t=1^T g(θ_π^*_t,x_π^*_t,t) - g(θ_π_t,x_π_t,t) subject to the state dynamics defined above, a notion known as either dynamic or tracking regret. Note that since the optimal policy may be quite different from the policy actually implemented, the resulting states and rewards observed at each time period may not necessarily be the same. There have been stronger definitions of dynamic regret proposed in the literature including strongly adaptive regret <cit.>. These definitions have been shown to not hold generally in the bandit feedback case even for models with optimal tracking regret rates <cit.>, and are applied in the context of full information convex on-line optimization with limited total variation. In this paper, we center our analysis on tracking regret as our setting includes both bandit feedback and is model-based without necessarily bounded total variation. §.§ Technical Assumptions on ROGUE Bandits In this paper, we will design a policy for ROGUE bandits that follow the assumptions described below:The rewards r_a,t are conditionally independent given x_a,0,θ_a (or equivalently the complete sequence of x_a,t, π_t and θ_a). This assumption states that for any two time points t,t' such that t ≠ t' we have that r_a,t|{x_a,t,θ} is independent of r_a,t'|{x_a,t',θ}, and it is a mild assumption because it is the closest analogue to the assumption of independence of rewards in the stationary MAB.The reward distribution ℙ_θ,x has a log-concave probability density function (p.d.f.) p(r|θ,x) for all x∈𝒳 and θ∈Θ.This assumption provides regularity for the reward distributions, and is met by many common distributions (e.g., Gaussian and Bernoulli).Now define f(·) to be L-Lipschitz continuous if |f(x_1) - f(x_2)| ≤ Lx_1-x_2_2 for all x_1,x_2 in the domain of f. We define f to be locally L-Lipschitz on a compact set 𝒮 if it has the Lipschitz property for all points of its domain on set 𝒮. Our next assumption is on the stability of the above distributions with respect to various parameters.The log-likelihood ratio ℓ(r;θ',x',θ,x) = logp(r|θ',x')/p(r|θ,x) associated with the distribution family ℙ_θ,x is locally L_f-Lipschitz continuous with respect to x,θon the compact set 𝒳×Θ for all values of θ',x'∈𝒳×Θ, and g is locally L_g-Lipschitz continuous with respect to x,θ on the compact set 𝒳×Θ.This assumption ensures that if two sets of parameters are close to each other in value then the resulting distributions will also be similar. We make the following additional assumption about the functional structure of the reward distribution family:The reward distribution ℙ_θ,x for all θ∈Θ and x∈𝒳 is sub-Gaussian with parameter σ, and either p(r|θ,x) has a finite support or ℓ(r;θ',x',θ,x) is locally L_p-Lipschitz with respect to r. This assumption (or a similar type of regularity) is needed to ensure that sample averages are close to their means, and it is satisfied by many distributions (e.g., a Gaussian location family with known variance). Last, we impose conditions on the dynamics for the state of each action: The dynamic transition function h is L_h Lipschitz continuous such that L_h ≤ 1. This assumption is needed to ensure the states of each action do not change too quickly, and it implies that the dynamics are stable. §.§ Instantiations of ROGUE Bandits The above assumptions are general and apply to many instantiations of ROGUE bandit models. To demonstrate the generality of these assumptions, we present two particular instances of ROGUE bandit models.§.§.§ ROGUE AgentOur first instantiation of a ROGUE bandit model consists of a dynamic version of a principal-agent model <cit.>, which is a model where a principal designs incentives to offer to an agent who is maximizing an (initally unknown to the principal) utility function that depends on the incentives.In particular, consider a setting with a single (myopic) agent to whom we would like to assign a sequence of behavioral incentives π_t ∈𝒜, and the states x_a,t and parameters θ_a are scalars.Given a particular incentive π_t at time t, the agent responds by maximizing the (random) utility functionr_t = _r ∈ [0,1] -1/2r^2 - (c_a,t+ ∑_a ∈𝒜x_a,tπ_t,a)r,where for fixed a∈𝒜 we have that c_a,t are i.i.d. random variables with a distribution ℙ_θ_a such that Var(c_a,t) = σ^2(θ_a) < ∞ and σ^2: ℝ→ℝ_+ is invertible. Moreover, the state dynamics arex_a,t+1 = proj_𝒳(α_a x_a,t + b_a (1- π_a,t) -k_a),Note the distribution of r_t is fully determined by x_a,t,θ_a,{π_k}_k=0^t, which means the rewards satisfy Assumption <ref>.We can further analyze the above ROGUE agent model.Solving the agent's optimization problem (<ref>) gives r_t|{x_a,t,θ_a} =0if c_a,t≤ - x_a,t,1if c_a,t≥ 1- x_a,t, c_a,t + x_a,t otherwise We can express the distribution of r_t|{x_a,t,θ_a} in terms of the cumulative distribution function (c.d.f.) F(·) and p.d.f. f(·) of c_a,t:p(r_t|{x_a,t,θ_a}) = F(-x_a,t)δ(r_t) +(1-F(1-x_a,t)) δ(1-r_t) + f(r_t -x_a,t) 1[r_t ∈ (0,1)].Though p(r_t|{x_a,t,θ_a}) is not an absolutely continuous function, it satisfies Assumptions <ref> and <ref>, whenever c_t has a log-concave p.d.f. that is Lipschitz continuous, if we interpret the above probability measure p(r_t|{x_a,t,θ_a}) as a p.d.f.§.§.§ ROGUE Generalized Linear Model (GLM) Dynamic logistic models and other dynamic generalized linear models <cit.> can be interpreted as non-stationary generalizations of the classical (Bernoulli reward) stationary MAB <cit.>. Here, we further generalize these models: Consider a setting where r_a,t|{θ_a,x_a,t} is an exponential family with mean parameterμ_a,t = 𝔼r_t = g(α_a^Tθ_a + β_a^Tx_a,t),for known vectors α_a,β_a, where the action states x_a,t have appropriate dynamics.In this situation, we can interpret g(·) as a link function of a generalized linear model (GLM). For example, if g is a logit function, then this model implies the rewards have a Bernoulli distribution with parameterμ_a,t = 1/1+exp(-(α_a^Tθ_a + β_a^Tx_a,t)).For the logistic case, the r_a,t is bounded and satisfies Assumptions <ref>-<ref>. These assumptions are also satisfied if r_a,t can be linked to a truncated exponential family distribution restricted to [0,1], meaning if the p.d.f. of r_a,t|{x_a,t,θ_a } is h(r)/F(1)-F(0)exp(T(r)g(α_a^Tθ_a + β_a^Tx_a,t) - A(α_a^Tθ_a + β_a^Tx_a,t)),where T(r) is a sufficient statistic h is the base measure (not to be confused with dynamics h_a), A is the log-partition function, and F is the full cdf of the un-truncated distribution.If instead we consider sub-Gaussian exponential families with infinite support, Assumption <ref> is satisfied if the sufficient statistic of the GLM is Lipschitz or bounded with respect to r. While we will mainly consider one-dimensional rewards (i.e., r_a,t∈ℝ), we note that this framework can also be extended to vector and array dynamic GLM's.§ PARAMETER ESTIMATION FOR ROGUE BANDITS Our approach to designing a policy for ROGUE bandits will involve generalizing the upper confidence bound policies <cit.> that have been developed for variants of stationary MAB's.As per the name of these policies, the key step involves constructing a confidence bound for the parameters θ_a, x_a,0 characterizing the distribution of each action a∈𝒜.This construction is simpler in the stationary case because the i.i.d. structure of the rewards allows use of standard Chernoff-Hoeffding bounds <cit.>, but we can no longer rely upon such i.i.d. structure for ROGUE bandits which are fundamentally non-stationary. This is because in ROGUE bandits the reward distributions depend upon states x_a,t, and so the structure of ROGUE bandits necessitates new theoretical results on concentration of measure in order to construct upper confidence bounds for the relevant parameters.For this analysis, let the variables {r_a,t}_t=1^T be the observed rewards for action a∈𝒜. It is important to note that the r_a,t here are no longer random variables, but are rather the actual observed values.Since the reward distributions for each action are mutually independent by the dynamics h_a, we can study the estimation problem for only a single action. Specifically, consider the likelihood p( {r_a,t}_t∈𝒯_a| θ_a,x_a,0 ), where 𝒯_a ⊂{1,...,T} is the set of times when action a was chosen (i.e., π_t = a for t∈𝒯_a).Let n(𝒯_a) denote the cardinality of the set 𝒯_a.Using Assumption <ref>, the likelihood can be expressed asp({r_a,t}_t∈𝒯_a|θ_a,x_a,0) = ∏_t∈𝒯_ap(r_a,t|θ_a,x_a,t) ∏_t∈𝒯_ap(x_t|θ_a,x_a,t_-).where t_- = max{s∈𝒯_a : s < t} is the latest observation before time t. Note the MLE of θ_a,x_a,0 is (θ̂_a,x̂_a,0) ∈∏_t∈𝒯_ap(r_a,t|θ_a,x_a,t) ∏_t∈𝒯_ap(x_t|θ_a,x_a,t_-). Observe that since the dynamics h_a are known the one step likelihood p(x_t|θ_a,x_a,t-1) is a degenerate distribution with all probability mass at x_a,t, by perpetuation of the dynamics h with initial conditions x_a,t-1. Thus we can express the MLE as the solution to the constrained optimization problem(θ̂_a,x̂_a,0) = min{-∑_t∈𝒯_alog p(r_a,t|θ_a,x_a,t) : x_a,t+1 = h_a(x_a,t,π_a,t)fort ∈{0,…,T}},where we have also taken the negative logarithm of the likelihood (<ref>). In this section, we will consider concentration properties of the solution to the above optimization problem. In particular, since our results concern the joint distributions of a trajectory of values, we require the definition of the following quantity which we call the trajectory Kullback–Leibler (KL) Divergence: For some input action sequence π_1^T and arm a ∈𝒜 with dynamics h_a, given starting parameter values (θ_a,x_a,0), (θ'_a,x'_a,0) ∈𝒳×Θ, define the trajectory KL–Divergence between these two trajectories with the same input sequence and different starting conditions as: D_a,π_1^T(θ_a,x_a,0||θ'_a,x'_a,0) = ∑_t∈𝒯_aD_KL(ℙ_θ_a,x_a,t||ℙ_θ'_a,x'_a,t) =∑_t∈𝒯_aD_KL(ℙ_θ_a,h_a^t(x_a,0)|| ℙ_θ'_a,h_a^t(x'_a,0)) where h^k_a represents the functional composition of h_a with itself k times subject to the given input sequence, ℙ_θ,x is the probability law of the system under parameters θ,x, and D_KL is the standard KL-Divergence. If θ^*_a,x^*_0,a for a∈𝒜 are the true parameter values of a ROGUE Bandit model, then we show thatFor any constant ξ >0 we haveP(1/n(𝒯_a)D_a,π_1^T(θ^*_a,x^*_a,0||θ̂_a,x̂_a,0) ≤ξ +c_f(d_x,d_θ)/√(n(𝒯_a))) ≥ 1 - exp(-ξ^2 n(𝒯_a)/2 L_p^2 σ^2)wherec_f(d_x,d_θ) = 8L_f(𝒳)√(π) + 48√(2)(2)^1/d_x+d_θL_f(𝒳×Θ)√(π(d_x+d_θ))is a constant that depends upon d_x (the dimensionality of 𝒳) and d_θ (the dimensionality of Θ), and D_a,π_1^T(θ_a,x_a,0||θ'_a,x'_a,0) is the trajectory Kullback–Leibler (KL) divergence between two different initial conditions.§.§ Conceptual Reformulation of MLEOur analysis begins with a reformulation of the MLE that removes the constraints corresponding to the dynamics through repeated composition of the function h_a.Let θ^*_a∈Θ and x^*_a,0∈𝒳 for a∈𝒜 be the true underlying parameters of the system, then the MLE is given by (θ̂_a,x̂_a,0) = _θ_a,x_a,0∈Θ×𝒳1/n(𝒯_a)∑_t∈𝒯_alog p(r_a,t|θ^*_a,h_a^t(x^*_a,0,θ^*_a,π_1^t))/p(r_a,t|θ_a,h_a^t(x_a,0,θ_a,π_1^t))where the notation h_a^k represents the repeated functional composition of h_a with itself k times, and π_1^t is the sequence of input decisions from time 1 to time t. The complete proof for this proposition is found in Appendix <ref>, and here we provide a sketch of the proof. Observe that this formulation is obtained by first adding constant terms equal to the likelihood of the true parameter values to the objective function and dividing by the total number of observations (which does not change the optimal solution), and then composing our system dynamics and writing them as explicit functions of the initial conditions. In practice, this reformulation is not practical to solve since clearly θ^*_a,x^*_a,0 are not known a priori and the composite function h_a^t may have a complex form. However, for theoretical analysis this reformulation is quite useful, since for fixed θ_a,x_a,0 taking the expected value of the objective under ℙ_θ^*_a,x^*_a,0 yields𝔼_θ^*_a,x^*_a,01/n(𝒯_a)∑_t∈𝒯_alog p(r_a,t|θ^*_a,h_a^t(x^*_a,0,θ^*_a,π_1^t))/p(r_a,t|θ_a,h_a^t(x_a,0,θ_a,π_1^t)) = 1/n(𝒯_a)∑_t∈𝒯_aD_KL(ℙ_θ^*_a,x^*_a,t|| ℙ_θ_a,x_a,t)=1/n(𝒯_a)D_a,π_1^T(θ^*_a,x^*_a,0||θ_a,x_a,0). Essentially, we have reformulated the MLE problem in terms of minimizing the KL divergence between the trajectory distribution of potential sets of parameters to the trajectory distribution of the true parameter set. Since we have clear interpretation for the expectation of our objective function we can now proceed to compute concentration inequalities. §.§ Uniform Law of Large Numbers for ROGUE Bandits Since our estimates are computed by solving an optimization problem, a pointwise law of large numbers is insufficient for our purposes since such a result would not be strong enough to imply convergence of the optimal solutions.To obtain proper concentration inequalities we must consider a uniform law of large numbers for the MLE problem. For any constant ξ > 0 we haveP(sup_θ_a,x_a,0∈Θ×𝒳|1/n(𝒯_a)∑_t∈𝒯_alog p(r_a,t|θ^*_a,h_a^t(x^*_a,0,θ^*_a,π_1^t))/p(r_a,t|θ_a,h_a^t(x_a,0,θ_a,π_1^t))- 1/n(𝒯_a)D_a,π_1^T(θ^*_a,x^*_a,0||θ_a,x_a,0) | > ξ + c_f(d_x,d_θ)/√(n(𝒯_a))) ≤exp(-ξ^2 n(𝒯_a)/2L_p^2σ^2)wherec_f(d_x,d_θ) = 8L_f(𝒳)√(π) + 48√(2)(2)^1/d_x+d_θL_f(𝒳×Θ)√(π(d_x+d_θ))is a constant. We will prove this result in several steps, the first of which uses the following lemma:Consider the mappingφ({r_t}_t=1^n(𝒯_a))=sup_θ_a,x_a,0∈Θ×𝒳|1/n(𝒯_a)∑_t∈𝒯_alog p(r_a,t|θ^*_a,h_a^t(x^*_a,0,θ^*_a,π_1^t))/p(r_a,t|θ_a,h_a^t(x_a,0,θ_a,π_1^t)) - 1/n(𝒯_a)D_a,π_1^T(θ^*_a,x^*_a,0||θ_a,x_a,0) |.The mapping φ is L_p-Lipschitz with respect to {r_t}_t=1^n(𝒯_a).A detailed proof is provided in Appendix <ref>, and the main argument of the proof relies on the preservation of Lipschitz continuity through functional composition and pointwise maximization. This result is necessary since showing that objective value variations are bounded is a prerequisite for the formalization of concentration bounds. Next we consider the Lipschitz constant of the log-likelihood with respect to the parameters.For any r∈ℝ, θ̅∈Θ, x̅∈𝒳, define the function ℓ: Θ×𝒳×{1,...,T}→ℝ such that ℓ(θ,x,t) = logp(r|θ̅,h_a^t(x̅,θ̅,π_1^t))/p(r|θ,h_a^t(x,θ,π_1^t)).Then for fixed t, the function ℓ is Lipshitz with constant L_f. Moreover, for all (x,θ) ∈𝒳×Θ and for all t,t' ∈{1,...,T} we have that |ℓ(θ,x,t) -ℓ(θ,x,t')| ≤ L_f(𝒳), where (𝒳) = max_x∈𝒳x_2.The result of this lemma can be derived using a similar argument to that of Lemma <ref>, by noting that the dynamics are bounded and Lipschitz, and then applying Assumption <ref>. The full proof of this lemma is in Appendix <ref>. Next we show the expected behavior of π is bounded.Let φ be defined as in Lemma <ref>. Then 𝔼φ({r_t}_t=1^n(𝒯_a)) ≤c_f(d_x,d_θ)/√(n(𝒯_a)), where c_f(d_x,d_θ) = 8L_f(𝒳)√(π) + 48√(2)(2)^1/d_x+d_θL_f(𝒳×Θ)√(π(d_x+d_θ)). The result of this lemma is derived by first using a symmetrization argument to bound the expectation by a Rademacher average and then using metric entropy bounds to derive the final result, and a complete proof is found in Appendix <ref>. Additional insight into these results is provided by the following remarks:The result of Lemma <ref> implies that 𝔼φ({r_a,t}_t=1^n(𝒯_a)) = 𝒪(√(d_x+d_θ/n(𝒯_a))) An improved constant can be achieved by using weaker metric entropy bounds (namely the union bound) however this would yield a bound of order 𝒪(√((d_x+d_θ)log n(𝒯_a)/n(𝒯_a))) Using the results of Lemmas <ref>–<ref>, we can complete the sketch of the proof for Theorem <ref>. Lemma <ref> says the mapping φ is L_p-Lipschitz, and combining this with Assumption <ref> implies that by Theorem 1 in <cit.> we have with probability at most exp(-ξ^2 n(𝒯_a)/2ϵ^2L_P^2σ^2) that the maximum difference between the empirical KL divergence and the true trajectory divergence is sufficiently far from its mean. Then using Lemma <ref> we obtain an upper bound on this expected value with the appropriate constants. For a complete proof of the theorem please refer to Appendix <ref>. This theorem is useful because it indicates the empirical KL divergence derived from the MLE objective converges uniformly in probability to the true trajectory KL divergence. §.§ Concentration of Trajectory DivergenceWe can complete the proof of Theorem <ref> using the results of Theorem <ref> and the definition of the MLE. First, Theorem <ref> implies that with high probability the trajectory divergence between the MLE parameters θ̂_a,x̂_a,0 and true parameters θ^*_a,x_a,0^* is within 𝒪(√(d_x+d_θ/n(𝒯_a))) of the empirical divergence between these two sets of parameters. Then, since θ̂_a,x̂_a,0 minimize the empirical divergence and the empirical divergence of θ^*_a,x^*_a,0 is zero, this means that the empirical divergence term is non-positive. Combining these two facts yields the concentration bound of Theorem <ref>, and the complete proof is given in Appendix <ref>.We conclude this section with an alternative statement of Theorem <ref>. For α∈ (0,1), with probability at least 1-α we have1/n(𝒯_a)D_a,π_1^T(θ^*_a,x^*_a,0||θ̂_a,x̂_a,0) ≤ B(α)√(log(1/α)/n(𝒯_a)).Where B(α) = c_f(d_x,d_θ)/√(log(1/α))+L_pσ√(2).This result can be obtained by making the substitution ξ = L_pσ√(log(1/α)/n(T_a)) into the expression in Theorem <ref>. This corollary is significant because it allows us to derive confidence bounds for our parameter estimates with regards to their trajectory divergence. Note that the term B(α) differs from the term that would be derived by Chernoff-Hoeffding bounds applied to i.i.d. random variables by the addition of c_f(d_x,d_θ)/√(log(1/α)) to the standard variance term. The reason for this addition is that since we are using MLE for our parameter estimation our estimates will be biased, and this bias must be accounted for in the confidence bounds. Though there may exist specific models where MLE can provide unbiased estimates, we will only present analysis for the more general case.§ POLICIES FOR OPTIMIZING ROGUE MODELS This section develops two different policies which could be used to achieve effective regret bounds for ROGUE bandit models. These are a ROGUE Upper Confidence Bounds (ROGURE-UCB) policy, and an ϵ-greedy policy (ϵ-ROGUE). While both policies exhibit good theoretical performance, each policy has different advantages during implementation, because of how they trade off exploration-exploitation and computation. ϵ-ROGUE is computationally efficient to implement, but during our experiments it tended to over-explore in the short term for many instances of ROGUE models. This may be because ϵ-greedy policies explore uniformly, and even in the stationary case ϵ-greedy policies are sensitive to their tuned parameters and do not perform well with multiple sub-optimal choices <cit.>.On the other hand, ROGUE-UCB is a more computationally intensive policy; however, in our experiments it was capable of finding optimal arms with fewer samples then ϵ-ROGUE. This makes ROGUE-UCBmore suitable for settings that require faster parameter identification such as those encountered in healthcare, while ϵ-ROGUE is better suited for low risk but high volume settings such as online advertising.Alhough several upper confidence bounds (UCB) policies have been proposed in the non-stationary setting <cit.>, these existing policies provide regret of order 𝒪(√(Tlog T)) in the general adversarial case, but can achieve 𝒪(log T) regret under certain model-based assumptions.In contrast, both the ROGUE-UCB and ϵ-ROGUE policies we construct achieve regret of order 𝒪(logT) under more general assumptions, which is optimal in the sense that it matches the lowest achievable rate for approximate policies in the stationary stochastic setting. We theoretically analyze the cumulative regret of both of these algorithms in the stochastic setting, and also develop a parameter free bound on regret for ROGUE-UCB. The algorithms we propose in this paper are designed for the stochastic ROGUE bandit setting.We leave the discussion of the adversarial ROGUE bandit setting for future research. §.§ ROGUE Upper Confidence Bounds (ROGUE-UCB) PolicyPseudocode for ROGUE-UCB is given in Algorithm <ref>, and the algorithm is written for the situation where the policy chooses actions over the course of T time periods labeled {1,...,T}.The upper confidence bounds used in this algorithm are computed using the concentration inequality from Theorem <ref>. Much like other UCB policies, for the first |𝒜| time steps of the algorithm each action a will be tried once. Then after this initialization, at each time step, we will first compute the MLE estimates of the parameters for each action (i.e., (θ_a,x̂_0,a) ∀ a∈𝒜) and then use Theorem <ref> to form the upper confidence bound on the value of g(θ_a,x_t,a), which we call g^UCB_a,t. Our approach for forming these bounds is similar to the method first proposed by <cit.> for the KL-UCB algorithm used for stationary bandits. Here, since we know that with high probability the true parameters belong to 𝒳 and Θ, we find the largest possible value of g(θ_a,x_t,a) within these sets. Finally, we choose the action that has the largest upper confidence bound, observe the result, and repeat the algorithm in the next time step. Methods for solving the component optimization problems effectively are highly dependent upon the underlying model and dynamics. In general when h_a is nonlinear these problems will be non-convex; however, the MLE problem can be computed using dynamic programming or filtering approaches while the UCB problem can be computed directly or through a convex relaxation as a relaxation solution will still be a valid UCB. In implementation, if d_x,d_θ are small, then enumeration and fine grid search methods can be employed. We leave the discussion of more efficient algorithms for specific ROGUE models for future research.The key theoretical result about the ROGUE-UCB algorithm concerns the regret R_Π(T) of the policy computed by the ROGUE-UCB algorithm. The expected regret 𝔼R_Π(T) for a policy Π computed by the ROGUE-UCB algorithm is𝔼R_Π(T) ≤ L_g(𝒳×Θ)∑_a∈𝒜(A(|𝒜|)^2 4log T/δ_a^2 + π^2/3).where A(x) = B(x^-4), and δ_a= min{1/n(𝒯_a)D_a,π_1^T(θ_a,x_a,0||θ_a',x_a',0): |g(h_a^t(x_a,0),θ_a) - g(h_a^t(x_a',0),θ_a')| ≥ϵ_a/2} ϵ_a= min_a'∈𝒜∖ a,t {|g(θ_a,h_a^t(x_a,0)) -g(θ_a',h_a^t(x_a',0))|: g(θ_a,h_a^t(x_a,0)) ≠ g(θ_a',h_a^t(x_a',0))}are finite and strictly positive constants. This corresponds to a rate of order 𝒪(log T) when lim inf_Tδ_a > 0.In fact, lim inf_Tδ_a > 0 for many settings such as (with appropriate choice of model parameter values) the ROGUE GLM and ROGUE agent defined in Section <ref>. To prove Theorem <ref>, we first present two propositions. The first proposition bounds the expected regret R_Π(T) by the number of times an action is taken while it is suboptimal.For a policy Π calculated using the ROGUE-UCB algorithm, if T̃_a = ∑_t=1^T 1{π_t = a, a ≠π^*_t}, then 𝔼R_Π(T) ≤ L_g(𝒳×Θ)∑_a∈𝒜𝔼T̃_a. For this proposition, we first use Assumption <ref> to upper bound the value of the regret with respect to the L_g and the diameter of the parameter set. Then since we are left with a finite sum of positive numbers, we can rearrange the summation term to obtain the expected number of suboptimal actions. For the detailed proof, please see Appendix <ref>. Next we proceed to prove a bound on the expected number of times a suboptimal action will be chosen. For a policy Π calculated using the ROGUE-UCB algorithm, we have that 𝔼T̃_a ≤ A(|𝒜|)^2 4log T/δ_a^2 + π^2/3, where A(t) = B(t^-4), δ_a = min{1/n(𝒯_a)D_a,π_1^T(θ_a,x_a,0||θ_a',x_a',0): |g(h_a^t(x_a,0),θ_a) - g(h_a^t(x_a',0),θ_a')| ≥ϵ_a/2}, and ϵ_a = min_a'∈𝒜∖ a,t {|g(θ_a,h_a^t(x_a,0)) -g(θ_a,h_a^t(x_a,0))|: g(θ_a,h_a^t(x_a,0)) ≠ g(θ_a,h_a^t(x_a,0))}.To prove this proposition, we proceed in a manner similar to the structure first proposed by <cit.>. We must show that if an action is chosen at a time when it is suboptimal, then this implies that either we have not properly estimated its parameters (i.e., have not explored enough) or the true values of the parameters x_a,0,θ_a or x_π^*_t,0,θ_π^*_t are not contained inside their confidence bounds. Using these facts, we use Theorem <ref> to show that the probability that all of these events occurring simultaneously is bounded, and then upper bound the expected number of times these events can occur.Combining the results of Propositions <ref> and <ref>, we thus prove the desired result of Theorem <ref>.The full proofs of Proposition <ref> and Theorem <ref> are provided in Appendix <ref>.Developing a regret bound in the ROGUE setting that is free of problem-dependent parameters is quite challenging. This is because in contrast to the stationary setting, if ϵ_a,δ_a are dependent on T and can arbitrarily approach zero in the long run, the bound presented in Theorem <ref> might not hold. This behavior, however, depends greatly on the rate at whichϵ_a,δ_a approach zero. For instance,it is clear that if ϵ_a = o(1/T), ∀ a ∈𝒜 then the bound from Theorem <ref> might in fact be loose, so to achieve the worst case rate ϵ_a,δ_a must approach zero quite slowly. In Section <ref> we provide additional discussion of how these rates may impact regret and can be computed in certain settings.In this section, we provide the following problem-dependent parameter free bound for the case when ϵ_a,δ_a are bounded from below away from zero:If there exists ϵ such that ∀ a ∈𝒜 and ∀ T ϵ_a ≥ϵ, then in the worst case ROGUE-UCB achieves𝔼R_Π(T) = 𝒪(T^4/5(|𝒜|A(|𝒜|)^2σ^2log T)^1/5). The proof of this corollary can be found in Appendix <ref>, but here we present a sketch. Essentially, if the ϵ_a are bounded from below by ϵ, then for small ϵ we would expect the regret to behave in the worst case linearly, namely at ϵ T, and not according to the 𝒪(log T) bound presented in Theorem <ref>. Thus, the result follows from finding an ϵ that minimizes both of these regret bounds and substituting into the given expressions. Unlike the stationary case, for which UCB1 has a worst case rate of 𝒪(√(Tlog T)), we note that this is a 𝒪((T^4log T)^1/5) worst case rate, and so is slightly worse but still sub-linear. This is not surprising, and can be attributed to the high variability of the problem parameters. In fact, this agrees with the bounds computed by <cit.>, who show that if total variation is proportional to T, we expect worst case regret of the form 𝒪(T^2 +β/3) for some β∈ (0,1], a rate our result meets up to a log factor. §.§ ϵ-Greedy Policy for ROGUE Bandits In this section we develop an ϵ-greedy policy that can be used to optimize ROGUE bandits. The pseudo code of this method can be found in Algorithm <ref>. At each time step t, the algorithm samples a standard uniform random variable u_t. If u_t is above some threshold ϵ_t the algorithm performs a pure exploration step (i.e. chooses an arm from 𝒜 uniformly randomly). Otherwise, the algorithm performs a greedy optimization step by computing the MLE valuesfor the expected rewards of each of the arms, and then playing the arm with the highest MLE expected reward. Note that ϵ_t decreases as the bandit is played – this is critical to ensure that as the MLE estimates improve the algorithm makes fewer unnecessary explorations. Additionally, if ϵ_t is held at some constant value ϵ, it is easy to see that this results in linear time regret, specifically a lower bound regret of Ω(ϵ T). We next show the following theoretical results for the cumulative regret performance of ϵ-ROGUE: For any time T> c|𝒜|/δ^2_min, the probability a sub-optimal arm is pulled for a policy Π computed by the ϵ-ROGUE algorithm is at most: 8L_p^2σ^2/δ^2_minexp(c_f(d_x,d_θ)^2/2L_p^2σ^2)(eTδ_min^2/|𝒜|c)^-c/8L_p^2σ^2 + c/δ_min^2(logeTδ_min^2/|𝒜|c)(eTδ_min^2/|𝒜|c)^-3c/28δ_min^2 + c/δ_min^2T Where δ_min≤min_a ∈𝒜δ_a. This instantaneous regret bound implies asymptotic cumulative regret of order 𝒪(log(T)) since the first several terms are o(1/T) for a sufficiently large choice of constant c, namely c ≥max{8L_p^2σ^2, 28δ_min^2/3}. A complete proof for Theorem <ref> can be found in Appendix <ref>, but here we provide a sketch of the proof. Essentially, to bound the probability of a sub optimal arm being pulled, we first use the union bound to bound it from above by the probability it was pulled randomly in an exploration step or during an exploitation step.We then bound the probability of subotimality in the exploration step by using the concentration bounds from Theorem <ref>.§.§ Discussion of the Behavior of ϵ_a and δ_a The constants ϵ_a and δ_a are critical for the results presented in the previous two sections. However, these quantities have a non-trivial dependence on the underlying model of the ROGUE bandit as well as the algorithm utilized, and are thus difficult to analyze in the most general case. Below we present two different examples meant to illustrate how ϵ_a and δ_a may behave for certain models and how this may impact performance.§.§.§ Exponentially Small ϵ_a,δ_aFor this section, consider a two-armed ROGUE bandit model with arms 𝒜:= {0,1} such that θ_1=θ_0=0 and r_a,t∼𝒩(x_a,t,1) ∀ a ∈𝒜, 𝒳:=[0,1], hence g(θ_a,x_a,t) = x_a,t. Define the dynamics for each arm as follows:h_0(x_0,t,π_0,t) = proj_[0,1]( 0 · x_0,t + 0 ·π_0,t) h_1(x_1,t,π_1,t) = proj_[0,1]( 0.5 · x_1,t + 0 ·π_1,t) Observe that here arm 0 stays at state x_0,t = 0 for all time periods t, while the state of arm one can be described by x_1,t = (1/2)^t x_1,0. So if x_1,0 > 0.As time progresses, the expected reward of each arm get exponentially close, and hence ϵ_a decreases exponentially. Moreover, using the KL divergence of normal distributions, it is clear thatδ_a = Ω((1/2)^2T x_1,0^2). This means that asymptotically the regret bounds provided in Sections <ref> and <ref> would no longer be of order 𝒪(log T). §.§.§ Constant Bounded ϵ_a,δ_aAs before, consider a two armed ROGUE bandit model with arms 𝒜 := {0,1} such that θ_1 = θ_0 = 0 and r_a,t∼𝒩(x_a,t,1) ∀ a ∈𝒜, 𝒳 := [0,1], and g(θ_a,t,x_a,t) =x_a,t. Furthermore define the dynamics of each arm as follows:h_0(x_0,t,π_0,t) = proj_[0,1](x_0,t + 2 ·π_0,t - 1)h_1(x_1,t,π_1,t) = proj_[0,1](x_1,t - 2 ·π_0,t + 1) Here, the dynamics are only with respect to the action taken on arm 0 since in a two armed bandit setting, by definition π_0,t = 1 - π_1,t.From these dynamics it is clear that regardless of which arm is played, whenever an action is taken the reward of of the played arm will decrease to 0 while the reward of the unplayed arm will increase to 1. Hence, for any horizon T, in this case ϵ_0 = ϵ_1 = min{1, |x_0,0 - x_1,0|} if x_0,0≠ x_1,0 and ϵ_0 = ϵ_1 = 1 otherwise. Using the KL divergence between normal distributions with different means and equal variances, this implies that δ_0 = δ_1 ≥1/8min{1, (x_0,0 - x_1,0)^2 } when x_0,0≠ x_1,0 and δ_0 = δ_1 = 1/8 otherwise. for any horizon T. This holds asymptotically, and so since δ_a,ϵ_a are bounded asymptotically, this indicates that the regret results from Sections <ref> and <ref> will hold and regret will be of order 𝒪(log T).§ NUMERICAL EXPERIMENTS In this section, we perform two numerical experiments where the policies computed by the ROGUE-UCB and ϵ-ROGUE algorithms are compared against the policies computed by other non-stationary bandit algorithms. The first experiment considers the ROGUE GLM described in Section <ref>, andspecifically looks at the logistic regression instantiation of ROGUE GLM. We use synthetically generated data for this first experiment. Next, we perform an experiment in the context of healthcare-adherence improving interventions to increase physical activity, which can be modeled using the ROGUE agent from Section <ref>. Using real world data from the mDPP trial <cit.>, we show how ROGUE-UCB and ϵ-ROGUE can be implemented to personalize messages for participants in this intervention. All experiments in this section were run using Python 3.5.2 and Anaconda on a laptop computer with a 2.4GHz processor and 16GB RAM.Our goal with these experiments is to show how each of the bandit algorithms performs in terms of long run reward and how quickly the algorithms can learn the underlying system state. While these problems are equivalent in the stationary bandit setting, in the non-stationary setting incorrect actions taken due to improper system identification may still have rewards that are quite close to optimal. As such, we consider two different performance measures for each algorithm, the tracking regret which measures how quickly the unknown model parameters are identified, and the long run reward. Our results show that both the Tuned ROGUE-UCB and ϵ-ROGUE algorithms outperform the other candidate algorithms both in terms of long run average reward and average regret. However, while both ROGUE algorithms have similar performance in the long run, ROGUE-UCB tends to identify model parameters faster then ϵ-ROGUE as it performs less exploration but at a slightly greater computational cost at each time step. This suggests that ROGUE-UCB is most useful for settings that require fast identification, such as settings that require higher consideration for risk and fairness (e.g., healthcare settings), while ϵ-ROGUE is ideal for low risk settings where computational resources are the main implementation constraint (e.g., targeted advertising).§.§ Tuned ROGUE-UCBAs has been noted for other UCB policies <cit.>, the high probability bounds derived theoretically for these methods are often too conservative. While the 𝒪(√(logt/n(𝒯_a))) is a tight rate, the term A(t) is too conservative. Drawing inspiration from <cit.> who used asymptotic bounds for Tuned UCB, we similarly construct a variant of our algorithm: This variant is described in Algorithm <ref> and called Tuned ROGUE-UCB. Using the results of <cit.>, we note that if the MLE θ̂_a,x̂_a,0 are in the interior of the feasible region and are consistent, then they are asymptotically normally distributed with a variance equal to their Fisher information. Using these results and the delta method <cit.>, we can derive the quantity 𝒮_a,π_1^T(θ_a,x_a,0||θ̂_a,x̂_a,0) = 1/n(𝒯_a)^2∇_θ',x'D_a,π_1^T(θ_a,x_a,0||θ',x')^T ℐ_{r_t}_t∈𝒯_a(θ',x')^-1∇_θ',x'D_a,π_1^T(θ_a,x_a,0||θ',x')|_θ',x'=θ̂_a,x̂_a,0, which is the asymptotic variance of the average trajectory KL-Divergence.Here, η is a constant that corresponds to the maximum value of the KL-divergence; ℐ_{r_t}_t∈𝒯_a(θ',x') represents the observed trajectory Fisher information, which can be calculated as ℐ_{r_t}_t∈𝒯_a(θ',x') = ∑_t∈𝒯_aℐ_r_t(θ',x'), due to Assumption <ref>. As an implementation note, if the empirical information matrix is singular, then the Moore-Penrose pseudoinverse should be used to achieve similar asymptotic results <cit.>. Note that although these asymptotic bounds work well in practice, they are not high probability bounds and do not provide the same theoretical guarantees as the ROGUE-UCB algorithm. A full analysis of regret for Tuned ROGUE-UCB is beyond the scope of this work. Instead, we only consider empirical analysis of this algorithm to show its strong performance. §.§ Experimental DesignWe examined two settings for our experiments, which correspond to the instantiations of ROGUE bandits presented in Sections <ref> and <ref>. For each of the scenarios, we compared the Tuned ROGUE-UCB algorithm and ϵ-ROGUE algorithms with ϵ_t = 1/t to policies determined by six alternative methods. For each scenario, we present two result metrics: cumulative regret of each algorithm in that scenario and the average reward to date of the algorithm. While these two measures are related, a key difference is that in the non-stationary setting sub-optimal actions may not have a significantly lower expected reward than the optimal action at all time periods. Hence, while an algorithm may incur a significant amount of regret it could still achieve a high amount of reward. The five alternative algorithms we used for comparison are as follows: * Pure Exploration: First, we considered a completely random, or “pure exploration” algorithm, which chooses an action uniformly at random from the set of available actions. * Stationary Upper Confidence Bound (UCB1): Next, we considered the UCB1 algorithm <cit.>, which is designed for stationary bandits. This approach uses the sample average as an estimate of the expected reward of each action and utilizes a padding upper confidence bound term derived from Hoeffding's bound. In our experiments, we implemented Tuned UCB1 <cit.>, which replaces the theoretical constants by the asymptotic variance of the sample average and a small constant that corresponds to the maximum variance of a Bernoulli random variable (since the rewards are bounded between 0 and 1).* Discounted Upper Confidence Bounds (D-UCB): D-UCB is an upper confidence bound approach designed for non-stationary systems. It utilizes an exponentially weighted average of the reward observations to estimate the expected reward at the current time period and a square root padding function to provide upper confidence bounds <cit.>. The weighted average is constructed with a positive discount factor that decreases the influence of older observations on the reward estimate to zero as time goes on. We implemented this algorithm with its optimal theoretical parameters, as described in <cit.>. * Sliding Window Upper Confidence Bounds (SW-UCB): The next approach we considered is the SW-UCB approach. This algorithm considers a fixed window size of how many action choices to “keep in memory”, and computes the estimate of the expected action rewards as the average of these choices <cit.>. We implemented this algorithm with its optimal theoretical parameters as proposed by <cit.>.* Exploration and Exploitation with Exponential Weights (EXP3): The next bandit algorithm we considered in our experiments is the EXP3 algorithm. Essentially, EXP3 is a modification of the exponential weights algorithm used in online optimization to the bandit setting where not all action rewards are observed <cit.>. Though EXP3 is designed for stationary bandits, unlike UCB approaches that assume a stochastic setting, it is meant foradversarial bandits, which makes it potentially robust to non-stationarity. The particular variant of EXP3 we utilized is EXP3.S proposed by <cit.>, which is designed for arbitrary reward sequences, using the theoretically optimal parameters as proposed by the authors. Specifically, the learning rate γ was set to min{1,√(|𝒜|/Tlog(|𝒜|T))} and prior weighting α was set to 1/T. Essentially, this parameter tuning assumes the full horizon the bandit is to be run for is known.* Restarting EXP3 (REXP3): The last bandit algorithm we considered in our experiments is the REXP3 algorithm proposed by <cit.>. This algorithm is a modification of the EXP3 algorithm for a non-stationary setting, where the none-stationarity has bounded total variation. This approachperforms EXP3 update steps for a preset number of arm pulls, and once this number is reached, the algorithm resets as if it is restarting. The length of the period before the reset depends on the total time horizon of the bandit, as well as the total variation of the underlying reward. In our experiments, we tuned this algorithm using various values for the total variation and found that the best results overall were achieved by setting this value to be proportional to the total horizon T, so we only present these results. §.§ ROGUE Logistic RegressionFor this experiment, we consider the logistic regression instantiation of the ROGUE GLM presented in Section <ref>.Our setup includes two actions whose rewards r_t,a are Bernoulli with a logistic link function of the form g(x,θ) = 1/1+ exp(-aθ - b x). The initial parameters and dynamics matrices for each of the actions are presented in Table <ref>. Here, the sets 𝒳 and Θ were set to [0,1]. Action 0 has a significant portion of its reward dependent on the time varying state x_t, and recovers its reward slowly but also decreases slowly. On the other hand, Action 1 has more of its expectation dependent on the stationary component θ, but it expectation decreases faster than that of Action 0. The experiments were run for 20,000 action choices and replicated 30 times for each of the candidate algorithms. Figure <ref> shows the cumulative regret accrued by each of the algorithms averaged across the replicates, and Figure <ref> shows the average reward per action for each algorithm averaged across the replicates. As expected in these experiments, the UCB1 algorithm achieves linear regret since it assumes a stationary model and thus converges to a single action, which causes a large gap between the expectations of the two actions. Interestingly, SW-UCB and D-UCB also perform worse than random choices. A key note here is that D-UCB and SW-UCB assume that action rewards do not change frequently and are independent of the choices. However, D-UCB outperforms SW-UCB since the weighted average contains more information about the trajectory of the expected reward of each action while data from earlier choices are removed from the estimates in the sliding window. EXP3, REXP3, and random action selection perform approximately the same in terms of both regret and expected reward. This is unsurprising because the weighting scheme in EXP3 emphasizes the rewards of the past action states as opposed to current action states, and high total variation essentially reduces the restart horizon of REXP3 which means it retains less information about the overall reward trajectory. In terms of both regret and reward, Tuned ROGUE-UCB and ϵ-Rogue substantially outperform the other approaches.In Figures <ref> and <ref> we compare the two model-based approaches in terms of their cumulative regret and average reward, respectively. While the other approaches seem to obtain linear regret, both ROGUE-UCB and ϵ-ROGUE do in fact have regret on the order of 𝒪(log T) in this experiment. Moreover, in this particular non-stationary setting we observe that these two algorithms have very similar long run average rewards while ROGUE-UCB obtains slightly better cumulative regret. This can be attributed to ROGUE-UCB performing less exploration then ϵ-ROGUE.§.§ Healthcare-Adherence Improving Intervention for Increasing Physical ActivityNext, we consider an experiment using real world data from the mobile diabetes prevention program (mDPP) <cit.>. This was a randomized control trial (RCT) that was conducted to evaluate the efficacy of a 5 month mobile phone based weight loss program among overweight and obese adults at risk for developing type 2 diabetes and was adapted from the Diabetes Prevention Program (DPP) <cit.>. Sixty one overweight/obese adults were randomized into an active control group that only received an accelerometer (n=31) or a treatment group that received the mDPP mobile app plus the accelerometer and clinical office visits (n=30). Changes in primary and secondary outcomes for the trial were clinically and statistically significant. The treatment group lost an average of 6.2 ± 5.9 kg (-6.8% ± 5.7%) between baseline and the 5 month follow up while the control group gained 0.3 ± 3.0 kg (0.3% ± 5.7 %) (p < 0.001). The treatment group's steps per day increased by 2551 ± 4712 compared to the control group's decrease of 734 ± 3308 steps per day (p < 0.001). Additional details on demographics and other treatment parameters are available in <cit.>.One key feature of the mDPP application was the ability for the clinicians to send daily messages to the participants to encourage that they adhere to the intervention and maintain a sufficiently increased activity level. Broadly speaking, there were 5 different message categories that the clinicians could choose to send to the patients. These categories are self-efficacy/confidence, motivation/belief/attitude, knowledge, behavior reinforcement, and social support. Each day the experimental group would receive a preprogrammed message from one of these categories, and all participants received the same messages each day. For our simulations, we used the data of what messages were sent to what participants, as well as their daily step counts.§.§.§ Patient ModelFor our experiment, we used a behavioral analytics model of patient behavior first proposed by <cit.>. Here, each patient is assumed to be a utility maximizing agent who chooses how many steps to take each day based on previous behavior and the intervention implemented. We defined each of the different message categories be one of the actions of the bandit, which forms a ROGUE agent model as described in Section <ref>. Using the notation of Section <ref>, let c_t be a sequence of i.i.d. Laplace random variables with mean zero and shape parameter θ. This means σ^2(θ) = 2θ^2. After normalizing the step counts to be in [0,1] (where 1 is equal 14,000 steps), we can then write the reward distribution of a particular message type a as p(r_t|{x_a,t,θ_a}) = 1/2exp(-x_a,t/θ_a) δ(r_t) + 1/2exp(x_a,t-1/θ_a) δ(1-r_t) + 1/2θ_aexp(-|r_t-x_t|/θ_a)1[r_t ∈ (0,1)], where the state x_a,t∈ [0,1] and θ_a ∈ [ϵ,1] for a small ϵ >0. This results in a reward function g(x,θ) = x + θ/2(exp(-x/θ)- exp(x-1/θ)). Using Laplace noise has the advantage of allowing commercial mixed integer programming solvers to be used for offline parameter estimation by solving inverse optimization problems <cit.>. Using this MILP reformulation and behavioral models, we estimated the respective trajectory parameters for each message group and each patient of the treatment group for which we had data. These initial parameters were found using the Gurobi Solver in Python <cit.>.§.§.§ Simulation ResultsThis simulation was conducted using the mDPP data described above. Each experiment consisted of 1,000 action choices, which would correspond to about two years of a message based physical activity intervention, and 10 replicates of the simulation were conducted per patient and algorithm. The results in Figures <ref> and <ref> represent averages across all patients and replicates. Since we are using real data, the interpretation of the y-axis of each of the plots corresponds to number of steps in units of 1,000 steps, and the x-axis corresponds to the day of the intervention.ROGUE-UCB and ϵ-ROGUE outperform all other algorithms both in terms of regret and average reward. In terms of regret, both ROGUE-UCB and ϵ-ROGUE obtain logarithmic-in-time regret; however, as before, ROGUE-UCB achieves lower cumulative regret than ϵ-ROGUE. While D-UCB is the only other algorithm that can outperform pure exploration, it only obtains linear regret. In terms of average reward, ROGUE-UCB, ϵ-ROGUE, and D-UCB are the only algorithms that outperform pure exploration. Interpreting these results in the healthcare context of this intervention, we find that the improved predictive model and use of MLE estimates within our specialized ROGUE Bandit algorithms results in an increase of 1,000 steps a day (approximately a half-mile more of walking per day) relative to the next best algorithm, which is a significant increase in activity. We note that in terms of average reward, both ROGUE-UCB and ϵ-ROGUE have similar performance; however, ROGUE-UCB has higher initial rewards due to fewer exploration steps. This property is of particular interest in this healthcare setting, as it insures that each participating individual receives the most appropriate recommendations at a faster rate.§ CONCLUSIONIn this paper, we defined a new class of non-stationary bandit models where the specific actions chosen influence the reward distributions of each action in subsequent time periods through a specific model. We conducted a finite sample analysis of the MLE estimates in this setting, and derived finite time concentration bounds. Using this analysis, we developed both the ROGUE-UCBand ϵ-ROGUE algorithms that provide policies for these bandit models. Our theoretical results show that in expectation ROGUE-UCB and ϵ-ROGUE achieve logarithmicin time regret. This is a substantial improvement over model-free algorithms, which can only achieve a square-root regret in the worst case scenario. We then showed through simulations using real and artificial data, that with minor modification, the ROGUE-UCB and ϵ-ROGUE algorithms significantly outperform state of the art bandit algorithms both in terms of cumulative regret and average reward. These results suggest that ROGUE bandits have strong potential for personalizing health care interventions, and in particular for healthcare-adherence improving interventions. The authors gratefully acknowledge the support of NSF Award CMMI-1450963, UCSF Diabetes Family Fund for Innovative Patient Care-Education and Scientific Discovery Award, K23 Award (NR011454), and the UCSF Clinical and Translational Science Institute (CTSI) as part of the Clinical and Translational Science Award program funded by NIH UL1 TR000004.informs2014 Yonatan Mintz: Yonatan is a Postdoctoral Research Fellow at the H. Milton Stewart School of Industrial and Systems Engineering at the Georgia Institute of Technology, previously he completed his PhD at the department of Industrial Engineering and Operations research at the University of California, Berkeley. His research interests focus on the application of machine learning and optimization methodology for personalized healthcare and fair and accountable decision making.Anil Aswani: Anil is an assistant professor in the Department of Industrial Engineering and Operations Research at the University of California, Berkeley. His research interests include data-driven decision making, with particular emphasis on addressing inefficiencies and inequities in health systems and physical infrastructure.Philip Kaminsky Philip is the Earl J. Isaac Professor in the Science and Analysis of Decision Making in the Department of Industrial Engineering and Operations Research at UC Berkeley, where he previously served as Executive Associate Dean of the College of Engineering, faculty director of the Sutardja Center for Entrepreneurship and Technology, and department chair of Industrial Engineering and Operations Research.His research focuses primarily on the analysis and development of robust and efficient tools and techniques for design, operation, and risk management in logistics systems and supply chains. He consults in the areas of production planning, logistics, and supply chain management., and prior to his graduate education, he worked in production engineering and control at Merck & Co.Elena Flowers Elena is an Associate Professor in the Department of Physiological Nursing and Institute for Human Genetics at the University of California, San Francisco. Her program of research is focused on the relationships between risk factors, risk-reduction interventions, and molecular biomarkers for chronic diseases, namely type-2 diabetes and preterm birth, in high risk racial groups.Yoshimi Fukuoka Yoshimi, PhD, RN, FAAN is a professor in the Department of Physiological Nursing at the University of California, San Francisco. Her research focuses on the primary prevention of cardiovascular disease and type 2 diabetes using digital technologies and artificial intelligence. § PROOFS OF PROPOSITIONS IN TEXT Proof of Proposition <ref>:To obtain this formulation first we can augment the objective function of the log-likelihood problem by adding the constant term log∑_t∈𝒯_ap(r_a,t|θ^*_a,x^*_a,t) and multiplying by the positive constant 1/n(𝒯_a) which does not change the value of the optimal solution. Next we use functional compositions to contract the dynamics and obtain an objective function which is explicitly a function of θ_a,x_0,a. Proof of Lemma <ref>:We can see that this is the case by noting that by Assumption <ref> we have that each of the log-likelihood ratios are Lipschitz with constant L_p. Since Lipschitz continuity is preserved by addition and averaging we note that the average of all of these log-likelihood ratios is also L_p-Lipschitz. Next we use the property that functional compositions of Lipschitz functions are Lipschitz with a constant equal to the product of their respective constants and the Lipschitz continuity is preserved through point wise maxima <cit.>. Since the absolute value function is 1-Lipschitz and we are performing maximization we have that ϕ is indeed L_p-Lipschitz with respect to the input sequence. Proof of Lemma <ref>: To show the first result we use a similar argument to that of the proof of Lemma <ref> by showing that the likelihood is Lipschitz and then using the preservation of Lipschitz continuity across functional compositions. First consider h_a^t(x), since by Assumption <ref> h_a is locally L_h Lipschitz such that L_h ≤ 1, hence we have that with respect to x,x' ∈𝒳 h_a^t(x) - h_a^t(x')_2 < x - x'_2.Hence h_a^t(x) is locally 1-Lipschitz continuous with respect to x,t. Next, applying Assumption <ref> shows that since the likelihood ratio is L_f-Lipschitz with respect to its two inputs we simply have a composition of Lipschitz functions and the result follows. To show the second result note that ℓ depends on t only through the composite dynamics mapping h_a^t. By definition h^t(x) ∈𝒳 which is a bounded set, we have that for any t,t' ∈{1,...,T} h_a^t(x) - h_a^t'(x)_2 ≤(𝒳), thus using Assumption <ref> we obtain the desired result.Proof of Lemma <ref> To prove this result we first bound the expectation by a Rademacher average <cit.> and then apply Dudley's Integral bound <cit.>. First let us consider the explicit form of 𝔼φ({r_a,t}_t=1^n(𝒯_a)). Using an identically distributed sequence of rewards {r'_a,t}_t=1^n(𝒯_a) which is independent of the observed sequence we see that𝔼sup_θ_a,x_a,0∈Θ×𝒳|1/n(𝒯_a)∑_t∈𝒯_alog p(r_a,t|θ^*_a,h_a^t(x^*_a,0,θ^*_a,π_1^t))/p(r_a,t|θ_a,h_a^t(x_a,0,θ_a,π_1^t)) - 1/n(𝒯_a)D_a,π_1^T(θ^*_a,x^*_a,0||θ_a,x_a,0) |=𝔼sup_θ_a,x_a,0∈Θ×𝒳|1/n(𝒯_a)𝔼[∑_t∈𝒯_alog p(r_a,t|θ^*_a,h_a^t(x^*_a,0,θ^*_a,π_1^t))/p(r_a,t|θ_a,h_a^t(x_a,0,θ_a,π_1^t)) - ∑_t∈𝒯_alog p(r'_a,t|θ^*_a,h_a^t(x^*_a,0,θ^*_a,π_1^t))/p(r'_a,t|θ_a,h_a^t(x_a,0,θ_a,π_1^t))|{r_a,t}_t=1^n(𝒯_a)] |≤𝔼sup_θ_a,x_a,0∈Θ×𝒳|1/n(𝒯_a)(∑_t∈𝒯_alog p(r_a,t|θ^*_a,h_a^t(x^*_a,0,θ^*_a,π_1^t))/p(r_a,t|θ_a,h_a^t(x_a,0,θ_a,π_1^t)) - ∑_t∈𝒯_alog p(r'_a,t|θ^*_a,h_a^t(x^*_a,0,θ^*_a,π_1^t))/p(r'_a,t|θ_a,h_a^t(x_a,0,θ_a,π_1^t))) |.Here the inequality follows from Jensen's Inequality <cit.>. Let {ϵ_t}_t=1^n(𝒯_a) be a sequence of i.i.d. Rademacher random variables, which are independent of the observations r_a,t,r'_a,t, then through a symmetrization argument its clear that𝔼φ({r_a,t}_t=1^n(𝒯_a)) ≤ 2𝔼sup_θ_a,x_a,0∈Θ×𝒳|1/n(𝒯_a)∑_t∈𝒯_aϵ_tlog p(r_a,t|θ^*_a,h_a^t(x^*_a,0,θ^*_a,π_1^t))/p(r_a,t|θ_a,h_a^t(x_a,0,θ_a,π_1^t))|.Since x^*_a,0,θ^*_a are constants we can use simplify the above expression using the notation introduced in Lemma <ref> to 2𝔼sup_θ_a,x_a ∈Θ×𝒳|1/n(𝒯_a)∑_t∈𝒯_aϵ_tℓ(θ_a,x_0,a,t)|.We can bound this expression as follows2𝔼sup_θ_a,x_a ∈Θ×𝒳|1/n(𝒯_a)∑_t∈𝒯_aϵ_tℓ(θ_a,x_0,a,t)|, = 2𝔼sup_θ_a,x_a ∈Θ×𝒳|1/n(𝒯_a)∑_t∈𝒯_aϵ_t(ℓ(θ_a,x_0,a,t)-ℓ(θ_a,x_0,a,0)+ℓ(θ_a,x_0,a,0))|, ≤ 2𝔼sup_θ_a,x_a ∈Θ×𝒳|1/n(𝒯_a)∑_t∈𝒯_aϵ_t(ℓ(θ_a,x_0,a,t)-ℓ(θ_a,x_0,a,0))|+ 2𝔼sup_θ_a,x_a ∈Θ×𝒳|1/n(𝒯_a)∑_t∈𝒯_aϵ_tℓ(θ_a,x_0,a,0)|. For our analysis we can consider each of these terms separately and bound them using Dudley's Integral Bound <cit.> and Lemmas <ref>,<ref>. Consider the first term, note that by Lemma <ref> we have that |ℓ(θ_a,x_0,a,t)-ℓ(θ_a,x_0,a,0)| ≤ L_f(𝒳) and is contained in an ℓ_2 ball of this radius, hence by Lemma <ref>2𝔼sup_θ_a,x_a ∈Θ×𝒳|1/n(𝒯_a)∑_t∈𝒯_aϵ_t(ℓ(θ_a,x_0,a,t)-ℓ(θ_a,x_0,a,0))|,≤ 8 ∫_0^L_f(𝒳)√(log𝒩(L_f(𝒳)B_2,α,_2)/n(𝒯_a))dα≤ 8L_f(𝒳)√(π/n(𝒯_a)).The last inequality follows from usinga volume bound on the covering number and using integration by parts. Next consider the second term in (<ref>), we can bound this term using a direct application of Dudley's entropy integral as follows2𝔼sup_θ_a,x_a ∈Θ×𝒳|1/n(𝒯_a)∑_t∈𝒯_aϵ_tℓ(θ_a,x_0,a,0)| ≤ 16√(2)∫_0^∞√(log2𝒩(α,ℓ(Θ×𝒳),_2)/n(𝒯_a))dα,≤16√(2)∫_0^∞√(log2𝒩(α/L_f,Θ×𝒳,_2)/n(𝒯_a))dα.Let v_ℓ B_2 be the ℓ_2 ball on ℝ^d_x+d_θ with radius v_ℓ = (𝒳×Θ), then(<ref>)≤ 16√(2)∫_0^∞√(log2𝒩(α/L_f,B_ℓ,_2)/n(𝒯_a))dα≤ 16√(2)∫_0^∞√(log2(3v_ℓ L_f/α)^d_x+d_θ/n(𝒯_a))dαSolving the integral shows that (<ref>) ≤ 48√(2)(2)^1/d_x+d_θL_fv_ℓ√(π(d_x+d_θ)/n(𝒯_a)). Hence the result follows. Proof of Theorem <ref>:Lemma <ref> guarantees that the mapping φ is Lispschitz continuous with respect to the observed rewards with parameter L_p, furthermore we have by Assumption <ref> that the reward distributions are sub-Gaussian with parameter σ^2. By applying Theorem 1 from <cit.> we obtain for ξ >0:ℙ(φ({r_t}_t=1^n(𝒯_a)) - 𝔼φ({r_t}_t=1^n(𝒯_a)) > ξ) ≤exp(-ξ^2n(𝒯_a)/2L_p^2σ^2). Hence, using the upper bound obtained from Lemma <ref>, we can substitute the result into the above equation giving the desired result. Proof of Theorem <ref>:Using Theorem <ref> we know that with probability at least 1-exp(-ξ^2n(𝒯_a)/2L^2_pσ^2) we have:1/n(𝒯_a)D_a,π_1^T(θ^*_a,x^*_a,0||θ̂_a,x̂_a,0) - 1/n(𝒯_a)∑_t∈ n(𝒯_a)logp(r_a,t|θ_a^*,h_a^t(x^*_a,0,θ^*_a,π_1^t))/p(r_a,t|θ̂_a,h_a^t(x̂_a,0,θ̂_a,π_1^t))≤c_f(d_x,d_θ)/√(n(𝒯_a)) +ξ.Also since θ̂_a,x̂_a are minimizers of the empirical trajectory divergence implies that1/n(𝒯_a)∑_t∈ n(𝒯_a)logp(r_a,t|θ_a^*,h_a^t(x^*_a,0,θ^*_a,π_1^t))/p(r_a,t|θ̂_a,h_a^t(x̂_a,0,θ̂_a,π_1^t))≤1/n(𝒯_a)∑_t∈ n(𝒯_a)logp(r_a,t|θ_a^*,h_a^t(x^*_a,0,θ^*_a,π_1^t))/p(r_a,t|θ^*_a,h_a^t(x^*_a,0,θ^*_a,π_1^t)) = 0.Hence the desired result follows. Proof of Proposition <ref>:Recall that by definition 𝔼R_Π(T) = ∑_t=1^Tg(θ_pi^*_t,x_pi^*_t) - g(θ_pi_t,x_pi_t). Since by Assumption <ref> we have that g is L_g-Lipschitz then we have ∀ t that g(θ_pi^*_t,x_pi^*_t) - g(θ_pi_t,x_pi_t) ≤ L_g(θ_pi^*_t,x_pi^*_t) - (θ_pi_t,x_pi_t)≤ L_g(𝒳×Θ)ℙ(π_t ≠π^*t). Hence𝔼R_Π(T)≤ L_g (𝒳×Θ) ∑_t=0^T ℙ(π_t ≠π_t^*) = L_g (𝒳×Θ) ∑_t=0^T ∑_a∈𝒜ℙ(π_t = a, a ≠π_t^*) = L_g (𝒳×Θ)∑_a∈𝒜∑_t=0^T ℙ(π_t = a, a ≠π_t^*) = L_g (𝒳×Θ) ∑_a∈𝒜𝔼T̃_aProof of Proposition <ref>:We proceed to prove this proposition in a similar method to that presented in <cit.>. Suppose that at time t, the ROGUE-UCB policy chooses a ≠π^*_t. If the upper confidence bounds hold then we observe that g_a,t^UCB≥ g_π^*_t,t^UCB≥ g_π^*_t,t. Also define the mapping ψ_a(γ) = max{|g(θ,h_a^t(x_0)) - g(θ̂_a,h_a^t(x̂_a,0))|: 1/n(𝒯_a) D_a,π_1^T(θ,x_0||θ̂_a,x̂_a,0) ≤γ}. Then clearly g_a,t^UCB - g(θ̂_a,h_a^t(x̂_a,0)) ≤ψ_a(A(t)√(4log(t)/n(𝒯_a))) and g(θ̂_a,h_a^t(x̂_a,0)) - g_a,t≤ψ_a(A(t)√(4log(t)/n(𝒯_a))). Hence we have that g_a,t^UCB≤ 2ψ(A(t)√(4log(t)/n(𝒯_a))) + g_a,t. Therefore ψ(A(t)√(4log(t)/n(𝒯_a))) ≥1/2(g_π^*_t,t - g_a,t). By definition of ϵ_a we thus have that ψ(A(t)√(4log(t)/n(𝒯_a))) ≥ϵ_a/2. Therefore, by definition of δ_a we observe that A(t)√(4log t/n(𝒯_a))≥δ_a and hence n(𝒯_a) ≤4A(t)^2log t/δ_a^2.Now, consider T̃_a:T̃_a = ∑_t=1^T 1{π_t = a, a ≠π^*_t} = ∑_t=1^T1{π_t = a, a ≠π^*_t, n(𝒯_a) ≤4A(t)^2log t/δ_a^2} + ∑_t=1^T1{π_t = a, a ≠π^*_t, n(𝒯_a) > 4A(t)^2log t/δ_a^2}≤∑_t=1^T1{π_t = a, a ≠π^*_t, n(𝒯_a) ≤4A(|𝒜|)^2log T/δ_a^2} +∑_t=1^T1{π_t = a, a ≠π^*_t, n(𝒯_a) > 4A(t)^2log t/δ_a^2}≤4log(T)/δ_a^2 A(|𝒜|)^2 + ∑_t=1^T1{π_t = a, a ≠π^*_t, n(𝒯_a) > 4A(t)^2log t/δ_a^2}Observe that if we play sub optimal action a at time t this means we either severely over estimate the value of g_a,t, severely under estimate the value of g_π_t^*,t, or the two values are very close to each other. Hence {π_t=a, a≠π^*_t, n(𝒯_a)> 4A(t)^2log t/δ_a^2}⊆{g_a,t^UCB - g_a,t > 2 ψ_a(A(t)√(4log(t)/n(𝒯_a))), n(𝒯_a) >4A(t)^2log t/δ_a^2}_(a) ∪{g_π^*_t,t >g^π_t^*,t_UCB, n(𝒯_a) >4A(t)^2log t/δ_a^2}_(b)∪{g_π^*_t,t - g_a,t≤ 2ψ_a(A(t)√(4log(t)/n(𝒯_a))), n(𝒯_a) >4A(t)^2log t/δ_a^2}_(c).However, as we established in the beginning of the proof the event (c) = ∅. Also note that for events (a),(b) to occur this would imply that θ_a,x_a,0 and θ_π_t^*,x_π_t^*,0 are not feasible points of their respective UCB deriving problems, hence {π_t=a, a≠π^*_t, n(𝒯_a)> 4A(t)^2log t/δ_a^2}⊆{∃ s<t : 1/sD_π^*_t,π_1^s(θ̂_π^*_t,x̂_π^*_t,0||θ_π^*_t,x_π^*_t,0) > A(t)√(4log(t)/s)}∪{∃ s'<t:1/s'D_a,π_1^s'(θ̂_a,x̂_a,0||θ_a,x_a,0) > A(t)√(4log(t)/s')}⊆⋃_1≤ s < t{1/sD_π^*_t,π_1^s(θ̂_π^*_t,x̂_π^*_t,0||θ_π^*_t,x_π^*_t,0) > A(t)√(4log(t)/s)}⋃_1≤ s' < t{1/s'D_a,π_1^s'(θ̂_a,x̂_a,0||θ_a,x_a,0) > A(t)√(4log(t)/s')}.Taking the expected value of T̃_a we obtain 𝔼T̃_a ≤4log(T)/δ_a^2 A(|𝒜|)^2 + 𝔼∑_t=1^T1{π_t = a, a ≠π^*_t, n(𝒯_a) > 4A(t)^2log t/δ_a^2}≤4log(T)/δ_a^2 A(|𝒜|)^2 + ∑_t=1^T∑_s=1^t∑_s'=1^tℙ(1/sD_π^*_t,π_1^s(θ̂_π^*_t,x̂_π^*_t,0||θ_π^*_t,x_π^*_t,0) > A(t)√(4log(t)/s))+ ∑_t=1^T∑_s=1^t∑_s'=1^tℙ(1/s'D_a,π_1^s'(θ̂_a,x̂_a,0||θ_a,x_a,0) > A(t)√(4log(t)/s'))≤4log(T)/δ_a^2 A(|𝒜|)^2 +2∑_t=1^T∑_s=1^t∑_s'=1^t t^-4≤4log(T)/δ_a^2 A(|𝒜|)^2 + π^2/3. Here the third inequality is derived by Theorem <ref> and the final inequality by utilizing the solution to the Basel Problem <cit.>. Hence we obtain the desired result. Proof of Theorem <ref>:Using Proposition <ref> we bound the expected regret as 𝔼R_Π(T) ≤ L_g(𝒳×Θ)∑_a∈𝒜𝔼T̃_a. Then applying the result of Proposition <ref> we obtain the desired result. Proof of Corollary <ref>:To obtain the desired rate we must first express the bound from Theorem <ref> in terms of ϵ. We can do this by noting that from Assumption <ref> we know that the reward distributions are sub-Gaussian with parameter σ, therefore using the property that exponential families achieve the maximum entropy given moment constraints <cit.>, we observe that ∀θ'_a,x'_a,0∈Θ×𝒳: 1/n(𝒯_a)D_a,π_1^T(θ_a,x_a,0||θ'_a,x'_a,0) ≥1/2σ n(𝒯_a)∑_t ∈𝒯(g(θ_a,x_a,t)- g(θ'_a,x'_a,t))^2 ≥1/2σϵ_a^2 Thus applying the definition of δ_a we observe that in fact δ_a ≥1/2σϵ_a^2. Hence substituting into the bound from Theorem <ref> we obtain the following weaker regret bound in terms of the ϵ_a: 𝔼R_Π(T) ≤ L_g(𝒳×Θ)∑_a∈𝒜(A(|𝒜|)^2 16σ^2log T/ϵ_a^4 + π^2/3). Note that this bound is convex and monotonically decreasing in the ϵ_a, and in fact is minimized if ϵ_a = max_(θ_1,x_1),(θ_2,x_2) ∈Θ×𝒳 |g(θ_1,x_1) - g(θ_2,x_2)| for all a ∈𝒜. Hence all ϵ_a will be equal to each other when this bound is minimized. However, this bound will go to infinity as the ϵ_a become small. Note though that since by assumption, ϵ_a ≥ϵ and hence as ϵ_a become small we would expect our regret to be at a rate of ϵ T. Using these two notions we can thus conclude that our regret can be bounded as: 𝔼R_Π(T) ≤min{ L_g(𝒳×Θ)|𝒜| (A(|𝒜|)^2 16σ^2log T/ϵ^4 + π^2/3),ϵ T} Since one of these expressions increases in ϵ and the other decreases in ϵ we can minimize the right hand side by setting both parts of the min to be equal to each other. This shows that the minimizing epsilon can be given as a zero to the following quintic polynomial: Tϵ^5 - L_g(𝒳×Θ)|𝒜|π^2/3ϵ^4 - L_g(𝒳×Θ)|𝒜|A(|𝒜|)^216σ^2log T Since the zeros of this polynomial are difficult to compute we can use the Fujiwara bound on polynomial zeros <cit.> to show that for sufficiently large T, ϵ = 𝒪(( |𝒜|A(|𝒜|)^2σ^2log T/T)^1/5), and hence the desired result follows from substituting this value into Equation (<ref>). Proof of Theorem <ref>:We proceed to prove this theorem using similar analysis to that provided in <cit.>. The probability that an arm a is played sub-optimally by the ϵ-ROGUE algorithm is given by:ℙ(π_t=a,a≠π^*_t) = ϵ_t/|𝒜| + (1-ϵ_t)ℙ(g(x̂_a,t,θ̂_a) ≥ g(x̂_π_t^*,t,θ̂_π_t^*))Where the first term coms from the probability of performing an exploration step and the second term corresponds to an exploration step. First lets consider the exploitation term, define Δ_a,t = g(x_π_t^*,t,θ_π^*_t) - g(x_a,t,θ_a), using this value we can upper bound the exploitation term as follows: ℙ(g(x̂_a,t,θ̂_a) ≥ g(x̂^*_a,t,θ̂^*_a))≤ℙ( (g(x̂_a,t,θ̂_a) - g(x_a,t,θ_a)) - (g(x̂_π*_t,t,θ̂_π^*_t) - g(x_π^*_t,t,θ_π^*_t)) ≥Δ_a,t) ≤ℙ(g(x̂_a,t,θ̂_a) - g(x_a,t,θ_a) ≥Δ_a,t/2) + ℙ( g(x̂_π*_t,t,θ̂_π^*_t) - g(x_π^*_t,t,θ_π^*_t) ≤-Δ_a,t/2) Using the definitions of ϵ_a,δ_a from Theorem <ref> we obtain the following:≤ℙ(g(x̂_a,t,θ̂_a) - g(x_a,t,θ_a) ≥ϵ_a/2) + ℙ( g(x̂_π*_t,t,θ̂_π^*_t) - g(x_π^*_t,t,θ_π^*_t) ≤-ϵ_a/2) ≤ℙ(D̅_a,π_1^t(θ_a,x_a,0||θ̂_a,x̂_a,0) ≥δ_a ) + ℙ( D̅_π^*_t,π_1^t(θ_π^*_t,x_π^*_t,0||θ̂_π^*_t,x̂_π^*_t,0) ≥δ_a ) Both of these terms can be bounded in a similar manner, as such we will present the arguments for bounding the first term and note that a similar procedure can be used to bound the second. We begin as follows:ℙ(D̅_a,π_1^t(θ_a,x_a,0||θ̂_a,x̂_a,0) ≥δ_a ) ≤∑_n=1^t ℙ(n(𝒯_a)=n, D̅_a,π_1^n(θ_a,x_a,0||θ̂_a,x̂_a,0) ≥δ_a ) ∑_n=1^t ℙ(n(𝒯_a)=n| D̅_a,π_1^n(θ_a,x_a,0||θ̂_a,x̂_a,0) ≥δ_a )ℙ(D̅_a,π_1^n(θ_a,x_a,0||θ̂_a,x̂_a,0) ≥δ_a) Applying Theorem <ref> we obtain:∑_n=1^t ℙ(n(𝒯_a)=n| D̅_a,π_1^n(θ_a,x_a,0||θ̂_a,x̂_a,0) ≥δ_a )exp(-(δ_a√(n)-c_f(d_x,d_θ))^2/2L_p^2σ^2) ≤∑_n=1^⌊ w ⌋ℙ(n(𝒯_a)=n| D̅_a,π_1^n(θ_a,x_a,0||θ̂_a,x̂_a,0) ≥δ_a ) + ∑_n= ⌈ w ⌉^∞exp(-(δ_a√(n)-c_f(d_x,d_θ))^2/2L_p^2σ^2) Where we define w = 1/2|𝒜|∑_n=1^t ϵ_t. We can bound the first term in the expression above by using the the total number of times arm a is chosen as part of an exploration step, we call this quantity n^R(𝒯_a). This can be done as follows:∑_n=1^⌊ w ⌋ℙ(n(𝒯_a)=n| D̅_a,π_1^n(θ_a,x_a,0||θ̂_a,x̂_a,0) ≥δ_a ) ≤∑_n=1^⌊ w ⌋ℙ(n^R(𝒯_a)≤ n |D̅_a,π_1^n(θ_a,x_a,0||θ̂_a,x̂_a,0) ≥δ_a) ≤ wℙ(n^R(𝒯_a)≤ w) The last two inequality follows since the amount of exploration steps is independent of how the parameter estimates are concentrating. Note that w = 1/2𝔼n^R(𝒯_a), and that (n^R(𝒯_a)) ≤𝔼n^R(𝒯_a). Hence we can employ the Bernstein bound <cit.> to obtain the following upper bound on the above expression:wℙ(n^R(𝒯_a)≤ w) ≤ w exp(-3w/14)We will bound the second term in the summation using an integral bound. To do this, first note that since -(δ_a√(n)-c_f(d_x,d_θ))^2 is concave in n, so we can upper bound it using an appropriate first order approximation. Namely, consider the first order Taylor expansion about n = (2c_f(d_θ,d_x)/δ_a)^2, then:-(δ_a√(n) - c_f(d_x,d_θ))^2 ≤-δ_a^2/2n + c_f(d_x,d_θ)^2 exp(-(δ_a√(n) - c_f(d_x,d_θ))^2/2L_p^2σ^2) ≤exp(-δ_a^2/2n + c_f(d_x,d_θ)^2/2L_p^2σ^2) Then using this bound we can upper bound the second summation term as follows:∑_n= ⌈ w ⌉^∞exp(-(δ_a√(n)-c_f(d_x,d_θ))^2/2L_p^2σ^2) ≤∑_n= ⌈ w ⌉^∞exp(-δ_a^2/2n + c_f(d_x,d_θ)^2/2L_p^2σ^2) ≤4L_p^2σ^2/δ_a^2exp(-δ_a^2/2⌊ w ⌋ + c_f(d_x,d_θ)^2/2L_p^2σ^2) Combining all of these equations together we obtain that:ℙ(π_t=a,a≠π^*_t) ≤ϵ_t/|𝒜| + 2w exp(-3w/14) + 8L_p^2σ^2/δ_a^2exp(-δ_a^2/2⌊ w ⌋ + c_f(d_x,d_θ)^2/2L_p^2σ^2) All that remains to complete the proof is to provide a lower bound on w. Let t': = c|𝒜|/δ_min^2 then:w = 1/2|𝒜|∑_n=1^t ϵ_n = 1/2|𝒜|∑_n=1^t'ϵ_n + 1/2|𝒜|∑_n=t'^t ϵ_n= c/2δ_min^2 + c/2δ_min^2∑_n=t'^t1/n≥c/2δ_min^2 + c/2δ_min^2logtδ_min^2/c|𝒜| = c/2δ_min^2logetδ_min^2/c|𝒜| Substituting thins bound into (<ref>) and expanding ϵ_t gives the desired result.§ TECHNICAL METRIC ENTROPY LEMMALet a ∈ A ⊆ℝ^n such that A is bounded and K = max_a∈ Ad(a,0)/n with respect to some metric d and ∀ a ∈ A, a_2 ≤ d(a,0) . Then for i.i.d Rademacher process {ϵ_i}_i=1^n :𝔼sup_a∈ A | 1/n∑_i=1^nϵ_i a_i | ≤ 4 ∫_0^K√(log2𝒩(α,A,d)/n) dα Proof: We proceed to prove this result in a similar technique to that used by <cit.>. LetA̅ = A ∪ A^- and {Â_i}_i=0^N be a sequence of successively finer covers of set A̅, such that Â_i is an α_i cover of set A̅ with respect to metric d and α_i = 2^-iK. Next, define a sequence ofapproximating vectors of a and denote these by â_i such that for any two successive approximations â_i ∈Â_i and â_i-1∈Â_i-1 we have that d(â_i,â_i-1) ≤α_i. Then observe we can rewrite a as follows:a = a + â_N - â_N = â_0 + ∑_i=1^N(â_i - â_i-1) + a - â_nObserve that we can set â_0 to the 0 vector since clearly a metric ball of radius K will form a K cover of set A. Hence we obtain:𝔼sup_a∈ A | 1/n∑_i=1^nϵ_i a_i |= 𝔼sup_a∈ A | 1/n⟨ϵ, a ⟩ | = 𝔼sup_a ∈A̅1/n⟨ϵ, a ⟩ = 𝔼sup_a∈A̅1/n⟨ϵ, ∑_j=1^N(â_i - â_i-1) + a - â_N ⟩≤𝔼∑_j=1^Nsup_â_j ∈Â_j, â_j-1∈Â_j-1⟨ϵ, â_j - â_j-1⟩ + 𝔼sup_a∈A̅⟨ϵ, a - â_N ⟩≤∑_j=1^N α_i √(2 log |Â_j||Â_j-1|/n) + α_NHere the final inequality is obtained by applying the finite class lemma <cit.>. Observe that |Â_j-1| ≤ |Â_j-1| = 𝒩(α_i,A̅,d) and that by construction α_j = 2(α_j - α_j+1). Hence:𝔼sup_a∈ A | 1/n∑_i=1^nϵ_i a_i | ≤∑_j=1^N 4(α_j - α_j+1) √(log𝒩(α_i,A̅,d)/n) + α_N ≤ 4 ∫_α_N+1^α_0√(log𝒩(α,A̅,d)/n) dα + α_N → 4 ∫_0^K √(log𝒩(α,A̅,d)/n) dαNote that 𝒩(α,A̅,d) ≤ 2𝒩(α,A,d) thus completing the proof. | http://arxiv.org/abs/1707.08423v3 | {
"authors": [
"Yonatan Mintz",
"Anil Aswani",
"Philip Kaminsky",
"Elena Flowers",
"Yoshimi Fukuoka"
],
"categories": [
"math.OC",
"cs.LG"
],
"primary_category": "math.OC",
"published": "20170726131440",
"title": "Non-Stationary Bandits with Habituation and Recovery Dynamics"
} |
Fermionic currents in AdS spacetime with compact dimensionsS. Bellucci^1 E-mail: [email protected] , A. A. Saharian^2 E-mail: [email protected] , V. Vardanyan^2,3,4 E-mail: [email protected] ^1 INFN, Laboratori Nazionali di Frascati,Via Enrico Fermi 40,00044 Frascati, Italy^2 Department of Physics, Yerevan State University,1 Alex Manoogian Street, 0025 Yerevan, Armenia^3 Lorentz Institute for Theoretical Physics, Leiden University,2333 CA Leiden, The Netherlands^4 Leiden Observatory, Leiden University, 2300 RA Leiden, The Netherlands December 30, 2023 ============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================== We derive a closed expression for the vacuum expectation value (VEV) of the fermionic current density in a (D+1)-dimensional locally AdS spacetime with an arbitrary number of toroidally compactified Poincaré spatial dimensions and in the presence of a constant gauge field. The latter can be formally interpreted in terms of a magnetic flux treading the compact dimensions. In the compact subspace, the field operator obeys quasiperiodicity conditions with arbitrary phases. The VEV of the charge density is zero and the current density has nonzero components along the compact dimensions only. They are periodic functions of the magnetic flux with the period equal to the flux quantum and tend to zero on the AdS boundary. Near the horizon, the effect of the background gravitational field is small and the leading term in the corresponding asymptotic expansion coincides with the VEV for a massless field in the locally Minkowski bulk. Unlike the Minkowskian case, in the system consisting an equal number of fermionic and scalar degrees of freedom, with same masses, charges and phases in the periodicity conditions, the total current density does not vanish. In these systems, the leading divergences in the scalar and fermionic contributions on the horizon are canceled and, as a consequence of that, the charge flux, integrated over the coordinate perpendicular to the AdS boundary, becomes finite. We show that in odd spacetime dimensions the fermionic fields realizing two inequivalent representations of the Clifford algebra and having equal phases in the periodicity conditions give the same contribution to the VEV of the current density. Combining the contributions from these fields, the current density in odd-dimensional C-,P- and T -symmetric models are obtained. As an application, we consider the ground state current density in curved carbon nanotubes described in terms of a (2+1)-dimensional effective Dirac model. PACS numbers: 04.62.+v, 03.70.+k, 98.80.-k, 61.46.Fg§ INTRODUCTION In a number of physical problems one needs to consider the model in background of manifolds with compact subspaces. The presence of extra compact dimensions is an inherent feature of fundamental theories unifying physical interactions, like Kaluza-Klein theories, supergravity and string theories. The compact spatial dimensions also appear in the low-energy effective description of some condensed matter systems. Examples for the latter are the cylindrical and toroidal carbon nanotubes and topological insulators.In quantum field theory, the periodicity conditions imposed along the compact dimensions modify the spectrum of the zero-point fluctuations of quantum fields. As a consequence of that, the vacuum expectation values (VEVs) of physical quantities are shifted by an amount depending on the geometry of the compact subspace. This general phenomenon, induced by the nontrivial topology, is the analog of the Casimir effect (for reviews see Ref. <cit.>) where the change in the spectrum of the vacuum fluctuations is caused by the presence of boundaries (conductors, dielectrics, branes in braneworld scenarios, etc.). It is known as the topological Casimir effect and has been investigated for different fields, bulk geometries and topologies. The corresponding vacuum energy depends on the lengths of the compact dimensions and the topological Casimir effect has been considered as a stabilization mechanism for the moduli fields related to extra dimensions. In addition, the vacuum energy induced by the compactification of spatial dimensions can serve as a model for the dark energy driving the accelerated expansion of the universe at a recent epoch <cit.>.For charged quantum fields, important characteristics for a given state are the expectation values of the charge and current densities. In the present paper we investigate the VEV of the current density for a massive fermionic field in the background of a locally anti-de Sitter (AdS) spacetime with an arbitrary number of toroidally compactified spatial dimensions (for a discussion of physical effects in models with toroidal dimensions, see for instance, Ref. <cit.>). The corresponding problem for a scalar field with general coupling to the Ricci scalar has been previously considered in Ref. <cit.> (see also Refs. <cit.> for additional effects induced by the presence of branes). Both the zero and finite temperature expectation values of the current density for charged scalar and fermionic fields in the background of flat spacetime with toral dimensions were investigated in Refs. <cit.>. The results were applied to the electronic subsystem of cylindrical and toroidal carbon nanotubes described in terms of a (2+1)-dimensional effective field theory. The vacuum current densities for charged scalar and Dirac spinor fields in de Sitter spacetime with toroidally compact spatial dimensions are considered in Ref. <cit.>. The influence of boundaries on the vacuum currents in topologically nontrivial spaces are studied in Refs. <cit.> for scalar and fermionic fields. The effects of nontrivial topology induced by the compactification of a cosmic string along its axis have been discussed in Ref. <cit.>. The vacuum energy and the VEV of the energy-momentum tensor in AdS spacetime with compact subspaces were investigated in Ref. <cit.>.Our choice of AdS spacetime as the background geometry is motivated by its importance in several recent developments of quantum field theory, gravity and condensed matter physics. The early interest to AdS spacetime as a bulk geometry in quantum field theory was motivated by principal questions of the quantization procedure on curved backgrounds. Because of the maximal symmetry of AdS spacetime, this procedure can be realized explicitly. Compared to the case of the Minkowski bulk, here principally new features arise related to the lack of global hyperbolicity and the presence of both regular and irregular modes. The main reason of the lack of hyperbolicity is that the AdS spacetime possesses a timelike boundary at spatial infinity through which the information may be lost or gained in finite coordinate time <cit.>. As a consequence, boundary conditions should be imposed at infinity to ensure a consistent quantum field theory. The natural appearance of AdS spacetime as a ground state in certain supergravity theories and also as the near horizon geometry of the extremal black holes and domain walls has stimulated further interest in quantum fields propagating on that background. Moreover, the AdS spacetime is a constant negative curvature manifold and the corresponding length scale can be used for the regularization of infrared divergences in interacting quantum field theories <cit.>. The dimension of the AdS isometry group is the same as that of the Poincaré group and the regularization is realized without reducing the symmetries.The renewed interest in physical models on AdS bulk is closely related to two rapidly developing fields in theoretical physics: the gauge/gravity duality and the braneworld scenario. The AdS spacetime played a crucial role in the original formulations of both these concepts in the form of the AdS/CFT correspondence <cit.> and the Randall-Sundrum type braneworlds <cit.>. The AdS/CFT correspondence (for reviews see Ref. <cit.>) is a type of holographic duality between two theories living in spacetimes with different dimensions: string theories or supergravity in the AdS bulk from one side and a conformal field theory localized on the AdS boundary from another. Among many interesting consequences, this duality opens an important opportunity to study quantum field theoretical effects in strongly coupled regime using a classical gravitational theory. It has also been used for the investigation of non-equilibrium phenomena in strongly coupled condensed matter systems. One of the applications is the holographic model for superconductors suggested in Ref. <cit.> (for a recent discussion with references, see e.g., Ref. <cit.>). In this model, the quantum physics of strongly correlated condensed matter system is mapped to the gravitational dynamics with black holes in one higher dimension.The braneworld scenario (see Ref. <cit.> for a review) provides an interesting alternative to the standard Kaluza-Klein compactification of extra dimensions. It uses the concept of brane as a submanifold embedded in a higher dimensional spacetime, on which the standard model fields are confined. Braneworlds naturally appear in the string/M-theory context and provide interesting possibilities to solve or to address from a different point of view various problems in cosmology and particle physics. In the model introduced by Randall and Sundrum the background geometry consists of two parallel branes, with positive and negative tensions, embedded in a five dimensional AdS bulk <cit.>. The fifth coordinate is compactified on orbofold and the branes are located at the two fixed points. The large hierarchy between the Planck and electroweak mass scales is generated by the large physical volume of the extra dimension. From the point of view of embedding the corresponding models into a more fundamental theory, such as string/M-theory, one may expect the presence of additional extra dimensions compactified on an internal manifold. Here we will consider a simple case of toroidal compactification of spatial dimensions in Poincaré coordinates.The plan of the paper is as follows. In the next section the problem is formulated and the complete set of fermionic modes is presented. By using these modes, in Section <ref>, we evaluate the VEV of the fermionic current density along compact dimensions. The asymptotics near the AdS boundary and near the horizon are investigated and limiting cases are discussed. In Section <ref>, we consider the current density in C-, P-, T-symmetric odd-dimensional models. The corresponding VEV is obtained by combining the results for two fermionic fields realizing the irreducible representations of the Clifford algebra. Applications are given to graphene made structures realizing the geometry under consideration. The main results of the paper are summarized in Section <ref>. In Appendix <ref> we consider an alternative representation of the Dirac matrices allowing for the separation of the equations for the upper and lower components of the fermionic mode functions in Poincaré coordinates. The main steps for the evaluation of the mode-sum for the current density are presented in Appendix <ref>.§ PROBLEM SETUP AND THE FERMIONIC MODES The dynamics of a fermionic field ψ (x) in a (D+1)-dimensional curved background with a metric tensor g_μν(x) and in the presence of an abelian gauge field A_μ(x) is described by the Dirac equationiγ ^μD_μψ -mψ =0,with the gauge extended covariant derivative D_μ=∂ _μ+Γ _μ+ieA_μ. Here, Γ _μ is the spin connection and the curved spacetime Dirac matrices γ ^μ are expressed in terms of the corresponding flat spacetime matrices γ ^(b) as γ ^μ=e_(b)^μγ ^(b), where e_(b)^μ are the tetrad fields. In this and in the next sections we consider a fermionic field realizing the irreducible representation of the Clifford algebra. In (D+1)-dimensional spacetime the corresponding Dirac matrices are N× N matrices, where N=2^[(D+1)/2] and [x] is the integer part of x (for the Dirac matrices in an arbitrary number of the spacetime dimension see, for example, Ref. <cit.>). In even-dimensional spacetimes the irreducible representation is unique up to a similarity transformation, whereas in odd-dimensional spacetimes there are two inequivalent irreducible representations (see Section <ref> below).As a background geometry we consider a locally AdS spacetime with the line elementds^2=e^-2y/aη _ikdx^idx^k-dy^2,where a is the curvature radius, i,k=0,1,… ,D-1 and η _ik= diag(1,-1,… ,-1). We assume that the subspace covered by the coordinates (x^p+1,… ,x^D-1) is compactified to a q-dimensional torus T^q with q=D-p-1. The length of the lth compact dimension will be denoted by L_l, 0⩽ x^l⩽ L_l, l=p+1,… ,D-1 . The subspace (x^1,… ,x^p) has trivial topology R^p with -∞ <x^l<+∞, l=1,… ,p, and for the coordinate y one has -∞ <y<+∞. Note that the compactification to the torus does not change the local AdS geometry with the scalar curvature R=-D(D+1)/a^2 . In terms of a new spatial coordinate z=ae^y/a, 0⩽ z<∞, the line element is presented in a conformally-flat formds^2=(a/z)^2(η _ikdx^idx^k-dz^2).The AdS boundary and horizon correspond to the hypersurfaces z=0 and z=∞, respectively. Note that L_l is the coordinate length of the compact dimension. For a given z, the proper length is given by L_(p)l=(a/z)L_l. The latter decreases with increasing z. In the conformal coordinates (x^1,… ,x^D-1,x^D=z), the tetrad fields can be chosen as e_(b)^μ=(z/a)δ _b^μ. For the corresponding components of the spin connection one gets Γ _k=η _klγ ^(D)γ ^(l)/(2z) for k=0,… ,D-1, and Γ _D=0.In the presence of compact dimensions, the field equation (<ref>) should be supplemented by the periodicity conditions on the field operator along those directions. Here, we will impose quasiperiodicity conditionsψ (t,x^1,… ,x^l+L_l,… ,x^D)=e^iα _lψ (t,x^1,… ,x^l,… ,x^D),with constant phases α _l, l=p+1,… ,D-1. The special cases α _l=0 and α _l=π correspond to untwisted and twisted fermionic fields, most frequently discussed in the literature. For the gauge field we will consider the simplest configuration A_μ=const. Though the corresponding field tensor vanishes, the nontrivial topology of the background spacetime gives rise the Aharonov-Bohm like effect on the VEVs of physical observables. For this special field configuration, the gauge field can be removed from the field equation by the gauge transformationψ =ψ ^'e^-ieχ, A_μ=A_μ^'+∂ _μχ ,with the function χ =A_μx^μ. In the new gauge A_μ^'=0. However, the gauge potential does not disappear from the problem completely. The gauge transformation of the field operator modifies the corresponding periodicity conditions. For the new field it takes the formψ ^'(t,x^1,… ,x^l+L_l,… ,x^D)=e^iα̃ ̃_̃l̃ψ ^'(t,x^1,… ,x^l,… ,x^D),with the new phasesα̃ ̃_̃l̃=α _l+eA_lL_l.The VEVs of physical observables will depend on the set {α _l,A_l} in the form of the combination (<ref>). Under the gauge transformation (<ref>) with χ =b_μx^μ and constant b_μ, we obtain a new set {α _l+eb_lL_l,A_l-b_l}. However, the combination (<ref>) remains invariant. Note that the phase shift in Eq. (<ref>), induced by the gauge field, can be presented as eA_lL_l=-2πΦ _l/Φ _0, where Φ _0=2π /e is the flux quantum. The quantity Φ _l can be formally interpreted as the magnetic flux enclosed by the l th compact dimension, Φ _l=-∮ dx^lA_l (no summation over l ). This flux acquires real physical meaning in models where the spacetime under consideration is embedded in a higher-dimensional manifold as a hypersurface (like branes in braneworld scenario) on which the fermionic field ψ (x) is localized. In what follows we will work in the gauge (A_μ^'=0,ψ ^'(x)) omitting the prime for the fermionic field. In this gauge, in Eq. (<ref>) we have D_μ=∂ _μ+Γ _μ. Of course, the VEVs of physical observables do not depend on the choice of the gauge.We are interested in the VEV of the fermionic current density ⟨ 0|j^μ(x)|0⟩≡⟨ j^μ(x)⟩ with the operator j^μ(x)=eψ̅(x)γ ^μψ (x), where |0⟩ stands for the vacuum state and ψ̅(x)=ψ ^†γ ^(0) . The VEV is presented as the coincidence limit⟨ j^μ(x)⟩ =-e/2lim_x^'→ x Tr(γ ^μS^(1)(x,x^')),with the two-point function S_ik^(1)(x,x^')=⟨ 0|[ψ _i(x),ψ̅_k(x^')]|0⟩ and with i, k being spinor indices. Expanding the field operator in terms of a complete set of mode functions {ψ _β^(+)(x),ψ _β^(-)(x)} for the Dirac equation and using the anticommutation relations for the annihilation and creation operators, one gets the mode-sum representation⟨ j^μ⟩ =e/2∑_β[ ψ̅_β^(-)(x)γ ^μψ _β^(-)(x)-ψ̅_β^(+)(x)γ ^μψ _β^(+)(x)] .Here, β is the set of quantum numbers specifying the fermionic modes and ∑_β is understood as summation for discrete subset of quantum numbers and integration over the continues ones. Following Ref. <cit.> (see also <cit.> for the geometry with a cosmic string perpendicular to the AdS boundary), in the discussion below we will take the Dirac matrices in the representationγ ^0=iz/a( [0 -1;10 ] ) , γ ^l=iz/a( [ -σ _l 0; 0σ _l ] ) ,with l=1,… ,D. The N/2× N/2 matrices σ _l obey the anticommutation relations σ _lσ _k+σ _kσ _l=2δ _lk. For hermitian σ _l one has γ ^0†=γ ^0 and γ ^l†=-γ ^l.The complete set of the modes for the problem under consideration can be found in the way similar to that used in Ref. <cit.> for the usual AdS bulk. The positive-energy modes are presented asψ _β^(+)=C_β^(+)z^D+1/2e^i𝐤𝐱 -iω t( [ Z_-(λ z)w^(σ ); 1/ωZ_+(λ z)( iλ +𝐤σ ) w^(σ ) ] ) ,where 0⩽λ <∞, ω =√(λ ^2+k^2), 𝐤𝐱=∑_l=1^D-1k_lx^l, 𝐤σ =∑_l=1^D-1k_lσ _l, andZ_±(x)=( [ J_ma± 1/2(x)0;0 J_ma∓ 1/2(x) ] ) ,with J_ν(x) being the Bessel function. In Eq. (<ref>), w^(σ ), σ = 1,… ,N/2, are one-column matrices having N/2 rows, with the elements w_l^(σ )=δ _lσ. For the negative-energy modes we getψ _β^(-)=C_β^(-)z^D+1/2e^i𝐤𝐱 -iω t( [ 1/ωZ_-(λ z)( iλ -𝐤σ ) w^(σ ); Z_+(λ z)w^(σ ) ] ) .The fermionic states are now specified by the set of quantum numbers β =(𝐤,λ ,σ ).The eigenvalues of the momentum components along the compact dimensions are found from the quasiperiodicity conditions (<ref>):k_l=2π n_l+α̃ ̃_̃l̃/L_l, l=p+1,… ,D-1,with n_l=0,± 1,± 2,…. For the components of the momentum along the noncompact dimensions one has -∞ <k_l<+∞, l=1,… ,p . The coefficients C_β^(± ) are found from the normalization condition ∫ d^Dx (a/z)^Dψ _β^(± )†ψ _β ^'^(± )=δ _ββ ^', where δ _ββ ^' is understood as the Kronecker delta for discrete quantum numbers and the Dirac delta function for the continuous ones. From that condition one gets| C_β^(± )| ^2=λ/2(2π )^pV_qa^D,where V_q=L_p+1⋯ L_D-1 is the volume of the compact subspace.In defining the mode functions (<ref>) and (<ref>), as a solution of the Bessel equation we have taken the Bessel function of the first kind (see Eq. (<ref>)). In the range ma⩾ 1/2 this choice is uniquely dictated by the normalizability condition of the mode functions: for the solution with the Neumann function Y_ma± 1/2(x) the normalization integral over z diverges at the lower limit z=0. For ma<1/2 both solutions of the Bessel equation are normalizable and, in general, we can take in Eq. (<ref>) the linear combination of the Bessel and Neumann functions. The first one of the coefficients in that linear combination is obtained from the normalization condition, whereas the second one is not uniquely determined. In the case ma<1/2, the normalized mode functions are obtained in a way similar to that we have presented above. They are still given by Eqs. (<ref>) and (<ref>) with the same coefficients C_β^(± ), where now in Eq. (<ref>) we need to make the replacementJ_ma± 1/2(x)→J_ma± 1/2(x)+B_βY_ma± 1/2(x) /√(1+B_β^2).Here, the coefficient B_β (which, in general, depends on quantum numbers) should be specified by an additional boundary condition on the AdS boundary (for a discussion of boundary conditions on fermionic fields in AdS see Refs. <cit.>). In what follows, for ma<1/2 we consider the boundary condition corresponding to B_β=0. It can be seen that this corresponds to the situation where the bag boundary condition is imposed on the fermionic field at z=δ >0 and then the limit δ→ 0 is taken (for the corresponding procedure in the case of the AdS bulk without compactification see Ref. <cit.>). This ensures the vanishing of the fermionic currents through the AdS boundary.§ FERMIONIC CURRENT By using the mode-sum formula (<ref>) with the Dirac matrices (<ref>) and the modes (<ref>), (<ref>), we can see that the VEVs of the charge density and of the components of the current density along noncompact dimensions vanish: ⟨ j^μ⟩ =0, μ =0,1,… ,p,D. For the current density along the lth compact dimension, l=p+1,… ,D-1, we find⟨ j^l⟩ =-(4π )^-p/2eNz^D+2/2Γ (p/2)V_qa^D+1∑_𝐧_q∫_0^∞dk_(p) k_(p)^p-1∫_0^∞dλ λ k_l/ ω[ J_ma+1/2^2(λ z)+J_ma-1/2^2(λ z)] ,where 𝐧_q=(n_p+1,… ,n_D-1), -∞ <n_i<+∞, k_(p)^2=∑_i=1^pk_i^2, ω ^2=λ ^2+k_(p)^2+k_(q)^2, andk_(q)^2=∑_i=p+1^D-1k_i^2=∑_i=p+1^D-1(2π n_i+ α̃_i)^2/L_i^2.The same expression for the VEV of the current density is obtained in Appendix <ref> by using another representation for the gamma matrices. In the AdS bulk without compactification the VEV of the current density vanishes (for a recent discussion of the finite temperature expectation value see Ref. <cit.>).For the further transformation of the current density, we write the VEV in the form⟨ j^l⟩ =-(4π )^-p/2eNz^D+2/2Γ (p/2)V_qa^D+1[ ℐ_ma(z)+ℐ_ma-1(z)] ,with the functionℐ_ν(z)=∑_𝐧_q∫_0^∞dk_(p) k_(p)^p-1∫_0^∞dλ λ k_l/ ωJ_ν +1/2^2(λ z).The transformation of this function is presented in Appendix <ref> with the final result given by Eq. (<ref>). As a result, for the current density we find⟨ j^l⟩ =-eNa^-D-1L_l/(2π )^(D+1)/2 ∑_n_l=1^∞n_lsin (α̃_ln_l)∑_𝐧 _q-1 cos (α̃_q-1·𝐧 _q-1)∑_j=0,1q_ma-j^( D+1) /2(b_𝐧_q),with 𝐧_q-1=(n_p+1,… ,n_l-1,n_l+1,… ,n_D-1), α̃_q-1·𝐧_q-1=∑_i=1,≠ l^D-1 α̃_in_i, andb_𝐧_q=1+g_𝐧_q^2/2z^2, g_𝐧 _q^2=∑_i=p+1^D-1n_i^2L_i^2.Here, the function q_ma-j^( D+1) /2(u) is defined by Eq. ( <ref>) or, equivalently, by the integral representation (<ref>). The VEV of the current density for a charged scalar field is also expressed in terms of this function <cit.>.The VEV of the component of the current density along the compact dimension x^l is an odd periodic function of α̃_l and an even periodic function of the remaining phases. In terms of the magnetic fluxes Φ _i, the period is equal to the flux quantum Φ _0. The current (<ref>) determines the charge flux through the spatial hypersurface x^l=const. Denoting by n_i^(l), n_i^(l)=δ _i^la/z, the normal to this hypersurface, for the charge flux one gets (no summation over l) n_i^(l)⟨ j^i⟩ =n_l^(l)⟨ j^l⟩. It depends on the lengths L_i and on the coordinate z through the ratio L_i/z. This ratio is the proper length of the ith compact dimension measured by an observer with a fixed z, in units of the curvature radius of the background geometry, L_i/z=L_(p)i/a. We could expect this feature from the maximal symmetry of the AdS spacetime. Of course, the current density (<ref>) obeys the covariant conservation equation ∂ _l(√(|g|)⟨ j^l⟩ )=0 with √(|g|) =(a/z)^D+1.In the model with a single compact dimension x^l with the length L_l=L and with the phase α̃_l=α̃, the general result (<ref>) is reduced to⟨ j^l⟩ =-eNa^-D-1L/(2π )^(D+1)/2 ∑_n=1^∞nsin (α̃n)∑_j=0,1q_ma-j^( D+1) /2( 1+n^2L^2/2z^2) .In Fig. <ref>, for D=4 and ma=1, we have plotted the corresponding charge flux measured in units of a^-D, namely, a^Dn_l^(l)⟨ j^l⟩, as a function of the phase α̃ and of the ratio z/L. The current density is a periodic function of α̃ with the period 2π. As it will be shown below by asymptotic analysis, the charge flux vanishes on the AdS boundary and diverges on the AdS horizon.Let us consider some special cases of the general formula (<ref>). For a massless field, m=0, the expressions for the functions q_0^( D+1) /2(u) and q_-1^( D+1) /2(u) are most easily found by using the representation (<ref>) and the expressions for the functions I_± 1/2(x). This givesq_j^( D+1) /2(u)=1/2Γ( D+1/2 ) [ 1/( u-1) ^(D+1)/2-(-1)^j/( u+1) ^(D+1)/2] ,for j=0,-1. Plugging into Eq. (<ref>), for the current density one gets ⟨ j^l⟩ =(z/a)^D+1⟨ j^l⟩ _M, where⟨ j^l⟩ _M=-eΓ ((D+1)/2)/π ^(D+1)/2 NL_l∑_n_l=1^∞n_lsin (α̃_ln_l)∑_ 𝐧_q-1 cos (α̃_q-1·𝐧 _q-1)/g_𝐧_q^D+1,is the VEV for a massless fermionic field in (D+1)-dimensional Minkowski spacetime with spatial topology R^p+1× T^q. The expression (<ref>) is obtained from the more general result from Ref. <cit.> for a massive fermionic field in the limit m→ 0 (with the replacement α̃_l→ -2πα̃_l). Note that, because of the boundary condition we have imposed on the fermionic modes on the AdS boundary, for a massless field the problem under consideration is conformally related to the problem in locally Minkowski spacetime with a boundary at z=0 on which the fermionic field obeys the bag boundary condition. The reason why the current density ⟨ j^l⟩ is conformally related to the current density ⟨ j^l⟩ _M in the boundary-free Minkowski case, is that the boundary-induced contribution in the latter problem vanishes for a massless field (see Ref. <cit.>).The current density on the Minkowski bulk is obtained in the limit a→∞ for fixed y. The conformal coordinate z is expanded as z=a+y+⋯. In the integral representation (<ref>) for the function q_ν^( D+1) /2(u) the order of the Bessel function is large and we use the corresponding uniform asymptotic expansion. To the leading order, this gives∑_j=0,1q_ma-j^( D+1) /2(u)=2m^(D+1)/2a^D+1 K_(D+1)/2(mu)/u^(D+1)/2,with u=b_𝐧_q and K_ν(x) being the Macdonald function. Substituting into Eq. (<ref>) we get lim_a→∞⟨ j^l⟩ =⟨ j^l⟩ _M with the Minkowskian result⟨ j^l⟩ _M=-2eNL_lm^(D+1)/2/(2π )^(D+1)/2∑_n_l=1^∞n_lsin (α̃_ln_l)∑_ 𝐧_q-1 cos (α̃_q-1·𝐧 _q-1)K_(D+1)/2(mg_𝐧_q)/g_𝐧_q^(D+1)/2.The latter coincides with the expression obtained in Ref. <cit.> (again, with the replacement α̃_l→ -2πα̃_l). In Fig. <ref>, for the model with D=4 and with a single compact dimension, we have plotted the ratio of the charge fluxes in locally AdS and Minkowski spacetimes, with the same proper lengths of the compact dimension, as a function of the proper length measured in units of the AdS curvature radius a. For the phase in the quasiperiodicity condition along the compact dimension we have taken α̃=π /2 and the numbers near the curves correspond to the values of ma. As is seen from the graphs, for large values of the proper length the charge flux in the AdS bulk is essentially larger than that for the Minkowski case. As it will be shown below, in the former case the decay of the current density for large values of the proper length goes like a power law, whereas in the Minkowski background and for a massive field the decay is exponential (see Eq. (<ref>)).Now we turn to the asymptotics of the current density for small and large values of z. Near the AdS boundary, z→ 0, the argument of the function q_ν^(D+1)/2(u) in Eq. (<ref>) is large. By using the corresponding asymptotic from Ref. <cit.>, we can see that the dominant contribution comes from the term q_ma-1^( D+1) /2(b_𝐧_q) and, to the leading order,⟨ j^l⟩ ≈ -eNL_lΓ (ma+(D+1)/2)/π ^D/2a^D+1Γ (ma+1/2)z^D+1+2ma ×∑_n_l=1^∞n_lsin (α̃_ln_l)∑_ 𝐧_q-1 cos (α̃_q-1·𝐧 _q-1)/g_𝐧_q^D+1+2ma.For a massless field this coincides with the exact result. As seen, the current density vanishes on the AdS boundary as z^D+1+2ma. For a special case D=4 with a single compact dimension this has been already demonstrated numerically in figure <ref>. The large values of z correspond to the near horizon limit. In this limit one has b_𝐧 _q-1≪ 1 and by using the asymptoticq_ν^( D+1) /2(u)≈Γ( (D+1)/2) /2( u-1) ^(D+1)/2[ 1-ν( ν +1) /D-1 ( u-1) ] ,valid for u-1≪ 1, to the leading order we find ⟨ j^l⟩≈ (z/a)^D+1⟨ j^l⟩ _M, with ⟨ j^l⟩ _M given by Eq. (<ref>). Near the horizon the dominant contribution comes from the fluctuations with small wavelengths, and the effects induced by the curvature and nonzero mass are small.Let us consider the behavior of the current density in asymptotic regions of the lengths for compact dimensions. If the length of the lth compact dimension is much smaller than the other lengths, L_l≪ L_i, i≠ l , and L_l≪ z, the leading contribution comes from the term with 𝐧_q-1=0 and by using the asymptotic expression for the function q_ν^(D+1)/2(u) for the values of the argument close to 1, one finds⟨ j^l⟩≈ -eNL_lΓ ((D+1)/2)/π ^(D+1)/2( aL_l/z) ^D+1∑_n_l=1^∞sin ( α̃_ln_l)/n_l^D.The contribution to the current density from the terms 𝐧_q-1≠ 0 is suppressed by the exponential factor e^-σ _l|𝐋·𝐧_q-1|/L_l, with σ _l=min(α̃ _l,2π -α̃_l), 0<α̃_l<2π. Note that Eq. (<ref>) coincides with the current density for a massless field in the model with a single compact dimension x^l.For large values of the proper length of the lth compact dimension compared with the AdS curvature radius one has L_(p)l/a=L_l/z≫ 1 and, hence, b_𝐧_q≫ 1. By using the asymptotic expression for the function q_ν^(D+1)/2(u) for large values of the argument, for the leading order term in the current density one gets the result (<ref>). If in addition L_l≫ L_i, i≠ l, the contribution from large values of |n_i| dominates and the corresponding summations can be replaced by the integration. Two cases should be considered separately. If all the phases α̃_i, i≠ l, are zero the leading term is given by⟨ j^l⟩≈ -NeΓ (p/2+ma+3/2)/π ^p/2+1Γ (ma+1/2)a^D+1V_qz^D+2ma+1/L_l^p+2ma+1 ∑_n_l=1^∞sin (α̃_ln_l)/n_l^p+2ma+2 .As is seen, in this case, for a given z, the decay of the current density as a function of the proper length L_(p)l follow a power law. For a massive field this behavior essentially differs from that for the current density in the Minkowski bulk. In the latter geometry and for a massive field, the current decays exponentially, as e^-mL_l. Provided at least one of the phases α̃_i, i≠ l, is different from zero, the current density is dominated by the term n_l=1 and one finds⟨ j^l⟩≈ -Nea^-1-Dsin (α̃ _l)z^D+2ma+1/2π ^(p+1)/2Γ (ma+1/2)V_qκ ^p/2+ma+1e^-κ L_l/(2L_l)^p/2+ma,with the notation κ ^2=∑_i=p+1,≠ l^D-1α̃ _i^2/L_i^2. Now we have an exponential decay as a function of L_l.For large values of the mass, ma≫ 1, the dominant contribution to the series in the right-hand side of Eq. (<ref>) comes from the term with n_i=0, i≠ l, and n_l=1. Introducing the notation u=L_l/(2z)=L_(p)l/(2a), for the leading order contribution one finds⟨ j^l⟩≈ -eNL_lsin (α̃_l)/2(4π )^D/2a^D+1(ma)^D/2u^-D/2-1(1+u^2)^-D/4/(1+2u^2+2u√( 1+u^2))^ma+1/2(u+√(1+u^2)),and the VEV is exponentially suppressed. In Fig. <ref>, the dependence of the charge flux along the compact dimension on the mass of the fermionic field is displayed for the background with D=4 with a single compact dimension. The graphs are plotted for α̃=π /2 and the numbers near the curves are the corresponding values of the ratio z/L.From the covariant conservation equation for the current density it follows that the charge flux through the hypersurface element d^D-1x=dz∏ _i=1,≠ l^D-1dx^i is given by √(|g|)j^ld^D-1x. For the expectation value of the total charge flux, per unit coordinate surface along spatial dimensions (x^1,… ,x^l-1,x^l+1,… ,x^D-1), integrated over z, one gets ∫_0^∞dz ( a/z) ^D+1⟨ j^l⟩. From the asymptotic analysis of the current density near the AdS boundary and horizon, given above, we can see that the integral is convergent in the lower limit and linearly diverges in the upper limit. Hence, similar to the Minkowskian case, the total charge flux diverges.Comparing the fermionic current density in the Minkowski bulk, Eq. (<ref>), with the corresponding expression from Ref. <cit.> for the current density ⟨ j^l⟩ _M^(s) of a charged scalar field, we see that⟨ j^l⟩ _M=-(N/2)⟨ j^l⟩ _M^ (s),provided the masses, charges and the phases in the periodicity conditions are the same for fermionic and scalar fields. In particular, in supersymmetric models with the same number of fermionic and scalar degrees of freedom, the total current in the Minkowski bulk vanishes. This is not the case for the AdS bulk (for a discussion of boundary conditions on the AdS boundary in supersymmetric models see, for example, Refs. <cit.> and references therein). Note that for supersymmetric models in AdS background the fields in the same multiplet do not necessarily have the same mass (see, e.g., Refs. <cit.>). By taking into account the expression of the current density for scalar fields <cit.>, for the total current in the system of a fermionic field and N/2 charged scalar fields (equal number of fermionic and scalar degrees of freedom) one gets⟨ j^l⟩ ^(t) = Nea^-1-DL_l/(2π )^(D+1)/2∑_n_l=1^∞n_lsin (α̃_ln_l)∑_ 𝐧_q-1 cos (α̃_q-1·𝐧 _q-1) ×[ 2q_ν _s-1/2^(D+1)/2(b_𝐧_q)-q_ma^ ( D+1) /2(b_𝐧_q)-q_ma-1^( D+1) /2(b_ 𝐧_q)] ,where ν _s=√(D^2/4-D(D+1)ξ +m^2a^2) and ξ is the curvature coupling parameter for scalar fields. Near the AdS horizon, the leading contribution from the scalar and fermionic parts in Eq. (<ref> ) cancel each other and we need to keep the next-to-leading term in the asymptotic (<ref>). This leads to the result⟨ j^l⟩ ^(t) ≈ D(D+1)NeL_l/4π ^(D+1)/2a^D+1Γ( D-1/2) ( ξ -ξ _D) z^D-1 ×∑_n_l=1^∞n_lsin (α̃_ln_l)∑_ 𝐧_q-1 cos (α̃_q-1·𝐧 _q-1)/g_𝐧_q^D-1,in the limit z→∞. Here, ξ _D=(D-1)/(4D) is the value of the curvature coupling parameter for conformal coupling. The leading term, given by Eq. (<ref>), does not depend on the mass. For ma<1/2 or ma⩾ 1/2 and D(D+1)( ξ -ξ _D) <ma, the fermionic part dominates near the AdS boundary and the total VEV ⟨ j^l⟩ ^(t) behaves as in Eq. (<ref>). In particular, the latter is the case for minimally coupled scalar fields. For ma⩾ 1/2 and D(D+1)( ξ -ξ _D) >ma, the total current density near the AdS boundary is dominated by the scalar contribution and ⟨ j^l⟩ ^(t)∝ z^D+2ν _s+2 for z→ 0.In Fig. <ref>, the dependence on the mass of the total charge flux is plotted in the (p,q)=(3,1) model with a fermionic field and N/2 scalar fields. The graphs are plotted for α̃=π /2 and the numbers near the curves correspond to the values of the ratio z/L. The left and right panels are for conformally and minimally coupled scalar fields. In both cases the total current is dominated by the fermionic contribution. In general, the current density is not a monotonic function of the mass.As seen from Eq. (<ref>), compared to the separate scalar and fermionic contributions, the divergence of the total current density on the horizon is weaker and, as a consequence of that, the charge flux integrated over the z-coordinate is finite. By using in Eq. (<ref>) the integral representation (<ref>) for the function q_ν^( D+1) /2(u), after the evaluation of the integral over z, the x -integral is reduced to ∫_0^∞dx e^-xg(x) with g(x)=∑_j=± 1/2I_ma+j(x)-2I_ν _s(x). Though this integral is convergent, the separate integrals with the modified Bessel functions diverge. In order to have the right to take the integrals separately, we can replace the integral by ∫_0^∞dx e^-bxg(x), b>1. After the evaluation of the separate integrals by using the formula from Ref. <cit.>, the limit b→ 1 is taken easily. In this way, for the integrated charge flux we get∫_0^∞dz ( a/z) ^D+1⟨ j^l⟩ ^ (t)=( ma-ν _s) Γ (D/2)eNL_l/2(4π )^D/2∑_n_l=1^∞n_lsin (α̃_ln_l)∑_ 𝐧_q-1 cos (α̃_q-1·𝐧 _q-1)/g_𝐧_q^D.It is of interest to note that the mass enters in the factor ( ma-ν _s) only.§ FERMIONIC CURRENT IN C-,P- AND T-SYMMETRIC ODD-DIMENSIONAL MODELS ON ADS BULK In the discussion above we have evaluated the VEV of the fermionic current density for a field realizing the irreducible representation of the Clifford algebra. It is known that (see, for example, Ref. <cit.>) in even values of the spatial dimension D (odd-dimensional spacetimes) the mass term mψ̅ψ in the Lagrangian density breaks P-invariance, C -invariance for D=4n, and T-invariance in D=4n+2 with n=0,1,2,… . P-, C-, and T-invariant models can be constructed combining two fermionic fields in the irreducible representations. In this section we assume that the flat spacetime Dirac matrices are taken in the representation (<ref>). For odd-dimensional spacetimes the matrix γ ^(D) can be expressed in terms of the product γ =γ ^(0)γ ^(1)⋯γ ^(D-1) in two inequivalent ways, namely, γ ^(D)=γ _(s)^(D)=sγ, s=± 1, for D=4n and γ ^(D)=γ _(s)^(D)=siγ for D=4n+2. The upper and lower signs correspond to two inequivalent representations of the corresponding Clifford algebra with the gamma matrices γ _(s)^(μ )=(γ ^(0),γ ^(1),⋯γ ^(D-1),γ _(s)^(D)) . The corresponding matrices in AdS spacetime can be taken as γ _(s)^μ=(a/z)γ _(s)^(μ ).Consider a system of two N-component fermionic fields, ψ _(+1) and ψ _(-1), with the combined Lagrangian density ℒ=∑_s=± 1ψ̅_(s)(iγ _(s)^μ∇ _μ^(s)-m)ψ _(s), ∇ _μ^(s)=∂ _μ+Γ _μ^(s). By suitable transformations of the fields (in general, mixing the separate fields) it can be seen that the Lagrangian density is invariant under the C-, P- and T-transformations (see, e.g., Ref. <cit.>). By taking into account the relations γ ^(0)γ ^†γ ^(0)γ =-1 and γ ^(0)γ ^†γ ^(0)γ _(+1)^μγ =γ _(-1)^μ, the Lagrangian density can also be rewritten asℒ=∑_s=± 1ψ̅_(s)^'(iγ ^μ∇ _μ-sm)ψ _(s)^'.with ψ _(+1)^'=ψ _(+1), ψ _(-1)^'=γψ _(-1), and γ ^μ=γ _(+1)^μ. As seen, the new field ψ _(-1)^' satisfies the same equation as ψ _(+1) with the opposite sign for the mass term. Introducing 2N× 2N Dirac matrices γ ^(2N)μ=σ _P3⊗γ ^μ, with σ _P3=diag(1,-1) being the Pauli matrix, the Lagrangian density is presented in terms of the 2N-component spinor Ψ =(ψ _(+1)^',ψ _(-1)^')^T in the formℒ=Ψ̅[iγ ^(2N)μ(∂ _μ+Γ _μ^(2N))-m]Ψ ,where Γ _μ^(2N)=I⊗Γ _μ.In the system of two fermionic fields realizing two inequivalent representations of the Clifford algebra, the total current density J^μ=eΨ̅γ ^(2N)μΨ is given by J^μ=∑_s=± 1j_(s)^μ, where j_(s)^μ=eψ̅_(s)γ _(s)^μψ _(s) are the current densities for separate fields. As it has been shown in Appendix <ref>, if the phases in the quasiperiodicity conditions along compact dimensions are the same for the fields ψ _(+1) and ψ _(-1) then the VEVs of the corresponding current densities are the same. In this case, the VEV ⟨ J^l⟩ is obtained from the expressions for ⟨ j^l⟩ given above with an additional coefficient 2. However, the phases α _l for the fields ψ _(+1) and ψ _(-1), in general, can be different.Among the most interesting physical realizations of fermionic systems in odd-dimensional spacetimes is the graphene. Graphene is an one-atom thick layer of graphite. The low-energy excitations of the corresponding electron subsystem can be described by a pair of two-component spinors, composed of the Bloch states residing on the two triangular sublattices A and B of the graphene honeycomb lattice. For the spatial dimension in the corresponding effective field theory one has D=2. For a given value of spin S=± 1, two spinors are combined in a 4-component spinor Ψ _S=(ψ _+,AS,ψ _+,BS,ψ _-,AS,ψ _-,BS)^T. The components ψ _± ,AS and ψ _± ,BS correspond to the amplitude of the electron wave function on sublattices A and B and the indices + and - correspond to inequivalent points, 𝐊_+ and 𝐊_- of the Brillouin zone (see Refs. <cit.>). The values of the parameter s=+1 and s=-1 in the discussion above, specifying the irreducible representations of the Clifford algebra, correspond to these points. Consequently, we can identify ψ _(s)=(ψ _s,AS,ψ _s,BS)^T. For the bulk geometry with vanishing 0th component of the spin connection, Γ _0=0, and for the zero scalar potential of the external electromagnetic field (these conditions are the case in the problem at hand) the combined Lagrangian density, in standard units, is presented asL=∑_S=± 1Ψ̅_S(iħγ ^(4)0∂ _t+iħ v_Fγ ^(4)lD_l-Δ )Ψ _S.Here, v_F≈ 7.9× 10^7 cm/s is the Fermi velocity of the electrons in graphene, D_l=(∇-ie𝐀/ħ c)_l, with l=1,2, is the spatial part of the gauge extended covariant derivative, and for electrons e=-|e|. The energy gap Δ can be created by a number of mechanisms (see, for instance, Ref. <cit.> and references therein) and it is related to the Dirac mass m through Δ =mv_F^2. For the analog of the Compton wavelength corresponding to the energy gap one has a_C=ħ v_F/Δ. The energy scale in the system is determined by the combination γ _F=ħ v_F/a_0 ( ≈ 2.51 eV), where a_0≈ 1.42 Å is the inter-atomic spacing of the graphene lattice. In terms of this combination one has a_ C=a_0γ _F/Δ. The Lagrangian densities (<ref> ) for separate S are the analog of Eq. (<ref>) for D=2.For a planar graphene sheet the spatial topology in the corresponding effective field theory is R^2. In the case of a sheet rolled into a cylinder (cylindrical carbon nanotubes) or torus (toroidal nanotubes) the topology becomes nontrivial: R^1× S^1 and S^1× S^1, respectively. In these systems, the magnetic fluxes Φ _l (l=2 for cylindrical nanotubes and l=1,2 for toroidal ones) we have introduced above, acquire real physical meaning. The carbon nanotubes are characterized by chiral vector 𝐂_h=n_a𝐚+n_b𝐛, where n_a, n_b are integers and 𝐚=(√(3),0)a_0, 𝐛 =(-√(3),3)a_0/2 are primitive translation vectors of the graphene hexagonal lattice (for general properties of carbon nanotubes, see, for example, Ref. <cit.>). Note that √(3)a_0 is the lattice constant. In the construction of the nanotube the hexagon at the origin is identified with the hexagon at 𝐂_h. For zigzag and armchair nanotubes one has n_b=0 and n_a=2n_b, respectively. The other pairs (n_a,n_b) correspond to chiral nanotubes. In the case n_a+n_b=3q_c, q_c∈ Z, the nanotube will be metallic and for n_a+n_b≠ 3q_c the nanotube will be semiconducting. In the latter case, the corresponding energy gap is inversely proportional to the diameter. In particular, the armchair nanotube is metallic and the (n_a,0) zigzag nanotube is metallic if and only if n_a is an integer multiple of 3. The chirality also determines the periodicity condition along the compact dimension for the fields ψ _(s). In the absence of the magnetic flux threading the nanotube, the periodicity conditions have the form ψ _(s)(𝐫+𝐂_h)=e^-2sπ ip_c/3ψ _(s)( 𝐫+𝐂_h), where, for a given nanotube, the parameter p_c=-1,0,+1 is defined by the relation n_a+n_b=3q_c+p_c (see, e.g., the discussion in Ref. <cit.>). Hence, for the phase we have introduced before, one has α =-2sπ ip_c/3. For metallic nanotubes one has the periodic boundary condition, α =0, and for semiconducting ones α =± 2π /3. As seen, the phases for the spinors corresponding to the points, 𝐊_+ and 𝐊_- of the Brillouin zone have opposite signs. As a consequence, in the absence of the magnetic flux, the total current density in cylindrical nanotubes vanishes.The problem under consideration is topologically equivalent to the case of cylindrical nanotubes, though the corresponding spatial geometry is curved (for the generation of curvature in graphene sheets and the corresponding effects on the properties of graphene see, for example, Ref. <cit.>). In Fig. <ref> we have plotted this geometry embedded into a 3-dimensional Euclidean space. The magnetic flux threading the compact dimension is shown as well. The proper length of the compact dimension is decreasing with increasing z. Note that, related to the graphene physics, a similar spatial geometry has been discussed in <cit.>. However, the spacetime geometry we consider here is different.Consequently, for a given S, the VEVs of the current densities for separate contributions coming from the points 𝐊_+ and 𝐊 _- are given by the expressions in previous sections with an additional factor v_F, and for the nonzero component of the total current one has ⟨ J^1⟩ =∑_s=± 1⟨ j_(s)^1⟩, where j_(± 1)^1 are the contributions from two valleys. In the problem under consideration separate spins S give the same contributions in the ground state currents. Assuming that the phases α for the contributions from s=+1 and s=-1 have opposite signs, one finds⟨ J^1⟩ =-√(2)ev_FL/π a^3∑_n=1^∞ncos (α _(+1)n)sin( 2π nΦ/Φ _0) h( a/a_C,1+n^2L^2/2z^2) ,with the functionh( ν ,x) =2^ν∂ _x[ 1/√(x-1)/ ( √(x+1)+√(x-1)) ^2ν] .In the absence of the magnetic flux the current density vanishes. In Eq. ( <ref>), the ratio a/a_C is the analog of the product ma in the discussion of the previous sections. If the curvature of the tube does not change the phases, then α _(+1)=0 for metallic tubes and α _(+1)=2π /3 for semiconducting ones. Fig. <ref> presents the current density in these two cases versus the magnetic flux treading the tube for different values of a/a_C (numbers near the curves). The left/right panels correspond to the metallic/semiconducting tubes. In the numerical evaluation we have taken z/L=2.§ CONCLUSION In the present paper we have investigated the VEV of the fermionic current density in (D+1)-dimensional AdS spacetime with a toroidally compactified subspace. The periodicity conditions for the field operator along compact dimensions contain arbitrary phases α _l and, in addition, the presence of a constant abelian gauge field is assumed. By a gauge transformation the problem is reduced to the one in the absence of the gauge field with the shifted phases (<ref>) in the quasiperiodicity conditions for the new field. The phase shift for the lth compact dimension is formally interpreted in terms of the magnetic flux enclosed by that dimension.For the evaluation of the vacuum currents we have used the direct summation over the complete set of fermionic modes (<ref>) and (<ref>). The same result is obtained by using the alternative set of fermionic mode functions (<ref>) and (<ref>). The VEVs of the charge density and of the components for the current density along uncompact dimensions vanish and the mode sum for the component of the current density along the lth compact dimension is presented as Eq. (<ref>). A more convenient representation for the renormalized VEV is given by Eq. (<ref>). The current density along the lth compact dimension is an even periodic function of the phases α̃_i, i≠ l, and an odd periodic function of the phase α̃_l. In particular, this means the periodicity in the magnetic flux with the period equal to the flux quantum. The current density is expressed in terms of the function (<ref>). An alternative integral representation is given by Eq. (<ref>). Note that the VEV of the current density for a charged scalar field is also expressed through the function (<ref>).In order to clarify the behavior of the current density, various limiting cases of the general result are considered. First of all, we have shown that, in the limit of the infinite curvature radius, the current density for a locally Minkowski bulk is obtained. For a massless field the problem under consideration is conformal to the one in Minkowski spacetime with toral dimensions in the presence of a boundary on which the fermionic field operator obeys the bag boundary condition. However, the boundary-induced contribution in the latter problem vanishes for a massless field and we have obtained a conformal relation with the boundary-free Minkowski case. On the AdS boundary the VEV of the current density vanishes as z^D+1+2ma, whereas on the AdS horizon it diverges as z^D+1. Near the horizon the dominant contribution comes from the fluctuations with small wavelengths and the effects induced by the curvature and nonzero mass are small. The influence of the curvature on the component of the current density along the lth compact dimension is also small, when the corresponding length L_l is much smaller than other length scales in the problem. The leading term in the corresponding asymptotic expansion coincides with the VEV for a massless field in the model with a single compact dimension x^l. In the opposite limit of large values for the proper length of the lth compact dimension, the behavior of the VEV ⟨ j^l⟩ is essentially different depending on the values of the phases in the quasiperiodicity conditions. If all the remaining phases vanish, α̃_i=0, i≠ l, the current density, as a function of the proper length, behaves as 1/L_(p)l^p+2ma+1. Unlike the case of the locally Minkowski bulk, the corresponding decay is power law for both massless and massive fields. If at least one of the phases α̃_i, i≠ l, is different from zero, the decay of the VEV ⟨ j^l⟩, as a function of L_l , is exponential.In the Minkowski bulk, for the system of a fermionic field and N/2 charged scalar fields, with the same masses, charges and the same phases in the quasiperiodicity conditions, the total current vanishes as a consequence of the cancellation between the fermionic and scalar contributions. This means that, in the corresponding supersymmetric models, no net current appears. In the AdS bulk the influence of the gravitational field on the VEVs for scalar and fermionic fields, in general, is different and there is no cancellation between the scalar and fermionic counterparts. The corresponding total current density is given by Eq. (<ref>). Near the horizon, the leading contributions from the scalar and fermionic parts are canceled, and the first term in the corresponding asymptotic expansion is presented as Eq. ( <ref>). It does not depend on the mass and vanishes for conformally coupled scalars. As a consequence of the weaker divergence on the horizon, the total charge flux integrated over the z-coordinate is finite.In odd spacetime dimensions the mass term in the Lagrangian density for a fermionic field realizing the irreducible representation of the Clifford algebra, in general, breaks C-, P-, and T-invariances. The models with these symmetries can be constructed combining two fermionic fields realizing two irreducible representations. In section <ref> we have considered the current density in this type of models. It has been shown that, if the phases α _l are the same for both the representations, their contributions to the total current coincide. However, the phases need not be the same. This type of situation arises in semiconducting cylindrical carbon nanotubes described by an effective Dirac theory in the long-wavelength approximation. In the corresponding Dirac model two irreducible representations correspond to two inequivalent points of the graphene Brillouin zone and, in the absence of the magnetic flux threading the tube, the corresponding phases have opposite signs. We have considered the fermionic current in the corresponding problem on the AdS bulk generated by a magnetic flux.§ ACKNOWLEDGMENTS A. A. S. was supported by the State Committee of Science Ministry of Education and Science RA, within the frame of Grant No. SCS 15T-1C110. The work was partially supported by the NATO Science for Peace Program under Grant No. SFP 984537. V. V. acknowledges support through De Sitter cosmology fellowship. A. A. S. gratefully acknowledges the hospitality of the INFN, Laboratori Nazionali di Frascati (Frascati, Italy), where a part of this work was done.§ ANOTHER CLASS OF FERMIONIC MODES In the evaluation of the current density we have used the representation ( <ref>) for the Dirac matrices and the corresponding fermionic modes ( <ref>), (<ref>). As it has been discussed in Ref. <cit.>, these modes are well adapted for the investigation of the effects induced by the presence of an additional brane, parallel to the AdS boundary, on which the field operator obeys the bag boundary condition. In this appendix we consider another representation of the Dirac matrices that allows for the separation of the equations for the upper and lower components of the fermionic mode functions. In the new representation the flat spacetime gamma matrices are given byγ ^(0)=( [0 χ _0; χ _0^†0 ] ) , γ ^(l)=( [ 0χ _l; -χ _l^† 0 ] ) ,where l=1,2,… ,D-1, and γ ^(D)=si diag(1,-1) with s=± 1. In odd-dimensional spacetimes, the values s=+1 and s=-1 correspond to two irreducible representations of the Clifford algebra. For the N/2× N/2 matrices χ _0, χ _l one gets the relations χ _lχ _n^†+χ _nχ _l^†=2δ _nl, χ _l^†χ _n+χ _n^†χ _l=2δ _nl for l,n=1,2,… ,D-1, and χ _0χ _l^†=χ _lχ _0^†, χ _0^†χ _l=χ _l^†χ _0, χ _0^†χ _0=1. As before, the curved spacetime gamma matrices are given by γ ^μ=(z/a)δ _b^μγ ^(b). In the special case D=2 we have χ _0=χ _1=1 and γ ^(0)=σ _P1, γ ^(1)=iσ _P2 , γ ^(2)=iσ _P3, where σ _Pμ are the Pauli matrices (see, for instance, Ref. <cit.>).For the positive-energy modes, decomposing the spinor into the upper and lower components, the separate equations are obtained for them with the solutionψ _β^(+)=z^D+1/2e^i𝐤𝐱-iω t( [J_ma+s/2(λ z)χ ^(σ ); 𝐤χ^†+ωχ _0^†/λ J_ma-s/2(λ z)χ ^(σ ) ] ) ,where χ ^(σ ), σ = 1,… ,N/2, are one-column matrices with N/2 rows and 𝐤χ^†=∑_l=1^D-1k_lχ _l^†. From the orthonormalization condition for the modes (<ref>) one obtainsχ ^(σ )†[( ω +𝐤χχ _0^†) ^2+λ ^2]χ ^(σ ^')=2λ ^2| C_β^(+)| ^2δ _σσ ^',where | C_β^(+)| ^2 is given by Eq. (<ref>). This shows that we can take( ω +𝐤χχ _0^†-iλ) χ ^(σ )=√(2)λ C_β^(+)w^(σ ),or invertingχ ^(σ )=C_β^(+)𝐤χχ _0^†+iλ -ω/√(2)iωw^(σ ),where the matrices w^(σ ) are the same as in Eq. (<ref>).As a result, for the positive-energy fermionic modes we getψ _β^(+)=C_β^(+)/√(2)z^D+1/2e^i 𝐤𝐱-iω t( [ 𝐤χχ _0^†+iλ -ω/ω J_ma+s/2(λ z)w^(σ ); iχ _0^†𝐤χχ _0^†+iλ +ω/ωJ_ma-s/2(λ z)w^(σ ) ] ) .In a similar way, for the negative-energy modes one finds the representationψ _β^(-)=C_β^(-)/√(2)z^D+1/2e^i 𝐤𝐱+iω t( [ iχ _0𝐤χ^†χ _0-iλ +ω/ ωJ_ma+s/2(λ z)w^(σ );𝐤χ^†χ _0-iλ -ω/ω J_ma-s/2(λ z)w^(σ ) ] ) .with C_β^(-) defined in Eq. (<ref>). As an additional check, it can be seen that the modes (<ref>) and (<ref>) are orthogonal.With the modes (<ref>) and (<ref>), we can evaluate the fermionic current density by using the mode-sum formula (<ref>). By taking into account that for a N/2× N/2 matrix M we have ∑_σw^(σ )†Mw^(σ )=tr M, the VEVof the component of the current density along the compact direction x^l is presented as⟨ j^l⟩ =-eNz^D+2/4(2π )^pV_qa^D+1∑_ 𝐧_q∫ d𝐤_(p) ∫_0^∞dλ k_lλ/ω∑_j=± 1J_ma+j/2^2(λ z).After the integration over the angular part of 𝐤_(p) this expression is reduced to Eq. (<ref>). As seen, the VEV of the current density does not depend on the parameter s in the definition of the Dirac matrix γ ^(D). In particular, in odd-dimensional spacetimes the current density is the same for two inequivalent irreducible representations of the Clifford algebra. From the derivation of the expression (<ref>) it follows that it is also valid for D=2 model with the compact dimension of the length L. The corresponding expression is obtained putting in Eq. ( <ref>) N=2, p=0, V_q=L, and omitting the integral over 𝐤_(p).§ EVALUATION OF THE MODE-SUM Here we evaluate the mode-sum in the definition (<ref>) for the function ℐ_ν(z). By using the integral representation1/ω=1/√(π)∫_0^∞ds/s^1/2e^-ω ^2s,the integral over λ in Eq. (<ref>) is expressed in terms of the modified Bessel function I_ν(z^2/2s) <cit.>. After the integration over k_(p) one findsℐ_ν(z)=Γ (p/2)/4√(π)∫_0^∞ ds/s^(p+3)/2e^-z^2/(2s)I_ν +1/2(z^2/2s)∑_𝐧 _qk_le^-k_(q)^2s.As the next step we employ the relation∑_n_j=-∞^+∞e^-sk_j^2=L_j/2√(π s) ∑_n_j=-∞^+∞e^in_jα̃ _je^-L_j^2n_j^2/(4s),which is a direct consequence of the Poisson resummation formula. From Eq. ( <ref>) we can see that∑_𝐧_qk_le^-k_(q)^2s=-iL_lV_q/2^q+1π ^q/2s^q/2+1∑_𝐧_qn_le^iα̃·𝐧_q-g_𝐧_q^2/(4s),with α̃·𝐧_q= ∑_i=p+1^D-1n_i α̃ ̃_̃ĩ and g_𝐧_q is defined by Eq. (<ref>).With these relations, the function (<ref>) is presented in the formℐ_ν(z)=-iL_lV_qΓ (p/2)/2^q-(D-3)/2π ^q/2+1z^D+2∑_𝐧_qn_le^iα̃·𝐧_qq_ν^(D+1)/2(b_𝐧_q),where b_𝐧_q is given by Eq. (<ref>). 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"authors": [
"S. Bellucci",
"A. A. Saharian",
"V. Vardanyan"
],
"categories": [
"hep-th",
"cond-mat.mes-hall",
"gr-qc",
"quant-ph"
],
"primary_category": "hep-th",
"published": "20170727142740",
"title": "Fermionic currents in AdS spacetime with compact dimensions"
} |
INT-PUB-17-028Institute for Nuclear Theory, University of Washington, Seattle, Washington 98195, USA Institut de Physique Nucléaire de Lyon, CNRS/IN2P3, Université de Lyon, Université Claude Bernard Lyon 1, F-69622 Villeurbanne Cedex, FranceInstitut de Physique Nucléaire, Université Paris-Sud, IN2P3-CNRS,Université Paris-Saclay, F-91406 Orsay Cedex, FranceCNRS/ENSICAEN/LPC/Université de Caen Basse Normandy,UMR6534, F-14050 Caen Cedex, FranceThe question of the competition between Λ and Ξ^0,- in the ground-state of multi-strange hypernuclei is addressed within a non-relativistic density functional approach, partially constrained by ab-initio calculationsand experimental data. The exploration of the nuclear chart for 10<Z<120 as a function of the strangeness number is performed by adding hyperons to a nuclear core imposing eitherconserved total charge Q or conserved proton number Z. We find that almost all Λ hypernuclei present an instability with respect to the strong interaction decay of Λ towards Ξ^0,- and that most of the instabilities generates Ξ^- (resp. Ξ^0) in the case of conserved total charge Q (resp. proton number Z).The strangeness number at which the first Ξ^0,- appear is generally lower for configurations explored in the case of conserved Q compared to the case of conserved Z, and corresponds to the crossing between the Λ and the neutron or proton chemical potentials. About two to three hundred thousands pure Λ hypernuclei may exist before the onset of Ξ^0,-. The largest uncertainty comes from the unknown ΛΞ interaction, since the NΛ and the NΞ ones can be constrained by a few experimental data. The uncertainty on the ΛΞ interaction can still modify the previous estimation by 30-40%, while the impact of the unknown ΞΞ interaction is very weak. Density Functional approach for multi-strange hypernuclei: competition between Λ and Ξ^0,- hyperons F. Gulminelli December 30, 2023 =====================================================================================================§ INTRODUCTION Since the discovery of the first hypernucleus in an emulsion exposed to cosmic rays <cit.>, single and double-Λ hypernuclei, as well as single-Ξ ones, have been synthesised and some of their ground state properties have been measured <cit.>.It is further expected from theoretical calculations that multi-strange hyperons remain bound up to a large number of hyperons <cit.>, but precise predictions require reliable hyperon interactions. The scarce amount of data however makes the hyperon interactions still rather unknown.Depending on the hyperon interaction, hyperon might or might not appear indense matter – hypernuclear matter – which exists in the inner core of neutron stars <cit.>. Finite hyper-nuclei and neutron stars are therefore the two systems which can provide constraints on the hyperon interactions.New dedicated experimental programs such as Japan Proton Accelerator Research Complex (J-PARC) in Japan and the proton antiproton detector array at GSI Facility for Antiproton and Ion Research (FAIR) are or will be providing new data which participate to a better understanding of the properties of hypernuclei <cit.>. The physics of hypernuclei opens a new direction in the exploration of the nuclear chart which is complementary to the direction towards more and more exotic nuclei.Hypernuclei are interesting finite nuclear systems since they allow to study the properties of bound strange hadrons and to test the behaviour of the baryon-baryon interaction. The representation of the nuclear chart, traditionally expressed in terms of the number of protons Z and neutrons N,acquires a new dimension associated to its strangeness number S. For a given strangeness number S, several configurations corresponding to different hyperons can be considered.The charge neutral and lightest hyperon Λ happens to be also the most bound, and single Λ hypernuclei havebeen synthesized through the nuclear chart, providing information such as global masses and single particle energies for most of them <cit.>. These data are important to reduce the uncertainties of the NΛ interaction, at least at very low density, as shown for instance in Refs. <cit.>. Multi-strange hypernuclei are still one of the least-explored, open questions in hypernuclear physics, from both experimental and theoretical viewpoints <cit.>. Data on double-Λ hypernuclei are very scarce, mostly because the production rates are low. A few of them are nevertheless known, such as ^6_ΛΛHe or ^11_ΛΛBe, allowing to extract the bond energy which is expected to be a measure of the ΛΛ interaction, here also at very low density <cit.>.The existence of an extra binding associated to a double-hyperon system implies that the ΛΛ interaction is at least marginally attractive, opening the possibility of multi-strange systems with a higher number of hyperons <cit.>.In particular, the production of multi-strange hypernuclei may be favoured during the cluster formation phase in relativistic heavy-ion collisions, since they usually lower the binding energy per particle <cit.>.From the theoretical point of view, there have been many relativistic and non-relativistic density functional approaches which were developed and applied to the prediction of the structure of hypernuclei, see Refs. <cit.> for a few of them. Multi-strange hypernuclei with more than two hyperons were first discussed in Ref. <cit.>, anda large variety of phenomena have been predicted for such nuclei during the 80's and the early 90's <cit.>.However, these studiesassumed very attractive hyperon-hyperon interactions, inspired by the first analyses of double-Lambda ^10_ΛΛBe and ^13_ΛΛB data which suggested a large bond energy Δ B_ΛΛ≈ 5 MeV <cit.>.Therefore, it may be interesting to check these predictions with a density functional approach including the latest phenomenological constraints. Since our knowledge on the hyperon interaction remains quite poor, it may also be interesting to evaluate to which extent this lack of knowledge impacts the predictions of multi-strange hypernuclear properties.In a previous work we have discussed hyperons and hypernuclear matter made of nucleons (N) and Λ particles <cit.>. It is however expected that multi-strange hypernuclei including also other hyperons, such asΞ^0,-, could be bound at large values of the strangeness numberS <cit.>. For a given S, these complex configurations might even correspond to the ground state of the multi-baryon system, meaning that the hypernuclear chart at large -S <cit.> should take into account all the possible hyperons. In this work, we want to investigate the properties of multi-strange hypernuclei with Λ and Ξ^0,- within non-relativistic density-functional theory which has proven to give a very good description of normal nuclei <cit.> and Λ-hypernuclei <cit.>, as well as Ξ-hypernuclei <cit.>.Supposing an initial hypernucleus made of nucleons and Λs,there are three kinds of possible strong interaction decays: i) reactions transforming Λ into Σ^±,0,Λ + n →Σ^0 + n, Λ + p →Σ^+ + n, Λ + n →Σ^- + p,with average free-reaction Q_Σ^free≈ -80 MeV; ii) reactions transforming two Λ into Ξ^0,-,Λ + Λ→Ξ^- + p, Λ + Λ→Ξ^0 + n,with average free-reaction Q_Ξ^free≈ -26 MeV; and finally iii) reactions transforming three Λ into Ω^-,Λ + Λ + Λ→Ω^- + n + p,with average free-reaction Q_Ω^free≈ -180 MeV. The Q^free-values make the previously listed decays non-favorable, and the hypernuclei with only Λ are usually preferred. However, the Q^free-values takes into account only the mass of the particles, while in dense matter as well as in finite nuclei there is an additional quantum effect induced by Pauli blocking: because of the Fermi energy, the total energy of hypernuclei with a large amount of Λ may become larger than the total energy of a system where some Λ are converted into other hyperons, leading to a positive Q^free-value in the medium. Moreover, the presence of other baryons in hypernuclei generates a potential field in non-relativistic approaches, or in-medium mass shift in relativistic approaches, which in turn shifts the Q^free-values. The Coulomb interaction contributes also to the mean-field, and shifts it down for negatively charged hyperons,making them more favored. Then, the minimum energy configuration for a fixed value of the quantum number set (A,Q,S), may have a finite amount of Ξ, or Σ, or Ω particles. The Q^free-values give however a reasonable hierarchy in the formation of new systems: it is expected that it will be easier to decay from Λ to Ξ, than to Σ and Ω. In this work, we therefore extend our previous analysis of the hypernuclear chart <cit.> considering the possible decay of Λ into Ξ^0,- (hereafter called the Ξ instability). The detailed study of the general properties of multi-strange hypernuclei is left to a future work.In the present work, we systematically look for the strangeness threshold associated to the appearance of Ξ^0,- in the hypernuclear ground state.To calculate the Ξ^0,- instability threshold, we consider a core-nucleus (A_core, Z_core) in between the drip-lines, and addstrangeness distributedover Λ and Ξ^0,- types of hyperons. Fixing the three conserved charges of the strong interaction, namely the baryon number A, the total charge Q and the strangeness number S, the ground state multi-strange hypernuclei is given by the one which minimizes the energy.This criterium corresponds to defining the stable configuration with respect to strong decays, and univocally defines the hypernuclear ground state.To compare to some results in the literature, we also consider another convention: fixing Z and adding strangeness on top of an core-nucleus (A_core, Z_core), as it was done for instance in Refs. <cit.>. It should be noted that a third strategy could be considered, which consists in adding strangeness at conserved total mass A, see Refs. <cit.> for instance.It should be stressed thatthese strategiesare convenient pictures to understand the effect of adding strangeness to ordinary nuclei, but none of them reflects exactly the present possibilities for the experimental production of multi-strange hypernuclei. In particular in HIC, the reactions forming multi-strange hypernuclei can certainly produce extra excited states which do not correspond to the minimum energy at conserved Q. The outline of the present work is as follows. In Sec. <ref>, we propose a non-relativistic density-functional approach to treat multi-strange hypernuclei.We first briefly recall in Sec. <ref> the formalism already used in our previous work <cit.>, and propose in Sec. <ref> a minimal extension to include the full baryonic octet with the inclusion of 10 additional coupling constants. The multi-strange hypernuclear chart is studied in Sec. <ref>.After a brief description of the numerical strategy (Sec. <ref>), the instability threshold corresponding to the onset of Ξ^0,- hyperons in the ground state of hypernuclei is computed in Sec. <ref> and the number of pure-Λ hypernuclei is calculated in Sec. <ref>.We show that the use of realistic NΛ, ΛΛ and NΞ interactions modifies the predictions with respect to previous results in the literature.The effect of the other largely unconstrained YY couplings is also analysed and we show that the ΛΞ interaction channel is the most influential one. Finally, conclusions and outlooks are presented in Sec. <ref>.§ DENSITY FUNCTIONAL THEORY FOR MULTI-STRANGE HYPERNUCLEIIn the present work, we consider the most general non-relativistic system composed of interacting nucleons N (neutrons and protons) and hyperons, hereafter noted Y forΛ, and Ξ^0,-. Notice that the extension to the other hyperons, Σ^0,±, and eventually Ω^-, is straight-forward, but will not be considered in this paper. The total Hamiltonian reads, Ĥ= ∑_i=N,Yt̂_i + ∑_i,j=N,Yv̂_ij^NY + 1/2∑_i=Nv̂_ii^NN + 1/2∑_i=Yv̂_ii^YY . In the following, we will consider the density functional theory which allows relating in a direct way the microscopic Brueckner-Hartree-Fock (BHF) theory for uniform matter based on the Nijmegen interactions, to the properties of hypernuclei.§.§ Energy-density functional for N and Λ hypernuclear matterIn a previous study of hypernuclei and nuclear matter <cit.> we used a density functional whichwas determined directly from the BHF theory including nucleons and single Λ-hyperons <cit.>. Here we recall the main equations and refer the reader to Ref. <cit.> for more details. The total energy density ϵ(ρ_N,ρ_Λ) is related to the energy per particle of infinite nuclear matter calculated within the BHF framework, e_BHF, as ϵ(ρ_N,ρ_Λ)=(ρ_N+ρ_Λ) e_BHF(ρ_N,ρ_Λ) and is decomposed in different terms,ϵ(ρ_N,ρ_Λ)= ħ^2/2m_Nτ_N+ ħ^2/2m_Λτ_Λ+ϵ_NN(ρ_N) +ϵ_NΛ(ρ_N,ρ_Λ) +F_Λϵ_ΛΛ(ρ_Λ) ,where τ_N and τ_Λ arethe kinetic energy densities, and the term ϵ_NΛ is parameterized in terms of the nucleon and hyperon densities as <cit.>,ϵ_NΛ(ρ_N,ρ_Λ)=-f_1^NΛ(ρ_N) ρ_Nρ_Λ + f_2^NΛ(ρ_N)ρ_Nρ_Λ^5/3,Here the first term physically corresponds to the attractive NΛ interaction, corrected by the presence of the medium given by the function f_1, and the second term is induced by the repulsive momentum dependent term of the Λ potential (considering the low-momentum quadratic approximation), also corrected by the medium through the function f_2.Some repulsion is indeed necessary at high density for the Λ in nuclear matter, as fits to single Λ hypernuclear data have revealed <cit.>. In the presence of the attractive ΛΛ interaction, the term ϵ_ΛΛ is solely determined by the hyperon density as <cit.>,ϵ_ΛΛ(ρ_Λ)=-f^ΛΛ(ρ_Λ)ρ_Λ^2 .To avoid self-interaction, the factor F_Λ in the functional Eq. (<ref>) is 0 if there is only one Λ and 1 for more.The functions f are given by the polynomial forms,f_1^NΛ(ρ_N)= α_1^NΛ-α_2^NΛρ_N+α_3^NΛρ_N^2 , f_2^NΛ(ρ_N)= α_4^NΛ-α_5^NΛρ_N+α_6^NΛρ_N^2 , f^ΛΛ(ρ_Λ)= α_1^ΛΛ-α_2^ΛΛρ_Λ+α_3^ΛΛρ_Λ^2. The values for the parameters α_1^NΛ-α_6^NΛ were determined in Refs. <cit.> from a fit of the BHF infinite nuclear matter calculations performed with different NΛ potentialswhich equally well fit the available NΛ phase shifts. In this workwe will use the models DF-NSC89, DF-NSC97a and DF-NSC97f, which parameters are given in Tabs. <ref> and <ref>.It should be noted that no direct experimental information is available on ΛΛ scattering, meaning that these phenomenological bare interactions are rather unconstrained in the ΛΛ channel.For this reason, NSC89 does not contain any ΛΛ interaction. The NSC97a-f models assume for this channel a simple SU(3) extension of the original Nijmegen potential models to multiple strangeness S=-2 <cit.>. For these models, the energy density associated to the ΛΛ interaction is expressed asϵ_ΛΛ=-(α_1^ΛΛ-α_2^ΛΛρ_Λ+α_3^ΛΛρ_Λ^2)ρ_Λ^2 .It turns out that these models do not lead to a satisfactory description of the bond energy of double-Λ hypernuclei <cit.>, which is the only empirical information that we have on ΛΛ couplings <cit.>.For this reason, in our previous work in Ref. <cit.>, we have empirically modified the α_1^ΛΛ-α_3^ΛΛ parameters such as to reproduce the measured binding energy of ^6_ΛΛHe. It should be noted that the parameters α_2^ΛΛ-α_3^ΛΛ control the high density behavior of the ΛΛ interaction.In that same work, we found that the global properties of Λ-hypernuclei were not impacted by the high density behavior of the ΛΛ interaction <cit.>. Therefore the parameters α_2^ΛΛ-α_3^ΛΛ have no impact in double-Λ hypernuclei, since the Λ density in these systems remains rather small. In the present work, by including additional hyperons, the Λ-density in multi-Y hypernuclei is expected to be even further reduced compared to the case of pure Λ-hypernuclei. We therefore simplify the ΛΛ interaction as expressed in Eq. (<ref>), to its first term as,ϵ_ΛΛ=- α_1^ΛΛρ_Λ^2 .In the following, we will refer to the modification of the ΛΛ interaction Eq. (<ref>) as the EmpC prescription.The parameter α_1^ΛΛ can still be approximately related to the average bond energy expected from a local density approximation Δ B_ΛΛ and the average density of Λ inside the nucleus, x_Λ=ρ_Λ/ρ_0 (see Ref. <cit.> for details), asα_1^ΛΛ=1/2Δ B_ΛΛ/ρ_0 x_Λ .A recent publication questions the validity of the local density approximation <cit.>, and points out the fact that the Λ potential obtained imposing Eq. (<ref>) with a constant value x_Λ=1/5 depends on the chosen functional, and so does the corresponding bond energy obtained by a direct HF hypernuclear calculation. If however the value of x_Λ is consistently obtained for each interaction model by a self-consistent HF calculation, we have shown in Ref. <cit.> that this simple prescription leads to very precise results. This point is demonstrated for the EmpC prescription in Table <ref>, whichshowsthe final values for α_1^ΛΛ for SLy4 and the ΛN potentials, DF-NSC89, DF-NSC97a and NSC97f, together with the HF results for the bond energy and the ratio of the average Λ-density to the saturation density in He obtained from our Hartree-Fock calculations.The resulting bond energy is very close to the value Δ B_ΛΛ=1 MeV imposed by Eq. (<ref>), provided the consistent value of x_Λ obtained in the ^6_ΛΛHe ground state, and given in Table <ref>, is used. In the nucleon sector, we use the SLy4 parametrization of the phenomenological Skyrme functional including non-local and spin-orbit terms, since itis the NN interaction which has been used to calibrate the ΛΛ interaction <cit.>, and it can correctly reproduce the properties of stable and exotic nuclei <cit.>. A three body YNN repulsive interaction has recently been proposed in relation with the hyperonization puzzle <cit.>: usual NN and NY interactions fitted on phase shifts cannot predict neutron star masses above about 1.6 M_⊙, while such objects were recently observed <cit.>. This YNN repulsive interaction was originally introduced to improve the agreement between the experimental Λ-separation energies and the one predicted from an Argonne like two-body potential <cit.>.It is interesting to remark that the functional we use does not include a bare NNY interaction, and still is able to well reproduce the experimental Λ-separation energies <cit.>. This seemingly contradictory result can be qualitatively explained by the fact thatthe Nijmegen functional (<ref>) has indeed a two-body induced NNY term, the α_2^NΛ term. It would however be interesting to have a functional form adjusted to the ab-initio calculations reported in Refs. <cit.> for future systematic applications to multi-hypernuclei such as in this work.§.§ Generalization of the energy functional for N and multi-Y hypernuclear matter In this section, we propose a general and simple density-functional considering the full hyperon octet. The functional form (<ref>) is generalized in order to include Y=Λ and Ξ^0,- hyperons, asϵ = ħ^2/2m_Nτ_N +ϵ_NN(ρ_N)+ ∑_Yħ^2/2m_Yτ_Y+∑_Yϵ_NY(ρ_N,ρ_Y) + ∑_Y_1,Y_2ϵ_Y_1Y_2(ρ_Y_1,ρ_Y_2) +∑_YF_Yϵ_YY(ρ_Y) .Using the isospin invariance of the strong interaction and neglecting spin dependence for simplicity, we suppose that the general-functional (<ref>) depends only on the following densities ρ_N, ρ_Λ, and ρ_Ξ=ρ_Ξ^0+ρ_Ξ^-. It should be noted that while the spin-orbit interaction between N and Λ is known to be small <cit.>, it is not certain that it is also the case for the interaction channels involving other hyperons. Due to the lack of data to set the spin-orbit interaction strength, we neglect the spin-orbit interactions in all NY and YY channels in the present work. In Eq. (<ref>), the parameters F_Y areintroduced in finite systems to avoid self-interactions: F_Y=1 if the associated number of Y, N_Y≥ 2, F_Y=0 otherwise. The mean-field potentials are deduced from the functional (<ref>) by using functional derivative <cit.>. Fig. <ref> displays the NΛ and the ΛΛ mean fields defined as v_XY^unif=∂ϵ_XY/∂ρ_Y for the functionals DF-NSC89, DF-NSC97a and DF-NSC97f. On the left panel the NΛ potential is shown for two cases: in the absence of Λ and for ρ_Λ=0.1 fm^-3. On the right panel, 30% of the baryons are taken as Λ. The NΛ mean field is consistent with the empirical expectation (box on Fig. <ref>), and a finite amount of Λ decreases the depth of the mean field, as expected <cit.>. There is a qualitative difference between the functional DF-NSC97a and the two others for a small amount of Λ: the functional DF-NSC97a is much more attractive than the two others, which are rather equivalent. We will see in the following that hypernuclei predicted by DF-NSC97a are consequently more bound. On the right panel, the prescriptions EmpB2 from Ref. <cit.> and the present simplified functional quadratic in density EmpC are compared for theΛΛ channel. It is shown that up to a large amount of Λ (30%) and at saturation density, there is almost no difference between the two prescriptions EmpB2 and EmpC for the ΛΛ interaction. This implies that the parameters α_2^ΛΛ-α_3^ΛΛ can indeed be ignored for hypernuclei as discussed above, and that the density dependence of the ΛΛ interaction cannot be constrained by hypernuclear physics, as concluded in our previous study <cit.>.To study this feature in more details, Fig. <ref> displays the evolution of the binding energies for a few illustrative nuclei as a function of the strangeness number -S: ^40-SΛCa, ^56-SΛNi, ^120-SΛSn, ^208-SΛPb. The calculations are stopped at the Ξ-instability. The similarity in the predictions for the ΛΛ interactions given by EmpB2 and its simplified version EmpC is demonstrated in Fig. <ref>. This figure clearly shows that the high density contribution of the ΛΛ interaction, namely the parametersα_2^ΛΛ-α_3^ΛΛ in Eq. (<ref>) which are neglected in the present parametrization EmpC, has no contribution to the mean field in multi-Λ-hypernuclei. This is consistent with our previous conclusions in Ref. <cit.> and confirms that the hyperon density in hypernuclei remains too low to provide information on the hyperon matter above saturation density.As expected from previous works (see Ref. <cit.> and references therein) the binding energy increases in absolute value when hyperons are added to normal nuclei (see Fig. <ref>) and for this reason it was suggested thatmulti-Λ hypernuclei may be formed in heavy-ion collisions <cit.>.As we have stressed in the introduction, this can be understood from the fact that adding hyperons opens new degree of freedom for which the Fermi energy is small.As the number of Λ increases this effect is less and less important, and finally, the binding energy saturates with the number of Λ, for a large number of Λ. For an even larger number of Λ, the total binding energy increases, and thus the chemical potential μ_Λ becomes positive indicating that theΛ drip-line is met (and the calculation is stopped at that point).For a large number of Λ one might expect that it could again be energetically favoured to add new types of hyperons, such as the Ξ^0,-. It should be noted however that when we are comparing different possible ground states configurations conserving A, S and Q (or Z),the final result might not be easy to anticipate because it depends on different competing effects. First, the highermasses ofΞ^0,-, Σ^0,± and Ω^- bring a penalty for the onset of an addition of these hyperons (see the discussion of the Q^free-values in the introduction of this work). In addition, the interaction of these new hyperons with the nucleons, which is the dominant contribution, might be attractive or repulsive, impacting their mean-field potential. The attractive Coulomb interaction for negatively charged hyperons could help the binding, as we will show in Sec. <ref>.Finally, the higher mass also induces a reduction of the kinetic energy of these particles,which could therefore slightly counterbalance the effect of a weaker interaction. All these phenomena are naturally included in our framework and in the following, we will study all their combined effects in more details.We now discuss the Ξ channel. This channel is much less known than the Λ one. The Ξ density is expected to remain quite low, even lower than the Λ-density in the case of pure Λ-hypernuclei <cit.>, since they shall be less numerous than the Λ, which is expected to be the more bound hyperon. In addition, the effective masses of the Ξ is assumed equal to their bare masses. This is justified from recent Bruckener-Hartree-Fock calculations <cit.> for the Ξ and it was also assumed in recent density-functional approaches <cit.>. Indeed, if we are only interested in ground-state properties, the effective masses can be incorporated in a deeper mean-field.Since we do not know much concerning the mean field or the effective masses of these hyperons, it is simpler to assume the effective mass equal to the bare mass, and eventually alter the depth of the mean-field. Following the previous arguments,theNY (Y=Ξ^0,-)terms of the potential energy density functional are givena quadratic density dependence, as obtained for a simple two-body effective interaction:ϵ_NY(ρ_N,ρ_Y) = -α^NYρ_Nρ_Y , and the same is assumed for theYY^' terms (Y^'=Λ and Ξ^0,-):ϵ_YY^'(ρ_Y,ρ_Y^')=-α^YY^'ρ_Yρ_Y^' ,leading to the definition of three additional constants. The parameter α^NΞ can be determined by imposing the Ξ-potential in uniform matter v_Ξ^unif=∂ϵ/∂ρ_Ξ to be equal to the empirical value U_Ξ at saturation density in the absence of Λ and Ξ, leading to:α^NΞ =-U_Ξ/ρ_0A value of U_Ξ≈ 14 MeV is deduced from the analysis of the spectrum of the (K^-,K^+) reaction on a ^12C target to produce ^12_ΞBe, assuming a Woods-Saxon potential for the Ξ^- potential <cit.>. This yields α^NΞ≈ 100 MeV fm^3. Another, and maybe more direct way, to determine the parameter α^NΞ is to calculate the Ξ^- removal energy B_Ξ^- defined as,B_Ξ^- = E_tot(N,Z)-E_tot(N,Z,N_Ξ^-),where E_tot is the total binding energy, that is the total energy with subtraction of the rest mass term <cit.>. This allows to compare to two experimental energies for ^12_Ξ sBe <cit.> (with N=6, Z=5) and^15_ΞC <cit.> (with N=7, Z=7), also called the "Kiso" event. The latter experimental data is however subject to two possible interpretations: 1) assuming that ^15_ΞC is produced in its ground-state (Ξ being in the 1s single particle state), or 2) assuming that ^15_ΞC is produced in its first excited-state (Ξ being in the 2p single particle state). A recent theoretical analysis based on mean-field theory has shown that interpretation 2) is also compatible with the removal energy of ^12_Ξ sBe <cit.>. We have performed a similar analysis with our density-functional as illustrated in Tab. <ref>.In our model, the Ξ^- removal energies B_Ξ^- is uniquely determined by the parameter α_1^NΞ, which we vary around 100 MeV fm^3. It is shown in Tab. <ref> that the value α^NΞ=109 MeV fm^3 which reproduces well the Ξ^- removal energies supposing ^15_ΞC in its ground-state is not compatible with the expected Ξ^- removal energies of ^12_Ξ sBe, while the value α^NΞ=125 MeV fm^3 which reproduces well the Ξ^- removal energies supposing ^15_ΞC in its first excited-state gives reasonable results for the expected Ξ^- removal energies of ^12_Ξ sBe and^15_Ξ sC <cit.>. In the following, we thus fix α_1^NΞ=125 MeV fm^3.In the case of the hyperon-hyperon couplings, since it is yet impossible to fix the values of these parameters from experimental data, they are normalized to better known parameters such as α^NΛ, α^NΣ α^NΞ and α^ΛΛ.The following dimensionless parameter are therefore introduced:β^YY^'=α^YY^'/α^NY^',Y≠ Y^'with Y=Ξ and Y^'=Λ and Ξ,β^YY=α^YY/α^ΛΛ. The parameters β^YY^' and β^YY are largely unknown. They are expected to be less influential than the parameters in the NY^' and ΛΛ channels since they act between minority species. The channels of interest in the present study are ΛΞ and ΞΞ. For the same argument related to the number of particles, we expect i) these channels to be rather weak, and ii) that the ΛΞ channel is more influential than the ΞΞ channel . Since the interaction in the ΛΞ and ΞΞ channels is unknown, it is difficult to do more than a sensitivity analysis. We have defined three models, VY0-2, for the sensitivity analysis which are given in table <ref>.The sign of the interaction parameters is not relevant for the sensitivity analysis, and it is arbitrarily chosen positive. The influence of these choices will be studied in Section <ref>. The mean field potentials in uniform matter v_Λ^unif=∂ϵ/∂ρ_Λ,and v_Ξ^unif=∂ϵ/∂ρ_Ξ are displayed in Fig. <ref> as a function of thenucleon density ρ_N (in units of the saturation density ρ_0) for the functional DF-NSC89+EmpC+VY0 and for various choices of the densities ρ_Λ and ρ_Ξ expressed in fm^-3 (a), and for various strength of the ΛΞ interaction (b).Let us first discuss the potential in uniform matter v_Λ^unif(ρ_N,ρ_Λ,ρ_Ξ) shown in Fig. <ref>(a). The addition of a finite amount of Λ for ρ_Λ=0.03 fm^-3 to standard nuclear matter, increases the value of the potential (by about 5 MeV at ρ_0), while the addition of the same amount of Ξ decreases the potential (by about -5 MeV at ρ_0). The larger the Λ N repulsion (ΛΞ attraction) the shallower (deeper) the potential. It should also be noted that the very same attractive term in the energy functional ϵ_ΛΞ is responsible for the decrease of v_Λ^unif(ρ_N,ρ_Λ,ρ_Ξ)and the decrease of v_Ξ^unif(ρ_N,ρ_Λ) by adding Ξ. If it was not for the cost in rest mass, it would therefore be preferable to addΞ-hyperons than Λ-hyperons, due to the gain in the Λ potential, as well as the reduced Pauli blocking of Ξ single-particle states in NΛ matter. If instead of being attractive, the ΛΞ channel is repulsive, the effect at the level of the potential would be opposite to the present case.The potential v_Ξ^unif(ρ_N,ρ_Λ) is displayed in Fig. <ref>(b) for various choices of the strength of the ΛΞ interaction (represented by the parameter β^ΛΞ) and for a fixed amount of Λ.This figure shows that the larger is the parameter β^ΛΞ, the more attractive is the potential v_Ξ^unif(ρ_N,ρ_Λ), as expected. § THE EXTENDED HYPER-NUCLEAR CHART Minimizing the total energy defined from the density functional (<ref>), and using theSkyrme model for the nucleonic part <cit.>, we obtain the usual Schrödinger equation (i=N,Y),[-∇·ħ^2/2 m^*_i(r)∇+V_i(r)-iW_i(r)(∇×σ)]φ_i,α(r)=-e_i,αφ_i,α(r),where W_i is the spin-orbit potential <cit.> and the nucleon potential V_N is defined as,V_N(r) = v_N^Skyrme+ ∂/∂ρ_N( m_Λ/m_Λ^*(ρ_N)) × ( τ_Λ/2m_Λ-3/5(3π^2)^2/3ħ^2/2m_Λρ_Λ^5/3) ,The Λ-hyperon potential V_Λ is given by,V_Λ(r) = v_Λ^unif -( m_Λ/m_Λ^*(ρ_N) -1) (3π^2)^2/3ħ^2/2m_Λρ_Λ^2/3 .and the Λ effective mass m_Λ^* determined from BHF calculations <cit.> is expressed asm_Λ^*(ρ_N)/m_Λ = μ_1^NΛ-μ_2^NΛρ_N+μ_3^NΛρ_N^2-μ_4^NΛρ_N^3 .The values for the parameters μ_1-4 for the functional considered here are given in Table <ref>.For the Ξ hyperon potentials, we have the following relationV_Ξ(r)=v_Ξ^unif=∂ϵ/∂ρ_Ξ. It should be noted that in the case of Ξ^-, an additional contribution to the Coulomb potential shall be considered. The Coulomb energy is generated by the Coulomb interaction among charged particles p and Ξ^-. It is decomposed into a direct term,E_Coul^D= ∑_i jsgn(i) sgn(j)e^2/2∫ d^3𝐫 d^3𝐫'ρ_i(r) 1/|𝐫-𝐫'|ρ_j(r') ,where i,j=p,Ξ^- and sgn(i) is the sign of the Coulomb charge of particle i. It should be noted that the pΞ^- channel is attractive with respect to the Coulomb direct interaction, which could favour the onset of the Ξ^- hyperon against Ξ^0.Considering the Slater approximation, the exchange term readsE_Coul^E=- e^2 3/4( 3/π)^1/3∫ d^3𝐫( ρ_p^4/3 + ρ_Ξ^-^4/3) .The exchange term is attractive for all charged particles, favouring again Ξ^- against Ξ^0.The contribution of the Coulomb interaction to the mean fields are obtained by functional derivation of Eqs. (<ref>) and (<ref>), giving from the proton direct term,u^D_Coul, p(r)=e^2 ∫ d^3𝐫'1/|𝐫-𝐫'|ρ_ch(r') .where the charge density is ρ_ch= ρ_p-ρ_Ξ^-. It should be noted that the direct Coulomb terms for all other particules are defined exactly thesame as for the proton case, with only a sign difference which refers to the charge of the considered hyperon. u^D_Coul, Ξ^-(r)=- u^D_Coul, p(r) .We consider the extension of the Slater approximation for multi-types of charges particles, giving for the exchange Coulomb potentialu^E_Coul, i(r)=- e^2( 3/π)^1/3∫ d^3𝐫'{ρ_i(r')}^1/3,where i=p, Ξ^-. As we noticed, the Coulomb interaction favour the onset of negatively charged particles over neutral ones. The Ξ^- hyperon could therefore be favoured against Ξ^0.§.§ Numerical strategy The HF equations are solved in coordinate representation assuming spherical symmetry. Deformations are known to play an important role in the structure of light hypernuclei <cit.>, however this approximation is expected to hold at the level of accuracy in our work. Indeed deformation induces corrections to energies which approximately scale as A^-1/6, and can be neglected when calculating energy differences of nuclei with A>20 as it is done in this work for the calculation of the Ξ-instability phenomenon.To correct for the spurious one-body center of mass energy,the mass m_i of each species i in the Schrödinger equation is replaced by the reduced mass m^'_i, defined as (m^'_i)^-1 = m^-1_i - (∑_j i N_j m_j)^-1, where i and j indexes run over N and Y=Λ, Ξ^0,-. The Numerov method is used to determine the wave-functions φ_i,α(r) for given potentials V_i(r) and W_i(r) as well as given effective mass m^*_i(r) and we consider the vanishing wave-function Dirichlet boundary condition.The coordinate space extends up to 30 fm and it is discretised with equal steps of 0.1 fm. Masses of particles are fixed to be their bare masses, except for neutrons, protons and Λ which acquire an effective mass in dense medium. As usual in HF solvers, the self-consistency is reached by successive iterations until the total energy converges within an accuracy of less than 10^-8 MeV. Further details about the implementation of the Hartree-Fock approach in the hypernuclear case can be found in Refs. <cit.> for instance. When considering nucleons and hyperons (here Λ and Ξ^0,-), the three conserved charges of the strong interaction are defined as:A = N_n+N_p+N_Y,Q = N_p-N_Ξ^-,-S = N_Λ+2N_Ξ,where the hyperon numbers areN_Y = N_Λ+N_Ξ,N_Ξ = N_Ξ^++N_Ξ^0In the following, hyperons are added on top of a core-nucleus (A_core, Z_core) with step in strangeness Δ S=2. The S-drip line (S-DL) is the drip line in the strangeness number S. This is the maximum value for -S before the chemical potential of any of the hyperon becomes positive.As discussed in the introduction, according to the free space Q-values, the hypernuclear ground state is expected to contain only Λ's for low strangeness -S, followed by Ξ^0,- when the number of Λ's is sufficiently high for the gain in kinetic energy to compensate the energy cost in rest mass.TheΞ^0,- instability is defined as the strangeness number -S at which the first Ξ^0,- appear in the hypernucleus ground state. This number is called in the following S_inst.. In the present work we limit the exploration in S up to S_inst.. The appearance ofΞ^0,- is given from the comparison of the energies of the Λ-hypernucleus to all other ground states formed by Λ and Ξ^0,- with same mass A, total charge Q (or proton charge Z, in the case where constant Z transformations are considered), and strangeness S numbers. Especially, we compare the energy of the systemA_core+N_Λ+N_Ξ to the energy of all the systems composed of (A_core+1)+(N_Λ-2)+(N_Ξ+1) , which correspond to the transformation of 2Λ into one Ξ and a N. An illustration of the search for the Ξ^0,- instability is shown in Fig. <ref>, for the case A=132, Q=50.The different colors correspond to different number of Ξ as indicated in the box. Up to the strangeness number -S≈ 20 the configuration with only Λ's corresponds to the hypernucleus ground state.The energy difference between the configurations with a given number of Ξ is due i) to the slight mass difference between Ξ^0 and Ξ^-, ii) to the Coulomb energy and iii) to the Pauli blocking effect. In this case, for same N_Ξ groups the lowest energy configurations are always the ones with the largest number of Ξ^-. The sharp energy drops reflect shell closures.Considering the lowest energy configuration for each N_Ξ groups, the energy hierarchy scales well with N_Ξ up to S≈ -20 but at larger values of -S, the different configurations are highly degenerated, and the composition of the actual ground state shall depend on the hypothesis for the unknown couplings.These unknown coupling are varied from VY0 to VY2, see Table <ref>. The Ξ^0,- instability is only weakly impacted by the choice for the unknown couplings, while the ground-state energy beyond the Ξ^0,- instability is largely impacted. The largest uncertainty comes from the ΛΞ interaction (VY1), while the ΞΞ interaction (VY2) seams to be weakly influential, even for a finite amount of Ξ. It is not surprising that the largest impact comes from the NΞ channel (VY0) since N is the dominant species, then comes the ΛΞ channel (VY1) and finally the weakest channel is the ΞΞ (VY2) one. For this same reason, the influence of the repulsive Coulomb interactionbetween Ξ^- turns out to be very small. In the absence of ΛΞ interaction (case VY0) it should be noted that the configurations with a same number of Ξ^-'s are almost degenerate beyond S=S_inst.. The introduction of ΛΞ interaction breaks this quasi-degeneracy.§.§ The Ξ^0,- instability over the nuclear chartWe now turn to a systematic exploration of the Ξ^0,- instability over the nuclear chart and compare the predictions of the different functionals. As already discussed in the introduction, the onset of Ξ^0 is slightly favoured over Ξ^- from the mass differences. However in dense matter, opposite effects coming from the kinetic energy term and the Coulomb interaction may play an important role. The final results of these contradictory tendencies reflect in the chemical potentials. We thus define the following quantities: Δμ(Ξ^-)≡μ_Ξ^-+μ_p-2μ_Λ and Δμ(Ξ^0)≡μ_Ξ^0+μ_n-2μ_Λ, where the chemical potentials are defined without rest mass. In order to evaluate the mean field contribution to the onset properties, the evolution of the chemical potentials is displayed in Fig. <ref> up to the S-DL for pure-Λ hypernuclei. The following typical core nuclei are considered: ^40Ca, ^132Sn, and ^208Pb, on top of which strangeness is added.The position of the S_inst. is indicated by the vertical arrows for conserved Q (purple arrow) or conserved Z (green arrow).As expected from the Coulomb interaction, the chemical potential of the Ξ^- is lower than that of the Ξ^0 for all hypernuclei. The difference between these chemical potentials is already of about 5 MeV for the lightest system shown in Fig. <ref>, and reaches about 40 MeV for the heaviest nuclei.This observation confirms the specific role played by the Ξ^-, especially for systems studied at conserved Q. It highlights the contribution of the Coulomb field in the correction to the mean field for negatively charged particles. We can therefore anticipate that Ξ^- will certainly appear as the first particle in most of the cases.Moreover, it should be noted that the Ξ-instability at conserved Q occurs when the Λ chemical potential crosses the neutron or proton chemical potential. At conserved Z, the Ξ-instability is observed for larger values of the strangeness number -S. There is no Ξ-instability at conserved Z in Ca, and as the mass increases, the Ξ-instability at conserved Z comes closer to that at conserved Q. The reason is because in light systems, there is a big gap between the onset of the Ξ^- (first appear at conserved Q) and of the Ξ^0 (single system allowed at conserved Z). This energy gap tends to become less and less important as the charge of the system increases. We now come to more systematics by analyzing Ca, Sn and Pb isotopes. Fig. <ref> displays a comparison of the S-drip line obtained for Λ-hypernuclei (black lines), as obtained in Ref. <cit.>, with the value of S_inst. associated to the onset of the first Ξ^0 and Ξ^- hyperons (red and blue lines). The onset of the first Ξ^0,- hyperon occurs before the S-DL for pure Λ multi-strange hypernuclei is reached, for all hypernuclei shown in Fig. <ref> except one,namely^236_56ΛPb. In this exceptional case the lowest energy state at the Ξ onset is composed of 1Ξ^0 and 1Ξ^-. The concurrent hypernucleus composed of Ξ^- has a lower energy, but its proton chemical potential is about 2 MeV higher.Since this nucleus is close to the proton drip line, this increase makes it proton-unstable. It should be noted that the next calculated nucleus of this isotopic chain, ^236_48Λ,2ΞPb, also exhibits a configuration with 1Ξ^0 and 1Ξ^- at the Ξ-instability. At constant Q, the presence of Ξ^- implies an extra proton, which explains whyfor extreme neutron deficient hypernuclei at the proton drip-line, Ξ^0 are favoured at the Ξ^0,- instability. The observed plateaus are due to strong shell effects for both the Ξ^0,- instability and the S-DL for pure Λ-hypernuclei. DF-NSC89 and DF-NSC97f predict similar results, while DF-NSC97a pushes up both the Ξ^0,- instability and the S-DL for pure Λ-hypernuclei. This is in agreement with the fact that DF-NSC97a predict Λ-matter more stable than the other functionals. The sensitivity of S_inst. on the NΛ interaction is quite large,but the global trend is an increasing value for -S_inst. with increasing nuclear mass. We can also compare to the other estimation of S_inst. <cit.> based on RMF Lagrangians. The data used in Ref. <cit.> to calibrate the model are roughly the same as ours, except for the ΛΛ channel which was consideredmore attractive: 5 MeV versus about 1 MeV now. For a core of ^208Pb, they have estimated -S_inst.=41 while we predict -S_inst.=36-40 depending on the NΛ interaction. The fact that the prediction in Ref. <cit.> is slightly higher than our is mostly explained by the different choice made for theΛΛ interaction. Extended predictions forS_inst. over the nuclear chart are displayed in Figs. <ref>-<ref>.We explore the nuclear chart delimited by the neutron and proton drip-lines, which are defined for ordinary nuclei (with only neutrons and protons). Calculations for Z_core between 10 and 120 are performed, with stepsΔ A_core=4 , Δ Z_core=2, andΔ S=2. As discussed above, the maximum number of strangeness S_inst. is reached in two cases: either the S-DL is reached before the onset of the Ξ^0,- instability (it occurs only for a few cases around Z_core=82 and A_core=182 in Figs. <ref>-<ref>), or the Ξ^0,- instability is reached before the S-DL and we indicate in Figs. <ref>-<ref> if the first Ξ to appear is a Ξ^0 (with x symbol) or Ξ^- (with + symbol). In Figs. <ref> and <ref>, we compare the predictions for the Ξ^0,- instability considering the functionals DF-NSC97a+EmpC and DF-NSC97f+EmpC. One of the main difference between these two functionals is that DF-NSC97a+EmpC predicts a higher strangeness at the Ξ^0,- instability -S_inst. than DF-NSC97f+EmpC. This feature is consistent with the fact that the NΛ mean field predicted by DF-NSC97a+EmpC is more attractive, leading to more boundΛ-hypernuclei. Despite some quantitative differences between the predictions for the Ξ^0,- instability shown inFigs. <ref>-<ref>, gross features emerge from the comparison of the results obtained with various Λ-interactions: * The instability with respect to the onset of Ξ^0,- is observed all along the nuclear chart. Only a very small region of proton rich nuclei around Z_core=82 may not be Ξ-unstable (only for DF-NSC89).* In most cases, the first hyperon to be formed is Ξ^-. The onset of Ξ^0 is predicted only for nuclei close to the proton drip-line and for Z_core<50.* As A_core increases the value of S_inst. increases by steps, showing some shell effects (also visible in Fig. <ref>).These predictions are very weakly influenced by the choice of the Λ interaction. The impact of the Λ interaction is only observed for the absolute value of S_inst.: the softer the Λ interaction, the higher S_inst.. For instance, on the stability valley the softer Λ interaction (DF-NSC97a+EmpC) predicts larger values for S_inst. for heavy hypernuclei (up to 60) than the others (40-50).§.§ Number of hypernucleiIn a previous work, we have counted the number of new multi-strange hypernuclei for system formed of N and Λ only <cit.>.Since the Ξ^0,- instability was not considered, we now proceed to a new counting of pure-Λ hypernuclei up to the Ξ^0,- instability. Table <ref> displays the counting of pure-Λ hypernuclei for two cases: first up to the Ξ^0,- instability, and then up to strange-drip line (unrestricted). The latter case is equivalent to our previous calculation in Ref. <cit.> but the counting is a bit different. In Ref. <cit.> only a few strangeness numbers were considered, -S=2,8,20,40,70, corresponding to Λ-shell closure without spin-orbit, and the position of the drip line was obtained by interpolation between these cases. We found in this previous work about 490 000 Λ-hypernuclei having a maximum of 70 Λ. In the present calculation, we systematically calculate the ground-state of hypernuclei for almost every strangeness number (step Δ S=2), and we do not limit the maximum strangeness number. Hypernuclei in-between shell-closure are still calculated within the spherical approximation. However,the effect of deformation cannot change the present estimations by more than a few percent. We found that the maximum strangeness number is about 56-58 below the Ξ^0,- instability and 140-180 in the unrestricted case. Since this maximum number is larger than the one considered in our previous work, we find a larger amount of hypernuclei in the unrestricted case: 600 000-800 000 Λ hypernuclei are presently predicted. Some differences are found between the predictions of the different V_NΛ functionals: as expected, the functional DF-NSC97a predicts a larger amount of hypernuclei since the NΛ interaction in this case is the more attractive.All these predictions for pure-Λ hypernuclei (unrestricted case) shall be revised since they do not consider the Ξ^0,- instability. Counting the number of pure-Λ hypernuclei below the Ξ^0,- instability, we now find that they are about 200 000 to 300 000hypernuclei. This number is about 1/3 to 1/2 of the unrestricted one, but it is however still very large. It offers a considerable potential of discovery of multi-strange hypernuclei which are expected in future hypernuclear facilities.We show in Fig. <ref> the number of pure-Λ hypernuclei below the Ξ^0,- instability (in black) and in the unrestricted case (in red). The predictions for the different NΛ functionals (DF-NSC89, DF-NSC97a, and DF-NSC97f) are shown with different line styles, see the legend in the figure. It should be noted that the shell effects which produce the steps corresponding to shell closures (magic numbers) or sub-shell closure. These shell closures are located at the same position for the various NΛ functionals. For the unrestricted case, most of the difference between DF-NSC97a and the two others is located for -S > 70. This is the reason why the difference in the counting between DF-NSC97a and the two others is larger in the present case compared to our previous estimation <cit.>. Finally, Tab. <ref> displays a comparison for the predictions of the number of multi-Λ hypernuclei below Ξ-instability and with10<Z_core<120, considering various choices for the unknown interaction channels such as VY0, VY1 and VY2 (Table <ref>). As expected, the largest corrections come from the unknown ΛΞ channel (VY1), and the ΞΞ channel (VY2) has almost no impact on the number of hypernuclei below S_inst.. This latest result is rather expected since the ΞΞ interaction can occur only if the number of Ξ at the onset threshold is at least 2, which rarely occurs. § CONCLUSIONS In this work we have presented the first extensive microscopic exploration of the nuclear chart along the strangeness dimension where the competition between the Λ and the Ξ hyperons is consistently treated.The exploration of the nuclear chart as function of the strangeness number S is performed by adding hyperons to a core (A_core,Z_core) imposing eitherconserved total charge Q or conserved proton number Z. This study, which is a continuation of our previous work detailed in Ref. <cit.>, is performed using realistic and microscopically rooted non-relativistic energy functionals.In particular, we use in the NΛ channel different functionals extracted from Brueckner-Hartree-Fock calculations with Nijmegen interactions.These effective interactions, fitted on all the available phase shifts, cover our present uncertainty on the interaction at least at low density, and have been successfully confronted to hypernuclear data in the past. We have proposed a phenomenological extension of these potentials to the whole baryonic octet. The experimental data on single and double Λ hypernuclei are used to constrain the NΛ interaction, and the mean-field analysis of the "Kiso" event is performed along the lineproposed in Ref. <cit.> to determine the NΞ interaction. Starting from a non-strange (A,Z) core and adding Λ hyperons, we have shown that the quasi-totality of the hypernuclei present an instability towards the decay into Ξ hyperons before the strangeness drip-line is met.The strangeness instability threshold increases by step with the mass of the system due to shell effects. It is approximately constant at a given Q for stable (A_core,Z_core) cores.A clear Coulomb effect is present, with Ξ^0 appearing in the proton rich side of the nuclear chart, and Ξ^- for the majority of hypernuclei (at conserved charge Q).At conserved charge Q, the onset of the first Ξ^0,- corresponds to the crossing between the Λ and the neutron or proton chemical potentials. We also show the impact of the different interacting channels on the results.The numerical value of the instability threshold largely depends on the NΛ and NΞ interaction model, which are the most important channels. The ΛΞ interaction has however a non-negligible impact: it can modify the number of pure-Λ hypernuclei by 30-40%. Finally, the ΞΞ interaction channel has almost no impact on the position of the Ξ^0,- instability, and therefore on the number of pure-Λ hypernuclei. It seems rather weakly impact multi-strange hypernuclei. In all cases the opening of the Ξ channel reduces the number of bound pure Λ-hypernuclei that we have previously estimated <cit.> by a factor of approximatively 1/3-1/2, to be about 200 000-300 000 hypernuclei.The detailed characteristics of multi-hypernuclei along the nuclear chart, as well as their excited states, will be addressed in a future study. We also plan to include Σ and Ω hyperons and perform a sensitivity study on their largely unknown coupling, in order to further asset the possible model dependence of the results. 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"authors": [
"J. Margueron",
"E. Khan",
"F. Gulminelli"
],
"categories": [
"nucl-th"
],
"primary_category": "nucl-th",
"published": "20170727040307",
"title": "Density Functional approach for multi-strange hypernuclei: competition between $Λ$ and $Ξ^{0,-}$ hyperons"
} |
Investigation of the commensurate magnetic structure in heavy fermion ]Investigation of the commensurate magnetic structure in heavy fermionusing magnetic resonant X-ray [email protected] Laboratory for Scientific Developments and Novel Materials, Paul Scherrer Institut, 5232 Villigen, SwitzerlandXMaS, UK-CRG, European Synchrotron Radiation Facility, BP220, F-38043 Grenoble Cedex, France Oliver Lodge Laboratory, Department of Physics, University of Liverpool, Oxford Street, Liverpool L69 7ZE, United KingdomLaboratory for Synchrotron Radiation, Paul Scherrer Institut, 5232 Villigen, SwitzerlandAdvanced Science Research Center, Japan Atomic Energy Agency, Tokai, Ibaraki, 319-1195, Japan[Present address: ]Condensed Matter Group, National High Magnetic Field Laboratory, Florida State University, Tallahassee, Florida 32310, USA Condensed Matter and Magnet Science, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USACondensed Matter and Magnet Science, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, [email protected] Laboratory for Neutron Scattering and Imaging, Paul Scherrer Institut, 5232 Villigen, Switzerland We investigated the magnetic structure of the heavy fermion compoundbelow T_N = 5.34(2) K using magnetic resonant X-ray diffraction at ambient pressure. The magnetic order is characterized by a commensurate propagation vector k_1/2 = ( 1/2 , 1/2, 1/2) with spins lying in the basal plane. Our measurements did not reveal the presence of an incommensurate order propagating along the high symmetry directions in reciprocal space but cannot exclude other incommensurate modulations or weak scattering intensities. The observed commensurate order can be described equivalently by either astructure or by a multi-k structure. Furthermore we explain how a commensurate-only ordering may explain the broad distribution of internal fields observed in nuclear quadrupolar resonance experiments (Sakai et al. 2011, Phys. Rev. B 83 140408) that was previously attributed to an incommensurate order. We also report powder X-ray diffraction showing that the crystallographic structure ofchanges monotonically with pressure up to P = 7.3 GPa at room temperature. The determined bulk modulus B_0 = 81.1(3) GPa is similar to the ones of the Ce-115 family. Broad diffraction peaks confirm the presence of pronounced strain in polycrystalline samples of . We discuss how strain effects can lead to different electronic and magnetic properties between polycrystalline and single crystal samples. 75.25.-j, 75.30.Mb, 61.05.cp [ Jonathan S. White December 30, 2023 ===================== § INTRODUCTIONElectrons can gain a large effective mass due to strong electronic correlations in crystals. Such materials are referred to as heavy fermion compounds and often have complex phase diagram due to the interplay of spin and electronic degrees of freedom. Of particular interest are the Ce-115 compounds CeMIn_5 (M = Co, Rh, Ir) that have been investigated for more than 15 years and yet their properties are still not completely understood.<cit.> These materials, which offer a unique playground to study quantum criticality,<cit.> are part of the larger family Ce_nM_mIn_3n+2m (M = Co, Rh, Ir, Pd, Pt) derived from the simple cubic CeIn_3: they are formed from CeIn_3 layers separated by MIn_2 layers. This separation of the Ce planes makes them generally more two-dimensional (2D) relative to the three-dimensional (3D) cubic CeIn_3.Furthermore, the hybridization of the Ce 4f-electrons with the conduction electron bands is controlled by the local environment of the In and M atoms.<cit.> It is therefore possible to investigate the effects of the dimensionality and the hybridization strength on the interplay between magnetism and superconductivity in these compounds.is a member of this family with n = 1 and m = 2. It is closely related to the Ce-115s and is obtained by adding a second MIn_2 plane in between the CeIn_3 planes. This larger separation of the planes containing Ce suggest that this system is more 2D than the Ce-115s.crystallizes in a body-centered tetragonal structure with space group I4/mmm and the magnetic Ce ion sits at theWyckoff 2b positions.<cit.> It has an antiferromagnetic (AFM) order with T_N≈ 5.5 K at ambient pressure. This order is suppressed with pressure and a superconductivity dome emerges around the AFM quantum critical point (QCP), with a maximum T_c = 2.1 K near the critical pressure P_c≈3.4 GPa, and which is also where an effective mass enhancement is observed.<cit.> This phase diagram is very similar to the analogous compound CeRhIn_5,<cit.> which is often described as a two-dimensional analogue of CeIn_3. Quantum oscillations reveal that the microscopic electronic properties ofare more closely related to CeIn_3 than CeRhIn_5, indicating thatis a better 2D analog of CeIn_3.<cit.> The 2D nature of the electronic properties is also suggested by specific heat measurements.<cit.> Optical measurements indicate a hybridization strength insimilar to the one in CeIn_3 and CeRhIn_5.<cit.>Nuclear quadrupolar resonance (NQR) measurements revealed the presence of two characteristic pressures in .<cit.> The first one at P^* = 2.4 GPa corresponds to a transition from localized to itinerant Ce 4f-electrons. The second one at P_c≈3.4 GPa corresponds to the AFM QCP. In CeRhIn_5, these characteristic pressures are very close to each other and it was suggested that the superconductivity emerges from the Kondo breakdown QCP.<cit.> Indeed, recent theoretical work proposes an enhancement of singlet superconductivity near a Kondo breakdown QCP,<cit.> which may explain the behaviour ofand CeRhIn_5.<cit.> The detailed understanding ofalso requires an accurate description of its magnetic order at ambient pressure and its evolution (or stability) under pressure. However, up until now only limited details of the nature of the magnetic order have been reported. NQR measurements on polycrystalline samples indicate a commensurate order and suggest a propagation vector (1/2,1/2) in the basal plane.<cit.> On the other hand, the results obtained using the same technique applied to single crystals were interpreted in terms of a coexistence of commensurate and incommensurate orders.<cit.> From muon spin rotation measurements, a commensurate order was proposed for polycrystalline samples.<cit.> A possible reason for these discrepancies is that the inherently larger surface strain of grains in polycrystalline samples provides a means to enhance the stability of the commensurate order.<cit.> It was also observed that the superconducting dome is broader for powders than for single crystals, suggesting a commensurate order to be more favourable for superconductivity.<cit.> However, both direct measurements of the magnetic order and its propagation, and evidence for the proposed crystallographic strain in powder samples are yet to be reported.Neutron scattering could clarify the bulk magnetic structure but it is challenging forbecause of the generally small size of single crystals, the large neutron absorption cross-section by In, and the small expected moment size. These limitations can be overcome by using magnetic resonant X-ray diffraction (MRXD) as an alternative scattering technique for determining the magnetic structure. We performed MRXD measurements onand we report here a model for the magnetic order at T = 1.8 K and ambient pressure. We also report the pressure dependence of its crystallographic structure at room temperature up to P = 7.3 GPa, which changes monotonically in the range of applied pressure. § EXPERIMENTAL DETAILSHigh purity single crystals ofwere synthesized as described previously.<cit.> The 0.38 mg sample used for the MRXD experiment was characterized by specific heat and magnetic susceptibility using a Quantum Design PPMS and MPMS, respectively. The results are in good agreement with the previously reported measurements.<cit.> The long range magnetic order is observed from a sharp peak in the specific heat at T_N = 5.36(2) K [Fig. [c]fig2] and the high purity of the sample is indicated by the absence of other peaks, compared to previous reports.<cit.> For the MRXD experiment, the plate-like sample with the c-axis perpendicular to the plate was fixed on a copper holder with silver Electrodag 1415 and mounted in a Joule Thomson cryostat on the bending magnet XMaS beamline, at the ESRF. The measurements were carried out using a Vortex Si Drift Diode detector. The (220) reflection of a LiF analyser crystal was used for the polarization analysis measurements. Except for photon energy dependent scans, all the measurements were carried out at E = 6.166 keV, the Ce-L_II absorption edge.The azimuthal scans presented in Fig. <ref> were corrected for X-ray absorption. The absorption correction was calculated by a finite element analysis assuming an absorption coefficient μ = 436.425 mm^-1 for , a beam size of 0.7×0.3 mm^2 and a sample size of 0.79×0.62×0.02 mm^3. The accuracy of this correction for the magnetic peaks was verified by comparison with azimuthal scans measured on structural peaks. Powder X-ray diffraction measurements under hydrostatic pressure were performed at the MS-X04SA beamline, Swiss Light Source at the Paul Scherrer Institut.<cit.> A 2D Pilatus 6M detector was used. LaB_6 was used as a standard for calibration of the detector position as well as the instrumental parameters. Single crystals ofwere finely ground, mixed with quartz powder and loaded in a diamond anvil pressure cell using methanol:ethanol 4:1 as a pressure medium. Quartz was used as an in-situ pressure calibrant.<cit.> Measurements were performed with a photon wavelength λ = 0.56491 Å in the angular range 1^∘<2θ<35^∘ at room temperature (T = 293 K) up to a maximal pressure P = 7.3 GPa. The data reduction was performed with the Dioptas software<cit.> and FullProf was used for Rietveld refinement of the one-dimensional diffraction patterns.<cit.> § EXPERIMENTAL RESULTS§.§ Magnetic Resonant X-ray DiffractionThe magnetic order ofwas successfully observed using MRXD, revealing unambiguously its commensurate propagation vector. Bragg peaks consistent with a propagation vector k_1/2 = ( 1/2 , 1/2, 1/2) were observed at T = 1.8 K. The magnetic origin of these Bragg peaks was verified by the resonance at the Ce-L_II absorption edge as well as polarization analysis. Q-scans around the magnetic Bragg peak Q = (0.5, -0.5, 6.5) are presented in Figs. [a]fig1-[c]fig1 at T = 1.8 K and can be compared with background scans done at 10 K. This magnetic Bragg peak has the same widths and shapes along H, K and L as the structural Bragg peak Q = (1,-1,6). This indicates that the magnetic peak widths are limited by the crystal mosaicity and that a 3D long range magnetic order is achieved. Several other peaks consistent with k_1/2 were measured. It was observed that all experimentally accessible magnetic Bragg peaks have non-zero intensity, indicating the absence of any selection rules of the magnetic structure. The fluorescence intensity of the sample was measured as function of the incident photon energy. It shows a maximum around E_i = 6.167 keV corresponding to the Ce-L_II absorption edge [Fig. [a]fig2]. The intensity of the magnetic Bragg peak Q = (0.5, -0.5, 6.5) is strongly enhanced around this edge, indicating a resonant magnetic effect.<cit.> In contrast, the intensity of the structural Bragg peak Q = (1,-1,6) shows a dip near this edge due to a larger absorption cross-section. The magnetic nature of the Bragg peak Q = (0.5, -0.5, 6.5) is further confirmed by the polarization analysis. The polarization σ is defined to be perpendicular to the scattering plane and the polarization π is parallel to it.<cit.> In the electric dipole approximation of MRXD, charge scattering, related to the crystallographic structure, is allowed in the σ-σ' channel and is forbidden in the σ-π' channel. Magnetic scattering has the opposite behaviour and appears in the σ-π' channel and not in the σ-σ' one.<cit.> The Bragg peak Q = (0.5, -0.5, 6.5) is present in the σ-π' channel and absent in the σ-σ' channel, clearly showing its magnetic nature [Fig. [b]fig2]. This observation combined with the peak resonance at the Ce-L_II edge establish unambiguously the magnetic origin of the Bragg peaks with the propagation vector k_1/2 = ( 1/2 , 1/2, 1/2). The temperature dependence of the magnetic Bragg peak Q = (0.5, -0.5, 6.5) has been measured from T = 1.8 K up to 7 K in the σ-π' channel. The width and position of this peak are temperature independent from T = 1.8 K to T_N. The integrated intensity indicates a Néel temperature of T_N = 5.34(2) K, as determined by a power law fit above 4.4 K [Fig. [c]fig2]. This transition temperature is in good agreement with the sharp peak observed in specific heat. The obtained critical exponent β = 0.31(4) corresponds to a 3D Ising model with β = 0.326 or a 3D XY model with β = 0.345.<cit.> Note that a beam injection occurred during the measurements at T = 4.1 K and that the intensity above and below this temperature can not be compared accurately. However, a previous temperature dependence of the Bragg peak Q = (0.5, -0.5, 6.5) without the polarization analysis (not shown) does not have any feature at T≈4 K.The magnetic structure of the propagation vector k_1/2 = ( 1/2 , 1/2, 1/2) was determined with the help of representation analysis performed with BasIreps.<cit.> Only two irreducible representations with non-zero basis functions are possible at the Ce position (0,0,0.5) in the space group I4/mmm. There is Γ_1, a two-dimensional irreducible representation with basis vector (M_x,M_y,0), and Γ_2, a one-dimensional irreducible representation with basis vector (0,0,M_z). Both representations do not have selection rules, in agreement with our observations, and hence can not be distinguished in this way.We have determined that the structure must be described by Γ_1 with moments in the ab plane by performing azimuthal scans. These scans measure the intensity variation when the sample is rotated by azimuthal angle Ψ around the scattering vector Q. In MRXD, the scattering intensity isproportional to | F(Q) ·k_f | ^2 where F(Q) is the magnetic structure factor and k_f is the scattered photon wavevector.<cit.> Azimuthal rotations change the moment direction, modifying F(Q) relative to a fixed k_f. The scattered intensity is thereforeexpected to change with Ψ and this can be compared with that expected according to a magnetic structure model. The azimuth Ψ is defined relative to a reference Bragg peak, here chosen to beQ = (-1,-1,0). The azimuthal angle is defined to be zero when the reference Bragg peak is in the scattering plane and forms the smallest angle with the incident photon wavevector k_i. Multiple datasets of azimuthal scans were collected and are represented by different symbols in Fig. <ref>. These datasets were collected in similar conditions (with and without optimizing the different rotation and translation motors) and all show the same general tendency. For both irreducible representations, the magnetic structure is collinear and the azimuthal scans correlate directly to the moment orientation. The theoretical azimuthal dependence curves for moments pointing along the a-axis (Γ_1^x), the b-axis (Γ_1^y) and the c-axis (Γ_2) are shown in Fig. <ref> for the magnetic Bragg peaks Q = (0.5, -0.5, 6.5) and Q = (1.5, -1.5, 10.5). Experimental results are overlaid and show that the system can be described by the coexistence of Γ_1^x and Γ_1^y domains with equal population. Since the axes a and b are equivalent, one would indeed expect that both domains are present. In general, if a domain exists with a moment pointing in a direction e within the ab plane, a domain with a moment pointing in a direction e' perpendicular to e in the ab plane is expected with an equal population. It can be shown that the azimuthal dependence of Γ_1^e+Γ_1^e' for any e in the ab plane is exactly the same one as the one of Γ_1^x+Γ_1^y. Therefore, our results indicate the moments are in the ab plane but do not allow us to determine their exact orientation. The magnetic structure for antiferromagnetically ordered moments pointing in the basal plane at an angle θ from the a-axis is schematized in Fig. [d]fig1. From previous NQR experiments, it was claimedthat at T = 1.6 K an incommensurate magnetic order coexists with the commensurate order, and that the volume fraction of commensurate:incommensurate order was 0.25:0.75. In addition the maximal internal field due to the incommensurate order is determined to be slightly larger than the one from the commensurate order, suggesting a similar moment size for both orders. For these two reasons, the magnetic peak intensities originating from the incommensurate order can be expected to be similar to the ones of the commensurate order. However, no evidence for incommensurate magnetic peaks was found in our MRXD experiment from scans along the high symmetry directions in reciprocal space. Measurements were carried out at T = 1.8 K for Q = (0.5,-0.5,L) from L = 6 to 8, Q = (H,H,7) from H = 0 to 1.2, Q = (H,H,6.5) from H = 0 to 1.2 and Q = (H,0,6.5) from H = 0 to 1.5. This rules out likely incommensurate propagation vectors similar to those of other incommensurate magnetic phases in Ce-based heavy-fermion compounds,<cit.> but we can not exclude the presence of incommensurate modulations propagating elsewhere in reciprocal space.§.§ Powder X-ray Diffraction Under Pressure Powder X-ray diffraction patterns ofare shown in Fig. <ref> at hydrostatic pressures P = 0.015(7) GPa and P = 5.09(3) GPa for a representative 2θ angular range. The general crystallographic structure, previously reported by Klimczuk et al.,<cit.> was confirmed by Rietveld refinement using FullProf.<cit.> Two strong diffraction peaks fromappear in the angular range 12.5^∘<2θ<15^∘ and this region has been excluded from the refinement to improve the sensitivity of the fit to weak features over the full angular range. Importantly, the diffraction peak profiles due to thesample are significantly broader than the instrumental resolution and this can be attributed to strain. The presence of strain in polycrystalline sample ofwas inferred previously in NQR measurements.<cit.> Our measurements remained in a hydrostatic regime up to the maximal applied pressure, as confirmed by the pressure independent widths of peaks due to scattering from quartz. However, the peak widths ofgradually broadened above P≈5 GPa, which show a loss of the structural integrity in terms of either a larger strain or breaking of crystallites into smaller particles. The refinement of the diffraction patterns was performed sequentially for increasing pressure and the results are presented in Fig. <ref>. We observe no obvious changes of the crystallographic structure related to the characteristic pressures P^* = 2.4 GPa and P_c≈3.4 GPa. The lattice constants a and c change monotonically up to the maximal applied pressure P = 7.3 GPa [Figs. [a]fig5-[c]fig5]. The Birch-Murnaghan equation of state was used to relate the crystal volume V to the applied pressure P:P(V)=3/2B_0( v^7 - v^5)(1-3/4(4-B_0'(v^3-1)),where B_0, and B_0' are respectively the initial bulk modulus and its derivative, and v = (V_0/V)^1/3.<cit.> By fitting this equation to the data shown in Fig. [d]fig5, we obtain B_0 = 81.1±0.3 GPa and B_0' = 5.8±0.1. Using the simple Murnaghan equation <cit.> results in the same fitted values for B_0 and B_0' within errors. These values are similar to those reported for other members of the Ce_nM_mIn_3n+2m family.<cit.> In these compounds, it was observed that adding MIn_2 layers stiffens the structure and increases the bulk modulus: B_0 = 67 GPa for CeIn_3, average B_0 = 70.4 GPa for Ce_2MIn_8 (2 layers CeIn_3 + 1 layer MIn_2) and average B_0 = 81.4 GPa for CeMIn_5 (1 layer CeIn_3 + 1 layer MIn_2). The addition of a second MIn_2 layer inrelative to CeMIn_5 could then be expected to stiffen the lattice further. However, the bulk moduli appear very similar forand the Ce-115s.In , the Ce and Pt atoms sit at Wyckoff positions 2b and 4e, respectively, and the In atoms are distributed on three different positions (In(1) at 2a, In(2) at 4d and In(3) at 8g). The only adjustable fractional coordinates in the structure ofare the Z positions of the Pt and In(3) atoms. The fractional coordinate Z of In(3) changes monotonically with pressure [Fig. [e]fig5] and the one of Pt is pressure independent [Fig. [f]fig5]. This indicates a non-uniform compression along the c-axis, with the strongest contraction occurring between the In(3)-planes and the Ce-In(1) planes [see Fig. [g]fig5]. The pressure dependence of various bond lengths is presented in Figs. [h]fig5-[j]fig5 and they all decrease monotonically with increasing pressure. Interestingly, the Ce-In(3) bond is more significantly affected by pressure than the Ce-In(1) bond [Fig. [h]fig5].Since the Ce-In coupling is expected to be the strongest with the out-of-plane In(3) atoms,<cit.> this change in distortion around the Ce atoms could modify significantly the ground-state Ce wavefunction.<cit.>§ DISCUSSIONAs mentioned previously, the pressure-temperature phase diagram of CeRhIn_5 is very similar to the one of . Their magnetic structures at ambient pressure also share similarities: both have an antiferromagnetic order in the basal plane with moments lying in that plane.<cit.> However, the ordering in CeRhIn_5 is incommensurate along the c-axis in contrast with the commensurate ordering in . While CeCoIn_5 and CeIrIn_5 do not order magnetically at ambient pressure and zero magnetic field, it is possible to induce magnetic order with doping. In particular, substituting the Co or Ir sites with Rh leads to the coexistence of an incommensurate order with k = (1/2,1/2,δ) and a commensurate order with k = (1/2,1/2,1/2) for a range of doping values.<cit.> It was shown for CeRh_0.7Ir_0.3In_5 specifically that the moments lie in the basal plane for both the commensurate and incommensurate orders. Doping the In site with Cd in CeCoIn_5 also stabilizes a commensurate order with k = (1/2,1/2,1/2).<cit.> On the other hand, substituting Ce by Nd in CeCoIn_5 leads to a propagation vector k = (1/2-δ,1/2-δ,1/2) with δ = 0.05,<cit.> suggesting a spin-density wave in the basal plane with fundamentally different properties from the localized moment magnetism in CeRhIn_5 and . In these systems, superconductivity emerges in the vicinity of an AFM QCP, suggesting a magnetically-driven pairing mechanism of superconductivity. The knowledge of the magnetic structure is therefore a crucial element for identifying the magnetic fluctuations responsible for this electron-electron coupling. The AFM order (1/2,1/2) in the basal plane prevails in these systems andappears as a new example where magnetic fluctuations associated with this AFM order are the pairing glue of the pressure-induced superconductivity. It is important to note that the magnetic structure ofmight change under pressure but it is unlikely to change the order in the basal plane. For example, the propagation vector in CeRhIn_5 changes under pressure but the order in the basal plane is conserved.<cit.>Based on NQR experiments, it was suggested that in single crystals ofthere is a coexistence of commensurate and incommensurate orders at ambient pressure.<cit.> Specifically, sharp peaks in the spectrum can be attributed to a basal plane AFM order with moments pointing along the a-axis or the b-axis. This was interpreted as a commensurate order. On the other hand, broad features are also observed in the spectrum and were attributed to a distribution of internal fields at the In(2) and In(3) sites. This was interpreted as an incommensurate order similar to the one of CeRhIn_5.<cit.> Our results presented in section <ref> confirm the presence of a commensurate order but do not reveal the presence of an incommensurate order along the high symmetry directions in reciprocal space indicated in section <ref>. The scenario involving the coexistence of both commensurate and incommensurate orders remains a possibility: we cannot rule out incommensurate modulations propagating elsewhere in reciprocal space, and the volume fraction and/or moment size could be too small to be detected under our current experimental conditions. On the other hand, we propose an alternative interpretation of the broad features observed in the NQR experiments that do not require the presence of an additional incommensurate order.With no restriction on the precise moment direction in the basal plane provided by our MXRD experiments, the distribution of internal fields observed by NQR could be generated if either the moment directions in the ab plane fluctuate, or there exist multiple domains with different moment orientations (different values of θ in Fig. <ref>). This commensurate-only scenario for the magnetic order inrequires a coexistence of domain-types; those with arbitrary moment orientations in the ab plane as outlined above, and those where the moments are rigidly aligned with the a- and b-axes. Here crystal strain could play an important role in stabilizing one type of domain over the other.In NQR experiments, different results for the reported spectra are obtained from polycrystalline and single crystal samples of .<cit.> These discrepancies are readily attributable to crystal/surface strain effects that vary in propensity with the sample crystallite size. Indeed, this is supported by the broad structural peaks in our high-resolution powder X-ray diffraction experiment on . In the NQR studies only sharp features are observed for powder samples, in contrast with the presence of broad features for single crystals. Furthermore, applied pressure on single crystals suppresses the contribution of the broad features.<cit.> Taken together, these two effects indicate that strain, either from surface strain from the grains in polycrystalline samples or stimulated by pressure, promotes the ordering with moments aligned along the a-axis or the b-axis. At the same time, in the absence of strain, the moments may align along an arbitrary direction in the ab plane. In this scenario, enhanced strain thus leads to an effective in-plane anisotropy that favors the alignment of the moments along the a-axis or b-axis.It is interesting then to note that the superconductivity is stabilized in a wider pressure range in powder samples and that it only appears in single crystals when the NQR signature interpreted in terms of incommensurate order is completely suppressed.<cit.> This suggests that domains with moments not aligned along the a-axis or the b-axis are detrimental to the formation of superconductivity in .Finally, we note that even if the magnetic structure presented in section <ref> is the simplest solution to explain the results, it is not the only possible one. Since the lattice ofis body-centered, the propagation vector k_1/2 is not equivalent to -k_1/2. This can lead either to two different k-domains, which was assumed in section <ref>, or a multi-k structure, as observed for example in the heavy fermion CeRh_2Si_2, which also has a body-centered tetragonal lattice.<cit.> A complete description of the multi-k structure inis given in the appendix. In such a multi-k structure, the moments between the nearest neighbouring Ce layers can be non-collinear while all the moments are collinear in a single-k structure. This non-collinearity suggests an effective decoupling of the nearest neighbour layers while keeping a coupling to the next-nearest neighbour planes, consequently forming two decoupled yet inter-penetrating sublattices. This scenario is plausible for the body-centered tetragonal lattice because of the presence of competing interactions. It was even suggested theoretically that the frustration in body-centered tetragonal lattices can destabilize long-range magnetic order and lead to spin liquid states in heavy fermion compounds.<cit.> The aforementioned discussion about the moment directions in the single-k model, and its application for consistently explaining previously reported NQR spectra, can also be done using the multi-k structure. Our results do not allow us to establish unambiguously if the single-k structure or the multi-k structure is the correct one. In fact, these two scenarios cannot be distinguished in a simple scattering experiment; doing so would require the application of either uniaxial strain or magnetic fields to control the magnetic domain formation in a single crystal sample.§ SUMMARY We have shown that the crystallographic structure ofchanges monotonically with pressure up to P = 7.3 GPa at room temperature. We also investigated the magnetic order ofat ambient pressure below T_N = 5.34(2) K by magnetic resonant X-ray diffraction. This order is characterized by a commensurate propagation vector k_1/2 = ( 1/2 , 1/2, 1/2). The magnetic origin of these diffraction peaks was confirmed by their resonance at the Ce-L_II absorption edge and by polarization analysis. Azimuthal scans confirm that the moments lie in the basal plane. The magnetic structure can be described by a single-k structure or by a multi-k structure. Both structures cannot be distinguished in a simple scattering experiment as reported here and the single-k structure is discussed for simplicity. The presence of incommensurate order inwas previously reported based on NQR experiments. Our measurements could not reveal the presence of such an order but are insufficient to exclude it completely. Using our results we propose a new scenario for the ambient pressure ground state ofthat is described only by commensurate magnetic order; namely a coexistence of domains wherein the moments are either rigidly aligned along the a- and b-axes, or arbitrarily aligned within the ab plane. Crystal strain is argued to be an effective tuning parameter for controlling the relative volume fractions of the two types of domain, thus providing a means for a consistent description of both the scattering data reported here, and previously reported NQR spectra obtained on both polycrystalline and single crystal samples.§ ACKNOWLEDGEMENTS The authors are thankful to M. Kenzelmann, D. G. Mazzone and F. Ronning for fruitful discussions. This research received support from the Natural Sciences and Engineering Research Council of Canada (Canada). Work at Los Alamos National Laboratory was performed under the auspices of the US Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering. XMaS is a UK mid-range facility supported by EPSRC.Note added. - During the preparation of this manuscript, we became aware of another report where the magnetic structure ofwas investigated using neutron diffraction.<cit.> In agreement with our results, they report a commensurate propagation vector k_1/2 and moments lying in the basal plane. The reported structure corresponds to a multi-k structure with non-collinear moments. * § SINGLE-K AND MULTI-K STRUCTURES§.§ Single-k structure The simplest magnetic structure model foris described by the single propagation vector k_1/2. This is represented in Fig. [a]fig6. In the unit cell, there are two Ce ions which are related by the body-centering symmetry. They are distinguished by the blue and green colors in Fig. <ref>. For a general single-k structure, the moments m_b and m_g at the blue and green sites, respectively, are expressed as:m_b =[ M cosθ; M sinθ;0 ]cosΦ m_g =[ M cosθ; M sinθ;0 ]cos (Φ±π/2).Here the parameter θ is the angle of the moment in the ab plane, which can take any value. To reproduce the data, the presence of two equally populated domains with θ and θ+90^∘ is assumed. The parameter Φ is a global phase that cannot be measured with scattering techniques. For physical reasons, we chose Φ=π/4 to generate equal moments for m_b and m_g. The single-k structure is therefore defined by:m_b =1/√(2)[ M cosθ; M sinθ;0 ] m_g = - 1/√(2)[ M cosθ; M sinθ;0 ].§.§ Multi-k structureDue to the body-centering symmetry, +k_1/2 and -k_1/2 are not equivalent and therefore, a magnetic structure can form that is composed of two propagation vectors. In a general way, the moments are defined at the blue and green sites, respectively, by:m_b^+k +m_b^-k =[ M cosθ_1; M sinθ_1;0 ]cosΦ_1 + [ M cosθ_2; M sinθ_2;0 ]cosΦ_2 m_g^+k+m_g^-k=[ M cosθ_1; M sinθ_1;0 ]cos (Φ_1 + π/2)+[ M cosθ_2; M sinθ_2;0 ]cos (Φ_2 - π/2)where θ_1 and Φ_1 are related to the propagation +k_1/2, and θ_2 and Φ_2 are related to -k_1/2. It is again assumed that there are two equally populated domains with {θ_1,θ_2 } and {θ_1+90^∘,θ_2 +90^∘}. Experimentally, this gives exactly the same scattering as the single-k structure. We must choose Φ_1 and Φ_2 to have equal moments on the blue and green sites for any θ_1 and θ_2. An elegant choice is Φ_1=n π/2 and Φ_2=(n+1) π/2 where n is an integer. It evidences the decoupling of the nearest-neighbour layers. For n=0, we obtain:m_b^+k +m_b^-k =[ M cosθ_1; M sinθ_1;0 ]andm_g^+k+m_g^-k =[ M cosθ_2; M sinθ_2;0 ].The structure is therefore defined by three parameters: the moment size M, the angle θ_1 of the first propagation vector and the angle θ_2 of the second propagation vector. While M is expected to be constant, θ_1 and θ_2 can take any value. Note that the single-k structure is obtained if θ=θ_1=θ_2. apsrev4-1 44 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[D. Thompson and Fisk(2012)]Thompson2012 author author J. D. Thompson and author Z. Fisk, 10.1143/JPSJ.81.011002 journal journal Journal of the Physical Society of Japan volume 81, pages 011002 (year 2012)NoStop [Gegenwart et al.(2008)Gegenwart, Si, and Steglich]Gegenwart2008 author author P. Gegenwart, author Q. Si, and author F. 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"authors": [
"Nicolas Gauthier",
"Didier Wermeille",
"Nicola Casati",
"Hironori Sakai",
"Ryan E. Baumbach",
"Eric D. Bauer",
"Jonathan S. White"
],
"categories": [
"cond-mat.str-el"
],
"primary_category": "cond-mat.str-el",
"published": "20170726153716",
"title": "Investigation of the commensurate magnetic structure in heavy fermion CePt2In7 using magnetic resonant X-ray diffraction"
} |
On-Chip Quantum Dot Light Source for Quantum State Readout J. R. Petta==========================================================A notion of delegated causality is introduced here. This subtle kind of causality is dual to interventional causality.Delegated causality elucidatesthe causal role of dynamical systems at the “edge of chaos", explicates evident cases of downward causation, and relates emergent phenomena to Gödel's incompleteness theorem. Apparently rich implications are noticedin biology and Chinese philosophy. § INTRODUCTION Living organisms, ecosystems, human minds, societies, economic markets are widely recognized as extraordinary complex systems.They are impressivelyorganized and possess properties that are hardly reducible to qualities of physical matter. Therebythey seem to contradict the reductionistic paradigm of fundamental causationfrom underlying physical processes. As yet,satisfying explanation of emerging coherent organization isa comparablechallenge for reductionist and holistic philosophies <cit.>, <cit.>. Even if the reductionist approach continues to deliveroutstandingresults in physics, chemistry, molecular biology, neuropsychology, much deeper understanding ofliving <cit.>, <cit.> and conscious <cit.>, <cit.>agencies may require an uneasy paradigm change<cit.>, after all. I introduce a concept that can simplify and unify analysisof intricate causal relations in complex systemsto a remarkable extent.This concept of delegated causality should clarify muchabout emergence of whole new phenomena <cit.>, spontaneous order <cit.>, synergy <cit.>, functionality <cit.>,purpose and intention <cit.>.If the new conceptionindeed refines established specialist perspectives, it will be worth revisiting sporadic revivals of Emergentism, post-Enlightenment rationalist skepticism ,classical Greek teleologies .The most rigorous contemporary relevance of the new perspective is to physics of emergence <cit.>, symmetry breaking <cit.>,<cit.>,thermodynamics <cit.>, <cit.>,and to information-theoretic measure of causal influence <cit.>, <cit.>. A comprehensive overview of the vast,growing literature on complex systems, self-organization,emergencewould not serve the purpose of this article to introduce delegated causality.This simple but subtle, overlooked kind of causality isprovoked (figuratively speaking) by critical dynamical systems with richbehavior and moderate sensitivity to the environment. The scope of my abstracted terminology will become clear withthe introduction of methodology M1–M3 in <ref> of analyzing causal interactions.Evident implications of delegated causality will be demonstratedby a brief accountof evolutionary biology (in <ref>) and a reference to Chinesephilosophy (in <ref>). This spirited article would be presentable toa scientific version of the TV show “The X-Factor" <cit.>.My argumentation is not deep formally, as the chief purpose is to justify the new concept by a few evocative arguments,agreeable examples, and linksto existing ideas.I start by reassessing contemporarymodeling of complex systems in <ref>.The fresh kind of causality is introduced formally in <ref>. Section 4 examines physical reductionism in the new light, and relates emergence, downward causation to Gödel's incompleteness theorem <cit.>.The later sections deliberatea few compelling(though not entirely comfortable) implications.All together, this article is gradually making a holistic argument for a new comprehensive view by building up the context for the integrating Section <ref>.§ COMPLEX SYSTEMSNatural complex systems are studied under manyframeworks: self-organization <cit.>, <cit.>,complex adaptive systems <cit.>, <cit.>,autopoiesis <cit.>, dissipative structures <cit.>,self-organized criticality <cit.>, <cit.>, etc. The models are often based on non-linear dynamical systems, their attractors, non-equilibrium thermodynamics <cit.>, <cit.>, phase transitions <cit.>, scaling analysis <cit.>, cellular automata <cit.>, <cit.>,variation and selection mechanisms <cit.>, <cit.>,systems theory <cit.>, <cit.>, developmental frameworks <cit.>, <cit.>, information dynamics <cit.>. Phenomenologyat the center of attention includes spontaneous increase of order <cit.>, <cit.>,emergence of coherent global behaviors from local interactions <cit.>, adaptation to environment perturbations.Applicable distinction between self-organization and emergence is put forward in <cit.>. Autonomy, decentralized control, interactive closure are often among defining features of complex systems <cit.>, <cit.>.But the autonomy assumption should not be idealized,especially when considering causation in actual complex phenomena. Reflections in this article suggest that behaviors (of constituting agents or the whole system) which amount to control sharing or transfer can be veryfar reaching.It will help clarity here to contrast various complex systems on the two-dimensional spectrum that combines interactivity with the environment and homogeneity of constituents; see Figure <ref>.Living organisms are highly non-homogeneous and experience variablepressure (often self-inflicted) from the environment.Their biological and physical organization is nested hierarchical <cit.>:[ organism[ organs [ tissues[ cells [ biomolecules [ atoms […] ] ] ] ] ] ];and they live in similarly nested hierarchical environments:[ [ [ [ [ [ [ organisms ]flocks ] habitats ] ecosystems ] biosphere ] planet ]…].General heterogeneous adaptive systems are mainly organized in a hierarchal way as outlined by Simon <cit.>. Intense, diverse interactivity between levels facilitates growth and adaptation. Let P denote the spectral corner representing these systems.Let Q denote the opposite spectral corner in Figure <ref>. Do we find phase transitions of homogeneous matter there? Their unfolding depends on a few macro-parameters such as temperature, and the environment influences them once it is included in a model. Similarly, chaotic dynamical systems are highly sensitive to perturbations. Hence deviations from deterministic trajectoriesare inevitable once a bit of environment exists.Dynamics with a finite time singularity <cit.> will inevitably change before the singularity. Exponentially growing dynamics are likely to meet boundary limitations as well.Are there deep causal implications of thispractically unavoidable environmental influence, particularly when the system happens to be fine-tuned to be influenced? Does this perceptive condition allow genuine downward causation from emergent entities? At least, can these queriesbe resolved for emergent phenomena near the corner Q,where reduction to basic physical causes seems to be assured? This article starts to address these questions.Physical reduction is practically ineffectual near the corner P. The renowned biologist Mayr writes <cit.>: “the physiochemical approach is totally sterile in evolutionary biology", and "analysis is continued downward only as long as it yields useful new information". Furthermore, Mayr cites Popper's harsh critique <cit.>:“as a philosophy, reductionism is a failure ... we live in a universe of emergent novelty; of a novelty which, as a rule, is not completely reducible to any of the preceding stages." If physical reductionism is tenableafter all, it is extremely deeply masked near the corner P. I reckon that this masking is doneby layers of delegated causality.§ DELEGATED CAUSATIONWe affirm our focus by the following three definitions. Theysuggest a non-reductive perspective where the focusis on potential interactions between dynamical processes rather than mathematical behavior of a single dynamical model.The first definition characterizes those (continuous or discrete) dynamical systems or their equilibria, bifurcations, critical phases, “edge of chaos" conditions<cit.>, <cit.>that are “waiting to be perturbed", figuratively speaking.Adynamical system (or its state) is primed if it can exhibit complex,potentially utilitarian behaviors dependingon moderate adjustmentofboundary conditions.This definition may seem unsatisfactory because of several subjective terms.However, the hint of subjectivity and external references are important features of the definition. Primed dynamical systems are to be considered not in isolation but under influence from each other and the environment.They passively, reactively follow the laws of physics, boundary conditions, perturbations. Most dynamical systems can be considered as primed if potential interactions with other systemsaresufficiently interesting. Definite examples of non-primed systems would be chaotic systems or those in a thermodynamic equilibrium. Once feedback loops between intricate reactions and triggersof various primed systems materialize, an elaborate cybernetic or evolutionarysystem may come out.Let us use an economic metaphor for the full spectrumof relative, relational configurations of this kind.A broad market is a set of primed dynamical systems, triggering influences and potentialreactions between them.Extending the economic metaphor, we may refer to sensitivities of a primed dynamical system as its demand, and to the driving effect as supply.A broad market may be structured into hierarchical <cit.>, <cit.> or cybernetic modules, or dominated by an exchange regime of that supply and demand. The causal relation between a primed dynamical system and an external, emergent or self-organized influence that drives the dynamical systemby a moderate force of interaction is called delegated causation.The “moderate force" here is defined in the capability context of driving influences. Their dynamic realizationis beside the main point.Just as operativefeatures of supply are fairly unimportant in marketing and commerce,the causal relation is not defined by the substance of influence factors. I discuss possible dynamic nature of driving influencesat the end of <ref>.These definitions make most sense for complex systems near the spectral corner P in Figure <ref>.Emergent wonders near the corner Q may play special roles in broad markets, as we can recognize them in computer hardware, inside smartphone screens. But the meaning of cause delegation within their internal homogeneous “markets" requires further interpretation, as I briefly discuss in the next section.Delegated causation ought toplay important roles in realistic complex systems, and it should be included in the modeling. Its radical openness <cit.>discourages modeling of complex phenomena by single dynamical systems.On the other hand, delegated causation allows to explicate cybernetic links, tipping points, feedback loops.The following methodological steps of analyzing a single delegated interactionmust be useful: M1Identify the primed dynamical system in the interaction. Determine its sensitivity to perturbations, and possible reactions to perturbations.M2Identify the perturbing influences; describe their mechanism.M3Describe the context, the broad market of the interaction.This methodology is illustrated in Table <ref> in <ref> by series of biological examples. The factor M2 can be identified as an interventional cause. I view M1 and M2as a dual pair of delegated and interventional causations.In the common language, “delegation" describes plausibly the caseswhen M1 is strong,while “intervention" describes the cases when M2 dominates.I use the language of delegation in the abstract and general senseof the causal factor M1 in any interaction. Layers ofdelegated causality signifysystems with great dynamical depth <cit.>. Measures of Kolmogoroviansophistication <cit.> arepromising for quantifying system complexity.Our perspective recommends perturbative analysisof dynamical systems and complexity measures, because significance of primed systems lies in their interactive potential.Delegated causality amounts to a significant case of Deacon'sspecific absence <cit.>, <cit.> as a “pulling" causal force.Causal roles of constrained absences, virtual demands, “pregnant" <cit.> opportunities merit good appreciation. For example, early stages of the British industrial revolution were much stimulated by specific needs of textile, iron, and mining industries <cit.>.On the other hand,dynamical laws and initial conditions enjoy the respectful statusof causes. But chaotic dynamics and causality delegation undermine that status.If an attractor, anequilibrium, or a self-organized state is reachedregardless of initial conditions, what is exactly a cause?§ PHYSICAL REDUCTIONISMOpenness of delegated causality conformswell with Pearl's empirical analysis <cit.> of causation in terms of interventions and counterfactuals. Interventions are “actions as external entities, originating from outside our theory,not as a mode of behavior within the theory" <cit.>. Causative interventions on primed dynamical systems(and possible malfunctions as reverse interventions) must be pivotal features of complex systems.Our first weighty thesis is this: delegated causation offers a conceptual mechanism how micro-scale dynamics results in empirical causation from macro-level agencies in terms of Pearl's causal calculus <cit.>and Hoel's causal emergence <cit.>.Here is one rationale formulated in the economic terminology: complex events are better temporally correlated with appearance of supply rather than with an onset of demand, commonly. A row of falling dominos can be interpreted as a prototypical example. It would be instructive to relate sensitivities of micro-level dynamics and effective information <cit.> measuresin a compelling example of a “smoothly" emergent phenomenon.As a mechanism of downward causation,delegated causality often entailsenvironmental influence.Other proposed mechanism of downward causation via environmental interactionis practopoietic cycle <cit.>. Our second weighty thesisrefers to Gödel's incompleteness theorem <cit.>. Delegated causality has a self-contradictory flavorof a Gödelian paradox <cit.> in the causality language. This messes up basic principlesfor physicalism <cit.>,and Kim's argument <cit.> against non-reductive physicalism. In particular,the causal closure principle says thatif a physical event has a cause, it has a physical cause <cit.>, and the exclusion principle states that no single event can have two independent sufficient causes <cit.>. Kim's conclusion is that non-physical (say, mental) events can have no causal power. In contrast, delegated causality provokes causal, informational contribution from external agents or from an emerging organization, even if delegation is not ideal. Sufficiency of physical causation is debatable then. Delegated causality clarifies inevitability of causal parity <cit.>and levels of explanation.It furnishes hierarchical dynamics, which in turn reinvigorates Aristotle's four categoriesof material, formal, efficient and final causes <cit.>, <cit.>. Incidentally, Popper invoked Gödel's theorem in his argument <cit.>against reductionism.By a similar reference to Gödel's incompleteness, Rosen <cit.> repudiated theoretic formalization of life. Kim's epiphenomenal implications have been countered by interventional <cit.>and counterfactual <cit.> argumentation.Causality and reduction in emergent phenomena are customarily analyzedin terms of supervenience <cit.>, <cit.>, <cit.>: the principle that entitieswith the same micro-level properties will have the same macro-level properties. A supervenience is often defined by a coarse-graining mapfrom physical micro-states to emergent macro-level states.Delegated causality amounts to engaging external information,possibly entailing an externalist <cit.>, <cit.> propensity to expand the supervenience base of micro-states beyond memorization, representation.This propensity can be quickly consummated in complex systems near the corner P, making a supervenience analysis doubtful.For example, an animal may habitually followcertain external clues, peer group behavior or “expert" guidance. Orthe neuro-physiological basis of its behaviormight be “eagerly" changing in live action. These are examples of situated, embodied cognition <cit.>. Human consciousness and free will are the most prominent emergent phenomena.Without more ado, they could be viewed as pinnacles of delegated causation in the known cosmos.From a utilitarian perspective, consciousness is a cognitive-behavioral characteristicthat is able to intervene on (sometimes particularly quiet) emotional, somatic drivers. Can emergent systems near the corner Q in Figure <ref> be interpreted as “eagerly" seeking outside influence,even if the outside is presumably negligible? In a sense, phase transitions delegate causality to dust particles, matter irregularities. Most dramatically, we can consider the whole Universe with no outside in principle. Can we then speculate that a collective behavior of the system isseeking to externalize its statistical parameters,thermodynamic “forces" <cit.>, thereby meeting Mach's principle <cit.>,renormalization group dynamics <cit.>, <cit.>,Heisenberg's indeterminacy, or an observer at the “boundary"? Can dynamic novelty be a valid extension of a supervenience basis? Does emergence itself reflect primordial waysof conceding causality?At least, these speculations could be simulated by a chaotic, fractalor hardly computable topology of the supervenience map, while the macro-dynamics would be described by smooth functions.The consideration of emergence as delegated causality issupported by the recently established correspondence between the renormalization group in theoretical physics and the deep learning approach in artificial intelligence <cit.>, <cit.>; I elaborate this in <ref>.The above questions ought to be addressed bytheories of everything <cit.>. Delegated causality gives an apparenttaste of physical implications ofGödel's incompleteness theorem<cit.>, <cit.>, <cit.>. Itmay even turn out to be a reformulation of Gödel's incompleteness in causal terms.The notion of delegated causality enhancesrather than renounces the reductionist paradigm by definingthe causal role of critical, “edge of chaos" systems. Possibilities of extravagant dynamics are wholly controlled by micro-level arrangements. But dynamical actualization is contingent toparticular instabilities or input from the environment.§ BIOLOGICAL CAUSALITYBiology is a great ground for testing explanatory power of delegated causality.We can expect lots of elaborate causative interventions, provocations. Mayr's influential article <cit.> distinguishes proximate (mainly physiological) and ultimate (mainlyevolutionary) causes of biological phenomena.Contrary to the reductionist template,the relatively more “teleological" levelof explanation by natural selection and adaptationis considered more fundamental in the conventional(neo-Darwinian) Modern Synthesis.Physiological development is guidedby the genetic code, which in turn is pressured by natural selection. Bya rigid interpretation of Modern Synthesis,biological functionality and organisms can be fully understoodonly from the evolutionary perspective,while comprehension of ontogenic developmentis principally unnecessary for that. This reduction of biological causesto statistical phenotype selection, genetic adaptation and driftcould be a deep reason of steady critique of the Modern Synthesis<cit.>, <cit.>, <cit.>, <cit.>, <cit.>.Laland and co-authors<cit.>, <cit.> suggestthat Mayr's proximate-ultimate dichotomy,although still vital, hinders a proper integration of evolution and development, recognition of multiple sources of evolutionary novelty.They advocate an intimate relation between developmental and evolutionary processes, with the former able to influence evolutionary change through phenotypic plasticity,developmental bias, epigenetic inheritance, behavioral changes,and ecological interactions such as niche construction.This is a good list to test scientific productivity of the delegated causality notion. Genes are masters of causal intervention in the biological world, enjoying vast biochemical infrastructure. But they themselves may beopen to interventions. Feedback fromdevelopmental and ecological conditions would bea powerful source of adaptation, with diversifiedagenciesand information forms as inevitable effects.The methodology M1–M3 in <ref>of explicating delegated causation is helpful tohighlight interventional forces M2 and signification M3 of interactions.This is illustrated in Table <ref>by several physiological, ecological, evolutionary, biochemical <cit.>,and developmental <cit.>, <cit.> examples. In many cases, the dual force of intervention dominates.But the central role of genes is compellinglya cybernetic hub of delegation, as I recount again in the middle of <ref>.In the biochemical context, the columns M1, M2delineate Rosen's rendition <cit.> of Aristotle's material and efficient causes, respectively. Systemic closure of efficient causes is underscored bytheories of biological autonomy <cit.>, <cit.>. In the context of causal stability and specificity <cit.>, <cit.>,the factor M1 tends to provide specificity, while M2 furnishes stability and permissiveness. The analysis M1–M3 must offer more clarity than the proposal in <cit.>, <cit.>to employ a terminology of reciprocal causation. The polarity between Mayr's proximate and ultimate causesshould rather stay.Other biological disciplines where explicit analysis of delegated causationought to be useful are microbiology, symbiosis <cit.>,communication <cit.>.The contentious subjects of group selection <cit.>,multilevel selection <cit.>, evolvability <cit.>,cooperation <cit.>, altruism <cit.>,longer term adaptationmight be greatly altered by the perspective of delegated causation as well.The criticized theory of Wynne-Edwards <cit.> of territorial and hierarchicalorganizations in various species regulating population growthbecomes more plausible,as game-theoretic explanations <cit.>turninto proximate causes relative to the “ultimate"adequacy between population size and resources. Mayr <cit.> acknowledges selection of cohesive social groups because their fitness values can be disproportionately largerthan the mean values of individual fitness. This is reminiscent to the emergence principle thatthe whole is greater than the sum of its parts.§ YIN AND YANGConceptions resonating with delegated causality can be found in Chinese philosophy. Provocation of some experiential apprehension from readers lookssensibleor frankly unavoidable in this presentation.That being the case, this section could be excused for being a little rhetorical.Certain cultural, social parallels may be promptly triggered. Carefully evaluative readers are encouraged tocontain social, emotional charges and judgements.The ancient Chinese concepts of Yin and Yang are customarilyevoked to affirm complementarity of opposing, interdependent forces.It is less widely known that Yin and Yang are defined <cit.> as two concrete complementary principles. The complementary harmony is better underscored by Taoism <cit.>, while the particular duality is more emphasized by Confucianism <cit.>.Recently, the Yin-Yang polarity has been interestingly related to holistic causality <cit.>,epistemology <cit.>, <cit.>, transformational change <cit.>, <cit.>, and evenmicrobiology <cit.>, molecular biology <cit.>.One of the defining complementarities is that Yin is anabstractionof passivity, inertness,while Yang is the active, generative principle.This particular duality captures the contrast between mechanical dynamical systems and emergent phenomena very well.A primed dynamical system of Definition <ref> would be a passively reactive Yin, while any effective influencein Definition <ref> would be Yang. It seems fittingly convenient to adoptthis language.Paraphrasing an earlierstatement, dynamic realization of Yang is beside the main point. Yang is the novelty, emergence beyond Yin's territory.Relative to underlying Yin dynamics,Yang is a hero <cit.>, a master, an artful trickster. Yang is the magic of actualizing synergetic possibilities. Yang “opposes" entropy increase by causative leadership, blocking unwelcome occurrences,forcing a decisive turn of action, claiming the language of communication. It is not necessaryto assume supernatural causes to explain Yang manifestations. Yin's ready anticipation of external perturbation and yielding to self-precipitated pressures are pulling causative forces already. Yang is defined by this anticipatory perception of Yin.We may refer to a finely triggered Yin's incipient reaction as Yin's satisfaction.In some cases, this satisfaction may be prompted straightforwardly;in other cases it may be an uncertain, rareevent. Both Yin and Yang of a particular interaction may evolve to sophisticated cybernetic levels,meetingcriteria of a broad market.With many Yin agents available as amenable resources,Yang competition to employ them emerges.Or who harnesses whom?Attribution of selection powers is relative in multifacetedinteractions of Yin and Yang.There is a certain mutual inclusion of Yin and Yang, symbolized by the taijitu sign:< g r a p h i c s > Forexample, a perceptual mechanism is Yin primed to be affected by the environment. But Yang should have perceptual capabilities to recognize peer Yin's dynamic demand and address it.The simplest mechanism of Yang (as external dynamics) is randomness. This means, Yin allows chance to direct its dynamics. For example, divination rituals were common in ancient human societies. In China of the Shang dynasty (circa 1600 BC-1046 BC), oracle bones of ox or turtle were prominently used <cit.>. Other simple mechanism for Yang is competition. That is, Yin sets up a contest who best performs or fakes a particular demand. Another manifestation of Yang is information.Generally, Yang can be identified in processes that are called “teleological" now and then.It is instructive to inspectMayr's classification of five teleologies <cit.>: “four of the five phenomena traditionally called teleological can be completely explained by science, while the fifth one,cosmic teleology, does not exist." The similar list of Ellis <cit.>, <cit.> of five downward causations is definitely germane as well.§ AN INTEGRATED VIEW The two weighty theses in<ref> relate delegated causalityto emergence and Gödel's incompleteness theorem. Emergence and Gödel's theorem are brought togetherin the literature at times <cit.>.For example, Jorgensen and Svirezhev write <cit.>: “In accordance with Gödel's Theorem, the properties of order and emergence cannot be observed and acknowledged from within the system, but only by an outside observer." It could be meaningful to consideremergent phenomena andselection modes as physical, biological, or socionomic <cit.>manifestations of Gödel's incompleteness. Monod's concept of gratuity<cit.> — i.e., independence between chemical qualities and function of biochemical processes — can be similarly related to Gödel's theorem as well. Delegated causality facilitates hierarchical dynamics and defines Nature's “joints" <cit.> along which dynamical levels are fused.Higher levels impose boundary conditions,constrains, selection regimes on dynamics of lower levels <cit.>.This downward causation is enabled by critical, primed organization of the lower levels.Intermediate “sub-wholes" in a dynamical hierarchy are Koestler's holons <cit.> — i.e., stable, integrated, largely autonomous, yet interactive entities.They are both primed dynamical systems and interventional forces,both consumers ofenergy and local sources of order. The constrains of a dynamical leveldefine distinct dynamics,bias, functionality, and a “behavior code" of the holons. Recently <cit.>, <cit.>,an equivalence between the renormalization technique in condensed matterphysics and the deep learning approach in artificial intelligence was established. This reinforces the interpretation that micro-dynamics of a phase transition (or a dissipative system such as Bénard's cells <cit.>, <cit.>) is organizing itself to “explore" and adjust to macro-boundaries.The scale-free dynamics extends mean free-path distancesand relaxation times of particle interactions by orders of magnitude,until limiting macro-dimensions are eventually met. Causality delegation becomes a paraphrase of this “deep learning" of macro-dimensions.This enriched metaphorical context echoesthe Santiago school view of living systems as cognitive <cit.>, sense making <cit.> systems, and Heylighen's view <cit.>of evolution and self-organization as cognitive processes.The action-centered ontology <cit.> assignsintentional stance <cit.> toincreasingly adaptive agents, and (in essence) recognizes interventions as basic constituents of reality. But the passive, incomplete kind of delegated causality should be recognized as well. Deacon's <cit.> cryptic notionof ententionalityis relatable here, if it characterizes being organized for some functioning. Elements of anticipation relate primed dynamical systems, deep learning, and enactive <cit.> cognition.Primed dynamical systems and delegated causality ought to be discernedin all phenomena and entities that are considered emergent or self-organizing. For example, a living cell is predominantly a product of developmental processes that delegate their coordination to genes.Biochemical reactions in the cytoplasm are orchestrated by genes in the form of nucleic acids,but epigenetic switching, energy input from mitochondria, and nutrient flow determine the mode of metabolism.Arrangements of tissues and organs are coordinated by genes again, but the behavior and fate of organic “vehicles" <cit.>are delegated to the nervous system, sociality, tribal customs,or eventually to democratic politics.Natural, symbiotic, artificial, and cultural selections blend into a rather continuous spectrum of efficient causes.Intuition on delegated causality can be further enhanced by reflecting on many regarded modes of economic causality<cit.>, <cit.>, <cit.>, <cit.>. Furthermore,it is worth to reflect on causality of avalanchesin self-organized critical systems <cit.>, <cit.>,where (in theory) only a scale-free statistical distribution has a predictive power. Scale-free dynamics in heterogeneous systemsnear the corner P in Figure <ref> can be self-reinforced by a few persistent motives of adaptation throughout the scale expanse.The Chinese concepts of Yin and Yang describe dynamical discontinuities fittingly.The interaction between Yang's constraints and Yin's demand or satisfaction should define a statistical partition for a relevant entropy tally, andthe semantics of what is caused by delegation.Yang's contingency suggests that universal principlesof self-organization or non-equilibrium thermodynamics(such as speculated laws of maximal entropy production <cit.>) would rather describe potentialities. Overwhelming interventions, catastrophes,dynamical collapses, black swans <cit.>fit the presented context of delegated causality as Yang forces. Resignation to them would not be called “delegation" in the common language. On the other hand, ignoring predictably unsustainabletrends (even if the timing and operation of a likely resolution is highly uncertain)amounts to delegation of responsibilities for consequences. For example, human conscious effort to address a climate change <cit.> is a potentially significant causal factor. Likelihood of a catastrophe indicatesa primed dynamical system, to be influenced possibly by its constituents. The perspective of Yin and Yang appliesto the cultural discord between conservative and post-modernist <cit.> views in the United States.As the American academic critic Bloom noted <cit.>, the post-modernist“relativity of truth is not a theoretical insight but a moral postulate". Post-modernism can be viewedas celebration of Yin birthrights and subjective preferences, as well asdenial of Yang authority. Firstly however, limitation of resources is not addressed operatively by the progressive optimism. That is left implicitly to social, political,or financial <cit.> hierarchies, be them patriarchal or not.Congruent resolution of oppression and sustaining social progressmay continue to behistorical challenges<cit.>, especially under a prolonged environmental stress. Secondly, Yang standards of leadership are in actuality yet appreciated by the progressiveelectorate. This is evident from the 2016 US presidential election <cit.>, where Hillary Clinton did not fullymotivate her expected voters.§ IN CONCLUSION Common experience, biology <cit.>, empirical causality <cit.>, information measures <cit.>, <cit.> tell that downward causation apparently exist. The interpretation of the ancientChinese concept of Yin as a primed dynamical systemintimatesthat downward causation does exist in a very strong, virtually mechanical sense. Instead of having“higher scales wrest the controls from lower scales" <cit.>, the lower scales can be organized to operatively concedea good deal of causality to some higher scales. This possibility shifts the reductionistic paradigm in a novel way <cit.>.The duality of delegated and interventional causations issubtly eminent in Yin-Yang philosophy.Letting things go their way for a matchedbalance is a part of this duality.Co-evolution of agencies with expandingcapabilitiesto provoke, perceive, impose causal relations can lead to understandably tremendous consequences. 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Model-Based Reasoning in Science, Technology, and Medicine.Springer, SCI64 (2007): 245–259. | http://arxiv.org/abs/1707.08905v4 | {
"authors": [
"Raimundas Vidunas"
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"categories": [
"nlin.AO",
"physics.hist-ph",
"03A10, 93A10"
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"primary_category": "nlin.AO",
"published": "20170727151854",
"title": "Delegated Causality of Complex Systems"
} |
[tue]Department of Applied Physics, Coherence and Quantum Technology Group, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, the Netherlands[fei]Thermo Fisher Scientific, Achtseweg Noord 5, 5651 GG Eindhoven, the Netherlands [icms]Institute for Complex Molecular Systems, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, the Netherlands tue]J.F.M. van Renscor [cor]Corresponding author [email protected] tue]W. Verhoeven tue,icms]J.G.H. Franssen tue]A.C. Lassisetue]X.F.D. Stragier fei]E.R. Kieft tue]P.H.A. Mutsaers tue,icms]O.J. LuitenWe present a theoretical description of resonant radiofrequency (RF) deflecting cavities in TM_110 mode as dynamic optical elements for ultrafast electron microscopy. We first derive the optical transfer matrix of an ideal pillbox cavity and use a Courant-Snyder formalism to calculate the 6D phase space propagation of a Gaussian electron distribution through the cavity. We derive closed, analytic expressions for the increase in transverse emittance and energy spread of the electron distribution. We demonstrate that for the special case of a beam focused in the center of the cavity, the low emittance and low energy spread of a high quality beam can be maintained, which allows high-repetition rate, ultrafast electron microscopy with 100 fs temporal resolution combined with the atomic resolution of a high-end TEM. This is confirmed by charged particle tracking simulations using a realistic cavity geometry, including fringe fields at the cavity entrance and exit apertures. Theory and particle tracking simulations of a resonant radiofrequency deflection cavity in TM_110 mode for ultrafast electron microscopy [ December 30, 2023 ========================================================================================================================================§ INTRODUCTION Since the introduction of the Ultrafast (Transmission) Electron Microscope (U(T)EM) by Ahmed Zewail <cit.>, the dynamics of various sorts of material properties have been studied using ultrafast electron techniques such as imaging <cit.> <cit.>, diffraction <cit.> and electron energy-loss spectroscopy (EELS) <cit.> <cit.> with picosecond to femtosecond temporal resolution. The research described in these references is all based on a pump-probe scheme in which the probing electron pulses are created from a flat photo-cathode using a fs laser system. This causes two limitations: First, the average current of the UTEM is limited by the repetition rate of the fs laser system, although long relaxation times of dynamical processes, or slow thermal diffusion, can also limit the maximal repetition times that can be used. Second, the relatively large area of the flat-photocathode limits the peak brightness of the generated electron pulses, hence the maximally achievable spatial resolution. A significant improvement in the peak brightness of laser-triggered electron sources has been achieved by sideways laser illumination of a nano-tip. The reduced dimensions of the photo-field emitter have resulted in a working UTEM with 200 fs electron pulses with a peak brightness comparable to continuous Schottky sources <cit.>. This technique has resulted in very impressive results <cit.> <cit.> <cit.>.An alternative approach requiring no laser at all, involves the chopping of a continuous beam of a high-end TEM into ultrashort electron pulses using a fast blanker in combination with a slit. Apart from the lack of need for an amplified laser system to create electron pulses, further advantages are that no alterations are needed to the gun and the fact that it allows easy switching between continuous mode and pulsed mode. The principle of chopping an electron beam has been realized in Scanning Electron Microscopes (SEMs) many years ago <cit.> <cit.> in the form of electrostatic blanking capacitors <cit.> <cit.> and cavity resonators <cit.> <cit.>. More recently, the use of a photo-conducting switch was proposed to create a laser-triggered, electrostatic beam blanker which can be used for pump-probe experiments in a SEM <cit.>. In parallel, advanced RF-laser synchronization techniques <cit.> <cit.> have reduced the timing jitter between electron pulses and laser pulses to levels below 100 fs <cit.> and even 5 fs <cit.>, making RF cavity-based pulsed beams also suitable for pump-probe experiments.RF cavities or resonators are specifically tailored metallic structures, in which electromagnetic energy can be stored in standing waves or modes. Because of resonant enhancement, RF cavities can be used to generate EM fields of high amplitudes with relatively low input power. Various types of RF cavities have been important elements of the standard toolbox for particle accelerators for many years. For example, a cavity in TM_010-mode supports an oscillating, electric field pointing along the beam axis, which is commonly used for the acceleration of relativistic charged particle pulses. Synchronized to a mode-locked laser system, a cavity in TM_010 mode can be used to for the compression of electron pulses in ultrafast electron diffraction experiments, resulting in pulses shorter than 100 fs <cit.>. A cavity in TM_110 mode supports a magnetic field oscillating perpendicular to the beam axis, transversely deflecting the beam. This mode has been used to chop the continuous beam of a 30 kV SEM into ultrashort pulses <cit.> and record time-of-flight electron energy loss spectra <cit.>. Synchronized to a mode-locked laser, a cavity in TM_110 mode can be used for pulse length measurements <cit.>, for example of non-relativistic, ultracold electron pulses extracted from laser cooled gases <cit.>. Note the same principles of pulse compression and metrology have also been applied with single-cycle THz fields instead of RF cavities <cit.>.In 2012, Lassise et al. showed that using a miniaturized RF cavity in TM_110 mode, it is possible to chop a 30 kV electron beam while fully maintaining the peak brightness <cit.>. Moreover he proposed to use this technique to chop the beam of a high-quality beam of a 200 kV TEM, also conserving the peak brightness of the Schottky field emission source <cit.>. Since then, such an RF cavity-based UTEM has been built at Eindhoven University of Technology (TU/e) and is currently operational <cit.>. Furthermore, alternative TEM beam chopping schemes involving multiple RF cavities are being investigated elsewhere <cit.>. For successful implementation of an RF cavity in a charged particle beam line, a thorough understanding of its effect on the beam dynamics is essential. If not used properly, the rapidly oscillating non-uniform and strong EM fields in RF cavities can have a detrimental effect on the beam quality. However, with proper settings of experimental parameters, such as the RF phase and the position of the beam crossover, the quality of the original beam can be fully maintained, essential for applications such as electron microscopy.In this paper we present the theoretical background of a resonant RF cavity in TM_110 mode as a dynamic optical element for UTEM. In section <ref> we explain the principle of deflection and chopping by calculating the trajectories of a charged particle propagating through an ideal TM_110 pillbox cavity. From these trajectories, we derive the optical transfer matrix of the cavity in section <ref>; and apply this in a Courant-Snyder formalism to calculate the 6D phase space propagation of a Gaussian electron distribution in section <ref>. In section <ref> we apply our findings to study the special case of a focused beam inside a 200 kV TEM column. We derive closed, analytic expressions for the increase in transverse emittance and energy spread. We show that using proper experimental parameter settings, the growth in transverse emittance can be fully eliminated and also the increase in energy spread can be minimized. In section <ref> we present charged particle tracking simulations using a realistic cavity geometry, including fringe fields at the cavity entrance and exit apertures, and a realistic electron beam. The simulations confirm our theoretical findings that an RF cavity in TM_110 mode can be used to chop an electron beam with negligible increase in both transverse emittance and energy spread. This property makes it a very interesting alternative for photo-emission based UTEM, especially in the light of the ever increasing brightness of continuous electron sources. § CHARGED PARTICLE TRAJECTORIES§.§ Beam brightness and emittanceFirst, we define a beam as a distribution of charged particles (charge q and mass m) with a collective motion along the z-axis with average speed v_z, for which each individual particle also has a small relative velocity δ𝐯=(v_x,v_y,δ v_z), withv_x,v_y,δ v_z ≪ v_z,see also figure <ref>. We describe the motion of each particle in the distribution by the 6D trace space coordinates x, x', y, y', z and z', in which x'≡v_x/v_z and y'≡v_y/v_z are the angles of the particle trajectory with respect to the mean trajectory of the total distribution. Analogously, z'≡δ v_z/v_z represents the relative difference in longitudinal velocity of an individual particle with respect to the mean velocity of the beam. Note that we have used the paraxial approximation of equation (<ref>). The 6D trace space volume occupied by this distribution is a measure for the quality of the beam. In accelerator physics, this is often expressed in terms of the root-mean-squared (rms) geometrical emittance in the j=x,y,z direction <cit.>ϵ_j≡√(< j^2> < j'^2> - < j j'>^2).Here the brackets indicate averaging over the entire distribution. The emittance ϵ_j is proportional to the area of the projection of the 6D trace-space density on the (j,j')-plane and is a measure for the focusability of the beam in the j-direction, given that the beam energy is fixed. The geometrical emittance ϵ_j is not a Lorentz-invariant quantity and therefore is not conserved during acceleration. To compare beams of different energy, the Lorentz-invariant rms normalized emittance in the j=x,y,z direction is defined asϵ_n,j≡√(< j^2> < p_j^2> - < j p_j >^2)/mc≈βγϵ_j,in which p_j=γ m v_j is the momentum in the j-direction, γ=1/√(1-β^2) is the relativistic Lorentz factor and β=v/c is the normalized speed <cit.>. In a beam waist, there are no correlations between x and x', hence the normalized transverse emittance is simply given byϵ_n,x^waist≡βγσ_x σ_x',in which σ_x=√(< x^2>) and σ_x'=√(< x'^2>) are the rms beam radius and rms semi-divergence angle in the beam waist. The quality of the beam in the z-direction is determined by the normalized longitudinal emittance ϵ_n,z. For an charged particle bunch with no chirp <z p_z>=0, so the normalized longitudinal emittance ϵ_n,z can be written asϵ_n,z=σ_z σ_p_z/mc≈σ_t σ_U/m c,in which σ_t is the rms pulse duration and σ_U is the rms energy spread <cit.>.Electron microscopists often describe the quality of the beam in terms of the reduced brightness, which is locally defined asB_r≡1/V^∗d^2 I/dAdΩ.The reduced brightness is proportional to the local current density d I/dA=d^2 I/dxdy per unit solid angle dΩ=dx'dy'. By dividing by the relativistically corrected beam potential V^∗≡ (1/2+γ/2)V with q V =(γ-1)m c^2, this quantity is also Lorentz-invariant. To define a measure for the overall quality of a charged particle beam, the practical reduced brightness was introduced by Bronsgeest et al. <cit.>B_pract≡1/V^∗I/A_practΩ=1/V^∗I/π(d_50/2)^2 πθ_x^2.It defines the amount of current I that can be focused into a waist with an area A_pract=π(d_50/2)^2, in which d_50 is the full width spot diameter in which 50% of the current is focused. Furthermore, it assumes a uniform angular distribution with semi divergence angle θ_x. We can now express the practical brightness in terms of the rms normalized transverse emittance ϵ_n,x=βγσ_x σ_x'. By assuming a uniform, angular distribution so that σ_x'=θ_x/2, and a Gaussian, spatial distribution in the beam waist so that σ_x=d_50/2√(2ln2), the practical reduced brightness can be expressed in terms of the rms, normalized transverse emittance asB_pract=q/m c^2I/4ln2·π^2 ϵ_n,x^2.§.§ Framework and assumptionsConsider an ideal, cylindrical cavity in TM_110 mode of length L_, aligned along the z-axis of a cartesian coordinate system, and with the entrance aperture positioned at (x,y,z)=(0,0,0). Close to the z-axis: kx, ky≪1, with k the wavenumber of the RF field, the magnetic field 𝐁 and the electric field 𝐄 of a cavity in TM_110 mode can be approximated by. 𝐁 =B_0cos(ϕ_0+ω t)𝐲̂ 𝐄 =-B_0 ω x sin(ϕ_0+ω t)𝐳̂}for 0<z<L_,in which B_0 is the magnetic field amplitude, ω=ck is the cavity resonance frequency and ϕ_0 is the phase of the RF field at t=0 <cit.>. The word 'ideal' refers to the top-hat profile of the magnetic field amplitude B_0(z)=B_0 as a function of z and the lack of fringe fields around the cavity apertures z=0 and z=L. The effect of both these non-ideal features are studied using particle tracking simulations in section <ref>.The motion of a charged particle described by position vector 𝐫(t)=(x,y,z) and velocity vector 𝐯(t)=(v_x,v_y,v_z)≡ (ẋ,ẏ,ż) will be affected by the Lorentz force 𝐅=q(𝐄+𝐯×𝐁) as the particle travels through the cavity. This is described by the equations of motiond𝐩/dt =𝐅 = -q B_0 [v_z cos(ϕ_0+ω t); 0; - v_x cos(ϕ_0+ω t)+ω x sin(ϕ_0+ω t) ] d𝐫/dt =𝐯=𝐩/γ m, with γ=1/√(1-|v|^2/c^2).Now consider a 6D charged particle trace space distribution traveling along the z-axis with an average velocity v_0𝐳̂. Figure <ref> shows the moment t=0 at which the center of the distribution enters the cavity, which is indicated by the dashed lines. The black dot in figure <ref> indicates a test particle that enters the cavity with trajectories𝐯(t) ≡ v_0(x_i',y_i',1+z_i'), 𝐫(t) ≡(x_i,y_i,z_i)+v_0(x_i',y_i',1+z_i')t.Here we define (x_i,y_i,z_i) as the t=0 position of the test particle with respect to the bunch center, and we define x_i'≡v_x/v_0,y_i'≡v_y/v_0 and z_i'≡δ v_z/v_0 as small deviations in propagation angles at t=0 in the frame of the traveling bunch center. Furthermore we define the 6D trace space coordinate 𝐱_i=(x_i,x_i',y_i,y_i',z_i,z_i') at t=0. To obtain an approximate solution of the equations of motion (<ref>) for the test particle of which the original motion is described by expression (<ref>), we use a perturbative approach based on the following two assumptions: * First, the charged particle gyrates only a small fraction of a full cyclotron orbit during one oscillation period of the EM field, henceω_c/ω≪ 1,in whichω_c≡q B_0/γ_0 mis the cyclotron frequency with γ_0=1/√(1-v_0^2/c^2) the Lorentz factor of the incident beam. This ensures that the charged particles remain close to the z-axis and we can use the paraxial approximation throughout the paper.* Secondly, the distances over which the individual particles move with respect to the bunch center are small compared to the length scales of the collective motion of the bunch, i.e.x_i',y_i',z_i',x_i/L,y_i/L,z_i/L≪ω_c/ω.§.§ Transverse trajectories of the bunch centerBased on assumption (<ref>), we first consider the motion of the bunch center, hence 𝐱_i=0. Therefore we substitute v_z=v_0 and v_x,x=0 in equation (<ref>) and integrate from t=0 to t. The momentum of the bunch center calculated to first order in ω_c/ω is then given by𝐩^(1)(t)=[ B_0 q v_0/ω(sinϕ_0-sin(ϕ_0+ω t));0;γ_0 m v_0 ]. Equation (<ref>) says that to first order, the bunch center is periodically deflected in the transverse direction while the longitudinal motion is unaffected. Because the transverse deflection is caused by the magnetic field, the kinetic energy remains unchanged, so that𝐯^(1)(t)=𝐩^(1)(t)/γ_0 m=[ v_0 ω_c/ω(sinϕ_0-sin(ϕ_0+ω t));0;v_0 ]and𝐫^(1)(t)=[ v_0 ω_c/ω^2(-cosϕ_0+cos(ϕ_0+ω t)+ω t sinϕ_0);0;v_0 t ],in which we have substituted equation (<ref>). The deflection angle at which the pulse exits the cavity is given byα=v_x/v_z=ω_c/ω(sinϕ_0-sin(ϕ_0+Λ)),where we have introduced the dimensionless cavity length parameterΛ≡ω L_/v_0.When a small aperture of diameter d is placed at a distance l≫ d behind the cavity, centered along the cavity symmetry axis, only the electrons go through for which |α(ϕ_0)|<d/2l≪ 1, ignoring a small offset in x. This condition is satisfied for values of the RF phase close toϕ_0=π-Λ/2.In this regime, the deflection of the charged particles by the cavity can be considered as a linear function of the RF phaseα≈ω_c/ω(ϕ_0-π-Λ/2)· 2 sinΛ/2,and the acceptance window of the slit in terms of the RF phase is given by-ω d/4ω_c lsin(Λ/2)< ϕ_0 - π-Λ/2<ω d/4ω_c lsin(Λ/2).The range of phases Δϕ_0 for which equation (<ref>) is satisfied, determines the pulse length τ of the charged particle bunch behind the slit. Hence, an ideal continuous beam of charged particles is chopped up in temporally uniform pulses of pulse lengthτ=Δϕ_0/ω=γ_0 m d/2 l q B_0sin(Λ/2).Equation (<ref>) shows that the pulse length of the resulting bunches for a given magnetic field amplitude can be minimized by choosing the cavity length parameter Λ = π, or L=π v_0/ω. For such a cavity length the transit time of the charged particle traveling through the cavity equals half an oscillation period of the RF field. For 200 kV electrons traveling through a cavity with a typical resonance frequency of ω=2π· 3 GHz, this corresponds to a cavity length of L_=35 mm. In combination with a d=10 μm slit, positioned a distance l=10 cm behind the cavity and with a typical magnetic field strength of B_0=3 mT <cit.>, this results in τ=100 fs pulses.The first order trajectories of the bunch center described by equations (<ref>) and (<ref>) for this situation are plotted in figure <ref> for various values of the RF phase ϕ_0. The red curve in figure <ref> shows the charged particles that have experienced RF phase ϕ_0=0, or more generally ϕ_0=(π-Λ)/2. These particles exit the cavity with zero deflection angle, but with a small shift in transverse position x. To chop these particles of the beam, the chopping slit would have to be positioned slightly off-axis. However, for easy switching between pulsed mode and continuous mode, in practice the chopping slit is placed on the optical axis. As a result, the created electron pulses that go through the slit have a finite transverse momentum. Standard TEM deflection coils can be used to redirect the pulses back to the optical axis.As a final remark, note that the spread in RF phase Δϕ_0 that defines the final pulses, also results in a spread of transverse momentum. Consequently, the transverse emittance is increased which reduces the focusability of the beam. In section <ref> it is shown how this can be prevented. §.§ Longitudinal trajectories of the bunch centerDue to the acquired motion in the x-direction, the bunch center now also starts to experience a Lorentz force in the z-direction. To calculate these second order effects, the first order expressions for v_z, v_x and x of equations (<ref>) and (<ref>) are substituted back into the equation of motion (<ref>). The momentum of the bunch center calculated to second order in ω_c/ω is then given by𝐩^(2)(t)=γ_0 m v_0[ ω_c/ω(sinϕ_0-sin(ϕ_0+ω t));0; 1 + ω_c^2/ω^2cos(ϕ_0+ω t)(-cosϕ_0+cos(ϕ_0+ω t)+ω t sinϕ_0) ].The work done by the electric field in the cavity results in a change in Lorentz factorΔγ =1/m c^2∫ q 𝐄·d𝐫=γ_0 ω_c^2/ω^2v_0^2/c^2[1-cosω t + ω t cos(ϕ_0 + ω t) sinϕ_0+cos(2 (ϕ_0 + ω t))-cos2 ϕ_0/4],in which we also have substituted the expression for x of equation (<ref>). With this change in Lorentz factor γ the second order longitudinal velocity and position are given byv_z^(2)(t) = p_z^(2)(t)/m (γ_0+Δγ)≈ p_z^(2)(t)/γ_0 m(1-Δγ/γ_0)=v_0 + v_0ω_c^2/ω^2{cos(ϕ_0 + ω t) (-cosϕ_0 + cos(ϕ_0 + ω t) +ω t sinϕ_0).. - v_0^2/c^2[1-cosω t + ω t cos(ϕ_0 +ω t) sinϕ_0-cos2 ϕ_0-cos(2 (ϕ_0 + ω t))/4]}andz^(2)(t) =v_0 t + v_0 ω_c^2/ω^3{ω t/2 - sinω t+ ω t sinϕ_0 sin(ϕ_0 + ω t) +sin(2 (ϕ_0 +ω t)) -sin2 ϕ_0/4. .-v_0^2/c^2[ ω t +sinϕ_0(-cosϕ_0 + cos(ϕ_0 +ω t) +ω t sin(ϕ_0 +ω t)).. ..-ω t cos2 ϕ_0+ (1 - cos(2 ϕ_0 +ω t)) sinω t/4] }in which we have assumed Δγ/γ_0≪ 1, and have omitted terms proportional to (ω_c/ω)^3 and higher, based on assumption (<ref>).Figure <ref> shows the longitudinal trajectories of equations (<ref>) and (<ref>) relative to a co-moving frame traveling with the initial velocity of the bunch v_0𝐳̂, for various values of the RF phase ϕ_0. In other words, figure <ref>a shows the relative change in longitudinal velocity of the bunch (v_z-v_0)/v_0 as a function of longitudinal position z in the cavity. Figure <ref>b shows the resulting deviation in longitudinal position relative to the moving frame. Again, the red curve shows the charged particles that have experienced RF phase ϕ_0=0, or more generally ϕ_0=(π-Λ)/2. Figure <ref> shows that these particles are first decelerated and subsequently accelerated. The spread in RF phase Δϕ_0 experienced in the bunch results in an increased energy spread. This is also addressed in section <ref>.§ OPTICAL TRANSFER MATRIXTo derive the optical transfer matrix of a RF cavity in TM_110 mode we follow the same perturbative approach as in section <ref> to calculate the trajectories of particles with 𝐱_i≠0, see figure <ref>. Because of assumptions (<ref>) and (<ref>), the additional coupling between the 𝐱_i terms and the fields in the cavity is regarded as a second order effect. Therefore, the first step is to solve the equation of motion for the test particle described by initial trajectories (<ref>), while substituting 𝐱_i=0 in the expression for the Lorentz force. Note that now the equations of motions must be integrated fromt_b≡-z_i/v_0(1+z_i')to t, because the test particle no longer enters the cavity at t=0. With this, the first order trajectories of the test particle are given by𝐯^(1)(t) =[ v_0 x_i'+v_0 ω_c/ω(sin(ϕ_0+ω t_b)-sin(ϕ_0+ω t));v_0 y_i';v_0 (1+z_i') ] and 𝐫^(1)(t) =[ x_i+v_0 x_i' t+ v_0 ω_c/ω^2(cos(ϕ_0+ω t)-cos(ϕ_0+ω t_b)+ω(t-t_b)sin(ϕ_0+ω t_b));y_i+v_0 y_i' t;z_i+v_0(1+z_i')t ].Subsequently, the first order expressions for v_z, v_x and x of equation (<ref>) are substituted back in equation (<ref>) to calculate the second order trajectories. This results in lengthy expressions for 𝐯^(2)(t) and 𝐫^(2)(t) with many cross-terms of the various initial particle coordinates 𝐱_i. However because of assumptions (<ref>) and (<ref>), only effects that are linearly dependent on initial particle coordinates 𝐱_i and up to second order in ω_c/ω are taken into account. This allows the definition of the optical transfer matrix M_cav via𝐱_f=M_cav𝐱_i,as the linear transformation that maps the initial 6D trace space coordinate 𝐱_i at t=0 onto the final 6D trace space coordinate 𝐱_f=(x_f,x_f',y_f,y_f',z_f,z_f'), defined at the time t_e at which the test particle exits the cavity:t_e=L-z_i/v_0(1+z_i').More specifically, we define[ x_f; y_f; z_f ]≡[ x^(2)(t_e); y^(2)(t_e); z^(2)(t_e)-v_0 t_e ] and [ x_f'; y_f'; z_f' ]≡1/v_0[ v_x^(2)(t_e); v_y^(2)(t_e); v_z^(2)(t_e)-v_0 ]as the position and propagation angle of the test particle at time t_e in the frame of the traveling bunch center. Note that in this step we have assumed the longitudinal velocity of the charged particle remains constant while traversing the cavity.After calculating the second order trajectories for the test particle, evaluating them at time t=t_e and expanding them to first order in 𝐱_i, we obtain the optical transfer matrix for a TM_110 cavityM_cav=[1 L_00C_1C_2;0100C_3C_4;001 L_00;000100;C_5C_6001+C_9 L_+ C_10;C_7C_800 C_11 1+C_12 ],in which the cavity constants C_1 through C_12 are given byC_1=-ω_c/ω(Λcosϕ_0 + sinϕ_0 - sin(Λ + ϕ_0)), C_2=L_/Λω_c/ω(cos(Λ+ϕ_0)-cosϕ_0+Λsin(Λ+ϕ_0)), C_3=-Λ/L_ω_c/ω(cosϕ_0 - cos(Λ + ϕ_0)), C_4=ω_c/ω(Λcos(Λ+ϕ_0)+sinϕ_0 -sin(Λ + ϕ_0)), C_5=C_1/γ_0^2, C_6=L_/Λω_c/ω[(1-2β_0^2)(cos(Λ+ϕ_0)-cosϕ_0)-β_0^2 Λsinϕ_0+Λ/γ_0^2sin(Λ+ϕ_0)], C_7=C_3/γ_0^2, C_8=ω_c/ω(Λ/γ_0^2cos(Λ+ϕ_0)+β_0^2(sin(Λ + ϕ_0)-sinϕ_0)), C_9= ω_c^2/ω^2[(1/2 - 5 /4β_0^2) cos( 2 ϕ_0) + (-1/2+β_0^2/4) cos(2 (Λ + ϕ_0))..+β_0^2 ( cos(Λ + 2 ϕ_0 )+ Λ/2sin(2 ϕ_0)) - Λ/γ_0^2sin(Λ + 2 ϕ_0)], C_10= Lω_c^2/ω^2[ 1/2(1-4β_0^2)cosΛ + β_0^2/4cos 2ϕ_0+1/2(1-β_0^2/2)cos(2(Λ+ϕ_0))..+1/2(1+β_0^2)[cos(Λ+2ϕ_0)-1]+(3β_0^2/Λ+Λ/2γ_0^2)sinΛ..+5β_0^2/4Λsin 2ϕ_0 -β_0^2/4Λsin(2(Λ+ϕ_0))], C_11= ω_c^2/ω^2Λ/L[ sin(2 (Λ + ϕ_0)) -sin(Λ +2ϕ_0)-(Λ/γ_0^2 + β_0^2 sinΛ)cos(Λ + 2 ϕ_0)]andC_12= ω_c^2/ω^2[Λ{sin(2 (Λ + ϕ_0)) -sin(Λ + 2 ϕ_0)+Λsinϕ_0sin(Λ + ϕ_0)}..- β_0^2( cos( Λ + ϕ_0)-cosϕ_0 + Λsinϕ_0 )Λsin(Λ + ϕ_0)..-2β_0^2 (1 - cosΛ+Λcos(Λ + ϕ_0) sinϕ_0+ 1/4[cos(2 (Λ + ϕ_0))-cos 2 ϕ_0 ] )],withβ_0≡v_0/cthe normalized initial speed of the bunch center.Note that cavity constants C_1-C_8 are proportional to the magnetic field amplitude and therefore scale linearly with ω_c/ω. The cavity constants C_9-C_12 represent second order effects scaling with (ω_c/ω)^2. In the limit where the fields are turned off (ω_c→ 0), all the cavity constants vanish and the matrix M_cav simply reduces to a drift over a distance L_. The determinant of the transfer matrix is unity to first order in ω_c/ω, which means that the 6D trace space density of a charged particle distribution is conserved during cavity transit. Furthermore, note that the transfer matrix M_cav simplifies significantly for an optimized cavity length of L_=π v_0/ω, i.e. Λ=π.§ COURANT-SNYDER TRACE-SPACE TRANSFORMATIONNow we have the optical transfer matrix of the TM_110 cavity, we use the Courant-Snyder formalism <cit.> to calculate the propagation of an entire 6D trace-space distribution of charged particles. For this purpose, we describe the distribution in terms of its rms ellipsoidal contours in trace space, of which the projections on the (j,j') planes (j=x,y,z) are given by the ellipsesϵ^i_j=γ̂_j j^2+2α̂_j j j'+β̂_j j^2 with β̂_jγ̂_j-α̂_j^2=1.Figure <ref> shows how the Courant-Snyder parameters α̂_j,β̂_j,γ̂_j are related to initial beam properties such as rms radius σ_j and rms angular divergence σ_j'. The rms area of the ellipse is given by πϵ^i_j, where ϵ^i_j is the (initial) projected rms emittance in the j,j'-plane.We can write equations (<ref>) in matrix notation𝐱^T A^-1𝐱=1,with A the real, positive definite, 6×6 beam matrix at t=0 given byA= [A_xx'00;0 A_y y'0;00 A_z z' ] with A_j j'=ϵ^i_j [β̂_j -α̂_j; -α̂_jγ̂_j ].Here we assume no correlations between spatial degrees of freedom. Note that ϵ^i_j=√((A_jj')) because of equation (<ref>). We can now propagate this charged particle distribution through an ideal TM_110 cavity. The beam matrix that describes the distribution at the exit of the cavity t=t_e is given byB=M_cavAM_cav^T≡[ B_xx' 0 cross terms; 0 B_yy' 0; cross terms 0 B_zz' ].The x,z-correlations are introduced by the non-zero, off-diagonal matrix elements of M_cav and will cause an exchange between the transverse and longitudinal emittance, hence energy spread. Both will decrease the focusability of the beam.The final normalized transverse emittance is calculated byϵ^f_n,x=β_0γ_0√(det(B_xx'))while the rms energy spread of the beam after cavity transit is given byσ_U^f=γ_0^3 m v_0 σ_v_∥^f,with σ_v_∥^f the rms spread in velocity along the propagation vector 𝐯^f of the deflected beam. To find σ_v_∥^f we consider the 2D beam ellipse that describes the final trace space distribution projected on the (x',z')-planeB_x'z'≡[ B_(2,2) B_(2,6); B_(6,2) B_(6,6) ].The diagonal matrix elements B_(2,2) and B_(6,6) are related to the velocity spread of the bunch in the x and z directions, via σ^f_v_x=v_0 √(B_(2,2)) and σ^f_v_z=v_0 √(B_(6,6)) respectively. Next, we consider this beam matrix in the (ξ',ζ')-coordinate system as illustrated in figure <ref>, which is rotated with respect to the (x',z')-coordinate system about the final propagation angle at which the bunch center exits the cavity α_f≡ v^f_x/v^f_z= x_f'/1+z_f'= ω_c/ω (sinϕ_0 -sin(Λ + ϕ_0))/{1+ω_c^2/ω^2[cos(Λ+ϕ_0)(cos(Λ+ϕ_0)+Λ/2sinϕ_0-cosϕ_0)....-v_0^2/c^2(1-cosΛ +Λcos(Λ+ϕ_0)sinϕ_0-sinΛsin(Λ+2ϕ_0)/2)]}.In this coordinate system, the ξ' and ζ' axes are perpendicular and parallel to the final velocity vector 𝐯^f of the bunch, respectively. The final beam matrix in the rotated coordinate system is given byB^rot_ξ'ζ'≡M_rotB_x'z'M_rot^T with M_rot≡[ 1-α^2_f-α_f; α_f 1-α^2_f ],of which the diagonal matrix elements B^rot_(2,2) and B^rot_(6,6) are related to the velocity spread parallel and perpendicular to 𝐯^f, via σ^f_v_⊥=v_0 √(B^rot_(2,2)) and σ^f_v_∥=v_0 √(B^rot_(6,6)) respectively. Now the final energy spread of the bunch after propagating through the cavity is given byσ_U^f=γ_0^3 m v_0 σ_v_∥^f=γ_0^3 m v_0^2 √(B^rot_(2,2)). So, using the Courant-Snyder formalism, we can derive analytical expressions for the final normalized transverse emittance (equation (<ref>)) and final energy spread (equation (<ref>)) of the beam after traversing an ideal TM_110 cavity. Moreover these expressions are derived as function of the initial transverse emittance and energy spread of the incident beam. To our knowledge, this is not possible in any other way.§ APPLICATION: FOCUSED BEAM IN A 200 KV TEMNow we apply the Courant-Snyder model to the special case of a focused beam in a 200 kV TEM. More specific, we calculate the increase in normalized transverse emittance and energy spread of a 200 kV 6D Gaussian charged particle distribution with a finite initial geometrical emittance and energy spread, focused to a crossover inside a TM_110 cavity. We are aware that Gaussian distributions are not realistic in electron microscopes, but they result in easy calculations for the rms quantities, required for the Courant-Snyder model. Furthermore, the functional dependencies in the final expressions are independent of the shape of distribution, the only difference is a proportionality factor. In section <ref>, we use charged particle tracking simulations to calculate actual numbers. Furthermore, the Courant-Snyder model obliges us to choose a finite initial pulse length, although in the experiment the initial beam is continuous. However, we are not interested in the electrons that are not part of the final (chopped) pulse. So by choosing an initial pulse length equal to the expected final pulse length we simply leave out the electrons of which we already know they will collide into the chopping aperture. In section <ref>, we use charged particle tracking simulations to test the validity of this approach.Figure <ref> shows an electron beam with initial geometrical emittance ϵ_x^i that is focused to a crossover at z=z_0 with rms divergence angle σ_x'. Therefore the rms beam radius at z=0 is given by σ_x=√((ϵ_x^i/σ_x')^2+(σ_x' z_0)^2). Furthermore the distribution has an initial rms pulse duration σ_t, initial rms energy spread σ_U^i and no initial chirp. The beam matrix A that describes this distribution at t=0 is defined by the Courant-Snyder parameters: β̂_x=σ_x^2/ϵ_x^i, γ̂_x=σ_x'^2/ϵ_x^i, α̂_x=√(β̂_xγ̂_x -1), ϵ_z^iβ̂_z=(v_0 σ_t)^2, ϵ_z^iγ̂_z=(σ_U^i/γ_0^3 m v_0^2)^2 and α̂_z=0. The beam matrix B that describes the beam at the exit of the cavity may be calculated using equation (<ref>). Then using equations (<ref>) and (<ref>) we can calculate the final normalized transverse emittance and energy spread of the beam. Figure <ref> shows (a) the final normalized transverse emittance ϵ^f_n,x and (b) the final energy spread σ_U^f of the 200 keV electron beam directly behind the TM_110 cavity as a function of the cavity RF phase ϕ_0 and the position of the crossover z_0 for typical beam parameters of a 200 kV (pulsed) TEM: ϵ_n,x=3 pm rad, σ_x' = 0.15 mrad, σ_t=200 fs, σ_U^i=0.5 eV; and cavity parameters: ω = 2π· 3 GHz, B_0=3 mT and L=35 mm. The figure shows a dark blue region in (ϕ_0,z_0) parameter space for which both quantities hardly increase and the quality of the incident beam is maintained. This will be investigated further in the next sections. §.§ Transverse emittanceThe expression for the final transverse normalized emittance ϵ_n,x^f that is calculated by equation (<ref>) and is plotted in figure <ref>a, is lengthy and does not provide much insight. However, if we assume an ideal incident electron beam: hence no initial transverse emittance ϵ_n,x^i=0 and energy spread σ_U^i=0, equation (<ref>) reduces to a closed, analytic expression for the final, normalized transverse emittance:ϵ_n,x^f=β_0γ_0σ_x'σ_t ω_c/ω| z_0/L_Λcosϕ_0 - (z_0/L_-1)Λcos(Λ+ϕ_0)+sinϕ_0-sin(Λ+ϕ_0)|.Note that the right-hand side of equation (<ref>) is zero forz_0/L_=Λcos(Λ+ϕ_0)-sin(Λ+ϕ_0)+sinϕ_0/Λ(cos(Λ+ϕ_0)-cosϕ_0),and for a cavity with optimized cavity length Λ=π, this reduces toz_0/L_=1/2-tanϕ_0/π.Equation (<ref>) describes the white dashed curve in figure <ref>a. It describes a region in parameter space where propagation of an ideal beam through an RF cavity in TM_110 mode results in zero increase in transverse emittance. In the situation of entrance phase ϕ_0=1/2(π-Λ), for which the shortest pulses are obtained, equation (<ref>) reduces to z_0/L=1/2, hence focusing the electron beam in the center of the cavity.Figure <ref> explains the principle of conjugate blanking for ϕ_0≈(π-Λ)/2 in more detail. It shows the real (x,z)-space (top), (x',x) phase-space (middle) and transverse emittance ϵ_x (bottom) as a function of longitudinal coordinate z for both a collimated and a focused beam propagating through a TM_110 cavity. The color coding shows the correlation with time, for which blue indicates the front and red indicates the back of the pulse. Furthermore, note that the transverse emittance is proportional to the (x,x') trace-space area. When a bunch of charged particles enters the cavity at z=0, the particles suddenly feel a force in the transverse direction, which results in a transverse deflection. This force varies with time, hence the front of the pulse feels a slightly different force than the back of the pulse. This results in an angular spread and therefore an increase in transverse emittance. The fields in the cavity are homogeneous along the z-axis, so during transit through the cavity, all the particles in the bunch experience the same forces and the emittance is unaffected. However, when the bunch arrives at the exit aperture, again there is a sharp step in the experienced Lorentz force, that changes in amplitude while the bunch travels past this gradient. Figure <ref>a shows that for a collimated beam, this results in a second increase in angular spread, hence a second emittance growth. Note that the increase in emittance at the apertures is not an effect of fringe fields.However, by focusing the beam in the center of the cavity (figure <ref>b) the angular spread that is obtained at the entrance aperture, can be canceled by the forces at the exit aperture. This is seen best in the (x,x') trace space plot. By focusing, an additional correlation is applied between x and x'. During passage through the cavity, the (x,x') trace space distribution is sheared parallel to the x-axis in such a way that at the exit of the cavity the trace space distribution is collapsed onto a line, thus canceling the emittance growth at the entrance of the cavity. This is called conjugate blanking.For an on-axis slit, hence ϕ_0=(π-Λ)/2, the focus point for conjugate blanking lies exactly in the center of the cavity (z_0=L/2). However, for any other phase than ϕ_0=(π-Λ)/2 the experienced Lorentz forces at the entrance and exit of the cavity are not symmetric. By focusing the beam at a different position given by equation (<ref>) the emittance growth at the entrance of the cavity can still be fully canceled at the exit. §.§ Pulse lengthFocusing the electron beam in the center of the cavity significantly increases the size of the beam at the position of the slit. As a result, the expression for the pulse length of equation (<ref>), which was done for an infinitely small beam, is no longer valid. The actual temporal profile of the pulses behind the slit is now proportional to the convolution of the tophat distribution of the chopping aperture and the approximately tophat distribution of the electron beam at the position of the chopping aperture.In a TEM, the rms divergence angle σ_x' is defined by the diameter of the C2-aperture and its distance to the center of the cavity. In the special case that the cavity is placed exactly in between the C2-aperture and a circular chopping aperture; and both apertures have the same diameter, the resulting rms pulse length is given byσ_t=√(2)γ_0 m σ_x'/q B_0sin(Λ/2).Equation (<ref>) shows that to maintain short pulses while focusing in the cavity, is it important to select the divergence angle as small as possible. Therefore, is it not only important to choose a small chopping aperture, but also a small C2-aperture, preferably of the same diameter.§.§ Energy spreadThe expression for the final energy spreadσ_U^f that describes figure <ref> is obtained by evaluating equation (<ref>) and is even more complicated than the general expression for the final transverse emittance ϵ_x^f. To gain insight in the different parameters contributing to a growth in energy spread, we consider three different situations in which we substitute three different sets of assumptions in equation (<ref>).*We start by assuming an ideal, focused beam, i.e. substituting ϵ_n,x^i=σ_U^i=σ_t=0 in equation (<ref>). We findσ_U^f=γ_0 m v_0 σ_x'ω_c/ω| z_0/L_Λcosϕ_0 - (z_0/L_-1)Λcos(Λ+ϕ_0)+sinϕ_0-sin(Λ+ϕ_0)|.This contribution is plotted in figure <ref>b as the white dashed curve. It has the same functional dependence on ϕ_0 and z_0 as the increase in transverse emittance in equation (<ref>), and can be fully eliminated using the same conjugate blanking scheme of equation (<ref>). *However, the energy spread also increases with the initial pulse duration σ_t≠0. By substituting σ_x'=0, Λ=π and z_0=L/2 into equation (<ref>), instead of σ_t=0, we findσ_U^f=γ_0 m v_0^2 πω_c^2/ωσ_t√(cos(2 ϕ_0^2)).This contribution describes the black dashed lines in figure <ref>b and explains the minima in σ_U^f at ϕ_0=±π/4. At the intersection points of the black dashed lines and the white dashed curve, both the increase in transverse emittance and energy spread are eliminated simultaneously. In principle, we can exploit these 'sweet spots' by placing the slit off-axis such that phase ϕ_0=±π/4 is chopped out of the beam, see also figure <ref>b, followed by choosing the correct position of the crossover. In practice, the deflection coils in a TEM could be used to redirect the beam back to the optical axis. *However, for easy switching between pulsed mode and continuous mode; and to obtain the shortest pulses, we choose to place the slit on-axis. For a beam with zero initial energy spread σ_U^i=0 and the chopping slit placed on-axis, hence ϕ_0=1/2(π-Λ), the final energy spread is given byσ_U^f=γ_0^3 m v_0^2 ω_c/ω√([4σ_x'^2(z_0/L_-1/2)^2Λ^2+k^2(ϵ_x^i)^2/β_0^2 σ_x'^2]sin^2(Λ/2)+ ω_c^2σ_t^2/γ_0^2(Λ-sinΛ)^2).Equation (<ref>) shows three terms: * The first term is proportional to the angular divergence σ_x' and can be eliminated by focusing in the center of the cavity, i.e. z_0=L_/2.* The second term scales with k ϵ_x^i/σ_x' and describes the sampling of small, off-axis electric fields due to the finite size of the beam in the crossover z=z_0. It can be minimized by decreasing the transverse emittance of the incident beam, for instance by using a smaller C2-aperture at the expense of average current. * The third term can be reduced by decreasing the cavity length parameter Λ. However, according to equation (<ref>) this goes at the expense of the pulse length. When we decrease the cavity length parameter from Λ=π to Λ=Λ' while we simultaneously increase the magnetic field by a factorω_c'/ω_c=sin(π/2)/sin(Λ'/2),the growth in energy spread can be decreased by a factorΔσ_U'(ω_c',Λ')/Δσ_U(ω_c,π) =Λ'-sinΛ'/πsin^2(Λ'/2)<1, for Λ'<πwhile the short pulse length is fully maintained. Of course, the increased magnetic field in the cavity requires a higher input power. A second way to decrease the third contribution to σ_U^f is to decrease the pulse length σ_t by decreasing the divergence σ_x' of the incident beam, see equation (<ref>). Figure <ref>a shows the final energy spread σ_U^f of equation (<ref>) for a beam with no initial energy spread focused in the center of the cavity (z_0=L/2) as a function of divergence angle σ_x' for varying cavity length parameter Λ and initial emittance ϵ_x^i. Here we have increased the magnetic field accordingly to keep the rms pulse length σ_t = 100 fs. This is done by solving equation (<ref>) for B_0 and substituting that in equation (<ref>). The magnetic field B_0 required to create σ_t=100 fs pulses for decreasing cavity length parameter Λ is plotted in figure <ref>bfor varying rms divergence angle σ_x'. Figure <ref>a demonstrates two methods that can be used to minimize the final energy spread of the chopped pulses. First, the final energy spread can be reduced significantly by decreasing the cavity length while simultaneously increasing the magnetic field B_0 to maintain short pulses. This is an effective strategy down to a cavity length parameter of Λ≈π/2, below which decreasing the cavity length further becomes very expensive in terms of B_0, see figure <ref>b. Secondly, decreasing the divergence angle σ_x' can reduce the remaining growth in energy spread even further. This is effective down to the point where the transverse emittance of the incident beam ϵ_n,x^i limits the minimum achievable final energy spread. § PARTICLE TRACKING SIMULATIONS In the previous sections, we have used an analytical approach to show that an ideal cylindrical cavity in TM_110 mode can be used to chop a relativistic electron beam into ultrashort electron pulses while maintaining the quality of the original beam. At this point, we want to investigate the limits of this technique in a scenario as realistic as possible. Therefore, we first implement the actual cavity geometry used in the TU/e UTEM in cst Microwave Studio <cit.> and numerically calculate the realistic B_y-field and the E_x-field along the cavity axis. The diameters of both the entrance and the exit aperture are 3 mm.Figure <ref> shows that the actual cavity geometry results in fringe fields near the entrance and exit apertures. Figure <ref>a shows the on-axis B_y-field, which is fitted with a double error-functionB_y(r=0,z)=B_y,0/erf(L_/2s)(1/2erf(z/s)- 1/2erf(z-L_/s)).Here B_y,0 is the maximum field strength at the center of the cavity and s is a fit-parameter that describes how the B_y-field falls off near the apertures of the cavity. Figure <ref>b shows the on-axis E_x-field, which is fitted with two Gaussians of opposite sign. When we substitute these fits into a fifth order power expansion of the solution of a cylindrical cavity, see equation (<ref>) in appendix A, we can reconstruct all the other (r,φ,z) components of the EM field close to the cavity axis. To investigate whether the fringe fields near the cavity apertures affect the beam quality, we implement the obtained field expansions of equation (<ref>) in the gpt-code <cit.> for realistic particle tracking simulations. These simulations also allow us to chop a continuous electron beam using the combination of a cavity and a slit, rather than ab initio assuming a Gaussian temporal distribution. Furthermore, we can simulate a more realistic initial beam with uniform spatial and angular distributions, rather than Gaussian distributions.In the simulations, we apply the practical lessons we learned from the Courant-Snyder model in the particle tracking simulations: * We select RF phase ϕ_0=1/2(π-Λ) by placing the slit on-axis to obtain the shortest pulses.* We focus in the center of the cavity z_0=L/2 to reduce the growth in transverse emittance and energy spread.* We choose the cavity length parameter Λ=π/2 to reduce the remaining growth in energy spread even further while still being able to make short pulses with a realistic cavity field amplitude.For the electron source we choose a typical 200 kV Schottky field-emission gun with a practical reduced brightness B_r=10^8 A/m^2 sr V and rms energy spread σ_U^i=0.5 eV. A DC current of I=10 nA then results in an initial normalized transverse emittance ϵ_n,x^i = 3 pm rad (rms). To test the validity of our theory for applications with higher currents, we also add a simulation series for ϵ_n,x^i = 100 pm rad (rms). Furthermore, we vary the magnetic field amplitude from B_0=1 mT to B_0=10 mT in steps of 1 mT and we vary the diameter of the C2- and chopping apertures between d=30 μm and d=10 μm. For these parameter settings, we measure the final rms pulse length σ_t, the rms normalized transverse emittance ϵ_n,x^f and rms energy spread σ_U^f.Figure <ref> shows both the results of these particle tracking simulations and the results of the Courant-Snyder model. The latter have been obtained by substituting the simulated rms values for σ_x' and σ_t in the non-simplified version of equation (<ref>). First of all we find excellent agreement between the theoretical model and the particle tracking simulations. This gives good confidence that the derived expressions for the optical transfer matrix correctly describe the actual particle trajectories, and therefore justifies the perturbative approach of describing a TM_110 cavity as a linear optical element for electrons based on the assumptions in section <ref>. Furthermore it shows that the fringe fields due to a non-idealized cavity geometry do not significantly affect the beam dynamics. Secondly, figure <ref> shows that there is a trade off between the final energy spread of the electron beam and the final pulse length. The amplitude of the magnetic field in the cavity can be used to shape the final time-energy phase space distribution, depending on the application. However above all, figure <ref> demonstrates the enormous potential for RF cavity based ultrafast electron microscopy. Especially when 10 μm apertures are used in combination with a low emittance input beam of 3 pm rad, extremely short pulses can be generated with hardly any increase in energy spread. Pulses of 200 fs (rms) with σ_U^f=0.5 are expected even for a magnetic field amplitude of only B_0=1 mT, which has already been realized at TU/e <cit.>. Ultimately, pulse lengths down to 20 fs (rms), combined with rms energy spread of σ_U^f=0.70 eV are expected for B_0=10 mT. Finally, we didn't observe any increase in normalized transverse emittance in any of the simulations in figure <ref>.As for any pulsed beam, a short pulse length results in a low average current of the pulsed beam. Although the 3 GHz repetition rate of an RF cavity based UEM is several orders of magnitude higher than in conventional UEMs based on photo-emission, the low charge per pulse: 0.006 e/pulse at I=10 nA for τ = 100 fs, limits the average current. However, because the peak brightness of the original beam is conserved after chopping with an RF cavity, any future improvements on the continuous source will directly improve a cavity-based pulsed source as well. For example, the recent developments of new LaB_6-emitters which promise a brightness up to B_r=10^10 A/m^2sr eV <cit.> are worth mentioning. Provided that they can be operated at sufficient current, the combination of such sources with an RF cavity could result in ultrafast electron microscopy with unprecedented spatial and temporal resolution at the average current of present-day continuous electron microscopes.§ CONCLUSIONSWe have developed a theoretical description of resonant radiofrequency deflecting cavities in TM_110 mode as dynamic optical elements for ultrafast electron microscopy. We have derived the optical transfer matrix of an ideal pillbox cavity and have calculated the 6D phase space propagation of a Gaussian electron distribution using a Courant-Snyder formalism. We have derived closed, analytic expressions for the increase in transverse emittance and energy spread that have resulted in practical insight that can be applied directly in an experiment. We have shown that the beam quality of the incident electron beam can by maintained by proper settings of the RF phase and the position of the crossover inside the cavity. In particular, we have explained the concept of conjugate blanking for fully eliminating increase in transverse emittance. The growth in energy spread can be minimized by decreasing the cavity length and the divergence angle. The correctness of our model and the potential of RF cavities for UEM are confirmed by charged particle tracking simulations using a realistic cavity geometry, that take into account fringe fields at the cavity entrance and exit apertures. In conclusion, RF cavities in TM_110 mode allow high-repetition rate, ultrafast electron microscopy with 100 fs temporal resolution combined with the atomic resolution of a high-end TEM.§ ACKNOWLEDGEMENTThis work is part of an Industrial Partnership Programme of the Netherlands Organisation for Scientific Research (NWO).§ REFERENCESunsrt § APPENDIX A: FIFTH ORDER POWER EXPANSION OF EM FIELDS IN A CYLINDRICAL CAVITY IN TM_110 MODE The fifth order power expansion of the (r,φ,z) components of the EM field in a cylindrical cavity TM_110 mode are given by:E_r = 1/192{(192-24k^2 r^2+k^4 r^4 ) E_x (0,z)+(-72r^2+6k^2 r^4 )∂^2 E_x (0,z)/∂^2 z+5r^4∂^4 E_x (0,z)/∂^4 z..+(-48r^2+4k^2 r^4 )kc ∂ B_y (0,z)/∂ z+4r^4 kc ∂^3 B_y (0,z)/∂^3 z}cosφcos(ω t+ϕ_0)E_φ = 1/192{(-192+72k^2 r^2-5k^4 r^4 ) E_x (0,z)+(24r^2-6k^2 r^4 )∂^2 E_x (0,z)/∂^2 z-r^4∂^4 E_x (0,z)/∂^4 z..+(-48r^2+4k^2 r^4 )kc ∂ B_y (0,z)/∂ z+4r^4 kc ∂^3 B_y (0,z)/∂^3 z}sinφcos(ω t+ϕ_0)E_z = 1/192cosφcos(ω t+ϕ_0) {(192r-24k^2 r^3+k^4 r^5 )(∂ E_x (0,z)/∂ z+kcB_y (0,z))..+(-24r^3+2k^2 r^5 )(∂^3 E_x (0,z)/∂^3 z+kc ∂^2 B_y (0,z)/∂^2 z)+r^5 (∂^5 E_x (0,z)/∂^5 z+kc ∂^4 B_y (0,z)/∂^4 z)}B_r = 1/192{(192-24k^2 r^2+k^4 r^4 ) B_y (0,z)+(-72r^2+6k^2 r^4 )∂^2 B_y (0,z)/∂^2 z+5r^4∂^4 B_y (0,z)/∂^4 z..+(48r^2-4k^2 r^4 )k/c∂ E_x (0,z)/∂ z-4r^4k/c∂^3 E_x (0,z)/∂^3 z}sinφsin(ω t+ϕ_0) B_φ = 1/192{(192-72k^2 r^2+5k^4 r^4 ) B_y (0,z)+(-24r^2+6k^2 r^4 )∂^2 B_y (0,z)/∂^2 z+r^4∂^4 B_y (0,z)/∂^4 z..+(-48r^2+4k^2 r^4 )k/c∂ E_x (0,z)/∂ z+4r^4k/c∂^3 E_x (0,z)/∂^3 z}cosφsin(ω t+ϕ_0)B_z = 1/192sinφsin(ω t+ϕ_0) {(192r-24k^2 r^3+k^4 r^5 )(∂ B_y (0,z)/∂ z-k/c E_x (0,z))..+(-24r^3+2k^2 r^5 )(∂^3 B_y (0,z)/∂^3 z-k/c∂^2 E_x (0,z)/∂^2 z)+r^5 (∂^5 B_y (0,z)/∂^5 z-k/c∂^4 E_x (0,z)/∂^4 z)},in which E_x(0,z) and B_y(0,z) are the on-axis electric and magnetic field amplitudes. Describing the cavity using these field expansions rather than a 6D field map results into 10-20 times faster particle tracking simulations. | http://arxiv.org/abs/1707.08835v2 | {
"authors": [
"J. F. M. van Rens",
"W. Verhoeven",
"J. G. H. Franssen",
"A. C. Lassise",
"X. F. D. Stragier",
"E. R. Kieft",
"P. H. A. Mutsaers",
"O. J. Luiten"
],
"categories": [
"physics.acc-ph",
"physics.optics"
],
"primary_category": "physics.acc-ph",
"published": "20170727123401",
"title": "Theory and particle tracking simulations of a resonant radiofrequency deflection cavity in TM$_{110}$ mode for ultrafast electron microscopy"
} |
[ V.P. Gachok July 14, 2017 =================§ INTRODUCTION §.§ Lorentz-violating Inflationary Cosmological Models Lorentz invariance is a fundamental symmetry of General Relativity (GR) and standard particle physics and has been tested to a very high degree of accuracy.However, there is growing evidence that some issues in quantum gravity (certain divergences, micro-causality) can be resolved by the removal of Lorentz invariance <cit.>. Furthermore, some approaches to quantum gravity may even desire a preferred rest frame in vacuum <cit.>.On larger scales, explanations of Dark Energy and Dark Matter in the current cosmological paradigms based on GR might also be explained using an alternative theory of gravity <cit.>, such as those in which the Lorentz invariance requirement is relaxed. Indeed, in cosmology there is a natural frame associated with the cosmic microwave background, and therefore it is possible the Lorentz invariance assumption may be relaxed at late times.The Einstein-Aether (AE) theory <cit.> is a proposed model of gravity which preserves general covariance and incorporates a violation of Lorentz invariance. The local space-time structure is determined by a dynamical time-like vector field, u^a (the aether), together with a metric tensor, g_ab. The field equations for this Einstein-Aether theory essentially consist of GR with a modified source due to the aether together with additional field equations describing the evolution of the aether vector field.In standard Einstein-Aether theory, it is commonly assumed that the violation of Lorentz invariance is only within the gravity sector of the theory, while the matter sector continues to be coupled only to the metric, and hence remains Lorentz invariant.However, it is natural to expect that Lorentz violations in the matter sector could also be permitted, albeit with the understanding that there are quite stringent constraints on such Lorentz violations <cit.>.Assuming matter is determined by a scalar field, some researchers have investigated the potential changes that arise as a result of a violation of Lorentz invariance in the matter sector of the Einstein-Aether theory <cit.>. Barrow <cit.> investigated the effect of scalar field/aether field coupling in which the dependence of the scalar field in the potential was exponential in nature.He found there are solutions with the possibility that the coupling parameter enables inflation in situations in which it would not otherwise occur. Sandin et al. used a similar ansatz for the scalar field potential in <cit.> and determined that there is a fundamental change in the future asymptotic behaviour when the coupling parameter becomes sufficiently large.Donnelly and Jacobson <cit.> considered a chaotic inflationary scenario and determined that the coupling of the scalar field to the aether field can either slow down or speed up the evolution of inflation. Solomon and Barrow <cit.> completed a detailed analysis with no prescribed coupling between the scalar field and the aether field and found conditions on the potential that must be satisfied if one is to have stable slow-roll inflationary solutions.Alhulaimi <cit.> generalized some of these results to include not only a coupling of the scalar field to the aether vector expansion, but also a coupling to the aether vector shear.Where most others have coupled the scalar field to the aether field through the scalar field potential, Kanno and Soda <cit.> took a very different approach. In their analysis, they assumed that the aether parameters c_i in equation (<ref>) are functions of the scalar field.They found that it is possible to have inflation without a scalar field potential,i.e., with a massless scalar field.Inflation has become a well accepted, but not yet proven, mechanism which attempts to explain many cosmological issues <cit.>. A finite period of accelerated expansion (inflation) in the early universe is desirable to help address the isotropy, spatial homogeneity, horizon, and flatness problems <cit.>. The standard inflationary model consists of a single massive scalar field that causes the universe to experience a period of exponential expansion early in its evolution.A common inflationary scenario assumes a convex potential, such as a harmonic scalar field potential, in which inflation takes place during a period of “slow roll”, when the scalar field is decreasing very slowly in comparison to the expansion of the universe.Not only is a period of accelerated expansion desirable at early times, but due to the Dark Energy phenomenon<cit.>, a period of accelerated expansion is also an attractive property to have at late times.In this paper we shall investigate the dynamical evolution of a class of isotropic and anisotropic spatially homogeneous Einstein-Aether cosmological models containing a scalar field that is coupled to the aether field through the scalar field potential.In particular, we explore the potential impact of Lorentz violation in the matter sector on the standard inflationary scenario <cit.>. More precisely, we study the inflationary scenario and investigate whether the inflationary solutions proposed <cit.> are stable when spatial curvature and anisotropy perturbations are considered. Further, we are also interested in the possibility of late time accelerated expansion in these models.§.§ Einstein-Aether Gravity The action under consideration contains a Lagrangian describing Einstein-Aether gravity together with a Lagrangian for a matter field or fields (M)S=∫ d^4x√(-g)[1/8π Gℒ^AE+ℒ^M] .The lagrangian ℒ^AE depends on the spacetime metric, g_ab, and the normalized aether vector field, u^a, and has the form<cit.>:ℒ^AE=1/2R - K^ab_ab cd∇_au^c∇_bu^d+ λ(u^au_a+1)whereK^ab_abcd≡c_1 g^ab g_cd + c_2δ_c^aδ_d^b+c_3δ_d^aδ_c^b+ c_4 u^a u^b g_cd.We note that the parameters c_i defined here are the same as those used in <cit.> which are equal to half of the values of the c_i employed in <cit.> with an opposite sign for the c_4. In comparison to <cit.>, the c_i used here are 8π G times the values of c_i. The metric signature is assumed to be +2.Letℒ^U= - K^ab_ab cd∇_au^c∇_bu^dthen the variation of the action (<ref>) with respect to the aether vector field u^a yields-2λ u_a =δℒ^U/δ u^a+8π G δℒ^M/δ u^a,which when contracted with u^a, provides an explicit expression for the Lagrange multiplier2λ=δℒ^U/δ u^au^a+8π G δℒ^M/δ u^au^a.Equation (<ref>) can then be used to eliminate the contribution of the Lagrange multiplier λ when calculating the effective energy momentum tensors.Variation of the action (<ref>) with respect to g^ab, λ and a generalized matter field or fields Ψ, yieldsG_ab =T_ab^U + 8π GT_ab^M,u^a u_a=-1,δℒ^M/δΨ =0.When the contributions from the Lagrange multiplier in equation (<ref>) are taken into account, expressions for the effective energy momentum tensors due to the aether vector field and the matter field becomeT_ab^U =-2δℒ^U/δ g^ab + g_abℒ^U+δℒ^U/δ u^cu^cu_au_b, T_ab^M=-2δℒ^M/δ g^ab + g_abℒ^M+δℒ^M/δ u^cu^cu_au_b. Given the form of the Lagrangian in equation (<ref>), the effective energy momentum tensor due to the Aether field <cit.> isT_ab^U =2∇_c(J_(a^acu^c_b) - J^c_c (au^c_b) -J_(ab)u^c )2c_1((∇_a u^c)(∇_b u_c) - (∇^c u_a)(∇_c u_b)) -2c_4u̇_au̇_b -2( u^d∇_c J^c_cd+c_4u̇_cu̇^c )u_au_b - g_ab(K^cd_cdef∇_c u^e ∇_d u^f),whereJ^a_ab =-K^ac_acbd∇_c u^d, u̇^a=u^b∇_b u^a. §.§ Matter as a Scalar Field Assuming that the matter component of the universe is a single scalar field having a potential that is assumed to be a function of the scalar field together with the expansion and shear scalars of the aether vector field, the matter Lagrangian becomes:L^M= -1/2 g^a b ∇_aϕ∇_bϕ - V(ϕ,θ,σ^2),where θ=∇_a u^a is the expansion scalar and σ^2=1/2σ_abσ^ab is the shear scalar.Again taking into account contributions from the Lagrange multiplier, equation (<ref>) yields the effective energy momentum tensor due to the scalar fieldT_ab^M = ∇_a ϕ∇_b ϕ - (1/2∇_aϕ∇^a ϕ + V )g_ab + θ V_θ g_ab+V̇_θ h_ab +(θ V_σ^2+V̇_σ^2)σ_ab+V_σ^2σ̇_ab-2σ^2V_σ^2u_au_bwhere the terms V_θ and V_σ^2 are the partial derivatives of the scalar field potential with respect to θ and σ^2, respectively, and h_ab≡ g_ab+u_au_b.If there is no coupling between the aether field and the scalar field via the potential, then V_θ=V_σ^2=0 and the energy momentum tensor reduces to the standard form for a minimally coupled scalar field. In addition, the field equation (<ref>) yields the Klein-Gordon equation for the scalar field:∇^a∇_aϕ-V_ϕ=0. § ISOTROPIC EINSTEIN-AETHER MODELS COUPLED TO A SCALAR FIELD §.§ The Spatially Homogeneous and Isotropic Model We shall assume that the spacetime is spatially homogeneous and isotropic having spacetime coordinates [t,r,θ,ψ] and a metric of the form:ds^2=-dt^2+a(t)^2(1/1-kr^2dr^2+r^2dθ^2+r^2sin^2(θ)dψ^2),where k takes on values {-1,0,1} for negative, zero and positive spatial curvature, respectively.In a spatially homogeneous and isotropic cosmological model with comoving time, the aether vector field necessarily coincides with the rest frame defined by the Hubble expansion.Specifically, this implies that in spatially homogeneous and isotropic models that the aether vector must be orthogonal to the three-dimensional spatial hyper-surfaces and takes the form u^a=(1,0,0,0).With the above assumptions on the metric and the aether vector, the shear, the vorticity and the acceleration of the aether vector are zero and the covariant derivative∇_b u_a=1/3θ (g_ab+u_au_b)is simply determined by the expansion scalarθ = ∇_au^a = 3ȧ/a. With the definition of T_ab^U in equation (<ref>), the effective energy density ρ^U and isotropic pressure, p^U, due to the aether field areρ^U= -1/3c_θθ^2 ,p^U=1/3c_θθ^2 +2/3c_θθ̇.Where a new parameter c_θ= (c_1 +3c_2 +c_3), defined before in <cit.>, allows for some efficiencies in notation since the field equations are independent of any other linear combinations of the c_i. The Einstein-aether field equations reduce to the following:0 =-1/3(1+c_θ)θ^2+8π Gρ^M-3k/a^2, 0=-(1+c_θ)θ̇-1/3(1+c_θ) θ^2-8π G/2(ρ^M+3p^M ),where there still exists the freedom to choose some appropriate units. Without loss of generality, new units can be chosen so that 8π G/1+c_θ=1 in which case the explicit dependence of the field equations on the aether parameter c_θ has been eliminated. §.§ The Scalar Field PotentialWe shall consider a class of quadratic scalar fieldpotentialsof the formV(ϕ, θ)=1/2 m^2 ϕ^2 + μθϕ,where the scalar field/Aether field coupling term μθϕ term can be interpreted as an incorporation of an external source, in this case the Aether, acting on the scalar field.The effective energy density ρ^M and isotropic pressure p^M from equation (<ref>), areρ^M=1/2ϕ̇^2+ 1/2 m^2 ϕ^2,p^M=1/2ϕ̇^2- 1/2 m^2 ϕ^2+μϕ̇.The Klein-Gordon equation becomes0=ϕ̈+θϕ̇+m^2ϕ+μθ,where we can more clearly see how μθ acts like an external source in the Klein-Gordon equation when compared to the usual non-coupled μ=0 version of the equation. §.§ The Dynamical System The Einstein-aether field equations and the Klein-Gordon equation can be expressed as the following system of ordinary differential equationsθ̇ =-1/3θ^2 +m^2/2ϕ^2-ψ^2-3/2μψ,ϕ̇ = ψ,ψ̇ =-θψ-m^2ϕ-μθ.with first integralθ^2/3 = m^2/2ϕ^2+1/2ψ^2-3k/a^2. Equations (<ref>)-(<ref>), therefore, yield a three dimensional dynamical system for the variables (θ, ϕ,ψ)depending on three parameters (k, m, μ) having a first integral given by equation (<ref>). Since the system of equations is invariant under the transformation (μ,ϕ,ψ) ↦ -(μ,ϕ,ψ), we can without loss of generality, assume that μ≥ 0.Given that the phase space for the dynamical system defined in equations (<ref>)-(<ref>) with first integral (<ref>) is notbounded, we employ dimensionless variables <cit.> which will transform the system into an autonomous system of differential equations on a bounded phase space.§.§ Qualitative Analysis §.§.§ Introducing Normalized Variables Introducing a time variableτdτ/dt=√(1+θ^2)and normalized variablesD≡ θ/√(1+θ^2),Φ ≡ √(3/2)(m ϕ/√(1+θ^2)), Ψ ≡ √(3/2)(ϕ̇/√(1+θ^2)),the evolution equations (<ref>)-(<ref>) becomeD^' =(1-D^2)𝒳, Φ^' =m Ψ√(1-D^2)-D Φ𝒳, Ψ^' =-D Ψ- √(1-D^2)( m Φ+√(3/2)μ D ) -Ψ D 𝒳,where the prime here indicates the differentiation with respect to τ and 𝒳 is given by the expression𝒳 = θ̇/θ^2+1=-1/3D^2-2/3Ψ^2+1/3Φ^2 -√(3/2)μΨ√(1-D^2) The Friedmann equation (<ref>) becomesD^2-Φ^2-Ψ^2= -9 k/ a^2(1+θ^2).Further, if k = -1,0, then it follows that0 ≤Φ^2+Ψ^2≤ D^2≤ 1.That is, D, Φ, Ψ are bounded in the flat and negatively curved scenarios and the phase space is a compact set.Hence forward, we shall restrict our analysis to k=0, -1 cases only.§.§.§ Invariant Sets and Monotonic Functions The phase space can be subdivided into four disjoint invariant sets according to the curvature of the model and whether D=1 (θ→∞) or not.A superscript “-” indicates that points in this set represent negatively curved models, while a superscript “0” indicates a flat model. The invariant sets are^-= {(D,Φ,Ψ)|D<1,Φ^2+Ψ^2< D^2}, ^0= {(D,Φ,Ψ)|D<1,Φ^2+Ψ^2= D^2}, ^-= {(D,Φ,Ψ)|D=1,Φ^2+Ψ^2< 1}, ^0= {(D,Φ,Ψ)|D=1,Φ^2+Ψ^2= 1},the dimensions of which are 3, 2, 2, and 1, respectively.We note the following closure properties of the sets^- = ^-∪^0∪^-∪^0, ^- = ^-∪^0, ^0 = ^0∪^0.Further the invariant set ^- can be divided into three distinct pieces depending on whether Φ<0, Φ=0 or Φ>0.If we define Λ_1 = D^2- Φ^2-Ψ^2 and Λ_2=D^2-1 thenΛ_1^'/Λ_1 =-2/3D( 3𝒳+1 ), Λ_2^'/Λ_2 =2D𝒳.in which case the non-negative function W=(Λ_1)^2(Λ_2)^2 has the derivativeW^' = -4/3WD.Since W> 0 and W^'<0in the set ^- we can conclude that there are no periodic orbits in this 3-dimensional invariant set. This implies that there are no equilibrium points in the set ^-, and any equilibrium points of the autonomous system of differential equations (<ref>)-(<ref>) will lie in the lower dimensional invariant sets ^0, ^-or ^0.We also note that in the invariant set ^-∩{Φ<0}, one can show that Φ^' <0, and similarly in the set ^-∩{Φ>0}, one can show that Φ^' >0.This shows that there are no closed or periodic orbits in these sets.The remaining portion ^-∩{Φ=0}, is 1 dimensional. No monotonic function has been found in the set ^0 and consequently the most interesting qualitative behaviour for this autonomous system of differential equations occurs in ^0.§.§.§ Equilibrium PointsThe equilibrium points and a non-isolated line of equilibria for the system (<ref>)-(<ref>) with the value of 𝒳and their stability are summarized in Table (<ref>). §.§.§ Stability of Equilibrium point P_0 Evaluating the linearization matrix of the system (<ref>)-(<ref>) at P_0 gives us the following eigenvaluesλ_1 =0,λ_2,3 = ±√(6)/2√(μ^2-μ_c^2).Note that, if μ > μ_c then P_0 is a saddle. But, ifμ < μ_c thenall the eigenvalueshavezero real part which implies that the local qualitative behaviour at P_0 is not determined by its linearization.However a perturbative solution near P_0can be found, and fortunately an analysis of the first order solution is sufficient to determine the local stability of P_0 when μ < μ_c. We first introduce new scaled variables (d,ϕ,ψ) such thatD=ϵ(d-μ/μ_cϕ),Φ=ϵϕ,Ψ=ϵψ,where ϵ is assumed to be small, to determine a leading order approximation to the solution of the equations near P_0.We note that the ϕ and ψ variables that are employed in this subsection are not the original variables used to describe the scalar field and its derivative.Using our new dependent variables (<ref>), and expanding (<ref>)-(<ref>) as a power series in ϵ we derive the followingd^' = ϵ/3(-d^2-2ψ^2+ ϕ^2+2μ/μ_cdϕ-μ^2/μ_c^2ϕ^2)+O(ϵ^2),ϕ^' = √(6)/2μ_cψ +O(ϵ^2),ψ^' = √(6)/2(- μ d -μ_cϕ +μ^2/μ_cϕ)+ ϵ(-d ψ+μ/μ_cϕψ)+O(ϵ^2),where we kept only terms up to linear order in ϵ.To proceed with the construction of a perturbative solution, we employ the method of multiple scales <cit.>. In the method of multiple scales with two time scales, the original fast time τ and a second slow time η=ϵτ, each dependent variable is expressed asx ≡x(τ, η)=x_0 (τ, η)+ϵx_1 (τ, η) + O(ϵ^2)and using the chain rule, derivatives become expanded asx^'=x_0τ+ ϵ (x_0η+ x_1τ )+ O(ϵ^2).The (') indicates the ordinary derivative of the variable with respect to time τ while the subscripts τ and η denote partial derivatives. A valid perturbative solution is obtained by ensuring that the solution remains bounded at all orders of ϵ.Using equation (<ref>) and (<ref>) for variables (d,ϕ,ψ) and substituting into(<ref>) and matching powers of ϵ yields the following system of partial differential equations for the zeroth order [O(ϵ^0)] termsd_0τ = 0, ϕ_0τ = √(6)/2μ_cψ_0, ψ_0τ = √(6)/2(-μ d_0 - μ_cϕ_0+μ^2/μ_cϕ_0),and the following system of partial differential equations for the first order [O(ϵ^1)] termsd_1τ = 1/3(-d_0^2-2ψ_0^2 + ϕ_0^2 +2μ/μ_cd_0ϕ_0-μ^2/μ_c^2ϕ_0^2)-d_0 η, ϕ_1τ = √(6)/2μ_cψ_1-ϕ_0 η,ψ_1τ = √(6)/2(-μ d_1-μ_cϕ_1 +μ^2/μ_cϕ_1) +(-d_0ψ_0+μ/μ_cϕ_0ψ_0)-ψ_0 η.Solving the partial differential equations for the Zeroth order terms yieldsd_0(τ,η)=B(η),ϕ_0(τ,η)=A(η)cos(λτ-Λ(η))-μμ_c/μ_c^2-μ^2B(η), ψ_0(τ,η)=-√(6)λ/3μ_cA(η)sin(λτ-Λ(η)),where λ=√(6)/2√(μ_c^2-μ^2) and A(η), B(η) and Λ(η) are as yet undetermined functions of the slow time η.Solving the partial differential equations for the first order terms, and restricting ourselves to only bounded solutions, determines a set of ordinary differential equations for the unknown functions A(η), B(η) and Λ(η),A_η = -1/2μ_c^2/μ_c^2-μ^2AB,B_η = -1/3(μ_c^2/μ_c^2-μ^2B^2 + μ_c^2-μ^2/2μ_c^2A^2), Λ_η =0. Therefore, in terms of the original variables the first term of the perturbative solution isD(τ)= ϵ(- μ/μ_cA(η)cos(λτ-Λ(η))+μ_c^2/μ_c^2-μ^2B(η) ),Φ(τ)= ϵ(A(η)cos(λτ-Λ(η))-μμ_c/μ_c^2-μ^2B(η)), Ψ(τ)= ϵ(-√(μ_c^2-μ^2)/μ_cA(η)sin(λτ-Λ(η))),where the functions A(η), B(η) and Λ(η) satisfy the differential equations (<ref>), and due to (<ref>) are bounded byB(η)^2≥(μ_c^2-μ^2)^2/μ_c^4A(η)^2,where we note that if B(η)→ 0 then we also have A(η)→ 0.We are interested in determining the asymptotic behaviour as τ→∞.We observe that the phase shift Λ(η) is a constant and has no effect on the future dynamics.The fast time τ essentially describes the oscillations of the scalar field, which to first order in ϵ has a period of T=2π/λ.We note that the period of these oscillations T ∼ 1/√(μ_c^2-μ^2), gets longer as the strength of the coupling parameter μ is increased towards μ_c.We also observe that the amplitude of the oscillations A(η), and the vertical shift B(η) are functions of the slow time η and consequently the amplitude and vertical shift drift slowly in comparison to the oscillatory changes. For initial values of B(η)>0 we see that both A(η),B(η)→ 0 as η→∞.That is, the amplitude of the oscillations and the vertical shift both slowly decrease to zero, indicating that the point P_0 is stable when μ<μ_c. §.§.§ Stability of Equilibrium points in ^-∪^0 Unfortunately, while we have an autonomous system of differential equations defined on a compact set, the system is not differentiable at any points in the invariant set D^-∪ D^0. We note that any equilibrium points in the invariant set D^-∪ D^0represent asymptotic states in which θ→∞.In order to determine the local qualitative behaviour at these equilibrium points, we replace variable D withT=1/√(1+θ^2)=√(1-D^2).The evolution equations (<ref>)-(<ref>) becomeT^' =-T√(1-T^2)𝒳,Ψ^' =-√(1-T^2)Ψ- T ( m Φ+√(3/2)μ√(1-T^2)) -Ψ√(1-T^2)𝒳, Φ^' =m Ψ T-√(1-T^2)Φ𝒳,with𝒳 = -1/3(1-T^2)-2/3Ψ^2+1/3Φ^2 -√(3/2)μΨ T.The value D=1 for the equilibrium points P_1,2,3,4,5 is simply replaced with T=0.With this transformation we are able to locally determine the qualitative behaviour of each the equilibrium points in the invariant set D^-∪ D^0.The eigenvalues of the points P_1 and P_2 are 1,4/3, 1 which implies these points are generally sources.The eigenvalues of the point P_5 is 1/3,1/3, -2/3 which implies this point is generally a saddle.Further, the eigen-directions that span the T=0 invariant set, are associated with one positive and one negative eigenvalue.Therefore this equilibrium point is a saddle in the T=0 set.The eigenvalues of the point P_3 and P_4 are 0,-1, -2/3 which implies that we cannot determine the general behaviour of this point without resorting to additional analysis.However, the eigen-directions that span the T=0 invariant set, are associated with the two negative eigenvalues.Therefore these equilibrium points are sinks in the T=0 set.To complete the analysis of the qualitative behaviour near P_3 and P_4 we calculate the center manifold <cit.>.In this case the center manifold is a one dimensional curve that must lie in the ^0 invariant set.The center manifold for P_3 can be parameterized asT=TΦ =1 - (1/2 + 3/4(μ+μ_c)^2)T^2 + O(T^4) Ψ =-√(6)/2(μ+μ_c)T+3/8√(6)μ_c(μ+μ_c)^2T^3+O(T^4)The leading order term of the dynamical system restricted to the center manifold reduces toT'= 3/2μ_c(μ+μ_c)T^3.Since T'>0 for T>0, P_3 is unstable along its center manifold. It is therefore a saddle in the full three dimensional phase space. The center manifold for P_4 can be parameterized asT=TΦ =-1 + (1/2 + 3/4(μ-μ_c)^2)T^2 + O(T^4) Ψ =-√(6)/2(μ-μ_c)T-3/8√(6)μ_c(μ-μ_c)^2T^3+O(T^4)The leading order term of the dynamical system restricted to the center manifold reduces toT'= -3/2μ_c(μ-μ_c)T^3.If μ<μ_c then T'>0 for T>0 and P_4 is unstable along its center manifold.However, if μ>μ_c then T'<0 for T>0 and P_4 is stable along its center manifold. Therefore P_4 is a saddle when μ<μ_c and a sink when μ>μ_c in the full three dimensional phase space.§.§.§ The Bifurcation Value When μ=μ_c, there is a line of equilibria given by (D,Φ,Ψ)=(s, -s,0) where 0<s<1 having endpoints P_0 and P_4.The eigenvalues of points on this line of equilibria are 0, -s, -2/3s which implies that points on this line are attractors when it exists. We observe that as the parameter μ increases towards its bifurcation value, μ_c, the stability of the point P_0 is transferred to the point P_4 via this line of equilibria.§.§.§ Heteroclinic Sequences Very often one is not only interested in the past and future behaviour of a system of differential equations, but one is also interested in the intermediate behaviour of the system.One technique to analyze the intermediate behaviour is to describe the heteroclinic sequences that are possible <cit.>.We note that for every heteroclinic sequence there exists a set of orbits that are arbitrarily close to that sequence.Figures (<ref>) describes the possible heteroclinic sequences that are possible. Again we see how the intermediate and future behaviour changes as the parameter μ changes.§.§ Inflation and Accelerated ExpansionAs an indicator of the accelerated expansion we introduce the deceleration parameterq ≡ - a ä/ȧ^2=- (3θ̇/θ^2+1).It follows that the deceleration parameter can also be expressed in terms of the normalized bounded variables as follows;q= -1/D^2(-2 Ψ^2 +Φ^2-3√(3/2)μ√(1-D^2)Ψ).The sign of the deceleration parameter indicates the nature of the expansionary evolution. If q > 0, then the cosmological expansion is decelerating, while negative values of q indicate an accelerating or inflationary dynamics. See Table (<ref>) for a summary of the values of q for each equilibrium point.§.§ Numerical Analysis It is constructive to illustrate a few numerical solutions for the three different regimes of future asymptotic behaviour, μ<μ_c [see Figure (<ref>)], μ=μ_c [see Figure (<ref>)], and μ>μ_c [see Figure (<ref>)]. In each case the integrations are done in the full 3-dimensional phase space. A few initial conditions are selected to show different past and future asymptotic behaviours. §.§ Observations In the spatially homogeneous and isotropic case analyzed here we observe that the past dynamics are independent of the strength of the coupling parameter μ. We find in the zero curvature and the negative curvature models that there are two possible asymptotic behaviours to the past, one in which (D,Φ,Ψ)→ (1,0,1), P_1, and one in which (D,Φ,Ψ)→ (1,0,-1), P_2.These past attractors represent a massless scalar field FRW model <cit.>. We also observe that the future asymptotic state depends on the strength of the coupling parameter μ.For weak coupling of the Aether field to the scalar field, i.e, , μ<μ_c=√(2/3)m, the dynamics are similar to that found when there is no coupling of the Aether field to the scalar field, i.e., when μ=0.If μ<μ_c then P_0 is the stable attractor: orbits oscillate and slowly decay in amplitude towards this final non-inflationary asymptotic state.For strong coupling of the Aether field to the scalar field, μ>μ_c, the dynamics are very different.When μ>μ_c the scalar field does not come to rest at the minimum of the potential: the strength of the Aether interaction forces a different final equilibrium state.If μ>μ_c then in both the zero curvature and the negative curvature models we find that the stable attractor in these models changes from P_0 to P_4.In this scenario, we find that the square of the scalar field and the square of the expansion scalar scale together asm^2/2ϕ^2 ∼1/3(1+θ^2)and consequently grow without bound.We also observe that all orbits (excepting for the exceptional orbits) experience some period of accelerating expansion as they evolve to their final asymptotic state, P_4.In the next section we will add anisotropy to these models to determine if the addition of anisotropy changes these observations.§ ANISOTROPIC EINSTEIN-AETHER MODELS COUPLED TO A SCALAR FIELD §.§ A Class of Diagonal Bianchi Models In order to investigate Einstein-Aether cosmological models with a scalar field having a potential with interaction terms that contain both the expansion and shear of the aether vector requires a broader class of space-time geometries which include anisotropy.For our purposes, the one parameter family of spatially homogeneous and anisotropic diagonal Bianchi type VI_h space-times provides an arena to determine the effect of adding a shear interaction term into the scalar field potential in the Einstein-Aether models studied in the previous section. The metric is assumed to have the formds^2=-dt^2+a(t)^2dx^2+b(t)^2e^2(h-1)xdy^2+c(t)^2e^2xdz^2.There are three special classes are worth mentioning: if the parameter h=2 then the metric is Bianchi type V which has the negatively curved isotropic models (see previous section) as a subcase, if h=1 then it is Bianchi type III, and if h=0 then it is Bianchi type VI_0.In a spatially homogeneous and isotropic cosmological model with comoving time, the aether vector field necessarily coincides with the rest frame defined by the Hubble expansion. Therefore in deviations from spatially homogeneity and isotropy, one could assume that the preferred frame for the aether approximately coincides with the cosmological rest frame defined by the Hubble expansion. However, if one relaxes only the isotropy assumption, then the aether vector would be tilted away from the hyper-surface normal of the spatially homogeneous hyper-surfaces which could allow for a richer set of dynamical behaviours. However, it has been argued in <cit.> that during cosmological evolution, the two frames will come into alignment. Given these arguments, we will assume that the aether vector is aligned with the hyper-surface unit normal to the surfaces of homogeneity and is of the form u^a=(1,0,0,0).With the above assumptions on the metric and the aether vector, the vorticity and the acceleration of the aether vector are zero, and the covariant derivative∇_b u_a=σ_ab+1/3θ(g_ab+u_a u_b),is simply determined by the expansion scalarθ = ∇_au^a = ȧ/a+ḃ/b+ċ/c,and the shear tensorσ_ab=u_(a;b)-1/3θ (g_ab+u_a u_b).The shear tensor has the form σ^a_ab=[0,σ_1,σ_2,-(σ_1+σ_2)] whereσ_1 = 1/3(2ȧ/a-ḃ/b-ċ/c), σ_2 = 1/3(-ȧ/a+2ḃ/b-ċ/c).We note that the shear scalar isσ^2=1/2σ_abσ^ab=σ_1^2+σ_2^2+σ_1σ_2. With the definition of T_ab^U in equation (<ref>), the effective energy density ρ^U, isotropic pressure, p^U, energy flux q_a^aU, and anisotropic stress π_ab^abU due to the aether field areρ^U= -1/3c_θθ^2 -2c_σσ^2,p^U=1/3c_θθ^2 +2/3c_θθ̇-2c_σσ^2,q_a^aU = 0, π_ab^abU=2c_σ (σ̇^a_ab+θσ^a_ab),Where new parameters c_θ= (c_1 +3c_2 +c_3) and c_σ=c_1+c_3, defined before in <cit.>, allow for some efficiencies in notation since the field equations are independent of any other linear combinations of the c_i.The Einstein-aether field equations reduce to the following set of linearly independent equations0 =-1/3(1+c_θ)θ^2+(1-2c_σ)σ^2+h^2-h+1/a^2+8π Gρ^M, 0=-(1+c_θ)θ̇-1/3(1+c_θ) θ^2 -2(1-2c_σ)σ^2 -8π G/2(ρ^M+3p^M ),0 =-(h+1)σ_1+(h-2)σ_2-8π G/a^2 q_2^aM0=(1-2c_σ)σ̇_1+(1-2c_σ)θσ_1 +(h-2)^2/3a^2 -8π Gπ^M_1, 0=(1-2c_σ)σ̇_2+(1-2c_σ)θσ_2 +(h-2)(h+1)/3a^2 -8π Gπ^M_2,where there still exists the freedom to choose some appropriate units.Without loss of generality, one can choose new units so that 8π G/1+c_θ=1, and a new parameter C=1-2c_σ/1+c_θ, in which case the explicit dependence of the field equations on the aether parameter c_θ has again been eliminated.Assuming that GR and Einstein Aether theory have equivalent PPN parameters and that there is stable positive energy modes but no vacuum Čerenkov radiation imposes some constraints on the values of c_θ and C.See Appendix <ref> for details. §.§ The Scalar Field PotentialWe shall consider a class of quadratic scalar field potentialsof the formV(ϕ, θ)=1/2 m^2 ϕ^2 + μθϕ+νσϕ.In this case, from equation (<ref>), the effective energy density ρ^M, isotropic pressure p^M, energy flux q_a^aM, and anisotropic stress π_ab^abM due to the scalar field areρ^M=1/2ϕ̇^2+ 1/2 m^2 ϕ^2,p^M=1/2ϕ̇^2- 1/2 m^2 ϕ^2+μϕ̇-νϕσ,q_a^aM= 0π_ab^abM=(νϕ/2σ) (σ̇^a_ab+θσ^a_ab)+d/dt(νϕ/2σ)σ^a_ab.where we define π^M_i such that π_ai^ibM=[0,π^M_1,π^M_2,-(π^M_1+π^M_2)]. The final field equation comes from the Klein-Gordon equation which becomes0=ϕ̈+θϕ̇+m^2ϕ+μθ+νσ. §.§ The Dynamical System Given that there is no energy flux either from the existence of the aether field or from the non-minimal coupling of the aether to the scalar field potential, we are able to use equation (<ref>) to rewrite all equations in terms of the anisotropy scalar σ:σ_1 = h-2/√(3)√(h^2-h+1)σ, σ_2 = h+1/√(3)√(h^2-h+1)σ.Hence, the final form of the Einstein-aether field equations and the Klein-Gordon equation yield the following dynamical systemθ̇ =-1/3θ^2 -2Cσ^2-ψ^2+m^2/2ϕ^2-3/2μψ+3/2νϕσ,σ̇ = -σθ+ν/2C(ψ+θϕ) +(h-2)/√(3)√(h^2-h+1)1/C(1/3θ^2-Cσ^2-1/2ψ^2-m^2/2ϕ^2),ϕ̇ = ψψ̇ =-θψ-m^2ϕ-μθ-νσ.with first integralθ^2/3 = m^2/2ϕ^2+1/2ψ^2+Cσ^2+h^2-h+1/a^2, Equations (<ref>)-(<ref>), therefore, yield a four dimensional dynamical system for the variables (θ,σ, ϕ,ψ)depending on five parameters (h, C, m, μ, ν) having a first integral given by equation (<ref>). We restrict our analysis to the diagonal Bianchi type V models (h=2).We note that even in this very simple anisotropic case when we compare the evolution equation for the shear with would would happen if the scalar field potential did not have an interaction term, we notice that the negatively curved FRW models are no longer an invariant subset of the Bianchi type V system (<ref>)-(<ref>).Since the system of equations when h=2 is invariant under the transformations (μ,σ,ϕ,ψ) ↦ -(μ,σ,ϕ,ψ), and (ν,σ) ↦ -(ν,σ), we can without loss of generality, assume that both μ≥ 0 and ν≥0.Given that the phase space for the dynamical system defined in equations (<ref>)-(<ref>) with first integral (<ref>) is notbounded, we employ dimensionless variables <cit.> which will transform the system into an autonomous system of differential equations on a bounded phase space. §.§ Qualitative Analysis §.§.§ Introducing Normalized Variables Introducing a time variable τd τ/dt≡√(1+θ^2),andnormalized variablesD≡ θ/√(1+θ^2) Σ ≡ √(3)σ/√(1+θ^2), Φ ≡ √(3/2)(m ϕ/√(1+θ^2)), Ψ ≡ √(3/2)(ϕ̇/√(1+θ^2)).then, the Bianchi type V evolution equations in equations (<ref>)-(<ref>) when h=2becomeD^' =(1-D^2)𝒳, Φ^' =m √(1-D^2)Ψ-D Φ𝒳 Ψ^' =-D Ψ(1+𝒳)- √(1-D^2)[m Φ+√(3/2)μ D+ ν/√(2)Σ], Σ^' =- Σ D(1+𝒳) + ν/√(2)C ( Ψ√(1-D^2) +D Φ/m),wherethe prime here indicates the differentiation with to respect to τ. 𝒳is given by theexpression𝒳 = θ̇/1+θ^2= -1/3D^2-2/3Ψ^2+1/3Φ^2 -2/3 C Σ^2 -√(3/2)μ√(1-D^2)Ψ + ν/√(2) mΣΦ.The Friedmann equation (<ref>) when h=2 becomesD^2-Φ^2-C Σ^2 -Ψ^2=9/a^2 ( 1+ θ^2).where if C≥0, then it follows that0 ≤Φ^2+Ψ^2 + C Σ^2≤ D^2≤ 1.That is, D, Φ, Ψ, Σ are bounded if C>0 and the phase space is a bounded set.Hence forward, we shall restrict our analysis to C≥0 case only.§.§.§ Invariant Sets and Monotonic FunctionsThe phase space can be subdivided into four disjoint invariant sets according to the curvature of the model and whether D=1 (θ→∞) or not.A superscript “-” indicates that points in this set represent negatively curved models, while a superscript “0” indicates a flat model. The invariant sets are^-= {(D,Φ,Ψ,Σ)|D<1, Φ^2+Ψ^2+CΣ^2< D^2}, ^0= {(D,Φ,Ψ,Σ)|D<1, Φ^2+Ψ^2+CΣ^2= D^2}, ^- = {(D,Φ,Ψ,Σ)|D=1, Φ^2+Ψ^2+CΣ^2< 1}, ^0 = {(D,Φ,Ψ,Σ)|D=1, Φ^2+Ψ^2+CΣ^2= 1}.The dimensions of which are 4, 3, 3, and 2, respectively.The invariant set ^- represents the Bianchi type V models while the invariant set ^0 represents the Bianchi type I models.Again, the shear interaction term in the scalar field potential plays a significant role, since Σ=0 will be an invariant set only if ν=0.Similar to the isotropic case, if we define Λ_3 = D^2- Φ^2-Ψ^2-CΣ^2 and Λ_4=D^2-1 thenΛ_3^'/Λ_3 =-2/3D( 3𝒳+1 ), Λ_4^'/Λ_4 =2D𝒳.in which case the non-negative function W=(Λ_3)^2(Λ_4)^2 has the derivativeW^' = -4/3WD.Since W> 0 and W^'<0in the set ^- we can conclude that there are no periodic orbits in this 4-dimensional invariant set. This also implies that there are no equilibrium points in the set ^-, and any equilibrium points of the autonomous system of differential equations (<ref>)-(<ref>) will lie in the lower dimensional invariant sets ^0, ^-or ^0.§.§.§ Equilibrium PointsThe equilibrium points forthe system (<ref>)-(<ref>) with the value of 𝒳and their stability are summarized in Table (<ref>). Comparing the equilibrium points in Table (<ref>) with those found in Table (<ref>) we see that the points P_0 and P_5 represent the same equilibrium states in both tables. The points P_1 and P_2 represents the same equilibrium state in both tables, however, in Table (<ref>) they are actually two special (isotropic) points on a non-isolated circle of generally non-isotropic equilibria given by C^*.The points P_3 and P_4 in Table (<ref>) reduce to P_3 and P_4 in Table (<ref>) when ν=0.Further, the dynamical behaviour and stability is analogous to the stability of P_3 and P_4 in Table (<ref>). Similar to the isotropic case, there is a line of equilibria L_04 when μ=μ_cν that connects P_0 and P_4.We note that the local stability of the equilibrium points depends on the bifurcation value μ_cν^2=μ_c^2 +ν^2/3C. Recall μ_c=√(2/3)m is the bifurcation value found in the isotropic case studied earlier, and so since C and ν are both positive, the bifurcation value for the anisotropic case, is always a bit larger than in the isotropic case. Further, by choosing smaller and smaller values of C, one can increase the value of bifurcation value μ_cν.§.§.§ Stability of Equilibrium Point P_0 Evaluating the linearization matrix of the system (<ref>)-(<ref>) at P_0 gives us the following eigenvaluesλ_1,2 =0,λ_3,4 = ±√(6)/2√(μ^2-μ_c ν ^2).Note that, if μ >μ_c ν then P_0 is a saddle. But, ifμ < μ_c νthenall the eigenvalueshavezero real part which implies that the local qualitative behaviour at P_0 is not determined by its linearization.However a perturbative solution near P_0can be found, and fortunately an analysis of the first order solution is sufficient to determine the local stability of P_0 when μ< μ_c ν . We first introduce new scaled variables (d,ϕ,ψ,σ) such thatD=ϵ(d-μ/μ_cϕ),Φ=ϵϕ,Ψ=ϵψ, Σ= ϵ(σ +ν/√(3) C μ_cϕ)where ϵ is assumed to be small, to determine a leading order approximation to the solution of the equations near P_0.We note that the ϕ,ψ and σ variables that are employed in this subsection are not the original variables used to describe the scalar field and its derivative.Using our new dependent variables (<ref>), and expanding (<ref>)-(<ref>) as a power series in ϵ we derive the followingd^' = ϵ/3(-d^2-2ψ^2+ ϕ^2+2μ/μ_cdϕ-μ^2/μ_c^2ϕ^2 +ν^2 / 3 C μ_c^2ϕ^2 -2 C σ^2 -√(3)ν/ 3 μ_cϕσ)+O(ϵ^2),ϕ^' = √(6)/2μ_cψ +O(ϵ^2),ψ^' = √(6)/2(- μ d -μ_cϕ +μ^2/μ_cϕ - ν/√(3)σ -ν^2/ 3 C μ_cϕ)+ ϵ(-d ψ+μ/μ_cϕψ)+O(ϵ^2), σ^'= ϵ( -σ d +μ/μ_cσϕ).where we kept only terms up to linear order in ϵ.To proceed with the construction of a perturbative solution, we employ the method of multiple scales <cit.>.Using equation (<ref>) and (<ref>) for variables (d,ϕ,ψ,σ) and substituting into(<ref>) and matching powers of ϵ yields the following system of partial differential equations for the zeroth order [O(ϵ^0)] termsd_0τ = 0, ϕ_0τ = √(6)/2μ_cψ_0, ψ_0τ = √(6)/2(- μ d_0 -μ_cϕ_0 +μ^2/μ_cϕ_0 - ν/√(3)σ_0 -ν^2/ 3 C μ_cϕ_0), σ_0τ = 0and the following system of partial differential equations for the first order [O(ϵ^1)] termsd_1τ = 1/3(-d_0^2-2ψ_0^2+ ϕ_0^2+2μ/μ_cd_0ϕ_0-μ^2/μ_c^2ϕ_0^2 +ν^2 / 3 C μ_c^2ϕ_0^2 -2 C σ_0^2 -√(3)ν/ 3 μ_cϕ_0σ_0)-d_0 η, ϕ_1τ = √(6)/2μ_cψ_1-ϕ_0 η,ψ_1τ = √(6)/2(- μ d_1 -μ_cϕ_1 +μ^2/μ_cϕ_1 - ν/√(3)σ_1 -ν^2/ 3 C μ_cϕ_1) +(-d_0ψ_0+μ/μ_cϕ_0ψ_0)-ψ_0 η،σ_1τ = (-d_0σ_0 +μ/μ_cσ_0ϕ_0) -σ_0 η.Solving the partial differential equations for the Zeroth order terms yieldsd_0(τ,η)=B(η),ϕ_0(τ,η)=A(η)cos(λτ-Λ(η))-μ_c/(μ_c ν^2 -μ^2)(μ B(η)+ν/√(3) S(η)), ψ_0(τ,η)=-√(6)λ/3μ_cA(η)sin(λτ-Λ(η)), σ_0(τ,η) =S(η).where λ=√(6)/2√(μ_c ν^2-μ^2) and A(η), B(η), S(η) and Λ(η) are as yet undetermined functions of the slow time η.Solving the partial differential equations for the first order terms, and restricting ourselves to only bounded solutions, determines a set of ordinary differential equations for the unknown functions A(η), B(η), S(η) and Λ(η).If we replace B(η) with the linear combinationB̅(η)=μ_cν^2B(η)+μν/√(3)S(η)then the resulting set of differential equations becomeA_η = -1/21/μ_c ν^2-μ^2 A B̅,B̅_η = -1/3(1/μ_cν^2-μ^2B̅^2 + μ_cν^2(μ_cν^2-μ^2)/2μ_c^2A^2 + 2μ_c C S^2 ),S_η = -1/μ_c ν^2-μ^2 S B̅,Λ_η =0. Therefore, in terms of the original variables the first term of the perturbative solution isD(τ)= ϵ[ B(η) - μ/μ_cA(η)cos(λτ-Λ(η))+μ/(μ_cν^2-μ^2)(μ B(η)+ν/√(3) S(η)) ], Φ(τ)= ϵ[ A(η)cos(λτ-Λ(η)) -μ_c/(μ_cν^2-μ^2)(μ B(η)+ν/√(3) S(η)) ],Ψ(τ)= ϵ[-√(μ_c ν^2-μ^2)/μ_cA(η)sin(λτ-Λ(η))],Σ( τ)= ϵ[ S(η)+ν/√(3) C μ_c A(η)cos(λτ-Λ(η)) -ν/√(3)C(μ_cν^2-μ^2)(μ B(η)+ν/√(3) S(η))],where the functions A(η), B(η), S(η), and Λ(η) satisfy the differential equations (<ref>), and due to (<ref>) are bounded byB̅(η)^2 ≥(μ_cν^2-μ^2/μ_c^2)A(η)^2+ μ_c^2/μ_cν^2CS(η)^2,where we note that if B̅(η)→ 0 then we also have A(η)→ 0 and S(η)→ 0 which then also implies that B(η)→ 0.We are interested in determining the asymptotic behaviour as τ→∞.We observe that the phase shift Λ(η) is a constant and has no effect on the future dynamics.The fast time τ essentially describes the oscillations of the scalar field, which to first order in ϵ has a period of T=2π/λ.We note that the period of these oscillations T ∼ 1/√(μ_c ν^2-μ^2), gets longer as the strength of the coupling parameter μ is increased.We also observe that the amplitude of the oscillations A(η), and the vertical shift B(η) and the shear term S(η) are functions of the slow time η and consequently the amplitude, vertical shift, and shear term drift slowly in comparison to the oscillatory changes. For initial values of B(η)>0 we see that A(η),B(η),S(η) → 0 as η→∞.That is, the amplitude of the oscillations, the vertical shift, and the shear term all slowly decrease to zero, indicating that the point P_0 is stable when μ< μ_c ν. §.§.§ Stability of Equilibrium Points in ^-∪^0Unfortunately, whilewe have an autonomous system of differential equations defined on a compact set, the system is not differentiableat any points in the invariant set ^-∪^0. In order to determine the local behaviour at thesepoints, we replace variable D withT=1/√(1+ θ^2)= √(1-D^2).The evolutionequations (<ref>)-(<ref>) becomeT^' =-T √(1-T^2)𝒳, Φ^' =m TΨ-√(1-T^2)Φ𝒳 ,Ψ^' =-√(1-T^2)Ψ(1+𝒳)- T (m Φ+√(3/2)μ√(1-T^2)+ ν/√(2)Σ),Σ^' =- Σ√(1-T^2)(1+𝒳) + ν/√(2)C( Ψ T +√(1-T^2)Φ/m),with𝒳=-1/3(1-T^2)-2/3Ψ^2+1/3Φ^2 -2/3C Σ^2 -√(3/2)μTΨ +√(3/2)ν/mΣΦ.The value D=1 for the equilibrium points P_1 P_2, P_3, P_4, P_5 and C^* is simply replaced with T=0. With this transformation we are able to locally determine the qualitative behaviour of each the equilibrium points.The eigenvalues of the linearization at the equilibrium points P_1, P_2, and C^* are 0, 1,1, 4/3 which implies these points are unstable and are sources. The zero eigenvalue indicates the non-isolated nature of the circle of equilibria C^*.The eigenvalues of the linearization at the equilibrium point P_5is 1/3, -2/3, 1/3, -2/3 which implies thatP_5 isgenerally saddle (unstable). Further, the eigen-directions that span the T=0invariant set, are associated with one positive and two negative eigenvalues. Therefore this equilibrium point is a saddle within theT=0 invariant set.The eigenvalues of the linearization at the equilibrium points P_3 and P_4 are 0, -1,-1, -2/3 which implies that we cannotdetermine the general behaviour of these points without resorting to additional analysis. However, the eigen-directions that span the T=0 invariant set, are associated with the three negative eigenvalues. Therefore, these equilibrium points are sinks in the T=0 set. One method to complete the determination of the general behaviour near P_3 and P_4 is to calculate the center manifold <cit.>.The center manifold for P_3 can be parameterized asT=TΦ = μ_c/μ_cν-μ/μ_cν(1+3/4(μ+μ_cν)^2)T^2+O(T^4),Ψ = -√(6)/2(μ+μ_cν)T+√(6)/2μ_cν(1/2+3/4(μ+μ_cν)^2)T^3+O(T^4),Σ=ν/√(3)C1/μ_cν-ν/√(3)C1/μ_cν(1+3/4(μ+μ_cν)^2)T^2+ O(T^4)The leading order term of the dynamical system restricted to the center manifold reduces toT'= 3/2μ_cν(μ+μ_cν) T^3.Since T'>0 for T>0, P_3 is unstable along its center manifold. Therefore P_3 is a saddle within the full four dimensional phase space. The center manifold for P_4 can be parameterized asT=TΦ =-μ_c/μ_cν+μ/μ_cν(1+3/4(μ-μ_cν)^2)T^2+O(T^4),Ψ = -√(6)/2(μ-μ_cν)T-√(6)/2μ_cν(1/2+3/4(μ-μ_cν)^2)T^3+O(T^4),Σ= -ν/√(3)C1/μ_cν+ν/√(3)C1/μ_cν(1+3/4(μ-μ_cν)^2)T^2+ O(T^4)The leading order term of the dynamical system restricted to the center manifold reduces toT'= -3/2μ_cν(μ-μ_cν) T^3.If μ > μ_c ν then T'<0 and P_4 is stable along its center manifold. It is therefore a sink when μ > μ_c ν, in the full four dimensional phase space and a saddle otherwise. §.§.§ The Bifurcation Value If μ=μ_cν then there is a line of equilibria given by (D,Φ,Ψ,Σ)=(s,-μ_c/μ_cνs,0,-ν/√(3)C1/μ_cνs) where 0<s<1 having endpoints P_0 and P_4.The eigenvalues of points on this line of equilibria are 0, -s,-s, -2/3s which implies that points on this line are attractors when it exists. We observe that as the parameter μ increases towards its bifurcation value, μ_cν, the stability of the point P_0 is transferred to the point P_4 via this line of equilibria.§.§.§ Heteroclinic Sequences Very often one is not only interested in the past and future behaviour of a system of differential equations, but one is also interested in the intermediate behaviour of the system.One technique to analyze the intermediate behaviour is to describe the heteroclinic sequences that are possible <cit.>.We note that for every heteroclinic sequence there exists a set of orbits that are arbitrarily close to that sequence.Figure (<ref>) describe the possible heteroclinic sequences.Again we see how the intermediate behaviour changes as the parameter μ changes.§.§ Inflation and Accelerated ExpansionIt follows that the deceleration parameter (<ref>) can also be expressed in terms of the normalized bounded variables in this case as follows;q= -1/D^2(-2 C Σ^2-2 Ψ^2 +Φ^2-3√(3/2)μ√(1-D^2)Ψ+3 ν/√(2)ΦΣ).The sign of the deceleration parameter indicates the nature of the expansionary evolution. If q > 0, then the cosmological expansion is decelerating, while negative values of q indicate an accelerating or inflationary dynamics. See Table (<ref>) for a summary of the sign of q for each equilibrium point. §.§ Numerical Analysis It is constructive to illustrate a few numerical solutions for the three different regimes of future asymptotic behaviour, μ<μ_cν [see Figure (<ref>)], μ=μ_cν [see Figure (<ref>)], and μ>μ_cν [see Figure (<ref>)]. In each case the integrations are done in the full 4-dimensional phase space. The initial conditions are selected to show different past and future asymptotic behaviours and are the same as those used in the isotropic case, in that here we initially set Σ(0)=0.We also do not show any phase portraits in this case as they are not as illustrative in higher dimensions as in the isotropic case. §.§ Observations In the spatially homogeneous and isotropic case analyzed here we observe that the past dynamics are independent of the strength of the coupling parameters μ and ν.We find in the zero curvature (Bianchi type I) and the negative curvature models (Bianchi type V) that the past asymptotic state is one which the anisotropy is non-trivial.The past solution for both the zero and negative curvature models is theJacobs’ Bianchi type I non-vacuum massless scalar field solution <cit.>.We also observe that the future asymptotic state depends on the strength of the coupling parameter μ and ν.For weak coupling of the Aether field to the scalar field, i.e., μ < μ_c ν, the dynamics are similar to but not the same as that found when there is no coupling of the Aether field to the scalar field, i.e., when μ=0 and ν=0.Having ν>0 drives the system towards intermediate states, P_3 and P_4, that are anisotropic in nature.If μ < μ_c ν then P_0 is the stable attractor: orbits oscillate and slowly decay in amplitude towards this final isotropic non-inflationary asymptotic state.Similar to the isotropic case analyzed in Section <ref>, for strong coupling of the Aether field to the scalar field, μ > μ_c ν, the dynamics are very different.When μ>μ_cν the scalar field does not come to rest at the minimum of the potential: the strength of the Aether interaction forces a different final equilibrium state. If μ > μ_c ν then we find that the stable equilibrium point in these models changes from the isotropic point P_0 to the anisotropic point P_4 if ν>0 which is isotropic if ν=0.In this case, we find that the square of the scalar field, the square of the shear scalar, and the square of the expansion scalar scale together asm^2/2ϕ^2∼ 1/3μ_2^2/μ_cν^2(1+θ^2) σ^2∼ 1/9ν^2/C^2μ_cν^2(1+θ^2)and consequently grow without bound.We also observe that all orbits (excepting for the exceptional orbits) experience some period of accelerating expansion as they evolve to their final asymptotic state which is consistent with the isotropic case. The fundamental difference in the anisotropic case when μ > μ_c ν is that the future asymptotic state need not be isotropic.Bianchi type V models in the standard inflationary scenario in GR (C=1,μ=0,ν=0) isotropize as a rule. In GR, having interaction terms in the scalar field potential, equation (<ref>), in which (C=1, μ>μ_cν,ν>0), changes this rule to one in which the future asymptotic behaviour has accelerated expansion but is not isotropic. Similarly, in the Einstein Aether theory in the standard inflationary scenario (0<C<1,μ=0,ν=0), one observes once again that the models will isotropize to the future.However, just as in GR,if (0<C<1,μ>μ_cν,ν>0) then the models also have a future asymptotic behaviour which has accelerated expansion but is not isotropic.§ DISCUSSION §.§ Slow Roll Inflation In the standard slow roll inflationary scenario in GR, inflation occurs at intermediate times during a period of slow roll (in which ϕ̈≪θϕ̇ and ϕ̇^2 ≪θ^2) and where the anisotropy, if present, is insignificant when compared to the expansion.However, the existence of a non-trivial coupling of the Aether field to the scalar field changes this scenario, and in particular if ν>0, then there can be a significant departure from the standard scenario.To find the slow roll inflationary attractor with scalar field/aether field coupling, we start off with all the same assumptions as above except one. The anisotropy is not assumed to be insignificant, but changes in the anisotropy are assumed to be small during slow roll inflation.With these assumptions the slow roll solutions take on the formθ = 3/2μ_cν|ϕ| σ = ν/2cϕ ϕ̇ =-μ_cν((ϕ)+μ/μ_cν)We see that if μ<μ_cν, then the slow roll solution is stable as ϕ and ϕ̇ have different signs (same for σ and σ̇). The number of e-foldings N that can take place are:N= 1/3∫_ϕ_i^ϕ_f θ/ϕ̇ dϕ= 1/4(1+(ϕ)μ/μ_cv)(ϕ_i^i2-ϕ_f^i2)We note that if (ϕ)<0, then N can be made large for fixed initial and final endpoints by choosing μ≲μ_cν. It appears that the existence of aether field/scalar field coupling terms of the nature studied here do not change the possibility of a period of slow-roll inflation at intermediate times.We do note, however, that if ν>0 then the inflation is anisotropic in nature.Indeed, slow roll inflation is possible even when the mass of the scalar field is zero by simply choosing the coupling parameters to satisfy μ < μ_cν|_m=0=ν/√(3C).In some sense, the slow roll inflationary expansion in this scenario is driven by the coupling to the shear.§.§ Final Comments We have investigated cosmological models in the Einstein-Aether theory in which scalar field matter is coupled to the aether through the scalar field potential. We have been especially interested in possible accelerated expansion and inflationary behaviour in a class of spatially homogeneous cosmological models. In particular, we have studied scalar field models in which the scalar-field potential depends on the time-like aether vector field through its expansion and shear.We have observed that in the isotropic case, by choosing appropriate units, the dynamics are independent of the aether parameters.The existence of the aether is to essentially re-normalize the gravitational constant G_c=G/(1+c_θ) in these cosmological settings. Further, in both the isotropic and anisotropic models studied here, we find that the past asymptotic state does not depend on the value of the aether parameters, c_i or on the value of the scalar field coupling parameters μ and ν.However, in the anisotropic model the intermediate states and in some cases the final states do depend on a single combination of Aether parameters c_θ and c_σ through the parameter C.Indeed, these intermediate and final states increasingly become more anisotropic as a result of decreasing the Aether parameter C from its maximum value C=1.We also note that the future asymptotic states in both the isotropic and anisotropic models depend on the value of the scalar field/aether field coupling parameters μ and ν in the scalar field potential.For sufficiently small values of the parameter μ, the both the isotropic and anisotropic models experience a period of slow-roll inflation at intermediate times, even when the scalar field is massless. For sufficiently large values of the parameter μ, the future asymptotic state changes to one which has accelerated expansion at late times.Indeed, it is possible to have an accelerated expansion at late times even when the mass of the scalar field is zero, provided μ>0 in the isotropic case, and μ>ν/√(3C) in the anisotropic case.In both of these isotropic and anisotropic models, the accelerated expansion at late times is a direct result of the scalar field/aether field coupling.Further, it must be noted that in the anisotropic case, having a non-zero coupling parameter ν causes the future asymptotic state to be anisotropic.The scalar field/aether field coupling parameters μ and ν in the scalar field potential modify the slow roll inflationary dynamics for a sufficiently small μ <μ_cν, which adds a driving force which can slow down or speed up (depending on the sign of scalar field initially) the slow roll inflation <cit.>.In the anisotropic case there are further refinements to the slow roll regime (which can occur for a sufficiently small non-zero parameter ν in the potential).Additionally, in the anisotropic case, the shear coupling causes the slow roll inflationary solution to be anisotropic in nature.Recall, that a period of accelerated expansion is desirable at early intermediate times for inflationary purposes, but a period of accelerated expansion is also an attractive feature to have at late times to describe the effects of Dark Energy.Here, in all cases (isotropic or anisotropic) or (zero curvature or negative curvature),if μ is sufficiently small then there is a period of slow roll inflation at intermediate times, and if μ is sufficiently large, there will be accelerated expansion at late times.Further, if ν>0, these statements are true even when the scalar field is massless.This project is supported in part by the Atlantic Association for Research in the Mathematical Sciences through a Collaborative Research Grant.We thank Theodore Kolokolnikov for his guidance on some technical points. 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D73 (2006) 064015, [https://arxiv.org/abs/gr-qc/0509083gr-qc/0509083].§ APPENDIX:CONSTRAINTS ON THE EINSTEIN AETHER PARAMETERS C_IAccording to <cit.> the PPN parameters for Einstein's General Relativity and Einstein Aether theory are identical ifc_2= -2c_1^2-c_1c_3+c_3^2/3c_1, c_4=-c_3^2/c_1.If one also assumes the squared speeds of massless modes relative to the Aether rest frame must be super-luminal to avoid vacuum Čerenkov radiation <cit.>, then two additional constraints must be satisfied0≤c_1+c_3 ≤1/2, 0≤c_1-c_3 ≤c_1+c_3/3[1-2(c_1+c_3)],which when combined with (<ref>) are sufficient to show both positive energy modes and linear stability.Recall that the c_i employed here are one-half of the values used in <cit.>. In terms of c_σ and c_θ used in this paper, assuming that Einstein Aether and GR are equivalent for weak fields, i.e., equation (<ref>), then these constraints (<ref>) become0≤c_σ≤1/2 c_σ/3c_σ-2 ≤c_θ≤ 0and in terms of c_σ and C0≤c_σ≤1/2, 1-2c_σ ≤C ≤ 1-3/2c_σ. Donnelly and Jacobson included the coupling parameter μ in their analysis of the parameter constraints and showed that these constraints are relaxed when μ>0 <cit.> and are automatically satisfied when the PPN parameters match those of GR. | http://arxiv.org/abs/1707.08911v1 | {
"authors": [
"B. Alhulaimi",
"R. J. van den Hoogen",
"A. A. Coley"
],
"categories": [
"gr-qc"
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"published": "20170727154705",
"title": "Spatially Homogeneous Einstein-Aether Cosmological Models: Scalar Fields with a Generalized Harmonic Potential"
} |
mylinkcolorrgb0,0,0.5 | http://arxiv.org/abs/1707.08426v2 | {
"authors": [
"Tanja Hinderer",
"Stanislav Babak"
],
"categories": [
"gr-qc"
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"primary_category": "gr-qc",
"published": "20170726132223",
"title": "Foundations of an effective-one-body model for coalescing binaries on eccentric orbits"
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SRF Theory Developments from the Center for Bright BeamsThis work was supported by the US National Science Foundation under Award OIA-1549132, the Center for Bright Beams. D. B. [email protected], T. Arias, D. L. Hall, M. Liepe, J. P. Sethna, N. Sitaraman, Cornell University, Ithaca, NY, United States of America A. Pack, M. K. Transtrum, Brigham Young University, Provo, UT, USA December 30, 2023 ================================================================================================================================================================================================================================We present theoretical studies of SRF materials from the Center for Bright Beams. First, we discuss the effects of disorder, inhomogeneities, and materials anisotropy on the maximum parallel surface field that a superconductor can sustain in an SRF cavity, using linear stability in conjunction with Ginzburg-Landau and Eilenberger theory. We connect our disorder mediated vortex nucleation model to current experimental developments of Nb_3Sn and other cavity materials. Second, we use time-dependent Ginzburg-Landau simulations to explore the role of inhomogeneities in nucleating vortices, and discuss the effects of trapped magnetic flux on the residual resistance of weakly- pinned Nb_3Sn cavities. Third, we present first-principles density-functional theory (DFT) calculations to uncover and characterize the key fundamental materials processes underlying the growth of Nb_3Sn. Our calculations give us key information about how, where, and when the observed tin-depleted regions form. Based on this we plan to develop new coating protocols to mitigate the formation of tin depleted regions.§ INTRODUCTIONThe fundamental limit to the accelerating E-field in an SRF cavityis the ability of the superconductor to resist penetration of the associated magnetic field H (or equivalently B). SRF cavities are routinely run at peak magnetic fields above the maximum field H_c1 sustainable in equilibrium; there is a metastable regime at higher fields due to an energy barrier at the surface <cit.>.marks the stability threshold of the Meissner state. In Fig. <ref> we show results from linear stability analysis <cit.>, valid near T_c, foras a function of the Ginzburg-Landau parameter κ, the ratio λ/ξ of the London penetration depth λ to the coherence length ξ. Niobium has κ≈ 1.5, most of the promising new materials have large κ. At lower temperatures, one must move tomore sophisticated Eliashberg theories <cit.>, for whichis known analytically for large κ; numerical studies at lower κ are in progress <cit.>. Broadly speaking, the results so far for isotropic materials appear similar to those of Ginzburg-Landau. This manuscript will briefly summarize theoretical work on(the threshold of vortex penetration and hence the quench field).First, we discuss the effect of materials anisotropy on <cit.>. Second, we discuss theoretical estimates of the effect of disorder <cit.>, and preliminary unpublished simulations of the effects of surface roughness and materials inhomogeneity. Third, we discuss key practical implications of theoretically calculated point defect energies, interactions, relaxation times, and mobilities in the promising new cavity material Nb_3Sn. Finally, some magnetic flux is trapped in cavities during the cooldown phase, and the response of these flux lines to the oscillating external fields appears to be the dominant source of dissipation in modern cavities. We model potentially important effects of multiple weak-pinning centers on this dissipation due to trapped flux.§ THE EFFECT OF MATERIALS ANISOTROPY ON THE MAXIMUM FIELDSome of the promising new materials are layered, with strongly anisotropic superconducting properties (MgB_2 and the pnictides, for example, but not Nb_3Sn or NbN). Fig. <ref> illustrates an anisotropic vortex (magnetized region blue, vortex core red) penetrating into the surface of a superconductor (grey). The anisotropy here is characteristic of MgB_2 at low temperatures, except that the vortex core is expanded by a factor of 30 to make it visible.Near T_c, we find in ref. <cit.> that a simple coordinate change and rescaling maps the anisotropic system onto the isotropic case(Fig. <ref> above, as studied in ref. <cit.>). We find, near T_c whereGinzburg-Landau theory is valid, thatis nearlyisotropic for large κ materials (Fig. <ref>. At lower temperatures, different heuristic estimates of the effects of anisotropy onyield conflicting results. Further work at lower temperatures could provide valuable insight into the possible role of controlled surface orientation for cavities grown from these new materials. § DISORDER-MEDIATED FLUX ENTRY AND MATERIALS ANISOTROPY Defect regions and inhomogeneity of superconductor properties can weaken the performance of SRF cavities. In ref. <cit.> we used simple estimates based on Bean and Livingston's energy barrier arguments <cit.>, to estimate the effects of disorder in loweringby providing flaws that lower the barrier to vortex penetration. Here we use these calculations to shed light about the relationship between tin depleted regions, low critical temperature profiles, defect sizes and quench fields.Consider an external magnetic field B, parallel to the surface of a semi-infinite superconductor occupying the half-space x>0. If B is larger than the lower critical field B_c1 (and smaller than B_c2), the vortex lattice phase is thermodynamically favored. However, if the field is not large enough, a newborn vortex line near the superconductor surface will have to surpass an energy barrier to penetrate the superconductor towards the bulk of the material. This instability typically is surmounted by the simultaneous entry of an entire array of vortices, whose interactions lower one another's barriers. Disorder, in contrast, will lead to a localized region allowing one vortex entry at a time. Bean and Livingston provided simple analytical calculations for the energy barrier felt by one vortex line; we extended their calculation to estimate the dirt needed to reduce this barrier tozero at a quench field H_q <. The new materials have larger κ, and in particular smaller vortex core sizes ξ; naively one would expect vortex penetration when flaws of size ξ arise. Are these new materials far more sensitive to dirt than niobium? Reassuringly, Fig. <ref> shows that the low values of the coherence length do not make these new materials substantially moresusceptible to disorder-induced vortex penetration <cit.>. We can use our model to estimate the suppressed superconducting transition temperature T_c^min and the flaw depth D_c needed to allow vortex penetration, as a function of H_q (or, in Tesla, B_q) (Fig. <ref>). For Nb_3Sn we find a flaw size of D_c∼ 100nm and T_c^min∼ 12K can allow vortex nucleation and quenches atfields of ∼ 77mT (Fig. <ref>), consistent with experimental results <cit.>. § TIME-DEPENDENT GINZBURG-LANDAU SIMULATIONS OF ROUGH SURFACES AND DISORDER To quantify the dependence ofon surface roughness and disorder,we have developed a time-dependent Ginzburg-Landau simulation.Fig. <ref> shows the density |ψ|^2 of superconducting electrons at a field just above (top left), showing the entry of several vortices for a 2D system with an irregular surface. On the bottom left, we show the corresponding supercurrent 𝐣; on the top right we show the magnetic field H (perpendicular to the plane of the simulation), and on the bottom right we show the effect of surface roughness on |ψ(θ)|^2 around theperimeter. Our initial results quantify how inward-curving regions in the plane perpendicular to the applied field on the perimeter can act as vortex nucleation sites in this geometry.An open question remains what the effect of curvature and surface roughness have when oriented parallel to the applied field. The effect of roughness in Fig. <ref> is to lowerby a few percent.By systematically varying the details of the roughness parameters, we can use this tool to identify at what scale roughness will have significant impact on vortex nucleation.SRF cavity roughness can be smoothed to varying degrees. Our TDGL environment can be used to find dangerous regimes or configurations that can have serious consequences for cavity performance.We can also use this tool as a way to explore vortex dynamics and the effects of pinning sites on trapped residual magnetic flux. Pinning sites originate from inhomogeneities in the material, such as grain boundaries or spatial inhomogeneities in the alloy stoichiometry.By incorporating this information into our TDGL environment we can try to better understand the mechanisms driving residual resistance for typical cavities.§ DFT CALCULATIONS Nb_3Sn cavities are created by depositing tin vapor on the surface of a niobium cavity, which reacts with the niobium to form an irregular surface layer of the compound. Of interest are regions of “tin depleted” Nb_3Sn, known to have a lower superconducting transition temperature than the surrounding Nb_3Sn. These regions may be the nucleation centers responsible for quenches observed well belowexpected for perfect Nb_3Sn <cit.>.Density functional theory (DFT) can be used to study layer growth, tin depletion, and other features of Nb_3Sn layers at the single-particle level. This information, combined with experimental data and accounting for the effects of grain boundaries and strain, makes it possible to build a multiscale model of layer growth. Our initial work uses in-house DFT software to calculate defect formation and interaction energies, impurity energies, and energy barriers in Nb_3Sn. We have found that antisite disorder (Figure <ref>), rather than impurities or vacancies, likely sets the electron mean free path in Nb_3Sn and may also be responsible for collective weak pinning. We have also found that under certain conditions during growth, it is energetically favorable for Nb_3Sn to form at tin-depleted stoichiometry, while during annealing existing Nb_3Sn near the surface or grain boundaries can become tin-depleted by diffusion (Figure <ref>). Either or both of these tin depletion mechanisms may result in quench nucleation centers; by understanding them we can for the first time make informed modifications to the coating process in an attempt to limit tin depletion and produce better cavities. § DYNAMICS OF TRAPPED VORTICES; POTENTIAL ROLE OF WEAK PINNING When the field is high enough for penetration of new vortices, one expects acascade of vortices leading to a quench. Vortices trapped during the cooling process, while not immediately fatal, do act as sources of residual resistance. Experiments show that the non-BCS surface resistance is proportional to the trapped flux, both for nitrogen-doped Nb cavities <cit.> and for Nb_3Sn <cit.>. This suggests that trapped vortices may bea dominant contribution to the quality factor of the cavity.Previous studies of the residual resistance due to a trapped flux line <cit.> focused on the Bardeen-Stephen viscous dissipation <cit.> of a free line pinned a distancebelow the surface, as the external field drags the line through a otherwise uniform superconducting medium. Experimental measurements in nitrogen-doped Nb cavities showed good agreement to this theory, except that the distance to the pinning center was presumed to change linearly with the mean-free path <cit.> as it changes due tonitrogen doping. Since nitrogen (or other contaminant gases <cit.>)should act as weak pinning centers (with many impurities per coherence length cubed), we have been modeling the role of weak pinning invortex dissipation. Line defects pulled through a disordered medium is one of the classical depinning transitions <cit.>. The disorder acts as a random potential, and macroscopically there is a threshold force per unit length f_pin needed to depin the line and start motion (Fig. <ref>). This depinning transition is preceded by avalanches of all sizes (local regions of vortex motion) and followed by fluctuations on all scales (jerky motion of the vortex line in space and time). For our initial estimates, we have ignored these fluctuations, using a `mean-field' model where our superconducting vortex line has a threshold supercurrent j_d ∝ f_pin^2/3 for motion. We presume also that theenergy dissipated is f_pin times the area swept out by the vortex as the external surface field pulls it to and fro (Fig. <ref>). The residual resistance measured in Nb_3Sn cavities shows a linear dependence on the peak RF field (Fig. <ref>, <cit.>). The scaling properties of the terms included in the earlier work <cit.> all predict no dependence on the strength of the external oscillating field. Our theory including weak pinning but ignoring the viscous dissipation produces a dissipation that is linear in this external field. Our estimates, however, suggest that our theory should be valid at MHz frequencies, but at the operating GHz frequencies the viscous term must be important for the energy dissipation. Our preliminary calculations suggest that incorporating both can provide a reasonable explanation of the experimental data, but we still do not obtain quantitative agreement.§ CONCLUSION The collaboration between scientists inside and outside traditional accelerator physicists made possible by the Center for Bright Beams has been immensely fruitful. This proceedings illustrates the richness of the science at the intersection of accelerator experimentalistsworking on SRF cavities with condensed-matter physicists with interests in continuum field theories and ab-initio electronic structure calculations of materials properties. (One must also note the important contributions of experimental condensed matter physicists in the collaboration.) Current SRF cavities are pushing fundamental limits of superconductors, and are a source of fascinating challenges for theoretical condensed-matter physics. Conversely, we find that theoretical calculations are remarkably fruitful in guiding and interpreting experimental findings.§ ACKNOWLEDGMENTWe thank Alex Gurevich for useful conversations. 99bardeen65 J. Bardeen, and M. J. Stephen, “Theory of the Motion of Vortices in Superconductors” Phys. Rev., vol. 140, p. A1197, 1965.bean64 C. P. Bean, J. D. Livingston, “Surface Barrier in Type-II Superconductors” Phys. Rev. Lett., vol. 12, p. 14, 1964.catelaniUnpublished G. Catelani, M. K. Transtrum, and J. P. Sethna, Unpublished. catelani08 G. Catelani, and J. P. Sethna, “Temperature dependence of the superheating field for superconductors in the high-κ London limit” Phys. Rev. B, vol. 78, p. 224509, 2008.fisher98 D. S. Fisher, “Collective transport in random media: from superconductors to earthquakes” Phys. Rep., vol. 301, p. 113, 1998.gonnella16 D. Gonnella, J. Kaufman, and M. Liepe, “Impact of nitrogen doping of niobium superconducting cavities on the sensitivity of surface resistance to trapped magnetic flux” J. Appl. Phys., vol. 119, p. 073904, 2016.gurevich13 A. Gurevich, and G. Ciovati, “Effect of vortex hotspots on the radio-frequency surface resistance of superconductors” Phys. Rev. B, vol. 87, p. 054502, 2013.hallIPAC17a D.L. Hall, P. Cueva, D. Liarte, M. Liepe, J.T. Maniscalco, D.A. Muller, R.D. Porter, and J.P. Sethna, “Quench Studies in Single-Cell Nb3Sn Cavities Coated Using Vapour Diffusion”,in Proc. 8th Int. Particle Accelerator Conf. (IPAC'17), Copenhagen, Denmark, May 2017, paper MOPVA116, pp. 1119–1122, <http://jacow.org/ipac2017/papers/mopva116.pdf>,<https://doi.org/10.18429/JACoW-IPAC2017-MOPVA116>, 2017.hallIPAC17bD.L. Hall, D. Liarte, M. Liepe, and J.P. Sethna,“Impact of Trapped Magnetic Flux and Thermal Gradients on the Performance of Nb3Sn Cavities”,in Proc. 8th Int. Particle Accelerator Conf. (IPAC'17), Copenhagen, Denmark, May 2017, paper MOPVA118, pp. 1127–1129, <http://jacow.org/ipac2017/papers/mopva118.pdf>,<https://doi.org/10.18429/JACoW-IPAC2017-MOPVA118>, 2017.koufalis17aP. N. Koufalis, D. L. Hall, M. Liepe, and J. T. Maniscalco, “Effects of interstitial carbon and oxygen on niobium superconducting cavities” arXiv:1612.08291, 2016.koufalis17bP. N. Koufalis, M. Liepe, and J. T. Maniscalco,“Low temperature doping of niobium cavities: what is really going on?”,in Proc. of the 18th Int. Conf. on RF Superconductivity, Beijing, China, 2017. liarte16 D. B. Liarte, M. K. Transtrum, and J. P. Sethna, “Ginzburg-Landau theory of the superheating field anisotropy of layered superconductors” Phys. Rev. B, vol. 94, p. 144505, 2016. liarte17 D. B. Liarte, S. Posen, M. K. Transtrum, G. Catelani, M. Liepe and J. P. Sethna, “Theoretical estimates of maximum fields in superconducting resonant radio frequency cavities: stability theory, disorder, and laminates”, Supercond. Sci. Technol., vol. 30, p. 033002, 2017.transtrum11 M. K. Transtrum, G. Catelani and J. P. Sethna, “Superheating field of superconductors within Ginzburg-Landau theory” Phys. Rev. B., vol. 83, p. 094505, 2011. | http://arxiv.org/abs/1707.09025v1 | {
"authors": [
"Danilo B. Liarte",
"Tomas Arias",
"Daniel L. Hall",
"Matthias Liepe",
"James P. Sethna",
"Nathan Sitaraman",
"Alden Pack",
"Mark K. Transtrum"
],
"categories": [
"physics.acc-ph"
],
"primary_category": "physics.acc-ph",
"published": "20170727195557",
"title": "SRF Theory Developments from the Center for Bright Beams"
} |
firstpage–lastpageSectoring in Multi-cell Massive MIMO Systems Shahram Shahsavari, Parisa Hassanzadeh, Alexei Ashikhmin, and Elza Erkip S. Shahsavari, P. HassanzadehandE. Erkip are with the ECE Department of New York University, Brooklyn, NY. Email: {shahram.shahsavari,ph990, elza}@nyu.eduA. Ashikhmin is with Bell Labs, Nokia, Murray Hill, NJ, USA. Email:[email protected] December 30, 2023 ========================================================================================================================================================================================================================================================================================================================================== Carbon enhanced metal poor (CEMP) stars are a particular class of low metalicity halo stars whose chemical analysis may provide important contrains to the chemestry evolution of the Galaxy and to the models of mass transfer and evolution of components in binary systems. Here, we present a detailed analysis of the CEMP star CD-50776, using high resolution optical spectroscopy. We found that CD-50776 has a metalicity [Fe/H] = -2.31 and a carbon abundance [C/Fe] = +1.21. Analyzing the s-process elements and the europium abundances, we show that this star is actually a CEMP-s star, based on the criteria set in the literature to classify these chemically peculiar objects. We also show that CD-50776 is a lead star, since it has a ratio [Pb/Ce] = +0.97.In addition, we show that CD-50776 develops radial velocity variations that may be attributed to the orbital motion in a binary system.The abundance pattern of CD-50776 is discussed and compared to other CEMP-s stars already reported in the literature to show that this star is a quite exceptional object among the CEMP stars, particularly due to its low nitrogen abundance. Explaining this pattern may require to improve the nucleosynthesis models, and the evolutionary models of mass transfer and binary interaction. nuclear reactions, nucleosynthesis — stars: abundances —stars: individual: CD-50776 — stars: chemically peculiar —stars: evolution — stars: fundamental parameters § INTRODUCTION During the last decades, considerable theoretical and observational efforts have been made to investigate the formation and evolution of the chemistry of the Galaxy through the study of its halo stars. The chemical composition of the halo stars is important because it can provide information not only about the early stages of the Galaxy formation, but also about some sites where nucleosynthesis of several elements took place, thus providing significant evidence to describe the initial stages of galactic nucleosynthesis. In order to carry on such studies, an adequate sample of halo stars must be selected. After the first surveys of low metallicity stars initiated by Bond et al. (1970, 1980), Bidelman (1981) and Bidelman & MacConnell (1973), where the metallicity limit was around -2.6 (Frebel & Norris, 2013, Beers et al. 2014), the surveys by Beers et al. (1985) and Christlieb et al. (2001) significantly increased the known number of low metalicity stars, including several candidates with metallicities less than -2.0.Following these surveys, spectroscopic studies revealed that some of the metal-poor stars from Beers et al. (1985) were also carbon-rich objects (Beers et al. 1992). Before 2005, high-resolution spectroscopic analysis of these stars confirmed the carbon-rich nature for some of them. In addition, it was noted that some of these stars were also enriched in either the r-, s- or r/s-procesess (McWilliam et al. 1995; Sneden et al. 1994, 1996, 2003a, 2003b; Barbuy et al. 1997, Norris et al. 1997a, 1997b, Bonifacio et al. 1998, Hill et al. 2000, 2002, Sivarani et al. 2004).In 2005, Rossi et al. (2005) using medium-resolution spectra of the stars in the samples of Beers et al. (1985) and Christlieb et al. (2001), noted the high frequency occurrence of the carbon-rich stars among the metal-poor stars, also known as CEMP (carbon enhanced metal poor) stars.According to Lucatello et al. (2006), 20% of the stars with metallicities down to -2.0 are CEMP stars. CEMP stars have been found in all the metallicity range from -2.0 to -4.0, with increasing frequency towards the lower metallicities (Lucatello et al. 2006). In view of this, several astrophysical sites for the origin of the carbon overabundances have been proposed (see Beers & Christlieb 2005 for a discussion).Beers & Christlieb (2005) (but see also Masseron et al. 2010) proposed that CEMP stars could be distinguished according to their barium and europium abundances, and also according to their [Ba/Eu] ratio. After their study, the CEMP stars were divided in CEMP-s, CEMP-r, CEMP-r/s and CEMP-no stars according to the heavy elements abundance pattern. The majority of CEMP stars are CEMP-s stars (Aoki et al. 2007). The most likely explanation for the observed excess of carbon and s-process elements in CEMP-s stars is the mass-transfer, just like in the CH stars and the barium stars. This conclusion is supported by the radial-velocity variations observed in several CEMP-s stars (Hansen et al. 2016). Therefore, detailed abundance analysis of CEMP-s stars is important to set observational constraints to the physics of mass-transfer (in the case of CEMP-s binaries), and also to the nucleosynthesis models.In this work we present the spectroscopic analysis of a new CEMP-s star: CD-50^∘776. CD-50^∘776 came to our attention during our high-resolution spectroscopy survey started in 1999, during the first agreement between Observatório Nacional and the European Southern Observatory, with the aim to search for halo chemically peculiar stars. Later on, we also searched for metal-poor hypervelocity candidate stars, following our analysis of CD-62^∘1346, a CH hypervelocity star candidate (Pereira et al. 2012), and the analysis of two metal-poor red-horizontal-branch stars: CD-41^∘115048 and HD 214362 (Pereira et al. 2013). To select these peculiar stars, we search over several surveys from the literature. In particular, CD-50^∘776 was selected from the work of Bidelman & MacConnel (1973), whose stars sample was later investigated by Norris et al. (1985) and Beers et al. (2014, 2017). In particular, Beers et al. (2014), based on a medium-resolution spectrum, determined the metallicity and the [C/Fe] ratio of this star, obtaining values of -2.23 and +1.81, respectively.Here, we show that CD-50^∘776 is in fact a CEMP star with an excess of the elements created by the s-process, without europium enrichment. Therefore it can be classified as a CEMP-s star. CD-50^∘776 is the second brightest CEMP-s star known to date, with V=10.05 (the brightest one is HD 196944 with V=8.4).The present work is based on the analysis of high-resolution spectra of CD-50^∘776 to determine its metallicity and abundance pattern.§ OBSERVATIONSThe high-resolution spectra analyzed in this work were obtained with the Feros (Fiberfed Extended Range Optical Spectrograph) spectrograph (Kaufer et al. 1999), that was initially coupled to the 1.52 m telescope and later to the 2.2 m telescope of ESO, at La Silla (Chile). Two observations were done for CD-50^∘776.One, on October 26, 1999 and another one on September 25, 2016.The exposures were 3600 and 2400 secs, respectively.Feros consists of a CCD detector of 2048× 4096 pixels having each pixel a size of 15 μ m. Feros has spectral coverage between 3 900Åand 9 200Ådistributed over 39 orders with a resolution of 48 000. The spectral reduction was made following a standard procedure, which includes bias subtraction, flat-fielding, spectral order extraction and wavelength calibration.All this procedure has been done using the MIDAS reduction pipeline.§ ANALYSIS & RESULTS The atomic absorption lines used for the determination of atmospheric parameters are basically the same as which were used in the study of other chemically peculiar stars (Pereira & Drake 2009). Table 1 shows the Fe i and Fe ii lines used to determine these parameters. The log gf values for the Fe i and Fe ii lines were taken from Lambert et al. (1996). §.§ Determination of the atmospheric parameters Our analysis was done using the spectral analysis code MOOG (Sneden 1973) and the model atmospheres of Kurucz (1993). The latest version of MOOG includes routines for the calculation of the Rayleigh-scattering contribution to the continuous opacity, as described in Sobeck et al. (2011). The temperature was obtained after searching for a zero slope of the relation between the iron abundances based on Fe i lines and the excitation potential while the microturbulent velocity was obtained after searching for a zero slope of the relation between the iron abundances based on the same Fe i lines and the reduced equivalent width (W_λ/λ). This procedure also provides the metallicity of the star.The surface gravity of the star was obtained by means of the ionization equilibrium, which means that we shouldfind a solution until the abundance of Fe i and Fe ii become equal.The final atmospheric parameters derived for CD-50^∘776 are given in Table 2. Table 2 also shows the values derived from previous spectroscopic observations of CD-50^∘776 conducted by Ryan & Deliyannis (1998) and Beers et al. (2014). The three atmospheric parameters given in Beers et al. (2014), labelled as 2a, 2b and 2c, differ according to the techniques used by these authors to obtain them.We note that our atmospheric parameters are in a good agreement either in temperature or surface gravity, depending on the specific technique used by Beers et al. (2014). A model with the highest temperature implies a change of +0.5 dex in the carbon abundance compared to our results. The model with log g=3.0 does not allow a good fit in the region of the C_2 molecule, at 5165Å.The errors reported in our effective temperature (T_ eff) and microturbulent velocity (ξ) were set from the uncertainty in the slope of the Fe i abundance versus. excitation potential and versus. W_λ/λ, respectively.For the gravity, the error was estimated until the mean abundances of Fe i and Fe iidiffer by 1σ of the standard deviation of the [Fe i/H] mean value.§.§ Abundance analysis The abundance pattern of CD-50^∘776 was determined using either equivalent width measurements of selected atomic lines and using the spectral synthesis technique. We used the solar abundances of Grevesse & Sauval (1998) as a reference. For iron it was used the solar abundance of logε(Fe) = 7.52.Table 3 shows the atomic lines used to derive the abundances of the elements, with their respective equivalent width measurements.The derived abundances are given in Table 4. For the elements whose abundances were derived using spectral synthesis technique they are labelled as syn.The abundances of the light elements, carbon and nitrogen, were determined by applying a spectrum synthesis technique in the local thermodynamic equilibrium (LTE). For carbon, we used the CH lines of the A^2Δ - X^2B system at ∼ 4365 Å, the C_2 (0,0) band head of the Swan system d^3Π_g - a^3Π_u at 5165 Å, and the C_2 (0,1) band head of the Swan system d^3Π_g - a^3Π_u at 5635 Å.For nitrogen we used the B^2Σ - X^2Σ violet system band head at 3883 Åwith line list provided by VALD. The (2, 0) band of the CN red system A^2Π - X^2Σ in the 7994–8020 Åoften used by us to determine the nitrogen abundance, is not visible in this star. We did not detect the oxygen forbidden line at 6300.0 ÅTherefore we assume that [O/Fe] = +0.50, which is a typical value for a star of this metallicity (Masseron et al. 2006).We also check our derived nitrogen abundance using a different linelist for the CN band at 3883 Ågiven by Jonsell et al. (2006) and Sneden et al. (2014), and the results were basically the same as using the linelist given by VALD.The abundances of barium, europium, cobalt, lead and praseodymium were also determined by means of spectral synthesis technique.The determination of barium abundance was obtained using the Ba ii lines at λ 4554.0, λ 4934.1, λ 5853,7, and λ 6141.7 Å. Hyperfine and isotope splitting were taken from McWilliam (1998).The europium abundance was found using the line of Eu ii at λ 4129.75 Åand the hyperfine splitting from Mucciarelli et al. (2008).The cobalt abundance was derived using the Co i line at λ 4121.33 Å, where the hyperfine splitting was taken from McWilliam et al. (1995).The lead abundance was derived from the Pb i line at λ 4057.81 Å.The line data, which include isotopic shifts and hyperfine splitting, were taken from van Eck et al. (2003).The abundance of praseodymium was obtained through spectral synthesis technique using the lines at 5259.73Åand 5322.77Å. The hyperfine splitting was taken from Sneden et al. (2009).Figures 1, 2, 3, 4 and 5 show the observed and synthetic spectra for the spectral regions where the abundances of carbon, the ^12C/^13C isotopic ratio, nitrogen, lead and europium were obtained. §.§ Abundance uncertainties The uncertainties in the abundances of CD-50^∘776 are given in Table 5.The uncertainties due to the errors of T_ eff, log g, ξ, and metallicity were estimated by changing these parameters one at a time by their standard errors given in Table 2. The final uncertainties of the abundances were calculated as the root squared sum of the individual uncertainties due to the errors in each atmospheric parameter and also in the equivalent widths under the assumption that these individual uncertainties are independent.For the elements analyzed via spectrum synthesis we used the same technique, varying the atmospheric parameters and then computing independently the abundance changes introduced by them.Uncertainties in the carbon abundances also result in variations of the nitrogen abundances, since the CN molecular lines were used for the nitrogen abundance determination. For carbon and nitrogen, typical uncertainties are 0.10 and 0.20, respectively.In Table 5, we see that the neutral elements are more sensitive to the temperature variations, while singly-ionized elements are more sensitive to the log g variations.For the elements whose abundance is based on stronger lines, such as strontium, the error introduced by the microturbulence is important.Finally, we observe that the abundances of carbon and nitrogen are weakly sensitive to the variations of the microturbulent velocity. § DISCUSSION§.§ The luminosity of CD-50^∘776 Once we estimated the temperature and gravity of CD-50^∘776, we are able to determine the luminosity considering the relationlog (L_⋆/L_⊙) =4log T_ eff ⋆ - log g_⋆ + M_⋆/M_⊙+10.61where we considered T_ eff ⊙ = 5 777 K and log g_⊙ = 4.44.Inserting the values of T_ eff=4 900 K, log g=2.1, and assuming a mass M_⋆=0.8M_⊙ for CEMP stars (Aoki et al. 2007), we obtain for the luminosity of CD-50^∘776 a value of log (L_⋆/L_⊙) = 1.95±0.3.Spectroscopic luminosities of low-metallicity giants derived from ionization balance may give higher values than those derived from stellar parallaxes or evolutionary models (Mashonkina et al. 2011; Ruchti et al. 2013). According to the recent work of Ruchti et al. (2013), the non-local thermodynamic equilibrium (NLTE) correction to the spectroscopic gravity is about +1.0 dex, and for the temperature the correction is around +400 K.Introducing these corrections in equation (1), we obtain a luminosity of log (L_⋆/L_⊙)=1.09±0.3.In Figure 6 we show the derived temperature and gravity of CD-50^∘776 in the log T_ eff - log g plane, together with the 12 and 14 Gyr Yale-Yonsei isochrones for a metallicity of [Fe/H] = -2.2 (Kim et al. 2002).As mentioned in Section 3.1, Beers et al. (2014) also determined the temperature, surface gravity and metallicity of CD-50^∘776 using three different techniques. However, their results using high-resolution spectroscopy provided a surface gravity +0.9 higher than the value obtained by us. §.§ CD-50^∘776 as a new CEMP star In Figure 7 we reproduce Figure 4 of Aoki et al. (2007), where the authors presented a new constraint for a star to be classified as a CEMP star. In Figure 7a, the position of CD-50776 is clearly above the lower limit for a star to be considered as a CEMP star. Figure 7b plots the [C/Fe] ratio versus. metallicity for the CEMP stars and again CD-50776 occupies in this diagram the same position as other CEMP stars. Therefore, based on these two diagrams, we can classify CD-50776 as a new CEMP star. In addition, the position of CD-50776 in figures 2 and 6 of Yoon et al. (2016) also supports our conclusion that CD-50776 is a CEMP star. We will show in Section 4.4.3 that, based on the abundance analysis, CD-50776 is actually a new CEMP-s star.§.§ Radial Velocity Table 6 shows the all known measurements of the radial velocity of CD-50776 available in the literature and determined in this work.It is clear that the radial velocity of CD-50776 presents variations due to orbital motion. Systematic radial velocity monitoring is necessary to confirm the possible binary nature of this new CEMP-s star. §.§ Abundances §.§.§ Nitrogen and ^12C/ ^13C isotopic ratio As shown in Table 4, the nitrogen abundance is low and the ^12C/^13C isotopic ratio is high. Combining these results with the high carbon abundance led us to conclude that the CN cycle was not efficient enough in the donor star of the binary system of CD-50776, and that significant carbon produced by the triple alpha process was transferred in the AGB stage. The sum (C+N) = 7.42 illustrates the fact that carbon is the actual responsible for the total sum of (C+N).If we assume that the nitrogen observed in the CEMP-s stars has the same origin as carbon, that is, it originates in the companion star during the AGB phase (Masseron et al.2010), then the low nitrogen abundance observed in CD-50776 may constrain, in principle, the mass of the donor star.In particular, for CD-50776, it is likely that the mass of the AGB star should not have been greater than 3.0 M_⊙. Models of AGB stars for a metallicity of Z = 0.0001 (Herwig 2004), which is the metallicity of CD-50776, show that the yields of carbon and nitrogen provide a ratio [C/N] of about 2.3 for a star of 2.0 M_⊙ at the end of the AGB phase. Threfore, nitrogen is not enhanced in such models. These models also predict a high ^12C/^13C isotopic ratio (figures 7 and 8 of Herwig 2004).For a star of 3.0 M_⊙, the ratio of [C/N] is 2.1 according to the yields given in Herwig (2004), allowing to conclude that low metallicity stars with masses between 2.0 and 3.0 M_⊙ should not be nitrogen enriched (Johnson et al. 2007).This would explain the low abundance of nitrogen observed in CD-50776, and implies that this star follows the evolution expected according to the models of Herwig (2004). However, for other CEMP-s stars, the high nitrogen abundance poses challenges to the evolutionary models. Masseron et al. (2010) considered that extra mixing mechanisms should be taken into account in order to explain the high abundance of nitrogen in CEMP-s stars. This is because a high nitrogen abundance is predicted by hot bottom burning, which occurs in stars with masses greater than 4.0 M_⊙ (Sackmann & Boothroyd 1992).Notwithstanding, since CEMP-s stars are members of the halo population, it is unlikely that their companions had masses larger than 4.0 M_⊙. This led Masseron et al. (2010) to conclude that the nitrogen abundance in CEMP-s stars should not be used to constrain the mass of the donor star.Figure 8 shows the [N/Fe] ratio versus. metallicity for CD-50776 (red star) compared to CEMP-s giants and dwarfs (squares), CH stars (polygons), one metal-poor barium star (HD 123396, (1)), one CEMP-no star (CS 22877-001, (2)) and one carbon star (HD187216, (3)).§.§.§ Sodium to Nickel Since CD-50776 is a new CEMP-s star (Section 4.4.3), we also compare its abundances to other CEMP-s stars. Figures 9 and 10 show the abundance ratios [X/Fe] versus.metallicity for Na, α-elements and iron-peak elements (Cr, Co, Ni and Zn) of CD-50776 compared to several previous abundance studies of stars of the thin and thick disks and the halo populations.CEMP-s giant stars and dwarfs are represented by filled and open squares, respectively. We also plot in these Figures the abundance ratios of barium stars and of some CH stars (except carbon, cobalt, zinc and barium), based on the recent analysis by de Castro et al. (2016).Sodium abundance in CEMP-s stars exhibits the same trend as for the other metal-poor field stars. Some CEMP-s stars display higher [Na/Fe] ratios than the stars with similar metallicity, however this can be caused by NLTE effects, which seem to be stronger in metal-poor stars (see Aoki et al. 2007 for a discussion of sodium abundance in CEMP stars). Our derived value of +0.07 for the [Na/Fe] ratio indicates that NLTE effects seem to be negligible in this star.In Figure 9, we verify that the abundances of α-elements (Mg, Ca and Ti) are those of other authors in the field stars of the same metallicity as CD-50776.The iron-group element nickel is expected to follow the iron abundance (Figure 10), as it actually does, with a [Ni/Fe] ratio equal to +0.07.Down to [Fe/H] <-2.0, the [Ni/Fe] ratio has a scatter around the mean [Ni/Fe] = 0.0. Chromium in CD-50776 has a negative [Cr/Fe] ratio (-0.17), following the same ratio observed in stars with equal metallicity, as well as in some CEMP-s stars.The other iron-peak elements, cobalt and zinc, do not deviate from the trend observed in the field giants of the same metallicity. Concerning the other CEMP-s stars, most of them also follow the same trend as the metal-poor field giants for the α-elements and the iron-peak elements. However, some of them present high [X/Fe] ratios for the Mg, Ca or Ti. Aoki et al. (2007) considered that some of these high [X/Fe] ratios, like calcium for example, “are possibly overestimated due to contamination by molecular features” since these stars have low temperatures. Another possibility is that the high [X/Fe] ratios would come from “faint supernovae” explosions (see Aoki et al. 2007 and the references therein). In addition some CEMP-s stars have abnormal low or high [X/Fe] ratios of the iron-peak elements, specially Cr and Ni. This is probably because the abundances of these elements in these stars were derived from one single line of each element (Cohen et al. 2006, Aoki et al. 2007).§.§.§ The heavy-elements: CD-50776 as a new CEMP-s star In Figures 11 and 12, we show the [X/Fe] ratios for the elements created by the r- and s-process: Sr, Y, Zr, Ba, La, Ce, Pr, Nd, Sm and Eu, in CD-50776 compared to other CEMP-s stars, field giants and barium stars, including barium stars and CH stars for several metallicities.Models of galactic chemical evolution do not predict the observed overabundances of the s-process elements observed in these plots (Travaglio et al. 1999, 2004). Since CD-50776 also follows the criteria given in Masseron et al. (2010) for a star to be considered as a CEMP-s star, that is [Ba/Fe] >1.0 and [Ba/Eu] >0.0 (our results are +1.01 and +1.27, respectively, see Table 4), we can finally classify CD-50776 as a new CEMP-s star.The mean abundance ratio of the s-process elements ([Sr/Fe], [Y/Fe], [Zr/Fe], [Ba/Fe], [La/Fe], [Ce/Fe], [Nd/Fe] and [Pb/Fe]) for CD-50776 is high: +0.77.If the radial velocity variation reported in Table 6 can be attributed to orbital motion, then the atmosphere of CD-50776 could have been contaminated by an extrinsic past event like in the mass-transfer hypothesis, which is the standard scenario to explain the excess of carbon and the overabundances of the s-process elements in these chemically peculiar stars (Hansen et al. 2016).Figures 11 and 12 also show that the abundance ratios [X/Fe] of the light elements of the s-process are lower than those of the heavy elements of the s-process.This is expected based on the s-process element production according to metallicity, since the first-peak elements (such as Sr, Y and Zr) are bypassed in favor of the second and third-peak elements (Busso, Gallino & Wasserburg 1999).Other CEMP-s stars show the same behavior (Aoki et al. 2007).For the elements of the r-process, the abundance ratios of [Pr/Fe], [Sm/Fe] and [Eu/Fe] in CD-50776 is similar to other CEMP-s stars previous analyzed.In addition, the low [Eu/Fe] ratio indicates that CD-50776 is a CEMP-s star.We note that CD-50776 is also a “lead star”.Figure 13 shows the [Pb/Ce] ratio as a function of metallicity for CD-50776 compared to the CH stars (blue polygons) the CEMP-s binary stars (red circles), the barium giants (red open squares) and the subgiant CH stars (red crosses).The position of CD-50776 in this diagram, close to the CH stars and other CEMP-s stars, indicates its lead star nature.§ CONCLUSIONS Based on high-resolution optical spectroscopic data, we present the first detailed analysis of the chemical abundances of the CEMP star CD-50776, including the light elements, Na, the α-elements, the iron-peak elements, and the s-process elements. We showed that CD-50776 is characterized by an enhancement of carbon, s-process elements, and lead.This pattern, together with its low metalicity ([Fe/H] =-2.31), indicates that it is CEMP-s star. CD-50776 is also a “lead star”, since its lead-to-cerium ratio +0.97 follows the theoretical predictions for a star of this metallicity.One way to verify that CD-50776 is indeed a CEMP-s star is to use the nucleosynthesis models for AGB stars calculated by Bisterzo et al. (2010). Using the tables given in this paper, we can compare the predicted surface abundance ratios, [X/Fe], with the observed abundances. The nucleosynthesis models forecast the theoretical [X/Fe] ratios for AGB stars with initial masses of 1.3M_⊙, 1.4M_⊙, 1.5M_⊙ and 2.0M_⊙, varying the number of thermal pulses and the quantity of ^13C pocket for a metallicity [Fe/H] = - 2.6, close to the metallicity of CD-50776.Figure 14 illustrates this comparison, and shows that the best nucleosynthesis model that fits the observations is that of a star with an initial AGB mass of 1.3M_⊙ for the ST/2 case. Inspecting another fits for the CEMP-s stars investigated in Bistezro et al. (2012), we verify that the abundance pattern of CD-50776 is similar to the pattern of the CEMP-s stars CS 22964-161, CS 22880-074, CS 22942-019, CS 30301-015, HD 196944, and BS 17436-058, where the abundance of lead was also determined.These stars were classified by Bistezro et al. (2012) as CEMP-sI, which means that the ratio [hs/Fe] (defined by Bistezro et al. (2012) as the mean [X/Fe] ratio given by ([La/Fe]+[Nd/Fe]+[Sm/Fe])/3.) is less than 1.5. In fact, CD-50776 has [hs/Fe] = 0.8.However, CD-50776 presents another chemical peculiarity rarely observed in the CEMP-s stars, that is, a low abundance of nitrogen. As far as we know, this peculiarity has also been observed in the extragalactic CEMP-s star Scl-1013644 (Salgado et al. 2016). It is worth noting that the nucleosynthesis models of Bisterzo et al. (2010) predict a high abundance of carbon and nitrogen for CD-50776, which is not supported by our observations.As mentioned in Bisterzo et al. (2011), the ratios of [C/Fe],[N/Fe] and the ^12C/^13C isotopic ratios areoverestimated in AGB models where the occurrence of mixingproduced by the 'Cool Bottom Processing' (CBP) has been accountedto explain the abundances of carbon and nitrogen in CEMP-s stars.However, the efficiency of this process is difficult to estimatedue to the influence of other physical phenomena such asrotation, thermohaline mixing, and magnetic fields.On the other hand, as discussed in Section 4.1.1, the low abundance of nitrogen could be explained assuming an initial mass of 2.0M_⊙ of the donor star without the occurrence of CBP. 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M., Rasmussen, K.C., Carollo, D. 2016, , 833, 20 Zhang, L., Ishigaki, M., Aoki, W., Zhao, G. & ; Chiba, M., 2009, , 706, 1095 | http://arxiv.org/abs/1707.09039v1 | {
"authors": [
"M. Roriz",
"C. B. Pereira",
"N. A. Drake",
"F. Roig",
"J. V. Sales Silva"
],
"categories": [
"astro-ph.SR"
],
"primary_category": "astro-ph.SR",
"published": "20170727204956",
"title": "High-resolution spectroscopic observations of the new CEMP-s star CD-50$^\\circ$776"
} |
[email protected] of Physics and International Centre for Condensed Matter Physics, University of Brasilia, 70910-900, Brasilia, Federal District, BrazilDepartamento de Física, Universidad de Guadalajara, Revolución 1500, Guadalajara, Jalisco 44420, MexicoDipartimento di Fisica e Chimica, Università degli Studi di Palermo, Via Archirafi 36, I-90123 Palermo, Italy I.N.F.N. Sezione di CataniaDipartimento di Fisica e Chimica, Università degli Studi di Palermo, Via Archirafi 36, I-90123 Palermo, Italy I.N.F.N. Sezione di Catania The antidynamical Casimir effect (ADCE) is a term coined to designate the coherent annihilation of excitations due to resonant external perturbation of system parameters, allowing for extraction of quantum work from nonvacuum states of some field. Originally proposed for a two-level atom (qubit) coupled to a single cavity mode in the context of nonstationary quantum Rabi model, it suffered from very low transition rate and correspondingly narrow resonance linewidth. In this paper we show analytically and numerically that the ADCE rate can be increased by at least one order of magnitude by replacing the qubit by an artificial three-level atom (qutrit) in a properly chosen configuration. For the cavity thermal state we demonstrate that the dynamics of the average photon number and atomic excitation is completely different from the qubit's case, while the behavior of the total number of excitations is qualitatively similar yet significantly faster. 42.50.Pq, 42.50.Ct, 42.50.Hz, 32.80-t, 03.65.Yz Speeding up antidynamical Casimir effect with nonstationary qutrits B. Militello December 30, 2023 =================================================================== § INTRODUCTION The broad term dynamical Casimir effect (DCE) refers to the generation of excitations of some field (Electromagnetic, in the majority of cases) due to time-dependent boundary conditions, such as changes in the geometry or material properties of the system <cit.> (see <cit.> for reviews; see also <cit.> for the related problem of a particle in a wall with moving boundaries). In the so called cavity DCE one considers nonadiabatic (periodic or not) modulation of the cavity natural frequency by an external agent, investigating the accumulation of intracavity photons or the photon emission outside the cavity <cit.>. The additional interaction of the cavity field with a stationary `detector' during the modulation (harmonic oscillator, few-level atom or a set of two-level atoms in the simplest examples) may dramatically alter the photon generation dynamics, for instance, altering the field statistics, shifting the resonance frequency and inhibiting the photon growth <cit.> (see <cit.> for a short review). Moreover, the degree of excitation of the detector varies according to the regime of parameters, and entanglement can be created between the cavity field and the detector, or between the set of atoms coupled to the field <cit.>.Over the past ten years a new path has attracted attention of the community working on nonstationary phenomena in cavity Quantum Electrodynamics (QED). Instead of changing the cavity frequency, different studies suggested the parametric modulation of the `detector' instead, promoting it from a passive to an active agent responsible for both the generation and detection of photons <cit.>. Beside eliminating the inconvenience of time-dependent Fock states of the field associated to time-varying cavity frequency <cit.>, this scheme makes full use of the counter-rotating terms in the light–matter interaction Hamiltonian and does not require the inclusion of additional parametric down-conversion terms in the formalism <cit.>. Moreover, it benefits from recent advances in the coherent control and readout of microscopic few-level quantum devices developed in the realm of the circuit QED for applications in Quantum Information Processing (see <cit.> for a recent review).The area of circuit QED investigates the interaction of artificial superconducting atoms, formed by a sophisticated array of Josephson Junctions, and the Electromagnetic field confined in increasingly complex microwave resonators, ranging from waveguide resonators or 3D cavities <cit.>. The advances in engineering allowed for implementation of multi-level atoms, with controllable transition frequencies and coupling strengths, that can interact with multiple cavities and other atoms controlled independently <cit.>. Moreover, circuit QED allows for unprecedented atom–field coupling strength, in what became known as ultrastrong and deep strong coupling regimes <cit.>. In the context of DCE, the exquisite control over the parameters of the Hamiltonian allows for multi-tone multi-parameter modulations <cit.>, while quantum optimal control strategies can be used to enhance the desired effects <cit.>.Photon generation is not the only phenomenon induced by parametric modulations in circuit QED. It was shown recently that the counter-rotating terms can also be employed to annihilate excitations of the Electromagnetic field from nonvacuum initial states, in what became known as antidynamical Casimir effect (ADCE) <cit.>. This effect was predicted in the context of the quantum Rabi model, which describes the interaction of the cavity field with a two-level atom <cit.>, and consists in the coherent annihilation of three photons accompanied by the excitation of the far-detuned atom <cit.> (four photons could be annihilated by employing a two-tone modulation <cit.>) . Thus an amount of energy ≲ 2ħω _0 could be extracted from the system due to resonant perturbation of some parameter, where ω _0 is the cavity frequency <cit.>. However, in the more accessible regime of weak atom–field interaction (beneath the ultrastrong coupling regime) the associated transition rate is quite small, so the modulation frequency must be finely tuned and the dissipation strongly affects the behavior <cit.>.In this paper we uncover that the ADCE rate can be enhanced by almost two orders of magnitude by employing artificial three-level atoms (qutrits) in the standard ladder configuration and weak coupling regime <cit.>. We obtain closed approximate description of the unitary dynamics when one or more atomic parameters undergo a low-amplitude multi-tone external perturbation, and assess the advantages and disadvantages of different regimes of parameters for the initial thermal state of the cavity field. We also discuss eventual complications that qutrits bring into the problem, such as adjustment of atomic energy levels with respect to the cavity frequency and two-tone driving with management of the modulation phases. Nevertheless it is argued that the substantial gain in the ADCE rate compensates for the additional technical issues.This paper is organized as follows. In Sec. <ref> we define our problem and derive the general mathematical formalism to obtain approximate expressions for the system dynamics in the dressed-states basis. In Sec. <ref> we discuss three specific configurations of the qutrit for which the overall behavior is most easily inferred: the double-resonant, dispersive and mixed regimes. In Sec. <ref> we identify the regimes of parameters and the transitions for which excitations can be annihilated from the cavity thermal state, assuming that the atom was initially in the lowest energy state. In Sec. <ref> we evaluate analytically the transition rates associated to ADCE between different dressed states and compare our predictions to the exact numerical solution of the Schrödinger equation, demonstrating that the ADCE rate can undergo almost 50-fold increase compared to the qubit's case while the amount of annihilated excitations is roughly the same. Our conclusions are summarized in Sec. <ref>.§ MATHEMATICAL FORMALISM We consider a three-level artificial atom (qutrit) interacting with a single cavity mode of constant frequency ω _0, as described by the Hamiltonian (we set ħ =1)Ĥ=ω _0n̂+∑_k=1^2E_kσ̂ _k,k+∑_k=0^1G_k(â+â^†)(σ̂_k+1,k+ σ̂_k,k+1).â (â^†) is the cavity annihilation (creation) operator and n̂=â^†â is the photon number operator. The atomic eigenenergies are E_0≡ 0,E_1 and E_2, with the corresponding states denoted as |0⟩ ,|1⟩ ,|2⟩; the atomic operators read σ̂ _k,j≡ |𝐤⟩⟨𝐣|. The parameters G_k (k=0,1) stand for the coupling strengths between the atomic states {| 𝐤⟩ ,|𝐤+1⟩} mediated by the cavity field.We assume that all the atomic parameters can be modulated externally as E_k(t)≡ E_0,k + ε _E,kf_E,k(t), G_k(t)≡ G_0,k+ε _G,kf_G,k(t), where {ε _E,k,ε _G,k} are the modulation depths and {E_0,k,G_0,k} are the corresponding bare values. The dimensionless functionsf_l(t)=∑_jw_l^(j)sin( η ^(j)t+ϕ _l^(j))represent the externally prescribed modulation, where the collective index l denotes {E;k=1,2} or {G;k=0,1}. Constants 0≤ w_l^(j)≤ 1 and ϕ _l^(j) are the weight and the phase corresponding to the harmonic modulation of l with frequency η ^(j), and the index j runs over all the imposed frequencies (in this paper at most 2-tone modulations will be examined). We normalize the weights so that ∑_jw_l^(j)=1 for any set l, so that ε _l characterizes completely the modulation strength (in our examples we shall set w_l^(j)=1 and ϕ _l^(j)=0 unless stated otherwise).To obtain a closed analytical description we first rewrite the Hamiltonian as Ĥ=Ĥ_0+Ĥ_c, whereĤ_0=ω _0n̂+∑_k=0^2[ E_0,kσ̂ _k,k+G_0,k(âσ̂_k+1,k+â^†σ̂ _k,k+1)]is the bare Hamiltonian in the absence of modulation and counter-rotating terms (to shorten the formulas we defined formally G_0,2=ε _G,2=0). For the realistic weak coupling regime (G_0,0,G_0,1≪ω _0) we expand the wavefunction corresponding to the total Hamiltonian Ĥ as|ψ (t)⟩ =∑_n=0^∞∑_𝒮(n)e^-itλ _n, 𝒮A_n,𝒮(t)|φ _n,𝒮⟩ ,where λ _n,𝒮 and |φ _n,𝒮⟩ are the n-excitations eigenvalues and eigenstates (dressed states) of the Hamiltonian Ĥ_0 and the index 𝒮 labels different states with a fixed number of excitations n, which is the quantum number associated to the operator N̂=n̂+|1⟩⟨1|+2|2⟩⟨2|. As shown in Sec.<ref>, the range of values of 𝒮 depends on n, and we denote such degeneration with g(n). Moreover, the number of excitations in the subspace coincides with the number of photons of the state having the atom in its ground (| 0,n⟩).Following the approach detailed in <cit.> we propose a change of variables that maps each group of g(m) variables A_m,𝒯 into another set b_m,𝒯, so that A_m,𝒯=∑_𝒯'α_𝒯𝒯' b_m,𝒯'. In particular, we consider the following transformation:A_m,𝒯 = e^iΦ _m,𝒯(t){ e^-itν _m, 𝒯b_m,𝒯(t).-1/2i∑_𝒮(m)≠𝒯e^-itν _m,𝒮b_m,𝒮(t) ×∑_j^'∑_k=0^2∑_L=E,GΥ _m,𝒯,𝒮^L,k,j ×. ∑_r=±e^riϕ _L,k^(j)e^it(λ _m,𝒯-λ _m, 𝒮+rη ^(j))-1/λ _m,𝒯-λ _m, 𝒮+rη ^(j)}Φ _m,𝒯(t)= ∑_j∑_k=0^2∑_L=E,GΥ_m,𝒯,𝒯^L,k,j/η ^(j) ×[ cos (η ^(j)t+ϕ _L,k^(j))-cosϕ _L,k^(j)] ,where we divided the sum in two parts: ∑_j^' runs over fast frequencies η ^(j' )∼λ _m+2,𝒮-λ _m,𝒯 and ∑_j^'' runs over the slow ones η ^(j'' )∼ |λ _m,𝒮-λ _m,𝒯|. The small frequency shift ν _m,𝒯 will be given in Eq. (<ref>) and we introduced constant coefficients (k=0,1,2)Υ_m,𝒯,𝒮^E,k,j≡ε _E,kw_E,k^(j)⟨φ _m,𝒯|σ̂ _k,k|φ _m,𝒮⟩ Υ_m,𝒯,𝒮^G,k,j≡ε _G,kw_G,k^(j)⟨φ _m,𝒯|(âσ̂ _k+1,k+â^†σ̂_k,k+1)|φ _m,𝒮⟩ . After substituting A_m,𝒯 into the Schrödinger equation and systematically eliminating the rapidly oscillating terms via Rotating Wave Approximation (RWA) <cit.>, to the first order in ε _E,k and ε _G,k we obtain the approximate differential equation for the effective probability amplitudeḃ_m,𝒯 = ∑_𝒮(m)≠𝒯ς _m,𝒯,𝒮e^it(λ̃_m,𝒯-λ̃_m,𝒮)b_m,𝒮+ ∑_j^''∑_𝒮(m)≠𝒯Ξ _m,𝒯,𝒮^(j)e^itϖ _m,𝒯,𝒮 (|λ̃_m,𝒯-λ̃_m,𝒮|-η ^(j))b_m,𝒮+ ∑_j^'[ ∑_𝒮(m+2)Θ _m+2, 𝒯,𝒮^(j)e^-it(λ̃_m+2,𝒮- λ̃_m,𝒯-η ^(j))b_m+2,𝒮.. -∑_𝒮(m-2)Θ _m,𝒮,𝒯^(j)∗e^it(λ̃_m,𝒯-λ̃_m-2, 𝒮-η ^(j))b_m-2,𝒮] .The time-independent transition rates between the dressed states areς _m,𝒯,𝒮 = i∑_k,l=0^1G_0,kG_0,l{∑_ℛ(m+2)Λ _k,m+2,𝒯,ℛΛ _l,m+2,𝒮, ℛ/λ _m+2,ℛ-λ _m,𝒮.. -∑_ℛ(m-2)Λ _k,m,ℛ,𝒯Λ _l,m,ℛ,𝒮/λ _m,𝒮-λ _m-2,ℛ} Ξ _m,𝒯,𝒮^(j)=ϖ _m,𝒯, 𝒮/2∑_k=0^2∑_L=E,GΥ _m,𝒯,𝒮^L,k,je^-iϖ _m,𝒯,𝒮ϕ _L,k^(j) Θ _m+2,𝒯,𝒮^(j) = ∑_k=0^1G_0,k/2 { -ε _G,k^(j)Λ _k,m+2,𝒯, 𝒮/G_0,k.+ ∑_l=0^2∑_L=E,G[ ∑_ℛ(m+2)Λ _k,m+2,𝒯,ℛΥ _m+2,ℛ,𝒮^L,l,je^iϕ _L,l^(j)/λ _m+2,ℛ-λ _m+2, 𝒮+η ^(j)]- . . ∑_ℛ(m)Λ _k,m+2,ℛ, 𝒮Υ _m,𝒯,ℛ^L,l,je^iϕ _L,l^(j)/λ _m,𝒯-λ _m,ℛ+η ^(j)] } Λ _k,m+2,𝒯,𝒮=⟨φ _m,𝒯| âσ̂_k,k+1|φ _m+2,𝒮⟩ .Here ϖ _m,𝒯,𝒮≡sign(λ̃_m,𝒯-λ̃_m,𝒮) and we introduced the complex modulation depth ε _l^(j)≡ε _lw_l^(j)exp (iϕ _l^(j)). Moreover, we defined the corrected eigenfrequenciesλ̃_m,𝒯≡λ _m,𝒯+ν _m, 𝒯+Δν ,where the correction due to counter-rotating terms readsν _m,𝒯 = [ ∑_𝒮(m-2)( ∑_k=0^1G_0,kΛ _k,m,𝒮,𝒯) ^2/λ _m,𝒯-λ _m-2,𝒮. . -∑_𝒮(m+2)( ∑_k=0^1G_0,kΛ _k,m+2,𝒯,𝒮) ^2/λ _m+2,𝒮-λ _m,𝒯]and Δν denotes the neglected contributions smaller than ν _m, 𝒯 and the terms of the order ∼ (Υ _m,𝒯, 𝒮^L,k,j)^2/ω _0,(ε _G,kΛ _k,m,𝒮, 𝒯)^2/ω _0.Throughout the derivation of the formula (<ref>) we have assumed the constraints|λ _m,𝒯-λ _m,𝒮| ,|Υ _m,𝒯,𝒮^L,k,j|,|G_0,kΛ _l,m,𝒮,𝒯/λ _m+2,𝒯-λ _m, 𝒮| G_0,l≪ω _0 G_0,k|Λ _k,m+2,𝒮,𝒯|≲ω _0 .Under these approximations we have |A_m,𝒯|≈ |b_m, 𝒯|, so from Eq. (<ref>) one can easily infer the evolution of populations of the dressed states. Besides, the generalization of our method for N-level atoms and second-order effects is straightforward <cit.>.It is worth noting that the occurrence of ADCE is essentially governed by the transition rates Θ^(j)_m,T, S that couple states belonging to subspaces with different numbers of excitations. Of course the whole dynamics is determined also by the transitions occurring inside each subspace, but the annihilation of (two) excitations is possible only in the presence of non negligible Θ-terms. § ANALYTICAL REGIMES We shall confine ourselves to three different regimes of parameters when the dressed states have simple analytical expressions. With the aid of these formulas we shall be able to evaluate analytically the coefficients Θ _m+2,𝒯,𝒮^(j) in the section <ref>.The ground state of Ĥ_0 is |φ _0⟩ =|0 ,0⟩ and the corresponding eigenenergy is λ _0=0. In this paper we denote |𝐤,n⟩≡ |𝐤⟩ _atom⊗ |n⟩ _field, where 𝐤 stands for the atomic level and n stands for the Fock state. Moreover, we define the bare atomic transition frequencies asΩ _01 = E_0,1-E_0,0≡ω _0-Δ _1 Ω _12 = E_0,2-E_0,1≡ω _0-Δ _2 ,where Δ _1 and Δ _2 are the bare detunings. §.§ Two-level atom (2L) We include this case (G_0,1=0) to compare the advantages and disadvantages of using qutrits instead of qubits. The exact expressions for m≥ 1 read λ _m,±D=ω _0m-Δ _1/2±D β _m/2 |φ _m,±D⟩ =1/√(β _m)[ √(β _m,±)|0,m⟩±D√(β _m,∓)|1,m-1⟩] ,where β _m=√(Δ _1^2+4G_0,0^2m), β _m,±=( β _m±|Δ _1|) /2 and we introduced the detuning symbol D=+1for Δ _1≥ 0 and D=-1 for Δ _1<0.For the qutrits we can use Eqs. (<ref>) – (<ref>) for the subspace containing a single excitation, m=1; the dressed states with m≥ 2 excitations are presented below. §.§ Double-resonant regime (RR) When both G_0,0 and G_0,1 are nonzero, first we consider the special case when Δ _2=-Δ _1, so that we have the double-resonance Ω _02=E_0,2-E_0,0=2ω _0. The exact formulas read (for m≥ 2)λ _m,0=mω _0 , λ _m,±D=mω _0±Dϱ _m,∓ |φ _m,0⟩ =𝒩_m,0^-1[ -G_0,1√(m-1)| 0,m⟩ +√(m)G_0,0|2,m-2⟩] |φ _m,±D⟩ = 𝒩_m,∓^-1[ √(m)G_0,0|0,m⟩±Dϱ _m,∓| 1,m-1⟩.. +√(m-1)G_0,1|2,m-2⟩] ,where we definedϱ _m=√(Δ _1^2/4+mG_0,0^2+( m-1) G_0,1^2) ϱ _m,±=ϱ _m±|Δ _1| /2 , ϱ _m,0=√(mG_0,0^2+( m-1) G_0,1^2) 𝒩_m,0=ϱ _m,0 , 𝒩_m,±=√(2ϱ _mϱ _m,±) .For example, if G_0,1∼ G_0,0 and |Δ _1|≫ G_0,0√(n) for all relevant values of n we have approximately |φ _m,-D⟩∼ |1,m-1⟩, |φ _m,D⟩∼ (|0,m⟩ +|2,m-2⟩ )/√(2), while for |Δ _1|≪ G_0,0,G_0,1 (near the atom–field resonance) we get |φ _m,±D⟩∼ (|0,m⟩±√(2)|1,m-1⟩ +| 2,m-2⟩ )/2. §.§ Dispersive regime (DR) Now we assume that both the atomic transition frequencies are far-detuned from the cavity frequency|Δ _1|,|Δ _2|,|Δ _1+Δ _2|≫ G_0,0√(m) ,G_0,1√(m-1). From the perturbation theory we obtain to the 4th order inG_0,0/Δ _1 and G_0,1/Δ _2λ _m,0=mω _0+δ _1m[ 1+G_0,1^2(m-1)/Δ _1(Δ _1+Δ _2)-G_0,0^2m/Δ _1^2] |φ _m,0⟩ = 𝒩_m,0^-1[ |0,m⟩ + ρ _m,0G_0,0√(m)/Δ _1|1,m-1⟩.. +r_m,0G_0,0G_0,1√(m(m-1))/Δ _1(Δ _1+Δ _2)|2,m-2⟩] λ _m,1 = mω _0-Δ _1-[ δ _1m-δ _2(m-1)]×[ 1-G_0,0^2m/Δ _1^2-G_0,1^2(m-1) /Δ _2^2] |φ _m,1⟩ = 𝒩_m,1^-1[ |1 ,m-1⟩ -ρ _m,1G_0,0√(m)/Δ _1|0 ,n⟩.. +r_m,1G_0,1√(m-1)/Δ _2|2,m-2⟩] λ _m,2 = mω _0-Δ _1-Δ _2-δ _2(m-1)×[ 1+G_0,0^2m/Δ _2(Δ _1+Δ _2)- G_0,1^2(m-1)/Δ _2^2] |φ _m,2⟩ = 𝒩_m,2^-1[ |2 ,m-2⟩ -ρ _m,2G_0,1√(m-1)/Δ _2|1 ,m-1⟩.. +r_m,2G_0,0G_0,1√(m(m-1))/Δ _2(Δ _1+Δ _2)|0,m⟩] ,where we defined the dispersive shifts δ _1≡ G_0,0^2/Δ _1 and δ _2≡ G_0,1^2/Δ _2. We adopted an intuitive notation in which the second index in |φ_m,𝒮⟩ represents the most probable atomic state in a given dressed state (for example, in the expansion of |φ_m,0⟩ the bare state |0,m⟩ appears with the highest weight). The parameters ρ _m,𝒮, r_m,𝒮 and 𝒩_m,𝒮 are equal to 1 to the first order in G_0,0/Δ _1, G_0,1/Δ _2 and are summarized in <cit.>. §.§ Mixed regime (MR) In the mixed regime we assume Δ _2=0 and|Δ _1|≫ G_0,0√(n),G_0,1√(n-1),i. e., the atomic transition |1⟩→ |2⟩ is resonant with the cavity mode, while the transition |0⟩→ |1⟩ is far-detuned. To the second order in G_0,0/Δ _1 we obtainλ _m,0=mω _0+Δ _1G_0,0^2m/Δ _1^2-G_0,1^2(m-1) |φ _m,0⟩ = 𝒩_m,0^-1{ G_0,1√(m-1)ρ _m,0|2,m-2⟩.. +ρ _m,0Δ _1|1,m-1⟩ +|0,m⟩} λ _m,±D = mω _0-D( |Δ _1|∓ G_0,1√(m-1)..+1/2G_0,0^2m/|Δ _1|∓ G_0,1√(m-1)) |φ _m,±D⟩ = 𝒩_m,±^-1{ (1-r_m,±)|2,m-2⟩.. ±D(1+r_m,±)|1,m-1⟩ +ρ _m,±| 0,m⟩} ,where we definedρ _m,±=G_0,0√(m)/G_0,1√(m-1)∓|Δ _1| , ρ _m,0=G_0,0√(m)/Δ _1^2-G_0,1^2(m-1) r_m,±=1/4G_0,0^2m/G_0,1√(m-1)(G_0,1√(m-1)∓|Δ _1| ) 𝒩_m,0=√(1+ρ _n,0^2[ Δ _1^2+(m-1)G_0,1^2] ) 𝒩_m,±=√(2+2r_m,±^2+ρ _m,±^2). § ADCE Our goal is to study the coherent annihilation of system excitations from the initial separable state ρ̂_0=|0⟩⟨0|⊗ρ̂_th, where ρ̂_th=∑_m=0^∞ρ _m|m⟩⟨ m| is the cavity thermal state with ρ _m=n̅^m/( n̅+1) ^m+1. Here n̅ =( e^ωβ-1) ^-1 is the average initial photon number, β ^-1=k_BT, T is the absolute temperature and k_B is the Boltzmann's constant. From Eq. (<ref>) it is clear that such process can be implemented via transition of the form |φ _m,𝒯⟩→ |φ _m-2,𝒮⟩ when the modulation frequency is η ^(res)= λ̃_m,𝒯- λ̃_m-2,𝒮. So first we must determine the dressed states for which the initial population of the state |φ _m,𝒯 ⟩, denoted as P_m,𝒯, is larger than P_m-2, 𝒮. We assume a small integer m (for the sake of illustration we choose m=4, although the overall behavior is similar for other values of m) and set the realistic parameters G_0,0=6× 10^-2ω _0 and n̅=1.5. We verified numerically that when G_0,1 is of the same order of G_0,0 the exact value of G_0,1 does not affect qualitatively the results, so in this paper we set G_0,1=1.2G_0,0. See <cit.> for an illustration of the quantitative differences in the results when G_0,1=G_0,0 or G_0,1=0.8G_0,0.In Fig. <ref> we plot the initial population difference P( m, 𝒯,𝒮) ≡ P_m,𝒯-P_m-2,𝒮 as function of |Δ _1| for m=4. Only positive values of P( 4, 𝒯,𝒮) are plotted and the values (𝒯, 𝒮) are indicated next to the curves, where the index stands for 2-level (2L), double-resonant (r), dispersive (d) and mixed (m) regimes. In the dispersive and mixed regimes we assume |Δ _1|/G_0,0≥ 4 in order to satisfy the approximations (<ref>) and (<ref>). Besides, throughout this paper we set Δ _2=6G_0,0sign( Δ _1) in the dispersive regime so that |Δ _1+Δ _2| never approaches zero, as required by the inequality (<ref>). One can see that large detuning |Δ _1| favors the implementation of ADCE; the transitions (1,2)_d and (D,-D)_m are not particularly useful since the population differences are always small and are inversely proportional to the detuning. As already known, for a qubit the ADCE relies on the transition ( D,-D) _2L. From Fig. <ref> we discover that for a qutrit we have the following candidates for the realization of ADCE; ( D,- D) _r and ( 0,-D) _r in the double-resonant regime; ( 0,1) _d and ( 0,2) _d in the dispersive regime; ( 0,D) _m and ( 0,- D) _m in the mixed regime.Now we are in position to evaluate the ADCE rate in different regimes according to Eq. (<ref>). For the transition |φ _m,𝒯⟩→ |φ _m-2,𝒮⟩ [denoted as ( T,S)] we evaluate analytically Θ _m,𝒯, 𝒮 under the resonant modulation frequency η ^(res )=λ̃_m,𝒯-λ̃_m-2,𝒮. In Fig. <ref>a we plot the dimensionless transition rate |Θ _m, 𝒯,𝒮| /ω _0 for m=4 assuming the harmonic modulation of E_1 with perturbative amplitude ε _E,1=5× 10^-2Ω _01. We disregard the region near Δ _1=0 since P( m,𝒯,𝒮) <0 in this case, so ADCE does not occur. We observe that for the qutrit in the dispersive or mixed regimes the transition rates can be slightly higher than for the qubit; the rate for the transition (D,-D)_m is substantially higher than for the qubit, however this transition is not useful for ADCE due to small population difference P( m,D,- D). We also note that in the dispersive regime one can induce the transition |φ _m,0⟩→ |φ _m-2,2⟩ for modulation frequency η ^(res)≈ 4ω _0-Ω _02, that corresponds approximately to thefour-photon transition |0,m⟩→ |2 ,m-4⟩. However the associated transition rate is even smaller than the ADCE rate for a qubit, hindering practical applications of such process.In the dispersive regime the transition rate and the population difference for the process |φ _m,0⟩→ |φ _m-2,1⟩ [denoted as (0,1)_d in the figures] is roughly the same as the process |φ _m,D⟩→ |φ _m-2,-D⟩ for a qubit [denoted as (D,-D)_2L]. Therefore, the behavior of multi-level atoms with respect to ADCE is similar to the one for a qubit, provided all the transitions are far detuned from the cavity frequency. Moreover, for the mixed regime and large detuning |Δ _1| the population differences P( m,0,D) and P( m,0,-D) are roughly the same as for the qubit, while the transition rates are several times larger, so the implementation of ADCE would be facilitated.The main finding of the paper is the observation that in the double-resonant regime the ADCE rate is at least one order of magnitude larger than for the qubit, and the difference increases for larger |Δ _1|, as can be seen from Fig. <ref>a. Besides, in this regime the population differences P( m,D,-D) and P( m,0,-D) also increase proportionally to |Δ _1|, achieving sufficiently large values for |Δ _1|∼ 8G_0,0 (see Fig. <ref>). Thus, it seems that one could speed up ADCE by at least one order of magnitude using three-level atoms in the double-resonant configuration instead of qubits, provided the detuning |Δ _1| is large enough.In real circuit QED setups it might be tricky to modulate only one parameter at a time, while keeping the other parameters constant. So in figure <ref>b we consider the simultaneous modulation of E_1 and E_2 (with the same modulation frequency η ^(res )=λ̃_m,𝒯-λ̃_m-2,𝒮) assuming parameters ε _E,1=5× 10^-2Ω _01, ε _E,2=5× 10^-2Ω _12, ϕ _E,1=0 and ϕ _E,2=π. Conveniently the ADCE transition rates increase even more when compared to an isolated modulation of either E_1 or E_2.In <cit.> we illustrate in details the transition rates and the population differences for different values of G_0,1 and isolated modulations of E_2, G_0 and G_1. It is found that the modulation of G_0 does not speed up significantly the transition rate in comparison to a qubit, whereas the modulation of E_2 or G_1 does increase the transition rate in the double-resonant regime by at least one order of magnitude. We also verified that under the simultaneous modulation of all the parameters (E_1, E_2, G_0 and G_1) the total transition rate is still substantially higher than for a qubit, provided the phases are properly adjusted. Hence, the simultaneous modulation of several parameters is not an issue from the experimental point of view, provided one can manage the phases ϕ _l^(j) corresponding to different modulation components. § NUMERICAL VERIFICATION Now we proceed to the numerical verification of the phenomenon predicted in the previous section, namely, the enhancement of the ADCE rate in the double-resonant regime. We solved numerically the Schrödinger equation for the Hamiltonian (<ref>) using the initial local thermal state ρ̂_0=|0⟩⟨0|⊗ρ̂_th and parameters m=4, G_0,0=6× 10^-2ω _0, G_0,1=1.2G_0,0 , n̅=1.5 and Δ _1=-Δ _2=-8G_0,0. One downside of using the double-resonant regime for qutrits is clear from Fig. <ref>: both the populations differences ( 0,-D) _r and ( D,-D) _r, involved in the ADCE, are roughly twice smaller than the population difference ( D,- D) _2L for the qubit. Hence, considering the connection between ADCE and quantum thermodynamic processes recently analyzed in Ref.<cit.>, we can say that the work extraction would be half smaller if one used qutrits instead of qubits. This nuisance can be readily surpassed by employing 2-tone modulation with frequencies η ^(1)= λ̃_m,0-λ̃_m-2,-D and η ^(2)= λ̃_m,D-λ̃_m-2,-D that drives simultaneously the transitions |φ _m,0⟩→ |φ _m-2,-D⟩ and |φ _m,D⟩→ |φ _m-2,-D⟩.In figure <ref>a we illustrate the dynamics of the average photon number n_ph=⟨n̂⟩, the average number of atomic excitations n_at=⟨∑_k=1^2kσ̂_k,k⟩ and the total average number of excitations n_tot=n_ph+n_at for a qubit (setting momentarily G_1=0) with modulation depth ε _E,1=5× 10^-2Ω _01. We observe the sinusoidal oscillation of n_ph, n_at and n_tot with typical period τ≈ 4× 10^3G_0,0^-1. The coherent annihilation of excitations does take place, but since the initial population of the state |φ _4,D⟩ was P_4, D≈ 5× 10^-2, the average number of annihilated excitations is ∼ 2P_m,D≈ 0.1, in agreement with the numerical data.In figure <ref>b we consider the qutrit under 2-tone modulation of E_1 with the previous amplitude ε _E,1=5× 10^-2Ω _01, weights w_E,1^(1)=10/17,w_E,1^(2)=7/17 and phases ϕ _E,1^(1)=0, ϕ _E,1^(2)=π (the weights were adjusted to equalize the two transition rates). We see that the total number of excitation exhibits the same qualitative behavior as for the qubit, but the transition rate undergoes a 30-fold enhancement. The behavior of n_ph and n_at differs drastically from the one observed for the 2-level atom partly due to the oscillations between the bare states |0 ,k⟩↔ |2,k-2⟩ for k≥ 2, and partly due to the oscillations between the dressed states |φ _k, D⟩↔ |φ _k,0⟩, as will be discussed shortly. In figure <ref>c we consider the simultaneous two-tone modulation of E_1 and E_2 with parameters ε _E,1=5× 10^-2Ω _01, ε _E,2=9× 10^-2Ω _12, w_E,1^(1)=w_E,2^(1)=10/17, w_E,1^(2)=w_E,2^(2)=7/17 and phases ϕ _E,1^(1)=ϕ _E,2^(2)=0, ϕ _E,1^(2)=ϕ _E,2^(1)=π. We see that the ADCE rate suffers an additional 50% enhancement compared to the sole modulation of E_1, while the average number of total annihilated excitations is roughly the same as in the previous cases.Finally, in Fig. <ref> we plot the probabilities of finding the system in the dressed states P(m,𝒮)= Tr[ρ̂(t)|φ _m,𝒮⟩⟨φ _m,𝒮|] as function of time for the 2-tone double-modulation discussed in Fig. <ref>c. As predicted by Eq. (<ref>) there is a simultaneous periodic transfer of populations from the states |φ _4,D⟩ and |φ _4,0⟩ to the state |φ _2,-D⟩, which correspond to the coherent annihilation of two system excitations. Other states |φ _k≠ 2,-D⟩ are not affected by the modulation, as illustrated for the state |φ _3,-D⟩ whichundergoes just minor fluctuations due to off-resonant couplings neglected under RWA. Moreover, one also observes periodic oscillations between the dressed states |φ _k,D⟩↔ |φ _k,0⟩ for k≥ 2. This occurs because for large |Δ _1| we have λ̃_k,0≈λ̃_k,D, as seen from Eq. (<ref>), hence the first term on the RHS of Eq. (<ref>) becomes nearly resonant and couples these states with the strength ∼ |ς _k,D,0| [this behavior is due solely to the counter-rotating terms in Eq. (<ref>) and is independent of modulation].§ CONCLUSIONS In conclusion, we showed that the resonant external modulation of a three-level artificial atom is highly advantageous for the implementation of the antidynamical Casimir effect (ADCE) in comparison to a two-level atom, since the transition rate can suffer almost 50-fold increase while the total amount of annihilated excitations is roughly the same. The strongest enhancement takes place in the double-resonant regime (when Δ _1=-Δ _2, so that Ω _02=2ω _0) and for large detuning |Δ _1|, though weaker enhancement may occur also in other regimes. Beside speeding up the ADCE, the use of qutrits also loosens the requirements for accurate tuning of the modulation frequency, and reproduces the characteristic ADCE behavior of a qubit when all the atomic transitions are largely detuned from the cavity field (and Ω _02≠ 2ω _0). However, for the optimum annihilation of excitations from a thermal state the usage of qutrits also brings some inconveniences, such as two-tone driving and the necessity of controlling the phase difference between different components of the modulation. Nevertheless, our results indicate that the substantial gain in the transition rate compensates for the additional complexity in the external control, favoring the experimental implementation of ADCE.A. V. 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"authors": [
"A V Dodonov",
"J J Díaz-Guevara",
"A Napoli",
"B Militello"
],
"categories": [
"quant-ph"
],
"primary_category": "quant-ph",
"published": "20170726205629",
"title": "Speeding up antidynamical Casimir effect with nonstationary qutrits"
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Interpolation and reproducing kernels]Reproducing kernels and choices of associated feature spaces, in the form of L^2-spaces(Palle E.T. Jorgensen) Department of Mathematics, The University of Iowa, Iowa City, IA 52242-1419, U.S.A. [email protected]://www.math.uiowa.edu/~jorgen/(Feng Tian) Department of Mathematics, Hampton University, Hampton, VA 23668, [email protected] by applications to the study of stochastic processes, we introduce a new analysis of positive definite kernels K, their reproducing kernel Hilbert spaces (RKHS), and an associated family of feature spaces that may be chosen in the form L^2(μ); and we study the question of which measures μ are right for a particular kernel K. The answer to this depends on the particular application at hand. Such applications are the focus of the separate sections in the paper.[2000]Primary 47L60, 46N30, 46N50, 42C15, 65R10, 05C50, 05C75, 31C20, 60J20; Secondary 46N20, 22E70, 31A15, 58J65, 81S25, 68T05.[ Feng Tian December 30, 2023 =====================§ INTRODUCTION The use of reproducing kernels and their reproducing kernel Hilbert spaces (RKHSs) was initially motivated by problems in classical analysis, and it was put into an especially attractive and useful form by Aronszajn in the 1950ties. Since then the applications of kernel theory has greatly expanded, both in pure and applied mathematics. An application of more recent vintage is machine learning. The number of applied areas include use of RKHSs in the study of stochastic processes, especially as a tool in Ito calculus; and in machine learning (ML). The last two are related, and they are the focus of our present paper. Dictated by a number of practical applications of the theory of ML, starting with a positive definite (p.d.) kernel K, it has proved useful to study both the associated RKHS itself, as well as a variety of choices of feature spaces (for details, see p1 inside the paper); and the interplay between them.Now motivated by related applications to the study of stochastic processes, it is of special significance to focus on the cases when the family of feature spaces may be chosen in the form L^2(μ); but this then raises the question of which measures μ are right for a particular kernel K, and its associated RKHS. The answer to this depends on the particular application at hand. Such applications are the focus of the separate sections below inside the paper. In our study of RKHSs and choices of feature spaces, we have focused on those of especial relevance to analysis of Gaussian calculus, but there are many others, for example, functional and harmonic analysis, boundary value problems, PDE, geometry and geometric analysis, operator algebras/theory, the theory of unitary representations, mathematical physics, and the study of fractals and fractal measures. Even this list is not exhaustive. Nonetheless, we have narrowed our scope, and our choice of applications, for the present paper. The reader will be able to follow up on the various other directions, not covered here, with the use of our cited references, see especially our discussion of the literature below.Discussion of the literature. The theory of RKHS and their applications is vast, and below we only make a selection. Readers will be able to find more cited there. As for the general theory of RKHS in the pointwise category, we find useful <cit.>. The applications include fractals (see e.g., <cit.>); probability theory <cit.>; and learning theory <cit.>.§ REPRODUCING KERNELS The present setting begins with a fixed positive definite (p.d.) kernel K, i.e., a function K:S× S⟶ℝ where S is a set, and satisfying∑ _i∑ _jα_iα_jK(s_i,s_j)≥0for all {α_i} _1^n, { s_i} _1^n, α_i∈ℝ, s_i∈ S, and n∈ℕ. Even though we shall state our definitions and results in the special case of real valued functions, the complex case will result from our present setting with only minor modifications. But in order to minimize technical points, we have restricted the present discussion to the real case. The two more general settings are as follows: (i) complex; and (ii) operator valued. * There the definition is as in (<ref>), but now K:S× S⟶ℂ, and the p.d. assumption is instead:∑ _i∑ _jα_iα_jK(s_i,s_j)≥0for all choices of {α_i} and { s_i}, α_i∈ℂ, s_i∈ S, 1≤ i≤ n. * Let H be a complex Hilbert space, and let ℬ(H)= the algebra of all bounded linear operators in H, i.e., H⟶ H. In this case, our setting for the kernel K is: K:S× S⟶ℬ(H), and now we assume instead that, for all s_i, h_i, with s_i∈ S, h_i∈ H, and 1≤ i≤ n; and all n∈ℕ, we have: ∑ _i∑ _j⟨ K(s_i,s_j)h_i,h_j⟩ _H≥0.Given a positive definite (p.d.) kernel K on S, we shall consider pairs (F,𝐇) where 𝐇 is a Hilbert space, and F:S⟶𝐇 is a function satisfying ⟨ F(s),F(t)⟩ _𝐇=K(s,t), ∀ s,t∈ S.If (F,𝐇) satisfies this, we say that 𝐇 is a feature space, or a feature Hilbert space. In a general setup, reproducing kernel Hilbert spaces (RKHSs) were pioneered by Aronszajn in the 1950s <cit.>; and subsequently they have been used in a host of applications. The key idea of Aronszajn is that a RKHS is a Hilbert space ℋ(K) of functions f on a set such that the values f(x) are “reproduced” from f and a vector K_x in ℋ(K), in such a way that the inner product ⟨ K_x,K_y⟩=:K(x,y) is a positive definite kernel. By a theorem of Kolmogorov, every Hilbert space may be realized as a (Gaussian) reproducing kernel Hilbert space (RKHS), see e.g., <cit.>, and the details below. Let (Ω,𝒞,ℙ) be a probability space. We will be interested in centered Gaussian processes (X_s)_s∈ S (see e.g., <cit.>), indexed by S, satisfying * X_s is Gaussian w.r.t. a probability space (Ω,𝒞,ℙ),* X_s∈ L^2(Ω,ℙ), and𝔼(X_s) =0, 𝔼(X_sX_t) =K(s,t), ∀ s,t∈ S,where 𝔼 denotes the expectation with respect to ℙ. Given a p.d. kernel K, it is well known that a Gaussian realization as in (<ref>)-(<ref>) always exists; in fact, we may choose ℙ such that Ω=ℝ^S= all functions on S, 𝒞= the corresponding cylinder σ-algebra of subsets of Ω; and X_s(ω):=ω(s), ∀ω∈Ω, s∈ S. A p.d. kernel of particular interest in the present paper will be as follows: Let (V,ℬ,μ) be a measure space, where μ is assumed positive and σ-finite. Set ℬ_fin:={ A∈ℬ;μ<∞}. On ℬ_fin×ℬ_fin, then define K^(μ) by K^(μ)(A,B):=μ(A∩ B), ∀ A,B∈ℬ_fin. It is immediate that K^(μ) is p.d., and there is therefore a canonical associated centered Gaussian process X=X^(μ), indexed by ℬ_fin, satisfying 𝔼(X_A^(μ)X_B^(μ))=μ(A∩ B), ∀ A,B∈ℬ_fin.We shall study this process in detail and show that it may be used to interpolate any Markov process built on (V,ℬ); see m1. A tool in our analysis will be reproducing kernel Hilbert spaces (RKHSs). Recall that every p.d. kernel K has an associated and unique RKHS ℋ(K). The reproducing axiom is as follows: K(·,s)∈ℋ(K), and F(s)=⟨ F,K(·,s)⟩ _ℋ(K), ∀ s∈ S, for . We now present two general lemmas, applied to any p.d. kernel K:S× S⟶ℂ. Let ℋ(K) be the corresponding RKHS. A function ψ on S is in ℋ(K) ⟺ ∃ C=C_ψ<∞, a finite constant, s.t. for ∀ n∈ℕ, ∀{ s_i} _1^n, s_i∈ S, ∀{α_i} _1^n, α_i∈ℂ, we have |∑ _iα_iψ(s_i)|^2≤ C_ψ∑ _i∑ _jα_iα_jK(s_i,s_j). This is standard Aronszajn theory <cit.>. If (<ref>) holds, then we may take the constant C_ψ=‖ψ‖ _ℋ^2, and it is the smallest choice of admissible constant. Let K_1 and K_2: S× S⟶ℂ both be p.d.; and let ℋ(K_i), i=1,2, be the corresponding RKHSs. Then the following are equivalent: * ℋ(K_1)⊆ℋ(K_2);* ∃ C<∞ such that for ∀ n∈ℕ, ∀{ s_i} _1^n, {α_i} _1^n, s_i∈ S, α_i∈ℂ, we have the estimate:∑ _i∑ _jα_iα_jK_1(s_i,s_j)≤ C∑ _i∑ _jα_iα_jK_2(s_i,s_j).Stated equivalently, CK_2-K_1 is positive definite (p.d.).This follows immediately from (<ref>) and rk1.§ APPLICATION TO WHITE NOISE ANALYSIS White noise analysis serves as a versatile framework for stochastic and infinite-dimensional analysis, with a growing number of applications to neighboring areas, probability, mathematical statistics, and quantum physics. The setting is that of (Gaussian, continuous parameter) white noise — a generalized random process indexed by elements in a σ-algebra and with independent values at disjoint sets; informally, we may view it as an infinite system of coordinates on which to base an infinite-dimensional calculus. More precisely, the starting point is the L^2-space of a white noise measure (Wiener measure). A common approach makes use of a certain choice of a Gelfand triples <cit.>. Our approach is both entirely different, and it is more general. The wider aim is an infinite-dimensional differential calculus, and calculus of variation. Let (V,ℬ) be a measure space, and let μ be a σ-finite measure on (V,ℬ); then define K=K^(μ) as follows: K^(μ):ℬ_fin×ℬ_fin⟶ℝ,K^(μ)(A,B)=μ(A∩ B), ∀ A,B∈ℬ_fin,where ℬ_fin={ A∈ℬ;μ(A)<∞}. K^(μ) as in (<ref>) is positive definite. We have ∑ _i∑ _jα_iα_jK^(μ)(A_i,A_j) =‖∑ _iα_iK^(μ)(·,A_j)‖ _ℋ(K^(μ))^2 =∫|∑ _iα_i1_A_i|^2dμ≥0,for ∀{α_i} _1^n, α_i∈ℝ, ∀{ A_i} _1^n, A_i∈ℬ_fin, ∀ n∈ℕ.Let (V,ℬ,μ) be a measure space, μ assumed σ-finite (positive). Let K^(μ)(A,B):=μ(A∩ B), A,B∈ℬ_fin, be the corresponding p.d. kernel; and let ℋ(K^(μ)) be the RKHS. Then ℋ(K^(μ)) ={ F signed measures on (V,ℬ) s.t.dF≪ dμ(abs. cont) with dF/dμ∈ L^2(μ)}; and ‖ F‖ _ℋ(K^(μ))=‖dF/dμ‖ _L^2(μ).We may use rk1 to show that F, as in (<ref>), is indeed in ℋ(K^(μ)). Assume F is as specified in (<ref>); and set φ=dF/dμ (∈ L^2(μ)), which is the condition from (<ref>) on the Radon-Nikodym derivative. We will show that, if n∈ℕ, (A_i)_1^n, A_i∈ℬ_fin (i.e., μ(A_i)<∞), α_i∈ℝ, then|∑ _iα_iF(A_i)|^2≤‖φ‖ _L^2(μ)^2∑ _i∑ _jα_iα_j=μ(A_i∩ A_j)K^(μ)(A_i,A_j)and so we conclude that F∈ℋ(K^(μ)), with ‖ F‖ _ℋ(K^(μ))≤‖φ‖ _L^2(μ). It is in fact “=”. See below. We now give the verification of (<ref>): Let n, (A_i)_1^n, (α_i)_1^n, and φ:=dF/dμ ∈ L^2(μ) be as stated in (<ref>), and the discussion above; then LHS_(<ref>)=|∑ _iα_iF(A_i)|^2=|∑ _iα_i∫_A_iφ dμ|^2 =|∫_Vφ·∑ _iα_i1_A_idμ|^2(Schwarz)≤ ‖φ‖ _L^2(μ)^2‖∑ _iα_i1_A_i‖ _L^2(μ)^2 =‖φ‖ _L^2(μ)^2∑ _i∑ _jα_iα_jμ(A_i∩ A_j)which is the desired conclusion (<ref>).Every F∈ℋ(K^(μ)) is a σ-additive signed measure, i.e., if A=∪_i=1^∞A_i, A_i∩ A_j=∅, i≠ j; (sets in ℬ_fin) then F(A)=∑ _i=1^∞F(A_i). Proof of (<ref>). LHS_(<ref>) =F(A)=⟨F,μ(·∩A)⟩_ℋ(K^(μ)), μ(·∩A)=K^(μ)(·,A), =⟨F,∑_i=1^∞μ(·∩A_i)⟩_ℋ(K^(μ)),μ(·∩A_i)=K^(μ)(·,A_i), =∑_i=1^∞⟨F,K^(μ)(·,A_i)⟩_ℋ(K^(μ)) =∑_i=1^∞F(A_i).We used K^(μ)(·,A_i)⊥ K^(μ)(·,A_j) for i≠ j. For A,B∈ℬ_fin, we have dK^(μ)(·,A)/dμ=1_A=the indicator function. Proof. For A,B∈ℬ_fin, we have K^(μ)(A,B)=∫_B1_A(x)dμ(x)=μ(B∩ A),and (<ref>) follows.If F∈ℋ(K^(μ)), then dF≪ dμ where dF is the signed measure in (<ref>).Proof. We show that [μ(A)=0] ⟹ [F(A)=0]. From the reproducing property in ℋ(K^(μ)), we have:F(A)=⟨ F,K^(μ)(·,A)⟩ _ℋ(K^(μ))=⟨ F,μ(·∩ A)⟩ _ℋ(K^(μ));hence, μ(A)=0 ⟹ F(A)=0, since μ(A)=0 ⟹ K^(μ)(·,A)=0, and so F(A)=⟨ F,K^(μ)(·,A)⟩ _ℋ(K^(μ))=0. The proof of n1 is complete. Let (V,ℬ,μ) be a fixed σ-finite measure space, and let ℋ(K^(μ)) be the RKHS of K^(μ)(A,B):=μ(A∩ B), A,B∈ℬ_fin (see (<ref>)). Define W^(μ) as an isometry: W^(μ):L^2(μ)∋ f⟼ fdμ∈ℋ(K^(μ)), where W^(μ)(f)=fdμ is a signed measure on (V,ℬ); then W^(μ):L^2(μ)≃ℋ(K^(μ))is an isometric isomorphism onto ℋ(K^(μ)). Let (V,ℬ) be a measure space, μ a σ-finite measure on ℬ , and set ℬ_fin={ A∈ℬ;μ(A)<∞}. Let K be a p.d. kernel on ℬ_fin×ℬ_fin, and let ℋ(K) be the corresponding RKHS. Suppose ℋ(K) consists of signed measures; and setℋ(μ):={ F∈ℋ(K);dF≪ dμ, and dF/dμ∈ L^2(μ)} .Then ℋ(μ)⊆ℋ(K^(μ)),where K^(μ)(A,B):=μ(A∩ B), ∀ A,B∈ℬ_fin;and therefore ∃ c(μ)<∞ such that c(μ)K^(μ)-Kis positive definite. Let F∈ℋ(μ), see (<ref>); and set φ^(F):=dF/dμ. Let n∈ℕ, { A_i} _1^n, A_i∈ℬ_fin, {α_i} _1^n, α_i∈ℝ; then |∑ _iα_iF(A_i)|^2=|∑ _iα_i∫_A_iφ^(F)(x)dμ(x)|^2 =|∫_V(∑ _iα_i1_A_i)φ^(F)dμ|^2(Schwarz)≤ ∑ _i∑ _iα_iα_jμ(A_i∩ A_j)‖φ^(F)‖_L^2(μ)^2.Hence by rk1, F∈ℋ(K^(μ)), see (<ref>); and ‖ F‖ _ℋ(K^(μ))≤‖φ_L^2(μ)^(F)‖. Conclusion (<ref>) now follows. Finally conclusion (<ref>) is immediate from rk2. If V=[0,∞), ℬ= Borel σ-algebra, μ=dx=λ, Lebesgue measure, then X^(μ)= standard Brownian motion. Let A=[0,t], B=[0,s], s,t∈[0,∞), then X_A^(μ)=W_[0,t], X_B^(μ)=W_[0,s] satisfying E(X_A^(μ)X_B^(μ))=𝔼(W_[0,t]W_[0,s])=λ([0,t]∩[0,s])=s∧ t,so the standard p.d. kernel which determines Brownian motion; and d[W_0,t]_2=dt, and [W_0,t]_2=t, referring to the quadratic variation, see also qa and qv. Recall that in general, if K is a p.d. kernel, ℋ(K) the RKHS, then whenever F:S⟶ℋ is a function from S into a Hilbert space ℋ s.t. K(t,s)=⟨ F(s),F(t)⟩ _ℋ, there is then a corresponding transform L=L_F:ℋ⟶ℋ(K), given by (Lh)(t)=⟨ h,F(t)⟩ _ℋ, ∀ t∈ S, ∀ h∈ℋ.In bm1, we may apply this to this to the kernel K=K^(μ), S=ℬ_fin, K^(μ)(A,B):=μ(A∩ B), A,B∈ℬ_fin, and let F(A):=X_A^(μ), A∈ℬ_fin.Let (V,ℬ,μ) be as in bm1. Let Ω:=ℝ^ℬ_fin, and set F:ℬ_fin⟶ L^2(Ω,𝒞,ℙ),F(A):=X_A^(μ), A∈ℬ_fin,where X_A^(μ)∈ L^2(Ω,ℙ) is the centered Gaussian process with covariance kernel 𝔼(X_A^(μ)X_B^(μ))=μ(A∩ B). Then the transform L:L^2(Ω,ℙ)⟶ℋ(K^(μ)) is (Lh)(A)=𝔼(hX_A^(μ)), ∀ A∈ℬ_fin, ∀ h∈ L^2(Ω,ℙ). Let ℱ_V:= all measurable functions in (V,ℬ), and f∈ L^2(μ)⊂ℱ_V (real valued), we get the Ito-integral ∫_VfdX:=lim∑_if(s_i)X_A_i^(μ),where the limit is taken over all measurable partitions of V, mesh →0. Then 𝔼(|∫_VfdX|^2)=∫|f|^2dμ. (sketch) For all partitions { A_i} on V, s_i∈ A_i, the Ito-isometry (<ref>) follows from the approximation:𝔼(|∑ _if(s_i)X_A_i^(μ)|^2)=∑ _i|f(s_i)|^2μ(A_i)∫_V|f|^2dμ.Using this version of Ito integral, we get the following conclusions. Given a fixed σ-finite measure space (V,ℬ,μ), set K=K^(μ), K^(μ)(A,B)=μ(A∩ B), A,B∈ℬ_fin,and let L^2(Ω,𝒞,ℙ) be the corresponding probability space s.t. X_A^(μ)(ω)=ω(A), ω∈Ω=ℝ^ℬ.Then 𝔼(X_A^(μ)X_B^(μ))=K^(μ)(A,B)=μ(A∩ B),and the Ito integral X_f^(μ)=∫_VfdX is well defined with 𝔼(|X_f^(μ)|^2)=∫_V|f|^2dμ,and μ=QV(X)=[X,X]=[X]_2;see also qa and qv. The correspondence {X_A^(μ)}_A∈ℬ⟷{X_f^(μ)}_f∈ L^2(μ) is bijective. Easy direction: given X_f^(μ) as above, A∈ℬ, set f=1_A. For details on Ito calculus and Brownian motion, see, e.g., <cit.>. Given (V,ℬ,μ) fixed, σ-finite measure space, we introduce the kernel K^(μ), and the associated centered Gaussian process X:=X^(μ). From our Ito-calculus, it follows that X^(μ) may be realized in two equivalent ways:* Ω=ℝ^ℬ= all functions from ℬ into ℝ, X_A^(μ)(ω)=ω(A), A∈ℬ;* Ω=ℝ^V= all functions from V into ℝ, X_f^(μ)(ω)=ω(f), f∈ L^2(μ). From standard Kolmogorov consistency theory <cit.>, in (<ref>) the probability measure ℙ is defined on the cylinder σ-algebra of ℝ^ℬ, and in case (<ref>) it is defined on the σ-algebra for ℝ^V. We also get two equivalent versions of the covariance function for X, which is indexed by ℬ or by L^2(μ):* 𝔼(X_AX_B)=μ(A∩ B), A,B∈ℬ,⇕ * 𝔼(X_fX_g)=∫_Vf(x)g(x)dμ(x)=𝔼((∫ fdX)(∫ gdX)), ∀ f,g∈ L^2(μ), real valued, where X_f=∫ fdX is the Ito integral formula which made the link from (<ref>) to (<ref>). Let X:=X^(μ) be the Gaussian process as above, andX_f:=∫_VfdX,then 𝔼(e^iX_f)=e^-1/2∫_V|f|^2dμ, ∀ f∈ L^2(μ). Direct proof from the power series expansions. See, e.g., <cit.>, and the papers cited there. Note that (<ref>) is analogous to the Gelfand triple construction, but more general. In the present setting, we do not need a Gelfand triple in order to make process (<ref>) above. If { f_n} _n∈ℕ_0 is an orthonormal basis (ONB) in L^2(μ), then { X_f_n} _n∈ℕ_0 is an i.i.d. N(0,1) system, i.e., X_f_n=Z_n∼ N(0,1), and the following Karhunen-Loeve decomposition holds: X_A=∑_n=0^∞(∫_Af_ndμ)Z_n, ∀ A∈ℬ_fin. Assume μ is non-atomic. Then the quadratic variation of X:=X^(μ) is μ itself, i.e., if B∈ℬ, d[X,X]=dμ. Let B∈ℬ with a partition { A_i} s.t. B=∪ A_i, A_i∩ A_j=∅, i≠ j. If μ is non-atomic, then =:[X,X]=[X]_2lim∑_i(X_A_i)^2=μ(B)1where 1 denotes the constant function in L^2(Ω,ℙ), and the limit is over the set of all partitions of B with mesh tending to 0. See qv for additional details. Let f:ℝ⟶ℝ, or ℂ, f∈ C^2, then df(X_s)=f'(X_s)dX_s+1/2f”(X_s)dμ(s),or equivalently, f(X_B)=∫_Bf'(X_s)dX_s+1/2∫_Bf”(X_s)dμ(s),for ∀ B∈ℬ_fin, where we used the Ito integral, and d[X,X](s)=dμ(s) for the quadratic variation. We can do most of the white noise analysis in the more general setting, i.e., w.r.t (<ref>). Let the setting be as in n2, i.e., (V,ℬ,μ), and K are specified as in (<ref>)-(<ref>). In particular, in addition to K, we also have the μ-kernel K^(μ)(A,B)=μ(A∩ B) as in (<ref>). Let X^(K) be the centered Gaussian process with kernel K, i.e., 𝔼(X_A^(K)X_B^(K))=K(A,B), ∀ A,B∈ℬ_fin.Let X^(μ) be the Gaussian process (<ref>) with Ito integral X_f^(μ)=∫_Vf(x)dX_x^(μ), f∈ L^2(μ),and 𝔼(|X_f^(μ)|^2)=∫_V|f|^2dμ;see et. Then there is a function G:ℬ_fin× V⟶ℝ,measurable in the second variable, such thatG(A,·)∈ L^2(μ), ∀ A∈ℬ_fin;K(A,B) =∫_VG(A,x)G(B,x)dμ(x)(compare with p1 below), and X_A^(K)=∫_VG(A,x)dX_x^(μ). The existence of G follows from n2, and the Hida-Cramer transform <cit.>. Hence, by (<ref>), we may define a Gaussian process X^(K) by (<ref>); and for A,B∈ℬ_fin, we have𝔼(X_A^(K)X_B^(K)) = K(A,B)(by (<ref>))= ∫_VG(A,x)G(B,x)dμ(x),which is the desired conclusion. § GAUSSIAN INTERPOLATION OF MARKOV PROCESSES Markov models, or hidden Markov models, are ubiquitous in model building, e.g., to models for speech and handwriting recognition, to software, and learning mechanisms in biological neural networks. Within the study of support vector machines, one use of Markov processes is to solve both the problem of classification, and that of clustering. The list of optimization tasks includes that of maximizing an “expected goodness of classification,” or a “goodness of clustering” criterion. This in turn leads to the study of specific kinds of probability distribution over sequences of vectors — for which we have good parameter estimation, and good marginal distribution algorithms.Hidden Markov models tend to be robust in many uses, for example, in determining the nature of an input signal, given the corresponding an output. The model aims to determine the most probable set of parameters which dictate input states, when based on an observed sequence of output states.The literature is quite large: Here we mention just <cit.>, and the papers cited there. §.§ The Markov processes In our previous work <cit.>, we already discussed applications of the family of Gaussian processes from wn. Our present aim is to use them in an interpolation algorithm for non-atomic Markov processes. Recall the Gaussian processes {X_A^(μ);A∈ℬ_fin}, such that 𝔼(X_A^(μ)X_B^(μ))=μ(A∩ B), ∀ A,B∈ℬ_fin;where (V,ℬ,μ) is a given measure space, and μ assumed positive and σ-finite. K^(μ)[rd] μ@ >[rr][ur]X^(μ) Below we consider a family of Gaussian processes corresponding to a given Markov process P(x,A), where (V,ℬ) is a measure space, x∈ V, P(x,·) is a non-atomic probability measure, i.e., P(x,V)=1. We shall denote P as the transition operator, defined for measurable functions f on (V,ℬ), by(Pf)(x)=∫_Vf(y)P(x,dy), ∀ x∈ V.Thus P(1)=1, and the constant function 1 is harmonic. (Also see <cit.>, and the papers cited there.)Every generalized Markov process P(x,·) induces a dual pairs of actions:* action on measurable functions f on (V,ℬ),f⟼∫ f(y)P(x,dy)=(Pf)(x), x∈ V; and * action on signed measures ν on (V,ℬ),ν⟼∫ P(x,·)dν(x)=P^*(ν),where (P^*(ν))(A)=∫ P(x,A)dν(x), ∀ A∈ℬ. As in standard Markov theory, P_2(x,A)=∫ P(x,dy)P(y,A)=P[P(·,A)](x),and inductivelyP_n+1(x,A)=∫ P_n(x,dy)P(y,A). For each of the measures P(x,·),P_2(x,·)⋯,P_n(x,·), there is a corresponding white noise process X^(x), i.e., an indexed family of Gaussian processes X_A^(x) ∼ P(x,A), where 𝔼_x(X_A^(x))=0, and 𝔼_x(X_A^(x)X_B^(x))=P(x,A∩ B), ∀ A,B∈ℬ.We now introduce a more general family of Ito integrals, and get a new process W_A^(x) which has P_2(x,A) as its covariance kernel. See m1 below.Let P(x,·) and X_A^(x) be as specified above, see (<ref>)-(<ref>). Set W_A^(x):=∫_VX_A^(y)dX^(x)(y),defined as an Ito integral. Then * {W_A^(x)} is a Gaussian process;* 𝔼_x(W_A^(x))=0;* 𝔼_x(W_A^(x)W_B^(x))=P_2(x,A∩ B), ∀ A,B∈ℬ, ∀ x∈ V. * By induction, with an n-fold Ito integral from (<ref>), we get W_A^(n)(x) such that 𝔼_x(W_A^(n)(x)W_B^(n)(x))=P_n(x,A∩ B),for n∈ℕ, x∈ V, and A,B∈ℬ.Let P(x,A) and X_A^(x) be as specified above. We then form the Ito integral X_(·)^(x) with P(x,·) as covariance. Note that for every y∈ V, X^(y) is a centered Gaussian process with covariance kernel 𝔼_y(X_A^(y)X_B^(y))=P(y,A∩ B).We shall show that W_(·)^(x) is also a centered (i.e., mean zero) Gaussian process, now with P_2(x,·) as covariance measure, i.e., that W^(x) from (<ref>) will satisfy:𝔼_x(W_A^(x)W_B^(x))=P_2(x,A∩ B), ∀ A,B∈ℬ; The idea is that the white noise process interpolates the Markov process. Aside from the induction, the key step in the argument is an analysis of the Ito integral (<ref>). By general Ito theory, we have𝔼(|W_A^(x)|^2) =∫𝔼(|W_A^(y)|^2)d[X^(x),X^(x)](y) =∫ P(y,A)P(x,dy)=P_2(x,A).In the last step, we used (<ref>) on the term 𝔼(|X_A^(y)|^2) in (<ref>); and used the formula for the quadratic variationd[X^(x),X^(x)](y)=P(x,dy).See also qv below.Note that (<ref>) is a special case of an analogous property of white noise, subject to a fixed measure μ. Assume μ is non-atomic, then d[X^(μ),X^(μ)](y)=dμ(y),for x fixed. We apply this to dμ(y)=P(x,dy), and X^(x)∼ P(x,·).Give (V,ℬ,μ) as in above, and let X:=X^(μ) be the corresponding Gaussian process, centered, with covariance given by (<ref>). Let d[X,X] be the quadratic variation measure, i.e., for B∈ℬ, QV(B)=lim∑ X_A_i^2, where the limit is taken over all measurable partitions π={ A_i} of B, as mesh(π)→0. Then μ(B)=QV(B), and dμ=d[X,X]. It is true in general that if (V,ℬ,μ) is a σ-finite non-atomic measure space, and X=X^(μ) is the white noise Gaussian process determined by 𝔼(X_AX_B)=μ(A∩ B), ∀ A,B∈ℬ;then for the quadratic variation measure [X,X]=[X]_2 , we have:[X]_2(B)=μ(B),and so d[X]_2(s)=dμ(s). To see this, fix B∈ℬ, and take a limit on all measurable partitions π={ A_i}, where A_i∩ A_j, i≠ j, ∪ A_i=B, and set α_i=μ(A_i). A direct calculation gives 𝔼(X_A_i^4)=3(𝔼(X_A_i^2))^2=3α_i^2, and 𝔼(|μ(B)-∑ _iX_A_i^2|^2) =𝔼(|∑ _iμ(A_i)-∑ _iX_A_i^2|^2) =∑ _i𝔼(|μ(A_i)-X_A_i^2|^2)+∑ _i≠ j𝔼((μ(A_i)-X_A_i^2)(μ(A_j)-X_A_j^2)) = 2∑ _iμ(A_i)^2=2∑ _iα_i^20,since ∑_iα_i=μ(B)>0 is fixed. In the proof of qv, we used the following fact:Let n∈ℕ be fixed, and let α_i>0 satisfying ∑_iα_i=1, then inf_n{∑ _1^nα_i^2;∑ _1^nα_i=1} =0. Fix n∈ℕ, and apply Lagrange multiplier to get α_i=1/n, for all i. Then ∑_1^nα_i^2=1/n0. §.§ Fourier duality Let (V,ℬ,μ)=(ℝ,ℬ,μ) where μ is a probability measure on ℝ. Let X=X^(μ) be the corresponding Gaussian process, and use the Ito integral to define X_e_t=∫_ℝe_t(x)dX_x, where e_t(x)=exp(i2π xt), x,t∈ℝ.Then 𝔼(X_e_tX_e_s)=μ̂(t-s), t,s∈ℝ,where μ̂ denotes the standard Fourier transform. Direct computation using (<ref>): 𝔼(X_e_tX_e_s)=∫ e_t(x)e_s(x)d[X]_2=∫ e_t-s(x)dμ(x)where we used that d[X]_2=dμ, see qv. Then ∫_ℝ∫_ℝe_t(x)e_s(x) use orthogonality𝔼(dX_x^(μ)dX_y^(μ))=∫_ℝe_t(x)e_s(x)dμ(x).Note that if A∩ B=∅, then 𝔼(X_B^(μ)X_B^(μ))=μ(A∩ B)=μ(∅)=0.The Fourier transform X_e_t^(μ)=∫ e_t(x)dX_x^(μ) is well defined, and it is a stationary process with covariance kernelK^F(s,t)=μ̂(t-s),where μ̂ is the standard Fourier transform of the measure μ. Application. In one of our earlier papers <cit.>, and papers cited there, we studied tempered measures μ on ℝ, and processes { Y_φ} indexed by φ∈𝒮 (= the Schwartz space), and we get 𝔼(|Y_φ|^2)=∫_ℝ|φ̂(x)|^2dμ(x),or equivalently, 𝔼(e^iY_φ)=e^-1/2∫|φ̂|^2dμ.But we can recover this setting from the case (ℝ,ℬ,μ) by setting Y_φ=∫φ̂(x)dX_x^(μ)as an Ito integral. (Note that the RHS in (<ref>) is a continuous positive definite function in φ∈𝒮 (the Schwartz space), and so Minlos' theorem applies; see <cit.>.) ThenY_φ=∫φ̂(x)dX_x^(μ)=∬φ(t)e_t(x)dt dX_x^(μ) =∫φ(t)(∫e_t(x)dX_x^(μ))dt,and 𝔼(|Y_φ|^2) =𝔼(|∫φ̂(x)dX_x^(μ)|^2) =𝔼(|∫φ(t)dX_e_t^(μ)dt|^2) =∬φ(t)φ(s)𝔼(X_e_tX_e_s)dtds =∬φ(t)μ̂(t-s)φ(s)dtdsby & =∫_ℝ|φ̂(x)|^2dμ(x).This is the desired conclusion (<ref>). The idea is that we get all these conclusions without Gelfand triples. The converse holds too. Suppose {Y_φ}_φ∈𝒮 is a Gaussian process (based on ℝ) computed from a tempered measure μ, ∫_ℝdμ(x)/1+x^2<∞,with the Gelfand triple 𝒮↪ L^2(ℝ)↪𝒮'where 𝒮 is the Schwartz space, and 𝒮' the dual of all tempered distributions. Then {Y_φ}_φ∈𝒮 is the transform of the process X^(μ) ⟺𝔼(|Y_φ|^2)=∫_ℝ|φ̂(x)|^2dμ(x). Here the process X^(μ) is determined by measure μ, 𝔼(X_A^(μ)X_B^(μ)) =μ(A∩ B),∀ A,B∈ℬ,X_f^(μ)=∫ fdX^(μ)∀ f∈ L^2(ℝ,μ), and Y_φ=∫φ̂(x)dX_x^(μ)∀φ∈𝒮.Indeed, we already proved that (<ref>) ⟹ (<ref>). Note that 𝔼(|Y_φ|^2)(Ito)=∫|φ̂(x)|^2dμ(x) =∭φ(x)φ(t)e_x(s-t)dμ(x)dsdt =∬φ(s)φ(t)μ̂(s-t)dsdt.Now set X_e_t=∫ e_t(x)dX_x^(μ), then 𝔼(X_e_tX_e_s) =μ̂(s-t), and∫φ̂(x)dX_x^(μ)=∫φ(t)X_e_tdt, ∀φ∈𝒮.§.§ A stochastic bilinear form Let (V,ℬ,μ) be a fixed σ-finite measure as above, and let (Ω,𝒞,ℙ) be the measure space which realizes the process {X_A^(μ)}_A∈ℬ, and set X_f^(μ)=∫ f(x)dX_x^(μ) (Ito integral), f∈ L^2(μ). Hence 𝔼(|X_f^(μ)|^2)=∫_Ω|X_f^(μ)|^2dℙ=∫_V|f|^2dμholds by the generalized Ito isometry, and we define the transform L:H⟶ℋ(K^(μ)),(Lh)(A)=𝔼(hX_A^(μ)), ∀ A∈ℬ,where H=L^2(Ω,ℙ), and ℋ(K^(μ))= the RKHS of K^(μ)(A,B):=μ(A∩ B). But using (<ref>) again, we get 𝔼(hX_A^(μ))=∫_Vh1_Adμ=∫_Ahdμfor h∈ L^2(Ω,ℙ), and where 1_A, ∀ A∈ℬ, is the indicator function on A, i.e.,1_A(x)=δ_x(A)= 1if 0else. We have realized ℋ(K^(μ)) as a Hilbert space of functions on (V,ℬ) viz ≃ L^2(μ). Note that since K^(μ)(A,B)=μ(A∩ B) is a p.d. kernel on ℬ_fin×ℬ_fin, initially ℋ(K^(μ)) is a Hilbert space of functions on ℬ_fin, but not functions on (V,ℬ). Let (V,ℬ,μ) satisfy the axioms from above, let K^(μ) be the p.d. kernel on ℬ_fin×ℬ_fin, and ℋ(K^(μ)) be the corresponding RKHS. Let X_f^(μ), f∈ L^2(μ), be the Gaussian process which extends {X_A^(μ)}_A∈ℬ_fin, and let H=L^2(Ω,ℙ) be the Gaussian Hilbert space with inner product ⟨ h_1,h_2⟩ _H=𝔼(h_1h_2), ∀ h_i∈ H. Then the bilinear mapping L^2(μ)× H∋(f,h)⟼𝔼(X_f^(μ)h)defines two operators in duality: L^2(μ)@/^1.3pc/[rr]^I_μ H@/^1.3pc/[ll]^ξ^(μ)The Ito isometry I_μ:L^2(μ)⟶ H,I_μ(f)=X_f^(μ), ∀ f∈ L^2(μ),and the co-isometry ξ^(μ):H⟶ L^2(μ), determined by ⟨ I_μ(f),h⟩ _H=⟨ f,ξ^(μ)(h)⟩_L^2(μ), ∀ f∈ L^2(μ), ∀ h∈ H.In particular, I_μ^*=ξ^(μ), (ξ^(μ))^*=I_μ.Duality. Given (V,ℬ,μ) for σ-finite measure μ, let K^(μ)(A,B):=μ(A∩ B), A,B∈ℬ_fin where ℬ_fin:={ A∈ℬ;μ(A)<∞}. Set ℋ(K^(μ)):=the RKHS of functions on ,s.t. for all F∈ℋ(K^(μ)), F(A)=⟨ F,K_A^(μ)⟩_ℋ(K^(μ))=⟨ F,μ(A∩·)⟩ _ℋ(K^(μ)).Set L(h)(A):=⟨ h,X_A^(μ)⟩=𝔼(hX_A^(μ)), L^*(K_A^(μ))=X_A^(μ),h∈ H:=L^2(Ω,𝒞,ℙ), X_A^(μ)= the Gaussian process of K^(μ), where X_f^(μ)=∫ f(x)dX_x^(μ), f∈ L^2(μ). A new operator ξ:=ξ^(μ):H⟶ L^2(μ), where H:=L^2(Ω,𝒞,ℙ), and L^2(μ):=L^2(V,ℬ,μ). With the setting as above, ξ^(h)(h)∈ L^2(μ), ∀ h∈ H, is determined uniquely by∫ξ^(μ)(h)fdμ=𝔼(hX_f^(μ)), ∀ h∈ H, ∀ f∈ L^2(μ). ξ^(h)(h) is determined from (<ref>) and Reisz since |𝔼(hX_f^(μ))|^2≤‖ h‖ _H^2∫_V|f|^2dμ.So we only need to show the estimate (<ref>), but it follows again from the Ito isometry, as follows: Let h∈ H, and f∈ L^2(μ), then |𝔼(hX_f^(μ))|^2 (Schwarz)≤ ‖ h‖ _L^2(ℙ)^2‖ X_f^(μ)‖_L^2(ℙ) =𝔼(|h|^2)𝔼(|X_f^(μ)|^2) =𝔼(|h|^2)=:‖ f‖ _L^2(μ)^2∫_V|f|^2dμwhere we used the Ito isometry in the last step.ξ^(μ) is contractive, ‖ξ^(μ)(h)‖_L^2(μ)≤‖ h‖ _H, ∀ h∈ H. The two operators I^(μ) and ξ^(μ) are specified as follows: L^2(μ)∋f[rr]_I^(μ) X_f^(μ)@/_1.3pc/[ll]_(⋯)^*∈ H=L^2(Ω,𝒞,ℙ) L^2(μ)∋ξ^(μ)(h)@/^1.2pc/[rr]^(ξ^(μ))^* h@/^1.2pc/[ll]^ξ^(μ)∈ H =L^2(Ω,𝒞,ℙ)We have ξ^(μ)=(f⟶ X_f^(μ))^*=the adjoint operator, and f⟶ X_f^(μ)=(ξ^(μ))^*, and f⟼ X_f^(μ)=I^(μ)(f) is isometric (Ito).Moreover, (I^(μ))^*I^(μ)=I_L^2(μ), while I^(μ)(I^(μ))^*=Q_μ=projection on ;, proj on. Or, we may rewrite (<ref>)-(<ref>) as ξ^(ξ)I^(μ)=identity operator on I^(μ)ξ^(μ)= the proj inonto the range of § PARSEVAL FRAMES IN THE MEASURE CATEGORY Let U be a set, and let K:U× U⟶ℝ be a positive definite (p.d.) kernel. We assume that the corresponding RKHS ℋ(K) is separable. (The result below will apply mutatis mutandis also to complex p.d. kernels K:U× U⟶ℂ, but for simplicity, we shall state our theorem only in the real case.) Let (S,ℬ,μ) be a measure space with μ assumed positive and σ-finite. We shall say that L^2(μ) is a feature space if there is a function r:U⟶ L^2(μ) such that K(x,y)=∫_Sr_x(s)r_y(s)dμ(s), ∀ x,y∈ U.(In the complex case, the RHS in <ref> will instead be ∫_Sr_x(s)r_y(s)dμ(s). See also fs.)The notion of feature space derives from the setting of machine learning <cit.>, where learning optimization is made precise with the use of a choice of reproducing kernel Hilbert space (RKHS). In practical terms, a choice of feature space refers to a specified collections of features that used to characterize data. For example, feature space might be (Gender, Height, Weight, Age). In a support vector machine (SVM), we might want to consider a different set of characteristics in order to describe such data as (Gender, Height, Weight) etc; and we will then arrive at mappings into other feature spaces. In finite dimension, we may have feature spaces referring to some fixed number n of dimensions. Since data is “large” it is useful to consider feature spaces to be function spaces, especially a choice of L^2-spaces. The term feature space is used often in the machine learning (ML) literature because a task in ML is feature extraction. Hence we view all variables as features.We always have two distinguished feature spaces in the L^2-category: l^2(ℕ) vs Gaussian. CASE 1. S=ℕ (note the separability assumption.), and μ:= the counting measure. If { h_n} _n∈ℕ is such that K(x,y)=∑_n=1^∞h_n(x)h_n(y), ∀ x,y∈ U;then h_n∈ℋ(K) for all n∈ℕ, and { h_n} _n∈ℕ is a Parseval frame in ℋ(K), i.e., ‖ F‖ _ℋ(K)^2=∑_n=1^∞|⟨ F,h_n⟩ _ℋ(K)|^2;and F(x)=∑_n=1^∞⟨ F,h_n⟩ _ℋ(K)h_n(x)is also strongly convergent, for all x∈ U, and F∈ℋ(K). The lemma follows from standard RKHS theory <cit.>. An important point is to note that if K and { h_n} _n∈ℕ satisfy the assumptions in CASE 1, then h_n∈ℋ(K) for ∀ n∈ℕ. We may get this as an application of rk1. Indeed, let n_0∈ℕ be fixed. Let k∈ℕ, {α_i} _1^k, { x_i} _1^k, α_i∈ℝ, x_i∈ U; then |∑ _iα_ih_n_0(x_i)|^2 ≤ ∑ _n∈ℕ|∑ _iα_ih_n(x_i)|^2 =∑ _i∑ _jα_iα_j∑ _n∈ℕh_n(x_i)h_n(x_j)by = ∑ _i∑ _jα_iα_jK(x_i,x_j);and so the premise in rk1 holds, and we conclude that h_n_0∈ℋ(K). CASE 2. Set S=ℝ^U, ℬ= the cylinder σ-algebra, and μ= the Gaussian probability measure on S determined by its finite samples: k∈ℕ, { x_i} _1^k, x_i∈ U. On ℝ^k, define the standard centered Gaussian, so with mean 0, and covariance matrix (K(x_i,x_j))_i,j=1^k. Then apply Kolmogorov consistency, and μ=ℙ_Kolm(K) will be the corresponding measure, also called the Wiener measure. Setting, for x∈ U, r_x(s)=s(x);and the desired conclusions follow:* Each r_x∈ L^2(μ) is Gaussian,* 𝔼(r_x)=∫ r_xdμ=0, * 𝔼(r_xr_y)=K(x,y), ∀ x,y∈ U. Let K:U× U⟶ℝ be a positive definite (p.d.) kernel, and let (S,ℬ,μ,r) be a feature space as specified in p1; in particular, we have r_x∈ L^2(μ) , ∀ x∈ U, and K(x,y)=∫_Sr_x(s)r_y(s)dμ(s), see (<ref>).* Set R(x,A)=∫_Ar_x(s)dμ(s),for x∈ U, A∈ℬ_fin, where ℬ_fin={ A∈ℬ;μ(A)<∞} ;then R in (<ref>) is a measure in the second variable, and it is measurable in x (∈ U). * For all F∈ℋ(K), we have ‖ F‖ _ℋ(K)^2=∫_S|⟨ F(·),R(·,ds)⟩ _ℋ(K)|^2,and F(x)=∫_Sr_x(s)⟨ F(·),R(·,ds)⟩ _ℋ(K).We study the the parallel between the present conclusions (<ref>)-(<ref>), and the more familiar ones (<ref>)-(<ref>) from standard Parseval frame-theory, see, e.g., <cit.>, and also see <cit.> for direct integrals.Note first that there is a natural isometry J defined by limits and closure as follows: J(∑ _iα_iK(·,x_i))=∑ _iα_ir_x_i.Indeed for finite sample k, α_i∈ℝ, x_i∈ U, 1≤ i≤ k, we have ‖∑ _iα_iK(·,x_i)‖ _ℋ(K)^2=‖∑ _iα_ir_x_i‖ _L^2(μ)^2since both sides in (<ref>) reduce to ∑_i∑_jα_iα_jK(x_i,x_j). As a result, in order to verify (<ref>)-(<ref>), we need only consider the case F(·)=K(·,y) when y∈ U is fixed. Then it is enough to show that (<ref>) holds, and (<ref>) will follow. Let y∈ U be fixed, and assume F(·)=K(·,y). Then LHS_(<ref>)=K(y,y), and =∫_S|r_y(s)|^2dμ=K(y,y),by (<ref>). Similarly, LHS_(<ref>)=K(x,y), and RHS_(<ref>)=∫_Sr_y(s)r_x(s)dμ(s)=K(x,y),again from an application of assumption (<ref>). § TRANSFORMS Let K:U× U⟶ℝ be a positive kernel. (We shall state the result below for the real case but extensions to p.d. functions with values in ℂ are straightforward; even to the case of operator valued kernels.) Now let (S,ℬ,μ) be a measure space with μ fixed and assume σ-finite. We shall further assume that L^2(μ) is a feature space; see p1 and p1, i.e., we assume that there is a function, UL^2(μ), F(x)=r_x(·)(∈ L^2(μ)) such that (<ref>) holds. * With the setting (K,U,μ,{ r_x} _x∈ U) as above, there is then a unique isometry J:ℋ(K)⟶ L^2(μ) specified by J(K(·,x))=r_x. * The adjoint operator of J is L:=J^*:L^2(μ)⟶ℋ(K), given by (Lh)(x)=∫_Sh(s)r_x(s)dμ(s),∀ h∈ L^2(μ), and x∈ U. * Q:=JJ^*=L^*L is the projection in L^2(μ) onto the closed subspace spanned by { r_x(·);x∈ U}(⊆ L^2(μ)).(<ref>) Let the setting be as specified in the Proposition. Define J as in (<ref>) and extend by linearity, first on finite linear combinations J(∑ _iα_iK(·,x_i))=∑ _iα_ir_x_i(·).It is isometric, since ‖∑ _iα_iK(·,x_i)‖ _ℋ(K)^2=∑ _i∑ _jα_iα_jK(x_i,x_j),and ∫_S|∑ _iα_ir_x_i(·)|^2dμ=∑ _i∑ _jα_iα_j∫_Sr_x_ir_x_jdμ =∑ _i∑ _jα_iα_jK(x_i,x_j),where we used (<ref>) in the last step. Since J is thus isometric on a dense subspace in ℋ(K), it extends uniquely by limits, to define an isometry J:ℋ(K)⟶ L^2(μ) as required in (<ref>).(<ref>) We now turn to the operator L:L^2(μ)⟶ℋ(K) as in (<ref>). The important point is that L maps into ℋ(K). This follows from an application of rk1 as follows. Let n∈ℕ, α_i∈ℝ, x_i∈ U, 1≤ i≤ n; then |∑ _iα_i(Lh)(x_i)|^2=|∫_S(∑ _iα_ir_x_i)hdμ|^2Schwarz≤ ∫_S|∑ _iα_ir_x_i|^2dμ∫_S|h|^2dμby = ∑ _i∑ _jα_iα_jK(x_i,x_j)‖ h‖ _L^2(μ)^2,and the conclusion now follows from rk1.(<ref>) We have, since Lh∈ℋ(K),⟨ K(·,x),Lh⟩ _ℋ(K)=(Lh)(x)by = ∫_Sr_xhdμ=⟨ r_x,h⟩ _L^2(μ),so L^*(K(·,x))=r_x, L^*=J, and J^*=L, now follow from (<ref>). Since J is isometric, Q=JJ^* is the projection specified in (<ref>) in the Proposition. § HILBERT SPACES OF SIGNED MEASURES, AND OF DISTRIBUTIONS Let K:U× U⟶ℝ be a fixed positive definite (p.d.) kernel, and let ℋ(K) be the corresponding RKHS. Suppose U is a metric space, and that K is continuous. (This is not a strict restriction since K automatically induces a metric d_K on U given by d_K(x,y) =‖ K(·,x)-K(·,y)‖ _ℋ(K) =(K(x,x)+K(y,y)-2K(x,y))^1/2,and |K(x_1,y)-K(x_2,y)|≤ d_K(x_1,x_2)K(y,y)^1/2, ∀ x_1,x_2,y∈ U.) Introduce the Dirac delta measures {δ_x}, x∈ U, we get δ_xKδ_y=K(x,y); or more precisely, ∫_U∫_UK(s,t)dδ_x(s)dδ_y(t)=K(x,y).If n∈ℕ, α_i∈ℝ, x_i∈ U, 1≤ i≤ n, set ξ:=∑ _i=1^nα_iδ_x_i,and we get ξ Kξ=∑ _i∑ _jα_iα_jK(x_i,x_j)=‖∑ _iα_iK(·,x_i)‖ _ℋ(K)^2.Hence, if we complete the measures from (<ref>) with respect to (<ref>), we arrive at a Hilbert space ℒ(K) consisting of signed measures, or of more general linear functionals, e.g., distributions.Let K, U, ℋ(K) and ℒ(K) be specified as above; then J(∑ _iα_iδ_i):=∑ _iα_iK(·,x_i)defines an isometry of ℒ(K) onto ℋ(K) . This follows from the definitions. In particular, for ξ∈ℒ(K), we have, by (<ref>)ξ Kξ=‖ Jξ‖ _ℋ(K)^2.Let K, U, ℋ(K) and ℒ(K) be as stated in hm1, and let (S,ℬ,μ) be a σ-finite measure space such that L^2(μ) is a feature space; see p1. Then a signed measure ξ is in ℒ(K) if and only if ∫_S|∫_Ur_x(s)dξ(x)|^2dμ(s)<∞;and in this case ξ Kξ= the RHS in (<ref>).Since ℒ(K)ℋ(K) and K(·,x)⟶ r_x extends by limiting and closure to an isometry ℋ(K)⟶ L^2(μ), by tr1, the following computation is valid when ξ∈ℒ(K), and vise versa: ∫_U∫_UK(x,y)dξ(x)dξ(y) =∫_U∫_U∫_Sr_x(s)r_y(s)dμ(s)dξ(x)dξ(y) =∫_S|∫_Ur_x(s)dξ(x)|^2dμ(s)=RHS_(<ref>).If U× Uℝ(or ) is C^∞, or analytic for suitable choices of Uand K, then we may have distribution solutions ξ to ξ Kξ<∞. A simple example illustrating this is U=(-1,1)= the interval, and K(x,y)=1/1-xy.For n∈ℕ, let ξ=δ_0^(n)= the n^th derivative of the Dirac measure at x=0. Then ∫_UK(x,y)dξ(y)=n!x^n/(1-xy)^n+1|_y=0=n!x^n;and∫_U∫_UK(x,y)dξ(x)dξ(y)=(n!)^2.In fact, δ_0^(n)Kδ_0^(m)=δ_n,m(n!)^2, ∀ n,m∈ℕ;so the system {δ_0^(n)}_n∈{ 0}∪ℕ is orthogonal and total in ℒ(K). If x∈ U\{ 0}, then δ_x=∑ _n=0^∞x^n/n!δ_0^(n),as an identity for compactly supported distributions. The co-authors thank the following colleagues for helpful and enlightening discussions: Professors Daniel Alpay, Sergii Bezuglyi, Ilwoo Cho, Myung-Sin Song, Wayne Polyzou, and members in the Math Physics seminar at The University of Iowa.amsalpha | http://arxiv.org/abs/1707.08492v1 | {
"authors": [
"Palle Jorgensen",
"Feng Tian"
],
"categories": [
"math.FA",
"math.PR",
"47L60, 46N30, 46N50, 42C15, 65R10, 05C50, 05C75, 31C20, 60J20\n (Primary), 46N20, 22E70, 31A15, 58J65, 81S25, 68T05 (Secondary)"
],
"primary_category": "math.FA",
"published": "20170726152505",
"title": "Reproducing kernels and choices of associated feature spaces, in the form of $L^{2}$-spaces"
} |
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