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abstract: 'We present a catalog of GALEX Near-UV (NUV) and Far-UV (FUV) photometry for the Palomar/MSU and SDSS DR7 spectroscopic M dwarf catalogs. The catalog contains NUV measurements matched to spectroscopically confirmed M dwarfs and FUV measurements matched to spectroscopically confirmed M dwarfs. Using these data, we find that NUV and FUV luminosities strongly correlate with H$\alpha$ emission, a typical indicator of magnetic activity in M dwarfs. We also examine the fraction of M dwarfs with varying degrees of strong line emission at NUV wavelengths. Our results indicate that the frequency of M dwarf NUV emission peaks at intermediate spectral types, with at least $\sim$30% of young M4-M5 dwarfs having some level of activity. For mid-type M dwarfs, we show that NUV emission decreases with distance from the Galactic plane, a proxy for stellar age. Our complete matched source catalog is available online.'
author:
- 'David O. Jones, Andrew A. West'
title: A Catalog of GALEX Ultraviolet Emission from Spectroscopically Confirmed M Dwarfs
---
Introduction
============
The physical processes underlying the magnetic dynamos of low-mass stars are of great importance to a comprehensive understanding of the Milky Way’s stellar populations. In addition to being of interest to the understanding of stellar interiors and the creation of stellar magnetic fields, stellar activity has ramifications for the transient population of the Galaxy and the potential habitability of attending planets.
Magnetic processes in low-mass dwarfs give rise to emission from the X-ray to the radio, but at many wavelengths the strength and frequency of this emission is poorly constrained, as is its dependence on spectral type. Magnetic activity is best understood at optical wavelengths, where large spectroscopic studies such as the Sloan Digital Sky Survey [@Y00; @A09 SDSS] have enabled a statistical treatment of M dwarf activity, particularly by using H$\alpha$ emission as a diagnostic ($L_{\textrm{H}\alpha}$/$L_{\textrm{bol}}$;[@W04; @W08; @W11]). @W08 showed that activity in the optical varies as a function of both spectral type and stellar age, with a peak in the active fraction occurring near a spectral type of M8 [@R95; @H96; @G00; @W04].
The manner in which these relationships extend to magnetic activity at X-ray, ultraviolet, and radio wavelengths is not as well-determined due to the small numbers of low-mass stars that have been observed at these wavelengths. However, several recent studies have examined the relationships between H$\alpha$ and non-optical emission using data from all-sky surveys. In the X-ray, @R06 combined ROSAT data [@V99] with sources in the Two Micron All Sky Survey (2MASS; [@S06]) to identify nearby M dwarfs for spectroscopic follow-up in order to examine the relationship between X-ray and H$\alpha$ luminosity. They found that X-ray luminosity “saturates” (that there is an upper limit to the amount of quiescent X-ray emission) at a value of log($L_X$/$L_{\mathrm{bol}})\sim-3$. @S09 used emission from ROSAT as a signature of youth to build a census of low-mass stars within 25 pc. They combined the ROSAT data with spectroscopic follow-up to identify spectroscopic binaries and estimate stellar ages, surface gravities, and magnetic activity. In addition, @C08b used archival data from the Extended *Chandra* Multiwavelength Project (ChaMP; [@K04; @K07]) to identify X-ray emitting stars and calculate a linear relationship between the H$\alpha$ and X-ray fractional luminosities in M dwarfs. @C08b also observed a decrease in the magnetic activity with age. Because the work of @C08b was limited by having H$\alpha$ measurements for only $\sim$100 stars, further analysis is required to fully constrain the relationship between X-ray and H$\alpha$ emission in low-mass stars. At other wavelengths, @Mclean12 used Very Large Array (radio wavelength) observations to derive a relation between radio emission and rotation for stars with spectral types M0-M6. @Harding13 found a correlation between radio emission and optical variability for late M and early L dwarfs, although the physical mechanism responsible for this phenomenon has yet to be determined. In the ultraviolet regime, @Wal08 used *Hubble Space Telescope* (*HST*) data to observe the near-ultraviolet (NUV; 1750-250Å) emission of 33 M dwarfs. Comparing NUV to optical and X-ray emission, they did not find a universal trend between the optical and UV emission. However, they did identify a clear correlation between flux and ROSAT X-ray flux. @France13 also used *HST* to image six M dwarf exoplanet hosts in the UV to characterize the radiation field incident on the planets. They found substantial line variability on scales of $\sim$100-1000 seconds ($\sim$50-500%), Ly$\alpha$ emission ($\sim$37-75% of 1150-3100Å emission), and hot H$_2$ gas of photospheric or possibly planetary origin.
These previous studies of M dwarf emission in the UV were limited by small sample sizes that restricted their statistical significance. However, data from the Galaxy Evolution Explorer (GALEX; [@M05; @M07]) present a unique opportunity to constrain the ultraviolet properties of a much larger sample of low-mass stars, albeit using only broad photometric filters. GALEX imaged $\sim$2/3 of the sky in its NUV and far-ultraviolet (FUV; 1350-1750 Å) bands. Although the primary purpose of GALEX was to examine the UV properties of extragalactic objects, several Galactic studies have used it to measure stellar UV emission. @W07 used GALEX data in order to examine the characteristics of M dwarf flares at UV wavelengths. @S11 and @R11 used GALEX to show that ultraviolet activity is an indication of youth in low-mass stars and can be used to search for members of young moving groups in the solar neighborhood. @Shkolnik14 expanded on this analysis, showing that NUV activity as a function of age remains constant for stars younger than $\sim$300 Myr, similar to what has been observed for X-ray emission. @F11 explored the use of GALEX data as a stellar activity indicator for a calibration sample of nearby stars within 50 pc. They presented preliminary relations between an optical Ca II activity indicator (R$'_{HK}$) and the GALEX UV flux and detected evidence for a subtle correlation between NUV flux and stellar age across a variety of stellar types. @F11 did not investigate other optical activity tracers such as H$\alpha$ or the fractions of UV-active stars.
Several recent studies have also examined the GALEX UV properties of M dwarfs within $\sim$50 pc of the sun. @Stelzer13 looked at the UV and X-ray properties of M dwarfs within 10 pc to find that UV chromospheric emission is connected to other activity indicators with a power-law dependence. They find the range in activity for a given spectral type is $\sim$2-3 dex, which peaks in M4 dwarfs. @Ansdell15 used the @Lepine11 magnitude-limited catalog of $J<10$ early-type M dwarfs and found a correlation between H$\alpha$ equivalent width and NUV$-$Ks magnitude, as well as an average activity level that remains constant until $\sim$200 Myr and then declines, a result that agrees with UV studies of young moving group members [@Shkolnik14]. While @Stelzer13 found that all M dwarfs in their 10 pc sample showed NUV emission above photospheric levels, @Ansdell15 identified a population of M dwarfs showing only basal emission, the likely source of which is NUV line emission from the upper chromosphere present in many or all M dwarfs. When examining UV activity, it is likely necessary to account for this emission. We note that while @Shkolnik14 and @Ansdell15 use “saturation” to refer to the plateau of activity as a function of age for young stars, in this work we discuss saturation in the context of an upper limit to the level of emission in our most optically-active stars (e.g. [@R06])
In this study, we take an extensive look at UV activity in both nearby and more distant M dwarfs. We match broad band GALEX data to stars in the Palomar/MSU nearby star spectroscopic survey [@R95], a local sample similar to that of @Ansdell15, and the SDSS Data Release 7 M dwarf spectroscopic sample (DR7; [@W11]), which probes an older stellar population. Using these catalogs, we were able to measure the UV properties of low-mass dwarfs across a wide range of distances and spectral types. For the first time, we have examined the fraction of chromospherically-emitting M dwarfs as a function of spectral type and age, as well as the correlation between fractional NUV luminosity and fractional H$\alpha$ luminosity. These diagnostics uncover a different set of physical relations than studies using R$'_{HK}$, H$\alpha$ equivalent width, and mean NUV luminosity as a function of age.
In studying the UV emission with GALEX broadband filters, we trace a diverse group of spectral features. Figure \[fig:spec\] shows the moderately active, GALEX-detected, M dwarf spectrum of GJ 876 from @France13 to demonstrate the UV line emission features (tracers of UV activity) that fall within the GALEX NUV and FUV passbands. GJ 876 has GALEX NUV and FUV fluxes of 31.64$\pm$2.52 $\mu$Jy and 3.44$\pm$1.14 $\mu$Jy, respectively. In the FUV, the continuum emission is low, and line emission dominates the filter. In the NUV, the continuum is present, along with and features. Therefore, the GALEX NUV and FUV bands contain, and are often dominated by, tracers of stellar activity (emission lines), allowing us to expect a strong correlation between GALEX UV luminosities and magnetic activity tracers in the optical regime (§4.1). In addition, we can be confident that stars with high fractional UV luminosities must have strong line-emission present in the GALEX bands, allowing us to examine how the frequency of active stars depends on spectral type and distance from the Galactic plane (a tracer of stellar age; §4.2). GJ 876 also lacks H$\alpha$ emission or absorption, demonstrating that this moderate level of UV activity does not always correlate with optical line emission.
We describe the GALEX data along with the PMSU and SDSS DR7 M dwarf catalogs in §2. We describe the process of matching the GALEX data to the M dwarf samples and our analysis in §3. We present our results in §4, and discuss our conclusions in §5.
Data
====
The high quality photometric and spectroscopic data from SDSS over large solid angles have been useful for many large Galactic and extragalactic surveys. For low-mass stars, the SDSS has produced photometric and spectroscopic samples of more than 30 million and 70,000 M dwarfs respectively [@B10; @W11]. Low-mass stars in SDSS have been used as tracers of Galactic dynamics [@B07a; @B11; @F09] Galactic extinction [@J11] and to investigate several aspects of the stars themselves, including but not limited to, the stellar mass and luminosity functions [@B10; @C08], properties of magnetic activity [@W04; @W08], and the low-metallicity population of the Galactic disk [@Savcheva14].
To compare magnetic activity in the optical with UV activity, we matched the SDSS Data Release 7 (DR7; [@A09]) M dwarf spectroscopic catalog [@W11] to GALEX data from Data Releases 6 and 7. The SDSS spectra were observed using twin fiber-fed spectrographs that collected 640 simultaneous observations. Individual exposure times of $\sim$15-20 minutes were co-added for total exposure times of $\sim$45 minutes, producing medium resolution spectra with R $\sim$ 2000 [@Y00]. The DR7 M dwarf catalog consists of 70,841 SDSS M dwarfs with spectral types verified by eye. To measure optical stellar activity, the catalog includes (among other quantities) fractional H$\alpha$ luminosities ($L_{\textrm H\alpha}$/$L_{\textrm{bol}}$) for each star in the sample and the magnetic (H$\alpha$) activity state of each star (with “active” and “inactive” flags). Distances were measured for all stars using the photometric parallax technique described in @B10 and corrected for dust [@J11], and have an accuracy of $\sim$18%. These uncertainties are distance-independent, as they result from photometric parallaxes and are dominated by variance in the M dwarf luminosity function. See @W11 for more details on the sample selection and the value added quantities measured for the SDSS DR7 M dwarf sample.
The SDSS sample contains a large number of M dwarfs. However, the closest stars, for which we have the best chance of observing UV emission, often saturate SDSS detectors. We therefore supplemented the SDSS sample with the Palomar/MSU Nearby-Star Spectroscopic Survey (PMSU; [@R95]), which contains 1,684 nearby low-mass stars (1,415 M dwarfs) as part of the northern sample ($\delta>-30^{\circ}$) and 282 nearby low-mass stars (228 M dwarfs) as part of the southern sample. These data contain spectral types, proper motions, magnitudes, distances, and spectra for most of the stars in the sample and provide a census of young, low-mass objects in the solar neighborhood. The vast majority of stars in the sample lie within 50 pc with a median distance of $\sim$20 pc. For active stars in the sample (dMe), photometry is given by @H96. PMSU distances come from a variety of sources; many were measured by the Hipparchos Satellite, while others were derived using spectrophotometric parallax. The trigonometric parallax-based distances in this sample have uncertainties $\lesssim$5%, while the spectrophotometric parallax distances have uncertainties of up to 30%.
We matched these two catalogs to the GALEX data, which consist mainly of two large-area surveys. The All-Sky Imaging Survey (AIS) has surveyed $\sim$2/3 of the sky to a depth of NUV $m_{AB}\sim20.5$ and the Medium Imaging Survey (MIS) has imaged $\sim$1/10 of the sky to a depth of NUV $m_{AB}\sim23$. The data contain images in a near-UV band from $\sim$1750-2750 Å and a far-UV band from $\sim$1350-1750 Å. More information about the GALEX data releases can be found in @M05 and @M07 as well as at the GALEX website, part of the Mikulski Archive for Space Telescopes (MAST).
Analysis
========
To match sources in the GALEX data with the SDSS DR7 M dwarf sample [@W11], we queried the GALEX Data Release 6/7 to identify UV sources within 5 arcsec of the stellar positions. We found matches to $\sim$3,511 SDSS sources. Some SDSS stars had multiple GALEX counterparts; to eliminate chance alignments and remove multiple detections, we used the proper motions in the DR7 M dwarf sample [@M04] to determine the position for each star on a given observation date. Using these positions, we were able to eliminate most of the multiple matches, keeping only the UV source that was closest to the position of the DR7 M dwarf at the date that GALEX observed it. We removed all GALEX matches with the nuv\_artifact flag (for NUV analysis) or fuv\_artifact flag (for FUV analysis) greater than 1 and stars located more than 0.55 degrees from the center of the field of view, which are near the edge of the detector and have less reliable photometry [@M07]. Only DR7 stars that reported good photometry (the “GOODPHOT” flag set to 1; see [@W11]) and that had colors inconsistent with possible M dwarf - White dwarf binaries, were used. Stars that reported FUV flux but no NUV detection were removed, due to the fact that M dwarfs are extremely unlikely to have FUV emission without having corresponding emission in the NUV (for example, see observed NUV and FUV fluxes in [@S11]). We also used the positional errors reported by GALEX to remove GALEX sources located more than 2 arcsec away from the corresponding SDSS M dwarf position to reduce the likelihood of spurious matches.
To analyze the activity of M dwarf - White dwarf binaries, we also investigated whether any of the 495 possible pairs identified in the full DR7 M dwarf sample could be found in GALEX. Possible M dwarf - White dwarf binaries were selected from among the visually identified M dwarfs in the DR7 M Dwarf sample using the color cuts of @S04. These binaries were then verified using the method of @Morgan12, which developed color cuts from GALEX, SDSS, UKIDSS [@Lawrence07], and 2MASS [@Skrutskie06] photometry to reliably identify binaries. Approximately 1/5, or $\sim$100, of the possible WD-dM pairs could be visually verified as binaries and 75 of these positions had been observed by GALEX. Out of the 75 positions, we found matches for 42 of these pairs in GALEX after sample cuts, a *much* higher detected fraction (56%) than the fraction of SDSS M dwarfs detected overall. Increased activity in WD-dM pairs has already been observed in the optical by @Morgan12, who proposed that such an increase in activity is a result of faster stellar rotation arising due to possible tidal effects, angular momentum exchange, or disk disruption.
To match GALEX to the PMSU sample, we first corrected the PMSU positions and proper motions using the SIMBAD Astronomical Database and the @Lepine11 M dwarf catalog. SIMBAD’s astrometry originates from a variety of sources, typically 2MASS [@S06] or Hipparcos (e.g. [@Hog00]), and @Lepine11 proper motions and astrometry originate from the SUPERBLINK survey [@Lepine02]. Both are much more accurate than the original PMSU catalog. We queried the GALEX DR6/7 CasJobs database to identify UV sources within 5 arcsec of the position of each PMSU star in each year of GALEX observations (2003-2013). We kept only the closest GALEX match to a given star’s positions during the years those positions were observed by GALEX. We removed GALEX matches with the nuv\_artifact flag (for NUV analysis) or fuv\_artifact flag (for FUV analysis) greater than 1 and stars located more than 0.55 deg from the center of the field of view. Although most PMSU positions and proper motions are accurate, $\sim$20% of our sample had somewhat uncertain proper motions. In spite of this, the positional uncertainties are typically accurate, allowing us to apply the same 2 arcsec angular separation cut that we used for SDSS M dwarfs. Using the DR7 and PMSU spectra, we derived several stellar properties that were useful to quantify the UV activity. First, we used a second-order polynomial relation to determine the absolute bolometric magnitude of our M dwarfs as a function of spectral type, which we then converted to stellar luminosity. This relation was created by fitting to the 8-parsec stellar sample of @RG97. Although most of the stars in this sample are inactive, and active stars may have higher bolometric luminosities, our derived values are $\sim$35% lower than those found for the inactive early-type dwarfs in @Mann13 and agree well with the values of @Dieterich14 for late-type dwarfs. We next calculated the H$\alpha$ luminosity of the DR7 and PMSU samples following @W11, which integrates over a line region of width 8Å, and subtracts a the nearby continuum (see Table 1 and §2.2 in [@W11]).
We used the stellar distances and GALEX bandpass data to convert the GALEX NUV and FUV fluxes, measured in $\mu$Jy, to luminosities in erg s$^{-1}$. Once the H$\alpha$, NUV, and FUV luminosities were measured, we divided these quantities by each star’s bolometric luminosity in order to find the fractional luminosities emitted by each of these three components.
Removing Contaminating Sources
------------------------------
[lcccc]{} Total Unique Matches&877&265&3511&968\
\
$-$ FOV radius $>$ 0.55&91&16&501& 108\
$-$ artifact flag $>$ 1&159&28&451&191\
$-$ FUV without NUV&&11&&205\
$-$ Suspect SDSS photometry&&&207&32\
$-$ MD-WD binary&&&146&74\
$-$ $>$2 arcsec from survey position&99&26&1525&198\
$-$ $<$3$\sigma$ detection&80&56&339&37\
$-$ Binary/nearby source in the literature&77&35&&\
$-$ non-M dwarf source within 5.5 arcsec&0&0&156&66\
\
Final Sample &371&93&186&57\
There are a variety of possible contaminants of our M dwarf sample. The most likely source is contaminating background sources, as well as blended sources due to the $\sim$5.5 arcsec GALEX PSF. Another possible contaminants is unresolved binaries (such as M dwarf - White dwarf binaries).
In the SDSS sample, most close binary sources have been discovered and removed from the sample via visual spectral classification. Additionally, possible M dwarf - White dwarf binaries were removed via the method of @Morgan12, as discussed above. In PMSU, we flagged all stars with known nearby sources using Digitized Sky Survey Images in conjunction with SExtractor [@Bertin96] to identify nearby companions in the images and remove those stars from our sample. Second, we queried SIMBAD for known binary companions in the literature and removed these from our sample. In total, only 4 DSS sources had companions within 5.5 arcsec, but 57 of our previously-identified M dwarf matches had companions in the literature.
We then estimated the probability of false matches (contaminating background sources) for both samples by querying GALEX for a series of false object positions within a few arcminutes of the true positions such that our false positions would likely have been observed with the same exposure times as the true positions. For both surveys, we found a $\sim$0.5% chance of detecting a source for a given random position. In PMSU, this is only a minor concern; because over 25% of PMSU stars are detected in GALEX, the false positive fraction is $\lesssim$2% for the sample. For active (dMe) stars, the detection rate is higher, and the false positive fraction is correspondingly lower.
For SDSS, contaminating background sources are much more of a concern. Because we initially detected $\sim$1,000 sources that passed all cuts out of our 70,000 initial positions, up to 30% of our matched sample could be false positives. Although we were unable to completely alleviate this concern, we were helped by the fact that the vast majority of contaminating background sources should be detected in SDSS. We therefore removed all stars from our sample that had SDSS-detected sources within 5.5 arcsec ($\sim$1 PSF FWHM). Our random source list detected 376 random NUV matches within 1000 pc that passed all our cuts and when we removed all known SDSS sources, just 170 matches remained. Finally, when we limited our results to 3$\sigma$ detections to remove noise fluctuation (we removed $<$3$\sigma$ detections from PMSU matches as well), 26 were left (compared to 186 stars in the full sample). We found 146 spurious FUV matches, and all were removed when we removed all known SDSS sources. For the NUV, our estimated contamination is 6% for H$\alpha$-active stars and 14% for H$\alpha$ inactive stars.
After performing these steps, we believe that the correlations discussed in this paper are robust, but note that the measured fraction of emitting stars is somewhat biased towards non-detections due to the 3$\sigma$ flux threshold and strict 2 arcsec matching radius, which is within the 2$\sigma$ positional uncertainty for some GALEX sources. Table \[table:samplecuts\] lists each cut we made on our sample, and the number of sources each step removed. Our final catalog had NUV-matched DR7 sources, FUV-matched DR7 sources, NUV-matched PMSU sources, and FUV-matched PMSU sources.
Examining the Changing Fractions of UV-Emitting M Dwarfs {#section:3.1}
--------------------------------------------------------
We examined the fractions of UV-emitting M dwarfs by determining the fraction of stars in our sample that showed NUV or FUV emission above specific threshold fractional luminosities. Any UV emission greater than the expected continuum emission should be due to line emission, and therefore magnetic activity. In §4, we examine three different thresholds of UV emission that span the range of line emission levels for which there are non-zero fractions of stars both above and below the threshold, and which are not so faint that the GALEX emission is likely coming from the UV continuum. The choice of which thresholds of UV emission to use is necessarily semi-arbitrary, as broad band photometry cannot precisely detect the presence or absence of emission lines, particularly for the stars in our sample with large distance uncertainties.
For a statistical understanding of the role that magnetic activity plays in M dwarfs, the quiescent, undetectable stars are as important as the active ones. To determine the emitting fractions of M dwarfs, we first assessed the sensitivities of the different GALEX data products, which have different exposure times and magnitude limits. We made certain that a given star was not detected merely because the exposure time was not long enough. To measure whether a star with a certain UV luminosity could be detected, we identified the minimum observable UV fluxes of each field in which a M dwarf was located. We first queried CasJobs to determine whether the positions had been observed by GALEX. Out of 70,841 DR7 stars, only 8,022 had positions that had not been observed by GALEX, while 20% ($\sim$395 stars) of the PMSU sample had not been observed. For the positions that had been observed (detections and non-detections), we estimated the detection threshold as the flux uncertainty of the faintest neighboring source within 2 arcminutes. Figure \[fig:min flux hist\] shows the distribution of the minimum detectable NUV fluxes for the DR7 sample, which has two notable peaks. The larger minimum fluxes are generally sources observed in the All-Sky Imaging Survey (AIS), while the smaller minimum fluxes are generally observed with the Medium Imaging Survey (MIS) or another smaller-area GALEX survey.
For some stars, the NUV stellar continua should be detectable. An M0 dwarf at 10 pc, for example, will have a GALEX NUV continuum flux of approximately $4 \times 10^{16}$ erg s$^{-1}$ based on an average temperature of 3,800 K [@R05], which could be detected in some of our deepest fields. The PMSU sample has 57 M0-M2 dwarfs within 10 pc, so we expect a handful of GALEX-detected dwarfs that don’t have significant NUV activity. When examining the fraction of M dwarfs with line emission, using broadband photometry alone, we must be sure to avoid stars such as these.
The sensitivity of GALEX allows us to examine the fraction of M dwarfs with $L_{\mathrm{NUV}}/L_{\mathrm{bol}} > 10^{-4}$, $>5\times10^{-4}$, and $>10^{-3}$. Higher NUV luminosity limits would eliminate the majority of M dwarfs from the sample, and lower luminosity limits begin to observe the M dwarf NUV continuum. We used the GALEX detection thresholds in each field to determine which M dwarfs were detectable at each UV emission level in conjunction with a Monte Carlo sampling of the NUV fractional luminosity uncertainties, which contribute to uncertainty in both the emission and the emitting threshold of a given star. We subtracted the @Ansdell15 basal NUV emission from our detections and minimum thresholds.
In addition to these criteria, positions that were 0.55 degrees or more from the center of the GALEX field of view or positions for which the closest match reported an nuv\_artifact flag (for NUV analysis) or fuv\_artifact flag (for FUV analysis) greater than one were not used, in agreement with the criteria we set above for reliable GALEX detections. We also excluded positions whose nearest neighbor was $>$ 0.55 degrees or more from the center of the GALEX field of view, due to the fact that the reported photometry of these neighbors could be erroneous. This slightly biases the measured activity fractions (tending to raise them), due to the fact that this quality cut was applied exclusively to non-detections, but because essentially every position within 0.55 degrees of field center had a neighbor within 0.55 degrees of the field center ($\sim$99%), the effect is likely negligible. In our matched source catalog, we include GALEX measurements for stars that don’t meet any of the fractional luminosity thresholds discussed above.
Results
=======
Our final matched sample consists of NUV-matched and FUV-matched DR7 M dwarfs and NUV-matched and FUV-matched PMSU M dwarfs. Tables \[table:DR7 data\] and \[table:PMSU data\] contain the first ten rows of our matched DR7 and PMSU samples, respectively, and include object IDs, positions, spectral types, and NUV, FUV and H$\alpha$ fractional luminosities for the M dwarfs in our samples. The full catalogs can be found online.
Figure \[fig:evplot\] shows H$\alpha$-emitting, GALEX-detected M dwarfs in our matched source catalog, with $L_{\mathrm{NUV}}/L_{\mathrm{bol}}$ as a function of $R_c - I_c$ color for the DR7 M dwarfs (top) and the PMSU sample (bottom). For the SDSS stars, we converted $r - i$ to $R - I$ color using the empirical relations of @Jordi06. We computed the median $R-I$ color for each spectral type and labeled these along the top axis. Note that spectral types M7-M9 are redder than the limit of the x axis; only a handful of active stars with spectral types of M7 and later were matched to GALEX due to the lower luminosities of these stars in SDSS and the small number of late-type M dwarfs contained in PMSU. We distinguished between stars with high FUV activity (red diamonds) and low FUV activity (green stars). As expected, stars with greater FUV activity had significantly greater NUV activity (NUV flux/FUV flux correlation coefficients for the PMSU and DR7 samples are 0.977 and 0.998, respectively). Stars with undetected FUV activity (omitted for visual clarity) appear to be spread throughout the sample, likely due to the greater difficulty of M dwarf FUV detections for more distant stars and the 2009 malfunction of the FUV detector. We also found that the stars in the DR7 sample flagged as M dwarf - White dwarf binaries (from [@Morgan12]; blue triangles) tended to be significantly more NUV-luminous on average than other stars. @S11 [their Figure 3] find similar results for stars in the NStars 25 pc sample [@R07] detected in GALEX. The dashed line in Figure \[fig:evplot\] shows the approximate NUV blackbody continuum emission for these stars as a function of color, based on M dwarf temperatures from @R05 and the typical colors for M dwarfs in the DR7 and PMSU samples. Some stars appear to emit less than the approximate NUV continuum, but this is likely due to our uncertainties. For these stars, we may have detected only the UV continuum, but stellar distance and temperature uncertainties make this impossible to determine with these data. We caution that distance has a significant effect on our sample. Because stars at greater distances become increasingly difficult to detect, we found that the NUV luminosities became increasingly larger as the distances increased. This is opposite of the expected effect for an unbiased sample, which would tend to probe older populations and potentially less magnetically active stars at greater distances, as these stars would tend to lie farther from the Galactic plane (because most SDSS sight-lines are near the north Galactic cap).
Figure \[fig:nuvdist\] shows the dependence of the observed $L_{\mathrm{NUV}}/L_{\mathrm{bol}}$ for both DR7 and PMSU M dwarfs as a function of distance. In this and future figures we used the median $L_{\mathrm{NUV}}/L_{\mathrm{bol}}$ values for the stars as the central value in each distance bin and we measured errors using the dispersion around the median value divided by the square root of the number of stars in each bin. We also show a non-parametric regression using spatial averaging, with 95% confidence intervals from bootstrap resampling. We found that both the NUV and FUV fractional luminosities for the DR7 sample increased with distance, indicating the presence of a selection effect (intrinsically brighter objects are easier to see at large distances). This indicated that most of the DR7 stars detected in GALEX needed to not be only active, but highly active to be detected. However, PMSU stars show no significant trend with distance due to the small range of distances probed by the sample. Regardless, we were able to examine the relationship between UV and H$\alpha$ fractional luminosities, with the caveat that the DR7 data (and the PMSU data to a lesser extent) could only be used to probe the highly UV-luminous part of the relation. Throughout this section, we have verified that our results are consistent using a variety of distance limits to ensure that our results were not affected by spurious matches beyond several hundred parsec (we nominally restrict our sample to $<$1000 pc).
Correlating Optical and UV Activity
-----------------------------------
Aside from creating a catalog of M dwarfs with UV emission, one of the principal goals of this study was to better understand how optical activity, traced by H$\alpha$, correlates with NUV and FUV activity. To investigate this correlation, we compared the NUV and FUV emission with the H$\alpha$ flux in stars with H$\alpha$ emission. @Ansdell15 discovered a strong correlation between NUV luminosity and H$\alpha$ equivalent width for M0-M3 dwarfs. Our results include later spectral types but are qualitatively similar, with the caveat that we use H$\alpha$ fractional luminosity to probe this relation, a quantity less dependent on spectral type or color than H$\alpha$ equivalent width (used in [@Ansdell15]). We subtracted the expected basal chromospheric emission following @Ansdell15 to remove the approximate contribution of chromospheric emission lines to the UV flux. For most stars in our sample, this basal emission is well below the detection threshold.
Figure \[fig:pmsu uvhalpha\] shows a strong correlation between NUV and FUV luminosities and the H$\alpha$ luminosities of the PMSU sample. However, we do not see a correlation with DR7 stars, likely because the close proximity of PMSU stars enable intrinsically fainter UV detections; the minimum detected UV flux is 1-2 orders of magnitude lower for the PMSU sample than for the DR7 stars, demonstrating that when we examine SDSS data, only the most active stars can be detected. The PMSU sample shows a strong correlation between NUV and H$\alpha$ (Figure \[fig:pmsu uvhalpha\]; left panel) that is independent of spectral type (although the PMSU sample is largely composed of spectral types M3 and M4). This correlation is expected, as it is thought that both UV and H$\alpha$ emission are good indicators of overall stellar activity. The best fit line of the PMSU NUV to H$\alpha$ luminosity relation is:
$$\textrm{log}(L_{\mathrm{NUV}}/L_{\mathrm{bol}}) = 0.67 \times \textrm{log}(L_{\mathrm{H\alpha}}/L_{\mathrm{bol}})-0.85,$$
where the formal 1-$\sigma$ uncertainties in the slope and Y-intercept are 0.12 and 0.57, respectively.
The right panel in Figure \[fig:pmsu uvhalpha\] shows the correlation between PMSU FUV and H$\alpha$ luminosities, which is less strong than the NUV-H$\alpha$ correlation but still indicates a trend (2.2$\sigma$ significance).
The smallest DR7 NUV luminosities measured are 1-2 orders of magnitude higher than the smallest PMSU luminosities, due to the smaller distances of the PMSU stars. Additionally, we observed no correlation between DR7 UV data and H$\alpha$ emission. This may indicate an upper bound to the amount of quiescent NUV emission that occurs in very active stars, which we will refer to as “saturation”. We separated the DR7 sample into groups of similar spectral types to see if our null result was caused by the grouping of spectral types, but found no correlations, even for late spectral types ($\sim$M6$-$M9), which can be observed at smaller distances without saturating the SDSS detectors and may be able to probe lower activity levels. Figure \[fig:DR7 nuvhalpha\] shows the null correlation of H$\alpha$ with NUV luminosity for all DR7 spectral types (left). The DR7 data also predict lower H$\alpha$ luminosities for a given NUV luminosity than is seen in the PMSU sample. Again, this is likely the result of the distance-based selection effect; H$\alpha$ activity is relatively easy to measure throughout the sample, but at large distances only extremely NUV-active stars can be found.
If we restrict our sample to low NUV detection thresholds (Figure \[fig:DR7 nuvhalpha\], right panel), there are $\sim$1$\sigma$ hints of a trend, showing consistency with our PMSU results. However, the saturated M dwarfs at log(L$_{\textrm{NUV}}$/L$_{\textrm{bol}}$) $>$ -3.0 cannot be explained entirely by our 14% fraction of spurious DR7 matches or a distance bias, and likely result from saturation. In DR7 it is not possible to draw conclusions with the FUV data due to the small number of stars with measurable FUV flux.
We further demonstrate the high-luminosity behavior of NUV emission by combining the PMSU and DR7 samples for spectral types M3-M4, the most common spectral types in the combined sample, to create a single NUV - H$\alpha$ relation (Figure \[fig:comb uvhalpha\]). The low end of NUV activity consists of PMSU stars, while the high end consists mainly of DR7 stars, of which only the most active can be observed. We over-plotted the best-fit line from PMSU data. The slope of the NUV-H$\alpha$ relation shown here is consistent with the data, but tends to under-predict the high NUV luminosities, perhaps due to distance selection effects or NUV saturation. The non-parametric regression is consistently above the median bins due to the effect of highly-active DR7 stars.
{width="7in"}
The Fraction of NUV-Emitting M Dwarfs {#sec:actfrac}
-------------------------------------
Figure \[fig:DR7 lnuv actfrac\] shows the fraction of DR7 stars (left) and PMSU stars (right) emitting certain percentages of their luminosity in the NUV as a function of spectral type. These trends are meant to probe UV magnetic activity, with the caveat that while these measurements are indicative of UV activity, we are unable to directly examine if emission lines were present. We show three different $L_{\mathrm{NUV}}/L_{\mathrm{bol}}$ thresholds to show that, although the data are somewhat correlated across different threshold levels, the general trends are independent of the specific NUV emission “cut-off” levels that we choose. These values span the range of useful probes of NUV magnetic activity; if we examine stars with $L_{\mathrm{NUV}}/L_{\mathrm{bol}}$ below $10^{-4}$, it becomes clear that we are observing the stellar continuum in early-type dwarfs, as these suddenly have a large NUV-detected fraction while mid- and late-type dwarfs stay the same. If we examine only stars with $L_{\mathrm{NUV}}/L_{\mathrm{bol}} > 10^{-3}$, there are so few stars with this level of UV emission that we cannot effectively probe the relationship of spectral type to line emission. Our error bars are computed using binomial statistics. The shaded regions add binomial statistics in quadrature to errors estimated from Monte Carlo sampling of the $L_{\mathrm{NUV}}/L_{\mathrm{bol}}$ uncertainties.
Figure \[fig:DR7 lnuv actfrac\] is noisy, but both data sets show that the fractions of emitting M dwarfs appear to reach a maximum at mid or later spectral types, similar to the trend in H$\alpha$ activity, which is maximized at a spectral type of M8 [@W04]. The emitting fractions show a significant bump between spectral types M3 and M4, where M dwarfs begin to become fully convective [@W08], mirroring the increase in H$\alpha$ activity at these types. We have excluded PMSU M7-M9 dwarfs due to lack of data, and have little data with which to draw conclusions for M5 and M6 dwarfs. For early- to mid-spectral types, the NUV-emitting fractions for DR7 stars follow the same general trend as PMSU stars, peaking in M4-M6 dwarfs. The data are consistent with the H$\alpha$ peak in active fraction, but the uncertainties are too large for late spectral types to determine the most active types. The lowest threshold shows that $\sim$30-40% of mid-type M dwarfs have some level of detectable NUV activity. The true fraction of NUV-active M dwarfs is likely higher, as even our lowest threshold is above the NUV-emitting level of some moderately active M dwarfs such as GJ 876 (Figure \[fig:spec\]; $L_{\mathrm{NUV}}/L_{\mathrm{bol}} = 1.9\times10^{-5}$) and our strict selection cuts remove some matches. We examined the FUV-emitting fractions as a function of spectral type, but found that the small samples and resulting error bars were too large to draw any significant conclusions.
We examined the dependence of these NUV-emitting fractions on distance for the most active DR7 spectral types, M3-M7 (Figure \[fig:DR7 lnuv dist\]), the spectral types for which we have a significant sample. We combined multiple spectral types to increase our statistics. Figure \[fig:DR7 lnuv dist\] shows a decrease in NUV-emitting fraction with absolute vertical distance from the Galactic plane for the higher two thresholds, although the evidence is marginal for early-type stars likely due to their lower active fraction overall. Due to dynamical heating, stars gravitationally scatter as they move through the Galaxy and thus, older stars are preferentially farther from the plane [@W06; @W08]. A typical decrease in H$\alpha$ activity for the same distance from the Galactic plane is a factor of 2-4 [@W08]; for M5-M7 dwarfs, the high level of UV activity examined in this figure (green triangles/blue circles) falls off at a similar rate. This figure also confirms the results of @S11, who found that UV stellar activity tends to be an indication of youth.
![The fraction of DR7 M dwarfs emitting certain fractions of their luminosity in the NUV as a function of absolute vertical height above the Galactic plane for two subsets of the most active spectral types (M2 - M4 and M5 - M7). We show three different NUV emission thresholds: $L_{\mathrm{NUV}}/L_{\mathrm{bol}} > 10^{-4}$ (red circles), $>5\times10^{-4}$ (green stars), or $>10^{-3}$ (blue triangles). For late-type stars, the data show that the NUV-emitting fractions fall by a factor of 2-4 as distance from the plane increases; for early type stars, we see some evidence for a decrease but our data are noisier likely due to the lower average activity of these stars. This behavior is roughly consistent with that of H$\alpha$ activity, which is also shown to decrease by a factor of $\sim$2-4 over distances of 0 to 200 pc from the Galactic Plane [@W08]. Too little data are available to investigate how NUV emission in individual or less active spectral types changes with age.[]{data-label="fig:DR7 lnuv dist"}](f9.png){width="3.4in"}
Conclusions
===========
We investigated the NUV and FUV emission of low-mass stars by identifying NUV and FUV matches between GALEX and the DR7 M dwarf sample and NUV and FUV matches between GALEX and the PMSU M dwarf sample. We found evidence for clear correlations between optical (H$\alpha$), NUV, and FUV emission strength in the PMSU M dwarf sample. For PMSU, we found that the relation between the logarithm of the NUV and H$\alpha$ fractional luminosities is well fit by a line of slope $\sim$0.6. The DR7 sample demonstrates that there may be evidence that highly NUV-active stars reach a “saturation” point where increased activity at other wavelengths does not correspond to a large increase in NUV luminosity, but further work is needed to verify this.
By examining the NUV-emitting fractions of these M dwarfs (above three semi-arbitrary thresholds), we found the fractions peak at mid spectral types, with at least $\sim$30-40% of young M dwarfs having some level of activity. For the most active spectral types in the DR7 sample (M5-M7), we determined that NUV emission is also a function of stellar age due to the fact that it falls off significantly with increasing distance from the Galactic plane. This result agrees with previous studies of activity in H$\alpha$ and other wavelength regimes, and is another indication of the effect of stellar age, the strong correlation between activity at various wavelengths across the stellar spectrum, and a confirmation of the conclusion of @S11 that young low-mass stars tend to be active in the ultraviolet.
Our results indicate that NUV and H$\alpha$ activity in M dwarfs are strongly correlated and exhibit many of the same characteristics. In the absence of UV spectra for a large sample of M dwarfs, these relations can be used to constrain stellar atmosphere models and the UV radiation incident on extrasolar planets (e.g. [@Rugheimer15]). Larger UV samples of mid- and late-type M dwarfs are needed to characterize the UV-active fraction and the UV age-activity relation before UV radiation can be characterized as a function of stellar age for these stars.
We are providing our full cross matched catalog to the Astronomical community for future use at [<http://www.pha.jhu.edu/~djones/GALEX.html>]{}. Future studies will be able to use these relations and data in order to gain a more comprehensive understanding of the stellar physics of low-mass stars and the potential habitability of their attending exoplanets.
We would like to thank the anonymous referee for many helpful comments. We also thank Evgenya Shkolnik, Saurav Dhital, Dylan Morgan, John Bochanski, Suvi Gezari, and Tamas Budavari for their many contributions to this study. AAW acknowledges the support of the NASA/GALEX grant program under Cooperative Agreement No. NNX10AM62G issued through the NASA Shared Services Center. A.A.W also acknowledges funding from NSF grants AST-1109273 and AST- 1255568 and the support of the Research Corporation for Science Advancement’s Cottrell Scholarship. This study is based on observations made with the NASA Galaxy Evolution Explorer. GALEX is operated for NASA by the California Institute of Technology under NASA contract NAS5-98034.
Funding for the Sloan Digital Sky Survey (SDSS) and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, and the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web site is http://www.sdss.org/.
The SDSS is managed by the Astrophysical Research Consortium (ARC) for the Participating Institutions. The Participating Institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, University of Basel, University of Cambridge, Case Western Reserve University, The University of Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the Japan Participation Group, The Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, Ohio State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory, and the University of Washington.
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[cccccccccc]{}
587728950122053760&2411926227589794816&10:06:52.421&+01:02:28.07&22.04$\pm$0.19&1.52e-02$\pm$2.60e-03&22.36$\pm$0.24&8.81e-03$\pm$1.97e-03&3.05e-05$\pm$2.52e-05&M1\
587724648192409856&2415409480426589184&13:23:09.313&-03:23:39.34&22.23$\pm$0.21&1.74e-02$\pm$3.32e-03&22.44$\pm$0.24&1.13e-02$\pm$2.46e-03&5.36e-05$\pm$2.11e-05&M6\
587725489987321984&2415866877263743488&16:56:51.488&+60:55:32.75&23.00$\pm$0.27&3.93e-03$\pm$9.74e-04&23.01$\pm$0.28&3.02e-03$\pm$7.70e-04&-1.21e-05$\pm$8.42e-06&M2\
587725578034413696&2415972430377917440&17:25:52.453&+63:29:06.62&21.73$\pm$0.10&1.89e-02$\pm$1.78e-03&&&-7.73e-06$\pm$7.15e-06&M2\
587731185135976704&2417450174007744512&00:53:19.471&-00:51:43.71&23.19$\pm$0.30&7.17e-03$\pm$1.96e-03&23.52$\pm$0.42&4.11e-03$\pm$1.58e-03&-4.17e-05$\pm$1.92e-05&M3\
587724198819921920&2419174208237998080&01:50:16.365&+14:05:28.68&23.17$\pm$0.31&2.11e-04$\pm$6.02e-05&&&1.91e-04$\pm$6.11e-05&M2\
588009368547033216&2421496376795863040&09:11:55.458&+54:01:26.79&20.45$\pm$0.04&3.11e-02$\pm$1.28e-03&&&-4.64e-05$\pm$1.33e-05&M0\
587724242309415168&2423853729725811712&04:08:57.875&-04:41:09.30&23.02$\pm$0.26&1.38e-02$\pm$3.32e-03&&&-1.68e-04$\pm$8.31e-05&M3\
588007005258907776&2431594291587456000&14:54:44.247&+59:27:53.81&22.45$\pm$0.23&3.77e-03$\pm$7.99e-04&23.59$\pm$0.26&1.02e-03$\pm$2.42e-04&1.56e-05$\pm$4.77e-06&M7\
587729228228067456&2468819357257443328&16:22:09.325&+50:07:52.51&19.10$\pm$0.02&5.06e-02$\pm$8.87e-04&18.72$\pm$0.03&5.55e-02$\pm$1.32e-03&1.53e-04$\pm$3.75e-05&M0\
\[table:DR7 data\]
[cccccccccc]{}
Gl 2&6372076894527948800&00:05:10.888&+45:47:11.64&18.73$\pm$0.06&4.70e-05$\pm$1.41e+00&&&-2.66e-06$\pm$9.61e+05&M1\
V351&6375982386675452928&00:08:27.279&+17:25:27.47&19.31$\pm$0.06&4.54e-05$\pm$5.66e+00&21.56$\pm$0.29&4.44e-06$\pm$5.66e+00&1.66e-06$\pm$1.74e+06&M0\
G131-026&6375982351239873536&00:08:53.919&+20:50:25.24&18.57$\pm$0.07&6.57e-04$\pm$4.24e+01&&&7.46e-05$\pm$3.60e+08&M4\
G131-047&6375982355536939008&00:16:56.779&+20:03:55.10&21.00$\pm$0.18&1.37e-04$\pm$4.24e+01&22.74$\pm$0.47&2.14e-05$\pm$4.24e+01&9.90e-06$\pm$1.82e+08&M3\
LHS1054&6376721218758771712&00:17:20.323&+29:10:58.81&20.64$\pm$0.25&7.24e-05$\pm$8.48e+00&&&2.67e-06$\pm$1.23e+07&M2\
G158-052&6380310079474761728&00:17:40.903&-08:40:56.15&19.22$\pm$0.06&1.27e-04$\pm$1.13e+01&21.52$\pm$0.27&1.20e-05$\pm$1.13e+01&7.66e-07$\pm$3.49e+06&M0\
Gl 16&6380978543887648768&00:18:16.589&+10:12:10.03&20.16$\pm$0.15&3.65e-05$\pm$4.24e+00&&&-2.14e-06$\pm$4.23e+06&M1\
G130-063&6376721273519606784&00:18:53.530&+27:48:50.00&21.27$\pm$0.21&1.31e-04$\pm$4.24e+01&&&-1.31e-05$\pm$2.57e+08&M4\
GJ 2003&6380415634736415744&00:20:08.380&-17:03:40.97&20.98$\pm$0.18&1.96e-05$\pm$9.90e+00&&&-3.24e-06$\pm$4.55e+06&M0\
LP149-56&6372076865539016704&00:21:57.81&+49:12:38.00&19.34$\pm$0.08&3.48e-04$\pm$4.10e+01&20.91$\pm$0.21&6.39e-05$\pm$4.10e+01&5.34e-05$\pm$8.61e+07&M2\
\[table:PMSU data\]
|
---
abstract: 'We introduce the notion of a continuous Schauder frame for a Banach space. This is both a generalization of continuous frames and coherent states for Hilbert spaces and a generalization of unconditional Schauder frames for Banach spaces. As a natural example, we prove that any wavelet for $L_p(\R)$ with $1<p<\infty$ generates a continuous wavelet Schauder frame. Furthermore, we generalize the properties shrinking and boundedly complete to the continuous Schauder frame setting, and prove that many of the fundamental James theorems still hold in this general context.'
address:
- |
Department of Mathematics\
University of Virginia\
Charlottesville, VA 22901 USA
- |
Department of Mathematics and Statistics\
St Louis University\
St Louis MO 63103 USA\
and\
Department of Mathematics\
Duke University\
Durham NC 27708 USA
author:
- Joseph Eisner
- Daniel Freeman
title: Continuous Schauder frames for Banach spaces
---
[^1]
[^2]
Introduction {#S:Intro}
============
In Hilbert spaces, frames and orthonormal bases give discrete ways to represent vectors using series, and continuous frames and coherent states give continuous ways to represent vectors using integrals. A [*frame*]{} for a Hilbert space $H$ is a collection of vectors $(x_j)_{j\in J}\subset H$ for which there exists constants $0\leq A\leq B$ such that for any $x\in H$, $A\|x\|^2\leq \sum_{j\in J}|\langle x, x_j\rangle|^2\leq
B\|x\|^2$. Given any frame $(x_j)_{j\in J}$ for a Hilbert space $H$, there exists a frame $(f_j)_{j\in J}$ for $H$, called a [*dual frame*]{}, such that $$\label{E:0}
x=\sum_{j\in J} \langle x, f_j\rangle x_j\quad \textrm{ for all $x\in H$}.$$ The equality in (\[E:0\]) allows the reconstruction of any vector $x$ in the Hilbert space from the sequence of coefficients $(\langle
x, f_j\rangle)_{j\in J}$. Continuous frames and coherent states are a generalization of frames, in that instead of summing over a discrete set, we integrate over a measure space. Coherent states were invented by Schrödinger in 1926 [@S] and were generalized to continuous frames by Ali, Antoine, and Gazeau [@AAG1]. The short time Fourier transform and the continuous wavelet transform are two particularly important examples of continuous frames. Let $(M, \Sigma, \mu)$ be a $\sigma$-finite measure space and let $H$ be a separable Hilbert space. A measurable function $\psi:M\rightarrow H$ is a [*continuous frame*]{} of $H$ with respect to $\mu$ if there exists constants $A,B>0$ such that for all $x\in H$, $A\|x\|^2\leq \int_M |\langle x, \psi(t)\rangle|^2 d\mu(t)\leq B \|x\|^2$. If $A=B=1$ then the continuous frame is called a [*coherent state*]{}. As is the case with frames, any continuous frame may be used to reconstruct vectors using a dual frame. That is, if $\psi:M\rightarrow H$ is a continuous frame, then there exists a [*dual frame*]{} $\phi:M\rightarrow H$ such that $$\label{E:c0}
x=\int_M \langle x, \phi(t)\rangle \psi(t) d\mu(t)\quad \textrm{ for all $x\in H$}.$$ Equation involves integrating vectors in a Hilbert space, and is defined weakly using the Pettis Integral. We will define the Pettis Integral and discuss it further in Section \[S:CSF\].
Frames for Hilbert spaces have been generalized to Banach spaces in multiple ways, such as atomic decompositions [@FG1], Banach frames [@G], framings [@CHL], and Schauder frames [@CDOSZ]. These particular methods are based on extending the reconstruction formula to Banach spaces. Given a Banach space $X$ with dual $X^*$, a sequence of pairs $(x_j,f_j)_{j=1}^\infty\subseteq X\times X^*$ is called a [*Schauder frame*]{} of $X$ if $$\label{E:SF}
x=\sum_{j=1}^\infty f_j(x) x_j\quad \textrm{ for all $x\in X$}.$$ Thus, Schauder frames are direct generalizations of the reconstruction formula for frames in Hilbert spaces. A Schauder frame is called [*unconditional*]{} or a [*framing*]{} if the series in converges in any order. Though coherent states and continuous frames in Hilbert spaces have long been studied and play important roles in mathematical physics and harmonic analysis, the natural extension to continuous Schauder frames has only been considered for certain Banach spaces using coorbit theory [@FG1][@FG2][@FG3][@FoR] and for complemented subspaces of $L_p$ using p-frames [@FO]. Given a Banach space $X$ with dual $X^*$ and a $\sigma$-finite measure space $(M, \Sigma, \mu)$, we call a measurable function $t\mapsto(x_t,f_t)\in X\times X^*$ a [*continuous Schauder frame*]{} of $X$ if for all $x\in X$, $$\label{E:cSFdef}
x=\int_M f_t(x) x_t d\mu(t).$$ As with continuous frames for Hilbert spaces, the integral in Equation involves integrating vectors and is defined weakly using the Pettis Integral which we define in Section \[S:CSF\]. Unlike series, there is no order for integration, and so all continuous Schauder frames are by necessity unconditional. In the case that the measure space $(M,\mu)$ is simply the natural numbers with counting measure, then $(x_n, f_n)_{n\in\N}$ is a continuous Schauder frame if and only if it is an unconditional Schauder frame. Thus, continuous frames are indeed generalizations of unconditional Schauder frames.
Many coherent states and continuous frames are of the form $(S^a f)_{a\in \Lambda}$ or $(S^aT^b f)_{(a,b)\in \Lambda}$ for some isometries $S$ and $T$ and continuous ring $\Lambda$. Many important bases and frames can be obtained by sampling the continuous frame at a discrete sub-lattice of $\Lambda$. For example, wavelet bases and Gabor frames can be obtained by sampling the continuous wavelet transform and short time Fourier transform respectively. Discrete frames and bases are well suited for computations, but the continuous frames they are sampled from are often very useful to work with directly themselves. For example, if one translates a function in $L_2(\R)$ then the resulting continuous wavelet frame coefficients are simply shifted by the amount translated, however if one translates a function by a non-integer value then the resulting discrete wavelet coefficients are completely different. In Section \[S:ex\] we show that any discrete wavelet for $L_p(\R)$ for $1<p<\infty$ gives rise to a continuous wavelet Schauder frame and thus much of the analysis that is done for the continuous wavelet transform in Hilbert spaces may be done in the Banach space setting as well.
We also consider how to generalize other properties of Schauder frames to continuous Schauder frames. In [@CL] and [@L], they define the properties shrinking and boundedly complete for Schauder frames and prove that many of James’ classic theorems [@Ja] on shrinking and boundedly complete Schauder bases can be extended to Schauder frames. In particular, a Schauder frame $(x_j,f_j)_{j=1}^\infty$ of $X$ is shrinking if and only if $(f_j,x_j)_{j=1}^\infty$ is a Schauder frame of $X^*$, and if $(x_j,f_j)_{j=1}^\infty$ is shrinking and boundedly complete then $X$ is reflexive. On the other hand, in [@BFL] they prove that every infinite dimensional Banach space which has a Schauder frame also has a non-shrinking Schauder frame, so unlike for Schauder bases the converse of the previous theorem for Schuader frames does not hold. In [@CLS], they prove that an unconditional Schauder frame is shrinking if and only if the Banach space does not contain $\ell_1$, and that an unconditional Schauder frame is boundedly complete if and only if the Banach space does not contain $c_0$. In Section \[S:sbc\] we define what it means for a continuous Schauder frame to be shrinking or boundedly complete, and we prove that the previous stated theorems are true for continuous Schauder frames as well. Sections \[S:ex\] and \[S:ex2\] are devoted to constructing examples of continuous Schauder frames in $L_p(\R)$ and $\ell_p$ for $1<p<\infty$. Our final section proposes an open problem on when continuous Schauder frames may be sampled to obtain a discrete Schauder frame.
The Pettis Integral and Continuous Schauder frames {#S:CSF}
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The main concept of the Pettis integral is to integrate vector valued functions by considering the Lebesgue integrals of the real valued functions formed by composing with linear functionals. This method allows one to transfer many of the fundamental properties of Lebesgue integration to the Banach space setting.
Let $(M, \Sigma, \mu)$ be a $\sigma$-finite measure space and $X$ a separable Banach Space. A vector-valued function $F: M \rightarrow X$ is said to be $\mu$-*Pettis integrable* (or *Pettis integrable*, or merely *integrable* if context is understood) if for any $E \in \Sigma$ there exists $x_E \in X$ such that $f(x_E) = \int_E f(F) d\mu$ for all $f \in X^{*}$ (where this latter integral is Lebesgue). Then we say $\int_E F d\mu = x_E$ and, in particular, $\int F d\mu = x_M$.
If the vector valued map takes values in a dual space $X^*$ then one can instead consider just using the weak\*-continuous linear functionals.
Let $(M, \Sigma, \mu)$ be a $\sigma$-finite measure space and $X$ a separable Banach Space with dual $X^{*}$. A functional-valued function $G: M \rightarrow X^{*}$ is said to be $\mu$-*Pettis\* integrable* (or *Pettis \*integrable*, or merely *\*integrable* if context is understood) if for any $E \in \Sigma$ there exists $f_E \in X^{*}$ such that $f_E(x) = \int_E G(x) d\mu$ for all $x \in X$. Then we say $\int_E^{*} G d\mu = f_E$ and, in particular, $\int^{*} G d\mu = f_M$.
Recall that we use the Pettis integral to define continuous Schauder frames, and we will use the Pettis\* integral to define continuous\* Schauder frames.
Given a separable Banach space $X$ with dual $X^*$ and a $\sigma$-finite measure space $(M, \Sigma, \mu)$, a measurable function $t\mapsto(x_t,f_t)\in X\times X^*$ is called a [*continuous Schauder frame*]{} of $X$ if for all $x\in X$, $$\label{E:cSF}
x=\int_M f_t(x) x_t d\mu(t).$$ The dual map $t\mapsto(f_t,x_t)\in X^*\times X$ is called a [*continuous\* Schauder frame*]{} of $X$ if for all $f\in X$, $$\label{E:cSF}
f=\int^*_M f(x_t) f_t d\mu(t).$$
A sequence of vectors $(x_j)_{j=1}^\infty$ is a Schauder basis for a Banach space $X$, if and only if the biorthogonal functionals $(x^*_j)_{j=1}^\infty$ are a $w^*$-Schauder basis for $X^*$. That is, for all $f\in X^*$, the series $\sum_{j=1}^\infty f(x_j) x^*_j$ converges $w^*$ to $f$. We will prove in Lemma \[L:DualFrame1\] that if $(x_t,f_t)_{t\in M}$ is a continuous Schauder frame of $X$ then $(f_t,x_t)_{t\in M}$ is a continuous\* Schauder frame of $X^*$. This relationship will be useful for us when using duality techniques. However, Example \[E:notCSF\] shows that the converse does not always hold for continuous Schauder frames.
By the definition of the Pettis integral, $t\mapsto(x_t,f_t)\in X\times X^*$ is a continuous Schauder frame of $X$ if and only if for all $x\in X$, $f\in X^*$, and $E\in \Sigma$ there exists $x_E\in X$ such that $x=x_X$ and $$\label{E:cSFP}
f(x_E)=\int_E f_t(x) f(x_t) d\mu(t).$$ We are primarily interested in the representation of $x$ as $x=\int_M f_t(x) x_t d\mu(t)$, and so it may feel tedious to check Equation for all measurable sets $E\in \Sigma$. However, the following example shows that it is necessary to check for all $E\in \Sigma$ and not just $E=M$.
\[E:notCSF\] Let $(e_j)_{j\in\N}$ be the unit vector basis for $c_0$ with biorthogonal functionals $(e_j^*)_{j\in\N}$. Consider the following sequence of pairs in $c_0\times \ell_1$, $$(x_n,f_n)_{n=1}^\infty=(e_1,e^*_1),(e_1,-e^*_1),(e_1,e^*_1),(e_2,e_2^*),(e_2,-e_1^*),(e_2,e_1^*),(e_3,e_3^*),(e_3,-e_1^*),(e_3,e_1^*),...$$ If we consider $\N$ with counting measure, then for all $x\in c_0$ and $f\in \ell_1$, $f(x)=\int_\N f_n(x) f(x_n)$. However, $(x_n,f_n)_{n\in\N}$ is not a continuous Schauder frame. Indeed, let $x=e_1$ and suppose $x_{3\N}\in c_0$ is such that for all $f\in \ell_1$, $f(x_{3\N})=\int_{3\N}f_n(e_1)f(x_n)$. Then for all $n\in\N$, $e_n^*(x_{3\N})=1$. Thus, $x_{3\N}=(1,1,1,...)\not\in c_0$ which is a contradiction.
What is particularly interesting about Example \[E:notCSF\] is that although $(x_n,f_n)_{n\in\N}$ is not a continuous Schauder frame of $c_0$, we do have that the dual $(f_n, x_n)_{n\in\N}$ is a continuous\* Schauder frame of $\ell_1$. Indeed, if $f\in \ell_1$ and $E\subseteq \N$ then $$f_E=\sum_{n\in M\cap(3\N-2)} f(e_{(n+2)/3})e_{(n+2)/3}^*+\sum_{n\in M\cap 3\N} f(e_{n/3})e_1^*-\sum_{n\in M\cap (3\N-1)} f(e_{(n+1)/3})e_1^*.$$ As $f\in\ell_1$, we have that all the above series converge in norm to an element of $\ell_1$. Thus, we have that it is possible to have a continuous\* Schauder frame $(f_t, x_t)_{t\in M}$ for a dual space $X^*$ such that $(x_t, f_t)_{t\in M}$ is not a continuous Schauder frame for $X$.
The following lemma shows that any continuous Schauder frame or continuous\* Schauder frame satisfies an unconditionality inequality.
\[L:suppression\] Let either $(x_t,f_t)_{t\in M}\in X\times X^*$ be a continuous Schauder frame of a Banach space $X$ or $(f_t,x_t)_{t\in M}\in X^*\times X$ be a continuous\* Schauder frame for $X^*$. Then, there exists a constant $B>0$ such that for every measurable set $E$ in $M$ and every $x\in X$ and $f\in X^*$ we have that $$|f(x_E)|=|f_E(x)|=\left|\int_E f_t(x) f(x_t) d\mu(t)\right|\leq B \|x\|\|f\|$$
Suppose that either $(x_t,f_t)\in X\times X^*$ is a continuous Schauder frame of $X$ or $(f_t,x_t)\in X^*\times X$ is a continuous\* Schauder frame for $X^*$. In either case, we have that $|f(x)|=|\int f_t(x) f(x_t) d\mu(t)|\leq \int |f_t(x) f(x_t)| d\mu(t)<\infty$ for all $x\in X$ and $f\in X^*$. Thus, the linear map, $(x,f)\mapsto f_t(x) f(x_t)$ is pointwise bounded from $X\times X^*$ to $L_1(\mu)$ and is hence uniformly bounded. Thus, there exists $B>0$ such that $\int |f_t(x) f(x_t)| d\mu(t) \leq B \|x\|\|f\|$ for all $(x,f)\in X\times X^*$. In particular, for all measurable $E$ we have, $|\int_E f_t(x) f(x_t) d\mu(t)|\leq \int |f_t(x) f(x_t)| d\mu(t) \leq B \|x\|\|f\|$.
Lemma \[L:suppression\] gives that for every continuous Schauder frame $(x_t,f_t)\in X\times X^*$ there exists a constant $B_{s}>0$ such that for every measurable set $E$ we have that $\|x_E\|\leq B_s\|x\|$. The least constant $B_s$ to satisfy Lemma \[L:suppression\] is called the [*suppression unconditionality constant*]{} of $(x_t,f_t)_{t\in M}$. Likewise, the [*unconditionality constant*]{} of $(x_t,f_t)_{t\in M}$ is the least constant $B_u$ to satisfy $\int |f_t(x) f(x_t)| d\mu(t)\leq B_u|\int f_t(x) f(x_t) d\mu(t)|$ for all $x\in X$ and $f\in X^*$. Just like unconditional bases, these constants satisfy $B_s\leq B_u\leq 2B_s$.
As duality techniques are ubiquitous in the theory of Banach spaces, it will be important to determine the relationship between a map $t\mapsto (x_t,f_t) \in X\times X^*$ and the dual map $t\mapsto (f_t,x_t) \in X^*\times X$. The following lemma states that if $(x_t,f_t)_{t\in M}$ is a continuous Schauder frame of $X$ then $(f_t,x_t)_{t\in M}$ is a continuous\* Schauder frame of $X^*$. In Section \[S:sbc\] we will characterize when $(f_t,x_t)_{t\in M}$ is a continuous Schauder frame of $X^*$ and not just a continuous\* Schauder frame.
\[L:DualFrame1\] Let $X$ be a separable Banach Space and let $t\mapsto(x_t,f_t)\in X\times X^*$ be a measurable map from a $\sigma$-finite measure space $M$ to $X\times X^*$. If $(x_t, f_t)_{t \in M}$ is a continuous Schauder frame for $X$ then the *dual frame* $(f_t, x_t)_{t \in M}$ is a continuous\* Schauder Frame for $X^{*}$.
We assume that $(x_t,f_t)_{t\in M}$ is a continuous Schauder frame of $X$. Fix $f \in X^{*}$ and let $E$ be measurable. We have by Lemma \[L:suppression\] that the map $x\mapsto \int_E f_t(x) f(x_t) d\mu(t)$ defines a bounded linear functional on $X$. Thus, there exists $f_E\in X^*$ such that $f_E(x)=\int_E f_t(x) f(x_t) d\mu(t)$ for all $x\in X$. Furthermore, $f(x)=\int f_t(x) f(x_t) d\mu(t)$ for all $x\in X$ and hence $f_M=f$ and $(f_t, x_t)_{t \in M}$ is a continuous\* Schauder Frame for $X^{*}$.
Shrinking and boundedly complete continuous Schauder frames {#S:sbc}
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The properties shrinking and boundedly complete give very nice structural results for Schauder bases. The properties are extended to atomic decompositions and Schauder frames in [@CL], [@CLS] and [@L], and they prove that many of the fundamental James Theorems for bases extend to Schauder frames. The goal for this section is to extend these results to continuous Schauder frames as well. The natural numbers are used to index Schauder bases and Schauder frames, and so it is very easy to work with properties using limits. Continuous frames however are indexed by arbitrary measure spaces, and so we will have to work with nets over a directed set instead.
Given a $\sigma$-algebra $\Sigma$ and measure $\mu$, we introduce $$\mathcal{D} = \{ E \in \Sigma : \mu(E) < \infty\textrm{ and }\sup_{t \in A} \|f_t\|\|x_t\| < \infty \}$$ We make $\mathcal{D}$ a directed set by defining $E \preceq F$ whenever $E \subseteq F$, and we refer to the elements of $\mathcal{D}$ as *extra finite*.
For $E\in\Sigma$, we define bounded operators $P_E,T_E:X\rightarrow X$ by for all $x\in E$ $$P_E(x)=\int_E f_t(x) x_t d\mu(t)\quad\textrm{ and }\quad T_E(x)=\int_{M\setminus E} f_t(x) x_t d\mu(t).$$ We consider $P_E$ as a generalization of the basis projection for a Schauder basis and $T_E$ as the tail projection. However, in the setting of Schauder frames and continuous Schauder frames these operators are no longer projections.
\[D:S\] A continuous Schauder Frame $(x_t, f_t)_{t \in M}$ for $X$ is called *shrinking* if $\lim_{E \in \mathcal{D}} \|T^*_{ E}f\| = 0$ for all $f \in X^{*}$.
\[L:uncS\] Let $(M,\Sigma,\mu)$ be a sigma finite measure space and let $(x_w,f_w)_{w\in M}$ be a continuous Schauder frame of a Banach space $X$. If $(x_w,f_w)_{w\in M}$ is shrinking then for all $H\in \Sigma$ we have that $$\lim_{E \in \mathcal{D}} \|T^*_{H \cup E}f\| = 0\quad\textrm{ for all }f \in X^{*}.$$
Let $f\in X^*$ and $\vp>0$. Assume that $(x_w,f_w)_{w\in M}$ is shrinking. Thus, there exists $E_0\in \mathcal{D}$ such that $\|T^*_E f\|<\vp$ for all $E\in \mathcal{D}$ with $E_0\subseteq E$. Let $x\in B_X$. We consider the two sets $$F^+=\{t\in M \,:\, f_t(x) f(x_t)\geq 0\}\quad\textrm{ and }\quad F^-=\{t\in M\,:\, f_t(x) f(x_t)< 0\}.$$ Choose increasing sequences $(F^+_n)_{n\in\N},(F^-_n)_{n\in\N}\subseteq \mathcal{D}$ such that $\cup F^+_n=F^+$ and $\cup F^-_n=F^-$. We now have the following estimate for each $x\in B_X$.
$$\begin{aligned}
\int_{E_0^c} |f_t(x) f(x_t)| d\mu&=\int_{(E_0\cup F^-)^c} f_t(x) f(x_t) d\mu-\int_{(E_0\cup F^+)^c} f_t(x) f(x_t) d\mu\\
&=\lim_{n\rightarrow \infty}\int_{(E_0\cup F_n^-)^c} f_t(x) f(x_t) d\mu-\int_{(E_0\cup F_n^+)^c} f_t(x) f(x_t) d\mu\\
&\leq \lim_{n\rightarrow \infty}\|T^*_{E_0\cup F_n^-} f\|+ \|T^*_{E_0\cup F_n^+}f\|\leq 2\vp\end{aligned}$$
Thus, for all $x\in B_X$ we have that $\int_{E_0^c} |f_t(x) f(x_t)| d\mu<2\vp$. Let $H\in M$ and $E\in\mathcal{D}$ with $E_0\subseteq E$. $$\|T^*_{E\cup H}f\|=\sup_{x\in B_X}\left|\int_{E^c\cap H^c} f_t(x)f(x_t)d\mu\right|\leq \sup_{x\in B_X}\int_{E_0^c} |f_t(x)f(x_t)|d\mu\leq 2\vp$$ Thus, $\lim_{E \in \mathcal{D}} \|T^*_{H \cup E}f\| = 0$.
A Schauder basis for a Banach space is shrinking if and only if its biorthogonal functionals form a Schauder basis for the dual. The following theorem proves this for continuous Schauder frames, which essentially means that we have the correct definition for what it means for a continuous Schauder frame to be shrinking.
\[T:AltShrink\] Let $(x_t, f_t)_{t \in M}$ be a continuous Schauder frame for a Banach space $X$. Then, $(x_t, f_t)_{t \in M}$ is shrinking if and only if its dual frame $(f_t, x_t)_{t \in M} \subseteq X^{*} \times X^{**}$ is a continuous Schauder Frame for $X^{*}$.
Let $f\in X^*$. By Lemma \[L:DualFrame1\], $(f_t, x_t)_{t \in M}$ is a continuous\* Schauder frame for $X$. Thus, there exists $f_H\in X^*$ with $f_H(x)=\int_H f_t(x) f(x_t) d\mu(t)$ for all $x\in X$ and all measurable $H\subseteq M$, and $f_M=f$. We now need to prove that $(x_t, f_t)_{t \in M}$ is shrinking if and only if $x^{**}(f_H)=\int_H x^{**}(f_t) f(x_t) d\mu(t)$ for all $x^{**}\in X^{**}$ and all measurable $H\subseteq M$.
$M$ is $\sigma$-finite so we can find a monotone sequence $(V_i)_{i = 1}^\infty$ of finite measure sets such that $M = \bigcup_{i = 1}^\infty V_i$. We also define $W_i = \{ t \in M : {\left\lVertf_t\right\rVert} {\left\lVertx_t\right\rVert} \leq i \}$ and note that $(W_i)_{i = 1}^\infty$ is monotone with $M = \bigcup_{i = 1}^\infty W_i$. Now let $U_i = V_i \cap W_i$. We see that $(U_i)_{i = 1}^\infty \subseteq \mathcal{D}$ is monotone and $M = \bigcup_{i = 1}^\infty U_i$.
We now fix $x^{**} \in X^{**}$ and choose $(y_n)_{n = 1}^\infty \subseteq X$ converging weak$^{*}$ to $x^{**}$ such that $\|y_n\|= \|x^{**}\|$ for all $n\in\N$. In particular, we have for all $n,k\in\N$ that $|x_t(f) y_n(f_t)|\leq k^2\|f\|\|x^{**}\|$ for all $t\in M$. Thus, the function $t\mapsto x_t(f) y_n(f_t)$ is bounded by the constant $k^2\|f\|\|x^{**}\|$ on the finite measure space $U_k$. Let $H\in \Sigma$. For each $k\in \N$, we have that
$$\begin{aligned}
x^{**}(f_H) =& \lim_{n \rightarrow \infty} f_H(y_n) \\
=& \lim_{n \rightarrow \infty} \int_H f(x_t) f_t(y_n) d\mu \\
=& \lim_{n \rightarrow \infty}\int_{H \cap U_k} f(x_t) f_t(y_n) d\mu + \lim_{n \rightarrow \infty} \int_{H \cap U_k^C} f(x_t) f_t(y_n) d\mu \\
=& \int_{H \cap U_k} f(x_t) x^{**}(f_t) d\mu + \lim_{n \rightarrow \infty} \int_{H \cap U_k^C} f(x_t) f_t(y_n) d\mu \quad \textrm{by dominated convergence.} \end{aligned}$$
Now as $x_t(f) x^{**}(f_t)\in L_1(H)$ we have that,
$$\begin{aligned}
x^{**}(f_H) =& \lim_{k\rightarrow\infty} \int_{H \cap U_k} f(x_t) x^{**}(f_t) d\mu +\lim_{k\rightarrow\infty} \lim_{n \rightarrow \infty} \int_{H \cap U_k^C} f(x_t) f_t(y_n) d\mu \\
=& \int_{H} f(x_t) x^{**}(f_t) d\mu +\lim_{k\rightarrow\infty} \lim_{n \rightarrow \infty} \int_{H \cap U_k^C} f(x_t) f_t(y_n) d\mu \quad \textrm{by dominated convergence.}
\end{aligned}$$
Thus, we have that $(f_t, x_t)_{t \in M}$ is a continuous Schauder Frame for $X^{*}$ if and only if $\lim_{k \rightarrow \infty}\lim_{n \rightarrow \infty} \int_{H \cap U_k^C} f(x_t) f_t(y_n) d\mu = 0$ for every $f \in X^{*}$, $H \in \Sigma$, and $w$-Cauchy $(y_n)_{n=1}^\infty$ in $X$.
We first assume that $(x_w,f_w)_{w\in M}$ is shrinking. Let $\varepsilon > 0$, $f \in X^{*}$ with $\|f\|=1$, and $H \in \Sigma$. By Lemma \[L:uncS\], there exists $E \in \mathcal{D}$ such that for all $F \succeq E$ we have that ${\left\lVertT^*_{H^C \cup F}f\right\rVert} \leq \varepsilon$. This $E$ is extra finite and so let $A = \sup_{t \in E} {\left\lVertf_t\right\rVert} {\left\lVertx_t\right\rVert} < \infty$. As $E$ has finite measure and $M=\cup_{j=1}^\infty U_j$, there exists $k \in \mathbb{N}$ so that $\mu(H \cap U_k^C \cap E) < \frac{\varepsilon}{A}$. Then:
$$\begin{aligned}
\lim_{n \rightarrow \infty} &\left |\int_{H \cap U_k^C} f(x_t) f_t(y_n) d\mu \right| \leq \|x^{**}\|\sup_{{\left\lVertx\right\rVert} = 1} \left|\int_{H \cap U_k^C} f(x_t) f_t(x) d\mu\right| \\
& \leq \|x^{**}\|\sup_{{\left\lVertx\right\rVert} = 1} \left|\int_{H \cap U_k^C \cap E^C} x_t(f) f_t(x) d\mu\right| + \|x^{**}\|\sup_{{\left\lVertx\right\rVert} = 1} \left|\int_{H \cap U_k^C \cap E} x_t(f) f_t(x) d\mu\right| \\
&\leq\|x^{**}\| {\left\lVertT^*_{H^C \cup U_k \cup E}f\right\rVert} + \|x^{**}\| \mu(H \cap U_k^C \cap E) A \\
&\leq \|x^{**}\| \varepsilon+\|x^{**}\|\varepsilon
\end{aligned}$$
Thus, $\lim_{k \rightarrow \infty}\lim_{n \rightarrow \infty} \int_{H \cap U_k^C} x_t(f) y_n(f_t) d\mu = 0$ and thus $(f_t, x_t)_{t \in M}$ is a continuous Schauder Frame for $X^{*}$.
We now assume that $(f_t, x_t)_{t \in M}$ is not a continuous Schauder frame of $X^*$. Thus, there exists $f \in X^{*}$, $H \in \Sigma$, $(m_k)_{k=1}^\infty\in[\N]^\omega$, $\vp>0$, and a $w^*$-convergent sequence $y_n\rightarrow x^{**}$ with $\|y_n\|=x^{**}$ for all $n\in\N$ such that $\lim_{n \rightarrow \infty} \int_{H \cap U_{m_k}^C} x_t(f) y_n(f_t) d\mu >\vp$ for all $k\in\N$.
Let $E\in \mathcal{D}$. let $A = \sup_{t \in E} {\left\lVertf_t\right\rVert} {\left\lVertx_t\right\rVert} < \infty$. As $E$ has finite measure and $M=\cup_{j=1}^\infty U_j$, there exists $k \in \mathbb{N}$ so that $\mu(H \cap U_{m_k}^C \cap E) < \frac{\varepsilon}{2A\|f\|\|x^{**}\|}$. Then:
$$\begin{aligned}
\|T^*_{H\cup U_k\cup E}f\|&=\sup_{\|x\|=1} \int_{H^c\cap U_{m_k}^c \cap E^c} f_t(x) f(x_t) d\mu\\
&\geq \lim_{n\rightarrow\infty} \frac{1}{\|x^{**}\|} \int_{H^c\cap U_{m_k}^c \cap E^c} f_t(y_n) f(x_t) d\mu\\
&= \lim_{n\rightarrow\infty} \frac{1}{\|x^{**}\|}\left( \int_{H^c\cap U_{m_k}^c} f_t(y_n) f(x_t) d\mu- \int_{H^c\cap U_{m_k}^c \cap E} f_t(y_n) f(x_t) d\mu\right)\\
&\geq \lim_{n\rightarrow\infty} \frac{1}{\|x^{**}\|}\left( \int_{H^c\cap U_{m_k}^c} f_t(y_n) f(x_t) d\mu- A\|x^{**}\|\|f\|\mu(H^c\cap U_{m_k}^c \cap E)\right)\\
&\geq \lim_{n\rightarrow\infty} \frac{1}{\|x^{**}\|}\left( \int_{H^c\cap U_{m_k}^c} f_t(y_n) f(x_t) d\mu- \vp/2\right)\\
&\geq \frac{1}{\|x^{**}\|}\frac{\vp}{2}\end{aligned}$$
Thus, $(x_w,f_w)_{w\in M}$ is not shrinking by Lemma \[L:uncS\] as $E\in\mathcal{D}$ is arbitrary and $E\preceq U_k\cup E$.
\[T:l1\]Let $(x_t, f_t)_{t \in M}$ be a continuous Schauder frame for a Banach space $X$. Then $(x_t, f_t)_{t \in M}$ is shrinking if and only if $\ell_1$ does not embed into $X$.
We first assume that $(x_t, f_t)_{t \in M}$ is shrinking. Thus, $(f_t, x_t)_{t \in M}$ is a continuous Schauder frame for $X^*$ by Theorem \[T:AltShrink\]. In particular, $X^*$ must be separable. However, the dual of $\ell_1$ is not separable, so $\ell_1$ cannot embed into $X$.
We now assume that $(x_t, f_t)_{t \in M}$ is not shrinking. There exists $f \in X^{*}$ with $\|f\|=1$ and $A>0$ such that, for every $E \in \mathcal{D}$ there is some $F \succeq E$ such that ${\left\lVertT^*_{F} f\right\rVert} > A$. Let $B$ be the suppression unconditionality constant of $(x_t, f_t)_{t \in M}$.
We will create by induction a sequence of extra finite sets $U_1 \subseteq V_1\subseteq U_2\subseteq V_2...$ and vectors $(y_n)_{n=1}^\infty$ such that for all $n\in\N$,
1. $\|y_n\|\leq 2$,
2. $P^*_{V_n\setminus U_n} f(y_n)> A/2$,
3. $|P^*_{V_j\setminus U_j} f(y_n)|<\frac{A}{9nB}$ for all $j<n$,
4. $|P^*_{V_n\setminus U_n} f(y_j)|<\frac{A}{92^{n}B}$ for all $j<n$.
We first choose $U_1=\emptyset$. There exists $E\in\mathcal{D}$ such that $\|T_E^* f\|>A$. Let $y_1\in X$ be a unit vector such that $T_E^*f(y_1)>A$. We choose $V_1\subseteq E^c$ such that $P_{V_1}f(y_1)>A$.
We now assume that $n\in\N$ and that $y_1,...y_n$ and $U_1 \subseteq V_1...\subseteq U_n\subseteq V_n$ have been chosen. As $(x_t, f_t)_{t \in M}$ is a continuous Schauder frame for $X$ there exists $U\in\mathcal{D}$ such that for all $E\subset U^c$, we have that $|P_E^*f(y_j)|<\frac{A}{92^{n+1}B}$ for all $j\leq n$. We have that for every $E \in \mathcal{D}$ there is some $F \succeq E$ and $z\in X$ with $\|z\|=1$ such that $|{T^*_{F} f}(z)| > A$. By compactness, we may stabilize the values $|P^*_{V_j\setminus U_j} f(z)|$ for $1\leq j\leq n$. That is, for all $E_1 \in \mathcal{D}$ there is some $F_1 \succeq E_1$ and $z_1\in X$ with $\|z_1\|=1$ such that for all $E_2 \in \mathcal{D}$ there is some $F_2 \succeq E_2$ and $z_2\in X$ with $\|z_2\|=1$ so the following is satisfied for all $1\leq j\leq n$, $$\label{E:ind}
|{T^*_{F_1} f}(z_1)| > A,\quad |{T^*_{F_2} f}(z_2)| > A,\quad\textrm{and}\quad |P^*_{V_j\setminus U_j} f(z_1-z_2)|<\frac{A}{9(n+1)B}.$$ We choose $F_1\succeq U \cup V_n$ and obtain $z_1$ satisfying . We choose $E_2\in\mathcal{D}$ so that $|{T^*_{F} f}(z_1)|<A/2$ for all $F\succeq E_2$. We now choose $F_2\succeq E_2$ and $z_2$ satisfying . Let $U_{n+1}=E_2$ and $V_{n+1}\succeq U_{n+1}$ such that $P^*_{V_{n+1}\setminus U_{n+1}} f(z_2)> A$. We now let $y_{n+1}=z_2-z_1$. These choices now satisfy our induction proof.
We have that $(y_n)_{n=1}^\infty$ is semi-normalized by (1) and (2). We now prove that $(y_n)_{n=1}^\infty$ is equivalent to the unit vector basis of $\ell_1$. Let $(a_n)_{n=1}^\infty\in\ell_1$ with $\sum |a_n|=1$. Without loss of generality, we assume there is a subsequence $(a_{k_n})_{n=1}^\infty$ such that $a_{k_n}> 0$ for all $n\in\N$ and $\sum_{n=1}^\infty a_{k_n}\geq \frac{1}{2}$.
$$\begin{aligned}
\|\sum_{n=1}^\infty a_n y_n\|&\geq B^{-1} (P^*_{\cup_{j\in\N} V_{k_j}\setminus U_{k_j}} f)(\sum_{n=1}^\infty a_n y_n)\\
&= B^{-1} \sum_{n=1}^\infty a_{k_n} P^*_{V_{k_n}\setminus U_{k_n}} f (y_n)+\sum_{j<n}a_n P^*_{V_{k_j}\setminus U_{k_j}} f( y_n)+
\sum_{j>n}a_n P^*_{V_{k_j}\setminus U_{k_j}} f( y_n)\\
&\geq B^{-1} \sum_{n=1}^\infty a_{k_n} A/2-\sum_{j<n} |a_n| n^{-1}B^{-1}9^{-1}A-
\sum_{j>n}|a_n| 2^{-j}B^{-1}9^{-j}A\\
&\geq B^{-1} \sum_{n=1}^\infty a_{k_n} A/2-\sum_{n\in\N} |a_n| B^{-1}9^{-1}A-
\sum_{n\in\N}|a_n| B^{-1}9^{-1}A\\
&\geq B^{-1} A/4-B^{-1}9^{-1}A-B^{-1}9^{-1}2^{-j}A=B^{-1}A36^{-1}\end{aligned}$$
Thus, $(y_n)_{n=1}^\infty$ dominates the unit vector basis of $\ell_1$ and is hence equivalent to it.
We now consider the generalization of boundedly complete to the continuous setting.
A continuous Schauder Frame $(x_t, f_t)_{t \in M}$ for $X$ is said to be *boundedly complete* if $\lim_{E \in \mathcal{D}} \int_E x^{**}(f_t) x_t d\mu \in X$ for all $x^{**} \in X^{**}$.
Frames for Hilbert spaces are nicely characterized as projections of Riesz bases for larger Hilbert spaces. Likewise, Schauder frames for Banach spaces are characterized as projections of Schauder bases for larger Banach spaces. Furthermore, a Schauder frame is shrinking or boundedly complete if and only if it is the projection of a shrinking or boundedly complete Schauder basis [@BFL]. This allows for constructing and studying frames by working directly with bases and then projecting onto a subspace. Essentially, a redundant frame may be dilated to a non-redundant basis. However, this concept of dilation is only possible for continuous frames over purely atomic measures. The following proposition shows that the reverse direction is still valid for continuous frames in that projecting continuous Schauder frames onto closed subspaces gives a continuous Schauder frame.
\[P:ProjF\] Let $(x_t, f_t)_{t}$ be a continuous Schauder frame for a Banach space $X$. Let $Y\subseteq X$ be a complemented subspace and let $P:X\rightarrow Y$ be a bounded projection.
1. $(P x_t, P^* f_t)_{t}$ is a Schauder frame for $Y$.
2. If $(x_t, f_t)_{t}$ is shrinking then $(P x_t, P^* f_t)_{t}$ is shrinking.
3. If $(x_t, f_t)_{t}$ is boundedly complete then $(P x_t, P^* f_t)_{t}$ is boundedly complete.
Let $y\in Y$ and $g\in Y^*$. Let $E\subseteq M$ be measurable. There exists $y_E\in X$ such that $y_E=\int f_t(y) x_t dt$. Thus, $$g(P y_E)=P^* g(y_E)=\int_E f_t(y) P^*g(x_t)dt=\int_E P^*f_t(y) g(Px_t)dt.$$ Thus, $P y_E= \int_E f_t(y) x_t dt$. Furthermore, $P y_M=Py=y$. This proves that $(P x_t, P^* f_t)_{t}$ is a Schauder frame for $Y$.
We now assume that $(x_t, f_t)_{t}$ is shrinking. Let $E\subseteq M$ be measurable. If $P_E$ is the projection operator for the Schauder frame $(x_t, f_t)_{t}$ of $X$ then we have that $P P_E$ is the projection operator for the frame $(Px_t, P^*f_t)_{t}$ of $Y$. Thus, if $T_E$ is the tail operator for the Schauder frame $(x_t, f_t)_{t}$ of $X$ then $P T_E$ is the tail operator for the frame $(Px_t, P^*f_t)_{t}$ of $Y$. This gives that for all $g\in Y^*$ that $$\lim_{E \in \mathcal{D}} \|(PT_E)^*g\|=\lim_{E \in \mathcal{D}} \|T_E^* (P^*g)\| = 0 .$$ Thus, $(Px_t, P^*f_t)_{t}$ is shrinking.
We now assume that $(x_t, f_t)_{t}$ is boundedly complete. Let $y^{**}\in Y^{**}$. Let $I_X:Y\rightarrow X$ be the inclusion operator of $Y$ into $X$. Thus, $I_X^{**}y^{**}\in X^{**}$ and there exists $x\in X$ such that $x=\int I_X^{**}y^{**}(f_t) x_t dt$. Let $E\subseteq M$ be measurable and $f\in X^{**}$. We have that, $$\begin{aligned}
g(P(x_E))&=\int_E I^{**}y^{**}(f_t) P^*g^*(x_t) dt \\
&=\int_E y^{**}(f_t|_Y) g^*(Px_t) dt \\
&=\int_E y^{**}(P^*f_t) g^*(Px_t) dt \\\end{aligned}$$ Thus, we have that $Px=\int y^{**}(P^*f_t) Px_t dt$. Hence, $(Px_t, P^*f_t)_{t}$ is boundedly complete.
\[L:inX\] Let $(M,\Sigma,\mu)$ be a $\sigma$-finite measure space and $(x_t, f_t)_{t \in M}$ be a continuous Schauder frame for $X$. For each $E \in \mathcal{D}$, the operator $P^{**}_E$ on $X^{**}$ defined by $P_E(x^{**})=\int_E x^{**}(f_t)x_t d\mu(t)$ is compact, has its range in $X$, and satisfies $\|P^{**}_E\|\leq B$ where $B$ is the suppression unconditionality constant of $(x_t, f_t)_{t \in M}$.
We denote $L(X^{**},X)$ to be the set of bounded linear operators from $X^{**}$ to $X$. We have for each $t\in M$ that $f_t \otimes x_t\in L(X^{**},X)$, where $f_t \otimes x_t(x^{**})=f_t(x^{**})x_t$ for all $t\in M$. Let $\psi:M\rightarrow L(X^{**},X)$ be the measurable map $t\mapsto f_t\otimes x_t$. As $(M,\Sigma,\mu)$ is $\sigma$-finite, if $(H_\alpha)$ is a measurable collection of pairwise disjoint subsets of $L(X^{**},X)$ then there exists a countable subset such that $\mu(\psi^{-1}(H_\alpha))=0$ for all $\alpha$ not in the subset. Thus, for each $\vp>0$, there exists a countable collection of sets $(H_j)_{j\in J}$ in $L(X^{**},X)$ with diameter at most $\vp$ such that $\cup_{j\in J} \psi^{-1}(H_j)=E$ almost everywhere and $\psi^{-1}(H_j)\neq\emptyset$ for all $j\in J$. For all $j\in J$ choose $t_j\in \psi^{-1}(H_j)$. Let $C=\sup_{t\in E}\|x_t\|\|f_t\|$. We have that $$\sum_{j\in J}\|f_{t_j}\otimes x_{t_j}\| \mu(\psi^{-1}(H_j))\leq \sum_{j\in J} C \mu(\psi^{-1}(H_j))= C\mu(E).$$ Thus, $\sum_{j\in J} f_{t_j}\otimes x_{t_j} \mu(\psi^{-1}(H_j))$ converges unconditionaly in norm. As, $ f_{t_j}\otimes x_{t_j}\in L(X^{**},X)$ for all $j\in J$, we have that $T_\vp:=\sum_{j\in J} f_{t_j}\otimes x_{t_j} \mu(\psi^{-1}(H_j))$ is an element of $L(X^{**},X)$. For each $x^{**}\in X^{**}$ and $f\in X^*$, we have that $$\begin{aligned}
|\int_E x^{**}(f_t) &f(x_t)d\mu - \sum_{j\in J} x^{**}(f_{t_j})f(x_{t_j})\mu(H_j)|\\
&=| \sum_{j\in J} \left(\int_{\psi^{-1}(H_j)} x^{**}(f_t) f(x_t)d\mu -x^{**}(f_{t_j})f(x_{t_j})\mu(H_j)\right)|\\
&\leq \sum_{j\in J} \int_{\psi^{-1}(H_j)} \|x^{**}\|_{X^{**}}\|f\|_{X^*}\|f_t\otimes x_t- f_{t_j}\otimes x_{t_j}\| d\mu\\
&\leq \sum_{j\in\N} \int_{\psi^{-1}(H_j)} \|x^{**}\|_{X^{**}}\|f\|_{X^*}\vp d\mu\\
&= \int_{E} \|x^{**}\|_{X^{**}}\|f\|_{X^*}\vp d\mu\\
&= \|x^{**}\|_{X^{**}}\|f\|_{X^*}\vp \mu(E)
\end{aligned}$$
Thus, $(T_{1/n})_{n=1}^\infty$ is Cauchy and it converges in norm to $P^{**}_E=\int_E f_t \otimes x_t d\mu$. As the range of $T_{1/n}$ is in $X$ for all $n\in\N$, we have that the range of $P^{**}_E$ is in $X$. Note that $P_E^{**}=(P_E)^{**}$, the double adjoint of the truncation operator for the continuous Schauder frame $(x_t, f_t)_{t \in M}$. Thus, $\|P_E^{**}\|=\|P_E\|\leq B$.
\[T:bc\] Let $(x_t, f_t)_{t\in M}$ be a continuous Schauder frame for a Banach space $X$. Then $(x_t, f_t)_{t\in M}$ is boundedly complete if and only if $c_0$ does not embed into $X$.
We first assume that $(x_t, f_t)_{t\in M}$ is not boundedly complete. Thus, there exists $x^{**}\in X^{**}$ such that $\lim_{E \in \mathcal{D}} \int_E x^{**}(f_t) x_t d\mu$ does not converge to an element of $X$. We have by Lemma \[L:inX\] that, $\int_E x^{**}(f_t) x_t d\mu$ is an element of $X$ for all $E\in\mathcal{D}$. Hence, the net $\lim_{E \in \mathcal{D}} \int_E x^{**}(f_t) x_t d\mu$ is not Cauchy. This gives that there exists $\delta > 0$ and extra finite sets $V_1 \subseteq W_1 \subseteq V_2 \subseteq W_2 \subseteq ...$ such that for $u_n =\int_{W_n - V_n} x^{**}_0(f_t) x_t d\mu$, we have ${\left\lVertu_n\right\rVert} > \delta$ for every $k \in \mathbb{N}$.
Let $f\in X^*$. As, $|\int_{M} x^{**}_0(f_t) f(x_t) d\mu|<\infty$ and $(W_n-V_n)_{k=1}^\infty$ is pairwise disjoint, we have that $$\lim_{n\rightarrow\infty}f(u_n)=\int_{W_n - V_n} x^{**}_0(f_t) f(x_t) d\mu=0.$$ Thus, $(u_n)_{n=1}^\infty$ is a semi-normalized weakly null sequence. After passing to a subsequence, we assume without loss of generality that $(u_n)_{n=1}^\infty$ is a basic sequence. We will now prove that $(u_n)_{n=1}^\infty$ is isomorphic to the unit vector basis of $c_0$. Let $C>0$ be the unconditionality constant of $(x_t, f_t)_{t\in M}$. Let $f\in X^*$ and $(a_n)_{n\in\N}\in c_{00}$. We have that $$|f(\sum a_n u_n)|=|\sum_{n\in\N}\int_{W_n-V_n} a_n x^{**}_0 (f_t) f(x_t) d\mu|\leq\sup |a_n| \int |x^{**}_0 (f_t) f(x_t)|d\mu \leq C\|f\|\|a_n\|_\infty$$ Thus, $(u_n)_{n\in\N}$ is equivalent to the unit vector basis of $c_0$.
We now assume that $c_0$ is isomorphic to a subspace of $X$, and for the sake of contradiction we assume that $X$ has a boundedly complete continuous Schauder frame. As $X$ is separable, every subspace isomorphic to $c_0$ is complemented in $X$ (see [@Go] for a nice proof of this fact). Thus, there exists a boundedly complete continuous Schauder frame $(x_t ,f_t)_{t\in M}$ of $c_0$ by Theorem \[P:ProjF\]. As $(x_t ,f_t)_{t\in M}$ is boundedly complete, we may define $P:\ell_\infty\rightarrow c_0$ by $$P(x^{**})=\lim_{E \in \mathcal{D}} \int_E x^{**}(f_t) x_t d\mu\quad\textrm{ for all }x^{**}\in \ell_\infty.$$ This gives a bounded linear projection from $\ell_\infty$ to $c_0\subseteq\ell_\infty$. This is a contradiction as $c_0$ is not complemented inside $\ell_\infty$.
If a Banach Space $X$ admits a continuous Schauder Frame $(x_t, f_t)_{t \in M}$ then the following are equivalent:
1. $(x_t, f_t)_{t \in M}$ is shrinking and boundedly complete,\
2. $X$ does not contain a copy of $c_0$ or $\ell_1$,\
3. $X$ is reflexive.
We have that (1) and (2) are equivalent by Theorem \[T:l1\] and Theorem \[T:bc\]. Furthermore, $(3) \implies (2)$ is clear since $c_0$ and $\ell_1$ are not reflexive. We now prove that $(1) \implies (3)$.
Since $(x_t, f_t)_{t \in M}$ is boundedly complete, for each $x^{**} \in X^{**}$ we have an $x \in X$ such that $x = \lim_{A \in \mathcal{D}} \int_A x^{**}(f_t) x_t d\mu$. As $(x_t,f_t)$ is shrinking, Theorem \[T:AltShrink\] gives that every $f \in X^{*}$ satisfies $f = \int x_t(f) f_t d\mu$. Then take arbitrary $f \in X^{*}$ and observe $$\begin{aligned}
x(f) =& f(x) = f(\lim_{A \in \mathcal{D}} \int_A x^{**}(f_t) x_t d\mu) \quad \textrm{by boundedly complete}, \\
=& \lim_{A \in \mathcal{D}} f( \int_A x^{**}(f_t) x_t d\mu) \quad \textrm{by continuity}, \\
=& \lim_{A \in \mathcal{D}} \int_A x^{**}(f_t) x_t(f) d\mu \quad \textrm{by definition of the Pettis Integral}, \\
=& \int x^{**}(f_t) x_t(f) d\mu \quad \textrm{by dominated convergence.}
\end{aligned}$$ On the other hand we have $$\begin{aligned}
x^{**}(f) =& x^{**}(\int x_t(f) f_t d\mu) \quad \textrm{by Theorem \ref{T:AltShrink},} \\
=& \int x^{**}(f_t) x_t(f) d\mu \quad \textrm{by definition of the Pettis Integral.}
\end{aligned}$$ Compiling the above, we have $x(f) = x^{**}(f)$ for all $f \in X^{*}$. So $x^{**} = x \in X$. But this was for arbitrary $x^{**} \in X^{**}$. So $X^{**} = X$ as desired.
Continuous wavelet frames for $L_p$ with $1<p<\infty$ {#S:ex}
=====================================================
Frame theory for Hilbert spaces developed concurrently with that of wavelets, and wavelets still provide some of the most useful examples of both continuous and discrete frames. Wavelets are important in Banach spaces as well, and the Haar basis is possibly the most commonly used Schauder basis for $L_p$ with $1\leq p<\infty$. One nice aspect of using a continuous wavelet frame as opposed to a discrete one is that when a function is translated or dilated, the continuous wavelet frame coefficients for the new function are the same (except occurring at different indexes). On the other hand, if we translate or dilate a function by a non-integer amount, then the discrete wavelet frame coefficients for the new function are completely different. It is well known that any discrete wavelet for a Hilbert space gives a continuous wavelet [@WW]. The goal of this section is to prove the corresponding result for $L_p$ for $1<p<\infty$. We must give a completely different proof than that was used in proving the $L_2$ result as the Fourier transform is not an isometry on $L_p$ for $p\neq2$.
Let $1< p<\infty$ and $\psi\in L_p(\R)$. For $a,b\in\R$, the dilation operator $D_a:L_p\rightarrow L_p$ and translation operator $T_b:L_p\rightarrow L_p$ are defined by $D_a(f)(t)=2^{a/p}f(2^at)$ and $T_b(f)(t)=f(t-b)$ for all $f\in L_p$ and $t\in \R$. We call $\psi\in L_p(\R)$ with $\|\psi\|=1$ a wavelet for $L_p$ if $(D_nT_k\psi)_{k,n\in Z}$ is an unconditional Schauder basis of $L_p$. If $\psi^*\in L_{p'}$ is the biorthogonal functional to $\psi=D_0T_0\psi$ in $(D_nT_k\psi)_{k,n\in Z}$. Then for all $k,n\in\Z$, the biorthogonal functional to $D_nT_k\psi$ is given by $D^*_{-n}T^*_{-k}\psi^*$. One can also think of $D^*_{-n}=D_{n}$ and $T^*_{-k}=T_k$ where $D_n$ and $T_k$ are now considered as the dilation and translation operators on $L_{p'}$ for $1/p+1/p'=1$. We say that $\psi$ is a continuous wavelet for $L_p$ if $(D_{a}T_b\psi,D^*_{-a}T^*_{-b}\psi^*)_{a,b\in\R}$ is a continuous frame of $L_p$.
The operators $T_a$ and $D_b$ have the following relationships. For all $a,b\in\R$, $T_a^{-1}=T_{-a}$, $D_a^{-1}=D_{-a}$, $T_aT_b=T_{a+b}$, $D_aD_b=D_{a+b}$, $D_aT_b=T_{2^{-a}b}D_a$ and $D^*_aT^*_b=T^*_{2^{a}b}D^*_a$. For the sake of convenience, we write $\psi_{a,b}=D_aT_b\psi$ and $\psi^*_{a,b}=D^*_{-a}T^*_{-b}\psi$ where $a,b\in\R$.
\[L:limit\] Let $(M,\Sigma,\mu)$ be a sigma finite measure space and $X$ be a Banach space. Let $t\mapsto (x_t,f_t)$ be a measurable function from $M$ to $X\times X^*$. If $((x^n_t,f^n_t)_{t\in M})_{n=1}^\infty$ is a sequence of continuous frames of $X$ with unconditionality constant $C$ such that there exists $D>0$ such that $\|x^n_t\|,\|f^n_t\|\leq D$ for all $t\in M$ and $n\in\N$. Suppose that for all $x\in X$ and $f\in X^*$ there exists an increasing sequence of finite measure sets $(E_N)_{N=1}^\infty \subseteq \Sigma$ with $\cup_{N=1}^\infty E_n=M$ so that for all $N\in\N$.
1. $\lim_{n\rightarrow\infty} \|x-\int_{E_N} f_t^n(x)x_t^n dt\|\leq 1/N$,
2. $\lim_{n\rightarrow\infty} \int_{E_N} |(f_t-f_t^n)(x)|dt=0$,
3. $\lim_{n\rightarrow\infty} \int_{E_N} | f(x^n_t-x_t)|dt=0$.
Then $(x_t,f_t)_{t\in M}$ is a continuous frame of $X$ with unconditionality constant $C$.
Let $x\in X$ and $f\in X^*$ . For $E\in\Sigma$ we have that $$\begin{aligned}
&|\int_E f_t(x)f(x_t)dt|\leq \int_E |f_t(x)f(x_t)dt|\\
&= \lim_{N\rightarrow\infty}\int_{E\cap E_N} |f_t(x)f(x_t)|dt\quad\textrm{ by continuity from below}\\
&\leq \lim_{N\rightarrow\infty}\lim_{n\rightarrow\infty}\int_{E_N} | f_t(x)f(x^n_t-x_t)|dt+ \int_{E_N} | (f_t-f_t^n)(x)f(x^n_t)|dt+\int_{E_N} | f_t^n(x)f(x^n_t)|dt\\
&\leq \lim_{N\rightarrow\infty} \lim_{n\rightarrow\infty} \|f_t\|\|x\|\int_{E_N} |f(x^n_t-x_t)|dt + \|f\|\|x^n_t\|\int_{E_N} |(f_t-f_t^n)(x)|dt+\int_{ E_N} | f_t^n(x)f(x^n_t)dt|\\
&\leq 0+0 + C\|f\|\|x\|\end{aligned}$$ This proves that $x\mapsto \int_E f_t(x) x_t dt$ defines a bounded linear functional on $X^*$ with norm at most $C\|x\|$. As $X$ is reflexive, there exists $x_E\in X$ with $\|x_E\|\leq C\|x\|$ so that $f(x_E)= \int_E f_t(x) f(x_t) dt$ for all $f\in X^*$.
We now check that $x=x_M$.
$$\begin{aligned}
&|f(x)\!-\!\int\! f_t(x)f(x_t)dt|=\lim_{N\rightarrow\infty} |f(x)\!-\!\int_{E_N}\! f_t(x)f(x_t)dt|\\
&\leq\lim_{N\rightarrow\infty} \lim_{n\rightarrow\infty}\! \int_{E_N}\! |f_t(x)f(x^n_t-x_t)|dt\!+\! \int_{E_N}\! | (f_t-f_t^n)(x)f(x^n_t)|dt\!+\!|f(x)\!-\!\int_{E_N}\! f_t^n(x)f(x^n_t)dt|\\
&\leq\lim_{N\rightarrow\infty} \lim_{n\rightarrow\infty} \|f_t\|\|x\|\int_{E_N} |f(x^n_t-x_t)|dt + \|f\|\|x^n_t\|\int_{E_N} |(f_t-f_t^n)(x)|dt+|f(x)-\int_{E_N} f_t^n(x)f(x^n_t)dt|\\
&=\lim_{N\rightarrow\infty} 0+0 + 1/N=0\end{aligned}$$
Let $1< p<\infty$. Suppose that $\psi$ is a wavelet for $L_p$ with unconditonality constant $C$. Then $\psi$ is a continuous wavelet for $L_p$ with unconditionality constant $C$.
For each $N\in\N$ and $a,b\in\R$ let $l_a,m_b\in\Z$ and $r_a,s_b\in\{0,1,...,N-1\}$ such that $l_a+r_a N^{-1}\leq a<l_a+(r_a+1)N^{-1}$ and $m_b+s_b N^{-1}\leq b<m_b+(s_b+1)N^{-1}$. We now set $$\psi^N_{a,b}=\psi_{l_a+\frac{r_a }{N},m_b+\frac{s_b 2^{l_a}}{N}}.$$
For all $f\in L_{p'}$ and $N\in\N$, we have that, $ |f(\psi^{N}_{a,b})-f(\psi_{a,b})| \leq 2\|\psi\|\|x\|.
$ The map $(a,b)\mapsto \psi_{a,b}$ is continuous in norm. Thus, for all $a,b\in\R$, $\psi^N_{a,b}\rightarrow \psi_{a,b}$ in norm (though we actually only need weak convergence). We have by the dominated convergence theorem that for all $M\in\N$, $$\label{E:conv2}
\lim_{N\rightarrow\infty}\int_{-M}^M \int_{-M}^M |f(\psi^{N}_{a,b})-f(\psi_{a,b})| dadb=0.$$ Likewise, for all $x\in L_p$ and $M\in\N$, $$\label{E:conv}
\lim_{N\rightarrow\infty}\int_{-M}^M \int_{-M}^M |\psi^{N*}_{a,b}(x)-\psi^*_{a,b}(x)| dadb=0.$$
Now, let $x\in L_p$ and $\vp>0$. The set $\{ D_aT_bx:|a|,|b|\leq 1\}$ is compact in $L_p$ and thus there exists $M\in\N$ such that $\|D_aT_bx-\sum_{n,k=-M}^{M-1} \psi^*_{n,k}(D_aT_bx)\psi_{n,k}\|<\vp$ for all $|a|,|b|\leq1$.
$$\begin{aligned}
&\|x-\int_{-M}^M\int_{-M}^M\psi^{N*}_{a,b}(x)\psi^N_{a,b}dadb\|=\|x-\sum_{r,s=0}^{N-1}\sum_{l,m=-M}^M N^{-2} \psi^{*}_{l+\frac{r}{N},m+\frac{s 2^{l}}{N}}(x)\psi_{l+\frac rN,m+\frac{s 2^{l}}{N}}\|\\
&=\|x-N^{-2}\sum_{r,s=0}^{N-1}\sum_{l,m=-M}^{M-1} D^*_{-l-\frac{r}{N}}T^*_{-m-\frac{s2^l}{N}}\psi^*(x) D_{l+\frac{r}{N}}T_{m+\frac{s2^l}{N}}\psi\|\\
&=\|x-N^{-2}\sum_{r,s=0}^{N-1}\sum_{l,m=-M}^{M-1} D^*_{-\frac{r}{N}}D^*_{-l}T^*_{-\frac{s2^l}{N}}T^*_{-m}\psi^*(x) D_{\frac{r}{N}}D_{l}T_{\frac{s2^l}{N}}T_{m}\psi\|\\
&=\|x-N^{-2}\sum_{r,s=0}^{N-1}\sum_{l,m=-M}^{M-1} D^*_{-\frac{r}{N}}T^*_{-\frac{s}{N}}D^*_{-l}T^*_{-m}\psi^*(x) D_{\frac{r}{N}}T_{\frac{s}{N}}D_{l}T_{m}\psi\|\\
&=\|x-N^{-2}\sum_{r,s=0}^{N-1}\sum_{l,m=-M}^{M-1} D^*_{-\frac{r}{N}}T^*_{-\frac{s}{N}}D^*_{-l}T^*_{-m}\psi^*(x) D_{\frac{r}{N}}T_{\frac{s}{N}}D_{l}T_{m}\psi\|\\
&\leq N^{-2}\sum_{r,s=0}^{N-1}\|x-\sum_{l,m=-M}^{M-1} D^*_{-\frac{r}{N}}T^*_{-\frac{s}{N}}D^*_{-l}T^*_{-m}\psi^*(x) D_{\frac{r}{N}}T_{\frac{s}{N}}D_{l}T_{m}\psi\|\\
&\leq N^{-2}\sum_{r,s=0}^{N-1}\|T_{-\frac{s}{N}}D_{-\frac{r}{N}}x-\sum_{l,m=-M}^{M-1} D^*_{-l}T^*_{-m}\psi^*(T_{-\frac{s}{N}}D_{-\frac{r}{N}}x) D_{l}T_{m}\psi\|\\
&< N^{-2} \sum_{r,s=0}^{N-1} \vp=\vp\end{aligned}$$
Thus, we have for all $N\in\N$ that $$\label{E:part1}
\|x-\int_{-M}^M\int_{-M}^M\psi^{N*}_{a,b}(x)\psi^N_{a,b}\|dadb<\vp$$
Let $x\in L_p$, $f\in L_{p'}$, and measurable $E\subset\R^2$. For each $l,m\in\Z$ and $r,s\in\{0,1,...,N-1\}$ we let $E_{l,r,m,s}=E\cap [l+\frac{r}{N},l+\frac{r+1}{N}]\times[m+\frac{s}{N},m+\frac{s+1}{N}]$.
We have that $$\begin{aligned}
|\int_{(a,b)\in E} &\psi^{N*}_{a,b}(x)f(\psi^N_{(a,b)})d(a,b)|=|\sum_{r,s=0}^{N-1}\sum_{l,m\in\Z} \psi^{*}_{l+\frac{r}{N},m+\frac{s 2^{l}}{N}}(x)f(\psi_{l+\frac rN,m+\frac{s 2^{l}}{N}})\lambda(E_{l,r,m,s})|\\
&\leq\|f\|\sum_{r,s=0}^{N-1}\|\sum_{l,m\in\Z} \psi^{*}_{l+\frac{r}{N},m+\frac{s 2^{l}}{N}}(x)\psi_{l+\frac rN,m+\frac{s 2^{l}}{N}}\lambda(E_{l,r,m,s})\|\\
&=\|f\|\sum_{r,s=0}^{N-1}\|\sum_{l,m\in\Z} D^*_{-l-\frac{r}{N}}T^*_{-m-\frac{s2^l}{N}}\psi^*(x) D_{l+\frac{r}{N}}T_{m+\frac{s2^l}{N}}\psi \lambda(E_{l,r,m,s})\|\\
&=\|f\|\sum_{r,s=0}^{N-1}\|\sum_{l,m\in\Z-M} D^*_{-l}T^*_{-m}\psi^*(T_{-\frac{s}{N}}D_{-\frac{r}{N}}x) D_{l}T_{m}\psi \lambda(E_{l,r,m,s})\|\qquad\textrm{ as before.}\\
&\leq\|f\|CN^{-2}\sum_{r,s=0}^{N-1}\|\sum_{l,m\in\Z-M} D^*_{-l}T^*_{-m}\psi^*(T_{-\frac{s}{N}}D_{-\frac{r}{N}}x) D_{l}T_{m}\psi\|\qquad\textrm{ by $C$-unconditionality.}\\
&=\|f\|CN^{-2}\sum_{r,s=0}^{N-1}\|T_{-\frac{s}{N}}D_{-\frac{r}{N}}x\|\\
&=\|f\|CN^{-2}\sum_{r,s=0}^{N-1}\|x\|\qquad\textrm{ as $T_{-\frac{s}{N}}$ and $D_{-\frac{r}{N}}$ are isometries.} \\
&=\|f\|C\|x\|
\end{aligned}$$
Thus, the map $f\mapsto \int_{(a,b)\in E} \psi^{N*}_{a,b}(x)f(\psi^N_{(a,b)})d(a,b)$ defines a bounded linear functional on $L_{p'}$ with norm at most $C\|x\|$. Hence, as $L_p$ is reflexive, there exists $x_E\in L_p$ with $\|x_E\|\leq C\|x\|$ so that for all $f\in L_{p'}$ we have that $f(x_E)=\int_{(a,b)\in E} \psi^{N*}_{a,b}(x)f(\psi^N_{(a,b)})d(a,b)$. By , we have that $x_{\R^2}=x$. Thus, $(\psi^N_{a,b},\psi^{N*}_{a,b})$ is a $C$-unconditional continuous Schauder frame of $L_p$. By Lemma \[L:limit\], we have that $(\psi_{a,b},\psi_{a,b}^*)_{(a,b)\in\R^2}$ is a $C$-unconditional continuous Schauder frame of $L_p$.
Continuous Schauder frames for $\ell_p$ for $1<p<\infty$ {#S:ex2}
========================================================
Given a Banach space $X$, constructing a Schauder basis for $X$ can be done by finding a dense linearly independent sequence $(x_n)_{n=1}^\infty$ in $X$ so that the projection operators are uniformly bounded. This is usually much easier done than proving that every vector in the space has a unique basis representation. On the other hand, we don’t have any other option than to show the reconstruction formula explicitly when proving that we have a continuous Schauder frame. This makes constructing continuous Schauder frames often much harder than constructing Schauder bases. In this section we give a general procedure to construct a large class of non-trivial continuous Schauder frames for $\ell_p$ with $1<p<\infty$.
The following lemma is very similar to Young’s inequality for estimating the $L_p(\R)$ norm for convolutions of functions, and we prove it in a similar way.
\[T:Young\] Let $f\in L_1(\R)$ such that $\|\|f(t-n)\|_{\ell_1(\Z)}\|_{L_\infty(\R)}:=\sup_{t\in\R}\sum_{n\in\Z}|f(t-n)|<\infty$. If $1< p<\infty$ and $(a_n)_{n\in\Z}\in \ell_p(\Z)$ then for $p'=p/(p-1)$, $$\int |\sum_{n\in\Z} f(t-n) a_n|^p \,dt\leq \|f\|_1 \|(a_n)\|_p^p \left(\sup_{t\in\R}\sum_{n\in\Z}|f(t-n)|\right)^{p/p'}$$
$$\begin{aligned}
\int |\sum_{n\in\Z} f(t-n) a_n|^p\,dt &\leq \int \left(\sum_{n\in\Z} |f(t-n) a_n|\right)^p\!\!\!dt\\
&=\int \left(\sum_{n\in\Z} |f(t-n)|^{1/p} |a_n||f(t-n)|^{1/p'}\right)^p\,dt\\
&\leq\int \left(\sum_{n\in\Z} |f(t-n)| |a_n|^p\right)\left(\sum_{n\in\Z}|f(t-n)|\right)^{p/p'}\!\!\!\!\!dt\quad\textrm{ by Holders}\\
&\leq \left(\int \sum_{n\in\Z} |f(t-n)| |a_n|^p\,dt\right) \left(\sup_{t\in\R}\sum_{n\in\Z}|f(t-n)|\right)^{p/p'} \\
&= \left(\sum_{n\in\Z} \int |f(t-n)| |a_n|^p\,dt\right) \left(\sup_{t\in\R}\sum_{n\in\Z}|f(t-n)|\right)^{p/p'} \quad\textrm{ by Fubini}\\
&= \|f\|_{L_1(\R)} \|(a_n)\|_{\ell_p(\Z)}^p \left(\sup_{t\in\R}\sum_{n\in\Z}|f(t-n)|\right)^{p/p'} \end{aligned}$$
Using the above lemma, we are now able to construct a large class of continuous Schauder frames for $\ell_p$ for all $1<p<\infty$.
\[T:cframe\] Let $f:\R\rightarrow \R$ be a measurable function such that the following are satisfied.
1. $f\in L_1(\R)$.
2. $(f(\cdot-n))_{n\in\Z}$ is an ortho-normal sequence in $L_2(\R)$.
3. $\|\|f(t-n)\|_{\ell_1(\Z)}\|_{L_\infty(\R)}=\sup_{t\in\R}\sum_{n\in\Z}|f(t-n)|<\infty$
Let $x_t=(f(t-n))_{n\in\Z}\in\ell_p(\Z)$ and $f_t=(f(t-n))_{n\in\Z}\in\ell_q(\Z)$. Then $\psi:\R\rightarrow \ell_p(\Z)\times\ell_q(\Z)$ given by $\psi(t)=(x_t,f_t)$ is a continuous Shauder frame of $\ell_p(\Z)$. Furthermore, $\psi$ has suppression unconditionality constant $C_s=\|f\|_1sup_{t\in\R}\sum_{n\in\Z}|f(t-n)|$.
Let $n\in\N$. We have that $$\int f_t(e_n)e^*_n(x_t) dt=\int f(t-n) f(t-n)dt=\int |f(t)|^2dt=1$$ and for all $m\in\Z$ with $n\neq m$ we have that $$\int f_t(e_n)e^*_m(x_t) dt=\int f(t-n) f(t-m)dt=\int f(t) f(t-(m-n))dt=0.$$ Thus, we just need to show that $\int f_t(x)x_t\,dt$ is Pettis integrable for all $x\in \ell_p$. Let $E\subseteq\R$ be measurable, $x=(a_n)_{n\in\Z}\in \ell_p$, and $x^*=(b_n)_{n\in\Z}\in\ell_q$. Let $g(t)=\sum_{n\in\Z}a_n 1_{[n,n+1)}$ $$\begin{aligned}
|\int_E f_t(x)x^*(x_t)dt|&=|\int_E (\sum_{n\in\Z} f(t-n) a_n)(\sum_{m\in\Z}f(t-m)b_m)dt|\\
&\leq (\int |\sum_{n\in\Z} f(t-n) a_n|^p dt)^{1/p} (\int |\sum_{m\in\Z} f(t-m) b_m|^q dt)^{1/q}\quad\textrm{ by Holder's inequality}\\
&\leq \left(\|f\|_1^{1/p}\|(a_n)\|_p \left(\sup_{t\in\R}\sum_{n\in\Z}|f(t-n)|\right)^{1/q}\right)\left(\|f\|_1^{1/q}\|(b_n)\|_q \left(\sup_{t\in\R}\sum_{n\in\Z}|f(t-n)|\right)^{1/p}\right)\\
&\qquad\qquad\textrm{ by Lemma \ref{T:Young}}\\
&=C_s \|x\|_p \|x^*\|_q\end{aligned}$$
Thus, we have that $|\int_E f_t(x)x^*(x_t)dt|\leq C_s \|x\|_p \|x^*\|_q$ for all $x\in \ell_p$ and $x^*\in \ell_q$. This gives that for each measurable $E$ and $x\in \ell_p$, the map $x^*\mapsto \int_E f_t(x)x^*(x_t)dt$ defines a bounded linear functional on $\ell_q$. Hence there exists unique $x_E\in \ell_p$ such that $x^*(x_E)=\int_E f_t(x)x^*(x_t)dt$ for all $x^*\in \ell_q$. Thus, $\int f_t(x)x_t\,dt$ is Pettis integrable for all $x\in \ell_p$.
The following is an example of using Theorem \[T:cframe\] to create a non-trivial continuous Schauder frame of $\ell_p$ for $1<p<\infty$.
Let $(r_n)_{n\in\Z}$ be a sequence of different Rademacher functions on the interval $[0,1]$. That is, $|r_n(x)|=1$ for all $x\in[0,1]$ and $\int_{0}^1 r_n(x) r_m(x) dx=0$ for all $n\neq m$. Let $(a_n)_{n\in\Z}\in\ell_1$ such that $\|(a_n)_{n\in\Z}\|_2=1$. Then $f=\sum_{n\in\Z} a_n T_n r_n$ satisfies the conditions of Theorem \[T:cframe\] where $T_n$ is the operator which translates a function to the right by $n$. The suppression unconditionality constant of the resulting continuous Schauder frame is $(\sum_{n\in\Z}|a_n|)^2$.
Sampling continuous Schauder frames {#S:O}
===================================
Many important frames for Hilbert spaces arrise as samplings of continuous frames. In particular, wavelet frames, Gabor frames, and Fourier frames are all samplings of different continuous frames. Futhermore, all the frames introduced by Daubechies, Grossmann, and Meyer [@DGM] in“Painless nonorthogonal expansions"are created by sampling different coherent states. Formally, if $(M,\Sigma,\mu)$ is a $\sigma$-finite measure space and $(x_t,f_t)_{t\in M}$ is a continuous frame of a Banach space $X$ and $(t_j)_{j=1}^\infty$ is a sequence in $M$ then $(x_{t_j},f_{t_j})_{j=1}^\infty$ is called a sampling of $(x_t,f_t)_{t\in M}$. The discretization problem, posed by Ali, Antoine, and Gazeau [@AAG2], asks when a continuous frame of a Hilbert space can be sampled to obtain a frame. A solution for certain types of continuous frames was obtained by Fornasier and Rauhut using the theory of co-orbit spaces [@FoR] and a complete solution was recently given by Speegle and the second author [@FS] using the solution of the Kadison Singer problem by Marcus, Spielman, and Srivastava [@MSS]. In particular, every bounded continuous frame on a Hilbert space may be sampled to obtain a discrete frame.
\[P\] What are some Banach spaces where every bounded continuous Schauder frame may be sampled to obtain a discrete Schauder frame? What are some Banach spaces where there exists a bounded continuous Schauder frame which cannot be sampled to obtain a discrete Schauder frame?
Note that the discretization problem was solved for continuous Hilbert space frames, and thus Problem \[P\] is even open for continuous Schauder frames for separable Hilbert spaces.
[CP]{}
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[^1]: The second author was supported by grant 353293 from the Simon’s foundation. This paper forms a portion of the first authors masters thesis which was prepared at St Louis University.
[^2]: 2010 *Mathematics Subject Classification*: 42C15, 81R30, 46B10
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abstract: 'The high-redshift radio galaxy MRC1138-262 (‘Spiderweb Galaxy’; $z = 2.16$), is one of the most massive systems in the early Universe and surrounded by a dense ‘web’ of proto-cluster galaxies. Using the Australia Telescope Compact Array, we detected CO(1-0) emission from cold molecular gas – the raw ingredient for star formation – across the Spiderweb Galaxy. We infer a molecular gas mass of M$_{\rm H2} = 6 \times 10^{10}$ M$_{\odot}$ (for M$_{\rm H2}$/L’$_{\rm CO}$=0.8). While the bulk of the molecular gas coincides with the central radio galaxy, there are indications that a substantial fraction of this gas is associated with satellite galaxies or spread across the inter-galactic medium on scales of tens of kpc. In addition, we tentatively detect CO(1-0) in the star-forming proto-cluster galaxy HAE 229, 250 kpc to the west. Our observations are consistent with the fact that the Spiderweb Galaxy is building up its stellar mass through a massive burst of widespread star formation. At maximum star formation efficiency, the molecular gas will be able to sustain the current star formation rate (SFR $\approx$ 1400 M$_{\odot}$yr$^{-1}$, as traced by Seymour et al.) for about 40 Myr. This is similar to the estimated typical lifetime of a major starburst event in infra-red luminous merger systems.'
author:
- |
B. H. C. Emonts$^{1}$[^1], I. Feain$^{1}$, H. J. A. Röttgering$^{2}$, G. Miley$^{2}$, N. Seymour$^{1}$, R. P. Norris$^{1}$, C. L. Carilli$^{3}$, M. Villar-Martín$^{4}$, M. Y. Mao$^{3}$, E. M. Sadler$^{5}$, R.D. Ekers$^{1}$, G.A. van Moorsel$^{3}$, R.J. Ivison$^{6,7}$, L. Pentericci$^{8}$, C.N. Tadhunter$^{9}$, D.J. Saikia$^{10,11}$\
$^{1}$CSIRO Astronomy and Space Science, Australia Telescope National Facility, PO Box 76, Epping NSW, 1710, Australia\
$^{2}$Leiden Observatory, University of Leiden, P.O. Box 9513, 2300 RA Leiden, Netherlands\
$^{3}$National Radio Astronomy Observatory, P.O. Box 0, Socorro, NM 87801-0387, USA\
$^{4}$Centro de Astrobiología (INTA-CSIC), Ctra de Torrejón a Ajalvir, km 4, 28850 Torrejón de Ardoz, Madrid Spain\
$^{5}$School of Physics, University of Sydney, NSW 2006, Australia\
$^{6}$UK Astronomy Technology Centre, Science and Technology Facilities Council, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ\
$^{7}$Institute for Astronomy, University of Edinburgh, Blackford Hill, Edinburgh EH9 3HJ\
$^{8}$INAF Osservatorio Astronomico di Roma, Via Frascati 33,00040 Monteporzio (RM), Italy\
$^{9}$Department of Physics and Astronomy, University of Sheffield, Sheffield S3 7RH, UK\
$^{10}$Cotton College State University, Panbazar, Guwahati 781 001, India\
$^{11}$National Centre for Radio Astrophysics, TIFR, Ganeshkhind, Pune 411 007, India
title: 'CO(1-0) detection of molecular gas in the massive Spiderweb Galaxy ($z$=2)[^2]'
---
\[firstpage\]
galaxies: active – galaxies: high-redshift – galaxies: clusters: individual: Spiderweb – galaxies: individual: MRC1138-262 – galaxies: formation – galaxies: ISM
Introduction {#sec:intro}
============
High-$z$ Radio Galaxies (HzRGs) are signposts of large over-densities in the early Universe, or proto-clusters, which are believed to be the ancestors of local rich clusters [e.g. @mil08; @ven07]. Historically, HzRGs were often identified by the ultra-steep spectrum of their easily detectable radio continuum, which served as a beacon for tracing the surrounding faint proto-cluster [@rot94; @cha96]. HzRGs are typically the massive central objects in these proto-clusters.
One of the most impressive HzRGs is MRC1138-262, also called the ‘Spiderweb Galaxy’ [$z=2.16$; @pen97; @mil06]. It is one of the most massive galaxies in the early Universe [M$_{\star}$$\sim$2$\times$10$^{12}$M$_{\odot}$; @sey07; @bre10]. The Spiderweb Galaxy is a conglomerate of star forming clumps [or ‘galaxies’, following @hat09] that are embedded in a giant ($>$200kpc) [Ly$\alpha$]{} halo, located in the core of the Spiderweb proto-cluster [@pen97; @car98; @car02]. The central galaxy hosts the ultra-steep spectrum radio source MRC1138-262 [$-1.2 \leq \alpha^{\rm 8.1\,GHz}_{\rm 4.5\,GHz} \leq -2.5$ for the various continuum components; @pen97; @car97]. This source exerts dramatic feedback onto the [Ly$\alpha$]{} gas [@nes06]. Emission-line surveys identified tens of galaxies to be associated with the Spiderweb Galaxy and its surrounding proto-cluster [@pen00; @kur04; @cro05; @doh10; @kui11]. These galaxies harbour a significant fraction of the system’s unobscured star formation [@hat09]. Over-densities of red sequence galaxies were found by @kod07 and @zir08. At Mpc scales, an overdensity of X-ray AGN trace a filamentary structure roughly aligned with the radio axis [@pen02]. @kui11 showed that the proto-cluster is dynamically evolved and a possible merger of two subclusters. @hat09 predicted that most of the central proto-cluster galaxies will merge with MRC1138-262 and double its stellar mass by $z=0$, but that gas depletion will have exhausted star formation long before, so that the Spiderweb Galaxy will evolve into a cD galaxy found in the centres of present-day clusters.
Sub-millimeter observations showed that the Spiderweb has massive star formation extended on scales of $>$200kpc [@ste03], in agreement with PAH (Polycylic Aromatic Hydrocarbon) emission from MRC 1138-262 and two H$\alpha$-emitting companions [@ogl12]. Despite significant jet-induced feedback [@nes06; @ogl12], @sey12 derived a high star formation rate of SFR $\approx$ 1400 M$_{\odot}$yr$^{-1}$ and an AGN accretion rate of 20$\%$ of the Eddington limit for MRC1138-262, indicating that it is in a phase of rapid growth of both black hole and host galaxy.
How long this phase will last and how much stellar mass will be added depends on the available fuel for the ongoing star formation. The raw ingredient for star formation (and potential AGN fuel) is molecular hydrogen (H$_{2}$). Extremely luminous mid-IR line emission from warm ($T>300$K), shocked H$_{2}$ gas has been detected in MRC1138-262 with [*Spitzer*]{} [@ogl12]. This indicates that the radio jets may heat large amounts of molecular hydrogen, possibly quenching star formation in the nucleus (see @ogl12 for a discussion). However, in order to fuel the observed large star formation rate, an additional extensive reservoir of cold molecular gas must be present. An excellent tracer of the cold component of H$_{2}$ is carbon-monoxide, CO([*J,J-1*]{}). Particularly efficient is the study of the ground-transition CO(1-0), which is the most robust tracer of the overall H$_{2}$ gas, including the widespread, low-density and sub-thermally excited component [@pap00; @pap01; @pap02; @dan09; @car10; @ivi11].
In this paper,we present the detection of CO(1-0) in the Spiderweb Galaxy. We assume H$_{0} = 71$[km s$^{-1}$Mpc$^{-1}$]{}, $\Omega_{\rm M} = 0.27$ and $\Omega_{\rm \Lambda} = 0.73$ (i.e. angular distance scale of 8.4 kpc arcsec$^{-1}$ and luminosity distance $D_{\rm L} = 17309$ Mpc).
Observations {#sec:observations}
============
CO(1-0) observations were performed with the Australia Telescope Compact Array (ATCA) during Aug 2011 - Mar 2012 in the compact hybrid H75 and H168 array configurations. The total on-source integration time was 22h [after discarding data taken in poor weather, i.e. atmospheric path length rms fluctuations $>$400$\mu$m; @mid06]. Both 2GHz ATCA bands were centred close to $\nu_{\rm obs}$=36.5GHz (T$_{\rm sys}$$\sim$$70-100$K), corresponding to the redshifted CO(1-0) line. The phases and bandpass were calibrated every 5-15 min with a 2 min scan on the nearby bright calibrator PKS1124-186. Fluxes were calibrated using Mars. For the data reduction we followed @emo11b. The relative flux calibration accuracy between runs was $\la$5$\%$, while the uncertainty in absolutely flux accuracy was up to $20\%$ based on the flux-model for Mars (version March 2012). The broad 2GHz band ($\Delta$v $\approx 16,000$ [km s$^{-1}$]{}) allowed us to separate the continuum from the line emission in the uv-domain by fitting a straight line to the line-free channels. We Fourier transformed the line data[^3] to obtain a cube with robust weighting +1 [@bri95], beam-size 9.54”$\times$5.31” (PA63.3$^{\circ}$) and channel width 8.6 [km s$^{-1}$]{}. The line data were binned by 15 channels and subsequently Hanning smoothed to a velocity resolution of 259 [km s$^{-1}$]{}, resulting in a noise level of 0.085 mJybm$^{-1}$ per channel.
The spectra presented in this paper were extracted against the central pixel in the regions descibed in the text (pixelsize $2.3'' \times 2.3''$), unless otherwise indicated. Total intensity images of the CO(1-0) emission were made by summing the channels across which CO(1-0) was detected. All estimates of $L'_{\rm CO}$ in this paper have been derived from these total intensity images. The data were corrected for primary beam attenuation (FWHM$_{\rm PrimBeam} = 77$arcsec) and are presented in optical barycentric velocity with respect to $z=2.161$.
Results {#sec:results}
=======
We detect CO(1-0) emission in the Spiderweb Galaxy (Fig. \[fig:map\]). The CO(1-0) profile appears double-peaked, with a firm 5$\sigma$ ‘red’ peak and tentative 3$\sigma$ ‘blue’ peak, separated by $\sim$1000 [km s$^{-1}$]{}(with $\sigma$ derived from the integrated line profile). Figure \[fig:map\] [*(left)*]{} shows a total intensity map of the red and blue component. The total (‘red+blue’) CO(1-0) emission-line luminosity that we derive is $L'_{\rm CO} = 7.2 \pm 0.6 \times 10^{10}$ ${\rm K~km~s^{-1}~pc^2}$ [following equation 3 in @sol05].[^4] Table \[tab:table\] summarises the CO(1-0) emission line properties.
{width="89.00000%"}
Figure\[fig:map\] [*(left)*]{} shows that the bulk of the CO(1-0) coincides with the radio galaxy (region ‘B’). However, there are strong indications from both the gas kinematics and distribution that a significant fraction of the CO(1-0) emission is spread across tens of kpc.
First, despite the limited spatial resolution of our observations, there appears to be a velocity gradient in the gas kinematics across the inner 30-40 kpc of the Spiderweb. As shown in Fig.\[fig:map\], the redshift of the CO(1-0) peak emission decreases when going from region ‘A’ to region ‘B’ to region ‘C’. Most prominent is the apparent spatial separation between the peak of the blue component of the double-peaked CO(1-0) profile in region C and the peak of the red component in region B. Fig.\[fig:chanmaps\] visualises that also between regions B and A there is a clear velocity gradient in the CO(1-0) emission. The decrease in the velocity of the CO(1-0) peak emission from region A $\rightarrow$ B $\rightarrow$ C is consistent with a decrease in redshift of optical line emitting galaxies found in these regions (see Fig.\[fig:map\], though note that the large ATCA beam prevents us from spatially resolving the individual galaxies in our CO data). Fig.\[fig:map\] also shows indications for an extension (‘tail’) of the CO(1-0) emission beyond region A (stretching up to 100 kpc NE of the radio core), but this tail is detected only at a 3$\sigma$ level and thus needs to be verified. We have started an observational program to verify the distribution and kinematics of the CO(1-0) emission at higher sensitivity and spatial resolution (results will be reported in a future paper).
{width="\textwidth"}
In addition, the double-peaked CO(1-0) profile spreads over 1700 [km s$^{-1}$]{} (FWZI). This is extreme compared to what is found for quasars and submm-galaxies [see @cop08; @wan10; @ivi11; @rie11; @bot12; @kri12 and references therein]. A few notable exceptions are high-$z$ systems in which the broad CO profiles arise from merging galaxies [@sal12 and references therein]. As can be seen in Fig.1 [*(bottom right)*]{}, the double-peaked CO profile resembles the velocity distribution of optical line emitters detected in the Spiderweb proto-cluster [@kui11], be it with a lower velocity dispersion of the CO gas, in particular on the blue-shifted side.
These results thus indicate that a significant fraction of the CO(1-0) detected in the Spiderweb Galaxy likely originates from (merging) satellites of the central radio galaxy, or the inter-galactic medium (IGM) between them.
---------------------- ----------------------------------------------- -- -- --------------------------------- ------------------- --------------------------------- ---------------- ----------------- -------------------
[**HAE229**]{}
SG region A SG region B SG region C
$z_{\rm CO(1-0)}$ 2.163 $\pm$ 0.001 2.161 $\pm$ 0.001 2.150 $\pm$ 0.001 2.147 $\pm$ 0.001
v$_{\rm CO(1-0)}$ ([km s$^{-1}$]{}) 175 $\pm$ 75 0 $\pm$ 30 -1060$^{+185}_{-40}$ -1355 $\pm$ 65
FWHM ([km s$^{-1}$]{}) 550$^{+165}_{-210}$$^{\dagger}$ 540 $\pm$ 65 550$^{+150}_{-300}$$^{\dagger}$ 395 $\pm$ 75
FWZI ([km s$^{-1}$]{}) 775 $\pm$ 130 520 $\pm$ 130
$S_{\nu}$(peak) (mJybeam) 0.44 $\pm$ 0.06 0.44 $\pm$ 0.06 0.30 $\pm$ 0.09 0.32 $\pm$ 0.05
$I_{\rm CO(1-0)}$ (Jy[km s$^{-1}$]{}) 0.03 $\pm$ 0.01 0.14 $\pm$ 0.01
${L'}_{\rm CO(1-0)}$ ($\times$ 10$^{10}$ K[km s$^{-1}$]{}pc$^{2}$) 0.7 $\pm$ 0.2 3.3 $\pm$ 0.2
M$_{\rm H2}$ ($\times$ 10$^{10}$ M$_{\odot}$) 0.6 $\pm$ 0.2 3 $\pm$ 1
---------------------- ----------------------------------------------- -- -- --------------------------------- ------------------- --------------------------------- ---------------- ----------------- -------------------
$^{\dagger}$ Values are quoted for a single Gaussian profile fit, with errors reflecting uncertainties due to assymmetry of the corresponding profile component.\
$^{\ddagger}$ Regions A and B are spatially unresolved and only marginally resolved kinematically, hence a single value is derived from the entire ‘red’ part of the total intensity image of Fig.\[fig:map\].
Our results also suggests that the redshift of the central radio galaxy is associated with the red peak of the CO(1-0) profile, giving $z_{\rm CO(1-0)} = 2.161 \pm 0.001$. @kui11 discuss that determining the redshift from optical and UV rest-frame emission lines is bound to a much larger uncertainty, but they derive $2.158 < z < 2.170$, which is in agreement with our estimated $z_{\rm CO(1-0)}$.
HAE 229
-------
Fig.\[fig:HAE229\] shows that also the dusty star-forming galaxy HAE 229 [M$_{\star}$$\sim$5$\times$10$^{11}$M$_{\odot}$; @kur04; @doh10] is detected in CO(1-0) at 3.7$\sigma$ significance. We derive $L'_{\rm CO} = 3.3 \pm 0.2 \times 10^{10}\ {\rm K~km~s^{-1}~pc^2}$ for HAE229. Table \[tab:table\] summarises the CO(1-0) properties. HAE229 is located 250 kpc (30”) west of MRC1138-262, i.e. outside the gaint [Ly$\alpha$]{} halo. The CO(1-0) signal peaks at $v = -1354$[km s$^{-1}$]{}, which agrees with the H$\alpha$ redshift from @kur04. None of the other line-emitting galaxies outside the [Ly$\alpha$]{} halo, but within the field-of-view of our observations, is reliably detected in CO(1-0).
Discussion {#sec:discussion}
==========
Molecular gas in the Spiderweb {#sec:mass}
------------------------------
We can estimate the mass of molecular gas by adopting a standard conversion factor $\alpha_{\rm x} = {\rm M}_{\rm H2}/{\rm L}'_{\rm CO} = 0.8$ \[M$_{\odot}$ (K [km s$^{-1}$]{} pc$^{2}$)$^{-1}$\] [where M$_{\rm H2}$ includes a helium fraction; e.g. @sol05]. This is consistent with $\alpha_{\rm x}$ found in ultra-luminous infra-red galaxies [$L_{\rm IR} > 10^{12} L_{\odot}$; @dow98], but we stress that the conversion from $L'_{\rm CO}$ into M$_{\rm H2}$ is not yet well understood [@tac08; @ivi11] and that $\alpha_{\rm x}$ crucially depends on the properties of the gas, such as metallicity and radiation field [@glo11]. Adopting $\alpha_{\rm x} = 0.8$ M$_{\odot}$ (K [km s$^{-1}$]{} pc$^{2}$)$^{-1}$ results in an estimated molecular gas mass in the Spiderweb Galaxy of M$_{\rm H2} \sim 6 \times 10^{10}$ M$_{\odot}$. We argue that this is likely a conservative estimate, based on the adopted conversion factor and the fact that a large amount of shock-heated molecular gas resides in the warm (T$>$100K) phase [see @ogl12]. The putative H$_{2}$ mass of HAE229 is M$_{\rm H2}$$\sim$3$\times$$10^{10}$M$_{\odot}$.
### Nature of the molecular gas {#sec:nature}
In Sect. \[sec:results\] we saw that, while the CO(1-0) distribution is concentrated on the central radio galaxy, the CO emission spreads across the inner 30-40 kpc (a region which is rich in satellite galaxies). Based on its distribution and kinematics, we argued that part of the molecular gas is thus most likely associated with these satellite galaxies or the IGM between them. The extreme FWZI of the CO(1-0) emission (Sect. \[sec:results\]) is another indication that the double-peaked profile is not likely caused by a nuclear disc in the central radio galaxy. This is consistent with earlier speculation by @ogl12 that the high star formation rates could be the result of the accretion of gas or gas-rich satellites (which were found by @hat09 to contain most of the dust-uncorrected, instantaneous star formation), while nuclear star formation may be quenched by jet-induced heating of the molecular gas.
The tentative NE tail (see Fig.\[fig:map\]) spreads further out, beyond region A (which is rich in satellite galaxies) into a region with no known companion galaxies or detectable [Ly$\alpha$]{} emission (Fig. \[fig:HAE229\]). However, a Mpc-scale filamentary structure exists in east-west direction [@pen02]. We speculate that – if confirmed – this tentative tail might indicate that cold gas is found, or being accreted along, this filament.
Two alternative scenarios that should be considered to explain the CO characteristics are AGN driven outflows and cooling flows. Cooling flows have been detected in CO in giant central cluster (cD) galaxies at $z$$<$0.4, some of which contain H$_{2}$ masses similar to that of the Spiderweb Galaxy. However, compared to the Spiderweb Galaxy, these cooling flow galaxies show much narrower typical line widths (FWHM$_{\rm CO} < 500$[km s$^{-1}$]{}, taking into account an uncertain $\alpha_{\rm x}$ and the use of narrow-band receivers; @edg01 [@sal03; @sal06]). Radio-jet driven outflows of optical emission-line gas were found on scales of tens of kpc in the Spiderweb Galaxy by @nes06. Similar to the CO distribution, these optical emission lines are significantly more redshifted NW compared to SE of the radio core, though the ambiguity in optical redshifts makes a direct comparison difficult. Still, there is an interesting alignment between the redshifted CO(1-0) emission in region A and the region in which @nes06 detect the fastest redshifted outflow velocities in the optical emission-line gas (their ‘zone 1’, which stretches in between region A and B). The FWHM of the optical emission-line gas is, however, significantly larger than that of the CO(1-0) emission, indicating that it has a much larger velocity dispersion. Both the cooling flow and the radio-jet feedback scenario deserve further investigation, once CO observations with higher resolution and sensitvity have confirmed the extent of the CO(1-0) emission.
Evolutionary stage {#sec:evolution}
------------------
{width="\textwidth"}
From fitting the mid- to far-IR spectral energy distribution, @sey12 derive a starburst IR-luminosity of L$_{\rm IR} = 8 \times 10^{12}$L$_{\odot}$ and star formation rate of SFR=1390 M$_{\odot}$yr$^{-1}$ for MRC1138-262. L$_{\rm IR}$/L’$_{\rm CO(1-0)}$ agrees well with correlations found in various types of low- and high-$z$ objects [e.g. @ivi11]. Assuming that all the H$_{2}$ is available to sustain the high SFR, we derive a minimum mass depletion time-scale of t$_{\rm depl} = \frac{{\rm M}_{\rm H2}}{\rm SRF} \approx 40$ Myr. This is comparable to the estimated typical lifetime of a major starburst episode in IR-luminous merger systems [@mih94; @swi06], though the bulk of the intense star formation in the Spiderweb Galaxy may occur on scales of tens of kpc (Sect. \[sec:nature\]). The mass depletion time-scale may be shorter if the cold molecular gas is more rapidly depleted by feedback processes, such as shock-heating [@ogl12] or jet-induced outflows (found to occur at rates of $\sim$400M$_{\odot}$yr$^{-1}$ in the optical emission line gas by @nes06). Nevertheless, both the current large star formation rate and cold molecular gas content imply that we are witnessing a phase of rapid galaxy growth though massive star formation, coinciding with the AGN activity.
The H$_{2}$ mass is much larger than the estimated mass of the emission-line gas in the [Ly$\alpha$]{} halo [M$_{\rm emis} = 2.5 \times 10^{8}$M$_{\odot}$; @pen97]. However, @car02 show that the radio source is enveloped by a region of hot, shocked X-ray gas of potentially M$_{\rm hot} = 2.5 \times 10^{12}$M$_{\odot}$. This suggests that, even when the current reservoir of cold molecular gas is consumed, there is a potential gas reservoir available for future episodes of starburst (and AGN) activity, provided that the gas can cool down to form molecular clouds [e.g. @fab94] and this process is not entirely counter-acted by ongoing AGN feedback [@car02; @nes06; @ogl12]. The merger of proto-cluster galaxies with the Spiderweb Galaxy may also trigger a new burst of star formation, depending on the available gas reservoir in these systems. For the dusty star-forming galaxy HAE229, $L'_{\rm CO(1-0)}$ is comparable to that of IR-selected massive star-forming galaxies at $z$=1.5 [@ara10] and some high-$z$ submm galaxies [e.g. @ivi11]. Our CO results are consistent with observations by @ogl12 that HAE229 is going through a major and heavily obscured starburst episode. From their calculated SFR$\sim$880M$_{\odot}$yr$^{-1}$, we derive t$_{\rm depl} \approx 30$ Myr, i.e. similar to that of the Spiderweb Galaxy.
CO(1-0) in HzRGs {#sec:coinhzrg}
----------------
MRC1138-262 is part of an ATCA survey for CO(1-0) in a southern sample of HzRGs [$1.4<z<3$; Emonts et al in prep, see also @emo11a; @emo11b]. So far, it is one of only very few secure CO(1-0) detections among HzRGs; two other examples being MRC0152-209 [$z=1.92$; @emo11a] and 6C1909+72 [$z=3.53$; @ivi12].
CO detections in HzRGs made with narrow-band receivers and/or higher transitions are also still limited in number [@sco97; @all00; @pap00; @pap01; @bre03; @bre03AR; @bre05; @gre04; @kla05; @nes09; @emo11b; @ivi08; @ivi12 also review by @mil08]. However, in some cases CO is resolved on tens of kpc scales [@ivi12], associated with various components [e.g. merging gas-rich galaxies; @bre05], or found in giant Ly$\alpha$ halos that surround the host galaxy [@nes09]. This shows that detectable amounts of cold molecular gas in HzRGs are not restricted to the central region of the radio galaxy. @ivi12 discuss that CO detected HzRGs (often pre-selected on bright far-IR emission from a starburst) are generally associated with merger activity. This is also the case for the Spiderweb Galaxy, which is located in an extreme merger environment.
Using the JVLA, @car11 mapped CO(2-1) emission throughout a $z$=4 proto-cluster associated with the sub-millimeter galaxy GN20 (tracing a combined mass of M$_{\rm H2} \sim 2 \times 10^{11}$M$_{\odot}$). Our results on the Spiderweb Galaxy and HAE229 are another example of the potential for studying the lowest CO transitions in proto-cluster environments with the ATCA and JVLA.
Conclusions {#sec:conclusions}
===========
We detect CO(1-0) emission from cold molecular gas across the massive Spiderweb Galaxy, a conglomerate of star forming galaxies at z=2.16. While the bulk of the CO(1-0) coincides with the central radio galaxy, part of the molecular gas is spread across tens of kpc. We explain that this gas is most likely associated with satellites of the central radio galaxy, or the IGM between them (though other scenarios are briefly discussed). The extensive reservoir of cold molecular gas likely provides the fuel for the widespread star formation that has been observed across the Spiderweb Galaxy. Continuous galaxy-merger and gas-accretion processes are the likely triggers for the observed high star formation rates. The total mass of cold gas ($M_{\rm H2} = 6 \times 10^{10}$$_{\,[\alpha_x=0.8]}$ M$_{\odot}$) is enough to sustain the current high star formation rate in the Spiderweb Galaxy for $\sim$40 Myr, which is similar to the typical lifetime of major starburst events seen in IR-luminous merger systems. Our CO results on the Spiderweb Galaxy show the potential for studying the cold gas throughout high-z proto-clusters with the ATCA, JVLA and ALMA.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank Ernst Kuiper for sharing his HST imaging and the anonymous referee for useful feedback that improved this paper. We thank the Narrabri observatory staff for their help. BE acknowledges the Centro de Astrobiología/INTA for their hospitality. NS is recipient of an ARC Future Fellowship. The Australia Telescope is funded by the Commonwealth of Australia for operation as a National Facility managed by CSIRO.
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[^1]: E-mail:bjorn.emonts@csiro.au
[^2]: From observations with the Australia Telescope Compact Array
[^3]: Similarly, we made a continuum map (10.39”$\times$6.55”, PA75.5$^{\circ}$). We detect the 36.6GHz radio continuum with an integrated flux of $S_{\rm 36.6\,GHz}$ = 10.7 mJy ($P_{\rm 36.6 GHz} = 3.6 \times 10^{26}$ WHz$^{-1}$) across three beam-sizes, following the morphology of high resolution 4.7/8.2GHz data of @car97. A detailed discussion on the 36.6GHz radio continuum is deferred to a future paper.
[^4]: The measurement error in $L'_{\rm CO}$ does not include a 20$\%$ uncertainty in the model of our used flux calibrator Mars (Sect. \[sec:observations\]).
|
---
abstract: 'In this paper, we made an extension to the convergence analysis of the dynamics of two-layered bias-free networks with one $ReLU$ output. We took into consideration two popular regularization terms: the $\ell_1$ and $\ell_2$ norm of the parameter vector $w$, and added it to the square loss function with coefficient $\lambda/2$. We proved that when $\lambda$ is small, the weight vector $w$ converges to the optimal solution $\hat{w}$ (with respect to the new loss function) with probability $\geq (1-\varepsilon)(1-A_d)/2$ under random initiations in a sphere centered at the origin, where $\varepsilon$ is a small value and $A_d$ is a constant. Numerical experiments including phase diagrams and repeated simulations verified our theory.'
author:
- Zhifeng Kong
title: 'Convergence Analysis of the Dynamics of a Special Kind of Two-Layered Neural Networks with $\ell_1$ and $\ell_2$ Regularization'
---
Introduction
============
A substantial issue in deep learning is the theoretical analysis of complex systems. Unlike multi-layer perceptrons, deep neural networks have various structures [@summary], which mainly come from intuitions, and they sometimes yield good results. On the other hand, the optimization problem usually turns out to be non-convex, thus it is difficult to analyze whether the system will converge to the optimal solution with simple methods such as stochastic gradient descent.
In Theorem 4 in [@related], convergence for a system with square loss with $\ell_2$ regularization is analyzed. However, assumption 6 in [@related] requires the activation function $\sigma$ to be three times differentiable with $\sigma'(x)>0$ on its domain. Thus the analysis cannot be applied to some popular activation functions such as $ReLU$ [@relu] and $PReLU$ [@prelu], where $ReLU(x)=\max(x,0)$ and $PReLU(x)=\max(x,\alpha x),\ 0<\alpha<1$.
Theorem 3.3 in [@work] provides another point of view to analyze the $\sigma=ReLU$ situation by using the Lyapunov method [@lyapunov]. The conclusion is weaker: the probability of convergence is less than $1/2$. However, this method successfully deals with this activation function. In this paper, we take into consideration $\ell_1$ and $\ell_2$ regularization and analyze the convergence of these two systems with an analogous method. Also, a similar conclusion is drawn in the end.
The square of the $\ell_1$ and $\ell_2$ norms of a vector $v$ are $$\|v\|_1^2=\left(\sum_{i=1}^n |v_i|\right)^2,~~
\|v\|_2^2=\sum_{i=1}^n v_i^2=v^{\top}v.$$ These two regularization terms are popular because they control the scale of $v$. Because there is an important difference between $\ell_1$ and $\ell_2$ regularization (usually it is possible to acquire an explicit solution of a system with $\ell_2$ regularization, but hard for a system with $\ell_1$ regularization), we need different tools to deal with the problems.
Preliminary
===========
In this paper a two-layered neural network with one $ReLU$ output is considered. Let $X=(x_1,x_2,\cdots,x_N)^{\top}$, an $N\times d$ matrix ($N\gg d$), be the input data. Assume that the columns of $X$ are identically distributed Gaussian independent random $d$-dimensional vector variables: $x_i$’s $(i.i.d.)$ $\sim \mathcal{N}(0,I_d)$. Let $w$, a vector with length $d$, be the vector of weights (parameters) to be learned by the model. Let $w^*$ be the optimal weight with respect to $X$. Let $\sigma=ReLU$ be the activation function. Then, the output with input vector $x$ and weight $w$ is $g(x,w)=\sigma(x^{\top}w)$. For convenience, define $g(X,w)$ an $N\times1$ vector with $i^{th}$ element $g(x_i,w)$. Now, we are able to write down the loss function with the regularization term $R(w)$: $$E(w)=\frac{1}{2N}\|g(X,w^*)-g(X,w)\|^2+\frac{\lambda}{2}R(w),$$ where $\lambda\geq0$ is a parameter. When $\lambda=0$, there is no regularization. In this paper, we focus on the situation where $R(w)=$ $\|w\|_1^2$ or $\|w\|_2^2$ and $\lambda$ is very small.
We have a easy way to represent $g(X,w)$ by introducing a new matrix function $D$ given by $D(w)=diag\{d_1,d_2,\cdots, d_N\}$ where $d_i=1$ if $(Xw)_i>0$ and $d_i=0$ if $(Xw)_i\leq0$. Then, $g(X,w)$ can be written in matrix form: $$g(X,w)=D(w)Xw.$$ Additionally, let $D^*=D(w^*)$ for convenience.
Now we introduce the gradient descent algorithm for the model. The iteration has the form $$w^{t+1}=w^t+\eta\Delta w^t,$$ where $\eta$ is the learning rate (usually small) and $\Delta w^t=-\nabla_w E(w^t)$ is the negative gradient of the loss function. According to [@work] $\Delta w$ has the closed form $$\Delta w=\frac1N X^{\top}D(w)\left(D^*Xw^*-D(w)Xw\right)+\frac{\lambda}{2}\frac{\partial R}{\partial w}.$$ Its expectation (corresponding to $X$) is given explicitly by $$\mathbb{E}\Delta w=\frac 12(w^*-w)+\frac{1}{2\pi}\left((\alpha\sin\theta) w-\theta w^*\right)+\frac{\lambda}{2}\frac{\partial R}{\partial w},$$ where $\alpha=\|w^*\|/\|w\|$ and $\theta\in(0,\pi/2]$ is the angle between $w$ and $w^*$.
Theoretical Analysis
====================
\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\# Now consider the Lyapunov function $$V(w)=\frac12\|w-w^*\|^2+\frac{\lambda}{2}\|w\|^2$$ Now, we are going to prove that $\dot{V}=(w-w^*)^{\top}\mathbb{E}\Delta w+\lambda w^{\top}\mathbb{E}\Delta w$ is negative-definite. Let $y=(\|w\|,\|w^*\|)^{\top}$, we can write $\dot{V}=-y^{\top}My$. The matrix $M$ is given in the following: $$M=\left(
\begin{array}{cc}
M_{11} & M_{12} \\
M_{21} & M_{22} \\
\end{array}
\right)$$
$$\begin{array}{rl}
\dot{V}=&((1+\lambda)w-w^*)^{\top}\left(\left(\lambda-\frac12\right)w+\left(\frac12-\frac{\theta}{2\pi}\right)w^*+\frac{\sin\theta}{2\pi}\frac{\|w^*\|}{\|w\|} w\right)\\
=& (1+\lambda)\left(\lambda-\frac12\right)\|w\|^2-\left(\frac12-\frac{\theta}{2\pi}\right)\|w^*\|^2
+\left((1+\lambda)\left(\frac12-\frac{\theta}{2\pi}\right)-\left(\lambda-\frac12\right)\right)\cos\theta\|w\|\|w^*\|\\
&+ (1+\lambda)\frac{\sin\theta}{2\pi}\|w\|\|w^*\|-\frac{\sin\theta}{2\pi}\cos\theta \|w^*\|^2
\end{array}$$
Thus, $$M_{11}=-(1+\lambda)\left(\lambda-\frac12\right)=(1+\lambda)\left(\frac12-\lambda\right)$$ $$M_{22}=\frac12-\frac{\theta}{2\pi}+\frac{\sin\theta}{2\pi}\cos\theta$$ $$M_{12}=M_{21}=-\frac12\left(\left((1+\lambda)\left(\frac12-\frac{\theta}{2\pi}\right)-\left(\lambda-\frac12\right)\right)\cos\theta+(1+\lambda)\frac{\sin\theta}{2\pi}\right)$$ \#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#
Usually, $w$ does not converge to $w^*$ because of the regularization term. Let $\hat{w}$ be the optimal weight vector that minimizes $E(w)$, i.e. $\frac{\partial E}{\partial w}(\hat{w})=0$. First, we’ll solve $\hat{w}$ for small $\lambda$, and then we prove that $w^t$ will converge to $\hat{w}$ in $\hat{\mathcal{B}}_{\|w^*\|}(w^*)=\mathcal{B}_{\|w^*\|}(w^*)\setminus \ell(w^*)$ using the Lyapunov method [@lyapunov], where $\mathcal{B}_{r}(y)=\{x\in\mathbb{R}^d:\|x-y\|^2\leq r^2\}$ and $\ell(y)$ is the line $\{ky:k\in\mathbb{R}\}$ .
We firstly provide three lemmas that help the analysis in sections 3.2 and 3.3. The lemmas show that extreme situations will happen with small probability, and provide some mathematical tricks that are useful in the theoretical analysis.
Preparation
-----------
**Lemma 1**: $A_k=Prob\left\{rank(D^*)\leq k\right\}=2^{-N}\sum_{i=0}^k\binom Ni$ for $k\in\{0,1,\cdots,N\}$.
*Proof*: Let $x\sim\mathcal{N}(0,I_{d})$, then $Prob\{x^{\top}w^*\leq0\}=\frac12$. Thus, $$Prob\{rank(D^*)=i\}=\binom Ni \prod_{j=1}^i Prob\{x_j^{\top}w^*>0\}\cdot \prod_{j=i+1}^N Prob\{x_j^{\top}w^*\leq0\}=2^{-N}\binom Ni$$ Finally, $$A_k=\sum_{i=0}^k Prob\{rank(D^*)=i\}=2^{-N}\sum_{i=0}^k\binom Ni$$ When $N\gg k$, $A_k$ is a small value bounded by $2^{-N}k\binom Nk$. $\Box$
**Lemma 2**: $Prob\{X^{\top}D^*X \mbox{ is positive definite}\}=1-A_d$.
*Proof*: First we show when $rank(D^*)>d$, $Prob\{X^{\top}D^*X\mbox{ is positive definite}\}=1$. Since $x_i$’s are $i.i.d.$, any $d$ rows of $X$ are linearly independent with probability 1. This implies that with probability 1 $Xr$ doesn’t contain more that $d$ $0$’s $\forall r\in \mathbb{R}^d\setminus\{0\}$. However, $D^*$ has more than $d$ $1$’s, so $r^{\top}X^{\top}D^*Xr=(Xr)^{\top}D^*(Xr)>0\ a.s.$ Then since $Prob\{rank(D^*)>d\}=1-A_d$, the probability that $X^{\top}D^*X$ is positive definite also equals to this amount. $\Box$
**Lemma 3**: For a positive definite matrix $B$ and a small value $\varepsilon$ $$(B-\varepsilon I)^{-1}=(I+\varepsilon B^{-1}+\textbf{o}(\varepsilon))B^{-1}$$ where $\textbf{o}(\varepsilon)$ refers to a matrix with every element $=o(\varepsilon)$.
*Proof*: Since $B$ is positive definite, $B^{-1}$ exists. Then, $$\begin{array}{ll}
(B-\varepsilon I)^{-1}&=\left(B(I-\varepsilon B^{-1})\right)^{-1}\\
&=(I-\varepsilon B^{-1})^{-1}B^{-1}\\
&=(I+\varepsilon B^{-1}+\textbf{o}(\varepsilon))B^{-1}
\end{array}$$ This shows that $B^{-1}$ and $(B-\varepsilon I)^{-1}$ are closed to each other. $\Box$
Convergence Area for the $\ell_2$ Regularization Case
-----------------------------------------------------
In this case, we have $R(w)=\|w\|_2^2=w^{\top}w$, and $\partial R/\partial w=2w$. Then, the loss function is given in the following equation: $$E(w)=\frac{1}{2N}\|g(X,w^*)-g(X,w)\|^2+\frac{\lambda}{2}\|w\|_2^2$$
**Theorem 1**: When $\lambda$ is small, $\hat{w}$ can be solved explicitly with probability $1-A_d$.
*Proof*: Let $\partial E/\partial w=0$, and according to equation (5), we have $$X^{\top}D(\hat{w})(D^*Xw^*-D(\hat{w})X\hat{w})+\lambda N \hat{w}=0$$ Let’s first assume that $D(\hat{w})=D^*$. Then the equation can be simplified as $$X^{\top}D^*Xw^*=(X^{\top}D^*X-\lambda NI_d)\hat{w}$$ Thus, we have $$\hat{w}=(X^{\top}D^*X-\lambda NI_d)^{-1}X^{\top}D^*Xw^*$$ The inverse exists with probability $1-A_d$ according to lemmas 2 and 3.
We now show that when $\lambda$ is small enough, this $\hat{w}$ ensures that $D(\hat{w})=D^*$. According to lemmas 2 and 3, we have $$\hat{w}=w^*+\lambda N(X^{\top}D^*X)^{-1}w^*+\textbf{o}_{d\times1}(\lambda)$$ It is sufficient to show that $X\hat{w}$ and $Xw^*$, two vectors in $\mathbb{R}^N$, share the same signs in the $N$ positions with probability 1. These two vectors are related by the equation $$X\hat{w}=Xw^*+\lambda NX(X^{\top}D^*X)^{-1}w^*+\textbf{o}_{N\times1}(\lambda)$$ Since $Xw^*$ doesn’t contain 0 with probability 1, we can exclude these cases. Then, all terms after $Xw^*$ above don’t influence the sign of $Xw^*$ when $$\lambda\leq\frac{1}{2N}\min_{1\leq i\leq N}\frac{|(Xw^*)_i|}{\left|(X(X^{\top}D^*X)^{-1}w^*)_i\right|}$$ The “2” on the denominator is used for eliminating the effects of $\textbf{o}_{N\times1}(\lambda)$. $\Box$
Now, we have shown that $\hat{w}$ is closed to $w^*$ when $\lambda$ is small. The next step is to show that $w$ converges to $\hat{w}$ in a certain area, which the Lyapunov method [@lyapunov] is very good at. In order to apply the Lyapunov method, we regard $t$ as a continuous index.
**Theorem 2**: With probability $1-A_d$, the following statement holds. When $N$ is large and $\lambda$ is small, consider the Lyapunov function $\mathcal{V}(w)=\frac12\|w-\hat{w}\|^2$. We have $\dot{\mathcal{V}}(=\partial V/\partial t)<0$ in $\hat{\mathcal{B}}_{\|w^*\|}(w^*)$, and thus the system is asymptotically stable. That is, $w=w^{t}\rightarrow \hat{w}$ as $t\rightarrow\infty$.
*Proof*: We can write $\dot{\mathcal{V}}$ as: $$\dot{\mathcal{V}}=(w-\hat{w})^{\top}\mathbb{E}\Delta w$$ In order to simplify, let $\hat{w}=w^*+\lambda T$, where $T$ is given by $$T=N(X^{\top}D^*X)^{-1}w^*+\textbf{o}_{d\times1}(1)\in\mathbb{R}^d.$$
Note $y=(\|w\|,\|w^*\|)^{\top}$; $\dot{\mathcal{V}}$ can be written as $-y^{\top}Ky$ where $$K=M+\lambda P$$ According to Lemma7.3 [@work], $M$ is given by the following: $$M=\frac{1}{4\pi}\left(
\begin{array}{cc}
2\pi & -(2\pi-\theta)\cos\theta-\sin\theta \\
-(2\pi-\theta)\cos\theta-\sin\theta & \sin2\theta+2\pi-2\theta \\
\end{array}
\right)$$ $P$ can also be divided into two parts: $P=P_1+P_2$, where $$P_1=-\left(
\begin{array}{cc}
1 & -\frac{\cos\theta}{2} \\
-\frac{\cos\theta}{2} & 0 \\
\end{array}
\right)$$ and $P_2$ satisfies that $y^{\top}P_2y=T^{\top}\mathbb{E}\Delta w$. From this, we see that $P$ is bounded. Since $M$ is positive definite for $\theta\in(0, \pi/2]$ according to Lemma7.3 [@work], when $\lambda$ is small, $K$ is also positive definite for $\theta\in(0, \pi/2]$. As a result, $\dot{\mathcal{V}}<0$, which leads to the result that the system is asymptotically stable in $\hat{\mathcal{B}}_{w^*}(w^*)$. $\Box$
Convergence Area for the $\ell_1$ Regularization Case
-----------------------------------------------------
In this case, we have $R(w)=\|w\|_1^2$, and $\partial R/\partial w=2\|w\|_1sign(w)$, where $sign(w)$ is the vector of signs of elements in $w$. Then, the loss function is given in the following equation: $$E(w)=\frac{1}{2N}\|g(X,w^*)-g(X,w)\|^2+\frac{\lambda}{2}\|w\|_1^2$$
**Theorem 3**: When $\lambda$ is small, $\hat{w}$ can be solved (not explicitly) with probability $1-A_d$.
*Proof*: Let $\partial E/\partial w=0$, and according to equation (5), we have $$X^{\top}D(\hat{w})(D^*Xw^*-D(\hat{w})X\hat{w})+\lambda N \|\hat{w}\|_1sign(\hat{w})=0$$ We still assume that $D^*=D(\hat{w})$ to simplify the problem. Then, the equation becomes $$f(\lambda,w^*,\hat{w})=X^{\top}D^*Xw^*-X^{\top}D^*X\hat{w}+\lambda N \|\hat{w}\|_1sign(\hat{w})=0$$ This problem is hard to solve, so we use the Implicit Function Theorem [@implicit] here. The key is to examine whether the Jacobian matrix $J$ is invertible, where $J(i,j)=\partial f_i/\partial \hat{w}_j$. The result is $$\begin{array}{ll}
J(i,j)&=-(X^{\top}D^*X)_{ij}+\lambda N sign(\hat{w}_i)sign(\hat{w}_j)\\
&\displaystyle =-\sum_{k=1}^NI(x_k^{\top}w^*>0)x_{ki}x_{kj}+\lambda N sign(\hat{w}_i)sign(\hat{w}_j)
\end{array}$$ Since $X^{\top}D^*X$ is positive definite with probability $1-A_d$ according to Lemma 2, and when $\lambda$ is small the second term doesn’t influence, we know that $J$ is then invertible. Thus, there exists a unique continuously differentiable function $g$ such that $\hat{w}=g(w^*,\lambda)$ is the solution. Notice that when $\lambda=0$, $\hat{w}=w^*$ is the solution. As a result, $\hat{w}=g(w^*,\lambda)$ can be extended as $w^*+\lambda u+\textbf{o}_{d\times1}(\lambda)$ for some vector $u$. Additionally, $u$ might be very large because there is an $N$ after $\lambda$ in equations (24)-(26).
Then, we show that for $\lambda$ small, we have $D(\hat{w})=D^*$. The analysis is quite similar to Theorem 1. When $$\lambda\leq\frac12\min_{(Xw^*)_i>0}\frac{(Xw^*)_i}{|(Xu)_i|}$$ we have that $D(\hat{w})=D^*$. $\Box$
**Remark**: In Theorem 3 the bound of $\lambda$ is given in equation (27), where there is an unknown vector $u$ on the denominator $(Xu)_i$. In fact, we are able to estimate its value from known quantities. When we apply the extension $\hat{w}=w^*+\lambda u+\textbf{o}(\lambda)$ to equation (25), we have $$X^{\top}D^*X(-\lambda u+\textbf{o}(\lambda))+\lambda N\|\hat{w}\|_1sign(\hat{w})=0,$$ which is equivalent to the following equation $$X^{\top}D^*Xu=N\|\hat{w}\|_1sign(\hat{w})+\textbf{o}(1).$$ As assumed in Theorem 3, $rank{D^*}=\delta>d$, and assume that $D^*_{ii}=1$ for $i=1,2,\cdots,\delta$. Let $X_{\delta}$ be the matrix consisting the first $\delta$ rows of $X$. Then, $X^{\top}D^*Xu=X_{\delta}^{\top}X_{\delta}u$. Thus, we have $$u=N\|\hat{w}\|_1(X_{\delta}^{\top}X_{\delta})^{-1}sign(\hat{w})+\textbf{o}(1).$$
Further more, we are able to estimate each element of $u$: $$|u_i|\leq 2N\|w^*\|_1\|(X_{\delta}^{\top}X_{\delta})^{-1}\|_{\infty}$$ for small $\lambda$ such that $\|\hat{w}\|_1\leq(2-\epsilon)\|w^*\|_1$ with small value $\epsilon$ that eliminates the effect of $\textbf{o}(1)$ in equation (30). The next step is to estimate $\max_{(Xw^*)_i>0} (Xu)_i=\max_{1\leq i\leq \delta} (Xu)_i$. Let $\Phi(\sigma)=\int_{-\infty}^\sigma \exp(-t^2/2)dt$. Then, since $x_i\sim\mathcal{N}(0,I_d)$, we know that $$Prob(|x_{ij}|<\sigma, 1\leq j\leq d)=(2\Phi(\sigma)-1)^d.$$ Thus, with probability at least $(2\Phi(\sigma)-1)^{\sigma d}$, we have $$\max_{1\leq i\leq \delta} (Xu)_i\leq 2N\sigma d\|w^*\|_1\|(X_{\delta}^{\top}X_{\delta})^{-1}\|_{\infty}.$$ Finally, the bound given in equation (27) can be modified by $$\lambda\leq\frac{\min_{(Xw^*)_i>0}(Xw^*)_i}{4N\sigma d\|w^*\|_1\|(X_{\delta}^{\top}X_{\delta})^{-1}\|_{\infty}}.$$ Then we have $$Xu = N\|\hat{w}\|_1X(X_{\delta}^{\top}X_{\delta})^{-1}sign(\hat{w})+\textbf{o}(1),$$ which indicates that $$\max_{(Xw^*)_i>0}(Xu)_i\leq 2N\|\hat{w}\|_1\|X(X_{\delta}^{\top}X_{\delta})^{-1}\|_{\infty}$$ for small $\lambda$ such that $\|\hat{w}\|_1\leq(2-\epsilon)\|w^*\|_1$ with small value $\epsilon$ that eliminates the effect of $\textbf{o}(1)$ in equation (31). Finally, we are able to modify the bound in equation (27) by using the upper bound of $(Xu)_i$ in equation (32) to substitute this amount. The explicit bound is then given by the following equation: $$\lambda\leq\frac{\min_{(Xw^*)_i>0}(Xw^*)_i}{4N\|w^*\|_1\|X(X_{\delta}^{\top}X_{\delta})^{-1}\|_{\infty}}.$$ $\Box$
Although the explicit solution of $\hat{w}$ can’t be found, we still draw the conclusion that $\hat{w}$ is closed to $w^*$ for small $\lambda$. This is enough for the Lyapunov method, because we are able to control $K$ in equation (20) with a similar way.
**Theorem 4**: The statement in Theorem 2 still holds for $\ell_1$ regularization.
*Proof*: Similar to the analysis in Theorem 2, we still have equation (20) in this case with a different $P$. Thus, when $\lambda$ is small enough $K$ is positive definite, and the conclusion is still correct here. $\Box$
The Final Result
----------------
Since it’s hard to draw samples from $\hat{\mathcal{B}}_{w^*}(w^*)$, we consider a small sphere $\mathcal{B}_0(r)$ centered at the origin. The analysis is in Theorem 7.4 (proof of Theorem 3.3) in [@work].
**Theorem 5**: For both $\ell_1$ and $\ell_2$ regularization, if the initial weight vector $w^1$ is sampled uniformly in $\mathcal{B}_0(r)$ with $r\leq\varepsilon\sqrt{\frac{2\pi}{d+1}}\|w^*\|$, $w$ converges to $\hat{w}$ with probability $\geq\frac{1-\varepsilon}{2}(1-A_d)$.
*Proof*: The proof is almost exactly the same as the proof of Theorem 7.4 in [@work]. The only thing to notice is that we exclude the line $\ell(w^*)$ because we need $\theta>0$. However, the line has measure zero and thus doesn’t change the conclusion. $\Box$
Now, we have proved that Theorem 3.3 in [@work] still applies for $\ell_1$ and $\ell_2$ regularization with small $\lambda$. And this result is consistent with the argument that initial weights should be small rather than being large [@course].
Experiment Results and Analysis
===============================
First, in Figure \[four\] we demonstrate all possibilities: the dynamics converge/do not converge with $\ell_1$/$\ell_2$ regularization. The parameters are: $N=10, d=2, \eta=0.05, \varepsilon=0.1,$ and $\lambda=0.01\mbox{(for cases with convergence)}, 0.1\mbox{(for cases that without convergence)}$. In Figure \[phase\] we show two phase diagrams (or vector fields, after normalized) of the dynamics with $\ell_1$ and $\ell_2$ regularization with randomly selected $X$ and parameters $N=10, d=2$, and $\lambda=0.01$. The big black point is $w^*=(1,1)$, the small black points are the grid points uniformly selected in the plane, and green lines refer to the orientations of $\delta w$ (from the end with a black point to the end without any point). Especially, when $w$ equals to $(0,0)$ in the $\ell_1$ case the dynamic is meaningless because $\partial R/\partial w$ does not exist.
![Four possible dynamics. The left 1 shows the dynamic that converges to $\hat{w}$ with $\ell_1$ regularization and $\lambda=0.01$. The left 2 shows the dynamic that converges to $\hat{w}$ with $\ell_2$ regularization and $\lambda=0.01$. The left 3 shows the dynamic that does not converge to $\hat{w}$ with $\ell_1$ regularization and $\lambda=0.1$. The left 4 shows the dynamic that does not converge to $\hat{w}$ with $\ell_2$ regularization and $\lambda=0.1$. []{data-label="four"}](l1_converge_new.png "fig:"){width="3.5cm" height="2.5cm"} ![Four possible dynamics. The left 1 shows the dynamic that converges to $\hat{w}$ with $\ell_1$ regularization and $\lambda=0.01$. The left 2 shows the dynamic that converges to $\hat{w}$ with $\ell_2$ regularization and $\lambda=0.01$. The left 3 shows the dynamic that does not converge to $\hat{w}$ with $\ell_1$ regularization and $\lambda=0.1$. The left 4 shows the dynamic that does not converge to $\hat{w}$ with $\ell_2$ regularization and $\lambda=0.1$. []{data-label="four"}](l2_converge_new.png "fig:"){width="3.5cm" height="2.5cm"} ![Four possible dynamics. The left 1 shows the dynamic that converges to $\hat{w}$ with $\ell_1$ regularization and $\lambda=0.01$. The left 2 shows the dynamic that converges to $\hat{w}$ with $\ell_2$ regularization and $\lambda=0.01$. The left 3 shows the dynamic that does not converge to $\hat{w}$ with $\ell_1$ regularization and $\lambda=0.1$. The left 4 shows the dynamic that does not converge to $\hat{w}$ with $\ell_2$ regularization and $\lambda=0.1$. []{data-label="four"}](l1_not_converge_new.png "fig:"){width="3.5cm" height="2.5cm"} ![Four possible dynamics. The left 1 shows the dynamic that converges to $\hat{w}$ with $\ell_1$ regularization and $\lambda=0.01$. The left 2 shows the dynamic that converges to $\hat{w}$ with $\ell_2$ regularization and $\lambda=0.01$. The left 3 shows the dynamic that does not converge to $\hat{w}$ with $\ell_1$ regularization and $\lambda=0.1$. The left 4 shows the dynamic that does not converge to $\hat{w}$ with $\ell_2$ regularization and $\lambda=0.1$. []{data-label="four"}](l2_not_converge_new.png "fig:"){width="3.5cm" height="2.5cm"}\
![Phase diagrams (or vector fields, after normalized) in the $(x,y)$ plane of the dynamics with $\ell_1$(left) and $\ell_2$(right) regularization.[]{data-label="phase"}](l1_phase.png "fig:"){width="40.00000%"} ![Phase diagrams (or vector fields, after normalized) in the $(x,y)$ plane of the dynamics with $\ell_1$(left) and $\ell_2$(right) regularization.[]{data-label="phase"}](l2_phase.png "fig:"){width="40.00000%"}\
Then, in order to examine the prediction given by Theorem 5, we made the following simulation. Under different values of $N$, $d$ and $\lambda$, we simulated the dynamics for 500 times and compared the experiment ratio of convergence to the theoretical ratio (that is, the probability) of convergence in Theorem 5. Specifically, for both $\ell_1$ and $\ell_2$ situation $N$ was selected in $\{10, 20, 100\}$, $d$ was selected in $\{2, 3, 5\}$, and $\lambda$ was selected in $\{0.001, 0.01, 0.1\}$. The learning rate $\eta$ was set to be 0.05 and $\varepsilon$ was set to be 0.1. Each time $X$ was sampled according to normal distribution and $w^1$ was sampled uniformly in $\mathcal{B}_0(r)$. The results for the $\ell_1$ and $\ell_2$ regularization case are demonstrated in Table \[result\].
--- ----- ------- ------- ------- ----------- ------- ------- -------
0.001 0.01 0.1 0.001 0.01 0.1
2 10 0.425 0.912 0.832 0.436 0.940 0.912 0.700
20 0.450 0.992 0.976 0.578 0.970 0.956 0.840
100 0.450 0.996 1 0.950 0.996 0.986 0.880
3 10 0.373 0.852 0.712 [0.170]{} 0.966 0.972 0.736
20 0.449 0.994 0.966 [0.342]{} 0.998 0.996 0.940
100 0.450 1 1 0.856 1 1 0.962
5 10 0.170 0.452 0.304 [0.016]{} 1 1 0.612
20 0.441 0.97 0.820 [0.112]{} 1 1 0.960
100 0.450 1 1 0.706 1 1 1
--- ----- ------- ------- ------- ----------- ------- ------- -------
: The comparison between theoretical ratio of convergence (the 3rd col.) and experiment ratio of convergence (the 4th-9th col.) under different parameters.[]{data-label="result"}
According to Table \[result\], we are able to make the following discussion. $(i)$ As shown in the table, there are four bold numbers, all of which lie in the $\ell_2$ regularization case when $\lambda=0.1$, indicating that for the $\ell_2$ situation $0.1$ is beyond the upper bound of $\lambda$ for Theorem 1 or Theorem 2. $(ii)$ In most situations, the experiment ratio of convergence decreases as $\lambda$ increases, and the gap between $\lambda=0.1$ and $0.01$ is much larger than the gap between $\lambda=0.01$ and $0.001$, which implies that $\lambda$ also plays an important role in the convergence probability in Theorem 5. $(iii)$ In most cases the experiment ratio is much larger than the theoretical ratio. This indicates that outside the sphere $\hat{\mathcal{B}}_{\|w^*\|}(w^*)$ in Theorem 2 and Theorem 4 there is still much area in which the initial weights will converge to $w^*$. $(iv)$ Under the same parameters, the experiment ratio of convergence in the $\ell_1$ case is always greater than that in the $\ell_2$ case. This shows that the $\ell_1$ regularization makes the dynamic easier to converge than the $\ell_2$ regularization does.
Conclusion and Future Work
==========================
In this paper, we presented our convergence analysis of the dynamics of two-layered bias-free networks with one $ReLU$ output, where the loss function includes the square error loss and $\ell_1$ or $\ell_2$ regularization on the weight vector. This is an extension to Theorem 3.3 in [@work]. We first solved the optimal weight vector $\hat{w}$ with small regularization coefficient for both cases, and then used the Lyapunov method [@lyapunov] to show that the system is asymptotically stable in certain area. In the final step, we claimed that Theorem 3.3 in [@work] is still correct in these two situations. We also verified our theory through numerical experiments including plotting the phase diagrams and making computer simulations.
Our work made a theoretical justification of convergence for two popular models. We started from the intuition that small regularization doesn’t change the system too much, and our conclusion is compatible with this intuition. In the future, we plan to analyze the system with larger regularization, since in real situations $\lambda$ is fixed to be, for example, 0.5, which may be larger than the bound in equations (17) and (27). This is more difficult since we won’t expect $D^*=D(\hat{w})$, and other advanced techniques may be applied. We also plan to consider other popular regularization terms, and provide a more general theory on this topic.
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|
.5cm One of biggest mysteries in the Universe is a cosmological constant, if it exists[@Weinberg]. Recent observation of Type Ia Super Novae suggests that the expansion of the Universe is accelerating now[@SNIa]. This acceleration could be explained by a cosmological constant with $\Omega_{\Lambda} \sim 0.7$. Although a cosmological constant seems to be preferred from a view point of the age of the Universe and a structure formation in the Universe, one may wonder why the cosmological constant is almost the same order of magnitude as the present mass density of the Universe. It is very difficult to explain such value from a view point of particle physics, which usually predicts much larger value of a vacuum energy. If the observation is confirmed, we will face on a serious problem in fundamental physics. One of the way out would be the so-called “quintessence" [@Caldwell_Dave_Steinhardt], in which a potential of a scalar field plays as a decaying cosmological constant[@Dolgov_Fijii; @Ratra_Peebles]. In the quintessence scenario, a scalar field with some specific potential shows an interesting behavior called “tracking" (or “scaling") in its evolution[@Zlatev_Wang_Steinhardt; @Ratra_Peebles]. The energy of a scalar field tracks the radiation energy (or matter energy) for rather long time and then eventually becomes dominant after matter dominant era. In order for this scenario to work well, a scalar field will catch up neither radiation nor matter so early. This naively means that the initial energy of a scalar field was very small. In the quintessence models, however, the final value of quintessence energy density is insensitive to the initial conditions because of its attractor property. For example, a potential of $V=\mu^{\alpha+4} Q^{-\alpha}$ will catch up matter density late in the evolution of the Universe for a wide range of initial conditions, if $\mu$ is suitably chosen. However, this suitable choice of $\mu$ may need a finetuning[@Weinberg2]. Some modified models have been proposed to solve this problem[@Armendariz-Picon_Chiba; @Fujii_Dodelson]. Here, in order to resolve this mystery, we propose a scenario based on a brane universe.
Recently, a new type of world view has been proposed, which is called a brane world based on a superstring or M-theory[@Arkani-Hamed; @Horava_Witten; @Randall_Sundrum]. Our 3-dimensional universe is described by a brane in a higher-dimension[@Rubakov_Shaposhinikov], and usual matter field and force except for gravity are confined on the brane. Among them, Randall and Sundrum’s second model[@Randall_Sundrum] gives an interesting picture of gravity, i.e. although the extra-dimension is not compact, four-dimensional Newtonian gravity is recovered in five-dimensional anti-de Sitter spacetime (AdS$_5$) in low energy limit. Since gravity in those brane world could be quite different from the 4-dimensional Einstein theory, many authors discussed interesting difference from a conventional cosmological model[@Arkani-Hamed2; @brane_cosmology; @brane_cosmology2; @brane_cosmology3]. The quadratic term of energy-momentum appears and may be important in the early stage of the universe, and dark “radiation", which is constrained by a successful nucleosynthesis, may also exist[@brane_cosmology2; @Shiromizu_Maeda_Sasaki]. In particular, the former will change the expansion law of the Universe in the very early stage, then we may expect some important difference in a “quintessence" scenario.
Here, we will show that the quadratic term will indeed change drastically the evolution of a scalar field and its density parameter will decrease in time until the quadratic term becomes unimportant. This provides us a successful and natural scenario for a conventional quintessence model.
In this letter, we shall analyze a cosmological solution based on the Randall-Sundrum model, although the similar result would be obtained in other brane world models. Assuming flat Friedmann-Robertson-Walker spacetime in our brane world, we find the effective Friedmann equation as follows[@brane_cosmology2; @Shiromizu_Maeda_Sasaki]:\
$$\begin{aligned}
H^2 = {\kappa_4^2 \over 3} \rho + {\kappa_5^4 \over 36}
\rho^2 +{{\cal C}\over a^4}\end{aligned}$$ where $\kappa_4^2 = 8\pi G_N$ and $\kappa_5^2$ are 4- and 5-dimensional gravitational constants, respectively, $H=\dot{a}/a$ is the Hubble parameter, and ${\cal C} $ is a constant, which denotes “dark" radiation[@brane_cosmology2]. The 4-dimensional Planck mass $m_4\,(=\kappa_4^{-1} = 2.4\times 10^{18}$GeV) and the 5-dimensional one $m_5\,(=\kappa_5^{-2/3})$ will be used in the following discussion. Here the 4-dimensional cosmological constant is set to zero. We also assume that all matter field including a scalar field $Q$ are confined on the brane. As for the potential, we consider one of typical quintessence-type potential, i.e. $V(Q) = \mu^{\alpha+4} Q^{-\alpha} $[@Ratra_Peebles], although the present mechanism may work for other potentials. It is worth noting that this potential may be naturally derived in some supersymmetric QCD with a fermion condensation. In that case, $\alpha$ is given by the numbers of colors and of flavors[@Binetruy]. Since the energy density decreases when the universe expands, the quadratic term ($\rho^2$) dominates in the early stage of the universe. The conventional Friedmann universe is recovered after when the quadratic and the linear terms are equal at $\rho
=\rho_c = 12\kappa_4^2 /\kappa_5^4 = 12 m_5^6/m_4^2$. Since we know well about the behavior of the scalar field in the linear-term dominant stage, which is the conventional cosmological model, we first study the behavior of the scalar field in the quadratic-term dominant stage. In order to study the dynamics of a scalar field, we discuss two cases separately; the radiation dominant era and the scalar-field dominant era.
First we analyze the case with radiation dominance. The Friedmann equation is approximated as $
H ={\kappa_5^2 \over 6} \rho_{\rm r} \sim a^{-4},
$ where we assume that the Universe is expanding ($H>0$). This gives the expansion law of the Universe as $a \propto t^{1/4}$. Then the equation of motion for the scalar field is $$\begin{aligned}
\ddot{Q} +{3\over 4t}\dot{Q} -\alpha \mu^{\alpha+4} Q^{-(\alpha+1)} =0 .\end{aligned}$$ We find an exact solution for $\alpha<6$, that is $$\begin{aligned}
Q \propto t^{2\over \alpha +2} ~~~~~{\rm and}~~~~~~
\rho_Q \propto t^{-{2\alpha\over \alpha +2}} \propto
a^{-{8\alpha\over \alpha +2}}.
\label{sol_scalar}\end{aligned}$$ The density parameter of the scalar field, which we denote as $\Omega_Q$, is $$\begin{aligned}
\Omega_Q = {\rho_Q \over \rho_Q+\rho_{\rm r}+\rho_{\rm m}}\sim
{\rho_Q
\over \rho_{\rm r}} \sim \beta \left({a\over a_s}\right)^{-{4(\alpha-2)\over
\alpha +2}},
\label{evol_Omega1}\end{aligned}$$ where $\beta\,(\leq 1)$ is a constant and $a_s$ is an “initial" scale factor. If $\alpha > 2$, $\Omega_Q $ [*decreases*]{} with time, just contrary to a tracking solution. The scalar field energy decreases faster than the radiation energy. This is a new interesting feature for a quintessence because smallness of a quintessence-field energy when the Universe enters the conventional stage could be dynamically obtained. We will show that it is really the case. If $\alpha = 2$, $\Omega_Q$ is constant until the linear term becomes dominant. This is the so-called “scaling" solution.
We also find that the above solution (\[sol\_scalar\]) is an attractor, and can show that any solutions in the radiation dominant era will eventually converge to this attractor solution[@Maeda_Yamamoto]. The constant $\beta$ then denotes $\Omega_Q$ when the Universe reaches this solution at $a=a_s$.
One may wonder if the scalar field initially dominates the radiation. If $\alpha <2$, the potential term will overcome the kinetic term, leading to an inflationary universe as $a \propto \exp[H_0t^{(2-\alpha)/2}]$, where $H_0$ is a constant determined by $\mu$ and $m_5$[@Maeda_Yamamoto]. For the case with $\alpha=2$, we find a power-law solution[@Maeda_Yamamoto], i.e. $a \propto t^p$ with $p={1\over 6} [1+{1\over
8}\left({\mu/ m_5}\right)^6 ] $ and $
Q = 2\sqrt{2}m_5^{3/2}t^{1/2} .$ If $p>1$ (i.e. $\mu {\mbox{\raisebox{-1.ex}{$\stackrel
{\textstyle>}{\textstyle\sim}$}}}1.85\, m_5$), we have a power-law inflationary solution, which is an attractor of the dynamical system. While, if $p<1/4$ (i.e. $\mu {\mbox{\raisebox{-1.ex}{$\stackrel
{\textstyle<}{\textstyle \sim}$}}}1.26\, m_5$), this solution is no longer an attractor, leading to the radiation dominant era discussed above. If $\alpha>2$, we can show that a kinetic term of the scalar field will always become dominant even if we start with a potential dominance[@Maeda_Yamamoto]. Since $\rho_Q \propto a^{-6}$ in the case of kinetic-term dominance, we find that the radiation will overcome the scalar-field energy and the Universe will evolve into the radiation dominant era discussed above. Since the solution (\[sol\_scalar\]) is a unique attractor in the radiation dominant era, any solution in $\rho^2$-dominant stage will approach to this attractor solution, if $2<\alpha<6$. Then the evolution of $\Omega_Q$ is given by Eq. (\[evol\_Omega1\]).
As the universe expands and the energy density decreases below $\rho_{\rm c}$, we find the conventional Friedmann universe, in which many authors studied quintessence models [@Caldwell_Dave_Steinhardt; @Zlatev_Wang_Steinhardt; @Armendariz-Picon_Chiba; @Fujii_Dodelson; @quintessence]. During the radiation or matter dominant era, when the Universe evolves as $a
\propto t^{1/2}$ or $t^{2/3}$, the tracking attractor solution in the present model was found, giving the evolution of $\Omega_Q$ as $$\begin{aligned}
\Omega_Q &\sim& \rho_Q/ \rho_{\rm r} \propto a^{8\over( \alpha
+2)}~~~~{\rm (radiation ~dominant)}
\nonumber \\
\Omega_Q &\sim& {\rho_Q / \rho_{\rm m}} \propto a^{6\over( \alpha
+2)}~~~~{\rm (matter ~dominant)}.
\label{evol_Omega2}\end{aligned}$$ In both radiation dominant and matter dominant cases, the energy density of the scalar field decreases slower than that of radiation or matter fluid, and will eventually overcome those, resulting in an accelerating Universe.
From (\[evol\_Omega1\]) and (\[evol\_Omega2\]), the present value of $\Omega_Q$ is estimated as $$\begin{aligned}
\Omega_{Q,0} &=&\beta \left({a_c\over
a_s}\right)^{-{4(\alpha-2)\over
\alpha +2}} \left({a_{\rm eq}\over
a_c}\right)^{{8\over
\alpha +2}}\left({a_0\over
a_{\rm eq}}\right)^{{6\over
\alpha +2}}\nonumber \\
&\sim& \beta \left({T_c\over T_s}\right)^{{4(\alpha-2)\over
\alpha +2}}\left({T_{\rm eq}\over T_c}\right)^{-{8\over \alpha
+2}}\left({T_0\over T_{\rm eq}}\right)^{-{6\over \alpha
+2}},\end{aligned}$$ where $a_c$, $a_{\rm eq}$, and $a_0$ are scale factors at the time when the quadratic term of energy density drops just below the linear term, when radiation density becomes equal to matter density, and the present time, and $T_c$, $T_{\rm eq}$, and $T_0$ are corresponding temperatures of the Universe, respectively. In the present brane quintessence model, $\Omega_Q$ first decreases during the quadratic-energy dominant stage, and the scalar field energy could be very small when the conventional cosmology is recovered. This may explain naturally why the scalar field energy was so small in the early stage of the Universe in the conventional quintessence scenario.
Now we discuss about constraints on parameters to find a successful quintessence scenario. Since we do not know about initial condition, we set our initial time when the attractor solution is reached ($a=a_s$). In this case, we have only one unknown parameter $\beta$ (the value of $\Omega_Q$ at $a=a_s$). Although we do not know the value of $\beta$, it should be smaller than unity because the attractor solution appears only in the radiation dominant era. If an equipartition for the energy of each particle is assumed at $a_s$, we expect $\beta \sim 1/g$, where $g$ is a degree of freedom of particles. $\beta$ could be smaller because the kinetic energy of the scalar field might be dominant before the attractor solution.
Setting $T_0 =2.73$ K and $T_{\rm eq} \sim 10^4$ K, and assuming $\Omega_{Q,0} \sim 0.7$ and equipartition at $a_s$ (i.e. $\beta=1/g=0.01$), we find a relation between $T_s$ and $T_c$, which is shown in Fig. 1 by a solid line for each $\alpha$.
=4.5cm 1em
One stringent constraint in any cosmological models is nucleosynthesis. It must take place in the conventional radiation dominant era to explain the present amount of light elements. Therefore, $T_c$ must be higher than the temperature at nucleosynthesis, $T_{\rm NS}
\sim
$ 1 MeV. This constraint implies $m_5 > 1.6 \times 10^4 \,(g/100)^{1/6} (T_{\rm NS}/$1MeV$)^{2/3}$ GeV, which is included in Fig. 1. The r.h.s. of the vertical line is the allowed region. If the second Randall-Sundrum model turns out to be a fundamental theory, in order to recover the Newtonian force above 1mm scale, the 5-dimensional Planck mass is constrained as $m_5
\geq 10^8$ GeV[@Randall_Sundrum], which is satisfied in the r.h.s. region of the dotted vertical line in Fig. 1.
Although we do not know the “initial" temperature $T_s$, if the 5-dimensional spacetime is fundamental and gravity is unified at the energy scale $m_5$, we expect $T_s {\mbox{\raisebox{-1.ex}{$\stackrel
{\textstyle<}{\textstyle \sim}$}}}m_5$. Even if the 5-dimensional theory is effective, $T_s {\mbox{\raisebox{-1.ex}{$\stackrel
{\textstyle<}{\textstyle \sim}$}}}m_5$ may be required to justify the present 5-dimensional analysis. This condition with the equipartition at $a_s$ gives a constraint on the scale of the potential as $\mu {\mbox{\raisebox{-1.ex}{$\stackrel
{\textstyle>}{\textstyle\sim}$}}}(0.2-0.3) \,m_5$. For reference, we have inserted three lines of $T_s=0.1\, m_5$ (the lower dotted line), $ m_5$ (solid line), and $10 \, m_5$ (the upper dotted line) in Fig. 1. If $\alpha {\mbox{\raisebox{-1.ex}{$\stackrel
{\textstyle>}{\textstyle\sim}$}}}4$, we find a successful quintessence scenario with natural conditions. On the other hand , if $\alpha<4$, either $\beta $ should be much smaller than the value expected from equipartition, or $T_s \gg m_5$, then we may need a finetuned or unnatural initial condition.
In Fig. 2, we show one example of the time variation of $\Omega_Q$ for the case of $\alpha=5$ with $\beta=0.01$ and $T_s
= m_5$. For these parameters, we find that $T_c \sim $280 MeV ($\gg T_{\rm NS}
$), and $T_s
(=
m_5) \sim 8.6 \times 10^6$ GeV.
1em
If $\alpha\geq 6$, the above solution (\[sol\_scalar\]) is no longer an attractor. We can show that the kinetic term always dominates in the quadratic-term dominant stage. Then $\Omega_Q $ drops as $a^{-2}$ until the conventional cosmology is recovered at $a_c$. However, we know that the kinetic term drops much faster than the potential term after $a=a_c$ and the tracking solution eventually will be reached. Hence, we could approximate $\Omega_Q
$ at $a=a_c$ by the potential term, i.e $\Omega_{Q,c} \sim
\beta(a_c/a_s)^{4-\alpha}$. Using this estimation, we find the relation between $T_s$ and $T_c$ to obtain $\Omega_{Q,0} \sim 0.7$. Those results are also included in Fig. 1.
In the brane universe, “dark" radiation ${\cal C}\, (>0)$ may exist. If this term dominates, the present scenario may not work because it evolves as $a^{-4}$ just as radiation, . The preliminary analysis shows that the quadratic energy of the scalar field will drops as the same as the present model, but the linear term will increases because the expansion law of the Universe is $a\propto
t^{1/2}$. It will depend on the initial condition whether we find a successful model or not.
In this letter, we have discussed a quintessence model in the context of a brane world scenario. We show that the energy of a scalar field decreases faster than the radiation when the quadratic energy density is dominant. As a result, when we recover a conventional cosmology, the density parameter of a scalar field is enough small for a scalar field to dominate just at present epoch. If our 3-dimensional brane universe starts at $T_s \sim m_5$, $\alpha
\, {\mbox{\raisebox{-1.ex}{$\stackrel
{\textstyle>}{\textstyle\sim}$}}}\, 4$ is required to find a natural quintessence scenario. We should note that the present model may not solve the coincidence problem, but will give a natural explanation of smallness of a scalar field energy in the early stage of the conventional cosmology. It is also important to point out that we could predict when the Universe is getting into an acceleration phase, if we know the values of fundamental parameters such as $m_5$ and $\mu$.
The present model prefers rather large value of $\alpha$ ($\alpha {\mbox{\raisebox{-1.ex}{$\stackrel
{\textstyle>}{\textstyle\sim}$}}}4$), but recent observation may force to the constraint of $\alpha {\mbox{\raisebox{-1.ex}{$\stackrel
{\textstyle<}{\textstyle \sim}$}}}2$[@Balbi_Baccigalupi_Matarrese_Perrotta_Vittorio]. If this is the case, we have to look for other type of quintessence potential[@quintessence]. For example, an exponential potential could be another candidate. The results for such a study will be published elsewhere.
The essential point in the present scenario is that the dynamics of the scalar field is completely modified in the quadratic energy dominant stage[@brane_cosmology; @brane_cosmology2; @Maeda_Yamamoto]. Although we assume the Randall-Sundrum model in the present analysis, the quadratic energy term will usually appear in any brane universe models[@brane_cosmology; @brane_cosmology2], then we expect the similar effect, which may provide us a successful quintessence. Since the dilaton field will also appear in a superstring or M-theory, it will also change the dynamics of the Universe[@brane_cosmology; @brane_cosmology3], which requires further analysis.
KM would like to thank T. Harada, S. Mizuno, A. Starobinsky and K. Yamamoto for useful comments and discussions. This work is supported partially by the Grant-in-Aid for Scientific Research Fund of the Ministry of Education, Science and Culture (Specially Promoted Research No. 08102010) and by the Yamada foundation.
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---
abstract: 'We discuss how to achieve mapping from large sets of imperfect simulations and observational data with unsupervised domain adaptation. Under the hypothesis that simulated and observed data distributions share a common underlying representation, we show how it is possible to transfer between simulated and observed domains. Driven by an application to interpret stellar spectroscopic sky surveys, we construct the domain transfer pipeline from two adversarial autoencoders on each domains with a disentangling latent space, and a cycle-consistency constraint. We then construct a differentiable pipeline from physical stellar parameters to realistic observed spectra, aided by a supplementary generative surrogate physics emulator network. We further exemplify the potential of the method on the reconstructed spectra quality and to discover new spectral features associated to elemental abundances.'
bibliography:
- 'refs.bib'
---
Introduction {#sec:intro}
============
In an effort to understand the nearby galactic dynamics, and how stellar populations form and evolve, large dedicated sky surveys [@buder2018; @holtzman2018] are currently accumulating millions of stellar spectra. Large statistical samples of survey data enable galactic archaeologists to construct tools to study the history of galaxy formation or stellar chemical evolution of the Milky Way and the nearby Universe. From the large collections of spectra, estimates of stellar properties such as temperature, surface gravity, metallicities or elemental abundances are routinely produced from data reduction pipelines. The production of these stellar parameter databases requires computationally demanding simulations [[*e.g.*]{}, @kurucz1970; @meszaros2012new], and implementing boutique data analysis pipelines [[*e.g.*]{}, @perez2016aspcap; @ting2019]. However simulations of synthetic spectra are naturally limited to the assumptions behind the stellar physics that we can currently model [[*e.g.*]{}, @bialek2020; @shetrone2015]. Imperfect corrections of the instrumental and earth-atmosphere signatures also are often left-over systematics from the data-reduction pipelines. It is thus difficult to have a clear understanding of the systematic uncertainties involved with either the data reduction, the synthetic spectral modelling, or from the algorithms limitations.
Thus, in this paper, we propose an unsupervised domain adaptation method, that learns these corrections without human intervention.
Methods {#sec:methods}
=======
Our work is largely based on the `UNIT` framework [@liu2017], briefly summarized here. The key `UNIT` idea is to share a latent representation between two unpaired domains of interest. In each domain, a Variational AutoEncoder (VAE) is trained to generate fake samples, sharing the common latent space. The generated samples are thereafter challenged with adversarial classifier networks, trained to discriminate between pairs of reconstructed samples originating from either domain.
We set the domains of interest to be a large sample of $\mathcal{X}_{synth}$ simulation data and a large sample $\mathcal{X}_{obs}$ of observed data. For our stellar physics purpose, domain data are synthetic and observed spectra. Applying `UNIT` requires technical modifications to adapt to noisy 1D data, but we also made important architectural choices. We summarize the architecture in the diagram in Fig. \[fig1\]. The main high level differences between the `UNIT` and our current method are: 1) a disentangled latent representation, and 2) stitching a supplementary generative surrogate physics emulator network to the main `UNIT` framework. Another minor difference is that we have our autoencoders to be deterministic.
0.2in
![The architecture of the unsupervised domain transfer between simulations and observations. Synthetic spectra are generated on the fly with a physics simulator with input parameters $\theta$. Both synthetic and observed spectra are encoded into a common latent space $\mathcal{Z}_{shared}$. The observed spectra are also encoded simultaneously to a split latent space $\mathcal{Z}_{split}$, coding other observation-specific factors. The rest of the architecture is similar as the UNIT [@liu2017] framework.[]{data-label="fig1"}](fig1.png){width="0.9\columnwidth"}
-0.3in
#### Disentangled Latent Representation
Even good physics simulators are capable of only partly representing the complexity of all the observations. Thus, it will always suffer from the gap between theory and practice. We model this gap by splitting the latent space in two: one shared latent space between simulations and observations, and one latent space unique to observations. In Fig. \[fig1\], we denote this shared latent space as $\mathcal{Z}_{shared}$ and the latent space unique to observation as $\mathcal{Z}_{split}$. In our experiments, we found that this simple observation was critical to ensure that the shared latent space variable covers the same distributions between the two domains, and contains only the information from stellar physics. To further facilitate sharing of latent space, unlike `UNIT`, we feed not only the domain translated samples and the decoded spectra, but also their corresponding latent spaces to the discriminators. If the domain transfer succeeds, we expect the shared latent representation to show well-mixed samples originating from both domains, coding physics explainable with the simulator. For our stellar spectra application, we show a t-SNE plot in Fig \[fig2\], where each point represents a spectrum in the respective original domains, in the shared latent space, and after domain adaptation. The t-SNE result shows that the domain-adapted synthetic spectra distribution are well mixed with the observations, closing the synthetic-observation gap.
![t-SNE where each point represent a spectrum. Left: synthetic and observed spectra. Middle: observed spectra and domain-adapted synthetic spectra by the network. Right: shared latent space.[]{data-label="fig2"}](fig2.png){width="1.1\columnwidth"}
-0.3in
#### Surrogate Physics Simulator
The samples in the synthetic domain are generated by a physical simulator. In our application, the physical simulator is computationally expensive, thus producing a stochastic simulator would be very costly. We replace it with a fast neural-network trained to emulate the physics. We carefully selected physical ranges and sampling strategies of the physical parameter space, within the same “expected” range of parameters covered by the observed samples. We then ran the costly simulations, and trained a network capable of emulating the generated samples from the physical parameters. In our particular case, a simple MLP was enough to ensure 0.1% accuracy at emulating the physical simulator over the full spectral range. Once the differentiable emulator network is trained, the main network can be trained from generating samples on-the-fly.
Besides on-the-fly generation of synthetic spectra, the introduction of the surrogate simulator network allows us to constitute an end-to-end system from interpretable physical parameters to realistic spectra which implicitly learns the non-modelled physics. Being end-to-end, the system is also fully differentiable, and allows one to differentiate the the domain-adapted synthetic spectra with respect to physical parameters. We, in fact, utilize this property to infer physics from spectra.
Experiments {#sec:experiments}
===========
We present two case studies to demonstrate how we generate systematic-corrected synthetic models from unlabelled observed spectra through domain adaptation.
Data
----
For these two experiments, we use the publicly available APOGEE [@holtzman2018] survey data, which comprises of 250,000 high-resolution $R \equiv \lambda/\Delta \lambda \sim 22,000$ infrared spectra, where $\lambda$ designates wavelength.
- For the synthetic domain, we use the Kurucz [atlas12]{}/[synthe]{} models [see @ting2019 for details]
- As for the observed domain, we adopt the APOGEE Data Release 14 spectra. The APOGEE spectra are wavelength calibrated to vacuum to be consistent with the Kurucz models. Multiple visits to the same object are pre-processed and co-added. We corrected the spectra for spectral redshift and self-consistently continuum normalized both the Kurucz models and APOGEE spectra using the same routine as laid out in @ting2019.
The data set provides estimated stellar parameters for both dwarf and giant stars. We further use the samples in the dataset in a random order to make sure spectra from the two domains are not paired.
APOGEE spectra are adopted from half of the objects as the observed domain, and for the other half, we generate synthetic Kurucz spectra via the surrogate emulator assuming only their stellar parameters. We inherit the 25 stellar parameters derived in @ting2019: $T_{\rm eff}$, $\log g$, microturbulence $v_{\rm turb}$, additional broadening $v_{\rm broad}$, and 20 elemental abundances, namely, C, N, O, Na, Mg, Al, Si, P, S, K, Ca, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, Ge, and the isotopic ratio, C12/C13. This procedure yields a training set of about 100,000 spectra in each domain. We adopt 80,000 spectra as the training set, 10,000 spectra as the validation set. The rest (7,000 spectra) are held out as the test set. All results below are based on the test set.
Stellar Spectra Translation
---------------------------
As a natural application, we investigate the quality of the translated spectra in the observed domain to that originated from the synthetic domain. Once the framework is trained, the generation is fast and allows us to perform a maximum-likelihood fit between the translated and the observed spectra to get the best-fit stellar parameters.
In Fig \[fig3\], we compare the average residuals between our transferred best-fit model (bottom) to a best-fit emulated (non-translated) physics model (top). Fitting spectra is a typical process for stellar parameter analysis. The residuals are normalised with the spectra uncertainties provided by the APOGEE survey data release. Well calibrated uncertainties with unbiased best-fit models would show a distribution of $z$-score residuals following $\mathcal{N}(0,1)$. We show the sample mean $\overline{m}$ and standard deviation $s$ in each case. The systematics-learned model clearly outperforms the stellar physics-only model.
![Residuals between physical (Kurucz) best-fit modelled spectra and the APOGEE observed spectra. The top panel shows the difference between the 10,000 APOGEE test spectra and their corresponding best-fit Kurucz models. The bottom panel is a similar comparison but with the transferred spectra by our network, showing smaller mean bias $\overline{x}$ and standard deviation $s$ of the residuals.[]{data-label="fig3"}](fig3.png){width="1.1\columnwidth"}
-0.3in
Spectral Lines
--------------
Here, we study the derivatives of the full network, in particular the derivatives with respect to elemental abundances. The derivatives ([*i.e.*]{}, flux “response”) of elemental abundances examine which spectral features are associated with a particular element. Unfortunately, for the APOGEE observed spectra, it is impossible to know the ground truth – we simply do not know what might be missing in the Kurucz models. Therefore, as a proof of concept, a mock “observed” data set is drawn from the Kurucz models instead, of which we know the ground truth derivatives. More advanced applications are deferred to future studies.
![Generated Kurucz spectral models but 30% masked of the spectral features in the “synthetic” domain, and original Kurucz models (with noise) as the observed domain. The top panel shows the Kurucz spectrum with missing spectral features, whereas the second panel shows the systematic-corrected synthetic spectra. The third panel shows the differences between the synthetic and “observed” spectrum, demonstrating the missing features; the missing features of Mg, Si, Fe, and C are annotated in blue, orange, green, and red respectively. The final panel shows the differences in the derivatives between the synthetic domain and the transferred models.[]{data-label="fig4"}](fig4.png){width="\columnwidth"}
-0.3in
For the observed domain, instead of adopting the APOGEE spectra, an “observed” data set is synthesized with our physics simulator. We further add noise to these observed mock spectra to mimic the same noise distribution of the APOGEE spectra. As for the synthetic domain, we mask approximately 30% of the spectral features randomly by setting them to the continuum level. In short, in this controlled experiment, we assume two sets of unpaired Kurucz models.
To demonstrate that we can learn actual physics, the goal here is to correctly identify the physical source ([*i.e.*]{}, which element) of these masked spectral features, which act as missing information in the synthetic models that we want to learn and recover from the data.
We calculate the difference between the flux derivatives of the synthetic emulator, $\partial G(\mathcal{Y})/\partial \mathcal{Y}$ and the derivatives of the domain transferred spectra, $\partial T_{synth\rightarrow obs}(G(\mathcal{Y}))/\partial \mathcal{Y}$, where $G$ is the surrogate emulator, and $T_{synth\rightarrow obs}$ the domain adaptation network and $\mathcal{Y}$ the physical stellar parameters. The former informs us about the original spectral features in the “synthetic” domain, which has about 30% missing spectral features. The latter reveals the “true”, unmasked, Kurucz line list learned from the “observed” data.
Fig. \[fig4\] shows the result of domain adaptation. We assume a typical K-giant star with Solar elemental abundances to be the reference point. The top panel shows the spectrum from the synthetic domain, and the second panel shows the transferred spectrum. By construction, the synthetic spectrum is missing many spectral features compared to the transferred spectrum. Although not shown, the transferred spectrum is almost identical to the observed spectrum in this case. When mapped to the observed domain, missing features in the synthetic domain are faithfully reconstructed.
The last two panels demonstrate that not only we can faithfully reconstruct the missing features, but it also correctly links them to their corresponding elements. The third panel shows differences between the synthetic spectrum and the transferred spectrum. The masked features are colour-coded with their associated elements and we focus on the three elements that have a prominent presence in the APOGEE spectra, namely Mg, Si, Fe, and C. The final panel shows the differences between the true and recovered derivatives. As shown in this final panel, the differences in the derivatives are the strongest when calculated with the correct input element. The missing features are recovered with accurate associations, even though the training was trained with noisy, unpaired, observed spectra that mimic the APOGEE observations.
Conclusion {#sec:discussion}
==========
In this study, we illustrated that domain-adaptation framework can have broad implications for simulation-based physics. As a proof of concept, we focused on stellar spectral modeling. Synthetic spectral models are auto-calibrated through a domain adaptation network, thereby reducing the gap between theory and observations. The same network can also predict the elemental sources of unknown spectral features in the synthetic models.
|
---
author:
- 'C. W. Engelbracht'
- 'L. K. Hunt'
- 'R. A. Skibba'
- 'J. L. Hinz'
- 'D. Calzetti'
- 'K. D. Gordon'
- 'H. Roussel'
- 'A. F. Crocker'
- 'K. A. Misselt'
- 'A. D. Bolatto'
- 'R. C. Kennicutt'
- 'P. N. Appleton'
- 'L. Armus'
- 'P. Beirão'
- 'B. R. Brandl'
- 'K. V. Croxall'
- 'D. A. Dale'
- 'B. T. Draine'
- 'G. Dumas'
- 'A. Gil de Paz'
- 'B. Groves'
- 'C.-N. Hao'
- 'B. D. Johnson'
- 'J. Koda'
- 'O. Krause'
- 'A. K. Leroy [^1]'
- 'S. E. Meidt'
- 'E. J. Murphy'
- 'N. Rahman'
- 'H.-W. Rix'
- 'K. M. Sandstrom'
- 'M. Sauvage'
- 'E. Schinnerer'
- 'J.-D. T. Smith'
- 'S. Srinivasan'
- 'L. Vigroux'
- 'F. Walter'
- 'B. E. Warren'
- 'C. D. Wilson'
- 'M. G. Wolfire'
- 'S. Zibetti'
title: 'Enhanced dust heating in the bulges of early-type spiral galaxies[^2]'
---
Introduction
============
The infrared emission from galaxies contains roughly half of the entire energy budget in the Universe (e.g., Hauser & Dwek [@hauser01]). In addition to providing information on the amount of attenuation suffered by the stellar light, the infrared emission provides clues to important physical quantities, such as the metal, dust, and cold gas content of galaxies (e.g., Draine et al. [@draine07a], Bernard et al. [@bernard08]). Infrared spectral energy distributions (SEDs), especially those extending into the submillimeter regime, can be used to measure the dust mass and temperature in galaxies (e.g., Dunne et al. [@dunne00; @dunne01]; Seaquist et al.[@seaquist04]; Vlahakis et al. [@vlahakis05]; Draine et al.[@draine07a]; Willmer et al. [@willmer09]; Liu et al. [@liu10]).
These temperature measurements have shown that the dust is warmer in the centers of galaxies than in the outskirts (e.g., Alton et al. [@alton98]; Radovich et al. [@radovich01]; Melo et al. [@melo02]; Dupac et al.[@dupac03]). Cool dust at roughly the same temperature in spiral disks is detected [*globally*]{} at longer wavelengths (850 $\mu$m and 1.3 mm; Siebenmorgen et al. [@siebenmorgen99]; Dunne et al.[@dunne00; @dunne01]; Vlahakis et al. [@vlahakis05]), but there is also some evidence of a warmer temperature component associated with the central regions. Warmer dust temperatures tend to be associated with significant star-formation activity, the resulting intense interstellar radiation field (Stevens et al. [@stevens05]) and the earlier Hubble type (Bendo et al. [@bendo03]).
Here we use the infrared SEDs, from 70 to 500 $\mu$m (a range which should be dominated by emission from grains in thermal equilibrium with the radiation field, e.g., Popescu et al. [@popescu00], Engelbracht et al.[@engelbracht08]), of a local sample of 13 galaxies spanning the range of spiral galaxies in the Hubble sequence, to derive the temperature of the cool (T$ \sim 20-30$ K) dust component of the central region and the disks separately, and investigate differences in the dust heating in the two regions. To achieve this goal, we use the 250, 350, and 500 $\mu$m *Herschel*/SPIRE (Spectral and Photometric Imaging REceiver; Griffin et al. [@griffin10]) images of the galaxies combined with the *Spitzer*/MIPS (Multiband Imaging Photometer for *Spitzer*; Rieke et al. [@rieke04]) 70 and 160 $\mu$m images. Eventually we will use PACS (Photodetector Array Camera and Spectrometer; Poglitsch et al.[@poglitsch10]) imaging for this study, but at the time of this writing, the data were not yet available. These galaxies have little nuclear activity that might heat the dust, with only NGC 1097 having a strongly active nucleus, so this is the first study to cleanly separate the far-infrared properties of central and disk regions in a sample of normal galaxies.
Until recently, little work has been done to dissect the dust emission in galaxies into sub-galactic components, owing to the general paucity of infrared images with the required angular resolution and to poor long-wavelength sensitivity. Some recent work includes a study of the galaxy pair NGC1512/1510 (a target also discussed in this paper), which finds that the dust temperature in the central region of NGC1512 is slightly higher than in the disk and that there is a significantly higher fraction of warm dust, in agreement with the center of NGC1512 being a starburst (Liu et al.[@liu10]). Work by Muñoz-Mateos et al. ([@munoz-mateos09]) examines radial trends in dust properties in a number of nearby galaxies.
The *Herschel* Space Telescope promises to yield a breakthrough in the study of subgalactic components in galaxies. This paper is the first investigation to leverage the longest wavelength *Herschel* data available for the KINGFISH (key insights on nearby galaxies: a far-infrared survey with *Herschel*; this program is largely derived from SINGS, the *Spitzer* infrared nearby galaxy survey by Kennicutt et al.[@kennicutt03]) sample of nearby galaxies, which will eventually total 61. Companion papers from the science demonstration phase for this program showcase the shorter wavelength imaging (Sandstrom et al.[@sandstrom10]) and the spectroscopic data (Beirão et al.[@beirao]).
Here we present new SPIRE images acquired during the first few months of *Herschel* operations, in the context of the KINGFISH Open-Time Key Project. We divide each galaxy into two spatially-resolved zones: the circumnuclear region and the surrounding disk. Then we compare central temperatures with those for the disk.
Observations and data reduction
===============================
We observed 33 nearby galaxies with the SPIRE instrument on *Herschel* (Pilbratt et al. [@pilbratt10]) in the scan map mode as part of the KINGFISH program over the period of November 2009 to January 2010. They were observed in the scan mode out to 1.5 optical radii, to depths of (3.2, 2.5, and 2.9) mJy/beam at (250, 350, and 500) $\mu$m. They were reduced using the standard calibration products and algorithms available in HIPE (the *Herschel* Interactive Processing Environment; Ott [@ott10]), version 1.2.5 or 2.0.0 (whichever version was the latest available when the target was observed, since the SPIRE calibration is not sensitive to this range of HIPE versions). We modified the offset-subtraction algorithm in the pipeline, by requiring it to measure the offset only outside bright objects in the field (usually the galaxy that we targeted)—this procedure reduced a small negative offset (6% in the first galaxy we observed, NGC 4559) in the background in the same rows and columns as the source to an undetectable level. Otherwise, the data reduction was performed using the default pipeline.
Analysis
========
Of the 33 galaxies observed, we selected 13 which had both a large extent (i.e., greater than several 160/500 $\mu$m beam diameters, or $\sim2$) and a bright disk (brighter than $\sim0.5$ Jy at 500 $\mu$m, excluding the nucleus). These galaxies are listed in Table \[tab:sample\]. For the wavelengths with significantly smaller beams (i.e., 70, 250, and 350 $\mu$m) than the largest used here, we convolved the data to the SPIRE 500 $\mu$m resolution using the kernels described by Gordon et al. ([@gordon08]) and updated by those authors for *Herschel*. For each galaxy, we computed masks that isolated the disk and central regions. The outer mask is an ellipse sized to encompass the disk where it contrasts strongly with the background, at 1.2, 6.6, 2.7, 1.4, and 0.8 MJy/sr at 70, 160, 250, 350, and 500 $\mu$m (where the SPIRE data have been converted to MJy/sr as described below), respectively. The inner mask is a circle centered on the galaxy peak, with a minimum size chosen to be as small as possible (i.e., set by the MIPS 160 $\mu$m beam) and with a radius inversely proportional to the distance to the galaxy. Thus, a similar physical region of $\sim3$ kpc diameter (in which one would expect to find structures like a bulge, bar, ring, and/or inner disk, e.g., see Erwin & Sparke [@erwin02], Athanassoula & Martinet [@athanassoula80], and Knapen [@knapen05]) is sampled in the central region of each galaxy. Our results are not sensitive to a modest change in the aperture size, for example, similar results are achieved with a fixed central aperture size of 40 for each galaxy. A sample mask is shown in Fig. \[fig:masking\].
--------- -------- ------- -------------- -------------------- ----------------------
Name Hubble D[^3] Spectral[^4] T$_{\rm disk}$[^5] T$_{\rm center}$$^c$
type (Mpc) type (K) (K)
NGC0628 SAc 7.3 HII? 22.7$\pm$0.4 24.2$\pm$0.5
NGC0925 SABd 9.0 HII 22.3$\pm$0.4 23.9$\pm$0.5
NGC1097 SBb 19.1 Sy1 23.4$\pm$0.5 28.7$\pm$0.7
NGC1291 SB0/a 10.4 … 18.3$\pm$0.3 27.6$\pm$0.6
NGC1512 SBa 14.4 HII 21.3$\pm$0.4 27.3$\pm$0.6
NGC3621 SAd 6.9 … 21.9$\pm$0.4 25.6$\pm$0.6
NGC4559 SABcd 8.5 HII 22.7$\pm$0.4 24.2$\pm$0.5
NGC4594 SAa 9.4 Sy1.9 21.3$\pm$0.4 22.9$\pm$0.4
NGC4631 SBd 7.6 HII 24.5$\pm$0.5 27.7$\pm$0.6
NGC5055 SAbc 10.2 Transition2 22.0$\pm$0.4 25.3$\pm$0.5
NGC5457 SABcd 7.1 HII 23.4$\pm$0.5 23.9$\pm$0.5
NGC6946 SABcd 6.8 HII 24.5$\pm$0.5 26.6$\pm$0.6
NGC7331 SAb 14.9 Transition2 23.2$\pm$0.4 26.2$\pm$0.6
--------- -------- ------- -------------- -------------------- ----------------------
: Galaxy sample and computed temperatures[]{data-label="tab:sample"}
{width="\textwidth"}
We measured flux densities in these apertures for each of the 3 *Herschel*/SPIRE bands, as well as the *Spitzer*/MIPS 70 and 160 $\mu$m bands. The *Herschel*/SPIRE data were calibrated in units of Jy/beam, which we converted to Jy by assuming beam sizes of 371, 720, and 1543 square arcseconds in the 250, 350, and 500 $\mu$m bands, respectively (Swinyard et al. [@swinyard10]). We assign uncertainties as the quadratic sum of the data-dependent scatter and (by far the larger term) the uncertainty in the absolute calibration, taken to be 5% and 12% for *Spitzer*/MIPS at 70 and 160 $\mu$m (Gordon et al. [@gordon07] and Stansberry et al. [@stansberry07], respectively) and 16% (the quadratic sum of the 15% absolute calibration uncertainty from Swinyard et al.[@swinyard10] and a 2% uncertainty in the size of the beam) for SPIRE data.
To each set of data, we fit a blackbody with a frequency-dependent emissivity of 1.5. Our conclusions do not change if values of 1 or 2 are used, although the temperatures increase or decrease, respectively. The uncertainties in the temperatures were computed via a Monte Carlo simulation, in which we performed 10000 trials where the photometric measurements were allowed to vary in a normal distribution with a standard deviation indicated by the photometric uncertainty. An example of fits and data points is shown in Fig. \[fig:fit\], where we can see that the 250 $\mu$m flux density is underpredicted by this simple model. This happens frequently in our dataset, and may be a sign that multiple temperature components are present in the regions we measured. The values of the *Herschel*/SPIRE uncertainty indicated by the uncertainty in the absolute calibration were allowed to vary together; i.e., the calibration uncertainty was assumed to be correlated among the three bands. In contrast, we allowed the MIPS uncertainties to vary independently; while not being strictly independent (see Stansberry et al. [@stansberry07]), the 160 $\mu$m calibration observations, in particular, are dominated by observational scatter and can be treated independently of the 70 $\mu$m band. The temperature of each component is taken to be the temperature of the best-fit model, with an uncertainty determined by the Monte Carlo simulation.
Results
=======
For each galaxy, we computed a ratio of the central to disk temperature (from data at wavelengths longer than 70 $\mu$m) of the cool dust component. The average ratio of central-to-disk temperatures in this sample is $1.15 \pm
0.03$. We have plotted this ratio against morphological type in Figure \[fig:ratios\], where the average ratio is $1.23 \pm 0.03$ for types earlier than Sc and $1.09 \pm 0.03$ for later types. The trend is poorly described by a line, because it has a correlation coefficient of 0.66. We plot the ratio against bar strength (as defined by the morphological type) in Fig. \[fig:ratios\_v\_bar\], where the average ratio is $1.29 \pm 0.04$ for strong bars and $1.09 \pm 0.03$ for weak bars.
![Far-infrared SEDs of the disk and central region of NGC 1512, each fit by a blackbody function modified by a $\lambda^{-1.5}$ emissivity.\[fig:fit\]](14677fg2.ps){width="49.00000%"}
![Ratio of central dust temperature to disk dust temperature as a function of morphological type. The dotted line is drawn at a ratio of 1 (i.e., at equal temperatures) to guide the eye.\[fig:ratios\]](14677fg3.ps){width="49.00000%"}
![Ratio of central dust temperature to disk dust temperature as a function of bar strength, as indicated by morphological type. The dotted line is drawn at a ratio of 1 (i.e., at equal temperatures) to guide the eye.\[fig:ratios\_v\_bar\]](14677fg4.ps){width="49.00000%"}
A significant ($3\sigma$) correlation between temperature ratio and Hubble type is observed in Fig. \[fig:ratios\], although the correlation is only marginal, at $2\sigma$, if the most significant point (NGC 1291) is removed from the correlation. This trend suggests that the dominant stellar bulges in early-type spirals are able to heat the cool dust in the ISM to greater temperatures than their late-type counterparts. The effect is most pronounced in the earliest type galaxy in our sample, NGC 1291, in which the dust in the central region is almost 50% warmer than the dust in the disk. In comparing this to the spectral types in Table \[tab:sample\], we have found no obvious correlation between the temperature ratio and the presence of a central nonthermal source in our galaxies.
Bars may play an important role in driving the correlation of temperature ratio with morphology. As shown in Fig. \[fig:ratios\_v\_bar\], a strong bar is correlated with a high center/disk temperature ratio, although the scatter in the strong-bar ratios is large. This could occur through bar-induced starbursts, since mid-infrared colors are redder in barred galaxies, implying warmer dust temperatures (Huang et al. [@huang96]; Roussel et al. [@roussel01]). Barred galaxies also tend to have higher star-formation rates (SFRs) than unbarred galaxies, possibly because bars drive gas into the inner regions where it can pile up in the inner dynamical resonances associated with rings (Hawarden et al. [@hawarden86]). The bar-driven gas transport has been shown in CO observations (e.g., Sakamoto et al. [@sakamoto99]). Stars will form quite readily in such conditions, where the gas transported by the bar fuels star formation (Erwin & Sparke [@erwin02]; B[ö]{}ker et al. [@boker08]; Comer[ó]{}n et al.[@comeron08]). In fact, the frequency of inner rings in early-type barred spirals (Hunt & Malkan [@hunt99]; Erwin & Sparke [@erwin02]) could conspire to produce the effect seen in our sample. Similarly, bars are more frequently found in bulge-dominated galaxies (e.g., Masters et al.[@masters10]), which would also contribute to the trends we see. The concentrations of CO gas in the centers have been observed in early-type spiral galaxies with intense star formation (Koda et al. [@koda05]). In addition to a dependence on stellar density, the higher temperature ratios in the early types shown here could therefore be associated with bar-induced star formation. This is consistent with the result we get if the 70 $\mu$m fluxes are not included in the modified blackbody fits. Without them, the correlation between temperature ratio and either Hubble type or bar strength disappears; i.e., measurements at wavelengths shorter than the peak emission are required to determine the dust temperature accurately.
Conclusion
==========
We used far-infrared data from *Spitzer* and submillimeter data from *Herschel* to compute separate SEDs for the center and disk regions of 13 nearby galaxies observed as part of the KINGFISH program. We fit those SEDs (at wavelengths longer than 70 $\mu$m) with blackbody functions (modified by a frequency-dependent emissivity) to compute temperatures. On average, the cool dust temperature of the central component is $15 \pm 3$% hotter than the disk. We find that the central temperature is higher than the disk by 20% to 50% in galaxies of type S0 to Sb, but only 9% higher in later types. This ratio is also higher (at $1.29 \pm 0.04$) in strongly barred galaxies than in weakly barred galaxies (at $1.09 \pm 0.03$).
The data therefore indicate that the large (or “classical”) grains that dominate the far-infrared and submillimeter emission are warmer in the centers of those galaxies with a substantial bulge and/or a strong bar. This may simply be caused by the higher density of the radiation field in the centers of early-type spirals, enhanced star formation due to the bar, or some combination of the two. A cleaner separation of morphological components (perhaps with larger samples and/or less distant galaxies) and a more thorough assessment of the density of starlight and star formation activity, plus an evaluation of the impact of central nonthermal sources, may help separate these effects.
The analysis presented here illustrates the power of [*Herschel*]{} observations in characterizating the spatially resolved distribution of dust in nearby galaxies. This power will grow with the use of better-resolved far-infrared SEDs as measured by PACS (Poglitsch et al. [@poglitsch10]), which will let us measure smaller and/or more distant galaxies and determine radial trends of dust temperature.
The following institutes have provided hardware and software elements to the SPIRE project: University of Lethbridge, Canada; NAOC, Beijing, China; CEA Saclay, CEA Grenoble, and OAMP in France; IFSI, Rome, and University of Padua, Italy; IAC, Tenerife, Spain; Stockholm Observatory, Sweden; Cardiff University, Imperial College London, UCL-MSSL, STFC-RAL, UK ATC Edinburgh, and the University of Sussex in the UK. Funding for SPIRE has been provided by the national agencies of the participating countries and by internal institute funding: CSA in Canada; NAOC in China; CNES, CNRS, and CEA in France; ASI in Italy; MCINN in Spain; Stockholm Observatory in Sweden; STFC in the UK; and NASA in the USA. Additional funding support for some instrument activities has been provided by ESA. We would also like to thank the anonymous referee whose comments helped improve this paper.
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[^1]: Hubble Fellow
[^2]: Herschel is an ESA space observatory with science instruments provided by the European-led Principal Investigator consortia and with important participation from NASA.
[^3]: As listed in NED: The NASA/IPAC Extragalactic Database (NED) is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.
[^4]: from NED or Ho et al. [@ho97]
[^5]: includes uncertainty in the photometric calibration
|
---
author:
- 'Zs. Sándor'
- 'W. Kley'
bibliography:
- 'kley8.bib'
date: 'Received ; accepted '
title: On the evolution of the resonant planetary system HD 128311
---
[A significant number of the known multiple exoplanetary systems are containing a pair of giant planets engaged in a low order mean motion resonance. Such a resonant condition protects the dynamics of these planets resulting in very stable orbits. According to recent studies the capture into a resonance is the result of a planetary migration process induced by the interaction of the planets with a protoplanetary disk. If the migration is slow enough (adiabatic) next to a mean motion resonance, the two planets will also be in apsidal corotation.]{} [The recently refined orbital parameters of the system HD 128311 suggest that the two giant planets are in a 2:1 mean motion resonance, however without exhibiting apsidal corotation. Thus the evolution of this system can not be described by an adiabatic migration process alone. We present possible evolution scenarios of this system combining migration processes and sudden perturbations. ]{} [We model migration scenarios through numerical integration of the gravitational N-body problem with additional non-conservative forces. Planet-planet scattering has been investigated by N-body simulations.]{} [We show that the present dynamical state of the system HD 128311 may be explained by such evolutionary processes.]{}
Introduction
============
Among the 19 multi-planet systems found to date about a third are engaged in a low order mean motion resonance [@2005ApJ...632..638V]. The most prominent case is the exact 2:1 resonance of the two outer planets in GJ 876. There the orbital elements are very well determined, due to the short periods of the planets of $\approx$ 30 and 60 days [@2005ApJ...622.1182L]. The formation of resonant configurations between planets must be due to dissipative processes altering the semi-major axis. For the system GJ 876 the orbits are in apsidal corotation and both resonant angles librate with small amplitudes, a condition which can best explained by a sufficiently slow and long lasting differential migration process induced by the interaction of the planets with a protoplanetary disk . Hence, the occurrence of resonances constitutes a very strong indication of migration in young planetary systems, in addition to the hot Jupiter cases.
Recent analysis of the planetary system HD 128311 suggests that two giant planets are engulfed in a 2:1 mean motion resonance, however without exhibiting apsidal corotation [@2005ApJ...632..638V]. On the other hand, according to @2002ApJ...567..596L [@2006MNRAS.365.1160B], in the case of a sufficiently slow migration process the resonant planets should also be in apsidal corotation. In order to resolve the above discrepancy we construct [*mixed*]{} evolutionary scenarios of migrating planetary systems incorporating migration and other additional perturbative effects. For our investigation we use N-body numerical integrations containing also non-conservative forces [@2002ApJ...567..596L], which have been tested on full hydrodynamic evolutions of embedded planets . In this letter we report our findings in modeling the behavior of the resonant exoplanetary system HD 128311.
Orbital solution and its stability
==================================
It is mentioned by @2005ApJ...632..638V that the best fit to the radial velocity curve of HD 128311, based only on the use of Keplerian ellipses, leads to orbital data resulting in unstable behavior. Therefore they present also an alternate fit for HD 128311 calculated by using three-body gravitational interactions. The orbital data determined by this alternate fit (listed in Table 1) result in stable orbits, where the two giants are in a protecting 2:1 mean motion resonance (MMR). We perform three-body numerical integrations using these initial conditions of @2005ApJ...632..638V, as listed in Table 1. Only the resonant angle $\Theta_1 = 2\lambda_2 - \lambda_1 - \varpi_1$ librates around $0^{\circ}$ with an amplitude of $\sim 60^{\circ}$ (Fig. \[Fig1\], top), while both $\Theta_2 = 2\lambda_2 - \lambda_1 - \varpi_2 $ and $\Delta \varpi = \varpi_2 - \varpi_1$ circulate. Here $\lambda_j$ are mean longitudes and $\varpi_j$ are longitudes of periapse, both numbered from the inside out. This implies that the systems, although engaged in a MMR, is not in apsidal corotation. Moreover, the eccentricities show also large oscillations (Fig. \[Fig1\], bottom).
As described by @2005ApJ...632..638V, due to the relatively high stellar jitter, there exist a large variety of dynamically distinct stable orbital solutions resulting in the current radial velocity curve. The authors present the results obtained by a self-consistent two-planet model (taking into account the mutual gravitational interactions) and a Monte-Carlo procedure. As final outcome, they have found that only very few samples exhibit stable behavior without resonance. The remaining stable solutions are in 2:1 MMR, in the majority of the cases ($\sim 60\%$) only $\Theta_1$ librates while $\Theta_2$ circulates, in the remaining cases $\Theta_2$ librates as well. Hence, we assume that their derived parameters in Table 1 are an accurate representation of the dynamics of the system.
planet mass \[M$_J$\] $a$ \[AU\] $e$ $M$ \[deg\] $\varpi$ \[deg\]
-------- ---------------- ------------ ------ ------------- ------------------
b 1.56 1.109 0.38 257.6 80.1
c 3.08 1.735 0.21 166.0 21.6
: Orbital data of HD 128311 as provided by @2005ApJ...632..638V. $M$ denotes the mean anomaly.[]{data-label="table:1"}
Evolution through a migration process
=====================================
Adiabatic migration {#sec:adiabat}
-------------------
As shown in the previous section, the planets in the system HD 128311 are presently engaged in a stable 2:1 MMR. Although the actual orbital parameters do not exhibit apsidal corotation, the existence of the resonance suggests that the system has in the past gone through a migration process.
The migration of a single planet can be characterized by the migration rate $\dot a/a$ and the eccentricity damping rate $\dot e/e$. Here, we use the corresponding e-folding times for the semi-major axes and eccentricities: $\tau_a$ and $\tau_e$, respectively. The relation between the damping rates and e-folding times is $\dot a/a = - 1/\tau_a$, and similarly for the eccentricities. Investigating the system GJ 876, @2002ApJ...567..596L have found that for a sufficiently slow migration the final state of the system depends only on the ratio $K$ of the e-folding times $K = \tau_a/\tau_e$.
From hydrodynamical calculations we know that the order of magnitude of $K$ is close to unity and it reflects the physical properties (i.e. mass and viscosity) of the protoplanetary disk . If a system is a subject to such an adiabatic migration, a given value of $K$ results in unique values of the final eccentricities [@2002ApJ...567..596L]. However, as shown above, in the case of HD 128311 the eccentricities are varying with quite large amplitudes.
We have performed a set of numerical integrations of the general co-planar three-body problem adding non-conservative drag forces, varying the value of $K$.
![[**Top**]{}: Time evolution of the resonant angle $\Theta_1$ obtained by numerical integration using initial conditions of Table 1. [**Bottom**]{}: The corresponding evolution of the eccentricities.[]{data-label="Fig1"}](Figure1.eps "fig:"){width="8cm"} ![[**Top**]{}: Time evolution of the resonant angle $\Theta_1$ obtained by numerical integration using initial conditions of Table 1. [**Bottom**]{}: The corresponding evolution of the eccentricities.[]{data-label="Fig1"}](Figure2.eps "fig:"){width="8cm"}
![The behavior of the semi-major axes and eccentricities during an adiabatic migration with $\tau_a=2\times10^3$ years and $K=5$. The migration is stopped between $2\times10^3$ and $3\times10^3$ years by applying a linear reduction.[]{data-label="fig:adiabat"}](Figure3.eps){width="8cm"}
In these simulations the planets move originally in quasi-circular orbits and we start them from $a_1=4$AU and $a_2=2$AU. Assuming that only the outer planet is forced to migrate inward, we implement a dissipative force which results in a migration characterized by the e-folding time $\tau_a=2\times10^3$ years in the semi-major axis, and $\tau_e=4\times10^2$ years in the eccentricity of the outer planet (corresponding to $K=5$). In order to obtain the present values of the semi-major axes we slow down the migration beginning from $t_1=2\times10^3$years and decreasing it linearly to zero until $t_2=3\times10^3$years. In this way we model the smooth dispersal of the protoplanetary disk. After this migration process, the system is locked into a deep 2:1 resonance, the resonant angles $\Theta_{1,2}$, and $\Delta \varpi$ librate around $0^{\circ}$ with small amplitudes. The eccentricities $e_1$ and $e_2$ are almost constant (showing only a small amplitude oscillations), where the index 1 refers to the inner and the index 2 to the outer planet. For the chosen $K=5$ we find $e_1=0.46$ and $e_2=0.15$, see Fig. \[fig:adiabat\]. Smaller/larger values of $K$ yield always systems deep in resonance and result in larger/smaller $e_{1,2}$, contradicting Fig. \[Fig1\].
[*Clearly, the present behavior of HD 128311 is not the result of such an adiabatic migration process alone.*]{} In the following, we shall present two additional mechanisms, which may be responsible for the large oscillations of the eccentricities and breaking the apsidal corotation.
Sudden stop of the migration of the outer planet
------------------------------------------------
Recent [*Spitzer*]{} observations of young stars confirm that the inner part of the protoplanetary disk may contain only very little mass @2005ApJ...621..461D [@2005ApJ...630L.185C], possibly due to photo-evaporation induced by the central star. Thus, upon approaching the inner rim of such a disk, the inward migration of a planet can be terminated rapidly.
In order to model this type of scenario, we perform additional simulations where the migration of the outer planet has been stopped abruptly reaching the actual value of its semi-major axis ($a_2=1.73$AU). We assume that the inner planet orbits in the empty region of the disk at $a_1=1.5$AU, and we start the outer planet from $a_2=4$AU forcing to migrate inward very fast ($\tau_a = 500$ years). We find that the present behavior of the eccentricities can be obtained by using $K=10$. The sudden stop of the migration results in a behavior of the eccentricities shown in Fig. \[fig:sudden\] which is very similar to the observed case (Fig. \[Fig1\]). We note that after the above sudden stop of the migration the planets remain in apsidal corotation but with increased libration amplitudes of the resonant angles.
![The behavior of the eccentricities obtained by a sudden stop of the migration of the outer planet.[]{data-label="fig:sudden"}](Figure4.eps){width="8cm"}
Planet-planet scattering event
------------------------------
![ Time evolution of the eccentricities ([**Top**]{}), resonant angles $\Theta_1$ ([**Middle**]{}) and $\Delta\varpi$ ([**Bottom**]{}) of the inward migrating giant planets, before and after a scattering event with an inner low mass planet. []{data-label="fig:inner1"}](Figure5.eps "fig:"){width="8cm"} ![ Time evolution of the eccentricities ([**Top**]{}), resonant angles $\Theta_1$ ([**Middle**]{}) and $\Delta\varpi$ ([**Bottom**]{}) of the inward migrating giant planets, before and after a scattering event with an inner low mass planet. []{data-label="fig:inner1"}](Figure6.eps "fig:"){width="8cm"} ![ Time evolution of the eccentricities ([**Top**]{}), resonant angles $\Theta_1$ ([**Middle**]{}) and $\Delta\varpi$ ([**Bottom**]{}) of the inward migrating giant planets, before and after a scattering event with an inner low mass planet. []{data-label="fig:inner1"}](Figure6b.eps "fig:"){width="8cm"}
The behavior of the eccentricities of the giant planets of the system HD 128311 is very similar to that observed in the system around $\upsilon$ Andromedae. @2005Natur.434..873F proposed that such a behavior in $\upsilon$ And is most likely the result of a planet-planet scattering event. Investigating the resonant system HD 73526, @tinney-2006- also suggested that the lack of the apsidal corotation can be the result of a dynamical scattering event. Therefore we investigate whether the present behavior of HD 128311 can be modeled by such an effect. We present two cases.\
[*Additional inner planet*]{}\
In this case we assume that an additional small mass planet is orbiting close to the central star in a quasi circular orbit. When the outer giants are far enough from the star the orbit of the small mass planet is stable. However, as the giants migrate inward approaching their present positions they perturb the orbit of the small planet making its motion chaotic, which in long term may result in an increase of its eccentricity. Finally, due to its high eccentricity, the small planet can suffer a close encounter with one of the giant planets. After the close encounter the small planet may be ejected from the system or pushed into an orbit with very large semi-major axis. Depending on the initial position of the small planet, it can also be captured into a mean-motion resonance with the inner giant, and forced to migrate inward. During the migration its eccentricity will increase in a much shorter timescale than by a “pure" chaotic evolution. Thus, in this case a close encounter with the inner giant is very likely as well.
In Fig. \[fig:inner1\] we show the behavior of the eccentricities and the resonant angles $\Theta_1$ and $\Delta\varpi$ when the giant planets are migrating inwards with initial conditions given in Sect. \[sec:adiabat\]. Starting the small planet ($m=0.03M_{\rm Jup}$) from a nearly circular orbit with $a=0.5$AU, it is captured into a resonance with the inner giant. During the migration the small planet’s eccentricity increases, and finally the small planet crosses the orbits of giant planets and suffers a close encounter with them. After the encounter at $t\approx 2500$yrs, the apsidal corotation of the giant planets breaks (i.e. $\Theta_2$ and $\Delta \varpi$ circulate), however the giant planets remain in the 2:1 resonance, since $\Theta_1$ still librates around $0^\circ$, see Fig. \[fig:inner1\].\
[*Additional outer planet*]{}\
In this second case we assume that the third planet ($m=0.03M_{\rm Jup}$) originates from the outer region, and that it approaches the giant planets through inward migration where we assume that the adiabatic migration of the two giant planets is already finished, as displayed in Fig. \[fig:adiabat\]. They orbit for instance in a gas-free environment while the small planet is still embedded in the outer protoplanetary disk. In Fig. \[fig:outer1\] we display the variation of the eccentricities of the giants after the scattering event. The small planet is started from $a=2.6$AU and migrates inward with an e-folding time $\tau_a=2\times10^3$ years. After scattering it is pushed into a very distant orbit $a\sim300$AU. We note that in this case the giant planets remain in apsidal corotation, however with a substantial increase in $\Theta_2$ and $\Delta\varpi$. In additional simulations we find that apsidal resonance is always preserved, however we cannot exclude the possibility of breaking the apsidal corotation entirely.
![The behavior of the eccentricities before and after a scattering with a small mass planet migrating inward from outside starting from $a=2.6$AU.[]{data-label="fig:outer1"}](Figure7.eps){width="8cm"}
Conclusions
===========
In this paper we investigate evolutionary scenarios, which may have led to the observed behavior of the resonant system HD 128311, where the two planets are engaged in a 2:1 mean motion resonance, but not in apsidal corotation. We assume that the two giant planets have been formed far from the central star and migrated inwards, due to gravitational interaction with the protoplanetary disk. During this differential migration, they have been locked into the 2:1 mean motion resonance. At the end of their migration we imagine that the system suffers a sudden perturbation.
As a first case we study the sudden stop of the migration, which may possibly be induced by an inner rim of the disk and an empty region inside of it, as indicated by some observations of young stars.
In the second scenario the sudden perturbation is caused by a planet-planet scattering event, similar to that suggested by @2005Natur.434..873F in the case of $\upsilon$ And. We analyze an encounter with a small ($\sim 10 M_{\oplus}$) planet, approaching the two massive planets either from outside or inside, which is thrown to a large $a$ orbit or directly ejected after the scattering event.
We find that the sudden perturbation caused by an encounter with an [*inner*]{} small planet is clearly able to break the apsidal resonance of the two planets. But also in the other cases the orbital behavior of the giants is very similar to the most recent orbital data of HD 128311 [@2005ApJ...632..638V].
The system HD 128311 constitutes another example which demonstrates the important interplay of migration and scattering processes in shaping the dynamics of exoplanetary systems.
The authors acknowledge the supports of the Hungarian Scientific Research Fund (OTKA) under the grant D048424 and the German Science Foundation (DFG) under grant 436 UNG 17/5/05. WK would like to thank Greg Laughlin for the many fruitful discussions.
|
---
abstract: 'We address the question of the dependence of the fragility of glass forming supercooled liquids on the ”softness” of an interacting potential by performing numerical simulation of a binary mixture of soft spheres with different power $n$ of the interparticle repulsive potential. We show that the temperature dependence of the diffusion coefficients for various $n$ collapses onto a universal curve, supporting the unexpected view that fragility is not related to the hard core repulsion. We also find that the configurational entropy correlates with the slowing down of the dynamics for all studied $n$.'
author:
- 'Cristiano De Michele${}^{1,2}$'
- Francesco Sciortino
- Antonio Coniglio
title: 'Scaling in soft spheres: fragility invariance on the repulsive potential softness'
---
When a liquid is cooled below its melting temperature, if crystallization does not take place, it becomes [*supercooled*]{}. In this supercooled region, the viscosity increases by more than 15 order of magnitude in a a small $T$-range. When the viscosity $\eta$ reaches a value of about $10^{13}$ Poise the liquid can be treated as an amorphous solid, i.e., a glass [@stillnaturerev; @torquatorevglass; @stillglassreviewscience; @angellnatureliqland] and the corresponding temperature is defined as the glass transition temperature (labelled $T_g$). The $T$-dependence of the viscosity $\eta$ differs for different glass formers. Angell has proposed a classification based on the behaviour of $\eta(T)$. Glasses are said to be [*fragile*]{} if they show large deviations from an Arrhenius law ($\eta(T)\propto \exp[E/T]$) or [*strong*]{} otherwise [@angellfrag]. The fragility $m$ of a glass forming liquid can be quantified by the slope of $\log\eta(T)$ vs $T_g/T$, evaluated at $T_g$, i.e. as $$m = \frac{{\it d}\log \eta}{{\it d}(T_g/T)}|_{T=Tg}
\label{eq:fragdlog}$$ While the original definition of fragility is based on a purely dynamic quantity, correlation between $m$ and other physical properties of glass forming liquids, both with dynamic and thermodynamic properties, have been reported . Recently, a remarkable correlation with vibrational properties of the glass state has been discovered [@tullioscience]. One of the main challenges in the physics of supercooled liquid and glasses is to understand the connection between dynamical properties of the liquid close to the glass transition, i.e. the fragility, and microscopic properties. Is the fragility most affected by the steepness of the repulsive potential or by the inter particle attraction? Is it controlled by other properties of the interaction potential? In the present letter we address this question calculating numerically the fragility of several models for liquids, differing only in the softness of the repulsive potential. We aim at understanding whether changing the softness of the repulsive potential the fragility changes accordingly. We show more generally that the diffusion coefficient $D$ can be scaled on a universal master curve by changing the softness of the repulsive potential. This implies that the fragility does not depend on the softness of the interaction potential. We complement this dynamical study with the evaluation of the configurational entropy to check the validity of the Adam-Gibbs[@AG; @antosconf; @schillingreview; @wolynes] relation. In this Letter we consider a simple glass former, a $80:20$ binary mixture of $N=1000$ soft spheres [@replicaBMLJPRL; @speedySSPEL; @depablosoft], that is an ensemble of spheres interacting via the following potential $$V_{\alpha\beta}(r) = 4\epsilon_{\alpha\beta} \left(\frac{\sigma_{\alpha\beta}}{r}\right)^n
\label{Eq:Vsoftsphere}$$ where $\alpha,\beta\in{A,B}$, $\sigma_{AA}=1.0$, $\sigma_{AB} = 0.8$, $\sigma_{BB} = 0.88$, $\epsilon_{AA}=1.0$, $\epsilon_{AB}=1.5$, $\epsilon_{BB}=0.5$ and $n$ is a parameter by which is possible to tune the ”softness” of the interaction [@hansenmcdonald]. This interaction potential is a Kob-Andersen potential [@kobandersenPRE] in which the attractive part of the potential has been dropped. In particular we investigate the values $n=6,8,12,18$. This choice for the binary mixture is motivated by the fact that such a system is not prone to crystallization, that is it can be easily supercooled below its melting temperature . Still, for $n < 6$, crystallization takes place within the simulation time, determining a lower limit to the range of investigated $n$ values. Reduced units will be used in the following, length will be in units of $\sigma_{AA}$, energy in units of $\epsilon_{AA}$ and time in units of $(M\sigma_{AA}^2/\epsilon_{AA})^{1/2}$, where $M$ is the mass of all particles. In physical units, assuming the atom $A$ is Argon, the units are a length of $3.4$$\AA$, an energy of $120K k_B$ and a time of $2.15 ps$.
The self-similar nature of the soft-sphere potential couples $T$ and $V$. It can be shown that all thermodynamic properties depend on the quantity $TV^{\frac{1}{n}}$[@softeos]. Dynamic properties can also be scaled accordingly [@japsoft]. Hence, it is sufficient to quantify the $T$-dependence or the $V$-dependence of any observable to fully characterize the behavior of the system. As a consequence the fragility does not change upon changing the density of the soft binary mixture.
\[Fig:DallMCT\]
Figure \[Fig:diffMCT\] shows the $T$-dependence of the diffusion coefficients, evaluated from the long time limit behavior of the mean square displacement, for all $n$ investigated and covering a window of about four order of magnitudes. In the attempt of compare the $n$-dependence of the diffusion coefficient, we report in Fig. \[Fig:scaleDT\] the data as a function of $T_n/T$, where $T_n$ is chosen in such a way to maximize the overlap between data of different $n$, i.e. to collapse all data onto a single master curve. Figure \[Fig:scaleDT\] shows that all curves can be successfully scaled onto the master curve ${\cal D}$ choosing a proper set of scaling parameters $T_n$ (whose $n$-dependence is plotted in the inset of this Figure). The very good quality of the resulting master curve $$D(T) = {\cal D}(T/T_n).
\label{Eq:Dscaling}$$ suggests that the $n$-dependence enters only via a rescaling of the temperature [@note3; @tarjusmossa].
The remarkable consequence of latter result is that the fragility of the system does not depend on the repulsive interaction potential. In fact according to Eq. (\[Eq:Dscaling\]) and from the definition of liquid’s fragility $m$ given in Eq. (\[eq:fragdlog\]), assuming $D\propto\tau^{-1}$ we get: $$m = \frac{T_g(n)}{T_n}\; \frac{1}{{\cal D}[T_g(n)/T_n]}\;
\frac{d{\cal D}(x)}{dx}|_{x=T_g(n)/T_n}
\label{Eq:scaledfrag}$$ where $T_g(n)$ is the glass transition temperature for the system with softness $n$, which can be defined as the temperature at which diffusivity reaches an arbitrary small value $10^{\cal K}$[@note], i.e. $$-\log D[T_g(n)] = -\log {\cal D}\left [ \frac{T_g(n)}{T_n}\right ]={\cal K}
\label{Eq:BSdefTg}$$
Eq.\[Eq:scaledfrag\] shows that the fragility index $m$ is a function only of the scaled variable $\frac{T_g(n)}{T_n}$ and hence, as far as the scaling reported in Fig. \[Fig:scaleDT\] keeps holding even at temperatures lower than the one we are able to equilibrate, the dynamic fragility $m$ is independent of $n$ as well. By fitting the master curve to a Vogel-Tamman-Fulcher fit, as shown in Fig. \[Fig:scaleDT\], an estimate of $\frac{T_g(n)}{T_n} = 10^{\cal K} $ can be calculated, resulting into a estimation of $m \approx 130$. This figure should be compared with the value $m=81$ for o-terphenyl (OTP), that is a typical fragile liquid and $m=20$ for the prototypical strong glass the liquid silica ($SiO_2$).
For completeness, we report also in Fig. \[Fig:scaleDT\] a fit of the master curve according to the prediction of mode-coupling theory, which has been shown to be consistent with numerical data for several models in the weak supercooling region. A best fit procedure requires the exclusion of the low $T$ points, for which deviations from the power-law fit are observed [@note2].
Recently, evidence has been presented that kinetic fragility strongly correlates with thermodynamic fragility[@angelmartinez]. In this respect, it is worth looking if the scaling observed in dynamical properties has a counterpart in thermodynamic properties. In particular, we evaluate the configurational entropy for the system, within the potential energy landscape framework as discussed in details in Refs.[@crifraJCP; @fraPELPRL; @stillweberPRA; @emiliaOTP; @sastryPELform; @sastryBMLJnature]. In brief, we estimate $S_{c}$ as difference between the liquid entropy (calculated via thermodynamic integration from the ideal gas) and of the vibrational entropy (calculated via thermodynamic integration, including anharmonic corrections, from the very low temperature harmonic dynamics of the disordered solid associated to the liquid configuration). We then focus on the ability of the Adam-Gibbs (AG) relation — which states that $$D(T) = A_{AG} e^{\frac{B_{AG}}{T S_c}},
\label{Eq:AdamGibbs}$$ — of modelling the temperature dependence of $D$. Fig. \[Fig:AGallANH\] shows the AG plot for the studied $n$ values. For all $n$, a satisfactory linear representation of $\log(D)$ vs. $1/TS_{c}(T)$ is observed. As discussed in more details in Ref.[@jchemphysme], the simultaneous validity of the VTF description of $D$ and of the AG relation requires the identity of the kinetic and thermodynamic fragilities. In this respect, the independence of $n$ discussed above for the case of kinetic fragility carries on also to thermodynamic fragility.
\[Fig:AGDTanh\]
A remarkable consequence of the validity of the AG relation (Eq. \[Eq:AdamGibbs\]), associated to the scaling with $n$ of $D$ (Eq.\[Eq:Dscaling\]) is that the configurational entropy can be written as $$S_c(T) = S_0(n) {\cal F}(T/T_n)
\label{Eq:Scscaled}$$ where $F(x)$ is a scaling function and $S_0(n) = B_{AG}/T_n$. To support such proposition, we show in Figure \[Fig:Scscaled\] $S_c$ multiplied by the factor $B_{AG}/T_n$ as a function of $T/T_n$, were $T_n$ are the values for which $D$ scaling is recovered (inset of Fig.\[Fig:scaleDT\]). Again, the quality of the data collapse stresses the validity of the scaling with $n$.
To conclude the relevant result that has been shown in this Letter is that in the case of soft sphere potentials, the dynamic fragility is independent on the power $n$ of the short range repulsion. This conclusion is based on the hypothesis that the scaling observed in the range of $T$ where simulations are feasible extends also to lower temperatures, down to the glass transition temperature. Indeed, a particular effort has been made to equilibrate configurations to temperatures lower than the MCT temperature, where dynamical processes different from the ones captured by MCT are active. If the scaling is indeed valid, the results presented in this Letter strongly support the possibility, that contrary to our common understanding, fragility in liquids is mostly controlled by other properties of the potential, more than by the hard core repulsion. Finally we note that one could be tempted to associate the fact that the diffusion coefficient data can be rescaled only by change the energy scale by $T_n$ to a simple overall rescaling of the landscape potential. The data in Fig.\[Fig:Scscaled\] suggest that this is not the case since $S_c(E)$ is not just scaling function of $T/T_n$ but it needs to be rescaled by a factor $S_0(n)$ and hence the number of distinct basins explored at the same $T/T_n$ changes with $n$. A non-trivial compensation mechanism between the scaling of the static properties ($S_c$) and the scaling of the kinetic coefficient $B_{AG}(n)$ (defined in Eq. (\[Eq:AdamGibbs\])) on $n$ must be present.
We thank L. Angelani and G. Ruocco for useful discussions. We acknowledge support from INFM Initiative Parallel Computing, Marie Curie Network and Miur FIRB and COFIN2002.
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abstract: |
Supersymmetric solution of PT-/non-PT-symmetric and non-Hermitian Morse potential is studied to get real and complex-valued energy eigenvalues and corresponding wave functions. Hamiltonian Hierarchy method is used in the calculations.\
PACS numbers: 03.65.-w; 03.65.Ge\
Keywords: PT-symmetry, non-Hermitian operators, Hamiltonian Hierarchy method
author:
- |
Metin Aktaş and Ramazan Sever$\thanks{Corresponding author:
sever@metu.edu.tr}$\
Department of Physics, Middle East Technical University\
06531 Ankara, Turkey
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Introduction
============
PT-symmetric Hamiltonians has acquired much interest in recent years\[1, 2, 3\]. Bender and Boettcher \[1\] suggested that a non-Hermitian complex potential with the characteristic of PT-invariance has real energy eigenvalue if PT-symmetry is not spontaneously broken. The other concept for a class of non-Hermitian Hamiltonians is pseudo-Hermiticity. This kind of operators satisfy the similarity transformation $\eta~\hat{H}~\eta^{-1}=\hat{H}^{\dag}$ \[3, 4, 5\]. PT-invariant operators have been analysed for real and complex spectra by using a variety of techniques such as variational methods \[7\], numerical approaches \[8\], semiclassical estimates \[9\], Fourier analysis \[10\] and group theoretical approach with the Lie algebra \[11\]. It is pointed out that PT-invariant complex-valued operators may have real or complex energy eigenvalues \[12\]. Many authors have studied on PT-symmetric and non-PT-symmetric non-Hermitian potential cases such as flat and step potentials with the framework of SUSYQM \[13-15\], exponential type potentials \[16-21\], quasi exactly solvable quartic potentials \[22-24\], complex Hénon-Heiles potential \[25\], and therein \[26-28\]. In the present work, the real and complex-valued bound-state energies of the q-deformed Morse potential are evaluated through the Hamiltonian Hierarchy method \[29\] by following the framework of PT-symmetric quantum mechanics. This method also known as the factorization method of the Hamiltonian introduced by Schrödinger \[30\], and later developed by Infeld and Hull \[31\], It is useful to discover for different potentials with equivalent energy spectra in non-relativistic quantum mechanics. Various aspects has been studied within the formalism of SUSYQM \[32\]. This paper is arranged as follows: In Sec. II we introduce the Hamiltonian Hierarchy method. In Sec. III we apply the method to solve the Schrödinger equation with PT-symmetric and non-PT-symmetric non-Hermitian forms of the q-deformed Morse potential. In Sec. IV we discuss the results.
SUSYQM and Hamiltonian Hierarchy Method
=======================================
Supersymmetric algebra allows us to write Hamiltonians as \[30\]
$$H_{\pm}=-\frac{\hbar^{2}}{2m}\frac{d^{2}}{dx^{2}}+
V_{\pm}(x),$$
where The supersymmetric partner potentials $V_{\pm}(x)$ in terms of the superpotential $W(x)$ are given by
$$V_{\pm}(x)=W^{2}\pm\frac{\hbar}{\sqrt{2m}}
\frac{dW}{dx}.$$
The superpotential has a definition
$$W(x)=-\frac{\hbar}{\sqrt{2m}}\left[\frac{d\ln\Psi_{0}^{(0)}(x)}{dx}\right],$$
where, $\Psi_{0}^{(0)}(x)$ denotes the ground state wave function that satisfies the relation
\_[0]{}\^[(0)]{}(x)=N\_[0]{} .
The Hamiltonian $H_{\pm}$ can also be written in terms of the bosonic operators $A^{-}$ and $A^{+}$
$$H_{\pm}=A^{\mp}~A^{\pm},$$
where
$$A^{\pm}=\pm\frac{\hbar}{\sqrt{2m}}\frac{d}{dx}+W(x).$$
It is remarkable result that the energy eigenvalues of $H_{-}$ and $H_{+}$ are identical except for the ground state. In the case of unbroken supersymmetry, the ground state energy of the Hamiltonian $H_{-}$ is zero $\left(E_{0}^{(0)}=0\right)$ \[30\]. In the factorization of the Hamiltonian, the Eqs. (1), (5) and (6) are used respectively. Hence, we obtain
$$\begin{aligned}
H_{1}(x)&=&-\frac{{\hbar}^{2}}{2m}\frac{d^{2}}{dx^{2}}+V_{1}(x)\nonumber\\[0.2cm]
&=&(A_{1}^{+}~A_{1}^{-})+E_{1}^{(0)}.\end{aligned}$$
Comparing each side of the Eq. (7) term by term, we get the Riccati equation for the superpotential $W_{1}\left(x\right)$
$$W_{1}^{2}-W_{1}^{'}=\frac{2m}{{\hbar}^{2}}\left(V_{1}(x)-E_{1}^{(0)}\right).$$
Let us now construct the supersymmetric partner Hamiltonian $H_{2}$ as
$$\begin{aligned}
H_{2}(x)&=&-\frac{{\hbar}^{2}}{2m}\frac{d^{2}}{dx^{2}}+V_{2}(x)\nonumber\\[0.2cm]
&=&\left(A_{2}^{-}~A_{2}^{+}\right)+E_{2}^{(0)},\end{aligned}$$
and Riccati equation takes $$W_{2}^{2}+W_{2}^{'}=\frac{2m}{{\hbar}^{2}}\left(V_{2}(x)-E_{2}^{(0)}\right).$$
Similarly, one can write in general the Riccati equation and Hamiltonians by iteration as
$$\begin{aligned}
W_{n}^{2}\pm
W_{n}^{'}&=&\frac{2m}{{\hbar}^{2}}\left(V_{n}(x)-E_{n}^{(0)}\right)\nonumber\\[0.2cm]
&=&\left(A_{n}^{\pm}~A_{n}^{\mp}\right)+E_{n}^{(0)},\end{aligned}$$
and
$$\begin{aligned}
H_{n}(x)&=&-\frac{{\hbar}^{2}}{2m}\frac{d^{2}}{dx^{2}}+V_{n}(x)\nonumber\\[0.2cm]
&=& A_{n}^{+}~A_{n}^{-}+E_{n}^{(0)},\quad\quad n=1,2,3,\ldots\end{aligned}$$
where $$A_{n}^{\pm}=\pm\frac{\hbar}{\sqrt{2m}}\frac{d}{dx}+\frac{d}
{dx}\left(\ln\Psi_{n}^{(0)}(x)\right).$$
Because of the SUSY unbroken case, the partner Hamiltonians satisfy the following expressions \[30\]
$$E_{n+1}^{(0)}=E_{n}^{(1)}, \quad with \quad E_{0}^{(0)}=0,\quad
n=0,1,2,\ldots$$
and also the wave function with the same eigenvalue can be written as \[30\]
$$\Psi_{n}^{(1)}=\frac{A^{-}~\Psi_{n+1}^{(0)}}{\sqrt {E_{n}^{(0)}}},$$
with
$$\Psi_{n+1}^{(0)}=\frac{A^{+}~\Psi_{n}^{(1)}}{\sqrt {E_{n}^{(0)}}}.$$
This procedure is known as the hierarchy of Hamiltonians.
Calculations
============
The General q-deformed Morse case
---------------------------------
Let us first consider the generalized Morse potential as \[16\] $$V_{M}(x)=V_{1}~e^{-2~a~ x}-V_{2}~e^{-a~x}.$$
where $V_{1}$ and $V_{2}$ are real parameters. By comparing the Eq. (17) with the following equation
$$V_{M}(x)=D(e^{-2~a~x}-2qe^{-a~x}),$$
we have $V_{1}=D$ and $V_{2}=2qD$. Therefore we can construct the hierarchy of Hamiltonians for Schrödinger equation with $\ell=0$,
$$[-\frac{d^{2}\Psi}{dx^{2}}+\mu^{2}(e^{-2~a~x}-2qe^{-a~x})]\Psi(x)=\varepsilon\Psi(x),$$
where $\ds{\mu^{2}=\frac{2mV_{1}}{a^{2}\hbar^{2}}}$ and $\ds{E=\varepsilon\frac{a^{2}\hbar^{2}}{2m}}$. We can also write the Riccati equation as $$W_{1}^{2}-W_{1}^{\prime}+\varepsilon_{0}^{(1)}=V_{1}(x).$$
Here $V_{1}(x)$ is the superpartner of the superpotential $W_{1}(x)$. Following by ansatz equation, we have
$$W_{1}(x)=-\mu~e^{-a~x}+q~\delta,$$
and inserting this into the Eq. (20), we get $$\delta=(\mu-\frac{a}{2q}),$$
with the first ground state energy $$\varepsilon_{0}^{(1)}=-q^{2}(\mu-\frac{a}{2q})^{2}.$$
In order to construct the other superpartner potential $V_{2}(x)$, we will solve the equation
$$W_{1}^{2}+W_{1}^{\prime}+\varepsilon_{0}^{(1)}=V_{2}(x).$$
Then we can find the second member superpotential as
$$W_{2}(x)=-\mu~e^{-a~x}+q~\kappa.$$
Now, putting this ansatz into the Eq. (20), we get
$$\kappa=(\mu-\frac{3a}{2q}),$$
with
$$\varepsilon_{0}^{(2)}=-q^{2}(\mu-\frac{3a}{2q})^{2}.$$
By similar iterations, one can get the general results
$$W_{n+1}(x)=-\mu~e^{-ax}+q\left[\mu-\frac{a}{q}(n+\frac{1}{2})\right],$$
$$V_{n+1}(x)=\mu^{2}(e^{-2ax}-2qe^{-ax})+2na\mu~e^{-ax},$$
$$E_{n+1}^{(\ell=0)}=-q^{2}\left[\mu-\frac{a}{q}(n+\frac{1}{2})\right]^{2},$$
and ground state wave function $$\Psi_{0}(x)=N~\exp\{-\widetilde{\mu}e^{-ax}+q\left[\mu
-\frac{a}{q}(n+\frac{1}{2})\right]x\}.$$
where we choose $\widetilde{\mu}=a\mu$ and set $(\hbar=2m=1)$ in Eq. (30).
Non-PT-symmetric and Non-Hermitian Morse case
---------------------------------------------
Let us now consider the Eq. (17) with respect to $V_{1}\rightarrow~D$ as real and $V_{2}\rightarrow~2iqD$ as complex parameters. Hence we construct the hierarchy of Hamiltonian of the Schrödinger equation for the complexified Morse potential as
$$[-\frac{d^{2}\Psi}{dx^{2}}+\mu^{2}(e^{-2~a~x}-2iqe^{-a~x})]\Psi(x)=\varepsilon\Psi(x),$$
where $\ds{\mu^{2}=\frac{2mV_{1}}{a^{2}\hbar^{2}}}$ and $\ds{E=\varepsilon\frac{a^{2}\hbar^{2}}{2m}}$.
Applying the hierarchy of Hamiltonians as in the previous section, the $(n+1)$-th member results will be
$$W_{n+1}(x)=-\mu~e^{-a~x}+q\left[i\mu-\frac{a}{q}(n+\frac{1}{2})\right],$$
$$V_{n+1}(x)=\mu^{2}(e^{-2ax}-2qe^{-ax})+2na\mu~e^{-ax},$$
$$E_{n+1}^{(\ell=0)}=-q^{2}\left[i\mu-\frac{a}{q}(n+\frac{1}{2})\right]^{2},$$
with $$\Psi_{0}(x)=N~\exp~\{-\widetilde{\mu}e^{-ax}+q\left[i\mu-\frac{a}{q}(n
+\frac{1}{2})\right]x\},$$ where $\widetilde{\mu}=a\mu$.
PT-symmetric and Non-Hermitian Morse Case
-----------------------------------------
Let us assume the potential parameters $V_{1}=(\alpha+i\beta)^{2}$ and $V_{2}=(2\gamma+1)(\alpha+i\beta)$ in Eq. (17). Here we choose $\alpha$ and $\beta$ and $\gamma=-\frac{1}{2}+q(\alpha+i\beta)$. When $a\rightarrow~i~a$ in Eq. (17) and choosing $V_{1}$ and $V_{2}$ as in the previous section, the potential form will be
$$V_{M}(x)=(\alpha+i\beta)^{2}(e^{-2iax}-2qe^{-iax}).$$
The ansatz equation is $$W_{1}(x)=\xi~e^{-ia~x}+iq~\delta.$$
As a result the Schrödinger equation can be written by using the Eq. (19) for $\ds{\mu^{2}=\frac{2m(\alpha+i\beta)^{2}}{a^{2}\hbar^{2}}}$ and $\ds{E=\varepsilon\frac{a^{2}\hbar^{2}}{2m}}~(if E<0)$. Applying the same procedure again, one can get
$$W_{n+1}(x)=\xi~e^{-ia~x}+iq\left[i\xi-\frac{a}{q}(n+\frac{1}{2})\right],$$
$$V_{n+1}(x)=\mu^{2}(e^{-2iax}-2qe^{-iax})+2in\xi~a~e^{-iax},$$
$$E_{n+1}^{(\ell=0)}=-q^{2}\left[i\xi-\frac{a}{q}(n+\frac{1}{2})\right]^{2},$$
with $$\Psi_{0}(x)=N~\exp~\{-\widetilde{\xi}e^{-iax}+iq\left[i\xi
-\frac{a}{q}(n+\frac{1}{2})\right]x\},$$ where $\widetilde{\mu}=ia\mu$.
Conclusions
===========
We have used the PT-symmetric formulation developed recently within non-relativistic quantum mechanics to a more general Morse potential. We have solved the Schrödinger equation in one dimension by applying Hamiltonian hierarchy method within the framework of SUSYQM. We discussed many different complex forms of this potential. Energy eigenvalues and corresponding eigenfunctions are obtained exactly. We also point out that the exact results obtained for the complexified Morse potential may increase the number of interesting applications in the study of different quantum mechanical systems. In the case of $\beta=0$ in Eq. (41), there is only real spectra, when $\alpha=0$, otherwise there exists a complex-valued energy spectra. This implies that broken PT-symmetry doesn’t occur spontaneously. Moreover, PT-/non-PT-symmetric non-Hermitian solutions have the same spectra. We also note that both real and imaginary part of the energy eigenvalues corresponds to the anharmonic and harmonic oscillator solutions. The $(n+1)-th$ member superpotential, its superpartner and also corresponding ground state eigenfunctions of PT-symmetric non-Hermitian potentials satisfy the condition of PT-invariance though the others are not.
[99]{}
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---
abstract: 'The manifold of coupling constants parametrizing a quantum Hamiltonian is equipped with a natural Riemannian metric with an operational distinguishability content. We argue that the singularities of this metric are in correspondence with the quantum phase transitions featured by the corresponding system. This approach provides a universal conceptual framework to study quantum critical phenomena which is differential-geometric and information-theoretic at the same time.'
author:
- Paolo Zanardi
- Paolo Giorda
- Marco Cozzini
title: 'The differential information-geometry of quantum phase transitions'
---
.– Suppose you are given with a set of quantum states associated to a family of Hamiltonians smoothly depending on a set of parameters, e.g., coupling constants. This parameter manifold – that can include temperature in the case the considered states are thermal ones – is partitioned in regions characterized by the fact that inside them one can “adiabatically" move from one point to the other and no singularities in the expectation values of any observables are encountered. The boundaries between these regular regions are in turn associated to the non-analytic behaviour of some observable and are referred to as critical points; crossing one of these points results in a [*phase transition*]{} (PT). States lying in different regions generally have some strong structural difference and are, in principle, easily distinguishable once somehow a preferred observable is chosen.
The standard machinery, i.e., the so-called Landau-Ginzburg paradigm, to deal with this phenomenon is based on the notions of symmetry breaking, order parameter, and correlation length [@huang]. On the other hand, some system fails to fall in this conceptual framework. This can be due to the difficulty of identifying the proper order parameter for systems whose symmetry breaking pattern is unknown or to the very absence of a local order parameter, e.g., quantum phase transitions (QPTs) involving different kinds of topological order [@toporder]. Even another standard characterization of QPTs, i.e., singularities in the ground state (GS) energy as a function of the coupling constant, misses to capture the boundaries between phases for some QPTs, e.g., those with matrix-product states [@wo-ci].
In the last few years ideas and tools borrowed from quantum information science [@qis] have been used to study quantum, i.e., zero temperature, phase transistions [@sachdev]; in particular the role of quantum entanglement in QPTs has been extensively investigated [@qpt-qis]. More recently an approach to QPTs based on the concept of [*quantum fidelity*]{} has been put forward [@za-pa] and applied to systems of quasi-free fermions [@za-co-gio; @co-gio-za], to the so-called matrix-product states [@co-ion-za], and extended to finite-temperature [@zhong-guo]. In the fidelity approach, QPTs are identified by studying the behavior of the amplitude of the overlap, i.e., scalar product, between two ground states corresponding to two slightly different set of parameters; at QPTs a drop of the fidelity with scaling behaviour is observed and quantitative information about critical exponents can be extracted [@co-gio-za; @co-ion-za]. The fidelity approach is not based on the identification of an order parameter – and therefore does not require a knowledge of symmetry breaking patterns – or more in general on the analysis of any distinguished observable, e.g., Hamiltonian, but it is a purely metrical one. All the possible observables are in a sense considered at once.
In this paper we shall unveil the universal differential-geometric structure underlying these observations. We shall show how QPTs can be associated to the singularities of a Riemannian metric tensor inherited by the parameter space from the natural Riemannian structure of the projective space of quantum states. This structure has an interpretation in terms of information-geometry [@woo; @bra-ca] providing the differential-geometric approach of this paper with an information-theoretic content.
[*Information-geometry and QPTs*]{}.– Let us consider a smooth family $H(\lambda),\,\lambda\in{\cal M}$(=the parameter manifold), of quantum Hamiltonians in the Hilbert-space $\cal
H$ of the system. If $|\Psi_0(\lambda)\rangle\in{\cal H}$ denotes the (unique for simplicity) ground-state of $H(\lambda),$ one has defined the map $\Psi_0\colon{\cal
M}\rightarrow{\cal H}/\lambda\rightarrow|\Psi_0(\lambda)\rangle$ associating to each set of parameters the ground-state of the corresponding quantum Hamiltonian. This map can be seen also as a map between ${\cal M}$ and the projective space $P{\cal H}$(=manifold of “rays" of $\cal H$). This space is a metric space being equipped with the so-called Fubini-Study distance $d_{FS}(\psi,\phi):= \cos^{-1}{\cal F}(\psi,\phi)$, where $${\cal F}(\psi,\phi):=|\langle\psi,\phi\rangle|
\label{fidelity}$$ and $\|\psi\|=\|\phi\|=1.$ In Ref. [@woo] Wootters showed that this metric has a deep operational meaning: it quantifies the maximum amount of statistical distinguishability between the pure quantum states $|\psi\rangle$ and $|\phi\rangle.$ More precisely, $d_{FS}(\psi,\phi)$ is the maximum over all possible projective measurements of the Fisher-Rao statistical distance between the probability distributions obtained from $|\psi\rangle$ and $|\phi\rangle$ [@fish]. Moreover, this result extends to mixed states as well by replacing the pure-state fidelity (\[fidelity\]) with the Uhlmann fidelity [@Uhlmann] and the projective measurements with generalized ones [@bra-ca].
These results are non-trivial and allow to identifty in a precise manner the Hilbert space geometry with a geometry in the information space: [*the bigger the Hilbert (or projective) space distance between $|\psi\rangle$ and $|\phi\rangle$ the higher the degree of statistical distinguishability of these two states.*]{} From this perspective it is clear that a single real number, i.e., the distance, virtually encodes information about infinitely many observables, e.g., order parameters, one may think to measure. This remark basically contains the main intuition at the basis of the metric approach to QPTs advocated in this paper: at the transition points, a small difference between the control parameters results in a greatly enhanced distinguishability of the corresponding GSs, which should be quantitatively revelead by the behavior of their distance.
For the purposes of this paper it is crucial to note that the projective manifold $P{\cal H}$, besides the structure of metric space, has a well-known structure of Riemannian manifold, i.e., it is equipped with a metric tensor. Here, for the sake of self-consistency, we briefly recall how this Riemannian metric is obtained starting from the Hilbert space structure of $\cal H.$ $P{\cal H}$ can be seen as the base manifold of a (principal) fiber bundle with total space given by the unit ball $S$ of $\cal H$, i.e., $S:=\{|\psi\rangle\in
{\cal H}\,/\, ||\psi||=1\}$, and projection $\pi\colon S\rightarrow
P{\cal H}\,/\, |\psi\rangle \rightarrow \{e^{i\theta}|\psi\rangle\,/\,\theta\in
[0,2\pi)\}.$ The tangent space to each point $|\psi\rangle$ of $S$ is isomorphic to a subspace of $\cal H$ and has therefore defined over it the Hermitean bilinear form $g_{|\psi\rangle}(u,v):=\langle u, v\rangle$ ($u$ and $v$ are tangent vectors, i.e., elements of $\cal H$). This defines a (complex) metric tensor field $g$ over $S.$ To project $g$ down to $P{\cal H}$ one has to introduce the notion of horizontal subspace for each tangent space of $S$ or equivalently that of parallel transport and the associated one of connection. In this case the Hilbert space structure of the tangent spaces provides a natural solution to this task: the horizontal subspace is simply the set of vectors $|u\rangle$ which are orthogonal to the fiber over $|\psi\rangle$, i.e., $\langle u,
\psi\rangle=0.$ It follows that the complex metric over $P{\cal H}$ is given by $\tilde{g}_{\pi(|\psi\rangle)}(u,v)=\langle u,
(1-|\psi\rangle\langle\psi|)v\rangle$, called the [*quantum geometric tensor*]{} [@pro]. The real (imaginary) part of this quantity defines a Riemannian metric tensor (symplectic form) on $P{\cal H}.$ Another, elementary way of getting the form of the Riemannian metric over $P{\cal H}$ is by means of Eq. (\[fidelity\]). For $\cal F$ very close to the unity, one can write $d_{FS}^2(\psi,\psi+\delta\psi)\simeq 2(1-{\cal F}).$ Since ${\cal
F}(\psi,\psi+\delta\psi)\simeq |1 +\langle\psi,\delta\psi\rangle +(1/2)
\langle\psi,\delta^2\psi\rangle|^2,$ using this expression and the normalization of $|\psi\rangle$ one finds $$\begin{aligned}
ds^2:&=&d_{FS}^2(\psi,\psi+\delta\psi)= \langle\delta\psi,\delta\psi\rangle -|\langle\psi,\delta\psi\rangle|^2
\nonumber\\
&=&\langle \delta\psi,(1-|\psi\rangle\langle\psi|) \delta\psi\rangle \ .
\label{FS}\end{aligned}$$
What we would like to do now is to see the metric in the parameter manifold $\cal M$ induced, i.e., “pulled-back" by the ground state mapping $\Psi_0$ introduced above. By writing $\delta|\Psi_0(\lambda)\rangle=\sum_\mu
|\partial_\mu \Psi_0\rangle d\lambda^\mu$, with $\partial_\mu:=\partial/\partial\lambda^\mu$, $\mu=1,\ldots,\rm{dim}\,{\cal M}$, and using Eq. (\[FS\]), one imediately obtains $ds^2=\sum_{\mu\nu} g_{\mu\nu}
d\lambda^\mu d\lambda^\nu,$ where $$g_{\mu\nu}=\Re \langle\partial_\mu \Psi_0|
\partial_\nu \Psi_0\rangle -\langle \partial_\mu
\Psi_0|\Psi_0\rangle\langle\Psi_0|
\partial_\nu \Psi_0\rangle \ .
\label{g_munu}$$ Now we provide a simple perturbative argument on why one should expect a singular behavior of the metric tensor at QPTs [@BPcomment]. By using the first order perturbative expansion $|\Psi_0(\lambda+\delta\lambda)\rangle\sim |\Psi_0(\lambda)\rangle +\sum_{n\neq
0}(E_0-E_n)^{-1} |\Psi_n(\lambda)\rangle\langle\Psi_n(\lambda)|\delta
H|\Psi_0(\lambda)\rangle$, where $\delta H:=
H(\lambda+\delta\lambda)-H(\lambda)$, one obtains for the entries of the metric tensor (\[g\_munu\]) the following expression $$g_{\mu\nu}= \Re \sum_{n\neq 0}\frac{\langle \Psi_0(\lambda)|\partial_\mu H|
\Psi_n(\lambda)\rangle\langle\Psi_n(\lambda)|\partial_\nu H| \Psi_0(\lambda)\rangle}
{[E_n(\lambda)- E_0(\lambda)]^2} \ .
\label{pert}$$ An analogous expression, with the real part replaced by the imaginary one, gives the antisymmetric tensor which describes the curvature two-form whose holonomy is the Berry phase [@BP]. Continuous QPTs are known to occur when, for some specific values of the parameters and in the thermodynamical limit, the energy gap above the GS closes. This amounts to a vanishing denominator in Eq. (\[pert\]) that may break down the analyticity of the metric tensor entries.
To get further insight about the physical origin of these singularities we notice that the metric tensor (\[g\_munu\]) can be cast in an interesting covariance matrix form [@pro]. In the generic case, by moving from $H(\lambda)$ to $H(\lambda+\delta\lambda)$ no level-crossings occur. In this case the unitary operator $O(\lambda,
\delta\lambda):=\sum_n|\Psi_n(\lambda+\delta\lambda\rangle\langle\Psi_n(\lambda)|$ “adiabatically" maps the eigenvectors at $\lambda$ onto those at $\lambda+\delta\lambda.$ Then by introducing the observables $X_\mu:=
i(\partial_\mu O)O^\dagger$ the metric tensor (\[g\_munu\]) takes the form $g_{\mu\nu}=(1/2) \langle \{ \bar{X}_\mu ,\, \bar{X}_\nu\} \rangle$ where $\bar{X}_\mu:=X_\mu -\langle X_\mu \rangle.$ Moreover, the line element $ds^2$ can be seen as the variance of the observable $X:=i(dO)O^\dagger$, i.e., $ds^2=\langle \bar{X}^2 \rangle.$ The operator $X$ is the generator of the map transforming eigenstates corresponding to different values of the parameter into each other. The smaller the difference between these eigenstates for a given parameter variation, the smaller the variance of $X.$ Intuitively, at the QPT one expects to have the maximal possible difference between $|\Psi_0(\lambda)\rangle$ and $|\Psi_0(\lambda+\delta\lambda)\rangle$, i.e., many “unperturbed” eigenstates $|\Psi_n(\lambda)\rangle$ are needed to build up the “new” GS; accordingly the variance of $X$ can get very large, possibly divergent. In a sense $ds^2$ can be seen as a sort of susceptibility of the “order parameter" $X.$
[*Quasi-Free fermionic systems*]{}.– In order to show explicitly how the singularities, i.e., divergencies of $g_{\mu\nu}$ arise, we will discuss the case of the $XY$ model in a detailed fashion; before doing that we would like to make some general considerations about the systems of quasi-free fermions on the basis of the results presented in Ref. [@za-co-gio]. Systems of quasi-free fermions are defined by the following quadratic Hamiltonian $$H=\sum_{i,j=1}^L c_i^\dagger A_{ij} c_j +
\frac{1}{2} \sum_{i,j=1}^L(c_i^\dagger B_{ij} c_j^\dagger +\rm{H.c.}) \ ,
\label{ham}$$ where: the $c_i$’s ($c_i^\dagger$’s) are the annihilation (creation) operators of $L$ fermionic modes, $A,B\in M_L(\mathbb{R})$ are $L\times L$ [*real*]{} matrices, symmetric and anti-symmetric respectively, i.e., $A^T=A,\, B^T=-B$. In Ref. [@za-co-gio] it has been shown that the set of GSs of Eq. (\[ham\]) is parametrized by orthogonal $L\times L$ real matrices $T$ giving the unitary part of the polar decomposition of the matrix $Z:=A-B.$ One can then prove that ${\cal F}(Z,Z^\prime):=|\langle
\Psi_Z|\Psi_{Z^\prime}\rangle|=\sqrt{|\det [(T+T^\prime)/2]|}$ [@za-co-gio]. With no loss of generality we can assume $\det(T)=1$ which identifies the GS manifold of the quasi-free systems (\[ham\]) with $SO(L,\mathbb{R}).$ Since $f(Z^\prime):={\cal F}(Z,Z^\prime)$ has a maximum equal to one at $Z^\prime=Z$ one has $\delta^2 f(Z^\prime)|_Z=\delta^2 \ln
f(Z^\prime)|_Z$; from this, the expansion for $Z^\prime\rightarrow Z$ of the above formula for ${\cal F}$ (Eq. (8) in Ref. [@za-co-gio]) and by defining $K:=\ln T \in so(L,\mathbb{R}),$ one finds an explicit form for the metric: $ds^2\simeq 2(1-{\cal F})=(1/8) {\mathrm{Tr}}(dK)^2.$ From this equation, if $K=K(\lambda)$, with $\lambda\in{\cal M}$, one obtains the following expression for the metric tensor induced over $\cal M$, i.e., $g_{\mu\nu}=(1/8) {\mathrm{Tr}} (\partial_\mu
K \partial_\nu K ).$ For translationally invariant Hamiltonians (\[ham\]) the anti-symmetric matrix $K$ can be always cast in the canonical form $K=i\oplus_k \theta_k \sigma_k^y$ where $k$ is a momentum label. Therefore in this important case one has $g_{\mu\nu}=(1/4) \sum_k(\partial\theta_k/\partial\lambda^\mu)
(\partial\theta_k/\partial\lambda^\nu).$
We see here that the connection established in Refs. [@za-co-gio; @co-gio-za]) between QPTs, e.g., due to the vanishing of a quasi-particle energy, and a singularity in the second order expansion of ${\cal F}$ can be directly read as a [*connection between QPTs in quasi-free systems and singularities in the metric tensor*]{} $g_{\mu\nu}.$
The nature of this connection will be now exemplified by considering the QPTs of the periodic antiferromagnetic $XY$ spin chain in a transverse magnetic field. By writing the spin operator in terms of Pauli matrices, i.e., $\bm{S}=\bm\sigma/2$, the Hamiltonian for an odd number of spins $L=2M+1$ reads $H=\sum_{j=-M}^M[-(1+\gamma)\sigma_j^x\sigma_{j+1}^x/4-
(1-\gamma)\sigma_j^y\sigma_{j+1}^y/4+h\sigma_j^z/2]$, where $\gamma$ is the anisotropy parameter in the $x$-$y$ plane and $h$ is the magnetic field. This Hamiltonian can be cast in the form (\[ham\]) by the Jordan-Wigner transformation. The critical points of this model are given by the lines ${h}=\pm1$ and by the segment $|{h}|<1,\gamma=0$. The single particle energies are $\Lambda_k=\sqrt{\epsilon_k^2+\gamma^2\sin^2(2\pi{}k/L)}$, where $\epsilon_k=\cos(2\pi{}k/L)-{h}$ and $k=-M,\dots,M$. For this model the $\theta_k$’s defined above have the form $\theta_k=\cos^{-1}(\epsilon_k/\Lambda_k)$ and $g_{\mu\nu}=(1/4)\sum_{k=1}^M
(\partial\theta_k/\partial{\lambda^\mu})
(\partial\theta_k/\partial{\lambda^\nu})$, where $\lambda^{1,2}=h,\gamma$. One finds $(\partial\theta_k/\partial{h})^2=\gamma^2\sin^2{x_k}/\Lambda_k^4$, $(\partial\theta_k/\partial\gamma)^2=
\sin^2{x_k}(\cos{x_k}-{h})^2/\Lambda_\nu^4$, and $(\partial\theta_k/\partial{h})(\partial\theta_k/\partial\gamma)=
\gamma\sin^2{x_k}(\cos{x_k}-{h})/\Lambda_k^4$, with $x_k=2\pi k/L$.
In the thermodynamic limit (TDL), the explicit calculation of $g_{\mu\nu}$ can be performed analytically. Indeed, except at critical points, for large $L$ one can replace the discrete variable $x_k$ with a continuous variable $x$ and substitute the sum with an integral, i.e., $\sum_{k=1}^M\to[L/(2\pi)]\int_0^\pi\mathrm{d}x$. At critical points this is not generally feasible due to singularities in some of the terms in the sums. Outside critical points, the resulting integrals, albeit non-trivial, yield simple analytical formulas, which differ depending on whether $|{h}|<1$ or $|{h}|>1$.
![Induced curvature $R$ scaled by the system size $L$ for the parameter space of the $XY$ model.[]{data-label="fig:1"}](XYcurvc.eps){width="8.5cm"}
For $|{h}|<1$ in the TDL one finds a diagonal metric tensor $$g= \frac{L}{16|\gamma|}\mathrm{diag}
\left(\frac{1}{1-{h}^2},\frac{1}{(1+|\gamma|)^2}\right)$$ Closed analytic formulas in the TDL can be obtained also for $|{h}|>1$, although in a less compact form, which we omit here for brevity. We only note that for $|{h}|>1$ also the off-diagonal elements of the metric tensor are non-zero. Having the induced metric tensor it is also possible to investigate the induced curvature of the parameter manifold. We therefore compute the scalar curvature $R$, which is the trace of the Ricci curvature tensor [@Nak]. We find $R(|{h}|<1)=-(16/L)(1+|\gamma|)/|\gamma|$ and $R(|{h}|>1)=(16/L)(|{h}|+\sqrt{{h}^2+\gamma^2-1})/\sqrt{{h}^2+\gamma^2-1}$. Note that the curvature diverges on the segment $|{h}|\leq1,\gamma=0$ and is discontinuous on the lines ${h}=\pm1$. Indeed, $\lim_{|{h}|\to1^+}R=-\lim_{|{h}|\to1^-}R$. The behaviour of the curvature $R$ is shown in Fig. \[fig:1\].
[*Mixed states and classical transitions.*]{}– In this section we would like to make some extentions of the idea developed in this paper to finite temperature. This will allow us to establish a connection between the present approach and the one for classical PTs developed in [@ruppy; @brody]. This latter formalism is in fact obtained in the special case of commuting density matrices which effectively turns the quantum problem into a classical one.
The fidelity approach to QPTs can be extended to finite-temperature, i.e., to mixed-states, by using the Uhlmann fidelity [@Uhlmann]: ${\cal F}(\rho_0,\rho_1):= \mathrm{Tr}[\rho_1^{1/2}\rho_0 \rho_1^{1/2}]^{1/2}.$ When $\rho_0$ and $\rho_1$ are commuting operators the fidelity takes the form ${\cal F}(\rho_0,\rho_1)=\sum_n \sqrt{p_n^0 p_n^1}$ where the $p_n^\alpha$ are the eigenvalues of the $\rho_\alpha$’s [@rel-ent]. In particular, when $\rho_\alpha=Z_\alpha^{-1} \exp(-\beta_\alpha H), Z_\alpha:=\mathrm{Tr} \exp(-\beta_\alpha H),\,(\alpha=0,1)$ one immediately finds that the fidelity has a simple expression in terms of partition functioms: ${\cal F}=Z(\beta_0/2+\beta_1/2)(Z(\beta_0) Z(\beta_1))^{-1/2}$ [@zhong-guo]. By expanding for $\beta_0=\beta, \beta_1=\beta+\delta\beta$ one obtains $${\cal F}(\beta,\beta+\delta\beta) \simeq
\exp\left[-\frac{\delta\beta^2}{8\beta^2} c_V(\beta) \right]
\label{fid-thermo}$$ where $c_V(\beta)$ denotes the specific heat [@huang]. This relation is remarkable in that it connects the distinguishability degree of two neighboring thermal quantum states directly to the macroscopic thermodynamical quantity $c_V.$ The line element of the parameter space, i.e., the $\beta$ axis, is then given by $ds^2\sim c_V(\beta)\beta^{-2} d\beta^2
= (\langle H^2\rangle_\beta -\langle H\rangle^2_\beta) d\beta^2.$ A closely related formula has been obtained in [@ruppy; @brody]. Since $PTs$ are associated to anomalies, e.g., divergences, in the behavior of $c_V(\beta)$, we see that also in this “classical" case the metric $ds^2$ induced on the parameter space contains signatures of the critical points. In this sense the information-geometrical approach to PTs seems able to put quantum and classical PTs under the same conceptual umbrella.
[*Conclusions.*]{}– In this paper we proposed a differential-geometric approach to study quantum phase transtions. The basic idea is that, since distance between quantum states quantitatively encodes their degree of distinguishability, crossing a critical point separating regions with structurally different phases should result in some sort of singular behaviour of the metric. This intuition, based on early studies of quantum fidelity, can be made rigorous in some simple yet important cases, e.g., quasi-free fermion systems. The manifold of coupling constants parameterizing the system’s Hamiltonian can be equipped with a (pseudo) Riemannian tensor $g$ whose singularities correspond to the critical regions. For the case of the $XY$ chain we explicitely computed the components of $g$ in the thermodynamic limit, showing that they are divergent, with universal exponents, at the critical lines. We also computed the scalar curvature of $g$ and analyzed its relation with criticality. The geometrical approach advocated in this paper does not depend on the knowledge of any order parameter or on the analysis of a distinguished observable, it is universal and information-theoretic in nature. The study of the physical meaning of the geometric invariants one can build starting from $g$ (e.g., the curvature), their finite-size as well as scaling behaviour, and their relations with the nature of the quantum phase transition are important questions to be addressed in future research.
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Given the complete set of one-dimensional projections $\{|i\rangle\langle i|\}_{i=1}^{\rm{dim}\,{\cal H}}$ one has the two probability distributions $p_i=|\langle i|\psi\rangle|^2$ and $q_i=|\langle
i|\phi\rangle|^2.$ For $p$ and $q$ infintesimally close to each other the Fisher distance is given by $d_F(p,q):=\sum_i (p_i-q_i)^2/p_i= \sum_i p_i (d\ln p_i)^2.$
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This heuristic argument is going to be parallel to the one used in Ref. [@BP-qpt] where the relation between QPTs an Berry phases is studied. While the Berry curvature can be identically vanishing, e.g., for real wavefunctions, this cannot be the case for the quantity (\[g\_munu\]).
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For $p_n^1=p_n^0+dp_n$ ($dp_n\approx0$) we have that ${\cal
F}(\rho_0,\rho_1)\approx 1-(1/8)\sum_n dp_n^2/p_n^0$; thus, to lowest non-zero order, the difference of the fidelity from 1 is proportional to the Fisher-Rao distance, that, for infinitesimal variations, coincides with the relative entropy between the two probability distributions $\{p_n^0\}$ and $\{p_n^1\}$.
|
---
abstract: 'A general analysis of the $n$-vertex loop amplitude in a strong magnetic field is performed, based on the asymptotic form of the electron propagator in the field. As an example, the photon-neutrino processes are considered, where one vertex in the amplitude is of a general type, and the other vertices are of the vector type. It is shown, that for odd numbers of vertices only the $SV_1 \ldots V_{n-1}$ amplitude grows linearly with the magnetic field strength, while for even numbers of vertices the linear growth takes place only in the amplitudes $PV_{1} \ldots V_{n-1}$, $VV_{1} \ldots V_{n-1}$ and $AV_{1} \ldots V_{n-1}$. The general expressions for the amplitudes of the processes $\gamma \gamma \to \nu \bar\nu$ (in the framework of the model with the effective $\nu \nu e e$ – coupling of a scalar type) and $\gamma \gamma \to \nu \bar\nu \gamma$ (in the framework of the Standard Model) for arbitrary energies of particles are obtained. A comparison with existing results is performed.'
author:
- |
A. V. Kuznetsov, N. V. Mikheev, D. A. Rumyantsev\
[*Division of Theoretical Physics, Yaroslavl State (P.G. Demidov) University,*]{}\
[*Sovietskaya 14, 150000 Yaroslavl, Russian Federation*]{}\
[E-mail: avkuzn@uniyar.ac.ru, mikheev@uniyar.ac.ru, rda@uniyar.ac.ru]{}
title: |
[Yaroslavl State University\
Preprint YARU-HE-02/09\
hep-ph/0210029]{}\
General amplitude of the $n$ – vertex\
one-loop process in a strong magnetic field
---
[*Talk presented at the 12th International Seminar\
“Quarks-2002”,\
Valday and Novgorod, Russia, June 1-7, 2002*]{}
Introduction
============
Nowadays, there exists a growing interest to astrophysical objects, where the strong magnetic fields with the strength $B>B_e$ can be generated $(B_e = m^2/e \simeq 4.41 \cdot 10^{13}$ Gs [^1] is the so-called critical field value).
The influence of a strong external field on quantum processes is interesting because it catalyses the processes, it changes the kinematics and it induces new interactions. It is especially important to investigate the influence of external field on the loop quantum processes where only electrically neutral particles in the initial and the final states are presented, such as neutrinos, photons and hypotetical axions, familons and so on. The external field influence on these loop processes is provided by the sensitivity of the charged virtual fermion to the field and by the change of the photon dispersion properties and, therefore, the photon kinematics.
The research of the loop processes of this type has a rather long history. The two-vertex loop processes (the photon polarization operator in an external field, the decays $\gamma\to\nu\bar{\nu}$, $\nu\to\nu\gamma$ and so on) were studed in the papers [@Tsai:1974; @Shabad:88; @Skobelev:1995; @Gvozdev:1996; @Ioannisian:1997]. The general expression for the two-vertex loop amplitude $j \to f \bar f
\to j'$ in the homogeneous magnetic and in the crossed field was obtaned in the paper [@Borovkov:1999], where all combinations of scalar, pseudoscalar, vector and axial-vector interactions of the generalized currents $j,\,j'$ with fermions were considered.
A loop process with three vertices is also intresting for theoreticians. For example, the photon splitting in a magnetic field $\gamma \to \gamma \gamma$ is forbidden in vacuum. The review [@Papanian:1986] and the recent papers [@Adler:1996; @Baier:1996; @Baier:1997; @Chistyakov:1998; @Kuznetsov:1999] were devoted to this process. One more three-vertex loop process is the conversion of the photon pair into the neutrino pair, $\gamma\gamma\to\nu\bar{\nu}$. This process is interesting as a possible channel of stellar cooling. A detailed list of references on this process can be found in our paper [@Kuznetsov:2002a].
It is well-known (the so-called Gell-Mann theorem [@Gell-Mann:1961]), that for massless neutrinos, for both photons on-shell and in the local limit of the standard-model weak interaction, the process $\gamma\gamma\to\nu\bar{\nu}$ is forbidden. Because of this, the four-vertex loop process with an additional photon $\gamma\gamma\to\nu\bar{\nu}\gamma$ was considered by some authors. In spite of the extra factor $\alpha$, this process has the probability larger than the two-photon process. The process $\gamma\gamma\to\nu\bar{\nu}\gamma$ was studied both in vacuum (from the first paper [@VanHieu:1963] to the recent Refs. [@Dicus:1997; @Dicus:1999; @Abada:1999a; @Abada:1999b; @Abada:1999c]), and under the stimulating influence of a strong magnetic field [@Loskutov:1987; @Skobelev:2001; @Kuznetsov:2002b].
So, the calculation of the amplitude of the $n$-vertex loop quantum processes ($\gamma \gamma \to \nu \bar\nu$, $\gamma \gamma \to \gamma \nu \bar\nu$, the axion and familon processes $\gamma \gamma \to \gamma a$, $\gamma \gamma \to \gamma \varPhi$ and so on) in a strong magnetic field is important, because these results can be useful for astrophysical applications.
The paper is organized as follows. A general analysis of the $n$-vertex one-loop process amplitude in a strong magnetic field is performed in Section 2. The amplitude, in which the one vertex is of a general type (scalar $S$, pseudoscalar $P$, vector $V$ or axial-vector $A$), and the other vertices are of the vector type (contracted with photons), is calculated in Section 3. This amplitude is the main result of the paper. The analitical expressions for the amplitudes of the processes $\gamma \gamma \to \nu \bar\nu$ and $\gamma \gamma \to \nu \bar\nu \gamma$ are presented in Sections 4 and 5.
General analysis of the $n$-vertex one-loop processes in a strong magnetic field
================================================================================
We use the effective Lagrangian for the interaction of the generalized currents $j$ with electrons in the form: $$\begin{aligned}
{\cal L}(x) \, = \, \sum \limits_{i} g_i
[\bar {\psi_e}(x) \Gamma_i \psi_e(x)] j_i,
\label{eq:L}\end{aligned}$$ where the generic index $i = S, P, V, A $ numbers the matrices $\Gamma_i$, e.g. $\Gamma_S = 1, \, \Gamma_P = \gamma_5, \, \Gamma_V = \gamma_{\alpha},
\, \Gamma_A = \gamma_{\alpha} \gamma_5 $, $j$ is the corresponding quantum object (the current or the photon polarisation vector), $g_i$ are the coupling constants. In particular, for the electron - photon interaction we have $g_V = e,\,\Gamma_V = \gamma_{\alpha},\,j_{V\alpha}(x) = A_\alpha(x)$.
A general amplitude of the process, corresponding to the effective Lagrangian (\[eq:L\]), is described by fig. \[fig:loop1\]. In the strong field limit, after integration over the coordinates, the amplitude takes the form $$\begin{aligned}
{\cal M}_n \simeq
\frac{i \,(-1)^n \,e B}{(2 \pi)^3}
\exp \! \left (-\frac{R_{\mprp n}}{2eB}\right )
\int d^2 p_{\mprl} \, \mbox{Tr} \, \big \{\prod \limits^{n}_{k=1} g_k
\Gamma_k j_k S_{\mprl}(p-Q_k) \big \},
\label{eq:M2}\end{aligned}$$
where $d^2 p_{\mprl} = dp_0 dp_z$, $S_{\mprl}(p) = \Pi_{-} ((p\gamma)_{\mprl} + m)/(p_{\mprl}^2 - m^2)$ is the asymptotic form of the electron propagator in the limit $eB /\vert m^2 - p_{\mprl}^2 \vert \gg 1$, $$R_{\mprp 2} = q_{\mprp 1}^2, \quad R_{\mprp 3} = q_{\mprp 1}^2 +
q_{\mprp 2}^2 + (q_1 \varphi \varphi q_2) + i(q_1 \varphi q_2),$$ $$R_{\mprp n}(n\ge 3) = \sum \limits^{n-1}_{k=1} Q_{\mprp k}^2 -
\sum \limits^{n-1}_{k=2} \sum \limits^{k-1}_{j=1} \left [(Q_k \varphi
\varphi Q_j) + i(Q_k \varphi Q_j)\right ],$$ $$\quad Q_k = \sum \limits^{k}_{i=1}
q_i, \quad Q_n = 0,$$ $q_{\mprl}^2 = (q \tilde \varphi \tilde \varphi q)$, $q_{\mprp}^2 = (q \varphi \varphi q)$, $\varphi_{\alpha \beta} = F_{\alpha \beta} /B$ is the dimensionless field tensor, ${\tilde \varphi}_{\alpha \beta} = \frac{1}{2} \varepsilon_{\alpha \beta
\mu \nu} \varphi_{\mu \nu}$ is the dual tensor, and the indices of the four-vectors and tensors standing inside the parentheses are contracted consecutively, e.g. $(a \varphi b) = a_\alpha \varphi_{\alpha \beta} b_\beta$.
As is seen from Eq. (\[eq:M2\]), the amplitude depends only on the longitudinal components of the momenta, if the magnetic field strength is the maximal physical parameter $e B \gg q_{\mprp}^2, q_{\mprl}^2$.
The photons processes
=====================
Let the vertices $\Gamma_1 \ldots \Gamma_{n-1}$ are of the vector type, and the vertex $\Gamma_n$ is arbitrary. It can be shown that in the limit $q_{\mprp}^2 \ll e B $, for odd numbers of vertices, only the $S V_1 \ldots V_{n-1}$ amplitude grows linearly with the magnetic field strength, while for even numbers of vertices the linear growth takes place only in the amplitudes $PV_{1} \ldots V_{n-1}$, $VV_{1} \ldots V_{n-1}$ and $AV_{1} \ldots V_{n-1}$.
It should be noted, that in the amplitude (\[eq:M2\]) the projecting operators $\Pi_-$ separate out the photons of only one polarization $(\perp)$ of the two possible (in Adler’s notation [@Ad71]) $$\begin{aligned}
\varepsilon^{(\mprl)}_{\alpha}
= \frac{\varphi_{\alpha\beta} q_{\beta}}{\sqrt{(q\varphi \varphi q)}},
\qquad
\varepsilon^{(\mprp)}_{\alpha}
= \frac{\widetilde \varphi_{\alpha\beta} q_{\beta}}
{\sqrt{(q \widetilde \varphi \widetilde \varphi q)}}.
\label{eq:vect}\end{aligned}$$ As can be deduced from the corresponding analysis, the calculation of any type of the amplitude can be reduced to the evaluation of the scalar integrals $$\begin{aligned}
S_{n}(Q_{1 \mprl},\ldots ,Q_{n \mprl}) = \int \, \frac{d^2 p_{\mprl}}{(2\pi)^2} \,
\prod \limits^{n}_{i=1} \frac{1}{(p - Q_i)^2_{\mprl} - m^2 + i\varepsilon}.
\label {eq:s1}\end{aligned}$$
Notice that the use of the standart method of Feynman parametrization in calculation of the integrals (\[eq:s1\]) can be non-optimal, because the number of integrations grows. For example, if $n=3$, the double integral (\[eq:s1\]) is transformed into the integral over the two Feynman variables. If $n=4$, the double integral (\[eq:s1\]) is transformed into the integral over the three Feynman variables and so on. Here we suggest another way. By integrating (\[eq:s1\]) over $dp_0dp_z$, we obtain $$\begin{aligned}
S_{n}(Q_{1 \mprl},\ldots ,Q_{n \mprl}) =
\, \frac{i}{8m^2\pi}
\sum \limits^{n}_{i=1} \underset{l \ne i}{\sum \limits^{n}_{l=1}}
\, \left [ H \left (\frac{d_{il}^2}{4m^2} \right) + 1 \right ]
Re \, \bigg \{ \underset{k \ne i,l}
{\prod \limits^{n}_{k=1}} \frac{1}{Y_{ilk}} \bigg \} ,
\label {eq:s4}\end{aligned}$$
where $$Y_{ilk} = (d_{lk} d_{ik}) \, + \, i (d_{lk} \tilde \varphi d_{ik})
\sqrt{\frac{4m^2}{d_{il}^2} - 1}, \quad
d_{il}^{\alpha} = Q_{\mprl \, i}^{\alpha} - Q_{\mprl \, l}^{\alpha}.$$
The function $H(z)$ is defined by the expressions $$\begin{aligned}
&&H(z) = \frac{1}{2\sqrt{-z(1 - z)}}
\ln{\frac{\sqrt{1 - z} + \sqrt{-z}}{\sqrt{1 - z} - \sqrt{-z}}} - 1,\quad z<0,
\nonumber \nonumber \\
&&H(z) = \frac{1}{\sqrt{z(1 - z)}}
\arctan{\sqrt{\frac{z}{1 - z}}} - 1,\quad 0<z<1,
\nonumber \nonumber \\
&&H(z) = - \frac{1}{2\sqrt{z(z - 1)}}
\ln{\frac{\sqrt{z} + \sqrt{z - 1}}{\sqrt{z} - \sqrt{z - 1}}}
- 1 + \frac{i\pi}{2\sqrt{z(z - 1)}} ,\quad z>1,
\nonumber \end{aligned}$$
and it has the asymptotics $$\begin{aligned}
&&H(z) \simeq \frac{2}{3}z + \frac{8}{15}z^2 + \frac{16}{35}z^3, \quad |z| \ll 1,
\label {eq:s5}\end{aligned}$$ $$\begin{aligned}
&&H(z) \simeq -1 - \frac{1}{2z}\ln{4|z|}, \quad |z| \gg 1.
\label {eq:s6}\end{aligned}$$
The process $\gamma \gamma \to \nu \bar\nu$
===========================================
Let us apply the results obtained to the calculation of some quantum processes. For the amplitude of the process $\gamma \gamma \to \nu \bar\nu$ in the framework of the model with the effective $\nu \nu e e$ – coupling of a scalar type we obtain from Eqs. (\[eq:M2\]), (\[eq:s1\]), (\[eq:s4\]) $$\begin{aligned}
{\cal M}_{3}^s \,& = &\,
\frac{2\alpha}{\pi} \, \frac{B}{B_e} \, g_{s} \, j_s \, m
\, \frac{(q_1 \widetilde \varphi \varepsilon^{(1)})
(q_2 \widetilde \varphi \varepsilon^{(2)})}
{4m^2 [(q_1 q_3)^2_{\mprl} - q^2_{1\mprl} q^2_{3\mprl}] +
q^2_{1\mprl} q^2_{2\mprl} q^2_{3\mprl}} \times
\nonumber \nonumber \\
&\times &\, \left \{ \left [q^2_{1\mprl} q^2_{3\mprl} -
2m^2 (q^2_{3\mprl} + q^2_{1\mprl} - q^2_{2\mprl})\right ]
H \left (\frac{q_{1\mprl}^2}{4m^2} \right ) \right. \, +
\nonumber \nonumber \\
& + & \, \left [q^2_{2\mprl} q^2_{3\mprl} -
2m^2 (q^2_{3\mprl} + q^2_{2\mprl} - q^2_{1\mprl})\right ]
H \left (\frac{q_{2\mprl}^2}{4m^2} \right ) \, +
\nonumber \nonumber \\
& + & \, \left. q^2_{3\mprl}(4m^2 -q^2_{3\mprl})
H \left (\frac{q_{3\mprl}^2}{4m^2} \right ) -
2q^2_{3\mprl}(q_1q_2)_{\mprl} \, \right \},
\label{eq:MS3}\end{aligned}$$
where $g_s = - 4 \; \zeta \; G_{\mbox{\normalsize{F}}}/\sqrt{2}$ is the effective $\nu \nu e e$ – coupling constant in the left-right-symmetric extension of the Standard Model, $\zeta$ is the small mixing angle of the charged $W$ bosons, $j_s = [\bar \nu_e(p_1) \nu_e(-p_2)]$ is Fourier transform of the scalar neutrino current, $q_3 = p_1 + p_2$ is the neutrino pair momentum.
Substituting the photon polarization vector $\varepsilon^{(\mprp)}_{\alpha}$ from Eq. (\[eq:vect\]) into (\[eq:MS3\]) and using (\[eq:s5\]) and (\[eq:s6\]), we obtain the asymptotics:
- at low photon energies, $\omega_{1,2} \lesssim m$ $$\begin{aligned}
{\cal M}_{3}^{s} \simeq \frac{8 \alpha}{3 \pi} \,
\frac{G_{\mbox{\normalsize{F}}}}{\sqrt{2}}\,
\frac{\zeta}{m}\;\frac{B}{B_e}\,
\left [\bar\nu_e (p_1) \, \nu_e (- p_2) \right ]\,
\sqrt{q_{1\mprl}^2 q_{2\mprl}^2} ;
\label{eq:M<}\end{aligned}$$
- at high photon energies, $\omega_{1,2} \gg m$, in the leading log approximation: $$\begin{aligned}
{\cal M}_{3}^{s} \simeq \frac{16 \alpha}{\pi} \,
\frac{G_{\mbox{\normalsize{F}}}}{\sqrt{2}}\,
\zeta\;\frac{B}{B_e}\,m^3 \,
\left [\bar\nu_e (p_1) \, \nu_e (- p_2) \right ]\,
\frac{1}{\sqrt{q_{1\mprl}^2 q_{2\mprl}^2}}\,
\ln \frac{\sqrt{q_{1\mprl}^2 q_{2\mprl}^2}}{m^2}.
\label{eq:M>}\end{aligned}$$
These expressions coincide with the results obtained in the paper [@Kuznetsov:2002a].
The process $\gamma \gamma \to \nu \bar\nu \gamma $
===================================================
The process of this type, where one initial photon is virtual, namely, the photon conversion into neutrino pair on nucleus was considered, in the framework of the Standard Model, in the papers [@Skobelev:2001; @Kuznetsov:2002b]. This process can be studied by using the amplitude of the transition $\gamma \gamma \to \nu \bar\nu \gamma $, which can be obtained from Eq. (\[eq:M2\]) in the form: $$\begin{aligned}
{\cal M}_{4}^{VA} \,& = &\, - \,
\frac{8 i e^3}{\pi^2} \, \frac{B}{B_e} \,
\frac{G_{\mbox{\normalsize{F}}}}{\sqrt{2}} \, m^2 \times
\nonumber \nonumber \\
&\times & (q_1 \widetilde \varphi \varepsilon^{(1)})
(q_2 \widetilde \varphi \varepsilon^{(2)})
(q_3 \widetilde \varphi \varepsilon^{(3)})
[g_V (j\widetilde \varphi q_4) +
g_A (j\widetilde \varphi \widetilde \varphi q_4)] \times
\nonumber \nonumber \\
&\times & \frac{1}{D} \left \{ I_4(q_{1\mprl},q_{2\mprl},q_{3\mprl})
+ I_4(q_{2\mprl},q_{1\mprl},q_{3\mprl}) +
I_4(q_{1\mprl},q_{3\mprl},q_{2\mprl}) \right \},
\label{eq:MVA4}\end{aligned}$$
where $g_V,\;g_A$ are the vector and axial-vector constants of the effective $\nu \nu e e$ Lagrangian of the Standard Model, $g_V = \pm 1/2 + 2 \sin^2 \theta_W, \; g_A = \pm 1/2$, here the upper signs correspond to the electron neutrino, and lower signs correspond to the muon and tau neutrinos; $j_{\alpha} = [\bar \nu_e(p_1) \gamma_{\alpha} (1 + \gamma_5) \nu_e(-p_2)]$ is the Fourier transform of the neutrino current; $q_4 = p_1 + p_2$ is the neutrino pair momentum; $$D = (q_1q_2)_{\mprl}(q_3q_4)_{\mprl} + (q_1q_3)_{\mprl}(q_2q_4)_{\mprl} +
(q_1q_4)_{\mprl}(q_2q_3)_{\mprl}.$$
The formfactor $I_4 (q_{1 \mprl} ,q_{2 \mprl} ,q_{3 \mprl})$ is expressed in terms of the integrals (\[eq:s1\]), (\[eq:s4\]) $$\begin{aligned}
&&I_4 (q_{1 \mprl}, q_{2 \mprl}, q_{3 \mprl}) \, = \,
S_3 (q_{1 \mprl} + q_{2 \mprl}, q_{4 \mprl},0) + S_3 (q_{1 \mprl}, q_{4 \mprl},0) +
\nonumber \nonumber \\
&& + S_3 (q_{1 \mprl} + q_{2 \mprl}, q_{1 \mprl},0) +
S_3 (q_{2 \mprl} - q_{3 \mprl}, q_{2 \mprl},0) +
\nonumber \nonumber \\
&& + [6 m^2 - (q_1 + q_2)_{\mprl}^2 - (q_2 - q_3)_{\mprl}^2]
S_4 (q_{1 \mprl}, q_{1 \mprl}+q_{2 \mprl}, q_{4 \mprl},0). \end{aligned}$$
Using the asymptotics of the functions $H(z)$, we obtain
- at low photon energies, $\omega_{1,2,3} \ll m$ $$\begin{aligned}
&&{\cal M}_{4}^{VA} \, \simeq \,
- \frac{2 e^3}{15\pi^2} \, \frac{B}{B_e} \,
\frac{G_{\mbox{\normalsize{F}}}}{\sqrt{2}} \, \frac{1}{m^4} \times
\nonumber \nonumber \\
&&\times (q_1 \widetilde \varphi \varepsilon^{(1)})
(q_2 \widetilde \varphi \varepsilon^{(2)})
(q_3 \widetilde \varphi \varepsilon^{(3)})
[g_V (j\widetilde \varphi q_4) +
g_A (j\widetilde \varphi \widetilde \varphi q_4)],
\label{eq:I_4_low}\end{aligned}$$
which is in agreement with the result of the paper [@Kuznetsov:2002b];
- at high photon energies, $\omega_{1,2,3} \gg m$, in the leading log approximation we obtain: $$\begin{aligned}
&&{\cal M}_{4}^{V,A} \, \simeq \,
- \frac{8 e^3}{3\pi^2} \, \frac{G_{\mbox{\normalsize{F}}}}{\sqrt{2}} \,
\frac{B}{B_e} \, m^4 \times
\nonumber \nonumber \\
&&\times (q_1 \widetilde \varphi \varepsilon^{(1)})
(q_2 \widetilde \varphi \varepsilon^{(2)})
(q_3 \widetilde \varphi \varepsilon^{(3)})
[g_V (j\widetilde \varphi q_4) +
g_A (j\widetilde \varphi \widetilde \varphi q_4)] \times
\nonumber \nonumber \\
&&\times \frac{1}{q^2_{1 \mprl}q^2_{2 \mprl}q^2_{3 \mprl}q^2_{4 \mprl}}
\ln \frac{\sqrt{q_{1 \mprl}^2 q_{2 \mprl}^2 q_{3 \mprl}^2}}{m^3}.
\label{eq:I_4_hig}\end{aligned}$$
To the best of our knowledge, this result is obtained for the first time.
Conclusions
===========
We have obtained the general expressions (\[eq:MS3\]) and (\[eq:MVA4\]) for the amplitudes of the processes $\gamma \gamma \to \nu \bar\nu$ (in the framework of the model with the effective $\nu \nu e e$ coupling of a scalar type) and $\gamma \gamma \to \nu \bar\nu \gamma$ (in the framework of the Standard Model) for arbitrary energies of particles. A comparison with the existing results has been performed.
[**Acknowledgements**]{}
We express our deep gratitude to the organizers of the Seminar “Quarks-2002” for warm hospitality.
This work was supported in part by the Russian Foundation for Basic Research under the Grant No. 01-02-17334 and by the Ministry of Education of Russian Federation under the Grant No. E00-11.0-5.
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[^1]: We use natural units in which $c = \hbar = 1$, $m$ is the electron mass, $e > 0$ is the elementary charge.
|
---
abstract: 'We construct mock galaxy catalogues to analyse clustering properties of a $\Lambda$ cold dark matter ($\Lambda$CDM) universe within a cosmological dark matter simulation of sufficient resolution to resolve structure down to the scale of dwarfs. We show that there is a strong age-clustering correlation for objects likely to host luminous galaxies, which includes the satellite halo (subhalo) population. Older mock galaxies are significantly more clustered in our catalog, which consists of satellite haloes as well as the central peaks of discrete haloes, selected solely by peak circular velocity. This age dependence is caused mainly by the age-clustering relation for discrete haloes, recently found by Gao [[et al.]{} ]{}, acting mostly on field members, combined with the tendency for older mock galaxies to lie within groups and clusters, where galaxy clustering is enhanced. Our results suggest that the clustering age dependence is manifested in real galaxies. At small scales (less than $\sim$5 $h^{-1}$Mpc), the very simple assumption that galaxy colour depends solely on halo age is inconsistent with the strength of the observed clustering colour trends, where red galaxies become increasingly more clustered than blue galaxies toward smaller scales, suggesting that luminosity weighted galaxy ages do not closely trace the assembly epoch of their dark matter hosts. The age dependence is present but is much weaker for satellite haloes lying within groups and clusters than for the global population.'
author:
- |
Darren S. Reed,$^{1,2}$[^1] Fabio Governato,$^{3,4}$ Thomas Quinn,$^3$ Joachim Stadel,$^5$ and George Lake$^5$\
$^1$Institute for Computational Cosmology, Dept. of Physics, University of Durham, South Road, Durham DH1 3LE, UK\
$^2$Theoretical Astrophysics Group, Los Alamos National Laboratory, PO Box 1663, MS 627, Los Alamos, NM, 87545 USA\
$^3$Astronomy Department, Box 351580, University of Washington, Seattle, WA 98195 USA\
$^4$INAF, Osservatorio Astronomico di Brera, via Brera 28, I-20131 Milano, Italy\
$^5$Institute for Theoretical Physics, University of Zurich, Winterthurerstrasse 190, 8057, Switzerland
title: The age dependence of galaxy clustering
---
\[firstpage\]
galaxies: haloes – galaxies: formation – methods: N-body simulations – cosmology: theory – cosmology:dark matter
introduction
============
A critical test of the $\Lambda$CDM model is whether it accurately predicts the clustering properties of galaxies formed within dark matter haloes and “subhaloes” that are satellite clumps withing larger host haloes. Subhaloes serve as hosts for visible galaxies within clusters, groups, or larger galaxies, and so provide a natural basis for constructing simulated mock catalogs, whose clustering properties can then be compared with observed galaxies.
Clustering of haloes depends on halo age, a phenomenon recently measured in $\Lambda$CDM simulations by Gao, Springel, & White (2005), who found that older haloes are more strongly clustered than younger haloes. The likely explanation is that haloes of a given mass generally form earlier within denser regions. Thus, older haloes tend to populate denser regions, which naturally leads to stronger clustering with halo age. Sheth & Tormen (2004) measured such a trend between mean halo formation epoch and local over-density in simulation data, which was recently confirmed in larger simulations by Harker [[et al.]{} ]{}(2005). The precise physical origin, however, of the age dependence is a subject of recent debate (see discussion by [[e.g. ]{}]{}Zentner 2006). Wang, Mo & Jing (2007), use numerical simulations to suggest that accretion onto low mass haloes in high density regions is inhibited by competition with massive neighbours via tidal interactions and local dynamical heating, creating a correlation with halo age and environment. Sandvik [[et al.]{} ]{}(2006) suggest that the formation history and the current epoch environment of low mass haloes may be affected by their presence in massive pancakes and filaments at high redshift. The clustering age dependence, which has been confirmed by a number of authors (Harker [[et al.]{} ]{}2005; Wechsler [[et al.]{} ]{}2006; Gao & White 2006; Jing, Suto & Mo 2006; Wang, Mo & Jing 2007), is strong for haloes of masses of 10$^{11-12}$ $h^{-1} \msun$ (Gao [[et al.]{} ]{}2005, Wechsler [[et al.]{} ]{}2006), which are likely to host galaxies, and decreases with halo mass, becoming insignificant for haloes more massive than $\sim$10$^{13}$ $h^{-1} \msun$ (Gao [[et al.]{} ]{}2005). There is evidence that the age-clustering dependence may reverse sign for larger haloes (Wetzel [[et al.]{} ]{}2006; Jing, Suto & Mo 2006; Gao & White 2006; Zentner 2006).
Previous studies have focussed on the age dependence of clustering of discrete virialized haloes, and did not consider directly the contribution of satellite populations to the age dependence of clustering. Because a large fraction of galaxies belongs to groups and clusters, the clustering of the general galaxy population could have a strong dependence on subhalo ages. Galaxies of a given circular velocity will have formed earlier if they lie in groups or clusters today. Thus, we can expect that the contribution of group and cluster members will increase the tendency for older objects to be more strongly clustered. Also, recent studies relating subhalo numbers and distribution to age and to host halo properties hint that subhalo clustering could depend on subhalo age. Recent simulations have shown that older haloes tend to host fewer subhaloes ([[e.g. ]{}]{}Gao [[et al.]{} ]{}2004; Zentner [[et al.]{} ]{}2005; Taylor & Babul 2005; Zhu [[et al.]{} ]{}2006). Additionally, the clustering strength of virialized haloes is correlated with the numbers of their satellite haloes (Wechsler [[et al.]{} ]{}2006). Furthermore, subhaloes tend to lie nearer their host centres if they were either formed earlier (Willman [[et al.]{} ]{}2004) or were accreted earlier (Gao [[et al.]{} ]{}2004; Taylor & Babul 2005; see however Moore, Diemand, & Stadel 2004).
In order to understand more fully the age dependence of subhalo clustering and its potential effects on observable galaxies, we analyze the relation between age and clustering within halo catalogs that include both the satellite haloes that populate group and cluster haloes as well as the discrete virialized haloes likely to host only a single galaxy. We construct a simple mock galaxy catalog wherein haloes are selected by peak circular velocity to roughly match the galaxy luminosities and abundances in large surveys. Our catalog is selected from a high resolution dark matter simulation that resolves structures within a cosmological volume down to the scale of dwarf galaxies. We stress that we are not attempting to create a realistic catalog of “simulated galaxies”, but rather that we are merely using observationally relevant circular velocities as a convenient means of assessing the potential dependence of clustering on age of haloes$+$subhaloes over a range that has the potential to host galaxies in the $\Lambda$CDM model. In § 2, we describe the simulations and the construction of the halo$+$subhalo catalog. In § 3, we detail the age dependence of clustering in our mock catalog, the implications of which we discuss in § 4.
numerical techniques
====================
the simulations
---------------
We use the parallel k-D (balanced binary) Tree (Bentley 1975) gravity solver [PKDGRAV]{} (Stadel 2001; Wadsley, Stadel & Quinn 2004) to model a 50 $h^{-1}$Mpc cube, consisting of 432$^{3}$ dark matter particles of equal mass (the CUBEHI run of Reed [[et al.]{} ]{}(2003; 2005ab). By modelling a relatively small cosmological volume, we are able to probe down to the small masses needed to resolve satellites within groups. The particle mass is 1.3$\times 10^{8} h^{-1} \msun$. A starting redshift of 69 and a force softening of 5 $h^{-1}$kpc (comoving) are used. This run adopts a $\Lambda$CDM cosmology with $\Omega_m=$ 0.3 and $\Lambda=$ 0.7, and the initial density power spectrum is normalised to $\sigma_{\rm 8}=$ 1.0, consistent with WMAP ([[e.g. ]{}]{}Bennett [[et al.]{} ]{}2003; Spergel et al. 2003). We use a Hubble constant of $h=0.7$, in units of 100 km s$^{-1}$ Mpc$^{-1}$, and assume no tilt (i.e. a primordial spectral index of 1). To set the initial conditions, we use the Bardeen [[et al.]{} ]{}(1986) transfer function with $\Gamma=\Omega_{\rm
m}\times h$.
mock catalog construction
-------------------------
Mock galaxies are chosen from a catalog of haloes selected by circular velocity using the Spline Kernel Interpolative [DENMAX]{} ([SKID]{}) halo finder (Stadel 2001; http://www-hpcc.astro.washington.edu/tools/skid.html). [SKID]{} haloes are identified using local density maxima to identify bound mass concentrations independently of environment. Note that [SKID]{} identifies discrete virialized haloes as well as subhaloes (satellite haloes). The radial extent of each [SKID]{} halo is determined by the distribution of bound particles, and no predetermined subhalo shape is imposed. The peak circular velocity of each subhalo, $v_{c,peak}$, is computed from the peak of the rotation curve ${\rm v_c(r) =
(GM(<r)/r)^{0.5}}$. The formation epoch is defined as the time at which $v_{c,peak}$ of the largest progenitor (amongst all branches of the merger tree at a given epoch) reaches 75$\%$ of its maximum value. A progenitor is defined as a halo with at least 30$\%$ of its particles incorporated into its descendent. Further detail on formation and accretion times of subhaloes can be found in a number of prior studies ([[e.g. ]{}]{}De Lucia [[et al.]{} ]{}2004; Gao [[et al.]{} ]{}2004).
The mock galaxies are selected to have a magnitude range similar to that of the Sloan Digital Sky Survey (SDSS) sample analyzed by Zehavi [[et al.]{} ]{}(2002), $-22 > M_r > -19$, though our results are not sensitive to the precise range. The absolute r-band magnitude $M_r$ of each [SKID]{} halo is estimated by applying $v_{c,peak}$ to the Tully & Pierce (2000) variant of the Tully-Fisher (Tully & Fisher 1977) relation: $$M_R = -21.12 - 7.65 (log W_R - 2.5),$$ where the linewidth $W_r$ is approximately twice $v_{c,peak}$ (Tully & Fouque 1985). We select approximately 6,000 mock “galaxies” with $84~km~s^{-1} < v_{c,peak} < 206~km~s^{-1}$. In practice, our faintest simulated galaxies are [SKID]{} haloes of several hundred particles. While the Tully-Fisher magnitude assignment is subject to a number of uncertainties, including the fact that we apply this to ellipticals as well as spirals (see [[e.g. ]{}]{}Desai [[et al.]{} ]{}2004), it provides a convenient method for building a catalog of mock galaxies with magnitudes comparable to those in galaxy surveys. The spatial abundance of the mock catalog is 4.8 $\times$ 10$^{-2}$ $h^{3}$Mpc$^{-3}$, which is 2.6 times that of the SDSS sample selected from the same magnitude range. However, we stress that our results are not sensitive to the abundance or to the limits used for inclusion into the mock catalog; [[i.e. ]{}]{}, we are able to detect a clustering-age dependence for a range of $v_{c,peak}$-selected catalogs in addition to the one presented here, as we show later. Thus, even though the objects in our catalog are not expected to describe precisely the galaxy population, we can still capture many of the important clustering properties of the dark hosts of galaxies.
correlation functions
---------------------
The spatial pairwise correlation function of galaxies is an important cosmological test, as it quantitatively measures basic clustering properties (see [[e.g. ]{}]{}Peebles 1980). The spatial correlation function is calculated using the direct estimator (as in [[e.g. ]{}]{}Governato [[et al.]{} ]{}1999): $$\xi(r)={2N_p(r)\over n_{c}^2 V (\delta V)} -1,$$ where $N_p(r)$ is the number of pairs in radial bins of volume $\delta
V$, centred at $r$; $n_c$ is the mean space density of the catalog; and $V$ is the volume of the simulation. We take into account our periodic boundary conditions when finding pairs. The correlation function is often approximated by a simple power law: $$\xi(r)=\left({r\over r_0}\right)^{-\gamma},$$ with $\xi(r_0) = 1$, where $r_0$ is the correlation length. The relative clustering amplitude between haloes and the mass distribution is referred to as bias: $$b^2 = {\xi_{halo-halo}(M, r, z) \over \xi_{dm}(r ,z))}.$$ For all error estimates, we use 1$\sigma$ poisson errors (equal to the square root of the number of pairs in each bin), which are likely to underestimate the true errors because they do not take into account clustering and sample variance ([[e.g. ]{}]{}Croft & Efstathiou 1994). However, because we are interested primarily in the relative clustering between age-selected objects, and not the true clustering strength, poisson errors are adequate for this study.
results
=======
correlation functions of mass and haloes
----------------------------------------
In Fig. \[cfhostvc\], we plot $\xi(r)$ and the bias factor ($b(r) \equiv \sqrt{\xi_{haloes}(r)/\xi_{mass}(r)}$) for [SKID]{} haloes with $v_{c,peak} >$ 50, 100, 150, and 200 ${\rm km~s^{-1}}$. Larger [SKID]{} haloes have steeper correlation functions and larger correlation lengths. The largest [SKID]{} haloes are “antibiased” ($b<1$) with respect to the mass on small scales, but on large scales they are slightly more clustered than the mass. In general, haloes are “antibiased” with respect to the mass, particularly on small scales for small haloes. This is consistent with small scale “antibias” found in previous simulations ([[e.g. ]{}]{}Jenkins [[et al.]{} ]{}1998; Colin [[et al.]{} ]{}1999; Kravtsov & Klypin 1999; Yoshikawa [[et al.]{} ]{}2001; Diemand [[et al.]{} ]{}2004; Reed [[et al.]{} ]{}2005b) and may be caused by merging or destruction of subhaloes in high density regions ([[e.g. ]{}]{}Jenkins [[et al.]{} ]{}1998; Klypin & Kravtsov 1999). However, previous similar studies have found that the correlation function slope does not become shallower at small scales (Colin [[et al.]{} ]{}1999; Kravtsov [[et al.]{} ]{}2004; Neyrinck, Hamilton & Gnedin 2004; Conroy, Wechsler & Kravtsov 2006). The reasons for the difference are not clear, but we we note that the correlation function is a combination of a number of non-powerlaw components (central-satellite, central-central, and satellite-satellite), so there is no [*a priori*]{} expectation that $\xi(r)$ should follow a power-law ([[e.g. ]{}]{}Benson [[et al.]{} ]{}2000; Berlind & Weinberg 2002; Kravtsov [[et al.]{} ]{}2004). In fact, the small scale departure from a power law that we see begins approximately where the satellite-satellite term begins to dominate $\xi(r)$. At these scales, $\xi(r)$ could be sensitive to a number of issues that affect the relative contributions of these components. For example, the number of massive clusters, which can dominate the satellite-satellite term, can be affected by run to run “sample variance” or box size (see [[e.g. ]{}]{}Reed [[et al.]{} ]{}2007 and references therein). Differences in halo finder behavior at small scales could also be important. Further study is warranted, though our conclusions are not dependent on the smallest scales. It is difficult to quantify the precise scale below which the correlation function will no longer be robust. However, Reed [[et al.]{} ]{}(2005b) indicate that the subhalo distribution is robust down to 100 $h^{-1}$kpc for simulations of similar resolution. For this reason, we have plotted all correlation functions only down to 100 $h^{-1}$kpc.
the age dependence of the mock galaxy catalog correlation function
------------------------------------------------------------------
We plot $\xi{(r)}$ for our mock galaxy sample in Fig. \[cfvcages\], binned according to formation times. Older catalog members are significantly more clustered for all pair separations. The oldest 10$\%$ is most preferentially clustered at small scales, with a clustering amplitude of $\sim$10$\times$ that of the full mock catalog for separations less than $\sim$1 $h^{-1}$Mpc. The differences between the clustering of the young samples and the full catalog are smaller, but are significant. There is little difference in the spatial correlations of the youngest 10$\%$ and the youngest 50$\%$. The striking visual appearance of the age-clustering dependence is seen in Fig. \[agepics\], which shows the redshift zero simulation snapshot divided into the youngest and oldest 20$\%$ subsets of the mock catalog.
Fig. \[cfvcgroupsages\] shows the correlation function for members of groups or clusters larger than $3.2 \times 10^{13} \msun h^{-1}$. There is some age dependence of the clustering of group members, but it is limited mainly to small pair separations, and is significantly weaker than that found for the full mock catalog. Clustering of field objects, shown in Fig. \[cfvcfieldages\], has a strong age dependence, though not as strong found in the complete sample. To determine group membership, groups are identified using [*friends-of-friends*]{} (FOF; Press & Davis 1982; Davis [[et al.]{} ]{}1985), wherein the FOF haloes consist of particles of separated by less than 0.2 times the mean inter-particle separation. The group extent is subsequently computed assuming a virial overdensity of approximately 100 times the critical density (Eke, Cole, & Frenk 1996), and mock catalog members whose centre of mass lies within this region are assigned membership to that group. The overall correlation amplitude is a significantly higher for group members than for field members, an unsurprising result given that group members are selected deliberately from within regions of high density, and belong to massive haloes, which are strongly clustered due to the well-known mass-clustering relation.
The age-clustering relation in our mock catalog is likely due to a combination of causes. For the field sample, the obvious mechanism is the Gao [[et al.]{} ]{}age-clustering correlation for discrete virialized haloes. For the full field plus group and cluster catalog (Fig. \[cfvcages\]), the age dependence is stronger than that of the field sample alone (Fig. \[cfvcfieldages\]) because group and cluster members, which are found in highly clustered environments, tend to be old. Even though group and cluster members comprise only $\sim 10\%$ of the full sample, their contribution to the clustering age-dependence is significant due to the strong age correlation with environment. For example, 80$\%$ of our group and cluster members are older than the median mock galaxy age; and group and cluster members are 10 times more likely to belong to the 10$\%$ oldest subset than to the 10$\%$ youngest subset.
On small scales (less than $\sim 1 h^{-1}$Mpc), dynamical interactions become important for group and cluster members. Upon accretion onto a group or cluster halo, the subhalo will spiral in via dynamical friction, undergoing tidal stripping in the process. This leads to a subhalo distribution where centrally located group and cluster members were accreted earlier and are older. This is the likely cause of the small scale age-dependence within the group and cluster subsample. Finally, for mock galaxies belonging to groups or cluster of similar mass, the age dependence of their host group-group clustering may produce some effect on the correlation function, but it should be mild because our group and cluster hosts are larger than the $\sim10^{13} \msun h^{-1}$ mass threshold above which the age-clustering relation of discrete haloes becomes weak (Gao [[et al.]{} ]{}2005).
$\begin{array}{c@{\hspace{0.15\textwidth}}c @{\hspace{0.2\textwidth}}c}
\mbox{\bf (a) youngest 20$\%$} & \mbox{\bf (b) full sample} & \mbox{\bf (c) oldest 20$\%$}
\end{array}$
can the age-clustering relation cause the observed colour-clustering dependence?
--------------------------------------------------------------------------------
In this section, we perform a simple test to determine whether the magnitude of the age-clustering dependence seen in simulations could be sufficient to account for the observed colour-clustering dependence under the simple assumption that halo age is a proxy for galaxy colour. Here, we have split the catalog into an “old” and a “young” subsample with a 2:1 ratio of old to young haloes, which matches the ratio of red to blue galaxies of the Zehavi [[et al.]{} ]{}(2002) SDSS sample. In Fig., \[cfredages\], we plot the redshift space two-point correlation function for these catalog subsamples. The magnitude of the age dependence in the mock catalog sample is comparable to the clustering colour-dependence in SDSS for pair separations larger than $\sim$5 $h^{-1}$Mpc; see Fig. 11 of Zehavi [[et al.]{} ]{}(2002) for the SDSS comparison. However, the age-effect is much weaker relative to the observed colour trends for smaller separations, and is insignificant (within the uncertainties) for pairs separated by 200 $h^{-1}$kpc or less. The observed galaxy clustering colour-dependence, however, extends down to $\sim$ 100 $h^{-1}$kpc ([[e.g. ]{}]{}Zehavi [[et al.]{} ]{}2002; 2005, Madgwick [[et al.]{} ]{}2003; Li [[et al.]{} ]{}2006). The redshift space correlation function suggests that the age dependence has the potential to account for the observed colour-clustering trends only at large galaxy pair separations. This provides an independent argument that galaxy luminosity-weighted ages, indicated by colour, are different from the ages of the host dark matter subhaloes in which they lie.
Qualitative differences between age selected simulated haloes and colour-selected observed haloes are also apparent in the projected-space two-point correlation function, $$w(r_p)={2N_p(r)\over \mu^2 A(\delta A)} -1,$$ where ${\rm w(r_p)}$ is the pair excess over random with projected separation ${\rm r_p}$ binned with area $\delta A$, and $\mu$ is the mean projected density of haloes over the projected simulation area $A$ (see [[e.g. ]{}]{}Peebles 1980). In Fig. \[cfprojages\], the differences between the overall slope and small scale amplitude of ${\rm
w_p(r_p)}$ are relatively small between the young and old simulation samples. However, observed red galaxies have a much steeper and larger amplitude ${\rm w_p(r_p)}$ than blue galaxies (see [[e.g. ]{}]{}Zehavi [[et al.]{} ]{}2002, Fig. 13; Zehavi [[et al.]{} ]{}2005, Fig. 13) for a wide range of luminosity selected samples ([[e.g. ]{}]{}Li [[et al.]{} ]{}2006).
We note that our overall correlation amplitude is significantly smaller than that of the SDSS sample. This is due, at least in part, to our finite box size, which means that large scale density fluctuations are not fully and accurately represented ([[e.g. ]{}]{}Bagla & Ray 2005; Sirko 2005; Power & Knebe 2006; Reed [[et al.]{} ]{}2007), and should not be interpreted as an indication of a conflict with observations. A further contribution to our lower clustering amplitude may be the higher spatial abundance of our mock catalog, which implies that we are selecting smaller, and hence less strongly clustered, objects than in the SDSS sample.
A caveat here is that the mock catalog is inherently different from the SDSS sample. Because the correlation function of the full mock catalog is not a power law, as is generally observed in real galaxies, one should question whether the relative correlations of age-selected samples will display the same properties as real galaxies. A truly realistic simulated galaxy sample would of course require modelling correctly all baryon physics, including star formation and feedback at small scales, a task that is not feasible at this time. However, it is prudent to consider how the details of our mock catalog construction could affect the measured age-clustering signal. To enable better statistics, our mock catalog was selected to contain smaller galaxies than the SDSS sample. It is clear from Fig. \[cfhostvc\] that a sample of mock galaxies with higher $v_{c,peak}$ results in a correlation function that is closer to a power law, better matching observations. We thus make a test to show whether the age-dependence of the correlation function is sensitive to $v_{c,peak}$ (or abundance). In Fig \[cf150projages\], we show that the clustering dependence on age has similar qualitative behavior, although the age dependence is somewhat weaker, for a sample of larger mock galaxies selected purely by $v_{c,peak} > ~150 ~km ~s^{-1}$. This indicates that the scale dependence of age-clustering relation in our simulation is relatively insensitive to our specific choice of $v_{c,peak}$ range.
age dependence of the pairwise velocity dispersion
--------------------------------------------------
Galaxy peculiar velocities, as measured by the pairwise velocity dispersion, also provide a valuable probe of galaxy clustering, as well as providing an important component for dynamical probes of the dark energy equation of state ([[e.g. ]{}]{}Governato [[et al.]{} ]{}1997; Baryshev, Chernin & Teerikorpi 2001). The 1-D pairwise velocity dispersion, $\sigma_{\parallel}$(r), is the velocity dispersion for particle pairs in the direction parallel to the line of separation. In Fig. \[cvages\]a, we plot $\sigma_{\parallel}(r)$ for our mock catalog and for a random subsample of particles. Old mock galaxies have substantially higher pairwise velocities than young mock galaxies, as expected given their higher degree of spatial clustering. The old galaxy $\sigma_{\parallel}$(r) is $\sim$ 50 $km~s^{-1}$ “hotter” than the combined sample, and the young sample is “cooler” by up to 200 $km~s^{-1}$. The lower pairwise velocities for the mock catalog with respect to the dark matter particles is consistent with its spatial “antibias”, shown in Fig. \[cfhostvc\].
age dependence in a mass-selected sample
----------------------------------------
One important difference between our $v_{c,peak}$-selected haloes and the Gao [[et al.]{} ]{}age dependence of clustering strength in friends-of-friends haloes is that our sample has a large range in masses whereas the Gao [[et al.]{} ]{}study considered clustering at fixed mass. Because there is both a mass-age dependence and a mass-clustering dependence in CDM models, it is useful to consider what is the relation between age and clustering for [SKID]{} haloes at fixed mass. It should be noted that there are nontrivial dependencies of [SKID]{} masses on environment; for example, less mass will be found to be gravitationally self-bound in high density environments due mainly to tidal stripping, but also affected to some degree by the addition of the external potential in the computation of self-bound mass. Thus, it is not obvious that there should be a similar age dependence among [SKID]{} haloes at fixed mass as there is for friends-of-friends haloes. We show in Fig. \[cfskidmages\] that the age dependence of clustering strength is indeed present for [SKID]{} haloes. This effect has a strong dependence on pair separation and on halo mass wherein the oldest 20$\%$ of the ${1.3-1.6 \times 10^{10} h^{-1} \msun}$ haloes have a clustering amplitude approximately 10 times larger than that of the the youngest 20$\%$ at scales of 0.5 $h^{-1}$Mpc. At ${10^{11} h^{-1}
\msun}$, the effect is much weaker, consistent with little or no age dependence, though our uncertainties are large in this higher mass range due to the smaller number of haloes.
discussion
==========
We have shown that the clustering age dependence found for discrete friends-of-friends haloes by Gao [[et al.]{} ]{}is also present in a mock galaxy catalog that consists of dark haloes plus satellites selected by $v_{c,peak}$ to correspond to probable galaxy hosts. This provides strong evidence that galaxy clustering properties depend on the assembly epoch of the dark matter hosts for a population that includes field galaxies as well as galaxies within groups.
It has been suggested by [[e.g. ]{}]{}Gao [[et al.]{} ]{}that the age dependence of the clustering amplitude may be a problem for the models of galaxy formation ([[e.g. ]{}]{}Kauffmann, Nusser, & Steinmetz 1997; Benson [[et al.]{} ]{}2000; Wechsler [[et al.]{} ]{}2001) and the models of galaxy clustering in the halo model ([[e.g. ]{}]{}Seljak 2000; Cooray & Sheth 2002; Berlind [[et al.]{} ]{}2003; van den Bosch, Yang, & Mo 2003) that assume that statistical galaxy properties depend only upon halo mass. The result that the age dependence on clustering is weaker among group members suggests that any impact on these models will likely be strongest for the low mass haloes that typically host $\simlt$1 luminous galaxy.
Though the clustering age dependence is very strong for the extremes of the mock galaxy population, the age dependence is generally weaker, and has a different scale dependence than the colour dependence observed in recent large surveys. We expect that these qualitative differences are not sensitive to the precise selection criteria of the simulated sample. The different behavior of the simulated clustering age dependence suggests that luminosity-weighted galaxy ages, [[i.e. ]{}]{}colour, do not trace halo age. An apparent observed lack of correlation of stellar ages and halo ages is evident by the general trend that massive (luminous) galaxies tend to have old (red) stellar populations, in apparent contradiction to the inverse mass-age relation present in hierarchical structure formation. This naively suggests that local environmental effects may have a strong influence on galaxy stellar populations. However, the different merger rates of haloes in regions of different density are also important because progenitor haloes of a given mass will have been assembled earlier in more massive present day haloes, leading to earlier star formation ([[e.g. ]{}]{}Mouri & Taniguchi 2006; Neistein, van den Bosch & Dekel 2006).
In any case, ages and colours of galaxies are expected to be influenced by a large number of astrophysical phenomena, including the suppression of star formation by winds, AGN, or ram pressure stripping (see [[e.g. ]{}]{}Berlind [[et al.]{} ]{}2005; Bower [[et al.]{} ]{}2005). It may thus be difficult to decrypt a halo age-clustering signal in the real universe, as many of the influences on galaxy properties may depend on mass, environment, or other parameters (see [[e.g. ]{}]{}Abbas & Sheth 2006; Cooray 2006). However, there may still exist some correlation between stellar population and halo age, if for example, major mergers trigger major starbursts, or if the age of the oldest stars in a galaxy is correlated with halo age. A recent semi-analytical study by Croton, Gao, & White (2007) suggests that that the clustering of group or cluster central galaxies should correlate with group host dark matter halo assembly age. Some evidence for an age-clustering trend of group haloes has recently been found by observing that groups of similar mass, whose central galaxies have more passive star formation, which may indicate earlier group assembly, are more strongly clustered (Yang, Mo & van den Bosch 2006). See however, an apparently opposite relation found by Berlind [[et al.]{} ]{}(2006), who find that massive groups tend to be less strongly clustered if they have redder central galaxies. To measure this effect in individual galaxies, one would need to measure accurately star formation histories of galaxies hosted by a narrow ranges of halo masses (see [[e.g. ]{}]{}Heavens [[et al.]{} ]{}2004 for discussion of age measurements in SDSS stellar populations). Comparisons between the clustering age dependence among simulated haloes and the age dependence among galaxy stellar populations could then provide clues to the physics of galaxy formation.
Summary
-------
Within a high resolution cosmological dark matter simulation, we have examined the clustering properties of a mock galaxy catalog selected by $v_{c,peak}$ to match approximately the luminosity range and number density of observable galaxies.
$\bullet$ A strong clustering age dependence is found for mock galaxy catalogs that include both central haloes and satellite haloes, and is reflected in both spatial and in kinematic clustering measures. It is caused primarily by 1) the age clustering relation for discrete virialized haloes, acting on field mock galaxies, and 2) the contribution of group and cluster members, which tend to be older, and are highly clustered due to their presence within massive dark matter hosts, thereby increasing the tendency for old members of the full sample to be highly clustered.
$\bullet$ The strength of the clustering age dependence implies that it is likely to be manifested in real galaxies. The clustering age dependence is weaker than the clustering colour dependence in 2dF and SDSS for pair separations less than $\sim$5 $h^{-1}$Mpc, and has a different scale dependence. This means that the observed clustering colour-dependence cannot be fully explained by assuming that stellar population ages trace halo ages. That is, one cannot simply assume that red galaxies lie in old haloes and blue galaxies lie in young haloes. The clustering colour dependence must be influenced by additional processes that affect the baryons.
$\bullet$ The clustering age dependence is weaker among group and cluster mock catalog members than for the general galaxy population.
Acknowledgments {#acknowledgments .unnumbered}
===============
DR has been supported by PPARC, and acknowledges early funding by a NASA GSRP fellowship while at the University of Washington. FG was supported as a David E. Brooks Research Fellow, and was partially supported by NSF grant AST-0098557 at the University of Washington. TQ was partially supported by the NSF grant PHY-0205413. We thank Geraint Harker, Carlos Frenk, and Richard Bower for useful discussion. Simulations were performed on the Origin 2000 at NCSA and NASA Ames, the IBM SP4 at the Arctic Region Supercomputing Center (ARSC), and the NASA Goddard HP/Compaq SC 45. We thank Chance Reschke for dedicated support of our computing resources, much of which were graciously donated by Intel.
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\[lastpage\]
[^1]: Email: reed@lanl.gov
|
[**Effective Schrödinger equations**]{}\
[**for nonlocal and/or dissipative systems**]{}\
*[**Arnold Neumaier**]{}*
*Institut für Mathematik, Universität Wien*
*Strudlhofgasse 4, A-1090 Wien, Austria*
*email: Arnold.Neumaier@univie.ac.at*
*WWW: http://www.mat.univie.ac.at/neum/*
[**Abstract.**]{}
The projection formalism for calculating effective Hamiltonians and resonances is generalized to the nonlocal and/or nonhermitian case, so that it is applicable to the reduction of relativistic systems (Bethe-Salpeter equations), and to dissipative systems modeled by an optical potential.
It is also shown how to recover [*all*]{} solutions of the time-independent Schrödinger equation in terms of solutions of the effective Schrödinger equation in the reduced state space and a Schrödinger equation in a reference state space.
For practical calculations, it is important that the resulting formulas can be used without computing any projection operators. This leads to a modified coupled reaction channel/resonating group method framework for the calculation of multichannel scattering information.
[**Keywords**]{}: Feshbach projection, effective Hamiltonian, nonlocal Schrödinger equation, Bethe-Salpeter equation, coupled reaction channels, resonating group method, dissipative quantum system, optical potential, form factor, doorway operator, time-independent perturbation theory, backward error analysis, multichannel scattering\
[**E-print Archive No.**]{}: hep-th/0201085
[**1998 PACS Classification**]{}: primary: 03.65.-w secondary: 31.15.Dv\
Introduction {#In}
============
In many applications, a quantum system of interest is part of a much bigger system, and the latter’s state influences the system state. If the big system is represented by solutions of a Schrödinger equation in a big state space, it is desirable to find an effective Schrödinger equation in a small state space that describes how the small system of interest is affected by the embedding into the big system. Under certain conditions, a Schrödinger equation in the big state space can indeed be reduced to an effective Schrödinger equation in the small state space, in such a way that the interesting part of any solution of the full Schrödinger equation satisfies exactly the effective Schrödinger equation.
Much effort has gone into solving this reduction problem, and in a sense it is well understood [@AdhK; @FadM; @Glo; @KukKH; @WilT]. Exact expressions for the effective Hamiltonian can be given. In its exact form, the effective Hamiltonian is energy dependent and usually acquires a nonhermitian part; its eigenvalues describe the bound states and resonances of the reduced system. There are approximation schemes that compute the effective Hamiltonian (at least in principle) to arbitrary accuracy.
Less known (but proved here) is that certain solutions of the full Schrödinger equation can be reconstructed from solutions of the effective Schrödinger equation, using little more than what is already available from the reduction process.
The reduction is usually done for bound state calculations by the variational principle discussed in every textbook on quantum mchanics. For resonance calculations, the reduction may be done within the [Feshbach]{} [@Fes1; @Fes2] projection formalism (for an exposition see, e.g., [Kukulin]{} et al. [@KukKH Chapter 4]). In both cases, the reduced state space is finite-dimensional. For the calculation of scattering states, the reduction may be done by means of coupled reaction channel equations, also called the resonating group method; a nice exposition is given in [Wildermuth & Tang]{} [@WilT]. In this case, the reduced state space is a direct sum of finitely many function spaces, one for each energetically admissible arrangement of particles into clusters, with states parameterized by coordinates or momenta of cluster centers only.
The coupled reaction channel equations are numerically easy to handle, but the approximations involved in their derivation make an assessment of their accuracy difficult. On the other hand, the projection formalism gives in principle exact results, limited in accuracy only by the approximations made in the calculation of the effective Hamiltonian. However, this involves projection operators, which are clumsy to use if an orthogonal basis is not easily available.
Moreover, if the full Hamiltonian is already nonhermitian, or if the Hamiltonian is energy-dependent (such as in relativistic calculations; cf. [@FriM; @Ish; @IvaLLR; @KleFM; @KviB; @LahA; @LuL; @MeuGP; @RobW]), the standard derivation of the effective Hamiltonian is no longer valid. In the following, we shall remedy both defects.
The paper is organized as follows. We first extend the projection formalism such that it applies to the reduction of nonlocal systems and of dissipative systems modeled by an optical potential. Thus we work throughout with complex symmetric Hamiltonians, introduced in Section \[s.sym\], and generalize in Sections \[s.red\] and \[s.pert\] the traditional Feshbach projection approach to this more general situation. We then show in Section \[s.schro\] that the formalism in fact allows to recover [*all*]{} solutions of the time-independent Schrödinger equation in terms of solutions of two Schrödinger equations, one in the reduced state space and the other in a reference state space. The resulting formulas are closely related to those of time-independent perturbation theory.
For practical calculations, the resulting formalism is revised in Section \[s.noproj\] so that it can be used without computing any projection operators. Section \[s.form\] shows how the use of doorway operators gives flexible approximation schemes for the exact formulas derived in Section \[s.noproj\]. A backward error analysis discussed in Section \[s.qual\] allows the estimation of the reliability of approximate solutions of Schrödinger equations obtained by this or any other method.
Specific choices of the embedding map lead in Section \[s.multi\] to a modified coupled reaction channel/resonating group method framework for the calculation of multichannel scattering information, in which solutions of the full Schrödinger equation can be obtained from solutions of coupled reaction channel equations for the effective Hamiltonian. In principle it is capable of arbitrarily accurate approximations to the full dynamics, and shares this feature with the two Hilbert space method of [Chandler & Gibson]{} [@ChaG; @ChaG2], which partly inspired the present investigations.
The theory is presented in a fully rigorous manner, allowing for unbounded operators by using in place of Hilbert spaces a pair of dual topological vector spaces. The most useful results are in Sections \[s.noproj\], \[s.form\], and \[s.multi\].
The reader interested in the results but not in mathematical rigor may omit all references to spaces and topology, may think of all spaces as finite-dimensional and of operators as matrices, and may skip all proofs. In particular, of Section \[s.sym\] introducing the basic terminology, only – are essential, and most of Sections \[s.red\]–\[s.schro\] can be skimmed.
Symmetric operators {#s.sym}
===================
$\Lin(\Vz,\Wz)$ denotes the space of continuous linear mappings between two topological vector spaces $\Vz$ and $\Wz$, $\Vz^*=\Lin(\Vz,\Cz)$ denotes the dual space of continuous, complex-valued linear functionals on $\Vz$, and $\Lin\Vz=\Lin(\Vz,\Vz)$ denotes the algebra of continuous linear transformations of $\Vz$.
In the following, $\Hz$ is a complex topological vector space with a definite, continuous symmetric bilinear form, providing a natural embedding of $\Hz$ into the dual space $\Hz^*$. We refer to $\Hz^*$ as a [**state space**]{}, and to the $\psi\in\Hz^*$ as [**states**]{}. We write the pairing and the bilinear inner product as $$\phi^T\psi=\psi^T\phi~~~\mbox{for }\phi\in\Hz,~\psi\in\Hz^*.$$ The notation is chosen such that it looks as closely as possible like standard finite-dimensional linear algebra.
We say that a sequence (or net) $\psi_l\in\Hz^*$ [**converges weakly**]{} to $\psi\in\Hz^*$, and write $\psi_l\tow \psi$, if $$\lim_{l\to\infty} \phi^T\psi_l=\phi^T\psi \forall \phi\in\Hz.$$ We extend the bilinear inner product to arbitrary pairs $(\phi,\psi)\in\Hz^*\times \Hz^*$ for which $$\phi^T\psi=\lim_{l\to\infty} \phi_l^T\psi_l$$ is defined and independent of the weakly converging sequences (or nets) $\phi_l\tow \phi$, $\psi_l\tow \psi$ of $\phi_l, \psi_l \in\Hz$. (In the applications, this allows to form the inner product of state vectors corresponding to bound states and resonances but not that of scattering states.)
Complex conjugation is denoted by a bar, and, with the notation $\psi^*=\bar\psi^T$, the Hermitian inner product on $\Hz$ is $$\<\phi|\psi\>=\phi^*\psi~~~\mbox{for $\phi,\psi\in\Hz$}.$$ The associated Euclidean norm is $$\|\psi\|=\sqrt{\psi^*\psi},$$ and $\bar \Hz$ is the closure of $\Hz$ in $\Hz^*$ with respect to the Euclidean norm. Thus $\Hz\subseteq \bar\Hz\subseteq \Hz^*$, and $\bar \Hz$ is a Hilbert space.
In the applications, $\Hz$ is a space of sufficiently nice functions (namely arbitrarily often differentiable, with compact support) on some finite- or infinite-dimensional manifold, the inner product of two functions is some integral of their pointwise product induced by a nonnegative measure on the manifold, and $\Hz\subseteq \bar\Hz\subseteq \Hz^*$ is a Gelfand triple (or rigged Hilbert space). (For these concepts, see [Gelfand & Vilenkin]{} [@GelV], [Maurin]{} [@Mau]. For an exposition in physicists’ terms see [Kukulin]{} [@KukKH Appendix A]; cf. also [Böhm]{} [@Boh].)
The [**transpose**]{} of a linear operator $A\in\Lin (\Hz,\Hz^*)$ is the linear operator $A^T\in\Lin(\Hz,\Hz^*)$ defined by $$(A^T\phi)^T\psi=\phi^TA\psi ~~~\mbox{for }\phi,\psi\in\Hz,$$ $A^*=\bar A^T$ defines the [**adjoint**]{} of $A$, and $$\re A=\frac{1}{2}(A+A^*),~~~\im A=\frac{1}{2i}(A-A^*)$$ define the [**real**]{} and [**imaginary part**]{} of $A$. Clearly, $(AB)^T=B^TA^T$ and $(AB)^*=B^*A^*$. The operator $A\in \Lin \Hz$ is called [**symmetric**]{} if $A^T=A$ on $\Hz$, [**Hermitian**]{} if $A^*=A$ on $\Hz$, and [**positive semidefinite**]{} if $$\psi^*A\psi\ge 0~~~\mbox{for all $\psi\in\Hz$}.$$ In particular, $A$ is Hermitian if and only if $\im A=0$. We extend symmetric operators $A\in \Lin \Hz$ to $\Lin \Hz^*$ by defining $A\psi\in\Hz^*$ for $\psi\in\Hz^*$ by $$\phi^TA\psi=(A\phi)^T\psi \forall\phi\in\Hz.$$
In the following, $\AC \in \Lin \Hz $ is always a symmetric operator such that . (In particular, this includes the case where $\AC $ is Hermitian since then $\im \AC =0$.)
Since $\psi^*\AC \psi=\psi^*(\re \AC )\psi+i \psi^*(\im \AC )\psi$ and both $\psi^*(\re \AC )\psi$ and $\psi^*(\im \AC )\psi$ are real, is equivalent to \^\*0 . In particular, the spectrum of $\AC $ is in the complex upper half plane. If $\AC $ has a spectral resolution then, since $|\lambda+i\eps|\ge \eps$ for $\im \lambda\ge 0$, we conclude that $(\AC +i\eps)^{-1}$ exists for all $\eps>0$ as a bounded operator on $\bar\Hz$ with spectral norm $$\|(\AC +i\eps)^{-1}\|\le \eps^{-1}~~~\mbox{for all } \eps>0.$$ The traditional situation is the one where $\AC =E-H$ with a complex energy $E$ satisfying $\im E\ge 0$ and a [**Hamiltonian**]{} $H=H_s-\frac{i}{2}\Gamma$. Here $H_s,\Gamma\in \Lin \Hz $ are symmetric and Hermitian, and $\Gamma$ is positive semidefinite. In the most important case of a conservative system, $\Gamma=0$ and $H$ is Hermitian. However, care is taken that all our results hold in the nonhermitian case, corresponding to dissipative systems with an optical potential that contributes to $\Gamma$.
The use of $\AC $ helps to avoid a multitude of expressions involving $E-H$ or $E-H+i\eps$. Since the (time-independent) [**Schrödinger equation**]{} $H\psi=E\psi$ takes the simple form =0, this makes the formal manipulations independent of energy and free of references to the Hamiltonian $H$, and thus much more readable. (Of course, in actual calculations, $E$ and $H$ reappear.)
More generally, also covers nonlocal problems with a nonlinear dependence of $\AC $ on $E$, and therefore can be used for the Bethe-Salpeter equations arising in bound state and resonance calculations for relativistic systems [@Ish; @IvaLLR; @LahA; @LuL; @RobW].
Reduction of the state space {#s.red}
============================
Let $\Hz$ and $\Hz_\eff$ be topological vector spaces with a definite, continuous bilinear inner product, related by the [**embedding map**]{} $P$, an injective, closed linear operator from $\Hz_\eff^*$ to $\Hz^*$ satisfying $\bar P=P$ and $$P\psi_\eff\in\Hz~~~\mbox{for all }\psi_\eff \in \Hz_\eff.$$ We want to relate a Schrödinger equation $\AC \psi=0$ in the full state space $\Hz^*$ to an effective Schrödinger equation $\AC _\eff\psi_\eff=0$ in the reduced state space $\Hz_\eff^*$.
$P^*=P^T$ maps $\Hz^*$ to $\Hz_\eff^*$ and $\Hz$ to $\Hz_\eff$. Moreover, $P^*P:\Hz_\eff^*\to\Hz_\eff^*$ is invertible since $P$ is closed and injective. The [**pseudo inverse**]{} P\^I:=(P\^\*P)\^[-1]{}P\^\* maps $\Hz^*$ to $\Hz_\eff^*$ and possesses the properties (PP\^I)\^T=PP\^I, P\^TPP\^I=P\^T, P\^IP=1. This implies that Q:=1-PP\^I=1-(P\^I)\^TP\^T\^\* satisfies P\^IQ=P\^TQ=0, QP=0, Q\^2=Q\^T=Q. and hence is the orthogonal projection to the orthogonal complement of the range of $P$.
A [**$Q$-resolvent**]{} of a symmetric operator $\AC \in \Lin \Hz $ is a symmetric operator $G\in \Lin \Hz $ satisfying GP=0, GQ=Q. Since $GQ=G(1-PP^I)=G$, the symmetry of $G$ implies QG=G=GQ, QG=Q=GQ. Formally, $G=Q(Q\AC Q)^{-1}Q$, but $(Q\AC Q)^{-1}$ is only defined on the range $Q\Hz^*$ of $Q$.
The following discussion generalizes the Feshbach projection formalism which is obtained in the special case where $\AC =E-H$ and $H$ is Hermitian and $G$ is the ordinary resolvent of $QHQ$ in $Q\Hz^*$ (cf. the development in [Kukulin]{} et al. [@KukKH Chapter 4]; in their notation, $G=G_Q(z)$ is called the ‘orthogonalized resolvent’).
\[p3.1\]
\(i) If $\AC $ has a $Q$-resolvent $G$, it is uniquely determined, the operator \_:=P\^TP-P\^TGP=P\^T(-G)P=P\^T(P-GP) is symmetric, and $\im \AC _\eff$ is positive semidefinite.
\(ii) For arbitrary $\psi_\eff\in\Hz_\eff^*$, the vector =(P-GP)\_\^\* satisfies P\^T=\_\_, Q=0, \_=P\^I.
If holds with $G'$ in place of $G$ then $$G'=G'Q=G'Q\AC G=G'\AC G=G'\AC QG=QG=G,$$ giving uniqueness. The vector satisfies $$Q\AC \psi=Q\AC (P-G\AC P)\psi_\eff=(Q-Q\AC G)\AC P\psi_\eff=0,$$ $$P^T\AC \psi=P^T\AC (P-G\AC P)\psi_\eff=\AC _\eff\psi_\eff,$$ and, since $P^IG=P^IQG=0$, $$P^I\psi=P^I(P-G\AC P)\psi_\eff=P^IP\psi_\eff=\psi_\eff.$$ This proves (ii). Since $\bar P=P$, we have $$\bary{lll}
\psi^*\AC \psi&=&\psi^*((P^I)^TP^T+Q)\AC \psi=\psi^*(P^I)^T\AC _\eff \psi_\eff\\
&=&\psi^*(P^I)^*\AC _\eff \psi_\eff=\psi_\eff^*\AC _\eff \psi_\eff.
\eary$$ Thus, for arbitrary $\psi_\eff\in\Hz_\eff^*$, we have $\im \psi_\eff^T\AC _\eff \psi_\eff=\im \psi^*\AC \psi \ge 0$. This proves (i).
The second term :=P\^TGP in the definition of $\AC _\eff$ is called the [**optical potential**]{} induced by the reduction process. (The name is explained in [Taylor]{} [@Tay p.385].)
In the special case where $P^TP=1$ and $\AC =E-H$, $G=G(E)$ and hence the optical potential $\Delta(E)=P^T(E-H)G(E)(E-H)P$ is energy-dependent (and nonlocal) and we have $\AC _\eff=E-H_\eff(E)$ with the [**effective Hamiltonian**]{} $$H_\eff(E)=P^THP+\Delta(E).$$ Thus the optical potential causes energy shifts in the eigenvalues of the [**projected Hamiltonian**]{} $P^THP$. We also note that the reduced Schrödinger equation is generally a [*nonlinear*]{} eigenvalue problem $$H_\eff(E)\psi=E\psi.$$ Frequently, the energy-dependence is ignored; however, nonlinear eigenvalue problems for nonlocal Schrödinger equations arising from Bethe-Salpeter equations for relativistic problems were actually solved, e.g., [@Ish; @IvaLLR; @LahA; @LuL; @RobW].
Note that the resonating group method ([Wildermuth & Tang]{} [@WilT]) works with coupled reaction channel equations derived from a projected Hamiltonian and hence misses the optical potential; an energy-independent term is added instead on a phenomenological basis [@WilT Chapter 8.2]. An alternative exact method is derived below.
We return to the general case.
The following identities hold: \_P\^I=P\^T(1-G), (P\^I)\^T\_=(P-GP), and, if $\AC $ and $\AC _\eff$ are invertible, P\^T\^[-1]{}P=P\^TP\_\^[-1]{}P\^TP.
Since $(\AC -\AC G\AC )Q=\AC (Q-G\AC Q)=0$, we have $$\AC _\eff P^I=P^T(\AC -\AC G\AC )PP^I=P^T(\AC -\AC G\AC )(1-Q)=P^T(\AC -\AC G\AC ).$$ This gives the first equation in , and the transpose gives the second equation. For invertible $\AC $, $\AC _\eff$, we conclude from and $GP=GQP=0$ that $$\AC _\eff P^I\AC ^{-1}P= P^T(1-\AC G)P=P^TP,$$ hence $P^TP\AC _\eff^{-1}P^TP=P^TPP^I\AC ^{-1}P=P^T\AC ^{-1}P$.
implies that for invertible $\AC $ and $\AC _\eff$, the part of $\AC ^{-1}$ accessible from $\Hz_\eff^*$ can be computed from the knowledge of $\AC _\eff$ alone. Similarly, our next result says that, if $\AC $ has a $Q$-resolvent, all solutions of the Schrödinger equation $\AC \psi=0$ can be computed from solutions of the reduced Schrödinger equation $\AC _\eff\psi_\eff=0$ and a knowledge of the [**correction operator**]{} R=GP occurring in and .
\[t0.1\] Suppose that $\AC $ has a $Q$-resolvent $G$.
\(i) For any $\psi_\eff\in\Hz_\eff^*$ with $\AC _\eff\psi_\eff=0$, the vector $\psi\in\Hz^*$ defined by satisfies $\AC \psi=0$ and \_=P\^I\_\^\*. (ii) For any $\psi\in\Hz^*$ with $\AC \psi=0$, the vector satisfies $\AC _\eff\psi_\eff=0$, and we can reconstruct $\psi$ from .
\(i) Multiplication of the second equation of with $\psi_\eff$ gives $$0=(P^I)^T\AC _\eff\psi_\eff=\AC (P-G\AC P)\psi_\eff=\AC \psi,$$ and follows from Proposition \[p3.1\](ii).
\(ii) Multiplication of the first equation of with $\psi$ gives $$\AC _\eff\psi_\eff=\AC _\eff P^I\psi=P^T(1-\AC G)\AC \psi=0.$$ Since $P\psi_\eff=PP^I\psi=(1-Q)\psi=\psi-Q\psi$, follows from $$\bary{lll}
(1-G\AC )\psi&=&(1-G\AC )\psi-(Q-G\AC Q)\psi=(1-G\AC )(1-Q)\psi\\
&=&(1-G\AC )P\psi_\eff=(P-G\AC P)\psi_\eff
\eary$$ since G=0.
Note that, while $\AC _\eff\psi_\eff=0$ looks like a Schrödinger equation, this is generally a [**nonlinear**]{} eigenvalue problem. Indeed, if $\AC =\AC (E)=E-H$ then $\AC _\eff(E)$ is generally a nonlinear analytic function of $E$, and the [**effective Hamiltonian**]{} $$H_\eff(E):=E-\AC _\eff(E)$$ has a nonlinear dependence on the energy.
In many cases of interest, the $Q$-resolvent does not exist for the Hamiltonian $H$ of interest. But frequently the $Q$-resolvents $G_\eps$ for the perturbed Hamiltonians $H-i\eps$ corresponding to $\AC _\eps=\AC +i\eps$ exist for all $\eps>0$, and are given by $$G_\eps=Q(Q\AC Q+i\eps)^{-1}Q.$$ If the limit R:=\_[0]{} G\_\_P exists, most of the preceding proof still goes through, with $R$ in place of $G\AC P$ and $R^T$ in place of $P^T\AC G$, and the effective Hamiltonian (usually) acquires a nonhermitian part.
The only exception is the second half of statement (ii), which must be modified. (This is most conspicuously seen when $P=0$, where $\AC _\eff=0$ and the reduced Schrödinger equation provides no information at all.) Inspection of the proof shows that fails. Thus, if $\psi\in\Hz^*$ satisfies $\AC \psi=0$, the expression \^:=-(P-R)\_need not vanish (as predicted by under the stronger assumptions), but only the much weaker equation $P^T\psi^\perp=0$ follows. Thus the reduction does no longer allow one to recover [*all*]{} solutions of $\AC \psi=0$. By Theorem \[t0.1\](i), $\psi^\perp$ is also a solution of the Schrödinger equation in $\Hz^*$, and since $P^T\psi=0$ implies $\psi_\eff=0$ and hence $\psi^\perp=\psi$ we have $$\{\psi^\perp \mid \AC \psi=0\}=\{\psi\mid \AC \psi=0,~P^T\psi=0\}.$$ We discuss later (Theorem \[t4.4\]) how to access this missing part.
Perturbation theory {#s.pert}
===================
The computation of the $Q$-resolvent is traditionally done using perturbation theory. It is assumed that the inhomogeneous Schrödinger equation for a related reference problem is explicitly solvable, and the problem of interest is considered as a perturbation of the reference Schrödinger equation.
We therefore assume that we have a symmetric operator $\AC _\iref\in \Lin \Hz $ satisfying \_Q=Q\_; this commutation relation can always be achieved by taking an arbitrary symmetric approximation $\AC _0$ to $\AC $ and putting \_=\_0-(1-Q)\_0Q-Q\_0(1-Q). Let $G$ be a $Q$-resolvent of $\AC $ and put V=\_-, T=V+VGV, :=1+GV. $V$ is called the [**interaction**]{}, $T$ the ($Q$-version of the) [**transition operator**]{} or [**T-matrix**]{}, and $\Omega$ the ($Q$-version of the) [**Möller operator**]{}. This terminology is justified by the close formal relations of the properties of $T$ and $\Omega$, derived below, with those of the T-matrix and the Möller operators of standard scattering theory. Indeed, for $P=0$, $Q=1$ (which is uninteresting from the point of view of effective Schrödinger equations), the results reduce to those of standard perturbation theory.
\[p0.3\] If holds then P=(1-G)P, \^TP=P, T=V=\^TV, P\^TP=\_=P\^T(\_-T)P.
Since $$G\AC _\iref P=GQ\AC _\iref P=G\AC _\iref QP=0$$ we have $$G\AC P=G\AC _\iref P-GVP=-GVP,$$ $$P^T\AC G=(G\AC P)^T=-(GVP)^T=-P^TVG,$$ hence $$\Omega P=P+GVP=(1-G\AC )P.$$ Together with $\Omega^TP=(1+VG)P=(1+VGQ)P=P$, this gives . follows directly from . Since $$\bary{lll}
P^T\AC \Omega P&=&P^T\AC (1-G\AC )P=\AC _\eff\\
&=&P^T\AC P-P^T\AC G\AC P=P^T\AC P+P^T\AC GVP\\
&=&P^T(\AC _\iref-V)P-P^TVGVP=P^T(\AC _\iref-T)P,
\eary$$ follows.
\[p0.3a\] If $\Omega$ is invertible then G\_=\^[-1]{}G is a $Q$-resolvent of $\AC _\iref$, and =(1-W)\^[-1]{}, W=G\_V.
$$\Omega(1-W)=\Omega-\Omega G_\iref V=\Omega-GV=1,$$ gives . Since $G_\iref P=\Omega^{-1}GP=0$ and $$G_\iref \AC _\iref Q=G_\iref \AC Q+G_\iref VQ=\Omega^{-1}G\AC Q +WQ
=(1-W)Q+WQ=Q,$$ $G_\iref$ is a $Q$-resolvent of $\AC _\iref$.
\[t3.4\] Suppose that $G_\iref$ is a $Q$-resolvent of a symmetric operator $\AC _\iref$ satisfying , and suppose that $\Omega$ with exists. Then $G=\Omega G_\iref$ is a $Q$-resolvent of $\AC =\AC _\iref-V$, and with T=V+VGV, – hold. Moreover, we have =1+W=1+W, T=V+TW=V+W\^TT, G=G\_+W G=G\_+GW\^T.
follows directly from $\Omega(1-W)=(1-W)\Omega=1$ and implies $\Omega=1+\Omega G_\iref V=1+GV$, hence . Since $GP=\Omega G_\iref P=0$ and $$G\AC Q=G\AC _\iref Q-GVQ=\Omega G_\iref \AC _\iref Q-GVQ =\Omega Q-GVQ=Q,$$ $G$ is a $Q$-resolvent of $\AC $. Thus Proposition \[p0.3\] applies, and gives –. is obvious. The first equality in follows from and , and the second follows by transposing the first equation. The first equation in follows from $$WG=G_\iref VG=G_\iref QVQG=G_\iref(Q\AC _\iref Q-Q\AC Q)G=G-G_\iref,$$ and the second follows again by transposing the first equation.
Inserting the definition of $W$ into gives the (Q-version of the) [**Dyson equation**]{} G=G\_+G\_V G=G\_+GV\^T G\_.
If the spectral norm of $W=G_\iref V$ is smaller than one, – can be used to calculate iteratively the Möller operator $\Omega$, the transition operator $T$ and the $Q$-resolvent $G$. To lowest order, we get $$\Omega\approx 1+G_\iref V,~~~T\approx V,~~~
G\approx G_\iref+G_\iref VG_\iref.$$ When $\Hz_\eff=\{0\}$, this is the [**Born approximation**]{}, and when $\Hz_\eff$ is finite-dimensional, this is a version of the [**distorted wave Born approximation**]{} (see, e.g., [Newton]{} [@New Section 9.1]). In the Born approximation, we simply get $$\AC _\eff \approx P^T(\AC _\iref -V)P = P^T\AC P;$$ in second order, \_P\^TP -P\^TVG\_VP, giving the approximation $\Delta\approx P^TVG_\iref VP$ for the optical potential . Further iteration gives the [**Born series**]{} $$T=V+VG_\iref V+VG_\iref VG_\iref V+\dots,$$ giving the exact optical potential $$\Delta= P^TVG_\iref VP+P^TVG_\iref VG_\iref VP+\dots .$$
Thus we have recovered a generalized version of traditional perturbation theory. For $\Hz_\eff=\{0\}$, $P=0$, $Q=1$, the equations obtained above reduce to those of standard perturbative scattering theory; the missing part – that one gets the scattering solutions of the Schrödinger equations – follows in the next section. Of course, from the point of view of effective Schrödinger equations, the case $P=0$ is completely uninteresting; however, it is instructive in that it shows that the methods used to solve scattering problems apply with small modifications to the problem of finding effective Hamiltonians.
Similarly, if $\Hz_\eff$ is the eigenspace of $H_\iref$ corresponding to a bound state of the reference system and $P$ the orthogonal projector to this space, we get the situation leading (in the Born approximation) to Fermi’s Golden Rule; cf. [Kukulin]{} et al. [@KukKH Section 4.4]. For a nondegenerate bound state, $\Hz_\eff$ is one-dimensional, and again the point of view of effective Schrödinger equations is empty. For degenerate bound states, however, we recover a nontrivial low-dimensional eigenvalue problem as effective Schrödinger equation.
Solving the Schrödinger equation {#s.schro}
================================
We are now ready to express all solutions of the Schrödinger equation in terms of the solutions of a reduced Schrödinger equation and special solutions of a reference Schrödinger equation.
\[p5.1\] If \_=P\^I, \_=Q(1-W)then P\^T\_=0, =(P\_+Q\_). If holds then \_\_=P\^T\^T, \_\_=Q, =(P\^I)\^T\_\_+\_\_- (1-Q)\^T\_\_.
Since $$QW=QG_\iref V=G_\iref V=W,$$ implies $$\bary{ll}
Q\psi_\iref &=Q(1-W)\psi=(Q-W)\psi\\
&=(1-PP^I-W)\psi=(1-W)\psi-P\psi_\eff,
\eary$$ hence $$\Omega(P\psi_\eff+Q\psi_\iref)=\Omega(1-W)\psi=\psi,$$ $$P^T\psi_\iref=P^TQ(1-W)\psi=0.$$ This gives . Now suppose that holds. By , P\^T\^T=(P)\^T=P\^T(1-G)=P\^T(1-G), and P\^T\^TQ=P\^T(Q-GQ)=0. Since $(1-QW)\Omega=(1-W)\Omega=1$, this implies P\^T\^T=P\^T\^T(1-QW)=P\^T\^T. Using , , , and , we find $$\bary{ll}
P^T\Omega^T\AC \psi &=P^T\Omega^T\AC \Omega(P\psi_\eff+Q\psi_\iref)
=P^T\Omega^T\AC (P\psi_\eff+Q\psi_\iref)\\
&=P^T\Omega^T\AC P\psi_\eff=P^T\AC (1-G\AC )P\psi_\eff=\AC _\eff\psi_\eff,
\eary$$ giving . Since $Q\AC _\iref W=Q\AC _\iref G_\iref V=QV$, we have $$Q\AC \Omega=(Q\AC _\iref-QV)\Omega=Q\AC _\iref(1-W)\Omega=Q\AC _\iref=\AC _\iref Q,$$ hence $$\bary{ll}
Q\AC \psi &=Q\AC \Omega(P\psi_\eff+Q\psi_\iref)=\AC _\iref
Q(P\psi_\eff+Q\psi_\iref)\\
&=\AC _\iref Q\psi_\iref=\AC _\iref\psi_\iref,
\eary$$ by , giving . Finally, $X:=1-(1-Q)\Omega^T$ satisfies $$XP=P-(1-Q)\Omega^TP=P-(1-Q)P=QP=0$$ by , hence $$\bary{lll}
XQ\AC =X(1-PP^I)\AC =X\AC =(1-(1-Q)\Omega^T)\AC =\AC -(P^I)^TP^T\Omega^T\AC ,
\eary$$ so that $$\AC \psi=((P^I)^TP^T\Omega^T+XQ)\AC \psi
=(P^I)^T\AC _\eff\psi_\eff+X\AC _\iref\psi_\iref$$ by and .
Since the formulas in Proposition \[p5.1\] and $\AC _\eff=P^T\AC \Omega P$ do not involve $G$, we can take limits and obtain:
\[t4.4\] Let $G_{\iref,\eps}$ be a $Q$-resolvent of $\AC _{\iref,\eps}$ with $\D\lim_{\eps\downto 0} \AC _{\iref,\eps}=\AC _\iref$. Suppose that $$W:=\lim_{\eps\downto 0} G_{\iref,\eps}V ~~\mbox{and}~~ \Omega=(1-W)^{-1}$$ exist. If $$\psi_\eff=P^I\psi,~~~\psi_\iref=Q(1-W)\psi$$ then $$P^T\psi_\iref=0,~~~\psi=\Omega(P\psi_\eff+Q\psi_\iref).$$ Moreover, with $\AC _\eff:=P^T\AC \Omega P$, we have $$\AC \psi=0 \iff \AC _\eff\psi_\eff=0,~~~\AC _\iref\psi_\iref=0.$$
State space reduction without projections {#s.noproj}
=========================================
Projection operators and the associated $Q$-resolvents are clumsy to use if an orthogonal basis is not easily available. We therefore revise the above formalism so that it can be used without computing any projection operators or $Q$-resolvents. Additional flexibility is gained by using in place of the reference operator $\AC _\iref\in \Lin \Hz $ an appropriate operator $P_0:\Hz_0^*\to\Hz^*$ from the dual of a reference space $\Hz_0$. For exact solutions, this reference space must be at least as big as $\Hz$; however, in Section \[s.form\], we choose $\Hz_0$ to be a smaller space in which numerical calculations are tractable, and obtain practical approximation schemes for the correction operator and the effective Hamiltonian. (This is related to the two Hilbert space method of [Chandler & Gibson]{} [@ChaG; @ChaG2], and indeed was inspired by their work.)
As before, $\AC \in \Lin \Hz $ is assumed to be symmetric, and $P:\Hz_\eff^*\to\Hz^*$ is assumed to be a closed injective linear mapping satisfying $\bar P=P$ and $$P\psi_\eff\in\Hz~~~\mbox{for all }\psi_\eff \in \Hz_\eff.$$ Let $P_0:\Hz_0^*\to\Hz^*$ be a closed, surjective linear mapping satisfying $$P_0\psi_0\in\Hz ~~~\mbox{for all }\psi_0 \in \Hz_0.$$ Then $P_0^T$ maps $\Hz^*$ to $\Hz_0^*$ and $\Hz$ to $\Hz_0$. Moreover, $P_0P_0^T:\Hz^*\to\Hz^*$ is invertible since $P_0$ is closed and surjective.
The pseudo inverse $P^I$ of the injective $P$ satisfies as before $$P^I=(P^TP)^{-1}P^T,~~~P^IP=1,~P^TPP^I=P^T$$ but the pseudo inverse $P_0^I$ of the surjective $P_0$ satisfies $$P_0^I=P_0^T(P_0P_0^T)^{-1},~~~P_0P_0^I=1,~P_0^IP_0P_0^T=P_0^T.$$
If there are linear mappings $R:\Hz_\eff^*\to\Hz^*$ and $\AC ':\Hz_\eff^*\to\Hz_\eff^*$ such that $$R\psi_\eff\in\Hz ~~\mbox{for all }\psi_\eff \in \Hz_\eff;~~~~~
\AC '\psi_0\in\Hz_\eff ~~\mbox{for all }\psi_0 \in \Hz_\eff$$ and P\_0\^TR+P\_0\^TP’=P\_0\^TP, P\^TR=0, then \_:=P\^TP-R\^TR is a symmetric operator satisfying \_=P\^T(P-R), (P\^I)\^T\_=(P-R). Moreover, if $\AC $ and $\AC _\eff$ are invertible then P\^T\^[-1]{}P=P\^TP\_\^[-1]{}P\^TP.
Symmetry is obvious. Multiplication of with $Q(P^I)^T$ gives $$Q\AC P=Q\AC R+QP\AC '=Q\AC R.$$ Now $$\bary{lll}
R^T\AC R&=&R^T\AC (Q+(P^I)^TP^T)R=R^T\AC QR\\
&=&(Q\AC R)^TR=(Q\AC P)^TR=P^T\AC QR=P^T\AC R
\eary$$ since $QR=(1-(P^I)^TP^T)R=R$, hence $$\AC _\eff=P^T\AC P-R^T\AC R=P^T\AC P-P^T\AC R=P^T\AC (P-R).$$ This is the first half of . By multiplication with $(P^I)^T$, we find $$\bary{lll}
(P^I)^T\AC _\eff&=&(P^I)^TP^T\AC (P-R)=(1-Q)\AC (P-R)\\
&=&\AC (P-R)-Q\AC P+Q\AC R=\AC (P-R)
\eary$$ giving the second half of . If $\AC $ and $\AC _\eff$ are invertible then implies $$P^T\AC ^{-1}(P^I)^T\AC _\eff=P^T(P-R)=P^TP,$$ hence $$P^TP\AC _\eff^{-1}P^TP=P^T\AC ^{-1}(P^I)^TP^TP=P^T\AC ^{-1}P,$$ giving .
It is not difficult to see that $R$ and $\AC _\eff$ from Section \[s.red\] are an instance of this construction, with $\Hz_0=\Hz$ and $P_0=1$.
\(i) For arbitrary $\psi\in \Hz^*$, the vectors \_=P\^I\_\^\*, \_0=P\_0\^I(-(P-R)\_)\_0\^\* satisfy P\^TP\_0\_0=0, =(P-R)\_+P\_0\_0. (ii) If holds for some $\psi_\eff\in \Hz_\eff^*$, $\psi_0\in \Hz_0^*$ then \_\_=(P-R)\^T, P\_0\^TP\_0\_0=(P\_0-(P-R)P\^IP\_0)\^T, =(P\^I)\^T\_\_+(P\_0\^I)\^T(P\_0\^TP\_0\_0).
\(i) follows from $$P_0\psi_0=P_0P_0^I(\psi-(P-R)\psi_\eff)=\psi-(P-R)\psi_\eff,$$ $$\bary{lll}
P^TP_0\psi_0&=&P^T\psi-P^T(P-R)\psi_\eff=P^T\psi-P^TP\psi_\eff\\
&=&P^T\psi-P^TPP^I\psi=0.
\eary$$ (ii) Since by $$\bary{lll}
R^T\AC \psi&=& R^T\AC P_0\psi_0=(P_0^T\AC R)^T\psi_0
=(P_0^T\AC P-P_0^TP\AC ')^T\psi_0\\
&=&P^T\AC P_0\psi_0-\AC '^TP^TP_0\psi_0=P^T\AC P_0\psi_0,
\eary$$ follows from and : $$\bary{lll}
\AC _\eff\psi_\eff&=&P^T\AC (P-R)\psi_\eff=P^T\AC (\psi-P_0\psi_0)\\
&=&P^T\AC \psi-P^T\AC P_0\psi_0=P^T\AC \psi-R^T\AC \psi=(P-R)^T\AC \psi.
\eary$$ holds since by , and , $$\bary{lll}
P_0^T\AC P_0\psi_0&=&P_0^T\AC \psi-P_0^T\AC (P-R)\psi_\eff
=P_0^T\AC \psi-P_0^T(P^I)^T\AC _\eff\psi_\eff\\
&=&P_0^T\AC \psi-P_0^T(P^I)^T(P-R)^T\AC \psi\\
&=&(P_0-(P-R)P^IP_0)^T\AC \psi.
\eary$$ holds since by and , $$\bary{lll}
(P_0^I)^TP_0^T\AC P_0\psi_0&=&\AC P_0\psi_0=\AC P_0P_0^I(\psi-(P-R)\psi_\eff)\\
&=&\AC (\psi-(P-R)\psi_\eff)=\AC \psi-(P^I)^T\AC _\eff\psi_\eff.
\eary$$
We now have the following projector-free and $Q$-resolvent-free version of Theorem \[t4.4\], constructing all solutions of the Schrödinger equation $\AC \psi=0$ in terms of two simpler Schrödinger equations.
\(i) If $\im \AC $ is positive semidefinite then $\im \AC _\eff$ is positive semidefinite. If also $\bar P_0=P_0$ then also $\im P_0^T\AC P_0$ is positive semidefinite.
\(ii) For arbitrary $\psi\in \Hz^*$, $\psi_\eff\in \Hz_\eff^*$, $\psi_0\in \Hz_0^*$ satisfying or , $$\AC \psi=0\iff \AC _\eff\psi_\eff=0,~P_0^T\AC P_0\psi_0=0.$$ In particular, to find all solutions of $\AC \psi=0$, it suffices to solve the two problems $$\AC _\eff\psi_\eff=0,$$ $$P_0^T\AC P_0\psi_0=0,~~~ P^TP_0\psi_0=0.$$
\(i) For arbitrary $\psi_\eff\in \Hz_\eff^*$, we define $\psi$ by with $\psi_0=0$. Then $P^T\psi=P^T(P-R)\psi_\eff=P^TP\psi_\eff$, hence $\psi_\eff=P^I\psi$. Now gives $$\AC \psi=(P^I)^T\AC _\eff\psi_\eff,$$ and since $(P^I)^*=(P^I)^T$, we get $$\psi^*\AC \psi=\psi^*(P^I)^T\AC _\eff\psi_\eff
=(P^I\psi)^*\AC _\eff\psi_\eff=\psi_\eff^*\AC _\eff\psi_\eff.$$ Hence $\psi_\eff^*(\im \AC _\eff)\psi_\eff=\psi^*(\im \AC )\psi\ge 0$, and $\im \AC _\eff$ is positive semidefinite. If also $\bar P_0=P_0$ then $P_0^*=P_0^T$ and $\im P_0^T\AC P_0$ is positive semidefinite since $$\psi_0^*(\im P_0^T\AC P_0)\psi_0=(P_0\psi_0)^*(\im \AC )(P_0\psi_0)\ge 0.$$ (ii) The forward implication follows directly from and , the reverse implication from .
If $\psi_0=0$ then $\phi^T\psi=\phi_\eff^T(P^TP+R^TR)\psi_\eff$, so that $$G_\eff=P^TP+R^TR$$ is the [**effective metric**]{} induced on $\Hz_\eff$. Note that it is generally not the original metric in $\Hz_\eff$, not even when $P$ is an orthogonal projection.
Form factors {#s.form}
============
In this section we show how to obtain efficiently approximations to the correction operator $R$. This leads in Section \[s.multi\] to a modified coupled reaction channel/resonating group method framework for the calculation of multichannel scattering information.
We emphasize that the formulas derived in this section no longer involve a pseudo inverse. In particular, they can be used even when $P$ is not injective and $P_0$ is not surjective. (Hovever, since the assumptions under which the formulas are derived are then violated, they lose some information and hence give only approximate effective Hamiltonians.)
\[t.form\] Let \_0:=P\_0\^TP\_0, P\_1:=P\^TP\_0, U:=P\_0\^TP-P\_0\^TP with a symmetric $\widetilde \AC \in \Lin \Hz_\eff$ (and hence in $\Lin \Hz_\eff^*$), and write $$\widehat \AC _\eps:=\left(\bary{cc}
\AC _\eps & P_1^T\\
P_1 &-i\eps
\eary\right)\in \Lin (\Hz_0\oplus\Hz_\eff)~~~
\mbox{with }\AC _\eps=\AC _0+i\eps.$$ (i) If the strong limit \_[0]{}\_\^[-1]{}[U0]{}=[F\_0F\_1]{} exists then is solved by R=P\_0F\_0, ’=+F\_1. (ii) Relation holds with F\_0=\_[0]{} F\_[0]{}, F\_1=\_[0]{} F\_[1]{}, where F\_[1]{}=(P\_1\_\^[-1]{}P\_1\^T+i)\^[-1]{}P\_1\_\^[-1]{}U, F\_[0]{}=\_\^[-1]{}(U-P\_1\^TF\_[1]{}), if these limits exist.
\(i) follows from $$\bary{ll}
\D{P_0^T\AC R+P_0^TP\AC '\choose P^TR}
&=\D{P_0^T\AC P_0F_0+P_0^TPF_1+P_0^TP\widetilde \AC \choose P^TP_0F_0}\\
&=\D\widehat \AC _0{F_0\choose F_1}+{P_0^TP\widetilde \AC \choose 0}\\
&=\D\lim_{\eps\downarrow 0}\widehat \AC _0\widehat \AC _\eps^{-1}
{U\choose 0}+{P_0^TP\widetilde \AC \choose 0}\\
&=\D{U\choose 0}+{P_0^TP\widetilde \AC \choose 0}={P_0^T\AC P\choose 0}
\eary$$ by looking at the upper and the lower part separately. (ii) follows from the equation $$\widehat \AC _\eps{F_{0\eps}\choose F_{1\eps}}={U\choose 0},$$ which is easily verified by substitution.
In principle, $\widetilde \AC $ in may be chosen arbitrarily. However, for numerical calculations it may be advisable to choose $\widetilde \AC $ in such a way that $U$ (which replaces the interaction $V$ in the projection approach) becomes small in some sense. This has the beneficial consequence that then the numerical approximation errors have a much smaller effect on the calculated solution.
In practice it is impossible to compute the exact correction operator and hence the exact effective Hamiltonian, since these tend to be exceedingly complicated. One therefore exploits physical intuition to select a space $\Hz_0$ of manageable complexity whose dual contains the [**doorway states**]{} believed to mediate the interaction of the reduced system and the unmodelled environment. $\Hz_0^*$ is embedded into $\Hz^*$ by means of a [**doorway operator**]{} $P_0$ that is now no longer surjective. Fortunately, the formulas and defining $R$ and $\AC _\eff$ do not depend on pseudo inverses, and hence make also sense in this case. and now only yield approximate solutions for the correction operator and an approximate effective Hamiltonian. However, these approximations become better and better as the range of $P_0$ covers a bigger and bigger part of $\Hz$.
The situation is fully analogous to numerical discretization schemes that are necessary to solve all but the simplest partial differential equations; the only difference is that in the present context it frequently makes sense to consider approximations in manageable function spaces, so that one does not discretize completely.
A proper choice of the doorway operator $P_0$ makes the computations more tractable. At the same time, it limits the formal complexity of the optical potential =R\^TR=F\_0\^T(P\_0\^TP\_0)F\_0, the second term in $\AC _\eff$. As one can see, $P_0$ specifies the allowed form of the optical potential while the [**form factor**]{} $F_0$ specifies the coefficients in the optical potential, and thus introduces energy-dependent [**running coupling constants**]{}.
The art in applying the reduction technique consists in finding embeddings $P$ that ‘dress’ the subsystem of interest in a sufficiently accurate way, a doorway operator $P_0$ embedding the relevant doorway states, and an operator $\widetilde \AC $ such that $U$ is small, and the limit exists and can be approximated efficiently.
Setting $P_0=0$ gives $R=0$, $\Delta=0$, and hence the trivial approximation \_P\^TP. This simply amounts to discarding the interaction of the subsystem with the rest of the system. The choice $\Hz_0=\Hz$, $P_0=P$ is not better since then $P^TPF_0=P^TP_0F_0=P^TR=0$ by , and hence $F_0=0$, $R=0$. This is not surprising since we expect that the doorway operator $P_0$ should incorporate [*additional*]{} information about doorway states not yet represented in the subsystem but significantly interacting with it.
If $\Hz_0$ and $\Hz_\eff$ are finite-dimensional then defines finite-dimensional matrices, and the computation of the form factor amounts to solving the matrix equation ( \_0 & P\_1\^T\
P\_1 & 0 )[F\_0F\_1]{}=[U0]{} with a complex symmetric (and for $\AC =\AC ^*$, $\bar P_0=P_0$ Hermitian) coefficient matrix. can be solved efficiently by sparse matrix methods (cf. [Duff]{} et al. [@ReiD; @DufGR]) if suitable localized basis functions are used to construct $P$ and $P_0$.
In practice, it may be useful to employ in combination with discretization methods a complex absorbing potential in place of the $+i\eps$. In particular, if one proceeds as in [Neumaier & Mandelshtam]{} [@NeuM] one gets a quadratic eigenvalue problem that can handle all energies in a certain range simultaneously using harmonic inversion ([Mandelshtam & Taylor]{} [@ManT]).
If $\Hz_\eff$ is finite-dimensional but $\Hz_0$ is a function space then $\AC _0$ is a differential or integral operator on $\Hz_0$. By solving suitable differential or integral equations we can find the vector-valued functions $$B_0:=\lim_{\eps\downto 0}\AC _\eps^{-1}P_0^T\AC P,~~~
B_1:=\lim_{\eps\downto 0}\AC _\eps^{-1}P_0^TP.$$ and the complex symmetric matrix $$G_1:=P_1B_1=\lim_{\eps\downto 0}(P_1\AC _\eps^{-1}P_1^T+i\eps).$$ Noting that $\AC '=F_1$ if $\tilde \AC =0$, the formula for the form factor becomes $$F_0=B_0-B_1\AC ',~~~\mbox{where } \AC '=G_1^{-1}P_1B_0.$$
The quality of approximate state vectors {#s.qual}
========================================
In practice, it is usually impossible to find exact solutions of a Schrödinger equation $\AC \psi=0$. On the other hand, in real applications, $\AC $ is never precisely known either. Hence it makes sense to assess the quality of an approximate state vector $\psi$ by trying to modify $\AC $ a little to an operator $\widetilde \AC $ that satisfies $\widetilde \AC \psi=0$ exactly. If the modification $\widetilde \AC -\AC $ is within the accuracy to which $\AC $ is known, we are confident that $\psi$ is a good approximation to the true but unknown $\AC $.
This way of assessing the quality of an approximate solution of a problem is widely used in numerical analysis (see, e.g., [Wilkinson]{} [@Wil]) and is known under the name of [**backward error analysis**]{}. Here we give a backward error analysis for the equation $\AC \psi=0$, and deduce guidelines for quality assessment of approximate state vectors.
Let $\AC \in \Lin \Hz $ be symmetric. Then, for arbitrary $\psi\in \Hz^*$ with finite $\psi^T\psi\ne 0$, the modified operator =(1-[\^T\^T]{}) (1-[\^T\^T]{}). satisfies $\widehat \AC \psi=0$ and (-)\^2=\_(), where \_():=2[()\^T()\^T]{} -([\^T\^T]{})\^2.
Since the formulas are invariant under scaling $\psi$ we may assume that $\psi$ is normalized to norm 1. Then $$\psi^T\psi=1,~~~\psi^T\AC \psi=:\lambda,~~~\psi^T\AC ^2\psi=:\mu,$$ with real $\lambda$, $\mu$. The operator $$\Delta:=\AC -\widehat \AC =\psi\psi^T\AC +\AC \psi\psi^T-\lambda\psi\psi^T$$ satisfies $\Delta\psi=\AC \psi$, hence $\widehat \AC \psi=0$. Since $$\Delta \AC \psi=\lambda \AC \psi+(\mu-\lambda^2)\psi,$$ we find $$\Delta^2=\AC \psi\psi^T\AC +(\mu-\lambda^2)\psi\psi^T.$$ $\Delta^2$ maps $\Hz$ to the two-dimensional space spanned by $\psi$ and $\AC \psi$, hence is trace class. Using the formula $\tr\phi\psi^T=\psi^T\phi$, we find $$\tr\Delta^2=\mu+(\mu-\lambda^2)=2\mu-\lambda^2=\tau_\AC (\psi),$$ giving .
In the conservative case where $\im \AC =0$, and hence $\AC $ is Hermitian, it is possible to show that the choice is best possible. Note that a state vector $\psi\in\Hz^*$ with $\bar\psi=\psi$ and finite $\psi^T\psi$ is now in the Hilbert space $\bar\Hz$.
Let $\AC \in \Lin \Hz $ be Hermitian, and suppose that $\psi\in\bar\Hz\backslash\{0\}$ satisfies $\bar\psi=\psi$ and $\AC \psi\in\bar\Hz$.
\(i) Any symmetric and Hermitian $\widetilde \AC \in \Lin \Hz $ with $\widetilde \AC \psi=0$ satisfies (-)\^2\_(). Equality in is achieved precisely when $\widetilde \AC =\widehat \AC $.
\(ii) We always have 0\_()2 [()\^T\^T]{}. In particular, $\tau_\AC (\psi)=0$ if and only if $\AC \psi=0$.
By the preceding theorem, $\widetilde \AC =\widehat \AC $ gives equality in , and is a good choice since $\widehat \AC \psi=0$. Hence suppose that $\widetilde \AC \neq\widehat \AC $. Without loss of generality, $(\widetilde \AC -\AC )^2$ is trace class (otherwise the trace is infinity and is trivially satisfied). Since $(\widetilde \AC -\widehat \AC )\psi=\widetilde \AC \psi-\widehat \AC \psi=0$ we have $$\bary{ll}
\tr(\AC -\widehat \AC )(\widetilde \AC -\widehat \AC )
&=\tr\Delta(\widetilde \AC -\widehat \AC )\\
&=\psi^T\AC (\widetilde \AC -\widehat \AC )\psi
+\psi^T(\widetilde \AC -\widehat \AC )\AC \psi
-\lambda\psi^T(\widetilde \AC -\widehat \AC )\psi=0.
\eary$$ Therefore $$\bary{lll}
\tr(\widetilde \AC -\AC )^2-\tau_\AC (\psi)
&=&\tr(\widetilde \AC -\AC )^2-\tr(\widehat \AC -\AC )^2\\
&=&\tr(\widetilde \AC +\widehat \AC -2\AC )(\widetilde \AC -\widehat \AC )\\
&=&\tr(\widetilde \AC -\widehat \AC )^2
-2\tr(\AC -\widehat \AC )(\widetilde \AC -\widehat \AC )\\
&=&\tr(\widetilde \AC -\widehat \AC )^2>0
\eary$$ since $\widetilde \AC \neq\widehat \AC $. Therefore, holds for $\widetilde \AC \neq\widehat \AC $ with strict inequality. This proves (i). Since $\bar \psi=\psi$, the second term in is nonnegative. This gives the upper bound in and implies the final assertion. The lower bound follows from the Cauchy-Schwarz inequality $(\psi^T \AC \psi)^2\leq\psi^T\psi\cdot(\AC \psi)^T\AC \psi$.
is the orthogonal projection of $\AC $ to the orthogonal complement of $\psi$, and $\sqrt{\tau_\AC (\psi)}$ measures, in a sense, its deviation from $\AC $. Therefore, $\tau_\AC (\psi)$ (or its square root) serves as a useful measure for the quality of an approximate solution $\psi$ of the Schrödinger equation with Hermitian $\AC $.
In the nonhermitian case, it seems possible that $\tau_\AC (\psi)=0$ even if $\AC \psi\ne 0$. (A finite-dimensional example is $\AC ={ \alpha~~i \choose i~~0}$, $\psi={1 \choose 0}$ which has $\AC ^2=0$ and $\tau_\AC (\psi)=0$ for $\alpha=\sqrt{2}$, and for $\alpha=1$ even $\tau_\AC (\psi)<0$.) Thus, unless $\AC ^*=\AC $, the measure $\tau_\AC (\psi)$ might be sometimes too optimistic. However, in practice $\AC $ is nearly Hermitian and the use of $\tau_\AC (\psi)$ should cause no problems.
Multichannel scattering {#s.multi}
=======================
We now apply the above to the multichannel approach discussed below. A projection approach to multichannel scattering leading to effective Hamiltonians is discussed, e.g., in [Newton]{} [@New Section 16.6], but the equations derived there appear not to be suitable to numerical approximation. A more useful formulation is given by the present equations from Theorem \[t.form\], with $P$ and $P_0$ as given below.
An [**arrangement**]{} is a partition $A$ of the system of particles into clusters $i\in A$, with correct assignment of distinguishability. At a fixed energy $E$, those arrangements are relevant that contain [**channels**]{} defined by cluster bound states with energies $E_i$ such that $$\sum_{i\in A} E_i \le E+\Delta E$$ where $\Delta E$ is zero or a small quantity. These open or nearly open channels are assumed to correspond approximately to states in a $n_A$-dimensional space $$\Hz_{A0}\subseteq \bigotimes_{i\in A} \Hz_i$$ with basis functions \_[Ak]{}(x)=\_[iA]{} \_[il\_[ik]{}]{}(x\_i), where $x_i$ is the vector of coordinates of particles in cluster $i$, and the $\phi_{il}(x_i)$ are translation invariant basis functions from the cluster Hilbert space $\Hz_i$, used in all possible combinations in . The motion of the clusters is described by a space $\Hz_A$ of functions of a system of relative coordinates $r_A$ between the cluster centers.
Consider, for example, a 3-particle reaction $XY+Z\rightleftharpoons X+YZ$. Then $\Hz_{XY+Z,0}$ consists of products of approximate bound states $\psi_{XY}$ and the ground state $\psi_Z$; $\Hz_{X+YZ,0}$ consists of products of the ground state $\psi_X$ and approximate bound states $\psi_{YZ}$, and $\Hz_{XYZ,0}$ consists of sufficiently many states localized in the transition region to resolve the resonances of interest. Usually, one would keep the arrangements $(XY,Z)$ and $(X,YZ)$ in the reduced description, and use the states belonging to the arrangement $(XYZ)$ as doorway states for the transition regime.
Let ${\cal C}$ be the set of arrangements considered relevant for the reduced description. The reduced [**multichannel state space**]{} is then the dual of the space $$\Hz_\eff=\bigoplus_{A\in {\cal C}_{\eff}}\Hz_A^{n_A}$$ or the properly symmetrized subspace in case of indistinguishable clusters. The inner product in $\Hz_\eff$ is given by $$\phi_\eff^T\psi_\eff=
\sum_{A\in{{\cal C}_{\eff}}}\int dr_A \phi_A(r_A)^T\psi_A(r_A).$$ The embedding map $P:\Hz_\eff^*\to\Hz^*$ is given by $$P\psi_\eff=\sum_{A\in{{\cal C}_{\eff}}}P_A\psi_A,$$ where the $k$th component of $P_A:(\Hz_A^{n_A})^*\to\Hz^*$ maps a function of $r_A$ to the $k$th basis vector of $\Hz_A$ modified to have the corresponding dependence on the center of mass $r_A(x)$ of the cluster coordinates $x$, $$(P_A\psi_A)_k(x)=\psi_A(r_A(x))\phi_{Ak}(x).$$ The transpose $P^T:\Hz^*\to\Hz_\eff^*$ is given by $$(P^T\psi)_A=P_A^T\psi~~~\mbox{for all}~A\in{{\cal C}_{\eff}}.$$ By construction, $P^T\AC P$ is a direct sum of contributions of the form $$P_A^T\AC P_A = \AC _A-H_A,$$ where $H_A$ is the free Hamiltonian for the motion of the cluster centers and $\AC _A$ is a symmetric $n_A\times n_A$-matrix. In a basis of cluster eigenstates with energies $E_i$, $\AC _A$ is the diagonal matrix formed by the energy differences $E-E_i$.
The construction of $(\Hz_0,P_0)$ is completely analogous, using a larger set of arrangements and/or channels that contain the doorway states.
Thus the calculations have the same complexity as those for the coupled reaction channel equations (or resonating group method), as described, e.g., by [Wildermuth & Tang]{} [@WilT]). However, the present scheme is more flexible in that it can incorporate information from doorway states. In principle, by increasing the size of the doorway state space, it is capable of arbitrarily accurate approximations to the full dynamics, and shares this feature with the two Hilbert space method of [Chandler & Gibson]{} [@ChaG; @ChaG2] and with a technique by [Goldflam & Kowalski]{} [@GolK].
To solve the reduced Schrödinger equation (and, if a similar construction is used for the doorway operator, the equations for the form factor), the whole arsenal of methods developed in the applications is available. Binary arrangements can be handled by Lippmann-Schwinger equations (see, e.g., [Wildermuth & Tang]{} [@WilT], [Adhikari & Kowalski]{} [@AdhK Chapter 3]), and 3-cluster arrangements by the [Faddeev]{} [@FadM] connected kernel approach (see, e.g., [Glöckle]{} [@Glo Chapter 3], [Kukulin]{} et al. [@KukKH], [Adhikari & Kowalski]{} [@AdhK Chapter 7]).
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---
abstract: 'In a previous work we developed a family of orbital-free tensor equations for DFT \[J. Chem. Phys. 124, 024105 (2006)\]. The theory is a combination of the coupled hydrodynamic moment equations hierarchy with a cumulant truncation of the one-body electron density matrix. A basic ingredient in the theory is how to truncate the series of equation of motion for the moments. In the original work we assumed that the cumulants vanish above a certain order (N). Here we show how to modify this assumption to obtain the correct susceptibilities. This is done for N=3, a level above the previous study. At the desired truncation level a few relevant terms are added, which, with the right combination of coefficients, lead to excellent agreement with the Kohn-Sham Lindhard susceptibilities for an uninteracting system. The approach is also powerful away from linear response, as demonstrated in a non-perturbative study of a jellium with a repulsive core, where excellent matching with Kohn-Sham simulations is obtained while the Thomas Fermi and von-Weiszacker methods show significant deviations. In addition, time-dependent linear response studies at the new N=3 level demonstrate our previous assertion that as the order of the theory is increased, new additional transverse sound modes appear mimicking the RPA transverse dispersion region.'
author:
- 'Igor V. Ovchinnikov'
- 'Lizette A. Bartell'
- Daniel Neuhauser
title: 'Hydrodynamic Tensor-DFT with correct susceptibility'
---
Introduction
============
The development of new methods for quantum dynamics based upon hydrodynamic representations is very promising. In hydrodynamics the kinetics of the system is defined by a lesser number of variables than the number of variables required to define the complete one-particle density matrix (which contains all the information on off-diagonal quantum coherence as in, *e.g.*, the Kohn-Sham approach). For stationary studies the hydrodynamics approach is related to orbital-free density-functional theory [@books; @Carter1; @Carter2; @Carter3; @GarciaGonzalez; @Smargiassi; @Wang; @Watson; @Nehete; @Aguado; @Govind; @Depristo; @Wang1; @Brack; @Frankcombe; @Sim; @Ayers; @Chan; @Choly; @Glossman; @Gross; @Lee; @Plindov; @Yang; @Trickey]. It is the reduced number of variables depicting the system that makes hydrodynamical theories applicable for numerical studies of relatively large systems.
The simplest hydrodynamical approach is similar to the de Broglie-Bohm formulation of one-particle quantum mechanics [@generalMethods9; @Derrickson; @Madelung; @Deb; @Lopreore; @Wyatt; @Burant; @DayMultyDim]. In this approximation the complete complex-valued one-particle density matrix is substituted by two real valued fields $\rho$ and $\phi$, which are combined in an order parameter $\psi=\sqrt{\rho}\exp(i\varphi)$. The equations of motion are obtained by minimizing a Ginzburg-Landau-like functional on $\psi$. In addition the density matrix is assumed to possess long-range off-diagonal one-particle correlations.
A more rigorous and asymptotically exact approach is an infinite hierarchy of coupled hydrodynamic moment (CHM) equations [@Moments0; @Moments2; @Moments3; @Moments4; @Moments5]. The moments come from a Taylor expansion of the one-particle density matrix with respect to the off-diagonal variable. To get a tractable system of equations the infinite hierarchy must be truncated. The most physically meaningful truncation is a cumulant expansion for the density matrix [@Moments0]. Specifically, one decides on an order to terminate the method at; a low order will be less numerically demanding but less accurate than a higher one. Then, at that order, labeled N, the (N+1)-th order moment is expanded in terms of the previous set of moments, through the use of the cumulant expansion.
The CHM theory and the accompanying cumulant truncation have been applied so far to systems where particle statistics does not play an important role. In Ref.[@ours] we have generalized the CHM theory and cumulant expansion to statistically degenerate fermions. The main point has been the modification of the unperturbed one-particle density matrix of a locally homogenous electron gas by using the cumulants. Since the approach uses successive tensors, we labeled it Hydrodynamic tensor DFT (HTDFT).
It turns out that the lowest level of truncation, $N = 1$, HTDFT corresponds to a de-Brogilie-Bohm quantum hydrodynamics and in addition naturally incorporates the Thomas-Fermi [@Thomas] kinetic energy term into the energy functional. At the next level, $N=2$, HTDFT starts reproducing the spectrum of a homogenous Fermi liquid, *i.e.*, it gets transverse excitations, rather than just classical plasmonic longitudinal excitations. The transverse sound mode mimics the elementary excitations’ density of states.
A crucial feature of HTDFT is the value of the cumulant used at the truncation. In Ref.[@ours] we assumed that the $(N+1)$-th order cumulant is zero. It turns out, however, as we show here, that this assumption leads to a wrong susceptibility for a homogenous electron gas, *i.e.*, to a wrong linear response to a perturbation, even for a non-interacting system of electrons. We show here how to remedy this problem. This is exemplified below for truncation at the $N = 3$ level, which is the first level where the method will yield different ground-state results from the Thomas-Fermi approach. Specifically, the 4’th order cumulant is written as a sum of terms involving the gradients of the previous moments. The coefficients of these terms are obtained by fitting to the exact susceptibility of a non-interacting set of electrons (the Lindhard function).
The balance of the paper is as follows. The general methodology is first developed in Section II. In Section III the derivation of a correct susceptibility is done. Section IV applies the methodology to a static non-perturbative numerical study of a jellium with a deep spherically symmetric hole, where we show that the agreement with Kohn-Sham results is excellent while the Thomas Fermi and the von-Weiszacker methods have significant errors. Section V is a linear-response time-dependent study of the approach for N=3 as a function of frequency and wavevector. This latter part is a direct continuation of our work in Ref.[@ours] for $N=2$, and proves that there is an additional sound mode with respect to the $N=2$ case, just as suggested in Ref.[@ours]. Conclusions follow in Section VI.
The system and Tensor-DFT formulation
=====================================
Coupled Hydrodynamic Moment Hierarchy
-------------------------------------
For completeness, we rederive the basic aspects of the theory (see Ref.[@ours]). We assume that the many electron system can be described by the one-particle density matrix, $\rho^{(1)}$. The one-electron Hamiltonian governing this system, $h$, is, as usual, composed of kinetic terms, and a local potential terms.
The one-particle density matrix is then expressed in terms of average and difference coordinates as: $$\begin{aligned}
\rho^{(1)}(\bm R, \bm s) = \langle \hat \psi^\dagger(\bm R - \bm
s/2) \hat \psi(\bm R + \bm s/2)\rangle.\end{aligned}$$ The time evolution of the one-particle density matrix is governed by the Heisenberg equation, $i\dot\rho=\left[h,\rho\right]$, which in those coordinates takes the form: $$\begin{aligned}
i\frac{\partial }{\partial t} \rho^{(1)}(\bm R, \bm s) &=& \hat
P_\alpha \hat p_\alpha
\rho^{(1)}\nonumber \\
&&+\left(\tilde V(\bm R+\bm s/2)-\tilde V(\bm R - \bm
s/2)\right)\rho^{(1)}.\label{basicequation}\end{aligned}$$ Here $\hat P_\alpha$ and $\hat p_\alpha$ stand for the derivatives over the coordinates $R_\alpha$ and $s_\alpha$ $$\begin{aligned}
\hat P_\alpha = -i \partial/\partial R_\alpha,&& \hat p_\alpha = -i \partial/\partial s_\alpha,\end{aligned}$$ and $\tilde V(R)$ is the effective potential, which also takes into account the two-body interactions: $$\begin{aligned}
\label{EffectivePotential} \tilde V(\bm R) = \int \frac{\rho(\bm
R')-\rho_0(\bm R')}{|\bm R - \bm R'|}d^3R' - \frac{\delta
E_{xc}}{\delta \rho(\bm R)} + V_{ext}(\bm R),\end{aligned}$$ where $\rho(\bm R)=\rho^{(1)}(\bm R,\bm 0)$ is the spatial electron density, $\rho_0(\bm R)$ is the positive nuclear charge density, $V_{ext}$ is any external potential, and $E_{xc}$ is the exchange-correlation energy and $\rho_0$ is the nuclei density. There are a variety of functions $V_{xc}\equiv\delta
E_{xc}/\delta\rho$ in the literature (see, *e.g.*, Ref.[@Carter1; @Carter2; @Carter3]). For us, however, the specific form of $V_{xc}$ is not important. \[In future works we will aim to derive a form of $V_{xc}$ which depends also on other moments in addition to $\rho(\bm R)$.\]
The particle kinetics in the system can be exactly described by the complete infinite set of hydrodynamic moments (dynamic tensors) [@Moments0; @Moments2; @Moments3; @Moments4; @Moments5; @ours], which are the derivatives of the one-particle density matrix with respect to the off-diagonal distance, $\bm s$, at $\bm s = \bm 0$: $$\begin{aligned}
\Phi^{(N)}_{l_1\dots l_N}(\bm R) = \hat p_{l_1}\dots\hat
p_{l_N}\left.\rho^{(1)}(\bm R, \bm s)\right|_{\bm s=\bm
0}.\label{Phiintroduction}\end{aligned}$$ The particle and the current spatial densities are merely the first two tensors in the family: $$\begin{aligned}
\Phi^{(0)}(\bm R) = \rho(\bm R)&,& \Phi^{(1)}_{i}(\bm R) = J_i(\bm R).\end{aligned}$$
By using Eq.(\[basicequation\]) one derives an infinite set of equations which connects the moments at different orders:
\[setofequations\] $$\begin{aligned}
\frac\partial{\partial t} \rho &=& -\nabla_\alpha J_\alpha,\\
\frac\partial{\partial t} J_i &=& -\nabla_\alpha \Phi^{(2)}_{i\alpha} -
\rho \nabla_i \tilde V,\\
\frac\partial{\partial t} \Phi^{(2)}_{ik}
&=& -\nabla_\alpha \Phi^{(3)}_{ik\alpha} - J_i \nabla_{k}\tilde V -J_k\nabla_i\tilde V,\\
\frac\partial{\partial t} \Phi^{(3)}_{ikl}
&=& -\nabla_\alpha \Phi^{(4)}_{ikl\alpha} - \Phi^{(2)}_{ik} \nabla_l\tilde V -
\Phi^{(2)}_{il} \nabla_k \tilde V \nonumber
\\&& - \Phi^{(2)}_{lk}\nabla_i \tilde V + \frac14\rho \nabla_i\nabla_k\nabla_l \tilde V,\\
\text{etc.}\nonumber\end{aligned}$$
This generic set of equations is correct for both fermions and bosons. For this set to be useful one should terminate it at some level. As usual, this termination is actually a method for factorizing a moment $\Phi^{(N+1)}$ at some $N$ into moments $\Phi^{k},k\le N$. In addition, this truncation reflects the Fermi statistics of the particles. The order $N$ at which one terminates controls the precision with which we treat the system.
Fermi-factorization of higher order dynamic tensors {#FermiFactorization}
---------------------------------------------------
In Ref.[@ours] we proposed a factorization procedure for the lowest order dynamic tensors ($N=2,3$). Here we describe in detail how the factorization of the higher order dynamic tensors works in the Fermi case. The method proposed is based on the following general parametrization of the one-particle density matrix:
$$\begin{aligned}
\rho^{(1)}(\bm R, \bm s) &=& \rho\exp\{\phi(\bm R,\bm s)\}f_0(\rho,\bm s),\\
\phi(\bm R, \bm s) &=& \sum\limits_{\alpha\ge1}
\frac1{\alpha!}\phi^{(\alpha)}_{i_1i_2\dots i_\alpha}(\bm R)(i
s_{i_1})(i
s_{i_2})\dots (i s_{i_\alpha}),\\ f_0(\rho,\bm s) &=& 3\frac{\sin(k_Fs)-(k_Fs)\cos(k_F s)}{(k_F s)^3},\\
k_F&=&(3\pi^2\rho)^{1/3}.\end{aligned}$$
Here, $f_0$ is the normalized one-particle density matrix of a free fermion liquid with density $\rho(\bm R)$ and $k_F$ is the local (density-dependent) Fermi wave-vector. All the cumulants, $\phi^{(\alpha)}$, are symmetric in all the indices because they are convolved with the symmetric tensors $s_{i_1}\dots s_{i_n}$ and all the $\phi^{(\alpha)}$ are real as the one-particle density matrix is hermitian. The same is true for the tensors $\Phi^{(\alpha)}$.
The physical meaning of this paramerization is as following. If $\phi\equiv 0$, then we end up with the Thomas-Fermi approximation of a locally homogeneous Fermi liquid. The $\phi$ function perturbs this steady liquid picture, and the tensors $\phi^{\alpha}$’s and/or $\Phi^{(\alpha)}$’s determine different dynamic characteristics of the flowing electron liquid. The function $f_0$ assures the Fermi statistics of the particles at the one-particle level.
For brevity we introduce below the tensors $$\begin{aligned}
{\cal F}^{(\alpha)}\equiv \Phi^{(\alpha)}/\rho,\end{aligned}$$ instead of $\Phi^{(\alpha)}$. The tensors ${\cal F}^{(\alpha)}$ and $\phi^{(\alpha)}$ are interrelated. The relations between ${\cal
F}^{(\alpha)}$’s and $\phi^{(\alpha)}$’s for the lowest order tensors are given below for the first four relations:
\[relations\] $$\begin{aligned}
{\cal F}^{(1)}&=&\phi^{(1)},\\
{\cal F}^{(2)}&=&\phi^{(2)}+ \overline{\phi^{(1)}\phi^{(1)}} + e^{(2)},\\
{\cal F}^{(3)}&=&\phi^{(3)}+ 3 \overline{\phi^{(1)} \phi^{(2)}}+3 \overline{\phi^{(1)}e^{(2)}}
+\overline{\phi^{(1)}\phi^{(1)}\phi^{(1)}},\\
{\cal F}^{(4)}&=&\phi^{(4)}+ 4 \overline{\phi^{(1)}\phi^{(3)}} +
3{\phi^{(2)}\phi^{(2)}} + 6
\overline{\phi^{(1)}\phi^{(1)}\phi^{(2)}} \nonumber \\&& + 6
\overline{\phi^{(2)} e^{(2)}} + 6
\overline{\phi^{(1)}\phi^{(1)}e^{(2)}}+
\overline{\phi^{(1)}\phi^{(1)}\phi^{(1)}\phi^{(1)}} + e^{(4)},\quad
\label{relations4}\end{aligned}$$
*etc.* Here all terms are absolutely symmetric tensors of their indices so that there is no need to write down the indices explicitly; a bar denotes here a complete symmetrization, *e.g.*, for a product of $a_{i_1\dots i_K}$ and $b_{i_1\dots
i_M}$: $$\begin{aligned}
\overline{a b}_{i_1\dots i_{K+M}} =
\frac1{(K+M)!}\sum\limits_{p}a_{p(i_1)\dots
p(i_K)}b_{p(i_{K+1})\dots p(i_{K+M})},\end{aligned}$$ where summation is assumed over all $(K+M)!$ permutations of the indices and $p$ denotes a permutation. The symmetrized multiplication is associative and it can be considered a multiplication on a ring of symmetric tensors (note that in Ref.[@ours] a somewhat different symmetrization was used). Here is an explicit example of the symmetrized multiplication: $$\begin{aligned}
\label{symmetrization}
\overline{\phi^{(1)}\phi^{(1)}\phi^{(2)}}&\equiv&
\frac16\left(\phi^{(1)}_{i}\phi^{(1)}_{j}\phi^{(2)}_{kl}+
\phi^{(1)}_{i}\phi^{(1)}_{k}\phi^{(2)}_{jl}+\phi^{(1)}_{i}\phi^{(1)}_{l}\phi^{(2)}_{jk}\right.\nonumber\\&&\left.
+\phi^{(1)}_{k}\phi^{(1)}_{l}\phi^{(2)}_{ij}
+\phi^{(1)}_{j}\phi^{(1)}_{l}\phi^{(2)}_{ik}+\phi^{(1)}_{j}\phi^{(1)}_{k}\phi^{(2)}_{il}\right).\end{aligned}$$ In Eqs. (\[relations\]) the tensors $e^{(2)}$ and $e^{(4)}$ come from differentiating the function $f_0$, so that they have the physical meaning of averaging the particle momenta products over the unperturbed Fermi sea (ufs): $$\begin{aligned}
e^{(2)}_{ij} &=& \rho^{-1} \left\langle p_i p_j
\right\rangle_{ufs}=c_{2}\delta_{ij},\\
e^{(4)}_{ijkl}&=&\rho^{-1} \left\langle p_i p_jp_k p_l
\right\rangle_{ufs}=3 c_{4}\overline{\delta\delta}_{ijkl}\equiv
c_{4}\left(\delta_{ij}\delta_{kl}+\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}\right),\label{etensors}\end{aligned}$$ where the kinetic coefficients are defined as $$\begin{aligned}
c_{2}&=&\frac15 k_F^2, \quad
c_{4}=\frac1{35}k_F^4.\label{definitionofc}\end{aligned}$$ All the odd order $e$’s vanish.
The general recipe for how to express ${\cal F}^{(N)}$ in terms of $\phi^{(\alpha)}$’s is as follows. ${\cal F}^{(N)}$ is the sum of all the different symmetrized (in the sense discussed above) products of $\phi^{\alpha}\text{ and } e^{(\alpha)},\alpha\le N$. An additional rule is that each term may include one (and only one) $e$-tensor.
The relations inverse to Eqs. (\[relations\]) are:
\[invrelations\] $$\begin{aligned}
{\cal F}^{(1)}-\phi^{(1)}&=&0\label{invrelations1},\\
{\cal F}^{(2)}-\phi^{(2)}&=&\overline{{\cal F}^{(1)}{\cal F}^{(1)}} + e^{(2)}\label{invrelations2},\\
{\cal F}^{(3)}-\phi^{(3)}&=&3\overline{{\cal F}^{(1)}{\cal F}^{(2)}}
-2\overline{{\cal F}^{(1)}{\cal F}^{(1)}{\cal F}^{(1)}}\label{invrelations3},\\
{\cal F}^{(4)}-\phi^{(4)}&=&4\overline{{\cal F}^{(1)}{\cal
F}^{(3)}}-12\overline{{\cal F}^{(1)}{\cal F}^{(1)}{\cal
F}^{(2)}}+6\overline{{\cal F}^{(1)}{\cal F}^{(1)}{\cal F}^{(1)}{\cal
F}^{(1)}}\nonumber \\ && + 3\overline{{\cal F}^{(2)}{\cal
F}^{(2)}}-3\overline{e^{(2)} e^{(2)}}+e^{(4)}\label{invrelations4}.\end{aligned}$$
The inversion of the infinite set of relations (\[relations\]) is possible since an expression for any ${\cal F}^{(N)}$ in terms of $\phi^{(\alpha)}$’s contains only $\phi^{(\alpha)}, \alpha\le N$. This means that if one knows the expressions for the first $N\quad$ $\phi^{(\alpha)}$ tensors (*e.g.*, Eqs.(\[invrelations1\]-\[invrelations3\])) for $N=3$) then by substituting all the lower order $\phi^{(\alpha)}$’s, $\alpha\le N$, in the relation for ${\cal F}^{(N+1)}$ (Eq.(\[relations4\])) with corresponding expressions in terms of ${\cal F}^{(\alpha)}$’s one gets the inverse relation for $\phi^{(N+1)}$ (Eq.(\[invrelations4\])).
The factorization for the tensor $\Phi^{(N+1)}$ is then simply given by the $(N+1)^\text{th}$ equation in Eqs.\[15\]. The expression for the $\Phi^{(N+1)}$ tensor contains only kinetic tensors of order $n
\le N$ , as well as the $(N+1)^\text{th}$ order cumulant. Once this cumulant is known the system of Eqs.(8) closes and one arrives at the $N^\text{th}$ order tensor DFT theory.
$N=3$ Hydrodynamic tensor-DFT
-----------------------------
At the $N=3$ level the factorization of $\Phi^{(4)}$ is given by Eq.(\[invrelations4\]) with $\phi^{(4)}$ set to zero, or: $$\begin{aligned}
\Phi^{(4)}&=&4\rho^{-1}\overline{J \Phi^{(3)}} -
12\rho^{-2}\overline{J J
\Phi^{(2)}}\nonumber\\&&+6\rho^{-3}\overline{J J J J}\nonumber\\&& +
3\rho^{-1}\overline{\Phi^{(2)} \Phi^{(2)}} -3\rho \overline{e^{(2)}
e^{(2)}} +\rho e^{(4)} + \rho \phi^{(4)} \label{Phiphi4}.\end{aligned}$$ In order to complete the theory we need to obtain $\phi^{(4)}$. For this, we study the static linear response of a homogeneous electron gas. In the ground state all the odd order $\Phi^{(\alpha)}$-tensors vanish (a ground state has no currents as its wave-function is real when there is no magnetic field and no degeneracy), so that the first three terms in Eq.(\[Phiphi4\]) would give only non-linear contributions and can be neglected. As a result, the required factorization for static studies simplifies as (here we restore the indices): $$\begin{aligned}
\Phi^{(4)}_{ijkl}&=& 3\rho^{-1}\overline{\Phi^{(2)}_{ij}
\Phi^{(2)}_{kl}}\nonumber\\&&+3
\rho(c_4(\rho)-c_2^2(\rho))\overline{\delta_{ij} \delta_{kl}}
+\phi^{(4)}_{ijkl}.\end{aligned}$$
Static linear response of homogeneous fermions and adjustment to the Lindhard structure factor {#StaticResponse}
==============================================================================================
![\[Figure1\] The bare structure factors, $\chi_0$, on a scaled momentum scale, $q/2k_F$, for the $1/9$-von Weiszacker approach, bare and adjusted $\Phi^{(3)}$ HTDFT theories, and the Lindhard function (free-electron gas static density-density correlator). The $\Phi^{(3)}$ HTDFT is fitted to have the three properties of the Lindhard function given in Eq.(\[threeconditions\]).](Figure1.EPS)
The static properties of a homogeneous electron liquid are determined by the structure factor, $\chi(\bm q)$. The structure factor is actually the static limit of the density-density correlation function $\chi(\bm q)=\left.\langle\hat\rho(-\bm q,
-\omega)\hat\rho(\bm q, \omega)\rangle\right|_{\omega\to i0}$. The physical meaning of $\chi(\bm q)$ is the ratio between the amplitude of the infinitesimal harmonic change in electron density, $\underline{\rho}(\bm q)$, and that of the external potential, $\underline{v}_{ext}(\bm q)$, which induces the change in the electron density:
$$\begin{aligned}
\delta v_{ext}(\bm R) &=& \underline{v}_{ext} \exp(i \bm q \cdot \bm R) + c.c.,\\
\delta\rho(\bm R) &=& \underline{\rho} \exp(i \bm q \cdot \bm R) + c.c.,\\
\underline{\rho}&=& \frac{\chi(\bm q)}{\rho_0} \underline{v}_{ext} +
C \underline{v}_{ext}^3 + \dots\end{aligned}$$
In the ground state of a homogeneous liquid the non-zero values at the $N=3$ level are the density $\rho_0$ and the second and the fourth order dynamic tensors, $\Phi^{(2)}_{ij}=\rho_0c_2(\rho_0)\delta_{ij}$, $\Phi^{(4)}_{ijkl}=\rho_0c_4(\rho_0)\overline{\delta\cdot\delta}_{ijkl}$. In a static linear response problem all the odd-order kinetic tensors remain zero. Therefore, to study the static linear response of the system we let the values of $\rho$, $\Phi^{(2)}$ and $\Phi^{(4)}$ vary harmonically in space around their stationary values:
$$\begin{aligned}
v_\text{ext}&=&\underline v_\text{ext}
e^{i \bm q\cdot \bm R}+c.c. ,\\
\rho&=&\rho_0+\left(\underline{\rho}
e^{i \bm q\cdot \bm R}+c.c.\right),\\
\Phi^{(2)}_{ij} &=& c_2 \rho_0\delta_{ij}+\left(\underline{\Phi}^{(2)}_{ij}e^{i \bm q\cdot \bm R}
+c.c.\right),\\
\Phi^{(4)}_{ijkl} &=& 3c_4 \rho_0
\overline{\delta_{ij}\delta_{kl}}+\left(\underline{\Phi}^{(4)}_{ijkl}
e^{i \bm q\cdot \bm R} +c.c.\right) \nonumber\\&&+ \rho_0\phi^{(4)}_{ijkl},\\
\nabla_i\tilde V &=& q_i \left(i (\tilde v(q)
\underline{\rho}+\underline{v}_{ext})e^{i \bm q\cdot \bm
R}+c.c.\right),\end{aligned}$$
where the underlined variables are the linear response coefficients, while $$\begin{aligned}
\tilde v(q) = \frac{4\pi}{q^2} - \frac{\partial
V_{xc}(\rho)}{\partial \rho}(\rho_0).\end{aligned}$$ With the use of Eq.(\[Phiphi4\]) the infinitesimal deviation of $\Phi^{(4)}$ has the following form: $$\begin{aligned}
&&\underline{\Phi}^{(4)}_{ijkl} = 3 D \underline{\rho}
\overline{\delta_{ij}\delta_{kl}} + 6 c_2
\overline{\delta_{ij}\underline{\Phi}^{(2)}_{kl}}+\rho_0\phi^{(4)}_{ijkl},\end{aligned}$$ where $$\begin{aligned}
D = \left.\frac{\partial(\rho (c_4 - c_2^2))}{\partial
\rho}\right|_{\rho=\rho_0} - c_2^2 = -\frac1{15}k_F^4.\end{aligned}$$ Next we consider what terms can be in $\phi^{(4)}$. Our purpose is to make sure that the static response in the non-interacting case would resemble the Lindhard function (static density response of free electrons). The terms added should include the derivative of the available quantities, *i.e.*, the density and the stress tensor, so that they will be vanishing for uniform densities. Further, since $\phi^{(4)}$ is a fourth-order tensor, it needs to be constructed from available tensors; the only ones available in the static limit are $\nabla_i$, $\rho$, $\Phi_{kl}$ and $\delta_{jl}$. It is easy to see by inspection that only the following local terms are available to first order in the perturbation and to lowest orders needed in $\nabla_i$:
$$\begin{aligned}
\label{additionalterms} \rho_0\phi^{(4)}_{ijkl}(\bm R) = \Lambda
\overline{\nabla_i\nabla_j\nabla_k\nabla_l} \rho - 6 f
\overline{\nabla_i\nabla_j \Phi^{(2)}_{kl}} - 6 c_2 h
\overline{\delta_{ij} \nabla_k\nabla_l}\rho,\end{aligned}$$
where $\Lambda, f$ and $h$ are dimensionless parameters. Even in linear response these terms can be augmented by terms involving further derivatives, e.g., terms involving a Laplacian of the components in Eq. (\[additionalterms\]), (i.e., $q^2$ in Fourier space) but as orbital-free methods should be primarily geared towards the long-wavelength limit, we do not consider here such higher order terms in $q$.
The additional terms yield the following relation between the linear response coefficients $\underline{\rho}$, $\underline{\Phi}^{(2)}$, and $\underline{\Phi}^{(4)}$: $$\begin{aligned}
\underline{\Phi}^{(4)}_{ijkl}(\bm q) = 6 \overline{\left( c_2
\delta_{ij}+f q_i q_j\right)\underline\Phi^{(2)}_{kl}} + 3 D
\underline\rho\overline{\delta_{ij}\delta_{kl}} + \Lambda
\underline\rho q_i q_j q_k q_l + 6 c_2 h \underline\rho
\overline{q_iq_j\delta_{kl}}. \label{Phi4linearizedvariation}\end{aligned}$$ Finally, the linearized equations read:
$$\begin{aligned}
&&q_\alpha \underline\Phi^{(2)}_{i\alpha} + q_i \rho_0\tilde v \underline\rho
+ q_i\rho_0 \underline v_{\text{ext}}=0,\end{aligned}$$
and $$\begin{aligned}
&&3\overline{(c_2\delta_{ij}+fq_iq_j)(q_\alpha\underline
\Phi^{(2)}_{k\alpha})} + 3(c_2 + f q^2)\overline{q_i
\underline\Phi^{(2)}_{kl}}\nonumber\\
&& + \left( 3\left(D + c_2 \rho_0\tilde v + h c_2
q^2\right)\overline{\delta_{ij} q_k} +\left(\frac14\rho_0\tilde
v + \Lambda q^2 + 3 h c_2\right)q_iq_jq_k \right) \underline\rho\nonumber \\
&&+\left(3 c_2 \overline{\delta_{ij} q_k} +
\frac14q_iq_jq_k\right)\rho_0\underline v_{\text{ext}}=0.\end{aligned}$$
where the index $\alpha$ is summed over. The only preferential direction in the problem is the momentum vector, $\bm q$, so that the dynamic tensor, $\underline\Phi^{(2)}$, can be decomposed into the two form-factors: $$\begin{aligned}
\underline\Phi^{(2)}_{ij} = \delta_{ij}\underline \Phi^{(0)} +
\frac{q_iq_j}{q^2}\underline \Phi^{(1)}.\end{aligned}$$ Upon substituting this resolution into the initial equations and equating independent spatial tensor components we arrive at three equations for $\underline\rho$, $\underline\Phi^{(0)}$ and $\underline\Phi^{(1)}$: $$\begin{aligned}
&&
\underline\Phi^{(0)}+\underline\Phi^{(1)}+\rho_0\tilde v \underline
\rho = - \rho_0 \underline v_{\text{ext}},\\
&&\left(-\rho_0 \underline v_{\text{ext}} - \rho_0\tilde v
\underline \rho\right) + \left(1 + \frac
f{c_2}q^2\right)\underline\Phi^{(0)} +
\left(-\frac{k_F^2}{3}+\rho_0\tilde v + hq^2\right)\underline \rho =
-\rho_0\underline v_{\text{ext}}, \\ &&3f\left(-\rho_0\underline
v_{\text{ext}} - \rho_0\tilde v \underline\rho\right) +
3\frac{c_2}{q^2}\left(1 + \frac f{c_2}q^2\right)\underline\Phi^{(0)}
+ \left( \frac14\rho_0\tilde v + \Lambda q^2 +
3hc_2\right)\underline \rho = -\frac14\rho_0\underline
v_{\text{ext}}.\end{aligned}$$
Upon solving these linear equations one gets:
$$\begin{aligned}
\underline\rho &=& -\chi \rho_0 \underline v_{\text{ext}},\\
-\chi^{-1} &=& -\chi_0^{-1} + \tilde v, \label{renormalization}\end{aligned}$$
where $$\begin{aligned}
-\frac{\pi^2}{k_F}\chi_0 &=& \frac{1+20(2f-1/12)\eta^2}{1 -24 h
\eta^2 - 80\Lambda \eta^4},\end{aligned}$$
and we introduced the dimensionless momentum: $$\begin{aligned}
\eta = \frac q{2k_F}.\end{aligned}$$ Eq.(\[renormalization\]) is the definition of the structure factor renormalized with respect to two-body interactions. Therefore $\chi_0$ should be the structure factor of non-interacting electrons. In order to adjust our theory to the realistic description of electrons one should compare $\chi_0$ to the Lindhard function: $$\begin{aligned}
-\frac{\pi^2}{k_F}\chi_{\text{Lind}} = \frac12 +
\frac{1-\eta^2}{4\eta}\ln\left|\frac{1+\eta}{1-\eta}\right|.\end{aligned}$$ The freedom in choosing the parameters $\Lambda, f$, and $h$ allows us to fit our structure factor to the Lindhard function. The Lindhard function has the following properties: $$\begin{aligned}
-\frac{\pi^2}{k_F}\chi_{\text{Lind}} = \left\{
\begin{matrix}
\frac13\eta^{-2}, & \eta\to \infty \\
1-\frac13\eta^2, & \eta\to 0 \\
\frac12, & \eta=1 \\
\end{matrix}. \label{threeconditions}
\right.\end{aligned}$$ In order for our function, $\chi_0$, to possess these properties we should choose: $$\begin{aligned}
\Lambda = -\frac1{80}, f = \frac1{20}, \text{ and } h = -
\frac1{36}. \label{coefficients}\end{aligned}$$ A comparison between the resulting structure factor of the proposed theory with the Lindhard function and the structure factor provided by the 1/9-von Weiszacker theory is given in Fig.1.
Application to the ground state problem {#nonhomogeneousstudy}
=======================================
We applied the $\Phi^{(3)}$-theory to a ground state study of a non-perturbative non-homogenous jellium. We chose a spherically symmetric infinite electron system in the following positive jellium background density profile: $$\begin{aligned}
\rho_0(r) &=& \rho_\infty + \Delta\rho(r),\label{density}\\
\Delta\rho(r) &=&
-\zeta\rho_\infty\left(1-\frac{r^2}{3r_0^2}\right)e^{-\frac{r^2}{2r_0^2}},\nonumber\end{aligned}$$ where we took $\rho_\infty=0.01$, $\zeta=0.9$, and $r_0=3\text{ and
}2$ (all in a.u.). The additional non-homogeneous part of jellium density $\Delta\rho$ integrates to zero so we avoid complications connected with an overall non-neutral system. Alternatively this system can be viewed as having constant jellium background density, $\rho_\infty$, but with an external Gaussian potential: $$\begin{aligned}
V_b(r) = r_0^2\zeta\rho_\infty\frac{4\pi}3e^{-\frac{r^2}{2r_0^2}},\end{aligned}$$ which is related to $\Delta\rho(r)$ by the Poisson equation: $$\begin{aligned}
\frac1{r^2}\frac\partial{\partial r}r^2\frac\partial{\partial r} V_b(r) = 4\pi\Delta\rho(r).\end{aligned}$$ The Dirac exchange is used here: $$\begin{aligned}
V_{xc}(\bm R) = \frac{(3\pi^2)^{1/3}}{\pi}\rho(\bm R)^{1/3}.\end{aligned}$$ No correlation energy was employed (its contribution is very small; it will be included in future studies).
 
The simulations were performed by adiabatical turning on the nonhomogeneous part of the jellium positive background density, $\Delta\rho$. Initially the electron and the jellium densities are homogeneous, $\rho_\infty$. The odd-order kinetic tensors, $J_i$ and $\Phi^{(3)}_{ijk}$, are zero and the even order tensors are those of a homogeneous electron liquid $\Phi^{(2)}_{ij}=\rho_\infty
e^{(2)}_{ij}$ and $\Phi^{(4)}_{ijkl}=\rho_\infty e^{(4)}_{ijkl}$ with the $e$-tensors given in Eq.(\[etensors\]).
We then propagate the set of Eqs.(\[setofequations\]) while the jellium density gradually changes from homogenous to the final $\rho_0(r)$; this ensures that the system remains at the ground-state for all times. We implemented the adiabatic density by setting $$\begin{aligned}
\rho_0(R,t) = (1-g(t))\rho_0(R) + g(t)\rho_\infty,\end{aligned}$$ where $g(t)$ is a smooth function rising from 0 to 1; we chose here, quite arbitrarily, $$\begin{aligned}
g(t) = \frac1{1+\exp\left(\left(\frac{t_0-t}{\tau}\right)^3\right)},\end{aligned}$$ and used $$\begin{aligned}
t_0=3\tau.\end{aligned}$$ The width parameter, $\tau$, was typically taken as $50 a.u.$; this value was more than sufficient for adiabatic convergence. $\rho_0(R,t)$ is then used for the definition of the time-dependent potential, Eq.(\[EffectivePotential\]).
The evolution of the system is then determined from the four first equations in (\[setofequations\]), with the $\Phi^{(4)}$ tensor given by Eq.(\[Phiphi4\]) and Eq.(\[additionalterms\]). The 3D equations were discretized and the derivative were evaluated by Fourier-transforms, as was the Coulomb integral. Grid spacings of 1.6 a.u. - 2 a.u. were sufficient to converge when the hole width parameter, $r_0$, was set at 2.0 or 3.0 a.u., respectively. A simple fixed step Runge-Kutta algorithm with dt=0.2 a.u. was used to evolve the equations in time.
We compared the results to Thomas-Fermi, von-Weiszacker, and plane-wave Kohn-Sham simulations. The latter were done by a standard plane wave code; interestingly, we found that the grid spacing needed to converge the Kohn-Sham plane wave simulations had to be smaller by about 20$\%$ than those needed in the HFDFT code, so that they were about 1.3 and 1.6 a.u. for $r_0$=2.0 and 3.0, respectively. The grids contained typically $(20)^3$ points.
Fig.2 shows that HTDFT gives essentially the same density as the Kohn-Sham approach, while the von-Weiszacker and Thomas-Fermi results deviate significantly. Since the two-body interaction is treated the same in all four simulations, this proves that the hydrodynamic approach yields, even for this system which is shifted strongly away from uniformity, the same densities as the essentially exact description of the kinetic energy in the Kohn-Sham approach.
Time-dependent linear response and the collective modes {#dynamicresponse}
=======================================================
In our previous paper [@ours] we studied the ground-state of a homogenous electron gas at the $N=2$ level, with the assumption that $\phi^{(N+1)}$ is zero. Here we extend the studies to $N=3$, with $\phi^{(4)}$, as given by Eqs.(\[additionalterms\]),(\[coefficients\]). We derive the governing formulae in general, and arrive at analytical limits in the long wavelength limit (where $\phi^4$ is not contributing), showing new kinds of excitations.
In the ground state of a homogeneous liquid the non-zero values at the $N=3$ level are the density $\rho_0$ and the second and the fourth order dynamic tensors, $\Phi^{(2)}_{ij}=\rho_0c_2(\rho_0)\delta_{ij}$, $\Phi^{(4)}_{ijkl}=\rho_0c_4(\rho_0)
\overline{\delta_{ij}\delta_{kl}}$. To study the linear response of the system we let all the values in the problem vary harmonically around their stationary values:
$$\begin{aligned}
\rho&=&\rho_0+\left(\underline\rho
e^{-i(\omega t - \bm q\cdot \bm R)}+c.c.\right),\\
J_i &=& \underline{J}_i
e^{-i(\omega t - \bm q\cdot \bm R)}+c.c.,\\
\Phi^{(2)}_{ij} &=& c_2
\rho_0\delta_{ij}+\left(\underline\Phi^{(2)}_{ij}
e^{-i(\omega t - \bm q\cdot \bm R)}+c.c.\right),\\
\Phi^{(3)}_{ijk} &=& \underline\Phi^{(3)}_{ijk}
e^{-i(\omega t - \bm q\cdot \bm R)}+c.c.,\\
\nabla_i\tilde V &=& q_i \tilde v(q) \left(i \underline\rho
e^{-i(\omega t - \bm q\cdot \bm R)}+c.c.\right),\end{aligned}$$
After linearizing Eqs.(\[setofequations\]) one gets:
\[linearizedeqs\] $$\begin{aligned}
\omega\underline{\rho} &=&q_\alpha\underline J_\alpha,\\
\omega \underline J_{i}&=&q_\alpha\underline\Phi^{(2)}_{i\alpha}+\rho_0 q_i \tilde v(q) \underline\rho,\\
\omega\underline\Phi^{(2)}_{ij}&=&q_\alpha\underline\Phi^{(3)}_{ij\alpha},\\
\omega\underline\Phi^{(3)}_{ijk}&=&
3\overline{(c_2\delta_{ij}+fq_iq_j)(q_\alpha\underline
\Phi^{(2)}_{k\alpha})} + 3(c_2 + f q^2)\overline{q_i
\underline\Phi^{(2)}_{jk}}\nonumber\\&& + \left( 3\left(D + c_2
\rho_0\tilde v + h c_2 q^2\right)\overline{\delta_{ij} q_k}
+\left(\frac14\rho_0\tilde v + \Lambda q^2 + 3 h c_2\right)q_iq_jq_k
\right) \underline\rho,\end{aligned}$$
where the linearized variation of $\Phi^{(4)}$ is taken from Eq.(\[Phi4linearizedvariation\]), and $\alpha$ is again summed over. This is a system of linear homogeneous equations, and to find its solutions we have to diagonalize it.
All the variables in Eqs.(\[linearizedeqs\]) could be expressed in terms of $\underline\rho$ and $\underline\Phi^{(2)}_{ij}$. Therefore, we can consider the equations on $\underline\Phi^{(2)}$ and $\underline{\rho}$ only without losing any solutions. In matrix form these equations read:
\[linearequations\] $$\begin{aligned}
\frac{\omega^2}{q^2} \underline{\rho} &=& \text{Tr}\left(\underline{\bm\Phi}^{(2)} \bm Q \right)+ \rho_0 \tilde v(q) \underline{\rho},\\
\frac{\omega^2}{q^2}\underline{\bm\Phi}^{(2)}&=&(c_2+fq^2)\left(\underline{\bm\Phi}^{(2)}+ 2\left\{\underline{\bm \Phi}^{(2)},\bm
Q\right\}\right) + (c_2\bm I+fq^2\bm Q) \text{Tr}\left(\underline{\bm\Phi}^{(2)} \bm Q \right) \nonumber
\\&& +\left(\bm I Z(q) + \bm Q Z'(q) \right)\underline{\rho},\end{aligned}$$
where $$\begin{aligned}
Z(q) & = & D + c_2\rho_0\tilde v + h c_2 q^2,\\
Z'(q)& = &2 Z + \frac14\rho_0\tilde v(q) q^2 + \Lambda q^4 +
3hc_2q^2,\end{aligned}$$ and $\text{Tr}$ means matrix trace, $I_{ij}\equiv\delta_{ij}$ is the $3\times 3$ unity matrix, $Q_{ij}=q_iq_j/q^2$, curly brackets denote an anticommutator, and capital bold face letters refer here to matrices. Without loss of generality we can always assume that the wave-vector $\bm q$ is directed along the x-axis ($\bm q
=(q,0,0)^T$), so that $$\begin{aligned}
Q= \begin{pmatrix}
1 & 0 & 0\\
0 & 0 & 0\\
0 & 0 & 0
\end{pmatrix}.\end{aligned}$$ There are several solutions for these equations. The first three solutions are decoupled from the density fluctuations so that $\underline \rho = 0$ for all of them. They are:
$$\begin{aligned}
\underline{\Phi}^{(2)}= \begin{pmatrix}
0 & 1 & 0\\
1 & 0 & 0\\
0 & 0 & 0
\end{pmatrix},
\begin{pmatrix}
0 & 0 & 1\\
0 & 0 & 0\\
1 & 0 & 0
\end{pmatrix},\label{TransverseSound}\end{aligned}$$
which corresponds to the dispersion relation $$\begin{aligned}
\omega^2 = 3/5 k_F^2 q^2, \label{disrelation1}\end{aligned}$$ and
\[thirdsolution\] $$\begin{aligned}
\underline{\Phi}^{(2)}= \begin{pmatrix}
0 & 0 & 0\\
0 & 0 & 1\\
0 & 1 & 0
\end{pmatrix},\end{aligned}$$ with dispersion $$\begin{aligned}
\omega^2=1/5k_F^2q^2.\label{disrelation2}\end{aligned}$$
The first two solutions, Eq.(\[TransverseSound\]), correspond to transverse sound as the current is given as: $$\begin{aligned}
\omega(q)\underline{J_i}=q_j\underline\Phi^{(2)}_{ij} =\begin{pmatrix} 0 \\ q \\
0 \end{pmatrix}, \begin{pmatrix} 0 \\ 0 \\ q \end{pmatrix}.\end{aligned}$$ Note, however, that the velocity of this transverse sound mode is different from the one found for the same mode within the $N=2$ theory [@ours]. The third solution in Eq.(\[thirdsolution\]) is a new sound mode. This mode involves neither density nor current fluctuations and corresponds to transverse quadrupole fluctuations of the Fermi sea.
![\[Figure3\] The elementary excitation spectrum provided by Quantum Hydrodynamics (QH), HTDFT $\Phi^{(2)}$ and HTDFT $\Phi^{(3)}$ theories. QH gives only a plasmon mode. HTDFT also gives transverse sound modes which mimic the RPA elementary excitations in Fermi liquid. $\Phi^{(3)}$ HTDFT gives additional sound modes with respect to $\Phi^{(2)}$ HTDFT confirming the conjecture made in Ref.[@ours] that when increasing the order of HTDFT new sound modes should appear, and they will gradually cover the entire continuous RPA density of states of Fermi liquid.](Figure3.EPS)
The next three solutions are found by representing the tensor $\Phi^{(2)}$ in terms of the remaining diagonal tensors ($I$ and $Q$): $$\begin{aligned}
\underline{\bm \Phi}^{(2)} = \alpha \bm Q +\beta (\bm I - \bm Q),\end{aligned}$$ which leads, upon insertion into Eqs.(\[linearequations\]), to the following equations for $\alpha,\beta,\underline{\rho}$:
$$\begin{aligned}
\frac{\omega^2}{q^2} \alpha &=& 6 \left( c_2 + f q^2 \right)\alpha + \left(Z(q) + Z'(q))\right)\underline\rho .\\
\frac{\omega^2}{q^2} \beta &=& c_2 \alpha +\left( c_2 + f q^2 \right)\beta + Z(q)\underline\rho , \\
\frac{\omega^2}{q^2}\underline{\rho} &=& \alpha + \rho_0 \tilde v(q)
\underline\rho .\end{aligned}$$
This set of equations has complicated solutions, which, however, could be simplified in low-wavelength limit. In this limit, we can leave only the leading terms in $q$; in the effective potential it is the divergent Fourier components of the Coulomb potential. In the long-wavelength limit the system of equations has the following form: $$\begin{aligned}
\begin{pmatrix}
6 & 0 & 3\\
1 & 1 & 1\\
a(q) & 0 & a(q)
\end{pmatrix}
\begin{pmatrix}
\alpha\\
\beta\\
\underline{\rho}'
\end{pmatrix}=
\omega'^2
\begin{pmatrix}
\alpha\\
\beta\\
\underline{\rho}'
\end{pmatrix},\end{aligned}$$ where $a(q)=4\pi \rho_0 /(q^2 c_2)$, $\omega'^2=\omega^2/(q^2c_2)$, $\underline{\rho}'=4\pi\rho_0\underline{\rho}/q^2$, and $a(q)\gg1$. Dropping the terms of order $a(q)^{-1}$ and smaller, the three eigenvalues and corresponding eigenvectors are:
$$\begin{aligned}
\omega^2&=&\frac15k_F^2 q^2,\quad (\alpha, \beta, \underline{\rho}') = ( 1, 2/3 , -1);\label{fourthmode}\\
\omega^2&=&\frac35k_F^2 q^2,\quad (\alpha, \beta, \underline{\rho}') = ( 1, 0 , -1);\label{fifthmode}\\
\omega^2&=&\omega_P^2+\frac35k_F^2 q^2,\quad (\alpha, \beta,
\underline{\rho}') = (0, 0, 1),\end{aligned}$$
where $\omega_P^2=4\pi\rho_0$ is the plasmon frequency. Note that the first two of the three modes (\[fourthmode\], \[fifthmode\]) have the same eigenvalues as the transverse modes in Eqs.(\[disrelation1\]) and (\[disrelation2\]).
The total spectrum given by $N=3$ HTDFT for elementary excitations in the homogeneous electron gas is given in Fig.(\[Figure3\]). The spectrum found differs from that of the $N=2$ approach by an additional sound mode with velocity $\sqrt{3/5} k_F$ and by shifting the previous sound modes from $\sqrt{3/5}k_F$ to $\sqrt{1/5}k_F$. This result confirms the conjecture made in Ref.[@ours], that with increasing $N$ new sound modes should appear, and that they will gradually cover the entire continuous RPA density of states in the Fermi liquid.
Conclusions
===========
In conclusions, we have shown that HTDFT can also be used for time-independent studies. We have supplanted our previous conjecture where we assumed that the terms in the equation of motion hierarchy should be terminated with the next relevant cumulant (*i.e.*, $\phi^{(N+1)}$) being zero; instead, we now derived $\phi^{(N+1)}$ from fitting the linear response to a HEG. The resulting set of equations (given at the $N=3$ level by Eqs. (\[additionalterms\]),(\[additionalterms\]),(\[setofequations\]), and (\[Phiphi4\]) ) is closed and can be propagated forward in time.
The linear response in the static limit is fitted to the Lindhard function HEG for both short, intermediate and long wavelengths (for comparison, the 1/9 in the von-Weiszacker approach is obtained to fit long wavelengths, while a fit to long wavelengths would have required replacing the 1/9 by 1 in the von-Weiszacker theory). We have then applied HTDFT away from equilibrium, for a case of a jellium density with a deep hole in the middle, and have shown excellent agreement with the Kohn-Sham results, in a case where more approximate theories such as Thomas Fermi and von-Weiszacker fail; this is directly due to the fact that their structure factor do not follow the Lindhard function except at low wavelengths.
The last part of the paper dealt with time-dependent linear response studies at the present level, $N=3$. The analytical studies have confirmed our previous assertion, that as the level of the theory increases, more and more transverse excitations are found. A new excitation at the N=3 level couples neither to the current nor to the density. All excitations, including the new ones, lie within the RPA density of stats of elementary excitations in a Fermi liquid.
Future work will study the applicability of the approach to covalent chemical systems, where the directionality of the tensors should enable a correct description even at a low N, possibly as low as $N=3$. Further, dynamic susceptibilities will be studied so that further terms, depending on $J$, $\Phi^{(3)}$, *etc.*, will be included in the terminating cumulant ($\phi^4$ here, $\phi^{(N+1)}$ in general) so that the theory will be valid over a wide range of frequencies and wavevectors. In addition, effects of magnetic fields are straightforward to incorporate. The basic formalism developed here and in forthcoming work will then enable the application to dynamical problems which straddle the transition between molecular and nanostructures.
We note that other applications to fermionic systems can also be envisioned. For example, by replacing the zeroth-order HEG density matrix with a temperature-dependent density matrix, and fitting the coefficients of the derivative terms in the cumulant to a temperature dependent Lindblad expression, we will get a temperature-dependent HTDFT theory which can be applied to plasmas and to studies of narrow conduction bands. Similarly, applications to nuclear systems can also be envisioned.
Other future improvements will include better methods to solve the time-dependent HTDFT equations. One approach will be to include external electric fields that will have dipole and quadruple (or higher) components that will be time-dependent. The electric fields will be chosen, at each time-instant, to remove energy from the system (*i.e.*, to reduce the trace of $\Phi^{(2)}$ plus the total potential).
Acknowledgements
================
We thank Roi Baer for his help in performing and analyzing the numerical and linear response studies. This work was supported by the NSF and PRF. We are thankful for Emily Carter and the Referee for helpful comments.
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|
---
abstract: |
The modal $\mu$-calculus, introduced by Dexter Kozen, is an extension of modal logic with fixpoint operators. Its axiomatization, $\mathsf{Koz}$, was introduced at the same time and is an extension of the minimal modal logic $\mathsf{K}$ with the so-called Park fixpoint induction principle. It took more than a decade for the completeness of $\mathsf{Koz}$ to be proven, finally achieved by Igor Walukiewicz. However, his proof is fairly involved.
In this article, we present an improved proof for the completeness of $\mathsf{Koz}$ which, although similar to the original, is simpler and easier to understand.
[**Keywords:**]{} The modal $\mu$-calculus, completeness, parity games, parity automata.
author:
- |
Kuniaki Tamura\
15-9-103, Takasago 3-chome, Katsushika, Tokyo 125-0054, Japan\
E-mail: `kuniaki.tamura@gmail.com`
bibliography:
- 'ref.bib'
title: ' **Completeness of Kozen’s Axiomatization for the Modal $\mu$-Calculus: A Simple Proof** '
---
Introduction {#sec: introduction}
============
The *modal $\mu$-calculus* originated with Scott and De Bakker [@bak-sco69] and was further developed by Dexter Kozen [@DBLP:journals/tcs/Kozen83] into the main version currently used. It is used to describe and verify properties of labeled transition systems (Kripke models). Many modal and temporal logics can be encoded into the modal $\mu$-calculus, including $\mathsf{CTL}^{\ast}$ and its widely used fragments – the linear temporal logic $\mathsf{LTL}$ and the computational tree logic $\mathsf{CTL}$. The modal $\mu$-calculus also provides one of the strongest examples of the connections between modal and temporal logics, automata theory and game theory (for example, see [@DBLP:conf/dagstuhl/2001automata]). As such, the modal $\mu$-calculus is a very active research area in both theoretical and practical computer science. We refer the reader to Bradfield and Stirling’s tutorial article [@Bradfield07modalmu-calculi] for a thorough introduction to this formal system.
The difference between the modal $\mu$-calculus and modal logic is that the former has the *least fixpoint operator* $\mu$ and the *greatest fixpoint operator* $\nu$ which represent the least and greatest fixpoint solution to the equation $\alpha(x) = x$, where $\alpha(x)$ is a monotonic function mapping some power set of possible worlds into itself.[^1] In Kozen’s initial work [@DBLP:journals/tcs/Kozen83], he proposed an axiomatization $\mathsf{Koz}$, which was an extension of the minimal modal logic $\mathsf{K}$ with a further axiom and inference rule – the so-called Park fixpoint induction principle: $$\infer[(\textsf{Prefix})]
{\alpha(\mu x.\alpha(x)) \vdash \mu x.\alpha(x)}
{}
\qquad
\infer[(\textsf{Ind})]
{\mu x.\alpha(x) \vdash \beta}
{\alpha(\beta) \vdash \beta}$$ The system $\mathsf{Koz}$ is very simple and natural; nevertheless, Kozen himself could not prove completeness for the full language, but only for the negations of formulas of a special kind called the *aconjunctive formula*. Completeness for the full language turned out to be a knotty problem and remained open for more than a decade. Finally, Walukiewicz [@Walukiewicz2000142] solved this problem, but his proof is quite involved.[^2]
The aim of this article is to provide an improved proof that is easier to understand. First, we outline Walukiewicz’s proof and explain its difficulties, and then present our improvement.
The completeness theorem considered here is sometimes called weak completeness and requires that the validity follows the provability; that is:
1. For any formula $\varphi$, if $\varphi$ is not satisfiable, then $\sim\!\varphi$ is provable in $\mathsf{Koz}$.
Here, $\sim\!\varphi$ denotes the negation of $\varphi$. Note that strong completeness cannot be applied to the modal $\mu$-calculus since it lacks compactness. The first step of the proof is based on the results of Janin and Walukiewicz [@conf/mfcs/JaninW95], in which they introduced the class of formulas called *automaton normal form*,[^3] and showed the following two theorems:
1. For any formula $\varphi$, we can construct an automaton normal form $\mathsf{anf}(\varphi)$ which is semantically equivalent to $\varphi$.
2. For any automaton normal form $\widehat{\varphi}$, if $\widehat{\varphi}$ is not satisfiable, then $\sim\!\widehat{\varphi}$ is provable in $\mathsf{Koz}$; that is, $\mathsf{Koz}$ is complete for the negations of the automaton normal form.
The above theorems lead to the following Claim (d) for proving:
1. For any formula $\varphi$, there exists a semantically equivalent automaton normal form $\widehat{\varphi}$ such that $\varphi\rightarrow\widehat{\varphi}$ is provable in $\mathsf{Koz}$.
Indeed, for any unsatisfiable formula $\varphi$, Claim (d) tells us that $\sim\!\widehat{\varphi}\rightarrow\sim\!\varphi$ is provable; on the other hand, from Theorem (c) we obtain that $\sim\!\widehat{\varphi}$ is provable; therefore $\sim\!\varphi$ is provable in $\mathsf{Koz}$ as required. Hence, our target (a) is reduced to Claim (d).
Another important tool is the concept of a *tableau*, which is a tree structure that is labeled by some subformulas of the primary formula $\varphi$ and is related to the satisfiability problem for $\varphi$. Niwinski and Walukiewicz [@Niwinski199699] introduced a game played by two adversaries on a tableau (called *tableau games* in this article) and, by analyzing these games, showed that:
1. For any unsatisfiable formula $\varphi$, there exists a structure called the *refutation* for $\varphi$ which is a substructure of tableau.
Importantly, a refutation for $\varphi$ is very similar to a proof diagram for $\varphi$; roughly speaking, the difference between them is that the former can have infinite branches while the latter can not. Walukiewicz shows that if the refutation for $\varphi$ satisfies a special *thin* condition, it can be transformed into a proof diagram for $\varphi$. In other words,
1. For any unsatisfiable formula $\varphi$ such that there exists a thin refutation for $\varphi$, $\sim\!\varphi$ is provable in $\mathsf{Koz}$.
Note that Claim (f) is a slight generalization of the completeness for the negations of the aconjunctive formula in the sense that the refutation for an unsatisfiable aconjunctive formula is always thin, and Claim (f) can be shown by the same method as Kozen’s original argument.
The proof is based on confirming Claim (d) by induction on the length of $\varphi$, using (b) and (f). The hardest step of induction is the case $\varphi = \mu x.\alpha(x)$. Suppose $\varphi = \mu x.\alpha(x)$ and that we could assume, by inductive hypothesis, $\alpha(x)\rightarrow\widehat{\alpha}(x)$ is provable in $\mathsf{Koz}$ where $\widehat{\alpha}(x)$ is an automaton normal form equivalent to $\alpha(x)$. For the inductive step, we want to discover an automaton normal form $\widehat{\varphi}$ equivalent to $\mu x.\alpha(x)$ such that $\mu x.\alpha(x)\rightarrow\widehat{\varphi}$ is provable. Note that since $\alpha(x)\rightarrow\widehat{\alpha}(x)$ is provable, $\mu x.\alpha(x)\rightarrow\mu x.\widehat{\alpha}(x)$ is also provable. Furthermore, $\mu x.\alpha(x)$ and $\mu x.\widehat{\alpha}(x)$ are equivalent to each other. Set $\widehat{\varphi} := \mathsf{anf}(\mu x.\widehat{\alpha}(x))$. Then, it is sufficient to show that $\mu x.\widehat{\alpha}(x)\rightarrow\widehat{\varphi}$ is provable, and thus, from the induction rule $(\mathsf{Ind})$, $\widehat{\alpha}(\widehat{\varphi})\rightarrow\widehat{\varphi}$ is provable. To show this, Walukiewicz developed a new utility called *tableau consequence*, which is a binary relation on the tableau and is characterized using game theoretical notations. The following two facts were then shown:
1. Let $\widehat{\alpha}(x)$ and $\widehat{\varphi}$ be formulas denoted above. Then the tableau for $\widehat{\varphi}$ is a consequence of the tableau for $\widehat{\alpha}(\widehat{\varphi})$.
2. For any automaton normal forms $\widehat{\beta}(y)$ and $\widehat{\psi}$, if the tableau for $\widehat{\psi}$ is a consequence of the tableau for $\widehat{\beta}(\widehat{\psi})$, then we can construct a thin refutation for $\sim\!(\widehat{\beta}(\widehat{\psi})\rightarrow\widehat{\psi})$.[^4]
The real difficulty appeared when proving Claim (g). To establish this claim, Walukiewicz introduced complicated functions across some tableaux and analyzed the properties of these functions very carefully. Finally, Claims (f), (g) and (h) together immediately establish that $\widehat{\alpha}(\widehat{\varphi})\rightarrow\widehat{\varphi}$ is provable in $\mathsf{Koz}$. Thus, he obtained a proof for Claim (d), confirming completeness.
This article’s main contribution is the simplification of the proof of Claim (g). For this purpose, we will introduce a new tableau-like structure called a *wide tableau* and provide a more suitable re-formulation of the concept of tableau consequence to prove Claim (g). This re-formulation will be defined similarly to the concept of *bisimulation* (instead of the game theoretical notations), which is one of the most fundamental and standard notions in the model theory of modal and its extensional logics. Consequently, although our proof of completeness does not include any innovative concepts, it is far more concise than the original proof.
The author hopes that the method given in this article may assist investigation of the modal $\mu$-calculus and related topics.
Outline of the article {#subsec: outline of the article}
----------------------
The remainder of this article is organized as follows: in the following subsection \[subsec: notation\], we will define some terminologies used within the article. Section \[sec: the modal mu-calculus\] gives basic definitions of the syntax and semantics of the modal $\mu$-calculus. Section \[sec: automata\] and \[sec: games\] introduce well known results concerning parity automata and parity games, respectively. Section \[sec: tableaux\] contains the principle part of this article – the proof of Claim (g). For this proof, Claim (b) and the techniques used for proving (b) are fundamental. Therefore, we recount the argument of Janin and Walukiewicz [@conf/mfcs/JaninW95] in detail. In Section 6, we prove the completeness of $\mathsf{Koz}$ by showing Claim (d).
Notation {#subsec: notation}
--------
Sets:
: Let $X$ be an arbitrary set. The *cardinality of $X$* is denoted $|X|$. The *power set of $X$* is denoted $\mathcal{P}(X)$. $\omega$ denotes the set of natural numbers.
Sequences:
: A finite sequence over some set $X$ is a function $\pi: \{1, \dots, n\} \rightarrow X$ where $1 \leq n$. An infinite sequence over $X$ is a function $\pi: \omega\setminus \{0\} \rightarrow X$. Here, a sequence can refer to either a finite or infinite sequence. The length of a sequence $\pi$ is denoted $|\pi|$. Let $\pi$ be a sequence over $X$. The set of $x \in X$ which appears infinitely often in $\pi$ is denoted $\mathsf{Inf}(\pi)$. We denote the $n$-th element in $\pi$ by $\pi[n]$ and the fragment of $\pi$ from the $n$-th element to the $m$-th element by $\pi[n, m]$. For example, if $\pi = \mathsf{aabbcddd}$, then $\pi[5] = \mathsf{c}$ and $\pi[2, 6] = \mathsf{abbcd}$. Note that when $\pi$ is a finite non-empty sequence, $\pi[|\pi|]$ denotes the tail of $\pi$.
Alphabets:
: Suppose that $\Sigma$ is a non-empty finite set. Then we may call $\Sigma$ an *alphabet* and its element $v \in \Sigma$ a *letter*. We denote the set of finite sequences over $\Sigma$ by $\Sigma^{\ast}$, the set of non-empty finite sequences over $\Sigma$ by $\Sigma^{+}$, and the set of infinite sequences over $\Sigma$ by $\Sigma^{\omega}$. As usual, we call an element of $\Sigma^{\ast}$ a *word*, an element of $\Sigma^{\omega}$ an *$\omega$-word*, a set of finite words $\mathcal{L} \subseteq \Sigma^{\ast}$ a *language* and, a set of $\omega$-words $\mathcal{L'} \subseteq \Sigma^{\omega}$ an *$\omega$-language*. The notion of the *factor* on words is defined as usual: for two words $u, v \in \Sigma^{\ast} \cup \Sigma^{\omega}$, $u$ is a factor of $v$ if $v = xuy$ for some $x, y \in \Sigma^{\ast} \cup \Sigma^{\omega}$.
Graphs:
: In this article, the term *graph* refers to a directed graph. That is, a graph is a pair $\mathcal{G} = (V, E)$ where $V$ is an arbitrary set of *vertices* and $E$ is an arbitrary binary relation over $V$, i.e., $E \subseteq V \times V$. A vertex $u$ is said to be an $E$-successor (or simply a successor) of a vertex $v$ in $\mathcal{G}$ if $(v, u) \in E$. For any vertex $v$, we denote the set of all $E$-successors of $v$ by $E(v)$. The sequence $\pi \in V^{\ast} \cup V^{\omega}$ is called an $E$-*sequence* if $\pi[n+1] \in E(\pi[n])$ for any $n < |\pi|$. $E^{\ast}$ denotes the reflexive transitive closure of $E$ and $E^{+}$ denotes the transitive closure of $E$.
Trees:
: The term *tree* is used to mean a *rooted direct tree*. More precisely, a tree is a triple $\mathcal{T} = (T, C, r)$ where $T$ is a set of *nodes*, $r \in T$ is a *root* of the tree and, $C$ is a *child relation*, i.e., $C \subseteq T \times T$ such that for any $t \in T \setminus \{r\}$, there is exactly one $C$-sequence starting at $r$ and ending at $t$. As usual, we say that $u$ is a child of $t$ (or $t$ is a parent of $u$) if $(t, u) \in C$. A node $t \in T$ is a *leaf* if $C(t) = \emptyset$. A *branch* of $\mathcal{T}$ is either a finite $C$-sequence starting at $r$ and ending at a leaf or an infinite $C$-sequence starting at $r$.
Unwinding:
: Let $\mathcal{G} = (V, E)$ be a graph. An *unwinding* of $\mathcal{G}$ on $v \in V$ is the tree structure $\mathsf{UNW}_{v}(\mathcal{G}) = (T, C, r)$ where:
- $T$ consists of all finite non-empty $E$-sequences that start at $v$,
- $(\pi, \pi') \in C$ if and only if; $|\pi|+1 = |\pi'|$, $\pi = \pi'[1, |\pi|]$ and $(\pi[|\pi|], \pi'[|\pi'|]) \in E$, and
- $r := v$.
This concept can be extended naturally into a graph with some additional relations or functions. For example, let $\mathcal{S} = (V, E, f)$ be a structure where $\mathcal{G} = (V, E)$ is a graph and $f$ is a function with domain $V$. Then we define $\mathsf{UNW}_{v}(\mathcal{S}) := (\mathsf{UNW}_{v}(\mathcal{G}), f')$ as $f'(\pi) := f(\pi[|\pi|])$ for any $\pi \in V^{+}$. Note that we use the same symbol $f$ instead of $f'$ in $\mathsf{UNW}_{v}(\mathcal{S})$ if there is no danger of confusion.
Functions:
: Let $f$ be a function from some set $X$ to some set $Y$. We define the new function $\vec{f}$ from $X^{+} \cup X^{\omega}$ to $Y^{+} \cup Y^{\omega}$ as: $$\vec{f}(\pi) := f(\pi[1])f(\pi[2])\cdots$$ where $\pi \in X^{+} \cup X^{\omega}$. It is obvious that for any $\pi \in X^{+} \cup X^{\omega}$, we have $|\pi| = |\vec{f}(\pi)|$.
The modal $\mu$-calculus {#sec: the modal mu-calculus}
========================
We will now introduce the syntax, semantics and axiomatization $\mathsf{Koz}$ of the modal $\mu$-calculus, and then present some additional concepts and results for use in the following sections.
Syntax {#subsec: syntax}
------
\[def: formula\]Let $\mathsf{Prop} = \{ p, q, r, x, y, z, \dots \}$ be an infinite countable set of *propositional variables*. Then the collection of the *modal $\mu$-formulas* is defined as follows: $$\varphi ::= (\top), (\bot), (p) \mid (\neg p) \mid (\varphi \vee \varphi) \mid
(\varphi \wedge \varphi) \mid (\Diamond \varphi) \mid (\square \varphi) \mid
(\mu x.\varphi) \mid (\nu x.\varphi)$$ where $p, x \in \textsf{Prop}$. Moreover, for formulas of the form $(\sigma x.\varphi)$ with $\sigma \in \{ \mu, \nu \}$, we require that each occurrence of $x$ in $\varphi$ is positive; that is, $\neg x$ is not a subformula of $\varphi$. Henceforth in this article, we will use $\sigma$ to denote $\mu$ or $\nu$. A formula of the form $p$ or $\neg p$ for $p \in \mathsf{Prop}$, $\top$ and $\bot$ is called *literal*. We use the term $\mathsf{Lit}$ to refer to the set of all literals, i.e., $\mathsf{Lit} := \{ p, \neg p, \bot, \top \mid p \in \mathsf{Prop} \}$. We call $\mu$ and $\nu$ *the least fixpoint operator* and *the greatest fixpoint operator*, respectively.
In Definition \[def: formula\], we confined the formula to a *negation normal form*; that is, the negation symbol may only be applied to propositional variables. However, this restriction can be inconvenient, and so we extend the concept of the negation to an arbitrary formula $\varphi$ (denoted by $\sim\! \varphi$) inductively as follows:
- $\sim\! \top := \bot$, $\sim\! \bot := \top$.
- $\sim\! p := \neg p$, $\sim\! \neg p := p$ for $p \in \mathsf{Prop}$.
- $\sim\! (\varphi \vee \psi) := ((\sim\! \varphi)\wedge(\sim\! \psi))$, $\sim\! (\varphi \wedge \psi) := ((\sim\! \varphi) \vee (\sim\! \psi))$.
- $\sim\! (\Diamond \varphi) := (\square (\sim\! \varphi))$, $\sim\! (\square \varphi) := (\Diamond (\sim\! \varphi))$.
- $\sim\! (\mu x.\varphi(x)) := (\nu x.(\sim\! \varphi(\neg x)))$, $\sim\! (\nu x.\varphi(x)) := (\mu x.(\sim\! \varphi(\neg x)))$.
We introduce *implication* $(\varphi \rightarrow \psi)$ as $((\sim\! \varphi) \vee \psi)$ and *equivalence* $(\varphi \leftrightarrow \psi)$ as $((\varphi \rightarrow \psi) \wedge (\psi \rightarrow \varphi))$ as per the usual notation. To minimize the use of parentheses, we assume the following precedence of operators from highest to lowest: $\neg$, $\sim$, $\Diamond$, $\square$, $\sigma x$, $\vee$, $\wedge$, $\rightarrow$ and $\leftrightarrow$. Moreover, we often abbreviate the outermost parentheses. For example, we write $\Diamond p \rightarrow q$ for $((\Diamond p) \rightarrow q)$ but not for $(\Diamond (p \rightarrow q))$.
As fixpoint operators $\mu$ and $\nu$ can be viewed as quantifiers, we use the standard terminology and notations for quantifiers. We denote the set of all propositional variables appearing free in $\varphi$ by $\mathsf{Free}(\varphi)$, and those appearing bound by $\mathsf{Bound}(\varphi)$. If $\psi$ is a subformula of $\varphi$, we write $\psi \leq \varphi$. We write $\psi < \varphi$ when $\psi$ is a proper subformula. $\mathsf{Sub}(\varphi)$ is the set of all subformulas of $\varphi$ and $\mathsf{Lit}(\varphi)$ denotes the set of all literals which are subformulas of $\varphi$. Let $\varphi(x)$ and $\psi$ be two formulas. The *substitution* of all free appearances of $x$ with $\psi$ into $\varphi$ is denoted $\varphi(x)[x/\psi]$ or sometimes simply $\varphi(\psi)$. As with predicate logic, we prohibit substitution when a new binding relation will occur by that substitution.
The following two definitions regarding formulas will be used frequently in the remainder of the article.
\[def: well-named formula\]The set of *well-named formulas* $\mathsf{WNF}$ is defined inductively as follows:
1. $\mathsf{Lit} \subseteq \mathsf{WNF}$.
2. Let $\alpha, \beta \in \mathsf{WNF}$ where $\mathsf{Bound}(\alpha) \cap \mathsf{Free}(\beta) = \emptyset$ and $\mathsf{Free}(\alpha) \cap \mathsf{Bound}(\beta) = \emptyset$. Then $\alpha \vee \beta, \alpha \wedge \beta \in \mathsf{WNF}$.
3. Let $\alpha \in \mathsf{WNF}$. Then $\Diamond \alpha, \square \alpha \in \mathsf{WNF}$.
4. Let $\alpha(x) \in \mathsf{WNF}$ where $x \in \mathsf{Free}(\alpha(x))$ occurs only positively, moreover, $x$ is in the scope of some modal operators. Then $\sigma x_{1}.\dots \sigma x_{k}.\alpha(x_{1}, \dots, x_{k}) \in \mathsf{WNF}$ where $\alpha(x) = \alpha(x_{1}, \dots, x_{k})[x_{1}/x, \dots, x_{k}/x]$, $x \notin \mathsf{Sub}(\alpha(x_{1}, \dots, x_{k}))$ and $x_{1}, \dots, x_{k} \notin \mathsf{Sub}(\alpha(x))$.
The formula $\sigma x_{1}.\dots \sigma x_{k}.\alpha(x_{1}, \dots, x_{k})$ which is mentioned above clause $4$ is sometimes abbreviated $\sigma \vec{x}.\alpha(\vec{x})$. If $\varphi$ is well-named and $x$ is bounded in $\varphi$, then there is exactly one subformula which binds $x$; this formula is denoted $\sigma_{x}x.\varphi_{x}(x)$.
\[def: alternation depth\]Given a formula $\varphi$,
1. Let $\preceq^{-}_{\varphi}$ be a binary relation on $\mathsf{Bound}(\varphi)$ such that $x \preceq^{-}_{\varphi} y$ if and only if $x \in \mathsf{Free}(\varphi_{y}(y))$. The *dependency order* $\preceq_{\varphi}$ is defined as the transitive closure of $\preceq^{-}_{\varphi}$.
2. A sequence $\langle x_{1}, x_{2}, \dots, x_{K}\rangle \in \mathsf{Bound}(\varphi)^{+}$ is said to be an alternating chain if: $$x_{1}\preceq^{-}_{\varphi}x_{2}\preceq^{-}_{\varphi}\dots\preceq^{-}_{\varphi}x_{K}$$ and $\sigma_{x_k} \neq \sigma_{x_{k+1}}$ for every $k \in \omega$ such that $1 \leq k \leq K-1$. The *alternation depth* of $\varphi$ (denoted $\mathsf{alt}(\varphi)$) is the maximal length of alternating chains in $\varphi$. That is, the alternation depth of $\varphi$ is the maximal number of alternations between $\mu$- and $\nu$-operators in $\varphi$.
For a formula $\varphi = \mu x.\nu y.(\Diamond x \vee (\mu z.(\Diamond z \wedge \square y)))$, we have $\mathsf{alt}(\varphi) = 3$ since $x\preceq^{-}_{\varphi}y\preceq^{-}_{\varphi}z$ with $\sigma_{x} \neq \sigma_{y}$ and $\sigma_{y} \neq \sigma_{z}$. Note that although $x \notin \mathsf{Free}(\varphi_{z}(z))$, we have $x\preceq_{\varphi}z$.
Semantics {#subsec: semantics}
---------
A *Kripke model* for the modal $\mu$-calculus is a structure $\mathcal{S}=(S, R, \lambda)$ such that:
- $S = \{ s, t, u, \dots \}$ is a non-empty set of *possible worlds*.
- $R$ is a binary relation over $S$ called the *accessibility relation*.
- $\lambda:\mathsf{Prop}\rightarrow \mathcal{P}(S)$ is a *valuation*.
Let $\mathcal{S} = (S, R, \lambda)$ be a Kripke model and let $x$ be a propositional variable. Then for any set of possible worlds $T \in \mathcal{P}(S)$, we can define a new valuation $\lambda[x \mapsto T]$ on $S$ as follows: $$\begin{aligned}
\lambda[x \mapsto T](p):=
\left\{\begin{array}{ll}
T&\text{if $p = x$,}\\
\lambda(p)&\text{otherwise.}\\
\end{array}\right.\end{aligned}$$ Moreover, $\mathcal{S}[x \mapsto T]$ denotes the Kripke model $(S, R, \lambda[x \mapsto T])$. A *denotation* ${\ensuremath{[\![\varphi]\!]}}_{\mathcal{S}} \in \mathcal{P}(S)$ of a formula $\varphi$ on $\mathcal{S}$ is defined inductively on the structure of $\varphi$ as follows:
- ${\ensuremath{[\![\bot]\!]}}_{\mathcal{S}} := \emptyset$ and ${\ensuremath{[\![\top]\!]}}_{\mathcal{S}} := S$.
- ${\ensuremath{[\![p]\!]}}_{\mathcal{S}} := \lambda(p)$ and ${\ensuremath{[\![\neg p]\!]}}_{\mathcal{S}} := S \setminus \lambda(p)$ for any $p \in \mathsf{Prop}$.
- ${\ensuremath{[\![\varphi \vee \psi]\!]}}_{\mathcal{S}} :=
{\ensuremath{[\![\varphi]\!]}}_{\mathcal{S}} \cup {\ensuremath{[\![\psi]\!]}}_{\mathcal{S}}$ and ${\ensuremath{[\![\varphi \wedge \psi]\!]}}_{\mathcal{S}} :=
{\ensuremath{[\![\varphi]\!]}}_{\mathcal{S}} \cap {\ensuremath{[\![\psi]\!]}}_{\mathcal{S}}.$
- ${\ensuremath{[\![\Diamond \varphi]\!]}}_{\mathcal{S}} :=
\{ s \mid \exists t \in S, (s, t) \in R \wedge t \in {\ensuremath{[\![\varphi]\!]}}_{\mathcal{S}}\}.$
- ${\ensuremath{[\![\square \varphi]\!]}}_{\mathcal{S}} := \{ s \mid \forall t \in S, (s, t) \in R
\Longrightarrow t \in {\ensuremath{[\![\varphi]\!]}}_{\mathcal{S}}\}.$
- ${\ensuremath{[\![\mu x.\varphi(x)]\!]}}_{\mathcal{S}} := \bigcap\{T \in \mathcal{P}(S) \mid
{\ensuremath{[\![\varphi(x)]\!]}}_{\mathcal{S}[x \mapsto T]} \subseteq T\}.$
- ${\ensuremath{[\![\nu x.\varphi(x)]\!]}}_{\mathcal{S}} := \bigcup\{ T \in \mathcal{P}(S) \mid
T \subseteq {\ensuremath{[\![\varphi(x)]\!]}}_{\mathcal{S}[x \mapsto T]}\}.$
In accordance with the usual terminology, we say that a formula $\varphi$ is *true* or *satisfied* at a possible world $s \in S$ (denoted $\mathcal{S}, s \models \varphi$) if $s \in {\ensuremath{[\![\varphi]\!]}}_{\mathcal{S}}$. A formula $\varphi$ is *valid* (denoted $\models \varphi$) if $\varphi$ is true at every world in any model.
Let $\mathcal{S} = (S, R, \lambda)$ be a Kripke model. A formula $\varphi(x)$ such that $x \in \mathsf{Free}(\varphi(x))$ can be naturally seen as the following function: $$\begin{xy}
(0,8) *{\mathcal{P}(S)},(20,8)*{\mathcal{P}(S)},
(0,4) *{\rotatebox[origin=c]{90}{$\in$}},(20,4) *{\rotatebox[origin=c]{90}{$\in$}},
(0,0) *{T}="T",(26,0) *{{\ensuremath{[\![\varphi(x)]\!]}}_{\mathcal{S}[x \mapsto T]}.}="[T]",
\ar (5,8);(15,8)
\ar @{|->} (3,0);(15,0)
\end{xy}$$ This function is *monotone* if $x$ is positive in $\varphi(x)$. Thus, by the Knaster-Tarski Theorem [@tarski1955], ${\ensuremath{[\![\mu x.\varphi(x)]\!]}}_{\mathcal{S}}$ and ${\ensuremath{[\![\nu x.\varphi(x)]\!]}}_{\mathcal{S}}$ are the least and greatest fixpoint of the function $\varphi(x)$, respectively.
Under this characterization of fixpoint operators, we find that many interesting properties of the Kripke model can be represented by modal $\mu$-formulas. For example, consider the formula $\varphi_{1} = \mu x.(\Diamond x \vee p)$. For every Kripke model $\mathcal{S}$ and its possible world $s$, we have $\mathcal{S}, s \models \varphi_{1}$ if and only if there is some possible world reachable from $s$ in which $p$ is true. Consider the formula $\varphi_{2} = \nu y.\mu x.((\Diamond y \wedge p)\vee(\Diamond x \wedge \neg p))$. Then $\mathcal{S}, s \models \varphi_{2}$ if and only if there is some path from $s$ on which $p$ is true infinitely often.
Axiomatization {#subsec: axiomatization}
--------------
We give the Kozen’s axiomatization $\mathsf{Koz}$ for the modal $\mu$-calculus in the Tait-style calculus.[^5] Hereafter, we will write $\Gamma$, $\Delta$, $\dots$ for a finite set of formulas. Moreover, the standard abbreviation in the Tait-style calculus are used. That is, we write $\alpha, \Gamma$ for $\{ \alpha \} \cup \Gamma$; $\Gamma, \Delta$ for $\Gamma \cup \Delta$; and $\sim\!\Delta$ for $\{ \sim\!\delta \mid \delta \in \Delta \}$ and so forth.
$\mathsf{Koz}$ contains basic tautologies of classical propositional calculus and the *pre-fixpoint axioms*: $$\infer[(\textsf{Bot})]
{\bot \vdash}
{}
\qquad
\infer[(\textsf{Tau})]
{\varphi, \sim\!\varphi \vdash}
{}
\qquad
\infer[(\textsf{Prefix})]
{\alpha(\mu x.\alpha(x)), \sim\!\mu x.\alpha(x) \vdash}
{}$$
In addition to the classical inference rules from propositional modal logic, for any formula $\varphi(x)$ such that $x$ appears only positively, we have the *induction rule* $(\mathsf{Ind})$ to handle fixpoints: $$\infer[(\vee)]{\alpha\vee\beta, \Gamma \vdash}
{\alpha, \Gamma \vdash \quad \beta, \Gamma \vdash} \qquad
\infer[(\wedge)]{\alpha\wedge\beta, \Gamma \vdash}
{\alpha, \beta, \Gamma \vdash} \qquad$$ $$\infer[(\mathsf{Weak})]{\alpha, \Gamma \vdash}
{\Gamma \vdash} \qquad
\infer[(\Diamond)]{\Diamond \psi, \Gamma \vdash}
{\psi, \{ \alpha \mid \square \alpha \in \Gamma\} \vdash}$$ $$\infer[(\mathsf{Cut})]{\Gamma, \Delta \vdash}
{\Gamma, \sim\!\alpha \vdash \quad \alpha, \Delta \vdash} \qquad
\infer[(\mathsf{Ind})]{\mu x.\varphi(x), \sim\!\psi \vdash}
{\varphi(\psi), \sim\!\psi \vdash}$$ Of course, the condition of substitution is satisfied in the $(\mathsf{Ind})$-rule; namely, no new binding relation occurs by applying the substitution $\varphi(\psi)$. As usual, we say that a formula $\sim\!\bigwedge\Gamma$ is *provable* in $\mathsf{Koz}$ (denoted $\Gamma \vdash$) if there exists a proof diagram of $\Gamma$. We frequently use notation such as $\Gamma \vdash \Delta$ to mean $\Gamma, \sim\!\Delta \vdash$.
The following two lemmas state some basic properties of $\mathsf{Koz}$. We leave the proofs of these statement as an exercise to the reader.
\[lem: basic properties of KOZ 01\] Let $\varphi$ be a modal $\mu$-formula and let $\alpha(x)$ and $\beta(x, x)$ be modal $\mu$-formulas where $x$ appears only positively. Then, the following holds:
1. $\vdash \sigma x.\alpha(x) \leftrightarrow \sigma y.\alpha(y)$ where $y \notin \mathsf{Free}(\alpha(x))$.
2. $\vdash \sigma x.\beta(x, x) \leftrightarrow \sigma x.\sigma y.\beta(x, y)$ where $y \notin \mathsf{Free}(\beta(x, x))$.
3. $\vdash \mu x. \alpha(x) \leftrightarrow \alpha(\bot)$, if no appearances of $x$ are in the scope of any modal operators.
4. $\vdash \nu x. \alpha(x) \leftrightarrow \alpha(\top)$, if no appearances of $x$ are in the scope of any modal operators.
5. We can construct a well-named formula $\mathsf{wnf}(\varphi) \in \mathsf{WNF}$ such that $\vdash \varphi \leftrightarrow \mathsf{wnf}(\varphi)$.
\[lem: basic properties of KOZ 02\] Let $\alpha$, $\beta$, $\varphi(x)$, $\psi(x)$, $\chi_{1}(x)$ and $\chi_{2}(x)$ be modal $\mu$-formulas where $x$ appears only positively in $\varphi(x)$ and $\psi(x)$. Further, suppose that $\chi_{1}(\alpha)$, $\chi_{1}(\beta)$ and $\chi_{2}(\alpha)$ are legal substitution; namely, a new binding relation does not occur by such substitutions. Then, the following holds:
1. If $\vdash \varphi(x) \rightarrow \psi(x)$ then $\vdash \sigma x.\varphi(x) \rightarrow \sigma x.\psi(x)$.
2. If $\vdash \alpha \leftrightarrow \beta$ then $\vdash \chi_{1}(\alpha) \leftrightarrow \chi_{1}(\beta)$.
3. If $\vdash \chi_{1}(x) \leftrightarrow \chi_{2}(x)$ then $\vdash \chi_{1}(\alpha) \leftrightarrow \chi_{2}(\alpha)$.
\[rem: substitution\]Let $\varphi(x)$ and $\psi$ be formulas where $\varphi(x) = \varphi(x_{1}, \dots, x_{k})[x_{1}/x, \dots, x_{k}/x]$ and $x \notin \mathsf{Free}(\varphi(x_{1}, \dots, x_{k}))$; i.e., $\varphi(x_{1}, \dots, x_{k})$ is a formula obtained by renaming all instances of $x$ in $\varphi(x)$. Let $\varphi'(x)$ be the formula obtained by renaming bound variables in $\varphi(x)$ and let $\psi_{i}$ with $1 \leq i \leq k$ be formulas obtained by renaming bound variables in $\psi$ so that; $$\begin{aligned}
\mathsf{Bound}(\varphi'(x))\cap \mathsf{Free}(\varphi'(x)) = \emptyset
\label{eq: substitutaion 01}\\
\mathsf{Bound}(\varphi'(x))\cap\mathsf{Free}(\psi_{i}) = \emptyset \quad (1 \leq \forall i \leq k)
\label{eq: substitutaion 02}\\
\mathsf{Free}(\varphi'(x))\cap\mathsf{Bound}(\psi_{i}) = \emptyset \quad (1 \leq \forall i \leq k)
\label{eq: substitutaion 03}\\
\mathsf{Bound}(\psi_{i})\cap\mathsf{Free}(\psi_{j}) = \emptyset \quad (1 \leq \forall i, \forall j \leq k)
\label{eq: substitutaion 04}\\
\mathsf{Bound}(\psi_{i})\cap\mathsf{Bound}(\psi_{j}) = \emptyset \quad
(1 \leq \forall i, \forall j \leq k, \: i \neq j)
\label{eq: substitutaion 05}\end{aligned}$$ Then the formula $\varphi'(\psi_{1}, \dots, \psi_{k})$ is termed well-named. Moreover, from Lemmas \[lem: basic properties of KOZ 01\] and \[lem: basic properties of KOZ 02\], we can assume that $\varphi'(\psi_{1}, \dots, \psi_{k})$ is syntactically (and thus semantically) equivalent to $\varphi(\psi)$. Hereafter, we will assume that $\varphi(\psi)$ is an abbreviation for $\varphi'(\psi_{1}, \dots, \psi_{k})$; this abbreviation is harmless as far as provability and satisfiability are concerned. Furthermore, we can write $\varphi(\psi)$ even if a new binding relation occurs by the substitution; in this case, we will regard it as merely an abbreviation for $\varphi'(\psi_{1}, \dots, \psi_{k})$.
Automata {#sec: automata}
========
The purpose of this section is to define the *parity automata* and introduce a classical result concerning the complement of an $\omega$-language characterized by some parity automaton, namely, the *Complementation Lemma*.
A parity automaton is a quintuple $\mathcal{A} = (Q, \Sigma, \delta, q_{I}, \Omega)$ where:
- $Q$ is a finite set of *states* of the automaton,
- $\Sigma$ is an *alphabet*,
- $q_{I} \in Q$ is a state called the *initial state*,
- $\delta : Q \times \Sigma \rightarrow \mathcal{P}(Q)$ is a *transition function*, and
- $\Omega : Q \rightarrow \omega$ is called the *priority function*.
Using the usual definitions, we say that $\mathcal{A}$ is *deterministic* if $|\delta(q, a)| = 1$ for every $q \in Q$ and $a \in \Sigma$. Let $\mathcal{A} = (Q, \Sigma, \delta, q_{I}, \Omega)$ be a parity automaton. A *run* of $\mathcal{A}$ on an $\omega$-word $\pi \in \Sigma^{\omega}$ is an infinite sequence $\xi \in Q^{\omega}$ of a state where $\xi[1] = q_{I}$ and $\xi[n+1] \in \delta(\xi[n], \pi[n])$ for any $n \geq 1$. An $\omega$-word $\pi \in \Sigma^{\omega}$ is *accepted* by $\mathcal{A}$ if there is a run $\xi$ of $\mathcal{A}$ on $\pi$ satisfying the following condition: $$\max \mathsf{Inf}(\vec{\Omega}(\xi)) = 0 \pmod 2.$$ The $\omega$-language of all $\omega$-words accepted by $\mathcal{A}$ is denoted by $\mathcal{L}(\mathcal{A})$.
Let $\mathcal{A} = (Q, \Sigma, \delta, q_{I}, \Omega)$ be a parity automaton and $\pi \in \Sigma^{\ast}$. If $\mathcal{A}$ is deterministic, then the state of $\mathcal{A}$ by reading $\pi$ is uniquely determined. We denote this state $\delta(q_{I}, \pi)$; in other words, $\delta(q_{I}, \pi)$ is defined inductively on the length of $\pi$ by $$\begin{aligned}
\delta(q_{I}, \pi) := \left\{\begin{array}{ll}
q_{I} & (|\pi| = 0)\\
\delta \big( \delta(q_{I}, \pi[1, n]), \pi[n+1] \big) & (|\pi| = n+1).
\end{array}\right.\end{aligned}$$ Moreover, for any $\pi \in \Sigma^{\ast} \cup \Sigma^{\omega}$, we denote the run of $\mathcal{A}$ on $\alpha$ by $\vec{\delta}(q_{I}, \pi)$, that is, $$\vec{\delta}(q_{I}, \pi) := q_{I}
\delta (q_{I}, \pi[1, 1])
\delta (q_{I}, \pi[1, 2])
\delta (q_{I}, \pi[1, 3])
\dots \in Q^{\ast} \cup Q^{\omega}.$$
The following lemma shows that the complement of the $\omega$-language characterized by a parity automaton is also characterized by some parity automaton. The proof of this lemma can be found in the literature, for example, see [@DBLP:conf/dagstuhl/2001automata].
\[lem:complemantation lemma\] For any parity automaton $\mathcal{A} = (Q, \Sigma, \delta, q_{I}, \Omega)$, we can construct a deterministic parity automaton $\bar{\mathcal{A}}$ such that $\mathcal{L}(\bar{\mathcal{A}}) = \Sigma^{\omega} \setminus \mathcal{L}(\mathcal{A})$ with $2^{\mathcal{O}(|Q|^2 \log |Q|^2)}$ states and priorities bounded by $\mathcal{O}(|Q|^2)$.
Games {#sec: games}
=====
It is well known that *Parity games* and *Evaluation games* are important tools in the modal $\mu$-calculus. They will also play a crucial role in this article. This section introduces these games.
Parity games {#subsec: Parity games}
------------
A *parity game* $\mathcal{G}$ is defined in terms of an *arena* $\mathcal{A}$ and a *priority function* $\Omega$. An arena is a (possibly infinite) directed graph $\mathcal{A} = \langle V_{0}, V_{1}, E \rangle$, where $V_{0} \cap V_{1} = \emptyset$ and the edge relation is $E \subseteq (V_{0} \cup V_{1}) \times (V_{0} \cup V_{1})$. We call each element of $V := V_{0} \cup V_{1}$ a *game position* of the arena. The priority function is $\Omega: V \rightarrow \omega$ where $\Omega(V)$ is a finite set.
A *play* in arena $\mathcal{A}$ can be finite or infinite. In the former case, the play is an $E$-sequence $\pi = v_{1}\cdots v_{n} \in V^{+}$ such that $E(v_{n}) = \emptyset$. In the later case, the play is simply an infinite $E$-sequence. Thus, a finite or infinite play in a game can be seen as the trace of a token moved on the arena by two players, Player $0$ and Player $1$, in such a way that if the token is in position $v \in V_{\delta}$ $(\delta \in \{ 0, 1 \})$, then Player $\delta$ has to choose a successor of $v$ to which to move the token. A play $\pi$ is *winning* for Player $0$ if:
- If $\pi$ is finite, then the last position $\pi[|\pi|]$ of the play is in $V_{1}$.
- If $\pi$ is infinite, then $\max \mathsf{inf}(\vec{\Omega}(\pi)) = 0 \pmod 2$.
A play is winning for Player $1$ if it is not winning for Player $0$.
Let $\mathcal{G} = \langle \langle V_{0}, V_{1}, E \rangle, \Omega \rangle$ be the parity game presented in Figure \[fig:An example of a parity game.\]. We have the $0$-vertices $V_{0} = \{ v_{1}, v_{5} \}$ (circles) and the $1$-vertices $V_{1} = \{ v_{2}, v_{3}, v_{4} \}$ (squares). The edge relation $E$ and priority function $\Omega$ may be derived from the figure, e.g., $\Omega(v_{1}) = 2$ and $\Omega(v_{2}) = 3$.
$$\xymatrix @C=15mm{
*+++[o][F-]{v_{1}, 2} \ar[r] \ar@/_/[d] & *++[F-]{v_{2}, 3} \ar[ld] \ar[d]&\\
*++[F-]{v_{3}, 1} \ar@/_/[r] \ar@/_/[u] & *++[F-]{v_{4}, 3} \ar@/_/[l] \ar[r] & *+++[o][F-]{v_{5}, 0}\\
}$$
A possible infinite play in this game is, for example, $\pi = v_{1}v_{2}(v_{3}v_{1})^{\omega}$. This play is winning for Player $0$ because $\vec{\Omega}(v_{1}v_{2}(v_{3}v_{1})^{\omega}) =
\langle 2, 3, 1, 2, 1, 2, \dots \rangle$ and: $$\max \mathsf{inf}(\vec{\Omega}(\pi)) = \max \mathsf{inf}(\langle 2, 3, 1, 2, 1, 2, \dots \rangle) =
\max \{ 1, 2 \} = 2 = 0 \pmod 2.$$ A finite $E$-sequence $\pi = v_{1}v_{2}v_{4}v_{5}$ is also a possible play since $v_{5}$ is a dead-end. This play is winning for Player $1$ because the last position $v_{5}$ is in $V_{0}$.
Let $\mathcal{A}$ be an arena. A *strategy* for Player $\delta$ with $\delta \in \{ 0, 1 \}$ is a partial function $f_{\delta}: V^{\ast}V_{\delta} \rightarrow V$ such that for any $\pi \in V^{\ast}V_{\delta}$, if $E(\pi[|\pi|]) \neq \emptyset$ then $f_{\delta}(\pi)$ is defined and satisfies $f_{\delta}(\pi) \in E(\pi[|\pi|])$. A play $\pi$ is said to be *consistent* with $f_{\delta}$ if for every $n \in \omega$ such that $1 \leq n < |\pi|$, $\pi[n] \in V_{\delta}$ implies $f_{\sigma}(\pi[1, n]) = \pi[n+1]$. The strategy $f_{\delta}$ is said to be a *winning strategy* for Player $\delta$ if every play consistent with $f_{\delta}$ is winning for Player $\delta$. A position $v \in V$ is winning for Player $\delta$ if there is a strategy $f_{\delta}$ such that every play consistent with $f_{\delta}$ which starts in $v$ is winning for Player $\delta$. A winning strategy $f_{\delta}$ is called *memoryless* if for all finite $E$-sequences $\pi$ and $\pi'$, $f_{\delta}(\pi) = f_{\delta}(\pi')$ whenever $\pi[|\pi|] = \pi'[|\pi'|]$. For parity games, we have a memoryless determinacy result.
\[the: memoryless strategy\] For any parity game, one of the Players has a memoryless winning strategy from each game position.
Considering this theorem, we will assume that all winning strategies are memoryless. In other words, a winning strategy in a parity game for Player $0$ is a function $f_{0}: V_{0} \rightarrow V$, and is denoted analogously for Player $1$.
Evaluation games {#subsec: evaluation games}
----------------
Given a well-named formula $\varphi$, a Kripke model $\mathcal{S} = (S, R, \lambda)$ and its world $s_{0}$, we define the *evaluation game* $\mathcal{EG}(\mathcal{S}, s_{0}, \varphi)$ as a parity game with Player $0$ and $1$ moving a token to positions of the form $\langle\psi, s\rangle \in \mathsf{Sub}(\varphi)\times S$. Intuitively, Player $0$ asserts that “the formula $\varphi$ is true at the possible world $s_{0}$” and Player $1$ asserts the opposite.
The initial game position is $\langle\varphi, s_{0}\rangle$. Table \[tab:rule of EG\] displays the rules of the game, that is, admissible moves from a given position, and the player supposed to make this move.
Position Player Admissible moves
----------------------------------------------------------------------------------------- -------- ---------------------------------------------------------
$\langle \bot, s\rangle$ 0 $\emptyset$
$\langle \top, s\rangle$ 1 $\emptyset$
$\langle p, s\rangle$ with $p \in \mathsf{Free}(\varphi)$ and $s \in \lambda(p)$ 1 $\emptyset$
$\langle p, s\rangle$ with $p \in \mathsf{Free}(\varphi)$ and $s \notin \lambda(p)$ 0 $\emptyset$
$\langle\neg p, s\rangle$ with $p \in \mathsf{Free}(\varphi)$ and $s \notin \lambda(p)$ 1 $\emptyset$
$\langle\neg p, s\rangle$ with $p \in \mathsf{Free}(\varphi)$ and $s \in \lambda(p)$ 0 $\emptyset$
$\langle\alpha \wedge \beta, s\rangle$ 1 $\{ \langle\alpha, s\rangle, \langle\beta, s\rangle \}$
$\langle\alpha \vee \beta, s\rangle$ 0 $\{ \langle\alpha, s\rangle, \langle\beta, s\rangle \}$
$\langle\square\alpha, s\rangle$ 1 $\{ \langle\alpha, t\rangle \mid (s, t) \in R\}$
$\langle\Diamond\alpha, s\rangle$ 0 $\{ \langle\alpha, t\rangle \mid (s, t) \in R\}$
$\langle\sigma x.\alpha, s\rangle$ 0 $\{ \langle\alpha, s\rangle \}$
$\langle x, s\rangle$ with $x \in \mathsf{Bound}(\varphi)$ 0 $\{ \langle\varphi_{x}(x), s\rangle \}$
: Admissible move of $\mathcal{EG}(\mathcal{S}, s_{0}, \varphi)$[]{data-label="tab:rule of EG"}
In order to define the priority function $\Omega_{e}: V \rightarrow \omega$, we define the function $\Omega_{\varphi}: \mathsf{Sub}(\varphi) \rightarrow \omega$ as follows: $$\begin{aligned}
\label{eq: priority of formulas}
\Omega_{\varphi}(\psi) := \left\{
\begin{array}{ll}
\mathsf{alt}(\sigma_{x}.\varphi_{x}(x))-1&
\text{if $\psi = \varphi_{x}(x)$, $\sigma_{x} = \mu$ and $\mathsf{alt}(\sigma_{x}.\varphi_{x}(x)) = 0 \pmod 2$,}\\
\mathsf{alt}(\sigma_{x}.\varphi_{x}(x))&
\text{if $\psi = \varphi_{x}(x)$, $\sigma_{x} = \mu$ and $\mathsf{alt}(\sigma_{x}.\varphi_{x}(x)) = 1 \pmod 2$,}\\
\mathsf{alt}(\sigma_{x}.\varphi_{x}(x))-1&
\text{if $\psi = \varphi_{x}(x)$, $\sigma_{x} = \nu$ and $\mathsf{alt}(\sigma_{x}.\varphi_{x}(x)) = 1 \pmod 2$,}\\
\mathsf{alt}(\sigma_{x}.\varphi_{x}(x))&
\text{if $\psi = \varphi_{x}(x)$, $\sigma_{x} = \nu$ and $\mathsf{alt}(\sigma_{x}.\varphi_{x}(x)) = 0 \pmod 2$,}\\
0&
\text{otherwise.}
\end{array}\right.\end{aligned}$$ Then we define $\Omega_{e}( \langle \psi, s \rangle ) := \Omega_{\varphi} (\psi) $ for each game position $\langle \psi, s \rangle$.
The following theorem was proved by Streett and Emerson [@journals/iandc/StreettE89].
\[the: fundamental theorem\] For any well-named formula $\varphi$, Kripke model $\mathcal{S}$ and its world $s$, we have $\mathcal{S}, s \models \varphi$ if and only if Player $0$ has a (memoryless) winning strategy for $\mathcal{EG}(\mathcal{S}, s, \varphi)$.
Tableaux {#sec: tableaux}
========
In this section, we introduce the concept of a *tableau* and investigate some of its characteristic properties. The main result of this section is Corollary \[cor: wide tableau\] in which we prove Claim (g) as foreshadowed in Section \[sec: introduction\]. This section is divided into the following three subsections.
In Subsection \[subsec: tableau games\], we introduce the tableau and *tableau games*, which originated in Niwinski and Walukiewicz [@Niwinski199699], with some modifications for our concept.
In Subsection \[subsec: automaton normal form\], the *automaton normal form* is introduced and Claim (b) is shown; namely, for any formula $\varphi$ we can construct an equivalent automaton normal form $\mathsf{anf}(\varphi)$. Although this result is not new, we will see the proof of it in detail since our argument relies on both the result and the process for proving (b).
In Subsection \[subsec: wide tableau\], we introduce the novel concept of a *wide tableau*, which is a generalization of tableaux and prove Claim (g) using this new resource.
Tableau games {#subsec: tableau games}
-------------
Let $\Phi$ be a finite set of formulas. Then $\triangledown \Phi$ denotes an abbreviation of the following formula: $$\big(\bigwedge\Diamond \Phi\big) \wedge \big(\square \bigvee\Phi\big).$$ Here, $\Diamond \Phi$ denotes the set $\{ \Diamond \varphi \mid \varphi \in \Phi\}$, and as always, we use the convention that $\bigvee\emptyset := \bot$ and $\bigwedge\emptyset := \top$. The symbol $\triangledown$ is called the *cover modality*.
Note that the both the ordinary diamond $\Diamond$ and the ordinary box $\square$ can be expressed in term of cover modality and the disjunction: $$\begin{aligned}
\Diamond \varphi \equiv \triangledown\{\varphi, \top\},\\
\square \varphi \equiv \triangledown\emptyset \vee \triangledown\{\varphi\}.\end{aligned}$$ Therefore, without loss of generality we restrict ourselves to using only $\triangledown$ instead of $\Diamond$ and $\square$. Hereafter, we exclusively use cover modality notation instead of ordinal modal notation; thus *if not otherwise mentioned, all formulas are assumed to be using this new constructor*. Moreover, the concepts from Section \[sec: the modal mu-calculus\] such as the well-named formula and the alternation depth extend to formulas using this modality.
Let $\Gamma$ be a set of formulas. We will say that $\Gamma$ is *locally consistent* if $\Gamma$ does not contain $\bot$ nor any propositional variable $p$ and its negation $\neg p$ simultaneously. On the other hand, $\Gamma$ is said to be *modal* (under $\varphi$) if $\Gamma$ does not contain formulas of the forms $\alpha\vee\beta$, $\alpha\wedge\beta$, $\sigma x.\alpha(x)$, or $x \in \mathsf{Bound}(\varphi)$. In other words, if $\Gamma$ is modal, then $\Gamma$ can possess only the elements of $\mathsf{Lit}(\varphi)$ and formulas of the form $\triangledown\Phi$.
\[def: tableau\]Let $\varphi$ be a well-named formula. A set of *tableau rules* for $\varphi$ is defined as follows: $$\infer[(\vee)]{\alpha\vee\beta, \Gamma}
{\alpha, \Gamma \;\mid\; \beta, \Gamma} \qquad
\infer[(\wedge)]{\alpha\wedge\beta, \Gamma}
{\alpha, \beta, \Gamma}$$ $$\infer[(\sigma)]{\sigma_{x}x.\varphi_{x}(x), \Gamma}
{\varphi_{x}(x), \Gamma} \qquad
\infer[(\mathsf{Reg})]{x, \Gamma}
{\varphi_{x}(x), \Gamma}$$ $$\infer[(\triangledown)]
{\triangledown\Psi_{1}, \dots, \triangledown\Psi_{i}, l_{1}, \dots, l_{j}}
{\{\psi_{k}\}\cup\{\bigvee\Psi_{n}\mid n \in N_{\psi_{k}}\}\:\mid\: \text{For every $k \in \omega$ with
$1 \leq k \leq i$ and $\psi_{k} \in \Psi_{k}$.}}$$ where in the $(\triangledown)$-rule, $l_{1}, \dots, l_{j} \in \mathsf{Lit}(\varphi)$ and $N_{\psi_{k}} := \{ n \in \omega \mid 1 \leq n \leq i,\; n \neq k\}$. Therefore, the premises of a $(\triangledown)$-rule is equal to $\sum_{1 \leq k\leq i}|\Psi_{k}|$.
A *tableau* for $\varphi$ is a structure $\mathcal{T}_{\varphi} = (T, C, r, L)$ where $(T, C, r)$ is a tree structure and $L: T \rightarrow \mathcal{P}(\mathsf{Sub}(\varphi))$ is a *label function* satisfying the following clauses:
1. $L(r) = \{\varphi\}$.
2. Let $t \in T$. If $L(t)$ is modal and inconsistent then $t$ has no child. Otherwise, if $t$ is labeled by a set of formulas which fulfills the form of the conclusion of some tableau rules, then $t$ has children which are labeled by the sets of formulas of premises of one of those tableau rules, e.g., if $L(t) = \{ \alpha \vee \beta\}$, then $t$ must have two children $u$ and $v$ with $L(u) = \{\alpha\}$ and $L(v) = \{\beta\}$.
3. The rule $(\triangledown)$ can be applied in $t$ only if $L(t)$ is modal; in other words, $(\triangledown)$ is applicable when no other rule is applicable.
We call a node $t$ a $(\triangledown)$-node if the rule $(\triangledown)$ is applied between $t$ and its children. The notions of $(\vee)$-node, $(\wedge)$-node, $(\sigma)$-node and $(\mathsf{Reg})$-node are defined similarly.
\[def: modal and choice nodes\]Leaves and $(\triangledown)$-nodes are called *modal nodes*. The root of the tableau and children of modal nodes are called *choice nodes*. We say that a modal node $t$ and choice node $u$ are *near* to each other if $t$ is a descendant of $u$ and between the $C$-sequence from $u$ to $t$, there is no node in which the rule $(\triangledown)$ is applied. Similarly, we say that a modal node $t'$ is a *next modal node* of a modal node $t$ if $t'$ is a descendant of $t$ and between the $C$-sequence from $t$ to $t'$, rule $(\triangledown)$ is applied exactly once between $t$ and its child. Note that, in some cases, a choice node may be also a modal node.
\[def: trace\]Let $\varphi$ be a well-named formula and $\mathcal{T}_{\varphi} = (T, C, r, L)$ be a tableau for $\varphi$. For each node $t \in T$ and its child $u \in C(t)$, we define the *trace function* $\mathsf{TR}_{tu}: L(t)\rightarrow\mathcal{P}(L(u))$ as follows:
- If $t$ is a $(\vee)$-node where the rule applied between $t$ and its children forms $$\infer[(\vee)]
{\alpha\vee\beta, \Gamma}
{\alpha, \Gamma \;\mid\; \beta, \Gamma}$$ then we set $\mathsf{TR}_{tu}(\gamma) := \{\gamma\}$ for every $\gamma \in \Gamma$. Further, we set $\mathsf{TR}_{tu}(\alpha\vee\beta) := \{\alpha\}$ when $L(u) = \{\alpha\}\cup\Gamma$ and set $\mathsf{TR}_{tu}(\alpha\vee\beta) := \{\beta\}$ when $L(u) = \{\beta\}\cup\Gamma$.
- If $t$ is a $(\wedge)$-node where the rule applied between $t$ and its child forms $$\infer[(\wedge)]
{\alpha\wedge\beta, \Gamma}
{\alpha, \beta, \Gamma}$$ then we set $\mathsf{TR}_{tu}(\gamma) := \{\gamma\}$ for every $\gamma \in \Gamma$, and set $\mathsf{TR}_{tu}(\alpha\wedge\beta) := \{\alpha, \beta\}$.
- If $t$ is a $(\sigma)$-node where the rule applied between $t$ and its child forms $$\infer[(\sigma)]
{\sigma_{x}x.\varphi_{x}(x), \Gamma}
{\varphi_{x}(x), \Gamma} \qquad$$ then we set $\mathsf{TR}_{tu}(\gamma) := \{\gamma\}$ for every $\gamma \in \Gamma$, and set $\mathsf{TR}_{tu}(\sigma_{x}x.\varphi_{x}(x)) := \{\varphi_{x}(x)\}$.
- If $t$ is a $(\mathsf{Reg})$-node where the rule applied between $t$ and its child forms $$\infer[(\mathsf{Reg})]{x, \Gamma}
{\varphi_{x}(x), \Gamma}$$ then we set $\mathsf{TR}_{tu}(\gamma) := \{\gamma\}$ for every $\gamma \in \Gamma$, and set $\mathsf{TR}_{tu}(x) := \{\varphi_{x}(x)\}$.
- If $t$ is a $(\triangledown)$-node where the rule applied between $t$ and its children forms $$\infer[(\triangledown)]
{\triangledown\Psi_{1}, \dots, \triangledown\Psi_{i}, l_{1}, \dots, l_{j}}
{\{\psi_{k}\}\cup\{\bigvee\Psi_{n}\mid n \in N_{\psi_{k}}\}\:\mid\: 1 \leq k \leq i,\;
\psi_{k} \in \Psi_{k}.}$$ Moreover, suppose $u$ is labeled by $\{\psi_{k}\}\cup\{\bigvee\Psi_{n}\mid n \in N_{\psi_{k}}\}$ for some $k \leq i$ and $\psi_{k} \in \Psi_{k}$. Then we set $\mathsf{TR}_{tu}(\triangledown \Psi_{k}) := \{\psi_{k}\}$, $\mathsf{TR}_{tu}(\triangledown \Psi_{n}) := \{\bigvee\Psi_{n}\}$ for every $n \in N_{\psi_{k}}$, and $\mathsf{TR}_{tu}(l_{n}) := \emptyset$ for every $n \leq j$.
Take a finite or infinite $C$-sequence $\pi$ of $\mathcal{T}_{\varphi}$. A *trace* $\mathsf{tr}$ on $\pi$ is a finite or infinite sequence of $\mathsf{Sub}(\varphi)$ satisfying the following two conditions;
- $\mathsf{tr}[1] = \varphi$.
- For any $n \in \omega\setminus\{0\}$, if $\mathsf{tr}[n]$ is defined and satisfies $\mathsf{TR}_{\pi[n]\pi[n+1]}(\mathsf{tr}[n]) \neq \emptyset$, then $\mathsf{tr}[n+1]$ is also defined and satisfies $\mathsf{tr}[n+1] \in \mathsf{TR}_{\pi[n]\pi[n+1]}(\mathsf{tr}[n])$.
Note that, from the definition, for any $n \in \omega$ such that $1 \leq n \leq |\mathsf{tr}|$, we have $\mathsf{tr}[n] \in L(\pi[n])$. The infinite trace $\mathsf{tr}$ is said to be *even* if $$\max \mathsf{Inf}(\vec{\Omega}_{\varphi}(\mathsf{tr})) = 0 \pmod 2.$$ Furthermore, an infinite branch $\pi$ is *even* if every trace on it is even. The set of all traces on $\pi$ is denoted by $\mathsf{TR}(\pi)$. $\mathsf{TR}(\pi[n, m])$ denotes the set $\{ \mathsf{tr}[n, m] \mid \mathsf{tr} \in \mathsf{TR}(\pi) \}$ and may also be written $\mathsf{TR}(\pi[n], \pi[m])$. For any two factors $\mathsf{tr}[n, m]$ and $\mathsf{tr'}[n', m']$, we say $\mathsf{tr}[n, m]$ and $\mathsf{tr'}[n', m']$ are *equivalent* (denoted $\mathsf{tr}[n, m] \equiv \mathsf{tr'}[n', m']$) if, by ignoring invariant portions of the traces, they can be seen as the same sequence. For example, let; $$\xymatrix @C=5mm@R=1mm{
*{\mathsf{tr}[n, n+3] =}&
*{\langle (\alpha\wedge\beta)\vee\gamma,} &
*{(\alpha\wedge\beta)\vee\gamma,} &
*{\alpha\wedge\beta,} &
*{\beta \rangle} &
&\\
*{\mathsf{tr'}[n', n'+4] =}&
*{\langle (\alpha\wedge\beta)\vee\gamma,} &
*{\alpha\wedge\beta,} &
*{\alpha\wedge\beta,} &
*{\alpha\wedge\beta,} &
*{\beta \rangle} &
}$$ then $\mathsf{tr}[n, n+3]$ and $\mathsf{tr'}[n', n'+4]$ are equivalent to each other. Let $X$ and $Y$ be the set of some factors of some traces. Then we write $X \Subset Y$ if for any $\mathsf{tr}[n, m] \in X$ there exists $\mathsf{tr'}[n', m'] \in Y$ such that $\mathsf{tr}[n, m] \equiv \mathsf{tr'}[n', m']$; and write $X \equiv Y$ if $X \Subset Y$ and $X \Supset Y$.
Let $\varphi$ be a formula. Since $\mathcal{P}(\mathsf{Sub}(\varphi))$ is a finite set, it can be seen as an alphabet. The next lemma shows that there is an automaton $\mathcal{A}_{\varphi}$ which precisely detects the evenness of a branch of the tableau.
\[lem: automaton\] Let $\varphi$ be a well-named formula and $\mathcal{T}_{\varphi} = (T, C, r, L)$ be a tableau for $\varphi$. Set $M = |\mathsf{Sub}(\varphi)|$. Then we can construct a deterministic parity automaton $$\mathcal{A}_{\varphi} = (Q, \mathcal{P}(\mathsf{Sub}(\varphi)), \delta, q_{I}, \Omega)$$ with $|Q| \in 2^{\mathcal{O}(M^{2} \log M^{2})}$ and priorities bounded by $\mathcal{O}(M^{2})$ such that for any infinite branch $\pi$, $\mathcal{A}_{\varphi}$ accepts $\vec{L}(\pi) \in \mathcal{P}(\mathsf{Sub}(\varphi))^{\omega}$ if and only if $\pi$ is even.
First, we construct a non-deterministic parity automaton $$\mathcal{B}_{\varphi} = (Q', \mathcal{P}(\mathsf{Sub}(\varphi)), \delta', q'_{I}, \Omega')$$ which only accepts sequences of labels of $\pi$ that are *not* even. Set $Q' := \mathsf{Sub}(\varphi)\uplus\{q'_{I}\}$, then $\mathcal{B}_{\varphi}$ has $(M+1)$ states. We define the transition function $\delta'$ so that $\delta'(q'_{I}, \{\varphi\}) := \{\varphi\}$ and $\delta'(\psi, \pi[n+1]) := \mathsf{TR}_{\pi[n]\pi[n+1]}(\psi)$ for any $n \geq 1$. The priority is defined as $\Omega'(q'_{I}) := 0$ and $\Omega'(\psi) := \Omega_{\varphi}(\psi)+1$ for every $\psi \in \mathsf{Sub}(\varphi)$.
Now, $\mathcal{B}_{\varphi}$ is defined in such a way that a run of the automaton on $\vec{L}(\pi)$ forms one trace on $\pi$ and the automaton accepts only *odd* traces. By applying the Complementation Lemma \[lem:complemantation lemma\], we obtain the required automaton.
Now, we define the *tableau games* introduced by Niwinski and Walukiewicz [@Niwinski199699]. To distinguish players of this game from players of the evaluation games defined in subsection \[subsec: evaluation games\], we assume that players of a tableau game have other popular names; say Player $2$ and Player $3$. Intuitively, Player $2$ asserts that “$\varphi$ is satisfiable” and Player $3$ asserts the opposite. This is justified by Lemmas \[lem: tableau game 01\] and \[lem: tableau game 02\].
\[def: tableau game\]Let $\varphi$ be a well-named formula, $\mathcal{T}_{\varphi} = (T, C, r, L)$ be a tableau for $\varphi$, and $\mathcal{A}_{\varphi} = (Q, \mathcal{P}(\mathsf{Sub}(\varphi)), \delta, q_{I}, \Omega)$ be an automaton given by Lemma \[lem: automaton\]. A *tableau game* for $\varphi$ (denoted $\mathcal{TG}(\varphi)$) is a parity game played by Player $2$ and Player $3$ defined as follows:
Positions
: Let $M \subseteq T$ be the set of all modal nodes which are consistent. The positions of Player $2$ are given by $V_{2} := (T\setminus M)$ and the positions of Player $3$ are given by $V_{3} := M$; therefore the set of game positions is $T$. The starting position of this game is the root $r$.
Admissible moves
: In a position $t \in V_{2}$, Player $2$ chooses the next position from $C(t)$. Note that when $t$ is modal and locally inconsistent, Player $2$ loses the game immediately since $C(t) = \emptyset$ and so she has no choice from $t$. In a position $t \in V_{3}$, Player $3$ chooses the next position from $C(t)$. Note that when $L(t)$ does not contain a formula of the form $\triangledown \Psi$, Player $3$ loses the game immediately since $C(t) = \emptyset$ and so he has no choice from $t$.
Priority
: For any tableau node $t \in T$, we define the *automaton states* of $t$ by $\mathsf{stat}(t) := \delta(q_{I}, \vec{L}(\pi))$ where $\pi$ is the $C$-sequence starting at $r$ and ending at $t$. Then, the priority of $t \in T$ is $\Omega(\mathsf{stat}(t))$.
\[lem: tableau game 01\] Let $\varphi$ be a well-named formula. If $\varphi$ is satisfiable, then Player $2$ has a winning strategy in the tableau game $\mathcal{TG}(\varphi)$.
Let $\mathcal{S} = (S, R, \lambda)$ be a model and $s_{0} \in S$ be a possible world such that $\mathcal{S}, s_{0} \models \varphi$. From Theorem \[the: fundamental theorem\], we can assume that there exists a memoryless winning strategy $f_{0}$ for Player $0$ in evaluation game $\mathcal{EG}(\mathcal{S}, s_{0}, \varphi)$. Now, we will construct a winning strategy for Player $2$ in $\mathcal{TG}(\varphi)$ inductively; in the process of the defining the strategy, we will also define the *marking function* $\mathsf{mark}: T \rightarrow S$ simultaneously such that
> $(\dag)$: If the current game position is in $t \in T$ and $\mathsf{mark}(t) = s$, then for any $\gamma \in L(t)$, Player $0$ can win at the position $\langle \gamma, s\rangle$ by using the strategy $f_{0}$.
Initially, we define $\mathsf{mark}(r) := s_{0}$. This marking indeed satisfies $(\dag)$. The remaining strategy and marking are divided into the following three cases:
- Suppose that the current position $t$ is a $(\vee)$-node where $t$ and its children are labeled $$\infer[(\vee)]{\alpha\vee\beta, \Gamma}
{\alpha, \Gamma \;\mid\; \beta, \Gamma}$$ Then Player $2$ must choose the next game position from these two children, say $u$ and $v$ which are labeled by $\{ \alpha \}\cup\Gamma$ and $\{ \beta \}\cup\Gamma$, respectively. By our induction assumption, we can assume that there exists a marking $\mathsf{mark}(t) = s$ which satisfies $(\dag)$. Then Player $2$ chooses $u$ if and only if $f_{0}(\alpha \vee \beta, s) = \langle \alpha, s\rangle$. Player $2$ also defines the new marking as $\mathsf{mark}(u) := \mathsf{mark}(t)$.
- Suppose the current position $t$ is a $(\triangledown)$-node where $t$ and its children are labeled $$\infer[(\triangledown)]
{\triangledown\Psi_{1}, \dots, \triangledown\Psi_{i}, l_{1}, \dots, l_{j}}
{\{\psi_{k}\}\cup\{\bigvee\Psi_{n}\mid n \in N_{\psi_{k}}\}\:\mid\: 1 \leq k \leq i,\;
\psi_{k} \in \Psi_{k}.}$$ Moreover, suppose that Player $3$ chooses $u \in C(t)$ which is labeled by $\{\psi_{k}\}\cup\{\bigvee\Psi_{n}\mid n \in N_{\psi_{k}}\}$. By our induction assumption, there is a marking $\mathsf{mark}(t) = s$ such that for any $m \in \omega$ with $1 \leq m \leq i$, $\langle\triangledown\Psi_{m}, s\rangle$ is a winning position for Player $0$ by using $f_{0}$. Since $\langle\triangledown\Psi_{k}, s\rangle$ is winning for Player $0$, the position $\langle\Diamond\psi_{k}, s\rangle$ is also winning for Player $0$ (because $\triangledown \Psi_{k} \equiv
\big(\bigwedge\Diamond \Psi_{k}\big) \wedge \big(\square \bigvee\Psi_{k}\big)$, and since $\langle \big(\bigwedge\Diamond \Psi_{k}\big) \wedge \big(\square \bigvee\Psi_{k}\big), s\rangle$ is winning for Player $0$ and Player $1$ can choose the position $\langle \Diamond \psi_{k}, s\rangle$ from this position, $\langle \Diamond \psi_{k}, s\rangle$ must be winning for Player $0$). Take the possible world $s'$ such that $f_{0}(\Diamond\psi_{k}, s) = \langle\psi_{k}, s'\rangle$. Note that for any $n \in N_{\psi_{k}}$, since $\langle\triangledown\Psi_{n}, s\rangle$ is winning for Player $0$, the position $\langle\square\bigvee\Psi_{n}, s\rangle$ is also winning, and thus, $\langle\bigvee\Psi_{n}, s'\rangle$ is winning for Player $0$. Finally, Player $2$ creates a new marking as $\mathsf{mark}(u) := s'$, and this marking satisfies $(\dag)$ as discussed above.
- In another position $t$, Player $2$ has at most one choice and so the strategy is determined automatically. Player $2$ sets the new marking as $\mathsf{mark}(u) := \mathsf{mark}(t)$ for $u \in C(t)$.
Every marking and game position consistent with this strategy satisfies $(\dag)$. In fact, it can be easily checked that our strategy satisfies the following stronger assertion;
> $(\ddag)$: Let $\pi$ be a finite or infinite play of $\mathcal{TG}(\varphi)$ consistent with our strategy, and let $\xi := \vec{\mathsf{mark}}(\pi) \in S^{+}\cup S^{\omega}$ be the corresponding sequence of possible worlds. Then, for any trace $\mathsf{tr}$ on $\pi$, $$\langle \mathsf{tr}[1], \xi[1]\rangle \langle \mathsf{tr}[2], \xi[2]\rangle
> \langle\mathsf{tr}[3], \xi[3]\rangle\cdots$$ is a play of $\mathcal{EG}(\mathcal{S}, s_{0}, \varphi)$ which is consistent with $f_{0}$.
From $(\ddag)$, we can confirm that the above strategy is winning. Take an arbitrary play $\pi$ of $\mathcal{TG}(\varphi)$ consistent with the strategy. Suppose $\pi$ is a finite branch. In this case, for any $l \in L(\pi[|\pi|])\cap\mathsf{Lit}(\varphi)$, by $(\ddag)$, we can assume that $\mathcal{S}, s\models l$ and thus $L(\pi[|\pi|])$ must be consistent. This means that the final position $\pi[|\pi|]$ belongs to $V_{3}$ and, thus, Player $2$ wins in this play. Suppose $\pi$ is an infinite branch. In this case, by $(\ddag)$, we can assume that every trace $\mathsf{tr}$ on $\pi$ is even and, thus, $\pi$ is also even so Player $2$ wins in this play. Hence, our strategy is a winning strategy for Player $2$.
\[lem: tableau game 02\] Let $\varphi$ be a well-named formula. If Player $2$ has a winning strategy in the tableau game $\mathcal{TG}(\varphi)$, then $\varphi$ is satisfiable.
Let $\mathcal{T}_{\varphi} = (T, C, r, L)$ be a tableau for $\varphi$, and let $f_{2}$ be a winning strategy for Player $2$ in the tableau game $\mathcal{TG}(\varphi)$. Consider the tree with label $\mathcal{T}_{\varphi}|f_{2} = (T_{f_{2}}, C_{f_{2}}, r, L_{f_{2}})$ which is obtained from $\mathcal{T}_{\varphi}$ by removing all nodes of $\mathcal{T}_{\varphi}$ except those used by $f_{2}$. Here, $C_{f_{2}}$ and $L_{f_{2}}$ are appropriate restrictions of $C$ and $L$, respectively. We call the structure $\mathcal{T}_{\varphi}|f_{2}$ a *winning tree* for Player $2$ derived by $f_{2}$. We also define a Kripke model $\mathcal{S} = (S, R, \lambda)$ as follows:
Possible worlds:
: $S$ consists of all modal positions belonging to $T_{f_{2}}$.
Accessibility relation:
: For any $s, s' \in S (\subseteq T_{f_{2}})$, we have $(s, s') \in R$ if and only if $s'$ is a next modal node of $s$.
Valuation:
: For any $p \in \mathsf{Prop}$ and $s \in S$, we have $s \in \lambda(p)$ if and only if $\neg p \notin L(s)$.
Note that for any $t \in T_{f_{2}}$, there exists exactly one modal node $s \in S$ which is near $t$, and so we can denote such an $s$ by $\mathsf{mark}(t)$. From now on, we construct a winning strategy for Player $0$ of the evaluation game $\mathcal{EG}(\mathcal{S}, \mathsf{mark}(r), \varphi)$. If we accomplish this task, then the Lemma follows since, from Theorem \[the: fundamental theorem\], we have $\mathcal{S}, \mathsf{mark}(r) \models \varphi$. Note that the strategy we will construct below is not necessarily memoryless.
First, Player $0$ brings on a token and stores $\langle \varphi, r \rangle$ in that token. Subsequently, some element $\langle \psi, t \rangle \in \mathsf{Sub}(\varphi) \times T_{f_{2}}$ is stored in the token at any time. Player $0$ will replace the content in the token according to the current game position of $\mathcal{EG}(\mathcal{S}, \mathsf{mark}(r), \varphi)$. It is always the case that:
> $(\dag)$: If $\langle \psi, t \rangle$ is in the token, then one of the following four conditions is satisfied:
>
> (C1)
>
> : Current game position is $\langle \psi, \mathsf{mark}(t) \rangle$ with $\psi \in L(t)$.
>
> (C2)
>
> : Current game position is $\langle \bigwedge \Diamond \Delta' , \mathsf{mark}(t) \rangle$ with $\psi = \triangledown \Delta \in L(t)$ and $\Delta' \subseteq \Delta$.
>
> (C3)
>
> : Current game position is $\langle \square \bigvee \Delta, \mathsf{mark}(t) \rangle$ with $\psi = \triangledown \Delta \in L(t)$.
>
> (C4)
>
> : Current game position is $\langle \bigvee \Delta', \mathsf{mark}(t) \rangle$ with $\triangledown \Delta \in L(u)$, $\Delta' \subseteq \Delta$ and $\psi \in \Delta'$ where $\mathsf{mark}(t)$ is a next modal node of a modal node $u \in S$.
>
The strategy satisfying Condition $(\dag)$ is straightforward. Suppose $\langle \psi, t \rangle$ is in the token and satisfies Condition $(\dag)$, and ${\bf (C1)}$. If $\psi = \alpha \vee \beta$, then Player $0$ proceeds accordingly on the $C_{f_{2}}$-path from $t$ to a $(\vee)$-node $u$ where $\alpha \vee \beta$ is reduced to $\alpha$ or $\beta$ between $u$ and $v \in C_{f_{2}}(u)$. Then, Player $0$ chooses $\langle \alpha, \mathsf{mark}(v) \rangle (= \langle \alpha, \mathsf{mark}(t) \rangle)$ as the next position if and only if $\alpha \vee \beta$ is reduced to $\alpha$ between $u$ and $v$ and, further, replaces the content in the token by $\langle \alpha, v \rangle$ or $\langle \beta, v \rangle$ according to her choice of position. If $\psi = \alpha \wedge \beta$, then Player $1$ chooses the next position from $\langle \alpha, \mathsf{mark}(t) \rangle$ or $\langle \beta, \mathsf{mark}(t) \rangle$. Player $0$ proceeds according on $C_{f_{2}}$-path from $t$ to a $(\wedge)$-node $u$ where $\alpha \wedge \beta$ is reduced to $\alpha$ and $\beta$ between $u$ and $v \in C_{f_{2}}(u)$. Then Player $0$ replaces the content in the token to $\langle \alpha, v \rangle$ or $\langle \beta, v \rangle$ according to Player $1$’s choice of position. The case of $\psi = x \in \mathsf{Bound}(\varphi)$ and $\psi = \sigma x.\varphi_{x}(x)$, Player $0$ replaces the content of the token similarly to the above cases. If $\psi = \triangledown \Delta$, then Player $1$ chooses the next position from $\langle \bigwedge \Diamond \Delta, \mathsf{mark}(t) \rangle$ or $\langle \square \bigvee \Delta, \mathsf{mark}(t) \rangle$. In both cases, Player $0$ replaces the context in the token to $\langle \triangledown \Delta, \mathsf{mark}(t)\rangle$. Therefore either Condition ${\bf (C2)}$ or ${\bf (C3)}$ is satisfied.
Suppose ${\bf (C2)}$ is satisfied. Then, Player $0$ does nothing until the position reaches the forms $\langle \Diamond \delta, \mathsf{mark}(t) \rangle$ with $\delta \in \Delta$. In the position $\langle \Diamond \delta, \mathsf{mark}(t) \rangle$, Player $0$ seeks the node $u \in C_{f_{2}}(\mathsf{mark}(t)) (= C(\mathsf{mark}(t)))$ in which $\triangledown \Delta$ is reduced to $\delta$. Then Player $0$ chooses the position $\langle \delta, \mathsf{mark}(u) \rangle$ and replaces the content of the token to $\langle \delta, u \rangle$. This game position and the content in the token satisfy ${\bf (C1)}$.
Suppose ${\bf (C3)}$ is satisfied. In this case, Player $1$ chooses the next position $\langle \bigvee \Delta, \mathsf{mark}(u) \rangle$ with $u \in C_{f_{2}}(\mathsf{mark}(t))$. If $\triangledown \Delta \in L(t)$ is reduced to $\bigvee \Delta$ in $u$, then Player $0$ replaces the content in the token to $\langle \bigvee \Delta, u \rangle$; therefore, ${\bf (C1)}$ is satisfied in this case. If $\triangledown \Delta \in L(t)$ is reduced to $\delta \in \Delta$ in $u$, then Player $0$ replaces the content in the token to $\langle \delta, u \rangle$; therefore, ${\bf (C4)}$ is satisfied in this case.
Suppose ${\bf (C4)}$ is satisfied. In this case, from the current game position $\langle \bigvee \Delta', \mathsf{mark}(t) \rangle$ Player $0$ chooses the next position $\langle \bigvee \Delta'', \mathsf{mark}(t) \rangle$ such that $\psi \in \Delta''$. By repeating this choice, Player $0$ can reach the position $\langle \psi, \mathsf{mark}(t)\rangle$. Then, the content in the token and the current game position satisfy ${\bf (C1)}$.
Let $\xi$ be a play of $\mathcal{EG}(\mathcal{S}, \mathsf{mark}(r), \varphi)$ consistent with our strategy. If $\xi$ is finite, then for $\xi[|\xi|] = \langle l, \mathsf{mark}(t) \rangle$, we have $l \in L(\mathsf{mark}(t))$ and, thus, from the definition of $\lambda$, we have $\mathcal{S}, \mathsf{mark}(t) \models l$. This means $\xi$ is winning for player $0$. Let $\xi$ be infinite. Then, from the construction of the strategy, we can find the branch $\pi$ of $\mathcal{T}|f_{2}$ and the trace $\mathsf{tr}$ on $\pi$ such that $$\label{eq: tableau game 01}
\xi =
\langle \mathsf{tr}[1], \mathsf{mark}(\pi[1])\rangle
\langle \mathsf{tr}[2], \mathsf{mark}(\pi[2])\rangle
\langle \mathsf{tr}[3], \mathsf{mark}(\pi[3])\rangle \dots$$ Since $\pi$ is a play of the tableau game $\mathcal{TG}(\varphi)$ consistent with $f_{2}$, $\pi$ is even, and so $\mathsf{tr}$ is also even. From $(\ref{eq: tableau game 01})$ we know that $\xi$ is even and, thus, winning for Player $0$. From the above argument, $\xi$ is winning for Player $0$ in either case and, thus, our strategy is winning for Player $0$.
Automaton normal form {#subsec: automaton normal form}
---------------------
For technical reasons, we now expand our language by adding *indexed tops* $\mathsf{Top} := \{ \top_{i} \mid i \in I\}$ where $I$ is an infinite countable set of indices. Each $\top_{i}$ is treated like $\top$, e.g., $\top_{i}$ belongs to the literal, $\sim\!\top_{i} := \bot$, and for any model $\mathcal{S}$ and its world $s$, we have $\mathcal{S}, s \models \top_{i}$.
\[def: automaton normal form\]The set of an *automaton normal form* $\mathsf{ANF}$ is the smallest set of formulas defined by the following clauses:
1. If $l_{1}, \dots, l_{i} \in \mathsf{Lit}$, then $\bigwedge_{1 \leq j \leq i} l_{j} \in \mathsf{ANF}$.
2. If $\alpha \vee \beta \in \mathsf{ANF}$, $\mathsf{Bound}(\alpha)\cap\mathsf{Free}(\beta) = \emptyset$ and $\mathsf{Free}(\alpha)\cap\mathsf{Bound}(\beta) = \emptyset$, then $\alpha\vee\beta \in \mathsf{ANF}$.
3. If $\alpha(x) \in \mathsf{ANF}$ where $x$ occurs only positively in the scope of some modal operator (cover modality), and $\mathsf{Sub}(\alpha(x))$ does not contain a formula of the form $x\wedge\beta$. Then, $\sigma \vec{x}.\alpha(\vec{x}) \in \mathsf{ANF}$ where $\sigma \vec{x}.\alpha(\vec{x})$ is the abbreviation of $\sigma x_{1}. \dots \sigma x_{k}.\alpha(x_{1}, \dots, x_{k})$ as stated in Definition \[def: well-named formula\].
4. If $\Phi \subseteq \mathsf{ANF}$ is a finite set such that for any $\varphi_{1}, \varphi_{2} \in \Phi$, we have $\mathsf{Bound}(\varphi_{1})\cap\mathsf{Free}(\varphi_{2}) = \emptyset$, then $(\triangledown \Phi) \wedge (\bigwedge_{1 \leq i \leq j}l_{i}) \in \mathsf{ANF}$ where $l_{1}, \dots, l_{j} \in \mathsf{Lit}\setminus
\bigcup_{\varphi \in \Phi}\mathsf{Bound}(\varphi)$ with $0 \leq j$.
5. If $\alpha \in \mathsf{ANF}$ then $\alpha \wedge \top_{i} \in \mathsf{ANF}$.
Note that the above clauses imply $\mathsf{ANF} \subseteq \mathsf{WNF}$.
\[rem: shape of anf\]For any automaton normal form $\widehat{\varphi}$, a tableau $\mathcal{T}_{\widehat{\varphi}} = (T, C, r, L)$ for $\widehat{\varphi}$ forms very simple shapes. Indeed, for any node $t \in T$, there exists at most one formula $\widehat{\alpha} \in L(t)$ which includes some bound variables. Note that for any infinite trace $\mathsf{tr}$, $\mathsf{tr}[n]$ must include some bound variables. Consequently, for any infinite branch of the tableau for an automaton normal form, there exists a unique trace on it.
\[def: tableau bisimulation\]Let $\mathcal{T}_{\alpha} = (T, C, r, L)$ and $\mathcal{T}_{\beta} = (T', C', r', L')$ be two tableaux for some well-named formulas $\alpha$ and $\beta$. Let $T_{m}$ and $T'_{m}$ be sets of modal nodes of $\mathcal{T}_{\alpha}$ and $\mathcal{T}_{\beta}$, respectively, and let $T_{c}$ and $T'_{c}$ be a set of choice nodes of $\mathcal{T}_{\alpha}$ and $\mathcal{T}_{\beta}$, respectively. Then $\mathcal{T}_{\alpha}$ and $\mathcal{T}_{\beta}$ are said to be *tableau bisimilar* (notation: $\mathcal{T}_{\alpha}\rightleftharpoons\mathcal{T}_{\beta}$) if there exists a binary relation $Z\subseteq (T_{m}\times T'_{m})\cup(T_{c}\times T'_{c})$ satisfying the following seven conditions:
Root condition:
: $(r, r') \in Z$.
Prop condition:
: For any $t \in T_{m}$ and $t' \in T'_{m}$, if $(t, t') \in Z$, then $$(L(t)\cap\mathsf{Lit}(\alpha))\setminus\mathsf{Top} =
(L'(t')\cap\mathsf{Lit}(\beta))\setminus\mathsf{Top}.$$ Consequently $L(t)$ is consistent if and only if $L'(t')$ is consistent.
Forth condition on modal nodes:
: Take $t \in T_{m}$, $u \in T_{c}$ and $t' \in T'_{m}$ arbitrarily. If $(t, t') \in Z$ and $u \in C(t)$, then there exists $u' \in C'(t')$ such that $(u, u') \in Z$ (See Figure \[fig: forth condition\]).
![The forth conditions.[]{data-label="fig: forth condition"}](fig_tableau_bisimulation.eps){width="12cm"}
Back condition on modal nodes:
: The converse of the forth condition on modal nodes: Take $t \in T_{m}$, $t' \in T'_{m}$ and $u' \in T'_{c}$ arbitrarily. If $(t, t') \in Z$ and $u' \in C'(t')$, then there exists $u \in C(t)$ such that $(u, u') \in Z$.
Forth condition on choice nodes:
: Take $u \in T_{c}$, $t \in T_{m}$ and $u' \in T'_{c}$ arbitrarily. If $(u, u') \in Z$ and $t$ is near $u$, then there exists $t' \in T'_{m}$ such that $(t, t') \in Z$ and $t'$ is near $u'$ (See Figure \[fig: forth condition\]).
Back condition on choice nodes:
: The converse of the forth condition on choice nodes: Take $u \in T_{c}$, $u' \in T'_{c}$ and $t' \in T'_{m}$ arbitrarily. If $(u, u') \in Z$ and $t'$ is near $u'$, then there exists $t \in T_{m}$ such that $(t, t') \in Z$ and $t$ is near $u$.
Parity condition:
: Let $\pi$ and $\pi'$ be infinite branches of $\mathcal{T}_{\alpha}$ and $\mathcal{T}_{\beta}$, respectively. We say that $\pi$ and $\pi'$ are *associated* with each other if the $k$-th modal nodes $\pi[i_{k}]$ and $\pi'[i'_{k}]$ satisfy $(\pi[j_{k}], \pi'[j'_{k}]) \in Z$ for any $k \in \omega\setminus\{0\}$. For any $\pi$ and $\pi'$ which are associated with each other, we have $\pi$ is even if and only if $\pi'$ is even.
If $\mathcal{T}_{\alpha}$ and $\mathcal{T}_{\beta}$ are tableau bisimilar with $Z$, then $Z$ is called a *tableau bisimulation* from $\mathcal{T}_{\alpha}$ to $\mathcal{T}_{\beta}$.
\[rem: tableau bisimulation\]As will be shown in Lemma \[lem: basic properties of tableau bisimulation\], if $\mathcal{T}_{\alpha}$ and $\mathcal{T}_{\beta}$ are tableau bisimilar, then, $\alpha$ and $\beta$ are semantically equivalent. However, the reverse is not applied. For example, consider the following two tableaux, say $\mathcal{T}_{1}$ and $\mathcal{T}_{2}$: $$\infer[(\wedge)]
{
(p \wedge (q \vee r)) \wedge (q \vee r)
}
{
\infer[(\wedge)]
{
p \wedge (q \vee r), q \vee r
}
{
\infer[(\vee)]
{
p, q \vee r
}
{
p, q \quad \mid \quad p, r
}
}
}
\qquad
\infer[(\wedge)]
{
(p \wedge (q \vee r)) \wedge (q \vee r)
}
{
\infer[(\vee)]
{
p \wedge (q \vee r), q \vee r
}
{
\infer[(\wedge)]
{
p \wedge (q \vee r), q
}
{
\infer[(\vee)]
{
p, q \vee r, q
}
{
p, q
\quad \mid \quad
p, q, r
}
}
\quad \mid \quad
\infer[(\wedge)]
{
p \wedge (q \vee r), r
}
{
\infer[(\vee)]
{
p, q \vee r, r
}
{
p, q, r
\quad \mid \quad
p, r
}
}
}
}$$ In this example, even $\mathcal{T}_{1}$ and $\mathcal{T}_{2}$ are tableaux for the same formula $(p \wedge (q \vee r)) \wedge (q \vee r)$, there does not exist a tableau bisimulation between them. Because, $\mathcal{T}_{2}$ has leaves labeled by $\{ p, q, r\}$ but $\mathcal{T}_{1}$ does not. Note that if $\widehat{\varphi}$ is an automaton normal form, then the tableau $\mathcal{T}_{\widehat{\varphi}}$ for $\widehat{\varphi}$ is uniquely determined.
\[lem: basic properties of tableau bisimulation\] Let $\alpha$, $\beta$ be well-named formulas. If $\mathcal{T}_{\alpha}\rightleftharpoons\mathcal{T}_{\beta}$, then $\models \alpha \leftrightarrow \beta$.
First, we will introduce the notion of a *marking relation*, which is a slight generalization of the marking function discussed in the proof of Lemmas \[lem: tableau game 01\] and \[lem: tableau game 02\]. Let $\mathcal{T}_{\varphi} = (T, C, r, L)$ be a tableau for some well-named formula $\varphi$, and $\mathcal{S} = (S, R, \lambda)$ be a model and $s_{0} \in S$ be its possible world. The marking relation $\mathsf{Mark} \subseteq T \times S$ between $\mathcal{T}_{\varphi}$ and $\langle \mathcal{S}, s_{0} \rangle$ is a relation satisfying the following clauses;
- $(r, s_{0}) \in \mathsf{Mark}$
- If $(t, s) \in \mathsf{Mark}$ and $t$ is a choice node, then there exists modal node $u \in C^{\ast}(t)$ such that $u$ is near $t$ and $(u, s) \in \mathsf{Mark}$.
- If $(t, s) \in \mathsf{Mark}$ and $t$ is a modal node, then for any $u \in C(t)$, there exists $s' \in R(s)$ such that $(u, s') \in \mathsf{Mark}$.
- If $(t, s) \in \mathsf{Mark}$, $t$ is a modal node and $C(t) \neq \emptyset$, then for any $s' \in R(s)$, there exists $u \in C(t)$ such that $(u, s') \in \mathsf{Mark}$.
- For any modal node $t \in T$ and possible world $s \in S$ such that $(t, s) \in \mathsf{Mark}$, if $l \in L(t)\cap\mathsf{Lit}(\varphi)$, then $\mathcal{S}, s \models l$.
- For any infinite branch $\pi$ such that $\{ n \in \omega \mid \exists s \in S; (\pi[n], s) \in \mathsf{Mark} \}$ is infinite, $\pi$ is even.
Then the following assertion holds:
> $(\dag)$: $\mathcal{S}, s_{0} \models \varphi$ if and only if there exists a marking relation between $\mathcal{T}_{\varphi}$ and $\langle \mathcal{S}, s_{0} \rangle$.
$(\dag)$ is provable in the same method as the proofs of Lemmas \[lem: tableau game 01\] and \[lem: tableau game 02\]. We leave the proof of $(\dag)$ as an exercise to the reader.
Suppose $\mathcal{T}_{\alpha}\rightleftharpoons\mathcal{T}_{\beta}$ and so there exists a bisimulation $Z$ from $\mathcal{T}_{\alpha}$ to $\mathcal{T}_{\beta}$. Then, the converse relation $Z^{-} := \{ (t', t) \mid (t, t') \in Z \}$ is a bisimulation from $\mathcal{T}_{\beta}$ to $\mathcal{T}_{\alpha}$, and thus $\mathcal{T}_{\beta}\rightleftharpoons\mathcal{T}_{\alpha}$. Therefore, it is enough to show that $\models \alpha \rightarrow \beta$. Take a model $\mathcal{S} = (S, R, \lambda)$ and its world $s_{0}$ such that $\mathcal{S}, s_{0} \models \alpha$. Then by $(\dag)$, there exists a marking relation $\mathsf{Mark'}$ between $\mathcal{T}_{\alpha}$ and $\langle \mathcal{S}, s_{0} \rangle$. Consider the composition $$\mathsf{Mark} := Z^{-}\mathsf{Mark'} = \{ (t, s) \mid (t, t') \in Z^{-}, \; (t', s) \in \mathsf{Mark'} \}.$$ Then $\mathsf{Mark}$ is a marking relation between $\mathcal{T}_{\beta}$ and $\langle \mathcal{S}, s_{0} \rangle$; thus, from $(\dag)$, we have $\mathcal{S}, s_{0} \models \beta$. Therefore, we obtain $\models \alpha \rightarrow \beta$.
\[the: automaton normal form\] For any well-named formula $\alpha$, we can construct an automaton normal form $\mathsf{anf}(\alpha)$ such that $\mathcal{T}_{\alpha}\rightleftharpoons\mathcal{T}_{\mathsf{anf}(\alpha)}$ for some tableau $\mathcal{T}_{\alpha}$ for $\alpha$.
Let $\mathcal{T}'_{\alpha} = (T, C, r, L)$ be a tableau for a given formula $\alpha$, $\mathcal{A}_{\alpha} = (Q, \mathcal{P}(\mathsf{Sub}(\alpha)), \delta, q_{I}, \Omega)$ be an automaton as given by Lemma \[lem: automaton\], and $\mathsf{stat}(t)$ be the automaton states of $t \in T$ as defined in Definition \[def: tableau game\].
First, we construct a tableau-like structure $\mathcal{TB}_{\alpha} =
(T_{b}, C_{b}, r_{b}, L_{b}, B_{b})$ called a *tableau with back edge* from $\mathcal{T}'_{\alpha}$ as follows:
- The node $t \in T$ is called a *loop node* if;
$(\spadesuit)$
: There is a proper ancestor $u$ such that $\langle L(t), \mathsf{stat}(t)\rangle = \langle L(u), \mathsf{stat}(u)\rangle$, and
$(\heartsuit)$
: for any $v \in T$ such that $v \in C^{\ast}(u)$ and $t \in C^{\ast}(v)$, we have $\Omega(\mathsf{stat}(v)) \leq \Omega(\mathsf{stat}(t)) (= \Omega(\mathsf{stat}(u)))$.
In this situation, the node $u$ is called a *return node* of $t$. Note that for any infinite branch $\pi$ of $\mathcal{T}_{\alpha}$, there exists a loop node on $\pi$. Indeed, take $N := \max \Omega(\mathsf{Inf}(\vec{\mathsf{stat}}(\pi)))$. Then, since $\mathcal{P}(\mathsf{Sub}(\alpha)) \times Q$ is finite, there exists $\langle \Gamma, q \rangle \in \mathcal{P}(\mathsf{Sub}(\alpha)) \times Q$ such that $\Omega(q) = N$ and $$\mathcal{N} := \{ n \in \omega \mid \langle \Gamma, q \rangle = \langle L(\pi[n]),
\mathsf{stat}(\pi[n]) \rangle \}$$ is an infinite set. Take a natural number $K$ such that for any $n > K$, we have $\Omega(\mathsf{stat}(\pi[n])) \leq N$. Moreover, take $n_{1}, n_{2} \in \mathcal{N}$ such that $K < n_{1} < n_{2}$. Then, from the definitions of $\mathcal{N}$ and $K$, we have $\langle L(\pi[n_{1}]), \mathsf{stat}(\pi[n_{1}])\rangle =
\langle L(\pi[n_{2}]), \mathsf{stat}(\pi[n_{2}])\rangle$ and for any $k \in \omega$ such that $n_{1} \leq k \leq n_{2}$, $\Omega(\mathsf{stat}(\pi[k])) \leq \Omega(\mathsf{stat}(\pi[n_{2}]))$. Therefore $\pi[n_{2}]$ is a loop node with return node $\pi[n_{1}]$.
We define the set $T_{b}$ of nodes as follows: $$T_{b} := \{ t \in T \mid
\text{for any proper ancestor $u$ of $t$, $u$ is \textit{not} a loop node}\}$$ Intuitively speaking, we trace the nodes on each branch from the root and as soon as we arrive at a return node, we cut off the former branch from the tableau.
- Set $C_{b} := C|_{T_{b}\times T_{b}}$, $r_{b} := r$ and $L_{b} := L|_{T_{b}}$.
- $B_{b} := \{(t, u) \in T_{b}\times T_{b} \mid
\text{$t$ is a loop node and $u$ is a return node of $t$}\}$. An element of $B_{b}$ is called *back edge*.
By König’s lemma, we can assume that $\mathcal{TB}_{\alpha}$ is a finite structure because it has no infinite branches. The tableau with back edge is very similar to the basic tableau. In fact, the unwinding $\mathsf{UNW}_{r_{b}}(\mathcal{TB}_{\alpha})$ is a tableau for $\alpha$. Therefore, we use the terminology and concepts of the tableau, such as the concept of the parity of the sequence of nodes. From the definition of loop and return nodes (particularly Condition $(\heartsuit)$), we can assume that
> $(\dag)$: Let $\pi$ be an infinite $(C_{b}\cup B_{b})$-sequence and let $t \in T_{b}$ be the return node which appears infinitely often in $\pi$ and is nearest to the root of all such return nodes. Then, $\pi$ is even if and only if $\Omega(\mathsf{stat}(t))$ is even.
Next, we assign an automaton normal form $\mathsf{anf}(t)$ to each node $t \in T_{b}$ by using top-down fashion:
Base step:
: Let $t \in T_{b}$ be a leaf. If $t$ is not a loop node, then $t$ must be a modal node with an inconsistent label or contain no formula of the form $\triangledown \Phi$. In both cases, we assign $\mathsf{anf}(t) := \bigwedge_{1 \leq k \leq i} l_{k}$ where $\{ l_{1}, \dots, l_{i}\} = L_{b}(t)\cap\mathsf{Lit}(\alpha)$. If $t$ is a loop node, we take $x_{t} \in \mathsf{Prop}\setminus\mathsf{Sub}(\varphi)$ uniquely for each such leaf and we set $\mathsf{anf}(t) := x_{t}$.
Inductive step I:
: Suppose $t \in T_{b}$ is a $(\triangledown)$-node where $t$ is labeled by $\{ \triangledown \Psi_{1}, \dots, \triangledown \Psi_{i}, l_{1}, \dots, l_{j} \}$ with $l_{1}, \dots, l_{j} \in \mathsf{Lit}(\alpha)$, and we have already assigned the automaton normal form $\mathsf{anf}(u)$ for each child $u \in C_{b}(t)$. In this situation, we first assign $\mathsf{anf}^{-}(t)$ to $t$ as follows: $$\begin{aligned}
\mathsf{anf}^{-}(t) &:=
\triangledown \{\mathsf{anf}(u)\mid u \in C_{b}(t)\}\wedge
\left(\bigwedge_{1\leq k\leq j}l_{k}\right) \notag \\
&=\left( \bigwedge_{u \in C_{b}(t)} \Diamond \mathsf{anf}(u) \right) \wedge \square
\left( \bigvee_{1 \leq k \leq i}
\left(
\bigvee_{u \in C^{(k)}_{b}(t)}\mathsf{anf}(u)
\right)
\right) \wedge
\left(\bigwedge_{1\leq k\leq j}l_{k}\right) \label{eq: automaton normal form 1}
\end{aligned}$$ where $C^{(k)}_{b}(t)$ denotes the set of all children $u \in C_{b}(t)$ such that $\triangledown \Psi_{k}$ is reduced to some $\psi_{k} \in \Psi_{k}$ between $t$ and $u$. That is, we designate the order of disjunction in $\mathsf{anf}^{-}(t)$ for technical reasons (see Remark \[rem: automaton normal form\]). If $t$ is not a return node, then we set $\mathsf{anf}(t) := \mathsf{anf}^{-}(t)$. Alternatively, if $t$ is a return node, then let $t_{1}, \dots, t_{n}$ be all the loop nodes such that $(t_{k}, t) \in B_{b}$ $(1 \leq k \leq n)$. We set $$\begin{aligned}
\label{eq: automaton normal form 2}
\sigma_{t} := \left\{\begin{array}{ll}
\mu & \text{If $\Omega(\mathsf{stat}(t)) (= \Omega(\mathsf{stat}(t_{1})) = \dots
= \Omega(\mathsf{stat}(t_{n}))) = 1 \pmod 2$}\\
\nu & \text{If $\Omega(\mathsf{stat}(t)) (= \Omega(\mathsf{stat}(t_{1})) = \dots
= \Omega(\mathsf{stat}(t_{n}))) = 0 \pmod 2$}
\end{array}\right.
\end{aligned}$$ In this case we define $\mathsf{anf}(t)$ as $\mathsf{anf}(t) := \sigma_{t}x_{t_{1}}.\dots\sigma_{t}x_{t_{n}}.\mathsf{anf}^{-}(t)$.
Inductive step II:
: Suppose $t \in T_{b}$ is a $(\vee)$-node where, for both children $u, v \in C_{b}(t)$, we have already assigned the automaton normal forms $\mathsf{anf}(u)$ and $\mathsf{anf}(v)$, respectively. If $t$ is not a return node, then we set $\mathsf{anf}(t) := \mathsf{anf}(u) \vee \mathsf{anf}(v)$. Suppose $t$ is a return node. Let $t_{1}, \dots, t_{n}$ be all the loop nodes such that $(t_{k}, t) \in B_{b}$ $(1 \leq k \leq n)$. In this case, $\sigma_{t}$ is defined in the same way as $(\ref{eq: automaton normal form 2})$ and we define $\mathsf{anf}(t)$ as $\mathsf{anf}(t) :=
\sigma_{t}x_{t_{1}}.\dots\sigma_{t}x_{t_{n}}.\big(\mathsf{anf}(u) \vee \mathsf{anf}(v)\big)$.
Inductive step III:
: Suppose $t \in T_{b}$ is a $(\wedge)$-, $(\sigma)$- or $(\mathsf{Reg})$-node where we have already assigned the automaton normal form $\mathsf{anf}(u)$ for the child $u \in C_{b}(t)$. If $t$ is not a return node, then we assign $\mathsf{anf}(t) := \mathsf{anf}(u) \wedge \top_{t}$ where $\top_{t}$ is an indexed top which is taken uniquely for each $t \in T_{b}$. If $t$ is a return node and $t_{1}, \dots, t_{n}$ are all the loop nodes such that $(t_{k}, t) \in B_{b}$ $(1 \leq k \leq n)$, then, $\sigma_{t}$ is defined in the same way as $(\ref{eq: automaton normal form 2})$, and we define $\mathsf{anf}(t)$ as $\mathsf{anf}(t) := \sigma_{t}x_{t_{1}}.\dots\sigma_{t}x_{t_{n}}.\;\mathsf{anf}(u)$.
We take $\mathsf{anf}(\alpha) := \mathsf{anf}(r_{b})$.
Consider the structure $(T_{b}, C_{b}, r_{b}, \mathsf{anf}, B_{b})$. We intuit that this structure is almost a tableau with back edge for $\mathsf{anf}(\alpha)$. To clarify this intuition, we give a structure $\mathcal{TB}_{\mathsf{anf}(\alpha)} = (\widehat{T}, \widehat{C}, \widehat{r}, \widehat{L},
\widehat{B})$ by applying the following four steps of procedure re-formatting $(T_{b}, C_{b}, r_{b}, \mathsf{anf}, B_{b})$ so that $\mathcal{TB}_{\mathsf{anf}(\alpha)}$ can be seen as a proper tableau with back edge. At the same time, we define the relation $Z^{+} \subseteq T_{b} \times \widehat{T}$.
Step I (insert $(\sigma)$-nodes)
: Initially, we set $(\widehat{T}, \widehat{C}, \widehat{r}, \widehat{L}, \widehat{B}) :=
(T_{b}, C_{b}, r_{b}, \widehat{L}, B_{b})$ where $\widehat{L}(t) := \{ \mathsf{anf}(t) \}$, and set $Z^{+} := \{ (t, t) \mid t \in T_{b} \}$. Let $t \in \widehat{T}$ be a return node where $t_{1}, \dots, t_{n}$ are all the loop nodes such that $(t_{k}, t) \in \widehat{B}$ $(1 \leq k \leq n)$. Then, we insert the $(\sigma)$-nodes $u_{1}, \dots, u_{n}$ between $t$ and its children in such a way that $$\mathsf{anf}(t) = \sigma_{t}x_{t_{1}}.\sigma_{t}x_{t_{2}}.\dots\sigma_{t}x_{t_{n}}.
\beta(x_{t_{1}}, \dots, x_{t_{n}})$$ is reduced to $\beta(x_{t_{1}}, \dots, x_{t_{n}})$ from $u_{1}$ to $u_{n}$.[^6] Moreover, we expand the relation $Z^{+}$ by adding $\{ (t, u_{k}) \mid 1 \leq k \leq n \}$. For example, if $t$ is a $(\vee)$-node in $\mathcal{TB}_{\alpha}$ such that $\{ v_{1}, v_{2} \} = C_{b}(t)$, then our procedure would be as follows: $$\infer[\quad \Rightarrow]{\sigma_{t}x_{t_{1}}.\sigma_{t}x_{t_{2}}.\dots\sigma_{t}x_{t_{n}}.\left(
\mathsf{anf}(v_{1}) \vee \mathsf{anf}(v_{2})
\right)}
{\mathsf{anf}(v_{1}) \;\mid\; \mathsf{anf}(v_{2})} \quad
\infer[(\sigma)]
{
\sigma_{t}x_{t_{1}}.\sigma_{t}x_{t_{2}}.\dots\sigma_{t}x_{t_{n}}.\left(
\mathsf{anf}(v_{1}) \vee \mathsf{anf}(v_{2})
\right)
}
{\infer*[(\sigma)]
{
\sigma_{t}x_{t_{2}}.\dots\sigma_{t}x_{t_{n}}.\left(
\mathsf{anf}(v_{1}) \vee \mathsf{anf}(v_{2})
\right)
}
{
\infer[(\vee)]
{\mathsf{anf}(v_{1}) \vee \mathsf{anf}(v_{2})}
{\mathsf{anf}(v_{1}) \:\mid\: \mathsf{anf}(v_{2})}
}
}$$
Step II (insert $(\wedge)$-nodes)
: Let $t \in \widehat{T}$ be a node which is labeled by; $$\triangledown \{\mathsf{anf}(u)\mid u \in \widehat{C}(t)\}\wedge
\left(\bigwedge_{1\leq k\leq j}l_{k}\right).$$ Then, we insert the $(\wedge)$-nodes $u_{0}, \dots, u_{i}$ between $t'$ and its children (i.e., the nodes of $\widehat{C}(t)$) and label such $u_{1}, \dots, u_{j}$ as below: $$\infer[\quad \Rightarrow]
{\triangledown \{\mathsf{anf}(u)\mid u \in \widehat{C}(t)\}\wedge\left(\bigwedge_{1\leq k\leq j}l_{k}\right)}
{\mathsf{anf}(u) \;\mid\; u \in \widehat{C}(t)} \quad
\infer[(\wedge)]
{
\triangledown \{\mathsf{anf}(u)\mid u \in \widehat{C}(t)\}\wedge\left(\bigwedge_{1\leq k\leq j}l_{k}\right)
}
{\infer*[(\wedge)]
{
\triangledown \{\mathsf{anf}(u)\mid u \in \widehat{C}(t)\}, \left(\bigwedge_{1\leq k\leq j}l_{k}\right)
}
{
\infer[(\triangledown)]
{\triangledown \{\mathsf{anf}(u)\mid u \in \widehat{C}(t)\}, l_{1}, \dots, l_{j}}
{\mathsf{anf}(u) \;\mid\; u \in \widehat{C}(t)}
}
}$$ Further, we expand the relation $Z^{+}$ by adding $\{ (t, u_{k}) \mid 1 \leq k \leq j \}$.
Step III (revise the back edges)
: Let $t_{k}$ with $1 \leq k \leq n$ be the loop node, and $t$ be the return node of $t_{k}$ such that $$\begin{aligned}
\mathsf{anf}(t_{k})& = x_{t_{k}}\\
\mathsf{anf}(t)& = \sigma_{t}x_{t_{1}}.\sigma_{t}x_{t_{2}}.\dots\sigma_{t}x_{t_{n}}.
\beta(x_{t_{1}}, \dots, x_{t_{n}}).\end{aligned}$$ If $2 \leq k$, then we delete $(t_{k}, t)$ from $\widehat{B}$ and add $(t_{k}, u_{k})$ into $\widehat{B}$ where $u_{k}$ is the unique nodes satisfying; $$\widehat{L}(u_{k}) = \{ \sigma_{t}x_{t_{k}}.\dots\sigma_{t}x_{t_{n}}.
\beta(x_{t_{1}}, \dots, x_{t_{n}})\}.$$ By this revising procedure, for any loop node $t$ and its return node $u$, $\widehat{L}(t)$ and $\widehat{L}(u)$ form the $(\mathsf{Reg})$-rule of $\mathsf{anf}(\alpha)$.
Step IV (add the indexed tops)
: Suppose $t \in \widehat{T}$ and its child $u$ are labeled as follows; $$\infer
{\mathsf{anf}(u)\wedge \top_{t}}
{\mathsf{anf}(u)}$$ Then, we add $\top_{t}$ to $\widehat{L}(v)$ where $v \in (\widehat{C}\cup\widehat{B})^{+}(t)$ such that, between the $(\widehat{C}\cup\widehat{B})$-path from $t$ to $v$, there does not exist a $(\triangledown)$-node. By this adding procedure, such a $t$ becomes a proper $(\wedge)$-node.
The structure $\mathcal{TB}_{\mathsf{anf}(\alpha)} = (\widehat{T}, \widehat{C}, \widehat{r}, \widehat{L},
\widehat{B})$ repaired by the above four procedures can be seen as a tableau with back edge for $\mathsf{anf}(\alpha)$ in the sense that the following two assertions hold:
$(\clubsuit)$
: The unwinding $\mathsf{UNW}_{\widehat{r}}(\mathcal{TB}_{\mathsf{anf}(\alpha)})$ is a tableau of $\mathsf{anf}(\alpha)$.
$(\diamondsuit)$
: Let $\widehat{\pi}$ be an infinite $(\widehat{C}\cup\widehat{B})$-sequence and let $\widehat{t} \in \widehat{T}$ be the return node which appears infinitely often in $\widehat{\pi}$ and is nearest to the root of all such return nodes. Then $\widehat{\pi}$ is even if and only if $\widehat{L}(\widehat{t})$ includes a $\nu$-formula.
Set $Z := Z^{+}|_{((T_{b})_{m} \times \widehat{T}_{m}) \cup ((T_{b})_{c} \times \widehat{T}_{c})}$. If we extend the relation $Z$ to the pair of nodes of $\mathsf{UNW}_{r}(\mathcal{TB}_{\alpha})$ and $\mathsf{UNW}_{\widehat{r}}(\mathcal{TB}_{\mathsf{anf}(\alpha)})$, then $Z$ clearly satisfies the root condition, prop condition, back conditions and forth conditions. Moreover, from $(\dag)$ and $(\diamondsuit)$, we can assume that $Z$ satisfies the Parity condition. Therefore, we have $\mathsf{UNW}_{r}(\mathcal{TB}_{\alpha})\rightleftharpoons\mathsf{UNW}_{\widehat{r}}(\mathcal{TB}_{\mathsf{anf}(\alpha)})$, and so $\mathcal{T}_{\alpha} := \mathsf{UNW}_{r}(\mathcal{TB}_{\alpha})$ and $\mathsf{anf}(\alpha)$ satisfy the required condition.
\[rem: automaton normal form\] Let $\mathsf{Sub}'(\mathsf{anf}(\alpha))$ be the set of subformulas of $\mathsf{anf}(\alpha)$ which contains some bound variables. From the relation $Z^{+}$ constructed in the proof of Lemma \[the: automaton normal form\], we can construct a function $f$ from $\mathsf{Sub}'(\mathsf{anf}(\alpha))$ to $\mathcal{P}(\mathsf{Sub}(\alpha))$ naturally because of the following:
- for any $\widehat{\beta} \in \mathsf{Sub}'(\mathsf{anf}(\alpha))$, there exists a unique $\widehat{t} \in \widehat{T}$ such that $\widehat{\beta} \in \widehat{L}(\widehat{t})$; and
- for any $\widehat{t} \in \widehat{T}$ there exists a unique $t \in T_{b}$ such that $(t, \widehat{t}) \in Z^{+}$.
Therefore, if we define $f(\widehat{\beta}) := L(t)$ where $\widehat{\beta} \in \widehat{L}(\widehat{t})$ and $(t, \widehat{t}) \in Z^{+}$, then the function $f$ is well-defined. Moreover, let $t \in T_{b}$ be a $(\triangledown)$-node such that $L_{b}(t) = \{ \triangledown \Psi_{1}, \dots, \triangledown \Psi_{i}, l_{1}, \dots, l_{j} \}$. Then, we expand $f$ to the formula $\chi_{1}$ and $\chi_{2}$ such that $$\mathsf{anf}(u) \leq \chi_{1} \leq
\bigvee_{u \in C^{(k)}_{b}(t)}\mathsf{anf}(u) \leq
\chi_{2} \leq
\left(
\bigvee_{1 \leq k \leq i}\left(\bigvee_{u \in C^{(k)}_{b}(t)}\mathsf{anf}(u)\right)
\right),$$ for every $k$ where $1 \leq k \leq i$ and for every $u \in C^{(k)}_{b}(t)$. Now, we define $f(\chi_{2})$ as $$f(\chi_{2}) := \left\{ \bigvee \Psi_{n} \mid 1 \leq n \leq i \right\}.$$ Next, we note that for any $u \in C^{(k)}_{b}(t)$ there is a unique $\psi_{k} \in \Psi_{k}$ such that $\triangledown \Psi_{k}$ is reduced to $\psi_{k}$. We denote such a $\psi_{k}$ by $\mathsf{cor}(u)$. Suppose $\chi_{1} = \bigvee_{u \in X^{(k)}}\mathsf{anf}(u)$ where $X^{(k)} \subseteq C^{(k)}_{b}(t)$. Then we define $f(\chi_{1})$ as; $$f(\chi_{1}) := \left\{ \bigvee \Psi_{n} \mid 1 \leq n \leq i, \; n \neq k \right\} \cup
\left\{ \bigvee_{u \in X^{(k)}}\mathsf{cor}(u) \right\}.$$ Recalling Equation $(\ref{eq: automaton normal form 1})$, the reason we designated the order of disjunction in $\mathsf{anf}(t)$ is that, in conjunction with above definition of $f$, we obtain the following useful property:
(Corresponding Property):
: Consider the section of the tableau which has the root labeled by $$\left(
\bigvee_{1 \leq k \leq i}\left(\bigvee_{u \in C^{(k)}_{b}(t)}\mathsf{anf}(u)\right)
\right),$$ and every leaf labeled by some $\mathsf{anf}(u)$. Then, for any node $u$ and its children $v_{1}$ and $v_{2}$ we have (i) $f(L(u)) = f(L(v_{1})) = f(L(v_{2}))$ or, (ii) $f(L(u))$, $f(L(v_{1}))$ and $f(L(v_{2}))$ forming a $(\vee)$-rule.
Let us confirm the above property by observing a concrete example as depicted in Figure \[fig: an example of the corresponding property\].
![An example of the corresponding property.[]{data-label="fig: an example of the corresponding property"}](fig_corresponding_property.eps){width="16cm"}
In this example, the root and its children satisfy (i), and the child of the root and its children form a $(\vee)$-rule. Thus, (ii) is satisfied.
The function $f$ will be used in the proof of Part $5$ of Lemma \[lem: basic properties of tableau consequence\].
For any well-named formula $\alpha$, we can construct an automaton normal form $\mathsf{anf}(\alpha)$ which is semantically equivalent to $\alpha$. Moreover, for any $x \in \mathsf{Free}(\alpha)$ which occurs only positively in $\alpha$, it holds that $x \in \mathsf{Free}(\mathsf{anf}(\alpha))$ and $x$ occurs only positively in $\mathsf{anf}(\alpha)$.
This is an immediate consequence of Lemma \[lem: basic properties of tableau bisimulation\] and Theorem \[the: automaton normal form\].
Wide tableau {#subsec: wide tableau}
------------
Let $\varphi$ be a well-named formula. The rule of a *wide tableau* for $\varphi$ is obtained by adding the following seven rules to the rule of tableau, which are collectively called the *wide rules*: $$\infer[(\epsilon_{1})]
{\Gamma}
{\Gamma} \qquad
\infer[(\epsilon_{2})]
{\Gamma}
{\Gamma \;\mid\; \Gamma}$$ $$\infer[(\vee_{w})]{\alpha\vee\beta, \Gamma}
{\alpha, \alpha\vee\beta, \Gamma \;\mid\; \beta, \alpha\vee\beta, \Gamma} \qquad
\infer[(\wedge_{w})]{\alpha\wedge\beta, \Gamma}
{\alpha, \beta, \alpha\wedge\beta, \Gamma}$$ $$\infer[(\sigma_{w})]{\sigma_{x}x.\varphi_{x}(x), \Gamma}
{\varphi_{x}(x), \sigma_{x}x.\varphi_{x}(x), \Gamma} \qquad
\infer[(\mathsf{Reg}_{w})]{x, \Gamma}
{\varphi_{x}(x), x, \Gamma}$$ $$\infer[(\triangledown_{w})]
{\triangledown\Psi_{1}, \dots, \triangledown\Psi_{i}, l_{1}, \dots, l_{j}}
{\{\psi_{k}\}\cup\{\bigvee\Psi_{n}\mid n \in N_{\psi_{k}}\}\:\mid\: \text{For every $k \in \omega$ with
$1 \leq k \leq i$ and $\psi_{k} \in \Psi_{k}$.}}$$ where in the $(\triangledown_{w})$-rule, $l_{1}, \dots, l_{j} \in \mathsf{Lit}(\varphi)$ and, for each $\psi_{k} \in \bigcup_{1 \leq k \leq i}\Psi_{k}$ we have $N_{\psi_{k}} = \{ n \in \omega \mid 1 \leq n \leq i,\; n \neq k\}$ or $N_{\psi_{k}} = \{ n \in \omega \mid 1 \leq n \leq i\}$. Therefore, the premises of the $(\triangledown_{w})$-rule is, as with the $(\triangledown)$-rule, equal to $\sum_{1 \leq k\leq i}|\Psi_{k}|$.
A *wide tableau* for $\varphi$ (notation: $\mathcal{WT}_{\varphi}$) is the structure defined as a tableau for $\varphi$, but satisfying the following additional clause:
4. For any infinite branch $\pi$ of $\mathcal{WT}_{\varphi}$, $\{ n \in \omega \mid \text{$\pi[n]$ is $(\triangledown)$-node or $(\triangledown_{w})$-node}\}$ is an infinite set.
Clause $4$ restrains a branch that does not reach any modal node eternally by infinitely applying the wide rules except $(\triangledown_{w})$-rule.
A tableau can be considered a special case of the wide tableau, in which the wide-rules are not used. The concepts of modal and choice nodes as per Definition \[def: modal and choice nodes\] naturally extend to the wide tableau. Let $t$ be a node of some wide tableau and $u$ be its child. Then, the trace function $\mathsf{TR}_{tu}$ as per Definition \[def: trace\] is extended as follows:
- If $t$ is a $(\epsilon_{1})$- or $(\epsilon_{2})$-node where $t$ and $u$ are labeled by $\Gamma$, then we set $\mathsf{TR}_{tu}(\gamma) := \{\gamma\}$ for every $\gamma \in \Gamma$.
- If $t$ is a $(\vee_{w})$-node where the rule applied between $t$ and its children forms $$\infer[(\vee)]{\alpha\vee\beta, \Gamma}
{\alpha, \alpha\vee\beta, \Gamma \;\mid\; \beta, \alpha\vee\beta, \Gamma}$$ then we set $\mathsf{TR}_{tu}(\gamma) := \{\gamma\}$ for every $\gamma \in \Gamma$. Furthermore, we set $\mathsf{TR}_{tu}(\alpha\vee\beta) := \{\alpha, \alpha\vee\beta\}$ when $L(u) = \{\alpha, \alpha\vee\beta\}\cup\Gamma$ and set $\mathsf{TR}_{tu}(\alpha\vee\beta) := \{\beta, \alpha\vee\beta\}$ when $L(u) = \{\beta, \alpha\vee\beta\}\cup\Gamma$.
- If $t$ is a $(\wedge_{w})$-node where the rule applied between $t$ and its child forms $$\infer[(\wedge_{w})]{\alpha\wedge\beta, \Gamma}
{\alpha, \beta, \alpha\wedge\beta, \Gamma}$$ then we set $\mathsf{TR}_{tu}(\gamma) := \{\gamma\}$ for every $\gamma \in \Gamma$, and we set $\mathsf{TR}_{tu}(\alpha\wedge\beta) := \{\alpha, \beta, \alpha\wedge\beta\}$.
- If $t$ is a $(\sigma_{w})$-node where the rule applied between $t$ and its child forms $$\infer[(\sigma_{w})]{\sigma_{x}x.\varphi_{x}(x), \Gamma}
{\varphi_{x}(x), \sigma_{x}x.\varphi_{x}(x), \Gamma} \qquad$$ then we set $\mathsf{TR}_{tu}(\gamma) := \{\gamma\}$ for every $\gamma \in \Gamma$, and we set $\mathsf{TR}_{tu}(\sigma_{x}x.\varphi_{x}(x)) := \{\varphi_{x}(x), \sigma_{x}x.\varphi_{x}(x)\}$.
- If $t$ is a $(\mathsf{Reg})$-node where the rule applied between $t$ and its child forms $$\infer[(\mathsf{Reg}_{w})]{x, \Gamma}
{\varphi_{x}(x), x, \Gamma}$$ then we set $\mathsf{TR}_{tu}(\gamma) := \{\gamma\}$ for every $\gamma \in \Gamma$, and we set $\mathsf{TR}_{tu}(x) := \{\varphi_{x}(x), x\}$.
- If $t$ is a $(\triangledown_{w})$-node where the rule applied between $t$ and its children forms $$\infer[(\triangledown_{w})]
{\triangledown\Psi_{1}, \dots, \triangledown\Psi_{i}, l_{1}, \dots, l_{j}}
{\{\psi_{k}\}\cup\{\bigvee\Psi_{n}\mid n \in N_{\psi_{k}}\}\:\mid\: 1 \leq k \leq i,\;
\psi_{k} \in \Psi_{k}.}$$ Moreover, suppose $u$ is labeled by $\{\psi_{k}\}\cup\{\bigvee\Psi_{n}\mid n \in N_{\psi_{k}}\}$. Then, we set $\mathsf{TR}_{tu}(\triangledown \Psi_{k}) := \{\psi_{k}\}$ when $N_{\psi_{k}} = \{ n \in \omega \mid 1 \leq n \leq i,\; n \neq k\}$, and we set $\mathsf{TR}_{tu}(\triangledown \Psi_{k}) := \{\psi_{k}, \bigvee \Psi_{k}\}$ when $N_{\psi_{k}} = \{ n \in \omega \mid 1 \leq n \leq i\}$. We set $\mathsf{TR}_{tu}(\triangledown \Psi_{n}) := \{\bigvee\Psi_{n}\}$ for every $n \in N_{\psi_{k}}\setminus \{ k \}$, and set $\mathsf{TR}_{tu}(l_{n}) := \emptyset$ for every $n \leq j$.
Under this extended definition of the trace, the automaton $\mathcal{A}_{\varphi}$ of Lemma \[lem: automaton\] and the bisimulation of Definition \[def: tableau bisimulation\] can also be naturally extended to the wide tableau. Thus, we apply these concepts and results freely to this new structure.
\[def: inserted trace\]Let $\mathcal{WT}_{\varphi} = (T, C, r, L)$ be a wide tableau for some well-named formula $\varphi$. Let $\pi$ be a finite or infinite branch of $\mathcal{WT}$ and let $\mathsf{tr}$ be a trace on $\pi$. For technical reasons, we will need an *inserted trace* (denotation: $\mathsf{tr^{+}}$) for each trace $\mathsf{tr}$ which is constructed by the following procedure $(\dag)$ (see also Figure \[fig: an inserted trace\]);
![An inserted trace.[]{data-label="fig: an inserted trace"}](fig_inserted_trace.eps){width="8cm"}
> $(\dag)$: Suppose $\Psi = \{\psi_{0}, \psi_{1}, \dots, \psi_{k}\}$ and that $\pi[n]$ is a $(\triangledown)$- or $(\triangledown_{w})$-node in which $\mathsf{tr}[n] = \triangledown \Psi$ is reduced into $\mathsf{tr}[n+1] = \psi_{0}$. Then, we insert the sequence $$\langle \bigvee \Psi, \bigvee (\Psi\setminus\{\ \psi_{1}\}),
> \bigvee (\Psi\setminus\{\ \psi_{1}, \psi_{2}\}),
> \dots, \bigvee \{ \psi_{0}, \psi_{k-1}, \psi_{k} \},
> \bigvee \{ \psi_{0}, \psi_{k} \} \rangle$$ between $\mathsf{tr}[n]$ and $\mathsf{tr}[n+1]$.
Note that $\mathsf{tr}$ is even if and only if $\mathsf{tr}^{+}$ is even because inserted formulas are all $\vee$-formulas and, thus, the priorities of these formulas are equal to $0$ (recall Equation $(\ref{eq: priority of formulas})$). The set of inserted traces $\mathsf{TR^{+}}(\pi)$ and the set of factors of inserted traces $\mathsf{TR^{+}}(\pi[n, m])$ or $\mathsf{TR^{+}}(\pi[n], \pi[m])$ are defined similarly.
\[def: tableau consequence\]Let $\mathcal{WT}_{\alpha} = (T, C, r, L)$ and $\mathcal{WT}_{\beta} = (T', C', r', L')$ be two wide tableaux for some well-named formula $\alpha$ and $\beta$. Let $T_{m}$ and $T'_{m}$ be the set of modal nodes of $\mathcal{WT}_{\alpha}$ and $\mathcal{WT}_{\beta}$, and let $T_{c}$ and $T'_{c}$ be the set of choice nodes of $\mathcal{WT}_{\alpha}$ and $\mathcal{WT}_{\beta}$, respectively. Then $\mathcal{WT}_{\beta}$ is called a *tableau consequence* of $\mathcal{WT}_{\alpha}$ (notation: $\mathcal{WT}_{\alpha}\rightharpoonup\mathcal{WT}_{\beta}$) if there exists a binary relation $Z\subseteq (T_{m}\times T'_{m})\cup(T_{c}\times T'_{c})$ satisfying the following six conditions (here, the condition of the tableau consequence is similar to the condition of tableau bisimulation so we have illustrated the differences between these two conditions using underlines):
Root condition:
: $(r, r') \in Z$.
Prop condition:
: For any $t \in T_{m}$ and $t' \in T'_{m}$, if $(t, t') \in Z$, then $$(L(t)\cap\mathsf{Lit}(\alpha))\setminus\mathsf{Top} \uwave{\;\supseteq\;}
(L'(t')\cap\mathsf{Lit}(\beta))\setminus\mathsf{Top}.$$ Consequently, $L(t)$ is consistent $L'(t')$ is consistent.
Forth condition on modal nodes:
:
Back condition on modal nodes:
: Take $t \in T_{m}$, $t' \in T'_{m}$ and $u' \in T'_{c}$ arbitrarily. If $(t, t') \in Z$ and $u' \in C'(t')$, then or there exists $u \in C(t)$ such that $(u, u') \in Z$.
Forth condition on choice nodes:
: Take $u \in T_{c}$, $t \in T_{m}$ and $u' \in T'_{c}$ arbitrarily. If $(u, u') \in Z$ and $t$ is near $u$, then there exists $t' \in T'_{m}$ such that $(t, t') \in Z$ and $t'$ is near $u'$.
Back condition on choice nodes:
:
Parity condition:
: Let $\pi$ and $\pi'$ be infinite branches of $\mathcal{WT}_{\alpha}$ and $\mathcal{WT}_{\beta}$ respectively. If $\pi$ and $\pi'$ are associated with each other, then $\pi$ is even $\pi'$ is even.
A relation $Z$ which satisfies the above six conditions called *tableau consequence relation* from $\mathcal{WT}_{\alpha}$ to $\mathcal{WT}_{\beta}$.
Let $\mathcal{T}_{1}$ and $\mathcal{T}_{2}$ be tableaux mentioned in Remark \[rem: tableau bisimulation\], then they are not bisimilar. However, we can assume that $\mathcal{T}_{2}\rightharpoonup\mathcal{T}_{1}$. Suppose $t$ is a node of some tableau labeled by $\{ \gamma \} \cup \Gamma$ and, $u$ is a its child labeled by $\{ \gamma' \} \cup \Gamma$. Then, there exists two possibilities; $\gamma' \in \Gamma$ or $\gamma' \notin \Gamma$. We say a collision occurred between $t$ and $u$ if $\gamma' \in \Gamma$. In Remark \[rem: tableau bisimulation\], we can find collisions in $\mathcal{T}_{1}$ but cannot in $\mathcal{T}_{2}$. In general, if we construct a tableau $\mathcal{T}_{\varphi}$ for a given formula $\varphi$ so that collisions occur as many as possible, then, we have $\mathcal{WT}_{\varphi}\rightharpoonup\mathcal{T}_{\varphi}$ for any wide tableau $\mathcal{WT}_{\varphi}$ for $\varphi$. To denote this fact correctly, we introduce the following definition and lemma.
A well-named formula $\varphi$ and a set $\Gamma \subseteq \mathsf{Sub}(\varphi)$ are given. For a formula $\gamma \in \Gamma$, a closure of $\gamma$ (denotation: $\mathsf{cl}(\gamma)$) is defined as follows:
- $\gamma \in \mathsf{cl}(\gamma)$.
- If $\alpha \circ \beta \in \mathsf{cl}(\gamma)$, then $\alpha, \beta \in \mathsf{cl}(\gamma)$ where $\circ \in \{ \vee, \wedge \}$.
- If $\sigma_{x}x.\varphi_{x}(x) \in \mathsf{cl}(\gamma)$, then $\varphi_{x}(x) \in \mathsf{cl}(\gamma)$.
- If $x \in \mathsf{cl}(\gamma) \cap \mathsf{Bound}(\varphi)$, then $\varphi_{x}(x) \in \mathsf{cl}(\gamma)$.
In other words, $\mathsf{cl}(\gamma)$ is a set of all formulas $\delta$ such that for any tableau $\mathcal{T}_{\varphi} = (T, C, r, L)$ and its node $t \in T$, if $\gamma \in L(t)$, then, there is a descendant $u \in C^{\ast}(t)$ near $t$ and a trace $\mathsf{tr}$ on the $C$-sequence from $t$ to $u$ where $\mathsf{tr}[1] = \gamma$ and $\mathsf{tr}[|\mathsf{tr}|] = \delta$. We say $\gamma$ is *reducible* in $\Gamma$ if, for any $\gamma' \in \Gamma \setminus \{ \gamma \}$, we have $\gamma \notin \mathsf{cl}(\gamma')$. A tableau $\mathcal{T}_{\varphi} = (T, C, r, L)$ is said *narrow* if for any node $t \in T$ which is not modal, the reduced formula $\gamma \in L(t)$ between $t$ and its children is reducible in $L(t)$.
\[lem: narrow tableau\] For any well-named formula $\varphi$, we can construct a narrow tableau for $\varphi$.
Let $\varphi$ be a well-named formula. Then, it is enough to show that for any $\Gamma \subseteq \mathsf{Sub}(\varphi)$ which is not modal, there exists a reducible formula $\gamma \in \Gamma$. Suppose, moving toward a contradiction, that there exists $\Gamma \subseteq \mathsf{Sub}(\varphi)$ which is not modal and does not include any reducible formula. Take a formula $\gamma_{1} \in \Gamma$ such that $\mathsf{cl}(\gamma_{1}) \supsetneq \{ \gamma_{1} \}$. Since $\gamma_{1}$ is not reducible in $\Gamma$, there exists $\gamma_{2} \in \Gamma \setminus \{ \gamma_{1} \}$ such that $\gamma_{1} \in \mathsf{cl}(\gamma_{2})$. Since $\gamma_{2}$ is not reducible in $\Gamma$, there exists $\gamma_{3} \in \Gamma \setminus \{ \gamma_{2} \}$ such that $\gamma_{2} \in \mathsf{cl}(\gamma_{3})$. And so forth, we obtain the sequence $\langle \gamma_{n} \mid n \in \omega \setminus \{ 0 \} \rangle$ such that $\gamma_{n+1} \in \Gamma \setminus \{ \gamma_{n} \}$ and $\gamma_{n} \in \mathsf{cl}(\gamma_{n+1})$ for any $n \in \omega \setminus \{ 0 \}$. Since $|\Gamma|$ is finite, there exists $i, j \in \omega$ such that $1 \leq i < j$ and $\gamma_{i} = \gamma_{j}$. Consider the tableau $\mathcal{T}_{\varphi} = (T, C, r, L)$ and its node $t \in T$ such that $\gamma_{j} \in L(t)$. Then, from the definition of the closure $\mathsf{cl}$, there exists a trace $\mathsf{tr}$ on $\pi$ such that:
$(\heartsuit)$
: $\pi$ is a finite $C$-sequence starting at $t$ where $(\triangledown)$-rule nor $(\triangledown_{w})$-rule do not applied between $\pi$.
$(\clubsuit)$
: $\mathsf{tr}[1] = \mathsf{tr}[|\mathsf{tr}|] = \gamma_{j}$.
On the other hand, since $\varphi$ is well-named, for any bound variable $x \in \mathsf{Bound}(\varphi)$, $x$ is in the scope of some modal operator (cover modality) in $\varphi_{x}(x)$. Thus we have:
$(\spadesuit)$
: For any trace $\mathsf{tr}$ on $\pi$, if $(\clubsuit)$ is satisfied, then $\pi$ includes a $(\triangledown)$-node or $(\triangledown_{w})$-node.
$(\heartsuit)$ and $(\spadesuit)$ contradict each other.
The next lemma states some basic properties of the tableau consequence.
\[lem: basic properties of tableau consequence\] Let $\alpha$, $\beta$, $\gamma$ and $\varphi(x)$ be well-named formulas where $x$ appears only positively and in the scope of some modality in $\varphi(x)$. Then, we have:
1. If $\mathcal{WT}_{\alpha}\rightleftharpoons\mathcal{WT}_{\beta}$, then $\mathcal{WT}_{\alpha}\rightharpoonup\mathcal{WT}_{\beta}$.
2. If $\mathcal{WT}_{\alpha}\rightharpoonup\mathcal{WT}_{\beta}$ and $\mathcal{WT}_{\beta}\rightharpoonup\mathcal{WT}_{\gamma}$, then $\mathcal{WT}_{\alpha}\rightharpoonup\mathcal{WT}_{\gamma}$.
3. If $\mathcal{T}_{\alpha}$ is narrow, then, for any wide tableau $\mathcal{WT}_{\alpha}$ we have $\mathcal{WT}_{\alpha}\rightharpoonup\mathcal{T}_{\alpha}$.
4. For any tableau $\mathcal{T}_{\varphi(\mu \vec{x}.\varphi(\vec{x}))}$, there exists a wide tableau $\mathcal{WT}_{\mu \vec{x}.\varphi(\vec{x})}$ such that $\mathcal{T}_{\varphi(\mu \vec{x}.\varphi(\vec{x}))}
\rightleftharpoons\mathcal{WT}_{\mu \vec{x}.\varphi(\vec{x})}$.
5. For any tableau $\mathcal{T}_{\varphi(\mathsf{anf}(\alpha))}$, there exists a wide tableau $\mathcal{WT}_{\varphi(\alpha)}$ such that $\mathcal{T}_{\varphi(\mathsf{anf}(\alpha))}\rightleftharpoons\mathcal{WT}_{\varphi(\alpha)}$.
Suppose $\mathcal{WT}_{\alpha}\rightleftharpoons\mathcal{WT}_{\beta}$. Then there exists a tableau bisimulation $Z$ from $\mathcal{WT}_{\alpha}$ to $\mathcal{WT}_{\beta}$. It is easily checked that $Z$ satisfies the conditions of the tableau consequence relation from $\mathcal{WT}_{\alpha}$ to $\mathcal{WT}_{\beta}$ and, thus, $\mathcal{WT}_{\alpha}\rightharpoonup\mathcal{WT}_{\beta}$.
Suppose $\mathcal{WT}_{\alpha}\rightharpoonup\mathcal{WT}_{\beta}$ and $\mathcal{WT}_{\beta}\rightharpoonup\mathcal{WT}_{\gamma}$. Then, there is a tableau consequence relation, $Z$, from $\mathcal{WT}_{\alpha}$ to $\mathcal{WT}_{\beta}$ and there is a tableau consequence relation, $Z'$, from $\mathcal{WT}_{\beta}$ to $\mathcal{WT}_{\gamma}$. The composition $ZZ' := \{ (t, t'') \mid (t, t') \in Z,\; (t', t'') \in Z'\}$ is a tableau consequence relation from $\mathcal{WT}_{\alpha}$ to $\mathcal{WT}_{\gamma}$ and, thus $\mathcal{WT}_{\alpha}\rightharpoonup\mathcal{WT}_{\gamma}$.
Let $T_{c}$ and $T'_{c}$ be the sets of choice nodes of $\mathcal{WT}_{\alpha}$ and $\mathcal{T}_{\alpha}$, and let $T_{m}$ and $T'_{m}$ be the sets of modal nodes of $\mathcal{WT}_{\alpha}$ and $\mathcal{T}_{\alpha}$, respectively. The tableau consequence relation $Z$ is constructed inductively in a bottom-up fashion. Our construction of $Z$ satisfies the following additional property:
> $(\dag)$ For any $t \in T_{c}\cup T_{m}$ and $t' \in T'_{c}\cup T'_{m}$, if $(t, t') \in Z$ then $L(t) \supseteq L'(t')$.
For the base step, add $(r, r')$ into $Z$. This expansion indeed satisfies $(\dag)$, since $L(r) = L'(r') = \{ \alpha \}$. The inductive step is divided into two cases.
For the first case, suppose that $u \in T_{c}$ and $u' \in T'_{c}$ satisfies $(u, u') \in Z$ and $(\dag)$. From the facts $L(u) \supseteq L'(u')$ and that $\mathcal{T}_{\alpha}$ is narrow, for any $t \in T_{m}$ which is near $u$, we can find $t' \in T'_{m}$ which is near $u'$ such that $\mathsf{TR}[u, t] \Supset \mathsf{TR}[u', t']$. We add such pairs $(t, t')$ into $Z$; this expansion indeed preserves $(\dag)$. Note that it is possible that, although $L(u) = L'(u')$, our extension yields $L(t)\supsetneq L'(t')$ due to collisions and the $(\vee_{w})$-rule. For example, consider a section of a wide tableau and a tableau as depicted in Figure \[fig: an extension of the tableau consequence relation\].
![An extension of the tableau consequence relation.[]{data-label="fig: an extension of the tableau consequence relation"}](fig_tableau_consequence.eps){width="13cm"}
In this example, if $(u, u') \in Z$, we must extend it so that $Z$ includes $$\{ (t_{1}, t'_{1}), (t_{2}, t'_{1}), (t_{3}, t'_{1}), (t_{2}, t'_{2}), (t_{3}, t'_{2}), (t_{4}, t'_{2})\}$$ because of, for example, $$\xymatrix @C=5mm@R=1mm{
*{\mathsf{TR}[u, t_{2}]}&
*{=} &
*{\{ \langle \triangledown \Psi_{1}\vee\triangledown \Psi_{2},} &
*{\triangledown \Psi_{1},} &
*{\triangledown \Psi_{1}\rangle,} &\\
&
&
*{\langle \triangledown \Psi_{1}\vee\triangledown \Psi_{2},} &
*{\triangledown \Psi_{1}\vee\triangledown \Psi_{2},} &
*{\triangledown \Psi_{2}\rangle \}} &\\
&
*{\Supset} &
*{\{\langle \triangledown \Psi_{1}\vee\triangledown \Psi_{2},} &
*{\triangledown \Psi_{1}\rangle \}} &\\
&
*{=} &
*{\mathsf{TR}[u', t'_{1}].}
}$$ Thus, we have $(t_{2}, t'_{1}) \in Z$. Consequently, although $L(u) = L'(u') = \{\triangledown \Psi_{1}\vee\triangledown \Psi_{2}\}$, we have $L(t_{2}) = \{\triangledown \Psi_{1}, \triangledown \Psi_{2}\}
\supsetneq \{\triangledown \Psi_{1}\} = L'(t'_{1})$.
For the second case, suppose that $t \in T_{m}$ and $t' \in T'_{m}$ satisfy $(t, t') \in Z$ and $(\dag)$. Let $$\begin{aligned}
{2}
L(t)
&=\triangledown \Psi_{1}, \dots, \triangledown \Psi_{a},
\triangledown \Psi_{a+1}, \dots, \triangledown \Psi_{b},
&\hspace{0.1cm} l_{1}, \dots, l_{c},
&\;l_{c+1}, \dots, l_{d},\label{eq: tableau consequence 1}\\
L'(t')
&=\triangledown \Psi_{1}, \dots, \triangledown \Psi_{a},
&\hspace{0.1cm} l_{1}, \dots, l_{c},
&\label{eq: tableau consequence 2}\end{aligned}$$ with $0 \leq a, b, c, d$. If $a = 0$, then we halt the expansion of $Z$ from $(t, t')$. This halting procedure does not conflict with the forth and back conditions on modal nodes $t$ and $t'$ since $C'(t') = \emptyset$. Similarly, if $\{ l_{1}, \dots, l_{d}\}$ is inconsistent, then we halt the expansion of $Z$ from $(t, t')$. This halting procedure does not conflict with the forth and back conditions on modal nodes $t$ and $t'$ since $C(t) = \emptyset$. Suppose $a > 0$ and $\{ l_{1}, \dots, l_{d}\}$ is consistent. Then, for the back condition on modal nodes, for any $u' \in C'(t')$, we must find $u \in C(t)$ such that $(u, u') \in Z$. For $u' \in C'(t')$ which is labeled by $\{ \psi_{k} \} \cup \{ \bigvee \Psi_{n} \mid n \in N'_{\psi_{k}} \}$, we add pairs $(u, u')$ into $Z$ where $u \in C(t)$ is labeled by $\{ \psi_{k} \} \cup \{ \bigvee \Psi_{n} \mid n \in N_{\psi_{k}} \}$. This expansion clearly preserves Condition $(\dag)$. For the forth condition on modal nodes, for any $u$ which is a next modal node of $t$ we must find $u'$ which is a next modal node of $t'$ such that $(u, u') \in Z$. From $(\ref{eq: tableau consequence 1})$, $(\ref{eq: tableau consequence 2})$ and the fact that $\mathcal{T}_{\alpha}$ is narrow, for any $u$ near $t$, there exists $u'$ near $t'$ such that $\mathsf{TR^{+}}[t, u] \Supset \mathsf{TR^{+}}[t', u']$. We add such pairs $(u, u')$ into $Z$. Again, this expansion preserves $(\dag)$.
Finally, we must prove that the relation $Z$ constructed above satisfies the parity condition. Let $\pi$ and $\pi'$ be infinite branches of $\mathcal{WT}_{\alpha}$ and $\mathcal{T}_{\alpha}$, respectively, such that $\pi$ and $\pi'$ are associated with each other. Then, by the construction of $Z$, we can assume that $$\mathsf{TR^{+}}(\pi) \Supset \mathsf{TR^{+}}(\pi') \label{eq: tableau consequence 3}$$ If $\pi'$ is *not* even, then there exists an odd trace $\mathsf{tr}$ on $\pi'$. From $(\ref{eq: tableau consequence 3})$, we can assume that $\mathsf{TR^{+}}(\pi)$ includes $\mathsf{tr}^{+}$ and, thus, there exists an odd trace on $\pi$ (this is because, remember that $\mathsf{tr}$ is even if and only if $\mathsf{tr^{+}}$ is even). This means $\pi$ is also *not* even and, therefore, the parity condition is indeed satisfied.
First, recall Remark \[rem: substitution\]. Since $\varphi(x)$ is well-named, we can assume that $\varphi(\mu \vec{x}.\varphi(\vec{x}))$ is an abbreviation of $$\varphi(\mu \vec{x}_{1}. \varphi(\vec{x}_{1}), \dots, \mu \vec{x}_{k}. \varphi(\vec{x}_{k}))$$ where $\varphi(x) = \varphi(x_{1}, \dots, x_{k})[x_{1}/x, \dots, x_{k}/x]$, $x \notin \mathsf{Free}(\varphi(x_{1}, \dots, x_{k}))$ and $\vec{\mu x_{i}}. \varphi(\vec{x_{i}}) =
\mu x^{(1)}_{i}.\dots \mu x^{(k)}_{i}.\varphi(x^{(1)}_{i}, \dots, x^{(k)}_{i})$ with $1 \leq i \leq k$ are appropriate renaming formulas of $\mu \vec{x}.\varphi(\vec{x})$ so that Equations $(\ref{eq: substitutaion 01})$ through $(\ref{eq: substitutaion 05})$ are satisfied. Then, we can divide $\mathsf{Sub}(\varphi(\mu \vec{x}.\varphi(\vec{x}))$ into the following three sets of formulas, each of them pairwise disjoint; $$\begin{aligned}
\mathsf{Sub}_{1} &:=
\big\{ \alpha(\mu \vec{x}_{1}. \varphi(\vec{x}_{1}), \dots, \mu \vec{x}_{k}. \varphi(\vec{x}_{k}))
\mid \alpha(\vec{x}) \in \mathsf{Sub}(\mu \vec{x}.\varphi(\vec{x})) \big\} \setminus
\mathsf{Sub}_{3}\\
\mathsf{Sub}_{2} &:=
\bigcup_{1 \leq i \leq k}\mathsf{Sub}(\mu \vec{x}_{i}. \varphi(\vec{x}_{i})) \setminus
\Big( \big\{ \vec{\mu x_{1}}. \varphi(\vec{x}_{1}), \dots, \vec{\mu x_{k}}. \varphi(\vec{x}_{k})) \big\}
\cup \mathsf{Sub}_{3}\Big)\\
\mathsf{Sub}_{3} &:=
\{ \psi \in \mathsf{Sub}(\varphi(\mu \vec{x}. \varphi(\vec{x}))) \mid \text{$\psi$ does not
contain any bound variable.} \}\end{aligned}$$ Next, we define the function $f: \mathsf{Sub}(\varphi(\mu \vec{x}.\varphi(\vec{x}))) \rightarrow
\mathsf{Sub}(\mu \vec{x}.\varphi(\vec{x}))$ by $$\begin{aligned}
f(\psi):=
\left\{\begin{array}{ll}
\alpha(\vec{x}) & \text{if $\psi =
\alpha(\vec{\mu x_{1}}. \varphi(\vec{x_{1}}), \dots, \vec{\mu x_{k}}. \varphi(\vec{x_{k}})) \in
\mathsf{Sub}_{1}$},\\
\beta(\vec{x}) & \text{if $\psi = \beta(\vec{x}_{i}) \in \mathsf{Sub}_{2}$
with $1 \leq i \leq k$,}\\
\psi & \text{if $\psi \in \mathsf{Sub}_{3}$.}\\
\end{array}\right.\end{aligned}$$ Let $\mathcal{T}_{\varphi(\mu \vec{x}.\varphi(\vec{x}))} = (T, C, r, L)$ be a tableau for $\varphi(\mu \vec{x}.\varphi(\vec{x}))$. Consider the structure $$\mathcal{WT}_{\mu \vec{x}.\varphi(\vec{x})} =
(T \uplus \{ r_{1}, \dots, r_{k} \}, C \uplus \{ (r_{n}, r_{n+1}), (r_{k}, r) \mid 1 \leq n < k\}, r_{1}, L')$$ where $L'(r_{n}) := \{ \mu x_{n}.\dots\mu x_{k}.\varphi(\vec{x}) \}$ with $1 \leq n \leq k$ and $L'(t) := f(L(t))$ for any $t \in T$. Then, we can assume $\mathcal{WT}_{\mu \vec{x}.\varphi(\vec{x})}$ is a wide tableau for $\mu \vec{x}.\varphi(\vec{x})$. Note that, in general, wide rules are necessary in $\mathcal{WT}_{\mu \vec{x}.\varphi(\vec{x})}$; for example, consider a part of $\mathcal{T}_{\varphi(\mu \vec{x}.\varphi(\vec{x}))}$ and the corresponding part of $\mathcal{WT}_{\mu \vec{x}.\varphi(\vec{x})}$ depicted in Figure \[fig: an initial example of a corresponding wide tableau\].
![An initial example of a corresponding wide tableau.[]{data-label="fig: an initial example of a corresponding wide tableau"}](fig_corresponding_wide_tableau.eps){width="13cm"}
In this example, we assume that considering the label of a node includes $$\begin{aligned}
\psi_{1} := \sigma y.\alpha(y, \mu \vec{x}_{1}.\varphi(\vec{x}_{1}), \dots,
\mu \vec{x}_{k}.\varphi(\vec{x}_{k})) \in \mathsf{Sub}_{1}\\
\psi_{2} := \sigma y.\alpha(y, \vec{x}_{1}) \in \mathsf{Sub}_{2}\end{aligned}$$ where $f(\psi_{1}) = f(\psi_{2}) = \sigma y.\alpha(y, \vec{x})$. If we reduce $\psi_{1}$, then the corresponding label of the node on $\mathcal{WT}_{\mu \vec{x}.\varphi(\vec{x})}$ includes $\alpha(y, \vec{x})$ and $\sigma y.\alpha(y, \vec{x})$. Therefore, this case requires the $(\sigma_{w})$-rule.
Take an infinite trace $\mathsf{tr}$ of $\mathcal{T}_{\varphi(\mu \vec{x}.\varphi(\vec{x}))}$ arbitrarily. Then, from the definition of $f$, we have;
> $(\ddag)$: $\mathsf{tr}$ is even if and only if $\vec{f}(\mathsf{tr})$ is even.
Set $Z := \{ (r, r_{1}), (t, t) \mid t \in (T_{m}\cup T_{c}) \setminus \{ r \}\}$, where $T_{m} \subseteq T$ is the set of modal nodes and $T_{c} \subseteq T$ is the set of choice nodes. This relation $Z$ satisfies the conditions of tableau bisimulation; we only have to confirm the parity condition since all the other conditions are obviously satisfied. Let $\pi$ be an infinite branch of $\mathcal{T}_{\varphi(\mu \vec{x}.\varphi(\vec{x}))}$ and let $\pi'$ be an associated infinite branch of $\mathcal{WT}_{\mu \vec{x}.\varphi(\vec{x})}$. Then, from the construction of $\mathcal{WT}_{\mu \vec{x}.\varphi(\vec{x})}$ and $Z$, we can assume that $\pi[n] = \pi'[n+k]$ for every $n \in \omega\setminus\{0, 1\}$. If $\pi$ is not even, then there exists a trace $\mathsf{tr}$ on $\pi$ which is not even. Consider the sequence $\langle \mu \vec{x}.\varphi(\vec{x}), \dots, \mu x_{k}.\varphi(\vec{x}) \rangle
\vec{f}(\mathsf{tr})$. From $(\ddag)$, we can assume that this sequence is a trace on $\pi'$, which is also not even. Therefore $\pi'$ is not even. Conversely, suppose that $\pi'$ is not even. Then, there exists a trace $\mathsf{tr}'$ on $\pi'$ which is not even. Take a trace $\mathsf{tr}$ on $\pi$ such that $\langle \mu \vec{x}.\varphi(\vec{x}), \dots, \mu x_{k}.\varphi(\vec{x}) \rangle\vec{f}(\mathsf{tr}) =
\mathsf{tr}'$. Then, $\mathsf{tr}$ is also not even and, therefore, $\pi$ is not even. The above implies the parity condition of $Z$.
First, as in the proof of Part $4$, we divide $\mathsf{Sub}(\varphi(\mathsf{anf}(\alpha)))$ into three sets of formulas, each of them pairwise disjoint; $$\begin{aligned}
\mathsf{Sub}_{1} &:=
\{ \beta(\mathsf{anf}(\alpha)_{1}, \dots, \mathsf{anf}(\alpha)_{k})
\mid \beta(x_{1}, \dots, x_{k}) \in \mathsf{Sub}(\varphi(\vec{x})) \} \setminus
\mathsf{Sub}_{3}\\
\mathsf{Sub}_{2} &:=
\bigcup_{1 \leq i \leq k}\mathsf{Sub}(\mathsf{anf}(\alpha)_{i}) \setminus
\Big(
\big\{ \mathsf{anf}(\alpha)_{1}, \dots, \mathsf{anf}(\alpha)_{k} \big\} \cup
\mathsf{Sub}_{3}
\Big)\\
\mathsf{Sub}_{3} &:=
\{ \psi \in \mathsf{Sub}(\varphi(\mathsf{anf}(\alpha))) \mid \text{$\psi$ does not
contain any bound variable.} \}\end{aligned}$$ where $\varphi(x) = \varphi(x_{1}, \dots, x_{k})[x_{1}/x, \dots, x_{k}/x]$, $x \notin \mathsf{Free}(\varphi(x_{1}, \dots, x_{k}))$ and $\mathsf{anf}(\alpha)_{i}$ with $1 \leq i \leq k$ are appropriate renaming formulas of $\mathsf{anf}(\alpha)$. Recall Remark \[rem: automaton normal form\]; there we had given the partial function $f: \mathsf{Sub}(\mathsf{anf}(\alpha))
\rightarrow \mathcal{P}(\mathsf{Sub}(\alpha))$. We define the function $f^{+}: \mathsf{Sub}(\varphi(\mathsf{anf}(\alpha))) \rightarrow
\mathcal{P}(\mathsf{Sub}(\varphi(\alpha)))$ by expanding $f$ as follows; $$\begin{aligned}
f^{+}(\psi):=
\left\{\begin{array}{ll}
\{ \beta(\alpha_{1}, \dots, \alpha_{k}) \} &
\text{if $\psi = \beta(\mathsf{anf}(\alpha)_{1}, \dots, \mathsf{anf}(\alpha)_{k})
\in \mathsf{Sub}_{1}$,}\\
f(\widehat{\gamma}) & \text{if $\psi = \widehat{\gamma}_{i} \in \mathsf{Sub}_{2}$
with $1 \leq i \leq k$,}\\
\{ \psi \} & \text{if $\psi \in \mathsf{Sub}_{3}$.}\\
\end{array}\right.\end{aligned}$$ where $\alpha_{i}$ with $1 \leq i \leq k$ are appropriate renaming formulas of $\alpha$ and $\widehat{\gamma}_{i}$ with $1 \leq i \leq k$ are appropriate renaming formula of $\widehat{\gamma} \in \mathsf{Sub}(\mathsf{anf}(\alpha))$. Let $\mathcal{T}_{\varphi(\mathsf{anf}(\alpha))} = (T, C, r, L)$ be a tableau for $\varphi(\mathsf{anf}(\alpha))$. Then, we can assume the corresponding structure $\mathcal{WT}_{\varphi(\alpha)} := (T, C, r, f^{+}\circ L)$ is a wide tableau for $\varphi(\alpha)$. Note that the wide rules $(\wedge_{w})$, $(\vee_{w})$, $(\sigma_{w})$, $(\mathsf{Reg}_{w})$ and $(\triangledown_{w})$ are needed when we reduce $\chi_{1}$ where the node under consideration includes $\chi_{1}$ and $\chi_{2}$ such that $f^{+}(\chi_{1}) \cap f^{+}(\chi_{2}) \neq \emptyset$. We observe this fact by confirming a constructed example depicted in Figure \[fig: a second example of a corresponding wide tableau\].
![A second example of a corresponding wide tableau.[]{data-label="fig: a second example of a corresponding wide tableau"}](fig_example_of_corresponding_structure_01.eps){width="15cm"}
In this example, the node of $\mathcal{T}_{\varphi(\mathsf{anf}(\alpha))}$ under consideration is a $(\vee)$-node which is labeled by $\{ \widehat{\alpha}_{1}\vee\widehat{\alpha}_{2}, \widehat{\beta}_{1}\vee\widehat{\beta}_{2}\}
\cup \Gamma$ where $\widehat{\alpha}_{1}\vee\widehat{\alpha}_{2}, \widehat{\beta}_{1}\vee\widehat{\beta}_{2} \in
\mathsf{Sub}_{2}$ such that $$\begin{aligned}
f^{+}(\widehat{\alpha}_{1}\vee\widehat{\alpha}_{2}) & = A\cup \{ \psi_{1}\vee\psi_{2} \}\\
f^{+}(\widehat{\beta}_{1}\vee\widehat{\beta}_{2}) & = B\cup \{ \psi_{1}\vee\psi_{2} \}\\
f^{+}(\widehat{\alpha}_{1}) & = A\cup \{ \psi_{1}\}\\
f^{+}(\widehat{\alpha}_{2}) & = A\cup \{ \psi_{2}\}\end{aligned}$$ Thus, the corresponding labels $f^{+}\circ L$ of such nodes form the $(\vee_{w})$-rule. Moreover, note that the wide rules $(\epsilon_{1})$ and $(\epsilon_{2})$ are needed when we reduce $\chi_{1}$ to $\chi_{2}$ such that $f^{+}(\chi_{1}) = f^{+}(\chi_{2})$.
Consider the relation $Z := \{ (t, t) \mid t \in T_{m} \cup T_{c} \}$ where $T_{m}$ is the set of modal nodes, and $T_{c}$ is the set of choice nodes of $\mathcal{T}_{\varphi(\mathsf{anf}(\alpha))}$. To complete the proof, we have to show that $Z$ is a bisimulation relation from $\mathcal{T}_{\varphi(\mathsf{anf}(\alpha))}$ to $\mathcal{WT}_{\varphi(\alpha)}$. It is obvious that $Z$ satisfies the root condition, prop condition, forth conditions and back conditions. Therefore we only have to confirm the parity condition of $Z$. Let $\pi$ be an infinite branch of $\mathcal{T}_{\varphi(\mathsf{anf}(\alpha))}$. We divide the set of traces $\mathsf{TR}(\pi)$ of $\mathcal{T}_{\varphi(\mathsf{anf}(\alpha))}$ into two sets; $\mathsf{TR}_{1}(\pi)$ consists of all traces $\mathsf{tr}$ such that $\mathsf{tr}[n] \in \mathsf{Sub}_{1}$ for every $n \in \omega$, and $\mathsf{TR}_{2}(\pi)$ consists of all traces $\mathsf{tr}$ such that $\mathsf{tr}[n] \in \mathsf{Sub}_{2}$ for some $n \in \omega$. Then $\vec{f^{+}}(\mathsf{TR}_{1}(\pi)) \cup \vec{f^{+}}(\mathsf{TR}_{2}(\pi))$ is the set of all traces of $\mathcal{WT}_{\varphi(\alpha)}$ on $\pi$. Since $$\Omega_{\varphi(\alpha)}(\beta(\alpha)) =
\Omega_{\varphi(\mathsf{anf}(\alpha))}(\beta(\mathsf{anf}(\alpha)_{1}, \dots, \mathsf{anf}(\alpha)_{k}))
\pmod 2$$ for any $\beta(\mathsf{anf}(\alpha)_{1}, \dots, \mathsf{anf}(\alpha)_{k}) \in \mathsf{Sub}_{1}$, we have
> $(\spadesuit):$ $\mathsf{TR}_{1}(\pi)$ includes an odd trace if and only if $\vec{f^{+}}(\mathsf{TR}_{1}(\pi))$ includes an odd trace.
On the other hand, for any $\mathsf{tr} \in \mathsf{TR}_{2}(\pi)$, from the construction of $f^{+}$, we have;
> $(\heartsuit):$ $\mathsf{tr}$ is odd if and only if $\vec{f^{+}}(\mathsf{tr})$ includes an odd trace.
From $(\spadesuit)$ and $(\heartsuit)$, we have that $\mathsf{TR}(\pi)$ is even if and only if $\vec{f^{+}}(\mathsf{TR}(\pi))$ is even, and so the parity condition is indeed satisfied. Therefore, Part $5$ of the Lemma is true.
\[cor: wide tableau\] Let $\widehat{\alpha}(x)$ be an automaton normal form in which $x \in \mathsf{Free}(\widehat{\alpha}(x))$ appears only positively. Set $\widehat{\varphi} := \mathsf{anf}(\mu \vec{x}.\widehat{\alpha}(\vec{x}))$. Then we have $\mathcal{T}_{\widehat{\alpha}(\widehat{\varphi})}\rightharpoonup\mathcal{T}_{\widehat{\varphi}}$.
This corollary is proved using four wide tableaux; Figure \[fig: the plan for the proof of the corollary\] depicts the plan of the proof.
![The plan for the proof of the corollary.[]{data-label="fig: the plan for the proof of the corollary"}](fig_wide_tableau.eps){width="15cm"}
First, we have $\mathcal{T}_{\widehat{\alpha}(\widehat{\varphi})}\rightleftharpoons
\mathcal{WT}_{\widehat{\alpha}(\mu \vec{x}.\widehat{\alpha}(\vec{x}))}$ from Part $5$ of Lemma \[lem: basic properties of tableau consequence\]. Second, take a narrow tableau $\mathcal{T}_{\widehat{\alpha}(\mu \vec{x}.\widehat{\alpha}(\vec{x}))}$, then, we have $\mathcal{WT}_{\widehat{\alpha}(\mu \vec{x}.\widehat{\alpha}(\vec{x}))}\rightharpoonup
\mathcal{T}_{\widehat{\alpha}(\mu \vec{x}.\widehat{\alpha}(\vec{x}))}$ from Part $3$ of Lemma \[lem: basic properties of tableau consequence\]. Third, we have $\mathcal{T}_{\widehat{\alpha}(\mu \vec{x}.\widehat{\alpha}(\vec{x}))}\rightleftharpoons
\mathcal{WT}_{\mu \vec{x}.\widehat{\alpha}(\vec{x})}$ from Part $4$ of Lemma \[lem: basic properties of tableau consequence\]. Fourth, take a narrow tableau $\mathcal{T}_{\mu \vec{x}.\widehat{\alpha}(\vec{x})}$, then, we have $\mathcal{WT}_{\mu \vec{x}.\widehat{\alpha}(\vec{x})}\rightharpoonup
\mathcal{T}_{\mu \vec{x}.\widehat{\alpha}(\vec{x})}$, again from Part $3$ of Lemma \[lem: basic properties of tableau consequence\]. Fifth, the equivalence $\mathcal{T}_{\mu \vec{x}.\widehat{\alpha}(\vec{x})}\rightleftharpoons
\mathcal{T}_{\widehat{\varphi}}$ is trivial by the definition of $\widehat{\varphi}$. Finally, by applying Part $1$ and $2$ of Lemma \[lem: basic properties of tableau consequence\] repeatedly, we obtain $\mathcal{T}_{\widehat{\alpha}(\widehat{\varphi})}\rightharpoonup\mathcal{T}_{\widehat{\varphi}}$.
Completeness {#sec: completeness}
============
In this section, we prove the completeness of $\mathsf{Koz}$. In Subsection \[subsec: refutatiuon\], we give the concept of *refutation* and show that every unsatisfiable formula has a refutation. We also introduce the concept of *thin refutation* and exhibit Claim (f). In Subsection \[subsec: proof of completeness\], we prove the completeness of $\mathsf{Koz}$ by proving Claim (h) and (d), in that order.
Refutation {#subsec: refutatiuon}
----------
\[def: refutation\]A well-named formula $\varphi$ is given. *Refutation rules* for $\varphi$ are defined as the rules of tableau, but this time, we modify the set of rules by adding an explicit weakening rule: $$\infer[(\mathsf{Weak})]{\alpha, \Gamma}
{\Gamma}$$ and, instead of the $(\triangledown)$-rule, we take the following $(\triangledown_{r})$-rule: $$\infer[(\triangledown_{r})]
{\triangledown\Psi_{1}, \dots, \triangledown\Psi_{i}, l_{1}, \dots, l_{j}}
{\{\psi_{k}\}\cup\{\bigvee\Psi_{n}\mid n \in N_{\psi_{k}}\}}$$ where in the $(\vee_{w})$-rule, we have $1 \leq k \leq i$, $\psi_{k} \in \Psi_{k}$, $N_{\psi_{k}} = \{ n \in \omega \mid 1 \leq n \leq i, \; n \neq k \}$ and $l_{1}, \dots, l_{j} \in \mathsf{Lit}(\varphi)$. Therefore the $(\triangledown_{r})$-rule has one premise.
A *refutation* for $\varphi$ is a structure $\mathcal{R}_{\varphi} = (T, C, r, L)$ where $(T, C, r)$ is a tree structure and $L: T \rightarrow \mathcal{P}(\mathsf{Sub}(\varphi))$ is a *label function* satisfying the following clauses:
1. $L(r) = \{\varphi\}$.
2. Every leaf is labeled by some inconsistent set of formulas.
3. Let $t \in T$. If $L(t)$ is modal and inconsistent, then $t$ has no child. Otherwise, if $t$ is labeled by the set of formulas which fulfils the form of the conclusion of some refutation rules, then $t$ has children which are labeled by the sets of formulas of premises of those refutation rules.
4. The rule $(\triangledown_{r})$ can be applied to $t$ only if $L(t)$ is modal.
5. For any infinite branch $\pi$, $\pi$ is *odd* (not even) in the sense of Definition \[def: trace\].
\[lem: refutation\] Let $\varphi$ be a well-named formula. If $\varphi$ is not satisfiable, then there exists a refutation for $\varphi$.
From Lemmas \[lem: tableau game 01\] and \[lem: tableau game 02\], we find that $\varphi$ is not satisfiable if and only if Player $3$ has the memoryless winning strategy $f_{3}$ for the tableau game $\mathcal{TG}(\varphi)$. If Player $3$ has the memoryless the winning strategy $f_{3}$, then winning tree $\mathcal{T}_{\varphi}|f_{3}$ derived by $f_{3}$ is a refutation for $\varphi$.
Let $\varphi$ be a well-named formula, and $\preceq_{\varphi}$ be its dependency order (recall Definition \[def: alternation depth\]). Then,
- For any $\psi \in \mathsf{Sub}(\varphi)$ and $x \in \mathsf{Bound}(\varphi)$, we say $x$ is *active* in $\psi$ if there exists $y \in
\mathsf{Sub}(\psi)\cap\mathsf{Bound}(\varphi)$ such that $x \preceq_{\varphi} y$.
- A variable $x \in \mathsf{Bound}(\varphi)$ is called *aconjunctive* if, for any $\alpha\wedge\beta \in \mathsf{Sub}(\varphi_{x}(x))$, $x$ is active in at most one of $\alpha$ or $\beta$.
- $\varphi$ is called *aconjunctive* if every $x \in \mathsf{Bound}(\varphi)$ such that $\sigma_{x} = \mu$ is aconjunctive.
\[def: thin refutation\] Let $\mathcal{R}_{\varphi}$ be a refutation for some well-named formula $\varphi$. We say that $\mathcal{R}_{\varphi}$ is *thin* if, whenever a formula of the form $\alpha\wedge\beta$ is reduced, some node of the refutation and some variable is active in $\alpha$ as well as $\beta$, then at least one of $\alpha$ and $\beta$ is immediately discarded by using the $(\mathsf{Weak})$-rule.
From Definition \[def: thin refutation\] and Lemma \[lem: refutation\], it is obvious that every unsatisfiable aconjunctive formula has a thin refutation (without the $(\mathsf{Weak})$-rule). The following Theorem \[the: thin refutation\] was first proved in Kozen [@DBLP:journals/tcs/Kozen83] for the refutation of an aconjunctive formula, and then extended in the following way in Walukiewicz [@Walukiewicz2000142]. We will omit its proof.
\[the: thin refutation\] Let $\varphi$ be a well-named formula. If there exists a thin refutation for $\varphi$, then $\sim\!\varphi$ is probable in $\mathsf{Koz}$.
\[cor: completeness for anf\] Let $\widehat{\varphi}$ be an automaton normal form. Then, we have
1. $\widehat{\varphi}$ is aconjunctive.
2. If $\widehat{\varphi}$ is not satisfiable, then $\vdash \sim\!\widehat{\varphi}$.
The first assertion of the Corollary is obvious from the observation of Remark \[rem: shape of anf\]. For the second assertion, suppose that $\widehat{\varphi}$ is not satisfiable. Then, from Lemma \[lem: refutation\], there exists a refutation for $\widehat{\varphi}$. Since $\widehat{\varphi}$ is aconjunctive, this refutation is thin and, thus, we have $\vdash \sim\!\widehat{\varphi}$ from Theorem \[the: thin refutation\].
In the next Lemma, we confirm that some compositions preserve aconjunctiveness.
\[lem: composition\] Let $\varphi$, $\psi$ and $\alpha(x)$ be aconjunctive formulas where $x \in \mathsf{Prop}$ appears only positively in $\alpha(x)$. Then $\varphi\wedge\psi$, $\alpha(\varphi)$ and $\nu \vec{x}.\alpha(\vec{x})$ are also aconjunctive.
We only prove the claim concerning $\alpha(\varphi)$ and the other two claims are left as exercises for the reader. As mentioned in Remark \[rem: substitution\], $\alpha(\varphi)$ is an abbreviation of $\alpha(\varphi_{1}, \dots, \varphi_{k})$ where $\varphi_{i}$ with $1 \leq i \leq k$ are appropriate renaming formulas of $\varphi$. For our purpose, the following assertions are fundamental; $$\begin{aligned}
\mathsf{Bound}(\alpha(x))\cap\mathsf{Bound}(\varphi_{i}) = \emptyset \quad
(1 \leq \forall i \leq k)
\label{eq: composition 01}\\
\mathsf{Bound}(\alpha(x))\cap\mathsf{Free}(\varphi_{i}) = \emptyset \quad
(1 \leq \forall i \leq k)
\label{eq: composition 02}\\
\mathsf{Bound}(\varphi_{i})\cap\mathsf{Bound}(\varphi_{j}) = \emptyset \quad
(1 \leq i, j \leq k, i \neq j)
\label{eq: composition 03}\end{aligned}$$ Let $y \in \mathsf{Bound}(\alpha(\varphi))$ be a variable such that $\sigma_{y} = \mu$. From $(\ref{eq: composition 01})$ and $(\ref{eq: composition 03})$, we have $y \in \mathsf{Bound}(\alpha(x))$ or $y \in \mathsf{Bound}(\varphi_{i})$ for some $i \in \omega$ such that $1 \leq i \leq k$. If $y \in \mathsf{Bound}(\alpha(x))$, then from $(\ref{eq: composition 02})$, for every $z \in \mathsf{Bound}(\alpha(\varphi))$ such that $y \preceq_{\alpha(\varphi)} z$, we have $z \in \mathsf{Bound}(\alpha(x))$. Hence, $y$ is aconjunctive in $\alpha(\varphi)$ if and only if $y$ is aconjunctive in $\alpha(x)$. By a similar argument, we can show that if $y \in \mathsf{Bound}(\varphi_{i})$, then $y$ is aconjunctive in $\alpha(\varphi)$ if and only if $y$ is aconjunctive in $\varphi_{i}$. From the above argument and the assumptions of the Lemma, we can assume that every bound variable $y$ is aconjunctive in $\alpha(\varphi)$ and thus $\alpha(\varphi)$ is indeed aconjunctive.
Proof of completeness {#subsec: proof of completeness}
---------------------
\[lem: completeness for tableau consequence\] Let $\alpha$ be an aconjunctive formula, and $\widehat{\varphi}$ be an automaton normal form. A tableau $\mathcal{T}_{\alpha} = (T_{\alpha}, C_{\alpha}, r_{\alpha}, L_{\alpha})$ for $\alpha$ and a tableau $\mathcal{T}_{\widehat{\varphi}} =
(T_{\widehat{\varphi}}, C_{\widehat{\varphi}}, r_{\widehat{\varphi}}, L_{\widehat{\varphi}})$ for $\widehat{\varphi}$ are given. If $\mathcal{T}_{\widehat{\varphi}}$ is a tableau consequence of $\mathcal{T}_{\alpha}$, then we can construct a thin refutation $\mathcal{R}$ for $\alpha \wedge \sim\!\widehat{\varphi}\; (\equiv \; \sim\!(\alpha \rightarrow \widehat{\varphi}))$.
Let $\mathcal{T}_{\alpha}$ and $\mathcal{T}_{\widehat{\varphi}}$ be the tableaux satisfying the condition of the Lemma. Then, there exists a tableau consequence relation $Z$ from $\mathcal{T}_{\alpha}$ to $\mathcal{T}_{\widehat{\varphi}}$. Now, we will construct a thin refutation $\mathcal{R} = (T, C, r, L)$ for $\alpha \wedge\!\sim\!\widehat{\varphi}$ inductively. To facilitate the construction, we define two correspondence functions $\mathsf{Cor}_{\alpha}: T \rightarrow T_{\alpha}$ and $\mathsf{Cor}_{\widehat{\varphi}}: T \rightarrow T_{\widehat{\varphi}}$. These functions are partial and, in every considered node $t$ of $\mathcal{R}$, the following conditions are satisfied: $$\begin{aligned}
L(t) = L_{\alpha}(\mathsf{Cor}_{\alpha}(t)) \cup
\left\{ \sim \bigvee (L_{\widehat{\varphi}}(\mathsf{Cor}_{\widehat{\varphi}}(t))
\setminus \mathsf{Top}) \right\}
\label{eq: thin refutation 01} \\
(\mathsf{Cor}_{\alpha}(t), \mathsf{Cor}_{\widehat{\varphi}}(t)) \in Z \label{eq: thin refutation 02}\end{aligned}$$ Of course, the root of $\mathcal{R}$ is labeled by $\{ \alpha \wedge \sim\!\widehat{\varphi} \}$ and its child, say $t_{0}$, is labeled by $\{ \alpha, \sim\!\widehat{\varphi} \}$. For the base step, set $\mathsf{Cor}_{\alpha}(t_{0}) := r_{\alpha}$ and $\mathsf{Cor}_{\widehat{\varphi}}(t_{0}) := r_{\widehat{\varphi}}$. Then, the Condition $(\ref{eq: thin refutation 01})$ and $(\ref{eq: thin refutation 02})$ are indeed satisfied. The remaining construction is divided into two cases; the second of which will be further divided into four cases.
Inductive step I
: Suppose we have already constructed $\mathcal{R}$ up to a node $t$ where $\mathsf{Cor}_{\alpha}(t)$ and $\mathsf{Cor}_{\widehat{\varphi}}(t)$ are choice nodes of appropriate tableaux and satisfy Conditions $(\ref{eq: thin refutation 01})$ and $(\ref{eq: thin refutation 02})$. In this case, we prolong $\mathcal{R}$ up to $u$ so that:
1. $\mathsf{Cor}_{\alpha}(u)$ is a modal node of $\mathcal{T}_{\alpha}$ near $\mathsf{Cor}_{\alpha}(t)$.
2. $\mathsf{Cor}_{\widehat{\varphi}}(u)$ is a modal node of $\mathcal{T}_{\widehat{\varphi}}$ near $\mathsf{Cor}_{\widehat{\varphi}}(t)$.
3. Conditions $(\ref{eq: thin refutation 01})$ and $(\ref{eq: thin refutation 02})$ are satisfied in $u$.
4. $\mathsf{TR}[t, u] \equiv \mathsf{TR}[\mathsf{Cor}_{\alpha}(t), \mathsf{Cor}_{\alpha}(u)] \cup
\left\{ \langle \sim\!\bigvee (L_{\widehat{\varphi}}(t_{1}) \setminus \mathsf{Top}), \cdots,
\sim\!\bigvee (L_{\widehat{\varphi}}(t_{k})\setminus \mathsf{Top}) \rangle \right\}$ where $t_{1}\cdots t_{k} \in
T^{+}_{\widehat{\varphi}}$ is the $C_{\widehat{\varphi}}$-sequence starting at $\mathsf{Cor}_{\widehat{\varphi}}(t)$ and ending at $\mathsf{Cor}_{\widehat{\varphi}}(u)$.
The idea of the prolonging procedure is represented in Figure \[fig: the prolonging procedure for inductive step i\].
![The prolonging procedure for Inductive step I.[]{data-label="fig: the prolonging procedure for inductive step i"}](fig_thin_refutation_01.eps){width="12cm"}
From $t$, we first apply the tableau rules to the formulas of $\mathsf{Sub}(L_{\alpha}(\mathsf{Cor}_{\alpha}(t)))$ in the same order as they were applied from $\mathsf{Cor}_{\alpha}(t)$ and its nearest modal nodes. Then, we obtain a finite tree rooted in $t$ which is isomorphic to the section of $\mathcal{T}_{\alpha}$ between $\mathsf{Cor}_{\alpha}(t)$ and its nearest modal nodes. Therefore, for each leaf $t'$ of this section of $\mathcal{R}$, we can take unique modal node $t'_{\alpha}$ of $\mathcal{T}_{\alpha}$ that is isomorphic to $t'$. Note that $L(t') = L_{\alpha}(t'_{\alpha}) \cup
\{ \sim\!\bigvee (L_{\widehat{\varphi}}(\mathsf{Cor}_{\widehat{\varphi}}(t))\setminus\mathsf{Top}) \}$. Now, the forth condition on the choice node of $Z$ is used. From $(\ref{eq: thin refutation 02})$, we can find $t'_{\widehat{\varphi}} \in T_{\widehat{\varphi}}$ which is near $\mathsf{Cor}_{\widehat{\varphi}}(t)$ and satisfies $(t'_{\alpha}, t'_{\widehat{\varphi}}) \in Z$. Let us look at the path from $\mathsf{Cor}_{\widehat{\varphi}}(t)$ to $t'_{\widehat{\varphi}}$ in $\mathcal{T}_{\widehat{\varphi}}$. Since $\widehat{\varphi}$ is an automaton normal form on this path only the $(\vee)$-, $(\sigma)$- and $(\mathsf{Reg})$-rules, and $(\wedge)$-rules reducing $\widehat{\psi}\wedge\top_{i}$ to $\{\widehat{\psi}, \top_{i}\}$ may be applied first. Then, we have zero or more applications of the $(\wedge)$-rule. Let us apply dual rules to $\sim\!\bigvee L_{\widehat{\varphi}}(\mathsf{Cor}_{\widehat{\varphi}}(t))$ (note that $(\mathsf{Reg})$ and $(\sigma)$ are self-dual).
For an application of the $(\vee)$-rule in $\mathcal{T}_{\widehat{\varphi}}$, we apply the $(\wedge)$-rule followed by the $(\mathsf{Weak})$-rule to leave only the conjunct which appears on the path to $t'_{\widehat{\varphi}}$. In this way, we ensure the resulting path of $\mathcal{R}$ will be thin.
For an application of the $(\wedge)$-rule reducing $\widehat{\psi}\wedge\top_{i}$ to $\{\widehat{\psi}, \top_{i}\}$ in $\mathcal{T}_{\widehat{\varphi}}$, we apply the $(\vee)$-rule in $\mathcal{R}$. Then, we have two children, say $v_{1}$ and $v_{2}$ such that $L(v_{1})$ includes $\sim\!\widehat{\psi}$ and $L(v_{2})$ includes $\sim\!\top_{i} = \bot$. Since $L(v_{2})$ is inconsistent, if we further prolong $\mathcal{R}$ from $v_{2}$ to its nearest modal nodes, such modal nodes also labeled inconsistent set. This means that the modal nodes can be leaves of a refutation. We therefore stop the prolonging procedure on such modal nodes.
After these reductions, we get a node $u$ which is labeled by $L_{\alpha}(t'_{\alpha}) \cup \{ \sim \bigvee (L_{\widehat{\varphi}}(t'_{\widehat{\varphi}})\setminus
\mathsf{Top})\}$. Setting $\mathsf{Cor}_{\alpha}(u) := t'_{\alpha}$ and $\mathsf{Cor}_{\widehat{\varphi}}(u) := t'_{\widehat{\varphi}}$ establishes Conditions $(\ref{eq: thin refutation 01})$ and $(\ref{eq: thin refutation 02})$. Conditions $1$ through $4$ follow directly from the construction.
Inductive step II
: Suppose we have already constructed $\mathcal{R}$ up to a node $t$ where $\mathsf{Cor}_{\alpha}(t)$ and $\mathsf{Cor}_{\widehat{\varphi}}(t)$ are modal nodes of appropriate tableaux and satisfy Conditions $(\ref{eq: thin refutation 01})$ and $(\ref{eq: thin refutation 02})$. Note that, since $\widehat{\varphi}$ is an automaton normal form, we can put $L_{\widehat{\varphi}}(\mathsf{Cor}_{\widehat{\varphi}}(t))\setminus \mathsf{Top}
= \{ \triangledown \Psi, l_{1}, \dots, l_{i} \}$ or $L_{\widehat{\varphi}}(\mathsf{Cor}_{\widehat{\varphi}}(t))\setminus \mathsf{Top}
= \{ l_{1}, \dots, l_{i} \}$ where $l_{1}, \dots, l_{i} \in \mathsf{Lit}(\widehat{\varphi})$. Moreover, observe that $$\begin{aligned}
\sim\!\left(\triangledown \Psi \wedge \bigwedge_{1 \leq k \leq i} l_{k}\right)&
\equiv\: \sim\!\triangledown \Psi \vee \left(\bigvee_{1 \leq k \leq i}\sim\!l_{k}\right)\\
&\equiv\: \sim\!\left( \left(\bigwedge \Diamond \Psi\right) \wedge \square
\left(\bigvee \Psi\right) \right)
\vee \left(\bigvee_{1 \leq k \leq i}\sim\!l_{k}\right)\\
&\equiv\: \left(\bigvee_{\psi \in \Psi} \square \sim\!\psi\right) \vee
\Diamond \left(\bigwedge \sim\!\Psi\right)
\vee \left(\bigvee_{1 \leq k \leq i}\sim\!l_{k}\right)\\
&\equiv\: \left(\bigvee_{\psi \in \Psi} (\triangledown \{\sim\!\psi\} \vee
\triangledown \emptyset)\right) \vee
\triangledown \left\{\left(\bigwedge \sim\!\Psi\right), \top\right\}
\vee \left(\bigvee_{1 \leq k \leq i}\sim\!l_{k}\right).\end{aligned}$$ Therefore, if we prolong $\mathcal{R}$ from $t$ up to its nearest modal nodes $u$ by applying the $(\vee)$-rule repeatedly, the label of $u$ can be categorized as one of following four cases:
(Case 1):
: $L(u) = L_{\alpha}(\mathsf{Cor}_{\alpha}(t))\cup
\{ \sim\!l_{k} \}$ for some $k$ such that $1 \leq k \leq i$.
(Case 2):
: $L(u) = L_{\alpha}(\mathsf{Cor}_{\alpha}(t))\cup\{ \triangledown \emptyset \}$.
(Case 3):
: $L(u) = L_{\alpha}(\mathsf{Cor}_{\alpha}(t))\cup\{ \triangledown \{\sim\!\psi\}\}$ for some $\psi \in \Psi$.
(Case 4):
: $L(u) = L_{\alpha}(\mathsf{Cor}_{\alpha}(t))\cup
\left\{ \triangledown \left\{\left(\bigwedge\!\sim\!\Psi\right), \top\right\} \right\}$.
In every cases, it is possible that $L_{\alpha}(\mathsf{Cor}_{\alpha}(t))$ is inconsistent and, thus, $L(u)$ is also inconsistent. If this is so, all $u$ can be a leaf of a refutation. Therefore, we stop the prolonging procedure on $u$ in this case. Now, we consider the case where $L_{\alpha}(\mathsf{Cor}_{\alpha}(t))$ is consistent.
In Case $1$, the prop condition is used; by Condition $(\ref{eq: thin refutation 02})$, we have $l_{k} \in L_{\alpha}(\mathsf{Cor}_{\alpha}(t))$. Thus, $L(u)$ includes $l_{k}$ and $\sim\!l_{k}$. This means that $L(u)$ is inconsistent and so $u$ can be a leaf of a refutation. We therefore stop the prolonging procedure on $u$ in this case.
In Case $2$, the back condition on modal nodes is used. Since $C_{\widehat{\varphi}}(\mathsf{Cor}_{\widehat{\varphi}}(t)) \neq \emptyset$, it must hold that $C_{\alpha}(\mathsf{Cor}_{\alpha}(t)) \neq \emptyset$. Take $v_{\alpha} \in C_{\alpha}(\mathsf{Cor}_{\alpha}(t))$ arbitrarily. We prolong $\mathcal{R}$ from $u$ to $v \in C(u)$ in such a way that $L(v) = L_{\alpha}(v_{\alpha}) \cup \{ \bigvee \emptyset (\equiv \bot) \}$. Since $L(v)$ is inconsistent, if we further prolong $\mathcal{R}$ from $v$ to its nearest modal nodes, such modal nodes are also inconsistent. This means that the modal nodes can be a leaves of a refutation. We therefore stop the prolonging procedure on such modal nodes in this case.
In Case $3$, the back condition on modal nodes is used. Let $v_{\widehat{\varphi}}$ be a child of $\mathsf{Cor}_{\widehat{\varphi}}(t)$ such that $L_{\widehat{\varphi}}(v_{\widehat{\varphi}}) = \{ \psi \}$. Then, by Condition $(\ref{eq: thin refutation 02})$, we can find $v_{\alpha} \in C_{\alpha}(\mathsf{Cor}_{\alpha}(t))$ such that $(v_{\alpha}, v_{\widehat{\varphi}}) \in Z$. We create a new child $v$ of $u$ which is labeled by $L_{\alpha}(\mathsf{Cor}_{\alpha}(v_{\alpha})) \cup \{ \sim\!\psi\}$. Moreover, we set $\mathsf{Cor}_{\alpha}(v) := v_{\alpha}$ and $\mathsf{Cor}_{\widehat{\varphi}}(v) := v_{\widehat{\varphi}}$. This prolonging procedure preserves Conditions $(\ref{eq: thin refutation 01})$ and $(\ref{eq: thin refutation 02})$. Note that, in this case, $\mathsf{Cor}_{\alpha}(v)$ and $\mathsf{Cor}_{\widehat{\varphi}}(v)$ are choice nodes of appropriate tableaux.
In Case $4$, the forth condition on modal nodes is used. The idea of the prolonging procedure is represented in Figure \[fig: the prolonging procedure for case 4\].
![The prolonging procedure for Case $4$.[]{data-label="fig: the prolonging procedure for case 4"}](fig_thin_refutation_02.eps){width="14cm"}
Let $L_{\alpha}(\mathsf{Cor}_{\alpha}(t)) =
\{ \triangledown \Delta_{1}, \dots, \triangledown \Delta_{i}, l_{1}, \dots, l_{j}\}$. In this case, we first create a new child $v$ of $u$ such that $$L(v) = \left\{
\bigvee \Delta_{1}, \dots, \bigvee \Delta_{i}
\right\} \cup
\left\{
\bigwedge\!\sim\!\Psi
\right\}.$$ From the choice node $v$, we further prolong $\mathcal{R}$ up to its nearest modal nodes $t'$ so that
5. $\mathsf{Cor}_{\alpha}(t')$ is a next modal node of $\mathsf{Cor}_{\alpha}(t)$.
6. $\mathsf{Cor}_{\widehat{\varphi}}(t')$ is a next modal node of $\mathsf{Cor}_{\widehat{\varphi}}(t)$.
7. Condition $(\ref{eq: thin refutation 01})$ and $(\ref{eq: thin refutation 02})$ are satisfied in $t'$.
8. $\mathsf{TR}[u, t'] \equiv \mathsf{TR}^{+}[\mathsf{Cor}_{\alpha}(t), \mathsf{Cor}_{\alpha}(t')] \cup
\{ \langle
\triangledown \left\{\left(\bigwedge\!\sim\!\Psi\right), \top\right\},
\bigwedge\!\sim\!\Psi, \dots,
\sim\!\psi =
\sim\!\bigvee (L_{\widehat{\varphi}}(t_{1})\setminus \mathsf{Top}), \cdots,
\sim\!\bigvee (L_{\widehat{\varphi}}(t_{k})\setminus \mathsf{Top})
\rangle \}$ where $t_{1}\cdots t_{k} \in T_{\widehat{\varphi}}^{+}$ is the $C_{\widehat{\varphi}}$-sequence starting at the child of $\mathsf{Cor}_{\widehat{\varphi}}(t)$ labeled by $\{ \psi \}$ and ending at $\mathsf{Cor}_{\widehat{\varphi}}(t')$.
Next, we apply $(\vee)$-rules to $\bigvee \Delta_{1}$ repeatedly until we arrive at the node $w$ such that $$L(w) =
\{ \delta_{1} \} \cup
\left\{ \bigvee \Delta_{2}, \dots, \bigvee \Delta_{i} \right\} \cup
\left\{ \bigwedge\!\sim\!\Psi \right\}$$ where $\delta_{1} \in \Delta_{1}$. Note that there exists $w_{\alpha} \in C_{\alpha}(\mathsf{Cor}_{\alpha}(t))$ such that $$L_{\alpha}(w_{\alpha}) =
\{ \delta_{1} \} \cup
\left\{ \bigvee \Delta_{2}, \dots, \bigvee \Delta_{i} \right\}$$ From $w$, we apply the tableau rules to formulas of $\mathsf{Sub}(L_{\alpha}(w_{\alpha}))$ in the same order as they were applied from $w_{\alpha}$ and its nearest modal nodes. Then, we obtain a finite tree rooted in $w$ which is isomorphic to the section of $\mathcal{T}_{\alpha}$ between $w_{\alpha}$ and nearest modal nodes. Therefore, for each leaf $u'$ of this section of $\mathcal{R}$, we can take a unique modal node $u'_{\alpha}$ of $\mathcal{T}_{\alpha}$ which is isomorphic to $u'$. Note that $L(u') = L_{\alpha}(u'_{\alpha}) \cup
\left\{
\bigwedge\!\sim\!\Psi
\right\}$. Since $u'_{\alpha}$ is a next modal node of $\mathsf{Cor}_{\alpha}(t)$, from Condition $(\ref{eq: thin refutation 02})$ and the forth condition on modal nodes, we can assume that there exists $u'_{\widehat{\varphi}}$ which is a next modal node of $\mathsf{Cor}_{\widehat{\varphi}}(t)$ and satisfies $(u'_{\alpha}, u'_{\widehat{\varphi}}) \in Z$. We will now look at the path from $\mathsf{Cor}_{\widehat{\varphi}}(t)$ to $t'_{\widehat{\varphi}}$ in $\mathcal{T}_{\widehat{\varphi}}$ and exploit $(\wedge)$-rules and $(\mathsf{Weak})$-rules so that the trace $\mathsf{tr}$ on this path satisfies Condition $8$. Finally, we get a node $t'$ which is labeled by $L_{\alpha}(u'_{\alpha})\cup \{ \sim \! \bigvee
L_{\widehat{\varphi}}(u'_{\widehat{\varphi}}) \}$. Setting $\mathsf{Cor}_{\alpha}(t') := u'_{\alpha}$ and $\mathsf{Cor}_{\widehat{\varphi}}(t') := u'_{\widehat{\varphi}}$ establishes Conditions $(\ref{eq: thin refutation 01})$ and $(\ref{eq: thin refutation 02})$. Then, Conditions $5$ through $8$ follow directly from the construction.
The above two procedures completely describe $\mathcal{R}$. All the leaves are labeled by an inconsistent set. Moreover, take an infinite branch $\pi$ of $\mathcal{R}$ arbitrarily. Let $\pi_{\alpha}$ be the branch of $\mathcal{T}_{\alpha}$ such that $\{ n \in \omega \mid \mathsf{Cor}_{\alpha}(\pi) = \pi_{\alpha}[n] \}$ is an infinite set. Let $\pi_{\widehat{\varphi}}$ be the branch of $\mathcal{T}_{\widehat{\varphi}}$ such that $\{ n \in \omega \mid \mathsf{Cor}_{\widehat{\varphi}}(\pi) = \pi_{\widehat{\varphi}}[n] \}$ is an infinite set. For any trace $\mathsf{tr} \in \mathsf{TR}(\pi)$, we have $\mathsf{tr}[1] = \alpha \wedge\!\sim\!\widehat{\varphi}$ and, $\mathsf{tr}[2] = \alpha$ or $\mathsf{tr}[2] = \sim\!\widehat{\varphi}$. $\mathsf{TR}_{1}(\pi)$ denotes the set of all the trace $\mathsf{tr} \in \mathsf{TR}(\pi)$ such that $\mathsf{tr}[2] = \alpha$. $\mathsf{tr}_{2} \in \mathsf{TR}(\pi)$ denotes the trace such that $\mathsf{tr}_{2}[2] = \sim\!\widehat{\varphi}$. Then, from the construction of $\mathcal{R}$, we have;
(T1)
: $\mathsf{TR}(\pi) = \mathsf{TR}_{1}(\pi) \cup \{ \mathsf{tr}_{2} \}$.
(T2)
: $\mathsf{TR}^{+}_{1}(\pi) \equiv \mathsf{TR}^{+}(\pi_{\alpha})$.
(T3)
: $\mathsf{tr}_{2}$ is even if and only if $\pi_{\widehat{\varphi}}$ is odd.
(T4)
: $\pi_{\alpha}$ and $\pi_{\widehat{\varphi}}$ are associated with each other.
Above conditions imply that $\pi$ is odd. Indeed, if $\pi_{\alpha}$ is odd, then, from [**(T2)**]{}, $\pi$ is also odd. If $\pi_{\alpha}$ is even, then, from [**(T4)**]{}, $\pi_{\widehat{\varphi}}$ is also even. Therefore, from [**(T3)**]{}, $\mathsf{tr}_{2}$ is odd. From [**(T1)**]{}, we can assume that $\pi$ is odd. $\mathcal{R}$ is also thin because $\alpha$ is aconjunctive and whenever we reduce a $\wedge$-formula originated from $\sim\!\widehat{\varphi}$, we leave only one conjunction and discard the other by applying $(\mathsf{Weak})$-rule. Therefore, $\mathcal{R}$ is a thin refutation as required.
\[lem: completeness\] For any well-named formula $\varphi$, there exists a semantically equivalent automaton normal form $\widehat{\varphi}$ such that $\varphi \rightarrow \widehat{\varphi}$ is provable in $\mathsf{Koz}$. Moreover, for any $x \in \mathsf{Free}(\varphi)$ which occurs only positively in $\varphi$, it hold that $x \in \mathsf{Free}(\widehat{\varphi})$ and $x$ occurs only positively in $\widehat{\varphi}$.
We prove the lemma by the induction on the structure of $\varphi$.
Case: $\varphi \in \mathsf{Lit}$.
: In this case, $\widehat{\varphi}$ is just $\varphi$.
Case: $\varphi = \alpha \vee \beta$.
: By the induction assumption, there exist automaton normal forms $\widehat{\alpha}$ and $\widehat{\beta}$ which are equivalent to $\alpha$ and $\beta$, respectively, such that $\vdash \alpha \rightarrow \widehat{\alpha}$ and $\vdash \beta \rightarrow \widehat{\beta}$. Set $\widehat{\varphi} := \widehat{\alpha} \vee \widehat{\beta}$. Then, we have $\vdash \alpha \vee \beta \rightarrow \widehat{\varphi}$.
Case: $\varphi = \triangledown \Psi$.
: This case is very similar to the previous one.
Case: $\varphi = \alpha \wedge \beta$.
: By the induction assumption, there exist automaton normal forms $\widehat{\alpha}$ and $\widehat{\beta}$ which are equivalent to $\alpha$ and $\beta$ respectively, such that $\vdash \alpha \rightarrow \widehat{\alpha}$ and $\vdash \beta \rightarrow \widehat{\beta}$; thus, we have $\vdash \alpha \wedge \beta \rightarrow \widehat{\alpha} \wedge \widehat{\beta}$. Set $\widehat{\varphi} := \mathsf{anf}(\widehat{\alpha} \wedge \widehat{\beta})$. Then, from Theorem \[the: automaton normal form\], we have $\mathcal{T}_{\widehat{\alpha} \wedge \widehat{\beta}} \rightleftharpoons \mathcal{T}_{\widehat{\varphi}}$ and, thus, $\mathcal{T}_{\widehat{\alpha} \wedge \widehat{\beta}} \rightharpoonup \mathcal{T}_{\widehat{\varphi}}$. On the other hand, by Lemma \[lem: composition\], we can assume that $\widehat{\alpha} \wedge \widehat{\beta}$ is aconjunctive. From Lemma \[lem: completeness for tableau consequence\] and Theorem \[the: thin refutation\], we have $\vdash \widehat{\alpha} \wedge \widehat{\beta} \rightarrow \widehat{\varphi}$. Therefore, we have $\vdash \alpha \wedge \beta \rightarrow \widehat{\varphi}$.
Case: $\varphi = \nu x_{1}.\dots\nu x_{k}.\alpha(x_{1}, \dots, x_{k})$.
: By the induction assumption, we have an equivalent automaton normal form $\widehat{\alpha}(x)$ of $\alpha(x)$ such that $\vdash \alpha(x) \rightarrow \widehat{\alpha}(x)$. Therefore, $\vdash \nu \vec{x}.\alpha(\vec{x}) \rightarrow \nu \vec{x}.\widehat{\alpha}(\vec{x})$. Set $\widehat{\varphi} := \mathsf{anf}(\nu\vec{x}.\widehat{\alpha}(\vec{x}))$. Then, from Theorem \[the: automaton normal form\], we have $\mathcal{T}_{\nu \vec{x}.\widehat{\alpha}(\vec{x})} \rightleftharpoons \mathcal{T}_{\widehat{\varphi}}$ and, thus, $\mathcal{T}_{\nu \vec{x}.\widehat{\alpha}(\vec{x})} \rightharpoonup \mathcal{T}_{\widehat{\varphi}}$. On the other hand, by Lemma \[lem: composition\], we can assume that $\nu \vec{x}.\widehat{\alpha}(\vec{x})$ is aconjunctive. From Lemma \[lem: completeness for tableau consequence\] and Theorem \[the: thin refutation\], we have $\vdash \nu \vec{x}.\widehat{\alpha}(\vec{x}) \rightarrow \widehat{\varphi}$. Therefore, $\vdash \nu \vec{x}.\alpha(\vec{x}) \rightarrow \widehat{\varphi}$.
Case: $\varphi = \mu x_{1}.\dots\mu x_{k}.\alpha(x_{1}, \dots, x_{k})$.
: By the induction assumption, we have an equivalent automaton normal form $\widehat{\alpha}(x)$ of $\alpha(x)$ such that $\vdash \alpha(x) \rightarrow \widehat{\alpha}(x)$. Therefore, $\vdash \mu \vec{x}.\alpha(\vec{x}) \rightarrow \mu \vec{x}.\widehat{\alpha}(\vec{x})$. Set $\widehat{\varphi} := \mathsf{anf}(\mu \vec{x}.\widehat{\alpha}(\vec{x}))$. Then, from Corollary \[cor: wide tableau\], we have $\mathcal{T}_{\widehat{\alpha}(\widehat{\varphi})} \rightharpoonup \mathcal{T}_{\widehat{\varphi}}$. On the other hand, by Lemma \[lem: composition\], we can assume that $\widehat{\alpha}(\widehat{\varphi})$ is aconjunctive. From Lemma \[lem: completeness for tableau consequence\] and Theorem \[the: thin refutation\], $\vdash \widehat{\alpha}(\widehat{\varphi}) \rightarrow \widehat{\varphi}$. By applying the $(\mathsf{Ind})$-rule, we obtain $\vdash \mu \vec{x}.\widehat{\alpha}(\vec{x}) \rightarrow \widehat{\varphi}$. Thus, $\vdash \mu \vec{x}.\alpha(\vec{x}) \rightarrow \widehat{\varphi}$.
Hence, we have proved the Lemma for all cases.
For any formula $\varphi$, if $\varphi$ is not satisfiable, then $\sim\!\varphi$ is provable in $\mathsf{Koz}$.
Let $\varphi$ be an unsatisfiable formula. By Part $5$ of Lemma \[lem: basic properties of KOZ 01\], we can construct a well-named formula $\mathsf{wnf}(\varphi)$ such that $$\label{eq: completeness 01}
\vdash \varphi \leftrightarrow \mathsf{wnf}(\varphi)$$ On the other hand, from Lemma \[lem: completeness\], there exists an automaton normal form $(\mathsf{wnf}(\varphi))\verb|^|$ which is semantically equivalent to $\mathsf{wnf}(\varphi)$ and thus to $\varphi$ such that $$\label{eq: completeness 02}
\vdash \mathsf{wnf}(\varphi) \rightarrow (\mathsf{wnf}(\varphi))\verb|^|$$ Since $(\mathsf{wnf}(\varphi))\verb|^|$ is not satisfiable, by Corollary \[cor: completeness for anf\] we have $$\label{eq: completeness 03}
\vdash (\mathsf{wnf}(\varphi))\verb|^| \rightarrow \bot$$ Finally by combining Equations $(\ref{eq: completeness 01})$ through $(\ref{eq: completeness 03})$ we obtain $\vdash \varphi \rightarrow \bot$ as required.
[^1]: In the modal $\mu$-calculus, the term *state* is preferred to *possible world* since it originated in the area of verification of computer systems. However, we do not use this terminology since it is reserved for *state of automata* in this article.
[^2]: The difficulties of the proof have been pointed out, e.g., see [@Bradfield07modalmu-calculi; @DBLP:journals/logcom/Alberucci09; @Bezhanishvili20121; @DBLP:conf/fossacs/CateF10; @Lenzi05]
[^3]: In the original article [@conf/mfcs/JaninW95], this class of formulas was called the *disjunctive formula*; however, the term *automaton normal form* is the currently used terminology, to the author’s knowledge.
[^4]: More precisely, this assertion must be stated more generally to be applicable in other cases of an inductive step, see Lemma \[lem: completeness for tableau consequence\].
[^5]: In Kozen’s original article [@DBLP:journals/tcs/Kozen83], the system $\mathsf{Koz}$ was defined as the axiomatization of the equational theory. Nevertheless we present $\mathsf{Koz}$ as an equivalent Tait-style calculus due to the calculus’ affinity with the tableaux discussed in the sequel.
[^6]: In other words, we add $u_{1}, \dots, u_{n}$ into $\widehat{T}$, add $(t, u_{1}), (u_{1}, u_{2}), \dots, (u_{n-1}, u_{n})$ and $\{ (u_{n}, u) \mid u \in \widehat{C}(t) \}$ into $\widehat{C}$, discard $\{ (t, u) \mid u \in \widehat{C}(t) \}$ from $\widehat{C}$, and expand $\widehat{L}$ to $u_{1}, \dots, u_{n}$ appropriately.
|
---
abstract: 'Recent observations of the Ly$\alpha$ forest show large-scale spatial variations in the intergalactic Ly$\alpha$ opacity that grow rapidly with redshift at $z>5$, far in excess of expectations from empirically motivated models. Previous studies have attempted to explain this excess with spatial fluctuations in the ionizing background, but found that this required either extremely rare sources or problematically low values for the mean free path of ionizing photons. Here we report that much – or potentially all – of the observed excess likely arises from residual spatial variations in temperature that are an inevitable byproduct of a patchy and extended reionization process. The amplitude of opacity fluctuations generated in this way depends on the timing and duration of reionization. If the entire excess is due to temperature variations alone, the observed fluctuation amplitude favors a late-ending but extended reionization process that was roughly half complete by $z\sim9$ and that ended at $z\sim6$. In this scenario, the highest opacities occur in regions that reionized earliest, since they have had the most time to cool, while the lowest opacities occur in the warmer regions that reionized most recently. This correspondence potentially opens a new observational window into patchy reionization.'
author:
- 'Anson D’Aloisio$^\dagger$, Matthew McQuinn, & Hy Trac'
title: |
Large opacity variations in the high-redshift Ly$\alpha$ forest:\
the signature of relic temperature fluctuations from patchy reionization
---
Introduction
============
When the first galaxies emerged $\approx100 - 500$ million years after the Big Bang, their starlight reionized and heated the intergalactic hydrogen that had existed since cosmological recombination. Much is currently unknown about this process, including what spatial structure it had, when it started and completed, and even which sources drove it. The Ly$\alpha$ forest provides one of the only robust constraints on this process, showing that it was at least largely complete by $z\approx6$, when the Universe was one billion years old [@fan06; @2008MNRAS.386..359G; @mcgreer15]. This Letter argues that there exists another, potentially groundbreaking signature of reionization in the Ly$\alpha$ forest data.
The amount of absorption in the Ly$\alpha$ forest can be quantified by the effective optical depth, ${\tau_{\mathrm{eff}}}\equiv-\ln\langle F\rangle_L$, where $F\equiv\exp[-\tau_{\rm Ly\alpha}(x)]$ is the transmitted fraction of a quasar’s flux, $\langle...\rangle_L$ indicates an average over a segment of the forest of length $L$, and $\tau_{\rm Ly\alpha}(x)$ is the optical depth in Ly$\alpha$ at location $x$ along a sightline. The optical depth, $\tau_{\rm Ly\alpha}(x)$, scales approximately as the [H[ i]{}]{} number density, which after reionization scales as $T^{-0.7}\Delta_b^2/\Gamma$. Here, $T$ is temperature, $\Delta_b(x)$ is the gas density in units of the cosmic mean, and $\Gamma(x)$ is the [H[ i]{}]{} photoionization rate, which scales with the amplitude of the local ionizing radiation background.
Observations of high-$z$ quasars show a steep increase in the dispersion of ${\tau_{\mathrm{eff}}}$ among coeval forest segments around $z=6$ [@fan06; @2015MNRAS.447.3402B]. In the limit of a uniform ionizing background, the well-understood fluctuations in $\Delta_b$ fall well short of producing the observed dispersion at $z\gtrsim5.5$, as shown recently by @2015MNRAS.447.3402B (hereafter B2015). Previous studies have attempted to explain this excess with spatial fluctuations in the ionizing background. The properties of spatial fluctuations in the background depend on the number density of sources and the mean free path of photons, ${\lambda_{\mathrm{mfp}}^{912}}$. While ${\lambda_{\mathrm{mfp}}^{912}}$ is well constrained at $z<5.2$, being too large to yield significant background fluctuations for standard source models [@2014MNRAS.445.1745W], B2015 showed that the excess ${\tau_{\mathrm{eff}}}$ dispersion at $z=5.6$ could be matched in a model where ${\lambda_{\mathrm{mfp}}^{912}}$ decreases by a factor of $\approx 5$ between $z=5.2$ and $z=5.6$ – a time scale of just $100$ million years. However, such rapid evolution in ${\lambda_{\mathrm{mfp}}^{912}}$ is inconsistent with extrapolations based on measurements at lower redshifts [@becker13; @2014MNRAS.445.1745W], and would imply that the emissivity of ionizing sources, in turn, increases by an unnatural factor of $\approx 5$ over the same cosmologically short time interval. Because of these issues, B2015 speculated that the excess dispersion was evidence for large spatial variations in the mean free path[^1]. Alternatively, fluctuations in the ionizing background could have been enhanced if the sources of ionizing photons were rarer than the observed population of galaxies. However, current models require half of the background to arise from bright sources with an extremely low space density of $\sim10^{-6}~{\mathrm{Mpc}}^{-3}$[@2015arXiv150501853C]. This scenario is a possibility of current debate [e.g. @2015arXiv150707678M but see D’Aloisio et al. in prep.].
In this Letter, we explore a source of dispersion in ${\tau_{\mathrm{eff}}}$ that has so far been neglected and that, unlike ionizing background fluctuations, has straightforward implications for the reionization process itself. In addition to $\Delta_b(x)$ and $\Gamma(x)$, the Ly$\alpha$ opacity depends on $T(x)$, mainly because the amount of neutral hydrogen after reionization is proportional to the recombination rate, which scales as $T^{-0.7}$. Previous attempts to model Ly$\alpha$ opacity fluctuations had not included the residual temperature fluctuations that must have been present if reionization were patchy and temporally extended. As ionization fronts propagated supersonically through the IGM, the gas behind them was heated to tens of thousands of degrees Kelvin by photoionizations of [H[ i]{}]{} and [He[ i]{}]{}. After reionization, the gas cooled mainly through adiabatic expansion and through inverse Compton scattering with cosmic microwave background (CMB) photons [@1994MNRAS.266..343M; @1997MNRAS.292...27H; @2003ApJ...596....9H; @2015arXiv150507875M]. Since different regions in the IGM were reionized at different times, these heating and cooling processes imprinted an inhomogeneous distribution of intergalactic temperatures that persisted after reionization [@2008ApJ...689L..81T; @2009ApJ...706L.164C; @furlanetto09; @2014ApJ...788..175L]. We will show that these residual temperature variations likely account for much of the observed dispersion in ${\tau_{\mathrm{eff}}}$ at $z\gtrsim 5.5$, and may even account for all of it – a scenario that would yield new information on the timing, duration, and patchiness of reionization.
The remainder of this Letter is organized as follows. In §\[SEC:methods\] and §\[SEC:toymodel\], we describe our simulations and methodology. In §\[SEC:results\], we present our main results. In §\[SEC:conclusion\], we offer concluding remarks. We use comoving units for distances and physical units for number densities.
Numerical Methods {#SEC:methods}
=================
{width="8.5cm"} {width="8.5cm"}
To model the impact of relic temperature fluctuations from reionization on the distribution of ${\tau_{\mathrm{eff}}}$, we ran a suite of 20 cosmological hydrodynamics simulations using a modified version of the code of [@2004NewA....9..443T]. The simulations were initialized at $z=300$ from a common cosmological initial density field. We used a matter power spectrum generated by CAMB [@Lewis:1999bs] assuming a flat $\Lambda$CDM model with $\Omega_m=0.3051$, $\Omega_b=0.04823$, $h=0.68$, $\sigma_8=0.8203$, $n_s=0.9667$, and $Y_{\mathrm{He}}=0.2453$, consistent with recent measurements [@2015arXiv150201589P]. Our production runs use a cubical box with side length $L_{\mathrm{box}}=12.5h^{-1}~{\mathrm{Mpc}}$, with $N_{\mathrm{dm}}=1024^3$ dark matter particles and $N_{\mathrm{gas}}=1024^3$ gas cells.
In each simulation, reionization was modeled in a simplistic manner by instantaneously ionizing and heating the gas to a temperature ${T_{\mathrm{reion}}}$ at a redshift of ${z_{\mathrm{reion}}^{\mathrm{inst}}}$. Subsequently, ionization was maintained with a homogeneous background with spectral index $\alpha=0.5$, consistent with recent post-reionization background models [@2012ApJ...746..125H]. Utilizing the periodic boundary conditions of our simulations, we trace skewers of length $L=50h^{-1}~{\mathrm{Mpc}}$ (following the convention of the B2015 ${\tau_{\mathrm{eff}}}$ measurements) at random angles through all of the hydro simulation snapshots. Each skewer is divided into $N_x=4096$ equally spaced velocity bins of size $\Delta v_\mathrm{skewer} = \dot{a} L/N_x$ (where $a$ is the cosmological scale factor), and Ly$\alpha$ optical depths are computed using the method of [@1998MNRAS.301..478T]. Although reionization occurs instantaneously at ${z_{\mathrm{reion}}^{\mathrm{inst}}}$ within each simulation box, we piece together skewer segments from simulations with different ${z_{\mathrm{reion}}^{\mathrm{inst}}}$ to model the effect of an inhomogeneous reionization process, as we describe further in the next section.
The post-reionization temperatures in the simulations are relatively insensitive to the spectrum of the ionizing background, but they are sensitive to the amount of heating that is assumed to occur at the time a gas parcel is reionized. Previous calculations [@1994MNRAS.266..343M; @2008ApJ...689L..81T; @mcquinn-Xray] have bracketed the range of possible reionization temperatures to ${T_{\mathrm{reion}}}\approx 20,000-30,000$ K. (We note that previous large-scale reionization simulations do not accurately capture ${T_{\mathrm{reion}}}$, as they do not resolve the $\sim0.3~$ physical kpc ionization fronts.) Thus, we have run two sets of ten simulations – one set with ${T_{\mathrm{reion}}}=20,000$ K and the other with ${T_{\mathrm{reion}}}=30,000$ K – where each set contains instantaneous reionization redshifts of ${z_{\mathrm{reion}}^{\mathrm{inst}}}=\{6,6.5,7,7.5,8,8.5,9,10,11,12\}$. This redshift range spans the likely duration of reionization. Simulations with ${z_{\mathrm{reion}}^{\mathrm{inst}}}\gtrsim12$ are driven to a common temperature by $z<6$, so they are well approximated by the ${z_{\mathrm{reion}}^{\mathrm{inst}}}=12$ simulation.
Figure \[FIG:physics\] shows the post-reionization thermal and associated hydrodynamic relaxation of intergalactic gas (and its effect on the Ly$\alpha$ forest opacity) in our simulations. The top-left panel shows the volume-weighted average gas temperature in the ten ${T_{\mathrm{reion}}}=30,000$ K simulations. The bottom-left panel shows this same average, but limited to gas cells with densities of $\Delta_b<0.3$, the deepest voids that dominate transmission in the highly-saturated $z\gtrsim5$ forest. Even at $z\sim5$, the $\Delta_b <0.3$ gas temperatures differ by up to a factor of five between the simulations that were reionized at different times. The right panels show the [H[ i]{}]{} number densities (top) and the transmission (bottom) at $z=5.8$ for the same skewer through our ${z_{\mathrm{reion}}^{\mathrm{inst}}}=12$ (blue/solid) and ${z_{\mathrm{reion}}^{\mathrm{inst}}}=6$ (red/dashed) simulations. The transmission is nearly zero in the ${z_{\mathrm{reion}}^{\mathrm{inst}}}=12$ case, owing to colder temperature and hence enhanced [H[ i]{}]{} densities, whereas there is significant transmission in the ${z_{\mathrm{reion}}^{\mathrm{inst}}}=6$ case.
Constructing the ${\tau_{\mathrm{eff}}}$ distribution: a toy model {#SEC:toymodel}
==================================================================
Reionization is a process that is inhomogeneous and temporally extended, unlike in our individual hydro simulations. Modeling the thermal imprint of patchy reionization on the Ly$\alpha$ forest thus requires an additional ingredient: a model for the redshifts at which points along our skewers are reionized. In this section, we present a simplified toy model to illustrate how we piece together skewer segments from our hydro simulations, and to provide insight into how the timing, duration and morphology of reionization affect the amplitude of Ly$\alpha$ opacity fluctuations.
For illustrative purposes, let us assume that the Ly$\alpha$ forest is made up of segments of equal length, $l_c$, where each segment is reionized at a single redshift. Let us further assume that the reionization redshift of each segment is drawn from a uniform probability distribution over the interval $[{z_{\mathrm{start}}}, {z_{\mathrm{end}}}]$, resulting in a global reionization history in which the mean ionized fraction, $\bar{x}_{\mathrm{HII}}(z)$, is linear in redshift (since $\bar{x}_{\mathrm{HII}}(z)$ is the cumulative probability distribution of the reionization redshift).
Assuming that the reionization redshift field is in the Hubble flow, each segment of length $l_c$ spans $\Delta N_x=4096\times l_c/(50h^{-1}{\mathrm{Mpc}})$ velocity bins of our hydro simulation skewers. For the first segment, with reionization redshift $z_1$, we take the initial $\Delta N_x$ spectrum velocity bins of a randomly drawn skewer that has ${z_{\mathrm{reion}}^{\mathrm{inst}}}$ closest to $z_1$. For the next segment, with reionization redshift $z_2$, we take the next $\Delta N_x$ velocity bins *from the same skewer* through the simulation that has ${z_{\mathrm{reion}}^{\mathrm{inst}}}$ closest to $z_2$ (note that all simulations were started from the same initial density field). We repeat this procedure until an entire $50h^{-1}~{\mathrm{Mpc}}$ sightline is filled.
In what follows, we compare these toy models against the B2015 measurements of the cumulative probability distribution function of ${\tau_{\mathrm{eff}}}$, ${P(<{\tau_{\mathrm{eff}}})}$. The measurements considered here span $z=5.1-5.9$ in bins of width $\Delta z=0.2$. We construct ${P(<{\tau_{\mathrm{eff}}})}$ from 4000 randomly drawn sightlines at the central redshift of each bin. For each redshift, we rescale the nominal post-reionization photoionization rate of our simulations ($\Gamma=10^{-13}~\mathrm{s}^{-1}$) by a constant factor, such that our model ${P(<{\tau_{\mathrm{eff}}})}$ is equal to the observed distribution at either ${P(<{\tau_{\mathrm{eff}}})}=0.15$ or ${P(<{\tau_{\mathrm{eff}}})}=0.3$, depending on which value provides the better visual fit. We have performed extensive numerical convergence tests using simulations of varying resolution and box size. We found excellent convergence of ${P(<{\tau_{\mathrm{eff}}})}$ for our production runs in both box size and resolution (especially for our patchy reionization models).
![Comparison of cumulative probability distribution functions of ${\tau_{\mathrm{eff}}}$ in our toy model (colored curves) compared to the B2015 measurements (black histograms) and to an instantaneous reionization model with ${z_{\mathrm{reion}}^{\mathrm{inst}}}=8.5$ (black/dotted curves). [**Top row:**]{} Varying the coherence length, ${l_\mathrm{c}}$, assuming a uniform ${z_{\mathrm{reion}}^{\mathrm{inst}}}$ probability between $z=5.9-13$. [**Bottom row:**]{} Varying the redshift interval over which the ${z_{\mathrm{reion}}^{\mathrm{inst}}}$ values are drawn, assuming ${l_\mathrm{c}}=25h^{-1}~{\mathrm{Mpc}}$. []{data-label="FIG:toymodel_CDFs"}](fig2.eps){width="9cm"}
Figure \[FIG:toymodel\_CDFs\] shows the ${P(<{\tau_{\mathrm{eff}}})}$ of these toy models for a range of $({z_{\mathrm{start}}},{z_{\mathrm{end}}},l_c)$, compared against the B2015 measurements in the two highest redshift bins (black histograms). The curves with shorter reionization durations, or with smaller $l_c$, fall closer to the black/dotted curves, which assume that reionization occurs instantaneously (at ${z_{\mathrm{reion}}^{\mathrm{inst}}}=8.5$, although it does not depend on this choice). Figure \[FIG:toymodel\_CDFs\] affords three insights: (1) The larger the coherence length, $l_c$, over which gas shares a similar reionization redshift, the larger the spread in ${\tau_{\mathrm{eff}}}$; (2) The width of the observed ${P(<{\tau_{\mathrm{eff}}})}$ can be fully accounted for only if reionization ended at $z\lesssim7$ and was well underway by $z\sim9$; (3) The maximal width is achieved by a late-ending (${z_{\mathrm{end}}}\sim6$) and extended reionization model in which large contiguous segments ($\gtrsim12.5h^{-1}~{\mathrm{Mpc}}$) of the Ly$\alpha$ forest were reionized at $z\gtrsim9$. In the next section, we apply these insights to construct more realistic models of reionization that reproduce the observed width of ${P(<{\tau_{\mathrm{eff}}})}$.
The Effect of Temperature Fluctuations on the High-Redshift Ly$\alpha$ Forest {#SEC:results}
=============================================================================
Results
-------
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We generate physically-motivated reionization redshift fields using simulations based on the excursion-set model of reionization (ESMR) [@2004ApJ...613....1F; @2007ApJ...654...12Z; @2009ApJ...703L.167A; @2010MNRAS.406.2421S; @2011MNRAS.411..955M], which has been shown to reproduce the ionization structure found in full radiative transfer simulations [@2005ApJ...630..657Z; @2011MNRAS.414..727Z; @2014MNRAS.443.2843M]. In particular, our ESMR uses a realization of the linear cosmological density field and a top hat in Fourier space filter to generate a realization of reionization in a cubical box with side $400h^{-1}~{\mathrm{Mpc}}$, sampled with $512^3$ cells. For details of the algorithm, see @2007ApJ...654...12Z. We use the simplest formulation of the ESMR, with two free parameters: (1) The minimum mass of halos that host galactic sources of ionizing photons, ${M_{\mathrm{min}}}$; (2) The ionizing efficiency of the sources, $\zeta$. We tune $\zeta$ to obtain a reionization history that is approximately linear in redshift, similar to those in our toy model.
The ESMR calculation yields a cube of reionization redshifts. We construct mock absorption spectra by first tracing $50h^{-1}~{\mathrm{Mpc}}$ skewers through this cube. We then piece together spectrum segments from our suite of hydro simulations, much like in our toy model, except here we match ${z_{\mathrm{reion}}^{\mathrm{inst}}}$ to the reionization redshifts along the ESMR skewers. This process does not account for how ${z_{\mathrm{reion}}^{\mathrm{inst}}}$ correlates with density, an effect that will underestimate (making our models conservative) the width of ${P(<{\tau_{\mathrm{eff}}})}$, which we address shortly.
The left panel of Figure \[FIG:zreionfield\] shows a 2D slice through our fiducial reionization redshift field in which reionization spans $z=6-13$. In the right panels, the blue/solid curves show the fraction of transmitted flux along four different sightlines through this field, selected to span a range of ${\tau_{\mathrm{eff}}}$. The red/dashed curves show the corresponding reionization redshifts along the sightlines. On average, regions that reionize at later times yield more transmission, while regions that reionize at $z\gtrsim9$ result in dark gaps in the Ly$\alpha$ forest. The variation among these mock spectra is similar to the well-known variation seen in observations of the $z\sim6$ Ly$\alpha$ forest [@fan06]. In our interpretation, this variation reflects differences in the reionization redshifts – and hence temperatures[^2] – between segments of the Ly$\alpha$ forest.
The leftmost four panels of Figure \[FIG:ePS\_CDFs\] show ${P(<{\tau_{\mathrm{eff}}})}$ in three ESMR models that take ${T_{\mathrm{reion}}}=30,000~\mathrm{K}$ and ${M_{\mathrm{min}}}=2\times10^9~{M_{\odot}}$, with reionization histories shown in the inset of the $5.5<z<5.7$ panel. The CMB electron scattering optical depths in these models are ${\tau_{\mathrm{es}}}= 0.054, 0.068$ and $0.080$, within uncertainties of the latest [*Planck*]{} measurement of $\tau_{\mathrm{es}}=0.066\pm0.016$ [@2015arXiv150201589P]. These models are compared against the B2015 measurements (black histograms) and against the homogeneous reionization reference model with ${z_{\mathrm{reion}}^{\mathrm{inst}}}=8.5$ (dotted curves). The blue/long-dashed and green/short-dashed curves correspond to scenarios in which reionization spans $z=6-10$ and $z=7-14$, respectively. While these two models produce significantly more width in ${P(<{\tau_{\mathrm{eff}}})}$ than the homogeneous reionization model, they fall short of producing the full range of ${\tau_{\mathrm{eff}}}$ required to match the observations. However, the fiducial $z=6-13$ reionization model (red/solid curves) generally provides a good match to the measurements. A significant success of the fiducial model is that the observed redshift evolution of the ${P(<{\tau_{\mathrm{eff}}})}$ width is reproduced without any additional tuning of parameters. (We have checked that this success also holds over $z=4-5$, lower redshifts than those shown where the effect is reduced.) Indeed, there are not any parameters that can be tuned in our model to change the post-reionization evolution of the width. The pink shaded regions indicate the 90% confidence levels of our fiducial model, estimated from bootstrap realizations, showing consistency with all the data aside from a single high-opacity point at $z=5.4$ and at $z=5.6$.
The discrepancy at the highest opacities may arise because our method of constructing mock absorption spectra does not capture correlations between temperature and density that should be present, since denser regions are more likely reionized earlier. Such correlations would act to increase the width of ${P(<{\tau_{\mathrm{eff}}})}$, as the denser regions around galaxies are ionized earlier in our models. One might naively think these correlations are small, because correlations between the density on the much larger scales of the [H[ ii]{}]{} bubbles and the smaller scales of voids in the forest should be weak, but calculations show that they may not be negligible [@furlanetto09; @2008ApJ...689L..81T; @2013ApJ...776...81B]
Effect of varying reionization model parameters
-----------------------------------------------
The right panels of Figure \[FIG:ePS\_CDFs\] show the effect of varying ${T_{\mathrm{reion}}}$ and ${M_{\mathrm{min}}}$. For ${T_{\mathrm{reion}}}=20,000~\mathrm{K}$, the distribution of ${\tau_{\mathrm{eff}}}$ is somewhat narrower than the case with ${T_{\mathrm{reion}}}=30,000~\mathrm{K}$. Hotter temperatures are likely achieved towards the end of reionization, when [H[ ii]{}]{} bubbles are larger and propagate at quicker speeds. A model with ${T_{\mathrm{reion}}}\sim30,000~\mathrm{K}$ near the end of reionization, and smaller temperatures earlier on, would likely produce more width than a model with ${T_{\mathrm{reion}}}\sim30,000~\mathrm{K}$ at all times. However, any conclusions about ${T_{\mathrm{reion}}}$ prior to modeling the density/reionization-redshift correlations are premature.
The blue/long-dashed and green/short-dashed curves in the right panels show the effect of varying ${M_{\mathrm{min}}}$. For the former atomic cooling halos (ACHs) curve, $M_{\mathrm{min}}$ is set to the minimum mass required to achieve a halo virial temperature of $10,000~\mathrm{K}$. For both curves, we tune $\zeta$ to match the reionization history of our fiducial model (red curve in the inset). We find that the effects of varying $M_{\mathrm{min}}$ are minor.
Conclusion {#SEC:conclusion}
==========
We have shown that residual temperature inhomogeneities from a patchy and extended reionization process likely account for much of the opacity fluctuations in the $z\gtrsim5$ Ly$\alpha$ forest. Inhomogeneities in the ionizing background may also contribute at a significant level, though current models in this vein have required very small mean free paths or extremely rare sources. We showed that residual temperature fluctuations alone could account for the entire spread of observed ${\tau_{\mathrm{eff}}}$. A significant success of this interpretation is that it is able to reproduce the rapid growth of ${\tau_{\mathrm{eff}}}$ fluctuations with redshift, despite having very little freedom in its post-reionization evolution. In this scenario, the observations favor a late but extended reionization process that is roughly half complete by $z\sim9$ and that ends at $z\sim6$.
Unlike ionizing background fluctuations, which do not necessarily signal the end of reionization, temperature fluctuations directly probe the timing, duration, and patchiness of this process. If most of the opacity variations owe to temperature, it would mean that, on average, the darkest $\gtrsim10~{\mathrm{Mpc}}$ segments of the $z\gtrsim 5$ Ly$\alpha$ forest were reionized earliest, and the brightest segments last – a potentially powerful probe of cosmological reionization.\
The authors acknowledge support from NSF grant AST1312724. H.T. also acknowledges support from NASA grant ATP-NNX14AB57G. Computations were performed with NSF XSEDE allocation TG-AST140087. We thank Paul La Plante, Jonathan Pober, Phoebe Upton Sanderbeck, George Becker, Adam Lidz, and Fred Davies for helpful discussions.
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[^1]: See also @2015arXiv150907131D, which appeared after we submitted this paper.
[^2]: We find that pressure smoothing plays an insignificant role in generating ${\tau_{\mathrm{eff}}}$ fluctuations.
|
---
abstract: 'Many processes of scientific importance are characterized by time scales that extend far beyond the reach of standard simulation techniques. To circumvent this impediment a plethora of enhanced sampling methods has been developed. One important class of such methods relies on the application of a bias that is function of a set of collective variables specially designed for the problem under consideration. The design of good collective variables can be challenging and thereby constitutes the main bottle neck in the application of these methods. To address this problem, recently we have introduced Harmonic Linear Discriminant Analysis, a method to systematically construct collective variables. The method uses as input information on the metastable states visited during the process that is being considered, information that can be gathered in short unbiased MD simulations, to construct the collective variables as linear combinations of a set of descriptors. Here, to scale up our examination of the method’s efficiency, we applied it to the folding of Chignolin in water. Interestingly, already before any biased simulations were run, the constructed one dimensional collective variable revealed much of the physics that underlies the folding process. In addition, using it in Metadynamics we were able to run simulations in which the system goes from the folded state to the unfolded one and back, where to get fully converged results we combined Metadynamics with Parallel Tempering. Finally, we examined how the collective variable performs when different sets of descriptors are used in its construction.'
author:
- Dan Mendels
- Giovannimaria Piccini
- 'Z. Faidon Brotzakis'
- 'Yi I. Yang'
- Michele Parrinello
bibliography:
- 'library.bib'
title: Folding a Small Protein Using Harmonic Linear Discriminant Analysis
---
Introduction {#introduction .unnumbered}
============
Simulations of complex processes such as drug binding, protein association, protein folding, phase transitions, etc. have proven to be of great value and are a pillar of contemporary scientific investigation. However, many such processes are characterized by very long time scales which prohibit their simulation using conventional simulation techniques. Hence, to circumvent this limitation, a plethora of enhanced sampling methods has been developed over the years including replica exchange based methods such as Parallel Tempering [@sugita1999replica] and bias based techniques such as Umbrella Sampling [@Torrie1977], Metadynamics [@laio_parrinello_2002] and Variationally Enhanced Sampling [@valsson_parrinello_2014]. The latter category relies on the use of collective variables (CVs) which describe the most essential degrees of freedom of the processes being considered. Constructing appropriate CVs however, can be challenging and time consuming. Thus, and in light of the expected continuing increase in the complexity and size of the systems being studied, devising techniques for the systematic construction of efficient CVs is regarded as an important objective of the enhanced sampling community. Also, finding good CVs is not only a technical issue, but is a way of encoding in a compact and transparent way the essence of the process being considered.
In the effort to address this challenge, in a recent publication [@mendels2018collective; @Piccini_Mendels2018] we have proposed a new scheme for constructing systematically viable CVs through the utilization of the supervised learning class classification paradigm, and in particular using a modification of Fisher’s Linear Discriminant Analysis (LDA), termed Harmonic Linear Discriminant Analysis (HLDA) (see also ref. [@2018arXiv180210510S]). The LDA assumes a multivariate normal distribution of the descriptors. For this reason we also examine how deviation from multivariate normality affect the CVs efficiency. The scheme of choice requires as input only short unbiased trajectories, for each metastable state. Using these data, HLDA can estimate the direction within an $N_d$ dimensional space of selected system descriptors upon which the projections of these sets of data are best separated. The linear combination corresponding to this direction is then utilized as the CV.
To test its applicability, HLDA has been used in ref. [@mendels2018collective] in two examples taken from the realm of Materials Science and Chemistry. In both cases HLDA was found to be able to generate good CVs, leading to biased runs characterized by high frequency of transitions between the metastable states of interest and to a rapidly converging sampling.
The application of HLDA in ref. [@mendels2018collective] was, however, still confined to a set of relatively simple problems. Hence, if its use (and the class classification paradigm underlying it) is to be adopted for scientific and technological problems of increasing complexity, it would need to prove effective in such systems. Here, we would like to investigate the performance of HLDA for a relatively more complex system. We consider the case of a small protein, Chignolin, for which extensive simulations on purpose build machines are available [@Lindorff-Larsen2011]. The procedure to determine the HLDA CVs requires the use of a convenient set of descriptors $d_{i}(R)$ that are capable of describing the initial and final states. This study will give us information on the effect of the choice of descriptors on the CVs quality.
We show also how the HLDA CVs bring out much of the physics even before performing the simulations. Moreover, we find that employing the HLDA CVs within Metadynamics simulations enables sampling numerous folding and unfolding path-ways and that through their incorporation in Parallel Tempered Metadynamics (PTMetaD) simulations [@bussi2006free], estimates for the system Free Energy Surface (FES) can be obtained.
Methods {#methods .unnumbered}
=======
Constructing the collective variable {#constructing-the-collective-variable .unnumbered}
------------------------------------
To construct CVs which describe the Chignolin folding process we utilize the paradigm introduced in ref. [@mendels2018collective] that estimates the direction $\mathbf{W}$ in an $N_d$ dimensional descriptor space in which the projections of the unbiased distributions of the folded and unfolded states are best separated. As in ref. [@mendels2018collective] we do this using HLDA, a modification of Fisher’s LDA. Thus, as in Fisher’s LDA, the estimation of $\mathbf{W}$ is done through the maximization of the ratio between the system’s so called between class $\mathbf{S}_b$ and within class $\mathbf{S}_w$ scatter matrices. Like LDA, the former is measured by the square of the distances between the projected means, and can be written as
$$\label{mean_transf_mat}
\mathbf{W}^T \mathbf{S}_b \mathbf{W}$$
with
$$\label{between_class}
\mathbf{S}_b = \left( \boldsymbol{\mu}_A - \boldsymbol{\mu}_B \right)\left( \boldsymbol{\mu}_A - \boldsymbol{\mu}_B \right)^T$$
where $\boldsymbol{\mu}_{A,B}$ are the expectation values of the two metastable states. In contrast to LDA, in which the within class matrix is estimated using the average of the two metastable states multivariate variances $\boldsymbol{\Sigma}_{A,B}$, here it is estimated from the spreads harmonic average
$$\label{cov_transf_mat}
\mathbf{W}^T \mathbf{S}_w \mathbf{W}$$
with
$$\label{harmonic_mean}
\mathbf{S}_w = \frac{1}{\frac{1}{\boldsymbol{\Sigma}_A} + \frac{1}{\boldsymbol{\Sigma}_B}}.$$
The HLDA objective function which has the form of a Rayleigh ratio
$$\label{fisher_ration}
\mathcal{J(\mathbf{W})} = \frac{\mathbf{W}^T \mathbf{S}_b \mathbf{W}}{\mathbf{W}^T \mathbf{S}_w \mathbf{W}}$$
is then maximized by
$$\label{maximizer}
\mathbf{W}^* = \mathbf{S}_w^{-1} \left( \boldsymbol{\mu}_A - \boldsymbol{\mu}_B \right).$$
which in turn yields the HLDA CV
$$\label{maximizer_harm}
s_{HLDA}(\mathbf{R}) = \left( \boldsymbol{\mu}_A - \boldsymbol{\mu}_B \right)^T \left( \frac{1}{\boldsymbol{\Sigma}_A} + \frac{1}{\boldsymbol{\Sigma}_B} \right) \mathbf{d}(\mathbf{R}).$$
Computational Details {#computational-details .unnumbered}
---------------------
Simulations of Chignolin (sequence TYR-TYR-ASP-PRO-GLU-THR-GLY-THR-TRP-TYR) were conducted using GROMACS 5.1.2 [@berendsen1995gromacs; @abraham2015gromacs] and the PLUMED 2.4 plugin [@tribello_2014]. The CHARMM22\* force field [@piana2011robust] and the three-site transferable inter-molecular potential (TIP3P) water model [@jorgensen1983comparison] were used to make direct comparisons with ref. [@lindorff2011fast]. ASP and GLU residues were simulated in their charged states, as were the N- and C-terminal amino acids. A time step of 2 fs was used for all systems and a constant temperature of 340 K (in agreement with ref. [@lindorff2011fast]) was maintained by the velocity rescaling thermostat of Bussi et al. [@bussi2007canonical]. All bonds involving H atoms were constrained with the linear constraint solver (LINCS) algorithm [@hess1997lincs]. Electrostatic interactions were calculated with the particle mesh Ewald scheme [@essmann1995smooth] and 1 nm cutoff was applied to all non-bonded interactions. Runs were all conducted with a box containing 1649 water molecules, along with 2 sodium ions to neutralize the system. Parallel Tempering simulations were run with 40 replicas, each at a different temperature. The replica with the lowest temperature was set to be at T=340 K while the rest of the temperatures were arranged in a geometrical series with a factor $a=1.011$. (Details regarding the utilized descriptor sets can be found in the Supplementary Information).
Results {#results .unnumbered}
=======
In accordance with the basic requirements of the HLDA approach, we began the study by acquiring two unbiased trajectories spanning roughly $2 \mu s$ each in the system’s folded and unfolded states. The first step in applying HLDA requires the selection of a set of system descriptors that can be instrumental in describing the folding and unfolding of the mini-protein. Ideally, since this step should not require an expert’s understanding of the system we proceeded by selecting fairly naively three different sets of descriptors to observe how in the present context this selection can influence the outcome of the method implementation. Here, we also chose to asses the HLDA ability to perform beyond its strict theoretical limitations, namely that the descriptor unbiased fluctuations are normal in form.
Thus, the first set $D_1$ consisted of 12 distances between different atomic sites on the protein. Six of these distances were selected between atoms situated on the backbone, while six more where taken between atoms situated on the protein’s side chains. The second set of descriptors $D_2$ consisted of 6 contacts placed on the protein’s backbone while the third $D_3$ consisted of $\alpha \beta$ functions corresponding to the protein’s backbone dihedral angles, i.e. $\alpha \beta=\frac{1}{2}(1+cos(\phi_i))$ with $i=1..18$. (For a detailed list of the atom pairs used for the construction of $D_1$ and $D_2$ and the contacts parameters see the SI). The two covariance matrices $\boldsymbol{\Sigma_{f}}$, $\boldsymbol{\Sigma_{u}}$ and two mean vectors $\boldsymbol{\mu_{f}}$, $\boldsymbol{\mu_{u}}$ corresponding to the folded and unfolded states respectively, were thus constructed for each of the descriptor sets.
Using this information and applying HLDA we could next obtain an estimation of the hyper planes that best separated the unbiased distributions corresponding to the folded and unfolded states within the space spanned by each set of descriptors. Concomitantly, the sought after CVs were obtained using Eq. \[maximizer\_harm\]. The weights of the HLDA CVs attained for each of the descriptor sets are plotted in Fig. \[fig:Illustration\_descriptors\] along with an illustrations of the utilized descriptors.
Interestingly, analysis of the weight distributions reveals much valuable information about the system showing that the main features of the folding process are encoded in the CVs themselves. Thus, we find that for both the sets $D_1$ and $D_2$ most of the weight is assigned to the descriptors $d_1$, $d_2$ and $d_3$ which correspond to the distances and contacts between facing amino acids located away from the backbone beta-turn. Similarly, in both $D_1$ and $D_2$ the distances/contacts located in the beta-turn are found to be comparably less important, alluding to the fact that in the unfolded state the beta-turn is intermittently formed. In the case of $D_1$ the distances between the side chains are assigned lower weights as well. Nevertheless, for the distances $d_{10}$, $d_{11}$ and $d_{12}$ non negligible weight is assigned, reflecting the associated side-chain’s contact formation in the folded state, due to their hydrophobic nature.
Inspecting the weight distribution obtained for $D_3$ reveals interesting trends as well. Thus, one can observe that by and large the higher weights of the CV are assigned to $\alpha\beta$s of the backbone dihedral angles situated in and near the backbone beta-turn, reflecting these angles’ importance in the folding process. In addition, we find that in comparison it is the $\alpha\beta(\Psi)$ that attain higher weights. Inspection here shows that while the $\alpha\beta(\Phi)$ fluctuations do not change much between the folded and unfolded states, clearly configurational changes of $\alpha\beta(\Psi)$ between the folded and unfolded states are present. Another interesting feature of the CV weight distribution is that with the exception of $d_6$, the weights of $\alpha\beta(\Psi)$ and $\alpha\beta(\Phi)$ tend to be in anti-phase. Here, inspection shows that this results from the correlation between the $\alpha\beta(\Psi)$ and $\alpha\beta(\Phi)$ fluctuations in the $\alpha$-helical basin that is visited in the unfolded state. Moreover, we attribute the single positive value of $d_6$ (which corresponds to the Glycine $\Phi$) to the fact that unlike the other $\Phi$ backbone dihedral angles it can also assume a left helix conformation [@Lovell2003]. Finally, examination of $d_{13}$, the descriptor assigned with the largest weight, shows that it corresponds to the $\alpha\beta(\Psi)$ of the Proline amino acid, coinciding with the fact that such angles are associated with a relatively high energetic rotational barrier [@KANG2004135].
![The weights assigned by HLDA to each of the descriptors for a) $D_1$ b) $D_2$ and c) $D_3$. d) Illustration of the distances between side-chain sites utilized in the set $D_1$. e) Illustration of the distances and contacts used between backbone sites in the sets $D_1$ and $D_2$ (the line thicknesses are set to indicate the descriptors’ importance in both sets). f) Illustration of the descriptor set $D_3$. Sphere colors correspond to the absolute values of the weights assigned to each of the utilized backbone dihedral angles. Small spheres represent the $\Phi$ angles whereas large ones the $\Psi$ angles. []{data-label="fig:Illustration_descriptors"}](Chignolin_illustration_high_res.pdf){width="1\columnwidth"}
Biased Simulations {#biased-simulations .unnumbered}
------------------
With the CVs at hand we could next launch Metadynamics simulations with the objective of sampling the system phase space. Monitoring these simulations, we found that several folding events could be observed with $D_1$ and $D_2$ taking the lead in the transition frequencies. Thus, with the little initial information with which we commenced, an assortment of folding and unfolding pathways could be harvested, thereby shedding light on the mechanisms underlying these events. Fig. \[fig:Paths\_discovery\] presents segments of three simulations, each run with a CV generated by a different descriptor set, showing the $C\alpha$ RMSD of the protein with respect to it’s folded crystal structure as function of Metadynamics simulations time. All three segments exhibit both folding and unfolding events.
Despite observing even multiple transitions between the folded and unfolded states in some of the simulations, attaining estimates of converged FES using Metadynamics alone was not possible. Observing the simulations dynamics we found this to be caused by the system’s phase space intricate multidimensional nature with a profusion of kinetic bottlenecks and free energy barriers. Thus, to circumvent this impediment we resorted to the utilization of Parallel Tempering Metadynamics which has previously been shown to be very effective for such problems. By running such simulations, now with the HLDA generated CVs, we could observe that for all three sets of descriptors effective sampling of the relevant system phase space was achieved. Moreover, estimates of the system FES could be obtained using the Well tempered Metadynamics (WTMD) [@Barducci2008] relation Eq. \[Eq:V\_to\_FES\]
$$\label{Eq:V_to_FES}
F(\boldsymbol{s})=-\Big( \frac{\gamma}{\gamma-1} \Big) V(\boldsymbol{s}).$$
where $\gamma$ is the WTMD bias factor and $V(s)$ is the simulation bias. Fig. \[fig:FES\_HLDA\_PT\_compare\] presents the FES obtained from three different simulations using the CVs generated by the three different sets of descriptors. For the sake of comparison, the corresponding FES obtained using a histogram analysis on a $100 \mu s$ unbiased simulations taken from the D.E. Shaw data bank [@lindorff2011fast] is presented as well. As can be seen in all three cases reasonable estimates of the FES could be obtained. However, the results obtained using the sets $D_1$ and $D_2$ are markedly more accurate. Additionally, differences in the convergence times between the different simulations were observed, namely $t_{conv}(D_1)\approx25 ns$, $t_{conv}(D_2)\approx35 ns$ and $t_{conv}(D_3)\approx45 ns$, where we defined convergence when the calculated FES fluctuations as function of simulation time reached their minimal amplitudes around their final average result.
![Free-energy profiles at $T=340 K$ obtained from PTMetaD simulations along the $S_{HLDA}$ CV constructed using the descriptor set (a) $D_1$, (b) $D_2$, (c) $D_3$. Shaded areas indicate the fluctuations in time of the FES curves during convergence. Dotted lines, represent the FES profiles obtained via a histogram analysis on the $100 \mu s$ unbiased trajectories taken from the D.E. Shaw database. []{data-label="fig:FES_HLDA_PT_compare"}](FES_comparison_to_DE_Shaw_high_resolution7.jpg){width="0.52\columnwidth"}
One probable reason for the differences in performance between the different CVs, is the extent to which their underlying descriptors unbiased fluctuations deviate from multivariate normality. To asses the possible influence of such deviations on the CVs quality we thus computed the Kurtosis and Skewness [@mardia1975assessment] of the covariance matrices, $\boldsymbol{\Sigma}_{f,u}$, eigenvectors. Fig. \[fig:Gaussinaity\] presents these data for each of the descriptor sets in both the folded and unfolded states. As can be seen the values obtained for $D_1$ largely match those that correspond to perfect normal distributions (indicated by dashed lines). In slight contrast, the set $D_2$ can be seen to exhibit larger deviations from normality, while the set $D_3$ seems to exhibit the most and largest instances of such deviations. Observation of these data thus indicates that while HLDA seems to be forgiving when applied to data which is not strictly multinormal, it is likely that a price is to be paid in their quality as deviations increase.
![Kurtosis and skewness of the unbiased distributions in the folded (left) and unfolded (right) states of the covariance matrices eigenvectors corresponding to the three utilized sets of descriptors. The expected values for perfect multinormal distributions are indicated by the dashed lines. []{data-label="fig:Gaussinaity"}](Multivariate_comparison.jpg){width="1.0\columnwidth"}
Conclusions {#conclusions .unnumbered}
===========
As the size and complexity of systems simulated using molecular dynamics increases, the need for systematic ways of constructing viable CVs for these systems which do not require an expert’s knowledge is becoming more evident. In the present study we have applied HLDA, a recently developed modification of LDA to develop one dimensional CVs for the folding problem of Chignolin. In doing so we have found that given a naive selection of descriptors the method is able to generate CVs that, when biased, are able to drive the system back and forth between the system’s folded and unfolded states. In addition, we have found that incorporating these CVs in PTMetaD can enable obtaining of good estimates of the systems FES. In both cases we found that some deviation from multivariate normality is tolerated by HLDA, yet increasing the amount of deviation may lead to a reduction in the constructed CVs’ quality. Finally, we found that within the weight distributions of the calculated HLDA CVs themselves reside abundant useful information and physical insight about the process being studied. We thus conclude that the present study suggests that HLDA can be applied to increasingly more complex systems for the systematic construction of CVs, a path which we wish to continue and explore in the near future.
We acknowledge D. E. Shaw Research for sharing data from the simulations of chignolin. This research was supported by the European Union Grant No. ERC-2014-AdG-670227/VARMET. Calculations were carried out on the Mönch cluster at the Swiss National Supercomputing Center (CSCS).
|
---
author:
- Anonymous ECCV submission
bibliography:
- 'egbib.bib'
title: 'High Resolution Zero-Shot Domain Adaptation of Synthetically Rendered Face Images - **Supplementary Material**'
---
Overview
========
We use the supplementary material to provide more detail on our method, specifically related to the control vectors and the CS-ANN algorithm. We also show more results (Figure \[fig:ours\_1\] and Figure \[fig:ours\_2\]), as well as some typical failure cases (Figure \[fig:ours\_failure\]). Our method is entirely implemented in tensorflow [@tensorflow2015-whitepaper].
Control Vectors
===============
The set of $33$ control vectors, $v_{control}$, is used in our method both during the sampling stage, and the CS-ANN algorithm. As with the centroids detailed in the main text, the vectors $v_{control}$ are selected purely empirically, based on what we perceived were problem cases for the fitting. We acknowledge that this could be automated in a more principled way in future. We show all control vectors used, added to the mean face of StyleGAN2 (SG2) [@karras2019analyzing] in Figure \[fig:control\_vectors\_by\_cat\]. These vectors were obtained by annotating $2000$ samples from the prior of SG2 based on eight roughly defined categories: (a) Gaze Direction, (b) Beard Style, (c) Head Orientation, (d) Light Direction, (e) Degree of Mouth Open, (f) Hair Style, (g) Hair Type, (h) Skin Texture. Note that we do not claim that these are orthogonal. Interestingly, bias in the FFHQ dataset [@DBLP:journals/corr/abs-1812-04948] can be observed from $v_{control}$. For example, the gaze direction (a) is strongly correlated with the head orientation because subjects tend to look at the camera in photographs.
Convex Set Approximate Nearest Neighbour Search
===============================================
After step 1 of the method returns $w^{s}$, we want to refine this latent code to map to an image better aligned to the input while staying in the space of plausible samples from $G$. The algorithm we propose and describe in the main text can be summarised as follows:
Initialise variables and fix random seeds
While it would be possible to return the *last* result of Algorithm \[alg:csanns\], we found it slightly more effective to return the one that achieves the best loss across all iterations. To do this, we simply store the best loss and corresponding $w$ at any point in the outer or inner loop. If not using a learnable $\beta$ and the corresponding control vectors, we found that convergence was substantially slower. Thus, while it carries the danger of producing implausible results (as it no longer guarantees the result is a combination of samples), we allow $\beta$ to be learned. We note that clamping the range of $\beta$ during the optimisation appeared to be a sufficient constraint in practise.
As pointed out in the text, we allow different weights $\alpha$ for each of the $18$ SG2 inputs. We also experimented with a single weight per sample, using different alphas for each of the actual $18\times512$ inputs, as well as mixtures of these combinations. The former was substantially slower to converge while the latter was quick to produce implausible results.
For matching synthetic images with illumination substantially different from the dataset of real images, we also found it helpful to include a single learnable parameter (clamped to be in the range of $0.7-1.3$) to multiply the reproduced images with, thereby changing brightness by a simple linear transformation. We acknowledge that additional transformations such as histogram equalisation could be explored in future.
Failure Cases
=============
We count any image for which our method fails to produce a relatively close match after $10$ runs with different random seeds as a failure case. As can be seen in Figure \[fig:ours\_failure\], we found very light hair, extreme poses / facial expressions and a lack of contrast between the face and hair challenging cases. We also observe that the further the synthetic image semantics are from the data space (i.e. FFHQ), the less likely our method is to produce satisfactory results. Cases (a) and (b) in Figure \[fig:ours\_failure\], combining beards and haircuts more typically associated with female faces, are examples of this. In such instances, the extension to our method that retains the hair from the synthetics can still produce plausible results as long as the facial shapes are approximately matched.
![Extended illustration of the different steps of our method. (a) synthetic render, (b) output of the sampling in step 1, (c) the result of CS-ANN Search in step 2, and (d-f) results from step 3. Please refer back to the method section of the main text.[]{data-label="fig:ours_1"}](ppt_figures/images/algorithm_progress_1.png){width="\textwidth"}
![All control vectors, added to the ’mean’ face of SG2. The categories are (a) Gaze Direction, (b) Beard Style, (c) Head Orientation, (d) Light Direction, (e) Degree of Mouth Open, (f) Hair Style, (g) Hair Type, (h) Skin Texture.[]{data-label="fig:control_vectors_by_cat"}](ppt_figures/images/control_vector_by_category.png){width="\textwidth"}
![More results of our method vs the *StyleGAN2 Baseline*. We note that while our method preserves most of the synthetic image semantics, smaller beards, and eye gaze can be lost. Best viewed digitally, as every image is at 1K resolution.[]{data-label="fig:ours_1"}](ppt_figures/images/ours_Results_1.png){width="\textwidth"}
![Yet more results of our method vs the *StyleGAN2 Baseline*. We note that while our method preserves most of the synthetic image semantics, smaller beards, and eye gaze can be lost. Best viewed digitally, and in colour.[]{data-label="fig:ours_2"}](ppt_figures/images/ours_Results_2.png){width="\textwidth"}
![Example failure cases (persistent after 10 random starts of our method). Note how retaining the hair from synthetics in *(ours (only face))* can help in such instances. It is of interest that the more unrealistic hair appearance in (e) is matched to a hat by our algorithm. The SG2 generator also appears to resist being steered with combinations unlikely to appear in FFHQ, such as (a) or (b).[]{data-label="fig:ours_failure"}](ppt_figures/images/failure_cases_0.png){width="\textwidth"}
|
---
abstract: 'Lithium abundances in a sample of halo dwarfs have been redetermined by using the new T$_{eff}$ derived by Fuhrmann et al (1994) from modelling of the Balmer lines. These T$_{eff}$ are reddening independent, homogeneous and of higher quality than those based on broad band photometry. Abundances have been derived by generating new atmospheric models by using the ATLAS-9 code by Kurucz (1993) with enhanced $\alpha$-elements and without the overshooting option. The revised abundances show a remarkably flat [*plateau*]{} in the Li-T$_{eff}$ plane for T$_{eff}$$>$ 5700 K with no evidence of trend with T$_{eff}$ or falloff at the hottest edge. Li abundances are not correlated with metallicity for \[Fe/H\]$<$ -1.4 in contrast with Thorburn (1994). All the determinations are consistent with the same pristine lithium abundance and the errors estimated for individual stars fully account for the observed dispersion. The weighted average Li value for the 24 stars of the plateau with T$_{eff}$$>$ 5700 K and \[Fe/H\]$\le$ -1.4, is \[Li\] = 2.210 $\pm$ 0.013, or 2.224 when non-LTE corrections by Carlsson et al (1994) are considered.'
author:
- 'Paolo Molaro$^1$, Francesca Primas$^2$, Piercarlo Bonifacio$^2$'
date: 'Accepted: January 1995'
title: Lithium Abundance of Halo Dwarfs Revised
---
Introduction
============
The lithium observed in the atmospheres of unevolved halo stars is generally believed to be an essentially unprocessed element which reflects the primordial yields. In the framework of the standard BBN it provides a sensitive measure of $\eta$=$n_{b}/n_{\gamma}$ at the epoch of the primordial nucleosynthesis and thus of the present baryon density $\Omega_{b}$. The primordial nature of the lithium of the halo dwarfs is inferred from the presence of a constant lithium abundance for all the halo dwarfs where convection is not effective (T$_{eff} \ge$ 5600 K). Such an uniformity is taken as evidence for the absence of any stellar depletion during the formation and the long life of the halo stars and also as evidence for the absence of any production mechanism acting either before or at the same time of the formation of the halo population.
The existence of a real [*plateau*]{} has been recently questioned by Thorburn (1994), Norris et al (1994) and Deliyannis et al (1993). Thorburn (1994) found trends of the Li abundance both with T$_{eff}$ and \[Fe/H\], while Norris et al (1994) found that the most extreme metal poor stars provide lower abundances by $\approx$ 0.15 dex, thus questioning their genuine primordial value. An intrinsic dispersion of Li abundances in the plateau was claimed by Deliyannis et al (1993) from the analysis of the [*observable*]{} EW and (b-y)$_{0}$ . These results open the possibility of substantial depletion by rotational mixing where a certain degree of dispersion is foreseen for different initial angular momenta of the stars and/or to a significant Galactic lithium enrichment within the first few Gyrs. Thus it appears rather problematic to pick up the precise primordial value from the observations of the Pop II stars. Thorburn (1994) has suggested to estimate it from the surface lithium abundances of the hottest and most metal-poor stars.
In this work we tackle these problems by recomputing the lithium abundances for a significant subset of those stars already studied in literature for which new and better effective temperatures are now available. A possible origin of the systematic differences in the lithium abundances resulting in the most recent determinations will also be discussed. Further details can be found in Molaro et al (1995).
Lithium Abundances
==================
The role of T$_{eff}$ and of atmospheric models
-----------------------------------------------
In the atmospheres of G dwarfs lithium is mainly ionized and the abundance determination is particularly sensitive to the effective temperature and to the T($\tau$) behaviour inside the photosphere since it requires large ionization corrections. Conversely, lithium abundance is not particularly sensitive to the stellar surface gravity and to the metallicity of the star. Also the LiI 6707 Å line is not generally saturated due to the intrinsically low lithium abundance of the halo stars, and therefore it is insensitive to the value of the microturbulent velocity. The effective temperature is by far the most important parameter and its accuracy determines the ultimate lithium abundance accuracy. Unfortunately, the determination of the effective temperature for cool stars and in particular for metal poor stars is rather poor, as discussed in detail in Fuhrmann et al (1994). Fuhrmann et al pointed out the severe limitations of the methods based on broad band photometry, which they considered inadequate to provide accurate effective temperatures for individual stars. The main arguments rely on the dependence of colour-based temperatures on the reddening corrections and on the particular color used.
Star T$_{B}\pm\sigma_{T_{B}}$ $\log g$ EW$\pm\sigma_{EW}$ \[Li\]$\pm\sigma_{Li}$
---------------------- -------------------------- ---------- -------------------- ---------------------------
HD 3567 5750$\pm$200 4.0 45$\pm$5.8 2.221$\pm$0.197
HD 19445 6040$\pm$52 4.2 33.6$\pm$0.5 2.253$\pm$0.039
HD 64090 5499$\pm$56 4.1 12.1$\pm$0.7 1.331$\pm$0.062
HD 74000 6211$\pm$44 4.5 25$\pm$3.2 2.215$\pm$0.073
HD 108177 6090$\pm$77 4.3 30$\pm$1.3 2.229$\pm$0.063
HD 116064 5822$\pm$72 3.6 30$\pm$2.5 2.015$\pm$0.073
HD 140283 5814$\pm$44 3.6 46.5$\pm$0.6 2.262$\pm$0.036
HD 160617 5664$\pm$84 3.5 42$\pm$3.8 2.103$\pm$0.087
HD 166913 5955$\pm$109 3.3 40$\pm$3.8 2.294$\pm$0.100
HD 188510 5500$\pm$220 4.0 18$\pm$3.4 1.516$\pm$0.222
HD 189558 5573$\pm$92 4.0 42$\pm$1.6 2.021$\pm$0.086
HD 193901 5700$\pm$109 4.0 30$\pm$3.3 1.962$\pm$0.111
HD 194598 5950$\pm$100 4.0 27$\pm$0.7 2.094$\pm$0.076
HD 200654 5522$\pm$119 3.2 8$\pm$1.7 1.129$\pm$0.143
HD 201889 5645$\pm$61 4.1 5$\pm$3.3 1.065$^{+0.230}_{-0.500}$
HD 201891 5797$\pm$57 4.4 24.3$\pm$0.8 1.925$\pm$0.048
HD 211998 5338$\pm$65 3.5 13$\pm$3.4 1.219$\pm$0.150
HD 219617 5815$\pm$76 4.2 40.2$\pm$0.8 2.198$\pm$0.062
BD 2$^{\circ}$ 3375 6034$\pm$60 4.0 31.5$\pm$2.1 2.198$\pm$0.058
BD 3$^{\circ}$ 740 6264$\pm$73 3.5 17.3$\pm$1.3 2.062$\pm$0.065
BD 9$^{\circ}$ 352 6285$\pm$77 4.5 34$\pm$6.7 2.429$\pm$0.130
BD 9$^{\circ}$ 2190 6452$\pm$60 4.0 18$\pm$3.4 2.200$\pm$0.107
BD 17$^{\circ}$ 4708 6100$\pm$110 4.1 25$\pm$1.7 2.139$\pm$0.084
BD 21$^{\circ}$ 607 6135$\pm$70 4.0 25$\pm$2.6 2.190$\pm$0.074
BD 23$^{\circ}$ 3130 5190$\pm$84 2.7 13$\pm$1.3 1.066$\pm$0.095
BD 24$^{\circ}$ 1676 6278$\pm$76 3.9 27$\pm$1.6 2.296$\pm$0.062
BD 26$^{\circ}$ 2606 6161$\pm$64 4.1 30$\pm$1.2 2.252$\pm$0.050
BD 29$^{\circ}$ 366 5760$\pm$64 3.8 14$\pm$2.9 1.641$\pm$0.105
BD 37$^{\circ}$ 1458 5451$\pm$59 3.5 11$\pm$2.1 1.226$\pm$0.107
BD 38$^{\circ}$ 4955 5337$\pm$73 4.5 8$\pm$3.3 1.016$\pm$0.250
BD 42$^{\circ}$ 3607 5836$\pm$66 4.4 47$\pm$4.5 2.298$\pm$0.083
BD 66$^{\circ}$ 268 5511$\pm$91 4.0 10$\pm$7.8 1.237$\pm$0.476
G 64-12 6356$\pm$75 3.9 25.8$\pm$2.4 2.318$\pm$0.074
G 64-37 6364$\pm$75 4.1 14$\pm$1.2 2.029$\pm$0.066
G 66-9 5885$\pm$83 4.6 29$\pm$2.9 2.047$\pm$0.081
G 206-34 6258$\pm$50 4.3 27$\pm$2.5 2.265$\pm$0.060
G 239-12 6260$\pm$70 4.1 24$\pm$3.1 2.214$\pm$0.083
G 255-32 5962$\pm$53 4.0 30$\pm$2.9 2.119$\pm$0.064
LP 608 62 6435$\pm$52 4.1 21.5$\pm$2.3 2.282$\pm$0.063
: Table 1
Fuhrmann et al (1994) derived T$_{eff}$ from the full spectral synthesis of the Balmer lines for a large sample of stars. These T$_{eff}$ are reddening independent and they show a high degree of internal consistency when the various members of the serie are used. Being obtained from absorption lines, they are particularly suitable for line abundance applications. Fuhrmann et al are also able to provide errors in the T$_{eff}$ for individual stars and in most cases they are as good as $\pm$ 50 K, which is a factor 2 smaller than the grossly estimated errors for photometric-based T$_{eff}$.
Out of the Fuhrmann et al’ sample, 39 have already been studied for lithium. They represent a significant fraction of the presently available lithium determinations, and form an unique sample with a good and homogeneous T$_{eff}$.
On average, the Fuhrmann et al temperatures are 125$\pm$120 K higher than those previously used in literature on lithium (cfr Fig. 1). The presence of considerable scatter together with a systematic offset suggest that there are intrinsic differences in the individual temperatures derived by various methods. The Li equivalent widths have been taken from the literature and the theoretically derived random errors from Deliyannis et al (1993) or computed following their prescriptions. In the case of multiple measurements of the lithium line of the same star we adopted the weighted average to minimize the errors. Atmospheric models may be important for the lithium abundance. The role played by different atmospheric models is illustrated in Fig. 2 where several curves of growth obtained with different atmospheric models are shown. The COGs are for T$_{eff}$ = 6000 K, $\log$g = 3.5, microturbulence $\xi$ = 2 km s$^{-1}$ and \[Fe/H\]=-3.0, but a similar behaviour is shown by other temperatures relevant for Li. The curves of growth for lithium obtained by using ATLAS-9 and ATLAS-8 are notably different. In Fig. 2 are also shown the COG used by Thorburn (1994) referred to unpublished models of Kurucz (1991) and to those of Bell and Gustaffsson. For a given EW the abundances of the most recent Kurucz codes are $\approx$ 0.1 dex higher than all the other curves. We have computed a grid of atmospheric models where the convection is treated with the mixing length theory but without the overshooting option. The corresponding COG, also shown in Fig. 2, is very close to the COG of the old Kurucz or Bell and Gustaffsson models. This shows that the implementation of the approximate treatment for overshooting in the ATLAS-9 code is likely to be responsible for the difference among the versions of the ATLAS code. The center of the convective bubbles stops at the top of the convective zone so that convective flux extends one bubble radius above the end of the convection zone. Overshooting rises the temperature in correspondence of the depths relevant for the formation of the lithium line, thus resulting in higher abundance for the same equivalent width. The effect is almost negligible at solar metallicities but it increases towards lower metallicities where the fraction of the total flux transported via convection increases.
The presence of abundances derived with different atmospheric models is also responsible for the systematic differences in the lithium abundances for common stars among different authors. This factor has been essentially overlooked and has introduced a spurious scatter in a straightforward compilation of the literature data. The comparison of the measurements of Thorburn (1994) with those of the literature shows that the equivalent widths are almost the same but the Thorburn abundances derived using Kurucz (1991) are systematically higher than those by other authors. The finding by Norris et al (1994) showing that Li abundances in stars with \[Fe/H\]$<$-3.0 are on average about 0.15 dex lower than the higher metallicity halo stars from Thorburn (1994) can be also understood as a model effect. Norris computed the Li abundances by using Bell and Gustaffsson atmospheric models which give abundances lower than those by Thorburn (1994). The comparison between the stars studied by both Norris et al (1994) and Thorburn (1994) shows nearly identical EWs for both, but lower abundances in the former authors.
Implementation of overshooting improves the reproduction of the solar spectrum, but the effects on metal poor stars have not been yet fully tested and it should be considered with caution until a deeper analysis of possible secondary effects on metal poor objects is carried out. Here we have not used this option and the new lithium abundances have been determined by computing Kurucz (1993) atmospheric models without overshooting. For all the stars the models have been computed with the temperatures available from Fuhrmann et al (1994). The gravities used are taken from the literature according to the compilation of Deliyannis et al (1993) or computed from [**]{} c$_{0}$ colors, and are reported in Table 1. Convection is treated with the mixing length theory with a scale height over pressure scale of 1.25. The choice of this parameter is not critical and a change from 0.5 to 2 produces almost negligible effects on the lithium abundance. If we use the Kurucz 1993 grid which includes overshooting the lithium abundance is increased by $\approx$ 0.09 dex, but all other conclusions are still valid, since they are not dependent on this assumption. In view of the important implications related to the primordial abundances, it is desirable that a deeper discussion may follow shortly.
Results
=======
The Li abundances in the Li-T$_{eff}$ plane with associate 1 $\sigma$ errors are shown in Fig. 3. Following Rebolo et al (1988) and Deliyannis et al (1990), stars with -1.4$<$\[Fe/H\]$<$-1.0 (open squares in Fig. 3) are taken off from sample of genuine halo stars since they may show some depletion. The errors shown in Fig. 3 are those given in the last column of Table 1. They have been estimated by summing under quadrature the errors in the lithium abundance produced by the uncertainties in T$_{eff}$ and in EW, also given in Table 1, namely $\sigma_{Li}^2 = \sigma_{Li}^2(EW) + \sigma_{Li}^2(T_{eff})$. We have considered negligible the effects produced by uncertainties in gravity, microturbulence and metallicity. The lithium abundances show a plateau extending up to nearly 6500 K, with no evidence of falloff at the hottest edge as expected by microscopic diffusion models, and with the depletion region bending at T$_{eff}$ $\approx$ 5700 in good agreement with what observed in the Hyades.
The existence of a tilt of the plateau has received considerable attention (Molaro 1987, Rebolo et al 1988, Thorburn 1994, Norris et al 1994) since the presence of a moderate stellar depletion in the cool edge [*plateau*]{} implies that only the highest values are close to the pristine value. Norris et al (1994) and Thorburn (1994) found a slope of 0.03 and 0.024 for 100 K, respectively. Thorburn (1994) claimed also that when this underlying trend with T$_{eff}$ is taken into account the increase of N(Li) with the metallicity becomes notable already at \[Fe/H\]$\approx$-2.0. Considering only stars with Teff $>$ 5700 K and \[Fe/H\]$\le$ -1.4 the weighted linear fit is \[Li\]$\propto 0.58(\pm 0.88) \cdot (T_{eff}/10^4)$. The slope we found is one order of magnitude lower than those of Norris et al (1994) or Thorburn (1994), and is consistent with a real [*plateau*]{} in the undepleted region of the Li-T$_{eff}$ plane. Non-LTE effects have been studied by Carlsson et al (1994), who provide corrections of 0.020, 0.015 and 0.010 at 5500, 6000 and 6500 K respectively, and for \[Fe/H\] between -1.0 and -3.0. Once we correct our abundances for nonLTE effects the slope of the fit becomes even smaller: 0.29($\pm$ 0.93).
The lithium abundances versus the stellar metallicity are shown in Fig. 4 and do not show any clear trend with iron over two orders of magnitude of increase in the stellar metallicity. Strictly speaking we observe a decrease in the lithium abundance with the increasing of the metallicity. The weighted regression analysis gives a negative slope with lithium decreasing by 0.008 dex for $\Delta$\[Fe/H\]=1. By contrast Thorburn (1994) found an increase in Li by 0.4 dex from the \[Li\]=2.20 at \[Fe/H\]=-3.5 up to 2.60 at \[Fe/H\]=-1.0. Our interception at \[Fe/H\]=-3.5 is of \[Li\]=2.215, fully consistent with the value of \[Li\]=2.221 obtained from the hottest halo dwarfs (T$_{eff}$=6400 K). In our data sample both the lowest metallicity stars, i.e. the oldest, and the hottest subdwarf, i.e. the less depleted, share the same Li abundance.
Fig. 5 shows a zoom of the plateau region with the errors in the Li abundances at the 3 $\sigma$ level plotted for each star. On the plateau all the Li abundances are consistent with a unique pristine Li abundance. In 10 out of 24 cases the consistency is achieved already at 1 $\sigma$ confidence level. For three stars (G 64-13, BD 3$^{\circ}$ 740, HD 116064) the full 3$\sigma$ error box is required to achieve the consistency, and they might show a real dispersion if errors can be further reduced. With the present estimated errors our analysis does not support the presence of real dispersion on the plateau region, and it seems likely that the dispersion claimed by Thorburn (1994), or the correlations of Li with metallicity or temperature, are artifacts caused by errors in the effective temperatures. We stress that both the absence of a tilt and dispersion in the plateau region are independent of the assumption we made on overshooting, since its inclusion would increase the Li abundance by about the same amount for all the stars. The weighted mean on the plateau of the 24 stars with T$_{eff}$ $\ge$ 5700 and \[Fe/H\]$\le$ -1.4, where each abundance is weighted inversely by its own variance in the sum, is \[Li\]=2.210 $\pm$ 0.013. When the non-LTE corrections of Carlsson et al (1994) are considered, the mean rises to 2.224 $\pm$ 0.013. This value is somewhat higher than the 2.08$\pm$0.1 previously estimated by Spite and Spite (1982), Hobbs and Duncan (1988), Rebolo et al (1988), and Molaro (1991). The increase in the value of the plateau results from the increase of the Fuhrmann et al effective temperatures compared to those previously used in the lithium literature.
The present analysis shows that when very precise effective temperatures and individual errors are considered, the Li abundances on the plateau show no trends either with T$_{eff}$ in a range of 600 K or with the stellar metallicity over two orders of magnitude. The lithium abundances are all closely gathered and are consistent with the same initial abundance, thus confirming that the lithium observed in these stars is essentially undepleted and very close to the primordial value as already put forward by Spite and Spite (1982).
Carlsson M., Rutten, R.J., Bruls, J.H.M.J., Shchukina N.G. 1994 AA 288, 860 Fuhrmann, K., Axer, M., Gehren T., 1994 AA 285, 585 Deliyannis C. P., Demarque, P., and Kawaler, S.D., 1990 ApJS 73, 21. Deliyannis C. P., Pinsonneault M. H., Duncan D. K., 1993 ApJ 414, 740 Hobbs, L.M., and Duncan D.K., 1988 ApJ 317, 796. Kurucz, R.L., 1993, CD-ROM No 11, 13, 18 Kurucz, R.L., 1991, private comunication in Thorburn (1994) Molaro, P., 1991 Mem. Soc. Astron. Ital., 62, 17 Molaro, P., 1987 phd Thesis I.S.A.S. Trieste, Italy. Molaro, P., Bonifacio, P., Primas, F. 1995 proc. IAU JCM on [*Stellar and Interstellar Lithium and Primordial Nucleosynthesis*]{} memorie SAIt in press. Norris, J. E., Ryan, S. G., Stringfellow G. S., 1994 ApJ 423, 386 Rebolo, R., Molaro, P., Beckman, J., 1988 AA 192, 192. Spite, F., Spite, M., 1982 AA 163, 140. Thorburn, J., 1994, ApJ 421, 318.
|
---
abstract: |
We address the statistics of continuous weak linear measurement on a few-state quantum system that is subject to a conditioned quantum evolution. For a conditioned evolution, both the initial and final states of the system are fixed: the latter is achieved by the post-selection in the end of the evolution. The statistics may drastically differ from the non-conditioned case, and the interference between initial and final states can be observed in the probability distributions of measurement outcomes as well as in the average values exceeding the conventional range of non-conditioned averages. We develop a proper formalism to compute the distributions of measurement outcomes, evaluate and discuss the distributions in experimentally relevant setups.
We demonstrate the manifestations of the interference between initial and final states in various regimes. We consider analytically simple examples of non-trivial probability distributions. We reveal peaks (or dips) at [*half-quantized*]{} values of the measurement outputs. We discuss in detail the case of zero overlap between initial and final states demonstrating anomalously big average outputs and [*sudden jump*]{} in time-integrated output.
We present and discuss the numerical evaluation of the probability distribution aiming at extending the analytical results and describing a realistic experimental situation of a qubit in the regime of resonant fluorescence.
author:
- 'A. Franquet'
- 'Yuli V. Nazarov'
bibliography:
- 'authorbibtex.bib'
title: Probability distributions of continuous measurement results for conditioned quantum evolution
---
Introduction {#sec:intro}
============
The concept of measurement is one of the most important, characteristic, and controversial parts of quantum mechanics. Due to the intrinsically probabilistic nature of the measurement and associated paradoxes,[@Leggett] it continues to attract research attention and stimulate new experiments. The ability to control a quantum system that is of increasing importance in the context of quantum information processing, requires an adequate yet sufficiently general description of the measurement process. Such description is provided by the theory of continuous weak linear measurement (CWLM), where a sufficiently weak coupling between the quantum system and multiple degrees of freedom of a detector mediates their entanglement and results in conversion of discrete quantum information into continuous time-dependent readings of the detector.[@CWLM0; @CWLM1; @CWLM2; @CWLM25; @NazWei; @CWLM3; @CWLM4] The description follows from the general linear response theory and gives an explicit connection between quantum measurement and quantum noise.[@QNoise]\
Recent experimental advances have made possible the efficient continuous measurement and monitoring of elementary quantum systems (qubits) giving the information on individual quantum trajectories.[@Devoret; @SiddiqiSingle; @SiddiqiEntanglement]The individual traces of quantum evolution can be post-selected by a projective measurement at the end of evolution, thus enabling the experimental investigation of conditioned quantum evolution where both initial and final states are known.[@Huard; @DiCarlo; @SiddiqiMapping; @SiddiqiMolmer]\
For experimentally relevant illustrations, we concentrate in this paper on a setup of resonance fluorescence.[@Huard] In this setup, a transmon qubit with ground state ${| g\rangle}$ and excited state ${| e\rangle}$ is enclosed in a non-resonant three-dimensional (3D) superconducting cavity connected to two transmission lines. A resonant field drives the qubit via the weakly coupled line, while most of the fluorescence signal exits via the other line which is coupled strongly. The amplitude of the signal is proportional to $\sigma_-$, the average of the lowering operator $\hat{\sigma}_-={| g\rangle}{\langle e |}$ of the qubit, and oscillates with the Rabi frequency $\Omega$ set by the resonant drive.\
A heterodyne detection setup is used to measure this signal. The measurement proceeds in many runs of equal time duration. At each run, the qubit is prepared in a state ${| e\rangle}$ or ${| g\rangle}$ and the signal is monitored at the time interval $0<t<{\cal T}$. At the end of the interval, $t={\cal T}$, one can projectively measure the qubit to find it either in the state ${| e\rangle}$ or ${| g\rangle}$ with high fidelity using a microwave tone at the bare cavity frequency. With such a setup, the fluorescence signal can be interpreted as a result of a weak continuous measurement, that can be conditioned not only on an initial state but also on a final state by post-selecting with the result of the projective measurement. The authors have concentrated on the conditioned signal at a given moment of time that is averaged over many runs. Its time traces reveal interference patterns interpreted in terms of weak values [@WeakValues] and associated with the interference of initial and final quantum states in this context.[@WisemanWeakValues; @paststates]
The concept of weak values has been introduced in [@WeakValues] to describe the average result of a weak measurement subject to post-selection in a simplified setup. The authors have shown that the average measurement results may be paradoxically large as compared to the outputs of corresponding projective measurements. Since that, the concept has been extended in various directions, e.g. to account for the intermediate measurement strength, the Hamiltonian evolution of the quantum states during the measurement, see [@WeakReview1; @WeakReview2] for review. In [@WisemanWeakValues], the average measurement outputs have been investigated in the context of continuous weak measurement, this has been further elaborated in [@ContWeak1; @ContWeak2; @ContWeak3]. As to the detailed statistics of the measurement outcomes, in this context it has been considered only for simplified meter setups that correspond to measuring the light intensities in quantum optics.[@WeakReview1; @WeakReview2] There is a tendency to term “weak value” a result of any weak measurement that involves post-selection. This may be confusing in general. For instance, the duration of a weak measurement can exceed the relaxation time of the system measured. The averaged measurement output in this case is not affected by post-selection and equals to the expectation value of the operator measured with the equilibrium density matrix. This is very far from the original definition of weak values[@WeakValues]. We prefer to stick to the original definition.
We notice that the experiment discussed gives access not only to the conditioned averages, but also to the conditioned statistics of the measurement results. For instance, at each run one can accumulate the output signal on a time interval that is $(0,{\cal T})$ or a part of it and record the results. After many runs, one makes a histogram of the records that depends on the initial as well as on the final state of the qubit.
This article elaborates on the method to evaluate the distribution of the accumulated signal and gives the detailed theoretical predictions of the conditioned statistics for examples close to the actual experimental situation, and in a wide range of parameters.
In this Article, we put forward and investigate two signatures of the conditioned statistics. First is the [*half-quantized*]{} measurement values. A non-conditioned CWLM distribution under favorable circumstances peaks at the values corresponding to quantized values of the measured operator, in full correspondence with a text-book projective measurement. We demonstrate that a conditioned distribution function displays peculiarities — that are either peaks or dips — at [*half-sums*]{} of the quantized values.
Second signature pertains the case of zero or small overlap between initial and final state and time intervals that are so short as the wave function of the system does not significantly change. In this case, we reveal unexpectedly large values of the cumulants of the distribution function of time-integrated outputs for such short intervals, that we term [*sudden jump*]{}. For the average value of the output, the fact that it may by far exceed the values of typical outcome of a projective measurement, can be understood from the weak value theory [@WeakValues]. We extend these results to the distributions of the output and reveal the role of decoherence at small time intervals.
We stress that the signatures by itself present no new phenomenon. Rather, the basic quantum phenomena like interference manifest themselves in these signatures in the context of CWLM statistics. As such, we permit a re-interpretation of these phenomena in the context considered.
Our approach to the CWLM statistics is based on the theory of full counting statistics in the extended Keldysh formalism.[@QTBook] The statistics of measurements of $\int dt \hat{V}(t)$, $V(t)$ being a quantum mechanical variable representing linear degrees of freedom of the environment, are generated via a characteristic function method and the use of counting field technique. It provides the required description of the whole system consisting of the measured system, the environment and detectors.\
Here we develop this formalism first introduced in,[@NazKin; @NazWei] to include the conditioned evolution. We focus on the pre- and post-selected measurements. In this case, a quantum system is initially prepared in a specific state. After that, it is subject of CWML during a time interval ${\cal T}$. The post-selection in a specific state takes place in the end of the procedure. We show that the evolution of a qubit whose past and future states are known can be inferred and understood from the measured statistics of measurement outcomes. The measurement of the statistics can reveal purely quantum features in experimentally relevant regimes.\
We show how interference arises even at relatively small time scales and how the information about the initial qubit state is lost during the time evolution making the interference to vanish at sufficiently long time scales. We exemplify how different features in the distributions can be understood as the manifestations of the qubit evolution during the measurement. And we numerically study various parameter regimes of interest in the case of a measurement of a single observable.
Actually, we show with our results that one can have very detailed theoretical predictions of CWLM distributions that can account for every detail of the experiment. This enables investigation and characterization of quantum effects even if the choice of parameters is far from the optimal one and these effects are small.
The structure of the article is as follows. We develop the necessary formalism in Section \[sec:method\], starting from a Bloch-master equation for the qubit evolution that is augmented with counting fields to describe the detector statistics, and explain how the post-selection is introduced in this scheme. The scheme can be applied to various experimental scenarios, in particular we focus on the setup described in [@Huard]. It is important to illustrate how the Cauchy-Schwartz inequalities impose restrictions on the parameters entering the Bloch-master equations, this resulting in several different time scales. In Section \[sec:simple\] we examine a measurement of a general observable and explain how the half-quantized peculiarities arise in the distributions of measurement outcomes depending on the initial and final state. In Section \[sec:suddenjump\] we concentrate on the case of zero overlap and take the Hamiltonian dynamics into account to arrive at essentially non-Gaussian probability distributions. In these Sections, we mostly concentrate on a simple limit where the time interval ${\cal T}$ is much smaller than the typical time scales of qubit evolution, this gives the opportunity for analytical results. Next, we extend our study to longer time intervals. In the Section \[sec:decscale\] we present numerical simulations at the scale of decoherence time for three relevant cases: the case of an ideal detector, and the experimentally relevant case with and without detuning. In Section \[sec:shortscale\], we concentrate on the time scales of Hamiltonian dynamics and experimentally relevant parameters. We conclude in the Section \[sec:conclusion\]
Method {#sec:method}
======
The description of CWLM can be achieved by several methods, all of them taking into account the stochastic nature of the measurement process. In simplest situations like non-demolition measurements [@CWLM1] one can use the quantum filtering equation [@Belavkin]. More sophisticated approaches include effective action method [@CWLM0; @CWLM4], path integral formulation[@NazWei; @CWLM3], past states formalism [@paststates]. A powerful numerical method of experimental significance is the stochastic update equation [@trajectory] that allows to monitor density matrix taking into account the measurement results. In this method, the distribution of outcomes is obtained numerically by collecting statistics of the realizations of “quantum trajectories”. In contrast to this, the method of[@NazWei] permits the direct computation of the generating function of the probability distribution.
The present goal is to formulate a method to compute probability distributions of a continuous measurement in the course of a conditioned quantum evolution. We will extend the method presented in [@NazWei] where the central object is a Bloch-master equation for the evolution of the measured quantum system that is augmented with the counting fields. Evaluating the trace of the extended density matrix from this equation as a function of the counting fields provides the generating function for the probability distribution of the detector output. To outline the formalism, we will focus first on the simplest setup where a single detector measures a single qubit variable $\hat{{\cal O}}$. In the end of the section we will give a generalization to the case of two variables.\
In general, the dynamics of an isolated quantum system are governed by a Hamiltonian $\hat{H}_{q}$. For a realistic system, weak interaction with an environment representing the outside world will generate decoherence and relaxation . In the CWLM paradigm, the quantum system is embedded in a linear environment described in the same manner by a Hamiltonian $\hat{H}_{d}$. The quantum system interacts with the environment via a coupling Hamiltonian $\hat{H}_{c}$,
$$\label{eq1}
\hat{H} = \hat{H}_{q} + \hat{H}_{c} + \hat{H}_{d}$$
with
$$\label{eq2}
\hat{H}_{c} = \hat{{\cal O}}\hat{Q},$$
$\hat{{\cal O}}$ being an operator in the space of the quantum system, that value is to be measured. Since $\hat{H}_{d}$ is a Hamiltonian of a linear system, it can generally be represented by a boson bath Hamiltonian. The input of the detector is characterized by an [*input*]{} variable $\hat{Q}$ that is linear in boson fields. The output of the detector is represented by the [*output*]{} variable $\hat{V}$ that is also linear in boson fields.\
The dynamics and statistics of the measurement process are fully characterized by the two-time correlators of the operators $\hat{Q}(t)$, $\hat{V}(t)$. If we assume the qubit dynamics is slower than a typical time scale of the environment, the four relevant quantities correspond to zero-frequency values of the correlators,
\[corr\] $$\begin{aligned}
S_{QQ} =& \frac{1}{2}{\displaystyle \int_{-\infty}^{t}}dt'\left\langle\left\langle\hat{Q}(t)\hat{Q}(t')+\hat{Q}(t')\hat{Q}(t)\right\rangle\right\rangle,\\*
S_{QV} =& \frac{1}{2}{\displaystyle \int_{-\infty}^{t}}dt'\left\langle\left\langle\hat{Q}(t)\hat{V}(t')+\hat{V}(t')\hat{Q}(t)\right\rangle\right\rangle,\\*
S_{VV} =& \frac{1}{2}{\displaystyle \int_{-\infty}^{t}}dt'\left\langle\left\langle\hat{V}(t)\hat{V}(t')+\hat{V}(t')\hat{V}(t)\right\rangle\right\rangle,\\*
a_{VQ} =& -\frac{i}{\hbar}{\displaystyle \int_{-\infty}^{t}}dt'\left\langle[\hat{V}(t),\hat{Q}(t')]\right\rangle,\\*
a_{QV} =& -\frac{i}{\hbar}{\displaystyle \int_{-\infty}^{t}}dt'\left\langle[\hat{Q}(t),\hat{V}(t')]\right\rangle.\end{aligned}$$
where $\langle\langle\hat{A}\hat{B}\rangle\rangle = \langle(\hat{A}-\langle\hat{A}\rangle)\rangle\langle(\hat{B}-\langle\hat{B}\rangle)\rangle$ for any pair of operators $\hat{A},\hat{B}$.\
These four quantities define the essential characteristics of the measurement process and have the following physical meaning. $S_{QQ}$ is the noise of the input variable. It is responsible for the inevitable measurement back action and associated decoherence of the qubit. $S_{VV}$ is the output variable noise: it determines the time required to measure the detector outcome with a given accuracy. The cross noise $S_{QV}$ quantifies possible correlations of these two noises. The response function $a_{VQ}$ determines the detector gain: it is the susceptibility relating the detector output to the qubit variable measured, ${\langle \hat{V} \rangle}=a_{VQ}{\langle \hat{{\cal O}} \rangle}$. The response function $a_{QV}$ is correspondingly the reverse gain of the detector: it gives the change of the qubit variable proportional to the detector reading. Conforming to the assumption of slow qubit dynamics, the noises are white and responses are instant.\
The values of these noises and responses are restricted by a Cauchy-Schwartz inequality, [@QNoise]
$$\label{ineq}
S_{QQ}S_{VV}-{| S_{QV} |}^{2}\geq\frac{\hbar^{2}}{4}{| a_{VQ}-a_{QV} |}^{2}.$$
For a simple system like a single qubit it is natural to make the measured operator dimensionless, with eigenvalues of the order of one, or, even better, $\pm 1$. With this, one can define and relate the dephasing rate $2 \gamma = 2S_{QQ}/\hbar^{2}$ and the acquisition time $t_{a}\equiv 4 S_{VV}/{| a_{VQ} |}^{2}$ required to measure the variable with ${\cal O}$ with a relative accuracy $\simeq 1$. If one further assumes the direct gain to be much larger than the reverse gain, $a_{VQ}\gg a_{QV}$, it is implied by the central equation of [@QNoise], Eq. , $$\gamma t_{a}\geq 1$$ This figure of merit shows that one cannot measure a quantum system without dephasing it.\
The statistics of the detector variable $\hat{V}$ can be evaluated with introducing a counting field $\chi(t)$ coupled to the output variable $\hat{V}$. This field plays the role of the parameter in the generating function $C({\chi(t)})$ of the probability distribution of the detector readings $V(t)$.\
This generating function is computed in the extended Keldysh scheme [@QTBook] where the evolution of the “ket” and “bra” wave functions is governed by different Hamiltonians, $\hat{H}^{+}$ and $\hat{H}^{-}$ respectively. The extra term describing interaction with the counting field reads $\hat{H}^{\pm} = \hat{H} \pm \hbar \chi(t)\hat{V}(t)/2$. The generating function has then the form $$\label{gen1}
C(\{\chi(t)\}) = \text{Tr}_{q}\left(\hat{\rho}(\{\chi(t)\})\right),$$ $\hat{\rho}$ being a quasi-density matrix of the qubit in the end of evolution, $$\label{gen2}
\hat{\rho}(\chi; t) = \text{Tr}_{d}\left(\overrightarrow{T}e^{-i/\hbar\int dt \hat{H}^{-}}\hat{\rho}(0)\overleftarrow{T}e^{+i/\hbar\int dt \hat{H}^{+}}\right).$$ Here, $\text{Tr}_{q}(\cdots)$ and $\text{Tr}_{d}(\cdots)$ denote the trace over qubit and detector variables, respectively, and $\overrightarrow{T}$($\overleftarrow{T}$) denotes time (reversed) ordering in evolution exponents. $\hat{\rho}(0)$ is the initial density matrix for both qubit and detector systems.\
Assuming white noises and instant responses, one can derive an evolution Bloch-master equation for the quasi-density matrix that is local in time, like Eq. (13) in [@NazWei]. For the simplest setup, under assumption of a single coupling operator $\hat{{\cal O}}$ it reads:
$$\begin{aligned}
\label{eq3}
\frac{\partial\hat{\rho}}{\partial t} =& -\frac{i}{\hbar}[\hat{H}_{q},\hat{\rho}] - \frac{S_{QQ}}{\hbar^{2}}\mathcal{D}[\hat{\cal O}]\hat{\rho} -\frac{\chi^{2}(t)}{2}S_{VV}\hat{\rho} \\*
\nonumber
& -\frac{S_{QV}}{\hbar}\chi(t)[\hat{\rho},\hat{\cal O}] + \frac{i a_{VQ}\chi(t)}{2}[\hat{\rho},\hat{\cal O}]_{+}.\end{aligned}$$
Here, $[,]$ and $[,]_{+}$ refer to commutator and anti-commutator operations respectively and $\mathcal{D}[\hat{A}]\hat{\rho}\equiv\left(\frac{1}{2}[\hat{A}^{\dagger}\hat{A},\hat{\rho}]_{+}-\hat{A}\hat{\rho}\hat{A}^{\dagger}\right)$. Here we have also assumed $a_{VQ}\gg a_{QV}$, a general condition for a good amplifier. A single coupling operator is an idealization, in a more realistic situation, the quantum system is also coupled to the environment with other degrees of freedom not related to the equation, this is manifested as intrinsic relaxation and decoherence. This modifies the above equation.
We give the concrete form of this equation for the experimental situation of [@Huard]. There is a qubit with two levels split in $z$-direction under conditions of strong resonant drive that compensates the splitting of the qubit levels. The effective Hamiltonian reads $$\hat{H}_{q}=\frac{\hbar}{2}\Omega\hat{\sigma}_{x}+ \frac{\hbar}{2}\Delta\hat{\sigma}_{z},$$ $\Omega$ being the Rabi frequency proportional to the amplitude of the resonant drive, and $\Delta$ being the detuning of the drive frequency from the qubit energy splitting. The interaction with the environment induces decoherence, excitation and relaxation of the qubit, with the rates $\gamma_{d},\gamma_{\uparrow},\gamma_{\downarrow}$ respectively. The measured quantity is the amplitude of the irradiation emitted from the qubit, so ${\cal O}$ is convenient to choose to be either $\sigma_x$ or $\sigma_y$. With this, the equation reads $$\begin{aligned}
\label{eq3exp}
\frac{\partial\hat{\rho}}{\partial t} =& -\frac{i}{\hbar}[\hat{H}_{q},\hat{\rho}] -\gamma_{d}\mathcal{D}[\hat{\sigma}_{z}]\hat{\rho}-\gamma_{\uparrow}\mathcal{D}[\hat{\sigma}_{+}]\hat{\rho}\\*
\nonumber
&-\gamma_{\downarrow}\mathcal{D}[\hat{\sigma}_{-}]\hat{\rho} -\frac{S_{QV}}{\hbar}\chi(t)[\hat{\rho},\hat{{\cal O}}]\\*
\nonumber
& + \frac{i a_{VQ}\chi(t)}{2}[\hat{\rho},\hat{{\cal O}}]_{+}-\frac{\chi^{2}(t)}{2}S_{VV}\hat{\rho},\end{aligned}$$ $\hat{\sigma}_{+}$ ($\hat{\sigma}_{-}$) being the rising and lowering operators of the qubit, and $\hat{\sigma}_{z}={| e\rangle}{\langle e |}-{| g\rangle}{\langle g |}$ the standard Pauli operator.\
The rates and noises are restricted by the following Cauchy-Schwartz inequality: $\frac{1}{4}\left(\gamma_{\uparrow}+\gamma_{\downarrow}\right)S_{VV}-{| S_{QV} |}^{2}\geq\frac{\hbar^{2}}{4}{| a_{VQ} |}^{2}$. All the parameters entering the equation can be characterized from experimental measurements. We provide an example of concrete values in Section \[sec:decscale\].\
We will concentrate on a single measurement during a time interval $(0,{\cal T})$. To define an output of such measurement, we accumulate the time-dependent detector output during this time interval and normalize it by the same interval, $V \equiv \frac{1}{{\cal T}}\int_{0}^{{\cal T}} V(t') dt'$. The counting field $\chi(t)$ corresponding to this output is conveniently constant , $\chi(t)\equiv \chi$ on the time interval and 0 otherwise. Our goal is to evaluate the probability distribution $P(V)$ of the measurement results, conditioned to an initial qubit state given by $\hat{\rho}(0)$, and to a post-selection of the qubit in a specific state ${| \Psi\rangle}$ at the time moment ${\cal T}$. This involves the projection on the state ${| \Psi\rangle}$, represented by the projection operator $\hat{P}_{\Psi}={| \Psi\rangle}{\langle \Psi |}$ .\
The probability distribution of the detector outcomes with no regard for the final qubit state can be computed from the generating function defined by Eq. ,
$$\label{eq5}
P(V) = \frac{{\cal T}}{2\pi}\int d\chi e^{-i\chi V{\cal T}}C(\chi; {\cal T}).$$
The joint statistics are extracted from the quasi-density matrix $\hat{\rho}(\chi; {\cal T})$ at the end of the interval.\
Upon the post-selection, the quasi-density matrix is projected on the final state measured, $\hat{P}_{\Psi}\hat{\rho}(\chi; {\cal T})$, so the conditioned generating function of the detector outcomes reads as $$\label{eq6}
\tilde{C}(\chi; {\cal T}) = \frac{\text{Tr}_{q}(\hat{P}_{\Psi}\hat{\rho}(\chi; {\cal T}))}{\text{Tr}_{q}(\hat{P}_{\Psi}\hat{\rho}(\chi=0; {\cal T}))}.$$ where the proper normalization is included.\
This is the second central equation in our method. Together with Eq. it permits an efficient evaluation of the conditioned probability distributions as the Fourier transform of this generating function.\
Sometimes it is convenient to normalize the time-integrated output introducing ${\cal O} = V/a_{VQ}$ that immediately corresponds to the eigenvalues of $\hat{\cal O}$ (We stress that ${\cal O})$ are coming from the averaging of an environmental operator rather than $\hat{\cal O}$.
In this Article, we will concentrate on the distributions of a single variable. For completeness, we mention that the approach can be extended to joint statistics of simultaneous measurement of two non-commuting observables, e.g. $\hat{\sigma}_{x}$ and $\hat{\sigma}_{y}$. For the case of identical but independent detectors with associated output variables $\hat{V}_{x}, \hat{V}_{y}$ and counting fields $\chi_{x}(t),\chi_{y}(t)$ the corresponding equation reads( $i$ labels $\lbrace x,y\rbrace$) $$\begin{aligned}
\label{eq3_2}
&\frac{\partial\hat{\rho}}{\partial t} = -\frac{i}{\hbar}[\hat{H}_{q},\hat{\rho}]-\sum_{i}\frac{S_{QQ}(i)}{\hbar^{2}}\mathcal{D}[\hat{\sigma}_{i}]\hat{\rho}\\*
\nonumber
& -\sum_{i}\left(\frac{S_{QV}}{\hbar}\chi_{i}(t)[\hat{\rho},\hat{\sigma}_{i}] +\frac{i a_{VQ}\chi_{i}(t)}{2}[\hat{\rho},\hat{\sigma}_{i}]_{+} -\frac{\chi_{i}^{2}(t)}{2}S_{VV}\hat{\rho}\right).\end{aligned}$$\
for the situation where the qubit decoherence is due to the detector back actions only. The parameters are restricted by inequalities similar to Eq. for each set of noise and response functions corresponding to a given detector.\
The form of this equation that can account for the realistic experimental situation [@Huard] is similar to Eq. : $$\begin{aligned}
\label{eq3_2exp}
&\frac{\partial\hat{\rho}}{\partial t} = -\frac{i}{\hbar}[\hat{H}_{q},\hat{\rho}] -\gamma_{d}\mathcal{D}[\hat{\sigma}_{z}]\hat{\rho}-\gamma_{\uparrow}\mathcal{D}[\hat{\sigma}_{+}]\hat{\rho}\\*
\nonumber
& -\gamma_{\downarrow}\mathcal{D}[\hat{\sigma}_{-}]\hat{\rho}-\sum_{i}\frac{S^{(i)}_{QV}}{\hbar}\chi_{i}(t)[\hat{\rho},\hat{\sigma}_{i}]\\*
\nonumber
& +\sum_{i}\frac{i a_{VQ}^{(i)}\chi_{i}(t)}{2}[\hat{\rho},\hat{\sigma}_{i}]_{+} -\sum_{i}\frac{\chi_{i}^{2}(t)}{2}S^{(i)}_{VV}\hat{\rho},\end{aligned}$$ where $i=x,y$ and we account for detector-dependent noises and response functions. Two inequalities put restrictions on the parameters involved:
\[ineq3\] $$\begin{aligned}
\frac{1}{4}\left(\gamma_{\uparrow}+\gamma_{\downarrow}\right)S^{(x)}_{VV}-{| {S_{QV}^{(x)}} |}^{2}\geq&\frac{\hbar^{2}}{4}{| a_{VQ}^{(x)} |}^{2},\\*
\frac{1}{4}\left(\gamma_{\uparrow}+\gamma_{\downarrow}\right)S_{VV}^{(y)}-{| S_{QV}^{(y)} |}^{2}\geq&\frac{\hbar^{2}}{4}{| a_{VQ}^{(y)} |}^{2}.\end{aligned}$$
Here, we have assumed an ideal and fast post-selection so that the system measured is projected on a known pure state $|\Psi\rangle$. This is the case of the experimental setup [@Huard]. In reality, there can be errors in the post-selection. We note that such errors can also be accounted for in the formalism outlined. To this end, one replaces the projection operator $\hat{P}_{\Psi}$ with a density matrix-like Hermitian operator $\hat{\rho}_f$ satisfying ${\rm Tr } [\hat{\rho}_f] = 1$. For instance, if after a faulty projection measurement with the result “1” the system is in a orthogonal state $|\Psi_2\rangle$ with probability $p_e$, the corresponding $\hat{\rho}_f$ reads $$\hat{\rho}_f = (1-p_e) |\Psi_1\rangle\langle\Psi_1| + p_e |\Psi_2\rangle\langle\Psi_2|$$
Half-quantization: a straightforward case {#sec:simple}
=========================================
The outcomes of an ideal projective measurement of a quantum variable $\hat{{\cal O}}$ are confined to the eigenvalues ${\cal O}_i$ of the corresponding operator. If a CWML approximates well this ideal situation, one expects the distribution of outcomes to peak near ${\cal O}_i$, and it is indeed so. In this Section, we argue that if the measurement outcomes are conditioned on a final state, the distribution also has peculiarities at [*half-sums*]{} $({\cal O}_i+{\cal O}_j)/2$ of the eigenvalues. We prove first this counter-intuitive statement for a restricting limiting case where the measurement interval ${\cal T}$ is much smaller than the typical time scales of the system dynamics. The results are summarized in Eq. \[eq:half-sums\]. The resulting distributions may formally correspond to negative probabilities in the limit of vanishing overlap between initial and final state. To correct for this, and to extend the limits of validity to larger time intervals, we concentrate further on a specific but constructive case of non-demolition measurement. With this, we investigate the influence of decoherence on half-quantization. The results are given by Eq. \[eq:results-nondemolition\].
To start, we take the measurement interval ${\cal T}$ to be much smaller than typical time scales of the quantum system dynamics. This immediately implies that the accuracy of the measurement will be too low to make it practically useful. However, the resulting distribution comes out of a straightforward calculation, since the state of the quantum system does not have time to change significantly during the measurement.
In Eq. we may then neglect all terms describing the dynamics and containing no $\chi(t)$ Let us also assume no correlation between the noises of the input and output variables of the detector, $S_{QV} = 0$.\
With this, Eq. can be simplified to the following form
$$\label{eq3simp}
\frac{\partial\hat{\rho}}{\partial t} = -\frac{\chi^{2}(t)}{2}S_{VV}\hat{\rho} + \frac{i a_{VQ}\chi(t)}{2}[\hat{\rho},\hat{{\cal O}}]_{+}.$$
Let us concentrate on a piecewise-constant $\chi(t)\equiv \chi\Theta(t)\Theta({\cal T} - t)$ corresponding to the accumulation of the signal during the measurement interval. We take $\hat{\rho}(\chi; 0)=\hat{\rho}(0)$ as the initial condition. After the time interval of the measurement ${\cal T}$, the quasi-density matrix becomes
$$\label{simp}
\hat{\rho}(\chi; {\cal T}) = e^{-\frac{S_{VV}}{2}\chi^{2}{\cal T}}e^{i\frac{a_{VQ}}{2}\chi{\cal T}\hat{{\cal O}}}\hat{\rho}(0)e^{i\frac{a_{VQ}}{2}\chi{\cal T}\hat{{\cal O}}}.$$
The generating function of the outcome distribution is given by Eq. and involves the projection $\hat{P}_{\Psi}$ on the final state $|\Psi\rangle$. The calculations are straightforward in the basis of the eigenstates of the operator $\hat{{\cal O}}$, $\hat{{\cal O}}|i\rangle = {\cal O}_i |i\rangle$. It is also convenient to normalize the output variable on the value of $\hat{{\cal O}}$ introducing a rescaled variable ${\cal O} \equiv V/a_{VQ}$. The resulting distribution is a linear superposition of shifted normal distributions $$g(x)=\frac{1}{\sigma \sqrt{2\pi}} \exp\left(-\frac{x^2}{2\sigma^2}\right)$$ with the same variance $\sigma^2 =S_{VV}/({\cal T} a^2_{VQ})= t_a/4{\cal T}$, $$\begin{aligned}
\tilde{P}({\cal O}) = \sum_i W_{ii} g({\cal O}-{\cal O}_i) + \nonumber \\
\sum_{i\ne j} W_{ij} g\left({\cal O}-\frac{{\cal O}_i+{\cal O}_j}{2}\right)
\label{eq:half-sums}\end{aligned}$$ and the weights $W_{ij}$ given by $$W_{ij} = \frac{\Psi_j \Psi^*_i \rho^{(0)}_{ij}}{\langle \Psi|\rho^{(0)}|\Psi\rangle}; \; \sum_{i,j}W_{ij}=1.$$
Let us discuss this result. The terms of the first group are normal distributions centered at the eigenvalues of ${\cal O}_i$. The coefficients in front of these terms are proportional to the product of the initial probability to be in the state $i$, $\rho(0)_{ii}$, and the probability to be found in final state after being in the state $i$, $|\Psi_i|^2$. If there would be no quantum mechanics, the system on its way from initial to final state should definitely pass one of the eigenstates of $\hat{{\cal O}}$ shifting the measurement output by the corresponding eigenvalue. The sum of the probabilities $W_{ii}$ would be $1$. In fact, it is not $1$: owing to quantum interference, the system does not have to pass a definite state $i$. One can say that “bras” and “kets” may pass the different states, and this shifts the output by a half-sum of the corresponding eigenvalues. These interference contributions disappear if there is no post-selection in the final state. Indeed, summing $W_{ij}$ over a complete basis of possible final states $|\Psi\rangle$ gives zero. These coefficients also disappear in case of diagonal $\hat{\rho}^{(0)}$ Although the form (\[eq:half-sums\]) suggests that real values $W_{i,j}+W_{j,i}$ could be interpreted as “probabilities” of “half-quantized” outcomes, this does not work since these values can be negative as well as positive, and the contributions centered at half-quantized values can be peaks as well as dips. This is typical for an interference effect. The double peak structure of the distribution has been discussed earlier in the context of CWLM [@CWLM1; @CWLM2; @CWLM26; @NazWei] The interpretation in terms of half-quantization is an innovation of the present article.
A double-peak probability distribution has been predicted in the context of post-selected measurements [@Inverse0; @Inverse1]. While this effect is also based on interference, it is clearly distinct from the half-quantization considered here since it is observed for an operator with continuous spectrum and in fact, in distinction from the effect described here, permits a classical interpretation[@Inverse1]. The half-quantization also does not bear any resemblance with the 3-box paradox [@completedescription] since the latter involves a third quantum state absent in our setup. Nevertheless, the interference signatures can be revealed by a close inspection of the probability distribution of the outcomes of the conditioned measurement. We notice that the limit of small ${\cal T}$ we presently concentrate on is not favorable for such inspection since the peaks (or dips) are hardly separated, ${\cal O}_i \ll \sqrt{\sigma}$, so that $P({\cal O}) \approx g({\cal O})$, that is, hardly depends on the quantum system measured. To enhance the effect, one would increase ${\cal T}$. However, at sufficiently large ${\cal T}$ the quantum system would relax to equilibrium, this suppresses the interference effects. Numerical calculations presented in Sections \[sec:decscale\] and \[sec:shortscale\] show that the interference contributions become quite pronounced in the case of intermediate ${\cal T}$.
{width="1.6\columnwidth"}
In this Section, we mention a special case where the interference effects become enhanced and significant even in the limit ${\cal T} \to 0$. This is the case of a small overlap between the initial state $\hat{\rho}(0)$ and the post-selected final state, ${| \Psi\rangle}$, $\langle \Psi |\hat{\rho}(0)|\Psi\rangle \to 0.$\
The coefficients $W_{ij}$ diverge upon approaching this limit, and Eq. \[eq:half-sums\] becomes invalid giving a negative probability density. To consider the case properly, we need to regularize Eq. \[simp\] taking into account the dephasing which comes at least from the detector back-action. The simplest way to provide such regularization is to include dephasing produced by interaction with the same operator $\hat{{\cal O}}$. The resulting equation reads
$$\label{eq3simpS}
\frac{\partial\hat{\rho}}{\partial t} = -\frac{\chi^{2}(t)}{2}S_{VV}\hat{\rho} + \frac{i a_{VQ}\chi(t)}{2}[\hat{\rho},\hat{{\cal O}}]_{+} - \gamma\mathcal{D}[\hat{{\cal O} }]\hat{\rho}.$$
It looks we have disregarded the Hamiltonian dynamics in Eq. \[eq3simpS\]. This does not seem consistent since usually $H_q \gg \hbar \gamma$, this provides a common separation between the fast time-scales of Hamiltonian dynamics and longer time-scales of the decoherence and relaxation. We note that we do not have to disregard it in an important case of non-demolition measurement when $\hat{H}_q$ and $\hat{\cal O}$ commute. In this case, the only effect of the Hamiltonian dynamics is to provide time-dependent phase factors for non-diagonal elements of the density matrix. These trivial phase factors can be compensated by a proper rotation of the final state and the Hamiltonian dynamics can be gauged away from Eq. \[eq3simpS\]. We address the relevant Hamiltonian dynamics in the next Section.
By virtue of the Cauchy-Schwartz inequality (\[ineq\]), $\gamma \ge a^2_{QV}/4 S_{VV}$. Therefore it is convenient to characterize the dephasing rate $\gamma$ with dimensionless $K \equiv 4 \gamma S_{VV}/a^2_{QV} = \gamma t_a$, $K\geq1$, that characterizes the quality of the detector.
The equation is easily solved in the basis of eigenvalues of $\hat{{\cal O}}$. In comparison with Eq. \[simp\], each non-diagonal element $\rho_{ij}$ of the quasi-density matrix acquires an extra time-dependent suppression factor $\exp\left(-\gamma t \frac{({\cal O}_i - {\cal O}_j)^2}{2}\right)$. With this, the probability distribution is given by Eq. \[eq:half-sums\] with modified coefficients $W_{ij} \to \tilde{W}_{ij}$, $$\begin{aligned}
\label{eq:results-nondemolition}
\tilde{W}_{ij} \equiv \frac{\Psi_j \Psi^*_i \rho_{ij} e^{-\gamma {\cal T}\frac{({\cal O}_i - {\cal O}_j)^2}{2}}}{\tilde{W}}; \\
\tilde{W} \equiv \sum_{i,j}\Psi_j \Psi^*_i \rho_{ij} e^{-\gamma {\cal T}\frac{({\cal O}_i - {\cal O}_j)^2}{2}} \nonumber\end{aligned}$$
At any non-zero overlap, $P({\cal O}) \to g({\cal O})$ in the limit of ${\cal T} \to 0$. Let us concentrate on a special case of zero overlap, $\langle \Psi |\hat{\rho}^{(0)}|\Psi\rangle$=0, and let us note that this also implies $\hat{\rho}^{(0)}|\Psi\rangle =0 $ by virture of positivity of the density matrix. In the limit of ${\cal T} \to 0$ the chance to find the system in the final state vanishes, $\tilde{W} \approx \gamma {\cal T}\langle \Psi |{\hat{\cal O}}\hat{\rho}^{(0)}{\hat{\cal O}}|\Psi\rangle$. This divergency should be compensated by the terms $\propto {\cal T}$ that come from expansion of $g({\cal O} - ({\cal O}_i +{\cal O}_j)/2)$ up to the second order in ${\cal O}_i$ as well as $\tilde{{W}}_{ij}$. The resulting distribution of the measurement outcomes for these rare events differs essentially from the normal one, $$P({\cal O})= \left(1 +\frac{({\cal O}/\sigma)^2-1}{K} \right) g({\cal O}) \ne g({\cal O})
\label{eq:zero-overlap}$$ For an ideal detector, $K=1$, the probability even vanishes at ${\cal O}=0$. For bigger decoherence exceeding the minimal one, $K\gg 1$, the interference term vanishes and $P({\cal O}) \approx g({\cal O})$.
We illustrate the content of this Section with some simple plots (Fig. \[fig:simple-plots\]). We consider a qubit that is initially prepared in $Z^+$ state, $\hat{\sigma}_z|Z^+\rangle = |Z^+\rangle$. The measurement accesses the $x$-component of the qubit spin, ${\cal O}=\hat{\sigma}_x$. After the measurement, the qubit is post-selected in either $Z^+$ or $Z^-$ state. As it follows from the preceding discussion, we expect the probability distribution of the outputs to be composed of the Gaussians centered at $\pm 1$, and also at the half sum of the eigenvalues, that is, at $0$.
For the first four plots, we choose a relatively big ${\cal T} = 0.5 \gamma^{-1}$. Although this choice is contrary to our assumptions, it permits an easy visual resolution of the Gaussian peaks. We assume ideal detector $K=1$ and use Eq. \[eq3simpS\] to evaluate the distributions. The distribution of the outcomes with no post-selection (Fig. \[fig:simple-plots\]a.) is composed from two Gaussian peaks centered at $\pm 1$ that are hardly separated. The post-selected distributions differ much from each other and the original one (Fig. \[fig:simple-plots\]b.) The distribution for $Z^-$ gives well-separated peaks while a single peak is seen in the distribution for $Z^{+}$. This is due to the negative or positive half-sum contribution as illustrated in Fig. \[fig:simple-plots\]c. an d.
The Fig. \[fig:simple-plots\]e. demonstrates the essential change of the conditioned distribution function for zero overlap. The distribution for ideal detector reaches zero, and approaches normal distribution upon increasing $K$.
To investigate in more detail the manifestations of the interference effects at longer time intervals $ \simeq t_a, \gamma^{-1}$ and in experimental conditions, in Section \[sec:decscale\] we numerically solve the evolution equations and compute the conditioned probability distributions. For this work, we concentrate on a single qubit.
Sudden jump: a simple consideration {#sec:suddenjump}
===================================
Let us now change the situation and consider the measurement of a variable that does not commute with the Hamiltonian. To simplify, we consider very small ${\cal T}$ such that the change of density matrix due to Hamiltonian dynamics is small. This is a more severe limitation than that used in the previous Section where ${\cal T}$ was only supposed to be smaller than the decoherence rate. Generally, this time interval is too small to measure anything and we expect the distribution to be close to $g({\cal O})$ thus to have a large spread. There is, however, an exceptional situation of zero overlap where after the measurement the state is projected on $|\Psi\rangle$ that is precisely orthogonal to the initial state $|i\rangle$, $\langle \Psi|i\rangle =0$. Let us concentrate on this situation and demonstrate a peculiarity of the output distribution which is best described as a [*sudden jump*]{} of the integrated output.
To give a clear picture, we first treat the situation completely disregarding the decoherence/relaxation terms, and take into account the Hamiltonian dynamics only. This seems relevant at such small ${\cal T}$. The general result is given by Eq. \[eq:suddengeneralresult\] while a constructive case is given by \[eq:Cshort\]. This gives a sudden jump of cumulants while the attempt to derive the distribution results in a negative probability in an interval of outputs that increases with decreasing ${\cal T}$. To improve on this, we will sophisticate the treatment by including the decoherence. We reveal that the decoherence becomes important at very small time intervals ${\cal T} \ll (\Omega^2 t_a)^{-1}$, that can be interpreted as a finite but small duration of the sudden jump. The resulting probability distribution is given by Eq. \[eq:distshorttimes\] and is positive at any ${\cal T}$. To start with, we disregard relaxation/decoherence terms in the evolution equation which seems relevant for such small ${\cal T}$ and owing to orthogonality, the projected $\rho(\chi)$ vanishes at ${\cal T} \to 0$ and is determined by the first-order corrections to bra- and ket wave functions, $$\text{Tr}(\hat{P}_{\Psi}\hat{\rho}(\chi) )= \hbar^{-2}{\cal T}^2 \langle \Psi | \hat{H}^+_q | i\rangle \langle i |\hat{H}^-_q|\Psi \rangle e^{-\chi^2 {\cal T} S_{VV}/2}$$ Here $H^{\pm} = H_q \pm \hbar \chi a_{VQ} \hat{\cal O}$.
The small factor ${\cal T}^2$ cancels upon normalization in Eq. \[eq6\] so that the generating function of the conditioned output reads $$\label{eq:suddengeneralresult}
\tilde{C}(\chi; {\cal T}) = \frac{\langle \Psi | \hat{H}^+_q | i\rangle \langle i |\hat{H}^-_q|\Psi \rangle}{|\langle \Psi | \hat{H}_q | i\rangle|^2} e^{-\chi^2 {\cal T} S_{VV}/2}$$ We note that $\tilde{C}(\chi; {\cal T}\to 0) \ne 1$. Since the derivatives of $\ln \tilde{C}$ at $\chi \to 0$ are related to the cumulants $\kappa_n$ of the distribution of the integrated output $\int_0^{{\cal T}} dt \hat{V}(t)$. This implies that the cumulants of the distribution of the integrated output do not vanish in the limit of short time interval: rather, there is a [*sudden jump*]{} of the integrated output not depending on the duration of the measurement. The jump occurs for the averaged output as well as for all cumulants. This is very counter-intuitive for a CWLM situation. In this case, one may expect that the integrated output in this limit is dominated by the detector noise, so that $ \int_0^{{\cal T}} dt \hat{V}(t) \simeq {\cal T}^{1/2}$ , $\kappa_n \simeq {\cal T}^{n/2}$, and thus vanishes at ${\cal T} \to 0$.
To see this in more detail, let us turn to a concrete example. We consider a situation corresponding to [@Huard]: a qubit with the Hamiltonian $\hat{H}_{q}=\frac{\hbar}{2}\Omega\hat{\sigma}_{x}$. The initial and projected states are $Z^+$ and $Z^-$, respectively, and we measure the projection of the qubit on Y-axis, $\hat{{\cal O}} = \hat{\sigma}_y$. In this case, $$\label{eq:Cshort}
\tilde{C}(\chi; {\cal T})= \left( 1-\frac{i\chi a_{VQ}}{\Omega}\right)^2 e^{-\chi^2 {\cal T} S_{VV}/2}$$ In the limit ${\cal T} \to 0$ we obtain for the cumulants: $$\kappa_n = \frac{\partial^{n}}{\partial (i\chi)^{n}} \ln \left( 1-\frac{i\chi a_{VQ}}{\Omega}\right)^2 = 2 (-1)^n\left(\frac{a_{VQ}}{\Omega}\right)^{n} (n-1)!$$ We see a sudden jump in the cumulants of the time-integrated output.
The average value of the output ($\kappa_1$) is given by $$\label{eq:suddenjump}
a_{VQ}^{-1}\int_0^{{\cal T}} dt \langle \hat{V}(t)\rangle = -\frac{2}{\Omega};\;\bar{{\cal O}} = -\frac{2}{\Omega {\cal T}}.$$ This corresponds to the time-averaged output $\propto {\cal T}^{-1}$ that can exceed by far the expected values of a projective measurement, $\pm 1$. Such anomalously big outputs are naturally associated with the weak values [@WeakValues]. Indeed, one can relate the above result with weak value conform to the definition [@WeakValues] if one takes into account the evolution of the quantum state during the measurement [@weakanddynamics]. However, we need to stress that the full distribution of the outputs cannot be obtained with the traditional weak value formalism and so far has not been obtained with its extensions [@ContWeak1; @ContWeak2; @ContWeak3] for continuous measurement. The method outlined here does not explicitly evoke the notion of weak values and provides a more elaborated description of a realistic measurement process.
![Probability distributions of outputs (Eq. \[eq:distshorttimes\]) in the sudden jump regime in case of an ideal detector. The alternating solid-dotted curves correspond to different ${\cal T} = (0.25,0.5,1.0,2.0,4.0) (\Omega^2 t_a)^{-1}$. Each curve consists of two peaks separated by a gap at ${\cal O}
=\Omega t_a/4$. The curves with bigger ${\cal T}$ are sharper, and the peaks become increasingly symmetric upon lowering ${\cal T}$.[]{data-label="fig:shortscales"}](shortscaledist.pdf){width="0.9\columnwidth"}
An attempt to derive from (\[eq:Cshort\]) the overall distribution of the time-averaged outputs yields $$P({\cal O}) = \left(1+ \frac{\partial_{\cal O}}{\Omega {\cal T}}\right)^{2} g({\cal O})= \left(\left(1-\frac{{4 \cal O}}{\Omega t_a}\right)^2-\frac{4}{\Omega^2 {\cal T} t_a}\right)g({\cal O})$$ There is a problem with this expression: it is negative in an interval of ${\cal O}$, and at sufficiently small ${\cal T} \lesssim (\Omega^2 t_a)^{-1}$ this interval encompasses the body of the “distribution”. This signals that the current approach must be corrected. As we have seen in the previous Section, such correction most likely requires a proper account of the detector back-action that causes the decoherence of the qubit.
It is unusual to expect a decisive role of decoherence at such small time scales. However, if we take into account the decoherence (second term in the r.h.s. of Eq. \[eq3\] ), we obtain $$\text{Tr}(\hat{P}_{\Psi}\hat{\rho}(\chi)) = \left(\gamma {\cal T}+ \frac{{\cal T}^2}{4} \left(\Omega - i a_{QV}\chi \right)^2\right) e^{-\chi^2 {\cal T} S_{VV}/2}$$ Here, $\gamma \equiv S_{QQ}/\hbar^2$ is the corresponding decoherence rate. We see that the decoherence term may indeed compete with the term coming from Hamiltonian dynamics at short time intervals. The Physical reason for this is that a decoherence term of this sort induces the relaxation in $Z$-basis. The relaxation brings the qubit to $Z^-$ faster than the Hamiltonian: The probability to find the system in $Z^-$ is thus proportional to ${\cal T}$ in contrast to the probability $\propto {\cal T}^2$ induced by the Hamiltonian dynamics.
The resulting characteristic function reads $$\tilde C(\chi)=\frac{ 4\gamma + {\cal T}\left(\Omega - i a_{QV}\chi \right)^2}{4\gamma+{\cal T}\Omega^2}e^{-\chi^2 {\cal T} S_{VV}/2}$$ and gives the average output $$\label{eq:shorterscales}
\bar {{\cal O}}= - \frac{2 \Omega}{4\gamma + {\cal T}\Omega^2}$$ The value of the average output thus saturates at $-\Omega/2\gamma \ll -1$ in the limit of small ${\cal T}\ll \gamma/\Omega^2$. So if the decoherence is taken into account, the change of the output averages is not really sudden. One can regard the small time scale $\gamma/\Omega^2$ of the saturation as a typical duration of the sudden jump of the time-integrated output.
The probability distribution valid at all time scales $\ll \Omega^{-1}$ is given by $$\label{eq:distshorttimes}
P({\cal O})=\frac{K-1 +({\cal T}/4t_a) \left(\Omega t_a - 4 {\cal O}\right)^2}{K+{\cal T}t_a \Omega^2/4} g({\cal O})$$ where we again introduce the dimensionless $K=\gamma t_a\geq 1$ that characterizes the quality of the detector. The distribution is illustrated in Fig. \[fig:shortscales\] for an ideal detector $K=1$ and various ${\cal T}$. In this case, the probability density is zero at ${\cal O}=\Omega t_a/4$.
If we compare the distributions (\[eq:zero-overlap\]) and (\[eq:distshorttimes\]), we see that the results of the previous Section are reproduced in the limit $\Omega \to 0$, as well as in the limit of ${\cal T} \ll (\Omega^2 t_a)^{-1}$ if we take $\sigma^2 = t_a/4 {\cal T}$. The distribution (\[eq:distshorttimes\]) thus generalizes (\[eq:zero-overlap\]) to the case where the Hamiltonian dynamics are relevant.
To extend the results on larger time intervals $\simeq \Omega$ and on realistic conditions, we numerically solve the evolution equations in Section \[sec:shortscale\] and compute the corresponding conditioned probability distributions.
Numerical results: long time scales {#sec:decscale}
===================================
In Section \[sec:simple\], we have presented an analytical solution in the limit of small ${\cal T}$ and shown that it remains qualitatively valid for bigger ${\cal T}$, at least in the case of ideal detectors. We will extend these results evaluating the conditioned distributions numerically. We concentrate on longer measurement times where the qubit dynamics become important. We will take into account the effects of decoherence and relaxation, as well as the effects of strong qubit drive or detuning, all being important in experimental situations.\
In this Section, we address the distributions of the CWLM outcomes of a single variable at the time scales of the order of coherence/relaxation times and $t_a$. Generally, one can associate it with the qubit variable $\hat{{\cal O}}=\hat{\sigma}_{x}$. To start with, we assume zero detuning, that is, a qubit Hamiltonian of the form $\hat{H}_{q}=\frac{\hbar}{2}\Omega\hat{\sigma}_{x}$. In principle, we are now in the situation of a non-demolition measurement.
To start with, let us assume an idealized situation where all the decoherence is brought by the detector back action and its rate $\propto S_{QQ}$ assumes the minimum value permitted by the inequality (). Since $\hat{H}_{q}=\frac{\hbar}{2}\Omega\hat{\sigma}_{x}$, the back-action does not interfere with free qubit dynamics causing transitions between the levels. In $\sigma_x$ representation, the diagonal elements of the density matrix remain unchanged keeping the initial probability to be in $X^{\pm}$ states while the non-diagonal ones oscillate with frequency $\Omega$ and decay with much slower rate $\Gamma_d \ll \Omega$.
If we keep the final state fixed to $Z^{\pm}$, the interference contribution to the conditioned distributions will exhibit fast oscillations as function of ${\cal T}$ with a period $2\pi/\Omega$. It is proficient from both theoretical and experimental considerations to quench these rather trivial oscillations. We achieve this by projecting the qubit after the measurement on the states $|\bar{Z}^{\pm}\rangle = e^{-i \hat{H}_q {\cal T}} |Z^{\pm} \rangle$ thereby correcting for the trivial qubit dynamics. In practice, such correction can be achieved by applying a short pulse rotating the qubit about $x$-axis right before the post-selection measurement. With this, the conditioned distribution of outcomes changes only at the time scale $t_a \simeq \Gamma^{-1}_d$, that is much longer than $\Omega^{-1}$, and the dynamics are described by Eq. with $\hat{\cal O}=\hat{\sigma}_x$.
In Fig. \[fig2\], we give the plots of the probability distributions conditioned on $\bar{Z}^{\pm}$ for a series of measurement time intervals ${\cal T}$. We see that (different curves) are shown, for two cases in which the visibility of the interference feature is stronger, the case of equal preparation and post-selection, (a), and the case of orthogonal preparation and post-selection states, (b).\
In this ideal situation, even for very small time intervals, the additional knowledge of the post-selection can lead to perfect resolution of the two eigenstates of the qubit variable (Fig. \[fig2\] (b)). While for small time intervals the middle peak results in less resolution for the opposite choice of post-selected qubit state (Fig. \[fig2\] (a)), at large time intervals, the detector back action has resulted in a complete decoherence of the qubit state and the interference signature disappears, making both distributions converge to two narrow peaks corresponding to either $+X$ or $-X$. This exemplifies how the knowledge of the qubit preparation is lost in time due to decoherence.
The fact that we see no difference between the distributions in this limit is a result of a symmetric choice we made with respect to the projections. Indeed, if we project on $\pm X$ instead, the distributions would consist of a single peak positioned at the value of ${\cal O}=\pm 1$. Generally, for projections on arbitrary pair of orthogonal superpositions of $X$ and $Z$, we expect in this limit different peak weights for two different projections. This difference, however, is of trivial origin and has nothing to do with the interference effects of interest. So we have made a symmetric choice to cancel it.
With this, the difference between the two distributions is due to interference only, that is, due to the half-quantized peak described in the previous Section. At smaller ${\cal T}$, the distributions take a very distinct shape: single-peak for that conditioned on $+Z$, and double-peak for that conditioned on $-Z$. The half-quantization is dumped on the scale of the decoherence time, so the difference is seen only for ${\cal T} <t_a$.
The separation of the distribution onto two peaks in the limit of ${\cal T} \gg t_a$ is a signature of the ideal situation of a quantum non-demolition measurement where neither measurement nor any other agent induces the relaxation rates causing the transitions between the qubit states. In this situation, the density matrix efficiently relaxes to its equilibrium value $\rho_{eq}$at time interval ${\cal T}$, and the distribution of the detector output tends to concentrate on the average value $\langle {\cal O}\rangle = {\rm Tr}[\hat{\cal O} \hat{\rho}_{eq}]$ with decreasing width $\simeq \sqrt{t_a/{\cal T}}$.
Let us now turn to the analysis of the experimental situation. We use the general evolution equation Eq. to compute the distributions and substitute the parameters $\gamma_{\downarrow} = (22.5 \mu\text{s})^{-1}, \gamma_{\uparrow} = (56 \mu\text{s})^{-1},\gamma_d = (15.6 \mu\text{s})^{-1}$ given in [@Huard]. The acquisition time comes from the measurement rate $2/t_{a}\approx(92 \mu\text{s})^{-1}$. This rate in fluorescence experiments can be characterized by two different methods both based on the estimation of the probability distribution for the integrated homodyne signal conditioned on the state of the qubit, see Appendix F in the supplementary material of [@QNoise]. The quality of the measurement setup is thus rather far from ideal, $K = t_a \gamma_d \approx 12$. Nevertheless we predict some measurable interference effects in the outcome distributions.
We plot in Fig. \[fig3\] the results for zero detuning. There is no visible difference between the distributions, so in distinction from Fig. \[fig2\], we give only a single set of curves in Fig. \[fig3\]. The curves for all ${\cal T}$ look dully Gaussian, no peak separation is visible. This is because of the low quality of the detector: the relaxation to the stationary density matrix $\hat{1}/2$ mainly takes place at a time interval shorter than the acquisition time, so most of the time the detector measures this featureless state. As to short ${\cal T}$, the distribution is too wide to manifest the features of the density matrix.
However, there are still observable signatures of interference. To reveal those, we plot in Fig. \[fig3\] the difference of the probability densities for two projections. We see that at smallest ${\cal T} = 0.2t_a$ the relative difference achieves $0.1$ at ${\cal O}\approx 0$ and can be thus revealed from the statistics of several hundreds individual measurements. The shape of the difference suggests that the $P_-$ is pushed on both positive and negative values of ${\cal O}$ in comparison with $P_+$, in agreement with the previous findings. The decoherence and relaxation quickly diminish the difference upon increasing ${\cal T}$.
At big values of ${\cal O}$, the difference quickly decreases together with the distributions. In this respect, it is instructive to inspect the difference normalized on the sum of the probability densities, $C({\cal O})\equiv(P_+(\mathcal{O}) - P_-(\mathcal{O}))/(P_+(\mathcal{O}) + P_-(\mathcal{O}))$. This quantity gives the certainty with which one can distinguish two distributions from each other given a reading ${\cal O}$. The values $C=\pm 1$ would imply that the measurement is [*certainly*]{} post-selected with $\pm Z$. As we see from Fig. \[fig3\] , the certainty saturates with increasing ${\cal O}$, reaches relatively large values at short ${\cal T}$, and fades away upon increasing ${\cal T}$.
Let us inspect the distributions at non-zero detuning. In this case, there is no reason to expect the ${\cal O} \to -{\cal O}$ symmetry in the distribution. We illustrate the situation in Fig. \[fig4\] assuming relatively large detuning $\Delta = 1.7 \Omega$. This value is chosen to maximize $\langle{\cal O}\rangle$ for the equilibrium density matrix. In the plots of Fig. \[fig4\]a, we see a shift of the distribution maximum that tends to $\langle{\cal O}\rangle \approx -0.1$ at ${\cal T} \gg t_a$. The value of the shift does depend on ${\cal T}$ as well as on the post-selection state.
If we concentrate on the difference of the probability distributions(Fig. \[fig4\]b), we see the same order of magnitude as at zero detuning. However, the difference does not vanish in the limit of big ${\cal T}$. Rather, it is concentrated in an increasingly narrow interval of ${\cal O}$ conform to the decreasing width of the distribution. As to the certainty (Fig. \[fig4\]c), it rather quickly converges upon increasing ${\cal T}$ to finite and rather big values in a wide interval of ${\cal O}$. This does not imply that the distributions $P^{\pm}$ are different in this limit, since they become concentrated with divergent probability density, and the values of ${\cal O}$ with high certainty occur with exponentially low probability, yet the finite limit of $P_+-P_-$ is worth noting and deserves an explanation.
We can qualitatively explain these features assuming that in this limit the probability distributions are the Gaussians with a shift that depends on the post-selection state and the variance $\sigma^2 = t_a/4 {\cal T}$, $P_{\pm} = g({\cal O}\pm s_{\pm}({\cal T}))$. In the limit of big ${\cal T}$ we expect the difference of the shifts to be proportional $({\cal T})^{-1}$, $s^{\pm} = \langle {\cal O}\rangle \pm S (t_a/{\cal T})$, $S \simeq 1$. This is because the effect of the post-selection is only felt during a time interval $\simeq \gamma^{-1}$ before the end of measurement, so that, at a fraction of the whole interval that is proportional to $ ({\cal T})^{-1}$. With this, at ${\cal O} \simeq \sigma$ the difference of the probabilities approaches a limit not depending on ${\cal T}$ $$P_+-P_- = \frac{S}{2} \frac{({\cal O} - \langle {\cal O}\rangle )}{\sigma \sqrt{2\pi}}\exp\left(-\frac{({\cal O} - \langle {\cal O}\rangle )^2}{2 \sigma^2}\right),$$ The maximum difference of probabilities $|P_+-P_-|_{{\rm max}} \approx 1.9 S$ is thus achieved at ${\cal O} = \langle {\cal O}\rangle \pm\sigma$.
As to the certainty, it approaches an alternative limit at ${\cal O} \simeq 1 \gg \sigma$ that also does not depend on ${\cal T}$ at ${\cal T} \to \infty$ $$C({\cal O}) = \frac{P_+(\mathcal{O}) - P_-(\mathcal{O})}{P_+(\mathcal{O}) + P_-(\mathcal{O})} = {\rm tanh}\left( 4 S (\mathcal{O} - \langle {\cal O}\rangle) \right)$$ As we see, the certainty reaches $\pm 1$ in the limit of large (exponentially improbable) $|{\cal O}| \gg 1$.
The numerical results presented are satisfactory fitted by above expressions with $S \approx 0.04$. However, the fits are not mathematically exact since, for the sake of simplicity, the shifts $s^{\pm}$ have been assumed not to depend on ${\cal O}$ while in general they do.
Our results show that the difference of the conditioned distributions can be detected under realistic experimental circumstances.
Although the interference signature seem to disappear for rather short ${\cal T}$ in a realistic experimental regime, the actual measurements are done [@Huard] for time intervals yet smaller than the time scale of qubit relaxation/decoherence. This correspond to the first several choices of short time intervals in Figures \[fig2\], \[fig3\], and \[fig4\] where the interference is still visible.\
Numerical results: short time scales {#sec:shortscale}
====================================
In the previous Section, we have considered the statistics at time-scales ${\cal T} \simeq \gamma^{-1},t_a$ extending the analytical results of Section \[sec:decscale\]. In this Section, we will extend the analytical results of Section \[sec:suddenjump\]. We present numerical solutions for the probability distributions at a larger time-scale $ {\cal T}\Omega \simeq 1$ of the Hamiltonian dynamics where the decoherence and relaxation does not play an important role. We also consider smaller ${\cal T}$ where the [*sudden jump*]{} behavior is manifested, and yet smaller ${\cal T}$ where the decoherence becomes important again and the time-averaged output saturates to the value $\simeq \Omega/\gamma \gg 1$. We restrict ourselves to the experimental circumstances and use for the computation the Eq. with the parameters specified in Section \[sec:decscale\].\
We will concentrate on the conditioned measurement statistics of the variable $\hat{\sigma}_y$, that anticommutes with the qubit Hamiltonian $\hat{H}_q=\frac{\hbar}{2}\Omega\hat{\sigma}_x+\frac{\hbar}{2}\Delta\hat{\sigma}_z$. The qubit is initially prepared in $Z^{+}$ state and post-selected in either $Z^{+}$ or $Z^{-}$. In Fig. \[fig5\], the probability distributions of the integrated output ${\cal O}$ are presented. The upper row plots (Figs. (a) and (b)) are for zero detuning ($\Delta = 0$), while the lower row plots (Figs. (c) and (d)) show the corresponding distributions when at the detuning $\Delta\approx1.7\Omega$ that maximizes $\langle \sigma_x \rangle$ .\
Left and right figures correspond to post-selection in $Z^{+}$ and $Z^{-}$, respectively.
For unconditioned distributions, the average output is given by $Y({\cal T}) = \frac{1}{\cal T}\int_0^{{\cal T}}dt \langle \Psi(t)|\sigma_y| \Psi(t)\rangle$, where $| \Psi(t)\rangle$ is obtained from $Z^+$ by Hamiltonian evolution. The function $Y({\cal T})$ is plotted in the insets of the right plots with a solid curve. We would expect the distributions to be shifted with respect to the origin by a value ${\cal O} \simeq 1$. This shift would be clearly seen in the plots since the width of the distribution $\simeq \sqrt{t_a\simeq {\cal T}} \simeq \sqrt{t_a \Omega}$ is not very big at experimental values of $\Omega t_a \approx 200$. However, the plots on the left are perfectly centered at the origin at any ${\cal T}$. Indeed, the zero average of the distributions conditioned at $Z^+$ can be proven analytically in the limit of Hamiltonian dynamics. The averages of the distributions conditioned at $Z^-$ (given by dashed curves in the insets of the plots) increase at small ${\cal T}$ as ${\cal T}^{-1}$, in agreement with Eq. \[eq:suddenjump\]. The ratio of this average to conditioned average is just the inverse probability to be found in $Z-$, $p_-({\cal T }) = \sin^2(\sqrt{\Omega^2+\Delta^2}){\cal T}/2) /(1+(\Delta/\Omega)^2)$, $p_- \propto {\cal T}^2$ at small ${\cal T}$.
These averages are visually manifested as the shifts of the distributions that are largely Gaussian. We do not see anything resembling a gap in the distribution predicted for an ideal detector (Fig. \[fig:shortscales\]). This is explained by relatively low detection efficiency (c.f. Eq. \[eq:distshorttimes\]).
In a separate Fig. \[fig6\] we present the distributions conditioned on $Z^{-}$ at yet smaller time-scales of the order of the sudden jump duration (see Eq. \[eq:shorterscales\]). In this regime, we see the saturation of the average $\bar{\cal O}$ at a value close to $-11$ in the limit ${\cal T}\to 0$. This gives the upper limit of anomalously big averages under experimental conditions of [@Huard]. The distributions can be well approximated by shifted Gaussians, smaller ${\cal T}$ corresponding to wider distributions.
Conclusion {#sec:conclusion}
==========
Recent experimental progress has enabled the measurements in course of the conditioned quantum evolution. The average signals have been experimentally studied in [@Huard; @DiCarlo; @SiddiqiMolmer]. The technical level of these experiments permits the characterization of the complete statistics of the measurement outputs.\
In this work, we have developed a proper theoretical formalism based on full counting statistics approach [@NazWei; @NazKin] to describe and evaluate these statistics. We illustrate it with several examples and prove that the interesting features in statistics can be seen in experimentally relevant regimes (Fig. \[fig3\] and \[fig6\]), for both short and relatively long measurement time intervals.
We reveal and investigate analytically two signatures of the conditioned statistics that are related to quantum interference effects. First is the [*half-quantized*]{} measurement values. We demonstrate that the conditioned distribution function may display peculiarities — that are either peaks or dips — at [*half-sums*]{} of the quantized values.
Second signature pertains the case of zero overlap between initial and final state and time intervals that are so short as the wave function of the system does not significantly change by either Hamiltonian or dissipative dynamics.
We reveal unexpectedly large values of the time-integrated output cumulants for such short intervals, that we term [*sudden jump*]{}. We show that the account for decoherence leads to a finite duration of the jump at ultra-short time-scale $\gamma/(\Omega^2)$ and saturation of the anomalous eigenvalues at $\Omega/\gamma$, $\Omega$ and $\gamma$ being the frequency scales of the Hamiltonian and dissipative dynamics, respectively.
Actually, we have shown with our results that one can have very detailed theoretical predictions of CWLM distributions that can account for every detail of the experiment. This enables investigation and characterization of quantum effects even if the choice of parameters is far from the optimal one and these effects are small. We emphasize once again that the interference signature in the distributions that we predict in this Article can be seen in realistic experimental regimes and hope the effects can be experimentally observed soon. The efficient recording of time traces for a weak continuous monitoring of one, or several, qubit variables, is a key ingredient for accessing these statistics. It has been achieved in several articles and applied for observation of single quantum “trajectories” or real time feedback.[@SiddiqiFeedback] High fidelity preparation and post-selection of the qubit is also required for experiments with conditioned evolution, yet this is a general requirement in most qubit experiments. We thus believe that it is possible to extract the interesting statistics from the existing records.\
This work was supported by the Netherlands Organization for Scientific Research (NWO/OCW), as part of the Frontiers of Nanoscience program.
|
---
abstract: |
State-of-the-art recommendation algorithms – especially the collaborative filtering (CF) based approaches with shallow or deep models – usually work with various unstructured information sources for recommendation, such as textual reviews, visual images, and various implicit or explicit feedbacks. Though structured knowledge bases were considered in content-based approaches, they have been largely neglected recently due to the availability of vast amount of data, and the learning power of many complex models.
However, structured knowledge bases exhibit unique advantages in personalized recommendation systems. When the explicit knowledge about users and items is considered for recommendation, the system could provide highly customized recommendations based on users’ historical behaviors. A great challenge for using knowledge bases for recommendation is how to integrated large-scale structured and unstructured data, while taking advantage of collaborative filtering for highly accurate performance. Recent achievements on knowledge base embedding sheds light on this problem, which makes it possible to learn user and item representations while preserving the structure of their relationship with external knowledge. In this work, we propose to reason over knowledge base embeddings for personalized recommendation. Specifically, we propose a knowledge base representation learning approach to embed heterogeneous entities for recommendation. Experimental results on real-world dataset verified the superior performance of our approach compared with state-of-the-art baselines.
author:
- Yongfeng Zhang
- Qingyao Ai
- Xu Chen
- Pengfei Wang
- 'Yongfeng Zhang$^{1}$, Qingyao Ai$^{2}$, Xu Chen$^3$, Pengfei Wang$^4$'
bibliography:
- 'paper.bib'
---
Introduction
============
Most of the existing Collaborative filtering (CF)-based recommendation systems work with various unstructured data such as ratings, reviews, or images to profile the users for personalized recommendation. Though effective, it is difficult for existing approaches to model the explicit relationship between different information that we know about users and items. In this paper, we would like to ask a key question, i.e., *“can we extend the power of collaborative filtering upon large-scale structured user behavior data?”*. The main challenge to answer this question is how to effectively integrate different types of user behaviors and item properties, while preserving the internal relationship betwen them to enhance the final performance of personalized recommendation.
Fortunately, the emerging success on knowledge base embeddings may shed some light on this problem, where heterogeneous knowledge entities and relations can be projected into a unified embedding space. By encoding the rich information from multi-type user behaviors and item properties into the final user/item embeddings, we can enhance the recommendation performance while preserving the internal structure of the knowledge.
Inspired by the above motivation, in this paper, we design a novel collaborative filtering framework over knowledge graph. The main building block is an integration of traditional CF and knowledge-base embedding technology. More specifically, we first define the concept of user-item knowledge graph, which encodes our knowledge about the users and items as a relational graph structure. The user-item knowledge graph focuses on how to depict different types of user behaviors and item properties over heterogenous entities and relations in a unified framework. Then, we extend the design philosophy of collaborative filtering (CF) to learn over the knowledge graph for personalized recommendation.
\[example\]
**Contributions.** The main contributions of this paper can be summarized as follows:
$\bullet$ We propose to directly reason over a structured knowledge-base with embeddings for recommendation. $\bullet$ We extend traditional collaborative filtering to learn over the heterogenous knowledge-base embeddings, which makes it possible to capture the user preferences more comprehensively.
$\bullet$ Extensive experiments verify that our model can consistently outperform many state-of-the-art baselines on real-world e-commerce datasets.
In the following part of the paper, we first illustrate our models in section 2, and verify its effectiveness with experimental results in section 3. At last, the related work and conclusions are presented in section 4 and 5, respectively.
Collaborative Filtering with Knowledge-Graph
============================================
In this section, we illustrate our model to make recommendations by learning knowledge-graph embeddings with collaborative filtering.
Model structure
---------------
Incorporating additional information has been a widely used method to enhance recommendation performance, however, the objects in a practical recommender system are usually very heterogeneous (*e.g.* product brands, categories, and user reviews, etc), and the relations between them can also be quite different (*e.g.* belong to, buy, view, etc). To tackle with such heterogenous data in a unified framework, we first define a user-item knowledge-graph structure specialized for recommender systems, then we conduct collaborative filtering on this graph to provide personalized recommendations.
### User-Item Knowledge Graph in Recommender system
In the context of recommender system, the user-item knowledge graph is built from a set of triplets. Each triplet $(i,j,r)$ is composed of a head entity $i$, a tail entity $j$, and the relation $r$ from $i$ to $j$. The semantic for a triplet $(i,j,r)$ is that *$i$ has a (directed) relation $r$ with $j$*. For example, $(user, item, buy)$ means that the *$user$* has *bought* the *$item$* before, and $(item, brand, belong\_to)$ means that the $item$ *belongs to* a particular $brand$.
Specifically, we define 5 types of entities and 6 types of relations in our system, where the entities include *user, item, word, brand* and *category*, while the relations include:\
$\bullet$ $buy$: the relation from a user to an item, meaning that the user has bought the item.\
$\bullet$ $belong\_to\_category$: the relation from an item to a category, meaning that the item belongs to the category.\
$\bullet$ $belong\_to\_brand$: the relation from an item to a brand, meaning that the item belongs to the brand.\
$\bullet$ $mention\_word$: relation from a user or an item to a word, meaning that the word is mentioned in the reviews of the user or item.\
$\bullet$ $also\_bought$: the relation from an item to another item, meaning that users who bought the first item also bought the second item.\
$\bullet$ $also\_view$: the relation from an item to another item, meaning that users who bought the first item also viewed the second item.\
An example of the constructed user-item knowledge graph can be seen in Figure \[example\].
Entity set $E$, relation set $R$, triplet set $S$, dimension $D$, number of negative samples $k$;\
Randomly initialize the embeddings for $r\in R$ and $e\in E$ $l \leftarrow 0, S^t \leftarrow \varnothing, S^h \leftarrow \varnothing$ $e_j^\prime \gets sample(E)$, $e_i^\prime \gets sample(E)$ $S^t \gets S^t \cup (e_i,e_j^\prime,r), S^h \gets S^h \cup (e_i^\prime,e_j,r)$ $l \gets l + 1$ Update embeddings according to $\nabla L$ Embeddings of the subjects and relations
[l|rrrr|rrrr|rrrr|rrrr]{} Dataset & & & &\
Measures(%) & NDCG & Recall & HT & Prec & NDCG & Recall & HT & Prec & NDCG & Recall & HT & Prec & NDCG & Recall & HT & Prec\
BPR & 2.009 & 2.679 & 8.554 & 1.085 & 0.601 & 1.046 & 1.767 & 0.185 & 1.998 & 3.258 & 5.273 & 0.595 & 2.753 & 4.241 & 8.241 & 1.143\
BPR\_HFT & 2.661 & 3.570 & 9.926 & 1.268 & 1.067 & 1.819 & 2.872 & 0.297 & 3.151 & 5.307 & 8.125 & 0.860 & 2.934 & 4.459 & 8.268 & 1.132\
VBPR & 0.631 & 0.845 & 2.930 & 0.328 & 0.560 & 0.968 & 1.557 & 0.166 & 1.797 & 3.489 & 5.002 & 0.507 & 1.901 & 2.786 & 5.961 & 0.902\
DeepCoNN & 4.218 & 6.001 & 13.857 & 1.681 & 1.310 & 2.332 & 3.286 & 0.229 & 3.636 & 6.353 & 9.913 & 0.999 & 3.359 & 5.429 & 9.807 & 1.200\
CKE & 4.620 & 6.483 & 14.541 & 1.779 & 1.502 & 2.509 & 4.275 & 0.388 & 3.995 & 7.005 & 10.809 & 1.070 & 3.717 & 5.938 & 11.043 & 1.371\
JRL & [5.378]{}$^*$ & [7.545]{}$^*$ & [16.774]{}$^*$ & [2.085]{}$^*$ & [1.735]{}$^*$ & [2.989]{}$^*$ & [4.634]{}$^*$ & [0.442]{}$^*$ & [4.364]{}$^*$ & [7.510]{}$^*$ & [10.940]{}$^*$ & [1.096]{}$^*$ & [4.396]{}$^*$ & [6.949]{}$^*$ & [12.776]{}$^*$ & [1.546]{}$^*$\
CFKG & **5.563** & **7.949** & **17.556** & **2.192** & **3.091** & **5.466** & **7.972** & **0.763** & **5.370** & **9.498** & **13.455** & **1.325** & **6.370** & **10.341** & **17.131** & **1.959**\
Improvement & 3.44 & 5.35 & 4.66 & 5.13 & 78.16 & 82.87 & 72.03 & 72.62 & 23.05 & 26.47 & 22.99 & 20.89 & 44.90 & 48.81 & 34.09 & 26.71\
\[tab:result\]
### Collaborative Filtering based on User-Item Knowledge Graph
The user-item knowledge graph provides us with the ability to access different information sources and multi-type behaviors in a unified manner. In this section, we conduct collaborative filtering on this graph for accurate user profiling and personalized recommendation. Inspired by [@bordes2013translating], we project each entity and relation into a unified low-dimensional embedding space. Intuitively, the embedding of a tail entity should be close to its translated head entity embedding. Formally, for a triplet $(i,j,r)$, suppose the embeddings of $i,j,r$ are $e_i,e_j,e_r$, respectively, then we want that $trans_{e_r}(e_i) \approx e_j$. Considering all the observed triplets $S$, we minimize a margin-based loss to learn the embeddings as follows: $$\begin{aligned}
L &= \sum_{(i,j,r)\in S} \Big\{\sum_{(i,j^\prime,r)\in S^t} \big[\gamma+d\big(trans_{e_r}(e_i), e_j\big)-d\big(trans_{e_r}(e_{i}), e_{j^\prime}\big)\big]_+ \\
&+ \sum_{(i^\prime,j,r)\in S^h} \big[\gamma+d\big(trans_{e_r}(e_i), e_j\big)-d\big(trans_{e_r}(e_{i^\prime}), e_{j}\big)\big]_+ \Big\}
\end{aligned}$$ where, $S^t$ is the set of negative triplets that replace the tail by a random entity, and $T^h$ is another set of negative triplets that replace the head by a random entity. $d(\cdot)$ is a metric function to measure the distance between two embeddings, where we select $\ell_2$-norm as its specific implementation. $trans_{e_r}(e_{i})$ is an arbitrary translation function, or even a neural network, for here, we adopt the addition function $trans_{e_r}(e_{i}) = e_r + e_i$ as in the transE model [@bordes2013translating], because it gives us better efficiency and effectiveness on our dataset. However, it is not necessarily restricted to this function and many other choices can be used in practice.
In the loss function $L$, we essentially try to discriminate the observed triplets from the corrupted ones by a hinge loss function, and the embeddings will be forced to recover the ground truth. Our model can be learned by stochastic gradient descent (SGD), and the model learning algorithm is shown in Algorithm \[alg:A\].
### Personalized Recommendation
We will obtain the embeddings for all entities and relations in the graph once our model is optimized. To generate personalized recommendations for a particular user, we take advantage of the relation type *buy*. Specifically, suppose the embedding of the relation $buy$ is $e_{buy}$, and the embedding of a target user is $e_u$, then we can generate recommendations for the user by ranking the candidate items $e_j$ in ascending order of the distance $d(trans_{e_{buy}}(e_i), e_j)$.
Experiments
===========
In this section, we evaluate our proposed models by comparing with many state-of-the-art methods. We begin by introducing the experimental setup, and then analyze the experimental results.
Experimental Setup {#set}
------------------
**Datasets**. Experiments are conducted on the Amazon e-commerce dataset [@he2016ups]. We adopt four representative sub-datasets in terms of size and sparsity, which are CD, Clothing, Cell Phone, and Beauty. Statistics of the four datasets are summarized in Table \[tb-dataset\].
[p[2cm]{}<|p[0.8cm]{}<|p[0.8cm]{}<|p[1.6cm]{}<|p[1cm]{}<]{} Datasets &\#Users &\#Items&\#Interactions &Density\
&75258&64421&1097592&0.0226%\
&39387&23033&278677&0.0307%\
&27879&10429&194493&0.0669%\
&22363&12101&198502&0.0734%\
\[tb-dataset\]
**Evaluation methods**. In our experiments, we leverage the widely used Top-N recommendation measurements including **Precision**, **Recall**, **Hit-Ratio** and **NDCG** to evaluate our model as well as the baselines. The former three measures aim to evaluate the recommendation quality without considering the ranking positions, while the last one evaluates not only the accuracy but also the ranking positions of the correct items in the final list.
**Baselines**. We adopt the following representative and state-of-the-art methods as baselines for performance comparison: $-$ **BPR:** The bayesian personalized ranking [@bpr] model is a popular method for top-N recommendation. We adopt matrix factorization as the prediction component for BPR.
$-$ **BPR\_HFT:** The hidden factors and topics model [@mcauley2013hidden] is a recommendation method leveraging textual reviews, however, the original model was designed for rating prediction rather than top-N recommendation. To improve its performance, we learn HFT under the BPR pair-wise ranking framework for fair comparison.
$-$ **VBPR:** The visual bayesian personalized ranking [@he2016vbpr] model is a state-of-the-art method for recommendation with images.
$-$ **DeepCoNN:** A review-based deep recommender [@zheng2017joint], which leverages convolutional neural network (CNN) to jointly model the users and items.
$-$ **CKE:** This is a state-of-the-art neural recommender [@zhang2016collaborativekdd] that integrates textual, visual information, and knowledge base for modeling, but it used knowledge base as regularizers and did not consider the heterogenous connection across different types of entities.
$-$ **JRL:** The joint representation learning model [@zhang2017joint] is a state-of-the-art neural recommender, which can leverage multi-model information for Top-N recommendation.
**Parameter settings**. All the embedding parameters are randomly initialized in the range of $(0,1)$, and then we update them by conducting stochastic gradient descent (SGD). The learning rate is determined in the range of $\{1,0.1,0.01,0.001\}$, and model dimension is tuned in the range of $\{10, 50, 100, 200, 300, 400, 500\}$. This gives us the final learning rate as 0.01 and dimension as 300. For the baselines, we also determine the final settings by grid search, and for fair comparison, the models designed for rating prediction (i.e. HFT and DeepCoNN) are learned by optimizing the ranking loss similar to BPR. When conducting experiments, 70% items of each user are leveraged for training, while the remaining are used for testing. We generate Top-10 recommendation list for each user in the test dataset.
[l|rrrr|rrrr|rrrr|rrrr]{} Relations & & & &\
Measures(%) & NDCG & Recall & HT & Prec & NDCG & Recall & HT & Prec & NDCG & Recall & HT & Prec & NDCG & Recall & HT & Prec\
*buy* & 3.822 & 5.185 & 12.828 & 1.628 & 1.019 & 1.754 & 2.780 & 0.265 & 3.387 & 5.806 & 8.548 & 0.848 & 3.658 & 5.727 & 10.549 & 1.305\
*buy+category* & 4.287 & 5.990 & 14.388 & 1.790 & 1.705 & 3.021 & 4.639 & 0.442 & 3.372 & 5.918 & 8.842 & 0.869 & 3.933 & 6.253 & 11.515 & 1.370\
*buy+brand* & 3.541 & 4.821 & 12.239 & 1.563 & 1.101 & 1.906 & 2.981 & 0.284 & 3.679 & 6.211 & 9.118 & 0.898 & 4.832 & 7.695 & 13.406 & 1.621\
*buy+mention* & 4.265 & 5.858 & 13.874 & 1.731 & 1.347 & 2.305 & 3.585 & 0.344 & 4.065 & 7.065 & 10.316 & 1.026 & 4.364 & 6.942 & 12.476 & 1.492\
*buy+also\_view* & 3.724 & 5.070 & 12.633 & 1.604 & 2.276 & 3.931 & 5.827 & 0.561 & 3.305 & 5.705 & 8.458 & 0.840 & 5.295 & 8.723 & 14.891 & 1.728\
*buy+also\_bought* & 5.055 & 7.094 & 16.216 & 2.032 & 1.799 & 3.078 & 4.634 & 0.446 & 5.018 & 8.707 & 12.375 & 1.220 & 5.058 & 8.118 & 13.907 & 1.643\
*all* (CFKG) & **5.563** & **7.949** & **17.556** & **2.192** & **3.091** & **5.466** & **7.972** & **0.763** & **5.370** & **9.498** & **13.455** & **1.325** & **6.370** & **10.341** & **17.131** & **1.959**\
\[tab:info-result\]
Performance Comparison
----------------------
Performance of our Collaborative Filtering with Knowledge Graph (CFKG) model as well as the baseline methods are shown in Table \[tab:result\]. Basically, the baseline methods can be classified according to the information source(s) used in the method, which are rating-based (BPR), review-based (HFT and DeepCoNN), image-based (VBPR), and heterogenous information modeling (CKE and JRL). Generally, the information sources used by our model include ratings (through the *buy* relation), reviews (through $mention$ relation), and our knowledge about the items (through the *belong\_to\_category, belong\_to\_brand, also\_view* and *also\_bought* relations).
From the experimental results we can see that, both of the review-based models can enhance the performance of personalized recommendation from rating-based methods, and by considering multiple heterogenous information sources, JRL and CKE outperform the other baselines, with JRL achieving the best performance among the baselines. It is encouraging to see that our collaborative filtering with knowledge graph (CFKG) method outperforms the best baseline (JRL) consistently over the four datasets and on all evaluation measures, which verifies the effectiveness of our approach for personalized recommendation. However, the performance improvement of our approach may benefit from two potential reasons – that we used more information sources, and that we used a better structure (i.e., structured knowledge graph) to model heterogenous information. For better understanding, we analyze the contribution of types of relation to our model in the following subsection.
Further Analysis on Different Relations
---------------------------------------
We experiment the performance of our model when using different relations. Because we eventually need to provide item recommendations for users, the CFKG approach would at least need the *buy* relation to model the user purchase histories. As a result, we test our model when using only the *buy* relation (which simplifies into the translation-based model [@he2017translation]), as well as the *buy* relation plus each of the other relations, as shown in Table \[tab:info-result\].
We see that when using only the *buy* relation, our CFKG\_*buy* model significantly outperforms the BPR approach on all measures and datasets. On the NDCG measure, the percentage improvement can be 90% (CDs), 70% (Clothing), 69% (Phone), and 33% (Beauty). Because both BPR and CFKG\_*buy* only used the user purchase information, this observation verifies the effectiveness of using structured knowledge graph embeddings for recommendation. Similarly, CFKG\_*buy+mention* significantly outperforms BPR\_HFT, and also outperforms DeepCoNN except for the recall on the CD dataset. Considering that CFKG\_*buy+mention*, BPR\_HFT and DeepCoNN all work with user purchase history plus textual reviews, this observation further verifies the advantage of using structured knowledge graph for user modeling and recommendation.
Furthermore, we see that adding any one extra relation to the basic *buy* relation gives us improved performance from CFKG\_*buy*. Finally, by modeling all of the heterogenous relation types, the final CFKG model outperforms all baselines and the simplified versions of our model with one or two types of relation, which implies that our knowledge base embedding approach to recommendation is scalable to new relation types, and it has the ability to leverage very heterogeneous information sources in a unified manner.
Related Work
============
Using knowledge base to enhance the performance of recommender system is an intuitive idea, which has attracted research attention since very early stages of the recommendation community [@trewin2000knowledge; @ghani2002building]. However, the difficulty of reasoning over the paths on heterogenous knowledge graphs prevent current approaches from applying collaborative filtering on very different entities and relation types [@zhang2016collaborativekdd; @catherine2017explainable], which further makes it difficult to take advantage of the wisdom of crowd.
Fortunately, recent years have witnessed the success of heterogenous knowledge base embedding techniques [@bordes2013translating; @wang2014knowledge; @lin2015learning], which can help to learn the embeddings of very different entities to support various application scenarios such as question answering [@bordes2014question] and relation extraction from text [@lin2015learning]. We believe that learning knowledge base embeddings while preserving the structure of knowledge for reasoning is vital for knowledge-enhanced AI in recommendation systems.
Conclusions and Future Work {#sec:conclusions}
===========================
In this paper, we propose to learn over heterogenous knowledge base embeddings for personalized recommendation. To do so, we construct the user-item knowledge graph to incorporate both user behaviors and our knowledge about the items. We further learn the knowledge base embeddings with the heterogenous relations collectively, and leverage the user and item embeddings to generate personalized recommendations. Experimental results on real-world datasets verified the superior performance of our approach, as well as its flexibility to incorporate multiple relation types.
|
---
abstract: 'We study the mean absorption spectrum of the Damped Lyman alpha population at $z\sim 2.6$ by stacking normalized, rest-frame shifted spectra of $\sim 27\,000$ DLAs from the DR12 of BOSS/SDSS-III. We measure the equivalent widths of 50 individual metal absorption lines in 5 intervals of DLA hydrogen column density, 5 intervals of DLA redshift, and overall mean equivalent widths for an additional 13 absorption features from groups of strongly blended lines. The mean equivalent width of low-ionization lines increases with $N_{\rm HI}$, whereas for high-ionization lines the increase is much weaker. The mean metal line equivalent widths decrease by a factor $\sim 1.1-1.5$ from $z\sim2.1$ to $z \sim 3.5$, with small or no differences between low- and high-ionization species. We develop a theoretical model, inspired by the presence of multiple absorption components observed in high-resolution spectra, to infer mean metal column densities from the equivalent widths of partially saturated metal lines. We apply this model to 14 low-ionization species and to AlIII, SIII, SiIII, CIV, SiIV, NV and OVI. We use an approximate derivation for separating the equivalent width contributions of several lines to blended absorption features, and infer mean equivalent widths and column densities from lines of the additional species NI, ZnII, CII${}^{*}$, FeIII, and SIV. Several of these mean column densities of metal lines in DLAs are obtained for the first time; their values generally agree with measurements of individual DLAs from high-resolution, high signal-to-noise ratio spectra when they are available.'
author:
- 'Lluís Mas-Ribas'
- 'Jordi Miralda-Escudé'
- 'Ignasi Pérez-Ràfols'
- 'Andreu Arinyo-i-Prats'
- Pasquier Noterdaeme
- Patrick Petitjean
- 'Donald P. Schneider'
- 'Donald G. York'
- Jian Ge
bibliography:
- 'dla\_lluis.bib'
title: 'The mean metal-line absorption spectrum of Damped Lyman Alpha Systems in BOSS'
---
Introduction
============
The existence of luminous quasars at high redshift is a gift of Nature. It allows us to explore in an unbiased way any population of gas clouds in the Universe by means of the absorption lines they produce in the spectra of the background sources. Without luminous quasars, we would not have sources at high redshift that are sufficiently bright to obtain spectra of high resolution and signal-to-noise ratio (hereafter, S/N) in which the [Ly$\alpha$ ]{}line, as well as numerous ultraviolet metal absorption lines, are shifted to the optical range and can easily be observed from the ground. Damped Lyman Alpha systems [DLAs, hereafter; @Wolfe1986] are generally defined to have hydrogen column densities $\rm{N_{HI}} > 2\times 10^{20} {\, {\rm cm}}^{-2}$. Systems of this high column density have two important characteristics: ([*i*]{}) they are self-shielded of the external cosmic ionizing background, implying that the hydrogen in these systems is mostly in atomic form [@Vladilo2001], and (*ii*) the damped profile of their hydrogen [Ly$\alpha$ ]{}line is clearly visible even in low-resolution spectra, therefore the column density can be measured from the absorption profile [see @Wolfe2005; @Barnes2014 for detailed reviews]. DLAs provide a reservoir of atomic gas clouds that were available at high redshift for the formation of galaxies. The mean cosmic density of baryons contained in these systems is directly obtained from the measurements of the column density distribution, and accounts for a small fraction of the critical density $\Omega_{\rm DLA}\simeq 10^{-3}$ at redshifts $2 < z < 3.5$ [e.g., @Peroux2003; @Peroux2005; @Prochaska2005; @Noterdaeme2012; @Zafar2013; @Padmanabhan2015; @Crighton2015; @Sanchez2015]. This value corresponds to $\sim$ 2% of all the baryons in the Universe, comparable to the fraction of baryons that had turned into stars at these redshifts [@Prochaska2005; @Prochaska2009; @Noterdaeme2009].
The metal absorption lines of damped [Ly$\alpha$ ]{}systems have been explored since the discovery of DLAs. High-resolution spectra reveal a diversity of velocity structures of the absorbers, characterized by multiple components. Sometimes a single component with a narrow velocity width close to the thermal value for photoionized gas clouds is observed, but often several components are seen over a typical velocity range $\sim 100-300 \,{\rm {\, {\rm km \, s}^{-1}}}$ [@Prochaska1997; @Zwaan2008]. The derived metallicities are generally low, distributed over a broad range of $10^{-3}$ to $10^{-1} Z_{\odot}$ [@Prochaska2002; @Prochaska2003c; @Kulkarni2005; @Ledoux2006], and on average declining with redshift [@Kulkarni2002; @Vladilo2002; @Prochaska2003; @Calura2003; @Khare2004; @Akerman2005; @Kulkarni2005; @Rafelski2012; @Jorgenson2013; @Neeleman2013; @Moller2013; @Rafelski2014; @Quiret2016]. The complex velocity profiles suggest a highly turbulent environment, and models of clouds moving in random orbits in galactic halos or thick disks can generally explain the observations . The fact that several absorbing components are typically seen along a given line of sight, each corresponding to clouds moving at different velocities within a larger halo, implies that these clouds are colliding with each other about once every orbital period [@McDonald1999]. For individual DLAs it can be difficult to model the column densities and velocity structure of the metal species due to a complex variety of gas phases arising from photoionisation, shock-heating and collisions leading to a broad range of temperatures and densities [e.g., @Fox2007a; @Berry2014; @Dutta2014; @Lehner2014; @Neeleman2015; @Cooke2015; @Rubin2015].
In the context of the Cold Dark Matter model of structure formation, the non-linear collapse of structure leads to hierarchical merging of dark matter halos. The cosmological theory, starting from a matter power spectrum that is now accurately determined from observations of the Cosmic Microwave Background [@Planck2015 and references therein], predicts the number density of halos as a function of halo mass that exist at any epoch, $n(M_h,z)$. The observed rate of DLAs per unit of redshift in any random direction due to halos of mass $M_h$ in the range $dM_h$ is then $n(M_h,z)\, \Sigma(M_h,z)\, c\, dt/dz\, dM_h$, where $\Sigma(M_h,z)$ is the mean cross section (or area) within which a DLA is observed in a halo of mass $M_h$. Although it has been generally believed that DLAs are associated with dwarf galaxies [e.g., @York1986; @Dessauges2004; @Khare2007; @Fumagalli2014; @Webster2015; @Bland2015; @Cooke2015], observations of the large-scale cross-correlation amplitude of DLAs with the [Ly$\alpha$ ]{}forest absorption have determined their mean bias factor $b_{\rm DLA}\simeq 2$ [@FontRibera2012], which is consistent with DLAs being distributed over a broad range of halo masses $10^9 M_{\odot} \lesssim M_h \lesssim 10^{13} M_{\odot}$, from dwarf galaxies to halos of massive galaxies and galaxy groups.
The Baryon Oscillation Spectroscopic Survey [BOSS; @Dawson2013] of the Sloan Digital Sky Survey-III [SDSS-III; @Eisenstein2011] has obtained spectra for more than $160\,000$ quasars at $z>2$, providing an unprecedentedly large sample of DLAs. Despite the relatively low resolution (R$\sim2000$) and S/N of the BOSS spectra, the large number of observed DLAs allows an accurate measurement of the mean metal-line absorption strength by stacking many systems, and studying the dependence of the equivalent widths of any line on the hydrogen column density. This approach has the advantage of directly providing mean properties of the DLA population, rather than properties of individual systems which have a large intrinsic random variation. Although the study of individual systems in detail obviously results in invaluable additional information that is lost in a stacked spectrum of the global DLA population, even a single DLA has absorption that probes a mixture of different gas phases and is difficult to model in practice. Moreover, absorption lines that are located in the [Ly$\alpha$ ]{}forest region can be accurately measured only after averaging over a large number of absorption systems, and they can provide important information that is not accessible from lines on the red side of [Ly$\alpha$ ]{}[see @Rahmani2010; @Khare2012; @Noterdaeme2014 for analysis of composite DLA spectra from BOSS].
This paper presents the mean absorption spectra of metal lines in DLAs that is derived from the Data Release 12 [@Alam2015] of BOSS, using the DLA catalog generated with the technique described in [@Noterdaeme2009]. The two main results are (*i*) the mean dependence of the equivalent width of each metal species on the hydrogen column density, and (*ii*) an analytical model of the mean equivalent widths of multiple absorption lines of the same metal species to account for the effect of saturation and derive mean column densities in our DLA sample. We also present for the first time the mean equivalent widths in DLAs of various species that are usually difficult to measure owing to the confusion with the Lyman forest, e.g., SIV, SIII, FeIII and NII, as well as accurate determinations of the mean equivalent width and inferred column densities of more commonly measured species like OVI.
In § \[sec:data\], we present the method to calculate the mean quasar continuum spectrum and the DLA stacked absorption spectrum. We also detail the corrections we apply to improve the mean quasar continuum which, in turn, results in a more reliable stacked spectrum. In § \[sec:results\] we compute the mean equivalent width of detected metal lines, their dependence on the hydrogen column density is assessed and presented in § \[sec:nhi\], and in § \[sec:z\] we address the dependence on DLA redshift. In § \[sec:model\], a simple model is proposed to correct for line saturation and is used to infer the mean column densities of several low- and high-ionization species for which the mean equivalent width of absorption lines has been measured. We discuss our results in § \[sec:discussion\], before summarizing and concluding in § \[sec:summary\].
All the equivalent widths in this paper are rest-frame.
Data Analysis {#sec:data}
=============
We use the spectra of quasars in the SDSS-III BOSS Data Release 12 Quasar Catalogue ‘DR12Q’ [@Paris2016]. The SDSS telescope and camera are described in detail in [@Gunn1998; @Gunn2006; @Ross2012], and the SDSS/BOSS spectrographs in [@Smee2013].
We use the DR12 DLA catalog, which is the expanded version of the catalog presented in [@Noterdaeme2012] for the DRQ9 [@Paris2012], and contains a total of $34\,593$ DLA candidates with a measured column density $\rm{N_{HI}} > 10^{20} {\, {\rm cm}}^{-2}$. The detection of these systems is performed by means of a fully-automatic procedure based on profile recognition using the Spearman correlation analysis, as described in [@Noterdaeme2009]. Only $19\,376$ of the DLAs in the catalog have $\log (N_{\rm HI}/{\rm cm}^{-2}) > 20.3$ and can therefore be designated DLAs if we strictly use the standard definition of a DLA. This column density threshold was, however, set for observational purposes, and to ensure that the hydrogen gas in the inner regions of the DLAs is mostly neutral due to self-shielding from the background radiation [@Wolfe1986]. We include systems down to $\log (N_{\rm HI}/{\rm cm}^{-2})> 20$ because we find them to be also useful to characterize the mean properties of the population and their dependence on column density. We demonstrate below (§ \[sec:curvenhi\]) that the inclusion of systems with $\log
(N_{\rm HI}/{\rm cm}^{-2}) < 20.3$ does not substantially change our results, although the mean equivalent widths do change with column density and redshift and, therefore, accurate comparisons with other stacked spectra in the future need to take into account our distribution of column densities and redshifts. A relatively low minimum continuum-to-noise ratio $C/N>2$ is required to include a DLA in the catalog, with the goal of maximizing the size of the catalog without having a large number of false DLA systems arising from spectral noise [@Noterdaeme2012]. This C/N threshold is specially important for systems with low column density since these have a higher probability to be false detections.
Most of the results presented in this paper are obtained from stacks using the whole DLA catalog, which is designated here as [**total sample**]{}. This sample should be unbiased, in the sense that only the HI [Ly$\alpha$ ]{}absorption line has been used to select the DLAs, and not the strength of the metal lines. However, DLA samples from optically-selected quasars may be biased against systems containing substantial amounts of absorbing dust [e.g., @Fall1993; @Boisse1998; @Ellison2001; @Smette2005; @Noterdaeme2015], although the presence of dust in DLAs is expected to be small . The presence of dust-biasing would have little impact on general HI studies [@Trenti2006; @Ellison2008], but it may significantly underestimate the metal content in DLAs [@Pontzen2009]. In addition, we also study a subsample of the DLA catalog, which we call [**metal sample**]{}, containing $12\,420$ DLA candidates where metal lines can be individually detected by using templates (of these, $8\,699$ have $\log (N_{\rm HI}/{\rm cm}^{-2}) > 20.3$). The metal detection results from a cross-correlation of the observed spectrum with an absorption template containing the most prominent low-ionisation metal absorption lines, which is done in addition to the previous Spearman correlation analysis for the [Ly$\alpha$ ]{}line. If one or more metal absorption lines are detected in the individual spectra, these are used to refine the absorption redshift of the DLA [see section 3.2 in @Noterdaeme2009 for further details]. The improved redshift of systems in the metal sample gives rise to more sharply defined lines in the resulting stack, and allows detection of the weakest metal lines and measurement of the effect of redshift uncertainties in the total sample. However, the metal sample is obviously biased in favor of DLA systems with strong metal lines and/or high S/N, and therefore cannot be used to obtain mean properties of the true population of DLAs. The absence of metal lines is never used to discard candidate DLAs at a low signal-to-noise ratio from the total sample. The mean stacked spectra of the DLAs used in each of these two samples are computed in a similar way, except for a few differences that are discussed below.
The left panels in Figure \[fig:distrib\] present the redshift and ${N_{\rm HI}}$ distributions of the total (blue histogram) and the metal sample (green histogram). The ratio of the number of DLAs in the two samples is also denoted as the red line, with the scale on the right axis. The metal sample contains a greater fraction of high column density systems because the strength of metal lines increases, on average, with ${N_{\rm HI}}$. The metal sample also has a smaller fraction of systems at high redshift, because of the declining mean metallicity of DLAs with redshift and the decreasing number of metal lines redwards of [Ly$\alpha$ ]{}that are observable with increasing redshift. In addition, the increased density of the [Ly$\alpha$ ]{}forest at high redshift may give rise to an increase of false positive DLA detections, specially in the total sample [see, e.g., @Rafelski2014]. The right panels display the same two distributions as before, but now considering only the systems that are finally used for the calculation of the composite spectra, after the application of cuts in the DLA sample and after weighting the spectra as described in the next subsection. The distributions on the right panels are the actual ones that give rise to the results of this paper, which should be used in order to precisely compare to any other future observational results or model predictions. The shape of the distributions in the right panels is similar to those in the left ones, but the metal sample has a higher contribution than in the left panels, in general. This difference is mostly because metal lines are more likely to be identified in high S/N spectra, which have higher weights, as explained in the next subsection, and because of the cuts that remove possible false DLAs in the catalog. Additionally, the redshift distribution of the total sample narrows when applying the cuts and weighting the spectra. The mean values for the total sample in the right (left) panels are $\log { (\bar N_{\rm HI}/{\rm cm}^{-2})}=20.49$ (20.70) for column density, and ${\bar z=2.59}$ (2.65) for redshift.
Continuum quasar spectrum calculation {#sec:continuum}
-------------------------------------
A crucial part of computing a mean stacked spectrum of the transmitted flux fraction for a sample of DLAs is the calculation of the quasar continuum. We use a method that is similar to that in [@PerezRafols2014], who measured the mean absorption by MgII around the redshift of a galaxy near the line of sight to a quasar. Some variations are necessary in our case, however, owing to the mean absorption by the [Ly$\alpha$ ]{}forest and the presence of the DLA metal lines themselves. We now describe in detail our procedure for estimating the continuum.
The method starts by computing a mean spectrum of the quasars used in both the total sample and the metal sample. First, each quasar spectrum is shifted to the quasar rest-frame wavelength, $\lambda_r = \lambda_{obs}/(1+z_q)$, where $\lambda_{obs}$ is the observed wavelength of every spectral bin, using the quasar redshift $z_q$ provided in the DR12 DLA catalog [this is the visual inspection redshift of the quasar catalog from @Paris2016]. The values and errors of the flux are rebinned into new pixels of width $1.0\, {\rm \AA}$ in the rest-frame by standard interpolation, averaging the values in the original pixels as they are projected, partly or fully, onto the new pixels. Any pixels affected by skylines, as reported in [@Natalie2013], are removed from the spectra and excluded from all the analysis.
The spectrum of each individual quasar is then normalized by computing the mean flux in two fixed rest-frame wavelength intervals: $1300 \, {\rm \AA} < \lambda_r < 1383\, {\rm \AA}$, and $1408 \, {\rm \AA} < \lambda_r < 1500\, {\rm \AA}$. These intervals are chosen to avoid the principal broad emission lines of quasars and the region of the [Ly$\alpha$ ]{}forest absorption, and to be roughly centered in the spectral range of interest for the DLA metal lines. The normalization factor for each quasar $j$ is defined as $$n_{j}= \sum_i{f_{ij} \over N_j} ~,$$ where $f_{ij}$ is the flux per unit wavelength in the pixel $i$ of the quasar $j$, and the sum is done over all the $N_j$ pixels that are comprised within the two wavelength intervals for the normalization. Some pixels in these two intervals are discarded because of the skylines that are removed or because of additional corrections discussed below (see § \[sec:correct\]). Any quasar for which more than 20% of the pixels in the normalizing intervals are discarded is removed from the sample. This results in the removal of $1\,074$ quasars for the total sample and $298$ for the metal sample.
{width="100.00000%"}
A mean S/N, $s_j$, is computed for each quasar spectrum using the same two rest-frame wavelength intervals, $$s_{j}=\frac{\sum_i{f_{ij}/N_j}}{\left(\sum_i{e_{ij}}^2/N_j\right)^{1/2}} ~,$$ where $e_{ij}$ is the uncertainty for the flux $f_{ij}$. The resulting distribution of the $s_j$ parameter is presented in Figure \[fig:snr\] for the two samples. This distribution peaks at $s_j\simeq 2$ for the total sample, and at a higher value for the metal sample (as expected, because metal lines are more difficult to detect for low signal-to-noise). We discard from our sample any spectra with $s_j<1.0$, because of the very poor quality of these spectra, resulting in $264$ and $5$ spectra from the total and metal sample, respectively, not being further considered for our calculations. These discarded spectra are a small fraction of the total because of the independent constraint of a continuum-to-noise $C/N>2$ in the [Ly$\alpha$ ]{}forest region that was already applied to the DLA catalog with the method of [@Noterdaeme2009].
With the S/N values we assign a weight $w_j$ to each quasar spectrum, defined as $$\label{eq:weight}
w_{j}=\frac{1}{{s_j}^{-2}+\sigma^2} ~,$$ where $\sigma$ is a constant that is introduced to prevent the quasars with highest $s_j$ contributing excessively to the mean in the presence of an intrinsic variability of quasar spectra, in addition to observational noise. We choose, somewhat arbitrarily, a value $\sigma = 0.1$, which represents our estimate that the typical intrinsic variability of quasar spectra is $10\%$. We ran the stacking using $\sigma = 0.2$ and saw that this difference do not produce any relevant effect in our resulting spectrum.
{width="100.00000%"}
Finally, the resulting mean normalized quasar spectrum is computed as a weighted mean, $$\bar{f_i}=\frac{\sum_jw_j(f_{ij}/n_j)}{\sum_jw_j} ~,
\label{eq:mnf}$$ where $\overline{f}_i$ is the normalized flux per unit wavelength of the mean quasar spectrum at the quasar rest-frame wavelength pixel $i$. This mean quasar spectrum is displayed as the [*black line*]{} in Figure \[fig:mean\], with the most prominent emission lines labelled. This is the mean spectrum of quasars that have (at least) a DLA absorption system. Therefore, the mean spectrum incorporates any mean modification that the DLA lines have produced. Some quasars ($\sim 20\% $ for both samples) have more than one DLA system in the catalog; in this case the same quasar contributes to the mean as many times as the number of DLAs it contains. The [*red region*]{} in Figure \[fig:mean\] illustrates the 68% contours of the distribution of normalized spectra around the mean. For comparison, we also plot the non-weighted mean, represented by the [*cyan line*]{}. This spectrum shows stronger emission lines compared to the weighted mean, because of the Baldwin effect: emission line equivalent widths decrease with quasar luminosity [@Baldwin1977]. A small difference is also present in the [Ly$\alpha$ ]{}forest region (barely visible in this plot), which is likely due to small variations of the distribution of S/N with redshift. Apart from this, we see that the [*black line*]{} is less noisy than the [*cyan*]{} one, which is our main reason to use the weights for computing the mean quasar spectrum.
Composite DLA spectrum calculation {#sec:stack}
----------------------------------
To compute a stacked DLA absorption spectrum, we start by shifting each quasar spectrum to the DLA rest-frame, rebinning now into a pixel width of $0.3\, {\rm \AA}$ to obtain better sampling. We also discard pixels affected by skylines, and then divide by the previously computed mean quasar spectrum shifted to the same DLA rest-frame, obtaining the transmission at the rebinned pixel $i$ of the quasar spectrum $k$, $$F_{ik}=\frac{f_{ik}/n_k}{\overline{f}_{ik}} ~,
\label{eq:trnorm}$$ where $\overline{f}_{ik}$ now has the subindex $k$ labeling each DLA only because the mean continuum $\bar f_i$ has been shifted and rebinned on the DLA rest-frame. By means of error propagation, uncertainties are normalized and rebinned in the same way to obtain the error, $E_{ik}$, of the transmission spectrum of each DLA, $F_{ik}$. The final composite spectrum and its error is again obtained from a weighted mean, $$\bar{F_i}=\frac{\sum_kF_{ik}w_k}{\sum_kw_k} ~, \qquad
\bar{E_i}^{-2}=\frac{\sum_kE_{ik}^{-2}w_k}{\sum_kw_k} ~.$$ We set the weight $w_k$ to be the same as in Eq. \[eq:weight\], with the same value of $\sigma=0.1$. This value does not really have to be the same for computing the mean quasar continuum and the mean DLA transmission spectrum; in general, we could choose a higher $\sigma$ for stacking the DLA transmission because we need to take into account the intrinsic variability of the DLA metal lines as well. As we did for the case of the continuum, we have tested that increasing to $\sigma=0.2$ does not substantially alter the results presented below; the two spectra have visually the same appearance.
The composite DLA transmission spectrum obtained after these calculations for the total sample is denoted by the black line in Figure \[fig:stack\]. The grey line in this figure indicates the fraction of DLA systems contributing to the estimated stacked spectrum at each rest-frame wavelength bin. This fraction is less than unity on both sides of the range considered because a fraction of the quasar spectra, depending on their redshift, do not extend along the entire observed wavelength. The mean of the Lyman series lines due to hydrogen and many metal lines of our sample of DLAs are clearly evident. However, there are also broad regions where the mean transmission deviates from unity, which are clearly not associated with the narrow DLA absorption lines. This deviation may be due to several effects: the spectra that contribute to a given wavelength for the quasar composite are not the same that contribute to a given wavelength in the DLA composite, thus the quasar continuum is not entirely cancelled out. In addition, the [Ly$\alpha$ ]{}forest causes an absorption both in the mean quasar spectrum and in the estimated transmission spectrum of each DLA, which does not exactly cancel when dividing by the mean quasar spectrum due to the redshift evolution of the [Ly$\alpha$ ]{}forest. In the next subsection, we assess and apply several corrections for these effects in order to obtain a better quasar continuum and composite spectrum for the two samples.
The mean metal line equivalent widths that we will obtain in this paper, which vary with column density and redshift, depend on the selection of our sample. In addition, some fraction of the DLAs may be false and result from a concentration of [Ly$\alpha$ ]{}forest lines that, in noisy spectra, may look like a DLA, while others may have large errors in redshift. This should cause a reduction of the mean equivalent widths measured in our stacked spectra, implying that our results may be sensitive to our adopted cut in the $C/N$ ratio and the way we choose to weight the contribution of DLAs depending on the signal-to-noise ratio from Eq. \[eq:weight\]. We have tested the effect of eliminating the weights when calculating the composite spectrum (keeping the same sample of DLAs that we use after our cuts), which results in a very similar continuum, but substantially weaker metal absorption lines, with equivalent widths reduced typically by $\sim 30\%$. We believe most of this reduction is due to the fraction of false DLAs in the lowest S/N spectra, together with increased errors in redshift which wash out the metal lines in the stacked spectrum. Some of this reduction may also be due to a lower mean column density and higher mean redshift of the unweighted sample: The mean column density drops from from $\log (\bar N_{\rm HI}/{\rm cm}^{-2})= 20.49$ to 20.47, and the mean redshift increases from $\bar z = 2.59$ to 2.68, when eliminating the weights, but as we shall see in our results (Figures \[fig:nhi\] and \[fig:z\], this accounts for only a small part of the reduction in mean equivalent widths when removing the weights.
However, by maintaining the weights, the contribution of false DLAs in low signal-to-noise spectra should be greatly reduced, and our systematic underestimate of the metal lines mean equivalent widths should be much less than 30%. We have tested this by examining variations with the minimum threshold in $C/N$ to accept DLAs in our sample. We find that the median relative increase of equivalent widths of metal lines analysed in this paper is $2.5\%$, $3.7\%$ and $5\%$ as the minimum $C/N$ is raised to 3, 4, and 5, respectively. These fractional variations are, we believe, a fair estimate of our systematic errors caused by impurity and large redshift errors in our DLA sample. Again, part of this increasing mean equivalent width with $C/N$ may be caused by an increasing mean column density and decreasing redshift with $C/N$. In any case, this suggests that our weighting scheme is useful to reduce the effect of impurity in the DLA catalog, and that the remaining systematic reduction of mean equivalent widths is at the level of $\sim$ 5%.
For illustrative purposes, we have created two movies displaying the evolution of the mean quasar and composite DLA spectra as the number of stacked objects is increased. The two movies are publicly available and can be found, together with a brief description of the calculations, in the url <https://github.com/lluism>.
Corrections on the continuum spectrum {#sec:correct}
-------------------------------------
We now present the corrections that we apply to improve our first version of the transmission spectrum in Figure \[fig:stack\]. Briefly, these corrections consist of detecting and removing bad pixels, correcting for the mean [Ly$\alpha$ ]{}forest absorption, and correcting for the average effect of the DLA lines on the continuum spectrum. In addition, we describe the procedure applied to spectra where Ly$\beta$ absorptions can be mistaken for Ly$\alpha$. All these corrections, described in detail below, are applied equally for the total and metal sample.
### Detection and removal of outliers {#sec:outliers}
A variety of effects, e.g., cosmic rays, may cause large deviations of the flux in a few pixels from the correct values that clearly set them as outliers from the normal noise distribution. We prefer to eliminate these outliers, rather that working with median values which reduce the sensitivity to outliers, because we want to obtain mean equivalent widths in the end, and a relation of median to mean values would be model-dependent. Outliers cannot be eliminated by simply setting a maximum noise deviation of the transmission from the expected range of zero to unity, because the intrinsic variability of the quasar spectra can be large, implying that a more generous transmission range should be allowed.
Therefore, for the purpose of eliminating outliers, we first obtain a fitted continuum, $C_{ik}$, to the transmission $F_{ik}$ computed previously for each quasar. The detailed method we use to fit this continuum is described below in § \[sec:wcalc\]; essentially, $C_{ik}$ is a smoothed version of $F_{ik}$ computed once the regions of the expected metal lines of the DLA have been excluded. We then eliminate all pixels with a transmission $F_{ik}$ that is outside the interval $[C_{ik}-2-3E_{ik}$ , $C_{ik}+2+3E_{ik}]$, where $E_{ik}$ is the transmission uncertainty in each pixel defined after Eq. \[eq:trnorm\]. This is a generously broad range, which allows for an uncertainty in the pixel flux of three times the estimated standard deviation $E_{ik}$, and adds an additional variation of twice the normalized mean quasar spectrum. Despite this broad range, it still excludes the most important outliers without eliminating any pixels that are not obviously bad. After the outliers are eliminated, we recalculate a new mean quasar continuum and a new stacked DLA absorption spectrum, which we adopt as the new composite spectrum in our analysis.
### Mean absorption of the [Ly$\alpha$ ]{}forest {#sec:lyaforest}
The [Ly$\alpha$ ]{}forest causes a systematic, redshift-dependent flux decrement bluewards of the [Ly$\alpha$ ]{}emission line of the quasar. The redshift evolution of this decrement means that if it is not corrected, its mean value at a certain wavelength in the DLA rest-frame in our composite absorption spectrum is generally not equal to the mean value of the decrement in the mean quasar spectrum that was used to obtain the transmission from the observed flux, leaving a residual effect. This residual effect consists in an increase of the transmission in the DLA composite spectrum bluewards of the [Ly$\alpha$ ]{}feature, more important for longer wavelengths.
We use the fit obtained by [@Faucher2008] for the mean fractional transmission as a function of redshift, $$\label{Fz}
F_{\alpha}(z) = {\rm exp} \left[ -0.0018(1+z)^{3.92} \right] ~.$$ We divide the normalized flux, $f_{ij}/n_j$ in Eq. \[eq:mnf\], in the spectrum of each quasar by $F_{\alpha}(z)$ at the redshift $z=\lambda_{obs}/\lambda_\alpha - 1$ (where $\lambda_{\alpha}=
1215.67 \, {\rm \AA}$), before stacking to obtain the mean quasar spectrum. We then divide again each spectrum containing a DLA by $F_{\alpha}$ at the same redshift, before dividing by the mean quasar spectrum in Eq. \[eq:trnorm\]. The net correction does not completely cancel, and depends on the probability distribution of the DLA and quasar redshifts in our catalog, which is sensitive to selection effects reflecting the DLA detection probability.
{width="90.00000%"}
Figure \[fig:meancorr\] displays the impact of the correction for the [Ly$\alpha$ ]{}forest transmission on the mean quasar continuum. The original spectrum (yellow line) is raised by $\sim$ 20% after this correction (green line), reflecting the mean decrement at the mean redshift of our quasar sample. The change of the composite DLA spectrum after applying this correction is presented in Figure \[fig:stackcorr\] (yellow and green lines for the original and corrected spectra, respectively). Both figures display the result for the total sample.
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Eq. \[Fz\] includes the effect of the [Ly$\alpha$ ]{}forest only. Metal lines are believed to increase the decrement by $\sim 5\%$ of that due to [Ly$\alpha$ ]{}[@Faucher2008]. We neglect these metal lines, and we also do not include the additional corrections for the forests of Ly$\beta$ and higher-order hydrogen lines, although we do correct for the effect of all the lines associated with the DLAs themselves that are stacked in the final composite spectrum, as we describe below.
### Effect of the DLA absorption lines {#sec:lines}
When we compute the mean quasar continuum, there is a mean incidence rate of hydrogen and metal lines of DLAs that are present in each quasar spectrum and contribute to lower the measured flux. This average flux decrement caused by the DLA lines is larger than in a random sample of quasars because we use only the quasars in the DLA catalog to obtain our mean quasar spectrum, where by construction, each quasar spectrum contains at least one DLA. We now describe the procedure to correct for the presence of these DLA lines, which is applied both to the total and metal sample.
For each DLA in the catalog, we redo the quasar continuum spectrum calculation after modifying the flux values in pixels that are comprised within predefined windows around each of the expected DLA absorption lines. We also remove some of these pixels where the DLA absorption is strongest. We account for all the DLA metal absorption lines listed in Appendix \[sec:ew\], in Tables \[ta:low\] and \[ta:high\], as well as for the blended lines listed in Table \[ta:blended\]. We include in addition the hydrogen Lyman transitions from [Ly$\alpha$ ]{}up to $n=9$, covering the range between $920$ Åand $3\,000$ Å. The wavelength windows around each line center used for this correction are the same as those used for the computation of the equivalent width of the metal lines, described in detail in § \[sec:wcalc\] below.
For each DLA absorption line, the flux in each pixel within its window is corrected as $$\label{line}
{f_{L}} = \frac{{f_{QSO}}}{{\overline{F}_D}},$$ where $f_{L}$ is the corrected flux, $f_{QSO}$ is the observed flux before correction in the quasar rest-frame at redshift $z_q$, and ${\overline{F}_D}$ is the transmission in the mean DLA composite spectrum, after shifting to the quasar rest-frame by multiplying the wavelength by $(1+z_{DLA})/(1+z_q)$. The DLA composite spectrum is rebinned in order to match the quasar rest-frame, in the same way as in § \[sec:stack\] when computing the mean quasar spectrum. This correction is applied only within each DLA absorption line window, when $1.0 > \overline{F}_D \ge 0.4$. For the strongest DLA lines, pixels where the DLA mean transmission is $\overline{F}_D < 0.4$ in the composite spectrum are removed instead of being corrected, and not taken into account for calculating the improved quasar continuum. We adopted this approach to avoid excessive noise from the regions that are highly absorbed. Small variations for the threshold value $0.4$ do not significantly change our results. This correction is applied without considering detecting any DLA line in the individual spectra, to correct for their mean expected absorption.
The new mean quasar continuum is used to recalculate the DLA composite spectrum. We can now iterate the same procedure, since the DLA composite spectrum is needed to compute the correction to each quasar continuum, until there is no significant improvement. This convergence is reached after three iterations. The black line in Figure \[fig:meancorr\] shows how the mean quasar continuum is further modified by these DLA lines in the region of the [Ly$\alpha$ ]{}forest (this is mostly the effect of the DLA [Ly$\alpha$ ]{}line), and Figure \[fig:stackcorr\] indicates how the DLA composite spectrum is improved after the first, second and third iterations (red, blue and black lines, respectively; the spectrum is displayed only at $\lambda < 2000$ Å, at longer wavelengths the corrections to the continuum are very small). A clear improvement is seen after the first iteration (red line), in the sense that the continuum between the DLA metal lines moves closer to unity over most of the regions, but a smaller improvement is achieved with subsequent iterations.
Despite the improvements that result from these corrections, Figure \[fig:stackcorr\] demonstrates that the continuum still deviates slightly from unity due to other uncorrected systematics. One contribution is probably the proximity effect, which accounts for the fact that the [Ly$\alpha$ ]{}forest near the quasar redshift has a lower mean decrement than far from the quasar redshift. There is also a small rise of the continuum above unity in the region longwards of the CIV line, which may be partly caused by the forest of CIV lines associated with the [Ly$\alpha$ ]{}forest and Lyman limit systems. Further improving our continuum model would clearly require more detailed work to correct for these effects and other systematics that are likely present. We have decided to stop here and to use a simple method to flexibly fit the continuum between the DLA lines in the next section.
The correction involving the effect of the DLA absorption lines assumes that there is only one DLA in each quasar spectrum. For each DLA in the catalog, the quasar spectrum is corrected for the presence of only that DLA, ignoring the possible presence of other DLAs in the same spectrum. A more accurate procedure would take into account all the detected DLAs in each quasar spectrum. As mentioned above, $\sim 80\%$ of the spectra in both samples contain only one DLA, so we expect spectra with more than one DLA to produce a small effect on the mean quasar continuum.
Finally, in order to compare our method for correcting the effect of the DLA lines, we adopt a different approach. We calculate the mean quasar spectrum, now using all the spectra in the DR12 quasar catalog from BOSS that do not contain DLAs. After our cuts, this results in the use of $\sim210\,000$ quasar spectra. The continuum of this mean quasar spectrum overlaps with our previously computed mean spectrum, except in the Ly$\alpha$ forest region. The difference between the two corresponds to a $\sim10\%$ increase of the flux in the forest for the sample without DLAs. Our correction ([*black line*]{} in Figure \[fig:meancorr\]) increases the flux depending on wavelength, from $\sim5\%$ at $\sim1050$ [ [Å]{}]{} to $\sim10\%$ at $\sim1150$ [ [Å]{}]{}. Therefore, our correction might simply account for a fraction of the total DLA effect but, because of other possible contributions to the observed difference (e.g., the strength of the emission lines are different in the two samples), we consider our original approach for further calculations. This will not affect our results because of the use of the additional continuum fit to the final composite spectrum.
### Contamination by DLA Ly$\beta$ lines mistaken for [Ly$\alpha$ ]{} {#sec:lyb}
When we first obtained the DLA composite spectrum, we noticed the presence of a few regions with unexplained anomalous absorption features. These can be seen in Figure \[fig:stack\], where the spectrum for the total sample has strange spectral features, for example, near $1440\,{\rm \AA}$ or $1520\,{\rm \AA}$ (just to the left of the SiII line), which do not appear in the spectrum of the metal sample. The source of these features is that some Ly$\beta$ absorption lines are incorrectly identified as DLA [Ly$\alpha$ ]{}lines in the DLA catalog. This error produces spurious absorption features in the stacked spectrum and other undesired effects (the features are not present in the metal sample because the DLAs identified at a wrong redshift obviously do not yield any metal line detections). To avoid this problem, we have removed all the DLA spectra with a redshift ${(1+z_{DLA})\le 27/32\,(1+z_q)}$, ensuring that the detected [Ly$\alpha$ ]{}lines cannot possibly be a Ly$\beta$ line of a higher redshift DLA. The amplitude of the fictitious features caused by these DLAs indicates that only $\sim 10\%$ of them are incorrect identifications of a Ly$\beta$ line, but we have not further attempted to separate these mistaken detections in order to avoid any other selection effects in our total sample. This criterion reduces the number of DLAs in our total sample to $26\,931$, and to $10\,766$ in our metal sample, to which the whole analysis described above has been applied. In the rest of the paper, we analyze the results obtained for these restricted catalogs.
Analysis of the Composite DLA Spectrum {#sec:results}
======================================
We measure the mean equivalent width of metal lines detected in the stacked DLA absorption spectra below. We also divide the two DLA samples into five column density bins to assess the dependence of these mean equivalent widths on the hydrogen column density, ${{ N_{\rm HI}}}$, in § \[sec:nhi\]. The results are tabulated in Table \[ta:blended\], and Tables \[ta:low\] to \[ta:highnhi76\] in Appendix \[sec:ew\].
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Figures \[fig:lines24\] and \[fig:lines76\] present an expanded version of the final composite absorption spectrum of DLAs, for the total and the metal sample, respectively, with labels denoting the detected absorption lines. Absorption lines in the metal sample (Figure \[fig:lines76\]) are usually stronger and present a sharper profile than those in the total sample, allowing for more detections of weak absorption features despite the noise. This is simply because the metal sample is selected to include only DLAs with individually detected metal lines, so DLAs of low metallicity and/or low $C/N$ spectra are usually excluded, reducing the noise in the final stacked spectrum.
The [Ly$\alpha$ ]{}absorption feature in Figure \[fig:lines24\] does not present a broad flat region with null flux at the position of the line center as observed in single systems. This effect is due to two reasons: First, there may be some zero-flux error with a mean transmission of $\sim 0.01$ that is systematically added to all the spectra. Second, the DLA sample is not totally pure, and there may be a few percent of the absorbers catalogued as DLAs that are actually arising as a combination of lower column density absorbers and spectral noise, contributing also a residual flux at the [Ly$\alpha$ ]{}line profile of the stacked spectrum. For the metal sample, where the percentage of false DLAs should be much smaller, there is a flatter bottom of the DLA mean profile that is consistent with the width of the damped profile expected for the lowest column densities present in the catalog. This may also affect the metal lines: as mentioned earlier, our equivalent widths may be systematically underestimated because of a fraction of DLAs in the catalog that are not real or have large redshift errors. The zero-flux error may also cause an additional underestimation of metal line equivalent widths, but the relative error should be similar to the fraction of zero flux present at the bottom of the [Ly$\alpha$ ]{}line in Figures \[fig:lines24\] and \[fig:lines76\], which is only a few percent. We ignore these systematic errors in this paper; future improvements on this work should use mocks of the [Ly$\alpha$ ]{}spectra in BOSS to correct for the fraction of false DLAs in our catalog, as well as the zero-flux error [see, e.g., @Bautista2015].
We detect transitions of elements rarely seen in DLAs, such as TiII, CII\*, PII, CoII, ArI, and several lines of CI, a species that is associated with ${\rm H_2}$ [@Srianand2005; @Ledoux2015]. We also observe several high-ionization lines, including NV, OVI and SIV, which are extremely difficult to detect in individual spectra; in fact, NV and SIV have been detected only in a few DLAs, and when they are detected they are hard to separate unambiguously from the Lyman forest lines, particularly for SIV and OVI lines which are always blended with the forest [@Fox2007a; @Lehner2008; @Lehner2014]. Here, we will analyse a total of 42 low-ionization lines, 8 high-ionization lines, and 13 absorption features that are the result of blends of several metal lines that are unresolved at the BOSS resolution. These lines are listed in Tables \[ta:low\] and \[ta:high\] in Appendix \[sec:ew\] and in Table \[ta:blended\]. The uncertainties in their equivalent widths indicate that not all of them are detected at a high confidence level, particularly in the total sample. We now describe how the equivalent widths and uncertainties are evaluated.
Line windows and fitted continuum {#sec:wcalc}
---------------------------------
We select the set of lines described above to measure their equivalent widths. These features were chosen simply from their visual appearance to have a detection in our composite spectrum of the metal sample. These lines, in addition to the first 8 hydrogen Lyman transitions (all of them with wavelengths within the range $920\, {\rm \AA} < \lambda < 3\,000\, {\rm \AA}$), are the ones used for the computation of the fitted continuum described below, as well as for the quasar continuum correction presented in Section \[sec:lines\].
Before equivalent widths can be measured, a set of windows around each line need to be defined over which the absorption fraction is to be computed. We generally choose a total window width of $7\, {\rm \AA}$ centered in the DLA rest-frame line center, which is wide enough to include all appreciable absorption for any unblended line. The line profiles have all nearly the same widths because they are unresolved, thus the equivalent width is, in practice, the only information that can be obtained from these line profiles. We have tested that the measured equivalent widths do not vary significantly under small variations of the window widths. The Ly$\alpha$ and Ly$\beta$ transitions are treated differently because they have a clearly resolved mean absorption profile. We use halfwidths of $40$ and $5\, {\rm \AA}$ for their windows, respectively. In addition, whenever several lines have overlapping windows, we define broader windows which include all the individual windows of the lines in the blend (this is described in detail below, in § \[sec:blended\]).
Before measuring the equivalent widths, we perform a final continuum fitting of the DLA composite spectrum to remove the residual variations left after the corrections discussed in the previous section. This method is similar to that applied by, e.g., [@Pieri2014; @Sanchez2015; @Berg2016]; we proceed in the following manner: for each pixel outside any of the line windows described above, we compute the mean value of the transmission within a $10\, {\rm \AA}$ width window centered on the pixel, excluding any pixels that are inside the absorption line windows (pixels belonging to skylines or outliers have already been removed and do not contribute to the fitting calculation). We then compute a standard cubic-spline fit over the range $900\,-\, 3100 {\rm \AA}$ in the DLA rest-frame, using only one (starting from the first) out of every $15$ of these mean values for the transmission (pixels in the DLA stacked spectrum have widths of $0.3{\, {\rm \AA}}$, therefore the averaging of the transmission is done over about 33 pixels, so all pixels contribute to the determination of this final continuum). The 15 pixels separation between successive points used for the spline fitting corresponds to a distance of $4.5\, {\rm \AA}$ when there are no absorption lines or other effects which can discard pixels in between. We have checked that using 20 or 10 pixels instead of 15 makes no substantial difference. This approach produces a smoother continuum compared to using the averaged flux in all pixels, which produces undesired ‘waves’ over the regions of the absorption lines.
Despite averaging the continuum in $10\, {\rm \AA}$ width windows, this new fitted continuum is still affected by the pixel noise near the window edges. The statistical error that this effect introduces is accounted for with the bootstrap method described in Section \[sec:bootstrap\], but any other possible systematic effects on equivalent widths introduced by our method are not included in our errors. The resulting cubic spline fit is used as the new continuum to calculate equivalent widths and limits on the detection of outliers described in § \[sec:outliers\].
The equivalent width calculation for the case of lines within the [Ly$\alpha$ ]{}window needs special attention. We determine a continuum that includes the [Ly$\alpha$ ]{}absorption, with the goal of being able to measure the equivalent widths of metal lines that are blended with the [Ly$\alpha$ ]{}line. For the purpose of computing the continuum in the [Ly$\alpha$ ]{}line region, we ignore the previously defined metal line windows, and select instead the following windows (in units of Å) that appear to be free of absorption by metal lines in both the total and metal samples stacked spectra: $[1175.67-1188],\,[1196-1198],\,[1202-1205],\,[1209-1235],\,
[1246-1248],\,[1255-1255.67]$. These intervals are selected as a compromise for maximizing the number of points used for the continuum and minimizing points near the metal absorption lines. We then use linear interpolation to connect all the pixels throughout this region, connecting also linearly the points at the window edges. This approach yields our fitted continuum over the region $[1175.67-1255.67\,{\rm \AA}]$. Outside this range, we use the previously described cubic spline fitted continuum.
The final fitted continuum is displayed as the red line in Figure \[fig:lyafit\] over the $1000-1700{\, {\rm \AA}}$ range. Green points are used to fit the linear continuum in the [Ly$\alpha$ ]{}line window.
We stress that our stacked [Ly$\alpha$ ]{}line does not have the shape of a single Voigt profile because it arises from a superposition of DLAs with different column densities. We have not attempted to fit the observed profile by modeling it with a column density distribution for our sample, in view of the zero flux error and other complicated effects (e.g., the cross-correlation of DLAs and the [Ly$\alpha$ ]{}forest). Ignoring this statement and forcing a fit to a single Voigt profile, we obtain a column density ${\rm \log (N_{HI}/cm^{-2})=
20.49}$, which coincides with the mean column density of our sample (with the weights applied in our stacked spectrum) as measured from the individual systems. In future work, it should be interesting to fit the mean profiles of all the Lyman series lines, which should contain valuable information on the distribution of velocity dispersions in the DLA systems.
Equivalent width estimator {#sec:wcalct}
--------------------------
We next fit a Gaussian optical depth line profile to each metal line within its window, $$\label{eq:fitting}
F = C_f\, {\rm exp} \left[ -b\, {\mathrm{exp}\left(\frac{-(\lambda-\lambda_{0})^2}{2a^2}\right)}
\right] ~,$$ where $b$ and $a$ are two free parameters, $\lambda$ is the pixel wavelength in the DLA rest-frame, $\lambda_0$ is fixed to the known central wavelength of the line, and $C_f$ is the value of the fitted continuum in each pixel. We perform a standard least-squares fit to the measured $F$ in the pixels of each line window with the two free parameters $a$ and $b$.
In practice, the parameter $a$ in Eq. \[eq:fitting\], reflecting the width of the lines, is essentially determined by the spectrograph resolution of BOSS, except in a few cases of blended lines. The BOSS resolution depends smoothly on the observed wavelength, but once this smooth variation is taken into account, the resolution should not vary among different metal lines. Therefore, the accuracy of the fit to the line equivalent widths should improve if we impose a fixed width parameter $a$ on the lines, assuming that the width is not affected by variable levels of saturation of the absorption lines. To examine the variation of $a$ with wavelength, Figure \[fig:param\] displays the values of $a$ obtained for all the metal lines, as a function of their rest-frame wavelength, for the case of the total sample. Yellow dots indicate blended lines, or lines that are apparently very weak and strongly affected by noise, so that they are deemed likely to present deviations of their width from any smooth dependence. In cases of lines forming part of an atomic doublet, the lines are jointly fitted and are required to have the same value of $a$, but different values of $b$.
![Fitted values of the $a$ parameter for each absorption line in the stacked spectrum for the total sample. Black data points are used to compute the linear regression, shown as the red line. Yellow points are considered outliers, affected by blended lines or low signal-to-noise ratio, and not used. The values for the $a$ parameter can be considered as an estimation of the spectrograph resolution assuming that the lines are unresolved. []{data-label="fig:param"}](param.pdf){width="48.00000%"}
It is apparent from Figure \[fig:param\] that there is a smooth increase of the width parameter $a$ with wavelength, with a small scatter for the black points corresponding to lines that are not affected by blends or a very weak signal-to-noise ratio. There is no evidence for any difference in the width between low-ionization and high-ionization lines, which might have indicated a different contribution from a physical velocity dispersion to the measured widths. We fit a linear regression to the values of $a$ as a function of wavelength using the black points only (with each point weighted equally), and we obtain the result $$a = 0.23\times 10^{-3} \lambda +0.25 \, {\rm \AA} ~,
\label{eq:afit}$$ shown as the red line in Figure \[fig:param\]. The same procedure for the metal sample yields a linear regression of the form $$a = 0.25\times 10^{-3} \lambda +0.18 \, {\rm \AA} ~.
\label{eq:afit76}$$ These equations can be considered as an estimation of the spectrograph resolution assuming that the lines are unresolved. The difference between the two samples is mostly due to the better accuracy of the redshifts in the metal sample, which makes the lines appear slightly narrower. Having tested that the deviations of the $a$ parameter from this linear regression arise from noise and not from physical differences among the lines, we now repeat the fits to each line with Eq. \[eq:fitting\], but keeping $a$ fixed to these linear regressions and fitting only the $b$ parameter. We then integrate the area below the continuum represented by the function in Eq. \[eq:fitting\] to compute the corresponding equivalent width for each line.
In some cases, a group of absorption lines are close enough for their wings to overlap, but the absorption maxima are still well separated. In this case, we use a common window including the windows of all the blended lines and we measure their equivalent widths in a single joint fit. The optical depths modeled as Gaussians are added, or equivalently, the transmissions from each line are multiplied, to obtain the total profile. The equivalent widths of these lines, marked with a superscript denoting overlap, are listed together with all the other lines for the total and metal samples in Tables \[ta:low\] and \[ta:high\] in Appendix \[sec:ew\]. Their errors are computed using the bootstrap method described in § \[sec:bootstrap\].
Figures \[fig:fit\] and \[fig:fit2\] in Appendix \[sec:linefit\] present the fit for all the absorption features analysed in the total sample. These figures display the line window, the continuum and the fitted absorption profile. In cases of overlap, each individual line is indicated separately in addition to the total absorption profile. Below, we discuss the case for lines that are strongly blended.
Strongly blended absorption lines {#sec:blended}
---------------------------------
[cccccc]{}\
Total W &Lines &$f$ &Fitted $W$ &Model $W$ &Inferred $W$\
$0.393\pm0.019$ & & & & &$0.524\pm0.009$\
&OI$\lambda988.58$ &0.0005 &$0.000\pm0.118$ &$0.034\pm0.002$ &\
&OI$\lambda988.65$ &0.008 &$0.283\pm0.135$ &$0.200\pm0.004$ &\
&OI$\lambda988.77$ &0.047 &$0.131\pm0.073$ &$0.330\pm0.004$ &\
$0.271\pm0.021$ & & & & &\
&NIII$\lambda989.80$ &0.123 &$0.089\pm0.108$ & &
------------------------------------------------------------------------
\
&SiII$\lambda989.87$ &0.171 &$0.191\pm0.110$ &$0.275\pm0.002$ &\
$0.157\pm0.019$ & & & & &\
&FeII$\lambda1062.15$ &0.003 &$0.004\pm0.012$ & $0.0115\pm0.0004$ &\
&SIV$\lambda1062.66$ &0.049 &$0.020\pm0.011$ & &$0.024\pm0.012$\
&FeII$\lambda1063.18$ &0.055 &$0.122\pm0.010$ &$0.110\pm0.003$ &\
&FeII$\lambda1063.97$ &0.005 &$0.013\pm0.010$ &$0.0113\pm0.0004$ &\
$0.115\pm0.009$ & & & & &\
&FeII$\lambda1121.97$ &0.029 &$0.065\pm0.007$ &$0.077\pm0.002$ &\
&FeIII$\lambda1122.53$ &0.054 &$0.051\pm0.007$ & &$0.042\pm0.008$\
$0.178\pm0.014$ & & & & &$0.133\pm0.010$\
&FeII$\lambda1133.67$ &0.006 &$0.031\pm0.011$ &$0.0203\pm0.0007$ &\
&NI$\lambda1134.17$ &0.015 &$0.013\pm0.020$ &$0.0214\pm0.0012$ &\
&NI$\lambda1134.41$ &0.029 &$0.075\pm0.020$ &$0.039\pm0.002$ &\
&NI$\lambda1134.98$ &0.042 &$0.063\pm0.011$ &$0.054\pm0.003$ &\
$0.071\pm0.010$ & & & & &$0.071\pm0.004$\
&FeII$\lambda1142.36$ &0.004 &$0.023\pm0.006$ &$0.0140\pm0.0005$ &\
&FeII$\lambda1143.23$ &0.019 &$0.048\pm0.005$ &$0.057\pm0.002$ &\
$0.411\pm0.006$ & & & & &$0.436\pm0.006$\
&SIII$\lambda1190.21$ &0.024 &$0.079\pm0.015$ &$0.056\pm0.013$ &\
&SiII$\lambda1190.42$ &0.292 &$0.345\pm0.012$ &$0.393\pm0.002$ &\
$0.296\pm0.007$ & & & & &$0.293\pm0.003$\
&NI$\lambda1199.55$ &0.133 &$0.135\pm0.006$ &$0.137\pm0.005$ &\
&NI$\lambda1200.22$ &0.087 &$0.104\pm0.008$ &$0.104\pm0.004$ &\
&NI$\lambda1200.71$ &0.043 &$0.063\pm0.007$ &$0.061\pm0.003$ &\
$0.623\pm0.008$ & & & & &$0.688\pm0.008$\
&SII$\lambda1259.52$ &0.017 &$0.083\pm0.006$ &$0.095\pm0.005$ &\
&SiII$\lambda1260.42$ &1.180 &$0.553\pm0.006$ &$0.550\pm0.003$ &\
&FeII$\lambda1260.53$ &0.024 &$-$ &$0.082\pm0.002$ &\
$0.012\pm0.007$ & & & & &\
&TiII$\lambda1910.61$ &0.104 &$0.000\pm0.004$ &$0.006\pm0.003$ &\
&TiII$\lambda1910.95$ &0.098 &$0.012\pm0.006$ &$0.006\pm0.003$ &\
$0.630\pm0.007$ & & & & &\
&CII$\lambda1334.53$ &0.128 &$0.597\pm0.006$ &$0.597\pm0.014$ &\
&CII\*$\lambda1335.71$ &0.115 &$0.037\pm0.005$ & &$0.037\pm0.005$\
$0.035\pm0.008$ & & & & &\
&ZnII$\lambda2026.14$ &0.501 &$0.033\pm0.009$ & &$0.025\pm0.007$\
&CrII$\lambda2026.27$ &0.001 &$0.002\pm0.009$ &$0.00030\pm0.00009$ &\
&MgI$\lambda2026.48$ &0.113 &$0.000\pm0.003$ &$0.010\pm0.002$ &\
$0.034\pm0.007$ & & & & &\
&CrII$\lambda2062.23$ &0.076 &$0.031\pm0.008$ &$0.022\pm0.007$ &\
&ZnII$\lambda2062.66$ &0.246 &$0.003\pm0.006$ & &$0.011\pm0.007$\
There are several blended absorption lines that do not present separate absorption maxima. We designate these groups ‘strongly blended lines’. These groups and their individual blended lines are listed in Table \[ta:blended\].
We use the same procedure described above to perform a joint fit to all the lines belonging to a blend: we fix all central wavelengths and Gaussian widths, and only allow the amplitude of each line to vary. Any lines that may be part of a blend with an equivalent width expected to contribute less than $1\%$ to the total equivalent width are ignored. In these fits, the only reliable measurement is usually the total equivalent width of each blended group, which is listed in the first column with its bootstrap uncertainty. The individual equivalent widths of each line (with wavelengths and oscillator strengths listed in the second and third columns, respectively), with highly correlated and larger errors, are listed in the fourth column. The fifth column reports a modeled estimate of individual equivalent widths, according to a theoretical model, described below in § \[sec:model\], that uses measurements of other absorption lines of the same species. Finally, the sixth column gives an inferred equivalent width for one of the blended lines once the modeled lines are taken into account, and in cases where all the lines in a blend are modeled, it lists the inferred total equivalent width of the blend. These results will be discussed in detail in Section \[sec:blend\].
We wish to warn here, however, that the results for the fitted equivalent widths of blended lines in the fourth column are subject to an important systematic error: we are simply assuming that the transmission of the individual lines modeled with the profile in Eq. \[eq:fitting\] can be multiplied to fit the entire blend. This situation is actually not true because the absorbing components in each individual line are not independent, but they have a highly correlated velocity structure in each DLA. When the blended lines are very close together (for example, for the first blend in Table \[ta:blended\] for OI lines), the combined equivalent widths of the various lines may hardly increase in DLAs with very narrow lines, which are highly saturated even if the blended line at the BOSS resolution appears far from saturation, and may increase to some extent in DLAs with broader velocity profiles. It is impossible to make a reliable estimate of how the ‘fitted’ equivalent widths of individual lines should be added to yield the total blend equivalent width without a complete model of the absorbing subcomponents width distribution and velocity correlations. This effort would be much more ambitious than the simple model presented below in § \[sec:curve\] for modeling single line equivalent widths in terms of mean column densities. We therefore choose to use the simple assumption of multiplying transmissions of all the individual lines. This approach produces approximately correct results only for blends of lines that are sufficiently separated or weak that they do not have frequently overlapping absorbing components that are saturated at full resolution.
The fits to the blended absorption profile in these cases are also presented in Figures \[fig:fit\] and \[fig:fit2\]. Due to the systematic errors mentioned above, individual line profiles are not separately shown for these strongly blended lines.
Bootstrap error computation {#sec:bootstrap}
---------------------------
The errors on the line equivalent widths cannot be obtained from the errors on the observed flux in individual pixels because they are often dominated by continuum uncertainties. We therefore use a bootstrap method: we randomly split our DLA sample (both the total and the metal one) into $100$ subsets containing roughly the same number of DLA candidates. We then generate 1000 new samples by randomly selecting $100$ of these subsets, allowing for repetition, and we recompute the composite spectrum, the continuum fit and the equivalent width calculations for each new sample. We use the same values of $a(\lambda)$ that were obtained in equations \[eq:afit\] and \[eq:afit76\], and we obtain the errors from the standard deviation of each metal line equivalent width among the bootstrap samples.
Dependence of equivalent widths on $\rm{N_{HI}}$ {#sec:nhi}
================================================
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We now examine the dependence of the mean equivalent width of the metal lines on the property that can be best measured for each individual DLA: the hydrogen column density ${ N_{\rm HI}}$, which determines the damped profile of the hydrogen [Ly$\alpha$ ]{}line. We divide the total and the metal samples into five intervals in $\log {\rm (N_{\rm HI}/cm^{-2})}$ chosen to contain a similar number of DLAs: $[20.0-20.13],(20.13-20.30],(20.30-20.50],(20.50-20.80],(20.80-22.50]$. The logarithm of the mean values of the column density in these ranges are $20.06$, $20.22$, $20.40$, $20.65$ and $21.13$, respectively. For each interval, we redo the previous calculations: a composite spectrum of transmission fraction is obtained for the DLAs that have a column density in the specified range, the continuum is fitted, and equivalent widths of our list of metal lines with bootstrap errors are obtained. The $a$ parameter in Eq. \[eq:fitting\] is kept fixed to the same values used for the entire samples, assuming that it is independent of $N_{\rm HI}$. The bootstrap errors are computed as before, using only the DLAs in every bootstrap subset which are in the $N_{\rm HI}$ bin of interest when doing the stacking. The bootstrap subsets are large enough to contain similar numbers of DLAs in each column density bin. Results for the mean equivalent widths are tabulated in Tables \[ta:lownhi24\] - \[ta:highnhi76\] in Appendix \[sec:ew\], separately for low-ionization and high-ionization species, and for the total and metal samples. The intermediate species CIII, SiIII, AlIII and SIII are included in the table of low-ionization species owing to the fact that the kinematic properties for these elements resemble those of low-ionization species [@Wolfe2005].
The dependence of the mean equivalent widths on column density is presented in Figure \[fig:nhi\]. Several low-ionization species are shown in the top panels (total sample on the left, metal sample on the right). Points are connected with dotted lines to help visualize the trend with ${N_{\rm HI}}$ for each species (they are slightly shifted horizontally from the true mean column density values specified previously to avoid overlapping errorbars). A general trend of increasing $W$ with increasing $N_{\rm HI}$ is apparent, as expected. If the mean metallicity and dust depletion in DLAs do not vary substantially with column density, hydrogen is mostly atomic, and the low-ionization species displayed in Figure \[fig:nhi\] are the dominant ionization stage of the respective elements, then we expect the metal column densities to increase linearly with $N_{\rm HI}$. This prediction, however, cannot be tested directly because a large fraction of lines may be saturated, and the degree of saturation depends on a complex distribution of velocity dispersion, metallicity and multi-component structure of the absorption systems.
The bottom panels show the dependence of the mean equivalent widths on column density for three high-ionization species, SiIV, OVI and CIV, in the total and metal samples (left and right panels, respectively), and for an intermediate species, AlIII. We also present the behaviour of the FeII$\,\lambda1608$ line for comparison (in soft grey). The total sample demonstrates that there is also an increase in the mean equivalent width with column density, in all cases. However, the increase is much smaller than for low-ionization lines. Over the column density range $20 < \log ({ N_{\rm HI}/{\rm cm}^{-2}}) < 20.7$, the mean equivalent width of the CIV and SiIV lines increases by a factor $\sim 1.25$, whereas the increase for relatively weak lines of low-ionization species in the top panels over the same range is a factor $\sim 3$, and even the strongest low-ionization line, CII (which is most saturated), increases by a factor $\sim 1.8$. The AlIII line increases by a factor that is intermediate between the low-ionization and high-ionization cases.
The metal sample presents a different behavior: the mean equivalent width is essentially independent of column density for the high-ionization lines, and has a weaker increase with $N_{\rm HI}$ than in the total sample for the low-ionization ones. There are two possible reasons for this difference. First, the DLAs in the metal sample are selected to be the ones with measured redshifts from the metal lines, and therefore where the metal lines have been individually detected in the BOSS spectra. This procedure selects DLAs with strong metal lines, and also spectra of high signal-to-noise ratio. The low column density DLAs are included in the metal sample with a lower frequency than the high column density ones, with an important selection in favor of systems with strong metal lines (either because of high metallicity, or because of high velocity dispersion which reduces the degree of line saturation). The second reason is that some fraction of the DLAs at low column density may be false detections, where a combination of noise and the presence of a cluster of Ly$\alpha$ forest lines may be confused with a DLA in low signal-to-noise ratio spectra. The low column density DLAs should have a higher impurity fraction, or fraction of false systems, and this impurity fraction is significantly reduced when selecting systems that have associated metal lines detected. Therefore, the $\sim$ 25% increase in the high-ionization mean equivalent width with $N_{\rm HI}$ may in part be real for the total sample (not suffering from metal selection effects), but may also be due to a worsening sample purity at low column density for the total sample. These effects illustrate how special care must be taken in the sample selection for evaluating mean properties of the DLA population.
Even though the metal sample results in a composite spectrum where weak lines can be detected at a greater statistical significance, it will not be used in the rest of this paper because it does not produce results that can be easily corrected for the sample selection effects. The question of the effect of the sample impurity (i.e., the rate of false DLAs in the total sample) needs to be addressed with simulations of DLA catalogs from mock spectra, and is left to be analyzed in future work when these simulations are available.
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Additional interesting information can be obtained from the mean equivalent width ratio of the doublets of CIV and SiIV. The equivalent width ratio stays constant with column density near a value of $1.4$ to $1.5$ for both CIV and SiIV. This factor is between the values of $2$ and $1$, corresponding to the two extreme cases of completely optically thin and completely saturated lines, respectively. The fact that this ratio remains constant as the mean equivalent width increases suggests that the velocity dispersion, or the mean number of subcomponents, should be increasing with column density, although our uncertainties are still too large to place much confidence on this interpretation. The ratio of $\sim 1.5$ indicates that the absorbers have a mixture of weak, unsaturated components, and strong saturated ones, that contribute about equally to the total equivalent widths. The doublet equivalent width ratio of OVI appears to be larger, but the errors here are also larger and systematics due to the blending of the weaker line of the doublet with CII and OI lines are likely present.
Dependence of equivalent widths on $z$ {#sec:z}
======================================
We here explore the evolution of the mean metal line equivalent widths with redshift of the DLAs. We split the total and the metal samples into five DLA redshift intervals, $[1.9-2.24],(2.24-2.4],(2.4-2.7],(2.7-3.0],(3.0-6.4]$, with mean redshift values $2.12$, $2.32$, $2.54$, $2.84$ and $3.50$, respectively. The mean values of the column densities for each bin are $20.50$, $20.50$, $20.50$, $20.49$ and $20.46$, respectively, showing little differences between them. We repeat the equivalent width and uncertainty calculations we did for the case of column densities, but this time for each redshift bin. The $a$ parameter in Eq. \[eq:fitting\] is kept fixed to the same values used for the entire samples, assuming that it is independent of DLA redshift. The mean equivalent widths in every redshift bin are tabulated in Tables \[ta:lowz24\] - \[ta:highz76\] in Appendix \[sec:ew\], separately for low-ionization and high-ionization species, and for the total and metal samples.
Figure \[fig:z\] displays the evolution of the mean equivalent widths with redshift. The top panels show several low-ionization species. The points are slightly shifted horizontally from their original positions to clarify the visualization. A general decrease of equivalent width by a factor $\sim1.3-1.5$ from $z\sim2.1$ to $z\sim3.5$ is apparent for the total sample ([*left panel*]{}). The nearly equal values of the mean column density at every redshift bin confirm that the observed trend is not driven by changes in $N_{\rm HI}$. However, the effect of false positive DLA detections, which result in lower equivalent widths, might be present, specially at the highest $z$ bins. The evolution of the metal sample, which should be less affected by false positives, presents a slightly smoother decrease, by a factor $\sim1.1-1.2$ in the same redshift range ([*right panel*]{}).
The bottom panels show the dependence of the mean equivalent widths on $z$ for two high-ionization species, SiIV and CIV, and for an intermediate species, AlIII. We also present the behaviour of the FeII$\,\lambda1608$ line for comparison (in soft grey). The doublet of OVI is not shown as its rest-frame wavelength is only covered by the three upper redshift bins which, in turn, are those most affected by noise. Here, as for the low-ionization species in the upper panels, the metal sample ([*right panel*]{}) suggests a smoother decrease with redshift than the total sample ([*left panel*]{}). However, for both samples, the equivalent width decreases by a factor $\gtrsim 1.5$ within the redshift range $z\sim2.1$ to $z\sim3.5$. Whether a different evolution between low- and high-ionization species exists or is an artifact driven by noise and/or systematics is unclear, and we defer more detailed examinations to future work.
Theoretical Model for line saturation {#sec:model}
=====================================
As previously described, the absorption profiles of the metal lines are mostly determined by the instrumental spectrograph resolution. The absorption system components are often much narrower than the BOSS resolution, and can therefore be saturated even though they appear to be weak [@York2000]. Although a fraction of DLAs are known to have velocity dispersions that are comparable to the BOSS resolution [@Prochaska1997], this intrinsic line width likely contributes only to extend the wings of the mean absorption profile that can be seen in the strongest lines (see, e.g., the CIV profile in Figure \[fig:fit2\]). Deconvolving the BOSS spectral point spread function from the mean profile is difficult because of its wavelength and fiber-to-fiber dependence, and inaccuracies in its determination. We therefore use only the mean equivalent widths of the lines as the quantity that can be determined from the BOSS stacked spectra we have created.
The physical quantities that the equivalent width depends on are the column density and the velocity distribution of each species. The velocity distribution is known to be generally complex, because several absorbing components are often resolved in the metal line profiles of DLAs [@Prochaska1997; @Zwaan2008], which means that there is not a simple relation between mean equivalent widths and column densities that can be expressed in terms of a single velocity dispersion, as for a Voigt profile. The scenario that is usually invoked to explain the observed absorption profiles is that a number of gas clouds are orbiting inside a halo or thick disk [@Haehnelt1998; @Wolfe1998; @McDonald1999], which randomly intercept the line of sight to the quasar. Every absorbing system is characterized by a halo velocity dispersion, which describes the random motions among the clouds seen as absorbing components in the spectra, and an internal cloud velocity dispersion, which can be partly thermal and partly turbulent, determining the width of the individual absorbing components. We shall use this scenario to construct a simple model for the relation of the mean equivalent width and column density for a large sample of DLAs similar to the one we are using here.
With this purpose in mind, we analyse in detail the transitions of FeII and SiII for the total sample, which are the two species with the largest number of measurable absorption lines, $12$ for FeII and $5$ for SiII (except for NiII, which has 6 transitions, but they are weaker lines with a narrower range of oscillator strengths), most of which are strong enough to yield a reliable measurement of the mean equivalent width, $W$. We plot in Figure \[fig:curve\] the dimensionless ratio $W/\lambda$ as a function of the product $f\lambda$ for each one of these transitions, where $f$ is the oscillator strength of the line at wavelength $\lambda$. For a completely optically thin absorption line, the equivalent width is given by $${W\over \lambda} = \pi f \lambda r_e N ~,$$ where $r_e = e^2/(m_e c^2)$ is the classical electron radius, $e$ is the electron charge and $m_e$ the electron mass. Therefore, in the optically thin regime, we expect $W/\lambda \propto f\lambda$ for the mean of a DLA sample with any distribution of column densities. As an increasing fraction of the DLA components become optically thick with increasing $f$, the mean relation should flatten until $W/\lambda$ becomes nearly constant with $f\lambda$ at a value determined by the mean maximum velocity range that is covered by the absorbing components in a DLA. This is indeed the behavior seen in Figure \[fig:curve\]. The precise functional dependence of $W/\lambda$ on $f\lambda$ cannot be reliably predicted from theory, because it depends on the detailed velocity and column density distribution and the internal velocity structure of the DLA absorbing components.
However, we can argue that all the low-ionization species measured in DLAs should follow the same functional form of $W/\lambda$ versus $f\lambda$, except for a horizontal rescaling that reflects the element abundance. This relation follows from assuming that all the low-ionization species have the same distribution of velocities in the DLA sample we are using, and that different ionization corrections do not lead to significant differences, so that they all have the same distribution of optical depths within the velocity range of the absorbing systems except for the rescaling reflecting the abundance. In general, ionization corrections in DLAs are found to be $<0.2$ dex [e.g., @Howk1999; @Vladilo2001; @Prochaska2002; @Kisielius2015], supporting the view that differences in the shape of the optical depth distribution among metal species should be minimal. A model is proposed below for the shape of $W/\lambda$ versus $f\lambda$, which is displayed as the blue curves in Figure \[fig:curve\]. This simple model allows us to infer mean column densities from the measured mean equivalent widths, up to the systematic errors arising from the model assumptions. We shall generally assume that the model can be applied with the same parameters to all low-ionization species. We will also apply the model to high-ionization species allowing for different fit parameters, since these absorbers are believed to reside in different regions and have different velocity distributions than the low-ionization species [e.g., @Wolfe2005]. We consider, however, this model to be less reliable for high-ionization species because of the lack of a test with many lines from the same species with different values of $f$, similar to the one provided by FeII and SiII for low-ionization.
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Before describing this model, we should address the large differences in the error bars of different lines in Figure \[fig:curve\]. Large errors are sometimes caused by blends with other absorption features that are difficult to separate (e.g., the SiII$\,\lambda1020$ transition appearing at $f\lambda=17.1\, {\rm \AA}$ in the right panel is blended with the Ly$\beta$ transition), or when the lines are very weak. The leftmost FeII line has an expected $W$ value smaller than its uncertainty, so the lower edge of the error bar is replaced by an arrow.
Modeling equivalent widths as a function of oscillator strength {#sec:curve}
---------------------------------------------------------------
For any metal line centered at wavelength $\lambda_0$, we define the rest-frame equivalent width in units of velocity as: $$W_v = {c\, W\over \lambda_0} = {c\over \lambda_0}
\int d\lambda\, [1- {\rm exp}(-\tau(\lambda))] ~,$$ where the integral is performed over a rest-frame wavelength interval around $\lambda_0$ that contains the whole absorption line. We define the [*linearized equivalent width*]{} as $$\label{eq:wtau}
{ W_\tau\over c} = \int {d\lambda\over\lambda_0}\, \tau(\lambda)
= \pi f \lambda_0 r_e N ~,$$ Although the mean of $W_\tau$ is the quantity that is directly related to the mean column density, only the mean of $W_v$ is measured from our BOSS stacked spectra.
For absorption lines that contain a single absorbing component with a Gaussian velocity dispersion (arising from thermal motions or from a turbulent velocity dispersion that also has a Gaussian distribution), the relation between $W_v$ and $W_\tau$ is the well known Voigt curve of growth [@Goody1964]. For a velocity dispersion $\sigma$, when $W_\tau \gg W_v$, we have $$W_v \simeq 2\sigma \sqrt{2\log [W_\tau/(\sqrt{2\pi}\sigma)]} ~.$$ The mean equivalent width resulting from the stacked metal lines of many DLAs is not related in the same simple way to the mean column density because each absorption system has a different distribution of optical depths. We search for a simple, physically motivated fitting formula to adjust the observed mean values of $W_v$ of the low-ionization lines as a function of the values of $W_\tau$ that are theoretically predicted from column densities.
We assume the absorption lines are composed of subcomponents with an internal velocity dispersion $\sigma$, given by $\sigma^2 = \sigma_0^2 (m_H/m_i) + \sigma_h^2$, where $\sigma_h$ is the turbulent (or hydrodynamic) velocity dispersion, assumed to be the same for all the lines of all the species, and $\sigma_0$ is the thermal velocity dispersion of hydrogen, which we shall fix to $\sigma_0=10{\, {\rm km \, s}^{-1}}$, corresponding to the typical equilibrium temperature of photoionized gas $T\simeq 10^4\, {\rm K}$. The mass of the atom of the species of each line is $m_i$, and $m_H$ is the hydrogen mass.
We assume that the subcomponents of a certain absorption line are contained within an interval of velocity width $V$, which is wide enough so that any absorption outside this interval due to the line can be neglected. Let the probability distribution of the optical depth at any pixel in the spectrum within this interval of width $V$ be $\psi(\tau)$. In other words, $\psi(\tau)\, d\tau$ is the fraction of pixels in the intervals of width $V$ around the central wavelength of the line in any DLA that have an optical depth $\tau$ within the range $d\tau$. The two measures of the equivalent widths are given by: $$\label{eq:wprob}
W_\tau = V \int d\tau\, \psi(\tau)\, \tau ~,$$ and $$W_v = V \int d\tau\, \psi(\tau)\, \left[ 1 - e^{-\tau} \right] ~.$$ As a model for the probability distribution of the optical depth, we choose the function $\psi(\tau)=A/\tau$ within a certain range $\tau_{min} < \tau < \tau_{max}$, and $\psi=0$ outside this range. The normalization constant is $A=1/\log(\tau_{max}/\tau_{min})$. This definition is chosen as a simple approximation that assumes that $\tau$ varies over a broad range in a scale-invariant way. More realistically, there should be a continuous distribution of $\tau$ reaching down to zero, but we simply assume that $\tau_{min}$ is small enough to neglect this detail. Let us now consider absorbers with a fixed $W_\tau$, i.e., a fixed column density of a certain species absorption line. The maximum optical depth is obtained when only one Gaussian absorbing subcomponent accounts for all the absorption, with $$\label{eq:taumax}
\tau_{max}= {W_\tau\over \sqrt{2\pi} \sigma} ~.$$ When several subcomponents are present, each one must have a smaller central optical depth to produce the same $W_\tau$, and our model assumes the distribution is flat in $\log\tau$. We now use the fact that the average number of absorbing subcomponents in DLAs is of order unity, to require that the mean $W_\tau$, from Eq. \[eq:wprob\], is equal to that due to the case of a single component reaching the maximum optical depth $\tau_{max}$, i.e., $$V A (\tau_{max} - \tau_{min})= W_\tau= \sqrt{2\pi} \sigma \tau_{max} ~.$$ We assume that the ratio $\tau_{max}/\tau_{min}$ is constant, independent of $W_\tau$. We then obtain the following relation between $V$, $\tau_{max}$ and $\tau_{min}$: $$\label{eq:sig1}
V = { \sqrt{2\pi} \sigma\, \log(\tau_{max}/\tau_{min}) \over
1-\tau_{min}/\tau_{max} } \equiv \sqrt{2\pi} \sigma_1 ~,$$ where $\sigma_1$ is defined by the last equality to replace $V$, and can be interpreted as a rough estimate of the velocity dispersion of the subcomponent absorbers, or DLA clouds, within their host halo.
Finally, we find for the equivalent width $W_v$ the expression $$\label{eq:Wfinal}
{W_v\over W_\tau} = {\log(\tau_{max}/\tau_{min}) -
\int_{\tau_{min}}^{\tau_{max}} d\tau\,
e^{-\tau} /\tau \over \tau_{max} - \tau_{min} } ~.$$
The model relating $W_v$ to $W_\tau$ has only two free parameters: $\sigma_h$ and $\sigma_1$. The thermal dispersion is fixed to $\sigma_0
= 10\, {\rm km\,s^{-1}}$. The fitting formula provides a value of $W_v$ for any input value of $W_\tau$: Eq. \[eq:taumax\] first reveals the value of $\tau_{max}$, the value of $\tau_{min}$ is derived from Eq. \[eq:sig1\], and we then obtain $W_v$ from Eq. \[eq:Wfinal\]. These two free parameters can be fixed to be the same for all the low-ionization species, if we believe that the subcomponent kinematics are an invariant characteristic of the absorption profiles that has little dependence on the column density. Under this assumption, the dependence of $W_v$ on $W_\tau$ is unique, and represents both the variation with absorption lines of different $f$ for fixed column density, and the variation with column density for a fixed line. In addition, once $\sigma_h$ and $\sigma_1$ are determined, the relation for each new species depends on one additional parameter only: the ratio of its column density to the hydrogen one.
![Parameter posterior probabilities of the joint fit of FeII and SiII transitions for the estimation of $\sigma_h$ and $\sigma_1$ and the two mean column densities in the total sample. The $\chi^2$ contours are at 0.5, 1, 1.5 and 2$\sigma$ levels (in some panels the 0.5$\sigma$ contour is too small to be visible). The blue dots and lines, plotted over the contours, indicate the best fit. Note the large uncertainty of the $\sigma_1$ parameter. This figure has been produced using the open source code *triangle.py* from [@triangle].[]{data-label="fig:triangle"}](triangleFeSi.pdf){width="48.00000%"}
We obtain the best fit parameters for low-ionization species by computing a joint fit to the above mentioned FeII and SiII transitions, with the four free parameters $N_{\rm FeII}$, $N_{\rm SiII}$, $\sigma_h$ and $\sigma_1$, which minimizes the sum of the two $\chi{^2}$ values. The column densities $N_{\rm FeII}$ and $N_{\rm SiII}$ are the mean values for our DLA sample used directly in Eq. \[eq:wtau\] to compute $W_\tau$ for any absorption line. The best fit and parameter uncertainties are calculated with the open-source code *EMCEE* [@emcee2013], using the Monte Carlo Markov Chain method (MCMC). Figure \[fig:triangle\] displays the one and two dimensional projections of the posterior probability distributions of the parameters; the contours are at 0.5, 1, 1.5 and 2$\sigma$ levels.
The best fit values and uncertainties, indicating the 16th and 84th percentiles (the mean of the two) in the marginalized distribution for $\sigma_1$ (for $\sigma_h$ and $\log N$), are $\sigma_h = 8.56\pm0.22\, {\rm km\,s^{-1}}$, $\sigma_1 = 90^{+205}_{-16}\, {\rm km\,s^{-1}}$, $\log N_{\rm FeII} = 14.538 \pm 0.011$ and ${\log N_{\rm SiII}} = 15.087 \pm 0.012$. The fit reproduces the trend of the measured points, as seen in Figure \[fig:curve\] where our best fit model is indicated by blue lines. The total ${\rm \chi^2}$ value of the fit is $\chi^2=14.98$, for an expected value of $13$ (17 data points and four free parameters; some of the oscillator strengths have substantial uncertainties, which are not taken into account and may contribute to some increase in the value of $\chi^2$). The halo velocity dispersion $\sigma_1$ is the parameter with the largest uncertainty: any value in the range $\sim 70 - 300\, \rm{km\,s^{-1}}$ is essentially equally good, with no correlation with any of the other parameters. Our formula provides a good fit as long a turbulent velocity dispersion of individual components provides rough equipartition of the thermal and turbulent kinetic energies ($\sigma_h\simeq 8.6\, {\rm km\,s^{-1}}$), and the well-determined normalizations of the curves in Figure \[fig:curve\] correspond to the values of $N_{\rm FeII}$ and $N_{\rm SiII}$.
For high-ionization species, we adopt the same assumptions as above and perform a joint fit to the two doublets CIV and SiIV, which have unblended, strong absorption lines. We use these species because they have the strongest lines, and are believed to be produced mostly by photoionization [e.g., @Fox2007a]. We assume the same fixed value for $\sigma_0=10\,{\rm km\,s^{-1}}$ because these species are thought to arise in warm gas regions with temperatures ${\rm T\sim10^4\, K}$ [@Lehner2008]. We obtain the values $\sigma_h = 14.51\pm0.64\, {\rm km\,s^{-1}}$, $\sigma_1 = 185^{+112}_{-85}\, {\rm km\,s^{-1}}$, $\log N_{\rm CIV} = 14.396 \pm 0.019$ and ${\log N_{\rm SiIV}} = 13.783 \pm 0.016$. The larger value for $\sigma_h$ compared to the low-ionization species is indicative of a more violently turbulent environment for the high-ionization species. Changing the thermal velocity dispersion to $\sigma_0 = 30 {\, {\rm km \, s}^{-1}}$ reduces $\sigma_h$ by $\sim 15\%$ and increases $\sigma_1$ by $\sim 5\%$, with variations of less than $1\%$ on the column densities.
As explained in the previous section, the use of a curve of growth for fitting the transitions of a given species and obtaining the column density is not accurate for our composite spectra, due to the large number of components and systems contributing to the mean absorption features. Although a detailed assessment of our theoretical model is beyond the scope of this work, we perform a simple comparison with the curve of growth approach, since the latter is broadly used in absorption line studies, e.g., in the recent work by [@Noterdaeme2014]. We use the publicly available package `linetools`[^1] to find the best-fit parameters of the curve of growth, column density, $N\,{\rm (cm^{-2})}$, and velocity dispersion, $b\,{\rm (km/s)}$, applied to the species for which we have more transitions, i.e., FeII, SiII and NiII. The [*dashed green lines*]{} in Figure \[fig:curve\] show the resulting curves for FeII and SiII. The best fit parameters for the three species are $\log N_{\rm FeII} = 14.452 \pm 0.010$, $\log N_{\rm SiII} =
14.957 \pm 0.095$ and $\log N_{\rm NiII} = 13.843 \pm 0.072$, with Doppler parameters $b_{\rm FeII} = 24.04 \pm 0.73$, $b_{\rm SiII} = 26.18 \pm 1.39$ and $b_{\rm NiII} = 3.21 \pm 0.49$. These results are significantly different from the ones obtained with our model, which allows for a distribution of component structure and velocity dispersions.
Mean column densities of low- and high-ionization species {#sec:deduction}
---------------------------------------------------------
We now assume $\sigma_h$ and $\sigma_1$ to be constant for all the low- and high-ionization species, fixing their values to the best fit obtained for the FeII and SiII transitions, $\sigma_h = 8.56\, \rm{km\,s^{-1}}$ and $\sigma_1 = 90\, \rm{km\,s^{-1}}$, and for CIV and SiIV, $\sigma_h = 14.51\,
\rm{km\,s^{-1}}$ and $\sigma_1 = 185\, \rm{km\,s^{-1}}$, respectively. For every species we fit the curve of this fixed model with the mean column density $N$ as the only free parameter, i.e., we assume the relation of $W_v$ and $W_\tau$ is fixed, and fit only the constant of proportionality relating $W_\tau$ to $f\lambda$, which is proportional to the mean column density of the species in our DLA sample through Eq. \[eq:wtau\]. This approach yields a ratio of the ion abundance to that of HI considering the logarithm of the mean hydrogen column density of the total sample, $\log {\bar N}_{\rm HI}=20.49$. If the ionization corrections can be neglected (i.e., assuming that the fraction of the species in question is the same as the neutral hydrogen fraction), this result is equal to the element abundance compared to hydrogen.
The results of this model are presented in the middle column in Table \[ta:N\]. Errors include equivalent width measurement uncertainties only, computed earlier from our bootstrap analysis of the total DLA sample, and do not include any systematic errors arising from our model assumptions. We use all the lines reported in Table \[ta:low\]. For species that have only one line, the error of the column density directly reflects the error of the equivalent width; when there are several lines the error is reduced by obtaining the best fit to all the lines.
We also report column densities for four intermediate ionization species from Table \[ta:low\]: AlIII, CIII, SIII and SiIII, using the same parameters as for the low-ionization species since they are believed to have similar velocity distributions [@Wolfe2005]. For the high-ionization species we list the results obtained for the fit to CIV and SiIV for the results in Table \[ta:high\] with their different parameters $\sigma_1$ and $\sigma_h$, and apply them also to NV and OVI. These more highly ionized species are believed to arise from higher temperature gas, [see @Fox2007a; @Lehner2008; @Fox2011 and § \[sec:discussion\]], with velocity distributions that are likely broader than for CIV and SiIV, implying perhaps an overestimate of the OVI column density, but not for NV which produces lines that are mostly optically thin.
The strongly blended lines in Table \[ta:blended\] are in general not used for the determination of column densities whenever we have other lines of the same species in Table \[ta:low\], because blending adds additional modeling uncertainties, as discussed in section \[sec:blended\]. However, there are several species that are only measurable using these blended lines. In the case of TiII and NI, we use the TiII blend at $ 1911 {\, {\rm \AA}}$ and the NI blend at $ 1200 {\, {\rm \AA}}$, containing only lines from a single species, to determine the mean column densities directly from the “fitted W” results of the lines. These lines are not very close together, so we believe that the systematic effects discussed in section \[sec:blended\] are not important in this case. The TiII line is in any case quite weak and has a large statistical error. The rest of the lines with column densities listed in Table \[ta:N\] derived from blended groups are for ZnII, CII${}^{\*}$, FeIII and SIV, which are inferred after correcting for lines of other species. We will discuss these more complex cases further in § \[sec:blend\].
The third column in Table \[ta:N\] denotes the abundances obtained using the expression ${\rm [X/HI]} = \log\left({N_{\rm X}}/{N_{\rm HI}}\right) -
\log\left(N_{\rm X_e}/N_{\rm H}\right)_{\odot}$, where $X$ denotes the ion of interest, $X_e$ the corresponding element, and $\odot$ denotes solar values. We use solar abundances for elements from the photospheric data in Table 1 of [@Asplund2009]. These results for $\rm [X/HI]$ can be interpreted as element abundances relative to solar values if ionization and dust depletion corrections can be neglected. We give the abundances only for species that are expected to have fractional columns compared to their elements similar to the neutral hydrogen fraction (often the first ionized species, but the neutral one for O).
Metal ion $\log \bar N$ \[X/HI\]
----------- -------------------- ------------------
AlII $13.806 \pm 0.016$ $-1.13 \pm 0.03$
CII $16.025 \pm 0.060$ $-0.90 \pm 0.08$
CII\* $13.351 \pm 0.060$
CrII $12.908 \pm 0.117$ $-1.22\pm0.12$
FeII $14.538 \pm 0.011$ $-1.45\pm0.04$
MgI $12.387 \pm 0.073$
MgII $14.943 \pm 0.041$ $-1.15 \pm 0.06$
MnII $12.300\pm 0.188$ $-1.62 \pm 0.19$
NI $14.128 \pm 0.025$
NII $14.497 \pm 0.051$
NiII $13.452 \pm 0.030$ $-1.26\pm0.05$
OI $15.959 \pm 0.025$ $-1.22\pm0.06$
PII $12.890\pm 0.141$ $-1.01\pm0.14$
SII $14.731 \pm 0.032$ $-0.88 \pm 0.04$
SiII $15.087 \pm 0.012$ $-0.91 \pm 0.03$
TiII $12.264\pm 0.193$ $-1.19\pm0.20$
ZnII $12.157\pm 0.111$ $-0.89 \pm 0.12$
AlIII $12.946 \pm 0.023$
CIII $16.696 \pm 0.121$
SIII $14.345 \pm 0.123$
SiIII $15.107 \pm 0.036$
FeIII $13.905\pm 0.095$
CIV $14.396 \pm 0.019$
NV $13.074 \pm 0.090$
OVI $14.771 \pm 0.030$
SIV $13.707 \pm 0.184$
SiIV $13.783 \pm 0.016$
: Derived mean column densities in the total sample.[]{data-label="ta:N"}
Doubly ionized species generally have column densities that are not much lower than the singly-ionized ones. In particular, the SiIII and SiII column densities are equal within the measurement error, and that of CIII is substantially higher than for CII. In the case of CIII, the column density we obtain is subject to a large systematic error due to our modeling assumptions. The CIII column density is derived only from the line at $977 {\, {\rm \AA}}$, which has the highest value of $W/\lambda\simeq 6.6\times 10^{-4}$ of any of our lines, and involves a large extrapolation of our model relating $W/\lambda$ to $f\lambda$ from the curve in Figure \[fig:curve\] which is not tested from the FeII and SiII lines in the figure. For the cases of SiIII, SIII and AlIII, there is no extrapolation in the values of $W/\lambda$; the velocity structure may still be different from the singly ionized species, but the overestimate of the column density this may induce is probably not large. These cases demonstrate that the doubly ionized species can have column densities close to the singly ionized species, implying a substantial ionization correction for the abundances. The ratios of double to single ionization species can be used to constrain photoionization models of DLAs [e.g., @Vladilo2001]. The column densities of more highly ionized species are, as expected, substantially lower than those of their low-ionization counterparts.
In the case of neutral species, the mean column density of NI is about half the mean column density of NII. Nitrogen, with an ionization potential of $14.5$ eV, should be completely neutral in the deeply self-shielded inner parts of DLAs, so in this case the ionization correction is quite important. The fraction of oxygen in the form of OI is probably even smaller than for nitrogen, owing to its lower ionization potential, so our derived nitrogen and oxygen abundances are highly dependent on these uncertain ionization corrections.
Without ionization and dust depletion corrections, the abundances for iron and silicon would be $\rm{[FeII/HI]}= -1.45 \pm 0.04$ and $\rm{[SiII/HI]}= -0.91 \pm 0.03$, respectively. In this case, the ionization corrections are likely to be similar for silicon and iron because of the similar ionization potentials. The measured ratio $\rm{[SiII/FeII]}\simeq 0.54 \pm 0.05$, significantly higher than the typical intrinsic $\alpha$-element enhancement in DLAs relative to the Sun [${\rm [\alpha/Fe]\sim0.3}$, e.g., @Prochaska2003b; @Rafelski2012; @Cooke2015], reflects the effect of the large dust depletion of iron compared to silicon [e.g., @Pettini1997; @Kulkarni1997; @Akerman2005; @Vladilo2002; @Vladilo2011; @Cooke2011; @Cooke2013].
Modelling Blended Lines {#sec:blend}
-----------------------
Many lines of high scientific interest are blended. As described in § \[sec:blended\], the equivalent widths for the groups of blended lines listed in Table \[ta:blended\] are measured by jointly fitting the parameter $b$ in Eq. \[eq:fitting\] for each line, while the $a$ parameter is, as usual, kept fixed. This approach leads to large, correlated errors for the equivalent widths of the individual lines contributing to the blend, as listed in the fourth column of Table \[ta:blended\], and to systematic errors caused by our assumption that the total transmission is simply the product of the individual line transmissions. In several cases, we are particularly interested in measuring the equivalent width of one line in one of these groups, where all the other lines of the group can be modelled from other lines of the same species measured independently. For example, the only line available for SIV is at $1062 {\, {\rm \AA}}$ and is blended with several FeII lines, which can be modelled from the set of all lines used in Figure \[fig:curve\]. Using the model prediction for the FeII lines can allow for a better estimate of the SIV line equivalent width.
We use this approach for five of the line groups listed in Table \[ta:blended\], in order to measure lines of SIV, FeIII, CII\* and ZnII (the latter in two different blended groups). The predictions for the modelled lines are listed in the fifth column of Table \[ta:blended\], obtained from the column densities listed in Table \[ta:N\] and the fixed values of the $\sigma_1$ and $\sigma_h$ parameters of all the low-ionization lines. The errors of these modelled equivalent widths are small because they are derived from the error in the column density of each element, without including model uncertainties. We repeat the fit to the measured profile of the blended group by fixing the $b$ parameter of all the modelled lines to reproduce these predicted equivalent widths, and leaving as a single free variable the $b$ parameter of the line that is inferred from this model prediction (the $a$ parameter of all lines is fixed as usual by Eq. \[eq:afit\]). The results for the equivalent widths of these inferred lines are listed in the sixth column, for the five mentioned groups. Errors are computed from bootstrap realizations that keep the modelled lines fixed. They therefore include only the observational noise and continuum modeling, but not the systematic uncertainty of the modelled lines. This process is not applied in the case of the NIII line at $990{\, {\rm \AA}}$, for which the equivalent width is strongly blended with a SiII line and partly blended also with the OI group, and is too small to be measured from our stacked spectrum.
As mentioned before, the TiII group at $1911 {\, {\rm \AA}}$ and the NI group at $1200 {\, {\rm \AA}}$ are used as the only measurements to infer the TiII and NI column densities. The inferred $W$ values in the fifth column are then derived from our model using these inferred column densities.
Finally, there are five absorption features in which all the lines can be modelled from other measured lines: the OI group, the FeII-NI group, the FeII group, the SIII-SiII group, and the SiII-FeII group. These modelled equivalent widths are also given in the fifth column of Table \[ta:blended\]. In addition, the table presents the total inferred equivalent width from these modelled lines in the sixth column, for these five blended groups. These values are often consistent with the measured total equivalent widths of the group, supporting the reliability of our model, with some exceptions. The main exception is the OI group, for which the inferred $W$ is larger than the total $W$ measured for the blend. This behavior can be attributed to the systematic error discussed in section \[sec:blended\]: the three OI lines are very close together, and while we assume that their absorbing components are independent and the transmission of the individual lines can simply be multiplied to obtain the total transmission, in reality the absorbing components are of course the same in all three lines in each individual DLA and are often much narrower than the BOSS spectrograph resolution and almost certainly highly saturated. Therefore the measured $W$ is smaller than the inferred $W$ when treating the three lines as independent. The other smaller discrepancies in the remaining blend groups can be attributed to the same systematic effect.
Our treatment of the blend groups provides mean equivalent widths for lines of three species that are in general extremely difficult to measure: SIV$\lambda1062$, FeIII$\lambda1122$, and CII\*$\lambda1335$. The first two are in the [Ly$\alpha$ ]{}forest, and blended with FeII and SiII lines. The lines are therefore doubly difficult to measure from individual DLA systems, even in spectra of high resolution and signal-to-noise ratio. The [Ly$\alpha$ ]{}forest contamination is automatically removed in our technique, where we measure only the mean equivalent width of a large sample of DLAs. The FeII or SiII contamination is reliably predicted by our model, as illustrated by the small scatter of the measured points in Figure \[fig:curve\] and the $\chi^2$ value of our three-parameter fit to these points. The systematic effects of correlated components are in this case less serious, because the line separations are comparable to the BOSS resolution so that overlap of narrow components is not frequent. Still, the errors we obtain in the end for these species are relatively high, especially for SIV because it accounts for only $\sim$ 15% of the total equivalent width of the blended line group.
Finally, we can derive the equivalent width of the ZnII lines from the blends at $2026{\, {\rm \AA}}$ and $2062{\, {\rm \AA}}$. The most reliable measurement originates from the $2026{\, {\rm \AA}}$ blend, for which the CrII and MgI contributions are a relatively small correction, so we use only this group to derive the ZnII column density included in Table \[ta:N\]. This result yields a mean abundance with no ionization correction of $\rm{[ZnII/HI]}= -0.89 \pm 0.12$. This element is a particularly good tracer of metallicity because it is one of the least affected elements by dust depletion [e.g., @Pettini1990; @Pettini1997; @Prochaska2002; @Jenkins2009; @Cooke2011; @Cooke2013]. We discuss and compare these abundance values in § \[sec:discussion\].
Behaviour of the model parameters with ${ N_{\rm HI}}$ {#sec:curvenhi}
------------------------------------------------------
We now explore the behaviour of our three model parameters ($N$, $\sigma_1$ and $\sigma_h$) for the FeII and SiII transition lines with ${ N_{\rm HI}}$. We make use of the total sample, now dividing it into the following three ${ N_{\rm HI}}$ intervals chosen to contain a similar number of DLAs: $[20.0-20.24], (20.24-20.60], (20.6-22.50]$, with mean values of ${\rm \log (N_{HI}/cm^{-2})}$ of $20.11$, $20.41$ and $20.97$, respectively. We compute the three composite spectra following the procedure explained in the above sections and calculate the equivalent widths of the metal lines. The [*left panel*]{} of Figure \[fig:cognhi\] presents the measurements of $W/\lambda$ versus $f\lambda$ of the FeII transition lines in these three $N_{\rm HI}$ intervals. When modeling these results, we ignore the FeII$\lambda1125$ line, with points at $f\lambda \simeq 18\,{\rm \AA}$ in this panel of Figure \[fig:cognhi\]. This line is, particularly for the lowest column density interval, strongly affected by the noise and continuum systematic errors, and is a clear outlier from the relation followed by the other lines; for some reason that we have not been able to determine, this is not properly captured by our bootstrap calculation of the uncertainties. The [*right panel*]{} displays the 5 transitions of SiII. For the lowest column density range we have considered 4 points since in one case we have not been able to obtain a measurement for the equivalent width, owing to excessive noise in this very weak transition line.
{width="50.00000%"}{width="48.00000%"}
We obtain first a fit to the data setting $\sigma_h$ and $\sigma_1$ to the fixed values found in § \[sec:curve\], with the FeII and SiII column densities as the only free parameter. The result is presented in Table \[ta:cognhi\], and the fitted curves are displayed as the solid lines in the left and right panels of Figure \[fig:cognhi\] for FeII and SiII, respectively. The expected value of $\chi^2$ is in this case equal to 10 for FeII and 4 for SiII, with 11 and 5 lines (4 in the lowest column density range due to noise effects) used for FeII and SiII, respectively, and only one free parameter for each species. The values of the fits in Table \[ta:cognhi\] demonstrate that the fits are not good except for the middle $N_{\rm HI}$ interval: the slope of the $W/\lambda$ versus $f\lambda$ curve is too shallow (steep) for the high (low) $N_{\rm HI}$ interval, compared to the data, as clearly seen in the left panel of Figure \[fig:cognhi\] for the case of FeII. A similar trend is observed for the SiII lines in the right panel.
Good fits are obtained when all three parameters are allowed to vary in each $N_{\rm HI}$ interval, as represented by the values of $\chi^2$ in Table \[ta:cognhi\] for the free cases. We have not been able to fit one of the SiII transitions, that at $f\lambda\sim18\,{\, {\rm \AA}}$, due to the effect of noise and for this reason we do not present the values for this free case. The general improvement is mainly related to an increase of $\sigma_h$ with $N_{\rm HI}$, which allows adjusting the slope of the curve to that shown by the data, as visible in Figure \[fig:cognhi\]. The value of $\sigma_1$ is lower than for the fixed case, but generally also with a large uncertainty for the allowed upper range. Surprisingly, for the upper density range of SiII, $\sigma_1$ is tightly constrained, contrary to the results in other cases. This behaviour is produced by the three rightmost data points, which have the smallest uncertainties and are well fitted by the model. Therefore, small departures from the best fit $\sigma_1$ value produce large changes in $\chi^2$, thus constraining the range of allowed values in the parameter space. The ratio of the metal column density to $N_{\rm HI}$ now decreases slightly with ${N_{\rm HI}}$.
${\rm \hspace{3.5cm} FeII}$ ${\rm \hspace{3.5cm} SiII}$
----------------------------------- ------------------ ------------------------------- -------------------- ----------------------------------- --------------------
$\log {\rm (N_{\rm HI}/cm^{-2})}$ Fixed Free Fixed Free
${\rm logN_{X}}$ $13.967 \pm 0.020$ $14.098 \pm 0.072$ $14.423 \pm 0.017$
$20.11$ ${\rm \sigma_h}$ $8.56$ $5.59\pm0.61$ $8.56$
${\rm \sigma_1}$ $90$ $45^{+211}_{-23}$ $90$
${\rm \chi^2}$ $33.36$ $13.20$ $23.55$
${\rm logN_{X}}$ $14.334 \pm 0.017$ $14.318 \pm 0.053$ $14.913 \pm 0.019$ $14.863 \pm 0.086$
$20.41$ ${\rm \sigma_h}$ $8.56$ $9.92\pm1.01$ $8.56$ $9.56\pm0.61$
${\rm \sigma_1}$ $90$ $31^{+191}_{-12} $ $90$ $46^{+202}_{-23} $
${\rm \chi^2}$ $4.92$ $3.96$ $3.49$ $1.25$
${\rm logN_{X}}$ $14.994\pm 0.015$ $14.773 \pm 0.076$ $15.540 \pm 0.017$ $15.320 \pm 0.036$
$20.97$ ${\rm \sigma_h}$ $8.56$ $12.03\pm0.88$ $8.56$ $12.15\pm0.66$
${\rm \sigma_1}$ $90$ $51^{+198}_{-9}$ $90$ $54^{+2}_{-2}$
${\rm \chi^2}$ $31.02$ $9.43$ $68.26$ $0.35$
We conclude that although our model works well as a fitting formula for our observational results of the equivalent width dependence on $f$ displayed in Figure \[fig:curve\], it should not be interpreted as a physical model in a straightforward way, because the values of $\sigma_h$ and $\sigma_1$ need to vary when we consider DLAs in different $N_{\rm HI}$ intervals. The increase of $\sigma_h$ with $N_{\rm HI}$ is probably not directly related to a real increase of the internal dispersion from turbulence of absorbing components, but to the need to fit a variable shape of the $W/\lambda$ versus $f\lambda$ curves that is caused by other effects. A more accurate model that is calibrated to precisely account for the properties of metal lines in high-resolution spectra would be required to improve the physical interpretation of these fits. In the meantime, the validity of the column densities in Table \[ta:N\] are subject to our assumption that the curve presented in Figure \[fig:curve\] has a unique shape for all the species we consider, and are furthermore not affected by any photoionization and dust depletion corrections.
The trend of the declining ratio $N_{\rm FeII}/N_{\rm HI}$ and $N_{\rm SiII}/N_{\rm HI}$ with column density in the results of Table \[ta:cognhi\] appears only when all three parameters are allowed to vary. Several works have suggested a decrease of metallicity with hydrogen column density in DLAs [e.g., @Boisse1998; @Kulkarni2002; @Khare2004; @Akerman2005; @Meiring2006], which could also extend to sub-DLAs . As seen in Figure \[fig:triangle\], the value of $\sigma_h$ and the derived metal column densities are anticorrelated. The trend of an increasing $\sigma_h$ with $N_{\rm HI}$ is therefore correlated with the decrease of the ratio of metal columns to the hydrogen column. In fact, when we fix the parameters $\sigma_1$ and $\sigma_h$, the column density ratios (see Table \[ta:cognhi\]) remain constant. We have further checked this point by dividing our sample of DLAs in only two groups: systems that are usually classified as sub-DLAs in the literature, with $N_{\rm HI} <
10^{20.3} {\, {\rm cm}}^{-2}$ (and a mean column density in our total sample of $10^{20.14} {\, {\rm cm}}^{-2}$), and standard DLAs, with $N_{\rm HI} > 10^{20.3}
{\, {\rm cm}}^{-2}$ (with a mean column density of $10^{20.72}{\, {\rm cm}}^{-2}$ in our total sample). We find results very similar to those in Table \[ta:cognhi\]: the ratios of $N_{\rm FeII}$ and $N_{\rm SiII}$ to $N_{\rm HI}$ are higher in sub-DLAs by $\sim 0.2$ dex compared to standard DLAs, but the value of $\sigma_h$ is $7{\, {\rm km \, s}^{-1}}$ for sub-DLAs compared to $10 {\, {\rm km \, s}^{-1}}$ for standard DLAs. These variations are consistent with the expected correlation of errors seen in Figure \[fig:triangle\], and we therefore conclude that the decline of column density ratios with $N_{\rm HI}$ is not necessarily a real effect.
We do not assess the evolution of our model parameters with redshift, because of the unknown systematic uncertainties and dependencies on the model, and the small redshift range covered by our DLA sample.
Discussion {#sec:discussion}
==========
This paper has presented a new technique to study mean properties of the metal lines of a sample of DLAs. After evaluating a continuum for the mean quasar spectrum in our sample of $34\,593$ detected DLAs, the composite DLA absorption spectrum presented in Figures \[fig:lines24\] and \[fig:lines76\] is obtained for the total and metal samples, respectively, from which we obtain mean equivalent widths of all the detectable metal lines. This is the largest sample of DLAs ever analyzed for this purpose. Previously, similar stacking techniques were applied by [@Khare2012] for the purpose of measuring the effect of dust reddening and determining mean equivalent widths. Here, we have focused on completing a more extensive analysis of mean equivalent widths of all the metal lines we can detect with our much larger sample, for which we are presenting detailed tables with bootstrap errors that include the uncertainty in the continuum determination. Specifically, we have analyzed the dependence of the mean equivalent width on $N_{\rm HI}$ for low- and high-ionization lines, developed a model for the effects of line saturation to relate equivalent widths to mean column densities, and used the model to separate the contributions of lines contributing to several blended groups listed in Table \[ta:blended\].
The advantage of using this stacking technique with a very large sample is that the superposition of the forest of Ly$\alpha$ and higher order Lyman series lines from the intergalactic medium with the metal lines associated with DLAs is automatically removed. We can therefore measure several lines at wavelengths which have never been previously measured without the ambiguity due to the contamination by the forests. Of course, the BOSS spectra are missing all the information on the rich velocity structure of the DLA metal lines that is observed in high-resolution, high signal-to-noise ratio spectra, but we must bear in mind that even in the highest quality spectra, the individual components in metal lines can be highly saturated and arise from gas at different densities and temperatures, with a variable degree of turbulence, a situation which is not essentially different from the situation we face when trying to model the mean equivalent widths we measure here.
Dependence of mean equivalent widths on $N_{\rm HI}$
----------------------------------------------------
As described in § \[sec:nhi\], whereas there is a clear increase of $W$ with $N_{\rm HI}$ for low-ionization lines, the equivalent widths of high-ionization lines stay practically constant as $N_{\rm HI}$ increases. This result is clear observational evidence in favor of the widely-believed picture that low-ionization metal lines arise in self-shielded gas that is centrally concentrated in the absorption systems in a similar way as the atomic hydrogen [e.g., @Wolfe2005 and references therein]. The high-ionization lines must, on the other hand, arise in a more extended gas distribution around the self-shielded gas to explain their weak dependence on $N_{\rm HI}$. This picture has also been supported in the past by the different velocity profiles of low-ionization and high-ionization species, indicating that they arise from different gaseous structures [@WolfeProchaska2000; @Prochaska2002; @Fox2007a], but our results uniquely demonstrate that the high-ionization gas arises in a more extended spatial region that surrounds the low-ionization gas, as this is the only way to understand the nearly uniform properties of high-ionization lines over DLAs with highly variable low-ionization column densities, combined with independent observations of the higher incidence rate of absorption systems selected from CIV and other high-ionization lines compared to DLAs [@Shull2012].
As explained in § \[sec:nhi\], it is not clear from our observations if the high-ionization lines have a weak increase of $W$ with $N_{\rm HI}$, owing to selection effects in the DLA samples we use. Our metal sample, which includes only DLAs with a detected presence of metal lines at the individual basis, is consistent with a constant $W$ with $N_{\rm HI}$, but this result may be affected by a stronger selection in favor of metal-rich systems at low $N_{\rm HI}$, compared to the mean. This interpretation is in fact consistent with the histograms in the top panel of Figure \[fig:distrib\], demonstrating a clear increase of the fraction of DLAs in the metal sample with $N_{\rm HI}$. However, the effect may also be due in part to a fraction of false DLAs in our sample that is higher at the low end of our column density range. The relative importance of these two effects can only be modelled with detailed mocks that simulate the entire process of DLA detection. Undertaking such a task is difficult because false DLAs will often be the result of clusters of [Ly$\alpha$ ]{}forest lines of high column density (not reaching DLA values) that may also have metals associated with them.
At present, we can only conclude that any correlation of the high-ionization lines with $N_{\rm HI}$ in DLAs cannot be stronger than the result we have found for our total sample. The general picture described above postulates that most of the atomic hydrogen in DLAs, together with low-ionization metal lines, should arise from relatively dense, self-shielded clouds, whereas the high-ionization species should mostly arise in lower density, more extended, unshielded gas. Therefore, in this picture, the column densities of the two types of species should present little correlation in any individual halo. However, a correlation may be induced by a dependence of the $N_{\rm HI}$ radial profile in the self-shielded clouds of DLAs on the metallicity or velocity dispersion of the halo (note that the velocity dispersion increases the metal equivalent widths, even at fixed metallicity, because of line saturation effects). Indeed, a correlation between the equivalent widths of the lines CIV$\,\lambda$1548 and SiII$\,\lambda$1526 was reported by @Prochaska2008 (their figure 8). If absorption systems with weak metal lines have broad cores and rarely reach the highest hydrogen column densities in our sample, whereas systems with stronger metal lines have more cuspy $N_{\rm HI}$ profiles with higher central values, that would induce an increase of the high-ionization lines $W$ with $N_{\rm HI}$ even if the high-ionization gas has a constant $W$ unrelated to $N_{\rm HI}$ for individual systems. This topic can motivate further work on correcting for any sample selection effects in the future.
We have also measured the dependence of $W$ on $N_{\rm HI}$ for the intermediate ionization species AlIII, CIII, SIII and SiIII. The results, given in Tables \[ta:lownhi24\] and \[ta:lownhi76\], and in Figure \[fig:nhi\] for the AlIII case, tend to show an increase of $W$ with $N_{\rm HI}$ that lies between the behavior of low- and high-ionization lines, although with fairly large uncertainties. These intermediate species can be formed via X-ray photoionization in self-shielded regions, and may arise in the same gas region as low-ionization species [@Wolfe2005], but may also have an important contribution from the more extended regions of the high-ionization gas. The results suggest that SIII and SiIII behave more similarly to the high-ionization lines, and AlIII and CIII behave more similarly to the low-ionization lines, but the uncertainties are too large to reach a clear conclusion at this point, and further modeling of the sample selections is required. The point we wish to stress here is that these observations have a high potential to constrain models for the distribution of gas in halos and their photoionization state, and to test theoretical results from simulations of galaxy formation.
Mean abundances of low-ionization species {#sec:abundances}
-----------------------------------------
In § \[sec:model\], we have presented our model for line saturation used to derive the column densities listed in Table \[ta:N\]. How reliable are these column densities? Our basic argument for believing they are approximately correct is that the curves presented in Figure \[fig:curve\] have reached the unsaturated regime at low $f\lambda$, i.e., they follow a linear relation $W/\lambda \propto f\lambda$ from which the column densities are derived. The problem is, of course, that the lines we can measure at low $f\lambda$ have large equivalent width errors, and the column densities directly derived from the weakest lines are also subject to these errors. Our model is used to reduce the error by fitting all the other lines at higher $f\lambda$ with a three-parameter curve that approaches the linear regime at low $f\lambda$ according to reasonable assumptions on the distribution and velocity dispersions of the individual absorption components. For internal bulk velocity dispersions of these components $\sigma_h\simeq 9 {\, {\rm km \, s}^{-1}}$, metal lines should reach the unsaturated regime roughly at $W/\lambda \sim
\sigma_h/c \simeq 3\times 10^{-5}$, which is consistent with our measurements shown in Figure \[fig:curve\]. For ions other than FeII and SiII, we assume that their lines follow the same curve, with variations only in the mean column density. If the lines for which our column densities are based on are very weak (i.e., in the unsaturated regime), they are not subject to any model dependence, but if they are saturated then our model assumption of a single curve for all the low-ionization ions is used to extrapolate to low values of $f\lambda$ and correct for the saturation effects.
In general, our derived abundances are slightly higher, although in broad agreement, than previous determinations. Our results for $\rm {[FeII/HI]}=-1.45 \pm 0.04$, $ \rm {[SiII/HI]}= -0.91 \pm 0.03$, $\rm {[CrII/HI]}=-1.22 \pm 0.12$, $\rm{[MnII/HI]}= -1.62 \pm 0.19$, $\rm {[NiII/HI]}=-1.26 \pm 0.05$ and $\rm{[ZnII/HI]}= -0.89\pm 0.11$ are within $0.5$ dex of those reported by [@Khare2012] (their table 5). We can also compare our results with high-resolution studies of smaller numbers of DLAs. Our zinc result is within $3\sigma$ of the mean value reported by [@Pettini2006], ${\rm [\langle Zn/H \rangle]=-1.2}$, at $1.8<z<3.5$. To compare with the results of [@Rafelski2012], we use our mean DLA redshift $z=2.59$ in their expression for the evolution of the mean DLA metallicity with redshift, $\langle Z\rangle \,=\,(-0.22\pm0.03)\,
z_{DLA}\,-\, (0.65\pm0.09)$, to find $\langle Z\rangle= -1.22\pm0.10$ for their result, which is in agreement within 3$\sigma$ of our $\rm{[ZnII/HI]}$ value and also with our result for sulfur, $\rm{[SII/HI]}= -0.88\pm 0.04$, both species being commonly used as metallicity indicators [@Wolfe2005]. Considering equation 4 in [@Quiret2016], $\langle Z \rangle_{\rm DLAs}= (-0.15\pm0.03)z_{\rm DLA}
-(0.60\pm0.13)$, we obtain a HI-weighted mean DLA metallicity of $\langle Z \rangle_{\rm DLAs}=-0.99\pm0.15$, consistent within 1$\sigma$ with our $\rm{[ZnII/HI]}$, $\rm{[SII/HI]}$ and $\rm{[SiII/HI]}$ results. Our mean abundances of titanium and phosphorus from Table \[ta:N\], $\rm{[TiII/HI]}=-1.19 \pm 0.20$ and $\rm{[PII/HI]}= -1.01 \pm 0.14$, are within 2$\sigma$ and 3$\sigma$, respectively, from a small number of detections reported by [@Prochaska2001] (their table 41), with mean values $\rm {[Ti/H]}\sim-1.45$ (13 data points) and $\rm {[P/H]} \sim -1.42$ (5 data points).
Our measurement of $\rm{[CrII/ZnII]}= -0.33 \pm 0.17$, which is the ratio normally used to estimate the dust content in DLAs, is in agreement with high-resolution studies of DLAs with abundances $\rm{[ZnII/HI]}\sim-1$ [@Pettini1997; @Prochaska2002; @Akerman2005]. We also find $\rm{[FeII/ZnII]}= -0.56 \pm 0.13$ and $\rm{[NiII/ZnII]}=
-0.37 \pm 0.14$, which agree with the values by, e.g., [@Prochaska2001; @Pettini2004; @Khare2004; @Rafelski2012]. Figure \[fig:dp\] illustrates that, in general, our \[X/Zn\] measurements are in broad agreement with the gas depletion pattern in the halo of the Milky Way reported by [@Welty1999], who analysed the absorption signatures of the Milky Way and LMC gas in the spectra of the supernova SN1987A. The value for \[FeII/ZnII\] indicates a modest dust depletion effect in DLAs which yields a correction for the abundances [e.g., @Vladilo2002; @Vladilo2011]. Silicon is more weakly depleted, and sulfur is extremely weakly depleted, and the values $\rm{[SiII/FeII]}=0.54\pm0.05$ and $\rm{[SII/FeII]}=0.57\pm0.06$ are in agrement, for example, with [@Prochaska2002; @Dessauges2006; @Rafelski2012; @Berg2015], when we use our measured value of ${\rm [ZnII/FeII]} \simeq 0.6$. Ratios of strongly depleted elements, such as ${\rm [CrII/FeII]} = 0.23 \pm 0.12$ and ${\rm [MnII/FeII]} = -0.17 \pm 0.19$, are also in agreement with various high-resolution studies [e.g., @Dessauges2006].
Adding the column densities of neutral and singly ionized nitrogen, our measured ratios of $\rm{[N/SII]}= -0.79 \pm 0.08$ and $\rm{[N/SiII]}= -0.76 \pm 0.08$ agree with the results found for evolved DLAs, ${\rm [N/{\alpha}] \sim -0.7}$ [@Prochaska2002; @Pettini2002; @Centurion2003; @Petitjean2008; @Zafar2014].
Doubly-ionized species {#sec:highlow}
----------------------
On the intermediate ionization species, our mean measured ratio for ${\rm [AlIII/AlII]} = -0.86\pm 0.03$ is in good agreement with the results of [@Vladilo2001], who reported this ratio to vary from $-0.2$ to $-1.2$ as $\log (N_{\rm HI}/{\rm cm}^{-2})$ varies from $20.3$ to $21.0$. We find higher values for the ratios ${\rm [SiIII/SiII]} = 0.02\pm 0.04$, ${\rm [SIII/SII]}= -0.39 \pm 0.12$, ${\rm [FeIII/FeII]} = -0.63\pm 0.10$, and ${\rm [CIII/CII]} = 0.67 \pm 0.12$, which indicate that doubly-ionized species often have comparable column densities to the singly ionized ones. As explained in § \[sec:deduction\], we do not consider the unusually high ratio obtained for the case of carbon to be reliable, because the CIII column density depends on an extrapolation of our model to a large value of $W/\lambda$ that is untested in our data. The other ratios, however, are probably not affected so severely by the systematic modeling uncertainty of saturated lines, and they illustrate the potential of our method to derive mean column densities for various species in DLAs. These results can be further developed and analyzed in the future to probe the photoionization state of DLAs.
High-ionization species {#sec:highlow}
-----------------------
We also report in this paper measurements of the mean column density of several lines of high-ionization species. The strongest features are the doublets of CIV and SiIV. The doublet ratios for the total sample are, from Table \[ta:high\], equal to $1.44\pm 0.02$ for CIV and $1.50 \pm 0.04$ for SiIV. The doublet ratio for both species denotes that the absorbers have a mixture of weak, unsaturated, and strong, saturated components, contributing almost equally to the total absorption feature. The fact that this ratio is constant with DLA column density implies that the velocity dispersion, or the mean number of subcomponents, should increase with column density.
Our result for the mean column density of CIV in our DLA sample is ${\rm \log N_{\rm CIV}=14.40\pm 0.02}$. We can compare that to the results on CIV absorption of DLAs reported by [@Fox2007a], who studied a sample of 63 DLAs and found a relation of CIV column density to the DLA metallicity of ${\rm \log N_{\rm CIV}=15.8 + 1.2 [Z/H]}$. Using our mean metallicity derived from the zinc abundance, ${\rm [ZnII/HI]=-0.89}$ (from our Table \[ta:N\]), there is a remarkably good agreement with our measured mean CIV column density. Our measured SiIV column density is lower than that of CIV by a factor $\sim 4$, also in agreement with the results of [@Fox2007b]. We also obtain for the first time a mean column density of SIV in DLAs, which is difficult to obtain because of the blend with FeII lines, and we find a value similar to the column density of SiIV.
The OVI lines at $1031$ and $1037 {\, {\rm \AA}}$ are clearly detected, thanks to the elimination in our stacked spectrum of the usual difficulty due to blending with Lyman forest lines. We also detect the much weaker lines of NV at $1238$ and $1242 {\, {\rm \AA}}$. It has been demonstrated that OVI is commonly present in DLAs [@Rahmani2010; @Fox2007a]. The mean column density we obtain, $N_{\rm OVI}=14.77 \pm 0.03$, should therefore be a typical value in halos hosting DLAs, which, as we have seen in Figure \[fig:nhi\], does not strongly depend on the HI column density. For NV, which has nearly optically thin lines, the mean column density we obtain is $N_{\rm NV}=13.07 \pm 0.09$, a remarkable measurement given the small number of cases where NV has been detected in individual DLAs [@Pettini1995; @Prochaska2002; @Centurion2003; @Henry2007; @Fox2007a; @Lehner2008; @Petitjean2008; @Fox2009]. This result confirms the presence of phases of highly ionized gas around DLAs, and provides observational constraints for clarifying the photoionized or collisionally ionized state of this gas.
Atomic and excited ionized carbon {#sec:lownhi}
---------------------------------
We report a detection of the average column of excited CII, or CII\*, from the absorption line at $1335.71 {\, {\rm \AA}}$. We are able to deblend this line from the CII line at $1334.53{\, {\rm \AA}}$, but, as can be seen in Figure \[fig:fit\], the detection is only made from a slight asymetry in the blended absorption profile and is therefore subject to systematic errors. Nevertheless, our mean inferred column density of $\log N_{\rm CII^{*}} = 13.35 \pm 0.06$ is in rough agreement with the results of [@Wolfe2003]. Measuring the ${\rm CII}^{*}$ column density yields estimates of the cooling rate in the DLA gas [@Wolfe2003; @Wolfe2004].
There is clear evidence for the presence of CI absorption lines in our stacked spectrum, a species that is usually a good tracer of molecular gas [@Srianand2005; @Ledoux2015]. These absorption lines are indicated in Figure \[fig:lines76\] at $\lambda \simeq 1278$, 1329, 1560 and $1657 {\, {\rm \AA}}$, but they are extremely weak and we have not attempted to measure them.
Possible contamination by broad absorption line systems {#sec:lownhi}
-------------------------------------------------------
Some of the DLAs in the catalog we use might be broad absorption line systems that have been incorrectly identified as DLAs in the low signal-to-noise ratio BOSS spectra. To test possible influence of this potential contamination, we remove from our total sample all DLAs at a velocity separation $v<5000\, {\rm {\, {\rm km \, s}^{-1}}}$ from the quasar. There are $3\,295$ such objects, which is $\sim10\%$ of the total sample. After calculating the stacked spectrum for the remaining systems, we visually confirm that there are no significant differences between this and the total sample spectrum.
Summary and conclusions {#sec:summary}
=======================
We have calculated DLA composite absorption spectra using the DLA catalog of [@Noterdaeme2012] of Data Release 12 of BOSS. We have measured the mean equivalent width of 50 absorption lines, 38 from low-ionization species (neutral or singly-ionized), 4 from doubly ionized species, and 8 from high-ionization species. In addition, we have measured the total equivalent widths of 13 groups of strongly blended lines. We have performed the same analysis with a subsample of DLAs with individually identified metal lines, called the metal sample, containing about a third of the total sample, which allows for the detection of fainter absorption lines but is not representative of the mean DLA properties. We have divided the two previous samples in 5 ${ N_{\rm HI}}$ and 5 $z$ ranges and have analysed the dependence of the metal equivalent widths on the DLA hydrogen column density and redshift, in § \[sec:nhi\] and § \[sec:z\], respectively.
The increase of the mean $W$ with $N_{\rm HI}$ for the low-ionization lines confirms that these species are closely associated with the self-shielded atomic hydrogen in DLAs. The much weaker dependence of the high-ionization lines on $N_{\rm HI}$, conversely, demonstrates that these species occur in a different gas phase at lower density that is more extended and surrounds the low-ionization region, in view of the fact that high-ionization lines are ubiquitous in DLAs. The equivalent widths decrease by a factor $\sim 1.1 - 1.5$ from redshift $z\sim2.1$ to $z\sim 3.5$, in general, with the high-ionization species showing a slightly steeper evolution. However, it is not clear whether this possible difference has a physical origin or it is simply driven by systematics.
We have presented a new simple model to correct for line saturation and derive column densities for all the species we measure, described in § \[sec:model\]. Our model is quite successful in fitting the available lines of FeII and SiII with different oscillator strengths, as displayed in Figure \[fig:curve\]. The inferred abundances for other species generally agree with the determinations that have been made from high-resolution spectra, when available. We have also been able to measure the mean column density of OVI associated with DLAs, which is otherwise difficult to do because of the superposition with the [Ly$\alpha$ ]{}forest. For the first time, we obtain also a mean equivalent width and inferred column density for NV and SIV for DLAs. We obtained inferred column densities of several doubly ionized species, like AlIII, SIII and SiIII, which we believe are generally reliable (although not in the case of CIII owing to the required extrapolation of our simple model for line saturation correction), and can be used to test models of photoionization of the various layers surrounding DLAs.
In conclusion, the techniques we have developed here to use stacked DLA absorption spectra demonstrate a promise to explore the photoionization state and abundances of the various heavy element species present in DLAs, with some advantages that can be exploited with very large samples of DLAs even when the resolution and signal-to-noise ratio of individual spectra are poor. Further refinement of our simple model for deriving column densities from partly saturated lines, measurement of correlations among different absorption lines, and correlations of the large-scale bias factor of DLAs [@FontRibera2012] with metal lines are all promising avenues for future research.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank the anonymous referee for a detailed and careful analysis which improved the quality of our work. We are very grateful to Mat Pieri, Hélion du Mas des Bourboux, Hadi Rahmani, Benjamine Racine, Pilar Gil-Pons, Signe Riemer-Sorensen, Xavier Prochaska, Sebastián López, Céline Péroux, George Becker and Max Pettini for their useful comments and suggestions during the different stages of this work. This research was partially supported by the Munich Institute for Astro- and Particle Physics (MIAPP) of the DFG cluster of excellence ‘Origin and Structure of the Universe’. LM and JM have been partly supported by Spanish grant AYA2012-33938. Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science. The SDSS-III web site is <http://www.sdss3.org/>. SDSS-III is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS-III Collaboration including the University of Arizona, the Brazilian Participation Group, Brookhaven National Laboratory, Carnegie Mellon University, University of Florida, the French Participation Group, the German Participation Group, Harvard University, the Instituto de Astrofisica de Canarias, the Michigan State/Notre Dame/JINA Participation Group, Johns Hopkins University, Lawrence Berkeley National Laboratory, Max Planck Institute for Astrophysics, Max Planck Institute for Extraterrestrial Physics, New Mexico State University, New York University, Ohio State University, Pennsylvania State University, University of Portsmouth, Princeton University, the Spanish Participation Group, University of Tokyo, University of Utah, Vanderbilt University, University of Virginia, University of Washington, and Yale University.
Appendix A: Rest Equivalent Width Measurement {#sec:ew}
=============================================
This appendix presents the tables for the calculated rest equivalent widths and their uncertainties. Tables \[ta:low\] and \[ta:high\] present the values for the low and high-ionization species in the metal and total samples, respectively. For the case of the $5$ $\rm{N_{HI}}$ ranges, we present Tables \[ta:lownhi24\], \[ta:highnhi24\], \[ta:lownhi76\] and \[ta:highnhi76\], which list the measurements of low- and high-ionization species, in the total and metal samples, respectively. Whenever the convergence of the fitting method is not reached or the measurements present negative values, the equivalent widths are set to zero while the uncertainties remain unchanged. The values for $\rm{N_{HI}}$ displayed in the tables are those computed taking the mean column density within the corresponding interval. Tables \[ta:lowz24\], \[ta:highz24\], \[ta:lowz76\] and \[ta:highz76\] correspond to the same cases for the $5$ redshift bins, $\bar z_{\rm DLA}$. Positions with no values indicate the cases where the corresponding line wavelength is not covered by the spectrum in that redshift bin. The oscillator strengths, $f$, are those tabulated by [@Morton2003] except for the NiII$\,\lambda$1317 line, where we use the value measured by [@Dessauges2006].
Total sample Metal sample
--------------------- ----------- ------------------------ ------------------------
Transition $f$ $W$ (${\, {\rm \AA}}$) $W$ (${\, {\rm \AA}}$)
AlII$\lambda1670$ $1.740$ $0.452\pm0.005$ $0.656\pm0.005$
AlIII$\lambda1854$ $0.559$ $0.117\pm0.006$ $0.172\pm0.006$
AlIII$\lambda1862$ $0.278$ $0.067\pm0.006$ $0.096\pm0.006$
CII$\lambda1036^*$ $0.118$ $0.418\pm0.012$ $0.579\pm0.020$
CIII$\lambda977^*$ $0.757$ $0.646\pm0.019$ $0.884\pm0.033$
CrII$\lambda2056$ $0.103$ $0.028\pm0.009$ $0.043\pm0.008$
CrII$\lambda2066^*$ $0.051$ $0.034\pm0.020$ $0.028\pm0.009$
FeII$\lambda1081^*$ $0.013$ $0.030\pm0.010$ $0.058\pm0.013$
FeII$\lambda1096$ $0.032$ $0.086\pm0.010$ $0.134\pm0.013$
FeII$\lambda1125^*$ $0.016$ $0.069\pm0.009$ $0.090\pm0.010$
FeII$\lambda1144^*$ $0.083$ $0.166\pm0.007$ $0.228\pm0.010$
FeII$\lambda1608$ $0.058$ $0.228\pm0.004$ $0.329\pm0.004$
FeII$\lambda2249$ $0.002$ $0.016\pm0.020$ $0.029\pm0.024$
FeII$\lambda2260$ $0.002$ $0.034\pm0.017$ $0.068\pm0.012$
FeII$\lambda2344$ $0.114$ $0.520\pm0.014$ $0.717\pm0.014$
FeII$\lambda2374$ $0.031$ $0.282\pm0.014$ $0.413\pm0.014$
FeII$\lambda2382$ $0.320$ $0.669\pm0.026$ $0.973\pm0.013$
FeII$\lambda2586$ $0.069$ $0.462\pm0.023$ $0.680\pm0.023$
FeII$\lambda2600$ $0.239$ $0.722\pm0.021$ $1.002\pm0.025$
MgI$\lambda2852$ $1.830$ $0.231\pm0.030$ $0.277\pm0.031$
MgII$\lambda2796^*$ $0.616$ $1.149\pm0.034$ $1.598\pm0.030$
MgII$\lambda2803^*$ $0.306$ $1.067\pm0.025$ $1.437\pm0.025$
MnII$\lambda2576$ $0.361$ $0.040\pm0.021$ $0.059\pm0.019$
NII$\lambda1084^*$ $$0.111$$ $0.168\pm0.010$ $0.250\pm0.011$
NiII$\lambda1317$ $0.057$ $0.032\pm0.005$ $0.040\pm0.005$
NiII$\lambda1370$ $0.077$ $0.027\pm0.005$ $0.037\pm0.005$
NiII$\lambda1454$ $0.032$ $0.016\pm0.002$ $0.041\pm0.003$
NiII$\lambda1709$ $0.032$ $0.023\pm0.004$ $0.032\pm0.004$
NiII$\lambda1741$ $0.043$ $0.029\pm0.005$ $0.042\pm0.005$
NiII$\lambda1751$ $0.028$ $0.024\pm0.005$ $0.034\pm0.005$
OI$\lambda1039^*$ $0.009$ $0.251\pm0.013$ $0.343\pm0.016$
OI$\lambda1302$ $0.048$ $0.460\pm0.006$ $0.621\pm0.007$
PII$\lambda1152$ $0.245$ $0.021\pm0.008$ $0.038\pm0.010$
SII$\lambda1250^*$ $0.005$ $0.044\pm0.005$ $0.064\pm0.006$
SII$\lambda1253^*$ $0.011$ $0.063\pm0.005$ $0.086\pm0.005$
SIII$\lambda1012$ $0.044$ $0.067\pm0.017$ $0.093\pm0.022$
SiII$\lambda1020^*$ $0.017$ $0.096\pm0.017$ $0.116\pm0.026$
SiII$\lambda1193^*$ $0.582$ $0.452\pm0.006$ $0.617\pm0.008$
SiII$\lambda1304^*$ $0.086$ $0.333\pm0.006$ $0.453\pm0.007$
SiII$\lambda1526$ $0.133$ $0.443\pm0.004$ $0.630\pm0.004$
SiII$\lambda1808$ $0.002$ $0.059\pm0.006$ $0.088\pm0.006$
SiIII$\lambda1206$ $1.630$ $0.555\pm0.008$ $0.755\pm0.007$
: Rest equivalent widths of **low-ionization** metal absorption lines for the two samples.[]{data-label="ta:low"}
Total sample Metal sample
-------------------- --------- ------------------------ ------------------------
Transition $f$ $W$ (${\, {\rm \AA}}$) $W$ (${\, {\rm \AA}}$)
CIV$\lambda1548^*$ $0.190$ $0.429\pm0.004$ $0.565\pm0.005$
CIV$\lambda1550^*$ $0.095$ $0.298\pm0.003$ $0.394\pm0.005$
NV$\lambda1238^*$ $0.156$ $0.023\pm0.006$ $0.031\pm0.007$
NV$\lambda1242^*$ $0.078$ $0.015\pm0.006$ $0.015\pm0.007$
OVI$\lambda1031$ $0.133$ $0.331\pm0.015$ $0.411\pm0.025$
OVI$\lambda1037^*$ $0.066$ $0.191\pm0.011$ $0.256\pm0.014$
SiIV$\lambda1393$ $0.513$ $0.292\pm0.004$ $0.400\pm0.005$
SiIV$\lambda1402$ $0.254$ $0.195\pm0.004$ $0.270\pm0.005$
$\log({\bar N}_{\rm HI} / {\rm {\, {\rm cm}}^{-2}}) $ 20.06 20.22 20.40 20.65 21.13
------------------------------------------------------- --------- ------------------------ ------------------------ ------------------------ ------------------------ ------------------------
Transition $f$ $W$ (${\, {\rm \AA}}$) $W$ (${\, {\rm \AA}}$) $W$ (${\, {\rm \AA}}$) $W$ (${\, {\rm \AA}}$) $W$ (${\, {\rm \AA}}$)
AlII$\lambda1670$ $1.740$ $0.278\pm0.012$ $0.345\pm0.008$ $0.412\pm0.011$ $0.526\pm0.011$ $0.673\pm0.012$
AlIII$\lambda1854$ $0.559$ $0.076\pm0.009$ $0.076\pm0.010$ $0.105\pm0.009$ $0.146\pm0.007$ $0.173\pm0.010$
AlIII$\lambda1862$ $0.278$ $0.039\pm0.011$ $0.045\pm0.010$ $0.060\pm0.010$ $0.085\pm0.008$ $0.097\pm0.008$
CII$\lambda1036$ $0.118$ $0.260\pm0.022$ $0.318\pm0.022$ $0.356\pm0.021$ $0.481\pm0.022$ $0.613\pm0.024$
CIII$\lambda977$ $0.757$ $0.553\pm0.048$ $0.699\pm0.042$ $0.551\pm0.048$ $0.701\pm0.043$ $0.811\pm0.045$
CrII$\lambda2056$ $0.103$ $0.000\pm0.014$ $0.000\pm0.014$ $0.022\pm0.015$ $0.043\pm0.013$ $0.079\pm0.013$
CrII$\lambda2066$ $0.051$ $0.011\pm0.010$ $0.094\pm0.089$ $0.016\pm0.011$ $0.024\pm0.012$ $0.042\pm0.012$
FeII$\lambda1081$ $0.013$ $0.030\pm0.018$ $0.000\pm0.017$ $0.005\pm0.018$ $0.027\pm0.019$ $0.087\pm0.015$
FeII$\lambda1096$ $0.032$ $0.025\pm0.018$ $0.057\pm0.018$ $0.064\pm0.017$ $0.096\pm0.017$ $0.179\pm0.019$
FeII$\lambda1125$ $0.016$ $0.056\pm0.015$ $0.015\pm0.013$ $0.048\pm0.017$ $0.076\pm0.014$ $0.154\pm0.015$
FeII$\lambda1144$ $0.083$ $0.096\pm0.015$ $0.101\pm0.013$ $0.150\pm0.013$ $0.184\pm0.013$ $0.286\pm0.012$
FeII$\lambda1608$ $0.058$ $0.110\pm0.007$ $0.147\pm0.006$ $0.186\pm0.006$ $0.264\pm0.007$ $0.414\pm0.009$
FeII$\lambda2249$ $0.002$ $0.033\pm0.020$ $0.028\pm0.021$ $0.000\pm0.020$ $0.031\pm0.015$ $0.003\pm0.068$
FeII$\lambda2260$ $0.002$ $0.011\pm0.029$ $0.018\pm0.021$ $0.027\pm0.020$ $0.050\pm0.017$ $0.049\pm0.055$
FeII$\lambda2344$ $0.114$ $0.279\pm0.027$ $0.342\pm0.026$ $0.463\pm0.026$ $0.621\pm0.023$ $0.816\pm0.024$
FeII$\lambda2374$ $0.031$ $0.101\pm0.027$ $0.158\pm0.028$ $0.264\pm0.023$ $0.315\pm0.020$ $0.544\pm0.022$
FeII$\lambda2382$ $0.320$ $0.441\pm0.024$ $0.406\pm0.101$ $0.616\pm0.072$ $0.778\pm0.023$ $1.041\pm0.023$
FeII$\lambda2586$ $0.069$ $0.212\pm0.046$ $0.300\pm0.035$ $0.414\pm0.062$ $0.484\pm0.031$ $0.846\pm0.030$
FeII$\lambda2600$ $0.239$ $0.451\pm0.035$ $0.529\pm0.040$ $0.644\pm0.060$ $0.911\pm0.040$ $0.997\pm0.042$
MgI$\lambda2852$ $1.830$ $0.185\pm0.053$ $0.083\pm0.065$ $0.100\pm0.048$ $0.282\pm0.062$ $0.364\pm0.036$
MgII$\lambda2796$ $0.616$ $0.700\pm0.080$ $1.014\pm0.048$ $1.256\pm0.078$ $1.258\pm0.095$ $1.500\pm0.050$
MgII$\lambda2803$ $0.306$ $0.760\pm0.053$ $0.793\pm0.063$ $1.062\pm0.048$ $1.215\pm0.042$ $1.491\pm0.045$
MnII$\lambda2576$ $0.361$ $0.017\pm0.033$ $0.001\pm0.039$ $0.004\pm0.034$ $0.051\pm0.056$ $0.134\pm0.025$
NII$\lambda1084$ $0.111$ $0.103\pm0.018$ $0.127\pm0.019$ $0.142\pm0.020$ $0.201\pm0.016$ $0.258\pm0.020$
NiII$\lambda1317$ $0.057$ $0.029\pm0.011$ $0.018\pm0.010$ $0.029\pm0.011$ $0.031\pm0.012$ $0.055\pm0.011$
NiII$\lambda1370$ $0.077$ $0.002\pm0.009$ $0.015\pm0.009$ $0.021\pm0.010$ $0.021\pm0.009$ $0.067\pm0.008$
NiII$\lambda1454$ $0.032$ $0.000\pm0.004$ $0.010\pm0.005$ $0.011\pm0.004$ $0.022\pm0.004$ $0.043\pm0.004$
NiII$\lambda1709$ $0.032$ $0.012\pm0.009$ $0.008\pm0.007$ $0.011\pm0.008$ $0.023\pm0.006$ $0.051\pm0.007$
NiII$\lambda1741$ $0.043$ $0.009\pm0.008$ $0.015\pm0.008$ $0.016\pm0.008$ $0.030\pm0.006$ $0.070\pm0.006$
NiII$\lambda1751$ $0.028$ $0.012\pm0.007$ $0.016\pm0.008$ $0.012\pm0.008$ $0.029\pm0.006$ $0.050\pm0.007$
OI$\lambda1039$ $0.009$ $0.114\pm0.029$ $0.163\pm0.022$ $0.211\pm0.024$ $0.287\pm0.025$ $0.414\pm0.025$
OI$\lambda1302$ $0.048$ $0.288\pm0.011$ $0.362\pm0.010$ $0.429\pm0.012$ $0.543\pm0.012$ $0.654\pm0.013$
PII$\lambda1152$ $0.245$ $0.010\pm0.015$ $0.027\pm0.013$ $0.025\pm0.015$ $0.030\pm0.013$ $0.000\pm0.016$
SII$\lambda1250$ $0.005$ $0.014\pm0.012$ $0.023\pm0.010$ $0.041\pm0.011$ $0.042\pm0.011$ $0.092\pm0.010$
SII$\lambda1253$ $0.011$ $0.039\pm0.011$ $0.022\pm0.010$ $0.046\pm0.010$ $0.081\pm0.009$ $0.116\pm0.010$
SIII$\lambda1012$ $0.044$ $0.074\pm0.032$ $0.020\pm0.031$ $0.065\pm0.037$ $0.029\pm0.037$ $0.121\pm0.036$
SiII$\lambda1020$ $0.017$ $0.000\pm0.032$ $0.093\pm0.031$ $0.076\pm0.032$ $0.048\pm0.033$ $0.240\pm0.029$
SiII$\lambda1193$ $0.582$ $0.333\pm0.018$ $0.362\pm0.010$ $0.433\pm0.014$ $0.529\pm0.012$ $0.605\pm0.014$
SiII$\lambda1304$ $0.086$ $0.177\pm0.012$ $0.251\pm0.010$ $0.315\pm0.011$ $0.390\pm0.010$ $0.513\pm0.012$
SiII$\lambda1526$ $0.133$ $0.275\pm0.008$ $0.337\pm0.008$ $0.407\pm0.008$ $0.514\pm0.009$ $0.656\pm0.010$
SiII$\lambda1808$ $0.002$ $0.023\pm0.010$ $0.019\pm0.009$ $0.042\pm0.010$ $0.061\pm0.009$ $0.135\pm0.009$
SiIII$\lambda1206$ $1.630$ $0.472\pm0.016$ $0.517\pm0.014$ $0.586\pm0.015$ $0.614\pm0.014$ $0.612\pm0.017$
$\log({\bar N}_{\rm HI} / {\rm {\, {\rm cm}}^{-2}}) $ 20.06 20.22 20.40 20.65 21.13
------------------------------------------------------- --------- ------------------------ ------------------------ ------------------------ ------------------------ ------------------------
Transition $f$ $W$ (${\, {\rm \AA}}$) $W$ (${\, {\rm \AA}}$) $W$ (${\, {\rm \AA}}$) $W$ (${\, {\rm \AA}}$) $W$ (${\, {\rm \AA}}$)
CIV$\lambda1548$ $0.190$ $0.375\pm0.010$ $0.410\pm0.008$ $0.429\pm0.009$ $0.459\pm0.010$ $0.471\pm0.010$
CIV$\lambda1550$ $0.095$ $0.263\pm0.009$ $0.291\pm0.007$ $0.296\pm0.007$ $0.319\pm0.008$ $0.319\pm0.008$
NV$\lambda1238$ $0.156$ $0.024\pm0.013$ $0.061\pm0.013$ $0.018\pm0.012$ $0.022\pm0.012$ $0.000\pm0.011$
NV$\lambda1242$ $0.078$ $0.010\pm0.013$ $0.025\pm0.014$ $0.028\pm0.012$ $0.018\pm0.013$ $0.000\pm0.013$
OVI$\lambda1031$ $0.133$ $0.250\pm0.028$ $0.281\pm0.028$ $0.305\pm0.028$ $0.323\pm0.025$ $0.433\pm0.030$
OVI$\lambda1037$ $0.066$ $0.156\pm0.021$ $0.154\pm0.024$ $0.122\pm0.022$ $0.200\pm0.021$ $0.265\pm0.022$
SiIV$\lambda1393$ $0.513$ $0.246\pm0.009$ $0.272\pm0.009$ $0.286\pm0.008$ $0.318\pm0.008$ $0.338\pm0.010$
SiIV$\lambda1402$ $0.254$ $0.172\pm0.008$ $0.191\pm0.008$ $0.179\pm0.008$ $0.208\pm0.008$ $0.227\pm0.009$
$\log({\bar N}_{\rm HI} / {\rm {\, {\rm cm}}^{-2}}) $ 20.06 20.22 20.40 20.65 21.13
------------------------------------------------------- --------- ------------------------ ------------------------ ------------------------ ------------------------ ------------------------
Transition $f$ $W$ (${\, {\rm \AA}}$) $W$ (${\, {\rm \AA}}$) $W$ (${\, {\rm \AA}}$) $W$ (${\, {\rm \AA}}$) $W$ (${\, {\rm \AA}}$)
AlII$\lambda1670$ $1.740$ $0.510\pm0.014$ $0.561\pm0.012$ $0.598\pm0.011$ $0.682\pm0.011$ $0.825\pm0.012$
AlIII$\lambda1854$ $0.559$ $0.143\pm0.014$ $0.139\pm0.009$ $0.145\pm0.009$ $0.184\pm0.008$ $0.217\pm0.009$
AlIII$\lambda1862$ $0.278$ $0.082\pm0.014$ $0.085\pm0.011$ $0.079\pm0.010$ $0.104\pm0.009$ $0.114\pm0.009$
CII$\lambda1036$ $0.118$ $0.505\pm0.033$ $0.513\pm0.030$ $0.504\pm0.024$ $0.595\pm0.022$ $0.731\pm0.025$
CIII$\lambda977$ $0.757$ $1.053\pm0.077$ $1.229\pm0.074$ $1.156\pm0.071$ $1.300\pm0.082$ $1.489\pm0.097$
CrII$\lambda2056$ $0.103$ $0.017\pm0.021$ $0.015\pm0.017$ $0.018\pm0.014$ $0.043\pm0.014$ $0.093\pm0.012$
CrII$\lambda2066$ $0.051$ $0.011\pm0.007$ $0.013\pm0.014$ $0.011\pm0.018$ $0.022\pm0.012$ $0.057\pm0.016$
FeII$\lambda1081$ $0.013$ $0.024\pm0.022$ $0.039\pm0.023$ $0.023\pm0.021$ $0.063\pm0.019$ $0.097\pm0.015$
FeII$\lambda1096$ $0.032$ $0.083\pm0.024$ $0.098\pm0.023$ $0.098\pm0.021$ $0.121\pm0.019$ $0.216\pm0.016$
FeII$\lambda1125$ $0.016$ $0.040\pm0.022$ $0.038\pm0.016$ $0.075\pm0.016$ $0.088\pm0.015$ $0.169\pm0.016$
FeII$\lambda1144$ $0.083$ $0.151\pm0.019$ $0.156\pm0.016$ $0.205\pm0.016$ $0.220\pm0.015$ $0.331\pm0.014$
FeII$\lambda1608$ $0.058$ $0.192\pm0.010$ $0.231\pm0.009$ $0.268\pm0.008$ $0.336\pm0.008$ $0.502\pm0.008$
FeII$\lambda2249$ $0.002$ $0.042\pm0.025$ $0.042\pm0.022$ $0.023\pm0.024$ $0.043\pm0.018$ $0.006\pm0.077$
FeII$\lambda2260$ $0.002$ $0.029\pm0.028$ $0.060\pm0.022$ $0.040\pm0.019$ $0.062\pm0.018$ $0.120\pm0.017$
FeII$\lambda2344$ $0.114$ $0.447\pm0.033$ $0.519\pm0.037$ $0.640\pm0.030$ $0.783\pm0.020$ $0.989\pm0.022$
FeII$\lambda2374$ $0.031$ $0.168\pm0.032$ $0.265\pm0.042$ $0.364\pm0.026$ $0.435\pm0.023$ $0.657\pm0.020$
FeII$\lambda2382$ $0.320$ $0.686\pm0.030$ $0.794\pm0.029$ $0.922\pm0.027$ $1.016\pm0.024$ $1.230\pm0.022$
FeII$\lambda2586$ $0.069$ $0.433\pm0.048$ $0.484\pm0.050$ $0.601\pm0.067$ $0.676\pm0.030$ $0.984\pm0.031$
FeII$\lambda2600$ $0.239$ $0.719\pm0.037$ $0.818\pm0.047$ $0.884\pm0.081$ $1.108\pm0.051$ $1.261\pm0.052$
MgI$\lambda2852$ $1.830$ $0.263\pm0.077$ $0.202\pm0.051$ $0.231\pm0.056$ $0.306\pm0.069$ $0.394\pm0.040$
MgII$\lambda2796$ $0.616$ $1.168\pm0.159$ $1.519\pm0.057$ $1.580\pm0.048$ $1.746\pm0.050$ $1.812\pm0.049$
MgII$\lambda2803$ $0.306$ $1.142\pm0.088$ $1.210\pm0.068$ $1.450\pm0.053$ $1.482\pm0.042$ $1.733\pm0.047$
MnII$\lambda2576$ $0.361$ $0.003\pm0.042$ $0.037\pm0.035$ $0.000\pm0.043$ $0.040\pm0.035$ $0.147\pm0.027$
NII$\lambda1084$ $0.111$ $0.183\pm0.023$ $0.224\pm0.025$ $0.211\pm0.022$ $0.268\pm0.019$ $0.311\pm0.019$
NiII$\lambda1317$ $0.057$ $0.051\pm0.014$ $0.028\pm0.012$ $0.032\pm0.014$ $0.032\pm0.012$ $0.057\pm0.010$
NiII$\lambda1370$ $0.077$ $0.017\pm0.012$ $0.029\pm0.012$ $0.035\pm0.009$ $0.020\pm0.010$ $0.071\pm0.008$
NiII$\lambda1454$ $0.032$ $0.000\pm0.006$ $0.020\pm0.005$ $0.011\pm0.005$ $0.021\pm0.004$ $0.042\pm0.004$
NiII$\lambda1709$ $0.032$ $0.012\pm0.013$ $0.018\pm0.009$ $0.017\pm0.008$ $0.031\pm0.008$ $0.059\pm0.006$
NiII$\lambda1741$ $0.043$ $0.019\pm0.011$ $0.019\pm0.009$ $0.032\pm0.008$ $0.040\pm0.008$ $0.084\pm0.007$
NiII$\lambda1751$ $0.028$ $0.023\pm0.011$ $0.021\pm0.011$ $0.020\pm0.009$ $0.031\pm0.007$ $0.063\pm0.007$
OI$\lambda1039$ $0.009$ $0.245\pm0.034$ $0.283\pm0.028$ $0.269\pm0.026$ $0.369\pm0.025$ $0.499\pm0.023$
OI$\lambda1302$ $0.048$ $0.471\pm0.017$ $0.533\pm0.013$ $0.573\pm0.013$ $0.658\pm0.014$ $0.771\pm0.013$
PII$\lambda1152$ $0.245$ $0.051\pm0.020$ $0.031\pm0.018$ $0.024\pm0.018$ $0.028\pm0.015$ $0.064\pm0.018$
SII$\lambda1250$ $0.005$ $0.037\pm0.014$ $0.034\pm0.012$ $0.063\pm0.014$ $0.051\pm0.012$ $0.109\pm0.013$
SII$\lambda1253$ $0.011$ $0.038\pm0.013$ $0.060\pm0.011$ $0.070\pm0.014$ $0.088\pm0.010$ $0.140\pm0.011$
SIII$\lambda1012$ $0.044$ $0.131\pm0.060$ $0.068\pm0.046$ $0.090\pm0.043$ $0.020\pm0.044$ $0.149\pm0.042$
SiII$\lambda1020$ $0.017$ $0.059\pm0.050$ $0.134\pm0.045$ $0.095\pm0.040$ $0.081\pm0.038$ $0.200\pm0.033$
SiII$\lambda1193$ $0.582$ $0.506\pm0.022$ $0.546\pm0.014$ $0.591\pm0.015$ $0.645\pm0.014$ $0.728\pm0.013$
SiII$\lambda1304$ $0.086$ $0.316\pm0.016$ $0.372\pm0.010$ $0.414\pm0.013$ $0.478\pm0.011$ $0.595\pm0.012$
SiII$\lambda1526$ $0.133$ $0.477\pm0.012$ $0.532\pm0.010$ $0.579\pm0.009$ $0.655\pm0.009$ $0.793\pm0.010$
SiII$\lambda1808$ $0.002$ $0.037\pm0.014$ $0.038\pm0.012$ $0.061\pm0.010$ $0.084\pm0.008$ $0.167\pm0.008$
SiIII$\lambda1206$ $1.630$ $0.740\pm0.018$ $0.757\pm0.018$ $0.769\pm0.015$ $0.751\pm0.015$ $0.762\pm0.019$
$\log({\bar N}_{\rm HI} / {\rm {\, {\rm cm}}^{-2}}) $ 20.06 20.22 20.40 20.65 21.13
------------------------------------------------------- --------- ------------------------ ------------------------ ------------------------ ------------------------ ------------------------
Transition $f$ $W$ (${\, {\rm \AA}}$) $W$ (${\, {\rm \AA}}$) $W$ (${\, {\rm \AA}}$) $W$ (${\, {\rm \AA}}$) $W$ (${\, {\rm \AA}}$)
CIV$\lambda1548$ $0.190$ $0.575\pm0.017$ $0.585\pm0.013$ $0.559\pm0.012$ $0.558\pm0.011$ $0.553\pm0.010$
CIV$\lambda1550$ $0.095$ $0.409\pm0.014$ $0.419\pm0.011$ $0.388\pm0.010$ $0.391\pm0.009$ $0.373\pm0.008$
NV$\lambda1238$ $0.156$ $0.045\pm0.019$ $0.077\pm0.015$ $0.020\pm0.016$ $0.044\pm0.012$ $0.000\pm0.013$
NV$\lambda1242$ $0.078$ $0.011\pm0.017$ $0.023\pm0.016$ $0.029\pm0.013$ $0.021\pm0.014$ $0.000\pm0.014$
OVI$\lambda1031$ $0.133$ $0.410\pm0.037$ $0.398\pm0.036$ $0.385\pm0.032$ $0.396\pm0.030$ $0.481\pm0.031$
OVI$\lambda1037$ $0.066$ $0.260\pm0.028$ $0.270\pm0.031$ $0.204\pm0.023$ $0.256\pm0.021$ $0.310\pm0.023$
SiIV$\lambda1393$ $0.513$ $0.406\pm0.013$ $0.410\pm0.011$ $0.391\pm0.010$ $0.395\pm0.010$ $0.399\pm0.010$
SiIV$\lambda1402$ $0.254$ $0.276\pm0.012$ $0.291\pm0.012$ $0.253\pm0.009$ $0.263\pm0.009$ $0.269\pm0.008$
$ \bar z_{\rm DLA}$ 2.12 2.32 2.50 2.49 2.46
--------------------- --------- ------------------------ ------------------------ ------------------------ ------------------------ ------------------------
Transition $f$ $W$ (${\, {\rm \AA}}$) $W$ (${\, {\rm \AA}}$) $W$ (${\, {\rm \AA}}$) $W$ (${\, {\rm \AA}}$) $W$ (${\, {\rm \AA}}$)
AlII$\lambda1670$ $1.740$ $0.490\pm0.009$ $0.474\pm0.012$ $0.462\pm0.010$ $0.422\pm0.011$ $0.296\pm0.010$
AlIII$\lambda1854$ $0.559$ $0.151\pm0.008$ $0.133\pm0.009$ $0.116\pm0.009$ $0.090\pm0.011$ $0.067\pm0.011$
AlIII$\lambda1862$ $0.278$ $0.091\pm0.009$ $0.077\pm0.009$ $0.064\pm0.009$ $0.054\pm0.012$ $0.035\pm0.011$
CIII$\lambda977$ $0.757$ $-$ $-$ $0.809\pm0.118$ $0.655\pm0.034$ $0.607\pm0.022$
CrII$\lambda2056$ $0.103$ $0.028\pm0.012$ $0.029\pm0.013$ $0.030\pm0.012$ $0.026\pm0.017$ $0.023\pm0.022$
FeII$\lambda1081$ $0.013$ $-$ $0.058\pm0.025$ $0.043\pm0.016$ $0.028\pm0.015$ $0.010\pm0.015$
FeII$\lambda1096$ $0.032$ $-$ $0.084\pm0.021$ $0.100\pm0.016$ $0.076\pm0.017$ $0.047\pm0.015$
FeII$\lambda1608$ $0.058$ $0.254\pm0.006$ $0.239\pm0.009$ $0.228\pm0.008$ $0.204\pm0.009$ $0.133\pm0.007$
FeII$\lambda2249$ $0.002$ $0.038\pm0.017$ $0.035\pm0.016$ $0.040\pm0.016$ $0.018\pm0.021$ $-$
FeII$\lambda2260$ $0.002$ $0.051\pm0.018$ $0.039\pm0.016$ $0.046\pm0.017$ $0.053\pm0.021$ $0.056\pm0.043$
FeII$\lambda2344$ $0.114$ $0.553\pm0.019$ $0.542\pm0.019$ $0.517\pm0.019$ $0.481\pm0.024$ $0.392\pm0.070$
FeII$\lambda2374$ $0.031$ $0.332\pm0.019$ $0.315\pm0.019$ $0.298\pm0.020$ $0.233\pm0.030$ $0.148\pm0.067$
FeII$\lambda2382$ $0.320$ $0.756\pm0.020$ $0.742\pm0.021$ $0.717\pm0.019$ $0.667\pm0.028$ $0.516\pm0.075$
FeII$\lambda2586$ $0.069$ $0.534\pm0.025$ $0.505\pm0.025$ $0.456\pm0.029$ $0.333\pm0.048$ $-$
FeII$\lambda2600$ $0.239$ $0.768\pm0.024$ $0.748\pm0.025$ $0.682\pm0.027$ $0.556\pm0.081$ $-$
MgI$\lambda2852$ $1.830$ $0.228\pm0.029$ $0.205\pm0.038$ $0.329\pm0.059$ $-$ $-$
MgII$\lambda2796$ $0.616$ $1.267\pm0.028$ $1.168\pm0.034$ $1.117\pm0.032$ $-$ $-$
MgII$\lambda2803$ $0.306$ $1.117\pm0.027$ $1.046\pm0.034$ $1.127\pm0.042$ $-$ $-$
MnII$\lambda2576$ $0.361$ $0.056\pm0.025$ $0.041\pm0.023$ $0.035\pm0.025$ $0.017\pm0.050$ $-$
NII$\lambda1084$ $0.111$ $-$ $0.207\pm0.025$ $0.187\pm0.016$ $0.174\pm0.013$ $0.105\pm0.014$
NiII$\lambda1317$ $0.057$ $0.036\pm0.009$ $0.041\pm0.010$ $0.031\pm0.009$ $0.019\pm0.011$ $0.003\pm0.015$
NiII$\lambda1370$ $0.077$ $0.019\pm0.008$ $0.035\pm0.010$ $0.023\pm0.009$ $0.019\pm0.008$ $0.018\pm0.010$
NiII$\lambda1709$ $0.032$ $0.020\pm0.007$ $0.029\pm0.008$ $0.019\pm0.007$ $0.021\pm0.009$ $0.015\pm0.007$
NiII$\lambda1741$ $0.043$ $0.034\pm0.007$ $0.038\pm0.008$ $0.023\pm0.006$ $0.027\pm0.009$ $0.017\pm0.008$
NiII$\lambda1751$ $0.028$ $0.026\pm0.007$ $0.033\pm0.009$ $0.027\pm0.008$ $0.020\pm0.010$ $0.010\pm0.009$
OI$\lambda1302$ $0.048$ $0.445\pm0.012$ $0.412\pm0.013$ $0.403\pm0.011$ $0.385\pm0.012$ $0.297\pm0.016$
SII$\lambda1250$ $0.005$ $0.031\pm0.010$ $0.045\pm0.010$ $0.045\pm0.010$ $0.019\pm0.013$ $0.037\pm0.015$
SII$\lambda1253$ $0.011$ $0.067\pm0.007$ $0.057\pm0.009$ $0.055\pm0.010$ $0.053\pm0.010$ $0.040\pm0.016$
SIII$\lambda1012$ $0.044$ $-$ $-$ $0.060\pm0.042$ $0.064\pm0.026$ $0.045\pm0.021$
SiII$\lambda1020$ $0.017$ $-$ $-$ $0.109\pm0.030$ $0.075\pm0.023$ $0.088\pm0.021$
SiII$\lambda1304$ $0.086$ $0.378\pm0.011$ $0.352\pm0.012$ $0.327\pm0.011$ $0.322\pm0.011$ $0.224\pm0.016$
SiII$\lambda1526$ $0.133$ $0.469\pm0.007$ $0.461\pm0.009$ $0.440\pm0.008$ $0.428\pm0.010$ $0.289\pm0.008$
SiII$\lambda1808$ $0.002$ $0.076\pm0.008$ $0.071\pm0.009$ $0.058\pm0.008$ $0.050\pm0.011$ $0.013\pm0.011$
SiIII$\lambda1206$ $1.630$ $0.576\pm0.013$ $0.541\pm0.011$ $0.540\pm0.011$ $0.529\pm0.015$ $0.371\pm0.019$
$ \bar z_{\rm DLA}$ 2.12 2.32 2.50 2.49 2.46
--------------------- --------- ------------------------ ------------------------ ------------------------ ------------------------ ------------------------
Transition $f$ $W$ (${\, {\rm \AA}}$) $W$ (${\, {\rm \AA}}$) $W$ (${\, {\rm \AA}}$) $W$ (${\, {\rm \AA}}$) $W$ (${\, {\rm \AA}}$)
CIV$\lambda1548$ $0.190$ $0.480\pm0.008$ $0.452\pm0.009$ $0.418\pm0.008$ $0.377\pm0.009$ $0.264\pm0.008$
CIV$\lambda1550$ $0.095$ $0.361\pm0.007$ $0.341\pm0.007$ $0.307\pm0.007$ $0.269\pm0.009$ $0.195\pm0.008$
NV$\lambda1238$ $0.156$ $0.029\pm0.010$ $0.008\pm0.012$ $0.025\pm0.011$ $0.030\pm0.013$ $0.005\pm0.019$
NV$\lambda1242$ $0.078$ $0.006\pm0.010$ $0.011\pm0.011$ $0.016\pm0.012$ $0.011\pm0.015$ $0.014\pm0.021$
SiIV$\lambda1393$ $0.513$ $0.325\pm0.008$ $0.298\pm0.008$ $0.272\pm0.008$ $0.263\pm0.007$ $0.180\pm0.011$
SiIV$\lambda1402$ $0.254$ $0.220\pm0.008$ $0.197\pm0.008$ $0.180\pm0.007$ $0.169\pm0.008$ $0.118\pm0.010$
$\bar z_{\rm DLA}$ 2.12 2.32 2.50 2.49 2.46
-------------------- --------- ------------------------ ------------------------ ------------------------ ------------------------ ------------------------
Transition $f$ $W$ (${\, {\rm \AA}}$) $W$ (${\, {\rm \AA}}$) $W$ (${\, {\rm \AA}}$) $W$ (${\, {\rm \AA}}$) $W$ (${\, {\rm \AA}}$)
AlII$\lambda1670$ $1.740$ $0.671\pm0.009$ $0.666\pm0.012$ $0.666\pm0.012$ $0.655\pm0.015$ $0.564\pm0.014$
AlIII$\lambda1854$ $0.559$ $0.206\pm0.009$ $0.198\pm0.010$ $0.169\pm0.010$ $0.140\pm0.012$ $0.128\pm0.011$
AlIII$\lambda1862$ $0.278$ $0.119\pm0.011$ $0.109\pm0.011$ $0.097\pm0.010$ $0.080\pm0.012$ $0.061\pm0.010$
CIII$\lambda977$ $0.757$ $-$ $-$ $1.198\pm0.159$ $0.904\pm0.040$ $0.896\pm0.031$
CrII$\lambda2056$ $0.103$ $0.031\pm0.015$ $0.047\pm0.014$ $0.047\pm0.013$ $0.035\pm0.017$ $0.056\pm0.021$
FeII$\lambda1081$ $0.013$ $-$ $0.082\pm0.027$ $0.071\pm0.017$ $0.031\pm0.018$ $0.029\pm0.016$
FeII$\lambda1096$ $0.032$ $-$ $0.116\pm0.021$ $0.136\pm0.018$ $0.132\pm0.019$ $0.090\pm0.021$
FeII$\lambda1608$ $0.058$ $0.341\pm0.008$ $0.330\pm0.012$ $0.335\pm0.010$ $0.309\pm0.011$ $0.257\pm0.009$
FeII$\lambda2249$ $0.002$ $0.053\pm0.021$ $0.056\pm0.017$ $0.056\pm0.018$ $0.042\pm0.022$ $-$
FeII$\lambda2260$ $0.002$ $0.073\pm0.021$ $0.061\pm0.018$ $0.068\pm0.017$ $0.074\pm0.024$ $0.039\pm0.034$
FeII$\lambda2344$ $0.114$ $0.750\pm0.020$ $0.751\pm0.019$ $0.737\pm0.021$ $0.715\pm0.026$ $0.583\pm0.086$
FeII$\lambda2374$ $0.031$ $0.451\pm0.021$ $0.438\pm0.019$ $0.443\pm0.022$ $0.382\pm0.030$ $0.254\pm0.079$
FeII$\lambda2382$ $0.320$ $1.016\pm0.021$ $1.020\pm0.022$ $1.004\pm0.021$ $0.966\pm0.030$ $0.786\pm0.098$
FeII$\lambda2586$ $0.069$ $0.716\pm0.027$ $0.713\pm0.026$ $0.674\pm0.029$ $0.619\pm0.058$ $-$
FeII$\lambda2600$ $0.239$ $1.048\pm0.026$ $1.049\pm0.024$ $0.992\pm0.030$ $0.844\pm0.129$ $-$
MgI$\lambda2852$ $1.830$ $0.295\pm0.035$ $0.282\pm0.040$ $0.389\pm0.065$ $-$ $-$
MgII$\lambda2796$ $0.616$ $1.688\pm0.029$ $1.656\pm0.037$ $1.604\pm0.042$ $-$ $-$
MgII$\lambda2803$ $0.306$ $1.501\pm0.029$ $1.493\pm0.033$ $1.550\pm0.045$ $-$ $-$
MnII$\lambda2576$ $0.361$ $0.073\pm0.029$ $0.078\pm0.025$ $0.043\pm0.028$ $-$ $-$
NII$\lambda1084$ $0.111$ $-$ $0.281\pm0.030$ $0.274\pm0.018$ $0.255\pm0.016$ $0.193\pm0.020$
NiII$\lambda1317$ $0.057$ $0.040\pm0.011$ $0.059\pm0.012$ $0.041\pm0.012$ $0.036\pm0.016$ $0.012\pm0.017$
NiII$\lambda1370$ $0.077$ $0.033\pm0.009$ $0.045\pm0.012$ $0.033\pm0.011$ $0.033\pm0.013$ $0.033\pm0.015$
NiII$\lambda1709$ $0.032$ $0.035\pm0.007$ $0.040\pm0.010$ $0.029\pm0.009$ $0.028\pm0.009$ $0.027\pm0.009$
NiII$\lambda1741$ $0.043$ $0.048\pm0.009$ $0.050\pm0.009$ $0.036\pm0.008$ $0.036\pm0.010$ $0.039\pm0.008$
NiII$\lambda1751$ $0.028$ $0.035\pm0.009$ $0.042\pm0.011$ $0.035\pm0.008$ $0.028\pm0.010$ $0.027\pm0.009$
OI$\lambda1302$ $0.048$ $0.568\pm0.012$ $0.555\pm0.015$ $0.562\pm0.013$ $0.567\pm0.017$ $0.508\pm0.019$
SII$\lambda1250$ $0.005$ $0.051\pm0.010$ $0.075\pm0.013$ $0.065\pm0.012$ $0.035\pm0.017$ $0.068\pm0.026$
SII$\lambda1253$ $0.011$ $0.093\pm0.009$ $0.076\pm0.012$ $0.079\pm0.011$ $0.094\pm0.014$ $0.047\pm0.021$
SIII$\lambda1012$ $0.044$ $-$ $-$ $0.083\pm0.061$ $0.104\pm0.032$ $0.075\pm0.027$
SiII$\lambda1020$ $0.017$ $-$ $-$ $0.156\pm0.040$ $0.128\pm0.027$ $0.133\pm0.024$
SiII$\lambda1304$ $0.086$ $0.492\pm0.012$ $0.478\pm0.013$ $0.457\pm0.012$ $0.458\pm0.016$ $0.391\pm0.019$
SiII$\lambda1526$ $0.133$ $0.634\pm0.009$ $0.634\pm0.010$ $0.624\pm0.011$ $0.634\pm0.012$ $0.557\pm0.013$
SiII$\lambda1808$ $0.002$ $0.102\pm0.010$ $0.099\pm0.010$ $0.085\pm0.009$ $0.077\pm0.010$ $0.051\pm0.011$
SiIII$\lambda1206$ $1.630$ $0.763\pm0.014$ $0.730\pm0.017$ $0.736\pm0.015$ $0.763\pm0.019$ $0.665\pm0.023$
$ \bar z_{\rm DLA}$ 2.12 2.32 2.50 2.49 2.46
--------------------- --------- ------------------------ ------------------------ ------------------------ ------------------------ ------------------------
Transition $f$ $W$ (${\, {\rm \AA}}$) $W$ (${\, {\rm \AA}}$) $W$ (${\, {\rm \AA}}$) $W$ (${\, {\rm \AA}}$) $W$ (${\, {\rm \AA}}$)
CIV$\lambda1548$ $0.190$ $0.608\pm0.010$ $0.582\pm0.011$ $0.550\pm0.011$ $0.512\pm0.013$ $0.444\pm0.015$
CIV$\lambda1550$ $0.095$ $0.463\pm0.008$ $0.438\pm0.011$ $0.404\pm0.009$ $0.358\pm0.012$ $0.317\pm0.013$
NV$\lambda1238$ $0.156$ $0.034\pm0.012$ $0.021\pm0.013$ $0.029\pm0.016$ $0.035\pm0.018$ $0.017\pm0.024$
NV$\lambda1242$ $0.078$ $0.004\pm0.013$ $0.004\pm0.013$ $0.014\pm0.017$ $0.007\pm0.018$ $0.040\pm0.025$
SiIV$\lambda1393$ $0.513$ $0.427\pm0.009$ $0.402\pm0.012$ $0.381\pm0.010$ $0.377\pm0.010$ $0.328\pm0.018$
SiIV$\lambda1402$ $0.254$ $0.291\pm0.008$ $0.269\pm0.011$ $0.260\pm0.007$ $0.248\pm0.012$ $0.207\pm0.014$
Appendix B: Absorption line fitting {#sec:linefit}
===================================
Figures \[fig:fit\] and \[fig:fit2\] display the result of the profile fits of every absorption feature analysed in this work.
![ Absorption line profile fits for the total sample, denoting transmission as a function of DLA rest-frame wavelength (in ${\rm \AA}$). *Black thin line*: final stacked spectrum, with points indicating the data pixels. *Green thick line*: fitted continuum. *Magenta thick line*: Total fitted profile. Other lines (*dotted thin blue and yellow, and solid red*) represent the fits of individual absorption features. Good fits are generally obtained for the central parts of the lines. In some of the strong lines, the observed tails are broader than the fits, possibly due to the intrinsic velocity dispersions of the absorption systems. The Ly$\beta$ and Ly$\gamma$ lines are strong and have substantial damped wings, which explains why the fit is substandard in these cases. The fit to the SiII$\lambda1020$ line does not account for the blend with the Ly$\beta$ line. The transmission scales vary with each panel to maximize the visibility of the lines.[]{data-label="fig:fit"}](fit_972v2.pdf "fig:"){width="25.00000%"}![ Absorption line profile fits for the total sample, denoting transmission as a function of DLA rest-frame wavelength (in ${\rm \AA}$). *Black thin line*: final stacked spectrum, with points indicating the data pixels. *Green thick line*: fitted continuum. *Magenta thick line*: Total fitted profile. Other lines (*dotted thin blue and yellow, and solid red*) represent the fits of individual absorption features. Good fits are generally obtained for the central parts of the lines. In some of the strong lines, the observed tails are broader than the fits, possibly due to the intrinsic velocity dispersions of the absorption systems. The Ly$\beta$ and Ly$\gamma$ lines are strong and have substantial damped wings, which explains why the fit is substandard in these cases. The fit to the SiII$\lambda1020$ line does not account for the blend with the Ly$\beta$ line. The transmission scales vary with each panel to maximize the visibility of the lines.[]{data-label="fig:fit"}](fit_988v2.pdf "fig:"){width="25.00000%"}![ Absorption line profile fits for the total sample, denoting transmission as a function of DLA rest-frame wavelength (in ${\rm \AA}$). *Black thin line*: final stacked spectrum, with points indicating the data pixels. *Green thick line*: fitted continuum. *Magenta thick line*: Total fitted profile. Other lines (*dotted thin blue and yellow, and solid red*) represent the fits of individual absorption features. Good fits are generally obtained for the central parts of the lines. In some of the strong lines, the observed tails are broader than the fits, possibly due to the intrinsic velocity dispersions of the absorption systems. The Ly$\beta$ and Ly$\gamma$ lines are strong and have substantial damped wings, which explains why the fit is substandard in these cases. The fit to the SiII$\lambda1020$ line does not account for the blend with the Ly$\beta$ line. The transmission scales vary with each panel to maximize the visibility of the lines.[]{data-label="fig:fit"}](fit_1012v2.pdf "fig:"){width="25.00000%"}![ Absorption line profile fits for the total sample, denoting transmission as a function of DLA rest-frame wavelength (in ${\rm \AA}$). *Black thin line*: final stacked spectrum, with points indicating the data pixels. *Green thick line*: fitted continuum. *Magenta thick line*: Total fitted profile. Other lines (*dotted thin blue and yellow, and solid red*) represent the fits of individual absorption features. Good fits are generally obtained for the central parts of the lines. In some of the strong lines, the observed tails are broader than the fits, possibly due to the intrinsic velocity dispersions of the absorption systems. The Ly$\beta$ and Ly$\gamma$ lines are strong and have substantial damped wings, which explains why the fit is substandard in these cases. The fit to the SiII$\lambda1020$ line does not account for the blend with the Ly$\beta$ line. The transmission scales vary with each panel to maximize the visibility of the lines.[]{data-label="fig:fit"}](fit_1020v2.pdf "fig:"){width="25.00000%"} ![ Absorption line profile fits for the total sample, denoting transmission as a function of DLA rest-frame wavelength (in ${\rm \AA}$). *Black thin line*: final stacked spectrum, with points indicating the data pixels. *Green thick line*: fitted continuum. *Magenta thick line*: Total fitted profile. Other lines (*dotted thin blue and yellow, and solid red*) represent the fits of individual absorption features. Good fits are generally obtained for the central parts of the lines. In some of the strong lines, the observed tails are broader than the fits, possibly due to the intrinsic velocity dispersions of the absorption systems. The Ly$\beta$ and Ly$\gamma$ lines are strong and have substantial damped wings, which explains why the fit is substandard in these cases. The fit to the SiII$\lambda1020$ line does not account for the blend with the Ly$\beta$ line. The transmission scales vary with each panel to maximize the visibility of the lines.[]{data-label="fig:fit"}](fit_1031v2.pdf "fig:"){width="25.00000%"}![ Absorption line profile fits for the total sample, denoting transmission as a function of DLA rest-frame wavelength (in ${\rm \AA}$). *Black thin line*: final stacked spectrum, with points indicating the data pixels. *Green thick line*: fitted continuum. *Magenta thick line*: Total fitted profile. Other lines (*dotted thin blue and yellow, and solid red*) represent the fits of individual absorption features. Good fits are generally obtained for the central parts of the lines. In some of the strong lines, the observed tails are broader than the fits, possibly due to the intrinsic velocity dispersions of the absorption systems. The Ly$\beta$ and Ly$\gamma$ lines are strong and have substantial damped wings, which explains why the fit is substandard in these cases. The fit to the SiII$\lambda1020$ line does not account for the blend with the Ly$\beta$ line. The transmission scales vary with each panel to maximize the visibility of the lines.[]{data-label="fig:fit"}](fit_1036v2.pdf "fig:"){width="25.00000%"}![ Absorption line profile fits for the total sample, denoting transmission as a function of DLA rest-frame wavelength (in ${\rm \AA}$). *Black thin line*: final stacked spectrum, with points indicating the data pixels. *Green thick line*: fitted continuum. *Magenta thick line*: Total fitted profile. Other lines (*dotted thin blue and yellow, and solid red*) represent the fits of individual absorption features. Good fits are generally obtained for the central parts of the lines. In some of the strong lines, the observed tails are broader than the fits, possibly due to the intrinsic velocity dispersions of the absorption systems. The Ly$\beta$ and Ly$\gamma$ lines are strong and have substantial damped wings, which explains why the fit is substandard in these cases. The fit to the SiII$\lambda1020$ line does not account for the blend with the Ly$\beta$ line. The transmission scales vary with each panel to maximize the visibility of the lines.[]{data-label="fig:fit"}](fit_1062v2.pdf "fig:"){width="25.00000%"}![ Absorption line profile fits for the total sample, denoting transmission as a function of DLA rest-frame wavelength (in ${\rm \AA}$). *Black thin line*: final stacked spectrum, with points indicating the data pixels. *Green thick line*: fitted continuum. *Magenta thick line*: Total fitted profile. Other lines (*dotted thin blue and yellow, and solid red*) represent the fits of individual absorption features. Good fits are generally obtained for the central parts of the lines. In some of the strong lines, the observed tails are broader than the fits, possibly due to the intrinsic velocity dispersions of the absorption systems. The Ly$\beta$ and Ly$\gamma$ lines are strong and have substantial damped wings, which explains why the fit is substandard in these cases. The fit to the SiII$\lambda1020$ line does not account for the blend with the Ly$\beta$ line. The transmission scales vary with each panel to maximize the visibility of the lines.[]{data-label="fig:fit"}](fit_1081v2.pdf "fig:"){width="25.00000%"} ![ Absorption line profile fits for the total sample, denoting transmission as a function of DLA rest-frame wavelength (in ${\rm \AA}$). *Black thin line*: final stacked spectrum, with points indicating the data pixels. *Green thick line*: fitted continuum. *Magenta thick line*: Total fitted profile. Other lines (*dotted thin blue and yellow, and solid red*) represent the fits of individual absorption features. Good fits are generally obtained for the central parts of the lines. In some of the strong lines, the observed tails are broader than the fits, possibly due to the intrinsic velocity dispersions of the absorption systems. The Ly$\beta$ and Ly$\gamma$ lines are strong and have substantial damped wings, which explains why the fit is substandard in these cases. The fit to the SiII$\lambda1020$ line does not account for the blend with the Ly$\beta$ line. The transmission scales vary with each panel to maximize the visibility of the lines.[]{data-label="fig:fit"}](fit_1096v2.pdf "fig:"){width="25.00000%"}![ Absorption line profile fits for the total sample, denoting transmission as a function of DLA rest-frame wavelength (in ${\rm \AA}$). *Black thin line*: final stacked spectrum, with points indicating the data pixels. *Green thick line*: fitted continuum. *Magenta thick line*: Total fitted profile. Other lines (*dotted thin blue and yellow, and solid red*) represent the fits of individual absorption features. Good fits are generally obtained for the central parts of the lines. In some of the strong lines, the observed tails are broader than the fits, possibly due to the intrinsic velocity dispersions of the absorption systems. The Ly$\beta$ and Ly$\gamma$ lines are strong and have substantial damped wings, which explains why the fit is substandard in these cases. The fit to the SiII$\lambda1020$ line does not account for the blend with the Ly$\beta$ line. The transmission scales vary with each panel to maximize the visibility of the lines.[]{data-label="fig:fit"}](fit_1121v2.pdf "fig:"){width="25.00000%"}![ Absorption line profile fits for the total sample, denoting transmission as a function of DLA rest-frame wavelength (in ${\rm \AA}$). *Black thin line*: final stacked spectrum, with points indicating the data pixels. *Green thick line*: fitted continuum. *Magenta thick line*: Total fitted profile. Other lines (*dotted thin blue and yellow, and solid red*) represent the fits of individual absorption features. Good fits are generally obtained for the central parts of the lines. In some of the strong lines, the observed tails are broader than the fits, possibly due to the intrinsic velocity dispersions of the absorption systems. The Ly$\beta$ and Ly$\gamma$ lines are strong and have substantial damped wings, which explains why the fit is substandard in these cases. The fit to the SiII$\lambda1020$ line does not account for the blend with the Ly$\beta$ line. The transmission scales vary with each panel to maximize the visibility of the lines.[]{data-label="fig:fit"}](fit_1133v2.pdf "fig:"){width="25.00000%"}![ Absorption line profile fits for the total sample, denoting transmission as a function of DLA rest-frame wavelength (in ${\rm \AA}$). *Black thin line*: final stacked spectrum, with points indicating the data pixels. *Green thick line*: fitted continuum. *Magenta thick line*: Total fitted profile. Other lines (*dotted thin blue and yellow, and solid red*) represent the fits of individual absorption features. Good fits are generally obtained for the central parts of the lines. In some of the strong lines, the observed tails are broader than the fits, possibly due to the intrinsic velocity dispersions of the absorption systems. The Ly$\beta$ and Ly$\gamma$ lines are strong and have substantial damped wings, which explains why the fit is substandard in these cases. The fit to the SiII$\lambda1020$ line does not account for the blend with the Ly$\beta$ line. The transmission scales vary with each panel to maximize the visibility of the lines.[]{data-label="fig:fit"}](fit_1142v2.pdf "fig:"){width="25.00000%"} ![ Absorption line profile fits for the total sample, denoting transmission as a function of DLA rest-frame wavelength (in ${\rm \AA}$). *Black thin line*: final stacked spectrum, with points indicating the data pixels. *Green thick line*: fitted continuum. *Magenta thick line*: Total fitted profile. Other lines (*dotted thin blue and yellow, and solid red*) represent the fits of individual absorption features. Good fits are generally obtained for the central parts of the lines. In some of the strong lines, the observed tails are broader than the fits, possibly due to the intrinsic velocity dispersions of the absorption systems. The Ly$\beta$ and Ly$\gamma$ lines are strong and have substantial damped wings, which explains why the fit is substandard in these cases. The fit to the SiII$\lambda1020$ line does not account for the blend with the Ly$\beta$ line. The transmission scales vary with each panel to maximize the visibility of the lines.[]{data-label="fig:fit"}](fit_1152v2.pdf "fig:"){width="25.00000%"}![ Absorption line profile fits for the total sample, denoting transmission as a function of DLA rest-frame wavelength (in ${\rm \AA}$). *Black thin line*: final stacked spectrum, with points indicating the data pixels. *Green thick line*: fitted continuum. *Magenta thick line*: Total fitted profile. Other lines (*dotted thin blue and yellow, and solid red*) represent the fits of individual absorption features. Good fits are generally obtained for the central parts of the lines. In some of the strong lines, the observed tails are broader than the fits, possibly due to the intrinsic velocity dispersions of the absorption systems. The Ly$\beta$ and Ly$\gamma$ lines are strong and have substantial damped wings, which explains why the fit is substandard in these cases. The fit to the SiII$\lambda1020$ line does not account for the blend with the Ly$\beta$ line. The transmission scales vary with each panel to maximize the visibility of the lines.[]{data-label="fig:fit"}](fit_1190v2.pdf "fig:"){width="25.00000%"}![ Absorption line profile fits for the total sample, denoting transmission as a function of DLA rest-frame wavelength (in ${\rm \AA}$). *Black thin line*: final stacked spectrum, with points indicating the data pixels. *Green thick line*: fitted continuum. *Magenta thick line*: Total fitted profile. Other lines (*dotted thin blue and yellow, and solid red*) represent the fits of individual absorption features. Good fits are generally obtained for the central parts of the lines. In some of the strong lines, the observed tails are broader than the fits, possibly due to the intrinsic velocity dispersions of the absorption systems. The Ly$\beta$ and Ly$\gamma$ lines are strong and have substantial damped wings, which explains why the fit is substandard in these cases. The fit to the SiII$\lambda1020$ line does not account for the blend with the Ly$\beta$ line. The transmission scales vary with each panel to maximize the visibility of the lines.[]{data-label="fig:fit"}](fit_1199v2.pdf "fig:"){width="25.00000%"}![ Absorption line profile fits for the total sample, denoting transmission as a function of DLA rest-frame wavelength (in ${\rm \AA}$). *Black thin line*: final stacked spectrum, with points indicating the data pixels. *Green thick line*: fitted continuum. *Magenta thick line*: Total fitted profile. Other lines (*dotted thin blue and yellow, and solid red*) represent the fits of individual absorption features. Good fits are generally obtained for the central parts of the lines. In some of the strong lines, the observed tails are broader than the fits, possibly due to the intrinsic velocity dispersions of the absorption systems. The Ly$\beta$ and Ly$\gamma$ lines are strong and have substantial damped wings, which explains why the fit is substandard in these cases. The fit to the SiII$\lambda1020$ line does not account for the blend with the Ly$\beta$ line. The transmission scales vary with each panel to maximize the visibility of the lines.[]{data-label="fig:fit"}](fit_1206v2.pdf "fig:"){width="25.00000%"} ![ Absorption line profile fits for the total sample, denoting transmission as a function of DLA rest-frame wavelength (in ${\rm \AA}$). *Black thin line*: final stacked spectrum, with points indicating the data pixels. *Green thick line*: fitted continuum. *Magenta thick line*: Total fitted profile. Other lines (*dotted thin blue and yellow, and solid red*) represent the fits of individual absorption features. Good fits are generally obtained for the central parts of the lines. In some of the strong lines, the observed tails are broader than the fits, possibly due to the intrinsic velocity dispersions of the absorption systems. The Ly$\beta$ and Ly$\gamma$ lines are strong and have substantial damped wings, which explains why the fit is substandard in these cases. The fit to the SiII$\lambda1020$ line does not account for the blend with the Ly$\beta$ line. The transmission scales vary with each panel to maximize the visibility of the lines.[]{data-label="fig:fit"}](fit_1238v2.pdf "fig:"){width="25.00000%"}![ Absorption line profile fits for the total sample, denoting transmission as a function of DLA rest-frame wavelength (in ${\rm \AA}$). *Black thin line*: final stacked spectrum, with points indicating the data pixels. *Green thick line*: fitted continuum. *Magenta thick line*: Total fitted profile. Other lines (*dotted thin blue and yellow, and solid red*) represent the fits of individual absorption features. Good fits are generally obtained for the central parts of the lines. In some of the strong lines, the observed tails are broader than the fits, possibly due to the intrinsic velocity dispersions of the absorption systems. The Ly$\beta$ and Ly$\gamma$ lines are strong and have substantial damped wings, which explains why the fit is substandard in these cases. The fit to the SiII$\lambda1020$ line does not account for the blend with the Ly$\beta$ line. The transmission scales vary with each panel to maximize the visibility of the lines.[]{data-label="fig:fit"}](fit_1250v2.pdf "fig:"){width="25.00000%"}![ Absorption line profile fits for the total sample, denoting transmission as a function of DLA rest-frame wavelength (in ${\rm \AA}$). *Black thin line*: final stacked spectrum, with points indicating the data pixels. *Green thick line*: fitted continuum. *Magenta thick line*: Total fitted profile. Other lines (*dotted thin blue and yellow, and solid red*) represent the fits of individual absorption features. Good fits are generally obtained for the central parts of the lines. In some of the strong lines, the observed tails are broader than the fits, possibly due to the intrinsic velocity dispersions of the absorption systems. The Ly$\beta$ and Ly$\gamma$ lines are strong and have substantial damped wings, which explains why the fit is substandard in these cases. The fit to the SiII$\lambda1020$ line does not account for the blend with the Ly$\beta$ line. The transmission scales vary with each panel to maximize the visibility of the lines.[]{data-label="fig:fit"}](fit_1260v2.pdf "fig:"){width="25.00000%"}![ Absorption line profile fits for the total sample, denoting transmission as a function of DLA rest-frame wavelength (in ${\rm \AA}$). *Black thin line*: final stacked spectrum, with points indicating the data pixels. *Green thick line*: fitted continuum. *Magenta thick line*: Total fitted profile. Other lines (*dotted thin blue and yellow, and solid red*) represent the fits of individual absorption features. Good fits are generally obtained for the central parts of the lines. In some of the strong lines, the observed tails are broader than the fits, possibly due to the intrinsic velocity dispersions of the absorption systems. The Ly$\beta$ and Ly$\gamma$ lines are strong and have substantial damped wings, which explains why the fit is substandard in these cases. The fit to the SiII$\lambda1020$ line does not account for the blend with the Ly$\beta$ line. The transmission scales vary with each panel to maximize the visibility of the lines.[]{data-label="fig:fit"}](fit_1302v2.pdf "fig:"){width="25.00000%"} ![ Absorption line profile fits for the total sample, denoting transmission as a function of DLA rest-frame wavelength (in ${\rm \AA}$). *Black thin line*: final stacked spectrum, with points indicating the data pixels. *Green thick line*: fitted continuum. *Magenta thick line*: Total fitted profile. Other lines (*dotted thin blue and yellow, and solid red*) represent the fits of individual absorption features. Good fits are generally obtained for the central parts of the lines. In some of the strong lines, the observed tails are broader than the fits, possibly due to the intrinsic velocity dispersions of the absorption systems. The Ly$\beta$ and Ly$\gamma$ lines are strong and have substantial damped wings, which explains why the fit is substandard in these cases. The fit to the SiII$\lambda1020$ line does not account for the blend with the Ly$\beta$ line. The transmission scales vary with each panel to maximize the visibility of the lines.[]{data-label="fig:fit"}](fit_1317v2.pdf "fig:"){width="25.00000%"}![ Absorption line profile fits for the total sample, denoting transmission as a function of DLA rest-frame wavelength (in ${\rm \AA}$). *Black thin line*: final stacked spectrum, with points indicating the data pixels. *Green thick line*: fitted continuum. *Magenta thick line*: Total fitted profile. Other lines (*dotted thin blue and yellow, and solid red*) represent the fits of individual absorption features. Good fits are generally obtained for the central parts of the lines. In some of the strong lines, the observed tails are broader than the fits, possibly due to the intrinsic velocity dispersions of the absorption systems. The Ly$\beta$ and Ly$\gamma$ lines are strong and have substantial damped wings, which explains why the fit is substandard in these cases. The fit to the SiII$\lambda1020$ line does not account for the blend with the Ly$\beta$ line. The transmission scales vary with each panel to maximize the visibility of the lines.[]{data-label="fig:fit"}](fit_1334v2.pdf "fig:"){width="25.00000%"}![ Absorption line profile fits for the total sample, denoting transmission as a function of DLA rest-frame wavelength (in ${\rm \AA}$). *Black thin line*: final stacked spectrum, with points indicating the data pixels. *Green thick line*: fitted continuum. *Magenta thick line*: Total fitted profile. Other lines (*dotted thin blue and yellow, and solid red*) represent the fits of individual absorption features. Good fits are generally obtained for the central parts of the lines. In some of the strong lines, the observed tails are broader than the fits, possibly due to the intrinsic velocity dispersions of the absorption systems. The Ly$\beta$ and Ly$\gamma$ lines are strong and have substantial damped wings, which explains why the fit is substandard in these cases. The fit to the SiII$\lambda1020$ line does not account for the blend with the Ly$\beta$ line. The transmission scales vary with each panel to maximize the visibility of the lines.[]{data-label="fig:fit"}](fit_1370v2.pdf "fig:"){width="25.00000%"}![ Absorption line profile fits for the total sample, denoting transmission as a function of DLA rest-frame wavelength (in ${\rm \AA}$). *Black thin line*: final stacked spectrum, with points indicating the data pixels. *Green thick line*: fitted continuum. *Magenta thick line*: Total fitted profile. Other lines (*dotted thin blue and yellow, and solid red*) represent the fits of individual absorption features. Good fits are generally obtained for the central parts of the lines. In some of the strong lines, the observed tails are broader than the fits, possibly due to the intrinsic velocity dispersions of the absorption systems. The Ly$\beta$ and Ly$\gamma$ lines are strong and have substantial damped wings, which explains why the fit is substandard in these cases. The fit to the SiII$\lambda1020$ line does not account for the blend with the Ly$\beta$ line. The transmission scales vary with each panel to maximize the visibility of the lines.[]{data-label="fig:fit"}](fit_1393v2.pdf "fig:"){width="25.00000%"}
{width="25.00000%"}{width="25.00000%"}{width="25.00000%"}{width="25.00000%"} {width="25.00000%"}{width="25.00000%"}{width="25.00000%"}{width="25.00000%"} {width="25.00000%"}{width="25.00000%"}{width="25.00000%"}{width="25.00000%"} {width="25.00000%"}{width="25.00000%"}{width="25.00000%"}{width="25.00000%"} {width="25.00000%"}{width="25.00000%"}{width="25.00000%"}{width="25.00000%"} {width="25.00000%"}{width="25.00000%"}{width="25.00000%"}{width="25.00000%"} {width="25.00000%"}{width="25.00000%"}
\[References\]
[^1]: <http://linetools.readthedocs.io/en/latest/index.html>
|
---
abstract: |
Assume that $(X,g)$ is an $n$-dimensional smooth connected Riemannian manifold without boundary and $Y$ is an $n$-dimensional compact connected $C^0$-submanifold in $X$ with nonempty boundary $\partial Y$ ($n \ge 2$). We consider the metric function $\rho_Y(x,y)$ generated by the intrinsic metric of the interior ${\mathop{\rm Int}}Y$ of $Y$ in the following natural way: $\rho_Y(x,y) =
\liminf\limits_{x' \to x,\,y' \to x;\,\,x',y' \in {\mathop{\rm Int}}Y}
\{\inf[l(\gamma_{x',y',{\mathop{\rm Int}}Y})]\}$, where $\inf[l(\gamma_{x',y',{\mathop{\rm Int}}Y})]$ is the infimum of the lengths of smooth paths joining $x'$ and $y'$ in the interior ${\mathop{\rm Int}}Y$ of $Y$. We study conditions under which $\rho_Y$ is a metric and also the question about the existence of geodesics in the metric $\rho_Y$ and its relationship with the classical intrinsic metric of the hypersurface $\partial Y$.
author:
- 'Anatoly P. Kopylov[^1], Mikhail V. Korobkov[^2]'
title: 'On Properties of the Intrinsic Geometry of Submanifolds in a Riemannian Manifold[^3]'
---
Let $(X,g)$ be an $n$-dimensional smooth connected Riemannian manifold without boundary and let $Y$ be an $n$-dimensional compact connected $C^0$-submanifold in $X$ with nonempty boundary $\partial Y$ ($n \ge 2$). A classical object of investigations (see, for example, [@al1]) is given by the intrinsic metric $\rho_{\partial Y}$ on the hypersurface $\partial Y$ defined for $x,y \in \partial Y$ as the infimum of the lengths of curves $\nu
\subset \partial Y$ joining $x$ and $y$. In the recent decades, an alternative approach arose in the rigidity theory for submanifolds of Riemannian manifolds (see, for instance, the recent articles [@kop1], [@kop2; @kor1], which also contain a historical survey of works on the topic). In accordance with this approach, the metric on $\partial Y$ is induced by the intrinsic metric of the interior ${\mathop{\rm Int}}Y$ of the submanifold $Y$. Namely, suppose that $Y$ satisfies the following condition[^4]:
\(i) if $x,y\in Y$, then $$\rho_Y(x,y)=\liminf_{x'\to x,\;y'\to y;\;x',\;y'\in{\mathop{\rm Int}}Y}
\{\inf\,[l(\gamma_{x',\;y',\;{\mathop{\rm Int}}Y})]\}<\infty, \label{1}$$ where $\inf\,[l(\gamma_{x',\;y',\;{\mathop{\rm Int}}Y})]$ is the infimum of the lengths $l(\gamma_{x',\;y',\;{\mathop{\rm Int}}Y})$ of smooth paths $\gamma_{x',\;y',\;{\mathop{\rm Int}}Y}:[0,1]\to{\mathop{\rm Int}}Y$ joining $x'$ and $y'$ in the interior ${\mathop{\rm Int}}Y$ of $Y$.
Note that the intrinsic metric of convex hypersurfaces in $\mathbb
R^n$ (i.e., a classical object) is an important particular case of a function $\rho_Y$. (To verify that, take as $Y$ the complement of the convex hull of the hypersurface.) However, here there appear some new phenomena. The following question is of primary interest in our paper: Is the function $\rho_Y$ defined by (\[1\]) a metric on $Y$? If $n = 2$ then the answer is ‘yes’ (see Theorem 1 below) and if $n > 2$ then it is ‘no’ (see Theorem 2). Moreover, we prove that if $\rho_Y$ is a metric (for an arbitrary dimension $n \ge 2$) then any two points $x,y \in Y$ may be joined by a shortest curve (geodesic) whose length in the metric $\rho_Y$ coincides with $\rho_Y(x,y)$ (Theorem 3).
We will begin with the following result.
[**Theorem 1.**]{} [*Let $n = 2$. Then, under condition [(i),]{} $\rho_Y$ is a metric on $Y$.*]{}
[Proof.]{} It suffices to prove that $\rho_Y$ satisfies the triangle inequality. Let $A$, $O$, and $D$ be three points on the boundary of $Y$ (note that this case is basic because the other cases are simpler). Consider $\varepsilon > 0$ and assume that $\gamma_{A_{\varepsilon} O^1_{\varepsilon}} : [0,1] \to {\mathop{\rm Int}}Y$ and $\gamma_{O^2_{\varepsilon} D_{\varepsilon}} : [2,3] \to {\mathop{\rm Int}}Y$ are smooth paths with the endpoints $A_{\varepsilon} =
\gamma_{A_{\varepsilon} O^1_{\varepsilon}}(0)$, $O^1_{\varepsilon}
= \gamma_{A_{\varepsilon} O^1_{\varepsilon}}(1)$ and $D_{\varepsilon} = \gamma_{O^2_{\varepsilon} D_{\varepsilon}}(3)$, $O^2_{\varepsilon} = \gamma_{O^2_{\varepsilon}
D_{\varepsilon}}(2)$ satisfying the conditions $\rho_X(A_{\varepsilon}, A) \le \varepsilon$, $\rho_X(D_{\varepsilon}, D) \le \varepsilon$, $\rho_X(O^j_{\varepsilon}, O) \le \varepsilon$ ($j = 1;\; 2$), $|l(\gamma_{A_{\varepsilon} O^1_{\varepsilon}}) - \rho_Y(A,O)| \le
\varepsilon$, and $|l(\gamma_{O^2_{\varepsilon} D_{\varepsilon}})
- \rho_Y(O,D)| \le \varepsilon$. Let $(U, h)$ be a chart of the manifold $X$ such that $U$ is an open neighborhood of the point $O$ in $X$, $h(U)$ is the unit disk $B(0, 1) = \{(x_1, x_2) \in
\mathbb R^2 : x_1^2 + x_2^2 < 1\}$ in $\mathbb R^2$, and $h(O) =
0$ ($0 = (0, 0)$ is the origin in $\mathbb R^2$); moreover, $h : U
\to h(U)$ is a diffeomorphism having the following property: there exists a chart $(Z,\psi)$ of $Y$ with $\psi(O) = 0$, $A,D \in U
\setminus {\mathop{\rm Cl}}_X Z$ (${\mathop{\rm Cl}}_X Z$ is the closure of $Z$ in the space $(X,g)$) and $Z = \widetilde{U} \cap Y$ is the intersection of an open neighborhood $\widetilde{U}$ ($\subset U$) of $O$ in $X$ and $Y$ whose image $\psi(Z)$ under $\psi$ is the half-disk $B_+(0, 1)
= \{(x_1, x_2) \in B(0, 1) : x_1 \ge 0\}$. Suppose that $\sigma_r$ is an arc of the circle $\partial B(0, r)$ which is a connected component of the set $V \cap \partial B(0, r)$, where $V = h(Z)$ and $0 < r < r^* = \min \{|h(\psi^{-1}(x_1, x_2))| : x_1^2 + x_2^2
= 1/4, \, x_1 \ge 0\}$. Among these components, there is at least one (preserve the notation $\sigma_r$ for it) whose ends belong to the sets $h(\psi^{-1}(\{-te_2 : 0 < t < 1\}))$ and $h(\psi^{-1}(\{te_2 : 0 < t < 1\}))$ respectively. Otherwise, the closure of the connected component of the set $V \cap B(0, r)$ whose boundary contains the origin would contain a point belonging to the arc $\{e^{i\theta}/2 : |\theta| \le \pi/2\}$ (here we use the complex notation $z = re^{i\theta}$ for points $z \in \mathbb
R^2$ ($= \mathbb C$)). But this is impossible. Therefore, the above-mentioned arc $\sigma_r$ exists.
It is easy to check that if $\varepsilon$ is sufficiently small then the images of the paths $h \circ
\gamma_{A_{\varepsilon}O^1_{\varepsilon}}$ and $h \circ
\gamma_{O^2_{\varepsilon}D_{\varepsilon}}$, also intersect $\sigma_r$, i.e., there are $t_1 \in ]0,1[$, $t_2 \in
]2,3[$ such that $\gamma_{A_\varepsilon O^1_\varepsilon}(t_1) =
x^1 \in Z$, $\gamma_{O^2_\varepsilon D_\varepsilon}(t_2) = x^2 \in
Z$ and $h(x^j) \in \sigma_r$, $j = 1,2$. Let $\widetilde\gamma_r :
[t_1,t_2] \to \sigma_r$ be a smooth parametrization of the corresponding subarc of $\sigma_r$, i.e., $\widetilde\gamma_r(t_j)
= h(x^j)$, $j = 1,2$. Now we can define a mapping $\gamma_\varepsilon : [0,3] \to {\mathop{\rm Int}}Y$ by setting $$\gamma_\varepsilon(t)=\left\{ \aligned
\gamma_{A_{\varepsilon} O^1_{\varepsilon}}(t) , & \ \ t\in [0,t_1]; \\
h^{-1}(\widetilde\gamma_r(t)), & \ \ t\in]t_1,t_2[; \\
\gamma_{O^2_{\varepsilon}D_{\varepsilon} }(t) , & \ \ t\in
[t_2,3].
\endaligned \right.$$ By construction, $\gamma_\varepsilon$ is a piecewise smooth path joining the points $A_\varepsilon = \gamma_\varepsilon(0)$, $D_\varepsilon = \gamma_\varepsilon(3)$ in ${\mathop{\rm Int}}Y$; moreover, $$l(\gamma_\varepsilon) \le l(\gamma_{A_\varepsilon O^1_\varepsilon}) +
l(\gamma_{O^2_\varepsilon D_\varepsilon}) + l(h^{-1}(\sigma_r)).$$ By an appropriate choice of $\varepsilon > 0$, we can make $r> 0$ arbitrarily small, and since a piecewise smooth path can be approximated by smooth paths, we have $\rho_Y(A,D) \le \rho_Y(A,O)
+ \rho_Y(O,D)$, q.e.d.
In connection with Theorem 1, there appears a natural question: Are there analogs of this theorem for $n \ge 3$? According to the following Theorem 2, the answer to this question is negative.
[**Theorem 2.**]{} [*If $n \ge 3$ then there exists an $n$-dimensional compact connected $C^0$-manifold $Y \subset
\mathbb R^n$ with nonempty boundary $\partial Y$ such that condition*]{} (i) ([*where now $X = \mathbb R^n$*]{}) [*is fulfilled for $Y$ but the function $\rho_Y$ in this condition is not a metric on $Y$.*]{}
[Proof.]{} It suffices to consider the case of $n = 3$. Suppose that $A$, $O$, $D$ are points in $\mathbb R^3$, $O$ is the origin in $\mathbb R^3$, $|A| = |D| = 1$, and the angle between the segments $OA$ and $OD$ is equal to $\frac{\pi}{6}$.
The manifold $Y$ will be constructed so that $O \in \partial Y$, and $]O,A] \subset {\mathop{\rm Int}}Y$, $]O,D] \subset {\mathop{\rm Int}}Y$. Under these conditions, $\rho_Y(O,A) = \rho_Y(O,D) =1$. However, the boundary of $Y$ will create “obstacles” between $A$ and $D$ such that the length of any curve joining $A$ and $D$ in ${\mathop{\rm Int}}Y$ will be greater than $\frac{12}{5}$ (this means the violation of the triangle inequality for $\rho_Y$).
Consider a countable collection of mutually disjoint segments $\{I^k_j\}_{j \in \mathbb N,\, k = 1,\dots,k_j}$ lying in the interior of the triangle $6 \Delta AOD$ (which is obtained from the original triangle $\Delta AOD$ by dilation with coefficient $6$) with the following properties:
$(*)$ every segment $I^k_j =[x^k_j,y^k_j]$ lies on a ray starting at the origin, $y^k_j = 11 x^k_j$, and $|x^k_j| = 2^{-j}$;
$(**)$ For any curve $\gamma$ with ends $A$, $D$ whose interior points lie in the interior of the triangle $4 \Delta AOD$ and belong to no segment $I^k_j$, the estimate $l(\gamma) \ge 6$ holds.
The existence of such a family of segments is certain: they must be situated chequerwise so that any curve disjoint from them be sawtooth, with the total length of its “teeth” greater than $6$ (it can clearly be made greater than any prescribed positive number). However, below we exactly describe the construction.
It is easy to include the above-indicated family of segments in the boundary $\partial Y$ of $Y$. Thus, it creates a desired “obstacle” to joining $A$ and $D$ in the plane of $\Delta AOD$. But it makes no obstacle to joining $A$ and $D$ in the space. The simplest way to create such a space obstacle is as follows: Rotate each segment $I^k_j$ along a spiral around the axis $OA$. Make the number of coils so large that the length of this spiral be large and its pitch (i.e., the distance between the origin and the end of a coil) be sufficiently small. Then the set $S^k_j$ obtained as the result of the rotation of the segment $I^k_j$ is diffeomorphic to a plane rectangle, and it lies in a small neighborhood of the cone of revolution with axis $AO$ containing the segment $I^k_j$. The last circumstance guarantees that the sets $S^k_j$ are disjoint as before, and so (as above) it is easy to include them in the boundary $\partial Y$ but, due to the properties of the $I^k_j$’s and a large number of coils of the spirals $S^k_j$, any curve joining $A,D$ and disjoint from each $S^k_j$ has length $\ge \frac{12}{5}$.
We turn to an exact description of the constructions used. First describe the construction of the family of segments $I^k_j$. They are chosen on the basis of the following observation:
Let $\gamma : [0,1] \to 4 \Delta AOD$ be any curve with ends $\gamma (0) = A$, $\gamma (1) = D$ whose interior points lie in the interior of the triangle $4 \Delta AOD$. For $j \in \mathbb
N$, put $R_j = \{x \in 4 \Delta AOD : |x| \in [8 \cdot 2^{-j}, 4
\cdot 2^{-j}]\}$. It is clear that $$4 \Delta AOD \setminus \{O\} = \cup_{j \in \mathbb N} R_j.$$ Introduce the polar system of coordinates on the plane of the triangle $\Delta AOD$ with center $O$ such that the coordinates of the points $A,D$ are $r = 1$, $\varphi = 0$ and $r = 1$, $\varphi
= \frac{\pi}{6}$, respectively. Given a point $x \in 6 \Delta
AOD$, let $\varphi_x$ be the angular coordinate of $x$ in $[0,\frac{\pi}{6}]$. Let $\Phi_j = \{\varphi_{\gamma(t)} :
\gamma(t) \in R_j\}$. Obviously, there is $j_0 \in \mathbb N$ such that $$\label{2}
\qquad\qquad\qquad\qquad\qquad\mathcal H^1(\Phi_{j_0}) \ge
2^{-j_0} \frac{\pi}{6},$$ where $\mathcal H^1$ is the Hausdorff $1$-measure. This means that, while in the layer $R_{j_0}$, the curve $\gamma$ covers the angular distance $\ge 2^{-j_0} \frac{\pi}{6}$. The segments $I^k_j$ must be chosen such that (\[2\]) together with the condition $$\gamma(t) \cap I^k_j = \varnothing \quad \forall t \in [0,1]\,\,
\forall j \in \mathbb N\,\, \forall k \in \{1,\dots,k_j\}$$ give the desired estimate $l(\gamma) \ge 6$. To this end, it suffices to take $k_j = [(2 \pi)^j]$ (the integral part of $(2 \pi)^j$) and $$I^k_j = \{x \in 6 \Delta AOD : \varphi_x = k (2 \pi)^{-j} \frac{\pi}{6},\,
|x| \in [11 \cdot 2^{-j},2^{-j}]\},$$ $k = 1,\dots,k_j$. Indeed, under this choice of the $I^k_j$’s, estimate (\[2\]) implies that $\gamma$ must intersect at least $(2 \pi)^{j_0} 2^{-j_0} = \pi^{j_0} > 3^{j_0}$ of the figures $$U_k = \{x \in R_{j_0} : \varphi_x \in (k(2 \pi)^{-j_0} \frac{\pi}{6},
(k + 1) (2 \pi)^{-j_0} \frac{\pi}{6})\}.$$ Since these figures are separated by the segments $I^k_{j_0}$ in the layer $R_{j_0}$, the curve $\gamma$ must be disjoint from them each time in passing from one figure to another. The number of these passages must be at least $3^{j_0} - 1$, and a fragment of $\gamma$ of length at least $2 \cdot 3 \cdot 2^{-j_0}$ is required for each passage (because the ends of the segments $I^k_{j_0}$ go beyond the boundary of the layer $R_{j_0}$ containing the figures $U_k$ at distance $3 \cdot 2^{-j_0}$). Thus, for all these passages, a section of $\gamma$ is spent of length at least $$6 \cdot 2^{-j_0} (3^{j_0} -1) \ge 6.$$ Hence, the construction of the segments $I^k_j$ with the properties ($*$)–($**$) is finished.
Let us now describe the construction of the above-mentioned space spirals.
For $x \in \mathbb R^3$, denote by $\Pi_x$ the plane that passes through $x$ and is perpendicular to the segment $OA$. On $\Pi_{x^k_j}$, consider the polar coordinates $(\rho, \psi)$ with origin at the point of intersection $\Pi_{x^k_j} \cap [O,A]$ (in this system, the point $x^k_j$ has coordinates $\rho =\rho^k_j$, $\psi = 0$). Suppose that a point $x(\psi) \in \Pi_{x^k_j}$ moves along an Archimedean spiral, namely, the polar coordinates of $x(\psi)$ are $\rho(\psi) = \rho^k_j - \varepsilon_j \psi$, $\psi
\in [0,2 \pi M_j]$, where $\varepsilon_j$ is a small parameter to be specified below, and $M_j \in \mathbb N$ is chosen so large that the length of any curve passing between all coils of the spiral is at least $10$.
Describe the choice of $M_j$ more exactly. To this end, consider the points $x(2 \pi)$, $x(2 \pi (M_j - 1))$, $x(2 \pi M_j)$, which are the ends of the first, penultimate, and last coils of the spiral respectively (with $x(0) = x^k_j$ taken as the starting point of the spiral). Then $M_j$ is chosen so large that the following condition hold:
$(*_1)$ [*The length of any curve on the plane $\Pi_{x^k_j},$ joining the segments $[x^k_j,x(2 \pi)]$ and $[x(2 \pi (M_j - 1)),x(2 \pi M_j)]$ and disjoint from the spiral $\{x(\psi) : \psi \in [0,2 \pi M_j]\},$ is at least $10$.*]{}
Figuratively speaking, the constructed spiral bounds a “labyrinth”, the mentioned segments are the entrance to and the exit from this labyrinth, and thus any path through the labyrinth has length $\ge 10$.
Now, start rotating the entire segment $I^k_j$ in space along the above-mentioned spiral, i.e., assume that $I^k_j(\psi) = \{y =
\lambda x(\psi) : \lambda \in[1,11]\}$. Thus, the segment $I^k_j(\psi)$ lies on the ray joining $O$ with $x(\psi)$ and has the same length as the original segment $I^k_j = I^k_j(0)$. Define the surface $S^k_j = \cup_{\psi \in [0, 2 \pi M_j]} I^k_j(\psi)$. This surface is diffeomorphic to a plane rectangle (strip). Taking $\varepsilon_j > 0$ sufficiently small, we may assume without loss of generality that $2 \pi M_j \varepsilon_j$ is substantially less than $\rho^k_j$; moreover, that the surfaces $S^k_j$ are mutually disjoint (obviously, the smallness of $\varepsilon_j$ does not affect property $(*_1)$ which in fact depends on $M_j$).
Denote by $y(\psi)=11x(\psi)$ the second end of the segment $I^k_j(\psi)$. Consider the trapezium $P^k_j$ with vertices $y^k_j$, $x^k_j$, $x(2 \pi M_j)$, $y(2 \pi M_j)$ and sides $I^k_j$, $I^k_j(2 \pi M_j)$, $[x^k_j,x(2 \pi M_j)]$, and $[y^k_j,y(2 \pi M_j)]$ (the last two sides are parallel since they are perpendicular to the segment $AO$). By construction, $P^k_j$ lies on the plane $AOD$; moreover, taking $\varepsilon_j$ sufficiently small, we can obtain the situation where the trapeziums $P^k_j$ are mutually disjoint (since $P^k_j \to I^k_j$ under fixed $M_j$ and $\varepsilon_j \to 0$). Take an arbitrary triangle whose vertices lie on $P^k_j$ and such that one of these vertices is also a vertex at an acute angle in $P^k_j$. By construction, this acute angle is at least $\frac{\pi}{2} - \angle
AOD = \frac{\pi}{3}$. Therefore, the ratio of the side of the triangle lying inside the trapezium $P^k_j$ to the sum of the other two sides (lying on the corresponding sides of $P^k_j$) is at least $\frac{1}{2} \sin \frac{\pi}{3} > \frac{2}{5}$. If we consider the same ratio for the case of a triangle with a vertex at an obtuse angle of $P^k_j$ then it is greater than $\frac{1}{2}$. Thus, we have the following property:
$(*_2)$ [*For arbitrary triangle whose vertices lie on the trapezium $P^k_j$ and one of these vertices is also a vertex in $P^k_j$, the sum of lengths of the sides situated on the corresponding sides of $P^k_j$ is less than $\frac{5}{2}$ of the length of the third side*]{} ([*lying inside*]{} $P^k_j$).
Let a point $x$ lie inside the cone $K$ formed by the rotation of the angle $\angle AOD$ around the ray $OA$. Denote by ${\mathop{\rm Proj_{rot}}}x$ the point of the angle $\angle AOD$ which is the image of $x$ under this rotation. Finally, let $K_{4 \Delta AOD}$ stand for the corresponding truncated cone obtained by the rotation of the triangle $4 \Delta AOD$, i.e., $K_{4 \Delta AOD} = \{x \in K :
{\mathop{\rm Proj_{rot}}}x \in 4 \Delta AOD\}$.
The key ingredient in the proof of our theorem is the following assertion:
$(*_3)$ [*For arbitrary space curve $\gamma$ of length less than $10$ joining the points $A$ and $D$, contained in the truncated cone $K_{4 \Delta AOD} \setminus \{O\}$, and disjoint from each strip $S^k_j$, there exists a plane curve $\tilde{\gamma}$ contained in the triangle $4 \Delta AOD \setminus
\{O\},$ that joins $A$ and $D$, is disjoint from all segments $I^k_j$ and such that the length of $\tilde{\gamma}$ is less than $\frac{5}{2}$ of the length of*]{} ${\mathop{\rm Proj_{rot}}}\gamma$.
Prove $(*_3)$. Suppose that its hypotheses are fulfilled. In particular, assume that the inclusion ${\mathop{\rm Proj_{rot}}}\gamma \subset 4
\Delta AOD \setminus \{O\}$ holds. We need to modify ${\mathop{\rm Proj_{rot}}}\gamma$ so that the new curve be contained in the same set but be disjoint from each of the $I^k_j$’s. The construction splits into several steps.
[**Step 1.**]{} If ${\mathop{\rm Proj_{rot}}}\gamma$ intersects a segment $I^k_j$ then it necessarily intersects also at least one of the shorter sides of $P^k_j$.
Recall that, by construction, $P^k_j = {\mathop{\rm Proj_{rot}}}S^k_j$; moreover, $\gamma$ intersects no spiral strip $S^k_j$. If ${\mathop{\rm Proj_{rot}}}\gamma$ intersected $P^k_j$ without intersecting its shorter sides then $\gamma$ would pass through all coils of the corresponding spiral. Then, by $(*_1)$, the length of the corresponding fragment of $\gamma$ would be $\ge 10$ in contradiction to our assumptions. Thus, the assertion of step 1 is proved.
[**Step 2.**]{} Denote by $\gamma_{P^k_j}$ the fragment of the plane curve ${\mathop{\rm Proj_{rot}}}\gamma$ beginning at the first point of its entrance into the trapezium $P^k_j$ to the point of its exit from $P^k_j$ (i.e., to its last intersection point with $P^k_j$). Then this fragment $\gamma_{P^k_j}$ can be deformed without changing the first and the last points so that the corresponding fragment of the new curve lie entirely on the union of the sides of $P^k_j$; moreover, its length is at most $\frac{5}{2}$ of the length of $\gamma_{P^k_j}$.
The assertion of step 2 immediately follows from the assertions of step 1 and $(*_2)$.
The assertion of step 2 in turn directly implies the desired assertion $(*_3)$. The proof of $(*_3)$ is finished.
Now, we are ready to pass to the final part of the proof of Theorem 2.
$(*_4)$ [*The length of any space curve $\gamma \subset \mathbb
R^3 \setminus \{O\}$ joining $A$ and $D$ and disjoint from each strip $S^k_j$ is at least*]{} $\frac{12}{5}$.
Prove the last assertion. Without loss of generality, we may also assume that all interior points of $\gamma$ are inside the cone $K$ (otherwise the initial curve can be modified without any increase of its length so that it have property $(*_4)$). If $\gamma$ is not included in the truncated cone $K_{4 \Delta AOD}
\setminus \{O\}$ then ${\mathop{\rm Proj_{rot}}}\gamma$ intersects the segment $[4A,4D]$; consequently, the length of $\gamma$ is at least $2(4
\sin \angle OAD - 1) = 2(4 \sin \frac{\pi}{3} - 1) = 2 (2 \sqrt 3
- 1) > 4$, and the desired estimate is fulfilled. Similarly, if the length of $\gamma$ is at least $10$ then the desired estimate is fulfilled automatically, and there is nothing to prove. Hence, we may further assume without loss of generality that $\gamma$ is included in the truncated cone $K_{4 \Delta AOD} \setminus \{O\}$ and its length is less than $10$. Then, by $(*_3)$, there is a plane curve $\tilde \gamma$ contained in the triangle $4 \Delta
AOD \setminus \{O\}$, joining the points $A$ and $D$, disjoint from each segment $I^k_j$, and such that the length of $\tilde
\gamma$ is at most $\frac{5}{2}$ of the length of ${\mathop{\rm Proj_{rot}}}\gamma$. By property $(**)$ of the family of segments $I^k_j$, the length of $\tilde \gamma$ is at least $6$. Consequently, the length of ${\mathop{\rm Proj_{rot}}}\gamma$ is at least $\frac{12}{5}$, which implies the desired estimate. Assertion $(*_4)$ is proved.
The just-proven property $(*_4)$ of the constructed objects implies Theorem 2. Indeed, since the strips $S^k_j$ are mutually disjoint and, outside every neighborhood of the origin $O$, there are only finitely many of these strips, it is easy to construct a $C^0$-manifold $Y \subset \mathbb R^3$ that is homeomorphic to a closed ball (i.e., $\partial Y$ is homeomorphic to a two-dimensional sphere) and has the following properties:
\(I) $O \in \partial Y$, $[A,O[ \cup [D,O[ \subset {\mathop{\rm Int}}Y$;
\(II) for every point $y \in (\partial Y) \setminus \{O\}$, there exists a neighborhood $U(y)$ such that $U(y) \cap \partial Y$ is $C^1$-diffeomorphic to the plane square $[0,1]^2$;
\(III) $S^k_j \subset \partial Y$ for all $j \in \mathbb N,\, k =
1,\dots,k_j$.
The construction of $Y$ with properties (I)–(III) can be carried out, for example, as follows: As the surface of the zeroth step, take a sphere containing $O$ and such that $A$ and $D$ are inside the sphere. On the $j$th step, a small neighborhood of the point $O$ of our surface is smoothly deformed so that the modified surface is still smooth, homeomorphic to a sphere, and contains all strips $S^k_j$, $k = 1,\dots,k_j$. Besides, we make sure that, at the each step, the so-obtained surface be disjoint from the half-intervals $[A,O[$ and $[D,O[$, and, as above, contain all strips $S^k_i$, $i \le j$, already included therein. Since the neighborhood we are deforming contracts to the point $O$ as $j \to
\infty$, the so-constructed sequence of surfaces converges (for example, in the Hausdorff metric) to a limit surface which is the boundary of a $C^0$-manifold $Y$ with properties (I)–(III).
Property (I) guarantees that $\rho_Y(A,O) = \rho_Y(A,D) = 1$ and $\rho_Y(O,x) \le 1 + \rho_Y(A,x)$ for all $x \in Y$. Property (II) implies the estimate $\rho_Y(x,y) < \infty$ for all $x,y \in Y \setminus \{O\}$, which, granted the previous estimate, yields $\rho_Y(x,y) < \infty$ for all $x,y \in Y$. However, property (III) and the assertion $(*_4)$ imply that $\rho_Y(A,D) \ge \frac{12}{5} > 2 = \rho_Y (A,O) + \rho_Y(A,D)$. Theorem 2 is proved.
In the case where $\rho_Y$ is a metric (the dimension $n$ ($\ge 2$) is arbitrary), the question of the existence of geodesics is solved in the following assertion, which implies that $\rho_Y$ is an [*intrinsic metric*]{} (see, for example, §6 from [@al1]). [**Theorem 3.**]{} [*Assume that $\rho_Y$ is a finite function and is a metric on $Y$. Then any two points $x,y \in Y$ can be joined in $Y$ by a shortest curve $\gamma : [0,L] \to Y$ in the metric $\rho_Y;$ i.e.[,]{} $\gamma(0) = x,$ $\gamma(L) = y$, and $$\qquad\qquad\rho_Y(\gamma(s),\gamma(t)) = t - s \quad \forall s,t
\in [0,L], \quad s < t. \label{3}$$* ]{}
[Proof.]{} Fix a pair of distinct points $x,y \in Y$ and put $L
= \rho_Y (x,y)$. Now, take a sequence of paths $\gamma_j : [0,L]
\to Y$ such that $\gamma_j(0) = x_j$, $\gamma_j(L) = y_j$, $x_j
\to x$, $y_j \to y$, and $l(\gamma_j) \to L$ as $j \to \infty$. Without loss of generality, we may also assume that the parametrizations of the curves $\gamma_j$ are their natural parametrizations up to a factor (tending to $1$) and the mappings $\gamma_j$ converge uniformly to a mapping $\gamma : [0,L] \to Y$ with $\gamma (0) = x$, $\gamma (L) = y$. By these assumptions, $$\qquad\qquad\lim_{j \to \infty} l(\gamma_j|_{[s,t]}) = t - s \quad
\forall s,t \in [0,L], \quad s < t. \label{4}$$
Take an arbitrary pair of numbers $s,t \in [0,L]$, $s < t$. By construction, we have the convergence $\gamma_j(s) \in {\mathop{\rm Int}}Y
\to \gamma(s)$, $\gamma_j(t) \in {\mathop{\rm Int}}Y \to \gamma(t)$ as $j \to
\infty$. From here and the definition of the metric $\rho_Y(\cdot,\cdot)$ it follows that $$\rho_Y(\gamma(s),\gamma(t)) \le \lim_{j \to \infty}
l(\gamma_j|_{[s,t]}).$$ By (\[4\]), $$\quad\qquad\qquad\rho_Y(\gamma(s),\gamma(t)) \le t - s \quad
\forall s,t \in [0,L],\,\, s < t. \label{5}$$ Prove that (\[5\]) is indeed an equality. Assume that $$\rho_Y(\gamma(s'),\gamma(t')) < t' -s'$$ for some $s',t' \in [0,L]$, $s' < t'$. Then, applying the triangle inequality and then (\[5\]), we infer $$\rho_Y(x,y) \le \rho_Y(x,\gamma(s')) +
\rho_Y(\gamma(s'),\gamma(t')) + \rho_Y(\gamma(t'),y)
< s' + (t' - s') + (L - t') = L,$$ which contradicts the initial equality $\rho_Y(x,y) = L$. The so-obtained contradiction completes the proof of identity (\[3\]).
[**Remark.**]{} Identity (\[3\]) means that the curve $\gamma$ of Theorem 3 is a geodesic in the metric $\rho_Y$, i.e., the length of its fragment between points $\gamma(s)$, $\gamma(t)$ calculated in $\rho_Y$ is equal to $\rho_Y(\gamma(s),\gamma(t)) = t - s$. Nevertheless, if we compute the length of the above-mentioned fragment of the curve in the initial Riemannian metric then this length need not coincide with $t-s$; only the easily verifiable estimate $l(\gamma|_{[s,t]}) \le t - s$ holds (see (\[4\])). In the general case, the equality $l(\gamma|_{[s,t]}) = t - s$ can only be guaranteed if $n = 2$ (if $n \ge 3$ then the corresponding counterexample is constructed by analogy with the counterexample in the proof of Theorem 2, see above). In particular, though, by Theorem 3, the metric $\rho_Y$ is always intrinsic in the sense of the definitions in [@al1 §6], the space $(Y,\rho_Y)$ may fail to be [*a space with intrinsic metric*]{} in the sense of \[ibid\].
Acknowledgements {#acknowledgements .unnumbered}
================
The authors were partially supported by the Russian Foundation for Basic Research (Grant 11-01-00819-a), the Interdisciplinary Project of the Siberian and Far-Eastern Divisions of the Russian Academy of Sciences (2012-2014 no. 56), the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grant NSh-921.2012.1), and the Exchange Program between the Russian and Polish Academies of Sciences (Project 2011–2013).
[99]{}
Intrinsic Geometry of Convex Surfaces \[English translation\], Chapman&Hall/CRC Taylor&Francis Group, Boca Raton (2006), 426 p.
A rigidity condition for the boundary of a submanifold in a Riemannian manifold, [*Doklady Mathematics*]{}, [**77**]{}, No. 3, pp. 340–341 (2008).
Unique determination of domains // In: Differential Geometry and its Applications. Proc. Conf., in Honour of L. Euler, Olomouc, August 2007. World Scientific Publishing Company, 2008, pp. 157–169.
A criterion for the unique determination of domains in Euclidean spaces by the metrics of their boundaries induced by the intrinsic metrics of the domains, [*Siberian Advances in Mathematics*]{}, [**20**]{}, No. 4, pp. 256–284 (2010).
[^1]: Sobolev Institute of Mathematics, pr. Akad. Koptyuga 4, 630090, Novosibirsk, Russia and Novosibirsk State University, ul. Pirogova 2, 630090, Novosibirsk, Russia; E-mail: apkopylov@yahoo.com
[^2]: Sobolev Institute of Mathematics, pr. Akad. Koptyuga 4, 630090, Novosibirsk, Russia and Novosibirsk State University, ul. Pirogova 2, 630090, Novosibirsk, Russia; E-mail: korob@math.nsc.ru
[^3]: [*Mathematical Subject classification*]{} (2010). 53C45; [*Key words*]{}: intrinsic metric, induced boundary metric, triangle inequality, geodesics.
[^4]: Easy examples show that if $X$ is an $n$-dimensional connected smooth Riemannian manifold without boundary then an $n$-dimensional compact connected $C^0$-submanifold in $X$ with nonempty boundary may fail to satisfy condition $(i)$. For $n = 2$, we have the following counterexample: Let $(X,g)$ be the space $\mathbb R^2$ equipped with the Euclidean metric and let $Y$ be a closed Jordan domain in $\mathbb R^2$ whose boundary is the union of the singleton $\{0\}$ consisting of the origin $0$, the segment $\{(1-t)(e_1 + 2e_2) +
t(e_1 + e_2) : 0 \le t \le 1\}$, and the segments of the following four types:$$\biggl\{\frac{(1-t)(e_1 + e_2)}{n} + \frac{te_1}{n + 1}\, :\,
0 \le t \le 1\biggr\} \quad (n = 1,2,\dots);$$ $$\biggl\{\frac{e_1 + (1-t)e_2}{n}\, :\, 0 \le t \le 1\biggr\}
\quad (n = 2,3,\dots);$$ $$\biggl\{\frac{(1-t)(e_1 + 2e_2)}{n} + \frac{2t(2e_1 +e_2)}{4n
+3}\, :\, 0 \le t \le 1\biggr\} \quad (n = 1,2,\dots);$$ $$\biggl\{\frac{(1-t)(e_1 + 2e_2)}{n + 1} + \frac{2t(2e_1 + e_2)}{4n
+3}\, :\, 0 \le t \le 1\biggr\} \quad (n = 1,2,\dots).$$ Here $e_1, e_2$ is the canonical basis in $\mathbb R^2$. By the construction of $Y$, we have $\rho_Y(0,\; E) = \infty$ for every $E \in Y\setminus\{0\}$.
|
---
abstract: 'We develop an exact Green-Kubo formula relating nonequilibrium averages in systems of interacting active Brownian particles to equilibrium time-correlation functions. The method is applied to calculate the density-dependent average swim speed, which is a key quantity entering coarse grained theories of active matter. The average swim speed is determined by integrating the equilibrium autocorrelation function of the interaction force acting on a tagged particle. Analytical results are validated using Brownian dynamics simulations.'
author:
- 'A. Sharma'
- 'J.M. Brader'
title: 'Green-Kubo approach to the average swim speed in active Brownian systems'
---
Assemblies of active, interacting Brownian particles (ABPs) are intrinsically nonequilibrium systems. In contrast to equilibrium, for which the statistical mechanics of Boltzmann and Gibbs enables the calculation of average properties, there is no analogous framework out-of-equilibrium. However, useful exact expressions exist, which enable average quantities to be calculated in the nonequilibrium system by integrating an appropriate time correlation function; the Green-Kubo formulae of linear response theory [@green; @kubo; @kubo_book]. Transport coefficients, such as the diffusion coefficient or shear viscosity, are thus conveniently related to [*equilibrium*]{} autocorrelation functions. Given the utility of the approach it is surprising that the application of Green-Kubo-type methods to active Brownian systems has received little attention [@seifert].
The primary aim of the present work is to extend Green-Kubo-type methods to treat ABPs. This approach has two appealing features. Firstly, information about the active system can be obtained from equilibrium simulations. Secondly, the exact expressions derived provide a solid starting point for the development of approximation schemes and first-principles theory. The method we employ is a variation of the integration-through-transients approach, originally developed for treating interacting Brownian particles subject to external flow [@evans_morriss; @fuchscates2002; @fuchscates2005; @brader2012].
A fundamental feature of ABPs is the persistent character of the particle trajectories. For strongly interacting many-particle systems the interplay between persistent motion and interparticle interactions can generate a rich variety of collective phenomena, such as motility-induced phase separation (see [@cates_tailleur_review] for a recent overview). A quantity which features prominantly in many theories of ABPs [@cates_tailleur_review; @cates_pnas; @fily_marchetti; @speck_lowen; @krinninger_schmidt_brader] is the density-dependent average swim speed, which describes how the motion of each particle is obstructed by its neighbours. Given the ubiquity of the average swim speed in the literature on ABPs we choose it as a relevant observable with which to illustrate our general Green-Kubo-type approach. We demonstrate that this quantity can be obtained from a history integral over the equilibrium autocorrelation of tagged-particle force fluctuations, which we investigate in detail using Brownian dynamics (BD) simulation.
We consider a three dimensional system of $N$ active, interacting, spherical Brownian particles with coordinate ${\boldsymbol{r}}_i$ and orientation specified by an embedded unit vector ${\boldsymbol{p}}_i$. A time-dependent self-propulsion of speed $v_0(t)$ acts in the direction of orientation. Allowing for time-dependence of this quantity both clarifies the general structure of the theory and leaves open the possibility to model physical systems for which the the amount of fuel available to the particles is not constant (see e.g. [@vt1; @vt2]). Omitting hydrodynamic interactions the motion can be modelled by the Langevin equations $$\begin{aligned}
\label{full_langevin}
&\!\!\!\!\!\!\dot{{\boldsymbol{r}}}_i = v_0(t)\,{\boldsymbol{p}}_i + \gamma^{-1}{\boldsymbol{F}}_i + {\boldsymbol{\xi}}_i\;\;,
\;\;\;
\dot{{\boldsymbol{p}}}_i = {\boldsymbol{\eta}}_i\times{\boldsymbol{p}}_i \,,\end{aligned}$$ where $\gamma$ is the friction coefficient and the force on particle $i$ is generated from the total potential energy according to ${\boldsymbol{F}}_i\!=\!-\nabla_i U_N$. The stochastic vectors ${\boldsymbol{\xi}}_i(t)$ and ${\boldsymbol{\eta}}_i(t)$ are Gaussian distributed with zero mean and have time correlations $\langle{\boldsymbol{\xi}}_i(t){\boldsymbol{\xi}}_j(t')\rangle=2D_t\boldsymbol{1}\delta_{ij}\delta(t-t')$ and $\langle{\boldsymbol{\eta}}_i(t){\boldsymbol{\eta}}_j(t')\rangle=2D_r\boldsymbol{1}\delta_{ij}\delta(t-t')$. The translational and rotational diffusion coefficients, $D_t$ and $D_r$, are treated in this work as independent model parameters.
It follows exactly from that the joint probability distribution, $\!P({\boldsymbol{r}}^N\!\!,{\boldsymbol{p}}^N\!\!,t)$, evolves according to [@gardiner] $$\begin{aligned}
\label{smol_eq}
\frac{\partial P(t)}{\partial t} = {\Omega}_{\rm a}(t) P(t).\end{aligned}$$ The time-evolution operator can be split into a sum of two terms, ${\Omega}_{\rm a}(t)={\Omega_{\rm eq}}+{\delta\Omega}_{\rm a}(t)$, where the equilibrium contribution is given by $$\begin{aligned}
\label{smol_op_eq}
{\Omega_{\rm eq}}= \sum_{i=1}^{N} {\boldsymbol{\nabla}}_{i}\!\cdot\!
\big[
D_\text{t}\!\left({\boldsymbol{\nabla}}_{i}\! - \beta{\boldsymbol{F}}_i\right)
\big] \!+\! D_\text{r}{\boldsymbol{R}}_i^2, \end{aligned}$$ with rotation operator ${\boldsymbol{R}}\!=\!{\boldsymbol{p}}\times\!\nabla_{\!{\boldsymbol{p}}}$ (see, e.g. [@morse_feshbach]) and the time-dependent, active part of the dynamics is described by the operator $$\begin{aligned}
\label{smol_op_del}
{\delta\Omega}_{\rm a}(t) =
-\sum_{i=1}^{N} v_0(t){\boldsymbol{\nabla}}_{i}\!\cdot{\boldsymbol{p}}_i. \end{aligned}$$
To solve we define a nonequilibrium part of the distribution function, $\delta P(t) = P(t) - P_{\rm eq}$ [@fuchs_cates], where $P_{\rm eq}$ is the equilibrium distribution of position and orientation. Using ${\Omega_{\rm eq}}P_{\rm eq}=0$ yields the equation of motion $$\begin{aligned}
\label{delP_eom}
\frac{\partial}{\partial t}\delta P(t) = {\Omega}_{\rm a}(t) \delta P(t)
+ {\delta\Omega}_{\rm a}(t) P_{\rm eq}\,. \end{aligned}$$ Treating the last term as an inhomogeneity and solving for $\delta P(t)$ we obtain a formal solution for the nonequilibrium distribution $$\begin{aligned}
\label{P_eom}
P(t) = P_{\rm eq} \,-\, \int_{-\infty}^{t}\!dt'
v_0(t')
\,e_{+}^{\int_{t'}^{t}ds\,{\Omega}_{\rm a}(s)}
\beta F^p P_{\rm eq}\,,\end{aligned}$$ where $e_{+}(\cdot)$ is a positively ordered exponential function (see the appendix in [@brader2012]) and we have used ${\delta\Omega}_{\rm a}(t)P_{\rm eq} = -\beta v_0(t)F^p P_{\rm eq}$, with ‘projected force’ fluctuation $$\begin{aligned}
F^p = \sum_i {\boldsymbol{p}}_i\cdot{\boldsymbol{F}}_i.\end{aligned}$$ The projected force emerges as a central quantity within our approach and indicates to what extent the interparticle interaction forces act in the direction of orientation, either assisting or hindering the self-propulsion. We will show that this quantity is closely related to the average swim speed in the active system.
Introducing a test function, $f$, on the space of positions and orientations and integrating by parts yields a formally exact expression for a nonequilibrium average $$\begin{aligned}
\label{nonlinear_average}
\langle f \rangle(t) = \langle f \rangle_{\rm eq} - \int_{-\infty}^{t}\!dt'\,
v_0(t')\langle \beta F^p e_{-}^{\int_{t'}^{t}ds\,{{\Omega}^{\dagger}}_{\rm a}(s)}f \rangle_{\rm eq}, \end{aligned}$$ where $e_{-}(\cdot)$ denotes a negatively ordered exponential [@brader2012] and $\langle\cdot\rangle_{\rm eq}$ is an equilibrium average over positional and orientational degrees of freedom. The adjoint operator is given by ${{\Omega}^{\dagger}}_{\rm a}(t)={{\Omega}^{\dagger}}_{\rm eq}-{\delta\Omega}_{\rm a}(t)$, where $$\begin{aligned}
{{\Omega}^{\dagger}}_{\rm eq} = \sum_{i}
D_\text{t}\!\left({\boldsymbol{\nabla}}_{i}\! + \beta{\boldsymbol{F}}_i\right)
\!\cdot\!{\boldsymbol{\nabla}}_{i} \!+\! D_\text{r}{\boldsymbol{R}}_i^2\end{aligned}$$ generates the equilibrium dynamics. The integrand appearing in involves the [*equilibrium*]{} correlation between the projected force at time $t'$ and the observable $f$, which evolves from $t'$ to $t$ according to the full dynamics. The average is nonlinear in $v_0(t)$, because of the activity dependence of the adjoint operator.
The response of the system to linear order in $v_0(t)$ is obtained by replacing the full time-evolution operator ${{\Omega}^{\dagger}}_{\rm a}(t)$ in by the time-independent equilibrium operator ${{\Omega}^{\dagger}}_{\rm eq}$. Further simplification occurs if the activity is constant in time, $v_0(t)\rightarrow v_0$, leading to $$\begin{aligned}
\label{linear_tindep}
\langle f \rangle_{\rm lin} = \langle f \rangle_{\rm eq} - v_0 \int_{0}^{\infty}\!dt\,
\langle \beta F^p e^{{{\Omega}^{\dagger}}_{\rm eq}t} f \rangle_{\rm eq}, \end{aligned}$$ which can be used to define a general active transport coefficient $\alpha\!=\!\lim_{v_0 \to 0}(\langle f \rangle_{\rm lin}\!-\! \langle f \rangle_{\rm eq})/v_0$. Equation is the desired Green-Kubo relation for calculating the linear response of ABPs to a time-independent activity. As mentioned previously, a quantity of current interest is the average, density-dependent swim speed, $v(\rho)$. This describes how the bare swim speed, $v_0$, is influenced by interparticle interactions and is an important quantity in many of the various theories addressing ABPs [@cates_tailleur_review; @cates_pnas; @fily_marchetti; @speck_lowen; @krinninger_schmidt_brader]. In particular, the tendency of the system to undergo motility-induced phase-separation is determined by the rate of decrease of $v(\rho)$ with increasing density; a positive feedback mechanism can result when increasing the local density leads to a sufficiently strong reduction of the local average swim velocity.
![\[fig:correlator\] (a) The correlator $H(t)$ for a system of soft spheres. The arrow indicates the direction of increasing density. Inset: The initial value $H(0)$ as a function of the density. Squares: BD simulation. Circles: using equation and the Percus-Yevick $g_{\rm eq}(r)$ as input. (b) The same data as in (a) on a log scale. The relaxation becomes faster with increasing density. ](Fig1a.pdf){width="0.31\linewidth" height="0.17\textheight"}
![\[fig:correlator\] (a) The correlator $H(t)$ for a system of soft spheres. The arrow indicates the direction of increasing density. Inset: The initial value $H(0)$ as a function of the density. Squares: BD simulation. Circles: using equation and the Percus-Yevick $g_{\rm eq}(r)$ as input. (b) The same data as in (a) on a log scale. The relaxation becomes faster with increasing density. ](Fig1b.pdf){width="0.31\linewidth" height="0.17\textheight"}
The average swim speed is defined as the nonequilibrium average $$\begin{aligned}
\label{swim_speed_definition}
v(\rho)=\frac{1}{N}\left\langle \sum_i {\boldsymbol{v}}_i \cdot {\boldsymbol{p}}_i \right\rangle\end{aligned}$$ where ${\boldsymbol{v}}_i$ is the velocity of particle $i$. Using to eliminate the velocity in favour of the forces and using the fact that the Brownian force ${\boldsymbol{\xi}}_i$ is uncorrelated with the orientation ${\boldsymbol{p}}_i$, it follows that $$\begin{aligned}
\label{swim_speed_Fp}
v(\rho) = v_0 + \frac{\gamma^{-1}}{N}\langle F^p \rangle. \end{aligned}$$ For a time-independent $v_0$ we can employ to calculate the average in to linear order $$\begin{aligned}
\label{GK_vrho}
v(\rho) = v_0\left(
1 - D_t\int_0^{\infty}\!dt\, H(t)
\right)\,,\end{aligned}$$ where the integrand is the equilibrium autocorrelation of projected force fluctuations $$\begin{aligned}
\label{correlator}
H(t) = \frac{1}{N}
\langle
\,\beta F^p e^{{{\Omega}^{\dagger}}_{\rm eq}t} \beta F^p
\,\rangle_{\rm eq}\,.\end{aligned}$$ Spatial and orientational degrees of freedom decouple in equilibrium, which enables the orientational integrals in to be evaluated exactly. This yields $$\begin{aligned}
\label{correlator_int}
H(t)
= \frac{1}{3}e^{-2D_r t}\beta^2
\left\langle {\boldsymbol{F}}\cdot e^{{{\Omega}^{\dagger}}_{\rm eq,s}t}{\boldsymbol{F}}\right\rangle_{\rm eq, s}, \end{aligned}$$ where ${\boldsymbol{F}}$ is the interaction force acting on an arbitrarily chosen (‘tagged’) particle, ${{\Omega}^{\dagger}}_{\rm eq,s} \!= \sum_{i} D_\text{t}\!\left({\boldsymbol{\nabla}}_{i}\! + \beta{\boldsymbol{F}}_i\right)
\!\cdot\!{\boldsymbol{\nabla}}_{i}$ is the spatial part of the time-evolution operator and $\langle\cdot\rangle_{\rm eq,s}$ indicates an equilibrium average over spatial degrees of freedom. The initial value is given by $H(0)\!=\!\beta^2\langle |{\boldsymbol{F}}|^2 \rangle_{\rm eq}/3$. If we consider pairwise additive interaction potentials, then the Yvon theorem [@HM] leads to $$\begin{aligned}
\label{yvon_result}
H(0)=\frac{1}{3}\,\rho\! \int \!d{\boldsymbol{r}}\,g_{\rm eq}(r)\nabla^2\beta u(r)\,,\end{aligned}$$ where $\rho$ is the number density, $u(r)$ is the passive pair potential and $g_{\rm eq}(r)$ is the corresponding equilibrium radial distribution function.
Equation shows that the nontrivial physics underlying the linear response of the system to activity is contained in the tagged-particle force-autocorrelation function. This function was encountered many years ago by Klein and coworkers [@klein] in a study of the velocity autocorrelation in overdamped Brownian systems. By manipulation of the operator it was shown that $$\begin{aligned}
\label{klein_result}
\left\langle {\boldsymbol{F}}(t)\cdot {\boldsymbol{F}}(0) \right\rangle_{\rm eq, s} \!\!=\!
\frac{3}{(\beta D_t)^2}\big(
D_t\,\delta(t) - Z_{\rm eq}(t)
\big) ,\end{aligned}$$ where $Z_{\rm eq}(t)$ is the velocity autocorrelation function, defined in terms of the tagged particle velocity, ${\bf v}(t)$, according to the familiar relation $$\begin{aligned}
Z_{\rm eq}(t)=\frac{1}{3}\left\langle {\bf v}(t)\cdot{\bf v}(0) \right\rangle_{\rm eq,s}. \end{aligned}$$ The velocity autocorrelation function is a quantity of fundamental interest in describing the dynamics of interacting liquids and is closely related to other important quantities (e.g. the mean-squared displacement and self diffusion coefficient). Substituting into yields $$\begin{aligned}
\label{H_vct}
H(t) = \frac{1}{D_t^2}e^{-2 D_r t}\big(
D_t\,\delta(t) - Z_{\rm eq}(t)
\big),\end{aligned}$$ thus providing, via , a direct connection between $Z_{\rm eq}(t)$ and $v(\rho)$. The latter can thus be determined to linear order in $v_0$ using a standard, equilibrium BD simulation. Finally, we note that $H(t)$ remains integrable in all spatial dimensions, because of the exponential in . There is thus no principal difficulty in calculating $v(\rho)$ in two dimensions, in contrast to the situation for transport coefficents, such as the self-diffusion coefficient, for which the relevant Green-Kubo time-integral diverges [@evans_morriss].
In a recent study of the pressure in active systems Solon [*et al.*]{} [@solonPRL] express the density-dependent average swim speed in the form $v(\rho)=v_0 + I_2/\rho$, where $\rho$ is the bulk number density. The interaction potential is encoded in the quantity $I_2$ via its dependence on a [*static*]{} structural correlation between density and polarization, which are given, respectively, by the first and second harmonic moments of the orientation-resolved single particle density. This leads to the identification $I_2=-D_t\rho\, v_0\int_0^{\infty}dt\, H(t)$. An advantage of the present Green-Kubo formulation over that of Solon [*et al.*]{} is that it enables identification of the relevant relaxation processes contributing to the decrease of $v(\rho)$. Moreover, we anticipate that will prove more convenient for the development of approximations.
![\[fig:projected\_velocity\] The scaled average swim speed as a function of density. Lines with symbols: data from direct calculation of using active BD simulations. Diamonds: the linear response result (independent of $v_0$) calculated using the equilibrium time correlation function data from Fig.\[fig:correlator\] as input to . []{data-label="fig1"}](vpredicted1.pdf){width="0.85\columnwidth"}
In order to test the range of validity of the linear response result we perform BD simulations on a three-dimensional system of $N\!=\!1000$ particles interacting via the pair-potential $\beta u(r) = 4\varepsilon((\sigma/r)^{12} - (\sigma/r)^6)$, where $\sigma$ sets the length scale and we set $\varepsilon=1$. The potential is truncated at its minimum, $r=2^{1/6}\sigma$ to yield a softly repulsive interaction. The system size $L$ is determined as $L\!=\!(N/\rho)^{1/3}$ in order to obtain the desired density. The integration time step is fixed to $dt\!=\!10^{-5} \tau_B$ where $\tau_B\!=\!d^2/D_t$ is the time-scale of translational diffusion. Measurements are made after a minimum time of $20\tau_B$ to ensure equilibration. In order to measure time-correlations the system is sampled every $\tau_p/100$ s, where $\tau_p = 1/2D_R$ is the rotational diffusion time scale. The total run time is $300\tau_B$. We choose the ratio of diffusion coefficients as $D_r/D_t\!=\!20$, although there is nothing special about this particular choice.
In Fig. \[fig:correlator\]a we show the correlator $H(t)$ as a function of time for a number of different densities, the largest of which is close to the freezing transition for our model interaction potential. Aside from the strong increase of $H(0)$ with increasing density (shown in the inset), the most striking aspect of the correlator is that the decay of $H(t)$ is much faster than the timescale of rotational diffusion (note that time is scaled with $\tau_p$ in the figure). Indeed, very large values of the ratio $D_r/D_t$ would be required for the exponential factor in to significantly influence the decay of $H(t)$. In the limit of large $D_r$ we obtain $H(t)=H(0)\exp(-2D_r t)$ and thus $v(\rho)/v_0 = 1 - H(0)D_t/(2D_r)$. We conclude that, provided the value of $D_r$ is not extremely large, the relevant relaxation process is the decorrelation of the tagged particle interaction force.
In the inset to Fig. \[fig:correlator\]a we show the initial value, $H(0)$, as a function of the density. To check the expression we have confirmed that using $g_{\rm eq}(r)$ from our equilibrium simulations to evaluate the r.h.s. indeed reproduces the $t\rightarrow 0$ limit of our dynamical $H(t)$ data. Moreover, we have also employed an approximate liquid-state integral equation theory (Percus-Yevick theory) [@HM] to calculate $g_{\rm eq}(r)$ and evaluate $H(0)$. Very good agreement of the predicted $H(0)$ with simulation data is obtained. In Fig. \[fig:correlator\]b we replot the data on a semi-logarithmic scale, with the initial value scaled out. This representation makes clear that $H(t)$ is non-exponential and that the decay occurs more rapidly as the density is increased, in contrast to the structural relaxation of the system, which slows down with increasing density. The latter observation can be rationalized by considering that small positional changes can give rise to large changes in the force for closely packed particles residing in regions of strong interaction-force gradient. The fact that $H(t)$ is non-exponential is not surprising, given that it can be expressed in terms of the velocity autocorrelation function, a quantity which famously exhibits power law asymptotic behaviour (‘long-time tails’) [@klein; @HM]. Klein [*et al.*]{} have shown analytically that for a dilute system of Brownian hard-spheres $\langle {\boldsymbol{F}}(t)\cdot {\boldsymbol{F}}(0) \rangle_{\rm eq, s}\sim t^{-\frac{5}{2}}$ for long times.
![\[fig:phase\] The region for which linear response agrees with the result of active-BD simulations to a relative error less than $5\%$. $\Delta v$ is the difference between linear response and the active BD simulation result. The breakdown of linear response is related to the onset of activity-induced phase separation. []{data-label="fig1"}](phasediagram1.pdf){width="0.85\columnwidth"}
In Fig. \[fig:projected\_velocity\] we show simulation data for the average swim speed as a function of density. The red diamonds show the linear response prediction obtained by using the data of Fig. \[fig:correlator\] in the integral expression . This yields a result for $v(\rho)/v_0$ which is independent of $v_0$. The remaining curves show data obtained by direct evaluation of using active BD simulations at three different values of $v_0$. As one might expect, deviations from linear response occur at lower density for larger values of $v_0$.
The above observation can be made more concrete by estimating a region in the $(v_0,\phi)$ plane where linear response breaks down. In Fig. \[fig:phase\] we use our simulation data to map the locus of points for which the error in the linear response result, relative to the full active BD simulations, equals $5\%$. Although the chosen criterion is somewhat arbitrary, it at least gives a visual impression of the range of validity of linear response within the space of our control parameters. The locus of points shown in Fig. \[fig:phase\] is correlated with the onset of strong spatial inhomogeneities and phase separation. However, an analysis of active phase separation would go beyond the scope of the present work. The linear response formula thus appears to be reliable for parameter values away from phase separation, but, beyond this, higher orders in $v_0$ will become important in determining $v(\rho)$.
To summarize our main findings: we have derived a formally exact expression for calculating averages in a system of interacting Brownian particles, subject to a time-dependent activity $v_0(t)$. From this we obtain the linear-response expression for a time-independent activity. Application of this result to calculate the average swim speed yields and identifies the relevant time-correlation function, $H(t)$, as given by . We find that linear response provides an accurate account of $v(\rho)$ over a large parameter range, except for those regions of parameter space where phase separation occurs.
Although we have focused our attention on the linear-response regime, our exact results could in principle be used to develop nonlinear theories in the spirit of Refs. [@fuchscates2002; @fuchscates2005; @brader2012], which address Brownian particles under external flow. It would also be interesting to use to investigate the transient dynamics arising from time-dependent activity, but we defer this line of enquiry until an experimentally relevant protocol can be identified. Aside from using an equilibrium integral equation theory to determine $H(0)$ (inset to Fig. \[fig:correlator\]a), all of the data presented comes from BD simulation. A clear next step is to investigate approximations to $H(t)$ which enable predictions to be made from first-principles, without simulation input. Given the relation it seems likely that existing approximations to the velocity autocorrelation function (e.g. projection operator approaches) could be usefully exploited.
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|
---
author:
- 'W. Ishibashi'
- 'T. J.-L. Courvoisier'
bibliography:
- 'biblio.bib'
date: 'Received; accepted'
title: 'X-ray variability time scales in Active Galactic Nuclei'
---
Introduction
============
Variability in Active Galactic Nuclei (AGN) covers a wide range of time scales and amplitudes over the entire electromagnetic spectrum. The observed light curves are characterized by aperiodic and featureless fluctuations, suggesting that the variability mechanism is of random nature.
Early studies of X-ray variability focused on the search of characteristic time scales (@McH_2001 and references therein). One of the methods used to study the temporal structure of the variations is the Power Spectral Density (PSD) analysis. PSD results are mainly derived from high quality X-ray light curves provided by the Rossi X-ray Timing Explorer (RXTE) and XMM-Newton. The observed power spectrum is generally modeled by a power law of the form $P_{\nu} \propto \nu^{\alpha}$, where $\nu = 1/T$ is the temporal frequency. At high frequencies (short time scales), the PSD presents a steep slope of $\alpha$$\sim$-2, with no characteristic time scale. At low frequencies (long time scales), the PSD is characterized by a slope of $\alpha$$\sim$-1, representing flicker noise. A characteristic time scale can be defined, associated with this bend in the power spectrum. Estimations of the PSD bend or break time scale are obtained by fitting broken power laws to the observed PSD.
Subsequent studies showed that the characteristic break time scale $T_{B}$ is related to the black hole mass $M_{BH}$: the larger the central mass, the longer the characteristic time scale, and a linear scaling (of the form $T_{B} \propto M_{BH}$) has been proposed (@M_et_2003, @P_2004). But a non negligible scatter has soon been noticed, with narrow-line objects not fitting into the general picture. Narrow-line Seyfert 1 galaxies (NLS 1) form a particular class of AGNs, characterized by rapid and large variability combined with other peculiar spectral properties. NLS 1 galaxies are usually believed to be powered by small black holes accreting at high rates, close to the Eddington limit. The fact that the break time scales are shorter in NLS 1 objects when compared with classical broad-line galaxies of the same mass, suggests that the break time scale depends on a second parameter, such as accretion rate or black hole spin. (@McH_et_2004, @U_McH_2005). Following this hypothesis, @McH_et_2006 proposed a sort of ‘fundamental plane’ relating variability time scale, black hole mass, and accretion rate. In particular, they obtained an observational scaling relationship in which the break time scale explicitly depends on both black hole mass and accretion rate.
The PSD break time scale is expected to provide insights into the emission mechanisms, and several physical interpretations have been proposed. Various accretion disc time scales have been compared with the measured PSD time scales, but with no conclusive result (@M_et_2003, @P_2004). The currently favored interpretation is based on the inner propagating fluctuation models (@L_1997, @C_et_2001).
Cooling processes responsible for the emitted radiation are known in considerable detail, while the process of transfer of gravitational energy into radiative energy is still poorly understood. In standard accretion disc models, it is provided by some form of viscosity parametrized by the $\alpha$ parameter (@S_S_1973). Another likely source for heating of the accreting gas is given by dissipative processes, such as shocks. In previous papers (@C_T_2005, @I_C_2009, hereafter Paper I), we discussed the cascades of shocks model in which radiation is emitted as a result of shocks between elements (clumps) forming the accretion flow. In this picture, clumps move with velocities determined by the gravitational field of the central black hole following different orbits depending on the different initial conditions. In the central regions, colliding clumps are characterized by high relative velocities. When such high-velocity flows converge, shocks will result. This process releases the bulk of the clump kinetic energy converting it into radiation. Optically thick and optically thin shocks then account for the observed optical/UV and X-ray emissions, respectively. Characteristic time scales can be derived within this framework. Here we compare model time scales with observed variability time scales and suggest a possible interpretation of the break seen in the power spectrum.
The present paper is organized as follows. Some elements of clumpy accretion flows are recalled in Sect. 2. We derive X-ray variability time scales and present model results in Sect. 3; comparison with observational results are given in Sect. 4. We then discuss some aspects of optical/UV variability in Sect. 5 and correlations between different wavelengths in Sect. 6. The main results are summarized in the Conclusion.
Cascades of shocks in clumpy accretion flows
============================================
In Paper I, we studied the properties of the inhomogeneous accretion flow formed by interacting clumps of matter, where shocks between elements and the subsequent evolution are at the origin of the radiation.
A collision between two clumps of several solar masses, $M_{c} = M_{33} \cdot 10^{33}$g, moving at the local free-fall velocity at a typical distance of $\sim$$100 R_{S}$ (parametrized in units of the Schwarzschild radius $R= \zeta \cdot R_{S}$ where $\zeta = \zeta_{UV} \cdot 100$) leads to an optically thick shock. Following this optically thick shock the resulting gas cloud expands rapidly, with a fraction $\eta_{rad} = \eta_{1/3} \cdot \frac{1}{3}$ of the kinetic energy being radiated at the photosphere. The time it takes for the collision energy to be radiated is given by the expansion time $$\tau_{exp} \cong 10^{6} M_{33}^{1/2} \zeta_{UV}^{1/4} \, \textrm{s} \, .$$
We assume a spherical expansion with the photospheric radius given by $$R_{max} \cong 3 \cdot 10^{15} M_{33}^{1/2} \zeta_{UV}^{-1/4} \, \textrm{cm} \, .$$
In the central regions, the expansion of the gas envelopes lead to interactions between envelopes originating from different events. Further shocks are then expected, with the expanding regions filling a volume $\sim R_{max}^{3}$.
In Paper I, we distinguished two classes of objects, Class Q and Class S, according to the importance of the volume filling factor of the configuration relative to the typical system size, defined as $$\epsilon = \left( \frac{R_{max}}{100\zeta_{UV} R_{S}} \right)^{3} \cong 10^{-3} \, M_{33}^{3/2} \zeta_{UV}^{-15/4} \left( \frac{M_{BH}}{10^{9} M_{\odot}} \right)^{-3} \, .$$
Class Q objects are characterized by a small filling factor ($\epsilon \ll 1$), a condition met for large black hole mass; Class S objects are characterized by a large filling factor ($\epsilon \sim 1$) associated with a small central mass. We identified Class Q objects with massive and luminous quasars, while Class S objects were identified with less luminous sources, such as Seyfert galaxies. In the following we focus on Class S objects, since timing analysis are mainly performed on local Seyfert 1 galaxies. Each class was further subdivided into two cases, Case A and Case B, depending on whether the radiation time is longer (Case A) or shorter (Case B) than the accretion time.
The second generation shocks occur in optically thin conditions. In these optically thin shocks, Coulomb collisions with ions, carrying most of the gravitational energy, energetize the electrons which cool through Compton radiation. The seed photons for Compton upscattering are provided by the optical/UV photons emitted in the optically thick event. In a stationary situation, the plasma temperature is set by the balance between heating and cooling rates. The electron temperature is then estimated assuming an equilibrium between Coulomb heating and Compton cooling: $L_{Coulomb} = L_{Compton}$. This gives a value for the average electron energy $E_{e}$ of the order of a few hundred keV, hence Compton cooling of the hot electrons is responsible for the observed X-ray emission. The resulting X-ray luminosities, $L_{X}$, in the different classes and sub-cases have been calculated in Paper I.
X-ray variability properties
============================
Characteristic time scale
-------------------------
We introduce the heating time scale of the electrons given by $$\tau_{heat} = \frac{E_{e}}{(\frac{dE}{dt})_{heat}} \, .$$
At equilibrium this is equal to the electron Compton cooling time $$\tau_{cool} = \frac{E_{e}}{(\frac{dE}{dt})_{cool}} \, .$$
The above time scales are directly linked to the physical process responsible for the X-ray emission and can therefore be considered as a characteristic X-ray time scale, defined as: $$\tau_{X} \sim \tau_{heat} \sim \tau_{cool} \, .$$
The numerical value is of the order of $\sim$days: $$\tau_{X} \cong 5 \, \eta_{1/3}^{-1} \zeta_{UV}^{3} \dot{M}_{0}^{-1} M_{8}^{2} \; \textrm{d} \, ,
\label{tau_value}$$ where the black hole mass is expressed in units of $M_{BH} = M_{8} \cdot 10^{8} M_{\odot}$, and the accretion rate in units of $\dot{M} = \dot{M}_{0} \cdot 1M_{\odot}/yr$.
The characteristic X-ray time scale is proportional to the square of the central mass and inversely proportional to the accretion rate. The shortest time scale is obtained in the case of a low mass object accreting at a high rate. We note that the dependence on mass is stronger than that on accretion rate. The dependence on black hole mass and accretion rate comes from the photon energy density given by the luminosity of the optically thick shocks, proportional to $\dot{M}$, and the average distance of the optically thick shocks at the origin of the UV photons, proportional to $M_{BH}^{2}$. Thus X-ray variability seems to be determined by the source parameters, black hole mass and accretion rate. This may account for the different variability properties observed in different AGN classes.
Luminosity variations
---------------------
In our picture the X-ray emission is produced by Comptonization of seed UV photons, emitted in the optically thick shocks, by energetic electrons created in an optically thin event. The X-ray luminosity then depends on the properties of both optically thick and optically thin shocks through the photon energy density ($u_{ph}$) and the electron number density ($n_{e}$), respectively. We therefore re-express the X-ray luminosities leaving the explicit dependences on $u_{ph}$ and $n_{e}$, using the photon energy density and the electron number density estimated in Paper I (eq. (13) and eq. (14), respectively).
Here we parametrize these two quantities by appropriate values ($u_{ph} = u_{\gamma} \cdot 35.4 \, \mathrm{erg/cm^{3}}$ and $n_{e} = n_{8} \cdot 1.2 \cdot 10^{8} \, \mathrm{cm^{-3}}$), to obtain: $$\textrm{Case A:} \quad
\left\langle L_{X} \right\rangle \cong 4.9 \cdot 10^{43} \, E_{p,MeV}^{4/7} \zeta_{UV}^{3} M_{8}^{3} u_{\gamma}^{5/7} n_{8}^{9/7} \; \textrm{erg/s}$$ $$\textrm{Case B:} \quad
\left\langle L_{X} \right\rangle \cong 2.1 \cdot 10^{43} \, E_{p,MeV}^{4/7} \zeta_{UV}^{3/2} M_{8}^{2} u_{\gamma}^{-2/7} n_{8}^{9/7} \; \textrm{erg/s}
\label{ratio_IB}$$
We observe that the dependence of the X-ray luminosity on the electron number density is stronger than that on the photon energy density. This already suggests that the resulting X-ray luminosity and its variations are more sensitive to variations of the upscattering medium than those in the seed photon population.
We can now relate variations in the X-ray luminosity to electron density modulations. Assuming that the main dependence is on the electron density, we suppose a relation of the form $L_{X}(n_{e}) = K \cdot n_{e}^{9/7}$ (where $K$ is a numerical factor). We then consider small density perturbations $\Delta n_{e}$ around a mean value $n_{e}$, and expanding to first order we obtain: $$\frac{\Delta L_{X}}{L_{X}} \sim \frac{9}{7} \frac{\Delta n_{e}}{n_{e}} \, .
\label{delta}$$
Expression ($\ref{delta}$) directly relates density fluctuations to luminosity variations. Modulations in the electron density can lead to significant X-ray variations of order unity, with density fluctuations being slightly amplified. Variations in the seed photon population would have a weaker effect.
Density modulations typically occur on a time scale associated with the inhomogeneous structure resulting from the succession of shocks. We introduce the inhomogeneity time scale $\tau_{inh} \sim l/V$, where $l$ is a typical inhomogeneity size and $V$ the medium velocity. This time scale is essentially determined by the local inhomogeneity properties and does not scale with the system parameters, such as the black hole mass. It can cover a broad range of time scales reflecting the inhomogeneities of the accretion flow. The slope of the luminosity variations as a function of the temporal frequency, $\frac{\Delta L_{X}}{L_{X}}(\nu=V/l)$, is related to that of the density modulations as a function of the inhomogeneity size, $\frac{\Delta n_{e}}{n_{e}}(l)$. Thus the shape of the power spectrum is given by the spectrum of density fluctuations. Electron density modulations can be transmitted, and thus observed as luminosity variations, only if the local inhomogeneity time is greater than the electron heating time ($\tau_{inh} > \tau_{heat}$). Fluctuations on time scales shorter than the heating time ($\tau_{inh} < \tau_{heat}$) cannot be propagated, they are smoothed out and effectively suppressed. This leads to a decline in variability power, which induces a break in the power spectrum. This is schematically illustrated in Figure 1.
![Sketch of a typical power spectrum in the $(\nu \cdot P_{\nu})$ vs. $\nu$ representation. Electron density modulations on time scales longer than the heating time ($\tau_{inh} > \tau_{heat}$) are effectively transmitted, leading to fluctuations in the emitted radiation. This corresponds to the flat portion in the observed power spectrum. Rapid modulations on time scales shorter than the heating time ($\tau_{inh} < \tau_{heat}$) are smoothed out, with the amplitude of fluctuations being considerably dampened. This corresponds to the steepening observed in the power spectrum. The suppression of fluctuations on time scales shorter than a critical value leads to a break in the variability power which may be associated with the observed PSD break. []{data-label="figure"}](dessin_02.pdf){width="50.00000%"}
Break time scale: comparison with observations
==============================================
Magnitude and functional dependence
-----------------------------------
The Power Spectral Density (PSD) is defined as the modulus-squared of the Fourier transform of the light curve, in units of light curve variance per Hz. The typical power spectrum is characterized by a slope of $\alpha$$\sim$$-1$, steepening to a slope of $\alpha$$\sim$$-2$ above a break frequency $\nu_{B}$. A break time scale $T_{B} = 1/\nu_{B}$ is associated with the bend in the power spectrum and can be considered as a characteristic X-ray variability time scale.
Break time scales in AGNs have been studied by many authors and PSD measurements have been obtained for around 20 sources (@U_2007). Here we summarize a number of recent observational results. @M_et_2003 analysed a sample of six Seyfert 1 galaxies and found a significant correlation between break time scale and black hole mass. The measured PSD break time scales were typically of the order of a few days, and the data could be fitted by a linear relation of the form $T_{B} (\textrm{days}) \approx M_{BH}/10^{6.5} M_{\odot}$. The luminosity-time scale correlation was found to be negligible compared with the mass-time scale relation. A similar result was obtained in the case of broad-line galaxies with the break frequency decreasing with increasing mass, following a relation of the form $\nu_{hfb} \approx 1.5 \cdot 10^{-6}/(M/10^{7}M_{\odot})$Hz (@P_2004). More recently, @M_et_2008 reported a break time scale of $\sim$34 days in Mrk 509, in agreement with the mass-break time scale relationship discussed in the previous works.
However, a significant scatter is observed in the linear black hole mass-break time scale correlation. Broad-line AGNs are consistent with a linear scaling of break time scales with central mass, but narrow-line galaxies are observed to lie systematically above this relation. This implies that, for a given black hole mass, the break time scale should be shorter in narrow-line galaxies than in broad-line objects. Following this supposition, @McH_et_2004 suggested that the break time scale might depend on additional parameters such as accretion rate and/or black hole spin. The difference in break time scales observed in different AGN classes has been confirmed by @U_McH_2005 who analysed the mass-time scale relation for all available objects with measured PSDs. They showed that for a given black hole mass, higher accretion rate objects indeed have shorter break time scales than lower accretion rate counterparts.
The main result, on which all observational works agree, is the existence of a scaling between characteristic time scale and black hole mass. The typical value of the break time scale is of the order of $\sim$days for a $10^{7}-10^{8} M_{\odot}$ object. At fixed central mass, the break time scale is shorter in higher accretion rate systems, such as NLS 1 galaxies. The numerical value of the X-ray time scale defined in eq. ($\ref{tau_value}$) thus matches the typically measured $T_{B}$ value. This correspondence leads us to associate the characteristic X-ray time scale $\tau_{X}$ with the PSD break time scale $T_{B}$. The predicted scaling of $\tau_{X}$ with black hole mass is in agreement with the observed mass-break time scale correlation. For a given black hole mass $\tau_{X}$ decreases with increasing accretion rate, a trend confirmed by the short time scales observed in narrow-line objects.
The general relation between break time scale, black hole mass, and accretion rate has been quantified by @McH_et_2006. Following the hypothesis that the break time scale depends on both black hole mass and accretion rate, they obtained a scaling relationship of the form $$T_{B} \approx \frac{M_{BH}^{1.12}}{\dot{m}_{E}^{0.98}} \, ,
\label{eq_McH}$$ where $\dot{m}_{E} \approx L_{bol}/L_{E}$, with $L_{bol}$ the bolometric luminosity and $L_{E}$ the Eddington luminosity. The Eddington luminosity scales with black hole mass ($L_{E} \propto M_{BH}$) and assuming that the bolometric luminosity is proportional to the accretion rate ($L_{bol} \propto \dot{M}$), the empirical relationship can be re-written as: $$T_{B} \propto \frac{M_{BH}^{2.1}}{\dot{M}^{0.98}} \, .
\label{T_B}$$
We recall that the predicted dependence of the characteristic X-ray time scale on central mass and accretion rate is of the form: $$\tau_{X} \propto \frac{M_{BH}^{2}}{\dot{M}} \, .
\label{tau_dependence}$$
The model dependence is therefore in excellent agreement with the observational relation obtained by @McH_et_2006.\
The empirical relation given in eq. ($\ref{eq_McH}$) is stated to be valid down to galactic black hole binary systems in which accretion is expected to be ruled by angular momentum dissipation in a disc. We will investigate whether our model results can be extended to this regime in a future work.
Physical interpretations
------------------------
Several authors have proposed possible interpretations of the X-ray characteristic time scale by comparing different model time scales with the observed break time scales. In this perspective, different accretion disc time scales have been compared with measured PSD break time scales (@M_et_2003, @P_2004). The orbital, thermal, and viscous time scales at different radial distances have been considered. Orbital time scales are too short; on the contrary, viscous time scales are much too long. The thermal time scale, at certain radii, gives the closest value to the measured break time scale. In such cases, thermal instabilities have been invoked as a possible source of variability.
The characteristic variability time scale has also been discussed in the framework of inner propagating accretion flow fluctuation models (@L_1997, @C_et_2001). In this picture, modulations produced at different outer radii in the accretion flow propagate inwards, until reaching the inner X-ray emitting region where they cause variations of the X-ray flux. These authors consider a geometry formed by a geometrically thin, optically thick disc surrounded by a geometrically thick corona extending out to large radial distances. The disc is assumed to be truncated at some radius from the centre. The characteristic X-ray time scale is then associated with the time scale of the modulations at the truncation radius of the disc. The difference in break time scale can be related to the location of the disc truncation radius: the smaller the truncation radius the shorter the characteristic time scale.
In our model, the PSD break time scale is associated with the characteristic X-ray time scale, $\tau_{X}$. Since $\tau_{X}$ represents the heating/cooling time of electrons, the break time scale is directly linked to the physical process at the origin of the observed X-ray emission. Its numerical value and functional dependence are in agreement with the observed PSD break time scale. Fluctuations on time scales shorter than the heating time cannot be transmitted and the resulting suppression of rapid variations translates into a break in the power spectrum. The observed PSD break then reflects the suppression of short time scale variations. We are thus able to predict, at least qualitatively, the overall shape of the power spectrum and in particular the appearance of the break.
Below the break frequency, variations are found to hold the same slope ($P_{\nu} \propto \nu^{-1}$, flicker noise) over several decades in frequency. For instance, the slope in NGC 4051 remains unchanged for over four decades (@McH_et_2004). The similar shape suggests that the same physical mechanism is responsible for the variations over this broad range of frequencies or equivalently time scales. In the inner propagating fluctuations model, this broad range of variability time scales is provided by the range of radii where modulations initially originate. Since different radial distances are associated with different time scales, the optically thin corona must be extended in the radial direction in order to account for the observed range of time scales. In the picture discussed here, the broad range of time scales can be provided by the range of time scales associated with local inhomogeneities in the accretion flow. The inhomogeneity time is independent of the system parameters, while the heating time is determined by the properties of the source. In particular the latter is shorter for small central mass and/or high accretion rate objects. Since the inhomogeneity time does not scale with the source parameters, the condition $\tau_{inh} > \tau_{heat}$ is easily met in such objects. Therefore the decline in variability (observed as a break in the power spectrum) occurs on a shorter time scale, explaining the large rapid variability characteristic of NLS 1 galaxies.
Optical/UV variability
======================
X-ray variability is also related to variations in the seed photon population. It is thus important to investigate possible sources of seed variations and to study the related optical/UV variability. In general, optical/UV light curves are characterized by irregular and aperiodic fluctuations. This suggests that variability is due to random processes and stochastic approaches seem appropriate. In the framework of the so-called discrete event models, variability is attributed to a superposition of independent and random events described by a Poisson distribution. The physical nature of the events can be represented by a variety of phenomena: supernova explosions, stellar collisions, and hot spots on the accretion disc surface. A characteristic variability time scale is given by the event duration, part of the observed UV variability is then directly related to the event rate.
Characteristic variability time scales may be obtained from the study of time series through power spectral densities and structure function analysis. @C_P_2001 derived a characteristic optical/UV time scale of $\sim$5-100 days in a sample of 10 AGNs, performing a structure function analysis. Studying the variability time scales in the framework of discrete-event models, @F_et_2005 showed a lack of (strong) dependence of the event duration on the source luminosity, hence on central mass. In their sample, the average luminosity of the sources covered four orders of magnitude, while the event duration varied by only two orders of magnitude. Physical interpretations based on accretion disc time scales, which scale linearly with black hole mass, can thus be excluded. In our model, the typical time scale for the radiation of the shock energy is given by the expansion time. A characteristic UV time scale can then be associated with this expansion time, which is of the order of $\tau_{exp} \cong 12 M_{33}^{1/2} \zeta_{UV}^{1/4} \, \textrm{d}$. The order of magnitude lies in the observed range and is determined by collision parameters. It has no explicit dependence on the black hole mass, in agreement with observations of @F_et_2005. Shocks can be described by a Poisson distribution with each event rising and decaying on its own characteristic time scale given by the expansion time. These long-term variations may be associated with large amplitude variations seen in the optical/UV range over hundreds of days.
Discrete event models, in which a more luminous object is obtained simply by increasing the number of identical events, predicts a power law slope of -1/2 for the variability versus luminosity relation ($\sigma(L) \propto L^{-1/2}$). But this value is found to be incompatible with the observed trend which seems to favor a flatter dependence, with a slope of -0.08 (@P_C_1997). Two possibilities are envisaged: either the variability process does not follow a Poisson distribution or the individual events are not identical, with the latter hypothesis being more plausible. Indeed, @F_et_2005 found a lack of correlation between event rate and object luminosity, suggesting that a larger total luminosity cannot only be explained by a greater number of identical events. This implies that the event energy cannot be the same for all events.\
In our picture, the average UV luminosity is given by the luminosity of a single event multiplied by the average number of collisions: $\left\langle L_{UV} \right\rangle = L_{UV} \cdot \left\langle N_{c} \right\rangle$. If the difference in total luminosity is not due to the difference in the number of events, then single events should produce different luminosities. In Paper I, we have derived the UV luminosity of a single event in terms of the collision parameters: $$L_{UV} \cong 3 \cdot 10^{45} \, \eta_{1/3} M_{33}^{1/2} \zeta_{UV}^{-5/4} \, \textrm{erg/s} \, .$$
The luminosity radiated in each shock is different, as each event is characterized by different collision parameters. In this case the difference in total luminosity is attributed to differences in the luminosity of individual events and not to the event rate. This can be reconciled with the observed flat dependence of variability on luminosity.
As the first shock occurs in optically thick conditions, the photospheric temperature can be estimated assuming blackbody emission. The post-shock temperature evolution has been studied by @C_T_2005. The predicted time delays between optical and UV light curves (increasing with the increase of the wavelength difference) correspond to the observed time lags. Moreover a very short lag between optical and UV is expected in our model since the optical/UV emissions share a common origin, arising from the same optically thick event. This property is in agreement with the observed quasi-simultaneity of optical and UV light curves (@C_C_1991). The underlying common origin also seems to be supported by the observed equivalence of the mean optical and ultraviolet PSDs (@C_P_2001).
An additional source of variability may be related to the geometry of the expansion following the optically thick shock. We have assumed a spherical expansion for simplicity, which is a very rough approximation. In reality, the expansion is expected to occur in a more irregular and asymmetric way, with different elements of the expanding envelopes having different speeds and thus providing a range of time scales. The inhomogeneous expansion may then account for shorter time scale variations, and this could explain the short-term variability sometimes observed even in the optical/UV.
The multiple contributions to the UV variability then affect the X-ray variability, since optically thin shocks arise from optically thick ones, providing a connection between the two energy bands.
Correlations between optical/UV and X-ray emissions
===================================================
The study of correlations between different energy bands are thought to provide information concerning emission processes and causal links relating different emission regions. Indeed, the UV and X-ray emissions are coupled through reprocessing of X-rays and/or upscattering of UV seed photons. Time lags with longer wavelengths leading shorter ones have been interpreted as fluctuations in the accretion disc, propagating from the outer optical/UV emitting regions toward the innermost X-ray emitting region, on the viscous or thermal time scale. Several multi-wavelength monitoring campaigns have searched for correlations and time lags between the two energy bands, in a number of AGNs. However, the sign of the time lag can be different from case to case, and even the existence of correlations is not always confirmed (@N_et_1998, @M_et_2002, @M_et_2008). The global picture is still quite confusing and the problem seems not definitively settled yet.
In our cascade model, the two emitting media are coupled through the succession of shocks, which also determines the sign of the time lag : optically thick shocks provide the seed photons that will later be upscattered by electrons heated in the optically thin shocks. Following the optically thick shock, UV photons are emitted and escape the region when the expanding clump reaches the size of the photosphere and becomes optically thin, i.e. after $\tau_{exp}$ from the shock event. The expansion continues until neighboring envelopes overlap, leading to optically thin shocks in which electrons gain energy on the characteristic heating time. X-rays are emitted as a result of the interaction between photons, generated in the current event, and hot electrons, created in a previous optically thin event. Two distinct time lags are then expected: a nearly zero lag due to the immediate Comptonization process, and a longer lag related to the temporal evolution of electrons, with the UV photons leading the X-rays. The first lag between UV and X-rays is very small, being of the order of the light travel time. In Class S, the second lag is expected to be of the order of the heating time, since optically thin shocks occur rapidly after the expanding sphere has become optically thin. A rough estimate of the order of magnitude of the time lag between the two energy bands is given by $\tau_{X} \sim 5 M_{8}^{2}/\dot{M}_{0}$ d, of the order of a few days for a $10^{8} M_{\odot}$ object. A shorter lag is expected in sources with a smaller black hole.
Significant correlations between X-rays and optical light curves with a delay consistent with zero lag have been recently reported in MR 2251-178 (@A_et_2008) and Mrk 79 (@B_et_2009). Such correlations at very short time lags between X-ray and optical emissions may be interpreted as a result of the Comptonization process of seed photons.
Several cases with time lags of the order of $\sim$days with the optical/UV leading the X-rays have also been observed (@S_et_2003, @A_et_2005, @M_et_2008). The direction of the time lag and the order of magnitude are thus compatible with our predicted lags. @M_et_2008 report a time lag of 15 days in Mrk 509, with the optical leading the X-rays. They attribute the shorter lags found in other objects (of 1-2-days for less massive Seyferts) to a lower black hole mass. This is consistent with the above discussion, if the measured lags are associated with lags related to the heating time.
In Class Q, where the filling factor is small, we need to take into account the different locations of optically thick and optically thin shocks, with the corresponding electron travel time. Matter has to travel from the 100 $R_{S}$ region where seed photons are emitted toward the innermost region where optically thin shocks take place. This is done on the local free-fall time, which is of the order of several months for a $10^{9} M_{\odot}$ source. We would then expect an order of magnitude longer time lags in massive objects compared with Class S sources. Cross-correlation analysis of 3C 273 show two peaks between UV and X-ray light curves, one at zero lag and the other with the UV leading the X-rays by $\sim$1.8 yr (@P_C_W_1998, @S_et_2008). This suggests that the first lag corresponds to the Comptonization process, while the second one may be interpreted as the travel time, from the 100$R_{S}$ region to the innermost region. Thus the overall picture seems to be supported by observations. However, one has to bear in mind that other physical processes might contribute to the observed properties (such as particular features possibly related to a jet component) leading to non negligible uncertainties.
Conclusion
==========
We have considered characteristic time scales derived in the framework of clumpy accretion flows. The value of the characteristic X-ray time scale $\tau_{X}$, associated with the electron heating/cooling process, corresponds to the typically observed value of the PSD break time scale, $T_{B}$. The predicted time scale is shorter for small black hole mass and/or high accretion rate, correctly reproducing the observed trend. The dependence on black hole mass and accretion rate we derived in eq. (\[tau\_value\]) remarkably agrees with the observational relation found by @McH_et_2006, without additional parameters.
In our picture, X-ray variability is attributed to both variations in the seed photon population and density fluctuations in the upscattering medium, with a greater contribution from the latter. A break is expected in the power spectrum, since fluctuations on time scales shorter than the electron heating time are not observable. The associated X-ray time scale $\tau_{X}$, directly related to the physical process of X-ray emission, may thus provide a possible interpretation of the PSD break time scale.
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abstract: 'We have recently shown that both passive and active gravitational masses of a composite body are not equivalent to its energy due to some quantum effects. We have also suggested an idealized and more realistic experiments to detect the above mentioned inequivalence for a passive gravitational mass. The suggested idealized effect is as follows. A spacecraft moves protons of a macroscopic ensemble of hydrogen atoms with constant velocity in the Earth’s gravitational field. Due to non-homogeneous squeezing of space by the field, electron ground state wave function experiences time-dependent perturbation in each hydrogen atom. This perturbation results in the appearance of a finite probability for an electron to be excited at higher energy levels and to emit a photon. The experimental task is to detect such photons from the ensemble of the atoms. More realistic variants of such experiment can be realized in solid crystals and nuclei, as first mentioned by us. In his recent Comment on our paper, Crowell has argued that the effect, suggested by us, contradicts the existing experiments and, in particular, astronomic data. We show here that this conclusion is incorrect and based on the so-called “free fall” experiments, where our effect does not have to be observed.'
author:
- '[**Andrei G. Lebed**]{}$^*$'
- 'Correspondence should be addressed to A.G. Lebed: lebed@physics.arizona.edu'
title: 'Reply to Comment on “Does the Equivalence between Gravitational Mass and Energy Survive for a Composite Quantum Body?”'
---
Introduction
============
Creation of the so-called Theory of Everything is well known to be one of the most important problems in physics. It is also known that development of the Quantum Gravitation theory is one of the most important steps in this direction. Nevertheless, the latter problem appears to be extremely difficult. One of the reasons for that is the fact that the foundations of General Relativity and Quantum Mechanics are very different. Another reason is the absence of the corresponding experimental data. We recall that, so far, quantum effects have been directly tested only in the Newtonian variant of gravitation (see, for example, Refs. \[1,2\]). In this complex situation, we have recently suggested two novel phenomena \[3-9\]. In particular, we have demonstrated that both passive and active gravitational masses of a composite body are not equivalent to its energy due to some quantum effects. We have also suggested two experimental ways \[3-9\] to test the above mentioned phenomena. If one of such experiments is done, it will be the first direct observation of quantum effects in General Relativity.
Goal
====
Very recently, Crowell has published a Comment \[10\] on our paper \[5\], which criticizes one of the suggested by us experiments, which can demonstrate inequivalence of passive gravitational mass of a composite quantum body and its energy. The idealized variant of the experiment is as follows. There is a macroscopic ensemble of hydrogen atoms with each of them being in ground state at $t=0$. Protons of all atoms are dragged by a spacecraft with constant velocity in the Earth’s gravitational field. Due to non-homogeneous squeezing of space by the gravitational field, the atoms are shown \[3-7,9\] become excited and emit photons. As mentioned in Ref.\[3\], the above described phenomenon is very general and have to be observed in solids, nuclei, and elementary particles. The main criticism of the experiment \[3-7,9\] in Comment \[10\] is the statement that the application of our theory to experiments on proton decay is not consistent with the existing experimental data. The goal of our Reply is three-fold. First, we pay attention that the discussed in Ref. \[10\] existing experimental data are obtained for free falling objects. On the other hand, the idealized experiment, suggested by us \[3-7,9\], corresponds to transportation of centers of masses of the hydrogen atoms (i.e., protons) by spacecraft with a constant velocity. We stress that these are two different types of experiments. Second, to strengthen our arguments, we derive the Hamiltonian for the transportation of a hydrogen atom with constant velocity, semi-quantitatively introduced in Refs. \[3-7,9\], from the Dirac equation in a curved spacetime of General Relativity. Third, we discuss “free fall” experiments for a hydrogen atom and make the conclusion that the effect, suggested by us for passive gravitational mass \[3-7,9\], does not have to be observed under such conditions. Thus, proton decay does not have to demonstrate our effect in “free fall” experiments too. So, we make a conclusion that, contrary to the statement of Comment \[10\], the existing experiments on proton decay do not contradict to our theoretical results.
Semi-Quantitative Hamiltonian
=============================
First, let us derive the Hamiltonian of Refs.\[3-7,9\] for a hydrogen atom in the Earth’s gravitational field, using semi-quantitative approach. Below, we consider the case of a weak gravitational field, therefore, we can write the standard interval, describing spacetime in a weak field approximation \[11\]: $$\begin{aligned}
&ds^2 = - \biggl( 1 + 2\frac{\phi}{c^2} \biggl)(c dt)^2 + \biggl(
1 - 2 \frac{\phi}{c^2} \biggl) (dx^2 +dy^2+dz^2 ), \nonumber\\
&\phi = - \frac{GM}{R}.\end{aligned}$$ \[Here $G$ is the gravitational constant, $c$ is the velocity of light, $M$ is the Earth mass, and $R$ is the distance between its center and proton.\] In accordance with General Relativity, we introduce the so-called local proper spacetime coordinates, $$\begin{aligned}
&x'=\biggl(1-\frac{\phi}{c^2} \biggl) x, \ \ \ y'=
\biggl(1-\frac{\phi}{c^2} \biggl) y,
\nonumber\\
&z'=\biggl(1-\frac{\phi}{c^2} \biggl) z , \ \ \ t'=
\biggl(1+\frac{\phi}{c^2} \biggl) t,\end{aligned}$$ where space coordinates do not depend on time and where the interval (1) has the Minkowski form.
In these local spacetime coordinates, we can approximately write the Schrödinger equation for electron in the atom in the standard form, $$i \hbar \frac{\partial \Psi({\bf r'},t')}{\partial t'} = \hat H_0
(\hat {\bf p'},{\bf r'}) \ \Psi({\bf r'},t') ,$$ $$\hat H_0 (\hat {\bf p'},{\bf r'})= m_ec^2 - \frac{\hat {\bf
p'}^2}{2 m_e} - \frac{e^2}{r'},$$ where proton is supposed to have a fixed position due to action of some non-gravitational force on it. \[Here $\hat{\bf p'} = -i \hbar
\partial/\partial {\bf r'}$; $m_e$ and $e$ are electron mass and charge, respectively.\] Let us discuss the approximation (1)-(4). First, in Eqs.(1),(2), we take into account only terms of the order of $|\phi|/c^2$, which can be estimated as $10^{-9}$ near the Earth. Second, in Eqs.(3),(4), we disregard the so-called tidal effects. This means that we do not differentiate gravitational potential, $\phi$, with respect to electron coordinates, ${\bf r}$ and ${\bf r'}$. In the next section, we estimate the tidal terms in the Hamiltonian, which, as will be shown, are of the order of $(r_B/R_0)|\phi/c^2| (e^2/r_B) \sim
10^{-17}|\phi/c^2| (e^2/r_B)$ in the Earth’s gravitational field. \[Here $r_B$ is a hydrogen atom typical “size” (i.e., the Bohr’s radius), $R_0$ is the Earth’s radius.\] Third, we consider proton as a classical particle with mass $m_p \gg m_e$, whose position is fixed and kinetic energy is negligible. As usual, we treat the weak gravitation (1),(2), as a perturbation in the inertial coordinate system, corresponding to the coordinates $(x,y,z,t)$ in Eq.(2). By substituting of these coordinates in the Hamiltonian (3),(4), it is easy to obtain the following effective electron Hamiltonian: $$\hat H(\hat {\bf p},{\bf r}) = m_e c^2 + \frac{\hat {\bf
p}^2}{2m_e}-\frac{e^2}{r} + m_e \phi + \biggl( 3 \frac{\hat {\bf
p}^2}{2 m_e} -2\frac{e^2}{r} \biggl) \frac{\phi}{c^2}$$ and to rewrite it in more convenient form: $$\hat H(\hat {\bf p},{\bf r}) = m_e c^2 + \frac{\hat {\bf
p}^2}{2m_e} -\frac{e^2}{r} + \hat m_g (\hat {\bf p},{\bf r}) \phi
\ .$$ We point out that, in Eq.(6), we introduce the following expression for electron passive gravitational mass operator: $$\hat m_g (\hat {\bf p},{\bf r}) = m_e + \biggl(\frac{\hat {\bf
p}^2}{2m_e} -\frac{e^2}{r}\biggl) \frac{1}{c^2} + \biggl(2
\frac{\hat {\bf p}^2}{2m_e}-\frac{e^2}{r} \biggl) \frac{1}{c^2} ,$$ which is equal to electron weight operator in the weak gravitational field (1). Note that, in Eq.(7), the first term is the bare electron mass, $m_e$, the second term corresponds to the expected electron energy contribution to the mass operator, whereas the third term is the non-trivial virial contribution to the gravitational mass operator. We recall that the Hamiltonian (6),(7) is derived for the case, where a hydrogen atom center of mass (i.e., proton) has a fixed position with respect to the Earth. In other words, it is supported in the gravitational field (1) by some non-gravitational force. Now, suppose that the proton is dragged with small and constant (with respect to the Earth) velocity, $u \ll \alpha c$, by a spacecraft, where $\alpha$ is the fine structure constant and $\alpha c$ is a characteristic value of electron velocity in a hydrogen atom. In this case, we can use adiabatic approximation \[3-7,9\], which results in the following perturbation for the electron Schrödinger equation: $$\hat V (\hat {\bf p},{\bf r}, R, t) = + \biggl(2 \frac{\hat {\bf
p}^2}{2m_e}-\frac{e^2}{r} \biggl) \frac{\phi(R+ut)}{c^2}.$$ Note that we are interested in electron excitations, therefore, in the electron Hamiltonian (8), we keep only the virial term, which does not commute with the Hamiltonian, taken in the absence of gravitational field. Since the Hamiltonian (8) is time dependent it cause to the appearance of electron excitations and, thus, to the appearance of photon emission from a macroscopic ensemble of the atoms. It is important that the Hamiltonians (6)-(8) are not valid for the free falling atoms, where we have to introduce the so-called normal Fermi coordinates \[13,14\]. As a result free falling atoms “feel” only second derivatives of the gravitational potential \[13,14\].
The Most General Hamiltonian
============================
To strengthen our arguments, in this section, we derive our Hamiltonian (6),(7) from the more general Hamiltonian of Ref.\[12\]. It is obtained from Dirac equation in curved spacetime of General Relativity. In Ref.\[12\], completely different physical effect - the mixing effect between even and odd wave functions in a hydrogen atom (i.e., the so-called relativistic Stark effect) - is studied. It is important that it is studied not for the free falling atoms but for the atom, whose center of mass is supported by non-gravitational force in the weak gravitational field (1). Note that the corresponding Hamiltonian is derived in $1/c^2$ approximation, as in our case. The peculiarity of the calculations of Ref.\[12\] is that not only terms of the order of $\phi/c^2$ are calculated, as in our case, but also terms of the order of $\phi'/c^2$, where $\phi'$ is a symbolic derivative of $\phi$ with respect to relative electron coordinates in the atom. Note that, in accordance with the existing tradition, we call the latter terms tidal ones. Obtained in Ref.\[12\] the Hamiltonian (3.24) for the corresponding Schrödinger equation can be expressed as a sum of the following four terms: $$\hat H (\hat {\bf P}, \hat {\bf p}, \tilde {\bf R},r)= \hat H_0 +
\hat H_1 + \hat H_2 + \hat H_3 \ ,$$ where $$\hat H_0 = m_e c^2 + m_p c^2 + \biggl[\frac{\hat {\bf P}^2}{2(m_e
+ m_p)} + \frac{\hat {\bf p}^2}{2 \mu} \biggl] - \frac{e^2}{r} ,$$
$$\begin{aligned}
&\hat H_1 = \biggl\{ m_e c^2 + m_p c^2 + \biggl[3 \frac{\hat {\bf
P}^2}{2(m_e + m_p)} + 3 \frac{\hat {\bf p}^2}{2 \mu} - 2
\frac{e^2}{r} \biggl]\biggl\}
\nonumber\\
&\times \biggl( \frac{\phi - {\bf g}\tilde
{\bf R}}{c^2} \biggl),\end{aligned}$$
$$\begin{aligned}
&\hat H_2 = \frac{1}{c^2} \biggl(\frac{1}{m_e}-\frac{1}{m_p}
\biggl)[-({\bf g}{\bf r}) \hat {\bf p}^2 + i \hbar {\bf g} \hat
{\bf p}]
\nonumber\\
&+\frac{1}{c^2} {\bf g} \biggl(\frac{\hat {\bf s_e}}{m_e} -
\frac{\hat {\bf s_p}}{m_p} \biggl) \times \hat {\bf p} + \frac{e^2
(m_p-m_e)}{2(m_e+m_p)c^2} \frac{{\bf g}{\bf r}}{r},\end{aligned}$$
$$\begin{aligned}
&\hat H_3 = \frac{3}{2}\frac{i \hbar {\bf g}{\bf
P}}{(m_e+m_p)c^2} +\frac{3}{2} \frac{{\bf g}{\bf(s_e+s_p)}\times
{\bf P}}{(m_e+m_p)c^2}
\nonumber\\
&- \frac{({\bf g}{\bf r})({\bf
P}{\bf p})+({\bf P}{\bf r})({\bf g}{\bf p})-i\hbar {\bf g}{\bf
P}}{(m_e+m_p)c^2},\end{aligned}$$
\[Here, ${\bf g}=-G \frac{M}{R^3} {\bf R}$\]. Let us describe notations in Eqs.(9)-(13). Note that $\tilde {\bf R}$ and ${\bf
P}$ are position and momentum of a center of mass of the atom, correspondingly. On the other hand, ${\bf r}$ and ${\bf p}$ are relative electron position and momentum in the center of mass coordinate system; $\mu = m_e m_p /(m_e + m_p)$ is the reduced electron mass. As seen from Eq.(10), $\hat H_0 (\hat {\bf P}, \hat
{\bf p}, r)$ corresponds to the Hamiltonian of a hydrogen atom in the absence of the external field. We point out that $\hat H_1
(\hat {\bf P}, \hat {\bf p}, \tilde {\bf R}, r)$ corresponds to couplings of the bare electron and proton masses as well as electron kinetic and potential energies with the gravitational field (1). The Hamiltonians $\hat H_2 (\hat {\bf p}, {\bf r})$ and $\hat H_3 (\hat {\bf P}, \hat {\bf p}, \tilde {\bf R}, r)$ describe the tidal effects.
Note that, in the previous section, we have semi-quantitatively derived the Hamiltonian (6),(7). Below, we strictly derive it from the more general Hamiltonian (9)-(13). First, we use the approximation, where $m_p/m_e \gg 1$, and, thus, we have $\mu =
m_e$. This allows us to consider proton as a heavy classical particle. We can fix its position, $\tilde {\bf R}$ = const, in coordinate system, corresponding to the source of the gravitational field (1), by putting ${\bf P}= 0$ in the Hamiltonian (9)-(13). Therefore, we can disregard center of mass momentum and center of mass kinetic energy. Moreover, as seen from Eq.(13), $\hat H_3 (\hat {\bf P}, \hat {\bf p}, \tilde {\bf R},
r)=0$ in this case. Moreover, let us estimate the first tidal term (12) in the Hamiltonian. We recall that $|{\bf g}| \simeq
|\phi|/R_0$. It is important that, $|{\bf r}| \sim \hbar / |{\bf
p}| \sim r_B$ and ${\bf p}^2/(2m_e) \sim e^2/r_B$ in a hydrogen atom. These allow us to evaluate the Hamiltonian (12) as $H_2 \sim
(r_B/R_0) (\phi/c^2) (e^2/r_B) \sim 10^{-17} (\phi/c^2)
(e^2/r_B)$, which is $10^{-17}$ smaller than $H_1 \sim (\phi/c^2)
(e^2/r_B)$. Therefore, we can also disregard the first tidal term (12) in the total Hamiltonian (9)-(13). As a result, the Hamiltonian (9)-(13) can be rewritten in a familiar way: $$\hat H (\hat {\bf p},r)= \hat H_0 (\hat {\bf p}, r) + \hat H_1
(\hat {\bf p}, r)$$ $$\hat H_0 (\hat {\bf p}, r) = m_e c^2 + \frac{\hat {\bf p}^2}{2m_e}
- \frac{e^2}{r} ,$$ $$\hat H_1 (\hat {\bf p}, r) = \biggl\{ m_e c^2 + \biggl[3
\frac{\hat {\bf p}^2}{2 m_e} - 2 \frac{e^2}{r} \biggl]\biggl\}
\biggl( \frac{\phi}{c^2} \biggl),$$ where we place the proton at the point $\tilde R = R$. Thus, we can make a conclusion that the Hamiltonian (14)-(16), derived in this section, exactly coincides with the Hamiltonian, semi-quantitatively derived by us earlier \[3-7,9\] \[see Eqs.(6),(7)\].
What is Right and What is Wrong?
================================
As earlier as in Ref.\[3\], we concluded that the suggested by us effect was very general. In particular, we proposed \[3\] to use it not only in atomic physics, but also in condensed matter physics \[3,9\], nuclear physics \[3,10\], and elementary particle physics \[3,10\]. Here, we recall the physical meaning of the effect. Some quantum macroscopic system is placed in spacecraft and dragged with small constant velocity in an external weak gravitational field. In this case, due to nonhomogeneous squeezing of space by the field, there appear some quantum excitations in the system, which result in emission of photons \[3-7\], phonons \[9\], pions \[10\] or some other particles. The experimental task is to detect these particles. We pay attention that, in all our previous works \[3-7,9\] as well as in the previous sections of the current paper, we consider the case, where center of mass of a composite quantum system is dragged by spacecraft. It is important that it is dragged by means of non-gravitational forces with constant velocity with respect to source of gravity. We claim that the extension of our effect to free falling bodies, performed in the Comment \[10\], is not legitimate. It is clear seen from papers \[13,14\], where examples of a free falling hydrogen atom is considered and the Fermi normal coordinates are used. As stressed in Refs.\[13,14\], the free falling atoms “feel” only second derivative of the metric (1) and, thus, cannot exhibit our effect. This is also true for nuclear versions of free falling experiment, considered in the Comment \[10\]. To summarize our effect does not have to be observed in “free fall” experiments, discussed in \[10\]. Therefore, the central statement of Comment \[10\], that considered there nucleus experiments contradict to our effect \[3-7,9\], is incorrect.
Acknowledgements {#acknowledgements .unnumbered}
================
We are thankful to N.N. Bagmet, V.A. Belinski, Steven Carlip, Douglas Singleton, Elias Vagenas, and V.E. Zakharov for useful discussions.
$^*$Also at: L.D. Landau Institute for Theoretical Physics, 2 Kosygina Street, Moscow 117334, Russia.
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author:
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Dominic Joyce\
Lincoln College, Oxford
title: '$\U(1)$-invariant special Lagrangian 3-folds. I. Nonsingular solutions'
---
\#1 \#1 \#1[[(\[\#1\])]{}]{} \[section\] \[thm\][Lemma]{} \[thm\][Proposition]{} \[thm\][Corollary]{} \[thm\][Definition]{} \[thm\][Example]{} \#1[\#1\^2]{} \#1[\#1 ]{} \#1[\#1 ]{} \#1\#2[\#1 \_[C\^[\#2]{}]{}]{} \#1[\#1]{}
Introduction {#un1}
============
Special Lagrangian submanifolds (SL $m$-folds) are a distinguished class of real $m$-dimensional minimal submanifolds in $\C^m$, which are calibrated with respect to the $m$-form $\Re(\d z_1\w\cdots\w\d z_m)$. They can also be defined in (almost) Calabi–Yau manifolds, are important in String Theory, and are expected to play a rôle in the eventual explanation of Mirror Symmetry between Calabi–Yau 3-folds.
This is the first of a suite of three papers [@Joyc6; @Joyc7] studying special Lagrangian 3-folds $N$ in $\C^3$ invariant under the $\U(1)$-action \^[i]{}:(z\_1,z\_2,z\_3)([e]{}\^[i]{}z\_1,[e]{}\^[-i]{}z\_2,z\_3) \[un1eq1\] These three papers and [@Joyc8] are surveyed in [@Joyc9]. Locally we can write $N$ as
N={(z\_1&,z\_2,z\_3)\^3: (z\_3)=u((z\_3),(z\_1z\_2)),\
&(z\_1z\_2)=v((z\_3),(z\_1z\_2)), -=2a},
\[un1eq2\] where $a\in\R$ and $u,v:\R^2\ra\R$ are differentiable functions. It will be shown that $N$ is a special Lagrangian 3-fold in $\C^3$ if and only if $u,v$ satisfy ==-2(v\^2+y\^2+a\^2)\^[1/2]{}. \[un1eq3\]
In fact we have to modify this a bit to allow $N$ to have singularities, which is one of the main things we are interested in. When $a\ne 0$ it turns out that $N$ is always nonsingular, and $u,v$ are always smooth and satisfy in the usual sense. However, when $a=0$, at points $(x,0)$ with $v(x,0)=0$ the factor $-2(v^2+y^2+a^2)^{1/2}$ in becomes zero, and then is no longer elliptic.
Because of this, when $a=0$ the appropriate thing to do is to consider [*weak solutions*]{} of , which may have [*singular points*]{} $(x,0)$ with $v(x,0)=0$. At such a point $u,v$ may not be differentiable, and $\bigl(0,0,x+iu(x,0)\bigr)$ is a singular point of the SL 3-fold $N$ in $\C^3$. Weak solutions of when $a=0$ and their singularities will be studied in the sequels [@Joyc6; @Joyc7], and this paper will focus on the nonsingular case when $a\ne 0$.
We begin in §\[un2\] with an introduction to special Lagrangian geometry, and then §\[un3\] summarizes some background material from analysis that we will need later, to do with Hölder spaces of functions and elliptic operators. Section \[un4\] considers special Lagrangian 3-folds invariant under the $\U(1)$-action , shows that they can locally be written in the form where $u,v$ satisfy , and gives an explanation of why is a [*nonlinear Cauchy–Riemann equation*]{} in terms of almost Calabi–Yau geometry. Examples of solutions $u,v$ of are given in §\[un5\], and the corresponding SL 3-folds $N$ in $\C^3$ described.
Section \[un6\] exploits the fact that is a nonlinear Cauchy–Riemann equation, and so $u+iv$ is a bit like a holomorphic function of $x+iy$. We prove analogues for solutions $u,v$ of of well-known results in complex analysis, in particular those involving multiplicity of zeroes, and formulae counting zeroes of a holomorphic function in terms of winding numbers.
As an application we show that if $S,T$ are domains in $\R^2$ and $(\hat u,\hat v):S\ra T$ are solutions of such that $\hat u,\hat v,\frac{\pd\hat v}{\pd x}$ and $\frac{\pd\hat v}{\pd y}$ take given values at a point, then there do not exist $(u,v):T\ra
S^\circ$ satisfying such that $u,v,\frac{\pd v}{\pd x}$ and $\frac{\pd v}{\pd y}$ take given values at a point. This will be used in [@Joyc6] to prove a priori estimates for derivatives of bounded solutions $u,v$ of on domains in $\R^2$, and these in turn will be important in proving the existence of weak solutions of when $a=0$.
In §\[un7\] we show that if $S$ is a domain in $\R^2$ and $u,v\in C^1(S)$ satisfy , then there exists $f\in C^2(S)$ with $\frac{\pd f}{\pd y}=u$ and $\frac{\pd f}{\pd x}=v$, unique up to addition of a constant, satisfying (()\^2+y\^2+a\^2 )\^[-1/2]{}+2=0. \[un1eq4\] This is a [*second-order quasilinear elliptic equation*]{}. Using results from analysis, we prove existence and uniqueness of solutions of the Dirichlet problem for on strictly convex domains when $a\ne 0$. Combining this with the results of §\[un4\] gives existence and uniqueness results for nonsingular $\U(1)$-invariant SL 3-folds in $\C^3$ satisfying certain boundary conditions.
Section \[un8\] takes a different approach to the same problem. We show that if $S$ is a domain in $\R^2$ and $u,v\in C^2(S)$ satisfy , then $v$ satisfies +2=0. \[un1eq5\] Again, this is a second-order quasilinear elliptic equation, and we can prove existence and uniqueness of solutions of the Dirichlet problem for on domains in $\R^2$ when $a\ne 0$. This gives existence and uniqueness results for nonsingular $\U(1)$-invariant SL 3-folds in $\C^3$ satisfying a different kind of boundary condition.
In the sequel [@Joyc6] we first prove a priori estimates for $\frac{\pd u}{\pd x},\frac{\pd u}{\pd y},\frac{\pd v}{\pd x}$ and $\frac{\pd v}{\pd y}$ when $u,v$ are bounded solutions of on a domain $S$ in $\R^2$, and $a\ne 0$. Using these we generalize Theorems \[un7thm2\], \[un7thm3\], \[un8thm1\] and \[un8thm2\] below to the case $a=0$, proving existence and uniqueness of [*weak*]{} solutions $f\in C^1(S)$ and $u,v\in C^0(S)$ to the Dirichlet problems for and on strictly convex domains when $a=0$. This gives existence and uniqueness results for [*singular*]{} $\U(1)$-invariant SL 3-folds in $\C^3$ satisfying certain boundary conditions.
The following paper [@Joyc7] studies these singular solutions $u,v$ of when $a=0$ in more detail. We show that under mild conditions $u,v$ have only isolated singularities, and these isolated singular points have a [*multiplicity*]{}, which is a positive integer, and one of two [*types*]{}. We also use our results to construct many [*special Lagrangian fibrations*]{} on open subsets of $\C^3$. In [@Joyc8] these are used as local models to study special Lagrangian fibrations of (almost) Calabi–Yau 3-folds, and to draw some conclusions about the [*SYZ Conjecture*]{} [@SYZ]. All four papers are reviewed briefly in [@Joyc9].
A fundamental question about compact special Lagrangian 3-folds $N$ in (almost) Calabi–Yau 3-folds $M$ is: [*how stable are they under large deformations*]{}? Here we mean both deformations of $N$ in a fixed $M$, and what happens to $N$ as we deform $M$. The deformation theory of compact SL 3-folds under [*small*]{} deformations is already well understood, and is described in [@Joyc4 §9] and [@Joyc5 §5]. But to extend this understanding to large deformations, one needs to take into account singular behaviour.
One possible moral of this paper and its sequels [@Joyc6; @Joyc7] is that [*compact SL $3$-folds are pretty stable under large deformations*]{}. That is, we have shown existence and uniqueness for (possibly singular) $\U(1)$-invariant SL 3-folds in $\C^3$ satisfying certain boundary conditions. This existence and uniqueness is [*entirely unaffected*]{} by singularities that develop in the SL 3-folds, which is quite surprising, as one might have expected that when singularities develop the existence and uniqueness properties would break down.
This is encouraging, as both the author’s programme for constructing invariants of almost Calabi–Yau 3-folds in [@Joyc1] by counting special Lagrangian homology 3-spheres, and proving some version of the SYZ Conjecture [@SYZ] in anything other than a fairly weak, limiting form, will require strong stability properties of compact SL 3-folds under large deformations; so these papers may be taken as a small piece of evidence that these two projects may eventually be successful.
[*Acknowledgements.*]{} Mark Gross discusses $\U(1)$-invariant special Lagrangian fibrations in [@Gros §4], and an idea of his helped me to make progress at a difficult stage in the composition of this paper and its sequels. I would also like to thank Rafe Mazzeo and Rick Schoen for helpful conversations. I was supported by an EPSRC Advanced Fellowship whilst writing this paper.
Special Lagrangian geometry {#un2}
===========================
We now introduce the idea of special Lagrangian submanifolds, in two different geometric contexts. First, in §\[un21\], we discuss special Lagrangian submanifolds in $\C^m$. Then §\[un22\] considers special Lagrangian submanifolds in [*almost Calabi–Yau manifolds*]{}, Kähler manifolds equipped with a holomorphic volume form which generalize the idea of Calabi–Yau manifolds. For an introduction to special Lagrangian geometry, see Harvey and Lawson [@HaLa] or the author [@Joyc4; @Joyc5].
Special Lagrangian submanifolds in $\C^m$ {#un21}
-----------------------------------------
We begin by defining [*calibrations*]{} and [*calibrated submanifolds*]{}, following Harvey and Lawson [@HaLa].
Let $(M,g)$ be a Riemannian manifold. An [*oriented tangent $k$-plane*]{} $V$ on $M$ is a vector subspace $V$ of some tangent space $T_xM$ to $M$ with $\dim V=k$, equipped with an orientation. If $V$ is an oriented tangent $k$-plane on $M$ then $g\vert_V$ is a Euclidean metric on $V$, so combining $g\vert_V$ with the orientation on $V$ gives a natural [*volume form*]{} $\vol_V$ on $V$, which is a $k$-form on $V$.
Now let $\vp$ be a closed $k$-form on $M$. We say that $\vp$ is a [*calibration*]{} on $M$ if for every oriented $k$-plane $V$ on $M$ we have $\vp\vert_V\le \vol_V$. Here $\vp\vert_V=\al\cdot\vol_V$ for some $\al\in\R$, and $\vp\vert_V\le\vol_V$ if $\al\le 1$. Let $N$ be an oriented submanifold of $M$ with dimension $k$. Then each tangent space $T_xN$ for $x\in N$ is an oriented tangent $k$-plane. We say that $N$ is a [*calibrated submanifold*]{} if $\vp\vert_{T_xN}=\vol_{T_xN}$ for all $x\in N$. \[un2def1\]
It is easy to show that calibrated submanifolds are automatically [*minimal submanifolds*]{} [@HaLa Th. II.4.2]. Here is the definition of special Lagrangian submanifolds in $\C^m$, taken from [@HaLa §III].
Let $\C^m$ have complex coordinates $(z_1,\dots,z_m)$, and define a metric $g$, a real 2-form $\om$ and a complex $m$-form $\Om$ on $\C^m$ by
g=++,&=(z\_1|z\_1++z\_m|z\_m),\
&=z\_1z\_m.
\[un2eq1\] Then $\Re\Om$ and $\Im\Om$ are real $m$-forms on $\C^m$. Let $L$ be an oriented real submanifold of $\C^m$ of real dimension $m$. We say that $L$ is a [*special Lagrangian submanifold*]{} of $\C^m,$ or [*SL $m$-fold*]{} for short, if $L$ is calibrated with respect to $\Re\Om$, in the sense of Definition \[un2def1\]. \[un2def2\]
As in [@Joyc1; @Joyc2] there is a more general definition of special Lagrangian $m$-fold involving a [*phase*]{} ${\rm e}^{i\th}$, but we will not use it here. Harvey and Lawson [@HaLa Cor. III.1.11] give the following alternative characterization of special Lagrangian submanifolds.
Let $L$ be a real $m$-dimensional submanifold of $\C^m$. Then $L$ admits an orientation making it into a special Lagrangian submanifold of $\C^m$ if and only if $\om\vert_L\equiv 0$ and $\Im\Om\vert_L\equiv 0$. \[un2prop1\]
An $m$-dimensional submanifold $L$ in $\C^m$ is called [*Lagrangian*]{} if $\om\vert_L\equiv 0$. Thus special Lagrangian submanifolds are Lagrangian submanifolds satisfying the extra condition that $\Im\Om\vert_L\equiv 0$, which is how they get their name.
Next we give a result characterizing SL 3-planes $\R^3$ in $\C^3$. Define an anti-bilinear cross product $\t:\C^3\t\C^3\ra\C^3$ by (r\_1,r\_2,r\_3)(s\_1,s\_2,s\_3)=(|r\_2|s\_3-|r\_3|s\_2, |r\_3|s\_1-|r\_1|s\_3,|r\_1|s\_2-|r\_2|s\_1). \[un2eq2\] It is equivariant under the $\SU(3)$-action on $\C^3$. Using this notation, we prove
Let ${\bf r},{\bf s}\in\C^3$ be linearly independent over $\R$, with $\om({\bf r},{\bf s})=0$. Then ${\bf r},{\bf s}$ and ${\bf r}\t{\bf s}$ are linearly independent over $\R$, and $\an{{\bf r},{\bf s},{\bf r}\t{\bf s}}_{\sst\mathbb R}$ is the unique special Lagrangian $3$-plane in $\C^3$ containing $\an{{\bf r},{\bf s}}_{\sst\mathbb R}$. \[un2prop2\]
Explicit calculation using shows that $$\begin{gathered}
g({\bf r},{\bf r}\t{\bf s})=g({\bf s},{\bf r}\t{\bf s})=0,
\label{un2eq3}\\
\om({\bf r},{\bf r}\t{\bf s})=\om({\bf s},{\bf r}\t{\bf s})=0,
\label{un2eq4}\\
\ms{{\bf r}\t{\bf s}}=\ms{{\bf r}}\ms{{\bf s}}
-g({\bf r},{\bf s})^2-\om({\bf r},{\bf s})^2,
\label{un2eq5}\\
\text{and}\quad (\Im\Om)({\bf r},{\bf s},{\bf r}\t{\bf s})=0,
\label{un2eq6}\end{gathered}$$ for all ${\bf r},{\bf s}\in\C^3$. When ${\bf r},{\bf s}$ are linearly independent and $\om({\bf r},{\bf s})=0$, equation shows that ${\bf r}\t{\bf s}$ is orthogonal to ${\bf r},{\bf s}$, and that $\md{{\bf r}\t{\bf s}}\ne 0$. Therefore ${\bf r},{\bf s}$ and ${\bf r}\t{\bf s}$ are linearly independent.
Also we have $\om({\bf r},{\bf s})=\om({\bf r},{\bf r}\t{\bf s})=
\om({\bf s},{\bf r}\t{\bf s})=0$ by , so that $\an{{\bf r},{\bf s},{\bf r}\t{\bf s}}_{\sst\mathbb R}$ is a [*Lagrangian*]{} 3-plane. Then shows that $\an{{\bf r},{\bf s},{\bf r}\t{\bf s}}_{\sst\mathbb R}$ is a [*special*]{} Lagrangian 3-plane, by Proposition \[un2prop1\]. It is easy to see that this is the only SL 3-plane in $\C^3$ containing $\an{{\bf r},{\bf s}}_{\sst\mathbb R}$.
Almost Calabi–Yau $m$-folds and SL $m$-folds {#un22}
--------------------------------------------
We shall define special Lagrangian submanifolds not just in Calabi–Yau manifolds, as usual, but in the much larger class of [*almost Calabi–Yau manifolds*]{}.
Let $m\ge 2$. An [*almost Calabi–Yau $m$-fold*]{}, or [*ACY $m$-fold*]{} for short, is a quadruple $(X,J,\om,\Om)$ such that $(X,J)$ is a $m$-dimensional complex manifold, $\om$ is the Kähler form of a Kähler metric $g$ on $X$, and $\Om$ is a non-vanishing holomorphic $(m,0)$-form on $X$.
We call $(X,J,\om,\Om)$ a [*Calabi–Yau $m$-fold*]{}, or [*CY $m$-fold*]{} for short, if in addition $\om$ and $\Om$ satisfy \^m/m!=(-1)\^[m(m-1)/2]{}(i/2)\^m|. \[un2eq7\] Then for each $x\in X$ there exists an isomorphism $T_xX\cong\C^m$ that identifies $g_x,\om_x$ and $\Om_x$ with the flat versions $g,\om,\Om$ on $\C^m$ in . Furthermore, $g$ is Ricci-flat and its holonomy group is a subgroup of $\SU(m)$. \[un2def3\]
This is not the usual definition of a Calabi–Yau manifold, but is essentially equivalent to it. (Usually one also assumes that $X$ is compact). Next, motivated by Proposition \[un2prop1\], we define special Lagrangian submanifolds of almost Calabi–Yau manifolds.
Let $(X,J,\om,\Om)$ be an almost Calabi–Yau $m$-fold with metric $g$, and $N$ a real $m$-dimensional submanifold of $X$. We call $N$ a [*special Lagrangian submanifold*]{}, or [*SL $m$-fold*]{} for short, if $\om\vert_N\equiv\Im\Om\vert_N\equiv 0$. \[un2def4\]
The properties of SL $m$-folds in almost Calabi–Yau $m$-folds are discussed by the author in [@Joyc4; @Joyc5]. The deformation and obstruction theory for [*compact*]{} SL $m$-folds in almost Calabi–Yau $m$-folds is well understood, and beautifully behaved.
In this paper we will focus exclusively on special Lagrangian 3-folds in $\C^3$, and the more general almost Calabi–Yau context will hardly enter our story at all. However, because SL $m$-folds in ACY $m$-folds are expected to behave locally just like SL $m$-folds in $\C^m$, our results tell us about SL 3-folds in ACY 3-folds, especially their singular behaviour.
Background material from analysis {#un3}
=================================
We now briefly summarize some background material we will need for later analytic results. Our principal reference is Gilbarg and Trudinger [@GiTr].
Banach spaces of functions on subsets of $\R^n$ {#un31}
-----------------------------------------------
We first define a special class of subsets of $\R^n$ called [*domains*]{}.
A closed, bounded, contractible subset $S$ in $\R^n$ will be called a [*domain*]{} if it is a disjoint union $S=S^\circ\cup\pd S$, where the [*interior*]{} $S^\circ$ of $S$ is a connected open set in $\R^n$ with $S=\overline{S^\circ}$, and the [*boundary*]{} $\pd S=S\sm
S^\circ$ is a compact embedded hypersurface in $\R^n$. \[un3def1\]
Here the assumption that $S$ is contractible is made for simplicity, and will not always be necessary. Note that as they are contractible, domains in $\R^2$ are automatically diffeomorphic to discs. Next we define some Banach spaces of real functions on $S$.
Let $S$ be a domain in $\R^n$. For each integer $k\ge 0$, define $C^k(S)$ to be the space of continuous functions $f:S\ra\R$ with $k$ continuous derivatives, and define the norm $\cnm{.}k$ on $C^k(S)$ by $\cnm{f}k=\sum_{j=0}^k\sup_S\bmd{\pd^jf}$. Then $C^k(S)$ is a Banach space. Define $C^\iy(S)=\bigcap_{k=0}^\iy C^k(S)$ to be the set of smooth functions on $S$. It is not a Banach space, with its natural topology. \[un3def2\]
Here $\pd$ is the vector operator $(\frac{\pd}{\pd x_1},\ldots,
\frac{\pd}{\pd x_n})$, where $(x_1,\ldots,x_n)$ are the standard coordinates on $\R^n$, so that $\pd^jf$ maps $S\ra\bigot^k(\R^n)^*$, and has components $\frac{\pd^jf}{\pd x_{a_1}\cdots\pd x_{a_j}}$ for $1\le a_1,\ldots,a_j\le n$. The lengths $\bmd{\pd^jf}$ are computed using the standard Euclidean metric on $\R^n$.
For $k\ge 0$ an integer and $\al\in(0,1]$, define the [*Hölder space*]{} $C^{k,\al}(S)$ to be the subset of $f\in C^k(S)$ for which $$[\pd^k f]_\al=\sup_{x\ne y\in S}
\frac{\bmd{\pd^kf(x)-\pd^kf(y)}}{\md{x-y}^\al}$$ is finite, and define the [*Hölder norm*]{} on $C^{k,\al}(S)$ to be $\cnm{f}{k,\al}=\cnm{f}k+[\pd^kf]_\al$. Again, $C^{k,\al}(S)$ is a Banach space. \[un3def3\]
Linear and quasilinear elliptic operators {#un32}
-----------------------------------------
We begin by defining [*second-order linear elliptic operators*]{} on functions.
Let $S$ be a domain in $\R^n$. A [*second-order linear differential operator*]{} $P$ mapping $C^{k+2}(S)\ra C^k(S)$ or $C^{k+2,\al}(S)\ra C^{k,\al}(S)$ or $C^\iy(S)\ra C^\iy(S)$ is an operator of the form (Pu)(x)= \_[i,j=1]{}\^na\^[ij]{}(x)(x) +\_[i=1]{}\^nb\^i(x)(x)+c(x)u(x), \[un3eq1\] where $a^{ij}$, $b^i$ and $c$ lie in $C^k(S)$, or $C^{k,\al}(S)$, or $C^\iy(S)$, respectively, and $a^{ij}=a^{ji}$ for all $i,j=1,\ldots,n$. We call $a^{ij},b^i$ and $c$ the [*coefficients*]{} of $P$, so that, for instance, we say $P$ has $C^{k,\al}$ coefficients if $a^{ij}$, $b^i$ and $c$ lie in $C^{k,\al}(S)$. We call $P$ [*elliptic*]{} if the symmetric $n\t n$ matrix $(a^{ij})$ is positive definite at every point of $S$. \[un3def5\]
There is a much more general definition of ellipticity for differential operators of other orders, or acting on vectors rather than functions, but we will not need it. One can also define ellipticity for [*nonlinear*]{} partial differential operators. We will not do this in general, but only for [*quasilinear*]{} differential operators, which are linear in their highest-order derivatives.
Let $S$ be a domain in $\R^n$. A [*second-order quasilinear operator*]{} $Q:C^2(S)\ra C^0(S)$ is an operator of the form (Qu)(x)= \_[i,j=1]{}\^na\^[ij]{}(x,u,u)(x) +b(x,u,u), \[un3eq2\] where $a^{ij}$ and $b$ are continuous maps $S\t\R\t(\R^n)^*\ra\R$, and $a^{ij}=a^{ji}$ for all $i,j=1,\ldots,n$. We call the functions $a^{ij}$ and $b$ the [*coefficients*]{} of $Q$. We call $Q$ [*elliptic*]{} if the symmetric $n\t n$ matrix $(a^{ij})$ is positive definite at every point of $S\t\R\t(\R^n)^*$. \[un3def6\]
Elliptic operators have good [*regularity properties*]{} in Hölder spaces.
Let $S$ be a domain in $\R^n$ and $Q:C^2(S)\ra C^0(S)$ a second-order linear or quasilinear elliptic differential operator. Suppose that $Qu=f$, with $u\in C^2(S)$ and $f\in C^0(S)$, and $u\vert_{\pd S}=\phi$, for $\phi\in C^2(\pd S)$. Then
- Let $k\ge 0$ and $\al\in(0,1)$, and suppose that $Q$ has $C^{k,\al}$ coefficients, $f\in C^{k,\al}(S)$, and $\phi\in C^{k+2,\al}(\pd S)$. Then $u\in C^{k+2,\al}(S)$.
- Suppose $Q$ has smooth coefficients, $f\in C^\iy(S)$, and $\phi\in C^\iy(\pd S)$. Then $u\in C^\iy(S)$.
- Suppose $f$ and the coefficients of $Q$ are real analytic in $S^\circ$. Then $u$ is real analytic in $S^\circ$.
\[un3thm1\]
The linear case of part (a) follows from [@GiTr Th. 6.19, p. 111]. For the quasilinear case, regarding $u$ as fixed, write $$Pv=\sum_{i,j=1}^na^{ij}(x,u,\pd u)\frac{\pd^2v}{\pd x_i\pd x_j}(x),$$ so that $P$ is a [*linear*]{} elliptic operator. Applying the linear case of (a) to the equation $Pu=f-b(x,u,\pd u)$, we can deduce the quasilinear case by induction on $k$. Part (b) follows from (a), and part (c) from Morrey [@Morr §5.7–§5.8].
Essentially the theorem says that solutions $u$ of an elliptic equation $Pu=f$ on $S$ are as smooth as possible, given the differentiability of $f$ and the boundary condition $\phi$. For linear elliptic operators $P$ involving only the derivatives of $u$ there is a [*maximum principle*]{} [@GiTr Th. 3.1, p. 32]:
Let $S$ be a domain in $\R^n$ and $P:C^2(S)\ra C^0(S)$ a second-order linear elliptic differential operator of the form , with $c(x)\equiv 0$. Suppose $u\in C^0(S)\cap C^2(S^\circ)$. If $Pu\ge 0$ in $S^\circ$ then the maximum of $u$ is achieved on $\pd S$, and if $Pu\le 0$ in $S^\circ$ then the minimum of $u$ is achieved on $\pd S$. \[un3thm2\]
Existence results for the Dirichlet problem {#un33}
-------------------------------------------
We shall now use results from Gilbarg and Trudinger [@GiTr] to prove existence results for the Dirichlet problem for two classes of quasilinear elliptic operators, that will be needed in §\[un7\] and §\[un8\]. We begin by defining [*strictly convex domains*]{} in $\R^2$.
A domain $S$ in $\R^2$ is called [*strictly convex*]{} if $S$ is convex and the curvature of $\pd S$ is nonzero at every point. So, for example, $x^2+y^2\le 1$ is strictly convex but $x^4+y^4\le 1$ is not, as its boundary has zero curvature at $(\pm 1,0)$ and $(0,\pm 1)$. \[un3def7\]
Here is our first existence result.
Let $S$ be a strictly convex domain in $\R^2$, and suppose (Pf)(x)=\_[i,j=1]{}\^2a\^[ij]{}(x,f,f) (x) \[un3eq5\] is a second-order quasilinear elliptic operator in $S$ with $a^{ij}\in C^\iy(S\t\R\t\R^2)$. Then whenever $k\ge 0$, $\al\in(0,1)$ and $\phi\in C^{k+2,\al}(\pd S)$ there exists a solution $f\in
C^{k+2,\al}(S)$ of the Dirichlet problem $Pf=0$ in $S$, $f\vert_{\pd S}=\phi$. Furthermore $\cnm{f}{1}\le C\cnm{\phi}{2}$ for some $C>0$ depending only on $S$. \[un3thm3\]
It is not difficult to show that as $S$ is strictly convex there exists $K>0$ depending only on $S$, such that if $\phi\in C^2(\pd S)$ then any three distinct points in the graph of $\phi$ in $\pd S\t\R\subset\R^2\t\R$ lie in a unique plane in $\R^2\t\R$ with slope ${\bf s}\in(\R^2)^*$ satisfying $\md{{\bf s}}
\le K\cnm{\phi}{2}$. In the notation of [@GiTr p. 310], the boundary data $\pd S,\phi$ satisfies a [*three point condition*]{}.
Now (noting the equivalence of the three point and bounded slope conditions, [@GiTr p. 314]), [@GiTr Th. 12.7, p. 312] is an existence result for the Dirichlet problem for an operator of the form with boundary data satisfying a three point condition. Strengthened as in [@GiTr Remark (4), p. 314], it implies that if $\phi\in C^{2,\al}(\pd S)$ then there exists $f\in C^{2,\al}(S)$ with $Pf=0$ in $S$ and $f\vert_{\pd S}=\phi$, which satisfies $\cnm{\pd f}{0}\le K\cnm{\phi}{2}$.
By the maximum principle, Theorem \[un3thm2\], the maximum of $f$ is achieved on $\pd S$. Thus $\cnm{f}{0}=\cnm{\phi}{0}
\le\cnm{\phi}{2}$. Hence $$\cnm{f}{1}=\cnm{f}{0}+\cnm{\pd f}{0}\le
(1+K)\cnm{\phi}{2}=C\cnm{\phi}{2},$$ where $C=1+K$ depends only on $S$. This establishes the case $k=0$ of the theorem. If $\phi\in C^{k+2,\al}(S)$ for $k>0$ then $\phi\in
C^{2,\al}(S)$, so by the $k=0$ case there exists $f\in C^{2,\al}(S)$ with $Pf=0$ and $f\vert_{\pd S}=\phi$. But then Theorem \[un3thm1\] shows that $f\in C^{k+2,\al}(S)$, and the proof is complete.
Combining [@GiTr Th. 15.12, p. 382] and Theorem \[un3thm1\] gives:
Let $S$ be a domain in $\R^n$, and suppose the quasilinear operator (Qv)(x)=\_[i,j=1]{}\^na\^[ij]{}(x,v) (x)+b(x,v,v) \[un3eq6\] is elliptic in $S$ with coefficients $a^{ij}\in C^\iy(S\t\R)$ and $b\in C^\iy(S\t\R\t\R^n)$ satisfying $\bmd{b(x,v,p)}\le C\ms{p}$ and $v\,b(x,v,p)\le 0$ for all $(x,v,p)\in S\t\R\t\R^n$ and some $C>0$. Then whenever $k\ge 0$, $\al\in(0,1)$ and $\phi\in C^{k+2,\al}(\pd S)$ there exists a solution $v\in C^{k+2,\al}(S)$ of the Dirichlet problem $Qv=0$ in $S$, $v\vert_{\pd S}=\phi$. \[un3thm4\]
Note that in both theorems, $Q$ is not a general second-order quasilinear elliptic operator of the form , but has some restrictions on its structure. In particular, has $n=2$ and no term $b(x,f,\pd f)$, and in the $a^{ij}$ depend on $x$ and $v$ but not on $\pd v$, and the sign of $b$ is restricted.
A class of $\U(1)$-invariant SL 3-folds in $\C^3$ {#un4}
=================================================
We will now study special Lagrangian 3-folds $N$ in $\C^3$ invariant under the $\U(1)$-action \^[i]{}:(z\_1,z\_2,z\_3)([e]{}\^[i]{}z\_1,[e]{}\^[-i]{}z\_2,z\_3) \[un4eq1\]
We shall assume that $N$ may be written
N={(z\_1&,z\_2,z\_3)\^3: (z\_3)=u((z\_3),(z\_1z\_2)),\
&(z\_1z\_2)=v((z\_3),(z\_1z\_2)), -=2a},
\[un4eq2\] where $a\in\R$ and $u,v:\R^2\ra\R$ are continuous functions, which are smooth except perhaps at certain singular points. Here is why we choose to write $N$ in this form. As the functions $\Re(z_1z_2),\Im(z_1z_2),
\ms{z_1}-\ms{z_2},\Re(z_3)$ and $\Im(z_3)$ involved in are $\U(1)$-invariant, $N$ is automatically $\U(1)$-invariant.
Also, as in [@Joyc2 Prop. 4.2], if $N$ is a connected Lagrangian submanifold of $\C^m$ invariant under a Lie subgroup $G$ of the automorphism group $\U(m)\lt\C^m$ of $\C^m$, then the moment map $\mu$ of $G$ is constant on $N$. Now the moment map of the $\U(1)$-action is $\ms{z_1}-\ms{z_2}$. Thus $\ms{z_1}-\ms{z_2}=2a$ for some $a\in\R$ on any $\U(1)$-invariant SL 3-fold $N$ in $\C^3$, which is why we have taken $\ms{z_1}-\ms{z_2}=2a$ to be one of the equations defining $N$.
In the other two equations $\Re(z_1z_2)=v\bigl(\Re(z_3),\Im(z_1z_2)\bigr)$ and $\Im(z_3)=u\bigl(\Re(z_3),\Im(z_1z_2)\bigr)$, what we are doing is regarding the functions $x=\Re(z_3)$ and $y=\Im(z_1z_2)$ as [*coordinates*]{} on $N/\U(1)$, and expressing the other two degrees of freedom $\Re(z_1z_2)$ and $\Im(z_3)$ as functions of $x$ and $y$. Thus we define $N$ as a kind of graph of the pair of functions $(u,v)$.
Note that not every $\U(1)$-invariant SL 3-fold $N$ in $\C^3$ may be written in the form . Locally this is generally possible, but globally the functions $u$ and $v$ would have to be multi-valued, branched covers of $\R^2$ for instance. However, we will see that the class of SL 3-folds of this form do have many nice properties, and are interesting both in themselves and for our later applications. So equation should be regarded as more than just an arbitrary choice of coordinate system.
Finding the equations on $u$ and $v$ {#un41}
------------------------------------
We now calculate the conditions on the functions $u(x,y)$, $v(x,y)$ for the 3-fold $N$ of to be special Lagrangian.
Let $S$ be a domain in $\R^2$ or $S=\R^2$, let $u,v:S\ra\R$ be continuous, and $a\in\R$. Define
N={(z\_1,z\_2,z\_3)\^3:& z\_1z\_2=v(x,y)+iy,z\_3=x+iu(x,y),\
&-=2a,(x,y)S}.
\[un4eq3\] Then
- If $a=0$, then $N$ is a (possibly singular) special Lagrangian $3$-fold in $\C^3$, with boundary over $\pd S$, if $u,v$ are differentiable and satisfy = =-2(v\^2+y\^2)\^[1/2]{}, \[un4eq4\] except at points $(x,0)$ in $S$ with $v(x,0)=0$, where $u,v$ need not be differentiable. The singular points of $N$ are those of the form $(0,0,z_3)$, where $z_3=x+iu(x,0)$ for $x\in\R$ with $v(x,0)=0$.
- If $a\ne 0$, then $N$ is a nonsingular SL $3$-fold in $\C^3$, with boundary over $\pd S$, if and only if $u,v$ are differentiable on all of $S$ and satisfy ==-2(v\^2+y\^2+a\^2)\^[1/2]{}. \[un4eq5\]
\[un4prop1\]
We shall give the proof for part (a). Part (b) is similar but more complicated, and will be left to the reader. Let $a=0$, let $N$ be defined by , and let ${\bf z}=(z_1,z_2,z_3)\in N$. For $\bf z$ to be a nonsingular point of $N$, we need $u$ and $v$ to be differentiable at $(x,y)=\bigl(\Re(z_3),\Im(z_1z_2)\bigr)$ in $S$, and for the derivatives of the three functions $$\Re(z_1z_2)-v\bigl(\Re(z_3),\Im(z_1z_2)\bigr),\;\>
\Im(z_3)-u\bigl(\Re(z_3),\Im(z_1z_2)\bigr),\;\>\ms{z_1}-\ms{z_2}$$ on $\C^3$ to be linearly independent at $\bf z$.
Now if $z_1=z_2=0$ then $\ms{z_1}-\ms{z_2}$ has zero derivative at $\bf z$. Thus points of the form $(0,0,z_3)$ in $N$ will be singular. Clearly, these occur exactly when $z_3=x+iu(x,0)$ for $x\in\R$ with $v(x,0)=0$. Also, as $\ms{z_1}-\ms{z_2}=0$, such points occur in $N$ only when $a=0$. We shall see that these are the only singular points in $N$, provided $u$ and $v$ are differentiable.
To prove part (a) we need to show that each ${\bf z}\in N$ not of the form $(0,0,z_3)$ is a nonsingular point of $N$, and the tangent space $T_{\bf z}N$ is a special Lagrangian 3-plane $\R^3$ in $\C^3$. As $N$ is $\U(1)$-invariant, it is enough to prove this for one point in each orbit of the $\U(1)$-action . Since $\md{z_1}=\md{z_2}$ on $N$, each $\U(1)$-orbit in $N$ contains one or two points $(z_1,z_2,z_3)$ with $z_1=z_2$.
Thus it is enough to show that $T_{\bf z}N$ exists and is special Lagrangian for points ${\bf z}=(z_1,z_1,z_3)$ in $N$ with $z_1\ne 0$. In our next lemma we identify $T_{\bf z}N$ at such a point. The proof is elementary, and is left as an exercise.
Let ${\bf z}=(z_1,z_1,z_3)\in N$, with $z_1\ne 0$. Set $x=\Re(z_3)$ and $y=\Im(z_1^2)$. Then $N$ is nonsingular at $\bf z$, and $T_{\bf z}N=\an{{\bf p}_1,{\bf p}_2,
{\bf p}_3}_{\sst\mathbb R},$ where \_1&=(iz\_1,-iz\_1,0), \[un4eq6\]\
[**p**]{}\_2&=((2z\_1)\^[-1]{}(x,y), (2z\_1)\^[-1]{}(x,y), 1+i(x,y)) \[un4eq7\]\
[**p**]{}\_3&=((2z\_1)\^[-1]{}((x,y)+i), (2z\_1)\^[-1]{}((x,y)+i), i(x,y)). \[un4eq8\] \[un4lem1\]
Now define $\t:\C^3\t\C^3\ra\C^3$ as in , and apply Proposition \[un2prop2\] with ${\bf r}={\bf p}_1$ and ${\bf s}={\bf p}_2$. Clearly ${\bf p}_1$ and ${\bf p}_2$ are linearly independent, and $\om({\bf p}_1,{\bf p}_2)=0$. So Proposition \[un2prop2\] shows that $\an{{\bf p}_1,{\bf p}_2,
{\bf p}_1\t{\bf p}_2}_{\sst\mathbb R}$ is the unique SL 3-plane in $\C^3$ containing $\an{{\bf p}_1,{\bf p}_2}_{\sst\mathbb R}$.
Therefore $\an{{\bf p}_1,{\bf p}_2,{\bf p}_3}_{\sst\mathbb R}$ is an SL 3-plane if and only if ${\bf p}_3\in\an{{\bf p}_1,{\bf p}_2,
{\bf p}_1\t{\bf p}_2}_{\sst\mathbb R}$. Combining equations , and gives \_1\_2=(|z\_1(+i), |z\_1(+i),-i). \[un4eq9\] So suppose ${\bf p}_3=\al{\bf p}_1+\be{\bf p}_2+\ga{\bf p}_1\t{\bf p}_2$. As the first two coordinates are equal in ${\bf p}_2,{\bf p}_3$ and ${\bf p}_1\t{\bf p}_2$ but not in ${\bf p}_1$, we see that $\al=0$. Taking real parts in the third coordinate gives $\be=0$. And comparing real multiples of $i\bar z_1$ in the first coordinate shows that $\ga=\ha\md{z_1}^{-2}$.
Thus $T_{\bf z}N$ is special Lagrangian if and only if ${\bf p}_1\t{\bf p}_2=2\ms{z_1}{\bf p}_3$. By and , this reduces to ==-2 \[un4eq10\] But $v=\Re(z_1^2)$ and $y=\Im(z_1^2)$ by , so that $\md{z_1}^4=v^2+y^2$, and $\ms{z_1}=(v^2+y^2)^{1/2}$. Substituting this into gives equation , which proves part (a) of Proposition \[un4prop1\]. Part (b) is left to the reader.
Equations and are [*nonlinear versions of the Cauchy–Riemann equations*]{}. For if we replace the factors $2(v^2+y^2)^{1/2}$ and $2(v^2+y^2+a^2)^{1/2}$ in and by 1, the equations become $$\frac{\pd u}{\pd x}=\frac{\pd v}{\pd y}\quad\text{and}\quad
\frac{\pd v}{\pd x}=-\,\frac{\pd u}{\pd y},$$ which are the conditions for $u+iv$ to be a holomorphic function of $x+iy$. We may therefore expect the solutions of and to have qualitative features in common with solutions of the Cauchy–Riemann equations.
Let $S$ be a domain in $\R^2$, let $a\ne 0$, and suppose $u,v\in C^1(S)$ satisfy . Then $u,v$ are real analytic in $S^\circ$, and satisfy +2(v\^2+y\^2+a\^2)\^[1/2]{} +2&=0 \[un4eq11\]\
(v\^2+y\^2+a\^2)\^[-1/2]{} +2+&=0 \[un4eq12\] \[un4prop3\]
One can show that $u,v$ are real analytic in $S^\circ$ following Harvey and Lawson [@HaLa Th. III.2.7]. Thus $v$ is twice continuously differentiable, so that $\frac{\pd}{\pd x}\bigl[
\frac{\pd v}{\pd y}\bigr]=\frac{\pd}{\pd y}\bigl[\frac{\pd v}{\pd x}
\bigr]$ in $S^\circ$. Using to substitute for $\frac{\pd v}{\pd y},\frac{\pd v}{\pd x}$ in terms of $\frac{\pd u}{\pd x},\frac{\pd u}{\pd y}$ gives . Equation follows in the same way.
Regarding the factors $(v^2+y^2+a^2)^{\pm 1/2}$ as part of the coefficients $a^{ij}(x),b^i(x)$, we see that and are second-order linear elliptic equations in $u$ and $v$ respectively, of the form , with $c(x)\equiv 0$. Therefore by the maximum principle, Theorem \[un3thm2\], we have:
Let $S$ be a domain in $\R^2$, let $a\ne 0$, and suppose $u,v\in C^1(S)$ satisfy . Then the maxima and minima of $u$ and $v$ are achieved on $\pd S$. \[un4cor\]
Interpretation using Kähler quotients {#un42}
-------------------------------------
We can use an idea due independently to Goldstein [@Gold §2] and Gross [@Gros §1] to interpret some features of the above construction. Let $(X,J,\om,\Om)$ be an almost Calabi–Yau $m$-fold, as in §\[un22\], and $G$ a $k$-dimensional Lie group acting on $X$ preserving $J,\om,\Om$, with Lie algebra $\mathfrak g$. Suppose the $G$-action admits a moment map $\mu:X\ra{\mathfrak g}^*$.
Then for each $c\in Z({\mathfrak g}^*)$, the quotient $M_c=\mu^{-1}(c)/G$ is nonsingular wherever $G$ acts freely, and has the structure of an almost Calabi–Yau $(m\!-\!k)$-fold on its nonsingular part. If $N$ is a connected, $G$-invariant SL $m$-fold in $X$, then $N\subset\mu^{-1}(c)$ for some $c\in Z({\mathfrak g}^*)$, and $L=N/G$ is an SL $(m\!-\!k)$-fold in $M_c$. Conversely, if $L$ is an SL $(m\!-\!k)$-fold in $M_c$ then $L$ pulls back to an SL $m$-fold $N$ in $X$, contained in $\mu^{-1}(c)$.
In our case, $X$ is $\C^3$ and $G$ is $\U(1)$, acting as in . Any $\U(1)$-invariant SL 3-fold $N$ in $\C^3$ lies in $\mu^{-1}(2a)$ for some $a\in\R$, where $\mu(z_1,z_2,z_3)
=\ms{z_1}-\ms{z_2}$, and pushes down to an SL 2-fold in $M_a=\mu^{-1}(2a)/\U(1)$.
Now SL 2-folds in an almost Calabi–Yau 2-fold $(M,I,\om,\Om)$ are the same thing as [*pseudoholomorphic curves*]{} in $M$ with respect to an alternative almost complex structure $J$ depending on $I,\om$ and $\Om$. Thus, finding $\U(1)$-invariant SL 3-folds $N$ in $\C^3$ is equivalent to finding pseudoholomorphic curves $\Si$ in a family of almost complex 2-folds $M_a$.
Therefore, it is not surprising that and are nonlinear versions of the Cauchy–Riemann equations. However, this almost complex point of view is not that helpful in understanding the [*singular points*]{} of $N$, which occur when $a=v=y=0$. For the $\U(1)$-action on $\mu^{-1}(0)$ is not free, and thus $M_0=\mu^{-1}(0)/
\U(1)$ is a [*singular*]{} almost complex 2-fold.
So the problem is not one of studying singular pseudoholomorphic curves in a nonsingular almost complex 2-fold, which are already very well understood, but of studying pseudoholomorphic curves in a singular almost complex 2-fold, where the almost complex structure itself has unpleasant, non-isolated singularities, which are not at all like the singularities of complex manifolds.
Examples {#un5}
========
By starting with known examples $N$ of SL 3-folds in $\C^3$ invariant under the $\U(1)$-action and solving for $u$ and $v$, we can construct examples of solutions $u,v$ to equations and .
We shall do this with a family of explicit SL 3-folds in $\C^3$ written down by Harvey and Lawson [@HaLa §III.3.A], and studied in more detail by the author [@Joyc1 §3]. Let $a\ge 0$. Define a subset $N_a$ in $\C^3$ by
N\_a={(z\_1&,z\_2,z\_3)\^3:-2a==,\
&(z\_1z\_2z\_3)=0,(z\_1z\_2z\_3) 0}.
\[un5eq1\] By [@HaLa §III.3.A] and [@Joyc1 §3], $N_a$ is a nonsingular SL 3-fold diffeomorphic to ${\mathcal S}^1\t\R^2$ when $a>0$, and $N_0$ is an SL $T^2$-cone with one singular point at $(0,0,0)$. We shall show that these SL 3-folds can be written in the form .
Let $a\ge 0$. Then there exist unique $u_a,v_a:\R^2\ra\R$ such that
N={(z\_1&,z\_2,z\_3)\^3: (z\_3)=u\_a((z\_3),(z\_1z\_2)),\
&(z\_1z\_2)=v\_a((z\_3),(z\_1z\_2)), -=2a}
\[un5eq2\] is the special Lagrangian $3$-fold $N_a$ of . Furthermore:
- $u_a,v_a$ are smooth on $\R^2$ and satisfy , except at $(0,0)$ when $a=0$, where they are only continuous.
- $u_a(x,y)<0$ when $y>0$ for all $x$, and $u_a(x,0)=0$ for all $x$, and $u_a(x,y)>0$ when $y<0$ for all $x$.
- $v_a(x,y)>0$ when $x>0$ for all $y$, and $v_a(0,y)=0$ for all $y$, and $v_a(x,y)<0$ when $x<0$ for all $y$.
- $u_a(0,y)=-y\bigl(\md{a}+\sqrt{y^2+a^2}\,\,\bigr)^{-1/2}$ for all $y$.
- $v_a(x,0)=x\bigl(x^2+2\md{a}\bigr)^{1/2}$ for all $x$.
\[un5thm\]
For simplicity, we first consider the case $a=0$. Let $N_0$ be as in , let $(z_1,z_2,z_3)\in N_0$, and set x=(z\_3),y=(z\_1z\_2),u=z\_3 v=(z\_1z\_2). \[un5eq3\] Then $z_3=x+iu$, and $z_1z_2=v+iy$. Thus the first condition $\ms{z_1}=\ms{z_2}=\ms{z_3}$ in becomes $$\ms{z_1}=\ms{z_2}=x^2+u^2.$$ Squaring gives $\ms{z_1z_2}=(x^2+u^2)^2$, so substituting for $z_1z_2$ yields v\^2+y\^2=(x\^2+u\^2)\^2. \[un5eq4\] Similarly, using the expressions for $z_1z_2$ and $z_3$ above, the second and third conditions on $(z_1,z_2,z_3)$ in become vu+yx=0 vx-yu0. \[un5eq5\]
We will use equations and to prove parts (b) and (c) of the theorem. First suppose $y=0$. Then gives $vu=0$, so $v=0$ or $u=0$. If $v=0$ then gives $x^2+u^2=0$, so $x=u=0$. Thus $y=0$ implies $u=0$. Similarly $u=0$ implies $y=0$, so $u=0$ if and only if $y=0$, as in part (b). In the same way $v=0$ if and only if $x=0$, as in part (c).
We claim that the two terms $vx$ and $-yu$ in are both nonnegative. If one is zero this is obvious. So suppose both are nonzero, so that $x,y,u$ and $v$ are all nonzero. From , the signs of three of these terms determine the sign of the fourth. It is easy to verify that for all eight sign possibilities, $vx$ and $-yu$ have the same sign. So both are nonnegative by . Hence $yu\le 0$, and $u=0$ if and only if $y=0$. Clearly, this proves part (b). Part (c) follows in the same way.
Next we shall show that for each pair $(x,y)$, there is exactly one pair $(u,v)$ satisfying and . Multiplying by $u^2$ and replacing $v^2u^2$ by $y^2x^2$ using , we get $u^6+2x^2u^4+(x^2-y^2)u^2-y^2x^2=0$. This is a sextic in $u$, independent of $v$. Putting $\al=u^2$, it becomes $$P(\al)=\al^3+2x^2\al^2+(x^2-y^2)\al-y^2x^2=0.$$
Thus $u^2$ is a real, nonnegative root of the cubic $P$. Divide into cases
- $x\ne 0$, $y\ne 0$ and $P$ has three real roots $\ga_1,\ga_2,\ga_3$, not necessarily distinct;
- $x\ne 0$, $y\ne 0$ and $P$ has one real root $\ga$ and a complex conjugate pair of non-real roots $\de,\bar\de$;
- $y=0$; and (iv) $x=0$ and $y\ne 0$.
We shall show that in cases (i)–(iii), the cubic $P$ has exactly one real nonnegative root, giving a unique value of $u^2$. In case (iv) there are two nonnegative roots, but one can be excluded.
In case (i) we have $\ga_1+\ga_2+\ga_3=-2x^2<0$, so at least one $\ga_j$ is negative. But $\ga_1\ga_2\ga_3=y^2x^2>0$, so an even number of $\ga_j$ are negative and an odd number positive. The only possibility is that one $\ga_j$ is positive and two negative. So $P$ has exactly one nonnegative root. In case (ii) we have $\ga\ms{\de}=y^2x^2>0$, proving that $\ga>0$, so $P$ has exactly one nonnegative root. In case (iii) we have $P(\al)=\al\bigl(\al+
x^2\bigr)^2$, with roots 0 and $-x^2$ (twice), so the only nonnegative root is 0.
In case (iv) we have $P(\al)=\al^3-y^2\al$, with roots $y,0$ and $-y$. Thus there are two nonnegative roots, $\md{y}$ and 0. However, if $\al=0$ then $u^2=0$, and $x^2=0$ by assumption, so the right hand side of is zero. But $y\ne 0$, so the left hand side is positive, a contradiction. Hence $\al\ne 0$, and there is one allowable value for $\al$, which is $\md{y}$.
We have shown that and determine $u^2$ uniquely, and that there is a solution $u^2$ for all $x,y$. This yields $u$ up to sign. But part (b) gives the sign of $u$, so $u$ is determined uniquely. If $u\ne 0$, equation determines $v$. If $u=0$ then $y=0$ by (b), so gives $v^2=x^2$, and $v=\pm x$. The sign of $v$ is given by (c). Therefore for all pairs $x,y$, there are unique solutions $u,v$ to and .
Let us review what we have proved so far. If $(z_1,z_2,z_3)\in
N_0$ and $x,y,u,v$ are defined by , then they satisfy and . Also, given any $x,y$ there exist unique $u,v$ satisfying and . So, putting $u_0(x,y)=u$ and $v_0(x,y)=v$ defines the functions $u_0,v_0$ in the theorem uniquely, and then $N_0$ is a subset of the 3-fold $N$ of . The converse, that $N\subseteq N_0$, follows easily by reversing the argument above, since if $(z_1,z_2,z_3)
\in N$ then and are equivalent to the equations defining $N_0$. Hence $N=N_0$.
It remains to prove parts (a), (d) and (e). The smoothness in (a) follows directly from and , or indirectly from the fact that $N_0$ is smooth except at $(0,0,0)$, and $u_0,v_0$ satisfy where they are smooth by Proposition \[un4prop1\]. For part (d), set $x=0$. Then $v=0$ by (c), so gives $u^4=y^2$. So $u_0(0,y)=\pm\md{y}^{1/2}$, and the sign is determined by (b). Part (e) follows in the same way. This completes the proof for $a=0$.
When $a\ne 0$, equation must be replaced by $$v^2+y^2=\bigl(x^2+u^2\bigr)\bigl(x^2+u^2+2\md{a}\bigr),$$ but the rest of the proof is more-or-less unchanged.
Here are some remarks on the theorem.
- Let $a>0$. As depends only on $a^2$, the functions $u_a,v_a$ also solve with $a$ replaced by $-a$. The corresponding SL 3-fold is $$\begin{aligned}
N_{-a}=\Bigl\{(z_1,z_2,z_3)\in\C^3:\,&\ms{z_1}=\ms{z_2}-2a=\ms{z_3},\\
&\Im\bigl(z_1z_2z_3\bigr)=0,\quad \Re\bigl(z_1z_2z_3\bigr)\ge 0\Bigr\}.\end{aligned}$$
- The SL 3-fold $N_0$ is a [*cone*]{} in $\C^3$, so that $tN_0=N_0$ for all $t>0$. It follows that the functions $u_0,v_0$ constructed above satisfy u\_0(tx,t\^2y)=tu\_0(x,y) v\_0(tx,t\^2y)=t\^2v\_0(x,y) \[un5eq6\] a kind of [*weighted homogeneity equation*]{}.
- The functions $u_0,v_0$ in the theorem are not smooth at $(0,0)$. Their behaviour helps us to guess properties of more general singular solutions to . For instance, $u_0(0,y)=y\md{y}^{-1/2}$ by (d), so $\frac{\pd u_0}{\pd y}$ is unbounded near $(0,0)$. This will be important when we consider the problem of finding [*a priori estimates*]{} for derivatives of solutions $u,v$ of in [@Joyc6].
Here are some other explicit examples of solutions to and .
Let $\al,\be,\ga\in\R$ and define $u(x,y)=\al x+\be$ and $v(x,y)=\al y+\ga$. Then $u,v$ satisfy for any value of $a$. \[un5ex1\]
Let $S=\R^2$, $u(x,y)=y\tanh x$ and $v(x,y)=\ha
y^2\sech^2x-\ha\cosh^2x$. Then $u$ and $v$ satisfy . Equation with $a=0$ defines an explicit nonsingular special Lagrangian 3-fold $N$ in $\C^3$. It can be shown that $N$ is ruled, and arises from Harvey and Lawson’s ‘austere submanifold’ construction [@HaLa §III.3.C] of SL $m$-folds in $\C^m$, as the normal bundle of a catenoid in $\R^3$. \[un5ex2\]
Let $S=\R^2$, $u(x,y)=\md{y}-\ha\cosh 2x$ and $v(x,y)=-y\sinh 2x$. Then $u,v$ satisfy , except that $\frac{\pd u}{\pd y}$ is not well-defined on the line $y=0$. So equation defines an explicit special Lagrangian 3-fold $N$ in $\C^3$. It turns out that $N$ is the union of two nonsingular SL 3-folds intersecting in a real curve, which are constructed in [@Joyc3 Ex. 7.4] by evolving paraboloids in $\C^3$. \[un5ex3\]
Results using ‘winding number’ techniques {#un6}
=========================================
We will now discuss some results based on the idea of [*winding number*]{}.
Let $C$ be a compact oriented 1-manifold, and $\ga:C\ra\R^2\sm\{0\}$ a differentiable map. Then the [*winding number of $\ga$ about $0$ along*]{} $C$ is $\frac{1}{2\pi}\int_C\ga^*(\d\th)$, where $\d\th$ is the closed 1-form $(x\,\d y-y\,\d x)/(x^2+y^2)$ on $\R^2\sm\{0\}$.
In fact the winding number is simply the [*topological degree*]{} of $\ga$. Thus it is actually well-defined for $\ga$ only [*continuous*]{}, and is invariant under [*continuous deformations*]{} of $\ga$. \[un6def1\]
The motivation for our results is the following theorem from elementary complex analysis:
Let $S$ be a domain in $\C$, and suppose $f:S\ra\C$ is a holomorphic function, with $f\ne 0$ on $\pd S$. Then the number of zeroes of $f$ in $S^\circ$, counted with multiplicity, is equal to the winding number of $f\vert_{\pd S}$ about $0$ along $\pd S$. \[un6thm1\]
As is a nonlinear version of the Cauchy–Riemann equations for holomorphic functions, it is natural to expect that similar results should hold for solutions of . We will prove such results.
Winding number results for solutions of {#un61}
----------------------------------------
Rather than considering with a single solution $u,v$ of , we shall get more general results by working with two solutions $u_1,v_1$ and $u_2,v_2$, and treating $(u_1,v_1)-(u_2,v_2)$ like a holomorphic function for which we wish to count the zeroes. Here is the definition of the multiplicity of a zero of $(u_1,v_1)-(u_2,v_2)$.
Let $S$ be a domain in $\R^2$, let $a\ne 0$, and suppose $(u_1,v_1)$ and $(u_2,v_2)$ are solutions of in $C^1(S)$. Let $k\ge 1$ be an integer and $(b,c)\in S^\circ$. We say that $(u_1,v_1)-(u_2,v_2)$ [*has a zero of multiplicity $k$ at*]{} $(b,c)$ if $\pd^ju_1(b,c)=\pd^ju_2(b,c)$ and $\pd^jv_1(b,c)=\pd^j
v_2(b,c)$ for $j=0,\ldots,k-1$, but $\pd^ku_1(b,c)\ne\pd^ku_2(b,c)$ and $\pd^kv_1(b,c)\ne\pd^kv_2(b,c)$. Here $\pd$ is the vector operator $(\frac{\pd}{\pd x},\frac{\pd}{\pd y})$. \[un6def2\]
The following lemma justifies this definition, by showing that every zero of $(u_1,v_1)-(u_2,v_2)$ has a unique multiplicity.
Let $S$ be a domain in $\R^2$, let $a\ne 0$, and let $(u_1,v_1)$ and $(u_2,v_2)$ be solutions of in $C^1(S)$, with $(u_1,v_1)\not\equiv(u_2,v_2)$. Suppose $(b,c)\in
S^\circ$ with $u_1(b,c)=u_2(b,c)$ and $v_1(b,c)=v_2(b,c)$. Then $(u_1,v_1)-(u_2,v_2)$ has a zero of multiplicity $k$ at $(b,c)$ for some unique $k$. \[un6lem1\]
Since $(u_1,v_1)=(u_2,v_2)$ at one point and $\pd u_j$ determines $\pd v_j$ by , it is easy to see that $u_1\equiv u_2$ if and only if $v_1\equiv v_2$. But $(u_1,v_1)\not\equiv(u_2,v_2)$ by assumption. Thus $u_1\not\equiv u_2$ and $v_1\not\equiv v_2$. By Proposition \[un4prop3\], $u_1,v_1$ and $u_2,v_2$ are real analytic in $S^\circ$, and so they are locally given by their Taylor series at $(b,c)$. Thus, if $\pd^ju_1(b,c)=\pd^ju_2(b,c)$ for all $j=0,1,2,\ldots$ then $u_1\equiv u_2$, a contradiction.
Hence, there exists a unique integer $k\ge 1$ such that $\pd^ju_1(b,c)=\pd^ju_2(b,c)$ for $j=0,\ldots,k-1$, and $\pd^ku_1(b,c)\neq\pd^ku_2(b,c)$. Similarly, there exists a unique $l\ge 1$ such that $\pd^jv_1(b,c)=\pd^jv_2(b,c)$ for $j=0,\ldots,l-1$, and $\pd^lv_1(b,c)\neq\pd^lv_2(b,c)$. But if $\pd^ju_1(b,c)=\pd^ju_2(b,c)$ and $\pd^jv_1(b,c)=\pd^jv_2(b,c)$ for $j=0,\ldots,m-1$, one can show from that $\pd^mu_1(b,c)=\pd^mu_2(b,c)$ if and only if $\pd^mv_1(b,c)=
\pd^mv_2(b,c)$. This implies that $k=l$, and the lemma follows.
Next we show that near a zero, $(u_1,v_1)-(u_2,v_2)$ can be modelled by a genuine holomorphic function, to highest order.
Let $S$ be a domain in $\R^2$, let $a\ne 0$, and let $(u_1,v_1)$ and $(u_2,v_2)$ be solutions of in $C^1(S)$. Suppose $(u_1,v_1)-(u_2,v_2)$ has a zero of multiplicity $k\ge 1$ at $(b,c)$ in $S^\circ$. Then there exists a nonzero complex number $C$ such that
u\_1(x,y)+iv\_1(x,y)=u\_2(x,y)&+iv\_2(x,y) +C((x-b)+i(y-c))\^k\
&+O(\^[k+1]{}+\^[k+1]{}),
\[un6eq1\] where $\la=\sqrt{2}\bigl(v_1(b,c)^2+c^2+a^2\bigr)^{1/4}$. \[un6prop1\]
Define polynomials $p(x,y)$, $q(x,y)$ of order $k$ by $$\begin{aligned}
&p(x,y)=\sum_{j=0}^k\frac{(x-b)^j(y-c)^{k-j}}{j!(k-j)!}\cdot
\frac{\pd^k(u_1-u_2)}{\pd x^j\pd y^{k-j}}(b,c)\\
\text{and}\quad
&q(x,y)=\sum_{j=0}^k\frac{(x-b)^j(y-c)^{k-j}}{j!(k-j)!}\cdot
\frac{\pd^k(v_1-v_2)}{\pd x^j\pd y^{k-j}}(b,c).\end{aligned}$$ Then as $(u_1,v_1)-(u_2,v_2)$ has a zero of multiplicity $k$ at $(b,c)$, we see that $p,q$ are nonzero and
u\_1(x,y)&=u\_2(x,y)+p(x,y)+O(\^[k+1]{}+\^[k+1]{}),\
v\_1(x,y)&=v\_2(x,y)+q(x,y)+O(\^[k+1]{}+\^[k+1]{}).
\[un6eq2\]
Taking the difference of equation for $u_1,v_1$ and $u_2,v_2$, the highest order terms at $(b,c)$ imply that $$\frac{\pd p}{\pd x}=\frac{\pd q}{\pd y}\quad\text{and}\quad
\frac{\pd q}{\pd x}=-\la^2\frac{\pd p}{\pd y}.$$ But these are the Cauchy–Riemann equations for $\la p+iq$ to be a holomorphic function of $\la x+iy$. Since $p,q$ are homogeneous of order $k$ in $(x-b),(y-c)$ it follows that $\la p(x,y)+iq(x,y)=C\bigl(\la(x-b)+i(y-c)\bigr)^k$ for some $C\in\C$, which is nonzero as $p,q$ are nonzero. Combining this with gives .
From we see that if $(x,y)$ is close to $(b,c)$ in $S^\circ$ but not equal to it then $(u_1,v_1)\ne(u_2,v_2)$ at $(x,y)$. This proves:
Let $S$ be a domain in $\R^2$, let $a\ne 0$, and let $(u_1,v_1)$ and $(u_2,v_2)$ be solutions of in $C^1(S)$, with $(u_1,v_1)\not\equiv(u_2,v_2)$. Then the zeroes of $(u_1,v_1)-(u_2,v_2)$ are isolated in $S^\circ$, that is, they have no limit points in $S^\circ$.
Hence, if $(u_1,v_1)\ne(u_2,v_2)$ at every point of $\pd S$, then $(u_1,v_1)-(u_2,v_2)$ has finitely many zeroes in $S$. \[un6cor\]
The last part follows because $S$ is compact, and the set of zeroes of $(u_1,v_1)-(u_2,v_2)$ in $S$ has no limit points. Here is the main result of this section.
Let $S$ be a domain in $\R^2$ and $(u_1,v_1)$, $(u_2,v_2)$ solutions of in $C^1(S)$ for some $a\ne 0$, with $(u_1,v_1)\neq(u_2,v_2)$ at every point of $\pd S$. Then $(u_1,v_1)-(u_2,v_2)$ has finitely many zeroes in $S$. Let there be $n$ zeroes, with multiplicities $k_1,\ldots,k_n$. Then the winding number of $(u_1,v_1)-(u_2,v_2)$ about $0$ along $\pd S$ is $\sum_{i=1}^nk_i$. \[un6thm2\]
Let $B_\ep(x,y)$ denote the open ball of radius $\ep$ about $(x,y)$ in $\R^2$, and $\,\overline{\!B}_\ep(x,y)$ its closure. Let $\ga_\ep(x,y)$ be the circle of radius $\ep$ about $(x,y)$, with the natural orientation, and $\bar\ga_\ep(x,y)$ the same circle with the reverse orientation.
By Corollary \[un6cor\] there are finitely many zeroes of $(u_1,v_1)-(u_2,v_2)$ in $S$. Let these be $(b_1,c_1),\ldots,(b_n,c_n)$, with multiplicities $k_1,\ldots,k_n$ respectively. From we see that if $\ep>0$ is sufficiently small then the winding number of $(u_1,v_1)-(u_2,v_2)$ about 0 along $\ga_\ep(b_i,c_i)$ is $k_i$. Choose $\ep_1,\ldots,\ep_n>0$ small enough that:
- $\,\overline{\!B}_{\ep_i}(b_i,c_i)$ lies in $S^\circ$ for all $i=1,\ldots,n$;
- $\,\overline{\!B}_{\ep_i}(b_i,c_i)\cap
\,\overline{\!B}_{\ep_k}(b_k,c_k)=\emptyset$ for all $1\le j<k\le n$; and
- the winding number of $(u_1,v_1)-(u_2,v_2)$ about $0$ along $\ga_{\ep_i}(b_i,c_i)$ is $k_i$.
Define $T=S\sm\bigcup_{i=1}^nB_{\ep_i}(b_i,c_i)$. Then $(u_1,v_1)-(u_2,v_2)$ has no zeroes in $T$. It follows that the winding number of $(u_1,v_1)-(u_2,v_2)$ about 0 along $\pd T$ is zero. This can be proved from the definition using Stokes’ Theorem, as $(u_1-u_2,v_1-v_2)^*(\d\th)$ is a closed 1-form on $T$, so $\int_{\pd T}(u_1-u_2,v_1-v_2)^*(\d\th)=0$.
Now $\pd T$ is the disjoint union of $\pd S$ and $\bar\ga_{\ep_i}(b_i,c_i)$ for $i=1,\ldots,n$. Thus the winding number of $(u_1,v_1)-(u_2,v_2)$ about 0 along $\pd T$ is the sum of its winding numbers along $\pd S$ and $\bar\ga_{\ep_i}(b_i,c_i)$ for $i=1,\ldots,n$. But the winding number along $\bar\ga_{\ep_i}(b_i,c_i)$ is $-k_i$, as the winding number along $\ga_{\ep_i}(b_i,c_i)$ is $k_i$. Hence the winding number of $(u_1,v_1)-(u_2,v_2)$ about 0 along $\pd S$ minus the sum of $k_1,\ldots,k_n$ is zero, as we want.
Inverse solutions {#un62}
-----------------
Recall from §\[un4\] that equation was derived by beginning with a $\U(1)$-invariant SL 3-fold $N$, and defining functions $x,y,u$ and $v$ on $N$ by $$x=\Re(z_3),\quad u=\Im(z_3), \quad y=\Im(z_1z_2)
\quad\text{and}\quad v=\Re(z_1z_2)$$ for each $(z_1,z_2,z_3)$ in $N$, which also satisfies $\ms{z_1}-\ms{z_2}=2a$. Locally we can regard $u,v$ as functions of $x,y$ (except at branch points), and then the condition that $N$ be special Lagrangian is equivalent to .
Consider the map $\si:\C^3\ra\C^3$ defined by $\si(z_1,z_2,z_3)=
(\bar z_1,i\bar z_2,i\bar z_3)$. This is an isometry with $\si^*(\Re\Om)=-\Re\Om$, and therefore takes SL 3-folds to SL 3-folds, reversing orientation, as SL 3-folds are calibrated w.r.t. $\Re\Om$. Also, $\si^*(x)=u$, $\si^*(u)=x$, $\si^*(y)=v$ and $\si^*(v)=y$, so that $\si$ swaps round $(x,y)$ and $(u,v)$, and $\si$ preserves the equation $\ms{z_1}-\ms{z_2}=2a$.
Therefore, if we regard the SL 3-fold $N$ as a kind of graph of the function $(x,y)\mapsto(u,v)$, the SL 3-fold $\si(N)$ is the ‘graph’ of the inverse function $(u,v)\mapsto(x,y)$. By Proposition \[un4prop1\], it follows that $(u,v)$ satisfies if and only if its inverse satisfies , provided a differentiable inverse exists. So we have proved:
Let $S$ be a domain in $\R^2$, let $a\ne 0$, and let $u,v\in C^1(S)$ satisfy . Define $T=(u,v)[S]$, and suppose $(u,v):S\ra T$ has a differentiable inverse $(u',v'):
T\ra S$, for $u',v'\in C^1(T)$. Then $u',v'$ satisfy , and the $\U(1)$-invariant SL $3$-folds $N,N'$ in $\C^3$ corresponding to $u,v$ and $u',v'$ are related by the involution $(z_1,z_2,z_3)\mapsto(\bar z_1,i\bar z_2,i\bar z_3)$. \[un6prop2\]
One can also easily prove the proposition directly, by expressing the derivatives of $u',v'$ in terms of those of $u,v$ by matrix inversion, and observing that for $u,v$ is equivalent to for $u',v'$. We can interpret the proposition as an analogue of the fact that the inverses of holomorphic functions are holomorphic.
Nonexistence of $u,v$ with given $u,v,\frac{\pd v}{\pd x},
\frac{\pd v}{\pd y}$ at $(x_0,y_0)$ {#un63}
----------------------------------------------------------
We shall use the ‘winding number’ results of §\[un61\] and the ‘inverse solution’ idea of §\[un62\] to show that when $S,T$ are domains in $\R^2$ and $(\hat u,\hat v):S\ra T$ is a solution of , then maps $(u,v):T\ra S^\circ$ satisfying cannot have certain values of $u,v,\frac{\pd v}{\pd x},\frac{\pd v}{\pd y}$ at points $(x_0,y_0)$ in $T^\circ$.
Let $S,T$ be domains in $\R^2$. Let $a\ne 0$, $(\hat x_0,\hat y_0)\in S^\circ$, $(\hat u_0,\hat v_0)\in T^\circ$, and $(\hat p_0,\hat q_0)\in\R^2\sm\{0\}$. Suppose $(\hat u,\hat v):
S\ra T$ is $C^1$ and satisfies and u(x\_0,y\_0)=u\_0, v(x\_0,y\_0)=v\_0, (x\_0,y\_0)=p\_0 (x\_0,y\_0)=q\_0. \[un6eq3\] Define
x\_0=u\_0,y\_0=v\_0, u\_0=x\_0,v\_0=y\_0,\
p\_0=- q\_0=.
\[un6eq4\] Then there does not exist $(u,v):T\ra S^\circ$ which is $C^1$ and satisfies and u(x\_0,y\_0)=u\_0, v(x\_0,y\_0)=v\_0, (x\_0,y\_0)=p\_0 (x\_0,y\_0)=q\_0. \[un6eq5\] \[un6thm3\]
Suppose for a contradiction that there exists $(u,v):T
\ra S^\circ$ which is $C^1$ and satisfies and . Suppose also that $(\hat u,\hat v):S\ra T$ is injective with nowhere vanishing first derivatives. Define $U=(\hat u,\hat v)(S)$. Then $U$ is a domain in $\R^2$, and $(\hat u,\hat v):S\ra U$ is an invertible map with differentiable inverse.
Let $(u',v'):U\ra S$ be the inverse map. Then by Proposition \[un6prop2\], $u',v'$ satisfy . As $(\hat u,\hat v)
(\hat x_0,\hat y_0)=(\hat u_0,\hat v_0)$ we see that $(u',v')(x_0,y_0)
=(u_0,v_0)$. Also, since $\hat u,\hat v$ satisfy we deduce from that $$\begin{pmatrix} \frac{\pd\hat u}{\pd x} & \frac{\pd\hat u}{\pd y} \\
\frac{\pd\hat v}{\pd x} & \frac{\pd\hat v}{\pd y} \end{pmatrix}
(\hat x_0,\hat y_0)=\begin{pmatrix} \hat q_0 &
-\ha(\hat v_0^2+\hat y_0^2+a^2)^{-1/2}\hat p_0 \\
\hat p_0 & \hat q_0\end{pmatrix}.$$ But as $(u',v')$ is the inverse map of $(\hat u,\hat v)$ and $(\hat u,\hat v)(\hat x_0,\hat y_0)=(x_0,y_0)$ we have $$\begin{pmatrix} \frac{\pd u'}{\pd x} & \frac{\pd u'}{\pd y} \\
\frac{\pd v'}{\pd x} & \frac{\pd v'}{\pd y} \end{pmatrix}
(x_0,y_0)=
\begin{pmatrix} \frac{\pd\hat u}{\pd x} & \frac{\pd\hat u}{\pd y} \\
\frac{\pd\hat v}{\pd x} & \frac{\pd\hat v}{\pd y} \end{pmatrix}^{-1}
(\hat x_0,\hat y_0).$$
Combining the last two equations and shows that $$\begin{pmatrix} \frac{\pd u'}{\pd x} & \frac{\pd u'}{\pd y} \\
\frac{\pd v'}{\pd x} & \frac{\pd v'}{\pd y} \end{pmatrix}
(x_0,y_0)=\begin{pmatrix} q_0 & -\ha(v_0^2+y_0^2+a^2)^{-1/2}p_0 \\
p_0 & q_0\end{pmatrix}.$$ Comparing this with and remembering that $u,v$ satisfy , we see that at $(x_0,y_0)$ we have $$\ts u=u',\;\> v=v',\;\>
\frac{\pd u}{\pd x}=\frac{\pd u'}{\pd x},\;\>
\frac{\pd u}{\pd y}=\frac{\pd u'}{\pd y},\;\>
\frac{\pd v}{\pd x}=\frac{\pd v'}{\pd x}\;\>\text{and}\;\>
\frac{\pd v}{\pd y}=\frac{\pd v'}{\pd y}.$$ Thus, $(u',v')-(u,v)$ has a zero of multiplicity at least 2 at $(x_0,y_0)$, in the sense of Definition \[un6def2\].
As $U=(\hat u,\hat v)(S)$ and $(\hat u,\hat v):S\ra T$ we see that $U\subseteq T$. Therefore $(u,v)$ and $(u',v')$ are both solutions of on the domain $U$. Since $(u',v')$ is an orientation-preserving diffeomorphism $U\ra S$, it takes $\pd U$ to $\pd S$, and $(u',v')\vert_{\pd U}$ winds once round $\pd S$ in the positive sense.
Now $(u,v)$ maps to $S^\circ$ by assumption, and $S$ is contractible. Therefore the winding number of $(u',v')-(u,v)$ about 0 along $\pd U$ is 1. So by Theorem \[un6thm2\] the sum of the zeroes of $(u',v')-
(u,v)$ in $U^\circ$, counted with multiplicity, is 1. However, we have already shown that $(u',v')-(u,v)$ has a zero of multiplicity at least 2 at $(x_0,y_0)$, and $(x_0,y_0)\in U^\circ$ as $(u_0,v_0)\in S^\circ$, a contradiction.
This proves the theorem under the additional assumption that $(\hat u,\hat v):S\ra T$ is injective with nowhere vanishing first derivatives. To complete the proof we need to explain how to remove this assumption. We can do this using the Kähler quotient point of view of §\[un42\]. Let $\Si$ be the graph of $(u,v)$ in $S\t T$, swapping round the factors $S,T$, and $\hat\Si$ the graph of $(\hat u,\hat v)$ in $S\t T$.
We can naturally identify $S\t T$ with a subset of the Kähler quotient $M_a$ discussed in §\[un42\]. Thus, $S\t T$ carries an almost complex structure $J$. Since $\Si,\hat\Si$ are both quotients of $\U(1)$-invariant SL 3-folds in $\C^3$, from §\[un42\] we see that $\Si,\hat\Si$ are [*pseudo-holomorphic curves*]{} with respect to $J$.
Now $\pd\Si\subset S^\circ\t\pd T$ and $\pd\hat\Si\subset\pd S\t T$, and $\pd\Si,\pd\hat\Si$ wind once round $\pd T$ and $\pd S$ respectively. Therefore the algebraic intersection number $\Si\cap\hat\Si$ is 1. By properties of pseudo-holomorphic curves it follows that $\Si,\hat\Si$ intersect at only one point, with multiplicity 1. However, the argument above shows that $\Si,\hat\Si$ intersect with multiplicity at least 2 at $(u_0,v_0,x_0,y_0)$, a contradiction, and the theorem is complete.
This theorem will be used in [@Joyc6] to construct a priori estimates for $\frac{\pd u}{\pd x},\frac{\pd u}{\pd y},\frac{\pd
v}{\pd x}$ and $\frac{\pd v}{\pd y}$ for bounded solutions $u,v$ of .
Rewriting in terms of a potential $f$ {#un7}
=====================================
Let $S$ be a domain in $\R^2$, as in Definition \[un3def1\], and fix $a\ne 0$ in $\R$. We shall study differentiable functions $u,v:S\ra\R$ satisfying equation in $S$, and also certain [*boundary conditions*]{} on $\pd S$. As $\frac{\pd u}{\pd x}=\frac{\pd v}{\pd y}$, we can write $u,v$ in terms of a [*potential*]{} $f:S\ra\R$ with $u=\frac{\pd f}{\pd y}$ and $v=\frac{\pd f}{\pd x}$.
Let $S$ be a domain in $\R^2$ and $u,v\in C^1(S)$ satisfy for $a\ne 0$. Then there exists $f\in C^2(S)$ with $\frac{\pd f}{\pd y}=u$, $\frac{\pd f}{\pd x}=v$ and P(f)=(()\^2+y\^2+a\^2 )\^[-1/2]{}+2=0. \[un7eq1\] This $f$ is unique up to addition of a constant, $f\mapsto f+c$. Conversely, all solutions of yield solutions of . \[un7prop1\]
Define a 1-form $\al$ on $S$ by $\al=v(x,y)\d x+u(x,y)\d y$. Then $\d\al=0$ as $\frac{\pd v}{\pd y}=\frac{\pd u}{\pd x}$, so $\al$ is closed. As $S$ is contractible, $\al$ is exact, and so $\al=\d f$ for some $f\in C^2(S)$, unique up to addition of a constant. Equating coefficients of $\d x$ and $\d y$ in $\al=\d f$ gives $\frac{\pd f}{\pd x}=v$, $\frac{\pd f}{\pd y}=u$. Equation follows by substituting these into the first equation of and multiplying by $(v^2+y^2+a^2)^{-1/2}$. The converse is easy.
Now is a [*second-order quasilinear elliptic equation*]{}, so Theorem \[un3thm1\] gives:
Let $S$ be a domain in $\R^2$, let $a\ne 0$, and suppose $f\in C^2(S)$ satisfies with $f\vert_{\pd S}=\phi\in C^2(\pd S)$. Then $f$ is real analytic in $S^\circ$, and if $\phi\in C^{k+2,\al}(\pd S)$ for $k\ge 0$ and $\al\in(0,1)$ then $f\in C^{k+2,\al}(S)$, and if $\phi\in C^\iy(\pd S)$ then $f\in C^\iy(S)$. \[un7thm1\]
As is of the form with $b^i\equiv c\equiv 0$, by the maximum principle, Theorem \[un3thm2\], we deduce:
Let $S$ be a domain in $\R^2$, let $a\ne 0$, and suppose $f\in C^2(S)$ is a solution of . Then the maximum and minimum of $f$ are achieved on $\pd S$. \[un7lem1\]
Equation may also be written P(f)=+2=0, \[un7eq2\] where $A(y,v)$ is defined to be A(y,v)=\_0\^v(w\^2+y\^2+a\^2)\^[-1/2]{}w, =(v\^2+y\^2+a\^2)\^[-1/2]{}. \[un7eq3\] Equation is equivalent to , but is in [*divergence form*]{}.
Calculation shows that we may write $A$ explicitly as $$A(y,v)=\log\biggl[\frac{(v^2+y^2+a^2)^{1/2}+v}{(y^2+a^2)^{1/2}}\biggr]
=\log\biggl[\frac{(y^2+a^2)^{1/2}}{(v^2+y^2+a^2)^{1/2}-v}\biggr].$$ Note that $A$ is undefined when $a=y=0$. That is, if $a=0$ then $A$ is undefined along the $x$-axis.
Expressing as an Euler–Lagrange equation {#un71}
----------------------------------------
We shall show that equation is in fact the [*Euler–Lagrange equation*]{} of a certain functional $I:C^{0,1}(S)\ra\R$. Fix $a\ne 0$, and define a function $B(y,v)$ by $B(y,v)=\int_0^vA(w,y)\d w$, so that $\frac{\pd B}{\pd v}(y,v)=A(y,v)$. Define a function $F$ on $S\t\R^2$ by F(x,y,u,v)=B(y,v)+u\^2, \[un7eq4\] and define a functional $I:C^{0,1}(S)\ra\R$ by I(f)=\_SF(x,y,,)xy. \[un7eq5\] The [*Euler–Lagrange*]{} equation for $I$ is $$\frac{\pd}{\pd x}\biggl[\frac{\pd F}{\pd v}\Bigl(x,y,
\frac{\pd f}{\pd x},\frac{\pd f}{\pd y}\Bigr)\biggr]+
\frac{\pd}{\pd y}\biggl[\frac{\pd F}{\pd u}\Bigl(x,y,
\frac{\pd f}{\pd x},\frac{\pd f}{\pd y}\Bigr)\biggr]=0.$$ From this becomes $$\frac{\pd}{\pd x}\biggl[\frac{\pd B}{\pd v}\Bigl(y,\frac{\pd f}{\pd x}
\Bigr)\biggr]+
\frac{\pd}{\pd y}\biggl[2\,\frac{\pd f}{\pd y}\biggr]=0,$$ and this is equivalent to , since $\frac{\pd B}{\pd v}(y,v)
=A(y,v)$. Thus we have proved:
Equations and are equivalent to the Euler–Lagrange equation of the functional $I:C^{0,1}(S)\ra\R$ defined in . \[un7prop2\]
We could use this to solve the Dirichlet problem for on $S$, by choosing a minimizing sequence $(f_n)_{n=1}^\iy$ for $I$ amongst all $f\in C^{0,1}(S)$ with $f\vert_{\pd S}=\phi$ for some $\phi\in C^{k+2,\al}(\pd S)$, and then showing that $f_n$ converges to a solution as $n\ra\iy$. But we will instead do it by more elementary methods in §\[un73\].
Super- and subsolutions of {#un72}
---------------------------
Let $S$ be a domain in $\R^2$, let $a\ne 0$, and let $P$ be the operator defined in . A function $f\in C^2(S)$ is called a [*supersolution*]{} of if $P(f)\ge 0$ on $S$, and a [*subsolution*]{} if $P(f)\le 0$. Sub- and supersolutions $f,f'$ with $f\le f'$ on $\pd S$ satisfy $f\le f'$ on $S$.
Let $S$ be a domain in $\R^2$ and $a\ne 0$. Suppose $f,f'\in C^2(S)$ satisfy $P(f)\ge 0$ on $S$ and $P(f')\le 0$ on $S$, where $P$ is defined in . If $f\le f'$ on $\pd S$ then $f\le f'$ on $S$, and if $f<f'$ on $\pd S$ then $f<f'$ on $S$. \[un7prop3\]
Applying the Mean Value Theorem to $F(z)=(z^2+y^2+a^2
)^{-1/2}$ on the interval $[\frac{\pd f}{\pd x},\frac{\pd f'}{\pd x}]$ we find that $$\ts\bigl((\frac{\pd f}{\pd x})^2\!+\!y^2\!+\!a^2\bigr)^{-1/2}\!-\!
\bigl((\frac{\pd f'}{\pd x})^2\!+\!y^2\!+\!a^2\bigr)^{-1/2}\!=\!
-w\bigl(w^2\!+\!y^2\!+\!a^2\bigr)^{-3/2}
\bigl(\frac{\pd f}{\pd x}\!-\!\frac{\pd f'}{\pd x}\bigr)$$ for some $w$ between $\frac{\pd f}{\pd x}$ and $\frac{\pd f'}{\pd x}$. Using to expand $P(f)-P(f')$ and rearranging then gives $$\begin{aligned}
P(f)-P(f')=&\bigl(({\ts\frac{\pd f}{\pd x}})^2+y^2+a^2\bigr)^{-1/2}
\frac{\pd^2}{\pd x^2}\bigl(f-f'\bigr)
+2\frac{\pd^2}{\pd y^2}\bigl(f-f'\bigr)\\
&-\bigl(w(w^2+y^2+a^2)^{-3/2}{\ts\frac{\pd^2f'}{\pd x^2}}\bigr)
\frac{\pd}{\pd x}\bigl(f-f'\bigr).\end{aligned}$$
We may regard the right hand side of this equation as $L(f-f')$, where $L$ is a [*linear elliptic operator*]{} of the form with $c\equiv 0$. Thus $L(f-f')=P(f)-P(f')\ge 0$, as $P(f)\ge 0$ and $P(f')\le 0$. So by the maximum principle, Theorem \[un3thm2\], the maximum of $f-f'$ is achieved on $\pd S$. Thus, if $f-f'\le 0$ on $\pd S$ then $f-f'\le 0$ on $S$, and if $f-f'<0$ on $\pd S$ then $f-f'<0$ on $S$.
In particular, if $P(f)=P(f')=0$ and $f\vert_{\pd S}=f'\vert_{\pd S}$, then the proposition implies that $f\le f'$ and (exchanging $f,f'$) that $f'\le f$, so that $f=f'$. This implies uniqueness of solutions of the Dirichlet problem for .
The Dirichlet problem for $f$ {#un73}
-----------------------------
Observe that is of the form . Therefore Theorem \[un3thm3\] applies to give existence for the Dirichlet problem for $f$, and an a priori bound for $\cnm{f}{1}$. Combining this with the real analyticity in Theorem \[un7thm1\] and the uniqueness following from Proposition \[un7prop3\] gives:
Let $S$ be a strictly convex domain in $\R^2$, and let $a\ne 0$, $k\ge 0$ and $\al\in(0,1)$. Then for each $\phi\in C^{k+2,\al}(\pd S)$ there exists a unique solution $f$ of in $C^{k+2,\al}(S)$ with $f\vert_{\pd S}=\phi$. This $f$ is real analytic in $S^\circ$, and satisfies $\cnm{f}{1}\le C\cnm{\phi}{2}$, for some $C>0$ depending only on $S$. \[un7thm2\]
Thus, the Dirichlet problem for is uniquely solvable in a strictly convex domain. Combining the theorem with Propositions \[un4prop1\] and \[un7prop1\], we get an existence and uniqueness result for $\U(1)$-invariant special Lagrangian 3-folds in $\C^3$ satisfying certain boundary conditions.
However, solving the Dirichlet problem in a general, nonconvex domain is more difficult, as to get an a priori estimate for $\md{\pd f}$ on $\pd S$ one needs to find super- and subsolutions of satisfying certain equalities and inequalities on $\pd S$, and this does not seem easy to do in an elementary way. The point about strictly convex domains is that one can use affine functions as super- and subsolutions to estimate $\md{\pd f}$.
An analogue of Theorem \[un7thm2\] for the case $a=0$ will be given in [@Joyc6 Th. 7.1], which shows that has a unique solution $f\in C^1(S)$ with weak second derivatives, and $f\vert_{\pd
S}=\phi$. But $f$ may have singular points, at which it is only once differentiable.
By looking closely at the proofs of existence and uniqueness of $f$, one can show that small changes in $\phi$ and $a$ result in small changes in $f$, where ‘small’ may be interpreted in the $C^{k+2,\al}$ sense. Hence we may prove:
Let $S$ be a strictly convex domain in $\R^2$, $k\ge 0$ and $\al\in(0,1)$. Then the map $C^{k+2,\al}(\pd S)\t\bigl(\R\sm\{0\}
\bigr)\ra C^{k+2,\al}(S)$ taking $(\phi,a)\mapsto f$ is continuous, where $f$ is the unique solution of with $f\vert_{\pd S}=\phi$ constructed in Theorem \[un7thm2\]. \[un7thm3\]
Presumably this map $(\phi,a)\mapsto f$ is also smooth. An extension of Theorem \[un7thm3\] to include the case $a=0$ is given in [@Joyc6 Th. 7.2], but with the $C^1$ rather than the $C^{k+2,\al}$ topology on $f$.
Winding number results for potentials {#un74}
-------------------------------------
We shall now extend some of the ‘winding number’ results of §\[un6\] to the situation of this section. We begin with a definition.
Let $S$ be a domain in $\R^2$, and $\phi\in C^2(\pd S)$. Choose a smooth parametrization /2S, (x(),y()) \[un7eq6\] and regard $\phi$ as a function of $\th$. We call $\phi$ a [*Morse function*]{} if $\frac{\d\phi}{\d\th}$ is zero at only finitely many points in $\pd S$, and $\frac{\d^2\phi}{\d\th^2}$ is nonzero at each of these points.
It can be shown that this definition is independent of the parametrization , and that the Morse functions are an open dense subset of $C^2(\pd S)$. Also, each stationary point of $\phi$ on $\pd S$ is either a local maximum or a local minimum, as $\frac{\d^2\phi}{\d\th^2}\ne 0$, and there are the same number of each, so $\phi$ has exactly $l$ local maxima and $l$ local minima for some $l\ge 1$. \[un7def1\]
If $f\in C^2(S)$ and $f\vert_{\pd S}$ is a Morse function, we can relate the winding number of $\pd f$ round $\pd S$ to the number of local maxima and minima of $f$ on $\pd S$.
Let $S$ be a domain in $\R^2$, and $f\in C^2(S)$ with $f\vert_{\pd S}=\phi$. Suppose that $\pd f\ne 0$ at each point of $\pd S$ and the winding number of $\pd f$ about $0$ along $\pd S$ is $k$, and that $\phi\in C^2(\pd S)$ is a Morse function with $l$ local maxima and $l$ local minima for some $l\ge 1$. Then $1-l\le k\le 1+l$. \[un7prop4\]
Choose a smooth, positively oriented parametrization for $\pd S$ as in . Let the $l$ local maxima of $\phi$ be at $\th=\al_j$ and the $l$ local minima at $\th=\be_j$, where $\al_1,\ldots,\al_l$ and $\be_1,\ldots,\be_l$ lie in $\R/2\pi\Z$, and are arranged in the cyclic order $\al_1,\be_1,\al_2,\be_2,\ldots,
\be_l,\al_{l+1}=\al_1$. Define $(\ga,\de):\R/2\pi\Z\ra\R^2\sm\{0\}$ by $(\ga,\de)=\frac{\d}{\d\th}\bigl(x(\th),y(\th)\bigr)$, so that $(\ga,\de)(\th)$ is tangent to $\pd S$ at $\bigl(x(\th),y(\th)\bigr)$.
Then $\frac{\d\phi}{\d\th}=\ga\frac{\pd f}{\pd x}+\de\frac{\pd f}{\pd y}$ on $\pd S$. Therefore we have +
=0, & =\_j =\_j, j=1,…,l,\
<0, & \_j<0, & \_j<0,
\[un7eq8\]\
\_j&=
-1, & (\_j)(x(\_j),y(\_j))- (\_j)(x(\_j),y(\_j))<0,\
0, & (\_j)(x(\_j),y(\_j))- (\_j)(x(\_j),y(\_j))>0.
\[un7eq9\]
Now we can use equations – to compare the winding numbers of $(\de,-\ga)$ and $\pd f$ about 0 along $\pd S$, as they tell us when the direction of $\pd f$ crosses that of $\pm(\de,-\ga)$. But the winding number of $(\de,-\ga)$ about 0 along $\pd S$ is 1, as it is an outward normal vector to $\pd S$. Using this it is easy to show that the winding number of $\pd f$ about 0 along $\pd S$ is $k=1+\sum_{j=1}^l\eta_j+\sum_{j=1}^l\ze_j$. As $\eta_j$ is 0 or 1 and $\ze_j$ is 0 or $-1$, we see that $1-l\le k\le 1+l$.
Here is the main result of this subsection:
Let $S$ be a domain in $\R^2$ and $f_1,f_2\in C^2(S)$ satisfy for $a\ne 0$ with $f_j\vert_{\pd S}=\phi_j$. Set $u_j=\frac{\pd f_j}{\pd y}$ and $v_j=\frac{\pd f_j}{\pd x}$, so that $u_j,v_j\in C^1(S)$ satisfy . Suppose $\phi_1-\phi_2$ is a Morse function on $\pd S$, with $l$ local maxima and $l$ local minima. Then $(u_1,v_1)-(u_2,v_2)$ has $n$ zeroes in $S^\circ$ with multiplicities $k_1,\ldots,k_n$ and $m$ zeroes on $\pd S$, where $\sum_{i=1}^nk_i+m\le l-1$. \[un7thm4\]
First suppose, for simplicity, that $(u_1,v_1)
\neq(u_2,v_2)$ at every point of $\pd S$. Then $m=0$, and the theorem in this case follows from Theorem \[un6thm2\] and Propositions \[un7prop1\] and \[un7prop4\], noting that $$\ts\pd(f_1-f_2)=\bigl(\frac{\pd}{\pd x}(f_1-f_2),
\frac{\pd}{\pd y}(f_1-f_2)\bigr)=(v_1-v_2,u_1-u_2),$$ so that the winding number of $\pd(f_1-f_2)$ about 0 along $\pd S$ is $-\sum_{i=1}^nk_i$, by Theorem \[un6thm2\]. It remains to prove the result in the case when $(u_1,v_1)=(u_2,v_2)$ at $m\ge 1$ points $(x_0,y_0)$ in $\pd S$.
Then $\pd(f_1-f_2)=0$ at $(x_0,y_0)$, so $(x_0,y_0)$ must be one of the $l$ local maxima or $l$ local minima of $\phi_1-\phi_2$. Thus $m$ is finite. Furthermore, as $\phi_1-\phi_2$ is a Morse function $\frac{\d^2}{\d\th^2}(\phi_1-\phi_2)\ne 0$ at $(x_0,y_0)$, which implies that $\pd(u_1,v_1)\ne\pd(u_2,v_2)$ at $(x_0,y_0)$, and therefore $(x_0,y_0)$ is an [*isolated*]{} zero of $(u_1,v_1)-(u_2,v_2)$. By Corollary \[un6cor\] and compactness of $S$ we deduce that $(u_1,v_1)-(u_2,v_2)$ has finitely many zeroes in $S^\circ$, so we can suppose there are $n$ zeroes, with multiplicities $k_1,\ldots,k_n$.
For $\ep\ge 0$, define $S_\ep$ to be the subset of $(x,y)\in S$ with distance at least $\ep$ from $\pd S$, so that $S_0=S$. Choose $\ep>0$ sufficiently small that $S_\ep$ is a domain, and $S_\ep^\circ$ contains all the $n$ zeroes of $(u_1,v_1)-(u_2,v_2)$ in $S^\circ$, and $f\vert_{\pd S_\ep}$ is also a Morse function with $l$ local maxima and $l$ local minima. It is easy to see that this is possible. Then $(u_1,v_1)\neq(u_2,v_2)$ at every point of $\pd S_\ep$, as the zeroes of $(u_1,v_1)-(u_2,v_2)$ in $S^\circ$ lie in $S_\ep^\circ$.
Let $k$ be the winding number of $\pd(f_1-f_2)$ about 0 along $\pd
S_\ep$. Then Proposition \[un7prop4\] shows that $1-l\le k\le 1+l$. However, we can improve the result in this case. Recall that $\pd(f_1-f_2)=0$ at $m$ out of the $2l$ local maxima and minima of $\phi_1-\phi_2$ on $\pd S$. Using we can show that if $\th=\al_j$ is one of these $m$ points then $\eta_j=1$ in at the corresponding local maximum in $\pd S_\ep$, and if $\th=\be_j$ is one of the $m$ points then $\ze_j=0$ in at the corresponding local minimum in $\pd S_\ep$. Thus, the proof of Proposition \[un7prop4\] shows that $1-l+m\le k\le 1+l$. But applying Theorem \[un6thm2\] gives $k=-\sum_{i=1}^nk_i$, and the theorem follows.
The theorem can be used in conjunction with Theorem \[un7thm2\], the solution of the Dirichlet problem for $f$ on a strictly convex domain. In this case, we would know $\phi_1,\phi_2$ explicitly, but would otherwise know little about the $f_j,u_j$ or $v_j$. The theorem tells us something about $(u_1,v_1)$ and $(u_2,v_2)$, using only the boundary data $\phi_1,\phi_2$.
Using Theorems \[un7thm2\] and \[un7thm3\] we can drop the condition that $\phi_1-\phi_2$ is Morse, requiring instead that it has only finitely many local maxima and minima.
Let $S$ be a strictly convex domain in $\R^2$, let $a\ne 0$, $\al\in(0,1)$, and $f_1,f_2\in C^{2,\al}(S)$ satisfy with $f_j\vert_{\pd S}=\phi_j$. Set $u_j=\frac{\pd f_j}{\pd y}$ and $v_j=\frac{\pd f_j}{\pd x}$, so that $u_j,v_j\in C^{1,\al}(S)$ satisfy . Suppose $\phi_1-\phi_2$ has exactly $l$ local maxima and $l$ local minima on $\pd S$. Then $(u_1,v_1)-(u_2,v_2)$ has $n$ zeroes in $S^\circ$ with multiplicities $k_1,\ldots,k_n$, where $\sum_{i=1}^nk_i\le l-1$. \[un7thm5\]
It is not difficult to construct a smooth family $\phi_1^t\in C^{2,\al}(\pd S)$ for $t\in(0,1]$, such that $\phi_1^t\ra\phi_1$ as $t\ra 0_+$, and $\phi_1^t$ is a Morse function with $l$ local maxima and $l$ local minima, at the same points as $\phi_1$. Let $f_1^t$ be the solution of given by Theorem \[un7thm2\] with $f_1^t\vert_{\pd S}=\phi_1^t$, and set $u_1^t=\frac{\pd f_1^t}{\pd y}$ and $v_1^t=\frac{\pd
f_1^t}{\pd x}$.
Then the sum of the zeroes of $(u_1^t,v_1^t)-(u_2,v_2)$ in $S^\circ$ with multiplicity is no more than $l-1$ for all $t\in(0,1]$, by Theorem \[un7thm4\]. Also $(u_1^t,v_1^t)
\ra(u_1,v_1)$ in $C^{1,\al}(S)$ as $t\ra 0_+$ by Theorem \[un7thm3\]. Combining these using a limiting argument we find that the sum of the zeroes of $(u_1,v_1)-(u_2,v_2)$ in $S^\circ$ with multiplicity is no more than $l-1$.
Note that the theorem does not limit the number of zeroes of $(u_1,v_1)-(u_2,v_2)$ on $\pd S$, which can appear at any stationary point of $\phi_1-\phi_2$.
Another approach to solving {#un8}
============================
In Proposition \[un4prop3\] we showed that if $S$ is a domain in $\R^2$ and $u,v\in C^1(S)$ satisfy then $v$ satisfies in $S^\circ$. Conversely, if $v\in C^2(S)$ satisfies then using to find $\frac{\pd u}{\pd x}$, $\frac{\pd u}{\pd y}$, it is easy to show that as $S$ is contractible there exists $u\in C^2(S)$, unique up to addition of a constant, such that $u,v$ satisfy . In this way we prove:
Let $S$ be a domain in $\R^2$ and $u,v\in C^2(S)$ satisfy for $a\ne 0$. Then Q(v)=+2=0. \[un8eq1\] Conversely, if $v\in C^2(S)$ satisfies then there exists $u\in C^2(S)$, unique up to addition of a constant $u\mapsto u+c$, such that $u,v$ satisfy . \[un8prop1\]
Equation is a [*second-order quasilinear elliptic equation*]{} upon $v$. It is also in [*divergence form*]{}. By elliptic regularity, Theorem \[un3thm1\], we get:
Let $S$ be a domain in $\R^2$, let $a\ne 0$, and suppose $v\in C^2(S)$ is a solution of with $v\vert_{\pd S}=\phi$ for some $\phi\in C^2(\pd S)$. Then $v$ is real analytic in $S^\circ$, and if $\phi\in C^{k+2,\al}(\pd S)$ for $k\ge 0$ and $\al\in(0,1)$ then $v\in C^{k+2,\al}(S)$, and if $\phi\in C^\iy(\pd S)$ then $v\in C^\iy(S)$. \[un8prop2\]
Taking the derivative in gives the equivalent (v\^2+y\^2+a\^2)\^[-1/2]{} - ()\^2+2=0. \[un8eq2\] This is of the form with $c=0$. Therefore by the maximum principle, Theorem \[un3thm2\], we have:
Let $S$ be a domain in $\R^2$, let $a\ne 0$, and suppose $v\in C^2(S)$ is a solution of . Then the maximum and minimum of $v$ are achieved on $\pd S$. \[un8lem1\]
Super- and subsolutions of {#un81}
---------------------------
We now carry out the programme of §\[un72\] for equation .
Let $T$ be a closed, bounded subset of $\R^2$ whose boundary $\pd T=T\sm T^\circ$ is a piecewise-smooth closed curve, and let $a\ne 0$. Suppose $v,v'\in C^2(T)$ satisfy $Q(v)\ge 0$, $Q(v')\le 0$ and $v\ge v'$ on $T$, where $Q$ is defined in , and $v=v'$ on $\pd T$. Then $v=v'$ on $T$. \[un8prop3\]
Choose $C>0$ such that $y^2\le C$ on $T$. Then we have
0&-\_T(C-y\^2)xy\
&=-\_T(C-y\^2)xy\
&=\_[T]{}(C-y\^2)(v\^2+y\^2+a\^2)\^[-1/2]{} (-(v-v’))y\
&+2\_[T]{}(C-y\^2)(v-v’)x +4\_T(v-v’)xy,
\[un8eq3\] using integration by parts, and the fact that $v=v'$ on $\pd T$.
We claim that all three integrals on the final line of are nonnegative. For the first integral, as $v-v'=0$ on $\pd T$ and $v-v'\ge 0$ on $T$, we see that if $(x,y)\in\pd T$ and $w$ is a vector in $\R^2$ pointing outwards from $T$ at $(x,y)$ then $\pd_{w}(v-v')\vert_{(x,y)}\le 0$, and if $w$ points inwards from $T$ then $\pd_{w}(v-v')\vert_{(x,y)}\ge 0$. But $w=
\frac{\pd}{\pd x}$ points outwards from $T$ at $(x,y)$ if and only if $\d y\vert_{\pd T}$ is a positive 1-form on $\pd T$ at $(x,y)$, with the natural orientation on $\pd T$.
Hence $-\frac{\pd}{\pd x}\bigl(v-v'\bigr)\d y\vert_{\pd T}$ is a nonnegative 1-form on $\pd T$, and the first integral on the final line of is nonnegative. Similarly $\frac{\pd}{\pd
y}\bigl(v-v'\bigr)\d x\vert_{\pd T}$ is a nonnegative 1-form, so the second integral is nonnegative, and the third integral is nonnegative as $v-v'\ge 0$. But the sum of the three is nonpositive by . Thus all three integrals are zero, and $\int_T(v-v')\d x\,\d y=0$. As $v-v'\ge 0$ and $v,v'$ are continuous, this implies that $v=v'$ on $T$.
Using this we can prove an analogue of Proposition \[un7prop3\] for . The restriction to real analytic $v,v'$ is not really necessary, but simplifies the proof.
Let $S$ be a domain in $\R^2$ and $a\ne 0$. Suppose $v,v'\in C^2(S)$ are real analytic in $S^\circ$ and satisfy $v\le v'$ on $\pd S$, $Q(v)\ge 0$ on $S$ and $Q(v')\le 0$ on $S$, where $Q$ is defined in . Then $v\le v'$ on $S$. \[un8prop4\]
Define $T^\circ$ to be the subset of $S^\circ$ on which $v>v'$, and $T$ to be the closure of $T^\circ$. Suppose for a contradiction that $T$ is nonempty. Then $v>v'$ on $T^\circ$ and $v=v'$ on $\pd T$. As $v,v'$ are real analytic in $S^\circ$ by assumption, it follows that $T$ has piecewise-smooth boundary. Applying Proposition \[un8prop3\] then shows that $v=v'$ on $T$, a contradiction. Hence $T$ is empty, and $v\le v'$ on $S$.
If $v,v'\in C^2(S)$ satisfy then $Q(v)=Q(v')=0$ and $v,v'$ are real analytic in $S^\circ$ by Proposition \[un8prop2\]. So we have:
Let $S$ be a domain in $\R^2$, let $a\ne 0$, and suppose $v,v'\in C^2(S)$ satisfy on $S$. If $v\le v'$ on $\pd S$ then $v\le v'$ on $S$. \[un8cor\]
In particular, if $v\vert_{\pd S}=v'\vert_{\pd S}$ this gives $v\le v'$ and $v'\le v$ on $S$, so that $v=v'$. This implies uniqueness of solutions of the Dirichlet problem for . Here is an analogue of Corollary \[un8cor\] but with strict inequalities, proved using a different method.
Let $S$ be a domain in $\R^2$, let $a\ne 0$, and suppose $v,v'\in C^2(S)$ satisfy on $S$. If $v<v'$ on $\pd S$ then $v<v'$ on $S$. \[un8prop5\]
Suppose for a contradiction that there exists $(b,c)\in S^\circ$ with $v(b,c)=v'(b,c)$. By Proposition \[un8prop1\] there exist $u,u'\in C^2(S)$, unique up to addition of constants, such that $u,v$ and $u',v'$ satisfy . Choose the constants such that $u(b,c)=u'(b,c)$.
Now apply Theorem \[un6thm2\] to $(u,v)$ and $(u',v')$. As $v<v'$ on $\pd S$, the winding number of $(u,v)-(u',v')$ about 0 along $\pd S$ is zero, since $(u,v)-(u',v')$ is confined to a half-plane and cannot go round $(0,0)$. But $(u,v)-(u',v')$ has at least one zero in $S^\circ$, at $(b,c)$. This is a contradiction. Therefore $v\ne v'$ in $S^\circ$, and by continuity and connectedness we have $v<v'$ on $S$.
The Dirichlet problem for $v$ {#un82}
-----------------------------
We now show that the Dirichlet problem for $v$ is uniquely solvable in arbitrary domains $S$ in $\R^2$.
Let $S$ be a domain in $\R^2$. Then whenever $a\ne 0$, $k\ge 0$, $\al\in(0,1)$ and $\phi\in C^{k+2,\al}(\pd S)$ there exists a unique solution $v\in C^{k+2,\al}(S)$ of with $v\vert_{\pd S}=\phi$. Fix a basepoint $(x_0,y_0)\in S$. Then there exists a unique $u\in C^{k+2,\al}(S)$ with $u(x_0,y_0)=0$ such that $u,v$ satisfy . Furthermore, $u,v$ are real analytic in $S^\circ$. \[un8thm1\]
Observe that the operator $Q$ of is of the form , with coefficients $a^{ij}$ depending on $y$ and $v$ but not on $\pd v$, and $$b\bigl((x,y),v,\pd v\bigr)=-\frac{v}{\bigl(v^2+y^2+a^2\bigr)^{3/2}}
\Bigl(\frac{\pd v}{\pd x}\Bigr)^2.$$ As $\bmd{v(v^2+y^2+a^2)^{-3/2}}\le a^{-2}$ the condition $\bmd{b(x,u,p)}\le C\ms{p}$ in Theorem \[un3thm4\] holds with $C=a^{-2}$, and the condition $v\,b\bigl((x,y),v,p\bigr)\le 0$ for all $\bigl((x,y),v,p\bigr)\in S\t\R\t\R^2$ also clearly holds.
Thus Theorem \[un3thm4\] applies, and there exists $v$ in $C^{k+2,\al}(S)$ satisying with $v\vert_{\pd S}=\phi$. Corollary \[un8cor\] shows that $v$ is unique. Using the condition $u(x_0,y_0)=0$ to fix the additive constant, Proposition \[un8prop1\] shows that there exists a unique $u\in C^2(S)$ with $u(x_0,y_0)=0$ such that $u,v$ satisfy . But shows that $\frac{\pd u}{\pd x},\frac{\pd u}{\pd y}\in C^{k+1,\al}(S)$ as $v\in C^{k+2,\al}(S)$, so $u\in C^{k+2,\al}(S)$. Finally, Proposition \[un4prop3\] shows that $u,v$ are real analytic in $S^\circ$.
Combining the theorem with Proposition \[un4prop1\], we again get an existence and uniqueness result for $\U(1)$-invariant SL 3-folds in $\C^3$ satisfying certain boundary conditions, but different boundary conditions to those in §\[un73\]. An analogue of Theorem \[un8thm1\] for the case $a=0$ will be given in [@Joyc6 Th. 6.1], which shows that has a unique [*weak*]{} solution $v\in C^0(S)$ with $v\vert_{\pd S}=\phi$.
In Theorem \[un7thm2\] we restricted $S$ to be a strictly convex domain, but Theorem \[un8thm1\] works for general domains. The basic reason for this is that in the Dirichlet problem for $v$ we automatically get an a priori estimate for $\cnm{v}{0}$, which implies positive upper and lower a priori bounds for $(v^2+y^2+a^2)^{-1/2}$.
Hence, in the Dirichlet problem for $v$ we know in advance that is [*uniformly elliptic*]{}. However, in the Dirichlet problem for $f$ we need an a priori bound for $\cnm{\frac{\pd f}{\pd
x}}{0}$ to make uniformly elliptic, and we assume $S$ is strictly convex to prove such a bound.
By analogy with Theorem \[un7thm3\], we can also prove:
Let $S$ be a domain in $\R^2$, $k\ge 0$, $\al\in(0,1)$ and $(x_0,y_0)\in S$. Then the map $C^{k+2,\al}(\pd D)\t\bigl(\R\sm
\{0\}\bigr)\ra C^{k+2,\al}(D)^2$ taking $(\phi,a)\mapsto(u,v)$ is continuous, where $(u,v)$ is the unique solution of with $v\vert_{\pd D}=\phi$ and $u(x_0,y_0)=0$ constructed in Theorem \[un8thm1\]. \[un8thm2\]
Presumably the map $(\phi,a)\mapsto(u,v)$ is also smooth. An extension of Theorem \[un8thm2\] to include the case $a=0$ is given in [@Joyc6 Th. 6.2], but with the $C^0$ rather than the $C^{k+2,\al}$ topology on $u,v$.
Winding number results for $v$ {#un83}
------------------------------
As in §\[un74\], we will now extend some of the ‘winding number’ results of §\[un6\] to the situation of this section. Here is the analogue of Morse function for $v$.
Let $S$ be a domain in $\R^2$, and let $v\in C^1(\pd S)$. Choose a smooth, positively oriented parametrization $\th$ for $\pd S$ as in , and regard $v$ as a function of $\th$. We call $v$ [*transverse*]{} if $v=0$ at only finitely many points in $\pd S$, and $\frac{\d v}{\d\th}\ne 0$ at each of these points.
This definition is independent of parametrization $\th$, and transverse functions are an open dense subset of $C^1(\pd S)$. Also, each zero of $v$ is either [*increasing*]{}, with $\frac{\d v}{\d\th}>0$, or [*decreasing*]{}, with $\frac{\d v}{\d\th}<0$, and there are the same number of each, so $f$ has exactly $l$ increasing and $l$ decreasing zeroes for some $l\ge 0$. \[un8def1\]
Here is the analogue of Theorem \[un7thm4\], with a similar proof.
Let $S$ be a domain in $\R^2$, let $a\ne 0$, and let $u_1,v_1\in C^1(S)$ and $u_2,v_2\in C^1(S)$ be solutions of . Suppose that $(v_1-v_2)\vert_{\pd S}$ is transverse with $2l$ zeroes. Then $(u_1,v_1)-(u_2,v_2)$ has $n$ zeroes in $S^\circ$ with multiplicities $k_1,\ldots,k_n$ and $m$ zeroes on $\pd S$, where $n,m\ge 0$ and $k_i\ge 1$, and $\sum_{i=1}^nk_i+m\le l$. \[un8thm3\]
Suppose $(u_1,v_1)=(u_2,v_2)$ at $(x_0,y_0)$ in $\pd S$. Then $v_1-v_2=0$ at $(x_0,y_0)$, so $(x_0,y_0)$ must be one of the $2l$ zeroes of $v_1-v_2$ on $\pd S$. Thus $m$ is finite. Let $m_1\ge 0$ of the $m$ zeroes of $(u_1,v_1)-(u_2,v_2)$ on $\pd S$ be [*increasing*]{} zeroes of $v_1-v_2$ on $\pd S$, and $m_2\ge 0$ be [*decreasing*]{} zeroes, where $m_1+m_2=m$. As in the proof of Theorem \[un7thm4\] we find that $(u_1,v_1)-(u_2,v_2)$ has finitely many zeroes in $S^\circ$, so let there be $n$ zeroes, with multiplicities $k_1,\ldots,k_n$.
Let $\ep>0$ be small. Then $(u_1+\ep,v_1)$ also satisfies , so we can consider the zeroes of $(u_1+\ep,v_1)-(u_2,v_2)$ in $S$. One can use the ideas of §\[un6\] to show that close to the $n$ zeroes of $(u_1,v_1)-(u_2,v_2)$ in $S^\circ$ there will be $n'\ge n$ zeroes of $(u_1+\ep,v_1)-(u_2,v_2)$ with multiplicities $k_1',\ldots,
k_{n'}'$, where $\sum_{i=1}^{n'}k_i'=\sum_{i=1}^nk_i$.
If $(x_0,y_0)$ is one of the $m_1$ increasing zeroes of $v_1-v_2$ on $\pd S$, as $\ep>0$ is small one can show that $(u_1+\ep,v_1)-
(u_2,v_2)$ has a zero in $S^\circ$ near $(x_0,y_0)$. If $(x_0,y_0)$ is one of the $m_2$ decreasing zeroes of $v_1-v_2$ on $\pd S$, there is no zero of $(u_1+\ep,v_1)-(u_2,v_2)$ in $S^\circ$ near $(x_0,y_0)$. So, by Theorem \[un6thm2\] the winding number of $(u_1+\ep,v_1)
-(u_2,v_2)$ about 0 along $\pd S$ is $k'=\sum_{i=1}^{n'}k_i'+m_1$.
Now $(u_1+\ep,v_1)-(u_2,v_2)$ crosses the $x$-axis exactly $2l$ times on $\pd S$, at the zeroes of $v_1-v_2$. However, at the $m_2$ decreasing zeroes $(u_1+\ep,v_1)-(u_2,v_2)$ crosses the $x$-axis at $(\ep,0)$ in the negative sense winding round 0. So it is not difficult to see that the winding number $k'$ satisfies $k'\le l-m_2$. The theorem then follows from the equations $k'=\sum_{i=1}^{n'}k_i'+m_1$, $\sum_{i=1}^{n'}k_i'=
\sum_{i=1}^nk_i$ and $m_1+m_2=m$.
The theorem can be used in conjunction with Theorem \[un8thm1\], the solution of the Dirichlet problem for $v$. In this case, we would know $v_1,v_2$ on $\pd S$ explicitly, but would otherwise know little about the $u_j$ or $v_j$. The theorem tells us something about $(u_1,v_1)$ and $(u_2,v_2)$, using only the known boundary values of $v_1,v_2$.
A maximum principle for $\frac{\pd v}{\pd x}$ {#un84}
---------------------------------------------
Finally we show that if $v$ satisfies on $S$ then $\bmd{\frac{\pd v}{\pd x}}$ is maximum on $\pd S$.
Let $S$ be a domain in $\R^2$, let $a\ne 0$, and suppose $v\in C^2(S)$ satisfies . Then the maximum of $\bmd{\frac{\pd v}{\pd x}}$ is achieved on $\pd S$. \[un8prop6\]
As $v$ satisfies it is real analytic in $S^\circ$ by Proposition \[un8prop2\], and satisfies . Taking the derivative $\frac{\pd}{\pd x}$ of in $S^\circ$ and rearranging gives $$\begin{aligned}
L\Bigl(\frac{\pd v}{\pd x}\Bigr)&=
\bigl(v^2+y^2+a^2\bigr)^{-3/2}\Bigl(\frac{\pd v}{\pd x}\Bigr)^3,
\quad\text{where}\\
L(g)&=\bigl(v^2+y^2+a^2\bigr)^{-1/2}\frac{\pd^2g}{\pd x^2}
+2\frac{\pd^2g}{\pd y^2}-3\bigl(v^2+y^2+a^2\bigr)^{-3/2}v
\frac{\pd v}{\pd x}\cdot\frac{\pd g}{\pd x}.\end{aligned}$$ Then $L$ is a linear elliptic operator of the form , with $c(x)\equiv 0$.
Suppose that $\frac{\pd v}{\pd x}$ has a positive maximum achieved at $(x_0,y_0)\in S^\circ$, with $\frac{\pd v}{\pd x}(x_0,y_0)=M>0$ say, and $\frac{\pd v}{\pd x}<M$ on $\pd S$. Let $\ep\in(0,M)$ be generic and small enough that $\frac{\pd v}{\pd x}<M-\ep$ on $\pd S$, and define $T=\bigl\{(x,y)\in S:\frac{\pd v}{\pd x}\ge M-\ep\bigr\}$. Then as $\ep$ is generic $T$ lies in $S^\circ$ and is compact with smooth boundary, and $\frac{\pd v}{\pd x}=M-\ep$ on $\pd T$. Also $L\bigl(\frac{\pd v}{\pd x}\bigr)>0$ on $T$, as $\frac{\pd v}{\pd x}
\ge M-\ep>0$ on $T$.
Applying Theorem \[un3thm2\] shows that the maximum of $\frac{\pd v}{\pd x}$ on $T$ is achieved on $\pd T$. This contradicts $(x_0,y_0)\in T^\circ$, $\frac{\pd v}{\pd x}(x_0,y_0)
=M$ and $\frac{\pd v}{\pd x}=M-\ep$ on $\pd T$. Thus, if $\frac{\pd v}{\pd x}$ has a positive maximum it is achieved on $\pd S$. Similarly, if $\frac{\pd v}{\pd x}$ has a negative minimum it is achieved on $\pd S$. Thus the maximum of $\bmd{\frac{\pd v}{\pd x}}$ is achieved on $\pd S$.
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---
abstract: 'We report the first operation of a rotating odd-parity Lorentz Invariance test in electrodynamics using a microwave Mach-Zehnder interferometer with permeable material in one arm. The experiment sets a direct bound to $ \kappa_{tr}$ of $-0.3\pm 3\times10^{-7}$. Using new power recycled waveguide interferometer techniques (with the highest spectral resolution ever achieved of $2\times10^{-11} rad/\sqrt{Hz}$) we show an improvement of several orders of magnitude is attainable in the future.'
author:
- 'Michael E. Tobar$^{1}$, Eugene N. Ivanov$^{1}$, Paul L. Stanwix$^{2}$, Jean-Michel G. le Floch$^{1}$,John G. Hartnett$^{1}$'
title: 'Rotating Odd-Parity Lorentz Invariance Test in Electrodynamics'
---
I. Introduction
===============
The Standard Model Extension (SME) extracts terms from additional Lorentz and CPT violating fields that preserve the local gauge symmetry of the usual standard model of particle physics. Thus, in the photon sector the SME leads naturally to an extension of Quantum Electrodynamics with extra Lorentz and CPT violating photon fields [@Kosto1_1; @Kosto1_2; @Kosto1_3]. This work focuses on renormalizable components of the SME in the photon sector as first described in [@KM; @Kosto], which involves operators of mass dimension four or less. Recent developments by the same authors allowing for operators of arbitrary mass dimension have not been considered in this work [@KM_arb]. Of the nineteen independent components, ten associated with vacuum birefringence have been constrained by astrophysical tests to no more than parts in $10^{37}$ [@KM2006]. This leaves nine components, including the scalar component $\kappa_{tr}$, the $3\times 3$ symmetric traceless $\kem^{jk}$ matrix with five degrees of freedom, and the $3\times 3$ antisymmetric $\kop^{jk}$ matrix with three degrees of freedom.
Even parity experiments have leading order sensitivity to the $\kem^{jk}$ Lorentz violating coefficients. Examples of these are the modern Michelson Morley experiments [@Lipa; @Stanwix; @Stanwix2; @Tobar2; @Herrmann; @Mueller:2007; @Antonini; @Peters09; @Schiller09], which have set limits on these coefficients of order $10^{-17}$. The odd parity coefficients $\kop^{jk}$ are only sensitive through the boost of the experiment with respect to the considered frame of reference (which in this case is a sun centered frame). This means the sensitivity is suppressed by the value of the boost, which in this case is of order of the Earth orbital speed ($10^{-4}$). The sensitivity to $\kappa_{tr}$ should be second order sensitive to boost for these type of experiments, even though the details are yet to be fully calculated one expects the sensitivity to be reduced by a factor of the boost squared with a suppression factor of order $10^{-8}$.
Examples of odd parity experiments have been recently shown to be of the Ives-Stilwell type [@Tobar; @Reinhardt; @Hoh; @Saathoff; @Bailey; @MP07; @Carone]. These type of experiments are leading order sensitive to the odd parity coefficients $\kop^{jk}$, with boost suppression sensitivity to $\kappa_{tr}$. Thus, these experiments have been used to set limits to the least well known parameter $\kappa_{tr}$, since Michelson Morley experiments have set limits on the even and odd parity coefficients of order $10^{-17}$ and $10^{-13}$ respectively. In this work we take this approach and perform the first rotating odd parity experiment to set a limit on $ \kappa_{tr}$.
It may seem counterintuitive that a rotating experiment is sensitive to the isotropic Lorentz violating coefficient, however because the odd parity experiment is leading order boost sensitive to $\kappa_{tr}$, a rotating experiment will detect a time varying signal as it changes its orientation with respect to the vector boost. Thus, for a constant rotating experiment, a signal will be expected at the rotation frequency for a non-zero value of $\kappa_{tr}$. However, due to the sidereal rotation of the Earth with respect to the sun centered frame and the orbital motion around the sun, mixing between these frequencies causes signals at sidereal and annual frequency offsets from the rotation frequency, which places the frequencies of interest away from the main frequency where systematics occur.
II. Odd parity microwave interferometer
=======================================
Interferometry is an important precision measurement technology that interferes two waves in and out of phase to create a bright port (BP) and a dark port (DP), where the output of the DP becomes a classical “null detector” and has underpinned a variety of applications over a long period of time [@Saulson]. The results presented here are based on the thermal noise limited microwave interferometer [@ITW98] shown in Fig. \[fig:coaxnms\]. Two systems are presented: 1). A coaxial magnetically asymmetric Mach-Zehnder interferometer system (as proposed in [@Tobar]), which is configured as the first ever odd parity rotating test of Lorentz invariance in electrodynamics: 2) The new waveguide power recycled interferometer [@Parker; @IT09], which is more sensitive than any laser interferometer operating at the same level of signal power[@laser02; @laser98] and improves the resolution of phase noise measurements by more than an order of magnitude with respect to current state-of-the-art[@IT02]. We show that the new interferometer has the potential to improve the sensitivity of the test of Lorentz Invariance by more than two orders of magnitude in the next generation of experiments.
![The microwave Inerferometer as a precision phase detector. The system consists of the microwave ($\mu W$) pump source, the interferometer, and the phase sensitive readout of the Dark Port (DP). The Bright Port (BP) is used to drive the mixer LO input, and a variable attenuator ($\alpha$) and phase shifter ($\phi$) are used to balance the interferometer.[]{data-label="fig:coaxnms"}](coaxnms.jpg){width="3.3in"}
The magnetic asymmetry of the Mach-Zehnder interferometer (necessary for sensitivity to $\kappa_{tr}$, see [@Tobar] for details) is created by introducing in one arm of the interferometer a Device Under Test (DUT) with a different permeability to the balancing arm. In this case the DUT is a string of six magnetically shielded ferrite isolators of effective length 20 cm (length of which the propagating field travels through the ferrite) and relative permeability 0.88 [@krupka]. A voltage controlled attenuator and phase shifter located in the other arm maintains balance of the interferometer via a closed loop feedback system of bandwidth 0.25 Hz. The bridge balance can be so exact that the DP is balanced to near zero ($\approx$ -120 dB), so that high gain amplification is useful before the down-mixing process, and overcomes the relatively high technical fluctuations in the mixing stage. This enables the effective noise temperature of the readout system to be close to its physical temperature [@ITW98]. A stabilized dielectric resonator-oscillator at 9.04 GHz was implemented as the microwave ($\mu W$) source . The amplified signal from the DP is mixed with that from the BP to convert the fluctuations inside the interferometer into voltage noise. The phase of the reference signal ($\phi_{ref}$ in Fig. \[fig:coaxnms\]) driving the mixer is adjusted during the calibration procedure to ensure that the voltage noise produced is synchronous with the phase fluctuations of the interferometer.
The interferometer was mounted inside a stainless steel vacuum can on a thermally controlled aluminum plate. Thermal control was necessary to limit phase drift and phase to amplitude conversions. The stainless steel can was located on the rotation table (previously used for a rotating cryogenic oscillator experiment [@Stanwix; @Stanwix2; @Tobar2]), powered via a rotating connector on top of the experiment. The phase variations of the interferometer were inferred from the voltage variations at the output of the mixing stage (IF port in Fig. \[fig:coaxnms\]) which, in turn, were measured with the digital voltmeter (DVM). Data was collected using a computer that rotated with the experiment. Central to the experiment was keeping track of the interferometer orientation with respect to a universal reference frame. This was achieved in two ways; firstly, by time stamping the DVM measurements with respect to UTC, and; secondly, by triggering the DVM measurements using the orientation of the experiment in the laboratory. The DVM was triggered at a rate of 2 Hz, while the rotation period was approximately 6 seconds, i.e. the DVM was triggered 12 times in one rotation period.
To maintain the balance of the interferometer the output of the mixer was fed back to the voltage controlled phase shifter via a low pass filter of corner frequency 0.25 Hz. The servo suppressed long term fluctuations while the sensitivity of the interferometer at the rotation frequency (0.17 Hz) was maintained at 16.3 v/rad (including filtering effects). From the DVM output we search for periodic signal at the rotation frequency offset by sidereal with the correct phase with respect to the sun centred frame as predicted by the SME [@Tobar].
![Phase noise spectra of the interferometric measurement systems. The noise spectrum of the conventional interferometer (see Fig. \[fig:coaxnms\]) was measured at Fourier frequencies $f < 1$ Hz. The solid line gives the fit to the noise of stationary experiment. Rotation adds bright lines at the harmonics of the rotation frequency plus some excess broad band noise above 0.1 Hz (highlighted by the dashed box). The noise spectrum of a power recycled interferometer is displayed at $f > 1$ Hz. No rotation was applied. The interferometer was based on low-loss waveguide components to enhance the efficiency of power recycling and boost its phase sensitivity (see description below). The accompanying solid line is the phase noise model of $1.7\times10^{-8}/f + 2\times10^{-11} rad/\sqrt{Hz} .$[]{data-label="fig:psd"}](PSD.jpg){width="3.3in"}
![Amplitude and standard error for the September 2007 data set as a function of offset frequency from the rotation frequency $\omega_R/2\pi$ in units of $\omega_{\oplus}/2\pi$.[]{data-label="fig:SysLeak"}](systleak.jpg){width="3.3in"}
Fig. \[fig:psd\] shows the spectra of phase fluctuations exhibited by the microwave noise measurement systems. Bright lines can be seen that correspond to the rotation frequency and its harmonics. Rotating the experiment also lifted the broadband noise to a level of about -127 dBc/Hz. For the LI test the noise at 0.17 Hz (the rotation frequency) is the most important. Modulated mechanical vibrations induced by the rotation system were found to be the primary source of systematic in this experiment. The DUT was shielded from stray magnetic fields using nu metal shielding. Oscillating magnetic fields of up to $2.5\times10^{-5}$ Tesla were applied with Helmholtz coils along the x-, y- and z-axis with no response visible.
III. Data Analysis
==================
To search for a Lorentz violation the data was analyzed using a Demodulated Least Squares (DLS) technique[@Stanwix2], which reduces the size of the data set by performing an initial demodulation of the data with respect to the first harmonic of the rotation frequency ($\omega_R$). Assuming only non-zero $\kappa_{tr}$ in the photon sector of the SME and short data set approximation[@Lipa; @Stanwix], i.e. data sets span much less than one year, the phase variations inside interferometer $\Delta\phi$ will be of the form given by Eqs.\[nuTest2\] to \[stage2DAC\]: $$\Delta\phi = A + B t + S(t) ~ {\rm sin}(\omega_{R} t + \varphi) + C(t) ~ {\rm cos}(\omega_{R} t + \varphi) \label{nuTest2}$$ $$S(t) = S_0 + S_{s} ~ {\rm sin}(\omega_\oplus t + \varphi_\oplus)
+ S_{c} ~ {\rm cos}(\omega_\oplus t + \varphi_\oplus)
\label{stage2DAS}$$ $$C(t) = C_0 + C_{s} ~ {\rm sin}(\omega_\oplus t + \varphi_\oplus)
+ C_{c} ~ {\rm cos}(\omega_\oplus t + \varphi_\oplus)
\label{stage2DAC}$$ where the coefficients are given by; $$S_{s}=\frac{4 \pi L \beta_{0} \kappa_{tr} (\mu_r-1) \cos \eta \cos \chi \cos \Phi_0 }{\lambda }
\label{coefSs}$$ $$S_{c}= \frac{4 \pi L \beta_{0} \kappa_{tr} (\mu_r-1) \sin \Phi_0 }{\lambda }
\label{coefSc}$$ $$C_{s}= -\frac{4 \pi L \beta_{0} \kappa_{tr} (\mu_r-1)\cos \chi \sin \Phi_0 }{\lambda }
\label{coefCs}$$ $$C_{c}= \frac{4 \pi L \beta_{0} \kappa_{tr} (\mu_r-1) \cos \eta \cos \Phi_0}{\lambda }
\label{coefCc}$$ Here, $L$ is the effective length of the interferometer, $\beta_{0}$ is the magnitude of the orbital boost of the Earth, $\mu_r$ is the relative permeability of the magnetic arm of the interferometer, $\lambda$ is the free space wavelength of the microwaves, $\chi$ is the colatitude (121.8 degrees for Perth), $\eta$ is the angle between the celestial equatorial plane and the ecliptic (23.3 degrees) and $\Phi_{0}$ is the phase of the orbit since the vernal equinox [@Tobar2].
The demodulation was achieved by simultaneously averaging the quadrature amplitudes ($S_{s}, S_{c}, C_{s}, C_{c}$) over a certain number of cycles ($n$) at $\omega_R$. Ordinary Least Squares is then applied to the demodulated data set to search for variations at the sidereal frequency $\omega_{\oplus}$. The conversion from radians to $\kappa_{tr}$ [@Tobar] for our experiment is given in Tab. \[sens\]. The optimum number of cycles is determined by the value that gave the minimum standard error, and typically varied between $n$ = 2 to 7 over the individual data sets. At this point the analysis approximates an optimal filter and contributions from fluctuations in the systematic (narrow band noise) will equal the noise contributed by the broad band noise.
Because, the expected Lorentz violating signal with respect to the sun-centered frame occurs at the sidereal offset (from the short data set approximation), we avoid systematic effects in the same way as reference [@Stanwix]. Fig. \[fig:SysLeak\] shows Ordinary Least Squares fit of the amplitude of phase variations (in rads) of the interferometer, as well as the standard error from a 13.2 day data set during September 2007. The data clearly shows that there is no leakage from the rotational frequency to the sidereal sidebands as the phase amplitudes are not significant. The estimate of $\kappa_{tr}$ as a function of time from 11 data sets are shown in Fig. \[fig:kappa\], which gives a final limit of $-0.3\pm3\times10^{-7}$ by calculating the weighted average over all data sets. Other experiments, of the Ives-Stillwell (IS) type are also sensitive to the Robertson Mansouri Sexl (RMS) Lorentz violating time dilation parameter $\alpha$ in the same way as $\kappa_{tr}$ (i.e. the limit put on $\alpha$ is the limit on $\kappa_{tr}$). However, this type of experiment is different in that it is sensitive to $\kappa_{tr}$ but not $\alpha$. Previous IS experiments have put an upper bound on the magnitude of $\kappa_{tr}$ and $\alpha$, of $3\times10^{-8}$ [@Carone] , $8.4\times10^{-8}$ [@Reinhardt] and $2.2\times10^{-7}$ [@Reinhardt; @Hoh; @Saathoff].
Coefficient Conversion rads to $\kappa_{tr}$
------------- ---------------------------------------------- -- -- --
$S_s$ $4.4\times10^{-4}\cos\Phi_{0}\ \kappa_{tr}$
$S_c$ $-9.0\times10^{-4}\sin\Phi_{0}\ \kappa_{tr}$
$C_s$ $-4.8\times10^{-4}\sin\Phi_{0}\ \kappa_{tr}$
$C_c$ $-8.3\times10^{-4}\cos\Phi_{0}\ \kappa_{tr}$
: \[sens\] Quadrature amplitudes conversion from output phase to $\kappa_{tr}$ using the short data set approximation. Here $\Phi_{0}$ is the phase of the orbit since the vernal equinox [@Tobar2]. The relationship is calculated from the parameters at Perth with respect to the sun centered frame and the experimental parameters of the interferometer[@Stanwix; @Tobar].
![The value of $\kappa_{tr}$ from 11 data sets of various lengths from Modified Julian Day 54356 to 54634. The value is extracted from fits to data using DLS at $\omega_R\pm\omega_{\oplus}$. The error bars represent the standard error, and the lengths represent the duration of the data set. In general the longer the duration the smaller the standard error. By taking the weighted average over the multiple data sets the value of $-0.3\pm3\times10^{-7}$ is obtained, which is shown to the right of the figure[]{data-label="fig:kappa"}](Kappa.jpg){width="3.3in"}
IV. Discussion
==============
Even though the determination of $\kappa_{tr}$ is a factor of 3.5 worse than the current best direct laboratory limit, the experiment is the first of its type and uses only a low power standard interferometer and is vibration limited. The accuracy of phase measurements reported above can be improved by at least two orders of magnitude by providing vibration isolation and switching from the coaxial to the waveguide ’magic tee’ based microwave interferometer and making use of a power recycling[@Parker; @IT09] (note, there is no point switching to the higher sensitive interferometer until the vibration isolation system is installed). Recently, we have built a microwave interferometer where the test sample is placed inside a distributed resonator formed by a short-circuited piece of a waveguide and inductive diaphragm. This extends the interaction time between the test sample and microwave carrier which, in its turn, enhances the random phase/amplitude modulation of the carrier signal by the non-thermal fluctuations in the test sample. The microwave signal reflected from the distributed resonator interferes destructively with a fraction of the incident signal at the dark port of the ’magic tee’, which cancels the carrier of the difference signal while preserving the noise modulation sidebands caused by fluctuations in the interferometer arms. In the same way as the coaxial system the noise sidebands are amplifed and demodulated to DC in the non-linear mixing stage resulting in a voltage noise spectral density.
The main reasons for choosing the waveguide components instead of micro-strip (used in [@ITW98; @IT02]) were $(i)$ to increase the efficiency of power recycling (by reducing the distributed loss in the interferometer arms) and $(ii)$ to minimize the effect of technical noise sources inside the interferometer (micro-strip power splitters could exhibit an excess noise due to poor adhesion of the metal film to dielectric substrate resulting in power-to-phase conversion phenomena). The phase sensitivity of the measurement system was optimized by adjusting the aperture of the inductive diaphragm and its distance from the symmetry plane of the Magic Tee. A piece of hollow waveguide approximately $10 cm$ long was used as the test sample. At $P_{inc} = 1W$ the highest value of phase sensitivity was measured to be $1.4 kV/rad$ with a recycling power enhancement of a factor of 16 (16 W circulating power).
The phase noise floor of the above measurement system is shown in Fig. \[fig:coaxnms\]. At Fourier frequencies $f>5kHz$, the phase noise floor was $2\times10^{-11} rad/\sqrt{Hz}$. This is almost an order of magnitude better than the phase resolution of a shot noise limited laser interferometer with power recycling reported in [@laser02; @laser98]. The above measurements were repeated with the input of the low-noise microwave amplifier terminated to evaluate the contribution of the readout ($LNA$ and $DBM$ assembly) to the overall uncertainty of phase measurements. At low Fourier frequencies the spectral density of phase fluctuations was $1.7\times10^{-8}/f$ $rad/\sqrt{Hz}$, which we believe is a ÒsignatureÓ of ambient temperature fluctuations, which requires further investigation.
Noise measurements with the coaxial interferometer below 1 kHz were sometimes inconsistent showing a large scatter from one experimental run to another. In the best case the noise floor of the coaxial systems was close to that of the waveguide system, while in others it was almost an order of magnitude higher. In this respect, the noise performance of the waveguide based interferometers was always highly reproducible and is one of the reason why the coaxial system had only a 37$\%$ duty cycle as noisy periods were vetoed from the analysis (the system was also out of operation for one month while systematics were investigated). It is not clear, yet, what causes the excess noise in the passive coaxial components, but an example of the noisy periods is compared when the system is stationary and rotating in Fig. \[fig:timetrace\].
![Left. The time trace of the mixer output voltage over a 12.5 day period of continuous operation while the experiment was stationary. Intermitent noisy periods are apparent. The phase noise spectral density of the quiet period shown between the dashed lines is shown in fig.\[fig:psd\]. Right. The time trace of the mixer output voltage over a 5.8 day period of continuous operation while the experiment was rotating. Intermitent noisy periods are apparent (dark regions), which were vetoed in the final analysis. The phase noise spectral density of the quiet period shown between the dashed lines is shown in fig.\[fig:psd\].[]{data-label="fig:timetrace"}](time.jpg){width="3.3in"}
To calculate the potential sensitivity we use the relationship between radians and phase given in Tab. \[sens\]. Given that we use all four coefficients to determine the standard error of $\kappa_{tr}$ ($\delta\kappa_{tr}$), then we can write the following simple relation to approximate the sensitivity. $$\begin{aligned}
\delta\kappa_{tr} \approx \frac{1}{10^{-3}}\frac {\delta\phi}{\sqrt{f_{R}\tau_{obs}}}
\label{eq:sens}\end{aligned}$$ Here $\delta\phi$ is the spectral density of $rms$ phase fluctuations in $rads/\sqrt{Hz}$, $\tau_{obs}$ is the total observation time in seconds and $f_{R}$ is the rotation frequency in Hz. The phase noise spectral density at 0.17 Hz was $10^{-7}$ $rads/\sqrt{Hz}$ (-140 dBc/Hz) for the recycled waveguide interferometer. Thus, for the parameters of our experiment, we have the possibility of determining $\kappa_{tr}$ to $5\times10^{-8}$ using the recycled interferometer. Faster rotation frequencies could also be used to improve performance. For example, by simply increasing the rotation rate by a factor of ten (1.7 Hz) the phase noise would be $10^{-8}$ $rads/\sqrt{Hz}$ (-160 dBc/Hz) and in the same amount of time a sensitivity of $10^{-9}$ could be achieved. It is also important to identify the source of low frequency$1/f$ phase fluctuations and of ways to reduce them. If one could achieve the Nyquist thermal noise limit, given by $2\times10^{-11}$ $rads/\sqrt{Hz}$, a sensitivity of order $10^{-12}$ could be achieved.
The recycled interferometer is also ideally suited for studying the noise phenomena in low-loss components and materials at microwave frequencies. We have used it to characterize the intrinsic phase fluctuations in ferrite circulators (used for this test). So far, we could only claim that there is no general rule describing the phase noise in such devices: in some cases the circulator phase fluctuations were easily observable, while in others no noise was detected. For the rotating experiment we chose isolators that exhibited no measurable phase noise. We also constructed a ferrite loaded waveguide impedance-matched on both ends, which exhibited no measurable noise. This will be used as a DUT in our future experiment based on the waveguide interferometer placed on top of the vibration isolated platform.
Finally we mention that a model dependent theory-based analysis has recently provided improved indirect bounds by observing gamma rays from cosmic sources [@Klink]. Also, new high energy experiments, which are non-sensitive to the odd parity terms have also set new bounds on $\kappa_{tr}$ [@Altschul; @MH1; @MH2]. These bounds are tighter, however are set at a very different energy scale.
The authorÕs thank Peter Wolf, Alison Fowler and Michael Miao for their assistance. This work was funded by the Australian Research Council.
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abstract: 'It is a tantalising possibility that quantum gravity (QG) states remaining coherent at astrophysical, galactic and cosmological scales could exist and that they could play a crucial role in understanding macroscopic gravitational effects. We explore, using only general principles of General Relativity, quantum and statistical mechanics, the possibility of using long-range QG states to describe black holes. In particular, we discuss in a critical way the interplay between various aspects of long-range quantum gravity, such as the holographic bound, classical and quantum criticality and the recently proposed quantum thermal generalisation of Einstein’s equivalence principle. We also show how black hole thermodynamics can be easily explained in this framework.'
author:
- |
M. Cadoni${}^{ab}$[^1], M. Tuveri${}^{b}$[^2] and A. P. Sanna${}$[^3]\
\
${}^a$*Dipartimento di Fisica, Università di Cagliari*\
[*Cittadella Universitaria, 09042 Monserrato, Italy*]{}\
\
${}^b$*I.N.F.N, Sezione di Cagliari*\
[*Cittadella Universitaria, 09042 Monserrato, Italy*]{}\
\
title: '**Long-Range Quantum Gravity**'
---
=1
Introduction
============
Conventional wisdom asserts that quantum gravity effects may be relevant only at scales of the order of the Planck length $l_p =\sqrt{\hbar G/ c^3}\sim 10^{-33} \ \text{cm}$. This simple result comes upon equating the Compton length associated with a self-gravitating object with its gravitational size, its Schwarzschild radius, $R_s$. On one hand, this would imply that only Planck-size black holes have an intrinsic quantum gravity nature. Conversely, astrophysical black holes, originated from the collapse of stars which, in turn, have proved their existence through gravitational waves detection [@Abbott:2016blz] and from the first photo of their light ring [@Akiyama:2019cqa], can be described by means of classical gravity only. On the other hand, starting from the pioneering works of Hawking and Bekenstein [@Bekenstein:1973; @Hawking:1974; @Hawking:1974sw], we know that black holes are thermal objects, whose thermodynamic behavior cannot be explained by classical gravity alone. From a microscopic point of view, black holes entropy and temperature are a manifestation of the existence of some (yet unknown) internal degrees of freedom (DOF), i.e. quanta, whose dynamics should be responsible for both the quantum and the thermodynamical properties of these objects [@Sakharov:1967pk; @Jacobson:1995ab; @Padmanabhan:2009vy; @Jacobson:2015hqa; @Padmanabhan:2016eld].
It is logically possible that the macroscopic behaviour of black holes is the result of a microscopic - Planck scale - QG theory, in the same way as black body emission and specific heat of solids are manifestation of microscopic quantum mechanics. However, there are strong indications that this may not be the case. The area-law, which encodes the so-called *holographic principle*, the information paradox [@Hawking:2015qqa; @Mathur:2009hf] and the fact that black holes have no hairs [@Cardoso:2016ryw], altogether they imply that the quantum characterization of a black hole must be done at the *horizon scale*, $R_s$. Indeed, the holographic principle tells us that the would-be DOF making up black hole entropy are localized on the horizon, the information paradox concerns only near-horizon physics and black hole hairs can be fully expressed in terms of the size of the black hole.
The logically simplest solution to this puzzle is to assume that the black hole is made up of a large number of QG states, which remain coherent at the horizon scale. This quantum portrait, which describes black holes similarly to Bose-Einstein condensates (BEC), has been first proposed in Ref. [@Dvali:2011aa] and has been extended to describe several macroscopic effects of gravity (the de Sitter (dS) universe, inflation, emergence of a dark force in galactic dynamics) as long-range QG effects [@Mueck:2013mha; @Dvali:2013eja; @Das:2014agf; @Oriti:2016acw; @Casadio:2016zpl; @Linnemann:2017hdo; @Das:2018udn; @Das:2018lwl; @Tuveri:2019zor; @De:2019fva; @Compere:2019ssx; @Cadoni:2017evg; @Casadio:2017cdv; @Cadoni:2018dnd; @Giusti:2019wdx; @Tuveri:2019zor; @Cadoni:2020izk]. In this description, black holes are considered as critical systems, saturating a *maximal packing condition*.
However, this simple solution produces a certain tension with the equivalence principle of General Relativity (GR), which sees the black hole horizon as a place with nothing special. This tension can be solved by the formulation of the quantum *generalized thermal equivalence principle (GTEP)* [@Tuveri:2019zor], which generalizes the classical equivalence principle of GR by postulating a fundamental relation between the temperature and the acceleration (the surface gravity).
In this essay we explore, using only general principles of GR, quantum and statistical mechanics, the possibility of using long-range QG states to describe black holes. In particular, we discuss in a critical way the interplay between various aspects of this description, namely the holographic principle, criticality and the peculiarities of the classical limit. We will show how the GTEP allows us to fit them in a nice and consistent description. Using a simple toy model to describe some aspects of black holes physics, we also show that the GTEP is a fundamental and unifying principle in QG. Indeed, we show that black hole thermodynamics can be fully derived from the combination of GTEP with another principle chosen from $1)$ the holographic bound, $2)$ classical and $3)$ quantum criticality. What is really interesting is that, whereas the GTEP appears to be a fundamental principle, the others give three complementary description of a black hole, respectively as $1)$ a state of maximal information, $2)$ a classical critical star and $3)$ a quantum critical state. We will be mainly concerned with black holes, but most of our considerations can be extended also to the case of the dS universe.
The structure of the paper is as follows. In Sect. \[LRQG\] we discuss the main features of our QG description, namely criticality, the holographic bound, the classical limit and the GTEP. In Sect. \[sec:toy\] we use a simple toy model for a black hole to see how they are intertwined among themselves and we show how the GTEP can be used as unifying principle to describe black holes. Finally, in Sect. \[sect:conc\] we present our conclusions.
Long-range properties of quantum gravity {#LRQG}
=========================================
Quantum mechanics can be responsible for the macroscopic behavior of several physical systems, like Bose-Einstein condensates, superfluids, superconductors and neutron stars. These are all quantum objects that can maintain quantum coherence even at macroscopic scales, so that both their macroscopic phenomenology and microscopic properties are a clear manifestation of quantum mechanics.
There is growing evidence that a similar description could hold true also for gravitational systems like black holes and the dS universe [@Witten:2001kn; @Binetruy:2012kx; @Mueck:2013mha; @Das:2014agf; @Oriti:2016acw; @Casadio:2016zpl; @Das:2018udn; @Das:2018lwl; @Tuveri:2019zor; @De:2019fva]. Thus, long-range QG effects could be relevant for explaining macroscopic gravitational effects, such as, for instance, a non-Newtonian component of the acceleration in galactic and cosmological dynamics [@Cadoni:2017evg; @Casadio:2017cdv; @Cadoni:2018dnd; @Giusti:2019wdx; @Tuveri:2019zor; @Cadoni:2020izk]. The existence of long-range QG states allows one to circumvent the usual argument that QG effects are only relevant at scales of the order of the Planck length $l_p$. As in BECs, where the scale of quantum effects is the size of the condensate itself, in principle we can think that, for QG effects, it coincides with the Schwarzschild radius $R_s$ and the cosmological horizon $L$ for black holes and the dS universe, respectively [@Verlinde:2016toy; @Tuveri:2019zor]. Moreover, in this picture, a *mesoscopic* length-scales for QG may be generated at galactic level, if an interaction between dark energy and baryonic matter is assumed [@Verlinde:2016toy; @Cadoni:2018dnd].
The existence of a long-range QG regime is thus related to the peculiar features of gravity and in particular to: a) the peculiarity of the *classical limit* in QG; b) the *criticality* of the quantum gravitational systems; c) the *holographic* character of gravity. All these features are ruled by a single length scale, representing the size of the system. In the following subsection, we will elucidate the interplay and the deep connection between these features, also arguing that a consistent description of them requires a generalization of Einstein’s equivalence principle.
The simplest QG model for a black hole of mass $M$ is that of a coherent state of $N$ quanta of energy $\varepsilon$, with: =,M[ c\^2]{}= N. In other words, a black hole is modeled as a cavity of length $\ell \sim R_S$, whose quantum degrees of freedom are quantum states in this cavity (see also [@Casadio:2020ueb; @Casadio:2020rsj] for a quantum description of black holes and gravity in terms of coherent states).
Similar expressions hold true also for the dS universe, with $R_s$ replaced by $L$ and $M$ by the total (dark) energy inside the cosmological horizon. In this picture, gravity emerges to sustain the coherence of a quantum critical system at large scales (see also [@Bruschi:2020xbm] for a discussion on this topic).
Criticality, the holographic bound and the classical limit
----------------------------------------------------------
For a generic gravitational system with radius $R$ and mass $M$, whose quantum description is given by , the *classical bound* 1, translates into a maximally packing condition [@Dvali:2010bf; @Dvali:2010jz; @Dvali:2011th] N, where $m_p$ is the Planck mass. This quantum bound tells us that black holes are critical objects (see Sect. \[sec:BHcritical\]), although its physical origin is not completely clear. It does not seem to be a consequence of the Heisenberg uncertainty principle alone, which only determines the relation between the size of the region and the energy of the quanta, $\varepsilon=\hbar c/R_s$ or, equivalently, $M_{BH}= m_p^2 c^2/(2 \varepsilon)$. Thus, Eq. seems to have a genuine long-range QG origin in the form of some effective repulsive interaction which, in turns, stabilizes the black hole.
The criticality bound can be written, upon using Eq. , as N, which is the well-known *holographic bound* [@Susskind:1994vu], limiting the quantity of information we can store inside a sphere with area $4\pi R_s^2$.
Although the two bounds and are equivalent, they are conceptually independent and have a completely different origin and interpretation. Whereas the former has an informational nature - its origin being related to the Bekenstein-Hawking formula for black hole entropy- the latter has a dynamical nature. Furthermore, Eq. is intrinsically holographic, whereas Eq. does not give any hint about the localization of the $N$ quanta - they may be as well localized inside the sphere or on its boundary. We will further discuss this point in Sect. \[sec:toy\].
Even if they can be described as quantum critical systems, black holes (and the dS universe) are solutions of Einstein’s GR and thus have also a classical interpretation, independent from their microscopic structure. However, the transition from the quantum to the classical description is highly non-trivial. It involves both the usual classical limit $\hslash \to 0$ and a [*classicalization*]{} process related to the number of quantum coherent states of the system, $N$, and it occurs when $N\to \infty$ [@Dvali:2011aa; @Dvali:2012en; @Dvali:2012rt], as in BECs. Macroscopic quantum phenomena are thus associated either to a system with a large number of states or to a quantum state occupied by a large number of particles. This is surely quite intuitive for a black hole, where quantum states have macroscopic size of the order of its Schwarzschild radius. From a technical point of view, this is a typical feature of systems where conformal invariance arises [@Cadoni:2006ww].
A key point is that, in a statistical mechanical description, any quantum system is characterized by thermal macroscopic quantities reflecting its microscopic structure, like its temperature and entropy. Thus, the bridge between the classical and quantum description is provided by black hole thermodynamics, which has been widely used to give a coarse grained thermal description of black holes.
Generalized Thermal Equivalence Principle (GTEP)
------------------------------------------------
The thermodynamic behavior of the Schwarzschild black hole is fully characterized by its temperature $T$, mass $M$ and entropy $S$: T=,M=,S= R\_s\^2, from which the first principle of thermodynamics $dM=TdS$ follows immediately by considering a classical process that changes the black hole radius $R_s$. Thus, black hole thermodynamics is fully defined in terms of fundamental constants ($\hbar,c,G$) and one single macroscopic quantity: the black hole radius $R_s$. This is the thermodynamic manifestation of black holes’ criticality, or, equivalently, of the maximally packing condition. More generally, Eqs. (\[tp\]) imply that black hole thermodynamics is a manifestation of quantum physics acting at horizon scale.
Furthermore, we see that $\hbar$ does not enter in the mass/radius relation. This latter can be thought as the analogous to compactness relation for a star and, in principle, can have a classical explanation. On the other hand, the entropy/radius relation depends on both $\hbar$ and $G$ and can therefore have only a QG origin. In terms of the coherent states building the black hole, it represents the maximal amount of information which can be codified in a spherical region of radius $R_s$ via the holographic bound . Finally, the relation between the temperature and $R_s$ is rather mysterious when considered in the classical limit. In terms of quantum mechanics, it reflects the fact that the temperature measures the average energy of the quanta, i.e. $T\sim \varepsilon=\frac{\hbar c}{R_s}$. This is particularly evident when one rewrites it in terms of the surface gravity $a$, T= a, where: a= . This temperature/acceleration relation is quite puzzling, particularly considering that it does not depend directly of the strength of the gravitational interaction $G$. Another puzzling point is the tension that exists between the equivalence principle of GR, which implies that the black hole horizon is not a special place in spacetime, and the long-range QG description aiming to explain the black hole in terms of quantum states with horizon size.
The simplest solution of these puzzles, which also allows to reconcile the classical and quantum perspectives in describing black hole physics, is to explain Eq. as a consequence of a *Generalized Thermal Equivalence Principle* (GTEP) [@Tuveri:2019zor]. It has been formulated as the generalization of Smolin’s quantum version of the universality of free fall (the thermal equivalence principle) [@Smolin:2017kkb], which is based on the Deser-Levin formula [@Narnhofer:1996zk; @Deser:1997ri; @Jacobson:1997ux]. The GTEP asserts that whenever we have a thermal ensemble at temperature $T$ of quantum gravity degrees of freedom, the macroscopic acceleration produced on a test mass is given by the Deser-Levin formula a=, where $T_{dS}$ is the dS temperature. One can easily check that Eq. simply follows from Eq. when $T\gg T_{dS}$. The GTEP has been used to explain galactic dynamics and in particular the Tully-Fisher relation, without assuming the existence of dark matter [@Tuveri:2019zor]. The connection with the QG description of black holes can be obtained using Eq. into Eq. to find a nice relation between the energy of the quanta and the acceleration: a=. The same result can be obtained in the corpuscular model of black holes of Refs. [@Dvali:2013eja; @Casadio:2016zpl; @Cadoni:2017evg; @Cadoni:2018dnd; @Dvali:2012rt].
Conversely, Eqs. and can be used to derive Eq. , i.e. to understand the peculiarity of the classical limit involved in describing black holes as classical objects. Indeed, we see that the surface gravity has a quantum origin, but its do not scale away in the limit $\hslash\to 0$ as it should be the case in standard quantum mechanics. This is because $\hslash$ appears in the numerator of $\varepsilon$ (see Eq. ) but in the denominator of Eq. .
Notice that Eq. is the result of [*both*]{} the GTEP and criticality. To see this, let us consider a *non-critical* classical gravitational system, i.e. a system for which the bound is not saturated. We consider a spherically symmetric configuration of energy $E=Mc^2$, inside a sphere of radius $R$, which may or not coincide with the physical radius of the system. We also assume that the gravitational system allows for a microscopic long-range QG description in terms of $N$ degrees of freedom characterized by the temperature $T$ (derived upon a suitable coarse grained operation). Notice that, in this situation, Eqs. do not necessarily hold.
Furthermore, we assume: a) the validity of GTEP, given by Eq. with $T\gg T_{dS}$, i.e. $a=\frac{2\pi c}{\hslash}T$; b) the saturation of the holographic bound , $N=R_s^2/l_p^2$; c) an equipartition rule for the energy inside the sphere: $ E=Mc^2= \frac{1}{2} N T$. One can easily check that Newton’s law $a_N = GM/R^2$ simply follows from putting together a), b) and c). Only when the criticality condition is saturated, $a_N$ takes the form .
The GTEP also applies to dS universe. In this case, Eq. gives the well-known relation between the dS temperature and the cosmological acceleration $H$ [@Narnhofer:1996zk; @Deser:1997ri]: $a=H=\frac{2\pi c}{\hbar }T_{dS}=\frac{c^2}{ L }$. The dS universe can be thought as an ensemble of quanta with typical energy $\varepsilon= \hbar c/L$, so that we find $a=\frac{c}{\hbar}\varepsilon$, which is similar to Eq. up to a factor of $2$.
A black hole toy model
======================
In the previous Section, we have seen how quantum and classical properties of black holes, including their thermodynamics, can be fully explained using three main principles: criticality, the holographic bound and the GTEP. Two important, albeit related, questions arise: are these principles logically independent one from the other or can one of them be derived from the others? Is one of them more fundamental than the others?
In this Section we will try to settle down these issues by building a simple toy model to describe black holes as quantum gravitational systems, with a large number of internal degrees of freedom, $N\gg 1$, with typical energy and mass given by Eq. .
Black hole as a critical star
-----------------------------
Let us assume, beyond the validity of GTEP, that a black hole has a classical description in term of GR. We define the Schwarzschild radius as $R_s=2GM/c^2$. This means that the classical criticality bound is saturated, namely the black hole is considered as a sort of critical star (see also [@Casadio:2019tfz; @Calmet:2019eof]). Putting together Eq. and the definition of the Schwarzschild radius, we find that the number of degrees of freedom in the black hole cavity scales as the area, i.e. it scales holographically in the sense previously described: $$\label{Ndof}
N\sim\frac{{ c^3}}{G\hslash}R_S^2.$$ This allows us to find that also the entropy of the black hole scales holographically as its area: $$\label{Sbh}
S\sim N\sim \frac{R_S^2}{l_P^2}.$$ By using the GTEP, we can easily find the black hole temperature and, more in general, characterize its thermodynamics. Since in the case under study we expect that $T\gg T_{dS}$, putting together Eq. and Eq. , we find the expected result: $$\label{Tbh}
T=\frac{\hslash c}{4\pi R_S}.$$ Thus, it is possible to derive black holes thermodynamics by considering them as critical stars in GR and assuming the validity of the GTEP. In this case, holography appears as a consequence of these twos rather than a first principle.
Black hole as a critical quantum system {#sec:BHcritical}
---------------------------------------
Let us now consider a black hole of mass $M$ as a quantum critical system. In this case, we assume the validity of the GTEP and the saturation of the bound : $$\label{N_LRQ}
N=\frac{m_p^2 { c^4}}{2}\frac{1}{\varepsilon^2}.$$ The total mass of the system is: $$\label{M_LRQ}
M=\frac{m_p^2 c^2}{2}\frac{1}{\varepsilon}.$$ Eq. states that for a given mass $M$, the number of quantum degrees of freedom cannot vary freely, rather it depends on the energy $\varepsilon$. In a system of finite size, we can pack many states of small energy or few states of large energy. This implies that, at Planck scales, black holes have $N\sim 1$, whereas astrophysical black holes have $N\gg 1$. This means that they satisfy the maximally packing condition described in Sect. \[LRQG\]. Thus, in a given region of size $R_S$ and corresponding quanta of energy $\varepsilon=\hslash{ c}/R_S$, we can put $N{\leqslant}m_P^2{c^4}/\varepsilon^2$ quanta. Once the bound $N=m_p^2{ c^4}/\varepsilon^2 $ is saturated we cannot put other quanta in that region without enlarging it. Thus, black holes are classically characterized by the condition $M=Rc^2/2G$, which is analogous to the condition $N\varepsilon^2=m_p^2 {c^4}/2$ in the quantum portrait [@Dvali:2011aa; @Dvali:2012en; @Dvali:2012rt; @Dvali:2013eja]. The saturation of the holographic bound easily follows from Eq. upon using Eq. and the Bekenstein-Hawking formula for the entropy in . On the other hand, the black hole temperature still follows from the GTEP, whereas the mass/radius relation is now obtained from .
In this derivation, black hole thermodynamics follows in a straightforward way from the GTEP and the quantum criticality condition (\[N\_LRQ\]).
Black holes as states of maximal information
--------------------------------------------
In the previous subsection we have seen that the holographic bound is equivalent, upon the use of Eq. , to the criticality bound . It follows that we can derive black hole thermodynamics from the GTEP and the requirement of the saturation of the holographic bound . The Bekenstein-Hawking entropy $S$ follows directly from the latter, the Hawking temperature follows from the former, whereas the expression of the mass $M$ follows from the second Eq. .
We stress again that, despite the equivalence between the holographic bound and the criticality bound , they have a completely different conceptual meaning. Whereas the latter has an informational nature, the former has a fully dynamical meaning.
It is interesting to see how, in the last two derivations of black hole thermodynamics, the black hole mass $M=R_sc^2/2G$ is obtained by summing up the energy of $N$ quanta (see Eq. ). The mass $M$ is a classical observable and cannot vanish in the $\hbar\to 0$ limit. Indeed, $\hbar$ cancels out by putting together Eqs. and . This is completely analogous to the $\hbar$ classical limit cancellation, which occurs for the surface gravity. We expect this $\hbar$ cancellation to hold not only for the mass but also for all black hole hairs.
Conclusions
===========
In this paper we have explored the possibility that quantum gravity states remaining coherent at astrophysical scales could be used to describe black holes. They can be seen as quantum macroscopic critical objects, which are characterized by a typical length scale (their radius) which determines all their features in terms of $N$ coherent states, i.e. quanta with a typical energy and temperature determined by the black hole radius. In the critical phase, it is not possible to put any further quanta in the system without changing its size. This is related to the saturation of the informational (entropy) bound which, in the case of black holes, appears as a holographic bound. One interesting point is that the same features are shared by the de Sitter universe.
We have also discussed the interplay between the various aspects of the long-range quantum gravity description of black holes, namely the holographic principle, criticality and the peculiarities of the classical limit. We have seen that a quantum, thermal extension of Einstein’s equivalence principle, which we called GTEP, allows for a nice, consistent and unifying description of black holes. In particular, we have seen how it is possible to derive black hole thermodynamics starting from three main principles: classical or quantum criticality, holography and the GTEP. The GTEP seems to be more fundamental than the others. Indeed, the choice of one instead of the other remaining principles leads to different complementary description of a black hole, i.e. as a state of maximum information, as a classical or a quantum critical state, respectively.
Let us conclude with the main caveats of our description of black holes in terms of long-range QG. The discussion we have presented in this paper has a rather speculative character. Although based on general features of General Relativity, quantum and statistical mechanics, it suffers from the fact that until now, we do not have a well-defined quantum theory of gravity. Only in that framework, notions like “QG long-range coherent states” we have used in this paper would have a well-defined meaning. On the other hand, it is quite clear that any formulation of a quantum theory of gravity requires ingredients bridging between GR and quantum mechanics. The basic principles on which we have built our discussion, namely the holographic principle, criticality and the GTEP, have to be considered as bridging principles in this direction. One of the main results of our discussion is that the GTEP seems to work at a more fundamental level than the others.
Another question we have in mind (probably strongly related to the previous one) is about the physical mechanism underlying the stability of black holes. Classicalization and criticality should prevent the formation of singularities allowing the formation of stable critical (quantum gravitational) macroscopic systems made by a large number of quantum coherent states with energies of the order of $1/\ell$, where $\ell$ is the typical size of the system. However, if black holes can be described by something similar to a BEC, why does the condensate remain stable? Gravity is an attractive force, and in principle, a condensate of gravitons should collapse to form a singularity if some repulsive potential does not arise to compensate the gravitational interaction and stabilize the system. Standard BECs are stabilized by repulsive forces typical of Coulombian interactions between atoms in the critical phase. This is not the case of gravitational systems, where no repulsive components of the force arise in the system and the only fundamental force governing them seems to be gravity.
A different way to formulate the previous question is to ask about the physical origin of the holographic and the criticality bounds, in Eqs. and , respectively. Have they a purely informational origin or have they a fundamental dynamical nature? It is quite clear that the answer to these questions is one of the main challenges of any quantum theory of gravity.
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[^1]: E-mail: mariano.cadoni@ca.infn.it
[^2]: E-mail: matteo.tuveri@ca.infn.it
[^3]: E-mail: asanna9564@yahoo.it
|
---
abstract: |
Accurate localization of brain regions responsible for language and cognitive functions in Epilepsy patients should be carefully determined prior to surgery. Electrocorticography (ECoG)-based Real Time Functional Mapping (RTFM) has been shown to be a safer alternative to the electrical cortical stimulation mapping (ESM), which is currently the clinical/gold standard. Conventional methods for analyzing RTFM signals are based on statistical comparison of signal power at certain frequency bands. Compared to gold standard (ESM), they have limited accuracies when assessing channel responses.
In this study, we address the accuracy limitation of the current RTFM signal estimation methods by analyzing the full frequency spectrum of the signal and replacing signal power estimation methods with machine learning algorithms, specifically random forest (RF), as a proof of concept. We train RF with power spectral density of the time-series RTFM signal in supervised learning framework where ground truth labels are obtained from the ESM. Results obtained from RTFM of six adult patients in a strictly controlled experimental setup reveal the state of the art detection accuracy of $\approx 78\%$ for the language comprehension task, an improvement of $23\%$ over the conventional RTFM estimation method. To the best of our knowledge, this is the first study exploring the use of machine learning approaches for determining RTFM signal characteristics, and using the whole-frequency band for better region localization. Our results demonstrate the feasibility of machine learning based RTFM signal analysis method over the full spectrum to be a clinical routine in the near future.
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title: 'Automatic Response Assessment in Regions of Language Cortex in Epilepsy Patients Using ECoG-based Functional Mapping and Machine Learning'
---
Epilepsy, Machine Learning, ECoG, RTFM, Random Forest
Introduction
============
Epilepsy is a neurological disorder characterized by unpredictable seizures. There are over 65 million people around the world who have epilepsy and an incidence rate of 150,000 new cases every year in just USA alone [@epilepsyStats]. Drug Resistant Epilepsy (DRE) (or intractable epilepsy) is defined when the seizures cannot be controlled by medications and about 25% of all epileptic cases are DRE [@shorvon2013longitudinal]. The only viable option in this case is to surgically remove the affected tissue. Epilepsy surgery is a curative option for pharmacoresistant epilepsy, but brain regions associated with language and cognitive functions can be affected by surgery. To do this accurately, unaffected regions of the brain must be identified (called “localized”). The motor and language comprehension are examples of functionally significant region localization. Accurate localization helps to prevent post-surgical loss of functionality.
### Clinical standard and the state-of-the-art method for RTFM evaluation {#clinical-standard-and-the-state-of-the-art-method-for-rtfm-evaluation .unnumbered}
The gold standard task localization, the Electro-Cortical Stimulation Mapping (ESM), utilizes electrodes that are placed on the surface of the brain by means of craniotomy. During the ESM, the current is delivered for a short duration to stimulate the region of interest. The behavioral response corresponding to changes in function are simultaneously recorded. The inherent drawback of this approach is that **the stimulation can cause the neurons in that region to uncontrollably discharge, i.e., cause seizure.** Recently, ElectroCorticography (ECoG)-based real-time functional mapping (RTFM) [@schalk2008real] has been proposed as a promising alternative to ESM. The typical RTFM based task localization and experimental setup is illustrated in Figure \[RTFM\]. Similar to the ESM, subdural grids on the cortical surface are utilized for signal collection, however, no external stimulus is provided and only the physiological changes corresponding to the processed stimuli are recorded via the electrodes. Hence, no seizure due to stimulation occurs.
### Research gap {#research-gap .unnumbered}
The results of RTFM are not always concordant with the gold standard due to the difficulty in understanding the brain signals without stimulation and lack of sufficient accuracy of the state-of-the-art method, ECoG-based functional mapping [@korostenskaja2015electrocorticography] (ECoG-EM from now on, where EM stands for expectation maximization). There is a need for a method that would improve RTFM signal classification accuracy and make it a strong and safer alternative to the ESM. Current approaches for detecting positive response channels in the eloquent cortex localization task, focus on the power of the signal in the $\alpha$, $\beta$ and primarily, the high-$\gamma$ (70Hz-170Hz) frequency bands [@schalk2008real; @prueckl2013cortiq]. In these approaches, a baseline recording of each channel at resting-state is used. The power of the signal during the tests is computed using an autoregressive (AR) spectral estimation approach and is then statistically compared to the baseline to calculate the probability whether the channel has a response that is significantly different from it’s resting-state (baseline) condition or not. This is repeated every 100 ms for the entire experiment. These approaches do not compare the channels to each other and also do not account for the signal in the frequency range beyond high-$\gamma$.
### Our contributions {#our-contributions .unnumbered}
We present for ECoG signal analysis with RF to accurately discriminate channels that respond positive and negative in regards to language functional mapping task. To the best of our knowledge, this is the first work comparing the different (positive and negative) responses rather than using a baseline approach. We show the superiority of our approach to the state of the art ECoG-based functional analysis using Expectation Maximization approaches (ECoG-EM), and demonstrate its strong potential to become an alternative to ESM. The rest of the paper is organized as follows: In Sec. \[methods\] we discuss the ECoG data collection, pre-processing of the data into the discriminative domain and the proposed classification approach. In Sec. \[expt\], we present our experimental results. In Sec. \[conclusion\], we summarize our findings.
Methods\[methods\]
==================
Data Collection and Experimental Setup
--------------------------------------
ECoG represents the electrical activity of the brain recorded directly from the cortical surface. ECoG-based functional mapping allows identification of brain activity correlated with certain task, e.g., language. The basic setup for ECoG-based functional mapping is shown in Figure \[RTFM\]. ECoG signals from the implanted subdural grids are split into two streams: one for continuous clinical seizure monitoring and the other for ECoG-based functional mapping. The tool used to record the incoming ECoG signal was BCI2000 [@bci2000]. A baseline recording of the cortical activity was first acquired to capture the “resting-state” neuronal activity of the regions.
\[1a\] \[1b\]\
\[1c\] \[1d\]
The literature on localization of motor function using ECoG-based functional mapping (such as RTFM) is vast [@roland2010passive][@kapeller2015cortiq]. Unlike good accuracies obtained from such studies, the localization of eloquent language cortex has proved to be more challenging [@arya2015electrocorticographic]. The language function in the brain is processed in several regions primarily, the **Wernicke’s area** and **Broca’s area** as demonstrated in Figure \[speechcontrol\].
The Wernicke’s area is located in the posterior section of the superior temporal gyrus and is responsible for the receptive language task i.e., language comprehension. The Broca’s area, on the other hand, is more involved in speech production. There exists an anatomical connection between these two regions, named the arcuate fasciculus, which could induce a response in one region owing to the other’s activation.
### Language comprehension task {#language-comprehension-task .unnumbered}
Following the baseline recording step, paradigms similar to those employed in ESM or functional Magnetic Resonance Imaging (fMRI) are also employed to record the task-related ECoG signal for functional mapping purposes [@korostenskaja]. Figure \[setup\] shows one such paradigm, mimicking experimental setup for the language comprehension task. Alternate 30 second blocks of ECoG data during “control" and “active" conditions are recorded continuously at a fixed sampling rate of 1200 Hz.
For the language comprehension task, the active condition implies listening to a story, while the control task involves listening to broadband noise [@korostenskaja2014real]. Another associated paradigm is the reading comprehension task where the subject reads sentences from a screen, and replies with a “True” or “False” response. The system records information from 128 ECoG channels as illustrated in Figure \[setup\].
Pre-processing
--------------
As a first step of preparing the data, non-task/control time points in the signal are eliminated. These correspond to the spontaneous activity recording before the 0-min in Figure \[setup\] and any trailing signals at the end of the experiment. The use of power spectral density (PSD) is proposed in [@schalk2008real] as a discriminating feature between the baseline and task signals. In a slightly different manner, we represent PSD with a number of coefficients extracted from an autoregressive (AR) model. Unlike conventional methods, we simplify signal representation with PSD coefficients only. Herein, the AR parameters, $\tilde{a}[n]$, are estimated by forward linear prediction coefficients and then, the spectral estimate is calculated as $$\begin{aligned}
\label{psd}
\tilde{P}(f) = \frac{T\tilde{\rho}}{\abs{1+\sum_{n=1}^{p}\tilde{a}[n]e^{-i2\pi fnT}}}\end{aligned}$$ where $T$ is the inverse of the sampling rate ($f_s$), $\tilde{\rho}$ is the estimated noise variance, and $p$ is the order of the AR process. This approach gives us $\frac{f_s}{2}+1$ frequency components. The PSD estimates are computed for each block (task/control) of each channel. Later, we use these components as features to determine RTFM characteristics.
Classification Model
--------------------
To differentiate positive response channels (PRC) from negative response channels (NRC), we identify structured signal patterns in signal blocks, which are not readily visible to the human eyes. We hypothesize that the features of the active and control tasks are globally similar between PRC and NRC but still include substantial differences. This hypothesis can be visually tested and partially confirmed in Figure \[freqBands\] where the PSD of the active and control blocks of PRC are larger than that of NRC.
To test our hypothesis and provide scientific evidences of ECoG signal separation between functionally positive and negative regions, we design a RF classifier [@breiman2001random] to model structured local signal patterns for challenging RTFM signal characterization. It has been shown in various different areas that RF is an efficient classifier with considerably good accuracies in classification tasks [@bromiley2016fully; @verhoeven2016using; @sarfaraz]. Its superiority to most other classifiers comes from its *generalization* property.
![Auto-classification workflow\[rf\]: First the signals are split into it’s contributing blocks. After, power spectral density (PSD) of the signal is estimated and the blocks are stacked from all channels. Finally, a random forest (RF) classifier is used for discriminating positive response channels (PRC) and negative response channels (NRC).](Picture1.jpg){width="0.7\columnwidth" height="9cm"}
In RF, briefly, each new tree is created and grown by first randomly sub-sampling the data with replacement. An ensemble of algorithms are used so that the sub-trees are learned differently from each other. For a feature vector $\mathbf{v}=(v_1,v_2,...,v_d)\in \mathbb{R}^d$, where $d$ represents feature dimension, RF trains multiple decision trees and the output is determined based on combined predictions. In each node of decision trees, there is a weak learner (or split function) with binary output: $h(\mathbf{v},\theta): \mathbb{R}^d \times \mathcal{T} \rightarrow \{0,1\}$, where $\mathcal{T}$ represents the space of all split parameters. Note that each node is assigned a different split function. RF includes hierarchically organized decision trees, in which data arriving at node $j$ is divided into two parameters.\
Overall, RF treats finding split parameters $\theta_j$ as an optimization problem $\theta_j=\operatorname*{arg\,max}_{\theta\in\mathcal{T}}{I(\mathbf{v},\theta)}$, where $I$ is the objective function (i.e., split function) and $v$ represents the PSD coefficients in this particular application. As the tree is grown (Figure \[rf\]), an information criterion is used to determine the quality of a split. Commonly used metrics are **Gini impurity** and **Entropy** for information gain. To overcome potential over-fittings, a random sample of features is input to the trees so that the resulting predictions have minimal correlation with each other (i.e., minimum redundancy is achieved). In our experiments, we have used linear data separation model of the RF.
In our experiments, we use full spectrum of RTFM signal (0-600 Hz) in frequency domain instead of restricted $\gamma$-band. Moreover, we stack the signal to enhance the frequency specific features rather than concatenating them. Each channel has 10 blocks (Figure \[setup\]) and the final channel classification is based on a majority voting (Figure \[rf\]) on the classified sub-blocks. For the tested data point (feature) $\mathbf{v}$, the output is computed as a conditional distribution $p(c|\mathbf{v})$ where $c$ represents the categorical labels (positive vs. negative response). Final decision (classification) is made after using majority voting over $K$ leafs: $p(c|\mathbf{v})=\frac{1}{K}\sum_{k=1}^{K}p_t(c|\mathbf{v})$.
### Model parameters {#model-parameters .unnumbered}
Number of trees, number of features, and data size fed to each tree with or without resampling and the information metric for data splitting are some of the RF parameters that need to be optimized. To achieve this, the model was repeatedly tested under different combinations of the above parameters. For the total number of trees, an incremental update approach was used where we increased the total number of trees till the increase in performance was negligible. Similarly, the number of features was set as the square root of the number of input variables. For the choice of splitting function, Gini impurity was used as for a binary classification problem, both measures yield similar results [@breiman1996technical].
Experiments and Results {#expt}
=======================
--------- ------- ----- ------------------ ----------- ------------- ------------------
Subject Age Sex Epilepsy Grid Epilepsy Channels Tested/
\# (yrs) Focus Placement Onset (yrs) PRC / NRC
1 19 M Frontal-Temporal Lateral 16 54 / 22 / 32
2 33 F Frontal-Temporal Lateral 10 32 / 5 / 27
3 20 M Frontal-Temporal Lateral 6 127 / 16 / 111
4 22 F Parietal Lateral 20 30 / 19 / 11
5 32 F Temporal Bilateral 26 48 / 10 / 38
6 52 M Temporal Lateral 30 48 / 5 / 43
--------- ------- ----- ------------------ ----------- ------------- ------------------
With IRB approval, ECoG data were recorded from six adult patients with intractable epilepsy. Table \[demographics\] summarizes the patient demographics and the number of channels tested per patient. The ESM results were served as gold standard for separating ECoG channels into two classes: “ESM-positive” and “ESM-negative” electrodes. The number of tested ESM electrodes varies based on the task in hand (the function that can be compromised during the surgery and therefore needs to be localized), patient’s status, possible after-discharges, location of the grid on the brain surface, the epilepsy focus and to a smaller extent on the specialist performing the test.
Except subject 4, all subjects were tested with the language comprehension paradigm as shown in Figure \[setup\]. Subject 4, on the other hand, underwent the reading comprehension test involving reading sentences presented on the screen and responding to questions as “True” or “False”. Since this test also incorporates speech which would incite a response from face/tongue sensory motor areas of the brain as well as the Broca’s area, channels corresponding to these specific regions were not included in our calculations.
There were $77$ PRCs and $262$ NRCs in total. Each data block in a channel was assigned the same label. For 5 minutes long recording, we had 5 blocks of control and active conditions each per channel and hence, 3390 data samples in total. Due to the large imbalance in data, $77$ NRCs were randomly chosen from the $262$. In total, we have $1540$ blocks of data. For unbiased evaluation of the RF based results, we used 10-fold cross-validation and the average over a 100-iteration was conducted.
### Time-domain analysis {#time-domain-analysis .unnumbered}
First, we tested whether the raw time signal data has sufficiently discriminating information. For this analysis, a RF model with 100 trees was used. The resulting classification accuracy was $61.79\%$ with sensitivity and specificity around $60\%$. While this is marginally better than the simple flip of a coin scenario, it is insufficient to encourage the use of ECoG-based functional mapping over ESM.
### Frequency-domain analysis {#frequency-domain-analysis .unnumbered}
Each block in a time-domain signal was transformed into the frequency-domain using the pre-processing step described in Section \[methods\] (i.e., PSD coefficient via AR model). The order of the AR process is set to $SamplingRate/10 = 120$. The PSD estimate is of length $f_s/2+1=$ 601. We then log normalized PSD coefficients to train a RF classifier. An ensemble of 200 bagged classification trees was trained on 9 folds of the data and tested on the last fold.
In order to validate the use of control & active task blocks for channel classification, we first performed block classification on the $1540$ blocks. The classification accuracy was found to be $94\%$ with sensitivity and specificity of $\approx 93\%$. These results validate the efficacy of the proposed block-based classification strategy.
### Frequency-band analysis {#frequency-band-analysis .unnumbered}
Three different experiments (E1, E2, E3) were performed to understand the contribution of the different frequency bands to the channel classification problem:
1. Classification using full signal spectrum
2. Classification using $\alpha$, $\beta$, high-$\gamma$ sub-bands
3. Classification using only the High-Gamma sub-band
![Classification scores on ECoG signal classification on Language Comprehension Task. E1 - Classification using full signal spectrum, E2 - Classification using $\alpha$, $\beta$, high-$\gamma$ sub-bands & E3 - Classification using only the high-$\gamma$ band. \[lang\]](lang_excel.jpg){width="\columnwidth"}
In these experiments, the blocks were classified and a majority voting was applied to classify a channel as PRC/NRC. Figure \[lang\] summarizes the results of the above experiments for the language comprehension task. In concordance to what was observed in the ECoG-EM approaches such as SIGFRIED [@schalk2008real] and CortiQ [@cortiq], we found that the lower frequency bands, specifically, $\alpha$ and $\beta$, did not contribute largely towards classification and the high-$\gamma$ band achieved good classification accuracy. In other words, the full signal spectrum based classification had higher classification accuracy, sensitivity and specificity than the sub-band approaches indicating that the full spectrum had more information to offer.
### Block-size analysis {#block-size-analysis .unnumbered}
We also tested the use of smaller blocks of data by further dividing each control/active task block into 10 sub-blocks. Each sub-block of data was the power spectrum representation of 3 seconds of the recording. The classification was done based on a majority voting of the classified sub-blocks within a channel. The resulting classification accuracy was reported to be $78\%$, higher than the block-based approach. This indicates that there was more local information to be extracted from the signal.
### Comparison to the state of the art {#comparison-to-the-state-of-the-art .unnumbered}
ECoG-EM has been extensively tested on motor localization tasks [@prueckl2013cortiq], but not as much on language localization. Still, ECoG-EM is considered to be the state of the art method. To have a fair comparison with ECoG-EM, we applied ECoG-EM on the frequently tested sub-bands - $\alpha,\beta$ and high-$\gamma$, as well as on the frequency bands beyond and upto 350 Hz. The results are shown in Figure \[sigfried\]. **While ECoG-EM approach provides a higher specificity, it has a much lower accuracy and sensitivity than the proposed RF based approach.** This is a strong validation of our hypothesis that discriminating PRCs and NRCs is a promising technique as compared to the baseline reference channel classification approach.
![Comparison of ECoG signal classification using proposed approach - Random Forest (RF) and conventionally used, ECoG-EM, on the language comprehension task.\[sigfried\]](cortiq_excel.jpg){width="\columnwidth"}
Conclusion
==========
Discriminating between the response in the eloquent language cortex regions based on the associated task is a challenging problem. In the current study, we developed a novel framework towards the ECoG-based eloquent cortex localization with promising results: 78% accuracy on channel classification in comparison to the 55% accuracy of the state of the art ECoG-based functional mapping. We showed the efficacy of machine learning based RTFM signal analysis as a strong alternative to the ESM.
Acknowledgments {#acknowledgments .unnumbered}
===============
Acknowledgment {#acknowledgment .unnumbered}
==============
The authors would like to acknowledge UCF-FH seed grant (PIs: M. Korostenskaja and U. Bagci) for supporting this study. The authors would also like to thank Drs. Schott Holland and Jennifer Vannest for sharing the story listening task developed in their Neuroimaging Center at Cincinnati Children’s Central Hospital. Special thanks to Drs. G. Schalk and P. Brunner for providing their in-house built version of BCI2000-based software for ECoG recording and for their continued support of our ECoG-related studies.
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---
abstract: 'Considering both the power Maxwell invariant source and the Einstein–Gauss–Bonnet gravity, we present a new class of static solutions yields a spacetime with a longitudinal nonlinear magnetic field. These horizonless solutions have no curvature singularity, but have a conic geometry with a deficit angle $\delta \phi$. In order to have vanishing electromagnetic field at spatial infinity, we restrict the nonlinearity parameter to $s>1/2$. Investigation of the energy conditions show that these solutions satisfy the null, weak and strong energy conditions simultaneously, for $s>1/2$, and the dominant energy condition is satisfied when $s \in \left( {\frac{1}{2},1}\right]$. In addition, we look for about the effect of nonlinearity parameter on the energy density and also deficit angle, and find that these quantities are sensitive with respect to variation of nonlinearity parameter. We find that for special values of nonlinearity parameter, two important subclass of solutions, so-called conformally invariant Maxwell and BTZ-like solutions, with interesting properties, emerge. Then, we generalize the static solutions to the case of spinning magnetic solutions and find that, when one or more rotation parameters are nonzero, the brane has a net electric charge which is proportional to the magnitude of the rotation parameters. We also use the counterterm method to compute the conserved quantities of these spacetimes such as mass, angular momentum, and find that these conserved quantities do not depend on the nonlinearity parameter.'
author:
- 'S. H. Hendi$^{1,2,3}$[^1], S. Kordestani$^{1}$ and S. N. Doosti Motlagh$^{1}$'
title: 'The effects of nonlinear Maxwell source on the magnetic solutions in Einstein-Gauss-Bonnet gravity'
---
Introduction
============
Among the theories of gravity with higher derivative corrections, the Gauss-Bonnet (GB) gravity is quite special. Indeed, in order to have a ghost-free action, the quadratic curvature corrections to the Einstein-Hilbert action should not contain derivatives of metrics of order higher than second, and should be proportional to the GB term [@Boulware]. This combination also appear naturally in the next-to-leading order term of the heterotic string effective action, and plays a fundamental role in some gravitational theories [@Cham]. Generally, in recent years, GB gravity has been studied by many authors (see [@Wil1; @DH; @Deh1; @Deh2; @DehMagGB; @Deh3; @Levi; @Vil; @Mukh; @Dias; @Dehghani; @NUT; @Od1] and references therein).
In the conventional, straightforward generalization of the Maxwell field to higher dimensions one essential property of the electromagnetic field is lost, namely, conformal invariance. The first black hole solution derived for which the matter source is conformally invariant is the Reissner-Nordström solution in four dimensions. Indeed, in this case the source is given by the Maxwell action which enjoys the conformal invariance in four dimensions. Maxwell theory can be studied in a gauge which is invariant under conformal rescalings of the metric, and firstly, has been proposed by Eastwood and Singer [@EasSin]. Recently, there exists a nonlinear extension of the Maxwell Lagrangian in higher dimensions, if one uses the Lagrangian of the $U(1)$ gauge field in the form [@HasMar; @HendiRastegar; @HendiPLB; @HendiEslamPanah] $$L=F^{s}, \label{IGMF}$$ where $F=F_{\mu \nu }F^{\mu \nu }$ is the Maxwell invariant, $F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }$ is the Maxwell tensor and $A_{\mu }$ is the vector potential. In what follows, we consider this Lagrangian as the matter source coupled to the Einstein-GB gravity. The first motivation is to take advantage of the conformal symmetry to construct the analogues of the four-dimensional Reissner-Nordström black hole solutions in higher dimensions, and the second motivation comes from the generalization of Maxwell field and investigation of their effects on the energy-momentum tensor.
In this paper we want to restrict ourself at most to the first three terms of Lovelock gravity. The first two terms are the Einstein-Hilbert term with cosmological constant, while the third term is known as the Gauss-Bonnet term. Because of the nonlinearity of the field equations, it is very difficult to find out nontrivial exact analytical solutions of Einstein’s equation with higher curvature terms. In most cases, one has to adopt some approximation methods or find solutions numerically. These facts provide a strong motivation for considering new exact solutions of the Einstein-Gauss-Bonnet gravity with nonlinear source. The main goal of this work is to present analytical solutions for a typical class of magnetic horizonless of GB-nonlinear Maxwell source and investigate their properties. These kinds of work have been investigated in many papers of Einstein gravity. Static uncharged cylindrically symmetric solutions of Einstein gravity in four dimensions were considered in [@Levi]. Similar static solutions in the context of cosmic string theory were found in [@Vil]. All of these solutions [@Levi; @Vil] are horizonless and have a conical geometry, which are everywhere flat except at the location of the line source. An extension to include the electromagnetic field has also been done [@Muk; @Lem1]. The generalization of the four-dimensional solution found in [@Lem1] to the case of $(n+1)$-dimensional solution with all rotation and boost parameters has been done in [@Deh4].
The outline of our paper is as follows. In next Section, we briefly present the basic field equations of the GB gravity and nonlinear Maxwell source. In section \[Long\], we present a new class of static magnetic solutions and consider the properties of the solutions as well as the energy condition. In section \[Rot\], we endow these spacetime with global rotations and then apply the counterterm method to compute the conserved quantities of these solutions. Finally, we finish our paper with some closing remarks.
Field Equations
===============
The gravitational and electromagnetic field equations of the Einstein-GB gravity in the presence of power of Maxwell invariant field may be written as $$\begin{aligned}
&&G_{\mu \nu }+\Lambda g_{\mu \nu }-\frac{\alpha }{2}\left[
8R^{\rho \sigma }R_{\mu \rho \nu \sigma }-4R_{\mu }^{\ \rho \sigma
\lambda }R_{\nu \rho \sigma \lambda }-4RR_{\mu \nu }+8R_{\mu
\lambda }R_{\text{ \ }\nu }^{\lambda
}+\right. \nonumber \\
&&\left. g_{\mu \nu }\left( R_{\mu \nu \gamma \delta }R^{\mu \nu
\gamma \delta }-4R_{\mu \nu }R^{\mu \nu }+R^{2}\right) \right]
=2\kappa \left( sF_{\mu \rho }F_{\nu }^{~\rho
}F^{s-1}-\frac{1}{4}g_{\mu \nu }F^{s}\right) , \label{Geq}\end{aligned}$$ $$\partial _{\mu }\left( \sqrt{-g}F^{\mu \nu }F^{s-1}\right) =0, \label{Maxeq}$$where $G_{\mu \nu }$ is the Einstein tensor, $\Lambda
=-n(n-1)/2l^{2}$ is the negative cosmological constant, $\alpha $ is the GB coefficient with dimension $(\mathrm{length})^{2}$, $R$, $R_{\mu \nu }$ and $R_{\mu \nu \gamma \delta }$ are Ricci scalar, Ricci and Riemann tensors. In addition, $ \kappa $ is a constant in which we set $\kappa =1$ without loss of generality and consequently the energy density (the $T_{\widehat{0}\widehat{0 }}$ component of the energy-momentum tensor in the orthonormal frame) is positive. In the limit $s=1$, the nonlinear electromagnetic field reduces to the standard Maxwell form, as it should be. It is easy to show that for $\alpha =0$, the equation (\[Geq\]) reduces to the Einstein gravity coupled with power Maxwell invariant source.
Static magnetic branes\[Long\]
==============================
Here we want to obtain the $(n+1)$-dimensional solutions of Eqs. (\[Geq\]) and (\[Maxeq\]) which produce longitudinal magnetic fields in the Euclidean submanifold spans by $x^{i}$ coordinates ($i=1,...,n-2$). We will work with the following ansatz for the metric [@Lem1]: $$ds^{2}=-\frac{\rho ^{2}}{l^{2}}dt^{2}+\frac{d\rho ^{2}}{f(\rho )}%
+l^{2}f(\rho )d\phi ^{2}+\frac{\rho ^{2}}{l^{2}}dX^{2},
\label{Met1a}$$ where $dX^{2}={{\sum_{i=1}^{n-2}}}(dx^{i})^{2}$ is the Euclidean metric on the $(n-2)$-dimensional submanifold. The angular coordinates $\phi $ is dimensionless as usual and ranges in $[0,2\pi ]$, while $x^{i}$’s range in $(-\infty ,\infty )$. The motivation for this metric gauge $[g_{tt}\varpropto -\rho ^{2}$ and $(g_{\rho \rho })^{-1}\varpropto g_{\phi \phi }]$ instead of the usual Schwarzschild gauge $[(g_{\rho \rho })^{-1}\varpropto
g_{tt}$ and $ g_{\phi \phi }\varpropto \rho ^{2}]$ comes from the fact that we are looking for a horizonless magnetic solution instead of electrical one. Also, one can obtain the presented metric (\[Met1a\]) with local transformations $ t\rightarrow
il\phi $ and $\phi \rightarrow it/l$ in the horizon flat Schwarzschild-like metric, $ds^{2}=-f(\rho )dt^{2}+\frac{d\rho
^{2}}{f(\rho ) }+\rho ^{2}d\phi ^{2}+\frac{\rho
^{2}}{l^{2}}dX^{2}$. Thus, the nonzero component of the gauge potential is $A_{\phi}$, which can be written as $$A_{\mu }=-2ql^{n-1}h(\rho )\delta _{\mu }^{\phi },$$where $h(\rho )$ is $\ln (\rho )$ for $s=n/2$, and for other values of $s$, we have $$h(\rho )=\rho ^{(2s-n)/(2s-1)},$$ therefore the non-vanishing component of electromagnetic field tensor is now given by $$F_{\rho \phi }=2ql^{n-1}\left\{
\begin{array}{cc}
\rho ^{-1}, & s=\frac{n}{2} \\
\frac{2s-n}{2s-1}\rho ^{-(n-1)/(2s-1)}, & \text{Otherwise}%
\end{array}%
\right. . \label{Ftr}$$ Because of vanishing the electromagnetic field for $s=0,1/2$, we ignore this cases. It is notable that for $s<\frac{1}{2}$, the electromagnetic field (\[Ftr\]) diverge as $\rho \longrightarrow
\infty $ and therefore we restrict our solutions to $s>\frac{1}{2}$. To find the function $f(\rho )$, one may use any components of Eq. (\[Geq\]). The solution of Eq. (\[Geq\]) can be written as $$f(\rho )=\frac{2\rho ^{2}}{(n-1)\gamma }\left(
1-\sqrt{1+\frac{2\gamma \Lambda }{n}+\frac{\gamma m}{\rho
^{n}}-\gamma \Gamma (\rho )}\right) , \label{f(r)}$$$$\begin{aligned}
\Gamma (\rho ) &=&\left\{
\begin{array}{cc}
2^{3n/2}(n-1)l^{n(n-2)}q^{n}\frac{\ln \rho }{\rho ^{n}}, & s=\frac{n}{2} \\
\frac{(2s-1)^{2}}{2s-n}\left( \frac{8l^{2(n-2)}q^{2}(2s-n)^{2}}{%
(2s-1)^{2}\rho ^{2(n-1)/(2s-1)}}\right) ^{s}, & s>\frac{1}{2},s\neq \frac{n}{%
2}%
\end{array}%
\right. , \label{GAMMA1} \\
\gamma &=&\frac{4\alpha (n-2)(n-3)}{(n-1)}, \nonumber\end{aligned}$$where mass parameter, $m,$ is related to integration constant. It is easy to show that for $\alpha \longrightarrow 0$, Eq. (\[f(r)\]) reduces to $$f_{E}(\rho )=\frac{-2\Lambda \rho ^{2}}{n(n-1)}-\frac{m}{(n-1)\rho ^{n-2}}+%
\frac{\rho ^{2}}{(n-1)}\Gamma (\rho ), \label{Einstein}$$where $f_{E}(\rho )$ is the Einstein solution of Eq. (\[Geq\])($\alpha =0$).
Energy conditions
-----------------
Here, we discuss the energy conditions for the power Maxwell invariant electromagnetic field in diagonal metric. For the energy momentum tensor written in the orthonormal contravariant basis vectors as $T^{\mu \nu }=diag(
%TCIMACRO{\U{3bc} }%
%BeginExpansion
\mu
%EndExpansion
,p_{r},p_{t_{1}},p_{t_{2}},$$)$, the null energy condition (NEC) is the assertion that $p_{r}+%
%TCIMACRO{\U{3bc} }%
%BeginExpansion
\mu
%EndExpansion
\geq 0$ and $p_{t_{i}}+%
%TCIMACRO{\U{3bc} }%
%BeginExpansion
\mu
%EndExpansion
\geq 0$, and the weak energy condition (WEC) implies $%
%TCIMACRO{\U{3bc} }%
%BeginExpansion
\mu
%EndExpansion
\geq 0$, $p_{r}+%
%TCIMACRO{\U{3bc} }%
%BeginExpansion
\mu
%EndExpansion
\geq 0$, and $p_{t_{i}}+%
%TCIMACRO{\U{3bc} }%
%BeginExpansion
\mu
%EndExpansion
\geq 0$, while the dominant energy condition (DEC) implies $
%TCIMACRO{\U{3bc} }%
%BeginExpansion
\mu
%EndExpansion
\geq 0$, $-%
%TCIMACRO{\U{3bc} }%
%BeginExpansion
\mu
%EndExpansion
\leq p_{r}\leq
%TCIMACRO{\U{3bc} }%
%BeginExpansion
\mu
%EndExpansion
$, and $-%
%TCIMACRO{\U{3bc} }%
%BeginExpansion
\mu
%EndExpansion
\leq p_{t_{i}}\leq
%TCIMACRO{\U{3bc} }%
%BeginExpansion
\mu
%EndExpansion
$, and strong energy condition (SEC)which implies $p_{r}+%
%TCIMACRO{\U{3bc} }%
%BeginExpansion
\mu
%EndExpansion
\geq 0$, $p_{t_{i}}+%
%TCIMACRO{\U{3bc} }%
%BeginExpansion
\mu
%EndExpansion
\geq 0$, and $%
%TCIMACRO{\U{3bc} }%
%BeginExpansion
\mu
%EndExpansion
+p_{r}+\sum_{i=1}^{n-1}p_{t_{i}}\geq 0$. The physical interpretations of $
%TCIMACRO{\U{3bc} }%
%BeginExpansion
\mu
%EndExpansion
$, $p_{r}$, and $p_{t_{i}}$ are energy density, radial pressure, and the tangential pressure, respectively. For our diagonal metric, using the orthonormal contravariant (hatted) basis vectors $$\mathbf{e}_{\widehat{t}}=\frac{l}{r}\frac{\partial }{\partial
t},\text{ \ \ } \mathbf{e}_{\widehat{r}}=f^{1/2}\frac{\partial
}{\partial r},\text{ \ \ } \mathbf{e}_{\widehat{\phi
}}=\frac{1}{lf^{1/2}}\frac{\partial }{\partial \phi },\text{ \
}\mathbf{e}_{\widehat{x^{i}}}=\frac{l}{r}\frac{\partial }{
\partial x^{i}},$$ the mathematics and physical interpretations become simplified. It is a matter of straight forward calculations to show that the stress-energy tensor is $$\begin{aligned}
T_{_{\widehat{t}\widehat{t}}}
&=&-T_{_{\widehat{i}\widehat{i}}}=\frac{1}{2}
\left( \frac{2F_{\phi r}^{2}}{l^{2}}\right) ^{s}, \\
&& \nonumber \\
\text{ \ \ }T_{_{\widehat{r}\widehat{r}}} &=&T_{_{\widehat{\phi
}\widehat{ \phi }}}=\frac{2s-1}{2}\left( \frac{2F_{\phi
r}^{2}}{l^{2}}\right) ^{s}, \label{EMtensor}\end{aligned}$$ so for satisfaction of the null and weak energy condition, we should justify $s>0$. $$T_{_{\widehat{t}\widehat{t}}}\geq
0,\hspace{1cm}T_{_{\widehat{t}\widehat{t}
}}+T_{_{\widehat{i}\widehat{i}}}\geq 0,\text{ \
}T_{_{\widehat{t}\widehat{t}
}}+T_{_{\widehat{r}\widehat{r}}}=T_{_{\widehat{t}\widehat{t}}}+T_{_{\widehat{
\phi }\widehat{\phi }}}\geq 0. \label{WEC}$$ One may show that for satisfaction of the dominant and strong energy conditions, we should set $0<s<1$ and $s\geqslant
\frac{n-1}{4}$, respectively. Since for Einstein gravity or (GB gravity) $n\geqslant 3$ or $(4)$ and also, we restrict our solutions to $s>\frac{1}{2}$, the presented solutions always satisfy the null, weak and strong energy conditions, simultaneously, and dominant energy condition is satisfied when $\frac{1}{2}<s\leqslant 1$.
In order to investigate the effect of the nonlinearity of the electromagnetic field on energy density of the spacetime, we plot the $T_{_{\widehat{t}\widehat{t}}}$ versus $r$ (for different values of nonlinearity parameter $s$) and $s$. Figs. \[T00r\] and \[T00s2\] show that for $s>\frac{n}{2}$, on one hand, for a fixed value of $r$, as nonlinearity parameter increases, the energy density of the spacetime increase too and on the other hand, in order to reduce the concentration area of the energy density, we should reduce the nonlinearity parameter. Also, Fig. \[T00s1\] shows that $T_{_{\widehat{t}\widehat{t}}}$ has a local maximum when the nonlinearity parameter changes from $\frac{1}{2}$ to $\frac{n}{2}$.
Conformally invariant electromagnetic field\[Conformally\]
----------------------------------------------------------
It is easy to show that the clue of the conformal invariance of Maxwell source lies in the fact that we have considered power of the Maxwell invariant, $F=F_{\mu \nu }F^{\mu \nu }$. Here we want to justify the nonlinearity parameter $s$, such that the electromagnetic field equation be invariant under conformal transformation ($g_{\mu \nu }\longrightarrow \Omega ^{2}g_{\mu \nu
}$ and $A_{\mu }\longrightarrow A_{\mu }$). The idea is to take advantage of the conformal symmetry to construct the analogues of the four dimensional Reissner-Nordström solutions in higher dimensions. It is easy to show that for Lagrangian in the form $L(F)$ in $(n+1)$ -dimensions, $T_{\mu }^{\mu }\propto \left[
F\frac{dL}{dF}-\frac{n+1}{4}L \right] $; so $T_{\mu }^{\mu }=0$ implies $L(F)=Constant\times F^{(n+1)/4}$. For our case, nonlinear Maxwell field, $L(F)\propto F^{s}$, we should set $s=(n+1)/4$ for conformally invariance condition. It is worthwhile to mention that Since $n\geq 3$ and therefore $s=(n+1)/4\geq 1$, one can show that the magnetic solutions with conformally invariant Maxwell source are asymptotically AdS in arbitrary dimensions. In this case the functions $f(\rho )$ and $h(\rho )$ reduce to $$\begin{aligned}
f(\rho ) &=&\frac{2\rho ^{2}}{(n-1)\gamma }\left(
1-\sqrt{1+\frac{2\gamma \Lambda }{n}+\frac{\gamma m}{\rho
^{n}}+\gamma g(\rho )}\right) ,
\label{f(r)Conf} \\
&&g(\rho )=2^{(n-3)/4}(n-1)\left( \frac{2l^{n-2}q}{\rho
^{2}}\right) ^{(n+1)/2}, \nonumber\end{aligned}$$ $$h(\rho )\propto \frac{1}{\rho }, \label{Em2}$$ and therefore the electromagnetic field is analogues of the four dimensional Reissner-Nordström solutions, $F_{\phi \rho
}\propto \rho ^{-2}$ in arbitrary dimensions.
The higher dimensional BTZ-like solutions
-----------------------------------------
The (2+1)-dimensional BTZ solution [@BTZ] have obtained a great importance in recent years because this provide a simplified model for exploring some conceptual issues, not only about black hole thermodynamics and magnetic solutions but also about quantum gravity and string theory [@BTZ1]. The line element of BTZ solution with negative cosmological constant $\Lambda =-1/l^{2}$ may be written as $$ds^{2}=-f(\rho )dt^{2}+\frac{d\rho ^{2}}{f(\rho )}+\rho ^{2}d\phi
^{2}, \label{BTZmetric}$$ where $$f(\rho )=-M+\frac{\rho ^{2}}{l^{2}}+\frac{Q^{2}}{2}\ln \rho ,$$ in which $M$ and $Q$ are the mass and the electric charge of the solution, respectively [@BTZ2].
The (2+1)-dimensional static subsection of the metric (\[Met1a\]) can be written as $$ds^{2}=-\frac{\rho ^{2}}{l^{2}}dt^{2}+\frac{d\rho ^{2}}{f(\rho )}
+l^{2}f(\rho )d\phi ^{2}, \label{MetBTZ}$$ One can obtain the presented magnetic metric (\[MetBTZ\]) with local transformations $t\rightarrow il\phi $ and $\phi \rightarrow
it/l$ in the electrical BTZ metric (\[BTZmetric\]) with the same metric function $f(\rho )$.
Comparing (\[MetBTZ\]) with (\[Met1a\]) help us to conclude that Eqs. (\[Ftr\]), (\[f(r)\]) and (\[Einstein\]) with metric (\[Met1a\]) may be interpreted as higher dimensional BTZ-like magnetic solutions for $s=\frac{n}{2}$. It is easy to show that in 3 dimension ($n=2$), the original magnetic BTZ solution emerge. It is notable that for $s=\frac{n}{2}$, BTZ-like solutions, the electromagnetic field $F_{\phi \rho }\propto \rho
^{-1}$ in arbitrary dimensions.
Properties of the solutions
---------------------------
At first, we investigate the effects of the nonlinearity on the asymptotic behavior of the Einstein and GB solutions. It is worthwhile to mention that for $s>\frac{1}{2}$ (including $s=\frac{n}{2}$), the asymptotic behavior of Einstein-(GB)-nonlinear Maxwell field solutions are the same as Einstein-(GB)-Born-Infeld and linear AdS case.
In order to study the general structure of these spacetime, we first look for the essential singularity(ies). After some algebraic manipulation, one can show that for the rotating metric (\[Met1a\]), the Kretschmann and Ricci scalars are $$\begin{aligned}
R_{\mu \nu \rho \sigma }R^{\mu \nu \rho \sigma } &=&f^{\prime
\prime 2}(\rho )+\frac{2(n-1)f^{\prime 2}(\rho )}{\rho
^{2}}+\frac{2(n-1)(n-2)f^{2}(\rho )}{
\rho ^{4}}, \label{RR} \\
R &=&-f^{\prime \prime }(\rho )-\frac{2(n-1)f^{\prime }(\rho
)}{\rho }-\frac{ (n-1)(n-2)f(\rho )}{\rho ^{2}}, \label{R}\end{aligned}$$ where prime and double prime are first and second derivative with respect to $\rho $ , respectively. Denoting the largest real root of $1+\frac{2\gamma \Lambda }{n}+\frac{\gamma m}{\rho ^{n}}-\gamma
\Gamma (\rho )=0$ (in the case that it has real root(s)) by $r_{1}$, Eq. (\[f(r)\]) show that $\rho $ should be greater than $r_{1}$ in order to have a real spacetime. By substituting the metric function (\[f(r)\]), It is easy to show that the Kretschmann invariant and Ricci scalar diverge at $r_{0}=Max\{0,r_{1}\}$ and they are finite for $\rho >r_{0}$. It is notable that as $\rho \rightarrow \infty $, we have $$\begin{aligned}
R_{\mu \nu \rho \sigma }R^{\mu \nu \rho \sigma }
&=&\frac{8(n+1)}{n(n-1)^{2}}
\Lambda ^{2}, \label{RR2} \\
R &=&\frac{2(n+1)}{(n-1)}\Lambda , \label{R2}\end{aligned}$$ which confirm that asymptotic behavior of the solutions is AdS. Considering the divergency of the Kretschmann and Ricci scalars, one might think that there is a curvature singularity located at $\rho =r_{0}$. Two cases happen. In the first case the function $f(\rho )$ has no real root greater than $r_{0}$, and therefore we encounter with a naked singularity which we are not interested in it. So we consider only the second case which the function has one or more real root(s) larger than $r_{0}$. In this case the function $f(\rho )$ is negative for $\rho <r_{+}$, and positive for $\rho >r_{+}$ where $r_{+}$ is the largest real root of $f(\rho )=0$. This leads to an apparent change of signature of the metric, and therefore indicates that $\rho $ should be greater than $r_{+}$. Thus the coordinate $\rho $ assumes the value $r_{+}\leq \rho <\infty $. The function $f(\rho )$ given in Eq. (\[f(r)\]) is positive in the whole spacetime and is zero at $\rho =r_{+}$, (while $f^{\prime }(\rho =r_{+})\neq 0$). Thus, one cannot extend the spacetime to $\rho <r_{+}$. To get rid of this incorrect extension, we introduce the new radial coordinate $r$ as $$r^{2}=\rho ^{2}-r_{+}^{2}\Rightarrow d\rho
^{2}=\frac{r^{2}}{r^{2}+r_{+}^{2}} dr^{2}.$$ With this new coordinate, the metric (\[Met1a\]) is $$ds^{2}=-\frac{r^{2}+r_{+}^{2}}{l^{2}}dt^{2}+\frac{r^{2}}{
(r^{2}+r_{+}^{2})f(r)}dr^{2}+l^{2}f(r)d\phi
^{2}+\frac{r^{2}+r_{+}^{2}}{l^{2} }dX^{2}, \label{Metr1b}$$ where the coordinate $r$ and $\phi $ assume the value $0\leq r<$ $\infty $ and $0\leq \phi <2\pi $. The function $f(r)$ is now given as $$f(r)=\frac{2(r^{2}+r_{+}^{2})}{(n-1)\gamma }\left(
1-\sqrt{1+\frac{2\gamma \Lambda }{n}+\frac{\gamma
m}{(r^{2}+r_{+}^{2})^{n/2}}-\gamma \Gamma (r)} \right) ,
\label{F2}$$ where $\Gamma (r)$ changes to $$\Gamma (r)=\left\{
\begin{array}{cc}
2^{(3n-2)/2}(n-1)l^{n(n-2)}q^{n}\frac{\ln (r^{2}+r_{+}^{2})}{
(r^{2}+r_{+}^{2})^{n/2}}, & s=\frac{n}{2} \\
\frac{(2s-1)^{2}}{2s-n}\left( \frac{8l^{2(n-2)}q^{2}(2s-n)^{2}}{
(2s-1)^{2}(r^{2}+r_{+}^{2})^{(n-1)/(2s-1)}}\right) ^{s}, &
s>\frac{1}{2} ,s\neq \frac{n}{2}
\end{array}
\right. ,$$ and $\gamma $ remains unchanged. The electromagnetic field equation in the new coordinate is $$F_{r\phi }=2ql^{n-1}\left\{
\begin{array}{cc}
(r^{2}+r_{+}^{2})^{-1/2}, & s=\frac{n}{2} \\
\frac{2s-n}{2s-1}(r^{2}+r_{+}^{2})^{-(n-1)/(4s-2)}, &
s>\frac{1}{2},s\neq \frac{n}{2}
\end{array}
\right. . \label{f33}$$ The function $f(r)$ given in Eq. (\[F2\]) is positive in the whole spacetime and is zero at $r=0$. One can easily show that the Kretschmann scalar does not diverge in the range $0\leq r<\infty
$. However, the spacetime has a conic geometry and has a conical singularity at $r=0$, since: $$\lim_{r\rightarrow 0}\frac{1}{r}\sqrt{\frac{g_{\phi \phi
}}{g_{rr}}}\neq 1. \label{limit}$$ For more explanations, using a Taylor expansion, in the vicinity of $r=0$, we can rewrite (\[F2\]) $$f(r)=f(r)\left\vert _{r=0}\right. +\left( \frac{df}{dr}\left\vert
_{r=0}\right. \right) r+\frac{1}{2}\left(
\frac{d^{2}f}{dr^{2}}\left\vert _{r=0}\right. \right)
r^{2}+O(r^{3})+...,$$ where $$\begin{aligned}
f(r)\left\vert _{r=0}\right. &=&\frac{df}{dr}\left\vert
_{r=0}\right. =0,
\label{ffp} \\
\frac{d^{2}f}{dr^{2}}\left\vert _{r=0}\right. &\neq &0.\end{aligned}$$
As a result, we can rewrite Eq. (\[Metr1b\]) $$ds^{2}=-\frac{r_{+}^{2}}{l^{2}}dt^{2}+\frac{2\left(
\frac{d^{2}f}{dr^{2}} \left\vert _{r=0}\right. \right)
^{-1}}{r_{+}^{2}}dr^{2}+\frac{l^{2}}{2} \left(
\frac{d^{2}f}{dr^{2}}\left\vert _{r=0}\right. \right) r^{2}d\phi
^{2}+ \frac{r_{+}^{2}}{l^{2}}dX^{2},$$ and since $\frac{d^{2}f}{dr^{2}}\left\vert _{r=0}\right. \neq
\frac{ 2}{lr_{+}}$, one can show that $$\lim_{r\longrightarrow 0}\frac{1}{r}\sqrt{\frac{g_{\phi \phi
}}{g_{rr}}} =\lim_{r\longrightarrow
0}\frac{r_{+}}{r^{2}}lf(r)=\frac{lr_{+}}{2}\left(
\frac{d^{2}f}{dr^{2}}\left| _{r=0}\right. \right) \neq 1.$$ which clearly shows that the spacetime has a conical singularity at $r=0$ since, when the radius $r$ tends to zero, the limit of the ratio circumference/radius is not $2\pi $. The canonical singularity can be removed if one identifies the coordinate $\phi$ with the period $$\text{Period}_{\phi }=2\pi \left( \lim_{r\rightarrow
0}\frac{1}{r}\sqrt{ \frac{g_{\phi \phi }}{g_{rr}}}\right)
^{-1}=2\pi (1-4\mu ),$$ where $\mu$ is given by $$\mu =\frac{1}{4}\left[ 1-\frac{2}{lr_{+}}\left(
\frac{d^{2}f}{dr^{2}} \left\vert _{r=0}\right. \right)
^{-1}\right] . \label{mu}$$
By the above analysis, one concludes that near the origin $r=0$ the metric (\[Metr1b\]) describes a spacetime which is locally flat but has a conical singularity at $r=0$ with a deficit angle $\delta \phi =8\pi \mu $. It is worthwhile to mention that the magnetic solutions obtained here have distinct properties relative to the electric solutions obtained in [@HendiEslamPanah]. Indeed, the electric solutions have curvature singularity and horizon(s) and interpreted as black hole (brane) solutions, while the magnetic horizonless solutions have conic singularity. In order to interpreted these solutions, we should mention that near the origin, this metric in $4$ dimensions is identical to the spacetime generated by a cosmic string, for which $\mu $ can be interpreted as the mass per unit length of the string. Thus, here we may interpret $\mu $ as the mass per unit volume of the brane. In order to investigate the effect of the nonlinearity of the magnetic field on $\mu $, we plot the deficit angle $\delta \phi$ versus the nonlinearity parameter $s$. This is shown in Figs. \[deficit1\] and \[deficit2\], which show that the deficit angle has a local maximum ($\delta \phi_{m}/{8\pi} \approx
0.2110$) for $\frac{1}{2}<s<\frac{n}{2}$. For $s>\frac{n}{2}$, the deficit angle is an increasing function, and for large values of nonlinearity parameter $s$, it goes to an asymptotic value ($\delta \phi_{asy}/{8\pi} \approx 0.2500$). It is easy to show that for $s=\frac{n}{2}$ with $n=5$, $l=1$, $q=2$, and $r_{+}=2$, we obtain $\delta \phi_{\frac{n}{2}}/{8\pi}=0.2487$. It is worthwhile to mention that for arbitrary choice of metric parameters, we have $$\delta \phi_{m}<\delta \phi_{\frac{n}{2}}<\delta \phi_{asy}$$ $$\lim_{s \longrightarrow \frac{1}{2}^{+}} {\delta \phi}=\lim_{s
\longrightarrow \frac{n}{2}^{-}} {\delta \phi}=\lim_{s
\longrightarrow \frac{n}{2}^{+}} {\delta \phi}\neq {\delta
\phi}|_{s=\frac{n}{2}}$$
One can find easily that the function $\left(
\frac{d^{2}f}{dr^{2}} \left\vert _{r=0}\right. \right)$ is an increasing function of nonlinearity parameter, $s$ (see Fig. \[d2F\]). Thus for large values of $s$, this function goes to infinity, second term in Eq. (\[mu\]) vanishes, and therefore, the asymptotic value for $\delta \phi/{8\pi}$ is $0.25$. One may conclude that since the nonlinearity parameter s, has an effect on the energy density $T_{_{\widehat{t}\widehat{t}}}$ and the metric function $f(r)$, so it can directly have an effect on the deficit angle of conic singularity.
Spinning Magnetic Branes\[Rot\]
===============================
Here, we desire to give rotation to our spacetime solutions (\[Metr1b\]). In order to add angular momentum to the spacetime, we perform the following rotation boost in the $t$-$\phi $ plane $$t\mapsto \Xi t-a\phi \ \ \ \ \ \ \ \ \ \ \phi \mapsto \Xi \phi
-\frac{a}{l^{2}}t, \label{tphi}$$ where $a$ is the rotation parameter and $\Xi
=\sqrt{1+a^{2}/l^{2}}$. Substituting Eq. (\[tphi\]) into Eq. (\[Metr1b\]) we obtain $$ds^{2}=-\frac{r^{2}+r_{+}^{2}}{l^{2}}\left( \Xi dt-ad\phi \right)
^{2}+\frac{ r^{2}dr^{2}}{(r^{2}+r_{+}^{2})f(r)}+l^{2}f(r)\left(
\frac{a}{l^{2}}dt-\Xi d\phi \right)
^{2}+\frac{r^{2}+r_{+}^{2}}{l^{2}}dX^{2}, \label{Metr2}$$ where $f(r)$ is the same as $f(r)$ given in Eq. (\[F2\]). The non vanishing electromagnetic field components become $$F_{rt}=-\frac{a}{\Xi l^{2}}F_{r\phi }=-\frac{2qal^{n-3}}{\Xi
}\left\{
\begin{array}{cc}
(r^{2}+r_{+}^{2})^{-1/2}, & s=\frac{n}{2} \\
\frac{2s-n}{2s-1}(r^{2}+r_{+}^{2})^{-(n-1)/(4s-2)}, &
s>\frac{1}{2},s\neq \frac{n}{2}
\end{array}
\right. .$$ The transformation (\[tphi\]) generates a new metric, because it is not a permitted global coordinate transformation. This transformation can be done locally but not globally. Therefore, the metrics (\[Metr1b\]) and (\[Metr2\]) can be locally mapped into each other but not globally, and so they are distinct. Again, this spacetime has no horizon and curvature singularity, However, it has a conical singularity at $r=0$.
Second, we study the rotating solutions with more rotation parameters. The rotation group in $n+1$ dimensions is $SO(n)$ and therefore the number of independent rotation parameters is $[n/2]$, where $[x]$ is the integer part of $x$. We now generalize the above solution given in Eq. (\[Metr1b\]) with $k\leq \lbrack
n/2]$ rotation parameters. This generalized solution can be written as $$\begin{aligned}
ds^{2} &=&-\frac{r^{2}+r_{+}^{2}}{l^{2}}\left( \Xi
dt-{{\sum_{i=1}^{k}}} a_{i}d\phi ^{i}\right) ^{2}+f(r)\left(
\sqrt{\Xi ^{2}-1}dt-\frac{\Xi }{\sqrt{
\Xi ^{2}-1}}{{\sum_{i=1}^{k}}}a_{i}d\phi ^{i}\right) ^{2} \nonumber \\
&&+\frac{r^{2}dr^{2}}{(r^{2}+r_{+}^{2})f(r)}+\frac{r^{2}+r_{+}^{2}}{
l^{2}(\Xi ^{2}-1)}{\sum_{i<j}^{k}}(a_{i}d\phi _{j}-a_{j}d\phi
_{i})^{2}+ \frac{r^{2}+r_{+}^{2}}{l^{2}}dX^{2}, \label{Metr5}\end{aligned}$$ where $\Xi =\sqrt{1+\sum_{i}^{k}a_{i}^{2}/l^{2}}$, $dX^{2}$ is the Euclidean metric on the $(n-k-1)$-dimensional submanifold with volume $V_{n-k-1}$ and $f(r)$ is the same as $f(r)$ given in Eq. (\[F2\]). The non-vanishing components of electromagnetic field tensor are $$F_{rt}=-\frac{(\Xi ^{2}-1)}{\Xi a_{i}}F_{r\phi
^{i}}=-\frac{2ql^{n-1}(\Xi ^{2}-1)}{\Xi a_{i}}\left\{
\begin{array}{cc}
(r^{2}+r_{+}^{2})^{-1/2}, & s=\frac{n}{2} \\
\frac{2s-n}{2s-1}(r^{2}+r_{+}^{2})^{-(n-1)/(4s-2)}, &
s>\frac{1}{2},s\neq \frac{n}{2}
\end{array}
\right. .$$
It is worthful to note that one can find a close relation between the Kerr-NUT-AdS solutions of Ref. [@KerrNUT] and the presented solutions, Eq. (\[Metr5\]) with metric function given in Eq. (\[F2\]) for vanishing both the Gauss-Bonnet parameter $\alpha$ and the nonlinearity parameter $s$.
Conserved Quantities \[Conserve\]
---------------------------------
Here, we present the calculation of the angular momentum and mass density of the solutions. Generally, in order to have finite conserved quantities for asymptotically AdS solutions of Einstein gravity, one may use of the counterterm method inspired by the anti-de Sitter/conformal field theory (AdS/CFT) correspondence [@Mal]. In addition, for asymptotically AdS solutions of Lovelock gravity with flat boundary, $\widehat{R}_{abcd}(\gamma
)=0$ (our solutions), the finite energy momentum tensor is [@DBSH; @DM1] $$T^{ab}=\frac{1}{8\pi }\{(K^{ab}-K\gamma ^{ab})+2\alpha
(3J^{ab}-J\gamma ^{ab})-\left( \frac{n-1}{l_{eff}}\right) \gamma
^{ab}\}, \label{Stress}$$ where $l_{eff}$ is $$\begin{aligned}
l_{eff} &=&3\sqrt{\frac{\zeta }{2}}\frac{\left( 1-\sqrt{1-\zeta
}\right)
^{1/2}}{\left( 1-\sqrt{1-\zeta }+\zeta \right) }l, \label{L} \\
\zeta &=&\frac{(n-1)\gamma }{l^{2}}. \nonumber\end{aligned}$$ It is notable that, when $\alpha $ goes to zero (Einstein solutions), $l_{eff}$ reduces to $l$, as it should be. In Eq. (\[Stress\]), $K^{ab}$ is the extrinsic curvature of the boundary, $K$ is its trace, $\gamma ^{ab}$ is the induced metric of the boundary, and $J$ is trace of $J^{ab}$ $$J_{ab}=\frac{1}{3}
(K_{cd}K^{cd}K_{ab}+2KK_{ac}K_{b}^{c}-2K_{ac}K^{cd}K_{db}-K^{2}K_{ab}).$$ To compute the conserved charges of the spacetime, we should write the boundary metric in Arnowitt-Deser-Misner form. When there is a Killing vector field $\mathcal{\xi }$ on the boundary, then the quasilocal conserved quantities associated with the stress tensors of Eq. (\[Stress\]) can be written as $$\mathcal{Q}(\mathcal{\xi )}=\int_{\mathcal{B}}d^{n-1}\varphi
\sqrt{\sigma } T_{ab}n^{a}\mathcal{\xi }^{b}, \label{charge}$$ where $\sigma $ is the determinant of the metric $\sigma _{ij}$, and $n^{a}$ is the timelike unit normal vector to the boundary $\mathcal{B}$[. ]{}For our case, the magnetic solutions of GB gravity, the first Killing vector is $\xi =\partial /\partial t$, therefore its associated conserved charge is the total mass of the brane per unit volume $V_{n-k-1}$, given by $$M=\int_{\mathcal{B}}d^{n-1}x\sqrt{\sigma }T_{ab}n^{a}\xi
^{b}=\frac{(2\pi )^{k}}{4}\left[ n(\Xi ^{2}-1)+1\right] m.
\label{Mas}$$ For the rotating solutions, the conserved quantities associated with the rotational Killing symmetries generated by $\zeta
_{i}=\partial /\partial \phi ^{i}$ are the components of angular momentum per unit volume $V_{n-k-1}$ calculated as $$J_{i}=\int_{\mathcal{B}}d^{n-1}x\sqrt{\sigma }T_{ab}n^{a}\zeta
_{i}^{b}= \frac{(2\pi )^{k}}{4}n\Xi ma_{i}. \label{Ang}$$ Finally, we calculate the electric charge of the solutions. To determine the electric field we should consider the projections of the electromagnetic field tensor on special hypersurfaces. Then the electric charge per unit volume $V_{n-k-1}$ can be found by calculating the flux of the electromagnetic field at infinity, yielding $$Q=\frac{(2\pi )^{k}}{32}\sqrt{\Xi ^{2}-1}\times \left\{
\begin{array}{cc}
2^{3n/2}l^{n-1}nq^{n-1}, & s=\frac{n}{2} \\
2^{3s+1}l^{2s-1}sq^{2s-1}, & s>\frac{1}{2},s\neq \frac{n}{2}
\end{array}
\right. , \label{elecch}$$ which show that the electric charge is proportional to the magnitude of rotation parameters and is zero for the static solutions ($\Xi =1$).
Closing Remarks
================
In this paper, we started with a new class of static magnetic solutions in Gauss–Bonnet gravity in the presence of power Maxwell invariant field. One may obtain this magnetic metric with transformations $t\rightarrow il\phi $ and $\phi \rightarrow it/l$ in the horizon flat Schwarzschild-like metric. Because of the periodic nature of $\phi $, this transformation is not a proper coordinate transformation on the entire manifold. Therefore, the magnetic and Schwarzschild-like metrics can be locally mapped into each other but not globally, and so they are distinct [@Sta]. Also, we found that these solutions have no curvature singularity and no horizon. The metric function $f(r)$ is nonnegative in the whole spacetime and is zero at $r_{+}$.
Then, we restricted the nonlinearity parameter to $s>1/2$, since electromagnetic field at spatial infinity should vanish. Investigation of the energy conditions showed that since $s>1/2$, the presented magnetic brane solutions satisfied, simultaneously, the null, weak and strong energy conditions, and only for $\frac{1}{2}<s \leq 1$, the dominant energy condition satisfied. Also, we plot the energy density for various $s$, and found that it has a local maximum when $\frac{1}{2}<s<\frac{n}{2}$, and for $s>\frac{n}{2}$ it is an increasing function.
In addition, we showed that for a special value of nonlinearity parameter, $s=(n+1)/4$, the energy–momentum tensor is traceless and the solutions are conformally invariant. In this case, the electromagnetic field $F_{\phi r}\propto r^{-2}$ in arbitrary dimensions and it means that the expression of the Maxwell field does not depend on the dimensions and its value coincides with the Reissner-Nordström solution in four dimension. Also, we discussed about the special choice of nonlinearity parameter, $s=n/2$, and interpreted these solutions as higher dimensional BTZ-like magnetic solutions [@BTZlike]. In this case, like BTZ solutions, the electromagnetic field $F_{\phi r}\propto r^{-1}$ in arbitrary dimensions.
Then we investigated other properties of the solutions and found that that for $s>\frac{1}{2}$ (including $s=\frac{n}{2}$), the asymptotic behavior of Einstein-(GB)-nonlinear Maxwell field solutions are AdS. Then, we encountered with a conic singularity at $r=0$ with a deficit angle $\delta \phi$ which is sensitive to the nonlinearity of the electromagnetic field. We plotted it with respect to the $s$, and found that, the deficit angle has a local maximum for $\frac{1}{2}<s<\frac{n}{2}$ and for $s>\frac{n}{2}$, the deficit angle is an increasing function, and for large values of nonlinearity parameter $s$, it goes to its asymptotic value, $\delta \phi=2\pi$.
Calculation of electric charge showed that for the spinning solutions, when one or more rotation parameters are nonzero, the solutions has a net electric charge density which is proportional to the magnitude of the rotation parameter given by $\sqrt{\Xi
^{2}-1}$. This electric charge is sensitive to the nonlinearity parameter, as it should be.
Finally, we calculated the conserved quantities of the magnetic branes such as mass, angular momentum and found that these conserved quantities do not depend on the nonlinearity parameter $s$. This can be understand easily, since at the boundary at infinity the effects of the nonlinearity of the electromagnetic fields vanish (since $s>\frac{1}{2}$).
Acknowledgements {#acknowledgements .unnumbered}
================
This work has been supported financially by Research Institute for Astronomy and Astrophysics of Maragha, Iran.
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[^1]: email address: hendi@mail.yu.ac.ir
|
---
abstract: 'Ly$\alpha$ absorption spectra of QSOs at redshifts $z\simeq6$ show complete Gunn-Peterson absorption troughs (dark gaps) separated by tiny leaks. The dark gaps are from the intergalactic medium (IGM) where the density of neutral hydrogen are high enough to produce almost saturated absorptions, however, where the transmitted leaks come from is still unclear so far. We demonstrate that leaking can originate from the lowest density voids in the IGM as well as the ionized patches around ionizing sources using semi-analytical simulations. If leaks were produced in lowest density voids, the IGM might already be highly ionized, and the ionizing background should be almost uniform; in contrast, if leaks come from ionized patches, the neutral fraction of IGM would be still high, and the ionizing background is significantly inhomogeneous. Therefore, the origin of leaking is crucial to determining the epoch of inhomogeneous-to-uniform transition of the the ionizing photon background. We show that the origin could be studied with the statistical features of leaks. Actually, Ly$\alpha$ leaks can be well defined and described by the equivalent width $W$ and the full width of half area $W_{\rm H}$, both of which are less contaminated by instrumental resolution and noise. It is found that the distribution of $W$ and $W_{\rm H}$ of Ly$\alpha$ leaks are sensitive to the modeling of the ionizing background. We consider four representative models: uniform ionizing background (model 0), the photoionization rate of neutral hydrogen $\GHI$ and the density of IGM are either linearly correlated (model I), or anti-correlated (model II), and $\GHI$ is correlated with high density peaks containing ionizing sources (model III). Although all of these models can match to the mean and variance of the observed effective optical depth of the IGM at $z\simeq 6$, their distribution of $W$ and $W_{\rm H}$ are very different from each other. Consequently, the leak statistics provides an effective tool to probe the evolutionary history of reionization at $z\simeq5-6.5$. Similar statistics would also be applicable to the reionization of He II at $z \simeq 3$'
author:
- |
Longlong Feng$^{1}$[^1], Hongguang Bi$^{1,2}$, Jiren Liu$^{2}$, Li-Zhi Fang$^{2}$\
$^{1}$ Purple Mountain Observatory,Nanjing, 210008, China\
$^{2}$ Department of Physics, University of Arizona, Tucson, AZ 85721, USA
title: 'Ly$\alpha$ Leaks and Reionization'
---
=7.4truecm =17.7truecm =6.0truecm
\[firstpage\]
cosmology: theory - intergalactic medium - large-scale structure of the universe
Introduction
============
In the last decade, the Ly$\alpha$ forests of QSO’s absorption spectra at redshifts $z \leq 5$ have played an important role in understanding the diffuse cosmic baryon gas and the UV ionizing photon background, and constraining cosmological models and parameters (e.g. Rauch et al. 1997; Croft et al. 2002; Bolton et al. 2005; Seljak et al. 2005; Jena et al. 2005; Viel et al. 2006). Recently, more and more UV photon sources, including QSOs, GRB, Lyman-break galaxies, and Ly$\alpha$-emitters at redshifts $z>5$ have been observed (see Ellis 2007 and reference therein). Due to the rapidly increase of Gunn-Peterson (GP) optical depth at $z > 5$, their absorption spectra show long dark gaps on scales of tens of Mpc separated by tiny transmitted leaks. It has been suggested that we are observing the end stage of reionization (Fan et al. 2006).
It has been known that the dark gaps are from the IGM where the density of neutral hydrogen are high enough to produce almost complete absorptions, however, where the transmitted leaks come from is still unclear. In photoionization equilibrium, the density of neutral hydrogen $n_{\rm
HI}\propto\alpha\rho^2$, here $\alpha$ is the recombination rate and $\rho$ is the density of IGM; therefore, even when most of the IGM are neutral enough to produce complete Ly$\alpha$ absorptions, it is still possible for the lowest density voids to provide prominent transmitted fluxes. On the other hand, the leaks can also come from ionized patches around ionizing sources where the intensity of UV radiation are higher than average.
The origin of leaking is crucial to understanding the history of reionization. According to commonly accepted scenario of the reionization, in the early stage, the ionized regions are isolated patches in the neutral hydrogen background (e.g., Ciardi et al. 2003; Sokasian et al. 2003; Mellema et al. 2006; Gnedin 2006; Trac & Cen 2006). and the subsequent growing and overlapping of the ionized patches lead to the ending of reionization (e.g., Ciardi et al. 2003; Sokasian et al. 2003; Mellema et al. 2006; Gnedin 2004). Thus, if leaks mostly come from ionized patches, reionization should happen in the early stage. In contrast, if they were produced in lowest density voids, the the UV ionizing background might has already underwent an evolution from highly inhomogeneous to uniform distribution.
A variety of statistics has been used to study the evolution of reionization, such as the mean and dispersion of GP optical depth, the probability distribution function (PDF) of the flux, and the size of dark gaps (e.g., Fan et al. 2002, 2006; Songaila & Cowie 2002; Paschos & Norman 2005; Kohler et al. 2007; Gallerani et al. 2006; Becker et al. 2006), but all of them seem to be ineffective to provide the information of leak’s origin and the inhomogeneity of UV ionizing background. The GP optical depth is an average, and not sensitive to details of reionization. The statistical properties (mean and variance) of the GP optical depth at $z\simeq 6$ can be well explained by either the fluctuation of ionizing background (Fan et al. 2006) or models with uniform ionizing background (Lidz et al. 2006; Liu et al. 2006, hereafter PaperI). The PDF of the flux is also insensitive to the geometry of reionization. In addition, the PDF is heavily contaminated by noise and distorted by resolution.
Dark gaps are defined to be continuous regions with optical depth above a threshold in spectra. Intuitively, the statistics of dark gap should contain the same information as leaks. However, the size of dark gaps are sensitive to the instrumental resolution, because higher resolution data contain more small leaks (e.g., Paschos & Norman 2005; PaperI), and they also are contaminated by observational noise. Moreover, dark gaps are from saturated absorptions, they are featureless and contains generally less information of non-saturated absorption.
In this paper we made a statistical approach to [Ly$\alpha$ ]{}leaks. The purpose is to show that the statistical features of [Ly$\alpha$ ]{}leaks would be effective tool to reveal the origin of [Ly$\alpha$ ]{}leaks, and to probe the evolution of reionization. Similar to [Ly$\alpha$ ]{}absorption lines, [Ly$\alpha$ ]{}leaks have a rich set of statistical properties, such as the width of leak profile. Unlike dark gaps, the properties of [Ly$\alpha$ ]{}leaks can be defined through integrated quantities, which are less contaminated by resolution and noise. Moreover, [Ly$\alpha$ ]{}leaks are from regions of non-saturated absorptions and encode more information of reionization; therefore, the [Ly$\alpha$ ]{}statistics would provide more underlying physics of reionization than all the above-mentioned statistics.
The paper is organized as follows. §2 describes the method to produce [Ly$\alpha$ ]{}absorption samples. §3 presents the statistical properties of Ly$\alpha$ leaks with a uniform ionizing background. §4 analyzes the effect of inhomogeneous ionizing background. Conclusion and discussion are given in §5.
SIMULATION SAMPLES OF HIGH REDSHIFT Ly$\alpha$ ABSORPTION SPECTRUM
==================================================================
Method
------
We simulate Lyman series absorption spectra of QSOs between $z=3.5$ and 6.5 using the same lognormal method as those for low redshifts $z \simeq 2 - 3$ (e.g., Bi et al. 1995; Bi & Davidsen 1997). In this model, the density field $\rho({\bf x})$ of the IGM is given by an exponential mapping of the linear density field $\delta_0 ({\bf
x})$ as $$\rho({\bf x}) = \bar{\rho}_0\exp[\delta_0 ({\bf x}) - \sigma_0^2/2],$$ where $\sigma^2_0=\langle \delta_0 ^2 \rangle $ is the variance of the linear density field on scale of the Jeans length. Obviously, the 1-point PDF of $\rho({\bf x})$ is lognormal. In this model, the velocity field of baryon gas is produced by considering the statistical relation between density and velocity field (Bi & Davidsen, 1997; Choudhury et al. 2001; Veil et al. 2002).
The dynamical bases of the lognormal model have gradually been settled in recent years. First, although the evolution of cosmic baryon fluid is governed by the Naiver-Stokes equation, the dynamics of growth modes of the fluid can be sketched by a stochastic force driven Burgers’ equation (Berera & Fang 1994). On the other hand, the lognormal field is found to be a good approximation of the solution of the Burgers’ equation (Jones 1999). The one-point distribution of the cosmic density and velocity fields on nonlinear regime are consistent with lognormal distribution (e.g. Yang et al. 2001, Pando et al. 2002). Especially, it has been shown recently that the velocity and density fields of the baryon matter of the standard $\Lambda$CDM model is well described by the so-called She-Lévĕque’s universal scaling formula, which is given by a hierarchical process with log-Poisson probability distribution (He et al. 2006, Liu & Fang 2007).
The simulation is performed in the concordance $\Lambda$CDM cosmological model with parameters $\Omega_{m} = 0.27$, $h=0.71$, $\sigma_8 = 0.84$, and $\Omega_{b} = 0.044$. The thermodynamic evolution in the IGM is actually a rather complex process, because the nonlinear clustering leads to a multi-phased IGM. As shown in cosmological hydrodynamic simulations (e.g., He et al. 2004), for a given mass density, the temperature of the IGM could have large scatters with differences up to two orders. Nevertheless, the equation of state in [Ly$\alpha$ ]{}clouds can be well approximated by a polytropic relation with $\gamma =
4/3$ ( Hui & Gnedin; He et al. 2004). The neutral fraction $f_{\rm
HI}$ is obtained by solving the photoionization equilibrium equation. The photoionization rate $\Gamma_{\rm HI}$ will be given in §2.2. We then construct synthetic absorption spectra by convoluting the neutral hydrogen density field with Voigt profiles. For each given $z$, the size of the simulation samples is $\Delta z=0.3$, and there are $2^{14}$ pixels in each simulation box.
Redshift-Dependence of Photoionization Rate
-------------------------------------------
If the distribution of the IGM is uniform and the UV ionizing background is independent of redshift, the mean GP optical depth of [Ly$\alpha$ ]{}absorption should approximately increase with redshift as $(1+z)^{4.5}$. The observations of dark gaps directly show that the GP optical depth undergoes a stronger evolution at $z \simeq 6$, and consequently, the UV ionizing background would decrease rapidly with redshift at $z\simeq 6$. The strong evolution scenario of the UV ionizing background is supported by a number of simulations or semi-analytical models of reionization (e.g., Razoumov et al. 2002; Gnedin 2004; Oh & Furlanetto 2005; Pascho & Norman 2005; Wyithe & Loeb 2005; Kohler et al. 2007; Gallerani et al. 2006). It has been found that an evolution of photoionization rate as follows can fit the strong redshift evolution of the GP optical depth (Paper I) : $$\Gamma_{\rm HI}(z) = \Gamma_0\exp\{-[(1+z)/(1+3.2)]^{2.4} \},$$ which is in units of 10$^{-12}$ s$^{-1}$. Note that the power index 2.4 in equation (2) is little different from the one used in Paper I because we use a different $T_0$ in this paper (also see below).
With eq.(2), we calculate the redshift dependencies of neutral hydrogen fraction $f_{\rm HI}$ and the effective optical depth, $\tau_{eff}\equiv -\ln(\overline{F})$, where $\overline{F}$ is the mean of transmitted flux. The results are plotted in Figure 1. The data points for $\tau_{eff}$ are taken from [Ly$\alpha$ ]{}observations of Fan et al. (2006). For best fitting, the amplitude $\Gamma_0$ is in the range 5-7, which can be considered as the allowed range of $\Gamma_{\rm HI}(z)$. In this paper, we will use $\Gamma_0=6$ as the fiducial photoionization rate. It is interesting to note that $f_{\rm HI}$ approaches to $\simeq 0.1$ at redshift $z\simeq 10$, which is consistent with the electron scattering optical depth given by the data of CMB polarization of WMAP III (Page et al. 2007).
It should be pointed out that the assumption of $T_0=2\times 10^{4}$ K (§2.1) is well reasonable at $z\leq 5$ (e.g. Hui & Gnedin 1997; He et al 2004) and may still be applicable at $z\simeq6$ if the mass averaged neutral fraction of hydrogen is not larger than 10$^{-3}$, and the photon heating rate is small. However, at higher redshift, say $z \geq 6$, the temperature $T_0$ might be redshift-dependent. Yet, no proper information on $T_0(z)$ is available at high redshift, and this leads to uncertainty of the model. Fortunately, in photoionization equilibrium, the neutral fraction $f_{\rm HI}$ depends mainly on a degenerate factor $\Gamma_{\rm
HI}(z)T^{0.75}_0(z)$. Thus, the problem with the uncertainty of $T_0(z)$ can be overcame if we use the combined parameter of $\Gamma_{\rm HI}(z)[T_0(z)/2\times 10^4]^{0.75}$ to fit the data. In the range $z\leq 6$, this parameter actually is $\Gamma_{\rm
HI}(z)$; in the range of $z>6$, it is different from $\Gamma_{\rm
HI}(z)$ by a factor of $[T_0(z)/2\times 10^4]^{0.75}$. Thus, the redshift-evolution of $\Gamma_{\rm HI}(z)$ would be slower than eq.(2) if $T_0(z)$ is less than $2\times 10^4$ K at higher redshifts.
An Example of Ly$\alpha$ Absorption Spectrum at $z=6$
-----------------------------------------------------
As an example of Ly$\alpha$ leaks, we plot a simulated sample of [Ly$\alpha$ ]{}absorption spectrum at $z=6$ with a uniform ionizing background in Figure 2, which shows the transmitted flux $F$, the density $\rho$ of baryon gas, the bulk velocity $v$, and the neutral hydrogen fraction $f_{\rm HI}$. As expected, the mean of transmitted flux is very small, about 0.004, and corresponds to an effective optical depth 5.5. Nevertheless, we see spiky features with the transmitted flux $F$ as large as $0.15$. They are leaks.
At low redshifts $z<5$, the [Ly$\alpha$ ]{}forests in QSO’s spectra have a spectral filling factor significantly less than one and can be decomposed into individual [Ly$\alpha$ ]{}absorption lines. At redshifts $z>5$, it is meaningless to decompose the spectra into individual lines since almost the whole spectra are absorbed completely. We note, however, the transmitted leaks look like emission features upon the dark background, and the absorption spectra can be decomposed into individual “emission lines”, i.e., [Ly$\alpha$ ]{}leaks.
Comparing the top, the second and bottom panels, we see that all the leaks comes from the regions with mass density less than 0.3 of the mean mass density of baryon gas. The neutral fraction for leaks is $f_{\rm HI}\sim 2\times10^{-5}$, which yields a GP optical depth $\sim2.5$ for overdensity 0.25 and a $F\sim0.1$, while the mean neutral fraction of the entire example is about $7\times 10^{-5}$, which is high enough to produce dark gaps. The column density of neutral hydrogen of the leaking features is mainly in the range of 10$^{13}$-10$^{14}$ cm$^{-2}$, which are the non-saturated absorption regions, and therefore, leaks can come from regions where no enough neutral hydrogen to produce complete absorptions.
Similar to very high density clouds, $\rho/\bar{\rho}\gg 1$, the regions with very low density $\rho/\bar{\rho}\ll 1$ are rare events in the cosmic clustering. Therefore, leaks may provide valuable test on models of clustering. For instance, the lognormal distributions are long tailed in both high and low density sides, and it contains more high density events as well as low density events than Gaussian model. It should be emphasized that, once the photoionization rate $\Gamma_{\rm HI}$ is determined from the GP optical depth, the statistical property of the sample shown in Figure 2 doesn’t contain free parameters. These samples have been successfully used to explain the following observations: 1.) the large dispersion of the GP optical depth; 2.)the PDF of the flux, and 3.) the evolution of the size of dark gaps (Paper I). Now we use them to study the statistical properties of Ly$\alpha$ leaks.
STATISTICAL PROPERTIES OF Ly$\alpha$ LEAKS
==========================================
The [Ly$\alpha$ ]{}absorption lines at low redshifts are described by well defined quantities, such as the equivalent width, FWHM (full width half maximum), and the Voigt profile, all of which are easily related to physical interpretations. Since the transmitted leaks look like emission features, one can decompose the spectra into individual “emission lines” and describes it by quantities similar to the [Ly$\alpha$ ]{}absorption lines. In this section we study the statistical distribution of [Ly$\alpha$ ]{}leaks in the model with a uniform ionizing background. The effects of inhomogeneous ionizing background will be discussed in next section.
Profile of Leaks
----------------
The center of a leak is identified as the maximum of transmitted flux, and the boundaries are two nearby positions around the center where the flux falls down to zero or to noise level (we take it to be $F = 0.001$ in this paper). The properties of leaks can be measured by two quantities: an equivalent width $W$, which is the total area under the profile of a leak, and a half width $W_{\rm
H}$, which is defined as the width around center within which the area under the profile of the leak is equal to half of the total area of the leak.
The equivalent width $W$ and the half width $W_{\rm H}$ are two independent measurements of [Ly$\alpha$ ]{}leaks. This point can be seen in Figure 3, which gives the value of $W$ and $W_{\rm H}$ for each leak at $z=6$. It does not indicate a significant correlation between $W$ and $W_{\rm H}$, especially in the region of $W>0.1$ [Å ]{}and $W_{\rm
H}>0.1$ Å. Figure 3 also shows that the distribution of $W_{\rm
H}$ has a lower limit $0.2$ Å, which is due to the Jeans scale used for smoothing the sample. On the other hand, the equivalent width $W$ distributes in the range from $0.001$ [Å ]{}to $5$ Å. Obviously, one is unable to introduce two independent quantities for characterizing dark gaps.
The profiles of 100 randomly sampled leaks with $W$ = 0.01, 0.1, and 1 Å are displayed in Fig.\[profile\]. The tails of the profiles in the three panels of Fig.\[profile\] look like the Lorentz profile, and of course, they do not have the meaning of the natural width of an absorption line. The profiles of leaks for a given equivalent width $W$ have very large dispersions; for example, for $W$=0.1Å, the flux covers a range from $F=0.03$ to 0.15, and $W_{\rm H}$ can be 0.4 to 4 Å. As the leaks are formed out of the two neighboring complete absorption troughs, which depend on inhomogeneities of density, velocity, and temperature fields, the large scatter of the profiles is expected.
Figure 4 also shows that some leaks may have multiple local maximums above the noise level. For clarity and easy-operating, we treat them as one leak. The current observational resolution is of the order of 10km/s, which corresponds to $\Delta\lambda \sim 0.28$[Å ]{}in observer’s frame, or about 13 pixels of simulation. Therefore, the observed [Ly$\alpha$ ]{}leaks would have a resolvable width.
Number Density Distributions of Ly$\alpha$ Leaks
------------------------------------------------
Similar to Ly$\alpha$ forests, we can define the cumulative number densities $n(>W,z)$ and $n(>W_{\rm H},z)$ of leaks as the number of leaks with widths larger than a given $W$ and $W_{\rm H}$ at $z$ per unit $z$, respectively. The differential number densities are $n(W,z)=dn(>W,z)/dW$ and $n(W_{\rm H},z)=dn(>W_{\rm H},z)/dW_{\rm
H}$. It should be pointed out that statistics of $W$ and $W_H$ are not the same as the largest peak width statistics proposed by Gallerani et al. (2007), which considered only the largest peak width. A peak may contains more than one leaks, i.e. leak statistics describe the details of the leaking area. The mean transmitted flux at $z$ within $dz$ is $\bar{F}=\int_0^\infty n(W,z)WdW$. We calculate the number densities of leaks in redshift range $z=5-6.5$. In each redshift region we produce 100 light-of-sight samples to calculate the density functions. The results are shown in Figure 5. The errors are estimated by Jackknife method, i.e., the variance over 5 subsamples, each of which contains 20 light-of-sight samples.
The number density $n(W,z)$ shown in the up-left panel of Figure 5 are similar to a Schechter function: they follow a power law at small $W$ and have a cut-off at large $W$. The distribution of $n(\WH,z)$ at $\WH < 0.5$[Å ]{}declines with decreasing $\WH$, this is because of the Jeans length smoothing.
The slope of $n(W,z)$ and $n(\WH,z)$ are smaller for small redshift. It means the lack of low density voids with small size. That is, the increase of voids of small size is less than voids of large size. This trend can also be seen from the flattening of the cumulative density distributions $n(>\WH,z)$ and $n(>W,z)$.
The redshift-evolution of the number densities $n(>W,z)$ and $n(>W_H,z)$ of leaks for $W$, $W_{\rm H}$ =0.5, 1.0, and 3.0 [Å ]{}are shown in Figure \[zdis\]. As has been seen in Figure \[ldis\], the number densities dramatically decrease at higher redshifts. The evolution is more rapidly for large leaks: the number density $n(>W=0.5$Å,$\ z)$ drops by a factor of $\sim$ 5 when redshift increasing from 5 to 5.8, while $n(>W=3$Å,$ \ z)$ drops by a factor of $\sim$ 60. The evolution trend is the same for the number density $n(\WH,z)$. From the error bars of Figures \[ldis\] and \[zdis\] we see that the predicted features of leaks would be able used to compare with observed data set containing 20 or more light-of-sight samples with the similar quality as simulation.
Effects of Resolution and Noise
--------------------------------
In this section, we study the observational and instrumental effects on the statistics of leaks. Since both $W$ and $W_{\rm H}$ are defined through the area under the profile of leaks, the effects of resolution and noise would be small. To simulate the observational effects, we bin the original data to a coarse grid corresponding to a resolution of 20000, and we add Gaussian noises with signal-to-noise ratio S/N=3 on binned pixels. The number densities of leaks for the noisy binned samples are shown in Figures \[noise\].
The effect of binning and noise is very small for $W$: the original plot of $W$ actually is the same as the plot of binned $W$, and the distribution of noisy $W$ is affected only when $W<0.01$Å. The effect of binning for $\WH$ is also small on scales larger than the binning length. As expected, the noise effects for $\WH$ are significant for $W_{\rm H}<0.4$ Å. The noise effects for $\WH$ are even smaller if we smooth the noisy sample. This is very different from the PDF of the flux and the size of dark gaps, both of which are heavily contaminated by instrumental resolution and observational noises.
One can compare the uncertainty of $\Gamma_0$ with the effect of noises. Figures \[gamma\] shows the number densities for the UV background amplitude $\Gamma_0$ = 5, 6, and 7 (eq.2), which represent the allowed range of $\Gamma_0$. Different from noises, the difference of $\Gamma_0$ will cause uncertainty in the whole ranges of $W$ and $W_{\rm H}$. The uncertainties of number densities are within a factor of 2 when the amplitude $\Gamma_0$ changes from 7 to 5. These uncertainties essentially are from the mass density perturbations with long wavelengths (Paper I). The error bars from the scattering of light-of-sight samples is also shown in Figure \[gamma\]. Therefore, the scattering of $\geq 20$ light-of-sight samples actually is less than the uncertainty of $\Gamma_0$.
[Ly$\alpha$ ]{}LEAKS OF INHOMOGENEOUS IONIZING BACKGROUND
=========================================================
In the early stage of reionization, ionizing photons are mainly in ionized patches around UV ionizing sources, and therefore, the spatial distribution of ionizing background is highly inhomogeneous and has patchy structures. When the ionized patches spread over the whole space, the ionizing background become uniform or quasi-uniform, and so the ionizing background underwent an inhomogeneous-to-uniform transition during the reionization. In this section we study the effects of inhomogeneous ionizing background on [Ly$\alpha$ ]{}leak statistics.
Models of Inhomogeneous Ionizing Background
-------------------------------------------
The first question is how to model the inhomogeneous spatial distribution of the ionizing photon field. To our problem, the most important property is the relation between the fields of mass density and ionizing photon background or photoionization rate. If the spatial fluctuation of photoionization rate $\Gxz$ is statistically uncorrelated with the density field $\rho({\bf x},z)$, the reionization of IGM will statistically be the same as a uniform ionizing background, regardless of the details of the fluctuation of $\Gxz$. In this case, the only effect of the fluctuations would be to yield a larger variance in relevant statistics. However, as shown in last sections, the uncertainty of the leak statistics is already large even when the ionizing background is uniform, and therefore, one would not be able to distinguish the fluctuating photon field from inhomogeneous density field if they are uncorrelated.
Although many simulations on the UV photon field at the epoch of reionization have been done, there still lack the results of the correlation between photon and density fields. In this context, we will consider the following four models on the statistical relation between the inhomogeneous fields of photon and density, which are mainly based on physical consideration of different stage of reionization.
[*Model 0*]{}. The photoionization rate is spatially uniform. It corresponds to the post-overlapping stage of reionization. This model has been used in last two sections.
[*Model I*]{}. The photoionization rate at a give redshift is assumed to be proportional to the density field of IGM, $\Gamma_{\rm
HI}=\Gamma_{\rm I} \rho $, $\Gamma_{\rm I}$ being a constant. This model is motivated by the so-called inside-out scenario: high density regions around UV sources are ionized first, and is most probable at the early stage of reionization.
[*Model II*]{}. Just opposite to model I, the photoionization rate at a give redshift is assumed to be inversely proportional to the density field of IGM, $\Gamma_{\rm HI}=\Gamma_{\rm II} \rho^{-1}$, $\Gamma_{\rm II}$ being a constant. This model comes from the so-called outside-in scenario: under-dense regions are ionized first (Miralda-Escude et al. 2000), which is applicable at the late stage of reionization (Furlanetto & Oh 2005).
In order to fit the observed effective optical depth $\tau_{eff}$ at redshift $z=6$, we take the following parameters: $\Gamma_{\rm
I}=\Gamma_0\times3.53$, $\Gamma_{\rm II}=\Gamma_0/2.77$ where $\Gamma_0$ is the photoionization rate at $z=6$ for uniform ionizing background \[eq.(2)\].
[*Model III or patch model*]{}. This model corresponds to the stage before the overlapping of ionized patches. The IGM is almost fully neutral except for some isolated ionizing patches. We model the ionized patch as a Stromgren sphere: the neutral hydrogen fraction is equal to 0 within the patch and equal to 1 outside the patch. The mean radius of the Stromgren sphere is assumed to be $R_s\simeq 1.8$ comoving Mpc, which corresponding to an UV photon source with luminosity $L= 5\times 10^{43}$ erg s$^{-1}$ with active time $10^7$ year and a $\nu^{-3}$ spectrum. To fit the observed optical depth, we found there should be 2 patches every simulated box ($\Delta
z=0.3$). This number is actually consistent with the following estimation $$\frac{dN}{dz}= \frac{\pi R_s^2\phi(L)c}{H(z)},$$ where $\phi(L)$ the comoving luminosity function, i.e. the 3-D number density of sources with luminosity $L$. Using $\phi(5\times 10^{43})\sim 1.5
\times 10^{-3}$ Mpc$^{-3}$ (Bouwens et al. 2006), we have $dN/dz\simeq 7$ and $(dN/dz)\Delta z\simeq 2$.
[Ly$\alpha$ ]{}Leaks of Inhomogeneous Ionizing Background
---------------------------------------------------------
For the four models described in the last section, we recalculate the transmitted flux with the same underlying density and velocity fields of fig2. The results are shown in Figure 9. As mentioned above, the mean of transmitted flux for all models takes approximately the same value of 0.004, which corresponds to an effective optical depth $\simeq 5.5$. However, Figure 9 shows clearly different behaviors of [Ly$\alpha$ ]{}leaks in various models.
For models I and II, the leaks appear exactly at the same positions as model 0 except some small leaks, which are more prominent for model I. In other words, all the leaks of models 0, I and II are from lowest density voids. Therefore, the three models have the same distribution of dark gaps, and it is impossible to discriminate among these models with the dark gap statistics.
However, the profiles of the leaks of model 0, I and II are statistically different from each other. For model I, the leaks generally have larger width and lower height than model 0, while for model II, the width of leaks generally is narrower than model 0, and the height of leaks is larger than model 0. The reason is straightforward. Comparing with model 0, model I gives a higher $f_{\rm HI}$ at low density and lower $f_{\rm HI}$ at high density. This leads to lower amplitude and broader width. For model II, the effect is just opposite to model I and yields higher amplitude and narrower width.
In the patch model, all the leaks come from ionized patches, within which ionizing sources are enclosed, and so the internal information of ionized patches can be inferred from the leak statistics. The size of dark gaps is actually given by the distance between ionized patches. Generally, [Ly$\alpha$ ]{}leaks in the patch model have a maximum flux $\simeq 0.6$, which is higher than the maximum flux of other models, 0.2. This is because the neutral fraction within the Stromgren sphere is much less that than other models. This behavior is similar to the so-called proximity effect of QSOs at low redshift (e.g., Rauch 1998). Thus, the statistics of the maximum flux of [Ly$\alpha$ ]{}leaks may be used to reveal the patchy origin of [Ly$\alpha$ ]{}leaks.
As we sample the size of patches along the sight of light according to the impact probability, the size of patches is smaller than $R_s$. If the size is too small, the ionized patch will be opaque to [Ly$\alpha$ ]{}photons due to the damping wing of the surrounding neutral hydrogen absorption (Miralda-Escude 1998). Obviously, it explains why only one leak is apparent in the patch model as displayed in Figure 9.
Number Densities of [Ly$\alpha$ ]{}Leaks of Inhomogeneous Ionizing Background
-----------------------------------------------------------------------------
We now calculate the number densities $n(W, z)$, $n(>W,z)$, $n(W_{\rm H},z)$ and $n(>W_{\rm H}, z)$ for three inhomogeneous ionizing background models at $z=6$, and the results are shown in Figures 10. We can see from Figure 10 that the effects of different models on the number densities of $W$ and $\WH$ are different. That is, although all the models of 0, I, II and III give the same mean effective optical depth, their leak distributions are different. For clarifying, in Figures 10 we show only error bars for the curves of model 0. It would be enough to show that these results can be tested with data set containing about 20 light-of-sight data.
First, for model I, both of the number densities $n(W,z)$ and $n(>W,z)$ have only small deviations from model 0. However, $n(W_{\rm H},z)$ are significantly different from their counterparts of model 0. The number density of leaks with $W_{\rm H}>1$[Å ]{}is much more than model 0. The basic feature of model I is to increase the number of leaks with large $W_{\rm H}$ as shown in Figure 9.
Second, for model II, we see once again that the number densities $n(W,z)$ and $n(>W,z)$ have only small deviations from model 0. Yet, the number densities $n(\WH,z)$ and $n(>W_{\rm H},z)$ of the model II are systematically lower than model 0. Therefore, the basic feature of model II is to keep the total area under the profile of leaks almost unchanged, but the widths of leaks are significantly narrowed.
Finally, the behavior of patch model is very different from models 0, I and II. As expected, the patch model yields more leaks with large $W$, and $n(W,z)$ shows a bump around $W=3-5$ Å, which characterize the area of the ionized patches. On the other hand, the number densities of $n(\WH,z)$ and $n(>W_{\rm H},z)$ are lower than model 0 because the characteristic size of ionized patches is less than that of low density voids. It is interesting to note that for the patch model, the tails of $n(W,z)$, $n(>W,z)$ and $n(W_{\rm H},
z)$, $n(>W_{\rm H}, z)$ are quite different from each other. Generally, the tails of $n(W,z)$ and $n(>W,z)$ can extend to as large as $W\simeq 5$Å, while for $n(W_{\rm H},z)$ and $n(>W_{\rm H},z)$, there are no tails higher than 3 Å. It results partially from the damping wing effect (Miralda-Escude 1998).
In summary, the statistical properties of leaks with respective to $W$ and $\WH$ are sensitive to the details of ionizing photon field. Combining the distribution of $W$ and $\WH$, the [Ly$\alpha$ ]{}leaks would be able to probe the origin of themselves, and thus reveal the ionization state of IGM, the inhomogeneity of ionizing background, and the evolution stage of reionization.
It should be pointed out that we considered only the patches of the HI regions around high redshift galax ies. The HI regions around quasars or the proximity effect would also be the patches of leaking. The mean luminosity of quasars probably is higher than galaxies, and therefore, the above-mentioned feature of $n(\WH,z)$ and $n(>W_{\rm H},z)$ would be more prominent for the patches of quasars. The damped Ly$\alpha$ absorption system is important for modeling low redshift Ly$\alpha$ absorption. Since these systems have high column density of neutral hydrogen, it will not contribute to leaking either in density void models or patch model.
DISCUSSION AND CONCLUSION
=========================
We show that the Ly$\alpha$ absorption spectra of UV photon emitters at redshifts around $z\sim6$ can be decomposed into [Ly$\alpha$ ]{}leaks, which come from either lowest density voids or ionized patches containing ionizing sources. The [Ly$\alpha$ ]{}leaks are well defined and described by the equivalent width $W$ and the width of half area $\WH$. Since both $W$ and $\WH$ are defined through integrated quantities, the distributions of [Ly$\alpha$ ]{}leaks in terms of $W$ and $\WH$ are stable with respect to observational noises and instrumental resolution. Although the number densities $n(W,z)$, $n(>W,z)$, $n(W_{\rm H},z)$, and $n(>W_{\rm H},z)$ evolve very rapidly at redshift $z\simeq 6$, these statistics are measurable up to $z=6.5$.
If the [Ly$\alpha$ ]{}leaks come from lowest density voids, the IGM should be still highly ionized and the ionizing background is almost uniform; in contrast, if the leaks come from isolated ionized patches, the ionizing background should be inhomogeneous, and the reionization is still in the overlapping stage. Therefore, the origin of [Ly$\alpha$ ]{}leaks is crucial to understand the history of reionization.
We show that the statistics with $W$ and $\WH$ are sensitive to the origin of [Ly$\alpha$ ]{}leaks, because the [Ly$\alpha$ ]{}leaks are sensitive to the correlation between photon and density fields. Based on physical consideration of reionization, we studied four phenomenological models of the photon field: the uniform ionizing background (model 0), the photoionization rate $\GHI$ is proportional to the density $\rho$ (model I), $\GHI$ and $\rho$ are anti-correlated (mode II), and ionized regions only given by Stromgren sphere around ionizing photo sources (patch model). We found that, although all the four models can fit the observed mean and variance of optical depth at $z\simeq 6$, the width distribution of [Ly$\alpha$ ]{}leaks show very different behaviors.
For model 0, I, and II, most of [Ly$\alpha$ ]{}leaks are from lowest density voids, and the distribution of dark gaps are similar. However, the properties of individual [Ly$\alpha$ ]{}leaks are different. Model I gives broader width $\WH$ than model 0; model II gives narrower width $\WH$ than model 0. For patch model, the [Ly$\alpha$ ]{}leaks are mainly from ionized sphere, and they generally have higher maximum of transmitted flux than other models. There is a bump in the distribution of the equivalent width $W$, which characterize the intensity of UV photons within ionized patches. [Ly$\alpha$ ]{}leaks from ionized patches will provide the information of ionized patches, such as their intensity and size, and constrain the properties of UV sources that contribute most of ionizing photons of the reionization.
Finally, we point out that similar analysis is also applicable to the reionization of HeII. The optical depth of HeII [Ly$\alpha$ ]{}evolves rapidly at $z\simeq2-3$. It reaches $\simeq 5$ at $z\simeq 3$. That is, the evolution of HeII reionization at $z\simeq 3$ would be similar to that of H at $z\simeq 6$. Observed sample of HeII [Ly$\alpha$ ]{}absorption indeed show structures similar to the leaks of hydrogen absorption spectrum (Zheng et al. 2004). One can also draw the information of background photons at the energy band of HeII ionization.
ACKNOWLEDGEMENT {#acknowledgement .unnumbered}
================
This work is supported in part by the US NSF under the grant AST-0507340. LLF acknowledges support from the National Science Foundation of China (NSFC).
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[^1]: E-mail: fengll@pmo.ac.cn
|
---
abstract: 'We establish a relation between the trace evaluation in ${\rm SO}(3)$ topological quantum field theory and evaluations of a topological Tutte polynomial. As an application, a generalization of the Tutte golden identity is proved for graphs on the torus.'
address:
- 'Paul Fendley All Souls College and Rudolf Peierls Centre for Theoretical Physics, Parks Rd, Oxford OX1 3PU, UK'
- 'Vyacheslav Krushkal Department of Mathematics, University of Virginia, Charlottesville, VA 22904-4137 USA'
author:
- Paul Fendley and Vyacheslav Krushkal
title: |
Topological quantum field theory and\
polynomial identities for graphs on the torus
---
Introduction
============
The Witten-Reshetikhin-Turaev topological quantum field theory (TQFT) associates invariants to ribbon graphs in $3$-manifolds. A part of this theory is an invariant of graphs on surfaces: given a graph $G\subset \Sigma$, the [*trace evaluation*]{} is the invariant associated to the embedding $G\subset {\Sigma}\times \{*\}\subset {\Sigma}\times S^1$. We study the trace evaluation for ${\rm SO}(3)$ TQFTs.
For [*planar*]{} graphs, the ${\rm SO}(3)$ quantum evaluation is known to equal the flow polynomial $F^{}_G$, or equivalently the chromatic polynomial ${\chi}^{}_{G^*}$ of the dual graph. In [@FK] we showed that this quantum-topological approach gives a conceptual framework for analyzing the relations satisfied by the chromatic and flow polynomials of planar graphs. In particular, we gave a proof in this setting of the Tutte golden identity [@T2]: given a planar triangulation $T$, $$\label{golden identity eq}
{\chi}^{}_T({\phi}+2)=({\phi}+2)\; {\phi}^{3\,V(T)-10}\,
({\chi}^{}_T({\phi}+1))^2,$$ where $V(T)$ is the number of vertices of the triangulation, and ${\phi}$ denotes the golden ratio, ${\phi}=\frac{1+\sqrt5}{2}$. The dual formulation in terms of the flow polynomial states that for a planar cubic graph $G$, $F^{}_G({\phi}+2) \, = \, {\phi}^E\, (F^{}_G({\phi}+1))^2.$ In [@FK] we also showed that the golden identity may be thought of as a consequence of level-rank duality between the ${\rm SO}(3)_4$ and the ${\rm SO}(4)_3$ TQFTs, and the isomorphism ${\mathfrak{so}}(4)\cong {\mathfrak{so}}(3)\times {\mathfrak{so}}(3)$. (Consequences of ${\rm SO}$ level-rank duality for link polynomials have also been studied in [@MPS].)
The main purpose of this paper is to formulate an extension of the results on TQFT and polynomial invariants to graphs on the torus; in particular we prove a generalization of the golden identity. Some of the motivation for this work has its origins in lattice models in statistical mechanics, where it has long been known (see e.g. [@DSZ], [@Pasquier]) that when deriving identities for partition functions [on the torus]{}, one must typically sum over “twisted” sectors. Twisted sectors are more complicated analogs of the spin structures (cf. [@CR]) familiar in field theories involving fermions. Such sectors are described naturally in TQFT, as for example can be seen in the study of lattice topological defects [@AMF]. We show how the golden identity is generalised to the torus precisely by considering such sums over analogous sectors. In a forthcoming paper [@FK3] we will elaborate further on the connections to statistical mechanics, in particular on the relation with the Pasquier height model [@Pasquier].
The chromatic and flow polynomials are $1$-variable specializations of the Tutte polynomial, known in statistical mechanics as the partition function of the Potts model. Relations between the ${\rm SO}(3)$ quantum evaluation of planar graphs, the chromatic and flow polynomial, and the Potts model are discussed in detail in [@FK2]. From the TQFT perspective, the case of graphs embedded in the plane (or equivalently in the $2$-sphere $S^2$) is very special in that the TQFT vector space associated to $S^2$ is ${\mathbb C}$. For surfaces $\Sigma$ of higher genus they are vector spaces (of dimension given by the Verlinde formula) which are part of the rich structure given by the $(2+1)$-dimensional TQFT. Multi-curves, and more generally graphs embedded in $\Sigma$, act as “curve operators” on the TQFT vector space, and our goal is to analyze the trace of these operators.
For the ${\rm SO}(3)$ TQFT, this invariant of graphs on surfaces satisfies the contraction-deletion rule, familiar from the study of the Tutte polynomial and the Potts model. A non-trivial feature on surfaces of higher genus is that the “loop value” depends on whether the loop is trivial (bounds a disk) in the surface, or whether it wraps non-trivially around the surface; in fact in the latter case the invariant is not multiplicative under adding/removing a loop. This behavior is familiar in generalizations of the Tutte polynomial which encode the topological information of the graph embedding in a surface. The study of such “topological” graph polynomials was pioneered by Bollobás-Riordan [@BR]; a more general version was introduced in [@Kr]. To express the ${\rm SO}(3)$ trace evaluation for graphs on the torus we need a further extension of the polynomial, defined in section \[graph poly section\]. In Theorem \[evaluations are equal\] we show that the quantum invariant equals a sum of values of the polynomial, where the sum is parametrized by labels corresponding to the TQFT level. Here individual summands (evaluations of the topological Tutte polynomial) correspond to TQFT sectors.
The identity (\[golden identity eq\]) for the chromatic polynomial and its analogue for the flow polynomial in general do [*not*]{} hold for non-planar graphs. In fact, it is conjectured [@AK] that the golden identity for the flow polynomial characterizes planarity of cubic graphs. Our result (Theorem \[golden torus theorem\]) involves invariants of graphs derived from TQFT, or equivalently evaluations of graph polynomials, and it recovers the original Tutte’s identity for planar graphs.
A different “quantum” version of the golden identity, for the Yamada polynomial of ribbon cubic graphs in ${\mathbb R}^3$, was established in [@AK2]. It is an extension of the Tutte identity from the flow polynomial of planar graphs to the Yamada polynomial of spatial graphs, which may be thought of as elements of the skein module of $S^2$ (isomorphic to ${\mathbb C}$). Our result manifestly involves graphs on the torus and the elements they represent in the associated TQFT vector spaces. We expect that an extension of our results holds on surfaces of genus $>1$ as well, although computational details on surfaces of higher genus are substantially more involved.
To date, the study of topological Tutte polynomials of graphs on surfaces has been carried out primarily in the context of topological combinatorics. While there are applications to quantum invariants of links (cf. [@DFKLS; @MM]) and to noncommutative quantum field theory [@KRTW], to our knowledge the results of this paper provide the first direct relation between TQFT and evaluations of graph polynomials on surfaces.
After briefly reviewing background information on TQFT in section \[TQFT section\], we define the relevant topological Tutte polynomial for graphs on the torus in section \[graph poly section\]. The relation between TQFT trace and evaluation of the graph polynomial is stated and proved in section \[relation section\]. The generalization of the golden identity for graphs on the torus is established in section \[golden torus section\].
Background on TQFT {#TQFT section}
==================
We refer the reader to [@FK] for a summary of the relevant facts about the Temperley-Lieb algebras ${\rm TL}_n$ and the Jones-Wenzl projectors $p_n$ [@Jo; @We], and to [@KL] for a more general introduction to the calculus of quantum spin networks.
To fix the notation, recall the definition of quantum integers $[n]$, and the evaluation of the $n$-colored unknot, ${\Delta}_n$: $$[n]=\frac{A^{2n}-A^{-2n}}{A^2-A^{-2}}\ , \qquad {\Delta}_n=[n+1]\ .$$
We will interchangeably use the parameters $q, A$, as well as the loop value $d$ and a graph parameter $Q$, related as follows: $$\begin{aligned}
\label{notation1}
q=A^4, \quad d=A^2+A^{-2}, \quad Q=d^2\ .\end{aligned}$$
A basic calculation for the colored Hopf link gives $$\label{LinkingLoop fig}
\vcenter{\hbox{\includegraphics[height=2cm]{LinkingLoop}
\put(-131,25){$m$}
\put(-96,50){$n$}
{ \put(-78,22){$= \, \frac{[(m+1)(n+1)]}{[n+1]}$} }
\put(2,50){$n$}
}}$$
We use the construction of ${\rm SU}(2)$ Witten-Reshetikhin-Turaev TQFTs, given in [@BHMV]. Given a closed orientable surface $\Sigma$, the ${\rm SU}(2)$, level $r-2$ TQFT vector space will be denoted $V_r({\Sigma})$, where $A=e^{2{\pi} i/4r}$. (Note that in the TQFT literature the notation $V_{2r}$ is sometimes used instead. Also note that in the physics literature, labels are often divided by 2 and called “spin”, so that odd and even labels correspond to half-integer and integer spins respectively.)
The main focus of this paper is on the torus case, ${\Sigma}={\mathbb T}$, and in this case the notation $V_r:=V_r({\mathbb T})$ will be used throughout the paper. Consider ${\mathbb T}$ as the boundary of the solid torus $H$. $V_r$ has a basis $\{ e_0, \ldots, e_{r-2}\}$, where $e_j$ corresponds to the core curve of $H$, labeled by the $j$-th projector $p_j$. This basis will be used in the evaluation of the trace in Section \[trace subsection\].
The discussion in the rest of this section applies to surfaces $\Sigma$ of any genus. A curve $\gamma$ in $\Sigma$ acts as a linear operator on $V_r({\Sigma})$, so associated to $\gamma$ is an element of $V^*_r({\Sigma})\otimes V_r({\Sigma})$. Given a graph $G\subset \Sigma$, we consider it as an ${\rm SU}(2)$ quantum spin network in $\Sigma$ by turning each edge into a “double line”. Namely, as in [@FK] we label edges by the second Jones-Wenzl projector, up to an overall normalization. Concretely, each edge $e$ of $G$ is replaced with a linear combination of curves as indicated in (\[phi\]), and the curves are connected without crossings on the surface near each vertex. $$\label{phi}
\vcenter{\hbox{
\includegraphics[width=12cm]{phi}
\put(-350,5){$e$}
\put(-322,29){${\Phi}$}
\put(-277,21){$=$}
\put(-244,21){$-\frac{1}{d}$}
\put(-85,29){${\Phi}$}
\put(-62,21){$d\;\cdot$}
}}$$ In this map, a factor $d^{(k-2)/2}$ is associated to each $k$-valent vertex, so that for example the $4$-valent vertex in (\[phi\]) is multiplied by $d$. The overall factor for a graph $G$ is the product of the factors $d^{(k(V)-2)/2}$ over all vertices $V$ of $G$. Therefore the total exponent equals half the sum of valencies over all vertices, minus the number of vertices, i.e. minus the Euler characteristic of $G$. (This count does not involve faces - so this is the Euler characteristic of the graph $G$, and not of the underlying surface $\Sigma$.) Using this map $\Phi$, the graph is mapped to a linear combination of multicurves in the surface $\Sigma$. We thus may consider graphs $G$ on the torus as elements ${\Phi}(G)\in{\rm Hom}(V_r, V_r)$.
Given a graph $G\subset{\mathbb T}$, consider the following local relations (1)–(3), illustrated in figures \[fig:rel1\], \[fig:rel23\].
![Relation (1)[]{data-label="fig:rel1"}](ChromaticRelation11 "fig:"){height="2cm"} (-175,28)[$=$]{} (-80,28)[$-$]{} [ (-212,26)[$e$]{} (-245,54)[$G$]{} (-152,54)[$G/e$]{} (-7,54)[$G\backslash e$]{}]{}
\(1) If $e$ is an edge of a graph $G$ which is not a loop, then $G=G/e-G\backslash e$, as illustrated in figure \[fig:rel1\].
\(2) If $G$ contains an edge $e$ which is a loop, then $G=(Q-1)\;
G\backslash e$, as in figure \[fig:rel23\]. (In particular, this relation applies if $e$ is a loop not connected to the rest of the graph.)
\(3) If $G$ contains a $1$-valent vertex as in figure \[fig:rel23\], then $G=0$.
![Relations (2), (3)[]{data-label="fig:rel23"}](ChromaticRelation21 "fig:"){height="2cm"} (3,25)[$=\; 0.$]{} (-218,25)[$=\; (Q-1)\;\cdot$]{} (-100,15)[$,$]{} (-275,33)[$e$]{} (-39,24)[$e$]{}
Replacing the edge labeled $e$ in each figure with the linear combination of curves, defined by the second JW projector, one checks that the relations (1)–(3) hold for ${\Phi}(G)$ in ${\rm Hom}(V_r, V_r)$, where the parameters $A, Q$ are related as in (\[notation1\]), and the value of $A$ in the definition of $V_r$ is $e^{2{\pi} i/4r}$.
[**Remark.**]{} In the planar case, the map gives a homomorphism from the [*chromatic algebra*]{} ${\mathcal
C}^{Q}_n$, defined in [@FK], to the Temperley-Lieb algebra $TL^d_{2n}$, with the parameters related by $Q=d^2$. This map is used to show [@FK Lemma 2.5] that for planar graphs $G$, up to a normalization the quantum evaluation equals the flow polynomial of $G$ or equivalently the chromatic polynomial of the dual graph $G^*$. Indeed, the relations (1)–(3) are sufficient for evaluating any graph [*in the plane*]{}, and these relations are precisely the defining relation for the chromatic polynomial (of the dual graph). On the torus, or any other surface of higher genus, the “evaluation” is not an element of ${\mathbb C}$ but rather an element of the higher-dimensional TQFT vector space ${\rm Hom}(V_r, V_r)$.
A crucial feature underlying the construction of TQFTs is that for each $r$, [*in addition to*]{} (1)–(3) there is another local relation corresponding to the vanishing of the corresponding Jones-Wenzl projector. For example, consider the case $r=5$, important in the proof of the golden identity below. In this case, graphs G$\subset {\Sigma}$, considered as elements of the vector space ${\rm Hom}(V_{5}({\Sigma}), V_{5}({\Sigma}))$, satisfy the local relation in (\[fig:graphp4\]). $$\label{fig:graphp4}
\vcenter{\hbox{
\includegraphics[height=1.6cm]{graphp4}
{\small
\put(-193,20){$\phi$}
\put(-130,20){$=$}
\put(-60,20){$+$}}
{\tiny
\put(-142,2){$G^{}_X$}
\put(-72,2){$G^{}_I$}
\put(-5,2){$G^{}_E$}
}
}}$$ This relation (discovered in the setting of the chromatic polynomial of planar graphs by Tutte [@T1]) corresponds to the $4$th JW projector, see [@FK Section 2] for more details.
[**Remark.**]{} In fact, the Turaev-Viro ${\rm SU}(2)$ TQFT associated to a surface $\Sigma$, isomorphic to ${\rm Hom}(V_{r}({\Sigma}), V_{r}({\Sigma}))$, can be defined as the vector space spanned by multi-curves on $\Sigma$, modulo the local relations given by “$d$-isotopy” and the vanishing of the JW projector. (See Theorem 3.14 and section 7.2 in [@FNWW].) The ${\rm SO}(3)$ theory may be built by considering the “even labels” subspace spanned by graphs modulo relations (1)–(3) above, and the JW projector.
Polynomial invariants of graphs on surfaces {#graph poly section}
===========================================
Let $H$ be a graph embedded in the torus ${\mathbb T}$. Let ${\rm n}(H)$ denote the nullity of $H$, that is the rank of the first homology group $H_1(H; {\mathbb Z})$. The rank $r(H\subset {\mathbb T})$ of the image of the map $i_*{\colon\thinspace}H_1(H; {\mathbb Z})\longrightarrow H_1({\mathbb T}; {\mathbb Z})\cong {\mathbb Z}^2$, induced by the inclusion $H\subset {\mathbb T}$, is either $0$, $1$ or $2$. Consider the following homological invariants: $${\rm s}(H) :=
\begin{cases}
1 & \text{if $r(H\subset {\mathbb T}) =2 $ (in other words, $H_1(H)\longrightarrow H_1({\mathbb T})$ is surjective),} \\
0 & \text{otherwise}.
\end{cases}$$ $${\rm s}^{\perp}(H) :=
\begin{cases}
1 & \text{if $r(H\subset {\mathbb T}) =0 $ (i.e. $H_1(H)\longrightarrow H_1({\mathbb T})$ is the zero map),} \\
0 & \text{otherwise.}
\end{cases}$$ Let ${\rm c}(H)$ be the number of rank $0$ connected components of $H$, that is the number of connected components $H^{(i)}$ such that $r(H^{(i)}\subset {\mathbb T})=0$.
In case $r(H\subset {\mathbb T}) =1$, let $\bar {\rm c}(H)$ be the number of “essential” components of $H$, i.e. the number of components $H^{(i)}$ such that $H_1(H^{(i)})\longrightarrow H_1({\mathbb T})$ is non-trivial. (Note that for each such component, the image of the map on homology is the same rank $1$ subgroup of $H_1({\mathbb T})$.) If $r(H\subset {\mathbb T})$ is $0$ or $2$, $\bar {\rm c}(H)$ is defined to be zero.
The most general polynomial, encoding the homological information of the embedding of a graph $G$ in the torus, is defined by the following state sum: $$\label{general poly definition}
\widetilde{P}^{}_G(X,Y,W,A,B)\, :=\, \sum_{H\subset G} (-1)^{E(G)-E(H)} X^{{\rm c}(H)}\, Y^{{\rm n}(H)}\, W^{\bar {\rm c}(H)}\, A^{{\rm s}(H)}\, B^{{\rm s}^{\perp}(H)},$$ where the summation is taken over all spanning subgraphs of $G$, $E(G)$ denotes the number of edges of $G$, and $(-1)^{E(G)-E(H)}$ provides a convenient normalization. Note that our convention for the sign and the variables differs from the usual convention for the Tutte polynomial. The usual proof (cf. [@Kr Lemma 2.2]) shows that this polynomial satisfies the contraction-deletion relation.
[**Remark**]{}. The polynomials defined in [@BR; @Kr] are specializations of $\widetilde P$ in the case of graphs embedded in the torus.
To establish a relation with the trace evaluation in TQFT, consider the specialization of $\widetilde{P}$ obtained by setting $X=B=1$: $$\label{poly definition}
P^{}_G(Y,W,A)\, :=\, \sum_{H\subset G} (-1)^{E(G)-E(H)} \, Y^{{\rm n}(H)}\, W^{\bar {\rm c}(H)}\, A^{{\rm s}(H)}.$$ This is a generalization of the flow polynomial, including variables $W$ and $A$ which reflect the topological information of how the graph $G$ wraps around the torus. In particular, if $G$ is homologically trivial on the torus (the rank $r(H\subset {\mathbb T})$ is zero), $P^{}_G(Y,W,A)$ recovers the flow polynomial $F^{}_G(Y)$. We thus name $P_G$ the “topological flow polynomial”.
Three key examples are illustrated in figure \[trace11 fig\]. For the graph consisting of $k$ disjoint, trivial loops on the torus in figure \[trace11 fig\]a, the polynomial $P=(Y-1)^k$.
![[]{data-label="trace11 fig"}](trace11 "fig:"){height="2.5cm"} (-290,-14)[$k$]{} (-172,-14)[$k$]{} (-335, -5)[(a)]{} (-215, -5)[(b)]{} (-95,-5)[(c)]{}
For the graph consisting of $k$ non-trivial loops in figure \[trace11 fig\]b, $P=(YW-1)^k$. The calculation is simple; for example the subgraphs $H$ are $$\vcenter{\hbox{\includegraphics[height=2.5cm]{trace12}
\put(-230,30){$+\; 2$}
\put(-108,30){$+$}
}}$$ and so here the polynomial is $$\label{trace12 fig}=\; Y^2W^2-2YW+1\; =\; (YW-1)^2.$$ Finally, the polynomial of the graph in figure \[trace11 fig\]c is computed as follows: $$\vcenter{\hbox{
\includegraphics[height=2.5cm]{trace4}
\put(-321,30){$+$}
\put(-214,30){$+$}
\put(-107,30){$+$}
}}$$ $$\label{poly example}
=\; AY^2- 2YW+1.$$
Duality {#duality section}
-------
The chromatic and flow polynomials ${\chi}, F$ are $1$-variable specializations of the Tutte polynomial, satisfying the relation $F^{}_G(Q)=Q^{-1}{\chi}^{}_{G^*}(Q)$, where $G$ is a planar graph and $G^*$ is its dual. We can extend this duality to the torus by defining the “topological chromatic polynomial’. Namely, we specialize $\widetilde{P}_G$ in to $$\label{chrom poly definition}
{C}^{}_G(X,U,B)\, :=\, \sum_{H\subset G} (-1)^{E(G)-E(H)} X^{{\rm c}(H)-{\rm c}(G)}\, U^{\bar {\rm c}(H)} B^{{\rm s}^{\perp}(H)}.$$ When $r(H\subset {\mathbb T}) =0$, ${C}^{}_G(X,U,B)$ is a renormalized version of the chromatic polynomial ${\chi}^{}_G(X)$. Analogously to the proof of [@Kr Theorem 3.1], one shows that for a cellulation $G\subset {\mathbb T}$ (i.e. when each face of the embedding is a $2$-cell), $$\label{duality eq}
{P}^{}_G(Y,W,A) \, =\, {C}^{}_{G^*}(Y,YW,AY^2),$$ where $G^*\subset {\mathbb T}$ is the dual graph. This topological chromatic polynomial is thus dual to the topological flow polynomial $P_G$ from (\[poly definition\]).
TQFT trace as an evaluation of a graph polynomial {#relation section}
=================================================
TQFT trace {#trace subsection}
----------
Given an odd $r$ and a multi-curve $\gamma\subset {\mathbb T}$, the trace of $\gamma$ is defined as $Z_r({\mathbb T}\times S^1, {\gamma})$, the quantum invariant of the banded link $\gamma$ in the $3$-manifold ${\mathbb T}\times S^1$ [@BHMV 1.2]. Concretely, it may be calculated as the trace of the curve operator in ${\rm Hom}( V_r, V_r)$ with respect to the basis discussed in Section \[TQFT section\]. This basis is given by the core circle of a fixed solid torus $H$, bounded by ${\mathbb T}$, labeled with an integer $0\leq j\leq r-2$. For a graph $G\subset{\mathbb T}$, the trace is defined by mapping $G$ to a linear combination of multicurves using ${\Phi}$ in (\[phi\]) and then computing the trace of ${\Phi}(G)$.
While one could work with the full ${\rm SU}(2)$ TQFT vector space $V_r$, the invariants of graphs obtained by putting the $2$nd Jones-Wenzl projector on the edges, as in (\[phi\]), naturally fit in the context of the ${\rm SO}(3)$ theory. This corresponds to taking the subspace $\overline V_r$ of $V_r$, spanned by even labels. For the remainder of the paper, ${\rm tr}_r(G)$ will be evaluated as the trace of $G$ considered as an operator in ${\rm Hom}(\overline V_r, \overline V_r)$. A basis of $\overline V_r$ is given by the core circle $e_j$ of a solid torus bounded by ${\mathbf T}$, labeled with an [*even*]{} integer $0\leq j\leq r-2$. The result of $G$ applied to a basis element $e_j$ may be computed by pushing $G$ (considered as a quantum spin network with edges labeled by $2$) into the solid torus and re-expressing the result as a linear combination of $\{e_j\}$ using the recoupling theory [@KL]. In the examples below the trace will be computed using the expansion ${\Phi}(G)$ of $G$ in terms of multi-curves; the multi-curves in question will act diagonally on $\overline V_r$ with respect to the basis $\{ e_j, 0\leq j\leq r-2, j \; {\rm even} \}$. We emphasize that while multi-curves (consisting of “spin $1/2$ loops”, or in the usual TQFT terminology loops labelled $1$) are elements of ${\rm SU}(2)$ and not ${\rm SO}(3)$ theory, the configurations of multi-loops considered below preserve the subspave $\overline V_5$, and they provide a convenient evaluation method.
We give several sample calculations of the trace ${\rm tr}^{}_5$, used in the proof below. First consider $k$ trivial loops (labelled $1$) on the torus. The usual $d$-isotopy relation states that removing a loop gives a factor $d={\phi}$, so ${\rm tr}^{}_5$ equals ${\phi}^k$ times the trace of the empty diagram. Since the dimension of the space (spanned by the core of the solid torus with labels $0$ and $2$) is $2$, the result is $2 {\phi}^k$. The trace evaluation of the [*graph*]{} consisting of $k$ trivial loops (figure \[trace11 fig\](a)) equals $2(d^2-1)^k$, which (precisely at this root of unity!) also equals $2 {\phi}^k$.
Next consider non-trivial (spin $1/2$) loops on the torus in (\[trace0 eq\]). Using (\[LinkingLoop fig\]), the action on $\overline V^{}_{5}$ is seen to be diagonal: $$\label{trace0 eq}
\vcenter{\hbox{\includegraphics[height=2.3cm]{trace1}\hspace{4.2cm} \includegraphics[height=2.3cm]{torus}
\put(-255,-14){$k$}
{\scriptsize
\put(-275,48){$j$}
\put(-68,48){$j$}
}
{\large
\put(-202,30){$=\, \left[ \frac{{\rm sin}(2(j+1){\pi}/5)}{{\rm sin}((j+1){\pi}/5)} \right]^k$}
}
}}
,$$ $j=0,2$. Therefore the trace of $k$ non-trivial loops with label $1$ on the torus equals $$\label{trace1 eq}
\vcenter{\hbox{\includegraphics[height=2.3cm, trim=15cm 0 0 0]{trace0}
\put(-47,-14){$k$}
{\large
\put(-103,27){${\rm tr}^{}_{5}$}
\put(3,30){$=\, \left[ \frac{{\rm sin}(2{\pi}/5)}{{\rm sin}({\pi}/5)} \right]^k+\left[ \frac{{\rm sin}(6{\pi}/5)}{{\rm sin}(3{\pi}/5)} \right]^k$}}
\put(145,30){$=\, {\phi}^k+(-{\phi}^{-1})^k.$}
}}$$
The analogous calculation for the [*graph*]{} consisting of $k$ non-trivial loops on the torus (or “spin $1$ loops”) gives $$\label{trace2 eq}
\vcenter{\hbox{\includegraphics[height=2.3cm, trim=15cm 0 0 0]{trace02}
\put(-47,-14){$k$}
{\large
\put(-103,27){${\rm tr}^{}_{5}$}
\put(3,30){$=\, \left[ \frac{{\rm sin}(3{\pi}/5)}{{\rm sin}({\pi}/5)} \right]^k+\left[ \frac{{\rm sin}(9{\pi}/5)}{{\rm sin}(3{\pi}/5)} \right]^k$}}
\put(145,30){$={\phi}^k+(-{\phi}^{-1})^k.$}
}}$$ The answer is again the same as for spin $1/2$ loops precisely at the $5$th root of unity. Note that the ${\rm SO}(3)$ trace is invariant under modular transformations of the torus, so (\[trace2 eq\]) gives the trace of [*any*]{} $k$ non-trivial, spin $1$ loops on the torus. The situation is a bit more subtle with spin $1/2$ loops: the calculations in (\[trace0 eq\]), (\[trace1 eq\]) work specifically for non-trivial loops which bound disks intersecting the core of the solid torus once. A single spin $1/2$ loop which wraps in some other way around the torus and acts as a curve operator $V_5\longrightarrow V_5$, does not have to preserve the subspace $\overline V_5$. Nevertheless, an [*even*]{} number of non-trivial curves preserve $\overline V_5$, and moreover the evaluation (\[trace1 eq\]) for $k$ even is in fact modular invariant: using (\[phi\]), a pair of parallel spin $1/2$ loops may be expressed as a spin $1$ loop plus a scalar multiple of a trivial loop. This property will be used in the following section to evaluate the trace of graphs on the torus in terms of surround loops.
[**Remark.**]{} There are two equivalent ways of computing the trace in (\[trace2 eq\]): one using the formula (\[LinkingLoop fig\]) directly, or alternatively the $2$nd JW projectors can be expanded into linear combinations of spin $1/2$ loops, reducing the calculation to (\[trace1 eq\]).
Finally, the trace of the graph in figure \[trace3 fig\] is obtained by expanding both second projectors, and applying (\[trace1 eq\]).
(-390,114)[${\rm tr}^{}_{5}$]{} (-285,114)[$= \; d\, {\rm tr}^{}_{5}$]{} (-155, 114)[$=$]{} (-435,31)[$\; d{\rm tr}^{}_{5}$]{} (-328,31)[$-\, {\rm tr}^{}_{5}$]{} (-222,31)[$-\, {\rm tr}^{}_{5}$]{} (-116,31)[$+\frac{1}{d}{\rm tr}^{}_{5}$]{} (-320,-30)[$=\, 2{\phi}^2-2({\phi}^2+(\frac{1}{\phi})^2)+2\, =\, 2-\frac{2}{{\phi}^2}$]{}
Trace and graph evaluations
---------------------------
\[poly eval def\] Given a graph $G\subset {\mathbb T}$, consider $$\label{P5 equ}
R^{}_5(G):=P^{}_G({\phi}^2, 1, {\phi}^{-2})+P^{}_G({\phi}^2, {\phi}^{-4}, {\phi}^{-2})$$
The two summands in the definition of $R^{}_5(G)$ are given by evaluations of the polynomial $P^{}_G$ in (\[poly definition\]). In both cases, $Y={\phi}^2$ and $A={\phi}^{-2}$. Note that the first summand corresponds to $YW={\phi}^2$, and the second one to $YW={\phi}^{-2}$.
\[equality at B5\] [*Given any graph $G\subset {\mathbb T}$, the ${\rm SO}(3)$ TQFT trace evaluation ${\rm tr}^{}_{5}(G)$ equals $R^{}_5(G)$.* ]{}
This result is the TQFT version on the torus of the loop evaluation of the flow polynomial of planar graphs at $Q={\phi}+1$, corresponding to $q=e^{2{\pi}i/5}$ (see (\[notation1\])).
[*Proof of Theorem*]{} \[equality at B5\]. We begin the proof by comparing calculations of ${\rm tr}^{}_{5}(G)$ and $R^{}_5(G)$ for the graphs in figure \[trace11 fig\], using results of Sections \[graph poly section\], \[trace subsection\].
\(a) $G$ consists of $k$ trivial loops, figure \[trace11 fig\]a. $${\rm tr}^{}_{5}(G)=2(d^2-1)^k= 2{\phi}^k. \; \; R^{}_5(G)=(Y-1)^k+(Y-1)^k.$$ The factor $2$ in the expression for ${\rm tr}^{}_{5}(G)$ comes from the dimension $2$ of the vector space spanned by the even labels $0, 2$. Since $Y={\phi}^2$, the two expressions coincide.
\(b) $k$ non-trivial loops, figure \[trace11 fig\]b. According to (\[trace2 eq\]), ${\rm tr}^{}_{5}(G)={\phi}^k+(-{\phi}^{-1})^k.$ By (\[trace12 fig\]), $R^{}_5(G)=(YW-1)^k|_{YW={\phi}^2}+(YW-1)^k|_{YW={\phi}^{-2}}.$ Individual terms match: $({\phi}^2-1)^k+({\phi}^{-2}-1)^k={\phi}^k+(-{\phi}^{-1})^k.$
\(c) The graph in figure \[trace11 fig\]c. The TQFT calculation in figure \[trace3 fig\] gives ${\rm tr}^{}_{5}(G)=2{\phi}^2-2({\phi}^2+(\frac{1}{\phi})^2)+2.$ By (\[poly example\]), $P^{}_G(Y, W, A)=AY^2- 2YW+1$. The two evaluations of $P^{}_G$, contributing to $R^{}_5(G)$, give ${\phi}^2-2{\phi}^2+1$ and ${\phi}^2-2(\frac{1}{\phi})^2+1$, adding up to the expression for ${\rm tr}^{}_{5}(G).$
The proof of Theorem \[equality at B5\] for an arbitrary graph $G\subset {\mathbb T}$ is obtained by expanding the second JW projectors for all edges. The resulting summands for the TQFT trace are in $1$-$1$ correspondence with spanning subgraph $H\subset G$. To be precise, given a spanning subgraph $H$, in this correspondence the first term in the expansion (\[phi\]) of the $2$nd projector is taken for each edge $e$ in $H$, and the second term is taken for each edge $e$ in $G\smallsetminus H$. The resulting loop configuration, called the [*surround loops*]{}, is the boundary of a regular neighborhood of $H$ on the surface. Each individual term in the trace evaluation equals the sum of two entries, corresponding to the two labels $0, 2$, and we show next that these entries precisely match the corresponding terms in the expansions $P^{}_G({\phi}^2, 1, {\phi}^{-2})$, $P^{}_G({\phi}^2, {\phi}^{-4}, {\phi}^{-2})$.
For each spanning subgraph $H$ there are three cases, analogous to the examples (a)-(c) above:
(${\mathcal A}$) $H$ is homologically trivial on the torus: $H_1(H)\longrightarrow H_1({\mathbb T})$ is the zero map. The exponents of the variables $W$ and $A$ in (\[poly definition\]) are zero in this case. The proof that loop evaluation in TQFT equals the summand in the definition of the graph polynomial (\[poly definition\]) is thus identical to the planar case [@FK Lemma 2.5]. Both quantities in the statement of the theorem have a factor $2$: for ${\rm tr}^{}_{5}(G)$ this is because ${\rm dim}(\overline V^{}_5)=2$; for $R_{5}(G)$ the factor is the result of adding two identical summands in (\[P5 equ\]).
(${\mathcal B}$) The image of $H_1(H)\longrightarrow H_1({\mathbb T})$ has rank $1$. In this case ${\rm s}(H)=0$. Recall that ${\bar {\rm c}(H)}$ denotes the number of connected components $H^{(i)}$ of $H$ such that $H_1(H^{(i)})\longrightarrow H_1({\mathbb T})$ of rank $1$ for each $i$. The term in the expansion of $R^{}_5(G)$ corresponding to $H$ is $$(-1)^{E(G)-E(H)} \, {\phi}^{2{\rm n}(H)}\, [1^{\bar {\rm c}(H)}+{\phi}^{-4{\bar {\rm c}(H)}}]\, = \, (-1)^{E(G)-E(H)} \, {\phi}^{2({\rm n}(H)-\bar{\rm c}(H))}\, [{\phi}^{2\bar {\rm c}(H)}+{\phi}^{-2\bar {\rm c}(H)}] .$$
Each $H^{(i)}$ has two surround loops which are non-trivial on the torus. In the calculation of ${\rm tr}^{}_5(G)$, these $\bar {\rm c}(H)$ non-trivial loops give a factor ${\phi}^{2\bar {\rm c}(H)}+{\phi}^{-2\bar {\rm c}(H)}$, matching the factor in square brackets in the calculation of $R^{}_5(G)$ above.
The last step is to check that the remaining factor $(-1)^{E(G)-E(H)} \, {\phi}^{2({\rm n}(H)-\bar{\rm c}(H))}$ above corresponds to the normalization and the trivial surround loops in the trace evaluation. As explained after (\[phi\]), the normalization factor in the definition of $\Phi$ is $d^{E(G)-V(G)}$. Moreover, each edge in $G\smallsetminus H$ gives rise to an additional factor $-d^{-1}$ coming from the second term of the JW projector. This gives the desired sign $(-1)^{E(G)-E(H)}$. Thus the overall normalization factor corresponding to $H$ is $d^{E(H)-V(H)}$, where $d={\phi}$. In addition, each trivial surround loop of $H$ gives a factor $\phi$ in the trace evaluation. An Euler characteristic count gives the equality $$E(H)-V(H)+{\rm number\; of\; trivial\; loops}= 2({\rm n}(H)-\bar{\rm c}(H)),$$ concluding the proof in case ${\mathcal B}$.
(${\mathcal C}$) $H_1(H)\longrightarrow H_1({\mathbb T})$ is surjective, so ${\rm s}(H)=1$ and $\bar {\rm c}(H)=0$. This case is similar to (${\mathcal A}$) since all surround loops are trivial on the torus. Because of the homological assumption, there are two fewer surround loops than in the planar case, expected from the nullity ${\rm n}(H)$. In the TQFT evaluation this undercount gives a factor $d^{-2}$. This factor precisely matches the factor $A^{s(H)}=A={\phi}^{-2}$ in (\[poly definition\]).
Recall that the vanishing of the Jones-Wenzl projector is built into the definition of the TQFT vector space at the corresponding root of unity, so the $4$-th JW projector gives a local relation in $\overline V^{}_5$.
\[corollary\] [*The graph evaluation $R^{}_5(G)$ satisfies the local relation (\[fig:graphp4\]), corresponding to the $4$th Jones-Wenzl projector. More precisely, given three graphs $G^{}_X, G^{}_I, G^{}_E$ on the torus, locally related as shown in figure (\[fig:graphp4\]), $${\phi}R^{}_5(G^{}_X)=R^{}_5(G^{}_I)+R^{}_5(G^{}_E).$$* ]{}
More generally, given $G\subset {\mathbb T}$ and odd $r$, consider $$\label{Pr equ}
R^{}_r(G):=\sum_{j=0, \, j\, {\rm even}}^{r-2} P^{}_G(d^2, W_{j,r} , d^{-2}),$$ where $d=q^{1/2}+q^{-1/2}$ is the TQFT loop value corresponding to the root of unity $q=e^{2{\pi} i/r}$, and $W_{j,r}$ is defined by $$YW_{j,r}-1= \frac{{\rm sin}(2(j+1){\pi}/r)}{{\rm sin}((j+1){\pi}/r)}.$$
The proof of the following result is directly analogous to that of Theorem \[equality at B5\], with TQFT sectors precisely corresponding to the summands in (\[Pr equ\]):
\[evaluations are equal\] *The ${\rm SO}(3)$ TQFT trace evaluation of $G$ at $q=e^{2{\pi}i/r}$ equals $R^{}_r(G)$.*
[**Remark.**]{} A generalization of the polynomial $P$ for links $L$ in ${\mathbb T}\times [0,1]$ (along the lines of [@Kr Section 6]) gives a similar expression for the ${\rm SU}(2)$ trace of $L$. A polynomial $P_L$ for more general links in a surface $\Sigma$ times the circle was formulated in [@MS]. It is an interesting question whether the polynomial of [@MS] can be defined for ribbon [*graphs*]{} in ${\Sigma}\times S^1$, and whether our results extend to this setting.
Golden identity for graphs on the torus {#golden torus section}
=======================================
Using TQFT methods developed above, in this section we formulate and prove an extension of the Tutte golden identity for graphs on the torus.
Proof of the Tutte golden identity (\[golden identity eq\]) for planar graphs {#planar subsection}
-----------------------------------------------------------------------------
We start by summarizing the ideas underlying the proof in the planar case. Following [@FK], we prove here the golden identity for the flow polynomial of cubic planar graphs $G$: $F^{}_G({\phi}+2) \, = \, {\phi}^E\, (F^{}_G({\phi}+1))^2.$ The version of the contraction-deletion rule for cubic graphs reads $$\label{trivalent figure}
\vcenter{\hbox{
\includegraphics[height=1.85cm]{trivalent}
\put(-209,25){$+$}
\put(-138,25){$=$}
\put(-66,25){$+$}
}}$$ Using induction on the number of edges, it suffices to show that if three of the graphs in (\[trivalent figure\]) for $F^{}_G({\phi}+2)$ satisfy the golden identity, then the fourth one does as well. Given a cubic graph $G$, consider ${\phi}^E\, (F^{}_G({\phi}+1))^2$. It is convenient to formally depict, as in (\[fig:tutte\]), two identical copies of $G$, each one evaluated at ${\phi}+1$, with an overall factor ${\phi}^E= {\phi}^{3V/2}$. $$\label{fig:tutte}
\vcenter{\hbox{
\includegraphics[height=1.7cm]{tutte}
{\scriptsize
\put(-232,25){${\Psi}$}
\put(-212,20){${\phi}^{3/2}\cdot$}
\put(-62,26){${\Psi}$}
\put(-40,20){${\phi}^3\cdot$}}
}}$$ Here $E$ and $V$ denote the number of edges and vertices, respectively, of $G$. This doubling of lines is [*not*]{} that in the map $\Phi$ using the projector $p_2$, but instead a map ${\Psi}(G)=G\times G$. Note that for $G=\,$circle, $F_{\rm circle}({\phi}+2)={\phi}+1$. The corresponding value for $\Psi$(circle) is ${\phi}^2={\phi}+1$, indeed the same.
The strategy is to check that the evaluation ${\phi}^E\, (F^{}_G({\phi}+1))^2$ satisfies the relation (\[trivalent figure\]) as a consequence of the [*additional*]{} local relation at $Q={\phi}+1$. This additional relation is the graph version (\[fig:graphp4\]) of the $4$th Jones-Wenzl projector. Using the contraction-deletion rule, one checks that (\[fig:graphp4\]) is equivalent to each of the two relations shown in figure \[fig:JW\].
![Local relations for the flow polynomial at $Q={\phi}+1$, equivalent to Tutte’s relation (\[fig:graphp4\]).[]{data-label="fig:JW"}](graphJW "fig:"){height="1.6cm"} [ (-427,20)[$\phi$]{} (-370,20)[$=$]{} (-312,20)[$+\,(1-{\phi})$]{} (-205,20)[$\phi$]{} (-144,20)[$=$]{} (-87,20)[$+\,(1-{\phi})$]{}]{} (-223,7)[$,$]{}
Consider the image of (\[trivalent figure\]) under $\Psi$: $$\label{tutte2}
\vcenter{\hbox{
\includegraphics[height=1.65cm]{tutte2}
\put(-310,21){${\phi}^3\, \cdot$}
\put(-241,21){$+$}
\put(-166,21){$= \;{\phi}^3\, \cdot$}
\put(-70,21){$+$}
}}$$ Applying the relation on the left in figure \[fig:JW\] to both copies of the graph on the left in figure (\[tutte2\]) yields figure (\[tutte3\]).
![[]{data-label="tutte3"}](tutte3 "fig:"){height="1.55cm"} (-412,19)[${\phi}^3\,\cdot$]{} (-348,19)[$+$]{} (-283,19)[$=\;{\phi}\,\cdot$]{} (-198,19)[$-$]{} (-131,19)[$-$]{} (-69,19)[$+\;{\phi}\,\cdot$]{}
The resulting expression on the right is invariant under 90 degree rotation, so must also be equal to the graph on the right of (\[tutte2\]). Thus (\[tutte2\]) holds, showing that the evaluation ${\phi}^E\, (F^{}_G({\phi}+1))^2$ satisfies the contraction-deletion relation (\[trivalent figure\]). This concludes the proof of the golden identity for the flow polynomial of planar cubic graphs.
Extension to graphs on the torus
--------------------------------
The expression (\[Pr equ\]) is defined for odd $r$. We generalize it with the following sum of evaluations of the graph polynomial $P^{}_G$ in (\[poly definition\]): $$\label{P10 equ}
R^{}_{10}(G):= P^{}_G(Y, W_1, A)+P^{}_G(Y, W_2, A)+2P^{}_G(Y, W_3, A),$$ where $Y={\phi}+2$, $A =(2({\phi}+2))^{-1}$, and the values $W_j$, $j=1,2,3$, are defined by $$YW_1={\phi}+2, \; YW_2=1+{\phi}^{-2}, \; YW_3=0.$$ The choice of these values may be thought of as a choice of particular sectors of the ${\rm SO}(3)$ TQFT vector space of the torus at $q=e^{2{\pi}i/10}$. (Our forthcoming work, relating these results to lattice models on the torus, will give further evidence for why this is a relevant invariant at this root of unity.) The value of $R^{}_{10}(G)$ for $G$ a trivial loop on the torus equals $4(Y-1)=4{\phi}^2$. Using (\[trace12 fig\]), the value of $R^{}_{10}(G)$ for the graph consisting of a single non-trivial loop is seen to be ${\phi}^2+{\phi}^{-2}-2$.
We are in a position to state the main result of this section.
\[golden torus theorem\] *Let $G\subset {\mathbb T}$ be a cubic graph. Then $$\label{golden torus eq}
R^{}_{10}(G)\, =\, {\phi}^E R^{}_{5}(G)^2,$$ where $E$ is the number of edges of $G$.*
The version of the contraction-deletion relation for cubic graphs is shown in (\[trivalent figure\]). As usual, we allow cubic graphs with disjoint loops; such loops do not count towards $E$. The proof in the planar case (see section \[planar subsection\]) showed that if three of the graphs in (\[trivalent figure\]) satisfy the golden identity, then the fourth one satisfies it as well. This fact holds for the identity for graphs on the torus as well. Indeed, (\[trivalent figure\]) holds for $R^{}_{10}(G)$ since it is defined as the sum (\[P10 equ\]) of polynomials satisfying the contraction-deletion rule. And (\[trivalent figure\]) holds for ${\phi}^E R^{}_{5}(G)^2$ for the same reason as in section \[planar subsection\], since by Corollary \[corollary\] $R^{}_{5}(G)$ obeys the local relation corresponding to the $4th$ JW projector.
For cubic graphs $G$ which are homologically trivial on the torus ($r(G\subset {\mathbb T}) =0$ in the notation of section \[graph poly section\]) the proof of (\[golden torus eq\]) follows from the planar case in section \[planar subsection\] since the polynomial $P^{}_G$ in (\[poly definition\]) equals the flow poynomial $F^{}_G$. (For example, calculations in section \[graph poly section\] show that in the special case of the graph consisting of $k$ trivial loops, $R^{}_5(G)=2{\phi}^k$, and $R^{}_{10}(G)= 4{\phi}^{2k}=R^{}_5(G)^2$.)
Next consider the case of cubic graphs $G$ of rank $r(G\subset {\mathbb T}) =2$. Consider a minimal cubic graph $G$ of this type, shown in (\[fig:TorusGraph\]), where the square with opposite sides identified is the usual representation of the torus. A direct calculation using (\[poly definition\]), or alternatively using the contraction-deletion rule to reduce this to calculations above, shows $R^{}_5(G)=R^{}_{10}(G)={\phi}^{-3}$, proving (\[golden torus eq\]) in this case. $$\label{fig:TorusGraph}
\vcenter{\hbox{
\includegraphics[height=2.7cm]{TorusGraph.pdf}
}}$$
The proof in general for rank $2$ cubic graphs is by induction on the number of edges. It is proved in [@D] that any two triangulations of the torus with the same number of vertices are related by diagonal flips, up to equivalence given by diffeomorphisms. (This was extended in [@Negami] to pseudo-triangulations of surfaces of any genus, where an embedding $\Gamma\subset {\mathbb T}$ is a [*pseudo-triangulation*]{} if each face is a three-edged $2$-cell, possibly with multiple edges and loops.) Formulated in terms of dual cubic graphs, two cellular embeddings of cubic graphs (or in other words rank $2$ graphs) on the torus are related by the $I-H$ move: $$\label{fig:flip}
\vcenter{\hbox{
\includegraphics[height=2cm]{flip.pdf}
}}$$ The relation (\[trivalent figure\]) accomplishes the $I-H$ move, while also introducing graphs with fewer edges. The theorem has been checked for a minimal cubic graph in (\[fig:TorusGraph\]), and the inductive step is achieved by a local modification (\[fig:local\]), which increases the number of edges and preserves (\[golden torus eq\]). $$\label{fig:local}
\vcenter{\hbox{
\includegraphics[height=1.9cm]{Local.pdf}
}}$$
Finally consider cubic graphs of rank $1$ (that is, $r(G\subset {\mathbb T}) =1$), or equivalently graphs on the cylinder. The fact that triangulations of the sphere with the same number of vertices are related by diagonal flips dates back to [@Wagner]. Using (\[trivalent figure\]) and (\[fig:local\]) as above, the proof in the rank $1$ case therefore follows from the calculation for $G$ consisting of $k$ non-trivial loops on the torus. In this case, using (\[trace12 fig\]) one has $R^{}_5(G)={\phi}^k+(-{\phi}^{-1})^{k}$, and $R^{}_{10}(G)= {\phi}^{2k}+{\phi}^{-2k}+2(-1)^k=R^{}_5(G)^2$.
[**Remark.**]{} Theorem \[golden torus theorem\] is an extension of the golden identity for the flow polynomial of planar cubic graphs. The focus of this paper is on the “topological flow polynomial” $P^{}_G$, since it is related to the TQFT trace evaluation, as stated in Theorem \[evaluations are equal\]. It is worth noting that an analogue of the invariants $R^{}_5(G), R^{}_{10}(G)$ may be defined using the polynomial $C^{}_G$ in place of $P^{}_G$. Using the duality relation (\[duality eq\]), the identity (\[golden torus eq\]) then gives rise to an extension of the original Tutte golden identity (\[golden identity eq\]) for the “topological chromatic polynomial” $C^{}_G$ of triangulations of the torus.
[**Acknowledgments.**]{} VK would like to thank Calvin McPhail-Snyder and Sittipong Thamrongpairoj for discussions about polynomial invariants of graphs on surfaces. VK was supported in part by NSF grant DMS-1612159; he also would like to thank All Souls College and the Department of Physics at the University of Oxford for hospitality and support.
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---
abstract: 'We use the next-to-leading-order (NLO) amplitude in an effective field theory (EFT) for ${{}^{3}\mathrm{He}}+{{}^{4}\mathrm{He}}\rightarrow {{}^{7}\mathrm{Be}}+ \gamma$ to perform the extrapolation of higher-energy data to solar energies. At this order the EFT describes the capture process using an s-wave scattering length and effective range, the asymptotic behavior of ${{}^{7}\mathrm{Be}}$ and its excited state, and short-distance contributions to the E1 capture amplitude. We use a Bayesian analysis to infer the multi-dimensional posterior of these parameters from capture data below 2 MeV. The total $S$ factor $S(0)= 0.578^{+0.015}_{-0.016}$ keV b at 68% degree of belief. We also find significant constraints on ${{}^{3}\mathrm{He}}$-${{}^{4}\mathrm{He}}$ scattering parameters.'
author:
- Xilin Zhang
- 'Kenneth M. Nollett'
- 'Daniel R. Phillips'
title: |
$S$-factor and scattering parameters from\
${}^3$He + ${}^4$He $\rightarrow {}^7$Be + $\gamma$ data
---
The solar-fusion reaction ${{}^{3}\mathrm{He}}+{{}^{4}\mathrm{He}}\rightarrow {{}^{7}\mathrm{Be}}+ \gamma$ has not been measured directly at solar energies, due to the exponential suppression of the cross section there. Solar models use cross sections for it based on extrapolants that are derived using potential models or R-matrix, and constrained by $S$-factor and ${{}^{3}\mathrm{He}}$-${{}^{4}\mathrm{He}}$ scattering data, as well as ${{}^{7}\mathrm{Be}}$ bound-state properties. Ref. [@Adelberger:2010qa] reviews the most prominent efforts before 2011; additional evaluations have emerged since [@deBoer:2014hha; @Iliadis:2016vkw].
Formalism for $E1$ capture {#sec:E1form}
==========================
We use Halo Effective Field Theory (EFT) [@Hammer:2017], treating ${{}^{3}\mathrm{He}}$ ($\frac{1}{2}^+$) and ${{}^{4}\mathrm{He}}$ ($0^+$) as fundamental degrees of freedom and ${{}^{7}\mathrm{Be}}$ (ground state, GS, $\frac{3}{2}^-$) and ${{}^{7}\mathrm{Be}}^\ast$ (excited state, ES, $\frac{1}{2}^-$) as shallow p-wave bound states of the two. From the breakup energies of ${{}^{3}\mathrm{He}}$ and ${{}^{4}\mathrm{He}}$ we infer an EFT breakdown scale $\Lambda$ of about $200$ MeV. The energy range $E \lsim 2$ MeV implies a low-momentum scale $Q$ of $70$–$80$ MeV, thus we have $Q/\Lambda \approx 0.4$. We systematically expand both scattering and reaction amplitudes in this small parameter. The NLO $S$-factor for E1 capture to the ${}^7$Be GS can then be written [@Zhang:2019] $$S_{_{{{}^{}P_{3/2}}}}(E) = \frac{e^{2\pi {\eta }}}{e^{2\pi{\eta }}-1} \frac{8\pi}{9} \left(e\, Z_{eff}\right)^{2} {k_{C}}\omega^3 C_{({{}^{}P_{3/2}})}^{2} \left(\mid \mathcal{S} \mid^{2}+2 \mid \mathcal{D} \mid^{2}\right) \ , \label{eqn:sfactormaster1}$$ with the same result, [*mutatis mutandis*]{} for capture to the ${{}^{}P_{1/2}}$ ES. This is analogous to our results for ${{}^{7}\mathrm{Be}}+ p \rightarrow {{}^{8}\mathrm{B}}+ {\gamma}$ [@Zhang:2015ajn; @Zhang:2017yqc]. Here, ${k_{C}}\equiv \alpha_\mathrm{em} Z^2 {M_{\mathrm{R}}}$ with ${M_{\mathrm{R}}}$ the reduced mass of the ${{}^{3}\mathrm{He}}$-${{}^{4}\mathrm{He}}$ system; $\eta\equiv {k_{C}}/p$ is the well-known Sommerfeld parameter; $\omega$ is the energy of the photon produced in the reaction; and the “effective charge" $Z_{eff} \equiv \left(Z/{M_{\mathrm{4}}}-Z/{M_{\mathrm{3}}}\right) {M_{\mathrm{R}}}$. The factors $C_{({{}^{}P_{3/2}})}^{2}$ ($C_{({{}^{}P_{1/2}})}^{2}$) are the squared p-wave asymptotic normalization coefficients (ANCs) of the GS (ES) [@Zhang:2017yqc]. The two reduced matrix elements, $\mathcal{S}$ and $\mathcal{D}$, are for the E1 transition from initial s- and d-wave states. At NLO, $\mathcal{S}$ consists of the well-known external capture contributions plus a short-distance piece similar to R-matrix internal capture. We parameterize the latter contribution to capture to the GS (ES) by a single number, $\overline{L}$ ($\overline{L}_\ast$). The d-wave reduced matrix element $\mathcal{D}$ is given by the standard asymptotic expression for external capture, but integrated all the way to zero radius. Explicit formulae for $\mathcal{S}$ and $\mathcal{D}$ can be found in Refs. [@Zhang:2019; @Zhang:2017yqc].
Capture reactions to the ground and excited state share the same initial state for s-waves ($\frac{1}{2}^+$), so $\mathcal{S}$ depends on the scattering length, $a_0$ and effective range, $r_0$. Up to NLO there are then 6 EFT parameters, henceforth denoted as the vector ${\bf g}$: $ C_{({{}^{}P_{3/2}})}^2$ ($\mathrm{fm}^{-1}$), $ C_{({{}^{}P_{1/2}})}^2$ ($\mathrm{fm}^{-1}$), $a_0$ (fm), $r_0$ (fm), $\overline{L}$ (fm), and $\overline{L}_\ast$ (fm).
Data, Bayesian analysis, and Results
====================================
There are six total $S$-factor data sets, here labeled Seattle, Weizmann, Luna, Erna, Notre Dame (ND), and Atomki. There are four branching-ratio data sets: Seattle, Luna, Erna, and Notre Dame. In order to ensure that the data used are within the domain of validity of the EFT we only employ data with $E \leq 2$ MeV. This, together with other data-selection criteria, yields 59 $S$-factor and 32 $Br$ data, see Fig. \[fig:SBrvsENLON4Lv2FullData\]. (Details, including original references and a full listing of these data, will appear in Ref. [@Zhang:2019].) To account for the common-mode errors we introduce normalization corrections, {$\xi_J$: $J=1 \ldots N_\mathrm{exp}$}, for the $S$-factor data. Such errors mostly cancel for $Br$ data, so this correction is not used for them.
We take the EFT expressions such as (\[eqn:sfactormaster1\]) and employ Bayesian analysis—implemented via Markov-Chain-Monte-Carlo (MCMC) sampling—to infer probability distribution functions (PDFs) for the EFT parameters ${\bf g}$ from these data. Taking box priors with ranges considerably larger than those suggested by naive dimensional analysis for ${\bf g}$, and gaussian priors for the $\xi_J$’s, we can write the desired PDF as $ {\rm pr} \left({\bf g},\{\xi_J\} \vert D;T; I \right) \equiv c\, \exp\left(-\chi^2/2\right) $. The $\chi^2$ is non-standard because it includes not only contributions from $S$-factor and branching-ratio measurements, but also the effect of the normalization corrections [@Zhang:2019; @Zhang:2017yqc].
![Total $S$-factor and branching-ratio results. The data is denoted in the legend, and summarized in Ref. [@deBoer:2014hha]. The green band shows the 68% interval for $S(E)$ and $Br(E)$ in our NLO Halo EFT analysis. The mean is denoted by the blue line.[]{data-label="fig:SBrvsENLON4Lv2FullData"}](S_BrvsEn_NLO_FullData.pdf){width="65.00000%"}
The MCMC sampling produces the full twelve-dimensional pdf for $\vec{g}$ and $\{\xi_J\}$. These samples can then be used to compute a histogram for $S(E)$ and $Br(E)$ at any energy, $E$. Fig. \[fig:SBrvsENLON4Lv2FullData\] shows the resulting 68% intervals: the mean is denoted by the blue line. The data is shown without re-scaling by the factors $1/(1-\xi_J)$, so Fig. \[fig:SBrvsENLON4Lv2FullData\] under-represents the quality of our final result. Adopting values for the $\xi_J$’s that maximize their posterior pdf produces a distribution of $\chi^2$’s in our MCMC sample peaks at 1.1 per degree of freedom.
At NLO we have $S(0)= 0.578^{+0.015}_{-0.016}$ keV b. The recommended value from Ref. [@Adelberger:2010qa] is $0.56\pm 0.02\mathrm{(exp)} \pm 0.02\mathrm{(theory)}$—consistent with our result, but with an uncertainty that is almost a factor of two larger. Other recent analyses are broadly consistent, but also have somewhat bigger errors [@deBoer:2014hha; @Iliadis:2016vkw]. We also find $Br(0)=0.406^{+0.013}_{-0.011}$. There are two essential differences between this paper and another, recent, EFT evaluation of the same reaction [@Higa:2016igc]. First, we employ Bayesian methods. Second, we do not not include existing scattering phase shift analyses in our constraints because their systematic errors are poorly quantified.
Fig. \[fig:a\_r\_Lt\_Corr\_NLOFullData\] displays a three-dimensional scatter plot of the NLO MCMC samples, projected to the $a_0$–$r_0$–$\overline{L}_T$ subspace. Projecting further onto the $a_0$-$r_0$ subspace shows that significant constraints on ${{}^{3}\mathrm{He}}$-${{}^{4}\mathrm{He}}$ scattering parameters can be obtained from the extant radiative capture data—in contrast to cases such as ${{}^{7}\mathrm{Be}}(p,\gamma)$ [@Zhang:2015ajn]. The corresponding effective-range function can be tested against future high-quality ${{}^{3}\mathrm{He}}$-${{}^{4}\mathrm{He}}$ scattering data at low energy.
We conclude that data on ${{}^{3}\mathrm{He}}+{{}^{4}\mathrm{He}}\rightarrow {{}^{7}\mathrm{Be}}+ \gamma$ already tightly constrain important aspects of the dynamics needed for extrapolation of this reaction’s $S$-factor: we find quite small uncertainties on the s-wave elastic scattering parameters and the ANCs of the final states. Better measurements of scattering cross sections will test the EFT approach to the reaction presented here.
#### Acknowledgments
This work is supported by the US Department of Energy under grant no. DE-FG02-93ER-40756 (DP), DE-FG02-97ER-41014 (XZ), DE-SC0019257 (KMN), and through MSU subcontract RC107839-OSU for the NUCLEI SciDAC collaboration (XZ), by the US National Science Foundation under Grant No. PHY-1614460 (XZ), and by the ExtreMe Matter Institute EMMI at the GSI Helmholtzzentrum für Schwerionenphysik (DP).
[45]{} Adelberger, E. G., [*et al.*]{}: Solar fusion cross sections II: the pp chain and CNO cycles. Rev. Mod. Phys. 83, 195 (2011). <doi:10.1103/RevModPhys.83.195>
deBoer, R. J., [*et al.*]{}: ‘Monte Carlo uncertainty of the He3($\alpha,\gamma$)Be7 reaction rate. Phys. Rev. C 90, 035804 (2014). <doi:10.1103/PhysRevC.90.035804>
Iliadis, C, [*et al.*]{}: Bayesian estimation of thermonuclear reaction rates. Astrophys J. 831, 107 (2016). <doi:10.3847/0004-637X/831/1/107>
Hammer H.-W., Ji C., Phillips D. R.: Effective field theory description of halo nuclei. J. Phys. G 44, 103002 (2017). <doi:10.1088/1361-6471/aa83db>
Zhang, X., Nollett, K. M., Phillips, D. R., in preparation.
Zhang, X., Nollett, K. M., Phillips, D. R.: Halo effective field theory constrains the solar $^7$Be + p $\rightarrow$ $^8$B + $\gamma$ rate. Phys. Lett. B 751, 535 (2015). <doi:10.1016/j.physletb.2015.11.005>
Zhang, X., Nollett, K. M., Phillips, D. R.: Models, measurements, and effective field theory: Proton capture on $^7$Be at next-to-leading order. Phys. Rev. C 98, 034616 (2018). <doi:10.1103/PhysRevC.98.034616>
Higa, R., Ruak, G., Vaghani, A.: Radiative$^{3}$He( $\alpha , \gamma$ )$^{7}$Be reaction in halo effective field theory. Eur. Phys. J. A 54, 89 (2018). <doi:10.1140/epja/i2018-12486-5>
|
---
author:
- 'Alexei Mishchenko, Alexander Treier'
title: Knapsack problem for nilpotent groups
---
Introduction
============
In the paper [@MNU:KnapsackProblemInGroups] A. Myasnikov, A. Nikolaev, and A. Ushakov stated a group version of the well known Knapsack problem. The motivation for our research and initial results in this direction may be found in that paper, and further results in [@MNU:KnapsackProblemInGroups2; @Lor:KPGg; @Lor:KPNilp].
We give a definition of Knapsack problem for groups following [@MNU:KnapsackProblemInGroups]. Let $G$ be an arbitrary group with a presentation $G = \langle X | R \rangle$ and solvable word problem. Let $g_1, \ldots, g_k, g$ be finite words in the alphabet $X \cup X^{-1}$. Then the Knapsack Problem for the group $G$ is stated in the following way.
**Knapsack Problem. ${{\mathrm{KP}}}$.** *Given input words ${{{g}}_{1}, \ldots, {{g}}_{{k}}}, g$, decide whether there exist integers ${{{\varepsilon}}_{1}, \ldots, {{\varepsilon}}_{{k}}}$ such that the equality $$\label{eq:KnapsackProblem}
{{{g}^{{\varepsilon}_1}_{1} \ldots {g}^{{\varepsilon}_{k}}_{k}}} = g$$ holds in the group $G$.*
There are several notable questions related to ${{\mathrm{KP}}}$. One such question is that of decidability of ${{\mathrm{KP}}}$ for a specific class of groups ${{\mathcal{K}}}$. In the case when ${{\mathrm{KP}}}$ is decidable for a class ${{\mathcal{K}}}$, another natural question is how computationally hard ${{\mathrm{KP}}}$ for class ${{\mathcal{K}}}$ is. In this regard, it is known that ${{\mathrm{KP}}}$ is decidable in polynomial time for abelian and hyperbolic groups. In this work we investigate decidability of ${{\mathrm{KP}}}$ for nilpotent groups.
The main results of the present paper are as follows. In *Theorem \[theorem:main\]* we prove that Knapsack problem (${{\mathrm{KP}}}$) is undecidable for any group of nilpotency class two if the number of generators (without torsion) of the derived subgroup is at least $322$. This theorem together with the fact that if ${{\mathrm{KP}}}$ is undecidable for a subgroup then it undecidable for the whole group allows us extend our result to certain classes of polycyclic groups, linear groups and nilpotent groups of higher nilpotency class ($\geq 3$).
We draw the reader’s attention to a result of Daniel König, Markus Lohrey, and Georg Zetzsche [@Lor:KPNilp] that ${{\mathrm{KP}}}$ is undecidable for a direct product of sufficiently many copies of the discrete Heisenberg group $H_3({{\mathbb{Z}}})$. This implies that ${{\mathrm{KP}}}$ is generally undecidable for nilpotent groups. We would like to point out that our approach is different from that of Daniel König, Markus Lohrey, and Georg Zetzsche. Moreover, our Theorem 1 provides an explicit bound, $322$, for the number of copies of $H_3({{\mathbb{Z}}})$ in a direct product that suffices for undecidable ${{\mathrm{KP}}}$. The paper [@Lor:KPNilp] also contains interesting results on Subset Sum Problem and Knapsack problem for nilpotent, polycyclic, and co-context-free groups.
The authors are grateful to A. Miasnikov and A. Nikolaev for their advice and discussions.
Preliminaries
=============
Nilpotent groups
----------------
Recall the definition and basic properties of nilpotent groups. A group $G$ is called a nilpotent group of class $c$ if it has a lower central series of length $c$: $$G = G_1 \trianglerighteq G_2 \trianglerighteq \ldots \trianglerighteq G_c \trianglerighteq G_{c+1} = \{1\},$$ where $G_{k+1} = [G_k, G]$, $k = {1, \ldots, {c}}$ and $G_1 = G$.
Let $X = \{{{{x}}_{1}, \ldots, {{x}}_{{n}}}\}$ be a set of letters, and $G = \langle X \rangle$ be a free nilpotent group of class $2$. By definition, the following identity holds for group $G$:
$$\label{eq:Identity}
\forall x,y,z \in G \ [x,[y,z]] = 1$$
Using identity [(\[eq:Identity\])]{}, the collection process in group $G$ is organized via the transformation $$\label{eq:Collecting}
yx = xy [x,y]^{-1},$$ where $x,y$ are any elements of $G$. Using the equality [(\[eq:Collecting\])]{} we can reduce any word $g$ in the alphabet $X \cup X^{-1}$ to the normal form for elements of the group $G$: $$\label{eq:NormalForm}
g = {{x}^{{\alpha}_1}_{1} \ldots {x}^{{\alpha}_{n}}_{n}} \prod_{i<j} [x_i,x_j]^{\beta_{ij}},$$ where $\alpha_i,\beta_{ij} \in {{\mathbb{Z}}}, i,j = {1, \ldots, {n}}, i < j$ and $[x_i,x_j] = x^{-1}_ix^{-1}_jx_ix_j$.
Using [(\[eq:Identity\])]{}, it is not hard to show that for any two elements $a,b$ of the group $G$ and $\alpha, \beta \in {{\mathbb{Z}}}$ we have the following equality:
$$\label{eq:RingMultiplication}
[a^\alpha,b^{\beta}] = [a,b]^{\alpha\beta}.$$
Knapsack problem {#section:KP}
----------------
We stated the Knapsack problem (${{\mathrm{KP}}}$) for groups in Introduction. Recall that the ${{\mathrm{KP}}}$ is called decidable for the class of groups ${{\mathcal{K}}}$ if for any group $G \in {{\mathcal{K}}}$ there exists an algorithm that, given any input ${{{g}}_{1}, \ldots, {{g}}_{{k}}}, g$, answers the question whether or not the exponential group equation [(\[eq:KnapsackProblem\])]{} has a solution in the group $G$. We can restrict the notion of decidability of ${{\mathrm{KP}}}$ and explore ${{\mathrm{KP}}}$ for single group or for some type of inputs of ${{\mathrm{KP}}}$. In our work we concentrate on decidability of ${{\mathrm{KP}}}$ for the class of nilpotent groups.
Let $G$ be a free nilpotent group of class $2$ and let ${{{g}}_{1}, \ldots, {{g}}_{{k}}}, g$ be presented in the form [(\[eq:NormalForm\])]{}. Using [(\[eq:Collecting\])]{} and [(\[eq:RingMultiplication\])]{} we can reduce the expression ${{{g}^{{\varepsilon}_1}_{1} \ldots {g}^{{\varepsilon}_{k}}_{k}}}$ to the form [(\[eq:NormalForm\])]{}. Thus, the following proposition holds:
\[prop:KPtoDiophantine\] Let $G$ be a free two-step nilpotent $n$-generated group. Then ${{\mathrm{KP}}}$ stated above for the group $G$ is equivalent to a system of Diophantine equations with unknowns ${{{\varepsilon}}_{1}, \ldots, {{\varepsilon}}_{{k}}}$ of degree $2$. Moreover, the number of linear equations in the system is not greater than $n$ and the number of quadratic equations is not greater than $\frac{n(n-1)}{2}$.
Diophantine equations and Hilbert’s Tenth problem {#section:DiophantineEquations}
-------------------------------------------------
*Proposition \[prop:KPtoDiophantine\]* shows that ${{\mathrm{KP}}}$ for nilpotent groups is closely related to Diophantine equations. This section is devoted to Diophantine equations.
A polynomial equation $D({{{x}}_{1}, \ldots, {{x}}_{{n}}}) = 0$ with integer coefficients is called Diophantine.
In 1900 at the Second International Congress of Mathematicians D. Hilbert presented his famous list of problems. The 10th problem is concerned with Diophantine equations. The problem statement is as follows: is there an algorithm that for any Diophantine equation answers the question whether or not this equation has a solution in integers? In 60-70th of previous century M. Davis, J. Robinson, H. Putnam, and Yu. Matyasevich proved that there is no algorithm to decide whether an arbitrary Diophantine equation has solution in integers or not. For more details on Hilbert’s Tenth Problem we refer the reader to the book of Yu. Matiyasevich [@Mat:Hilbert10], which, in addition to the solution of the problem, provides a historical survey and describes a number of applications of negative solution of Hilbert’s Tenth Problem.
In some cases of Diophantine equations there exists an algorithm to decide whether the equation has a solution. In [@Siegel:QuadraticDecidable] C. Siegel gives an algorithm for a single Diophantine equation of degree $\leq 2$. So, if we have $2$-generated free two-step nilpotent group $G$ (which is known as Heisenberg group) by *Proposition \[prop:KPtoDiophantine\]* the ${{\mathrm{KP}}}$ for any input is equivalent to a system of two linear equations and one quadratic equation. Such a system may be reduced to a single quadratic equation (for example, this is shown in [@DLS:EquationsInNilpotentGroups]), and therefore, the following proposition holds:
\[prop:Heisenberg group\] The Knapsack problem for Heisenberg group is decidable on any input.
Now we return to the question of undecidability of Diophantine equations. From papers of Julia Robinson, Martin Davis, Hilary Putnam [@DPR:ExpDiophantine] and Yu. Matiyasevich [@Mat:EnumerableAreDiophantine] every recursive enumerable set $W$ can be presented in Diophantine form: $$x \in W \iff \exists {{{x}}_{1}, \ldots, {{x}}_{{n}}}\ P(x, {{{x}}_{1}, \ldots, {{x}}_{{n}}}) = 0,$$ where the variables ${{{x}}_{1}, \ldots, {{x}}_{{n}}}$ are positive integers and $P(x, {{{x}}_{1}, \ldots, {{x}}_{{n}}})$ is a Diophantine polynomial. Since there exist recursively enumerable but non recursive sets then there is no algorithm to decide for arbitrary Diophantine equation whether it has a solution. Moreover, if $W_1, W_2, \ldots$ is a list all recursively enumerable sets, then there is a polynomial $U$ such that for any $k \in {{\mathbb{N}}}$ $$\label{eq:UniversalEquation}
x \in W_k \iff \exists {{{x}}_{1}, \ldots, {{x}}_{{n}}}\ U(x, k, {{{x}}_{1}, \ldots, {{x}}_{{n}}}) = 0.$$
The polynomial $U(x,k,{{{x}}_{1}, \ldots, {{x}}_{{n}}})$ has fixed degree and fixed number of variables. Such polynomial $U$ is called a universal polynomial. J.P. Jones in [@Jon:UndecidableDEquations; @Jon:UniversalDEquations] constructed a universal system of equations that can be reduced to a universal polynomial of degree $4$ with $58$ unknowns. To reduce the Jones system to a single equation we need to prepare this system (because some equations have degree greater than $2$) by transformations and substitutions which are described by Jones. After that we introduce several new variables which are tied by linear relations to lower the number of generators of two step nilpotent group $G$ for building an input for ${{\mathrm{KP}}}$ (see the next sections for details). We are not aware of any published work that provides an explicit version of the universal system of equations of degree $\leq 2$, so we give this system in the present paper. In the next sections we use this system for constructing a universal ${{\mathrm{KP}}}$ input and calculating rank of nilpotent groups with undecidable ${{\mathrm{KP}}}$.
Any letter symbols in system below are variables except $x, {\blacksquare}_z, {\blacksquare}_y, {\blacksquare}_u$ which are positive integer parameters of $U$. The constants ${\blacksquare}_z, {\blacksquare}_y, {\blacksquare}_u$ encode a r.e. set which determines the universal system. So if we put ${\blacksquare}_z, {\blacksquare}_y, {\blacksquare}_u$ that encode a non-recursive set $W$, then there is no algorithm for any $x \in W$ to answer the question whether the equation have a solution. After applying transformations to Jones system we obtain the following universal system:
Equivalence between system of Diophantine equations and Knapsack Problem for nilpotent groups
=============================================================================================
In this section we show that any finite system of Diophantine equations is equivalent to ${{\mathrm{KP}}}$ for some two step nilpotent group $G$ on some input. This means that for any finite system $S$ of Diophantine equations there exists a group $G = \langle {{{x}}_{1}, \ldots, {{x}}_{{n}}} \rangle$ and input ${{{g}}_{1}, \ldots, {{g}}_{{k}}}, g$ which are words of alphabet $X \cup X^{-1}$ such that ${{\mathrm{KP}}}$ for group $G$ has solution if and only if the system $S$ has solution.
Let $S = \{{{{s}}_{1}, \ldots, {{s}}_{{r}}}\}$ be a finite system of Diophantine equations with variables ${{{x}}_{1}, \ldots, {{x}}_{{n}}}$, where $s_{i} := (f_i({{{x}}_{1}, \ldots, {{x}}_{{n}}}) = c_i) $ is a Diophantine equation. Since any finite system of Diophantine equations is equivalent to finite system of equations of degree less or equal than $2$, we may assume that every equation in $S$ written in the form $$\label{eqn:quadratic}
s_i := \left(\sum^{n}_{i=1} \alpha_ix_i + \sum^{n}_{i,j = 1} \beta_{ij}x_ix_j = \gamma\right),$$ where $\alpha_i,\beta_{ij}, \gamma \in {{\mathbb{Z}}}$.
We start by showing how to construct an input for ${{\mathrm{KP}}}$ equivalent to a single quadratic Diophantine equation [(\[eqn:quadratic\])]{}. Let $a,b$ be generators of the group $G$ and $[a,b]$ a nontrivial basic commutator in $G$. Below we pick elements ${{{g}}_{1}, \ldots, {{g}}_{{r}}}\in G$ such that the ${{{g}^{{\varepsilon}_1}_{1} \ldots {g}^{{\varepsilon}_{r}}_{r}}}$ is equal to $[a,b]^{\sum^{n}_{i=1} \alpha_ix_i + \sum^{n}_{i,j = 1} \beta_{ij}x_ix_j}$, then we put $g = [a,b]^\gamma$. ${{\mathrm{KP}}}$ on the obtained input will be equivalent to [(\[eqn:quadratic\])]{}.
Consider the linear part of [(\[eqn:quadratic\])]{}. For every summand $\alpha_ix_i, \ i = {1, \ldots, {n}}$ we put $g_i = [a,b]^{\alpha_i}$ and get ${{{g}^{{\varepsilon}_1}_{1} \ldots {g}^{{\varepsilon}_{n}}_{n}}} = [a,b]^{\sum^{n}_{i=1} \alpha_i\varepsilon_i}$. Thus, we assume that $x_i = \varepsilon_i$.
Turn to the quadratic part of [(\[eqn:quadratic\])]{}. For every summand $\beta_{ij}x_ix_j$ we assign four new elements of input (we assume that in previous steps we constructed $r$ elements of input): $$\begin{aligned}
g_{r+1} & = & a^{-\beta_{ij}} \cdot c_1, \\
g_{r+2} & = & b^{-1} \cdot c_2, \\
g_{r+3} & = & a^{\beta_{ij}}\cdot c^{-1}_1, \\
g_{r+4} & = & b \cdot c^{-1}_2,\end{aligned}$$ where $c_1,c_2 \in [G,G]$ are non-trivial commutators that have not appeared previously in construction of the input. Then $$K = g^{\varepsilon_{r+1}}_{r+1} g^{\varepsilon_{r+2}}_{r+2} g^{\varepsilon_{r+3}}_{r+3} g^{\varepsilon_{r+4}}_{r+4} = a^{-\beta_{ij}\varepsilon_{r+1}}b^{-\varepsilon_{r+2}}a^{\beta_{ij}\varepsilon_{r+3}}b^{\varepsilon_{r+4}} c^{\varepsilon_{r+1} - \varepsilon_{r+3}}_1 c^{\varepsilon_{r+2} - \varepsilon_{r+4}}_2.$$
Setting that the exponents of commutators $c_1$ and $c_2$ are equal to zero in element $g$ is equivalent to the condition $\varepsilon_{r+1} = \varepsilon_{r+3}$ and $\varepsilon_{r+2} = \varepsilon_{r+4}$. As a result we have $K = [a,b]^{\beta_{ij}\varepsilon_{r+1}\varepsilon_{r+2}}$. Now we need to tie the values of $\varepsilon_{r+1}$ to $\varepsilon_i$ and $\varepsilon_{r+2}$ to $\varepsilon_j$. To do that we apply the same trick as in the previous case. Let $c_3$ be a non-trivial commutator that we have never used before. We put $g'_i = g_i c_3$ and $g'_{r+1} = g_{r+1} c^{-1}_3$, then we replace $g_{i}$ by $g'_{i}$ and $g_{r+1}$ by $g'_{r+1}$ in the input. The imposed restrictions give us $\varepsilon_{r+1} = \varepsilon_i = x_i$. Then we repeat the same with $\varepsilon_{r+2}$ and $\varepsilon_j$. Proceeding in the same way with all other quadratic summands we finally get the following exponential expression: $${{{g}^{{\varepsilon}_1}_{1} \ldots {g}^{{\varepsilon}_{k}}_{k}}} = [a,b]^{\sum^{n}_{i=1} \alpha_ix_i + \sum^{n}_{i,j = 1} \beta_{ij}x_ix_j},$$ where $x_i = \varepsilon_i, \ i = {1, \ldots, {n}}$. Then we set $g = [a,b]^\gamma$ and obtain the exponential equation ${{{g}^{{\varepsilon}_1}_{1} \ldots {g}^{{\varepsilon}_{k}}_{k}}} = g$ in group $G$ equivalent to Diophantine equation [(\[eqn:quadratic\])]{}.
It is easy to see that if we have an arbitrary finite system $S$ of $l$ quadratic Diophantine equations we can build an input for ${{\mathrm{KP}}}$ that realizes all equations in the system $S$ as powers of $l$ basic commutators ($[a,b], [a,c], [c,d]$, e.t.c., where $a,b,c,d, \ldots$ are generators of $G$) as described above. Thus, for any finite system $S$ and any nilpotent group $G$ with sufficiently many basic commutators (recall that, besides $l$ basic commutators for equations of $S$, we need more commutators to realize bindings between variables of ${{\mathrm{KP}}}$) we can construct an input on which ${{\mathrm{KP}}}$ for the group $G$ is equivalent to system $S$.
From the above we have the following
\[prop:equivalence\] For any finite system of Diophantine equations exists a finitely generated free group $G$ of nilpotency class $2$ and an input ${{{g}}_{1}, \ldots, {{g}}_{{k}}}, g \in G$ such that ${{\mathrm{KP}}}$ on this input has solution in $G$ if and only if the system $S$ has solution in ${{\mathbb{N}}}\cup \{0\}$.
Now we briefly describe another, more general, approach to establishing equivalence between ${{\mathrm{KP}}}$ for nilpotent groups and decidability of Diophantine equations. This reduction may be more convenient than the one described above in case of an arbitrary Diophantine equation (or any finite system of equations) of degree greater than $2$.
We begin by defining the notion of a Diophantine term by induction as follows.
1. Every constant is a term.
2. Every variable is a term.
3. For every two terms $t_1$ and $t_2$ the $t_1 + t_2$ and $t_1 t_2$ are terms.
A term is called [*simple*]{} if it is a constant or a variable.
We can present any Diophantine equation as equality of two terms $t_1 = t_2$. There are many ways to express a given polynomial as a combination of sums and products of Diophantine terms. For example, we may present the polynomial $f(x) = x^2 - 1$ as sum of two terms: $x^2$ and $-1$ and then $x^2$ is a product of $x$ and $x$, or we may look at $f(x)$ as the product of $x-1$ and $x+1$. We can represent computation scheme of a term as a binary tree where leafs are simple terms and internal vertices are symbols of multiplication “$\cdot$” or addition “$+$”.
Let $t_1 = t_1({{{\varepsilon}}_{1}, \ldots, {{\varepsilon}}_{{n}}})$ and $t_2 = t_2({{{\varepsilon}}_{1}, \ldots, {{\varepsilon}}_{{n}}})$ be Diophantine terms such that ${{{g}^{{\varepsilon}_1}_{1} \ldots {g}^{{\varepsilon}_{r}}_{r}}} = h \cdot [a,b]^{t_1}[c,d]^{t_2}$ and the powers of $[a,b]$ and $[c,d]$ in $g$ and $h$ are equal to zero. Thus, to describe how to construct an input for ${{\mathrm{KP}}}$ equivalent to a given Diophantine polynomial we need to show, for two terms $t_1, t_2$, how to extend the input to realize the following: terms $t_1 + t_2$, $t_1 \cdot t_2$ and equations $t_1 = t_2$, $t_1 = \gamma$, where $\gamma \in {{\mathbb{Z}}}$.
- : to satisfy this condition we introduce one new input element: $$\begin{aligned}
g_{r+1} & = & [a,b] c_1,\end{aligned}$$ where $c_1$ is a basic commutator in $G$ which has not been used before, and set $g' = g c^{\gamma}_1$.
- : in this case we introduce two new input elements: $$\begin{aligned}
g_{r+1} & = & [a,b]^{-1} c_1, \\
g_{r+2} & = & [c,d]^{-1} c^{-1}_1, \end{aligned}$$ then ${{{g}^{{\varepsilon}_1}_{1} \ldots {g}^{{\varepsilon}_{r+2}}_{r+2}}} = [a,b]^{t_1 - \varepsilon_{r+1}}[c,d]^{t_2 - \varepsilon_{r+2}} c^{\varepsilon_{r+1} - \varepsilon_{r+2}}_1$, which gives us $t_1 = \varepsilon_{r+1} = \varepsilon_{r+2} = t_2$ (provided that the powers of $[a,b], [c,d]$, and $c_1$ in $g$ are $0$ ).
- : $$\begin{aligned}
g_{r+1} & = & [a,b]^{-1} c_1, \\
g_{r+2} & = & [c,d]^{-1} c_1, \end{aligned}$$ then ${{{g}^{{\varepsilon}_1}_{1} \ldots {g}^{{\varepsilon}_{r+2}}_{r+2}}} = [a,b]^{t_1 - \varepsilon_{r+1}}[c,d]^{t_2 - \varepsilon_{r+2}} c^{\varepsilon_{r+1} + \varepsilon_{r+2}}_1$, which gives us $c^{t_1 + t_2}_1$ provided that the powers of $[a,b]$ and $[c,d]$ in the element $g$ are $0$.
- : $$\begin{aligned}
g_{r+1} & = & [a,b] x^{-1} \cdot c_1, \\
g_{r+2} & = & [c,d] y^{-1} \cdot c_2, \\
g_{r+3} & = & x \cdot c^{-1}_1, \\
g_{r+4} & = & y \cdot c^{-1}_2,\end{aligned}$$ then $$\begin{aligned}
{{{g}^{{\varepsilon}_1}_{1} \ldots {g}^{{\varepsilon}_{r+2}}_{r+2}}} =
& [a,b]^{t_1 - \varepsilon_{r+1}}[c,d]^{t_2 - \varepsilon_{r+2}} \\
& c^{\varepsilon_{r+1} - \varepsilon_{r+3}}_1 c^{\varepsilon_{r+2} - \varepsilon_{r+4}}_2 \\
& x^{-\varepsilon_{r+1}}y^{-\varepsilon_{r+2}}x^{\varepsilon_{r+3}}y^{\varepsilon_{r+4}}\end{aligned}$$ which gives us $[x,y]^{t_1 \cdot t_2}$ provided that the powers of $[a,b], [c,d]$, $c_1,c_2$ in $g$ are $0$.
Nilpotent groups with undecidable ${{\mathrm{KP}}}$
===================================================
In previous section we described two reductions of any Diophantine equation or a system of Diophantine equations to ${{\mathrm{KP}}}$ in a nilpotent group $G$ with sufficient number of generators. Now we want to give a lower bound for the number of basic commutators in $G'$ of a torsion free two step nilpotent group $G$ with undecidable ${{\mathrm{KP}}}$. We do not aim to get the lowest possible bound for the number of commutators in a group $G$, but we note some simple transformations of the original Jones system of equations to reduce the number of generators. We omit a full description of the input for ${{\mathrm{KP}}}$ (because it contains $334$ elements of input) which is equivalent to the system of equations [(\[system:first\])]{} – [(\[system:last\])]{}. However, we give an example that clarifies the process of input construction.
Consider an equation [(\[system:example\])]{}: $$(K - \Gamma_{18})(K + \Gamma_{18}) + \Gamma^2_{19} = 1.$$ Let a,b be generators of $G$ such that the commutator $[a,b]$, along with commutators ${{{c}}_{1}, \ldots, {{c}}_{{7}}}$, have never been used before. Then we put $$\begin{aligned}
g_{1} & = & a^{-1} c_1 c_3, (\text{for } K)\\
g_{2} & = & a^{-1} c_1 c_4, (\text{for } -\Gamma_{18}) \\
g_{3} & = & b^{-1} c_2 c^{-1}_3, (\text{for } K)\\
g_{4} & = & b^{-1} c_2 c_4, (\text{for } \Gamma_{18})\\
g_{5} & = & a c^{-1}_1,\\
g_{6} & = & b c^{-1}_2.\end{aligned}$$ Thus, the elements ${{{g}}_{1}, \ldots, {{g}}_{{6}}}$ are used to construct the term that corresponds to $(K - \Gamma_{18})(K + \Gamma_{18})$. The next input elements $g_7, g_8, g_9, g_{10}$ serve in a similar capacity for $\Gamma^{2}_{19}$, $$\begin{aligned}
g_{7} & = & a^{-1} c_5 c_7,\\
g_{8} & = & b^{-1} c_6 c^{-1}_7,\\
g_{9} & = & a c^{-1}_5,\\
g_{10} & = & b c^{-1}_6.\end{aligned}$$ Finally, the right hand side of ${{\mathrm{KP}}}$ expression is given by $$g = [a,b]^{1},$$ and all commutators ${{{c}}_{1}, \ldots, {{c}}_{{7}}}$ have zero power in the element $g$.
$$\begin{aligned}
{{{g}^{{\varepsilon}_1}_{1} \ldots {g}^{{\varepsilon}_{6}}_{6}}} & = & a^{-\varepsilon_{1} - \varepsilon_{2}}b^{-\varepsilon_{3} - \varepsilon_{4}}a^{\varepsilon_{5}}b^{\varepsilon_{6}} c^{\varepsilon_1 + \varepsilon_2 - \varepsilon_5}_1 c^{\varepsilon_3+ \varepsilon_4 - \varepsilon_6}_2 c^{\varepsilon_1 - \varepsilon_3}_3 c^{\varepsilon_2 + \varepsilon_4}_4 = \\
& = & [a,b]^{(\varepsilon_1 + \varepsilon_2)(\varepsilon_3+\varepsilon_4)}c^{\varepsilon_1 - \varepsilon_3}_3 c^{\varepsilon_2 + \varepsilon_4}_4 = \\
& = & [a,b]^{(\varepsilon_1 + \varepsilon_2)(\varepsilon_1 - \varepsilon_2)} = (\text{put } \varepsilon_1 = K, \ \varepsilon_2 = \Gamma_{18} )\\
& = & [a,b]^{(K - \Gamma_{18})(K + \Gamma_{18})}.\end{aligned}$$
$$\begin{aligned}
g^{\varepsilon_7}_7 g^{\varepsilon_8}_8 g^{\varepsilon_9}_9 g^{\varepsilon_{10}}_{10} & = & a^{-\varepsilon_{7}}b^{-\varepsilon_{8}}a^{\varepsilon_{9}}b^{\varepsilon_{10}} c^{\varepsilon_7 - \varepsilon_9}_5 c^{\varepsilon_8 - \varepsilon_{10}}_6 c^{\varepsilon_7-\varepsilon_8}_7 = \\
& = & [a,b]^{\varepsilon_7\varepsilon_8} c^{\varepsilon_7 - \varepsilon_8}_7 = \\
& = & [a,b]^{\varepsilon^2_7} = (\text{put } \varepsilon_7 = \Gamma_{19})\\
& = & [a,b]^{\Gamma^2_{19}}.\end{aligned}$$
The two latter expressions are equivalent to the following system:
$$\left\{\begin{array}{l}\varepsilon_1 + \varepsilon_2 = \varepsilon_5; \\
\varepsilon_3 + \varepsilon_4 = \varepsilon_6; \\
\varepsilon_1 = \varepsilon_3; \\
\varepsilon_2 = -\varepsilon_4; \\
\varepsilon_7 = \varepsilon_8 = \varepsilon_9 = \varepsilon_{10}; \\
(\varepsilon_1 + \varepsilon_2) (\varepsilon_1 - \varepsilon_2) + \varepsilon_7^2 = 1;
\end{array}\right.$$ Combining everything together we get $${{{g}^{{\varepsilon}_1}_{1} \ldots {g}^{{\varepsilon}_{10}}_{10}}} = [a,b]^{(K - \Gamma_{18})(K + \Gamma_{18}) + \Gamma^2_{19}} = [a,b],$$ which gives us the desired equation [(\[system:example\])]{}.
Finally, we need $167$ basic commutators in the group $G$ to interpret all equations [(\[system:first\])]{}–[(\[system:last\])]{}. If any variable occurs $n+1$ times in our system, then we need another $n$ commutators to tie these variables. Additionally, we need $155$ commutators to tie the same variables in the equations. Hence the total number of commutators to realize the system [(\[system:first\])]{}–[(\[system:last\])]{} is $167 + 155 = 322$. The input for ${{\mathrm{KP}}}$ is given by elements $g_1, \ldots, g_{334},g$, which depend on four integer parameters $x, {\blacksquare}_z, {\blacksquare}_y, {\blacksquare}_u$.
Based on previous computations we have the following
\[lemma:main\] Let $G$ be a torsion free group of nilpotency class $2$ with ${{\mathrm{rank}}}([G, G]) > {322}$, then for every recursively enumerable set $W$ exists an input $I_W = \{g_1, \ldots, g_{334}, g\}$ such that $$x \in W \text{ iff } {{\mathrm{KP}}}\text{ has a solution in the group } G \text{ for the input } I_W.$$
For every recursively enumerable set $W$ there exist parameters ${\blacksquare}_z, {\blacksquare}_y, {\blacksquare}_u$ such that an integer $x$ lies in $W$ if and only if the system $S_W(x,{\blacksquare}_z, {\blacksquare}_y, {\blacksquare}_u)$ has a solution. Since ${{\mathrm{rank}}}([G, G]) > {322}$ we can construct an input $I_W = \{g_1, \ldots, g_{334}, g\}$ for ${{\mathrm{KP}}}$ such that the corresponding instance of ${{\mathrm{KP}}}$ for $G$ has a solution if and only if the system $S_W$ has a solution. $\square$
\[theorem:main\] Let $G$ be a torsion free group of nilpotency class $2$ and ${{\mathrm{rank}}}([G, G]) > {322}$, then group $G$ has undecidable ${{\mathrm{KP}}}$ problem.
There is set a $W$ that is recursively enumerable but is not enumerable. The statement follows by applying Lemma \[lemma:main\] to this set $W$. $\square$
Corollaries
===========
In this section we give corollaries of *Theorem \[theorem:main\]*.
\[corollary:freenilp\] Let $G$ be a free group of nilpotency class $2$ with $n$ generators. If $n$ is at least ${26}$ then the group $G$ has undecidable ${{\mathrm{KP}}}$.
Note that $G$ has $\frac{n(n-1)}{2}$ basic commutators. Since it is enough to have ${322}$ basic commutators, we see that ${26}$ generators suffice. $\square$
Let $G$ be a group of nilpotency class $2$, $H$ be its torsion subgroup, $G_1 = G / H$ be the corresponding quotient group. If ${{\mathrm{rank}}}([G_1, G_1]) > {322}$, then the group $G$ has undecidable ${{\mathrm{KP}}}$.
If $n \geq 53$ then ${{\mathrm{KP}}}$ is undecidable for groups ${UT_{n}(\mathbb{Z})}$, $GL_n({{\mathbb{Z}}})$, $SL_n({{\mathbb{Z}}})$.
Denote by $F_{k}$ the free 2-step nilpotent group of rank $k$ with generators $X = \{{{{x}}_{1}, \ldots, {{x}}_{{k}}}\}$. By Corollary \[corollary:freenilp\] the $KP$ is undecidable for the group $F_{{26}}$. By the theorem of Jennings every finitely generated torsion-free nilpotent group can be embedded into ${UT_{n}(\mathbb{Z})}$. Willem A. De Graaf and Werner Nickel [@GN:ConstructingRepresentation] give the algorithm that constructs this embedding. Hence the ${{\mathrm{KP}}}$ is undecidable for ${UT_{n}(\mathbb{Z})}$ and we only need to get an estimate of $n$. The algorithm described by De Graaf and Nickel embeds the group $F_k$ in ${UT_{n}(\mathbb{Z})}$, where $n = k + C^2_k$. We construct an embedding $\rho$ which embeds $F_n$ into ${UT_{2n+1}(\mathbb{Z})}$.
For every generator $x_i$ of the group $F_n$ we define an $(n+1)\times (n+1)$ matrix $M_i$, $$M_i =
\bordermatrix{
~& & & & i+1 & & \cr
&0 & & & & & \cr
&\vdots & & & & & \cr
i &1 & & & & & \cr
&\vdots & & & & & \cr
&0 & & & & & \cr
&1 & 0 & \cdots & 1 & \cdots & 0 \\
}.$$ Then we define the images of all $x_i$ as the following $(2n+1) \times (2n+1)$ matrices, $$\rho(x_i) =
\left(
\begin{array}{ccc|cccc}
1 & & & & & & \\
& \ddots & & & & M_i & \\
& & 1 & & & & \\
& & & & & & \\ \hline
& & & & 1 & & \\
& & & & & \ddots & \\
& & & & & & 1 \\
\end{array}
\right).$$
Now we show that the map $\rho$ extends to an embedding of $F_n$ into ${UT_{2n+1}(\mathbb{Z})}$. Denote by $U$ the image of $F_n$. Images of all generators $x_i$ are denoted by $m_i = \rho(x_i)$. It is easy to see that for any distinct $i$ and $j$ we have $[m_i, m_j] \neq E, \ [m_i,m_j] = [m_j,m_i]^{-1}$, and $[[m_i,m_j], m_k] = E$ for any $i,j,k = {1, \ldots, {n}}$, where $E$ is the $(2n+1) \times (2n+1)$ identity matrix. Thus an image under the map $\rho$ of any word in the alphabet $X \cup X^{-1}$ can be reduced to an expression ${{m}^{{\alpha}_1}_{1} \ldots {m}^{{\alpha}_{n}}_{n}} \prod y^{\beta_{ij}}_{ij}$ in the group $U$, where $\alpha_i, \beta_{ij} \in {{\mathbb{Z}}}, \ i < j, y_{ij} = [m_i, m_j]$, so the group $U$ is a two step nilpotent group with generators ${{{m}}_{1}, \ldots, {{m}}_{{n}}}$. To claim that the map $\rho: F_n \rightarrow UT_{2n+1}({{\mathbb{Z}}})$ is embedding, it remains to prove that the map $\rho$ has a trivial kernel. In other words, it suffices to show that ${{m}^{{\alpha}_1}_{1} \ldots {m}^{{\alpha}_{n}}_{n}} \prod y^{\beta_{ij}}_{ij} = E$ iff $\alpha_i = 0$ and $\beta_{ij} = 0, \ i,j = {1, \ldots, {n}}, i < j$.
Let ${{m}^{{\alpha}_1}_{1} \ldots {m}^{{\alpha}_{n}}_{n}} \prod y^{\beta_{ij}}_{ij} = E$, then ${{m}^{{\alpha}_1}_{1} \ldots {m}^{{\alpha}_{n}}_{n}} = \prod y^{-\beta_{ij}}_{ij}$. Since every $y_{ij}$ commutes with $m_i, \ i = {1, \ldots, {n}},$ we get the following, $$\begin{array}{l}
[{{m}^{{\alpha}_1}_{1} \ldots {m}^{{\alpha}_{n}}_{n}}, m_i] = [\prod y^{\beta_{ij}}_{ij}, m_i],
\end{array}$$ $$\begin{array}{l}
\prod y^{\alpha_j}_{ji} = E.
\end{array}$$
Recall that $U' = [U,U]$ is an abelian subgroup of ${UT_{2n+1}(\mathbb{Z})}$, so $U'$ is torsion free and the latter equality holds iff $\alpha_i = 0, \ i = {1, \ldots, {n}}$. Similarly, $\prod y^{\beta_{ij}}_{ij} = E$ iff $\beta_{ij} = 0, i,j = {1, \ldots, {n}}, \ i < j$. Therefore, $F_{{26}}$ is embeddable into ${UT_{53}(\mathbb{Z})}$, so ${UT_{r}(\mathbb{Z})}, r \geq 53$, has undecidable ${{\mathrm{KP}}}$. Since ${UT_{53}(\mathbb{Z})}$ is a subgroup of $GL_n({{\mathbb{Z}}})$, $SL_n({{\mathbb{Z}}}), \ n \geq 53$, then $GL_{n}({{\mathbb{Z}}})$, $SL_{n}({{\mathbb{Z}}})$ have undecidable ${{\mathrm{KP}}}$ for $n \geq 53$.
\[lemma:KPUndecidableInQuotientGroup\] Let $G$ be a finitely generated polycyclic group and $H$ be a normal subgroup of $G$ such that the quotient group $G/H$ has undecidable ${{\mathrm{KP}}}$. Then group $G$ has undecidable ${{\mathrm{KP}}}$.
Assume that the group $G$ has decidable ${{\mathrm{KP}}}$, that is there is an algorithm that solves ${{\mathrm{KP}}}$ problem in $G$. Let $A$ denote the quotient group $G/H$. Suppose we have an input for ${{\mathrm{KP}}}$ in the group $A$: $a_1H, a_2H, \ldots, a_kH, aH$, where $a_i, a \in G$. To solve ${{\mathrm{KP}}}$ we are required to find numbers ${{{\epsilon}}_{1}, \ldots, {{\epsilon}}_{{n}}} \in {{\mathbb{Z}}}$ such that $$\label{eq:KPinGroupA}
(a_1H)^{\epsilon_1}(a_2H)^{\epsilon_2}\ldots (a_kH)^{\epsilon_k} = a H.$$ This equation is equivalent to the following: $$a_1^{\epsilon_1}H a_2^{\epsilon_2}H \ldots a_k^{\epsilon_k}H = a H,$$ $$a_1^{\epsilon_1} a_2^{\epsilon_2} \ldots a_k^{\epsilon_k} H = a H,$$ $$\exists h \in H \ a_1^{\epsilon_1} a_2^{\epsilon_2} \ldots a_k^{\epsilon_k} h = a.$$
If $H$ is a finitely generated polycyclic group then there exists $b_1, \ldots, b_m \in H$ such that for any $h \in H$ there are integers $k_1, \ldots, k_m$ that $h = b_1^{k_1} \ldots b_m^{k_m}$. Hence if we solve ${{\mathrm{KP}}}$ problem $a_1^{\epsilon_1} a_2^{\epsilon_2} \ldots a_k^{\epsilon_k} b_1^{k_1} \ldots b_m^{k_m} = a$ in the group $G$, we get solution of ${{\mathrm{KP}}}$ (\[eq:KPinGroupA\]) in the group $A$. This contradicts the assumption that the group $A$ has undecidable ${{\mathrm{KP}}}$. $\square$
Let $G$ be a polycyclic group and $Fit(G)$ have rank of derived subgroup greater or equals than $322$. Then ${{\mathrm{KP}}}$ is undecidable in $G$.
Since $G$ is a polycylcic group then $F = Fit(G)$ is a nilpotent group. Thus $F' = F / [[F,F],F]$ is a nilpotent class two group with rank of derived subgroup greater or equal to $322$. By *Theorem \[theorem:main\]* the ${{\mathrm{KP}}}$ is undecidable for $F'$ and by *Lemma \[lemma:KPUndecidableInQuotientGroup\]* the ${{\mathrm{KP}}}$ is undecidable for the group $G$.
Let $G$ be a nilpotent group of class $c \geq 3$ with lower central series $$G = G_1 \trianglerighteq G_2 \trianglerighteq \ldots \trianglerighteq G_c \trianglerighteq G_{c+1} = \{1\},$$ where $G_{k+1} = [G_k, G]$, $k = {1, \ldots, {c}}$. Let $N$ be the quotient group $G/G_3$. If $rank([N,N]) > {322}$ then the group $G$ has undecidable ${{\mathrm{KP}}}$.
The group $N$ has undecidable ${{\mathrm{KP}}}$ by Corollary \[corollary:freenilp\]. Hence, the group $G$ has undecidable ${{\mathrm{KP}}}$ problem by Lemma \[lemma:KPUndecidableInQuotientGroup\]. $\square$
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|
---
abstract: 'We consider nonequilibrium systems such as the Edwards-Anderson Ising spin glass at a temperature where, in equilibrium, there are presumed to be (two or many) broken symmetry pure states. Following a deep quench, we argue that as time $t \to \infty$, although the system is usually in some pure state locally, either it [*never*]{} settles permanently on a fixed lengthscale into a single pure state, or it does but then the pure state depends on [*both*]{} the initial spin configuration [*and*]{} the realization of the stochastic dynamics. But this latter case can occur only if there exists an uncountable number of pure states (for each coupling realization) with almost every pair having zero overlap. In both cases, almost no initial spin configuration is in the basin of attraction of a single pure state; that is, the configuration space (resulting from a deep quench) is all boundary (except for a set of measure zero). We prove that the former case holds for deeply quenched $2D$ ferromagnets. Our results raise the possibility that even if more than one pure state exists for an infinite system, time averages don’t necessarily disagree with Boltzmann averages.'
author:
- |
[**C. M. Newman**]{}[^1]\
[newman@cims.nyu.edu]{}\
[*Courant Institute of Mathematical Sciences*]{}\
[*New York University*]{}\
[*New York, NY 10012, USA*]{}
- |
[**D. L. Stein**]{}[^2]\
[dls@physics.arizona.edu]{}\
[*Depts. of Physics and Mathematics*]{}\
[*University of Arizona*]{}\
[*Tucson, AZ 85721, USA*]{}
title: '[Equilibrium Pure States and Nonequilibrium Chaos]{}'
---
6.5in 8.5 in
[**KEY WORDS:**]{} spin glass; nonequilibrium dynamics; deep quench; stochastic Ising model; broken ergodicity; coarsening; persistence; damage spreading.
Introduction {#sec:intro}
============
In this paper we study nonequilibrium dynamics of spin systems. While our approach is general, and covers both ordered and disordered, Ising and non-Ising systems, in the presence or absence of a magnetic field, we will for specificity here focus mostly on the dynamics of the Edwards-Anderson$^{\cite{EA}}$ Ising spin glass in zero field. We make no assumptions about the real- or state-space structure of the low-temperature spin glass phase, but instead derive several general principles and then explore their consequences.
Our results indicate that equilibrium pure state structure plays an important role in nonequilibrium dynamics. E.g., we will show that (at fixed temperature) a system with many pure states may have very different dynamical behavior than one with only a single pair[^3] — so nonequilibrium dynamics can serve as an important probe of the equilibrium pure state structure.
That the number of and relationships among pure states can affect nonequilibrium dynamics may seem surprising in light of a general supposition (see, for example, Refs. [@BY; @MPV] and many of the references therein) that the time evolution of an infinite system is confined within a single pure state for all finite times. A consequence of this supposition is the prevalent notion that, if there exists at some temperature $T$ more than one pure state (e.g., due to broken symmetry), then necessarily the limits $t\to\infty$ (time) and $N\to\infty$ (system size) do not commute (when, say, measuring the state of some observable). An equivalent statement is that time averages (as performed in the lab) and Boltzmann averages (as performed on paper or the computer) will give differing results. (One proposal$^{\cite{HJY}}$ for avoiding this problem in theoretical treatments of spin glass dynamics is to start with a Boltzmann distribution of initial configurations, rather than a single one.) We will argue, however, that these are not [*necessary*]{} consequences of broken symmetry or multiple pure states; they may be true in some cases, but not in general. We examine here in which contexts they are true and in which they are not[^4]. We begin by constructing dynamical probability measures on spin configurations, which will enable us to clarify notions such as time evolution within a pure state.
Dynamical Measures {#sec:measures}
==================
We will mostly, but not exclusively, consider the EA Ising spin glass in zero field. Its Hamiltonian is given by: $$\label{eq:EA}
{\cal H}_{\cal J} = -\sum_{<xy>} J_{xy}\sigma_x \sigma_y\, ,$$ where the sites $x,y\in {\bf Z}^d$ and the sum is taken over nearest neighbors only. The couplings $J_{xy}$ are independent random variables, whose common probability density is symmetric about zero; we let ${\cal J}$ denote a particular realization of the couplings.
We consider Glauber dynamics of an [*infinite*]{} EA Ising spin glass starting from an initial (infinite-volume) spin configuration $\sigma^0$. We regard $\sigma^0$ as chosen from the (infinite temperature) ensemble where the individual spins are independent random variables equally likely to be $+1$ or $-1$. This corresponds to the experimental situation following a deep quench, and also (for smaller systems) most numerical simulations. We denote by $\omega$ a given realization of the dynamics. So there are three sources of randomness in the problem, with realizations given by ${\cal J}$, $\sigma^0$, and $\omega$. All three are needed to determine the spin configuration at time $t$ (we take the starting time to be 0). From here on we take ${\cal J}$ to be fixed. We note that the exact choice of spin flip rates plays no role in our analysis at positive temperature, as long as detailed balance is satisfied. At $T=0$ we consider only the widely used dynamical rule where flips that are energy lowering (or neutral or raising) occur at rate $1$ (or $1/2$ or $0$). In all cases, $\omega$ can be regarded in the usual way as a collection of random times $(t_{x,i}:\,x \in {\bf Z}^d,\,i=1,2,\dots)$ when spin flips are considered (forming a Poisson process for each $x$) along with uniformly distributed random numbers $u_{x,i}$ that determine if the flips are taken.
We now define a dynamical probability measure $\nu_{t^*,\tau(t^*)}$ on the spin configurations, for $0 \le \tau(t^*) \le t^*$. Given some ${\sigma^0}$, we let the system evolve according to some $\omega$ up to a time $t^*-\tau(t^*)$, after which we average over the dynamics up to time $t^*$. That is, if we denote the dynamical (Markov) process as $\sigma^t = \sigma^t(\omega)$ for $t \ge 0$, then $\nu_{t^*,\tau(t^*)}$ is the conditional distribution of $\sigma^{t^*}$ conditioned on (${\cal J}$ and ${\sigma^0}$ and) all $(t_{x,i},u_{x,i})$’s with $t_{x,i} \le t^*-\tau(t^*)$. So $\nu_{t^*} \equiv
\nu_{t^*,t^*}$ represents a complete averaging over the dynamics (and corresponds to the usual dynamical measure—i.e., to the distribution of $\sigma^{t^*}$ for fixed ${\cal J}$ and ${\sigma^0}$), while $\nu_{t^*,0}$ represents no averaging at all (and hence a single spin configuration). To avoid awkward notation, we generally suppress the dependence, which is understood, of $\nu$ on ${\cal J}$, $\sigma^0$, $\omega$ (up to time $t^* - \tau$), and $T$. We also note that neither $t^*$ nor $\tau$ depend on ${\cal J}$, $\sigma^0$ or $\omega$. In Section \[sec:fixed\] we will briefly discuss the construction of measures based instead on time averaging for [*fixed*]{} dynamics.
We can now begin to answer the question, what does it mean for the system to evolve or settle into (or “spend all its time inside”) a single pure state? Consider the cube $\Lambda_L$ of linear size $L$ and volume $|\Lambda_L|$ (which may be arbitrarily large) centered at the origin. When $T>0$ (but not at $T=0$), we expect that for sufficiently large $t^*$ (depending on $L$) and for almost every $\sigma^0$, the measure $\nu_{t^*}$ approximates a (possibly mixed, possibly $t^*$-dependent) Gibbs state [*restricted to the cube $\Lambda_L$*]{}. By this we mean that there is some (infinite volume) Gibbs state $\rho_{t^*}$ such that for any $L$ and any spin configuration $\sigma^{(L)}$ in $\Lambda_L$, the probability assigned to $\sigma^{(L)}$ by the dynamical measure $\nu_{t^*}$ and that assigned by the equilibrium Gibbs measure $\rho_{t^*}$ approach each other as $t^*\to\infty$; that is, $\nu_{t^*}(\sigma^{(L)}) - \rho_{t^*}(\sigma^{(L)})\to 0$ as $t^* \to \infty$. More generally, we expect that for almost every $\sigma^0$ and $\omega$, $\nu_{t^*,\tau(t^*)}$ approximates some Gibbs state $\rho_{t^*,\tau(t^*)}$ providing only that $\tau(t^*) \to \infty$.
The notion that, as $t^*$ increases, $\nu_{t^*,\tau(t^*)}$ is increasingly well approximated by some infinite volume Gibbs state (possibly depending on $t^*$), may seem surprising — especially in view of frequent assertions (see, e.g., Ref. [@CK]) that equilibrium states are of little relevance for the nonequilibrium dynamics of infinite systems. In fact, this notion is nothing more than the property that for any $L$ and any two spin configurations $\sigma^{(L)}, \sigma'^{(L)}$, defined within $\Lambda_L$ and agreeing on its (internal) boundary, the ratio $\nu_{t^*,\tau}(\sigma^{(L)})/\nu_{t^*,\tau}(\sigma'^{(L)})$ converges to the usual [*finite*]{} volume Gibbs expression coming from the EA Hamiltonian. This kind of convergence may be expected as it is similar to the conjectured property of Glauber dynamics that [*only*]{} Gibbs states are stationary.
To answer the question posed above about the meaning of settling into a single pure state, we now note that if $\tau(t^*)$ also grows sufficiently slowly, then $\nu_{t^*,\tau(t^*)}$ should[^5] (for most $\omega$’s) approximate a [*pure*]{} (i.e., extremal) Gibbs state $\rho^{\alpha(t^*)}$ depending on $\sigma^{0}$ and $\omega$ (up to time $t^* -\tau$). If on every fixed (and arbitrarily large) lengthscale $L$ this $\rho^{\alpha}$ eventually becomes independent of time (after a timescale depending on $L$), then the system has settled into the pure state $\alpha$.
We now present our main results. Unless otherwise indicated, all are for $T>0$ and are independent of space dimension. Our first result concerns the fully averaged dynamical measure $\nu_{t^*}$.
Evolution of the Dynamical Measure {#sec:evolution}
==================================
[*Theorem 1.*]{} Given some ${\cal J}$, assume that for almost every $\sigma^0$, $\nu_{t^*}$ converges to a limiting [*pure*]{} Gibbs state $\nu_{\infty}$ as $t^*\to\infty$. Then $\nu_{\infty}$ is the same pure state for almost every $\sigma^0$.
[*Proof.*]{} We use a coupling argument where a single $\omega$ is used with two starting spin realizations $\sigma^0$ and $\sigma'^{\,0}$ that differ at only finitely many sites. Then, by the nature of Glauber dynamics, there is a positive probability (with respect to the $\omega$’s) that the two spin configurations merge after a finite time. Let $\theta>0$ represent the probability that the difference in configurations disappears by time $t_0$. So for $t^* \ge t_0$, $$\label{eq:one}
\nu_{t^*}^{\sigma^0}=\theta\mu_{t^*}+(1-\theta)\tilde\mu_{t^*}\, ,\,
\nu_{t^*}^{\sigma'^0}=\theta\mu_{t^*}+(1-\theta)\tilde\mu'_{t^*}\,,$$ where $\mu_{t^*}$, $\tilde\mu_{t^*}$, and $\tilde\mu'_{t^*}$ are some probability measures. It follows that for $A$ any (measurable) set of spin configurations, $$\label{eq:coupling}
|\nu_{t^*}^{\sigma^0}(A) - \nu_{t^*}^{\sigma'^0}(A)|\, \le \,1-\theta\,.$$ The same is true with ${t^*}$ replaced by $\infty$, first for $A$ a locally defined event and then, by approximation, for general $A$. The strict positivity of $\theta$ then implies that $\nu_{\infty}^{\sigma^0}$ and $\nu_{\infty}^{\sigma'^0}$ cannot be mutually singular measures (i.e., living on completely disjoint regions of configuration space) and hence$^{\cite{Georgii}}$, as pure states, they must be identical. So a change of finitely many spins in $\sigma^0$ doesn’t change $\nu_{\infty}^{\sigma^0}$, and we can then use the Kolmogorov zero-one law$^{\cite{Feller}}$ to conclude that the final pure state must be independent of $\sigma^0$. $\, \diamond$
Despite the innocuous look of the theorem, it has important consequences. Its conclusion is obvious if there exists only one pure state, but it applies equally to the situation where many pure states exist. Of course, it could logically be that only one pure state is “present” (in the sense that the conclusion of the theorem is valid) while other pure states exist but are not physically relevant in our dynamical context. (This might even be the case, for example, in the EA spin glass with a small nonzero field.) But if the conclusion of the theorem is not valid, then only one of two possibilities can occur[^6]: either (1) $\nu_{\infty}$ is a mixed Gibbs state (which may or may not depend on $\sigma^0$), or (2) $\nu_{t^*}$ does not converge as $t^* \to \infty$. We note that the latter case is already known to occur$^{\cite{FIN}}$ in some $1D$ disordered ferromagnets at $T=0$; on the other hand, if $\nu_{\infty}$ exists and does not depend on $\sigma^0$, then it would be analogous to (and perhaps the same as) $\rho_{\cal J}$, the average over the metastate discussed in Ref. [@NS97].
Let’s consider each of these two possibilities, taking into account that $\nu_{t^*}$ is the average over $\omega$’s of $\nu_{t^*,\tau(t^*)}$. Possibility (1) implies that although $\nu_{t^*,\tau(t^*)}$ (with properly chosen $\tau$) is approximately a pure state $\rho^{\alpha(t^*)}$, that pure state depends not only on $\sigma^0$ (as expected) but also on $\omega$. This allows (but doesn’t require — see Subsection \[subsec:ctd\]) the system always to “land” in a pure state in the sense that $\nu_{t^*,\tau(t^*)} \to \rho^{\alpha(\sigma^0,\omega)}$ — but then the pure state is (almost) never determined solely by $\sigma^0$.
Now, the basin of attraction of a pure state $\bar{\alpha}$ may be defined as the set of $\sigma^0$’s such that $\alpha(\sigma^0,\omega) = \bar{\alpha}$ for almost every $\omega$ (see Ref. [@vEvH] for related discussions). We claim that if $\sigma^0$ is in the basin of attraction of some pure state, then by a modified version (see below) of the coupling argument used in the proof of Theorem 1, the same will be true after a change of finitely many spins in $\sigma^0$ and so, by the Kolmogorov zero-one law, the set of $\sigma^0$’s that are in some basin of attraction has probability either zero or one. Therefore, if many pure states are present (i.e., if $\alpha(\sigma^0,\omega)$ is not the same for almost all $\sigma^0$ and $\omega$), then [*the union of all their basins of attraction must form a set of measure zero in the space of $\sigma^0$’s*]{}; i.e., the configuration space resulting from a deep quench is all “boundary” in the sense that almost every initial configuration will land in one of several (or many) pure states depending on the realization of the dynamics (if it lands at all).
The modified coupling argument is as follows. Let $\sigma^0$ and $\sigma'^{\,0}$ be fixed and let $D$ denote the finite set of $x$’s where they differ. Rather than using the same $\omega$ for the coupled processes, we take an $\omega = (t_{x,i},u_{x,i})$ and an $\omega' = (t'_{x,i},u'_{x,i})$ that are identical for $x$ outside $D$ but for $x$ inside $D$, they are identical only for times after $\sigma^t$ and $\sigma'^{\,t}$ merge. For earlier times, $\omega$ and $\omega'$ inside $D$ are independent of each other. If $A'$ is an event defined in terms only of $\omega'$ with Prob$(A') > 0$ and $M$ denotes the event of eventual merger of the two processes, then since $\omega$ inside $D$ may (with small but strictly positive probability) force a merger by a small time $\varepsilon$, it follows that Prob$(A' \cap M) > 0$. Assuming $\alpha(\sigma^0,\omega) = \bar{\alpha}(\sigma^0)$ for almost every $\omega$, we take $A'$ to be the complement of the event that $\alpha(\sigma'^{\,0},\omega') = \bar{\alpha}(\sigma^0)$ so that Prob$(A' \cap M) > 0$ would yield the contradiction that Prob$(\alpha(\sigma^0,\omega) = \bar{\alpha}(\sigma^0)) < 1$. It follows that Prob$(A') = 0$ and so $\alpha(\sigma'^{\,0},\omega') = \bar{\alpha}(\sigma^0)$ for almost every $\omega'$, which is exactly the claim made in the previous paragraph.
It has been speculated$^{\cite{KL}}$ that slow relaxation in spin glasses may be due to points in (high-dimensional) state space always being “near” a boundary. What we’ve shown here differs in fundamental respects: our conclusion is that almost every point in state space is actually [*on*]{} a boundary, and therefore the dynamical consequences are not restricted to very low temperatures.
We note finally that Theorem 1 may be relevant to damage spreading$^{\cite{Bag,Grass,JR}}$, where one asks whether the damage (i.e., discrepancy) between $\sigma^t$ and $\sigma'^{\,t}$ (with a single $\omega$) grows as $t \to \infty$. Theorem 1 suggests that if damage spreading occurs, then $\nu_{t^*}$ doesn’t converge to a single pure state (e.g., it might converge to a mixed state, as above).
Local Non-Equilibration {#sec:lne}
=======================
Before discussing possibility (2), let us consider the physical picture implied by Theorem 1. Roughly speaking, some time after an initial quench the system will form domains, whose average size increases with time, corresponding to the different pure states. This scenario has been analyzed for the two-state droplet picture$^{\cite{FH,KH,TH}}$. It is also a well-known scenario for coarsening in a ferromagnet following a deep quench$^{\cite{Bray}}$. (Of course, in contrast to the spin glass case, one [*does*]{} know how to prepare a ferromagnet in a pure state; for a general discussion, see Ref. [@Palmer].)
In Ref. [@Bray] it was stated that the infinite homogeneous ferromagnet never reaches equilibrium in any finite time (following a deep quench) because the domain sizes (in this case, of positive and negative magnetization) increase with time but are never infinite on any finite timescale. We do not consider this by itself to be nonequilibration because it does not preclude the possibility that on any [*finite*]{} lengthscale, the system equilibrates after some finite time, in the sense that after that time domain walls cease to move across the region. Instead, we now propose a much stronger version of nonequilibration — the possibility of [*local non-equilibration*]{} (LNE) on any [*finite*]{} lengthscale, which is implied by possibility (2) and could also occur with possibility (1). We will discuss the difference between these two cases of LNE in Subsection \[subsec:ctd\], but for now will not distinguish between the two. We will also discuss below the relation between LNE and persistence exponents$^{\cite{BDG,DBG,DG,MBCS}}$.
By LNE we mean that in any fixed finite region, the system never settles down into a pure state. That is, domain walls do not simply move farther from the region as time progresses, but continually return and sweep across it, changing the state within. More precisely, LNE is said to occur unless there is some choice of $\tau(t^*)$ such that for almost all $\sigma^0$ and $\omega$, $\nu_{t^*,\tau(t^*)} \to \rho^{\alpha(\sigma^0,\omega)}$, a [*pure*]{} state. If LNE occurs, it would force us to revise the usual dynamical definition$^{\cite{BY}}$ of, e.g., the EA order parameter. It could also mean that, for infinite systems, time averages and Gibbs averages could agree, despite the presence of many pure states. We will return to this in Section \[sec:fixed\] after investigating LNE in more detail by means of the next theorem, which applies to both homogeneous and disordered systems.
[*Theorem 2.*]{} If only a single pair (or countably many, including a countable infinity) of pure states exists (with fixed ${\cal J}$) and these all have nonzero EA order parameter, then LNE occurs.
[*Proof.*]{} Suppose that there exists at $T$ (and for the given ${\cal J}$) only a single pair of pure states, and assume that LNE does [*not*]{} occur so that for each ${\sigma^0}$ and $\omega$, there is a limiting pure state $\alpha(\sigma^0,\omega)$. The overlap of $\alpha(\sigma^0,\omega)$ and $\alpha(\sigma'^0, \omega')$ is $$\label{eq:overlap}
Q({\cal J},\sigma^0,\omega,\sigma'^0, \omega') \,=\,
\lim_{L \to \infty} |\Lambda_L|^{-1} \sum_{x \in \Lambda_L}
<\sigma_x>_{\alpha(\sigma^0,\omega)}<\sigma_x>_{\alpha(\sigma'^0, \omega')}\, ,$$ where $<\cdot>_{\alpha}$ denotes the (thermal) average with respect to the pure state $\rho^{\alpha}$. The possible overlap values can be only $\pm q_{EA}$ and both of these outcomes must have positive probability of occurring (as $\sigma^0, \omega, \sigma'^0, \omega'$ vary independently to yield a pair of replicas). But the translation-ergodicity of each of the distributions from which ${\cal J}$, $\sigma^0, \omega, \sigma'^0, \omega'$ are chosen implies the same for the joint (product) distribution of $({\cal J},\sigma^0,\omega,\sigma'^0, \omega')$. Thus, the fact that the overlap $Q$ is a translation-invariant function implies that Q must be constant for almost all realizations, leading to a contradiction. This argument can be immediately extended to any countable number of pure state pairs. $\, \diamond$
This proof also shows that if LNE does not occur (and the limiting pure states $\alpha(\sigma^0,\omega)$ have nonzero $q_{EA}$), then almost every (as $\sigma^0,\omega,\sigma'^0, \omega'$ vary) overlap of the pair of pure states, $\alpha(\sigma^0,\omega)$ and $\alpha(\sigma'^0, \omega')$, is zero. This leads to:
[*Corollary*]{}. If LNE does not occur and $q_{EA} \ne 0$, then there must be an [*uncountable*]{} number of pure states, with almost every pair (in the above sense) having overlap zero.
This shows that, as stated in the introduction, nonequilibrium dynamics provides important information on the structure of equilibrium pure states. It also suggests a dynamical test of the two-state picture: search for chaotic time dependence in $\nu_{t^*,\tau(t^*)}$. If LNE does not occur, then the two-state picture has been ruled out: there must be an uncountable number of states, almost all of which have overlap zero (consistent with the results of Ref. [@NS97]). If LNE does occur then neither the two-state nor the many-state pictures have been ruled out.
How might one go about observing LNE in a spin glass, where, unlike the ferromagnet, one doesn’t know what a domain wall looks like? Here one can use the clustering property that characterizes pure states in general. E.g., a truncated 2-point correlation function of the form $<\sigma_x \sigma_0>-<\sigma_x><\sigma_0>$ approaches 0 as $|x|\to\infty$ if the averaging is done in a pure state, but not otherwise. In the current context a pure state average corresponds to a dynamical average using $\nu_{t^*,\tau}$ with $\tau\ll t^*$. In principle one could evaluate this correlation function numerically for $t^*\gg\tau\gg|x|\gg 1$; if it does not approach zero for increasing values of these parameters, that would constitute a clear signal of the occurrence of LNE.
An important consequence of Theorem 2 is that LNE must also occur at positive temperature in the $2D$ uniform Ising ferromagnet and the random Ising ferromagnet for $d<4$. (However, the argument for LNE in random ferromagnets is not entirely rigorous as there is no complete analysis of interface pure states there$^{\cite{For}}$. There is though a rigorous proof that for the SOS approximation, these states do exist for $d \ge 4$ and do not for $d < 4$; see Refs. [@BK; @BK2]).
Moreover, in the $2D$ homogeneous ferromagnet (on the square lattice) results on LNE can be extended to $T=0$, in the sense there that $\sigma^t$ does not converge as $t\to\infty$; in fact, [*every*]{} spin flips infinitely often. The proof (by contradiction) is based on showing that if some spin (say at the origin) remained fixed forever, then (by translation ergodicity and spin flip symmetry) so would two spins of opposite sign on the $x$ and $y$ axes. But then there would be a domain wall passing through the rectangle determined by these three spins and “cutting off” one of them. In every time interval there would be a nonzero probability of the domain wall moving to flip that spin, and so with probability one, it eventually [*would*]{} flip. For the full proof, see Ref. [@NN].
It is also shown in Ref. [@NN] that for many systems $\sigma^t$ [*does*]{} converge to some limit at $T=0$. This is based on a very general argument that there can be only finitely many flips at any site that strictly lower the energy. Examples include spin glasses and random ferromagnets where the common distribution of the $J_{xy}$’s is continuous, and either has a finite mean (as in most ordinary models) or else is sufficiently “spread out”, as in the highly disordered spin glass or ferromagnet$^{\cite{NS94}}$. Other examples are homogeneous ferromagnets (or homogeneous antiferromagnets or $\pm J$ spin glasses) on lattices with an odd number of nearest neighbors, such as the $2D$ hexagonal lattice or a double-layered $2D$ square lattice. (There are also systems, such as the $\pm J$ spin glass on the square lattice, where some spins flip only finitely many times and some spins flip infinitely often$^{\cite{GNS}}$.) In light of these results, we restrict the term LNE to $T>0$, since in the zero-temperature situations where $\sigma^t$ converges, the limit configuration is typically only metastable rather than a ground state and so equilibration has not really occurred. In these systems one can define a dynamical order parameter, related to the autocorrelation, that does [*not*]{} decay to zero.
To further clarify the discussion of LNE for the ferromagnet, we note that it is a phenomenon separate from the spontaneous formation (at positive temperature) of domains of, say, the minus phase within the plus phase. That is, on a timescale exponential in $L$, there will form a domain containing the origin (of size $L$) of the minus phase. Similarly, for a finite system of size $L$, the entire system will flip back and forth between the plus and minus phases on a timescale exponential in $L$. However, this is different from LNE, which presumably takes place on time scales of some power of $L$. Also, the spontaneous formation of droplets described above cannot occur at $T=0$, but as already discussed, in the $2D$ ferromagnet the phenomenon of domain walls forever sweeping across any finite region persists at zero temperature.
Since the existence of LNE for all $T<T_c$ in the $2D$ Ising ferromagnet may seem surprising, we present a possible physical mechanism for this case which may also shed light on LNE in general. The initial spin configuration has (with probability one) no infinite domains. As the configuration evolves, some domains shrink and others coalesce. So the origin should always be contained in a finite domain, whose size will usually be slowly decreasing, but sporadically will have a large change either by coalescing or because a domain wall passes through the origin and the identity of the domain changes. Thus LNE is primarily the result of nonequilibrium domain wall motion (driven by mean curvature) combined with the complex domain structure resulting from the original quench. It is also consistent with phase separation (as would be expected from equilibrium roughening arguments). In particular, the occurrence of LNE does not preclude the divergence with $t$ of the [*mean*]{} scale of the domain containing the origin (although for [*fixed*]{} $\sigma^0$ and $\omega$, there is a much more complex behavior, as indicated above).
LNE and chaotic time dependence {#subsec:ctd}
-------------------------------
We noted earlier in this section that LNE can occur in the context of either possibility (1) (the fully averaged dynamical measure $\nu_{t^*}$ has a limit, which is a mixed state) or (2) ($\nu_{t^*}$ does not converge). That is, LNE [*must*]{} occur if possibility (2) holds, but may or may not occur if possibility (1) holds. We now explore further the distinctions between the two cases of LNE.
As described earlier, one way for possibility (1) to occur is if, for some choice of $\tau(t^*)$, $\nu_{t^*,\tau(t^*)} \to \rho^{\alpha(\sigma^0,\omega)}$ for almost all $\sigma^0$ and $\omega$, where $\rho^{\alpha(\sigma^0,\omega)}$ is some pure state. But it could also happen that for any choice of $\tau(t^*)<<t^*$, $\nu_{t^*,\tau(t^*)}$ never settles down to a single pure state — so the system is usually in a pure state locally, but the pure state forever changes. Nevertheless, a full average over the dynamics (i.e., letting $\tau = t^*$) still yields a single limit. This is to be contrasted with possibility (2), where even the fully averaged measure never settles down.
To clarify these statements, we use the illustration of the $2D$ homogeneous ferromagnet below $T_c$, where we know LNE to occur by Theorem 2. Suppose that possibility (1) occurs. Then (for fixed $\sigma^0$ and large time) for approximately half of the dynamical realizations, a region of fixed lengthscale $L$ surrounding the origin is in the up state (i.e., the pure Gibbs state $\rho^+$), and for most of the other half the same region is in the down state (the pure Gibbs state $\rho^-$), and this one-to-one ratio remains essentially fixed after some timescale depending on $L$. Then as $t^*\to\infty$, $\nu_{t^*}\to\overline\rho$, where $\overline\rho$ is the mixed Gibbs state $(1/2)\rho^+ + (1/2)\rho^-$. Nevertheless, in any given dynamical realization (with averaging done as usual after time $t^*-\tau$, with $1<<\tau<<t^*$), the region never settles permanently into either $\rho^+$ or $\rho^-$.
By contrast, if possibility (2) occurs, then even the fully averaged dynamical measure $\nu_{t^*}$ forever changes. This could happen (again for fixed $\sigma^0$) if the random dynamics fails to sufficiently “mix” the states (in which case one has, given $\sigma^0$, some amount of predictive power for determining from $\sigma^0$ the likely state of the system in the region for arbitrarily large times $t^*$). This is conceivable because even though $\sigma^0$ is globally unbiased between the plus and minus states, it does have fluctuations in favor of one or the other state of order $\sqrt{(L^*)^2}$ on lengthscale $L^*$; with $L^*$ taken as an appropriate power of $t^*$, these fluctuations could (partially) predict the sign of the phase at the origin at time $t^*$. In possibility (1) on the other hand, there is a greater capability of the random dynamics to “mix” the states which eventually destroys the predictive power contained in the fluctuations of the initial state.
So there are really two kinds of non-equilibration, corresponding either to LNE in the framework of possibility (1) (“weak LNE”) or else to LNE resulting from the stronger possibility (2). Because $\nu_{t^*}$ evolves deterministically according to an appropriate master equation, its lack of a limit in possibility (2) corresponds to the usual notion of deterministic chaos and can thus legitimately be called chaotic time dependence (CTD)$^{\cite{FIN}}$. If weak LNE occurs, this term is not appropriate because here the effect is due to the random dynamics.
In our discussion of LNE in the previous section, we do not yet know which of the cases correspond to CTD and which to weak LNE. This remains a problem for future investigation. However, we note that the occurrence of LNE (but not CTD) in homogeneous ferromagnets (on ${\bf Z}^d$) is implicit in the (nonrigorous for $d \neq 1$) analysis of persistence exponents$^{\cite{BDG,DBG,DG,MBCS}}$ in the sense that the fraction of sites that remain in the same phase from time $t_1$ to time $t_2$ tends to zero for $1 << t_1 << t_2$. On the other hand, the fact mentioned above that the $T=0$ analogue of this phenomenon is [*lattice dependent*]{} appears to have gone unnoticed.
Time-averaged dynamical measures {#sec:fixed}
================================
We return to one issue that needs discussion, and will be treated in greater detail elsewhere. If LNE occurs (e.g., because only a single pair of pure states is present), would a (long) time average of, say, the spin at the origin (or, for the ferromagnet, the magnetization in a finite region), give zero? The answer is: not necessarily. It could be that, after long times, the system has spent roughly equal amounts of time in both states, in which case the usual time average$^{\cite{BY}}$ (or a discrete average over equally spaced times) [*would*]{} approach zero. But it could also happen that, after any long time, the system has spent significantly more of its life in one or the other state (which itself would change with the observational timescale). In other words (still using the example of a two-state system), at any long time the weights of the two states, as defined by a dynamical measure involving a fixed ${\omega}$ and an average over uniformly spaced times, could be different from $1/2$, and moreover they may change with time. This is analogous to an equilibrium phenomenon discovered by Külske$^{\cite{Kuelske}}$ and seems to be exactly what occurs in homogeneous ferromagnets, as reported in recent numerical studies$^{\cite{DG}}$. To get a zero average in this situation one would need to average over a [*sparse*]{} sequence of increasingly separated times.
Summary {#sec:summary}
=======
We have presented a rigorous approach to the dynamics of infinite spin systems which introduced various dynamical measures on the spin configurations, and considered whether they evolve into pure states. We showed that in the case of the EA Ising spin glass with broken spin-flip symmetry, one (or both) of two interesting things must happen: either LNE occurs, where the system never settles into any pure state (i.e., domain walls forever pass through any finite region, causing it forever to change its pure state)[^7], or else there exist uncountably many pure states, with almost every pair having zero overlap.
We proved that the union of the basins of attraction of all pure states (again, if broken symmetry occurs) forms a set of measure zero in configuration space; i.e., almost every starting configuration is on a boundary between (several or possibly all) pure states. While this is also true for the ferromagnet, it obviously is still easy to prepare that system in a pure state. But this result has serious dynamical consequences for the spin glass, and not only for deep quenches. Because of the possibility of chaotic temperature dependence [@BM; @FH88], it is potentially relevant even for small temperature changes made slowly. Experimentally observed slow relaxation and long equilibration times in spin glasses may therefore be a consequence of small (relative to the system) domain size and slow (possibly due to pinning) motion of domain walls.
More generally, we have argued against a common viewpoint that pure state multiplicity is irrelevant to the dynamics of infinite (or very large) systems on finite timescales. A system need not — and in several cases, does not — spend all of its time in a single pure state, even locally. Because of this, it is also not necessarily true that “absolutely broken ergodicity” — i.e., the presence of more than one pure state separated by infinite barriers — implies that time averages and Boltzmann averages must disagree (or equivalently, that the limits $N\to\infty$ and $t\to\infty$ cannot commute). Both averages can be zero if for each state the (infinite) system locally spends (roughly) equal amounts of time in it and its global flip, as discussed in Section \[sec:fixed\]. The other possibility is that after almost any long time, the system has spent more of its life in one or the other state (which itself would change with the observational timescale). If this is the case, the averages should disagree, but due to a mechanism different from the standard one. Further development of these ideas, and a discussion of their application to experiment, will be presented in a future paper.
[*Acknowledgments.*]{} We thank Anton Bovier, David Huse and Christof Külske for useful comments. CMN thanks WIAS, Berlin for its hospitality and DLS thanks the Aspen Center for Physics.
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[^1]: Partially supported by the National Science Foundation under grant DMS-95-00868 and DMS-98-02310.
[^2]: Partially supported by the U.S. Department of Energy under grant DE-FG03-93ER25155 and by the National Science Foundation under grant DMS-98-02153.
[^3]: We emphasize that we are talking here about the equilibrium pure states present [*at*]{} the temperature at which the dynamics is observed, as distinguished from the proposal (see, e.g., Ref. [@HLOOV]) that metastable states with $O(1)$ free energy barriers affecting dynamics at a given temperature are the precursors of equilibrium pure states at lower temperatures
[^4]: Except in some comments about other papers, we will not refer to metastable states, often proposed as responsible for the anomalous dynamical behavior of spin glasses. While we have no argument with this, we question the usefulness of the usual practice of inserting metastability by hand, requiring a guess as to the structure (usually in state space) and nature of the metastable states. In the treatment presented here, metastability may emerge naturally.
[^5]: The tracking of $\nu_{t^*,\tau(t^*)}$ by the [*pure*]{} state $\rho^{\alpha(t^*)}$ is in general weaker than the tracking by $\rho_{t^*,\tau(t^*)}$ in that $\alpha(t^*)$ may not have a limit and $\nu_{t^*,\tau(t^*)}(\sigma^{(L)}) - \rho^{\alpha(t*)}(\sigma^{(L)})$ may tend to zero only in probability rather than for almost all $\omega$’s. We do expect, however, that for almost all $\omega$’s, $\rho^{\alpha(t^*)}$ tracks $\nu_{t^*,\tau(t^*)}$ for [*most*]{} large $t^*$’s (i.e., when domain walls are far from the origin).
[^6]: Logically, there is a third possibility that $\nu_{\infty}$ is not a Gibbs state, but this would violate the expected behavior discussed in Section \[sec:measures\] and hence we disregard it (except when $T=0$ — see Section \[sec:lne\] and Ref. [@NN]).
[^7]: LNE may seem somewhat analogous to “weak ergodicity breaking” (see, e.g., Ref. [@Bouchaud]) in which the system forever moves to different metastable states with increasing lifetimes. There are important differences, however: that scenario explicitly requires the system to remain within a single pure state for all time (and unlike LNE is supposed to be able to occur even if there is only one pure state), and further makes no reference to what is occurring in real space.
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---
abstract: 'In the absence of a light Higgs boson, the mechanism of electroweak symmetry breaking will be best studied in processes of vector boson scattering at high mass. Various models predict resonances in this channel. Scalar and vector resonances have been investigated in the $WW$, $WZ$ and $ZZ$ channels. The ability of ATLAS to measure the di-boson cross-section over a range of centre-of-mass energies has been studied with particular attention paid to the reconstruction of jet pairs with low opening angle resulting from the decays of highly boosted vector bosons.'
author:
- Adam Davison
bibliography:
- 'VBS-CSC-note.bib'
title: Vector Boson Scattering at High Mass with ATLAS
---
[ address=[on behalf of the ATLAS Collaboration]{} ,altaddress=[Department of Physics & Astronomy, University College London]{} ]{}
In the absence of a light Higgs boson, the Standard Model predicts unphysical cross-sections for vector boson scattering at the TeV scale [@Dawson:1998yi]. Some mechanism for Electroweak Symmetry Breaking must be observed at the LHC to resolve this inconsistency. The Standard Model Higgs is just one of many possible solutions to this problem [@Lane:2002sm][@Csaki:2003dt; @Csaki:2003zu; @Cacciapaglia:2004rb][@Csaki:2004sz; @Birkedal:2005yg; @Sekhar_Chivukula:2006cg][@Casalbuoni:2000gn] and measurements of vector boson scattering cross-sections will likely be a key tool in distinguishing between different models. It is therefore desirable to understand the main issues with performing such an analysis with ATLAS and also have some feel for the required luminosity for such measurements to be feasible.
Experimental Signature
======================
The category of processes described here as vector boson scattering includes a range of possible final states. We are scattering a pair of vector bosons, which can be $WW$, $WZ$ or $ZZ$. Furthermore, each boson may decay leptonically or hadronically. Obviously some channels are more challenging than others. Usually channels with all-hadronic decays are inaccessible due to large backgrounds from QCD multijet processes. Channels with both bosons decaying leptonically tend to be extremely clean but require higher luminosities due to low branching ratios. Our attention then turns to channels where one boson decays hadronically and one leptonically. These provide a challenging but promising arena in which to study vector boson scattering and also an opportunity to take advantage of ATLAS’ comparatively fine hadronic calorimetry.
The main backgrounds to observation of these processes are $W/Z+jets$ and $t\overline{t}$ production. In $W/Z+jets$ the jets may simulate the presence of a hadronic vector boson. $t\overline{t}$ events are an issue for $WW$ channels due to the presence of two real $W$ bosons, although the signal has a significantly different event topology. The presence of these backgrounds means these channels are hard to observe in the low mass region. As a general guide, the $p_\perp$ of the vector bosons is usually required to be above $200$ GeV, which presents unique challenges for identifying the hadronic system, as will be discussed in more detail below.
The experimental signature of semi-leptonic vector boson scattering events is a leptonic system reconstructible as either a $W$ or $Z$, a hadronic system from the hadronic decay and two “tag” jets from the original boson production. The emission of high-$p_\perp$ vector bosons from the incoming partons tends to be balanced by the presence of jets at very low angle. While other processes may produce similar signatures in the center of the detector they are not correlated with the presence of these “tag” jets, making this a powerful additional discriminator.
Hadronic Vector Boson Identification
====================================
For the first time at the LHC, we will have an opportunity to observe the production of $O(100)$ GeV particles with significant boosts. Decay products in this scenario tend to be much closer together which presents a challenge in terms of identifying separate objects in a detector. However, combinatoric effects are reduced or even eliminated in these scenarios. Specifically for hadronic decays, particles start to be resolved as a single jet which means that standard techniques such as dijet mass cuts are no longer applicable.
The first technique at our disposal is single jet mass. By analogy with traditional dijet mass cuts, if the decay products are all boosted into a single jet, then the mass of that jet should be of the order of the mass of the parent particle. By contrast, jet mass in QCD processes tends to be much lower. This provides a powerful tool for identifying jets containing highly boosted heavy particle decays. The ability of the ATLAS detector to measure these quantities is explored through plots such as Figure \[fig:mres\], showing a resolution of $O(10)$ GeV, enough to possibly even separate $W$ and $Z$ bosons with high enough statistics.
However, it is also useful to have some additional discriminating power. Specifically, when using the $k_\perp$ jet finder, it is possible to explore the energy scales at which the jet can be decomposed into smaller sub-jets [@Butterworth:2002tt]. These “y-scales” are an additional useful discriminator because while QCD processes can produce high mass jets, often the source is multiple smaller showering splittings, whereas with the decay of a heavy particle there must be a hard splitting. Therefore we can examine the scale at which a jet can be subdivided into two sub-jets and for vector bosons we find it to be $O(m_{W/Z})$ while for QCD jets it tends to be much lower. These “y-scales” are highly correlated with the jet mass but some significant additional rejection is still provided by a cut on this value.
![Resolution of single jet mass in a sample of boosted $W$ bosons from a vector boson scattering signal as observed in ATLAS simulation [@CSC].[]{data-label="fig:mres"}](masses.eps){width="\linewidth"}
These techniques enable heavy jets which are likely to have come from vector boson decay to be effectively selected. Recent advances in jet reconstruction propose that other related methods may be of value for this channel but this has yet to be studied in this context.
ATLAS Simulation
================
This process has now been studied for the first time with full detector simulation of the ATLAS detector [@CSC]. Monte Carlo samples of signals and backgrounds have been simulated and a sample analysis performed to explore the sensitivity of the ATLAS experiment.
Due to the broad range of theoretical scenarios of interest in this channel, this study attempted to produce model independent results which can then be easily interpreted in many contexts. Specifically, the analysis is designed to produce invariant mass spectra of the vector boson scattering system, where resonances or other more generic predictions can be compared to data, like those in Figure \[fig:signals\].
![Parton level differential cross-sections for a selection of possible vector boson scattering signal samples.[]{data-label="fig:signals"}](signals.eps){width="\linewidth"}
Signals were generated by modifying couplings in the EWChL model [@Kilian:2003pc; @Kilian:2003yw] and applying the Padé unitarization protocol [@Dobado:1996ps]. Depending on the choice of coupling constants, this produces scalar or vector resonances. This was implemented using the [Pythia]{} [@pythia] generator. Backgrounds of $W/Z+jets$ and $t\overline{t}$ events were produced with MadGraph [@Maltoni:2002qb] and MC@NLO [@mcatnlo2] respectively.
Sensitivity
===========
The sensitivity to a selection of vector and scalar resonances, with masses in the range $500$ GeV to $1.1$ TeV was explored. The requirements of the analysis were broadly the following:
- 1 hadronic $W$ or $Z$ with $p_\perp > 200$ GeV and $|\eta| < 2$,
- 1 leptonic $W$ or $Z$ with $p_\perp > 200$ GeV and $|\eta| < 2$,
- 2 “tag” jets present in opposite hemispheres of the detector with $|\eta| > 2$,
- No top candidates,
- No additional jets with $|\eta| < 2$,
where the hadronic vector boson has been identified as either a single jet with high mass (and “y-scale” when the $k_\perp$ algorithm is used) as described above or as a pair of jets with high dijet mass.
Using this selection over several channels, the sensitivities observed in Table \[table:sens\] were extracted for observation of peaks due to resonances. The fully leptonic modes perform better for lower mass, higher cross-section resonances whereas the semi-leptonic modes perform better for the higher mass resonances. These results imply that ATLAS is sensitive to some types of resonances with as little as a few tens of fb$^{-1}$ of well understood data. However models which predict multiple resonances or resonance production with higher cross-sections or in multiple channels could be observed sooner.
**Channel** **Resonance Mass**
---------------------------- -------------------- -------------
$WW/WZ\rightarrow l\nu jj$ $500$ GeV $85fb^{-1}$
$WW/WZ\rightarrow l\nu jj$ $800$ GeV $20fb^{-1}$
$WW/WZ\rightarrow l\nu jj$ $1.1$ TeV $85fb^{-1}$
$ZW/ZZ\rightarrow ll jj$ $500$ GeV $30fb^{-1}$
$ZW/ZZ\rightarrow ll jj$ $800$ GeV $30fb^{-1}$
$ZW/ZZ\rightarrow ll jj$ $1.1$ TeV $90fb^{-1}$
$ZZ\rightarrow \nu \nu ll$ $500$ GeV $20fb^{-1}$
: Estimated sensitivities of the ATLAS experiment for various resonances in different channels.[]{data-label="table:sens"}
Conclusions
===========
Vector boson scattering has been presented as an important channel for study at the LHC. Although in many ways challenging, especially in terms of requirements in hadronic calorimetry, it may be the key to understanding particle physics at the TeV scale.
This channel has now been studied in some detail using the ATLAS detector simulation and has been found to have potential for discovery within a few years of design luminosity delivered to ATLAS from the LHC.
|
---
abstract: |
In this paper, we present lower bounds on the rank of the split closure, the multi-branch closure and the lattice-free closure for packing sets as a function of the integrality gap. We also provide a similar lower bound on the split rank of covering polyhedra. These results indicate that whenever the integrality gap is high, these classes of cutting planes must necessarily be applied for many rounds in order to obtain the integer hull.\
\
**Keywords.** Integer programming, packing, covering, split rank, multi-branch split rank, lattice-free rank
author:
- 'Merve Bodur[^1]'
- 'Alberto Del Pia[^2]'
- 'Santanu S. Dey[^3]'
- 'Marco Molinaro[^4]'
bibliography:
- 'LatticeFreeRank.bib'
title: 'Lower bounds on the lattice-free rank for packing and covering integer programs'
---
Introduction {#sec:intro}
============
*Split cuts* are a very important class of cutting planes in integer programming both from a theoretical and computational perspective (see for example [@balas:1979; @BalasS08; @cook:ka:sc:1990]). Recently, many generalizations of split cuts have been studied, such as the *multi-branch split cuts* [@dash2014lattice; @dash2013t; @li:ri:2008] and the *lattice-free cuts* [@andersen2007inequalities; @basu2015geometric; @borozan:2007; @RichardDey]. In order to study the strength of the cutting plane procedures, a very useful concept is the notion of *rank* which represents the minimum rounds of cuts needed to obtain the integer hull. The notion of rank was first studied in the context of Chvátal-Gomory (CG) cuts [@Schrijver80]. Many lower bounds on the rank of the above mentioned closures have been proven; see [@BasuCM12; @bodur2017cutting; @cook:ka:sc:1990; @dey:lowerbnd:2009; @DeyL11; @li:ri:2008] for the split rank, see [@dash2013t] for the multi-branch rank, and see [@averkov2017approximation] for the lattice-free rank.
A standard notion describing the [difficulty]{} of an integer program is the *integrality gap* which in this paper refers to the ratio between the optimal objective function values of the integer program and its linear programming relaxation. While it is natural to expect that the rank of a cutting plane procedure should increase with the increase in the integrality gap, only a few results of this nature exist in the literature [@bodur2016aggregation; @PokuttaS11].
In this paper, we present lower bounds on the rank of the split closure, the multi-branch closure and the lattice-free closure for *packing sets* as a function of the integrality gap. We also provide a similar lower bound on the split rank of *covering polyhedra*. These results indicate that whenever the integrality gap is high, these classes of cutting planes must necessarily be applied for many rounds in order to obtain the integer hull.
The rest of the paper is organized as follows. We provide all necessary definitions in Section \[sec:Prelim\]. We state all our main results in Section \[sec:Statements\]. Finally, in Section \[sec:Packing\] and Section \[sec:Covering\] we present the proofs for results concerning the packing and covering cases, respectively.
Preliminaries {#sec:Prelim}
=============
For an integer $t \geq 1$, we use $[t]$ to describe the set $\{1, \dots, t\}$. Also, we represent the $j^{\text{th}}$ unit vector, the vector of ones and the vector of zeros in appropriate dimension by $e_j$, ${\boldsymbol{1}}$ and ${\boldsymbol{0}}$, respectively. Given a set of vectors $v^1, \hdots, v^t$, we denote the linear subspace spanned by these given vectors as $\spann ( \{ v^j \}_{j \in [t]} )$.
#### Sets.
In this paper, we work with *covering [polyhedra]{}* and *packing [polyhedra]{}*, which are of the form $$P_C = \{x \in {\mathbb{R}}^n_+ \mid Ax \ge b \} \quad \text{and} \quad P_P = \{x \in {\mathbb{R}}^n_+ \mid Ax \le b \},$$ respectively, where all the data $(A,b) \in {\mathbb{Q}}_+^{m \times n} \times {\mathbb{Q}}_+^m$. [Thus, an inequality of packing type (respectively covering type) is one of the form $a^\top x \le b$ (respectively $a^\top \ge b$) for non-negative $(a,b) \in {\mathbb{Q}}_+^n \times {\mathbb{Q}}_+$.]{} If it is obvious from the context that the polyhedron is of covering (resp. packing) type, we may drop the subscript $C$ (resp. $P$). For the packing case, we also work with more general sets. We call $Q \ {{\color{black}}\subseteq} \ {\mathbb{R}}_+^n$ a *packing set* if $x \in Q$ and ${\boldsymbol{0}}\leq y \leq x$ imply that $y \in Q$.
Throughout this paper, we make a technical assumption regarding the sets under consideration that we call as *well-behavedness*. The set $P_C$ is *well-behaved* if $A_{ij} \leq b_i$ for all $i \in [m], j \in [n]$. Notice that this is a natural assumption since if $A_{ij} > b_i$ for some $i \in [m], j \in [n]$, then we can replace the coefficient $A_{ij}$ by $b_i$ to obtain a tighter linear programming relaxation with the same set of feasible integer points. A packing set $Q$ is *well-behaved* if $e_j \in Q$ for all $j \in [n]$. This is not a restrictive assumption since if $e_j \notin Q$ for some $j \in [n]$, then we replace $Q$ with the packing set $\{ x \in Q \mid x_j = 0\}$, which provides a tighter linear relaxation with the same set of feasible integer points. Note that if $Q$ is the polytope $P_P$, then [the]{} well-behavedness definition is equivalent to $A_{ij} \leq b_i$ for all $i \in [m], j \in [n]$.
[A *relaxation* of a set $P$ is any superset $\tilde{P} \supseteq P$.]{} Let $\alpha > 0$ be a scalar. If a given covering polyhedron $\tilde{P}_C$ is a relaxation of $P_C$ and satisfies $$\min\{c^\top x \mid x \in \tilde{P}_C\} \ge \frac{1}{\alpha} \cdot \min\{c^\top x \mid x \in P_C\}, \ \ \forall c \in {\mathbb{R}}_+^n,$$ then $\tilde{P}_C$ is an *$\alpha$-approximation* of $P_C$. Similarly, given a packing set $P_P$ and one of its relaxation $\tilde{P}_P$ of packing type, $\tilde{P}_P$ is an *$\alpha$-approximation* of $P_P$ if $$\max\{c^\top x \mid x \in \tilde{P}_P\} \le \alpha \cdot \max\{c^\top x \mid x \in P_P \}, \ \ \forall c \in {\mathbb{R}}_+^n.$$ For a set $P \subseteq {\mathbb{R}}^n$, we define $\alpha P:= \{\alpha x \mid x \in P\}$. The equivalent definitions of $\alpha$-approximation for covering and packing cases are provided in [@bodur2016aggregation] as $$\frac{1}{\alpha} \tilde{P}_C {\subseteq}P_C \quad \text{and} \quad \tilde{P}_P {\subseteq}\alpha P_P,$$ respectively.
Given a polyhedron $P {\subseteq}{\mathbb{R}}^n$, we denote its integer hull by $P^I := \operatorname{conv}( \{x \mid x \in P \cap {\mathbb{Z}}^n\})$ where $\operatorname{conv}(\cdot)$ is the convex hull operator. We let ${z^{LP}(c)}$ and ${z^{I}(c)}$ to denote the optimal value of a given objective function $c^\top x$ over $P$ and $P^I$, respectively. For convenience, we will sometimes refer to ${z^{LP}(c)}$ and ${z^{I}(c)}$ as $z^{LP}$ and $z^{I}$, respectively.
#### Closures.
We call a set ${M}\in {\mathbb{R}}^n$ a *strict lattice-free set* if $M \cap {\mathbb{Z}}^n = \emptyset$. Note that the set ${M}$ need not to be convex. Given a set $Q$, one can obtain a relaxation of $Q^I$ as $$Q^{M}:= \operatorname{conv}(Q \setminus {M}).$$ Given a collection of strict lattice-free sets ${\mathcal{{M}}}$, we define the corresponding closure as $${\mathcal{{M}}}(Q) = \bigcap_{{M}\in {\mathcal{{M}}}} Q^{M}.$$ For convenience, we sometimes refer to ${\mathcal{{M}}}$ as the closure operator or just as closure.
Next, we define three special cases of the strict lattice-free closures, namely the split closure, the multi-branch closure and the lattice-free closure.
We denote the *split set* associated with $(\pi,\pi_0) \in {\mathbb{Z}}^n \times {\mathbb{Z}}$ by $$S(\pi,\pi_0) := \{ x \in {\mathbb{R}}^n \mid \pi_0 < \pi^\top x < \pi_0+1\}.$$ [Denoting]{} the collection of all split sets by $${{\color{black}}{\mathcal{S}}= \{S(\pi,\pi_0) \mid (\pi,\pi_0) \in {\mathbb{Z}}^n \times {\mathbb{Z}}\},}$$ the split closure of $Q$, denoted as ${\mathcal{S}}(Q)$, is defined to be $${\mathcal{S}}(Q) = \bigcap_{S \in {\mathcal{S}}} Q^S.$$ For convenience, we denote $Q^{S(\pi,\pi_0)}$ by $Q^{\pi,\pi_0} $ which is explicitly defined as $$Q^{\pi,\pi_0} = \operatorname{conv}(Q \setminus S(\pi,\pi_0)) = \operatorname{conv}\big( (Q \cap \{ \pi^\top x \leq \pi_0 \}) \cup (Q \cap \{ \pi^\top x \geq \pi_0+1 \}) \big).$$ A generalization of split closure, called the *$k$-branch split closure*, which is defined by [@li:ri:2008], is obtained by removing the union of *at most* $k$ split sets simultaneously. Letting $$\begin{aligned}
Q^{\pi^1,\dots, \pi^k;\pi^1_0, \dots, \pi^k_0} :&= \operatorname{conv}\Big( Q \setminus \bigcup_{i \in [k]} S(\pi^i,\pi^i_0) \Big) \\
&= \operatorname{conv}\left( \bigcap_i (Q \cap \{ (\pi^i)^\top x \leq \pi^i_0 \}) \cup (Q \cap \{ (\pi^i)^\top x \geq \pi^i_0+1 \}) \right),\end{aligned}$$ the $k$-branch split closure of $Q$, denoted by ${{\mathcal{S}}^{k}}(Q)$, can be written as $${{\mathcal{S}}^{k}}(Q) = \bigcap_{(\pi^i,\pi^i_0) \in {\mathbb{Z}}^n \times {\mathbb{Z}},~i \in [k]} Q^{\pi^1,\dots, \pi^k;\pi^1_0, \dots, \pi^k_0}.$$ Note that the 1-branch split closure is equivalent to the split closure, i.e, ${\mathcal{S}}^{1}(Q)={\mathcal{S}}(Q)$.
A further generalization of the split closure is the so-called *lattice-free closure*, which is obtained by considering convex sets having no integer point in their interior; see [@dash2012two; @dash2014lattice] for relations to the $k$-branch split closure. A set $L {\subseteq}{\mathbb{R}}^n$ is called a *lattice-free set* if $\int(L) \cap {\mathbb{Z}}^n = \emptyset$ where $\int(\cdot)$ is the interior operator. For each integer $k \geq 2$, we define ${\mathcal{L}}^k$ as the family of full-dimensional lattice-free polyhedra $L \subset {\mathbb{R}}^n$ defined by *at most* $k$ inequalities. (Note that it is not possible to have lattice-free sets defined by only one inequality.) We denote the *$k$-lattice-free closure* of $P$ by ${\mathcal{L}}^k(Q)$, i.e., $${\mathcal{L}}^k(Q) = \bigcap_{L \in {{\color{black}}{\mathcal{L}}^k}} Q^L,$$ where $$Q^L := \operatorname{conv}(Q \setminus \int(L)).$$ Given a closure operator ${\mathcal{{M}}}$ and a nonnegative objective function $c \in {\mathbb{R}}^n_+$, we use $z^{{\mathcal{{M}}}}$ to denote the optimal value of the minimization (or maximization) of $c^\top x$ over the closure ${\mathcal{{M}}}(Q)$. Lastly, we define the *rank* of the closure ${\mathcal{{M}}}$, denoted by $\operatorname{rank}_{{\mathcal{{M}}}}(Q)$, as the minimum number of iterative applications of ${\mathcal{{M}}}$ to obtain the integer hull of $Q$. We note that the split rank, thus the multi-branch rank and the lattice-free rank, are finite whenever $Q$ is a rational polyhedron or is a bounded set [@Schrijver80].
Main results {#sec:Statements}
============
Packing
-------
The main proof strategy to prove lower bounds on ranks of various cutting plane closures is presented in the proposition below.
\[thm:thm1\] Let ${\mathcal{{M}}}$ be a collection of strict lattice-free sets which satisfies the following two conditions:
1. Packing invariance: For any packing set $Q$, ${\mathcal{{M}}}(Q)$ is a packing set.
2. Constant approximation: There exists $\alpha_{\mathcal{{M}}}\geq 1$ such that $Q {\subseteq}\alpha_{\mathcal{{M}}}{\mathcal{{M}}}(Q)$ for every well-behaved packing set $Q$.
Then, for any well-behaved packing set $Q$, $$\operatorname{rank}_{{\mathcal{{M}}}}(Q) \geq {{\color{black}}\sup_{c \in {\mathbb{R}}_+^n}} \left\lceil \frac {\log_2 \left( \frac{{z^{LP}(c)}}{{z^{I}(c)}}\right)}{\log_2 \alpha_{\mathcal{{M}}}}\right\rceil.$$
The proof of Proposition \[thm:thm1\] is based on a simple iterative argument, which is provided in Section \[subsec:4.1\].
### Tools to prove the assumptions of Proposition \[thm:thm1\]
In order to use Proposition \[thm:thm1\], we need to verify the packing invariance and constant approximation properties. The next tool is very helpful in proving packing invariance.
\[thm:thm2\] Let ${\mathcal{{M}}}$ be a collection of strict lattice-free sets. For ${T}{\subseteq}[n]$, define $H[{T}] := \{ x \in {\mathbb{R}}^n \mid x_j = 0, \ \forall j \in T\}$. Given ${M}\in {\mathcal{{M}}}$, let $$M[{T}] := ( M \cap H[{T}] ) + \spann ( \{ e_{j} \}_{j \in T} ).$$ Suppose that ${\mathcal{{M}}}$ satisfies the following property: For any ${M}\in {\mathcal{{M}}}$ and $T {\subseteq}[n]$, $M[{T}] \neq \emptyset$ implies that $M[{T}] \in {\mathcal{{M}}}$. Then ${\mathcal{{M}}}$ is packing invariant.
Note that it is straightforward to see that the set ${M}[T]$ in Theorem \[thm:thm2\] is guaranteed to be a strict lattice-free set by construction. The proof of Theorem \[thm:thm2\] is essentially based on the fact that a cut generated using a strict lattice-free set $M$ is dominated by a packing type inequality that is obtained using the strict lattice-free set $M[T]$ for a specifically chosen set $T$. The details of the proof of Theorem \[thm:thm2\] are given in Section \[subsec:4.2\].
We observe here that in order to use Proposition \[thm:thm1\], we must prove [the]{} constant approximation property for general well-behaved packing sets, rather than just for polyhedra. The reason is that the closures of some cutting plane families we consider are not known to be polyhedral. In order to prove [the]{} constant approximation property for general well-behaved packing sets, we will find it convenient to prove this property first for well-behaved packing polyhedra. It turns out that this is sufficient to prove constant approximation property for any well-behaved packing set as the next theorem states.
\[thm:thm3\] Let ${\mathcal{{M}}}$ be a collection of strict lattice-free sets with the following property: There exist $\alpha_{\mathcal{{M}}}\geq 1$ such that $P_P {\subseteq}\alpha_{\mathcal{{M}}}{\mathcal{{M}}}(P_P)$ for every well-behaved packing polyhedron $P_P$. Then, $Q {\subseteq}\alpha_{\mathcal{{M}}}{\mathcal{{M}}}(Q)$ for every well-behaved packing set $Q$.
Theorem \[thm:thm3\] is proven by first constructing a well-behaved packing polyhedron which is an inner approximation of $Q$ and is arbitrarily close to $Q$. We then show how to “transfer" the $\alpha_{\mathcal{{M}}}$ factor from this polyhedron to $Q$. The details of the proof of Theorem \[thm:thm3\] are provided in Section \[subsec:4.3\].
### Applications of Proposition \[thm:thm1\] to split, multi-branch split and lattice-free closures
We use Theorem \[thm:thm2\] to verify the following result.
\[thm:thm4\] ${\mathcal{{M}}}$ is packing invariant for ${\mathcal{{M}}}\in \{ {\mathcal{S}}, {\mathcal{S}}^k, {\mathcal{L}}^k \}$.
Theorem \[thm:thm4\] is proven in Section \[subsec:4.4\].
\[thm:thm5\] For ${\mathcal{{M}}}\in \{ {\mathcal{S}}, {\mathcal{S}}^k, {\mathcal{L}}^k \}$, ${\mathcal{{M}}}$ satisfies the constant approximation property, where [we can choose]{} $\alpha_{{\mathcal{S}}} = 2$, [$\alpha_{{\mathcal{S}}^k} = \min \{ 2^k,n \}+1$, and $\alpha_{{\mathcal{L}}^k} = \min \{ k,n \}+1$]{}.
Moreover, the factor $\alpha_{{\mathcal{S}}}$ is tight, i.e., for every $\epsilon >0$, there exists a well-behaved packing polyhedron $\tilde{P}_P$ such that $\tilde{P}_P \not\subseteq (2-\epsilon) {\mathcal{S}}(\tilde{P}_P)$.
Observe that the split cuts are a special case of multi-branch split cuts. However, we have stated their constant approximation result separately since the general factor for multi-branch split closure is not tight for the split closure. Indeed, proving the factor of 2 in the case of split cuts involves more careful analyses. Moreover, this factor of 2 for the split case is tight as stated in the theorem. The proof of Theorem \[thm:thm5\] for the split, multi-branch split and lattice-free cases are given in Sections \[subsubsec:4.5.1\], \[subsubsec:4.5.2\] and \[subsubsec:4.5.3\], respectively.
Note that a factor of $2$ is proven in [@bodur2016aggregation] as an approximation factor of the *aggregation closure*, which is very similar to the result [for the]{} split closure in Theorem \[thm:thm5\]. However, the split closure result of Theorem \[thm:thm5\] is not implied by the result of [@bodur2016aggregation] since for packing polyhedra, split cuts are not [always]{} dominated by *aggregation cuts*, see the example given in Observation \[obs:PackingSplitAgg\] in Appendix \[sec:appendix\].
Proposition \[thm:thm1\], Theorem \[thm:thm4\] and Theorem \[thm:thm5\] lead us to the following lower bounds on the rank of [the]{} split closure, $k$-branch split closure and $k$-lattice-free closure of packing sets. As Corollary \[cor:cor1\] is a direct application of Proposition \[thm:thm1\], we omit its proof.
\[cor:cor1\] Let $Q$ be a well-behaved packing set. Then
1. $\operatorname{rank}_{{\mathcal{S}}}(Q) \geq {{\color{black}}\sup_{c \in {\mathbb{R}}_+^n}} \left\lceil \log_2 \left( \frac{{z^{LP}(c)}}{{z^{I}(c)}}\right)\right\rceil$.
2. $\operatorname{rank}_{{\mathcal{S}}^k}(Q) \geq {{\color{black}}\sup_{c \in {\mathbb{R}}_+^n}} \left\lceil\frac{\log_2 \left( \frac{{z^{LP}(c)}}{{z^{I}(c)}}\right)}{\log_2 (\min \{ 2^k,n \}+1)}\right\rceil$ for any $k \in {\mathbb{Z}}_+, k \geq 1$.
3. $\operatorname{rank}_{{\mathcal{L}}^k}(Q) \geq {{\color{black}}\sup_{c \in {\mathbb{R}}_+^n}} \left\lceil\frac{\log_2 \left( \frac{{z^{LP}(c)}}{{z^{I}(c)}}\right)}{\log_2 (\min \{ k,n \}+1)}\right\rceil$ for any $k \in {\mathbb{Z}}_+, k \geq 2$.
Corollary \[cor:cor1\] shows that if the integrality gap is high, then we cannot expect the split rank, the multi-branch split rank or the lattice-free rank of a well-behaved packing set to be low.
To the best of our knowledge, the only other paper analyzing the rank of general lattice-free closures is [@averkov2017approximation], and the only papers presenting lower bounds on the rank of multi-branch split closure for very special [kinds]{} of polytopes are [@dash2013t] and [@li:ri:2008]. We note that none of these bounds are related to the [integrality]{} gap.
There have been a number of papers giving lower bounds on split ranks such as [@BasuCM12; @bodur2017cutting; @cook:ka:sc:1990; @dey:lowerbnd:2009; @DeyL11] and bounds on a closely related concept, the [reverse]{} split rank [@ConfortiPSFSpli15]. To the best of our knowledge, this is the first work connecting the integrality gap to the split rank. We note that the first part of Corollary \[cor:cor1\] can be seen as a generalization of the result given in [@PokuttaS11] for the CG rank.
The lower bound on the split rank given in Corollary \[cor:cor1\] is tight within a constant factor as formally stated below.
\[prop:prop1\] There exists a well-behaved packing polyhedron $Q$ and a nonnegative objective function $c$ such that $\operatorname{rank}_{{\mathcal{S}}}(Q) \le O\left(\textup{log}_2\left( \frac{{z^{LP}(c)}}{{z^{I}(c)}}\right)\right)$.
The proof of Proposition \[prop:prop1\] is given in Section \[subsubsec:4.5.4\].
Covering
--------
We now state our results for covering [polyhedra]{}. All the proofs regarding the covering case are given in Section \[sec:Covering\].
\[thm:mainAppCovering\] Let $P_C$ be well-behaved. Then, the followings hold:
- ${\mathcal{S}}(P_C)$ is a well-behaved covering polyhedron.
- $\frac{1}{2} P_C {\subseteq}{\mathcal{S}}(P_C) $.
Moreover, the bound given in (ii) is tight, i.e., for every $\epsilon >0$, there exists a well-behaved covering polyhedron $\tilde{P}_C$ such that $\frac{1}{2-\epsilon} \tilde{P}_C \not {\subseteq}{\mathcal{S}}(\tilde{P}_C)$.
Regarding part (i) of Theorem \[thm:mainAppCovering\], ${\mathcal{S}}(P_C)$ is known to be a rational polyhedron since $P_C$ is assumed to be a rational polyhedron [[@cook:ka:sc:1990]]{}, and it is straightforward to show that the split closure is of covering type (Proposition \[prop:CoveringPoly\]); whereas its well-behavedness can be proven by showing that each split cut that violates the well-behavedness property is dominated by a well-behaved split cut (Proposition \[prop:CoveringSplitWell\]). Proof of part (ii) follows from a case analysis that gives the correct factor of [$\frac{1}{2}$]{} (Proposition \[prop:covering2approx\]). For the last statement in the theorem, we provide a tight example in Proposition \[prop:LBAC\].
Note that a [result similar]{} to Theorem \[thm:mainAppCovering\] is proven in [@bodur2016aggregation] with respect to the [aggregation closure]{}. However, Theorem \[thm:mainAppCovering\] is not implied by the result of [@bodur2016aggregation] since for covering polyhedra, split cuts are not dominated by [aggregation cuts]{}, see the example given in Observation \[obs:CoveringSplitAgg\] in Appendix \[sec:appendix\].
Similar to the proof of Proposition \[thm:thm1\] in the packing case, Theorem \[thm:mainAppCovering\] yields the following lower bound on the split rank of covering polyhedra.
\[cor:SplitRankPolyCovering\] Let $P_C$ be well-behaved. Then, $$\operatorname{rank}_{{\mathcal{S}}}(P_C) \geq {{\color{black}}\sup_{c \in {\mathbb{R}}_+^n}} \left\lceil \textup{log}_2\left( \frac{{z^{I}(c)}}{{z^{LP}(c)}}\right)\right\rceil.$$
Unlike the packing case, we are unable to generalize the result of Corollary \[cor:SplitRankPolyCovering\] for the case of $k$-lattice-free rank. The key technical argument that is a roadblock is to prove the well-behavedness of the $k$-lattice-free closure of covering polyhedron. [We do not know if the $k$-lattice-free closure of covering polyhedron is well-behaved. Note that in contrast]{} in the packing case, if we start from a well-behaved set and the closure is of packing type, then trivially the closure is also well-behaved.
Proofs for packing problems {#sec:Packing}
===========================
We use the following observation, from [@bodur2016aggregation], in some of the proofs.
\[obs:bijection\] Let $\phi:\mathbb{R}^n \rightarrow \mathbb{R}^n$ be a bijective map, let $\{S^i\}_{i \in I}$ be a collection of subsets of $\mathbb{R}^n$ and [for $S {\subseteq}\mathbb{R}^n$]{} let $\phi(S):= \{ \phi(x) \,|\, x \in S\}$. Then $\phi\left(\bigcap_{i \in I} S^i\right) = \bigcap_{i \in I} \phi(S^i)$.
Proof of Proposition \[thm:thm1\] {#subsec:4.1}
---------------------------------
Let ${\mathcal{{M}}}$ be a collection of strict lattice-free sets which satisfies the packing invariance and constant approximation properties. Let $Q {\subseteq}{\mathbb{R}}^n$ be a well-behaved packing set. Since $Q^I {\subseteq}{\mathcal{{M}}}(Q)$, we have that $e_j \in {\mathcal{{M}}}(Q)$ for all $j \in [n]$. Therefore, by the packing invariance property, ${\mathcal{{M}}}(Q)$ is also a well-behaved packing set.
[Assume that the rank of the closure ${\mathcal{{M}}}$ is finite, as there is nothing to prove otherwise.]{} Let $t = \operatorname{rank}_{{\mathcal{{M}}}}(Q)$ and let $c \in {\mathbb{R}}_+^n$ be a given objective vector. Define $z^{i}$ to be the optimal objective function value of maximizing $c^\top x$ over the $i^{\text{th}}$ closure with respect to ${\mathcal{{M}}}$ of $Q$. Since, ${\mathcal{{M}}}(Q)$ is a well-behaved packing set, by induction, the $i^{\text{th}}$ closure with respect to ${\mathcal{{M}}}$ of $Q$ is a well-behaved packing set. Therefore, the constant approximation property guarantees that $z^i \le {{\color{black}}\alpha_{\mathcal{{M}}}} z^{i+1}$. Thus, $$\begin{aligned}
\frac{{z^{LP}(c)}}{{z^{I}(c)}} = \frac{{z^{LP}(c)}}{z^1} \frac{z^1}{z^2}\dots\frac{z^{t-1}}{z^{t}} \leq {(\alpha_{\mathcal{{M}}})}^{t}.\end{aligned}$$ This implies the inequality $$\begin{aligned}
t = \operatorname{rank}_{{\mathcal{{M}}}}(Q) \geq \left\lceil \frac{\textup{log}_2\left( \frac{{z^{LP}(c)}}{{z^{I}(c)}}\right)}{\textup{log}_2 \alpha_{\mathcal{{M}}}} \right\rceil,\end{aligned}$$ which is the required result.
Proof of Theorem \[thm:thm2\] {#subsec:4.2}
-----------------------------
Let ${\mathcal{{M}}}$ be a collection of strict lattice-free sets with the following property. For ${T}{\subseteq}[n]$, define $H[{T}] := \{ x \in {\mathbb{R}}^n | x_j = 0, \ \forall j \in T\}$. Given ${M}\in {\mathcal{{M}}}$, let $$M[{T}] := ( M \cap H[{T}] ) + \spann ( \{ e_{j} \}_{j \in T} ).$$ Assume that for any ${M}\in {\mathcal{{M}}}$ and $T {\subseteq}[n]$, if $M[{T}] \neq \emptyset$, then $M[{T}] \in {\mathcal{{M}}}$. We will show that ${\mathcal{{M}}}$ is packing invariant.
Let $Q$ be a packing set. If $Q$ is empty, then there is nothing to prove. Therefore, assume that $Q$ is nonempty.
Let ${M}\in {\mathcal{{M}}}$. Let ${\beta}^\top x \le \delta$ be a valid inequality for $Q^{M}$. We will show that this inequality is dominated by a packing type inequality valid for ${\mathcal{{M}}}(Q)$. Let $T = \{ j \in [n] : {\beta}_j < 0\}$. If $T = \emptyset$, there is nothing to prove. So, assume that $T \neq \emptyset$.
For convenience, we define an operator $\breve{(\cdot)}$ as follows: For a given vector $u \in {\mathbb{R}}^n$, $\breve{u} \in {\mathbb{R}}^n$ is $$\begin{aligned}
\breve{u}_j = \left \{
\begin{array}{rl}
u_j, & \text{if} \ j \in [n] \setminus T \\
0, & \text{if} \ j \in T
\end{array}
\right. . \label{eq:opBreve}
\end{aligned}$$
We will show that $\breve {\beta}^\top x \le \delta$ is a valid inequality for ${\mathcal{{M}}}(Q)$. Since $\breve {\beta}\in {\mathbb{R}}^n_+$ and $\{x \in {\mathbb{R}}^n_+ : \breve {\beta}x \le \delta \} \subseteq \{x \in {\mathbb{R}}^n_+ : {\beta}^\top x \le \delta \}$, we obtain the required result.
Let $\bar Q := Q \cap H[T]$. As $\bar Q \subseteq Q$, we have that ${\beta}^\top x \le \delta$ is a valid inequality for ${\bar Q}^{M}$. Since $\breve {\beta}^\top x = {\beta}^\top x$ for every $x \in H[T]$, we obtain that $$\label{betaineqValidForQbarM}
\breve {\beta}^\top x \le \delta \ \text{is a valid inequality for} \ {\bar Q}^{M}.$$
Now, we [distinguish]{} two cases:
- $H[{T}] \cap {M}= \emptyset$: In this case, we know that $\bar Q = {\bar Q}^{M}$, thus, using , we have that $\breve {\beta}x \leq \delta$ is valid for $\bar Q$. We show that $\breve {\beta}^\top x \le \delta$ is valid for $Q$, and therefore trivially for ${\mathcal{{M}}}(Q)$. Assume by contradiction that there is a point $x \in Q$ such that $\breve {\beta}^\top x > \delta$. We have $\breve {\beta}^\top \breve x = \breve {\beta}^\top x > \delta$. As $Q$ is a packing set, we have $\breve x \in Q$. Moreover, since $\breve x \in H[T]$, we have $\breve x \in \bar Q$. Thus $\breve x$ is a vector in $\bar Q$ with $\breve {\beta}^\top \breve x > \delta$, a contradiction since $\breve {\beta}x \leq \delta$ is valid for $\bar Q$. Therefore, in this case the statement is trivially satisfied.
- $H[{T}] \cap {M}\neq \emptyset$: By the definition of $M[T]$, we have that ${M}[{T}] \neq \emptyset$ and $$H[{T}] \cap {M}= H[{T}] \cap {M}[T].$$ Therefore, $\bar Q \setminus {M}= \bar Q \setminus {M}[T]$, which together with imply that $$\label{factForContradictionMT}
\breve {\beta}^\top x \le \delta \ \text{is a valid inequality for} \ {\bar Q}^{{\mathcal{{M}}}[T]}.$$ We now show that $\breve {\beta}^\top x \le \delta$ is a valid inequality for $Q^{{M}[T]}$. Assume by contradiction that there is a point $x \in Q \setminus {M}[T]$ such that $\breve {\beta}^\top x > \delta$. We have $\breve {\beta}^\top \breve x = \breve {\beta}^\top x > \delta$. As $Q$ is a packing set, we have $\breve x \in Q$. Moreover, since $\breve x \in H[T]$, we have $\breve x \in \bar Q$. Finally, since $x \notin {M}[T]$, we obtain that also $\breve x \notin {M}[T]$ by definition of ${M}[T]$. Thus $\breve x$ is a vector in $\bar Q \setminus {M}[T]$ with $\breve {\beta}^\top \breve x > \delta$, a contradiction to .
Proof of Theorem \[thm:thm3\] {#subsec:4.3}
-----------------------------
Let ${\mathcal{{M}}}$ be a collection of strict lattice-free sets with the following property: There exist $\alpha_{\mathcal{{M}}}\geq 1$ such that $P_P {\subseteq}\alpha_{\mathcal{{M}}}{\mathcal{{M}}}(P_P)$ for every well-behaved packing polyhedron $P_P$. Let $Q$ be a well-behaved packing set. We will show that $Q {\subseteq}\alpha_{\mathcal{{M}}}{\mathcal{{M}}}(Q)$.
Our strategy to prove this statement is to first construct, in Lemma \[prop:SantanuLemma\], a well-behaved packing polyhedron which is an inner approximation of $Q$ and [can be chosen]{} arbitrarily close to $Q$.. Then, we apply the $\alpha_{\mathcal{{M}}}$ factor to this polyhedral approximation and “transfer" it to $Q$.
\[prop:SantanuLemma\] Let $\epsilon > 0$. Then, there exists a well-behaved packing polyhedron $P_\epsilon$ such that $\frac{1}{1+\epsilon} Q \subseteq P_\epsilon \subseteq Q$.
[First, consider the case that $Q$ is bounded.]{} Let $\sigma_Q$ be the support function of $Q$, i.e., $$\sigma_Q (u) = \sup \{ u^\top x | x \in Q \},$$ and $$C^n := \{ u \in {\mathbb{R}}^n_+ \, | \, \|u\|_2 = 1 \}.$$ Also, let $\tilde{Q} = \frac{1}{1+\epsilon} Q$. We first show that there exists $M > 0$ such that $$\label{eq:sigmaLB}
\sigma_{\tilde{Q}}(u) \geq M \ \text{for all} \ u \in C^n.$$ Let $S = \{ x \in {\mathbb{R}}^n_+ ~|~ {\boldsymbol{1}}^\top x \leq 1 \}$ and $\tilde{S} = \frac{1}{1+\epsilon} S$. Since $Q$ is well-behaved, we have that $S \subseteq Q$, thus $\tilde{S} \subseteq \tilde{Q}$. Therefore, $\sigma_{\tilde{S}}(u) \leq \sigma_{\tilde{Q}}(u)$ for all $u \in C^n$. Since [$\sigma_{\tilde{S}}(u) \geq \frac{1}{\sqrt{n}(1+\epsilon)}$]{} for all $u \in C^n$, holds.
Let $\bar{M} = \max \{ \|x\|_\infty ~|~ x \in \tilde{Q} \}$. It is well-known that $\sigma_{\tilde{Q}} (\cdot)$ is continuous since $\tilde{Q}$ is a compact convex set [@rockafellar:1970]. Moreover, as $\|\cdot\|_2$ is also continuous, [for any $u \in C^n$ and $\epsilon > 0$ there exists a neighborhood $N_u$ of $u$ (in the topology of the sphere)]{} such that for all $v \in N_u$ we have [$$\label{eq:contsigma}
|\sigma_{\tilde{Q}}(u) - \sigma_{\tilde{Q}}(v)| \le \frac{\epsilon M}{4}$$ ]{} and $$\label{eq:cont2norm}
\| u - v\|_2 \leq \frac{\epsilon M}{4 \bar{M}\sqrt{n}}.$$ Since $C^n$ is a compact set, there exists a finite list of vectors $v_1,\hdots,v_\ell $ such that $C^n = \cup_{i=1}^\ell N_{v_i}$. Define $$\label{eq:Psetdef}
P^1_\epsilon := \{ x \in {\mathbb{R}}^n_+ ~|~ (v_i)^\top x \leq \sigma_{\tilde{Q}}(v_i), \forall i=1,\hdots,\ell, \ \text{and} \ x_i \leq \bar{M}, \forall i=1,\hdots,n \}$$
We now show that $$\label{eq:innerapprox}
\tilde{Q} \subseteq P^1_\epsilon \subseteq (1+\frac{\epsilon}{2}) \tilde{Q} \subseteq Q.$$ Note that the first and the last containments are straightforward. In order to show the second containment, we [show]{} that $\sigma_{P^1_\epsilon}(u) / \sigma_{\tilde{Q}}(u) \leq 1+\frac{\epsilon}{2}$ for all $u \in C^n$. For a given $u \in C^n$, let $i \in \{ 1, \hdots,\ell\}$ such that $u \in N_{v_i}$. Observe that $$\begin{aligned}
\sigma_{P^1_\epsilon}(u) & \leq \sigma_{P^1_\epsilon}(v_i) + \sigma_{P^1_\epsilon}(u-v_i) \nonumber \\
& \leq \sigma_{P^1_\epsilon}(v_i) + \| u - v_i \|_2 \cdot \max_{x \in P^1_\epsilon} \{ \| x \|_2 \} \nonumber \\
& \leq \sigma_{P^1_\epsilon}(v_i) + \frac{\epsilon M}{{{\color{black}}4} \sqrt{n} \bar{M}} \sqrt{n} \bar{M} \nonumber \\
& = \sigma_{P^1_\epsilon}(v_i) + \frac{\epsilon M}{4} \nonumber \\
& \leq \sigma_{\tilde{Q}}(v_i) + \frac{\epsilon M}{4} \nonumber \\
& \leq \sigma_{\tilde{Q}}(u) + \frac{\epsilon M}{4} + \frac{\epsilon M}{4} \nonumber \\
& = \sigma_{\tilde{Q}}(u) + \frac{\epsilon M}{2}, \label{eq:fifthineq} \end{aligned}$$ where the first inequality is due to the subadditivity property of the support functions [@rockafellar:1970], the second is due to the Cauchy-Schwartz inequality, the third follows from and , the fourth inequality is implied by the constraints defining $P_\epsilon$ in , and the last inequality is satisfied by . Inequality can be written as $$\frac{\sigma_{P^1_\epsilon}(u)}{\sigma_{\tilde{Q}}(u)} \leq 1+ \frac{\epsilon M}{2 \sigma_{\tilde{Q}}(u)} \leq 1+ \frac{\epsilon}{2},$$ [the second inequality]{} follows from .
Due to , $P^1_\epsilon$ achieves almost all the required conditions except the fact that it may not be well-behaved. Therefore, let $$P_\epsilon = \operatorname{conv}{(P^1_\epsilon \cup S)}.$$ First, note that $\tilde{Q} \subseteq P^1_\epsilon \subseteq P_\epsilon \subseteq Q$ where the first two containments are straightforward, and the last containment follows from the fact that $S \subseteq Q$ and $P^1_\epsilon \subseteq Q$. It remains to verify that $P_\epsilon$ is a packing polyhedron which would imply that it is [a]{} well-behaved packing polyhedron since $S \subseteq P_\epsilon$. However, observe that $P_\epsilon$ is the convex hull of the union of two packing polyhedra, and therefore it is straightforward to verify that it is a packing polyhedron.
Now suppose $Q$ is not bounded. Then we can decompose it as $Q = B + R$, where $B$ is a bounded packing set and $R$ is the recession cone of $Q$; explicitly, let $I = \{i \mid \operatorname{cone}(e_i) \subseteq Q\}$, so $B = Q \cap \{x \mid x_i \le 0~\forall i \in I\}$ and $R = \operatorname{cone}(\{e_i\}_{i \in I})$. Furthermore, let $\bar{B} = \operatorname{conv}(B \cup \bigcup_{i \in I} e_i)$, so that $\bar{B}$ is a *well-behaved* bounded packing set; notice $Q = B + R = \bar{B} + R$.
Applying the proof above to the bounded set $\bar{B}$, we obtain a well-behaved packing *polyhedron* $P_\epsilon$ satisfying $\frac{1}{1+\epsilon} \bar{B} \subseteq P_\epsilon \subseteq \bar{B}$. Then $P_\epsilon + R$ is a well-behaved packing polyhedron (the polyhedrality follows from the fact $\bar{B}$ is a polytope and $R$ is finitely generated, see for example Theorem 3.13 of [@ConCorZam14b]). Finally, since $\alpha P_\epsilon + R = \alpha (P_\epsilon + R)$ for all $\alpha > 0$, $$\frac{1}{1+\epsilon} (\bar{B} + R) = \frac{1}{1 + \epsilon} \bar{B} + R \subseteq P_\epsilon + R \subseteq \bar{B} + R.$$ Since $\bar{B} + R = Q$, $P_\epsilon + R$ is the desired polyhedral approximation. This concludes the proof.
Noting that $${\mathcal{{M}}}(Q) = \bigcap_{{M}\in {\mathcal{{M}}}} Q^{M},$$ it is sufficient to prove that $$\label{eq:Pkplusone}
Q \subseteq (\alpha_{\mathcal{{M}}}) \, Q^{M},$$ [for an arbitrary $M \in {\mathcal{{M}}}$]{} (see Observation \[obs:bijection\]).
Let $\epsilon > 0$ and $P_\epsilon$ be the well-behaved packing polyhedron satisfying the conditions of Lemma \[prop:SantanuLemma\]. Then, observe that $$\label{eq:generalContainment}
\frac{1}{1+\epsilon} Q \subseteq P_\epsilon \subseteq (\alpha_{\mathcal{{M}}}) (P_\epsilon)^{M}\subseteq (\alpha_{\mathcal{{M}}}) \, Q^{M},$$ where the first and the last containments follow due to $\frac{1}{1+\epsilon} Q \subseteq P_\epsilon \subseteq Q$, whereas the second one holds by assumption and the fact that $P_\epsilon$ is well-behaved.
Note that can be written as $Q \subseteq (1+\epsilon) (\alpha_{\mathcal{{M}}}) \, Q^{M}$. Since $\epsilon$ can be arbitrarily small, we obtain that $Q \subseteq (\alpha_{\mathcal{{M}}}) \, Q^{M}$.
Proof of Theorem \[thm:thm4\] {#subsec:4.4}
-----------------------------
Note that it is sufficient to prove the statement for ${\mathcal{{M}}}\in \{ {\mathcal{S}}^k, {\mathcal{L}}^k \}$ since ${\mathcal{S}}$ is a special case of ${\mathcal{S}}^k$. We will use Theorem \[thm:thm2\] to prove this statement. That is, letting $T {\subseteq}[n]$, we will show that for every ${M}\in {\mathcal{{M}}}$, we have ${M}[{T}] \in {\mathcal{{M}}}$ as well. [Recall the operator $\breve{(\cdot)}$ from equation .]{}
- Consider an arbitrary element of ${\mathcal{S}}^k$ as $$M = \bigcup_{i \in [k]} S(\pi^i,\pi^i_0).$$ Observe that $$M[{T}] = \bigcup_{i \in [k]} S(\breve \pi^i,\pi^i_0).$$ If $\breve \pi^i = {\boldsymbol{0}}$, then $S(\breve \pi^i,\pi^i_0) = \emptyset$. Therefore, $M[T]$ is also a $k$-branch split set since $k$-branch split is defined to be the union of at most $k$ split sets.
- Consider an arbitrary element of ${\mathcal{L}}^k$ as $$M = \{ x \in {\mathbb{R}}^n | (\pi^i)^\top x < \pi^i_0, \ i = 1,\hdots,k \} .$$ Observe that $$M[{T}] = \{ x \in {\mathbb{R}}^n | (\breve \pi^i)^\top x < \pi^i_0, \ i = 1,\hdots,k \} .$$ If $\breve \pi^i = {\boldsymbol{0}}$, then either the inequality $(\breve \pi^i)^\top x < \pi^i_0$ is trivially satisfied, or $M[{T}] = \emptyset$. Therefore, $M[T]$ is also a $k$-lattice-free set since $k$-lattice-free set is defined to be the union of at most $k$ lattice-free sets.
Proof of Theorem \[thm:thm5\]
-----------------------------
In order to make the proofs more self-contained, we record here standard bounds on the integrality gap of well-behaved packing polyhedra, which are essentially Proposition 6 of [@bodur2016aggregation].
\[prop:intGapPack\] Consider a well-behaved packing polyhedron $P_P = \{x \in {\mathbb{R}}^n_+ \mid (a^i)^{\top} x \le b_i,~\forall i \in [m]\}$. Then $P_P$ is a $(\min\{m,n\} + 1)$-approximation of the integer hull $P_P^I$.
It is equivalent to show that $P_P$ is both an $(m+1)$- and an $(n+1)$-approximation of $P_P^I$. The former is precisely Proposition 6 of [@bodur2016aggregation], and the latter follows from similar arguments, reproduced here. Given a cost vector $c \in {\mathbb{R}}^n_+$, we need to show that $\max\{c^\top x \mid x \in P_P\} \le (n + 1) \max\{c^\top x \mid x \in P_P^I\} =: (n+1) z^I$. Let $x^{LP}$ be an optimal solution for the left-hand side, and let $\hat{x}$ be the integer solution obtained by rounding down each component of $x^{LP}$. Since each component of the difference $x^{LP} - \hat{x}$ is in $[0,1]$, we have $$\begin{aligned}
z^I \ge c^\top \hat{x} \ge c^{\top} x^{LP} - \|c\|_{\infty}.
\end{aligned}$$ Moreover, since $P_P$ is well-behaved, all canonical vectors $e_i$ belong to $P_P^I$ and hence $z^I \ge \|c\|_{\infty}$. Adding $n$ times this lower bound to the displayed equation, we obtain $(n+1)z^I \ge c^{\top} x^{LP}$, thus proving the desired result. This concludes the proof.
### Case of ${\mathcal{S}}$ {#subsubsec:4.5.1}
We show that $\alpha_{\mathcal{S}}= 2$ in Proposition \[prop:packing2approx\]. The proof of Proposition \[prop:packing2approx\] involves a reduction to analyzing split closure of a packing polyhedron in ${\mathbb{R}}^2$, and a case analysis in ${\mathbb{R}}^2$ gives the correct factor of 2 (Lemma \[lem:pack\_2approx\]). For the last statement in the theorem, we provide a tight example in Proposition \[prop:pack\_TightEx\].
[We start with the proof of the result for the two-dimensional case.]{}
\[lem:pack\_2approx\] Let $P_P \subseteq {\mathbb{R}}^2$ be well-behaved. Then $P_P {\subseteq}2 {\mathcal{S}}(P_P)$.
[By [@cook:ka:sc:1990] and by Theorem \[thm:thm4\], the set $S(P_P)$ is a well-behaved packing polyhedron. To show that $P_P {\subseteq}2 {\mathcal{S}}(P_P)$, we just need to show that for all facet-defining inequalities ${\beta}^\top x \le \delta$ of $S(P_P)$, the inequality ${\beta}^\top x \le 2 \delta$ is valid for $P_P$. This is trivially satisfied for the facet-defining inequalities of $S(P_P)$ of the type $x_i \ge 0$, thus it remains to be shown for the other facet-defining inequalities of $S(P_P)$. Since $S(P_P)$ is of packing type, such facet-defining inequalities are of the form ${\beta}^\top x \le \delta$ with ${\beta}\in {\mathbb{R}}^2_+$. Since the split closure is finitely generated [@Ave12], each facet-defining inequality ${\beta}^\top x \le \delta$ of $S(P_P)$ defines a facet of a set $P_P^T := \operatorname{conv}(P_P \setminus \int T)$, where $T$ is a split set. (Note that the polyhedra $P_P^T$ are not necessarily of packing type.) Therefore, to complete the proof of the lemma, it suffices to show that for every split set $T$, and for every inequality ${\beta}^\top x \le \delta$ with ${\beta}\in {\mathbb{R}}^2_+$ valid for $P_P^T$, the inequality ${\beta}^\top x \le 2 \delta$ is valid for $P_P$. We show that for every split set $T$, and for every ${\beta}\in {\mathbb{R}}^2_+$, there exists $\hat x \in P_P^T$ that satisfies $\max\{{\beta}^\top x \mid x \in P_P\} \le 2{\beta}^\top \hat x$. This completes the proof because ${\beta}^\top \hat x \le \delta$ implies that every point in $P_P$ satisfies ${\beta}^\top x \le \max\{{\beta}^\top x \mid x \in P_P\} \le 2{\beta}^\top \hat x \le 2 \delta$.]{}
[Now, fix a split set $T$ and a vector]{} ${\beta}\in {\mathbb{R}}^2_+$, and let $\bar x$ be a vector in $P_P$ that achieves $\max\{{\beta}^\top x \mid x \in P_P\}$. Since $P_P$ is a packing polyhedron, we have $\bar x \ge 0$. We divide the proof in three main cases based on the position of vector $\bar x$.
1\. In the first case we assume that $\bar x \ge (1,1)$, and we define $\hat x := \lfloor \bar x \rfloor$. Since $P_P$ is a packing polyhedron, we have that $\hat x \in P_P \cap {\mathbb{Z}}^2 \subseteq P_P^T$. As $\bar x \ge (1,1)$, we have $2 \hat x \ge \bar x$. Finally, ${\beta}\ge 0$ implies $2 {\beta}^\top \hat x \ge {\beta}^\top \bar x$ as desired.
2\. In the second case we assume that $\bar x \le (1,1)$. Since $P_P$ is well-behaved, we have that points $(1,0)$ and $(0,1)$ are in $P_P$ and therefore in $P_P^T$. If ${\beta}_1 \ge {\beta}_2$, we define $\hat x := (1,0)$. Then $2{\beta}^\top \hat x = 2 {\beta}_1 \ge {\beta}_1 + {\beta}_2$. Since ${\beta}\ge 0$ and $\bar x \le (1,1)$, we have ${\beta}_1 + {\beta}_2 \ge {\beta}^\top \bar x$, which implies $2{\beta}^\top \hat x \ge {\beta}^\top \bar x$ as desired. Symmetrically, if ${\beta}_2 \ge {\beta}_1$, we define $\hat x := (0,1)$, and obtain $2{\beta}^\top \hat x \ge {\beta}^\top \bar x$.
3\. In the third case we assume that $\bar x_1 < 1$ and $\bar x_2 > 1$. (The same argument works for the symmetric case $\bar x_2 < 1$ and $\bar x_1 > 1$.) Assume first that $T$ is *not* a vertical split set [$\{x \mid t \le x_1 \le t+1\}$]{} for some integer $t$. Define now $\hat x^1 := \lfloor \bar x \rfloor = (0,{\lfloor\bar x_2\rfloor})$. Since $P_P$ is [a packing polyhedron]{}, [the]{} vector $\hat x^1$ is in $P_P \cap {\mathbb{Z}}^2$, and therefore in $P_P^T$. If $2{\beta}^\top \hat x^1 = 2 {\beta}^\top (0,{\lfloor\bar x_2\rfloor}) \ge {\beta}^\top \bar x$, then we are done, thus we now assume $2 {\beta}^\top (0,{\lfloor\bar x_2\rfloor}) \le {\beta}^\top \bar x$.
Let $\hat x^2 := (\bar x_1, \bar x_2-1)$. It can be shown that, since $T$ is not a vertical split set, the vector $\hat x^2$ is in $P_P^T$. We show $2 {\beta}^\top \hat x^2 = 2{\beta}^\top (\bar x_1,\bar x_2 -1) \ge {\beta}^\top \bar x$. Since ${\lfloor\bar x_2\rfloor} \ge 1$ and ${\beta}_2 \ge 0$, we have ${\beta}^\top \bar x \ge 2 {\beta}_2 {\lfloor\bar x_2\rfloor} \ge 2 {\beta}_2$. By adding ${\beta}^\top \bar x$ to both sides we obtain $2 {\beta}^\top \bar x - 2 {\beta}_2 \ge {\beta}^\top \bar x$, thus $2{\beta}^\top (\bar x_1,\bar x_2 -1) \ge {\beta}^\top \bar x$.
Finally, assume that $T$ is a vertical split set [$\{x \mid t \le x_1 \le t+1\}$]{} for some integer $t$. Define now $\hat x^1 := (1,0)$. Since $P_P$ is well-behaved, [the]{} vector $\hat x^1$ is in $P_P \cap {\mathbb{Z}}^2$, and therefore in $P_P^T$. If $2{\beta}^\top \hat x^1 = 2 {\beta}^\top (1,0) \ge {\beta}^\top \bar x$, then we are done, thus we now assume $2 {\beta}^\top (1,0) \le {\beta}^\top \bar x$. Define $\hat x^2 := (0, \bar x_2)$ and note that $\hat x^2 \in P_P^T$ since $T$ is a vertical split set. We show $2{\beta}^\top\hat x^2 = 2{\beta}^\top (0, \bar x_2) \ge {\beta}^\top \bar x$. Since ${\beta}_1 \ge 0$ and $\bar x_1 < 1$, we have $2 {\beta}_1 \bar x_1 < 2 {\beta}_1$. By summing the latter with $2 {\beta}_1 \le {\beta}^\top \bar x$ we obtain $2 {\beta}_1 \bar x_1 \le {\beta}^\top \bar x$. By adding and subtracting $2{\beta}_2 \bar x_2$ to the left-hand, we get $2 {\beta}^\top \bar x - 2{\beta}_2 \bar x_2 \le {\beta}^\top \bar x$ which implies $2 {\beta}_2 \bar x_2 \ge {\beta}^\top \bar x$ as desired.
\[prop:packing2approx\] Let $Q {\subseteq}{\mathbb{R}}^n$ be a well-behaved packing set. Then, $Q {\subseteq}2 {\mathcal{S}}(Q).$
It is sufficient to prove this proposition for a packing polyhedron, $P_P$, due to Theorem \[thm:thm3\]. Let $(\pi,\pi_0) \in {\mathbb{Z}}^n \times {\mathbb{Z}}$ and let ${\beta}^\top x \leq \delta$ be a valid inequality for $(P_P)^{\pi,\pi_0}$. Note that, due to Observation \[obs:bijection\], it is sufficient to show that ${\beta}^\top x \leq 2 \delta$ is valid for $P_P$. Due to Farkas’ Lemma [(e.g., Theorem 3.22 in [@ConCorZam14b])]{}, there exist ${{\color{black}}\lambda^1, \lambda^2} \in {\mathbb{R}}_+^m, \ {{\color{black}}\mu_1, \mu_2 \in {\mathbb{R}}_+}$ and ${{\color{black}}\sigma^1, \sigma^2} \in {\mathbb{R}}_+^n$ such that for any $j \in [n]$, we have $${\beta}_j = \sum_{i=1}^m \lambda^1_i A_{ij} + \mu_1 \pi_j - \sigma^1_j = \sum_{i=1}^m \lambda^2_i A_{ij} - \mu_2 \pi_j - \sigma^2_j.$$ Let $$Q := \{ x \mid (\lambda^1)^\top Ax \leq (\lambda^1)^\top b, ~ (\lambda^2)^\top Ax \leq (\lambda^2)^\top b, ~ x \geq 0\}.$$ Now, observe that $Q \supseteq P_P$. Therefore, it is sufficient to show that ${\beta}^\top x \leq 2 \delta$ is valid for $Q$. We will prove that the following holds: $$\label{eq:QclaimPacking}
Q \subseteq 2 Q^{\pi,\pi_0}.$$ Since ${\beta}^\top x \leq \delta$ is valid for $Q^{\pi,\pi_0}$ by the definition of $Q$, this will imply that ${\beta}^\top x \leq 2 \delta$ is valid for $Q$. In order to show that holds, we verify that $$\label{eq:minPacking}
\max \{ c^\top x \mid x \in Q\} \leq 2 \max \{ c^\top x \mid x \in Q^{\pi,\pi_0} \},$$ for any objective vector $c \in {\mathbb{R}}_+^n$. Let $x^*$ be a vertex of $Q$ that maximizes $c^\top x$ over $Q$. As $Q$ is defined by two linear inequalities, together with non-negativities, we know that at least $n-2$ components of $x^*$ are zero, say $x^*_j = 0$ for all $j=3,\hdots,n$. We will focus on the restriction of $Q$ to the first two variables, which we denote by $Q |_{{\mathbb{R}}^2}$.
Observe that $$\label{eq:qc1}
\max \{ c^\top x \mid x \in Q\} = \max \{ c_1 x_1 + c_2 x_2 | (x_1,x_2) \in Q |_{{\mathbb{R}}^2} \}.$$ Moreover, we have $$\label{eq:qc2}
\max \{ c^\top x \mid x \in Q^{\pi,\pi_0} \} \geq \max \{ c_1 x_1 + c_2 x_2 | (x_1,x_2) \in (Q |_{{\mathbb{R}}^2})^{\pi,\pi_0} \}$$ because [$(Q |_{{\mathbb{R}}^2})^{\pi,\pi_0} \subseteq Q^{\pi,\pi_0} |_{{\mathbb{R}}^2}$.]{} Due to and , in order to prove , it is sufficient to only prove in ${\mathbb{R}}^2$. Since $Q |_{{\mathbb{R}}^2}$ is well-behaved, this immediately follows from Lemma \[lem:pack\_2approx\].
\[prop:pack\_TightEx\] For every $\epsilon >0$, there exists a well-behaved packing polyhedron $\tilde{P}_P$ such that $\tilde{P}_P \not\subseteq (2-\epsilon) {\mathcal{S}}(\tilde{P}_P)$.
Let $\epsilon > 0$ and $M = \max\{1,\lceil \frac{2}{\epsilon}-1 \rceil \}$. Consider the instance $\textup{max} \{ x_1 + x_2 \,|\, x \in \tilde{P}_P\}$, where $$\begin{aligned}
\tilde{P}_P = \{x \in {\mathbb{R}}^2_+ \,|\, x_1 + Mx_2 \leq M, \ Mx_1 + x_2 \leq M \}.\end{aligned}$$ Note that $\tilde{P}_P$ is well-behaved. It is sufficient to show that $\frac{z^{LP}}{z^{{\mathcal{S}}}} \ge 2 - \epsilon$ for this instance.
1. $z^{LP} \ge \frac{2M}{M + 1}$: It can be checked that the point $\bar x_1 = \bar x_2 = \frac{M}{M + 1}$ is in $\tilde{P}_P$. Thus, $z^{LP} \ge \frac{2M}{M + 1}$.
2. $z^{{\mathcal{S}}} \le1$: Adding the two constraints defining $\tilde{P}_P$ we obtain the valid inequality $$\begin{aligned}
x_1 + x_2 \le \frac{2M}{M + 1}\end{aligned}$$ The corresponding CG cut is $x_1 + x_2 \leq 1$. Since each CG cut is also a split cut we obtain $z^{{\mathcal{S}}} \le 1$.
Thus, $\frac{z^{LP}}{z^{{\mathcal{S}}}} \ge \frac{2M}{M + 1}$; and our choice of $M$ completes the proof.
We note that the example given in Proposition \[prop:pack\_TightEx\] is the same as the one used in [@bodur2016aggregation] to show that the $2$-approximation bound for the CG closure of a well-behaved packing polyhedron is tight.
### Case of ${\mathcal{S}}^k$ {#subsubsec:4.5.2}
We will show that [we can choose $\alpha_{{\mathcal{S}}^k} = \min \{ 2^k,n \}+1$]{}. It is sufficient to prove this proposition for a packing polyhedron, $P_P$, due to Theorem \[thm:thm3\].
Let $P_P = \{ x \in {\mathbb{R}}^n | Ax \leq b, x \geq 0\}$ and $\pi^i \in {\mathbb{Z}}^n, \pi^i_0 \in {\mathbb{Z}}\textup{ for all } i \in [k]$. It is sufficient to prove that $(\min\{2^k,n\}+1) \, (P_P)^{\pi^1,\dots, \pi^k;\pi^1_0, \dots, \pi^k_0} \supseteq P_P$.
Let ${\beta}^\top x \leq \delta$ be a valid inequality for $(P_P)^{\pi^1,\dots, \pi^k;\pi^1_0, \dots, \pi^k_0}$. Since\
${\boldsymbol{0}}\in (P_P)^{\pi^1,\dots, \pi^k;\pi^1_0, \dots, \pi^k_0}$, we have $\delta \geq 0$. Therefore, it is sufficient to prove that $$(\min\{2^k,n\}+1) \left( \{ x | {\beta}^\top x \leq \delta \} \right) \supseteq P_P.$$ Let $\mathcal{G} = \{ G {\subseteq}{{\color{black}}[k]} : (P_P)_G^{{\pi}^1,\dots, {\pi}^k;\pi^1_0, \dots, \pi^k_0} \neq \emptyset \}$, where $(P_P)_G^{{\pi}^1,\dots, {\pi}^k;\pi^1_0, \dots, \pi^k_0}$ is defined as $$(P_P)_G^{\pi^1,\dots, \pi^k;\pi^1_0, \dots, \pi^k_0} = P_P {\mathbin{\scalebox{1.5}{\ensuremath{\cap}}}}\left(\bigcap_{i \in G} \{ (\pi^i)^\top x \geq \pi^i_0 + 1)\} \right){\mathbin{\scalebox{1.5}{\ensuremath{\cap}}}}\left(\bigcap_{i \in {{\color{black}}[k]} \setminus G} \{ (\pi^i)^\top x \leq \pi^i_0)\}\right).$$
By Farkas’ Lemma, we know that ${\beta}^\top x \leq \delta$ is valid for $$\label{eq:aggPPG}
\{ x \in {\mathbb{R}}_+^n | (\lambda^G)^\top A x \leq (\lambda^G)^\top b, (\pi^i)^\top x \geq \pi^i_0 + 1, \ \forall \ i \in G, \ (\pi^i)^\top x \leq \pi^i_0, \ \forall \ i \in {{\color{black}}[k]} \setminus G \},$$ for some $\lambda^G \in {\mathbb{R}}_+^m$.
Let $$Q = \{ x \in {\mathbb{R}}_+^n | (\lambda^G)^\top A x \leq (\lambda^G)^\top b, \ \forall G \in \mathcal{G} \}$$ which is well-behaved since $P_P$ is assumed to be well-behaved. Now, observe that $$\begin{aligned}
(\min\{2^k,n\}+1) \left( \{ x | {\beta}^\top x \leq \delta \} \right) & \supseteq (\min\{|\mathcal{G}|,n\}+1) \left( \{ x | {\beta}^\top x \leq \delta \} \right) \\
& \supseteq (\min\{|\mathcal{G}|,n\}+1) Q^I \supseteq Q \supseteq P_P,\end{aligned}$$ where the second containment follows from , the third one follows from [Proposition \[prop:intGapPack\]]{} since $Q$ is well-behaved, and the last one is straightforward.
### Case of ${\mathcal{L}}^k$ {#subsubsec:4.5.3}
We show that [we can choose $\alpha_{{\mathcal{L}}^k} = \min \{ k,n \}+1$]{}. It is sufficient to prove this proposition for a packing polyhedron, $P_P$, due to Theorem \[thm:thm3\].
Let $P_P = \{ x \in {\mathbb{R}}^n | Ax \leq b, x \geq 0\}$ and [let]{} $$L = \{ x \in {\mathbb{R}}^n | (\pi^j)^\top x \leq \pi^j_0, \ j = 1,\hdots,k \}$$ [be lattice-free.]{} Then, observe that $$\label{eq:convPPL}
(P_P)^L = \operatorname{conv}(P_P \setminus \int(L)) = \operatorname{conv}\left(\bigcup_{j=1}^k \left\{ x \in P_P ~|~ (\pi^j)^\top x \geq \pi^j_0 \right\} \right).$$ Without loss of generality, assume that the set $\{ x \in P_P ~|~ (\pi^j)^\top x \geq \pi^j_0 \}$ is non-empty if $j \leq r$, and empty otherwise, for some $r$ with $1 \leq r \leq k$.
Let ${\beta}^\top x \leq \delta$ be a valid inequality for $(P_P)^L$. Since the origin is contained in $(P_P)^L$, we have $\delta \geq 0$. Therefore, it is sufficient to prove that $$(\min\{k,n\}+1) \left( \{ x | {\beta}^\top x \leq \delta \} \right) \supseteq P_P.$$ By equation and Farkas’ Lemma, we know that ${\beta}^\top x \leq \delta$ is valid for $$\label{eq:aggPP}
\{ x \in {\mathbb{R}}_+^n | (\lambda^j)^\top A x \leq (\lambda^j)^\top b, (\pi^j)^\top x \geq \pi^j_0 \},$$ for $j = 1,\hdots,r$ where $\lambda^j \in {\mathbb{R}}_+^m$.
Let $$Q = \{ x \in {\mathbb{R}}_+^n | (\lambda^j)^\top A x \leq (\lambda^j)^\top b, \ j = 1,\hdots,r \}$$ which is well-behaved since $P_P$ is assumed to be well-behaved. Now, observe that $$(\min\{k,n\}+1) \left( \{ x | {\beta}^\top x \leq \delta \} \right) \supseteq (\min\{k,n\}+1) Q^I \supseteq Q \supseteq P_P,$$ where the first containment follows from and $L$ being a lattice-free set, the second one follows from [Proposition \[prop:intGapPack\]]{} since $Q$ is well-behaved, and the last one is straightforward.
### Proof of Proposition \[prop:prop1\] {#subsubsec:4.5.4}
Let $P_P$ be the standard relaxation of the stable set polytope: $$\begin{aligned}
P_P = \{ x \in {\mathbb{R}}_+^{n} \,|\, x_i + x_j \le 1 \ \forall i,j \in [n], \ i < j\}.\end{aligned}$$ Corresponding to the clique inequality ${\boldsymbol{1}}^\top x \le 1$, we optimize the all ones vector over $P_P$ and $(P_P)^I$, and obtain $z^{LP}=n/2$ and $z^{I}=1$, respectively. The CG rank of the clique inequality is known to be $\lceil \log_2 (n-1) \rceil$ [@hartmann], therefore it also constitutes an upper bound on the split rank.
Proofs for covering problems {#sec:Covering}
============================
\[prop:CoveringPoly\] ${\mathcal{S}}(P_C)$ is a covering polyhedron.
If $P_C$ is empty, then there is nothing to prove. So, assume that $P_C$ is not empty. It is known that the split closure of a polyhedron is also a polyhedron [@cook:ka:sc:1990]. Let ${\beta}^\top x \geq \delta$ be a valid inequality for ${\mathcal{S}}(P_C)$. Then, there exists $(\pi,\pi_0) \in {\mathbb{Z}}^n \times {\mathbb{Z}}$ such that ${\beta}^\top x \geq \delta$ is valid for $P_C^{\pi,\pi_0}$. If one side of the disjunction is empty, then we already know that ${\beta}^\top x \geq \delta$ is of covering type as it is a CG cut. Now, assume that both sides are nonempty. Then, due to Farkas’ Lemma, there exist multipliers $\lambda^1, \lambda^2 \in {\mathbb{R}}_+^m, \ \mu_1, \mu_2 \in {\mathbb{R}}_+$ and $\sigma^1, \sigma^2 \in {\mathbb{R}}_+^n$ for the aggregation $$\begin{aligned}
{4}
& (\lambda^1) \quad && Ax \geq b \qquad \qquad \qquad && (\lambda^2) \quad && Ax \geq b \\
& (\mu_1) && - \pi^\top x \geq -\pi_0 && (\mu_2) && \pi^\top x \geq \pi_0 +1 \\
& (\sigma^1_j) && x_j \geq 0 && (\sigma^2_j) && x_j \geq 0 \qquad \qquad j=1,\hdots,n\end{aligned}$$ such that, for any $j=1,\hdots,n$, $$\label{eq:cSplit}
{\beta}_j = \sum_{i=1}^m \lambda^1_i A_{ij} - \mu_1 \pi_j + \sigma^1_j = \sum_{i=1}^m \lambda^2_i A_{ij} + \mu_2 \pi_j + \sigma^2_j.$$ This implies that, for any $j=1,\hdots,n$, we have ${\beta}_j \geq 0$ (based on the sign of $\pi_j$, either the middle or the last expression [witnesses non-negativity]{}). Lastly, note that if $\delta < 0$, then ${\beta}^\top x \geq \delta$ is dominated by ${\beta}^\top x \geq 0$, which concludes the proof.
\[prop:covering2approx\] Let $P_C$ be well-behaved, i.e., $A_{ij} \leq b_i$ for all $i \in [m], j \in [n]$. Then, $$z^{LP} \geq \frac{1}{2} z^{{\mathcal{S}}}.$$
Let ${\beta}^\top x \geq \delta$ be a facet-defining inequality for ${\mathcal{S}}(P_C)$. Note that, due to Observation \[obs:bijection\], it is sufficient to show that ${\beta}^\top x \geq \frac{\delta}{2}$ is valid for $P_C$. [If ${\beta}^\top x \geq \delta$ is valid for $P_C$, then there is nothing to show. So, assume that it is a non-trivial inequality.]{}
If ${\beta}^\top x \geq \delta$ is a [non-trivial]{} CG cut, then [$\delta \geq 1$. We know that the strict inequality ${\beta}^\top x > \delta - 1$ is valid for $P_C$. If $\delta \geq 2$, then $\delta - 1 \geq \frac{\delta}{2}$, which implies that ${\beta}^\top x \geq \frac{\delta}{2}$ is valid for $P_C$. So, now assume that $\delta = 1$. Let $\lambda \in {\mathbb{R}}_+^m$ such that the CG cut is obtained by rounding up the coefficients and the right-hand-side of the base inequality $\lambda^\top A x \geq \lambda^\top b$, i.e., $\beta = {\lceil\lambda^\top A\rceil}$ and $\delta = {\lceil\lambda^\top b\rceil}$. As $\delta = 1$, we have $0 < \lambda^\top b \leq 1$. Thus, the scaled base inequality $(\lambda^\top A / \lambda^\top b) x \geq 1$ is valid for $P_C$. In addition, since $P_C$ is well-behaved, $\lambda^\top A_{\cdot j} \leq \lambda^\top b \ (\leq 1)$ for all $j \in [n]$ (where $A_{\cdot j}$ denotes the $j^\text{th}$ column of $A$. Then, for any $x \in P_C$, we have $$\beta^\top x = \sum_{j \in [n]} {\lceil\lambda^\top A_{\cdot j}\rceil} x_j \geq \sum_{j \in [n]} \frac{\lambda^\top A_{\cdot j}}{\lambda^\top b} x_j \geq 1 \geq \frac{{\lceil\lambda^\top b\rceil}}{2} = \frac{\delta}{2},$$ which implies that is $\beta^\top x \geq \frac{\delta}{2}$ valid for $P_C$. ]{}
Otherwise, let $(\pi,\pi_0) \in {\mathbb{Z}}^n \times {\mathbb{Z}}$ be a corresponding vector such that ${\beta}^\top x \geq \delta$ is valid for $P_C^{\pi,\pi_0}$. Let $$Q := \{ x \mid (\lambda^1)^\top Ax \geq (\lambda^1)^\top b, ~ (\lambda^2)^\top Ax \geq (\lambda^2)^\top b, ~ x \geq 0\},$$ where $\lambda^1$ and $\lambda^2$ are the multipliers that satisfy . Now, observe that $Q \supseteq P_C$. Therefore, it is sufficient to show that ${\beta}^\top x \geq \frac{\delta}{2}$ is valid for $Q$. We will prove that the following holds: $$\label{eq:Qclaim}
Q \subseteq \frac{1}{2} Q^{\pi,\pi_0}.$$ Since ${\beta}^\top x \geq \delta$ is valid for $Q^{\pi,\pi_0}$ by the definition of $Q$, this will imply that ${\beta}^\top x \geq \frac{\delta}{2}$ is valid for $Q$. In order to show that holds, we verify that $$\label{eq:min}
\min \{ c^\top x \mid x \in Q\} \geq \frac{1}{2} \min \{ c^\top x \mid x \in Q^{\pi,\pi_0} \},$$ for any objective vector $c \in {\mathbb{R}}_+^n$. Let $x^*$ be a vertex of $Q$ that minimizes $c^\top x$ over $Q$. If $x^*$ belongs to $Q^{\pi,\pi_0}$, we are done. Thus, assume that $x^* \notin Q^{\pi,\pi_0}$. We will prove by showing that there exists a point $\hat{x} \in Q^{\pi,\pi_0}$ such that $c^\top \hat{x} \leq 2 c^\top x^*$. As $Q$ is defined by two linear inequalities, together with non-negativities, we know that at least $n-2$ components of $x^*$ are zero, say $x^*_j = 0$ for all $j=3,\hdots,n$. We will focus on this restriction of $Q$ in ${\mathbb{R}}_+^2$ in order to identify $\hat{x}$. Without loss of generality, assume that $c_1 \geq c_2$. A key observation that follows from the definition of split cuts is $$\label{eq:vee}
(x^*_1+1,x^*_2,{\boldsymbol{0}}) \in Q^{\pi,\pi_0} \vee (x^*_1,x^*_2+1,{\boldsymbol{0}}) \in Q^{\pi,\pi_0}.$$ [Moreover, if $\pi \neq e_1$, then $(x^*_1,x^*_2+1,{\boldsymbol{0}}) \in Q^{\pi,\pi_0}.$\
]{} Now, we will consider two cases to prove .\
\
*Case 1*. $x^*_1 \geq 1$: [Using (\[eq:vee\]), there are two subcases based on whether $(x^*_1+1,x^*_2,{\boldsymbol{0}}) \in Q^{\pi,\pi_0} $ or $ (x^*_1,x^*_2+1,{\boldsymbol{0}}) \in Q^{\pi,\pi_0}$. If $\hat{x} = (x^*_1+1,x^*_2,{\boldsymbol{0}}) \in Q^{\pi,\pi_0}$, then it is sufficient to show that $$c^\top x^* \geq \frac{1}{2} (c^\top x^* + c_1),$$ which is equivalent to $c^\top x^* \geq c_1$, which holds because $x^*_1 \geq 1$ and $c_1, c_2, x^*_2 \geq 0$. If $\hat{x} = (x^*_1,x^*_2 + 1,{\boldsymbol{0}}) \in Q^{\pi,\pi_0}$, then it is sufficient to show that $$c^\top x^* \geq \frac{1}{2} (c^\top x^* + c_2),$$ which is equivalent to $c^\top x^* \geq c_2$, which holds because $x^*_1 \geq 1$, $c_1 \geq c_2$ and $c_2, x^*_2 \geq 0$. ]{}\
\
*Case 2*. $0 \leq x^*_1 < 1$: Note that by construction, $Q$ is a well-behaved covering polyhedron. Now, consider the following two subcases:\
\
*Case 2a*. $(\pi,\pi_0) \neq (e_1,0)$: [In this case as discussed above]{}, [$\hat{x} = (x^*_1,x^*_2 + 1,{\boldsymbol{0}}) \in Q^{\pi,\pi_0}$]{}. It is sufficient to show that $$c^\top x^* \geq \frac{1}{2} (c^\top x^* + c_2),$$ which is equivalent to $c^\top x^* \geq c_2$. This holds as we have $x^*_1+x^*_2 \geq 1$ since $Q$ is well-behaved [and because $c_1 \geq c_2$]{}.\
\
*Case 2b*. $(\pi,\pi_0) = (e_1,0)$: Let $\hat{x} = (x^*_1,x^*_2 + x^*_1 x^*_2,{\boldsymbol{0}})$. We will first show that $\hat{x} \in Q^{\pi,\pi_0}$.
Figure \[fig:1pic\] illustrates the restriction of $Q$ to the first two variables, which we denote by $Q |_{{\mathbb{R}}^2}$. Observe that $Q |_{{\mathbb{R}}^2}$ is a well-behaved covering polyhedron because $Q$ is a well-behaved covering polyhedron. Note that $x_1^* > 0$ due to the assumption $x^* \notin Q^{e_1,0}$. [Since $0 < x^*_1 < 1$, and $Q |_{{\mathbb{R}}^2}$ is a well-behaved covering polyhedron, we have that $x_1 \geq \gamma$ cannot be a valid inequality for $Q |_{{\mathbb{R}}^2}$ where $0< \gamma \leq x^*_1$. In other words, all non-trivial inequalities $\alpha_1 x_1 + \alpha_2 x_2 \geq \theta$ defining $Q |_{{\mathbb{R}}^2}$ must have $\alpha_2 > 0$. Therefore, there must exist a vertex of the form $(0,y)$ of $Q |_{{\mathbb{R}}^2}$.]{} Since $(x^*_1,x^*_2)$ is a vertex of $Q |_{{\mathbb{R}}^2}$ [with $x_1^* > 0$]{}, we know that $(0,y) \neq (x^*_1,x^*_2)$.
Let $(h,0)$ be the intercept of the line passing through the vertices $(0,y) $ and $(x^*_1,x^*_2)$. [We first claim that $h \geq 1$. Note that the supporting hyperplane corresponding to non-trivial facet-defining inequality at $(0, y)$ (i.e., different from $x_1 \geq 0$) intersects the $x_1$-axis at a point $(\tilde{h}, 0)$ with $\tilde{h} \geq 1$ due to well-behavedness of $Q |_{{\mathbb{R}}^2}$. Since the line passing through the vertices $(0,y) $ and $(x^*_1,x^*_2)$ has an intercept at least as large as $\tilde{h}$, we obtain that $h \geq 1$.]{}
Then, we have
(0,0) – (0,4.5); (0,0) – (4.5,0); (0,3.5) – (1.5,1.5); (1.5,1.5) – (4,0); (0,4.5) – (0,3.5) – (1.5,1.5) – (4,0) – (4.5,0) – (4.5,4.5) – cycle; (1.5,1.5) – (2.635,0); at (4,0) ; at (1.5,1.5) ; (xstar) at (2.1,1.7) [$(x^*_1,x^*_2)$]{}; at (0,3.5) ; (xstar) at (0.58,3.65) [$(0,y)$]{}; (h) at (2.635,-0.3) [$(h,0)$]{}; at (2.635,0) [x]{};
$$\label{eq:heq}
\frac{y}{h} = \frac{y - x^*_2}{x^*_1} \Rightarrow h = \frac{y x^*_1}{y-x^*_2} \ .$$
[Since $h \geq 1$, we have:]{} $$y x^*_1 \geq y - x^*_2 \iff \displaystyle y \leq \frac{x^*_2}{1-x^*_1}
\Rightarrow \displaystyle (0, \frac{x^*_2}{1-x^*_1},{\boldsymbol{0}}) \in Q^{e_1,0}.$$ The last implication follows from the fact that $Q$ is a covering polyhedron, $(0,y,{\boldsymbol{0}}) \in Q$ and $(\pi,\pi_0) = (e_1,0)$. Similarly, we have $(1,x^*_2,{\boldsymbol{0}}) \in Q^{e_1,0}$. The following convex combination of these two points yields $\hat{x}$ as $$(1-x^*_1) (0, \frac{x^*_2}{1-x^*_1},{\boldsymbol{0}}) + x^*_1 (1,x^*_2,{\boldsymbol{0}}) = (x^*_1,x^*_2 + x^*_1 x^*_2,{\boldsymbol{0}}) = \hat{x}.$$ Finally, observe that [since $c_1 \geq 0$ and $0 \leq x^*_1 \leq 1$]{} $${{\color{black}}c^\top x^* \geq c_2 x^*_2 \geq c_2 x^*_1 x^*_2 \Rightarrow c^\top x^* \geq \frac{1}{2} (c^\top x^* + c_2 x^*_1 x^*_2) = \frac{1}{2}c^\top\hat{x}},$$ which completes the proof.
We now show that Proposition \[prop:covering2approx\] is tight. In order to do so, we exhibit an instance of a well-behaved covering polyhedron and a nonnegative objective function such that LP is not better than a 2-approximation of ${\mathcal{S}}$. The construction given in the following example is the same that we used in [@bodur2016aggregation] to show that our $2$-approximation bounds for *1-row closure* and *1-row CG closure* are tight.
\[prop:LBAC\] For every $\epsilon >0$, there exists a well-behaved covering polyhedron $\tilde{P}_C$ such that $\frac{1}{2-\epsilon} \tilde{P}_C \not {\subseteq}{\mathcal{S}}(\tilde{P}_C)$.
Let $\epsilon >0$ and $n = \textup{max}\{2,\lceil \frac{1}{\epsilon}\rceil\}$. Consider the instance $\textup{min} \{ \sum_{j = 1}^n x_j \,|\, x \in \tilde{P}_C\}$, where $$\begin{aligned}
\tilde{P}_C = \{x \in {\mathbb{R}}^n_+ \,|\, x_i + \displaystyle \sum_{j \in [n]\setminus \{i\}} 2x_j \geq 2, \ \forall i \in [n]\}.\end{aligned}$$ Note that $\tilde{P}_C$ is well-behaved. It is sufficient to show that $\frac{z^{{\mathcal{S}}}}{z^{LP}} \geq 2 - \epsilon$ for this instance.
1. $z^{LP} \le \frac{2n}{2n - 1}$: It can be checked that the point $\bar x_j = \frac{2}{2n - 1}$ for each $j \in [n]$ is in $\tilde{P}_C$. Thus, $z^{LP} \le \frac{2n}{2n - 1}$.
2. $z^{{\mathcal{S}}} \geq 2$: Adding all the constraints defining $\tilde{P}_C$ we obtain the valid inequality $$\begin{aligned}
\sum_{j \in [n]} x_j \geq \frac{2n}{2n - 1}.\end{aligned}$$ The corresponding CG cut is $\sum_{j \in [n]} x_j \geq 2$. Since each CG cut is also a split cut we obtain $z^{{\mathcal{S}}} \geq 2$.
Thus, $\frac{z^{{\mathcal{S}}}}{z^{LP}} \geq 2 - \frac{1}{n}$; and our choice of $n$ completes the proof.
\[prop:CoveringSplitWell\] Let $P_C$ be well-behaved, i.e., $A_{ij} \leq b_i$ for all $i \in [m], j \in [n]$. Then, ${\mathcal{S}}(P_C)$ is well-behaved.
Let ${\beta}^\top x \geq \delta$ be a facet-defining (i.e., nondominated) inequality for ${\mathcal{S}}(P_C)$. If it is a CG cut, then we are done. Thus, we assume that it is a non-CG cut. For a contradiction, suppose that ${\beta}_1 > \delta$. [This is because any CG cut can be written as $\sum_j \lceil \sum_{i = 1}^m \lambda_i A_{ij} \rceil x_j \geq \lceil \sum_{i = 1}^m \lambda_i b_i \rceil$, where $\lambda_i \geq 0 $ for $i \in [m]$. Therefore $\sum_{i = 1}^m \lambda_i A_{ij} \leq \sum_{i = 1}^m \lambda_i $ for all $j \in [n]$, which implies that $\lceil \sum_{i = 1}^m \lambda_i A_{ij} \rceil \leq \lceil \sum_{i = 1}^m \lambda_i b_i \rceil$ for all $j \in [n]$.]{}
We know that ${\beta}^\top x \geq \delta$ is valid for $P_C^{\pi,\pi_0}$ for some $(\pi,\pi_0) \in {\mathbb{Z}}^n \times {\mathbb{Z}}$. Then, there exist multipliers $\lambda^1, \lambda^2 \in {\mathbb{R}}_+^m, \ \mu_1, \mu_2 \in {\mathbb{R}}_+$ and $\sigma^1, \sigma^2 \in {\mathbb{R}}_+^n$ such that $$\begin{aligned}
\label{eqn:CovWell_SideOne}
& ({\beta},\delta) = \lambda^1 (A,b) + \mu_1 (-\pi,-\pi_0) + \sigma^1 ({\boldsymbol{1}},0) \\
\label{eqn:CovWell_SideTwo}
& ({\beta},\delta) = \lambda^2 (A,b) + \mu_2 (-\bar{\pi},-\bar{\pi}_0) + \sigma^2 ({\boldsymbol{1}},0)\end{aligned}$$ where $(\bar{\pi},\bar{\pi}_0) = (-\pi,-\pi_0-1)$. Note that if $\sigma^1_1 > 0$ and $\sigma^2_1 > 0$, then we can obtain another split cut by decreasing both $\sigma^1_1$ and $\sigma^2_1$ by $\min \{ \sigma^1_1,\sigma^2_1\}$, which dominates the given split cut ${\beta}^\top x \geq \delta$. Therefore, we assume, WLOG, that $\sigma^2_1 = 0$. Then, we make the following two cases:\
*Case 1*. $- \bar{\pi}_0 \geq - \bar{\pi}_1$: This implies that $\lambda^2 A_{\cdot 1} + \mu_2 (-\bar{\pi}_1) \leq \lambda^2 b + \mu_2 (-\bar{\pi}_0)$ (where $A_{\cdot 1}$ denotes the first column of $A$), equivalently ${\beta}_1 \leq \delta$, which is a contradiction.\
*Case 2*. $- \bar{\pi}_0 \ {{\color{black}}<} - \bar{\pi}_1$: This condition is equivalent to $1-\pi_1 < -\pi_0$. We first claim that $\sigma^1_1 > \mu_1$. From and ${\beta}_1 > \delta$, we have $$\lambda^1 A_{\cdot 1} - \mu_1 \pi_1 + \sigma^1_1 > \lambda^1 b + \mu_1 (-\pi_0) > \lambda^1 b + \mu_1 (1-\pi_1),$$ which implies that $$(\lambda^1 A_{\cdot 1}-\lambda^1 b) + \sigma^1_1 - \mu_1 > 0.$$ As $P_C$ is well-behaved, we have $\lambda^1 A_{\cdot 1}-\lambda^1 b \leq 0$, thus we get $\sigma^1_1 > \mu_1$. Next, we let $$\tilde{\pi} := \pi - e_1, \ \tilde{\sigma}^1 := \sigma^1 - \mu_1 e_1, \ \tilde{\sigma}^2 := \sigma^2 + \mu_2 e_1.$$ Note that $\tilde{\sigma}^1_1 > 0$. Also, due to and , we have $$\label{eq:CovWellBothSides}
({\beta},\delta) = \lambda^1 (A,b) + \mu_1 (-\tilde{\pi},-\pi_0) + \tilde{\sigma}^1 ({\boldsymbol{1}},0) = \lambda^2 (A,b) + \mu_2 (\tilde{\pi},-\bar{\pi}_0) + \tilde{\sigma}^2 ({\boldsymbol{1}},0).$$ Note that $\mu_2 > 0$ since otherwise, i.e., when $\mu_2 = 0$, the equation and $\sigma^2_1 = 0$ give the contradiction ${\beta}_1 = \lambda^2 A_{\cdot 1} \leq \lambda^2 b = \delta$. Therefore, we have $\tilde{\sigma}^2_1 > 0$ as well. If we reduce both $\tilde{\sigma}^1_1$ and $\tilde{\sigma}^2_1$ by a sufficiently small $\epsilon > 0$, so that they are still nonnegative, from we obtain another valid split cut $({\beta}- 2 \epsilon e_1)^\top x \geq \delta$ which dominates ${\beta}^\top x \geq \delta$, hence a contradiction.
**Acknowledgements.** Santanu S. Dey would like to acknowledge the support of the NSF grant CMMI\#1149400. Marco Molinaro would like to acknowledge the support of CNPq grants Universal \#431480/2016-8 and Bolsa de Produtividade em Pesquisa \#310516/2017-0; this work was partially while the author was a Microsoft Research Fellow at the Simons Institute for the Theory of Computing. [We would like to thank the reviewers for their careful and constructive comments that have significantly improved the paper.]{}
Additional proofs {#sec:appendix}
=================
\[obs:PackingSplitAgg\] For packing polyhedra, split cuts are not necessarily aggregation cuts.
An example, where there exists a split cut that cannot be obtained as an aggregation cut, is provided in Figure \[fig:PackingSplitAgg\].
(0,0) – (1.8,0) – (0.9,1.9) – (0,1.9) – cycle; (0,0) – (0,4.2); (0,0) – (4.2,0); (0,3.8) – (1.8,0); (0,1.9) – (4,1.9); at (0,2.7) ; at (1.8,1.4) ; (A) at (-0.2,3.8) [7]{}; (A) at (-0.2,2.7) [2]{}; (A) at (-0.35,1.9) [7/4]{}; (A) at (1.8,-0.2) [1]{}; (0,1.4) – (1.8,1.4); (A) at (-0.152,1.4) [1]{}; (1.8,3.8) – (1.8,0); (0,1.9) – (1.8,0);
In the figure, the shaded region represents the packing polyhedron $P = \{ x \in {\mathbb{R}}_+^2 \mid 7 x_1 + x_2 \leq 7, 4 x_2 \leq 7\}$. It is easy to see that $7 x_1+4 x_2 \leq 7$ (the dashed line in the figure) is a split cut obtained by using the split set $S(e_1,0)$. Note that this cut separates both of the points $(0,2)$ and $(1,1)$. We next show that this cut is not an aggregation cut by proving that $(0,2)$ and $(1,1)$ are not separated at the same time by any aggregation cut. An inequality is an aggregation cut for $P$ if it is valid for the set $P(\alpha) := \operatorname{conv}( \{ x\in {\mathbb{R}}_+^2 \mid (7-7\alpha) x_1 + (3 \alpha +1) x_2 \leq 7 \})$ for some $\alpha \in [0,1]$. It can be easily verified that if $\alpha \leq 5/6$, then $(0,2) \in P(\alpha)$, and if $\alpha \geq 1/4$, then $(1,1) \in P(\alpha)$.
\[obs:CoveringSplitAgg\] For covering polyhedra, split cuts are not necessarily aggregation cuts.
An example, where there exists a split cut that cannot be obtained as an aggregation cut, is provided in Figure \[fig:CoveringSplitAgg\].
(0,3.8) – (0.7108,1.1) – (4,1.1) – (4,3.8) – cycle; (0,0) – (0,4.2); (0,0) – (4.2,0); (0,3.8) – (1,0); (0,1.1) – (4,1.1); at (0,2.7) ; at (1,0.7) ; (A) at (-0.2,3.8) [7]{}; (A) at (-0.2,2.7) [6]{}; (A) at (-0.35,1.1) [7/4]{}; (A) at (1,-0.2) [1]{}; (0,0.7) – (1,0.7); (A) at (-0.152,0.7) [1]{}; (1,3.8) – (1,0); (0,3.8) – (1.222,0.5);
In the figure, the shaded region represents the covering polyhedron $P = \{ x \in {\mathbb{R}}_+^2 \mid 7 x_1 + x_2 \geq 7, 4 x_2 \geq 7\}$. It is easy to see that $21 x_1+4 x_2 \geq 28$ (the dashed line in the figure) is a split cut obtained by using the split set $S(e_1,0)$. Note that this cut separates both of the points $(0,6)$ and $(1,1)$. We next show that this cut is not an aggregation cut by proving that $(0,6)$ and $(1,1)$ are not separated at the same time by any aggregation cut. An inequality is an aggregation cut for $P$ if it is valid for the set $P(\alpha) := \operatorname{conv}( \{ x\in {\mathbb{R}}_+^2 \mid (7-7\alpha) x_1 + (3 \alpha +1) x_2 \geq 7 \})$ for some $\alpha \in [0,1]$. It can be easily verified that if $\alpha \geq 1/18$, then $(0,6) \in P(\alpha)$, and if $\alpha \leq 1/4$, then $(1,1) \in P(\alpha)$.
[^1]: bodur@mie.utoronto.ca
[^2]: delpia@wisc.edu
[^3]: santanu.dey@isye.gatech.edu
[^4]: mmolinaro@inf.puc-rio.br
|
---
abstract: 'There is an observational indication of extragalactic magnetic fields. No known astrophysical process can explain the origin of such large scale magnetic fields, which motivates us to look for their origin in primordial inflation. By solving the linearized Einstein equations, we study metric perturbations sourced by magnetic fields that are produced during inflation. This leads to a simple but robust bound on the inflation models by requiring that the induced metric perturbation should not exceed the observed value $10^{-5}$. In case of the standard single field inflation model, the bound can be converted into a lower bound on the Hubble parameter during inflation.'
---
RESCEU-9/12
.5in
[**Metric perturbation from inflationary magnetic field and generic bound on inflation models** ]{}
.45in
[Teruaki Suyama$^1$ and Jun’ichi Yokoyama$^{1,2}$ ]{}
.45in
[*$^1$ Research Center for the Early Universe (RESCEU), Graduate School of Science,\
The University of Tokyo, Tokyo 113-0033, Japan* ]{}\
[*$^2$ Institute for the Physics and Mathematics of the Universe (IPMU),\
The University of Tokyo, Kashiwa, Chiba, 277-8568, Japan* ]{}
.4in
Introduction
============
It has been well known that magnetic fields of $\sim \mu {\rm G}$ exist in galaxies (see for example, [@Kronberg:1993vk; @Grasso:2000wj; @Widrow:2002ud; @Govoni:2004as]). These fields are thought to be an outcome of the amplification of seed magnetic fields of unknown nature through the dynamo mechanism. Magnetic fields are also known to exist in cluster of galaxies, too [@Kronberg:1993vk]. Although such seed magnetic fields are expected to still reside in the intergalactic medium and in voids, there had been no observational support of the seed magnetic fields until quite recently. In 2010, observational data taken by Fermi and High Energy Stereoscopic System (HESS) gamma-ray telescopes provided a strong support of the existence of extragalactic magnetic fields of at least $B \simeq 10^{-17}~{\rm G}$ on ${\rm Mpc}$ scales [@Neronov:1900zz; @Taylor:2011bn; @Vovk:2011aa]. The lower bound on $B$ was derived from detection by HESS of ${\rm TeV}$ gamma rays coming from TeV blazars and non-detection by Fermi of ${\rm GeV}$ scale cascade emission, which is in contradiction with the assumption of zero magnetic field.
Explaining the origin of the extragalactic magnetic fields of $B \simeq 10^{-17}~{\rm G}$ on ${\rm Mpc}$ scales is challenging and remains to be addressed. No promising astrophysical processes are known to generate the suggested amplitude of magnetic fields on such large length scales [^1]. In light of this situation, the best alternative we can think of is to make use of inflation in the early Universe [@Guth:1980zm; @Sato:1980yn; @Starobinsky:1980te] which causally connects length scales far beyond the Hubble radius through superluminal expansion [@Ratra:1991bn; @Gasperini:1995dh; @Bamba:2003av; @Bamba:2004cu; @Bamba:2006ga; @Martin:2007ue; @Subramanian:2009fu; @Kandus:2010nw; @Barnaby:2012tk]. It is well known that to realize magnetogenesis by inflation the action for the electromagnetic field needs to be modified such that the action breaks conformal invariance. The most-used Lagrangian for this purpose is a form $f^2(\phi) F^{\mu \nu} F_{\mu \nu}$, where $\phi$ is some scalar field (e. g. an inflaton or a dilaton) that varies during inflation. It is not easy, however, to obtain a successful model, because typical models encounter either breakdown of perturbation theory due to the strong coupling that occurs in the earlier stage of inflation [@Demozzi:2009fu], or generation of excessive electric fields whose energy density may cause serious backreaction [@Ratra:1991bn; @Bamba:2003av]. The purpose of this paper is to discuss the latter effect caused by primordial magnetic fields without respect to its specific generation mechanisms. As already discussed in the literatures [@Kanno:2009ei; @Finelli:2011cw; @Byrnes:2011aa; @Urban:2011bu; @Demozzi:2012wh; @Barnaby:2012tk], if the energy density of the magnetic field exceeds the background inflaton energy density, its backreaction onto the background dynamics becomes significant, which generally destroys the homogeneity and isotropy of the Universe and puts a constraint on model building. In addition to this, even if the backreaction effect is small and is not problematic for background evolution, there is another issue that one must take into account. The energy-momentum tensor of the magnetic field also induces metric perturbations and distorts the homogeneous and isotropic FLRW Universe [@Barrow:2006ch; @Kojima:2009ms; @Caldwell:2011ra]. Degree of the distortion depends on the inflation models and also on the electromagnetic models that achieve magnetogenesis. But whatever be the model of inflationary magnetogenesis, the resultant amplitude of metric perturbation must not exceed the observed value $\sim 10^{-5}$ [@Komatsu:2010fb]. Put it in another way, in addition to the standard constraints of explaining the observed feature of primordial perturbations, we can derive an additional generic bound that must be imposed on any inflation model by analyzing metric perturbations arising from the inflationary magnetic fields, which we work out in this paper.
To achieve this in quantitative manner, we study evolution of metric perturbations induced by magnetic field which is supposed to be generated during inflation. We will treat the energy-momentum tensor of the magnetic field as first order perturbation and solve the linearized Einstein equations on the FLRW Universe throughout the inflationary era and the subsequent radiation dominated era. This paper reports the results of such a calculation and derives a new constraint on inflation models. It is found that the slow-roll parameter that characterizes how slowly the Hubble parameter changes during inflation needs to be bounded from below in order to avoid large metric perturbation from magnetic field. This is basically equivalent to the constraint $\delta \rho_{\inf} \gtrsim \delta \rho_B$ which tells that the fluctuation of inflaton energy density must be larger than the energy density of the magnetic field.
Metric perturbation from magnetic field
=======================================
In this section, we consider the evolution of the metric perturbation during inflation and subsequent radiation dominated Universe in the presence of magnetic field generated quantum mechanically by breaking the conformal invariance. Since magnetic field is assumed to be absent at the background level, we will treat its energy-momentum tensor as linear perturbation. Perturbations of inflaton field and radiation/matter energy-momentum tensor are also treated as linear perturbations.
Let us first write down the perturbed metric having only scalar-type perturbations in general gauge (for example, see [@Kodama:1985bj]); $$ds^2=-(1+2A) dt^2+2a^2 \partial_i B dx^i dt+a^2(t) \left[ (1+2\psi) \delta_{ij}+2 \partial_i \partial_j E \right] dx^i dx^j.$$ Here $A,~B,~\psi$ and $E$ are metric perturbations. We denote the perturbed energy-momentum tensor including both matter such as inflaton or radiation and magnetic field as $$\delta T^0_{~0}=-\delta \rho,~~~\delta T^0_{~i}=(\rho+P) a^2 \partial_i v,~~~\delta T^i_{~0}=-(\rho+P) \partial_i (v-B),~~~\delta T^i_{~j}=\delta^i_{~j} \delta P+\pi^i_{~j},$$ where $v$ is defined by $u_i \equiv a^2 \partial_i v$ and $\pi^i_{~i}=0$ is the anisotropic stress. Since in many inflation models the anisotropic stress of matter is second order in perturbation, we will assume that only the magnetic field yields the first order anisotropic stress. If the time evolution of the magnetic field is solely due to the cosmological expansion, $\pi^i_{~j}$ decays in proportional to $a^{-4}$ just in the same way as radiation. On the other hand, if the electromagnetic action is modified, for example, by having the electromagnetic field tensor couple to the inflaton like $$S = -\frac{1}{4} \int d^4x~\sqrt{-g} f(\phi) F^{\mu \nu}F_{\mu \nu}, \label{ele-action}$$ where $\phi$ is inflaton, then $\pi^i_{~j}$ can be made to decay much slower than $a^{-4}$. In particular, in order to have scale invariant power spectrum of magnetic field on cosmological scales, $\pi^i_{~j}$ needs to be almost constant during inflation. This is indeed possible for some suitable choices of the functional form of $f(\phi)$.
For later convenience, we will write down the explicit transformation properties of perturbation variables under the gauge transformation, $t \to {\bar t}=t-T,~x^i \to {\bar x}^i=x^i-\partial_i L$; $$\begin{aligned}
&&{\bar A}=A+{\dot T},~~~{\bar B}=B+{\dot L}-\frac{T}{a^2},~~~{\bar \psi}=\psi+HT,~~~{\bar E}=E+L,~~~{\bar \sigma_g}=\sigma_g-\frac{T}{a^2}, \\
&&{\overline {\delta \rho}}=\delta \rho+{\dot \rho}T,~~~{\overline {\delta P}}=\delta P+{\dot P}T,~~~{\bar v}=v-\frac{T}{a^2},\end{aligned}$$ where $\sigma_g \equiv B-{\dot E}$ denotes the shear of the four-velocity $u^\mu$.
The linearized Einstein equations in Fourier space are given by $$\begin{aligned}
&&\frac{k^2}{a^2} \psi+Hk^2 \sigma_g-3H^2 A+3H {\dot \psi}=4\pi G \delta \rho, \\
&&HA-{\dot \psi}=-4\pi G(\rho+P) a^2 v, \\
&&-\frac{k^2}{3a^2} (A+\psi)+(3H^2+2{\dot H}) A-{\ddot \psi}-3H {\dot \psi}+H{\dot A}-\frac{k^2}{3} ( {\dot \sigma_g}+3H\sigma_g )=4\pi G\delta P, \\
&&\frac{k^2}{a^2} (A+\psi)+k^2 ( {\dot \sigma_g}+3H\sigma_g )=-8\pi G \Pi,\end{aligned}$$ and $\Pi$ is defined by $$\pi^i_{~j}=\left( \frac{1}{3} \delta_{ij}-\frac{k_i k_j}{k^2} \right) \Pi.$$ The linearized conservation laws $\nabla_\mu T^\mu_{~\nu}=0$ are given by $$\begin{aligned}
&&{\dot {\delta \rho}}+3H (\delta \rho+\delta P)+3(\rho+P) {\dot \psi}-(\rho+P) k^2 (v-\sigma_g)=0, \\
&&\partial_t \left[ (\rho+P)a^2v \right]+\delta P-\frac{2}{3} \Pi+3H(\rho+P)a^2v+(\rho+P)A=0.\end{aligned}$$
To see how the curvature perturbation is affected by the presence of magnetic field, let us consider the curvature perturbation on the uniform energy density hyper-surface on which $\delta \rho=0$ is satisfied. Denoting the curvature perturbation on this slice by $\zeta$, we can derive its evolution equation by using both Einstein equations and conservation laws given above. On superhorizon scales $k/(aH) \ll 1$, we find that the evolution equation reduces to $${\ddot \zeta}+3H {\dot \zeta}+\frac{1}{a^3}{\left( \frac{a^3 H}{\rho+P} \delta P_{\rm rel} \right)}^{\cdot}-\frac{8\pi G}{3} \Pi=0,$$ where $\delta P_{\rm rel} \equiv \delta P_{\rm em}-\frac{\dot P}{\dot \rho} \delta \rho_{\rm em}$ ($\delta \rho_{\rm em}$ and $\delta P_{\rm em}$ is the energy density and pressure perturbation for the electromagnetic field, respectively) is the nonadiabatic pressure perturbation due to the relative entropy perturbation [@Kodama:1985bj; @Malik:2002jb] between the magnetic field and the component dominating the Universe [^2]. Inhomogeneous solution of this differential equation is given by $$\zeta (t)=-\int_{t_*}^t dt_1 \frac{H(t_1)}{\rho(t_1)+P(t_1)} \delta P_{\rm rel}(t_1)+\frac{8\pi G}{3} \int_{t_*}^t \frac{dt_1}{a^3 (t_1)} \int_{t_*}^{t_1} dt_2~a^3(t_2) \Pi (t_2), \label{zeta-mag}$$ where we have imposed an initial condition that $\zeta (t_*)=0$. In the actual situation, $t_*$ may be taken to be a horizon crossing time. Equation (\[zeta-mag\]) represents the magnetic field contribution to the curvature perturbation.
Determining a precise value of the time integral in Eq. (\[zeta-mag\]) requires both model specification and numerical computation, which is not what we want to do in this paper. Instead, without referring to the specific inflation model, we can estimate the order of magnitude on the basis of some generic properties of inflation. Let us first evaluate the first term containing $\delta P_{\rm rel}$ in the radiation dominated Universe achieved after reheating. Noting that the electromagnetic tensor is traceless [^3], $\delta P_{\rm rel}$ during inflation can be approximated as $$\delta P_{\rm rel} \simeq \frac{4}{3} \delta \rho_{\rm em}, \label{P-rho}$$ where we have used an approximate relation ${\dot \rho} \simeq -{\dot P}$. In the radiation dominated Universe, the energy density of magnetic field decays in the same way as that of radiation and we have $\delta P_{\rm rel}=0$. Therefore, the first term yields a constant: $$-\int_{t_*}^t dt_1 \frac{H(t_1)}{\rho(t_1)+P(t_1)} \delta P_{\rm rel}(t_1) \simeq -\frac{2 {\cal N}}{\epsilon} \frac{\delta \rho_{\rm em}}{\rho_{\rm inf}}, \label{integral-1st}$$ where $\epsilon \equiv -{\dot H}/H^2$ is the slow-roll parameter, $\rho_{\rm inf}$ is the energy density of the inflaton and ${\cal N}$ is the number of e-fold measured from the time when the mode of interest crossed the horizon to the end of inflation. Equation (\[integral-1st\]) becomes exact if all the quantities are constant during inflation but may deviate from the correct one if one of the quantities significantly changes during inflation. While it is a good approximation to treat $\rho_{\rm inf}$ and $\epsilon$ as constants in many inflation models, this does not necessarily hold for $\delta \rho_{\rm em}$. In order to be conservative as possible as we can, let us replace Eq. (\[integral-1st\]) by $$\bigg| \int_{t_*}^t dt_1 \frac{H(t_1)}{\rho(t_1)+P(t_1)} \delta P_{\rm rel}(t_1) \bigg| > \frac{1}{\epsilon} \bigg| \frac{\delta \rho_{\rm em}(t_{\rm end})}{\rho_{\rm inf}} \bigg|, \label{ineq-1st}$$ where $t_{\rm end}$ is the time of inflation end. A merit of this replacement is that, as we will see later, $| \delta \rho_{\rm em}(t_{\rm end})/ \rho_{\rm inf} |$ can be connected to the observed magnetic field strength and duration of the dust-like Universe after inflation but before the reheating. If $\delta \rho_{\rm em}$ is a growing function like $\delta \rho_{\rm em} \propto a^p~(p>0)$, then left-hand side and right-hand side of the above inequality are almost the same. If it is a decreasing function, then left-hand side becomes much larger than the right-hand side. In either case, the left-hand side must not exceed the observed value of the primordial curvature perturbation which is about $10^{-5}$. If we further use another observational constraint that the curvature perturbation is almost Gaussian [@Komatsu:2010fb; @Smidt:2010ra], the bound can be made by a few orders of magnitude tighter. This is because the curvature perturbation from Eq. (\[ineq-1st\]) is proportional to the magnetic field squared and hence is highly non-Gaussian [@Caprini:2009vk; @Seery:2008ms; @Barnaby:2012tk]. But since we do not know quantitatively how largely the non-Gaussianity from the magnetic field is allowed by observations, we decide to adopt the conservative bound that the curvature perturbation from the magnetic field is smaller than $10^{-5}$. This puts a non-trivial lower bound on $\epsilon$ as $$\epsilon > 10^5 \times \bigg| \frac{\delta \rho_{\rm em}(t_{\rm end})}{\rho_{\rm inf}} \bigg|. \label{bound}$$ As we will check later, inclusion of the second term of Eq. (\[ineq-1st\]) does not alter this bound. If one resorts to the inflationary magnetogenesis to explain the observed magnetic fields on ${\rm Mpc}$ scales, such inflationary model needs to satisfy this bound.
In case of the standard single field inflation model, the generic bound (\[bound\]) can be rewritten into the bound on combination of Hubble parameter during inflation $H_{\rm inf}$ and the so-called reheating parameter $R_{\rm rad}$ defined by, $$R_{\rm rad}=\frac{a_{\rm end}}{a_{\rm reh}} {\left( \frac{\rho_{\rm end}}{\rho_{\rm reh}} \right)}^{1/4}=\exp \bigg[ \frac{\Delta N}{4}(-1+3w) \bigg],$$ where $\Delta N$ is the number of $e$-fold between the end of inflation and the time of reheating and $w=P/\rho$ is the equation of state parameter during that era. This parameter was introduced and used in [@Martin:2006rs; @Ringeval:2007am; @Martin:2010kz; @Demozzi:2012wh] and roughly measures the degree of deviation from the radiation dominated expansion of and duration of that era. Basic requirement that the inflation energy scale is less than ${(10^{-5}M_P)}^4$ and the reheating occurs well before the Big-Bang Nucleosynthesis [@Kawasaki:1999na] restricts the allowed range of $R_{\rm rad}$ as $-35< \log R_{\rm rad} <12$. Using the standard formula for the curvature perturbation [@mukhanov] and observational data [@Komatsu:2010fb]: $${\cal P}_\zeta \simeq \frac{H_{\rm inf}^2}{8\pi^2 M_P^2 \epsilon} \simeq 2.4 \times 10^{-9},$$ we can replace $\epsilon$ in Eq. (\[bound\]) by $H_{\rm inf}$. Assuming no entropy is produced after reheating and using the reheating parameter, Eq. (\[bound\]) can then be written as $$\frac{H_{\rm inf}}{M_P} R_{\rm rad}^2 > 0.14 \times {\left( \frac{g_{*,{\rm reh}}}{g_{*,0}} \right)}^{1/6} {\left( \frac{\rho_{B,0}}{\rho_{\gamma,0}} \right)}^{1/2}
\simeq 9.0 \times 10^{-13} ~{\left( \frac{g_{*,{\rm reh}}}{200} \right)}^{1/6} \left( \frac{B}{10^{-17}~{\rm G}} \right), \label{bound2}$$ where $g_*$ represents the relativistic degrees of freedom and $\rho_{\gamma,0}(\rho_{B,0})$ is the energy density of radiation(magnetic field) today. This basically puts a lower bound on the energy scale of inflation. In particular, if the inflation end is immediately followed by reheating or $w=\frac{1}{3}$, in which case we have $R_{\rm rad}=1$, the bound becomes $$H_{\rm inf} > 2.2 \times 10^6~{\rm GeV} ~{\left( \frac{g_{*,{\rm reh}}}{200} \right)}^{1/6} \left( \frac{B}{10^{-17}~{\rm G}} \right).$$ In the case the energy density of the magnetic field is diluted by $\xi <1$ compared to the dominant energy density due to the entropy injection after the reheating, a factor $\xi^{-1/2}$ must be multiplied on the right-hand side of Eq. (\[bound2\]). Therefore, if future observations detect tensor-to-scalar ratio, which uniquely fixes $H_{\rm inf}$, and provide an decisive answer that the extragalactic magnetic fields of $B \simeq 10^{-17}~{\rm G}$ are of inflationary origin, we get a lower bound on $\xi$, namely, the maximum amount of entropy injection.
So far, we have focused on the first term on the right-hand side of Eq. (\[zeta-mag\]). As for the second term, it does not dominate over the first term. To see this, let us first notice that the integrand of the first term contains $\rho+P$ in the denominator which enhances the integrand by $\epsilon^{-1}$ while the second term does not have such an enhancement factor. As a result, the contribution from inflationary era to the second term is suppressed by the slow-roll parameter $\epsilon$ compared to the first term. The second term receives an enhancement $\propto \log a$ in the radiation dominated era, but this boosts the second term by at most $N_{\rm rad} < 100$, the number of $e$-folds measured from the time of reheating. The second term can marginally reach the first term if $\epsilon$ is as large as ${\cal O}(0.01)$ but becomes much smaller for smaller value of $\epsilon$. Therefore, apart from ${\cal O}(1)$ modification that may arise for inflation models giving large $\epsilon$, inclusion of the second term does not alter our bound (\[bound\]).
Finally, before closing this section, we provide expressions for other perturbation variables on the uniform density slice. The Einstein equations and conservation laws can be recast into equations that give $A$, $\sigma_g$ and $v$ in terms of $\zeta$. Using the solution (\[zeta-mag\]) for $\zeta$, on super-horizon scales, they are given by $$\begin{aligned}
&&A=-\frac{\delta P_{\rm rel}}{\rho+P}, \\
&&k^2 \sigma_g = -\frac{8\pi G}{a^3} \int_{t_*}^t dt_1~a^3(t_1) \Pi (t_1), \label{uniform-sigma_g}\\
&&v =\frac{2}{3a^5 (\rho+P)} \int_{t_*}^t dt_1~a^3(t_1) \Pi (t_1).\end{aligned}$$ These are at most the order of $\zeta$ or smaller than $\zeta$.
Calculation in the Newtonian gauge
==================================
In this section, we again consider the evolution of metric perturbations in the Newtonian gauge which is widely used in the literature including [@Bonvin:2011dt; @Bonvin:2011dr] which obtained a physically incorrect result. In the Newtonian gauge, the perturbed metric is given by $$ds^2=-(1+2\Phi)dt^2+a^2(t) (1-2\Psi) \delta_{ij} dx^i dx^j.$$ There is no gauge degree of freedom left in this coordinate system. In this gauge, we find that the evolution equation of $\Psi$ can be written as $${\ddot \Psi}+\left( 4+\frac{3 {\dot P}}{\dot \rho} \right) H {\dot \Psi}+3H^2 \left( \frac{\dot P}{\dot \rho}-\frac{P}{\rho} \right) \Psi+c_s^2 \frac{k^2}{a^2} \Psi
=S,$$ where $c_s^2$ is the sound speed of the dominant component (e. g. , $c_s^2=\frac{\dot P}{\dot \rho}$ for the perfect fluid with barotropic equation of state and $c_s^2=1$ for the canonical scalar field) and $S$ is defined by $$S=\frac{8\pi G a^2 H}{\rho k^2} \bigg[ (\rho+P) {\left( \frac{\rho}{\rho+P} \right)}^\cdot \Pi+\rho \left( {\dot \Pi}+2H \Pi \right) \bigg]+4\pi G \left( \delta P_{\rm rel}-\frac{2}{3} \Pi \right). \label{def-S}$$ Following the standard procedure (e. g. , the one given in [@mukhanov]), the inhomogeneous solution in the superhorizon regime ($k \ll aH$) is found to be $$\begin{aligned}
\Psi = \frac{\sqrt{\rho(t)}}{a(t)} \bigg[ -&&\int_{t_*}^t dt_1 S(t_1) \frac{\sqrt{\rho(t_1)}}{\rho (t_1)+P(t_1)}
\int_{t_*}^{t_1} dt_2 ~\frac{a(t_2) \left( \rho (t_2)+P(t_2) \right)}{\rho (t_2)} \nonumber \\
&&+\int_{t_*}^t dt_1 ~\frac{a(t_1) \left( \rho (t_1)+P(t_1) \right)}{\rho (t_1)}
\int_{t_*}^t dt_1 S(t_1) \frac{\sqrt{\rho(t_1)}}{\rho (t_1)+P(t_1)} \bigg]. \label{sol-Psi}\end{aligned}$$ Substituting Eq. (\[def-S\]) and performing integration by parts, this can be further recast into another form which makes its time dependence more transparent: $$\begin{aligned}
\Psi = &&\frac{8\pi GH(t)}{a(t) k^2} \bigg[ \int_{t_*}^t dt_1~a^3(t_1) \Pi (t_1) -\frac{a^2 (t_*) \rho (t_*) \Pi (t_*)}{\rho (t_*)+P(t_*)}\int_{t_*}^t dt_1 ~\frac{a(t_1) \left( \rho (t_1)+P(t_1) \right)}{\rho (t_1)}\bigg] \nonumber \\
&&+\frac{H(t)}{a(t)} \int_{t_*}^t dt_1~\frac{H(t_1)}{\rho(t_1)+P(t_1)} \left( \delta P_{\rm rel}(t_1)-\frac{2}{3} \Pi(t_1) \right) \left( \frac{a(t)}{H(t)}-\int_{t_1}^t dt_2~a(t_2) \right). \label{final-Psi}\end{aligned}$$ This is our final form of $\Psi$ induced by magnetic field.
Now, let us first consider the behavior of this solution during inflationary era. Just for simplicity, we assume the single field inflation model and treat $\Pi,~\rho,~P$ and hence $\epsilon$ too as very slowly changing variables. Then, the first term of Eq. (\[final-Psi\]) gives the dominant contribution. Neglecting the other terms, Eq. (\[final-Psi\]) during inflation becomes $$\Psi \simeq \frac{8\pi G a^2}{3k^2} \Pi ={\left( \frac{aH}{k} \right)}^2 \frac{\Pi}{\rho}. \label{Psi-inf}$$ This shows that the curvature perturbation in the Newtonian gauge grows very rapidly in proportion to $\propto a^2$. Since $\Pi$ has the same order of magnitude as the energy density of the magnetic field, $\Psi$ is suppressed by a small number, which is the fraction of the magnetic field energy density to the total one. But it is also enhanced by another big number which is square of the ratio of the wavelength of the mode to the horizon radius. As a demonstration, if we approximate $\Pi/\rho$ at the end of inflation by the ratio of magnetic field energy density to the CMB energy density today and assume modes corresponding to $1~{\rm Mpc}$ today have expanded 50 $e$-folds after the horizon exit, $\Psi$ at the end of inflation becomes as large as $3\times 10^{20}$ for $B = 10^{-17}~{\rm G}$ today. This rough estimate is enough to see that $\Psi$ induced by inflationary magnetic field corresponding to $10^{-17}~{\rm G}$ today becomes quite large at the end of inflation.
Let us next consider the time evolution of $\Psi$ in the radiation dominated era, assuming that reheating completes instantly just after inflation. Using the conservation law $\Pi a^4 ={\rm const.}$ in the radiation dominated Universe, we have $$\int_{t_*}^t dt_1~a^3 (t_1) \Pi (t_1) \simeq \frac{\Pi (t) a^3 (t)}{H(t)}.$$ Therefore, the contribution from the first term to $\Psi$ in Eq. (\[final-Psi\]) becomes $$\Psi_{\rm first} \simeq 3 {\left( \frac{aH}{k} \right)}^2 \frac{\Pi}{\rho}. \label{Psi-first}$$ Apart from the numerical factor of ${\cal O}(1)$, this takes the same form as that in the inflationary era (see Eq. (\[Psi-inf\])). However, their time dependence is quite different. Indeed, in the radiation dominated Universe, $\Psi_{\rm first}$ decays in proportion to $a^{-2}$, which significantly reduces the amplitude of $\Psi_{\rm first}$ that was quite large at the time of reheating. In particular, at the time of horizon reentry, $\Psi_{\rm first}$ becomes as small as $\Pi /\rho$. For the magnetic field corresponding to $B=10^{-17}~{\rm G}$ today, this becomes $\Pi /\rho \sim 10^{-23}$, which is negligibly small. Therefore, contributions from the remaining terms may eventually dominate over $\Psi$ at late time, especially at the time of horizon reentry. Given that the anisotropic stress and the energy density of magnetic field are of the same order of magnitude, we can show that magnitude of $\Psi$ at the time of horizon reentry is roughly given by Eq. (\[integral-1st\]). Since what we observe is $\Psi$ after the horizon reentry, the observational bound $\Psi \lesssim 10^{-5}$ must be imposed at the time of horizon reentry, which yields the same bound as Eq. (\[bound\]). Therefore, as it should be, we have confirmed that consideration in the Newtonian gauge yields the same bound as the one in the uniform density slice.
The feature $\Psi \propto \Pi a^2/k^2$ during inflation was also found in [@Bonvin:2011dt; @Bonvin:2011dr]. It was then claimed in [@Bonvin:2011dt; @Bonvin:2011dr] that the large amplitude of $\Psi$ at the end of inflation generically enters a constant mode in the radiation dominated era, making the Universe eventually highly inhomogeneous and spoiling the success of standard cosmology based on the FLRW metric. This observation was derived by using matching conditions formulated in [@Deruelle:1995kd] under an assumption of instant transition from the end of inflation to reheating. However, as we have shown above, we do not find such contamination to the constant mode. Instead, we observe that the growing magnetic mode $\Psi \propto \Pi a^2/k^2$, which was significantly enhanced by the end of inflation, is taken over only by the decaying mode in the radiation dominated era. Therefore, the breaking of the FLRW metric in the radiation dominated era is not a generic consequence of the inflationary magnetogenesis.
We can also rederive the time evolutionary behavior of $\Psi$ by connecting it with perturbations evaluated in the uniform density slice using the gauge transformation, which enables us to understand intuitively the origin of the rapid growth $\Psi \propto a^2$ during inflation. Noting that the shear $\sigma_g$ is identically zero in the Newtonian gauge, time translation $T$ connecting the uniform density slice and the Newtonian gauge is uniquely determined by $\sigma_g$ evaluated in the uniform density slice (see Eq. (\[uniform-sigma\_g\])); $$T=-\frac{8\pi G}{ak^2} \int_{t_*}^t dt_1~a^3(t_1) \Pi (t_1). \label{time-translation}$$ Assuming $\Pi$ is almost constant during inflation, we see that $T$ grows in proportion to $a^2$. This means that $t={\rm const.}$ hypersurface in the uniform density slice and that in the Newtonian gauge deviate more and more with time. Using the gauge transformation rule for $\psi$, the curvature perturbation $\Psi$ is then given by $$\Psi=-\zeta+\frac{8\pi G \Pi}{3} \frac{a^2}{k^2}.$$ The second term grows rapidly in time and eventually dominates over $\Psi$. The Universe becomes apparently inhomogeneous with time in the Newtonian gauge, but this is simply because the time translation (\[time-translation\]) becomes very large. From a geometric point of view, the rapid growth of $T$ reflects mismatch between spacetime having anisotropic stress even on large scales and the Newtonian gauge in which the spacial metric looks isotropic. Therefore, the rapid growth of inhomogeneity does not occur on the slicing such as uniform density slicing where the shear $\sigma_g$ does not vanish.
In the radiation dominated era, calculation of Eq. (\[time-translation\]) shows that $T$ becomes constant in time. Since $H$ decays in proportion to $a^{-2}$, the gauge transformation rule for the curvature perturbation tells us that difference between $\Psi$ and $\zeta$ becomes smaller and smaller with time. In particular, $\Psi$ and $\zeta$ become of the same order of magnitude at the time of horizon reentry.
Summary
=======
We have studied time evolution of metric perturbation induced by magnetic field that is supposed to be produced during inflation. Treating the energy-momentum tensor of the magnetic field as first order perturbation, we solved the linearized Einstein equations on FLRW background spacetime. The resultant metric perturbation is at least of the order of $\delta \rho_B/(\epsilon \rho_{\rm inf})$, where $\delta \rho_B$ is the typical amplitude of the magnetic field at the end of inflation, $\epsilon$ is the slow-roll parameter and $\rho_{\rm inf}$ is the total energy density during inflation. This perturbation must not exceed the observed amplitude $\sim 10^{-5}$, from which we could obtain a generic bound on inflation models, namely, inequalities (\[bound\]) and (\[bound2\]). Any inflationary model that achieves magnetogenesis must satisfy these constraints.
We also performed perturbation analysis in the Newtonian gauge in which perturbations were found to rapidly grow during inflation on super-horizon scales. This weird behavior is simply an artifact of mismatch of the Newtonian gauge which uses isotropic coordinate and the anisotropic stress of magnetic field that persists even on superhorizon scales. Thus we do not have to worry about the breakdown of the FLRW background contrary to the claim of [@Bonvin:2011dt; @Bonvin:2011dr]. The apparently enhanced perturbation by the end of inflation in this gauge starts to attenuate in the subsequent radiation dominate Universe. The requirement that the perturbation must be smaller than $\sim 10^{-5}$ at the time of horizon reentry leads to the same bound as above.\
[**Acknowledgments:**]{} This work was supported by a Grant-in-Aid for JSPS Fellows No. 1008477(TS), JSPS Grant-in-Aid for Scientific Research No. 23340058 (JY), and the Grant-in-Aid for Scientific Research on Innovative Areas No. 21111006 (JY).
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[^1]: For magnetic field generated from standard cosmological perturbations, see [@Takahashi:2005nd; @Ichiki:2006cd; @Kobayashi:2007wd; @Maeda:2008dv]. For the effect of the primordial magnetic field on CMB temperature anisotropy and its observational constraints, see [@Kosowsky:2004zh; @Kahniashvili:2005xe; @Kristiansen:2008tx; @Kahniashvili:2010wm; @Ichiki:2011ah; @Shiraishi:2012rm].
[^2]: Precisely speaking, intrinsic entropy perturbation for the dominant component $P_{\rm intr}$ will be correlated with $\delta P_{\rm rel}$. But since the electromagnetic field is subdominant, it does not produce $P_{\rm intr}$ comparable to $\delta P_{\rm rel}$, or almost equal to $-P_{\rm rel}$. Therefore, we do not need to consider the intrinsic entropy perturbation for the dominant component.
[^3]: Traceless nature of the electromagnetic tensor persists even when the electromagnetic action is modified to a typical one given by Eq. (\[ele-action\]). This may not be true for other modification of the Maxwell equations. Eq. (\[P-rho\]) becomes an overestimate only if the effective equation of state parameter $w_m=P_m/\rho_m$ for the electromagnetic field becomes very close to $-1$, which is unlikely.
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abstract: 'The coefficient of restitution of a spherical particle in contact with a flat plate is investigated as a function of the impact velocity. As an experimental observation we notice non-trivial (non-Gaussian) fluctuations of the measured values. For a fixed impact velocity, the probability density of the coefficient of restitution, $p(\varepsilon)$, is formed by two exponential functions (one increasing, one decreasing) of different slope. This behavior may be explained by a certain roughness of the particle which leads to energy transfer between the linear and rotational degrees of freedom.'
author:
- Marina Montaine
- Michael Heckel
- Christof Kruelle
- Thomas Schwager
- Thorsten Pöschel
title: The Coefficient of Restitution as a Fluctuating Quantity
---
#### Introduction.
The dissipative collision of a solid particle with a hard plane may be described by the coefficient of normal restitution $$\varepsilon = -\dfrac{{\bm{v}}^{\,\prime}\cdot{\bm{n}}}{{\bm{v}}\cdot{\bm{n}}}\,,$$ relating the normal components of the relative velocity before and after a collision at the point of contact. The unit vector ${\bm{n}}$ indicates the relative particle positions at the instant of the collision. Obviously, $\varepsilon=1$ stands for elastic collisions whereas $\varepsilon=0$ indicates the complete dissipation of the energy of the relative motion. There are several techniques for measuring the coefficient of restitution, including high-speed video analysis, e.g., [@Labous:1997] and sophisticated techniques, where the particle is attached to a compound pendulum with the axis of rotation very close to the center of mass [@Bridges:1984; @Hatzes:1988] which makes this method particularly suitable for the measurement of the coefficient of restitution for very small impact velocities up to mm/sec and below. In the presence of gravity, the coefficient of restitution can be measured by determining the time lag between consecutive impacts of a particle bouncing on a hard plane, using a piezoelectric force sensor, e.g., [@Koller:1987; @Falcon:1998] or an accelerometer mounted to the plate which detects elastic waves excited by the impact, e.g., [@King:2005]. When both particle and plate are metallic, the time of the impacts can be determined by applying a DC voltage between ball and plate and determining the instant when the circuit closes [@King:2005]. In [*many*]{} papers, the time lag and, thus, $\varepsilon$ is determined by recording the sound emitted from a spherical particle bouncing on an underlying flat plane, e.g., [@Bernstein:1977; @Stensgaard:2001] and many others. From the times $t_i$ of impacts one can compute the coefficient of restitution via $$\label{eq:epsmeas}
\varepsilon(v_i)=\frac{v_i^\prime}{v_i} = \frac{t_{i+1}-t_{i}}{t_{i}-t_{i-1}}\,,~~~v_i=\frac{g}{2}(t_i-t_{i-1})\,,$$ where $v_i$ and $v_i^\prime$ are the normal pre- and postcollisional velocities of the impact taking place at time $t_i$ and $g$ is the acceleration due to gravity.
Although it is frequently assumed that the coefficient of restitution is a material constant (e.g., this assumption is common in many textbooks and widely used in simulations), numerous experimental studies have revealed that it depends on many parameters: impact velocity, material characteristics of the impacted bodies, particle size, shape and roughness, and adhesion properties. Here we restrict ourselves to the investigation of $\varepsilon$ of a single dry steel ball bouncing on a hard plane, such that the coefficient of restitution is only a function of the impact velocity $\varepsilon=\varepsilon(v)$.
In several experimental investigations, using either photographic techniques [@Sorace:2009] or an acoustic emission analysis [@Falcon:1998], an extraordinary high fluctuation level was noticed whose origin remained obscure and cannot be attributed to the imperfections of the experimental setup. By means of large-scale experiments as well as micromechanical modeling we make an attempt to characterize the fluctuations of the coefficient of restitution and to explain the mechanism leading to these fluctuations.
#### Experiments.
Our experimental setup consists of a robot to move a small vertical tube to a desired position $\{x,y,z\}$. In the beginning of each experimental trial a stainless steel bearing ball is suspended at the end of the tube by means of a vacuum pump. Switching off the pump the sphere is released to bounce repeatedly off the ground where the sound is recorded by a piezoelectric sensor. When the ball eventually comes to rest, it is pushed to a defined position by a fan where it is picked up by the robot who moves the ball again to the start position for the next trial. In each cycle the initial $\{x,y\}$-position is chosen randomly within the central region of the ground plate such that edge-effects [@Sondergaard:1990] are not noticeable. The dropping height is chosen randomly from $z \in [9, 10]$cm, corresponding to impact velocities $v\in[1.33, 1.4]$m/s which allows for the observation of 90-100 bounces of the ball. A massive hard glass plate of size 30$\times$20$\times$1.9cm$^3$ can either serve directly as a ground plate for the experiment or serve as a carrier for other ground plates. For our experiments we used an extremely hard SiC disk and a Vitreloy disk. The measuring process is fully automatized, this allows us to perform a large-scale experiment collecting data from thousands of bouncing ball trials.
From the analysis of the sound-sensor signal we obtain the impact times $t_i$ and, thus, via Eq. the normal impact velocities $v_i$ and the coefficients of restitution $\varepsilon(v_i)$. Figure \[fig:Exp\] displays the abundance of data, $\varepsilon(v)$, for a stainless steel ball (radius $R=3.0$ mm, Young modulus $Y=200$ GPa, Poisson ratio $\nu=0.30$, density $\rho_s=7.90 $ g/cm$^3$). Besides the expected decay of $\varepsilon$ with increasing impact velocity, e.g., [@Brilliantov:1996; @SchwagerPoeschel:2008], we observe a large scatter of the data. This scatter becomes apparent only when analyzing a large number of measurements (here 220,000) and was, therefore, not noticed in earlier, similar experiments [@Koller:1987; @Falcon:1998].
![(color online). Experimental results. Top: Coefficient of restitution $\varepsilon$ plotted against the normal impact velocity $v$. The data are colored according to the normalized frequency of occurrences. Bottom: Histograms of $\varepsilon$ for impact velocities from small intervals, centered around $v=0.3\:\ldotp\ldotp1.0$ m/s. The lines are exponential fits.[]{data-label="fig:Exp"}](Exp.png){width="0.9\columnwidth"}
![(color online). Experimental results. Top: Coefficient of restitution $\varepsilon$ plotted against the normal impact velocity $v$. The data are colored according to the normalized frequency of occurrences. Bottom: Histograms of $\varepsilon$ for impact velocities from small intervals, centered around $v=0.3\:\ldotp\ldotp1.0$ m/s. The lines are exponential fits.[]{data-label="fig:Exp"}](P_exp.png){width="0.9\columnwidth"}
The scatter of $\varepsilon$ is asymmetric, that is, the deviation of the data with $\varepsilon$ lower than the mean is noticeably larger. Figure \[fig:Exp\] (bottom) shows the normalized frequency $p(\varepsilon)$ of measurements of a certain value of $\varepsilon$ for several small intervals of the impact velocity $v$. Thus, if we consider $\varepsilon$ as a fluctuating quantity, $p(\varepsilon)$ may be considered as its probability density function. This function reveals strongly non-Gaussian behavior, but the distribution is shaped by a combination of two exponential functions (one increasing, one decreasing) of different slope. We believe that uncommon statistical properties are not due to imperfections of the experimental setup but are a consequence of microscopic asperities at the sphere’s surface. This hypothesis is checked by means of a numerical simulation.
#### Numerical simulations.
The procedure of our simulation is analogue to the experiment; the rigid ball is dropped from a certain height $h$ and repeatedly collides with a smooth, hard plate. The air drag is neglected and the ball is subjected only to gravity.
The ball is modeled as a composite multi-sphere particle (Fig. \[fig:ball\]).
![Sketch of the particle model and closeup of its surface.[]{data-label="fig:ball"}](Ball1.png "fig:"){width="0.45\columnwidth"} ![Sketch of the particle model and closeup of its surface.[]{data-label="fig:ball"}](Ball2.png "fig:"){width="0.45\columnwidth"}
The large central sphere of radius $R$ is randomly covered by many ($N\sim 10^6$) tiny spheres of different microscopic size, representing asperities. The center of each small sphere $i$ of radius $R_i$ is located at the surface of the central sphere. Since $R_i/R\sim 10^{-3}$ (mass ratio $m_i/m\sim 10^{-9}$) we can safely neglect the contribution of the small particles for the computation of the moment of inertia of the ball. Thus, the ball is characterized by its mass $m$, $R$, center of mass velocity ${\bm{v}}_G$, angular velocity ${\bm{\omega}}$, the set of the radii, $R_i$, of the asperities and the set of their position vectors, ${\bm{r}}_i$, pointing from the center of the large sphere to the asperities. When bouncing, the ball may come in contact with the plane either through the central sphere or through one of the asperities. In both cases the collision is assumed instantaneous and inelastic, characterized by coefficients of restitution.
In the intervals between collisions the ball follows a ballistic trajectory $$\label{eq:rG}
{\bm{r}}_G(t)={\bm{r}}_G(t_0) + {\bm{v}}_G(t_0)\,(t-t_0) + \dfrac{(t-t_0)^2}{2}{\bm{g}}, \quad t\geq t_0$$ where $t_0$ is the time of the preceding collision and ${\bm{g}}$ is gravity while the angular velocity ${\bm{\omega}}\equiv \omega{\bm{e}}_\omega$ remains constant. The evolution of a vector ${\bm{p}}$ fixed to the particle is then given by $$\begin{split}
{\bm{p}}_\text{rot} & ={\bm{p}} \cos\omega t + {\bm{e}}_\omega({\bm{e}}_\omega \cdot {\bm{p}}\,) (1 - \cos\omega t) +({\bm{e}}_\omega \times {\bm{p}}\,)\sin\omega t\\
&=\hat{A}(t){\bm{p}},
\end{split}
\label{eq:r_rot}$$ which defines the rotation matrix $\hat{A}$ [@Goldstein:2001]. Therefore, the particle contacts the ground when the center of the first asperity $j$ reaches the height $R_j$, that is, the time of contact $t_\text{c}$ follows from the condition $$\label{eq:mincondition}
\min_{j=1,N}\left[{\bm{r}}_G(t_c) + \hat{A}(t_c){\bm{r}}_j \right]_z = R_j\,,$$ where $\left[...\right]_z$ means the vertical component of the argument. The vectorial impact velocity at the point of contact ${\bm{r}}_c$ is then $$\label{eq:vc}
{\bm{v}}_c = \left({\bm{v}}_c\cdot {\bm{n}}\right){\bm{n}} + \left({\bm{v}}_c\cdot {\bm{t}}\,\,\right){\bm{t}} = {\bm{v}}_G+{\bm{\omega}}\times {\bm{r}}_c \,,$$ where ${\bm{n}}$ and ${\bm{t}}$ are the unit vectors in normal and tangential directions, see Fig. \[fig:sketch\].
![Sketch of a particle collision. For simplicity only the impacting asperity (of exaggerated size) is drawn.[]{data-label="fig:sketch"}](ColMech.png){width="0.8\columnwidth"}
The post-collisional velocity at the contact point is given by $$\label{eq:epsdef}
{\bm{v}}^{\,\prime}_c\cdot{\bm{n}} = -\varepsilon \left({\bm{v}}_c\cdot{\bm{n}}\right)\,,~~~~~
{\bm{v}}^{\,\prime}_c\cdot{\bm{t}} = \beta \left({\bm{v}}_c\cdot{\bm{t}}\,\,\right)\,,$$ with the coefficients of normal and tangential restitution $\varepsilon$ and $\beta$. Finally, we compute the post-collisional velocity ${\bm{v}}_G^{\,\prime}$ and angular velocity ${\bm{\omega}}^\prime$ $$\label{eq:P}
\begin{split}
{\bm{v}}_G^{\,\prime} -{\bm{v}}_G & = \Delta {\bm{v}}_G = \frac{1}{m}{\bm{P}}\\
{\bm{\omega}}^\prime - {\bm{\omega}} & = \Delta{\bm{\omega}} = \frac{1}{\hat{J}}{\bm{r}}_c\times {\bm{P}}\,,
\end{split}$$ where the transferred momentum ${\bm{P}}$ is obtained from Eqs. and : $$\label{eq:deltaVc}
{\bm{v}}_c^{\,\prime}-{\bm{v}}_c = \Delta{\bm{v}}_G+\Delta {\bm{\omega}}\times {\bm{r}}_c=\frac{1}{m}{\bm{P}}+\frac{1}{\hat{J}}\left({\bm{r}}_c\times {\bm{P}}\right)\times {\bm{r}}_c$$ with the mass $m$ and the moment of inertia $\hat{J}$ of the particle.
Using Eqs. and we can compute the dynamics of the bouncing particle and, in particular, the times and velocities of the impacts, while the coefficients of restitution, $\varepsilon$ and $\beta$ are given. For $\varepsilon$ as a function of the normal impact velocity ${v}_c$ we use the expression for viscoelastic spheres [@SchwagerPoeschel:2008] $$\label{eq:epsvisco}
\varepsilon({v}_c)=1+\sum_{i=1}^\infty C_iA_i({v}_c)^{i/10},$$ with $C_i$ being known constants ($C_1=C_3=0$; $C_2=-1.153$; $C_4=0.798$; $C_5=0.267$…, see [@SchwagerPoeschel:2008] for details) and $A_i$ being material constants. We determined the first three non-trivial constants, $A_2\approx 0.0467$; $A_4\approx 0.1339$; $A_5\approx -0.2876$, by fitting Eq. to the experimental data shown in Fig. \[fig:Exp\].
The coefficient of tangential restitution $\beta$ depends on both bulk material properties and surface properties. Therefore, $\beta$ cannot be analytically derived from material properties, except for the limiting case of pure Coulomb friction [@SchwagerBecker:2008]. Here we use $\beta=1$.
#### Simulation results,
On the macroscopic level (neglecting the microscopic asperities at the surface of the particle), the simulation results can be analyzed in the same way as the experimental data. We introduce the [*macroscopic*]{} coefficient of restitution $\tilde{\varepsilon}$ as the ratio of the post-collisional to pre-collisional center of mass velocities in normal direction $$\label{eq:eps_macro}
\tilde{\varepsilon}=-\dfrac{{\bm{v}}^{\,\prime}_G\cdot{\bm{n}}}{{\bm{v}}_G\cdot{\bm{n}}}.$$ Surprisingly, the macroscopic interpretation of our simulation results depicted in Fig. \[fig:Sim\]
![(color online). Simulation results. Top: The [*macroscopic*]{} coefficient of restitution $\tilde{\varepsilon}$ plotted against the normal center of mass velocity $v_G$ in moment of impact. The data are colored according to the normalized frequency of occurrences. Bottom: Histograms of $\tilde{\varepsilon}$ for velocities from small intervals, centered around $v_G=0.3\:\ldotp\ldotp1.0$ m/s. The lines are exponential fits.[]{data-label="fig:Sim"}](Sim.png){width="0.9\columnwidth"}
![(color online). Simulation results. Top: The [*macroscopic*]{} coefficient of restitution $\tilde{\varepsilon}$ plotted against the normal center of mass velocity $v_G$ in moment of impact. The data are colored according to the normalized frequency of occurrences. Bottom: Histograms of $\tilde{\varepsilon}$ for velocities from small intervals, centered around $v_G=0.3\:\ldotp\ldotp1.0$ m/s. The lines are exponential fits.[]{data-label="fig:Sim"}](P_sim.png){width="0.9\columnwidth"}
shows a striking resemblance with the experimental data. While the actual coefficient of normal restitution $\varepsilon$ is a function of the normal impact velocity described by Eq. , the macroscopic coefficient of restitution $\tilde{\varepsilon}$ computed via Eq. reveals strong fluctuations in agreement with the experiment.
The probability densities, $p(\tilde{\varepsilon})$, (bottom panel of Fig. \[fig:Sim\]) are also close to the distribution obtained in the experiment. Their shape is excellently approximated by a combination of two exponential functions, one increasing and one decreasing. The peaks of $p(\tilde{\varepsilon})$ are in line with the values of $\varepsilon(v_G)$ obtained from Eq. for the corresponding velocities.
For the case of an absolutely smooth particle (no asperities), the impact velocity ${\bm{v}}_c$ at the point of contact is the same as ${\bm{v}}_G$ and $\tilde{\varepsilon}$ is equal to the coefficient of normal restitution $\varepsilon$. Therefore, in this case, there would be no scatter of the measured values of $\tilde{\varepsilon}$. Consequently a scatter of these data must be attributed to the asperities at the surface of the particle. From the agreement of the experimental data, Fig. \[fig:Exp\], and the simulation data, Fig. \[fig:Sim\], it became apparent that the tiny microscopic imperfections of the surface of the otherwise macroscopically smooth particle lead to the characteristic fluctuations of the coefficient of normal restitution. The slight asymmetry in the shape of the probability density in case of the experimental results, i.e., $p(\varepsilon)$ compared to $p(\tilde{\varepsilon})$, may be caused by some additional dissipative forces that, however, have not be taken into account in our model.
Let us consider the role of the sphere’s rotational degrees of freedom which may be interpreted as internal degrees of freedom since they do not enter the computation of the coefficient of restitution, neither in the experiment, Eq. , nor in the simulation, Eq. . Initially, the particle had no spin and only one degree of freedom in its translational motion. However, due to eccentric impacts caused by asperities, the particle gains some rotation and acquires velocity in horizontal direction. The partition of total kinetic energy in rotational and translational degrees of freedom constantly varies from one impact to another. Thus, the kinetic energy of the linear vertical motion (the only component which enters $\varepsilon$) just before a collision is transformed into energy of the rebound vertical velocity, dissipated energy due to the coefficient of restitution, and changes of the horizontal and rotational velocities. The latter two contribution may be positive or negative, leading to a reduced or increased values of the [*measured*]{} coefficients of restitution, according to Eqs. and . Therefore, the particle rotation may be considered as a reservoir of internal energy, leading to fluctuations of the measured coefficient of restitution.
The substantial increase of data scattering with the decrease of impact velocity can be attributed to the growing role of the rotational degrees of freedom in the energy partition. In consecutive collisions the translational energy decreases appreciably due to dissipation, but the amount of energy concentrated in rotation varies only slightly. Thus, the proportion of rotational energy to translational energy increases with the number of particle bounces. The kinetic energy transfers from rotational to translational mode and vice-versa result in a more apparent fluctuation of the coefficient of restitution. As a consequence, the scattering of the restitution coefficient increases as the impact velocity decreases.
#### Conclusion.
We have performed an experimental and numerical study of the coefficient of normal restitution as a function of impact velocity. From about $2.2\times 10^5$ experiments of [*the same*]{} stainless steel sphere bouncing on a massive horizontal plate we determined experimentally the probability distribution of the fluctuations of the coefficient of normal restitution, $\varepsilon$. We found that $\varepsilon$ increases as the impact velocity decreases. For fixed impact velocity, the probability density of the coefficient of restitution $p(\varepsilon)$ is non-Gaussian. It consists of two exponential functions (one increasing, one decreasing) of different slope. We modelled the particle used in the experiment by a mathematical sphere whose surface is covered by a large number of much smaller spheres (asperities) to simulate a certain roughness. The simulations revealed the same properties of the fluctuations. Since the asperities are the only origin of scatter, we conclude that the experimental observed fluctuations of the coefficient of restitution coefficient are due to microscopic surface roughness of the ball, causing energy transfer between the translational and rotational degrees of freedom.
We thank Knut Reinhardt for technical assistance in setting up the robot and Deutsche Forschungsgemeinschaft (DFG) for funding through the Cluster of Excellence Engineering of Advanced Materials.
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abstract: 'We report an experimental study of diffusion in a quasi-one-dimensional (q1D) colloid suspension which behaves like a Tonks gas. The mean squared displacement as a function of time is described well with an ansatz encompassing a time regime that is both shorter and longer than the mean time between collisions. This ansatz asserts that the inverse mean squared displacement is the sum of the inverse mean squared displacement for short time normal diffusion (random walk) and the inverse mean squared displacement for asymptotic single-file diffusion (SFD) ($\frac{1}{<x^2(t)>}=\frac{1}{2D_{o}t}+\frac{1}{2Ft^{1/2}}$ where $D_o$ is the q1D self-diffusion coefficient and $F$ is the single-file 1D mobility). The dependence of $F$ on the concentration of the colloids agrees quantitatively with that derived for a hard rod model, which confirms for the first time the validity of the hard rod SFD theory. We also show that a recent SFD theory by Kollmann [@970] leads to the hard rod SFD theory for a Tonks gas.'
author:
- Binhua Lin
- Mati Meron
- Bianxiao Cui
- 'Stuart A. Rice'
- Haim Diamant
title: 'From random walk to single-file diffusion'
---
The diffusion of particles in quasi-one-dimensional (q1D) pores and channels is a basic feature of ion transport in cell membranes, molecular motion in zeolites, and particle flows in microfluidic devices (see references in [@879]). The unique feature that separates q1D diffusion from diffusion in higher dimensions is the geometric confinement that forces the particles into a single file with a fixed spatial sequence. This confinement generates a self-diffusion mechanism that has different time dependences of the mean squared particle displacement in different time domains.
For time intervals shorter than the time between particle collisions, in the presence of a randomizing background fluid (e.g. a colloid particle in a solvent), the probability density for the particle displacement is $$P_{S}(x,t)=\frac{1}{\sqrt{4\pi D_{o}t}}\exp{\Bigg\{-\frac{x(t)^{2}}{4D_{o}t}}\Bigg\},
\label{eq:short t prob}$$where $x$ is the displacement during time interval $t=t_{1}-t_{0}$, and $D_{o}$ is the q1D self-diffusion coefficient. However, the fixed spatial sequence of the particles severely restricts the possibility for large single particle displacements and, therefore, drastically reduces the diffusion rate at long time. An analytic description of 1D diffusion in a system of hard rods with stochastic background forces was first reported by Harris [@978]. Several other 1D systems have been examined with a similar approach [@880; @882; @881; @883; @856; @857; @854]; the results obtained converge to the same solution. For an infinite 1D system the long time behavior of the probability density for displacement is $$P_{L}(x,t)=\frac{1}{\sqrt{4\pi Ft^{1/2}}}\exp{\Bigg\{-\frac{x(t)^{2}}{4Ft^{1/2}}}\Bigg\},
\label{eq:long t prob}$$where $F$ is a 1D mobility defined by $$F=F^{HR}=l\sqrt{\frac{D_{o}}{\pi}}=\frac{1-\rho\sigma}{\rho}\sqrt{\frac{D_{o}}{\pi}}=D_{o}\sqrt{\frac{2t_{c}}{\pi}}.
\label{eq:SFD}$$We denote the 1D mobility of the hard rods by $F^{HR}$. In Eq.(\[eq:SFD\]) $\sigma$ is the particle length, $\rho$ is the 1D number density, $l$ is the mean spacing between the particles, and $t_{c}=l^2/2D_{o}$ is the mean time between collisions in the system. Equations (\[eq:long t prob\]) and (\[eq:SFD\]) draw a remarkably simple picture of 1D diffusion at long time: the self-diffusion process, determined by the width of the probability density, is proportional to $t^{1/2}$ (i.e. $<x(t)^2>\sim t^{1/2}$) , and the proportionality constant is determined by the short time *individual particle* dynamics.
Recently Kollmann reported an analysis of the long time behavior of 1D diffusion that is valid both for atomic and colloid systems [@970]. For colloid systems he finds the asymptotic particle density function displayed in Eq.(\[eq:long t prob\]) with the 1D mobility, denoted by $F^q$, $$F^q=\frac{S(q)}{\rho}\sqrt{\frac{D_{c}(q)}{\pi}}\Bigg|_{q \ll 4\pi/\sigma},
\label{eq:Fq}$$where $q$, $S(q)$, $D_{c}(q)$ are the momentum transfer, static structure factor, and the short time *collective*-diffusion coefficient in $q$-space, respectively. The small $t$, small $q$ approximation for the dynamic structure factor, $S(q,t)$ [@907] , yields the relation $$S(q,t)=S(q)\exp{\Bigg\{-q^{2}D_{c}(q)t}\Bigg\}\Bigg|_{t \ll t_{c},q \ll 4\pi/\sigma}.
\label{eq:sqt}$$Kollmann’s analysis predicts that the long time character of 1D diffusion is determined by the short time *collective* dynamics of the system.
Although theoretical analyses of 1D diffusion have been reported for the past four decades, the first experimental studies were reported only in the past decade. Studies of molecular diffusion in zeolites, and of colloid particles confined in a channel lead to the result $<x(t)^2>\sim t^{1/2}$ at long time [@879; @868; @946]. Very recently, Lutz, Kollmann and Bechinger [@979] reported the results of an experimental study of single-file diffusion in a strongly interacting colloid suspension. The 1D mobility, determined from $<x(t)^2>=2Ft^{1/2}$ at long time, agrees with that determined from Eq.(\[eq:Fq\]) at short time, as predicted [@970]. However, the 1D mobility they find is only qualitatively similar to $F^{HR}$.
The main difficulty encountered in the study of single-file diffusion is to obtain data at long time; this difficulty is most pronounced for low concentration samples. To obtain the required data one needs a long-lived experimental system and stable instruments, such as those cleverly devised for the studies reported in references [@879] and [@979].
In this Letter, we report an experimental study of q1D diffusion in a weakly interacting colloid suspension confined in a narrow straight groove. We establish an ansatz that accurately approximates the q1D diffusion process from the short time region to the long time region, thereby allowing us to study the long time single-file diffusion within a reasonable time frame (requiring a sample lifetime of $\sim$1 hour), as well as diffusion in the cross-over time region. The experimentally determined q1D mobility of the system agrees quantitatively with $F^{HR}$.
We note that Kollmann states that $F^q$ is not equivalent to $F^{HR}$, and the experimental results in [@979] support this statement. However, we show that these two theories are equivalent when applied to a system, such as ours, which obeys the Tonks equation of state [@982].
Our experimental system consists of silica colloid spheres (density $2.2 g/cm^3$) suspended in water and confined in straight and narrow grooves. The grooves are printed on a polydimethysiloxane substrate from a master pattern fabricated lithographically on a Si wafer (Stanford Nanofabrication Facility). The small width of the groove ($<2\sigma$) prevents the spheres from passing one other, and gravity keeps them from escaping the groove. The spheres are very weakly attractive ($<0.4k_{B}T$); the short-range attraction is derived from surface tension effects [@941]. Digital video microscopy is used to directly track the time-dependent trajectories of the spheres along the groove (the motion transverse to the groove is very limited and, therefore, is not considered here). Details relevant to sample preparation and data analysis have been described elsewhere [@946; @941].
We have studied q1D diffusion at various colloid concentrations, characterized by a line packing fraction $\eta=\rho\sigma=N\sigma/L$, where $L$ is the length of the groove in the field of view, and $N$ the number of spheres within $L$. We used two different silica colloid suspensions. For $\eta$=0.09, 0.17, 0.20, 0.38, 0.57, and 0.70 the samples had silica spheres with diameter $\sigma_{1}=1.58\mu m\pm0.04\mu m$ in a groove that was $3\mu m\pm0.3\mu m$ wide and deep, and $2mm$ long. For $\eta$=0.73 and 0.986 we used silica spheres with diameter $\sigma_{2}=3.7\mu m\pm0.1\mu m$ in a groove that was $5\mu m\pm0.1\mu m$ wide, $4\mu m\pm0.5\mu m$ deep, and $10mm$ long. Care has been taken to assure that there were no blockages in the grooves. We used the large spheres for the higher concentration samples because the small spheres could not be contained inside the grooves when $\eta>0.7$.
![Typical probability density of particle displacement evolving with time (for $\eta$=0.57). The solid lines are fits of the data to Eq.(\[eq:inter t prob\]).[]{data-label="fig:gauss"}](fig1-gaus.eps){width="3.1in"}
The self-diffusion process is usually described by the self-part of the van Hove function, $G_{s}(x,t)$, which is the probability density for finding a particle at a point $x_{0}+x$ at time $t_{0}+t$ given that it was at $x_{0}$ at $t_{0}$ ($G_{s}(x,t)=\frac{1}{N}\Big<\sum_{i=1}^N \delta[x+x_{i}(t_{0})-x_{i}(t_{0}+t)] \Big>$[@743]). Figure \[fig:gauss\] shows a typical $G_{s}(x,t)$ for our system, derived from time-dependent trajectories. The deviation of $G_{s}(x,t)$ from a Gaussian, characterized by $\alpha_{2}(t)=\Big(\frac{<x(t)^4>}{3<x(t)^2>^2}-1\Big)$, is found to be negligible ($\alpha_{2}(t)\lesssim0.1$). We therefore assume $G_{s}(x,t)$ to have the Gaussian form $$G_{s}(x,t)=\frac{1}{\sqrt{2\pi <x^2(t)>}}\exp{\Bigg\{-\frac{x(t)^2}{2<x^2(t)>}}\Bigg\}.
\label{eq:inter t prob}$$Figure \[fig:gauss\] also shows the Gaussian fits to $G_{s}(x,t)$. The mean squared displacement determined from the fitting is sensibly the same as that determined from $<x(t)^2>=\frac{1}{N}\Big<\sum_{i=1}^N [x_{i}(t_{0}+t)-x_{i}(t_{0})] ^2\Big>$.
Figure \[fig:msd\] shows $<x(t)^2>$ as a function of $t$ at various concentrations, extracted from fitting $G_{s}(x,t)$ to Eq.(\[eq:inter t prob\]). Qualitatively, $<x(t)^2>$ is proportional to $t$ at short time, changes smoothly to $<x(t)^2>\sim t^{\gamma}$ ($\gamma<$1) at later time, and reaches $<x(t)^2>\sim t^{1/2}$ at long time for the higher concentrations. Because of the expected trend in the behavior of $<x(t)^2>$ as a function of $\eta$, it is reasonable to postulate that with long enough time the low concentration samples will also exhibit $<x(t)^2>\sim t^{1/2}$. Accordingly, we use the following ansatz to describe $<x(t)^2>$ over the entire time range: $$\frac{1}{<x^2(t)>}=\frac{1}{2D_{o}t}+\frac{1}{2Ft^{1/2}}.
\label{eq:new width}$$Equation (\[eq:new width\]) leads to $$<x^2(t)>=\frac{2D_{o}t}{1+(D_{0}/F)t^{1/2}} =\frac{2D_{o}t}{1+(t/t_{x})^{1/2}} .
\label{eq:inter MSD}$$ By construction, Eq. (\[eq:inter MSD\]) satisfies both the short and long time limits, and it provides a characteristic cross-over time, $t_{x}=(F/D_{o})^2$. If $F=F^{HR}$, then $t_{x}=t^{HR}_{x}\equiv2t_{c}/\pi$ (see Eq.(\[eq:SFD\])), so that for hard rods $t_{x}$ is, essentially, the mean time between collisions. The fits of $<x(t)^2>$ to Eq.(\[eq:inter MSD\]) shown in Fig.\[fig:msd\] indicate that Eq.(\[eq:inter MSD\]) is a reasonable approximation for all time, and the fitting yields three pertinent parameters describing the q1D diffusion: the short time self-diffusion coefficient, $D_{o}$, the long time q1D mobility, $F$, and the cross-over time, $t_{x}$.
![ Mean squared displacement as a function of $t$ at different concentrations. Note that $<x(t)^2>$ for large spheres is scaled by the factor $\sigma_{2}/\sigma_{1}$. The data (symbols) are shifted downward a factor of 3 from one another for clarity. The error bars are smaller than the symbols used. For $t\leq1s$ the movies were grabbed at $30frames/s$, and for $t>1s$ the images were grabbed at $4frames/s$ and $5frames/s$ for small and large spheres, respectively (only a subset of the data are plotted for clarity). The solid lines are fits of the data to Eq.(\[eq:inter MSD\]).[]{data-label="fig:msd"}](fig2-MSD.eps){width="3.1in"}
When $\eta \leq 0.4$ the fitted values for $D_{o}$ are $D_{o1}=0.11\pm0.005 \mu m^2/s$ and $D_{o2}=0.036\pm0.005 \mu m^2/s$, respectively, for the small and large spheres (the lower concentration data for large spheres are not shown here). These values are, within the experimental precision, the same as those calculated for isolated colloids confined by the three walls of the groove [@946] and, therefore, are used to determine $F^{HR}$ in Eq.(\[eq:SFD\]) and $t_{x}$ in Eq.(\[eq:inter MSD\]). At higher concentrations the fitted self-diffusion coefficient is slightly smaller ($\sim 70\%-80\% D_{o})$, suggesting that hydrodynamic interaction between colloid particles comes into play even at the shortest time accessible in our experiment [@946; @948].
![Quasi-1D mobility (solid circles) determined with the empirical expression (Eq.(\[eq:inter MSD\])) as a function of concentration ($F$ for the large colloids is scaled by the factor $(\sigma_{1}/\sigma_{2})\sqrt{D_{o1}/D_{o2})}$. The solid line represents $F^{HR}$. Other symbols represent $F^q$ determined from Eq.(\[eq:Fq\]). The inset zooms into the data at higher $\eta$.[]{data-label="fig:F"}](Fig3_F.eps){width="3.1in"}
Figure \[fig:F\] shows the fitted $F$ as a function of $\eta$. When $0.17\leq\eta \leq 0.57$, $F=F^{HR}$ within the experimental precision. However, $F\approx F^{HR}/2$ when $\eta =0.09$ and $F\lesssim 2F^{HR}$ when $\eta\geq 0.7$. Using the fitted $F$ and $D_{o}$ we find $t_{x}=205, 150, 107, 17, 4.5, 5.2, 3.8, 0.006s$ for $\eta=0.09, 0.17, 0.19, 0.38, 0.57, 0.70, 0.73, 0.986$, correspondingly, to be compared with $t^{HR}_{x}=955, 172, 116, 19, 4.1, 1.3, 0.7, 0.001s$, respectively (note for large spheres $t_{x}$ and $t^{HR}_{x}$ are scaled by a factor $(\sigma_{1}/\sigma_{2})^{2}(D_{o2}/D_{o1})$). For $\eta\geq0.7$ we can force $F=F^{HR}$ by replacing the colloid diameter in $F^{HR}$ with a larger effective diameter. We speculate that at higher concentration the colloid-colloid interaction, though weak, must be accounted for. Since the first peak of the pair correlation function is at a separation slightly larger than the sphere diameter [@941], the effective sphere diameter is thereby increased.
The accuracy of the fitted $F$ and $D_{o}$ depends on the range of $t$ relative to $t^{HR}_{x}$. If the range of $t$ extends both to $t\ll t^{HR}_{x}$ and $t\gg t^{HR}_{x}$ we can extract $F$ and $D_{o}$ accurately from Eq.(\[eq:inter MSD\]); if not the values obtained are less accurate, as shown by the discrepancies between the fitted $F$ and $F^{HR}$ for $\eta=0.09$, and the fitted $D_{o}$ and expected $D_{o}$ at higher $\eta$.
![ The experimentally determined static structure factor $S(q)$ for small $q$ as a function of concentration, compared with that derived from the equation of state for a Tonks gas.[]{data-label="fig:chi"}](Fig4Chi.eps){width="3.1in"}
We now show that, as is to be expected, $F^{HR}=F^q$ for a q1D system that is described by Tonks equation of state $f(1-\rho\sigma)=\rho k_{B}T$ ($f$ is the linear force) [@982]. The relative isothermal compressibility of a Tonks gas is $$\chi_{T}/\chi_{To}=(1-\rho\sigma)^2=(1-\eta)^2=S(q)\Bigg|_{q=0},
\label{eq:S0}$$where $\chi_{T}=-\frac{1}{L}\frac{\partial f}{\partial L}$ for 1D, and $\chi_{To}=(k_{B}T\rho)^{-1}$. Figure \[fig:chi\] shows $S(q)$ for our system for small $q$ ($2\pi/q \gg \sigma/2$) as a function of $\eta$; the agreement with Eq.(\[eq:S0\]) clearly indicates that our system behaves like a Tonks gas. The slight shift of experimental values of $S(q)$ to larger $\eta$ from that of a Tonks gas is consistent with the weak colloid-colloid attraction [@886].
Kollmann’s theory relates $F^q$ to the collective diffusion coefficient, $D_{c}(q)$, and the relative isothermal compressibility, $S(0)$. Substituting $S(0)$ given in Eq.(\[eq:S0\]) and $D_{c}(q)=D_{o}H(q)/S(q)$ [@907] ($H(q)$ is the hydrodynamic factor) into Eq.(\[eq:Fq\]), we obtain $F^q=l\sqrt{\frac{D_{o}}{\pi}}=F^{HR}$, if $H(q)=1$, i.e. if hydrodynamic interaction is negligible.
We have calculated $F^q$ as follows. First, $S(q,t)$ was determined from the trajectories using $S(q,t) =\frac{1}{N}\Big<\rho_{q}(t)\rho_{-q}(0)\Big>$, where $\rho_{q}(t)=\frac{1}{\sqrt{N}}\int dx \exp(-iqx) \sum_{k=1}^N \delta[x-x_{k}(t)]$. Then $S(q,t)$ was fitted to the short time approximation (Eq.(\[eq:sqt\])) to extract $D_{c}(q)$ at small $q$ ($q\ll 4\pi/\sigma$). Within the short time range $0.03s\leq t\leq 1s$, $S(q,t)$ is well described by Eq.(\[eq:sqt\]) except for the case $\eta=0.986$. Finally, $F^q$ was calculated using Eq.(\[eq:Fq\]) for all the concentrations except $\eta=0.986$. As shown in Fig.\[fig:F\], $F^q$ agrees with $F^{HR}$ within the experimental precision. The data in Fig.\[fig:F\] also show that $F^q$ depends on $q$, which we attribute to hydrodynamic interaction in the system. In a q1D system hydrodynamic interaction is screened on the length scale of the channel width, so it can be treated as generating a pair-interaction [@948]. Then the effect of $H(q)$ on $F^q$ is not significant. A full discussion of $H(q)$ in the q1D system will be published separately.
It is worth noting that $<x(t)^2>$ in Eq.(\[eq:inter MSD\]) is the width of a Gaussian which is the product of the short time probability density (Eq.(\[eq:short t prob\])) and the long time probability density (Eq.(\[eq:long t prob\])). The success of the approximation given in Eq.(\[eq:inter MSD\]) suggests that the van Hove function displayed in Eq.(\[eq:inter t prob\]) is valid for all time for our system. It is further implied that the randomizing background that determines the short time behavior and the correlated motion that determines the long time single-file diffusion are coexisting independent processes with time dependent weights. For $t\ll t_{x}$ and $t\gg t_{x}$ the system exhibits normal diffusion and single-file diffusion, respectively. However, for $t\sim t_{x}$, the motion of a particle in 1D is hindered by its neighbors and the short time displacement distribution is modified by the long time distribution.
We thank Tom Witten and Sidney Nagel for helpful discussions. This research was supported by the NSF (CTS-021174 and CHE-9977841), the Israel Science Foundation (77/03), and the NSF-funded MRSEC laboratory at The University of Chicago.
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---
abstract: 'We investigate in this paper a Bickel-Rosenblatt test of goodness-of-fit for the density of the noise in an autoregressive model. Since the seminal work of Bickel and Rosenblatt, it is well-known that the integrated squared error of the Parzen-Rosenblatt density estimator, once correctly renormalized, is asymptotically Gaussian for independent and identically distributed (i.i.d.) sequences. We show that the result still holds when the statistic is built from the residuals of general stable and explosive autoregressive processes. In the univariate unstable case, we prove that the result holds when the unit root is located at $-1$ whereas we give further results when the unit root is located at $1$. In particular, we establish that except for some particular asymmetric kernels leading to a non-Gaussian limiting distribution and a slower convergence, the statistic has the same order of magnitude. We also study some common unstable cases, like the integrated seasonal process. Finally we build a goodness-of-fit Bickel-Rosenblatt test for the true density of the noise together with its empirical properties on the basis of a simulation study.'
address:
- 'Institut de Mathématiques de Toulouse; UMR5219. Université de Toulouse; CNRS. UT2J, F-31058 Toulouse, France.'
- 'Ho Chi Minh City University of Science, 227 Nguyen Van Cu, Phuong 4, Ho Chi Minh, Vietnam'
- 'Laboratoire Angevin de REcherche en MAthématiques (LAREMA), CNRS, Université d’Angers, Université Bretagne Loire. 2 Boulevard Lavoisier, 49045 Angers cedex 01.'
author:
- Agnès Lagnoux
- Thi Mong Ngoc Nguyen
- Frédéric Proïa
bibliography:
- 'BR-AR.bib'
title: 'On the Bickel-Rosenblatt test of goodness-of-fit for the residuals of autoregressive processes '
---
[^1]
Introduction and Motivations {#SecIntro}
============================
For i.i.d. sequences of random variables, there is a wide range of goodness-of-fit statistical procedures in connection with the underlying true distribution. Among many others, one can think about the Kolmogorov-Smirnov test, the Cramér-von Mises criterion, the Pearson’s chi-squared test, or more specific ones like the whole class of normality tests. Most of them have become of frequent practical use and directly implemented on the set of residuals of regression models. For such applications the independence hypothesis is irrelevant, especially for time series where lagged dependent variables are included. Thus, the crucial issue that naturally arises consists in having an overview of their sensitivity facing some weakened assumptions. This paper focus on such a generalization for the *Bickel-Rosenblatt* statistic, introduced by the eponymous statisticians [@BickelRosenblatt73] in 1973, who first established its asymptotic normality and gave their names to the associated testing procedure. The statistic is closely related to the $L^2$ distance on the real line between the Parzen-Rosenblatt kernel density estimator and a parametric distribution (or a smoothed version of it). Namely it takes the form of $$\int_{{\mathbb{R}}} \big( \widehat{f}_{n}(x) - f(x) \big)^2 a(x)\, {\mathrm{d}}x$$ with notation that we will detail in the sequel. Some improvements are of interest for us. First, Takahata and Yoshihara [@TakahataYoshihara87] in 1987 and later Neumann and Paparoditis [@NeumannPaparoditis00] in 2000 extended the result to weakly dependent sequences (with mixing or absolute regularity conditions). As they noticed, these assumptions are satisfied by several processes of the time series literature. Then, Lee and Na [@LeeNa02] showed in 2002 that it also holds for the residuals of an autoregressive process of order 1 as soon as it contains no unit root (we will explain precisely this fact in the paper). Such a study leads to a goodness-of-fit test for the distribution of the innovations of the process. Bachmann and Dette [@BachmannDette05] went further in 2005 by putting the results obtained by Lee and Na into practice. Their study also enables to get an asymptotic normality of the correctly renormalised statistic under some fixed alternatives.
The main purpose of this paper is the generalization of the results of Lee and Na to autoregressive processes of order $p$ ($p \geq 1$) besides refining the set of hypotheses, to discuss on the effect of unit roots on the statistic and to derive a goodness-of-fit test in the same way. On finite samples, it has been observed that the Gaussian behavior is difficult to reach and that, instead, an asymmetry occurs for dependent frameworks (see, *e.g.*, Valeinis and Locmelis [@ValeinisLocmelis12]). In the simulation study, we will use the configurations suggested in this previous paper, and the ones of Fan [@Fan94] and Ghosh and Huang [@GhoshHuang91] to try to minimize this effect. In the end of this section, we introduce the context of our study and present both notation and vocabulary used in the sequel. Moreover, we recall the well-known asymptotic behavior of the Bickel-Rosenblatt statistic for i.i.d. random variables. In Section 2, we give an overview of the existing results about the least-squares estimation of the autoregressive parameter, depending on the roots of its characteristic polynomial, since it is of crucial interest for our reasonings. Section 3 is dedicated to our results that are proved in the Appendix. To sum up, we establish the asymptotic behavior of the Bickel-Rosenblatt statistic applied to the residuals of stable and explosive autoregressive processes of order $p$ ($p \geq 1$), together with some results related to common unstable autoregressions (like the random walks and the seasonally integrated processes). We also suggest some considerations to deal with general unstable processes or mixed processes (for example, the unstable ARIMA($p-1$,1,0) process would deserve a particuliar attention due to its widespread use in the econometric field). In Section 4, we build a goodness-of-fit test in keeping with our context and discuss on some empirical bases.
To start with, let us consider an autoregressive process of order $p$ (AR($p$)) defined by $$\label{AR}
X_{t} = \theta_1\, X_{t-1} + \hdots + \theta_{p}\, X_{t-p} + {\varepsilon}_t$$ for any $t \geq 1$ or equivalently, in a compact form, by $$X_{t} = \theta{^{T}}\Phi_{t-1} + {\varepsilon}_t$$ where $\theta = (\theta_1, \hdots, \theta_{p}){^{T}}$ is a vector parameter, $\Phi_0$ is an arbitrary initial random vector, $\Phi_{t} = (X_{t}, \hdots, X_{t-p+1} ){^{T}}$ and $({\varepsilon}_t)$ is a strong white noise having a finite positive variance $\sigma^2$ and a marginal density $f$ (positive on the real line). The corresponding characteristic polynomial is defined, for all $z \in {\mathbb{C}}$, by $$\label{PolCar}
\Theta(z) = 1 - \theta_1\, z - \hdots - \theta_{p}\, z^{p}$$ and the companion matrix associated with $\Theta$ (see, *e.g.*, [@Duflo97 Sec. 4.1.2]) is given by $$\label{CompMat}
C_{\theta} = \begin{pmatrix}
\theta_1 & \theta_2 & \hdots & \theta_{p-1} & \theta_p \\
1 & 0 & \hdots & 0 & 0 \\
0 & 1 & \hdots & 0 & 0 \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
0 & 0 & \hdots & 1 & 0
\end{pmatrix}.$$ It follows that the process may also be written as $$\label{VAR}
\Phi_{t} = C_{\theta}\, \Phi_{t-1} + E_{t}$$ where $E_{t} = ({\varepsilon}_{t}, 0, \hdots, 0){^{T}}$ is a $p$–dimensional noise. It is well-known that the stability of this $p$–dimensional process is closely related to the eigenvalues of the companion matrix that we will denote and arrange like $$\rho(C_{\theta}) = \vert \lambda_1 \vert \geq \vert \lambda_2 \vert \geq \hdots \geq \vert \lambda_{p} \vert.$$ In particular, according to [@Duflo97 Def. 2.3.17], the process is said to be *stable* when $\vert \lambda_1 \vert < 1$, *purely explosive* when $\vert \lambda_{p} \vert > 1$ and *purely unstable* when $\vert \lambda_1 \vert = \vert \lambda_{p} \vert = 1$. Among the purely unstable processes of interest, let us mention the seasonal model admitting the complex $s$–th roots of unity as solutions of its autoregressive polynomial $\Theta(z) = 1 - z^{s}$ for a season $s \in {\mathbb{N}}\backslash\{ 0, 1\}$. In the paper, this model will be shortened as *seasonal unstable of order $p=s$*, it satisfies $\theta_1 = \hdots = \theta_{s-1} = 0$ and $\theta_{s} = 1$. In a general way, it is easy to see that $\det(C_{\theta}) = (-1)^{p+1}\, \theta_{p}$ so that $C_{\theta}$ is invertible as soon as $\theta_{p} \neq 0$ (which will be one of our hypotheses when $p > 0$). In addition, a simple calculation shows that $$\det(C_{\theta} - \lambda\, I_{p}) = (-\lambda)^{p}\, \Theta(\lambda^{-1})$$ which implies (since $\Theta(0) \neq 0$) that each zero of $\Theta$ is the inverse of an eigenvalue of $C_{\theta}$. Consequently, the stability of the process may be expressed in the paper through the eigenvalues of $C_{\theta}$ as well as through the zeroes of $\Theta$. We will also consider that $$z^{p}\, \Theta(z^{-1}) = z^{p} - \theta_1\, z^{p-1} - \ldots - \theta_{p}$$ is the minimal polynomial of $C_{\theta}$ (which, in the terminology of [@Duflo97], means that the process is *regular*). Now, assume that $X_{-p+1}, \hdots, X_0, X_1, \hdots, X_{n}$ are observable (for $n \gg p$) and let $$\label{OLS}
{\widehat{\theta}_{n}}= \left(\sum_{t=1}^n \Phi_{t-1} \Phi_{t-1}{^{T}}\right)^{\!-1} \sum_{t=1}^n \Phi_{t-1}\, X_{t}$$ be the least-squares estimator of $\theta$ (for $p>0$). The associated residual process is $$\label{ResSet}
\widehat{{\varepsilon}}_{t} = X_{t} - {\widehat{\theta}_{n}}^{\: T}\, \Phi_{t-1}$$ for all $1 \leq t \leq n$, or simply $\widehat{{\varepsilon}}_{t} = X_{t}$ when $p=0$. Hereafter, ${\mathbb{K}}$ is a kernel and $(h_{n})$ is a bandwidth. That is, ${\mathbb{K}}$ is a non-negative function satisfying $$\int_{{\mathbb{R}}} {\mathbb{K}}(x)\, {\mathrm{d}}x = 1, {\hspace{0.5cm}}\int_{{\mathbb{R}}} {\mathbb{K}}^2(x)\, {\mathrm{d}}x < +\infty {\hspace{0.5cm}}\text{and} {\hspace{0.5cm}}\int_{{\mathbb{R}}} x^2\, {\mathbb{K}}(x)\, {\mathrm{d}}x < +\infty,$$ and $(h_{n})$ is a positive sequence decreasing to 0. The so-called Parzen-Rosenblatt estimator [@Parzen62; @Rosenblatt56] of the density $f$ is given, for all $x \in {\mathbb{R}}$, by $$\label{PREst}
\widehat{f}_{n}(x) = \frac{1}{n\, h_{n}}\, \sum_{t=1}^n {\mathbb{K}}\left( \frac{x - \widehat{{\varepsilon}}_{t}}{h_{n}} \right).$$ The local behavior of this empirical density has been well studied in the literature. However, for a goodness-of-fit test, we focus on the global fitness of $\widehat{f}_{n}$ to $f$ on the whole real line. From this viewpoint, we consider the Bickel-Rosenblatt statistic that we define as $$\label{BRStat}
\widehat{T}_{n} = n\, h_{n} \int_{{\mathbb{R}}} \big( \widehat{f}_{n}(x) - ({\mathbb{K}}_{h_{n}} * f)(x) \big)^2 a(x)\, {\mathrm{d}}x$$ where ${\mathbb{K}}_{h_{n}} = h_{n}^{-1}\, {\mathbb{K}}(\cdot/h_{n})$, $a$ is a positive piecewise continuous integrable function and $*$ denotes the convolution operator, *i.e.* $(g*h)(x) = \int_{{\mathbb{R}}} g(x-u)\, h(u)\, {\mathrm{d}}u$. A statistic of probably greater interest and easier to implement is $$\label{BRStatF0}
\widetilde{T}_{n} = n\, h_{n} \int_{{\mathbb{R}}} \big( \widehat{f}_{n}(x) - f(x) \big)^2 a(x)\, {\mathrm{d}}x.$$ Bickel and Rosenblatt show in [@BickelRosenblatt73] that under appropriate conditions, if $T_{n}$ is the statistic given in built on the strong white noise $({\varepsilon}_{t})$ instead of the residuals, then, as $n$ tends to infinity, $$\label{AsNormBR}
\frac{T_n - \mu}{\sqrt{h_{n}}} {~ \overset{{\mathcal{D}}}{\longrightarrow} ~}{\mathcal{N}}(0, \tau^2)$$ where the centering term is $$\label{CenterBR}
\mu = \int_{{\mathbb{R}}} f(s)\, a(s)\, {\mathrm{d}}s\, \int_{{\mathbb{R}}} {\mathbb{K}}^2(s)\, {\mathrm{d}}s$$ and the asymptotic variance is given by $$\label{AsVarBR}
\tau^2 = 2\, \int_{{\mathbb{R}}} f^{\, 2}(s)\, a^2(s)\, {\mathrm{d}}s\, \int_{{\mathbb{R}}} \left( \int_{{\mathbb{R}}} {\mathbb{K}}(t)\, {\mathbb{K}}(t+s)\, {\mathrm{d}}t \right)^{\! 2}\, {\mathrm{d}}s.$$ The aforementioned conditions are summarized in [@GhoshHuang91 Sec. 2], they come from the original work of Bickel and Rosenblatt later improved by Rosenblatt [@Rosenblatt75]. In addition to some technical assumptions that will be recalled in (A$_0$), we just notice that a choice of a continuous and positive kernel defined on ${\mathbb{R}}$ and a bandwidth $h_{n} = h_0\, n^{-\kappa}$ must coincide with $0 < \kappa < 1/4$, $$\int_{\vert x \vert\, \geq\, 3} \vert {\mathbb{K}}^{\prime}(x) \vert\, \sqrt{ \vert x \vert^3\, \ln \ln \vert x \vert}\, {\mathrm{d}}x < +\infty {\hspace{0.5cm}}\text{and} {\hspace{0.5cm}}\int_{{\mathbb{R}}} ({\mathbb{K}}^{\prime}(x))^2\, {\mathrm{d}}x < +\infty.$$ If ${\mathbb{K}}$ is chosen to be bounded on a compact support, then $0 < \kappa < 1$ in the bandwidth is a sufficient condition. We can find in later references (like [@TakahataYoshihara87], [@NeumannPaparoditis00] or [@BachmannDette05]) some alternative proofs of the asymptotic normality with $a(x)=1$ and appropriate assumptions. However in this paper we will keep $a$ as an integrable function, so as to make the calculations easier and to follow the original framework of Bickel and Rosenblatt.
It is convenient to have short expressions for terms that converge in probability to zero. In the whole study, the notation $o_{\P}(1)$ (resp. $O_{\P}(1)$) stands for a sequence of random variables that converges to zero in probability (resp. is bounded in probability) as $n \to \infty$.
Preliminary results {#SecPrelim}
===================
We start by giving some (already known) preliminary results related to the behavior of $(X_{t})$, depending on the eigenvalues of $C_{\theta}$ and, in each case, the asymptotic behavior of the least-squares estimator. In the sequel, we assume that $\Phi_0$ shares the same assumptions of moments as $({\varepsilon}_{t})$. We only focus on results that will be useful for our reasonings.
Asymptotic behavior of the process {#SecPrelimProcess_1}
----------------------------------
\[PropStable\] Assume that $({\varepsilon}_{t})$ is a strong white noise such that, for some $\nu \geq 1$, ${\mathbb{E}}[ \vert {\varepsilon}_1 \vert^{\nu}]$ is finite. If $(X_{t})$ satisfies and is stable (that is $\vert \lambda_1 \vert < 1$ or, equivalently, $\Theta(z) \neq 0$ for all $\vert z \vert \leq 1$), then as $n$ tends to infinity, $$\sum_{t=1}^{n} \vert X_{t} \vert^{\nu} = O(n) {\hspace{0.3cm} \textnormal{a.s.}}{\hspace{0.5cm}}\text{and} {\hspace{0.5cm}}\sup_{1\, \leq\, t\, \leq\, n} \vert X_{t} \vert = o(n^{1/\nu}) {\hspace{0.3cm} \textnormal{a.s.}}$$ See [@BercuProia13 Lem. A.2] and [@Proia13 Lem. B.2]. In addition if $\nu \geq 2$, then $$\sum_{t=1}^{n} \Phi_{t} = O_{\P}(\sqrt{n}).$$
The last result is deduced from the invariance principle of [@DedeckerRio00 Thm. 1], since $(X_{t})$ is (asymptotically) stationary in this case. Indeed, we clearly have $$\frac{1}{\sqrt{n}}\, \sum_{t=1}^{n} \Phi_{t} = \frac{1}{\sqrt{n}}\, \sum_{t=1}^{n} \Phi_{t}^{*} + \frac{1}{\sqrt{n}}\, \bigg( \sum_{t=1}^{n} C_{\theta}^{\, t} \bigg) ( \Phi_0 - \Phi_0^{*} )$$ where $(\Phi_{t}^{*})$ is the (second-order) stationary version of the process for $\nu \geq 2$. We conclude that $(\Phi_{t})$ and $(\Phi_{t}^{*})$ share the same invariance principle (with rate $\sqrt{n}$), since $\rho(C_{\theta}) < 1$.
\[PropExplo\] Assume that $({\varepsilon}_t)$ is a strong white noise having a finite variance $\sigma^2$. If $(X_t)$ satisfies – and is purely explosive (that is $\vert \lambda_{p} \vert > 1$ or, equivalently, $\Theta(z) \neq 0$ for all $\vert z \vert \geq 1$), then $$\lim_{n \rightarrow +\infty} C_{\theta}^{-n}\, \Phi_{n} = \Phi_0 + \sum_{k=1}^{\infty} C_{\theta}^{-k}\, E_{k} = Z {\hspace{0.3cm} \textnormal{a.s.}}$$ and $$\lim_{n \rightarrow +\infty} C_{\theta}^{-n} \left(\sum_{t=1}^n \Phi_{t-1} \Phi_{t-1}{^{T}}\right) (C_{\theta}^{-n}){^{T}}= \sum_{k=1}^{\infty} C_{\theta}^{-k}\, Z\: Z{^{T}}(C_{\theta}^{-k}){^{T}}= G {\hspace{0.3cm} \textnormal{a.s.}}$$ In addition, $G$ is (a.s.) positive definite. See [@LaiWei83 Thm. 2].
\[PropUnstable\] Assume that $({\varepsilon}_t)$ is a strong white noise having a finite variance $\sigma^2$. If $(X_t)$ satisfies – with $p=1$ and is unstable (that is $\vert \lambda \vert = 1$ or, equivalently, $\Theta(z) \neq 0$ for all $\vert z \vert \neq 1$), then for $k \in {\mathbb{N}}$, $$\sum_{t=1}^{n} X_{t}^{2\, k} = O_{\P}(n^{k+1}) {\hspace{0.5cm}}\text{and} {\hspace{0.5cm}}\vert X_{n} \vert = O_{\P}(\sqrt{n}).$$ In addition, $$\sum_{t=1}^{n} X_{t} = \left\{
\begin{array}{ll}
O_{\P}(n^{3/2}) & \mbox{for } \lambda=1 \\
O_{\P}(\sqrt{n}) & \mbox{for } \lambda=-1.
\end{array}
\right.$$ The same rates are reached in the seasonal case for $p = s$.
These results are detailed in Lemma \[LemUnstable1\] and then proved.
Asymptotic behavior of the least-squares estimator {#SecPrelimProcess_2}
--------------------------------------------------
In this subsection, $({\varepsilon}_{t})$ is supposed to have a finite moment of order $2+\gamma$, for some $\gamma > 0$. The consistency results have been established in [@LaiWei83 Thm. 1], while the weak convergences are proved in [@BrockwellDavis06], [@Stigum74 Thm. 2] and [@ChanWei88 Thm. 3.5.1] respectively.
\[PropOLSStable\] In the stable case ($\vert \lambda_1 \vert < 1$), the least-squares estimator ${\widehat{\theta}_{n}}$ of $\theta$ is strongly consistent. In addition, $$\sqrt n\, ({\widehat{\theta}_{n}}- \theta) {~ \overset{{\mathcal{D}}}{\longrightarrow} ~}{\mathcal{N}}(0, \sigma^2\, \Gamma_{p}^{-1})$$ where $\Gamma_{p}$ is the $p \times p$ asymptotic covariance matrix of the process.
\[PropOLSExplo\] In the purely explosive case ($\vert \lambda_{p} \vert > 1$), the least-squares estimator ${\widehat{\theta}_{n}}$ of $\theta$ is strongly consistent. In addition, $$C_{\theta}^{n}\, ({\widehat{\theta}_{n}}- \theta) {~ \overset{{\mathcal{D}}}{\longrightarrow} ~}U$$ where $U$ is a nondegenerate random vector given in (1.8)–(1.9)–(1.10) of [@Stigum74].
\[PropOLSUnstable\] In the purely unstable case ($\vert \lambda_1 \vert = \vert \lambda_{p} \vert = 1$), the least-squares estimator ${\widehat{\theta}_{n}}$ of $\theta$ is strongly consistent. In the univariate case ($p=1$), we have in addition $$n\, ({\widehat{\theta}_{n}}- \theta) {~ \overset{{\mathcal{D}}}{\longrightarrow} ~}\textnormal{sgn}(\theta)\, \frac{\frac{1}{2}\, (W^2(1)-1)}{\int_0^1 W^2(u)\, {\mathrm{d}}u}$$ where $(W(t),\, t \in [0,1])$ is a standard Wiener process and $\textnormal{sgn}(\theta)$ stands for the sign of $\theta$.
In the seasonal case ($p = s$), since $\Theta(z) = 0$ is an equation admitting the complex $s$–th roots of unity as solutions, $$n\, ({\widehat{\theta}_{n}}- \theta) {~ \overset{{\mathcal{D}}}{\longrightarrow} ~}S(W_{s})$$ where $S(W_{s})$ is a functional of a standard Wiener process $(W_{s}(t),\, t \in [0,1])$ of dimension $s$ that can be explicitly built following the given reference.
The Bickel-Rosenblatt statistic {#SecBR}
===============================
In this section, we derive the limiting distribution of the test statistics $\widehat{T}_{n}$ and $\widetilde{T}_{n}$ given by and , based on the residuals in the stable, some unstable and purely explosive cases. For the whole study, we make the following assumptions.
1. The strong white noise $({\varepsilon}_{t})$ has a bounded density $f$ which is positive, twice differentiable, and the second derivative $f^{\prime \prime}$ is itself bounded. The weighting function $a$ is positive, piecewise continuous and integrable. The kernel ${\mathbb{K}}$ is bounded, continuous on its support. The kernel ${\mathbb{K}}$ and the bandwidth $(h_{n})$ are chosen to satisfy the hypotheses of Bickel and Rosenblatt summarized at the end of Section \[SecIntro\].
Some additional hypotheses are given below, not simultaneously needed.
1. The kernel ${\mathbb{K}}$ is such that ${\mathbb{K}}^{\prime \prime \prime}$ exists and is bounded on the real line, $$\int_{{\mathbb{R}}} \big\{ \vert {\mathbb{K}}^{\prime}(x) \vert + \vert {\mathbb{K}}^{\prime \prime}(x) \vert \big\}\, {\mathrm{d}}x < +\infty, {\hspace{0.5cm}}\int_{{\mathbb{R}}} ( {\mathbb{K}}^{\prime \prime}(x) )^2\, {\mathrm{d}}x < +\infty,$$ $$\int_{{\mathbb{R}}} \vert x\, {\mathbb{K}}^{\prime}(x) \vert\, {\mathrm{d}}x < +\infty {\hspace{0.5cm}}\text{and} {\hspace{0.5cm}}\int_{{\mathbb{R}}} x^2\, \vert {\mathbb{K}}^{\prime}(x) \vert\, {\mathrm{d}}x < +\infty.$$
2. The kernel ${\mathbb{K}}$ satisfies $$\int_{{\mathbb{R}}} \vert {\mathbb{K}}(x+\delta) - {\mathbb{K}}(x) \vert\, {\mathrm{d}}x \leq B\, \delta$$ for some $B > 0$ and all $\delta >0$.
3. The bandwidth $(h_{n})$ satisfies $${\lim_{n\, \rightarrow\, +\infty}}n\, h_{n}^{\alpha} = +\infty {\hspace{0.5cm}}\text{and/or} {\hspace{0.5cm}}{\lim_{n\, \rightarrow\, +\infty}}n\, h_{n}^{\beta} = 0.$$
4. The noise $({\varepsilon}_{t})$ and the initial vector $\Phi_0$ have a finite moment of order $\nu$.
To be rigorous in the technical tools used in the proofs, we have in addition to consider that any kernel with compact support must be approximated arbitrarily well by kernels satisfying the above hypotheses, around the discontinuities. Lee and Na [@LeeNa02] also made this observation in their simulation study.
\[ThmStable\] In the stable case ($\vert \lambda_1 \vert < 1$), assume that (A$_0$), (A$_1$), (A$_3$) with $\alpha = 4$, and (A$_4$) with $\nu = 4$ hold. Then, as $n$ tends to infinity, $$\frac{\widehat{T}_n - \mu}{\sqrt{h_{n}}} {~ \overset{{\mathcal{D}}}{\longrightarrow} ~}{\mathcal{N}}(0, \tau^2)$$ where $\mu$ and $\tau^2$ are given in and , respectively. In addition, the result is still valid for $\widetilde{T}_{n}$ if (A$_3$) with $\beta = 9/2$ holds.
See Section \[SecProofStable\].
Theorem \[ThmStable\] holds for $h_{n} = h_0\, n^{-\kappa}$ as soon as $2/9 < \kappa < 1/4$. A standard choice may be $h_{n} = h_0\, n^{-1/4+\epsilon}$ for a small $\epsilon > 0$.
\[ThmExplo\] In the purely explosive case ($\vert \lambda_{p} \vert > 1$), assume that (A$_0$), (A$_2$), (A$_3$) with $\alpha=1$ and (A$_4$) with $\nu = 2+\gamma$ hold, for some $\gamma > 0$. Then, as $n$ tends to infinity, $$\frac{\widehat{T}_n - \mu}{\sqrt{h_{n}}} {~ \overset{{\mathcal{D}}}{\longrightarrow} ~}{\mathcal{N}}(0, \tau^2)$$ where $\mu$ and $\tau^2$ are given in and , respectively. In addition, the result is still valid for $\widetilde{T}_{n}$ if (A$_3$) with $\beta = 9/2$ holds.
See Section \[SecProofExplo\].
Theorem \[ThmExplo\] holds for $h_{n} = h_0\, n^{-\kappa}$ as soon as $2/9 < \kappa < 1$ if ${\mathbb{K}}$ has a compact support, the usual bandwidth $h_{n} = h_0\, n^{-1/4}$ is appropriate. For a kernel positive on ${\mathbb{R}}$, we must restrict to $2/9 < \kappa < 1/4$ and a standard choice may be $h_{n} = h_0\, n^{-1/4+\epsilon}$ for a small $\epsilon > 0$.
We now investigate the asymptotic behavior of the statistics for some unstable cases. We establish in particular that the analysis of [@LeeNa02] is only partially true.
\[PropUnivUnstable\] In the unstable case for $p=1$ ($\lambda = \pm 1$), assume that (A$_0$), (A$_1$), (A$_3$) with $\alpha = 4$, and (A$_4$) with $\nu = 4$ hold. If $\lambda=- 1$ then, as $n$ tends to infinity, $$\frac{\widehat{T}_n - \mu}{\sqrt{h_{n}}} {~ \overset{{\mathcal{D}}}{\longrightarrow} ~}{\mathcal{N}}(0, \tau^2)$$ where $\mu$ and $\tau^2$ are given in and , respectively. If $\lambda=1$ and $\int_{{\mathbb{R}}} {\mathbb{K}}^{\prime}(s)\, {\mathrm{d}}s = 0$ then $$\frac{\widehat{T}_n - \mu}{\sqrt{h_{n}}} = O_{\P}(1).$$ The results are still valid for $\widetilde{T}_{n}$ if (A$_3$) with $\beta = 9/2$ holds. Finally, if $\lambda=1$ and $\int_{{\mathbb{R}}} {\mathbb{K}}^{\prime}(s)\, {\mathrm{d}}s \neq 0$ then, as $n$ tends to infinity, $$h_{n} (\widehat{T}_{n} - \mu) {~ \overset{{\mathcal{D}}}{\longrightarrow} ~}\sigma^2 \left( \frac{\frac{1}{2}\, (W^2(1)-1)}{\int_0^1 W^2(u)\, {\mathrm{d}}u}\, \int_0^1 W(u)\, {\mathrm{d}}u \right)^{\! 2}\! \int_{{\mathbb{R}}} f^2(s)\, a(s)\, {\mathrm{d}}s \left( \int_{{\mathbb{R}}} {\mathbb{K}}^{\prime}(s)\, {\mathrm{d}}s\right)^{\! 2}$$ where $(W(t),\, t \in [0,1])$ is a standard Wiener process. In the seasonal case with $p = s$, we also reach $$\frac{\widehat{T}_n - \mu}{\sqrt{h_{n}}} = O_{\P}(1)$$ or $$h_{n} (\widehat{T}_{n} - \mu) {~ \overset{{\mathcal{D}}}{\longrightarrow} ~}\sigma^2 \big( S(W_{s})^{T}\, H(W_{s}) \big)^2\! \int_{{\mathbb{R}}} f^2(s)\, a(s)\, {\mathrm{d}}s \left( \int_{{\mathbb{R}}} {\mathbb{K}}^{\prime}(s)\, {\mathrm{d}}s\right)^{\! 2}$$ depending on whether $\int_{{\mathbb{R}}} {\mathbb{K}}^{\prime}(s)\, {\mathrm{d}}s = 0$ or $\int_{{\mathbb{R}}} {\mathbb{K}}^{\prime}(s)\, {\mathrm{d}}s \neq 0$, respectively, where $S(W_{s})$ is defined in Proposition \[PropOLSUnstable\], $H(W_{s})$ will be clarified in the proof and $(W_{s}(t),\, t \in [0,1])$ is a standard Wiener process of dimension $s$.
See Section \[SecProofUnstable\].
This last result needs some observations.
1. Proposition \[PropUnivUnstable\] holds for $h_{n} = h_0\, n^{-\kappa}$ as soon as $2/9 < \kappa < 1/4$ $\lambda=-1$. A standard choice may be $h_{n} = h_0\, n^{-1/4+\epsilon}$ for a small $\epsilon > 0$.
2. One can observe that the value of $\int_{{\mathbb{R}}} {\mathbb{K}}^{\prime}(s)\, {\mathrm{d}}s$ is crucial to deal with the unstable case. In fact, all usual kernels are even, leading to $\int_{{\mathbb{R}}} {\mathbb{K}}^{\prime}(s)\, {\mathrm{d}}s = 0$. It seems that the unstable case always holds in applications (except for some very particular asymmetric kernels having different bound values).
3. At the end of the associated proof, we show that, for $\lambda=1$ and a choice of kernel such that $\int_{{\mathbb{R}}} {\mathbb{K}}^{\prime}(s)\, {\mathrm{d}}s \neq 0$, we also need $\int_{{\mathbb{R}}} s\, {\mathbb{K}}(s)\, {\mathrm{d}}s = 0$ to translate the result from $\widehat{T}_{n}$ to $\widetilde{T}_{n}$ under (A$_3$) with $\beta = 6$. In this case, the result holds for $h_{n} = h_0\, n^{-\kappa}$ as soon as $1/6 < \kappa < 1/4$.
4. In [@LeeWei99 Thm. 3.2], Lee and Wei had already noticed that only the unit roots located at 1 affect the residual process. Our result is coherent from that point of view, and the link between both studies lies in the fact that $$\widehat{T}_{n} = n\, h_{n}\, \int_{{\mathbb{R}}} \left( \frac{1}{h_{n}\, \sqrt{n}} \int_{{\mathbb{R}}} {\mathbb{K}}\left( \frac{x-s}{h_{n}} \right) {\mathrm{d}}\widehat{H}_{n}(F(s)) \right)^{\, 2} a(x)\, {\mathrm{d}}x$$ where $F$ is the cumulative distribution function of the noise and where, for all $0 \leq u \leq 1$, $$\widehat{H}_{n}(u) = \frac{1}{\sqrt{n}} \sum_{t=1}^{n} \big( {\mathbbm{1}}_{\{ F(\widehat{{\varepsilon}}_{t})\, \leq\, u \}} - u \big).$$
To conclude this part, we draw the reader’s attention to the fact that the purely unstable case is not fully treated. The general results may be a challenging study due to the phenomenon of compensation arising through unit roots different from 1. Lemma \[LemUnstable2\] at the end of the Appendix is not used as part of this paper but may be a trail for future studies. It illustrates the compensation *via* the fact that $(\sum_{k=1}^{t} X_{k})$ is of the same order as $(X_{t})$ in a purely unstable process having no unit root located at 1. Mixed models ($\vert \lambda_1 \vert \geq 1$ and $\vert \lambda_{p} \vert \leq 1$) should lead to similar reasonings, they also have to be handled to definitively lift the veil on the Bickel-Rosenblatt statistic for the residuals of autoregressive processes. As a priority, it seems that unstable ARIMA($p-1$,1,0) processes would deserve close investigations due to their widespread use in the econometric field. The difficulty arising here is that the estimator converges at the slow rate of stability while the process grows at the fast rate of instability: a compensation will be needed.
A goodness-of-fit Bickel-Rosenblatt test {#SecTest}
========================================
Our objective is now to derive a goodness-of-fit testing procedure from the results established in the previous section. First, one can notice that the choice of $$a(x) = \left\{
\begin{array}{ll}
1/f(x) & \mbox{for } x \in [-\delta\,;\,\delta] \\
0 & \mbox{for } x \notin [-\delta\,;\,\delta]
\end{array}
\right.$$ for any $\delta > 0$ leads to the simplifications $$\widetilde{T}_{n} = n\, h_{n} \int_{-\delta}^{\delta} \frac{\big( \widehat{f}_{n}(x) - f(x) \big)^2}{f(x)}\, {\mathrm{d}}x, {\hspace{0.5cm}}\mu = 2\, \delta \int_{{\mathbb{R}}} {\mathbb{K}}^2(s)\, {\mathrm{d}}s$$ and $$\tau^2 = 4\, \delta \int_{{\mathbb{R}}} \left( \int_{{\mathbb{R}}} {\mathbb{K}}(t)\, {\mathbb{K}}(t+s)\, {\mathrm{d}}t \right)^{\! 2}\, {\mathrm{d}}s.$$ Bickel and Rosenblatt [@BickelRosenblatt73] suggest a similar choice for the weight function $a$, with $[0\,;\,1]$ for compact support. Nevertheless, it seems more reasonable to work on a symmetric interval in order to test for the density of a random noise. In addition, $\mu$ and $\tau^2$ become independent of $f$, which will be useful to build a statistical procedure based on $f$. For a couple of densities $f$ and $f_0$ such that $f_0$ does not cancel on $[-\delta\,;\,\delta]$, let us define $$\label{DistDens}
\Delta_{\delta}(f,f_0) = \int_{-\delta}^{\delta} \frac{\big( f(x) - f_0(x) \big)^2}{f_0(x)}\, {\mathrm{d}}x.$$ Hence, $\Delta_{\delta}(f,f_0) = 0$ means that $f$ and $f_0$ coincide almost everywhere on $[-\delta\,;\,\delta]$, and everywhere under our usual continuity hypotheses on the densities. On the contrary, $\Delta_{\delta}(f,f_0) > 0$ means that there exists an interval $I \subseteq [-\delta\,;\,\delta]$ with non-empty interior on which $f$ and $f_0$ differ. Accordingly, let $${\mathcal{H}}_0: ``\Delta_{\delta}(f,f_0) = 0" \quad \text{vs.} \quad {\mathcal{H}}_1: ``\Delta_{\delta}(f,f_0) > 0".$$ The natural test statistic is therefore given by $$\label{StatTest}
\widetilde{Z}_{n}^{\, 0} = \frac{\widetilde{T}_{n}^{\, 0} - \mu}{\tau\ \sqrt{h_{n}}}$$ where $\widetilde{T}_{n}^{\, 0}$ is the statistic $\widetilde{T}_{n}$ reminded above built using $f_0$ instead of $f$.
\[PropGof\] Consider the set of residuals from one of the following AR$(p)$ models:
- a stable process with $p \geq 1$, under the hypotheses of Theorem \[ThmStable\],
- an explosive process with $p \geq 1$, under the hypotheses of Theorem \[ThmExplo\],
- an unstable process with $p = 1$ and $\lambda=-1$, under the hypotheses of Proposition \[PropUnivUnstable\].
Then, under ${\mathcal{H}}_0: ``\Delta_{\delta}(f,f_0) = 0"$ where $f_0$ is a continuous density which does not cancel on $[-\delta\,;\,\delta]$ for some $\delta>0$, $$\widetilde{Z}_{n}^{\, 0} {~ \overset{{\mathcal{D}}}{\longrightarrow} ~}{\mathcal{N}}(0,1).$$ In addition, under ${\mathcal{H}}_1: ``\Delta_{\delta}(f,f_0) > 0"$, $$\widetilde{Z}_{n}^{\, 0} {~ \overset{{\mathbb{P}}}{\longrightarrow} ~}+\infty.$$
The proof is immediate using our previous results. The consistency under ${\mathcal{H}}_1$ is reached using the fact that $$\widetilde{Z}_{n}^{\, 0} = \frac{\widetilde{T}_{n} - \mu}{\tau\ \sqrt{h_{n}}} + \frac{\widetilde{T}_{n}^{\, 0} - \widetilde{T}_{n}}{\tau\ \sqrt{h_{n}}}.$$
For any level $0 < \alpha < 1$, we reject ${\mathcal{H}}_0$ as soon as $$\widetilde{Z}_{n}^{\, 0} > u_{1-\alpha}$$ where $u_{1-\alpha}$ stands for the $(1-\alpha)$–quantile of the ${\mathcal{N}}(0,1)$ distribution. For our simulations, we focus on a normality test (probably the most useful in regression, for goodness-of-fit). We have trusted the observations of [@GhoshHuang91], [@Fan94] or [@ValeinisLocmelis12]. In particular, only the ${\mathcal{N}}(0,1)$ and the ${\mathcal{U}}([-1\,;\,1])$ kernels are used (in fact a smoothed version of the latter, to satisfy the hypotheses), with obviously $\int_{{\mathbb{R}}} {\mathbb{K}}^{\prime}(s)\, {\mathrm{d}}s = 0$. The bandwidth is $h_{n} = h_0\, n^{-1/4+\epsilon}$ for a small $\epsilon > 0$, where $h_0$ is calibrated to reach an empirical level close to $\alpha=5\%$ for the neutral model ($p=0$), and the true distribution of the noise is ${\mathcal{N}}(0,1)$. We only give an overview of the results in Table \[TabH0\] for some typical models:
- M$_0$ – neutral model ($p=0$),
- M$_1$ – stable model ($p=3$, $\theta_1 = -{1}/{12}$, $\theta_2 = {5}/{24}$, $\theta_3 = {1}/{24}$),
- M$_2$ – stable but almost unstable model ($p=1$, $\theta_1 = {99}/{100}$),
- M$_3$ – unstable model with negative unit root ($p=1$, $\theta_1 = -1$),
- M$_4$ – unstable model with positive unit root ($p=1$, $\theta_1 = 1$),
- M$_5$ – explosive model ($p=2$, $\theta_1 = 0$, $\theta_2 = {121}/{100}$).
${\mathbb{K}}$
---------------- ------- ------- --------------- ------- ------- ---------------
$n$ 50 100 500 50 100 500
$h_0$ 0.10 0.14 0.14 0.20 0.25 0.32
M$_0$ 0.051 0.049 0.049 0.052 0.048 0.048
M$_1$ 0.059 0.051 0.047 0.058 0.050 0.050
M$_2$ 0.055 0.046 0.051 0.044 0.046 0.051
M$_3$ 0.054 0.054 0.047 0.059 0.051 0.048
M$_4$ 0.048 0.051 0.047 0.050 0.045 0.050
M$_5$ 0.049 0.047 0.058$^{(*)}$ 0.046 0.049 0.054$^{(*)}$
: Empirical level of the test under ${\mathcal{H}}_0$, for the configurations described above. We used $\delta=2$, $n \in \{ 50, 100, 500 \}$ and $1000$ replications. $^{(*)}$Simulations that needed more than one numerical trial, due to the explosive nature of the process and the large value of $n$.[]{data-label="TabH0"}
For model M$_4$, it is important to note that Proposition \[PropGof\] may not hold. Nevertheless, we know by virtue of Proposition \[PropUnivUnstable\] that the statistic has the same order of magnitude, thus it seemed interesting to look at its empirical behavior in comparison with the other models (and we observe that it reacts as well). We now turn to the empirical power of the test, for the configuration $n=100$ and the ${\mathcal{N}}(0,1)$ kernel. In Figures \[FigH1moy\]–\[FigH1var\] below, we represent the percentage of rejection of ${\mathcal{H}}_0$ against the ${\mathcal{N}}(m,1)$ and ${\mathcal{N}}(0,\sigma^2)$ alternatives, for different values of the parameters, to investigate the sensitivity towards location and scale. We also make experiments in Figure \[FigH1dist\] with different distributions as alternatives (Student, uniform, Laplace and Cauchy). First of all, the main observation is that all models give very similar results (all curves are almost superimposed) even if $n$ is not so large. That corroborates the results of the paper: residuals from stable, explosive or some (univariate) unstable models satisfy the Bickel-Rosenblatt original convergence. Our procedure is roughly equivalent to the Kolmogorov-Smirnov one to test for location or scale in the Gaussian family (in fact it seems to be slightly less powerful for location and slightly more powerful for scale). However, it appears that our procedure better managed to discriminate some alternatives with close but different distributions. The objective of the paper is mainly theoretical and of course, a much more extensive study is needed to give any permanent conclusion about the comparison (role of $n$, $h_0$, $\kappa$, ${\mathbb{K}}$, $\delta$, the true distribution of $({\varepsilon}_{t})$, etc.).
![Empirical power of the test under ${\mathcal{H}}_1$, for $n=100$, ${\mathbb{K}}={\mathcal{N}}(0,1)$, $h_0=0.14$, $\delta=2$ and $1000$ replications. The alternatives are ${\mathcal{N}}(m, 1)$ for some $m \in [ -1\,;\,1]$ with true value $m=0$. Results obtained from M$_0$, $\hdots$, M$_5$ are superimposed in blue. The dotted line in red corresponds to the Kolmogorov-Smirnov test for M$_0$.[]{data-label="FigH1moy"}](H1moy "fig:"){width="12cm"}
![Empirical power of the test under ${\mathcal{H}}_1$, for $n=100$, ${\mathbb{K}}={\mathcal{N}}(0,1)$, $h_0=0.14$, $\delta=2$ and $1000$ replications. The alternatives are ${\mathcal{N}}(0, \sigma^2)$ for some $\sigma^2 \in [ 0.2\,;\,3.5]$ with true value $\sigma^2=1$. Results obtained from M$_0$, $\hdots$, M$_5$ are superimposed in blue. The dotted line in red corresponds to the Kolmogorov-Smirnov test for M$_0$.[]{data-label="FigH1var"}](H1var "fig:"){width="12cm"}
![Empirical power of the test under ${\mathcal{H}}_1$, for $n=100$, ${\mathbb{K}}={\mathcal{N}}(0,1)$, $h_0=0.14$, $\delta=2$ and $1000$ replications. The alternatives are from left to right the Student $t(1)$, $t(2)$, $t(5)$ distributions, the uniform ${\mathcal{U}}([-2\,;\,2])$ distribution, the centered Laplace ${\mathcal{L}}(1)$, ${\mathcal{L}}(1.5)$ and ${\mathcal{L}}(2)$ distributions and the centered Cauchy ${\mathcal{C}}(1)$ distribution. Results obtained from M$_0$, $\hdots$, M$_5$ are superimposed in blue. The red points correspond to the Kolmogorov-Smirnov test for M$_0$.[]{data-label="FigH1dist"}](H1dist "fig:"){width="12cm"}
In the unstable case with $p=1$ and a positive unit root (namely, the random walk), and a kernel satisfying $\int_{{\mathbb{R}}} {\mathbb{K}}^{\prime}(s)\, {\mathrm{d}}s \neq 0$, even if it is of lesser statistical interest, it is also possible to exploit Proposition \[PropUnivUnstable\] to derive a statistical procedure. Indeed, let $$\sigma^2_0 = \int_{{\mathbb{R}}} s^2\, f_0(s)\, {\mathrm{d}}s {\hspace{0.5cm}}\text{and} {\hspace{0.5cm}}F_0 = \int_{-\delta}^{\delta} f_0(s)\, {\mathrm{d}}s$$ with an adjustment of $\sigma^2_0$ if $f_0$ is not centered. Then, we can choose $$\widetilde{Z}_{n}^{\, 0} = \frac{h_{n} \big( \widetilde{T}_{n}^{\, 0} - \mu \big) }{\sigma_0\, \sqrt{F_0} \int_{{\mathbb{R}}} {\mathbb{K}}^{\prime}(s)\, {\mathrm{d}}s}$$ and compare it with the quantiles associated with the distribution of $$Z = \left( \frac{\frac{1}{2}\, (W^2(1)-1)}{\int_0^1 W^2(u)\, {\mathrm{d}}u}\, \int_0^1 W(u)\, {\mathrm{d}}u \right)^{\! 2}.$$
**Acknowledgments.** The authors warmly thank Bernard Bercu for all his advices and suggestions during the preparation of this work.
Appendix. Proofs of the main results {#SecProof .unnumbered}
====================================
In this section, we prove our results.
Proof of Theorem \[ThmStable\] {#SecProofStable}
------------------------------
We follow the same lines as in the proof in [@LeeNa02 Thm. 2.1], and we only consider the case where the kernel satisfies hypothesis (A$_1$). The difference between the statistics can be expressed like $$\label{DecompStable}
\widehat{T}_{n} - T_{n} = n\, h_{n}\, \int_{{\mathbb{R}}} \big( \widehat{f}_{n}(x) - f_{n}(x) \big)^2 a(x)\, {\mathrm{d}}x + R_{n}$$ with $$R_{n} = 2\, n\, h_{n}\, \int_{{\mathbb{R}}} \big( \widehat{f}_{n}(x) - f_{n}(x) \big)\, \big( f_{n}(x) - ({\mathbb{K}}_{h_{n}}*f)(x) \big)\, a(x)\, {\mathrm{d}}x.$$ Using the mean value theorem, we can write for all $1 \leq t \leq n$, $${\mathbb{K}}\left( \frac{x-\widehat{{\varepsilon}}_{t}}{h_{n}} \right) - {\mathbb{K}}\left( \frac{x-{\varepsilon}_{t}}{h_{n}} \right) = \frac{{\varepsilon}_{t} - \widehat{{\varepsilon}}_{t}}{h_{n}}\, {\mathbb{K}}^{\prime}\left( \frac{x-{\varepsilon}_{t}}{h_{n}} \right) + \frac{({\varepsilon}_{t} - \widehat{{\varepsilon}}_{t})^2}{2\, h_{n}^2}\, {\mathbb{K}}^{\prime \prime}\left( \Delta_{t x}\right)$$ where $$\Delta_{t x} = \frac{ x - {\varepsilon}_{t} + \zeta ({\varepsilon}_{t} - \widehat{{\varepsilon}}_{t})}{h_{n}}$$ for some $0 < \zeta < 1$. Note that $${\varepsilon}_{t} - \widehat{{\varepsilon}}_{t} = ({\widehat{\theta}_{n}}- \theta){^{T}}\Phi_{t-1} = \langle {\widehat{\theta}_{n}}- \theta, \Phi_{t-1} \rangle$$ and obviously that $$\big\vert \langle {\widehat{\theta}_{n}}- \theta, \Phi_{t-1} \rangle \big\vert \leq \Vert {\widehat{\theta}_{n}}- \theta \Vert\, \Vert \Phi_{t-1} \Vert.$$ On the one hand, we consider the first term (say, $I_{n}$) of the right-hand side of that can be bounded like $$\begin{aligned}
I_{n} & = & n\, h_{n}\, \int_{{\mathbb{R}}} \left\{ \frac{1}{n\, h_{n}}\, \sum_{t=1}^{n} \left( {\mathbb{K}}\left( \frac{x-\widehat{{\varepsilon}}_{t}}{h_{n}} \right) - {\mathbb{K}}\left( \frac{x-{\varepsilon}_{t}}{h_{n}} \right) \right) \right\}^{\! 2} a(x)\, {\mathrm{d}}x \nonumber \\
& \leq & \frac{2}{n\, h_{n}^3}\, \int_{{\mathbb{R}}} \left\{ \sum_{t=1}^{n} \langle {\widehat{\theta}_{n}}- \theta, \Phi_{t-1} \rangle\, {\mathbb{K}}^{\prime}\left( \frac{x-{\varepsilon}_{t}}{h_{n}} \right) \right\}^{\! 2} a(x)\, {\mathrm{d}}x \nonumber \\
& & {\hspace{0.5cm}}{\hspace{0.5cm}}+ ~ \frac{1}{2\, n\, h_{n}^5}\, \int_{{\mathbb{R}}} \left\{ \sum_{t=1}^{n} \langle {\widehat{\theta}_{n}}- \theta, \Phi_{t-1} \rangle ^2\, {\mathbb{K}}^{\prime \prime}\left( \Delta_{t x} \right) \right\}^{\! 2} a(x)\, {\mathrm{d}}x. \label{DecompStable2T}\end{aligned}$$ At this step, we need two technical lemmas.
\[LemStable1\] We have $$I_{1,n} = \int_{{\mathbb{R}}} \left\{ \sum_{t=1}^{n} \langle {\widehat{\theta}_{n}}- \theta, \Phi_{t-1} \rangle\, {\mathbb{K}}^{\prime}\left( \frac{x-{\varepsilon}_{t}}{h_{n}} \right) \right\}^{\! 2} a(x)\, {\mathrm{d}}x = O_{\P}(h_{n}).$$
One clearly has $$\begin{aligned}
I_{1,n} & \leq & 2\, \Bigg\{ \int_{{\mathbb{R}}} \left( \sum_{t=1}^{n} \langle {\widehat{\theta}_{n}}- \theta, \Phi_{t-1} \rangle\, v_{t}(x) \right)^{\! 2} a(x)\, {\mathrm{d}}x \nonumber \\
& & {\hspace{0.5cm}}{\hspace{0.5cm}}+ ~ \int_{{\mathbb{R}}} \left( \sum_{t=1}^{n} \langle {\widehat{\theta}_{n}}- \theta, \Phi_{t-1} \rangle\, e(x) \right)^{\! 2} a(x)\, {\mathrm{d}}x \Bigg\} ~ = ~ 2\, ( J_{1,n} + J_{2,n} ) \label{DecompStableIn1}\end{aligned}$$ where $$v_{t}(x) = {\mathbb{K}}^{\prime}\left( \frac{x-{\varepsilon}_{t}}{h_{n}} \right) - {\mathbb{E}}\left[ {\mathbb{K}}^{\prime}\left( \frac{x-{\varepsilon}_1}{h_{n}} \right) \right] {\hspace{0.5cm}}\text{and} {\hspace{0.5cm}}e(x) = {\mathbb{E}}\left[ {\mathbb{K}}^{\prime}\left( \frac{x-{\varepsilon}_1}{h_{n}} \right) \right].$$ Now let us consider $J_{1,n}$ and $J_{2,n}$. First, $$\begin{aligned}
J_{1,n} & \leq & \Vert\, {\widehat{\theta}_{n}}- \theta \Vert^2 \int_{{\mathbb{R}}} \bigg\Vert \sum_{t=1}^{n} \Phi_{t-1}\, v_{t}(x) \bigg\Vert^2 a(x)\, {\mathrm{d}}x \nonumber \\
& = & \Vert {\widehat{\theta}_{n}}- \theta \Vert^2\, \sum_{i=1}^{p} \int_{{\mathbb{R}}} \left( \sum_{t=1}^{n} X_{t-i}\, v_{t}(x) \right)^{\! 2} a(x)\, {\mathrm{d}}x. \label{DecompStableJ1}\end{aligned}$$ Let $i \in \{1, \hdots, p\}$. Then, $$\begin{aligned}
{\mathbb{E}}\left[ \int_{{\mathbb{R}}} \left( \sum_{t=1}^{n} X_{t-i}\, v_{t}(x) \right)^{\! 2} a(x)\, {\mathrm{d}}x \right] & = & \int_{{\mathbb{R}}} {\mathbb{E}}\left[ \left( \sum_{t=1}^{n} X_{t-i}\, v_{t}(x) \right)^{\! 2} \right]\, a(x)\, {\mathrm{d}}x \\
& = & \int_{{\mathbb{R}}} \sum_{t=1}^{n} {\mathbb{E}}\big[ X_{t-i}^{\, 2} \big]\, {\mathbb{E}}\big[ v_{t}^{\, 2}(x) \big] a(x)\, {\mathrm{d}}x \\
& = & \sum_{t=1}^{n} {\mathbb{E}}\big[ X_{t-i}^{\, 2} \big]\, \int_{{\mathbb{R}}} {\mathbb{E}}\big[ v_{t}^{\, 2}(x) \big]\, a(x)\, {\mathrm{d}}x.\end{aligned}$$ But, under our assumptions, we recall that ${\mathbb{E}}[X_{n}^{\, 2}] = O(1)$ from the asymptotic stationarity of the process and $$\begin{aligned}
\int_{{\mathbb{R}}} {\mathbb{E}}\big[ v_{t}^{\, 2}(x) \big]\, a(x)\, {\mathrm{d}}x & \leq & \int_{{\mathbb{R}}} {\mathbb{E}}\left[ \left( {\mathbb{K}}^{\prime}\left( \frac{x-{\varepsilon}_1}{h_{n}} \right) \right)^{\! 2} \right] a(x)\, {\mathrm{d}}x \\
& = & h_{n} \int_{{\mathbb{R}}} ({\mathbb{K}}^{\prime}(z))^2\, \int_{{\mathbb{R}}} f(x - h_{n}\, z)\, a(x)\, {\mathrm{d}}x\, {\mathrm{d}}z ~ = ~ O(h_{n})\end{aligned}$$ since $\Var(Z)\leq \E[Z^2]$ for some random variable $Z$ and by Assumptions (A$_0$) and (A$_1$). Thus $J_{1,n} = O_{\P}(h_{n})$ *via* Proposition \[PropOLSStable\]. Then, by a direct calculation, $$\begin{aligned}
J_{2,n} & = & \left( ({\widehat{\theta}_{n}}- \theta ){^{T}}\, \sum_{t=1}^{n} \Phi_{t-1} \right)^{\! 2} \int_{{\mathbb{R}}} e^{\, 2}(x)\, a(x)\, {\mathrm{d}}x \nonumber \\
& = & h_{n}^2 \left( ({\widehat{\theta}_{n}}- \theta ){^{T}}\, \sum_{t=1}^{n} \Phi_{t-1} \right)^{\! 2} \int_{{\mathbb{R}}} \left( \int_{{\mathbb{R}}} {\mathbb{K}}^{\prime}(z)\, f(x - h_{n}\, z)\, {\mathrm{d}}z \right)^{\! 2} a(x)\, {\mathrm{d}}x. \label{DecompStableJ2}\end{aligned}$$ Using Proposition \[PropStable\], it follows that $J_{2,n} = O_{\P}(h_{n}^2)$.
\[LemStable2\] We have $$I_{2,n} = \int_{{\mathbb{R}}} \left\{ \sum_{t=1}^{n} \langle {\widehat{\theta}_{n}}- \theta, \Phi_{t-1} \rangle ^2\, {\mathbb{K}}^{\prime \prime}\left( \Delta_{t x} \right) \right\}^{\! 2} a(x)\, {\mathrm{d}}x = O_{\P}\!\left( h_{n}^2 + \frac{1}{n\, h_{n}^2} \right).$$
We directly get $$\begin{aligned}
I_{2,n} & \leq & \Vert {\widehat{\theta}_{n}}- \theta \Vert^4\, \int_{{\mathbb{R}}} \left\{ \sum_{t=1}^{n} \Vert \Phi_{t-1} \Vert^2\, {\mathbb{K}}^{\prime \prime}\left( \Delta_{t x} \right) \right\}^{\! 2} a(x)\, {\mathrm{d}}x \nonumber \\
& \leq & 3\, \Vert {\widehat{\theta}_{n}}- \theta \Vert^4\, \Bigg( \int_{{\mathbb{R}}} \left\{ \sum_{t=1}^{n} \Vert \Phi_{t-1} \Vert^2\, \left( {\mathbb{K}}^{\prime \prime}\left( \Delta_{t x} \right) - {\mathbb{K}}^{\prime \prime}\left( \frac{x - {\varepsilon}_{t}}{h_{n}} \right) \right) \right\}^{\! 2} a(x)\, {\mathrm{d}}x \nonumber \\
& & {\hspace{0.5cm}}{\hspace{0.5cm}}+ ~ \int_{{\mathbb{R}}} \left\{ \sum_{t=1}^{n} \Vert \Phi_{t-1} \Vert^2\, \left( {\mathbb{K}}^{\prime \prime}\left( \frac{x - {\varepsilon}_{t}}{h_{n}} \right) - {\mathbb{E}}\left[ {\mathbb{K}}^{\prime \prime}\left( \frac{x - {\varepsilon}_1}{h_{n}} \right) \right] \right) \right\}^{\! 2} a(x)\, {\mathrm{d}}x \nonumber \\
& & {\hspace{0.5cm}}{\hspace{0.5cm}}+ ~ \int_{{\mathbb{R}}} \left\{ \sum_{t=1}^{n} \Vert \Phi_{t-1} \Vert^2\, {\mathbb{E}}\left[ {\mathbb{K}}^{\prime \prime}\left( \frac{x - {\varepsilon}_1}{h_{n}} \right) \right] \right\}^{\! 2} a(x)\, {\mathrm{d}}x \Bigg) \nonumber \\
& = & 3\, \Vert {\widehat{\theta}_{n}}- \theta \Vert^4\, ( K_{1,n} + K_{2,n} + K_{3,n} ). \label{DecompStableIn2}\end{aligned}$$ From the mean value theorem, under our hypotheses, $$\begin{aligned}
{\mathbb{K}}^{\prime \prime}\left( \Delta_{t x} \right) - {\mathbb{K}}^{\prime \prime}\left( \frac{x - {\varepsilon}_{t}}{h_{n}} \right) & = & {\mathbb{K}}^{\prime \prime}\left( \frac{ x - {\varepsilon}_{t} + \zeta ({\varepsilon}_{t} - \widehat{{\varepsilon}}_{t})}{h_{n}} \right) - {\mathbb{K}}^{\prime \prime}\left( \frac{x - {\varepsilon}_{t}}{h_{n}} \right) \\
& = & \frac{\zeta({\varepsilon}_{t} - \widehat{{\varepsilon}}_{t})}{h_{n}}\, {\mathbb{K}}^{\prime \prime \prime}\!\left(\frac{x - {\varepsilon}_{t} + \xi\, \zeta ({\varepsilon}_{t} - \widehat{{\varepsilon}}_{t})}{h_{n}} \right)\end{aligned}$$ for some $0 < \zeta, \xi < 1$. We deduce that $$\left\vert{\mathbb{K}}^{\prime \prime}\left( \Delta_{t x} \right) - {\mathbb{K}}^{\prime \prime}\left( \frac{x - {\varepsilon}_{t}}{h_{n}} \right) \right\vert \leq \frac{\vert {\varepsilon}_{t}-\widehat{{\varepsilon}}_{t} \vert}{h_{n}}\, \left\vert {\mathbb{K}}^{\prime \prime \prime}\!\left(\frac{x - {\varepsilon}_{t} + \xi\, \zeta ({\varepsilon}_{t} - \widehat{{\varepsilon}}_{t})}{h_{n}} \right) \right\vert.$$ Consequently, since ${\mathbb{K}}^{\prime \prime \prime}$ is bounded, $$\begin{aligned}
K_{1,n} ~ \leq ~ \frac{C}{h_{n}^2}\, \left( \sum_{t=1}^{n} \Vert \Phi_{t-1} \Vert^2\, \vert {\varepsilon}_{t} - \widehat{{\varepsilon}}_{t} \vert \right)^{\! 2} & \leq & \frac{C}{h_{n}^2}\, \sum_{t=1}^{n} \Vert \Phi_{t-1} \Vert^4\, \sum_{t=1}^{n} ({\varepsilon}_{t} - \widehat{{\varepsilon}}_{t} )^2 \nonumber \\
& \leq & \frac{C}{h_{n}^2}\, \Vert {\widehat{\theta}_{n}}- \theta \Vert^2\, \sum_{t=1}^{n} \Vert \Phi_{t-1} \Vert^4\, \sum_{t=1}^{n} \Vert \Phi_{t-1} \Vert^2 \label{DecompStableK1}\end{aligned}$$ for some arbitrary constants. Then, $K_{1,n} = O_{\P}(n\, h_{n}^{-2})$ as soon as we suppose that $({\varepsilon}_{t})$ has a finite moment of order $\nu=4$ (by virtue of Proposition \[PropStable\]). Now we proceed as for $J_{1,n}$ to get $$\begin{aligned}
{\mathbb{E}}[K_{2,n}] & \leq & \sum_{t=1}^{n} {\mathbb{E}}\big[ \Vert \Phi_{t-1} \Vert^4 \big]\, \int_{{\mathbb{R}}} {\mathbb{E}}\left[ {\mathbb{K}}^{\prime \prime} \left( \frac{x - {\varepsilon}_1}{h_{n}} \right) ^{\! 2} \right] a(x)\, {\mathrm{d}}x \nonumber \\
& = & h_{n}\, \sum_{t=1}^{n} {\mathbb{E}}\big[ \Vert \Phi_{t-1} \Vert^4 \big]\, \int_{{\mathbb{R}}} ({\mathbb{K}}^{\prime \prime}(z))^2\, \int_{{\mathbb{R}}} f(x - h_{n}\, z)\, a(x)\, {\mathrm{d}}x\, {\mathrm{d}}z ~ = ~ O(n\, h_{n}) \label{DecompStableK2}\end{aligned}$$ which shows $K_{2,n} = O_{\P}(n\, h_{n})$, since ${\mathbb{E}}\big[ X_{n}^{\, 4} \big] = O(1)$ under the hypotheses of stability and fourth-order moments. Finally, $$\begin{aligned}
K_{3,n} & \leq & h_{n}^2\, \int_{{\mathbb{R}}} \left( \int_{{\mathbb{R}}} \vert {\mathbb{K}}^{\prime \prime}(z) \vert\, f(x - h_{n}\, z)\, {\mathrm{d}}z \right)^2 a(x)\, {\mathrm{d}}x\, \left( \sum_{t=1}^{n} \Vert \Phi_{t-1} \Vert^2 \right)^{\! 2} \nonumber \\
& \leq & C\, h_{n}^2\, \left( \sum_{t=1}^{n} \Vert \Phi_{t-1} \Vert^2 \right)^{\! 2} ~ = ~ O_{\P}(n^2\, h_{n}^2) \label{DecompStableK3}\end{aligned}$$ for a constant $C$. Whence we deduce that $I_{2,n} = O_{\P}(h_{n}^2 + n^{-1} h_{n}^{-2})$.
We are now ready to conclude the proof of Theorem \[ThmStable\]. If we come back to $I_{n}$ in , then the combination of Lemmas \[LemStable1\] and \[LemStable2\] leads to $$\label{CvgInStable}
\frac{I_{n}}{\sqrt{h_{n}}} = O_{\P}\!\left( \frac{1}{n\, h_{n}^{5/2}} + \frac{1}{n\, h_{n}^{7/2}} + \frac{1}{n^2\, h_{n}^{15/2}} \right) = o_{\P}(1)$$ as soon as $n\, h_{n}^{15/4} \rightarrow +\infty$. Now by Cauchy-Schwarz inequality, $$\label{CvgRnStable}
\frac{\vert R_{n} \vert}{\sqrt{h_{n}}} \leq 2\, \sqrt{\frac{I_{n}\, T_{n}}{h_{n}}} = O_{\P}\bigg( \sqrt{\frac{I_{n}}{h_{n}}} \bigg) = o_{\P}(1)$$ from the previous reasoning, the asymptotic normality and Assumption (A$_3$) with $\alpha=4$. It follows from , and that $$\frac{\widehat{T}_{n} - \mu}{\sqrt{h_{n}}} = \frac{T_{n} - \mu}{\sqrt{h_{n}}} + o_{\P}(1)$$ which ends the first part of the proof. The second part makes use of a result of Bickel and Rosenblatt [@BickelRosenblatt73]. Indeed, denote by $\bar{T}_{n}$ the statistic given in built on the strong white noise $({\varepsilon}_{t})$ instead of the residuals. They show that $$\begin{aligned}
\label{CvgUnStable}
\frac{\vert T_{n} - \bar{T}_{n} \vert}{\sqrt{h_{n}}} = o_{\P}(1)\end{aligned}$$ as soon as $n\, h_{n}^{9/2} \rightarrow 0$. A similar calculation leads to $$\widetilde{T}_{n} = I_{n} + \bar{T}_{n} + \bar{R}_{n}$$ where $$\bar{R}_{n} = 2\, n\, h_{n}\, \int_{{\mathbb{R}}} \big( \widehat{f}_{n}(x) - f_{n}(x) \big)\, \big( f_{n}(x) - f(x) \big)\, a(x)\, {\mathrm{d}}x.$$ We deduce that $$\frac{\vert \widetilde{T}_{n} - \bar{T}_{n} \vert}{\sqrt{h_{n}}} \leq \frac{I_{n}}{\sqrt{h_{n}}} + \frac{\vert \bar{R}_{n} \vert}{\sqrt{h_{n}}} = O_{\P}\bigg( \sqrt{\frac{I_{n}}{h_{n}}} \bigg) = o_{\P}(1)$$ as soon as $\bar{T}_{n}$ satisfies the original Bickel-Rosenblatt convergence and and hold, that is $n\, h_{n}^{9/2} \rightarrow 0$ and $n\, h_{n}^4 \rightarrow +\infty$. It only remains to note that $$\begin{aligned}
\frac{\vert \widehat{T}_{n} - \widetilde{T}_{n} \vert}{\sqrt{h_{n}}}
& \leq & \frac{\vert \widehat{T}_{n} - T_{n} \vert}{\sqrt{h_{n}}} + \frac{\vert T_{n} - \bar{T}_{n} \vert}{\sqrt{h_{n}}} + \frac{\vert \bar{T}_{n} - \widetilde{T}_{n} \vert}{\sqrt{h_{n}}},\end{aligned}$$ each term being $o_{\P}(1)$.
Proof of Theorem \[ThmExplo\] {#SecProofExplo}
-----------------------------
Let $I_{n}$ be the first term of the right-hand side of , like in the last proof, and note that $$\begin{aligned}
\label{DecompExplo}
I_{n} ~ = ~ n\, h_{n} \int_{{\mathbb{R}}} \big( \widehat{f}_{n}(x) - f_{n}(x) \big)^2 a(x)\, {\mathrm{d}}x & \leq & 2\, (I_{n, 1} + I_{n, 2}),\end{aligned}$$ where for an arbitrary rate that we set to $v_{n}^{\, 2} = \ln(n\, h_{n})$, $$\begin{aligned}
I_{n, 1} & = & \frac{1}{n\, h_{n}} \int_{{\mathbb{R}}} \left\{ \sum_{t=1}^{n-[v_{n}]} \left({\mathbb{K}}\left(\frac{x-\widehat{{\varepsilon}}_{t}}{h_{n}} \right) - {\mathbb{K}}\left(\frac{x-{\varepsilon}_{t}}{h_{n}} \right)\right) \right\}^{\! 2} a(x)\, {\mathrm{d}}x, \\
I_{n, 2} & = & \frac{1}{n\, h_{n}} \int_{{\mathbb{R}}} \left\{ \sum_{t=n-[v_{n}]+1}^{n} \left({\mathbb{K}}\left(\frac{x-\widehat{{\varepsilon}}_{t}}{h_{n}} \right) - {\mathbb{K}}\left(\frac{x-{\varepsilon}_{t}}{h_{n}} \right) \right) \right\}^{\! 2} a(x)\, {\mathrm{d}}x.\end{aligned}$$ Under our hypotheses, this choice of $(v_{n})$ ensures $$\label{PropVn}
{\lim_{n\, \rightarrow\, +\infty}}v_{n} = +\infty {\hspace{0.5cm}}\text{and} {\hspace{0.5cm}}{\lim_{n\, \rightarrow\, +\infty}}\frac{v_{n}^{\, 2}}{n \sqrt{h_{n}}} = 0.$$ Let us look at $I_{n, 1}$. By Cauchy-Schwarz, $$\begin{aligned}
I_{n, 1} & \leq & \frac{n-[v_{n}]}{n\, h_{n}} \int_{{\mathbb{R}}} \sum_{t=1}^{n-[v_{n}]} \left[ {\mathbb{K}}\left(\frac{x-\widehat{{\varepsilon}}_{t}}{h_{n}} \right) - {\mathbb{K}}\left(\frac{x-{\varepsilon}_{t}}{h_{n}} \right) \right]^2 a(x)\, {\mathrm{d}}x \\
& \leq & \frac{2\, \Vert {\mathbb{K}}\Vert_{\infty}\, (n-[v_{n}])}{n\, h_{n}} \int_{{\mathbb{R}}} \sum_{t=1}^{n-[v_{n}]} \left\vert {\mathbb{K}}\left(\frac{x-\widehat{{\varepsilon}}_{t}}{h_{n}} \right) - {\mathbb{K}}\left(\frac{x-{\varepsilon}_{t}}{h_{n}} \right) \right\vert a(x)\, {\mathrm{d}}x \\
& \leq & \frac{2\, \Vert a \Vert_{\infty}\, \Vert {\mathbb{K}}\Vert_{\infty}\, (n-[v_{n}])}{n} \int_{{\mathbb{R}}} \sum_{t=1}^{n-[v_{n}]} \left\vert {\mathbb{K}}\left(u + \frac{{\varepsilon}_{t} - \widehat{{\varepsilon}}_{t}}{h_{n}} \right) - {\mathbb{K}}(u) \right\vert\, {\mathrm{d}}u \\
& \leq & \frac{2\, B\, \Vert a \Vert_{\infty}\, \Vert {\mathbb{K}}\Vert_{\infty}\, (n-[v_{n}])}{n\, h_{n}} \sum_{t=1}^{n-[v_{n}]} \left\vert {\varepsilon}_{t} - \widehat{{\varepsilon}}_{t} \right\vert \\
& = & \frac{2\, B\, \Vert a \Vert_{\infty}\, \Vert {\mathbb{K}}\Vert_{\infty}\, (n-[v_{n}])}{n\, h_{n}} \sum_{t=1}^{n-[v_{n}]} \big\vert ({\widehat{\theta}_{n}}- \theta){^{T}}\, \Phi_{t-1} \big\vert \\
& = & \frac{2\, B\, \Vert a \Vert_{\infty}\, \Vert {\mathbb{K}}\Vert_{\infty}\, (n-[v_{n}])}{n\, h_{n}} \sum_{t=1}^{n-[v_{n}]} \big\vert ({\widehat{\theta}_{n}}- \theta){^{T}}\, C_{\theta}^{n}\, C_{\theta}^{-[v_{n}]}\, C_{\theta}^{-n+[v_{n}]}\, \Phi_{t-1} \big\vert. \\\end{aligned}$$ Here we recall that $\vert \lambda_{p} \vert > 1$. Consequently, $\rho(C_{\theta}^{-1}) = 1/\vert \lambda_{p} \vert < 1$. It follows that there exists a matrix norm $\Vert \cdot \Vert_{*} = \sup(\vert \cdot\, u \vert_{*}\,;\, u \in {\mathbb{C}}^{p},\, \vert u \vert_{*} = 1)$ satisfying $\Vert C_{\theta}^{-1} \Vert_{*} < 1$ (see *e.g.* [@Duflo97 Prop. 2.3.15]), and a constant $k^{*}$ such that, with $C^{*} = 2\, B\, k^{*}\, \Vert a \Vert_{\infty}\, \Vert {\mathbb{K}}\Vert_{\infty}$, we obtain $$\begin{aligned}
I_{n, 1} & \leq & \frac{C^{*}\, (n-[v_{n}])}{n\, h_{n}} \sum_{t=1}^{n-[v_{n}]} \big\vert ({\widehat{\theta}_{n}}- \theta){^{T}}\, C_{\theta}^{n} \big\vert_{*} ~ \big\vert C_{\theta}^{-[v_{n}]}\, C_{\theta}^{-n+[v_{n}]}\, \Phi_{t-1} \big\vert_{*} \\
& \leq & \frac{C^{*}\, (n-[v_{n}])}{n\, h_{n}} ~ \big\vert ({\widehat{\theta}_{n}}- \theta){^{T}}\, C_{\theta}^{n} \big\vert_{*} ~ \Vert C_{\theta}^{-1} \Vert_{*}^{[v_{n}]} \sum_{t=1}^{n-[v_{n}]} \big\vert C_{\theta}^{-n+[v_{n}]}\, \Phi_{t-1} \big\vert_{*} \\
& \leq & \frac{C^{*}\, (n-[v_{n}])}{n\, h_{n}} ~ \big\vert ({\widehat{\theta}_{n}}- \theta){^{T}}\, C_{\theta}^{n} \big\vert_{*} ~ \Vert C_{\theta}^{-1} \Vert_{*}^{[v_{n}]} \sup_{0\, \leq\, t\, \leq\, n-[v_{n}]-1} \big\vert C_{\theta}^{-t}\, \Phi_{t} \big\vert_{*} \sum_{\ell=1}^{n-[v_{n}]} \Vert C_{\theta}^{-1} \Vert_{*}^{\ell}.\end{aligned}$$ But Proposition \[PropExplo\] ensures that $\sup_{t} \vert C_{\theta}^{-t}\, \Phi_{t} \vert_{*}$ is a.s. bounded for a sufficiently large $n$, and we also have $$\begin{aligned}
({\widehat{\theta}_{n}}- \theta){^{T}}\, C_{\theta}^{n} & = & \left( \sum_{t=1}^n \Phi_{t-1}{^{T}}\, {\varepsilon}_{t} \right) ( C_{\theta}^{-n} ){^{T}}~ ( C_{\theta}^{n} ){^{T}}\left(\sum_{t=1}^n \Phi_{t-1} \Phi_{t-1}{^{T}}\right)^{\! -1} C_{\theta}^{n} \\
& = & \sum_{t=1}^n ( C_{\theta}^{-n}\, \Phi_{t-1}\, {\varepsilon}_{t} ){^{T}}G_{n}^{-1}\end{aligned}$$ where $$G_{n} = C_{\theta}^{-n} \left(\sum_{t=1}^n \Phi_{t-1} \Phi_{t-1}{^{T}}\right) ( C_{\theta}^{-n} ){^{T}}.$$ We deduce that, for some constant $k^{*}$, $$\big\vert ({\widehat{\theta}_{n}}- \theta){^{T}}\, C_{\theta}^{n} \big\vert_{*} \leq k^{*}\, {\varepsilon}_{n}^{\sharp}\, \Vert G_{n}^{-1} \Vert_{*} \sup_{0\, \leq\, t\, \leq\, n-1} \big\vert (C_{\theta}^{-t}\, \Phi_{t}){^{T}}\big\vert_{*} \sum_{\ell=1}^{n} \Vert C_{\theta}^{-1} \Vert_{*}^{\ell}$$ where [@Duflo97 Cor. 1.3.21] shows that ${\varepsilon}_{n}^{\sharp} = \sup_{t} \vert {\varepsilon}_{t} \vert = o(\sqrt{n})$ a.s. under our conditions of moments on $({\varepsilon}_{t})$. Proposition \[PropExplo\] and the fact that $\Vert C_{\theta}^{-1} \Vert_{*} < 1$ lead to $$\label{DecompExplo1}
I_{n, 1} = o\left( \frac{n - [v_{n}]}{\sqrt{n}\, h_{n}} ~ R^{\, [v_{n}]} \right) {\hspace{0.3cm} \textnormal{a.s.}}$$ for some $0 < R < 1$. Let us now turn to $I_{n, 2}$ for which the same strategy gives $$\begin{aligned}
I_{2,n} & \leq & \frac{[v_{n}]}{n\, h_{n}} ~ \sum_{t=n-[v_{n}]+1}^{n} \int_{{\mathbb{R}}} \left[ {\mathbb{K}}\left( \frac{x - \widehat{{\varepsilon}}_{t}}{h_{n}} \right) - {\mathbb{K}}\left( \frac{x - {\varepsilon}_{t}}{h_{n}} \right) \right]^2 a(x)\, {\mathrm{d}}x \\
& \leq & \frac{2\, [v_{n}]}{n\, h_{n}} ~ \sum_{t=n-[v_{n}]+1}^{n} \int_{{\mathbb{R}}} \left[ {\mathbb{K}}^2\!\left( \frac{x - \widehat{{\varepsilon}}_{t}}{h_{n}} \right) + {\mathbb{K}}^2\!\left( \frac{x - {\varepsilon}_{t}}{h_{n}} \right) \right] a(x)\, {\mathrm{d}}x \\
& \leq & \frac{4\, \Vert a \Vert_{\infty}\, [v_{n}]}{n} ~ \sum_{t=n-[v_{n}]+1}^{n} \int_{{\mathbb{R}}} {\mathbb{K}}^2(u)\, {\mathrm{d}}u \\
& = & \frac{4\, \Vert a \Vert_{\infty}\, [v_{n}]^2}{n} ~ \int_{{\mathbb{R}}} {\mathbb{K}}^2(u)\, {\mathrm{d}}u.\end{aligned}$$ Thus, $$\label{DecompExplo2}
I_{n, 2} = O\left( \frac{[v_{n}]^2}{n} \right) {\hspace{0.3cm} \textnormal{a.s.}}$$ Finally, from , and , we have $$\label{DecompExploFin}
I_{n} = O\left( \frac{n - [v_{n}]}{\sqrt{n}\, h_{n}} ~ R^{\, [v_{n}]} + \frac{[v_{n}]^2}{n} \right) = o(\sqrt{h_{n}}) {\hspace{0.3cm} \textnormal{a.s.}}$$ for some $0 < R < 1$, as a consequence of the properties of $(v_{n})$ in . The cross term $R_{n}$ is treated in the same way as in the proof of Theorem \[ThmStable\]. Indeed, since our choice of $(v_{n})$ also ensures that $I_{n} = o(h_{n})$ a.s., the same reasoning leads to the conclusion. It follows from and that $$\frac{\widehat{T}_{n} - \mu}{\sqrt{h_{n}}} = \frac{T_{n} - \mu}{\sqrt{h_{n}}} + o(1) {\hspace{0.3cm} \textnormal{a.s.}}$$ which ends the first part of the proof. The second part merely consists in noting that still holds.
Proof of Proposition \[PropUnivUnstable\] {#SecProofUnstable}
-----------------------------------------
In this proof, the notation $\Longrightarrow$ refers to the weak convergence of sequences of random elements in $D([0,1])$, the space of right continuous functions on $[0,1]$ having left-hand limits, equipped with the Skorokhod topology (see Billingsley [@Billingsley99]). One can find in Thm. 2.7 of the same reference the statement and the proof of the continous mapping theorem. In Lemmas \[LemUnstable1\] and \[LemUnstable2\] below, $(Y_{n})_{n \geq 1}$ is a sequence of random variables satisfying, for some $\delta > 0$, $$\label{InvPrinc}
\frac{1}{n^{\delta}} \sum_{t\, \in\, {\mathcal{P}}_{[n\boldsymbol{\cdot}]}} Y_{t} {~ \Longrightarrow ~}L_{{\mathcal{P}}}(\boldsymbol{\cdot}) {\hspace{0.5cm}}\text{and} {\hspace{0.5cm}}\frac{1}{n^{\delta}} \sum_{t=1}^{[n\boldsymbol{\cdot}]} (-1)^{[n\boldsymbol{\cdot}]-t}\, Y_{t} {~ \Longrightarrow ~}\Lambda(\boldsymbol{\cdot})$$ where $L_{{\mathcal{P}}}$ and $\Lambda$ are random paths in $D([0,1])$ and where ${\mathcal{P}}_{[n\boldsymbol{\cdot}]}$ stands either for $\{ 1, \hdots, [n\boldsymbol{\cdot}] \}$ or for the sets of even/odd integers in $\{ 1, \hdots, [n\boldsymbol{\cdot}] \}$. We use the convention $\sum_{\varnothing} = 0$ and the notation $[x]$ to designate the integer part of any $x \geq 0$.
The first convergence obviously holds for a strong white noise having a finite variance and $\delta=1/2$, *via* Donsker theorem [@Billingsley99 Thm. 8.2]. In this case, $L_{{\mathcal{P}}}$ is a Wiener process and ${\mathcal{P}}_{[n\boldsymbol{\cdot}]}$ changes its variance (half as many terms leads to a doubled variance). If the distribution is symmetric, then $\Lambda \equiv L_{{\mathcal{P}}}$ for ${\mathcal{P}}_{[n\boldsymbol{\cdot}]} = \{ 1, \hdots, [n\boldsymbol{\cdot}] \}$, whereas an invariance principle for stationary processes (see, *e.g.*, [@DedeckerRio00 Thm. 1]) enables to identify $\Lambda$ in the nonsymmetric case.
\[LemUnstable1\] Let $X_0 = 0$ and, for $1 \leq t \leq n$, consider $$X_{t} = \theta\, X_{t-1} + Y_{t}, {\hspace{0.5cm}}\theta = \pm 1,$$ where $(Y_{t})$ satisfies . Then, $$\sum_{t=1}^{n} X_{t} = \left\{
\begin{array}{ll}
O_{\P}(n^{\delta+1}) & \mbox{for } \theta=1 \\
O_{\P}(n^{\delta}) & \mbox{for } \theta=-1
\end{array}
\right.$$ and, in both cases, for $k \in {\mathbb{N}}$, $$\sum_{t=1}^{n} X_{t}^{\, 2 k} = O_{\P}(n^{2 k \delta+1}).$$
Let $\theta=1$. Then, $$\begin{aligned}
\frac{1}{n^{\delta+1}} \sum_{t=1}^{n} X_{t} & = & \frac{1}{n} \sum_{t=1}^{n} \frac{1}{n^{\delta}} \sum_{s=1}^{t} Y_{s} \\
& = & \sum_{t=1}^{n} \int_{\frac{t}{n}}^{\frac{t+1}{n}} \frac{1}{n^{\delta}} \sum_{s=1}^{[n u]} Y_{s}\, {\mathrm{d}}u ~ {~ \overset{{\mathcal{D}}}{\longrightarrow} ~}~ \int_{0}^{1} L(u)\, {\mathrm{d}}u\end{aligned}$$ from the continuous mapping theorem, where $L \equiv L_{{\mathcal{P}}}$ for ${\mathcal{P}}_{[n\boldsymbol{\cdot}]} = \{ 1, \hdots, [n\boldsymbol{\cdot}] \}$. Following the same lines, $$\begin{aligned}
\frac{1}{n^{2 k \delta+1}} \sum_{t=1}^{n} X_{t}^{\, 2 k} & = & \frac{1}{n} \sum_{t=1}^{n} \left( \frac{1}{n^{\delta}} \sum_{s=1}^{t} Y_{s} \right)^{\! 2 k} \\
& = & \sum_{t=1}^{n} \int_{\frac{t}{n}}^{\frac{t+1}{n}} \left( \frac{1}{n^{\delta}} \sum_{s=1}^{[n u]} Y_{s} \right)^{\! 2 k} {\mathrm{d}}u ~ {~ \overset{{\mathcal{D}}}{\longrightarrow} ~}~ \int_{0}^{1} L^{2 k}(u)\, {\mathrm{d}}u.\end{aligned}$$ Now let $\theta=-1$. Then, $$\sum_{t=1}^{n} X_{t} = \sum_{t\, \in\, {\mathcal{P}}_{n}} Y_{t}$$ where ${\mathcal{P}}_{n} \subset \{ 1, \hdots, n \}$ is the set of even integers (resp. odd integers) between 1 and $n$, for an even (resp. odd) $n$. Then we conclude to the first result. Finally, $$\begin{aligned}
\frac{1}{n^{2 k \delta+1}} \sum_{t=1}^{n} X_{t}^{\, 2 k} & = & \frac{1}{n} \sum_{t=1}^{n} \left( \frac{1}{n^{\delta}} \sum_{s=1}^{t} (-1)^{t-s}\, Y_{s} \right)^{\! 2 k} \\
& = & \sum_{t=1}^{n} \int_{\frac{t}{n}}^{\frac{t+1}{n}} \left( \frac{1}{n^{\delta}} \sum_{s=1}^{[n u]} (-1)^{[n u]-s}\, Y_{s} \right)^{\! 2 k} {\mathrm{d}}u ~ {~ \overset{{\mathcal{D}}}{\longrightarrow} ~}~ \int_{0}^{1} \Lambda^{2 k}(u)\, {\mathrm{d}}u.\end{aligned}$$
**Proof of Proposition \[PropUnstable\] – Seasonal case**. Let $X_{-s+1} = \hdots = X_0 = 0$ and ${\mathcal{S}}_{k} = \{ \ell \in \{ 1, \hdots, n \},\, \ell[s] = k \}$, for $k \in \{ 0, \hdots, s-1 \}$, the notation $\ell[s]$ refering to the remainder of the Euclidean division of $\ell$ by $s$. Then the subsets ${\mathcal{S}}_{k}$ form a partition of $\{ 1, \hdots, n \}$, the path $(X_1, \hdots, X_{n})$ is made of $s$ uncorrelated random walks and Donsker theorem directly gives the $O_{\P}(\sqrt{n})$ rate for $\vert X_{n} \vert$. In addition, $$\sum_{t=1}^{n} X_{t}^{\, a} = \sum_{k=0}^{s-1} \sum_{t\, \in\, {\mathcal{S}}_{k}} X_{t}^{\, a}$$ so that, for $a \in \{1, 2, 4 \}$, the results follow from Lemma \[LemUnstable1\]. $\square$\
We now turn to the proof of Proposition \[PropUnivUnstable\]. Following the idea of [@LeeNa02], we make once again the decomposition $$\label{DecompUnstable}
\frac{\widehat{T}_{n} - \mu}{\sqrt{h_{n}}} = \frac{I_{n}}{\sqrt{h_{n}}} + \frac{R_{n}}{\sqrt{h_{n}}} + \frac{T_{n} - \mu}{\sqrt{h_{n}}}$$ where $\mu$ is the centering term of the statistic, $I_{n}$ is given in , $T_{n}$ is the original Bickel-Rosenblatt statistic and $R_{n}$ is the cross term in such that $$\label{CvgRnUnstable}
\frac{\vert R_{n} \vert}{\sqrt{h_{n}}} = O_{\P}\bigg( \sqrt{\frac{I_{n}}{h_{n}}} \bigg)$$ as we have seen earlier. Going on with the decomposition of $I_{n}$ in and using the same notation, we establish the two following lemmas. In all the sequel, the case $p=s$ refers to the seasonal process $\theta_1 = \hdots = \theta_{s-1} = 0$ and $\theta_{s}=1$.
\[LemUnstable3\] When $p=1$ or $p=s$, Lemma \[LemStable2\] still holds in the unstable case.
First, with $p=1$ and $\theta = \pm 1$, we note that $${\mathbb{E}}[X_{n}^2] = \sum_{1\, \leq\, t, s\, \leq\, n} \theta^{\, 2n - t-s}\, {\mathbb{E}}[{\varepsilon}_{t}\, {\varepsilon}_{s}] = O(n)$$ and $${\mathbb{E}}[X_{n}^4] = \sum_{1\, \leq\, t, s, u, v\, \leq\, n} \theta^{\, 4n - t-s-u-v}\, {\mathbb{E}}[{\varepsilon}_{t}\, {\varepsilon}_{s}\, {\varepsilon}_{u}\, {\varepsilon}_{v}] = O(n^2)$$ under our assumptions on the noise, and the same results obviously hold for $p=s$ where $X_{n}$ still has a random walk behavior. Thus, $K_{2,n}$ in is $O_{\P}(n^3\, h_{n})$. Lemma \[LemUnstable1\] and the second part of Proposition \[PropUnstable\] also give $$\sum_{t=1}^{n} X_{t}^2 = O_{\P}(n^2) {\hspace{0.5cm}}\text{and} {\hspace{0.5cm}}\sum_{t=1}^{n} X_{t}^4 = O_{\P}(n^3)$$ and, accordingly with Proposition \[PropOLSUnstable\], $K_{1,n}$ in is $O_{\P}(n^3\, h_{n}^{-2})$ and $K_{3,n}$ in is $O_{\P}(n^4\, h_{n}^2)$. Finally, concludes the proof.
\[LemUnstable4\] Let $p=1$. Then, Lemma \[LemStable1\] holds for $\lambda=\theta=-1$. On the contrary, for $\lambda=\theta=1$ and $\int_{{\mathbb{R}}} {\mathbb{K}}^{\prime}(s)\, {\mathrm{d}}s \neq 0$, $$I_{1,n} = O_{\P}(n\, h_{n}^2)$$ whereas for $\int_{{\mathbb{R}}} {\mathbb{K}}^{\prime}(s)\, {\mathrm{d}}s = 0$, $$I_{1,n} = O_{\P}(n\, h_{n}^4).$$ The last two results also hold for $p=s$.
On the one hand, let $\theta=-1$. The reasoning of $\eqref{DecompStableJ1}$ shows that $J_{1,n}$ is still $O_{\P}(h_{n})$, using Proposition \[PropOLSUnstable\] and ${\mathbb{E}}[X_{n}^2] = O(n)$. From Lemma \[LemUnstable1\], we know that $$\sum_{t=1}^{n} X_{t} = O_{\P}(\sqrt{n}).$$ Then, $J_{2,n}$ in must be $O_{\P}(n^{-1}\, h_{n}^2)$ and $I_{1,n} = O_{\P}(h_{n})$. On the other hand, we have to investigate the case where $\theta=1$. Since ${\mathbb{E}}[X_{n}^2]$ does not depend on $\theta$, $J_{1,n}$ has exactly the same behavior. For a better readability, let $$L_{n} = \int_{{\mathbb{R}}} \left( \int_{{\mathbb{R}}} {\mathbb{K}}^{\prime}(z)\, f(x - h_{n}\, z)\, {\mathrm{d}}z \right)^{\! 2} a(x)\, {\mathrm{d}}x.$$ Under our assumptions, it is easy to see that $$\label{UnstableLn1}
{\lim_{n\, \rightarrow\, +\infty}}L_{n} = \left(\int_{{\mathbb{R}}} f^2(s)\, a(s)\, {\mathrm{d}}s\right) \left( \int_{{\mathbb{R}}} {\mathbb{K}}^{\prime}(s)\, {\mathrm{d}}s\right)^{\! 2}$$ as soon as $\int_{{\mathbb{R}}} {\mathbb{K}}^{\prime}(s)\, {\mathrm{d}}s \neq 0$, and that $$\label{UnstableLn2}
{\lim_{n\, \rightarrow\, +\infty}}\frac{L_{n}}{h_{n}^{2}} = \left(\int_{{\mathbb{R}}} f^{\prime}(s)^2\, a(s)\, {\mathrm{d}}s\right) \left( \int_{{\mathbb{R}}} s\, {\mathbb{K}}^{\prime}(s)\, {\mathrm{d}}s\right)^{\! 2}$$ as soon as $\int_{{\mathbb{R}}} {\mathbb{K}}^{\prime}(s)\, {\mathrm{d}}s = 0$. Lemma \[LemUnstable1\] also gives $$\sum_{t=1}^{n} X_{t} = O_{\P}(n^{3/2})$$ which shows that $J_{2,n}$ is $O_{\P}(n\, h_{n}^2)$ or $O_{\P}(n\, h_{n}^4)$, depending on $\int_{{\mathbb{R}}} {\mathbb{K}}^{\prime}(s)\, {\mathrm{d}}s$. The latter reasoning still applies for $p=s$, *via* the second part of Proposition \[PropUnstable\]. The proof is achieved using and the hypotheses of Proposition \[PropUnivUnstable\].
The combination of Lemmas \[LemUnstable3\] and \[LemUnstable4\] is sufficient to establish that $$\frac{\widehat{T}_n - \mu}{\sqrt{h_{n}}} {~ \overset{{\mathcal{D}}}{\longrightarrow} ~}{\mathcal{N}}(0, \tau^2)$$ holds for $\theta=-1$, despite the instability, and replacing $\widehat{T}_{n}$ by $\widetilde{T}_{n}$ is possible without disturbing the asymptotic normality, as a consequence of for $n\, h_{n}^{9/2} \rightarrow 0$. When $\theta=1$ or $p=s$ and $\int_{{\mathbb{R}}} {\mathbb{K}}^{\prime}(s)\, {\mathrm{d}}s = 0$, Lemmas \[LemUnstable3\] and \[LemUnstable4\] show that $I_{n} = O_{\P}(h_{n}) = o_{\P}(\sqrt{h_{n}})$. It follows from and that $$\frac{\vert R_{n} \vert}{\sqrt{h_{n}}} = O_{\P}(1)$$ which unfortunately prevents us from concluding to the Bickel-Rosenblatt convergence in this case. However, we still have the order of magnitude $$\frac{\widehat{T}_n - \mu}{\sqrt{h_{n}}} = O_{\P}(1) {\hspace{0.5cm}}\text{and} {\hspace{0.5cm}}\frac{\widetilde{T}_n - \mu}{\sqrt{h_{n}}} = O_{\P}(1).$$ Finally for $\int_{{\mathbb{R}}} {\mathbb{K}}^{\prime}(s)\, {\mathrm{d}}s \neq 0$, the same lines also imply, together with , Lemmas \[LemUnstable3\] and \[LemUnstable4\] and the asymptotic normality , that $$h_{n} (\widehat{T}_{n} - \mu) = O_{\P}(1).$$ It remains to study the asymptotic behavior of the correctly renormalized statistic using the explicit expression of $J_{2,n}$ in . From the proof of Lemma \[LemUnstable1\], Proposition \[PropOLSUnstable\] and the continuous mapping theorem, we get for the univariate unstable case with $\theta=1$, $$\big( n\, ({\widehat{\theta}_{n}}- \theta ) \big)^2 \left( \frac{1}{n^{3/2}} \sum_{t=1}^{n} X_{t-1} \right)^{\! 2} {~ \overset{{\mathcal{D}}}{\longrightarrow} ~}\sigma^2 \left( \frac{\frac{1}{2}\, (W^2(1)-1)}{\int_0^1 W^2(u)\, {\mathrm{d}}u}\, \int_0^1 W(u)\, {\mathrm{d}}u \right)^{\! 2}$$ where $(W(t),\, t \in [0,1])$ is a standard Wiener process. Now for the seasonal case, reusing the same notation as above, $$\label{SeasUnstable}
\frac{1}{n^{3/2}} \sum_{t=1}^{n} X_{t} = \sum_{k=0}^{s-1} \frac{1}{n^{3/2}} \sum_{t\, \in\, {\mathcal{S}}_{k}} X_{t} {~ \overset{{\mathcal{D}}}{\longrightarrow} ~}\sigma\, \sum_{k=0}^{s-1} \int_0^\frac{1}{s} W_{s}^{(k+1)}(u)\, {\mathrm{d}}u$$ where $W_{s}^{(i)}$ stands for the $i$–th component of the standard Wiener process $(W_{s}(t),\, t \in [0,1])$ of dimension $s$, since the path can be seen as the concatenation of $s$ uncorrelated random walks. Using the Cramèr-Wold and the continuous mapping theorems, it follows that $$\left( n\, ({\widehat{\theta}_{n}}- \theta ){^{T}}\, \frac{1}{n^{3/2}} \sum_{t=1}^{n} \Phi_{t-1} \right)^{\! 2} {~ \overset{{\mathcal{D}}}{\longrightarrow} ~}\sigma^2 \big( S(W_{s})^{T}\, H(W_{s}) \big)^2$$ with the notation of Proposition \[PropOLSUnstable\], where $H(W_{s})$ is a random vector of size $s$ containing the limit variable on each component. The proof is then ended using the limiting behavior of $L_n$.
To go beyond, note that for $\theta=1$ and $\int_{{\mathbb{R}}} {\mathbb{K}}^{\prime}(s)\, {\mathrm{d}}s \neq 0$, we get $$h_{n}\, \vert \widehat{T}_{n} - \widetilde{T}_{n} \vert \leq 2\, h_{n}\,\sqrt{\widehat{T}_{n}\, U_{n}} + h_{n}\, U_{n}$$ where $$U_{n} = n\, h_{n} \int_{{\mathbb{R}}} \big( ({\mathbb{K}}_{h_{n}} * f)(x) - f(x) \big)^2\, a(x)\, {\mathrm{d}}x.$$ A straightforward calculation shows that $U_{n} = O(n\, h_{n}^3)$ if $\int_{{\mathbb{R}}} s\, {\mathbb{K}}(s)\, {\mathrm{d}}s \neq 0$ whereas $U_{n} = O(n\, h_{n}^5)$ if $\int_{{\mathbb{R}}} s\, {\mathbb{K}}(s)\, {\mathrm{d}}s = 0$. In the second case, $\widehat{T}_{n}$ may be replaced by $\widetilde{T}_{n}$ as soon as $n\, h_{n}^6 \rightarrow 0$. But it the first case, one needs $n\, h_{n}^4 \rightarrow 0$ which contradicts Assumption (A$_3$) with $\alpha=4$.
This last lemma is not used as part of this paper, but it may be a trail for future studies. It is related to the conclusion of Section \[SecBR\] and illustrates the phenomenon of compensation in purely unstable processes.
\[LemUnstable2\] Let $X_{-1} = X_0 = 0$ and, for $1 \leq t \leq n$, consider $$X_{t} = 2 \cos \theta\, X_{t-1} - X_{t-2} + Y_{t}, {\hspace{0.5cm}}0 \leq \theta \leq \pi,$$ where $(Y_{t})$ satisfies . Then, $$\sum_{t=1}^{n} X_{t} = \left\{
\begin{array}{ll}
O_{\P}(n^{\delta+2}) & \mbox{for } \theta=0 \\
O_{\P}(n^{\delta}) & \mbox{for } 0 < \theta \leq \pi.
\end{array}
\right.$$ Moreover, if the process is generated by $$X_{t} = X_{t-2} + Y_{t},$$ then $$\sum_{t=1}^{n} X_{t} = O_{\P}(n^{\delta+1}).$$
For $\theta = 0$, let $Z_{t} = X_{t}-X_{t-1}$. Then, $Z_{t}-Z_{t-1} = Y_{t}$ and from Lemma \[LemUnstable1\], $$\sum_{t=1}^{n} Z_{t} = O_{\P}(n^{\delta+1}).$$ A second application of Lemma \[LemUnstable1\] gives the result. If $0 < \theta \leq \pi$, we obtain $$\frac{2\, (1 - \cos \theta)}{n^{\delta}}\, \sum_{t=1}^{n} X_{t} = \frac{X_{n-1} + (1-2 \cos \theta)\, X_{n}}{n^{\delta}} + \frac{1}{n^{\delta}} \sum_{t=1}^{n} Y_{t}.$$ In the last case, let $Z_{t} = X_{t}-X_{t-1}$ and note that $Z_{t} = -Z_{t-1} + Y_{t}$. Then, Lemma \[LemUnstable1\] gives $$\sum_{t=1}^{n} Z_{t} = O_{\P}(n^{\delta}).$$ We conclude by applying again Lemma \[LemUnstable1\] to the relation $X_{t} = X_{t-1} + Z_{t}$.
In the previous lemma, $\theta=0$ corresponds to the unit roots $\{ 1, 1 \}$ in an AR(2) process generated by $(Y_{t})$ whereas $0 < \theta \leq \pi$ corresponds to the unit roots $\{ {\mathrm{e}}^{{\mathrm{i}}\, \theta}, {\mathrm{e}}^{-{\mathrm{i}}\, \theta} \}$, including $\{-1,-1\}$ for $\theta=\pi$. The last case corresponds to the unit roots $\{ -1,1 \}$, in other words this is the seasonal model for $s=2$.
[^1]:
|
---
abstract: 'Knot colorings are one of the simplest ways to distinguish knots, dating back to Reidemeister, and popularized by Fox. In this mostly expository article, we discuss knot invariants like colorability, knot determinant and number of colorings, and how these can be computed from either the coloring matrix or the Goeritz matrix. We give an elementary approach to this equivalence, without using any algebraic topology. We also compute knot determinant, nullity of pretzel knots with arbitrarily many twist regions.'
address: |
School of Mathematics\
Georgia Institute of Technology\
Atlanta, GA 30332, USA
author:
- Sudipta Kolay
title: 'Knot Colorings: Coloring and Goeritz matrices'
---
Introduction
============
The purpose of this note is to give an exposition of knot colorings through coloring and Goeritz matrices, and to discuss of $n$-colorings of pretzel knots. Although the coloring and Goeritz matrices can be combinatorially defined given a diagram of the knot, the classical proofs of this equality of their determinants, and related results in the literature [@L Chapter 9] require some background on the part of reader. We give an elementary proof of these results without using any algebraic topology. Our approach here is somewhat similar to that of [@T] (building on [@C]), although more elementary. While most of this article is expository, the results (and our methods of computation) in the last section for the determinant and number of colorings of pretzel knots with arbitrarily many twist regions are new. It is hoped that this article would be helpful to people at all levels interested in learning about knots and knot colorings.
A knot is a smooth embedding of a circle $S^1$ in three-space $\mathbb{R}^3$. Here are two examples:
![The trefoil and the figure eight knots[]{data-label="knotegs"}](knoteg.pdf){width="8"}
The fundamental question in knot theory is given two knots, can we tell them apart? One of the simplest ways of telling knots apart is *tricolorability*, whether we can non-trivially (i.e. using at least two colors) color the knot with three colors so that at any crossing the three strands coming together has either all the same colors, or three different colors. This is the simplest invariant which tells the above two knots, the trefoil and the figure eight knot, are different (not isotopic, meaning one cannot be deformed into another).
The notion of tricolorability goes back at least to Reidemeister, at the beginning of the nineteenth century. Ralph Fox generalized this notion to $n$-colorability, and popularized this way of studying knots. The property of being $n$-colorable (and related invariants, like total number of $n$-colorings) is a fairly powerful knot invariant, which can be developed with minimal amount of mathematical machinery, namely linear algebra.
There is a related integer valued knot invariant called the determinant of a knot whose divisors $n$ are exactly those for which the given knot is $n$-colorable. It is well known that this invariant determinant is the absolute value of the determinant of any coloring matrix of the knot and it is also the determinant of any Goeritz matrix for the knot. We will give an elementary proof of this result in Sections 6 and 7, after reviewing necessary background materials up to Section 5. We will use ideas from Section 6 to set up an equivalent linear system for pretzel knots in the final section.
*Acknowledgements*. The author would thanks John Etnyre for making helpful comments on earlier drafts of this paper. This work is partially supported by NSF grants DMS-1608684 and DMS-1906414.
Knots and links
===============
A link is a smooth embedding (or in other words an injective map which is differentiable at each point) of disjoint union of circles in $\mathbb{R}^3$ (so a knot is just a link with one component). Given any such embedding, we can orthogonally project everything with respect to the $z$-axis, and we get a projection of the knot in the $xy$-plane, which we will call the link diagram. Generically[^1], we may assume that the only non-embedded points in the diagram are isolated double points. The advantage of knot diagrams is that it is much easier to draw on a piece of paper and manipulate it. In fact, all the examples of knots and links given here (or any other paper based format) is a diagram. While it is clear that given any link in $\mathbb{R}^3$ we have a diagram for it, one may wonder if we can say when two diagrams represent the same knot. The answer is yes, as proven by Alexander-Briggs [@AB] and independently Reidemeister [@R].
Any two knot diagrams of the same link, up to planar isotopy, can be related by a sequence of the following three moves (called the Reidemeister moves). Each move operates on a small region of the diagram and is one of three types (see Figure \[RM\]):
1. Twist (or untiwst) a strand.
2. Isotope a strand over (or under) another strand.
3. Isotope a strand completely over (or under) a crossing.
![Reidemeister moves[]{data-label="RM"}](rm.pdf){width="14"}
The above result shows it is enough to understand links diagrams to understand links. Moreover, if we can define an object from a diagram, and show it remains unchanged under any of the Reidemeister moves, it follows that it is in fact an invariant of the link (and not the diagram).
Coloring of knots and links
===========================
Now we are ready to introduce $n$-colorings of links. Given any knot diagram $D$, an $n$-**coloring** of a link is is an assignment to each strand an element of $\mathbb{Z}/n\mathbb{Z}$, such that whenever we have a crossing with the overstrand associated to $y$, and the understrands associated to $x$ or $z$ as illustrated by Figure \[kcolor\], we have $x+z=2y$ in $\mathbb{Z}/n\mathbb{Z}$.
![Coloring equation: $2y=x+z$.[]{data-label="kcolor"}](kcol.pdf){width="6"}
Show that given any knot diagram with such a coloring, if a Reidemeister move is applied the new diagram has a unique coloring which is the same outside the small region where the move was applied. Conclude that an $n$-coloring is in fact an invariant of the link (and not just the diagram).
If $n$ is small it is customary to color the strands with $n$ distinct colors, and this is where the name colorability comes from. For example when $n=3$, we use three colors customarily red, blue and green; and $3$-colorability is also known as tricolorability.
![Examples of a 3-coloring and a 5-coloring[]{data-label="CE"}](coleg.pdf){width="9"}
When $n$ is large, it is not practical to use different colors, and we simply use the labelling (perhaps this invariant could also have been called $n$-labelling) on each of the strands. Note that if we color (or label) each strand by constant element $s\in\mathbb{Z}/n\mathbb{Z}$, then all the above constraints are trivially satisfied. Such a coloring is called a *trivial* coloring, and any link has $n$ such trivial $n$-colorings. Hence we will say a link is $n$-colorable if it has a non-trivial $n$-coloring (because otherwise every link has a coloring and this notion would not be able to distinguish between links). We will see in the next section that the set of all $n$-colorings (including the trivial ones) has additional structure, and this is also an invariant (see Exercise \[ex2\]). Had we removed the trivial colorings from this collection, the remaining set does not canonically get such a structure.
For the rest of this article, we will restrict to the case of knots, because some of the statements that follow would get very technical otherwise. The technical issues come from the fact that one has more trivial colorings for split links (i.e. links which can be split apart to lie inside disjoint three balls in $\mathbb{R}^3$), and to get appropriate count of number of different colorings we need to divide out by the number of trivial colorings. All of the statements hold for non-split links (i.e. links which not split links). If we had a split link, we could separate them into different regions and work out coloring invariants for each of the separate non-split links. This would suggest that there is no loss of generality to restricting to the case of non-split links, except for the caveat that given a link diagram, it is not easy to determine if it is a diagram of a split link or not; or if it is split, how many components it has.
Coloring matrix
===============
Notice that the constraints we had at each crossing for $n$-colorings is a linear equation, which suggests that we can consider all such equations together and use tools from linear algebra to gain a better understanding of what is going on.
Given a knot diagram $D$ with $c$ crossings, we have exactly $c$ strands (with the only exception being the standard diagram of the unknot, which has one strand and no crossings). For each strand let us assign a variable $x_i$, and for each crossing we can write down an equation $x_i+x_k=2 x_j$, which has to be satisfied if the $x_i$’s gave rise to a valid coloring. If we consider the set of linear equations, and write out the matrix form, we get $$\tilde{C}\vec x=\vec 0,$$ where $\tilde{C}$ is a $c\times c$ matrix, and $\vec x$ is a column vector with $i$-th entry $x_i$. We define $\tilde{C}$ to be the *pre-coloring matrix* for the diagram $D$.
![Coloring equations for the figure eight knot.[]{data-label="fig8"}](colm.pdf){width="12"}
\[eg1\] In this example we work out the pre-coloring matrix of the figure eight knot, see Figure \[fig8\]. The system of equations we obtain are: $$\spalignsys{x_1-2x_2+x_3=0;-2x_1+x_2+x_4=0;x_1-2x_3+x_4=0;x_2+x_3-2x_4=0}$$ Hence the pre-coloring matrix $\tilde{C}$ equals $$\spalignmat{1 -2 1 0; -2 1 0 1; 1 0 -2 1; 0 1 1 -2}.$$ Notice that we made some choices, we could have labelled the strands in a different order, or permuted the equations in the linear system, and we would have obtained a different pre-coloring matrix.
We observe that the set of all possible $n$-colorings is nothing but the solution space of the above equation reduced modulo $n$, i.e. the Null Space of the pre-coloring matrix, and as such gets the structure of a vector subspace.
\[ex1\] Go back to your solution of Exercise \[ex1\] and check that the linear structure of the set of all $n$-colorings is preserved under each of the Reidemeister moves. Conclude that the solution space is an invariant of the knot.
We note that the sum of the columns of these matrices are $\vec 0$, and this is in fact true more generally (possibly with a signed sum, depending on if we write the equation as $x_i+x_k-2 x_j=0$ or $-x_i-x_k+2 x_j=0$). In other words the constant vector $\vec s$ (with all entries $s$) is a solution to the system $\tilde{C}\vec x=\vec 0,$ and we will call such solutions *semitrivial*. Hence if $\vec x$ is any solution, $\vec x+ \vec s$ is again a solution for any scalar $s$. We can fix this redundancy, by requiring one of the $x_i$’s is zero. Since the pre-coloring matrix always has a non-trivial solution, it is singular, we know that there is some linear dependence among the rows as well. So let us delete some row and some column from the pre-coloring matrix, and we will call this resulting square matrix $C$ *a* **coloring matrix** for the diagram $D$. While the matrix $C$ is not an invariant (even for the knot diagram $D$), it turns out the absolute value of the determinant of $C$ is an invariant of the knot (not just a diagram!), which we will call *the* **determinant** of the knot.\
We digress to some linear algebra to see why various cofactors of pre-coloring matrices are the same. Recall, for any square matrix $A$, the $(i,j)$ cofactor $C_{i,j}$ is $(-1)^{ij}$ times the determinant of the matrix $A_{i,j}$ obtained by removing the $i$-th row and $j$-th column from A.\
\[prop1\] Suppose $A$ is an $n\times n$ matrix with real entries (or more generally with entries in some commutative ring) so that sum of all the columns is the column vector of all zeros, and sum of all the rows is the row vector of all zeros (equivalently, the sum of entries of each row and each column add up to zero). Then all the $(i,j)$ cofactors of $A$ are the same.
\[ex2\]
Work out the following outline to prove the above proposition:
1. For any such matrix $A$, show that you can reconstruct $A$ if you know the submatrix $A_{i,j}$, and the indices $i,j$.
2. Using $\vec R_1+...+\vec R_n=\vec 0$, and properties of the determinant, relate $C_{1,1}$ and $C_{1,2}$.
3. Relate $C_{1,1}$ with $C_{1,j}$.
4. Relate $C_{1,1}$ with $C_{i,j}$.
In order to use this result from linear algebra, we need to know that for an arbitrary knot diagram we can choose a pre-coloring matrix in the specified form, just as we saw in the example.
\[ex3\] Given any knot diagram $D$, show that one can number the crossings and strands in such a way so that hypothesis of Proposition \[prop1\] holds.
Another approach is to find a modified version of Proposition \[prop1\] which can be applied to any pre-coloring matrix.
\[ex4\] Suppose $A$ is an $n\times n$ matrix with real entries so that a linear combination (with each weight being $\pm 1$) of every column is the column vector of all zeros, a linear combination (with each weight being $\pm 1$) of every row is the row vector of all zeros. Then all the absolute values of cofactors (or equivalently, minors) of $A$ are the same.
The aforementioned results from linear algebra tell us that the determinant is indeed an invariant of the diagram, but we still need to justify that it does not depend on the diagram. One way (which is commonly used in the literature) to go about this is to show that the knot determinant is invariant under the Reidemeister moves. We will take a slightly different approach here, by invoking the fact (as seen in the last section) that $n$-colorability is a knot invariant.
\[prop2\] The following are equivalent:
1. A pre-coloring matrix $\tilde{C}$ has a non-semitrivial solution in $\mathbb{Z}/n\mathbb{Z}$.
2. A coloring matrix ${C}$ has a non-trivial solution in $\mathbb{Z}/n\mathbb{Z}$.
3. $\det C=0$ in $\mathbb{Z}/n\mathbb{Z}$.
4. $n$ divides $\det C$.
Suppose $C$ is obtained from $\tilde{C}$ by deleting the $i$-th row and $j$-th column.\
$(i)\Longleftrightarrow(ii)$: Given a solution $\vec x$ of the pre-coloring matrix $\tilde{C}$, we can add a constant vector $\vec s$ to get another solution $\vec y$ of $\tilde{C}$ with $y_j=0$ (note that this operation preserves if the solution vector was semitrivial or not). The column vector obtained by deleting $y_j$ from $\vec y$ is a solution to the coloring matrix $C$, and moreover one can go backwards by adding $0$ to the $j$-th spot. This process describes a bijection between all solutions of $\tilde{C}$ with $j$-th entry zero, and all solutions of $C$, and further the non-semitrivial solutions of $\tilde{C}$ correspond exactly to the non-trivial solutions of $C$.\
$(ii)\Longleftrightarrow(iii)$ is a standard fact in linear algebra; and $(iii)\Longleftrightarrow(iv)$ follows from the definition of the quotient set $\mathbb{Z}/n\mathbb{Z}$.
Suppose $D_1$ and $D_2$ are two diagrams for a knot $K$, with coloring matrices $C_1$ and $C_2$. Then for every $n$ dividing $\det(D_1)$, we see by Proposition 6 that the diagram $D_1$ is $n$-colorable, and hence (since $n$-colorability is a knot invariant) the diagram $D_2$ is $n$-colorable as well, and again by Proposition 6 we have $n$ divides $\det(D_2)$. So we see that $\det(D_1)$ divides $\det(D_2)$, and by symmetry $\det(D_2)$ divides $\det(D_1)$. Thus the absolute values of $\det(D_1)$ and $\det(D_2)$ are the same, so knot determinant is invariant of knot.
The determinant of knot $K$ determines for which $n$, the knot $K$ is $n$-colorable. However the reader may have noticed that in order to find the knot determinant directly from definition, we need to compute the determinant of a matrix of size $(c-1)\times (c-1)$, which in some cases may not be computationally efficient. In the next sections, we will talk about an alternate, easier approach to compute the knot determinant and related invariants.
Goeritz matrix
==============
Given a knot diagram we can checkerboard color the regions, for example see Figure \[GM\]. Suppose we assign to each crossing $c$ the sign $\eta(c)\in\{\pm 1 \}$ according to Figure \[SC\]:
![Signs at crossings. Note that these signs are independent of an orientation on the knot, but will flip if we switch the shaded and unshaded regions in the checkerboard coloring.[]{data-label="SC"}](sign.pdf){width="7"}
Suppose the shaded regions are enumerated $R_1,...,R_k$. Let us define the $k\times k$ pre-Goeritz matrix $\tilde{G}$ for the diagram $D$ by the following: For the off diagonal entries ($i\neq j$), we set $$\tilde{G}_{i,j}:= \sum \eta (c),$$ where the sum ranges over all crossings $c$ where regions $R_i$ and $R_j$ come together. Let us now define the diagonal entries by $$\tilde{G}_{i,i}:=-\sum_{j\neq i}\tilde{G}_{i,j}.$$ We now note that $\tilde{G}$ is a square symmetric matrix whose columns (respectively rows) sum to zero.
![A knot diagram with a checkerboard coloring.[]{data-label="GM"}](gm.pdf){width="10"}
\[eg2\] The pre-Goeritz matrix for the knot diagram in Figure \[GM\] with the given checkerboard coloring is: $$\spalignmat{3 -1 -2;-1 4 -3;-2 -3 5}$$
Just as we saw earlier for the pre-coloring matrices, the determinant of $\tilde{G}$ is zero, however if we form a Goeritz matrix by deleting any row and any column, the absolute value of determinant the resulting matrix is well defined for the diagram, which we will call *the* **Goeritz determinant** of the diagram. We get a similar result regarding the Goeritz matrices as we had for the coloring matrices, and the same proof works.
\[prop3\] Let us pick any pre-Goeritz matrix $\tilde{G}$ for a knot diagram $D$, and obtain a Goeritz matrix $G$ by deleting some row and some column. The following are equivalent:
1. $\tilde{G}$ has a non-semitrivial solution in $\mathbb{Z}/n\mathbb{Z}$.
2. ${G}$ has a non-trivial solution in $\mathbb{Z}/n\mathbb{Z}$.
3. $\det G=0$ in $\mathbb{Z}/n\mathbb{Z}$.
4. $n$ divides $\det G$.
As the reader can probably expect, the Goeritz determinant does not depend on the knot diagram. In fact, we will show it is exactly the same as the determinant of the knot (coming from coloring matrix)! This result also means that determinant of a Goeritz matrix obtained from the shaded regions is same in absolute value to the one coming from the unshaded regions. While it is not too difficult to see that the Goeritz determinant is invariant under the Reidemeister moves, it is not quite clear from this perspective if the determinants corresponding to the unshaded and shaded regions are related, or if they are related to the determinant of a coloring matrix. In the next two sections, we will describe another approach where we will create a bijection between the solution spaces of the pre-coloring and pre-Goeritz matrices, and this will give us the above result about determinants, and a bit more.
Difference
==========
Let us note that if we have a colored knot diagram with a collection of half twists as illustrated in Figure \[df\],
![Difference[]{data-label="df"}](df.pdf){width="12"}
the difference between the colors on top and the bottom is the same for every generic vertical slice (i.e. does not pass through a crossing or [touch]{}[^2] the knot diagram), since the coloring equation is equivalent to $x_i-x_j= x_j-x_k$. In other words, if we drew any generic arc between the regions $R_1$ and $R_2$ in the part of the diagram illustrated above, and took the difference of the colors, then we would get the same value. One may wonder if something similar works for any two crossing-adjacent shaded (or unshaded) regions. Indeed, this will be the case more generally (that is for non adjacent regions, even with different checkerboard colors), once we formulate the right generalization for this difference.
Given a knot diagram $D$ with regions $R_1,...,R_r$ (ignoring checkerboard coloring for now), of a knot $K$ with an $n$-coloring, let us define the **difference** $d(R_i,R_j)\in\mathbb{Z}/n\mathbb{Z}$ between any two regions by taking any generic (i.e. not passing through a crossing, and intersecting $D$ transversely) oriented simple arc between points in the interior of regions $R_i$ and $R_j$, and computing the alternate (beginning with positive) sum of the colors on the strand the arc crosses.
\[wd\] The difference $d(R_i,R_j)$ is well defined, that is it does not depend on the arc we chose to compute the alternating sum.
We sketch two proofs of this well definedness below, after we discuss an example.
\[egdiff\] Let us work out the differences in 5-coloring of the figure eight knot we saw earlier, see Figure \[fig8diff\]. Notice that for the arc $\gamma_1$ between regions $R_1$ and $R_6$, there are three intersection points, and the alternating sum is $2-4+1=-1=4$ in $\mathbb{Z}/5\mathbb{Z}$, where as there is only one intersection for the arc $\gamma_2$ and here the alternating sum in 4. Let us assume Claim \[wd\] for now and find the various values of the differences $d(R_1,R_j)$ for different $j$. $$d(R_1,R_2)=2 ,\quad d(R_1,R_3)=3 ,\quad d(R_1,R_4)=0,\quad d(R_1,R_5)=4,\quad d(R_1,R_6)=4.$$
\[exdiff\] Compute the other differences $d(R_i,R_j)$ for the knot coloring in Figure \[fig8diff\].
![Differences in a coloring of the figure eight knot.[]{data-label="fig8diff"}](diffB.pdf){width="7"}
Note that in order to show this alternating sum is same for any two arcs with same endpoints, it is equivalent to show the alternating sum along any generic closed loop in the plane is zero (concatenate one arc and the other’s reverse). Let us begin by considering a generic simple closed loop $\gamma$ (i.e. no self intersections, it can intersect $D$). We can think of $D$ together with the loop $\gamma$ as being a diagram of a link of two components, one of them being the original knot $K$, and the other knot lying entirely below $K$. The latter is easily seen to be the unknot (as the diagram has no crossings), unlinked from $K$, by the way we chose crossings. Hence we can isotope the unlinked unknot away from the original knot diagram, we can color it arbitrarily by some color. Since $n$-colorability is an invariant of the link, we know that the link diagram we started off with also has an $n$-coloring. Suppose the color of the strand of this new unknot in some region $R_k$ is $c$, if we use the coloring equations check that the color we end up with once we traverse along $\gamma$ is the sum of $c$ and the alternating sum of the colors the $\gamma$ crosses, and so this alternating sum has to be zero. Now if $\gamma$ was not simple, if $\gamma_1$ is a subloop of $\gamma$ which is simple then by our above discussion we see that the alternating sum along $\gamma_1$ is zero. Hence, we can we can start deleting innermost loops from $\gamma$ without changing the alternating sum, and hence we get the alternating sum along any generic closed loop is zero. Note that we can vary the endpoints of the arcs in the same region by concatenating with an arc that does not cross $D$, and hence the alternating sum remains the same. Hence, it follows that $d(R_i,R_j)$ is well defined.
An alternate approach to check well definedness would be to use the fact that any two such arcs are related by the following two moves: going over a crossing, and encountering a birth/death (think about the non-generic instances and then the various ways we can perturb them to get to a generic arc), as illustrated in Figure \[moves\].
![Moves relating a pair of isotopic arcs, when the knot diagram remains fixed.[]{data-label="moves"}](move.pdf){width="14"}
We leave it to the reader to verify that the alternating sum remains the same under these moves.
Let us now consider some checkerboard coloring of the knot diagram. For any triple $i,j,k$; we can concatenate an arc from $R_i$ to $R_j$; and an arc from $R_j$ to $R_k$ to obtain an arc from $R_i$ to $R_j$. It follows that for any triple $i,j,k$, the differences satisfy $$d(R_i,R_j)+d(R_j,R_k)=d(R_i,R_k) \text{ if } R_i \text{ and } R_j \text{ have same checkerboard coloring, and}$$ $$d(R_i,R_j)-d(R_j,R_k)=d(R_i,R_k) \text{ if } R_i \text{ and } R_j \text{ have different checkerboard coloring.}$$ Conversely, if we are given such a collection $\{d(R_i,R_j)\}$ for all pairs of regions satisfying the above relations, then we obtain an $n$-coloring on the knot $K$: pick any two regions adjacent along a strand of the knot, and the difference between the two regions tells us the coloring on the strand. In other words, the data of an $n$-coloring is equivalent to the data of the differences.
Since there are relations among the various $d(R_i.R_j)'s$, there is some redundancy; we note that having $d(R_i.R_j)'s$ for adjacent regions is enough to recover the coloring. In fact, as we will see in the next section, to recover the coloring (up to translation by constant colorings), it is enough to only know the differences $d(R_i.R_j)'s$ among the crossing-adjacent shaded regions; and the nullspace of the (pre-)Goeritz matrix stores the information of the differences in a compact way.\
Bijections between solutions of coloring and Goeritz matrices
=============================================================
In this section we demonstrate a bijection between solutions of the coloring matrix and Goeritz matrix of a diagram $D$. Suppose the shaded regions of $D$ are enumerated $R_1,...,R_s$. Suppose we have an element $\vec v$ in the nullspace of the pre-Goertiz matrix $\tilde{G}$, i.e. $\tilde{G}\vec{v}=\vec{0}$. Let us define $d(R_i,R_j):=v_j-v_i$, and we see that $$d(R_i,R_j)+d(R_j,R_k)=v_j-v_i+v_k-v_j=v_k-v_i=d(R_i,R_k).$$
If these numbers $d(R_i,R_j)$ are differences coming from an actual coloring (this is true as we shall see) of the knot diagram, if we know a color on one of the strands, we can use these differences to figure out the colors on all the other strands of the knot diagram. We will use this observation and assign a color to one strand, and given the numbers $d(R_i,R_j)$, we will assign colors on all the other strands, and verify that this actually gives rise to a valid coloring. Just like in our discussion of the difference, we will need to make some choices and we need to check that they are all give consistent coloring on a strand.
To this end, let us define an auxiliary knot diagram to be a modification of a knot diagram where we break up the overstrands at each crossing, and we will draw a small rectangle at each crossing to keep track of which strand was the overcrossing, see Figure \[auxkd\].
![Auxiliary knot diagram for the knot diagram in Figure \[GM\].[]{data-label="auxkd"}](aux.pdf){width="9"}
We will define an **auxiliary $n$-coloring** on a knot diagram to be an assignment of colors to each of strands of the auxiliary knot diagram, where at each crossing if the overstrands are labeled $w$ and $y$; and the understrands are labelled $x$ and $z$, then $w+y=x+z$, i.e. the overstrands need not have the same labelling, see Figure \[auxkcolor\].
![Auxiliary coloring equation: $w+y=x+z$.[]{data-label="auxkcolor"}](auxkcol.pdf){width="6"}
If it turns out that at each crossing both the segments of the overstrands have the same labelling, then this auxiliary coloring gives us an actual coloring of the knot diagram. Note that we can define the difference for an auxiliary knot coloring exactly the same way we did for a knot coloring, and the second proof of Claim \[wd\] carries over to show that this difference is well defined.
Let us pick a strand $\alpha_1$ in the auxiliary knot diagram and color it $x_1$. It will be part of the boundary of exactly one shaded region, let us call it $R_1$. As we go around the boundary of $R_1$ in the counterclockwise direction, if we come across a crossing $c$ with the other shaded region being $R_j$; we will add $\eta(c)d(R_1,R_j)$ to the coloring of the subsequent strand, see Figure \[auxcolor\].
![Assigning colors to each strand bounding a region in the auxiliary knot diagram.[]{data-label="auxcolor"}](auxcol.pdf){width="10"}
We need to make sure that when we go all the way across and come back to the strand $\alpha_1$ we still get the color $x_1$. Note that the color we get by going all the way across is $$x_1+\sum_{c\text{ is a crossing of }R_1} \eta(c)d(R_1,R_{j(c)}),$$ where $R_{j(c)}$ denotes the other shaded region which has crossing $c$. We observe that $$\sum_{c\text{ crossing of }R_1} \eta(c)d(R_1,R_{j(c)})=\sum_{c\text{ crossing of }R_1} \eta(c)(v_1-v_{j(c)}) =\sum_{j=2}^k\tilde{G}_{1,j}(v_j-v_1)$$ $$=(\sum_{j=2}^k-\tilde{G}_{1,j})v_1 + \sum_{j=2}^k\tilde{G}_{1,j}v_j=\tilde{G}_{1,1}v_1 + \sum_{j=2}^k\tilde{G}_{1,j}v_j=\sum_{j=1}^k\tilde{G}_{1,j}v_j=\tilde{G}\vec v=\vec 0$$ since $\vec v$ is in the null space of the pre-Goeritz matrix $\tilde{G}$.
We can do the exact same thing for each of the regions by picking one strand and arbitrarily assigning a color, and coloring all the other strands by adding $\eta(c)d(R_i,R_j)$ to the coloring as we pass across a crossing $c$ with shaded regions $R_i$ and $R_j$, and we obtain an auxiliary coloring of the knot diagram.
Let us now choose a collection of $s-1$ crossings joining the $s$ shaded regions $R_1,...,R_s$, i.e. given any $i$ and $j$, we can draw an arc from a point in the interior of region $R_i$ to the a point in the interior of region $R_j$ so that the arc passes between regions only in arbitrarily small neighborhoods of the chosen crossings. Starting with a color $x_1$ on a strand $\alpha_1$ in the region $R_1$, we obtain an auxiliary $n$-coloring on the entire knot diagram by choosing to color the overstrands at the chosen crossings by the same element of $\mathbb{Z}/n\mathbb{Z}$, and extending this to each of the strands partially bounding a region by the procedure explained in the preceding two paragraphs. By construction, the colorings on each of the overstrands agree on the chosen $s-1$ crossings, and we show below they agree on every other crossing as well, thereby giving us an actual knot coloring.
Let us denote by $\delta(R_p,R_q)$ the difference between the regions $R_p$ and $R_q$ of this auxiliary $n$-coloring. Suppose $c$ is any crossing in the knot diagram, with the shaded regions $R_i$ and $R_j$ incident on it. We have an arc $\gamma$ from a point in the interior of region $R_i$ to the a point in the interior of region $R_j$, traversing the regions $R_{l_1}=R_i,R_{l_2},...,R_{l_r}=R_j$, so that it goes between regions $R_{l_k}$ and $R_{l_{k+1}}$ in an arbitrarily small neighborhoods of the chosen crossings. Then we have $$\delta(R_i,R_j)=\sum_{k=1}^{r-1} \delta(R_{l_k},R_{l_{k+1}}) \quad\text{ (by choosing the arc $\gamma$ to compute the difference)}$$ $$=\sum_{k=1}^{r-1} d(R_{l_k},R_{l_{k+1}}) \quad\text{ (by construction of the auxiliary coloring)}$$ $$=d(R_i,R_j) \quad\quad\text{ (telescoping sum since $d(R_p,R_q)=v_q-v_p$).}$$ It follows that at the crossing $c$ the colorings on the overstrands agree, and since $c$ is arbitrary, this is true at any crossing. Thus, we have obtained a valid knot coloring with its difference for shaded regions $\delta(R_i,R_j)$ agreeing with $d(R_i,R_j)$ we defined earlier. We should note that this implies we would obtain the same knot coloring independent of the choice of which of the $s-1$ crossings we chose earlier. Also, if instead of coloring the strand $\alpha_1$ with $x_1$, we colored $\alpha_1$ with the color $x_1+a$, then the above procedure would give a new coloring where every strand would get the color $a$ plus the original coloring.
Conversely, we note that if we start with a valid coloring, we can look at the differences $d(R_i,R_j)$ among the shaded regions, and they would satisfy the Goeritz relations $$\sum_{c\text{ crossing
of }R_i} \eta(c)d(R_i,R_{j(c)})=0$$ for each region $R_i$ , which in turn would correspond to a solution of the pre-Goeritz matrix, once we choose a value for a region, say $R_1$.
Hence, under this procedure semitrivial (respectively non-semitrivial) solutions of the pre-Goeritz matrix correspond to the semitrivial (respectively non-semitrivial) solutions of the pre-coloring matrix. More concretely, we have:
\[thmA\]
Suppose we have a knot diagram $D$ for a knot $K$ with $c$ crossings and $s$ shaded regions, let us choose a strand $\alpha_1$ of the knot diagram partially bounding a shaded region $R_1$. Let’s suppose we delete the first column and first row of the pre-coloring matrix $\tilde{C}$ (respectively pre-Goeritz matrix $\tilde{G}$) to obtain coloring matrix $C$ (respectively Goeritz matrix $G$). Then for any $n\in \mathbb{N}$ there is a bijection among the collection of:
1. Non-trivial solutions $\tilde{v}\in (\mathbb{Z}/n\mathbb{Z})^c$ of $\tilde{C}$ with first entry 0.
2. Non-trivial solutions $v\in (\mathbb{Z}/n\mathbb{Z})^{c-1}$ of $C$.
3. Non-trivial solutions $w\in (\mathbb{Z}/n\mathbb{Z})^{s-1}$ of $G$.
4. Non-trivial solutions $\tilde{w}\in (\mathbb{Z}/n\mathbb{Z})^s$ of $\tilde{G}$ with first entry 0.
Moreover the above bijection between Nul($C$) and Nul($G$) is linear, and so we get an isomorphism of these null spaces of $\mathbb{Z}/n\mathbb{Z}$-modules (for $n$ prime, this means a vector space isomorphism). In particular, for prime $n$, the mod $n$ nullity (i.e. dimension of the null space) of both these matrices are equal.
\[exA\] Complete the proof of the above theorem by checking linearity of the bijection between the solution spaces.
For a prime $n$, the mod $n$ nullity of either the coloring or Goeritz matrix completely determines how many $n$-colorings there are of a knot $K$, and is an invariant of the knot, which we will refer to as the $n$-**nullity** of $K$. We remark that if $n$ is composite, then knowing the cardinality of Nul($C$) is not enough to understand the structure of Nul($C$) as a $\mathbb{Z}/n\mathbb{Z}$-module. But when $n$ is prime, Nul($C$) is a vector space, and we completely understand it once we know the dimension.
We note a few consequences of Theorem \[thmA\] (combined with earlier propositions):
1. $n$ divides $\det C \Longleftrightarrow$ ${C}$ has a non-trivial solution in $\mathbb{Z}/n\mathbb{Z}\Longleftrightarrow$ $K$ has a non-trivial $n$-coloring ${G}\Longleftrightarrow$ ${G}$ has a non-trivial solution in $\mathbb{Z}/n\mathbb{Z}\Longleftrightarrow$ $n$ divides $\det G$.
2. The Goeritz determinant is an invariant of the knot, and moreover equals the coloring determinant.
3. For any $n$, the null-space of a Goeritz matrix as a $\mathbb{Z}/n\mathbb{Z}$-module is a knot invariant, which is isomorphic to the null space of any coloring matrix of $K$.
Thus, we can find the determinant of knot; the space of $n$-colorings determinant from a Goeritz matrix, which is frequently smaller than a coloring matrix.
Colorings of Pretzel knots
==========================
In this final section, we will use ideas from the last few sections and find determinant and $n$-nullity of pretzel knots. They have been previously computed pretzel knots with up to four twist regions [@P1; @P2] using coloring matrices.\
Recall that a pretzel knot $P(q_1, q_2,...,q_m)$ has $m$ twist regions joined up as illustrated in Figure \[pret\], where there are $q_i$ (which can be both positive and negative) half-twists in the $i$-th region. See Figure \[p33m3\] for the diagram of the pretzel knot $P(3,3,-3)$.
![Pretzel Knot $P(q_1,...,q_m)$.[]{data-label="pret"}](pretA.pdf){width="8"}
Note that for the pretzel knot $P(q_1, q_2,...,q_m)$ the coloring matrix from the above diagram will be of the order $Q\times Q$, where $Q=|q_1|+...+|q_m|$, where as a Goeritz matrix will be a $(m-1)\times (m-1)$ matrix. It turns out that it seems it is easier to compute the determinant and nullity of pretzel knots from another linear system, constructed with the differences, as indicated below. An interested reader may work out determinant and $n$-nullity from a Goeritz matrix (and even from a coloring matrix, if feeling particularly adventurous).
Let $d_i$ denote the difference $d(R_i,R_{i+1})$ between shaded regions $R_i$ and $R_{i+1}$. Note that for each twist region if there are $q$ half-twists, and the difference is $d$, then we can figure out the colors on all the strands if we know a color on the leftmost strand. We observe that the difference between the colors of the top left (respectively right) and the bottom left (respectively right) strand in the $i$-th twist region is $q_id_i$. Also, note that the top (respectively bottom) right strand of the $i$-th twist region is exactly the same as the top (respectively bottom) leftt strand of the $(i+1)$-th twist region. Thus, for adjacent twist regions we must have the increase in the vertical direction must be the same, so we have $q_id_i=q_{i+1}d_{i+1}$ for all $i$. Moreover if we drew an arc horizontally traversing just below (see figure) each of the twist region regions we see that $-(d_1+d_2+...+d_m)=0$ , since the leftmost and rightmost strands are the same and must have the same color. We illustrate the above discussion with the explicit example of of the pretzel knot $P(3,3,-3)$ in Figure \[p33m3\].
![Pretzel Knot $P(3,3,-3)$.[]{data-label="p33m3"}](p33m3.pdf){width="9"}
Thus we obtain the following collection of linear equations in $\mathbb{Z}/n\mathbb{Z}$: $$d_1+d_2+...+d_m=0$$ $$-q_1d_1+q_2d_2=0$$ $$...$$ $$-q_1d_1+q_md_m=0$$
Let us write out the above system in matrix form (with $d_i$’s being the variables) $A\vec d=\vec 0$, where the matrix $A$ is:
$$\spalignmat{1 1 1 1 . . . 1 ; -q_1 q_2 0 0 . . . 0; -q_1 0 q_3 0 . . . 0;
. . . . . , , , .; . . . . , , . , , .;. . . . , , , . , .;
-q_1 0 0 0 . . . q_m}$$
\[expr\] For a pretzel knot diagram of $P(q_1, q_2,...,q_m)$, pick a strand and name it $\alpha_1$. Given a solution $\vec d$ to the linear system described above, and any labelling of the strand $\alpha_1$ by some $x_1 \in (\mathbb{Z}/n\mathbb{Z})$, show that there is a unique extension to coloring on the entire knot diagram by using the solution $\vec d$. Moreover show that this defines a linear bijective correspondence between the nullspace of the matrix $A$ reduced modulo $n$, and the space of $n$-colorings of the knot diagram (i.e. nullspace of any coloring matrix).
It follows that we can use the matrix $A$ to compute the determinant and $n$-nullity of $P(q_1, q_2,...,q_m)$, and this method seems to be the easiest way of computing them.
\[det\] The determinant of the $m\times m$ matrix $A$ is given by $$q_1q_2...q_m (\frac{1}{q_1}+\frac{1}{q_2}+...+\frac{1}{q_m})$$
We will prove the claim by induction on $m$.
Check the base cases for $m=2,3.$
In order to compute the determinant of $A$, we use Laplace expansion along the last row. $$\det A=(-1)^{m+1} q_1 \det \spalignmat{ 1 1 1 . . . 1 ; q_2 0 0 . . . 0; 0 q_3 0 . . . 0;
. . . . , , , .; . . . , , . , , .; . . . , , , . , .;
0 0 0 . . q_{m-1} 0}+ q_m \det \spalignmat{1 1 1 . . . 1 ; -q_1 q_2 0 . . . 0; -q_1 0 q_3 . . . 0;
. . . . . , , .; . . . . , , . , .;. . . . , , . . ;
-q_1 0 0 . . . q_{m-1}}$$
We observe that by expanding along the last column, we can compute the determinant of the first matrix easily since we get a diagonal matrix: $$\det \spalignmat{ 1 1 1 . . . 1 ; q_2 0 0 . . . 0; 0 q_3 0 . . . 0;
. . . . , , , .; . . . , , . , , .; . . . , , , . , .;
0 0 0 . . q_{m-1} 0}= (-1)^{m-1}\det \spalignmat{ q_2 0 . . . 0; 0 q_3 . . . 0;
. . . , , , .; . . , , . , , .; . . , , , . , .;
0 0 . . . q_{m-1}}= (-1)^{m-1} q_2...q_{m-1}$$
Inductively, we know what the second determinant in the above expansion is. $$\det \spalignmat{1 1 1 . . . 1 ; -q_1 q_2 0 . . . 0; -q_1 0 q_3 . . . 0;
. . . . . , , .; . . . . , , . , .;. . . . , , , . ;
-q_1 0 0 . . . q_{m-1}}= q_1q_2..q_{m-1} (\frac{1}{q_1}+\frac{1}{q_2}+...+\frac{1}{q_{m-1}})$$
Combining them we obtain $$\det A= (-1)^{m+1} q_1 (-1)^{m-1}q_2...q_{m-1} + q_mq_1q_2...q_{m-1} (\frac{1}{q_1}+\frac{1}{q_2}+...+\frac{1}{q_{m-1}})$$ $$=q_1q_2...q_{m-1}+ q_1q_2...q_m(\frac{1}{q_1}+\frac{1}{q_2}+...+\frac{1}{q_{m-1}})=q_1q_2...q_m (\frac{1}{q_1}+\frac{1}{q_2}+...+\frac{1}{q_m})$$
Let us now assume $n$ is a prime and compute mod $n$ nullity of $A$. We make a few observations:
\[colpr\] If all the $q_i$’s are coprime to $n$ then the mod $n$ nullity of $A$ is either 1 or 0, depending on whether n divides $\det A= q_1q_2...q_m(\frac{1}{q_1}+\frac{1}{q_2}+...+\frac{1}{q_m})$ or not.
Note that the submatrix $A_{1,1}$ obtained by deleting the first row and first column is a full rank diagonal matrix and so has mod $n$ rank $m-1$, since we are assuming $n$ does not divide any of $q_2,...,q_n$. It follows that the mod $n$ rank of $A$ is either $m-1$ or $m$ (which by the rank nullity theorem is equivalent to saying the mod $n$ nullity is 1 or 0). The proof is completed by using the Invertible Matrix Theorem, that mod $n$ nullity of $A$ is 0 iff $\det A=0$ modulo $n$.
\[nullpr\] If some of the $q_i$’s are divisible by $n$, then the mod $n$ nullity of $A$ is the total number of $q_i$’s divisible by $n$, minus $1$.
If some of the $q_i $’s are divisible by $n$, we may assume $n$ divides $q_1$ (note that there is cyclic symmetry for pretzel knots, $P(q_1,q_2,...,q_{m-1},q_m)$ is isotopic to $P(q_2,q_3,...,q_{m},q_1)$). In this case, the claim is equivalent to:\
If $n$ divides $q_1$,then the mod $n$ nullity of $A$ is the number of $q_i$’s (apart from $q_1$) divisible by $n$.
When $A$ is reduced modulo $n$, it becomes a upper triangular matrix, and the rank is the number of pivots (i.e. the number of $q_i$’s not divisible by $n$, plus 1 ), and the nullity is the number of 0’s in the main diagonal (the number of “other” $q_i$’s divisible by $n$).
\[exnull\] Give a direct proof Claim \[nullpr\] (without using that $n$-nullity is invariant of the knot), i.e. show the result is true when $q_1$ is coprime to $n$. (Hint: Use row operations).
To summarize, if none of the $q_i$’s are divisible by $n$, the pretzel knot can have at most one coloring and the determinant determines whether there is one. In cases that some of the $q_i$’s are divisible by $n$, for any $n$-coloring
1. the colors of all the strands in the $i$-th twist region will be the same when the corresponding $q_i$ is coprime to $n$;
2. for those $q_i$ which are divisible by $n$, the colors of the strands in the $i$-th twist region can be different, and these sort of correspond to the free variables, except one has to remember that the colors of the very last such twist region is determined (by the coloring on the first twist region), and this constraint is where we get the “minus 1” in the formula for $n$-nullity.
[10]{}
J.W. Alexander and G.B. Briggs. On types of knotted curves. , 28 : 2 (1927/28) pp. 563–586.
K. Brownell, K. O’Neil and L. Taalman. Counting m-coloring classes of knots and links. 12 (5), 265-278, 2005. J. S. Carter, D. S. Silver and S. G.Williams. Three Dimensions of Knot Coloring. , Vol. 121, No. 6 (June–July 2014), pp. 506-514.
W. B. R. Lickorish. An Introduction to Knot Theory. , 1997.
Robert Ostrander. P-Coloring of pretzel knots. [ Masters Thesis](http://scholarworks.csun.edu/handle/10211.2/5500), 2014.
K. Reidemeister. Elementare Begrundung der Knotentheorie. , 5 (1927) pp. 24–32
Lorenzo Traldi. Link colorings and the Goeritz matrix. , Vol. 26, No. 08, 2017.
[^1]: It may happen that some knot diagram has triple (or higher order) crossings, however we can isotope some of the strands so that the only crossings that remains are double points.
[^2]: By two curves touching at a point we will mean there is a point of intersection where the tangent lines agree.
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---
abstract: 'We give an apparently new proof of Stirling’s original asymptotic formula for the behavior of $\ln z!$ for large $z$. Stirling’s original formula is not the formula widely known as “Stirling’s formula", which was actually due to De Moivre. We also show by experiment that this old formula is quite effective for numerical evaluation of $\ln z!$ over $\mathbb{C}$, when coupled with the sequence acceleration method known as Levin’s $u$-transform. As an *homage* to Stirling, who apparently used inverse symbolic computation to identify the constant term in his formula, we do the same in our proof.'
author:
-
bibliography:
- 'bib.bib'
title: 'Stirling’s Original Asymptotic Series from a formula like one of Binet’s and its evaluation by sequence acceleration'
---
Introduction
============
Stirling’s original formula for the asymptotics of $\ln z!$ has been obscured by the formula popularly known as “Stirling’s formula", namely $$\begin{aligned}
\label{1}
\ln z! \sim (z+\frac{1}{2})\ln z -z + \ln \sqrt{2\pi} + z \sum_{n\geq 1}\dfrac{B_{2n}}{2n(2n-1)}\cdot\dfrac{1}{z^{2n}} \\
\sim (z+\frac{1}{2})\ln z -z + \ln \sqrt{2\pi} + \dfrac{1}{12z}-\dfrac{1}{360z^{3}}+\mathcal{O}(\dfrac{1}{z^5})\>,\end{aligned}$$ which was actually found by De Moivre after Stirling had found his (see, *e.g.*, [@bellhouse2011]). Stirling’s original formula is $$\begin{aligned}
\label{2}
\ln z! \sim Z\ln Z- Z +\ln\sqrt{2\pi}-Z\sum_{n\geq 1}\dfrac{(1-2^{1-2n})B_{2n}}{2n(2n-1)Z^{2n}} \\
\sim Z\ln Z- Z +\ln\sqrt{2\pi} - \dfrac{1}{24Z}+\dfrac{7}{2880Z^3}-\mathcal{O}(\dfrac{1}{Z^5})\>.\end{aligned}$$ where $Z=z+\frac{1}{2}$.\
\
As you can see here, the formulae are quite similar. Stirling’s original formula in equation has been rediscovered several times. Some people call it De Moivre’s formula! It seems to have been known to both Gauss and to Hermite (see *e.g.* [@richard]). There is a discussion in [@tweddle1984] of one such rediscovery in the physics literature; for a particularly ironic case where the rediscoverer claims the formula is “both simpler and more accurate" than “Stirling’s formula", look at [@spouge1994]. For a thorough exposition of Stirling’s actual work see the original, as masterfully translated and annotated by Tweddle [@tweddle1984].\
In this present work we give a short proof of equation , which we believe to be new, by deriving an apparently new formula that is similar to the following formula of Binet: $$\ln z!=(z+\frac{1}{2})\ln z-z+\ln\sqrt{2\pi}+\int_{t=0}^{\infty}\dfrac{1}{t} \left( \dfrac{1}{t}-\dfrac{1}{e^t-1}\right) e^{-tz}dt$$ which [@whittaker] claims is valid for $\Re z>0$. We will see later that this is not quite true in the modern context. This classical formula is proved in, for example, [@whittaker] and in [@sasvari]. The new formula is quite similar, again using $Z=z+\frac{1}{2}$: $$\label{6}
\ln z!=Z\ln Z-Z +\ln\sqrt{2\pi}-\int_{t=0}^{\infty}\dfrac{1}{t}\left( \dfrac{1}{t}-\dfrac{1}{2\sinh\frac{t}{2}}\right) e^{-tZ}dt,$$ and is valid for $\Re z>-\dfrac{1}{2}$ (again, we will adjust this caveat later). Formula appears as “Theorem $2$", without proof, in [@BorweinCorless1999].\
In the modern computational world, a new proof of an old mathematical result is rarely of interest for its own sake, but see for instance [@nplus1]. Indeed Stirling’s original proof of equation was algorithmic in nature and, apart from the use of “recognition" to identify $\sqrt{2\pi}$ and the lack of a “closed formula"— *i.e.* a relationship to other numbers, the Bernoulli numbers— Stirling’s proof was entirely satisfactory. So why record these results?\
We believe this formula is interesting for the following reasons. First, the rediscovery was identified as such by tracing patterns and citations in Google Scholar, and now there is some hope that the obscurity of the original formula can be lifted[^1]. Of course the mathematics history literature has it right, owing to the work of Tweddle, but still. Second, Stirling’s original proof used what is now called “Inverse Symbolic Computation," illustrating that a modern experimental technique worth investigation has significant historical roots. As an *homage* to Stirling we use the same technique in our ‘new’ proof below. Finally, we test Stirling’s original formula in a modern computational context by trying a nonlinear sequence acceleration technique, namely Levin’s $u$-transform; this gives a surprisingly viable method, comparable in cost (for a given accuracy) to the methods discussed in [@schmelzer]. The separate issue of the complexity of the computation of $\Gamma(1+z)$, $z!$, or $n!$ for $n \in \mathbb{N}$, is not addressed here. See for instance [@peter], [@richard] for entry into that literature. See also [@hare] for the computation of $\Gamma(z)$.\
Basic references for $\Gamma$ include the DLMF (chapter $5$), the Dynamic Dictionary, and [@andrews1999].
Notation
========
Here we use $z!$ and $\Gamma(z+1)$ interchangeably. As mentioned in [@robgamma] the “notation wars" and the annoyance of the continual nuisance of shifting by $1$ are amusing but not possible nowadays of resolution. We use $\ln$ for the natural logarithm because it’s unambiguous and ingeniously, as pointed out by David Jeffrey offers a free location for a subscript, which we use as follows $$\ln_k z=\ln z + 2\pi i k \>.$$ The unsubscripted $\ln z$ has range $-\pi < \Im \ln z \leq \pi$, the principal branch in universal usage nowadays in computers. We write $\ln z!$ for $\ln(z!)$; *i.e.* the factorial has higher precedence. We discuss the function $\ln\Gamma(z)$ in detail below as the analytic continuation of $\ln(z-1)!$. This modern notation is in contrast to Stirling’s, where he used $\ell,z$ to mean $\log_{10}z$. The factorial notation $!$ was apparently invented by Christian Kramp in $1808$; the $\Gamma$ notation was invented by Legendre, and although the shift by $1$ as apparently due to Euler himself [@gronau2003gamma], Legendre gets the blame for that, too.
Divergent asymptotic series
===========================
For a given sequence $\lbrace \phi_i(x) \rbrace$ where the $\phi_i(x)$’s are defined over a domain, one can define a formal series $\sum_{i= 1}^\infty a_i\phi_i(x)$. The idea of asymptotic series is to define a formal series with special property on the underlying sequence such that its partial sums approximate a given function over the same domain even more closely as $x\rightarrow x_0$.\
\
Assume $R$ is a domain and $\lbrace \phi_i(x) \rbrace$ is a sequence of functions defined over $R$. The sequence $\phi_i(x)$ is called an asymptotic sequence for $x \rightarrow x_0$ in $R$ if for each $i$, $\phi_{i+1}(x)= o(\phi_i(x))$ as $x \rightarrow x_0$. A simple example is $\lbrace (x-x_0)^i \rbrace$ for $x \rightarrow x_0$. Recall that $f(x)=o(g(x))$ as $x\rightarrow\infty$ if $$\forall c > 0 \;\; \exists N>0 \;\;\; \text{s.t.} \;\;\; |f(x)|<c|g(x)| \;\;\; \text{for} \;\;\; x>N \>.$$ or (if $x_0$ is finite) $$\forall c > 0 \;\; \exists \delta>0 \;\;\; \text{s.t.} \;\;\; |f(x)|<c|g(x)| \;\;\; \text{for} \;\;\; |x-x_0|<\delta \>.$$ Now suppose $\lbrace \phi_i(x) \rbrace$ is an asymptotic sequence which is defined over a domain $R$ and $f(x)$ is defined over $R$ as well. The formal series $\sum_{i= 1}^\infty a_i \phi_i(x)$ is said to be an asymptotic expansion (series) to $n$ terms of $f(x)$ as $x \rightarrow x_0$ if $$f(x) = \sum_{i= 1}^n a_i \phi_i(x) + o(\phi_{n+1}(x)) \,\,\, \text{as} \,\,\, x\rightarrow x_0 \>.$$ The formal series $\sum a_i \phi_i$ will be called an asymptotic series. An asymptotic series can be divergent or convergent itself as $n\rightarrow \infty$. For more details see *e.g.* [@erdelyi Chapter 1].
Tools
=====
We will use Fubini’s theorem, which justifies the interchange of order of iterated integrals of continuous functions, and we will use Watson’s Lemma. Loosely speaking, Watson’s Lemma allows the interchange of order of summation of a series and of integration even though the radius of convergence of the series is violated (leaving us with a divergent asymptotic series).\
[@bender1999] and [@copson] Assume $\alpha > -1$, $\beta >0$ and $b>0$. If $f(t)$ is a continuous function on $[0,b]$ such that it has asymptotic series expansion $$f(t) \sim t^{\alpha} \sum_{n=0}^\infty a_nt^{\beta n}, \,\,\, t \rightarrow 0^+ \>,$$ (and if $b=+\infty$ then $f(t) <k\cdot e^{ct}\, (t\rightarrow +\infty)$ for some positive constants $c$ and $k$), then $$\int_0^b f(t)e^{-xt}dt \sim \sum_{n=0}^\infty \dfrac{a_n \Gamma (\alpha + \beta n +1)}{x^{\alpha +\beta n+1}}, \,\,\, x \rightarrow +\infty$$
For a proof of Watson’s lemma, see [@bender1999].\
We will also use Gauss’ formula $$\dfrac{\Gamma'(z+1)}{\Gamma(z+1)}=\int_{t=0}^{\infty}\dfrac{e^{-t}}{t}-\dfrac{e^{-tz}}{e^t-1}dt \;\;\; \text{for} \;\;\; \Re z>0$$ a proof of which can be found for example in [@whittaker]. Alternatively, a more elementary proof can be found in [@sasvari].\
The next mathematical tool we need comes from a Laplace transform; using $\xi+\dfrac{1}{2}$ instead of the more common symbol $s$, the Laplace transform of $1$ is $$\int_{t=0}^{\infty}e^{-t(\xi+\frac{1}{2})}dt=\dfrac{1}{\xi+\frac{1}{2}}$$ by direct integration. The integral converges if $\Re(\xi)>-\frac{1}{2}$. We can then prove the following lemma:\
\[lemma\] For $\Re z > -1/2,$ $$\ln (z+\frac{1}{2})=\int_{t=0}^{\infty}\dfrac{e^{-t}}{t}-\dfrac{e^{-t(z+\frac{1}{2})}}{t}dt \>.$$
Integrate the Laplace transform with respect to $\xi$ from $\xi=\frac{1}{2}$ to $\xi=z$: $$\int_{\xi=\frac{1}{2}}^{z}\dfrac{d \xi}{\xi+\frac{1}{2}}=\int_{\xi=\frac{1}{2}}^{z}\int_{t=0}^{\infty}e^{-t(\xi+\frac{1}{2})}dtd\xi$$ Interchange the order of integration—by Fubini’s Theorem this is valid—and since $\int e^{-t(\xi+\frac{1}{2})}d\xi=-\dfrac{e^{-t(\xi+\frac{1}{2})}}{t}$, we have $$\ln(z+\frac{1}{2})-\ln(\frac{1}{2}+\frac{1}{2})=\int_{t=0}^{\infty}-\dfrac{e^{-t(z+\frac{1}{2})}}{t}+\dfrac{e^{-t(\frac{1}{2}+\frac{1}{2})}}{t}dt$$ which proves the lemma.
The formula like Binet’s
========================
\[2.3\] If $z> - \frac{1}{2}$, $$\label{formula17}
\ln z!=(z+\frac{1}{2})\ln (z+\frac{1}{2})-(z+\frac{1}{2})+\ln\sqrt{2\pi}-\int_{t=0}^{\infty}\dfrac{1}{t}\left( \dfrac{1}{t}-\dfrac{1}{2\sinh \frac{t}{2}}\right) e^{-t(z+\frac{1}{2})}dt$$
We start with Gauss’ formula and switching to $\Gamma$ notation because the derivative $d\Gamma/dz$ is easily written $\Gamma'$,\
$$\dfrac{\Gamma'(z+1)}{\Gamma(z+1)}=\int_{t=0}^{\infty}\dfrac{e^{-t}}{t}-\dfrac{e^{-tz}}{e^t-1}dt$$ (see *e.g.* [@whittaker]), and Lemma \[lemma\].\
Rearranging Gauss’ formula using $e^{t/2}-e^{-t/2}=2\sinh\frac{t}{2}$,\
$$\dfrac{\Gamma'(z+1)}{\Gamma(z+1)}=\int_{t=0}^{\infty}\dfrac{e^{-t}}{t}-\dfrac{e^{-t(z+\frac{1}{2})}}{2\sinh\frac{t}{2}}dt$$ Subtracting Lemma \[lemma\],\
$$\dfrac{\Gamma'(\xi+1)}{\Gamma(\xi+1)}-\ln (\xi+\frac{1}{2})=\int_{t=0}^{\infty}\dfrac{e^{-t(\xi+\frac{1}{2})}}{t}-\dfrac{e^{-t(\xi+\frac{1}{2})}}{2\sinh\frac{t}{2}}dt$$ Integrating from $\xi=\alpha> -\frac{1}{2}$ to $\xi=z> -\frac{1}{2}$ and interchanging the order of integration using Fubini’s theorem, we find (except for a branch issue that we take up later) that $$\ln \Gamma(z+1)-\ln\Gamma(\alpha+1)-(z+\frac{1}{2})\ln (z+\frac{1}{2})+(z+\frac{1}{2})+(\alpha+\frac{1}{2})\ln(\alpha+\frac{1}{2})-(\alpha+\frac{1}{2})=$$ $$\int_{t=0}^{\infty}\dfrac{1}{t}\left(\dfrac{1}{t}-\dfrac{1}{2\sinh\frac{t}{2}}\right)e^{-t(\alpha+\frac{1}{2})}dt-\int_{t=0}^{\infty}\dfrac{1}{t}\left(\dfrac{1}{t}-\dfrac{1}{2\sinh\frac{t}{2}}\right) e^{-t(z+\frac{1}{2})}dt\>.$$ We now need to evaluate the $\alpha$ integral. At $\alpha=0$ Maple and Mathematica can only find a numerical approximation; likewise at $\alpha=\frac{1}{2}$. The numerical approximation can be identified by (for instance) the Inverse Symbolic Calculator at CARMA[^2] (a proof is supplied in Remarks \[rmk5.3\] and \[rmk5.4\].) $$\int_{t=0}^{\infty}\dfrac{1}{t}\left(\dfrac{1}{t}-\dfrac{1}{2\sinh\frac{t}{2}}\right)e^{-t/2}dt=\frac{1}{2}\ln(\frac{\pi}{e})$$ Simplification then yields our formula.\
According to [@tweddle1984], this may have been the method Stirling used to identify $\log_{10}\sqrt{2\pi}$, except of course all calculations were done by hand. Apparently, he simply recognized the number $0.39908$. Nowadays very few people could do that unaided, but with the ISC it’s easy.
\[rmk5.3\] In [@sasvari] we find a trick that could be used to do this integral analytically; we leave this as an exercise.\
If one desires an actual proof, one can use “Stirling’s formula" (by De Moivre) and leverage the tricky identification of $\sqrt{2\pi}$, as follows.
As $z\rightarrow \infty$, $$\ln \Gamma(z+1)-(z+\frac{1}{2})\ln (z+\frac{1}{2})+(z+\frac{1}{2})\sim \ln\sqrt{2\pi}+\mathcal{O}(\frac{1}{z})\>.$$ Therefore (since the second integral goes to $0$ as $z\rightarrow \infty$) $$\ln\sqrt{2\pi}-\ln \Gamma(\alpha+1)+(\alpha+\frac{1}{2})\ln(\alpha+\frac{1}{2})-(\alpha+\frac{1}{2})$$ $$=\int_{t=0}^{\infty}\dfrac{1}{t}\left(\dfrac{1}{t}-\dfrac{1}{2\sinh\frac{t}{2}}\right) e^{-t(\alpha+\frac{1}{2})}dt$$ But this is, in fact, our desired theorem with $z=\alpha$.
\[rmk5.4\] This looks like a circular argument, but it is not. We have here used the $\sqrt{2\pi}$ from the formula popularly known as Stirling’s formula, for which there are many proofs analytically (see *e.g.* [@whittaker]).
[@levinson1970complex p. 399] By analytic continuation, formula holds for $\Re z\geq -1/2$, since the integral is convergent there.
Evaluation of $\Gamma$ using this divergent series
==================================================
First attempts
--------------
It has long been known that “Stirling’s approximation” leads to a viable method to evaluate $\ln\Gamma(z)$. The basic idea is to use the asymptotic series to evaluate $\ln\Gamma(z+n)$ for some large $n$ (large enough that the series gives some accuracy) and then work down with the recursive formula $$\ln\Gamma(z+n-1)= -\ln(z+n-1)+\ln\Gamma(z+n)$$ until we have reached $\ln\Gamma(z)$. This naive idea is surprisingly effective. The point of discussion is just how large $n$ should be, and how many terms in “Stirling’s series” one should retain, in order to make an effective formula.\
Given that we now have a different asymptotic formula under consideration (the original, more accurate, but certainly not “new” formula) all of the discussion points are necessarily changed. Just as an example, take (say), $z=11+i/2$. If we want $\ln((11+i/2)!)$ then Stirling’s original series gives $$\begin{gathered}
\ln\sqrt{2\pi}+(11.5+i/2)\ln(11.5+i/2)-(11.5+i/2)-\dfrac{1}{24(11.5+i/2)}+\mathcal{O}(\dfrac{1}{z^3}) \\
=17.4914469445+1.22148819106i \end{gathered}$$ Wolfram Alpha confirms this, giving $$\ln((11+i/2)!)\doteq 17.4914485209+1.22148798i \>.$$ Rather than get into the minutiae of how many terms to take, and how far to push the argument to the right, we take a different tack: we look at automatic sequence acceleration of the original divergent series. If $$S=\ln\sqrt{2\pi}+Z\ln Z-Z-Z\sum_{n\geq 1}\dfrac{(1-2^{1-2n})B_{2n}}{2n(2n-1)Z^{2n}}\>,$$ then we wonder if simple execution of the Maple command $$\texttt{evalf}\texttt{(Sum(}\texttt{a(n)}\texttt{,n=1..}\texttt{infinity))};$$ where $a(n)$ is defined as $\dfrac{(1-2^{1-2n})B_{2n}}{2n(2n-1)Z^{2n}}$ will automatically produce an accurate result.
“Sometimes Maple knows things that you don’t know. And then you wonder just what.”
–Jon Borwein.
Levin’s $u$-transform
---------------------
What Maple knows here is called Levin’s $u$-transform. This is a method to accelerate convergence of the sequence of partial sums $$S_n=\sum_{j=1}^{n}a_j$$ of the series we consider. For an introduction to sequence acceleration, see [@henrici1982] and [@henrici1964]. For an introduction to Levin’s $u$-transform, see [@weniger].\
\
The basic idea is to replace the sequence $S_0, S_1, S_2,\cdots$ with a new one that has the same limit but which converges faster. More precisely, Levin’s $u$-transform for $S_n$ is given as: $$u_{k}^{(n)}(\beta , S_{n}) = \dfrac{\sum_{j=0}^{k} (-1)^{j} {k \choose j} \dfrac{(\beta + n + j)^{k-2}}{(\beta + n + k)^{k-1}}\dfrac{S_{n+j}}{a_{n+j}}}{\sum_{j=0}^{k} (-1)^{j} {k \choose j} \dfrac{(\beta + n + j)^{k-2}}{(\beta + n + k)^{k-1}}\dfrac{1}{a_{n+j}}}$$ The parameter $\beta > 0$ is “in principle completely arbitrary” [@weniger]. In practice, Maple’s routine chooses $\beta=1$.\
For *irregular* sequence transforms such as Levin’s $u$-transform, this may even transform divergent series into rapidly convergent ones. The price, however, is that it doesn’t always work. It works well enough, though, that it is the default method coded in Maple [@geddes1992]. It is accessed most simply by applying the “evalf" command to an inert sum (denoted by capital-letter `Sum`). For instance,\
$$\texttt{evalf}\texttt{(Sum((}\texttt{-2)}^\texttt{n}\texttt{,n=0..}\texttt{infinity))};$$ yields $0.3333333333$[^3].\
Other sequence acceleration methods or quadratures could be used (see for example chapter $28$ of [@Trefethenbook]), but we wanted to show the capabilities of some (under-appreciated) off-the-shelf tools.\
If we issue the command (with a numerical value for $z$, say $z=11+i/2$)
> evalf(-(z+1/2)*(Sum((1-2^(1-2*n))*$\mathrm{bernoulli}$(2*n)/
(2*n*(2*n-1)*(z+1/2)^(2*n)), n = 1 .. infinity))+
ln(sqrt(2*Pi))+ln(z+1/2)*(z+1/2)-(z-1/2);
we get $\ln((11+i/2)!)$ with full accuracy: $14$ digits if Digits $:=15$, $28$ digits if Digits $:=30$, $58$ digits if Digits $:=60$, and so on. This divergent series is being accurately, and quickly, summed by Maple’s built-in sequence acceleration using the Levin $u$-transformation method above.\
If we test this summation by looking at the error $$\ln\Gamma(z+1)-\ln S(z)$$ over a range $-20\leq \Re z\leq 20$, $-20\leq \Im z \leq 20$, we get the curious result in Figure \[relerrRedBlue1\].
![The region of utility for Levin’s $u$-transform without an unwinding number.[]{data-label="relerrRedBlue1"}](relerrRedBlue1-eps-converted-to)
\
Everywhere in the red region (which includes the real axis for $x$ larger than about $2.1$) has full accuracy, whatever the setting of Digits. The region in white, in the middle, with its scalloped edges, is the region where Levin’s $u$-transform fails and Maple returns an unevaluated Sum, as one can see in the example below:
> Digits := 20:
> z := 1+.1*I:
> evalf(-(z+1/2)*(Sum((1-2^(1-2*n))*$\mathrm{bernoulli}$(2*n)/
(2*n*(2*n-1)*(z+1/2)^(2*n)), n = 1 .. infinity))+
ln(sqrt(2*Pi))+ln(z+1/2)*(z+1/2)-(z-1/2);
The boundary of this region is very curious, and we return to the proof of theorem \[2.3\] to try to understand why. After staring at it for some time, we realize that the transition from $$\dfrac{\Gamma'(z+1)}{\Gamma(z+1)} \;\;\; \text{to} \;\;\; \ln\Gamma(z+1)$$ depends on the path that $\Gamma(\xi+1)$ takes as $\xi$ goes from $\xi=1/2$ to $\xi=z$ (a straight line in the $\xi$ variable). But $\Gamma(\frac{1}{2}+t(z-\frac{1}{2}))$ may cross the negative real axis (the branch cut for logarithm) several times as $t$ goes from $0$ to $1$. Writing our answers, as we do, as $$\ln z! \sim Z\ln Z-Z+\ln\sqrt{2\pi}+Z\sum_{n\geq 1}\dfrac{(1-2^{1-n})B_{2n}}{2n(2n-1)(Z)^{2n}}$$ obscures the fact that the imaginary part of the logarithm on the left is in $(-\pi,\pi]$ while the imaginary part on the right might be anything. To make this equation actually true, we must subtract a multiple of $2\pi i$. To force the imaginary part of $S$ into $(-\pi,\pi]$ there is only one choice: replace $S$ by $$S-2\pi i {\ensuremath{\mathcal{K}}}(S)$$ where ${\ensuremath{\mathcal{K}}}(z)=\left\lceil \dfrac{\Im z-\pi}{2\pi} \right\rceil$ is the unwinding number of $z$ (see [@unwindHighm], [@CorlessUnwind1] and [@CorlessUnwind2]). This means that $\ln z! \sim S-2\pi i{\ensuremath{\mathcal{K}}}(S)$ not $\sim S$.\
As pointed out by a referee, this is because the sum $S$ is “really" asymptotic to the analytic function $\ln\Gamma(z+1)$, obtained by analytic continuation of the function compostion $\ln(\Gamma(z+1))$ for $z>0$. See *e.g.* [@hare] for details and for some simple formulae for ${\ensuremath{\mathcal{K}}}(S)$ in special cases.
When we plot the error $\ln z! - \ln S + 2\pi i {\ensuremath{\mathcal{K}}}(S)$ as in Figure \[relerrRedb\] we see that whenever the Levin’s $u$-transform actually returns an answer, we have only roundoff error. We get essentially perfect accuracy[^4] everywhere to the right of the scalloped boundary in Figure \[relerrRedb\]. So far as we know, this result is new. Of course, the detailed accuracy needs a proof: we have only provided experimental evidence, here. What every mathematician wants is a guarantee that the acceleration will work, or a perfect description of just when it will fail. We do not have this.\
However, when we plot the contours of the error $\ln z! - \ln S + 2\pi i{\ensuremath{\mathcal{K}}}(S)$ as in Figure \[relerrContour2\] we see that the Levin’s $u$-transform works as well as could possibly be expected: the visible contours are all less than $10^{-28}$, when we work in $30$ Digits; clearly the error is zero up to roundoff. We have computed the error at ten thousand locations in the region $[0-1000i, 1000+1000i]$ and the maximum error was $10^{-27}$ (on a $100 \times 100$ grid).\
![The region of utility for Levin’s $u$-transform. We have essentially perfect accuracy (up to roundoff error) outside the region around the negative real axis and the “lozenge of failure”. Curiously, the error increases gradually near the negative real axis.[]{data-label="relerrRedb"}](relerrRedb-eps-converted-to)
![$3D$ plot looking straight down of the error of $\ln z! - \ln S + 2\pi i{\ensuremath{\mathcal{K}}}(S)$. The errors are everywhere less than $10^{-27}$. We work in $30$ digits of precision.[]{data-label="relerrContour2"}](relerrContour2-eps-converted-to)
Truncating the series without Levin’s $u$-transform
---------------------------------------------------
In this section, we plot the absolute estimate error of the truncated series $T$ (not using Levin’s $u$-transform) $T-\ln(Z-1/2)!$ where $$\begin{aligned}
T=u-2\pi i{\ensuremath{\mathcal{K}}}(u)\end{aligned}$$ and $u=Z\ln(Z)-Z+\ln(\sqrt{2\pi})-\dfrac{1}{24Z}$. For different contours ($10^{-3}$ and $10^{-6}$), we get a very curious result as one can see in Figure \[truncseries\]. The error is small outside the keyhole contour. This is more the kind of error we expect from truncated asymptotic series. We see good accuracy even with very few terms. It may be surprising to see that the error is small even in parts of the left half plane, although not near the negative real axis.
![The absolute estimate error of the $T-\ln(Z-1/2)!$. The inner contour is at level $10^{-3}$, and the outer is $10^{-6}$. The truncation error is smaller outside each contour. We used Digits $=30$ and grid $=[600,600]$ in the construction of this figure. The “bubbles” and “wiggles” in this figure are unexplained.[]{data-label="truncseries"}](truncseries-eps-converted-to)
Concluding Remarks
==================
The Gamma function and the factorial function, invented in the $1700$’s, have been very thoroughly studied. Richard Brent’s article [@brentarxiv] points out some facts, known to Hermite and to Gauss, that were not covered in the survey [@robgamma], which looked at about $100$ references. One learns therefore that it is difficult to claim a result (formula or proof) is truly new; we are worried in particular that Gauss knew of our Binet–like formula proved here.\
Nonetheless we believe the proof and numerical experiments have some value in the modern literature. The appearance of the unwinding number in the asymptotic series (either Stirling’s or De Moivre’s) may also be of value for people who write programs to compute $z!$.
[^1]: Of course, there is no hope of changing the popular meaning of the name “Stirling’s formula”.
[^2]: **https://isc.carma.newcastle.edu.au**. Remark: The ISC is currently down because a security flaw was found. Discussion is under way as to how or if this can be resolved.
[^3]: Correctly, in the sense of Euler summation, taking $1+r+r^2+\cdots=1/(1-r)$ even if $|r|>1$ by redefining what the infinite sum actually means: see *e.g.* [@Hardy], for more classical work on making sense of divergent series.
[^4]: Except of course for rounding error. We do not attempt a numerical analysis here, which appears involved. The main difficulty is predicting the number of arithmetic operations.
|
---
abstract: 'We introduce an analogue of the inflation technique of Lalonde-McDuff, allowing us to obtain new symplectic forms from symplectic surfaces of negative self-intersection in symplectic four-manifolds. We consider the implications of this construction for the symplectic cones of Kähler surfaces, proving along the way a result which can be used to simplify the intersections of distinct pseudoholomorphic curves via a perturbation.'
address:
- |
School of Mathematics\
University of Minnesota\
Minneapolis, MN 55455
- |
Department of Mathematics\
Princeton University\
Princeton, NJ 08540
author:
- 'Tian-Jun Li & Michael Usher'
title: Symplectic forms and surfaces of negative square
---
[^1]
Introduction
=============
Given an embedded symplectic surface $C$ of non-negative self-intersection in a symplectic 4-manifold $(M,\omega)$, the inflation process in [@LM] gives rise to new symplectic forms in the class $[\omega]+tPD[C]$ for arbitrary $t>0$. In this paper we show that there is an analogous construction in the case of an embedded symplectic surface of negative self-intersection.
\[main\] Suppose $C$ is an embedded connected symplectic surface representing a class $e$ with $e\cdot e=-k<0$ and $a=\omega(e)$. Let $h=k$ if $C$ has positive genus or $C$ is a sphere with $k$ even, and $h=k+1$ if $C$ is a sphere with $k$ odd. Then there are symplectic forms $\omega_t$ representing the classes $[\omega] +tPD(e)$ for any $t\in
[0, {2a\over h})$.
This is achieved by the normal connected sum construction (see [@Go], [@MW]). In fact the inflation process can be viewed this way as well. However there are two distinct features from the inflation process. The first is the upper bound on $t$. The second is that the surface $C$ is not symplectic with respect to the forms $\omega_t$ when $t > {a\over k}$ (such values of $t$ occur as long as $C$ is not a ($-1$)-sphere). Indeed, the symplectic area of $C$ is non-positive for these values of $t$.
From the known characterization of the symplectic cones of $S^2-$bundles [@Mc3], for any triple $(g,k,a)$ with $g\geq 0$ and $k,a>0$ and any $\operatorname{\epsilon}>0$ it is a routine exercise to find a symplectic $4$-manifold $(M,\omega)$ containing a symplectic surface $\Sigma_{\operatorname{\epsilon}}$ of genus $g$, square $-k$, and area $a$ such that $[\omega]+(2a/h+\operatorname{\epsilon})PD[\Sigma_{\operatorname{\epsilon}}]$ does *not* admit symplectic forms, where $h$ is as in the statement of Theorem \[main\]. In this regard, Theorem \[main\] may be considered a best possible result for the generality in which it is stated.
Using the pairwise normal connected sum construction we will also show how to apply the construction of Theorem \[main\] to a configuration of surfaces intersecting each other positively and transversally.
To apply such a construction we need to locate configurations of surfaces. Such configurations sometimes appear as pseudo-holomorphic curves. It is shown in [@Mc4] that any irreducible simple pseudo-holomorphic curve can be perturbed to a pseudo-holomorphic immersion, possibly after a $C^1$-small change in the almost complex structure. We show how to further perturb such an immersion to an embedding. In fact we are able to show that any configuration of simple $J$-holomorphic curves can be perturbed to a configuration of symplectic surfaces which intersect each other positively and transversally and which are $J'$-holomorphic for an almost complex structure arbitrarily $C^1$-close to $J$.
Holomorphic curves of negative self-intersection actually characterize the Kähler cone by (the extension of) the Nakai-Moishezon criterion. Thus it is interesting to apply this construction to Kähler surfaces. Let $(M,J)$ be a Kähler surface and $H_J^{1,1}$ denote the real part of the $(1,1)-$subspace of $H^2(M;{\mathbb C})$ determined by $J$. The classical Hodge index theorem then asserts that the restriction of the intersection form to $H^{1,1}_{J}$ is a bilinear form of type $(1,h^{1,1}-1)$. The *positive cone* of $H^{1,1}_{J}$ is then by definition the set of classes in $H^{1,1}_{J}$ which have positive square and pair positively with the class of the given Kähler form. Buchdahl and Lamari have recently independently proven the following result:
\[bl\]*(*[@B],[@L]*)* For a Kähler surface $(M,J)$, any class in the positive cone of $H_{J}^{1,1}$ is represented by a Kähler form if it is positive on each holomorphic curve with negative self-intersection.
(Note that the Hodge index theorem implies that any class in the positive cone of $H^{1,1}_{J}$ is automatically positive on each curve of *non*-negative self-intersection.) Applying Theorem \[main\] to a curve $C$ of negative self-intersection, the Kähler cone can be enlarged across the “wall” consisting of cohomology classes which vanish on $[C]$ unless $C$ is a $(-1)$-sphere. This suggests the following symplectic Nakai-Moishezon criterion:
\[q\] For a Kähler surface $(M,J)$, is every class in the positive cone of $H^{1,1}_{J}$ which is positive on each $(-1)$-sphere (possibly reducible) represented by a symplectic form?
To motivate this, note that by the Riemann-Roch theorem and the adjunction formula the expected dimension of the space of embedded pseudoholomorphic genus $g$ curves in the class $[C]$ is $$d([C])=2(g-1+\langle c_1(M),[C]\rangle)=[C]\cap [C]-\langle
c_1(M),[C]\rangle=[C]\cap [C]+1-g,$$ which as the last expression above demonstrates is negative if $[C]$ is the class of any negatively self-intersecting curve other than a $(-1)$-sphere. Thus for generic almost complex structures $J'$ close to $J$, there will be no $J'$-holomorphic curves in the class $C$. The theory of pseudoholomorphic curves hence does not provide any obstruction to deforming the symplectic form to one which pairs negatively with $C$. If $C$ is the class of a $(-1)$-sphere, on the other hand, Gromov-Taubes theory shows that any symplectic form deformation equivalent to the Kähler form must pair positively with $C$.
When $p_g=0$ Question \[q\] has an affirmative answer. In this case we have $b^+=1$, so every class of positive square which is positive on $-1$ symplectic spheres is realized by a symplectic form ([@LL]). In addition, for a minimal surface of general type, the canonical class $K$ has been shown to be in the symplectic cone ([@S], [@Ca]).
In a more general setting, the answer to Question \[q\] seems elusive. Our methods do, however, enable us to progress somewhat farther on the following related question:
\[q2\] Let $\{C_1,\ldots,C_n\}$ be reduced irreducible holomorphic curves of negative square, none of which is a $(-1)$-sphere, such that there exist classes $\alpha$ in the positive cone of $H^{1,1}_{J}$ satisfying $\langle \alpha,
[C_i]\rangle<0$ for each $i$. Do some of these classes $\alpha$ admit symplectic forms?
In Section 4 we outline methods for using Theorem \[main\] to answer this question in certain situations, and we illustrate these methods by applying them in detail to all of the subsets of a particular set of 21 negative-square curves in a rigid surface $K$ that was introduced in [@KK]. We choose a rigid surface as our primary example in order to ensure that the curves in question cannot be made to disappear by an integrable variation in the complex structure; as such, we may state with certainty that the new symplectic forms that we construct are not directly obtainable by considerations of Kähler geometry.
The methods of Section 4 can be applied to a wide variety of configurations of the curves $C_1,\ldots,C_n$ in Question \[q2\], but there are also many configurations to which these methods do not apply. It seems unlikely that there is any necessary and sufficient condition on the configuration that can be expressed at all concisely, but we provide an example of a moderately general sufficient condition in Theorem \[ADE\].
We would like to thank D. McDuff for her valuable suggestions on how to extend her result in [@Mc4] to our situation. The first author is also grateful to Y. Ruan for discussions on the 6-dimensional symplectic minimal model program which inspired Theorem \[main\].
The construction
================
Theorem \[main\] is an application of the normal connected sum construction with symplectic $S^2-$bundles. So let us collect some facts about symplectic structures on such manifolds and embedded symplectic surfaces in them.
Up to diffeomorphisms, there are two orientable $S^2-$bundles over a Riemann surface $\Sigma$: the trivial one $\Sigma\times S^2$, and the non-trivial one $M_{\Sigma}$. By [@LM], symplectic forms on $S^2-$bundles are determined by their cohomology classes up to isotopy. Thus we can pick any convenient symplectic form in a fixed cohomology class.
We begin with the easier case: the product bundle. In this case we use split forms as our model forms. Clearly every class of the positive cone is represented by a split symplectic form. And for a split symplectic form, the vertical fibers and horizontal sections are symplectic. The class of any section with (even) positive square is then represented by an immersed symplectic surface with only positive transverse self-intersections, which can then be smoothed to an embedded symplectic surface.
Now let us deal with the non-trivial bundle $M_{\Sigma}$ over a positive genus surface. We use Kähler forms as our model forms. The following result is essentially contained in [@Mc3] and [@H] (we present it here since it may not be very well-known).
\[stablebundle\] Let ${\mathcal E}$ be a holomorphic rank 2 bundle over $\Sigma$ with $g(\Sigma)>0$ and $c_1({\mathcal E})=-1$. Let $(M,J_{\mathcal E})$ be the complex ruled surface $P({\mathcal E})$. Then the Kähler cone is the positive cone if and only if ${\mathcal
E}$ is stable. Furthermore, for appropriately chosen holomorphic structures on ${\mathcal E}$, the class of any section with (odd) positive square can be represented by an embedded surface which is symplectic with respect to any Kähler form.
Notice that the slope of ${\mathcal E}$ is $-{1\over 2}$. Therefore the stability of ${\mathcal E}$ is equivalent to the statement that every holomorphic line subbundle ${\mathcal L}$ of ${\mathcal E}$ has $c_1({\mathcal L})\leq -1$. Observe that any holomorphic line bundle ${\mathcal L}\subset
{\mathcal E}$ gives rise a to a holomorphic section $Z({\mathcal
L})$ of $P({\mathcal E})$, and vice versa. Since the normal bundle to $Z({\mathcal L})$ is ${\mathcal L}^*\otimes {\mathcal
E}/{\mathcal L}$, all sections of $P({\mathcal E})$ have positive self-intersection if and only if $\mathcal{E}$ is stable. The statement about Kähler cone now follows from the arguments in Proposition 3.1 in [@Mc3] (see also [@H]).
For the second statement, it suffices to show that the class $[s^+]$ of a section with square $+1$ is symplectic. As all the fibers are holomorphic and hence symplectic and the classes of sections with higher squares have form $[s^+]+m[fiber]$ for $m>0$, these classes are represented by positively immersed symplectic surfaces, which can be smoothed to embedded ones. We may take the holomorphic structure on ${\mathcal E}$ to be that on a non-trivial extension of ${\mathcal L}$ by the trivial line bundle ${\mathcal
O}$, where ${\mathcal L}$ is a degree $-1$ holomorphic line bundle. The section $Z({\mathcal L})$ is then a holomorphic, and so in particular symplectic, $+1$ section.
Finally, the non-trivial bundle over a sphere is diffeomorphic to the blow up of ${\mathbb CP}^2$ at a point, and the exceptional divisor is a section with square $-1$. As is well-known, either using the standard symplectic reduction picture or algebraic geometry, we can construct symplectic forms in every class in the positive cone which is positive on the class of a section with square $-1$, such that, for every odd $k\geq -1$, there are symplectic sections with square $k$.
Now we are ready to prove Theorem \[main\].
Let $R$ be the trivial sphere bundle over the surface of genus $g(C)$ if $k$ is even, and the non-trivial one if $k$ is odd. Let $s^{\pm k}$ be the class of a section with square $\pm k$. Then $s^{+k}$ and $s^{-k}$ form a basis for $H_2(R;{\mathbb
Z})$. Since $s^{+k}\cdot s^{-k}=0$, a cohomology class of the form $$c^+PD(s^{+k})+c^-PD(s^{-k})$$ has positive square if and only if $c^+>|c^-|$.
Suppose first that $C$ is not a sphere with $k$ odd. By Proposition \[stablebundle\] and the discussions preceding it, there exists a symplectic form $\tau_t$ on $R$ in the class $${a\over
k}PD(s^{+k})+(t-{a\over k})PD(s^{-k})$$ for any $t\in (0, {2a\over
k})$. By Proposition \[stablebundle\], there is a $\tau_t-$symplectic section $S^{+k}$ in the class $s^{+k}$. Notice that the symplectic surfaces $C$ and $S^{+k}$ have opposite self-intersection and equal symplectic area $a$. Thus we can perform the symplectic sum construction to $(M,C,\omega)$ and $(R, S^{+k},\tau_t)$ to obtain a new symplectic manifold $(X,\omega_t)$. As observed in [@Go], $X$ and $M$ are diffeomorphic. Moreover, because the surface $S^{-k}$ is disjoint from surface $S^{+k}$ in $R$, it becomes a surface in $M$ which is homologous to $C$. Thus we have $$\omega_t(e)=\tau_t(s^{-k})=a-tk.$$ Therefore $[\omega_t]=[\omega]+tPD(e)$.
In the case that $C$ is a sphere with $k$ odd, by the discussions after Proposition \[stablebundle\], there exists a symplectic form in the class $c^+ PD(s^{+1})+ c^- PD(s^{-1})$ if and only if $c^+>-c^->0$. We would like to express the condition in terms of the basis $s^{+k}$ and $s^{-k}$. Since $s^m\cdot s^n={m+n\over 2}$, we see that a class $\alpha$ contains a symplectic form if and only if $${\alpha(s^{-k})\over \alpha(s^{+k})}>{1-k\over k+1}.$$ Thus the allowed values of $t$ are those between $0$ and ${2a\over
k+1}$, as claimed.
Notice that $$\omega_t\cdot \omega_t=\omega^2+2t\omega(e)+t^2e\cdot e=-t^2k+2ta +\omega^2=\omega^2+k[{a^2\over k^2}-(t-{a\over k})^2].$$ So the volume of the symplectic manifold $(M,\omega_t)$ is greater than that of $(M,\omega)$ for each $t\in (0, 2a/k)$.
We can generalize Theorem \[main\] to a configuration of transversally intersecting symplectic surfaces.
\[configuration\] Suppose $C_1,..., C_l$ is a set of embedded symplectic surfaces with self-intersection $C_i\cdot C_i=-k_i<0$ and intersecting positively and transversally. Let $e_i$ be the class of $C_i$. Then there are symplectic forms in the class $[\omega]+\sum_i t_i PD(e_i)$ for any $0<t_i<2\omega(e_i)/h_i$, where $h_i$ is as in Theorem \[main\].
We prove the theorem in the case that there are only two curves $C_1$ and $C_2$. The idea for the general case is the same.
The key point is that the symplectic sum construction in Theorem \[main\], when applied to $C_1$, can be done in a way such that $C_2$, possibly after an isotopy, is still symplectic with respect to the new symplectic structures $\omega_1$, which is in the class $[\omega]+t_1PD(e_1)$ with $t_1\in (0, 2\omega(e_i)/h_1)$. This is possible due to the pairwise sum feature in [@Go].
First, by applying Lemma 2.3 of [@Go], perturb $C_2$ such that $C_2$ intersects $C_1$ orthogonally with respect to $\omega$. Since the fiber spheres in $R$ are symplectic and intersect the symplectic section $S^{+k_1}$ transversally, we can likewise assume that the symplectic section $S^{+k_1}$ intersects a total of $k_{12}=C_1\cdot
C_2$ fibers, all orthogonally. Denote this union of the fibers by $F$. Now apply pairwise sum to $(M, C_1, C_2)$ to $(R, S^{+k_1}, F)$ to get a symplectic surface $C_2'$. Finally, apply the symplectic sum construction to $C_2'$ as in the proof of Theorem \[main\].
Notice that since $e_1\cdot e_i\geq 0$ for $i\geq 2$, $[\omega]+t_iPD(e_1)$ is positive on $e_2$ for $t_1$ positive. One has $S_i^+=S_i^-+k_if$ where $f$ is the homology class of the fiber in $R$, so $$\int_{S_i^-}\tau=\int_{C_i}\omega+t_iPD(e_i)=a_i-t_ik_i=\int_{S_i^+}\tau-k_i\tau(f).$$ Thus $\tau(f)=t_i$. This is consistent with the normal connected sum picture. The area of the surface $C_j$ increases by $(e_j\cdot
e_i)\tau(f)$, which is indeed equal to $t_i(e_i\cdot e_j)$.
If these surfaces actually intersect, then some of the values of $t_i$ can be taken larger than in the statement of the theorem.
Configurations of embedded symplectic surfaces and pseudo-holomorphic curves
============================================================================
In attempting to answer questions such as Question \[q\], we might wish to apply Theorem \[configuration\] to some finite set of holomorphic curves. However, the proof of Theorem \[configuration\] depends on the assumption that the symplectic submanifolds being considered intersect positively and transversely, which is a property that our set of holomorphic curves might not be known to have. Assume that we are given a collection of distinct $J$-holomorphic curves $C_1,\ldots,C_k$ in the symplectic $4$-manifold $M$ (we adopt the convention that a $J$-holomorphic curve is the image of a generically injective $J$-holomorphic map from some irreducible compact Riemann surface). Corollary 4.2.1 of [@Mc4] asserts that, at the possible cost of $C^1$-slightly changing the almost complex structure $J$, we may perturb any one of these curves to a pseudo-holomorphic immersion. We first give a simple modification of McDuff’s argument to show that, in fact, we may perturb all of the curves and the almost complex structure simultaneously so that $C_1,\ldots,C_k$ become immersed.
\[immerse\] Let $u_i\co \Sigma_i\to M$ be $J$-holomorphic maps with images $C_i$. Then given $\operatorname{\epsilon}>0$ there are an almost complex structure $\tilde{J}$ and $\tilde{J}$-holomorphic immersions $\tilde{u}_i\co\Sigma_i\to M$ such that $\|\tilde{u}_i-u_i\|_{C^2}<\operatorname{\epsilon}$ and $\|\tilde{J}-J\|_{C^1}<\epsilon$.
Let $p\in M$ be a critical value for one or more of the $u_i$. It is shown in [@Mc4] that the various $u_i$ each have just finitely many critical points, so denote the various critical points in $\cup \Sigma_i$ having image $p$ by $z_1,\ldots, z_m$. For $j=1,\ldots,m$, if $z_j\in \Sigma_i$ let $D_j\subset \Sigma_i$ be a disc around $z_j$, and let $v_j=u_i|_{D_j}$. By shrinking the various $D_j$, we assume that the $D_j$ are disjoint and that $z_j$ is the only critical point of the restriction $v_j$. Since the intersections (and self-intersections) of the various $C_i$ are isolated, let $U\subset M$ be a coordinate neighborhood of $p$ in which the $C_i$ meet each other and themselves only at $p$ and such that for each $j$ $v_{j}^{-1}(U)\subset D_j$ is a connected component of $\cup u_{i}^{-1}(U)$. Shrinking $U$ if necessary, assume also that $U$ contains no critical values of the various $u_i$ other than $p$. Now fix neighborhoods $W_m\subset
U_m\subset\cdots\subset W_1\subset U_1\subset U$ of $p$. By Theorem 4.1.1 of [@Mc4], there is a family $v_{1}^{\delta}$ ($\delta>0$) of $J$-holomorphic immersions $D_1\to M$, converging in $C^2$ norm to $v_1$ as $\delta\to 0$. For $\delta$ small, define $\tilde{v}_1(z)=\chi(z) v_{1}^{\delta}(z)+(1-\chi)(z)v_1(z)$, where $\chi$ is a smooth cutoff function which is $1$ on a neighborhood of $v_{1}^{-1}(W_1)$ and $0$ on a neighborhood of the complement of $v_{1}^{-1}(U_1)$. $\tilde{v}_1$ is then $C^2$-close to $v_1$, so there is an almost complex structure $J'_1$ which agrees with $J$ away from $U_1\setminus W_1$, makes $\tilde{v}_1$ $J'_1$ holomorphic, and is $C^1$-close to $J$ everywhere (see the proof of Corollary 4.2.1 of [@Mc4], or the proof of Proposition \[pert\] below). Furthermore, if $U\cap Im v_1$ is a distance at least $K$ from $\cup C_i\setminus Im v_1$, then for $\delta$ small enough $U\cap Im
\tilde{v}_1$ will be a distance $K/2$ from $\cup C_i\setminus Im
v_1$, and so using a cutoff function supported in a $(K/3)$-neighborhood of $Im (\tilde{v}_1)\cap (U_1\setminus W_1)$, we can patch together $J$ and $J'_1$ to obtain an almost complex structure $\tilde{J}_1$ which is $C^1$-close to $J$, agrees with $J$ outside $U_1\setminus W_1$ and on a neighborhood of $\cup
C_i\setminus Im(\tilde{v}_1)$, and makes $\tilde{v}_1$ pseudo-holomorphic.
With this done, we now apply the same procedure sequentially to $v_2,\ldots,v_m$, obtaining almost complex structures $\tilde{J}_j$ which are $C^1$-close to $J$ globally and which agree with $\tilde{J}_{j-1}$ both near $\cup C_i\setminus Im(v_j)$ and outside $U_j\setminus W_j$, and $\tilde{J}_{j}$-holomorphic immersions $\tilde{v}_j$ which are $C^2$-close to $v_j$. Modifying the original maps $u_i\co \Sigma_i\to M$ by replacing the restrictions $v_j\co D_j\to M$ by $\tilde{v}_j$, we get $\tilde{J}_m$-holomorphic maps $\tilde{u}_i$ which have no critical values inside $U$ and agree with the $u_i$ outside $U$. So we have reduced the number of critical values by 1, and repeating the process at each critical value gives the almost complex structure $\tilde{J}$ and the $\tilde{J}$-holomorphic immersions $\tilde{u}_i$ that we desire.
Applying Lemma \[immerse\], we may assume that we now have a set of distinct immersed $J$-holomorphic curves $C_i$, and we aim now to show that these curves may be perturbed further to a set of symplectic surfaces $C'_i$ whose intersections are all transverse and positive with $C'_i\cap C'_j\cap C'_k=\varnothing$ when $i$,$j$,$k$ are all distinct. In fact, our perturbed curves $C'_i$ will agree with $C_i$ outside an arbitrarily small neighborhood of the initial intersection points; will be arbitrarily $C^1$-close to $C_i$ (from which it immediately follows that they are symplectic); and will be made simultaneously pseudoholomorphic by an almost complex structure $J'$ arbitrarily $C^1$-close to $J$.
We start by finding a nice coordinate system near any given intersection point of our curves. In the case where $J$ is integrable, any given holomorphic coordinate chart can be modified by an element of $GL(2,\mathbb{C})$ to satisfy the conditions we need, so the arguments below are only needed in the non-integrable case.
\[coord\] Given immersed $J$-holomorphic curves $C_0,\ldots,C_m\subset M$ all having an isolated intersection at the point $p$, there is a coordinate chart $U$ around $p$ with coordinates $z,w$ such that:
- $C_0\cap U=\{(z,w)\in U|w=0\}$,
- Each set $\{(z,w)\in U|w=const\}$ is $J$-holomorphic, and
- For $i\geq 1$ there are smooth functions $g_i$ of the form $g_i(z)=a_iz^{k_i}+O(|z|^{k_i+1})$ $(a_i\neq 0$,$k_i\geq 1)$ such that $C_i\cap U=\{(z,g_i(z))\}$
A coordinate chart $U_0=\{(z',w')\}$ satisfying (i) and (ii) may be constructed by using Lemmas 5.4 and 5.5(d) of [@Ta]. To obtain condition (iii), first note that for a generic linear change of coordinates $(z',w')\mapsto (z'+cw',w')$ we retain properties (i) and (ii) and additionally ensure that $\{z'=0\}$ is transverse to each of the $C_i$. Now condition (ii) implies that the antiholomorphic tangent space of our almost complex structure $J$ is given in these coordinates by $$T^{0,1}_{J}=\langle
\partial_{\bar{z'}}+\alpha(z',w')\partial_{z'},v(z',w')\rangle$$ for a certain function $\alpha$ and complexified vector field $v$. By Ahlfors-Bers’ Riemann mapping theorem with smooth dependence on parameters [@AB], the equation $u_{\bar{z'}}+\alpha(z',w')u_{z'}=0$ can be solved for a smooth function $u(z',w')$ with $u(0,w')=0$.
Changing coordinates to $(z,w)=(z'+u(z',w'),w')$, we have that $\{z=0\}=\{z'=0\}$ is transverse to each of the $C_i$, so that after possibly shrinking the coordinate chart $U$ we have $C_i\cap
U=\{(z,g_i(z))\}$ for some smooth functions $g_i$. In terms of the coordinates $(z,w)$, we have for certain functions $a$ and $b$ both vanishing at the origin,$$T^{0,1}_{J}=\langle
\partial_{\bar{z}},\partial_{\bar{w}}+a(z,w)\partial_w+b(z,w)\partial_z\rangle.$$ It is then a simple matter to check that a curve $\{(z,g(z))\}\subset U$ is $J$-holomorphic exactly if$$b(z,g(z))=\frac{g_{\bar{z}}-a(z,g(z))g_z}{|g_{\bar{z}}|^2-|g_z|^2}.$$ But then the fact that $a(z,g(z))$ and $b(z,g(z))$ are smooth functions of $z$ and vanish at $z=0$ implies that the lowest-order terms in the Taylor expansion of $g$ are functions only of $z$ and not of $\bar{z}$. Of course, our functions $g_i$ can’t be constants (since the $C_i$ ($i\geq 1$) have an isolated intersection point with $C_0=\{w=0\}$ at the origin), so it follows that the $g_i$ all have the form specified in condition (iii).
\[pert\] In the situation of Lemma \[coord\], given any sufficiently small $\delta>0$ there is a surface $C^{\delta}_{0}$ such that, where $B_{\delta}=\{(z,w)\in U||z|< \delta\}$, $C^{\delta}_{0}\cap(X\setminus B_{\delta})=C_0\cap(X\setminus
B_{\delta})$, while all intersection points of $C^{\delta}_{0}$ with $C_i$ ($i>0$) that are contained in $B_{\delta}$ are in fact contained in $B_{\delta^2}$ and are transverse, positive, and distinct from $p$ and from each other as $i$ varies. Further there is a constant $A$ depending on the curves $C_i$ but not on $\delta$ such that $dist_{C^2}(C^{\delta}_{0},C_0)\leq A\delta^2$, and there is an almost complex structure $J'$ agreeing with $J$ near $C_i$ ($i>0$) and making $C^{\delta}_{0}$ holomorphic with $\|J'-J\|_{C^1}\leq A\delta^2$.
Work in coordinates provided by the conclusion of Lemma \[coord\], and let $c_i$ be constants such that for $i>0$ $$|g_i(z)-a_iz^{k_i}|<c_i|z^{k_i+1}|.$$ Given $\operatorname{\epsilon}>0$ write $$R_{\operatorname{\epsilon}}=\max_{i\geq 1}\left(\frac{2\operatorname{\epsilon}}{|a_i|}\right)^{1/k_i};$$ we will only work with $\operatorname{\epsilon}$ so small that $$\sqrt{R_{\operatorname{\epsilon}}}<\min_{i\geq 1}\frac{|a_i|}{2c_i}$$ Then for any such $\operatorname{\epsilon}$, if $R_{\operatorname{\epsilon}}\leq |z| \leq \sqrt{R_{\operatorname{\epsilon}}}$, we have$$|g_i(z)|\geq (|a_i|-c_i|z|)|z|^{k_i}>\frac{|a_i|}{2}|z|^{k_i}\geq
\operatorname{\epsilon}.$$
Write $\delta=\sqrt{R_{\operatorname{\epsilon}}}$; note that we may alternatively express $\operatorname{\epsilon}$ in terms of an arbitrary $\delta >0$, and then for $\delta$ small enough $\operatorname{\epsilon}$ is bounded by a constant times $\delta^2$. Fix a cutoff function $\chi(z)$ with image $[0,1]$ equal to one for $|z|\leq \delta^2$ and zero for $|z|\geq\delta$, with $\|\chi\|_{C^2}<\frac{4}{\delta^2}$. Let $C_{0}^{\delta}=\{(z,\operatorname{\epsilon}^2\chi(z))\}$; obviously $C^{\delta}_{0}$ agrees with $C_0$ outside $B_{\delta}$. Since $\operatorname{\epsilon}^2\chi(z)\leq
\operatorname{\epsilon}^2<\operatorname{\epsilon}$ while each $|g_i(z)|>\operatorname{\epsilon}$ for $|z|\in[\delta^2,\delta]$, evidently the intersection points of $C_i$ with $C^{\delta}_{0}$ contained in $B_{\delta}$ are just those points $(z,\operatorname{\epsilon}^2)$ with $|z|<\delta^2=R_{\operatorname{\epsilon}}$ such that $g_{i}(z)=\operatorname{\epsilon}^2$.
Write $\tilde{g}_i(z)=\operatorname{\epsilon}^{-2}g_i\left(
\left(\frac{\operatorname{\epsilon}^2}{a_i}\right)^{1/k_i}z\right)$; then $$\tilde{g}_i(z)=1\Leftrightarrow
g_i\left(\left(\frac{\operatorname{\epsilon}^2}{a_i}\right)^{1/k_i}z\right)=\operatorname{\epsilon}^2,$$ so the intersections of $C_{i}$ with $C^{\delta}_{0}$ are of just the same type as the intersections of the graph of $\tilde{g}_i(z)$ with $\{w=1\}$. Now we see that $\tilde{g}_i(z)=z^{k_i}+\tilde{r}_i(z)$ where $|\tilde{r}_i(z)|\leq\tilde{c}_i\operatorname{\epsilon}^{2/k_i}|z|^{k_i+1}$. Hence the graph of $\tilde{g}_i(z)$ is $O(\operatorname{\epsilon}^{2/k_i})$ away in $C^1$ norm from that of $z\mapsto z^{k_i}$, so since the latter’s only intersections with $\{w=1\}$ are positive and transverse at the $k_i$th roots of unity, for $\operatorname{\epsilon}$ small enough the graph of $\tilde{g_i}$ will also have just $k_i$ distinct positive transverse intersections with $\{w=1\}$, each at a point a distance $O(\operatorname{\epsilon}^{2/k_i})$ from a different one of the $k_i$th roots of unity. Scaling back, we conclude that the intersections of $C_i$ with $C^{\delta}_{0}$ that are contained in $B_{\delta}$ are in fact contained in $B_{\delta^2}$ and are transverse, positive, and located at points a distance $O(\operatorname{\epsilon}^{4/k_i})$ from the various $(\operatorname{\epsilon}^2/a_i)^{1/k_i}\eta$ for $\eta$ a $k_i$th root of unity.
Obviously for any given $i$ the points of $C_i\cap C^{\delta}_{0}$ are all distinct for small enough $\operatorname{\epsilon}$. For small enough $\operatorname{\epsilon}$ these intersections vary continuously in $\operatorname{\epsilon}>0$, so if it weren’t the case that the sets $C_i\cap C_j\cap C^{\delta}_{0}$ were all eventually empty for $\operatorname{\epsilon}$ small enough and $i,j$ distinct, we would then, by varying $\operatorname{\epsilon}$, obtain a continuous family of points in $C_i\cap C_j$, which is impossible since $C_i$ and $C_j$ are distinct holomorphic curves and so have isolated intersections.
Finally, note that $\|\operatorname{\epsilon}^2\chi\|_{C^2}\leq\operatorname{\epsilon}^2(4/\delta^2)\leq
A\delta^2$ for a certain constant $A$ and $\delta$ sufficiently small, so that $C^{\delta}_{0}$ is indeed less than $A\delta^2$ away from $C_0$ in $C^2$ norm. Letting $\beta (w)$ be a cutoff function which is $1$ for $|w|<2\operatorname{\epsilon}^2$ and $0$ for $|w|\geq \operatorname{\epsilon}$, if $J$ is defined by $T^{0,1}_{J}=\langle
\partial_{\bar{z}},v\rangle$ then setting $$T^{0,1}_{J'}=\langle
\partial_{\bar{z}}+\beta(w)\left((\operatorname{\epsilon}^2\chi)_{\bar{z}}\partial_w+(\overline{\operatorname{\epsilon}^2\chi})_{\bar{z}}\right)\partial_{\bar{w}},v\rangle$$ defines an almost complex structure $J'$ which makes $C^{\delta}_{0}$ holomorphic and which (since $|g_i(z)|>\operatorname{\epsilon}$ whenever $\nabla(\operatorname{\epsilon}\chi)\neq 0$) agrees with $J$ near $C_i$ for $i>0$. Further one easily sees that $\|J'-J\|_{C^1}=O(\operatorname{\epsilon})\leq
O(\delta^2)$.
\[tvs\] Any set of distinct $J$-holomorphic curves $C_0,\ldots,C_m$ can be perturbed to symplectic surfaces $C'_0,\ldots,C'_m$ whose intersections are all transverse and positive, with $C'_i\cap
C'_j\cap C'_k=\varnothing$ when $i,j,k$ are all distinct. Furthermore, there is an almost complex structure $J'$ arbitrarily $C^1$-close to $J$ such that the $C'_i$ are $J'$-holomorphic.
Assume that the process used in the proof of the above proposition has been repeated to yield surfaces $C^{\delta_0}_{0},\ldots,C^{\delta_i}_{i}$ each missing $p$ and hitting the other $C_j$ transversely and positively. Let our neighborhood $U$ and the parameter $\delta_{i+1}$ be so small that each $C^{\delta_{j}}_{j}$ ($j\leq i$) misses $B_{\delta_{i+1}}$ (this is possible since the $C^{\delta_{j}}_{j}$ all miss $p$); then since $C_{i+1}^{\delta_{i+1}}\cap (X\setminus
B_{\delta_{i+1}})=C_{i+1}\cap (X\setminus B_{\delta_{i+1}})$, the intersection points of $C^{\delta_{i+1}}_{i+1}$ with $C_{j}^{\delta_j}$ ($j\leq i$) are the same as those of $C_{i+1}$ and $C_{j}^{\delta_j}$, and so are transverse, positive, and away from $p$. By the proposition, we have the same conclusion for the intersection points of $C^{\delta_{i+1}}_{i+1}$ with $C_j$ ($j>i+1$). So by induction we may perturb all of the $C_i$ to $C'_i=C^{\delta_i}_{i}$ with the desired intersection configuration. Moreover by choosing $\delta_0>\delta_1>\cdots \delta_{m-1}>0$ small enough, the $C^{\delta_i}_{i}$ can be made arbitrarily $C^2$-close to the $C_i$, so since the property of being a symplectic submanifold persists under $C^1$-small perturbations, the $C^{\delta_i}_{i}$ can be taken to all be symplectic. Repeating this local construction at all of the intersection points of two or more of the $C_i$ gives the global result.
Towards a symplectic Nakai-Moishezon criterion
==============================================
In this subsection let $(M,J)$ be a minimal Kähler surface and $H_J^{1,1}$ denote the real part of the $(1,1)-$subspace of $H^2(M;{\mathbb C})$ determined by $J$. We apply Theorem \[main\] to study the symplectic classes in $H_J^{1,1}$.
Given a homology class $e$, we define the reflection along $e$ to be $$R_e(\alpha)= \alpha-2{\alpha(e)\over e\cdot e}PD(e).$$ Notice that this is an automorphism of $H^2(M;{\mathbb Q})$ preserving the intersection form. But it is an automorphism of $H^2(M;{\mathbb Z})$ only if $e\cdot e=-1$ or $-2$. Geometrically, the annihilator of $e$ is a hyperplane in $H^2(M;{\mathbb R})$ which we call the “$e$-wall,” and $R_e$ is the reflection across this hyperplane.
A homology class $e$ is called small and effective if it is represented by a reduced irreducible holomorphic curve with negative self-intersection.
Notice that there is only one holomorphic curve $C$ representing a small and effective class.
\[refl\] Let $e$ be a small and effective class which is not represented by a curve of zero arithmetic genus and odd self-intersection. Then the reflection of the Kähler chamber along the $e-$wall is contained in the symplectic cone.
Let $x$ be a point in the Kähler cone. The Kähler cone is open in $H^{1,1}_{J}$, since the sum of a small closed real $(1,1)$ form and a Kähler form on a closed manifold is still a closed positive $(1,1)$ form, hence a Kähler form. Thus, for small $\epsilon$, $x-\epsilon e$ is also in the Kähler cone, and hence represented by a Kähler form $\omega$. By Proposition \[pert\], we can perturb $C$ to get an embedded $\omega-$symplectic surface, still denoted by $C$. Applying Theorem \[main\] to $\omega$ and $C$, we see that $R_e(x)=[\omega_t]$ for some $t$.
For an embedded $-2$ rational curve $C$, there is a diffeomorphism whose induced action on cohomology is $R_{[C]}$. Pulling the Kähler form back by this diffeomorphism gives an alternative way of enlarging the Kähler cone by reflection. However, this latter method, unlike Theorem \[main\], does not allow us to obtain symplectic forms in classes which vanish on the $(-2)$-curve.
We mention a simple case where the symplectic Nakai-Moishezon criterion can be established.
Suppose that $H_2(M;{\mathbb Z})$ contains only one small and effective class, $e$, and that $e$ is not represented by a sphere of odd square. Then every class $\alpha$ in the positive cone which is negative on $e$ lies in the image of the Kähler chamber under $R_e$. Therefore the symplectic Nakai-Moishezon criterion holds in this case.
Suppose $e\cdot e=-k$ and $\alpha$ is as in the statement of the proposition. Choose $s>0$ such that $$\alpha^2+2s|\alpha(e)|>s^2k>2s|\alpha(e)|.$$ Let $\beta=\alpha-sPD(e)$. Then $$\beta(e)=\alpha(e)+sk>|\alpha(e)|, \beta^2=\alpha^2+2s|\alpha(e)|-s^2k>0, \beta\cdot \alpha=\alpha^2+|\alpha(e)|>0.$$ Therefore $\beta$ is in the Kähler cone by Theorem \[bl\]. Now apply Theorem \[main\] to $\beta$.
The much more common situation in which $M$ contains more than one small and effective class is more difficult to analyze. We begin by establishing the following finiteness result, which might be known to experts.
\[finite\] For any (1,1) class $\alpha$ with positive square and in the positive cone, there are only finitely many classes which are represented by reduced irreducible holomorphic curves and pair non-positively with $\alpha$. Further, the intersection form on $M$ is negative definite on the subspace of $H^{1,1}_{J}$ spanned by the Poincaré duals of these classes.
Suppose $e_i$ are distinct such classes with negative square which are represented by reduced irreducible holomorphic curves. Notice that $e_i\cdot e_j\geq 0$ if $i\ne j$.
Then if a finite positive linear combination of $e_i$, say $\sum_i
a_ie_i$, has non-negative square, it must be in the positive cone or its boundary, as $\omega$ is positive on each $e_i$, $a_i\geq
0$, and $\omega$ itself in the positive cone. By the Hodge index theorem, as $\alpha$ is also in the positive cone, $\alpha$ is strictly positive on $\sum_i a_ie_i$. But $\alpha$ is non-positive on each $e_i$, so $\alpha$ is non-positive on $\sum_i a_ie_i$ as $a_i\geq 0$.
This contradiction shows that any positive linear combination of the $e_i$ has negative square. But this implies that for *any* $a_i\in {\mathbb R}$ not all zero we have, using positivity of intersections between the distinct $e_i$, $$\begin{aligned}
\left(\sum a_i e_i\right)^2 &=&\sum_i
a_{i}^{2}e_{i}^{2}+2\sum_{i<j} a_i a_je_i\cdot e_j\leq \sum_i
|a_{i}|^{2}e_{i}^{2}+2\sum_{i<j} |a_i| |a_j|e_i\cdot e_j\\
&\leq&\left(\sum |a_i| e_i\right)^2<0.\end{aligned}$$ Thus the $e_i$ are linearly independent, and they span a negative definite subspace of $H_2(X;{\mathbb{Z}})$. In particular, there are at most $h^{1,1}-1=b^-$ many $e_i$.
In view of the lemma above, we make the following definition. Here ${\mathcal P}$ denotes the positive cone in $H^{1,1}_{J}(X;{\mathbb
R})$.
A finite set of small and effective classes $G=\{e_1,...,e_l\}$ is called admissible if they are linearly independent, and the intersection form on the subspace of $H_2(M;\mathbb{Z})$ generated by these $e_i$ is negative definite. Given an admissible set $G$, the $G-$chamber is $${\mathcal C}(G)=\{\alpha\in{\mathcal P}|\alpha(e_i)\leq 0 \hbox{ if $e_i\in G$}, \quad \alpha(e)> 0
\hbox{ if $e\not \in G$} \}.$$ The $G-$corner is $${\mathcal C}^c(G)=\{\alpha\in{\mathcal P}|\alpha(e_i)= 0 \hbox{ if $e_i\in G$}, \quad \alpha(e)> 0
\hbox{ if $e\not \in G$} \}.$$
The following simple observation will be useful.
\[allplus\] Let $M$ be a symmetric negative definite matrix such that $M_{ij}\geq 0$ if $i\neq j$. Then every entry of $-M^{-1}$ is non-negative.
By multiplying $M$ by a scalar assume without loss of generality that all diagonal entries and all eigenvalues of $M$ are greater than $-1$. Then, where $I$ is the identity, $I+M$ has all its entries nonnegative and all its eigenvalues between 0 and 1. The latter condition implies that we have a convergent Taylor series expansion $$-M^{-1}=(I-(I+M))^{-1}=\sum_{n=0}^{\infty}(I+M)^n,$$ and the proposition follows from the fact that the set of matrices with all entries nonnegative is closed under addition and multiplication.
\[chambers\] The chambers ${\mathcal C}(G)$ for admissible sets $G$ form a partition of the positive cone and are all nonempty, as are the $G-$corners $\mathcal{C}^c(G)$. Each $G-$chamber and each $G-$corner is convex and hence connected.
That the $\mathcal{C}(G)$ form a partition of the positive cone follows directly from Lemma \[finite\]. Convexity is obvious from the definitions.
To see that each $\mathcal{C}(G)\neq \varnothing$, let $G=\{e_1,\ldots,e_n\}$ be an admissible set and denote by $M$ the matrix representing the restriction of the intersection form to the span of $G$, so that $M$ is negative definite. Pick an arbitrary $\alpha$ in the Kähler cone, and let $v_i=\langle
\alpha,e_i\rangle$, so that each $v_i>0$. Then where $\vec{t}=-M^{-1}\vec{v}$ and $\alpha'=\alpha+\sum t_iPD(e_i)$, we have $\langle \alpha',e_j\rangle=v_j-v_j=0$ for each $j$, and $$(\alpha')^2=\alpha^2+2\vec{v}\cdot\vec{t}+(M\vec{t})\cdot\vec{t}=\alpha^2-(M\vec{t})\cdot\vec{t}\geq
\alpha^2>0$$ since $M$ is negative definite, so $\alpha'$ is in the positive cone. Also, by Proposition \[allplus\], we have each $t_i>0$ since each $v_i>0$, so if $e$ is small and effective with $e\notin G$ then by positivity of intersections $\langle
\alpha',e\rangle\geq \langle \alpha,e\rangle>0$. Thus $\alpha'\in
\mathcal{C}^c(G)$, and $\mathcal{C}^c(G)$ is nonempty. Where $s_i=-\sum (M^{-1})_{ij}$, $\alpha'+\operatorname{\epsilon}\sum s_iPD(e_i)$ will evaluate as $-\operatorname{\epsilon}$ on each $e_i$, will be positive on each $e\notin
G$ (noting that each $s_i>0$), and will remain in the positive cone for small $\operatorname{\epsilon}>0$, so $\mathcal{C}(\{e_1,\ldots,e_n\})$ is also nonempty.
By Theorem \[bl\], the Kähler cone is just ${\mathcal C}(\emptyset)$. Within the positive cone, the *boundary* of the Kähler cone is the disjoint union of the $\mathcal{C}^c(G)$ over the admissible sets $G$.
Applying Theorem \[main\] with $\omega$ equal to a Kähler form, $e=[C]$, and $t$ between $a/k$ and $2a/h$ shows that each chamber ${\mathcal C}(e)$ contains symplectic classes. Iterating Theorem \[main\], the same can be said for any $G-$chamber ${\mathcal
C}(e_1,\ldots,e_n)$ with $e_i\cdot e_j=0$ for $i\ne j$.
We can apply Theorem \[configuration\] to show that more general $G-$chambers contain symplectic classes. To do this, it suffices to show that the corresponding $G$-corner contains symplectic classes, since as in the proof of Lemma \[chambers\] suitably chosen arbitrarily small perturbations of these will lie in $\mathcal{C}(G)$ and will remain symplectic. Under suitable hypotheses on the set $G$, we shall see that every class in the $G$-corner $\mathcal{C}^c(G)$ contains symplectic forms.
Accordingly, let $\alpha\in H^{1,1}_{J}(M;\mathbb{R})$ be an arbitrary class in the boundary of the Kähler cone and have positive square, so that $\alpha$ satisfies $\langle
\alpha,D\rangle\geq 0$ for every effective divisor $D$. $\alpha$ is then in some $G-$corner; say $G=\{e_1,\ldots,e_n\}$, so that $\alpha$ vanishes only on the $e_i$ and the $PD(e_i)$ span a negative definite subspace of $H^{1,1}_{J}(M;\mathbb{R})$. Our strategy for attempting to show that $\alpha$ contains symplectic forms consists of two steps:
- Find $t_i>0$ such that $\alpha-\sum t_i PD(e_i)$ lies in the Kähler cone.
- Beginning with a Kähler form in the class $\alpha-\sum t_i PD(e_i)$, apply the inflation procedure sequentially to the $e_i$ (and/or smoothings of unions thereof) to obtain a symplectic form in class $\alpha$.
We shall show presently that step (i) can always be completed.
\[bdrytoint\] If $\alpha$ and $e_i$ are as above, and if $s_i>0$ are such that $\sum_i s_i e_i\cdot e_j<0$ for every $j$, then for $r>0$ sufficiently small, $\alpha-\sum rs_iPD(e_i)$ admits Kähler forms.
Multiplying the $s_i$ by a small constant if necessary, assume that $\beta:=\alpha-\sum s_i PD(e_i)$ is in the positive cone. By Lemma \[finite\], there are then just finitely many curves on which $\beta$ is non-positive; denote them by $f_1,\ldots,f_m$ (note that the assumption on the $s_i$ implies that none of the $f_j$ is among the $e_i$). Now for each $f_j$ we have $\langle \alpha,f_j\rangle>0$, so since there are only finitely many $f_j$, for $r>0$ small enough $\alpha-\sum rs_i e_i=(1-r)c+rd$ will be positive on each $f_j$. Meanwhile $\langle \alpha,e_i\rangle =0$ and $\langle \beta,e_i\rangle >0$, and if $C$ is any curve not among the $e_i$ and $f_j$ both $\alpha$ and $\beta$ are positive on $[C]$, so for $r>0$ $(1-r)\alpha+r\beta$ is also positive on all curves other than those represented by the $f_j$. Hence by Theorem \[bl\] $(1-r)\alpha+r\beta$ admits Kähler forms if $r>0$ is small enough.
If $\alpha\in H^{1,1}_{J}$ has positive square and lies in the boundary of the Kähler cone, and if $e_1,\ldots,e_n$ are the homology classes of the finitely many curves on which $\alpha$ vanishes, then there are $t_i>0$ such that $\alpha-\sum t_i PD(e_i)$ contains Kähler forms.
By Lemma \[bdrytoint\] it suffices to find $s_i>0$ such that $\sum_i s_i e_i\cdot e_j<0$ for every $j$; we then set $t_i=rs_i$ for $r$ small. Define the $n\times n$ matrix $M$ by $M_{ij}=e_i\cdot e_j$. $M$ is negative definite by Lemma \[finite\], and its off-diagonal entries are nonnegative by positivity of intersections, so $-M^{-1}$ has all nonnegative entries by Proposition \[allplus\]. Then for any $v_i>0$ ($i=1,\ldots,n$), the $s_i=\sum -M^{-1}_{ik}v_k$ will each be positive, and we have $\sum_i s_i e_i\cdot
e_j=-\sum_{i,k}M_{ji}M^{-1}_{ik}v_k=-v_j<0$, as desired.
Carrying out step (ii) of our strategy is more difficult (and often impossible). As we allude to above, instead of applying inflation sequentially to curves $C_i$ representing the $e_i$, we will sometimes wish to smooth the union of the $C_i$ into an embedded symplectic submanifold $C$ (as is always possible since the $C_i$ may be assumed to meet positively and transversely by Corollary \[tvs\]) and then apply the inflation procedure to $C$. Now $C$ will no longer be symplectic after we do this, and in the smoothing construction $C$ will contain all but a small subset of each $C_i$, so the $C_i$ won’t be symplectic either. As such, it will not be possible to apply inflation to $C_i$ after we apply inflation to $C$. The following trick allows us to evade this issue in certain circumstances.
\[disjoin\] Let $C_0,\ldots,C_k$ be symplectic surfaces such that $C_0$ has only positive transverse intersections with the $C_i$ ($i>0$). Assume that $$\#\left(C_0\cap\left(\bigcup_{i\geq 1}
C_i\right)\right)\geq -[C_0]^2$$ Then there exist symplectic surfaces $\tilde{C_0}$ and $C$, homologous to $C_0$ and $\cup_{r\geq
0} C_r$ respectively, such that all intersections between $\tilde{C_0}$ and $C$ are positive and transverse.
Where $m=-[C_0]^2$, assume that, for some points $p_1,\ldots,p_m$, $C_0$ meets the surface $C_{i_j}$ at $p_j$; in complex coordinates $(z,w)$ in a neighborhood $U_j$ around $p_j$ we may assume $C_0\cap
U_j=\{z=0\}$ and $C_{i_j}\cap U_j=\{w=0\}$. By exponentiating a small scalar multiple of a smooth section of the normal bundle to $C_0$ which vanishes negatively precisely at the $m=-[C_0]^2$ points $p_j$, we take for $\tilde{C_0}$ a surface such that $\tilde{C_0}\cap C_0=\{p_1,\ldots,p_m\}$ and, for each of the above neighborhoods $U_j$, $\tilde{C_0}\cap U_j=\{(z,\epsilon \bar{z})\}$. For $\epsilon$ small enough, $\tilde{C_0}$ will be sufficiently $C^1$-close to $C_0$ as to guarantee that $\tilde{C_0}$ is symplectic and (like $C_0$) only meets the $C_i$ ($i>0$) positively and transversely. For $C$, we take a surface which coincides with $\cup_{r\geq 0} C_r$ outside the $U_j$ and whose intersection with $U_j$ is given by $$C\cap U_j=\{(z,w)|zw=\delta f_{j}(z,w)\}$$ where $f_j$ is a *real-valued* function supported on $U_j$ with $f(p_j)\neq 0$ and $\delta$ is a complex constant chosen small enough as to guarantee that $C$ is symplectic. Now for any $(z,w)\in \tilde{C_0}\cap U_j$, we have $zw\in \mathbb{R}\operatorname{\epsilon}$, while for any $(z,w)\in C\cap U_j$ we have $zw\in \mathbb{R}\delta$, so as long as we choose $\epsilon,\delta\in \mathbb{C}$ to have different phases we ensure that $C$ and $\tilde{C_0}$ have no intersections within $\cup_{j\geq 1} U_j$. By construction, any intersections of $\tilde{C_0}$ with $C$ outside $\cup_{j\geq 1} U_j$ are positive and transverse, proving the result.
There are many examples of intersection patterns of curves $C_1,\ldots,C_n$ for which our methods give rise to symplectic classes on the Kähler cone, but it does not seem possible at this juncture to give a concise yet anywhere-near-exhaustive list of the assumptions on the $[C_i]$ which are sufficient. Instead, we shall demonstrate the techniques on a particular complex surface, which we believe illustrates nicely both the subtleties involved and the fact that our construction gives rise to symplectic forms that cannot be obtained by classical methods.
The Kharlamov–Kulikov surface
-----------------------------
If $(M,J)$ is a complex surface admitting Kähler structures and $\mathcal{C}_J\subset H^{1,1}_{J}$ is the Kähler cone as given by the Buchdahl-Lamari theorem, then every class in $\mathcal{C}_J+Re\,H^{2,0}_{J}$ is of course represented by symplectic forms. Although our method gives seemingly new symplectic forms in classes $c$ outside $\mathcal{C}_J$ in the presence of ($J$-holomorphic) curves of negative square, a skeptic might imagine that if we were to vary the complex structure on $M$ to some other (integrable) $J'$, then the negative-square curves might disappear, and so these classes $c$ might lie in $\mathcal{C}_{J'}+Re\,H^{2,0}_{J'}$, in which case our method would not have been necessary to obtain the new forms.
Now the list of underlying manifolds $M$ of complex surfaces for which the effective cone is known for *every* complex structure on $M$ is rather short, so for most complex surfaces it is difficult to tell whether our new forms could have been obtained by algebro-geometric considerations. In the case that $M$ is *rigid*, though, there is no room to vary $J$, and so we can confidently assert that our main theorems give genuinely new forms as soon as we know that there are curves of negative square in the surface. We present here an example of a rigid surface $K$, borrowed from [@KK], which contains several (21) curves of negative square intersecting each other in a nontrivial fashion, and on which we can find symplectic forms in all classes in the positive cone which are nonnegative on each of these 21 curves. It seems likely (though we shall not attempt to prove this) that all curves of negative square in $K$ lie in the cone generated by these 21 special curves; if this is indeed the case then it would follow that the entire boundary of the Kähler cone of $K$ is contained in the symplectic cone. In any event, our results show that at least a rather substantial portion of the boundary of the Kähler cone of $K$ is contained in the symplectic cone, even though the standard methods of Kähler geometry alone seem to provide no reason to expect this to be the case.
We now recall the construction of $K$ from Section 2 of [@KK]. Begin with an arbitrary smooth cubic curve in $\mathbb{C}P^2$, and consider its 9 inflection points. Since these inflection points are each 3-torsion under the group law of the cubic, any line through two of them also passes through a third which is distinct from the first two; as such we obtain 12 lines each passing through precisely 3 of the inflection points. The dual arrangement provides us with 9 lines $L_1,\ldots,L_9$ and 12 points $p_{\{i,j,k\}}$ ($\{i,j,k\}\in$ $\{\{1,2,3\},$ $\{1,4,7\},$ $\{1,5,9\},$ $\{1,6,8\},$ $\{2,4,9\},$ $\{2,5,8\},$ $\{2,6,7\},$ $\{3,4,8\},$ $\{3,5,7\},$ $\{3,6,9\},$ $\{4,5,6\},$ $\{7,8,9\}\}$) in (the dual plane) $\mathbb{C}P^{2}$, with $p_{\{i,j,k\}}\in L_l$ iff $l\in\{i,j,k\}$. Let $\sigma\co\tilde{\mathbb{P}}^2\to\mathbb{C}P^2$ denote the blowup at the various $p_{\{i,j,k\}}$; let $E_{\{i,j,k\}}$ denote the corresponding exceptional divisors, and let $L'_i$ denote the strict transform of $L_i$. As is seen in [@KK], for suitable choices of a homomorphism $\phi\co
H_1(\tilde{\mathbb{P}}^2\setminus
\sigma^{-1}(\cup_{i=1}^{9}L_i);\mathbb{Z})\to
(\mathbb{Z}/5\mathbb{Z})^2$, the total space of the Galois cover branched over $\cup_{i=1}^{9}L_i$ associated to $\phi$ will be smooth. Call this total space $K$ and the covering map $g\co K\to
\tilde{\mathbb{P}}^2$.
Write $C_i=g^{-1}(L'_i)$, $D_{\{i,j,k\}}=g^{-1}(E_{\{i,j,k\}})$. Lemma 2.1 of [@KK] shows that each $C_i$ is a square-$(-3)$ curve of genus 4 and each $D_{\{i,j,k\}}$ is a square-$(-1)$ curve of genus 2. Further the canonical class of $K$ is ample and is given by $$K_K=\frac{1}{3}PD(7\sum [C_i]+12\sum [D_{\{i,j,k\}}]);$$ we have $K_{K}^{2}=333$ and $e(K)=111$, so $K$ is the quotient of the unit ball in $\mathbb{C}^2$ by a famous result of Miyaoka [@M] and Yau [@Y]; a theorem of Siu [@Siu] then shows that $K$ is rigid as promised.
Let $\alpha$ be any class in the positive cone of $H^{1,1}$ which is nonnegative on all holomorphic curves in $K$, and positive on all curves whose homology classes are not in the cone spanned by the $[C_i]$ and $[D_{\{i,j,k\}}]$. Then $\alpha$ is represented by symplectic forms.
First, note that the intersections of the distinct $C_i$ and $D_{\{i,j,k\}}$ are given by $$[C_i]\cdot [C_j]=0;\quad [D_{\{i,j,k\}}]\cdot [D_{\{l,m,n\}}]=0; \quad
[C_{l}]\cdot [D_{\{i,j,k\}}]=\left\{\begin{array}{ll}1 & l\in\{i,j,k\}\\
0 & l\notin\{i,j,k\}\end{array}.\right.$$
Let $\Gamma$ denote the dual graph to the subset of $\{[C_i],[D_{\{i,j,k\}}]\}$ on which $\alpha$ vanishes (in other words, $\Gamma$ has a vertex for each element of this set, and the number of edges connecting two distinct vertices of $\Gamma$ is the intersection number of the corresponding pair of classes). If $\Gamma$ were to contain a loop, then by virtue of the intersection pattern of the $C_i$ and $D_{\{i,j,k\}}$ that loop would consist of some number (say $a$) of curves $A_0=C_{i_0},\ldots,A_{a-1}=C_{i_{a-1}}$ and an equal number of curves $B_0=D_{\{i_0,j_0,k_0\}},\ldots,
B_{a-1}=D_{\{i_{a-1},j_{a-1},k_{a-1}\}}$ such that $[A_m]\cdot
[B_m]=[A_m]\cdot [B_{m+1}]=1$ for each $m$ (where $m\in
\mathbb{Z}/a\mathbb{Z}$). Hence since $[A_{m}]^{2}=-3$ and $[B_{m}]^{2}=-1$, $$\left(\sum_{m=0}^{a-1}[A_m]+\sum_{m=0}^{a-1}[B_m]\right)^2\geq
-3a-a+2(2a)=0,$$ which is impossible since $\alpha$ lies in the positive cone and vanishes on $\sum [A_m]+\sum [B_m]$.
In general if $\Gamma$ contains a connected component with at least 3 distinct $[B_{m}]=[D_{\{i_m,j_m,k_m\}}]$ ($1\leq m\leq 3$), then it contains a subgraph consisting of vertices $\{[B_1],[C_{i_1}],[B_2],[C_{i_2}],[B_3]\}$ where $[B_1]\cdot
[C_{i_1}]=[B_2]\cdot [C_{i_1}]=[B_2]\cdot [C_{i_2}]=[B_3]\cdot
[C_{i_2}]=1$. But then $$([B_1]+[C_{i_1}]+2[B_2]+[C_{i_2}]+[B_3])^2=-1-3-4-3-1+2+4+4+2=0,$$ which is again a contradiction since $\alpha$ is in the positive cone. Likewise, if $\Gamma$ contains a connected component with three distinct $[C_i]$ (say $[C_i]$, $[C_j]$, $[C_k]$), then it must also contain some $[D_{\{i,j,l\}}]$ and $[D_{\{j,k,m\}}]$ and we see $$([C_i]+3[D_{\{i,j,l\}}]+2[C_j]+3[D_{\{j,k,m\}}]+[C_k])^2=-3-9-12-9-3+6+12+12+6=0,$$ again a contradiction.
Now it will suffice to consider the case in which $\Gamma$ is connected, since if it is not we can apply our argument successively to each component. Assuming $\Gamma$ is connected, then, the above shows that it contains at most two $[C_i]$ and at most two $[D_{\{i,j,k\}}]$, so after relabeling it is a subgraph of the graph $\Gamma_0$ with vertices $[C_1]$, $[B_1]:=[D_{\{1,2,3\}}]$, $[C_2]$, and $[B_2]:=[D_{\{2,4,9\}}]$, with just one edge each connecting $[C_1]$ to $[B_1]$, $[B_1]$ to $[C_2]$, and $[C_2]$ to $[B_2]$. Suppose that $\Gamma=\Gamma_0$. Since $\alpha$ is positive on all curves represented by classes which are not in the span of $[C_1]$, $[B_1]$, $[C_2]$, and $[B_2]$, by taking $t>0$ small enough we ensure that $$\alpha_0=\alpha-tPD(8[C_1]+21[B_1]+12[C_2]+14[B_2])$$ will have the same property; we calculate $$\langle \alpha_0,[C_1]\rangle=3t,\,\langle
\alpha_0,[B_1]\rangle=t,\, \langle \alpha_0,[C_2]\rangle=t,\mbox{
and }\langle \alpha_0,[B_2]\rangle=2t,$$ so $\alpha_0$ is represented by Kähler forms. Apply Proposition \[disjoin\] twice: first to get a symplectic surface $\tilde{C}$ representing $[C_1]+[B_1]$ and disjoint from a symplectic representative of $[B_1]$, and then to get a symplectic surface $S$ representing $[C_2]+[\tilde{C}]+[B_1]+[B_2]=[C_1]+2[B_1]+[C_2]+[B_2]$ which is disjoint from $C_2$, $B_1$, and $B_2$. $S$ then has positive genus and square $-1$, so we can apply inflation to $S$ to get a symplectic form in the class $\alpha_0+sPD[S]$ for any parameter $s$ less than $2\langle \alpha_0,[S]\rangle=16t$. Take $s=8t$ to get a symplectic form $\omega_1$ representing $$\alpha_1=\alpha-tPD(5[B_1]+4[C_2]+6[B_2])$$ with respect to which $[B_1]$, $[C_2]$, and $[B_2]$ are symplectic. Now use Proposition \[disjoin\] to obtain a positive-genus $\omega_1$-symplectic surface $S'$ representing $[B_1]+[C_2]+[B_2]$ and meeting $[B_1]$ and $[B_2]$ transversely and positively. $[S']^2=-1$, and $\langle
\alpha_1,[S']\rangle=4t$, so inflation using $S'$ gives a symplectic form $\omega_2$ in the class $$\alpha_1+4tPD[S]=\alpha-tPD([B_1]+2[B_2]).$$ Since $B_1\cdot B_2=0$, we can now apply Theorem 1.1 rather directly to get the desired symplectic form in $\alpha$, by first inflating using (say) $B_1$ and then inflating using $B_2$.
In each case that $\Gamma$ is a *proper* subgraph of $\Gamma_0$, the desired symplectic representative of $\alpha$ can be obtained by similar (but easier) arguments, which we leave to the reader.
A more general criterion
------------------------
As a more general example of the circumstances in which our methods can be used to show that a class in the boundary of the Kähler cone admits symplectic representatives, we present the following theorem. Note that while condition (b) below is rather subtle, condition (a) is occasionally easy to check; for instance it holds for the canonical class in a minimal surface of general type and for any class in the positive cone of a minimal surface of Kodaira dimension 0 (though in both of these cases there exist other methods to prove that such a class is in the symplectic cone).
\[ADE\] Let $(M,\omega,J)$ be a Kähler surface and $\alpha\in
H^{1,1}_{J}$ any class in the positive cone such that
- If $e\in H_2(M;\mathbb{Z})$ is represented by a reduced, irreducible holomorphic curve of negative square, then $\langle \alpha,e\rangle\geq 0$, with equality only if $e^2=-2$ or $e^2=-1$ and $g(e)>0$; and
- There are no $E_6$-trees of holomorphic curves of square $-2$ on which $a$ vanishes.
Then $\alpha$ is represented by symplectic forms deformation equivalent to $\omega$.
(Sketch) Using negative-definiteness as in the case of the Kharlamov–Kulikov surface, one first shows that each connected component of the dual graph of the curves on which $\alpha$ is negative either
- contains just one curve of square $-1$ and (say) $n-1$ curves of square $-2$, in which case the dual graph is the Dynkin diagram $A_n$, with the square-$(-1)$ curve as one of the univalent vertices; or
- consists entirely of square-$(-2)$ curves, in which case it is one of the ADE Dynkin diagrams.
Now assumption (b) in the statement of the theorem restricts the Dynkin diagrams that can appear to $A_n$ and $D_n$, and is imposed because our methods do not seem strong enough to apply to the cases of $E_6$, $E_7$, or $E_8$. In the cases of $A_n$ and $D_n$, an approach parallel to that used in the case of the Kharlamov–Kulikov surface provides the desired form; the details of this are left as a mildly amusing exercise to the interested reader.
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[^1]: The first author is supported in part by NSF grant 0435099 and the McKnight fellowship. The second author is supported by an NSF postdoctoral fellowship.
|
---
author:
- Chelsea Zhang
- 'Sean J. Taylor'
- Curtiss Cobb
- Jasjeet Sekhon
title: Active Matrix Factorization for Surveys
---
Amid historically low response rates, survey researchers seek ways to reduce respondent burden while measuring desired concepts with precision. We propose to ask fewer questions of respondents and impute missing responses via probabilistic matrix factorization. The most informative questions per respondent are chosen sequentially using active learning with variance minimization. We begin with Gaussian responses standard in matrix completion and derive a simple active strategy with closed-form posterior updates. Next we model responses more realistically as ordinal logit; posterior inference and question selection are adapted to the nonconjugate setting. We simulate our matrix sampling procedure on data from real-world surveys. Our active question selection achieves efficiency gains over baselines and can benefit from available side information about respondents.
Introduction
============
Response burden in surveys
--------------------------
Literature spanning decades has established the negative impact of survey length on completion rate and response quality. In a 1978 meta-analysis of 98 published studies involving mailed questionnaires, Heberlein and Baumgartner show longer instruments are associated with lower return rates [@heberlein]. Fan and Yan cite evidence for this association in mail and web surveys, noting that differing definitions of survey length could result in heterogeneous effect sizes [@fan]. In addition to nonresponse, a longer instrument may also introduce measurement error. Respondents may avoid the cognitive burden of surveys by taking mental shortcuts, such as selecting “don’t know” or arbitrary responses, a behavior called satisficing [@krosnick]. Herzog and Bachman find high school seniors are more likely to give straight-line answers on a long version of a survey compared to a short version [@herzog]. More recently, Tourangeau et al review evidence that both interviewers and interviewees deliberately shortcut interviews to reduce burden, such as answering initial questions in the negative to avoid follow-up questions [@tourangeau].
In an earlier era of survey research, the shift from in-person to phone surveys increased nonresponse rates, as it became more acceptable for contacts to refuse a survey. Instruments administered via phone grew shorter [@groves]. Meanwhile, the development of imputation methods facilitated analyses with missing data. In the sequel, working with an incomplete matrix of survey responses, we impute missing responses using matrix completion. Common in the recommender systems literature, matrix completion has recently found use in causal inference, to estimate counterfactual outcomes in panel data [@athey] and to estimate latent confounders from an incomplete matrix of covariates [@kallus].
Recent efforts to reduce respondent burden try to shorten surveys by adaptively prioritizing questions. Gonzalez and Eltinge modify the Consumer Expenditure Interview Survey to use a matrix sampling format [@gonzalez]; each respondent is randomly assigned to one of six sub-questionnaires measuring a single expenditure category, rather than having to answer all six. They set assignment probabilities to minimize the variances of category means, using pilot estimates from an initial questionnaire. Early et al develop dynamic question-ordering strategies for two goals: minimizing predictive uncertainty and collecting maximal information before the respondent drops out of the survey [@early]. For the former goal they choose the question that minimizes prediction interval width in a measurement error model; for the latter goal they choose the question with maximum conditional entropy, traded off with dropout probability. They also review the literature on respondent burden in detail and cite adaptive information-maximizing strategies in other fields.
Optimal design and active learning
----------------------------------
Classical research on surveys addresses how to configure survey parameters to minimize variance of estimates. Each member of the population is sampled with some inclusion probability; optimal inclusion probabilities have been derived for a variety of survey designs [@sarndal]. More broadly, guidance for variance minimization comes from the classical field of optimal experimental design. Optimal design proceeds by minimizing a measure of inverse Fisher information, which is the asymptotic variance of the maximum likelihood estimator and a lower bound on the variance of unbiased estimators. Different measures of the inverse information matrix define different design criteria: A-optimality minimizes the trace, D-optimality minimizes the determinant and E-optimality minimizes the maximum eigenvalue [@settles]. Gonzalez and Eltinge use A- and D-optimality to obtain a scalar objective function involving all the variances they want to control [@gonzalez].
Closely related to optimal design for surveys is optimal subsampling, useful when training a model on the full dataset is too computationally demanding. For instance, prior to logistic regression, Wang et al derive optimal subsampling weights using A-optimality to minimize the variance of the subsample maximum likelihood estimate [@wang].
Optimal design occupies a place in the broader field of active learning, which encompasses many strategies for selecting informative training points when labeling them is expensive. Sequential active learning strategies iteratively select the next query point. A common baseline is uncertainty sampling, which simply chooses the point with greatest predictive uncertainty [@settles]. Uncertainty sampling is simple but myopic; it does not consider the global effect of item selection on the model or the goal at hand. MacKay distinguishes between various goals of active learning, such as obtaining maximal information about model parameters, or maximizing model performance in a region of input space [@mackay]. For the latter objective, a principled approach is to choose the point that minimizes the variance component of generalization error – the predictive variance integrated over the input distribution. The seminal work by Cohn et al derives, for several models, closed-form expressions for this integrated variance given a new training point, which can be optimized to suggest the next query point [@cohn].
The extension to a multi-step search horizon is studied by Garnett et al [@garnett]. They consider how to optimally query points in a dataset to estimate the proportion of a binary class. Casting the problem in a Bayesian decision-theoretic framework, they use a branch-and-bound strategy to prune the search space. In their experiments, a multi-step lookahead provides marginal gains over the greedy one-step lookahead; both outperform random and uncertainty sampling.
Theoretical work has established the lower sample complexity of active learning relative to passive sampling in certain settings, usually within binary classification [@settles]. Dasgupta et al show this for data distributed uniformly on the unit sphere, where the base learner is a linear separator through the origin [@dasgupta]. However, the advantage of active learning disappears when the learner is inhomogeneous, and is recovered by weakening the definition of sample complexity [@balcan]. In addition, Attenberg and Provost point out several practical challenges to the adoption of active learning [@attenberg]. These challenges include choosing an initial base learner and query selection strategy within the label budget; poor query selection by non-robust strategies, especially with rare classes or concepts; and artificial advantages given to active learning in research experiments.
Active learning for matrix factorization
----------------------------------------
The spectrum of active learning strategies has been specialized to matrix factorization to select the most informative entries of the response matrix. Often research asks, in a recommender systems context, which movies to prompt users to rate; the researcher seeks accurate predictions of user ratings or rankings of unseen movies. One baseline strategy simply prompts for the most popular items, since users are more likely familiar with them and more likely to remain attentive [@elahi]. Many variants of uncertainty sampling have been proposed. They represent unobserved matrix entries with disparate models, such as the graphical lasso and ensembles [@chakraborty]. Sutherland et al select matrix elements with highest posterior variance in a probabilistic matrix factorization model [@sutherland].
Other strategies quantify the global effect of item selection and try to maximize this effect. Rubens and Sugiyama find the item with largest influence empirically by perturbing ratings and computing the change in predictions [@rubens07]. Karimi et al seek the item that would change user factors most if its rating were known [@karimi11a]. Instead of influence, Silva and Carin maximize mutual information between selected and unobserved instances [@silva]. They also suggest an efficient alternative: sort both user and item factors by posterior variance, match them into user-item pairs, and query the corresponding matrix entries [@silva]. Our eventual strategy recalls elements of theirs, as they approximate posteriors with variational Bayes and reduce approximate posterior variance to the trace.
Still other strategies directly target the predictive error of matrix factorization [@golbandi; @karimi11b]. As responses are not known before querying, direct minimization of RMSE or MAE relies on assumptions about the empirical rating distribution, such as stationarity. Beyond prediction, active learning for recommender systems could target objectives like profitability or user satisfaction [@rubens15; @sutherland].
Computerized adaptive testing
-----------------------------
Active learning for matrix factorization could be posed as adaptive item selection that places respondents on latent scales with maximal precision. The largely separate literature of item response theory has long pursued this goal. Computerized adaptive testing (CAT) algorithms customize the questions asked of individual test-takers adaptively, to optimally determine their latent ability parameters with some question budget. Montgomery and Cutler advocate for applying CAT methods to public opinion surveys [@montgomery]. They model answers to political knowledge questions using logistic regression with a one-dimensional latent ability parameter. Their item selection strategy, which minimizes expected posterior variance of ability, can shorten a battery by 40% while retaining the measurement accuracy of the fixed battery.
The generalization of latent ability to higher dimensions is handled by Segall’s work on multidimensional adaptive testing (MAT) in a Bayesian setting [@segall09]. The probability of a correct response is a logistic function of the inner product of multivariate normal ability parameters and fixed ability discrimination parameters. Since the logistic form prevents a closed-form exact posterior update, Segall uses a Laplace approximation to the posterior of user ability parameters. Segall selects the question that maximizes the determinant of the precision matrix (D-optimal), equivalently minimizing the size of the posterior credibility region.
Approaching the same MAT problem with optimal design, Mulder and Van der Linden directly optimize the Fisher information of ability parameters [@mulder09a]. They note that the trace of inverse information includes the determinant as a factor, so A- and D-optimality should act similarly. Their simulations show A- and D-optimality outperform a random selection baseline, while E-optimality is worse than random. In a separate paper addressing the Bayesian MAT framework, Mulder and Van der Linden analyze additional item selection criteria based on KL divergence and mutual information [@mulder09b].
Like [@montgomery], we argue the item response theory literature can inform optimal design of adaptive surveys. The principled item selection for MAT in [@segall09] is possibly the closest work to ours. This work relies on the latent low-rank structure of the response matrix, and is clarified by the formalism of matrix factorization. In turn, we offer to matrix factorization an active learning technique that demonstrably outperforms random sampling.
Active matrix completion
========================
Matrix completion methods
-------------------------
Given $n$ users and $k$ questions, let $R$ denote the $n \times k$ response matrix. Matrix factorization finds a low-rank decomposition of $R$ – a set of user factors $U = [u_1, \ldots, u_n]^T$ and question factors $V^T = [v_1, \ldots, v_k]$ such that $R \approx U V^T$. Let $r$ be the dimensionality of latent space; then $u_i \in \mathbb{R}^r$ and $v_j \in \mathbb{R}^r$ for all $i$ and $j$, and $r$ is small.
Matrix completion performs matrix factorization on a response matrix with some unobserved entries. Let $I$ be an indicator matrix for whether the corresponding responses in $R$ exist, so that $I_{ij} = 1$ implies user $i$ responded to question $j$ with value $R_{ij}$. Matrix completion finds $U$ and $V$ that minimize the reconstruction error $u_i^T v_j$ for $R_{ij}$ on observed entries, where $I_{ij} = 1$.
Matrix completion that enforces a hard rank constraint is nonconvex and generally intractable [@fithian]. It is common to work with a convex relaxation that instead regularizes the nuclear norm, or sum of singular values [@srebro05]. This optimization problem seeks a matrix $Z$, in place of $UV^T$, that minimizes reconstruction error, and encourages a low-rank solution by favoring sparsity in the singular values.
$$\min_Z \frac{1}{2} \sum_{i=1}^n \sum_{j=1}^k I_{ij}(R_{ij}-Z_{ij})^2 + \lambda {\left\lVertZ\right\rVert}_*
\label{eqn:soft-impute}$$
This nuclear norm regularized problem enjoys theoretical guarantees: recovery of the complete matrix occurs with high probability when $O(n\, \text{polylog}(n))$ entries are observed at random, with or without noise [@recht11; @negahban]. Moreover, (\[eqn:soft-impute\]) has an efficient solution in the SoftImpute algorithm by Mazumder et al [@mazumder10]. SoftImpute iteratively computes the SVD of $Z$, soft-thresholds the singular values, and updates the entries where $I_{ij} = 0$ with the prediction from the soft-thresholded SVD, until convergence. Hence the solution can be expressed as $Z = UDV^T$ for some matrices $U, D, V$.
An alternate formulation of matrix completion, introduced by [@rennie05], penalizes the Frobenius norm of $U$ and $V$:
$$\min_{U, V} \frac{1}{2} \sum_{i=1}^n \sum_{j=1}^k I_{ij}(R_{ij}-u_i^T v_j)^2 + \frac{\lambda_U}{2}{\left\lVertU\right\rVert}_F^2 + \frac{\lambda_V}{2}{\left\lVertV\right\rVert}_F^2
\label{eqn:frobenius}$$
Nonconvex in $U$ and $V$, (\[eqn:frobenius\]) is solved via gradient descent. This formulation is useful for large-scale problems with low rank, since it is less expensive to operate on $U$ ($n \times r$) and $V$ ($k \times r$) than $Z$ ($n \times k$). Due to an identity relating the nuclear norm and sum of Frobenius norms, if $\lambda_U = \lambda_V$ and the solution to (\[eqn:soft-impute\]) has rank at most $r$, then this is also a solution to (\[eqn:frobenius\]) [@fithian; @mazumder10].
However, the user and question factors resulting from either optimization are point estimates, as are the imputed survey responses. We would like some measure of uncertainty over the imputed responses. Further, we seek a strategy for actively selecting users and questions to survey next based on the uncertainty reduction achieved. For this uncertainty quantification, we turn to Bayesian probabilistic matrix factorization (BPMF) [@salakh08].
Bayesian probabilistic matrix factorization
-------------------------------------------
BPMF, introduced by Salakhutdinov and Mnih, models user and question factors as independent and normally distributed around a prior mean and covariance. Responses add zero-mean, constant-variance Gaussian noise to the inner product of user and question factors. Following the notation in [@salakh08], $$\begin{aligned}
u_i &{\stackrel{iid}{\sim}}\mathcal{N}(\mu_U, \Lambda_U^{-1}) \\
v_j &{\stackrel{iid}{\sim}}\mathcal{N}(\mu_V, \Lambda_V^{-1}) \\
R_{ij} \mid U, V &{\stackrel{ind}{\sim}}\mathcal{N}(u_i^T v_j, \alpha^{-1})\end{aligned}$$
The authors of [@salakh08] note an interesting connection to the Frobenius norm regularized problem: for prior hyperparameters $\mu_U = \mu_V = 0$, $\Lambda_U = \alpha_U I$ and $\Lambda_V = \alpha_V I$, the MAP estimate of $U$ and $V$ conditional on $R$ is the solution to (\[eqn:frobenius\]) with $\lambda_U = \alpha_U / \alpha$ and $\lambda_V = \alpha_V / \alpha$.
The authors perform posterior inference via Gibbs sampling. They derive the following complete conditional for $u_i$ (the form for $v_j$ is analogous): $$\begin{aligned}
P(u_i \mid R, V, \mu_U, \Lambda_U, \alpha) &\propto P(u_i, R_{i\cdot} \mid V, \mu_U, \Lambda_U, \alpha) \\
&= \prod_{j=1}^k \left[P(R_{ij} \mid u_i^T v_j, \alpha^{-1})\right]^{I_{ij}} P(u_i \mid \mu_U, \Lambda_U^{-1})\end{aligned}$$ The complete conditional is conjugate normal with mean $\mu_i^*$ and precision $\Lambda_i^*$: $$\begin{aligned}
u_i \mid R, V, \mu_U, \Lambda_U, \alpha &\sim \mathcal{N}(\mu_i^*, [\Lambda_i^*]^{-1}) \\
\Lambda_i^* &= \Lambda_U + \alpha \sum_{j=1}^k I_{ij} v_j v_j^T \\
\mu_i^* &= \left[\Lambda_i^*\right]^{-1} \left( \alpha \sum_{j=1}^k I_{ij} R_{ij} v_j + \Lambda_U \mu_U \right)\end{aligned}$$ This can be recognized as the posterior for Bayesian linear regression with a Gaussian prior and Gaussian noise: $u_i$ are the coefficients, $V$ is the design matrix and $\alpha^{-1}$ is the noise variance. The expression for $\mu_i^*$ also arises in MAP estimation for BPMF, as the coordinate ascent update for $u_i$, for prior hyperparameters $\mu_U = 0, \Lambda_U = \alpha_U I$ (see section \[sec:bpmf-map-relation\]).
Active learning formulation
---------------------------
Assume for simplicity we have a fixed question bank with known factors $v_1, \ldots, v_k$, learned from abundant existing data. A new user $i$ enter the survey pool, and we want to select questions optimally for learning $u_i$. So far we assume we have no side information about users.
The BPMF model admits a convenient online formulation for updating our knowledge about $u_i$ given responses from user $i$. Suppose at time $t$, $u_i \sim \mathcal{N}(\mu_i^{(t)}, \left[\Lambda_i^{(t)}\right]^{-1})$. At time $t+1$ we gather the response by user $i$ to question $j$. Then the posterior for $u_i$ is $$\begin{aligned}
u_i^{(t+1)} \mid R^{(t+1)}, V, \mu_U, \Lambda_U, \alpha &\sim \mathcal{N}(\mu_i^{(t+1)}, \left[\Lambda_i^{(t+1)}\right]^{-1}) \\
\Lambda_i^{(t+1)} &= \Lambda_i^{(t)} + \alpha v_j v_j^T \\
\mu_i^{(t+1)} &= \left[\Lambda_i^{(t+1)}\right]^{-1} \left( \alpha R_{ij} v_j + \Lambda_i^{(t)} \mu_i^{(t)} \right)\end{aligned}$$ We now consider how to choose question $j$ optimally. One standard approach in active learning is to minimize posterior variance; here the posterior variance of $u_i$ is a matrix. Optimal design proposes minimizing various measures of the inverse information matrix, including the trace, determinant and maximum eigenvalue (A-, D- and E-optimality, respectively). Other optimality criteria are concerned with minimizing predictive variance over a certain input distribution.
For simplicity, let $\mu, \Sigma$ denote the posterior mean and variance of $u_i$. Inspired by the A-optimality criterion, we consider minimizing ${\text{tr} \;}\Sigma$, the sum of the component-wise variances of $u_i$. This criterion can be related to predictive variance as follows. Let $\mathbb{P}$ be the uniform distribution on the unit sphere $\{v: {\left\lVertv\right\rVert}_2 = 1\}$, and suppose $\tilde v \sim \mathbb{P}$ independently of $u_i$. We care about the prediction $u_i^T \tilde v$. The predictive variance is (see \[sec:trace-predvar-relation\] for details) $${\text{Var} \;}(u_i^T \tilde v) = \frac{1}{r} E {\left\lVertu_i\right\rVert}_2^2 = \frac{1}{r}\left({\text{tr} \;}\Sigma + {\left\lVert\mu\right\rVert}_2^2\right)$$
For any $\mu$, minimizing ${\text{tr} \;}\Sigma$ corresponds to minimizing a lower bound on the predictive variance along uniform latent directions. We cannot treat $\mu$ as a constant since both $\mu$ and $\Sigma$ are affected by the choice of question $j$.
Having justified our A-optimality strategy, we solve the following optimization at time $t$: $$\min_{j} {\text{tr} \;}\left[\Lambda_i^{(t+1)}\right]^{-1}$$ This is equivalent to solving the following, where $\lambda_\ell(M)$ denotes the $\ell$th eigenvalue of $M$: $$\min_j \sum_{\ell=1}^r \lambda_\ell\left(\left[\Lambda_i^{(t+1)}\right]^{-1}\right) = \min_j \sum_{\ell=1}^r \left[\lambda_\ell \left(\Lambda_i^{(t+1)}\right) \right]^{-1}$$
This variance criterion penalizes small eigenvalues of the precision matrix, along the directions in which it has least information. Information is acquired by sampling $v_j$ that lie in those directions. The optimal sampling strategy chooses questions as a function of their informativeness and their contribution to less explored directions. Provided questions exist in many directions with similar magnitudes, the strategy prefers new questions with factors $v_j$ roughly orthogonal to those of older questions.
Algorithm \[algo:active\] summarizes our active strategy. Note some limitations of this simple version. The algorithm is greedy with a one-step horizon; we could easily extend the active strategy to select multiple questions to ask in the next timestep. The optimal sequence of questions can be computed offline, as the optimality criterion does not depend on the actual responses. This results from the assumption that $V$ is fixed and the Gaussian assumptions inherent in BPMF, but the implication is unrealistic. There is one fixed question order for all respondents. Finally, nonresponse is assumed ignorable; in other words, responses are missing at random. This assumption is unrealistic in practice, and modeling the missingness pattern could reduce bias.
$\mathcal{O} \leftarrow \emptyset,\; \mathcal{U} \leftarrow \{1, \ldots, k\}$ $\mu_0 \leftarrow \mu_U,\; \Lambda_0 \leftarrow \Lambda_U$
Data collection and analysis
============================
Datasets
--------
Dataset Number of respondents Number of questions
----------------------------- ----------------------- --------------------- --
Facebook on-platform survey 11793 53
CCES 2012 54535 31
CCES 2016 64600 43
CCES 2016 (full) 64600 66
: Dataset characteristics. CCES 2016 (full) refers to CCES 2016 with extra covariates. \[tbl:datasets\]
We simulate the active strategy on multiple datasets, summarized in Table \[tbl:datasets\]. The Facebook survey is a survey of Facebook users, administered on the app or web interface, with a variety of questions about their experiences with the product and the company. The Facebook on-platform survey was administered in random order.
The Cooperative Congressional Election Survey (CCES) is a national Internet survey of adult U.S. citizens conducted by YouGov [@cces16] that seeks to gauge voter opinions about prevailing political issues and elected officials, before and after an election. Respondents are selected by matching an opt-in respondent pool to a stratified random sample from the American Community Survey. Our main results use the pre-election surveys from 2012 and 2016, limiting consideration to Common Content questions that ask respondents to evaluate national political issues or entities on a binary or ordinal scale. For robustness checks we expand the question set to include voter demographics, party identification and other characteristics; we refer to this as the “full” CCES dataset. We exclude questions about voter actions in the past year and opinions of state or local representatives, as well as questions with a majority of responses missing.
For each survey question, allowable responses are rescaled to $[-1,1]$. Some responses will be missing, either because they were not present in the original dataset, or because we dropped response values that violated the ordinal assumption. CCES has low overall missingness rates: about 4% in 2012 and less than 1% in 2016. The missingness distributions by question and by user are shown in Figure \[fig:dataset-sparsity\].
[0.5]{} ![Missingness distribution of datasets at the question and user levels \[fig:dataset-sparsity\]](all/dataset_sparsity_question_level.png "fig:"){width="\textwidth"}
[0.5]{} ![Missingness distribution of datasets at the question and user levels \[fig:dataset-sparsity\]](all/dataset_sparsity_user_level.png "fig:"){width="\textwidth"}
Simulating the active strategy \[sec:sim-procedure\]
----------------------------------------------------
Our simulations of the active strategy begin by randomly splitting the respondent set into a training half and a simulation half. On the responses for the training half, we perform SoftImpute, learning the fixed $V$ for the active strategy. We choose SoftImpute for its efficiency and empirical stability of $V$ across simulations. The regularization parameter $\lambda$ is selected by grid search with warm starts as recommended in [@mazumder10], based on mean absolute error on a 20% validation set within the training half.
On the simulation half, we hold out a random 20% of each user’s responses, and simulate running the survey on the remaining 80% of responses. We select the next question per respondent using the active strategy, reveal available responses to that question, and perform the BPMF posterior update for each user. Using the MAP estimate for all user factors along with the fixed $V$, we compute predictions on the holdout set and evaluate error. We repeat this process until all questions have been asked.
Error measures include mean squared error, mean absolute error, and bias of predictions, averaged over all questions. For interpretability, we also compute the proportion of predictions with the wrong sign. All simulations compare the active strategy to a baseline of asking questions in a random order per respondent and, in the case of CCES, existing question order.
We experimented with variations on the simulation procedure. For instance, rather than estimating question factors $v_j$ from SoftImpute, we attempted to use the columns of $V^T$ from solving the Frobenius norm regularized problem (\[eqn:frobenius\]), reasoning that these correspond to the MAP estimate for BPMF for certain simple hyperparameters. However, this nonconvex optimization yielded highly variable question factors and orderings (Figure \[fig:cces16-question-rank-frobenius\]). With SoftImpute, accounting for nonidentifiability, question factors are relatively stable across simulations. We also tried minimizing measures of posterior variance other than the trace, like the determinant and maximum eigenvalue, which correspond to D- and E-optimality respectively. These optimal design criteria lead to similar conclusions, with greater variability in predictive error for E-optimality (Figures \[fig:cces16-compare-metrics-d-optimality\], \[fig:cces16-question-rank-d-optimality\], \[fig:cces16-compare-metrics-e-optimality\], \[fig:cces16-question-rank-e-optimality\]).
Our main results use a rank-4 matrix decomposition ($r=4$); this results in lower predictive error than $r=2$, while keeping the dimensionality of latent space manageable. In the active strategy, we set the prior mean $\mu_U$ and prior precision $\Lambda_U$ using empirical Bayes. Specifically, we set $\mu_U$ and $\Lambda_U^{-1}$ to the sample mean and covariance of the rows of $UD$, the implied user factors from SoftImpute. It remains to set the noise variance $\alpha^{-1}$. By Popoviciu’s inequality and the prior rescaling of responses to $[-1,1]$, we know $\alpha^{-1} \leq 1$. Our main results use the upper bound $(\alpha^{-1} = 1)$, though we experimented with smaller values. Future work should estimate $\alpha$ from the training half of responses.
Results
=======
Efficiency
----------
Below we showcase simulated survey strategies on the 2016 CCES; similar results for the 2012 CCES and the Facebook survey appear in Appendix \[sec:cces12-results\] and \[sec:facebook-results\], respectively. In Figure \[fig:cces16-compare-metrics\] we see the active strategy outperforms the random strategy and existing question order in terms of mean squared and mean absolute prediction error. The active strategy also correctly predicts the sign of survey responses more often than the baselines. Over the course of the survey, the active and sequential strategies suffer worse bias than the random strategy, as they select questions in a deterministic order.
Figure \[fig:cces16-relative-complexity\] clarifies the improved efficiency of the active strategy. It measures the question complexity of the active strategy relative to the random strategy – the number of questions each needs to attain a certain error level. The active strategy almost always requires fewer questions. Suppose we ask 20 questions in a random order for each respondent; the active strategy attains the same error with half as many questions.
We verify the active strategy works as intended in Figure \[fig:cces16-objective\] – it minimizes the active learning objective. While the random strategy decreases the objective function with every new question by virtue of gathering information and reducing posterior variance, it only catches up to the information gain of the active strategy at the end of the simulated survey.
![We plot the error attained by each sampling strategy on the holdout set for the 2016 CCES, averaged over all questions. Error metrics include mean squared error, mean absolute error, the proportion of predictions with the wrong sign, and mean signed error or bias. The solid line depicts average error across 10 simulations of each strategy; shaded regions represent two standard deviations. In 500 simulations of each strategy, across survey questions and metrics, such intervals have reasonable coverage (approximately 95%, always between 93-98%) of data points. \[fig:cces16-compare-metrics\]](cces16/error_comparison_avg_over_questions.png)
![We depict the complexity of the active strategy relative to random-order questions for the 2016 CCES. For the error metrics in Figure \[fig:cces16-compare-metrics\], the solid line plots the number of questions required by each strategy to attain the same level of error. For instance, the active strategy requires 20 questions to reach the same MSE as the random strategy with 30 questions. When the curve lies above the dashed $45^{\circ}$ line, active sampling outperforms random. These curves are obtained by fitting, for each strategy, a loess smoother $f_{strategy}(\epsilon)$ to predict number of questions asked for a given error level. The curves simply plot $(f_{random}(\epsilon), f_{active}(\epsilon))$ for the range of $\epsilon$ attained by both strategies. \[fig:cces16-relative-complexity\]](cces16/relative_sample_complexity_active_vs_random.png)
Question order
--------------
To better understand how the active strategy operates, we examine the factors assigned to individual questions and their resultant order in the active survey. Figure \[fig:cces16-position\] visualizes question factors as vectors in latent space. For ease of visualization, these results use a rank-2 decomposition, keeping all other simulation settings the same. The 2016 CCES questions are distributed unevenly in latent space. Longer vectors indicate questions that feature more prominently in the principal components. (As SoftImpute is solved via soft-thresholded SVD, it effectively performs PCA on some complete version of the incomplete responses.)
Based on our earlier analysis of how to sample latent space efficiently, we might expect the active strategy to begin with longer, relatively orthogonal vectors. However, it front-loads medium-length vectors in the second and fourth quadrants – the direction where the user factors have greatest prior variance. The active strategy spends its initial question budget sweeping around this direction of latent space, gaining precision. The longest question vectors get asked midway through the survey, and the shortest vectors toward the end. With a higher $\alpha$, which increases the relative information conveyed in each response, the active strategy reaches the longest question vectors in the first and third quadrants earlier (Figure \[fig:cces16-question-rank-alpha4\]).
Visualizing the posterior of user factors during the survey reinforces this story. We can see from just 10 user trajectories in Figure \[fig:cces16-user-mean\] that the initial questions in the active ordering lie in the direction of greatest user variance. The active ordering first places users along this direction before seeking information along the orthogonal direction. Most precision gains occur along this direction, early in the survey, as the narrowing confidence region in Figure \[fig:cces16-user-posterior\] shows. The confidence region provides another measure of error over the survey duration.
Cross-referencing Figure \[fig:cces16-position\] against Figure \[fig:cces16-question-rank-rank2\], which shows the top questions across simulations and their rank distribution, we can locate concepts in latent space. Questions 2 and 3 concerning support for abortion restrictions point in almost the same direction; question 6 concerning abortion rights points in the opposite direction. This general direction broadly indicates partisanship: questions in the top left quadrant address policies favored by Democrats, whereas questions in the bottom right quadrant address policies favored by Republicans. It makes sense that most prior variance lies along this latent direction. In the orthogonal direction, we have a “none-of-the-above” question about immigration policies (12), which was answered affirmatively by 5% of respondents. Opposite that are questions about longer sentences for felons who have committed violent crimes (14) and background checks for all gun purchases (17), which are broadly popular policies supported by 84% and 90% of respondents, respectively. This direction seems to indicate bipartisan policies. With a 2-dimensional decomposition, the second direction is predetermined, and the latent concepts are limited to partisanship and bipartisanship.
To capture more meaningful concepts, our main results expand the dimensionality of latent space. In Figure \[fig:cces16-question-rank\] we examine the active ordering for a rank-4 decomposition. Across simulations, the order of the first 15 questions is almost constant. The foremost question is: “If you were in Congress would you vote FOR or AGAINST...\[repealing\] the Affordable Care Act of 2009 (also known as Obamacare)?” Questions about environment and abortion policies are prioritized over questions about the economy, crime and gun control. Unlike with the rank-2 decomposition, there is room in latent space to capture approval of elected representatives (6, 9, 12). (It must be noted that with a rank-2 decomposition for the 2012 CCES, approval of Congress is captured well – in the negatively bipartisan direction, with 79% of respondents somewhat or strongly disapproving. See Figure \[fig:cces12-position\].)
[0.6]{} {width="\textwidth"}
{width="\textwidth"}
![Active ordering for 2016 CCES using a rank-4 matrix decomposition. We show the rank of each question across 10 simulations of the active strategy as a box plot. The top 15 questions are relatively stable. \[fig:cces16-question-rank\]](cces16/question_rank__row-norm__base.png)
[0.49]{} {width="\textwidth"}
[0.49]{} {width="\textwidth"}
To confirm these results are not simply an artifact of our question inclusion criteria for the CCES, we perform a series of robustness checks. We progressively add questions about respondent political affiliation, demographics, education and other characteristics. Questions with categorical responses, like race, are converted into indicators for each common response. This one-hot encoding artificially creates a separate survey question per response value, so we lose some fidelity with the CCES. See Appendix \[sec:question-order-robustness\] for box plots of question rank with the additional questions included.
The actively chosen ordering with augmented questions remains largely faithful to the active ordering in Figure \[fig:cces16-question-rank\]. While questions about gender, party identification, parenthood and home ownership slot into the first 15 positions, the same questions about political issues dominate the top 15 in largely the same order. This provides more support for earlier inclusion of questions concerning the environment, abortion, and Obamacare. Respondent covariates displace questions about approval of senators. Interestingly, the active strategy leaves questions about race and education until later in the survey, possibly because these one-hot-encoded variables are not well captured by a low-rank matrix decomposition.
Changes in question order over time
-----------------------------------
We also compute the active ordering for the 2012 CCES using the same simulation parameters ($r=4, \alpha=1$); see Figure \[fig:cces12-question-rank\]. As in 2016, the Obamacare question comes first, followed by a question about Obamacare repeal in third place to gather more information along this latent direction. In 2016 the active strategy does not address immigration until the 10th question; in 2012 four of the top 10 questions are about harsher immigration policies. Gay marriage also features more prominently in the 2012 active ordering, coming in 4th versus 14th in 2016.
Inspecting the 4-dimensional question factors for both CCES years (Table \[tbl:cces16-pcs\] and \[tbl:cces12-pcs\]), the first two principal components seem to indicate partisanship and popularity, as in our earlier analysis of 2-dimensional question factors. However, in 2012 the third principal component aligns with support for isolationist or xenophobic policies, while no separate principal component loading heavily on the immigration questions appears in 2016. One possible explanation is that immigration, which was a separate enough concept in 2012 to command its own principal component, became absorbed into partisanship in 2016 as Trump claimed the immigration issue. Hence, the active strategy asks immigration questions early to determine user position along this concept in 2012 but not 2016.
The fourth principal component in 2012 may indicate fiscal conservatism and social liberalism, as it correlates support for gay marriage, ending the “Don’t ask, don’t tell” policy, and the Ryan and Simpson-Bowles budget plans reducing federal spending. In 2016, the third principal component seems to indicate approval of elected officials. The fourth principal component correlates support for greener environmental policies, support for abortion restrictions and opposition to gay marriage. This suggests a group of socially conservative or religious respondents who are concerned about the environment. The active strategy doubles down on both environmental and abortion questions in order to ascertain membership in this group, in addition to partisanship. Going beyond two latent dimensions helps to identify parts of the electorate that do not behave according to conventional partisan wisdom; the active strategy actively seeks information along these more subtle directions.
Side information
================
Active question ordering delivers a similar improvement for the Facebook survey – asking 20 questions achieves the same error as asking 30 questions in a random order – though the results are more variable. When we incorporate side information about respondents, both active and random strategies may see efficiency gains. Theoretical results have established that sufficiently informative side information improves the sample complexity of matrix completion [@xu; @chiang]. We give a simple proof-of-concept of the value of side information, by subgrouping respondents based on two covariates: country and length of time since joining Facebook. For each subgroup, we compute the mean and covariance of user factors in the training half, and use these as the empirical Bayes prior for all subgroup users in the simulation half. It is unclear whether the strategies using subgroup information attain lower predictive error than their fully pooled counterparts throughout the simulated survey. Future work should aim for shrinkage across subgroups, perhaps with a hierarchical model, and incorporate more covariate information.
As an alternative way of incorporating side information, we revisit the full CCES 2016 dataset used for robustness checks on question order. We reveal responses to all questions about respondent covariates before simulating a survey with the remaining questions, so that BPMF can update each user’s prior to include these “free” covariates. In terms of survey design, this can be regarded as requiring a set of covariate questions upfront, before applying an active ordering to opinion questions. Early in the survey, free covariates reduce error metrics for both random and active strategies, at the cost of introducing bias (Figure \[fig:cces16-free-covariates-compare-metrics\]). The information advantage of free covariates disappears as more questions are asked, so that strategies making use of free covariates end up at a higher final error than their agnostic counterparts. The point where free covariates are no longer useful depends on the error metric. For MSE the active strategy breaks even around 8 questions; the random strategy breaks even around 17. The active ordering with free covariates is similar to that without (Figure \[fig:cces16-question-rank-free-covariates\]).
![We plot error on the holdout set for the 2016 CCES, averaged over all questions. This time, we also make responses to covariate questions available for free (“free-cov”). At the beginning of the survey, information from free covariates narrows down user position in latent space, driving down predictive error. The free covariates participate in matrix factorization; the SoftImpute loss function now includes terms for reconstruction error on free covariates. This changes the question factors estimated from the training half. Meanwhile, predictive error is still evaluated on held-out survey responses only, which do not include covariates. Thus, we have biased question factors away from those that would optimize predictive error, in exchange for variance reduction from knowing the free covariates. This bias manifests as higher predictive error for the “free-cov” strategies at the end of the survey. \[fig:cces16-free-covariates-compare-metrics\]](cces16/error_comparison_with_free_covariates.png)
Ordinal logit response model
============================
In this section we explore adjusting the model to better capture binary and ordinal response values. We replace the Gaussian likelihood for responses with an ordinal (ordered) logit likelihood. As we will see, question selection now depends on the respondent’s previous answers; the active question order becomes nondeterministic. The determinism of the active question order for BPMF is an artifact of the Gaussian likelihood.
To model quantized outputs in the response matrix, prior work has used the ordinal logit likelihood [@cao] and similar link functions [@klopp]. These works formulate matrix completion as maximum likelihood with nuclear norm regularization. We continue with a Bayesian probabilistic matrix factorization approach. Posterior inference for $U, V$ in the ordinal logit model involves non-conjugacy, so we resort to variational inference, also used for matrix completion in [@lim; @seeger]. We forfeit the closed-form posterior update exploited by the active strategy for BPMF, but optimal design via Fisher information offers a way forward.
Probabilistic matrix factorization model
----------------------------------------
The model becomes: $$\begin{aligned}
u_i &{\stackrel{iid}{\sim}}\mathcal{N}(\mu_U, \Lambda_U^{-1}) \\
v_j &{\stackrel{iid}{\sim}}\mathcal{N}(\mu_V, \Lambda_V^{-1}) \\
R_{ij} \mid U, V &{\stackrel{ind}{\sim}}\text{OrdLogit}(u_i^T v_j, \beta_j)\end{aligned}$$
By modeling $R_{ij}$ as ordinal logit, we allow $R_{ij}$ to take values in the range $\{1, 2, \ldots, M_j+1\}$ with probabilities $\{\pi_{j,1}, \pi_{j,2}, \ldots, \pi_{j,M_j+1}\}$. Note response frequencies vary across questions, and questions need not have the same number of response values. The probabilities are defined by the logistic link and a series of cutpoints $\beta_j = (\beta_{j,1}, \ldots, \beta_{j,M_j})$. For simplicity of presentation, we drop the indexing for question $j$. Thus $R_{ij}$ takes values in $\{1, 2, \ldots, M+1\}$ with probabilities $\{\pi_1, \pi_2, \ldots, \pi_{M+1}\}$, parameterized by cutpoints $\beta = (\beta_1, \ldots, \beta_M)$ as follows: $$\begin{aligned}
\text{logit}\left(\sum_{k=1}^\ell \pi_k \right) &= u_i^T v_j + \beta_\ell \qquad (\ell = 1, \ldots, M) \\
\pi_{M+1} &= 1-\sum_{k=1}^{M} \pi_k\end{aligned}$$
Inference
---------
We perform posterior inference on $U, V$ in the above model when estimating user and question factors from the training half, and again after each iteration of the simulation, when the number of questions increases by one. From the updated posteriors per iteration, we compute predictive error on held-out survey responses. See Algorithm \[algo:ordlogit\].
Given responses $R$, we obtain approximate posteriors for $U$ and $V$ in the above model using mean-field variational inference in `edward` [@edward]. Our variational distributions are fully factorized Gaussian: $$\begin{aligned}
q(U) &= \prod_{i=1}^n q(u_i) = \prod_{i=1}^n \prod_{j=1}^r \mathcal{N}(\mu_{ij}, \sigma_{ij}) \\
q(V) &= \prod_{j=1}^k q(v_j) = \prod_{j=1}^k \prod_{i=1}^r \mathcal{N}(\nu_{ji}, \tau_{ji})\end{aligned}$$ Variational inference finds parameters $\{\mu_{ij}, \sigma_{ij}, \nu_{ji}, \tau_{ji}\}_{i,j}$ that maximizes the evidence lower bound, or equivalently minimizes the KL divergence between the variational distribution and the true posterior. To predict $R_{ij}$, we set $u_i$ and $v_j$ equal to their variational means $\mu_i$ and $\nu_j$ and compute the mean of the resulting ordinal logit random variable.
As inputs to variational inference, we employ priors $\mu_U = \mu_V \equiv 0$ and $\Lambda_V = \Lambda_V \equiv \mathbf{I}_r$. To derive cutpoints $\beta$, we follow the inverse approach in the `rstanarm` package [@rstanarm]. For each question, we obtain a length-$(M+1)$ vector of probabilities $\pi = (\pi_1, \ldots, \pi_M, 1-\sum_{i=1}^M \pi_i)$ from the simplex, corresponding to the ordinal response values. We then apply the logit transform to the first $M$ entries of $\texttt{cumsum}(\pi) = \left(\pi_1, \pi_1+\pi_2, \ldots, \sum_{i=1}^M \pi_i, 1 \right)$, obtaining $$\left\{\beta_j\right\}_{j=1}^M = \log \frac{\sum_{i=1}^j \pi_i}{1-\sum_{i=1}^j \pi_i}$$ We draw $\pi \sim \text{Dirichlet}(c_1, c_2, \ldots, c_{M+1})$, where the concentration parameters are prior counts of the response values. That is, we set $c_\ell$ equal to the number of times a respondent answers $\ell$ to this question in the training half.
Active learning formulation
---------------------------
We also update our item selection strategy for the ordinal logit response model. In this situation, we consider $V$ to be fixed, using the posterior mean of $q(V)$ given by $\{\nu_{j}\}_{j=1}^k$. We are administering the survey to user $i$; we seek the question $j$ that maximizes information about $u_i$.
We simplify the ordinal logit model to a single user: $$\begin{aligned}
u_i &{\stackrel{iid}{\sim}}\mathcal{N}(\mu_U, \Lambda_U^{-1}) \\
R_{ij} \mid U, V &{\stackrel{ind}{\sim}}\text{OrdLogit}(u_i^T v_j, \beta)\end{aligned}$$
Unlike in the case of Gaussian likelihood, we do not have a conjugate, closed-form update for the posterior of $u_i$, so we cannot minimize a measure of posterior variance directly. Instead, we work with the Fisher information, whose inverse is the asymptotic variance of the maximum likelihood estimate. This approach follows the optimal design literature, notably Segall [@segall09], who applies it to logistic likelihood for binary responses. Our approach can be considered an ordinal logit generalization of [@segall09].
Fisher information is computed around a value of $u_i$. As our latest estimate of $u_i$, we use the mean of the user variational distribution, $\mu_i$, from the previous iteration of probabilistic matrix factorization on revealed responses. Repurposing this provisional estimate of $u_i$ is more computationally efficient than computing a MAP estimate of $u_i$ from the single-user model.
We denote the Fisher information gained from a response to question $j$ as $\mathcal{I}^j(u_i)$, and the *observed* Fisher information from observing response $x$ to question $j$ as $\mathcal{J}^j(u_i; x)$. Then
$$\mathcal{J}^j(u_i; x) = -\frac{\partial^2}{\partial u_i \partial u_i^T} \log \text{Pr}(R_{ij} = x \mid u_i, v_j, \beta)
\equiv -\frac{\partial^2}{\partial u_i \partial u_i^T} \log \pi_{ijx}$$ and $$\mathcal{I}^j(u_i) = E\left[ \mathcal{J}^j(u_i; R_{ij}) \right] = \sum_{x=1}^{M+1} \pi_{ijx} \mathcal{J}^j(u_i; x)$$
In the ordinal logit model, $\mathcal{J}^j(u_i; x)$ and $\mathcal{I}^j(u_i)$ involve complicated but closed-form expressions. The Hessians are computed with autodifferentiation in `edward`.
Let $\mathcal{O}$ contain the indices of past questions and $\mathcal{U}$ the indices of unasked questions. We compute the sum of observed information over $\mathcal{O}$, and consider adding a Fisher information term for question $j \in \mathcal{U}$. We find the question that minimizes our optimal design criterion, that is, which potential response would contribute the most information to $u_i$ in expectation. Formally, this question is $$\min_{j \in \mathcal{U}} {\text{tr} \;}\left[ \Lambda_U + \sum_{\ell \in \mathcal{O}} \mathcal{J}^\ell(u_i; R_{i\ell}) + \mathcal{I}^j(u_i) \right]^{-1}$$
As before, each iteration of the simulation solves this optimization once per user, widening the survey by one question. The user-specific problems can be solved separately in parallel. This subprocedure is placed in context in Algorithm \[algo:ordlogit\].
$\{\mu_{ij}, \sigma_{ij}, \nu_{ji}, \tau_{ji}\}_{i,j} \leftarrow \OrdLogitPMF(R^{train}, \beta)$ $(v_1, \ldots, v_k) \leftarrow (\nu_1, \ldots, \nu_k)$ $\Lambda_U^{-1} \leftarrow \EmpiricalCov(\mu_1, \ldots, \mu_n)$ $\mathcal{O}_i \leftarrow \emptyset,\; \mathcal{U}_i \leftarrow \{1, \ldots, k\} \; \forall i$
Results
-------
Simulations with the ordinal logit likelihood largely uphold prior BPMF results using the Gaussian likelihood, while showing improved modeling flexibility. In Figure \[fig:cces16-ordlogit-compare-metrics\], the active strategy continues to outperform random in terms of predictive error. This comes at the cost of higher bias in the early stages of the survey. Matrix completion with the ordinal logit model attains lower final error levels than with BPMF, as a comparison with Figure \[fig:cces16-compare-metrics\] shows. The improvement comes from a combination of better modeling of ordinal responses and introducing additional parameters in the form of question-specific cutpoints. In addition, the advantage of the active strategy is more pronounced than before; see Figure \[fig:cces16-ordlogit-relative-complexity\].
In Figure \[fig:cces16-ordlogit-question-rank\] we plot question position across randomly sampled users in one simulation of the active strategy. The nondeterministic question order creates more variability in question rank within a single ordinal logit simulation than there is across BPMF simulations in Figure \[fig:cces16-question-rank\]. Still, the active orderings for both models prioritize questions about abortion and environmental policies. While ACA repeal is the top question for BPMF, the ordinal logit model places it anywhere in the first 25 questions. Other questions with highly variable positions concern approval of Obama, the national economy, and background checks for gun sales, all of which appear with little variability after the 25th question in the BPMF ordering. The ordinal logit model favors job approval questions more, possibly having detected greater prior variance in this latent direction.
Figure \[fig:cces16-ordlogit-question-paths\] plots the active ordering more granularly for this subset of users, as individual paths through survey questions. We see that the variability in question rank in Figure \[fig:cces16-ordlogit-question-rank\] arises not from a few common question orderings, but rather from diverse, personalized paths dependent on responses to previous questions. Using the ordinal logit likelihood, we have made our active strategy truly adaptive.
Order effects
=============
Algorithmic ordering of survey questions may produce biased responses (compared to random ordering) when responses vary according to their position in the survey instrument – a phenomenon known as *order effects*. In this section we estimate the magnitude of order effects in the randomly-ordered Facebook survey in order to understand how large the bias introduced by the active strategy is likely to be.
In order to estimate position effects, we fit a linear regression per survey question with the relative position of the question in the order as a predictor. In this model we use data from only completed surveys (about 30% of the surveys) in order to preclude attrition bias. Using this model, we estimate the difference in standardized response for each question appearing at the end of the survey compared to the beginning. As a null distribution, we randomly re-order the survey and fit the same model $200$ times. The results are presented in Figure \[fig:question-position-effects\](a). We find evidence for a number of survey questions exhibiting position effects, where responses vary significantly depending on whether they are asked early or late in the survey. The worst-case bias appears to be about $0.3$ standard deviations on the response scale.
We also estimate whether the previous survey question a user answered affects the response on the following question. We fit an L1-penalized regression with a parameter for all pairs of survey questions and previous questions, using 10-fold cross-validation to select the optimal penalty parameter. We visualize the results of this model in Figure \[fig:question-position-effects\](b). About 10% of the possible question pairs exhibit a non-zero interaction effect. Some questions tend to be influential on the following question (columns with multiple points) while others tend to be more likely to be affected by the prior question (rows with multiple points). Similarly to position effects, the effects we observe are usually less than $0.2$ standard deviations on the response scale.
While these estimates are rudimentary, they provide some sense of the size of the bias introduced by the active learning algorithm – it should be a small contribution to the total error compared to the variance reduction we achieve through receiving more informative responses. A promising area for future work would be to test the active learning strategy online compared to a random ordering and to directly estimate the bias.
[0.44]{} ![Estimated order effects in the Facebook survey. \[fig:question-position-effects\]](facebook/order_position.pdf "fig:"){width="\textwidth"}
[0.50]{} ![Estimated order effects in the Facebook survey. \[fig:question-position-effects\]](facebook/order_interactions.pdf "fig:"){width="\textwidth"}
Discussion and future work
==========================
With active learning that seeks to reduce variance in the latent space of concepts, we have identified survey orderings that achieve the same error with a smaller question budget. Exploring the CCES questions preferred by the active strategy, we develop insights about the most informative set of questions for predicting political opinion. The active strategy affords a new notion of feature importance for domains with wide item sets and low-dimensional latent structure. Matrix factorization gives us dimension reduction; active question order makes this dimension reduction interpretable.
However, the deterministic active question order arising from BPMF is not ideal; it is non-robust to estimation error of question factors. With the ordinal logit model, we have made the active ordering dependent on collected responses. This is a side effect of changing the likelihood from Gaussian to ordinal logit, but our active strategy still assumes away uncertainty in question factors. Our variational inference for matrix factorization already quantifies uncertainty in question factors approximately. Future work should incorporate this uncertainty in the variance being minimized. The active strategy could take other forms of uncertainty into account. For instance, it is natural to consider concepts in latent space as changing over time. By explicitly modeling parameter drift, we could encourage the active ordering to evolve. Another way of inducing different question orderings is, broadly, better managing the exploration-exploitation tradeoff. Currently the active strategy always minimizes the optimality criterion. We could experiment with bandit algorithms that make use of estimated uncertainty in question factors.
Future work should devote special attention to the logistics of survey administration. The difficulty of implementing an active ordering depends on the variant used. The active question order from BPMF is relatively straightforward to use across survey modes, as it can be computed offline. An adaptive order computed on the fly using previous responses, as with the ordinal logit model, would require more infrastructure. Web and computer-assisted telephone surveys could lean on backend software to suggest the next question. In-person field surveys would need a mechanism of inputting responses and quickly receiving the next question, such as a mobile app that calls a low-latency API for the active ordering. It would be productive to integrate with existing survey platforms to make the active strategy available. Additionally, suggesting questions in batch may be more practical than sequentially. This would compel us to move beyond greedy question selection to optimal design with a multi-step horizon. It is also an opportunity to design logical groupings of questions, informed by domain knowledge of practitioners.
Our evaluation of the active strategy focuses on measures of predictive error. This is motivated by the wide range of downstream uses of survey data. Survey researchers may be interested in the distribution of responses to individual questions; the relationship between one question and another across individuals; the reduction of multiple questions to a single construct; or any of these analyses conditional on arbitrary subgroups of respondents. Our predictive error measures, which assign equal weight to the loss from each user-question pair, allow us to evaluate our imputations from various strategies agnostic to the downstream use case. In some cases, individual responses to a wide set of questions are equally useful. For instance, using the CCES, researchers have investigated whether alignment between the policy preferences of constituents and the votes of their legislators predicts constituent support for their legislators [@ansolabehere]. This research relies on knowing individual responses to “roll call” questions about policy preferences – the same questions we have included from the CCES. As another example, in survey experiments, responses from a baseline survey could serve as pre-treatment covariates for estimation of heterogeneous treatment effects [@broockman]. A priori, the analyst cares equally about the predictive accuracy of imputed responses across respondents and questions. Still, future work should assess the frequency properties of common estimators applied to imputed responses, and explore how to specialize the active strategy to different estimators.
We have incorporated side information in two simple ways, but many avenues exist for more sophisticated modeling of side information. For instance, we could create individual user priors based on more covariates using multilevel regression. Alternatively, we could learn a Gaussian process that maps covariates into latent space, as a prior for user factors [@adams; @zhou]. A different tack is to add terms for user and question covariates to the response specification; they slot naturally into the nuclear norm minimization in [@athey] and the BPMF model in [@porteous]. From a cost perspective, we could explore the tradeoff between gathering side information and survey responses, incorporating the information gain and acquisition cost of both. Our experiments including responses to covariate questions in the matrix factorization suggest an exchange rate between side information and question responses.
Finally, it remains to address the bias of our matrix completion procedure. Standard matrix completion assumes entries are missing at random. This assumption is violated when users withhold responses that are noncommital or unpopular. By tailoring questions to a user’s inferred position in latent space, the active strategy can exacerbate bias. One bias correction method for matrix completion is explicitly modeling the missing-data mechanism [@marlin]. More common are weighting approaches that regularize a weighted nuclear norm [@srebro10] or de-bias the loss terms with inverse propensity weights [@schnabel; @athey]. All of these approaches increase the importance of correctly predicting rare users and items. Another source of bias is model misspecification: a low-rank linear decomposition does not capture all of the response variance. Thus, predictive error plateaus after a certain survey length. It could be worth exploring nonlinear matrix factorization in the form of Gaussian process latent variable models, which replace the dot product of user and question factors with a function of user factors, endowed with a Gaussian process prior [@lawrence].
Acknowledgements
================
We thank Karan Aggarwal, Eytan Bakshy, George Berry, Dennis Feehan, Avi Feller, Will Fithian, Mike Jordan, Frauke Kreuter, Jacob Montgomery, Adam Obeng, Rebecca Powell and Alex Theodoridis for their valuable input. CZ is supported by an NSF Graduate Research Fellowship. CZ and JS are supported by Office of Naval Research (ONR) Grant N00014-15-1-2367.
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abstract: 'Known bright S stars, recognized as such by their enhanced s-process abundances and C/O ratio, are typically members of the asymptotic giant branch (AGB) or the red giant branch (RGB). Few modern digital spectra for these objects have been published, from which intermediate resolution spectral indices and classifications could be derived. For published S stars we find accurate positions using the Two-Micron All Sky Survey (2MASS), and use the FAST spectrograph of the Tillinghast reflector on Mt. Hopkins to obtain the spectra of 57 objects. We make available a digital S star spectral atlas consisting of 14 spectra of S stars with diverse spectral features. We define and derive basic spectral indices that can help distinguish S stars from late-type (M) giants and carbon stars. We convolve all our spectra with the Sloan Digital Sky Survey (SDSS) bandpasses, and employ the resulting $gri$ magnitudes together with 2MASS $JHK_s$ mags to investigate S star colors. These objects have colors similar to carbon and M stars, and are therefore difficult to distinguish by color alone. Using near and mid-infrared colors from IRAS and AKARI, we identify some of the stars as intrinsic (AGB) or extrinsic (with abundances enhanced by past mass-transfer). We also use $V$ band and 2MASS magnitudes to calculate a temperature index for stars in the sample. We analyze the proper motions and parallaxes of our sample stars to determine upper and lower limit absolute magnitudes and distances, and confirm that most are probably giants.'
author:
- Elizabeth Otto
- 'Paul J. Green'
- 'and Richard O. Gray'
title: 'Galactic S Stars: Investigations of Color, Motion, and Spectral Features'
---
INTRODUCTION {#sec:introduction}
============
When a low to intermediate mass star exhausts its core supply of hydrogen, contraction and heating of the core causes the outer layers of the star to expand and cool, whereby the star becomes a red giant (RG) with an average effective temperature of 5000K or lower. Once the core temperature rises to about 3$\times 10^8$K, helium burning begins, and the stellar surface contracts and heats creating the horizontal branch (for Population II) or red clump (for Population I) stars on the color-magnitude diagram. Completion of core helium burning and the start of helium shell burning returns the star to a luminous, cooler phase as it evolves onto the asymptotic giant branch (early- or E-AGB) where the star greatly expands and the hydrogen shell is (almost) extinguished. Later, as the He-burning shell approaches the H-He discontinuity, a phase of double shell burning begins. Recurrent thermal instabilities in this thermally-pulsing asymptotic giant branch (TP-AGB) phase are driven by helium shell flashes, periodic thermonuclear runaway events in the He-shell The energy released by these pulses expand the star, whereby the hydrogen shell is again basically extinguished during some time. A third dredge up might then occur, and afterward the hydrogen shell will re-ignite (and a new thermal pulse will occur). During these short-lived pulses, nucleosynthesis products from combined H-shell and He-shell burning are dredged up to the outer layers of the star. The complex details of the dredge-up and its outcome are strongly mass-dependent ( and references therein), but generally speaking, surface abundance enhancements result in C, He, and the $s$-process elements, including Ba, La, Zr, and Y.
S and Carbon (C) stars are traditionally thought to be members of the TP-AGB. In addition to possibly elevated C/O ratios, these stars exhibit other unique spectral characteristics as a result of the dredge-up. While M giants often exhibit titanium oxide (TiO) absorption bands, both carbon and zirconium have a higher affinity to free oxygen. Therefore, a higher C/O ratio also results in the disappearance of TiO bands and the appearance of zirconium oxide (ZrO) bands. S stars are therefore distinguished primarily by their ZrO bands, while C stars show strong C$_2$ and CN bands. The S star spectral class is divided into three subtypes (in order of increasing carbon abundance): MS, S, and SC stars. Classification into one particular class is difficult, but is generally based on the C/O ratio and the relative strengths of TiO, ZrO, and CN molecular absorption bands. Overall, a star in the S star class is generally defined as having a C/O ratio (r) of $0.5 < r < 1.0$ and strong ZrO absorption features . Pure S stars are those that show only ZrO bands, and no TiO bands [@1978MNRAS.184..127W]. Classification is complicated by the fact that increased intensity of the ZrO bands may be due to an excess of Zr, rather than being exclusively tied to the C/O ratio [@2003IAUS..210P..A2P]. Furthermore, many AGB S and C giants are highly variable, and since the strength of molecular bands depends on the effective temperature and surface gravity of the star, spectral type can vary significantly with time.
S stars whose spectra show lines of the short-lived element $s$-process technetium (Tc) are known as “intrinsic” S stars, and are most likely be members of the TP-AGB. $^{99}$Tc is only produced by the $s$-process, while the star is on the AGB. Since the half-life of $^{99}$Tc is only 2.13 x 10$^5$ years, and the average duration of the AGB phase of stellar evolution is roughly 1 Myr, stars exhibiting significant abundances of technetium are almost certainly members of the AGB . By contrast, technetium-poor S stars are known as “extrinsic” S stars. These stars are likely to be part of binary systems and their unusual chemical abundances probably originated in a past mass-transfer episode . The present extrinsic S star likely once accreted $s$-process rich material from its companion, a TP-AGB star at the time that has since evolved into a white dwarf. Extrinsic S stars therefore display enriched $s$-process elements, despite having never produced these elements themselves. Most known extrinsic S stars are assumed to be members of the red giant branch (RGB). They can sometimes be distinguished from intrinsic S stars on the basis of color, because they do not necessarily show the red excess characteristic of stars on the AGB. While all currently known extrinsic S stars are thought to be giants, these definitions open up the possibility of extrinsic S stars on the main sequence (hereafter referred to as dwarf S stars) that have previously accreted $s$-process material and carbon from a companion but whose Tc has since decayed. Analogous to the dwarf C stars [@Green91], a faint S star could be shown to be a dwarf if it has a sufficiently large parallax or proper motion; a large proper motion predicts a tangential velocity greater than the Galactic escape speed unless the object is faint and nearby.
S stars are very similar to C stars, with the major difference being a slightly lower C/O ratio. Faint high galactic latitude carbon (FHLC) stars were once assumed be giant stars at large distances. In 1977, however, G77-61 was discovered to have high proper motion and an upper limit absolute magnitude of +9.6, therefore making it the first known carbon dwarf star [@1977ApJ...216..757D]. In this case, the term dwarf refers to a star on the main sequence. @Green91 discovered four other dwarf carbon (dC) stars using techniques that selected known carbon stars for high proper motion. Now, well over 100 dwarf carbon stars are known [@2004AJ....127.2838D]. The local space density of dCs is far higher than all types of C giants combined [@1992ApJ...400..659G]. Therefore, contrary to previous assumptions, dwarf C stars are the numerically dominant type of carbon star in the Galaxy.
Since dC stars are now known to be so common, we have begun a search for dwarf S stars. There is little reason to believe that dwarf S stars should not exist. In fact, they may be more abundant than dC stars, since less carbon enhancement is required to produce an S dwarf as compared to a C dwarf. On the other hand, if the range of abundance ratios that produces the characteristic features of S stars is narrow, they may be quite rare. Furthermore, the range of C/O ratios that produces S star spectral features may be different for giants and dwarfs, given the higher gravities and larger temperature ranges in dwarfs.
We use medium-resolution digital spectra of known S stars (which are probably giants) to generate corresponding Sloan Digital Sky Survey (SDSS) colors, to see whether their colors might be sufficiently distinctive to search for additional S stars in the SDSS with reasonable efficiency. We also use the FAST spectra to investigate molecular band spectral indices for classification.
Our paper is organized as follows: in §\[sec:obs\] we briefly review the observations and our data reduction and processing procedures. In section §\[sec:spatlas\] we describe the production of a spectral atlas of S stars, and derive spectral indices from the molecular band strengths in §\[sec:specind\]. In §\[sec:colors\] we describe the generation of SDSS colors and analyze the overall colors of the FAST sample in order to determine whether an efficient color selection for S stars exist. In §\[sec:intext\] we discuss techniques for distinguishing intrinsic and extrinsic S stars and apply them to the FAST sample. In §\[sec:motion\] we analyze the parallaxes and proper motions of S stars in the sample to characterize upper and lower limit magnitudes and distances. We use colors to generate approximate temperature indices for S stars in §\[sec:temp\]. In §\[sec:sum\] we summarize our results and present our conclusions.
OBSERVATIONS {#sec:obs}
============
Sample Selection {#sec:obs1}
----------------
The largest spectral catalog of S stars is Stephenson’s second edition (1984), which includes 1347 objects with poorly known positions. The stars in this catalog were originally identified from blue, red, and infrared objective prism plates. The majority of the S stars were discovered on the basis of the red system of ZrO absorption bands, with a bandhead around 6474[Å]{}. Since the Stephenson catalog has a limiting V magnitude of approximately 11.5, most of the objects are highly saturated in the SDSS and reliable u, g, r, i, and z magnitudes have not been found for any known S stars. We assembled a sample of known S giants in order to generate likely g and r colors for S giants and dwarfs. These constraints on colors of S giants and dwarfs could allow us to search the SDSS and 2MASS catalogs for potential dwarf S star candidates.
To choose the stars making up the sample, we selected high Galactic latitude ($|b| > 20$ deg), northern ($\delta > -05 $ deg) S stars from Stephenson’s catalog. This ensures that our sample is accessible from Mt. Hopkins, and also decreases problems of object confusion or reddening. We then correlated the sample from Stephenson’s catalog with 2MASS to find improved positions for most objects; accuracy of the original positions is typically $\sim 4\arcsec$, but commonly as poor as 20. The final sample list, which includes 57 objects, can be found in Table\[tab:stars\]. Each object is accompanied by its identification in both the Stephenson and 2MASS catalogs, as well as its position. Since many of these stars are known to be variable, we include the date of observation. We include other common identifiers for each target and a published spectral type, if available. We include our calculated temperature index for the star, which is discussed further in §\[sec:temp\]. Spectral types and temperature indices of the S stars in the sample may vary as a function of time, because the stars themselves do. When spectral types for different epochs were available, we match the $V$ magnitudes of the epochs to those listed in Table\[tab:mags\] to find the spectral type. Finally, we include identifications of some stars as intrinsic or extrinsic (denoted respectively by ’i’ and ’e’) from @2006AJ....132.1468Y or from our own calculations based on AKARI magnitudes. For more information regarding the intrinsic/extrinsic identifications, see §\[sec:intext\].
Observations and Data Reduction {#sec:obs2}
-------------------------------
Spectra were obtained using the FAST instrument on the 1.5 m Tillinghast reflector on Mt. Hopkins. FAST produces medium-resolution optical wavelength spectra, spanning a wavelength range from 3474 to 7418 [Å]{}. All data were taken using the 3" slit and a 300linesmm$^{-1}$ grating, producing a resolution of $\lambda / \Delta\lambda\sim$1,000 at 6,000Å. Exposure times varied widely based on the magnitude of the star, but ranged from $\sim$3 to 300 s. The two-dimesional spectra were bias-subtracted and flat-fielded by the observatory standard data reduction procedure. After extraction, and the one-dimensional spectra were wavelength calibrated.
We used standard star spectra taken each night of the observations to produce sensitivity curves and flux calibrated spectra. An extinction correction file from Kitt Peak National Observatory was also applied to correct for major atmospheric effects. The standard calibration stars used to create the sensitivity curve are BD+26 2606, BD+33 2642, BD+17 4708, Feige 34, and HD 19445. Flux calibration and extinction corrections were performed using the *onedspec* package and the *calibrate* task in IRAF. We used a combination of observatory logs and night-to-night comparison of the sensitivity function to determine whether the calibrated target spectra were reliable enough to use in color analysis. We rejected some of the initial data sample based on concerns about non-photometric conditions and atmospheric distortion. Other nights are included in the final data for analysis because while the data may not be entirely photometric, it remains reliable enough for color analysis because of the non-wavelength dependent nature of most remaining distortions. We discuss the errors in our derived SDSS colors further in §\[sec:colors1\].
Many of the targets are luminous enough in red wavelengths that longer exposure times cause parts of the spectrum to saturate. The wavelengths from 6000[Å]{} onward are of particular concern. However, shorter exposure times limited the amount of information we could derive from wavelengths below 4500[Å]{} because of low signal-to-noise ratios. To create an S star spectral atlas, when possible we interpolated between two different exposure times: a shorter exposure time that produced accurate, unsaturated spectra at the longer wavelengths, and a longer exposure that resulted in higher signal-to-noise at shorter wavelengths. Generally, we determined by hand which spectra were reliable in a particular wavelength region by comparing multiple exposure times and examining long exposures for nonlinear behavior in areas from 6000[Å]{} onward. For stars with only one available exposure time, we rejected a star as saturated if the counts exceeded 60,000. After interpolating between spectra to eliminate saturated regions and rejecting some data, we were left with 46 S star spectra.
Creating the Spectral Atlas {#sec:spatlas}
===========================
We use S star line identifications [@1978MNRAS.184..127W] to identify major spectral features and select objects with unique characteristics to include in the final spectral atlas, which includes 14 objects marked in Table\[tab:stars\]. The objects were chosen to be a representative sample of the larger FAST sample, which includes many fundamentally similar objects. Two sample spectra with associated molecular line identifications are plotted in Figure\[fig:speclines\]. While the spectra display the same molecular bands, most notably those associated with ZrO, they have different overall colors, which may be due to differences in abundances and/or effective temperatures. We also note that these spectra show some TiO bands, indicating that they are not pure S stars, but instead probably belong to the MS classification. The appearance of TiO bands indicates the possibility of a C/O ratio slightly less than 0.95. The creation of a spectral atlas allows for greater investigation into the similarities and differences within the S star class. For instance, some members of the spectral atlas exhibit, in addition to prominent ZrO features, H$\alpha$, H$\beta
$, and H$\gamma$ emission lines. The presence of these emission lines indicates that these objects are almost certainly AGB stars, e.g., Miras. The spectral atlas is made available digitally as 14 individual FITS files via this journal.
Spectral Indices of S Stars {#sec:specind}
===========================
We can immediately use the first digital spectral atlas of S stars to derive some medium-resolution spectral indices, potentially useful to classify stars as M, S, or C even where they may have very similar broadband colors. Apart from our own FAST spectra of 48 S stars, we use M giants from the Indo-US spectral atlas [@Valdes04]. We use eight carbon star spectra also obtained with the FAST for a separate project (Green et al. 2011, in preparation). These latter spectra span a wide range of spectral band strengths, but are selected from the SDSS color wedge defined by @Margon02.
We defined three indices, one each for TiO, ZrO, and C$_2$. We follow the basic premise of using ratios of mean flux per Ångström across key molecular bands, so that spectra of differing resolution should yield similar results. We use a neighboring comparison region for each band, so that sensitivity to broadband flux calibration is minimized. For the ZrO index, we divide the mean flux Å across 6400 – 6460Å (ZrO off-band) by that between 6475 – 6535Å (ZrO band). For the TiO index, we use mean Å$^{-1}$ flux from 7065 – 7175Å (TiO band) divided by 6965 – 7028Å (TiO off-band). The C index is derived from 5025 – 5175Å (C$_2$ band) and 5238 – 5390Å (off-band). Thus, lower flux in the denominator caused by strong molecular bands yields a larger spectral index value. We note that the C index near 5100Å can also be substantially contaminated by TiO, so we label it “CT51” hereafter. At least within this representative sample of stars, a combination of these indices yields a preliminary S star classification, as shown in Figure \[fig:specind\]. The objects most reliably classified as S stars would have ZrO index $>1.2$ and C$_2$ index $<1.4$.
Color Selection of S Stars {#sec:colors}
==========================
Generating SDSS and 2MASS Colors {#sec:colors1}
--------------------------------
To use the SDSS database to find likely candidates for both S giants and dwarfs, we need reliable colors for our objects in as many SDSS bandpasses as possible. The spectra from FAST cover the bandpasses for the $g$ and $r$ filters completely, and we are thus able to convolve our spectra with these bandpasses and generate SDSS colors. Each of the FAST spectra contains 2681 data points, with approximate wavelength separation of 1.47 Å. To convolve the spectra with the filter transmission curves, we use linear interpolation between transmission values to match a transmission constant to each data point in the spectrum. Once we find a transmission constant for each wavelength, we normalize the transmission curve and then convolve the spectrum with the transmission curve. We use the IRAF task *calcphot* in the *synphot* package of *stsdas* to convolve the spectra with the bandpasses and obtain SDSS colors from our spectra. The SDSS bandpass curves are taken from information made available by the SDSS consortium [@1996AJ....111.1748F]. As with all SDSS magnitudes, the resultant magnitudes are in the AB magnitude system. By including both SDSS and 2MASS bandpasses, we are able to calculate 10 unique colors.
We also conduct error analysis on the derived SDSS colors using the standard star spectra and the known g-r colors and r magnitudes for BD+26 2606, BD+33 2642, BD+28 4211, and BD+174708. These colors are published by the SDSS consortium, and allow us to constrain our errors in both the magnitudes and colors [@2002AJ....123.2121S]. We calculate the difference between measured and standard $g$ and $r$ magnitudes, and find a standard deviation of $ \sigma{_g} = 0.191 $ mag for the $g$ band and $ \sigma{_r} = 0.192 $ mag for the $r$ band. While these seem high and would suggest, via propagation of errors, a correspondingly high error in our colors (on the order of 0.4 mag), we find a standard deviation of only $ \sigma{_{color}} = 0.029 $ mag for the color. The value was calculated by comparing the derived $g-r$ colors for the standard stars to the known $g-r$ colors from the SDSS Data Release 5 standard star network [@2002AJ....123.2121S]. The low value in the uncertainty of our colors is probably due to the fact that while our spectra may not be absolutely photometrically calibrated, many of the remaining distortions are not wavelength dependent and therefore cancel out during color calculation. Examples of remaining distortions include slit losses, an incomplete cancellation of the telluric spectrum or a thin cloud layer that dims the overall flux at all wavelengths. So while individual magnitudes remain unreliable at best, the errors in the derived color are comparable to color errors in 2MASS and are thus reliable enough for our analysis. The 2MASS magnitude errors for our sample have a wide range dependent upon the reliability of the photometry for a particular night, but generally range from 0.02 magnitudes up to about 0.2 magnitudes. Colors calculated from 2MASS magnitudes thus have errors comparable to the derived SDSS colors.
Color Selection Analysis and Criteria {#sec:colors2}
-------------------------------------
The initial aim of the project was to select for S stars based on their SDSS and 2MASS colors. To achieve this goal, we derive $g$ and $r$ magnitudes for the stars making up our sample and combine with the known 2MASS magnitudes to construct a total of 10 unique colors. The $g$, $r$, and 2MASS magnitudes, along with $V$ band, IRAS, and AKARI photometry where available, are presented in Table\[tab:mags\]. Many of these colors, however, are ultimately unhelpful for analysis purposes because the colors of known S stars cover several magnitudes. Such a wide spread in magnitudes does not allow for efficient discovery of the S stars based on color. In other color combinations, the S stars are not well distinguished from the stellar locus, so selection based on color is contaminated by large numbers of stars from the main sequence. We find that the most useful color combinations are $g-r$, $J-H$, and $H-K_s$ because in these colors, the FAST S stars sample is well distinguished from the stellar locus. We compare these colors with those found for general SDSS stars, by overplotting some 300,000 SDSS/2MASS matches of [@2007AJ....134.2398C], which they used to parametrize the stellar locus in terms of color. In Figure\[fig:2MASS\_SDSScolors\], we present two color-color diagrams in which the S giants from the FAST sample are best separated from the stellar locus. The first shows the comparison of $g-r$ color and $J-H$ color, while the second shows the comparison $r-H$ and $J-K_s$ color.
However, the S giants that comprise the FAST sample, by virtue of their location on the TP-AGB or RGB, have similar colors and effective temperatures to M giants. In Figure\[fig:2MASS\_SDSScolors\], we also examine colors for M giants and dwarfs matched in the SDSS and 2MASS surveys. While the M giants are very few in number (due to the depth of both of these surveys, many nearby M giant targets are saturated), we can immediately see that S stars and M stars lie in essentially the same area of color space. We also include C stars on the color-color diagrams, since they follow evolutionary paths similar to S stars and also have enhanced abundances. The S stars of the FAST sample, while well-distinguished from the overall stellar locus on both color-color diagrams, are particularly difficult to distinguish from C stars on the $g-r$ versus $J-H$ diagram. On the plot comparing $r-H$ to $J-K_s$, S stars can be distinguished reasonably well from the bulk of the M star population using $J-K_s > 1.2$ and $r-H > 3.5$. Given the biases in the current sample of S stars, it is difficult to determine the efficiency of this selection, but this region on the color-color plot appears to be the least contaminated by M and C stars. Spectroscopy for a sample of perhaps 100 stars in this color region could directly test the efficiency of this color selection.
Distinguishing Intrinsic and Extrinsic S Stars {#sec:intext}
==============================================
Techniques {#sec:intext1}
----------
Distinguishing intrinsic and extrinsic S stars is difficult without high-resolution spectroscopy, where direct detection of Tc is possible. However, as presented in Van Eck and Jorrissen (2000), there are other measurements which serve to segregate the two types with reasonable statistical accuracy. Since extrinsic S stars are generally on the RGB (or early AGB), they cluster closer to the main sequence than intrinsic S stars in terms of color. Stars on the AGB generally show a red excess which is absent on the RGB. However, Van Eck and Jorissen (2000) find that blue excursions of Mira variables, which make up a significant proportion of intrinsic S stars, make the two classes difficult to distinguish based on distance from the main sequence alone. Additionally, intrinsic and extrinsic S stars have been shown to segregate well in a ($K-[12]$, $K-[25]$) diagram . These colors are based on IRAS photometry, which is available for many of the stars in the FAST sample. IRAS magnitudes in these bands are presented in Table\[tab:mags\] for 37 stars (with two bands available for 29). Identifications of many of the S stars in the FAST sample as intrinsic or extrinsic are included in Table\[tab:stars\]. All such identifications not marked with an asterisk are from @2006AJ....132.1468Y. Classification of most of the S stars in our sample is inconclusive because of their intermediate positions in the color-color diagrams. However, this IR technique remains the most efficient photometric classification method known to determine whether S stars are intrinsic or extrinsic.
We also use results from Vanture [[ et al. ]{}]{}(private communication), which indicate that S stars presenting Li in their spectra are intrinsic S stars. Lithium in stars is destroyed at fairly low temperatures [@1965ApJ...142..451B] but can be produced by the Cameron-Fowler mechanism, in high luminosity ($M_{Bol}<-6$) TP-AGB stars via hot bottom burning. Since, as mentioned above, extrinsic S stars are often less evolved along the giant branch than intrinsic S stars, they generally lack any evidence for the resonance line Li 6707Å (Vanture et al. 2011, in preparation). We compare the list of S stars with and without Li detections to our FAST sample and analyze them for the Li 6707[Å]{} feature. Unfortunately, we find that this line cannot be reliably detected with spectra of our resolution and S/N.
Identification Using AKARI Magnitudes {#sec:intext2}
-------------------------------------
The AKARI satellite surveyed much of the sky in both near and far infrared bands. We use flux data collected by AKARI in the S9W and L18W bandpasses to find infrared magnitudes for six objects in the FAST sample without available IRAS colors . The S9W and L18W bandpasses are qualitatively similar to the IRAS 12 and 25 micron bandpasses. All available AKARI magnitudes are presented in Table\[tab:mags\], for 52 stars (with two bands available for 35). Since the zero magnitude fluxes for these bandpasses are not well established, all magnitudes are presented on the AB system. In Figure\[fig:nearMidIRcolors\], we show two color-color diagrams comparing intrinsic and extrinsic stars using $K-[9]$, $K-[18]$, and $[9]-[18]$ colors. S stars previously classified as intrinsic vs. extrinsic are shown in black, with others plotted in red.
From the AKARI magnitudes and plots, we are able to identify an additional two stars in the sample as intrinsic or extrinsic. We identify 22315839+0201206 as an extrinsic S star. This star has a $K-[18]$ value of $-5.92$ mag and a $[K]-[9]$ value of $-4.23$ mag. This places it in the extreme lower left corner of the color-color diagram comparing $K-[9]$ color to $K-[18]$ color, far away from any known intrinsic stars but close to many known extrinsic stars. Similarly, we are able to identify 19364937+5011597 as an intrinsic S star on the basis of color. This star has a $K-[9]$ value of only $-3.09$ mag, and a $K-[18]$ value of $-4.01$ mag. This places it in the upper right-hand corner of the plot comparing $K-[9]$ and and $K-[18]$ color, far away from the cluster of extrinsic S stars. The four other stars for which we have no classification (03505704+0654325, 10505517+0429583, 13211873+4359136, and 16370314+0722207) cluster close to the transition between intrinsic and extrinsic S stars in all three color comparisons, meaning that we cannot confidently identify these with either class of stars. We include our additional identifications in Table\[tab:stars\], marked with an asterisk to distinguish them from those from @2006AJ....132.1468Y.
Calculating Temperature Indices {#sec:temp}
===============================
Using Colors to Determine Temperature {#sec:temp1}
-------------------------------------
As mentioned before, S stars cover a wide range of spectral types and effective temperatures. However, they generally have effective temperatures similar to those of M giants. The starting point for classifying the spectral type and temperatures of our S stars, therefore, is to calculate a temperature index using M-giant criteria [@Gray09]. @2000AJ....119.1424H use a grid of stellar models to calibrate a relation between color and temperature index for M stars. Their analysis uses CIT/CTIO colors $V-K$ and $J-K$. We find $V$ magnitudes for some of the FAST sample using Simbad. We also convert the 2MASS K$_s$ magnitudes and $J-K_s$ colors using the following relations from @2001AJ....121.2851C:[^1] $$\begin{aligned}
&K_s = K_{CIT} + (-0.019\pm0.004) + (0.001\pm0.005)(J-K)_{CIT} \notag \\
&J-K_s = (1.068\pm0.009)(J-K)_{CIT} + (-0.020\pm0.007)\end{aligned}$$ We then use the temperature index/color relation to assign a preliminary temperature index to the stars in the FAST sample. Where possible, we use the $(V-K)_{CIT}$ relation, since this is specified by the authors as the most temperature sensitive. When there is no $V$ magnitude available, we use the $(J-K)_{CIT}$ relation [@2000AJ....119.1424H].
Results of Temperature Analysis {#sec:temp2}
-------------------------------
Results of this analysis are presented in Table\[tab:stars\]. These temperature indices should be treated with caution, since many of the stars in the FAST sample are highly variable. Furthermore, differences in C/O ratios, may lead to temperature errors up to 400K [@2010arXiv1011.2092V]. However, the preliminary classification gives us a rough idea of the relative effective temperatures of the S stars. We find that the average temperature index of the stars in the sample is 5, which in M giants corresponds to an effective temperature of roughly 3500 K [@2000AJ....119.1424H]. In Table\[tab:stars\], we also present known spectral types from @1980ApJS...43..379K, which include temperature indices for the S stars. We find reasonably good agreement between our calculated temperature indices and those presented as part of the spectral type. Differences between our temperature index and those included as part of the published spectral types are within the range of variability of the star. We also note that since we do not have a reliable color/temperature relation for stars with a temperature index greater than 7, we present all of these classes as 7+. We also find reasonably good agreement (within one temperature index) between the values using $(V-K)_{CIT}$ and those derived using $(J-K)_{CIT}$.
We also conduct a preliminary analysis on possible correlations between the intrinsic/extrinsic distinction and the temperature indices. We find that the average temperature index of the intrinsic stars is $5.6\pm2.3$, while the average temperature index of extrinsic stars is $4.3\pm2.2$. Therefore, while the mean temperature index shows some differences, this measurement alone is not statistically significant. A larger sample would be needed to determine the validity of differences between average effective temperature between intrinsic and extrinsic S stars. The temperature indices of S stars in general are insufficient to classify them as intrinsic or extrinsic, except for the reddest objects.
Parallax, Proper Motion, and Absolute Magnitude {#sec:motion}
===============================================
Parallax Analysis {#sec:motion1}
-----------------
Four of the stars in the original FAST sample have reliable parallax measurements (defined as a parallax detection at the $3\,\sigma$ level) available from the Hipparcos survey , so calculation of the absolute magnitude of these stars in the $g$ and $r$ bands is possible. We calculate the error in our absolute magnitudes using the presented error in the parallax measurements and assuming apparent magnitude error values of $ \sigma{_g} = 0.191 $ and $ \sigma{_r} = 0.192 $. Results of this analysis are given in Table\[tab:plx\]. We note the calculated absolute $g$ and $r$ magnitudes, as well as the associated errors. Objects are identified using the 2MASS Identifier, and the parallax and associated error as given by the Hipparcos catalog are also included. We can immediately identify each of these four objects as asympotic or red giant branch members: if they were dwarf stars of the same spectral type, we would expect magnitudes in the $g$ and $r$ of +5 or higher, as opposed to the derived values, which center around 0.
Proper Motion Analysis {#sec:motion2}
----------------------
We also find that some stars in the original FAST sample display significant (again, $3\,\sigma$) proper motions. The proper motions are available in the Tycho-2 catalogue . Of these, we have reliable $g$ and $r$ photometry for 16 objects. Given the proper motion and assuming that all objects are moving at total space velocities less than the Galactic escape velocity, we can derive an upper limit on the distance, and thus also constrain the absolute magnitudes of the objects. To derive an approximate distance ($d$), we use the equation: $$\begin{aligned}
d &= \frac{v_{trans}}{\mu} < \frac{v_{esc}}{\mu} \\
d &< 0.0056\,pc/yr \left(\frac{\mu}{1\,rad/yr}\right)^{-1} \\
d &< 1.16\,kpc/yr \left(\frac{\mu}{1\,mas/yr}\right)^{-1}\end{aligned}$$ where $\mu$ is the proper motion of the object measured in radians per year, $v_{trans}$ is the transverse velocity of the object, and $v_{esc}$ is the Galactic escape velocity. We take the Galactic escape velocity to be 545 km/s [@2007MNRAS.379..755S], which is equivalent to 0.0056 parsecs per year. The distance constraint is only an upper limit because we neglect radial velocities. However, we note that our [rvsao]{} measurements generally yield small radial velocities ($\lax$50km/s) for the S star sample. We then use the same formulas presented above to derive $g$ and $r$ absolute magnitudes and associated errors. These $g$ and $r$ absolute magnitudes essentially represent the brightest the object concerned could be, considering its proper motion and assuming that all observed objects are gravitationally bound members of the Milky Way. Results of the analysis are presented in Table\[tab:pm\]. We can immediately see from these lower bound magnitudes that these objects could be RGB or AGB stars - even considering that the actual magnitudes are probably significantly fainter. For the sample of 16 stars, we find that the average lower limit on the absolute $g$ magnitude $M_g$ is $-5.0$ mag, correspondingly $M_r \sim -6.7$ mag. Cool dwarf stars generally have $g$ and $r$ absolute magnitudes greater than +8 mag We conclude that all 16 objects displaying significant proper motions could be giants.
Lower Limit Distances {#sec:motion3}
---------------------
To further bear out this analysis, we also determine a lower limit on the distance for these objects by assuming (conversely to above) that they [*do*]{} have a dwarf magnitude. We take the approximate absolute $g$-band magnitude for an M dwarf star to be $M_g$ = 10 mag and the absolute $r$-band magnitude to be $M_r$ = 9 mag . This method can be applied to the entire sample, because it does not rely on proper motion or parallax measurements. We find that most of these objects, were they to have typical dwarf magnitudes, would be within 25 pc of us. This distance would almost guarantee that these objects would have detectable parallax and proper motion. For instance, assuming extrinsic S stars are a mostly spheroid population (like dC or CH stars: see @1994ApJ...434..319G and @Bergeat02) with velocity of $-220$km/s relative to the Sun [@2004AJ....127..914S], the typical proper motion for a star 25 pc away would be on the order of 2.25 mas/yr. Additionally, such stars would have a parallax on the order of 40 mas. Since such a parallax would be easily detectable by Hipparcos, these lower limit distances therefore also suggest that the stars of the FAST sample are likely to be giants or AGB stars.
Summary and Conclusions {#sec:sum}
=======================
Using a sample of 57 medium-resolution S star spectra taken with the FAST spectrograph, we create a spectral atlas of S stars comprised of 14 objects that span a range of spectral types within the MS, S, and SC classes. The atlas is published as a collection of 1-D FITS files via this journal. After generating $g$ and $r$ SDSS magnitudes from the spectra, we find that the SDSS magnitudes in the $g$ and $r$ bands, combined with $J$, $H$, and $K_s$ magnitudes from the 2MASS catalog for S stars may with further confirmation allow for reasonably efficient color selection of these objects from the SDSS and 2MASS catalogs. We use previously published data to identify some of the stars in the sample as intrinsic or extrinsic stars, and find that the fraction of extrinsic S stars in the sample is approximately $54\%$. We also assign temperature indices to the stars in the sample based on the M star scale of temperature indices. We find that much of the S star sample falls at temperature index 4 or above, meaning that the effective temperatures of most of these stars are well below 5000 K. We also analyze objects in the FAST sample with detectable parallaxes and proper motions to generate absolute magnitude limits, as well as lower limit distances based on assumed dwarf magnitudes. This analysis bears out our initial assumption that the FAST sample is primarily composed of giant stars, either on the AGB or RGB.
Acknowledgements {#sec:ack}
================
Many thanks to the referee for a thorough reading. We are grateful to Kevin Covey for the use of the high-quality sample of SDSS/2MASS matches. Also, thanks to Warren Brown and Mukremin Kilic, who provided many helpful discussions and hints on color selection and the SDSS photometry. Many thanks to Doug Mink for all his invaluable help with *rvsao* and SDSS correlations, and to Bill Wyatt for his help with accessing SDSS DR-7 spectra, all of which we hope to use in upcoming publications. We gratefully acknowledge Bob Kurucz, Andrew Vanture and George Wallerstein for illuminating discussions. Thanks also to Verne Smith and GW for compiling an initial list of Li 6707Ådetections in S stars. I am grateful to the SAO REU program organizers - Marie Machacek, Christine Jones, and Jonathan McDowell - for all their support. Finally, This work is supported in part by the National Science Foundation Research Experiences for Undergraduates (REU) and Department of Defense Awards to Stimulate and Support Undergraduate Research Experiences (ASSURE) programs under Grant no. 0754568 and by the Smithsonian Institution.
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{height="4in" width="6in"}
{width="3.2in" height="3.2in"}
\[fig:specind\]
{width="3.2in" height="3.2in"} {width="3.2in" height="3.2in"}
{width="2.7in" height="2.7in"} {width="2.7in" height="2.7in"}
\[fig:nearMidIRcolors\]
[cccclclcc]{} 9& 006.008240& +38.577049& 00240197+3834373$^A$& HD 1967/R And&09/15/09 &S5e Zr5 Ti2 & 7& i\
22& 016.300978& +19.197842& 01051223+1911522& HD 6409/CR Psc& 09/15/09& & 4/5 & e\
32& 021.413226& +21.396074& 01253917+2123458& RX Psc& 09/15/09 & & 0/1 & i\
45& 028.582236& +21.889090& 01541973+2153207& BD+21 255& 09/15/09&S3 Zr1 Ti3 & 3/4 & e\
57& 036.476487& +38.122776& 02255435+3807219& BI And& 09/15/09 &S8 Zr7 Ti4 & 7 & i\
73& 052.040388& +17.679628& 03280969+1740466& & 09/20/09& & 4/5 &\
74& 052.936921& +04.695488& 03314486+0441437$^A$& & 09/20/09& & 6 &\
80& 055.889249& +22.437023& 03433341+2226132& BD+21 509& 09/16/09& & 5/6 & e\
83& 057.737708& +06.909048& 03505704+0654325& & 09/16/09& & 6 &\
94& 066.090802& -02.532759& 04242179-0231579& BD-02 891& 09/20/09&S2 Zr2- Ti2 & 4 & e\
106& 073.937675& +79.999931& 04554504+7959597& BD+79 156& 09/20/09&S4 Zr2- Ti3- & 4/5 & e\
134& 080.836187& -04.570626& 05232068-0434142& HD 35273/V535 Ori&09/20/09& & 7+ & i\
312& 108.717017& +68.804321& 07145208+6848155$^A$& HD 54587/AA Cam&11/10/09&M5S$\dagger$ & 4/5 & i\
339& 111.489470& +62.591026& 07255747+6235276$^A$& & 11/10/09& & 4 &\
347& 112.048398& +45.990597& 07281161+4559261& HD 58521/Y Lyn& 11/10/09&M6S$\dagger$ & 7+ & i\
403& 117.325749& +23.734451& 07491817+2344040& HD 63334/T Gem& 11/10/09&S3e Zr2.5 Ti2 & 4/5 & i\
405& 117.681704& +47.003967& 07504360+4700142& & 11/10/09& & 4/5 &\
413& 118.221777& +34.614128& 07525322+3436508& BD+34 1698& 11/10/09& & 6 & i\
418& 118.369741& +17.780506& 07532873+1746498& HD 64209& 11/10/09& & 4/5 & e\
431& 119.233374& +31.167286& 07565600+3110022$^A$& AO Gem& 11/10/09 & & 7 & i\
460& 121.836937& +11.321875& 08072086+1119187& & 03/10/10& & 4/5 &\
471& 122.761565& +08.139297& 08110277+0808214& & 03/10/10& & 5/6 & e\
494& 125.428453& +17.285120& 08214282+1717064& HD 70276/V Cnc& 03/10/10&S0.5e Zr0 & 1 & i\
589& 137.661696& +30.963114& 09103880+3057472& HD 78712/RS Cnc&03/10/10&M6S$\ddagger$ & 7+ & i\
612& 143.901826& +69.157043& 09353643+6909253& BD+69 524& 03/11/10& & 3/4 & e\
707& 162.729912& +04.499532& 10505517+0429583& & 03/11/10& & 6 &\
722& 166.970132& +68.366440& 11075283+6821591& HD 96360/HL UMa&03/11/10& & 4/5 & e\
803& 190.986134& +61.093254& 12435667+6105357& HD 110813/S UMa&03/11/10&S6e Zr6 & 5/6 & i\
819& 200.328046& +43.987118& 13211873+4359136& AV CVn& 03/11/10 &S3 Zr2 Ti1 & 3/4 &\
833& 207.141756& +31.999107& 13483402+3159567& & 03/11/10& & 0 &\
836& 208.005660& -03.480082& 13520135-0328482& HD 120832& 03/11/10& & 4/5 & e\
856& 214.462975& +83.831696& 14175111+8349541$^A$& HD 1272266/R Cam&06/24/09&S2e Zr2 & 3 & e\
855& 216.950026& -03.079711& 14274800-0304469& BD-02 3848& 05/03/09& & 4 & e\
892& 232.384097& +00.189076& 15293218+0011206& & 05/03/09& & 2/3 &\
902& 238.100360& -23.352097& 15522408-2321075& & 05/03/09& & 6 & e\
903& 237.694254& +48.483078& 15504662+4828590$^A$& HD 142143/ST Her&05/03/09&M6.5S Zr1 Ti6+ & 5/6 & i\
926& 245.401029& +56.877048& 16213624+5652373$^A$& HD 147923& 05/03/09& & 2 & e\
932& 249.263106& +07.372439& 16370314+0722207$^A$& BD+07 3210& 05/03/09& & 4/5 &\
958& 256.694247& -02.017769& 17064661-0201039& & 05/03/09& & 6 &\
981& 260.529640& +23.817570& 17220711+2349032& BD+23 3093& 05/03/09&S3.5 Zr3+ Ti4 & 4/5 & e\
986& 261.200788& +33.918953& 17244818+3355082& & 05/03/09& & 3 &\
1002&267.252799& +21.710087& 17490067+2142363$^A$& & 05/03/09& & 4 & e\
1065&279.398924& +43.256588& 18373574+4315237& & 05/03/09& & 3 & e\
1087&282.875526& +48.911934& 18513012+4854429$^A$& TU Dra& 05/03/09 & & 3/4 & i\
1150&294.205739& +50.199917& 19364937+5011597& HD 185456/R Cyg&05/03/09&S5e Zr5 Ti0 & 7 & i\*\
1152&294.345155& +67.188873& 19372283+6711199& & 05/03/09&S5- Zr4.5 & 5/6 & i\
1165&297.641369& +32.914139& 19503392+3254509$^A$& HD 187796/chi Cyg&05/03/09&S6+e Zr2 Ti6.5 & 7+ & i\
1222&307.913302& +04.061802& 20313919+0403424$^A$& & 05/23/09& & 5 &\
1247&313.355676& +06.539369& 20532536+0632217& & 05/23/09& & 2/3 &\
1262&317.601674& +01.606119& 21102440+0136220& & 05/27/09& & 4/5 &\
1264&318.232437& +09.548880& 21125578+0932559& & 07/17/09& & 3/4 &\
1290&331.995242& +29.665623& 22075885+2939562& & 07/17/09& & 4/5 & e\
1299&337.993328& +02.022399& 22315839+0201206& & 07/17/09& & 5 & e\*\
1304&340.954936& +33.926231& 22434918+3355344& HD 215336& 07/17/09& & 1/2 & e\
1315&343.648463& +16.941910& 22543563+1656308& HR Peg& 07/19/09 &S4+ Zr1.5 Ti4 & 4/5 & i\
1328&349.035110& +28.863674& 23160842+2851492$^A$& & 07/19/09& & 3 & e\
1340&356.540618& +34.783325& 23460974+3446599& & 07/19/09& & 3 &\
[crrrrrrrrrr]{} 00240197+3834373&8.40&6.37&7.39&2.02&0.77&0.12&&$-2.66$&3.17&$-2.3$\
01051223+1911522&9.38&7.79&&3.67&2.76&2.48&6.68&1.75&8.28&1.63\
01253917+2123458&16.53&14.40&8.80&6.27&5.39&4.98&8.90&3.91&10.16&\
01541973+2153207&10.18&8.60&9.00&5.29&4.48&4.29&8.57&3.59&10.26&3.61\
02255435+3807219&11.90&9.63&&3.50&2.43&1.92&6.14&1.10&7.54&0.95\
03280969+1740466&12.81&11.14&&7.91&7.02&6.72&11.09&&&\
03314486+0441437&12.68&11.05&&6.93&6.02&5.65&9.92&&&\
03433341+2226132&10.74&9.05&9.69&5.03&4.03&3.80&8.14&3.12&9.72&3.23\
03505704+0654325&12.40&10.54&&6.42&5.47&5.12&9.32&4.32&10.52&\
04242179-0231579&10.42&8.78&9.33&5.62&4.78&4.44&8.80&3.96&10.24&3.85\
04554504+7959597&10.68&9.04&9.66&5.67&4.87&4.55&8.97&4.04&10.18&3.99\
05232068-0434142&11.23&9.04&10.00&3.90&2.80&2.36&&0.38&6.47&$-0.50$\
07145208+6848155&8.85&7.33&&2.60&1.67&1.39&5.74&0.70&6.92&0.08\
07255747+6235276&12.49&10.92&&7.67&6.83&6.53&10.79&&&\
07281161+4559261&&&7.37&0.65&$-0.38$&$-0.69$&3.43&$-1.59$&4.15&$-0.67$\
07491817+2344040&13.37&11.10&8.00&4.12&3.22&2.71&7.16&2.20&8.60&2.05\
07504360+4700142&11.83&10.26&&7.32&6.43&6.17&10.43&&&\
07525322+3436508&11.34&9.79&&4.99&3.99&3.71&7.99&2.95&9.38&2.81\
07532873+1746498&&&8.56&4.82&3.84&3.47&7.76&2.79&9.23&2.83\
07565600+3110022&16.96&14.34&&6.60&5.66&5.13&8.81&3.73&10.11&2.97\
08072086+1119187&10.93&9.43&&7.37&6.52&6.22&10.43&&&\
08110277+0808214&10.19&8.30&&5.45&4.58&4.17&8.42&3.52&9.84&3.35\
08214282+1717064&&&7.50&5.08&4.00&3.59&6.67&1.46&8.07&1.46\
09103880+3057472&6.88&4.94&6.08&$-0.71$&$-1.56$&$-1.87$&&$-3.07$&2.72&$-1.87$\
09353643+6909253&10.24&8.78&9.30&5.55&4.69&4.42&8.84&3.84&10.08&3.89\
10505517+0429583&12.65&10.90&&5.36&4.54&4.05&8.28&&9.46&\
11075283+6821591&9.13&7.73&8.10&4.07&3.17&2.77&7.36&2.37&8.81&2.26\
12435667+6105357&9.69&7.69&8.87&4.46&3.43&3.02&7.21&2.06&8.39&1.71\
13211873+4359136&11.00&9.37&9.78&6.28&5.43&5.16&9.53&&10.79&\
13483402+3159567&13.30&11.80&&9.03&8.43&8.22&&&&\
13520135-0328482&&&9.59&5.50&4.76&4.35&8.75&3.78&10.35&\
14274800-0304469&10.26&8.77&9.47&5.74&4.93&4.59&9.01&4.05&10.66&\
14175111+8349541&10.58&8.81&6.97&3.82&2.90&2.46&6.90&1.83&8.38&1.66\
15293218+0011206&&&&7.47&6.70&6.40&10.77&&&\
15522408-2321075&&&&6.67&5.76&5.38&9.68&4.79&&\
15504662+4828590&8.41&6.50&&0.74&$-0.14$&$-0.54$&3.09&$-2.12$&3.83&$-2.9$\
16213624+5652373&8.50&7.05&7.59&4.71&3.65&3.46&7.66&2.82&9.18&2.64\
16370314+0722207&10.50&8.95&9.53&5.50&4.71&4.38&8.64&&9.98&\
17064661-0201039&12.73&10.94&&7.35&6.36&6.07&&&&\
17220711+2349032&10.91&9.37&10.10&5.73&4.98&4.59&8.98&4.09&10.40&3.99\
17244818+3355082&12.28&10.73&&7.73&6.92&6.63&10.98&&&\
17490067+2142363&12.71&11.22&&7.78&6.94&6.64&10.95&5.98&&\
18373574+4315237&12.42&10.93&&7.46&6.61&6.36&10.61&5.86&&\
18513012+4854429&17.02&14.59&10.00&5.82&5.14&4.74&7.99&2.52&8.78&2.01\
19364937+5011597&12.21&9.58&8.15&2.25&1.38&0.86&3.94&&4.87&\
19372283+6711199&11.18&9.23&9.70&5.07&4.19&3.76&7.88&2.79&9.30&2.64\
19503392+3254509&12.55&9.79&6.80&0.17&$-1.10$&$-1.70$&&$-4.44$&&$-1.68$\
20313919+0403424&12.43&10.90&&7.48&6.61&6.25&10.64&&&\
20532536+0632217&12.78&11.29&&7.99&7.17&6.90&11.14&&&\
21102440+0136220&&&&7.60&6.79&6.44&10.71&&&\
21125578+0932559&&&&7.94&7.09&6.81&11.15&&&\
22075885+2939562&&&&6.56&5.71&5.39&9.64&4.90&10.87&\
22315839+0201206&&&&6.46&5.56&5.21&9.44&&11.13&\
22434918+3355344&&&7.82&4.98&4.07&3.79&8.10&3.18&9.60&3.02\
22543563+1656308&7.02&5.43&6.47&2.31&1.24&1.04&5.09&$-0.02$&6.63&$-0.08$\
23160842+2851492&13.62&12.10&&9.11&8.32&8.03&&&&\
23460974+3446599&12.02&10.53&&7.49&6.63&6.40&10.70&&&\
[ccccccccc]{} 09103880+3057472&8.06&0.98&125&15&1.41&0.33&$-0.53$&0.33\
15504662+4828590&3.07&0.75&326&80&0.85&0.56&$-1.06$&0.56\
16213624+5652373&2.31&0.62&433&117&0.32&0.61&$-1.13$&0.61\
22543563+1656308&3.31&0.93&302&85&0.02&0.64&$-1.97$&0.64\
[clcccccc]{} 00240197+3834373&35.89&2.46&3.22&8.4&6.37&$-$4.14&$-$6.17\
01051223+1911522&12.65&0.95&9.13&9.38&7.79&$-$5.42&$-$7.01\
03505704+0654325&10.66&2.50&10.8&12.4&10.55&$-$2.77&$-$4.62\
04242179-0231579&7.50&1.27&15.4&10.41&8.78&$-$5.53&$-$7.16\
07145208+6848155&18.44&1.07&6.26&8.85&7.33&$-$5.13&$-$6.65\
09103880+3057472&34.22&0.85&3.38&6.8&4.94&$-$5.84&$-$7.70\
09353643+6909253&8.54&1.67&13.5&10.24&8.78&$-$5.42&$-$6.88\
11075283+6821591&22.65&1.15&5.10&9.13&7.73&$-$4.41&$-$5.81\
12435667+6105357&13.62&1.31&8.48&9.69&7.69&$-$4.95&$-$6.95\
13211873+4359136&11.31&2.18&10.2&10.99&9.37&$-$4.06&$-$5.68\
14175111+8349541&7.26&1.62&15.9&10.58&8.82&$-$5.43&$-$7.19\
14274800-0304469&13.18&1.89&8.77&10.26&8.77&$-$4.45&$-$5.94\
15504662+4828590&22.38&1.11&5.16&8.41&6.5&$-$5.15&$-$7.06\
16213624+5652373&12.32&1.14&9.37&8.5&7.05&$-$6.36&$-$7.81\
19372283+6711199&9.75&2.35&11.8&11.18&9.24&$-$4.19&$-$6.13\
22543563+1656308&15.75&0.99&7.33&7.02&5.43&$-$7.31&$-$8.90\
[^1]: Updated at http://www.astro.caltech.edu/jmc/2mass/v3/transformations
|
---
abstract: 'We present a sample of 33 galaxies for which we have calculated (i) the average rate of shear from publish rotation curves, (ii) the far–infrared luminosity from IRAS fluxes and (iii) The K–band luminosity from 2MASS. We show that a correlation exists between the shear rate and the ratio of the far–infrared to K–band luminosity. This ratio is essentially a measure of the star formation rate per unit mass, or the specific star formation rate. From this correlation we show that a critical shear rate exists, above which star formation would turn off in the disks of spiral galaxies. Using the correlation between shear rate and spiral arm pitch angle, this shear rate corresponds to the lowest pitch angles typically measured in near-infrared images of spiral galaxies.'
author:
- |
Marc S. Seigar[^1]\
Department of Physics & Astronomy, University of California Irvine, 4129 Frederick Reines Hall, Irvine, CA 92697-4575, USA
date: 'Accepted 2005 April 24. Received 2005 April 22; in original form 2005 March 30.'
title: The connection between shear and star formation in spiral galaxies
---
\[firstpage\]
galaxies: fundamental parameters – galaxies: spiral – infrared: galaxies
Introduction
============
Jeans (1929) took up the question of self–gravitating gas and found that under certain conditions it could be unstable enough to collapse under its own self–gravity. In order to show this he considered an adiabatic gas. A similar calculation can be applied to stellar systems.
The idea of instabilities is essentially a condensation of material, so in order for material to condense it is necessary to find out if gravity will cause collapse before velocity dispersion causes expansion. A characteristic time can be calculated for each process and compared to see which process is dominant. It turns out that on small scales velocity dispersion is dominant, and on large scales gravity is dominant. Jeans (1929) found a critical length above which gravitational instability becomes dominant, now known as the Jeans length, $L_{Jeans}$.
The situation in the disks of galaxies is different from the problem formulated by Jeans, due to the flatness of the system (instead of a spheroid assumed in the Jeans analysis) and more importantly, differential rotation. Velocities due to differential rotation are approximately proportional to $\Delta R$ and might even prevent the collapse from taking place, even at distances greater than the Jeans length. Differential rotation inhibits the gravitational collapse on large scales. The question is what happens in between? Toomre (1964) attempted to answer this question. He investigated the balance between differential rotation and self–gravitation. Differential rotation manifests itself physically from the fact that a contracting region conserves angular momentum. This spins up and causes a centrifugal force that might inhibit further collapse. Toomre (1964) found that differential rotation implies that disks are stabilised at lengths greater than a critical length, which we will call the rotation length, $L_{rot}$. It turns out that $L_{rot}$ is inversely proportional to the average angular velocity with respect to a fixed system. The higher the average angular velocity, the greater the rate of shear, $A/\omega$. In other words as the rate of shear increases, the value of $L_{rot}$ decreases, until it reaches the limiting case where $L_{rot}=L_{Jeans}$, and the entire disk is stable against gravitational collapse, i.e no clouds will collapse and therefore no star formation can occur.
In reality most disks have a region where they are unstable to gravitational instabilities and this region has a length scale between the Jeans length and the rotation length, i.e. $L_{Jeans} < L < L_{rot}$. However, the faster a disk rotates, the stronger its differential rotation. The stronger differential rotation becomes, the more inhibited star formation becomes, and thus we expect a correlation between star formation rate and shear rate.
Toomre (1964) derived a value of the velocity dispersion for the limiting regime where $L_{rot}=L_{Jeans}$, i.e. a critical velocity dispersion. The stability of disks can then be quoted as the ratio of of the actual velocity dispersion to the critical velocity dispersion, $Q$. Thus if $Q > 1$ then the velocity dispersion is high enough to prevent gravitational collapse, and if $Q < 1$ then gravitational collapse occurs. This condition is known as the [*Toomre stability criterion*]{} and the parameter $Q$ is known as [*Toomre’s parameter*]{}.
Since differential rotation acts to shear features in the disk, the shear rate is a viable quantity for measuring the differential rotation in the disks of spiral galaxies. Shear rate has been measured by several authors (e.g. Block et al. 1999; Seigar, Block & Puerari 2004; Seigar et al. 2005) and can be measured directly from a galaxy rotation curve. The sequence of low shear rates to high shear rates also follows the sequence from late-type spirals to early-type spirals (Seigar et al. 2005). It has been shown that as one progresses along the Hubble sequence, the specific star formation rate (measured as the H$\alpha$ equivalent width) in galaxies increases (e.g. James et al. 2004). If one takes this along with the correlation between pitch angle and shear rate in Seigar et al. (2005), then the existence of a correlation between shear rate and specific star formation rate is implied, assuming that the transition from tightly wound spiral structure to losely wound spiral structure follows the Hubble sequence from early- to late-type spiral galaxies.
This letter is arranged as follows. Section 2 describes the data we have used. Section 3 describes how shear rates were calculated using rotation curves. Section 4 describes a method for determining the star formation rate per unit surface area. Finally, section 5 discusses the results.
Data
====
In order to calculate shear rates, rotation curves are required. Burstein & Rubin (1985) presented data for 60 galaxies for which they had observed rotation curves. The rotation curve data was available in a series of papers from the early 1980s (Rubin, Ford & Thonnard 1980; Rubin et al. 1982, 1985). Of this sample of 60 galaxies, 40 were detected by IRAS at 60$\mu$m and 100$\mu$m, and by 2MASS in the $K_s$-band. 6 of these galaxies were rejected from our sample, as they were classified as HII or starburst and may therefore have external physical processes affecting their star formation rates. A further object did not have a rotation curve that covered a sufficient radial range and was also not included in the sample. The remaining 33 galaxies are presented here.
----------- -------- --------------- ---------- ----------- ------- ---------------- -------------------------- ---------------- ------------------------------- -------------------- --
Galaxy Hubble Shear $S_{60}$ $S_{100}$ $D$ $D_{25}$ $L_{FIR}$ $K_T$ $L_{K}$ SFR
name Type rate (Jy) (Jy) (Mpc) (kpc) ($\times 10^9$L$_\odot$) ($\times 10^{10}$L$_{\odot}$) ($M_{\odot}/year$)
NGC 753 SABbc 0.38$\pm$0.03 3.36 11.40 65.4 47.79$\pm$2.25 32.17$\pm$2.25 9.37$\pm$0.02 17.88$\pm$0.33 8.30$\pm$1.71
NGC 801 Sc 0.53$\pm$0.03 1.45 5.06 76.9 70.73$\pm$5.05 19.47$\pm$1.27 9.51$\pm$0.03 21.82$\pm$0.59 6.20$\pm$1.19
NGC 1024 SAab 0.65$\pm$0.02 0.57 2.62 47.1 53.30$\pm$2.54 3.40$\pm$0.49 8.74$\pm$0.02 16.49$\pm$0.30 2.31$\pm$0.43
NGC 1035 SAc? 0.37$\pm$0.03 3.57 11.12 16.6 10.81$\pm$0.77 2.09$\pm$0.13 9.13$\pm$0.01 1.42$\pm$0.01 0.68$\pm$0.14
NGC 1085 SAbc 0.52$\pm$0.04 0.88 3.16 90.5 77.69$\pm$7.50 16.71$\pm$1.25 9.80$\pm$0.03 23.40$\pm$0.64 6.90$\pm$1.40
NGC 1357 SAab 0.56$\pm$0.03 0.93 4.67 26.8 21.97$\pm$1.57 1.90$\pm$0.21 8.42$\pm$0.03 7.18$\pm$0.20 1.78$\pm$0.34
NGC 1417 SABb 0.45$\pm$0.03 1.59 5.82 54.2 42.43$\pm$3.03 10.95$\pm$0.60 9.14$\pm$0.03 15.28$\pm$0.42 5.81$\pm$1.15
NGC 1421 SABbc 0.31$\pm$0.04 8.48 21.32 27.8 28.69$\pm$1.35 12.55$\pm$0.69 8.40$\pm$0.02 7.91$\pm$0.14 4.34$\pm$1.05
NGC 1620 SABbc 0.44$\pm$0.03 1.31 5.31 46.8 39.26$\pm$2.80 7.15$\pm$0.43 8.92$\pm$0.03 13.91$\pm$0.38 5.46$\pm$1.08
NGC 2590 SAbc 0.44$\pm$0.03 2.01 6.03 66.6 43.38$\pm$3.08 18.68$\pm$0.93 9.38$\pm$0.03 18.43$\pm$0.50 7.24$\pm$1.44
NGC 2608 SBb 0.45$\pm$0.04 2.25 5.77 28.5 18.99$\pm$0.90 3.51$\pm$0.23 9.33$\pm$0.03 3.51$\pm$0.10 1.33$\pm$0.28
NGC 2639 SAa? 0.54$\pm$0.03 1.99 7.06 44.5 23.56$\pm$2.87 9.04$\pm$0.36 8.40$\pm$0.03 20.25$\pm$0.55 5.51$\pm$1.07
NGC 2742 SAc 0.40$\pm$0.03 3.08 10.49 17.2 15.11$\pm$1.08 2.04$\pm$0.10 8.81$\pm$0.01 2.06$\pm$0.02 0.91$\pm$0.18
NGC 2775 SAab 0.63$\pm$0.03 1.80 9.46 18.1 22.46$\pm$1.06 1.74$\pm$0.12 7.04$\pm$0.02 11.60$\pm$0.21 1.90$\pm$0.36
NGC 2815 SBb 0.60$\pm$0.03 1.04 4.79 33.9 34.19$\pm$1.03 3.21$\pm$0.27 8.25$\pm$0.03 13.44$\pm$0.37 2.68$\pm$0.52
NGC 2844 SAa 0.63$\pm$0.03 0.41 1.91 19.8 8.92$\pm$0.64 0.44$\pm$0.06 9.89$\pm$0.03 1.01$\pm$0.03 0.16$\pm$0.04
NGC 2998 SABc 0.41$\pm$0.03 1.58 4.63 63.8 53.53$\pm$2.52 13.28$\pm$0.93 9.93$\pm$0.04 10.23$\pm$0.37 4.39$\pm$0.89
NGC 3223 SAbc 0.61$\pm$0.03 3.79 14.76 38.6 45.75$\pm$2.16 13.71$\pm$1.51 7.58$\pm$0.02 32.22$\pm$0.59 6.06$\pm$1.16
NGC 3281 SABa 0.45$\pm$0.02 6.86 7.51 42.7 41.13$\pm$2.94 17.21$\pm$1.20 8.31$\pm$0.03 20.19$\pm$0.55 7.68$\pm$1.44
NGC 3495 Sd 0.42$\pm$0.02 1.82 7.28 15.2 21.66$\pm$1.02 1.03$\pm$0.08 8.93$\pm$0.02 1.43$\pm$0.03 0.60$\pm$0.11
NGC 3672 SAc 0.43$\pm$0.02 7.33 20.80 24.8 30.08$\pm$1.42 9.18$\pm$0.78 8.27$\pm$0.01 7.08$\pm$0.06 2.86$\pm$0.53
NGC 4378 SAa 0.69$\pm$0.03 0.36 1.45 34.1 28.61$\pm$1.35 1.04$\pm$0.18 8.51$\pm$0.02 10.71$\pm$0.20 1.00$\pm$0.18
NGC 4682 SABcd 0.51$\pm$0.03 0.69 2.38 31.1 23.25$\pm$1.66 1.51$\pm$0.14 9.60$\pm$0.02 3.25$\pm$0.06 1.00$\pm$0.20
NGC 6314 SAa 0.60$\pm$0.03 0.51 2.85 88.4 37.16$\pm$3.60 12.18$\pm$1.10 9.81$\pm$0.03 22.12$\pm$0.60 4.43$\pm$0.84
NGC 7083 SABc 0.43$\pm$0.03 5.02 17.19 41.5 49.96$\pm$2.36 19.41$\pm$0.87 8.42$\pm$0.03 17.17$\pm$0.47 6.95$\pm$1.39
NGC 7171 SBb 0.47$\pm$0.03 0.89 3.33 36.3 27.77$\pm$1.31 2.77$\pm$0.18 9.31$\pm$0.04 5.78$\pm$0.21 2.06$\pm$0.41
NGC 7217 SAab 0.66$\pm$0.02 4.96 18.45 12.7 14.37$\pm$0.68 1.89$\pm$0.17 6.83$\pm$0.01 6.93$\pm$0.06 0.89$\pm$0.16
IC 467 SABc 0.50$\pm$0.04 0.89 3.16 27.2 25.60$\pm$1.21 1.52$\pm$0.10 10.04$\pm$0.05 1.66$\pm$0.07 0.53$\pm$0.11
IC 724 Sa 0.69$\pm$0.03 0.34 1.27 79.6 54.28$\pm$5.23 5.09$\pm$0.81 9.39$\pm$0.04 26.10$\pm$0.94 2.40$\pm$0.44
UGC 3691 SAcd 0.33$\pm$0.03 1.36 3.79 29.4 18.71$\pm$1.80 2.36$\pm$0.21 10.31$\pm$0.07 1.51$\pm$0.09 0.80$\pm$0.16
UGC 10205 Sa 0.48$\pm$0.03 0.39 1.54 87.4 36.74$\pm$4.50 7.32$\pm$1.02 9.89$\pm$0.03 19.98$\pm$0.54 6.88$\pm$1.35
UGC 11810 SABbc 0.42$\pm$0.02 0.50 2.04 63.0 33.35$\pm$1.57 4.93$\pm$0.57 10.82$\pm$0.06 4.38$\pm$0.24 1.81$\pm$0.34
UGC 12810 SABbc 0.44$\pm$0.02 0.78 1.78 108.2 58.61$\pm$5.63 16.59$\pm$1.99 10.47$\pm$0.06 18.04$\pm$0.97 7.08$\pm$1.33
----------- -------- --------------- ---------- ----------- ------- ---------------- -------------------------- ---------------- ------------------------------- -------------------- --
\[landtable\]
Calculation of shear rate
=========================
The rotation curves presented by Rubin et al. (1980, 1982, 1985) are of good quality and can be used to derive shear rates. The rates of shear are derived from their rotation curves as follows, $$\label{shear}
\frac{A}{\omega}=\frac{1}{2}\left(1-\frac{R}{V}\frac{dV}{dR}\right)$$ where $A$ is the first Oort constant, $\omega$ is the angular velocity and $V$ is the measured line–of–sight velocity at a radius $R$. The value of $A/\omega$ is the shear rate.
Using equation (\[shear\]), we have calculated mean shear rates for these galaxies, over a radial range with the inner limit near the turnover radius, and the outer limit at the $D_{25}$ radius. The dominant source of error on the shear rate is the spectroscopic errors (i.e. a combination of the intrinsic spectroscopic error and the error associated with folding the two sides of the galaxy), which Rubin et al. (1980, 1982, 1985) claim is quite small, typically $<10$ per cent. In order to calculate the shear rate, the mean value of $dV/dR$ measured in km s$^{-1}$ arcsec$^{-1}$ is calculated by fitting a line of constant gradient to the outer part of the rotation curve (i.e. past the radius of rotation). This results in a mean shear rate. This is essentially the same method used by other authors to calculate shear rate (e.g. Block et al. 1999; Seigar, Block & Puerari 2004; Seigar et al. 2005). The shear rates for these galaxies are listed in Table 1.
IRAS fluxes as an indicator of star formation rates
===================================================
The IRAS 60$\mu$m and 100$\mu$m fluxes have been used to calculate the far–infared luminosity for the galaxies in this sample for which IRAS data was available (see Table 1). The far–infrared luminosity $L_{FIR}$ can be used as a quantitative indicator of star formation rates (Spinoglio et al. 1995; Seigar & James 1998a). This was divided by the K-band luminosity $L_{K}$ of the galaxies to compensate for differences in their overall size. The $K$ band luminosity can be used to estimate the stellar mass in galaxies (e.g. Brinchmann & Ellis 2000; Gavazzi et al. 2002) and so the $L_{FIR}/L_{K}$ ratio can be interpretted as a star formation rate per unit stellar mass, i.e. a specific star formation rate. This is closely related to the birthrate parameter (Gavazzi et al. 2002), i.e. the fraction of young to old stars. The far–infared luminosity in terms of the 60$\mu$m and 100$\mu$m flux is given by Lonsdale et al. (1985) as, $$\label{lonsdale}
L_{FIR}=3.75 \times 10^{5}D^{2}(2.58S_{60}+S_{100})$$ where $L_{FIR}$ is the far–infrared luminosity in solar units, $D$ is the distance to the galaxy in Mpc, $S_{60}$ is the 60$\mu$m flux in Jy and $S_{100}$ is the 100$\mu$m flux in Jy. As these galaxies are nearby, for the calculattion of distance $D$, a simple Hubble flow is assumed and a Hubble constant, $H_{0}=75$ km s$^{-1}$ Mpc$^{-1}$, is adopted.
Given an apparent K-band magnitude it is possible to calculate a K-band luminosity using a relationship from Seigar & James (1998a), $$\label{seigar}
\log_{10}{L_{K}}=11.364-0.4K_{T}+\log_{10}{(1+z)}+2\log_{10}{D}$$ where $L_{K}$ is the K-band luminosity in solar units, $K_{T}$ is the K-band apparent magnitude and $z$ is the redshift of the galaxy. The $\log_{10}{(1+z)}$ term is a first order K-correction. In order to calculate this total K-band luminosity, apparent K-band magnitudes from the 2MASS survey were used.
Discussion
==========
Fig. 1 shows a plot of the ratio of far–infrared luminosity to K-band luminosity versus shear rate. A good correlation is shown (correlation coefficient = 0.71; significance = 99.91%). This is essentially a correlation between the star formation rate per unit stellar mass (or the specific star formation rate) and shear rate. This correlation has been used to determine the relationship between shear rate and the far–infrared luminosity as follows, $$\label{correlation}
\frac{L_{FIR}}{L_{K}}=(0.269\pm0.020)-(0.386\pm0.040)\left(\frac{A}{\omega}\right)$$ Where $L_{FIR}$ is the far–infrared luminosity, $L_K$ is the K–band luminosity and $A/\omega$ is the shear rate. The relationship between star formation rate (in $M_{\odot}/$year) and the far–infrared luminosity (in ergs/s) for normal spiral galaxies is given by Buat & Xu (1996) as $$\label{kenn1}
SFR(M_{\odot}/year)=8\times 10^{-44} L_{FIR} (ergs/s)$$ This has been used to calculate the star formation rates listed in Table 1.
Galaxies with higher shear rates have tighter spiral structure. These are the early–type galaxies (Seigar, Block & Puerari 2004; Seigar et al. 2005). Those with low shear rates have losely wound spiral structure. These are the late–type galaxies. The transition from early–type to late–type spiral galaxies also follows a transition from galaxies with low amounts of gas to gas–rich galaxies (Bertin 1991; Block et al. 1994; Bertin & Lin 1996). Therefore later–type spirals have more gas with which to form stars and it therefore follows that late–type spiral galaxies might have larger specific star–formation rates than early–type spiral galaxies. Such a correlation has been shown and discussed by James et al. (2004), who show that a correlation exists between H$\alpha$ equivalent width (which is related to the specific star formation rate and the birthrate parameter) and galaxy type. Is it therefore possible that the correlation shown in Fig. 1 is affected by a selection bias such as the one described here? In an attempt to answer this we have investigated the relationship between shear rate and morphological type (Fig. 2), and $L_{FIR}/L_K$ and morphological type (Fig. 3). Fig. 2 shows a weak and insignificant correlation (correlation coefficient=0.24; significance=70.90 per cent). Although it is insignificant the correlation is in the expected sense, with early type galaxies have higher rates of shear. The weakness of the correlation may be attributed to the problems associated with assigning galaxies with a Hubble type, which is usually a process that is done by eye and sometimes bears little resemblance to the underlying stellar mass distribution (Seigar & James 1998a, b; Seigar et al. 2005). Fig. 3 also shows a weak and insignificant correlation (correlation coefficient=0.20; significance=83.29 per cent), although once again the correlation is in the expected sense. Given the weakness, and the low significance of the correlations shown in Fig. 2 and Fig. 3, it is unlikely that the good correlation between shear rate and $L_{FIR}/L_{K}$ is a selection affect.
We believe that the correlation shown in Fig. 1 is a useful diagnostic tool. Given a rotation curve and far-infrared data for a galaxy, it should be possible to measure the stellar mass in any given galaxy, and this could be a very powerful tool. We now investigate the affect of galaxy size on the overall star formation rate.
Before going any further, one factor should be taken into account when interpreting the results of this analysis. The psf of IRAS was not capable of resolving most nearby external galaxies, and this is certainly the case for this sample. In such an analysis, we are really only interested in the star formation in the disks of spiral galaxies. However, the use of IRAS fluxes means that the star formation rates presented in this letter may be contaminated by bulge and/or nuclear star formation. Furthermore, we have used the total 2MASS K magnitude, which will also have a significant bulge contribution. One may expect these two affects to cancel to some degree, although one should also consider that the bulge contribution to the K-band light is probably more significant than the bulge contribution to the far-infrared light. However, since the correlation in Fig. 1 is good, one can only assume that this affect is small for the current sample. It should be noted that any object with known nuclear activity was excluded from the sample.
Using equation \[kenn1\] we have calculated the star formation rates in these galaxies. These are listed in Table 1. Since we have measurements of $D_{25}$ for all of these galaxies, which have been taken from de Vaucouleurs et al. (1991), it is possible to measure the star formation rate per unit surface area, or the far-infrared luminosity per unit area, if we assume perfectly circular disks. One might expect that larger galaxies will also be more massive, at least in terms of their stellar mass. As a result, one might expect to see a correlation between shear rate and the far-infrared luminosity per unit area. Fig. 4 shows a plot of shear rate versus the far-infrared luminosity per unit area. Only a very weak, insignificant correlation is shown (correlation coefficient=0.13; significance=47.37 per cent). As a result, we decided to investigate the relationship between size, parameterized as $D_{25}$ and K-band luminosity (or stellar mass) in disk galaxies, and this is shown in Fig. 5. There seems to be a weak, yet significant, correlation between these two parameters (correlation coefficient=0.35; significance=99.50 per cent). As a result it seems that while $K$ band luminosity is a good means for estimating stellar mass (e.g. Brinchmann & Ellis 2000), it is not so good for the overall size of galaxies. Larger galaxies are not necessarily brighter in the near-infrared. However, on the average, galaxy size and mass do correlate, but for specific cases, it may not be possible to estimate either mass from size or size from mass. Also, the lack of a correlation in Fig. 4 may be a result of the weak correlation shown in Fig. 5.
From Fig. 1, it can be seen that there is a shear rate at which the star formation in disk galaxies switching off. In fact, from equation 4, this [*critical*]{} shear rate is $\left(\frac{A}{\omega}\right)_{c}=0.70\pm0.09$. Using the correlation between shear rate and spiral arm pitch angle (Seigar et al. 2005), it can be shown that this critical shear rate would result in spiral structure with a pitch angle, $P_{K}=10^{\circ}\hspace*{-1.3mm}.6\pm2^{\circ}\hspace*{-1.3mm}.5$, which is consistent with the tightest spiral structure typically seen in the near-infrared (e.g. Block et al. 1999). This also suggests that there would be a central mass concentration for which spiral galaxy disks are stabilised against gravitational collapse and subsequent star formation. One would expect that this critical shear rate corresponds to the regime, where the Toomre stability parameter, $Q=1$. This result implies that the correlation shown in Fig. 1 does contain some star formation physics, as it seems to accurately predict the tightest wound spiral arms observed in disk galaxies.
Acknowledgments {#acknowledgments .unnumbered}
===============
This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. The author wishes to thank the referee for useful suggestions which improved the content of this article.
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[^1]: E-mail: mseigar@uci.edu
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---
abstract: 'In this paper we address the problem of pool based active learning, and provide an algorithm, called UPAL, that works by minimizing the unbiased estimator of the risk of a hypothesis in a given hypothesis space. For the space of linear classifiers and the squared loss we show that UPAL is equivalent to an exponentially weighted average forecaster. Exploiting some recent results regarding the spectra of random matrices allows us to establish consistency of UPAL when the true hypothesis is a linear hypothesis. Empirical comparison with an active learner implementation in Vowpal Wabbit, and a previously proposed pool based active learner implementation show good empirical performance and better scalability.'
author:
- |
Ravi Ganti, Alexander Gray\
School of Computational Science & Engineering, Georgia Tech\
gmravi2003@gatech.edu, agray@cc.gatech.edu
bibliography:
- 'upal\_arxiv.bib'
title: 'UPAL: Unbiased Pool Based Active Learning'
---
Introduction
============
In the problem of binary classification one has a distribution $\cD$ on the domain $\cX\times\cY\subseteq \bbR^d\times\{-1,+1\}$, and access to a sampling oracle, which provides us i.i.d. labeled samples $\cS=\{(x_1,y_1),\ldots,(x_n,y_n)\}$. The task is to learn a classifier $h$, which predicts well on unseen points. For certain problems the cost of obtaining labeled samples can be quite expensive. For instance consider the task of speech recognition. Labeling of speech utterances needs trained linguists, and can be a fairly tedious task. Similarly in information extraction, and in natural language processing one needs expert annotators to obtain labeled data, and gathering huge amounts of labeled data is not only tedious for the experts but also expensive. In such cases it is of interest to design learning algorithms, which need only a few labeled examples for training, and also guarantee good performance on unseen data.
Suppose we are given a labeling oracle $\cO$, which when queried with an unlabeled point $x$ returns the label $y$ of $x$. Active learning algorithms query this oracle as few times as possible and learn a provably good hypothesis from these labeled samples. Broadly speaking active learning (AL) algorithms can be classified into three kinds, namely membership query (MQ) based algorithms, stream based algorithms and pool based algorithms. All these three kinds of AL algorithms query the oracle $\cO$ for the label of the point, but differ in the nature of the queries. In MQ based algorithms the active learner can query for the label of a point in the input space $\cX$, but this query might not necessarily be from the support of the marginal distribution $\cD_{\cX}$. With human annotators MQ algorithms might work poorly as was demonstrated by Lang and Baum in the case of handwritten digit recognition [-@baum1992query], where the annotators were faced with the awkward situation of labeling semantically meaningless images. Stream based AL algorithms [@cohn1994improving; @chu2011unbiased] sample a point $x$ from the marginal distribution $\cD_{\cX}$, and decide on the fly whether to query $\cO$ for the label of $x$? Stream based AL algorithms tend to be computationally efficient, and most appropriate when the underlying distribution changes with time. Pool based AL algorithms assume that one has access to a large pool $\cP=\{x_1,\ldots,x_n\}$ of unlabeled i.i.d. examples sampled from $\cD_{\cX}$, and given budget constraints $B$, the maximum number of points they are allowed to query, query the most informative set of points. Both pool based AL algorithms, and stream based AL algorithms overcome the problem of awkward queries, which MQ based algorithms face. However in our experiments we discovered that stream based AL algorithms tend to query more points than necessary, and have poorer learning rates when compared to pool based AL algorithms.
Contributions.
--------------
1. In this paper we propose a pool based active learning algorithm called UPAL, which given a hypothesis space $\cH$, and a margin based loss function $\phi(\cdot)$ minimizes a provably unbiased estimator of the risk $\bbE[\phi(y h(x))]$. While unbiased estimators of risk have been used in stream based AL algorithms, no such estimators have been introduced for pool based AL algorithms. We do this by using the idea of importance weights introduced for AL in Beygelzimer et al. [-@beygelzimer2009importance]. Roughly speaking UPAL proceeds in rounds and in each round puts a probability distribution over the entire pool, and samples a point from the pool. It then queries for the label of the point. The probability distribution in each round is determined by the current active learner obtained by minimizing the importance weighted risk over $\cH$. Specifically in this paper we shall be concerned with linear hypothesis spaces, i.e. $\cH=\bbR^d$.
2. In theorem \[thm:ewa\] (Section \[sec:ewa\]) we show that for the squared loss UPAL is equivalent to an exponentially weighted average (EWA) forecaster commonly used in the problem of learning with expert advice [@cesa2006prediction]. Precisely we show that if each hypothesis $h\in\cH$ is considered to be an expert and the importance weighted loss on the currently labeled part of the pool is used as an estimator of the risk of $h\in\cH$, then the hypothesis learned by UPAL is the same as an EWA forecaster. Hence UPAL can be seen as pruning the hypothesis space, in a soft manner, by placing a probability distribution that is determined by the importance weighted loss of each classifier on the currently labeled part of the pool.
3. In section \[sec:consistency\] we prove consistency of UPAL with the squared loss, when the true underlying hypothesis is a linear hypothesis. Our proof employs some elegant results from random matrix theory regarding eigenvalues of sums of random matrices [@hsu2011analysis; @hsu2011dimension; @tropp2010user]. While it should be possible to improve the constants and exponent of dimensionality involved in $n_{0,\delta},T_{0,\delta},T_{1,\delta}$ used in theorem \[thm:main\], our results qualitatively provide us the insight that the the label complexity with the squared loss will depend on the condition number, and the minimum eigenvalue of the covariance matrix $\Sigma$. This kind of insight, to our knowledge, has not been provided before in the literature of active learning.
4. In section \[sec:expts\] we provide a thorough empirical analysis of UPAL comparing it to the active learner implementation in Vowpal Wabbit (VW) [@langford2010vowpal], and a batch mode active learning algorithm, which we shall call as BMAL [@hoi2006batch]. These experiments demonstrate the positive impact of importance weighting, and the better performance of UPAL over the VW implementation. We also empirically demonstrate the scalability of UPAL over BMAL on the MNIST dataset. When we are required to query a large number of points UPAL is upto 7 times faster than BMAL.
Algorithm Design {#sec:alg_design}
================
A good active learning algorithm needs to take into account the fact that the points it has queried might not reflect the true underlying marginal distribution. This problem is similar to the problem of dataset shift [@quinonero2008dataset] where the train and test distributions are potentially different, and the learner needs to take into account this bias during the learning process. One approach to this problem is to use importance weights, where during the training process instead of weighing all the points equally the algorithm weighs the points differently. UPAL proceeds in rounds, where in each round $t$, we put a probability distribution $\{p_i^t\}_{i=1}^n$ on the entire pool $\cP$, and sample one point from this distribution. If the sampled point was queried in one of the previous rounds $1,\ldots,t-1$ then its queried label from the previous round is reused, else the oracle $\cO$ is queried for the label of the point. Denote by $\Qit\in\{0,1\}$ a random variable that takes the value 1 if the point $x_i$ was queried for it’s label in round $t$ and 0 otherwise. In order to guarantee that our estimate of the error rate of a hypothesis $h\in\cH$ is unbiased we use importance weighting, where a point $x_i\in\cP$ in round $t$ gets an importance weight of $\frac{\Qit}{\pit}$. Notice that by definition $\bbE[\Qit|\pit]=1$. We formally prove that importance weighted risk is an unbiased estimator of the true risk. Let $\cDn$ denote a product distribution on ${(x_1,y_1),\ldots,(x_n,y_n)}$. Also denote by $Q_{1:n}^{1:t}$ the collection of random variables $Q_{1}^1,\ldots,Q_{n}^1,\ldots, Q_{n}^t$. Let $\langle \cdot,\cdot \rangle$ denote the inner product. We have the following result.
\[thm:unbiased\] Let $\hatLth{\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}\frac{1}{nt}\sum_{i=1}^n \sum_{\tau=1}^t \frac{\Qitau}{\pitau} \phi(y_i\langle h,x_i\rangle),$ where $p_i^{\tau}>0$ for all $\tau=1,\ldots,t$. Then $$\bbE_{Q_{1}^{1},\ldots,Q_{n}^{t},\cDn} \hatLth=L(h).$$
$$\begin{gathered}
\bbE_{Q_{1:n}^{1:t},\cDn}\hatLth =\bbE_{Q_{1:n}^{1:t},\cDn} \frac{1}{nt}\sum_{i=1}^n\sum_{\tau=1}^t
\frac{\Qitau}{\pitau} \phi(y_i\langle h,x_i\rangle)\nonumber
=\bbE_{Q_{1:n}^{1:t},\cDn} \frac{1}{nt}\sum_{i=1}^n\sum_{\tau=1}^t \bbE_{Q_{i}^\tau|Q_{1:n}^{1:\tau-1},\cDn} \frac{\Qitau}{\pitau} \phi(y_i\langle h,x_i\rangle)
=\\\bbE_{\cDn} \frac{1}{nt}\sum_{i=1}^n\sum_{\tau=1}^t \phi(y_i\langle h,x_i\rangle)=L(w).\qedhere
\end{gathered}$$
The theorem guarantees that as long as the probability of querying any point in the pool in any round is non-zero $\hat{L}_{t}(h)$, will be an unbiased estimator of $L(h)$. How does one come up with a probability distribution on $\cP$ in round $t$? To solve this problem we resort to probabilistic uncertainty sampling, where the point whose label is most uncertain as per the current hypothesis, $h_{A,t-1}$, gets a higher probability mass. The current hypothesis is simply the minimizer of the importance weighted risk in $\cH$, i.e. $h_{A,t-1}=\arg\min_{h\in\cH} \hat{L}_{t-1}(h)$. For any point $x_i\in\cP$, to calculate the uncertainty of the label $y_i$ of $x_i$, we first estimate $\eta(x_i){\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}\bbP[y_i=1|x_i]$ using $h_{A,t-1}$, and then use the entropy of the label distribution of $x_i$ to calculate the probability of querying $x_i$. The estimate of $\eta(\cdot)$ in round $t$ depends both on the current active learner $h_{A,t-1}$, and the loss function. In general it is not possible to estimate $\eta(\cdot)$ with arbitrary convex loss functions. However it has been shown by Zhang [-@zhang2004statistical] that the squared, logistic and exponential losses tend to estimate the underlying conditional distribution $\eta(\cdot)$. Steps 4, 11 of algorithm \[alg:poolal\] depend on the loss function $\phi(\cdot)$ being used. If we use the logistic loss i.e $\phi(yz)=\ln(1+\exp(-yz))$ then $\hat{\eta_t}(x)=\frac{1}{1+\exp(-yh_{A,t-1}^Tx)}$. In case of squared loss $\hat{\eta_t}(x)=\min\{\max\{0,w_{A,t-1}^Tx\},1\}$. Since the loss function is convex, and the constraint set $\cH$ is convex, the minimization problem in step 11 of the algorithm is a convex optimization problem.
1\. Set num\_unique\_queries=0, $h_{A,0}=0$, $t=1$. 2. Set $\Qit=0$ for all $i=1,\ldots,n$. 3. Set $p_{\text{min}}^{t}=\frac{1}{nt^{1/4}}$. 4. Calculate $\hat{\eta_t}(x_i)=\bbP[y=+1|x_i,h_{A,{t-1}}]$. 5. Assign $\pit=p_{\text{min}}^{t}+(1-np_{\text{min}}^t)\frac{\hat{\eta}_t(x_i)\ln(1/\hat{\eta}_t(x))+(1-\hat{\eta}_t(x_i))\ln(1/(1-\hat{\eta}_t(x_i)))}{\sum_{j=1}^n\hat{\eta}_t(x_j)\ln(1/\hat{\eta}_t(x_j))+(1-\hat{\eta}_t(x_j))\ln(1/(1-\hat{\eta}_t(x_j)))}$. 6. Sample a point (say $x_j$) from $p^t(\cdot)$. 7. Reuse its previously queried label $y_j$. 8. Query oracle $\cO$ for its label $y_j$. 9. . 10. Set $Q_j^t=1$. 11. Solve the optimization problem: $h_{A,t}=\arg\min_{h\in \cH} \sum_{i=1}^n\sum_{\tau=1}^t \frac{Q_{i}^{\tau}}{p_i^{\tau}}\phi(y_ih^Tx_i)$. 12. $t\leftarrow t+1$. 13. Return $h_{A}{\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}h_{A,t}$
By design UPAL might requery points. An alternate strategy is to not allow requerying of points. However the importance weighted risk may not be an unbiased estimator of the true risk in such a case. Hence in order to retain the unbiasedness property we allow requerying in UPAL.
The case of squared loss {#sec:ewa}
------------------------
It is interesting to look at the behaviour of UPAL in the case of squared loss where $\phi(yh^Tx)=(1-yh^Tx)^2$. For the rest of the paper we shall denote by $\ha$ the hypothesis returned by UPAL at the end of $T$ rounds. We now show that the prediction of $\ha$ on any $x$ is simply the exponentially weighted average of predictions of all $h$ in $\cH$.
\[thm:ewa\] Let $$\begin{aligned}
z_i{\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}&\sum_{t=1}^T\frac{\Qit}{\pit} &\Sighz&{\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}\sum_{i=1}^n z_ix_ix_i^T\\
v_z{\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}& \sum_{i=1}^{n} z_iy_ix_i &c{\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}& \sum_{i=1}^n z_i.
\end{aligned}$$ Define $w\in\bbR^d$ as $$\label{eqn:w}
w=\frac{\int_{\bbR^d} \exp(-\hat{L}_{T}(h))h~\mathrm{d}h}{\int_{\bbR^d}\exp(-\hat{L}_{T}(h))~\mathrm{d}h}.$$ Assuming $\Sighz$ is invertible we have for any $x_0\in \bbR^d$, $w^Tx_0=h_A^Tx_0$.
By elementary linear algebra one can establish that $$\begin{aligned}
\ha&=\Sighzi v_z\label{eqn:ha}\\
\hat{L}_{T}(h)&=(h-\Sighzi v_z)\Sighz(h-\Sighzi v-z).
\end{aligned}$$ Using standard integrals we get $$Z{\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}\int_{\bbR^d}\exp(-\hat{L}_{T}(h))~\mathrm{d}h=
\exp(-c-v_z^T\Sighzi v_z)\sqrt{\pi^d}\sqrt{\det(\Sighzi)}.$$ In order to calculate $w^Tx_0$, it is now enough to calculate the integral $$I{\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}\int_{\bbR^d} \exp(-\hat{L}_{T}(h))~h^Tx_0~\mathrm{d}w.$$ To solve this integral we proceed as follows. Define $I_1=\int_{\bbR^d}\exp(-\hat{L}_{T}(h))~ h^Tx_0~\mathrm{d}h$. By simple algebra we get $$\begin{aligned}
I&=\int_{\bbR^d} \exp(-w^T\Sighz w+2w^Tv_z-c)~ w^Tx_0~\mathrm{d}w\\
&=\exp(-c-v_z^T\Sighzi v_z)I_1.\label{eqn:eqn_I}
\end{aligned}$$ Let $a=h-\Sighzi v_z$. We then get $$\begin{aligned}
I_1&=\int_{\bbR^d}h^Tx_0\exp\left(-(h-\Sighzi v_z)\Sighz(h-\Sighzi v_z)\right)~\mathrm{d}h\nonumber\\
&=\int_{\bbR^d} (a^Tx_0+v_z^T\Sighzi x_0)\exp(-a^T\Sighz a)~\mathrm{d}a\nonumber\\
&=\underbrace{\int_{\bbR^d} (a^Tx_0)\exp(-a^T\Sighz a)~\mathrm{d}a}_{I_2}+\nonumber
\underbrace{\int_{\bbR^d} v_z^T\Sighzi x_0\exp(-a^T\Sighz a)~\mathrm{d}a}_{I_3}.\label{eqn:eqn_2}\end{aligned}$$ Clearly $I_2$ being the integrand of an odd function over the entire space calculates to 0. To calculate $I_3$ we shall substitute $\Sighz=SS^T$, where $S\succ 0$. Such a decomposition is possible since $\Sighz\succ 0$. Now define $z=S^Ta$. We get $$\begin{aligned}
I_3&=v_z^T\Sighzi x_0\int \exp(-z^Tz)~\det(S^{-1})~\mathrm{d}z\\
&=v_z^T\Sighzi x_0 \det(S^{-1})\sqrt{\pi^{d}}.\label{eqn:eqn_for_I3}\end{aligned}$$ Using equations (\[eqn:eqn\_I\], \[eqn:eqn\_2\], \[eqn:eqn\_for\_I3\]) we get $$\begin{gathered}
I=(\sqrt{\pi})^dv_z^T\Sighzi x_0 ~\det(S^{-1})\exp(-c-v_z^T\Sighzi v_z).
\end{gathered}$$ Hence we get $$w^Tx_0=v_z^T\Sighzi x_0\frac{\det(S^{-1})}{\sqrt{\det(M^{-1})}}=v_z^T\Sighzi x_0=\ha^Tx_0,$$ where the penultimate equality follows from the fact that $\det(\Sighzi)=1/\det(\Sighz)=1/(\det(SS^T))=1/(\det(S))^2$, and the last equality follows from equation \[eqn:ha\].
Theorem \[thm:ewa\] is instructive. It tells us that assuming that the matrix $\Sighz$ is invertible, $\ha$ is the same as an exponentially weighted average of all the hypothesis in $\cH$. Hence one can view UPAL as learning with expert advice, in the stochastic setting, where each individual hypothesis $h\in\cH$ is an expert, and the exponential of $\hat{L}_{T}$ is used to weigh the hypothesis in $\cH$. Such forecasters have been commonly used in learning with expert advice. This also allows us to interpret UPAL as pruning the hypothesis space in a soft way via exponential weighting, where the hypothesis that has suffered more cumulative loss gets lesser weight.
Bounding the excess risk {#sec:consistency}
========================
It is natural to ask if UPAL is consistent? That is will UPAL do as well as the optimal hypothesis in $\cH$ as $n\rightarrow \infty,T\rightarrow \infty$? We answer this question in affirmative. We shall analyze the excess risk of the hypothesis returned by our active learner, denoted as $h_{A}$, after $T$ rounds when the loss function is the squared loss. The prime motivation for using squared loss over other loss functions is that squared losses yield closed form estimators, which can then be elegantly analyzed using results from random matrix theory [@hsu2011analysis; @hsu2011dimension; @tropp2010user]. It should be possible to extend these results to other loss functions such as the logistic loss, or exponential loss using results from empirical process theory [@vandegeer2000empirical].
Main result
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\[thm:main\] Let $(x_1,y_1),\ldots (x_n,y_n)$ be sampled i.i.d from a distribution. Suppose assumptions A0-A3 hold. Let $\delta\in(0,1)$, and suppose $n\geq n_{0,\delta},T\geq \max\{T_{0,\delta},T_{1,\delta}\}$. With probability atleast $1-10\delta$ the excess risk of the active learner returned by UPAL after $T$ rounds is $$L(h_A)-L(\beta)= O\left(\frac{1}{n}+\frac{n}{\sqrt{T}}(d+2\sqrt{d\ln(1/\delta)}+2\ln(1/\delta))\right).$$
Assumptions, and Notation.
--------------------------
1. (Invertibility of $\Sigma$) The data covariance matrix $\Sigma$ is invertible.
2. (Statistical leverage condition) There exists a finite $\gamma_0\geq 1$ such that almost surely $$||\Sig^{-1/2}{x}||\leq \gamma_0\sqrt{d}.$$
3. There exists a finite $\gamma_1\geq 1$ such that $\bbE[\exp(\alpha^Tx)]\leq \exp\left(\frac{||\alpha||^2\gamma_1^2}{2}\right)$.
4. (Linear hypothesis) We shall assume that $y=\beta^Tx+\xi(x)$, where $\xi(x)\in [-2,+2]$ is additive noise with $\bbE[\xi(x)|x]=0$.
Assumption A0 is necessary for the problem to be well defined. A1 has been used in recent literature to analyze linear regression under random design and is a Bernstein like condition [@rokhlin2008fast]. A2 can be seen as a softer form of boundedness condtion on the support of the distribution. In particular if the data is bounded in a d-dimensional unit cube then it suffices to take $\gamma_1=1/2$. It may be possible to satisfy A3 by mapping data to kernel spaces. Though popularly used kernels such as Gaussian kernel map the data to infinite dimensional spaces, a finite dimensional approximation of such kernel mappings can be found by the use of random features [@rahimi2007random]. **Notation.**
1. $h_A$ is the active learner outputted by our active learning algorithm at the end of $T$ rounds.
2. $$\begin{aligned}
\forall i=1,\ldots,n: z_i{\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}\sum_{t=1}^T\frac{\Qit}{\pit} &\hspace{30pt} \Sighz{\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}\sum_{i=1}^n z_ix_ix_i^T\\
\psi_z{\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}\sum_{i=1}^nz_i\xi(x_i)x_i& \hspace{30pt}\Sigh{\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}\frac{1}{n}\sum_{i=1}^n x_ix_i^T\\
\Sig{\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}\bbE[xx^T]& \hspace{30pt}\Sighz{\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}\sum_{i=1}^n z_ix_ix_i^T\\
n_{0,\delta}{\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}7200d^2\gamma_0^4(d\ln(5)+\ln(10/\delta))&\hspace{30pt}
T_{1,\delta}{\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}12+512\sqrt{2}d^{8/3}\gamma_0^{16/3}\ln^{4/3}(d/\delta)
\end{aligned}$$
$$T_{0,\delta}{\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}\gamma_1^{16/3}d^{8/3}\ln^{4/3}(d/\delta)\ln^{8/3}(n/\delta)\lambdamin^{8/3}(\Sigma)+4\ln(d/\delta)\frac{\lambdamax(\Sigma)}{\lambdamin(\Sigma)},$$ where $\delta\in(0,1)$.
Overview of the proof
---------------------
The excess risk of a hypothesis $h\in\cH$ is defined as $L(h)-L(\beta)=\bbE_{x, y\sim\cD} [(y-h^Tx)^2-(y-\beta^Tx)^2]$. Our aim is to provide high probability bounds for the excess risk, where the probability measure is w.r.t the sampled points $(x_1,y_1),\ldots,(x_n,y_n), Q_1^1,\ldots,Q_{n}^T$. The proof proceeds as follows.
1. In lemma \[lem:decompose\], assuming that the matrices $\Sighz,\Sigh$ are invertible we upper bound the excess risk as the product $||\Sig^{1/2}\Sighzi\Sig^{1/2}||^2||\Sig^{-1/2}\Sigh^{1/2}||^2 ||\Sigh^{-1/2}\psi_z||^2$. The prime motivation in doing so is that bounding such “squared norm” terms can be reduced to bounding the maximum eigenvalue of random matrices, which is a well studied problem in random matrix theory.
2. In lemma \[lem:1\] we provide an upper bound for $||\Sig^{-1/2}\Sigh^{1/2}||^2$. To do this we use the simple fact that the matrix 2-norm of a positive semidefinite matrix is nothing but the maximum eigenvalue of the matrix. With this obsercation, and by exploiting the structure of the matrix $\Sigh$, the problem reduces to giving probabilistic upper bounds for maximum eigenvalue of a sum of random rank-1 matrices. Theorem \[thm:litvak\] provides us with a tool to prove such bounds.
3. In lemma \[lem:2\] we bound $||\Sig^{1/2}\Sighzi\Sig^{1/2}||^2$. The proof is in the same spirit as in lemma \[lem:1\], however the resulting probability problem is that of bounding the maximum eigenvalue of a sum of random matrices, which are not necessarily rank-1. Theorem \[thm:mat\_bern\] provides us with Bernstein type bounds to analyze the eigenvalues of sums of random matrices.
4. In lemma \[lem:3\] we bound the quantity $||\Sigh^{-1/2}\psi_z||^2$. Notice that here we are bounding the squared norm of a random vector. Theorem \[thm:quadratic\] provides us with a tool to analyze such quadratic forms under the assumption that the random vector has sub-Gaussian exponential moments behaviour.
5. Finally all the above steps were conditioned on the invertibility of the random matrices $\Sigh,\Sighz$. We provide conditions on $n,T$ (this explains why we defined the quantities $n_{0,\delta},T_{0,\delta},T_{1,\delta}$) which guarantee the invertibility of $\Sigh,\Sighz$. Such problems boil down to calculating lower bounds on the minimum eigenvalue of the random matrices in question, and to establish such lower bounds we once again use theorems \[thm:litvak\], \[thm:mat\_bern\].
Full Proof
----------
We shall now provide a way to bound the excess risk of our active learner hypothesis. Suppose $\ha$ was the hypothesis represented by the active learner at the end of the T rounds. By the definition of our active learner and the definition of $\beta$ we get $$\begin{aligned}
\ha&=\arg\min_{h\in\cH} ~\sum_{i=1}^n \sum_{t=1}^T\frac{\Qit}{\pit} (y_i-h^Tx_i)^2=\sum_{i=1}^n z_i (y_i-h^Tx_i)^2=\Sighzi v_z\\
\beta&=\arg\min_{h\in\cH}\bbE(y-\beta^Tx)^2=\Sigi\bbE[yx].\end{aligned}$$
\[lem:decompose\] Asumme $\Sighz,\Sigh$ are both invertible, and assumption A0 applies. Then the excess risk of the classifier after $T$ rounds of our active learning algorithm is given by $$\label{eqn:decompose}
L(h_{A})-L(\beta)\leq ||\Sig^{1/2}\Sighzi\Sig^{1/2}||^2||\Sig^{-1/2}\Sigh^{1/2}||^2 ||\Sigh^{-1/2}\psi_z||^2.$$
$$\begin{aligned}
L(\ha)-L(\beta)&=\bbE[(y-\ha^Tx)^2-(y-\beta^Tx)^2]\nonumber\\
&=\bbE_{x,y}[\ha^Txx^T\ha-2y\ha^Tx-\beta^Txx^T\beta+2y\beta^Tx]\nonumber\\
&=\ha^T\Sigma \ha-2\ha^T\bbE[xy]-\beta^T\Sigma\beta+2\beta^T\Sigma\beta
\text{~[Since $\Sig\beta=\bbE[yx]$]}\nonumber\\
&=\ha^T\Sigma \ha-\beta^T\Sigma\beta-2\ha^T\Sigma\beta+2\beta^T\Sigma\beta \nonumber\\
&=\ha^T\Sigma \ha+\beta^T\Sigma\beta-2\ha^T\Sigma\beta \nonumber\\
&=||\Sigma^{1/2}(\ha-\beta)||^2\label{eqn:exrisk1}.
\end{aligned}$$
We shall next bound the quantity $||\ha-\beta||$ which will be used to bound the excess risk in Equation ( \[eqn:exrisk1\]). To do this we shall use assumption A3 along with the definitions of $\ha,\beta$. We have the following chain of inequalities. $$\begin{aligned}
\ha&=\Sighzi v_z\nonumber\\
&=\Sighzi\sum_{i=1}^n z_i y_ix_i\nonumber\\
&=\Sighzi\sum_{i=1}^n z_i(\beta^Tx_i+\xi(x_i))x_i\nonumber\\
&=\Sighzi\sum_{i=1}^n z_i x_ix_i^T\beta+z_i\xi(x_i)x_i\nonumber\\
&=\beta+\Sighzi\sum_{i=1}^n z_i\xi(x_i)x_i=\beta+\Sighzi\psi_z.\label{eqn:habd}\end{aligned}$$ Using Equations \[eqn:exrisk1\],\[eqn:habd\] we get the following series of inequalities for the excess risk bound $$\begin{aligned}
L(\ha)-L(\beta)&=||\Sig^{1/2}\Sighzi\psiz||^2\nonumber\\
&=||\Sig^{1/2}\Sighzi\Sigh^{1/2}\Sigh^{-1/2}\psiz||^2\nonumber\\
&=||\Sig^{1/2}\Sighzi\Sig^{1/2}\Sig^{-1/2}\Sigh^{1/2}\Sigh^{-1/2}\psiz||^2\\
&\leq ||\Sig^{1/2}\Sighzi\Sig^{1/2}||^2||\Sig^{-1/2}\Sigh^{1/2}||^2 ||\Sigh^{-1/2}\psi_z||^2.\qedhere\end{aligned}$$
The decomposition in lemma \[lem:decompose\] assumes that both $\Sighz,\Sigh$ are invertible. Before we can establish conditions for the matrices $\Sighz,\Sigh$ to be invertible we need the following elementary result.
\[prop:expmoments\_statlev\] For any arbitrary $\alpha\in \bbR^d$, under assumption A1 we have $$\bbE[\exp(\alpha^T\Sig^{-1/2}x)]\leq 5\exp\left(\frac{3d\gamma_0^2||\alpha||^2}{2}\right).$$
From Cauchy-Schwarz inequality and A1 we get $$-||\alpha||\gamma_0\sqrt{d} \leq -||\alpha||~||\Sig^{-1/2}x|| \leq \alpha^T\Sig^{-1/2}x\leq ||\alpha||~||\Sig^{-1/2}x||\leq ||\alpha||\gamma_0\sqrt{d} .$$ Also $\bbE[\alpha^T\Sig^{-1/2}x]\leq ||\alpha||\gamma_0\sqrt{d}$. Using Hoeffding’s lemma we get $$\begin{aligned}
\bbE[\exp(\alpha^T\Sig^{-1/2}x)]&\leq \exp\left(||\alpha||\gamma_0\sqrt{d}+\frac{||\alpha||^2d\gamma_0^2}{2}\right)\\
&\leq 5\exp(3||\alpha||^2d\gamma_0^2/2).\qedhere\end{aligned}$$
The following lemma will be useful in bounding the terms $||\Sig^{1/2}\Sighzi\Sig^{1/2}||$, $||\Sig^{-1/2}\Sigh^{1/2}||^2$.
\[lem:J\] Let $J{\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}\sum_{i=1}^n \Sigma^{-1/2}x_ix_i^T\Sigma^{-1/2}$. Let $n\geq n_{0,\delta}$. Then the following inequalities hold separately with probability atleast $1-\delta$ each $$\begin{aligned}
\lambdamax(J)\leq n+6dn\gamma_0^2\left[\sqrt{\frac{32(d\ln(5)+\ln(10/\delta))}{n}}+\frac{2(d\ln(5)+\ln(10/\delta))}{n}\right]\leq 3n/2\\
\lambdamin(J)\geq n-6dn\gamma_0^2\left[\sqrt{\frac{32(d\ln(5)+\ln(10/\delta))}{n}}+\frac{2(d\ln(5)+\ln(10/\delta))}{n}\right]\geq n/2.
\end{aligned}$$
Notice that $\bbE[\Sig^{-1/2}x_ix_i^T\Sig^{-1/2}]=I.$ From Proposition \[prop:expmoments\_statlev\] we have $\bbE[\exp(\alpha^T\Sig^{-1/2}x)]\leq 5\exp(3||\alpha||^2 d\gamma_0^2/2)$. By using theorem \[thm:litvak\] we get with probability atleast $1-\delta$: $$\lambdamax\left(\frac{1}{n}\sum_{i=1}^n (\Sig^{-1/2}x_i)(\Sig^{-1/2}x_i)^T\right)\leq 1+6d\gamma_0^2\left[\sqrt{\frac{32(d\ln(5)+\ln(2/\delta))}{n}}+\frac{2(d\ln(5)+\ln(2/\delta))}{n}\right].$$ Put $n\geq n_{0,\delta}$ to get the desired result. The lower bound on $\lambdamin$ is also obtained in the same way.
\[lem:inv\_sigh\] Let $n\geq n_{0,\delta}$. With probability atleast $1-\delta$ separately we have $\Sigh\succ 0$, $\lambdamin(\Sigh)\geq \frac{1}{2}\lambdamin(\Sigma)$, $\lambdamax(\Sigh)\leq \frac{3}{2}\lambdamax(\Sigma)$.
Using lemma \[lem:J\] we get for $n\geq n_{0,\delta}$ with probability atleast $1-\delta$, $\lambdamin(J)\geq 1/2$ and with probability atleast $1-\delta$, $\lambdamax(\Sigma)\leq 3/2$. Finally since $\Sigma^{1/2}J\Sigma^{1/2}=\Sigh$, and $J\succ 0,\Sigma\succ 0$, we get $\Sigh\succ 0$. Further we have the following upper bound with probability atleast $1-\delta$: $$\begin{aligned}
\label{eqn:sig_ub}
\lambdamax(\Sigh)&=||\Sigma^{1/2}J\Sigma^{1/2}||\\
&\leq ||\Sigma^{1/2}||^2~||J|| \\
&\leq ||\Sigma||~||J||\\
&=\lambdamax(\Sigma)\lambdamax(J)\\
&\leq \frac{3}{2} \lambdamax(\Sigma),\end{aligned}$$ where in the last step we used the upper bound on $\lambdamax(J)$ provided by lemma \[lem:J\]. Similarly we have the following lower bound with probability atleast $1-\delta$ $$\begin{aligned}
\label{eqn:sig_lb}
\lambdamin(\Sigh)&=\frac{1}{\lambdamax(\Sig^{-1/2}J^{-1}\Sig^{-1/2})}\\
&=\frac{1}{||\Sig^{-1/2}J^{-1}\Sig^{-1/2}||}\\
&\geq \frac{1}{||\Sig^{-1}||~||J^{-1}||~||\Sig^{-1/2}||}\\
&=\lambdamin(\Sigma)\lambdamin(J)\\
&\geq\frac{\lambdamin(\Sigma)}{2},\end{aligned}$$ where in the last step we used the lower bound on $\lambdamin(J)$ provided by lemma \[lem:J\].
The following proposition will be useful in proving lemma \[lem:inv\_sighz\].
\[prop:normxibound\] Let $\delta\in(0,1)$. Under assumption A2, with probability atleast $1-\delta$, $\sum_{i=1}^n||x_i||^4\leq 25\gamma_1^4d^2\ln^2(n/\delta)$
From A2 we have $\bbE[\exp(\alpha^Tx)]\leq \exp(\frac{||\alpha||^2\gamma_1^2}{2})$. Now applying theorem \[thm:quadratic\] with $A=I_{d}$ we get $$\bbP[||x_i||^2\leq d\gamma_1^2+2\gamma_1^2\sqrt{d\ln(1/\delta)}+2\gamma_1^2\ln(1/\delta)]\geq 1-\delta.$$ The result now follows by the union bound.
\[lem:inv\_sighz\] Let $\delta\in(0,1)$. For $T\geq T_{0,\delta}$, with probability atleast $1-4\delta$ we have $\lambdamin(\Sighz)\geq \frac{nT\lambdamin(\Sigma)}{4}>0$. Hence $\Sighz$ is invertible.
The proof uses theorem \[thm:mat\_bern\]. Let $M_t'{\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}\sum_{i=1}^n \frac{\Qit}{\pit} x_ix_i^T$, so that $\Sighz=\sum_{t=1}^T M_t'$. Now $\bbE_tM_t'=n\Sigh$. Define $R_t'{\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}n\Sigh-M_t'$, so that $\bbE_t R_t'=0$. We shall apply theorem \[thm:mat\_bern\] to the random matrix $\sum R_t'$. In order to do so we need upper bounds on $\lambdamax (R_t')$ and $\lambdamax (\frac{1}{T}\sum_{t=1}^T \bbE_t R_t'^2)$. Let $n\geq n_{0,\delta}$. Using lemma \[lem:inv\_sigh\] we get with probability atleast $1-\delta$ $$\begin{aligned}
\lambdamax(R_t')=\lambdamax(n\Sigh-M_t')\leq \lambdamax(n\Sigh)\leq \frac{3n\lambdamax(\Sig)}{2}{\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}b_2.\end{aligned}$$ $$\begin{aligned}
\lambdamax\left[\frac{1}{T}\sum_{t=1}^T \bbE_t R_t'^2\right]&=\frac{1}{T}\lambdamax\left[\sum_{t=1}^T \bbE_t(n\Sigh-M_t')^2\right]\label{eqn:jump-2}\\
&=\frac{1}{T}\lambdamax(-n^2T\Sigh^2+\sum_{t=1}^T\bbE_t\sum_{i=1}^n \frac{\Qit}{(\pit)^2}(x_ix_i^T)^2)\label{eqn:jump-1}\\
&=\frac{1}{T}\lambdamax(-n^2T\Sigh^2+\sum_{t=1}^T\sum_{i=1}^n \frac{1}{\pit}(x_ix_i^T)^2)\label{eqn:jump0}\\
&\leq \frac{1}{T}\lambdamax(\sum_{i=1}^n\sum_{t=1}^T\frac{1}{\pit}(x_ix_i^T)^2)-n^2\lambdamin^2(\Sigh)\label{eqn:jump1}\\
&\leq nT^{1/4}\lambdamax(\sum_{i=1}^n (x_ix_i^T)^2)\label{eqn:jump2}\\
&\leq nT^{1/4}\sum_{i=1}^n\lambdamax^2(x_ix_i^T)\label{eqn:jump3}\\
&=nT^{1/4}\sum_{i=1}^n ||x_i||^4\label{eqn:jump4}\\
&\leq 25\gamma_1^4d^2n^2T^{1/4}\ln^2(n/\delta){\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}\sigma_2^2\label{eqn:jump5}.\end{aligned}$$ Equation \[eqn:jump-1\] follows from Equation \[eqn:jump-2\] by the definition of $M_t'$ and the fact that at any given $t$ only one point is queried i.e. $\Qit Q_{j}^t=0$ for a given $t$. Equation \[eqn:jump0\] follows from equation \[eqn:jump-1\] since $E_{t}\Qit=\pit$. Equation \[eqn:jump1\] follows from Equation \[eqn:jump0\] by Weyl’s inequality. Equation \[eqn:jump2\] follows from Equation \[eqn:jump1\] by substituting $p_{\text{min}}^t$ in place of $\pit$. Equation \[eqn:jump3\] follows from Equation \[eqn:jump2\] by the use of Weyl’s inequality. Equation \[eqn:jump4\] follows from Equation \[eqn:jump3\] by using the fact that if $p$ is a vector then $\lambdamax(pp^T)=||p||^2$. Equation \[eqn:jump5\] follows from Equation \[eqn:jump4\] by the use of proposition \[prop:normxibound\]. Notice that this step is a stochastic inequality and holds with probability atleast $1-\delta$.
Finally applying theorem \[thm:mat\_bern\] we have $$\begin{aligned}
\bbP\left[\lambdamax(\frac{1}{T}\sum_{t=1}^T R_t')\leq \sqrt{\frac{2\sigma_2^2\ln(d/\delta)}{T}}+\frac{b_2\ln(d/\delta)}{T}\right]\geq 1-\delta\\
\implies \bbP\left[\lambdamax(n\Sigh-\frac{1}{T}\sum_{t=1}^TM_t')\leq \sqrt{\frac{2\sigma_2^2\ln(d/\delta)}{T}}+\frac{b_2\ln(d/\delta)}{T}\right]\geq 1-\delta\\
\implies \bbP\left[\lambdamin(n\Sigh)-\frac{1}{T}\lambdamin\left(\sum_{t=1}^TM_t'\right)\leq \sqrt{\frac{2\sigma_2^2\ln(d/\delta)}{T}}+\frac{b_2\ln(d/\delta)}{T}\right]\geq 1-\delta\end{aligned}$$ Substituting for $\sigma_2,b_2$, rearranging the inequalities, and using lemma \[lem:inv\_sigh\] to lower bound $\lambdamin(\Sigh)$ we get $$\begin{aligned}
\bbP\left[\lambdamin(\sum_{t=1}^TM_t')\geq T\lambdamin(n\Sigh)-\sqrt{2T\sigma_2^2\ln(d/\delta)}-b_2\ln(d/\delta)\right]\geq 1-\delta\\
\implies \bbP\left[\lambdamin(\sum_{t=1}^TM_t')\geq \frac{nT\lambdamin(\Sigma)}{2}-\sqrt{2T\sigma_2^2\ln(d/\delta)}-b_2\ln(d/\delta)\right]\geq 1-2\delta\\
\implies \bbP\left[\lambdamin(\sum_{t=1}^TM_t')\geq\frac{nT\lambdamin(\Sigma)}{2}-5\sqrt{2}\gamma_1^2dnT^{5/8}\sqrt{\ln(d/\delta)}\ln(n/\delta)-\frac{n\ln(d/\delta)\lambdamax(\Sigma)}{2}\right]\geq 1-4\delta\end{aligned}$$ For $T\geq T_{0,\delta}$ with probability atleast $1-4\delta$, $\lambdamin\sum_{t=1}^TM_t'=\lambdamin(\Sighz)\geq\frac{nT\lambdamin(\Sigma)}{4}$.
\[lem:1\] For $n\geq n_{0,\delta}$ with probability atleast $1-\delta$ over the random sample $x_1,\ldots,x_n$ $$||\Sig^{-1/2}\Sigh^{1/2}||^2\leq 3/2.$$
$$\begin{aligned}
||\Sig^{-1/2}\Sigh^{1/2}||^2&=||\Sigh^{1/2}\Sig^{-1/2}||^2\\
&=\lambdamax(\Sig^{-1/2}\Sigh\Sig^{-1/2})\\
&=\lambdamax\left(\frac{1}{n}\sum_{i=1}^n (\Sig^{-1/2}x_i)(\Sig^{-1/2}x_i)^T\right)\\
&=\lambdamax\left(\frac{J}{n}\right)\\
&\leq 3/2
\end{aligned}$$
where in the first equality we used the fact that $||A||=||A^T||$ for a square matrix $A$, and $||A||^2=\lambdamax(A^TA)$, and in the last step we used lemma \[lem:J\].
\[lem:2\] Suppose $\Sighz$ is invertible. Given $\delta\in (0,1)$, for $n\geq n_{0,\delta}$, and $T\geq \max\{T_{0,\delta}, T_{1,\delta}\}$ with probability atleast $1-3\delta$ over the samples $$||\Sig^{1/2}\Sighzi\Sig^{1/2}||^2\leq\frac{400}{n^2T^2}.$$
The proof of this lemma is very similar to the proof of lemma \[lem:inv\_sighz\]. From lemma \[lem:inv\_sighz\] for $n\geq n_{0,\delta}, T\geq T_{0,\delta}$ with probability atleast $1-\delta$, $\Sighz\succ 0$. Using the assumption that $\Sig\succ 0$, we get $\Sig^{1/2}\Sighzi\Sig^{1/2}\succ 0$. Hence $||\Sig^{1/2}\Sighzi\Sig^{1/2}||=\lambdamax(\Sig^{1/2}\Sighzi\Sig^{1/2})=\frac{1}{\lambdamin(\Sig^{-1/2}\Sighz\Sig^{-1/2})}$. Hence it is enough to provide a lower bound on the smallest eigenvalue of the symmetric positive definite matrix $\Sig^{-1/2}\Sighz\Sig^{-1/2}$. $$\begin{aligned}
\lambdamin(\Sig^{-1/2}\Sighz\Sig^{-1/2})&=\lambdamin\left(\sum_{i=1}^n z_i\Sig^{-1/2} x_ix_i^T\Sig^{-1/2}\right)\\
&=\lambdamin(\sum_{t=1}^T\underbrace{\sum_{i=1}^n \frac{\Qit}{\pit}\Sig^{-1/2} x_ix_i^T\Sig^{-1/2}}_{{\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}M_t})\\
&=\lambdamin\left(\sum_{t=1}^T M_t\right).\end{aligned}$$ Define $R_t{\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}J-M_t$. Clearly $\bbE_{t}[M_t]=J$, and hence $\bbE[R_t]=0$. From Weyl’s inequality we have $\lambdamin(J)+\lambdamax\left(\frac{-1}{T}\sum_{t=1}^T M_t\right)\leq \lambdamax(\frac{1}{T}\sum_{t=1}^T R_t)$. Now applying theorem \[thm:mat\_bern\] on $\sum R_t$ we get with probability atleast $1-\delta$ $$\label{eqn:matbern}
\lambdamin(J)+\lambdamax\left(\frac{-1}{T}\sum_{t=1}^T M_t\right)\leq \lambdamax\left(\frac{1}{T}\sum_{t=1}^T R_t\right)\leq \sqrt{\frac{2\sigma_
1^2\ln(d/\delta)}{T}}+\frac{b_1\ln(d/\delta)}{3T},$$ where $$\begin{aligned}
\lambdamax\left(\frac{1}{T}\sum_{t=1}^T J-M_t\right)\leq b_1\\
\lambdamax\left(\frac{1}{T}\sum_{t=1}^T \bbE_{t}(J-M_t)^2\right)\leq \sigma_1^2\end{aligned}$$ Rearranging Equation (\[eqn:matbern\]) and using the fact that $\lambdamax(-A)=-\lambdamin(A)$ we get with probability atleast $1-\delta$, $$\label{eqn:matbernrear}
\lambdamin\left(\sum_{t=1}^T M_t\right)\geq T\lambdamin(J)-\sqrt{2T\sigma_
1^2\ln(d/\delta)}-\frac{b_1\ln(d/\delta)}{3}.$$ Using Weyl’s inequality [@horn90matrix] we have $\lambdamax(\frac{1}{T}\sum_{t=1}^T J-M_t)\leq \lambdamax(J)\leq \frac{3n}{2}$ with probability atleast $1-\delta$, where in the last step we used lemma (\[lem:J\]). Let $b_1{\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}\frac{3n}{2}$. To calculate $\sigma_1^2$ we proceed as follows.
$$\begin{aligned}
\lambdamax\left(\frac{1}{T}\sum_{t=1}^T \bbE_{t}(J-M_t)^2\right)&=\frac{1}{T}\lambdamax\left(\sum_{t=1}^T \bbE_{t}(M_t^2)-J^2\right)\label{eqn:here0}\\
&\leq \frac{1}{T}\lambdamax\left(\sum_{t=1}^T \bbE_{t}M_t^2\right)\label{eqn:here1}\\
&=\frac{1}{T}\lambdamax\left(\sum_{t=1}^T \bbE_t \left(\sum_{i=1}^n \frac{\Qit}{\pit}\Sigma^{-1/2}x_ix_i^T\Sigma^{-1/2}\right)^2\right)\label{eqn:here2}\\
&=\frac{1}{T}\lambdamax\left(\sum_{t=1}^T \bbE_t \sum_{i=1}^n \frac{\Qit}{(\pit)^2}(\Sigma^{-1/2}x_ix_i^T\Sigma^{-1/2})^2\right)\label{eqn:here3}\\
&=\frac{1}{T}\lambdamax\left(\sum_{t=1}^T \sum_{i=1}^n \frac{1}{\pit}(\Sigma^{-1/2}x_ix_i^T\Sigma^{-1/2})^2\right)\label{eqn:here4}\\
&\leq \frac{1}{T}\sum_{t=1}^T\sum_{i=1}^n \frac{1}{\pit}||\Sig^{-1/2}x_i||^4\label{eqn:here5}\\
&\leq \frac{d^2\gamma_0^4}{T}\sum_{i=1}^n\sum_{t=1}^T \frac{1}{\pit}\label{eqn:here6}\\
&\leq \frac{nd^2\gamma_0^4}{T}\sum_{t=1}^T \frac{1}{p_{\text{min}}^t}\label{eqn:here7}\\
&\leq n^2d^2\gamma_0^4 T^{1/4}{\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}\sigma_1^2\label{eqn:here8}.\end{aligned}$$
Equation \[eqn:here1\] follows from Equation \[eqn:here0\] by using Weyl’s inequality and the fact that $J^2\succeq 0$. Equation \[eqn:here3\] follows from Equation \[eqn:here2\] since only one point is queried in every round and hence for any given $t,i\neq j$ we have $\Qit Q_{j}^t=0$, and hence all the cross terms disappear when we expand the square. Equation (\[eqn:here4\]) follows from Equation (\[eqn:here3\]) by using the fact that $\bbE_{t}Q_t=p_t$. Equation (\[eqn:here5\]) follows from Equation (\[eqn:here4\]) by Weyl’s inequality and the fact that the maximum eigenvalue of a rank-1 matrix of the form $vv^T$ is $||v||^2$. Equation (\[eqn:here6\]) follows from Equation (\[eqn:here5\]) by using assumption A1. Equation \[eqn:here8\] follows from Equation (\[eqn:here7\]) by our choice of $p_{min}^t=\frac{1}{n\sqrt{t}}$. Substituting the values of $\sigma_1^2, b_1$ in \[eqn:matbernrear\], using lemma \[lem:J\] to lower bound $\lambdamin(J)$, and applying union bound to sum up all the failure probabilities we get for $n\geq n_{0,\delta},T\geq \max\{T_{0,\delta},T_{1,\delta}\}$ with probability atleast $1-3\delta$, $$\begin{gathered}
\lambdamin\left(\sum_{t=1}^T M_t\right)\geq T\lambdamin(J)-\sqrt{2T^{5/4}n^2d^2\gamma_0^4\ln(d/\delta)}-3n/2\\
\geq \frac{nT}{2}-\sqrt{2}T^{5/8}nd\gamma_0^2\sqrt{\ln(d/\delta)}-3n/2\geq nT/4.\qedhere\end{gathered}$$
The only missing piece in the proof is an upper bound for the quantity $||\Sigh^{-1/2}\psi_z||^2$. The next lemma provides us with an upper bound for this quantity.
\[lem:3\] Suppose $\Sigh$ is invertible. Let $\delta\in (0,1)$. With probability atleast $1-\delta$ we have $$||\Sigh^{-1/2}\psi_z||^2\leq (2nT^2+56n^3T\sqrt{T})(d+2\sqrt{d\ln(1/\delta)}+2\ln(1/\delta)).$$
Define the matrix $A\in \bbR^{d\times n}$ as follows. Let the $i^{\text{th}}$ column of $A$ be the vector $\frac{\Sigh^{-1/2}x_i}{\sqrt{n}}$, so that $AA^T=\frac{1}{n}\Sigh^{-1/2}x_ix_i^T\Sigh^{-1/2}=I_d$. Now $||\Sigh^{-1/2}\psi_z||^2=||\sqrt{n}Ap||^2$, where $p=(p_1,\ldots,p_n)\in \bbR^n$ and $p_i=\xi(x_i)z_i$ for $i=1,\ldots,n$. Using the result for quadratic forms of subgaussian random vectors (threorem \[thm:quadratic\]) we get $$\begin{gathered}
\label{eqn:norm_Ap2}
||Ap||^2\leq \sigma^2(\operatorname{tr}(I_d)+2\sqrt{\operatorname{tr}(I_d)\ln(1/\delta)}+2||I_d||\ln(1/\delta))=\sigma^2(d+2\sqrt{d\ln(1/\delta)}+2\ln(1/\delta)), \end{gathered}$$ where for any arbitrary vector $\alpha$, $\bbE[\exp(\alpha^Tp)]\leq \exp(||\alpha||^2\sigma^2)$.
Hence all that is left to be done is prove that $\alpha^Tp$ has sub-Gaussian exponential moments. Let $$D_t{\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}\sum_{i=1}^n \frac{\alpha_i\xi(x_i)\Qit}{\pit}-\alpha^T\xi~~~\forall t=1,\ldots,T.$$ With this definition we have the following series of equalities $$\begin{aligned}
\label{eqn:decompose_condexpec}
\bbE[\exp(\alpha^Tp)]=\bbE[\exp(\sum D_t+T\alpha^T\xi)]=\bbE\left[\exp(T\alpha^T\xi)\bbE[\exp(\sum D_t)|\cDn]\right].\end{aligned}$$ Conditioned on the data, the sequence $D_1,\ldots,D_T$, forms a martingale difference sequence. Let $\xi=[\xi(x_1),\ldots,\xi(x_n)]$. Notice that $$\label{eqn:bound_Dt}
-\alpha^T\xi-\frac{2||\alpha||}{p_{\text{min}}^t}\leq D_t\leq -\alpha^T\xi+\frac{2||\alpha||}{p_{\text{min}}^t}.$$ We shall now bound the probability of large deviations of $D_t$ given history up until time $t$. This allows us to put a bound on the large deviations of the martingale sum $\sum_{t=1}^T D_t$. Let $a\geq 0$. Using Markov’s inequality we get $$\begin{aligned}
\bbP[D_t\geq a|Q_{1:n}^{1:t-1},\cDn]&\leq \min_{\gamma>0}~\exp(-\gamma a)\bbE[\gamma D_t|Q_{1:n}^{1:t-1},\cDn]\\
&\leq\min_{\gamma>0}\exp\left(\frac{2\gamma^2||\alpha||^2}{(p_{\text{min}}^t)^2}-\gamma a\right)\\
& \leq \exp\left(\frac{-a^2}{8||\alpha||^2n^2\sqrt{t}}\right).\end{aligned}$$ In the second step we used Hoeffding’s lemma along with the boundedness property of $D_t$ shown in equation \[eqn:bound\_Dt\]. The same upper bound can be shown for the quantity $\bbP[D_t\leq a|Q_{1:n}^{1:t-1},\cDn]$. Applying lemma \[lem:modazuma\] we get with probability atleast $1-\delta$, conditioned on the data, we have $$\frac{1}{T}\sum_{t=1}^T D_t\leq \sqrt{\frac{448||\alpha||^2n^2\ln(1/\delta)}{\sqrt{T}}}\\\implies
\sum_{t=1}^T D_t\leq \sqrt{112||\alpha||^2n^2T^{3/2}\ln(1/\delta)}.$$ Hence $\sum_{t=1}^T D_t$, conditioned on data, has sub-Gaussian tails as shown above. This leads to the following conditional exponential moments bound $$\label{eqn:sumdt}
\bbE[\exp(\sum_{t=1}^T D_t)|\cD_n]=\exp\left(56||\alpha||^2n^2T\sqrt{T}\ln(1/\delta)\right).$$ Finally putting together equations \[eqn:decompose\_condexpec\], \[eqn:sumdt\] we get $$\bbE[\exp(\alpha^Tp)]\leq \bbE\exp(T\alpha^T\xi)\exp(56||\alpha||^2n^2T\sqrt{T})\leq \exp((2T^2+56n^2T\sqrt{T})||\alpha||^2),$$ In the last step we exploited the fact that $-2 \leq \xi(x_i)\leq 2$, and hence by Hoeffding lemma $\bbE[\exp(\alpha^T\xi)]\leq \exp(2||\alpha||^2)$. This leads us to the choice of $\sigma^2=2T^2+56n^2T\sqrt{T}$. Substituting this value of $\sigma^2$ in equation \[eqn:norm\_Ap2\] we get $$||Ap||^2\leq (2T^2+56n^2T\sqrt{T})(d+2\sqrt{d\ln(1/\delta)}+2\ln(1/\delta)),$$ and hence with probability atleast $1-\delta$, $$||\Sigh^{-1/2}\psi_z ||^2=n||Ap||^2\leq (2nT^2+56n^3T\sqrt{T})(d+2\sqrt{d\ln(1/\delta)}+2\ln(1/\delta)).$$ We are now ready to prove our main result.
\[**Proof of theorem \[thm:main\]**\] For $n\geq n_{0,\delta}$ and $T\geq \max\{T_{0,\delta},T_{1,\delta}\}$ from lemma \[lem:inv\_sigh\], \[lem:inv\_sighz\], both $\Sighz$, and $\Sigh$ are invertible with probability atleast $1-\delta, 1-4\delta$ respectively. Conditioned on the invertibility of $\Sighz,\Sigma$ we get from lemmas \[lem:1\]-\[lem:3\], $||\Sigi\Sigh^{1/2}||^2\leq 3/2$ and $||\Sig^{1/2}\Sighzi\Sig^{1/2}||^2\leq400/n^2T^2$, and $||\Sigh^{-1/2}\psi_z||^2\leq (2nT^2+56n^3T^{3/2})(d+2\sqrt{d\ln(1/\delta)+2\ln(1/\delta)})$ with probability atleast $1-\delta,1-3\delta,1-\delta$ respectively. Using lemma \[lem:decompose\] and the union bound to add up all the failure probabilities we get the desired result.
Related Work
============
A variety of pool based AL algorithms have been proposed in the literature employing various query strategies. However, none of them use unbiased estimates of the risk. One of the simplest strategy for AL is uncertainty sampling, where the active learner queries the point whose label it is most uncertain about. This strategy has been popularl in text classification [@lewis1994sequential], and information extraction [@settles2008analysis]. Usually the uncertainty in the label is calculated using certain information-theoretic criteria such as entropy, or variance of the label distribution. While uncertainty sampling has mostly been used in a probabilistic setting, AL algorithms which learn non-probabilistic classifiers using uncertainty sampling have also been proposed. Tong et al. [-@tong2001support] proposed an algorithm in this framework where they query the point closest to the current svm hyperplane. Seung et al. [-@seung1992query] introduced the query-by-committee (QBC) framework where a committee of potential models, which all agree on the currently labeled data is maintained and, the point where most committee members disagree is considered for querying. In order to design a committee in the QBC framework, algorithms such as query-by-boosting, and query-by-bagging in the discriminative setting [@Abe1998query], sampling from a Dirichlet distribution over model parameters in the generative setting [@mccallumzy1998employing] have been proposed. Other frameworks include querying the point, which causes the maximum expected reduction in error [@zhu2003combining; @guo2007optimistic], variance reducing query strategies such as the ones based on optimal design [@flaherty2005robustdesign; @zhang2000value]. A very thorough literature survey of different active learning algorithms has been done by Settles [-@settlestr09]. AL algorithms that are consistent and have provable label complexity have been proposed for the agnostic setting for the 0-1 loss in recent years [@dasgupta2007general; @beygelzimer2009importance]. The IWAL framework introduced in Beygelzimer et al. [-@beygelzimer2009importance] was the first AL algorithm with guarantees for general loss functions. However the authors were unable to provide non-trivial label complexity guarantees for the hinge loss, and the squared loss.
UPAL at least for squared losses can be seen as using a QBC based querying strategy where the committee is the entire hypothesis space, and the disagreement among the committee members is calculated using an exponential weighting scheme. However unlike previously proposed committees our committee is an infinite set, and the choice of the point to be queried is randomized.
Experimental results {#sec:expts}
====================
We implemented UPAL, along with the standard passive learning (PL) algorithm, and a variant of UPAL called RAL (in short for random active learning), all using logistic loss, in matlab. The choice of logistic loss was motivated by the fact that BMAL was designed for logistic loss. Our matlab codes were vectorized to the maximum possible extent so as to be as efficient as possible. RAL is similar to UPAL, but in each round samples a point uniformly at random from the currently unqueried pool. However it does not use importance weights to calculate an estimate of the risk of the classifier. The purpose of implementing RAL was to demonstrate the potential effect of using unbiased estimators, and to check if the strategy of randomly querying points helps in active learning.
We also implemented a batch mode active learning algorithm introduced by Hoi et al. [-@hoi2006batch] which, we shall call as BMAL. Hoi et al. in their paper showed superior empirical performance of BMAL over other competing pool based active learning algorithms, and this is the primary motivation for choosing BMAL as a competitor pool AL algorithm in this paper. BMAL like UPAL also proceeds in rounds and in each iteration selects $k$ examples by minimizing the Fisher information ratio between the current unqueried pool and the queried pool. However a point once queried by BMAL is never requeried. In order to tackle the high computational complexity of optimally choosing a set of $k$ points in each round, the authors suggested a monotonic submodular approximation to the original Fisher ratio objective, which is then optimized by a greedy algorithm. At the start of round $t+1$ when, BMAL has already queried $t$ points in the previous rounds, in order to decide which point to query next, BMAL has to calculate for each potential new query a dot product with all the remaining unqueried points. Such a calculation when done for all possible potential new queries takes $O(n^2t)$ time. Hence if our budget is $B$, then the total computational complexity of BMAL is $O(n^2B^2)$. Note that this calculation does not take into account the complexity of solving an optimization problem in each round after having queried a point. In order to further reduce the computational complexity of BMAL in each round we further restrict our search, for the next query, to a small subsample of the current set of unqueried points. We set the value of $p_{\text{min}}$ in step 3 of algorithm 1 to $\frac{1}{nt}$. In order to avoid numerical problems we implemented a regularized version of UPAL where the term $\lambda||w||^2$ was added to the optimization problem shown in step 11 of Algorithm 1. The value of $\lambda$ is allowed to change as per the current importance weight of the pool. The optimal value of $C$ in VW [^1] was chosen via a 5 fold cross-validation, and by eyeballing for the value of $C$ that gave the best cost-accuracy trade-off. We ran all our experiments on the MNIST dataset(3 Vs 5) [^2], and datasets from UCI repository namely Statlog, Abalone, Whitewine. Figure \[fig:expt\_results\] shows the performance of all the algorithms on the first 300 queried points.
Sample size
------------- ------ ------- -------- -------
Time Error Time Error
1200 65 7.27 60 5.67
2400 100 6.25 152 6.05
4800 159 6.83 295 6.25
10000 478 5.85 643.17 5.85
: \[tab:fixed\_B\]
Budget Speedup
-------- ------ ------- --------- ------- -----
Time Error Time Error
500 859 5.79 1973 5.33 2.3
1000 1919 6.43 7505 5.70 3.9
2000 4676 5.82 32186 5.59 6.9
: \[tab:fixed\_n\]
On the MNIST dataset, on an average, the performance of BMAL is very similar to UPAL, and there is a noticeable gap in the performance of BMAL and UPAL over PL, VW and RAL. Similar results were also seen in the case of Statlog dataset, though towards the end the performance of UPAL slightly worsens when compared to BMAL. However UPAL is still better than PL, VW, and RAL.
Active learning is not always helpful and the success story of AL depends on the match between the marginal distribution and the hypothesis class. This is clearly reflected in Abalone where the performance of PL is better than UPAL atleast in the initial stages and is never significantly worse. UPAL is uniformly better than BMAL, though the difference in error rates is not significant. However the performance of RAL, VW are significantly worse. Similar results were also seen in the case of Whitewine dataset, where PL outperforms all AL algorithms. UPAL is better than BMAL most of the times. Even here one can witness a huge gap in the performance of VW and RAL over PL, BMAL and UPAL.
One can conclude that VW though is computationally efficient has higher error rate for the same number of queries. The uniformly poor performance of RAL signifies that querying uniformly at random does not help. On the whole UPAL and BMAL perform equally well, and we show via our next set of experiments that UPAL has significantly better scalability, especially when one has a relatively large budget $B$.
Scalability results
-------------------
Each round of UPAL takes $O(n)$ plus the time to solve the optimization problem shown in step 11 in Algorithm 1. A similar optimization problem is also solved in the BMAL problem. If the cost of solving this optimization problem in step $t$ is $c_{opt,t}$, then the complexity of UPAL is $O(nT+\sum_{t=1}^T c_{opt,t})$. While BMAL takes $O(n^2B^2+\sum_{t=1}^Tc'_{t,opt})$ where $c'_{t,opt}$ is the complexity of solving the optimization problem in BMAL in round $t$. For the approximate implementation of BMAL that we described if the subsample size is $|S|$, then the complexity is $O(|S|^2B^2+\sum_{t=1}^Tc'_{t,opt})$.
In our first set of experiments we fix the budget $B$ to 300, and calculate the test error and the combined training and testing time of both BMAL and UPAL for varying sizes of the training set. All the experiments were performed on the MNIST dataset. Table \[tab:fixed\_B\] shows that with increasing sample size UPAL tends to be more efficient than BMAL, though the gain in speed that we observed was at most a factor of 1.8.
In the second set of scalability experiments we fixed the training set size to 10000, and studied the effect of increasing budget. We found out that with increasing budget size the speedup of UPAL over BMAL increases. In particular when the *budget was 2000, UPAL is arpproximately 7 times faster than BMAL.* All our experiments were run on a dual core machine with 3 GB memory.
Conclusions and Discussion
==========================
In this paper we proposed the first unbiased pool based active learning algorithm, and showed its good empirical performance and its ability to scale both with higher budget constraints and large dataset sizes. Theoretically we proved that when the true hypothesis is a linear hypothesis, we are able to recover it with high probability. In our view an important extension of this work would be to establish tighter bounds on the excess risk. It should be possible to provide upper bounds on the excess risk in expectation which are much sharper than our current high probability bounds. Another theoretically interesting question is to calculate how many unique queries are made after $T$ rounds of UPAL. This problem is similar to calculating the number of non-empty bins in the balls-and-bins model commonly used in the field of randomized algorithms [@motwani1995ra], when there are $n$ bins and $T$ balls, with the different points in the pool being the bins, and the process of throwing a ball in each round being equivalent to querying a point in each round. However since each round is, unlike standard balls-and-bins, dependent on the previous round we expect the analysis to be more involved than a standard balls-and-bins analysis.
Some results from random matrix theory
======================================
\[thm:quadratic\](Quadratic forms of subgaussian random vectors [@litvak2005smallest; @hsu2011analysis]) Let $A\in \bbR^{m\times n}$ be a matrix, and $H{\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}AA^T$, and $r=(r_1,\ldots,r_n)$ be a random vector such that for some $\sigma\geq0$, $$\bbE[\exp(\alpha^Tr)]\leq \exp\left(\frac{||\alpha||^2\sigma^2}{2}\right)$$ for all $\alpha\in \bbR^n$ almost surely. For all $\delta\in(0,1)$, $$\bbP~\left[||Ar||^2>\sigma^2\operatorname{tr}(H)+2\sigma^2\sqrt{\operatorname{tr}(H^2)}\ln(1/\delta)+2\sigma^2||H||\ln(1/\delta)\right]\leq \delta.$$
The above theorem was first proved without explicit constants by Litvak et al. [@litvak2005smallest] Hsu et al [@hsu2011analysis] established a version of the above theorem with explicit constants.
\[thm:litvak\](Eigenvalue bounds of a sum of rank-1 matrices) Let $r_1,\ldots r_n$ be random vectors in $\bbR^d$ such that, for some $\gamma>0$, $$\begin{aligned}
\bbE[r_ir_i^T|r_1,\ldots,r_{i-1}]&=I\\
\bbE[\exp(\alpha^Tr_i)|r_1,\ldots,r_{i-1}]&\leq \exp(||\alpha||^2\gamma/2) ~\forall \alpha \in \bbR^d.
\end{aligned}$$ For all $\delta \in (0,1)$, $$\bbP\left[\lambdamax\left(\frac{1}{n}\sum_{i=1}^n r_ir_i^T\right)>1+2\epsilon_{\delta,n} \vee \lambdamin\left(\frac{1}{n}\sum_{i=1}^n r_ir_i^T\right)<1-2\epsilon_{\delta,n}\right]\leq \delta,$$ where $$\epsilon_{\delta,n}=\gamma\left(\sqrt{\frac{32(d~\ln(5)+\ln(2/\delta))}{n}}+\frac{2(d\ln(5)+\ln(2/\delta))}{n}\right).$$
We shall use the above theorem in Lemma \[lem:inv\_sigh\], and lemma \[lem:J\].
\[thm:mat\_bern\](Matrix Bernstein bound) Let $X_1\ldots,X_n$ be symmetric valued random matrices. Suppose there exist $\bar{b},\bar{\sigma}$ such that for all $i=1,\ldots,n$ $$\begin{aligned}
\bbE_i[X_i]&=0\\
\lambdamax(X_i)&\leq \bar{b}\\
\lambdamax\left(\frac{1}{n}\sum_{i=1}^n \bbE_{i}[X_i^2]\right)&\leq \bar{\sigma}^2.\end{aligned}$$ almost surely, then $$\begin{aligned}
\bbP\left[\lambdamax\left(\frac{1}{n}\sum_{i=1}^n X_i\right)> \sqrt{\frac{2\bar{\sigma}^2\ln(d/\delta)}{n}}+\frac{\bar{b}\ln(d/\delta)}{3n}\right]\leq \delta.\end{aligned}$$ A dimension free version of the above inequality was proved in Hsu et al [@hsu2011dimension]. Such dimension free inequalities are especially useful in infinite dimension spaces. Since we are working in finite dimension spaces, we shall stick to the non-dimension free version.
[@shamir2011variant]\[lem:modazuma\] Let $(Z_1,\cF_1),\ldots,(Z_T,\cF_T)$ be a martingale difference sequence, and suppose there are constants $b\geq 1,c_t>0$ such that for any $t$ and any $a>0$, $$\max\{\bbP[Z_t\geq a|\cF_{t-1}],\bbP[Z_t\leq -a|\cF_{t-1}]\}\leq b\exp(-c_ta^2).$$ Then for any $\delta>0$, with probability atleast $1-\delta$ we have $$\frac{1}{T}\sum_{t=1}^T Z_t\leq \sqrt{\frac{28b\ln(1/\delta)}{\sum_{t=1}^Tc_t}}.$$
The above result was first proved by Shamir [@shamir2011variant]. Shamir proved the result for the case when $c_1=\ldots=c_{T}$. Essentially one can use the same proof with obvious changes to get the above result.
[see @cesa2006prediction page 359] Let $X$ be a random variable with $a\leq X\leq b$. Then for any $s\in\bbR$ $$\bbE[\exp(sX)]\leq \exp\left(s\bbE[X]+\frac{s^2(b-a)^2}{8}\right).$$
Let $A, B$ be positive semidefinite matrices. Then $$\lambdamax(A)+\lambdamin(B)\leq \lambdamax(A+B)\leq \lambdamax(A)+\lambdamax(B).$$ The above inequalities are called as Weyl’s inequalities [see @horn90matrix chap. 3]
[^1]: The parameters initial\_t, $l$ were set to a default value of 10 for all of our experiments.
[^2]: The dataset can be obtained from <http://cs.nyu.edu/~roweis/data.html>. We first performed PCA to reduce the dimensions to 25 from 784.
|
---
abstract: 'We present a consistent self-contained and pedagogical review of the CMB Gibbs sampler, focusing on computational methods and code design. We provide an easy-to-use CMB Gibbs sampler named `SLAVE` developed in C++ using object-oriented design. While discussing why the need for a Gibbs sampler is evident and what the Gibbs sampler can be used for in a cosmological context, we review in detail the analytical expressions for the conditional probability densities and discuss the problems of galactic foreground removal and anisotropic noise. Having demonstrated that `SLAVE` is a working, usable CMB Gibbs sampler, we present the algorithm for white noise level estimation. We then give a short guide on operating `SLAVE` before introducing the post-processing utilities for obtaining the best-fit power spectrum using the Blackwell-Rao estimator.'
author:
- 'Nicolaas E. Groeneboom'
title: 'A self-contained guide to the CMB Gibbs sampler'
---
Introduction {#sec:introduction}
============
In recent years, increased resolution in the measurement of the cosmic microwave background (CMB) have driven the need for more accurate data analysis techniques. During the early years of CMB experiments, data was so sparse and noise levels so high that error bars in general overshadowed the observed signal. With the COBE experiment, [@smoot:1992] posteriors were mapped out by brute force, and the statistical methods employed were simplistic. This was sufficient, as advanced statistical methods weren’t needed for analyzing crude data. However, all this changed with the Wilkinson Microwave Anisotropy Probe (WMAP) experiment [@bennett:2003; @hinshaw:2007]. Suddenly, cosmological data became much more detailed, vastly improving our knowledge of the universe, but also introduced new problems. Which parts of the signal were pure CMB, and which were not? The need for knowledge about instrumental noise, point sources, dust emission, synchrotron radiation and other contaminations were required in order to estimate the pure CMB signal from the data. And, how does one properly deal with the the sky cut, the contamination from our galaxy? Even harder, how does one maximize the probability that the resulting signal really is the correct CMB signal? A new era of cosmological statistics emerged.
An important event was the introduction of Bayesian statistics in cosmological data analysis. Bayesian statistics differs from the frequentist thought by quantizing ignorance: what one knows and not knows are intrinsic parts of the analysis. The goal of any Bayesian analysis is to go from the prior $P(\theta)$, or what is known about the model, to the posterior $P(\theta|\textrm{data})$, the probability of a model given data. This is summarized via Bayes’ famous theorem: $$\label{eq:bayes}
P(\theta | \textrm{data}) = \frac{P(\textrm{data} | \theta) P(\theta)}{P(\textrm{data})}.$$ The posterior $P(\theta | \textrm{data})$ tells us something about how well a model $\theta$ fits the data, and is obtained by multiplying the prior $P(\theta)$, our assumption of the model, with the likelihood $P(\textrm{data} | \theta)$, the probability that the data fits the model.
The need for Bayesian statistics becomes evident when considering that we only have data from one single experiment to analyze. Bayesian statistics merges with frequentist statistics for large number of samples. And, in a cosmological context, we are stuck with only one sample, a sample that we are constantly measuring to higher accuracies. This sample is one realization of the underlying universe model, and we are unable to obtain data from another sample. In a standard Metropolis-Hastings (MH) Monte Carlo Markov chain-approach (MCMC), one samples from the joint distribution by letting chains of “random walkers” transverse the parameter space. The posterior is obtained by calculating the normalized histogram of all the samples in the chains. The posterior will eventually resemble the underlying joint distribution, or the likelihood surface. This is a simple and easy-to-understand approach, but not without drawbacks. For one, each MH step is required to test the likelihood value of the chain at the current position in parameter space up against a new proposed position. Many of these steps will be rejected, and this is where the computational costs usually reside. The Gibbs sampler provides something new: one never needs to reject samples, and every move becomes accepted and usable for building the posterior. This is done by assuming that we have prior knowledge of the conditional distributions. These are then sampled from, each in turn yielding accepted steps.
However, the main motivation for introducing the CMB Gibbs sampler is the drastically improvement in scaling. With conventional MCMC methods, one needs to sample from the joint distribution, which results in an $\mathcal O(n^3)$ operation. For a white noise case, the Gibbs sampler splits the sampling process into independently sampling from the two conditional distributions, which together yields a $\mathcal O(n^{1.5})$ operation. In other words, the Gibbs sampler enables sampling the high-$\ell$ regime much more effective than previous MCMC methods.
The problem of estimating the cosmological signal $\mathbf s$ from the full signal by Gibbs sampling was first addressed in @jewell:2004, @wandelt:2004 and @eriksen:2004b. The ultimate goal of the Gibbs sampler is to estimate the CMB signal $s$ from the data $d$, eliminating noise $n$, convolution $A$, all while including the sky cut. Today, a great number of papers have employed the Gibbs sampler since the introduction of the method [@eriksen:2008a; @eriksen:2008b; @Dunkley:2008; @cumberbatch:2009; @groeneboom:2008b; @groeneboom:2009a; @eriksen:2006; @rudjord:2009; @jewell:2009; @dickinson:2007; @chu:2005b; @dickinson:2009; @larson:2007].
In this paper, we review the basics of the CMB Gibbs sampler, and provide a simple, intuitive non-parallelized CMB Gibbs software bundle named `SLAVE`. `SLAVE` is written in C++, and employs object-oriented design in order to simplify mathematical implementation. The OOP design of `SLAVE` is presented in figure \[fig:diagram\]. For instance, assuming $A, B$ and $C$ are instances of the “real alm” class (they contain a set of real $a_{\ell m}$s), operator overloading enables us to directly translate the expression $A = (B+C)^{-1}$ by writing
A = (B+C).Invert();
This yields fast code that closely resembles equations, without having optimized too much for parallel computing, multiple data sets and other complexities.
The Master algorithm {#sec:master}
--------------------
One method of likelihood-estimator for obtaining the best-fit power spectrum for masked CMB data is given by the `MASTER` algorithm [@hivon:2002]. While Gibbs sampling estimates the full CMB signal $s$, the `MASTER` method only estimates the power spectrum. This method does not allow for variations in the estimated signal, except for the natural variations from simulating different realizations from the same power spectrum. However, the master algorithm estimates the power spectrum with cost scaling as $\mathcal O(n^3)$, which is slow for high-$\ell$ operations.
What do I need the CMB Gibbs sampler for?
-----------------------------------------
Often, people misunderstand the concepts behind the CMB Gibbs sampler, and what the Gibbs sampler can be used for. In this section, we try to explain in simple terms when you should consider employing the CMB Gibbs sampler.
Assume that you have a theoretical universe model $M(\theta)$, where $\theta = \{\theta_i\}$ is a set of cosmological parameters. This model might give rise to some additional gaussian effects in the CMB map, either as fluctuations, altered power, anisotropic contributions, dipoles, ring structures or whatever. You now wish to test whether existing CMB data contains traces of your fabulous new model, and how significant those traces are. Or maybe you are just interested in ruling out the possibility that this model could be observed at all. In any case, you need to implement some sort of numerical library that generates CMB maps based on your model. These maps will be “pure”, in the sense that you have complete control over its generation process and systematics. Assume that your model has 1 free parameter. You could now loop over the 1-dimensional parameter space and calculate the $\chi^2$ between a pure CMB signal map and the map from your model. This would have to be done for each step in parameter space, before obtaining the minimum. Even better, you could implement a Monte Carlo Markov chain framework, letting random walkers traverse a likelihood surface, yielding posteriors. This would enable support for a larger number of parameters, and is superior to the slow brute force approach.
In real-life however, things are not this simple. Data from any CMB experiment is contaminated by noise and foregrounds, most notably our own galaxy. This means that estimating the signal $s$ from the data is not trivial - one needs to “rebuild”, or make an assumption of what the fluctuations are within the sky cut and noise limits. This implies that it really isn’t possible to obtain “the correct” CMB map, all we can know is that there exist a statistical range of validity where a simulated map agrees with the true CMB signal. Therefore, the consideration that that the estimated CMB signal $s$ is a statistical random variable and not a fixed map should be included in the analysis. Hence, if you have implemented the `MASTER` method mentioned in section \[sec:master\], you should test your model map against a set of realizations from the `MASTER`-estimated signal power spectrum.
This is where the Gibbs sampler enters the stage. As previously mentioned, the Gibbs sampler will estimate the CMB signal given data, and not only the power spectrum. The Gibbs sampler also ensures that every step in parameter space is always valid, so one never needs to discard samples. And even better, each of these independent steps provide an operation cost for obtaining samples that are much lower than more conventional MCMC methods. In order to test whether your model $m$ fits the data, you therefore include the uncertainty in data by varying the signal. For example:
initialize Cl
do
s = the CMB signal given the
power spectrum Cl
m = the CMB signal of your model given
the estimated CMB signal s
Cl = the CMB power spectrum given m
save s, m and Cl
repeat until convergence
In the end, you calculate the statistical properties of s, m and Cl. Your model parameters have now been estimated, and the process included the intrinsic uncertainties in the signal. This method is not the most rapid - but it will always yield correct results.
The CMB Gibbs sampler
=====================
Throughout this paper, we assume that the data can be expressed as $$d = As +n$$ where $s$ is the CMB signal, $A$ the instrument beam and $n$ uncorrelated noise.
The `MASTER` algorithm estimates the the power spectrum $\langle \hat C_\ell \rangle$ and the standard deviation $\Delta C_\ell$. However, this method is a approximation to a full likelihood that can be expressed as follows: $$P(C_\ell | d) = \frac{1}{\sqrt{|S+N|}} e^{-\frac{1}{2}d^T(S+N)^{-1}d}.
\label{eq:fulldist}$$ where $S$ and $N$ are the signal and noise covariance matrices, respectively. While it is fully possible to use MCMC-methods to sample from this distribution, the calculation of the $(S+N)^{-1}$-matrix scales as $n^3$, where $n$ is the size of the $n \times n$ matrix. This is therefore an extremely slow operation, and is not feasible for large $\ell$s. If we demand that we sample the sky signal $s$ as well, the joint distribution becomes $P(C_\ell, s | d)$. This might seem unnecessary complicated, as one most of the time doesn’t need the signal $s$. But when feeding this distribution through the Gibbs sampler - that is, calculating the conditional distributions $P(C_\ell | s,d) $ and $P(s | C_\ell, d)$, we find that sampling from both are computationally faster than sampling from the full distribution in equation \[eq:fulldist\]. The derivations of the conditional distributions are presented in section \[sec:methods\].
Review of the Metropolis-Hastings algorithm
-------------------------------------------
The Gibbs sampler is a special case of the Metropolis-Hastings algorithm. We therefore review the basics of Monte Carlo Markov (MCMC) chain methods. The Metropolis-Hastings algorithm is a MCMC method for sampling directly from a probability distribution. This is done by letting “random walkers” transverse a parameter space, guided by the likelihood function, the probability that the data fits the model for the given parameter configuration. If a proposal step yields a likelihood greater than the current likelihood, then random walker accepts the step immediately. If the likelihood is less, then the walker will with a certain probability step “down” the likelihood surface. Eventually, the histogram of all the random walkers will converge to the posterior, the full underlying distribution.
Assume you have a model with $n$ parameters, $\theta = \{\theta_k\}$ and you wish to map out a joint distribution from $P(\theta)$. Usually, one calculates the ratio $R$ between the posteriors at the two steps $P(\theta^{i+1})$ and $P(\theta^{i})$, such that $$\label{eq:asymmprop}
R = \frac{P(\theta^{i+1})}{P(\theta^{i})} \cdot
\frac{T(\theta^{i} | \theta^{i+1})}{T(\theta^{i+1} | \theta^{i})}$$ where $T(\theta^{i} | \theta^{i+1})$ is the proposal distribution for going left or right. If the proposal distribution is symmetric (i.e. the probability of going left-right is equal for all $\theta_k$), then $T(\theta^{i} | \theta^{i+1}) = T(\theta^{i+1} | \theta^{i})$ such that: $$R = \frac{P(\theta^{i+1})}{P(\theta^{i})}$$ The MH acceptance rule now states: if $R$ is larger than 1, accepted the step unconditionally. If $R>1$, then accept the step if a random uniform variable $x =
U(0,1) < R$.
Review of the Gibbs algorithm {#sec:method_gibbs}
-----------------------------
Assume you have a model with two parameters, $\theta_1$ and $\theta_2$, and you wish to map out a joint distribution from $P(\theta_1, \theta_2)$. Now, also presume that you have prior knowledge of the conditional distributions, $P(\theta_1 | \theta_2)$ and $P(\theta_2 | \theta_1)$. A general proposal density is not necessary symmetric, and one must therefore consider the asymmetric proposal term as described in equation \[eq:asymmprop\]. However, we now define the proposal density $T$ for $\theta_2$ to be the conditional distributions: $$T(\theta^{i+1}_1, \theta_2^{i+1} | \theta^i_1, \theta^i_2) =
\delta(\theta^{i+1}_1 - \theta^{i}_1) P(\theta^{i+1}_2 | \theta^i_1).
\label{eq:prop}$$ In words, the proposal is only considered when $\theta^{i+1}_1 =
\theta^{i}_1$, which means that $\theta_1$ is fixed while $\theta_2$ can vary. If so, the acceptance is then given as the conditional distribution $P(\theta^{i+1}_2 | \theta^i_1)$, which we must have prior knowledge of. The reason for choosing such a proposal density becomes clear when investigating the Metropolis Hastings acceptance rate: $$R = \frac{P(\theta^{i+1}_2 , \theta^{i+1}_1 )}{P(\theta^{i}_2 , \theta^i_1)} \cdot
\frac{T(\theta^{i}_1,\theta^{i}_2 | \theta^{i+1}_1,\theta^{i+1}_2 )}
{T(\theta^{i+1}_1,\theta^{i+1}_2 | \theta^{i}_1,\theta^{i}_2)}$$ Using the conditional sampling proposal (\[eq:prop\]) one obtains $$R = \frac{P(\theta^{i+1}_2 | \theta^{i+1}_1 ) P(\theta^{i+1}_1)}{P(\theta^{i}_2 | \theta^i_1)P(\theta^i_1)} \cdot
\frac{P(\theta^i_2 | \theta^{i+1}_1)}{P(\theta_2^{i+1} | \theta_1^i)}
\frac{\delta}{\delta}$$ We now enforce the delta-function such that $\theta_1^{i+1} =
\theta_1^i$. This sampling from the conditional distributions is the crucial step in the Gibbs sampler, such that all terms cancel out: $$R = 1.$$ This implies that all steps are valid, and none are ever rejected. Hence one alternates between sampling from the known conditional distributions, where each step is independently accepted and can be performed as many times as needed.
\[sampling\]
The conditional distributions {#sec:methods}
=============================
In section \[sec:method\_gibbs\], it was explained how the Gibbs sampler requires previous knowledge about the underlying conditional distributions. The CMB Gibbs sampler will alternate between sampling power spectra $C_\ell$ and CMB signal $s$, where each proposed step will always be valid. In order to enable sampling from the joint distribution, we therefore need to derive the analytical properties of the conditional distributions: $$P(C_\ell |s, d) \hspace{7mm} \textrm{and} \hspace{7mm} P(s | C_\ell, d).$$ The derivations described here were first presented in @jewell:2004, @wandelt:2004 and @eriksen:2004b.The full, joint distribution is expressed as $$\begin{aligned}
P(C_\ell, s | d) & \propto & P(d|C_\ell, s)P(C_\ell, s) \\
& = & P(d|C_\ell,s )P(s | C_\ell)P(C_\ell)\end{aligned}$$ where $P(C_\ell)$ is a prior on $C_\ell$, typically chosen to be flat. The first term, $-2\ln P(d|C_\ell,s )$, is nothing but the $\chi^2$. The $\chi^2$ measures the goodness-of-fit between model and data, leaving only fluctuations in noise. As $n = d-s$ is distributed accordingly to a Gaussian with mean 0 and variance $N$, we find that $$P(d | C_\ell, s) \propto
e^{-\frac{1}{2}(d-s)^tN^{-1}(d-s)}.
\label{eq:chisq}$$
As we now assume that the signal $s$ is known and fixed, the data $d$ becomes redundant and $P(C_\ell | s,d ) = P(C_\ell| s) \propto P(s | C_\ell)$. We therefore first need to obtain an expression for $P(C_\ell | s, d)$.
Deriving $P(C_\ell | s, d)$
---------------------------
Assuming that the CMB map consists of Gaussian fluctuations, we can express the conditional probability density for a power spectrum $C_\ell$ given a sky signal $s$ as follows: $$P(C_\ell | s, d) = \frac{e^{-\frac{1}{2}s^TC^{-1}s}}{\sqrt{|C|}}
\label{eq:pscl}$$ where $C=C(C_\ell)$ is the covariance matrix. We now perform a transformation to spherical harmonics space, where $s = \sum_{\ell m}a_{\ell m}Y_{lm}$ and $C_{ij} = \sum_i \sum_j Y_{\ell' m'}^iC_{\ell 'm',\ell m}Y_{\ell m}^j$. Then equation (\[eq:pscl\]) transforms to $$s^TC^{-1}s = \sum_{\ell m}\sum_{\ell' m'} a_{\ell m}^*Y_{\ell m}^* Y_{\ell' m'}C^{-1} Y_{\ell m}^* Y_{\ell' m'} a_{\ell' m'}.$$ As the spherical harmonics are orthogonal, they all cancel out and leave delta functions for $\delta_{\ell \ell'}
\delta_{m m'}$ such that $$s^TC^{-1}s = \sum_{\ell m}a_{\ell m}^*C_\ell^{-1} a_{\ell m} =
\sum_{\ell m}a_{\ell m}^* \frac{1}{C_\ell} a_{\ell m}.$$ We now define a power spectrum $\sigma_\ell = \frac{1}{2\ell +1}\sum_m |a_{\ell m}|^2$ such that $$s^TC^{-1}s = \sum_{\ell} (2\ell+1) \frac{\sigma_l}{C_\ell}.$$
Similarly, the determinant is given as the product of the diagonal matrix $C$, which for each $\l$ has $2\ell +1$ values of $C_\ell$. The determinant is thus $|C| = \prod_\ell C_\ell^{2\ell +1}$. Expression (\[eq:pscl\]) can now be written as $$P(C_\ell |s)
= \prod_\ell \frac{e^{-\frac{(2\ell +1)}{2} \frac{\sigma_\ell}{C_\ell}}}{\sqrt{C_\ell^{2\ell +1}}}
\label{eq:invgamma}$$ which by definition means that the $C\ell$’s are distributed as an inverse Gamma function. In the computational section, we will discuss how to draw random variables from this distribution.
Deriving $P(s | C_\ell, d)$
---------------------------
Again, we begin with the full, joint distribution: $$P(C_\ell, s | d) \propto P(d|C_\ell, s)P(C_\ell| s).
\label{eq:fulls}$$ We now know from equation \[eq:invgamma\] and \[eq:chisq\] that the joint distribution can be expressed as $$P(C_\ell, s | d) \propto
e^{-\frac{1}{2}(d-s)^tN^{-1}(d-s)}
\prod_\ell \frac{e^{-\frac{2\ell+1}{2}\frac{\sigma_\ell}{C_\ell}
}}{C_\ell^{\frac{2\ell+1}{2}}}
\label{eq:fullpost}$$ omitting the prior $P(C_\ell)$. Again, note that it would be nearly impossible to sample directly from the full distribution. We now investigate what happens with equation \[eq:fullpost\] when $C_\ell$ becomes a fixed quantity. As the $C_\ell$s in the denominator vanishes, we use equation \[eq:pscl\] to obtain $$P(s | C_\ell, d) \propto e^{-\frac{1}{2}(d-s)^TN^{-1}(d-s)} e^{-\frac{1}{2}s^TC^{-1}s}.
\label{eq:scldbefore}$$
We now introduce a residual variable $r = d - s$, such that $r$ roughly consist of noise. As noise was uncorrelated, we can expect that $r$ follows a Gaussian distribution with zero mean and $N$ variance. Also, if $s$ is known, then $C_\ell$ is redundant. We complete the square, and introduce $\hat s = (S^{-1} + N^{-1})^{-1}N^{-1}d$. Equation (\[eq:scldbefore\]) can now be rewritten as $$P(s | C_\ell, d) \propto e^{-\frac{1}{2}(s-\hat s)^T(C^{-1} + N^{-1})(s-\hat s)}.
\label{eq:scomplete}$$ Hence $P(s | C_\ell, d)$ is a Gaussian distribution with mean $\hat s$ and covariance $(C^{-1} + N^{-1})^{-1}$. In the computational section, we will discuss how to draw random variables from this distribution.
Numerical implementation
========================
In its utter simplicity, the mechanics of the Gibbs sampler can be summarized as follows:
load data
initialize s and cl
loop number of chains
s = generate from p(s | cl, d)
cl = generate from p(cl | s, d)
save s and cl
end loop
We now present the computational methods for drawing from $P(s |
C_\ell, d)$ and $P(C_\ell| s, d)$.
$P(C_\ell| s, d)$
-----------------
We show that equation \[eq:invgamma\] is an inverse Gamma distribution. A general gamma-distribution is proportional to $$P_\Gamma(x; k,\theta) \propto x^{k-1}e^{-\frac{x}{\theta}}.$$ Equation \[eq:invgamma\] can be expressed as $$P(C_\ell |s) = C_\ell^{-\frac{2l +1}{2}}e^{-\beta/C\ell}$$ where $\beta = \frac{2l +1}{2}\sigma_i$. If we now perform a substitution $y = 1/C_\ell$, we see that $$P(y|s) = y^{\frac{2l +1}{2}}e^{-\beta y} \cdot y^{-2}$$ where the last term is the Jacobian. Hence $$P(y|s) = y^{\frac{2l -1}{2} -1}e^{-\beta y}$$ which is a gamma-distribution for $k = \frac{2l -1}{2}$. We now show that this particular distribution also happens to be a special case of the $\chi^2$ distribution: $$\chi(x; k) = x^{k'/2-1}e^{-\frac{x}{2}}.$$ Letting $z = 2\beta y$ and ignoring the constants, we find that $$P(z|s) = z^{k -1}e^{-z/2}$$ such that if $k' = 2k = 2l-1$, $z$ is distributed according to a $\chi^2$ distribution with $2l-1$ degrees of freedom. A random variable following such a distribution can be drawn as follows: $$z_\chi = \sum_{i=0}^{2l-1}|N_i(0,1)|^2$$ where $N_i(0,1)$ are random Gaussian variables with mean $0$ and variance $1$. Since $z = 2\beta y = 2\beta / C_\ell$, we find that $$C_\ell = (2l+1) \sigma_i /z_\chi.$$ Numerically, one can implement this as
for each l
z = 0
for i=0 to 2l-1
z = z+ rand_gauss()^2
end
C(l) = (2l+1)*sigma(l)/z
end
An example of this method can be found in the `SLAVE` libraries, within class “powerspectrum” method “draw\_gamma”.
$P(s | C_\ell, d)$
------------------
From equation \[eq:scomplete\], it is easy to see that $P(s |
C_\ell, d)$ is a Gaussian distribution with mean $\hat s$ and variance $(C^{-1} + N^{-1})^{-1}$. Instead of deriving a method for drawing a random variable from this distribution, we present the solution and show that this solution indeed has the necessary properties [@jewell:2004]. Let $$s = (C^{-1} + N^{-1})^{-1}(N^{-1}d + N^{-\frac{1}{2}}\omega_1 + C^{-\frac{1}{2}} \omega_2)
\label{eq:draws}$$ where $\omega_1$ and $\omega_2$ are independent, random $N(0,1)$ variables. We now show that the random variable $s$ indeed has mean $\hat s$ and variance $(C^{-1} + N^{-1})$. First, $$\langle s \rangle = (C^{-1} + N^{-1})^{-1}(N^{-1}\langle d\rangle +
N^{-\frac{1}{2}}\langle \omega_1 \rangle + C^{-\frac{1}{2}} \langle
\omega_2\rangle ).$$ As $\langle \omega_1\rangle = \langle \omega_2\rangle = 0$, $$\langle s \rangle = (C^{-1} + N^{-1})^{-1}N^{-1}\langle d\rangle =
\hat s$$ by definition.
The covariance is then $$\langle (s-\hat s)(s - \hat s)^T \rangle.$$ Note that in the term $s-\hat s$, we have $(C^{-1} + N^{-1})^{-1}(N^{-1}d - N^{-1}d)=0$, so we are only left with the terms with the random variables $\omega$: $$\begin{aligned}
\langle (s-\hat s)(s - \hat s)^T \rangle &= &
(C^{-1} + N^{-1})^{-2} \cdot \\
& &
\langle (N^{-\frac{1}{2}} \omega_1 + C^{-\frac{1}{2}}\omega_2 )(
\omega_1^TN^{-\frac{T}{2}} + \omega_2^T C^{-\frac{T}{2}}) \rangle\end{aligned}$$ But, as $\omega_1$ and $\omega_2$ are independently drawn from a $N(0,1)$ distribution, then $\langle \omega_i \omega_j \rangle = \delta_{ij}I$, and we end up with $$\langle (s-\hat s)(s - \hat s)^T \rangle = (C^{-1} + N^{-1})^{-1}$$ which shows that a random variable drawn using equation \[eq:draws\] has the desired properties of being drawn from $P(s|C_\ell, d)$.
Having implemented a “real alm” class in `SLAVE` with operator overloading, it is possible to directly translate equation \[eq:draws\] into code:
omega1.gaussian_draw(0, 1, rng);
omega2.gaussian_draw(0, 1, rng);
calculate_CNI();
S = CNI* (NI*D + NI.square_root()*omega1
+ CI.square_root()*omega2);
where the code has been slightly optimized: both $C^{-1}$, $N^{-1}$ and $(C^{-1} + N^{-1})^{-1}$ has been pre-calculated for efficiency. Note that this is only possible to do when assuming full-sky coverage with constant RMS noise. If the noise isn’t constant on the sky, then $N$ is a dense off-diagonal matrix, nearly impossible to calculate directly for large $\ell$. However, it is still possible to perform the calculation in pixel space, but this requires that we assume $N$ to be an operator instead of a matrix. We will address this issue in section \[sec:skycut\].
We have now presented the main simplified Gibbs-steps for calculating $P(s|C_\ell, d)$ and $P(C_\ell | s, d)$, without convolution, uniform noise and no sky cut. Sampling from these two distributions is then done alternating between the two Gibbs steps, and the chain output - $s$ and $C_\ell$ - are saved to disk during each step.
We now investigate the behavior of these fields, as each have special properties.
Field properties
----------------
Equation \[eq:draws\] can be broken into two separate parts: the Wiener filter $(C^{-1} + N^{-1})^{-1}(N^{-1}d)$ and the fluctuation map $(C^{-1} + N^{-1})^{-1} (N^{-\frac{1}{2}}\omega_1 +
C^{-\frac{1}{2}} \omega_2)$. In figure \[fig:wiener\], each of these maps are depicted. The Wiener filter map determines the fluctuations outside the sky cut - where they are heavily constrained by the known data, given cosmic variance and noise. However, within the sky cut, large-scale fluctuations are possible to pin down statistically while small-scales are repressed. The fluctuation map determines the small-scale fluctuations within the unknown sky cut, and are constrained by cosmic variance and noise effects. Outside the sky cut, the fluctuation map is constrained by the data, yielding very low small-scale fluctuations. The sum of these two parts make up the full CMB signal sample.
Verifying the sample signal: the $\chi^2$ test {#sec:chisq}
----------------------------------------------
When the signal is being sampled, it is vital to check that the input parameters/data maps are correctly set up. For instance, if you use `SLAVE` to start a large job, say, estimating the CMB signal $s$ for a $n_{pix}=512$ map, it can be very frustrating when realizing that one of the input parameters were incorrect, for instance beam convolution or noise RMS. The software will continue to run without errors, but the resulting output files will be incorrect. We therefore adopt a simple and useful method for verifying that the estimated CMB signal $s$ for each Gibbs step really is close to what one would expect.
The trick lies with the noise. As $d = As + n$, then $n =
d-As$. Uniform white noise is assumed to be $N(0,\sigma^2_{\textrm{RMS}})$-distributed, so $$N(0,1) \sim \frac{d-As}{\sigma_{\textrm{RMS}}}.$$ A $\chi^2$ distribution is nothing but a sum of squared Gaussian distributions. Hence $$\chi_{n_{\textrm{pix}}}^2 \sim \sum_{n_{\textrm{pix}}}\large(\frac{d-As}{\sigma_{\textrm{RMS}}}\large)^2$$ and the $\chi^2$ should be close to the number of pixels in the map plus minus $\sqrt{2n}$. Usually, when an incorrect parameter is used, the $\chi^2$ comes out far away from the expected value.
Calculating the $\chi^2$ is not particularly time-consuming, but it has other uses as well: the $\chi^2$ is used in the estimation of noise, as presented in section \[sec:noiseestimation\].
Convolution
-----------
A thing we did not address in the previous section was the inclusion of the instrumental beam convolution $A$. Including this in equation \[eq:draws\], we obtain $$(C^{-1} + A^TN^{-1}A)s = AN^{-1}d + AN^{-\frac{1}{2}}\omega_1 + C^{-\frac{1}{2}} \omega_2.$$ In `SLAVE`, the beam is loaded directly from a fits file, or generated as a Gaussian beam given a full width half-maximum (FWHM) range. The beam is then multiplied with the corresponding pixel window, and stored in the $a_{\ell m}$-object $A$ throughout the code.
The sky cut {#sec:skycut}
-----------
Until now, we have only assumed full-sky data sets contaminated by constant noise. However, in order to be able to investigate real data, we need to take into account both the foreground galaxy and anisotropic noise. The galaxy contributes to almost 20% of the WMAP data, and needs to be removed with a mask. This means that the usable pars of the maps becomes anisotropic, giving rise to correlations in the spherical harmonics $a_{\ell m}$s. In other words, all the previously diagonal and well-behaved matrices now have off-diagonal elements, which for large $\ell_{\textrm{max}}$ is an impossible feat to perform for dense matrices.
One way to get around these problems is to perform the calculations containing the sky cut mask in pixel space. This means that every time one needs to take into account the sky cut, one transforms from harmonic to pixel space, performs the operation including the sky cut before transforming back to harmonic space. While this operation in itself is trivial, equation \[eq:draws\] provides a few other problems: $$(C^{-1} + A^TN^{-1}A)s = AN^{-1}d + AN^{-\frac{1}{2}}\omega_1 +
C^{-\frac{1}{2}} \omega_2.
\label{eq:draws2}$$ The right-hand side can easily be calculated, letting $N^{-1}$ be an operator acting on $d$ and $\omega_1$, switching from spherical harmonics to pixel space and back. However, the left-hand side is troublesome - one cannot solve this equation explicitly. First, we need to rewrite \[eq:draws2\] a bit: $$\begin{aligned}
(1 +
C^{\frac{1}{2}}A^TN^{-1}AC^{\frac{1}{2}})(C^{-\frac{1}{2}}s) = \\
C^{\frac{1}{2}}AN^{-1}d + C^{\frac{1}{2}}AN^{-\frac{1}{2}}\omega_1 +
\omega_2 = b
\label{eq:draws3}\end{aligned}$$ The first thing one should note about equation \[eq:draws3\] is that the left-hand term is proportional to $(1 + S/N)$, where the diagonal parts are just the signal-to-noise ratios of the corresponding mode. Another nice feature about this form is that the variance of the signal is kept constant, that is, $\textrm{Var}(s)
\sim \ell^{-2}$, but $\textrm{Var}(C^{-1/2}s) \sim I$. Hence we obtain better numerical stability. In order to solve the equation $(1+S/N)x = b$, we implement a direct-from-textbook Conjugate Gradient (CG) algorithm presented on page 40 in [@Shewchuk]. The code looks like this:
b = L*( A*NI(D) + A*NI(map_work2,true)) + omega2;
MI = setup_preconditioner();
x = mult_by_A(x);
r = b - x;
d = MI*r;
r0 = r.norm_L1(r);
do {
Ad = mult_by_A(d);
alpha = r.dot(MI*r) / (d.dot(Ad));
x = x + d*alpha;
rn = r - Ad*alpha;
beta = rn.dot(MI*rn) / (r.dot(MI*r));
d = MI * rn + d*beta;
r = rn;
norm = r.norm_L1(r);
}
while (norm>r0*epsilon);
S = L*x;
C++ enables the CG algorithm to be translated almost directly from mathematical syntax to code. Here, the sky cut mask is taken into account in the $NI$-method - one only needs the mask when multiplying with the inverse noise matrix. The only other “initial condition” is the preconditioner. The preconditioner cannot affect the result, that is, it has nothing to do with the estimated signal $s$. The preconditioner only affects the number of iterations needed for the equation $Ax = b$ to be solved, and corresponds to a “best guess” of $A$. Without going into details, the standard preconditioner in `SLAVE` is proportional to $(1+S/n)$, but there exists many other suggestions for better pre-conditioners, yielding quicker convergence. See [@eriksen:2004] or [@smith:2007] for more examples.
When the CG search has completed, the signal $S$ has been obtained, including the sky cut and anisotropic noise.
Low signal-to-noise regime
--------------------------
A final thing we need to take into account is the low signal-to-noise regime. When the noise starts dominating the signal, the estimated $s$ will fluctuate wildly on small scales. In addition, the deconvolution will add to this effect, blowing up noise to extreme values. In itself, this isn’t a bad thing as we really cannot say exactly what is going in this regime, but it will affect the overall correlations between chains. In order to reduce this effect, we present a simple way to bin multipoles together on large l, reducing noise variance.
Let $N_\ell = \sigma^2_{\textrm{RMS}}4\pi/n_{pix}$ be the noise RMS in harmonic space. The variance is then given as $$Var(N_\ell) = \frac{2}{2l+1}N_l^2.$$ For a single binned set with $n$ multipoles ranging from $\ell_{\textrm{low}}$ to $\ell_{\textrm{high}}$, the average value of the power spectrum is given as $$D_\ell = \frac{1}{n}\sum_{\ell_{\textrm{low}}}^{\ell_{\textrm{high}}}C_\ell.$$ Similarly for the noise power spectrum, $$N_b = \frac{1}{n}\sum_{\ell_{\textrm{low}}}^{\ell_{\textrm{high}}}N_\ell.$$ Thus, the variance of the noise is given as $$\sigma_N^2 = \textrm{Var}(N_b) =
\frac{1}{n^2}\sum_{\ell_{\textrm{low}}}^{\ell_{\textrm{high}}} \textrm{Var}(N_\ell).$$ Obviously, $\sigma_N$ is reduced as the number of multipoles in the bin $n$ is increased. We now select bins such that the noise variance in a single bin is always less than three times the value of the angular power spectrum, or $\sigma_n<3 D_\ell$.
The only affected part of the code is where one determines $P(C_\ell |
s,d)$. Instead of generating a power spectrum $C_\ell$ given a set of $\sigma_l$, the calculation is now performed via a binning class that calculates the binned power spectrum $C_b$. That is, $$P(C_b |\sigma) = \prod_{\ell_{\textrm{low}}}^{\ell_{\textrm{high}}} (
\frac{e^{-{\frac{2l+1}{2}{\frac{\sigma_\ell}{C_b}}}}}{C_b^{{\frac{2\ell+1}{2}}}}
).$$ Absorbing the product into the exponential, this becomes $$P(C_b |\sigma) = \frac{e^{-\frac{1}{2C_b}\sum_\ell(2l+1)\sigma_\ell}}{C_b^{\frac{1}{2}\sum_\ell(2\ell+1)}}.$$ We now sample the signal with flat bins in $\ell(\ell+1)/(2\pi)$, not in $\ell$.
Generalizing the model: Noise estimation {#sec:noiseestimation}
========================================
In this section, we give a direct example of how one could extend the data model to the `SLAVE` Gibbs sampler. We derive the necessary conditional distribution, explain how this was integrated, and present some results from [@groeneboom:2009a], where a full analysis of the noise levels in the WMAP data was performed using the `SLAVE` framework.
Traditionally, the noise properties used in the Gibbs sampler [e.g., @eriksen:2004] have been assumed known to infinite precision. In this section, however, we relax this assumption, and introduce a new free parameter, $\alpha$, that scales the fiducial noise covariance matrix, $N^{\textrm{fid}}$, such that $N = \alpha
N^{\textrm{fid}}$. Thus, if there is no deviation between the assumed and real noise levels, then $\alpha$ should equal 1. The full analysis of the 5-yr WMAP data was presented in [@groeneboom:2009a], with interesting results. For the foreground-reduced 5-year WMAP sky maps, we find that the posterior means typically range between $\alpha=1.005\pm0.001$ and $\alpha=1.010\pm0.001$ depending on differencing assembly, indicating that the noise level of these maps are underestimated by 0.5-1.0%. The same problem is not observed for the uncorrected WMAP sky maps.
The full joint posterior, $P(s,C_\ell, \alpha \,|\, d)$, now includes the amplitude $\alpha$. We can rewrite this as follows: $$\label{eq:postnoise}
P(s,C_\ell, \alpha \,|\,d) = P(d \,|\,s, \alpha) \cdot P(s, C_\ell) \cdot P(\alpha)$$ where the first term is the likelihood, $$P(d \,| \,s, \alpha) = \frac{e^{-\frac{1}{2}(d-s)(\alpha N)^{-1}(d-s)
}}{\sqrt{|\alpha N|}},$$ the second term is a CMB prior, and the third term is a prior on $\alpha$. Note that the latter two are independent, given that these describe two a-priori independent objects. In this paper, we adopt a Gaussian prior centered on unity on $\alpha$, $P(\alpha) \sim
N(1,\sigma_\alpha^2)$. Typically, we choose a very loose prior, such that the posterior is completely data-driven.
The conditional distribution for $\alpha$ can now be expressed as $$P(\alpha \,|\, s,C_\ell, d) \propto
\frac{e^{-\frac{\beta}{2\alpha}}}{\alpha^{n/2}}
\cdot P(\alpha)$$ where $n=N_{\textrm{pix}}$ and $\beta = (d-s)N^{-1}(d-s)$ is the $\chi^2$. (Note that the $\chi^2$ is already calculated within the Gibbs sampler, as it is used to validate that the input noise maps and beams are within a correct range for each Gibbs iteration. Sampling from this distribution within the Gibbs sampler represent therefore a completely negligible extra computational cost.) For the Gaussian prior with unity mean and standard deviation $\sigma_\alpha$, we find that $$P( \alpha \,|\, s, C_\ell, d ) \propto
\frac{e^{-\frac{1}{2}(\frac{\beta}{\alpha} + \frac{ (\alpha-1)^2 }{
\sigma_\alpha^2 })}}{\alpha^{n/2}}
\label{eq:finally}$$
For large degrees of freedom, $n$, the inverse gamma function converges to a Gaussian distribution with mean $\mu = b/(k+1)$, where we have defined $k = n_{\textrm{pix}}/2 -1$, and variance $\sigma^2 =
b^2/((k-1)(k-1)(k-2))$. A good approximation is therefore letting $\alpha_{i+1}$ be drawn from a product of two Gaussian distributions, which itself is a Gaussian, with mean and standard deviation $$\mu = \frac{\mu_1\sigma_2^2 + \mu_2\sigma_1^2}{\sigma_1^2 + \sigma_2^2}$$ $$\sigma = \frac{\sigma_1^2\sigma_2^2}{\sigma_1^2 + \sigma_2^2}.$$
This sampling step has been implemented in `SLAVE` and we have successfully tested it on simulated maps. With $N_{\textrm{side}}=512$ and $l_{\textrm{max}}=1300$ and full sky coverage, we find $\alpha = 1.000 \pm
0.001$. The chains for the noise amplitude $\alpha$ are shown in figure \[fig:alpha\_simulated\]. Note that with such high resolution, the standard deviation on $\alpha$ is extremely low, and any deviation from the exact $\alpha =
1.0$ will be detected.
Running SLAVE
=============
In this section, we quickly review how to use `SLAVE`. For a more detailed usage, please see the `SLAVE` documentation (when the framework will be released).
`SLAVE` requires the `HEALPIX` [@gorski:2005] CXX-libraries installed. Please see the `HEALPIX` documentation on this topic. `SLAVE` is run command-line, and requires a parameter file as command-line parameter. The most important options in the parameter file are listed in table \[tab:main\].
[lll]{} `seed` & int & Initial random seed\
`verbosity` & int & Text output level (0=none)\
`healpix_dir` & string & HEALPIX home directory\
`output_sigmas` & bool & Output $\sigma_\ell$ or not\
`output_cls` & bool & Output $C_\ell$s or not\
`output_directory` & string & Output file directory\
`output_chisq` & bool & Output the $\chi^2$ or not\
`output_beam` & bool & Output the beam or not\
`output_beam_file` & string & Beam output filename\
`method` & string & Analysis type: brute force\_fullsky or CG (normal)\
`CG_convergence` & double & CG Convergence criteria (type $10^{-6}$)\
`preconditioner` & string & Pre-conditioner type: none, static or 3j\
`init_powerspectrum_power` & double & Initialized flat power spectrum value\
`init_powerspectrum_use_file` & bool & Use file instead of flat power spectrum\
`init_powerspectrum_file` & string & Initial power spectrum file\
`samples` & int & Number of Gibbs samples to produce\
`burnin` & int & Number of burn-in samples to reject\
`datasets` & int & Number of data sets (only 1 allowed yet..)\
`data_nsideN` & int & $n_{\textrm{side}}$ for data set $N = \{1,2,3,\dots\}$\
`data_mapN` & string & FITS map for data set $N = \{1,2,3,\dots\}$\
`data_rmsN` & string & FITS rms map for data set $N = \{1,2,3,\dots\}$\
`data_maskN` & string & FITS mask for data set $N = \{1,2,3,\dots\}$\
`beam_fileN` & string & FITS beam for data set $N = \{1,2,3,\dots\}$\
`lmax` & int & $\ell_{\textrm{max}}$ for the analysis\
`constant_rms` & bool & Use constant rms or not\
`constant_rms_value` & double & Value of constant rms\
`gaussian_beam` & bool & Use a Gaussian beam or not\
`gaussian_beam_fwhm` & double & Value of Gaussian beam\
`enable_noise_amplitude_sampling` & bool & Enable noise estimation or not\
`noise_sampling_sigma` & double & The noise prior sigma\
`noise_amplitude_filename` & string & Output noise filename\
`noise_alpha_init_val` & double & Initial value for $\alpha$\
`use_binning` & bool & Enable binning of power spectrum\
`binning_powerspectrum` & string & Power spectrum used for binning\
`bins_filename` & string & Text output the bins
Post-processing
---------------
After the Gibbs sampler has been cooking for a while, it is time to investigate the results. The main output of `SLAVE` are the estimated power spectra $C_\ell$’s and the signals $s$. However, as the signal is assumed to be statistically isotropic, we instead output the signal power spectra $\sigma_\ell$ defined as: $$\sigma_\ell \equiv \frac{1}{2\ell +1} \sum_{m=-\ell}^{m=\ell}
|s_{\ell m}|^2.$$ The text-files may be plotted directly through software such as `XMGRACE`, as presented in figure \[fig:result\_cls\].
In addition, `SLAVE` outputs the $\sigma_\ell$’s as a binary file for each chain. These binary files can be combined through the main post-processing software utility for `SLAVE` called `SLAVE_PROCESS`. This software will combine the binary chains into a single file, in addition to removing burn-in samples. To combine the sigmas into one file, type
slave_process 1 [no_chains] [no_samples]
[burnin] [output sigma_l file]
$C_\ell$ likelihoods
--------------------
The first important step is to verify that the output $C_\ell$s follow the desired inverse-Gamma distribution for low $\ell$, but converges to Gaussians for larger $\ell$. The `SLAVE` processing utility `SLAVE_PROCESS` can generate a set of $C_\ell$s from the $\sigma_\ell$s and output the corresponding values for a single $\ell$. It is then straight-forward to use a graphical utility such as `XMGRACE` to obtain the histogram. Such histograms are plotted together with the analytical likelihoods in figure \[fig:BR\_likelihoods\]. Note the good match between the histogram of the $C_\ell$s and the likelihoods obtained from the Blackwell-Rao estimator. The analysis for producing these plots was performed on simulated high-detail data, in order to verify the validity of the BR-estimator.
To save the cls for a specific $\ell$, type
./process 4 [sigma_l file] [l] [generate no cls]
[output textfile]
The Blackwell-Rao estimator
---------------------------
Our primary objective is obtaining the best-fit power spectrum from the estimated signal power spectra. If the $C_\ell$s were completely distributed according to a Gaussian, one would only need to select the maximum of the distribution for each $C_\ell$. However, as we saw in equation \[eq:invgamma\], this is not the case, and we need a better way to obtain the likelihood $\mathcal L(C_\ell)$ for each $\ell$.
Luckily, we can obtain an analytical expression of the likelihood for the $C_\ell$s via the Blackwell-Rao (BR) estimator, as presented in [@chu:2005]. By using prior knowledge of the distributions of the $C_\ell$s, we can build an analytical expression for the distribution for each $C_\ell$ given the signal power spectrum $\sigma_\ell$, or $P(C_\ell | \sigma_l)$.
Note that since the power spectrum only depends on the data through the signal and thus $\sigma_\ell$, then $$P(C_\ell \, | \, s,d) = P (C_\ell \,|\,s) = P(C_\ell \, | \, \sigma_\ell).$$ It is therefore possible to approximate the distribution $P(C_\ell \,
| \,d )$ as such: $$\begin{aligned}
P(C_\ell \, | \,d ) & = & \int P(C_\ell , s \, | \, d)\, ds \\
& = & \int P(C_\ell \, | \, s,d) P(s\,|\,d) \,ds \\
& = & \int P(C_\ell \, | \, \sigma_\ell) P(\sigma_\ell\,|\,d) \,D\sigma_\ell \\
& \approx & \frac{1}{N_G} \sum_{i=1}^{N_G} P(C_\ell \, | \, \sigma_\ell^i) \end{aligned}$$ where $N_G$ is the number of Gibbs samples in the chain. This method of estimating the $P(C_\ell \, | \,d )$ is called the Blackwell-Rao estimator. Now, for a Gaussian field,
$$P(C_\ell \, | \, \sigma_\ell ) \propto
\prod_{\ell=0}^\infty \frac{1}{\sigma_\ell} \Big(
\frac{\sigma_\ell}{C_\ell} \Big) e^{\frac{2\ell+1}{2}\frac{\sigma_\ell}{C_\ell}}.$$
Taking the logarithm, we obtain a nice expression $$\textrm{ln} P(C_\ell | \sigma_l) = \sum \Big( \frac{2\ell +1}{2}
\Big[-\frac{\sigma_\ell}{C_\ell} + \textrm{ln} \big( \frac{\sigma_\ell}{C_\ell}
\big) \Big] - \textrm{ln} \sigma_l \Big)$$ which is straight-forward to implement numerically. To output the BR-estimated likelihood for one $\ell$, type
./process 3 [sigma_l file] [l]
[output likelihood]
Power spectrum estimation
-------------------------
The best-fit BR-estimated power spectrum is obtained by choosing the maximum likelihood value of $C_\ell$ for each $\ell$. To do so, type
./process 2 [sigma_l file]
[output power spectrum file]
An example of a BR-estimated power spectrum can be seen in figure \[fig:BR\_powerspectrum\]. In addition, both the input-and noise power spectra are shown. Note how the BR-estimated power spectrum is exact on small scales (low $\ell$), while the convolution and noise dominated on higher scales.
Conclusions {#sec:results}
===========
We have presented a self-contained guide to a CMB Gibbs sampler, having focused on both deriving the conditional probability distributions and code design. We described in detail how one can draw samples from the conditional distributions, and saw how the Gibbs sampler is numerically superior to conventional MCMC methods, scaling as $\mathcal O(n^{1.5})$. We have also introduced a new object-oriented CMB Gibbs framework, which employs the existing `HEALPix` [@gorski:2005] C++ package. We presented a small guide to the usage of `SLAVE`, including post-processing tools and the Blackwell-Rao estimator for obtaining the likelihoods and the best-fit power spectrum. We also reviewed a new way of estimating noise levels in CMB maps, as presented in [@groeneboom:2009a]. The software package `SLAVE` will hopefully be released when it is completed during 2009, and will run on all operating systems supporting the GNU C++ compiler. Please see `http://www.irio.co.uk` for release details and information.
Nicolaas E. Groeneboom acknowledges financial support from the Research Council of Norway. Nicolaas especially wishes to thank Hans Kristian Eriksen, but also Jeffrey Jewell, Kris Gorski, Benjamin Wandelt and the whole “Gibbs team” at Jet Propulsion Laboratories (JPL) for useful discussions, comments and input. The computations presented in this paper were carried out on Titan, a cluster owned and maintained by the University of Oslo and NOTUR. We acknowledge use of the `HEALPix` [^1] software [@gorski:2005] and analysis package for deriving the results in this paper. We acknowledge the use of the Legacy Archive for Microwave Background Data Analysis (LAMBDA). Support for LAMBDA is provided by the NASA Office of Space Science.
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Wandelt, Benjamin D. and Larson, David L. and Lakshminarayanan, Arun 70,8
[^1]: http://healpix.jpl.nasa.gov
|
---
abstract: 'A strontium iodide crystal doped by europium (SrI$_2$(Eu)) was produced by using the Stockbarger growth technique. The crystal was subjected to a characterization that includes relative photoelectron output and energy resolution for $\gamma$ quanta. The intrinsic radioactivity of the SrI$_2$(Eu) crystal scintillator was tested both by using it as scintillator at sea level and by ultra-low background HPGe $\gamma$ spectrometry deep underground. The response of the SrI$_2$(Eu) detector to $\alpha$ particles ($\alpha/\beta$ ratio and pulse shape) was estimated by analysing the $^{226}$Ra internal trace contamination of the crystal. We have measured: $\alpha/\beta=0.55$ at $E_\alpha=7.7$ MeV, and no difference in the time decay of the scintillation pulses induced by $\alpha$ particles and $\gamma$ quanta. The application of the obtained results in the search for the double electron capture and electron capture with positron emission in $^{84}$Sr has been investigated at a level of sensitivity: $T_{1/2}\sim 10^{15}-10^{16}$ yr. The results of these studies demonstrate the potentiality of this material for a variety of scintillation applications, including low-level counting experiments.'
address:
- 'INFN sezione Roma “Tor Vergata”, I-00133 Rome, Italy'
- 'Dipartimento di Fisica, Università di Roma “Tor Vergata”, I-00133, Rome, Italy'
- 'INFN, Laboratori Nazionali del Gran Sasso, I-67100 Assergi (AQ), Italy'
- 'Institute for Nuclear Research, MSP 03680 Kyiv, Ukraine'
- 'Institute for Scintillation Materials, 61001, Kharkiv, Ukraine'
- 'INFN sezione Roma, I-00185 Rome, Italy'
- 'Dipartimento di Fisica, Università di Roma “La Sapienza”, I-00185 Rome, Italy'
- 'National University of Kyiv-Mohyla Academy, 04655 Kyiv, Ukraine'
author:
- 'P. Belli'
- 'R. Bernabei'
- 'R. Cerulli'
- 'F.A. Danevich'
- 'E. Galenin'
- 'A. Gektin'
- 'A. Incicchitti'
- 'V. Isaienko'
- 'V.V. Kobychev'
- 'M. Laubenstein'
- 'S.S. Nagorny'
- 'R.B. Podviyanuk'
- 'S. Tkachenko'
- 'V.I. Tretyak'
title: 'Radioactive contamination of SrI$_2$(Eu) crystal scintillator'
---
SrI$_2$(Eu) crystal scintillator ,Radioactive contamination ,Double beta decay
0.2cm
29.40.Mc Scintillation detectors ,23.40.-s $\beta$ decay; double $\beta$ decay; electron and muon capture
Introduction
============
The strontium iodide was discovered as scintillator by Hofstadter in 1968 [@Hofstadter:1968]. The interest in this material increased in the last few years because of the high light output ($> 100~000$ photons/MeV) and of the good energy resolution ($\approx3\%$ at 662 keV), recently reported in refs. [@Cherepy:2008; @Cherepy:2009a; @Loef:2009]. The main properties of SrI$_2$(Eu) crystal scintillators are presented in Table \[properties\].
------------------------------------------------ ---------------------- -----------------------------------------------------------------------
Property Value Reference
Density (g/cm$^3$) $4.5-4.6$ [@Cherepy:2008; @Loef:2009; @Alekhin:2011]
Melting point ($^\circ$C) 515 [@Cherepy:2008]
Structural type Orthorhombic [@Cherepy:2008]
Index of refraction $1.85$ [@Tan:2011]
Wavelength of emission
maximum (nm) $429-436$ [@Cherepy:2008; @Loef:2009; @Alekhin:2011]
Light yield (photons/MeV) $(68-120)\times10^3$ [@Cherepy:2008; @Loef:2009; @Glodo:2010]
Energy resolution (FWHM, %)
for 662 keV $\gamma$ of $^{137}$Cs $2.6-3.7$ [@Cherepy:2008; @Loef:2009; @Glodo:2010; @Sturm:2011; @Cherepy:2009b]
Scintillation decay time ($\mu$s)
under X ray / $\gamma$ ray excitation at 300 K $0.6-2.4$ [@Cherepy:2008; @Loef:2009; @Glodo:2010; @Alekhin:2011]
------------------------------------------------ ---------------------- -----------------------------------------------------------------------
: Properties of SrI$_2$(Eu) crystal scintillators
\[properties\]
An important advantage of SrI$_2$(Eu) in comparison to other high resolution scintillators, like for instance LaCl$_3$(Ce), LaBr$_3$(Ce), Lu$_2$SiO$_5$(Ce), LuI$_3$(Ce), is the absence of natural long-living radioactive isotopes (as $^{138}$La in lanthanum and $^{176}$Lu in lutetium). It makes SrI$_2$(Eu) scintillators promising in various applications, in particular for low counting experiments as e.g. those searching for double $\beta$ decay.
The main aim of our study was to test the internal radioactive contamination of a SrI$_2$(Eu) crystal scintillator. We have also estimated the response of the detector to $\alpha$ particles by using the data of low background measurements where events of $^{214}$Po decays (daughter of $^{226}$Ra from the $^{238}$U chain) were recorded. As a by-product of the measurements, we have derived limits on double $\beta$ processes in $^{84}$Sr.
Scintillator, measurements, results and discussion
==================================================
Development of SrI$_2$(Eu) crystal scintillators
------------------------------------------------
A single crystal of strontium iodide doped by 1.2% of Eu[^1] was grown in a quartz ampoule using the vertical Stockbarger method [@Stocke]. Anhydrous strontium iodide activated by europium was obtained by the reaction of the strontium carbonate and europium oxide with the hydroiodic acid as described in [@Hofstadter:1968]. After drying, the obtained hydrate was placed in the quartz ampoule for the crystal growth and slowly heated for five days up to $150^\circ$ C with permanent vacuum pumping. As a next step the ampoule was welded and placed in the Stockbarger growing set-up, the temperature was increased up to $538^\circ$ C. The crystal was grown with a speed of 20 mm per day as described in [@Loef:2009]. The crystal boule was cut in a dry box filled by pure nitrogen to obtain a near to cylindrical scintillator 13 mm in diameter and 11 mm length (see Fig. 1, left). The crystal was wrapped with PTFE tape and encapsulated using epoxy glue in an oxygen-free high thermal conductivity (OFHC) copper container with a quartz window, all the materials with low level of radioactive contamination. It is shown in Fig. 1, right; as one can see, the crystal scintillator is neither milky nor cracked. However, the scintillator is not of exact cylindrical shape.
Energy resolution and relative pulse amplitude
----------------------------------------------
In order to investigate its scintillation properties, the SrI$_2$(Eu) crystal scintillator was coupled to a 3" Philips XP2412 photomultiplier (PMT) with a bialkali photocathode using Dow Corning Q2-3067 optical couplant. The detector was irradiated with $\gamma$ quanta from $^{60}$Co, $^{137}$Cs, $^{207}$Bi, $^{232}$Th and $^{241}$Am $\gamma$ sources. The measurements were carried out using an ORTEC 572 spectrometric amplifier with $10~\mu$s shaping time and a peak sensitive analog-to-digital converter. Fig. 2 shows the pulse amplitude spectra measured by the SrI$_2$(Eu) scintillator with $^{60}$Co, $^{137}$Cs, $^{207}$Bi and $^{241}$Am $\gamma$ sources, respectively. The energy resolution FWHM for the 662 keV $\gamma$ line of $^{137}$Cs is 5.8%; it is worse than the best reported results (FWHM $=2.6\%-3.7\%$ at 662 keV) [@Cherepy:2008; @Loef:2009; @Glodo:2010; @Cherepy:2009b]. This fact is probably due to not enough high level of the initial purity of the used powder, to not perfect technology of the crystal production (which is under development now), to a lower amount of the Eu dopant and to a possible concentration gradient of the Eu in the crystal. Besides, some degradation of the energy resolution can be due to the irregular shape of the crystal (a clear effect of the shape of the SrI$_2$(Eu) scintillators on the energy resolution is reported in [@Sturm:2011]). Moreover, we have used a bialkali PMT, while PMTs with a super-bialkali photocathodes have been applied in the works [@Cherepy:2008; @Loef:2009; @Glodo:2010; @Sturm:2011; @Cherepy:2009b].
The energy resolution of the SrI$_2$(Eu) crystal scintillator measured in the ($60-2615$) keV energy range is presented in Fig. 3. According to [@Dorenbos:1995; @Moszynski:2003] the data were fitted (by the chi-square method; $\chi^2/n.d.f.=9.5/5=1.9$, where n.d.f. is number of degrees of freedom) by the function FWHM$(\%)=\sqrt{a+b/E_{\gamma}}$ (where $E_{\gamma}$ is the energy of the $\gamma$ quanta in keV) with parameters $a=(10\pm2)$ and $b=(14200\pm1500)$ keV. The relative pulse amplitude of the SrI$_2$(Eu) detector was found to be 87% of a commercial NaI(Tl) scintillator ($\oslash40$ mm $\times~40$ mm) (see Fig. 4).
It is rather difficult to derive a photon yield value from the measurements; indeed, one should know both the light collection and the PMT quantum efficiency for the scintillation detectors. It should be stressed that the calculations of the light collection in scintillation detectors is a rather complicated problem. In addition, we do not know an emission spectrum of our sample, while in literature there are different data on this [@Cherepy:2008; @Loef:2009; @Alekhin:2011]. Nevertheless, taking into account: i) the comparable emission spectra of SrI$_2$(Eu) and of NaI(Tl) (maximum at $429-436$ nm and at 415 nm, respectively); ii) the relatively flat behaviour of the PMT spectral sensitivity in the region $(400-440)$ nm; iii) the typical light yield of NaI(Tl) in literature: $\approx40\times10^3$ photons/MeV, we can conclude that the light yield of the sample under study is still far from the best reported $(68-120)\times 10^3$ photons/MeV [@Cherepy:2008; @Loef:2009; @Glodo:2010].
Low background measurements in scintillation mode at sea level
--------------------------------------------------------------
The radioactive contamination of the crystal was measured in the low background set-up installed at sea-level in the Institute for Nuclear Research (INR, Kyiv, Ukraine). In the set-up, a SrI$_2$(Eu) crystal scintillator was optically connected to a 3" photomultiplier tube Philips XP2412 through a high purity polystyrene light-guide ($\oslash 66\times120$ mm). The optical contact between the scintillation crystal, the light-guide and the PMT was provided by Dow Corning Q2-3067 optical couplant. The light-guide was wrapped with aluminised Mylar. The detector was surrounded by a passive shield made of OFHC copper (5-12 cm thick), and lead (5 cm thick). After the first run of measurements over 52 h, an anti-muon veto counter was installed above the set-up. The counter consists of polystyrene based plastic scintillator $50\times50\times8$ cm viewed by a low background PMT FEU-125 (Ekran Optical Systems, Russia) with a diameter of the photocathode equal to 15 cm. The anti-muon shield suppressed the background caused by cosmic rays by a factor $\approx3$ (at the energy $\approx 4$ MeV).
An event-by-event data acquisition system has recorded the pulse shape of the SrI$_2$(Eu) scintillator over a time window of 100 $\mu$s (by using a 20 MS/s 12 bit transient digitizer [@Fazzini:1998]), the arrival time of the signals (with an accuracy of 0.3 $\mu$s), and the signals amplitude by a peak sensitive analog-to-digital converter.
The energy scale and the energy resolution of the detector were determined in calibration runs by $^{60}$Co, $^{137}$Cs and $^{207}$Bi $\gamma$ ray sources. The energy resolution becomes slightly worse due to the light-guide used in the low background set-up. It can be fitted by a function: FWHM(%)$~=\sqrt{a+b/E_{\gamma}}$ with parameteres $a=(7.4\pm4.2)$ and $b=(28100\pm8000)$ keV, where $E_{\gamma}$ is the energy of the $\gamma$ quanta in keV.
A search for the fast chain $^{214}$Bi ($Q_{\beta}=3272$ keV, $T_{1/2}=19.9$ m) $\rightarrow $ $^{214}$Po ($Q_{\alpha}=7833$ keV, $T_{1/2}=164~\mu$s) $\rightarrow $ $^{210}$Pb of the $^{238}$U family was performed by analysing the double pulses (see the technique of the double pulse analysis e.g. in [@Danevich:2003; @Belli:2007]); the result of the analysis is presented in Fig. 5. The obtained energy spectra of the first and second events, as well as the time distribution between the signals can be explained by the fast $^{214}$Bi – $^{214}$Po decay sequence. Taking into account the detection efficiency in the time window $(15-77)~\mu$s (it contains 21.6% of $^{214}$Po decays), the mass of the crystal 6.6 g, the measuring time 101.52 h and the number of selected events (52), one can estimate the activity of $^{226}$Ra in the SrI$_2$(Eu) crystal as 100(14) mBq/kg.
By using the result of the double pulses analysis, we have estimated the response of the SrI$_2$(Eu) crystal scintillator to $\alpha$ particles. The quenching of the scintillation light yield can be expressed through the so called $\alpha/\beta$ ratio, which is the ratio of the position of an $\alpha$ peak in the energy scale measured with $\gamma$ quanta to the energy of the $\alpha$ particles. Considering the spectrum presented in Fig. 5(b) as given by $\alpha$ particles of $^{214}$Po with energy 7687 keV, we can estimate the $\alpha/\beta$ ratio as 0.55. A similar quenching was observed in ref. [@Sysoeva:1998] for NaI(Tl) ($\alpha/\beta=0.66$) and CsI(Tl) ($\alpha/\beta=0.67$) crystal scintillators with the $\alpha$ particles of a $^{241}$Am source with energy 5.48 MeV; in ref. [@DAMA] similar values have been measured in ultra low background NaI(Tl) for $\alpha$ trace contaminants internal to the crystals. It should be stressed that the energy resolution for the $\alpha$ peak of $^{214}$Po (FWHM$_{\alpha}~=12\%$) is worse than that expected at the energy $\approx 4.2$ MeV according to the calibration with $\gamma$ sources (FWHM$_{\gamma}~\approx3\%$). A similar effect was observed in crystal scintillators with anisotropic crystal structure, as for instance in CdWO$_4$ [@Danevich:2003] and ZnWO$_4$ [@Belli:2009].
A search for internal contamination of the crystal by $^{228}$Th (daughter of $^{232}$Th) was realized with the help of the time-amplitude analysis[^2]. To determine the activity of $^{228}$Th, the following sequence of $\alpha $ decays was selected: $^{220}$Rn ($Q_\alpha $ = $6405$ keV, $T_{1/2}$ = $55.6$ s) $ \to $ $^{216}$Po ($Q_\alpha $ = $6907$ MeV, $T_{1/2}$ = $0.145$ s) $\to $ $^{212}$Pb. Assuming that the $\alpha/\beta$ ratio for the $\alpha$ particles of ($6.3-7.7$) MeV is in the range of $0.5-0.6$, the energy interval for both $\alpha$ particles was chosen as ($2.8-4.5$) MeV. The result of the selection is presented in Fig. 6.
Despite the low statistics (only 12 pairs were found), the positions of the selected events and the distribution of the time intervals between the events do not contradict the expectations for the $\alpha $ particles of the chain. Taking into account the efficiency in the time window $(0.01-0.5)$ s to select $^{216}$Po $\to $ $^{212}$Pb events (86.2%), the activity of $^{228}$Th in the crystal can be calculated as 6(2) mBq/kg.
The energy spectrum measured with the SrI$_2$(Eu) scintillator over 101.52 h is presented in Fig. 7. There is a peak in the spectrum at the energy of $(665\pm 5)$ keV, which can be explained by contaminations of the SrI$_2$(Eu) detector or/and of the set-up by $^{137}$Cs (probably as a result of pollution after the Chernobyl accident). Taking into account that our set-up is installed at sea level, a significant part of the background above 2.6 MeV (the edge of the $\gamma$ quanta energy from the natural radioactivity) can be attributed to cosmic rays.
Peculiarities in the spectrum in the energy region $(2.5-4.5)$ MeV can be explained by the decays of $\alpha$ active U/Th daughters present in the crystal as trace contamination. To estimate the activity of the $\alpha$ active nuclides from the U/Th families in the crystal, the energy spectrum was fitted in the energy interval $(1.8-4.7)$ MeV by using fourteen Gaussian functions to describe $\alpha$ peaks of $^{232}$Th, $^{228}$Th (and daughters: $^{224}$Ra, $^{220}$Rn, $^{216}$Po, $^{212}$Bi), $^{238}$U (and daughters: $^{234}$U, $^{230}$Th), $^{226}$Ra with daughters ($^{222}$Rn, $^{218}$Po, $^{214}$Po, $^{210}$Po)[^3] plus an exponential function to describe background[^4]. A fit of the spectrum is shown in the Inset of Fig. 7. The main contribution to the $\alpha$ activity in the scintillator gives an activity $(91\pm8)$ mBq/kg for the $^{226}$Ra and daughters ($^{222}$Rn, $^{218}$Po and $^{214}$Po). This estimate is in agreement with the result of the double pulse analysis ($100\pm14$; mBq/kg, see above) and with the measurements performed deep underground with ultra-low background HPGe $\gamma$ ray spectrometry (Section 2.5). Because of the low statistics and of the relatively poor energy resolution for $\alpha$ particles, we conservatively give limits on activities of $^{232}$Th, $^{238}$U and $^{210}$Po in the SrI$_2$(Eu) scintillator. The data obtained from the fit are presented in Table \[rad-cont\].
To estimate the contamination of the scintillator by $^{40}$K, $^{60}$Co, $^{90}$Sr$-^{90}$Y, $^{137}$Cs, $^{138}$La, $^{152}$Eu, $^{154}$Eu, $^{176}$Lu, $^{210}$Pb$-^{210}$Bi, their decays in the SrI$_2$(Eu) detector were simulated with the GEANT4 package [@GEANT4] and the event generator DECAY0 [@DECAY4]. The radioactive contamination of the set-up, in particular, the radioactivity of the PMT can contribute to the background, too. Therefore we have also simulated the contribution from the contamination of the PMT by $^{232}$Th, $^{238}$U (with their daughters) and $^{40}$K. An exponential function was adopted to describe the contribution of the cosmic rays in the sea level installation where these measurements were carried out.
Apart from the peak of $^{137}$Cs and the $\alpha$ peaks in the energy region $(2.5-4.3)$ MeV, there are no other peculiarities in the spectrum which could be ascribed to internal trace radioactivity. However, even in the case of $^{137}$Cs we cannot surely distinguish the contribution of internal and external contamination. Therefore, only limits on contaminations of the crystal by the possible radionuclides were set on the basis of the experimental data. With this aim the spectrum was fitted in the energy interval $(0.2-4.65)$ MeV by the model composed by the background components ($^{40}$K, U and Th in PMT, pollution of the set-up surface by $^{137}$Cs, the $\alpha$ peaks of $^{226}$Ra with daughters, and cosmic rays) plus a distribution of possible internal radioactive contamination to be estimated. The result of the fit is presented in Fig. 7 together with the main components of the background.
The summary on activities (or limits) obtained by the analysis of the experimental data accumulated at the sea level low background scintillation set-up is presented in Table \[rad-cont\].
Despite the sea level location and the modest shield of the scintillation set-up, the measurements allowed the detection of the internal contamination of the scintillator by $^{226}$Ra and $^{228}$Th; besides, we have estimated limits on activities of $^{238}$U, $^{232}$Th, $^{210}$Pb, $^{210}$Po, $^{90}$Sr. It should be stressed that these radionuclides are rather hard to analyse with the help of low background HPGe $\gamma$ spectrometry due to the absence of noticeable $\gamma$ rays. Moreover, the measurements allowed us to estimate the $\alpha/\beta$ ratio and to measure the pulse shape for $\alpha$ particles (see the next Section) by recording the pulse profiles of $^{214}$Po $\alpha$ events inside the scintillator.
Pulse shape of scintillation for $\gamma$ quanta ($\beta$ particles) and $\alpha$ particles from the sea level measurements
---------------------------------------------------------------------------------------------------------------------------
The pulse profiles of 41 $\alpha$ events of $^{214}$Po were selected with the help of the double pulse analysis from the data of the low background measurements (see Section 2.3 and Fig. 5). The sum of the pulses is presented in Fig. 8 where also the sum of approximately two thousands of background $\gamma$ ($\beta$) events with energies $\approx1.5$ MeV is drawn.
The distributions were fitted in the time interval $(0-10)~\mu$s by the following function:
$f(t)=A(e^{-t/\tau}-e^{-t/\tau _{0}})/(\tau-\tau_{0}),\qquad t>0$,
where $A$ is the intensity (in arbitrary units), and $\tau$ is the decay constant of the light emission; $\tau_{0}$ is the integration constant of the electronics ($\approx 0.08~\mu$s). The fit gives the scintillation decay times in the SrI$_2$(Eu) crystal scintillator: $\tau_{\gamma}=(1.75\pm0.01)~\mu$s and $\tau_{\alpha}=(1.73\pm0.02)~\mu$s for $\gamma$ quanta ($\beta$ particles) and $\alpha$ particles, respectively. Therefore we have not observed any clear indication on differences in the kinetics of the scintillation decay in the SrI$_2$(Eu) crystal scintillator under $\gamma$ quanta ($\beta$ particles) and $\alpha$ particles irradiation.
Measurements with ultra-low background HPGe $\gamma$ ray spectrometry deep underground
--------------------------------------------------------------------------------------
The SrI$_2$(Eu) crystal scintillator was measured for 706 h with the ultra-low background HPGe $\gamma$ ray spectrometer GeCris. The detector has a volume of 468 cm$^3$ and a 120% efficiency relatively to a 3 in. $\times$ 3 in. NaI(Tl). This detector has a rather thin Cu window of 1 mm thickness. The passive shield of the detector consists of 15 cm of OFHC copper and 20 cm of low radioactive lead. The whole set-up is sealed in an air-tight plexiglass box continuously flushed with high purity nitrogen gas to avoid the presence of residual environmental radon. The facility is located deep underground in the Gran Sasso National Laboratories of the I.N.F.N. (average overburden of 3600 m water equivalent) [@Arpesella:2002; @Laubenstein:2004]. The background data were accumulated over 1046 h (see Fig. 9).
In order to determine the radioactive contamination of the sample, the detection efficiencies were calculated using a Monte Carlo simulation based on the GEANT4 software package [@GEANT4]. The peaks in the measured spectra are due to the naturally occurring radionuclides of the uranium and thorium chains, $^{40}$K and $^{137}$Cs. We have detected contaminations by $^{137}$Cs and $^{226}$Ra (the $\gamma$ lines of $^{137}$Cs, $^{214}$Bi and $^{214}$Pb were observed) in the crystal scintillator at the level of 53(11) mBq/kg and 120(50) mBq/kg, respectively, while limits were obtained for other potential contaminations. The measured activities and the limits are presented in Table \[rad-cont\].
In addition, we have observed the 1157 keV peak of $^{44}$Sc in the data accumulated with the SrI$_2$(Eu) crystal (at a rate of $6.1(9)\times10^{-2}$ counts/h). However, this peak was due to a contamination of the used HPGe detector (not of the crystal scintillator sample) by $^{44}$Ti.
------------ ----------------------- -------------------- ------------------
Chain Nuclide
(Sub-chain) Measured in Measured by HPGe
scintillation mode
$^{40}$K $\leq 200$ $\leq 255$
$^{60}$Co $\leq 540$ $\leq 16$
$^{90}$Sr$-^{90}$Y $\leq 90$
$^{137}$Cs $\leq 140$ $53\pm11$
$^{138}$La $\leq 1100$ $\leq 20$
$^{152}$Eu $\leq 840$ $\leq 108$
$^{154}$Eu $\leq 910$ $\leq 67$
$^{176}$Lu $\leq 970$ $\leq 143$
$^{232}$Th $^{232}$Th $\leq 3$
$^{228}$Ac $\leq 68$
$^{228}$Th $6\pm 2$ $\leq 52$
$^{238}$U $^{238}$U $\leq 40$
$^{226}$Ra $100\pm14$ $120\pm50$
$^{210}$Pb$-^{210}$Bi $\leq 180$
$^{210}$Po $\leq 60$
------------ ----------------------- -------------------- ------------------
: Radioactive contamination of the SrI$_2$(Eu) scintillator. The upper limits are given at 90% C.L., and the uncertainties of the measured activities at 68% C.L.
\[rad-cont\]
The radioactive purity of the SrI$_2$(Eu) scintillator is still far from that of NaI(Tl) and CsI(Tl) scintillators (especially those developed with high radiopurity for dark matter search [@DAMA; @KIMS]). At the same time it is much better than the typical purity of LaCl$_3$(Ce), LaBr$_3$(Ce), Lu$_2$SiO$_5$(Ce) and LuI$_3$(Ce) crystal scintillators (see Table \[rc-comp\], where the radioactive contamination of the SrI$_2$(Eu) crystal is compared with that of NaI(Tl), CsI(Tl) and scintillators containing La or Lu).
------------------------------- ----------- ----------------- ----------------- ------------ --------------
Scintillator
$^{40}$K $^{138}$La $^{176}$Lu $^{226}$Ra $^{228}$Th
SrI$_2$(Eu)$^a$ $\leq255$ $\leq20$ $\leq143$ 120 $\leq11$
NaI(Tl) [@DAMA] $< 0.6$ $\sim0.02$ $\sim 0.009$
CsI(Tl) [@KIMS] 0.009 0.002
LaCl$_3$(Ce) [@Bernabei:2005] $4.1\times10^5$ $\leq 35$ $\leq0.36$
LaBr$_3$(Ce) $3.0\times10^5$
Lu$_2$SiO$_5$(Ce) $3.9\times10^7$
LuI$_3$(Ce) $1.6\times10^7$
$^a$ This work
------------------------------- ----------- ----------------- ----------------- ------------ --------------
\[rc-comp\]
Search for 2$\beta$ decay of $^{84}$Sr
======================================
The data of the low background measurements with the HPGe detector can be used to search for double $\beta$ processes in $^{84}$Sr accompanied by the emission of $\gamma$ quanta. The decay scheme of $^{84}$Sr is presented in Fig. 10. The energy of double $\beta$ decay of $^{84}$Sr is comparatively high: $Q_{2\beta}=1787(4)$ keV [@Audi:2003], however the isotopic abundance is rather low: $\delta = 0.56(1)\%$ [@Berglund:2011].
We do not observe any peaks in the spectrum accumulated with the sample of the SrI$_2$(Eu) scintillator which could indicate double $\beta$ activity of $^{84}$Sr. Therefore, only lower half-life limits ($\lim T_{1/2}$) can be set according to the formula: $\lim
T_{1/2} = N \cdot \eta \cdot t \cdot \ln 2 / \lim S$, where $N$ is the number of $^{84}$Sr nuclei in the sample, $\eta$ is the detection efficiency, $t$ is the measuring time, and $\lim S$ is the number of events of the effect searched for which can be excluded at given confidence level (C.L.; all the limits obtained in the present study are given at 90% C.L.). The efficiencies of the detector for the double $\beta$ processes in $^{84}$Sr were calculated with the GEANT4 code [@GEANT4] and DECAY0 event generator [@DECAY4].
One positron can be emitted in the $\varepsilon\beta^+$ decay of $^{84}$Sr with energy up to $(765\pm4)$ keV. The annihilation of the positron will give rise to two 511 keV $\gamma$’s leading to an extra rate in the annihilation peak. The part of the spectrum in the energy interval $(450-550)$ keV is shown in Fig. 11.
There are peculiarities in both the spectra accumulated with the SrI$_2$(Eu) sample \[$(111\pm14)$ counts at $(510.9\pm0.2)$ keV\] and in the background \[($12\pm5)$ counts at $(510.8\pm0.3)$ keV\], which can be ascribed to annihilation peaks. The main contribution to the 511 keV peak \[($108 \pm 22)$ counts\] is coming from decays of $^{44}$Sc (daughter of $^{44}$Ti) present in the HPGe detector as contamination (see Section 3.4), $(8\pm3)$ counts corresponds to the background of the detector before the contamination. The difference in the areas of the annihilation peak: ($-5\pm26$) counts, which can be attributed to electron capture with positron emission in $^{84}$Sr, gives no indication on the effect. In accordance with the Feldman-Cousins procedure [@Feldman:1998] (here and hereafter we use this approach to estimate the values of $\lim
S$ for all the processes searched for) we should take $\lim S=38$ counts which can be excluded at 90% C.L. Taking into account the number of $^{84}$Sr nuclei in the sample ($6.5\times10^{19}$) and the detection efficiency ($\eta=7.2\%$), we have calculated the following limit on the half-life of $^{84}$Sr relatively to $\varepsilon\beta^+$ decay:
$T_{1/2}^{(2\nu+0\nu)\varepsilon\beta^{+}}($g.s.$~\rightarrow~$g.s.$)\geq
6.9\times10^{15}$ yr.
We cannot study the $2\nu2K$ capture in $^{84}$Sr to the ground state of $^{84}$Kr because the energies of the expected X rays after the decay are too low in energy (the binding energy of electrons at $K$ shell of krypton atom is only 14.3 keV [@ToI98] while the energy threshold of the HPGe detector is $\approx20$ keV).
In the neutrinoless double electron capture to the ground state of the daughter nucleus, in addition to the X rays, some other particle(s) must be emitted to take away the rest of the energy. Usually one bremsstrahlung $\gamma$ quantum is assumed. The energy of the $\gamma$ quantum is expected to be equal to $E_\gamma=Q_{2\beta}-E_{b1}-E_{b2}$, where $E_{b1}$ and $E_{b2}$ are the binding energies of the first and of the second captured electrons on the atomic shell. The binding energies on the $K$, $L_1, L_2$ and $L_3$ shells in Kr are equal to $E_K=14.3$ keV, $E_{L_1}=1.9$ keV, $E_{L_2}\approx E_{L_3}=1.7$ keV, respectively [@ToI98]. Therefore, the expected energies of the $\gamma$ quanta for the $0\nu2\varepsilon$ capture in $^{84}$Sr to the ground state of $^{84}$Kr are in the intervals: i) $E_\gamma=(1754-1762)$ keV for the $0\nu 2K$; ii) $E_\gamma=(1767-1775)$ keV for the $0\nu KL$; iii) $E_\gamma=(1779-1788)$ keV for the $0\nu 2L$ process.
No events are detected (see Fig. 12, a) in the energy intervals $(1754-1762)$ and $(1779-1788)$ keV, where the g.s. $\rightarrow$ g.s. $0\nu2K$ and $0\nu2L$ decay of $^{84}$Sr is expected.
According to [@Feldman:1998] we should take 2.4 events as $\lim S$. Therefore, taking into account the detection efficiencies of the effects (4.0% and 3.9%, respectively), we can set the following limits on the processes:
$T_{1/2}^{0\nu 2K}$(g.s.$~\rightarrow~$g.s.$)\geq~6.0\times10^{16}$ yr,
$T_{1/2}^{0\nu 2L}$(g.s.$~\rightarrow~$g.s.$)\geq~5.9\times10^{16}$ yr.
There are 3 events at energy $\approx1770$ keV (due to $^{207}$Bi in the background) where the $0\nu
KL$ decay of $^{84}$Sr is expected. Taking in this case $\lim S=7.4$ counts, while the detection efficiency is 3.9%, one can obtain the following half-life limit on the $0\nu KL$ process in $^{84}$Sr:
$T_{1/2}^{0\nu KL}$(g.s.$~\rightarrow~$g.s.$)\geq~1.9\times10^{16}$ yr.
To search for the double electron capture of $^{84}$Sr to the excited level $2^+$ of $^{84}$Kr, the experimental data were fitted in the energy interval $(868-896)$ keV by a Gaussian function (to describe the gamma peak with the energy of 881.6 keV) and a polynomial function of second degree (to approximate the background, see Fig. 12, b). The fit gives an area of $S=(-1.2\pm 4.9)$ counts for the double $\beta$ process searched for, giving no evidence for the effect ($\lim S=6.9$ counts). Taking into account the detection efficiency for $\gamma$ quanta with energy 882 keV (5.8%), we set the following limit on the process:
$T_{1/2}^{2\nu 2\varepsilon}$(g.s.$~\rightarrow~881.6$ keV$)\geq 3.1\times 10^{16}$ yr.
In the neutrinoless $2\varepsilon$ capture to the $2^+$ level, two $\gamma$ quanta should be emitted. The interaction of the additional $\simeq
0.9$ MeV $\gamma$ quantum with the HPGe detector slightly decreases the efficiency for the 882 keV peak (5.0%) leading to the limit:
$T_{1/2}^{0\nu 2\varepsilon}$(g.s.$~\rightarrow~881.6$ keV$)\geq 2.6\times 10^{16}$ yr.
All the half-life limits on $2\beta$ decay processes in $^{84}$Sr, obtained in the present experiment, are summarized in Table 4. Previously, only one limit on $0\nu\varepsilon\beta^+$ mode was known; it was derived in [@DBD-tab] on the basis of the data of an old experiment with photoemulsions [@Fremlin:1952], and is two orders of magnitude lower than the one obtained in this work. It should be also noted that an experiment to search for $2\beta$ decays in $^{84}$Sr with SrCl$_2$ crystal scintillator ($\oslash 2 \times 1.5$ cm) with $4\pi$ CsI(Tl) active shielding is in progress in the Yang-Yang underground laboratory [@Roo08] which has a potential to improve the limits presented here.
Conclusions
===========
The radioactive contamination of the SrI$_2$(Eu) crystal scintillator obtained using a Stockbarger growth technique was estimated with the help of two approaches: by low background measurements in scintillation mode at sea level, and with the help of ultra-low background HPGe $\gamma$ ray spectrometry deep underground. We have found a contamination of the scintillator by $^{137}$Cs, $^{226}$Ra and $^{228}$Th on the level of 0.05 Bq/kg, 0.1 Bq/kg and 0.01 Bq/kg, respectively. Only limits were set on the contamination of the detector by $^{138}$La at level of $\leq 0.02$ Bq/kg, while the activities of $^{40}$K, $^{90}$Sr, $^{152}$Eu, $^{154}$Eu, $^{176}$Lu are below the detection limits of $(0.1-0.3)$ Bq/kg. The intrinsic radiopurity of the SrI$_2$(Eu) scintillator is still far from NaI(Tl) and CsI(Tl) scintillators developed for low counting experiments, while it is three orders of magnitude better than that of the scintillation materials containing La, and five orders of magnitude better than that of the scintillators containing Lu.
The response of the SrI$_2$(Eu) crystal scintillator to $\alpha$ particles was estimated by using the trace contamination of the crystal by $^{226}$Ra. The $\alpha/\beta$ ratio was measured as 0.55 for 7.7 MeV $\alpha$ particles of $^{214}$Po. No difference in pulse shapes of scintillation for $\gamma$ quanta and $\alpha$ particles was observed (the decay time was estimated to be: $\approx 1.7~\mu$s).
Applicability of SrI$_2$(Eu) crystal scintillators to the search for the double beta decay of $^{84}$Sr was demonstrated for the first time. New improved half-life limits were set on double electron capture and electron capture with positron emission in $^{84}$Sr at level of $T_{1/2}\sim 10^{15}-10^{16}$ yr.
The results of these studies demonstrate the possible perspective of the SrI$_2$(Eu) highly efficient scintillation material in a variety of applications, including low counting measurements.
An R&D of SrI$_2$(Eu) crystal scintillators is in progress. We are going to study radioactive contaminations of larger volume SrI$_2$(Eu) crystal scintillators both by ultra-low background HPGe $\gamma$ spectrometry and low background scintillation counting at the Gran Sasso National Laboratory.
Acknowledgments
===============
The work of the INR Kyiv group was supported in part by the Project “Kosmomikrofizyka-2” (Astroparticle Physics) of the National Academy of Sciences of Ukraine.
[99]{}
R. Hofstadter, U.S. Patent No. 3,373,279 (March 12, 1968).
N.J. Cherepy et al., Appl. Phys. Lett. 92 (2008) 083508. N.J. Cherepy et al., IEEE Trans. Nucl. Sci. 56 (2009) 873. E.V. van Loef et al., IEEE Trans. Nucl. Sci. 56 (2009) 869. M.S. Alekhin et al., IEEE Trans. Nucl. Sci. 58 (2011) 2519. H. Tan, W.K. Warburton, Nucl. Instr. Meth. A 652 (2011) 221. J. Glodo et al., IEEE Trans. Nucl. Sci. 57 (2010) 1228. B.W. Sturm et al., Nucl. Instr. Meth. A 652 (2011) 242. N.J. Cherepy et al., “SrI$_2$ scintillator for gamma ray spectroscopy”, Proceedings of SPIE – The International Society for Optical Engineering, vol. 7449 (2009) art. no. 74490F. D. C. Stockbarger, Rev. Sci. Instrum. 7 (1936) 133. P. Dorenbos, J.T.M. de Haas, C.W.E. van Eijk, IEEE Trans. Nucl. Sci. 42 (1995) 2190. M. Moszy$\acute{n}$ski, Nucl. Instr. Meth. A 505 (2003) 101. T. Fazzini et al., Nucl. Instr. Meth. A 410 (1998) 213. F.A. Danevich et al., Phys. Rev. C 67 (2003) 014310. P. Belli et al., Phys. Rev. C 76 (2007) 064603. R.B. Firestone et al., [*Table of Isotopes*]{}, 8-th ed., John Wiley, New York, 1996 and CD update, 1998. E.V. Sysoeva et al., Nucl. Instr. Meth. A 414 (1998) 274. R. Bernabei et al., Nucl. Instr. Meth. A 592 (2008) 297. P. Belli et al., Nucl. Phys. A 826 (2009) 256. F.A. Danevich et al., Phys. Lett. B 344 (1995) 72. F.A. Danevich et al., Nucl. Phys. A 694 (2001) 375. J.H. Reeves et al., IEEE Trans. Nucl. Sci. NS-31 (1984) 697. N. Kamikubota et al., Nucl. Instr. Meth. A 245 (1986) 379. C. Arpesella, Appl. Radiat. Isot. 47 (1996) 991. T. Iwawaki et al., Natural Science Research 11 (1998) 1. S. Agostinelli et al., Nucl. Instr. Meth. A 506 (2003) 250;\
J. Allison et al., IEEE Trans. Nucl. Sci. 53 (2006) 270. O.A. Ponkratenko et al., Phys. At. Nucl. 63 (2000) 1282;\
V.I. Tretyak, to be published. C. Arpesella et al., Astropart. Phys. 18 (2002) 1. M. Laubenstein et al., Appl. Radiat. Isot. 61 (2004) 167. H.S. Lee et al., Nucl. Instr. Meth. A 571 (2007) 644. M. Berglund and M.E. Wieser, Pure Appl. Chem. 83 (2011) 397. R. Bernabei et al., Nucl. Instr. Meth. A 555 (2005) 270. G. Audi, A.H. Wapstra, C. Thibault, Nucl. Phys. A 337 (2003) 729. G.J. Feldman, R.D. Cousins, Phys. Rev. D 57 (1998) 3873. V.I. Tretyak, Yu.G. Zdesenko, At. Data Nucl. Data Tables 61 (1995) 43; 80 (2002) 83. J.H. Fremlin, M.C. Walters, Proc. Phys. Soc. A 65 (1952) 911. G. Rooh et al., IEEE Trans. Nucl. Sci. 55 (2008) 1445.
[^1]: We present data on the nominal concentration of Eu in the initial powder used for the crystal growth. The concentration of Eu in the crystal may be lower due to unknown segregation of Eu in SrI$_2$. Moreover we cannot exclude some nonuniformity of Eu distribution in the crystal volume and, therefore, presence of concentration gradient of Eu.
[^2]: The method of the time-amplitude analysis is described in detail in [@Danevich:1995; @Danevich:2001].
[^3]: We assume a broken equilibrium of the U/Th chains in the scintillator.
[^4]: At sea level the energy spectrum of cosmic ray induced background in different nuclear detectors, including scintillating, has a monotonic character, which can be approximated in reasonable narrow energy regions (a few MeV) by an exponential function [@Reeves:1984; @Kamikubota:1986; @Arpesella:1996; @Iwawaki:1998].
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abstract: 'In this paper some sufficient conditions are given for when two bounded rank-one transformations are isomorphic or disjoint. For commensurate, canonically bounded rank-one transformations, isomorphism and disjointness are completely determined by simple conditions in terms of their cutting and spacer parameters. We also obtain sufficient conditions for bounded rank-one transformations to have minimal self-joinings. As an application, we give a proof of Ryzhikov’s theorem that totally ergodic, non-rigid, bounded rank-one transformations have minimal self-joinings of all orders.'
address:
- |
Department of Mathematics\
University of North Texas\
1155 Union Circle \#311430\
Denton, TX 76203\
USA
- |
Department of Mathematics\
University of Louisville\
Louisville, KY 40292\
USA
author:
- Su Gao
- Aaron Hill
title: 'Disjointness between Bounded Rank-One Transformations'
---
Introduction
============
The research in this paper is motivated by the observation of Foreman, Rudolph, and Weiss [@FRW], based on King’s Weak Closure Theorem for rank-one transformations [@King1], that the isomorphism problem for rank-one transformations is a Borel equivalence relation. Our objective has been to identify a concrete algorithm to determine when two rank-one transformations are isomorphic.
The broader context of this research is the isomorphism problem in ergodic theory, originally posed by von Neumann, that asks how to determine when two (invertible) measure-preserving transformations are isomorphic.
Recall that a [*measure-preserving transformation*]{} is an automorphism of a standard Lebesgue space. Formally, it is a quadruple $(X, \mathcal{B}, \mu, T)$, where $(X, \mathcal{B}, \mu)$ is a measure space isomorphic to the unit interval with the Lebesgue measure on all Borel sets, and $T$ is a bijection from $X$ to $X$ such that $T$ and $T^{-1}$ are both $\mu$-measurable and preserve the measure $\mu$. When the algebra of measurable sets is clear, we refer to the transformation $(X, \mathcal{B}, \mu,T)$ simply by $(X, \mu, T)$.
Two measure-preserving transformations $(X, \mathcal{B}, \mu, T)$ and $(Y, \mathcal{C}, \nu, S)$ are [*isomorphic*]{} if there is a measure isomorphism $\varphi$ from $(X, \mathcal{B}, \mu)$ to $(Y, \mathcal{C}, \nu)$ such that $\varphi\circ T=S\circ \varphi$ a.e.
Halmos and von Neumann showed that two ergodic measure-preserving transformations with pure point spectrum are isomorphic if and only if they have the same spectrum. Ornstein’s celebrated theorem states that two Bernoulli shifts are isomorphic if and only if they have the same entropy. These are successful answers to the isomorphism problem for subclasses of measure-preserving transformations. For each of them, there is a concrete algorithm, which can be carried out at least in theory, to determine when two given measure-preserving transformations are isomorphic.
Foreman, Rudolph, and Weiss [@FRW] showed that the isomorphism problem for ergodic measure-preserving transformations is a complete analytic equivalence relation, and in particular not Borel. Intuitively, this rules out the existence of a satisfactory answer to the original isomorphism problem of von Neumann. However, in the same paper they showed that the isomorphism relation becomes much simpler when restricted to the generic class of rank-one transformations. Although their method does not yield a concrete algorithm for the isomorphism problem for rank-one transformations, it gives hope that the isomorphism problem has a satisfactory solution for a generic class of measure-preserving transformations. Since rank-one transformations are given by their cutting and spacer parameters $(r_n:n\in\N)$ and $(s_n: n\in\N)$ (more details are given in the next section), a satisfactory solution to the isomorphism problem would correspond to a simple algorithm that yields a yes or no answer with these parameters as input.
In this paper we make some progress toward such a satisfactory solution. Under the assumption that the cutting and spacer parameters are bounded, we investigate the isomorphism problem and yield some conditions to guarantee isomorphism and non-isomorphism. For the class of canonically bounded rank-one transformations, we are able to give a simple algorithm to determine isomorphism when the rank-one transformations are commensurate or just eventually commensurate (Corollary \[CBmain\] and Theorem \[mainisogen\]). The basic techniques of this investigation come from the recent [@Hill] by the second author.
In addition to the isomorphism problem, we investigate in this paper a stronger notion of non-isomorphism, namely, that of disjointness between measure-preserving transformations. Two measure-preserving transformations $(X, \mu, T)$ and $(Y, \nu, S)$ are [*disjoint*]{} if $\mu\times\nu$ is the only measure on $X\times Y$ that is $T\times S$-invariant and has $\mu$ and $\nu$ as marginals. A main result of this paper (Theorem \[maindisjoint\]) gives a condition for when two bounded rank-one transformations are disjoint. For the class of canonically bounded rank-one transformations, we again yield a simple algorithm to determine when two commensurate transformations are disjoint (Corollary \[CBmain\]).
Our results on isomorphism and disjointness for canonically bounded rank-one transformations extend what was already known for a class of Chacon-like transformations. Chacon’s transformation is a prototypical example of canonically bounded rank-one transformations; it can be described by the cutting parameter that is constantly equal to 3 and the spacer parameter that is constantly equal to (0,1)–i.e., there are no spacers inserted at the first opportunity and a single spacer inserted at the second opportunity. Given any sequence $e = (e_n : n \in \N)$ of 0s and 1s, we can build a Chacon-like transformation $T_e$ as follows. The cutting parameter for the transformation will be constantly equal to 3 and the spacer parameter at stage $n$ will be $(0,1)$ if $e_n =1$ and $(1,0)$ if $e_n = 0$. Fieldsteel [@Fieldsteel] showed that transformations $T_e$ and $T_{e^\prime}$ that are constructed in this way are isomorphic iff $e$ and $e^\prime$ eventually agree, i.e., there is some $N \in \N$ such that $e_n = e^\prime_n$ for all $n \geq N$. It is an exercise in Rudolph’s book [@RudolphBook] to show that in the case that $e$ and $e^\prime$ do not eventually agree, then $T_e$ and $T_{e^\prime}$ are in fact disjoint.
The notion of canonically bounded rank-one transformations was defined in [@GaoHill] and was used in [@GaoHill1] to characterize non-rigidity for bounded rank-one transformations. In our study of disjointness of commensurate, canonically bounded rank-one transformations, the basic method follows that of del Junco, Rahe, and Swanson [@delJuncoRaheSwanson] in which they showed that Chacon’s transformation–in fact, any Chacon-like transformation–has minimal self-joinings of all orders. Further exploration of the method gives us a generalization of the theorem of del Junco, Rahe, and Swanson (Theorem \[mainMSJ\]), which gives a general condition when bounded rank-one transformations have minimal self-joinings of all orders. Applying this general condition to canonically bounded rank-one transformations, and combining the results of [@GaoHill1], we are able to obtain a proof of Ryzhikov’s theorem [@Ryzhikov] that totally ergodic, non-rigid, bounded rank-one transformations have minimal self-joinings of all orders.
The rest of the paper is organized as follows. In Section 2 we provide further background and define the basic notions used throughout the paper. In Section 3 we state and prove the main results on non-isomorphism (Theorem \[mainiso\]) and disjointness (Theorem \[maindisjoint\]), and derive a satisfactory solution to the isomorphism and disjointness problems for commensurate, canonically bounded rank-one transformations (Corollary \[CBmain\]). In Section 4 we state the prove the main result on minimal self-joinings (Theorems \[mainMSJ\] and \[allorders\]), and derive Ryzhikov’s theorem (Corollary \[Ryzh\]) from our methods. In the final section, we give some concluding remarks and explain how the main results can be generalized to the broader context of eventually commensurate constructions.
Preliminaries
=============
Throughout this paper we let $\N$ be the set of all natural numbers $0, 1, 2, \dots$. Let $\N_+$ be the set of all positive integers. Let $\Z$ be the set of all integers.
Finite sequences, finite functions, and finite words
----------------------------------------------------
Let $\mathcal{S}$ be the set of all finite sequences of natural numbers. We will introduce some operations and relations on $\mathcal{S}$. We view each element of $\mathcal{S}$ from three different perspectives, that is, as a finite sequence, as a function with a finite domain, and as a finite word. For each $s\in S$, let ${{\mbox{\rm lh}}}(s)$ denote the [*length*]{} of $s$. Let $()$ denote the unique (empty) sequence with length $0$. A nonempty sequence in $\mathcal{S}$ is of the form $s=(a_1, \dots, a_n)$ where $n={{\mbox{\rm lh}}}(s)$ and $a_1, \dots, a_n\in\N$. We also view $()$ as the unique (empty) function with the empty domain, and view each nonempty $s\in \mathcal{S}$ as a function from $\{1, \dots, {{\mbox{\rm lh}}}(s)\}$ to $\N$. In addition, we refer to each $s\in \mathcal{S}$ as a [*word*]{} of natural numbers. When $s\in S$, the different points of view give rise to different notation for $s$; for example, we have $s=(s(1), \dots, s({{\mbox{\rm lh}}}(s)))=s(1)\dots s({{\mbox{\rm lh}}}(s))$.
For $s\in\mathcal{S}$ and $k\leq l\in{{\mbox{\rm dom}}}(s)$ (i.e. $1\leq k\leq l\leq {{\mbox{\rm lh}}}(s)$), define $s\!\upharpoonright\![k,l]$ to be the unique $t\in\mathcal{S}$ with ${{\mbox{\rm lh}}}(t)=l-k+1$ such that for $1\leq i\leq {{\mbox{\rm lh}}}(t)$, $t(i)=s(k+i-1)$. Also define $s\!\upharpoonright\! k=s\!\upharpoonright\![1,k]$ and $s\!\upharpoonright\!0=()$.
For $s, t\in \mathcal{S}$, $t$ is a [*subword*]{} of $s$ if there are $1\leq k\leq l\leq {{\mbox{\rm lh}}}(s)$ such that $t=s\!\upharpoonright\![k,l]$. When $t$ is a subword of $s$, we also say that $t$ [*occurs*]{} in $s$. If $t$ is a subword of $s$ and $1\leq k\leq {{\mbox{\rm lh}}}(s)$, then we say that there is an [*occurrence of $t$ in $s$ at position $k$*]{} if $t=s\!\upharpoonright\![k,k+{{\mbox{\rm lh}}}(t)-1]$. We say that $t$ is an [*initial segment*]{} of $s$, denoted $t\sqsubseteq s$, if $t=s\!\upharpoonright\![1,{{\mbox{\rm lh}}}(t)]$.
For $s_1,\dots, s_n\in \mathcal{S}$, we define the [*concatenation*]{} $s_1^\smallfrown\dots {}^\smallfrown s_n$ to be the unique word $t\in\mathcal{S}$ with length $\sum_{j=1}^n {{\mbox{\rm lh}}}(s_j)$ such that for all $1\leq j\leq n$, $s_j=t\!\upharpoonright\!\left[1+\sum_{i=1}^{j-1}{{\mbox{\rm lh}}}(s_i), \sum_{i=1}^j {{\mbox{\rm lh}}}(s_i)\right]$.
For $s\in \mathcal{S}$, define $s^0=()$ and, if $n\in\mathbb{N}_+$, define $s^n$ to be the word $s_1^\smallfrown \dots^\smallfrown s_n$ where $s_1=\dots=s_n=s$. Words of the form $s^n$, with ${{\mbox{\rm lh}}}(s)=1$, are called [*constant*]{}. For $s, t\in\mathcal{S}$ of the same length, we say that $s$ and $t$ are [*incompatible*]{}, denoted $s\perp t$, if $t$ is not a subword of $s^\smallfrown(c)^\smallfrown s$ for any $c\in\N$. It is easy to check that $\perp$ is a symmetric relation for words of the same length. We say that $s$ and $t$ are [*compatible*]{} if it is not the case that $s\perp t$.
Let $\mathcal{F}$ be the set of all binary words that start and end with $0$. Again, each element of $\mathcal{F}$ can be equivalently viewed as a finite sequence of 0s and 1s, as a finite function with codomain $\{0,1\}$, or most often, as a finite $0,1$-word.
Infinite and bi-infinite sequences
----------------------------------
We will consider infinite sequences of natural numbers as well as infinite binary sequences. Again, they will be equivalently viewed as sequences, functions, and infinite words. We tacitly assume that an infinite sequence has domain $\N$, unless explicitly specified otherwise.
For an infinite word $V$ and natural numbers $k\leq l$, define $V\!\upharpoonright\![k,l]$ to be the unique $s\in \mathcal{S}$ such that ${{\mbox{\rm lh}}}(s)=l-k+1$ and for all $1\leq i\leq {{\mbox{\rm lh}}}(s)$, $s(i)=V(k+i-1)$. Also define $V\!\upharpoonright\!k=V\!\upharpoonright\![0,k]$. In the same fashion as for finite words, we may speak of when a finite word $s$ is a [*subword*]{} of $V$ or $s$ [*occurs*]{} in $V$, of there being an [*occurrence of $s$ in $V$ at position $k$*]{} for $k\in\N$, and of $s$ being an [*initial segment*]{} of $V$, which is denoted as $s\sqsubseteq V$.
If $v_0\sqsubseteq v_1\sqsubseteq \dots \sqsubseteq v_n\sqsubseteq \dots$ is an infinite sequence of elements of $\mathcal{S}$ each of which is an initial segment of the next, then there is a unique infinite sequence $V$ such that $v_n\sqsubseteq V$ for all $n\in\N$. We call this unique infinite sequence the [*limit*]{} of $(v_n: n\in\N)$ and denote it by $\lim_n v_n$. Specifically, for each $n\in\N$ and $1\leq i\leq {{\mbox{\rm lh}}}(v_n)$, $(\lim_n v_n)(i)=v_n(i+1)$. The infinite words we consider will arise as limits of such sequences of finite words.
A [*bi-infinite*]{} sequence (or word) is an element of $\N^\Z$. A [*bi-infinite binary*]{} sequence is an element of $\{0,1\}^\Z$. The relations of [*subword*]{} and [*occurrence*]{} can be defined similarly between finite words and bi-infinite words. With $\{0,1\}$ equipped with the discrete topology and $\{0,1\}^\Z$ equipped with the product topology, $\{0,1\}^\Z$ becomes a compact metric space. The [*shift map*]{} $\sigma$ on $\{0,1\}^Z$ is defined as $$\sigma(x)(i)=x(i+1)$$ for all $x\in\{0,1\}^\Z$ and $i\in\Z$. With $\{0,1\}$ equipped with any probability measure and $\{0,1\}^\Z$ equipped with the product measure, $\sigma$ is a measure-preserving automorphism on $\{0,1\}^\Z$.
Symbolic rank-one systems and rank-one transformations\[s1\]
------------------------------------------------------------
Both symbolic rank-one systems and rank-one transformations are constructed from the so-called cutting and spacer parameters $(r_n:n\in\N)$ and $(s_n:n\in\N)$. The [*cutting parameter*]{} $(r_n:n\in\N)$ is an infinite sequence of natural numbers with $r_n\geq 2$ for all $n\in\N$. The [*spacer parameter*]{} $(s_n:n\in\N)$ is a sequence of finite sequences of natural numbers with ${{\mbox{\rm lh}}}(s_n)=r_n-1$ for all $n\in\N$.
Given cutting and spacer parameters $(r_n:n\in\N)$ and $(s_n:n\in\N)$, a symbolic rank-one system is defined as follows. First, inductively define an infinite sequence of finite binary words $(v_n:n\in\N)$ as $$v_0=0,\ \ v_{n+1}=v_n1^{s_n(1)}v_n\dots v_n1^{s_n(r_n-1)}v_n.$$ We call $(v_n:n\in\N)$ a [*generating sequence*]{}. Noting that each $v_n\in \mathcal{F}$ (that is, $v_n$ starts and ends with 0) and $v_n$ is an initial segment of $v_{n+1}$, we may define $$V=\lim_n v_n.$$ $V$ is said to be an [*infinite rank-one word*]{}. Finally, let $$X=X_V=\{ x\in \{0,1\}^\Z\,:\, \mbox{every finite subword of $x$ is a subword of $V$}\}.$$ Then $X$ is a closed subspace of $\{0,1\}^\Z$ invariant under the shift map $\sigma$, i.e., $\sigma(x)\in X$ for all $x\in X$. For simplicity we still write $\sigma$ for $\sigma\!\upharpoonright\! X$. We call $(X, \sigma)$ a [*symbolic rank-one system*]{}.
With cutting and spacer parameters $(r_n:n\in\N)$ and $(s_n:n\in\N)$, one can also define a rank-one measure-preserving transformation $T$ by a cutting and stacking process as follows. First, inductively define an infinite sequence of natural numbers $(h_n: n\in\N)$ as $$\label{h} h_0=1,\ \ h_{n+1}=r_n h_n+\sum_{i=1}^{r_n-1} s_n(i).$$ Next, define sequences $(B_n:n\in\N)$, $(B_{n, i}: n\in\N, 1\leq i\leq r_n)$ and $(C_{n,i,j}: n\in\N, 1\leq i\leq r_n-1, 1\leq j\leq s_n(i))$, all of which are subsets of $[0,+\infty)$, by induction on $n$. Define $B_0=[0,1)$ and $B_{n+1}=B_{n,1}$ for all $n\in\N$. Let $B_n$ be given and inductively assume that $T^k[B_n]$ are defined for $0\leq k<h_n$ so that $T^k[B_n]$, $0\leq k<h_n$, are all disjoint. Let $\{B_{n,i}: n\in\N, 1\leq i\leq r_n\}$ be a partition of $B_n$ into $r_n$ many sets of equal measure and let $\{C_{n,i,j}: n\in\N, 1\leq i\leq r_n-1, 1\leq j\leq s_n(i)\}$ be disjoint sets each of which is disjoint from $B_n$ and has the same measure as $B_{n,1}$. Then define $T$ so that for all $1\leq i\leq r_n-1$, $$T^{h_n}[B_{n,i}]=\left\{\begin{array}{ll} C_{n,i,1} & \mbox{if $s_n(i)>0$} \\ B_{n,i+1} & \mbox{if $s_n(i)=0$;}\end{array}\right.$$ and for $1\leq j\leq s_n(i)$, $$T[C_{n,i,j}]=\left\{\begin{array}{ll} C_{n,i, j+1} &\mbox{if $1\leq j<s_n(i)$} \\
B_{n,i+1} & \mbox{if $j=s_n(i)$}.\end{array}\right.$$ We have thus defined $T^k[B_{n+1}]$ for $0\leq k<h_{n+1}$ so that all of them are disjoint. Finally, let $$Y=\bigcup\{ T^k[B_n]: n\in\N, 0\leq k<h_n\}.$$ Then $T$ is a measure-preserving automorphism of $Y$. If $Y$ has finite Lebesgue measure, or equivalently if $$\label{eqn:a} \sum_{n=0}^\infty \displaystyle\frac{h_{n+1}-h_nr_n}{h_{n+1}}<+\infty,$$ then $(Y,\lambda)$, where $\lambda$ is the normalized Lebesgue measure on $Y$, is a probability Lebesgue space. Clearly $T$ is still a measure-preserving automorphism of $(Y, \lambda)$. Such a $T$ is called a [*rank-one (measure-preserving) transformation*]{}.
Connecting the symbolic and the geometric constructions, we can see that $h_n={{\mbox{\rm lh}}}(v_n)$ for all $n\in\N$. When $(\ref{eqn:a})$ holds, there is a unique probability Borel measure $\mu$ on $X$, and $(X,\mu,\sigma)$ and $(Y,\lambda, T)$ are isomorphic measure-preserving transformations. In particular, the isomorphism type of $(Y, \lambda, T)$ does not depend on the numerous choices one has to make in the process to construct $T$ (e.g. how the sets $B_{n, i}$ and $C_{n,i,j}$ are picked and how $T$ is defined on them).
Throughout the rest of the paper we tacitly assume that (\[eqn:a\]) is satisfied for all rank-one transformations under our consideration.
For more information on the basics of rank-one transformations, particularly on the connections between the symbolic and geometric constructions, c.f. [@Ferenczi] [@GaoHill0] and [@GaoHill].
Canonical generating sequences
------------------------------
The notion of the canonical generating sequence was developed in [@GaoHill1] in the study of topological conjugacy of symbolic rank-one systems. In [@GaoHill], however, we used the notion to characterize non-rigid rank-one transformations among all bounded rank-one transformations. We will use this notion later in this paper again.
For $u, v\in \mathcal{F}$ we say that $u$ is [*built from*]{} $v$, denoted $v\prec u$, if for some $n\geq 1$ there are $a_1, \dots, a_n\in\N$ such that $$u=v1^{a_1}v\dots v1^{a_n}v.$$ If in addition $a_1=\dots =a_n$, the we say that $u$ is [*simply built from*]{} $v$, and denote $v\prec_s u$. It is easy to see that $\prec$ is a transitive relation, and $\prec_s$ is not. If $V$ is an infinite word, we also say $V$ is [*built from*]{} $v$, and denote $v\prec V$, if there is an infinite sequence $(a_n: n\in\N_+)$ of natural numbers such that $$V=v1^{a_1}v\dots v1^{a_n}v\dots.$$ Similarly, we say that $V$ is [*simply built from*]{} $v$ if $a_1=\dots=a_n=\dots$. We say that $V$ is [*non-degenerate*]{} if $V$ is not simply built from any word in $\mathcal{F}$.
Let $V$ be an infinite rank-one word. As in Subsection \[s1\] a generating sequence for $V$, $(v_n:n\in\N)$, is a sequence of elements of $\mathcal{F}$ such that $v_0=0$, $v_n\prec v_{n+1}$ for all $n\in\N$, and $V=\lim_n v_n$. If follows that $v_n\prec V$ for each $n\in\N$. In general, if $v\in\mathcal{F}$ and $v\prec V$, then there is a unique way to express $V$ as an infinite concatenation of $v$ as above, and in this case the demonstrated occurrence of $v$ are called the [*expected occurrences*]{} of $v$.
The [*canonical generating sequence*]{} of $V$ is a sequence enumerating in increasing $\prec$-order the set of all $v\in\mathcal{F}$ such that there do not exist $u,w \in \mathcal{F}$ satisfying $u \prec v \prec w \prec V$ and $u \prec_s w$. In [@GaoHill1] it was shown that, if $V$ is non-degenerate, the canonical generating sequence is infinite.
Throughout the rest of this paper we consider only non-degenerate infinite rank-one words. There exist cutting and spacer parameters correspondent to each generating sequence. Thus we may speak of the [*canonical cutting and spacer parameters*]{} given any non-degenerate infinite rank-one word. A rank-one transformation $T$ is [*bounded*]{} if some cutting and spacer parameters $(r_n: n\in\N)$ and $(s_n:n\in\N)$ giving rise to $T$ are bounded, i.e., there is $B>0$ such that for all $n\in\N$ and $1\leq i\leq r_n-1$, $r_n<B$ and $s_n(i)<B$. Similarly, $T$ is [*canonically bounded*]{} if some canonical cutting and spacer parameters giving rise to $T$ are bounded. A canonically bounded rank-one transformation is necessarily bounded, but the converse is not true. The following theorem characterizes exactly which bounded rank-one transformations are canonically bounded.
\[TC\] Let $T$ be a bounded rank-one transformation. Then $T$ is non-rigid, i.e. $T$ has trivial centralizer, if and only if $T$ is canonically bounded.
Replacement schemes and topological conjugacy
---------------------------------------------
Given infinite rank-one words $V$ and $W$, a [*replacement scheme*]{} is a pair $(v, w)$ of elements of $\mathcal{F}$, such that $v\prec V$, $w\prec W$, and for all $k\in\N$, there is an expected occurrence of $v$ in $V$ at position $k$ if and only if there is an expected occurrence of $w$ in $W$ at position $k$. This notion is closely related to the topological conjugacy between symbolic rank-one systems.
In fact, if $v\prec V$, then every $x\in X_V$ can be uniquely expressed as $$x=\dots v1^{a_{-i}}v\dots v1^{a_0}v\dots v1^{a_i}v\dots$$ for $\dots a_{-i}, \dots, a_0, \dots, a_i,\dots \in \N$. We say that $x$ is [*built from*]{} $v$. The demonstrated occurrences of $v$ are again said to be [*expected*]{}. When $(v, w)$ is a replacement scheme for $V$ and $W$, we may define a map $\phi: X_V\to X_W$ so that $$\phi(x)=\dots w1^{b_{-i}}w\dots w1^{b_0}w\dots w1^{b_i}w\dots$$ i.e., $\phi(x)$ is built from $w$, and so that for all $k\in\Z$, there is an expected occurrence of $v$ in $x$ at position $k$ if and only if there is an expected occurrence of $w$ in $\phi(x)$ at position $k$. Intuitively, $\phi(x)$ is obtained from $x$ by replacing every expected occurrence of $v$ in $x$ by $w$, adding or deleting 1s as necessary. It is easy to see that $\phi$ is a topological conjugacy between $X_V$ and $X_W$. We showed in [@GaoHill1] that all topological conjugacies essentially arise this way.
Let $V$ and $W$ be non-degenerate infinite rank-one words. Then $(X_V, \sigma)$ and $(X_W, \sigma)$ are topologically conjugate if and only if there exists a replacement scheme for $V$ and $W$.
For the subject of this paper it is important to note that $\phi$ is also a measure-preserving isomorphism. This follows from the unique ergodicity of symbolic rank-one systems. Thus the existence of replacement schemes is a sufficient condition for two symbolic rank-one systems to be isomorphic.
We say that the two pairs of cutting and spacer parameters, $(r_n: n\in\N), (s_n:n\in\N)$ and $(q_n:n\in\N), (t_n:n\in\N)$, are [*commensurate*]{} if for all $n\in\N$, $r_n=q_n$ and $\sum_{i=1}^{r_n-1}{{\mbox{\rm lh}}}(s_i)=\sum_{i=1}^{r_n-1}{{\mbox{\rm lh}}}(t_i)$.
In the case of commensurate parameters, there is a straightforward way to identify replacement schemes and therefore it is easy to determine topological conjugacy.
\[topiso\] Let $(r_n:n\in\N)$ and $(s_n:n\in\N)$ be cutting and spacer parameters giving rise to non-degenerate infinite rank-one word $V$. Let $(r_n:n\in\N)$ and $(t_n:n\in\N)$ be cutting and spacer parameters giving rise to non-degenerate infinite rank-one word $W$. Suppose the two sets of parameters are commensurate. Then $(X_V,\sigma)$ and $(X_W, \sigma)$ are topologically conjugate if and only if there is $N\in\N$ such that for all $n\geq N$, $s_n=t_n$.
As mentioned above, this also gives an explicit sufficient condition for two symbolic rank-one systems to be measure-theoretically isomorphic.
Isomorphism and disjointness
============================
Non-isomorphism
---------------
First we state a theorem from [@Hill], Proposition 2.1, which is relevant to our results in this paper.
\[mainiso\] Let $(r_n: n \in \N)$ and $(s_n: n \in \N)$ be cutting and spacer parameters giving rise to symbolic rank-one system $(X, \mu, \sigma)$. Let $(r_n: n \in \N)$ and $(t_n: n \in \N)$ be cutting and spacer parameters giving rise to symbolic rank-one system $(Y, \nu, \sigma)$. Suppose the following hold.
1. The two sets of parameters are commensurate, i.e., for all $n$, $$\sum_{i=1}^{r_n-1} s_{n}(i) = \sum_{i=1}^{r_n-1} t_{n}(i).$$
2. There is an $S \in \N$ such that for all $n$ and all $1\leq i \leq r_n-1$, $$s_n(i) \leq S \textnormal{ and } t_n(i) \leq S.$$
3. There is an $R \in \N$ such that for infinitely many $n$, $$r_n \leq R \textnormal{ and } s_n \perp t_n.$$
Then $(X, \mu, \sigma)$ and $(Y, \nu, \sigma)$ are not isomorphic.
Disjointness
------------
\[maindisjoint\] Let $(r_n: n \in \N)$ and $(s_n: n \in \N)$ be cutting and spacer parameters giving rise to symbolic rank-one system $(X, \mu, \sigma)$. Let $(r_n: n \in \N)$ and $(t_n: n \in \N)$ be cutting and spacer parameters giving rise to symbolic rank-one system $(Y, \nu, \sigma)$. Suppose the following hold.
1. The two sets of parameters are commensurate, i.e., for all $n$, $$\sum_{i=1}^{r_n-1} s_{n}(i) = \sum_{i=1}^{r_n-1} t_{n}(i).$$
2. There is an $S \in \N$ such that for all $n$ and all $1\leq i \leq r_n-1$, $$s_n(i) \leq S \textnormal{ and } t_n(i) \leq S.$$
3. There is an $R \in \N$ such that for infinitely many $n$, $$r_n \leq R \textnormal{ and } s_n \perp t_n.$$
4. For each $k >1$, either $(X, \mu, \sigma^k)$ or $(Y, \nu, \sigma^k)$ is ergodic.
Then $(X, \mu, \sigma)$ and $(Y, \nu, \sigma)$ are disjoint.
The only difference between the hypotheses of Theorems \[mainiso\] and \[maindisjoint\] is condition (d) above. Condition (d) is necessary for disjointness. In fact, if for some $k>1$, both $(X, \mu, \sigma^k)$ and $(Y, \nu, \sigma^k)$ are not ergodic, then they have a common factor which is a cyclic permutation on an $k$-element set, and thus the two transformations are not disjoint. It will be clear from the proof below that the theorem still holds if condition (d) is weakened to the following:
1. For each $1<k\leq S$, where $S$ is the bound from condition (b), either $(X, \mu,\sigma^k)$ or $(Y,\nu, \sigma^k)$ is ergodic.
The rest of this subsection is devoted to a proof of Theorem \[maindisjoint\]. We will follow the approach of del Junco, Rahe, and Swanson [@delJuncoRaheSwanson] in their proof of minimal self-joinings for Chacon’s transformation, as presented by Rudolph in his book [@RudolphBook], Section 6.5.
The setup of the proof is standard. Let $\overline{\mu}$ be an ergodic joining of $\mu$ and $\nu$ on $X\times Y$. We need to show that $\overline{\mu}=\mu\times\nu$. By Lemma 6.14 of [@RudolphBook] (or Proposition 2 of [@delJuncoRaheSwanson]), it suffices to find some $k\geq 1$ such that $(X, \mu, \sigma^k)$ is ergodic and $\overline{\mu}$ is $(\sigma^k\times {{\mbox{\rm id}}})$-invariant, where ${{\mbox{\rm id}}}$ is the identity transformation on $Y$. For this let $(x, y)\in X\times Y$ satisfy the ergodic theorem for $\overline{\mu}$, i.e., for all measurable $A\subseteq X\times Y$, $$\displaystyle\lim_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1} \chi_A(\sigma^i(x), \sigma^i(y))=\overline{\mu}(A)$$ and $$\displaystyle\lim_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1} \chi_A(\sigma^{-i}(x), \sigma^{-i}(y))=\overline{\mu}(A).$$ Such $(x, y)$ exists by the ergodicity of $\overline{\mu}$. Lemma 6.15 of [@RudolphBook] gives a sufficient condition to complete the proof. We state it below in our notation.
\[tech\] Suppose there are integers $a_n, b_n, c_n, d_n, e_n\in\Z$ for all $n\in\N$, a positive integer $k\geq 1$ and a real number $\alpha>0$ such that for all $n\in\N$,
1. $a_n\leq 0\leq b_n$ and $\lim_n (b_n-a_n)=+\infty$;
2. $a_n\leq c_n\leq d_n\leq b_n$ and $a_n\leq c_n+e_n\leq d_n+e_n\leq b_n$;
3. $d_n-c_n\geq \alpha(b_n-a_n)$;
4. for all $c_n\leq i\leq d_n$, $x(i)=x(i+k+e_n)$ and $y(i)=y(i+e_n)$; and
5. $(X, \mu, \sigma^k)$ is ergodic.
Then $\overline{\mu}$ is $(\sigma^k\times {{\mbox{\rm id}}})$-invariant, and so $\overline{\mu}=\mu\times\nu$.
Note that Lemma \[tech\] has several valid variations. One variation is a symmetric version with the spaces $X$ and $Y$ switched. This version is obviously true since the setup is entirely symmetric for $X$ and $Y$. Another variation is the version in which $k\leq -1$ is a negative integer. Note that $(X, \mu, \sigma^{k})$ is ergodic if and only if $(X, \mu,\sigma^{-k})$ is ergodic. This version can be obtained by applying Lemma \[tech\] to $(X, \mu, \sigma^{-1})$ and $(Y, \nu, \sigma^{-1})$. Finally, we also have the variation in which both $k$ is negative and $X$ and $Y$ are switched.
Now we claim that a slightly weaker construction already suffices: it is enough to find $a_n, b_n, c_n, d_n, e_n\in\Z$ for all $n\in\N$, a positive integer $K\geq 1$ and a real number $\alpha$ so that (i)–(iii) hold and for each $n\in\N$, (iv) holds for some nonzero $k\in\Z$ with $k\in[-K,K]$. In fact, since there are only finitely many integers between $-K$ and $K$, we get some nonzero integer $k\in[-K,K]$ and infinitely many $n$ for which the conditions (i)–(iv) of Lemma \[tech\] are satisfied. If $k>0$ and $(X, \mu, \sigma^k)$ is ergodic then we are done by Lemma \[tech\]. If $k>0$ but $(X, \mu, \sigma^k)$ is not ergodic, then by condition (d), $(Y, \nu, \sigma^k)$ is ergodic. It follows that $(Y, \nu, \sigma^{-k})$ is ergodic. Now we are done by the variation of Lemma \[tech\] in which both $k$ is negative and $X$ and $Y$ are switched. If $k<0$ we similarly apply other variations of the lemma.
We now begin our construction. Let $K=S$ where $S$ is the bound in condition (b). Let $$\alpha= \displaystyle\frac{1}{12(R+1)}$$ where $R$ is the bound in condition (c). Note that $R\geq 2$ because $r_n\geq 2$ for all $n\in\N$. Let $$D=\{n\in\N: r_n\leq R \mbox{ and } s_n\perp t_n\}.$$ Then $D$ is infinite by condition (c).
Let $(v_n: n\in\N)$ be the generating sequence given by the cutting and spacer parameters $(r_n:n\in\N)$ and $(s_n:n\in\N)$. Then for each $n\in\N$, $x$ is built from $v_n$. Let $(w_n:n\in\N)$ be the generating sequence given by the cutting an spacer parameters $(r_n:n\in\N)$ and $(t_n:n\in\N)$. Then for each $n\in\N$, $y$ is built from $w_n$. By the commensurability condition (a), we have ${{\mbox{\rm lh}}}(v_n)={{\mbox{\rm lh}}}(w_n)$ for all $n\in\N$.
Fix an $n_0$ such that ${{\mbox{\rm lh}}}(v_{n_0})>3RS\geq 6S$. For any $n\in D$ with $n\geq n_0$, we define $$a_n, b_n, c_n, d_n, e_n\in\N$$ to satisfy (i)–(iii) and (iv) with some nonzero $k_n\in[-S, S]$. Define $$a_n=-2{{\mbox{\rm lh}}}(v_{n+1}) \mbox{ and } b_n=2{{\mbox{\rm lh}}}(v_{n+1}).$$ It is clear that (i) is satisfied.
Before defining $c_n, d_n, e_n$ and $k_n$ we need to analyze the expected occurrences of $v_{n+1}$ in $x$ and the expected occurrences of $w_{n+1}$ in $y$. Since $y$ is built from $w_{n+1}$, by condition (b) the interval $[-S, S]$ has a nonempty intersection with some expected occurrence of $w_{n+1}$ in $y$. Fix one such expected occurrence of $w_{n+1}$ and suppose the occurrence begins at position $l$ and finishes at position $m$. Thus $a_n\leq -{{\mbox{\rm lh}}}(v_{n+1})-S\leq l\leq S$ and $m=l+{{\mbox{\rm lh}}}(v_{n+1})-1\leq S+{{\mbox{\rm lh}}}(v_{n+1})\leq b_n$.
Note that $x$ is built from $v_n$. We can then define an integer $j\in\Z$ where $|j|$ is the least such that there is an expected occurrence of $v_n$ at position $l+j$. A moment of reflection shows that $|j|\leq \frac{1}{2}({{\mbox{\rm lh}}}(v_n)+S)$. Since ${{\mbox{\rm lh}}}(v_n)>6S$, the occurrence of $w_n$ at position $l$ and the occurrence of $v_n$ at position $l+j$ overlap for at least $\frac{1}{3}{{\mbox{\rm lh}}}(v_n)$ many positions.
Starting from the expected occurrence of $v_n$ at position $l+j$ in $x$, we examine the next $r_n$ many consecutive expected occurrences of $v_n$ in $x$. Suppose there is an occurrence of the following word in $x$ starting at position $l+j$: $$v_n1^{p(1)}v_n\dots 1^{p(r_n-1)}v_n$$ where $p\in \mathcal{S}$ with ${{\mbox{\rm lh}}}(p)=r_n-1$. Because $x$ is also built from $v_{n+1}$, and each expected occurrence of $v_{n+1}$ contains $r_n$ many expected occurrences of $v_n$, the above word is contained in an occurrence of $v_{n+1}1^qv_{n+1}$ for some $q\in\N$, where each demonstrated occurrence of $v_{n+1}$ is expected. Note that $$v_{n+1}1^qv_{n+1}=v_n1^{s_n(1)}v_n\dots 1^{s_n(r_n-1)}v_n1^qv_n1^{s_n(1)}v_n\dots 1^{s_n(r_n-1)}v_n.$$ By comparison, we get that $p$ is a subword of $s_n^\smallfrown(q)^\smallfrown s_n$.
Since $n\in D$ and therefore $s_n\perp t_n$, we conclude that $p\neq t_n$. Let $i_0$ be the least such that $1\leq i_0\leq r_n-1$ and $p(i_0)\neq t_n(i_0)$. Let $$h=(i_0-1){{\mbox{\rm lh}}}(v_n)+\sum_{i=1}^{i_0-1}t_n(i).$$ Then $l+h$ is the beginning position of an expected occurrence of $w_n$ in $y$, and $l+j+h$ is the beginning position of an expected occurrence of $v_n$ in $x$. There is an occurrence of $w_n1^{t_n(i_0)}w_n$ in $y$ beginning at position $l+h$, and there is an occurrence of $v_n1^{p(i_0)}v_n$ in $x$ beginning at position $l+j+h$.
Now we define $[c_n, d_n]$ to be the interval of overlap between the occurrence of $w_n$ in $y$ at position $l+h$ and the occurrence of $v_n$ in $x$ at position $l+j+h$. Since $|j|\leq \frac{1}{2}({{\mbox{\rm lh}}}(v_n)+S)$ and ${{\mbox{\rm lh}}}(v_n)>6S$, we get that $$d_n-c_n\geq \frac{1}{3}{{\mbox{\rm lh}}}(v_n).$$ Define $$e_n={{\mbox{\rm lh}}}(v_n)+t_n(i_0)$$ and $$k_n= p(i_0)-t_n(i_0).$$ Since $[c_n, d_n]$ is contained in the occurrence of $v_n$ at position $l+j+h$, and since $k_n+e_n={{\mbox{\rm lh}}}(v_n)+p(i_0)$, we have that $x\!\upharpoonright\![c_n,d_n]$ and $x\!\upharpoonright\! [c_n+k_n+e_n, d_n+k_n+e_n]$ are the same words. Similarly, $[c_n, d_n]$ is also contained in the occurrence of $w_n$ at position $l+h$, and it follows that $y\!\upharpoonright\![c_n,d_n]$ and $y\!\upharpoonright\![c_n+e_n, d_n+e_n]$ are the same words. This means that (iv) is satisfied.
Since $[c_n, d_n], [c_n+e_n, d_n+e_n]\subseteq [l,m]\subseteq[a_n, b_n]$, we know that (ii) is satisfied. Finally, $$\displaystyle\frac{d_n-c_n}{b_n-a_n}\geq \frac{{{\mbox{\rm lh}}}(v_n)}{3\cdot 4{{\mbox{\rm lh}}}(v_{n+1})}\geq\frac{{{\mbox{\rm lh}}}(v_n)}{12(R{{\mbox{\rm lh}}}(v_n)+RS)}\geq\alpha.$$ This shows that (iii) is satisfied.
The proof of Theorem \[maindisjoint\] is complete.
Applications to canonically bounded transformations
---------------------------------------------------
Theorems \[mainiso\] and \[maindisjoint\], in combination with results in [@GaoHill1], give combinatorial criteria for isomorphism and disjointness for certain bounded rank-one transformations. These criteria in terms of the cutting and spacer parameters are, in principle, easy to check.
Let $(r_n:n\in\N)$ and $(s_n:n\in\N)$ be the cutting and spacer parameters for a rank-one transformation $T$. For integer $d>1$, consider the statement $$\begin{aligned}
\tag{$\mbox{E}_d$} \forall N\in\N\ \exists n, i\in\N\ [\, n\geq N, 1\leq i\leq r_n-1, \mbox{and } h_N+s_n(i)\not\equiv 0 \mbox{ mod } d\,] \end{aligned}$$ where $(h_n:n\in\N)$ is the sequence defined in equation (\[h\]).
The following fact has been proved in [@GaoHill1].
Let $T$ be a bounded rank-one transformation with cutting and spacer parameters $(r_n:n\in\N)$ and $(s_n: n\in\N)$. Then for any integer $d>1$, $T^d$ is ergodic if and only if [($\mbox{E}_d$)]{} holds.
We can now state our main result about commensurate, canonically bounded rank-one transformations.
\[CBmain\] Let $T$ be a rank-one transformation with bounded canonical cutting and spacer parameters $(r_n:n\in\N)$ and $(s_n: n\in\N)$. Let $S$ be a rank-one transformation with bounded canonical cutting and spacer parameters $(q_n:n\in\N)$ and $(t_n: n\in\N)$. Suppose the parameters for $T$ and $S$ are commensurate. Then the following hold.
1. $T$ and $S$ are isomorphic if and only if there is $N\in\N$ such that for all $n\geq N$, $s_n=t_n$.
2. $T$ and $S$ are disjoint if and only if for infinitely many $n\in\N$, $s_n\neq t_n$ and for every integer $d>1$, either $T^d$ is ergodic or $S^d$ is ergodic.
As in Theorem \[maindisjoint\], if $D$ is an upper bound for the sequences $(s_n:n\in\N)$ and $(t_n:n\in\N)$, then clause (2) can be strengthened to
1. $T$ and $S$ are disjoint if and only if for infinitely many $n\in\N$, $s_n\neq t_n$ and for every integer $1<d\leq D$, either $T^d$ is ergodic or $S^d$ is ergodic.
The rest of this subsection is devoted to a proof of Corollary \[CBmain\].
Let $(v_n:n\in\N)$ be the generating sequence given by the cutting and spacer parameters $(r_n:n\in\N)$ and $(s_n:n\in\N)$. Let $V=\lim_n v_n$. Then $T$ is isomorphic to the symbolic rank-one system $(X_V, \mu, \sigma)$ for a uniquely ergodic Borel probability measure $\mu$. So we will assume that $T$ is $(X, \mu, \sigma)$. Let $(w_n:n\in\N)$ be the generating sequence given by the cutting and spacer parameters $(q_n: n\in\N)$ and $(t_n:n\in \N)$. Let $W=\lim_n w_n$. We will similarly assume that $S$ is the symbolic rank-one system $(X_W, \nu, \sigma)$ for a suitable measure $\nu$. By commensurability, we have that for all $n\in\N$, $q_n=r_n$ and ${{\mbox{\rm lh}}}(v_n)={{\mbox{\rm lh}}}(w_n)$.
First consider isomorphism. The condition is sufficient since it gives a replacement scheme, which in turn gives rise to a topological conjugacy which is also a measure-theoretic isomorphism. More specifically, if for all $n\geq N$, $s_n=t_n$, then $(v_N, w_N)$ is a replacement scheme, and the topological conjugacy it induces is an isomorphism between $T$ and $S$.
For the necessity, assume that for infinitely many $n\in\N$, $s_n\neq t_n$. Before proceeding with the proof we prove a basic fact about compatibility.
\[perp\] Let $s, t, s', t'\in \mathcal{S}$. Suppose $s\neq t$, ${{\mbox{\rm lh}}}(s)={{\mbox{\rm lh}}}(t)=l>0$, and ${{\mbox{\rm lh}}}(s')={{\mbox{\rm lh}}}(t')=m>0$. Assume the following two words are compatible: $$\label{s*}
{s}^\smallfrown ({s'(1)})^\smallfrown {s}^\smallfrown\dots {}^\smallfrown{s}^\smallfrown (s'(m))^\smallfrown s$$ $$\label{t*}
{t}^\smallfrown (t'(1))^\smallfrown {t}^\smallfrown\dots {}^\smallfrown{t}^\smallfrown (t'(m))^\smallfrown t.$$ Then $s'$ and $t'$ are both constant words.
Let $u$ be the word in (\[s\*\]) and $z$ be the word in (\[t\*\]). Suppose $z$ is a subword of $u^\smallfrown (c)^\smallfrown u$ for some $c\in\N$. Since $s\neq t$, the first occurrence of $t$ in $z$ cannot line up with any occurrence of $s$ in $u$, i.e., in the occurrence of $z$ in $u^\smallfrown(c)^\smallfrown u$, the starting position of the first occurrence of $t$ is not the same as the starting position of any demonstrated occurrence of $s$. Since ${{\mbox{\rm lh}}}(s)={{\mbox{\rm lh}}}(t)=l>0$, this implies that there is $1\leq j\leq l$ such that $t'(1)=s(j)$. But then it follows that $t'(2)=\dots=t'(m)=s(j)$. Thus $t'$ is constant. By symmetry, $s'$ is also constant.
Now back to the proof of Corollary \[CBmain\] (1). We have assumed that there are infinitely many $n\in\N$ with $s_n\neq t_n$. We inductively define an infinite sequence $(n_k: k\in\N)$ of natural numbers as follows. Define $n_0=0$. In general, assume $n_k$, $k\geq 0$, has been defined. Define $n_{k+1}=n_k+1$ if $s_{n_k}=t_{n_k}$. Otherwise, $s_{n_k}\neq t_{n_k}$, and we define $n_{k+1}=n_k+2$ if $s_{n_k+1}$ is not constant, and define $n_{k+1}=n_k+3$ otherwise. Let $v'_k=v_{n_k}$ and $w'_k=w_{n_k}$ for all $k\in\N$. Then $(v'_n:n\in\N)$ is a subsequence of $(v_n:n\in\N)$ giving rise to $T$ and $(w'_n:n\in\N)$ is a subsequence of $(w_n:n\in\N)$ giving rise to $S$. Let $(r'_n:n\in\N)$ and $(s'_n:n\in\N)$ be the cutting and spacer parameters correspondent to $(v'_n:n\in\N)$. Let $(q'_n:n\in\N)$ and $(t'_n:n\in\N)$ be the cutting and spacer parameters correspondent to $(w'_n:n\in\N)$. It is clear that the newly defined parameters are commensurate. We claim that the newly defined parameters for $T$ and $S$ satisfy all the other hypotheses of Theorem \[mainiso\]. Thus $T$ and $S$ are not isomorphic.
To verfity the claim, first note that $n_k< n_{k+1}\leq n_k+3$ for all $k\in\N$. This implies boundedness of the newly defined cutting and spacer parameters. In fact, if $R$ is a bound for $(r_n:n\in\N)$, then $R^3$ is a bound for $(r'_n:n\in\N)$. If $S$ is a bound for $(s_n:n\in\N)$, then $S$ is still a bound for $(s'_n:n\in\N)$.
It remains to verify that for infinitely many $k\in\N$, $s'_k\perp t'_k$. By our construction of the sequence $(n_k:k\in\N)$, there are infinitely many $k$ such that either $n_{k+1}=n_k+2$ or $n_{k+1}=n_k+3$. We claim that for each of these $k$ we have $s'_k\perp t'_k$. First suppose $k$ is such that $n_{k+1}=n_k+2$. By our construction this means that $s_{n_k}\neq t_{n_k}$ and $s_{n_k+1}$ is not constant. In this case, we have $$s'_k={s_{n_k}}^\smallfrown (s_{n_k+1}(1))^\smallfrown {s_{n_k}}^\smallfrown \dots {}^\smallfrown (s_{n_k+1}(r_{n_k+1}-1))^\smallfrown s_{n_k}$$ and $$t'_k={t_{n_k}}^\smallfrown (t_{n_k+1}(1))^\smallfrown{t_{n_k}}^\smallfrown \dots {}^\smallfrown (t_{n_k+1}(r_{n_k+1}-1))^\smallfrown t_{n_k}.$$ By Lemma \[perp\], $s'_k\perp t'_k$. Next suppose $k$ is such that $n_{k+1}=n_k+3$. By our construction this means that $s_{n_k}\neq t_{n_k}$ and $s_{n_{k+1}}$ is constant. A similar application of Lemma \[perp\] will complete the proof, provided that we verify the word $${s_{n_k+1}}^\smallfrown (s_{n_k+2}(1))^\smallfrown {s_{n_k+1}}^\smallfrown \dots {}^\smallfrown (s_{n_k+2}(r_{n_k+2}-1))^\smallfrown s_{n_k+1}$$ is not constant. Assume it is. Note that this sequence correspond to the way $v_{n_k+3}$ is built from $v_{n_k+1}$. Thus $v_{n_k+1}\prec_s v_{n_k+3}$ and $v_{n_k+2}$ is not on the canonical generating sequence. This contradicts our assumption that $(v_n: n\in\N)$ is a canonical generating sequence.
We have thus shown Corollary \[CBmain\] (1). For Corollary \[CBmain\] (2), the necessity of the condition is clear (c.f. the remarks after the statement of Theorem \[maindisjoint\]). For the sufficiency, it is enough to construct new pairs of cutting and spacer parameters as above, and apply Theorem \[maindisjoint\].
Minimal self-joinings and Ryzhikov’s theorem
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Minimal self-joinings
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\[mainMSJ\] Let $(r_n: n \in \N)$ and $(s_n : n \in \N)$ be cutting and spacer parameters giving rise to symbolic rank-one system $(X, \mu, \sigma)$. Suppose the following hold.
1. For some $R$ and all $n$, $r_n \leq R$.
2. For some $S$ and all $n$ and all $0 < i< r_n$, $s_n(i) \leq S$.
3. For all $n$ and all $c \in \N$, there are only two occurrence of $s_n$ in ${s_n}^\smallfrown(c)^\smallfrown s_n$.
4. $(X, \mu, \sigma)$ is totally ergodic.
Then $(X, \mu, \sigma)$ has minimal self-joinings of all orders.
First we note a well-known fact that for rank-one transformations, having minimal self-joinings of order 2 implies minimal self-joinings of all orders. We thank Eli Glasner for providing us the references and for allowing us to include the argument here for the benefit of the reader.
\[allorders\] If a rank-one transformation has minimal self-joinings of order 2, then it has minimal self-joinings of all orders.
An inductive argument (c.f. [@GlasnerBook] Theorem 12.16) shows that for any weakly mixing transformation, having minimal self-joinings of order 3 implies minimal self-joinings of all orders. A theorem of Ryzhikov [@Ryzhikov0] states that a 2-mixing measure-preserving transformation with minimal self-joinings of order 2 has minimal self-joinings of all orders. It follows that if a transformation has minimal self-joinings of order 2 but not order 3, then it is mixing but not 2-mixing (c.f. [@GlasnerBook] Corollary 12.22). A theorem of Kalikow [@Kalikow] states that any mixing rank-one transformation is also $2$-mixing (and in fact $k$-mixing for all $k>1$). Thus one concludes that a rank-one transformation with minimal self-joinings of order 2 also has minimal self-joinings of order 3. Since having minimal self-joinings of order 2 implies weakly mixing, such a transformation has minimal self-joinings of all orders.
The above theorem is well-known to experts in the field and the references provided here are not meant to be exhaustive. For instance, the theorem was mentioned in [@Ryzhikov] (without proof or further references). A weaker form of the theorem was mentioned in [@King88], which is sufficient for our purpose since we only consider bounded rank-one transformations, which are not mixing.
As in Theorem \[maindisjoint\] and Corollary \[CBmain\], condition (d) of Theorem \[mainMSJ\] can be weakened to
1. For each $1<k\leq S$, where $S$ is the bound from condition (b), $(X, \mu, \sigma^k)$ is ergodic.
This will be clear from the proof below.
The rest of this subsection is devoted to a proof of Theorem \[mainMSJ\] for minimal self-joinings of order 2. We again follow the approach of del Junco, Rahe, and Swanson [@delJuncoRaheSwanson] in their proof of minimal self-joinings for Chacon’s transformation, as presented by Rudolph in his book [@RudolphBook], Section 6.5.
Let $(v_n:n\in\N)$ be the generating sequence given by the cutting and spacer parameters $(r_n:n\in\N)$ and $(s_n:n\in\N)$.
\[cut\] Without loss of generality, we may assume $r_n\geq 3$ for all $n\in\N$.
Simply consider the subsequence $(v'_n:n\in\N)$ defined as $v'_n=v_{2n}$ for all $n\in\N$. Then $r'_n=r_{2n}r_{2n+1}\geq 4$ is the new cutting parameter, and the new spacer parameter $s'_n$ is $$\label{s} {s_{2n}}^\smallfrown(s_{2n+1}(1))^\smallfrown {s_{2n}}^\smallfrown(s_{2n+1}(2))^\smallfrown\dots ^\smallfrown{s_{2n}}^\smallfrown (s_{2n+1}(r_{2n+1}-1))^\smallfrown s_{2n}.$$ If $R$ is the bound for $r_n$ in condition (a), then $r'_n\leq R^2$. If $S$ is the bound for all $s_n(i)$ in condition (b), $S$ is still a bound for all $s'_n(j)$. Since $\lim_n v_n=\lim_n v'_n$, condition (d) continues to hold. It remains only to verify that condition (c) continues to hold for $s'_n$.
Towards a contradiction, suppose $s'_n$, which is in the form given by (\[s\]), occurs in ${s'_n}^\smallfrown (c)^\smallfrown s'_n$ not as demonstrated. We refer to this occurrence of $s'_n$ as the [*hidden occurrence*]{}. Note that $s'_n$ starts with an occurrence of $s_{2n}$. Thus the hidden occurrence of $s'_n$ must start at a position where an expected occurrence of $s_{2n}$ in ${s'_n}^\smallfrown (c)^\smallfrown s'_n$ begins, because otherwise we get that $s_{2n}$ occurs in some ${s_{2n}}^\smallfrown (d)^\smallfrown s_{2n}$ not as demonstrated, contradicting our condition (c). In other words, all expected occurrence of $s_{2n}$ in the hidden occurrence of $s'_n$ must be already demonstrated in the form given by (\[s\]). By comparison, we get that $s_{2n+1}$ occurs in ${s_{2n+1}}^\smallfrown (c)^\smallfrown s_{2n+1}$ not as demonstrated, again contradicting condition (c).
For the rest of the proof we assume that $r_n\geq 3$ for all $n\in\N$.
Let $E_0$ be the set of all $x\in X$ for which there is $n\in\N$ such that the position 0 is contained in an expected occurrence of $v_n$ in $x$. Let $E=\bigcap_{k\in\Z} \sigma^k[E_0]$. Then $\mu(E)=1$. In fact, by condition (b), $X\setminus E_0$ is finite. Thus $X\setminus E$ is at most countable.
We define a labeling function $\lambda_n: E\to \{1, \dots, r_n, \infty\}$ for each $n\in\N$. Let $n\in\N$ and $x\in E$ be given. If the position $0$ is not contained in an expected occurrence of $v_n$ in $x$, put $\lambda_n(x)=\infty$. Otherwise, the position $0$ is contained in an expected occurrence of $v_n$ in $x$, and it follows that the expected occurrence of $v_n$ (containing the position 0) is in turn contained in an expected occurrence of $v_{n+1}$ in $x$. Since there are exactly $r_n$ many expected occurrence of $v_n$ in $v_{n+1}$, we may speak of the $i$-th occurrence of $v_n$ in $v_{n+1}$ for $1\leq i\leq r_n$. Now put $\lambda_n(x)=i$ if the expected occurrence of $v_n$ containing position 0 is the $i$-th occurrence of $v_n$ in the expected occurrence of $v_{n+1}$ in $x$ containing the position 0. For any $x\in E$, $\lambda_n(x)<\infty$ for large enough $n$. We prove some basic facts about the labeling functions.
\[orbit\] If $x, y\in E$ are such that $\lambda_n(x)=\lambda_n(y)$ for all $n\geq N$ for some $N\in\N$, then $x$ and $y$ are in the same $\sigma$-orbit, i.e., there is $k\in\Z$ such that $\sigma^k(x)=y$.
We may assume without loss of generality that $\lambda_N(x)=\lambda_N(y)<\infty$. For $n\geq N$, let $l^x_n$ be the beginning position of the expected occurrence of $v_n$ in $x$ containing the position 0, and $l^y_n$ be the beginning position of the expected occurrence of $v_n$ in $y$ containing the position 0. Let $k=l^x_N-l^y_N$. Then by an easy induction on $n\geq N$ we have that for all $n\geq N$, $k=l^x_n-l^y_n$. This implies that $\sigma^k(x)=y$.
\[same\] Let $x, y\in E$ and $n\in\N_+$. Suppose that $\lambda_{n-1}(x)=\lambda_{n-1}(y)<\infty$. Let $[c,d]$ be the interval of overlap between the expected occurrence of $v_n$ in $x$ containing the position 0 and the expected occurrence of $v_n$ in $y$ containing the position 0. That is, letting $l^x_n$ be the beginning position of the expected occurrence of $v_n$ in $x$ containing the position 0 and $l^y_n$ be the beginning position of the expected occurrence of $v_n$ in $y$ containing the position 0, then $[c,d]=[l^x_n, l^x_n+{{\mbox{\rm lh}}}(v_n)]\cap [l^y_n, l^y_n+{{\mbox{\rm lh}}}(v_n)]$. Then $d-c\geq {{\mbox{\rm lh}}}(v_{n-1})$.
Suppose $\lambda_{n-1}(x)=\lambda_{n-1}(y)=i$. Then the $i$-th occurrence of $v_{n-1}$ in the expected occurrence of $v_n$ in $x$ containing $0$ has a nonempty overlap with the $i$-th occurrence of $v_{n-1}$ in the expected occurrence of $v_n$ in $y$ containing $0$. This implies that for all $1\leq j\leq r_{n-1}$, the $j$-th occurrence of $v_{n-1}$ in the expected occurrence of $v_n$ in $x$ containing $0$ has a nonempty overlap with the $j$-th occurrence of $v_{n-1}$ in the expected occurrence of $v_n$ in $y$ containing $0$. It follows that the length of $[l^x_n, l^x_n+{{\mbox{\rm lh}}}(v_n)]\setminus [c,d]$ cannot be greater than ${{\mbox{\rm lh}}}(v_{n-1})$. Since $r_{n-1}\geq 2$, we have $d-c\geq {{\mbox{\rm lh}}}(v_n)-{{\mbox{\rm lh}}}(v_{n-1})\geq {{\mbox{\rm lh}}}(v_{n-1})$.
Define another labeling function $\kappa_n: E\to \{-1, 0, +1, \infty\}$ for all $n\in\N$ as follows: $$\kappa_n(x)=\left\{ \begin{array}{ll} -1 & \mbox{ if $\lambda_n(x)=1$}, \\
0 & \mbox{ if $2\leq \lambda_n(x)\leq r_n-1$}, \\
+1 & \mbox{ if $\lambda_n(x)=r_n$}, \\
\infty & \mbox { if $\lambda_n(x)=\infty$.}\end{array}\right.$$
\[density\] For $\mu$-a.e. $x\in X$, the set $\{n\in\N: \kappa_n(x)=0\}$ has density at least 1/3. In particular, for $\mu$-a.e. $x\in X$, there are infinitely many $n\in \N$ such that $\kappa_n(x)=0$.
For each $N\in\N_+$ let $E_N=\{ x\in E: \kappa_N(x)<\infty\}$. Then $E_N\subseteq E_{N+1}$ for all $N\in\N_+$ and $E=\bigcup_{N\in\N_+} E_N$. For each $n\in\N_+$ and $\iota\in\{-1, 0, +1\}$, let $E_{n,\iota}=\{x\in E_n: \kappa_n(x)=\iota\}$. Then $\mu(E_{n,0})\geq \mu(E_n)/3\geq \mu(E_N)/3$ if $n\geq N$. Also, on each $E_N$ the functions $\kappa_N$, $\kappa_{N+1}$, $\dots$, are independent. By the law of large numbers, for each $N\in\N_+$ and $\mu$-a.e. $x\in E_N$, $\{ n\geq N: \kappa_n(x)=0\}$ has density at least $1/3$. It follows that for $\mu$-a.e. $x\in X$, $\{n\in\N: \kappa_n(x)=0\}$ has density at least $1/3$.
\[center\]Let $x, y\in E$ and $n\in\N_+$. Suppose that $\kappa_{n-1}(x)=0$ and $\kappa_{n-1}(y)<\infty$. Let $[c, d]$ be the interval of overlap between the expected occurrence of $v_n$ in $x$ containing the position 0 and the expected occurrence of $v_n$ in $y$ containing the position 0. Then $d-c\geq {{\mbox{\rm lh}}}(v_{n-1})$.
Suppose $\lambda_{n-1}(x)=i$. Then $1<i<r_n$. A moment of reflection gives that, in the expected occurrence of $v_n$ in $x$ containing the position 0, either the first expected occurrence of $v_{n-1}$ overlaps with the expected occurrence of $v_n$ in $y$ containing the position 0, or the last expected occurrence of $v_{n-1}$ overlaps with the expected occurrence of $v_n$ in $y$ containing the position 0. This shows that $d-c\geq {{\mbox{\rm lh}}}(v_{n-1})$.
We now proceed to set up the proof for minimal self-joinings of order 2. Let $\overline{\mu}$ be an ergodic joining on $X\times X$ with marginals $\mu$. Suppose $\overline{\mu}$ is not an off-diagonal measure. We need to show that $\overline{\mu}=\mu\times\mu$. Again by Lemma 6.14 of [@RudolphBook] it suffices to find some nonzero $k\in\Z$ such that $\overline{\mu}$ is $(\sigma^k\times {{\mbox{\rm id}}})$-invariant, since by our condition (d), $(X, \mu, \sigma^k)$ is ergodic. We let $(x, y)\in X\times X$ be a $\overline{\mu}$-generic pair in the sense that the following hold:
- $(x, y)$ satisfies the ergodic theorem for $\overline{\mu}$;
- $x, y\in E$ are not in the same $\sigma$-orbit; and
- the set $\{n\in\N: \kappa_n(x)=0\}$ has positive density.
Each of these properties are satisfied by $\overline{\mu}$-a.e. pairs in $X\times X$.
As in the proof of Theorem \[maindisjoint\] it suffices to find $a_n, b_n, c_n, d_n, e_n, k_n\in\Z$ for all $n\in\N$, a positive integer $K\geq 1$ and a real number $\alpha>0$ so that for all $n\in\N$,
1. $0<|k_n|\leq K$;
2. $a_n\leq 0\leq b_n$ and $\lim_n (b_n-a_n)=+\infty$;
3. $a_n\leq c_n\leq d_n\leq b_n$ and $a_n\leq c_n+e_n\leq d_n+e_n\leq b_n$;
4. $d_n-c_n\geq \alpha(b_n-a_n)$;
5. for all $c_n\leq i\leq d_n$, $x(i)=x(i+k_n+e_n)$ and $y(i)=y(i+e_n)$.
Applications of Lemma \[tech\] and its variations will give that $\overline{\mu}$ is $(\sigma^k\times {{\mbox{\rm id}}})$-invariant, and so $\overline{\mu}=\mu\times\mu$.
Let $K=S$ where $S$ is the bound in condition (b). Let $$\alpha=\displaystyle\frac{1}{2(R+1)^2}$$ where $R$ is the bound in condition (a). Fix an $n_0\in\N$ such that ${{\mbox{\rm lh}}}(v_{n_0})>RS$. Let $$D=\{ n\in\N: n> n_0, \lambda_n(x), \lambda_n(y)<\infty \mbox{ and } \lambda_n(x)\neq \lambda_n(y)\}.$$ Since $x$ and $y$ are not in the same $\sigma$-orbit, $D$ is infinite by Lemma \[orbit\].
\[cases\] There is an infinite $D'\subseteq D$ such that for all $n\in D'$, either $\lambda_{n-1}(x)=\lambda_{n-1}(y)<\infty$, or both $\kappa_{n-1} (x)=0$ and $\kappa_{n-1}(y)<\infty$.
If $\N\setminus D$ is infinite, then $$D'=\{n\in\N:n> n_0, \lambda_{n-1}(x)=\lambda_{n-1}(y)<\infty \mbox{ and } \lambda_n(x)\neq \lambda_n(y)\}$$ is infinite and $D'\subseteq D$. If $\N\setminus D$ is finite, then $$D'=\{n\in\N: n> n_0, \kappa_{n-1}(x)=0,\ \kappa_{n-1}(y)<\infty \mbox{ and } \lambda_n(x)\neq\lambda_n(y)\}$$ has positive density and therefore is infinite.
Fix an infinite $D'\subseteq D$ as in the above lemma. It suffices to define $a_n, b_n, c_n, d_n, e_n, k_n\in\Z$ for all $n\in D'$ as required. For the rest of the proof fix $n\in D'$.
Let $[c, d]$ be the interval of overlap between the expected occurrence of $v_n$ in $x$ containing the position 0 and the expected occurrence of $v_n$ in $y$ containing the position $0$. By Lemmas \[cases\], \[same\] and \[center\], we have that $$d-c\geq {{\mbox{\rm lh}}}(v_{n-1})\geq \frac{{{\mbox{\rm lh}}}(v_{n-1})}{R{{\mbox{\rm lh}}}(v_{n-1})+RS}{{\mbox{\rm lh}}}(v_n)\geq \frac{1}{R+1}{{\mbox{\rm lh}}}(v_n).$$
Define $$a_n=-{{\mbox{\rm lh}}}(v_{n+1}) \mbox{ and } b_n={{\mbox{\rm lh}}}(v_{n+1}).$$ Let $l=l^y_{n+1}$. Then the expected occurrence of $v_{n+1}$ in $y$ containing the position 0 starts at the position $l$. Suppose this occurrence finishes at position $m$. Then $a_n\leq l\leq 0\leq m\leq b_n$.
Let $i_y=\lambda_n(y)$. Then in $y$, the position 0 is contained in the $i_y$-th occurrence of $v_n$ in the expected occurrence of $v_{n+1}$ from position $l$ to position $m$. Correspondingly in $x$, we examine the $r_n$ many consecutive expected occurrences of $v_n$ so that the position 0 is contained in the $i_y$-th occurrence of $v_n$. Suppose the following word is observed: $$v_n1^{p(1)}v_n1^{p(2)}\dots v_n1^{p(r_n-1)}v_n.$$ Since $\lambda_n(x)\neq\lambda_n(y)$, this observed word is not contained in a single expected occurrence of $v_{n+1}$. Rather, it is contained in a subword of $x$ of the form $v_{n+1}1^qv_{n+1}$, where each demonstrated occurrence of $v_{n+1}$ is expected. By comparison, we obtain that $p$ is a subword of $s_n^\smallfrown (q)^\smallfrown s_n$, and that $p$ does not coincide with any of the two demonstrated occurrences of $s_n$. By our condition (c), this implies that $p\neq s_n$.
Let $i_0$ be such that $1\leq i_0\leq r_n-1$ and $p(i_0)\neq s_n(i_0)$ and so that $|i_0-i_y|$ is the least. For definiteness first assume that $i_0\geq i_y$. In this case let $$h=(i_0-i_y){{\mbox{\rm lh}}}(v_n)+\sum_{i=i_y}^{i_0-1}s_n(i).$$ Then in $x$ there is an occurrence of the word $v_n1^{p(i_0)}v_n$ beginning at the position $l^x_n+h$. Similarly, in $y$ there is an occurrence of the word $v_n1^{s_n(i_0)}v_n$ beginning at the position $l^y_n+h$. Define $[c_n, d_n]$ to be the interval of overlap between the these first demonstrated occurrences of $v_n$ in $x$ and in $y$. Then we have in fact $c_n=c+h$ and $d_n=d+h$. So $$d_n-c_n=d-c\geq \frac{1}{R+1}{{\mbox{\rm lh}}}(v_n).$$ Define $$e_n={{\mbox{\rm lh}}}(v_n)+s_n(i_0)$$ and $$k_n=p(i_0)-s_n(i_0).$$ We have that $x\!\upharpoonright\![c_n,d_n]=x\!\upharpoonright [c_n+k_n+e_n, d_n+k_n+e_n]$ and $y\!\upharpoonright\![c_n,d_n]=y\!\upharpoonright\![c_n+e_n, d_n+e_n]$. Since $[c_n, d_n], [c_n+e_n, d_n+e_n]\subseteq [l,m]\subseteq [a_n, b_n]$ and $$\displaystyle\frac{d_n-c_n}{b_n-a_n}\geq \frac{{{\mbox{\rm lh}}}(v_n)}{(R+1)\cdot 2{{\mbox{\rm lh}}}(v_{n+1})}\geq \frac{1}{2(R+1)^2}=\alpha,$$ our proof is complete in this case.
The alternative is the case $i_0<i_y$. In this case we let instead $$h=(i_0-i_y+1){{\mbox{\rm lh}}}(v_n)-\sum_{i=i_0+1}^{i_y-1}s_n(i)\leq 0.$$ Then in $x$ there is an occurrence of the word $v_n1^{p(i_0)}v_n$ where the beginning of the second demonstrated occurrence is at the position $l^x_n+h$. Similarly, in $y$ there is an occurrence of the word $v_n1^{s_n(i_0)}v_n$ where the beginning of the second demonstrated occurrence is at the position $l^y_n+h$. We similarly let $[c_n, d_n]$ be the interval of overlap of these second occurrences of $v_n$ in $x$ and in $y$. Then $c_n=c+h$ and $d_n=d+h$. Define $$e_n=-{{\mbox{\rm lh}}}(v_n)-s_n(i_0)$$ and $$k_n=-p(i_0)+s_n(i_0).$$ We still have that $d_n-c_n\geq {{\mbox{\rm lh}}}(v_n)/(R+1)$, and the proof is similarly completed.
We have thus shown that $(X, \mu, \sigma)$ has minimal self-joinings of order 2, and therefore minimal self-joinings of all orders.
Ryzhikov’s theorem
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As a corollary to Theorem \[mainMSJ\], we obtain the following theorem of Ryzhikov [@Ryzhikov] on minimal self-joinings for non-rigid, totally ergodic, bounded rank-one transformations.
\[Ryzh\] Let $T$ be a bounded rank-one transformation. Then $T$ has minimal self-joinings of all orders if and only if $T$ is non-rigid and totally ergodic.
It is easy to verify that having minimal self-joinings implies mild mixing (having no rigid factors), which implies non-rigidity. Having minimal self-joinings also implies weak mixing, which implies total ergodicity. Thus the two conditions are necessary.
For the sufficiency, let $T$ be a bounded rank-one transformation with cutting and spacer parameters $(r_n:n\in\N)$ and $(s_n: n\in\N)$. Assume that $T$ is non-rigid and totally ergodic. By Theorem \[TC\], $T$ is canonically bounded. Thus we may assume without loss of generality that $(r_n:n\in\N)$ and $(s_n:n\in\N)$ are canonical cutting and spacer parameters, which are also bounded. Let $(v_n: n\in\N)$ be the canonical generating sequence given by $(r_n:n\in\N)$ and $(s_n:n\in\N)$. We inductively define an infinite sequence $(n_k:k\in\N)$ of natural numbers as follows. Define $n_0=0$. In general, assume $n_k$, $k\geq 0$, has been defined. Define $n_{k+1}=n_k+2$ if $s_{n_k+1}$ is not constant, and define $n_{k+1}=n_k+3$ otherwise. Let $v_k'=v_{n_k}$ for all $k\in\N$. Then $(v'_n:n\in\N)$ is a subsequence of $(v_n:n\in\N)$, which still generates $T$. Let $(r'_n:n\in\N)$ and $(s'_n:n\in\N)$ be the cutting and spacer parameters corresponding to $(v'_n:n\in\N)$. Since $n_k<n_{k+1}\leq n_k+3$ for all $k\in\N$, these newly defined cutting and spacer parameters are still bounded.
To prove the corollary, we will apply Theorem \[mainMSJ\] to $(r'_n:n\in\N)$ and $(s'_n:n\in\N)$. The only condition to verify is (c), that is, for all $n\in\N$ and $c\in\N$, there are only two occurrences of $s'_n$ in ${s'_n}^\smallfrown(c)^\smallfrown s'_n$. Note that for every $k>0$, $s'_k$ is of the form $${s_{n_k}}^\smallfrown (u(1))^\smallfrown {s_{n_k}}^\smallfrown\cdots {}^\smallfrown (u(m))^\smallfrown{s_{n_k}}$$ where $u$ is either $s_{n_k+1}$ or $${s_{n_k+1}}^\smallfrown (s_{n_k+2}(1))^\smallfrown {s_{n_k+1}}^\smallfrown\cdots {}^\smallfrown(s_{n_k+2}(r_{n_k+2}-1))^\smallfrown {s_{n_k+1}}.$$ As in the proof of Corollary \[CBmain\], $u$ is not constant in either cases: in the former case $s_{n_k+1}$ is assumed not to be constant, and in the latter case $u$ corresponds to the way $v_{n_k+2}$ is built from $v_{n_k}$, and therefore is not constant since $v_{n_k+1}$ is assumed to be on the canonical generating sequence. Now if there is $c\in\N$ so that $s'_n$ occurs in ${s'_n}^\smallfrown (c)^\smallfrown {s'_n}$ not as demonstrated, then by a similar argument as the proof of Lemma \[perp\], it would follow that $u$ is constant, a contradiction.
This completes the proof of Corollary \[Ryzh\].
Concluding remarks
==================
Some results of this paper are applicable in a broader context than stated. We have noted that Theorems \[maindisjoint\], \[mainMSJ\] and Corollary \[CBmain\] can be strengthened with “partial total ergodicity” assumptions replacing the total ergodicity assumptions, which we denoted by (d’) and (2’) respectively. Here we note that Theorems \[mainiso\], \[maindisjoint\] and Corollary \[CBmain\] can be further strengthened with an “eventual commensurability” assumption replacing the commensurability assumption. For instance, Theorem \[mainiso\] can be strengthened as follows.
\[mainisogen\] Let $(r_n: n \in \N)$ and $(s_n: n \in \N)$ be cutting and spacer parameters giving rise to symbolic rank-one system $(X, \mu, \sigma)$. Let $(v_n:n\in\N)$ be the generating sequence given by $(r_n:n\in\N)$ and $(s_n:n\in\N)$.
Let $(q_n: n \in \N)$ and $(t_n: n \in \N)$ be cutting and spacer parameters giving rise to symbolic rank-one system $(Y, \nu, \sigma)$. Let $(w_n:n\in\N)$ be the generating sequence given by $(q_n:n\in\N)$ and $(t_n:n\in\N)$.
Suppose the following hold.
1. The two sets of parameters are “eventually commensurate”, i.e., there are $N, M\in\N$ such that ${{\mbox{\rm lh}}}(v_N)={{\mbox{\rm lh}}}(w_M)$ and for all $n\in\N$, $r_{N+n}=q_{M+n}$ and $$\sum_{i=1}^{r_{N+n}-1} s_{N+n}(i) = \sum_{i=1}^{q_{M+n}-1} t_{M+n}(i).$$
2. There is an $S \in \N$ such that for all $n$ and all $1\leq i \leq r_n-1$, $$s_n(i) \leq S \textnormal{ and } t_n(i) \leq S.$$
3. There is an $R \in \N$ such that for infinitely many $n$, $$r_n \leq R \textnormal{ and } s_n \perp t_n.$$
Then $(X, \mu, \sigma)$ and $(Y, \nu, \sigma)$ are not isomorphic.
Theorem \[maindisjoint\] and Corollary \[CBmain\] allow similar generalizations. It should be clear that the proofs of these generalizations are identical to the proofs given in [@Hill] and this paper.
It is, however, not clear how to determine if two rank-one transformations allow eventually commensurate cutting and spacer parameters. Of course, if two rank-one transformations do not allow eventually commensurate parameters, then they are not isomorphic. We conjecture that there is a Borel procedure for this determination.
Acknowledgments {#acknowledgments .unnumbered}
===============
The first author acknowledges the US NSF grant DMS-1201290 for the support of his research. He also acknowledges the support of the Issac Newton Institute (INI) for Mathematical Sciences at the University of Cambridge for a research visit during which a substantial part of this paper was written. He was a Visiting Fellow to the Mathematical, Foundational and Computational Aspects of the Higher Infinite (HIF) program at the INI, and he thanks the organizers of the program and the Scientific Advisory Committee for this opportunity. Both authors would like to thank Eli Glasner for useful discussions on the topics of the paper and for providing the references related to Theorem \[allorders\]. Both authors also benefit from discussions with Matt Foreman, Cesar Silva, and Benjy Weiss as a part of a SQuaRE program at the American Institute of Mathematics (AIM) focusing on the isomorphism problem of rank-one transformations.
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, An uncountable family of prime transformations not isomorphic to their inverses, unpublished manuscript.
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, [*Fundamentals of Measurable Dynamics. Ergodic Theory on Lebesgue Spaces*]{}. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1990.
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|
---
abstract: 'We present a comprehensive multi-wavelength analysis of the young cluster NGC 1624 associated with the region Sh2-212 using optical $UBVRI$ photometry, optical spectroscopy and GMRT radio continuum mapping along with the near-infrared (NIR) $JHK$ archival data. From optical observations of the massive stars, reddening $E(B-V)$ and distance to the cluster are estimated to be 0.76 - 1.00 mag and $6.0 \pm 0.8$ kpc, respectively. Present analysis yields a spectral class of O6.5V for the main ionizing source of the region and the maximum post-main-sequence age of the cluster is estimated as $\sim$ 4 Myr. Detailed physical properties of the young stellar objects (YSOs) in the region are analyzed using a combination of optical/NIR colour-colour and colour-magnitude diagrams. The distribution of YSOs in $(J-H)/ (H-K)$ NIR colour-colour diagram shows that a majority of them have $A_V$ $\le$ 4 mag. However, a few YSOs show $A_V$ values higher than 4 mag. Based on the NIR excess characteristics, we identified 120 probable candidate YSOs in this region which yield a disk frequency of $\sim$ 20%. However, this should be considered as a lower limit. These YSOs are found to have an age spread of $\sim$ 5 Myr with a median age of $\sim$ 2-3 Myr and a mass range of $\sim$ 0.1 - 3.0 $M_\odot$. A significant number of YSOs are located close to the cluster centre and we detect an enhanced density of reddened YSOs located/projected close to the molecular clumps detected by Deharveng et al. (2008) at the periphery of NGC 1624. This indicates that the YSOs located within the cluster core are relatively older in comparison to those located/projected near the clumps. From the radio continuum flux, spectral class of the ionizing source of the ultra-compact (UC ) region at the periphery of Sh2-212 is estimated to be $\sim$ B0.5V. From optical data, slope of the mass function (MF) $\Gamma$, in the mass range $1.2 \le M/M_{\odot}<27$ can be represented by a single power law with a slope -1.18 $\pm$ 0.10, whereas the NIR data in the mass range $0.65 \le M/M_{\odot}<27$ yields $\Gamma$ = -1.31 $\pm$ 0.15. Thus the MF agrees fairly with the Salpeter value. The slope of the $K$-band luminosity function (KLF) for the cluster is found to be 0.30 $\pm$ 0.06 which is in agreement with the values obtained for other young clusters.'
author:
- |
Jessy Jose $^{1}$[^1], A.K. Pandey$^{1}$, K. Ogura$^2$, D.K. Ojha$^3$, B.C. Bhatt$^4$, M.R. Samal$^{1}$, N. Chauhan$^{1}$, D.K. Sahu$^4$ and P.S. Rawat$^5$\
\
$^1$ Aryabhatta Research Institute of observational sciencES (ARIES), Manora Peak, Naini Tal, 263129, India\
$^2$ Kokugakuin University, Higashi, Shibuya-ku, Tokyo, 150-8440, Japan\
$^3$ Tata Institute of Fundamental Research, Mumbai (Bombay), 400 005, India\
$^4$ CREST, Indian Institute of Astrophysics, Koramangala, Bangalore, 560 034, India\
$^5$ Department of Physics, D.S.B. Campus, Kumaun University, Naini Tal, India\
title: 'A multi-wavelength census of stellar contents in the young cluster NGC 1624'
---
\[firstpage\]
stars: formation $-$ stars: luminosity function, mass function $-$ stars: pre$-$main$-$sequence $-$ open clusters and associations: individual: NGC 1624.
Introduction {#intro}
============
The study of the star formation process and the origin of stellar initial mass function (IMF), defined as the distribution of stellar masses at the time of birth, are key issues in astrophysics. Since majority of stars tend to form in clusters or groups, young star clusters are considered to be the fundamental units of star formation (Lada & Lada 2003). Young star clusters are useful tool to study the IMF as they contain statistically significant number of young stars of rather similar age spanning a wide range of masses. Since these objects are not affected by the dynamical evolution as the ages of these objects are significantly less in comparison to their dynamical evolution time, the present day mass function (MF) of these objects can be considered as the IMF. However, a recent study by Kroupa (2008) argues that even in the youngest clusters, it is difficult to trace the IMF, as clusters evolve rapidly and therefore eject a fraction of their members even at a very young age.
In the last decade, there have been a large number of studies in great detail in several young clusters within 2 kpc of the Sun investigating these issues (e.g., Lada & Lada 2003, Pandey et al. 2008, Jose et al. 2008). Although the theoretical expectation is that the IMF of a cluster should depend on the location, size, metallicity, density of the star forming environment and other conditions such as temperature or pressure (Zinnecker 1986; Larson 1992; Price & Podsiadlowski 1995), for clusters located within 2 kpc, there is no compelling evidence for variation in the stellar IMF above the solar mass (e.g. Meyer et al. 2000; Kroupa 2002; Chabrier 2005).
With the aim of understanding the star formation process and IMF in/around young star clusters, we selected an young cluster NGC 1624 ($\alpha_{2000}$ = $04^{h}40^{m}38^{s}.2$; $\delta_{2000}$ = $+50^{\circ}27^{\prime}36^{\prime\prime}$; l=155.36; b=+2.62) associated with the bright optical region Sh2-212 (Sharpless 1959). A colour composite image using the bands $B$, blue; [ ]{}, green; and [ ]{}, red for an area $ \sim 10\times10$ arcmin$^2$ centered at NGC 1624 is shown in Fig. \[cfht\] (left panel), where the cluster seems to be embedded in the region. The cluster is located significantly above the formal galactic plane ([*Z*]{} $\sim$ 250 pc) for an estimated distance of 6.0 kpc (cf. Sect. \[distance\]). The kinematic and spectrophotometric distances to NGC 1624 vary from 4.4 kpc (Georgelin & Georgelin 1970) to 10.3 kpc (Chini & Wink 1984). An IRAS point source (IRAS 04366+5022) with colours similar to that of the ultra-compact (UC) region (Wood & Churchwell 1989) is located at the periphery of Sh2-212. The molecular gas distribution of this region was mapped by CO observations (Blitz et al. 1982; Leisawitz et al. 1989; Deharveng et al. 2008).
Particularly, Deharveng et al. (2008) studied the region using $J = 2-1$ lines of $^{12}$CO and $^{13}$CO and reported a bright and thin semi-circular structure of molecular gas (in the velocity range -34.0 kms$^{-1}$ to -32.7 kms$^{-1}$) in $^{13}$CO at the rear side of Sh2-212 along with a filamentary structure (-36.8 kms$^{-1}$ to -35.9 kms$^{-1}$) extending from southeast to northwest. The semi-circular ring itself contains several molecular clumps, the most massive of which (-36.1 kms$^{-1}$ to -35.1 kms$^{-1}$) contains a massive young stellar object (YSO) which is the exciting source of the associated UCregion (see Fig. 1). They concluded that Sh2-212 is a good example of massive-star formation triggered via the collect and collapse process. They also reported the flow of ionized gas and suggested that this may be the indication of ‘Champagne flow’ towards the north of Sh2-212. A careful view of Fig. 1 (right panel) reveals that the central region of NGC 1624 is relatively devoid of gas and dust, whereas the outer regions, particularly east, south-east and west seem to be obscured by molecular gas. However, it is to be noted that the semi-circular structure containing clumps is located at the rear side of the cluster.
The present study is an attempt to understand the stellar content, young stellar population and the form of IMF/ $K$-band luminosity function (KLF) of the cluster NGC 1624 associated with Sh2-212 using our optical and radio continuum observations along with the near-infrared (NIR) archival data. In Sections 2 and 3, we describe the observations, data reductions and archival data used in the present work. Sections 4 to 8 describe various cluster parameters and young stellar properties derived using optical, NIR and radio continuum data. Sections 9 and 10 describe the IMF and KLF of the region and in section 11 we have summarized the results.
OBSERVATIONS AND DATA REDUCTIONS
================================
In the following sections we describe the observations and data reductions carried out in order to have a detailed study of NGC 1624.
Optical CCD Photometry {#obs}
----------------------
The CCD $UBVRI$ observations of NGC 1624 were carried out using Hanle Faint Object Spectrograph and Camera (HFOSC) of the 2-m Himalayan Chandra Telescope (HCT) of Indian Astronomical Observatory (IAO), Hanle, India on 2004 November 3. The 2048 $\times$ 2048 CCD with a plate scale of 0.296 arcsec pixel$^{-1}$ covers an area of $\sim$ 10$\times$10 arcmin$^2$ on the sky. We took short and long exposures in all filters to avoid saturation of bright stars. PG 0231 field from Landolt (1992) was observed to determine atmospheric extinction as well as to photometrically calibrate the CCD frames on the same night. The log of observations is tabulated in Table \[obslog\].
The CCD frames were bias-subtracted and flat-field corrected in the standard manner using various tasks available under IRAF[^2]. Aperture photometry was done for the standard stars of PG 0231 field and the following calibration equations were derived using a least-squares linear regression:\
$(U-B) = (1.269\pm 0.020) (u-b) - (2.617\pm 0.026)$,\
$(B-V)=(0.915\pm 0.016) (b-v) - (0.284\pm0.012)$,\
$(V-R) = (1.056\pm 0.013) (v-r) - (0.011\pm0.010)$,\
$(V-I) = (1.022\pm 0.009) (v-i) + (0.188\pm0.008)$,\
$V = v+(0.024\pm 0.011) (V-I) - (0.495\pm0.013)$,\
where, $u,b,v,r,i$ are the instrumental magnitudes corrected for the atmospheric extinctions and $U,B,V,R,I$ are the standard magnitudes. The standard deviations of the residuals, $\Delta$, between standard and transformed $V$ magnitudes, $(U-B)$, $(B-V)$, $(V-R)$ and $(V-I)$ colours of standard stars were 0.020, 0.045, 0.018, 0.014 and 0.021 mag, respectively. Different frames of the cluster region having same exposure time and observed with the same filters were averaged. Photometry of cleaned frames was carried out using the DAOPHOT-II (Stetson 1987) profile-fitting software.
We repeated the observations of NGC 1624 in $V$ and $I_c$ filters to get deeper photometry on 2006 December 12 using the 104-cm Sampurnanand Telescope (ST) of Aryabhatta Research Institute of observational sciencES (ARIES), Naini Tal, India. Log of the observations is given in Table \[obslog\]. The 2048 $\times$ 2048 CCD with a plate scale of 0.37 arcsec pixel$^{-1}$ covers a field of $\sim 13\times13$ arcmin$^2$ on the sky. To improve the signal to noise ratio (S/N), the observations were carried out in binning mode of $2\times2$ pixel. Secondary standards from the HCT observations were used to calibrate the data taken with ST. A combined photometry catalog is made using these two observations and this catalog has typical photometric errors of the order of $\sim$ 0.01 mag at brighter end ($V\sim$ 15), whereas the errors increase towards the fainter end ($\sim$ 0.04 at $V$ $\sim$ 21). The catalog is available in electronic form and a sample table is given in Table \[optdata\].
In order to check the accuracy of the present photometry, we compared our photometry with the $UBV$ photometry of 14 stars carried out by Moffat et al. (1979). The mean and standard deviation of the difference between Moffat’s and our photometry in $V$, $U-B$ and $B-V$ are $0.008 \pm
0.006$, $0.005 \pm 0.015$ and $0.004 \pm 0.006$, respectively, suggesting that the two photometries are in good agreement.
To study the luminosity function (LF)/MF, it is necessary to take into account the incompleteness of the present data that could occur due to various factors (e.g., crowding of the stars). We used ADDSTAR routine of DAOPHOT-II to determine the completeness factor (CF). The procedure has been outlined in detail in our earlier work (see e.g., Pandey et al. 2001). Briefly, we randomly added artificial stars to both $V$ and $I$ images taken with ST in such a way that they have similar geometrical locations but differ in $I$ brightness according to mean $(V-I)$ colour ($\sim 1.5$ mag) of the data sample. Luminosity distribution of artificial stars was chosen in such a way that more number of stars were inserted towards the fainter magnitude bins. The frames were reduced using the same procedure used for the original frames. The ratio of the number of stars recovered to those added in each magnitude interval gives the CF as a function of magnitude. Minimum value of the CF of the pair (i.e., $V$- and $I$-bands ) for the cluster region and field region (outside the cluster region), given in Table \[cf\_opt\], is used to correct the data incompleteness.
Spectroscopic observations
--------------------------
Low resolution optical spectroscopic observations of 4 optically bright sources of NGC 1624 were made using HFOSC of HCT. The log of observations is given in Table \[obslog\]. The spectra in the wavelength range 3800-6840 $\AA$ with a dispersion of 1.45 $\AA$ pixel$^{-1}$ were obtained using low resolution grism 7 with a slit having width 2$^{\prime\prime}$. One-dimensional spectra were extracted from the bias-subtracted and flat-field corrected images using the optimal extraction method in IRAF. Wavelength calibration of the spectra were done using FeAr and FeNe lamp sources. Spectrophotometric standard (Feige 110) was observed on 2006 September 08 and flux calibration was applied to the star observed on the same night.
Radio Continuum Observations
----------------------------
Radio continuum observations at 1280 MHz were carried out on 2007 July 17 using the Giant Metrewave Radio Telescope (GMRT), India. GMRT has a ‘Y’ shaped hybrid configuration of 30 antennae, each of 45 m diameter. Details of the GMRT antennae and their configurations can be found in Swarup et al. (1991). For the observations, the primary flux density calibrators used were 3C48 and 3C286. NRAO Astronomical Image Processing System (AIPS) was used for the data reduction. The data were carefully checked for radio frequency interference or other problems and suitably edited. Self calibration was carried out to remove the residual effects of atmospheric and ionospheric phase corruptions and to obtain the improved maps.
Archival data
=============
Near-infrared data from 2MASS
-----------------------------
NIR $JHK_s$ data for point sources within a radius of 10 arcmin around NGC 1624 have been obtained from Two Micron All Sky Survey (2MASS) Point Source Catalog (PSC) (Cutri et al. 2003). To improve photometric accuracy, we used photometric quality flag (ph$\_$qual = AAA) which gives a S/N $\ge$ 10 and a photometric uncertainty $ <$ 0.10 mag. This selection criterion ensures best quality detection in terms of photometry and astrometry as given on the 2MASS website. The $JHK_s$ data were transformed from 2MASS system to the California Institute of Technology (CIT) system using the relations given by Carpenter (2001). We used this data set to calibrate the NIR archival data from Canada-France-Hawaii Telescope (CFHT) (see Sect. \[cfhtdata\]) and also to produce the radial density profile of NGC 1624 (see Sect. \[rd\]).
Near-infrared data from CFHT {#cfhtdata}
----------------------------
NIR data for the region were obtained from the Canadian Astrophysical Data Centre’s (CADC) archive program. The NIR observations of the region were taken on 2002 October 20 (PI: L. Deharveng) using the instrument CFHT-IR at the 3.56-m CFHT. The 1024 $\times$ 1024 pixel HgCdTe detector with a plate scale of 0.211 arcsec/pixel was used for the observations. The catalog by Deharveng et al. (2008) lists a total of 891 sources in $JHK$ bands. Since our aim was to study the KLF of the region, where the estimation of the completeness of the photometry (ref. Sect. \[obs\]) was necessary, we re-reduced the CFHT observations. We used dithered images at 9 different locations having 10 frames at each position around the UC region of this field. Flat frames and sky frames were made from the median combined object frames. The sky subtracted and flat field corrected dithered images in each band were aligned and then combined to achieve a higher S/N. The final mosaic image covers an area of $5^{\prime}.2 \times 5^{\prime}.2$ with the UCregion at the centre and is shown in Fig. \[cfht\].
Photometry of the processed images were obtained using the DAOPHOT-II package in IRAF. Since the region was crowded, we performed PSF photometry on the images. The 2MASS counterparts of the CFHT sources were searched within a match radius of 1 arcsec. The CFHT instrumental magnitudes were compared to the selected 2MASS magnitudes to define a slope and zero point for the photometric calibration. The rms scatter between the calibrated CFHT and 2MASS data (i.e., $2MASS - CFHT$ data) for the $J, H$ and $K$-bands were 0.07, 0.08 and 0.06, respectively. In order to check the photometric accuracy, we compared our photometry with the photometry reported by Deharveng et al. (2008). The average dispersion between these two samples was $\sim$ 0.1 mag in $JHK$ bands with absolutely no shift, which shows that the present photometry is in agreement with the previous study. To ensure good photometric accuracy, we limited our sample with those stars having error $<$ 0.15 mag in all three bands and thus we obtained photometry for 951 sources in $J, H$ and $K$-bands. Additional 31 sources detected only in the $H$ and $K$ bands ($J$ drop out sources) having error $<$ 0.15 mag are also included in our analysis. Data of three saturated sources have been taken from the 2MASS catalog. The detection limits were 19.0, 18.4 and 18.0 mag for $J$, $H$ and $K$-bands, respectively. We combined the optical and NIR catalog within a match radius of 1 arcsec and the final catalog used in the present analysis is available in electronic form and a sample table is shown in Table \[optdata\]. We estimated the completeness limit of the data using the ADDSTAR routine of DAOPHOT-II. The procedure was the same as mentioned for the optical images (see Sect. \[obs\]). Completeness was greater than 90$\%$ for magnitudes brighter than 17.0 and reduced to 80 $\%$ for the magnitude range 17.0 - 17.5 in $K$-band. We did not find any significant spatial variation of the completeness factor within the entire area of $5^{\prime}.2 \times 5^{\prime}.2$ and hence we used an average completeness factor of the region for our analysis.
Structure of the cluster
========================
Two dimensional surface density distribution
--------------------------------------------
The initial stellar distribution in star clusters may be governed by the structure of parental molecular cloud and also how star formation proceeds in the cloud (Chen et al. 2004, Sharma et al. 2006). Later evolution of the cluster may then be governed by internal gravitational interaction among member stars and external tidal forces due to the Galactic disk or giant molecular clouds.
To study the morphology of the cluster, we generated isodensity contours for stars in $K$-band from CFHT data and is shown in Fig. \[ssnd\]. The contours are plotted above 3-sigma value of the background level as estimated from the control field. The star mark in Fig. \[ssnd\] represents the location of the cluster centre (Sect. \[rd\]). The surface density distribution of the CFHT data reveals prominent sub-structures which seem to be distributed symmetrically around the cluster centre at a radial distance of $\sim$ 35 arcsec. Interestingly, these sub-structures are lying just inside the thin molecular layer shown in Fig. \[cfht\].
Radial stellar surface density and cluster size {#rd}
-----------------------------------------------
The radial extent of a cluster is one of the important parameters used to study the dynamical state of the cluster. We used the star count technique to study the surface density distribution of stars in the cluster region and to derive the radius of the cluster. To determine the cluster centre, we used the stellar density distribution of stars in a $\pm$ 30 pixel wide strip along both X and Y directions around an eye estimated centre. The point of maximum density obtained by fitting a Gaussian curve was considered as the centre of the cluster. The coordinates of the cluster centre were found to be $\alpha_{2000}$ = $04^{h}40^{m}38^{s}.2 \pm 1^{s}.0$; $\delta_{2000}$ = $+50^{\circ}27^{\prime}36^{\prime\prime} \pm 15^{\prime\prime}$.
To investigate the radial structure of the cluster, we derived the radial density profile (RDP) using the ST observations for $V \le $ 20 mag and 2MASS $K_s$-band data ($K_s \le$ 14.3 mag). Sources were counted in concentric annular rings of 30 arcsec width around the cluster centre and the counts were normalized by the area of each annulus. The densities thus obtained are plotted as a function of radius in Fig. \[rad\], where, one arcmin at the distance of the cluster (6.0 kpc, cf. Sect. \[distance\]) corresponds to $\sim$ 1.8 pc. The upper and lower panels show the RDPs obtained from optical and 2MASS $K_s$-band data, respectively. The error bars are derived assuming that the number of stars in each annulus follows Poisson statistics.
Radius of the cluster $(r_{cl})$ is defined as the point where the cluster stellar density merges with the field stellar density. The horizontal dashed line in Fig. \[rad\] shows the field star density. For the optical RDP, the field star density is determined from the corner of our optical CCD image, whereas for the NIR RDP, the field star density is determined from an area which is 10 arcmin away from the cluster centre. The error limits in the field density distribution are shown using dotted lines.
To parametrize the RDP, we fitted the observed RDP with the empirical model of King (1962) which is given by $$\hspace{20mm}{\rho (r) = {{\rho_0} \over
\displaystyle {1+\left({r\over r_c}\right)^2}}}$$ where $r_c$ is the core radius at which the surface density $\rho(r)$ becomes half of the central density, $\rho_0$. The best fit to the observed RDPs obtained by a $\chi^2$ minimization technique is shown in Fig. \[rad\]. The core radii thus estimated from optical and NIR RDPs are 0.50 $\pm$ 0.06 and 0.48 $\pm$ 0.05 arcmin, respectively. Within errors, the King’s profile (Fig. \[rad\], solid curve) seems to be merging with the background field at $\sim$ 2.0 arcmin both for the optical and 2MASS data. Hence, we assign a radius of 2.0 arcmin for NGC 1624. Here we would like to point out that the core radius and boundary of the cluster are estimated assuming a spherically symmetric distribution of stars within the cluster. This approach is frequently used to estimate the extent of a cluster.
Analysis of optical data
========================
Reddening in the cluster {#reddening}
------------------------
To study the nature of the extinction law towards NGC 1624, we used two-colour diagrams (TCDs) as described by Pandey et al. (2003). The TCDs of the form of ($V-\lambda$) versus ($B-V$), where $\lambda$ is one of the broad-band filters ($R,I,J,H,K,L$), provide an effective method for separating the influence of normal extinction produced by the diffuse interstellar medium from that of the abnormal extinction arising within regions having a peculiar distribution of dust sizes (cf. Chini & Wargau 1990; Pandey et al. 2000). The ${E(V-\lambda)}\over {E(B-V)}$ values in NGC 1624 are estimated using the procedure as described in Pandey et al. (2003). The slopes of the distributions $m_{cluster}$ are found to be identical to the normal values as given in Pandey et al. (2003). Thus we adopt a normal reddening law ($R_V=3.1$) for NGC 1624.
In the absence of spectroscopic observations, the interstellar extinction $E(B -− V)$ towards the cluster region can be estimated using the $(U −- B )/(B -− V )$ colour-colour (CC) diagram. The CC diagram of NGC 1624 ($r \le 2^\prime$) is presented in Fig. \[ubbv\], where, continuous curves represent the empirical zero-age-main-sequence (ZAMS) locus by Girardi et al. (2002). The ZAMS locus is reddened by $E(B-V)$ = 0.76 and 1.00 mag along the normal reddening vector (i.e., $E(U - B) /E(B - V )$ = 0.72). Fig. \[ubbv\] indicates that majority of the $O-A$ type stars have $E(B - V)$ in the range of 0.76 - 1.00 mag. The stars lying within the reddened ZAMS may be probable members of NGC 1624. Using $K/ (J-K)$ colour-magnitude diagram (CMD), Deharveng et al. (2008) have also reported $A_V \sim 3$ mag for the whole region. A careful inspection of the CC diagram indicates the presence of further reddened population which could be the probable background population of the region. The theoretical ZAMS, shown by dashed line, is further shifted to match the reddened sequence. The $E(B - V)$ value for the background population comes out to be $\sim$ 1.15 mag.
Reddening of individual stars having spectral types earlier than A0 have also been computed by means of the reddening free index $Q$ (Johnson $\&$ Morgan 1953). Assuming a normal reddening law we can construct a reddening-free parameter index $Q = (U-B) - 0.72\times (B-V)$. For stars earlier than A0, value of $Q$ will be $<$ 0. For main-sequence (MS) stars, the intrinsic $(B-V)_0$ colour and colour-excess can be obtained from the relation $(B-V)_0 = 0.332\times Q$ (Johnson 1966; Hillenbrand et al. 1993) and $E(B-V) = (B-V) - (B-V)_0$, respectively. The individual reddening of the massive stars down to A0 spectral class within NGC 1624 ($r \le 2^\prime$) are found to vary in the range $E(B-V)$ $\simeq$ 0.76 - 1.05 mag implying the presence of differential reddening within the cluster. The $A_V$ values thus calculated for stars up to A0 spectral class have been given in Table \[optdata\]. Assuming the standard deviation of the residuals (cf. Sect. \[obs\]) as typical errors in photometry, we estimate a typical error in estimation of $E(B-V)$ as $\sim$ 0.05 mag.
Spectral classification of the bright sources in NGC 1624 {#slitspec}
---------------------------------------------------------
We carried out low resolution spectroscopy of four optically bright sources within 2 arcmin radius of NGC 1624. These sources are referred as M2, M4, M9 and M8 (see Fig. 6 of Deharveng et al. 2008). The brightest source M2 is the probable ionizing source of Sh2-212 (Moffat et al. 1979). This star was identified as an emission line star of class O5e by Hubble (1922). Moffat et al. (1979) classified this object as O5.5V star, whereas Chini & Wink (1984) classified it as O6I type star. To determine the spectral type of this star, we extracted low-resolution, one dimensional spectrum. In the top panel of Fig. \[spec\], we show the flux calibrated, normalized spectrum of the ionizing source M2 with important lines identified and labeled. Among the Balmer lines, $H{\alpha}$ and $H_{\beta}$ are relatively strong in emission compared to $H{\gamma}$, which is weak in emission. The $H{\delta}$ and $H{\epsilon}$ are in absorption. The other lines found in emission are $\lambda$ 4686 and [ ]{}$\lambda\lambda$ 4647-50.
In the case of early type stars, the ratio of $\lambda$ 4471/$\lambda$ 4542 is a primary indicator of the spectral type. This ratio is found to vary from less than 1 to 1 and greater than 1 as we move from O5 to O7 and later types. The presence of strong $\lambda$ 4542 in absorption which is often accompanied by weak [ ]{}$\lambda\lambda$ 4634-42 emission indicate a MS luminosity class denoted by ((f)). The absorption strength of $\lambda$ 4686 weakens while [ ]{}emission strength increases in intermediate luminosity classes, denoted by (f) category. Finally, the Of super giants show both and [ ]{}in strong emission (Walborn & Fitzpatrick 1990).
The ratio of $\lambda$ 4471/$\lambda$ 4542 for M2 is found to be (i.e., Log EW = Log (EW($\lambda$ 4471)/EW($\lambda$ 4542)) -0.15, implying that this star is likely to be of spectral type earlier to O7. Following Conti & Alschuler (1971) we assign O6.5 $\pm$ 0.5 spectral type to this star. The weak nature of [ ]{}$\lambda\lambda$ 4634-42 indicates that this star is likely to be in MS. Thus we assign a spectral class of O6.5 $\pm$ 0.5 V for the ionizing source of Sh2-212.
The bottom panel of Fig. \[spec\] shows the low resolution spectrum for the star M4. The absence of $\lambda$ 4200, $\lambda$ 4686 and $\lambda$ 4481 indicates that the spectral class of M4 is between B1-B2 (Walborn & Fitzpatrick 1990). The lack of spectral lines $\lambda$ 4481 and $\lambda$ 4552 rules out the possibility of it being an evolved star. A comparison with the low resolution stellar spectra of Jacoby et al. (1984) and Walborn & Fitzpatrick (1990) suggests this star as a spectral class of B1.5 $\pm$ 0.5 V.
The reddening slope E(B-V)/E(U-B) has also been obtained using the spectral types of the M2 (06.5V ) and M4 (B1.5V) stars. The value of the slope using the intrinsic values from Koorneef (1984) / Johnson (1966) comes out to be 0. 86 / 0.83 and 0.75 / 0.73 for M2 and M4, respectively. The reddening slope for the B type star agrees well the value obtained in Sect. §\[reddening\]. We adopt a normal reddening law in the region as mentioned in Sect. §\[reddening\] for further analysis of the data.
We also extracted the low resolution spectra (not shown here) for the stars M8 and M9. Presence of the spectral lines $\lambda$ 5893, $\lambda$$\lambda$ 6122, 6162, $\lambda$ 6456 and the line strength of , $\lambda$ 6497 put these two stars in the mid F giant category based on the spectral atlas given by Torres-Dodgen & Weaver (1993) and Jacoby et al. (1984).
Optical colour-magnitude diagrams : Distance and age {#distance}
----------------------------------------------------
The optical colour-magnitude diagrams (CMDs) are useful to derive the cluster fundamental parameters such as age, distance etc. Fig. \[q\] shows dereddened $V_0/(B-V)_0$ CMD for probable cluster members (Sect. \[reddening\]) lying within $r \le 2^{\prime}$ of NGC 1624. The stars having spectral type earlier than A0 were dereddened individually using $Q$ method as discussed in Sect. \[reddening\]. The stars labeled as M2, M4, M8 and M9 (following the nomenclature by Deharveng et al. 2008) have spectroscopic observations as discussed in Sect. \[slitspec\]. The spectral class of the ionizing source (M2; see Sect. \[slitspec\]) yields intrinsic distance modulus of 14.05 which corresponds to a distance of 6.5 kpc, whereas the spectral class of M4 yields intrinsic distance modulus of 13.8 which corresponds to a distance of 5.8 kpc. The average distance from these two spectroscopically identified cluster members comes out to be 6.15 kpc. We also calculated the individual distance modulus of the remaining 12 probable MS stars (shown as filled circles in Fig. \[q\]). The intrinsic colours for each star were estimated using the $Q$ method as discussed in Sect. \[reddening\]. Corresponding $M_V$ values have been estimated using the ZAMS by Girardi et al. (2002). The average value of the intrinsic distance modulus obtained from the 14 stars (2 from spectroscopy and 12 from photometry) comes out to be 13.9 $\pm$ 0.3 which corresponds to a distance of $6.0 \pm 0.8$ kpc. In Fig. \[q\] we have also plotted the theoretical isochrone of 2 Myr ($Z=0.02$; log age = 6.3) by Girardi et al. (2002), shifted for the distance modulus of $(m-M_V)_0$ = 13.90 $\pm$ 0.3, which seems to be matching well with the distribution of the probable MS members of the cluster. Present distance estimate is in agreement with that obtained by Moffat et al. (1979; 6.0 $\pm$ 0.5 kpc), whereas Chini & Wink (1984) have reported a distance of 10.4 kpc to NGC 1624. The distance estimates by Moffat et al. (1979) and Chini & Wink (1984) were based on the assumed spectral class of the ionizing source M2 (i.e., O5.5V and O6I, respectively). Here, it is worthwhile to mention that the $M_V$ value for an O6V star in the literature varies significantly; e.g., $M_V$ = -5.5 (Schmidt-Kaler 1982) to -4.9 (Martins et al. 2005). Hence, the distance estimation based on the O-type star alone may not be reliable. However, the present distance estimation is carried out using the O-type star as well as all the probable members earlier to A0 spectral type. The kinematic distance (6.07 kpc) to the region derived by Caplan et al. (2000) is in agreement with the present distance estimation. Since this cluster is located in the outer galactic disk, the possibility of a low metallicity for the region cannot be ruled out, which would imply bluer intrinsic colour for the members and hence a closer distance of NGC 1624. However, in the absence of any metallicity measurements towards this region, we have considered solar metallicity for the region and the distance of NGC 1624 is taken as 6.0 kpc for the present study.
The ages of young clusters are typically derived from the post-main-sequence evolutionary tracks for the earliest members if significant evolution has occurred and/or by fitting the low-mass contracting population with theoretical PMS isochrones. Since the most massive member of NGC 1624 seems to be a O6.5 MS star, the maximum age of the cluster should be of the order of the MS life time of the massive star i.e., $\sim$ 4.4 Myr (Meynet et al. 1994). In Fig. \[q\] we have also shown the isochrone of 4 Myr age by Girardi et al. (2002), which suggests that the maximum post-main-sequence age of the cluster could be $\sim$ 4 Myr. Stars which deviate significantly from the isochrone are likely field stars and are shown by open circles in Fig. \[q\], which include stars M8 and M9. Spectroscopic observations of these two stars indicate that they are of mid F giant spectral category (see Sect. \[slitspec\]) and hence cannot be the cluster members at this assumed distance and age.
$V/(V - I)$ CMD for the stars lying within the core of the cluster ($r \le 0^\prime$.5) is shown in Fig. \[cmd\]a and CMD for the stars outside the core ($0^\prime.5 \le r \le 2^\prime$) is shown in Fig. \[cmd\]b. In order to find out the field star contamination in the cluster region, we selected a control field having same area as that of the cluster from the corner of our CCD image. $V/(V - I)$ CMD for the control field is shown in Fig. \[cmd\]c. Assuming $E(B-V)_{min} =0.76$ mag, $E(B-V)_{max}$ =1.0 mag and using the relations $A_{V}=3.1\times E(B-V)$; $E(V-I)=1.25\times E(B-V)$, we have plotted theoretical isochrone of 2 Myr by Girardi et al. (2002) and pre-main-sequence (PMS) isochrone of 0.5 and 5 Myr (Siess et al. 2000) in Fig. \[cmd\]. It is evident from this figure that the MS ($V \le $ 16.5) is rather free from field star contamination. Although the CMDs of the cluster region show a significant number of stars towards the right of the 2 Myr isochrone at $(V-I) >2.5 $ and $ V >18 $ mag, a comparison between the cluster and field regions clearly reveals the contamination due to field star population in the CMD of the cluster region. However, the $V/(V - I)$ CMD of the core (Fig. \[cmd\]a) reveals uncontaminated population of PMS stars having ages 0.5 - 5 Myr.
As discussed in Sect. \[reddening\], there is indication for a population in the background of the cluster which is apparent in Figs. \[cmd\]b and \[cmd\]c. Assuming the average $E(B-V)$ = 1.15 mag, we estimate that the distance of the background population is $\sim$ 8 kpc. The study by Pandey et al. (2006) also indicates a background population at a distance of $\sim$ 8 kpc in the second galactic quadrant.
Emission from ionized gas {#ionized gas}
-------------------------
Fig. \[1280\] shows GMRT radio continuum map of Sh2-212 at 1280 MHz made with a resolution of $\sim$ 4$^{\prime\prime}$.9 $\times$ 3$^{\prime\prime}$.2. In the high resolution map, most of the extended diffuse emission associated with the region appears quite faint. However, a compact intense emission can be seen at the position of UCregion (04$^{\rm h}$40$^{\rm m}$27$^{\rm s}$.5, +50$^\circ$28$^\prime$28$^{\prime\prime}$) located at the periphery of Sh2-212 and is marked using an arrow. The UCregion is associated with the IRAS point source IRAS 04366+5022. The overall morphology of the map agrees well with that of our optical colour composite image shown in Fig. \[cfht\]. Fig. \[610\] shows an enlarged version of the UCregion at 1280 MHz. The integrated flux densities from the radio continuum contour maps for the evolved region (i.e., Sh2-212) and UCregion are estimated to be 3.6 $\pm$ 0.4 Jy and 16.5 $\pm$ 0.5 mJy, respectively. Assuming the ionized regions to be spherically symmetric and neglecting absorption of ultraviolet radiation by dust inside the region, the above flux densities together with assumed distance, allow us to estimate the number of Lyman continuum photons (N$_{Lyc}$) emitted per second, and hence the spectral type of the exciting stars. Using the relation given by Martín-Hernández et al. (2003) for an electron temperature of 10000 K, we estimated log N$_{Lyc}$ = 48.29 and log N$_{Lyc}$ = 45.96 for the evolved and UCregion, respectively, which corresponds to MS spectral types of $\sim$ O7 and $\sim$ B0.5, respectively (Vacca et al. 1996). On the basis of optical spectroscopy, we estimated spectral type of the ionizing source of Sh2-212 as O6.5V (see Sect. \[slitspec\]) which is in fair agreement with the above spectral type estimation from integrated radio continuum flux. Using the spectral energy distribution, Deharveng et al. (2008) have found that the source associated with the UCregion is a massive YSO of $\sim$ B0 type ($\sim$ 14 $M_\odot$), which is in agreement with the spectral type $\sim$ B0 obtained in the present work.
Analysis of near-infrared data {#nir}
==============================
NIR data are very useful tools to study the nature of young stellar population within the star forming regions (SFRs). Discriminating young stars in clusters from field stars is difficult. Young stars with strong infrared (IR) excess from disks and envelopes can be identified using the NIR and mid-IR (MIR) observations. We used the CFHT deep NIR photometry to study the PMS contents and KLF of NGC 1624. The CFHT $K$-band mosaic image centered on the UCregion covering an area of $5^{\prime}.2 \times 5^{\prime}.2$ is shown in Fig. \[cfht\] (right panel), where the ionizing source is marked with a white circle. A very rich cluster is apparent around the ionizing source. Since the centre of NGC 1624 is located towards the eastern edge of the CFHT frame, eastern half of the cluster is covered partially. The observations covered an area $\sim$ 9.6 arcmin$^2$ of NGC 1624 and is shown using a partial circle in Fig. \[cfht\]. A region covering an area $\sim$ 3.1 arcmin$^2$ towards north of the cluster shown by a box in Fig. \[cfht\], is considered as the control field. In the following sections, we discuss the NIR CC diagram and CMDs.
Colour-Colour Diagrams {#nircc}
----------------------
NIR and MIR photometry are useful tools to investigate the fraction of YSOs in a SFR. In the absence of ground based $L$-band observations or [*Spitzer*]{} based MIR observations, we used $(J-H)$/$(H-K)$ CC diagram to identify the young stellar population in NGC 1624 (Hunter et al. 1995; Haisch et al. 2000; 2001; Sugitani et al. 2002; Devine et al. 2008; Chavarría et al. 2010). The $(J-H)$/$(H-K)$ CC diagrams for the cluster region (area $\sim$ 9.6 arcmin$^2$) and the control field (area $\sim$ 3.1 arcmin$^2$) are shown in Fig. \[jhhk\]. The thin and thick solid curves are the locations of unreddened MS and giant stars (Bessell $\&$ Brett 1988), respectively. The dotted and dotted-dashed lines represent the locus of unreddened and reddened ($A_V$ = 4.0 mag) classical T Tauri stars (CTTSs; Meyer et al. 1997). The two long parallel dashed lines are the reddening vectors for the early MS and giant type stars (drawn from the base and tip of the two branches). One more reddening vector is plotted from the tip of the unreddened CTTS locus. The crosses on the reddening vectors are separated by an $A_{V}$ value of 5 mag. The extinction ratios, $A_J/A_V = 0.265, A_H/A_V = 0.155$ and $A_K/A_V=0.090$, are adopted from Cohen et al. (1981). The magnitudes, colours and the curves are in CIT system.
Presently YSOs are classified as an evolutionary sequence spanning a few million years as: Class 0/Class I - the youngest embedded protostars surrounded by infalling envelopes and growing accretion disks; Class II - PMS stars with less active accretion disks and Class III - PMS stars with no disks or optically thin remnant disk (Adams et al. 1987). Following Ojha et al. (2004a), we classified sources according to their locations in $(J-H)/(H-K)$ CC diagrams. The ‘F’ sources are those located between the reddening vectors projected from the intrinsic colours of MS and giant stars. These sources are reddened field stars (MS and giants) or Class III/Class II sources with little or no NIR excess (viz., weak-lined T Tauri sources (WTTSs) but some CTTSs may also be included). The sources located redward of region ‘F’ are considered to have NIR excess. Among these, the ‘T’ sources are located redward of ‘F’ but blueward of the reddening line projected from the red end of the CTTS locus. These sources are considered to be mostly CTTSs (Class II objects) with large NIR excesses (Lada & Adams 1992). There may be an overlap in NIR colours of Herbig Ae/Be stars and T Tauri stars in the ‘T’ region (Hillenbrand et al. 1992). The ‘P’ sources are those located in the region redward of region ‘T’ and are most likely Class I objects (protostellar-like) showing large amount of NIR excess. Here it is worthwhile to mention that Robitaille et al. (2006) have shown that there is a significant overlap between protostellar-like objects and CTTSs in the CC diagram.
A comparison of the colour distribution of the sources in the cluster and control field (Fig. \[jhhk\]) suggests that there is an appreciable difference between them. Significant fraction of sources in the cluster region are concentrated between the unreddened and reddened CTTS locus, whereas majority of sources in the control field are mainly concentrated in the ‘F’ region. Statistically, we can safely assume that majority of sources of the cluster region located between the unreddened and reddened CTTS locus are most likely to be cluster members. The comparison also indicates that the sources located in the ‘F’ region could be the reddened field stars but a majority of them are likely candidate WTTSs or CTTSs with little or no NIR excess. The sources lying towards the right side of the reddening vector at the boundary of ‘F’ and ‘T’ regions and above the unreddened CTTS locus can be safely considered as YSO/NIR excess sources. A total of 120 such sources have been detected within a $5^{\prime}.2 \times 5^{\prime}.2$ region which fall in the ‘T’ region and above the unreddened CTTS locus. However, this number is certainly a lower limit for the population of YSOs, as several of the cluster members detected in the $H$ and $K$ bands have not been detected in the $J$-band. Moreover, $L$-band or MIR observations would further increase the detection of YSOs in the region. Hence the present $JHK$ photometry provides only a lower limit to the population of YSOs in NGC 1624. The distribution of YSOs in Fig. \[jhhk\] manifests that majority of them have $A_V$ $\le$ 4 mag. Some of the sources in ‘F’ and ‘T’ regions, which might be the candidate WTTSs/CTTSs, show $A_V$ values higher than 4 mag. The $A_V$ for each star lying in ‘T’ region has been estimated by tracing back to the intrinsic CTTS locus along the reddening vector. The $A_V$ for stars within the cluster region (area $\sim$ 9.6 arcmin$^2$) and located in the ‘F’ region is estimated by tracing them back to the extension of the intrinsic CTTS locus (see Ogura et al. 2007; Chauhan et al. 2009 for details). The $A_V$ values thus calculated for the sources in ‘F’ and ‘T’ regions are given in Table \[optdata\]. Twenty one sources are found to have $A_V$ $\ge$ 6.0 mag, indicating that significant number of cluster members in the region may still be embedded.
The colour-magnitude diagram
----------------------------
Fig. \[jhj\] shows $J/(J-H)$ distribution of sources within $\sim$ 9.6 arcmin$^2$ area of NGC 1624. The encircled are the NIR excess sources in this region. The thick solid curve denotes the locus of 2 Myr PMS isochrone from Siess et al. (2000), which is the average age of NIR excess sources (see Sect. \[pms\], Fig. \[yso\]) and the thin curve is the 2 Myr isochrone from Girardi et al. (2002). Both the isochrones are shifted for the cluster distance and reddening. The continuous oblique lines denote the reddening trajectories up to $A_V$ = 10 mag for PMS stars of 2 Myr age having masses 0.1, 2.0 and 3.0 $M_\odot$, respectively. For the assumed age $\sim$ 2 Myr, reddening $A_V$ = 2.5 mag and distance = 6.0 kpc, the $J$-band detection limit of present observations corresponds to $M$ $\sim$ 0.1 $M_\odot$. In Fig. \[jhj\] majority of NIR excess sources ($\sim$ 98 %) are seen to have masses in the range 0.1 to 3.0 $M_\odot$.
The CMD indicates that the stellar population in NGC 1624 significantly comprises of low mass PMS stars similar to other SFRs studied by Ojha et al. (2004a), Sharma et al. (2007), Pandey et al. (2008) and Jose et al. (2008). These results further support the scenario that the high mass star forming regions are not devoid of low mass stars (e.g., Lada & Lada 1991; Zinnecker et al. 1993; Tapia et al. 1997; Ojha et al. 2004a). The distribution of stars located below the CTTS locus (cf. Fig. \[jhhk\]) is shown by crosses in Fig. \[jhj\] which indicates that a majority of these sources are likely to be field stars.
The brightest NIR excess source marked as a star symbol in Fig. \[jhj\] is the candidate ionizing source of the UCregion. The extinction to this star is estimated by tracing it back to the ZAMS along the reddening vector and found to be $A_V$ $\sim$ 10.6 mag. This extinction should be considered as an upper limit, as the star shows NIR excess, therefore, $J$ and $H$ magnitudes might have been affected by the NIR excess emission. The photometric spectral type of this star comes out to be $\sim$ B0 which is in agreement with the spectral type estimation based on our radio continuum observations (see Sect. \[ionized gas\]).
Field star decontamination {#field}
===========================
Distinguishing cluster members from field stars is a significant challenge for photometric surveys of clusters. To study the LF/MF, it is necessary to remove field star contamination from the cluster region. Membership determination is also crucial for assessing the presence of PMS stars because both PMS and dwarf foreground stars occupy similar positions above the ZAMS in the CMDs. As discussed in Sect. \[nir\], some of the YSOs can be identified with the help of NIR excess, however this is not true for the diskless YSOs. An alternative is to study the statistical distribution of stars in the cluster and field regions. Because proper motion studies are not available for the stars in the cluster region, we used following statistical criteria to estimate the number of probable members of NGC 1624.
To remove contamination due to field stars from the MS and PMS sample, we statistically subtracted the contribution of field stars from the observed CMD of the cluster region using the following procedure. For any star in the $V/(V-I)$ CMD of the control field (Fig. \[cmd\]c), the nearest star in the cluster’s $V/(V-I)$ CMD (Figs. \[cmd\]a and b) within $V$ $\pm$ 0.125 and $(V-I)$ $ \pm$ 0.065 was removed. The statistically cleaned $V/(V-I)$ CMD (SCMD) of the cluster region is shown in Fig. \[calone\], which clearly shows a sequence towards red side of the MS. PMS isochrones by Siess et al. (2000) for ages 0.5 and 5 Myr (dashed lines) and 2 Myr isochrone by Girardi et al. (2002) (continuous line) are shown in Fig. \[calone\]. The evolutionary tracks by Siess et al. (2000) for different masses are also shown which are used to determine the masses of PMS cluster members. Here we would like to remind the readers that the points shown by filled circles in Fig. \[calone\] may not represent the actual members of the clusters. However, the filled circles should represent the statistics of PMS stars in the region and the statistics has been used to study the MF of the cluster region (cf. Sect. \[imf\]).
We followed the above technique for the field star decontamination of the NIR data as well. Since the area of the selected field region is smaller in comparison to the cluster region, we subdivided the cluster region in to three sub regions having area equal to the field region. The field star contamination from $J/ (J-H)$ CMD of the cluster sub regions was subtracted using the $J/ (J-H)$ CMD of the field region in a similar manner as in the case of $V/(V-I)$ CMD.
Young stellar population in NGC 1624 {#pms}
------------------------------------
It is found that nineteen percent of the candidate PMS stars located above the intrinsic CTTS locus (cf. Fig. \[jhhk\]) have optical counterparts in $V$-band within 9.6 arcmin$^2$ area. The $V/(V-I)$ CMD for these sources is shown in Fig. \[yso\]. The encircled are the NIR excess sources which are the likely candidate YSOs (see Sect. \[nircc\]). PMS isochrones by Siess et al. (2000) for 0.5, 2, 5 Myr (dashed curves) and isochrone for 2 Myr by Girardi et al. (2002; continuous curve) corrected for cluster distance and reddening are also shown. Fig. \[yso\] reveals that majority of the sources have ages $\le$ 5 Myr with a possible age spread of $\sim 0.5 - 5$ Myr and $\sim$ 75$\%$ of the NIR excess sources show ages $\le$ 2 Myr. Since the reddening vector in $V/(V-I)$ CMD (see Fig. \[yso\]) is nearly parallel to the PMS isochrone, the presence of variable extinction in the region will not affect the age estimation significantly. Therefore the age spread indicates a possible non-coeval star formation in this region.
The membership of the YSOs shown in Fig. \[yso\] is calculated using the following procedure. Each YSO is corrected for its reddening calculated in the Sect. \[nircc\]. The intrinsic $(V-I)$ colour thus obtained is then compared with the PMS isochrones of varying ages from 5 Myr to 0.1 Myr. The $M_V$ value of each YSO is obtained from the best matching isochrone and hence the distance modulus. The sources lying within $3\sigma$ of the distance modulus obtained in Sect. \[distance\] are considered as the probable cluster members. It is found that three sources do not satisfy the above criteria and has been considered as non-members. These three sources are marked using box in Fig. \[yso\].
A comparison of Fig. \[yso\] with the field star decontaminated CMD shown in Fig. \[calone\] reveals a nice resemblance, suggesting that the statistics of PMS sources selected on the basis of SCMD can be used to study the IMF of PMS population of NGC 1624. As most of the sources in Fig. \[yso\] are located in the PMS region, it can be safely assumed that the sources lying above the unreddened CTTS locus of Fig. \[jhhk\] are likely cluster members. Thus sources falling in the ‘F’ region (see Fig. \[jhhk\]) are likely to be WTTSs or CTTSs with little or no NIR excess and those in the ‘T’ region are the candidate CTTSs with NIR excess. However, Fig. \[yso\] does not show any trend in age distribution between these sources. A comparison of Figs. \[jhj\] and \[calone\] confirms that most of the YSOs have masses $\le$ 3.0 $M_\odot$.
The fraction of NIR excess sources in a cluster is also an age indicator because the disks/envelopes become optically thin with age (Haisch et al. 2001; Carpenter et al. 2006; Hernández et al. 2007). For young embedded clusters having age $\le$ $ 1 \times 10^6$ yr, the disk fraction obtained from $JHK$ photometry is $\sim$ 50% (Lada et al. 2000; Haisch et al. 2000). Whereas the fraction reduces to $\sim$ 20% for the clusters with age $\sim$ 2 - 3 $\times$ $10^6$ yr ( Lada & Lada 1995; Haisch et al. 2001; Teixeira et al. 2004; Oliveira et al. 2005). After correcting for the field star contamination and photometric incompleteness, the fraction of NIR excess sources in an area $\sim$ 9.6 arcmin$^2$ of NGC 1624 is estimated to be $\sim$ 20%. There are 31 $J$ drop-out sources falling within our error criteria. Based on the colour and spatial distribution of these $J$ drop-out sources (see Sect. \[distribution\]), we presume that they can be included in the list of candidate YSOs and hence the NIR excess fraction increases to $\sim$ 25%. This suggests an age of $\sim$ 2 - 3 $\times$ $10^6$ yr for this cluster which is in agreement with the age estimation derived using the PMS evolutionary tracks in the optical CMD (cf. Fig. \[yso\]). This NIR excess fraction is to be considered as a lower limit to the actual YSO fraction of the cluster as we do not have $L$-band observations for this cluster. However, Yasui et al. (2009) point out that the disk fraction from only $JHK$ data are about 0.6 of those from $JHKL$ data and the lifetime estimation from $JHK$ data is basically identical to that from $JHKL$ data. Therefore, despite a little larger uncertainty, the disk fraction from $JHK$ data alone should still be effective even without $L$-band data. Here it is worthwhile to point out that in the case of Cep OB3B, Getman et al. (2009) have shown that the disk frequency depends on the distance from the exciting stars, as massive stars can photo-evaporate the disk around young stars. Also, Carpenter et al. (2006) have found evidence for mass dependent circumstellar disk evolution in the sense that the mechanism for disk dispersal operates less efficiently for low mass stars. Hence, keeping in mind the uncertainties mentioned above, the age estimation based on the disk frequency must be considered as an approximate estimation.
In order to check if there is any mass dependence of the NIR excess fraction, we divided the optically identified PMS members (shown in Fig. \[yso\]) in to three mass bins i.e., 2.5 - 1.5 $M_{\odot}$, 1.5 - 1.0 $M_{\odot}$ and 1.0 - 0.6 $M_{\odot}$ using the evolutionary tracks by Siess et al. (2000). After applying the completeness correction in each magnitude bin, we obtained the NIR excess fraction as 23%, 24% and 37%, respectively for the above mentioned mass bins. Hence, there is an evidence of mass dependent evolution of circumstellar disk as explained by Carpenter et al. (2006). However, this estimation has to be considered as a lower limit, as only 19% of the identified NIR PMS stars have the optical counterparts.
Deharveng et al. (2005; 2008) have identified signs of recent star formation in Sh2-212. They estimated the age of the massive star associated with the UCregion located at the periphery of Sh2-212 as $\sim$ 0.14 Myr on the basis of dynamical size of the UCregion. This indicates that the UCregion is relatively young as compared to the YSOs within the cluster region. The bright rim feature at one end of the UCregion (see Fig. 2 of Deharveng et al. 2008) also suggests that the UCregion might have formed at a later evolutionary stage of the region as a second generation object.
Spatial distribution of YSOs {#distribution}
============================
Fig. \[co\] displays the spatial distribution of YSOs (blue circles; likely Class II sources) identified on the basis of NIR excess characteristics (cf. Fig. \[jhhk\]) along with the CO emission contour map from Deharveng et al. (2008) for four condensations and filament. The $J$ drop-out sources are shown using red triangles. The molecular condensations make a semi-circular ring towards the southern side of Sh2-212. Fig. \[co\] reveals that majority of YSOs are located close to the cluster centre within a radius of 0.5 arcmin (i.e., within the cluster core radius of $\sim$ 0.9 pc; cf. Sect. \[rd\]), however, several other YSOs are found to be distributed outside of this radius along the thin semi-circular ring and filamentary structure. Interestingly, there is an apparent concentration of YSOs just at the boundary of the clump C2.
In Fig. 15 we have shown the $ K/(H-K) $ CMD for all the sources detected in this region. The encircled are the YSOs and the red triangles are the $J$ drop-out sources. It is evident from the CMD that majority of YSOs have $(H-K)$ colour in the range $\sim$ 0.6 - 0.8 mag. However a significant number of sources appear to be redder ($H-K$ $ \ge 1.0$ mag). The spatial distribution of sources having $(H-K) \ge 1.0$ mag has been shown in Fig. \[co\] with filled circles (i.e., YSOs) and triangles ($J$ drop-out sources), respectively and this figure reveals a higher density of reddened sources near the clump C2. The larger value of $(H-K)$ ($\ge$ 1.0 mag) could be either due to higher extinction, as most of these sources are lying within/very close to the CO distribution, or could be their intrinsic colour due to large NIR excess.
If the origin of this colour excess is merely from the interstellar extinction, then one must expect an increment in the value of $A_V$ by $\sim$ 12 mag as compared to the sources located close to the cluster center. In order to investigate the spatial distribution of extinction in the region, we plot radial variation of $A_V$ in Fig. \[radial\] (left panel). It is evident from the Fig. \[radial\] that $A_V$ is almost constant within an 80 arcsec cluster radius. Hence, we can presume that the origin of colour excess could be intrinsic in nature. This fact indicates an age sequence in the sense that YSOs located/projected over the semi-circular ring of molecular condensations are younger than those lying within the core of the cluster.
To further elucidate the youth of the YSOs located/projected over the semi-circular ring of molecular condensations, we plot radial variation of NIR excess, $\Delta(H-K)$, defined as the horizontal displacement from the reddening vector at the boundary of ‘F’ and ‘T’ regions (see Fig. \[jhhk\]). NIR excess is considered to be a function of age. An enhancement in the mean value of $\Delta(H-K)$ at $\sim$ 45 arcsec, i.e., near the periphery of the semi-circular ring is apparent in Fig. \[radial\] (middle panel). In the right panel we plot the radial variation of $(H - K)$ colour of YSOs and $J$ drop-out sources using dashed and solid histogram, respectively. The enhancement in the mean $(H - K)$ value at the same location is apparent in this figure as well. However, we have to keep in mind the possibility of photo-evaporation of the disk around YSOs lying within the core of the cluster due to stellar radiation of massive star at the centre of the cluster.
The above facts indicate that the sources near the molecular material are intrinsically redder and support the scenario of possible sequential star formation towards the direction of molecular clumps. It is interesting to mention that the distribution of YSOs in the NGC 1624 region is rather similar to the distribution of Class II sources in other star forming regions. e.g., RCW 82 (Pomarès et al. 2009), RCW 120 (Zavagno et al. 2007) and Sh2-284 (Puga et al. 2009). Majority of Class I sources in the case of RCW 82 and RCW 120 are found to be associated with the molecular material at their periphery and none are found around the ionizing source. The association of Class I sources with the molecular material manifests the recent star formation at their periphery. If star formation in Sh2-212 region is similar in nature to RCW 82 and RCW 120, one would expect a significant number of Class I sources in the surrounding molecular material. Unfortunately, the absence of MIR observations hampers the detailed study of the probable young sources lying towards the collected molecular material. However, the YSOs having $(H-K) \ge 1.0$ mag, which are expected to be the youngest sources of the region, are found to be distributed around the molecular clumps detected by Deharveng et al. (2008). It is interesting to mention that in the case of RCW 82, the YSOs having $(H-K) \ge 1.0$ mag are found to be associated with the molecular emission surrounding the region. Many of these sources are not observed in the direction of molecular emission peaks, but are located on the borders of the condensations (Pomarès et al. 2009). A similar distribution of YSOs (having $H-K \ge 1.0$) can be seen in the present study at the border of the clump C2.
According to Deharveng et al. (2008), the massive YSO associated with the UCH II region (clump C1) might have formed as a result of the collect and collapse process due to the expansion of the region. If the sources lying towards the molecular clump C2 and along the filament are formed as a result of the collect and collapse process, these sources must be younger than the ionization source by about 2 - 3 Myr as the model calculation by Deharveng at al. (2008) predicts the fragmentation of the collected layer after 2.2 - 2.8 Myr of the formation of the massive star in Sh2-212. Since the ionization source is an O6.5 $\pm$ 0.5 MS star, the maximum age of the ionization source should be of the order of its MS life time, i.e., $\sim$ 4.4 Myr (cf. Sect. \[distance\]). On the basis of the present analysis we can indicate that the sources with $(H-K) \ge 1.0$ seem to have a correlation with the semi-circular ring of molecular condensations and should be younger than the age of the ionization source of the region. However in the absence of optical photometry, the reliable age estimation of these YSOs is not possible. Since the distribution of youngest YSOs on the border of clump C2 has a resemblance to the distribution of Class I/ II YSOs in RCW 82, the formation of these YSOs could be due to the result of small-scale Jeans gravitational instabilities in the collected layer, or interactions of the ionization front with the pre-existing condensations as suggested by Pomarès et al. (2009) cannot be ignored.
Initial Mass Function {#imf}
=====================
The distribution of stellar masses that form in a star formation event in a given volume of space is called IMF and together with star formation rate, the IMF dictates the evolution and fate of galaxies and star clusters (Kroupa 2002). Young clusters are important tools to study IMF since their MF can be considered as IMF as they are too young to loose significant number of members either by dynamical or stellar evolution. To study the IMF of NGC 1624 we used the data within $r \le 2^\prime$.
The MF is often expressed by the power law, $N (\log m) \propto m^{\Gamma}$ and the slope of the MF is given as
$$\Gamma = d \log N (\log m)/d \log m$$
where $N (\log m)$ is the number of stars per unit logarithmic mass interval. For the mass range $0.4 < M/M_{\odot} \le 10$, the classical value derived by Salpeter (1955) for the slope of MF is, $\Gamma = -1.35$.
Since the NIR data is deeper, we expect to have a better detection of YSOs towards the fainter end in comparison to the optical data. Therefore we estimated the IMF using the optical and NIR data independently.
IMF from optical data
---------------------
With the help of SCMD shown in Fig. \[calone\], we can derive the MF using theoretical evolutionary models. A mass$-$luminosity relation is needed to convert the derived magnitude for each star to a mass. For the MS stars (see Fig. \[q\]), LF was converted to MF using the theoretical model by Girardi et al. (2002) for 2 Myr (cf. Pandey et al. 2001; 2005). The MF for PMS stars was obtained by counting the number of stars in various mass bins (shown as evolutionary tracks in Fig. \[calone\]). Necessary corrections for data incompleteness were taken into account for each magnitude bin to calculate the MF. The MF of NGC 1624 is plotted in Fig. \[mf\]. The slope, $\Gamma$ of the MF in the mass range $1.2 \le M/M_{\odot}<27$ can be represented by a power law. The slope of the MF for the mass range $1.2 \le M/M_{\odot}<27$ comes out to be, $\Gamma$ = $-1.18\pm0.10$, which is slightly shallower than the Salpeter value (-1.35). We conclude that within an acceptable margin, the slope of IMF for the cluster NGC 1624 is comparable to the Salpeter (1955) value.
IMF from NIR data
-----------------
We also estimate the IMF using J -band luminosity function (JLF). We preferred $J$-band over $K$-band as the former is least affected by the NIR excess. After removing the field star contamination using the statistical subtraction as explained in Sect. \[field\], we applied the completeness correction to the $J$-band data. Assuming an average age of 2 Myr for the PMS stars, distance 6.0 kpc and average reddening $A_V$ = 2.5 mag, the $J$ magnitudes were converted to mass using the 2 Myr PMS isochrone by Siess et al. (2000). For MS stars, the mass-luminosity relation is taken from Girardi et al. (2002). Completeness of the $J$-band data was $\sim$ 90 % at $J$ = 18 mag ($\sim$ 0.65 $M_\odot$). In Fig. \[mf\_ir\], we have shown the MF derived for NGC 1624 (within the area of $\sim$ 9.6 arcmin$^2$) in the mass range $0.65 \le M/M_{\odot}<27$. The linear fit gives a slope $\Gamma$ = $-1.31\pm0.15$ which is in agreement with the Salpeter (1955) value. The MF ($\Gamma$ = $-1.18\pm0.10$) derived using optical data is slightly shallower than that of IR data. However both the slopes are within error and can be considered to be in agreement.
Here we would like to point out that the estimation of IMF depends on the models used. We are pursuing studies of few young clusters, hence a comparative study of IMFs of various young clusters obtained using similar techniques will give useful information about IMFs. Our recent studies on young clusters (age $\sim 2 - 4 $ Myr), viz., NGC 1893 (Sharma et al. 2007), Be 59 (Pandey et al. 2008) and Stock 8 (Jose et al. 2008) have yielded the value of $\Gamma$ for stars more massive than $\sim $ 1 - 2 $M_\odot$ as -1.27 $\pm$ 0.08, -1.01 $\pm$ 0.11 and -1.38 $\pm$ 0.12, respectively. A comparison of the MF in the case of NGC 1624 and the clusters mentioned above indicates that the MF slope towards massive end (i.e., M $\ge 1 M_\odot$) in general, is comparable to the Salpeter value (-1.35).
K-band luminosity function
==========================
The KLF is frequently used in studies of young clusters as a powerful tool to constrain its age and IMF. Pioneering work on the interpretation of KLF was presented by Zinnecker et al. (1993). During the last decade several studies have been carried out with the aim of determining the KLF of young clusters (e.g., Muench et al. 2000; Lada & Lada 2003; Ojha et al. 2004b; Sanchawala et al. 2007; Sharma et al. 2007; Pandey et al. 2008; Jose et al. 2008). We have used CFHT $K$-band data to study the KLF of NGC 1624. Because the CFHT observations did not include the entire cluster region, we restricted the KLF study to a region within $\sim$ 9.6 arcmin$^2$ area of NGC 1624 (see Sect. \[nir\]).
In order to convert the observed KLF to the true KLF, it is necessary to correct the data incompleteness and field star contamination. We applied the CF (see Sect. \[cfhtdata\]) for the data incompleteness. The control field having an area $\sim$ 3.1 arcmin$^2$ shown in Fig. \[cfht\] has been used to remove the field star contribution. We applied a correction factor to take into account the different areas of cluster and control field regions. The field star population towards the direction of NGC 1624 is also estimated by using the Besançon Galactic model of stellar population synthesis (Robin et al. 2003) using a similar procedure as described by Ojha et al. (2004b). The star counts were predicted using the Besançon model towards the direction of the control field. An advantage of using this model is that we can simulate foreground ($d<6.0$ kpc) and background ($d>6.0$ kpc) field star populations separately. The use of this model allows us to apply the extra cloud extinction to the background stars. The foreground population was simulated using the model with $A_V$ = 2.36 mag ($E(B-V) = 0.76$ mag; ref. Sect. \[reddening\]) and $d < 6.0$ kpc. The background population ($d>6.0$ kpc) was simulated with an extinction value $A_V$ = 4.0 mag (see Sect. \[nircc\]). Thus we determined the fraction of the contaminating stars (foreground + background) over the total model counts. The scale factor we obtained to the control field direction was close to 1.0 in all the magnitude bins. This indicates that the moderate extinction of $A_V$ $\sim$ 4.0 mag is unlikely to have any significant effect on the field star distribution at this distance. Hence, we proceeded our analysis of KLF with the field star counts obtained from the observed control field. The completeness corrected and field star subtracted KLF for NGC 1624 is shown in Fig. \[klf\].
The KLFs of young embedded clusters are known to follow power-law shapes (Lada et al. 1991; 1993) which is expressed as:
${{ \rm {d} N(K) } \over {\rm{d} K }} \propto 10^{\alpha K}$
where ${ \rm {d} N(K) } \over {\rm{d} K }$ is the number of stars per 0.5 mag bin and $\alpha$ is the slope of the power law. The KLF for NGC 1624 shown in Fig. \[klf\] (solid line), yields a slope $0.30\pm0.06$ for the range $K$ = 13.5 - 17.5 mag, which is slightly lower than the average value of slopes ($\alpha \sim 0.4$) for young clusters of similar ages (Lada et al. 1991; Lada & Lada 1995; Lada & Lada 2003). However, a break in the power law can be noticed at $K$ = 15.75 mag and the KLF seems to be flat in the magnitude range 15.75 - 17.5. The slope of the KLF in the magnitude range 13.5 - 15.75 (dahsed line in Fig. \[klf\]) comes out to be 0.44 $\pm$ 0.11 which is comparable with the average value of slopes for young clusters. A turn off in the KLF has also been observed in a few young clusters. e.g., at $K \sim$ 14.5 mag and $K \sim$ 16.0 mag in the case of Tr 14 (distance $\sim$ 2.5 Kpc; Sanchawala et al. 2007) and NGC 7538 (distance $\sim$ 2.8 Kpc; Ojha et al. 2004), respectively.
KLF slope is an age indicator of young clusters. For clusters up to 10 Myr old, the KLF slope gets steeper as the cluster gets older (Ali & Depoy 1995; Lada & Lada 1995). However, there is no precise age - KLF relationship in the literature due to huge uncertainty in their correlation (Devine et al. 2008). There are many studies on KLF of young clusters. The studies by Blum et al. 2000; Figuerêdo et al. 2002; Leistra et al. 2005; 2006; Devine at al. 2008 indicate that the KLF slope varies from 0.2 -0.4 for clusters younger than 5 Myr. The KLF of NGC 1624 is worth comparing with the recent studies of young clusters viz; NGC 1893 (Sharma et al. 2007), Be 59 (Pandey et al. 2008) and Stock 8 (Jose et al. 2008), since all the KLFs are obtained using a similar technique. The slope of the KLF ($\alpha = 0.30\pm0.06$) obtained for NGC 1624 in the magnitude range 13.5 - 17.5 is comparable with those obtained for NGC 1893 ($\alpha = 0.34\pm0.07$), Stock 8 ($\alpha
= 0.31\pm0.02$) and Be 59 ($\alpha = 0.27\pm0.02$).
Summary
=======
We have carried out a comprehensive multi-wavelength study of the young cluster NGC 1624 associated with the region Sh2-212. Sh2-212 is thought to have experienced ‘Champagne flow’ and the molecular clumps along with the UCregion at the periphery are suggested as the possible outcome of the collect and collapse phenomena. In our present study, an attempt has been made to determine the basic properties of NGC 1624 as well as to study the nature of stellar contents in the region using optical $UBVRI$ photometry, optical spectroscopy of four stars, radio continuum observations from GMRT along with NIR $JHK$ archival data from 2MASS and CFHT.
From optical observations of massive stars, reddening ($E(B-V)$) in the direction of NGC 1624 is found to vary between 0.76 to 1.00 mag and distance is estimated to be $6.0 \pm 0.8$ kpc. The maximum post-main-sequence age of the cluster is estimated as $\sim$ 4 Myr. Present spectroscopic analysis of the ionizing source indicates a spectral class of O6.5V. We used $JHK$ colour criteria to identify sources with NIR excess and found 120 candidate YSOs in the region. Majority of the YSOs have $A_V \le$ 4.0 mag and masses in the range $\sim$ 0.1 - 3.0 $M_\odot$. Distribution of these YSOs on the CMD indicates an age spread of $\sim$ 0.5 - 5 Myr with an average age of $\sim$ 2-3 Myr, suggesting non-coeval star formation in NGC 1624. The lower limit for the NIR excess fraction on the basis of $JHK$ data is found to be $\sim$ 20% which indicates an average age $\sim$ 2 - 3 Myr for YSOs in NGC 1624. From the radio continuum flux, spectral type of the ionizing source of the UCregion is estimated to be $\sim$ B0.5V.
A significant number of YSOs are located close to the cluster centre and a few YSOs are seen to be located/projected over the molecular clumps detected by Deharveng et al. (2008), as well as farther away from the clumps. We detect an enhanced density of reddened YSOs located/projected close to the molecular clump C2. The NIR excess and $(H-K)$ colour distribution of these sources show indication of an age sequence in the sense that the YSOs located/projected near the clump C2 are younger than those located within the cluster core.
The slope of the MF, $ \Gamma$, derived from optical data, in the mass range $1.2 \le M/M_{\odot}<27$ can be represented by -1.18 $\pm$ 0.10. Whereas NIR data, in the mass range $0.65 \le M/M_{\odot}<27$ yields $ \Gamma$ = -1.31 $\pm$ 0.15. Thus MF fairly agrees with the Salpeter value (-1.35). Slope of the KLF for NGC 1624 in the magnitude range 13.5 - 17.5 is found to be 0.30 $\pm$ 0.06 which is smaller than the average value ($\sim$0.4) obtained for young clusters of similar ages (Lada et al. 1991; Lada & Lada 1995; Lada & Lada 2003), however, agrees well with the values 0.27 $\pm$ 0.02 for Be 59 (Pandey et al. 2008); 0.34 $\pm$ 0.07 for NGC 1893 (Sharma et al. 2007) and 0.31 $\pm$ 0.02 for Stock 8 (Jose et al. 2008). However, there is a clear indication of break in the power law at $K$ =15.75 mag. The KLF slope in the magnitude range 13.5 - 15.75 can be represented by $\alpha = 0.44 \pm 0.11$ and the KLF slope is found to be flat in the magnitude range 15.75 - 17.5.
Acknowledgments
===============
Authors are thankful to the referee Dr. Antonio Delgado for his useful comments which has improved contents and presentation of the paper significantly. We thank the staff of IAO, Hanle and its remote control station at CREST, Hosakote, ARIES, Naini Tal, and GMRT, Pune, India for their assistance during observations. This publication makes use of data from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. This research used the facilities of the Canadian Astronomy Data Centre operated by the National Research Council of Canada with the support of the Canadian Space Agency. We thank Annie Robin for letting us use her model of stellar population synthesis. JJ is thankful for the financial support for this study through a stipend from the DST and CSIR, India.
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------------------- ---------------------------------------- ------------- ---------- ----------------------------------------
$\alpha_{(2000)}$ $\delta_{(2000)}$ Date of Filter Exposure time
(h:m:s) ($^{\circ}:^{\prime}:^{\prime\prime}$) observation (s)$\times$no. of frames
[*HCT$^1$*]{}
04:40:38 +50:27:36 2004.11.03 $U$ 600$\times$3
04:40:38 +50:27:36 2004.11.03 $B$ 300$\times$3, 60$\times$1, 20$\times$1
04:40:38 +50:27:36 2004.11.03 $V$ 120$\times$3, 10$\times$1
04:40:38 +50:27:36 2004.11.03 $R$ 60$\times$3, 10$\times$1
04:40:38 +50:27:36 2004.11.03 $I$ 60$\times$3, 10$\times$1, 5$\times$1
04:40:38 +50:27:36 2007.01.26 [ ]{} 450$\times$1
04:40:38 +50:27:36 2007.01.26 [ ]{} 450$\times$1
04:40:37 +50:27:41 2006.09.08 Gr7/167l 900$\times$1
04:40:39 +50:27:18 2007.01.26 Gr7/167l 600$\times$1
04:40:35 +50:28:44 2007.01.26 Gr7/167l 750$\times$1
04:40:32 +50:27:54 2007.01.26 Gr7/167l 750$\times$1
[*ST$^2$*]{}
04:40:38 +50:27:36 2006.12.12 $V$ 300$\times$10
04:40:38 +50:27:36 2006.12.12 $I_c$ 300$\times$5
------------------- ---------------------------------------- ------------- ---------- ----------------------------------------
: Log of observations[]{data-label="obslog"}
$^1$ 2-m Himalayan Chandra Telescope, IAO, Hanle\
$^2$ 104-cm Sampurnanand Telescope, ARIES, Naini Tal\
------ ------------------- ---------------------------------------- -------- --------- --------- --------- --------- -------- -------- -------- -------
star $\alpha_{(2000)}$ $\delta_{(2000)}$ $V$ $(U-B)$ $(B-V)$ $(V-R)$ $(V-I)$ $J$ $H$ $ K$ $A_V$
ID (h:m:s) ($^{\circ}:^{\prime}:^{\prime\prime}$)
1 04:39:46.271 +50:30:00.70 18.415 - - - 1.611 - - - -
2 04:39:46.320 +50:22:03.89 21.320 - - - 1.865 - - - -
3 04:39:46.320 +50:22:23.00 20.893 - - - 1.887 - - - -
... ..... ..... ..... ..... ..... ..... ..... ..... ...... .....
... ..... ..... ..... ..... ..... ..... ..... ..... ...... .....
1155 04:40:32.181 +50:27:53.40 13.067 0.396 0.917 0.542 1.055 11.172 10.838 10.728 3.1\*
... ..... ..... ..... ..... ..... ..... ..... ..... ...... .....
------ ------------------- ---------------------------------------- -------- --------- --------- --------- --------- -------- -------- -------- -------
: $UBVRI_cJHK$ photometric data of sample stars. The complete table is available in electronic form only.[]{data-label="optdata"}
\
$A_V$ for the $\star$ marked sources have been obtained using optical photometry\
--------- ------------------ ----------------------
V range NGC 1624 Field region
(mag) $r < 2^{\prime}$ r $\ge$ $3^{\prime}$
11 - 12 1.00 1.00
12 - 13 1.00 1.00
13 - 14 1.00 1.00
14 - 15 1.00 1.00
15 - 16 1.00 1.00
16 - 17 0.98 0.98
17 - 18 0.98 0.97
18 - 19 0.90 0.95
19 - 20 0.90 0.93
20 - 21 0.80 0.89
21 - 22 0.55 0.61
--------- ------------------ ----------------------
: Completeness Factor of photometric data in the cluster and field regions.[]{data-label="cf_opt"}
[^1]: E-mail: jessy@aries.res.in
[^2]: IRAF is distributed by National Optical Astronomy Observatories, USA
|
---
abstract: 'We pose counting problems related to the various settings for Coxeter-Catalan combinatorics (noncrossing, nonnesting, clusters, Cambrian). Each problem is to count “twin” pairs of objects from a corresponding problem in Coxeter-Catalan combinatorics. We show that the problems all have the same answer, and, for a given finite Coxeter group $W$, we call the common solution to these problems the $W$-biCatalan number. We compute the $W$-biCatalan number for all $W$ and take the first steps in the study of Coxeter-biCatalan combinatorics.'
address: 'Department of Mathematics, North Carolina State University, Raleigh, NC, USA'
author:
- Emily Barnard
- Nathan Reading
title: 'Coxeter-biCatalan combinatorics'
---
Introduction {#intro sec}
============
This paper considers enumeration problems closely related to Coxeter-Catalan combinatorics. (For background on Coxeter-Catalan combinatorics, see for example [@Armstrong; @rsga]). Each enumeration problem can be thought of as counting pairs of “twin” Coxeter-Catalan objects—twin sortable elements or twin nonnesting partitions, etc. Many of the terms used in this introductory section are new to this paper and will be explained in Section \[defs sec\].
In the setting of sortable elements and Cambrian lattices/fans, the enumeration problem is to count the following families of objects:
- maximal cones in the bipartite biCambrian fan (the common refinement of two bipartite Cambrian fans);
- pairs of twin $c$-sortable elements for bipartite $c$;
- classes in the bipartite biCambrian congruence (the meet of two bipartite Cambrian congruences);
- elements of the bipartite biCambrian lattice;
- $c$-bisortable elements for bipartite $c$.
In type A, $c$-bisortable elements for bipartite $c$ are in bijection with permutations avoiding a set of four bivincular patterns in the sense of [@BCDK Section 2] and with alternating arc diagrams, as will be explained in Sections \[pat av sec\]–\[alt sec\]. In type B, similar bijections exist with certain signed permutations and with centrally symmetric alternating arc diagrams, as described in Section \[type B sec\].
In the setting of nonnesting partitions (antichains in the root poset), the enumeration problem is to count two families of objects:
- antichains in the doubled root poset;
- pairs of twin nonnesting partitions.
In the setting of clusters of almost positive roots (in the sense of [@ga]), the problem is to count two families of objects:
- maximal cones in the bicluster fan (the common refinement of the cluster fan, in the original bipartite sense of Fomin and Zelevinsky, and its antipodal opposite);
- pairs of twin clusters, again in the bipartite sense.
In the setting of noncrossing partitions, the problem is to count the following families of objects:
- pairs of twin bipartite $c$-noncrossing partitions;
- pairs of twin bipartite $(c,c^{-1})$-noncrossing partitions.
The main result of this paper is the following.
\[main thm\] For each finite Coxeter group/root system, all of the enumeration problems posed above have the same answer.
In all of the settings above except the nonnesting setting, the objects described above can be defined for arbitrary choices of a Coxeter element. However, the enumerations depend on the choice of Coxeter element, and we emphasize that Theorem \[main thm\] is an assertion about the enumeration in the case where the Coxeter element is chosen to be bipartite. See Section \[bicamb sec\] for the definition of Coxeter elements and bipartite Coxeter elements.
The enumeration problems in the nonnesting setting require a crystallographic root system, but Theorem \[main thm\] still holds in the other settings for noncrystallographic types. See also Remark \[H I remark\].
We will see in Section \[defs sec\] that within each group of bullet points above, the various enumeration problems have the same answer essentially by definition. Using known uniform correspondences from the usual Coxeter-Catalan combinatorics, it is straightforward to give (in Theorems \[bi cl\] and \[bi nc\]) uniform bijections connecting the Cambrian/sortable setting to the noncrossing and cluster settings. The difficult part of the main result is the following theorem which connects the nonnesting setting to the other settings.
\[hard part\] For crystallographic $W$, $c$-bisortable elements for bipartite $c$ are in bijection with antichains in the doubled root poset.
More specifically, we have the following refined version of Theorem \[hard part\].
\[hard part finer\] For crystallographic $W$ and for any $k$, the number of bipartite $c$-bisortable elements with $k$ descents equals the number of $k$-element antichains in the doubled root poset.
Our proof of Theorems \[hard part\] and \[hard part finer\] in Section \[double pos sec\] would be uniform if a uniform proof were known connecting the nonnesting setting to the other settings of the usual Coxeter-Catalan combinatorics. Indeed, the opposite is true: A well-behaved uniform bijection proving Theorem \[hard part\] or Theorem \[hard part finer\] would imply a uniform proof of the analogous Coxeter-Catalan statement. (See Remark \[uniform uniform\] for details.) However, the proofs of these theorems are far from a trivial recasting of Coxeter-biCatalan combinatorics in terms of Coxeter-Catalan combinatorics. Instead, it requires a count of antichains in the doubled root poset indirectly in terms of the Coxeter-Catalan numbers and a nontrivial proof that the same formula holds for bipartite $c$-bisortable elements. The formula uses a notion of “double-positive” Catalan and Narayana numbers, which already appeared in [@Ath-Sav] as the local $h$-polynomials of the positive cluster complex. (See Remark \[Ath connection\].)
We propose the terms [***$W$-biCatalan number***]{} and [***$W$-biNarayana number***]{} and the symbols ${\operatorname{biCat}}(W)$ and ${\operatorname{biNar}}_k(W)$ for the numbers appearing in Theorems \[main thm\] and \[hard part finer\].
\[enum thm\] The $W$-biCatalan numbers for irreducible finite Coxeter groups are: [$$\begin{array}{c||c|c|c|c|c|c|c|c|c|c}
\!W\!&\!A_n\!&\!B_n\!&\!D_n\!&\!E_6\!&\!E_7\!&\!E_8\!&\!F_4\!&\!H_3\!&\!H_4\!&\!I_2(m)\!\\\hline
&&&&&&&&&&\\[-9pt]
\!{\operatorname{biCat}}(W)\!&\!\binom{2n}n\!&\!2^{2n-1}\!&\!6\cdot4^{n-2} - 2\binom{2n-4}{n-2}\!&\!1700\!&\!8872\!&\!54066\!&\!196\!&\!56\!&\!550\!&\!2m\!
\end{array}\,.$$]{}
The type-A and type-B cases of Theorem \[enum thm\] are proved, in the nonnesting setting, in Section \[nn sec\] by recasting the antichain count as a count of lattice paths. The same cases can also be established in the setting of $c$-bisortable elements by recasting the problem in terms of alternating arc diagrams. Although the latter approach is more difficult, we carry out the type-A and type-B enumeration by the latter approach in Section \[type A sec\], because the combinatorial models for bipartite $c$-bisortable elements in types A and B are of independent interest, and because the enumeration of alternating arc diagrams provides the crucial insight which leads to the recursive proof of Theorem \[main thm\]. (See Remark \[type A insight\].) The type-D case of Theorem \[enum thm\] is much more difficult, and involves solving the type-D case of the recursion used in the proof of Theorem \[hard part\]. The formula in type D was first guessed using the package `GFUN` [@GFUN]. The enumerations in the exceptional types were obtained using Stembridge’s `posets` and `coxeter/weyl` packages [@StembridgePackages].
We also obtain formulas for the $W$-biNarayana numbers outside of type D.
\[biNar thm\] The biNarayana numbers of irreducible finite Coxeter groups, except in type D, are given by the following generating functions.
----------------- -----------------------------------------------------------------------------------
$W$ $\sum_{k=0}^n{\operatorname{biNar}}_k(W)\,q^k$
\[2pt\]
\[-9pt\]
\[-10pt\] $A_n$ \[10pt\]\[5pt\][$\sum_{k=0}^n\binom n k^2q^k$]{}
$B_n$ \[10pt\]\[5pt\][$\sum_{k=0}^n\binom{2n}{2k}q^k$]{}
$E_6$ \[10pt\]\[4pt\][$1+66q+415q^2+736q^3+415q^4+66q^5+q^6$]{}
$E_7$ \[10pt\]\[4pt\][$1+119q+1139q^2+3177q^3+3177q^4+1139q^5+119q^6+q^7$]{}
$E_8$ \[10pt\]\[4pt\][$1+232q+3226q^2+13210q^3+20728q^4+13210q^5+3226q^6+232q^7+q^8$]{}
$F_4$ \[10pt\]\[4pt\][$1+44q+106q^2+44q^3+q^4$]{}
$G_2$ \[10pt\]\[4pt\][$1+10q+q^2$]{}
$H_3$ \[10pt\]\[4pt\][$1+27q+27q^2+q^3$]{}
$H_4$ \[10pt\]\[4pt\][$1+116q+316q^2+116q^3+q^4$]{}
$I_2(m)$ \[10pt\]\[4pt\][$1+(2m-2)q+q^2.$]{}
----------------- -----------------------------------------------------------------------------------
Generating functions for biNarayana numbers for some type-D Coxeter groups are shown here. At present we have no conjectured formula for the $D_n$-biNarayana numbers. See Section \[type D biNar sec\] for a modest conjecture.
$D_4$ \[10pt\]\[4pt\][$1+20q+42q^2+20q^3+q^4$]{}
---------- --------------------------------------------------------------------------------
$D_5$ \[10pt\]\[4pt\][$1+35q+136q^2+136q^3+35q^4+q^5$]{}
$D_6$ \[10pt\]\[4pt\][$1+54q+343q^2+600q^3+343q^4+54q^5+q^6$]{}
$D_7$ \[10pt\]\[4pt\][$1+77q+731q^2+2011q^3+2011q^4+731q^5+77q^6+q^7$]{}
$D_8$ \[10pt\]\[4pt\][$1+104q+1384q^2+5556q^3+8638q^4+5556q^5+1384q^6+104q^7+q^8$]{}
$D_9$ \[10pt\]\[4pt\][$1+135q+2402q^2+13314q^3+29868q^4$]{}
\[10pt\]\[4pt\][$+29868q^5+13314q^6+2402q^7+135q^8+q^9$]{}
$D_{10}$ \[10pt\]\[4pt\][$1+170q+3901q^2+28624q^3+87874q^4+126336q^5$]{}
\[10pt\]\[4pt\][$+87874q^6+28624q^7+3901q^8+170q^9+q^{10}$]{}
Naturally, one would like a uniform formula for the $W$-biCatalan number, but we have not found one. A tantalizing near-miss is the *non-formula* $\prod_{i=1}^n\frac{h+e_i-1}{e_i}$, where $h$ is the Coxeter number and the $e_i$ are the exponents. This expression captures the $W$-biCatalan numbers for $W$ of types $A_n$, $B_n$, $H_3$, and $I_2(m)$—the “coincidental types” of [@Williams]—but fails to even be an integer in some other types. In every case, the expression is a surprisingly good estimate of the $W$-biCatalan number.
Section \[defs sec\] is devoted to filling in definitions and details for the discussion above and proving the easy parts of Theorem \[main thm\]. In Section \[type A sec\], we explain why, in type A, the bipartite bisortable elements are in bijection with alternating arc diagrams and carry out the enumeration of alternating arc diagrams. We carry out a similar enumeration in type B, in terms of centrally symmetric alternating arc diagrams. We conjecture that the bipartite biCambrian fan is simplicial (and thus that its dual polytope is simple), and prove the conjecture in types A and B. In Section \[double pos sec\], we discuss double-positive Coxeter-Catalan numbers and establish a formula counting antichains in the doubled root poset in terms of double-positive Coxeter-Catalan numbers. We then show that bipartite $c$-bisortable elements satisfy the same recursion, thus proving Theorem \[hard part finer\] and completing the proof of Theorem \[main thm\]. Finally, we establish some additional formulas involving double-positive Coxeter-Catalan numbers, Coxeter-Catalan numbers, and Coxeter-biCatalan numbers and use them to prove the formula for ${\operatorname{biCat}}(D_n)$ and thus complete the proof of Theorem \[enum thm\].
BiCatalan objects {#defs sec}
=================
In this section, we fill in the definitions and details behind the enumeration problems discussed in the introduction. An exposition in full detail would require reviewing Coxeter-Catalan combinatorics in full detail, so we leave some details to the references.
Antichains in the doubled root poset and twin nonnesting partitions {#nn sec}
-------------------------------------------------------------------
The [***root poset***]{} of a finite crystallographic root system $\Phi$ is the set of positive roots in $\Phi$, partially ordered by setting $\alpha\le\beta$ if and only if $\beta-\alpha$ is in the nonnegative span of the simple roots. Recall that the dual of a poset $(X,\le)$ is the poset $(X,\ge)$. That is, the dual has the same ground set, with $x\leq y$ in the dual poset if and only if $x\ge y$ in the original poset. The [***doubled root poset***]{} consists of the root poset, together with a disjoint copy of the dual poset, identified on the simple roots. Figure \[doubled\] shows some doubled root posets.
----------------------------------------------------------------------- -- ----------------------------------------------------------------------- -- ----------------------------------------------------------------------- -- -- -- --
![Some doubled root posets[]{data-label="doubled"}](doubledA5 "fig:") ![Some doubled root posets[]{data-label="doubled"}](doubledB3 "fig:") ![Some doubled root posets[]{data-label="doubled"}](doubledD4 "fig:")
$A_5$ $B_3$ $D_4$
----------------------------------------------------------------------- -- ----------------------------------------------------------------------- -- ----------------------------------------------------------------------- -- -- -- --
----------------------------------------------------------------------- -- -- -------------------------------------------------------------------------------------------
![Some doubled root posets[]{data-label="doubled"}](doubledD6 "fig:") \[190pt\]\[0pt\][![Some doubled root posets[]{data-label="doubled"}](doubledF4 "fig:")]{}
$D_6$ $F_4$
----------------------------------------------------------------------- -- -- -------------------------------------------------------------------------------------------
The antichain counts in types A and B are easy and known, in the guise of lattice path enumeration. Antichains in the doubled root poset of type $A_n$ are in an easy bijection with lattice paths from $(0,0)$ to $(n,n)$ with steps $(1,0)$ and $(0,1)$. The bijection can be made so that the number of elements in the antichain corresponds to the number of right turns in the path (the number of times a $(1,0)$-step immediately follows a $(0,1)$-step). To specify a path with $k$ right turns, we need only specify where the right turns are. This means choosing $0\le x_1<\cdots<x_k\le n-1$ and $1\le y_1<\cdots<y_k\le n$ and placing right turns at $(x_1,y_1),\ldots,(x_k,y_k)$. Thus, as is well-known, there are $\binom nk^2$ paths with $k$ right turns.
Antichains in the doubled root poset of type $B_n$ are similarly in bijection with lattice paths from $(-2n+1,-2n+1)$ to $(2n-1,2n-1)$ with steps $(2,0)$ and $(0,2)$ that are symmetric with respect to the reflection through the line $y=-x$. The $k$-element antichains correspond to paths with either $2k$ right turns, ($k$ of which are to the left of the line $y=-x$) or $2k-1$ right turns ($k-1$ of which are left of the line $y=-x$ and one of which is on the line $y=-x$). Each path is uniquely determined by its first $2n-1$ steps, whereupon the path intersects the line $y=-x$. Thus, the paths map bijectively to words of length $2n-1$ in the letters $N$ and $E$ (for North steps $(0,2)$ and East steps $(2,0)$). Appending the letter $E$ to the end of each word, the $k$-element antichains correspond to the words having exactly $k$ positions where an $E$ appears immediately after an $N$. (The number of right turns in the path is odd if and only if one of these is position $2n$.) The $2n$-letter words ending in $E$ and having exactly $k$ instances of an $E$ following an $N$ are in bijection with $2k$-element subsets of $\set{1,\ldots,2n}$. (Given such a word, take the set of positions where the letter changes, with the convention that an $N$ in the first position is a change but an $E$ in the first position is not. So, for example, $ENNEEE$ gives the subset $\set{2,4}$ and $NEEENE$ gives $\set{1,2,5,6}$.) We see that there are $\binom{2n}{2k}$ $k$-element antichains, and $2^{2n-1}$ total antichains, in the doubled root poset of type $B_n$.
\[H I remark\] It is not clear in general how one should define a “root poset” for a noncrystallographic root system. See [@Armstrong Section 5.4.1] for a discussion. In type $I_2(m)$, there is an obvious way to define an unlabeled poset generalizing the root posets of types $A_2$, $B_2$, and $G_2$. We say “unlabeled” here because it is obvious how the poset should look but not obvious how the poset elements should correspond to roots. There is also an unlabeled type-$H_3$ root poset suggested in [@Armstrong Section 5.4.1]. For these choices of root posets, one can verify that Theorem \[main thm\] holds in these types as well.
\[distr\] The doubled root poset, and similar posets, were probably first considered by Proctor (see [@StembridgeQuasi Remark 4.8(a)]) and then by Stembridge, as a tool for counting reduced expressions for certain elements of finite Coxeter groups. In the simply-laced types (A, D, and E), the doubled root poset corresponds to the [***smashed Cayley order***]{} defined by Stembridge in [@StembridgeQuasi Section 4]. In the non-simply laced types, the smashed Cayley order is disconnected and is a strictly weaker partial order than the doubled root poset. Stembridge [@StembridgeQuasi Theorem 4.6] shows that the component whose elements are short roots is a distributive lattice. Thus in particular the doubled root posets of types A, D, and E are distributive lattices. One can easily check distributivity in the remaining crystallographic types B, F, and G (and in fact in types $H_3$ and $I_2(m)$). By the Fundamental Theorem of Distributive Lattices [@EC1 Theorem 3.4.1], the doubled root poset is isomorphic to the poset of order ideals in its subposet of join-irreducible elements. These posets of join-irreducible elements are shown in Figure \[IrrDRP\] for several types. An explicit root-theoretic description of the poset of join-irreducible elements in the simply-laced types also appears in [@StembridgeQuasi Theorem 4.6].
---------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------- -- -- -- -- -- -- -- -- -- -- -- -- -- -- --
![Some posets of join-irreducibles of doubled root posets[]{data-label="IrrDRP"}](IrrDRPA6 "fig:") ![Some posets of join-irreducibles of doubled root posets[]{data-label="IrrDRP"}](IrrDRPB6 "fig:") ![Some posets of join-irreducibles of doubled root posets[]{data-label="IrrDRP"}](IrrDRPD6 "fig:") ![Some posets of join-irreducibles of doubled root posets[]{data-label="IrrDRP"}](IrrDRPH3 "fig:")
$A_6$ $B_6$ $D_6$ $H_3$
---------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------- -- -- -- -- -- -- -- -- -- -- -- -- -- -- --
---------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------- -- -- -- -- --
![Some posets of join-irreducibles of doubled root posets[]{data-label="IrrDRP"}](IrrDRPF4 "fig:") ![Some posets of join-irreducibles of doubled root posets[]{data-label="IrrDRP"}](IrrDRPE6 "fig:") ![Some posets of join-irreducibles of doubled root posets[]{data-label="IrrDRP"}](IrrDRPE7 "fig:") \[0pt\]\[0pt\][![Some posets of join-irreducibles of doubled root posets[]{data-label="IrrDRP"}](IrrDRPE8 "fig:")]{}
$F_4$ $E_6$ $E_7$ $E_8$
---------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------- -- -- -- -- --
The [***support***]{} of a root $\beta$ is the set of simple roots appearing with nonzero coefficient in the expansion of $\beta$ in the basis of simple roots. The support of a set of roots is the union of the supports of the roots in the set. We write $\Delta$ for the simple roots and, given a set $A$ of roots, we write $A^\circ$ for the set of non-simple roots in $A$. If $A_1$ and $A_2$ are nonnesting partitions (i.e. antichains in the root poset), then $(A_1,A_2)$ is a pair of [***twin nonnesting partitions***]{} if and only if $A_1\cap\Delta=A_2\cap\Delta$, and ${\operatorname{supp}}(A_1^\circ)\cap{\operatorname{supp}}(A_2^\circ)=\emptyset$.
Given an antichain $A$ in the doubled root poset, define ${\operatorname{top}}(A)$ to be the intersection of $A$ with the root poset that forms the top of the doubled root poset. Define ${\operatorname{bottom}}(A)$ to be the intersection of $A$ with the dual root poset that forms the bottom of the doubled root poset. Both ${\operatorname{top}}(A)$ and ${\operatorname{bottom}}(A)$ are sets of positive roots. The following proposition is an immediate consequence of the observation that a root $\beta$ in the top part of the doubled root poset is related to a root $\gamma$ in the bottom part of the doubled root poset if and only if the supports of $\beta$ and $\gamma$ overlap.
\[anti twin nn\] The map $A\mapsto({\operatorname{top}}(A),{\operatorname{bottom}}(A))$ is a bijection from antichains in the doubled root poset to pairs of twin nonnesting partitions.
We pause to observe that the first biNarayana number (the number of elements of the doubled root poset) is the number of roots minus the rank of $W$.
\[biNar 1\] If $W$ is an irreducible finite Coxeter group with Coxeter number $h$ and rank $n$, then ${\operatorname{biNar}}_1(W)=n(h-1)$.
BiCambrian fans {#bicamb sec}
---------------
The Cambrian fan is a complete simplicial fan whose maximal faces are naturally in bijection [@sortable; @camb_fan] with seeds in an associated cluster algebra of finite type and with noncrossing partitions. Furthermore, the Cambrian fan is the normal fan [@HohLan; @HLT] to a simple polytope called the [***generalized associahedron***]{} [@gaPoly; @ga], which encodes much of the combinatorics of the associated cluster algebra. More directly, the Cambrian fan is the ${\mathbf{g}}$-vector fan of the cluster algebra. (This was conjectured, and proved in a special case, in [@camb_fan Section 10] and then proved in general in [@YZ].)
The defining data of a Cambrian fan is a finite Coxeter group $W$ and a Coxeter element $c$ of $W$. We emphasize that the results discussed in Section \[intro sec\] concern a special “bipartite” choice of $c$, as explained below, but for now we proceed with a discussion for general $c$. A [***Coxeter element***]{} is the product of a permutation of the simple generators of $W$, or equivalently it is an orientation of the Coxeter diagram of $W$. Given a choice of $W$, we will assume the usual representation of $W$ as a reflection group acting with trivial fixed subspace. The collection of reflecting hyperplanes in this representation is the [***Coxeter arrangement***]{} of $W$. The hyperplanes in the Coxeter arrangement cut space into cones, which constitute a fan called the [***Coxeter fan***]{} ${\mathcal{F}}(W)$. The maximal cones of the Coxeter fan are in bijection with the elements of $W$. The Cambrian fan ${\operatorname{\mathbf{Camb}}}(W,c)$ is the coarsening of the Coxeter fan obtained by gluing together maximal cones according to an equivalence relation on $W$ called the $c$-Cambrian congruence. Further details on the $c$-Cambrian congruence appear in Section \[twin sec\]. For fixed $W$, all choices of $c$ give distinct but combinatorially isomorphic Cambrian fans.
For each Coxeter element $c$, the inverse element $c^{-1}$ is also a Coxeter element, corresponding to the opposite orientation of the diagram. We define the [***biCambrian fan***]{} ${\operatorname{\mathbf{biCamb}}}(W,c)$ to be the coarsest common refinement of the Cambrian fans ${\operatorname{\mathbf{Camb}}}(W,c)$ and ${\operatorname{\mathbf{Camb}}}(W,c^{-1})$. Since ${\operatorname{\mathbf{Camb}}}(W,c)$ and ${\operatorname{\mathbf{Camb}}}(W,c^{-1})$ are coarsenings of ${\mathcal{F}}(W)$, so is ${\operatorname{\mathbf{biCamb}}}(W,c)$. Naturally, ${\operatorname{\mathbf{biCamb}}}(W,c^{-1})={\operatorname{\mathbf{biCamb}}}(W,c)$.
\[B2 example\] To illustrate the definition, take $W$ of type $B_2$ with simple generators $s_1$ and $s_2$. Figure \[boring\] shows, from left to right, the $s_1s_2$-Cambrian fan, the $s_2s_1$-Cambrian fan, and the $s_1s_2$-biCambrian fan. Observe that the $s_1s_2$-biCambrian fan coincides with the $B_2$ Coxeter fan. In general, when $W$ is rank 2, the $c$-biCambrian fan for any choice of Coxeter element $c$ is equal to the Coxeter fan ${\mathcal{F}}(W)$.
\[A3 biCamb example\] For $W$ of type $A_3$, there are two non-isomorphic $c$-biCambrian fans, shown in Figures \[A3 biCamb linear\] and \[A3 biCamb bipartite\] respectively. Each figure can be understood as follows: Intersecting the $c$-biCambrian fan with a unit sphere about the origin, we obtain a decomposition of the sphere into spherical convex polygons. The picture shows a stereographic projection of this polygonal decomposition to the plane. In each case, the walls of one Cambrian fan are shown in red and the walls of the opposite Cambrian fan are shown in blue. Walls that are in both Cambrian fans are shown dashed red and blue.
\[common walls\] We observe that in Examples \[B2 example\] and \[A3 biCamb example\] that the common walls of ${\operatorname{\mathbf{Camb}}}(W,c)$ and ${\operatorname{\mathbf{Camb}}}(W,c^{-1})$ are exactly the reflecting hyperplanes for the simple generators of $W$. This fact true in general, and the simplest proof involves [***shards***]{}. We will not define shards here, but definitions and results can be found, for example, in [@shardint]. Assuming for a moment that background, we sketch a proof. First, recast [@typefree Theorem 8.3] as the statement that the $c$-Cambrian congruence removes all but one shard from each reflecting hyperplane of $W$. As explained in the argument for [@sort_camb Proposition 1.3] (located in [@sort_camb Section 3] just after the proof of [@sort_camb Theorem 1.1]), the antipodal map sends the shard that is not removed by the $c$-Cambrian congruence to the shard that is not removed by the $c^{-1}$-Cambrian congruence. The only shards that are fixed by the antipodal map are shards that consist of an entire reflecting hyperplane, and [@shardint Lemma 3.11] says that these are exactly the reflecting hyperplanes for the simple generators.
The $c^{-1}$-Cambrian fan ${\operatorname{\mathbf{Camb}}}(W,c^{-1})$ coincides with $-{\operatorname{\mathbf{Camb}}}(W,c)$, the image of the $c$-Cambrian fan under the antipodal map. This is an immediate corollary of [@sort_camb Proposition 1.3], which is a statement about the $c$-Cambrian congruence. See also [@afframe Remark 3.26]. Thus we have the following proposition which amounts to an alternate definition of the biCambrian fan.
\[biCamb alt def\] The biCambrian fan ${\operatorname{\mathbf{biCamb}}}(W,c)$ is the coarsest common refinement of ${\operatorname{\mathbf{Camb}}}(W,c)$ and $-{\operatorname{\mathbf{Camb}}}(W,c)$.
Since ${\operatorname{\mathbf{Camb}}}(W,c)$ and ${\operatorname{\mathbf{Camb}}}(W,c^{-1})$ are the normal fans of two generalized associahedra, a standard fact (see [@Ziegler Proposition 7.12]) yields the following result.
\[is poly\] For any $W$ and $c$, the fan ${\operatorname{\mathbf{biCamb}}}(W,c)$ is the normal fan of a polytope, specifically, the Minkowski sum of the generalized associahedra dual to ${\operatorname{\mathbf{Camb}}}(W,c)$ and ${\operatorname{\mathbf{Camb}}}(W,c^{-1})$.
The definition of ${\operatorname{\mathbf{biCamb}}}(W,c)$ seems strange *a priori*, but it is well-motivated *a posteriori* by enumerative results. The first such result is Theorem \[Baxter thm\] below. When $W$ is the symmetric group $S_n$ (i.e. when $W$ is of type $A_{n-1}$), the Coxeter diagram of $W$ is a path. A [***linear Coxeter element***]{} of $S_n$ is the product of the generators in order along the path.
\[Baxter thm\] When $W$ is the symmetric group $S_n$ and $c$ is the linear Coxeter element, the number of maximal cones in ${\operatorname{\mathbf{biCamb}}}(W,c)$ is the Baxter number.
For more on the Baxter number, see [@Baxter; @CGHK]. Theorem \[Baxter thm\] was observed empirically (in the language of lattice congruences) in [@con_app Section 10] and then proven by J. West [@West; @pers]. See also [@Giraudo; @rectangle]. The theorem is also related to the observation by Dulucq and Guibert [@DGStack] that pairs of twin binary trees are counted by the Baxter number.
Once one sees that the Baxter number counts maximal cones of ${\operatorname{\mathbf{biCamb}}}(W,c)$ for $W$ of type A and for a particular $c$, it is natural to look at other types of finite Coxeter group $W$, with the idea of defining a “$W$-Baxter number” for each finite Coxeter group $W$. Indeed, there is a good notion of a “type-B Baxter number” discovered by Dilks [@Dilks]. The Coxeter diagram of type B is also a path, and taking $c$ to be a linear Coxeter element, the maximal cones of ${\operatorname{\mathbf{biCamb}}}(W,c)$ are counted by the type-B Baxter number. Despite the nice type-B result, there seems to be little hope for a reasonable definition of the $W$-Baxter number, because some types of Coxeter diagrams are not paths and thus it is not clear how to generalize the notion of a *linear* Coxeter element.
There is, however, a choice of Coxeter element that can be made uniformly for all finite Coxeter groups. Since the Coxeter diagram of any finite Coxeter group is acyclic, the diagram is in particular bipartite. Thus we can fix a bipartition $S_+\cup S_-$ of the diagram and orient each edge of the diagram from its vertex in $S_-$ to its vertex in $S_+$. The resulting Coxeter element is called a [***bipartite Coxeter element***]{}, and if $c$ is a bipartite Coxeter element of $W$, we call ${\operatorname{\mathbf{biCamb}}}(W,c)$ a [***bipartite biCambrian fan***]{}.
Proposition \[is poly\] says that ${\operatorname{\mathbf{biCamb}}}(W,c)$ is the normal fan of a polytope, but does not guarantee that this polytope is simple (equivalently, that this fan is simplicial). In fact, simpleness fails for the linear Coxeter element of $S_n$, and this failure can be seen already in $S_4$. (See Figure \[A3 biCamb linear\], and also [@rectangle Figure 13]. The latter shows the $1$-skeleton of this polytope disguised as the Hasse diagram of a certain lattice.) We conjecture that the situation is better in the bipartite case.
\[simple poly\] If $W$ is a bipartite Coxeter element, then ${\operatorname{\mathbf{biCamb}}}(W,c)$ is a simplicial fan. (Equivalently, its dual polytope is simple.)
We have verified Conjecture \[simple poly\], with the aid of Stembridge’s packages [@StembridgePackages], up to rank 6. Also, in Section \[A B simp sec\], we prove the following theorem using alternating arc diagrams, by appealing to some results of [@IRRT] linking the lattice theory of the weak order to the representation theory of finite-dimensional algebras, and then applying a folding argument.
\[simple A B\] Conjecture \[simple poly\] holds in types A and B.
In Section \[twin sec\], we will prove the following theorem.
\[simple h\] If Conjecture \[simple poly\] holds for a Coxeter group $W$, then the $h$-vector of the simplicial sphere underlying ${\operatorname{\mathbf{biCamb}}}(W,c)$, for $c$ bipartite, has entries ${\operatorname{biNar}}_k(W)$.
In light of the evidence for Conjecture \[simple poly\] and in light of Theorem \[simple h\], we propose the term [***simplicial $W$-biassociahedron***]{} for the polytope whose face fan is ${\operatorname{\mathbf{biCamb}}}(W,c)$ *for $c$ bipartite*, and [***simple $W$-biassociahedron***]{} for the polytope whose normal fan is ${\operatorname{\mathbf{biCamb}}}(W,c)$ *for $c$ bipartite*.
\[A biassoc\] Theorems \[biNar thm\], \[simple A B\], and \[simple h\] imply that the $A_n$-biassociahedron has the same $h$-vector as the $B_n$-associahedron (also known as the cyclohedron). One is naturally led to ask whether these two polytopes are combinatorially isomorphic. The answer is no already for $n=3$. The normal fan to the $A_3$-biassociahedron is shown in Figure \[A3 biCamb bipartite\]. The dual graph to this fan has a vertex that is incident to two hexagons and a quadrilateral. The graph of the $B_3$-associahedron (shown for example in [@rsga Figure 3.9]) has no such vertex.
The biCambrian congruence, twin sortable elements, and bisortable elements {#twin sec}
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A [***congruence***]{} $\Theta$ on a lattice $L$ is an equivalence relation respecting the meet and join operations. In this paper, we consider only finite lattices, and some results quoted in this section can fail for infinite lattices. On a finite lattice, congruences are characterized by three properties: congruence classes are intervals; the projection \[0pt\]\[0pt\][${\pi_\downarrow}^\Theta$]{}, mapping each element to the bottom element of its congruence class, is order preserving; and the projection \[0pt\]\[0pt\][${\pi^\uparrow}_\Theta$]{}, mapping each element to the top element of its congruence class, is order preserving. The $\Theta$-classes are exactly the fibers of ${\pi_\downarrow}^{\Theta}$. The quotient $L/\Theta$ of a finite lattice $L$ modulo a congruence $\Theta$ is a lattice isomorphic to the subposet induced by the set \[0pt\]\[0pt\][${\pi_\downarrow}^\Theta(L)$]{} of elements that are the bottoms of their congruence classes. The congruence $\Theta$ is determined by the set ${\pi_\downarrow}^\Theta(L)$: Specifically $x\equiv y$ modulo $\Theta$ if and only if the unique maximal element of ${\pi_\downarrow}^\Theta(L)$ below $x$ equals the unique maximal element of ${\pi_\downarrow}^\Theta(L)$ below $y$.
The map ${\pi_\downarrow}^{\Theta}$ is a lattice homomorphism from $L$ onto the subposet \[0pt\]\[0pt\][${\pi_\downarrow}^\Theta(L)$]{}, but care must be taken to avoid misinterpreting this fact. Literally, the fact that ${\pi_\downarrow}^{\Theta}$ is a lattice homomorphism means that for any $U\subseteq L$, we have ${\pi_\downarrow}^{\Theta}({\bigvee}U) = {\bigvee}_{x\in U} {\pi_\downarrow}^{\Theta}(x)$ and ${\pi_\downarrow}^{\Theta}({\bigwedge}U) = {\bigwedge}_{x\in U} {\pi_\downarrow}^{\Theta}(x)$, but in each identity, the join on the left side occurs in $L$ while the join on the right side occurs in ${\pi_\downarrow}^\Theta(L)$. It is easy to check that ${\pi_\downarrow}^\Theta(L)$ is also a join-sublattice of $L$, so the distinction between the join in $L$ and the join in ${\pi_\downarrow}^\Theta(L)$ is unnecessary. However, in general, ${\pi_\downarrow}^{\Theta}(L)$ need not be a meet-sublattice of $L$, so in interpreting the identity ${\pi_\downarrow}^{\Theta}({\bigwedge}U) = {\bigwedge}_{x\in U} {\pi_\downarrow}^{\Theta}(x)$, it is crucial to be clear on where the meets occur.
The maximal cones of the Coxeter fan ${\mathcal{F}}(W)$, partially ordered according to a suitable linear functional, form a lattice isomorphic to the weak order on $W$. (This fact is true either for the right or left weak order. We will work with the right weak order.) Any lattice congruence $\Theta$ on the weak order on $W$ defines a fan ${\mathcal{F}}_\Theta(W)$ coarsening ${\mathcal{F}}(W)$. (See [@con_app Theorem 1.1] and [@con_app Section 5].) Specifically, for each $\Theta$-class, the union of the corresponding maximal cones in ${\mathcal{F}}(W)$ is itself a convex cone, and the collection of all these convex cones and their faces is the fan ${\mathcal{F}}_\Theta(W)$. Each Coxeter element $c$ specifies a congruence $\Theta_c$ on the weak order called the [***$c$-Cambrian congruence***]{}. (See [@cambrian] for the definition.) The fan ${\mathcal{F}}_{\Theta_c}(W)$ is the $c$-Cambrian fan ${\operatorname{\mathbf{Camb}}}(W,c)$ described earlier.
The set ${\operatorname{Con}}(L)$ of all congruences on a given lattice $L$ is itself a sublattice of the lattice of set partitions of $L$. In particular, the meet of two congruences is the coarsest set partition of $L$ refining both congruences. We define the [***$c$-biCambrian congruence***]{} to be the meet, in ${\operatorname{Con}}(W)$, of the Cambrian congruences $\Theta_c$ and $\Theta_{c^{-1}}$. The fan ${\mathcal{F}}_\Theta(W)$ for $\Theta=\Theta_c{\wedge}\Theta_{c^{-1}}$ is the coarsest common refinement of ${\mathcal{F}}(\Theta_c(W))$ and ${\mathcal{F}}(\Theta_{c^{-1}}(W))$. Thus the $c$-biCambrian fan ${\operatorname{\mathbf{biCamb}}}(W,c)$ is the fan ${\mathcal{F}}_\Theta(W)$ for $\Theta=\Theta_c{\wedge}\Theta_{c^{-1}}$. In particular, the $c$-biCambrian congruence classes are in bijection with the maximal cones of ${\operatorname{\mathbf{biCamb}}}(W,c)$. We define the [***$c$-biCambrian lattice***]{} to be the quotient of the weak order modulo the $c$-biCambrian congruence. The elements of the $c$-biCambrian lattice are thus in bijection with the maximal cones of ${\operatorname{\mathbf{biCamb}}}(W,c)$.
We write \[0pt\]\[0pt\][${\pi_\downarrow}^c$]{} for the projection taking each element of $W$ to the bottom element of its $c$-Cambrian congruence class, and similarly ${\pi_\downarrow}^{c^{-1}}$. (That is, ${\pi_\downarrow}^c$ stands for ${\pi_\downarrow}^\Theta$ where $\Theta=\Theta_c$.) Consider the map that sends each $c$-biCambrian congruence class to the pair $({\pi_\downarrow}^c(w),{\pi_\downarrow}^{c^{-1}}(w))$, where $w$ is any representative of the class. Because the $c$-biCambrian congruence $\Theta$ is the meet $\Theta_c{\wedge}\Theta_{c^{-1}}$, two elements $u$ and $v$ are congruent in the $c$-biCambrian congruence if and only if \[0pt\]\[0pt\][${\pi_\downarrow}^c(u)={\pi_\downarrow}^c(v)$]{} and \[0pt\]\[0pt\][${\pi_\downarrow}^{c^{-1}}(u)={\pi_\downarrow}^{c^{-1}}(v)$]{}. Thus, the map from classes to pairs is a well-defined bijection from $c$-biCambrian congruence classes to its image.
The bottom elements of the $c$-Cambrian congruence are called [***$c$-sortable elements***]{}. (In fact $c$-sortable elements have an independent combinatorial definition [@sortable Section 2], but were shown to be the bottom elements of $c$-Cambrian congruences in [@sort_camb Theorems 1.1 and 1.4].) Given elements $u$ and $v$ of $W$, we define the pair $(u,v)$ to be a pair of [***twin $(c,c^{-1})$-sortable elements***]{} of $W$ if there exists $w\in W$ such that $u={\pi_\downarrow}^c(w)$ and $v={\pi_\downarrow}^{c^{-1}}(w)$. The map considered in the previous paragraph is a bijection between $c$-biCambrian congruence classes and pairs of twin $(c,c^{-1})$-sortable elements of $W$. The twin sortable elements are similar in spirit to the twin binary trees of [@DGStack], which were already mentioned in connection with Theorem \[Baxter thm\]. Indeed, for $W$ of type A and $c$ linear, the connection is implicit in the construction in [@rectangle] of a diagonal rectangulation from a pair of binary trees. (See also [@rectangle Remark 6.6].) Also in type A, but for general $c$, the twin binary trees are generalized in [@ChaPil] to [***twin Cambrian trees***]{}, which correspond explicitly to pairs of twin $(c,c^{-1})$-sortable elements. Indeed, [@ChaPil Proposition 57] amounts to another computation of the type-A biCatalan number, quite different from the two given here (in Sections \[nn sec\] and \[alt\_arc\_diagrams\]).
Another set of objects naturally in bijection with $c$-biCambrian congruence classes are the bottom elements of $c$-biCambrian congruence classes. We coin the term [***$c$-bisortable***]{} elements for these bottom elements. Although the $c$-sortable elements have a direct combinatorial characterization [@sortable Section 2], we currently have no direct combinatorial characterization of $c$-bisortable elements. We do offer the following indirect characterization of $c$-bisortable elements in terms of $c$-sortable elements and $c^{-1}$-sortable elements.
\[c join cinv\] For any $c$, an element $w\in W$ is $c$-bisortable if and only if there exists a $c$-sortable element $u$ and a $c^{-1}$-sortable element $v$ such that $w=u{\vee}v$ in the weak order. When $w$ is $c$-bisortable, we can take $u={\pi_\downarrow}^c(w)$ and $v={\pi_\downarrow}^{c^{-1}}(w)$.
Given $c$-bisortable $w$, take $u={\pi_\downarrow}^c(w)$ and $v={\pi_\downarrow}^{c^{-1}}(w)$. Then $u\le w$ and $v\le w$. Since Cambrian congruence classes are intervals, any upper bound $w'$ for $u$ and $v$ with $w'\le w$ is congruent to $u$ modulo $\Theta_c$ and congruent to $v$ modulo $\Theta_{c^{-1}}$. Thus $w'$ is congruent to $w$ in the $c$-biCambrian congruence. Since $w$ is the bottom element of its $c$-biCambrian congruence class, we conclude that $w'=w$. We have shown that $w=u{\vee}v$.
Suppose $w=u{\vee}v$ for some $c$-sortable element $u$ and some $c^{-1}$-sortable element $v$. Since ${\pi_\downarrow}^c(w)$ is the unique maximal $c$-sortable element below $w$, we have ${\pi_\downarrow}^c(w)\ge u$. Similarly, ${\pi_\downarrow}^{c^{-1}}(w)\ge v$. If there exists $w'<w$ in the same $c$-biCambrian congruence class as $w$, then $w'\ge{\pi_\downarrow}^c(w')={\pi_\downarrow}^c(w)\ge u$ and $w'\ge{\pi_\downarrow}^{c^{-1}}(w')={\pi_\downarrow}^{c^{-1}}(w)\ge v$. This contradicts the fact that $w=u{\vee}v$, and we conclude that $w$ is $c$-bisortable.
Recall that for any congruence $\Theta$ on a finite lattice $L$, the set ${\pi_\downarrow}^\Theta(L)$ is a join-sublattice of $L$. The Cambrian congruences have a stronger property: For any Coxeter element $c$, the $c$-sortable elements constitute a sublattice [@sort_camb Theorem 1.2] of the weak order on $W$. It is natural to ask whether the same is true for $c$-bisortable elements, but the answer is no. We give an example for $W=S_5$ and bipartite $c$: The permutations $45312$ and $53142$ are both $c$-bisortable but their meet $31452$ is not. (To check this example, Proposition \[avoidance\] will be very helpful.)
Each $c$-bisortable element $v$ covers some number of elements in the $c$-biCambrian lattice. By a general fact on lattice quotients (see for example [@shardint Proposition 6.4]), $v$ covers the same number of elements in the weak order on $W$. This number is ${\operatorname{des}}(v)$, the number of descents of $v$. (We will define descents in Section \[sortable formula sec\].) The [***descent generating function of $c$-bisortable elements***]{} is the sum $\sum x^{{\operatorname{des}}(v)}$ over all $c$-bisortable elements $v$. We will show that its coefficients are the $W$-biNarayana numbers. Its coefficients are the $W$-biNarayana numbers. A general fact about lattice quotients of the weak order [@con_app Proposition 3.5] implies that, when ${\operatorname{\mathbf{biCamb}}}(W,c)$ is simplicial, the descent generating function of $c$-bisortable elements equals the $h$-polynomial of ${\operatorname{\mathbf{biCamb}}}(W,c)$. In the bipartite case, Theorem \[simple h\] follows immediately.
Twin clusters and bicluster fans {#clus sec}
--------------------------------
Clusters of almost positive roots were introduced in [@ga], where they were used to define the generalized associahedra. In [@ca2], clusters of almost positive roots were used to model cluster algebras of finite type. Here, we will not need the cluster-algebraic background, which can be found in [@ca2]. Instead, we define almost positive roots and $c$-compatibility and quote some results about $c$-clusters and their relationship to $c$-sortable elements. We will also not need the more refined notion of “compatibility degree.”
In a finite root system, the [***almost positive roots***]{} are those roots which either are positive, or are the negatives of simple roots. The definition of compatibility in [@ga] is a special case (namely the bipartite case) of what we here call $c$-compatibility. The general definition was given in [@MRZ], but here we give a rephrasing found in [@sortable Section 7], translated into the language of almost positive roots.
We write $\set{\alpha_1,\ldots,\alpha_n}$ for the simple roots and $\set{s_1,\dots,s_n}$ for the simple reflections. For each $i$ in $\set{1,\ldots,n}$, we define an involution $\sigma_i$ on the set of almost positive roots by $$\label{sigma def}
\sigma_i(\beta):=\left\lbrace\begin{array}{ll}
\beta&\text{if }\beta=-\alpha_j\text{ with }j\neq i,\mbox{ or}\\
s_i\beta&\text{otherwise}.
\end{array}\right.$$ We write $[\beta:\alpha_i]$ for the coefficient of $\alpha_i$ in the expansion of $\beta$ in the basis of simple roots. A simple reflection $s_i$ is [***initial***]{} in a Coxeter element $c$ if $c$ has a reduced word starting with $s_i$. If $s_i$ is initial in $c$, then $s_ics_i$ is another Coxeter element.
The [***$c$-compatibility***]{} relations are a family of symmetric binary relations ${\parallel}_c$ on the almost positive roots. They are the unique family of relations with
1. For any $i$ in $\set{1,\ldots,n}$, and Coxeter element $c$, $$-\alpha_i{\parallel}_c\beta\mbox{ if and only if }[\beta:\alpha_i]=0.$$
2. For each pair of almost positive roots $\beta_1$ and $\beta_2$, each Coxeter element $c$, and each $s_i$ initial in $c$, $$\beta_1{\parallel}_c \beta_2\mbox{ if and only if }\sigma_i(\beta_1)\,{\parallel}_{s_ics_i}\,\sigma_i(\beta_2).$$
The [***$c$-clusters***]{} are the maximal sets of pairwise $c$-compatible almost positive roots. By [@ga Theorem 1.8] and [@MRZ Proposition 3.5], for fixed $W$, all $c$-clusters are of the same size, and furthermore, each is a basis for the root space (the span of the roots). Write ${\mathbb{R}}_{\ge0} C$ for the nonnegative linear span of a $c$-cluster $C$. Then [@ga Theorem 1.10] and [@MRZ Theorem 3.7] state that the cones ${\mathbb{R}}_{\ge0} C$, for all $c$-clusters $C$, are the maximal cones of a complete simplicial fan. We call this fan the [***$c$-cluster fan***]{}.
We define the [***$c$-bicluster fan***]{} to be the coarsest common refinement of the $c$-cluster fan and its antipodal opposite. A pair $(C_1,C_2)$ of $c$-clusters is called a pair of [***twin $c$-clusters***]{} if the cones ${\mathbb{R}}_{\ge0} C_1$ and $-{\mathbb{R}}_{\ge0} C_2$ (the nonpositive linear span of $C_2$) intersect in a full-dimensional cone. It is immediate that maximal cones in the $c$-bicluster fan are in bijection with pairs of twin $c$-clusters.
\[A3 biclus example\] For $W$ of type $A_3$, up to symmetry there are two different $c$-bicluster fans: one for linear $c$ and one for bipartite $c$, shown in Figures \[A3 biclus linear\] and \[A3 biclus bipartite\] respectively. These are again stereographic projections as explained in Example \[A3 biCamb example\].
The two fans in Example \[A3 biclus example\] are combinatorially isomorphic. Despite this tantalizing fact, in this paper, we only explore bicluster fans in the special case of bipartite Coxeter elements (the original setting of [@ga; @ca2]), where they are easily related to biCambrian fans. For the bipartite choice of $c$, [@camb_fan Theorem 9.1] says that the $c$-Cambrian fan is *linearly* isomorphic to the cluster fan. Combining this fact with Proposition \[biCamb alt def\], we have the following theorem.
\[bi cl fans\] For all finite Coxeter groups $W$ and for *bipartite* of $c$, the $c$-bicluster fan is linearly isomorphic to the $c$-biCambrian fan.
Because of the bijection between $c$-bisortable elements and maximal cones in ${\operatorname{\mathbf{biCamb}}}(W,c)$ and the bijection between maximal cones in the $c$-bicluster fan and pairs of twin $c$-clusters, we have the following immediate consequence of Theorem \[bi cl fans\].
\[bi cl\] For all finite Coxeter groups $W$, $c$-bisortable elements for *bipartite* $c$ are in bijection with pairs of twin $c$-clusters.
Combining Theorems \[simple h\] and \[bi cl fans\], we obtain the following theorem.
\[simple h cl\] If Conjecture \[simple poly\] holds for a Coxeter group $W$, then the bipartite $c$-bicluster fan is simplicial and the $h$-vector of the underlying simplicial sphere has entries ${\operatorname{biNar}}_k(W)$.
Twin noncrossing partitions {#nc sec}
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The absolute order on a finite Coxeter group $W$ is the prefix order (or equivalently the subword order) on $W$ relative to the generating set $T$, the set of reflections in $W$. (By contrast, the prefix order relative to the simple reflections $S$ is the weak order, while the subword order relative to $S$ is the Bruhat order.) We will use the symbol $\le_T$ for the absolute order. The [***$c$-noncrossing partitions***]{} in a finite Coxeter group $W$ are the elements of $W$ contained in the interval $[1,c]_T$ in the absolute order on $W$. For details on the absolute order and noncrossing partitions, see for example [@Armstrong Chapter 2]. For our purposes, the key fact is a theorem of Brady and Watt.
Let $W$ be a finite Coxeter group of rank $n$ represented as a reflection group in ${\mathbb{R}}^n$ and let $T$ be the set of reflections of $W$. For each reflection $t\in T$, let $\beta_T$ be the corresponding positive root. Given $w\in[1,c]_T$, define a cone $$F_c(w)=\set{{\mathbf{x}}\in{\mathbb{R}}^n:{\mathbf{x}}\cdot\beta_t\le0\,\,\forall\,t\le_T w,\,\,{\mathbf{x}}\cdot\beta_t\geq0\,\,\forall\,t\le_T cw^{-1}}.$$ The following theorem combines [@BWassoc Theorem 1.1] with [@BWassoc Theorem 5.5].
\[BWthm\] For $c$ bipartite, the map $F_c$ is a bijection from $[1,c]_T$ to the set of maximal cones in the $c$-Cambrian fan.
The astute reader will notice a difference between our definition of $F_c$ and the definition appearing in [@BWassoc Section 1]. The set of reflections $t$ such that $t\le_Tw$ is the intersection of $T$ with some (non necessarily standard) parabolic subgroup of $W$. The definition in [@BWassoc] imposes inequalities ${\mathbf{x}}\cdot\beta_t\le0$ only for those $\beta_t$ that are simple roots for that parabolic subgroup. Our definition imposes additional inequalities, all of which are implied by the inequalities for the simple roots. We similarly add additional redundant inequalities of the form $x\cdot\beta_t\geq0$.
Theorem \[BWthm\] suggests a definition of twin noncrossing partitions. In fact, given Proposition \[biCamb alt def\], two natural definitions suggest themselves. Given $u,v\in[1,c]_T$, we call $(u,v)$ a pair of [***twin $c$-noncrossing partitions***]{} if $F_c(u)\cap(-{\mathcal{F}}_c(v))$ is full-dimensional. Similarly, given $u\in[1,c]_T$ and $v\in[1,c^{-1}]_T$, we call $(u,v)$ a pair of [***twin $(c,c^{-1})$-noncrossing partitions***]{} if $F_c(u)\cap{\mathcal{F}}_{c^{-1}}(v)$ is full-dimensional. Theorem \[BWthm\] now immediately implies the following theorem.
\[bi nc\] For all $W$ and bipartite $c$, the $c$-bisortable elements are in bijection with pairs of twin $c$-noncrossing partitions and with pairs of twin $(c,c^{-1})$-noncrossing partitions.
Bipartite $c$-bisortable elements and alternating arc diagrams {#type A sec}
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In this section, we show how bipartite $c$-bisortable elements of type A are in bijection with certain objects called alternating arc diagrams. We then prove the type-A enumeration of bipartite $c$-bisortable elements in Theorem \[main thm\] by counting alternating arc diagrams and prove the type-B enumeration by counting centrally symmetric alternating arc diagrams.
Pattern avoidance {#pat av sec}
-----------------
The Coxeter group of type $A_n$ is the symmetric group $S_{n+1}$. We will write permutations $x$ in $S_{n+1}$ in their one-line notations $x_1\cdots x_{n+1}$. In the weak order on permutations in $S_{n+1}$, there is a cover $x_1\cdots x_{n+1}{{\,\,<\!\!\!\!\cdot\,\,\,}}y_1\cdots y_{n+1}$ if and only if there exists $i$ such that $y_i=x_{i+1}>x_i=y_{i+1}$ and $y_j=x_j$ for $j\not\in\set{i,i+1}$. We say that $x$ is covered by $y$ via a swap in positions $i$ and $i+1$.
The Cambrian congruences on $S_{n+1}$ are described in detail in [@cambrian]. We quote part of the description here. The simple generator $s_i$ for $A_n$ is the transposition $(i\,\,\,i\!+\!1)$, for $i = 1, 2, \ldots n$. Each Coxeter element $c$ can be encoded by a coloring of the elements $2,\ldots,n$ that we call a [***barring***]{}. Each element $i$ is either [***overbarred***]{} and marked $\overline{i}$ if $s_i$ occurs before $s_{i-1}$ in every reduced word for $c$, or [***underbarred***]{} and marked $\underline{i}$ if $s_i$ occurs after $s_{i-1}$ in every reduced word for $c$. Passing from $c$ to $c^{-1}$ means swapping overbarring with underbarring.
We say $x$ is obtained from $y$ by a [***$\overline231\to\overline213$ move***]{} if $x$ is covered by $y$ via a swap in positions $i$ and $i+1$, for some $i$, and if there exists an *overbarred* element $x_j$ with $j<i$ and $x_i<x_j<x_{i+1}$. Similarly, $x$ is obtained from $y$ by a [***$31\underline2\to13\underline2$ move***]{} if $x$ is covered by $y$ via a swap in positions $i$ and $i+1$, for some $i$, and if there exists an *underbarred* element $x_j$ with $i+1<j$ and $x_i<x_j<x_{i+1}$. Combining [@cambrian Proposition 5.3] and [@cambrian Theorem 6.2], we obtain the following proposition:
\[same class\] Suppose $x$ and $y$ are permutations in $S_{n+1}$ with $x{{\,\,<\!\!\!\!\cdot\,\,\,}}y$ in the weak order, and assume that the numbers $2,\ldots,n$ have been barred according to $c$. Then $x$ and $y$ are in the same $c$-Cambrian congruence class if and only if $x$ is obtained from $y$ by a $\overline231\to\overline213$ move or a $31\underline2\to13\underline2$ move.
As an immediate corollary, we see that a permutation $y$ is the bottom element of its $c$-Cambrian congruence class (i.e. is $c$-sortable) if and only if none of the permutations covered by $y$ are obtained from $y$ by a $\overline231\to\overline213$ move or a $31\underline2\to13\underline2$ move. In other words, there is no subsequence $\overline bca$ of $y$ with $a<b<c$, with $c$ immediately preceding $a$, and with $b$ overbarred and no subsequence $ca\underline b$ of $y$ with $a<b<c$, with $c$ immediately preceding $a$, and with $b$ underbarred. In this case, we say that $y$ [***avoids***]{} $\overline231$ and $31\underline2$.
We can similarly describe bottom elements of $c$-biCambrian congruence classes (the $c$-bisortable elements), keeping in mind that passing from $c$ to $c^{-1}$ means swapping overbarring with underbarring: An element $y$ is the bottom element of its $c$-biCambrian congruence class if and only if none of the permutations covered by $y$ are obtained from $y$ by a $\overline231\to\overline213$ or $31\underline2\to13\underline2$ move that is *also* a $\underline231\to\underline213$ or $31\overline2\to13\overline2$ move. Thus $c$-bisortable permutations are described by a complicated pattern-avoidance condition that we will only describe, later, for the case of bipartite $c$, where it becomes much simpler.
Noncrossing arc diagrams {#arc sec}
------------------------
We now review the notion of [***noncrossing arc diagrams***]{} from [@arcs]. Beginning with $n+1$ distinct points on a vertical line, numbered $1,\ldots,n+1$ from bottom to top, we draw some (or no) curves called [***arcs***]{} connecting the points. Each arc moves monotone upwards from one of the points to another, passing either to the left or to the right of each point in between. Furthermore no two arcs may intersect in their interiors, no two arcs share the same upper endpoint, and no two arcs may share the same lower endpoint. We consider arc diagrams only up to their combinatorics, i.e. which pairs of points are joined by an arc and which points are left and right of each arc.
Given a permutation $x_1\cdots x_{n+1}$ in $S_{n+1}$, we define a noncrossing arc diagram $\delta(x_1\cdots x_{n+1})$. Each descent $x_i>x_{i+1}$ becomes an arc $\alpha$ in $\delta(x_1\cdots x_{n+1})$ with lower endpoint $x_{i+1}$ and upper endpoint $x_i$. For each integer $j$ with $x_{i+1}<j<x_i$ that occurs to the *left* of $x_i$ in $x_1\cdots x_{n+1}$, the point $j$ is *left* of the arc $\alpha$. For each integer $j$ with $x_{i+1}<j<x_i$ that occurs to the *right* of $x_{i+1}$ in $x_1\cdots x_{n+1}$, the point $j$ is *right* of the arc $\alpha$. It was shown in [@arcs Theorem 3.1] that $\delta$ is a bijection from permutations to noncrossing arc diagrams. More specifically, for each $k$, the map $\delta$ restricts to a bijection from permutations with $k$ descents to noncrossing arc diagrams with $k$ arcs.
A [***$c$-sortable arc***]{} is an arc that belongs to $\delta(v)$ for some $c$-sortable permutation $v$. The following characterization of $c$-sortable arcs in terms of the barring associated to $c$ is immediate from the pattern-avoidance description above. (Compare [@arcs Example 4.9].)
\[csort arc\] For $W=A_n$ and any $c$, the $c$-sortable arcs are the arcs that do not pass to the left of any underbarred element of $\set{2,\ldots,n}$ and do not pass to the right of any overbarred element of $\set{2,\ldots,n}$.
In particular, since $c$ and $c^{-1}$ correspond to opposite barrings, the only arcs that are both $c$ and $c^{-1}$-sortable are the arcs that connect adjacent endpoints $i$ and $i+1$.
Combining the above descriptions of $c$-sortable and $c$-bisortable elements in terms of overbarred and underbarred elements, we obtain the following proposition.
\[c-bisort arcs\] For $W=A_n$ and any $c$, the map $\delta$ restricts to a bijection from $c$-bisortable permutations with $k$ descents to noncrossing arc diagrams on $n+1$ vertices with $k$ arcs, each of which is either $c$ or $c^{-1}$-sortable.
Suppose $x=x_1\cdots x_n$ is a permutation such that $\delta(x)$ has an arc that is neither $c$-sortable nor $c^{-1}$-sortable. This arc has upper endpoint $x_i$ and lower endpoint $x_{i+1}$ for some $i$ and it fails the conclusion of Proposition \[csort arc\] for $c$ and for $c^{-1}$. That is, it either passes left of an underbarred element or right of an overbarred element *and* it either passes left of an overbarred element or right of an underbarred element. Thus, switching $x_i$ with $x_{i+1}$ is both a $\overline231\to\overline213$ or $31\underline2\to13\underline2$ move *and* a $\underline231\to\underline213$ or $31\overline2\to13\overline2$ move. Therefore, $x$ is not $c$-bisortable. The argument is easily reversed to prove the converse.
Alternately, Proposition \[c-bisort arcs\] follows from the description of the $c$-biCambrian congruence as the meet of the $c$-Cambrian and $c^{-1}$-Cambrian congruences.
Alternating arc diagrams {#alt sec}
------------------------
We now consider the case where $c$ is bipartite. Let $c_{+}$ be the product of the simple generators $s_i$ where $i$ is even, and $c_{-}$ be the product of the simple generators $s_i$ where $i$ is odd. The bipartite Coxeter elements in $A_n$ are $c_+c_-$ and its inverse $c_-c_+$. The barring associated to $c_+c_-$ has all even numbers overbarred and all odd numbers underbarred. A [***right-even alternating arc***]{} is an arc that passes to the right of even vertices and to the left of odd vertices. A [***left-even alternating arc***]{} is an arc that passes to the left of even vertices and to the right of odd vertices. A [***right-even alternating arc diagram***]{} is a noncrossing arc diagram all of whose arcs are right-even alternating, and [***left-even alternating arc diagrams***]{} are defined analogously. The following proposition is an immediate consequence of Proposition \[csort arc\].
\[c-arcs\] Suppose $W=A_n$ and $c$ is the bipartite Coxeter element $c_+c_-$.
1. The map $\delta$ restricts to a bijection from $c$-sortable permutations to right-even alternating arc diagrams.
2. The map $\delta$ restricts to a bijection from $c^{-1}$-sortable permutations to left-even alternating arc diagrams.
In each case, $\delta$ restricts further to send permutations with $k$ descents bijectively to arc diagrams with $k$ arcs.
An [***alternating arc***]{} is an arc that is either right-even alternating or left-even alternating or both. We call a noncrossing arc diagram consisting of alternating arcs an [***alternating arc diagram***]{}. Figure \[some\] shows several alternating noncrossing arc diagrams. From left to right, they are $\delta(5371624)$, $\delta(6473125)$, and $\delta(4275136)$.
![Some alternating noncrossing arc diagrams[]{data-label="some"}](somediagrams)
The following proposition is the bipartite case of Proposition \[c-bisort arcs\].
\[avoid alt\] For $W=A_n$ and $c$ bipartite, the map $\delta$ restricts to a bijection from $c$-bisortable permutations with $k$ descents to alternating arc diagrams on $n+1$ points with $k$ arcs.
Observe that an arc fails to be alternating if and only if it passes on the same side of two consecutive numbers. Thus, we obtain the following simpler description of the pattern avoidance condition defining bipartite $c$-bisortable elements.
\[avoidance\] If $c$ is the bipartite Coxeter element $c_+c_-$ of $A_n$, a permutation $x=x_1\cdots x_{n+1}$ is $c$-bisortable if and only if, for every descent $x_i>x_{i+1}$, there exists no $k$ with $x_{i+1}<k<k+1<x_i$ such that $k$ and $k+1$ are on the same side of the descent (i.e. $k$ and $k+1$ both left of $x_i$ or both right of $x_{i+1}$).
The condition in Proposition \[avoidance\] is that $x$ avoids subsequences $dabc$, $dacb$, $bcda$, and $cbda$ with $a<b<c<d$, with $d$ and $a$ adjacent in *position*, and with $b$ and $c$ being adjacent in *value*. This is an instance of [***bivincular***]{} pattern avoidance in the sense of [@BCDK Section 2]. We will not review the notation for bivincular patterns from [@BCDK], but we restate Proposition \[avoidance\] in that notation as follows:
\[avoidance bivinc\] For $c$ bipartite, a permutation is $c$-bisortable if and only if it avoids the bivincular patterns $(2341,\set{3},\set{2})$, $(3241,\set{3},\set{2})$, $(4123,\set{1},\set{2})$, and $(4132,\set{1},\set{2})$.
Counting alternating arc diagrams {#alt_arc_diagrams}
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Let $[n]$ denote the set $\{1,2,\ldots , n\}$. To prove the type-A enumeration of bipartite $c$-bisortable elements in Theorem \[main thm\], we give a bijection $\pi$ from noncrossing alternating arc diagrams on $n+1$ vertices with $k$ arcs to pairs $(S,T)$ of subsets of $[n]$ with $|S|=|T|=k$.
To describe the bijection, we begin with the case $k=1$. Recall that we number the endpoints in a diagram $1,\ldots,n+1$ from bottom to top. Suppose $\Sigma$ is an alternating arc diagram whose only arc connects $i$ to $j$ with $i<j$. If the arc is right-even alternating, define $\pi(\Sigma)$ to be $(\set{i},\set{j-1})$. If the arc is left-even alternating, define $\pi(\Sigma)$ to be $(\set{j-1},\set{i})$. (Any arc that is both right-even alternating and left-even alternating has $j=i+1$. The bijection sends this arc to $(\set{i},\set{i})$.)
Now suppose that $\Sigma$ is a noncrossing arc diagram with more than one arc. Whenever we encounter a right-even alternating arc in $\Sigma$ with endpoints $i<j$, we put $i$ into $S$ and $j-1$ into $T$; whenever we encounter a left-even alternating arc with endpoints $i<j$ we put $j-1$ into $S$ and $i$ into $T$. More precisely, suppose that $\Sigma$ is an alternating arc diagram with $k$ arcs. Let $S'$ denote the set of numbers $i$ such that $i$ is bottom endpoint of a right-even alternating arc in $\Sigma$ and let $S''$ denote the set of numbers $j-1$ such that $j$ is the top endpoint of a left-even alternating arc in $\Sigma$. Let $T'$ denote the set of numbers $j'-1$ such that $j'$ is the top endpoint of a right-even alternating arc in $\Sigma$ and let $T''$ denote the set of numbers $i'$ such that $i'$ is the bottom endpoint of a left-even alternating arc. The map $\pi$ sends $\Sigma$ to the pair $(S'\cup S'', T'\cup T'')$.
\[alt arc bij\] The map $\pi$ is a bijection from the set of alternating arc diagrams on $n+1$ points to the set of pairs of subsets of $[n]$ of the same size. For each $k$, the bijection restricts to a bijection from alternating arc diagrams with $k$ arcs to pairs of subsets of size $k$.
In preparation for the proof of Theorem \[alt arc bij\], we will break each alternating diagram into smaller pieces. Two alternating arcs with endpoints $i<j$ and $i'<j'$ [***overlap***]{} if the intersection of the sets $\{i, \ldots, j-1\}$ and $\{i',\ldots, j'-1\}$ is nonempty. Informally, the arcs overlap if some part of one arc passes along side of the other arc. (If they only touch at their endpoints but don’t pass along side one another, then they do *not* overlap). Given a collection ${\mathcal{E}}$ of arcs, we can define an “overlap graph” with vertices ${\mathcal{E}}$ and edges given by overlapping pairs in ${\mathcal{E}}$. We say that the collection ${\mathcal{E}}$ is [***overlapping***]{} if this overlap graph is connected. Each noncrossing diagram can be broken into overlapping components, maximal overlapping collections of arcs. The definition of alternating arc diagrams and the definition of right-even and left-even alternating arcs let us immediately conclude that two distinct arcs appearing in the same alternating arc diagram, one right-even alternating and one left-even alternating, cannot overlap. We have proved the following fact.
\[overlap comp\] Each overlapping component of an alternating arc diagram fits exactly one of the following descriptions: (1) It consists of right-even alternating arcs that are not left-even alternating; (2) It consists of left-even alternating arcs that are not right-even alternating; or (3) it consists of a single arc that is right-even and left-even alternating (and thus connects two adjacent points).
Proposition \[overlap comp\] implies that, on each overlapping component, the map $\pi$ collects all of the top endpoints of the arcs into one set, and all of the bottom endpoints into the other set.
Now we describe how to break an alternating diagram $\Sigma$ into its overlapping components. Let $P(\Sigma)$ be the set of numbers $p\in[n+1]$ such that no arc in $\Sigma$ passes left or right of $p$. (A point $p\in P(\Sigma)$ may still be an endpoint of one or two arcs.) Write $P(\Sigma)=\set{p_0,\ldots,p_m}$ with $p_0<\cdots<p_m$. In every case, $p_0=1$ and $p_m=n+1$. For each $i$, we claim that an arc in $\Sigma$ has its lower endpoint in $\{p_{i-1}, p_{i-1}+1,\ldots, p_{i}-1\}$ if and only if it has its upper endpoint in $\{p_{i-1}+1, p_{i-1}+2,\ldots, p_i\}$. Indeed, if an arc has a lower endpoint in $\{p_{i-1},p_{i-1}+1,\ldots, p_{i}-1\}$, then since it cannot pass on either side of $p_i$, it must end at a number in the set $\{p_{i-1}+1, p_{i-1}+2,\ldots, p_i\}$. A similar argument proves the converse, so we have established the claim. Let $\Sigma_i$ denote the set of arcs with lower endpoints in $\{p_{i-1},p_{i-1}+1,\ldots, p_{i}-1\}$ (and thus with upper endpoints in $\{p_{i-1}+1, p_{i-1}+2,\ldots, p_i\}$). By construction, $\Sigma_i$ is an overlapping component, and all overlapping components are $\Sigma_i$ for some $i$. Let $(S_i,T_i)$ be the image of $\Sigma_i$ under $\pi$, so that $\pi(\Sigma) = (\bigcup_{i=1}^mS_i\,,\,\bigcup_{i=1}^mT_i)$.
We say that two arcs are [***compatible***]{} if there is a noncrossing arc diagram containing both arcs. Our next task is to understand for which pairs $(s,t)$ and $(s',t')$ there exists an overlapping pair of compatible *alternating arcs*, one with endpoints $s$ and $t+1$ and one with endpoints $s'$ and $t'+1$. Since the arcs must overlap but may not share the same bottom endpoint and may not share the same top endpoint, and taking without loss of generality $s<s'$, there are only two cases. These cases are covered by the following two lemmas, which are easily verified.
\[arc compat 1\] Suppose $s<s'\le t<t'$. Then there exist two compatible alternating arcs, one with endpoints $s$ and $t+1$ and one with endpoints $s'$ and $t'+1$ if and only if $s'$ and $t$ have the same parity. The pair of arcs can be chosen in exactly two ways, either both as right-even alternating arcs or both as left-even alternating arcs.
\[arc compat 2\] Suppose $s<s'< t'<t$. Then there exist two compatible alternating arcs, one with endpoints $s$ and $t+1$ and one with endpoints $s'$ and $t'+1$ if and only if $s'$ and $t'$ have opposite parity. The pair of arcs can be chosen in exactly two ways, either both as right-even alternating arcs or both as left-even alternating arcs.
Given a pair $(S,T)$ of $k$-subsets of $[n]$, we will always write $S=\set{s_1,\ldots,s_k}$ with $s_1<\cdots<s_k$ and $T=\set{t_1,\ldots,t_k}$ with $t_1<\cdots<t_k$. Define $Q(S,T)$ to be the set of numbers $q\in[n+1]$ such that, for all $j$ from $1$ to $k$, neither $s_j<q\le t_j$, nor $t_j<q\le s_j$.
\[P Q lemma\] Let $\Sigma$ be an alternating arc diagram. Then $Q(\pi(\Sigma))=P(\Sigma)$.
Write $(S,T)$ for $\pi(\Sigma)$. If $p\in P(\Sigma)$, then no arc passes left or right of $p$. Thus there exists $k$ such that $s_j$ and $t_j$ are less than $p$ for all $j\le k$ and $s_j$ and $t_j$ are greater than or equal to $p$ for all $j>k$. We see that $p\in Q(S,T)$.
Suppose that $q\in (Q,S)$, and there exists some arc $\alpha$ that passes to the left or right of $q$. The arc $\alpha$ belongs to some overlapping component of $\Sigma$, and each pair $s_i, t_i$ in the image of a different component satisfies $s_i, t_i < q$ or $s_i, t_i> q$. Thus, we may as well assume that $\Sigma$ consists of a single overlapping component. Write $\pi(\Sigma) = (\set{s_1,\ldots,s_k},\set{t_1,\ldots,t_k})$ with $s_1<\cdots < s_k$ and $t_1<\cdots<t_k$. Lemma \[overlap comp\] says that $\Sigma$ consists of either right-even overlapping arcs or left-even overlapping arcs. Without loss of generality, we assume that $\Sigma$ consists of only right-even overlapping arcs, so that $\{s_1,\ldots, s_k\}$ is the set of bottom endpoints of those arcs. Thus, $s_i \le t_i$ for each $i=1,2,\ldots, k$. Let $s_i$ be the bottom endpoint of $\alpha$, and let $l$ be the largest number such that $s_l <q$. We make two observations. First, $\alpha$ must connect $s_i$ with $t_j+1$, where $j$ is strictly greater than $i$ (otherwise $s_i< q< t_j+1\le t_i$), and $j$ is strictly greater than $l$ (otherwise $s_j < q \le t_j$). Second, $t_{l+1} \ge q> t_{l}$, because $t_{l+1} \ge s_{l+1} \ge q > t_l\ge s_l$. We conclude that each number in the set of bottom endpoints $\{s_{l+1}, s_{l+2}, \ldots, s_k\}$ must connect with a number in the set $\{t_{l+1}+1,\ldots, t_k+1\}$. Since $t_j +1$ is already connected to $s_i$, there is some number in the set $\{t_{l+1}+1, \ldots, t_k+1\}$ that is the top endpoint of two arcs, and that is a contradiction.
We are now prepared to prove the main theorem of this section.
We first show that $\pi$ is well-defined. Since each arc in $\Sigma$ contributes exactly one of its endpoints to $S'\cup S''$ and the other to $T'\cup T''$, both $S'\cup S''$ and $T'\cup T''$ have size $k$ as long as each contribution to $S'\cup S''$ is distinct and each contribution to $T'\cup T''$ is distinct. Each contribution to $S'$ is distinct because no two arcs share the same lower endpoint, and each contribution to $S''$ is distinct because no two arcs share the same upper endpoint. Proposition \[overlap comp\] implies that a right-even alternating arc with bottom endpoint $i$ and a *distinct* left-even alternating arc with top endpoint $i+1$ are not compatible. Thus the only elements of $S'\cap S''$ come from arcs that are both right-even alternating and left-even alternating, and we see that each contribution to $S'\cup S''$ is distinct. The symmetric argument shows that each contribution to $T'\cup T''$ is distinct. We have shown that $\pi$ is a well-defined map from alternating arc diagrams with $k$ arcs to pairs of $k$-element subsets of $[n]$.
We complete the proof by exhibiting an inverse $\eta$ to $\pi$. Let $(S,T)$ be a pair of $k$-element subsets of $[n]$. Write $Q(S,T)=\set{q_0,\ldots,q_m}$ with $q_0<\cdots<q_m$. For each $i$ from $1$ to $m$, define $S_i=S\cap\set{q_{i-1},q_{i-1}+1,\ldots,q_i-1}$ and $T_i=T\cap\set{q_{i-1},q_{i-1}+1,\ldots,q_i-1}$. We claim that $|S_i|=|T_i|$, and more specifically, that $s_j\in S_i$ if and only if $t_j\in T_i$. Indeed, suppose $s_j\in S_i$, so that $q_{i-1}\le s_j<q_i$. If $t_j<q_{i-1}$, then $t_j<q_{i-1}\le s_j$, contradicting the fact that $q_{i-1}\in Q(S,T)$. If $t_j\ge q_i$, then $s_j<q_i\le t_j$, contradicting the fact that $q_i\in Q(S,T)$. We conclude that $t_j\in T_i$. The symmetric argument completes the proof of the claim.
Now, in light of Lemma \[P Q lemma\] and the definition of $\pi$, by subtracting $q_{i-1}-1$ from each element of $S_i$ and $T_i$, we reduce to the case where $m=1$ and thus $Q=\set{1,n+1}$ and $(S_1,T_1)=(S,T)$. In particular, all of the arcs in the diagram $\eta(S,T)$ are right-even alternating, or all of the arcs are left-even alternating. If $n=1$, then either $(S,T)=(\emptyset,\emptyset)$, in which case $\eta(S,T)$ has no arc, or $(S,T)=(\set{1},\set{1})$, in which case $\eta(S,T)$ has an arc connecting $1$ and $2$.
If $n>1$, then we observe that the element $1$ must be in $S$ or in $T$ but must not be in both. Indeed, if $1$ is in neither set or in both, we see that $2\in Q(S,T)$, and this is a contradiction. In particular, we will need to construct an arc whose lower endpoint is $1$ and whose upper endpoint is above $2$. This arc will pass by $2$, and so it is either right-even alternating or left-even alternating (but not both). If $1\in S$, then the corresponding arc is right-even alternating, and if $1\in T$ this arc is left-even alternating. Without loss of generality, we assume $1\in S$, so that each $i$ in $S$ is a bottom endpoint and for each $j$ in $T$, $j+1$ is a top endpoint of a right-even alternating arc in $\eta(S,T)$. To complete the proof, we show that there is a unique way to pair off each bottom endpoint in $S$ with a top endpoint in $T$ so that the union of the resulting arcs is a noncrossing arc diagram. Since the arcs in the diagram are all right-even alternating, we must pair each element of $S$ with a larger element of $T$.
We first decide which element of $T$ we should pair with $s_k$. Because $s_k$ is the maximum element of $S$, Lemma \[arc compat 1\] implies that we must pair $s_k$ with some $t'$ such that $\set{t\in T:s_k<t<t',\, t-s_k\text{ odd}}$ is empty. Similarly, Lemma \[arc compat 2\] implies that we must either pair $s_k$ with $t_k$ or pair $s_k$ with some $t'$ such that $t'-s_k$ is odd. Furthermore, if we choose $t'$ according to those two rules, no matter how we pair the remaining elements of $S$ and $T$, the arcs produced will be compatible with the arc whose bottom endpoint is $s_k$. We are forced to pair $s_k$ with $\min\set{t\in T:t\geq s_k,\, t-s_k\text{ odd}}$, or with $t_k$ if $\set{t\in T:t\geq s_k,\, t-s_k\text{ odd}}=\emptyset$. By induction on $k$, there is a unique way to pair the elements of $S\setminus\set{s_k}$ with the elements of $T\setminus\set{t'}$ to make a noncrossing alternating diagram. Putting in the pair $(s_k,t')$ we obtain the unique pairing of elements of $S$ with elements of $T$ to make a noncrossing alternating diagram. The base of the induction is where $k=1$. Here existence of a pairing is trivial and uniqueness comes from the requirement that the arc whose bottom endpoint is $1$ must be right-even alternating.
\[type A insight\] The proof of Theorem \[alt arc bij\] provides key insights that lead to our proof of Theorems \[hard part\] and \[hard part finer\]. When we generalize beyond type A, the role of the arcs in an alternating arc diagram will be played by the *canonical joinands* of a bipartite $c$-bisortable element. (The latter are defined in Section \[can sec\]. For the connection between arcs and canonical join representations, see [@arcs Section 3].) The fact that distinct right-even alternating and left-even alternating arcs do not overlap translates into the fact that, for $c$ bipartite, $c$ and $c^{-1}$-sortable join-irreducible permutations have disjoint support—a fact that we will prove uniformly in Section \[sortable formula sec\].
In light of the proof of Theorem \[alt arc bij\], we might count alternating arc diagrams $\Sigma$ with $n+1$ points in the following way: First, choose the set $P(\Sigma)=\set{p_0,\ldots,p_m}$ with $p_0<\cdots<p_m$. When $p_{i+1}=p_i+1$, choose either to connect $p_i$ to $p_i+1$ with an arc or not. When $p_{i+1}>p_i+1$, choose either to use right-even alternating or left-even alternating arcs, and construct a diagram on the points $p_i,\ldots,p_{i+1}$, such that for every $j$ with $p_i+1\le j\le p_{i+1}-1$, some arc passes left or right of $j$. Once we fix the type of arc (right-even or left-even alternating), the number of such diagrams on $p_i,\ldots,p_{i+1}$ is the number that in Section \[double pos sec\] will be called the double-positive Catalan number ${\operatorname{Cat}}{^{+\!\!+}}(A_m)$ for $m=p_{i+1}-p_i$. The proof of Theorems \[hard part\] and \[hard part finer\] generalizes this method of counting and shows that a corresponding method also counts antichains in the doubled root poset.
Looking ahead to Section \[double pos sec\], the previous remark implies an interpretation of the type-A double-positive Narayana number which—after some combinatorial manipulations that amount to changing from a bipartite Coxeter element to a linear Coxeter element—coincides with the interpretation given in .
Enumerating bipartite $c$-bisortable elements in type B {#type B sec}
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In this section, we use certain alternating arc diagrams to prove the enumeration of bipartite $c$-bisortable elements of type B given in Theorem \[main thm\]. We first analyze the $c$-biCambrian congruence on the weak order for $B_n$. In order to reuse much of our work from Section \[alt\_arc\_diagrams\], we realize the weak order on $B_n$ as a sublattice of the weak order on $A_{2n-1}$, through the usual signed-permutation model.
In any finite Coxeter group, the map $y\mapsto w_0 y w_0$ is a rank-preserving automorphism of weak order (where $w_0$ is the longest element). It is a well-known fact (and an easy exercise) that for any lattice automorphism, the set of fixed points of the automorphism is a sublattice. When $W$ is $A_{2n-1}$ it is easy to check that the set of fixed points of this map is a Coxeter group isomorphic to $B_n$. (For example, take as simple generators the fixed points $s_is_{2n-1-i}$ for $i<n$, and $s_n$.)
Writing each $x$ in $A_{2n-1}$ as a permutation of the set $\{\pm 1,\pm2, \ldots, \pm n\}$ with full one-line notation $x_{-n}x_{-n+1}\cdots x_{-1}x_1\cdots x_{n-1}x_n$, conjugation by $w_0$ acts by negating all of the entries of $x$ and reversing its order. The fixed points of this automorphism are the signed permutations on $\{\pm 1,\pm2, \ldots, \pm n\}$, meaning the permutations which satisfy $x_i = -x_{-i}$. The subposet of the weak order on $A_{2n-1}$ induced by the signed permutation is a sublattice, isomorphic to the weak order on $B_n$. Because $x_i=-x_{-i}$, it is convenient to write signed permutations in an abbreviated notation as $x_1x_2\cdots x_n$.
It is easy to check that the signed permutation $y_1\ldots y_n$ (written in abbreviated notation) covers $x_1\ldots x_n$ in the weak order on $B_n$ if and only if one of the two following conditions is satisfied: Either $y_i = x_{i+1} > x_i = y_{i+1}$ for $i,i+1\in [n]$ and $y_j = x_j$ for each $j\not\in \{i,i+1\}$, or $0<x_1=-y_1$ and $x_j = y_j$ for all $j \in \{2, 3,\ldots, n\}$. In the former case, the symmetry $y_i=-y_{-i}$ implies that $y_{-i-1}=x_{-i}> x_{-i-1} = y_{-i}$, so that the full one-line notation of $y_{-n},\ldots,y_n$ has two descents: $y_i>y_{i+1}$ and $y_{-i-1}>y_{-i}$. (For more information on this realization of the weak order on the type-$B$ Coxeter group see [@CombCoxeter Section 8.1]).
To define noncrossing diagrams of type B, we place $2n$ points on a vertical line, labeled from bottom to top by the integers $-n,-n+1,\ldots-1,1,\ldots,n-1,n$ such that there is a central symmetry that, for each $i$, maps the point labeled $i$ to the point labeled $-i$. A [***centrally symmetric noncrossing diagram***]{} is a noncrossing arc diagram that is fixed by the central symmetry. The map $\delta$ restricts to a bijection from signed permutations to centrally symmetric noncrossing diagrams. We use the term [***centrally symmetric arc***]{} to describe either an arc that is fixed by the central symmetry or a pair of arcs that form an orbit under the symmetry. For each $k$, the map $\delta$ restricts further to a bijection between signed permutations with $k$ descents and centrally symmetric noncrossing diagrams with $k$ centrally symmetric arcs.
The simple generators of $B_n$, are $s_0=(-1\,\,\,1)$ and $s_i=(-i\!-\!1\,\,\,-\!i)(i\,\,\,i\!+\!1)$ for $i=1,\ldots,n-1$, written in cycle notation as permutations of $\set{\pm 1, \ldots, \pm n}$. A [***symmetric Coxeter element***]{} of ${\mathcal{A}}_{2n-1}$ is a Coxeter element that is fixed by the automorphism $y\mapsto w_0yw_0$. Equivalently, the Coxeter element can be written as a product of some permutation of the elements $s_0,\ldots,s_{n-1}$ defined above. This product in $A_{2n-1}$ can be interpreted as a Coxeter element of $B_n$, which we denote by $\tilde c$. A Coxeter element is symmetric if and only if it corresponds to a barring of $\set{\pm1,\ldots,\pm(n-1)}$ with the property that $i$ is overbarred if and only if $-i$ is underbarred. Thus, a signed permutation avoids the pattern $\overline231$ if and only if it also avoids the pattern $31\underline2$ (in its full one-line notation). The signed permutations avoiding $\overline231$ (and equivalently $31\underline2$) in their full notation are exactly the $\tilde c$-sortable elements by [@cambrian Theorem 7.5]. Comparing with the description of $c$-sortable permutations following Proposition \[same class\], we obtain the following proposition.
\[b\_sort\] Suppose $c$ is a symmetric Coxeter element of $A_{2n-1}$ and suppose $\tilde c$ is the corresponding Coxeter element of $B_n$. A signed permutation is $\tilde c$-sortable in $B_n$ if and only if it is $c$-sortable as an element of $A_{2n-1}$.
The analogous result holds for $\tilde c$-bisortable elements.
\[typeB\_bisort\] Suppose $c$ is a symmetric Coxeter element of $A_{2n-1}$ and suppose $\tilde c$ is the corresponding Coxeter element of $B_n$. A signed permutation is $\tilde c$-bisortable in $B_n$ if and only if it is $c$-bisortable as an element of $A_{2n-1}$.
Suppose $w$ is a signed permutation. If $w$ is $\tilde c$-bisortable, then Proposition \[c join cinv\] says that $w = u {\vee}v$ for some $\tilde c$-sortable signed permutation $u$ and some signed permutation $v$. Proposition \[b\_sort\] says that, as elements of $A_{2n-1}$, $u$ is a $c$-sortable permutation and $v$ is a $c^{-1}$-sortable permutation. Since the weak order on $B_n$ is a sublattice of the weak order on $A_{2n-1}$, the join $u {\vee}v$ is the same in $A_{2n-1}$ as in $B_n$, and thus Proposition \[c join cinv\] implies that $w$ is $c$-bisortable.
On the other hand, if $w$ is $c$-bisortable as an element of $A_{2n-1}$, then as in Proposition \[c join cinv\], we can write $w$ as $u {\vee}v$, where $u$ is the $c$-sortable permutation ${\pi_\downarrow}^c(w)$ and $v$ is the $c^{-1}$-sortable permutation ${\pi_\downarrow}^{c^{-1}}(w)$. Since conjugation by $w_0$ is a lattice automorphism fixing $w$, we obtain $w = (w_0 u w_0) {\vee}(w_0 v w_0)$. But $w_0 u w_0$ is $c$-sortable and below $w$, so $w_0 u w_0\le u$. Since conjugation by $w_0$ is order preserving, we conclude that $w_0 u w_0 = u$. Similarly $w_0 v w_0= v$. Thus, by Proposition \[b\_sort\], $u$ is $\tilde c$-sortable and $v$ is $\tilde c^{-1}$-sortable in $B_n$. Since the weak order on $B_n$ is a sublattice of the weak order on $A_{2n-1}$, Proposition \[c join cinv\] says that $w$ is $\tilde c$-bisortable.
A bipartite Coxeter element $\tilde c$ of $B_n$ is a symmetric, bipartite Coxeter element of $A_{2n-1}$, so combining Propositions \[avoid alt\] and \[typeB\_bisort\], we immediately obtain the following proposition.
\[avoid alt B\] For $W=B_n$ and $\tilde c$ a bipartite Coxeter element, the map $\delta$ restricts to a bijection from $\tilde c$-bisortable signed permutations with $k$ descents to centrally symmetric alternating arc diagrams on $2n$ points with $k$ centrally symmetric alternating arcs.
Thus, to count the bipartite $c$-bisortable elements in $B_n$, it remains only to count centrally symmetric alternating arc diagrams. The points in the noncrossing arc diagram for a permutation in $S_n$ are labeled $1,\ldots,2n$ from bottom to top. If we instead label the points $-n,\ldots,-1,1,\ldots,n$ from bottom to top, we can interpret the map $\pi$ as returning an ordered pair of subsets of $\set{-n,\ldots,-1,1,\ldots n-1}$. Define $\pi_B$ to be the map on centrally symmetric alternating arc diagrams with $2n$ vertices that first does the map $\pi$ to obtain $(S,T)$ and then ignores $T$ and outputs only $S$. The following theorem shows that the number of centrally symmetric alternating arc diagrams with $k$ centrally symmetric arcs is $\binom{2n-1}{2k}+\binom{2n-1}{2k-1}=\binom{2n}{2k}$ as desired.
\[type b-alt arc bij\] For each $k$, the map $\pi_B$ restricts to a bijection from centrally symmetric alternating arc diagrams with $k$ centrally symmetric arcs to subsets of $\set{-n,\ldots,-1,1,\ldots n-1}$ of size $2k$ or $2k-1$.
We first show that $\pi_B$ is a bijection from centrally symmetric alternating arc diagrams to subsets of $\set{-n,\ldots,-1,1,\ldots n-1}$. Given $S\subseteq\set{\pm1,\ldots,\pm n}$, we write $-S-1$ for the set $\set{-i-1:i\in S}$, where we interpret $1-1$ to mean $-1$ in order to make $-S-1$ a subset of $\set{\pm1,\ldots,\pm n}$. Showing that $\pi_B$ is a bijection is equivalent to showing that an alternating diagram $\Sigma$ is centrally symmetric if and only if $\pi(\Sigma)=(S,-S-1)$ for some $S$.
The terms “right-even alternating” and “left-even alternating” should be understood in terms of the labeling of points as $1,\ldots,2n$. These terms become problematic when we label points as $-n,\ldots,-1,1,\ldots,n$. (For example, whether a right-even alternating arc passes left or right of the point labeled $i$ depends on the sign of $i$, the parity of $i$, and the parity of $n$.) Without worrying about these details, we make two easy observations: First, an alternating arc is right-even alternating if and only if its image under the central symmetry is right-even alternating. Second, the central symmetry swaps top with bottom endpoints and positive with negative endpoints. These observations immediately imply that $\pi$ maps centrally symmetric alternating arc diagrams to pairs of the form $(S,-S-1)$.
These observations also immediately imply that if $\pi$ maps an alternating arc diagram $\Sigma$ to $(S,T)$ and $\Sigma'$ is the image of $\Sigma$ under the central symmetry, then $\pi$ maps $\Sigma'$ to $(-T+1,-S-1)$, where $-T+1$ is the set $\set{-i+1:i\in T}$, where we interpret $-1+1$ to mean $1$. In particular, if $\pi$ maps $\Sigma$ to $(S,-S-1)$, then $\pi$ also maps $\Sigma'$ to $(S,-S-1)$. Since we already know that $\pi$ is a bijection, we conclude that in this case $\Sigma$ must be centrally symmetric. We have shown that $\Sigma$ is centrally symmetric if and only if $\pi(\Sigma)$ is of the form $(S,-S-1)$. Therefore $\pi_B$ is a bijection.
It is now immediate that $\pi_B$ maps a centrally symmetric alternating arc diagrams with $k$ centrally symmetric arcs to a $(2k-1)$-element set if the diagram has an arc that is fixed by the central symmetry or to a $2k$-element set if all of the arcs in the diagram come in symmetric pairs.
Simpliciality of the bipartite biCambrian fan in types A and B {#A B simp sec}
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We now prove Theorem \[simple A B\], which states that the bipartite biCambrian fan is simplicial in types A and B. The proof of the type-A case of Theorem \[simple A B\] proceeds by combining results of [@IRRT] and [@arcs].
Some collections of noncrossing arc diagrams (including, we will see, the alternating arc diagrams), correspond to lattice quotients of the weak order. More specifically, a collection of noncrossing arc diagrams may be the image, under $\delta$, of the bottom elements of congruence classes of some congruence. To describe when and how such a situation arises, we need the notion of a subarc. For $i<j$ and $i'<j'$, an arc $\alpha$ connecting $i$ to $j$ is a [***subarc***]{} of an arc $\alpha'$ connecting $i'$ to $j'$ if $i'\le i$ and $j'\ge j$ and if $\alpha$ and $\alpha'$ pass to the same side of every point between $i$ and $j$. It follows from [@arcs Theorem 4.1] and [@arcs Theorem 4.4] that a subset $D$ of the noncrossing arc diagrams on $n+1$ points is the image, under $\delta$, of the set of bottom elements for some congruence $\Theta$ if and only if all of the following conditions hold.
1. There exists a set $U$ of arcs such that a noncrossing diagram $\Sigma$ is in $D$ if and only if all arcs in $\Sigma$ are in $U$.
2. \[subarc req\] If an arc $\alpha$ is not in $U$ and $\alpha$ is a subarc of some arc $\alpha'$, then $\alpha'$ is also not in $U$.
We will call $U$ the set of [***unremoved arcs***]{} of the congruence $\Theta$. If $C$ is any set of arcs and $U$ is the maximal set such that $U\cap C=\emptyset$ and condition above holds, then we say that the congruence $\Theta$ is [***generated by removing the arcs $C$***]{}.
An element $j$ of a finite lattice $L$ is join-irreducible if it covers exactly one element $j_*$. A lattice congruence on $L$ [***contracts***]{} a join-irreducible element $j$ if the congruence has $j\equiv j_*$. A congruence is uniquely determined by the set of join-irreducible elements it contracts. The join-irreducible elements of the weak order on $A_n$ are the permutations in $S_{n+1}$ with exactly one descent. In particular, the map $\delta$ restricts to a bijection between join-irreducible elements in $S_{n+1}$ and noncrossing arc diagrams with exactly one arc. (We will think of this restriction as mapping join-irreducible elements to arcs, rather than to singletons of arcs.) Under this bijection, the join-irreducible elements *not* contracted by a congruence $\Theta$ correspond to the arcs in $U$, where $U$ is the set of unremoved arcs of $\Theta$. The congruence is [***generated by contracting***]{} a set $J$ of join-irreducible elements if and only if it is generated by removing the arcs $\delta(J)$.
We call $j$ a [***double join-irreducible***]{} element if it is join-irreducible and if the unique element $j_*$ covered by $j$ is either the bottom element of the lattice or is itself join-irreducible. The following is part of the main result of [@IRRT].
\[one\] Suppose $\Theta$ is a lattice congruence on the weak order on $A_n$. Then the following three conditions are equivalent.
1. The undirected Hasse diagram of the quotient lattice $A_n/\Theta$ is a regular graph.
2. ${\mathcal{F}}_\Theta(A_n)$ is a simplicial fan.
3. $\Theta$ is generated by contracting a set of double join-irreducible elements.
We now apply these considerations to alternating arc diagrams. First, it is apparent that the set of alternating arc diagrams is the image of $\delta$ restricted to the set of bottom elements of a congruence. (Indeed, this is the bipartite $c$-biCambrian congruence.) It is also apparent that the congruence is generated by removing the arcs that connect $i$ to $i+3$ and that *do not* alternate. (That is they pass to the same side of $i+1$ and $i+2$.) Applying the inverse of $\delta$, we see that the congruence is generated by contracting the join-irreducible elements $$1\cdots (i-1)(i+1)(i+2)(i+3)i(i+4)\cdots (n+1)$$ and $$1\cdots (i-1)(i+3)i(i+1)(i+2)(i+4)\cdots (n+1)$$ for $i=1,\ldots,n-2$. These are both double join-irrreducible elements, and thus Theorem \[one\] implies the type-A case of Theorem \[simple A B\].
We now move to the type-B case of Theorem \[simple A B\]. Just as in type-A, there is a correspondence between congruences on the weak order and certain sets of (centrally symmetric) noncrossing diagrams. However, there is currently no analogue to Theorem \[one\] in type B. Therefore, instead of arguing the type-B case as we argued the type-A case, we will use a folding argument to show that the type-A case implies the type-B case.
Say a lattice congruence of the weak order on $A_{2n-1}$ is [***symmetric under conjugation by $w_0$***]{} if for all $x,y\in A_{2n-1}$ we have $x\equiv y$ modulo $\Theta$ if and only if $w_0xw_0\equiv w_0yw_0$ modulo $\Theta$.
\[sym bottoms\] If $\Theta$ is a lattice congruence of the weak order on $A_{2n-1}$ that is symmetric under conjugation by $w_0$, then its restriction to the sublattice $B_n$ is a congruence $\Theta'$. An element of $B_n$ is the bottom element of its $\Theta'$-class if and only if it is the bottom element of its $\Theta$-class.
It is also a well-known and easy fact that the restriction of a lattice congruence to any sublattice is a congruence on the sublattice, and the first assertion of the proposition follows. One implication in the second assertion is immediate. For the other implication, suppose $x\in B_n$ is the bottom element of its $\Theta'$-class and let $y={\pi_\downarrow}^\Theta(x)$, so that in particular $x\equiv y$ modulo $\Theta$. Then because $\Theta$ is symmetric under conjugation by $w_0$, also $x=w_0xw_0\equiv w_0yw_0$ modulo $\Theta$. Since $y$ is the bottom element of its $\Theta$-class, $y\le w_0 y w_0$. Since conjugation by $w_0$ is order preserving, also $w_0yw_0\le y$, so $y=w_0 y w_0$. Thus $y$ is in the $\Theta'$-class of $x$, and we conclude that $y=x$, so that $x$ is also the bottom element of its $\Theta$-class.
\[typeB simplicial\] Suppose that $\Theta$ is a lattice congruence of the weak order on $A_{2n-1}$ and let $\Theta'$ denote its restriction to the weak order on $B_n$. If ${\mathcal{F}}_{\Theta}(A_{2n-1})$ is simplicial and $\Theta$ is symmetric under conjugation by $w_0$, then ${\mathcal{F}}_{\Theta'}(B_n)$ is simplicial.
Before we proceed with the proof of Proposition \[typeB simplicial\] we define some useful terminology. Recall that there is a linear functional $\lambda$ that orients the adjacency graph on maximal cones in ${\mathcal{F}}(W)$ to yield a partial order isomorphic to the weak order on $W$. A facet of a maximal cone is a [***lower wall***]{} (with respect to $\lambda$) if passing through it to an adjacent maximal cone is the same as moving down by a cover in the weak order. [***Upper walls***]{} are defined dually. The maximal cones of ${\mathcal{F}}_{\Theta}(W)$ similarly have lower and upper walls with respect to $\lambda$; passing from one cone to an adjacent cone through a lower wall corresponds to moving down by a cover in the lattice quotient induced by $\Theta$. The lower walls of a maximal cone in ${\mathcal{F}}_{\Theta}(W)$ are the lower walls of the smallest element in the corresponding $\Theta$-congruence class. (Recall that each maximal cone in ${\mathcal{F}}_{\Theta}(W)$ is the union of the set of maximal cones in ${\mathcal{F}}(W)$ in the same $\Theta$-congruence class.) Dually, the upper walls of a maximal cone in ${\mathcal{F}}_{\Theta}(W)$ are the upper walls of the cone corresponding to the largest element in the $\Theta$-congruence class.
We begin by considering type $A_{2n-1}$ in the usual geometric representation in ${\mathbb R}^{2n}$. However, to prepare for the type-B construction, we index the standard unit basis vectors of ${\mathbb R}^{2n}$ as $-n,\ldots,-1,1,\ldots,n$. In this representation, there is a reflecting hyperplane $H_{ji}$, with normal vector $e_j-e_i$, for each $i<j$ with $i,j\in\set{\pm1,\ldots\pm n}$. The maximal cone corresponding to the permutation $x_{-n}\cdots x_{-1}x_1\cdots x_n$ has a lower (respectively upper) wall contained in $H_{ji}$ if and only if there exists $r\in\set{-n,\ldots,-1,1,\ldots n-1}$ such that $x_r=j$ and $x_{r+1}=i$ (respectively, $x_{r+1}=j$ and $x_{r}=i$). As the price for our choice of indices, when $r=-1$, we must interpret $r+1$ here to mean $1$.
Recall that the signed permutations of $B_n$ are exactly the permutations in $A_{2n-1}$ that are fixed under conjugation by $w_0$ and that the restriction of weak order to these $w_0$-fixed permutations is weak order on $B_n$. As an abuse of terminology, the linear map on ${\mathbb R}^{2n}$ that sends each vector $(v_{-n},\ldots,v_{-1},v_1,\ldots,v_n)$ to $-(v_n,\ldots,v_1,v_{-1},\ldots,v_{-n})$ will be called the conjugation action of $w_0$ on ${\mathbb R}^{2n}$. Let $L$ be the linear subspace of ${\mathbb R}^{2n}$ consisting of vectors fixed by this action. These are the vectors with $v_i=-v_{-i}$ for all $i$. A permutation in $A_{2n-1}$ is fixed under conjugation by $w_0$ if and only if its corresponding cone in ${\mathcal{F}}(A_{2n-1})$ intersects $L$ in its relative interior, in which case the cone is also fixed under conjugation by $w_0$. Thus, we obtain ${\mathcal{F}}(B_n)$ as the fan induced on $L$ by ${\mathcal{F}}(A_{2n-1})$, and the weak order on $B_n$ arises from that induced fan, ordered by the same linear functional $\lambda$ as ${\mathcal{F}}(A_{2n-1})$. Moreover, ${\mathcal{F}}_{\Theta'}(B_n)$ is the fan induced on $L$ by ${\mathcal{F}}_{\Theta}(A_{2n-1})$.
Almost all of the lower walls of a $w_0$-fixed maximal cone $C$ in ${\mathcal{F}}_{\Theta}(A_{2n-1})$ intersect $L$ in pairs. Specifically, Proposition \[sym bottoms\] implies that any such cone is associated to a signed permutation $x=x_{-n}\cdots x_{-1}x_1\cdots x_n$ that is the bottom element of its $\Theta$-class. A descent $x_{-1}x_1$ of $x$ contributes a single lower wall to $C$, and thus a single lower wall to $C\cap L$. We will say that such a lower wall is centrally symmetric. All other descents of $x$ come in symmetric pairs $x_{-i-1}x_{-i}$ and $x_ix_{i+1}$, contributing two lower walls to $C$. However, these two walls have the same intersection with $L$ and thus contribute only one lower wall to $C\cap L$. Similar dual statements hold for the upper walls. Most importantly, among all of the walls of $C\cap L$, there are at most two that are centrally symmetric: at most one among the set of lower walls, and at most one among set of upper walls.
Since ${\mathcal{F}}_{\Theta}(A_{2n-1})$ is simplicial, $C$ has an odd number of walls. In particular, this implies that among all of the walls for $C$, there is exactly one that is centrally symmetric wall. Suppose that this wall is a lower wall. Then, $C$ has an odd number of lower walls, say $2k-1$, and their intersection with $L$ yields $k$ lower walls for the corresponding cone $C\cap L$ in ${\mathcal{F}}_{\Theta'}(B_n)$. Since ${\mathcal{F}}_{\Theta}(A_{2n-1})$ is simplicial, there are $2n-2k$ upper walls, which intersect $L$ in pairs, to form $n-k$ upper walls in ${\mathcal{F}}_{\Theta'}(B_n)$. Thus the cone associated to $C$ in ${\mathcal{F}}_{\Theta'}(B_n)$ has a total of $n$ walls. The same argument (switching lower walls with upper walls) shows that if the centrally symmetric wall is an upper wall, the cone associated to $C$ in ${\mathcal{F}}_{\Theta'}(B_n)$ has $n$ walls. We conclude that ${\mathcal{F}}_{\Theta'}(B_n)$ is simplicial.
Let $c$ be a bipartite Coxeter element in $A_{2n-1}$ and let $\tilde c$ be the same element thought of as a Coxeter element of $B_n$. Recall that $\tilde c$ is also bipartite.
Using the bipartite case of Proposition \[same class\] (with $n$ replaced by $2n-1$), it is easily checked that $x\equiv y $ modulo $\Theta_c$ if and only if $w_0xw_0\equiv w_0yw_0$ modulo $\Theta_{c}$. It follows that the $c$-biCambrian congruence is symmetric under conjugation by $w_0$. Since a congruence is uniquely determined by the set of bottom elements of its classes, Proposition \[typeB\_bisort\] implies that the restriction of the $c$-biCambrian congruence to $B_n$ is the $\tilde c$-biCambrian congruence. Thus the type-B case of the theorem follows from Proposition \[typeB simplicial\] and the type-A case of the theorem.
Double-positive Catalan numbers and biCatalan numbers {#double pos sec}
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For each finite Coxeter group $W$, the positive $W$-Catalan number ${\operatorname{Cat}}^+(W)$ is defined from the $W$-Catalan number ${\operatorname{Cat}}(W)$ by inclusion-exclusion. In this section, we review the definition of the positive $W$-Catalan number and define the double-positive $W$-Catalan number ${\operatorname{Cat}}{^{+\!\!+}}(W)$ from the positive $W$-Catalan number by inclusion-exclusion. We then prove Theorems \[hard part\] and \[hard part finer\] by showing how to count both antichains in the doubled root poset and bipartite $c$-bisortable elements by the same formula involving double-positive Catalan numbers. Recall that these two theorems in particular establish that the terms “biCatalan number” and “biNarayana number” make sense. As we prove these theorems, we obtain as a by-product a formula for the $W$-biCatalan numbers in terms of the double-positive Catalan numbers of parabolic subgroups of $W$. This formula leads to a recursion for the $W$-biCatalan numbers. Using a similar recursion for the $W$-Catalan numbers and a few other enumerative facts, we solve that recursion for ${\operatorname{biCat}}(D_n)$ to complete the proof of Theorem \[enum thm\]. The recursions discussed here all have Narayana $q$-analogues, but we are not at this time able to solve the recursion to find a formula for ${\operatorname{biCat}}(D_n;q)$. See Section \[type D biNar sec\] for a brief discussion of the type-D biNarayana numbers.
The positive $W$-Catalan and positive $W$-Narayana numbers have interpretations in each setting of Coxeter-Catalan combinatorics. (See for example [@Ath; @ABMW; @Ath-Tzan; @ga; @Haiman; @Panyushev; @sortable; @Sommers].) In this paper, we give the usual interpretations in the settings of nonnesting partitions and $c$-sortable elements, specifically in Sections \[antichain formula sec\] and \[sortable formula sec\]. We give interpretations of the double-positive $W$-Catalan and $W$-Narayana numbers in the settings of nonnesting partitions and $c$-sortable elements.
The double-positive $W$-Narayana numbers appeared in [@Ath-Sav] as the local $h$-vector of the positive part of the cluster complex. (See Remark \[Ath connection\].) As far as we know, [@Ath-Sav] was the first appearance of the double-positive $W$-Catalan/Narayana numbers and the only appearance before the current paper.
Double-positivity {#double pos subsec}
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We write $S$ for the set of simple reflections generating $W$. Given $J\subseteq S$, the notation $W_J$ stands for the subgroup of $W$ generated by $J$. The subgroup $W_J$ is called a [***standard parabolic subgroup***]{} of $W$ and is a Coxeter group in its own right with simple reflections $J$. In particular, each $W_J$ has a Catalan number. As usual, we define the [***positive $W$-Catalan number***]{} to be $$\label{Cat+ def}
{\operatorname{Cat}}^+(W)=\sum_{J\subseteq S}(-1)^{|S|-|J|}{\operatorname{Cat}}(W_J).$$ We define the [***double-positive $W$-Catalan number***]{} to be $$\label{Cat++ def}
{\operatorname{Cat}}{^{+\!\!+}}(W)=\sum_{J\subseteq S}(-1)^{|S|-|J|}{\operatorname{Cat}}^+(W_J).$$
We will prove the following formula for the biCatalan numbers.
\[bicat d-p\] For any finite Coxeter group $W$ with simple generators $S$, $$\label{bicat d-p formula}
{\operatorname{biCat}}(W)=\sum2^{|S|-|I|-|J|}{\operatorname{Cat}}{^{+\!\!+}}(W_{I}){\operatorname{Cat}}{^{+\!\!+}}(W_{J}),$$ where the sum is over all ordered pairs $(I,J)$ of disjoint subsets of $S$.
We can prove a refinement of Theorem \[bicat d-p\] using the usual notion of positive Narayana numbers and a notion of double-positive Narayana numbers. The [***positive $W$-Narayana numbers***]{} are $$\label{Nar+ def}
{\operatorname{Nar}}^+_k(W)=\sum_{J\subseteq S}(-1)^{|S|-|J|}{\operatorname{Nar}}_k(W_J).$$ We define the [***double-positive $W$-Narayana number***]{} to be $$\label{Nar++ def}
{\operatorname{Nar}}{^{+\!\!+}}_k(W)=\sum_{J\subseteq S}(-1)^{|S|-|J|}{\operatorname{Nar}}^+_{k-|S|+|J|}(W_J).$$ In all of the settings where the Narayana numbers appear, it is apparent that ${\operatorname{Nar}}_k(W)=0$ whenever $k<0$ or $k$ is greater than the rank of $W$. These definitions establish that ${\operatorname{Nar}}^+_k(W)={\operatorname{Nar}}{^{+\!\!+}}_k(W)=0$ as well for those values of $k$.
Defining ${\operatorname{Cat}}^+(W;q)=\sum_k{\operatorname{Nar}}^+_k(W)q^k$ and ${\operatorname{Cat}}{^{+\!\!+}}(W;q)=\sum_k{\operatorname{Nar}}{^{+\!\!+}}_k(W)q^k$, equations and correspond to $$\label{q Cat+ def}
{\operatorname{Cat}}^+(W;q)=\sum_{J\subseteq S}(-1)^{|S|-|J|}{\operatorname{Cat}}(W_J;q).$$ and $$\label{q Cat++ def}
{\operatorname{Cat}}{^{+\!\!+}}(W;q)=\sum_{J\subseteq S}(-q)^{|S|-|J|}{\operatorname{Cat}}^+(W_J;q).$$ Taking ${\operatorname{biCat}}(W;q)=\sum_k{\operatorname{biNar}}_k(W)q^k$, we will prove the following $q$-analog of Theorem \[bicat d-p\].
\[biCat GF d-p\] For any finite Coxeter group $W$ with simple generators $S$, $$\label{GF d-p formula}
{\operatorname{biCat}}(W;q)=\sum\, q^{|M|}{\operatorname{Cat}}{^{+\!\!+}}(W_{I};q){\operatorname{Cat}}{^{+\!\!+}}(W_{J};q),$$ where the sum is over all ordered triples $(I,J,M)$ of pairwise disjoint subsets of $S$.
The following theorem is equivalent to Theorem \[biCat GF d-p\].
\[binar d-p\] For any finite Coxeter group $W$ with simple generators $S$ and any $k$, $$\label{binar d-p formula}
{\operatorname{biNar}}_k(W)=\sum\,\sum_{i=0}^{k-|M|}{\operatorname{Nar}}{^{+\!\!+}}_i(W_{I}){\operatorname{Nar}}{^{+\!\!+}}_{k-|M|-i}(W_{J}),$$ where the outer sum is over all ordered triples $(I,J,M)$ of pairwise disjoint subsets of $S$. (If $|M|>k$, then the inner sum is interpreted to be zero.)
To prove these theorems, as well as Theorems \[hard part\] and \[hard part finer\], we establish (in Propositions \[q antichain\] and \[q bisortable formula\]) that the right side of counts antichains $A$ in the doubled root poset with weight $q^{|A|}$ and also counts bipartite $c$-bisortable elements $v$ with weight $q^{{\operatorname{des}}(v)}$. Once these counts are established, Theorems \[hard part\] and \[hard part finer\] follow, and in particular the definitions of the biCatalan and biNarayana numbers are validated. Also, Theorem \[biCat GF d-p\] holds, leading immediately to Theorems \[bicat d-p\] and \[binar d-p\].
Counting twin nonnesting partitions {#antichain formula sec}
-----------------------------------
We now recall the interpretations of the positive Catalan and Narayana numbers and give the interpretations of double-positive Catalan and Narayana numbers in the nonnesting setting. (Results in [@Ath-Tzan; @Sommers] give the same interpretations, but accomplish much more, by establishing bijections and counting formulas. By contrast, here we are only making simple assertions about inclusion-exclusion.) After giving these interpretations, we prove that the formula in Theorem \[binar d-p\] counts $k$-element antichains in the doubled root poset.
Since it is customary to talk about the “$W$-Catalan number” rather than the “$\Phi$-Catalan number,” we will make statements about “the root poset of $W$,” when $W$ is a crystallographic Coxeter group. This is harmless because, although the map from crystallographic root systems to Coxeter groups is not one-to-one, for each crystallographic Coxeter group, all corresponding crystallographic root systems have isomorphic root posets. Correspondingly, when $W_J$ is a standard parabolic subgroup of $W$, we will say that a root or set of roots is “contained in $W_J$” if it is contained in the subset of $\Phi$ forming a root system for $W_J$. An antichain that is not contained in any proper parabolic $W_J$ has full support, in the sense of Section \[nn sec\].
For any $J\subseteq S$, the number of antichains in the root poset for $W$ that are contained in $W_J$ is ${\operatorname{Cat}}(W_J)$. By inclusion-exclusion, we conclude that:
\[i-e nn\] The number of antichains in the root poset for $W$ with full support is ${\operatorname{Cat}}^+(W)$. The number of $k$-element antichains in the root poset for $W$ with full support is ${\operatorname{Nar}}^+_k(W)$.
For $J\subseteq S$, the map $A\mapsto A\setminus\set{\alpha_i:i\in J}$ is a bijection from the set of antichains containing the simple roots $\set{\alpha_i:i\in J}$ to the set of antichains in the root poset for $W_{S\setminus J}$.
Using this bijection, we prove the following proposition.
\[i-e nn 2\] The number of antichains in the root poset for $W$ containing no simple roots is ${\operatorname{Cat}}^+(W)$. The number of $k$-element antichains in the root poset for $W$ containing no simple roots is ${\operatorname{Nar}}^+_{n-k}(W)$.
The bijection mentioned above implies that the generating function for antichains containing the simple roots $\set{\alpha_i:i\in J}$ (and possibly additional simple roots) is $q^{|J|}{\operatorname{Cat}}(W_{S\setminus J};q)$. By inclusion-exclusion, the generating function for $k$-element antichains containing no simple roots is $\sum_{J\subseteq S}(-q)^{|S|-|J|}{\operatorname{Cat}}(W_J;q)$. On the other hand, starting with , replacing $q$ by $q^{-1}$, multiplying through by $q^{|S|}$ (i.e. $q^n$), and using the known symmetry $q^{|J|}{\operatorname{Cat}}(W_J;q^{-1})={\operatorname{Cat}}(W_J;q)$ of the coefficients of ${\operatorname{Cat}}(W_J;q)$, we obtain $$\sum_k{\operatorname{Nar}}^+_{n-k}(W)q^k=\sum_{J\subseteq S}(-q)^{|S|-|J|}{\operatorname{Cat}}(W_J;q).\qedhere$$
The bijection described above restricts to a bijection from the set of antichains *with full support* containing the simple roots $\set{\alpha_i:i\in J}$ to the set of antichains *with full support* in the root poset for $W_{S\setminus J}$. Thus, a similar inclusion-exclusion argument yields the following proposition.
\[i-e nn 3\] The number of antichains in the root poset for $W$ with full support containing no simple roots is ${\operatorname{Cat}}{^{+\!\!+}}(W)$. The number of $k$-element antichains in the root poset for $W$ with full support containing no simple roots is ${\operatorname{Nar}}{^{+\!\!+}}_k(W)$.
\[Ath connection\] The polynomials ${\operatorname{Cat}}{^{+\!\!+}}(W;q)$ appeared in [@Ath-Sav], where Athanasiadis and Savvidou showed that ${\operatorname{Cat}}{^{+\!\!+}}(W;q)$ is the local $h$-vector of the positive part of the cluster complex, as we now explain. We refer to [@Ath-Sav] for the relevant definitions, which we will not need here. In light of [@Ath-Tzan Theorem 1.5] and Proposition \[i-e nn 2\], the polynomial $h(\Delta_+(\Phi),x)$ appearing in [@Ath-Sav] is $x^{|S|}{\operatorname{Cat}}^+(W;x^{-1})$, where $(W,S)$ is the Coxeter system associated to $\Phi$. Thus the assertion of [@Ath-Sav Proposition 2.5] is that the local $h$-vector of the positive part of the cluster complex is $\sum_{J\subseteq S}(-1)^{|S|-|J|}x^{|J|}{\operatorname{Cat}}^+(W;x^{-1})$. But since the local $h$-vector is symmetric by [@Stanley Theorem 3.3], we can replace $x$ by $x^{-1}$ and multiply by $x^{|S|}$ to show that the local $h$-vector is $\sum_{J\subseteq S}(-x)^{|S|-|J|}{\operatorname{Cat}}^+(W;x)={\operatorname{Cat}}{^{+\!\!+}}(W;x)$.
We now prove the key result on antichains in the doubled root poset.
\[q antichain\] For any finite Coxeter group $W$ with simple generators $S$, the generating function $\sum_Aq^{|A|}$ for antichains $A$ in the doubled root poset is $$\label{q antichain formula}
\sum\, q^{|M|}{\operatorname{Cat}}{^{+\!\!+}}(W_{I};q){\operatorname{Cat}}{^{+\!\!+}}(W_{J};q),$$ where the sum is over all ordered triples $(I,J,M)$ of *pairwise disjoint* subsets of $S$.
In light of Proposition \[i-e nn 3\], the proposition amounts to the following assertions: First, there is a bijection from antichains $A$ in the doubled root poset to triples $(B,C,M)$ such that $B$ and $C$ are antichains in the root poset for $W$, each containing no simple roots, and the sets $I={\operatorname{supp}}(B)$, $J={\operatorname{supp}}(C)$ and $M$ are pairwise disjoint. Second, under this bijection, $|B|+|C|+|M|=|A|$. Every antichain $A$ in the doubled root poset consists of some set $B$ of positive non-simple roots in the top root poset, some set $C$ of positive non-simple roots in the bottom root poset, and some set $M$ of simple roots. The sets $I$, $J$, and $M$ are pairwise disjoint because $A$ is an antichain. The map $A\mapsto(B,C,M)$ is the desired bijection.
It will be useful to have a similar formula for antichains in the (not doubled) root poset, which are known to be counted by ${\operatorname{Cat}}(W)$.
\[Cat GF d-p\] For any finite Coxeter group $W$ with simple generators $S$. $$\label{Cat GF d-p formula}
{\operatorname{Cat}}(W;q)=\sum\, q^{|J|}{\operatorname{Cat}}{^{+\!\!+}}(W_{I};q),$$ where the sum is over all ordered pairs $(I,J)$ of *disjoint* subsets of $S$.
Every antichain $A$ in the root poset consists of some set $B$ of positive non-simple roots and some set $C$ of simple roots. Writing $I$ and $J$ for the supports of $B$ and $C$, again $I$ and $J$ are disjoint. By Proposition \[i-e nn 3\], each pair $(I,J)$ of disjoint subsets of $S$ contributes $q^{|J|}{\operatorname{Cat}}{^{+\!\!+}}(W_{I};q)$ to the count.
The following is an immediate consequence of Proposition \[i-e nn 3\] and will also be useful.
\[Cat++ reducible\] If $W$ is reducible as $W_1\times W_2$, then $$\label{Cat++ reducible formula}
{\operatorname{Cat}}{^{+\!\!+}}(W;q)={\operatorname{Cat}}{^{+\!\!+}}(W_1;q){\operatorname{Cat}}{^{+\!\!+}}(W_2;q).$$
Canonical join representations and lattice congruences {#can sec}
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To count bipartite $c$-bisortable elements, we will use a canonical factorization in the weak order called the canonical join representation. In this section, we focus exclusively on the lattice-theoretic tools that we will use in the following sections to complete the proof of Theorem \[binar d-p\].
The canonical join representation is a “minimal” expression for an element as a join of join-irreducible elements. The construction is somewhat analogous to prime factorizations of integers. Indeed, in the divisibility poset for positive integers, where $p\le q$ if and only if $p|q$, the canonical join representation coincides with prime factorization. For our purposes, the canonical join representation is useful because of how it interacts with lattice congruences. Recall that a lattice congruence $\Theta$ contracts a join-irreducible element $j$ if $j$ is equivalent modulo $\Theta$ to the unique element that it covers. Each congruence $\Theta$ of a finite lattice is determined by the set of join-irreducible elements that it contracts. In particular, we can see which elements of $W$ are $c$-sortable or $c$-bisortable by looking at their canonical join representations (much as we looked at the arcs in their arc diagrams in types A and B).
The canonical join representation of an element $a$ is an expression $a={\bigvee}A$ such that $A$ is minimal in two senses, among sets joining to $a$. First, the join ${\bigvee}A$ is [***irredundant***]{}, meaning that there is no proper subset $A'\subset A$ with ${\bigvee}A'={\bigvee}A$. Second, $A$ has the smallest possible elements (in terms of the partial order on $L$). Specifically, a subset $A$ of $L$ [***join-refines***]{} a subset $B$ of $L$ if for each $a\in A$ there is an element $b\in B$ such that $a\le b$. Join-refinement is a preorder on the subsets of $L$ that restricts to a partial order on the set of antichains. The [***canonical join representation***]{} of $a$, if it exists, is the unique minimal antichain $A$, in the sense of join-refinement, that joins irredundantly to $a$. We sometimes write ${\operatorname{Can}}(a)$ for $A$. The elements of $A$ are called the [***canonical joinands***]{} of $a$. It follows immediately that each canonical joinand is join-irreducible.
Not every finite lattice admits a canonical join representation for each of its elements. For example, in the diamond lattice $M_3$, which has five elements, three of which are atoms, the largest element does not have a canonical join representation. Many interesting lattices do admit canonical join representations, including all finite distributive lattices and, as we will see, the weak order on finite Coxeter groups. The next proposition establishes the promised connection between canonical join representations and lattice congruences. (The last assertion in the proposition also follows from [@shardint Proposition 6.3].)
\[can cong\] Suppose $L$ is a finite lattice such that each element in $L$ has a canonical join representation, and suppose that $\Theta$ is a lattice congruence on $L$. If $j$ is a canonical joinand of $a\in L$ and $j$ is not contracted by $\Theta$, then $j$ is a canonical joinand of ${\pi_\downarrow}^\Theta(a)$ in $L$. Moreover, if ${\pi_\downarrow}^{\Theta}(a)= a$ then none of the canonical joinands of $a$ are contracted by $\Theta$.
The assertion that $j$ is a canonical joinand of ${\pi_\downarrow}^\Theta(a)$ in $L$ implies also that $j$ is a canonical joinand of ${\pi_\downarrow}^\Theta(a)$ in ${\pi_\downarrow}^\Theta (L)$. (Since ${\pi_\downarrow}^\Theta(L)$ is a join-sublattice of $L$, every join-representation of ${\pi_\downarrow}^\Theta(a)$ in ${\pi_\downarrow}^\Theta(L)$ is also a join-representation of ${\pi_\downarrow}^\Theta(a)$ in $L$.)
Throughout the proof, we write $\set{j_1,\ldots j_k}$ for ${\operatorname{Can}}(a)$ with $j=j_1$. Recall that the lattice quotient $L/\Theta$ is isomorphic to the subposet of $L$ induced by the set ${\pi_\downarrow}^{\Theta}(L)$. Suppose $j$ is not contracted by $\Theta$, so that ${\pi_\downarrow}^{\Theta}(j)= j$. Recall that ${\pi_\downarrow}^{\Theta}$ is a lattice homomorphism, so ${\pi_\downarrow}^\Theta(a)={\bigvee}_{i=1}^k{\pi_\downarrow}^\Theta(j_1) = j{\vee}\left({\bigvee}_{i=2}^k{\pi_\downarrow}^\Theta(j_i)\right)$, (where the joins are all taken in the lattice quotient $L/\Theta$). Since $L/\Theta$ is also a join-sublattice of $L$, the join in $L/\Theta$ coincides with the join in $L$. Thus ${\pi_\downarrow}^\Theta(a)$ is equal to $j{\vee}\left({\bigvee}_{i=2}^k{\pi_\downarrow}^\Theta(j_i)\right)$ in $L$. Write $B$ for the set ${\operatorname{Can}}({\pi_\downarrow}^\Theta(a))$. Thus $B$ join-refines $\set{j}\cup\set{{\pi_\downarrow}^\Theta(j_2),\ldots,{\pi_\downarrow}^\Theta(j_k)}$. If no element of $B$ is less or equal to $j$, then this join-refinement implies that each element of $B$ is below some element of $\set{{\pi_\downarrow}^\Theta(j_2),\ldots,{\pi_\downarrow}^\Theta(j_k)}$, so that ${\pi_\downarrow}^\Theta(a)\le{\bigvee}_{i=2}^k{\pi_\downarrow}^\Theta(j_i)$. Since also ${\pi_\downarrow}^\Theta(a)$ is equal to $j{\vee}\left({\bigvee}_{i=2}^k{\pi_\downarrow}^\Theta(j_i)\right)$, we see that $j \le {\bigvee}_{i=2}^k{\pi_\downarrow}^\Theta(j_i)$. Recall that ${\pi_\downarrow}^\Theta (j_i) \le j_i$ for each $i$, so we have $j\le {\bigvee}_{i=2}^k j_i$. This contradicts the fact that ${\bigvee}_{i=1}^k j_i$ is irredundant. We conclude that there is some $j'\in B$ with $j'\le j$. Observe that $\left({\bigvee}B\right){\vee}\left({\bigvee}_{i=2}^k j_i\right) = a$ because ${j_1=j\le{\pi_\downarrow}^\Theta(a)\le a}$. Thus, $\{j_1,\ldots j_k\}$ join-refines $B\cup \{j_2,\ldots j_k\}$. Since $j$ is incomparable to each $j_i$, there is some $j''\in B$ such that $j\le j''$. But $B$ is an antichain, so $j'=j''=j$, and thus $j\in B$ as desired.
Now suppose that ${\pi_\downarrow}^{\Theta}(a)= a$. Then $a={\bigvee}_{i=1}^n {\pi_\downarrow}^{\Theta}(j_i)$, so $\set{j_1,\ldots j_k}$ join-refines $\set{{\pi_\downarrow}^\Theta(j_1),\ldots,{\pi_\downarrow}^\Theta(j_k)}$. Thus, for each $j_i$, there is some $j_m$ with $j_i\le {\pi_\downarrow}^{\Theta}(j_m)$. But ${\pi_\downarrow}^{\Theta}(j_m) \le j_m$, and since $\set{j_1,\ldots j_k}$ is an antichain, we have $j_i=j_m$, and thus also $j_i={\pi_\downarrow}^{\Theta}(j_i)$.
We will use the following easy proposition, which appears as [@arcs Proposition 2.2].
\[can cplx\] Suppose $L$ is a finite lattice and $J\subset L$. If ${\bigvee}J$ is the canonical join representation of some element of $L$ and if $J'\subseteq J$, then ${\bigvee}J'$ is the canonical join representation of some element of $L$.
Next we consider canonical join representations in the weak order. Before we begin, we briefly review some relevant terminology. For each $w\in W$, the [***length***]{} of $w$, denoted $l(w)$, is the number of letters in a reduced (that is, a shortest possible) word for $w$ in the alphabet $S$. The covers in the (right) weak order on $W$ are $w{{\,\,\,\cdot\!\!\!\! >\,\,}}ws$ whenever $w\in W$ and $s\in S$ have $l(ws) < l(w)$. In this case, the simple generator $s$ is a [***descent***]{} of $w$. Let $T$ denote the set of reflections in $W$. An [***inversion***]{} of $w$ is a reflection $t$ such that $l(tw)<l(w)$. We denote the set of inversions of $w$ by ${\operatorname{inv}}(w)$. A [***cover reflection***]{} of $w$ is an inversion $t$ of $w$ such that $tw=ws$ for some $s\in S$. Thus, the cover reflections of $w$ are in bijection with the descents of $w$. We write ${\mathrm{cov}}(w)$ for the set of cover reflections of $w$. The following proposition is quoted from [@typefree Theorem 8.1].
\[coxcjr\] Fix a finite Coxeter group $W$, and an element $w\in W$. The canonical join representation of $w$ exists and is equal to ${\bigvee}j_t$ where $t$ ranges over the set of cover reflections of $w$, and $j_t$ is the unique smallest element below $w$ that has $t$ as an inversion. In particular, $w$ has ${\operatorname{des}}(w)$ many canonical joinands.
Recall that the [***support***]{} of $w$, written ${\operatorname{supp}}(w)$, is the set of simple reflections appearing in a reduced word for $w$, and is independent of the choice of reduced word for $w$. The following lemma is an immediate consequence of the fact that every standard parabolic subgroup $W_J$ is a lower interval in the weak order on $W$.
\[supp\] For each $w\in W$, the support of $w$ equals $\bigcup_{j\in{\operatorname{Can}}(w)}{\operatorname{supp}}(j)$.
For each element $w$ and standard parabolic subgroup $W_J$, there is a unique largest element below $w$ that belongs to $W_J$. We write $w_J$ for this element and ${\pi_\downarrow}^J$ for the map that sends $w$ to $w_J$. In [@congruence Corollary [6.10]{}], it was shown that the fibers of ${\pi_\downarrow}^J$ constitute a lattice congruence of the weak order. We write $\Theta_J$ for this congruence. Since ${\pi_\downarrow}^J$ sends each element to the bottom if its fiber, it is a lattice homomorphism from $W$ to ${\pi_\downarrow}^J(W)$, which equals $W_J$.
\[disjoint joinands\] Suppose that $A_1$ and $A_2$ are antichains with disjoint support such that ${\bigvee}A_1$ and ${\bigvee}A_2$ are both canonical join representations in the weak order on $W$. Then ${\bigvee}(A_1\cup A_2)$ is a canonical join representation.
We write $A$ for $A_1\cup A_2$. First we show that ${\bigvee}A$ is irredundant. By way of contradiction, assume that there is some $j\in A$ such that ${\bigvee}A = {\bigvee}(A\setminus \{j\})$. We may as well take $j\in A_1$. We write $J$ for the support of $A_1$. Since the support of each join-irreducible element $j'$ in $A_2$ is disjoint from $J$, and since support decreases weakly in the weak order, we conclude that ${\pi_\downarrow}^{J}(j')$ is the identity element. Since ${\pi_\downarrow}^J$ is a lattice homomorphism, ${\pi_\downarrow}^{J}({\bigvee}A) = {\bigvee}A_1$ and ${\pi_\downarrow}^{J}({\bigvee}(A\setminus \{j\}))= {\bigvee}(A_1\setminus \{j\})$. We conclude that ${{\bigvee}A_1 = {\bigvee}(A_1\setminus \{j\})}$, contradicting the fact that ${\bigvee}A_1$ is a canonical join representation.
Next we show that ${\operatorname{Can}}({\bigvee}A)$ is contained in $A$. Assume that $j''$ is a canonical joinand of ${\bigvee}A$. There is some $j\in A$ such that $j''\le j$. Assume that $j\in A_1$, so that ${\operatorname{supp}}(j'')\subset J$. Thus, ${\pi_\downarrow}^{J}(j'') = j''$. Proposition \[can cong\] says $j''$ is a canonical joinand of ${\pi_\downarrow}^{J}({\bigvee}A) = {\bigvee}A_1$. Because $A$ is an antichain, $j''= j$. Since ${\bigvee}A$ is irredundant, and $A$ contains ${\operatorname{Can}}({\bigvee}A)$, we conclude that $A$ is equal to ${\operatorname{Can}}({\bigvee}A)$.
Observe that if $s\in S$ is a cover reflection of $w$ then Proposition \[coxcjr\] implies that $s$ is also a canonical joinand of $w$ because simple reflections are atoms in the weak order. We immediately obtain the following useful fact.
\[simple\_cjr\] Each $w\in W$ has ${\operatorname{Can}}(w)\cap S=cov(w)\cap S$.
In much of what follows, for $s\in S$, we will use the abbreviation ${{\langle s \rangle}}$ to mean $S\setminus\set{s}$. It is known (see for example [@sort_camb Lemma 2.8]) that if $w\in W_{{\langle s \rangle}}$, then ${{\mathrm{cov}}(w{\vee}s)} = {\mathrm{cov}}(w)\cup \{s\}$. We close this section with a lemma extends this statement to canonical join representations.
\[cover\_ref\] If $w\in W_{{\langle s \rangle}}$, then ${\operatorname{Can}}(w{\vee}s) ={\operatorname{Can}}(w)\cup \{s\}$.
Since support is weakly decreasing in the weak order, each $j\in {\operatorname{Can}}(w)$ has support contained in ${{\langle s \rangle}}$. Lemma \[disjoint joinands\] says that ${\bigvee}\left({\operatorname{Can}}(w)\cup \{s\}\right)$ is a canonical join representation.
Canonical join representations of $c$-bisortable elements {#canonical bisort sec}
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In this section we focus on canonical join representations of $c$-sortable elements and $c$-bisortable elements. Our goal is to prove the following result:
\[disjoint\_simple\_support\] Fix a bipartite $c$-bisortable element $w$ and the corresponding twin $(c,c^{-1})$-sortable elements $(u,v)=({\pi_\downarrow}^c(w),{\pi_\downarrow}^{c^{-1}}(w))$. Then
1. ${\operatorname{Can}}(w)\cap S={\operatorname{Can}}(u)\cap {\operatorname{Can}}(v)$
2. ${\operatorname{Can}}(w)$ is the disjoint union $({\operatorname{Can}}(u)\setminus S)\uplus({\operatorname{Can}}(v)\setminus S)\uplus({\operatorname{Can}}(w)\cap S)$
3. \[pw dis\] The sets ${\operatorname{supp}}({\operatorname{Can}}(u)\setminus S)$, ${\operatorname{supp}}({\operatorname{Can}}(v)\setminus S)$ and ${\operatorname{Can}}(w)\cap S$ are pairwise disjoint.
We begin with an easy application of Proposition \[can cong\] (the first item below can also be found as [@typefree Proposition 8.2]).
\[c or cinv\] For any Coxeter element $c$ and $w\in W$:
1. $w$ is $c$-sortable if and only if each of its canonical joinands is $c$-sortable.
2. $w$ is $c$-bisortable if and only if each of its canonical joinands is either $c$- or $c^{-1}$-sortable.
The first assertion follows immediately from Proposition \[can cong\]. Recall the notation $\Theta_c$ for the $c$-Cambrian congruence and write $\Theta$ for the $c$-biCambrian congruence. Since $\Theta = \Theta_c{\wedge}\Theta_{c^{-1}}$, a join-irreducible element in $W$ is contracted by $\Theta$ if and only if it is contracted by $\Theta_c$ *and* by $\Theta_{c^{-1}}$. The second assertion follows.
Recall from Section \[clus sec\] that a simple reflection $s$ is initial in a Coxeter element $c$ if there is a reduced word $a_1\ldots a_n$ for $c$ with $a_1=s$. Similarly $s$ is [***final***]{} in $c$ if there is a reduced word $a_1\ldots a_n$ for $c$ with $a_n=s$. In much of what follows, the key property of a bipartite Coxeter element is that every $s\in S$ is either initial or final in $c$.
The following lemma is the combination of [@typefree Propositions 3.13, 5.3, and 5.4]. Recall that $v_{{\langle s \rangle}}$ is the largest element in $W$ below $v$ that belongs to $W_{{\langle s \rangle}}$.
\[s\_initial\_or\_final\] Fix a $c$-sortable element $v$ in $W$ and a simple reflection $s\in S$.
1. If $s$ is final in $c$ and $v\ge s$, then $v_{{\langle s \rangle}}$ is $cs$-sortable and $v=s{\vee}v_{{\langle s \rangle}}$.
2. If $s$ be initial in $c$ and $s\in{\mathrm{cov}}(v)$, then $v_{{\langle s \rangle}}$ is $sc$-sortable and $v=s{\vee}v_{{\langle s \rangle}}$.
Observe that if $v$ satisfies the conditions of either item in Lemma \[s\_initial\_or\_final\], then by Lemma \[cover\_ref\], ${\operatorname{Can}}(v) = \{s\}\cup{\operatorname{Can}}(v_{{\langle s \rangle}})$. The following two lemmas are a straightforward application of Lemma \[s\_initial\_or\_final\].
\[j eq s lemma\] If $j$ is a $c$-sortable join-irreducible element and $s$ is final in $c$ with $j\ge s$, then $j=s$.
The first assertion of Lemma \[s\_initial\_or\_final\] says that $j= s{\vee}j_{{\langle s \rangle}}$. Since $j$ is join-irreducible and not equal to $j_{{\langle s \rangle}}$, we conclude that $j=s$.
\[only ji\] If $c$ is a bipartite Coxeter element and $j$ is a join-irreducible element that is both $c$-sortable and $c^{-1}$-sortable, then $j$ is a simple reflection.
Because $j$ is join-irreducible, it is not the identity, so there is some $s\in S$ such that $j\ge s$. Since $c$ is bipartite, we can assume without loss of generality that $s$ is final in $c$. (If not, then replace $c$ with $c^{-1}$.) Thus $j=s$ by Lemma \[j eq s lemma\].
Putting together Lemma \[j eq s lemma\] and Lemma \[only ji\], we obtain an explicit description of ${\pi_\downarrow}^{c^{-1}}(j)$, for bipartite $c$-sortable join-irreducible elements.
\[pidown\] Suppose that $c$ is a bipartite Coxeter element and $j$ is a $c$-sortable join-irreducible element. Let $S'$ denote the set of simple reflections $s$ such that $j\ge s$. Then ${\pi_\downarrow}^{c^{-1}}(j) = {\bigvee}S'$, which equals $\prod S'$, the product in $W$. Moreover, this join is a canonical join representation.
The statement of the lemma is obvious if $j$ is a simple reflection, so we assume that $j$ is not simple. Thus, Lemma \[only ji\] implies that $j$ is not $c^{-1}$-sortable, so ${\pi_\downarrow}^{c^{-1}}(j)$ is strictly less than $j$.
If any $s\in S'$ is final in $c$, then Lemma \[j eq s lemma\] says that $j=s$, contradicting our assumption. Thus, since $c$ is bipartite, each $s\in S'$ is initial. In particular, the elements of $S'$ pairwise commute, so that the notation $\prod S'$ makes sense and equals ${\bigvee}S'$. Moreover, since ${\bigvee}S'$ is an irredundant join of atoms, it is a canonical join representation. Since each simple reflection is both $c$- and $c^{-1}$-sortable, Proposition \[c or cinv\] says that this element is $c^{-1}$-sortable. We conclude that ${\pi_\downarrow}^{c^{-1}}(j)\ge {\bigvee}S'$.
Suppose that $j'$ is a canonical joinand of ${\pi_\downarrow}^{c^{-1}}(j)$. There is some simple reflection $s$ such that $j'\ge s$. Since also $j'\le {\pi_\downarrow}^{c^{-1}}(j)\le j$, we conclude that $s\in S'$. Every element of $S'$ is initial in $c$ and thus final in $c^{-1}$, so again by Lemma \[j eq s lemma\], $j'=s$. We conclude that ${\operatorname{Can}}({\pi_\downarrow}^{c^{-1}}(j))\subseteq S'$. Thus ${\pi_\downarrow}^{c^{-1}}(j) = {\bigvee}S'$.
Recall that Lemma \[disjoint joinands\] says that if $j$ and $j'$ are join-irreducible elements with disjoint support, then $j{\vee}j'$ is canonical. In Lemma \[simpler\_disjoint\_support\] below, we prove that when $j$ is bipartite $c$-sortable and $j'$ is bipartite $c^{-1}$-sortable, the converse is also true. We begin with the case when $j'$ is a simple reflection.
\[simple\_covers\] Given a bipartite Coxeter element $c$, a $c$-sortable join-irreducible element $j$ and a simple reflection $s\in{\operatorname{supp}}(j)$, there exists no element $w\in W$ with both $s$ and $j$ in ${\operatorname{Can}}(w)$.
In light of Proposition \[can cplx\], to prove this proposition, it is enough to show that no element can have $s{\vee}j$ as its canonical join representation. Suppose to the contrary that there is an element $v$ with canonical join representation $s{\vee}j$. By Proposition \[c or cinv\], $v$ is $c$-sortable. Also $s{\vee}j$ is irredundant, so $j$ and $s$ are incomparable. Since $c$ is bipartite, $s$ is either initial or final in $c$, so Lemma \[s\_initial\_or\_final\] says that $v = s{\vee}v_{{\langle s \rangle}}$. Since $v=s{\vee}j$ is a canonical join representation, we see that $j\le v_{{\langle s \rangle}}$, contradicting the hypothesis that $s$ is in the support of $j$.
\[simpler\_disjoint\_support\] Fix a bipartite Coxeter element $c$ in $W$. Suppose that $j$ is a $c$-sortable join-irreducible element and that $j'$ is a $c^{-1}$-sortable join-irreducible element. Suppose that $j {\vee}j'$ is a canonical join representation for some element of $W$. Then $j$ and $j'$ have disjoint support.
Suppose that $s\in {\operatorname{supp}}(j)\cap{\operatorname{supp}}(j')$, and assume without loss of generality that $s$ is initial in $c$. It is immediate from the definition of $c$-sortable elements that $s\le j$. (See for example [@typefree Proposition 2.29].) Since $s$ is a $c^{-1}$-sortable element, also $s\le{\pi_\downarrow}^{c^{-1}}(j{\vee}j')$. By Lemma \[s\_initial\_or\_final\](1) and Lemma \[cover\_ref\], $s$ is a canonical joinand of ${\pi_\downarrow}^{c^{-1}}(j{\vee}j')$. But also Proposition \[can cong\] says that $j'$ is a canonical joinand of ${\pi_\downarrow}^{c^{-1}}(j{\vee}j')$. We have reached a contradiction to Lemma \[simple\_covers\], and we conclude that ${\operatorname{supp}}(j)\cap{\operatorname{supp}}(j')=\emptyset$.
Finally, we prove Proposition \[disjoint\_simple\_support\].
Lemma \[only ji\] implies that ${\operatorname{Can}}(w)\setminus S$ is the disjoint union $({\operatorname{Can}}(w)\cap S)\uplus J_+\uplus J_-$ such that $J_+$ is the set of $c$-sortable join-irreducible elements in ${\operatorname{Can}}(w)\setminus S$ and $J_-$ is the set of $c^{-1}$-sortable join-irreducible elements in ${{\operatorname{Can}}(w)\setminus S}$. Moreover, by Lemma \[simpler\_disjoint\_support\], these sets have pairwise disjoint support. For each $j\in J_-$, write $S'_j$ for the set of simple reflections $s$ such that $s\le j$, and $S'= \bigcup S'_j$, where the union ranges over all $j\in J_-$. Lemma \[pidown\] says that ${{\pi_\downarrow}^c(j) = {\bigvee}S'_j}$. Since ${\pi_\downarrow}^c$ is a join-homomorphism, ${\pi_\downarrow}^c({\bigvee}J_-) = {\bigvee}S'$. Thus, applying the map ${\pi_\downarrow}^{c}$ to the join ${\bigvee}[({\operatorname{Can}}(w)\cap S)\uplus J_+\uplus J_-]$, we see that ${{\bigvee}[({\operatorname{Can}}(w)\cap S)\uplus J_+ \uplus S']}$ is a join representation of $u$. Since $S'$ is contained in the support of $J_-$, the sets ${\operatorname{Can}}(w)\cap S$, $J_+$, and $S'$ also have pairwise disjoint support. Proposition \[can cplx\] says that both ${\bigvee}{\operatorname{Can}}(w)\cap S$ and ${\bigvee}J_+$ are canonical join representations. Since ${\bigvee}S'$ is an irredundant join of atoms, it is also a canonical join representation. Thus, by Lemma \[disjoint joinands\], ${\bigvee}[({\operatorname{Can}}(w)\cap S)\uplus J_+ \uplus S']$ is the canonical join representation of $u$. The symmetric argument gives the canonical join representation of $v$. We conclude that ${\operatorname{Can}}(w)\cap S = {\operatorname{Can}}(u)\cap {\operatorname{Can}}(v)$, ${J_+={\operatorname{Can}}(u)\setminus S}$, and $J_-={\operatorname{Can}}(v) \setminus S$. The proposition follows.
Counting bipartite $c$-bisortable elements {#sortable formula sec}
------------------------------------------
In this section, we prove that the formulas in Theorem \[binar d-p\] counts bipartite $c$-bisortable elements, thus completing the proofs of Theorems \[hard part\], \[hard part finer\], \[bicat d-p\], \[biCat GF d-p\] and \[binar d-p\]. We begin by interpreting the double-positive Catalan and Narayana numbers in the $c$-sortable setting. We define [***positive $c$-sortable elements***]{} to be the set of $c$-sortable elements not contained in any standard parabolic subgroup of $W$. Equivalently, these are the $c$-sortable elements whose support is not contained in any proper subset of $S$. As the name suggests, positive $c$-sortable elements are counted by the positive Catalan numbers. The following analogue of Proposition \[i-e nn\] is the combination of [@sortable Corollary 9.2] and [@sortable Corollary 9.3].
\[positive\_sortable\] For any Coxeter element $c$ of $W$, the number of positive $c$-sortable elements in $W$ is ${\operatorname{Cat}}^+(W)$. The number positive $c$-sortable elements with $k$ descents is ${\operatorname{Nar}}^+_k(W)$.
We define [***clever $c$-sortable elements***]{} to be $c$-sortable elements which have no simple canonical joinands. We continue to let ${{\langle s \rangle}}$ stand for $S\setminus \{s\}$. To count clever $c$-sortable elements we will use Lemma \[s\_initial\_or\_final\] to define a map from $c$-sortable elements $v$ with simple cover reflection $s$ to $c'$-sortable elements in the standard parabolic subgroup $W_{{\langle s \rangle}}$, where $c'$ is the restriction of $c$ to $W_{{\langle s \rangle}}$. Our next task is to show that, for bipartite $c$, clever $c$-sortable elements are analogous, enumeratively, to antichains in the root poset having no simple roots:
\[i-e sortable\] Fix a bipartite Coxeter element $c$ of $W$.
1. The number of clever $c$-sortable elements is ${\operatorname{Cat}}^+(W)$.
2. The number of positive, clever $c$-sortable elements is ${\operatorname{Cat}}{^{+\!\!+}}(W)$.
3. The number of positive, clever $c$-sortable elements with exactly $k$ descents is ${\operatorname{Nar}}{^{+\!\!+}}_k(W)$.
We emphasize that while Proposition \[positive\_sortable\] holds for arbitrary $c$, Proposition \[i-e sortable\] holds only for bipartite $c$. The proof of Proposition \[i-e sortable\] will use inclusion-exclusion and the following technical lemma.
\[tech lemma\] For bipartite $c$ and $J\subseteq S$, let $c'$ be the restriction of $c$ to $W_{S\setminus J}$.
1. The map ${\pi_\downarrow}^{S\setminus J}:v\mapsto v_{S\setminus J}$ is a bijection from $c$-sortable elements of $W$ with ${J\subseteq{\operatorname{Can}}(v)}$ to $c'$-sortable elements of $W_{S\setminus J}$. Also, ${\operatorname{Can}}(v_{S\setminus J})={\operatorname{Can}}(v)\setminus J$.
2. The map restricts to a bijection from positive $c$-sortable elements of $W$ with $J\subseteq{\operatorname{Can}}(v)$ to positive $c'$-sortable elements of $W_{S\setminus J}$.
3. The map restricts further to a bijection from positive $c$-sortable elements of $W$ with $J\subseteq{\operatorname{Can}}(v)$ and with exactly $k$ descents to positive $c'$-sortable elements of $W_{S\setminus J}$ with exactly $k-|J|$ descents.
Suppose that $v$ is $c$-sortable, and $J\subseteq{\operatorname{Can}}(v)$. Lemma \[simple\_covers\] says that the support of each canonical joinand $j$ in ${\operatorname{Can}}(v)\setminus J$ is contained in $S\setminus J$. (Lemma \[simple\_covers\] applies to the non-simple elements of ${\operatorname{Can}}(v)$. Clearly, each simple reflection $s\in {\operatorname{Can}}(v)\setminus J$ is supported on $S\setminus J$.) On the one hand, ${\pi_\downarrow}^{S\setminus J}(j) = j$ for each $j\in {\operatorname{Can}}(v)\setminus J$. On the other hand, ${\pi_\downarrow}^{S\setminus J}(s)$ is the identity element for each $s$ in $J$. Since ${\pi_\downarrow}^{S\setminus J}$ is a lattice homomorphism, ${\pi_\downarrow}^{S\setminus J}({\bigvee}{\operatorname{Can}}(v)) = {\bigvee}[{\operatorname{Can}}(v) \setminus J]$. Proposition \[can cong\] implies that ${\bigvee}[{\operatorname{Can}}(v) \setminus J]$ is the canonical join representation of ${\pi_\downarrow}^{S\setminus J}(v) = v_{S\setminus J}$. Lemma \[c or cinv\] says that $v_{S\setminus J}$ is $c'$-sortable.
To complete the proof of the first assertion, we construct an inverse map. Suppose that $v'$ is a $c'$-sortable element in $W_{S\setminus J}$. Lemma \[supp\] says that the support of each canonical joinand $j\in {\operatorname{Can}}(v')$ is contained in $S\setminus J$. Lemma \[disjoint joinands\] says that the join ${\bigvee}[{\operatorname{Can}}(v')\cup J]$ is a canonical join representation for some element $v\in W$. Lemma \[c or cinv\] says that $v$ is $c$-sortable. We conclude that the map sending $v'$ to ${\bigvee}[{\operatorname{Can}}(v')\cup J]$ is a well-defined inverse.
Lemma \[supp\], Lemma \[simple\_covers\], and the fact that ${\operatorname{Can}}(v_{S\setminus J})={\operatorname{Can}}(v)\setminus J$ imply that $v$ is positive in $W$ if and only if $v_{S\setminus J}$ is positive in $W_{S\setminus J}$. The second assertion follows. The third assertion then follows from Proposition \[coxcjr\] and the fact that ${\operatorname{Can}}(v_{S\setminus J})={\operatorname{Can}}(v)\setminus J$.
Finally, we complete the proof of that bipartite $c$-bisortable elements are counted by the formula in Theorem \[binar d-p\].
\[q bisortable formula\] For any finite Coxeter group $W$ with simple generators $S$, the generating function $\sum_vq^{{\operatorname{des}}(v)}$ for bipartite $c$-bisortable elements is $$\sum\, q^{|M|}{\operatorname{Cat}}{^{+\!\!+}}(W_{I};q){\operatorname{Cat}}{^{+\!\!+}}(W_{J};q),$$ where the sum is over all ordered triples $(I,J,M)$ of *pairwise disjoint* subsets of $S$.
Similarly to the proof of Proposition \[q antichain\], the proposition amounts to establishing a bijection from bipartite $c$-bisortable elements $w$ to triples $(u',v',M)$ such that $u'$ is a clever $c$-sortable element, $v'$ is a clever $c^{-1}$-sortable element, and the sets $I={\operatorname{supp}}(u')$, $J={\operatorname{supp}}(v')$, and $M$ are disjoint subsets of $S$, and then showing that ${\operatorname{des}}(w)={\operatorname{des}}(u')+{\operatorname{des}}(v')+|M|$.
Given a bipartite $c$-bisortable element $w$, write $(u,v)$ for the corresponding pair $({\pi_\downarrow}^c(w),{\pi_\downarrow}^{c^{-1}}(w))$ of twin $(c,c^{-1})$-sortable elements. Proposition \[disjoint\_simple\_support\](2) says that ${\operatorname{Can}}(w)$ is the disjoint union $\left({\operatorname{Can}}(u)\setminus S\right)\uplus \left({\operatorname{Can}}(v)\setminus S\right)\uplus\left( {\operatorname{Can}}(w)\cap S\right)$. Proposition \[disjoint\_simple\_support\](\[pw dis\]) says that the sets $I={\operatorname{supp}}({\operatorname{Can}}(u)\setminus S)$, $J={\operatorname{supp}}({\operatorname{Can}}(v)\setminus S)$, and $M={\operatorname{Can}}(w)\cap S$ are pairwise disjoint subsets of $S$. By Proposition \[can cplx\], ${\bigvee}{\operatorname{Can}}(u)\setminus S$ is the canonical join representation of a positive, clever $c$-sortable element $u'$ in $W_I$. Similarly, ${\bigvee}{\operatorname{Can}}(v)\setminus S$ is the canonical join representation of a positive, clever $c^{-1}$-sortable element $v'$ in $W_J$. Applying Proposition \[coxcjr\] several times, we see that ${\operatorname{des}}(w) = {\operatorname{des}}(u')+{\operatorname{des}}(v')+|M|$.
We will show that this map $w\mapsto(u',v',M)$ is a bijection by showing that the map $(u',v',M)\mapsto u'{\vee}v'{\vee}({\bigvee}M)$ is the inverse. On one hand, given $w$, construct $(u',v',M)$ as above. Then $w$ equals ${\bigvee}{\operatorname{Can}}(w)$, which equals $$\left({\bigvee}{\operatorname{Can}}(u)\setminus S\right){\vee}\left({\bigvee}{\operatorname{Can}}(v)\setminus S\right){\vee}\left({\bigvee}{\operatorname{Can}}(w)\cap S\right)=u'{\vee}v'{\vee}({\bigvee}M).$$ On the other hand, given a triple $(u',v',M)$ satisfying the description above, set ${w=u'{\vee}v'{\vee}({\bigvee}M)}$. Since $u'$, $v'$ and $M$ have pairwise disjoint support, we conclude that ${\operatorname{Can}}(u')$, ${\operatorname{Can}}(v')$, and $M$ also have pairwise disjoint support. Lemma \[disjoint joinands\] says that ${\bigvee}{\operatorname{Can}}(u')\uplus {\operatorname{Can}}(v') \uplus M$ is the canonical join representation of $w$. By Lemma \[c or cinv\](1), each canonical joinand of $u'$ is $c$-sortable and each canonical joinand of $v'$ is $c^{-1}$-sortable. Since each simple generator is both $c$- and $c^{-1}$-sortable, we conclude that each canonical joinand of $w$ either either $c$- or $c^{-1}$-sortable. By Lemma \[c or cinv\](2), $w$ is $c$-bisortable. Thus, the map $(u',v', M) \mapsto u'{\vee}v'{\vee}({\bigvee}M)$ is a well-defined.
Lemma \[only ji\] says that ${\operatorname{Can}}(u')\uplus M$ is equal to the set of $c$-sortable canonical joinands of $w$. Since $u'$ is clever, ${\operatorname{Can}}(u')$ is equal to the set of $c$-sortable canonical joinands in ${\operatorname{Can}}(w)\setminus S$. Similarly, ${\operatorname{Can}}(v')$ is the set of $c^{-1}$-sortable canonical joinands in ${\operatorname{Can}}(w)\setminus S$, and ${\operatorname{Can}}(w)\cap S = M$. Define $u={\pi_\downarrow}^c(w)$ and $v={\pi_\downarrow}^{c^-1}(w)$. Proposition \[disjoint\_simple\_support\](2) says that ${\operatorname{Can}}(w)=({\operatorname{Can}}(u)\setminus S)\uplus ({\operatorname{Can}}(v) \setminus S)\uplus ({\operatorname{Can}}(w)\cap S)$. Comparing this to the expression ${\operatorname{Can}}(w)={\operatorname{Can}}(u')\uplus{\operatorname{Can}}(v')\uplus M$, we see that ${\operatorname{Can}}(u)\setminus S={\operatorname{Can}}(u')$, that ${\operatorname{Can}}(v)\setminus S={\operatorname{Can}}(v')$, and that ${\operatorname{Can}}(w)\cap S=M$. Thus the map described above takes $w$ back to $(u',v',M)$.
\[uniform uniform\] The proof given here that twin nonnesting partitions are in bijection with bipartite $c$-bisortable elements would be uniform if there were a uniform proof connecting $c$-sortable elements and nonnesting partitions. The opposite is true as well: Suppose one proved uniformly that a given map $\phi$ is a bijection from antichains in the doubled root poset to bipartite $c$-bisortable elements and also that $\phi$ preserves the triples $(I,J,M)$ appearing in Propositions \[q antichain\] and \[q bisortable formula\]. Then the restriction of $\phi$ to antichains in the root poset (i.e. those with $J=\emptyset$) is a bijection from antichains in the root poset to $c$-sortable elements.
\[other c\] The methods of this section don’t apply well to the case where $c$ is not bipartite, because the main structural results of the section, Propositions \[disjoint\_simple\_support\] and \[i-e sortable\], can fail when $c$ is not bipartite. This can already be seen in $A_3$ for the linear Coxeter element.
BiCatalan and Catalan formulas {#recursions sec}
------------------------------
In this section and the next, we prepare to prove the formula for ${\operatorname{biCat}}(D_n)$ in Theorem \[enum thm\], thus completing the proof of that theorem. Specifically, the proof requires combining a very large number of identities relating $q$-analogs of biCatalan numbers, Catalan numbers, and double-positive Catalan numbers that we quote or prove here. In this section, we give recursions for the $q$-analogs of $W$-biCatalan and $W$-Catalan numbers for irreducible finite Coxeter groups, in which $q$-analogs of double-positive Catalan numbers appear as coefficients.
\[q biCat recursion\] For an irreducible finite Coxeter group $W$ and a simple generator $s\in S$, the $q$-analog of the $W$-biCatalan number satisfies $$\begin{gathered}
\label{q biCat recursion formula}
{\operatorname{biCat}}(W;q)=(1+q){\operatorname{biCat}}(W_{S\setminus\set{s}};q)\\
+2\sum_{S_0}{\operatorname{Cat}}{^{+\!\!+}}(W_{S_0};q)\prod_{i=1}^m\left[\frac12{\operatorname{biCat}}(W_{S_i};q)+\frac{1+q}2{\operatorname{biCat}}(W_{S_i\setminus\set{s_i}};q)\right],\end{gathered}$$ where the sum is over all connected subgraphs $S_0$ of the diagram for $W$ with $s\in S_0$, the connected components of the complement of $S_0$ in the diagram are $S_1,\ldots,S_m$, and each $s_i$ is the unique vertex in $S_i$ that is connected by an edge to a vertex in $S_0$.
For fixed $s$, we break the formula in Theorem \[biCat GF d-p\] into four sums, according to whether $s$ is in $S\setminus(I\cup J\cup M)$, in $M$, in $I$, or in $J$. The sum of terms with $s\in S\setminus(I\cup J\cup M)$ equals ${\operatorname{biCat}}(W_{S\setminus\set{s}};q)$. The sum of terms with $s\in M$ equals $q\cdot{\operatorname{biCat}}(W_{S\setminus\set{s}};q)$.
Consider next the sum of terms with $s\in I$, and in each term let $S_0$ be the connected component of the diagram containing $s$. Using , we can reorganize the sum according to $S_0$ to obtain $$\sum_{S_0}{\operatorname{Cat}}{^{+\!\!+}}(W_{S_0};q)\sum q^{|M|}{\operatorname{Cat}}{^{+\!\!+}}(W_{I'};q){\operatorname{Cat}}{^{+\!\!+}}(W_J;q),$$ where the $S_0$-sum is as described in the statement of the proposition and the inner sum is over all ordered triples $(I',J,M)$ of disjoint subsets of $S\setminus S_0$ such that no element of $I'$ is connected by an edge of the diagram to an element of $S_0$. Again using , we factor the inner sum further to obtain $$\sum_{S_0}{\operatorname{Cat}}{^{+\!\!+}}(W_{S_0};q)\prod_{i=1}^m\left[\sum q^{|M_i|}{\operatorname{Cat}}{^{+\!\!+}}(W_{I_i};q){\operatorname{Cat}}{^{+\!\!+}}(W_{J_i};q)\right],$$ where the $S_i$ and $s_i$ are as in the statement of the proposition and the inner sum runs of over all ordered triples $(I_i,J_i,M_i)$ of pairwise disjoint subsets of $S_i$ with $s_i\not\in I_i$. The sum for each $i$ can be broken up into a sum over terms with $s_i \in J_i$ and terms with $s_i \not\in J_i$. Splitting the sum over terms with $s_i \not\in J_i$ in half, we obtain three sums: $$\begin{gathered}
\sum_{s_i\in J_i} q^{|M_i|}{\operatorname{Cat}}{^{+\!\!+}}(W_{I_i};q){\operatorname{Cat}}{^{+\!\!+}}(W_{J_i};q)\\
+\frac12\sum_{s_i\not\in J_i} q^{|M_i|}{\operatorname{Cat}}{^{+\!\!+}}(W_{I_i};q){\operatorname{Cat}}{^{+\!\!+}}(W_{J_i};q)\\
+\frac12\sum_{s_i\not\in J_i} q^{|M_i|}{\operatorname{Cat}}{^{+\!\!+}}(W_{I_i};q){\operatorname{Cat}}{^{+\!\!+}}(W_{J_i};q)\end{gathered}$$ The symmetry between $I$ and $J$ on the right side of Theorem \[biCat GF d-p\] lets us recognize the sum of the first two terms as $\frac12{\operatorname{biCat}}(W_{S_i};q)$, recalling that $s\not\in I_i$ throughout. The third term is $\frac{1+q}2{\operatorname{biCat}}(W_{S_i\setminus\set{s_i}};q)$. We see that the sum of terms with $s\in I$ is the sum in the proposed formula, without the factor $2$ in front. By symmetry, the sum of terms with $s\in J$ is the same sum, so we obtain the factor $2$ in the sum and we have established the desired formula.
We obtain the following recursion for ${\operatorname{biCat}}(D_n;q)$ from Proposition \[q biCat recursion\]. The notation $D_2$ means $A_1\times A_1$ and $D_3$ means $A_3$.
\[q biCat recursion D\] For $n\ge 3$, $$\begin{gathered}
\label{q biCat recursion D formula}
{\operatorname{biCat}}(D_n;q)=(1+q){\operatorname{biCat}}(D_{n-1};q)\\
+\sum_{i=1}^{n-3}{\operatorname{Cat}}{^{+\!\!+}}(A_i;q)\left({\operatorname{biCat}}(D_{n-i};q)+(1+q){\operatorname{biCat}}(D_{n-i-1};q)\right)\\
+2(1+q)^2{\operatorname{Cat}}{^{+\!\!+}}(A_{n-2};q)+4(1+q){\operatorname{Cat}}{^{+\!\!+}}(A_{n-1};q)+2{\operatorname{Cat}}{^{+\!\!+}}(D_n;q)\end{gathered}$$
In Proposition \[q biCat recursion\], take $s$ to be a leaf of the $D_n$ diagram whose removal leaves the diagram for $D_{n-1}$. The sum over $S_0$ splits into several pieces. First, the $S_0$ for which the diagram on $S\setminus\set{S_0}$ is of type $D_k$ for $k\ge 3$ give rise to terms $\sum_{i=1}^{n-3}{\operatorname{Cat}}{^{+\!\!+}}(A_i;q)\left({\operatorname{biCat}}(D_{n-i};q)+(1+q){\operatorname{biCat}}(D_{n-i-1};q)\right)$. Next, the term for which the diagram on $S\setminus\set{S_0}$ is of type $D_2$ is $2{\operatorname{Cat}}{^{+\!\!+}}(A_{n-2})(\frac12(1+q)+\frac{1+q}2\cdot1)^2$, which simplifies to $2(1+q)^2{\operatorname{Cat}}{^{+\!\!+}}(A_{n-2})$. The two terms for which the diagram on $S\setminus\set{S_0}$ is of type $A_1$ *each* contribute $2(1+q){\operatorname{Cat}}{^{+\!\!+}}(A_{n-1})$. Finally, the term with $S_0=S$ is $2{\operatorname{Cat}}{^{+\!\!+}}(D_n;q)$.
We obtain the following recursion for ${\operatorname{biCat}}(B_n;q)$ from Proposition \[q biCat recursion\] similarly. Here and throughout the paper, we interpret $B_0$ and $B_1$ to be synonyms for $A_0$ and $A_1$.
\[q biCat recursion B\] For $n\ge1$, $$\begin{gathered}
\label{q biCat recursion B formula}
{\operatorname{biCat}}(B_n;q)=\\(1+q){\operatorname{biCat}}(B_{n-1};q)+2{\operatorname{Cat}}{^{+\!\!+}}(B_n;q)+2(1+q){\operatorname{Cat}}{^{+\!\!+}}(A_{n-1};q)\\
+\sum_{i=1}^{n-2}{\operatorname{Cat}}{^{+\!\!+}}(A_i;q)\left[{\operatorname{biCat}}(B_{n-i};q)+(1+q){\operatorname{biCat}}(B_{n-i-1};q)\right].\end{gathered}$$
In Proposition \[q biCat recursion\], take $s$ to be a leaf of the $B_n$ diagram whose removal leaves the diagram for $B_{n-1}$. The terms with $|S_0|$ from $1$ to $n-2$ are in the summation in , but we separate out the terms with $|S_0|=n-1$ and $|S_0|=n$. For the term with $|S_0|=n-1$, we use the facts that ${\operatorname{biCat}}(B_1;q)=(1+q)$ and that ${\operatorname{biCat}}(B_0;q)=1$.
Similarly, we obtain the following recursion for ${\operatorname{biCat}}(A_n)$ by taking $s$ to be either leaf of the diagram.
\[q biCat recursion A\] For $n\ge1$, $$\begin{gathered}
\label{q biCat recursion A formula}
{\operatorname{biCat}}(A_n;q)=(1+q){\operatorname{biCat}}(A_{n-1};q)+2{\operatorname{Cat}}{^{+\!\!+}}(A_n;q)\\
+\sum_{i=1}^{n-1}{\operatorname{Cat}}{^{+\!\!+}}(A_i;q)\left[{\operatorname{biCat}}(A_{n-i};q)+(1+q){\operatorname{biCat}}(A_{n-i-1};q)\right].\end{gathered}$$
Next we gather some formulas involving the $q$-Catalan numbers. We begin with the usual recursion for the type-A Catalan numbers, although this $q$-version may be less widely familiar. It is easily obtained through the interpretation of ${\operatorname{Cat}}(A_n;q)$ as the descent generating function for $231$-avoiding permutations in $S_{n+1}$, by breaking up the count according to the first entry in the permutation. We omit the details.
\[Cat A q\] For $n\ge1$, $$\label{Cat A q formula}
{\operatorname{Cat}}(A_n;q)=(1+q){\operatorname{Cat}}(A_{n-1};q)+q\sum_{i=1}^{n-1}{\operatorname{Cat}}(A_{i-1};q){\operatorname{Cat}}(A_{n-i-1};q).$$
Furthermore, using known formulas for the Narayana numbers, we obtain a recursion that relates the $q$-Catalan number in types A and D.
\[Cat D A q\] For $n\ge2$, $$\label{Cat D A formula}
{\operatorname{Cat}}(D_n;q) = \frac{n+1}2(1+q){\operatorname{Cat}}(A_{n-1};q)-\Bigl(\frac{n-1}2+q+\frac{n-1}2q^2\Bigr){\operatorname{Cat}}(A_{n-2}).$$
Taking the coefficient of $q^k$ on both sides, we see that is equivalent to $$\begin{gathered}
\label{Nar D A formula}
{\operatorname{Nar}}_k(D_n) = \frac{n+1}2\left({\operatorname{Nar}}_k(A_{n-1})+{\operatorname{Nar}}_{k-1}(A_{n-1})\right)\\-\frac{n-1}2{\operatorname{Nar}}_k(A_{n-2})-{\operatorname{Nar}}_{k-1}(A_{n-2})-\frac{n-1}2{\operatorname{Nar}}_{k-2}(A_{n-2}).\end{gathered}$$ This can be verified using the known formulas for the type-A and type-D Narayana numbers. (See for example in [@gcccc (9.1)] and [@gcccc (9.3)], putting $m=1$ in both formulas).
Next, we give a recursion for ${\operatorname{Cat}}(W;q)$ analogous to . The proof follows the outline of the proof of Proposition \[q biCat recursion\], using Theorem \[Cat GF d-p\] instead of Theorem \[biCat GF d-p\]. This proof is simpler than the proof of Proposition \[q biCat recursion\], so we omit the details.
\[q Cat recursion\] For an irreducible finite Coxeter group $W$ and a simple generator $s$, the $q$-analog of the $W$-Catalan number satisfies $$\begin{gathered}
\label{q Cat recursion formula}
{\operatorname{Cat}}(W;q)=(1+q){\operatorname{Cat}}(W_{S\setminus\set{s}};q)\\
+\sum_{S_0}{\operatorname{Cat}}{^{+\!\!+}}(W_{S_0};q)\prod_{i=1}^m(1+q){\operatorname{Cat}}(W_{S_i\setminus\set{s_i}};q),\end{gathered}$$ where the sum is over all connected subgraphs $S_0$ of the diagram for $W$ with $s\in S_0$, the connected components of the complement of $S_0$ in the diagram are $S_1,\ldots,S_m$, and each $s_i$ is the unique vertex in $S_i$ that is connected by an edge to a vertex in $S_0$.
The following three propositions give the type-A, type-B, and type-D cases of .
\[q Cat recursion A\] For $n\ge 0$, $$\begin{gathered}
\label{q Cat recursion A formula}
{\operatorname{Cat}}(A_n;q)={\operatorname{Cat}}{^{+\!\!+}}(A_n;q)+(1+q)\sum_{i=0}^{n-1}{\operatorname{Cat}}{^{+\!\!+}}(A_i;q){\operatorname{Cat}}(A_{n-i-1};q).\end{gathered}$$
If $n=0$, then the formula is ${\operatorname{Cat}}(A_0;q)={\operatorname{Cat}}{^{+\!\!+}}(A_0;q)$, which says ${1=1}$. Otherwise, taking $s$ to be a leaf of the $A_n$ diagram in , the sum over $S_0$ has terms $\sum_{i=1}^{n-1}{\operatorname{Cat}}{^{+\!\!+}}(A_i;q)(1+q){\operatorname{Cat}}(A_{n-i-1};q)$ and ${\operatorname{Cat}}{^{+\!\!+}}(A_n;q)$. Because ${\operatorname{Cat}}{^{+\!\!+}}(A_0;q)=1$, we can merge the first term into the sum.
\[q Cat recursion B 2\] For $n\ge 0$, $$\begin{gathered}
\label{q Cat recursion B 2 formula} {\operatorname{Cat}}(B_n;q)={\operatorname{Cat}}{^{+\!\!+}}(B_n;q)+(1+q)\sum_{i=0}^{n-1}{\operatorname{Cat}}{^{+\!\!+}}(A_i;q){\operatorname{Cat}}(B_{n-i-1};q) \end{gathered}$$
The formula holds for $n=0$ and $n=1$. For $n>1$, take $s$ to be the leaf whose deletion leaves a diagram of type $B_{n-1}$ in , and rearrange the formula as in the proof of Proposition \[q Cat recursion A formula\].
\[q Cat recursion D\] For $n\ge3$, $$\begin{gathered}
\label{q Cat recursion D formula}
{\operatorname{Cat}}(D_n;q)=\\
(1+q){\operatorname{Cat}}(A_{n-1};q)+(1+q){\operatorname{Cat}}{^{+\!\!+}}(A_{n-1};q)+{\operatorname{Cat}}{^{+\!\!+}}(D_n;q)\\
+(1+q)^2\sum_{i=1}^{n-2}{\operatorname{Cat}}{^{+\!\!+}}(A_i;q){\operatorname{Cat}}(A_{n-i-2};q)\\
+(1+q)\sum_{i=3}^{n-1}{\operatorname{Cat}}{^{+\!\!+}}(D_i;q){\operatorname{Cat}}(A_{n-i-1};q).\\\end{gathered}$$
Start with Proposition \[q Cat recursion\], taking $s$ to be a leaf of the $D_n$ diagram whose removal leaves the diagram for $A_{n-1}$. For $S_0$ not containing the leaf symmetric to $s$, we get terms $(1+q)^2\sum_{i=1}^{n-2}{\operatorname{Cat}}{^{+\!\!+}}(A_i;q){\operatorname{Cat}}(A_{n-i-2};q)$ and $(1+q){\operatorname{Cat}}{^{+\!\!+}}(A_{n-1};q)$. (The $i=1$ term in the sum would be wrong, except that ${\operatorname{Cat}}{^{+\!\!+}}(A_1)=0$.) For $S_0$ containing the leaf symmetric to $s$, we get $(1+q)\sum_{i=3}^{n-1}{\operatorname{Cat}}{^{+\!\!+}}(D_i;q){\operatorname{Cat}}(A_{n-i-1};q)$ and ${\operatorname{Cat}}{^{+\!\!+}}(D_n;q)$.
The double-positive Catalan numbers
-----------------------------------
In this section, we consider the double-positive Catalan numbers for the classical reflection groups, and establish some identities for ${\operatorname{Cat}}{^{+\!\!+}}(A_n;q)$, ${\operatorname{Cat}}{^{+\!\!+}}(B_n;q)$, and ${\operatorname{Cat}}{^{+\!\!+}}(D_n;q)$ that will be useful for proving the type-D case of Theorem \[enum thm\].
Athanasiadis and Savvidou, in [@Ath-Sav Theorem 1.2], gave formulas for the polynomials ${\operatorname{Cat}}{^{+\!\!+}}(W;q)$ for each $W$ of finite type by explicitly determining coefficients $\xi_i$ such that ${\operatorname{Cat}}{^{+\!\!+}}(W;q)=\sum_{i=0}^{\lfloor n/2\rfloor}\xi_iq^i(1+q)^{n-2i}.$ Similar formulas for the relevant polynomials ${\operatorname{Cat}}(W;q)$ are known [@PRW Propositions 11.14–11.15], so the identities we need can in principle be obtained by manipulating the formulas from [@Ath-Sav; @PRW]. Indeed, Proposition \[Cat++ D q\] is easily obtained in this way, but such proofs of Propositions \[Cat++ A q\] and \[Cat++ Bn-1\] appear to be more complicated.
Below we list some examples of the double-positive Catalan numbers for the classical reflection groups. [$$\begin{array}{c||c|c|c|c|c|c|c||c|c|c|c|c||c|c|c|c}
\!W\!&\!A_0\!&\!A_1\!&\!A_2\!&\!A_3\!&\!A_4\!&\!A_5\!&\!A_6\!&\!B_2\!&\!B_3\!&\!B_4\!&\!B_5\!&\!B_6\!&\! D_4\!&\!D_5\!&\!D_6\!&\! D_7\\\hline
&&&&&&&&&&&&&&&&\\[-9pt]
\!{\operatorname{Cat}}{^{+\!\!+}}(W)\!&\!1\!&\!0\!&\!1\!&\!2\!&\!6\!&\!18\!&\!57\!&\!2\!&\!6\!&\!22\!&\!80\!&\!296\!&\!10\!&\!42\!&\!168\!&\!660
\end{array}\,$$ ]{}
From inspection of these numbers, several interesting relationships appear. First, the data suggests that $2{\operatorname{Cat}}{^{+\!\!+}}(A_n)+{\operatorname{Cat}}{^{+\!\!+}}(A_{n-1}) = {\operatorname{Cat}}(A_{n-1})$. Below, we establish a $q$-analog of this identity.
\[Cat++ A q\] For $n\ge1$, $$\label{Cat++ A q formula}
(1+q){\operatorname{Cat}}{^{+\!\!+}}(A_n;q)+q{\operatorname{Cat}}{^{+\!\!+}}(A_{n-1};q)=q{\operatorname{Cat}}(A_{n-1};q).$$
If $n=1$, then the identity is $(1+q)\cdot0+q\cdot1=q\cdot1$. If $n>1$, then by induction, we can replace $(1+q){\operatorname{Cat}}{^{+\!\!+}}(A_i;q)$ with $q({\operatorname{Cat}}(A_{i-1};q)-{\operatorname{Cat}}{^{+\!\!+}}(A_{i-1};q))$ in the terms $i>1$ of and observe that ${\operatorname{Cat}}{^{+\!\!+}}(A_0;q)=1$ to obtain $$\begin{gathered}
{\operatorname{Cat}}(A_n;q)={\operatorname{Cat}}{^{+\!\!+}}(A_n;q)+(1+q){\operatorname{Cat}}(A_{n-1};q)\\
+q\sum_{i=1}^{n-1}{\operatorname{Cat}}(A_{i-1};q){\operatorname{Cat}}(A_{n-i-1};q)\\
-q\sum_{i=1}^{n-1}{\operatorname{Cat}}{^{+\!\!+}}(A_{i-1};q){\operatorname{Cat}}(A_{n-i-1};q).\end{gathered}$$ The first sum, by Proposition \[Cat A q\], is $({\operatorname{Cat}}(A_n;q)-(1+q){\operatorname{Cat}}(A_{n-1};q))$. The second sum can be reindexed to $q\sum_{i=0}^{n-2}{\operatorname{Cat}}{^{+\!\!+}}(A_{i};q){\operatorname{Cat}}(A_{n-i-2};q)$, which, by Proposition \[q Cat recursion A\], equals $\frac{q}{1+q}({\operatorname{Cat}}(A_{n-1};q)-{\operatorname{Cat}}{^{+\!\!+}}(A_{n-1};q))$. We obtain $$\begin{gathered}
{\operatorname{Cat}}(A_n;q)={\operatorname{Cat}}{^{+\!\!+}}(A_n;q)+(1+q){\operatorname{Cat}}(A_{n-1};q)\\
+{\operatorname{Cat}}(A_n;q)-(1+q){\operatorname{Cat}}(A_{n-1};q)\\
-\frac{q}{1+q}({\operatorname{Cat}}(A_{n-1};q)-{\operatorname{Cat}}{^{+\!\!+}}(A_{n-1};q)),\end{gathered}$$ which simplifies to the desired identity.
The data also suggests that ${\operatorname{Cat}}{^{+\!\!+}}(D_n)=(n-2){\operatorname{Cat}}(A_{n-2})$. Indeed, the following is a $q$-analog.
\[Cat++ D q\] For $n\ge2$, $$\label{Cat++ D q formula}
{\operatorname{Cat}}{^{+\!\!+}}(D_n;q) = (n-2)q{\operatorname{Cat}}(A_{n-2};q).$$
For $n=2$, the identity is $q+q^2=(3-2)q(1+q)$. If $n\ge3$, then we start with . The first summation in the formula can be rewritten, using , as $$\label{firstsum}
(1+q)\bigl({\operatorname{Cat}}(A_{n-1};q)-(1+q){\operatorname{Cat}}(A_{n-2};q)-{\operatorname{Cat}}{^{+\!\!+}}(A_{n-1};q)\bigr).$$ By induction, the second summation can be rewritten as $$\label{secondsum}
(1+q)q\sum_{i=3}^{n-1}(i-2){\operatorname{Cat}}{^{+\!\!+}}(A_{i-2};q){\operatorname{Cat}}(A_{n-i-1};q).$$ To further simplify , we use to calculate $$\begin{aligned}
&(n-3)\bigl({\operatorname{Cat}}(A_{n-1};q)-(1+q){\operatorname{Cat}}(A_{n-2};q)\bigr)\\
&\qquad=(n-3)q\sum_{i=1}^{n-2}{\operatorname{Cat}}(A_{i-1};q){\operatorname{Cat}}(A_{n-i-2};q)\\
&\qquad=q\sum_{i=1}^{n-2}\bigr((i-1){\operatorname{Cat}}(A_{i-1};q){\operatorname{Cat}}(A_{n-i-2};q)\\&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+(n-i-2){\operatorname{Cat}}(A_{i-1};q){\operatorname{Cat}}(A_{n-i-2};q)\bigr)\\
&\qquad=q\sum_{i=1}^{n-2}(i-1){\operatorname{Cat}}(A_{i-1};q){\operatorname{Cat}}(A_{n-i-2};q)\\&\qquad\qquad\qquad\qquad\qquad\qquad\qquad+q\sum_{i=1}^{n-2}(n-i-2){\operatorname{Cat}}(A_{i-1};q){\operatorname{Cat}}(A_{n-i-2};q)\end{aligned}$$ Both sums can be reindexed to agree with , except for the initial factor $(1+q)$. Thus equals $\frac{n-3}2(1+q)({\operatorname{Cat}}(A_{n-1};q)-(1+q){\operatorname{Cat}}(A_{n-2};q))$. Finally, we use to rewrite the ${\operatorname{Cat}}(D_n;q)$. We obtain $$\begin{gathered}
\frac{n+1}2(1+q){\operatorname{Cat}}(A_{n-1};q)-\Bigl(\frac{n-1}2+q+\frac{n-1}2q^2\Bigr){\operatorname{Cat}}(A_{n-2})=\\
(1+q){\operatorname{Cat}}(A_{n-1};q)+(1+q){\operatorname{Cat}}{^{+\!\!+}}(A_{n-1};q)+{\operatorname{Cat}}{^{+\!\!+}}(D_n;q)\\
+(1+q)\bigl({\operatorname{Cat}}(A_{n-1};q)-(1+q){\operatorname{Cat}}(A_{n-2};q)-{\operatorname{Cat}}{^{+\!\!+}}(A_{n-1};q)\bigr)\\
+\frac{n-3}2(1+q)({\operatorname{Cat}}(A_{n-1};q)-(1+q){\operatorname{Cat}}(A_{n-2};q)).\end{gathered}$$ This can be rearranged to say ${\operatorname{Cat}}{^{+\!\!+}}(D_n;q) = (n-2)q{\operatorname{Cat}}(A_{n-2};q)$.
In order to establish a needed identity for double-positive Catalan numbers of type B, we need a recursion for the $q$-Catalan number that comes from a completely different direction. The $q$-Catalan numbers ${\operatorname{Cat}}(W;q)$ encode the $h$-vector of the generalized associahedron for $W$. (See, for example, [@rsga Section 5.2].) For each Coxeter group $W$ of rank $n$ and each $i$ from $0$ to $n$, define $f_i$ to be the number of simplices in the simplicial generalized associahedron having exactly $i$ vertices (and thus dimension $i-1$). Define a polynomial $$f(W;x)=\sum_{k=0}^nf_k(W)x^k.$$ The following is [@ga Proposition 3.7].
\[ga prop\] If $W$ is reducible as $W_1\times W_2$, then $f(W;x)=f(W_1;x)f(W_2;x)$. If $W$ is irreducible with Coxeter number $h$, then $$\label{ga prop formula}
\dod{f(W;x)}{x}=\frac{h+2}2\sum_{s\in S}f(W_{S\setminus\set{s}};x)$$
Since $f(W)$ encodes the $f$-vector of the generalized associahedron and ${\operatorname{Cat}}(W;q)$ encodes the $h$-vector, implies a formula for ${\operatorname{Cat}}(W;q)$. Since $f(W)$ is has coefficients reversed from the $f$-polynomial usually used to define $h$-vectors, the formula for ${\operatorname{Cat}}(W;q)$ is somewhat more complicated than .
\[q Cat FZ\] For an irreducible Coxeter group $W$ with rank $n\ge0$ and Coxeter number $h$, the $q$-analog of the Catalan number satisfies $$\label{q Cat FZ formula}
n{\operatorname{Cat}}(W;q)+(1-q)\dod{}{q}{\operatorname{Cat}}(W;q)=\frac{h+2}2\sum_{s\in S}{\operatorname{Cat}}(W_{S\setminus\set{s}};q).$$
We begin with the right side of and replace $q$ by $x+1$ throughout. The result is $\frac{h+2}2\sum_{s\in S}{\operatorname{rev}}(f(W_{S\setminus\set{s}};x))$, where ${\operatorname{rev}}$ is the operator that reverses the coefficients of a polynomial. In other symbols: $x^{n-1}\frac{h+2}2\sum_{s\in S}f(W_{S\setminus\set{s}};x^{-1})$ Using , the quantity becomes $x^{n-1}\dod{f(W;x^{-1})}{(x^{-1})}$.
Similarly, ${\operatorname{Cat}}(W;x+1)={\operatorname{rev}}(f(W;x))=x^nf(W;x^{-1})$, so $f(W;x^{-1})=x^{-n}{\operatorname{Cat}}(W;x+1)$. Thus the right side of equals $$\begin{aligned}
&x^{n-1}\dod{}{(x^{-1})}\left[x^{-n}{\operatorname{Cat}}(W;x+1)\right]\\
&=x^{n-1}\dod{}{x}\left[x^{-n}{\operatorname{Cat}}(W;x+1)\right](-x^2)\\
&=-x^{n+1}\left[-nx^{-n-1}{\operatorname{Cat}}(W;x+1)+x^{-n}\dod{}{x}{\operatorname{Cat}}(W;x+1)\right]\\
&=n{\operatorname{Cat}}(W;x+1)-x\dod{}{x}{\operatorname{Cat}}(W;x+1)\\
&=n{\operatorname{Cat}}(W;x+1)-x\dod{}{(x+1)}{\operatorname{Cat}}(W;x+1)\end{aligned}$$ Replacing $x$ by $q-1$ throughout, we obtain the left side of .
The type-B version of is the following recursion:
\[q Cat FZ B\] For $n\ge 0$, $$\label{q Cat FZ B formula}
n{\operatorname{Cat}}(B_n;q)+(1-q)\od{}{q}{\operatorname{Cat}}(B_n;q)=(n+1)\sum_{i=1}^n{\operatorname{Cat}}(A_{i-1};q){\operatorname{Cat}}(B_{n-i};q).$$
The following formula is obtained using known formulas for Narayana numbers of types A and B.
\[mystery prop\] For $n\ge0$, $$\label{mystery formula} \sum_{i=1}^n{\operatorname{Cat}}(A_{i-1};q){\operatorname{Cat}}(B_{n-i};q)=n{\operatorname{Cat}}(A_{n-1};q).$$
By , the assertion is equivalent to $$\label{Cat B q ODE formula}
n{\operatorname{Cat}}(B_n;q)+(1-q)\od{}{q}{\operatorname{Cat}}(B_n;q)=n(n+1){\operatorname{Cat}}(A_{n-1};q).$$ Taking the coefficient of $q^k$ on both sides, we see that is equivalent to $$\label{Nar B q ODE formula}
(n-k){\operatorname{Nar}}_k(B_n)+(k+1){\operatorname{Nar}}_{k+1}(B_n)=n(n+1){\operatorname{Nar}}_k(A_{n-1}).$$ This can be verified using the formulas for the type-A and type-B Narayana numbers, found for example in [@gcccc (9.1)] and [@gcccc (9.2)] (setting $m=1$ in both formulas from [@gcccc]).
Using and the observation that ${\operatorname{biCat}}(A_n;q)={\operatorname{Cat}}(B_n;q)$, then applying twice, (where, in the first instance $n$ is replaced by $n+1$ in ), we obtain the following formula.
\[gratuitous prop\] For $n\ge1$, $$\begin{gathered}
\label{gratuitous formula}
{\operatorname{Cat}}(B_n;q)=(1+q){\operatorname{Cat}}(B_{n-1};q)-(1+q){\operatorname{Cat}}{^{+\!\!+}}(A_{n-1};q)\\+{\operatorname{Cat}}{^{+\!\!+}}(B_n;q)+(1+q){\operatorname{Cat}}{^{+\!\!+}}(B_{n-1};q).\end{gathered}$$
Next, we obtain the following formula.
\[Cat B q combination\] For $n\ge2$, $$\begin{gathered}
\label{Cat B q combination formula} (1+q){\operatorname{Cat}}(B_n;q)=\\(1+q+q^2){\operatorname{Cat}}(B_{n-1};q)
+(n-1)q(1+q){\operatorname{Cat}}(A_{n-2};q)\\
+q{\operatorname{Cat}}{^{+\!\!+}}(B_{n-1};q)
+ (1+q){\operatorname{Cat}}{^{+\!\!+}}(B_n;q).\end{gathered}$$
Using to replace each instance of $(1+q){\operatorname{Cat}}{^{+\!\!+}}(A_i;q)$ in with $q({\operatorname{Cat}}(A_{i-1};q)-{\operatorname{Cat}}{^{+\!\!+}}(A_{i-1};q))$ for $i>0$ and splitting into two sums, we obtain: $$\begin{gathered}
{\operatorname{Cat}}(B_n;q)=(1+q){\operatorname{Cat}}(B_{n-1};q)+q\sum_{i=1}^{n-1}{\operatorname{Cat}}(A_{i-1};q){\operatorname{Cat}}(B_{n-i-1};q)\\
-q\sum_{i=1}^{n-1}{\operatorname{Cat}}{^{+\!\!+}}(A_{i-1};q){\operatorname{Cat}}(B_{n-i-1};q) + {\operatorname{Cat}}{^{+\!\!+}}(B_n;q)\end{gathered}$$ We use with $n$ replaced by $n-1$ to evaluate the first sum. We reindex the second sum and evaluate it using with $n$ replaced by $n-1$. $$\begin{gathered}
{\operatorname{Cat}}(B_n;q)=(1+q){\operatorname{Cat}}(B_{n-1};q)+q(n-1){\operatorname{Cat}}(A_{n-2};q)\\
-\frac{q}{1+q}({\operatorname{Cat}}(B_{n-1};q)-{\operatorname{Cat}}{^{+\!\!+}}(B_{n-1};q))
+ {\operatorname{Cat}}{^{+\!\!+}}(B_n;q).\end{gathered}$$ We multiply through by $(1+q)$ and simplify to obtain .
Solving both and for $(1+q){\operatorname{Cat}}{^{+\!\!+}}(B_n;q)$ and combining them, then solving for $(1+q+q^2){\operatorname{Cat}}{^{+\!\!+}}(B_{n-1};q)$, we obtain the key result for ${\operatorname{Cat}}{^{+\!\!+}}(B_{n-1})$.
\[Cat++ Bn-1\] $$\begin{gathered}
\label{Cat++ Bn-1 formula} (1+q+q^2){\operatorname{Cat}}{^{+\!\!+}}(B_{n-1};q)=-q{\operatorname{Cat}}(B_{n-1};q)\\+(n-1)q(1+q){\operatorname{Cat}}(A_{n-2};q)
+(1+q)^2{\operatorname{Cat}}{^{+\!\!+}}(A_{n-1})\end{gathered}$$
The Type D biCatalan number {#type D sec}
---------------------------
We now complete the proof of Theorem \[enum thm\] by proving the following theorem.
\[D biCat\] For $n\ge 2$, the $D_n$-biCatalan number is $$\label{D biCat formula}
{\operatorname{biCat}}(D_n)=6\cdot4^{n-2} - 2\binom{2n-4}{n-2}.$$
Since we have already established the type-A and type-B cases of Theorem \[enum thm\], Theorem \[D biCat\] is the assertion that ${\operatorname{biCat}}(D_n) = 3{\operatorname{biCat}}(B_{n-1}) - 2{\operatorname{biCat}}(A_{n-2})$. In preparation for the proof, we let $X=X(q)$ and $Y=Y(q)$ be any rational functions of $q$ and define, for each $n\ge2$, a rational function $Z_n=Z_n(q)$ given by $$Z_n={\operatorname{biCat}}(D_n;q)-X{\operatorname{biCat}}(B_{n-1};q)+Y{\operatorname{biCat}}(A_{n-2};q).$$ Combining , , and , we obtain the following recursion for $Z_n$ for $n\ge3$. $$\begin{gathered}
\label{Zn recursion formula}
Z_n=(1+q)Z_{n-1}+\sum_{i=1}^{n-3}{\operatorname{Cat}}{^{+\!\!+}}(A_i;q)\left(Z_{n-i}+(1+q)Z_{n-i-1}\right)\\
+2\bigl((1+q)^2-X(1+q)+Y\bigr){\operatorname{Cat}}{^{+\!\!+}}(A_{n-2})+4(1+q){\operatorname{Cat}}{^{+\!\!+}}(A_{n-1})\\
+2{\operatorname{Cat}}{^{+\!\!+}}(D_n;q)-2X{\operatorname{Cat}}{^{+\!\!+}}(B_{n-1};q)\end{gathered}$$
One way to obtain a formula for $q$-biCatalan numbers ${\operatorname{biCat}}(D_n;q)$ would be to find a choice of $X$ and $Y$ that makes this recursion for $Z_n$ into something that can be solved. We have thus far been unable to find a choice of $X$ and $Y$ that works. Instead, we will prove Theorem \[D biCat\] by showing that if $X(1)=3$ and $Y(1)=2$, then $Z_n(1)=0$ for all $n\ge2$. In the proof that follows, we take convenient choices of $X$ and $Y$ but delay specializing $q$ to $1$ until the end, because specializing earlier does not make the manipulations much easier, and because we hope that perhaps we are still getting closer to a formula for ${\operatorname{biCat}}(D_n;q)$.
Substituting and into , taking $X$ to be ${1+q+q^2}$, and taking $Y$ to be $2q-q^2+q^3$, we obtain $$\begin{gathered}
\label{Zn recursion formula specialized}
Z_n=(1+q)Z_{n-1}+\sum_{i=1}^{n-3}{\operatorname{Cat}}{^{+\!\!+}}(A_i;q)\left(Z_{n-i}+(1+q)Z_{n-i-1}\right)\\
+2q(1-q){\operatorname{Cat}}{^{+\!\!+}}(A_{n-2};q)+2(1-q)(1+q){\operatorname{Cat}}{^{+\!\!+}}(A_{n-1};q)\\
-2q\bigl(1+(n-1)q\bigr){\operatorname{Cat}}(A_{n-2};q)+2q{\operatorname{Cat}}(B_{n-1};q)\end{gathered}$$ We next apply to rewrite the two double-positive $q$-Catalan numbers in as a single $q$-Catalan number. $$\begin{gathered}
\label{Zn recursion formula specialized more}
Z_n=(1+q)Z_{n-1}+\sum_{i=1}^{n-3}{\operatorname{Cat}}{^{+\!\!+}}(A_i;q)\left(Z_{n-i}+(1+q)Z_{n-i-1}\right)\\
+2q(1-q){\operatorname{Cat}}(A_{n-1};q)-2q\bigl(1+(n-1)q\bigr){\operatorname{Cat}}(A_{n-2};q)+2q{\operatorname{Cat}}(B_{n-1};q)\end{gathered}$$
Finally specializing $q$ to $1$ and using the fact that ${\operatorname{Cat}}(B_{n-1})=n{\operatorname{Cat}}(A_{n-2})$ for $n\ge3$ (which is immediate from the well-known formulas for the type-A and type-B Catalan numbers), we see that $$\label{Zn recursion formula specialized even more}
Z_n(1)=2Z_{n-1}(1)+\sum_{i=1}^{n-3}{\operatorname{Cat}}{^{+\!\!+}}(A_i)\left(Z_{n-i}(1)+2Z_{n-i-1}(1)\right)$$ We easily verify that $Z_2(1)=0$, and thus we have a simple inductive proof that $Z_n(1)=0$ for all $n\ge2$. Since we chose $X$ and $Y$ to have $X(1)=3$ and $Y(1)=2$, we obtain the desired identity ${\operatorname{biCat}}(D_n) = 3{\operatorname{biCat}}(B_{n-1}) - 2{\operatorname{biCat}}(A_{n-2})$.
Type-D biNarayana numbers {#type D biNar sec}
-------------------------
Computational evidence suggests the following modest conjecture on the type-D biNarayana number ${\operatorname{biNar}}_k(D_n)$.
\[D biNar conj\] The type-D biNarayana number ${\operatorname{biNar}}_k(D_n)$ is a polynomial in $n$ (for $n\ge2$) of degree $2k$ and leading coefficient $\displaystyle\frac{4^k}{(2k)!}$.
If Conjecture \[D biNar conj\] is true, then the following table shows $\frac{(2k)!}{2^k}\cdot{\operatorname{biNar}}_k(D_n)$ for small $k$. $$\displaystyle\begin{array}{ll|ll}
k&&&\displaystyle\frac{(2k)!}{2^k}\cdot{\operatorname{biNar}}_k(D_n)\\[7pt]\hline&&\\[-9pt]
0&&&1\\[2pt]\hline&\\[-9pt]
1&&&2n^2-3n\\[2pt]\hline&\\[-9pt]
2&&&4n^4-20n^3+35n^2-7n-24\\[2pt]\hline&\\[-9pt]
3&&&8n^6-84n^5+365n^4-705n^3+212n^2+1104n-1080\\[2pt]\hline&\\[-9pt]
4&&&16n^8-288n^7+2268n^6-9576n^5+20349n^4\\
&&&\qquad\qquad\qquad\qquad\qquad\qquad-8022n^3-54133n^2+104826n-60480
\end{array}$$ The $k=1$ case is verified by Proposition \[biNar 1\], and with some effort, the $k=2$ case can be proved as well.
Acknowledgments {#sec:ack .unnumbered}
===============
Bruno Salvy’s and Paul Zimmermann’s package `GFUN` [@GFUN] was helpful in guessing a formula for the $D_n$-Catalan number. John Stembridge’s packages `posets` and `coxeter/weyl` [@StembridgePackages] were invaluable in counting antichains in the doubled root poset, in checking the distributivity of the doubled root poset, and in verifying the simpliciality of the bipartite biCambrian fan. The authors thank Christos A. Athanasiadis, Christophe Hohlweg, Richard Stanley, Salvatore Stella, and Bernd Sturmfels for helpful suggestions and questions.
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|
---
abstract: 'Word embeddings have been widely used in sentiment classification because of their efficacy for semantic representations of words. Given reviews from different domains, some existing methods for word embeddings exploit sentiment information, but they cannot produce domain-sensitive embeddings. On the other hand, some other existing methods can generate domain-sensitive word embeddings, but they cannot distinguish words with similar contexts but opposite sentiment polarity. We propose a new method for learning domain-sensitive and sentiment-aware embeddings that simultaneously capture the information of sentiment semantics and domain sensitivity of individual words. Our method can automatically determine and produce domain-common embeddings and domain-specific embeddings. The differentiation of domain-common and domain-specific words enables the advantage of data augmentation of common semantics from multiple domains and capture the varied semantics of specific words from different domains at the same time. Experimental results show that our model provides an effective way to learn domain-sensitive and sentiment-aware word embeddings which benefit sentiment classification at both sentence level and lexicon term level.'
author:
- |
Bei Shi[^1^]{}, Zihao Fu[^1^]{}, Lidong Bing[^2^]{} and Wai Lam[^1^]{}\
[[^1^]{}Department of Systems Engineering and Engineering Management\
The Chinese University of Hong Kong, Hong Kong]{}\
[[^2^]{}Tencent AI Lab, Shenzhen, China]{}\
[`{bshi,zhfu,wlam}@se.cuhk.edu.hk`]{}\
[`lyndonbing@tencent.com`]{}\
bibliography:
- 'acl2018.bib'
title: 'Learning Domain-Sensitive and Sentiment-Aware Word Embeddings[^1]'
---
Introduction
============
Sentiment classification aims to predict the sentiment polarity, such as “positive” or “negative”, over a piece of review. It has been a long-standing research topic because of its importance for many applications such as social media analysis, e-commerce, and marketing [@liu2012sentiment; @pang2008opinion]. Deep learning has brought in progress in various NLP tasks, including sentiment classification. Some researchers focus on designing RNN or CNN based models for predicting sentence level [@kim2014convolutional] or aspect level sentiment [@li2018transformation; @chen2017recurrent; @wang2016attention]. These works directly take the word embeddings pre-trained for general purpose as initial word representations and may conduct fine tuning in the training process. Some other researchers look into the problem of learning task-specific word embeddings for sentiment classification aiming at solving some limitations of applying general pre-trained word embeddings. For example, Tang et al. develop a neural network model to convey sentiment information in the word embeddings. As a result, the learned embeddings are **sentiment-aware** and able to distinguish words with similar syntactic context but opposite sentiment polarity, such as the words “good” and “bad”. In fact, sentiment information can be easily obtained or derived in large scale from some data sources (e.g., the ratings provided by users), which allows reliable learning of such sentiment-aware embeddings.
Apart from these words (e.g. “good” and “bad”) with consistent sentiment polarity in different contexts, the polarity of some sentiment words is **domain-sensitive**. For example, the word “lightweight” usually connotes a positive sentiment in the electronics domain since a lightweight device is easier to carry. In contrast, in the movie domain, the word “lightweight” usually connotes a negative opinion describing movies that do not invoke deep thoughts among the audience. This observation motivates the study of learning domain-sensitive word representations [@yang2017simple; @bollegala2015unsupervised; @bollegala2014learning]. They basically learn separate embeddings of the same word for different domains. To bridge the semantics of individual embedding spaces, they select a subset of words that are likely to be domain-insensitive and align the dimensions of their embeddings. However, the sentiment information is not exploited in these methods although they intend to tackle the task of sentiment classification.
In this paper, we aim at learning word embeddings that are both domain-sensitive and sentiment-aware. Our proposed method can jointly model the sentiment semantics and domain specificity of words, expecting the learned embeddings to achieve superior performance for the task of sentiment classification. Specifically, our method can automatically determine and produce domain-common embeddings and domain-specific embeddings. Domain-common embeddings represent the fact that the semantics of a word including its sentiment and meaning in different domains are very similar. For example, the words “good” and “interesting” are usually domain-common and convey consistent semantic meanings and positive sentiments in different domains. Thus, they should have similar embeddings across domains. On the other hand, domain-specific word embeddings represent the fact that the sentiments or meanings across domains are different. For example, the word “lightweight” represents different sentiment polarities in the electronics domain and the movie domain. Moreover, some polysemous words have different meanings in different domains. For example, the term “apple” refers to the famous technology company in the electronics domain or a kind of fruit in the food domain.
Our model exploits the information of sentiment labels and context words to distinguish domain-common and domain-specific words. If a word has similar sentiments and contexts across domains, it indicates that the word has common semantics in these domains, and thus it is treated as domain-common. Otherwise, the word is considered as domain-specific. The learning of domain-common embeddings can allow the advantage of data augmentation of common semantics of multiple domains, and meanwhile, domain-specific embeddings allow us to capture the varied semantics of specific words in different domains. Specifically, for each word in the vocabulary, we design a distribution to depict the probability of the word being domain-common. The inference of the probability distribution is conducted based on the observed sentiments and contexts. As mentioned above, we also exploit the information of sentiment labels for the learning of word embeddings that can distinguish words with similar syntactic context but opposite sentiment polarity.
To demonstrate the advantages of our domain-sensitive and sentiment-aware word embeddings, we conduct experiments on four domains, including books, DVSs, electronics, and kitchen appliances. The experimental results show that our model can outperform the state-of-the-art models on the task of sentence level sentiment classification. Moreover, we conduct lexicon term sentiment classification in two common sentiment lexicon sets to evaluate the effectiveness of our sentiment-aware embeddings learned from multiple domains, and it shows that our model outperforms the state-of-the-art models on most domains.
Related Works
=============
Traditional vector space models encode individual words using the one-hot representation, namely, a high-dimensional vector with all zeroes except in one component corresponding to that word [@baeza1999modern]. Such representations suffer from the curse of dimensionality, as there are many components in these vectors due to the vocabulary size. Another drawback is that semantic relatedness of words cannot be modeled using such representations. To address these shortcomings, Rumelhart et al. propose to use distributed word representation instead, called word embeddings. Several techniques for generating such representations have been investigated. For example, Bengio et al. propose a neural network architecture for this purpose [@bengio2003neural; @bengio2009learning]. Later, Mikolov et al. propose two methods that are considerably more efficient, namely skip-gram and CBOW. This work has made it possible to learn word embeddings from large data sets, which has led to the current popularity of word embeddings. Word embedding models have been applied to many tasks, such as named entity recognition [@turian2010word], word sense disambiguation [@collobert2011natural; @iacobacci2016embeddings; @Zhang.Hasan.3; @vachik_zhang_arxiv], parsing [@roth2016neural], and document classification [@tang2014building; @tang2014learning; @shi2017jointly].
Sentiment classification has been a long-standing research topic [@liu2012sentiment; @pang2008opinion; @chen2017recurrent; @moraes2013document]. Given a review, the task aims at predicting the sentiment polarity on the sentence level [@kim2014convolutional] or the aspect level [@li2018transformation; @chen2017recurrent]. Supervised learning algorithms have been widely used in sentiment classification [@pang2002thumbs]. People usually use different expressions of sentiment semantics in different domains. Due to the mismatch between domain-specific words, a sentiment classifier trained in one domain may not work well when it is directly applied to other domains. Thus cross-domain sentiment classification algorithms have been explored [@pan2010cross; @li2009knowledge; @glorot2011domain]. These works usually find common feature spaces across domains and then share learned parameters from the source domain to the target domain. For example, Pan et al. propose a spectral feature alignment algorithm to align words from different domains into unified clusters. Then the clusters can be used to reduce the gap between words of the two domains, which can be used to train sentiment classifiers in the target domain. Compared with the above works, our model focuses on learning both domain-common and domain-specific embeddings given reviews from all the domains instead of only transferring the common semantics from the source domain to the target domain.
Some researchers have proposed some methods to learn task-specific word embeddings for sentiment classification [@tang2014building; @tang2014learning]. Tang et al. propose a model named SSWE to learn sentiment-aware embedding via incorporating sentiment polarity of texts in the loss functions of neural networks. Without the consideration of varied semantics of domain-specific words in different domains, their model cannot learn sentiment-aware embeddings across multiple domains. Some works have been proposed to learn word representations considering multiple domains [@yang2017simple; @bach2016cross; @bollegala2015unsupervised]. Most of them learn separate embeddings of the same word for different domains. Then they choose pivot words according to frequency-based statistical measures to bridge the semantics of individual embedding spaces. A regularization formulation enforcing that word representations of pivot words should be similar in different domains is added into the original word embedding framework. For example, Yang et al. use S[ø]{}rensen-Dice coefficient [@sorensen1948method] for detecting pivot words and learn word representations across domains. Even though they evaluate the model via the task of sentiment classification, sentiment information associated with the reviews are not considered in the learned embeddings. Moreover, the selection of pivot words is according to frequency-based statistical measures in the above works. In our model, the domain-common words are jointly determined by sentiment information and context words.
Model Description
=================
We propose a new model, named DSE, for learning **D**omain-sensitive and **S**entiment-aware word **E**mbeddings. For presentation clarity, we describe DSE based on two domains. Note that it can be easily extended for more than two domains, and we remark on how to extend near the end of this section.
Design of Embeddings
--------------------
We assume that the input consists of text reviews of two domains, namely $\mathcal{D}^p$ and $\mathcal{D}^q$. Each review $r$ in $\mathcal{D}^p$ and $\mathcal{D}^q$ is associated with a sentiment label $y$ which can take on the value of $1$ and $0$ denoting that the sentiment of the review is positive and negative respectively.
In our DSE model, each word $w$ in the whole vocabulary $\Lambda$ is associated with a domain-common vector $U_w^c$ and two domain-specific vectors, namely $U_w^p$ specific to the domain $p$ and $U_w^q$ specific to the domain $q$. The dimension of these vectors is $d$. The design of $U_w^c$, $U_w^p$ and $U_w^q$ reflects one characteristic of our model: allowing a word to have different semantics across different domains. The semantic of each word includes not only the semantic meaning but also the sentiment orientation of the word. If the semantic of $w$ is consistent in the domains $p$ and $q$, we use the vector $U_w^c$ for both domains. Otherwise, $w$ is represented by $U_w^p$ and $U_w^q$ for $p$ and $q$ respectively.
In traditional cross-domain word embedding methods [@yang2017simple; @bollegala2015unsupervised; @bollegala2016cross], each word is represented by different vectors in different domains without differentiation of domain-common and domain-specific words. In contrast to these methods, for each word $w$, we use a latent variable $z_w$ to depict its domain commonality. When $z_w = 1$, it means that $w$ is common in both domains. Otherwise, $w$ is specific to the domain $p$ or the domain $q$.
In the standard skip-gram model [@mikolov2013distributed], the probability of predicting the context words is only affected by the relatedness with the target words. In our DSE model, predicting the context words also depends on the domain-commonality of the target word, i.e $z_w$. For example, assume that there are two domains, e.g. the electronics domain and the movie domain. If $z_w = 1$, it indicates a high probability of generating some domain-common words such as “good”, “bad” or “satisfied”. Otherwise, the domain-specific words are more likely to be generated such as “reliable”, “cheap” or “compacts” for the electronics domain. For a word $w$, we assume that the probability of predicting the context word $w_t$ is formulated as follows: $$\label{eq:context}
p(w_t | w) = \sum_{k \in{\{0, 1\}}}p(w_t|w, z_w=k)p(z_w=k)$$
If $w$ is a domain-common word without differentiating $p$ and $q$, the probability of predicting $w_t$ can be defined as: $$\label{eq:w_t}
p(w_t|w, z_w=1)=\frac{\exp(U_w^c \cdot V_{w_t})}{\sum_{w^{\prime}\in\Lambda}\exp(U_w^c \cdot V_{w^{\prime}})}$$ where $\Lambda$ is the whole vocabulary and $V_{w^{\prime}}$ is the output vector of the word $w^{\prime}$.
If $w$ is a domain-specific word, the probability of $p(w_t | w, z_w=0)$ is specific to the occurrence of $w$ in $\mathcal{D}^p$ or $\mathcal{D}^q$. For individual training instances, the occurrences of $w$ in $\mathcal{D}^p$ or $\mathcal{D}^q$ have been established. Then the probability of $p(w_t|w, z_w=0)$ can be defined as follows: $$\label{eq:w_t_z}
\begin{split}
p(w_t|w, z_w=0)=
\begin{cases}
\frac{\exp(U_w^p \cdot
V_{w_t})}{\sum_{w^{\prime}\in\Lambda}\exp(U_w^p \cdot
V_{w^{\prime}})}, \text{if } w \in \mathcal{D}^p \\
\\
\frac{\exp(U_w^q \cdot
V_{w_t})}{\sum_{w^{\prime}\in\Lambda}\exp(U_w^q \cdot
V_{w^{\prime}})}, \text{if } w \in \mathcal{D}^q \\
\end{cases}
\end{split}$$
Exploiting Sentiment Information
--------------------------------
In our DSE model, the prediction of review sentiment depends on not only the text information but also the domain-commonality. For example, the domain-common word “good” has high probability to be positive in different reviews across multiple domains. However, for the word “lightweight”, it would be positive in the electronics domain, but negative in the movie domain. We define the polarity $y_w$ of each word $w$ to be consistent with the sentiment label of the review: if we observe that a review is associated with a positive label, the words in the review are associated with a positive label too. Then, the probability of predicting the sentiment for the word $w$ can be defined as: $$\label{eq:senti}
p(y_w | w) = \sum_{k \in{\{0, 1\}}}p(y_w|w, z_w=k)p(z_w=k)$$ If $z_w = 1$, the word $w$ is a domain-common word. The probability $p(y_w = 1|w, z_w=1)$ can be defined as: $$p(y_w = 1|w, z_w=1) = \sigma(U_w^c \cdot \mathbf{s})$$ where $\sigma(\cdot)$ is the sigmoid function and the vector $\mathbf{s}$ with dimension $d$ represents the boundary of the sentiment. Moreover, we have: $$p(y_w = 0|w, z_w=1) = 1 - p(y_w=1|w, z_w=1)$$
If $w$ is a domain-specific word, similarly, the probability $p(y_w=1 | w, z_w = 0)$ is defined as: $$\begin{split}
p(y_w=1|w, z_w=0) =
\begin{cases}
\sigma(U_w^p \cdot \mathbf{s}) & \text{if } w \in
\mathcal{D}^p \\
\sigma(U_w^q \cdot \mathbf{s}) & \text{if } w \in \mathcal{D}^q \\
\end{cases}
\end{split}$$
Inference Algorithm
-------------------
Initialize $U_w^c$, $U_w^p$, $U_w^q$, $V$, $\mathbf{s}$, $p(z_w)$ Sample negative instances from the distribution P. Update $p(z_w|w, c_w, y_w)$ by Eq. \[eq:post\] and Eq. \[eq:sim\] respectively. Update $p(z_w)$ using Eq. \[eq:z\_w\] Update $U_w^c$, $U_w^p$, $U_w^q$, $V$, $\mathbf{s}$ via Maximizing Eq. \[eq:qu\]
We need an inference method that can learn, given $\mathcal{D}^p$ and $\mathcal{D}^q$, the values of the model parameters, namely, the domain-common embedding $U_w^c$, and the domain-specific embeddings $U_w^p$ and $U_w^q$, as well as the domain-commonality distribution $p(z_w)$ for each word $w$. Our inference method combines the Expectation-Maximization (EM) method with a negative sampling scheme. It is summarized in Algorithm \[alg:em\]. In the E-step, we use the Bayes rule to evaluate the posterior distribution of $z_w$ for each word and derive the objective function. In the M-step, we maximize the objective function with the gradient descent method and update the corresponding embeddings $U_w^c$, $U_w^p$ and $U_w^q$.
With the input of $\mathcal{D}^p$ and $\mathcal{D}^q$, the likelihood function of the whole training set is: $$\label{eq:obj}
\mathcal{L} = \mathcal{L}^p + \mathcal{L}^q$$ where $\mathcal{L}^p$ and $\mathcal{L}^q$ are the likelihood of $\mathcal{D}^p$ and $\mathcal{D}^q$ respectively.
For each review $r$ from $\mathcal{D}^p$, to learn domain-specific and sentiment-aware embeddings, we wish to predict the sentiment label and context words together. Therefore, the likelihood function is defined as follows: $$\mathcal{L}^p = \sum_{r \in \mathcal{D}^p}\sum_{w \in r}\log p(y_w, c_w|w)$$ where $y_w$ is the sentiment label and $c_w$ is the set of context words of $w$. For the simplification of the model, we assume that the sentiment label $y_w$ and the context words $c_w$ of the word $w$ are conditionally dependent. Then the likelihood $\mathcal{L}^p$ can be rewritten as: $$\begin{split}
\mathcal{L}^p = &\sum_{r \in \mathcal{D}^p}\sum_{w \in
r}\sum_{w_t \in c_w}\log p(w_t|w) +\\
&\sum_{r \in \mathcal{D}^p}\sum_{w \in
r}\log p(y_w|w)
\end{split}$$ where $p(w_t|w)$ and $p(y_w|w)$ are defined in Eq. \[eq:context\] and Eq. \[eq:senti\] respectively. The likelihood of the reviews from $\mathcal{D}^q$, i.e $\mathcal{L}^q$, is defined similarly.
For each word $w$ in the review $r$, in the E-step, the posterior probability of $z_w$ given $c_w$ and $y_w$ is: $$\small
\label{eq:post}
\begin{split}
&p(z_w=k|w, c_w, y_w) = \\
&\frac{p(z_w=k)p(y_w|w, z_w=k)\prod \limits_{w_t \in
c_w}p(w_t|w, z_w=k)}{\sum \limits_{k^{\prime}
\in \{0, 1\}}p(z_w=k^{\prime})p(y_w|w,z_w=k^{\prime})\prod \limits_{w_t \in
c_w}p(w_t|w, z_w=k^{\prime})}
\end{split}
\normalsize$$ In the M-step, given the posterior distribution of $z_w$ in Eq. \[eq:post\], the goal is to maxmize the following Q function: $$\label{eq:q}
\begin{split}
\mathbf{Q} =& \sum \limits_{r\in\{\mathcal{D}^p, \mathcal{D}^q\}}\sum
\limits_{w \in r}\sum_{z_w} p(z_w|w, y_w, w_{t+j})\\
&\times
\log (p(z_w) p(c_w, y|z, w_t))\\
= &\sum \limits_{r\in\{\mathcal{D}^p, \mathcal{D}^q\}}\sum
\limits_{w \in r}\sum_{z_w} p(z_w|w, y_w, c_w) \\
& [\log p(z_w) + \log(y_w|z, w) + \\& \sum_{w_t \in c_w}\log p(w_t|z_w, w)] \\
\end{split}
\normalsize$$
Using the Lagrange multiplier, we can obtain the update rule of $p(z_w)$, satisfying the normalization constraints that $\sum_{z_w \in {0, 1}} p(z_w)=1$ for each word $w$: $$\label{eq:z_w}
p(z_w) = \frac{\sum_{r \in \{\mathcal{D}^p, \mathcal{D}^q\}}\sum_{w
\in r}p(z_w|w, y_w, c_w)}{\sum_{r \in \{\mathcal{D}^p,
\mathcal{D}^q\}} n(w, r)}$$ where $n(w, r)$ is the number of occurrence of the word $w$ in the review $r$.
To obtain $U_w^c$, $U_w^p$ and $U_w^q$, we collect the related items in Eq. \[eq:q\] as follows: $$\label{eq:qu}
\begin{split}
\mathbf{Q_U} = \sum \limits_{r\in\{\mathcal{D}^p, \mathcal{D}^q\}}\sum
\limits_{w \in r}\sum_{z_w} p(z_w|w, y_w, w_{t+j}) \\
[\log(y_w|z_w, w) + \sum_{w_t \in c_w}\log p(w_t|z_w,
w)] \\
\end{split}$$
Note that computing the value $p(w_t|w, z_w)$ based on Eq. \[eq:w\_t\] and Eq. \[eq:w\_t\_z\] is not feasible in practice, given that the computation cost is proportional to the size of $\Lambda$. However, similar to the skip-gram model, we can rely on negative sampling to address this issue. Therefore we estimate the probability of predicting the context word $p(w_{t}|w,z_w=1)$ as follows: $$\label{eq:sim}
\begin{split}
\log p(w_{t}|w, z_w=1) & \propto \log \sigma(U_{w}^c\cdot V_{w_t})\\
&+ \sum_{i=1}^{n}\mathbf{E}_{w_i\sim P}[\log \sigma(-U_{w}^c\cdot V_{w_i})]
\end{split}$$ where $w_i$ is a negative instance which is sampled from the word distribution $P(.)$. Mikolov et al. have investigated many choices for $P(w)$ and found that the best $P(w)$ is equal to the unigram distribution $\textit{Unigram}(w)$ raised to the $3/4rd$ power. We adopt the same setting. The probability $p(w_{t}|w,z_w=0)$ in Eq. \[eq:w\_t\_z\] can be approximated in a similar manner.
After the substitution of $p(w_t|w, z_w)$, we use the Stochastic Gradient Descent method to maximize Eq. \[eq:qu\], and obtain the update of $U_w^c$, $U_w^p$ and $U_w^q$.
More Discussions
----------------
In our model, for simplifying the inference algorithm and saving the computational cost, we assume that the target word $w_t$ in the context and the sentiment label $y_w$ of the word $w$ are conditionally independent. Such technique has also been used in other popular models such as the bi-gram language model. Otherwise, we need to consider the term $p(w_t|w, y_w)$, which complicates the inference algorithm.
We define the formulation of the term $p(w_t|w, z)$ to be similar to the original skip-gram model instead of the CBOW model. The CBOW model averages the context words to predict the target word. The skip-gram model uses pairwise training examples which are much easier to integrate with sentiment information.
Note that our model can be easily extended to more than two domains. Similarly, we use a domain-specific vector for each word in each domain and each word is also associated with a domain-common vector. We just need to extend the probability distribution of $z_w$ from Bernoulli distribution to Multinomial distribution according to the number of domains.
Experiment
==========
-------------------- -------------------------------- -------------------------------- -------------------------------- -------------------------------- ---------------- ------------------------ -------------------------------- -------------------------------- -------------------------------- -------------------------------- -------------------------------- --------------------------------
*Acc.* F1 *Acc.* F1 *Acc.* F1 *Acc.* F1 *Acc.* F1 *Acc.* F1
[BOW]{} 0.680 0.653 0.738 0.720 0.734 0.725 0.705 0.685 0.706 0.689 0.739 0.715
[EmbeddingP]{} 0.753 0.740 0.752 0.745 0.742 0.741 0.740 0.746 0.707 0.702 0.761 0.760
[EmbeddingQ]{} 0.736 0.732 0.697 0.697 0.706 0.701 0.762 0.759 0.758 0.759 0.783 0.780
[EmbeddingCat]{} 0.769 0.731 0.768 0.763 0.763 0.763 0.787 0.773 0.770 0.770 0.807 0.803
[EmbeddingAll]{} 0.769 0.759 0.765 0.740 0.775 0.767 0.783 0.779 0.779 0.776 0.819 0.815
[Yang]{} 0.767 0.752 0.775 0.766 0.760 0.755 0.791 0.785 0.762 0.760 0.805 0.804
[SSWE]{} 0.783 0.772 0.791 0.780 [**0.801**]{} 0.792 0.825 0.815 0.795 0.790 0.835 0.824
[$\text{DSE}_c$]{} 0.773 0.750 0.783 0.781 0.775 0.773 0.797 0.792 0.784 0.776 0.806 0.800
[$\text{DSE}_w$]{} [**0.794**]{}$^{\dag\natural}$ [**0.793**]{}$^{\dag\natural}$ [**0.806**]{}$^{\dag\natural}$ [**0.802**]{}$^{\dag\natural}$ 0.797$^{\dag}$ [**0.793**]{}$^{\dag}$ [**0.843**]{}$^{\dag\natural}$ [**0.832**]{}$^{\dag\natural}$ [**0.829**]{}$^{\dag\natural}$ [**0.827**]{}$^{\dag\natural}$ [**0.856**]{}$^{\dag\natural}$ [**0.853**]{}$^{\dag\natural}$
-------------------- -------------------------------- -------------------------------- -------------------------------- -------------------------------- ---------------- ------------------------ -------------------------------- -------------------------------- -------------------------------- -------------------------------- -------------------------------- --------------------------------
Experimental Setup
------------------
We conducted experiments on the Amazon product reviews collected by Blitzer et al. . We use four product categories: books (**B**), DVDs (**D**), electronic items (**E**), and kitchen appliances (**K**). A category corresponds to a domain. For each domain, there are 17,457 unlabeled reviews on average associated with rating scores from 1.0 to 5.0 for each domain. We use unlabeled reviews with rating score higher than 3.0 as positive reviews and unlabeled reviews with rating score lower than 3.0 as negative reviews for embedding learning. We first remove reviews whose length is less than 5 words. We also remove punctuations and the stop words. We also stem each word to its root form using Porter Stemmer [@porter1980algorithm]. Note that this review data is used for embedding learning, and the learned embeddings are used as feature vectors of words to conduct the experiments in the later two subsections.
Given the reviews from two domains, namely, $\mathcal{D}^p$ and $\mathcal{D}^q$, we compare our results with the following baselines and state-of-the-art methods:
SSWE
: The SSWE model[^2] proposed by Tang et al. can learn sentiment-aware word embeddings from tweets. We employ this model on the combined reviews from $\mathcal{D}^p$ and $\mathcal{D}^q$ and then obtain the embeddings.
Yang’s Work
: Yang et al. have proposed a method[^3] to learn domain-sensitive word embeddings. They choose pivot words and add a regularization item into the original skip-gram objective function enforcing that word representations of pivot words for the source and target domains should be similar. The method trains the embeddings of the source domain first and then fixes the learned embedding to train the embedding of the target domain. Therefore, the learned embedding of the target domain benefits from the source domain. We denote the method as Yang in short.
EmbeddingAll
: We learn word embeddings from the combined unlabeled review data of $\mathcal{D}^p$ and $\mathcal{D}^q$ using the skip-gram method [@mikolov2013distributed].
EmbeddingCat
: We learn word embeddings from the unlabeled reviews of $\mathcal{D}^p$ and $\mathcal{D}^q$ respectively. To represent a word for review sentiment classification, we concatenate its learned word embeddings from the two domains.
EmbeddingP and EmbeddingQ
: In EmbeddingP, we use the original skip-gram method [@mikolov2013distributed] to learn word embeddings only from the unlabeled reviews of $\mathcal{D}^p$. Similarly, we only adopt the unlabeled reviews from $\mathcal{D}^q$ to learn embeddings in EmbeddingQ.
BOW
: We use the traditional bag of words model to represent each review in the training data.
For our DSE model, we have two variants to represent each word. The first variant $\text{DSE}_c$ represents each word via concatenating the domain-common vector and the domain-specific vector. The second variant $\text{DSE}_w$ concatenates domain-common word embeddings and domain-specific word embeddings by considering the domain-commonality distribution $p(z_w)$. For individual review instances, the occurrences of w in $\mathcal{D}_p$ or $\mathcal{D}_q$ have been established. The representation of $w$ is specific to the occurrence of $w$ in $\mathcal{D}_p$ or $\mathcal{D}_q$. Specifically, each word $w$ can be represented as follows: $$\begin{split}
U_w =
\begin{cases}
\text{if
} w \in \mathcal{D}^p \\
\quad U_w^c \times p(z_w) \oplus U_w^p \times (1.0-p(z_w)) \\
\text{if
} w \in \mathcal{D}^q \\
\quad U_w^c \times p(z_w) \oplus U_w^q \times (1.0-p(z_w))\\
\end{cases}
\end{split}$$ where $\oplus$ denotes the concatenation operator.
For all word embedding methods, we set the dimension to 200. For the skip-gram based methods, we sample 5 negative instances and the size of the windows for each target word is 3. For our DSE model, the number of iterations for the whole reviews is 100 and the learning rate is set to 1.0.
------------------ --------------- --------------- --------------- --------------- --------------- --------------- --------------- --------------- --------------- --------------- --------------- ---------------
HL MPQA HL MPQA HL MPQA HL MPQA HL MPQA HL MPQA
[EmbeddingP]{} 0.740 0.733 0.742 0.734 0.747 0.735 0.744 0.701 0.745 0.709 0.628 0.574
[EmbeddingQ]{} 0.743 0.701 0.627 0.573 0.464 0.453 0.621 0.577 0.462 0.450 0.465 0.453
[EmbeddingCat]{} 0.780 0.772 0.773 0.756 0.772 0.751 0.744 0.728 0.755 0.702 0.683 0.639
[EmbeddingAll]{} 0.777 0.769 0.773 0.730 0.762 0.760 0.712 0.707 0.749 0.724 0.670 0.658
[Yang]{} 0.780 0.775 0.789 0.762 0.781 0.770 0.762 0.736 0.756 0.713 0.634 0.614
[SSWE]{} [**0.816**]{} [**0.801**]{} 0.831 0.817 0.822 [**0.808**]{} [**0.826**]{} 0.785 0.784 0.772 0.707 0.659
[$\text{DSE}$]{} 0.802 0.788 [**0.833**]{} [**0.828**]{} [**0.832**]{} 0.799 0.804 [**0.797**]{} [**0.796**]{} [**0.786**]{} [**0.725**]{} [**0.683**]{}
------------------ --------------- --------------- --------------- --------------- --------------- --------------- --------------- --------------- --------------- --------------- --------------- ---------------
Review Sentiment Classification
-------------------------------
For the task of review sentiment classification, we use 1000 positive and 1000 negative sentiment reviews labeled by Blitzer et al. for each domain to conduct experiments. We randomly select 800 positive and 800 negative labeled reviews from each domain as training data, and the remaining 200 positive and 200 negative labeled reviews as testing data. We use the SVM classifier [@fan2008liblinear] with linear kernel to train on the training reviews for each domain, with each review represented as the average vector of its word embeddings.
We use two metrics to evaluate the performance of sentiment classification. One is the standard accuracy metric. The other one is Macro-F1, which is the average of F1 scores for both positive and negative reviews.
We conduct multiple trials by selecting every possible two domains from books (**B**), DVDs (**D**), electronic items (**E**) and kitchen appliances (**K**). We use the average of the results of each two domains. The experimental results are shown in Table \[table:sentiment\].
From Table \[table:sentiment\], we can see that compared with other baseline methods, our $\text{DSE}_w$ model can achieve the best performance of sentiment classification across most combinations of the four domains. Our statistical t-tests for most of the combinations of domains show that the improvement of our $\text{DSE}_w$ model over Yang and SSWE is statistically significant respectively (p-value $< 0.05$) at 95% confidence level. It shows that our method can capture the domain-commonality and sentiment information at the same time.
Even though both of the SSWE model and our DSE model can learn sentiment-aware word embeddings, our $\text{DSE}_w$ model can outperform SSWE. It demonstrates that compared with general sentiment-aware embeddings, our learned domain-common and domain-specific word embeddings can capture semantic variations of words across multiple domains.
Compared with the method of Yang which learns cross-domain embeddings, our $\text{DSE}_w$ model can achieve better performance. It is because we exploit sentiment information to distinguish domain-common and domain-specific words during the embedding learning process. The sentiment information can also help the model distinguish the words which have similar contexts but different sentiments.
Compared with EmbeddingP and EmbeddingQ, the methods of EmbeddingAll and EmbeddingCat can achieve better performance. The reason is that the data augmentation from other domains helps sentiment classification in the original domain. Our DSE model also benefits from such kind of data augmentation with the use of reviews from $\mathcal{D}^p$ and $\mathcal{D}^q$.
We observe that our $\text{DSE}_w$ variant performs better than the variant of $\text{DSE}_c$. Compared with $\text{DSE}_c$, our $\text{DSE}_w$ variant adds the item of $p(z_w)$ as the weight to combine domain-common embeddings and domain-specific embeddings. It shows that the domain-commonality distribution in our DSE model, i.e $p(w_z)$, can effectively model the domain-sensitive information of each word and help review sentiment classification.
Lexicon Term Sentiment Classification
-------------------------------------
To further evaluate the quality of the sentiment semantics of the learned word embeddings, we also conduct lexicon term sentiment classification on two popular sentiment lexicons, namely **HL** [@hu2004mining] and **MPQA** [@wilson2005recognizing]. The words with neutral sentiment and phrases are removed. The statistics of **HL** and **MPQA** are shown in Table \[table:lexicon\].
Lexicon Positive Negative Total
--------- ---------- ---------- -------
HL 1,331 2,647 3,978
MPQA 1,932 2,817 3,075
: Statistics of the sentiment lexicons.[]{data-label="table:lexicon"}
We conduct multiple trials by selecting every possible two domains from books (**B**), DVDs (**D**), electronics items (**E**) and kitchen appliances (**K**). For each trial, we learn the word embeddings. For our DSE model, we only use the domain-common part to represent each word because the lexicons are usually not associated with a particular domain. For each lexicon, we select 80% to train the SVM classifier with linear kernel and the remaining 20% to test the performance. The learned embedding is treated as the feature vector for the lexicon term. We conduct 5-fold cross validation on all the lexicons. The evaluation metric is Macro-F1 of positive and negative lexicons.
Table \[table:lexicon\_result\] shows the experimental results of lexicon term sentiment classification. Our DSE method can achieve competitive performance among all the methods. Compared with SSWE, our DSE is still competitive because both of them consider the sentiment information in the embeddings. Our DSE model outperforms other methods which do not consider sentiments such as Yang, EmbeddingCat and EmbeddingAll. Note that the advantage of domain-sensitive embeddings would be insufficient for this task because the sentiment lexicons are not domain-specific.
Case Study
==========
Term Domain $p(z=1)$ Sample Reviews
------ ----------- ----------- ----------------
B & D 0.999
B & E 0.404
B & K 0.241
D & E 0.380
D & K 0.013
E & K 0.696
[B & E]{} [0.297]{}
B & D 0.760
B & E 0.603
B & K 0.628
D & E 0.804
D & K 0.582
E & K 0.805
Table \[table:prob\] shows the probabilities of “lightweight”, “die”, “mysterious”, and “great” to be domain-common for different domain combinations. For “lightweight”, its domain-common probability for the books domain and the DVDs domain (“B & D”) is quite high, i.e. $p(z=1)=0.999$, and the review examples in the last column show that the word “lightweight” expresses the meaning of lacking depth of content in books or movies. Note that most reviews of DVDs are about movies. In the electronics domain and the kitchen appliances domain (“E & K”), “lightweight” means light material or weighing less than average, thus the domain-common probability for these two domains is also high, 0.696. In contrast, for the other combinations, the probability of “lightweight” to be domain-common is much smaller, which indicates that the meaning of “lightweight” varies. Similarly, “die” in the domains of electronics and kitchen appliances (“E & K”) means that something does not work any more, thus, we have $p(z=1)=0.712$. While for the books domain, it conveys meaning that somebody passed away in some stories. The word “mysterious” conveys a positive sentiment in the books domain, indicating how wonderful a story is, but it conveys a negative sentiment in the electronics domain typically describing that a product breaks down unpredictably. Thus, its domain-common probability is small. The last example is the word “great”, and it usually has positive sentiment in all domains, thus has large values of $p(z=1)$ for all domain combinations.
Conclusions
===========
We propose a new method of learning domain-sensitive and sentiment-aware word embeddings. Compared with existing sentiment-aware embeddings, our model can distinguish domain-common and domain-specific words with the consideration of varied semantics across multiple domains. Compared with existing domain-sensitive methods, our model detects domain-common words according to not only similar context words but also sentiment information. Moreover, our learned embeddings considering sentiment information can distinguish words with similar syntactic context but opposite sentiment polarity. We have conducted experiments on two downstream sentiment classification tasks, namely review sentiment classification and lexicon term sentiment classification. The experimental results demonstrate the advantages of our approach.
[^1]: This work was partially done when Bei Shi was an intern at Tencent AI Lab. This project is substantially supported by a grant from the Research Grant Council of the Hong Kong Special Administrative Region, China (Project Code: 14203414).
[^2]: We use the implementation from <https://github.com/attardi/deepnl/wiki/Sentiment-Specific-Word-Embeddings>.
[^3]: We use the implementation from <http://statnlp.org/research/lr/>.
|
---
abstract: 'The mechanism which discriminates the pattern classes at the same $\lambda$, is found. It is closely related to the structure of the rule table and expressed by the numbers of the rules which break the strings of the quiescent states. It is shown that for the N-neighbor and K-state cellular automata, the class I, class II, class III and class IV patterns coexist at least in the range, $\frac{1}{K} \le \lambda \le 1-\frac{1}{K} $. The mechanism is studied quantitatively by introducing a new parameter $F$, which we call quiescent string dominance parameter. It is taken to be orthogonal to $\lambda$. Using the parameter F and $\lambda$, the rule tables of one dimensional 5-neighbor and 4-state cellular automata are classified. The distribution of the four pattern classes in ($\lambda$,F) plane shows that the rule tables of class III pattern class are distributed in larger $F$ region, while those of class II and class I pattern classes are found in the smaller $F$ region and the class IV behaviors are observed in the overlap region between them. These distributions are almost independent of $\lambda$ at least in the range $0.25 \leq \lambda \leq 0.75$, namely the overlapping region in $F$, where the class III and class II patterns coexist, has quite gentle $\lambda$ dependence in this $\lambda$ region. Therefore the relation between the pattern classes and the $\lambda$ parameter is not observed.'
author:
- Sunao Sakai
- Megumi Kanno
- Yukari Saito
title: ' Quiescent String Dominance Parameter F and Classification of One-Dimensional Cellular Automata'
---
Introduction
============
Cellular automata (CA) has been one of the most studied fields in the research of complex systems. Various patterns has been generated by choosing the rule tables. Wolfram[@wolfram] has classified these patterns into four rough categories: class I (homogeneous), class II (periodic), class III (chaos) and class IV (edge of chaos). The class IV patterns have been the most interesting target for the study of CA, because it provides us with an example of the self-organization in a simple system and it is argued that the possibility of computation is realized by the complexity at the edge of chaos[@wolframs; @langton].\
Much more detailed classifications of CA, have been carried out mainly for the elementary cellular automata (3-neighbor and 2-state CA)[@hanson; @wuensche], in which the patterns are studied quite accurately for each rule table. And the classification of the rule tables are studied by introducing some parameters[@binder; @wuensche; @oliveira]. However, in this case, there are some confusions in the classification of the class IV CA. The result on the so to speak $\rho=1/2$ task from Packard[@packard] have been different from that of Mitchell, Crutchfield and Hraber[@mitchell]. In this case, the number of the independent rule tables are so small to treat them statistically and the symmetry of the interchange of the states ”0” and ”1” make the classifications of the the pattern classes more delicate than those of other CA with $K \geq 3$. Therefore, it may be worth to start with other models, in order to find the general properties of the CA.\
However the number of rule tables in N-neighbor and K-state cellular automata CA(N,K) grows like $K^{K^{N}}$. Therefore except for a few smallest combinations of the $N$ and $K$, the numbers of the rule tables become so large that studies of the CA dynamics for all rule tables are impossible even with the fastest supercomputers. Therefore it is important to find a set of parameters by which the pattern classes could be classified , even if it is a qualitative one.\
Langton has introduced $\lambda$ parameter and argued that as $\lambda$ increases the pattern class changes from class I to class II and then to class III. And class IV behavior is observed between class II and class III pattern classes[@langton0; @langton; @langton2]. The $\lambda$ parameter represents rough behavior of CA in the rule table space, but finally does not sufficiently classify the quantitative behavior of CA. It is well known that different pattern classes coexist at the same $\lambda$. Which of these pattern classes is chosen, depends on the random numbers in generating the rule tables. The reason or mechanism for this is not yet known; we have no way to control the pattern classes at fixed $\lambda$. And the transitions from a periodic to chaotic pattern classes are observed in a rather wide range of $\lambda$. In Ref.[@langton2], a schematic phase-diagram was sketched. However a vertical axis was not specified. Therefore, it has been thought that new parameters are necessary to arrive at more quantitative understandings of the rule table space of the CA [^1].\
In this article, we will clarify the mechanism which discriminate the class I, class II, class III and class IV pattern classes at fixed $\lambda$. It is closely related to the structure of the rule tables; numbers of rules which breaks strings of quiescent states. It is studied quantitatively by introducing a new parameter $F$, which we will call quiescent string dominance parameter. It is taken to be orthogonal to $\lambda$. In the region $1/K \le \lambda \le 1-1/K$, the maximum of $F$ corresponds to class III rule tables while minimum of $F$, to class II or class I rule tables. Therefore the transition of the pattern classes takes place somewhere between these two limits without fail. By the determination of the region of $F$, where the change of the pattern classes takes place, we could obtain the phase diagram in ($\lambda$,F) plane, and classify the rule table space.\
The determination of the phase diagram is carried out for CA(5,4). It is found that the rule tables are not separated by a sharp boundaries but they are represented by probability densities. Therefore we define the equilibrium points of two phases where the two probability densities of the pattern classes become equal, and define the transition region where the probability densities of the both pattern classes are not too much different from each other. By using the equilibrium points and the transition region, we draw a phase diagram in ($\lambda$,F) plane.\
It is found that $\lambda$ dependences of the equilibrium points and transition region are very gentle, and they continued to be found at least over the range $0.125 \le \lambda
\le 0.75$. It means that all the four pattern classes do coexist over the wide range in $\lambda$. Our results for the distributions of these pattern classes in ($\lambda,F$) plane do not support the well known relation between the pattern classes and the $\lambda$ parameter proposed in the Ref.[@langton]. It will be shown that the results there, are due to the methods to generate the rule tables with probability $\lambda$.\
In section II, we briefly summarize our notations and present a key discovery, which leads us to the understanding of the relation between the structure of the rule table and pattern classes. It strongly suggested that the rules which break strings of the quiescent states play an important role for the pattern classes.\
In section III, the rule tables are classified according to the destruction and construction of strings of the quiescent states and we find the method to change the chaotic pattern class into periodic one and vice versa while keeping $\lambda$ fixed. We will show that by using the method, the change of the pattern classes takes place without fail, in the region $1/K \le \lambda \leq 1-1/K$.\
In section IV, the result obtained in section III is studied quantitatively by introducing a new quiescent string dominance parameter $F$. It is determined by using the distribution of class IV rule tables, which we call optimal $F$ parameter.\
In section V, using $F$ and $\lambda$, we classify the rule tables of CA(5,4) in ($\lambda$,F) plane. It will be shown that all the four pattern classes do coexist in wide range in $\lambda$, contrary to the result of Ref. [@langton] by Langton. The reason why he obtained his result will be discussed.\
In section VI, the rule tables of CA(5,4) are classified in the ($\lambda$,F) plane, which provides us with the phase diagram.\
Section VII is devoted to discussions and conclusions, where a possibility of the transmission of the initial state information and the classification of the rule tables by the another intuitive $F$ parameter, will be discussed.\
SUMMARY OF CELLULAR AUTOMATA AND A KEY DISCOVERY
================================================
Summary of cellular automata
-----------------------------
In order to make our arguments concrete, we focus mainly on the one-dimensional 5-neighbor and 4-state CA (CA(5,4)) in the following, because this is a model in which Langton had argued the classification of CA by the $\lambda$ parameter. However the qualitative conclusions in this article, hold true for general CA(N,K). These points will be discussed in the subsections III B.\
We will briefly summarize our notation of CA[@wolfram; @langton]. In our study, the site consists of 150 cells having the periodic boundary condition. The states are denoted as $s(t,i)$. The $t$ represents the time step which takes an integer value, and the $i$ represents the position of cells which range from $0$ to $149$. The $s(t,i)$ takes values $0,1,2,$ and $3$, and the state $0$ is taken to be the quiescent state. The set of the states $s(t,i)$ at a time $t$ is called the configuration.\
The configuration at time $t+1$ is determined by that of time $t$ by using following local relation,
$$s(t+1,i) = T(s(t,i-2),s(t,i-1),s(t,i),s(t,i+1),s(t,i+2)).
\label{eq:table}$$
The set of the mappings $$T(\mu,\nu,\kappa,\rho,\sigma)=\eta,(\mu,\nu,etc.= 0,1,2,3)
\label{eq:r_table}$$ is called the rule table. The rule table consists of $4^5$ mappings, which is selected from a total of $4^{1024}$ mappings.\
The $\lambda$ parameter is defined as[@langton0; @langton] $$\lambda=\frac{N_{h}}{1024},
\label{eq:lambda}$$ where $N_{h}$ is the number in which $\eta$ in Eq. (\[eq:r\_table\]) is not equal to $0$. In other words the $\lambda$ is the probability that the rules do not select the quiescent state in next time step. Until section III, we set the rule tables randomly with the probability $\lambda$. Our method is to choose $1024-N_{h}$ rules randomly, and set $\eta=0$ in the right hand side of Eq. (\[eq:r\_table\]), and for the rest of the $N_{h}$ rules, the $\eta$ picks up the number 1,2 and 3 randomly. The initial configurations are also set randomly.\
The time sequence of the configurations is called a pattern. The patterns are classified roughly into four classes established by Wolfram[@wolfram]. It is widely accepted that pattern classes are classified by the $\lambda$[@langton]; as the $\lambda$ increases the most frequently generated patterns change from homogeneous (class I) to periodic (class II) and then to chaotic (class III), and at the region between class II and class III, the edge of chaos (class IV) is located.
Correlation between pattern classes and rules which break strings of quiescent states
-------------------------------------------------------------------------------------
In order to find the reason why the different pattern classes are generated at the same $\lambda$, we have started to collect rule tables of different pattern classes, and tried to find the differences between them. We fix at $N_{h}=450$ ($\lambda=0.44$), because we find empirically that around this point the chaotic, edge of chaos, and periodic patterns are generated with similar ratio. By changing the random number, we have gathered a few tens of the rule tables and classified them into chaotic, edge of chaos, and periodic ones.\
In this article, a pattern is considered the edge of chaos (class IV) when its transient length[@langton] is longer than $3000$ time steps.\
First, we study whether or not the pattern classes are sensitive to the initial configurations. We fix the rule tables and change the initial configurations randomly. For most of the rule tables, the details of the patterns depend on the initial configurations, but the pattern classes are not changed[@wolfram]. The exceptions will be discussed in the subsection VII A, in connection with the transmission of the initial state information. Thus the differences of the pattern classes are due to the differences in the rule tables, and the target of our inquiry has to do with the differences between them.\
For a little while, we do not impose a quiescent condition (QC), $T(0,0,0,0,0)=0$, because without this condition, the structure of the rule table becomes more transparent. This point will further be discussed in section IV.\
After some trial and error, we have found a strong correlation between the pattern classes and the QC. The probability of the rule table, which satisfies the QC is much larger in class II patterns than that in the class III patterns. This correlation suggest that the rule $T(0,0,0,0,0)=h,\hspace{0.2cm} h \neq 0$, which breaks the string of the quiescent states with length 5, pushes the pattern toward chaos. We anticipate that the similar situation will hold for the length 4 strings of quiescent states.\
We go back to the usual definitions of CA; in the following we discuss CA under QC, $T(0,0,0,0,0)=0$. We study the correlation between the number of the rules which breaks the length four strings of the quiescent states. These rules are given by, $$\begin{array}{ll}
T(0,0,0,0,i)=h,\\
T(i,0,0,0,0)=h,(i,h=1,2,3).
\label{eq:d4}
\end{array}$$ They will also push the pattern toward chaos. Similar properties of the rule tables had been noticed by Wolfram and Suzudo with the arguments of the unbounded growth[@wolfram] and expandability[@suzudo].\
We denote the total number of rules of Eq. (\[eq:d4\]) in a rule table as $N_4$. In order to study the correlation between the pattern classes and the number $N_4$, we have collected 30 rule tables and grouped them by the number $N_4$. We have 4 rule tables with $N_4 \ge 4$ , 13 rule tables with $N_4=3$, 9 rule tables with $N_4=2$ and 4 rule tables with $N_4 \le 1$. When $ N_4 \ge 4$, all rule tables generate chaotic patterns, while when $N_4 \le 1$, only periodic ones are generated. At $N_4=3$ and $N_4=2$, chaotic, edge of chaos, and periodic patterns coexist. Examples are shown in the Fig. \[fig:pattern\_at\_lambda0.44\]. The coexistence of three pattern classes at $N_4=3$ is seen in Fig. \[fig:pattern\_at\_lambda0.44\](b), Fig. \[fig:pattern\_at\_lambda0.44\](c) and Fig. \[fig:pattern\_at\_lambda0.44\](d) and that of $N_4=2$ is exhibited in Fig. \[fig:pattern\_at\_lambda0.44\](e), Fig. \[fig:pattern\_at\_lambda0.44\](f) and Fig. \[fig:pattern\_at\_lambda0.44\](g).\
As anticipated, the strong correlation between $N_4$ and the pattern classes has been observed in this case too. These discoveries have provided us with a key hint leading us to the hypothesis that the rules, which break strings of the quiescent states, will play a major role for the pattern classes.
Structure of rule table and pattern classes
===========================================
Structure of rule table and replacement experiment
--------------------------------------------------
In order to test the hypothesis of the previous section, we classify the rules into four groups according to the operation on strings of the quiescent states. In the following, Greek characters in the rules represent groups $0,1,2,3$ while Roman, represent groups $1,2,3$.\
Group 1: $T(\mu,\nu,0,\rho,\sigma)=h$.\
The rules in this group break strings of the quiescent states.\
Group 2: $T(\mu,\nu,0,\rho,\sigma)=0$.\
The rules of this group conserve them.\
Group 3: $T(\mu,\nu,i,\rho,\sigma)=0$.\
The rules of this group develop them.\
Group 4: $T(\mu,\nu,i,\rho,\sigma)=l$.\
The rules in this group do not affect string of quiescent states in next time step.\
Let us denote the number of the group 1 rules in a rule table as $N(g1)$. Similarly for the number of other groups. They satisfy the following sum rules, when $N_h$ is fixed. $$\begin{array}{ll}
N(g1)+N(g2)=256,\\
N(g3)+N(g4)=768,\\
N(g2)+N(g3)=1024-N_h,\\
N(g1)+N(g4)=N_h.
\end{array}
\label{eq:group}$$ In the methods of generating the rule tables randomly using $N_h$ ($\lambda$), these numbers are determined mainly by the probability $\lambda$; namely $N(g1) \simeq 256 \lambda$, $N(g2) \simeq 256(1-\lambda)$ $N(g3) \simeq 768(1-\lambda)$ and $N(g4) \simeq 768 \lambda$ respectively. Therefore they suffer from fluctuation due to randomness.\
The group 1 rules are further classified into five types according to the length of string of quiescent states, which they break. These are shown in Table \[tab:destruc\]. The rule D5 is always excluded from rule tables by the quiescent condition.\
------------------------- ----- ---- --------- -- -- --
$T(0,0,0,0,0)=h$ 1 D5 RP5,RC5
$T(0,0,0,0,i)=h$ 3 D4 RP4,RC4
$T(i,0,0,0,0)=h$ 3
$T(0,0,0,i,\sigma)=h$ 12
$T(i,0,0,0,m)=h$ 9 D3 RP3,RC3
$T(\mu,j,0,0,0)=h$ 12
$T(\mu,j,0,0,m)=h$ 36 D2 RP2,RC2
$T(i,0,0,l,\sigma)=h$ 36
$T(\mu,j,0,l,\sigma)=h$ 144 D1 RP1,RC1
------------------------- ----- ---- --------- -- -- --
Our hypothesis presented at the end of the section II is expressed more quantitatively as follows; the numbers of the D4, D3, D2, and D1 rules shown in Table \[tab:destruc\] will mainly determine the pattern classes.\
In order to test this hypothesis, we artificially change the numbers of the rules in Table \[tab:destruc\] while keeping the $N_h$ ($\lambda$) fixed. For D4 rules, we carry out the replacements defined by the following equations, $$\begin{array}{ll}
T(0,0,0,0,i)=h \rightarrow T(0,0,0,0,i)=0, \\
or \hspace {0.1cm}T(i,0,0,0,0)=h \rightarrow T(i,0,0,0,0)=0, \\
T(\mu,\nu,j,\rho,\sigma)=0 \rightarrow T(\mu,\nu,j,\rho,\sigma)=l,\\
\label{eq:toperio}
\end{array}$$ where except for $h$, the groups $\mu$, $\nu$, $\rho$, $\sigma$, $j$ and $l$ are selected randomly. Similarly the replacements are generalized for D3, D2, and D1 rules , which are denoted as RP4 to RP1 in Table \[tab:destruc\]. They change the rules of group 1 to that of group 2 together with group 3 to group 4 and are expected to push the rule table toward the periodic direction.\
The reverse replacements for D4 are $$\begin{array}{ll}
T(0,0,0,0,i)=0 \rightarrow T(0,0,0,0,i)=h, \\
or \hspace {0.1cm}T(i,0,0,0,0)=0 \rightarrow T(i,0,0,0,0)=h, \\
T(\mu,\nu,j,\rho,\sigma)=l \rightarrow T(\mu,\nu,j,\rho,\sigma)=0,\\
\label{eq:tochaos}
\end{array}$$ which will push the rule table toward the chaotic direction. In this case, the groups $h$, $\mu$, $\nu$, $j$, $\rho$, and $\sigma$ are selected randomly. Similarly we introduce the replacements for D3, D2, and D1, which will be called RC4 to RC1 in the following.\
By the replacement of RP4 to RP1 or RC4 to RC1, we change the numbers of the rules in Table \[tab:destruc\] while keeping the $N_h$ fixed. We denote these numbers $N_4$, $N_3$, $N_2$, and $N_1$ for D4, D3, D2, and D1 rules, respectively. By applying these replacements, we could study the rule tables which are difficult to obtain using only $N_h$ ($\lambda$).\
Examples of the replacement experiments are shown in Fig. \[fig:replace1\].
-------------------------- --- ---- ---- ---- --- ---- --- --- -- -- -- --
Fig. \[fig:replace1\](a) 3 22 53 96 0 0 0 0
Fig. \[fig:replace1\](b) 0 14 53 96 3 8 0 0
Fig. \[fig:replace1\](c) 0 13 53 96 3 9 0 0
Fig. \[fig:replace1\](d) 0 12 53 96 3 10 0 0
Fig. \[fig:replace1\](e) 1 7 53 96 2 15 0 0
Fig. \[fig:replace1\](f) 1 6 53 96 2 16 0 0
Fig. \[fig:replace1\](g) 2 1 53 96 1 21 0 0
Fig. \[fig:replace1\](h) 2 0 53 96 1 22 0 0
-------------------------- --- ---- ---- ---- --- ---- --- --- -- -- -- --
In this replacements, the RP4s are always carried out first, after that RP3s are done. The rule table of Fig. \[fig:replace1\](a) is obtained randomly with $N_h=615$ ($\lambda=0.6)$. The numbers of the D4, D3, D2 and D1 are shown in line Fig. \[fig:replace1\](a) of Table \[tab:table2\]. At $N_h=615$ most of the randomly obtained rule tables generate chaotic patterns. We start to make RP4 tree times, then number of the D4 becomes $N_4=0$. At this stage, the rule table still generate chaotic patterns. Then we proceed to carry out RP3. The chaotic patterns continues from $N_3=22$ to $N_3=15$, and when $N_3$ becomes $14$, the pattern changes to edge of chaos behavior, which is shown in the Fig. \[fig:replace1\](b). Fig. \[fig:replace1\](c) is obtained by one more RP3 replacements for Fig. \[fig:replace1\](b) rule table. It shows a periodic pattern with a rather long transient length. The pattern with one more replacement of RP3 for the Fig. \[fig:replace1\](c), is shown in Fig. \[fig:replace1\](d), where the transient length becomes shorter. These numbers of D4 rule ($N_4$) and D3 rule ($N_3$), and numbers of the replacements RP4s and RP3s for the Fig. \[fig:replace1\](a) rule table, are summarized in Table \[tab:table2\].\
Similarly replacement experiments in which the RP4s are stopped at $N_4=1$ and $N_4=2$ are shown in Fig. \[fig:replace1\](e), Fig. \[fig:replace1\](f), and Fig. \[fig:replace1\](g), Fig. \[fig:replace1\](h), respectively and the numbers of group 1 rules and the replacements are also summarized in the corresponding lines in the Table \[tab:table2\]. We should like to notice that the transitions to class III to class IV pattern classes take place at ($N_4=0$, $N_3=13$), ($N_4=1$,$N_3=6$) and ($N_4=2$,$N_3=0$). Therefore the effects to push the rule table toward chaos is stronger for D4 than D3 of Table \[tab:destruc\]. This point will be discussed more quantitatively in the next section.\
At $N_h=$819, $768$, $717$, $615$, $512$, $410$, $307$ and $205$ ($\lambda=0.8$, 0.75, 0.7, 0.6, 0.5, 0.4, 0.3 and 0.2), we have carried out replacements experiments for 119, 90, 90, 117, 107, 92, 103 and 89 rule tables, which are generated randomly using $N_h$ ($\lambda$). At all the $N_h$ points, we have succeeded in changing the the patten classes from class III to class II or class I or vice versa, by changing the numbers of the group 1 rules. And in many cases, the edge of chaos behaviors are observed between them.
Chaotic and periodic limit of general CA(N,K)
---------------------------------------------
Let us study the effects of the replacements theoretically in general cellular automata CA(N,K). In the general case too, the rule tables are classified into four groups as shown in subsection III A. We have denoted these numbers as $N(g1)$, $N(g2)$, $N(g3)$ and $N(g4)$. When $N_h$ is fixed, these numbers satisfy the following sum rules, which are the generalization of Eq. \[eq:group\]. $$\begin{array}{ll}
N(g1)+N(g2)=K^{N-1},\\
N(g3)+N(g4)=K^{N-1}(K-1),\\
N(g2)+N(g3)=K^{N}-N_h,\\
N(g1)+N(g4)=N_h.
\label{eq:sum_rule1}
\end{array}$$ However the individual number $N(gi)$ suffers from the fluctuations due to randomness. They distribute with the mean given by Table \[tab:N-K\]. In this subsection, let us neglect these fluctuations.\
--------- -------------------------------------- --------------------------- -- -- -- --
group 1 $T(\mu_1,\mu_2,...,0,...,\mu_{N})=h$ $K^{N-1}\lambda$
group 2 $T(\mu_1,\mu_2,...,0,...,\mu_{N})=0$ $K^{N-1}(1-\lambda)$
group 3 $T(\mu_1,\mu_2,...,i,...,\mu_{N})=0$ $K^{N-1}(K-1)(1-\lambda)$
group 4 $T(\mu_1,\mu_2,...,i,...,\mu_{N})=l$ $K^{N-1}(K-1)\lambda$
--------- -------------------------------------- --------------------------- -- -- -- --
The replacements to decrease the number of the group 1 rule while keeping the $\lambda$ fixed are given by, $$\begin{array}{ll}
N(g1) \rightarrow N(g1)-1,\hspace {0.5cm} N(g2) \rightarrow N(g2)+1,\\
N(g3) \rightarrow N(g3)-1,\hspace {0.5cm} N(g4) \rightarrow N(g4)+1.\\
\label{eq:del_num_RP}
\end{array}$$ In CA(5,4), they correspond to RP4 to RP1 of section III A.\
These replacements stop either when $N(g1)=0$ or $N(g3)=0$ is reached. Therefore when $N(g1) \le N(g3)$, which corresponds to $\lambda \le
(1-\frac{1}{K})$ in $\lambda$, all the group 1 rules are replaced by the group 2 rules. In this limit, quiescent states at time $t$ will never be changed, because there is no rule which converts them to other states, while the group 3 rules have a chance to create a new quiescent state in the next time step. Therefore the number of quiescent states at time t is a non-decreasing function of t. Then, the pattern class should be class I (homogeneous) or class II (periodic), which we call periodic limit. Therefore the replacements of Eq. (\[eq:del\_num\_RP\]) push the rule table toward the periodic direction.\
Let us discuss the reverse replacements of Eq. (\[eq:del\_num\_RP\]). In these replacements, if $N(g2) \le N(g4)$, which corresponds to $\frac{1}{K} \le \lambda$, all group 2 rules are replaced by the group 1 rules, except for the quiescent condition. In this extreme reverse case, almost all the quiescent states at time t are converted to other states in next time step, while group 3 rules will create them at different places. Then this will most probably develop into chaotic patterns. This limit will be called chaotic limit.\
We should like to say that there are possibilities that atypical rule tables and initial conditions might generate a periodic patterns even in this limit. But in this article, these atypical cases are not discussed.\
Therefore in the region, $$\frac{1}{K} \le \lambda \le 1-\frac{1}{K},
\label{eq:2-limit}$$ all the rule tables are located somewhere between these two limit, and by the successive replacements of Eq. (\[eq:del\_num\_RP\]) and their reverse ones, the changes of the pattern classes take place without fail. This is the theoretical foundation of the replacement experiments of previous subsection and also explains why in this region the four pattern classes coexist.\
The Eq. (\[eq:del\_num\_RP\]) provide us with a method to control the pattern classes at fixed $\lambda$. The details of the replacements of Eq. (\[eq:del\_num\_RP\]) depend on the models. In the CA(5,4), they have been RP4 to RP1 and RC4 to RC1. They will enable us to obtain a rule tables which are difficult to generate by the method ”random-table method” or ”random-walk-through method” and lead us to the new understandings on the structure of CA rule tables in section V.
Quiescent String Dominance Parameter $F$ in CA(5,4)
===================================================
In the previous section, we have found that each rule table is located somewhere between chaotic limit and periodic limit, in the region $\frac{1}{K} \le \lambda \le 1-\frac{1}{K}$. In order to express the position of the rule table quantitatively, we introduce a new quiescent string dominance parameter F, which provides us with a new axis ($F$-axis) orthogonal to $\lambda$. Minimum of $F$ is the periodic limit, while maximum of it corresponds to chaotic limit. In this section, we will determine the parameter $F$ for CA(5,4).\
As a first approximation, the parameter $F$ is taken to be be a function of the numbers of the rules D4, D3, D2 and D1, which have been denoted as $N_4$, $N_3$, $N_2$ and $N_1$, respectively. We proceed to determine $F(N_4,N_3,N_2,N_1)$ by applying simplest approximations and assumptions\
We apply Taylor series expansion for $F$, and approximate it by the linear terms in $N_4$, $N_3$ $N_2$ and $N_1$.\
$$F(N_4,N_3,N_2) \simeq c_4 N_4 + c_3 N_3 + c_2 N_2 + c_1 N_1
\label{eq:taylor}.$$ where $c_4= \partial F/\partial N_4$, similar for $c_3$, $c_2$ and $c_1$. They represent the strength of the effects of the rules D4, D3, D2 and D1 to push the rule table toward chaotic direction. These definitions are symbolic, because $N_{i}$ is discrete.\
The measure in the $F$ is still arbitrary. We fix it in the unit where the increase in one unit of $N_4$ results in the change of $F$ in one unit. This corresponds to divide $F$ in Eq. (\[eq:taylor\]) by $c_4$, and to express it by the ratio $c_3/c_4$ ($r_3$), $c_2/c_4$ ($r_2$) and $c_1/c_4$ ($r_1$).\
Before we proceed to determine $r_3$, $r_2$ and $r_1$, let us interpret the parameter $F$ geometrically. Most generally, the rule tables are classified in 1024-dimensional space in CA(5,4). The location of the rule table of each pattern classes forms a hyper-domain in this space. We map the points in the hyper-domain into 4-dimensional $(N_4,N_3,N_2,N_1)$ space, in which they will also be located in some region. We introduce a surface $S(N_4,N_3,N_2,N_1)=\Phi$ in order to line up these points. $F$-axis is a normal line of this surface. In Eq. (\[eq:taylor\]), we approximate it by a hyper plane.\
In order to determine $r_3$, $r_2$ and $r_1$, we apply an argument that the class IV rule tables are located around the boundary of the class II and class III rule tables. Our strategy to determine $r_3$, $r_2$ and $r_1$ is to find the regression hyper plane of class IV rule tables on four dimensional space, ($N_4$,$N_3$, $N_2$ $N_1$). It is equivalent to fix the $F$-axis in such a way that the projection of the distribution of class IV rule tables on $F$-axis, looks as narrow as possible. The quality of our approximations and assumptions reflects the width of the distribution of class IV rule tables.\
In the least square method, our problem is formulated to find $r_3$, $r_2$ and $r_1$, which minimize the quantity,
$$s(r_3,r_2,r_2)=
\frac{1}{c_{4}^{2}}\sum_{i,j}(F_{class IV}^i(N_4^i,N_3^i,N_2^i,N_1^i)
- F_{class IV}^j(N_4^j,N_3^j,N_2^j,N_1^j))^{2}.
\label{eq:ansatz},$$
where $i$ and $j$ label the class IV rule tables. We solve the equations, $\partial S/\partial r_3=0$, $\partial S/\partial r_2=0$ and $\partial
S/\partial r_1=0$, which are
$$\begin{array}{llll}
\displaystyle
r_3 \sum_{i,j} (\delta N_3^{i,j})^2 +
r_2 \sum_{i,j} \delta N_2^{i,j}\delta N_3^{i,j} +
r_1 \sum_{i,j} \delta N_1^{i,j}\delta N_3^{i,j}
=- \sum_{i,j} \delta N_4^{i,j}\delta N_3^{i,j}, \\
\displaystyle
r_3 \sum_{i,j} \delta N_3^{i,j} \delta N_2^{i,j} +
r_2 \sum_{i,j} (\delta N_2^{i,j})^2+
r_1 \sum_{i,j} \delta N_1^{i,j} \delta N_2^{i,j}
=- \sum_{i,j} \delta N_4^{i,j}\delta N_2^{i,j}, \\
\displaystyle
r_3 \sum_{i,j} \delta N_3^{i,j} \delta N_1^{i,j} +
r_2 \sum_{i,j} \delta N_2^{i,j} \delta N_1^{i,j}+
r_1 \sum_{i,j} (\delta N_1^{i,j})^2
=- \sum_{i,j} \delta N_4^{i,j}\delta N_1^{i,j}. \\
\label{eq:sol_r3}
\end{array}$$
where $\delta N_4^{i,j}=N_{4}^{i}-N_{4}^{j}$, similar for $\delta N_3^{i,j}$ and $\delta N_2^{i,j}$.\
In order to collect class IV rule tables, we have generated rule tables randomly both for $N_h$ in the region, $205 \le N_h \le 819$ ($0.2 \le \lambda \le 0.8$) and for the numbers of the group 1 rules in the ranges, $0 \le N_4 \le 6$, $0 \le N_3 \le 33$, $0 \le N_2 \le 72$ and $0 \le N_1 \le 144$.\
This is realized by the two step method. In the first step, we generate rule table randomly using the number $N_h$, which are explained in subsection II B. We should like to notice that under this methods, the numbers of the group 1 rules, $N_4$, $N_3$, $N_2$ and $N_1$ are distributed around, $6 \lambda$, $33 \lambda$, $72 \lambda$ and $144 \lambda$ respectively. They are denoted as $N_i^{\lambda}$. Then in second step, $N_i$s are determined randomly between zero and their maximum. The $N_i^{\lambda}$’s, which are obtained in the first step are changed to their random value by the replacements RC4 to RC1 or RP4 to RP1.\
We have generated about 14000 rule tables, and classify them into four pattern classes according to their transient length. There are 483 class I, 3169 class II, 10248 class III and 329 class IV rule tables, respectively. From the 329 Class IV rule tables, the coefficients $r_i$ are determined by solving the Eq. (\[eq:sol\_r3\]). They are summarized in the Table \[tab:coefficient-r\], where the errors are estimated by the Jackknife method.
------- -------- -------- --------- -- -- --
$r_3$ 0.1563 0.0013 0.18182
$r_2$ 0.0506 0.0007 0.08333
$r_1$ 0.0195 0.0002 0.04167
------- -------- -------- --------- -- -- --
: The optimal and intuitive coefficients $r_i$.[]{data-label="tab:coefficient-r"}
The results show that the coefficients are positive, and satisfy the order, $$c_4 > c_3 > c_2 > c_1.
\label{eq:order_force}$$ It means that the effects to move the rule table toward chaotic limit on the $F$-axis are stronger for the rules which break longer strings of the quiescent states [^2].\
The order in Eq. (\[eq:order\_force\]) is understood by the following intuitive arguments. If six D4 rules are included in the rule table, the string of the quiescent states with length 5 will not develop. Similarly, if 33 D3 rules are present in the rule table, there is no chance that the length 4 string of the quiescent states could be made. These are roughly similar situations for the formation of pattern classes. Thus the strength of the D3 rules $r_3$ will be roughly equal to $6/33$ of that of D4 rules, similar for the strength of the D2 and D1 rules. These coefficients are also shown in the Table \[tab:coefficient-r\]. We call the $F$ parameter with these $r_{i}$’s as intuitive $F$ parameter and those determined by solving the Eq. (\[eq:sol\_r3\]) as optimal one. It is found that the differences between them are not so large. The classification of the rule tables with intuitive $F$ parameter will be discussed in the subsection VII B.\
Distribution of the Rule Tables in ($\lambda$,F) Plane
======================================================
Using the optimal $F$ parameter we plot the position of the rule tables of each pattern classes in ($\lambda$,F) plane. They are shown in Fig. \[fig:lambda-F-distribution\].\
The Figs. \[fig:lambda-F-distribution\] shows that the class III rule tables are located in the larger $F$ region; about $F \geq 4$, while class I, class II rule tables, in the smaller $F$ region; about $F \leq 9$, and the class IV rule tables are found in the overlap region of class II and class III rule tables; about $4 \leq F \leq 9$. These results support the chaotic limit and periodic limit discussed in the subsection III B, and shows that at least in the $0.2 \leq \lambda \leq 0.75$ range, all four pattern classes coexist.\
These distributions of rule table in $F$, are almost independent of $\lambda$. It means that the CA pattern classes are not classified by $\lambda$, contrary to the results of Ref. [@langton], but are classified rather well by the quiescent string dominance parameter $F$.\
Let us discuss the reason why Langton had obtained his results. If the rule tables are generated using the probability $\lambda$ by the ”random-table method” or ”random-walk-through method”[@langton], the numbers of the group 1 rule tables $N_4$, $N_3$, $N_2$ and $N_1$ are also controlled by the probability $\lambda$. They distribute around $N^{\lambda}_{i}$ of section IV. Then the $F$ parameters are also distributed around, $$F^{\lambda} = (6+33r_3+72r_2+144r_1)\lambda.
\label{eq:F_with_lambda}$$ Therefore in these methods, $\lambda$ and $F$ are strongly correlated. The probabilities to obtain the rule tables, which are far apart from the line given by Eq. (\[eq:F\_with\_lambda\]) are very small. When $\lambda$ is small, the rule table with small $F$ are mainly generated, which are class I and class II CAs. On the other hand in the large $\lambda$ region, the rule tables with large $\lambda$ are dominantly generated, which are class III CAs. The line of Eq. (\[eq:F\_with\_lambda\]) crosses the location of class IV pattern classes around $0.4 \leq \lambda \leq 0.55$. This might be a reason Langton has obtained his results. But the distribution of rule tables in all ($\lambda$,F) plane, show the global structure of the CA rule table space as in Fig. \[fig:lambda-F-distribution\] and lead us to deeper understanding of the structure of CA rule tables.\
We should like to stress again that four pattern classes do coexist in a rather wide range in $\lambda$, which are rather well classified by the parameter $F$ not by $\lambda$.
Classification of Rule tables in ($\lambda$,F) Plane
====================================================
In the Fig. \[fig:lambda-F-distribution\], it is found that rule tables are not separated by sharp boundaries. And they seem to have some probability distributions. We denote the probability densities of class I, class II, class III and class IV pattern classes as $P^{I}$, $P^{II}$, $P^{III}$ and $P^{IV}$, respectively and proceed to classify the rule tables by using them.\
The equilibrium points $F^{II-III}_{E}(\lambda)$ of class II and class III rule tables are defined by the point where the relation $P^{II}(F)=P^{III}(F)$ is satisfied. The region in ($\lambda$,$F$) plane where $P^{II}$ and $P^{III}$ coexist in a similar ratio is defined as transition region. The upper points of the transition region $F^{II-III}_{U}$, are defined by the points, $P^{II}$=$\frac{P^{III}}{e}$ and similarly for the lower points of the transition region $F^{II-III}_{L}$, where $P^{II}$ and $P^{III}$ are interchanged. By these three points, $F^{II-III}_{E}$, $F^{II-III}_{U}$ and $F^{II-III}_{L}$, we define the phase boundary of the rule tables.\
The distributions of the rule tables in the Fig. \[fig:lambda-F-distribution\] show the qualitative probability distributions. However in order to study the $\lambda$ dependences of $F^{II-III}_{E}$, $F^{II-III}_{U}$ and $F^{II-III}_{L}$ more quantitatively, we generate a rule tables at fixed $\lambda$s. The $\lambda$ points and numbers of the rule tables are shown in Table \[tab:phtr\_data\].
--------- ----- ----- ------ ------ ----- ---- ----- ----- ----- ----- ----- -----
$0.125$ 128 202 1081 1052 42 16 0.0 2.0 3.2 3.0 6.6 9.0
$0.15$ 154 99 499 545 23 17 0.7 2.0 3.6 3.5 6.5 9.1
$0.2$ 205 141 728 1021 39 18 1.2 2.0 3.6 4.7 6.3 8.5
$0.25$ 256 98 479 949 19 8 0.0 1.7 3.8 4.4 5.9 7.5
$0.3$ 307 83 364 878 13 6 1.2 2.0 3.9 4.0 6.0 7.3
$0.4$ 410 117 385 1169 23 7 3.7 4.5 5.7 7.0
$0.5$ 512 53 341 884 23 7 4.8 6.0 7.2
$0.6$ 615 38 488 1316 61 10 4.8 6.1 7.3
$0.7$ 717 4 333 974 89 7 4.9 5.8 6.8
$0.75$ 768 2 256 960 108 7 4.5 5.4 6.6
$0.8$ 819 2 81 1064 12 5 4.3
--------- ----- ----- ------ ------ ----- ---- ----- ----- ----- ----- ----- -----
At each fixed $\lambda$, we divide the region in $F$ into bin sizes of $\delta F=1$, and count the number of rule tables of each classes in these bins. From these results, we estimate the probability densities $P(F_i)$, where $F_i$ is a middle point of that bin.\
Let us proceed to the classification of the class II and class III rule tables.
Classification of rule tables in $\frac{1}{4} \le \lambda \le
1-\frac{1}{4}$
-------------------------------------------------------------
. \[fig:Phtr\_0.3\]
The probability distributions of $P^{II}(F)$ and $P^{III}(F)$ at $\lambda=0.3$ are shown in Fig. \[fig:Phtr\_0.3\](a), and the determination of the $F^{II-III}_{E}$, $F^{II-III}_{U}$ and $F^{II-III}_{L}$ are demonstrated in Fig. \[fig:Phtr\_0.3\](b). For the other $\lambda$ points of Table \[tab:phtr\_data\], $F_{E}$, $F_{U}$ and $F_{L}$ are determined in the similar way. They are summarized in the Table \[tab:phtr\_data\].\
As already seen in Fig. \[fig:lambda-F-distribution\], the $\lambda$ dependences of the $F^{II-III}_{E}$, $F^{II-III}_{U}$ and $F^{II-III}_{L}$ are small, in the region $0.25 \le \lambda \le 0.75$. This is also confirmed by the studies at fixed $\lambda$s as shown in in the Table \[tab:phtr\_data\].\
Classification of rule tables in larger and smaller $\lambda$ region
--------------------------------------------------------------------
The $P^{II}$ and $P^{III}$ at $\lambda=0.8$ are shown in Fig. \[fig:L8\].
It should be noticed that there is no region of $F$ where $P^{II} \geq
P^{III}$. This means that $F^{II-III}_{E}$ and $F^{II-III}_{L}$ disappears. Only $F^{II-III}_{U}$ is determined.\
In order to understand what has changed at $\lambda=0.8$, we have studied the $\lambda$ dependences of $P^{II}$ and $P^{III}$ in the region $\lambda \geq 0.7$. They are shown in Fig. \[fig:Large\_L\].
It is found that $P^{III}$ gradually increases as $\lambda$ becomes larger but the increase is quite small, while $P^{II}$ decreases abruptly between $\lambda=0.75$ and $\lambda=0.8$. As a result $P^{II}$ becomes less than $P^{III}$ in all $F$ regions. This tendency could already be observed in Fig. \[fig:lambda-F-distribution\](b), but it is quantitatively confirmed by the studies at fixed $\lambda$s.\
In the smaller $\lambda$ region ($\lambda \le 0.3$), the behavior of $P^{II}$ and $P^{III}$ are shown in Fig. \[fig:Small\_L\]. In this case, $P^{II}$ is gradually increasing as $\lambda$ decreases but the change is small. On the contrary, the decrease in $P^{III}$ is larger. As a consequence of these changes the transition region of the class II and class III pattern classes spread over wider range in $F$.
These results are also summarized in Table \[tab:phtr\_data\] and shown in Fig. \[fig:Phase\_diagram\].\
In the same way, the classification of the class I and class II rule tables could be carried out. The preliminary results are shown in the column $F^{I-II}_{E}$, $F^{I-II}_{U}$ and $F^{I-II}_{L}$ of the Table \[tab:phtr\_data\]. In this case too, it is seen in Fig. \[fig:lambda-F-distribution\](a), (b) that density of class I rule tables decreases as $\lambda$ increases, while that of class II rule tables stays almost constant in $\lambda \leq 0.75$. This feature is more quantitatively confirmed by the studies at fixed $\lambda$s. At $\lambda=0.4$, there disappears the region of $F$, where $P^{I}$ is larger than $P^{II}$ and $F^{I-II}_{L}$ and $F^{I-II}_{E}$ could not be determined, just as in the same way as $P^{II}$ and $P^{III}$ distributions at $\lambda=0.8$. These results are also shown in Table \[tab:phtr\_data\].\
However we should like to say that the numbers of the class I rule tables, and those of class II rule tables in the region $F \leq 3$ are not large. Therefore the results may suffer from large statistical fluctuations. We think that the classification of class I and class II rule table needs more data to get quantitative conclusions, however the qualitative properties will not be changed.\
We proceed to the investigation of the classification of rule tables outside of these $\lambda$ region. In $\lambda < 0.25$ region, not all the group 2 rules could be replaced by the group 1 rules. Therefore the maximum numbers of group 1 rules $N(g1)_{Max}$ could not become 256, and it decreases to zero as $\lambda$ approaches to zero. Then the maximum of $F$, ($F_{Max}$) also decreases to zero toward $\lambda=0$.\
Conversely in $\lambda > 0.75$ region, not all the group 1 rules could be replaced by the group 2 rules. The minimum of N(g1) and therefore the minimum of $F$, ($F_{Min}$) could not becomes 0. The line $F_{Min}$ increases until its maximum at $\lambda=1$. In Fig. \[fig:Phase\_diagram\], we have schematically shown the $F_{Max}$ and $F_{Min}$ lines with dotted lines. We should like to stress that the dotted line should have some width due to fluctuations of $N_4$, $N_3$, $N_2$ and $N_1$ caused by the randomness.\
Discussions and Conclusions
===========================
Transmission of initial state informations
------------------------------------------
The computability of the CA is discussed very precisely mainly for elementary CA (CA(3,2)) in series of paper from Santa Fe Institute[@hanson]. In this subsection we discuss on the simplest problem of transmission of the initial state information to the later configurations.\
We have found some examples where class II and class IV patterns appear with similar probability by changing the initial configurations randomly. An example is shown in the Fig. \[fig:L75comp\], which is the transmission of initial state information to later configurations and is similar to the $\rho=1/2$ problem in the CA(3,2). It is interesting to investigate under what condition the changes of the pattern classes are taken place.\
We have focus on the difference of patten classes between class II and class IV, because in this case differences of the patterns are obvious. These rule tables are found in the wide region in $\lambda$, $0.125 \leq \lambda \leq
0.8$. The numbers of the rule tables of this property at fixed $\lambda$s are also shown in the column ”Comp” of Table \[tab:phtr\_data\].\
In addition, there are cases where the difference of patters seems to be realized within the same pattern classes. In these cases careful studies are necessary to distinguish the difference of these patterns. In this article, we have not studied these cases.
Classification of rule table by the intuitive $F$ parameter
-----------------------------------------------------------
The methods to determine the coefficients $r_{i}$ in Eq. (\[eq:taylor\]) are not unique. In the section IV, in order to determine them we have used regression hyper plane of class IV rule tables, and in order to obtain 329 class IV rule tables, we have generated totally about 14000 rule tables. It is a rather tedious task. However optimal set of $r_{i}$ has been close to the intuitive set of $r_{i}$.\
In this subsection we study the classification of rule tables by the intuitive $F$ parameter. The same analyses as sections V and VI are carried out and as the similar figures are obtained in this case too, we will show only the distributions of the rule tables in ($\lambda$,F) plane in Fig. \[fig:intuitive\].
The Figs. \[fig:intuitive\] are very similar to the Figs. \[fig:lambda-F-distribution\]. Therefore the classification of the rule table space are almost same as Fig. \[fig:Phase\_diagram\], except that the $F_{Max}$ changes from 17.6 to 24.\
If intuitive $F$ parameter could successfully classify the rule tables for general CA(N,K) it would be very convenient, because it reflect the structure of the CA rules and there is no need to gather a lot of class IV rule tables, in order to determine $r_i$s. Whether it is correct or not must be concluded after the studies of other CA(N,K)[^3].
Conclusions and discussions
---------------------------
We have started to find the mechanism which distinguishes the pattern classes at same $\lambda$s, and have found that it is closely related to the structure of the rule tables. The patten classes of the CA are mainly controlled by the numbers of the group 1 rules, which has been denoted by $N(g1)$.\
In the CA(N,K),in the region $\frac{1}{K} \le \lambda \le
1-\frac{1}{K}$, the maximum of $N(g1)_{Max}$ corresponds to a chaotic limit, and its minimum $N(g1)=0$, to a periodic limit. Therefore in this $\lambda$ region, we could control the patten classes by changing $N(g1)$ without fail. The method for it is the replacements of Eq. (\[eq:del\_num\_RP\]). Using the replacements, we could study the rule tables which are difficult to obtain by the ”random-table method” or ”random-walk-through method” of Ref.[@langton]. This property could be studied quantitatively by introducing a quiescent string dominance parameter $F$.\
In this article, a quantitative studies are carried out for CA(5,4). In this case, the group 1 rules are further classified into 5 types as shown in Table \[tab:destruc\], and the classification of rule tables is carried out in ($\lambda$,F) plane as shown in Fig. \[fig:Phase\_diagram\]. It is seen that the $\lambda$ dependences of the transition region are very gentle, and rule tables are classified better by the $F$ parameter rather than by $\lambda$. It is interesting whether or not the $\lambda$ dependences of the transition region depend on the models.\
In the replacement experiments, we have found the edge of chaos (very long transient lengths) behavior in many cases. The examples are shown in Fig. \[fig:replace1\]. Sometimes they are observed in some range in $N_3$ or $N_2$. This indicates that in many cases, the transitions are second-order like. But the widths in the ranges of $N_3$ or $N_2$ are different from each other, and there are cases where the widths are less than one unit in the replacement of RP2 (first-order like). It is very interesting to investigate under what condition the transition becomes first-order like or second-order like. The mechanism of the difference in the transitions is an open problem and it may be studied by taking into account effects of group 3 and 4 rules. In these studies another new parameters might be found and a more quantitative phase diagram might be obtained.\
These issues together with finding the points where the transition region crosses $F_{Max}$ and $F_{Min}$ lines (dotted lines) in Fig. \[fig:Phase\_diagram\], and the nature of the transition at these points will be addressed in the forthcoming publications.
[9]{} S. Wolfram, Physica D 10(1984) 1-35. S. Wolfram, Physica Scripta T9(1985) 170-185. C.G.Langton, Physica D 42(1990) 12-37. J.E.Hanson and J.P.Crutchfield, Physica D 103(1997),169-189, and references therein. A. Wuensche, Complexity Vol.4(1999) 47-66. P.M. Binder Complex System 7(1993),241-247 G.M.B.Oliveira, P.P.B.de Oliveira, Nizam Omar Artificial Life 7(2001),277-301 N.H.Packard, Adaption toward the edge of chaos. In ”Dynamic Patterns in Complex System”(1988), 293-301, edited by J. A. S. Kelso, A. J. Mandel,and M. F. Shlesinger,World Scientific,Singapore. M. Mitchell, J.P.Crutchfield and P.T. Hraber, Santa Fe Institute Studies in the Science of Complexity, Proceedings Volume 19. Reading, MA:, Addison-Wesley. online paper, http://www.santafe.edu/ mm/ paper-abstracts.html\#dyn-comp-edge. C.G.Langton, Physica D 22(1986) 120-149. W.LI, N.H.Packard and C.G.Langton, Physica D45(1990) 77-94. T. Suzudo, Crystallisation of Two-Dimensional Cellular Automata, Complexity International, Vol. 6(1999). on line journal,\
http://www.csu.edu.au/ci/vol06/suzudo/suzudo.html. See appendix.
[^1]: In this article, according to the previous authors, phase diagram and phase transition will be used in analogy with the statistical physics.
[^2]: If the quiescent condition is not imposed, Eq. (\[eq:order\_force\]) will becomes $$c_5 > c_4 > c_3 > c_2 > c_1.
$$ Therefore the correlation between pattern classes and the existence of D5 rule is stronger than that between those and the number of D4 rules. If we start our study within the quiescent condition, we may make a longer detour to find the hypothesis of section II and get the qualitative conclusion of section III.
[^3]: The preliminary studies on the CA(5,3) using the intuitive $F$ parameter show that the qualitative results are very similar to those of CA(5,4). The $\lambda$ dependences of the transition region are very weak in the region $\frac{1}{3} \leq \lambda \leq
\frac{2}{3}$, and four pattern classes coexist there. The detailed studies on general CA(N,K) will be reported in the forthcoming publications.
|
---
abstract: 'We present the results of XMM-Newton X-ray observations of the Mira AB binary system, which consists of a pulsating, asymptotic giant branch primary and nearby ($\sim0.6''''$ separation) secondary of uncertain nature. The EPIC CCD (MOS and pn) X-ray spectra of Mira AB are relatively soft, peaking at $\sim1$ keV, with only very weak emission at energies $> 3$ keV; lines of Ne [ix]{}, Ne [x]{}, and O [viii]{} are apparent. Spectral modeling indicates a characteristic temperature $T_X \sim 10^7$ K and intrinsic luminosity $L_X \sim 5\times10^{29}$ erg s$^{-1}$, and suggests enhanced abundances of O and, possibly, Ne and Si in the X-ray-emitting plasma. Overall, the X-ray spectrum and luminosity of the Mira AB system more closely resemble those of late-type, pre-main sequence stars or late-type, magnetically active main sequence stars than those of accreting white dwarfs. We conclude that Mira B is most likely a late-type, magnetically active, main-sequence dwarf, and that X-rays from the Mira AB system arise either from magnetospheric accretion of wind material from Mira A onto Mira B, or from coronal activity associated with Mira B itself, as a consequence of accretion-driven spin-up. One (or both) of these mechanisms also could be responsible for the recently discovered, point-like X-ray sources within planetary nebulae.'
author:
- 'Joel H. Kastner and Noam Soker'
title: 'X-rays from the Mira AB Binary System'
---
Introduction
============
The origin and nature of X-ray emission from highly evolved, post-main sequence stars remains uncertain. In late-type main-sequence stars, X-ray emission is generally assumed to trace surface magnetic activity ultimately derived from stellar magnetic dynamos. The X-ray detection of several first-ascent red giant stars by ROSAT (e.g., Schröder, Hünsch, & Schmitt 1998; Hünsch et al. 2003) and, more recently, Chandra (Hünch 2001) therefore suggests that low-mass, post-main sequence stars can be magnetically active (although it is also possible that the X-rays originate from active, main-sequence companion stars). On the other hand, it appears that — despite maser measurements of large local magnetic fields in asymptotic giant branch (AGB) star winds (e.g., Vlemmings, Diamond, & van Langevelde 2002) — single AGB stars are, at best, only weak X-ray sources (Kastner & Soker 2004). This indicates that the surface magnetic fields of AGB stars are locally rather than globally strong (Soker & Kastner 2003, hereafter Paper I, and references therein).
The lack of X-ray emission from AGB stars is intriguing and puzzling, given that several planetary nebulae, as well as certain ionized, bipolar nebulae associated with symbiotic stars, are now known to harbor point-like X-ray emission at their cores (Chu et al.2001; Guerrero et al. 2001; Kellogg et al. 2001; Kastner et al. 2003). These X-ray sources might be ascribed to PN central stars whose magnetic fields are sufficiently powerful to launch and/or collimate their mass outflows (Blackman et al. 2001a). Alternatively, the X-rays from these systems may originate with binary systems, in which the companion is either accreting material from the mass-losing primary or is itself magnetically active (Guerrero et al. 2001; Soker & Kastner 2002, hereafter SK02). In such systems the presence of an accreting companion, as opposed to magnetic fields on the primary, would explain the collimation of outflows (Soker & Rappaport 2000 and references therein). Blackman et al. (2001b) have proposed that both mechanisms might operate in certain systems that display multipolar symmetry.
The Mira (omicron Ceti) system provides a nearby ($D\sim128$ pc) example of a binary system consisting of a mass-losing AGB star and nearby ($0.6''$ separation) companion (e.g., Karovska et al.1997). The nature of the companion is uncertain, as its optical through UV spectrum appears to be dominated by emission from an accretion disk that presumably is accumulated from Mira A’s wind (Reimers & Cassatella 1985; Bochanski & Sion 2001; Wood, Karovska & Raymond 2002). The Mira AB system was detected as a weak X-ray source by the Einstein and ROSAT X-ray observatories (Jura & Helfand 1984, hereafter JH84; Karovska et al. 1996) but the origin of this X-ray emission is unknown, given the uncertainty concerning the nature of Mira B. If the X-rays arise from accretion onto Mira B, then the modest X-ray luminosity of the system appears to rule out the possibility that Mira B is a white dwarf (JH84). However, the early Einstein observations of the system lacked the sensitivity and X-ray spectral response to provide a definitive test, in this regard.
In Paper I, we analyzed the archival ROSAT data obtained for Mira AB in the context of the possibility that the X-rays arise in magnetic activity on the AGB star (Mira A). While fits of coronal plasma models to the ROSAT Position Sensitive Proportional Counter spectrum were consistent with such a hypothesis, these results left open alternative possibilities; for example, the X-ray emission might arise from magnetic activity on Mira B, or from an accretion disk around this companion star. Furthermore, ROSAT lacked hard ($>2.5$ keV) X-ray sensitivity.
Here, we report XMM-Newton observations of Mira AB. The sensitivity, energy coverage, and spectral resolution of XMM-Newton far surpass those of Einstein and ROSAT. Hence, these observations allow us to further constrain the various models for the Mira AB system.
Observations and Data Reduction
===============================
XMM-Newton (Jansen et al. 2001) observed the Mira system for 12.22 ks on 2003 July 23. The integration times with the European Photon Imaging Camera (EPIC) MOS and EPIC pn CCD detector systems were 11.97 ks and 10.34 ks, respectively. The spectral resolution of these CCD systems range from $\sim50$ eV to $\sim150$ eV over the energy range $0.1-10$ keV. The thick blocking filter was used to suppress optical photons from the Mira system. Standard X-ray event pipeline processing was performed by the XMM-Newton Science Center using version 5.4.1 of the XMM-Newton Science Analysis System (SAS[^1]).
The observation resulted in the detection by EPIC of $\sim60$ sources in a $\sim25'\times25'$ field centered near the position of Mira A (= HD 14386; SIMBAD position $\alpha_{J2000} =$ 02:19:20.7927, $\delta_{J2000} =$ -02:58:39.513). These detections include one source (at $\alpha_{J2000} =$ 02:19:20.81, $\delta_{J2000} =$ -02:58:41.1) that is consistent with the coordinates of the Mira AB system, given the astrometric precision of XMM-Newton (rms pointing uncertainty $>3''$ with a median pointing error of $\sim1''$[^2]). We are unable to establish whether this emission arises from Mira A or Mira B, as the latter is found only $0.58''$ from Mira A at position angle $108^\circ$ (Karovska et al. 1997).
We used SAS and the Interactive Data Language to extract spectra and light curves of this source from the MOS 1, MOS 2, and pn event data within circular regions of radius 20$''$. Background was determined from an annulus with inner and outer radii of 20$''$ and 40$''$, respectively (the X-ray count rates in these background regions were comparable to those obtained for regions farther off source). The resulting, background-subtracted count rates for the Mira AB system were 0.027$\pm0.003$ s$^{-1}$ for MOS 1 and MOS 2 (combined) and 0.031$\pm0.003$ s$^{-1}$ for pn.
Results
=======
XMM/EPIC X-ray Spectra
----------------------
In Fig. 1 we display the combined EPIC (MOS 1, MOS2, and pn) counts spectrum of the Mira system. The spectrum peaks at $\sim0.9$ keV (top panel), and emission lines of Ne [ix]{}, Ne [x]{}, and O [viii]{} appear to be present (bottom panel). There is little emission at energies $> 3$ keV.
We fit the EPIC spectra of the Mira system using XSPEC version 11.2 (Arnaud 1996). SAS was used to construct response matrices and effective area curves for the specific source spectral extraction regions. Motivated by the apparent emission lines of Ne and O in the merged EPIC spectrum, we used a variable-abundance MEKAL model (Kaastra et al. 1996) to fit the spectra (Fig. 2). In our spectral fits, the intervening absorbing column ($N_H$) and X-ray emission temperature ($T_X$) were taken as free parameters as the abundances of individual elements were systematically varied. This procedure was applied during simultaneous fits to the MOS 1, MOS2, and pn spectra, and the results were confirmed via independent fits to each of these three spectra. The best-fit model has an oxygen abundance that is enhanced by a factor $23\pm6$ relative to solar. The fit results further suggest that the Ne and Si abundances may be somewhat enhanced ($\sim4$ and $\sim2.5$ times solar, respectively, with large uncertainties), while the abundance of Fe is solar (to within the fit uncertainties). These results for the abundances in the X-ray-emitting gas are somewhat tentative, however; the best-fit, variable-abundance model yields $\chi^2 = 0.81$, whereas a model with all elemental abundances fixed at solar (for which the best-fit values of $N_H$ and $kT_X$ are similar to those of the variable-abundance model) yields $\chi^2 = 1.27$.
From the variable-abundance model, we find best-fit values for the intervening absorbing column of $N_H \approx 4.5\times10^{21}$ cm$^{-2}$ and X-ray emission temperature of $T_X \approx 10^7$ K, with formal uncertainties of $\sim20$%. These results are not very sensitive to the precise values of the abundances in the model. The best-fit model flux is $6.4\times10^{-14}$ erg cm$^{-2}$ s$^{-1}$ (0.3–3.0 keV), and the intrinsic (unabsorbed) luminosity derived from the model is $L_X \approx 5\times10^{29}$ erg s$^{-1}$. The values for $T_X$ and source flux derived from the EPIC data are very similar to those derived from model fitting of the ROSAT data for the Mira system (Paper I), while the values of $N_H$ and $L_X$ obtained here are somewhat larger than those obtained from the ROSAT data. Fixing $N_H = 2\times10^{21}$ cm$^{-2}$ (the value determined from the ROSAT PSPC data) does not change appreciably the EPIC fit results for $T_X$, but would imply that the overabundance of O is much more modest ($\sim1.5$ times solar) and, in addition, that Fe may be depleted ($\sim0.2$ times solar) in the X-ray-emitting plasma.
We also attempted to fit the EPIC CCD spectra with two-component thermal plasma models wherein $N_H$ is fixed at $2\times10^{20}$ cm$^{-2}$, the value derived from UV spectral modeling (Wood et al. 2002). We find that the fit in this case essentially reverts to an isothermal model — that is, the “cool” component contributes negligibly to the emission — but with an unrealistically high plasma temperature for the “hot” component. The result is a very poor fit, particularly in the 1 keV region where the Ne lines appear. We conclude that the X-ray-derived value of $N_H \sim (2-5)\times10^{21}$ cm$^{-2}$ is relatively robust.
XMM/EPIC X-ray Light Curve
--------------------------
In Fig. 3, we display the combined EPIC (MOS 1 + MOS 2 + pn) light curve of the Mira system. The EPIC count rate is observed to rise rather abruptly, by a factor $\sim2$, within the first 3 ks of the observation. The X-ray flux then more or less steadily declines, such that by the end of the $\sim10$ ks period during which all 3 CCD cameras were active, the count rate had returned approximately to a value at or below that at observation start. The shape of the X-ray light curve is thus suggestive of a magnetic flare or enhanced accretion rate event, although the time interval appears too short to ascertain the quiescent X-ray count rate and, hence, the characteristic flare timescale and amplitude.
Discussion
==========
Before considering the most likely sources of the X-ray emission from the Mira AB system, we first mention two mechanisms that are unlikely to contribute to this emission. One potential source of X-ray emission is that of collisions between winds from components A and B. It is difficult to estimate the likely X-ray luminosity due to such wind shocks, as the mass loss rate ($\dot M_B$) and wind speed ($v_B$) of Mira B appear to be strongly variable (Wood et al.2002). Recent HST UV observations indicate that $\dot M_B \simeq 5 \times 10^{-13} M_\odot \yr^{-1}$ and $v_B = 250 \km \s^{-1}$ whereas earlier IUE observations suggest $\dot M_B \simeq 10^{-11} M_\odot \yr^{-1}$ and $v_B = 400 \km \s^{-1}$ (see Fig. 12 in Wood et al. 2002). Nevertheless, adopting these values as representative of high and low states of mass loss from Mira B, we estimate that the shocked wind X-ray luminosity is only $L_x({\rm wind})=5\times 10^{27}-2.5 \times 10^{29} \erg
\s^{-1}$, where these estimates are obtained under the assumption that the half of the shocked wind that is expelled toward Mira A emits X-rays at $100\%$ efficiency. Therefore, as the conversion of wind to radiant energy is probably quite inefficient, the wind shock mechanism is unlikely to account for the measured X-ray luminosity of Mira AB (§3.1).
In addition, the Bondi-Hoyle accretion radius (measured from the center of Mira B) is $R_{\rm acc} = 2 G M_B/v_r^2= 18
\AU$ for reasonable values of the relevant parameters. This is much larger than the stagnation distance of the two winds (measured from the center of Mira B) as given by Wood et al. (2002) for the strong Mira B wind state, i.e., $R_s=3.7
\AU$ (this value will be lower for the weak Mira B wind state). Thus, the wind from Mira A is more likely to be accreted by Mira B than to collide with the wind blown by Mira B. While this conclusion leads us to propose that the wind from Mira B is a bipolar outflow (perhaps in the form of a collimated fast wind, of the type proposed by Soker & Rappaport 2000), it also casts further doubt on colliding winds as the origin for the X-ray emission. In addition, the apparent X-ray flaring detected here (§3.2) is more consistent with some form of magnetic and/or accretion activity than with wind shocks.
A second possibility is that the X-rays from the system originate with magnetic activity on Mira A (Paper I). In light of our recent XMM-Newton nondetections of X-ray emission from the (apparently single) Mira variables TX Cam and T Cas (Kastner & Soker 2004), such an explanation appears dubious. In addition, the $\sim2$ ks timescale of variation in the X-ray emission from the Mira system is much shorter than the characteristic dynamical timescales of AGB stars.
In the remaining discussion, therefore, we only consider processes in which the X-ray emission originates with Mira B or its immediate environment, as is well established in the case of the UV emission from the system (Karovska et al.1997; Wood et al. 2001). [*Chandra*]{} observations of Mira AB at high spatial resolution will provide a crucial test of this hypothesis.
The X-ray source: implications for the nature of Mira B
-------------------------------------------------------
The basic physics of accretion of the Mira A wind by Mira B was discussed by JH84. Those authors convincingly argued that the accreting star, Mira B, must be a main sequence star, with a probable mass of $M_B \simeq 0.5 M_\odot$, and radius $R_B \simeq 0.6 R_\odot$. The JH84 argument was based in large part on the relatively meager X-ray luminosity of the Mira system; from [*Einstein*]{} observations, JH84 derive an X-ray luminosity $L_x \simeq 3 \times 10^{29} \erg
\s^{-1}$ ($0.15-2.5~$keV), where we scale their $L_x$ result for a distance $D=128$ pc. This is several orders of magnitude lower than the $L_x$ expected from the Bondi-Hoyle model of mass accretion onto a white dwarf (WD) companion, given the mass loss rate and wind speed of Mira (for which JH84 assumed $\dot M_A = 4 \times 10^{-7} M_\odot
\yr^{-1}$ and $v=5$ km s$^{-1}$, respectively, both of which are consistent with more recent observational results; Knapp et al. 1998; Ryde & Schöier 2001).
The results presented in §3.1 confirm the early JH84 results for the $L_X$ of the Mira system. In particular, although the XMM-Newton results demonstrate that the Mira X-ray source is variable (§3.2), the EPIC data also indicate that there is no appreciable hard ($E > 2.5$ keV) X-ray emission from the system. Since the $L_X$ we and JH84 derive, $3-5 \times 10^{29} \erg
\s^{-1}$, is 1-3 orders of magnitude smaller than the X-ray luminosities typical of accreting WDs in binary (cataclysmic variable) systems (e.g., Pandel et al. 2003; Ramsay et al.2004) and the X-ray spectrum of Mira AB evidently lacks the high-temperature ($kT_X > 2$ keV) component characteristic of such systems, our results strongly support the contention of JH84 that Mira B is exceedingly unlikely to be a WD and is, instead, a low-mass, main sequence star. This conclusion, in turn, has important implications for the nature of the point-like X-ray sources within planetary nebulae. Indeed, such X-ray sources may be very similar in nature to the Mira AB binary and, therefore, also could be powered via one of two alternative, accretion-related mechanisms, as we now describe.
X-rays derived from accretion onto Mira B
-----------------------------------------
Although Mira B is probably not a WD, the X-rays from the Mira system might still be generated through accretion of AGB wind material onto Mira B. The accretion process can produce the X-rays directly, via star-disk interactions, or indirectly, via the spin-up and resulting enhanced magnetic activity of Mira B. Inserting reasonable parameters into equation (4) of SK02, we indeed find that an accretion disk is likely to be formed around a main sequence companion to Mira A, such that either mechanism is viable. We now discuss each process, in turn.
### The direct process: magnetospheric accretion
If Mira B is a late-type star with a magnetically active, convective envelope (see below), then the process of accretion of wind material from Mira A may closely resemble that of magnetospheric accretion onto low-mass, pre-main sequence stars. For such (classical T Tauri) stars, it is generally thought that material flows from accretion disk to star along magnetic field lines (or “funnels”) that are rooted to the stellar surface at high latitudes (e.g., Hayashi et al. 1996; Matt et al. 2002; Kastner et al. 2004; and references therein). In the case of the Mira system, a magnetospheric origin for X-ray emission via star-disk interactions is favored by the temperature derived from X-ray spectral fitting. This temperature ($T_X \sim 10^7$ K) is somewhat high to be due to accretion shocks, given the likely free fall velocity of matter onto the surface of a late-type main sequence star (see, e.g., Appendix A of JH84).
The abundance anomalies suggested by the model fits (§3.1) also point to Mira A’s AGB wind material as a likely source of the X-ray-emitting gas. Specifically, the enhanced abundances of O and (possibly) Ne are consistent with the origin of this gas in nuclear processed material dredged up from AGB interior layers (Marigo et al. 1996; Herwig 2004). It is unclear that Mira A is sufficiently massive and/or evolved to have generated excess O and Ne in thermal pulses, however; its 330 d period (Kukarkin et al. 1971) suggests a progenitor mass in the range 1.0–1.2 $M_\odot$ (Jura & Kleinmann 1992), whereas substantial O and Ne production likely requires a progenitor of mass $>2$ $M_\odot$ (Marigo et al.). Another argument in support of the origin of the X-ray emission in accretion processes is the result $N_H
> 2\times10^{21}$ cm$^{-2}$, derived from model fitting of both ROSAT and XMM data (Paper I and §3.1). This X-ray-derived $N_H$ is a factor $>20$ larger than the neutral H absorbing column determined from analysis of the H I Ly $\alpha$ line (Wood et al. 2002). This discrepancy suggests that the UV and X-ray emission arise in different zones around Mira B and, specifically, that the X-ray-emitting region may be embedded within accretion streams that effectively attenuate the X-rays, as described by JH84.
If the X-rays are indeed generated directly by accretion onto Mira B, then the similarity of the X-ray fluxes as measured in 1993 (by ROSAT) and in 2003 (by XMM) would suggest that the rate of accretion has recovered from the relatively low levels measured (via UV observations) in 1999-2001 (see Wood & Karovska 2004 and references therein).
### The indirect process: spin-up of Mira B
Alternatively, the X-ray emission from the Mira system may be derived from the spin-up — and resulting increase in magnetic activity — of Mira B, caused by accretion of mass and angular momentum from the AGB wind of Mira A. Such a process has been described by Jeffries, Burleigh, & Robb (1996), Jeffries & Stevens (1996) and SK02. In SK02 (§2.1.1), we conclude that to generate an X-ray luminosity of $L_x \simeq 5 \times 10^{29} \erg \s^{-1}$ via magnetic activity, a main sequence star of spectral type M4 to F7 (i.e., in the mass range $0.3 M_\odot \lesssim M_B \lesssim
1.3 M_\odot$) has to be spun up to a period of $P \lesssim
3~$days, corresponding to an equatorial rotation speed of $v_{\rm rot} \gtrsim 15-20 \km \s^{-1}$. Because we expect an accretion disk to be formed around Mira B, we can use equation (6) of SK02 to estimate the rotation velocity of Mira B. That equation neglects the initial angular momentum of the accreting main sequence star, and assumes that the entire angular momentum of the star comes from mass accreted from an accretion disk, with a Bondi-Hoyle mass accretion rate (including initial angular momentum will increase the rotation rate). Scaling with the parameters mentioned above, and an orbital separation $a=100 \AU$ (somewhat larger than the projected separation of $70 \AU$; Karovska et al.1997), we find $$v_{\rm rot} \simeq 20 \frac {\Delta M_{\rm AGB}}{0.3 M_\odot}
\left( \frac {M_B}{0.5 M_\odot} \right)^{3/2} \left( \frac
{R_B}{0.6 R_\odot} \right)^{-1/2} \left( \frac {a}{100 \AU}
\right)^{-2} \left( \frac {v_r}{7 \km \s^{-1}} \right)^{-4} \km
\s^{-1},$$ where $\Delta M_{\rm AGB}$ is the total mass lost by Mira A during its AGB phase (when its wind speed remains low). Hence, Mira B could have accreted sufficient angular momentum to spin fast enough to be magnetically active at levels sufficient to explain the observed $L_x \sim 5 \times 10^{29} \erg \s^{-1}$. The lack of significant emission at energies $E > 3$ keV would make Mira B somewhat unusual among highly active coronal sources, however (e.g., Huenemoerder et al. 2003).
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[^1]: http://xmm.vilspa.esa.es/sas/
[^2]: http://xmm.vilspa.esa.es/docs/documents/CAL-TN-0018-2-1.pdf
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---
abstract: 'We present a sublinear randomized algorithm to compute a sparse Fourier transform for nonequispaced data. Suppose a signal $S$ is known to consist of $N$ equispaced samples, of which only $L<N$ are available. If the ratio $p=L/N$ is not close to 1, the available data are typically non-equispaced samples. Then our algorithm reconstructs a near-optimal $B$-term representation $R$ with high probability $1-\delta$, in time and space $poly(B,\log(L),\log p,\log(1/\delta),$ $\epsilon^{-1})$, such that $\Vert S-R\Vert^{2}\leq(1+\epsilon)\Vert S-R_{opt}^{B}\Vert^{2}$, where $R_{opt}^{B}$ is the optimal $B$-term Fourier representation of signal $S$. The sublinear $poly(\log L)$ time is compared to the superlinear $O(N\log N+L)$ time requirement of the present best known Inverse Nonequispaced Fast Fourier Transform (INFFT) algorithms. Numerical experiments support the advantage in speed of our algorithm over other methods for sparse signals: it already outperforms INFFT for large but realistic size $N$ and works well even in the situation of a large percentage of missing data and in the presence of noise.'
author:
- 'Jing Zou [^1]'
title: 'A Sublinear Algorithm of Sparse Fourier Transform for Nonequispaced Data[^2]'
---
Introduction
============
We consider the problem in which the recovery of a discrete time signal $S$ of length $N$ is sought when only $L$ signal values are known. In general, this is of course an insoluble problem; we consider it here under the additional assumption that the signal has a sparse Fourier transform. Let us fix the notations: the signal is denoted by $S=(S(t))_{t=0,\ldots,N-1}$, but we have at our disposal only the $(S(i))_{i\in T}$, where the set $T$ is a subset of $\{0,\ldots,N-1\}$ and $|T|=L$. The Fourier transform of signal $S$ is $\hat{S}=(\hat{S}(0),\ldots,\hat{S}(N-1))$, defined by $\hat{S}(\omega)=\frac{1}{\sqrt{N}}\sum_{t=0}^{N-1}S(t)e^{-2\pi i\omega t/N}$. In terms of the Fourier basis functions $\phi_{\omega}(t)=\frac{1}{\sqrt{N}}e^{2\pi i\omega t/N}$, $S$ can be written as $S=\sum_{\omega=0}^{N-1}\hat{S}(\omega)\phi_{\omega}(t)$; this is the (discrete) Fourier representation of $S$. A signal $S$ is said to have a $B$-sparse Fourier representation, if there exists a subset $\Omega\subset\{0,\ldots,N-1\}$ with $|\Omega|=B$, and values $c(\omega)\neq0$ for $\omega\in\Gamma$, such that $S(t)=\sum_{\omega\in\Omega}c(\omega)\phi_{\omega}$. For a signal that does not have a $B$-sparse Fourier representation, we denote by $R_{opt}^{B}(S)$ the optimal $B$-term Sparse Fourier representation of $S$.
This paper presents a sublinear algorithm to recover a $B$-sparse Fourier representation of a signal $S$ from incomplete data. Our algorithm also extends to the case where the Fourier transform $\hat{S}$ is not $B$-sparse, where we aim to find a near-optimal $B$-term Fourier representation, i.e. $R=\sum_{\omega\in\Gamma}c(\omega)\phi_{\omega}$, such that $$\| S-R\|=\Vert S-\sum_{\omega\in\Gamma}c(\omega)\phi_{\omega}\Vert_{2}^{2}\leq(1+\epsilon)\Vert S-R_{opt}^{B}(S)\Vert_{2}^{2}.$$
A typical situation where our study applies is the observation of non-equispaced data, where the samples are nevertheless all elements of $\tau\mathbb{Z}$ for some $\tau>0$. For a signal with evenly spaced data, the famous Fast Fourier Transform (FFT) computes all the Fourier coefficients in time $O(N\log N)$. However, the requirement of equally distributed data by FFT raises challenges for many important applications. For instance, because of the occurrence of instrumental drop-outs, the data may be available only on a set of non-consecutive integers. Another example occurs in astronomy, where the observers cannot completely control the availability of observational data: a telescope can only see the universe on nights when skies are not cloudy. In fact, computing the Fourier representation from irregularly spaced data has wide applications [@Ware] in processing astrophysical and seismic data, the spectral method on adaptive grids, the tracking of Lagrangian particles, and the implementation of semi-Lagrangian methods.
In many of these applications, a few large Fourier coefficients already capture the major time-invariant wave-like information of the signal, and we can thus ignore very small Fourier coefficients. To find a small set of the largest Fourier coefficients and hence a (near) optimal $B$-sparse Fourier representation of a signal that describes most of the signal characteristics is a fundamental task in applied Fourier Analysis.
An equivalent version of this problem is as follows: define the matrix $A:=(e^{2\pi ikt_{j}})_{k=0,\ldots,N;}$ $_{j=0\ldots,L-1}$, where the $t_{j}$ are the locations of the available samples. Given $S(t_{j})$, we want to reconstruct the signal $S$, or equivalently, its Fourier coefficients $\hat{S}_{k}$, so that $A\hat{S}=S$. This linear system is over-determined. Several algorithms [@Bjork][@Hanke] [@KP] have provided efficient approaches to solve this problem. Among all INFFT algorithms, the iterative CGNE approach of [@FGS] in the benchmark software NFFT 2.0 is one of the fastest methods; it takes time $O(L^{1+(d-1)/\beta}\log L)$, where $L$ is the number of available points, $d$ is the number of dimensions, and $\beta>1$ is the smoothness for the original signal. The super-linearity relationship between the running time and $N$ (recall $L=pN$, where $p$ is the percentage of available data) poses difficulties in processing large dimensional signals, which have nothing to do with the unequal spacing. It follows that identifying a sparse number of significant modes and amplitudes is expensive for even fairly modest $N$. Our goal in this paper is to discuss much faster (sublinear) algorithms that can identify the sparse representation or approximation with coefficients $a_{1},\ldots,a_{B}$ and modes $\omega_{1},\ldots,\omega_{B}$ for unevenly spaced data. These algorithms will not use all the samples $S(0),\ldots,S(N-1)$, but only a very sparse subset of them.
Our approach is based on the paper [@GGIMS] that shows how to construct the Fourier representation for a signal $S$ with $B$-sparse Fourier representation in time and space $poly(B,\log N,$ $1/\epsilon,\log(1/\delta))$ on equal spacing data. The algorithm contains some random elements (which do not depend on the signal); their approach guarantees that the error of estimation is of order $\epsilon\Vert S\Vert^{2}$ with probability exceeding $1-\delta$. The ideas in [@GGIMS] have also been applied by its authors to sparse wavelet, wavelet packet representation, and histograms [@GGIKMS]. We have dubbed the whole family of algorithms RA$\ell$STA (for Randomized Algorithm for Sparse Transform Approximation); when dealing only with Fourier Transforms, as is the case here, we specialize it to RA$\ell$SFA (F for Fourier). Zou, Gilbert, Strauss and Daubechies [@Zou] improved and implemented the algorithm greatly. It convincingly beats FFT when the number of grid points $N$ is reasonably large. The crossover point lies at $N\simeq25,000$ in one dimension, and at $N\simeq460$ for data on a $N\times N$ grid in two dimensions for a two-mode signal. When $B=13$, RA$\ell$SFA surpasses $FFT$ at $N\geq300,000$ for one dimensional signals and $1100$ for two dimensional signals.
In this paper, we modify RA$\ell$SFA to solve the irregularly spaced data problem. The new NERA$\ell$SFA (Nonequispaced RA$\ell$SFA) uses sublinear time and space $poly(B,\log L,\epsilon,\log(1/\delta),$ $\log p)$ to find a near-optimal $B$-term Fourier representation, such that $\Vert S-R\Vert^{2}\leq(1+\epsilon)\Vert S-R_{opt}\Vert^{2}$ with high probability $1-\delta$. Similar to the RA$\ell$SFA algorithm, it outperforms existing INFFT algorithms in processing sparse signals of large size.
**Notation and Terminology** Denote by $\chi_{T}$ a signal that equals 1 on a set $T$ and zero elsewhere in the time domain. We say a signal $H$ is $q$ percent pure, if there exists a frequency $\omega$ and a signal $\rho$, such that $H=ae^{2\pi i\omega t/N}+\rho$, with $|a|^{2}\geq(q\%)\| H\|^{2}$. To quantify the unevenness of the data, introduce a parameter $p=L/N$ to be the percentage of the available data over all the data, where $L$ is the number of available data. Obviously a larger $p$ corresponds to more information about the signal. We use $L^{2}$-norm throughout the paper, which is denoted by $\|.\|$. The convolution $F*G$ is defined as $F*G(t)=\sum_{s}F(s)G(t-s)$. It follows that $\widehat{F*G}(\omega)=\sqrt{N}\hat{F}(\omega)\hat{G}(\omega)$.
A Box-car filter with width $2k+1$ is defined as follows: $$\begin{aligned}
\chi_{k}(t) & = & \left\{ \begin{array}{cc}
\frac{\sqrt{N}}{2k+1} & \,\,\,\,\,\,\,\textrm{$if\,\,\,\,\,-k\leq t\leq k$ },\\
0 & \,\,\,\,\,\,\,\,\,\, if\,\,\, t>k\,\, or\,\, t<-k\end{array}\right.\end{aligned}$$ In the frequency domain, this filter is in the form of $$\hat{\chi}_{k}(\omega)=\left\{ \begin{array}{c}
\frac{sin((2k+1)\pi\omega/N)}{(2k+1)sin(\pi\omega/N)}\,\,\,\,\,\,\,\,\, if\,\,\omega\neq0\\
\,\,\,\,\,\,\,\,\,\,1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, if\,\,\,\omega=0\end{array}\right.$$
A dilation operation on signal $H$ with a dilation factor $\sigma$ is defined as $H^{(\sigma)}(t)=H(\sigma t)$ for every points $t$.
**Organization** The paper is organized as follows. In Section 2, we give the outline of the RA$\ell$SFA algorithm. Section 3 presents the modification of RA$\ell$SFA that deals with the unavailability of some samples by a greedy method. In Section 4, an interpolation technique is introduced for better performance. Finally, we compare numerical results with existing algorithms in Section 5.
Set-up of RA$\ell$SFA
=====================
Given a signal $S$ of length $N$, the optimal $B$-term Fourier representation $R_{opt}^{B}(S)$ uses only $B$ frequencies; it is simply a truncated version of the Fourier representation of $S$, retaining only the $B$ largest coefficients. The following theorem is the main result of [@GGIMS].
Let an accuracy factor $\epsilon$, a failure probability $\delta$, and a sparsity target $B \in \mathbb{N}, B \ll N$ be given. Then for an arbitrary signal $S$ of length $N$, RA$\ell$SFA will find a $B$-term approximation $R$ to $S$, at a cost in time and space of order $poly(B,\log(N), 1/\epsilon, \log(1/\delta) )$ and with probability exceeding $1-\delta$, so that $\|S-R\|^2 \leq (1+\epsilon)\|S-R_{opt}^B(S)\|^2_2$.
The striking fact is that RA$\ell$SFA can build a near-optimal representation $R$ in sublinear time $poly(\log N)$ instead of the $O(N\log N)$ time requirement of other algorithms. Its speed surpasses FFT as long as the length of a signal is sufficiently large. If a signal is composed of only $B$ modes, RA$\ell$SFA constructs $S$ without any error.
The main procedure is a Greedy Pursuit with the following steps:
\[alg:total1\][<span style="font-variant:small-caps;">Total Scheme [@Zou]</span>]{}
1. Initialize the representation signal $R$ to 0. Set the maximum number of iterations $ITER=B\log(N)\log(1/\delta)/\epsilon^{2}$.
2. Test whether $\Vert S-R \Vert$ appears to be less than some user threshold, $\iota$. If yes, return the representation signal $R$ and the whole algorithm ends; else go to step 3..
3. Locate Fourier Modes $\omega$ for the signal $S-R$ by isolation and group test procedures.
4. Estimate Fourier Coefficients at $\omega$: $\widehat{(S-R)}(\omega)$.
5. Update the representation signal $R\leftarrow R+\widehat{(S-R)}(\omega) \phi_{\omega}(t)$.
6. If the total number of iterations is less than $ITER$, go to 2; else return the representation $R$.
\[alg:total1\]
The basic idea of Algorithm \[alg:total1\] is to identify significant frequencies and then estimate their corresponding coefficients. In order to locate those nonzero frequencies, we first construct a new signal where a previous significant frequency becomes predominant. Then a recursive approach called group test finds the exact label of this predominant mode, by splitting intervals, comparing energies, and keeping only intervals with large energies. After the frequency is located, coefficient estimation procedures give a good estimation by taking means and medians of random samples.
NERA$\ell$SFA with Greedy Technique
===================================
RA$\ell$SFA samples from a signal, implicitly assuming that uniform and random sampling is possible, with a fixed cost per sample. This raises challenges for processing unevenly spaced data. Specifically speaking, Fourier coefficients and norms can not be estimated properly. Thus one has to modify steps 3 and 4 accordingly. In this section, NERA$\ell$SFA, a modified version of RA$\ell$SFA with greedy technique, is introduced to overcome these problems.
The basic idea is a greedy pursuit for an available data point. Whenever the algorithm samples at a missing data point, it searches some other random indices $t$ until it finds one available data point $S(t)$ as the substitute. This technique is used in estimating both Fourier coefficients and norms.
A good data structure is important to save running time cost. We denote the availability of a data point by a label, say +1 for available and 0 for unavailable. Hence, the label is tested to see if its corresponding sample is valid. An alternative solution is to store all the sorted labels of available data in a long list. However, each search takes time $O(\log(N))$, which introduces a $O(\log N)^{2}$ factor into the whole computation. As the empirical results show, the running time of NERA$\ell$SFA algorithm is linear to $\log N$. For this reason, we selected the first method.
We now give a more detailed discussion of the different procedures used in steps 3 and 4 of Algorithm \[alg:total1\].
Estimating Fourier Coefficients
-------------------------------
First, we give the procedure for estimating Fourier coefficients for unevenly spaced data as follows.
[<span style="font-variant:small-caps;">Estimating Individual Fourier Coefficients</span>]{} \[estcoef\]\
Input a signal $S$, a frequency $\omega$, $n=2\log(1/\delta)$, $m=8/\epsilon^2$.
1. For $i=1, \ldots, n$
2. For $j=1,\ldots, m$\
Randomly generate the index $t$ until $S(t)$ is available.\
Then let $t_{ij}=t$. Evaluate $k(t_{ij})=<S(t_{ij}), \phi_{\omega}(t_{ij})>$.
3. Take the means of $m$ samples $k(t_{ij})$, i.e. $p(i)=\sum_{j=1}^{m} {k(t_{ij})}$, where $i=1,\ldots, n$.
4. Take the median of $n$ samples $c=median_i (p(i))$, where $i=1,\ldots,n$.
5. Return $c$ as the estimation of the Fourier coefficient $\hat{S}(\omega)$.
\[alg:coeff\]
Next, we show that using unevenly spaced data leads to a very good approximation to the true coefficient. The first lemma is one of most fundamental theorems in randomized algorithms. It essentially states that by repeating an experiment enough times, a small probability event will happen eventually.
\[lm:rept\] If an event happens with probability $p$, then in the first $k> \log \delta / \log(1-p)$ iterations, it happens at least once with success probability $1-\delta$. \[lm:rept\]
In our case, only $p=L/N$ percentage of the data is available, so that $k>\log\delta/\log(1-L/N)$ trials are needed to generate one available data point with success probability at least $1-\delta$.
In fact, most of the Fourier coefficients of a characteristic function on a typical set $T$ are small, under some conditions. The following lemma makes this more explicit.
\[lm:smallfilter\] Suppose the components $X_j$ of a discrete random variable $X=(X_j)_{j=0} ^{N-1}$ are identically and independently distributed in $ \{0,1\} $, with $p = Prob(X_j=1)$. Define the random set $T=\{j \in \{0, \ldots, N-1 \} |X_j=1 \}$ to be the set of all available data; $\hat{\chi}_T(\omega)$ is the Fourier transform of $\chi_T(t)= \sum_{j=0}^{N-1}X_j$. If $p \geq \frac{1}{1+(N-1) \lambda \tau^2}$, then\
$$Prob(|\hat{\chi}_T(\omega)|^2 \geq \lambda) \leq \tau^2.$$ \[lm:smallfilter\]
First, we claim that $E(|\hat{\chi}_T(\omega)|^2) \leq \frac{(1-p)}{p(N-1)}$.\
Since $\hat{\chi}_T(\omega) = \frac{1}{pN}\sum_{j \in T}(e^{2 \pi i \omega j /N})$, we have $$\begin{aligned}
|\hat{\chi}_{T}(\omega)|^{2}=\frac{1}{p^{2}N^{2}}\sum_{j,k\in T}e^{2\pi i\omega(j-k)/N} \\
=\frac{1}{p^{2}N^{2}}\sum_{j\in T}1+\frac{1}{p^{2}N^{2}}\sum_{j,k\in T,j\neq k}e^{2\pi i\omega(j-k)/N}.
\nonumber \end{aligned}$$ It follows that $$E(|\hat{\chi}_{T}(\omega)|^{2})=\frac{1}{pN}+\frac{1}{p^{2}N^{2}}p\frac{pN-1}{N-1}\sum_{j,k=0,j\neq k}^{N-1}e^{2\pi i\omega(j-k)/N}.
\nonumber$$ Observe that $\sum_{j,k=0,j\neq k}^{N-1}e^{2\pi i\omega(j-k)/N}=|\sum_{j=0}^{N-1}e^{2\pi i\omega j/N}|^{2}-\sum_{j=0}^{N-1}1=(N\delta_{\omega,0})^{2}-N$, hence $$\begin{aligned}
E(|\hat{\chi}_{T}(\omega)|^{2}) = \frac{1}{pN}+\frac{1}{pN^{2}}\frac{pN-1}{N-1}(N^{2}\delta_{\omega,0}-N)
=\frac{1}{pN}\left\{ 1+\frac{pN-1}{N-1}(N \delta_{\omega,0}-1)\right\} \nonumber \\
=\frac{1}{pN(N-1)}\left\{ N-1+(pN-1)(N\delta_{\omega,0}-1)\right\}. \nonumber\end{aligned}$$ By Markov’s Inequality, when $\omega \neq 0$, we have $$Prob(|\hat{\chi}_{T}(\omega)|^{2}\geq\lambda)\leq\frac{E(|\hat{\chi}_{T}(\omega)|^{2})}{\lambda}=\frac{1-p}{p(N-1)\lambda}.\nonumber$$ Since $p\geq\frac{1}{1+(N-1)\lambda \tau^{2}}$, it follows that $$Prob(|\hat{\chi}_{T}(\omega)|^{2}\geq\lambda)\leq\tau^{2}. \nonumber$$ That is , for any $\omega\neq 0$, with probability at least $1-\tau^{2}$ $$|\hat{\chi}_{T}(\omega)|\leq \sqrt{\lambda}.$$
In particular, we want both $\lambda$ and $\tau$ to be small, meaning that $p$ cannot be too small itself.
Next, we consider the conditions for the two coefficients $\hat{S}(\omega)$ and $\hat{S}_{1}(\omega)=\widehat{S\cdot\chi_{T}}(\omega)$ to be close.
\[lm:diffest\] Suppose the parameters $T$, $S$, $\chi_T(t)$, $\lambda$, $\tau$, $p$ are as stated in Lemma \[lm:smallfilter\], and define $S_1(t) = S(t)\chi_T(t)$. If $ p\geq \frac{1}{1+(N-1)\lambda \tau^2}$, and $\tau \leq \sqrt{1-(1-\delta)^\frac{1}{B}}$, then, for any $\omega$, $$|\hat{S}(\omega) - \hat{S}_1(\omega)| \leq \sqrt{B \lambda} \|S\|_2.$$ with probability exceeding $1-\delta$. \[lm:diffest\]
Suppose the significant terms of signal $S$ are $\omega_i$, where $i=1, \ldots, B$.\
Since $S_1(t) = S(t) \chi_T(t)$ and thus $\hat{S}_1(\omega) =\hat{S}(\omega)* \hat{\chi}_T(\omega)$, then $$\begin{split}
\hat{S}_{1}(\omega_{j}) = \sum_{i=1}^{B}\hat{S}(\omega_{i})\hat{\chi}_T(\omega_{j}-\omega_{i})
=\hat{S}(\omega_{j})\hat{\chi}_{T}(0)+\sum_{i=1,\omega_{j}\neq\omega_{i}}^{B}\hat{S}(\omega_{i})\hat{\chi}_T(\omega_{j}-\omega_{i}) \\
=\hat{S}(\omega_{j})+\sum_{i=1,\omega_{j}\neq\omega_{i}}^{B}\hat{S}(\omega_{i})\hat{\chi}_T(\omega_{j}-\omega_{i}). \nonumber
\end{split}$$ Therefore $$|\hat{S}_{1}(\omega_{j})-\hat{S}(\omega_{j})| = |\sum_{i=1,\omega_{j}\neq\omega_{i}}^{B}\hat{S}(\omega_{i})\hat{\chi}_T(\omega_{j}-\omega_{i})|$$ $$\leq\sqrt{\sum_{i=1,\omega_{j}\neq\omega_{i}}^{B}|\hat{S}(\omega_{i})|^{2}}\sqrt{\sum_{i=1,\omega_{j}\neq\omega_{i}}^{B}|\hat{\chi}_{T}(\omega_{j}-\omega_{i})|^{2}} \nonumber \\
\leq\| S\|_2 \sqrt{\sum_{i=1,\omega_{j}\neq\omega_{i}}^{B}|\hat{\chi}_{T}(\omega_{j}-\omega_{i})|^{2}}.
\nonumber$$ Because $p\geq\frac{1}{1+(N-1)\lambda \tau^{2}}$, we have $|\hat{\chi}_{T}(\omega)|^{2}\leq\lambda$ with probability at least $1-\tau^2$ for any $\omega\neq0$. This implies that $|\hat{S}_{1}(\omega_{j})-\hat{S}(\omega_{j})|\leq\| S\|_2 \sqrt{B \lambda}$ with probability at least $(1-\tau^2)^B \geq (1-\delta)$\
Then $$|\hat{S}_{1}(\omega_{j})-\hat{S}(\omega_{j})|\leq \sqrt{B \lambda} \| S\|_2.$$ For those $\omega\notin\{\omega_{i},i=1,\ldots,B\}$, $$\begin{aligned}
\hat{S}_{1}(\omega)=\sum_{i=1}^{B}\hat{S}(\omega)\hat{\chi}_T(\omega-\omega_{i}), \nonumber \\
\end{aligned}$$ and we conclude similarly that $|\hat{S}_{1}(\omega)-\hat{S}(\omega)|\leq \sqrt{B \lambda} \| S\|_2.$, with probability at least $1-\delta$.
We shall use Algorithm \[alg:coeff\] to estimate $\hat{S}_{1}(\omega)$; we now look at how close the approximation $A$ (i.e. the output of Algorithm \[alg:coeff\]) of $\hat{S}_{1}(\omega)$ is to the true coefficient $\hat{S}(\omega)$.
\[lm:coefftot\] For a set of parameters $T$, $S$, $\chi_T(t)$, $\lambda$, $\tau$, $p$ as stated in Lemma \[lm:smallfilter\], if $ p\geq \frac{1}{1+(N-1)\lambda \tau^2}$, and $\tau \leq \sqrt{1-(1-\delta)^{1/B} }$, then Algorithm \[alg:coeff\] for signal $S_1(t)=S(t) \* \chi_T(t)$ gives a good estimation $A$ of $\hat{S}(\omega)$, such that $$|A-\hat{S}(\omega)|\leq (\sqrt{\lambda} + \sqrt{B \lambda})\|S\|_2.$$ with high probability.\[lm:coefftot\]
Lemma 4.2 in [@Zou] says that the coefficient estimation algorithm returns $A$, such that $$|A-\hat{S}_1(\omega)|\leq \sqrt{\lambda} \|S\|_2.$$ By Lemma \[lm:diffest\] $$|\hat{S}_{1}(\omega)-\hat{S}(\omega)|\leq \sqrt{B \lambda} \| S\|_2.$$ Thus $$|A-\hat{S}(\omega)| \leq |A-\hat{S}_1(\omega)|+|\hat{S}_{1}(\omega)-\hat{S}(\omega)|\leq(\sqrt{\lambda}+\sqrt{B \lambda})\| S\|_2.$$
Finally, we derive the conclusion about estimating coefficients.
\[lm:mycoeff2\] For a set of parameters $T$, $S$, $\chi_T(t)$, $\lambda$, $\tau$, $p$ as stated in Lemma \[lm:smallfilter\], if $\lambda \leq \frac{\epsilon}{2(B+1)}$ and $ p\geq \frac{1}{1+(N-1)\lambda \tau^2}$, then every application of Algorithm \[estcoef\] produces, for each frequency $\omega$ and each signal $S$, and each $\lambda>0$, with high probability, an output $A$ (after inputting $(S, \omega, \epsilon)$ ), such that $|A-\hat{S}(\omega)|^2 \leq \epsilon \|S\|_2^2$.
By Lemma \[lm:coefftot\], $$|A-\hat{S}(\omega)| \leq (\sqrt{\lambda} + \sqrt{B \lambda}) \|S\|_2.$$ Thus we have $$|A-\hat{S}(\omega)|^2 \leq 2(\lambda + B \lambda) \|S\|_2^2.$$ From the conditions $2(\lambda + B \lambda) \leq \epsilon $, it follows that $$|A-\hat{S}(\omega)|^2 \leq \epsilon \|S\|_2^2.$$
When we are able to get most of the data, the computational cost for estimating Fourier coefficients on unevenly spaced data is only slightly more than for the evenly spaced data case. The time to compute the signal value remains almost the same as for the evenly spaced data case. The extra time, in the worst case $O(\frac{\log\delta}{\epsilon_{1}^{2}p\log(1-p)})$, comes from visiting unavailable data. Fortunately, the visit operation is very fast and therefore contributes little to the total time, especially when most of the data are available.
Moreover, as in [@Zou], one can speed up the algorithm by using multi-step coarse-to-fine coefficient estimation procedures, which turns out to be more efficient than single-step accurate estimation; the proof is entirely analogous to Lemma 4.3 in [@Zou].
Estimating Norms
----------------
The basic idea for locating the label of a significant frequency is to compare the energies (i.e. the $L^{2}$ norm) of signals restricted in different frequency intervals. If the energy of some interval is relatively large, the significant mode is in that region with higher probability. We construct the following new signals to focus on certain intervals $$H_{j}(t)=\chi_{1}(t)e^{\frac{2\pi ijt}{16}}\ast\chi_{[-q_{1},q_{1}]}(\sigma t)e^{\frac{2\pi it\theta}{N}}\ast S$$ where 2$q_{1}+1$ is the filter width, $j=0,\ldots,15$, $\sigma$ and $\theta$ are random dilation and modulation factors. (Please see [@Zou] for an explanation of the role of $\sigma$ and $\theta$). For convenience, we denote $H_{j}(t)$ by $H(t)$.
We need to evaluate values $H(t)$ for random indices $t\in\{0,\ldots,N-1\}$. Note that the signal $H$ results from the convolutions of two finite bandwidth Box-car filters with the original signal $S$. Therefore, any missing point needed by the two convolutions would lead to a failure of computing $F(t)$. The total number of signal points involved depends on the number of nonzero taps in these two filters. Moreover, random dilation and modulation factors of the second Box-car filter make computation more tricky.
One naive way is to dive into the two convolutions and sample each signal point. If it is not available, stop evaluating this $F(t)$ and start with a new index $t$. This definitely increases time cost by wasting abundant computation. For example, suppose five data are needed and only one of them is missing, then the algorithm may compute four data in vain in the worst case, where the missing data point is visited last in the sequence of 5.
To avoid the above situation, we first compute the locations of all the points that will be needed for the convolution; only if they are all available will we start the computation. The locations related to the convolution are given in the following lemma.
\[lm:location\] Suppose we have a signal $H(t)=( \chi_1^{(\sigma_1)} * ( \chi_{q_1}^{(\sigma_2)} * S)^{(\sigma_3)} )^{(\sigma_4)})(t)$, where $\sigma_1$, $\sigma_2$, $\sigma_3$, and $\sigma_4$ are dilation factors. From the definition of Box car filter, the taps for $\chi_1$ lies in the interval $[-1, 1]$, the taps for $\chi_{q_1}$ in $[-q_1, q_1]$, then in order to evaluate $H(t)$, we need values of $S$ with indices at $\sigma_3 \sigma_4 t - \sigma_3 \sigma_1 i - j \sigma_2$, where integers $i=-1,\ldots, 1$, $j=-q_1,\ldots,q_1$.
To evaluate H(t), first let signal $ r=( \chi_{q_1}^{(\sigma_2)} * S)^{(\sigma_3)} $, then $$H(t)=(\chi_1^{(\sigma_1)} * r)^{(\sigma_4)}(t) =\sum_{i=-1}^{ 1} \chi_1(\sigma_1 i) r(\sigma_4 t-\sigma_1 i)$$ $$\begin{aligned}
r(\sigma_4 t-\sigma_1 i)=( \chi_{q_1}^{(\sigma_2)} * S)^{(\sigma_3)}(\sigma_4 t-\sigma_1 i)
= ( \chi_{q_1}^{(\sigma_2)} * S)(\sigma_3 \sigma_4 t - \sigma_3 \sigma_1 i) \nonumber \\
= \sum_{j=-q_1}^{q_1} \chi_{q_1}(\sigma_2 j) S(\sigma_3 \sigma_4 t- \sigma_3 \sigma_1 i - \sigma_2 j).
\end{aligned}$$ Thus, in order to get the value of $H(t)$, we need values of all $S(t^{'})$, where $t^{'}=\sigma_3 \sigma_4 t- \sigma_3 \sigma_1 i - \sigma_2 j$, with $i=-1,\ldots, 1$ and $j=-q_1,\ldots, q_1$.
The scheme of the norm estimation algorithm is as follows.
\[alg:norm\]
[<span style="font-variant:small-caps;">Norm Estimation</span>]{} \[estnorm\]\
Input: signal $H$, $k=0$, the number of iterations $M=1.2\ln(1/\delta)$.\
While $k<M$:
1. Randomly generate the index $t_k$.
2. Compute all indices needed by the two convolutions: $\Upsilon=\{t^{'}, t^{'}=\sigma_3 \sigma_4 t- \sigma_3 \sigma_1 i - \sigma_2 j \}$, where $i=-1,\ldots, 1$ and $j=-q_1,\ldots, q_1$.
3. If all the points $ t^{'}\in \Upsilon$ are available, then compute $H(t_k)$ else go to step 1 and generate another index $t_k$.
4. estimate = 60-th percentile of the sequence $\{|H(t_k)|^2 N\} $, where $k=0,\ldots,M-1$.
\[estnorm\]
If there exist satisfactory data groups, although maybe very few, the norm estimation will eventually find them. However, when most data are unavailable, the program may struggle in a long loop and take a huge amount of time. We introduce some tricks to avoid this. For example, set an upper bound MAX on the number of the loops. If it is reached, just use the sample points generated so far to estimate the norms. This technique may lead to a larger error, and thus hamper our frequency identification. However, by repeating the calculation, as stipulated by Lemma 3.2, we reduce the inaccuracy. Anyway we cannot hope to recover the signal, if $p$ is too small.
The following lemma investigates the number of repetitions to get a satisfactory data group for estimating norms.
Suppose $\chi_{q_1}$ and $\chi_{q_2}$ are two Box-car filters with numbers of taps $2q_1+1$ and $2q_2+1$ respectively. Define $D_{q_1,q_2} = \chi_{q_1} * \chi_{q_2}$. Then $D_{q_1,q_2}$ has $2q_1+2q_2+1$ nonzero taps in the time domain.
Randomly choose an index for signal $H(t)$, then after $k>\log \delta /
\log(1-(1-p)^{2q_1+2q_2+1})$ iterations, we can get at least one satisfactory index with high probability $1-\delta$.
It is easy to prove by Lemma \[lm:rept\].
Here is a new scheme for estimating norms, which uses much fewer samples than the original one and still achieves good estimation. In [@Zou], we propose a lemma that enabled us to achieve a good norm estimation by only a few samples. The following lemma is its adaption to the case of unevenly spaced data.
If a signal $H$ is 95% pure and if $r>1.2 \ln (1/\delta)$, the output of Algorithm \[estnorm\] gives an estimation of its energy which exceeds $\|H\|^2/3$ with probability exceeding $1-\delta$.
The proof is very similar to that of Lemma 4.5 in [@Zou]. We shall present only the difference of these two proofs. Suppose we sample $r$ times for the signal $H$. Let $\kappa=\{t:N|H(t)|^2<\|H\|^2/3 \}$, with $\kappa^c$ as its complement, we have $$\left |\sum_{t \in \kappa}H(t) \right |^2 \leq |\kappa| \sum_{t \in \kappa}|H(t)|^2 \leq |\kappa|^2 \frac{1}{N}\frac{1}{3} \|H\|^2.$$ On the other hand, we know that the signal is 95$\%$ pure, i.e. $|\hat{H}(\omega_0)|^2 \geq 0.95\|H\|^2$ for some $\omega_0$. By modulating, $\omega_0$ can be moved to 0; therefore, we can, without loss of generality, suppose most of the energy concentrates at the frequency 0; then $$\left |\frac{1}{\sqrt{N}} \sum_{t=1}^N H(t) \right |^2 = |\hat{H}(0)|^2 \geq 0.95 \|H\|^2.$$ So we have $$\begin{aligned}
\left |\sum_{t \in \kappa^C}H(t) \right | \geq \sqrt{0.95N} \|H\| - |\kappa| \frac{1}{\sqrt{3N} } \|H\|.\end{aligned}$$ On the other hand,$|\sum_{t\in\kappa^{C}}H(t)|\leq|\kappa^{C}|\| H\|=(N-|\kappa|)\| H\|$, so that $$N-|\kappa| \geq \left (\sqrt{0.95N} - \frac{|\kappa|}{\sqrt{3N}} \right )^2.$$ Let $\alpha = \frac{|\kappa|}{N}$; the above inequality becomes $$\alpha^2 + \left( 3-2 \sqrt{0.95*3}\right) \alpha -0.15 \leq 0.$$ Thus $0 \leq \alpha \leq 0.075 $. Define now a random variable $X_{\kappa}= \left (\sum_{i=1}^N \chi_{\kappa}(i) \right )$; it will be useful to estimate $$E(X_{\kappa})=\frac{|\kappa|}{N} \leq 0.075,$$ and the expectation of the random variable $e^{z X_{\kappa}}$, $$E(e^{X_{\kappa} z}) = e^0 Prob(\chi_{\kappa}(i)=0) + e^z Prob(\chi_{\kappa}(i)=1) = 1-\alpha + \alpha e^z.$$ Suppose now we sample the signal $H$ $r$ times, and take the percentile of the numbers $N|H(t_1)|^2, \ldots, N|H(t_r)|^2$. By Chernoff’s standard argument and similar procedure of Lemma 4.5 in [@Zou], we have for $z>0$, $$\begin{aligned}
Prob \left (\mbox{\textit{{60-th}}} \, percentile < \frac{1}{3} \|H\|^2 \right ) = \left [ (1-\alpha) e^{-0.6z} + \alpha e^{0.4 z} \right ]^r. \nonumber \end{aligned}$$ Take $z=\ln (1.25(1-\alpha)/ \alpha)$, then $$(1-\alpha)e^{-0.6z} + \alpha e^{0.4z} = 1.97 \alpha^{0.6} (1-\alpha)^{0.4}.$$ The right hand side of (35) is increasing in $\alpha$ on the interval $[0, 0.075]$; since $\alpha \leq 0.075$, we obtain an upper bound by substituting $0.075$ for $\alpha$: $$\begin{aligned}
\left [ (1-\alpha) e^{-0.6z} + \alpha e^{0.4 z} \right ]^r = \left [ 1.97 \alpha^{0.6} (1-\alpha)^{0.4} \right ]^r \leq e^{-0.90 r}.\end{aligned}$$
For $Prob \left (\mbox{\textit{{60-th}}} \, percentile < \frac{1}{3} \|H\|^2 \right ) \leq \delta$, we need $r \geq 1.2 \ln (1/\delta)$, we have $$Prob(Output \geq \|H\|^2/3) = Prob(\mbox{\textit{{60-th}}}\, percentile\,of\, N|H(t)|^2 \geq \|H\|^2/3) \geq 1-\delta.$$
This norm estimation procedure will be used repeatedly in the group testing step below.
Isolation
---------
For a significant frequency in signal $S$, isolation aims to construct a series of new signals, such that this significant frequency becomes predominant in at least one of the new isolation signals.
Given signals $S$, $S_1$, and the parameters as stated in Lemma \[lm:smallfilter\]. Suppose $F_1(t) = S_1(t)*\chi_1(t) = (\chi_T(t) S(t))*\chi_1(t)$, $F(t)= S(t) * \chi_1(t)$. If $ p\geq \frac{1}{1+(N-1)\lambda \tau^2}$, then for each $\omega$ with $|\hat{S}(\omega)|^2 > B \lambda \|S\|^2$, isolation algorithm can create a signal $F_1^{*}$, such that $$|\hat{F}_1^{*}(\omega)|^2 \geq 0.98\|F_1^{*}\|^2.$$
\[lem:iso\]
Since $ |\hat{S}(\omega)|^2 > B \lambda \|S\|^2$, we have $|\hat{S}(\omega)| > \sqrt{B \lambda} \|S\|$. Then there exists some $\eta>0$, such that $|\hat{S}(\omega)| \geq (\sqrt{\eta}+\sqrt{B \lambda})\|S\|.$ Lemma \[lm:diffest\] states that $|\hat{S}_1(\omega)-\hat{S}(\omega)| \leq \sqrt{B \lambda} \|S\|$. Therefore $$|\hat{S}_1(\omega)|\geq \sqrt{\eta}\|S\|\geq \sqrt{\eta}\|S_1\|.$$ Isolation algorithm returns $F_1^{(0)}, \ldots, F_1^{(2k)}$ with $k<O(\frac{1}{\eta})$, as described in [@GGIMS]. For any $\omega$ with $|\hat{S}_1(\omega)|^2 \geq \eta \|S_1\|^2$, there exists some $j$, such that $$|\hat{F}_1^{(j)}(\omega)|^2 \geq 0.98 \|F_1^{(j)}\|^2.$$
Let $F_1^{*}=F_1^{(j)}$, then
$$|\hat{F}_1^{*}(\omega)|^2 \geq 0.98\|F_1^{*}\|^2.$$
Theoretically, in order to capture a significant mode, we need $O(1/\eta)$ signals. However, in practice, much fewer signals is enough to achieve this goal.
Group Testing
-------------
Isolation has produced several signals, one of which contains the most significant frequency. Group testing uses repeated zoom-ins on one of the signals, and norm testing to select where to zoom in, in order to determine the frequency. The goal of group testing is thus to find the most significant mode of the signal $F_{1}^{*}$ from isolation. It uses recursive procedures MSB (Most Significant Bit) to approach this mode gradually.
*Definition*: Denote a set $\{\omega:\,\,(2l-1)N/32\leq\omega\leq(2l+1)N/32\}$ by $interval_{l}$.
Group test algorithm is given as follows.
\[alg:grouptest\][<span style="font-variant:small-caps;">Group Testing</span>]{}\
Input isolation signal $F_1^{*}$ to $F_1^{(0)}$, $i=0$, $q=1$\
While $q<N$, in the $i$-th iteration,
1. Find the most significant bit $v$ and the number of significant intervals $c$ by the procedure MSB.
2. Update $i=i+1$, modulate the signal $F_1^{(i)}$ by $ \lfloor (v+0.5)N/16 \rfloor $ and dilate it by a factor of $ \lfloor 16/c \rfloor$. Store it in $F_1^{(i+1)}$.
3. Call Group Test again with the new signal $F_1^{(i)}$, denote its output by $g$.
4. Update the accumulation factor $q = q * \lfloor 16/c \rfloor $.
5. If $g> N/2$, then $g = g -N$.
6. return $ \lfloor g/\lfloor 16/c \rfloor + (v+1/2)N/16+0.5 \rfloor(mod\,\, N)$;
The MSB procedure is as follows.
\[alg:msb\][<span style="font-variant:small-caps;">MSB (Most Significant Bit)</span>]{}\
Input: signal $F_1^{(i)}$ with length $N$, a threshold $0<\eta<1$.
1. Get a series of new signals $H_j(t) =F_1^{(i)}(t) \star (e^{2 \pi i j t/16} \chi_1 )$, $j=0, \ldots, 15$.
2. Estimate the energies $e_j$ of $H_j$, $j=0, \ldots, 15$.
3. for $l=0,\ldots,15$, compare the energies $e_l$ with all other energies $e_j$, where $j=(l+4)mod\,16, (l+5)mod\,16, \ldots,(l+12)mod\,16$. If $e_l > e_j$ for all these $j$, label it as an interval with large energy.
4. Find the longest consecutive intervals of large energies. Take their center as $v $, and the number of those intervals as $c $.
5. If $c<8$, then do the original MSB in [*[@GGIMS]*]{} to get $v$ and set $c=8$;
6. Return the dilation-related factor $c$ and the most significant bit $v$.
\[alg:msb\]
For convenience, we denote $F_{1}^{(i)}$ by $\mathbf{F_{1}}$.
Given a $ 98\%$ pure signal $\mathbf{F_1}$, suppose $G_j(t) = e^{2 \pi i j t /16} \chi_1(t)$. Then Algorithm \[alg:grouptest\], with Algorithm \[alg:msb\] as its subroutine, can find the significant frequency $\omega_1$ of the signal $\mathbf{F_1}$ with high probability.
The proof is similar to that of Lemma 5 in [@GGIMS], with some changes:
Since the signal $\mathbf{F_1}$ is $98\%$ pure, there exist a frequency mode $\omega_1$ and a signal $\rho$, such that $\mathbf{F_1}=a\phi_{\omega_1}+\rho$, where $|a|^2 \geq 0.98\|\mathbf{F_1}\|^2$ and $\|\rho\|^2 \leq 0.02\|\mathbf{F_1}\|^2$. Without loss of generality, assume $\omega_1 \in [-N/32, N/32]$. The whole region is divided into 16 subintervals $[jN/16-N/32, jN/16+N/32]$, where $j=0,\ldots, 15$. To estimate $\widehat{\mathbf{F_1}*G_0}(\omega_1)$ for $|\omega_1|\leq N/32$, we use that $|\hat{G}_0(\omega_1)|=|\hat{\chi}_1(\omega_1)|\geq 0.987$ for $|\omega_1|\leq N/32$. It follows that $$\begin{aligned}
|\widehat{\mathbf{F_1} \ast G_{0}}(\omega_1)|^{2} = N \left|\hat{\mathbf{F}}_1(\omega_1)\hat{G}_0(\omega_1)\right|^2 \geq N 0.987^{2}|\hat{\mathbf{F}}_1(\omega_1)|^{2} \geq N 0.987^{2}0.98\| \mathbf{F_1}\|^{2} \nonumber \\
\geq 0.954N\|\hat{\mathbf{F}}_1\|^{2} \geq 0.954N\|\hat{\mathbf{F}}_1\hat{G_{0}}\|^{2}=0.954\|\mathbf{F_1}\ast G_{0}\|^{2}.\end{aligned}$$ Therefore the estimation $X$ of $ \|\mathbf{F_1} * G_0\|$ satisfies: $$\begin{aligned}
X \geq \|\mathbf{F_1} * G_0\|^2/3 = \|\widehat{\mathbf{F_1}*G_0}\|^2/3 = \sum_{\omega} |\widehat{\mathbf{F_1}*G_0}(\omega)|^2 /3 \geq |\widehat{\mathbf{F_1}*G_0}(\omega_1)|^2/3 \nonumber \\
\geq 0.954N\|\mathbf{F_1}\|^2/3 \geq 0.318 N\|\mathbf{F_1}\|^2.\end{aligned}$$ Next consider the energy of $\mathbf{F_1}*G_{4}$. $$\begin{aligned}
\|\hat{\rho}\hat{G_{4}}\|^{2} =\sum_{\omega}|\hat{\rho}(\omega)\hat{G_{4}}(\omega)|^{2} \nonumber \\
\leq \sum_{\omega}|\hat{\rho}(\omega)|^{2} = \| \rho\|^{2}\leq 0.02 \| \mathbf{F_1}\|^{2}.\end{aligned}$$ Since $|\hat{G}_4(\omega_1)|<0.464$, we have $$\begin{aligned}
|\hat{\mathbf{F}}_1(\omega_1)\hat{G}_{4}(\ \omega_1)|\leq|\hat{\mathbf{F}}_1(\omega_1)||\hat{G}_{4}(\ \omega_1)| \leq |\hat{\mathbf{F}}_1(\omega_1)|0.464 \leq 0.464 \| \mathbf{F_1}\|\end{aligned}$$ Also $ \|\hat{\mathbf{F}}_1 \hat{G}_{4}\|^{2}-|\hat{\mathbf{F}}_1(\omega_1) \hat{G}_{4}(\omega_1)|^{2} \leq 0.02\| \mathbf{F_1}\|^{2}$. Thus
$$\| \hat{\mathbf{F}}_1 \hat{G}_4 \|^2 \leq 0.464^2\| \mathbf{F_1}\|^{2}+0.02\| \mathbf{F_1}\|^{2}=0.24\| \mathbf{F_1}\|^{2}.$$ It follows that $$\| \mathbf{F_1} \ast G_4 \|^2 =\| \widehat{\mathbf{F_1}*G_4}\|^2 = N \|\hat{\mathbf{F}}_1 \hat{G}_4\| \leq 0.24N\| \mathbf{F_1}\|^{2}.$$ Then we compare $\| \mathbf{F_1}\ast G_{4}\|^{2}$ with the lower bound of the estimation of $\| \mathbf{F_1}\ast G_{0}\|^{2}$, which is $$0.24N \| \mathbf{F_1}\|^{2} \leq 0.318N \| \mathbf{F_1}\|^{2},$$ which is less than the estimation for $\| \mathbf{F_1}\ast G_{0}\|^{2}.$ In general, $\omega\in interval_{j}$, for $j$ not necessarily 0. Therefore we compare $\| \mathbf{F_1}\ast
G_{j^{'}}\|^{2}$with $\| \mathbf{F_1}\ast G_{j}\|^{2}$, where $|j-j^{'}|\geq4$. If there is some $j$ with $\| \mathbf{F_1}\ast G_{j}\|^{2}$ apparently larger than $\| \mathbf{F_1}\ast G_{j^{'}}\|^{2}$, then we conclude $\omega_1 \notin interval_{j^{'}}$. Otherwise, possibly $\omega_1\in interval_{j^{'}}$. By the above argument, we can always eliminate 9 consecutive interval regions out of 16, leaving a cyclic interval of length at most $7N/16$. The remaining proof is exactly the same as Lemma 8 in paper [@GGIMS].
Remark: In [@Zou], we showed that group testing works for a Box-car filter with width more than $21$, i.e. $k>10$. In that case, $2k+1$ intervals are sufficient. A similar conclusion still holds in the unevenly spaced data case. However, the lemma above proves the success of group testing under different conditions. In our proof, we use a Box-car filter with much shorter width, namely 3 in time domain; this works well if 16 intervals are taken. In practice, we use these shorter filters; we can usually (if $B$ is small) get away with using much fewer intervals as well (e.g. 3 instead of 16).
Adaptive Greedy Pursuit
-----------------------
In summary, given a signal $S$, for an accuracy $\epsilon$ and for $B$ modes, we can find a very good approximation of the signal $S$ by using Algorithm \[alg:total1\].
\[lm:totalcost\] Given a signal $S$, an accuracy $\epsilon$, success probability $1-\delta$, Algorithm \[alg:total1\] can output a $B$-term representation $R$ with sum-square-error $\|S-R\|^2\leq (1+\epsilon) \|S-R_{opt}\|^2$, where $R_{opt}$ is the $B$-term representation for $S$ with the least sum-square-error, with time and space cost $poly(B,\log(N), \frac{1}{\epsilon}, \log(1/\delta))$ for computing and $\frac{B \log M \log N \log \delta}{\lambda log(1-(1-p)^{2q_1+2q_2+1})}$ $+\frac{\log (1/\delta) \log M}{\lambda \log p}$ for just visiting samples.
NERA$\ell$SFA with Interpolation Technique
==========================================
The greedy algorithm described above is fast. When $p$ is sufficiently large (e.g. $p>0.7$), the approach proposed and discussed in the previous section works well. For smaller $p$, the amount of time wasted to find available sample groups becomes unacceptably long. For example, when $B=2$, $N=100$, $p=0.4$, the algorithm couldn’t find the signal within 200 greedy pursuit iterations. For this reason, we introduced an interpolation technique to get an approximate value of the missing point in the norm estimation procedure. This algorithm is efficient even in smaller $p$ cases.
Lagrange Interpolation Technique
--------------------------------
The task of interpolation is to estimate $S(t)$ for arbitrary $t$ by drawing a smooth curve through all the known points [@PTVF]. It is called interpolation when the desired $t$ is between the largest and smallest of these $t_{i}$’s. We use Lagrange Polynomial Interpolation, one of the simplest and most popular interpolation techniques.
Generally, the number of interpolation points determines the degree of a polynomial. A polynomial of higher degree is smoother with smaller approximation errors at the expense of more computation. Thus we choose a second degree polynomial, as a balance between computational complexity and accuracy. It is given explicitly by Lagrange’s classical formula. If the three nearest neighbors are $(t_{1},S(t_{1}))$, $(t_{2},S(t_{2}))$, $(t_{3},S(t_{3}))$, the polynomial is $$P(t)=\frac{(t-t_{2})(t-t_{3})}{(t_{1}-t_{2})(t_{1}-t_{3})}S(t_{1})+\frac{(t-t_{1})(t-t_{3})}{(t_{2}-t_{1})(t_{2}-t_{3})}S(t_{2})+\frac{(t-t_{2})(t-t_{1})}{(t_{3}-t_{2})(t_{3}-t_{1})}S(t_{3})$$
If $S(t)$ is three times differentiable in an interval $[a,b]$, and the points $t_{1},t_{2},t_{3}\in[a,b]$ are different, then there exists some $v\in[a,b]$, such that the approximation error is $S(t)-P(t)=\frac{S^{(3)}(v)}{3!}(t-t_{1})(t-t_{2})(t-t_{3})$.
Estimate Norms with Interpolation
---------------------------------
We introduce the interpolation scheme into estimating norms. The idea is to estimate the value of a missing point by the Lagrange interpolation. The detailed algorithm for estimating norms is as follows.
[<span style="font-variant:small-caps;">Estimate Norm with interpolation technique</span>]{}\
Input: signal $H$, $k=0$, the maximum number of samples $M$.
1. Randomly generate the index $t_k$, where $k=0,\ldots, M-1$.
2. For each $k$, if $H(t_k)$ is not available, estimate $ H(t_k) $ by Lagrange interpolation; else compute $H(t_k)$ directly.
3. Estimation = 60-th percentile of the sequence $\{|H(t_k)|^2 N\} $, where $k=0,\ldots,M-1$.
Note that we use interpolation *only* in norm estimation steps, where precision is less critical. With less precise norm estimation, the localization of important modes could still work well when iterated. For coefficient estimation, which needs to be more precise, we always search for available samples.
Numerical Results
=================
In this section, we present striking numerical results of NERA$\ell$SFA, comparing to the Inverse Non-equispaced Fast Fourier Transform (INFFT) algorithms. The popular benchmark software NFFT version 2.0 is used to give performance of INFFT, with default CGNE\_R method and Dirichlet kernel. Its time cost excludes the precomputation of samples values, which takes $O(L)$. Numerical experiments show the advantage of our NERA$\ell$SFA algorithm in processing large amount of data. We begin in Section 5.1 with comparing NERA$\ell$SFA with INFFT for some one and two dimensional examples with different length. In Section 5.2, the performance for different number of modes is shown. Finally, we test the capability of NERA$\ell$SFA to recover the signal in the situation with a large amount of missing data and in presence of large noise.
All the experiments were run on an AMD Athlon(TM) XP1900+ machine with Cache size 256KB, total memory 512 MB, Linux kernel version 2.4.20-20.9 and compiler gcc version 3.2.2. The numerical data is an average of 10 runs of the code; errors are given in the $L^{2}$ norm.
Experiments with Different Length of Signals
--------------------------------------------
We ran the comparison for a 8-mode superposition signal $S(t)=\sum_{i=1}^{B}\phi_{\omega_{i}}$, plus white noise $\nu$ with the standard deviation $\sigma=0.5$, damped by a factor of $1/\sqrt{N}$, ( so that $\Vert\nu\Vert^{2}=\sigma^{2}=0.25$; since $\Vert S\Vert^{2}=8$, this implies $SNR=20\log_{10}32\thickapprox30.1dB$). Other parameters are $B=8$, $\epsilon=0.02$, $\delta=0.01$, and $p=70\%$. The missing data are randomly and uniformly distributed. NERA$\ell$SFA outperforms INFFT in speed when $N$ is large; see Table \[tab:B13\] and Figure \[fig:diffN1d\]. The corresponding crossover point is $N\geq2^{15}=32768$ . For example, to process $2^{19}=524,288$ data, more than nineteen minutes (estimated) are needed for INFFT versus approximately one second for NERA$\ell$SFA. Experiments support the theoretical conclusion that NERA$\ell$SFA would be faster than INFFT after some $N$ for a sparse signal; whatever the sparsity, i.e. whatever the value of $B$, there always exists some crossover $N$.
----------------- ------- --------------- ----------------
N INFFT NERA$\ell$SFA NERA$\ell$SFA
(+sampling) (w/o sampling)
$2^{9}$=512 0.01 0.63 0.31
$2^{11}$=2048 0.03 0.77 0.37
$2^{13}$=8192 0.17 0.90 0.46
$2^{15}$=32768 0.83 0.93 0.49
$2^{17}$=131072 4.30 1.03 0.51
$2^{19}$=524288 19.94 1.20 0.61
----------------- ------- --------------- ----------------
: \[tab:B13\]Experiments with fixed $B=8$, $p=0.7$, $d=1$ (one dimension), and varying length $N$ of signals; an i.i.d. white noise is added with $\sigma=0.5$, or $SNR\simeq30dB$ (see text). For each length of the signal, 10 different runs were carried out; the average result is shown. We did all the tests for NERA$\ell$SFA with Lagrange interpolation, as explained in the text. Two kinds of time costs for NERA$\ell$SFA are provided. One is the total running time and another is the running time excluding the sampling time. The time of INFFT does not include the precomputation time for samples.
In two dimensions, we test a noisy 6-mode superposition signal $S(t)=\sum_{i=1}^{B}\phi_{\omega_{xi}}\phi_{\omega_{yi}}+\nu$, with $B=6$, $\epsilon=0.02$, $\delta=0.01$, $p=80\%$, and $\sigma=0.1$. Missing data are randomly and uniformly distributed. As the number of grid points $N$ in each dimension grows, two dimensional NERA$\ell$SFA outperforms two dimensional INFFT at $N\geq512$, as Table \[tab:2dB13\] and Figure \[fig:diffN2d\] show. The crossover point becomes much smaller in high dimensions situation. It would not be surprising that for recovering a 6-mode three dimensional signal, NERA$\ell$SFA surpasses INFFT at a hundred sampling grid points in each dimension.
-------- ------- --------------- ----------------
N INFFT NERA$\ell$SFA NERA$\ell$SFA
(+sampling) (w/o sampling)
$128$ 0.13 2.86 1.57
256 0.73 2.60 1.46
512 3.00 3.70 2.13
1024 11.59 4.31 2.94
$2048$ 54.94 6.56 4.90
-------- ------- --------------- ----------------
: \[tab:2dB13\]Experiments with fixed $B=6$, $p=0.8$, $d=2$ (two dimensions), and varying length $N$ of signals; an i.i.d white noise is added with $\sigma=0.1$, or $SNR\simeq56dB$ (see text). For each length of the signal, 10 different runs were carried out; the average result is shown. We did all the tests for NERA$\ell$SFA with two dimensional interpolation techniques as shown in the appendix. Again, two kinds of time costs for NERA$\ell$SFA, the one with and without sampling time is provided. The time of INFFT excludes the sampling time.
Experiments with Different Number of Modes
------------------------------------------
The number of modes has an important influence on the running time since the crossover point varies for signals with different $B$. To investigate this, we did the experiments with fixed $N=2^{18}=262144$, $p=0.6$ and varying $B$. As before, we take $S$ to be a superposition of exactly $B$ modes with white noise, i.e. $S(t)=\sum_{i=1}^{B}c_{i}\phi_{\omega_{i}}+\nu$, with standard deviation of noise $\sigma=0.05$. Available data are uniformly and randomly distributed. Table \[tab:diffB\] and Figure \[fig:diffB1d\] compare the running time for different $B$ using INFFT and NERA$\ell$SFA. At first, NERA$\ell$SFA takes less time because $N$ is so large. However, the execution time of INFFT keeps constant for different number of modes $B$, while that of modified RA$\ell$SFA is polynomial of higher order. INFFT is faster than NERA$\ell$SFA when $B\geq10$. The regression techniques shows empirically that the order of $B$ in NERA$\ell$SFA is greater than quadratic. This is one of the characteristics of this version of the RA$\ell$SFA algorithms and irrelevant to the nonequispaceness of the data. (A different version of RA$\ell$SFA in [@GMS] is linear in $B$, but maybe less easily used when not all equispaced data are available. )
----------------- ------ --------------- ---------------- -------
number of modes SNR NERA$\ell$SFA NERA$\ell$SFA INFFT
$B$ (dB) (+sampling) (w/o sampling)
2 58 0.06 0.01 1.35
4 64 0.24 0.06 1.35
6 68 0.61 0.23 1.35
8 70 1.44 0.69 1.35
10 72 2.45 1.39 1.35
13 74 5.78 3.64 1.35
16 76 10.03 7.17 1.35
----------------- ------ --------------- ---------------- -------
: \[tab:diffB\]Experiments with fixed $N=2^{18}$, $p=0.6$, $d=1$ (one dimension), $\sigma=0.05$, and varying number of modes $B$ of signals. For each length of the signal, 10 different runs were carried out; the average result is shown. We did all the tests for NERA$\ell$SFA with interpolation techniques. We present two different time costs of NERA$\ell$SFA, with and without sampling.
Experiments for Different Percentage of Missing Data
----------------------------------------------------
The advantage of interpolation techniques is to recover a signal even when a large percentage of data is missing. Table \[tab:B2rec\] shows the recovery effect for a two-mode pure signal $c_{1}\phi_{\omega_{1}}+c_{2}\phi_{\omega_{2}}$, $N=10^{6}$ with all the other parameters $\epsilon$ and $\delta$ the same as before. When the percentage of available data is large, both algorithms recover the signal well with similar running time.
----------- ----------------------- ------------- ----------------------- -------------
p Time of NERA$\ell$SFA success Time of NERA$\ell$SFA success
(with interpolation) probability (w/o interpolation) probability
1 0.03 100 $\%$ 0.03 100 $\%$
0.8 0.04 100 $\%$ 0.06 100 $\%$
0.6 0.05 100 $\%$ 0.49 100 $\%$
0.4 0.05 100 $\%$ 0.45 100 $\%$
0.3 0.06 100 $\%$ - 0 $\%$
0.2 0.06 100 $\%$ - 0 $\%$
0.1 0.07 100 $\%$ - 0 $\%$
$10^{-2}$ 0.11 100 $\%$ - 0 $\%$
$10^{-3}$ 0.51 100 $\%$ - 0 $\%$
$10^{-4}$ 4.58 100 $\%$ - 0 $\%$
$0.00002$ 758.22 97 $\%$ - 0 $\%$
----------- ----------------------- ------------- ----------------------- -------------
: \[tab:B2rec\]Experiments with fixed $B=2$, $N=10^{6}$, no noise, and varying percentage of available data. Each entry is based on the average of 10 different runs. In each run, the number of iterations is limited to 200; (this also corresponds to a fixed limit to the number of samples taken.) the success probability indicates the number of runs in which all 6 modes were found. When only $30\%$ of data is available, the NERA$\ell$SFA without interpolation cannot find all two significant modes within 200 iterations.
We tried another example of signal when $N=100$. NERA$\ell$SFA without interpolation techniques fails to recover the signal with high probability if more than $45\%$ data are unavailable. In contrast, with the help of interpolation technique, the NERA$\ell$SFA can always recover the signal with only $25\%$ available data.
Experiments also show that for NERA$\ell$SFA with interpolation technique, the total number of available data, instead of the percentage of available data determines the success probability. On the contrary, The success of NERA$\ell$SFA without interpolation is determined by the percentage.
Experiments to Recover Noisy Signals
------------------------------------
To recover a signal from very noisy data is a challenging problem. The following tests are done for $S(t)=\sum_{i=1}^{B}c_{i}\phi_{\omega_{i}}+\nu$, $B=6$, $\epsilon=0.02$, $N=2^{17}$, $p=0.6$, and different standard deviation $\sigma$ for noise. The amplitude of noise is still multiplied by a factor of $1/\sqrt{N}$. As Table \[tab:noise\] shows, NERA$\ell$SFA excels at extracting information from noisy data even in the case of small signal to noise ratio.
---------- ------- ----------------------- ----------------------- ---------------- -------------
$\sigma$ SNR Time of NERA$\ell$SFA Time of NERA$\ell$SFA Relative Error Success
(dB) (+sampling) ( w/o sampling) ($\%$) probability
0 - 0.48 0.21 0.02 100%
0.5 27.60 0.56 0.22 2.00 100%
1.0 15.56 0.87 0.32 4.50 90%
1.5 8.53 3.94 1.59 5.83 80%
2.0 3.52 4.78 1.86 7.67 50%
2.5 -0.35 7.96 2.14 8.50 30%
---------- ------- ----------------------- ----------------------- ---------------- -------------
: \[tab:noise\]Experiments with fixed $B=6$, $N=2^{17}$, $p=0.6$, and varying noise levels. For each noise level, 10 different runs were carried out; the average result is shown. In each run, the number of iterations is limited to 200; (this also corresponds to a fixed limit to the number of samples taken.) the success probability indicates the number of runs in which all 6 modes were found. The average relative error is the error of reconstructed signal with respect to the original signal.
Conclusion
==========
We provide a sublinear sampling algorithm that recovers, with high probability, a $B$-term Fourier representation for an unevenly spaced signal. It is faster than any existed methods for processing sparse signals of large size. Moreover, it recovers the signal in the situation of large percentage of missing data or small signal to noise ratio.
Acknowledgments
===============
For many helpful suggestions and discussions, I would thank my adviser Ingrid Daubechies. In addition, I thank Weinan E, Anna Gilbert, Martin Strauss for their suggestions.
Appendix {#appendix .unnumbered}
========
How to interpolate the two dimensional data to get values for missing points {#how-to-interpolate-the-two-dimensional-data-to-get-values-for-missing-points .unnumbered}
----------------------------------------------------------------------------
In one dimension, values of missing points can be interpolated by its few nearest left and right available neighbors. The idea can be extended to higher dimensional cases with more techniques.
For instance, in two dimensions, we first find four nearest available neighbors of a missing point in each quadrant. Suppose a missing point is $(x,y)$, its four neighbors are $(x_{1},y_{1})$, $(x_{2},y_{2})$, $(x_{3},y_{3})$, $(x_{4},y_{4})$. The weights of neighbors can be derived by solving the following linear system of equations.
$$\left(\begin{array}{cccc}
x_{1} & x_{2} & x_{3} & x_{4}\\
y_{1} & y_{2} & y_{3} & y_{4}\\
x_{1}y_{1} & x_{2}y_{2} & x_{3}y_{3} & x_{4}y_{4}\\
1 & 1 & 1 & 1\end{array}\right)\left(\begin{array}{c}
w_{1}\\
w_{2}\\
w_{3}\\
w_{4}\end{array}\right)=\left(\begin{array}{c}
x\\
y\\
xy\\
1\end{array}\right)\label{array1}$$
However, the matrix in (\[array1\]) could be singular. In this case we choose the three nearest neighbors in different quadrants and use the following equations:
$$\left(\begin{array}{ccc}
x_{1} & x_{2} & x_{3}\\
y_{1} & y_{2} & y_{3}\\
1 & 1 & 1\end{array}\right)\left(\begin{array}{c}
w_{1}\\
w_{2}\\
w_{3}\end{array}\right)=\left(\begin{array}{c}
x\\
y\\
1\end{array}\right)\label{array2}$$
\[array2\]
The time to locate those nearest neighbors and compute corresponding weights is considered a part of precomputation and excluded from total running time.
Note that we can use geometrical arguments to simplify the pre-computation of the weights. One easily sees that the system of equations (\[array1\]) is translation invariant: the two linear system of equations
$$\left(\begin{array}{cccc}
x_{1}+l & x_{2}+l & x_{3}+l & x_{4}+l\\
y_{1}+p & y_{2}+p & y_{3}+p & y_{4}+p\\
(x_{1}+l)(y_{1}+p) & (x_{2}+l)(y_{2}+p) & (x_{3}+l)(y_{3}+p) & (x_{4}+l)(y_{4}+p)\\
1 & 1 & 1 & 1\end{array}\right)\left(\begin{array}{c}
w_{1}\\
w_{2}\\
w_{3}\\
w_{4}\end{array}\right)=\left(\begin{array}{c}
l\\
p\\
lp\\
1\end{array}\right)$$ and $$\left(\begin{array}{cccc}
x_{1} & x_{2} & x_{3} & x_{4}\\
y_{1} & y_{2} & y_{3} & y_{4}\\
x_{1}y_{1} & x_{2}y_{2} & x_{3}y_{3} & x_{4}y_{4}\\
1 & 1 & 1 & 1\end{array}\right)\left(\begin{array}{c}
w_{1}\\
w_{2}\\
w_{3}\\
w_{4}\end{array}\right)=\left(\begin{array}{c}
0\\
0\\
0\\
1\end{array}\right)$$ have the same solutions for any $l$ and $p$. That means the location of the missing points does not influence the weights. Only the geometrical shape and relative distance of the available neighbors of a missing point matters.
Thus, we compute weights for the geometrical shapes of available neighboring points which occur most often. As we go through every missing point, we check if the shape of its neighboring available points matches those popular ones; if it does, we can directly get the weights without computation. This saves a huge amount of work, especially when $p$ is large.
For example, if the four neighboring points are located in the shape of a cross with the missing point as their center, as the left side of Figure \[fig:shape\] shows, then all of the weights are equal to one quarter. This situation happens with probability $p^{4}$, which is almost $2/3$ when $p=0.9$. Another often occurring case typically has one of the four neighbors of the previous configuration moved off to the diagonal (see the right side of Figure \[fig:shape\]), which happens with probability $4p^{4}(1-p)(2-p)$, i.e. about $28\%$ when $p=0.9$. In this case, the two neighbors on the same line as the mirroring points have a weight 0.5 respectively; the other two points have weight zero. Table \[tab:shape\] shows the probabilities of these two situations as $p$ varies.
$p$ $p^{4}$ $4p^{4}(1-p)(2-p)$ sum:$p^{4}+4p^{4}(1-p)(2-p)$
----- --------- -------------------- ------------------------------
1 $100\%$ 0 $100\%$
0.9 $65\%$ $29\%$ $94\%$
0.8 $41\%$ $39\%$ $80\%$
0.7 $24\%$ $37\%$ $61\%$
0.6 $13\%$ $29\%$ $42\%$
0.5 $6\%$ $19\%$ $25\%$
: \[tab:shape\]Two possibilities corresponding to the geometrical shapes in Figure 9. The parameter $p$ is the percentage of available data. The left side of Figure 9 happens with probability $p^{4}$; the right side appears with probability $4p^{4}(1-p)(2-p)$.
[10]{} <span style="font-variant:small-caps;">R. Bass</span> <span style="font-variant:small-caps;">and</span> <span style="font-variant:small-caps;">K. Gr</span>[<span style="font-variant:small-caps;">ö</span>]{}<span style="font-variant:small-caps;">chenig</span>, *Random sampling of multivariate trigonometric polynomials*, SIAM J. Math. Anal., Vol. 36 (2004), pp. 773-795. A. B[<span style="font-variant:small-caps;">jörck</span>]{}. *Numerical Methods for Least Squares Problems*. SIAM, Philadelphia, 1996. <span style="font-variant:small-caps;">J.P. Boyd</span>, *A fast algorithm for Chebyshev, Fourier and Sinc interpolation onto an irregular grid*, J. Comput. Phys., 103 (1992), pp. 243-257. E. C*[<span style="font-variant:small-caps;">andes</span>]{}*, J. R[<span style="font-variant:small-caps;">omberg</span>]{}, and T. T[<span style="font-variant:small-caps;">ao</span>]{}, Robust Uncertainty Principles: Exact Signal Reconstruction from Highly Incomplete Frequency Information, http://arxiv.org/pdf/math.CA/0411273 H. F[<span style="font-variant:small-caps;">assbender</span>]{}, *On numerical methods for discrete least-squares approximation by trigonometric polynomials*, Math. Comput., 66(1997), pp719-741. H. [<span style="font-variant:small-caps;">Feichtinger, K</span>]{}*[<span style="font-variant:small-caps;">.</span>]{}* [<span style="font-variant:small-caps;">Gröchenig</span>]{} <span style="font-variant:small-caps;">[<span style="font-variant:small-caps;">and</span>]{}</span> [<span style="font-variant:small-caps;">T. Strohmer</span>]{}, *Efficient numerical methods in non-uniform sampling theory*, Numer. Math., 69 (1995), pp423-440. A. C. G[<span style="font-variant:small-caps;">ilbert</span>]{}, S. G[<span style="font-variant:small-caps;">uha</span>]{}, P. I[<span style="font-variant:small-caps;">ndyk</span>]{}, Y. K[<span style="font-variant:small-caps;">otidis</span>]{}, S. M[<span style="font-variant:small-caps;">uthukrishnan</span>]{}, M. S[<span style="font-variant:small-caps;">trauss</span>]{}, *Fast, small-space algorithms for approximate histogram maintenance*. STOC 2002: 389-398. <span style="font-variant:small-caps;">A.C. Gilbert, S. Guha, P. Indyk, S. Muthukrishnan and M. Strauss</span>, *Near-Optimal Sparse Fourier Representations via Sampling*, STOC, 2002 <span style="font-variant:small-caps;">A.C. Gilbert, S. Muthukrishnan and M. Strauss</span>, Improved Time Bounds for Near-Optimal Sparse Fourier Representation, to appear. <span style="font-variant:small-caps;">L. Greengard</span> <span style="font-variant:small-caps;">and J. Lee</span>. Accelerating the Nonuniform Fast Fourier Transform, SIAM Review, 46 (2004), pp. 443-454. <span style="font-variant:small-caps;">G. Grimmett and D. Stirzaker</span>. *Probability and Random Processes*. Oxford University Press, 2001. M. H[<span style="font-variant:small-caps;">anke</span>]{}. Conjugate gradient type method for ill-posed problems. Wiley, New York, 1995. <span style="font-variant:small-caps;">S. Kunis and D. Potts</span>, *Stability results for scattered data interpolation by trigonometric polynomials*, preprint. <span style="font-variant:small-caps;">S. Kunis, D. Potts</span>, *NFFT, Software, C subroutine library,* http://www.math.uni-luebeck.de/potts/nfft, 2002-2004. <span style="font-variant:small-caps;">S. Kunis, D. Potts, G. Steidl</span>, *Fast Fourier transform at nonequispaced knots: A user’s guide to a C-library*, Manual of NFFT 2.0 software. <span style="font-variant:small-caps;">Y. Mansour</span>, *Randomized interpolation and approximation of sparse polynomials* , SIAM Journal on Computing 24:2 (1995). <span style="font-variant:small-caps;">A. Oppenheim, A. Willsky with S. Nowab</span>. *Signals and Systems*. Prentice Hall, 1998. <span style="font-variant:small-caps;">W. Press, S. Teukolsky, W. Vetterling and B. Flannery</span>. *Numerical Recipes in C: the art of scientific computing*. Cambridge University Press, 1992. <span style="font-variant:small-caps;">L. Reichel, G. S. Ammar, and</span> <span style="font-variant:small-caps;">W. B. Gragg</span>. *Discrete least squares approximation by trigonometric polynomials.* Math. Comput., 57(1991), pp. 273-289. <span style="font-variant:small-caps;">A. F. Ware</span>, *Fast Approximate Fourier Transforms for Irregularly Spaced Data*, SIAM Rev., 40 (1998), pp. 838–856. <span style="font-variant:small-caps;">J. Zou, A.C. Gilbert, M. Strauss and I. Daubechies</span>, *Theoretical and Experimental Analysis of a Randomized Algorithm for Sparse Fourier Transform Analysis*, submitted to Journal of Computational Physics.
[^1]: Program of Applied and Computational Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544, (`jzou@math.princeton.edu`)
[^2]: This work was partially supported by NSF grant DMS-03168875 and AFOSR grant 109-6047.
|
---
address: |
$^1$Department of Physics, Indian Institute of Technology Kanpur, Kanpur 208016\
$^2$Centre for Lasers and Photonics, Indian Institute of Technology Kanpur, Kanpur 208016
author:
- 'Garima Bawa$^1$, Indrajeet Kumar$^2$, Saurabh Mani Tripathi$^{1,2}$'
title: 'Simultaneous Existence of Ultra-high and Ultra-low Spectral Sensitivity for Directional Couplers'
---
**Abstract**\
We present the experimental evidence for the existence of two exciting wavelengths, termed $critical$ and $cross-over$ wavelength in the transmission spectrum of fiber-optic directional coupler, whose properties are conjugate to each other. The spectral shift associated with the transmission maxima/minima suddenly flip around these wavelengths and the spectrum shows largest (nil) spectral shift for the transmission maxima/minima closest to the $critical$ ( at the $cross-over$) wavelength, corresponding to the same perturbation parameter. A theoretical explanation of the observed experimental behavior has also been presented, highlighting that the underlying mechanisms for the existence of these wavelengths are entirely different. The knowledge of the precise spectral location of these wavelengths is necessary to avoid false alarms.
Introduction
============
Spectral and power interrogation techniques are among the most widely used interrogation techniques employed in photonic systems using modal coupling in optical waveguides. In the spectral interrogation technique, the wavelength shift associated with the transmission (reflection) reference maxima/minima are measured with respect to the changes in the external perturbation parameters. In contrast, in the power interrogation technique, the changes in the external perturbation parameters are quantified in terms of the changes in the power level recorded at the fixed reference maxima/minima. Both of these techniques have their inherent advantages and drawbacks. Spectral interrogation technique, for example, is immensely accurate and is immune to input power fluctuations and/or connector losses to which the power interrogation technique suffers badly. The power interrogation technique, on the other hand, is a much cheaper alternative and does not require expensive optical spectrum analyzer and broadband sources. It is the accuracy of the detection system, and absence of false alarms, which dictates the choice of the detection technique, and therefore, spectral interrogation technique is almost always preferred over the power interrogation technique. A number of highly sensitive sensors for detection of a variety of bio [@Dandapat], physical [@Huang] and chemical [@Aray] parameters, based on spectral interrogation technique have been reported. Although offering extremely high sensitivity, these sensors are also prone to false alarms arising due to erroneous association of a particular type of wavelength shift (blue or red) with increase/decrease in measurand quantity whereas the sensor records it conversely. For example, the experimental findings reported by different groups show contrasting results (entirely opposite nature of wavelength shift) with respect to the same perturbation applied to the sensor. Employing directional couplers (DCs) fabricated using SMF-28 (Corning, New York, 14831, USA), Shanshan et al. (- 1.13 nm/$^\circ C$ over 17$ ^\circ C$ - 31.6$ ^\circ C$) [@Wang] and Yuxuan et al. (- 5.3 nm/$^\circ C$ over 35$ ^\circ C$ - 45$ ^\circ C$) [@Jiang] have reported a negative spectral shift with increase in ambient temperature while Ming et al. (+ 11.96 pm/$^\circ C$ over 247$ ^\circ C$ - 1283$ ^\circ C$ ) [@Ding] and Pengfei et al. (+ 36.59 pm/$^\circ C$ over 700$ ^\circ C$ - 1000$ ^\circ C$) [@Wang2] have reported a positive spectral shift. However, no explanation of the opposite spectral shifts is available. Similar contrasting results have also been reported for multimodal-interference-effect based sensors [@SMT1; @Garima1; @Garima2; @Talata; @Li].
In this letter, we explain this ambiguity by demonstrating the existence of two unique wavelengths in the transmission spectrum of a DC, whose properties are conjugate to each other. These wavelengths are termed as $critical$ and $cross-over$ wavelengths. The spectral shift in the transmission spectrum suddenly flip around these wavelengths and the spectrum shows largest (nil) spectral shift for the transmission maxima/minima closest to the $critical$ (at the $cross-over$) wavelength, corresponding to the same perturbation parameter. A theoretical explanation of the observed experimental behavior has also been presented, along with an explanation of the origin of these unique wavelengths.
Experimental Details
====================
In our experiments, we fabricated several DCs using the heat and fuse method [@Ghatak1]. Small sections ($\sim$7 cm) of single-mode optical fibers were unjacketed and properly cleansed in ethanol ($\geqslant$99.8$\%$ (GC), Merck Millipore); after that, the cleansed portions were twisted together. To ensure a uniform elongation, a pre-calibrated dynamic tension (7 - 22mN) was applied to both the fibers; this also prevents any kink in the twisted region while fabricating the DC. The twisted portion was then heated using a butane burner set up for 8-10 minutes with optimized fiber to flame separation, blow rate, and angle of gas flow to get a continuous softening of the fiber glass without melting the fibers. The light was launched into one of the fiber using a supercontinuum source (LEUKOS SM-30-450) and the transmission spectra through the cross port were continuously monitored using an optical spectrum analyzer (YOKOGAWA AQ6370D). Fiber fusion was stopped once a power loss of $\sim$15 dB was achieved at various minima across the transmission spectrum. The experimental setup and optical microscopic image of the fabricated DC are shown in Figs. \[setup\]($a$), and \[setup\]($b$), respectively.
![(a) Photograph of the experimental setup, and (b) optical microscopic image of the fabricated DC.[]{data-label="setup"}](Fig1.pdf){width="10cm"}
Results and Discussion
======================
The temperature response of the fabricated DCs was studied by heating the DC through a heating plate. To maintain a uniform temperature throughout the coupling region and stabilize its transmission response, the couplers were kept at fixed temperatures for $\sim$20 min before making observations. Similar to the fabrication process, bends across the coupling region were avoided by maintaining constant tension throughout the experiments by fixing the fibers to a stationary stage near launching end and applying a fixed force near the other end.
The transmission spectra were recorded at various temperatures starting from room temperature (RT) i.e., 25 $^\circ C$ till 60$^\circ C$ with an interval of 5 $^\circ C$, for a DC fabricated using two identical single-mode optical fibers (Corning SMF-28, New York, 14831, USA). To clearly depict the spectral shifts in Fig. \[cross-over\]($a$) we have plotted the transmission spectra recorded at two extreme temperatures T = 25 $^\circ C$ (RT) (black curve) and T = 60 $^\circ C$ (red curve). The interaction length of the coupler was $\sim$8 mm and coupler width was 9.8$\pm$2 $\mu m$. We observe the existence of a point of inflection, which we term as $cross-over$ wavelength, at 1.167 $\mu$m: maxima/minima lower to this wavelength show a blue spectral shift with increasing temperature whereas maxima/minima on the higher side of $cross-over$ wavelength show a red spectral shift with increasing temperature. This explains why experimental results reported by different groups showed contrasting results (entirely opposite nature of wavelength shift) with respect to the same perturbation applied to the sensor [@Wang; @Jiang; @Wang2; @Ding]. In all probability the results reported by different groups were carried out with their reference wavelengths lying on different sides of the cross-over wavelength, e.g., in [@Wang; @Jiang] the experiments were carried out near 1470 nm whereas in [@Ding] the experiments were carried out around a reference wavelength of 1220 nm. More interesting observations are made once we plot the spectral shifts corresponding to various reference maxima/minima ($\Delta\lambda_m$) in Fig. \[cross-over\]($b$). It shows ($i$) an exponential dependence of $\Delta\lambda_m$ on the location of the reference maxima/minima, and ($ii$) spectral shifts are of opposite nature around $cross-over$ wavelength and increase further as we move away from the $cross-over$ wavelength with $\Delta\lambda_m$ = 0 at 1.167 $\mu$m ($cross-over$ wavelength). The exponential dependence of the spectral shift, and hence the sensitivity (= $ \frac{d\lambda_m}{d\chi} $, $\chi$ is the perturbation parameter), on the location of maxima/minima suggests the existence of a $critical$ wavelength on the lower wavelength side similar to the one observed in the transmission spectrum of single-multi-single mode structures [@SMT1]. The $critical$ wavelength for this structure, however, does not fall in the observable wavelength range (600 nm - 1700 nm) of the optical spectrum analyzers used in our experiments. Therefore, we could not record direct evidence of $critical$ wavelength. However, the trails of it were quite nicely captured by our experiment in terms of the exponentially increasing spectral shifts (0.45 $ nm / ^{\circ} C$ at the lowest minima).
In order to get direct evidence of the $critical$ wavelength, which corresponds to maximum sensitivity, we fabricated several DCs with different interaction lengths using identical fibers: Corning SMF-28 or Fibercore$^{TM}$ PS1250/1500 photosensitive fiber, and a combination of these. Using identical fibers we observed a similar behavior of the exponential dependence of spectral shift on the spectral location of various maxima/minima in the transmission spectrum. Interestingly, however, using DCs fabricated with non-identical fibers (Fig. \[crit-cross\](a)) we observed the existence of $critical$ wavelength at 0.850 $\mu$m whose properties are conjugate to $cross-over$ wavelength observed at 1.266 $\mu$m for the same DC. To illustrate this further, we have plotted the spectral shift versus the position of the reference maxima/minima in Fig. \[crit-cross\](b). This figure clearly shows that while $\Delta\lambda_m$ is largest for the maxima/minima around the $critical$ wavelength, it is zero for maxima/minima at the $cross-over$ wavelength. Also, nature of the flip of spectral shifts around the $critical$ and $cross-over$ wavelengths are opposite to each other. We would like to mention here that the spectral shifts (54.56 pm/$^{\circ}C$ considering the first minima on both sides together) observed for this structure are not large even for sufficiently large temperature variation ($\sim$ 230 $^\circ C$), which is mainly due to the use of non-identical fibers with dopants (GeO$_2$ and B$_2$O$_3$) of opposite thermo-optic coefficients ($dn/dt$ $>$ 0 for GeO$_2$ and $dn/dt$ $<$ 0 for B$_2$O$_3$) constituting the DC.
![($a$) Experimental transmission spectra and ($b$) Spectral shifts corresponding to various transmission minima of the DC employing two identical fibers (Corning SMF-28) at T = $25\ ^\circ C$ (RT), and $60\ ^\circ C$. Cross-over wavelength is found to exist at 1.167 $\mu$m.[]{data-label="cross-over"}](Fig2.pdf){width="10cm"}
{width="15cm"}
To understand the origin of $cross-over$ $\&$ $critical$ wavelengths and their conjugate properties theoretically, we consider a DC consisting of two optical fibers (core radii being 1.75 $\mu$m over the coupler region) kept very close to each other (center to center separation of 4.2 $\mu$m), such that the evanescent fields of their propagating modes overlap, causing a periodic energy exchange between the fibers. Since the exact dopant and their concentrations in the core region of the optical fibers used in our experiments were not disclosed by the manufacturers, in our simulations we consider the cladding region of fibers to be made of fused SiO$_2$, and the core region of Corning SMF-28 made of 3.1 mole$\%$ GeO$_2$ in SiO$_2$ (henceforth Fiber-1) and that of photosensitive Fibercore$^{TM}$ 1250/1500 made of 4.03 mole$\%$ GeO$_2$ and 9.7 mole$\%$ B$_2$O$_3$ in SiO$_2$ host (henceforth Fiber-2). To check the proximity of the opto-geometric parameters used in our simulations with the experimental fibers we simulated the mode field diameter (MFD) at 1550 nm using the aforementioned doping concentrations and found an excellent agreement with the MFD of experimental fibers. We further, found that the effective indices of the fundamental mode (HE$_{11}$) for both the fibers are very close to each other, matching till the fourth decimal place, as has been shown in Fig. \[neff\](a) and \[neff\](b). The modal field distribution of the fundamental mode supported by these fibers are therefore nearly identical to each other. For ease of calculation of the coupling coefficient ($\kappa$), we have used an analytical expression employing identical fibers given below [@Ghatak1] [@Snyder] $$\kappa(d)=\dfrac{\lambda_{0}}{2\pi n_{1}}\dfrac{U^2}{a^2 V^2} \dfrac{K_{0}(Wd/a)}{K^{2}_{1}(W)}.
\label{6}$$ Here $\lambda_{0}$ is the free space wavelength, $n_1$ is the core refractive index, $a$ is the core radius, $d$ is the separation between fiber axes, and $K_\nu (x)$ represents the modified Bessel function of order $\nu$. $V$, $U$, and $W$ have their usual meaning. The periodic powers carried by the first ($ \mid a(L) \mid^2$) and the second ($ \mid b(L) \mid^2$) SMFs of the DC fabricated using identical fibers are given by [@Ghatak2] $$\mid a(L)\mid^2=1-sin^2(\kappa L)\ \ \ ;\ \ \ \mid b(L)\mid^2=sin^2( \kappa L)$$
![Spectral Variation of ($ a $) effective indices ($n_{eff}$) of fundamental mode of Fiber 1 and Fiber 2 and ($ b $) difference in effective indices of the fundamental modes of the two fibers. []{data-label="neff"}](Fig4.pdf){width="10"}
In our study, Sellmeier relation [@Adams] has been used to incorporate the wavelength dependence of the refractive indices of the fibers. Temperature-dependent refractive indices of the core, as well as cladding regions, have been obtained using the relation $n=n_{0}+(dn/dT)(T-T_0)$, where $n_{0}$ is the refractive index at room temperature $T_0$. The thermo-optic coefficient $(dn/dT)$ for fused silica, 100 $\%$ GeO$_2$ and 100 $\%$ B$_2$O$_3$ are known to be 1.06 $ \times $ 10$^{-5}/^{\circ}C$ [@SMT2], 1.94 $ \times $ 10$^{-5}/^{\circ}C$, and -3.5 $ \times $ 10$^{-5}/^{\circ}C$, respectively. Since the core regions of the SMFs are not heavily doped, considering a linear dependence of $dn/dT$ on the respective concentrations [@thermo], the value of $dn/dT$ for 3.1 mole$\%$ GeO$_2$ in SiO$_2$ was calculated to be 1.0972 $ \times $ 10$^{-5}/^{\circ}C$. The change in the core radius, a of the SMF and the interaction length, L of the DC with temperature were obtained using $\Delta a = \alpha a \Delta T$ and $\Delta L = \alpha L \Delta T$ [@SMT2].
In our simulations, we have fixed the interaction length equal to the tenth coupling length i.e., 140 $\mu m$ at $\lambda =$ 1.55 $\mu m$; and studied the transmission spectra at various temperature differences. The transmission spectra at four different temperature variations i.e. $\Delta T=0\ ^\circ C$, $100\ ^\circ C$, $200\ ^\circ C$ and $230\ ^\circ C$ are shown in Fig. \[simulated transmission\]($ a $). From Fig. \[simulated transmission\]($ a $), we observe that around the wavelength 0.879 $\mu m$ the transmission maxima/minima show ($i$) opposite spectral shifts with an increase in temperature, and ($ii$) the spectral shift of the maximas/minimas closest to it are largest. Both of these properties correspond to the $critical$ wavelength observed in our experiments. From Fig. \[simulated transmission\]($a$) we also observe that around 1.034 $\mu$m, the spectrum once again flips its nature of spectral shift similar to our observations of $cross-over$ wavelength.
{width="16"}
The overall transmission of the sensor depends upon the phase, $\phi=\kappa L$ supported by the coupler. To explain these peculiar behaviors, we first note that the variation in the phase, $\phi$ obtained through the Taylor series expansion, which, retaining the first-order terms only, gives $$\Delta \phi = \dfrac{\partial \phi}{\partial \lambda} \Delta \lambda + \dfrac{\partial\phi}{\partial T} \Delta T .
\label{8}$$
Unlike $L$, $\kappa$ is a function of both the operating wavelength ($\lambda$) as well as the perturbation parameter (here temperature $T$). For constant phase points ( i.e. $\Delta \phi =0$) we get $$\frac{\Delta \lambda}{\Delta T} = -\dfrac{\partial \phi}{\partial T}\dfrac{1}{L} \Big(\dfrac{\partial\kappa}{\partial \lambda}\Big)^{-1} \\
=-\Big[\dfrac{\partial \kappa}{\partial T} + \kappa \alpha \Big] \Big(\dfrac{\partial\kappa}{\partial \lambda}\Big)^{-1}
\label{11}$$
Since the thermal expansion coefficient $(\alpha)$ and the coupling coefficient $(\kappa)$ are essentially positive quantities, the nature of the spectral shift of the transmission spectrum will be governed by $\dfrac{\partial \kappa}{\partial T}$ and $\dfrac{\partial \kappa}{\partial \lambda}$. In Fig. \[simulated transmission\]($b$), we have plotted the spectral variation of the coupling coefficient, $\kappa$, over a wavelength range of 0.63 - 1.65 $\mu m$ at two different temperatures. The zoomed transmission spectra and the corresponding spectral variation of the coupling coefficient, $\kappa$, over a wavelength range of 0.75 - 1.05 $\mu m$ have been plotted in Fig. \[simulated transmission\]($c$) and Fig. \[simulated transmission\]($d$), respectively, clearly showing the existence of $critical$ wavelength at 0.879 $\mu m$. Fig. \[simulated transmission\]($d$) shows that while $\dfrac{\partial \kappa}{\partial T}$ is monotonically negative throughout this wavelength range, $\dfrac{\partial \kappa}{\partial \lambda}$ is positive for $\lambda < 0.879$ $\mu m$ whereas it is negative for $\lambda > 0.879$ $\mu m$. Therefore, as per equation (\[11\]) the transmission spectrum should show a red spectral shift for $\lambda < 0.879$ $\mu m$ and a blue spectral shift for $\lambda > 0.879 $ $\mu m$. This agrees excellently with nature of the spectral shift shown in Fig. \[simulated transmission\]($c$) and the experimental behavior observed in our experiment (Fig. \[crit-cross\] (c)). Similarly, the spectral variation of $\kappa$ over a shorter wavelength range of 0.9-1.1 $\mu m$ is plotted in Fig. \[simulated transmission\]($f$). We see that in contrast to Fig. \[simulated transmission\]($d$), the slope $\dfrac{\partial \kappa}{\partial \lambda}$ is negative throughout this wavelength range whereas $\dfrac{\partial \kappa}{\partial T}$ is negative for $\lambda < 1.034$ $\mu m$ and positive for $\lambda > 1.034$ $\mu m$. Thus, as per equation (\[11\]), we should get opposite spectral shifts around this wavelength, with lower wavelength side showing a blue spectral shift with increasing temperature. This again agrees excellently with the spectral shifts of the transmission spectrum plotted in Fig. \[simulated transmission\]($e$) and the experimental behavior observed in our experiment (Fig. \[crit-cross\] (d)).
We would like to mention here that while the flip of the spectral shift is common for both; the $critical$ as well as $cross-over$ wavelength, the physical mechanism responsible for the flip is entirely different for both the cases. It is also interesting to note that one wavelength corresponds to ultra-high sensitivity ($critical$ wavelength) whereas the other with ultra-low sensitivity ($cross-over$ wavelength) towards temperature.
Conclusion
==========
In conclusion, we have demonstrated and explained the existence of two unique wavelengths – the $critical$ and the $cross-over$ wavelengths in the transmission spectrum of a DC, whose properties are conjugate to each other. The $critical$ wavelength has largest and opposite spectral shifts for the transmission maxima/minima closest to and on either side of it, whereas the $cross-over$ wavelength corresponds to zero spectral shift. We have also shown that the physical mechanisms responsible for the existence of these wavelengths are completely different. Knowledge of the precise spectral location of both the wavelengths are necessary to avoid false positives/negatives in the measurand quantity and to maximize the spectral sensitivity.
Acknowledgement
===============
The work was financially supported by Science and Engineering Research Board, Government of India through projects PDF/2017/002679 and EMR/2016/007936.
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|
---
abstract: 'The only allowed Higgs superpotential term at stringy tree level in the string derived Singlet Extensions of the Minimal Supersymmetric Standard Model (SEMSSM) is $h S H_d H_u$, which leads to an additional global $U(1)$ symmetry in the Higgs potential. We propose the string inspired SEMSSM where the global $U(1)$ symmetry is broken by the additional superpotential terms or supersymmetry breaking soft terms that can be obtained naturally due to the instanton effects or anomalous $U(1)_A$ gauge symmetry. In these models, we can solve the $\mu$ problem and the fine-tuning problem for the lightest CP-even Higgs boson mass in the MSSM, generate the baryon asymmetry via electroweak baryogenesis, and predict the new Higgs physics which can be tested at the LHC and ILC.'
author:
- Tianjun Li
title: String Inspired Singlet Extensions of the Minimal Supersymmetric Standard Model
---
[**Introduction –**]{} The Minimal Supersymmetric Standard Model (MSSM) can solve the gauge hierarchy problem elegantly due to supersymmetry, has neutralino as cold dark matter candidate, and accommodates the gauge coupling unification [@Langacker:1990jh; @Amaldi:1991cn]. So, it is the most natural extension of the Standard Model (SM). However, there are a few problems within the MSSM. The bilinear supersymmetric Higgs mass term $\mu H_d H_u$ in the superpotential, where $H_d$ and $H_u$ are one pair of Higgs doublets, does not violate supersymmetry and gauge symmetry. Then the natural scale for $\mu$ is about Planck scale but not the TeV scale, which leads to the $\mu$ problem. Moreover, in order to have the lightest CP-even Higgs boson mass larger than the low bound 114 GeV from the LEP experiment, there exists a few percent fine-tuning [@Okada:1990vk].
To solve the $\mu$ problem, the Next to the Minimal Supersymmetric Standard Model (NMSSM) was proposed in which a SM singlet $S$ and a $Z_3$ discrete symmetry are introduced [@Nilles:1982dy]. The $\mu H_d H_u$ term is forbidden by the $Z_3$ symmetry, and the superpotential in the NMSSM is $$\begin{aligned}
W &=& h S H_d H_u + {{\kappa}\over {3!}} S^3 ~,~\, \label{NMSSM-SP}\end{aligned}$$ where $h$ and $\kappa$ are Yukawa couplings. After $S$ obtains a vacuum expectation value (VEV), the effective $\mu$ term $\mu_{\rm
eff}=h \langle S \rangle$ is generated. Also, the F-term of $S$ will give additional contribution to the Higgs quartic coupling, and then can increase the lightest CP-even Higgs boson mass. In addition, the lightest CP-even Higgs boson in the NMSSM can have mass around 100 GeV because of its invisible decay [@Dermisek:2005ar], and the above fine-tuning problem for its mass can be solved. Moreover, the observed baryon asymmetry can be generated via electroweak baryogenesis because there are extra CP violating phases in the supersymmetry breaking soft parameters and the trilinear soft term $A_h h S H_d H_u$ can give us strong first order electroweak phase transition [@Pietroni:1992in]. Similar results hold for the nearly MSSM and the $U(1)'$-extended supersymmetric Standard Models [@UMSSM]. Therefore, the Singlet Extensions of the Minimal Supersymmetric Standard Model (SEMSSM) is very interesting from phenomenological point of view.
On the other hand, string theory may be the only known theory which can correctly describe the quantum gravity. In string models, we may solve the $\mu$ problem in the MSSM [@Blumenhagen:2006xt], and the doublet-triplet splitting problem in the Grand Unified Theories (GUTs) [@Braun:2005ux; @Chen:2006ip]. Thus, how to indirectly test string models at the Large Hadron Collider (LHC) and the International Linear Collider (ILC) is a pretty interesting question.
As we know, in the string model building, the renormalizable terms in the superpotential, which arise from the Chern-Simmons terms in the heterotic string compactification [@Li:1997sk] or instanton effects (triangles formed by the intersections of D6-branes) in Type IIA intersecting D6-brane models [@Cremades:2003qj], have the following generic trilinear form at stringy tree level $$\begin{aligned}
W &=& y_{\phi} \phi_1 \phi_2 \phi_3 ~,~\,\end{aligned}$$ where $y_{\phi}$ is the Yukawa coupling, and $\phi_i$ are different fields. Thus, the $\mu H_d H_u$ term in the MSSM and the $\kappa
S^3/3!$ term in the NMSSM do not exist at stringy tree level in the string derived models. And only the first term $h S H_d H_u$ in the superpotential in Eq. (\[NMSSM-SP\]) is allowed where $S$ is a modulus. With only $h S H_d H_u$ term in the superpotential, we have two global $U(1)$ symmetries in the Higgs potential in which one of them is $U(1)_Y$ gauge symmetry. So, there is one global $U(1)$ symmetry, and then there exists one massless Goldstone boson with $S$, $H_d^0$ and $H_u^0$ mixing components, which is excluded from the known experiments.
In this letter, we propose the string inspired SEMSSM. The global $U(1)$ symmetry in the Higgs potential is broken by the additional superpotential terms or supersymmetry breaking soft terms. The extra superpotential terms can be realized in the string derived models via instanton effects [@Blumenhagen:2006xt]. With anomalous $U(1)_A$ gauge symmetry [@Dreiner:2003yr], we construct four simple and concrete SEMSSM. In these models, we can naturally solve the $\mu$ problem and the fine-tuning problem for the lightest CP-even Higgs boson mass in the MSSM. We also calculate the Higgs boson masses, chargino masses and neutralino masses at tree level, and predict the new Higgs physics which can be tested at the LHC and ILC. A more detail discussions will be presented elsewhere [@Tianjun].
[**Model Building –**]{} Let us consider the most general SEMSSM. The generic superpotential is $$\begin{aligned}
W &=& h S H_d H_u + \mu H_d H_u + m^2 S +
{{\mu'}\over {2!}} S^2 +
{{\kappa}\over {3!}} S^3 ~,~\, \label{Superpotential-General}\end{aligned}$$ where $\mu$, $m^2$, and $\mu'$ are mass parameters. The corresponding $F$-term scalar potential is $$\begin{aligned}
V_F &=& |h H_d H_u +m^2+ \mu' S+{{\kappa}\over {2!}} S^2|^2
\nonumber\\&& +|h S +\mu|^2 |H_u|^2 +|h S +\mu|^2 |H_d|^2 ~.~\,\end{aligned}$$ And the $D$-term scalar potential is $$\begin{aligned}
V_D &=& {{G^2}\over 8} \left(|H_u|^2 - |H_d|^2\right)^2~,~\,\end{aligned}$$ where $G^2=g_Y^{2} +g_2^2$; $g_Y$ and $g_2$ are respectively the coupling constants for $U(1)_Y$ and $SU(2)_L$. Moreover, we introduce the supersymmetry breaking soft terms $V^I_{soft}$ and $V^{II}_{soft}$ as follows $$\begin{aligned}
V^I_{soft} &=& m_{H_d}^2 |H_d|^2 + m_{H_u}^2 |H_u|^2 + m_S^2
|S|^2~,~
\label{vsoftI} \\
V^{II}_{soft} &=& -\left(A_h h S H_d H_u + B \mu_B H_d H_u
+ A_X m_X^2 S \right.\nonumber\\&& \left.
+ {1\over {2!}} B^{\prime} \mu^{\prime}_B S^2 +{1\over {3!}}
A_{\kappa} \kappa_X S^3 + {\rm H.C.} \right) ~,~\,
\label{vsoftII}\end{aligned}$$ where $m^2_{H_d}$, $m^2_{H_u}$, and $m^2_{S}$ are supersymmetry breaking soft masses, $A_h$, $B$, $\mu_B$, $A_X$, $m_X^2$, $B'$, $\mu^{\prime}_B$, and $A_{\kappa}$ are supersymmetry breaking soft mass parameters, and $\kappa_X$ is the coupling constant. In addition, if $\mu \not=0$, $m^2\not=0$, $\mu'\not=0$, or $\kappa\not=0$, we assume $\mu_B=\mu$, $m_X^2=m^2$, $\mu^{\prime}_B=\mu^{\prime}$, or $\kappa_X=\kappa$, respectively. However, even if $\mu=0$, $m^2=0$, $\mu'=0$, or $\kappa=0$, we can show that $\mu_B$, $m_X^2$, $\mu^{\prime}_B$, or $\kappa_X$ might not be zero in general, so the global $U(1)$ symmetry in the Higgs potential can be broken by the supersymmetry breaking soft terms [@Tianjun].
In the string derived models, the terms $\mu H_d H_u$, $\mu'
S^2/2!$, and $\kappa S^3/3!$ in superpotential in Eq. (\[Superpotential-General\]), which are forbidden at stringy tree level, might be generated due to the instanton effects. And the effective $\mu$ is about $M_{\rm string} e^{-A}$ where $M_{\rm
string}$ is the string scale around $10^{17}~{\rm GeV}$. So, the $\mu$ problem in the MSSM is solved if $A \sim
33$ [@Blumenhagen:2006xt]. Similar result holds for $\mu'$. However, the $\kappa S^3/3!$ term generated from instanton effects might be highly suppressed.
To construct the string inspired SEMSSM, we consider the models with an anomalous $U(1)_A$ gauge symmetry [@Dreiner:2003yr]. In string model building, there generically exists one anomalous $U(1)$ gauge symmetry in the heterotic string model building [@Dreiner:2003yr] or up to four in the Type II orientifold model building [@Ibanez:2001nd]. The corresponding anomalies are cancelled by the (generalized) Green-Schwarz mechanism [@Green:1984sg]. We introduce a SM singlet field $\phi$ with $U(1)_A$ charge $-1$. To cancel the Fayet-Iliopoulos term of $U(1)_A$, we assume that $\phi$ obtains a VEV so that the $U(1)_A$ D-flatness and supersymmetry can be preserved. Interestingly, $\langle \phi \rangle/M_{\rm Pl}$ is about 0.22, where $M_{\rm Pl}$ is the Planck scale [@Dreiner:2003yr]. Moreover, to break the supersymmetry, we introduce a hidden sector superfield $Z$ whose F component acquires a VEV around $10^{21}~{\rm
GeV}^2$.
We assume that the $U(1)_A$ charges for $S$ and $Z$ are $n+p/q$ and $m+p'/q'$, respectively, where $m$ and $n$ are integers, ($p$, $q$) and ($p'$ and $q'$) are relatively prime positive integers, or $p/q$ or $p'/q'$ is zero. To have the $h S H_d H_u$ term in superpotential, we also assume that the total $U(1)_A$ charges for $H_d$ and $H_u$ are $-n-p/q$, but we do not give the explicit charges for $H_d$ and $H_u$ which are irrelevant to our discussions. Moreover, if $m+p'/q'$ is non-zero, the gaugino masses can not be generated via F-terms $ZW^{\alpha} W_{\alpha}/M_{\rm Pl}$. To have the gaugino masses, we can introduce another $U(1)_A$-uncharged hidden-sector superfield $Z'$ whose F component acquires a VEV. In fact, in the string model building, both dilaton and moduli fields can break the supersymmetry due to their F-component VEVs. In addition, the supersymmetry breaking soft mass terms in $V^I_{soft}$ can be generated via D-term operators $$\begin{aligned}
\int d^4x d^2\theta d^2\overline{\theta} {{\overline{Z} Z} \over
{M_{\rm Pl}^2}} \left( |S|^2+ |H_d|^2+ |H_u|^2 \right)~,~\,\end{aligned}$$ where for simplicity we neglect the coefficients of these operators in such kind of discussions in this paper. The first term $A_h h S
H_d H_u$ in $V^{II}_{soft}$ can be generated via the following F-term operator $$\begin{aligned}
\int d^4x d^2\theta {{Z ~({\rm or} ~Z')}\over {M_{\rm Pl}}} h S H_d
H_u + {\rm H. C.}~.~\end{aligned}$$
[*Model A –*]{} We choose the following $U(1)_A$ charges for $S$ and $Z$ $$\begin{aligned}
m+n~=~47, ~~p/q~=~1/5 ~,~~p'/q'=4/5~.~\,\end{aligned}$$ Then the $U(1)_A$ allowed renormalizable superpotential is $$\begin{aligned}
W &=& h S H_d H_u ~.~\, \label{Superpotential-Model-A}\end{aligned}$$ The additional supersymmetry breaking soft term $V^{II}_{soft}$ can be generated via the following operator (the other operators are forbidden by $U(1)_A$ or negligible) $$\begin{aligned}
\int d^4x d^2\theta M_{\rm string} Z S \left({{\phi}\over {M_{\rm
Pl}}}\right)^{48} + {\rm H. C.}~.~\,\end{aligned}$$ So, we have $$\begin{aligned}
V^{II}_{soft} &=& -\left(A_h h S H_d H_u
+ A_X m_X^2 S + {\rm H.C.} \right) ~,~\,
\label{vsoftII-MA}\end{aligned}$$ where $A_h \sim A_X \sim 10^2~{\rm GeV}$, and $m_X^2 \sim
10^{4-6}~{\rm GeV}^2$. Interestingly, the global $U(1)$ symmetry in the Higgs potential is indeed broken by the supersymmetry breaking soft term $A_X m_X^2 S$.
[*Model B –*]{} We choose the following $U(1)_A$ charges for $S$ and $Z$ $$\begin{aligned}
n=-22, ~~p/q~=~0 ~,~~m=0,~~p'/q'=0~.~\,\end{aligned}$$ The additional relevant F-term and D-term operators are $$\begin{aligned}
\int d^4x d^2\theta \left(M_{\rm string} H_d H_u + ZH_dH_u \right)
\left({{\phi}\over {M_{\rm Pl}}}\right)^{22} \nonumber
\\
+ \int d^4x d^2\theta d^2{\overline{\theta}} \left(\overline{Z} S +
{{\overline{Z} ZS}\over {M_{\rm Pl}}} \right)
\left({{\overline{\phi}}\over {M_{\rm Pl}}}\right)^{22} + {\rm H.
C.}~.~\end{aligned}$$ Thus, the superpotential in Model B is $$\begin{aligned}
W &=& h S H_d H_u + \mu H_d H_u + m^2 S ~,~\,
\label{Superpotential-Model-B}\end{aligned}$$ where $\mu \sim 10^{2-3}~{\rm GeV}$ and $m^2 \sim 10^{4-6}~{\rm
GeV}^2$. And the supersymmetry breaking soft terms in $V^{II}_{soft}$ are $$\begin{aligned}
V^{II}_{soft} &=& -\left(A_h h S H_d H_u
+B \mu_B H_d H_u \right.\nonumber\\&& \left.
+ A_X m_X^2 S
+ {\rm H.C.} \right),\,
\label{vsoftII-MB}\end{aligned}$$ where $A_h \sim B \sim \mu_B \sim A_X \sim 10^{2-3}~{\rm GeV}$, and $m_X^2 \sim 10^{4-6}~{\rm GeV}^2$.
[*Model C –*]{} In Model B, we consider the gauge mediated supersymmetry breaking scenario where the VEV of F component of $Z$ can be about $10^{10}~{\rm GeV}^2$. And then the tadpole term $m^2 S$ in the superpotential can be neglected. Thus, the superpotential in Model C is $$\begin{aligned}
W &=& h S H_d H_u + \mu H_d H_u ~.~\,
\label{Superpotential-Model-C}\end{aligned}$$ And the supersymmetry breaking soft terms in $V^{II}_{soft}$ are $$\begin{aligned}
V^{II}_{soft} &=& -\left(A_h h S H_d H_u
+B \mu_B H_d H_u \right).\,
\label{vsoftII-MC}\end{aligned}$$
Model C can also be considered as the string derived model with $h S
H_d H_u$ superpotential term where the extra $\mu H_d H_u$ term arises from instanton effects [@Blumenhagen:2006xt].
[*Model D –*]{} We choose the following $U(1)_A$ charges for $S$ and $Z$ $$\begin{aligned}
n=11, ~~p/q~=~1/2 ~,~~m=0,~~p'/q'=0~.~\,\end{aligned}$$ The additional relevant F-term operators are $$\begin{aligned}
\int d^4x d^2\theta \left(M_{\rm string} S^2 + ZS^2\right)
\left({{\phi}\over {M_{\rm Pl}}}\right)^{23} + {\rm H. C.} ~.~\end{aligned}$$ So, the superpotential is $$\begin{aligned}
W &=& h S H_d H_u + {{\mu'}\over {2!}} S^2~,~\,
\label{Superpotential-Model-D}\end{aligned}$$ where $\mu' \sim 10^2~{\rm GeV}$. And the supersymmetry breaking soft terms in $V^{II}_{soft}$ are $$\begin{aligned}
V^{II}_{soft} = -\left(A_h h S H_d H_u
+ {1\over {2!}} B^{\prime} \mu^{\prime}_B S^2
+ {\rm H.C.} \right),\,
\label{vsoftII-MD}\end{aligned}$$ where $A_h \sim B' \sim \mu'_B \sim 10^{2-3}~{\rm GeV}$.
Model D can be considered as the string derived model with $h S
H_d H_u$ superpotential term where the extra $\mu' S^2/2!$ term arises from instanton effects [@Blumenhagen:2006xt]. However, there exists a $Z_4$ symmetry in Model D, where $H_d$ and $H_u$ have charge 1, and $S$ has charge 2. To avoid the domain wall problem after symmetry breaking, we can turn on tiny instanton effects to break the $Z_4$ symmetry by generating small high-dimensional operators, and then we can dissolve the domain wall.
Model $\langle H_d^0 \rangle $ $\langle H_u^0 \rangle $ $\langle S \rangle $ $H^{\pm}$ $H_1^0$ $H_2^0$ $H_3^0$ $A_1^0$ $A_2^0$
------- -------------------------- -------------------------- ---------------------- ----------- --------- --------- --------- --------- ---------
A 119 127 213 205 67 196 210 127 251
B 123 123 188 179 45 184 206 142 214
C 123 123 161 165 66 148 171 31 214
D 120 126 167 176 67 145 181 39 225
: The Higgs VEVs, and the charged, CP-even, and CP-odd Higgs boson masses in GeV at tree level. \[tab:Higgs\]
[**Phenomenological Consequences –**]{} We shall calculate the Higgs boson masses, the chargino and neutralino masses at tree level in our models where we neglect the loop corrections for simplicity. The input parameters with dimensions of mass or mass-squared are chosen in arbitrary units. After finding an acceptable minimum they are rescaled so that $\sqrt{\langle H_d^0 \rangle^2+ \langle H_u^0
\rangle^2} \simeq 174.1$ GeV. For Model A, we choose: $h=0.7$, $m_{H_d}^2 =-0.1$, $m_{H_u}^2 =-0.2$, $m_{S}^2 = 0.1$, $A_h = 1.0$, $A_X=0.68$, $m_X^2=0.6$. And the VEVs for the Higgs fields at the minimum are $\langle H_d^0 \rangle =0.7031$, $\langle H_u^0 \rangle
=0.75$, and $\langle S \rangle =1.2563$. For Model B, we choose: $h=0.7$, $\mu=-0.2$, $m^2=-0.3$, $m_{H_d}^2 =-0.1$, $m_{H_u}^2
=-0.1$, $m_{S}^2 = 0.1$, $A_h = 0.6$, $B=-0.1$, $\mu_B=-0.2$, $A_X=-1.9$, $m_X^2=-0.3$. And the VEVs for the Higgs fields are $\langle H_d^0 \rangle =0.8625$, $\langle H_u^0 \rangle =0.8625$, and $\langle S \rangle =1.3156$. For Model C, we choose: $h=0.7$, $\mu=-0.1$, $m_{H_d}^2 =-0.1$, $m_{H_u}^2 =-0.1$, $m_{S}^2 = -0.6$, $A_h = 2.0$, $B=-0.6$, $\mu_B=-0.1$. And the VEVs for the Higgs fields are $\langle H_d^0 \rangle =1.5875$, $\langle H_u^0 \rangle
=1.5875$, and $\langle S \rangle =2.075$. For Model D, we choose: $h=0.7$, $\mu'=-0.3$, $m_{H_d}^2 =-0.1$, $m_{H_u}^2 =-0.4$, $m_{S}^2
= -0.68$, $A_h = 2.0$, $B'=-0.6$, $\mu'_B=-0.3$. And the VEVs for the Higgs fields are $\langle H_d^0 \rangle =1.6375$, $\langle H_u^0
\rangle =1.7203$, and $\langle S \rangle =2.275$.
We present the Higgs VEVs, the charged Higgs boson ($H^{\pm}$) mass, CP-even Higgs boson ($H_1^0$, $H_2^0$, and $H_3^0$) masses, and CP-odd Higgs boson ($A_1^0$ and $A_2^0$) masses in Table \[tab:Higgs\]. Interestingly, the couplings of the CP-even Higgs boson $H_1^0$ and $H_2^0$ with $Z^0$ gauge boson almost vanish, and only the heaviest CP-even Higgs boson $H_3^0$ can couple to $Z^0$ [@Tianjun]. Thus, the fine-tuning problem for the lightest CP-even Higgs boson mass in the MSSM from the LEP constraints can be completely relaxed, and we have new Higgs physics at the LHC and ILC because $H_3^0$ has mass around 190 GeV. Moreover, to calculate the chargino and neutralino masses, we choose the positive and negative gaugino masses $M_1$ and $M_2$ for $U(1)_Y$ and $SU(2)_L$: (1) $M_1=150$ GeV, and $M_2=300$ GeV; (2) $M_1=-150$ GeV, and $M_2=-300$ GeV. The masses for charginos ($\tilde \chi_1^{\pm}$ and $\tilde
\chi_2^{\pm}$) and neutralinos ($\tilde \chi_i^{0}$ with $i=1, 2,
..., 5$) are given in Table \[tab:char-neut\].
Model $M_i$ $\tilde \chi_1^{\pm}$ $\tilde \chi_2^{\pm}$ $\tilde \chi_1^{0}$ $\tilde \chi_2^{0}$ $\tilde \chi_3^{0}$ $\tilde \chi_4^{0}$ $\tilde \chi_5^{0}$
------- -------- ----------------------- ----------------------- --------------------- --------------------- --------------------- --------------------- ---------------------
A $> 0$ 115 334 68 88 175 217 336
A $ < 0$ 163 314 68 156 169 217 314
B $> 0$ 75 328 56 81 167 184 330
B $ < 0$ 118 315 81 125 156 184 316
C $> 0$ 76 329 58 80 167 185 330
C $ < 0$ 120 315 80 127 156 185 316
D $> 0$ 87 330 60 68 169 200 331
D $ < 0$ 132 315 61 138 156 200 315
: The chargino and neutralino masses in GeV. \[tab:char-neut\]
[**Conclusions –**]{} In the string derived SEMSSM, $h S H_d H_u$ is the only allowed superpotential term at stringy tree level. Then there exist an additional global $U(1)$ symmetry in the Higgs potential, and the axion problem. We propose the string inspired SEMSSM in which the global $U(1)$ symmetry is broken by the additional superpotential terms or supersymmetry breaking soft terms. The extra superpotential terms can be obtained via instanton effects in the string derived models. With anomalous $U(1)_A$ gauge symmetry, we present four simple and concrete SEMSSM. In these models, we can naturally solve the $\mu$ problem and the fine-tuning problem for the lightest CP-even Higgs boson mass in the MSSM, and generated the observed baryon asymmetry via electroweak baryogenesis. In addition, we calculate the Higgs boson masses, chargino masses and neutralino masses at tree level, and predict the new Higgs physics which can be tested at the LHC and ILC. Confirmation one of these models at the future colliders might give us indirect implication of string theory.
[**Acknowledgments –**]{} This research was supported in part by the Cambridge-Mitchell Collaboration in Theoretical Cosmology.
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---
abstract: |
Data Acquisition and Control Systems used in high energy physics experiments, such as those which will take place in the Large Hadron Collider (LHC) at CERN, require the specification of data formats and transmission protocols as well as the use of high speed links and interfaces.
In this context, a new Frame Segmentation process will be presented and discussed, based on data formats adopted by the LHCb experience for the interconnection of two standardized systems: S-link and Gigabit Ethernet. Simulation results of the transfer capacity of the proposed mechanism will be also reported, together with guidelines for its physical implementation.
author:
- 'Joaquim E. Neves'
- Richard Jacobsson
- Niko Neufeld
- Beat Jost
title: |
S-Link to Gigabit Ethernet Adapter\
New Frame Segmentation for LHCb Data Acquisition System
---
INTRODUCTION {#s1}
============
The Large Hadron Collider beauty experiment for precise measurements of CP violation [@SLG1] and rare decays (LHCb) [@SLG2] requires High Speed Interconnect (HSI) Systems in order to transport the large amount of data generated by the detectors connected to the Front-end Motherboards (FEMB) to the storage devices connected to the Read-out Motherboards (ROMB).
In the opposite direction, only a small volume of data generally has to be transferred between those parts for the control and management of the whole system.
The LHCb experiment adopted Gigabit Ethernet (GbE) [@SLG3] as the link technology from the output of the FEMB electronics boards to the input of the ROMB Sub-Farm Controllers (SFCs), and S-Link [@SLG4], as a standard interface between the FEMB and the Data Acquisition (DAQ) system [@SLG2].
The Architecture of the LHCb Data Acquisition, based both on full S-link transmission and on the S-Link to Gigabit Ethernet Adapter (SGbEA) System, is depicted in Figure \[f1\]. Since the Readout Network already supports the GbE, one S-Link card for the FEMB, based on GbE, was designed and built within the scope of the collaboration with the Atlas LHC experiment [@SLG5].
Next, in section \[s1\], the technical specification of the LHCb Data Acquisition System is discussed, while section \[s3\] describes the new frame segmentation mechanism of the SGbEA. Simulation results for SGbEA throughput, using two different operation modes and data packet lengths, are presented in section \[s4\]. The Section \[s5\] deals with the physical implementation of the New Frame Segmentation and Section \[s6\] summarizes the main conclusions.
{width="140mm"}
SYSTEM ARCHITECTURE {#s2}
===================
The S-LINK specification defines a simple FIFO-like user interface at both ends of the transmission link which remains independent of the technology used to implement the physical link and which provides the transfer of event data and control words, error detection, optional flow control and test facilities [@SLG6]. The S-link specification also describes the interfaces between the FEMB and the Link Source Card (LSC) and between the Link Destination Card (LDC) and the ROMB, in either simplex or duplex version.
In the simplex version, since there is no communication path from the LDC to the LSC, the transmission is unidirectional, while, in the duplex version, the return channel between the LDC and the LSC allows the transmission of flow control commands from the ROMB to the FEMB. In both versions a single, high-density, 64-pin connector is used to connect the FEMB to the LSC and the LDC to the ROMB, allowing, per clock cycle, the transport of a 32 bit word at the frequency of 40 MHz, which corresponds to a throughput of 1.28 Gbit/s.
There is more than one solution for the implementation of the Front-End Multiplexers (FEMs) and the Readout Units (RUs). This is because, in the LHCb experiment, the main difference between these modules results from the fact that the RUs have to interface to the Readout Network, and hence must respect the GbE flow control protocol, which is not true for the links between FEMs and RUs [@SLG2].
The SGbEA is a possible solution to interconnect FEMs and RUs, as is shown in Figure \[f1\]. This module has to implement the Ethernet functionalities of the Physical and Media Access Control (MAC) Layers, and the S-link specifications. The protocol conversion between S-link and GbE can be implemented within a FPGA, together with memories (FIFO, RAM and Registers), as can be seen in Figure \[f2\].
The SGbEA has to generate the Start Of Packet (SOP) and the End Of Packet (EOP) signals for the MAC device and to stop and restart the transmission when receiving watermark flags from the MAC or S-link FIFOs. Optionally, it can process the fragmentation of the S-link Packets on Ethernet frames, as is explained in the next section.
NEW FRAME SEGMENTATION {#s3}
======================
Since within the communication links, between LHCb FEMs and RUs, the data can be transported with two different formats, the SGbEA has to support two operational modes: Short Packet Mode (SPM) and Long Packet Mode (LPM).
In SPM, the length of the packets generated by the FEMB is variable, but always smaller than the maximum length of the Ethernet MAC frames. On the other hand, since the header of the LHCb data format already includes the header of the MAC Frames, no protocol conversion is necessary and the S-link packet is forwarded directly from the S-link FIFO to the MAC FIFO.
In LPM, the FEMB can generate data packets ranging in length from 52 Bytes to 32 Kbytes.
In this operational mode the length of the S-link packet can be higher than the maximum length of the Ethernet MAC frames. For this reason, the header of each S-link packet, which contains a field with its length, has to be memorized in the SGbEA RAM, before calculating the number of fragments into which it must be split in order to be transmitted.
{width="145mm"}
In this mode, the SGbEA inserts, within each fragment header, the Type/length of the frame on the Ethernet header field, together with the number of fragments and the current fragment number of the S-link packet.
Optionally, in both operational modes, the Ethernet Sources and Destination Address can be generated by the FEMB (on a packet-by-packet basis), or can be previously inserted in the SGbEA Registers, by the control system and then transferred to each packet fragment header.
SIMULATION RESULTS {#s4}
==================
A VHSIC Hardware Description Language (VHDL) model of the SGbEA module was developed for simulation purposes on the VisualHDL platform on a Computer-aided Engineering (CAE) system, together with an S-link packet generator.
Packets with different lengths and data formats were generated by this model at the S-link interface in either Long or Short Packet Mode, in order to be processed by the SGbEA and transmitted over the Gigabit Ethernet.
In both cases the S-link overhead, introduced by the control words used for signaling the beginning and the end of the LHCb data packets, was reduced to a minimum: one word for the start and another one for the end of packet. In another interface, at the Ethernet side, in addition to the MAC overhead, a minimum inter frame gap of 12 bytes was also guaranteed.
Short Packet Mode
-----------------
Since in SPM there is no frame segmentation, the inclusion of a MAC frame header at the beginning of the packet is the only constraint imposed on the data format.
Figures \[f3\] and \[f4\] report some results for the SGbEA simulation in SPM. The maximum throughput for different packet lengths is present in Figure \[f3\], while Figure \[f4\] shows the relationship between the effective throughput reached with those packets, and Gigabit Ethernet capacity.
As the figures show, due to the overhead of the Ethernet frames, when short packets with a size to the order of tens of bytes are transferred, the throughput is clearly lower than the transmission capacity, both in the S-link and in the GbE interfaces. On the other hand, for longer packets with over a hundred bytes, the transfer rate begins to be limited by the effective GbE capacity.
It is interesting to note that, for 64 byte packets, the reference value for packet length on several interfaces of the LHCb data acquisition system, the throughput is near 1.4 M Packet/s, allowing the use of SGBE on these interfaces with a trigger rate of about 1.1 MHz.
{width="160mm"}
{width="150mm"}
Long Packet Mode
----------------
Figures \[f5\] and \[f6\] present simulation results for Frame Segmentation in LPM. The maximum throughput of the system is shown in Figure \[f5\], while Figure \[f6\] shows the relationship between the effective throughput achieved on the transmission of those packets and Gigabit Ethernet capacity.
In this mode, the LHCb data packets presented on the S-link interface are of variable length, as in SPM, but now the packet header has 52 bytes, including the header of the MAC frame.
{width="150mm"}
{width="150mm"}
As was expected, Figure 5 shows that the throughput achieved for packets of a length of less than 64 bytes is lower than that achieved in SPM, but is still higher than 1,2 M Packet/s, which means that the LPM can also be used with these packet lengths on the interfaces with a trigger rate of about 1,1 MHz.
For longer packets, the decrease of the throughput is proportional to the increase of packet length, as the overhead introduced by the fragmentation process becomes negligible. As is depicted in Figure 6, the effective occupancy of the Ethernet payload increases with the length of the packet for short length packets - as was observed in the SPM simulations - and remains constant with longer packets.
PHYSICAL IMPLEMENTATION {#s5}
=======================
The Frame Segmentation mechanism for LHCb Data Acquisition System described previously has been implemented within a FPGA on the S-Link to Gigabit Ethernet Adapter board itself, developed at the Argonne National Laboratory for Atlas LHC Experience. This prototyping board was implemented on a small daughter board over a PCI [^1] Mezzanine Card (PMC), according to the IEEE Common Mezzanine Card standard [@SLG7].
A new version of the SGbEA has been specified for implementation in the near future. Since this new board will already be connected to the FEMB, the new Frame Segmentation can be fully tested, at the maximum S-link throughput, without the present constrains of the PCI Bus.
The single port MAC Controller [@SLG8] could be replaced by a dual port [@SLG9] that supports the Gigabit Media Independent Interface (GMII) and the Ten-Bit Interfaces (TBI), allowing the Ethernet transmission over copper and/or fiber media. The Physical Layer module could also be implemented with a single dual port transceiver, either for copper or fiber media [@SLG10]. While one serial port is used in the copper interface, by connecting this port to a magnetic device [@SLG11], the other is used in the fiber interface, by connecting this port to the optical module.
CONCLUSIONS {#s6}
===========
A new Frame Segmentation mechanism, with flow control capabilities, has been implemented over the S-Link to Gigabit Ethernet Adapter for the LHCb Data Acquisition System.
This mechanism is not only able to halt the transmission, in case of overflow, but can also optionally insert the Ethernet Source and Destination Addresses in the MAC frames.
As the simulation results show, this technique permits the transmission of the Short and/or Long Packets onto High Speed Interconnect Systems, which will be used in LHC experiments at CERN.
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|
---
abstract: 'Quantum coherence and nonlocality capture nature of quantumness from different aspects. For the two-qubit states with diagonal correlation matrix, we prove strictly a hierarchy between the nonlocal advantage of quantum coherence (NAQC) and Bell nonlocality by showing geometrically that the NAQC created on one qubit by local measurement on another qubit captures quantum correlation which is stronger than Bell nonlocality. For general states, our numerical results present strong evidence that this hierarchy may still hold. So the NAQC states form a subset of the states that can exhibit Bell nonlocality. We further propose a measure of NAQC that can be used for a quantitative study of it in bipartite states.'
author:
- 'Ming-Liang Hu'
- 'Xiao-Min Wang'
- Heng Fan
title: Hierarchy of the nonlocal advantage of quantum coherence and Bell nonlocality
---
Introduction {#sec:1}
============
Quantum correlations in states of composite systems can be characterized from different perspectives. From the applicative point of view, they are also invaluable physical resources which are recognized to be responsible for the power of those classically impossible tasks involving quantum communication and quantum computation [@Nielsen]. Stimulated by this realization, there are a number of quantum correlation measures being put forward up to date [@nonlocal; @entangle; @steer; @discord]. Some of the extensively studied measures include Bell nonlocality (BN) [@nonlocal], quantum entanglement [@entangle], Einstein-Podolski-Rosen steering [@steer], and quantum discord [@discord]. For two-qubit states, a hierarchy of these quantum correlations has also been identified [@new01; @new02; @hierarchy; @Angelo1; @Angelo2; @Angelo3]. This hierarchy reveals different yet interlinked subtle nature of correlations, and broadens our understanding about the physical essence of quantumness in a state.
Quantum coherence is another basic notion in quantum theory, and recent years have witnessed an increasing interest on pursuing its quantification [@Plenio; @Hu]. In particular, based on a seminal framework formulated by Baumgratz *et al.* [@coher], there are various coherence measures being proposed [@meas6; @asym1; @co-ski2; @meas4; @dist2; @measjpa; @new1]. This stimulates one’s enthusiasm to understand them from different aspects, as for instance the distillation of coherence [@dist2; @distill], the role of coherence played in quantum state merging [@qsm], and the characteristics of coherence under local quantum operations [@create1; @create2; @mc1; @mc2] and noisy quantum channels [@fro1; @fro2]. Moreover, some fundamental aspects of coherence such as its role in revealing the wave nature of a system [@path1; @path2], its tradeoffs under the mutually unbiased bases [@comple1] or incompatible bases [@comple2], have also been extensively studied.
Conceptually, coherence is thought to be more fundamental than various forms of quantum correlations, hence it is natural to pursue their interrelations for bipartite and multipartite systems. In fact, it has already been shown that coherence itself can be quantified by the entanglement created between the considered system and an incoherent ancilla [@meas1]. There are also several works which linked coherence to quantum discord [@Yao; @Ma; @Hufan] and measurement-induced disturbance [@Huxy].
In a recent work, Mondal *et al.* [@naqc] explored the interrelation of quantum coherence and quantum correlations from an operational perspective. By performing local measurements on qubit $A$ of a two-qubit state $AB$, they showed that the average coherence of the conditional states of $B$ summing over the mutually unbiased bases can exceed a threshold that cannot be exceeded by any single-qubit state. They termed this as the nonlocal advantage of quantum coherence (NAQC), and proved that any two-qubit state that can achieve a NAQC (we will call it the NAQC state for short) is quantum entangled. As there are many other quantum correlation measures, it is significant to purse their connections with NAQC. We explore such a problem in this paper. For two-qubit states with diagonal correlation matrix, we showed strictly that quantum correlation responsible for NAQC is stronger than that responsible for BN, while for general states this result is conjectured based on numerical analysis. We hope this finding may shed some light on our current quest for a deep understanding of the interrelation between quantum coherence and quantum correlations in composite systems.
Technical preliminaries {#sec:2}
=======================
We start by recalling two well-established coherence measures known as the $l_1$ norm of coherence and relative entropy of coherence [@coher]. For a state described by density operator $\rho$ in the reference basis $\{|i\rangle\}$, they are given, respectively, by $$\label{eq2-1}
C_{l_1}(\rho)=\sum_{i\neq j}|\langle i| \rho |j\rangle|, \;
C_{re}(\rho)=S(\rho_{\mathrm{diag}})-S(\rho),$$ where $S(\cdot)$ denotes the von Neumann entropy, and $\rho_{\mathrm{diag}}$ is an operator comprised of the diagonal part of $\rho$.
Using the above measures, Mondal *et al.* presented a “steering game” in Ref. [@naqc]: Two players, Alice and Bob, share a two-qubit state $\rho$. They begin this game by agreeing on three observables $\{\sigma_1,\sigma_2,\sigma_3\}$, with $\sigma_{1,2,3}$ being the usual Pauli operators. Alice then measures qubit *A* and informs Bob of her choice $\sigma_i$ and outcome $a\in\{0,1\}$. Finally, Bob measures coherence of qubit *B* in the eigenbasis of either $\sigma_j$ or $\sigma_k$ ($j,k \neq i$) randomly. By denoting the ensemble of his conditional states as $\{p(a|\sigma_i), \rho_{B|\sigma_i^a}\}$, the average coherence is given by $$\label{eq2-2}
\bar{C}_\alpha^{\sigma_j}(\{p(a|\sigma_i),\rho_{B|\sigma_i^a}\})
= \sum_a p(a|\sigma_i) C_\alpha^{\sigma_j}(\rho_{B|\sigma_i^a}),$$ where $p(a|\sigma_i)= \mathrm{tr}(\Pi_i^a \rho)$, $\rho_{B|\sigma_i^a}= \mathrm{tr}_A(\Pi_i^a \rho)/p(a|\sigma_i)$, $\Pi_i^a= [I_2+ (-1)^a \sigma_i]/2$, $I_2$ is the identity operator, and $C_\alpha^{\sigma_j}$ ($\alpha=l_1$ or $re$) is the coherence defined in the eigenbasis of $\sigma_j$.
By further averaging over the three possible measurements of Alice and the corresponding possible reference eigenbases chosen by Bob, Mondal *et al.* [@naqc] derived the criterion for achieving NAQC, which is given by $$\label{eq2-3}
C_\alpha^{na}(\rho)= \frac{1}{2}\sum_{i,j,a \atop i \neq j} p(a|\sigma_i)
C^{\sigma_j}_\alpha(\rho_{B|\sigma_i^a})>C_\alpha^m,$$ where $C_{l_1}^m=\sqrt{6}$, $C_{re}^m = 3H(1/2+\sqrt{3}/6) \simeq
2.2320$, and $H(\cdot)$ stands for the binary Shannon entropy function.
In fact, the above critical values are also direct results of the complementarity relations of coherence under mutually unbiased bases [@comple1]. To be explicit, by Eq. (4) of Ref. [@comple1] and the mean inequality (the arithmetic mean of a list of nonnegative real numbers is not larger than the quadratic mean of the same list) one can obtain the critical value $C_{l_1}^m$, while from Eq. (24) of Ref. [@comple1] one can obtain the critical value $C_{re}^m$.
To detect nonlocality in $\rho$, one can use the Bell-CHSH inequality $|\langle B_{\mathrm{CHSH}}\rangle_\rho| \leqslant 2$, where $B_{\mathrm{CHSH}}$ is the Bell operator [@chsh]. Violation of this inequality implies that $\rho$ is Bell nonlocal. The maximum of $|B_{\mathrm{CHSH}} \rangle_\rho|$ over all mutually orthogonal pairs of unit vectors in $\mathbb{R}^3$ is given by [@chsh2] $$\label{eq2-4}
B_{\max}(\rho) =2\sqrt{M(\rho)},$$ where $M(\rho) = u_1+u_2$, with $u_i$ ($i=1,2,3$) being the eigenvalues of $T^\dag T$ arranged in nonincreasing order, and $T$ stands for the matrix formed by elements $t_{ij} = \mathrm{tr}(\rho
\sigma_i \otimes \sigma_j)$. Clearly, $M(\rho)>1$ is also a manifestation of BN in $\rho$.
It has been shown that any $\rho$ that can achieve a NAQC is entangled, while the opposite case is not always true [@naqc]. This gives rise to a hierarchy of them. To further establish the hierarchy between NAQC and BN, and based on the consideration that the BN is local unitary invariant, we first consider the representative class of two-qubit states $$\label{eq2-5}
\tilde{\rho}= \frac{1}{4}\Bigl(I_4+\vec{r}\cdot\vec{\sigma}\otimes I_2
+I_2\otimes\vec{s}\cdot\vec{\sigma}
+\sum_{i=1}^3 v_{i}\sigma_i\otimes\sigma_i\Bigr),$$ where $\{\vec{r},\vec{s},\vec{v}\}\in \mathbb{R}^3$ satisfy the physical requirement $\tilde{\rho} \geqslant 0$. For $\vec{r}=
\vec{s}=0$, it reduces to the Bell-diagonal state $\rho_{\mathrm{Bell}}$ which is characterized by the tetrahedron $\mathcal{T}$ \[see Fig. \[fig:1\](a)\], and the region of separable $\rho_{\mathrm{Bell}}$ is the octahedron $\mathcal{O}$ [@Horodecki]. For $\vec{r} \cdot \vec{s} \neq 0$, physical $\tilde{\rho}$ shrinks to partial regions of $\mathcal{T}$. For this case, while the separable region is still inside $\mathcal{O}$, the entangled ones may not be limited to the four regions outside $\mathcal{O}$.
Hierarchy of NAQC and BN {#sec:3}
========================
The hierarchy of entanglement, steering, and BN shows that while entanglement clearly reveals the nonclassical nature of a state, steering and BN exhibit even stronger deviations from classicality [@new01; @new02; @hierarchy; @Angelo1; @Angelo2; @Angelo3]. Here, we show that NAQC may be viewed as a quantum correlation which is even stronger than BN.
To begin with, we prove the convexity of NAQC, $$\label{eq3-1}
C^{na}_\alpha\bigg({\sum_k q_k \rho_k}\bigg) \leqslant \sum_k q_k C^{na}_\alpha(\rho_k),$$ that is, the NAQC is nonincreasing under mixing of states. By combining Eqs. and , one can see that the NAQC is convex provided $\bar{C}^{\sigma_j}_\alpha$ is convex. For $\rho=\sum_k q_k \rho_k$, the conditional state of $B$ after Alice’s local measurements is $$\label{eq3-2}
\rho_{B|\sigma_i^a}= \frac{\sum_k q_k \mathrm{tr}_A(\Pi_i^a\rho_k)}
{\sum_k q_k \mathrm{tr}(\Pi_i^a\rho_k)}
= \frac{\sum_k q_k p_k(a|\sigma_i)\rho_{B|\sigma_i^a}^k}{p(a|\sigma_i)},$$ where $\rho_{B|\sigma_i^a}^k=\mathrm{tr}_A (\Pi_i^a \rho_k)/
p_k(a|\sigma_i)$, $p_k(a|\sigma_i)=\mathrm{tr}(\Pi_i^a \rho_k)$, and we have denoted by $p(a|\sigma_i)=\sum_k q_k p_k(a|\sigma_i)$. Then $$\label{eq3-3}
\begin{aligned}
\bar{C}^{\sigma_j}_\alpha \big(\{p(a|\sigma_i), \rho_{B|\sigma_i^a}\}\big)
&= \sum_a p(a|\sigma_i) C^{\sigma_j}_\alpha(\rho_{B|\sigma_i^a}) \\
&\leqslant \sum_{k,a} p(a|\sigma_i) \frac{q_k p_k(a|\sigma_i)}{p(a|\sigma_i)}C^{\sigma_j}_\alpha\bigl(\rho_{B|\sigma_i^a}^k\bigr) \\
&= \sum_{k,a} q_k p_k(a|\sigma_i) C^{\sigma_j}_\alpha\bigl(\rho_{B|\sigma_i^a}^k \bigr) \\
&=\sum_k q_k \bar{C}^{\sigma_j}_\alpha\big(\{p_k(a|\sigma_i),\rho_{B|\sigma_i^a}^k\}\big),
\end{aligned}$$ where the first inequality is due to convexity of the coherence measure. This completes the proof of Eq. .
Next, we give the level surface $\mathcal{S}$ of constant BN $M(\tilde{\rho})= 1$. It can be divided into four parts, corresponding to the four vertices of $\mathcal{T}$. For convenience of later presentation, we denote by $\mathcal{S}_A$ the part near vertex $A$ (see Fig. \[fig:2\]). It is described by $$\label{eq3-4}
\begin{aligned}
& v_i=\sin \theta,~ v_j=\cos \theta, \\
& v_k\in[\max\{\sin\theta,\cos\theta\},1+\sin \theta+\cos \theta],
~ \theta\in[\pi, 1.5\pi],
\end{aligned}$$ where $(i,j,k)=(1,2,3)$, $(2,3,1)$, and $(3,1,2)$. The equations for the other three parts of $\mathcal {S}$ can be obtained directly by their symmetry about the coordinate origin $O$. The corresponding results are showed in Fig. \[fig:1\](b).
In the following, we denote by $\mathcal{N}$ the set of NAQC states and $\mathcal{B}$ the set of Bell nonlocal states. We will prove the inclusion relation $\mathcal {N}\subset\mathcal{B}$ for any $\tilde{\rho}$, meaning that the existence of NAQC implies the existence of BN.
$l_1$ norm of of NAQC {#sec:3a}
---------------------
First, we consider the class of Bell-diagonal states. Without loss of generality, we assume $|v_1|\geqslant |v_2| \geqslant |v_3|$, then $$\label{eq3a-1}
C_{l_1}^{na}(\rho_{\mathrm{Bell}})=\sum_i|v_i|,~
M(\rho_{\mathrm{Bell}})=v_1^2+v_2^2,$$ from which one can obtain $|v_1|> \sqrt{6} /3$ and $|v_2|>(\sqrt{6}
-1)/2$ when $C_{l_1}^{na}(\rho_{\mathrm{Bell}})> \sqrt{6}$. This further gives rise to $M(\rho_{\mathrm{Bell}})>1$. That is, any $\rho_{\mathrm{Bell}}$ that can achieve a NAQC is Bell nonlocal. But the converse is not true, e.g., if $v_{1,2,3} \in [-\sqrt{6}/3,
-1/\sqrt{2})$, we have $M(\rho_{\mathrm{Bell}})>1$ and $C_{l_1}^{na}
(\rho_{\mathrm{Bell}}) \leqslant \sqrt{6}$. With all this, we arrived at the inclusion relation $\mathcal{N} \subset \mathcal{B}$. The level surfaces of $C_{l_1}^{na} (\rho_{\mathrm{Bell}})=
\sqrt{6}$ can be found in Fig. \[fig:1\](c).
Second, we consider $\tilde{\rho}$ sitting at the edges of $\mathcal{T}$ with general $\vec{r}$ and $\vec{s}$. We take the edge $AB$ as an example (see Fig. \[fig:2\]), the cases for the other edges are similar. Along this edge, we have $v_1=v_3$ and $v_2=-1$, then one can determine analytically the constraints imposed by $\tilde{\rho}\geqslant 0$ on the involved parameters as $r_{1,3}=
s_{1,3}=0$, $r_2=-s_2$, and $s_2^2\leqslant 1-v_1^2$ (see Appendix \[sec:A\]). Thus we have $$\label{eq3a-2}
C_{l_1}^{na}(\tilde{\rho}) = 1+|v_1|+\sqrt{v_1^2+s_2^2}.$$ It is always not larger than $\sqrt{6}$ in the region of $|v_1|
\leqslant \sqrt{6}-2$. On the other hand, the states located at the edge $AB$ other than its midpoint are Bell nonlocal. Hence, the inclusion relation $\mathcal{N} \subset \mathcal{B}$ holds for all $\tilde{\rho}$ located at the edges of $\mathcal{T}$.
Next, we consider $\tilde{\rho}$ associated with $v_{1,2,3}= v_0 =
-1/\sqrt{2}$. As $C_{l_1}^{na}$ is an increasing function of $|s_i|$ ($i=1,2,3$), one only needs to determine the maximal $|s_i|$ for which $\tilde{\rho}\geqslant 0$. Without loss of generality, we assume $s_3= w_0 s_1$ and $s_2= w_1 s_1$, then a detailed analysis shows that the resulting maximum NAQC states belong to the set of $\tilde{\rho}$ with $r_3= w_0 r_1$ and $r_2= w_1 r_1$. Under this condition, one can obtain analytically the eigenvalues $\epsilon_k$ of $\tilde{\rho}$. Then from $\epsilon_k \geqslant 0$ ($\forall k$) one can obtain $$\label{eq3a-3}
\begin{aligned}
& |s_1+r_1|\leqslant c_1= \frac{1+v_0}{\sqrt{1+w_0^2+w_1^2}}, \\
& |s_1-r_1|\leqslant c_2= \sqrt{\frac{1-2v_0-3v_0^2}{1+w_0^2+w_1^2}}.
\end{aligned}$$ For state $\tilde{\rho}$ with fixed $v_0$, $w_0$, and $w_1$, $C_{l_1}^{na}$ takes its maximum when the above inequalities become equalities. That is, when $$\label{eq3a-4}
s_1=\pm\frac{1}{2}(c_1+c_2), ~
r_1=\pm\frac{1}{2}(c_1-c_2),$$ then by further maximizing the resulting $C_{l_1}^{na}$ over $w_0$ and $w_1$, we obtain $C_{l_1,\max}^{na} \simeq 2.4405$ at the critical points $w_{0,1}=\pm 1$ (we have also checked the validity of this result with $10^7$ randomly generated $\tilde{\rho}$ for which $v_{1,2,3}= -1/\sqrt{2}$, and no violation was observed). As this maximum is smaller than $\sqrt{6}$, any $\tilde{\rho}$ with $v_{0}= -1/\sqrt{2}$ cannot achieve a NAQC.
To proceed, we introduce a polyhedron $\mathcal {P}$ with the set of its vertices near the vertex $A$ being given by $(v_0,v_0,v_0)$, $(-1,\gamma,\gamma)$, $(\gamma,-1,\gamma)$, $(\gamma,\gamma,-1)$, and its other vertices can be obtained by using their symmetry with respect to the point $O$ (see Fig. \[fig:2\]). One can show that when $|\gamma|< \sqrt{2}-1$, the surface $\mathcal {S}_A$ is always inside $\mathcal{P}$ (see Appendix \[sec:B\]). Finally, as $C_{l_1,\max}^{na}\simeq 2.4405$ at the point $(v_0,v_0,v_0)$, we choose $\gamma= 2-\sqrt{6}$ for which $C_{l_1}^{na}$ is also smaller than $\sqrt{6}$ at the other three points of $\mathcal{P}$ near vertex $A$ \[see Eq.\], then as any physical state with $\vec{v}$ inside $\mathcal {P}$ can be written as a convex combination of states with $\vec{v}$ at the vertices of $\mathcal
{P}$, we complete the proof of the inclusion relation $\mathcal{N}
\subset \mathcal{B}$ for general $\tilde{\rho}$ by using the convexity of NAQC.
In fact, for $\tilde{\rho}$ at the line $AO$ with fixed $w_0$ and $w_1$, one can obtain the critical $v_0^c$ at which $C_{l_1}^{na}=
\sqrt{6}$. As $C_{l_1}^{na}$ and $v_0^c$ considered here are invariant under the substitution $w_0\leftrightarrow w_1$, we showed in Fig. \[fig:3\](a) an exemplified plot of the $w_0$ dependence of $v_0^c$ with fixed $w_1=0$ and 1. It first increases to a peak value at $w_0=1$, then decreases gradually with the increase of $|\omega_0|$. By optimizing over $w_0$ and $w_1$, one can further obtain the region of $v_0^c \in (-0.7519, -0.7142)$, where the lower and upper bounds correspond to $w_{0,1}=0$ and $w_{0,1}=\pm 1$, respectively. Clearly, the point $(v_0^c,v_0^c,v_0^c)$ is always outside the surface $\mathcal{S}$.
Relative entropy of NAQC {#sec:3b}
------------------------
In this subsection, we consider NAQC measured by the relative entropy. First, for Bell-diagonal states, the corresponding NAQC can be obtained as [@naqc] $$\label{eq3b-1}
C_{re}^{na}(\rho_{\mathrm{Bell}})= 3-\sum_i H\Biggl(\frac{1+v_i}{2}\Biggr).$$ Then by imposing $C_{re}^{na}(\rho_{\mathrm{Bell}})>C_{re}^m$ with the assumption $|v_1|\geqslant |v_2| \geqslant |v_3|$, one can obtain $$\label{eq3b-2}
H\Biggl(\frac{1+v_1}{2}\Biggr) < \frac{3-C_{re}^m}{3},~
H\Biggl(\frac{1+v_2}{2}\Biggr) < \frac{3-C_{re}^m}{2},$$ which yields $M(\rho_{\mathrm{Bell}}) > 1$. Moreover, we have $M(\rho_{\mathrm{Bell}})>1$ and $C_{re}^{na}(\rho_{\mathrm{Bell}}) <
C_{re}^m$ for $v_{1,2,3} \in (-0.9140, -1/\sqrt{2})$. So $\mathcal{N} \subset \mathcal{B}$ holds for $\rho_\mathrm{Bell}$. The corresponding level surfaces were showed in Fig. \[fig:1\](d). Clearly, the region of NAQC states shrinks compared with that captured by the $l_1$ norm.
For $\tilde{\rho}$ sitting at the edges of $\mathcal {T}$ with general $\vec{r}$ and $\vec{s}$, we take the edge $AB$ as an example. Based on the results of Sec. \[sec:3a\], one can obtain $$\label{eq3b-3}
C_{re}^{na}= 2+H\Biggl(\frac{1+s_2}{2}\Biggr)-
2H\left(\frac{1+\sqrt{v_1^2+s_2^2}}{2}\right),$$ then it is direct to show that $C_{re}^{na}$ is always smaller than $C_{re}^m$ for $|v_1|<-b_0\simeq 0.3813$. So the inclusion relation $\mathcal{N}\subset \mathcal{B}$ holds for any $\tilde{\rho}$ at the edges of $\mathcal {T}$.
Based on the above preliminaries, we now consider $\tilde{\rho}$ at the surface $\mathcal {S}_A$ (the cases for the other parts of $\mathcal {S}$ are similar). We will show that for these $\tilde{\rho}$ the inequality $C_{re}^{na} < C_{re}^m$ holds. Then by further employing the convexity of NAQC and the fact that $\{\tilde{\rho}\}$ is a convex set, one can complete the proof of $\mathcal{N} \subset \mathcal{B}$. In fact, due to the structure of $\mathcal {S}_A$ \[see Eq. \], it suffices to prove that we always have $C_{re}^{na} < C_{re}^m$ at the boundary of $\mathcal{S}_A$.
First, we introduce the polygon line $EFG$ over $(-1,b_0,b_0)$, $(a_0, a_0, 1+2a_0)$, and $(b_0,-1, b_0)$. One can prove that there is no intersection of this line and the boundary of $\mathcal{S}_A$ at the facet $ABC$ when $a_0\lesssim -0.7082$ (Appendix \[sec:B\]). Moreover, along the line $AO'$, $\tilde{\rho}\geqslant
0$ yields $r_{1,2}= -s_{1,2}$ and $r_3=s_3$ (Appendix \[sec:A\]), then one can obtain that at the point $F$ with $a_0= -0.7082$, $C_{re}^{na}$ maximized over $\vec{r}$ and $\vec{s}$ is of about 1.4956. As $C_{re}^{na}$ is also smaller than $C_{re}^m$ at the points $E$ and $G$ \[see Eq. \], we have $C_{re}^{na}<
C_{re}^m$ for any $\tilde{\rho}$ at this boundary.
Second, if we make the substitutions $v_0=-0.7082$ and $\gamma=b_0$ to the vertices of $\mathcal {P}$, then one can show that the boundary of $\mathcal{S}_A$ inside $\mathcal {T}$ is also inside $\mathcal{P}$ (see Appendix \[sec:B\]). For $\tilde{\rho}$ at the point $(v_0,v_0,v_0)$, our numerical results showed that with fixed $v_0$, $w_0$, and $w_1$, $C_{re}^{na}$ also takes its maximum when $s_1$ and $r_1$ are given by Eq. . Then by further maximizing it over $w_0$ and $w_1$, we obtain $C_{re,\max}^{na}
\simeq 2.0041$ at $w_{0,1}=\pm 1$. As $C_{re}^{na}$ is also smaller than $C_{re}^m$ for $\tilde{\rho}$ at the vertices of $\mathcal {P}$ with $\gamma=b_0$ \[see Eq. \], we have $C_{re}^{na}<
C_{re}^m$ for any $\tilde{\rho}$ at this boundary.
Similar to the $l_1$ norm of NAQC, one can obtain $v_0^c$ at which $C_{re}^{na}= C_{re}^m$ with fixed $w_0$ and $w_1$. It is $v_0^c\in
(-0.8278, -0.8266)$, where the lower and upper bounds are obtained with $w_{0,1}=0$ and $w_{0,1}=\pm 1$, respectively. As is showed in Fig. \[fig:3\], $v_0^c$ for the two NAQCs exhibits qualitatively the same $w_0$ dependence.
Before ending this section, we would like to mention here that although for the set of Bell-diagonal states, one detects a wider region of NAQC states by using the $l_1$ norm as a measure of coherence than that by using the relative entropy (see Fig. \[fig:1\]), this is not always the case. A typical example is that for $\tilde{\rho}$ at the edge $AB$ of $\mathcal {T}$ with $|v_{1,3}|\in (0.3813, \sqrt{6}-2)$, one may have $C_{l_1}^{na}<
\sqrt{6}$ and $C_{re}^{na}>C_{re}^m$.
An explicit application of NAQC {#sec:4}
===============================
As it is a proven fact that all Bell nonlocal states are useful for quantum teleportation [@telep], the hierarchy we obtained implies that any NAQC state $\tilde{\rho}$ can serve as a quantum channel for quantum teleportation. That is, it always gives rise to the average fidelity $F_{av}>2/3$. In fact, $F_{av}$ achievable with the channel state $\tilde{\rho}$ is given by [@telep] $$\label{eq4-1}
F_{av}(\tilde{\rho})=\frac{1}{2}+ \frac{1}{6} \sum_i |v_i|.$$ Using this equation and the results of Sec. \[sec:3\], one can obtain that for any NAQC state $\tilde{\rho}$ captured by $C_{l_1}^{na}(\tilde{\rho})$, we always have $F_{av}> \sqrt{6}/3$, while for any NAQC state $\tilde{\rho}$ captured by $C_{re}^{na}
(\tilde{\rho})$, we always have $F_{av}\gtrsim 0.7938$. Both the two critical values are larger than $2/3$, so any NAQC state $\tilde{\rho}$ can serve as a quantum channel for nonclassical teleportation.
If we focus only on the class of NAQC Bell-diagonal states, the average fidelity $F_{av}$ can be further improved. More specifically, Eqs. and imply that $F_{av}>(3+\sqrt{6})/6$ for any NAQC state $\rho_\mathrm{Bell}$ captured by $C_{l_1}^{na}(\rho_\mathrm{Bell})$, and $F_{av} \gtrsim
0.9501$ for any NAQC state $\rho_\mathrm{Bell}$ captured by $C_{re}^{na} (\rho_\mathrm{Bell})$.
Summary and discussion {#sec:5}
======================
In summary, we have explored the interrelations of NAQC achievable in a two-qubit state under local measurements and BN detected by violation of the Bell-CHSH inequality. There are two different scenarios of NAQC being considered: one is characterized by the $l_1$ norm of coherence, and another one is characterized by the relative entropy of coherence. For both scenarios, we showed geometrically that the inclusion relation $\mathcal{N} \subset
\mathcal{B}$ holds for the class of states $\tilde{\rho}$ that have diagonal correlation matrix $T$. This extends the known hierarchy in quantum correlation, viz., BN, steerability, entanglement, and quantum discord to include NAQC.
One may also concern whether the obtained hierarchy holds for $\rho$ with nondiagonal $T$. As such $\rho$ is locally unitary equivalent to $\tilde{\rho}$, that is, $\rho=U_{AB} \tilde{\rho} U_{AB}^\dag$ with $U_{AB}=U_A\otimes U_B$, the proof can be completed by showing that for any $\tilde{\rho}$ with $M(\tilde{\rho})\leqslant 1$, we have $C_{\alpha}^{na} (U_{AB} \tilde{\rho} U_{AB}^\dag) \leqslant
C_{\alpha}^m$ for all unitaries $U_{AB}$. But due to the so many number of state parameters involved, it is difficult to give such a strict proof. For special cases, a strict proof may be available, e.g., for the locally unitary equivalent class of $\tilde{\rho}$ with $|\vec{v}|^2+2|\vec{s}|^2 \leqslant 2$, we are sure that $C_{l_1}^{na}\leqslant\sqrt{6}$, while for the locally unitary equivalent class of $\rho_\mathrm{Bell}$, we are sure that $C_{re}^{na} \leqslant C_{re}^m$ (see Appendix \[sec:C\]). Moreover, for $\tilde{\rho}$ with reduced number of parameters, we performed numerical calculations with $10^7$ equally distributed local unitaries generated according to the Haar measure [@Haar1; @Haar2], and found that $C_{\alpha}^{na}$ is always smaller than $C_\alpha^m$ (see Appendix \[sec:C\]). These results presented strong evidence that the hierarchy may hold for any two-qubit state, though a strict proof is still needed.
Moreover, one may argue that NAQC can be recognized as a quantum correlation. It is stronger than BN in the sense that the NAQC states form a subset of the Bell nonlocal states. But it is asymmetric, that is, in general $C^{na}_\alpha$ defined with the local measurements on $A$ does not equal that defined with the local measurements on $B$. This property is the same to steerability and quantum discord. The NAQC is also not locally unitary invariant. Its value may be changed by performing local unitary transformation to the mutually unbiased bases. To avoid this perplexity, one can define $$\label{eq5-1}
\tilde{C}_\alpha^{na}(\rho)= \frac{1}{2} \max_{\{U_A\otimes U_B\}} \sum_{i,j,a\atop i\neq j}
p(a|\sigma_{i,U_A})C^{\sigma_{j,U_B}}_\alpha (\rho_{B|\sigma_{i,U_A}^a}),$$ with $\sigma_{i,U_A} = U_A\sigma_i U_A^\dag$, and likewise for $\sigma_{j,U_B}$. As BN is locally unitary invariant, we have $\tilde{\mathcal{N}} \subset \mathcal {B}$ provided $\mathcal{N}
\subset \mathcal {B}$, where $\tilde{\mathcal {N}}$ is the set of NAQC states captured by $\tilde{C}_\alpha^{na} (\rho)> C_\alpha^m$ .
Finally, in light of those measures of steerability based on the maximal violation of various steering inequalities and the similar measure of Bell nonlocality [@Angelo1; @Angelo2], it is natural to quantify the degree of NAQC in a bipartite state $\rho$ by $$\label{eq5-2}
\tilde{Q}_{\alpha}(\rho)= \max\left\{0,\frac{\tilde{C}_\alpha^{na}(\rho)-C_\alpha^m}
{\tilde{C}_{\alpha,\max}^{na}- C_\alpha^m}\right\},$$ where $\tilde{C}_{\alpha,\max}^{na}= \max_\rho \tilde{C}_\alpha^{na}
(\rho)$, and the factor $\tilde{C}_{\alpha,\max}^{na}- C_\alpha^m$ was introduced for normalizing $\tilde{Q}_{\alpha}(\rho)$. For two-qubit states, we have $\tilde{C}_{\alpha,\max}^{na}=3$ ($\alpha=l_1$ or $re$), which are obtained for the Bell states $|\Phi^\pm\rangle=(|00\rangle\pm|11\rangle)/\sqrt{2}$ and $|\Psi^\pm\rangle=(|01\rangle\pm|10\rangle)/\sqrt{2}$. Moreover, we have used the fact that $C_\alpha^m$ cannot be increased by any unitary transformation in the above definition.
Of course, one may propose to define the NAQC-based correlation measure \[denoted $Q_{l_1}(\rho)$\] by replacing $\tilde{C}_\alpha^{na}(\rho)$ in Eq. with $C_\alpha^{na}(\rho)$. But if so, $Q_{l_1}(\rho)$ will not be locally unitary invariant, thus makes it violates the widely accepted property of a quantum correlation measure (e.g., Bell nonlocality, steerability, entanglement, and quantum discord) which should be locally unitary invariant.
As an example, we calculated numerically the NAQC-based correlation measure of the following state $$\label{eq5-3}
\rho_1=x|\Phi^+\rangle\langle\Phi^+|+(1-x)|\Psi^-\rangle\langle\Psi^-|,~~ x\in[0,1],$$ for which $\tilde{Q}_{l_1}(\rho_1)$ is symmetric with respect to $x=0.5$. As was showed in Fig. \[fig:4\], $\tilde{Q}_{l_1}
(\rho_1)> Q_{l_1}(\rho_1)$ in the region of $0\lesssim x\lesssim
0.141$. In particular, we have $\tilde{Q}_{l_1}(\rho_1) >0$ and $Q_{l_1}(\rho_1)=0$ when $0.138 \lesssim x\lesssim 0.141$, that is, $\tilde{Q}_{l_1}$ captures a wider region of NAQC states than $Q_{l_1}$.
ACKNOWLEDGMENTS {#acknowledgments .unnumbered}
===============
This work was supported by National Natural Science Foundation of China (Grants No. 11675129, No. 91536108, and No. 11774406), National Key R & D Program of China (Grants No. 2016YFA0302104 and No. 2016YFA0300600), the New Star Project of Science and Technology of Shaanxi Province (Grant No. 2016KJXX-27), the Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDB28000000), and the New Star Team of XUPT.
Constraints imposed on the parameters of $\tilde{\rho}$ {#sec:A}
=======================================================
At the edge $AB$ of $\mathcal {T}$, we have $v_1=v_3$ and $v_2=-1$. Then the positive semidefiniteness of $\tilde{\rho}$ requires $$\label{eqa-1}
\begin{aligned}
& \tilde{\rho}_{11}\tilde{\rho}_{44}- |\tilde{\rho}_{14}|^2=-(r_3+s_3)^2\geqslant 0,\\
& \tilde{\rho}_{22}\tilde{\rho}_{33}- |\tilde{\rho}_{23}|^2=-(r_3-s_3)^2\geqslant 0,
\end{aligned}$$ from which one can obtain $r_3=s_3=0$.
Moreover, all the [*i*]{}th-order principal minors of $\tilde{\rho}$ should be nonnegative. Under the constraint $r_3=s_3=0$ obtained above, the second- and third-order leading principal minors $D_{2,3}$ and the principal minor $\Delta_3$ (determinant of the matrix obtained by removing from $\tilde{\rho}$ its third row and third column) are $$\label{eqa-2}
\begin{aligned}
& D_2=1-v_1^2-s_1^2-s_2^2, \\
& D_3=(v_1-1)[(r_1+s_1)^2+ (r_2+s_2)^2], \\
& \Delta_3=- (v_1+1)[(r_1-s_1)^2+(r_2+s_2)^2],
\end{aligned}$$ which, together with Eq. , yields the following requirements $$\label{eqa-3}
r_{1,3}=s_{1,3}=0,~ r_2=-s_2,~ s_2^2\leqslant 1-v_1^2.$$ Similarly, one can obtain constraints imposed on the parameters of $\tilde{\rho}$ at the other edges of $\mathcal {T}$. They are $$\label{eqa-4}
\begin{aligned}
&AC\mathrm{:}~ r_{2,3}=s_{2,3}=0,~ r_1=-s_1,~ s_1^2\leqslant 1-v_3^2, \\
&AD\mathrm{:}~ r_{1,2}=s_{1,2}=0,~ r_3=-s_3,~ s_3^2\leqslant 1-v_2^2, \\
&CD\mathrm{:}~ r_{1,3}=s_{1,3}=0,~ r_2=s_2,~ s_2^2\leqslant 1-v_1^2, \\
&BD\mathrm{:}~ r_{2,3}=s_{2,3}=0,~ r_1=s_1,~ s_1^2\leqslant 1-v_3^2, \\
&BC\mathrm{:}~ r_{1,2}=s_{1,2}=0,~ r_3=s_3,~ s_3^2\leqslant 1-v_2^2.
\end{aligned}$$ For $\tilde{\rho}$ associated with $\vec{v}$ at the line $AO'$, we have $v_{1,2}=a_0$ and $v_3=1+2a_0$ ($-1\leqslant a_0\leqslant 0$), then a similar derivation gives $$\label{eqa-5}
\begin{aligned}
& r_{1,2}=-s_{1,2},~ r_3=s_3\in[-1-a_0, 1+a_0], \\
& |s_1|\leqslant \min\Big\{1+\frac{1}{2}a_0, \frac{1}{2}(1-a_0)\Big\},\\
& s_1^2+s_2^2 \leqslant -4a_0(1+a_0).
\end{aligned}$$
Intersection of two surfaces {#sec:B}
============================
Due to the symmetry, one only needs to consider the intersections of the level surface $\mathcal {S}_A$ described by Eq. and the facet of $\mathcal {P}$ with the vertices $(v_0,v_0, v_0)$, $(-1,\gamma, \gamma)$, $(\gamma,-1,\gamma)$. The plane equation for this facet is $$\label{eqb-1}
a v_1+a v_2+c v_3+1=0,$$ where $$\label{eqb-2}
a=\frac{v_0-\gamma}{v_0(1+\gamma)},~c=-\frac{1}{v_0}-2a.$$ Without loss of generality, we fix $(i,j,k)=(1,2,3)$ in Eq. . Then by plugging $v_1=\sin\theta$ and $v_2=\cos\theta$ into Eq. , we obtain $$\label{eqb-3}
v_3=\frac{(v_0-\gamma)(\sin\theta+\cos\theta)+v_0(1+\gamma)}
{1+2v_0-\gamma},$$ and for given $v_0$ and $\gamma$, one can check whether there are intersections for the two surfaces by checking whether $v_3$ obtained in Eq. belongs to the region $[\max\{\sin\theta, \cos\theta\}, 1+\sin\theta+\cos\theta]$. If there exists such $v_3$, then there are intersection of $\mathcal{S}_A$ and $\mathcal{P}$. Otherwise, $\mathcal {S}_A$ is totally inside or outside of $\mathcal{P}$.
One can also determine whether there are intersections of $\mathcal{S}_A$ and $\mathcal {P}$ by plugging Eq. into Eq. , and checking the resulting $\mathrm{sgn}( a
v_1+a v_2+c v_3+1)$. The surface $\mathcal{S}_A$ is inside $\mathcal {P}$ if it is always nonnegative. In fact, here one only needs to check the points at the boundary of $\mathcal {S}_A$.
Based on the above methods, it is direct to show that when $v_0=-1/\sqrt{2}$ and $|\gamma|<\sqrt{2}-1$, the level surface $\mathcal {S}_A$ is always inside $\mathcal{P}$. When $v_0\lesssim
-0.7082$ and $\gamma= b_0$, the boundary of $\mathcal {S}_A$ inside the tetrahedron $\mathcal{T}$ is also inside the polyhedron $\mathcal{P}$.
Similarly, by substituting $v_1=\sin\theta$, $v_2=\cos\theta$, and $v_3=1+\sin\theta+\cos\theta$ into the equation of the straight line $FG$ (see Fig. \[fig:2\]), one can obtain $$\label{eqb-4}
(1+a_0)\sin\theta+(b_0-a_0)\cos\theta=a_0(1+b_0).$$ For given $a_0$ and $b_0$, if there are solutions for Eq. in the region of $\theta\in [\pi,1.5\pi]$, there are intersections of $FG$ and the boundary of $\mathcal{S}_A$ described by $v_3=1+\sin\theta+ \cos\theta$. In this way, one can check that when $b_0\simeq -0.3813$ and $a_0\lesssim -0.7082$, there are no intersections of $FG$ and the boundary of $\mathcal{S}_A$.
NAQC of general two-qubit states {#sec:C}
================================
Suppose $U_{AB}= U_A \otimes U_B$ gives the map $\vec{r} \mapsto
\vec{x}$, $\vec{s} \mapsto \vec{y}$, and $\vec{v} \mapsto
T=(t_{ij})$, then the transformed state of $\tilde{\rho}$ is given by $$\label{eqc-1}
\rho= \frac{1}{4}\Big(I_4+\vec{x}\cdot\vec{\sigma}\otimes I_2
+I_2\otimes\vec{y}\cdot\vec{\sigma}
+\sum_{i,j=1}^3 t_{ij}\sigma_i\otimes\sigma_j\Big),$$ and we have the following equalities $$\label{eqc-2}
|\vec{r}|=|\vec{x}|,~ |\vec{s}|=|\vec{y}|,~
|\vec{v}|^2=\sum_{ij}t_{ij}^2.$$ By further using the mean inequality and the analytical solution of $C_{l_1}^{na}(\rho)$ given in Ref. [@naqc], we obtain $$\label{eqc-3}
\begin{aligned}
C_{l_1}^{na}(U_{AB}\tilde{\rho} U_{AB}^\dag)
& \leqslant \sqrt{\frac{3}{2}\Big(|\vec{v}|^2+ \sum_{i} t_{ii}^2 \Big)+6|\vec{s}|^2}, \\
& \leqslant \sqrt{3|\vec{v}|^2+6|\vec{s}|^2},
\end{aligned}$$ hence for the class of $\tilde{\rho}$ with $|\vec{v}|^2+2|\vec{s}|^2
\leqslant 2$, we are sure that $C_{l_1}^{na}(U_{AB}\tilde{\rho}
U_{AB}^\dag) \leqslant \sqrt{6}$. This class of $\tilde{\rho}$ includes (but not limited to) all $\tilde{\rho}$ with $|\vec{s}|^2\leqslant 1/4$ as we have $|\vec{v}|^2\leqslant 3/2$ for $M(\tilde{\rho})\leqslant 1$.
For the relative entropy of NAQC, due to its complexity, we consider only the case of $\rho_\mathrm{Bell}$, for which we have $$\label{eqc-4}
\begin{aligned}
C_{re}^{na}(U_{AB} \rho_\mathrm{Bell} U_{AB}^\dag)
=& \frac{1}{2}\sum_{i\neq j}H \Biggl(\frac{1+t_{ij}}{2}\Biggr) \\
& -\sum_i H \left(\frac{1+\sqrt{\sum_j t_{ij}^2}}{2} \right),
\end{aligned}$$ then by using $|\vec{v}|^2\leqslant 3/2$ when $M(\tilde{\rho})
\leqslant 1$, one can show that the maximum of the right-hand side of Eq. is of about 1.1974, which is achieved when $T=\mathrm{diag}\{v_0,v_0,v_0\}$, with $v_0=- 1/\sqrt{2}$. Hence $C_{re}^{na}(U_{AB} \rho_\mathrm{Bell} U_{AB}^\dag)< C_{re}^m$ for this class of $\tilde{\rho}$.
For general $\tilde{\rho}$ inside the level surface $\mathcal {S}$, it is hard even to give a numerical simulation as the derivation of the constraints imposed on $\vec{r}$ and $\vec{s}$ is also a difficult task. But if the number of the involved parameters can be reduced, a numerical verification may also be possible. Several examples where such a verification can be performed are as follows:
\(1) For the class of $\tilde{\rho}$ at the vertex $(0,-1,0)$ of $\mathcal {O}$ (the cases for the other vertices of $\mathcal {O}$ are similar), we have $r_{1,3}=s_{1,3}=0$ and $r_2=-s_2$, i.e., there is only one variable. We performed numerical calculation with $10^7$ equally distributed local unitaries generated according to the Haar measure [@Haar1; @Haar2], and found that the maximal $C_{l_1}^{na}$ and $C_{re}^{na}$ achievable by optimizing over $U_A\otimes U_B$ increase with the increase of $|s_2|$. When $|s_2|=1$, their maximal values are $\sqrt{6}$ and $C_{re}^m$, respectively. The corresponding optimal $U_{AB}\tilde{\rho}
U_{AB}^\dag$ is of the form of Eq. , with $$\label{eqc-5}
\vec{x}=-\vec{y}=\biggl(\pm \frac{1}{\sqrt{3}},\pm \frac{1}{\sqrt{3}},
\pm \frac{1}{\sqrt{3}}\biggr),~~
t_{ij}=-\frac{1}{3}~(\forall i,j).$$ (2) For the class of $\tilde{\rho}$ associated with $v_{1,2,3}=
-1/\sqrt{2}$ (the cases for $v_i=v_j=-v_k=1/\sqrt{2}$ are similar), the parameter regions can be reduced via $r_3^2+s_3^2 \leqslant
1-v_1^2$ and $|r_{1,3}\pm s_{1,3}|\leqslant 1\pm v_1$. The numerical results show that $C_{\alpha}^{na}(U_{AB} \tilde{\rho}U_{AB}^\dag)$ is still smaller than $C_{\alpha}^m$ ($\alpha=l_1$ or $re$). Specifically, when $w_{0,1}=\pm 1$, $s_1$ and $r_1$ take the values of Eq. , the NAQC of $\tilde{\rho}$ cannot be enhanced by $U_{AB}$, i.e., $C_{l_1}^{na}(\tilde{\rho}) \simeq 2.4405$ and $C_{re}^{na} (\tilde{\rho})\simeq 2.0026$ are already the maximum values.
\(3) For the class of $\tilde{\rho}$ with $v_{1,2}=-1/\sqrt{2}$ and $v_3=1-\sqrt{2}$ \[an intersection of $AO'$ and the curve of $M(\tilde{\rho})=1$\], one can obtain $|\vec{s}|^2\leqslant
\sqrt{2}-1$ by using Eq. . Hence $|\vec{v}|^2+
2|\vec{s}|^2 < 2$, and $C_{l_1}^{na}(U_{AB} \tilde{\rho}
U_{AB}^\dag)$ cannot exceed $\sqrt{6}$ due to Eq. . For NAQC characterized by the relative entropy, we performed numerical calculation with $10^3$ equally distributed $\tilde{\rho}$ of this class, while every $\tilde{\rho}$ is further optimized over $10^7$ equally distributed local unitaries. From these calculation we still have not found the case for which $C_{re}^{na} (U_{AB} \tilde{\rho}
U_{AB}^\dag)>C_{re}^m$.
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|
---
abstract: 'We give a new proof that there are infinitely many primes, relying on van der Waerden’s theorem for coloring the integers, and Fermat’s theorem that there cannot be four squares in an arithmetic progression. We go on to discuss where else these ideas have come together in the past.'
author:
- Andrew Granville
title: Squares in arithmetic progressions and infinitely many primes
---
Infinitely many primes
======================
Levent Alpoge recently gave a rather different proof [@Alp] that there are infinitely many primes. His starting point was the famous result of van der Waerden (see, e.g., [@TV]):
**van der Waerden’s Theorem.** *Fix integers $m\geq 2$ and $\ell\geq 3$. If every positive integer is assigned one of $m$ colors, in any way at all, then there is an $\ell$-term arithmetic progression of integers which each have the same color.*
Using a clever coloring in van der Waerden’s theorem, and some elementary number theory, Alpoge deduced that there are infinitely many primes. We proceed from van der Waerden’s theorem a little differently, employing a famous result of Fermat (see, e.g., [@Sil]):
**Fermat’s Theorem.** *There are no four-term arithmetic progressions of distinct integer squares.*
From these two results we deduce the following:
There are infinitely many primes.
If there are only finitely many primes $p_1,\ldots, p_k$, then every integer $n$ can be written as $p_1^{e_1}\cdots p_k^{e_k}$ for some integers $e_1,e_2,\ldots,e_k\geq 0$. We can write each of these exponents $e_j$ as $$e_j=2q_j+r_j, \ \text{where} \ r_j \ \text{ is the ``remainder'' when dividing} \ e_j \ \text{by 2,}$$ and equals 0 or 1. Therefore if we let $$R=p_1^{r_1}\cdots p_k^{r_k}$$ then $R$ is a squarefree integer that divides $n$, and $$n/R \ \text{is the square of an integer}.$$ (In fact, $n/R=Q^2$ where $Q=p_1^{q_1}\cdots p_k^{q_k}$.)
We will use $2^k$ colors to color the integers: Integer $n$ is colored by the vector $(r_1,\ldots,r_k)$. By van der Waerden’s theorem there are four integers in arithmetic progression $$A, A+D, A+2D, A+3D, \ \text{with} \ D\geq 1,$$ which all have the same color $(r_1,\ldots,r_k)$. Now $R=p_1^{r_1}\cdots p_k^{r_k}$ divides each of these numbers, so also divides $D=(A+D)-A$. Letting $a=A/R$ and $d=D/R$, we see that $$a,a+d,a+2d,a+3d \ \text{are four squares in arithmetic progression,}$$ contradicting Fermat’s theorem.
These ideas have come together before to make a rather different, not-too-obvious deduction:
The number of squares in a long arithmetic progression
======================================================
Let $Q(N)$ denote the maximum number of squares that there can be in an arithmetic progression of length $N$. A slight refinement of the Erdős–Rudin conjecture states that the maximum number is attained by the arithmetic progression $$\{ 24n+1:\ 0\leq n\leq N-1 \}$$ which contains $\sqrt{ \frac 83 N}$ squares, plus or minus one. From Fermat’s theorem one easily sees that $$Q(N) \leq \frac{3N+3} 4,$$ but it is difficult to see how to improve the bound to, say, $Q(N)\leq \delta N+b$ for some constant $\delta <\frac 34$.
It was this problem that inspired one of the most influential results [@Sz2] in combinatorics and analysis (see, e.g., [@Gow]):
**Szemer' edi’s Theorem.** *Fix $\delta>0$ and integer $\ell\geq 3$. If $N$ is sufficiently large (depending on $\delta$ and $\ell$) then any subset $A$ of $\{ 1,2,\ldots,N\}$ with $\geq \delta N$ elements, must contain an $\ell$-term arithmetic progression.*
van der Waerden’s theorem is a consequence of Szemer' edi’s theorem, because if we let $\delta=1/m$ and we color the integers in $\{ 1,2,\ldots,N\}$ with $m$ colors, then at least one of the colors is used for at least $N/m$ integers. We apply Szemer' edi’s theorem to this subset $A$ of $\{ 1,2,\ldots,N\}$, to obtain an $\ell$-term arithmetic progression of integers which each have the same color.
In [@Sz1], Szemer' edi applied his result to the question of squares in arithmetic progressions:
\[Szemer' edi\] For any constant $\delta>0$, if $N$ is sufficiently large, then $Q(N)<\delta N$.
Suppose that there are at least $\delta N$ squares in the arithmetic progression $\{ r+ns: \ n=1,2,\ldots ,N\}$ with $s\geq 1$; that is, there exists a subset $A$ of $\{ 1,2,\ldots,N\}$ with at least $\delta N$ elements for which $$r+ns \ \text{is a square, whenever} \ a\in A.$$ Szemer' edi’s theorem with $\ell=4$ then implies that $A$ contains a four-term arithmetic progression, say $u+jv$ for $j=0,1,2,3$. For these values of $n$, we have $r+ns=a+jd$, where $a=r+us$ and $d=vs>1$. That is, we have shown that $$a,a+d,a+2d,a+3d \ \text{are four squares in arithmetic progression,}$$ contradicting Fermat’s theorem.
More heavy machinery
====================
One day over lunch, in late 1989, Bombieri showed me a completely different proof of Theorem 2, this time relying on one of the most influential results in algebraic and arithmetic geometry, Faltings’ theorem [@BGU]. Faltings’ theorem is not easy to state, requiring a general understanding of an algebraic curve and its genus. The basic idea is that an equation in two variables with rational coefficients has only finitely many rational solutions (that is, solutions in which the two variables are rational numbers), unless the equation “boils down to” an equation of degree 1, 2 or 3. To be precise about “boiling down” involves the concept of [*genus*]{}, which is too complicated to explain here (see, e.g., [@BGU]). Here we only need a simple consequence of Faltings’ theorem.
**Corollary to Faltings’ Theorem.** *Let $b_1,b_2,\ldots,b_k$ be distinct integers with $k\geq 5$. Then there are only finitely many rational numbers $x$ for which $$(x+b_1)(x+b_2)\cdots (x+b_k) \ \text{is the square of a rational number}.$$*
\[Another proof of Theorem 2\] Fix an integer $M>6/\delta$. Let $B(M)$ be the total number of rational numbers $x$ and integer 6-tuples $ b_1=0< b_2<\ldots < b_6\leq M-1$ for which $(x+b_1)(x+b_2)\cdots (x+b_6)$ is the square of a rational number. Faltings’ theorem implies that $B(M)$ is some finite number, as there are only finitely many choices for the $b_j$. We let $N$ be any integer $\geq M(B(M)+5)$.
The interval $[0,N-1]$ is covered by the sub-intervals $ I_j$ for $j=0,1,2,\ldots,k-1$, where $I_j$ denotes the interval $[jM, (j+1)M)$, and $kM$ is the smallest multiple of $M$ that is greater than $N$.
Let $\mathcal N:=\{ n:\ 0\leq n\leq N-1 \ \text{and} \ a+nd \ \text{ is a square}\}$, where the arithmetic progression is chosen so that $|\mathcal N| =Q(N)$. Let $\mathcal N_j=\{ n\in \mathcal N:\ n\in I_j\}$ for each integer $j$. Let $J$ be the set of integers $j$ for which $\mathcal N_j$ has six or more elements.
Now if $n_1<n_2<\ldots<n_6$ all belong to $\mathcal N_j$, write $x=a/d+n_1$ and $b_i=n_i-n_1$ for $i=1,\ldots ,6$, so that $$b_1=0< b_2<\ldots < b_6\leq M-1$$ and each $x+b_i=a/d+n_i=(a+n_id)/d$, which implies that $$(x+b_1)(x+b_2)\cdots (x+b_6) = \frac{(a+n_1d)(a+n_2d)\cdots (a+n_6d)}{d^6}$$
is the square of a rational number.
This gives rise to one of the $B(M)$ solutions counted above, and all the solutions created in this way are distinct (since given $x,d,b_1,\ldots,b_6$ we have each $a+n_jd=d(x+b_j)$). Therefore the set $\mathcal N_j$ gives rise to $\binom{|\mathcal N_j|}6$ such solutions, and so in total we have $$\sum_{j\in J} \binom{|\mathcal N_j|}6 \leq B(M).$$
It is easy to verify that $r\leq 5+\binom r6$ for all integers $r\geq 1$, and so $$Q(N)=|\mathcal N| = \sum_{j=0}^{k-1}|\mathcal N_j| \leq \sum_{j=0}^{k-1} 5 + \sum_{j\in J} \binom{|\mathcal N_j|}6 \leq 5k+B(M),$$ as $|\mathcal N_j| \leq 5$ if $j\not\in J$. Finally, as $k\leq N/M+1$ we have $$Q(N) \leq 5k+B(M) \leq \frac{5N}M + (B(M)+5) \leq \frac{6N}M < \delta N,$$ as desired.
Bombieri [@BGP] went on, together with Granville and Pintz, to combine these two proofs (along with much more arithmetic geometry machinery), to prove that $$Q(N)<N^c$$ for any $c>\frac 23$, for sufficiently large $N$. Bombieri and Zannier [@BZ] improved this to $c>\frac 35$ with a rather simpler proof. The conjecture that $Q(N)$ behaves more like a constant times $N^{1/2}$ remains open.
[1]{}
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J.H. Silverman, *The arithmetic of elliptic curves*, Springer Verlag, New York, 1986. E. Szemerédi, The number of squares in an arithmetic progression, *Studia Sci. Math. Hungar.* [**9**]{} (1974), 417.
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Département de mathématiques et de statistique, Université de Montréal, CP 6128 succ. Centre-Ville, Montréal, QC H3C 3J7, Canada\
andrew@dms.umontreal.ca\
\
Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, United Kingdom\
a.granville@ucl.ac.uk
|
---
abstract: 'For many music analysis problems, we need to know the presence of instruments for each time frame in a multi-instrument musical piece. However, such a frame-level instrument recognition task remains difficult, mainly due to the lack of labeled datasets. To address this issue, we present in this paper a large-scale dataset that contains synthetic polyphonic music with frame-level pitch and instrument labels. Moreover, we propose a simple yet novel network architecture to jointly predict the pitch and instrument for each frame. With this multitask learning method, the pitch information can be leveraged to predict the instruments, and also the other way around. And, by using the so-called pianoroll representation of music as the main target output of the model, our model also predicts the instruments that play each individual note event. We validate the effectiveness of the proposed method for frame-level instrument recognition by comparing it with its single-task ablated versions and three state-of-the-art methods. We also demonstrate the result of the proposed method for multi-pitch streaming with real-world music. For reproducibility, we will share the code to crawl the data and to implement the proposed model at: <https://github.com/biboamy/instrument-streaming/>.'
address: |
$^{1}$ Research Center for IT Innovation, Academia Sinica, Taiwan\
$^{2}$ KKBOX Inc., Taiwan\
{biboamy,yang}@citi.sinica.edu.tw, annchen@kkbox.com
bibliography:
- 'refs.bib'
title: 'Multitask learning for frame-level instrument recognition'
---
Instrument recognition, pitch streaming
Introduction
============
Pitch and timbre are two fundamental properties of musical sounds. While the pitch decides the notes sequence of a musical piece, the timbre decides the instruments used to play each note. Since music is an art of time, for detailed analysis and modeling of the information of a musical piece, we need to build a computational model that predicts the pitch and instrument labels for each time frame. With the release of several datasets [@bittner2014medleydb; @thickstun2017learning] and the development of deep learning techniques, recent years have witnessed great progress in frame-level pitch recognition, a.k.a., multi-pitch estimation (MPE) [@Bittner2017DeepSR; @Thickstun2018InvariancesAD]. However, this is not the case for the instrument part, presumably due to the following two reasons.
![Architecture of the proposed model, which employs three loss functions for predicting the (multitrack) pianoroll, the pitch roll, and the instrument roll. The pitch and instrument predictions are computed directly from the predicted pianoroll, which is a tensor of {frequency, time, instrument}.[]{data-label="fig: streaming"}](img/inst_streaming.pdf){width="48.00000%"}
First, manually annotating the presence of instruments for each time frame in a multi-instrument musical piece is a time-consuming and labor-intensive process. As a result, most datasets available to the public only provide instrument labels on the *clip level*, namely, labeling which instruments are present over an entire audio clip of possibly multi-second long [@joder09taslp; @bosch12ismir; @openmic; @AUDIOSET]. Such clip-level labels do not specify the presence of instruments for each short-time frame (e.g., multiple milliseconds, or for each second). Datasets with frame-level instrument labels emerge only over the recent few years [@thickstun2017learning; @bittner2014medleydb; @Duan2010MultipleFF; @Gururani2017MixingS]. However, as listed in Table \[tab:dataset\] (and will be discussed at length in Section \[sec:background\]), these datasets contain at most a few hundred songs and some of them contain only classical musical pieces. The musical diversity found in these datasets might therefore not be sufficient to train a deep learning model that performs well for different musical pieces.
Pitch labels Instrument labels Real or Synth Genre Number of Songs
-------------------------------------- --------------------------------------------- ----------------------------------------------------------- --------------- ----------- -----------------
MedleyDB [@bittner2014medleydb] $\triangle$ [@Bittner2017DeepSR; @kimSLB18] ${\surd}$ [@Li2015AutomaticIR; @Gururani2018InstrumentAD] Real Variety 122
MusicNet [@thickstun2017learning] ${\surd}$ [@Thickstun2018InvariancesAD] ${\surd}$ [@Hung2018FramelevelIR] Real Classical 330
Bach10 [@Duan2010MultipleFF] ${\surd}$ [@Duan2010MultipleFF] ${\surd}$ [@Giannoulis2014ImprovingIR] Real Classical 10
Mixing Secret [@Gururani2017MixingS] ${\surd}$ [@Gururani2018InstrumentAD] Real Variety 258
MuseScore (this paper) ${\surd}$ ${\surd}$ Synthetic Variety 344,166
Second, we note that most recent work that explores deep learning techniques for frame-level instrument recognition focuses only on the instrument recognition task itself and adopts the *single-task* learning paradigm [@liu2018weakly; @Gururani2018InstrumentAD; @Hung2018FramelevelIR]. This has the drawback of neglecting the strong relations between pitch and instruments. For example, different instruments have their own pitch ranges and tend to play different parts in a polyphonic musical composition. Proper modeling of the onset and offset of musical notes may also make it easier to detect the presence of instruments [@Hung2018FramelevelIR]. From a methodological point of view, we see a potential gain to do better than these prior arts by using a *multitask* learning paradigm that models timbre and pitch jointly. This requires a dataset that contains both frame-level pitch and instrument labels.
In this paper, we introduce a new large-scale dataset called *MuseScore* to address these needs. The dataset contains the audio and MIDI pairs for 344,166 musical pieces downloaded from the official website (<https://musescore.org/>) of MuseScore, an open source and free music notation software licensed under GPL v2.0. The audio is synthesized from the corresponding MIDI file, usually using the sound font of the MuseScore synthesizer. Therefore, it is not difficult to temporally align the audio and MIDI files to get the frame-level pitch and instrument labels for the audio. Although the dataset only contains synthesized audio, it includes a variety of performing styles in different musical genres.
Moreover, we propose to transform each MIDI file to the *multitrack pianoroll* representation of music (see Fig. \[fig: streaming\] for an illustration) [@pypianoroll], which is a binary tensor representing the presence of notes over different time steps for each instrument. Then, we propose a multitask learning method that learns to predict from the audio of a musical piece its (multitrack) pianoroll, frame-level pitch labels (a.k.a., the *pitch roll*), and the instrument labels (a.k.a., the *instrument roll*). While the latter two can be obtained by directly summing up the pianoroll along different dimensions, the three involved loss functions would work together to force the model learn the interactions between pitch and timbre. Our experiments show that the proposed model can not only perform better than its task-specific counterparts, but also existing methods for frame-level instrument recognition [@liu2018weakly; @Gururani2018InstrumentAD; @Hung2018FramelevelIR].
Background {#sec:background}
==========
To our knowledge, there are four public-domain datasets that provide frame-level instrument labels, as listed in Table \[tab:dataset\]. Among them, MedleyDB [@bittner2014medleydb], MusicNet [@thickstun2017learning] and Bach10 [@Duan2010MultipleFF] are collected originally for MPE research, while Mixing Secret [@Gururani2017MixingS] is meant for instrument recognition. When it comes to building “clip-level” instrument recognizers, there are other more well-known datasets such as the ParisTech [@joder09taslp] and IRMAS [@bosch12ismir] datasets. Still, there are previous work that uses these datasets for building either clip-level [@Li2015AutomaticIR; @Giannoulis2014ImprovingIR] or frame-level [@Hung2018FramelevelIR; @Gururani2018InstrumentAD] instrument recognizers.
There are three recent works on frame-level instrument recognition. The model proposed by Hung and Yang [@Hung2018FramelevelIR] is trained and evaluated on different subsets of MusicNet [@thickstun2017learning], which consists of only classical music. This model considers the pitch labels estimated by a pre-trained model (i.e. [@Bittner2017DeepSR]) as an additional input to predict instrument, but the pre-trained model is fixed and not further updated. The model presented by Gururani *et al.* [@Gururani2018InstrumentAD] is trained and evaluated on the combination of MedleyDB [@bittner2014medleydb] and Mixing Secrets [@Gururani2017MixingS]. Both [@Hung2018FramelevelIR] and [@Gururani2018InstrumentAD] use frame-level instrument labels for training. In contrast, the model presented by Liu *et al.* [@liu2018weakly] uses only clip-level instrument labels associated with YouTube videos for training, using a weakly-supervised approach. Both [@liu2018weakly] and [@Gururani2018InstrumentAD] do not consider pitch information. As the existing datasets are limited in genre coverage or data size, prediction models trained on these datasets may not generalize well, as shown in [@Bittner2017DeepSR] for pitch recognition. Unlike these prior arts, we explore the possiblity to train a model on large-scale synthesized audio dataset, using a multitask learning method that considers both pitch and timbre.
OpenMIC-2018 [@openmic] is a new large-scale dataset for training clip-level instrument recognizers. It contains 20,000 10-second clilps of Creative Commons-licensed music of various genres. But, there is no frame-level labels.
*Multi-pitch streaming* has been referred to as the task that assigns instrument labels to note events [@Duan2014MultipitchSO]. Therefore, it goes one step closer to full transcription of musical audio than MPE. However, as the task involves both frame-level pitch and instrument recognition, it is only attempted sporadically in the literature (e.g., [@Duan2014MultipitchSO; @arora2015multiple]). By predicting the pianorolls, the proposed model actally performs multi-pitch streaming.
Proposed Dataset {#section_db}
================
The MuseScore dataset is collected from the online forum of the MuseScore community. Any user can upload the MIDI and the corresponding audio for the music pieces they create using the software. The audio is therefore usually synthesized by the MuseScore synthesizer, but the user has the freedom to use other synthesizers. The audio clips have diverse musical genres and are about two mins long on average. More statistics of the dataset can be found from our GitHub repo. While the collected audio and MIDI pairs are usually well aligned, to ensure the data quality we further run the dynamic time warping (DTW)-based alignment algorithm proposed by Raffel [@raffel2016learning] over all the data pairs. We then compute from each MIDI file the groundtruth pianoroll, pitch roll and instrument roll using `Pypianoroll` [@pypianoroll].
The dataset contains 128 different instrument categories as defined in the MIDI spec. A main limitation is that there is no singing voice. This can be made up by datasets with labels of vocal activity [@kyungyun18ismir], such as the Jamendo dataset [@ramona08icassp]. Due to copyright issues, we cannot share the dataset itself but the code to collect and process the data.
![The network architecture of the proposed model. It has a simple U-net structure [@ronneberger2015u] with four residual convolution layers and four residual up-convolution layers.[]{data-label="fig: model"}](img/network.png){width="40.00000%"}
Proposed Model {#section_model}
==============
As Fig. \[fig: streaming\] shows, the proposed model learns a mapping $f(\cdot)$ (i.e., the ‘Model’ block in the figure) between an audio representation $\mathbf{X}$, such as the constant-Q transform (CQT) [@schorkhuber2010constant], and the pianoroll $\mathbf{Y}_{roll} \in \{0,1\}^{F \times T \times M}$, where $F$, $T$ and $M$ denote the number of pitches, time frames and instruments, respectively. Namely, the model can be viewed as a multi-pitch streaming model. The model has two by-products, the pitch roll $\mathbf{Y}_{p} \in \{0,1\}^{F \times T}$ and the instrument roll $\mathbf{Y}_{i} \in \{0,1\}^{M \times T}$. As Fig. \[fig: streaming\] shows, from an input audio, our model computes $\widehat{\mathbf{Y}}_{p}$ and $\widehat{\mathbf{Y}}_{i}$ directly from the pianoroll $\widehat{\mathbf{Y}}_{roll}$ predicted by the model. Therefore, $f(\cdot)$ contains all the learnable parameters of the model.
We train the model $f(\cdot)$ with a multitask learning method by using three cost functions, $L_{roll}$, $L_p$ and $L_i$, as shown in Fig. \[fig: streaming\]. For each of them, we use the binary cross entropy (BCE) between the groundtruth and the predicted matrices (tensors). The BCE is defined as: $$L_* = - \textstyle{\sum} ~[\mathbf{Y}_*\cdot \ln\sigma(\widehat{\mathbf{Y}_*}) + (1-\mathbf{Y}_*) \cdot \ln(1-\sigma(\widehat{\mathbf{Y}_*})) ] \,,
\label{eq:lt}$$ where $\sigma$ is the sigmoid function that scales its input to $[0,1]$. We weigh the three cost terms so that they have the same range, and use their weighted sum to update $f(\cdot)$.
In sum, pitch and timbre are modeled jointly with a shared network by our model. This learning method is designed for music and, to our knowledge, has not been used elsewhere.
Network Structure
-----------------
The network architecture of our model is shown in Fig. \[fig: model\]. It is a simple convolutional encoder/decoder network with symmetric skip connections between the encoding and decoding layers. Such a “U-net” structure has been found useful for image segmentation [@ronneberger2015u], where the task is to learn a mapping function between a dense, numeric matrix (i.e., an image) and a sparse, binary matrix (i.e., the segment boundaries). We presume that the U-net structure can work well for predicting the pianorolls, since it also involves learning such a mapping function. In our implementation, the encoder and decoder are composed of four residual blocks for convolution and up-convolution. Each residual block has three convolution, two batchNorm and two leakyReLU layers. The model is trained with stochastic gradient descent with 0.005 learning rate. More details can be found from our GitHub repo.
Model Input {#section_model_io}
-----------
We use CQT [@schorkhuber2010constant] to represent the input audio, since it adopts a log frequency scale that better aligns with our perception of pitch. CQT also provides better frequency resolution in the low-frequency part, which helps detect the fundamental frequencies. For the convenience of training with mini-batches, each audio clip in the training set is divided into 10-second segments. We compute CQT by `librosa` [@mcfee2015librosa], with 16 kHz sampling rate, 512-sample hop size, and 88 frequency bins.
Method Instrument Pitch Pianoroll
--------------------------- ------------ ------- -----------
$L_{roll}$ only (ablated) — — 0.623
$L_i$ only (ablated) 0.896 — —
$L_p$ only (ablated) — 0.799 —
all (proposed) 0.947 0.803 0.647
: Performance comparison of the proposed multitask learning method (‘all’) and 3 single-task ablated versions, for frame-level instrument recognition (in F1-score), frame-level pitch recognition (Acc), and pianoroll prediction (Acc) using the triaining and test subsets of MuseScore, for 9 instruments.[]{data-label="tab:task_comp"}
Method Training set Piano Guitar Violin Cello Flute Avg
----------------------------- ------------------------------------------------------------------------- ------- -------- -------- ------- ------- -------
[@liu2018weakly] YouTube-8M [@youtube8m] 0.766 0.780 0.787 0.755 0.708 0.759
[@Gururani2018InstrumentAD] Training split of ‘MedleyDB+Mixing Secrets’ [@Gururani2018InstrumentAD] 0.733 0.783 0.857 0.860 0.851 0.817
[@Hung2018FramelevelIR] MuseScore training subset 0.690 0.660 0.697 0.774 0.860 0.736
Ours MuseScore training subset 0.718 0.819 0.682 0.812 0.961 0.798
Experiment {#section_exp}
==========
Ablation Study
--------------
We report two sets of experiments for frame-level instrument recognition. In the first experiment, we compare the proposed multitask learning method with its single-task versions, using two non-overlapping subsets of MuseScore as the training and test sets. Specifically, we consider only the 9 most popular instruments[^1] and run a script to pick for each instrument 5,500 clips as the training set and 200 clips as the test set. We consider three ablated versions here: using the U-net architecutre shown in Fig. \[fig: streaming\] to predict the pianoroll with only $L_{roll}$, to predict directly the instrument roll (i.e. only considering $L_{i}$), and to preidct directly the pitch roll (i.e. only $L_{p}$).
Result shown in Table \[tab:task\_comp\] clearly demonstrates the superiority of the proposed multitask learning method over the single-task counterparts, especially for instrument prediction. Here, we use `mir_eval` [@raffel2014mir_eval] to calculate the ‘pitch’ and ‘pianoroll’ accuracies. For ‘instrument’, we report the F1-score.
![The predicted pianoroll (best viewed in color) for the first 30 seconds of three real-world music. We paint different instruments with different colors: *Black*—piano, *Purple*—guitar, *Green*—violin, *Orange*—cello, *Yello*—flute.[]{data-label="fig: result"}](img/result.pdf "fig:"){width="48.00000%"} \[fig: all\_of\_me\]
Comparison with Existing Methods
--------------------------------
In the second experiment, we compare our method with three existing methods [@liu2018weakly; @Hung2018FramelevelIR; @Gururani2018InstrumentAD]. Following [@Gururani2018InstrumentAD], we take 15 songs from MedleyDB and 54 songs from Mixing Secret as the test set, and consider only 5 instruments (see Table \[tab:method\_comp\]). The test clips contain instruments (e.g., singing voice) that are beyond these five. We evaluate the result for per-second instrument recognition in terms of area under the curve (AUC). As shown in Table \[tab:method\_comp\], these methods use different training sets. Specifically, we retrain model [@Hung2018FramelevelIR] using the same training subset of MuseScore as the proposed model. The model [@liu2018weakly] is trained on the YouTube-8M dataset [@youtube8m]. The model [@Gururani2018InstrumentAD] is trained on a training split of ‘MedleyDB+Mixing Secret’, with 100 songs from each of the two datasets. The model [@Gururani2018InstrumentAD] therefore has some advantages since the training set is close to the test set. The result of [@liu2018weakly] and [@Gururani2018InstrumentAD] are from the authors of the respective papers.
Table \[tab:method\_comp\] shows that our model outperforms the two prior arts [@Hung2018FramelevelIR; @liu2018weakly] and is behind model [@Gururani2018InstrumentAD]. We consider our model compares favorably with [@Gururani2018InstrumentAD], as our training set is quite different from the test set. Interestingly, our model is better at the flute, while [@Gururani2018InstrumentAD] is better at the violin. This might be related to the difference between the real and synthesized sounds for these instruments, but future work is needed to clarify.
Multi-pitch Streaming
---------------------
Finally, Fig. \[fig: result\] demonstrates the predicted pianorolls for the first 30 seconds of three randomly-selected real-world songs.[^2] In general, the proposed model can predict the notes and instruments pretty nicely, especially for the second clip, which contains only a guitar solo. This is promising, since the model is trained with synthetic audio only. Yet, we also see two limitations of our model. First, it cannot deal with sounds that are not included in the training data—e.g., for the 5th–10th seconds of the third clip, our model mistakes the piano for the flute, possibly because the singer hums in the meanwhile. Second, it cannot predict the onset times accurately—e.g., the violin melody of the first clip actually plays the same note for several times, but the model mistakes them for long notes.
Conclusion {#section_conclusion}
==========
In this paper, we have presented a new synthetic dataset and a multitask learning method that models pitch and timbre jointly. It allows the model to predict instrument, pitch and pianorolls representation for each time frame. Experiments show that our model generalizes well to real music.
In the future, we plan to improve the instrument recognition by re-synthesizing the MIDI files from Musescore dataset to produce more realistic instrument sound. Moreover, we also plan to mix the singing voice clips from [@bittner2014medleydb] with our training data (for data augmentation) to deal with singing voices.
[^1]: Piano, acoustic guitar, electric guitar, trumpet, sax, violin, cello & flute.
[^2]: The three songs are, from top to bottom: *All of Me* violin & guitar cover (https://www.youtube.com/watch?v=YpYQh7eQULc), *Ocean* by Pur-dull (https://www.youtube.com/watch?v=5Lb9GvEO-sA) and *Beautiful* by Christina Aguilera (https://www.youtube.com/watch?v=eAfyFTzZDMM).
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***Charles de IZARRA***\
*Groupe de Recherche sur l’Énergetique des Milieux Ionisés,\
UMR6606 Université d’Orléans - CNRS, Faculté des Sciences, Site de Bourges,\
rue Gaston Berger, BP 4043, 18028 BOURGES Cedex, France.\
Charles.De$\_$Izarra@univ-orleans.fr\
Tél : 33 (0)2 48 27 27 31*\
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**Résumé**\
*L’objet du travail présenté est la calibration d’une lampe à filament de tungstène à partir de mesures électriques à la fois simples et précises, et qui permettent de déterminer la température du filament de tungstène en fonction de l’intensité du courant d’alimentation, puis de mesurer la surface du filament. Ces données permettent alors de calibrer la lampe en terme de luminance, directement utilisable pour étalonner des capteurs comme des pyromètres optiques ou tout autre capteur photométrique. La procédure de calibration proposée a été appliquée sur une lampe à filament standard (lampe utilisée dans l’éclairage automobile), puis sur une lampe étalon à ruban de tungstène. La procédure de calibration mise au point permet de retrouver les points de calibration de la lampe (NIST) avec une précision de l’ordre de 2%.*\
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*Mots-Clés : Lampe à ruban de tungstène, Photométrie, Rayonnement thermique*\
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**Introduction**\
Les capteurs photométriques tels que les pyromètres optiques, destinés à mesurer la température de corps chauffés nécessitent d’être calibrés en enregistrant le signal qu’ils délivrent en analysant un corps chauffé dont on connaît à la fois la température et l’émissivité. Le corps de référence choisi est souvent un corps noir de laboratoire, dont le principal inconvénient est d’être limité à des températures assez basses (inférieures à 2000 K pour les plus courants). Une autre possibilité est l’utilisation d’une lampe étalon (mère ou fille) à ruban de tungstène, dont le prix est relativement élevé (de l’ordre de 7000 €), dont on connaît la température pour une valeur de l’intensité du courant continu utilisé pour alimenter la lampe. Dans cette communication, on expose une technique de calibration d’une lampe à filament de tungstène avec des mesures électriques alliant simplicité et précision, et avec lesquelles on détermine la température du filament de tungstène en fonction de l’intensité du courant d’alimentation.\
**1. L’émission radiative des corps opaques chauffés**\
La surface d’un corps que nous choisirons opaque, à la température $T$, est la source d’une émission de rayonnement continu dont la répartition de l’énergie en fonction de la longueur d’onde est donnée par la loi de Planck \[Chéron, 1999\], basée sur le modèle du corps noir, qui, par définition est capable d’absorber toutes les longueurs d’onde $\lambda$ qu’il reçoit. Quantitativement, on utilise la luminance spectrale $L_\lambda ^0$ donnée par : $$L_\lambda ^0 = \frac{2 h c ^2}{\lambda ^5}
\frac{1}{\exp\left(\frac{hc}{\lambda kT}\right) -1} \label{Planck}$$ avec :
- $h$ : constante de Planck ($h$ = 6.6 10 $^{-34}$ J.s)
- $k$ : constante de Boltzmann ($k$ = 1.38 10 $^{-23}$ J/K)
- $c$ : célérité de la lumière dans le vide ($c$ = 3 10 $^{8}$ m/s)
et qui est le flux émis par unité de surface apparente, par unité d’angle solide et par unité de longueur d’onde (Unité : W m$^{-2}$ Sr$^{-1}$ m$^{-1}$).
Dans le cas où le corps émissif est une surface $S$ de corps noir, la puissance émise dans le demi-espace est donnée par la relation de Stefan-Boltzmann : $$P^0 = S \sigma T^4 \label{Stefan}$$ où $\sigma$ est la constante de Stephan ($\sigma = 5.67 10 ^{-8}$ W.K$^{-4}$.m$^{-2}$). Lorsque le corps considéré n’est pas un corps noir, on parle alors de corps gris ou de corps réel, et les données précédentes doivent être réduites en les multipliant par l’émissivité $\varepsilon(\lambda, T)$ inférieure à 1.\
**2. Les données thermophysiques du tungstène**\
Le tungstène est le métal possédant la température du point de fusion la plus élevée ($T_F$=3695 K). De ce fait, il a été largement étudié depuis le début du XX$^{\grave{e}me}$ siècle afin de permettre la production de filaments de lampes à incandescence \[Forsythe and Worthing, 1916\].
Sachant que le filament de la lampe est essentiellement une résistance morte chauffée par effet Joule, les données nécessaires au calcul de la résistance en fonction de la température sont la résistivité $\rho$ et le coefficient de dilatation thermique $\beta$. Ces deux grandeurs sont tabulées en fonction de la température, et $\beta$ est donné en % de la longueur $\ell_0$ du filament à la température de 300 K \[Lide, 1991-1992\].
Pour une température $T$ donnée, la résistance $R$ d’un fil de tungstène est donnée par la relation : $R(T)=\rho(T)
{\ell(T)}/{S_f(T)}$ où $S_f$ est la section du fil, que sous supposons constante sur sa longueur. La section, pas nécessairement circulaire, peut être exprimée en fonction d’une longueur caractéristique $d$. Nous avons $S=C d^2$, où $C$ est une constante de proportionnalité qui dépend de la forme géométrique de la section. Pour fixer les idées, $C=\pi/4$ dans le cas d’une section circulaire en choisissant le diamètre du fil comme longueur caractéristique $d$.
Calculons la variation de la résistance $R(T)$ relativement à la résistance $R_0$ à la température de 300 K prise comme référence. Nous avons : $R_0=\rho_0 {\ell_0}/{{S_f}_0}$ ou encore, en introduisant la longueur $d_0$ à 300 K : $R_0= {\rho_0 \ell_0}/{C
d_0 ^2}.$ Pour une température $T$, la résistance $R(T)$ est : $$R(T)= \frac{\rho(T) \ell(T)}{C d^2(T)} \Rightarrow
R(T)= \frac{\rho(T) \left[ \ell_0 + \frac{\beta \ell_0}{100}
\right]}{\left[ d_0 + \frac{\beta d_0}{100} \right]^2} .$$
Le rapport $R(T)/R_0$ est alors : $$R(T)/R_0 = \frac{100 \rho(T)}{\left[100+\beta\right] \rho_0}.
\label{rapport}$$
La relation (\[rapport\]), permet, à partir des données disponibles dans la littérature, de calculer le rapport $R(T)/R_0$ en fonction de la température $T$. La courbe représentative de $R(T)/R_0$ en fonction de $T$ est donnée sur la figure \[COURBE1\]; on remarque que lorsque la température augmente de 3000 K, la résistance est multipliée par 20.
Nous avons choisi de déterminer par une méthode des moindres carrés l’équation d’une parabole passant par les couples de points ($R(T)/R_0,T)$ avec une erreur relative inférieure à 0.1% pour les températures élevées (voir équation (\[FIT\])).
$$R(T)/R_0=-0,52427113+0,00466128 T +2,8420718 10^{-7}T^2
\label{FIT}$$
**3. Procédure de calibration d’une lampe à filament de tungstène**\
Les mesures ont été réalisées en utilisant une lampe Philips de type E4-2DT W21W, prévue pour fonctionner dans des conditions nominales sous une tension continue de 12 V, pour une puissance de 21 W. Le montage (figure \[MONTAGE\]) comprend une alimentation de courant continu de marque Convergie/Fontaine (type ASF1000/20.50) permettant une lecture directe de l’intensité du courant $I$ délivré, et un voltmètre numérique de marque METRIX avec lequel on mesure la chute de tension $U$ aux bornes de la lampe.
![*Lampe à filament de tungstène utilisée (image de gauche) et montage électrique utilisé (image de droite).* []{data-label="MONTAGE"}](photolamp.eps "fig:")![*Lampe à filament de tungstène utilisée (image de gauche) et montage électrique utilisé (image de droite).* []{data-label="MONTAGE"}](montage.eps "fig:")
En faisant varier l’intensité du courant $I$ depuis de très faibles valeurs (proches de zéro) jusqu’à environ 2.5 A, la mesure de la chute de tension $U$ aux bornes de la lampe permet de déterminer la résistance $R$ de la lampe pour chaque valeur de $I$ en appliquant la loi d’Ohm ($R=U/I$). Lors de l’expérience, il est nécessaire d’attendre que la lampe se stabilise en température avant de relever le couple de valeurs $(U,I)$.
Un soin tout particulier est nécessaire pour déterminer la résistance de la lampe à température ambiante $R_0$ qui conditionne la qualité et la précision de la procédure de calibration. Les points de mesures $(R, I)$ à très faible intensité ont été extrapolés à courant nul en utilisant un modèle parabolique de la forme : $R(I) = R_0 + A I^2$. Une procédure d’ajustage par la méthode des moindres carrés permet de déterminer le paramètre $A$ et la valeur de la résistance à température ambiante $R_0$.
Connaissant la valeur expérimentale du rapport $R(T)/R_0$ pour chaque valeur de l’intensité du courant, on est en mesure de déterminer la température $T$ du filament en utilisant la relation (\[FIT\]) (résolution d’une équation du second degré). Au final, on obtient une courbe de calibration sous la forme de la température $T$ du filament en fonction de l’intensité du courant $I$ (figure \[TfoncdeI\]).
D’un point de vue purement énergétique, le filament de la lampe chauffé est le siège de la transformation de l’énergie électrique sous forme de chaleur et sous forme de rayonnement. En considérant qu’un état stationnaire est atteint, on a l’égalité entre la puissance $P=UI$ dissipée par effet Joule, et la somme des puissances mises en jeu par transfert thermique entre le filament à température $T$ et le milieu ambiant à température $T_a$ et par rayonnement (loi de Stefan-Boltzmann). En appelant $\alpha$ le coefficient d’échange thermique incluant les phénomènes de conduction et de convection thermiques, $\varepsilon$ l’émissivité du tungstène, et $S$ la surface émissive du filament nous avons : $$UI = \alpha \left( T- T_a \right) + S \varepsilon \sigma T^4.
\label{EQUATIONUI}$$ À partir des mesures de $U$ et $I$, il est simple de calculer la puissance mise en jeu, puis de considérer la quantité $UI/T^4$. En effet, selon l’équation (\[EQUATIONUI\]), nous avons : $$\frac{UI}{T^4} = \alpha \frac{\left( T- T_a \right)}{T^4} + S
\varepsilon \sigma. \label{EQUATIONUISURT4}$$ L’équation (\[EQUATIONUISURT4\]) comporte deux termes ; le premier est prépondérant à basse température , et le second terme est constant, sous réserve que $\varepsilon$ soit constant en fonction de la température.
Sur la figure \[RAPPORTUIsurT4PHILIPS\], on a représenté le graphe de la quantité $UI/T^4$ en fonction de $T$, *qui est tout à fait conforme aux prévisions annoncées plus haut, et qui prouvent la validité des mesures*. Pour les températures élevées, la courbe est horizontale et permet de déterminer la surface émissive du filament de tungstène.
**4. Procédure appliquée à une lampe étalon**\
La procédure de calibration en température présentée plus haut a été appliquée à une lampe étalon à ruban de tungstène (lampe OSRAM WI 17G figure \[photoosram\]) étalonnée sur une valeur unique de l’intensité du courant $I$. L’étalonnage par le constructeur indique une température de 2689 K pour une intensité égale à 14.1 A.
Les courbes donnant la température de la lampe en fonction de l’intensité du courant et le rapport $UI/T^4$ en fonction de $T$ sont présentées sur la figure \[COURBESlampeOSRAM\]. La valeur de la température déterminée pour l’intensité de 14.1 A (intensité de calibration) est égale à 2760 K, ce qui correspond à une erreur relative de 2.6% par rapport à la température de calibration annoncée. Compte tenu du phénomène de vieillissement de cette lampe, avec une évaporation du tungstène et son dépôt sur la paroi interne de l’ampoule visible sur la figure \[photoosram\], il est certain que la calibration n’est plus optimale et permet d’expliquer l’écart entre la température mesurée pour 14.1 A et la température de calibration.
![*Photographie de la lampe étalon à ruban de tungstène.* []{data-label="photoosram"}](photoosram.eps)
**5. Conclusion et perspectives**\
La procédure de calibration d’une lampe à filament de tungstène présentée dans cette communication est relativement simple à mettre en [œ]{}uvre, et à la portée de tout laboratoire de recherche et développement. La validation de la procédure est réalisée grâce à l’évaluation du rapport $UI/T^4$ en fonction de $T$, dont le comportement est tout à fait conforme à ce que prévoit la théorie.
Enfin, pour être en mesure de calibrer un capteur photométrique à partir du rayonnement émis par un filament en tungstène dont on connaît la température de surface, il est nécessaire de considérer la relation (\[Planck\]) donnant la luminance spectrale du corps noir multipliée par l’émissivité du tungstène fonction de la température $T$ et de la longueur d’onde $\lambda$ (voir tableau \[TABLO1\]).
----------------- ------- ------- ------- ------- ------- ------- ------- ------- ------- -------
Température (K) 0.25 0.3 0.35 0.4 0.5 0.6 0.7 0.8 0.9 1
1600 0.448 0.482 0.478 0.481 0.469 0.455 0.444 0.431 0.413 0.390
1800 0.442 0.748 0.476 0.744 0.465 0.452 0.440 0.425 0.407 0.385
2000 0.436 0.474 0.473 0.747 0.462 0.448 0.436 0.419 0.401 0.381
2200 0.429 0.470 0.470 0.471 0.458 0.445 0.431 0.415 0.896 0.378
2400 0.422 0.465 0.466 0.468 0.455 0.441 0.427 0.408 0.391 0.372
2600 0.418 0.461 0.464 0.464 0.451 0.437 0.423 0.404 0.386 0.369
2800 0.411 0.456 0.461 0.461 0.448 0.434 0.419 0.400 0.383 0.367
----------------- ------- ------- ------- ------- ------- ------- ------- ------- ------- -------
: Emissivité du tungstène en fonction de la température et de la longueur d’onde \[Lide, 1991-1992\].\[TABLO1\]
[99]{}
Chéron B., 1999, *Transferts thermiques*, Ellipses.\
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Forsythe W.E. and Worthing A. G., 1916, *The properties of tungsten and the characteristics of tungsten lamps*, Physical Review, **7**,302, 146-185.\
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Lide R.D., Editor-in-Chief, 1991-1992, *CRC Handbook of Chemistry and Physics*, 72nd Ed., pp.**10**-285,**10**-286.
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abstract: 'Small separation between reactants, not exceeding $10^{-8}-10^{-7}cm$, is the necessary condition for various chemical reactions. It is shown that random advection and stretching by turbulence leads to formation of scalar-enriched sheets of [*strongly fluctuating thickness*]{} $\eta_{c}$. The molecular-level mixing is achieved by diffusion across these sheets (interfaces) separating the reactants. Since diffusion time scale is $\tau_{d}\propto \eta_{c}^{2}$, the knowledge of probability density $Q(\eta_{c},Re)$ is crucial for evaluation of chemical reaction rates. In this paper we derive the probability density $Q(\eta_{c},Re,Sc)$ and predict a transition in the reaction rate behavior from ${\cal R}\propto \sqrt{Re}$ ($Re\leq 10^{4}$) to the high-Re asymptotics ${\cal R}\propto Re^{0}$. The theory leads to an approximate universality of transitional Reynolds number $Re_{tr}\approx 10^{4}$. It is also shown that if chemical reaction involves short-lived reactants, very strong anomalous fluctuations of the length-scale $\eta_{c}$ may lead to non-negligibly small reaction rates.'
author:
- Victor Yakhot
date: '?? and in revised form ??'
title: ' Dissipation Scale Fluctuations and Chemical Reaction Rates in Turbulent Flows. '
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Introduction
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Efficiency of chemical reactions and combustion crucially depends upon number of reactant moles mixed on a molecular scale. Indeed, for a non negligibly small reaction rate, the separation between reacting species must not exceed $r\approx 10^{-8}-10^{-7}cm$, which is the main reason for immense importance of mixing process \[1\]. Slow diffusion in laminar media leads to an extremely poor mixing and, being the most common mixing accelerator, hydrodynamic turbulence plays vital role in natural and man-made processes like heat transfer, chemical transformations, combustion, meteorology and astrophysics. The mixing process in turbulent flows involves three main steps: 1. entrainment, creating pockets of material $B$ in a turbulent flow enriched by a substance $A$; 2. advection and stretching leading to formation of thin convoluted sheets of the thickness $\eta_{c}$ separating the reactants; [*This process is often related to as ‘mixing by random stirring’.*]{} 3. molecular diffusion across these ‘dissipation sheets ’ on a time scale $\tau_{d}\approx \eta_{c}^{2}/D$. If chemical reaction and turbulent mixing processes are fast - molecular diffusion is the reaction rate - determining step ( Dimotakis (2005), (1993)). Recently, investigation of the role of scalar dissipation and dissipation sheets has become an extremely active field (see, for example, excellent reviews by Bilger (2004), Sreenivasan (2004), Peters (2000), Dimotakis (2005) and recent papers by Celani et al (2005), Villermaux et al (2003) and Bush et al (1998). )
Kolmogorov - Batchelor phenomenology.
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Based on a classic Kolomogorov’s cascade concept, we illustrate the main qualitative features of the mixing of two liquids (chemical components) $A$ and $B$. (A quantitative dynamic description will be developed below). Consider a turbulent flow of a fluid $A$. The integral scale of turbulence, separating the energy and inertial ranges, is $L$ and the large-scale Reynolds number is $Re=u_{rms}L/\nu$. At a time $t=0$, a blob of the same fluid enriched with chemical component $B$ is placed in the flow. For simplicity we assume the linear dimension of a blob $r_{0}=L$. At this point, the area of the interface, separating $A$ and $B$ substances, is $O(L^{2})$. The life- (turn-over) time of this eddy is $T_{0}\equiv \tau_{eddy}\approx L/u_{rms}\approx L^{\frac{2}{3}}{\cal E}^{\frac{1}{3}}$, where $u_{rms}$ is the rms value of turbulent velocity. The magnitude of the energy flux across scales is ${\cal E}=\alpha_{{\cal E}} u_{rms}^{3}/L$ with $\alpha_{{\cal E}}\approx 0.8-0.9$. In accord with Kolmogorov’s phenomenology, after $\approx T_{0}$ seconds, by nonlinear interactions, this eddy is transformed into another one of the linear dimensions $L_{x,1}=L/2$ and $L_{y,1}=L_{z,1}\approx \sqrt{2}L$. Since the gradient in the $x$-direction is largest, the characteristic time-scale of this, “daughter” structure is: $T_{1}\approx \frac{L^{\frac{2}{3}}}{{\cal E}^{\frac{1}{3}}}2^{-\frac{2}{3}}$. Then, after $n\gg 1$ steps
$$L_{x,n}= 2^{-n}L;~~~T_{n}\approx 2^{-\frac{2n}{3}}\frac{L^{\frac{2}{3}}}{{\cal E}^{\frac{1}{3}}};~~~\tau_{v,n}\approx 2^{-2n}\frac{L^{2}}{\nu}$$
where the viscous time is denoted as $\tau_{v}$. We see that the viscous time of a structure strongly decreases with the number of ‘cascade steps’. The time needed to form the smallest Kolmogorov eddies on the scale $\eta_{K}\approx L Re^{-\frac{3}{4}}$ is thus:
$$T_{K}=T_{0}+T_{1}+\cdot\cdot\cdot T_{n}\approx 2.7 T_{0}\approx \frac{L^{\frac{2}{3}}}{{\cal E}^{\frac{1}{3}}}$$
where $n\approx \frac{3}{4}\frac{\ln Re}{\ln 2}\approx \ln Re\gg1$. It is important that the Reynolds number $Re=u_{rms}L/\nu$ used in the above relations is to be distinguished from $Re_{U}=UL/\nu$ where $U$ is the mean velocity in the flow. Typically, numerically, $Re<Re_{U}$. The fact that Kolmogorov’s scale $\eta_{K}$ and dissipation rate ${\cal E}$ are formed on the time-scale of a single large-scale turn-over time ($\tau_{eddy}$) has been tested in various numerical experiments.
If the Schmidt number $Sc=\nu/D\approx 1$, in accord with the cascade picture, the Kolmogorov scale is the smallest legth-scale created by turbulence and at times $t>T_{K}$, the mixing proceeds by molecular diffusion.
If however, $Sc\gg1$ and the scalar diffusion is extremely inefficient, then, after formation of the Kolmogorov scale $\eta_{K}$, the stretching process by the large-scale ($r<<\eta_{K}$) velocity field leads to generation of the ever thinner scalar-enriched sheets until $\eta_{c}(t)\approx \eta_{B}\approx \eta_{K}/\sqrt{Sc}\ll \eta_{K}$. It is only after that, the scalar diffusion takes over the mixing process. The mean width of these sheets is: $\eta_{B}\approx \eta_{K}/\sqrt{Sc}$, where $\eta_{B}$ is called the Batchelor (1959) scale. At this stage, the substances $A$ and $B$, separated by the distance $\eta_{B}$, can mix only by molecular diffusion on a time scale $\tau_{d}\approx \eta_{B}^{2}/D\approx \eta_{K}^{2}/\nu\approx \frac{L^{2}}{\nu} Re^{-\frac{3}{2}}$ .
To get a feel for “real life charm”, we consider a simple numerical example. In gases where $Sc\approx 1$, the length-scales $\eta_{K}\approx \eta_{B}$. In liquids, the situation is different and $Sc\approx 600-3000$ (Dimotakis (2005), (1993)) which means that $\eta_{K}/\eta_{B}\approx 25~-~50$. Thus, while the scales $\eta_{B}$ and $\eta_{K}$ can be very different, the corresponding diffusion and viscous times scales $\tau_{d}\approx \eta_{B}^{2}/D\approx \eta_{K}^{2}/\nu\approx \tau_{v}$ are of the same order. Moreover, it has been shown in various numerical and experimental studies (see for example Monin and Yaglom (1975)) that the relevant viscous scale $\eta_{\nu}$ is numerically larger than Kolmogorov’s one: $\eta_{\nu}=\alpha_{\nu} \eta_{K}$ with $\alpha_{\nu}\approx 10-20$. Thus, the diffusion time is, in fact, $\tau_{d}=\eta_{\nu}^{2}/\nu \approx \tau_{v}$. (The origin of the large factor $\alpha_{\nu}$ will be discussed below.)
Low and large Reynolds number behavior of reaction rates.
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If $Sc>>1$, the process consists of two steps: 1. Formation of structures (sheets) on the viscous scale $\eta_{\nu}\approx \alpha_{\nu} LRe^{-\frac{3}{4}}$; 2. Further stretching of the scalar fieled toward $\eta_{c}\approx \alpha_{\nu}\eta_{K}/\sqrt{Sc}$. It can be shown readily (see Monin and Yaglom (1975)) that, after initial formation of the dissipation scale $\eta_{\nu}$, the distance between two particles across the sheets, stretched by the large-scale velocity field decreases with time as $r(t)=\eta_{\nu}\exp(-\gamma t)$. Thus, the scalar dissipation scale is formed on a time - scale $\tau_{c}$ given by the relation:
$$\eta_{\nu}e^{-\gamma \tau_{c}}\approx \eta_{\nu}/\sqrt{Sc}$$
where $\gamma=a_{\gamma}\sqrt{\frac{{\cal E}}{\nu}}$ with $\frac{1}{\sqrt{3}}>a_{\gamma}>\frac{1}{2\sqrt{3}}$. This gives:
$$\tau_{c}= \frac{L}{2a_{\gamma} u_{rms}}Re^{-\frac{1}{2}} \ln Sc$$
The diffusion time across scalar dissipation sheets is:
$$\tau_{d}\approx \frac{\alpha_{\nu}^{2} L^{2} Re^{-\frac{3}{2}}}{\nu}\approx \alpha_{\nu}^{2}\frac{L}{u_{rms}}Re^{-\frac{1}{2}}$$
Comparing the above relations we come to a non-trivial conclusion: although the characteristics times scale with the Reynolds number as $\tau_{c}\propto \tau_{d}\propto \sqrt{Re}$, the stretching process leading to both sheet thinning and increase of the scalar gradient competes with the simultaneous concentration-gradient-decreasing diffusion across the developing interfaces. However, due to the large magnitude of numerical factor $\alpha_{\nu}\approx 10-20$, the process of the sheet-thinning by the large-scale stretching is numerically much faster than diffusion across the sheets. This means that during the stretching stage, the scalar diffusion across the interfaces can be neglected. [*The factor $\alpha_{\nu}\approx 10-20$ will be calculated below and it will become clear that its large magnitude is a consequence of complex small-scale dynamics of intermittent turbulence.*]{}
Thus, the ratio of the mixing (inviscid) and diffusion times can be estimated as:
$$\frac{\tau_{eddy}}{\tau_{d}}\approx \frac{\sqrt{Re}}{\alpha_{\nu}^{2}}$$
and $\tau_{eddy}/\tau_{d}\approx 1$ for $Re\approx \alpha_{\nu}^{2}\approx 10^{4}$.
In the flows with $Re<10^{4}$, molecular diffusion across the dissipation sheets is the longest, rate-determining, process. In the wall flows, this Reynolds number corresponds to $Re_{U}\leq 2-3 \times 10^{5}$. It is only when $Re_{U}\gg 10^{5}-10^{6}$, the rapid diffusion through extremely thin interfaces, is dynamically irrelevant for the mixing process and inviscid mixing time $\tau_{eddy}$ is the rate-determining step. Based on the above considerations, one can expect a transition from the diffusion- to advection - dominated mixing at
$$Re_{tr}\approx \alpha_{\nu}^{4}$$
If mixing is the reaction rate ( ${\cal R}$ ) determining process, we expect
$$\begin{aligned}
{\cal R}\propto \frac{\nu}{\eta_{K}^{2}}\approx \frac{\nu}{L^{2} }Re^{\frac{3}{2}} = \tau_{eddy}\sqrt{Re}
~~~~~Re< Re_{tr}\nonumber \\
{\cal R} \propto \frac{u_{rms}}{L}\approx \frac{\nu}{L^{2}}Re^{1}= \tau_{eddy}Re^{0}~~~~~Re>Re_{tr}\end{aligned}$$
In accord with (1.4), the transitional Reynolds number depends only upon coefficient $\alpha_{\nu}$, characterizing small-scale properties of turbulence. Thus, we can conclude that, [*since the small-scale property of turbulence $\alpha_{\nu}\approx 10-20$ is a more or less universal number, independent upon type of the flow, the derived $Re_{tr}\approx 10^{4}$ must be approximately universal.* ]{} A mixing transition leading to the Reynolds - number - independent reaction rate at approximately universal Reynolds number $Re_{tr}\approx 10^{4}$, has been observed in experiments by Dimotakis (2005), (1993).
Statistical Description of Disipation Structures. Evaluation of $\alpha_{\nu}$.
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The Kolmogorov theory (K41), treating the u.v. cut-off $\eta_{K}\approx L Re^{-\frac{3}{4}}=const$, completely disregarded the non-trivial dynamics of the dissipation range fluctuations. It became clear recently that the dissipation scale is not a constant number but a random field defined as :
$$\eta\approx \frac{\nu}{\delta_{\eta}u}\equiv \frac{\nu}{|(u(x+\eta)-u(x))|}$$
where $\delta_{y} u_{i}=u_{i}(x+y)-u_{i}(x)$. In this form the relation for $\eta$ was derived (Yakhot (2003, (2006)), Yakhot and Sreenivasan (2004), (2005)) from the dissipation anomaly, first introduced by Polyakov (1995) for the case of Burgers turbulence and later generalized to the Navier-Stokes turbulence by Duchon and Robert (2000), Eyink (2003). Even earlier, the fluctuating dissipation scale was used by Paladin and Vulpiani (1987) in the context of multifractal theory The local value of the Reynolds number $Re\approx u_{\eta}\eta/\nu\approx 1$, with $\delta_{\eta}u\equiv u_{\eta}$ as a typical speed of an eddy of linear dimension $\eta$, was mentioned in Landau and Lifshitz as a criterion for the onset of viscous dissipation, as early as 1959.
The physical meaning of the dissipation scale $\eta$ is understood as follows. As $r\rightarrow 0$, the velocity field is analytic, so that $\delta_{r}u\approx \partial_{x}u(x)r$. Thus, in this limit $S_{n}=\overline{(\delta_{r}u)^{n}}\propto r^{n}$. On the other hand, in the inertial range $S_{3}\propto r$ which, contradicting small-scale analyticity of velocity field, cannot be valid in the limit $r\rightarrow 0$. Moreover, in isotropic and homogeneous turbulence, the dissipation rate ${\cal E}\propto \nu\overline{(\partial_{x}u_{x})^{2}}=-\nu\lim_{r\rightarrow 0}\frac{\partial^{2}}{\partial r^{2}}S_{2}(r)\propto \nu \lim_{r\rightarrow 0} r^{-\frac{4}{3}}=O(1)$ is regularized by viscosity $\nu\rightarrow 0$. This leads to the definition of the dissipation scale $\eta_{K}$ : ${\cal E}^{\frac{1}{3}}\propto \nu \eta_{K}^{-\frac{4}{3}}$ and, since ${\cal E}\approx u_{rms}^{3}/L$, to Kolmogorov’s estimate for $\eta_{K}\propto Re^{-\frac{3}{4}}$.
Based on the above considerations, defined by (2.1), $\eta$ is a scale separating analytic ($r\ll \eta$) and singular ($r\gg \eta$) contributions to turbulent velocity field. The relation (2.1) is an order of magnitude estimate of the “dissipation scale” and, in general, $\eta\approx \frac{a \nu}{\delta_{\eta} u}$ where $a$ is a velocity- field -independent factor which was investigated in numerical simulations by J. Schumacher et. al. (2007) . Interested in qualitative aspects of the mixing process, we, for now, neglect this coefficient and use expression (2.1).
According to the theory (Yakhot (2003)), there exist an infinite number of “dissipation scales ” $\eta_{n}$ separating smooth $S_{n}\propto r^{n}$ ($r\ll \eta_{n}$) and singular $S_{n}\propto r^{\xi_{n}}$ ($r\gg \eta_{n}$) intervals of the moments $S_{n}=\overline{(\delta_{r} u)^{n}}$. This fact has been decisively demonstrated in numerical experiments of Schumacher et al (2007). According to the analytic theory (Yakhot (2003)), in general
$$L Re^{-\beta}\leq \eta_{n}\leq L Re^{-\frac{1}{2}}$$
where $\beta\approx 1$ and the Kolmogorov scale $\eta_{K}\approx \eta_{2}$ is only one of the possible dissipation scales. This property of turbulence will be important in what follows.
Let us consider an equation for concentration $c$ of a passive scalar advected by velocity ${\bf u}$:
$$\frac{\partial c}{\partial t}+{\bf u \cdot \nabla}c=D\nabla^{2}c; \hspace{0.5in} {\bf \nabla\cdot u}=0$$
We assume that the velocity field ${\bf u}$ is governed by the Navier-Stokes equations and ${\bf u=U+u'}$ with ${\bf U}$ and ${\bf u'}$ corresponding to quasy-regular (sometimes time-dependent) and chaotic (turbulent) contributions, respectively. The fluctuations of the scalar dissipation field $N=D(\nabla c)^{2}$, governed by (2.2), have been investigated in great detail for the case of the $\delta$-correlated in time, large-scale, velocity field (Kraichnan-Batchelor problem), where the stretched exponential tail of the distribution function $P(N)$ was derived in the range $N>>\overline{N}$ ( Chertkov et. al (1998), Gamba et al (1999)). In this paper we are interested in statistical properties of “molecular diffusion sheets”, which are not directly related to the tails of the scalar dissipation rate distribution.
The scalar field is analytic, so that for $r\rightarrow 0$, $c(x+r)-c(x)\approx \frac{\partial c(x)}{\partial x}r$. In addition, in the scalar “inertial range” $r\gg \eta_{K}$ (Monin and Yaglom (1975):
$$S_{3}^{u,c}=\overline{(u(x+r)-u(x))(c(x+r)-c(x))^{2}} =-\frac{4}{3}\overline{N}r$$
where $N=D\overline{(\frac{\partial c}{\partial x_{i}})^{2}}=O(1)$. In the case $Sc>>1$, there exist an additional scalar “rough” range $\eta_{B} \leq r\leq \eta_{K}$, where $S_{3}(r)\propto \ln r$. It is only at the scales $r\ll \eta_{B}$, the scalar field is smooth. By definition, the length scale $r\approx \eta_{c}$ is the scale separating analytic and singular contributions to the scalar field $c({\bf r},t)$. As follows from (2.3), in the inertial range, the scalar field $c({\bf x})$ is not differentiable and in the limit $D\rightarrow 0$, one has to be careful with evaluation of spatial derivatives of the scalar field.
From the equaion (2.2) we have
$$\frac{\partial c^{2}}{\partial t}+{\bf u \cdot \nabla}c^{2}=2Dc\nabla^{2}c$$
and introducing the “point - splitting” $c(\pm)=c(x\pm y)$ and ${\bf u}(\pm)={\bf u}(x\pm y)$, derive: $$\frac{\partial (c(+)c(-)}{\partial t}+(\nabla_{+}{\bf u}(+)+\nabla_{-}{\bf u}(-))c(+)c(-)=D(\nabla_{+}^{2}+\nabla_{-}^{2})c(+)c(-)$$
[*Equation (2.5) involves derivatives of singular ( in the inertial range) functions. Thus, in the limit $y\rightarrow 0$, the exact equation (2.4) for the scalar variance can appear from (2.5) only if singular and regular contributions balance separately.*]{} Indeed, in the limit $y\rightarrow \eta_{c}\rightarrow 0$, taking into account that $\frac{\partial}{\partial y}=\frac{\partial}{\partial x_{+}}=-\frac{\partial}{\partial x_{-}}$ and repeating all steps presented in detail in (Yakhot (2006)), we derive:
$$\frac{\partial}{\partial y_{i}}(\delta_{y}u_{i}(\delta_{y}c)^{2})+2\nabla_{+}{\bf u}(+)c^{2}(-)+2\nabla_{-}{\bf u}(-)c^{2}(+)=-4D\delta_{y}c\frac{\partial^{2}}{\partial y^{2}}\delta_{y}c$$
The relation (2.6) is exact locally in space and time. Since in incompressible isotropic turbulence the velocity- scalar correlation function $\overline{u_{i}(x)c^{2}(x')}=0$ (Monin and Yaglom (1975)), it is clear that averaging over a “ball” of radius $\eta_{c}$ (Duchon (2000), Eyink (2003), the second and third terms in the left side of (2.6) disappear giving the locally valid estimate for the scalar dissipation scale, independent upon specific model of turbulence:
$$\eta_{c}\approx \frac{D}{\delta_{\eta_{c}}u}$$
As follows from this relation, [*the random variable*]{} $\eta_{c}$ depends upon local values of velocity fluctuations. In the most interesting and important case $Sc=\nu/D\gg1$, on the scale $\eta_{c}\ll \eta$ the velocity field is analytic, giving:
$$\eta_{c}^{2}\approx \frac{D}{\frac{\partial u(x)}{\partial x}}\approx \frac{D \eta}{\delta_{\eta} u}\approx \frac{D\eta^{2}}{\nu}=\eta^{2}/Sc$$
Thus, the probability of the scalar dissipation scale $Q(\eta_{c})$ is evaluated readily from the PDF $Q(\eta)$ calculated in Yakhot (2006). This result leads to some important consequences: If
$$\tau_{d}\approx \eta^{2}_{\nu}/\nu=\alpha_{\nu}^{2}\eta_{K}^{2}/\nu\approx \alpha_{\nu}^{2}\tau_{eddy}/\sqrt{Re}$$
to evaluate the reaction rate and the proportionality coefficient $\alpha_{\nu}$, we need the probability density function $Q(\eta_{c})=\sqrt{Sc}Q(\eta \sqrt{Sc})$.
Probability densities.
======================
In what follows we set $L=1$, so that $\frac{r}{L}\equiv r<1$ and if the moments of velocity increments $S_{n,0}=\overline{(\delta_{r}u)^{n}}=A(n)r^{\xi_{n}}$, then the probability density function can be found from the Mellin transform:
$$P(\delta_{r}u,r)=\frac{1}{\delta_{r} u}\int_{-i\infty}^{i\infty} A(n)r^{\xi(n)}(\delta_{r}u)^{-n}dn$$
where we set the integral scale $L$ and the dissipation rate ${\cal E}$ equal to unity. Indeed, multiplying (3.1) by $(\delta_{r}u)^{k}$ and evaluating a simple integral, gives $S_{k,0}=A(k)r^{\xi_{k}}$. Under different name this transformation has been used in Tcheou et.al (1998) in the context of multifractal theory of turbulence . With the Gaussian large-scale boundary condition for the probability density at $r=1$, the amplitudes $A(n)=(2n-1)!!$ and, for the values of $n<1/b$, we can use the Taylor expansion of the function $\xi_{n}$ giving $\xi_{n}\approx( an-bn^{2})$. The detailed theory for this case has been recently developed in Yakhot (2006) with the result ($\delta_{\eta}u\equiv u$):
$$P(u,r)=\frac{2}{\pi u\sqrt{4b| \ln r|}}\int_{-\infty}^{\infty}e^{-x^{2}}
exp[-\frac{ (\ln \frac{u}{r^{a}\sqrt{2}x})^{2}}{4b|\ln r|}]dx$$
The PDF $P(u/r^{a},r)$ is plotted on Fig. 1 for a few values of the displacement $r$. Here $a\approx 0.38$ and $b=0.017$. On Fig. 1b we see the broad tails which, as a sign of strong intermittency, cannot be collapsed on a single curve.
[{height="4cm"}]{} [{height="4cm"}]{}
According to its definition, the dissipation scale is a linear dimension of a structure defined by the local value of the Reynolds number $Re_{\eta}=\eta \delta_{\eta}u/\nu=O(1)$. Experimentally, probability density $Q(\eta,Re,Re_{\eta})$ is found by fixing the displacement $r=\eta$ and counting the events with $\eta\delta_{\eta}u/\nu=Re_{\eta}=a$. This algorithm has been used by Schumacher (2007) in his numerical investigations (see Fig. 3). From the formula (3.1) we have:
$$P(\delta_{\eta} u)\equiv P(u_{\eta})=\frac{1}{u_{\eta}}\int_{-i\infty}^{i\infty} A(n)\nu^{\xi(n)}u_{\eta}^{-\xi(n)-n}dn$$
Fixing $L={\cal E}=u_{rms}=1$ gives the large-scale Reynolds number $\nu=1/Re$ and taking into account that by virtue of (2.1) $ u_{\eta}Re/u_{rms}=\frac{L\delta_{\eta}u}{\nu}\approx L/\eta$ gives for the probability density $Q(\eta)$ (in what follows we denote $\frac{\eta}{L}\equiv \eta$):
$$Q(\eta,Re)=\frac{1}{\eta}\int_{-\infty}^{\infty}e^{-x^{2}}dx\int_{-\infty}^{\infty}dn
e^{in \ln(\eta^{a+1}\sqrt{2}xRe)-bn^{2}\ln\eta}$$
and:
$$Q(\eta,Re)=\frac{1}{\eta\sqrt{4b\ln\eta}}\int_{-\infty}^{\infty}
e^{-x^{2}}dx e^{-\frac{\ln^{2}(\eta^{a+1}\sqrt{2}xRe)}{4b\ln \eta}}=
\frac{1}{\eta \sqrt{4b|\ln\eta|}}\int_{-\infty}^{\infty}
e^{-x^{2}}dx e^{-\frac{\ln^{2}((\frac{\eta}{\eta_{0}})^{a+1}\sqrt{2}x)}{4b|\ln \eta|}}$$
[{height="6cm"}]{}
where $\eta_{0}=LRe^{-\frac{1}{1+a}}$. As expected, the probability density of dissipation scales is expressed in terms of the ratio $\eta'=\eta/\eta_{0}$ and the width of the distribution is the weak function of the Reynolds number. The PDFs $Q(\eta/\eta_{0},Re)$ for $Re=10^{4}$ is shown on Fig. 2 for two values of the Scmidt number $Sc=1$ and $c=25$.
Mean dissipation scale and diffusion time.
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Now we can evaluate the moments of the dissipation scale: $$\overline{e^{n}}\equiv \overline{(\frac{\eta}{\eta_{0}})^{n}}=\int_{0}^{\infty}( \frac{\eta}{\eta_{0}})^{n}Q(\eta, \eta_{0})d\eta$$ and mean diffusion time: $$\overline{\tau_{d}}=\overline{\eta^{2}}/\nu=\int_{0}^{\infty}\eta^{2}Q(\eta,\eta_{0},Re)\ d \eta /\nu$$
The numerical results slightly vary with position of the maximum of PDF $Q(\eta/\eta_{0})$, which depends upon the magnitude of parameter $a$ in the expression $a\eta u_{\eta}/\nu=1$. If we choose $a$ so that, in accord with numerical simulation of Schumacher (2007), the maximum is set at $\eta_{max}/\eta_{0}\approx 2$, then numerical integration (4.1) yields $\overline{\eta}=\alpha_{\nu}\eta_{0}\approx 7\eta_{0}$ and $\overline{\eta^{2}}\approx 120\eta_{0}$. If however, $\eta_{max}\approx 5\eta_{0}$, we derive $\overline{\eta }=\alpha_{\nu}\eta_{0}\approx 13.5\eta_{0}$, which is close to the outcome of Dimotakis’s (2005) physical and Gotoh-Nakano’s (2003) numerical experiments. In this case, $\overline{\eta^{2}}\approx 500\eta_{0}$. In general, based on (4.1), (3.5), $\overline{\eta^{2}}\gg \alpha_{\nu}^{2}\overline{\eta}^{2}$. This result leads to important conclusion: [*Due to strong intermittency of the dissipation scales, the $O(\overline{\eta^{2}}/\nu)$ scalar diffusion time is much longer than $\tau_{d}\approx \alpha_{\nu}^{2}\eta_{K}^{2}$, calculated on the basis of Kolmogorov’s phenomenology.*]{} Therefore, molecular diffusion, as a reaction rate determining process, is even more restrictive than previously thought. We can also conclude that these fluctuations are responsible for the “large” magnitude of a constant $\alpha_{\nu}\approx 10-20$.
Mixing reactants having finite life-time .
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One can define dimensionless Damköller number $Da=(\tau_{c}+\tau_{d})/\tau_{reaction}$, which is the ratio of hydrodynamic mixing time scale to $\tau_{reaction}$, characterizing the reaction rate between perfectly mixed reactants $A$ and $B$. In case of fast reactions, we are interested in here, $Da>>1$. Below, we will show that in some cases, to describe chemical reactions, Kolmogorov’s cascade picture, though elegant and illuminating, is not sufficient.
In what follows we consider a simple example of a model photo-chemical reaction $A^{*}+B=AB+h\nu$ where $A^{*}$ is a component $A$ in an initially prepared electronically excited state characterized by a life-time $\tau_{e}$. By definition, the finite life-time $\tau_{e}$ implies time-dependence of concentration of excited states $c_{A^{*}}=c(t=0)\exp(-\frac{t}{\tau_{e}} )$. As above, we are interested in a diffusion-dominated limit $Re \leq 10^{4}$. A chemical reaction is possible only if diffusion time $\tau_{d} \approx \alpha_{\nu}^{2}\eta_{B}^{2}/D \leq \tau_{e}$, $\alpha_{\nu}\eta_{B}\leq \sqrt{\tau_{e}D}$ and $\eta_{\nu}\leq \sqrt{\tau_{e}\nu}$. This gives, in addition to the Damköhller mumber, a dimensionless reaction criterion:
$$Y=\alpha_{\nu}\sqrt{\frac{\tau_{eddy}}{\tau_{e}}}Re^{-\frac{1}{4}}\leq 1$$
When the life-time $\tau_{e}$ is very small, according to the relation (4.3), which is the outcome of Kolmogorov’s phenomenology, the photo-chemical reaction is impossible. Now we will show that due to strong fluctuations of the scalar dissipation scale $\eta_{c}$, this conclusion must be modified.
Let us consider the advection-diffusion equation for concentration of a passive scalar $c\equiv c_{A^{*}}$ undergoing a chemical reaction with another one of concentration $c_{B}$. If the maximum separation, for which reaction is still possible (reaction radius) is $\Delta$, we can define the probability to find a molecule $B$ within a sphere of radius $\Delta$ surrounding the reactant $A^{*}$ as ${\cal P}(r<\Delta)$ and write the balance equation for the reactant $A$ as: $$\frac{\partial c}{\partial t}+{\bf u \cdot \nabla}c=D\nabla^{2}c-c {\cal P}(r\leq \Delta)w_{AB}-\frac{c}{\tau_{e}}; \hspace{0.5in} {\bf \nabla\cdot u}=0$$
where $w_{AB}$ is the reaction rate of perfectly mixed reactants separated by the distance $r\leq \Delta$. The $O(1/\tau_{e})$ term in (4.4) accounts for the finite life -time of one of one the reactants $A^{*}$. [*It is clear that ${\cal P}(r\leq \Delta)$ depends upon concentration $c_{B}$.* ]{} As was mentioned above, the reaction radius $\Delta$, depending on the overlap of molecular orbitals, is very small and, during the mixing stage, when $r(t)>>\Delta$, the probability ${\cal P}=0$ and the $O(w_{AB})$ chemical contribution to the balance equation (4.4) can be neglected. The relation (4.4) illustrates importance of the molecular-level mixing process in chemical kinetics.
By substitution $c_{A}\equiv c\rightarrow ce^{-\frac{t}{\tau_{e}}}$, the remaining equation is transformed into (2.2). It is clear that the reaction rate is not negligibly small only if the mixing time $\tau_{d}<\tau_{e}$.
We illustrate the qualitative features of the process on a numerical example. If $\overline{\eta}/\eta_{0}\approx 13$, (Gotoh/Nakano (2003)), then, based on the PDFs computed above, $\overline{\tau_{d}}/\tau_{0}\approx 500$. For $\tau_{e}/\overline{\tau_{d}}\approx g\ll 1$, the naive (mean-field ) reaction yield, proportional to the concentration $c\propto \exp(-\overline{\tau_{d}}/\tau_{e})$ , is negligibly small. However, defining $\tau_{e}\approx \eta_{e}^{2}/\nu$, gives $\frac{\eta_{e}}{\eta_{0}}\approx \sqrt{500 g}$. This result means that a finite a fraction
$$F=\int_{0}^{\eta_{e}}Q(\eta,\eta_{0},Re)d\eta$$
of dissipation sheets with $\eta\leq \eta_{e}\approx \sqrt{500 g}\eta_{0}$ does contribute to the nonzero reaction rate. Since $\tau_{d}\approx \tau_{v}$, the integral is evaluated using the probability density of velocity dissipation scales.
Taking, for example, $g\approx 0.02$, we see that only the sheets of the thickness $\eta<\sqrt{10}\eta_{0}\approx 3\eta_{0}$ contribute to this reaction. The fraction of the dissipation structures satisfying this condition is $F\approx 0.3$. If $g\approx 0.01$, we find $F\approx 5\times 10^{-4}$. We can see that, due to strong fluctuations of the dissipation scale, the reaction is not negligible even when $\tau_{e}\ll \overline{\tau_{d}}$.
Conclusions.
============
To conclude: the scalar and velocity dissipation scales $\eta$ and $\eta_{c}$ in turbulent flows are not constant numbers but describe random fields with $\eta_{c}\approx D/\frac{\partial u}{\partial x}$ and $\eta\approx \nu/\delta_{\eta}u$, respectively. In an important case $Sc\gg1$, these scales are related as $\eta_{c}\approx \eta/\sqrt{Sc}$. Based on the Mellin transform and Taylor expansion of the scaling exponents of velocity structure functions, the probability density of the scalar dissipation scale has been derived. Two main results of this paper are: 1. Due to strong small-scale intermittency, the calculated mean thickness of a dissipation sheet is $\alpha_{\nu}\eta_{K}$ where $\alpha_{\nu}\approx 10-13$. Extremely strong intermittency leads to a long scalar- transport time $\overline{\tau_{d}}$ across the sheets and, in the flows with $Re\leq \alpha_{\nu}^{4}\approx 10^{4}$, to diffusion as a reaction rate- determining step . Therefore, the reaction rate is: ${\cal R}\propto \sqrt{Re}$ for $Re<10^{4}$, and ${\cal R}\propto Re^{0}$ in the interval $Re>10^{4}$. 2. Even when the life - time of the reactants is very short, due to the dissipation scale fluctuations, the reaction can proceed via diffusion across thinnest dissipation sheets $\eta_{c}<<\overline{\eta_{c}}$. In this case, the fluctuations lead to the non-negligibly small reaction rates. This result may be of importance for reactions involving short-lived radicals, excimers and other cases. We believe that experimental investigation of the sub-Batchelor scale dynamics of the mixing process is an extremely interesting and urgent task.
[{height="14cm"}]{}
The theory presented in this paper is based on the dissipation scale definitions (2.1), (2.7),(2.8), derived from the dissipation anomaly. This algorithm was numerically compared by Schumacher (2007) with the one based on the isosurfaces of the scalar dissipation rate. The obtained PDFs $Q(\eta)$, though qualitatively similar, had quite substantial quantitative differences (see Fig. 3). Since the relations based on dissipation anomaly (2.1) have been derived directly from equations of motion, we believe they are much better justified.
Interesting and stimulating discussions with N. Peters, U. Frisch, A. Kerstein, J. Schumacher, E. Villermaux and K.R. Sreenivasan are gratefully aknowledged.
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---
abstract: 'In 1992, following earlier conjectures of Lichtman and Makar-Limanov, Klein conjectured that a noncommutative domain must contain a free, multiplicative, noncyclic subsemigroup. He verified the conjecture when the center is uncountable. In this note we consider the existence (or not) of free subsemigroups in associative $k$-algebras $R$, where $k$ is a field not algebraic over a finite subfield. We show that $R$ contains a free noncyclic subsemigroup in the following cases: (1) $R$ satisfies a polynomial identity and is noncommutative modulo its prime radical. (2) $R$ has at least one nonartinian primitive subquotient. (3) $k$ is uncountable and $R$ is noncommutative modulo its Jacobson radical. In particular, (1) and (2) verify Klein’s conjecture for numerous well known classes of domains, over countable fields, not covered in the prior literature.'
address: |
Department of Mathematics\
Temple University\
Philadelphia, PA 19122
author:
- 'Edward S. Letzter'
title: |
On free subsemigroups\
of associative algebras
---
In 1977, Lichtman conjectured that the group of units of a noncommutative division algebra always contains a noncyclic free subgroup [@Lic-one]. Since then, an extensive, broad, and ongoing literature has developed on the existence of free subobjects of associative algebras. The reader is referred, e.g., to [@Gon-Shi] for an overview of this literature and, e.g., to [@Fer-For-Gon] and references therein for more recent results.
Our focus in this note is on free subsemigroups of associative algebras. (Henceforth, references to free subsemigroups and free subgroups will assume them to be multiplicative and noncyclic.) In 1984, Makar-Limanov conjectured that a noncommutative division algebra must contain a free subsemigroup, and he proved the same for division algebras over uncountable fields [@Mak]. In 1992, Klein proved that a noncommutative domain with an uncountable center must contain a free subsemigroup, and he conjectured the same for all noncommutative domains [@Kle]. Chiba proved in 1995 that a polynomial extension of a division algebra must contain a free subsemigroup (and also that a division algebra over an uncountable field must contain a free subgroup) [@Chi]. In 1996, Reichstein showed that an algebra over an uncountable field contains a free subsemigroup (or subalgebra) if the same holds true after an extension of the scalar field [@Rei] (cf. Smoktunowicz’s constructions [@Smo]).
Our primary aim, then, is to establish the existence of free subsemigroups for some well-known classes of algebras not covered in the prior literature.
*Setup.* Throughout, $k$ will denote a field not algebraic over a finite subfield. All mention of algebras, rings, subrings, and subalgebras will assume them to be associative and unital. Finitely generated $k$-algebras will be referred to as *$k$-affine*.
We begin with an elementary but useful “specialization” lemma, adapted from Passman [@Pas-one §2].
\[subfactor\] Let $A$ be a subring of a ring $R$, and suppose there exists a surjective ring homomorphism $\varphi\colon A \rightarrow \A$. If $\A$ contains a free subsemigroup then so does $R$.
Choose $\x, \y \in \A$ that generate a free subsemigroup $\Sem$ of $\A$. Choose $x \in \varphi^{-1}(\x)$ and $y \in \varphi^{-1}(\y)$. Letting $S$ denote the subsemigroup of $A$ generated by $x$ and $y$, we see that $\varphi$ restricts to a surjective semigroup homomorphism, from $S$ onto $\Sem$, mapping $x$ to $\x$ and $y$ to $\y$. By universality, $S$ must be isomorphic to $\Sem$, and so $S$ is a free subsemigroup of $R$.
\[alternative\] Let $n$ be an integer $> 1$. In view of the Tits Alternative [@Tit], and recalling that $k$ is not algebraic over a finite subfield, we can conclude that $GL_n(k)$ contains a free subgroup. Therefore, the (full) matrix algebra $M_n(k)$ contains a free subgroup, as does the algebra of $\nxn$ matrices over any $k$-algebra. Lichtman [@Lic-two] and Gonçalves [@Gon] used the Tits Alternative to verify that a noncommutative division algebra finite dimensional over its center must contain a free subgroup. We can now conclude that a noncommutative central simple algebra, finite dimensional over a field extension of $k$, must contain a free subgroup.
The proof of the following employs a reduction to the $k$-affine case, and I am grateful to Ken Brown for this approach.
\[PI\] Let $R$ be a $k$-algebra satisfying a polynomial identity, and suppose that $R$ is noncommutative modulo its prime radical. Then $R$ contains a free subsemigroup.
By (\[subfactor\]), we may assume without loss of generality that $R$ is a noncommutative semiprime PI algebra over $k$. Choose $x, y \in R$ such that $b := xy-yx \ne 0$.
Next, since $R$ is semiprime and PI, it follows that $RbR$ cannot be a nil ideal (see, eg., [@McC-Rob 13.2.6i]). Consequently, there exist $a_1, c_1, a_2, c_2, \ldots, a_t, c_t \in R$ such that $$d := a_1 b c_1 + a_2 b c_2 + \cdots + a_t b c_t$$ is not nilpotent.
Set $R' = k\langle x, y, b, a_1, c_1, \ldots, a_t, c_t \rangle$, a $k$-affine (not necessarily semiprime) PI algebra. Since $d$ is not nilpotent, the ideal $R'bR'$ of $R'$ is not nil. But it was proved by Amitsur that the Jacobson radical of a $k$-affine PI algebra must be nil [@Ami]. (Nilpotency was later established by Braun [@Bra].) Therefore, $R'bR'$ cannot be contained in the Jacobson radical of $R'$. Hence $xy-yx$ is not contained in the Jacobson radical of $R'$, and we can conclude that $R'$ is noncommutative modulo its Jacobson radical.
We now know that there exists at least one primitive ideal $P'$ of $R'$ such that $R'/P'$ is noncommutative, and so by Kaplansky’s Theorem (see, e.g., [@McC-Rob 13.3.8]), $R'/P'$ must be a noncommutative central simple algebra, finite dimensional over a field extension of $k$. Therefore, by (\[alternative\]), $R'/P'$ contains a free subsemisubgroup and, by (\[subfactor\]), $R$ contains a free subsemigroup.
\[nonart\] Let $R$ be a $k$-algebra with at least one nonartinian primitive subquotient. Then $R$ contains a free subsemigroup.
To start, assume that $R$ is both left primitive and not artinian. Let $M$ be a simple faithful left $R$-module, and let $E = \text{End}_RM$. Since $R$ is not artinian, it follows that $M$ has infinite length as a right $E$-module. Also, for any given integer $n > 1$, it follows from the Jacobson Density Theorem that there exists a $k$-subalgebra $A$ of $R$ equipped with a surjective homomorphism $\varphi\colon A \rightarrow M_n(E)$; see, e.g., [@Lam 11.19]. Again by (\[alternative\]), the subalgebra $M_n(k)$ of $M_n(E) = \varphi(A)$ must contain a free subsemigroup. The theorem now follows from (\[subfactor\]).
Observe that the preceding two results (\[PI\] and \[nonart\]) verify Klein’s conjecture when $R$ is a nonartinian primitive domain or when $R$ is a noncommutative PI domain. Moreover, those two results (especially when $k$ is countable) establish the existence of free subsemigroups in numerous well known classes of algebras not covered in the prior literature. Indeed, algebras with a nonartinian primitive subquotient or that are semiprime and PI are commonplace among iterated Ore extensions (cf. [@McC-Rob]), enveloping algebras of Lie algebras (cf. [@Dix]), group algebras (cf. [@Pas-two]), quantum groups (cf. [@Bro-Goo]), and algebras arising in nocommutative algebraic geometry (cf. [@BRSS]).
We can also use (\[nonart\]) to show that Makar-Limanov’s conjecture, over $k$, is equivalent to an *a priori* more general statement.
\[equiv\] The following two statements are equivalent: [(i)]{} Every noncommutative division algebra over $k$ contains a free subsemigroup. [(ii)]{} Every $k$-algebra noncommutative modulo its Jacobson radical contains a free subsemigroup.
It suffices to prove that (i) implies (ii). So assume (i), and let $R$ be a $k$-algebra noncommutiative modulo its Jacobson radical. To prove the corollary it will suffice to show, as follows, that $R$ contains a free subsemigroup: First, by (\[subfactor\]) we can assume without loss of generality that $R$ is primitive and noncommutative. Next, by (\[nonart\]), we can further reduce to the case when $R$ is simple artinian. Now, if $R$ has rank $n > 1$, then $R$ contains a copy of $M_n(k)$ and so contains a free subsemigroup by (\[alternative\]). It remains only to consider the case when $R$ is a division ring, which is exactly (i).
We conclude with a modest application to algebras over uncountable fields.
\[uncount\] Assume that $k$ is uncountable and that $R$ is noncommutative modulo its Jacobson radical. Then $R$ contains a free subsemigroup.
As noted above, it was proved in [@Mak] that a division algebra over an uncountable field contains a free subsemigroup. The corollary now follows from (\[equiv\]).
I am grateful to Ken Brown for his very useful comments on an earlier draft of this note, and in particular for his suggestion to (and how to) reduce to the affine case in (\[PI\]).
[99]{}
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|
---
abstract: 'We report on the synthesis and structural characterization of the magnetocaloric FeMnP$_{0.75}$Si$_{0.25}$ compound. Two types of samples (as quenched and annealed) were synthesized and characterized structurally and magnetically. We have found that minute changes in the degree of crystallographic order causes a large change in the magnetic properties. The annealed sample, with higher degree of order is antiferromagnetic with a zero net moment. The as-quenched sample has a net moment of 1.26 $\mu_B$/f.u. and ferrimagnetic-like behavior. Theoretical calculations give rather large values for the Fe and Mn magnetic moments, both when occupied on the tetrahedral and pyramidal lattice site. The largest being the Mn moment for the pyramidal site reaches values as high as 2.8 $\mu_B$/atom.'
author:
- Matthias Hudl
- Per Nordblad
- Torbjörn Björkman
- Olle Eriksson
- Lennart Häggström
- Martin Sahlberg
- Levente Vitos
- Yvonne Andersson
title: 'Order–disorder induced magnetic structures of FeMnP$_{0.75}$Si$_{0.25}$'
---
Introduction
============
Compounds based on Fe$_2$P gain increased interest due to a possible application in magnetocaloric refrigeration. Recent publications by Brück et al.[@Bruck:1; @Bruck:2], Dagula et al. [@Dagula:1] and Cam Thanh et al. [@Thanh:1] showed a huge magnetocaloric effect in FeMnP$_{1-x}$As$_{x}$, FeMnP$_{0.5}$As$_{0.5-x}$Si$_{x}$, and FeMnP$_{1-x}$Si$_{x}$ respectively close to room temperature. The compound FeMnP$_{1-x}$Si$_{x}$ is of particular interest since it consists of non-toxic elements. One drawback of FeMnP$_{1-x}$Si$_{x}$ in regard to applications is a strong thermal hysteresis when undergoing a first order para- to ferromagnetic phase transition. An explanation for the occurrence of a first order transition in Fe$_2$P is given by Yamada and Terao [@Yamada:1].
![Fe$_{2}$P structure with iron atom positions 3f (dark grey) and 3g (black), and phosphorus atom positions 1b (light grey) and 2c (white).[]{data-label="Fe2P:lattice"}](Fe2P-lattice.eps){width="40.00000%"}
The Fe$_2$P compound has been intensely studied during the last 5 decades. Fe$_2$P crystallizes in a hexagonal structure with space group D$^{3}_{3h}$ (P$\bar{6}$2m) [@Rundqvist:1]. The iron atoms occupy two different crystal sites, the 3f-site with four phosphorus atoms surrounding one iron atom (referred to as type-I or tetrahedral site) and 3g-site with 5 phosphorus atoms surrounding one iron atom (referred to as type-II or pyramidal site). The phosphorus atoms occupy the two dissimilar sites 2c (type-I) and 1b (type-II). Each Fe(I) site is surrounded by two P(I) and two P(II) atoms whereas Fe(II) is surrounded by four P(I) and one P(II) atoms (Fig. \[Fe2P:lattice\]). As regards the magnetic properties Fe$_2$P undergoes a first order para- to ferromagnetic phase transition with a Curie temperature of T$_{C}$ $\approx$ 216 K (see e.g. Wäppling et al. [@Wappling:1], Fujii et al. [@Fujii:1], and Lundgren et al.[@Lundgren:1]). It is worth to notice that prior to this investigation structural and magnetic studies on (Fe$_{1-y}$Mn$_{y}$)$_{2}$P and Fe$_{2}$P$_{1-x}$Si$_{x}$ were published by Srivastava et al. [@Srivastava:1] and Jernberg et al. [@Jernberg:1]. Due to its interesting magnetic properties, Fe$_2$P has also attracted theoretical interest, e.g. as revealed in Refs. [@Eriksson:1; @Unknown:1].
The preliminary phase diagram for FeMnP$_{1-x}$Si$_{x}$ is given by Cam Thanh et al. [@Thanh:1] and indicates a structural phase transition from orthorhombic to hexagonal for a silicon content of approximately x=0.25. In spite of prior studies, the importance of the iron to manganese ratio as well as the distribution of those atoms within the Fe$_2$P structure seem rather undefined. The present manuscript is a first attempt towards answering such questions. In our study the magnetic, structural and electronic properties of the FeMnP$_{0.75}$Si$_{0.25}$ alloy have been investigated, using XRD, Mössbauer spectroscopy, and magnetic measurements combined with theoretical calculations. We observe a significant difference in the magnetic order depending on the applied sample treatment which is shown to be reversible. The remarkable change in the magnetic order is supposedly caused by the degree of crystallographic order of iron and manganese atoms.
Experimental Details and Methods
================================
The FeMnP$_{0.75}$Si$_{0.25}$ sample was prepared by a drop synthesis method using a high frequency induction furnace [@Rundqvist:2]. The synthesis was done under argon atmosphere and temperatures of approx. 1350 $^{\circ}$C. The first sample was directly taken from the cooled melt and investigated by X-Ray diffraction (XRD). Thereafter some phosphorus was added and the fabricated material was annealed for 10 days at 1000 $^{\circ}$C. All subsequent heat treatments did not involve changes in the element composition.
The XRD measurements were performed using a focusing Bragg-Brentano type powder diffractometer with CuK$_{\alpha1}$ radiation. The magnetic properties of all samples where investigated by means of DC magnetization measurements mainly using a commercial vibrating sample magnetometer (Quantum Design PPMS). Zero field cooled (ZFC) and field cooled (FC) measurement protocols for different fields were applied. The annealed sample was remelted using an arc melting furnace and quenched. This remelt sample was characterized by magnetic measurements and room temperature Mössbauer spectroscopy. The Mössbauer spectra were recorded in the absorption mode with constant-acceleration drive and a $^{57}$CoRh source.
Finally the remelt sample was annealed again and characterized by magnetic measurements and Mössbauer spectroscopy. The magnetic properties of the later two samples were the same as those of the originally prepared samples before and after annealing, respectively.
Calculations using the local density approximation (LDA) have been performed for three different phases of FeMnP. We assumed that all these phases have the hexagonal Fe$_{2}$P structure, with the same crystallographic parameters as observed experimentally for FeMnP$_{0.75}$Si$_{0.25}$. In this structure there are two possible completely ordered phases, one with Mn atoms occupying the pyramidal (high moment) position and one where Mn occupies the tetrahedral (low moment) position. These structures are labeled Mn pyramidal and Fe pyramidal, respectively. In the third phase the Mn and Fe atoms are completely randomly distributed on the two Fe positions, and we will refer to this phase as Disordered.
Results and Discussion
======================
X-ray diffraction
-----------------
The X-ray data at room temperature confirmed a hexagonal Fe$_{2}$P-type structure with unit cell dimensions a = 5.973 Å, and c = 3.498 Å for the annealed sample as seen in Fig. \[FeMnPSi:XRD\] a. The unit cell dimensions for the quenched sample were a = 5.974 Å, and c = 3.493 Å. A weak cubic-structure with cell dimension a = 5.653 Å was detected in the raw synthesized material, possibly of Fe$_{3}$Si type. In order to eliminate this fractional phase and to compensate a possible loss of phosphorus during the synthesis some phosphorus was added and the sample was annealed. After the annealing no trace of the Fe$_{3}$Si impurity phase could be deduced from the XRD data. A composition analysis carried out using a electron probe micro-analyzer (WDS-EPMA) as well as energy dispersive spectroscopy (EDS) indicates an excess of iron to manganese with a Fe/Mn ratio of $\sim$ 1.24. In addition our analysis revealed a new impurity phase which amount to $\sim$ 5 % of our sample consisting of Fe and Si. This impurity phase is also present in the XRD dataset and was determined to be of FeSi type.
![(Color online) XRD patterns of a) an annealed and b) an as-quenched FeMnP$_{0.75}$Si$_{0.25}$ sample. The blue lines mark the indexed peaks of the FeMnP$_{0.75}$Si$_{0.25}$ phase.[]{data-label="FeMnPSi:XRD"}](XRD.eps){width="45.00000%"}
Mössbauer measurements
----------------------
The quenched sample and the annealed sample were probed by room temperature Mössbauer spectroscopy. The distribution of iron atoms on the two inequivalent atomic positions 3f and 3g was investigated. In a Fe$_{2}$P structure the pyramidal 3g site is preferentially occupied by the less electro-negative atom and for small differences in the electronegativity by the atom with larger radius [@Furchart:1]. For FeMnP the tetrahedral site is occupied by iron and the pyramidal site by manganese [@Srivastava:1]. In the case of FeMnP$_{0.75}$Si$_{0.25}$ with the same stoichiometric number of iron and manganese atom one therefore would expect the iron atoms to be on the tetrahedral site and the manganese atoms on the pyramidal site. The spectral intensities from the Mössbauer analysis are modified from the site abundancies due to the so called thickness effect. In the present case the thickness effect for the two spectra can be assumed to be very similar since the total absorbance is almost the same (same Mössbauer thickness) and also due to that the lines emanating from site Fe(I) and Fe(II) are not well resolved. The site abundancies of iron atoms on the pyramidal 3g site for the quenched sample is found to 17(1)%. After annealing the iron concentration on the pyramidal site decreased to 12(1)% (Fig. \[FeMnPSi:Moessbauer\]). The separation of Fe and Mn atoms on the two sites accords with the expected preferences. Already in the quenched sample a significant deviation from random occupany is found and after annealing an added fraction of atoms has diffused to their preferred positions. Any ordering of the pnictide elements on the two phosphorus sites 1b and 2c could not be investigated, in the present study, but may also take place as an effect of the annealing. An elemental P(I)/As substitution has e.g. been found for the closely related compound Fe$_{2}$P$_{1-x}$As$_{x}$ [@Catalano:1]. The isomer shifts $\delta$ (mm/s) vs. $\alpha$-Fe and electric quadrupole splittings $\Delta$ (mm/s) do not change significantly due to the metal element ordering, being ($\delta$,$\Delta$)=(0.27(1),0.24(1)) for Fe(I) and (0.55(1),0.53(1)) for Fe(II), respectively.
![(Color online) Room temperature Mössbauer spectra of FeMnP$_{0.75}$Si$_{0.25}$. Spectrum a) corresponds the annealed sample and spectrum b) to the quenched sample. The shaded doublets emanate from the pyramidally coordinated Fe nucleus at the 3g site.[]{data-label="FeMnPSi:Moessbauer"}](Moessbauer.eps){width="45.00000%"}
First principles theory
-----------------------
![(Color online) Total energy (upper frame) and ordered magnetic moments per formula unit (lower frame) for ordered and disordered phases of hexagonal FeMnP as functions of the lattice parameter. The ordered phases, correspond to the Mn atom occupying the pyramidal (high moment) and tetrahedral (low moment, labeled Fe pyramidal) positions, respectively. In the disordered phase the two positions are randomly occupied by Mn and Fe atoms. The dashed-dotted vertical line indicates the experimental lattice parameter.[]{data-label="FeMnP:eos_mom"}](FeMnP-2.eps){width="45.00000%"}
![(Color online) The site-projected moments of the ordered and disordered phases of FeMnP as a function of the lattice parameter. The dashed-dotted vertical line indicates the experimental lattice parameter.[]{data-label="FeMnP:sitemoms"}](FeMnP_sitemom-2.eps){width="45.00000%"}
In Fig. \[FeMnP:eos\_mom\] we show the calculated total energy as a function of the lattice parameter, $a$, for three different phases, Mn pyramidal, Fe pyramidal and Disordered (upper panel). As is clear from the figure the Mn pyramidal phase has lowest energy for all considered volumes (or lattice parameters). The energy difference between the different phases is actually rather significant (5-10 mRy/formula unit), and the phase with lowest energy is the one which is also observed experimentally, with Mn preferentially occupying the high moment site. The Mn pyramidal phase has a shallow energy minimum for a lattice constant of 5.83 Å<a<5.97Å, whereas our experimental value is 5.97 Å for the hexagonal FeMnP$_{0.75}$Si$_{0.25}$.
All three curves in the upper panel of Fig. \[FeMnP:eos\_mom\] have a small kink at a lattice constant close to 5.8 Å. This kink is intimately connected to a sharp change in the calculated total magnetic moment per formula unit, as shown in the lower panel of Fig. \[FeMnP:eos\_mom\], due to what is known as the magneto-volume effect. For the Mn pyramidal phase, the magnetic moment drops from a considerable value at a lattice constant larger than 5.8 Å, to vanish completely for lower lattice constants. For the Fe pyramidal phase the transition is not as sharp and the moment does not disappear until a lattice constant of $\sim$5.75 Å is reached, and for the Disordered phase the moment disappears at a lattice constant of $\sim$5.62 Å. Since the change in the magnetic moment is sharpest for the Mn pyramidal phase, it is natural that the kink in the total energy curve is most pronounced for this phase. The disappearance of the magnetism for lower volumes is a consequence of the competition between kinetic energy of the electron states, which is always lower for a spin-degenerate system, and the exchange energy, which is lower for a spin-polarized system. With decreasing volume the band-width becomes broader, and consequently the kinetic energy becomes the dominating term in the total energy. Hence, lower volumes favor the spin-degenerate state with a vanishing magnetic moment. The competition between kinetic and exchange energy depends intricately on the details of the electronic structure and since the different phases considered in Fig. \[FeMnP:eos\_mom\] have different electronic structures, the transition to a spin-degenerate, non-magnetic state is different for the different phases.
In Fig. \[FeMnP:sitemoms\] we display the site projected magnetic moments of the Fe and Mn atoms, for the Disordered phase (upper panel) and the Mn pyramidal and Fe pyramidal phases (lower panel). For the Disordered phase we observe that for large volumes all moments are ferromagnetically coupled, but for lower volumes the Mn moment on the low moment site (tetrahedral site) couples antiferromagnetically to the other moments. The transition from ferromagnetic coupling to antiferromagnetic coupling for this atomic moment occurs at a volume very close to the experimental volume. At this volume the Mn-Mn distance on the tetrahedral sites is 2.65 Å. The distance between the tetrahedral and pyramidal site is similar.
The same behavior is actually exhibited by the Fe pyramidal phase, in that the Mn moment on the tetrahedral site changes from ferromagnetic to antiferromagnetic with decreasing volume (see the lower panel of Fig. \[FeMnP:sitemoms\]). However, for the Mn pyramidal phase this does not happen, all moments are ferromagnetically coupled. This phase corresponds to Mn atoms occupying the pyramidal, high moment site, with Mn moments approaching 3 $\mu_B$/atom. In the Fe pyramidal phases the Mn moments are always lower than 1 $\mu_B$/atom for a<6.1Å, and it is tempting to explain the stabilization of the Mn pyramidal phase to be due to the exchange energy of the larger Mn moment of the pyramidal site.
Magnetization measurements
--------------------------
DC magnetization measurements on the quenched and annealed samples are displayed in Fig. \[FeMnPSi:MM\]. The as-quenched sample show a broad para- to ferromagnetic phase transition at approx. 250 K accompanied by a strong thermal hysteresis starting already around 280 K. The observed thermal hysteresis is a indicator of a first order nature of the phase transition.
![(Color online) Susceptibility (M/H) vs. temperature for FeMnP$_{0.75}$Si$_{0.25}$ measured on the quenched sample in an applied field of 50 Oe (black circles) and on the annealed sample under an applied field of 500 Oe (red triangles, inset). The open symbols indicate measurements using a zero field cold (ZFC) protocol and the filled symbols using a field cooled (FC) protocol.[]{data-label="FeMnPSi:MM"}](MMM.eps){width="45.00000%"}
The susceptibility vs. temperature curve for the annealed sample shows a para- to antiferromagnetic phase transition at approx. 160 K. Below the maximum signaling the antiferromagnetic transition at 160 K, the susceptibility slightly increases due to not fully compensated antiferromagnetism. Additionally, there is a significant thermal hysteresis between the ZFC and FC curves; the hysteresis first appears around 280 K, i.e well above the Néel temperature, but at a temperature that coincides with the temperature for onset of thermal hysteresis on the quenched ferromagnetic sample. It is also of interest to note that this temperature marks the onset of a frequency dependent ac-susceptibility, that remains frequency dependent only down to the antiferromagnetic ordering temperature at 160 K. This indicates that clusters of ferromagnetic order starts to form around 280 K (the same temperature range in which the quenched sample starts to form a long ranged ferromagnetic phase). On further cooling global antiferromagnetic interaction forces the sample in into a long ranged antiferromagnetic order below about 160 K.
It is worth to be mentioned that the closely related compound FeMnP (orthorhombic Co$_{2}$P structure) also show an antiferromagnetic structure for 176K<T<265K with a doubling of the crystallographic c axis. Below 175 K a complicated modulated helical antiferromagnetic structure is developed [@Sjostrom:1]. FeMnP is a fully ordered compound with tetrahedral Me(I) site and the pyramidal Me(II) site fully occupied by Fe and Mn atoms, respectively.
The magnetic moment of both samples as a function of the applied field at 30 K is shown in Fig. \[FeMnPSi:MM-Hyst\]. At an applied field of 3 T and 30 K the measured magnetic moment is 1.26 $\mu_{B}$ per f.u. for the quenched sample and 0.05 $\mu_{B}$ per f.u. for the annealed sample. The figure distinctly pictures the transformation of the low temperature state of the material from ferromagnetic to antiferromagnetic by only an almost marginal change of the site occupancy of the Fe and Mn atoms.
![(Color online) Magnetization vs. applied field for an as-quenched sample at 30 K (black circles) and an annealed sample at 30 K (red triangles).[]{data-label="FeMnPSi:MM-Hyst"}](MM-Hyst.eps){width="45.00000%"}
![(Color online) Experimental magnetic moments/f.u. in comparison with calculated moments assuming ferro–, ferri– and antiferromagnetic ordering.[]{data-label="FeMnPSi:Exp-Theo"}](Exp-Theo.eps){width="45.00000%"}
In Fig. \[FeMnPSi:Exp-Theo\] a comparison between calculated and measured magnetic moments is shown. The calculated magnetic moments are obtained by averaging the calculated ordered moments for the Mn pyramidal, Fe pyramidal, and Disordered phase in proportion to the Fe/Mn ratio and measured disorder from Mössbauer spectroscopy. For the ferrimagnetic case an Fe/Mn ratio of 1 and full crystallographic order was assumed. It can be seen that the experimental results for the as-quenched sample and the annealed sample do not coincide with ferromagnetic ordering. Both samples, annealed and as quenched exhibit a more complex magnetic structure such as antiferromagnetism or ferrimagnetism.
In the case of Fe$_{1-x}$S a rather similar experimental result was explained by the ordering of vacancies on magnetic sublattices, see Takayama et al.[@Takayama:1]. The appearance of an antiferromagnetic phase in Fe$_{2-x}$P due to heat treatment as seen for FeMnP$_{0.75}$Si$_{0.25}$ was not observed, see e.g. Lundgren et al. [@Lundgren:1]. The influence of vacancies due to non-full site occupation was therefore not further studied.
Summary and Conclusions
=======================
In this manuscript we report on the synthesis and structural characterization of FeMnP$_{0.75}$Si$_{0.25}$, a compound which crystallizes in the hexagonal Fe$_{2}$P-type structure. Two types of samples (as quenched and annealed) were synthesized and characterized structurally (using Mössbauer spectroscopy) and magnetically. It is found that marginal changes in the degree of crystallographic order causes a large change in the magnetic properties. Annealing causes a larger degree of order compared to the rapidly quenched sample, and our analysis from the Mössbauer data suggests that for the annealed sample $\sim$ 12 % of the pyramidal (high moment) site is occupied by Fe atoms and $\sim$ 88 % by Mn atoms. This correspond to almost full order for the measured Fe/Mn ratio of $\sim$ 1.24. Assuming full site occupation this means that the tetrahedral site has $\sim$ 88 % Fe atoms and $\sim$ 12 % Mn atoms. For the quenched sample the corresponding numbers are $\sim$ 17 % of the pyramidal site is occupied by Fe atoms and $\sim$ 83 % by Mn atoms. These small changes, which eventually may be accompanied by a pnictide element ordering, result in drastically different magnetic behaviour.
The annealed sample, with higher degree of order is antiferromagnetic with a zero net moment. The as-quenched sample has a net moment of 1.26 $\mu_B$/f.u. Our experimental finding that the magnetism depends very delicately upon the degree of order is in qualitative agreement with the theoretical first principles results.
The theoretical calculations give rather large magnetic moments for the Fe and Mn atoms, both when occupied on the tetrahedral and pyramidal site. The largest being the Mn moment for the pyramidal site, which reaches values as high as 2.8 $\mu_B$/atom. We do not have atomic resolved experimental values to compare these theoretical values with, but we can note that normally theory and experiment agree with each other for atomic projected moments of magnetic materials (see e.g. Ref.[@Eriksson:1]) with an error being less than 10 %. If we assume that this is also the case for the currently studied system, we must conclude that the as-quenched sample is a ferrimagnet or a non-collinear magnetic structure, possibly involving a spin-spiral state, since a ferromagnetic coupling of the calculated atomic moments would result in a net moment of $\sim 3.5$ $\mu_B$/f.u., a value much larger than the measured value (see Fig. \[FeMnPSi:Exp-Theo\]).
The observed magnetic response of FeMnP$_{0.75}$Si$_{0.25}$ is strongly dependent on the proportion of Fe and Mn atoms occupying the tetrahedral and pyramidal sites. A very small increase of the Fe concentration on the pyramidal site, only a few percent, causes a major change from an antiferromagnet to a ferrimagnet with a rather large saturation moment. It is unclear if a modification of the stoichiometry would cause a similar change in the magnetic response, but it is tempting to speculate that this may be the case. We also find from our theory that an ordered phase with all Mn atoms on the pyramidal site and all Fe atoms on the tetrahedral site has a significantly lower energy compared to the disordered phase.
The here studied material, FeMnPSi, has been characterized structurally and magnetically, with a range of experimental techniques and by first principles theory. As member of a family of materials which are relevant for magnetocaloric refrigeration. Our study indicates that the influence of crystallographic order and disorder on the magnetocaloric properties is important and should be studied in more detail. This involves both varying the concentration of Fe and Mn as well as different annealing conditions. It is also desirable to undertake neutron scattering experiments to detect atom projected moments. Such studies are underway.
Acknowledgments
===============
Financial support from the Swedish Energy Agency (STEM) and the Swedish Research Council (VR) is acknowledged. Calculations done on supercomputer resources provided by SNAC. We are grateful to Ece Gülşen for the sample fabrication, Rebecca Bejhed, Hans Harrysson, and Hans Annersten for composition analysis, and Roland Mathieu for valuable suggestions and advices.
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---
author:
- Olivio Chiatti
- Christian Riha
- Dominic Lawrenz
- Marco Busch
- Srujana Dusari
- 'Jaime Sánchez-Barriga'
- Anna Mogilatenko
- 'Lada V. Yashina'
- Sergio Valencia
- 'Akin A. Ünal'
- Oliver Rader
- 'Saskia F. Fischer'
title: '2D layered transport properties from topological insulator Bi$_2$Se$_3$ single crystals and micro flakes'
---
Introduction {#introduction .unnumbered}
============
A tremendous interest in the electronic properties of topological insulators (TIs),[@kon07; @kan05] such as Bi$_2$Se$_3$, stems from the fact that they have robust topological surface states (TSS).[@kan05; @fu07; @moore07; @has10] The existence of TSS in Bi$_2$Se$_3$ was observed by angle-resolved photoemission spectroscopy and scanning tunneling spectroscopy.[@hsi08; @xia09; @che09; @alp10; @pan11] While TSS are predicted to have peculiar properties of great interest for future electronic devices in spintronics and quantum computation,[@Fu09; @Akh09; @has11; @sem12; @wang12; @sch12; @che12] the direct access in transport experiments still remains to be demonstrated. The defect chemistry in Bi$_2$Se$_3$ is dominated by charged selenium vacancies, which act as electron donors, and increase the conductivity of bulk states dramatically.[@Hor] However, it was shown that the TSS coexist with the Se vacancies and that in ARPES measurements a single Dirac cone can be observed, even though Se vacancies exist.[@Felser] In particular, recent reports[@Analytis; @Petrushevsky-2012-prb; @caoPRL; @Yan] on highly doped Bi$_2$Se$_3$ additionally show indications of two-dimensional (2D) layered transport, which hampers the unambiguous identification of the TSS. For high electron densities ($\sim10^{19}$ cm$^{-3}$) a rich variety of magnetoresistance phenomena in high-magnetic fields were detected, such as Shubnikov-de Haas (SdH) oscillations and quantum Hall resistances, and attributed to the behavior of stacked 2D electron systems. However, their interpretations are controversial.[@Analytis; @Petrushevsky-2012-prb; @caoPRL]
In this work, we report on TSS, which were probed by angle-resolved photoemission spectroscopy (ARPES), and significant 2D layered transport properties from both, the high- and low-field magnetoresistance of a high quality topological insulator Bi$_2$Se$_3$ single crystal. By exfoliation macro and micro flakes were prepared to investigate bulk and surface contributions (see Sec. Methods). Comprehensive combined structural, electronic and low-temperature magnetotransport investigations show: Quantum corrections to the electronic conduction are dominated by the TSS carriers in the semiconducting regime, but in the metallic-like regime additional 2D layers contribute. In low-magnetic fields a weak-antilocalization (WAL) cusp in the conductivity exists in both cases, and we discuss the analysis employing the Hikami-Larkin-Nagaoka (HLN) model.[@hik80]. As support for the 2D layered transport in the metallic-like case, we find SdH oscillations in the longitudinal bulk conductivity and quantization of the transversal (Hall) resistivity in high-magnetic fields.
Results {#results .unnumbered}
=======
Structural characterization {#structural-characterization .unnumbered}
---------------------------
A high-quality single crystal of nominally undoped Bi$_2$Se$_3$ was grown by the Bridgman technique.[@sht09] The typical high-resolution transmission electron microscopy (HRTEM) image in Fig.\[fig1\](a) from a homogeneously thick region of the flake, viewed in the direction perpendicular to the surface, shows a 2D arrangement of lattice fringes with 6-fold symmetry and a lattice spacing of $\sim$0.21 nm. This proves that the surface normal of the exfoliated flake is parallel to the \[00.1\] Bi$_2$Se$_3$ zone axis.
The selected area electron diffraction (SAED) pattern shown in Fig. \[fig1\](b), obtained from a large region of the flake, reveals the single crystal nature of the sample. Interestingly, the SAED pattern shows the presence of weak superstructure reflections, indicating possible ordering effects appearing at the $\{1\bar{1}00\}$ planes. Similar diffraction patterns were reported for Bi$_2$Se$_3$ and Bi$_2$Te$_3$ nanoribbons and nanoplates.[@kon10; @xiu11; @wan03] For well-ordered Bi$_2$Se$_3$ single crystals the structure factor of these additional reflections is zero. Furthermore, these reflections cannot appear in the \[00.1\] Bi$_2$Se$_3$ zone axis due to multiple scattering. According to the positions of the superstructure reflections, in the (0001) plane the Bi$_2$Se$_3$ supercell is three times larger than the Bi$_2$Se$_3$ unit cell, which is built by $\mathbf{a}\times\mathbf{b}$ unit vectors. For example, a similar superstructure has been previously assumed in Cu$_x$Bi$_2$Se$_3$ single crystals, which raises the question if a similar effect can be caused by bulk dopants.[@han11] Indeed, such ordering effects can be caused by a strictly periodic arrangement of point defects at the corresponding atomic planes (vacancies, substitutional atoms, excess of Bi or Se).
![(a) HRTEM image of the flake region of about 40 nm in thickness, viewed in the direction of the surface normal. (b) SAED pattern viewed in \[00.1\] direction of the Bi$_2$Se$_3$. The diamond indicates the $\{1\bar{1}00\}$ reflection.[]{data-label="fig1"}](Figure1.eps){width="0.6\columnwidth"}
A second possible reason for the appearance of the $\{1\bar{1}00\}$ reflections can be a certain structural stacking disorder along the $c$-axis direction. For example, diffraction pattern simulations we carried out for a Bi$_2$Se$_3$ unit cell with one missing quintuple layer show the appearence of these reflections. This might appear at the Bi$_2$Se$_3$ surface and can become especially pronounced for thin Bi$_2$Se$_3$ layers and nanostructures, where the volume fraction of disordered surface regions is larger. Our HRTEM analysis did not reveal any visible superstructure in the $[00.1]$ zone axis of Bi$_2$Se$_3$ (Fig. \[fig1\](a)), which further supports this explanation. Therefore, we attribute this diffraction effect to a contribution by the surface. A detailed electron diffraction analysis of bulk Bi$_2$Se$_3$ single crystals and flakes of different thicknesses is required to unambiguously identify the origin.
Electronic structure {#electronic-structure .unnumbered}
--------------------
To verify the characteristics of the surface states of the Bi$_2$Se$_3$ bulk single crystal, high resolution ARPES experiments were performed at different photon energies. The original $(0001)$ surface was characterized to determine its electronic structure and the Fermi level position or intrinsic doping level of the samples. Fig. \[fig2\] shows high-resolution ARPES dispersions of the TSS, bulk conduction band (BCB) and bulk valence band (BVB) states, measured at different photon energies and as a function of the electron wave vector $k_\parallel$ parallel to the surface. A gapless Dirac cone representing the TSS with a Dirac point located at a binding energy of $\sim$0.35 eV is clearly observed. The binding energy position of the BCB crossing the Fermi level indicates that the crystals are intrinsically $n$-type, in agreement with our Hall measurements. At binding energies higher than the Dirac point, the lower half of the Dirac cone overlaps with the BVB. In ARPES, the photon energy selects the component of the electron wave vector $k_\mathrm{z}$ perpendicular to the surface. Since the lattice constant of Bi$_2$Se$_3$ is very large along the z direction ($c\!=\!28.64$ Å), the size of the first bulk Brillouin zone is very small ($\sim$0.5 Å$^{-1}$). Therefore, Figs. \[fig2\](a)-\[fig2\](f) show a range of low photon energies between 16 to 21 eV, where we practically cross the complete Bi$_2$Se$_3$ first bulk Brillouin zone enhancing the sensitivity to the out-of-plane dispersion of the bulk bands. We note that the maximum ARPES intensity changes with the photon energy as well due to the $k_{\mathrm{z}}$-dependence of the photoemission transitions.
Unlike the BCB or the M-shaped dispersion of the BVB, the TSS has no $k_{\mathrm{z}}$-dependence, a fact which confirms its 2D nature. Its dispersion remains very clearly the same when varying the photon energy, as can be seen in Figs. \[fig2\](a)-\[fig2\](f). In contrast, the BVB maximum is reached around at $h\nu\!=\!18$ eV ($k_{\mathrm{z}}\!=\!2.65$ Å$^{-1}$) and the BCB minimum near 21 eV ($k_{\mathrm{z}}\!=\!2.8$ Å$^{-1}$) at binding energies of $\sim$0.452 eV and $\sim$0.154 eV, respectively. This is consistent with a bulk band gap of about $\sim$0.3 eV. These facts unambiguously identify the existence of both a single-Dirac-cone surface state and a well-defined bulk band gap in our samples, two of the most important attributes of the topological insulator Bi$_2$Se$_3$. The data in Fig. \[fig2\] allow us to estimate the bulk carrier concentration $n_\mathrm{3D}$ from the size of the bulk Fermi wave vector $k_{\rm F,3D}$ according to $n_\mathrm{3D}\!=\!k_{\rm F,3D}^3/(3\pi^2)$. The limited accuracy of $k_{\rm F, 3D}\!=\!(0.064\pm0.01)$ Å$^{-1}$ yields $n_\mathrm{3D}\!=\!(8.8\pm0.4)\cdot10^{18}$ cm$^{-3}$.
![$k_{\mathrm{z}}$-dependence of the electronic structure of the Bi$_2$Se$_3$ bulk single crystal before mechanical exfoliation. Each panel shows high resolution ARPES $E(k_\parallel)$ dispersions measured at 12 K and at photon energies $h\nu$ of (a) 16 eV, (b) 16.5 eV, (c) 17.5 eV, (d) 18 eV, (e) 18.5 eV and (f) 21 eV across the first bulk Brillouin zone. In panel (a), the topological surface state (TSS), the bulk conduction band (BCB) and the bulk valence band (BVB) are labelled. The surface state does not show dispersion with photon energy, while bulk states exhibit a clear dependence.[]{data-label="fig2"}](Figure2.eps){width="100.00000%"}
Core-level spectroscopy {#core-level-spectroscopy .unnumbered}
-----------------------
![PEEM characterization of a contacted Bi$_2$Se$_3$ flake. (a) Overview image with vacuum UV light of the flake and Au contact area. (b) Corresponding confocal microscopy image of the same flake revealing thickness homogeneity. (c), (d) and (e): Core-level spectromicroscopy of the flake area and of the Au contact area. On the left, spatially resolved images taken at the kinetic energies of the (c) Se 3d$_{5/2}$, (d) Bi 5d$_{5/2}$ and (e) Au 4f$_{5/2}$ core-levels. On the right, the corresponding core-level spectra are shown. The spectra are extracted from the small areas indicated in each PEEM image on the left with a kinetic energy resolution of $\sim$0.2 eV. The flakes are stable to atmosphere and to the lithography process. The measurements were performed using soft x-rays of 400 eV photon energy and horizontal polarization. The PEEM field of view is 25 $\mu$m.[]{data-label="fig3"}](Figure3.eps){width="0.49\columnwidth"}
In order to investigate the chemistry of the micro flake surface after exfoliation and lithographic processing of electrical contacts for transport measurements, we performed spatially resolved core-level X-ray photoelectron emission microscopy (PEEM). It serves the following purposes: a test of stability of the micro flake surface in high-electric fields, a test of disturbance by charging of the SiO$_2$ substrate, and in particular, confirmation of the chemical composition against ambient atmosphere providing useful information about the oxidation state of the flakes. The experiments were performed at a photon energy of 400 eV and horizontal polarization. The kinetic energy resolution was $\sim$0.2 eV and the lateral resolution about 70 nm. Fig. \[fig3\](a) shows an overview PEEM image of a Bi$_2$Se$_3$ micro flake with a Ti/Au contact acquired in vacuum with UV light. Fig. \[fig3\](b) shows the corresponding confocal microscopy image, which zooms into the same flake and reveals its thickness homogeneity. The core-level spectromicroscopy results of the flake area and of the Au contact area are shown in Figs. \[fig3\](c)-\[fig3\](e). Spatially resolved images acquired at the kinetic energies of the Se 3d$_{5/2}$ (Fig. \[fig3\](c)), Bi 5d$_{5/2}$ (Fig. \[fig3\](d)) and Au 4f$_{5/2}$ (Fig. \[fig3\](e)) core-levels are shown together with the corresponding core-level spectra. The spectra are extracted from the small areas indicated in each PEEM image.
Our results demonstrate that the flakes and their composition are stable to ambient atmosphere and to the lithography process. However, we do note that the Se 3d core-levels show a small component separated by $\sim$2 eV to higher binding energy. The Bi 5d peaks exhibit two small additional components which are located $\sim$1 eV higher in binding energy with respect to the main peaks and are not well-resolved due to the experimental energy resolution. These findings indicate that a minor oxidation of the Bi$_2$Se$_3$ flake surface exists. Our estimations reveal that oxygen is adsorbed in less than 15% of a quintuple layer. Moreover, this oxygen adsorbate layer was easily removed by moderate annealing under ultra-high vacuum (UHV) conditions at $\sim120-180\,^{\circ}{\rm C}$ during 5 minutes.
Recently, we observed a similar surface reactivity by means of ARPES experiments in topological insulator bulk single crystals and films grown by molecular beam epitaxy.[@Yashina-ACSNano-2013] We found that the topological surface state in our samples is robust against the effects of surface reactions, which typically lead to doping and extra quantization effects in the ARPES dispersions due to band bending.[@Bianchi-NatComm-2011; @King-PRL-2011; @Bianchi-PRL-2011] By means of ARPES and core-level photoemission experiments at different pressures, we demonstrated that our Bi$_2$Se$_3$ bulk single crystals exhibit a negligible surface reactivity toward oxygen and water.[@Yashina-ACSNano-2013]
Temperature-dependent longitudinal resistances {#temperature-dependent-longitudinal-resistances .unnumbered}
----------------------------------------------
![(a) Temperature dependent resistivity $\rho_{\mathrm{xx}}$ of two Bi$_2$Se$_3$ micro flakes with thicknesses of $t\!=\!130$ nm (black squares) and 340 nm (brown triangles) and a Bi$_2$Se$_3$ macro flake with a thickness of $t\!=\!110$ $\mu$m (green diamonds). Insets in (a) show an optical microscope image of the Bi$_2$Se$_3$ micro flake with a thickness of $t\!=\!130$ nm with Ti/Au contacts, and the low temperature range of the measured resistivity $\rho_{\mathrm{xx}}$ of this micro flake (with a logarithmic temperature axis). Solid curves represent best Bloch-Grüneisen fits. (b) Temperature dependent resistivity $\rho_{\mathrm{xx}}$ of a semiconducting Bi$_2$Se$_3$ micro flake with a thickness of $t\!=\!220$ nm. Inset in (b) shows the optical microscope image of this micro flake with Ti/Au contacts.[]{data-label="fig4"}](Figure4.eps){width="0.47\columnwidth"}
Bi$_2$Se$_3$ flakes were contacted in Hall bar geometries for transport measurements. Two examples are shown in the insets of Fig. \[fig4\](a) and (b) by optical microscope images for thicknesses of $t\!=\!200$ nm (inset of Fig. \[fig4\](a)) and $t\!=\!220$ nm (inset of Fig. \[fig4\](b)), respectively. In the temperature-dependent longitudinal resistance we observe two cases: metallic-like behavior as shown Fig. \[fig4\](a) and semiconducting behavior as shown Fig. \[fig4\](b). In Fig. \[fig4\](a) the metallic-like behavior of the resistivity $\rho_{\mathrm{xx}}$ as function of temperature from 4.2 K up to 290 K for two typical Bi$_2$Se$_3$ micro flakes with thicknesses of $t\!=\!130$ nm and 340 nm and for a macro flake with a thickness of $t\!=\!110\;\mu$m is shown. The resistivity $\rho_{\mathrm{xx}}$ remains practically constant up to 30 K (see right inset in Fig. \[fig4\](a)), presumably due to a combination of surface states and static disorder scattering, as observed in the previous reports.[@kim11; @ban12; @che10; @he12; @he11] The residual resistivity ratio $\mathrm{RRR}\!=\!\rho_{\mathrm{xx}}(300~\mathrm{K})/\rho_{\mathrm{xx}}(2~\mathrm{K})\!=\!1.74$ for the Bi$_2$Se$_3$ macro flake indicates the high crystalline quality.[@Hyde] The RRR of the exfoliated micro flakes have similar values within $\pm20$%, and for a 80 nm thin micro flake we found $\mathrm{RRR}\!=\!\rho_{\mathrm{xx}}(290~\mathrm{K})/\rho_{\mathrm{xx}}(0.3~\mathrm{K})\!=\!1.99$ indicating a decrease of bulk defects. In the temperature range between 100 K and room temperature Bloch-Grüneisen fits can be performed. Such metallic-like behavior is in agreement with previous reports[@cao12] of exfoliated Bi$_2$Se$_3$ micro flakes with high electron densities of about $10^{19}$ cm$^{-3}$. However, as shown in Fig. \[fig4\](b) semiconducting behavior in the temperature dependent resistivity $\rho_{\mathrm{xx}}$ can be also observed. This occurs if the fabrication and storage procedures of the micro flakes minimize any exposure to air (see Sec. Methods). Otherwise, metallic-like behavior is induced. Semiconducting behavior was also found for MBE grown Bi$_2$Se$_3$ thin films.[@Hirahara]
Longitudinal and Hall conductivities {#longitudinal-and-hall-conductivities .unnumbered}
------------------------------------
While low-magnetic field conductivities were measured in order to determine the electron densities and mobilities, high-magnetic field conductivities show typical signatures of 2D transport, such as SdH oscillations and quantum Hall effect in macro flakes. The low-field conductivity $\sigma_\mathrm{xx}$ and the Hall conductivity $\sigma_\mathrm{xy}$ of the macro flake as a function of the perpendicular magnetic field $B$ at a temperature $T\!=\!4.1$ K are shown in Fig. S3 in the supplemental information. The conductivity curves were determined from the longitudinal resistivity $\rho_\mathrm{xx}$ and Hall resistivity $\rho_\mathrm{xy}$ as follows:[@Ando-2013-JPSJ] $\sigma_\mathrm{xx}\!=\!\rho_\mathrm{xx}/(\rho_\mathrm{xx}^2+\rho_\mathrm{xy}^2)$ and $\sigma_\mathrm{xy}\!=\!-\rho_\mathrm{xy}/(\rho_\mathrm{xx}^2+\rho_\mathrm{xy}^2)$. Both the $\sigma_\mathrm{xx}(B)$ and $\sigma_\mathrm{xy}(B)$ curves can be fitted within the Drude model and the results of the fits are in agreement with each other and yield an electron density $n_\mathrm{3D}\!=\!1.92\cdot10^{19}$ cm$^{-3}$ and mobility $\mu_\mathrm{3D}\!=\!970$ cm$^2$/(Vs). These values are consistent with those reported in a variety of highly-doped Bi$_2$Se$_3$ bulk crystals of similar resistivity.[@Hyde; @Koehler; @Petrushevsky-2012-prb; @Ge-2015-ssc]
![(a) Resistivity $\rho_{\mathrm{xx}}$ (red curve, left axis) and Hall resistivity $\rho_{\mathrm{xy}}$ (blue curve, right axis) vs magnetic field $B$ of the Bi$_2$Se$_3$ macro flake with a thickness of $t\!=\!110$ $\mu$m at $T\!=\!0.3$ K. (b) Hall conductance per 2D layer $\widetilde{G}_\mathrm{xy}\!=\!G_\mathrm{xy}/Z^*$ in units of e$^2$/h (red curve, left axis), with measured conductance $G_\mathrm{xy}\!=\!1/R_\mathrm{xy}$ and $Z^*\!=\!57500$, and $\mathrm{d}\sigma_{\mathrm{xx}}/\mathrm{d} B$ (blue curve, right axis) vs inverse magnetic field $1/B$ at $T\!=\!0.3$ K. The black arrows indicate the QHE plateaux. (c) Landau level (LL) fan diagram at $T\!=\!0.3$ K. The $1/B$-positions of the minima and maxima of $\sigma_\mathrm{xx}(B)$ are shown as a function of the corresponding LL level indices $N$ and $N-0.5$, respectively. The dashed line represents a linear fit to the data, yielding a slope $B_{\mathrm{f}}\!=\!151$ T and an intercept close to zero.[]{data-label="fig5"}](Figure5.eps){width="0.5\columnwidth"}
SdH oscillations in the resistivity $\rho_\mathrm{xx}$ and the onset of quantum Hall effect (QHE) plateaux in the Hall resistivity $\rho_\mathrm{xy}$ can be observed at high-magnetic fields. Fig. \[fig5\](a) shows $\rho_\mathrm{xx}$ (red curve) and $\rho_\mathrm{xy}$ (blue curve) of the macro flake with a thickness of $t\!=\!110$ $\mu$m as a function of the perpendicular magnetic field $B$ at a temperature of $T\!=\!0.3$ K. The slope of $\rho_\mathrm{xy}$ yields an electron density $n_\mathrm{3D}\!=\!1.84\cdot10^{19}$ cm$^{-3}$ and the onset field $B\!\approx\!7.5$ T yields a mobility $\mu_\mathrm{3D}\!\approx\!1300$ cm$^2$/(Vs), both values in fair agreement with the results of the low-field transport. For a comparison with the values from ARPES measurements, one has to consider the non-spherical Fermi surface which plays a role in 3D bulk measurements.[@Ando-2013-JPSJ] This shows that for micron thick crystals both the low-field and the high-field magnetotransport are dominated by carriers from the bulk. However, low-temperature transport measurements with a magnetic field parallel to the current show no clear signs of SdH oscillations or QHE plateaux. This is consistent with 2D transport, instead of 3D transport, and has been reported previously in $n$-type Bi$_2$Se$_3$.[@caoPRL; @cao12] Therefore, 2D layered transport plays an important role.
In Ref. [@caoPRL] the steps in Hall conductance $\Delta G_\mathrm{xy}\!=\!\Delta(1/R_\mathrm{xy})$ were found to scale with the sample thickness and yield a conductance of $\sim$e$^2$/h per QL. A similar scaling was also found in Fe-doped Bi$_2$Se$_3$ bulk samples,[@Ge-2015-ssc] where transport by TSS can be excluded. From this it is concluded that the bulk transport occurs over a stack of 2D layers. Fig. \[fig5\](b) shows such analysis performed on our high-field $R_\mathrm{xy}(B)$ data. We find a scaling of $\Delta G_\mathrm{xy}$ with the thickness and define $Z^*\!=\!\Delta G_\mathrm{xy}/(\mathrm{e}^2/\mathrm{h})$ as the number of 2D layers contributing to the transport, with $Z^*\!\approx\!0.5\times$ the number of QLs. Then $\widetilde{G}_\mathrm{xy}\!=\!G_\mathrm{xy}/Z^*\!=\!N\mathrm{e}^2/\mathrm{h}$ is the conductance per 2D layer, where $N$ is the Landau level index, and is shown in Fig. \[fig5\](b). Therefore, a conductance of $\sim$e$^2$/h can be associated with an effective thickness of one 2D layer of about 2 QLs thickness, i.e. $\sim2$ nm. Note that this is similar the length scale of the extention of the volume unit cell of Bi$_2$Se$_3$ ($c$-axis, 2.86 Å).[@Nakajima] The present results suggest that transport in macro flakes is consistent with a metallic-like 2D layered transport. Unfortunately, in our measurement setup for micro flakes the high-magnetic field ($>\!5$ T) magnetoresistance was not well enough resolved due to non-ideal contacts and time-varying thermovoltages. These effects may mask possible 2D transport signatures.
The resistivity curves were used to determine the conductivity $\sigma_\mathrm{xx}$ and Hall conductivity $\sigma_\mathrm{xy}$ at high fields. Fig. \[fig5\](c) shows the Landau level (LL) fan diagram determined from the measurements in Fig. \[fig5\](a). The $1/B$-positions of the minima of the $\sigma_\mathrm{xx}$ curve are shown as a function of the corresponding LL level indices $N$. The LL indices have been attributed to the $\sigma_\mathrm{xx}$ minima as in Fig. \[fig5\](b). Taking into account that the LL index of a maximum in $\sigma_\mathrm{xx}$ can be written as $N-0.5$,[@Ando-2013-JPSJ] the $1/B$-positions of the minima and the maxima collapse onto the same line. The slope of the line is the SdH frequency $B_{\mathrm{f}}\!=\!151$ T and it yields a Fermi wavevector $k_{\mathrm{F}}\!=\!0.0677$ Å$^{-1}$, which is in fair agreement with $k_{\mathrm{F,3D}}$ from our ARPES results (see Fig. \[fig2\]), and does not agree with that of $k_{\mathrm{F,TSS}}\!=\!0.086$ Å$^{-1}$. The intercept on the $x$-axis of the line yields the phase-factor $\beta$, which indicates whether fermions ($\beta\!=\!0$) or Dirac fermions ($\beta\!=\!0.5$) are responsible for the transport.[@Ando-2013-JPSJ] Within the experimental error the distinction between an intercept of 0 or 0.5 cannot be drawn unambiguously from the fan diagram. For this, higher magnetic fields such as 30 T are required. However, within the present high-field data the linear fit strongly suggests that the main contribution comes from the bulk. Conclusively, our present high-field results support a major contribution by 2D layered transport, additionally to the existence of TSS as proven by ARPES.
Weak-antilocalization effects {#weak-antilocalization-effects .unnumbered}
-----------------------------
As quantum correction to the classical magnetoresistance, the WAL effect is a signature of TSS originating from the Berry phase,[@Berry] which is associated with the helical states.[@Ando-2013-JPSJ] In the low-magnetic field longitudinal conductivity of exfoliated micro flakes we observe the typical WAL cusp. In macro flakes the low-magnetic field WAL signals are not well resolved, which strongly indicates that the contribution by a 3D conductivity in the bulk hampers its observation. However, in micro flakes the 3D bulk contribution to the conductivity is strongly reduced as the surface-to-volume ratio is increased by a factor of up to 1000. We clearly observe the WAL conductivity cusps around zero magnetic field. For metallic-like micro flakes the low-temperature electron density is about that of the bulk (macro flakes), but the mobility can be increased, e.g. for 130 nm thin flake $n_\mathrm{3D}\!=\!1.2\cdot10^{19}$cm$^{-3}$ and $\mu_\mathrm{3D}\!=\!2320$ cm$^2$/(Vs). In the semiconducting case we find a strongly reduced density but a bulk-like mobility, e.g. for a 220 nm thick flake $n_\mathrm{3D}\!=\!1.2\cdot10^{17}$cm$^{-3}$ and $\mu_\mathrm{3D}\!=\!675$ cm$^2$/(Vs). In the metallic-like case a higher density ($10^{19}$ cm$^{-3}$ vs. $10^{17}$ cm$^{-3}$) may lead to two effects which enhance the mobility: first, a screening of potential fluctuations and therefore enhanced scattering times, and second, an increase of the Fermi level and hence additional kinetic energy. In order to identify contributions of the TSS to the transport, we studied the change of the magnetoconductivity $\Delta\sigma_\mathrm{xx}$ in perpendicular low-magnetic fields as shown in Fig. \[fig6\](a) and (b). A WAL maximum is clearly visible near zero magnetic field and is suppressed for higher temperatures. At higher magnetic fields the total magnetoresistance has additional contributions $\propto\!B^2$.
Typically, the low-field behavior is associated with WAL originating from either strong spin-orbit interaction in the bulk and/or spin-momentum locking in the topological surface states.[@Ando-2013-JPSJ; @Altshuler; @Matsuo] In thin Bi$_2$Se$_3$ films or micro flakes the assumption usually is made that these can be considered as 2D systems with strong spin-orbit interaction, and the HLN model[@hik80] is used to analyze the 2D magnetoconductivity. Such a model can be applied as long as the dephasing time $\tau_\phi$ is much smaller than the spin-orbit time $\tau_{\mathrm{SO}}$ and the inelastic scattering (energy relaxation) time $\tau_{\mathrm{e}}$, i.e. $\tau_\phi\!\ll\!\tau_{\mathrm{SO}}$ and $\tau_\phi\!\ll\!\tau_{\mathrm{e}}$. We applied the HLN model to the temperature-dependent magnetoconductance data plotted in Fig. \[fig5\](a) and (b) by considering: $$\begin{aligned}
\Delta\sigma_\mathrm{xx}(B)=\sigma_\mathrm{xx}(B)-\sigma_\mathrm{xx}(B\!=\!0)=\alpha\frac{e^{2}}{2\pi^{2}\hslash}\left[\psi\left(\frac{1}{2}+\frac{B_\phi}{B}\right)-\ln\left(\frac{B_\phi}{B}\right)\right]\,,\end{aligned}$$ where $\psi(x)\!=\!\Gamma^{\prime}(x)/\Gamma(x)\!=\!\mathrm{d}\ln\Gamma(x)/\mathrm{d}x$ represents the digamma function. The prefactor $\alpha$ can be used to estimate the number of independent channels contributing to the interference: $\alpha\!=\!-0.5$ for a single coherent topological surface channel contributing to the WAL cusp, and $\alpha\!=\!-1$ for two independent coherent transport channels.[@lan13] $B_\phi\!=\!\hslash/(4e\ell_\phi^2)$ is the characteristic magnetic field and $\ell_\phi$ is the phase coherence length. In the analysis $\alpha$ and $\ell_\phi$ are fitting parameters. The symbols in Fig. \[fig6\](a) and (b) represent experimental data and the curves correspond to fits to the HLN model (see Eq. (1)).
{width="0.8\columnwidth"}
This analysis can be performed in a straightforward manner in the case of the semiconducting flake (see Fig. \[fig6\]b). As TSS are directly visible in the ARPES spectra, our evaluation is in agreement with the conventional interpretation of TSS contributions to WAL in Bi$_2$Se$_3$. However, this analysis cannot be applied directly to the conductivity data of the metallic-like micro flakes (see Fig. \[fig6\]a). In these, the change of the magnetoconductivity with the applied perpendicular magnetic field is by a factor of up to 100 too large, so that $\alpha$ cannot be obtained in a valid range within the HLN model. Due to the QHE observed in the metallic-like macro flakes, we identify the observed low-magnetic field conductivity as the result of 2D layered transport. Therefore, the HLN fit is applied to the magnetoconductivity divided by the number $Z^*$ of contributing 2D layers, which is about half the number of QLs, i.e. $Z^*\!=\!0.5\times t/(\mathrm{1~nm})$. Therefore, we define a magnetoconductivity per 2D layer: $\Delta\widetilde{\sigma}_\mathrm{xx}\!=\!\Delta\sigma_\mathrm{xx}/Z^*$ which is shown in Fig. \[fig6\](a).
In Fig. \[fig6\](c) and (d), $\alpha$ and $\ell_\phi$ of the HLN fits for two metallic-like and one semiconducting case at different temperatures are shown. At the lowest temperature of 300 mK the $\alpha$-values for the semiconducting ($t\!=\!220$ nm) and thin ($t\!=\!130$ nm) metallic-like micro flakes are about $-$0.5 and decrease in magnitude with increasing temperature, which is in accordance with previous reports.[@tas12; @ste11; @che11] This indicates that in the semiconducting case two coupled surface states dominate the WAL behavior in accordance with the HLN model. Furthermore, in the metallic-like case the assumption of $Z^*$ 2D layers allows a fit by the HLN model. For the thicker metallic-like case $\alpha\!=\!-0.25$ is obtained in the above manner. This lowered value could indicate more contributions from the bulk. The application of the HLN model appears feasible under the assumption that mainly $Z^*$ distinct 2D layers contribute equally - which would resemble the case of parallel conducting layers. This is valid if the 2D layer contribution dominates the WAL cusp.
The values of $\ell_\phi$ remain largely unaffected by such interpretations and are given in Fig. \[fig6\](d). At 300 mK $\ell_\phi$ is in agreement with the flake thickness of the metallic-like cases, which in general indicates that an assumption of a 2D system may be applied. For the semiconducting case $\ell_\phi$ is with 40 nm about one fifth of the flake thickness which may indicate an effective depth of about 20 nm in which the topological surface states extend into the bulk.
In conclusion, from application of the HLN model to the low-temperature and low-magnetic field conductivity we find that for a semiconducting flake the topological surface states can be identified from the WAL, as expected from our ARPES measurements. However, in the metallic-like micro flakes the strong decrease in magnetoconductivity with a magnetic field finds its explanation in the HLN model only if effectively more 2D layers than the two surface layers contribute. Combining the ARPES and QHE results from the macro flakes, we find that the simple HLN model can be applied if $Z^*$ coupled 2D layers contribute. In general, band bending at the surface could lead to additional 2D layers, however, we found no indications for band bending by different mobilities from Hall measurements and signatures in ARPES. Instead, the indications from the TEM analysis on the structural stacking disorder along the $c$-axis direction may be indicative for the origin of additional 2D layers in our transport experiments.
Discussion {#discussion .unnumbered}
==========
In summary, TSS as detected from ARPES measurements on Bi$_2$Se$_3$ single crystals are confirmed by transport measurements on semiconducting micro flakes. TEM and PEEM analysis ensures that micro flakes have the same structural and chemical composition as in the bulk. Additionally, our study reveals uniquely metallic-like behavior, which shows pronounced 2D effects in the [*high- and low-field*]{} magnetotransport. Most probably this occurs by contact to ambient atmospheres (air) and annealing steps during the micro contact preparation (see section Methods), leading to selenium vacancies in the near surface region. The comprehensive experiments suggest that 2D layered transport plays a decisive role in highly conductive Bi$_2$Se$_3$. Therefore, contributions to the magnetoresistance cannot be simply classified as originating only from the TSS or 3D bulk. Instead additional 2D layered transport [*in the bulk*]{} are important in $n$-type Bi$_2$Se$_3$ in the high electron-density regime , which behaves in a more complex manner than has been stated before. Both TSS at the surface and 2D layered transport in the bulk contribute to the localization phenomena. Based on our high- and low-magnetic field data we conclude that the 2D layers need not necessarily be homogenously distributed across the bulk cross section, but may contribute mostly from the near-surface regions. One reason may be the effective thickness of a Selen-depleted surface region, which hosts the additional 2D transport layers. The HLN fit can be successfully performed under the assumption of $Z^*$ stacked 2D conducting layers. This clearly indicates that a significant number of conducting 2D layers contribute, in the order of the number of quintuple layers in the micro flake (i.e. $\sim\!100$). This is much more than two possible TSS states alone. Therefore, we conclude that additionally to the (maximum) two TSS - and possibly other 2D channels due to band bending (maximum a few), there are up to the number of quintuple layers 2D channels which contribute. Our transport results in correlation with the ARPES experiments prove the coexistence of TSS and 2D layered transport. This may lead to a broad range of novel phenomena in highly conductive 3D topological insulator materials.
Methods {#methods .unnumbered}
=======
Sample preparation and characterization {#sample-preparation-and-characterization .unnumbered}
---------------------------------------
High-quality single crystalline Bi$_2$Se$_3$ were prepared from melt with the Bridgman technique.[@sht09] The growth time, including cooling was about 2 weeks for a $\sim$50 g crystal. The carrier density of the resulting samples was about $1.9\cdot10^{19}$ cm$^{-3}$, as determined by Hall measurements. The whole crystal was easily cleaved along the \[00.1\] growth direction, indicating crystal perfection. Macro and micro flakes were prepared by cleaving the single crystal. Macro flakes were prepared with a thickness of around 110 $\mu$m to investigate bulk properties. The micro flakes with a thickness in a range from 40 nm up to 300 nm were prepared using a mechanical exfoliation technique similar to the one used for graphene. The exfoliation procedure was carried out on top of a 300 nm thick SiO$_2$ layer grown on a boron-doped Si substrate. The preparation of the micro flakes involved in general the following steps: (i) Prior to sample preparation, the substrates were cleaned with acetone in an ultrasonic bath for about 3$-$4 minutes. After sonication in acetone, drops of ethanol were placed on the substrates and subsequently blow-dried with nitrogen. (ii) A $5\!\times\!5\!\times\!1$ mm$^3$ piece of the initial Bi$_2$Se$_3$ crystal was glued on a separate Si substrate with a GE7031 varnish. Using adhesive tape, a thin layer of Bi$_2$Se$_3$ was cleaved and then folded and unfolded back several times into the adhesive tape, resulting in a subsequent cleaving of the layer into thinner and thinner flakes. (iii) The SiO$_2/$Si substrate was then placed on top of the adhesive tape in a region uniformly covered by Bi$_2$Se$_3$ micro flakes. By pressing gently on top of the substrate with tweezers, the Bi$_{2}$Se$_{3}$ micro flakes were placed on the surface of the Si wafer due to the applied mechanical stress. In Fig. S2 in the supplemental information we show an atomic force microscopy (AFM) image of a selected Bi$_2$Se$_3$ flake homogeneous in thickness, drop-cast on the SiO$_2$ substrate. While the metallic-like macro flakes were prepared at room temperature without any annealing, during the micro contact preparation for the metallic-like micro flakes several annealing steps with photoresist were applied. For the semiconducting micro flakes the annealing was reduced to 1 minute at $\sim100\,^{\circ}{\rm C}$.
We explore the structural properties of the flakes with AFM, STEM and HRTEM. The flake composition and surface stability are investigated using energy-dispersive x-ray spectroscopy (EDX), see supplemental information, and spatially resolved core-level X-ray PEEM. Structural analysis using HRTEM and STEM was carried out at a JEOL JEM2200FS microscope operated at 200 kV. The sample preparation for HRTEM characterization consisted of ultrasonic separation of the flakes from the substrate, followed by their transfer onto a carbon-coated copper grid. Using adhesive tape, the surface was prepared by cleavage of the crystal along its trigonal axis in the direction perpendicular to the van-der-Waals-type $(0001)$ planes. ARPES measurements were performed at a temperature of 12 K in UHV at a pressure of $\sim5\cdot10^{-10}$ mbar with a VG Scienta R8000 electron analyzer at the UE112-PGM2a beamline of BESSY II using p-polarized undulator radiation. The micro flakes were characterized by spatially-resolved core-level microspectroscopy using an Elmitec PEEM instrument at the UE49-PGMa microfocus beamline of BESSY II.
After the micro flake selection micro-laser lithography was used to fabricate the contacts for the transport measurements. Positive photoresist AZ ECI 3027 was coated, and the samples were spun in air for 45 s at speeds varying from 2500 to 3000 rpm. After micro-laser lithography Ti/Au (10 nm/40 nm) were sputtered and lift-off processing was used to prepare the contacts. Semiconducting behavior is only observed in micro flakes if contact to air is avoided and heating procedures are minimized during the device processing. For macro flakes contacts were prepared with Ag paint and Au wires. All samples were stored in dry N$_2$ atmosphere.
Using a He-3 cryostat, temperature-dependent four-terminal resistance measurements were performed in a range from 300 K down to 0.3 K. The magnetoresistivities $\rho_\mathrm{xx}$ and $\rho_\mathrm{xy}$ were measured in a temperature range from 0.3 K up to several K. The macro flakes were measured with the Van der Pauw method[@Vanderpauw-1958-PRR], using a *Keithley* 6221 current source and a *Keithley* 2182A nanovoltmeter in “Delta mode” with a current $|I|\!=\!100~\mu$A. The micro flakes were measured with Hall-bar configurations, using either DC techniques (i.e., using a *Keithley* 6221 current source and a *Keithley* 2182A nanovoltmeter) with currents up to $|I|\!=\!1~\mu$A or low-frequency AC techniques (i.e. using *Signal Recovery* 7265 lock-in amplifiers) with currents $|I|$ between 2 nA and 20 nA.
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A method of measuring specific resistivity and Hall effect of dics of arbitrary shape. [*Philips Res. Rep.*]{} [**13**]{}, 1 (1958).
Acknowledgements {#acknowledgements .unnumbered}
================
Financial support from the Deutsche Forschungsgemeinschaft within the priority program SPP1666 (Grant No. FI932/7-1 and RA1041/7-1) and the Bundesministerium für Bildung und Forschung (Grant No. 05K10WMA) is gratefully acknowledged.
Author contributions statement {#author-contributions-statement .unnumbered}
==============================
L.V.Y. conducted the bulk crystal growth, O.C., C.R., D.L., M.B., S.D., and S.F.F. contributed to structural characterization and transport experiments, A.M. and S.D. performed the HRTEM, STEM and EDX analysis, J.S.-B., S.V., A.A.Ü., and O.R. conducted ARPES and PEEM experiments, O.C., O.R., J.S.-B., M.B., and S.F.F. conceived the experiments, analyzed the data and wrote the manuscript. All authors contributed to the discussion and reviewed the manuscript.
Additional information {#additional-information .unnumbered}
======================
**Competing financial interests:** The authors declare no competing financial interests.
|
---
abstract: 'Polarized flavor asymmetry $\Delta \bar u/\Delta \bar d$ is investigated in a meson-cloud model. A polarized nucleon splits into a $\rho$ meson and a baryon, then the polarized $\rho$ meson interacts with the virtual photon. Because of the difference between the longitudinally polarized distributions $\Delta \bar u$ and $\Delta \bar d$ in $\rho$, the polarized flavor asymmetry is produced in the nucleon. In addition, we show that the $g_2$ part of $\rho$ contributes to the asymmetry especially at medium $x$ with small $Q^2$.'
address: |
Department of Physics, Saga University, Saga, 840-8502, Japan\
Email: kumanos@cc.saga-u.ac.jp\
URL: http://hs.phys.saga-u.ac.jp
author:
- 'S. KUMANO'
title: |
A MODEL PREDICTION\
FOR POLARIZED ANTIQUARK FLAVOR ASYMMETRY
---
\
\
\
\
[(talk on Oct. 11, 2001) ]{}\
------------------------------------------------------------------------
\
[\* Email: kumanos@cc.saga-u.ac.jp; URL: http://hs.phys.saga-u.ac.jp.]{}\
Introduction {#intro}
============
Spin structure of the nucleon has been investigated extensively for the last ten years, and now we have a rough idea on the internal spin structure. The experimental information comes mainly from inclusive lepton scattering experiments. Although there are polarized semi-inclusive data, they are not accurate enough to pose a strong constraint, for example, on the polarized flavor asymmetry $\Delta \bar u/\Delta \bar d$. However, it will be clarified experimentally in the near future by $W$ production experiments at RHIC and also semi-inclusive measurements by the COMPASS collaboration.
Most theoretical papers on the unpolarized asymmetry $\bar u/\bar d$ are written after the NMC discovery on Gottfried-sum-rule violation. Therefore, the unmeasured $\Delta \bar u/\Delta \bar d$ is an appropriate quantity for testing various theoretical models. In this sense, it is desirable to present model predictions before the experimental data will be taken. There are already some model predictions by Pauli-exclusion, chiral-soliton, and meson-cloud models. The meson models are successful in explaining the unpolarized asymmetry,[@skpr] so that we try to investigate the details of the model in the polarized asymmetry.[@fs]$^-$[@cw] The following discussions are based on Ref. 4.
$\rho$ meson contributions {#summary}
==========================
We explain the outline of the formalism for calculating $\rho$ meson contributions to the flavor asymmetric distribution $\Delta \bar u - \Delta \bar d$. The polarized nucleon splits into a $\rho$ meson and a baryon, then the virtual photon interacts with the polarized $\rho$ meson. The $\rho$ meson is a spin-1 hadron, and $\rho^+$ or $\rho^-$ has difference between $\bar u$ and $\bar d$. Therefore, it affects the polarized flavor asymmetry $\Delta \bar u - \Delta \bar d$ in the nucleon.
The contribution to the nucleon tensor $W_{\mu \nu}$ from the splitting process into a vector meson $V$ and a baryon $B$ is expressed as[@km] $$W_{\mu\nu} (p_N, s_N, q) = \int \frac{d^3 p_B}{(2\pi)^3}
\, \frac{2 m_V m_B}{E_B}
\sum_{\lambda_V,\lambda_B} | J_{VNB} |^2
\, W_{\mu\nu}^{(V)} (k, s_V, q) .
\label{eqn:w-meson}$$ Here, $m_V$ and $m_B$ are the meson and baryon masses, $p_N$, $p_B$, $k$, and $q$ are the nucleon, baryon, meson, and virtual photon momenta, $s_N$ and $s_V$ are the nucleon and meson spins, $J_{VNB}$ is the $VNB$ vertex multiplied by the meson propagator, and $W_{\mu\nu}^{(V)}$ is the meson tensor. Polarized structure functions $g_1$ and $g_2$ are defined in the antisymmetric part of the nucleon tensor: $$W^A_{\mu \nu} (p_N, s_N, q) =
i \, \varepsilon_{\mu\nu\rho\sigma} \, q^{\, \rho}
\bigg [ \, s_N^{\, \sigma} \, \frac{g_1}{p_N \cdot q}
+ ( p_N \cdot q \, s_N^{\, \sigma} - s_N \cdot q \, p_N^{\, \sigma} )
\, \frac {g_2}{(p_N \cdot q)^2} \,
\bigg ] .
\label{eqn:g1g2}$$ In order to separate $g_1$ from $g_2$, the projection operator $$P^{\mu \nu} = - \frac{m_N^2}{2 \, p_N \cdot q}
\, i \, \varepsilon^{\mu\nu\alpha\beta} \, q_\alpha \, s_{N \, \beta} ,$$ is multiplied in both sides of Eq. (\[eqn:g1g2\]), then longitudinal and transverse polarizations are considered. The derivation is too lengthy to be written here, so that the details should be found in Ref. 4. As a result, we obtain $$\begin{aligned}
g_1 (x, Q^2) = \frac{1}{1+\gamma^2} \int_x^1 \frac{dy}{y}
\, \big [ \, & \big \{ \Delta f_{1L} (y) + \Delta f_{1T} (y) \big \}
\, g_1^{V} (x/y, Q^2)
\nonumber \\
- & \big \{ \Delta f_{2L} (y) + \Delta f_{2T} (y) \big \}
\, g_2^{V} (x/y, Q^2)
\, \big ] ,
\label{eqn:g1v}\end{aligned}$$ where the function $\Delta f_i^{VN}(y)$ with $i$=$1L$, $2L$, $1T$, or $2T$ is defined by $$\Delta f_i (y) = f_i^{\lambda_V=+1} (y)
- f_i^{\lambda_V=-1} (y) .
\label{eqn:dfy}$$ The factor $\gamma^2$ is defined by $\gamma^2 = 4 \, x^2 \, m_N^2 /Q^2$. The function $f_{1L}^{\lambda_V} (y)$ is the ordinary meson momentum distribution with the momentum fraction $y$ in the longitudinally polarized nucleon. There are, however, new contributions from the $2L$, $1T$, and $2T$ terms. The function $f_{1T}^{\lambda_V} (y)$ is the distribution in the transversely polarized nucleon. The functions $f_{2L}^{\lambda_V} (y)$ and $f_{2T}^{\lambda_V} (y)$ are the distributions associated with $g_2$ of the vector meson. Explicit expressions of $f_i^{\lambda_V}$ are given in the appendix of Ref. 4. Our studies are intended to investigate a role of the additional terms, $2L$, $1T$, and $2T$.
In order to estimate the $g_2^V$ effects, we approximate it by the Wandzura-Wilczek (WW) relation: $$g_2^{V (WW)} (x,Q^2)
= - g_1^V(x,Q^2) + \int_x^1 \frac{dy}{y} g_1^V(y,Q^2) .$$ Then, the leading-order expression of $g_1^V$ is used for obtaining $$g_2^{V (WW)} (x,Q^2) =
\frac{1}{2} \sum_i \, e_i^2 \, [ \, \Delta q_i^{V (WW)} (x,Q^2)
+ \Delta \bar q_i^{V (WW)} (x,Q^2) \, ] ,
\label{eqn:ww}$$ where the WW distributions are defined by $$\Delta \bar q_i^{V (WW)} (x,Q^2)
= - \Delta \bar q_i^V (x,Q^2)
+ \int_x^1 \frac{dy}{y}
\, \Delta \bar q_i^V (y,Q^2) ,$$ and the same equation for $\Delta q_i^{V (WW)} (x,Q^2)$. In this way, the vector-meson contribution to $\Delta \bar q_i$ in the nucleon becomes $$\begin{aligned}
\Delta \bar q_i^{VNB} (x, Q^2) = \frac{1}{1+\gamma^2} \int_x^1 \frac{dy}{y}
\, \big [ \, & \big \{ \Delta f_{1L} (y) + \Delta f_{1T} (y) \big \}
\, \Delta \bar q_i^V (x/y, Q^2)
\nonumber \\
- & \big \{ \Delta f_{2L} (y) + \Delta f_{2T} (y) \big \}
\, \Delta \bar q_i^{V (WW)} (x/y, Q^2)
\, \big ] .
\label{eqn:dPi}\end{aligned}$$ This equation is used for calculating the $\rho$ meson contributions to $\Delta \bar u - \Delta \bar d$. The new terms $2L$, $1T$, and $2T$, are proportional to $\gamma^2$, namely $1/Q^2$, so that they vanish in the limit $Q^2 \rightarrow \infty$. Then, Eq. (\[eqn:dPi\]) agrees on those in Refs. 2 and 3.
Results
=======
For calculating the obtained expression numerically, the splitting processes $N \rightarrow \rho N$ and $N \rightarrow \rho \Delta$ are included with the vertex couplings $$\begin{aligned}
V_{VNN} & = \widetilde \phi_V^* \cdot \widetilde T \,
F_{VNN} (k) \, \,
\overline u_{N'} \, \epsi^{\mu \, *}
\, \bigg[ \, g_V \gamma_\mu
- \frac{f_V}{2 m_N} \, i \, \sigma_{\mu\nu}
\khat^\nu \, \bigg ] \, u_N ,
\label{eqn:vvnn}
\\
V_{V N \Delta} & = \widetilde \phi_V^* \cdot \widetilde T \,
F_{VN \Delta} (k) \, \,
\overline U_{\Delta,\nu}
\, \frac{f_{VN\Delta}}{m_V} \, \gamma_5 \, \gamma_\mu
\, \big [ \, \khat^\mu \, \epsi^{\nu \, *}
- \khat^\nu \, \epsi^{\mu \, *} \, \big ] \, u_N .
\label{eqn:vvnd}\end{aligned}$$ Here, $\widetilde \phi_V^* \cdot \widetilde T$ indicates the isospin coupling, $F_{VNN} (k)$ and $F_{VN \Delta} (k)$ are form factors, $u_N$ is the Dirac spinor, $U_\Delta^\mu$ is the Rarita-Schwinger spinor, $\epsi^\mu$ is the polarization vector of $\rho$, $g_V$, $f_V$, and $f_{VN\Delta}$ are coupling constants, and $\khat^\mu$ is a vertex momentum. The momentum $\khat^\mu$ could be taken either (A) $(E_V, \vec k)$ or (B) $(E_N - E_B, \vec k)$; however, the prescription (B) is used in the following results. For $F_{VNN}$ and $F_{VN \Delta}$, exponential form factors are used with the 1 GeV cutoff. Using these vertices, we calculate the polarized meson momentum distributions in Eq. (\[eqn:dPi\]).
Obtained meson momentum distributions are convoluted with the polarized distributions in $\rho$. The charge symmetry is used for relating the valence quark distributions in $\rho^-$, $\rho^0$, and $\rho^+$: $$(\Delta \bar u)_{\rho^-}^{val}
= (\Delta \bar d)_{\rho^+}^{val}
= 2 (\Delta \bar u)_{\rho^0}^{val}
= 2 (\Delta \bar d)_{\rho^0}^{val}
= \Delta V_\rho .$$ Actual parton distributions are not known in $\rho$, so that they are assumed as $\Delta V_\rho = 0.6 \, V_\pi$ by considering a lattice QCD estimate. The distribution in the pion is taken from the GRS (Glück, Reya, and Schienbein) parametrization. Taking into account the isospin factors at the $\rho NN$ and $\rho N \Delta$ vertices, we obtain $$\begin{aligned}
(\Delta \bar u - \Delta \bar d )_{p \rightarrow \rho B}
= & \bigg [ \, -2 \, \Delta f^{\rho NN}_{1L+1T}
+ \frac{2}{3} \, \Delta f^{\rho N \Delta}_{1L+1T} \, \bigg ]
\otimes \Delta V_\rho
\nonumber \\
- & \bigg [ \, -2 \, \Delta f^{\rho NN}_{2L+2T}
+ \frac{2}{3} \, \Delta f^{\rho N \Delta}_{2L+2T} \, \bigg ]
\otimes \Delta V_\rho^{WW}
\, ,
\label{eqn:ubdb}\end{aligned}$$ where $\otimes$ indicates the convolution integral.
The $\rho NN$ and $\rho N \Delta$ are separately calculated and numerical results are shown in Figs. \[fig:rhonn\] and \[fig:rhond\], respectively, at $Q^2$=1 GeV$^2$. The ordinary longitudinal contributions are denoted as $1L$, and other new contributions are denoted as $2L$, $1T$, and $2T$. Among the new terms, $2L$ is the largest one, which becomes comparable magnitude with $1L$ at medium $x$ ($x>0.2$). We notice that the distributions from $\rho N \Delta$ are fairly small in comparison with those from $\rho NN$. All the distributions in these figures are mainly negative, which means that $\Delta \bar d$ excess is produced over $\Delta \bar u$ by the meson-cloud mechanism. Because $W$ production and semi-inclusive processes will be investigated experimentally, our prediction should be tested in the near future.
Summary
=======
The $\rho$ meson contributions to the polarized antiquark flavor asymmetry $\Delta \bar u - \Delta \bar d$ have been investigated in a meson-cloud picture. We pointed out especially the existence of additional terms from the $g_2$ part of $\rho$. The additional terms become important at medium $x$ with small $Q^2$. The obtained $\Delta \bar u - \Delta \bar d$ distributions are mostly negative, namely the model indicates $\Delta \bar d$ excess over $\Delta \bar u$, and it should be tested by future experiments.
Acknowledgments {#acknowledgments .unnumbered}
===============
S.K. was supported by the Grant-in-Aid for Scientific Research from the Japanese Ministry of Education, Culture, Sports, Science, and Technology. He also thanks his collaborator, M. Miyama, for discussions.
[0]{} S. Kumano, Phys. Rep. [**303**]{}, 183 (1998). R. J. Fries and A. Schäfer, Phys. Lett. [**B443**]{}, 40 (1998); hep-ph/9805509 (v3). F.-G. Cao and A. I. Signal, Eur. Phys. J. [**C21**]{}, 105 (2001). S. Kumano and M. Miyama, hep-ph/0110097 (Phys. Rev. [**D65**]{} in press). F.-G. Cao, M. Wakamatsu, talks at this conference.
|
---
abstract: 'Recently, it is increasingly popular to equip mobile RGB cameras with Time-of-Flight (ToF) sensors for active depth sensing. However, for off-the-shelf ToF sensors, one must tackle two problems in order to obtain high-quality depth with respect to the RGB camera, namely 1) online calibration and alignment; and 2) complicated error correction for ToF depth sensing. In this work, we propose a framework for jointly alignment and refinement via deep learning. First, a cross-modal optical flow between the RGB image and the ToF amplitude image is estimated for alignment. The aligned depth is then refined via an improved kernel predicting network that performs kernel normalization and applies the bias prior to the dynamic convolution. To enrich our data for end-to-end training, we have also synthesized a dataset using tools from computer graphics. Experimental results demonstrate the effectiveness of our approach, achieving state-of-the-art for ToF refinement.'
author:
- |
Di Qiu^1,2^[^1] Jiahao Pang^1^[ ^fnsymbol[1]{}^]{} Wenxiu Sun^1^ Chengxi Yang^1^\
^1^ SenseTime Research ^2^ The Chinese University of Hong Kong\
[sylvesterqiu@gmail.com, jpang@connect.ust.hk, {sunwenxiu,yangchengxi}@sensetime.com ]{}
bibliography:
- 'ref.bib'
title: 'Deep End-to-End Alignment and Refinement for Time-of-Flight RGB-D Module'
---
[ inter inter]{} [ widthdef widthdef]{}
Introduction {#sec:intro}
============
Related Work {#sec:related}
============
Alignment and Refinement {#sec:framework}
========================
Datasets and Augmentation {#sec:data}
=========================
Experimentation {#sec:exp}
===============
Conclusion {#sec:conclude}
==========
More Details on Framework {#sec:framework}
=========================
More Details on Data Generation and Pre-processing {#sec:data}
==================================================
More Experimental Results {#sec:results}
=========================
[^1]: Both authors contributed equally. Jiahao Pang is the corresponding author, this work was done while he was with SenseTime.
|
---
abstract: |
The world is becoming more interconnected every day. With the high technological evolution and the increasing deployment of it in our society, scenarios based on the Internet of Things (IoT) can be considered a reality nowadays. However, and before some predictions become true (around 75 billion devices are expected to be interconnected in the next few years), many efforts must be carried out in terms of scalability and security.
In this study we propose and evaluate a new approach based on the incorporation of Blockchain into current IoT scenarios. The main contributions of this study are as follows: *i)* an in-depth analysis of the different possibilities for the integration of Blockchain into IoT scenarios, focusing on the limited processing capabilities and storage space of most IoT devices, and the economic cost and performance of current Blockchain technologies; *ii)* a new method based on a novel module named BIoT Gateway that allows both unidirectional and bidirectional communications with IoT devices on real scenarios, allowing to exchange any kind of data; and *iii)* the proposed method has been fully implemented and validated on two different real-life IoT scenarios, extracting very interesting findings in terms of economic cost and execution time. The source code of our implementation is publicly available in the Ethereum testnet.
author:
- |
Oscar Delgado-Mohatar, Ruben Tolosana, Julian Fierrez and Aythami Morales\
School of Engineering, Universidad Autonoma de Madrid, Spain\
[{oscar.delgado, ruben.tolosana, julian.fierrez, aythami.morales}@uam.es]{}
bibliography:
- 'biblio.bib'
- 'blockchain\_iot.bib'
- 'blockchain\_biblio.bib'
title: |
Blockchain in the Internet of Things:\
Architectures and Implementation
---
Blockchain, Internet of Things, Security, Privacy
Introduction
============
The Internet of Things (IoT) has the potential to change the world, just as the Internet did [@ashton2009internet]. This term is referred to the set of objects, sensors, and everyday items that are equipped with computing capability and network connectivity to send/receive data through the Internet[@atzori2010internet]. As a result, IoT devices can generate and manage an autonomous ecosystem without any human intervention or supervision.
Scenarios based on the IoT can already be considered a reality nowadays, for example: *i)* smart homes where the electric light, heating, and kitchen equipment such as the fridge or washing machine are automatically operating, reporting continuosly to the User/Client, and *ii)* autonomous electric vehicles searching for a charging station so that as soon as the car is running out of battery, it automatically drives to the cheapest or nearest point, and starts the charging process. Once completed, the car conducts the payment[@Wachenfeld2016]. These are just some of the many applications of the IoT. With the increasing evolution and deployment of the technology in our lives, it is estimated that between 50 and 75 billion devices will be interconnected by 2025 [@statista2020; @ceo_ericcson].
Undoubtedly, and before this comes true, a lot of efforts must be carried out in order to manage such volume of information in a scalable and secure way. Many recent studies have focused on the IoT security [@atzori2010internet]. Additionally, key aspects of these low-cost IoT devices such as the limited processing capability and storage space must be further studied [@Morchon2019]. Also, most IoT devices do not usually include protection against physical attacks, so they can be compromised easily. This is exacerbated by the fact that IoT devices almost never have maintenance/upgrade capabilities to reduce production costs.
In order to mitigate these problems, different approaches have been proposed in the literature such as lightweight cryptography [@Lee2014; @Dutta2019], reinforcement of the perimeter security through the use of firewalls [@Gupta2017; @Sari2019], and zero-trust approaches [@Samaniego2018; @Luca2018]. Furthermore, recent studies have put their eyes on other technologies such as Blockchain to overcome some of the limitations existing in IoT scenarios [@dai2019blockchain].
Blockchain is essentially a decentralized public ledger of all data and transactions that have ever been executed in the system [@swan2015blockchain]. These transactions are recorded in blocks that are created and added to the Blockchain in a linear, chronological order (immutable). Each participating node in the network has the task of relaying transactions, and has a copy of the Blockchain. Other nodes, called miners, are also in charge of validating transactions, performing an expensive computational process, for which they are economically rewarded.
Blockchain was originally created and applied as an auxiliar technology of Bitcoin [@Nakamoto], providing a secure record of the economic transactions between users of the system. Nevertheless, a Blockchain could store any kind of digital information, providing its certification and guaranteeing its authenticity and integrity. As a result, from its origin up to now, Blockchain has been deployed in many different scenarios such as: biometrics, certification of documentation, mortgages, securities and any other official documents, assets and intelligent objects that can make decisions based on the information stored in the Blockchain, distributed markets without central authority, deposit and custody services that can resolve disputes between customers and merchants, savings accounts, voting systems, and improvements in the distribution chain for all kind of products [@swan2015blockchain; @REYNA2018173; @2019_ICBA_BlockchainBiometrics; @buchmann2017enhancing].
{width="\textwidth"}
However, and despite these opportunities, the current Blockchain technology suffers from some potential limitations that must be carefully studied and characterized before the adequate integration of Blockchain into IoT scenarios.
In this study we propose and evaluate a new method for incorporating Blockchain into current IoT scenarios. As a first approximation, Blockchain technology could provide IoT scenarios with some desirable properties such as immutability, accountability, availability, and universal access. These properties enabled by Blockchain may be very useful for the IoT, among other things, to improve security [@Huynh2019], transaction reliability [@Gervais2018], transparency [@REYNA2018173] or privacy levels [@Gervais2018].
The main contributions of this study can be summarized as follows:
- An in-depth analysis of the different possibilities for the integration of Blockchain into current IoT scenarios, focusing on: *i)* the limited processing capabilities and storage space of most IoT devices, and *ii)* the economic cost and performance of using a Blockchain.
- A novel method based on a new module named BIoT Gateway that allows both unidirectional and bidirectional communications with IoT devices on real scenarios, allowing to exchange any kind of data.
- The proposed method has been implemented and validated on two different real-life IoT scenarios, extracting very interesting findings in terms of economic cost and execution time. The source code of our implementation is publicly available in the Ethereum testnet[^1], and can be verified using explorers such as Etherscan.
The remainder of the paper is organized as follows. Sec. \[proposedApproach\] describes our proposed methods for the incorporation of Blockchain into IoT scenarios. Sec. \[sec:experimentalSetup\] describes all details of our experimental setup in order to validate our proposed approach on practical scenarios. Sec. \[experimentalResults\] describes the experimental results achieved using our proposed approach. Finally, Sec. \[conclusions\] draws the final conclusions and points out some future research lines.
Proposed Methods {#proposedApproach}
================
Fig. \[fig:configurations\] shows a graphical representation of our two proposed architectures. They comprise four main modules: User/Client, Blockchain, our proposed BIoT Gateway, and the IoT Device. These two proposed architectures differ in which module (Blockchain or BIoT Gateway) is reached first from the User/Client: *i)* Client-Blockchain-Gateway (CBG), and *ii)* Client-Gateway-Blockchain (CGB). It is important to highlight that our proposed architectures allow unidirectional and bidirectional communications between both the User/Client and the IoT Device, expanding their use to many different practical scenarios. We now describe in detail each module involved in the architecture and also each proposed configuration.
Modules {#architecture}
-------
### **User/Client** {#client}
It can take any form, typically a web or mobile application that serves as the interface with the final user, e.g., to turn on/off a smart light. This Client App is typically in charge of sending commands to the IoT devices, and receiving responses or readings from them. In our proposed architectures, the User/Client App can communicate with the IoT Device through both the BIoT Gateway or the Blockchain, depending on the selected configuration (Sec. \[configurations\]).
### **Blockchain** {#Blockchain}
We assume a Blockchain capable of storing data and running code through smart contracts, a well-known concept inside the cryptographic community [@Szabo1996]. A smart contract is, essentially, a piece of code executed in a secure environment that controls digital assets. This concept has not been popular until its inclusion in the Ethereum Blockchain platform [@Dannen2017]. In essence, Ethereum could be seen as a distributed computer, with capability to execute programs written in Turing-complete, high-level programming languages.
In our proposed method, we have developed a smart contract in order to enable a secure two-way communication, in which the integrity of the exchanged data is guaranteed by the smart contract and the underlying Blockchain. As a result, we keep track of the incoming and outgoing messages between the User/Client and the IoT Device, and also control the devices registered in the BIoT Gateway to prevent unauthorized use.
Essentially, two operations are implemented in the smart contract to read and store the data coming from or going to the IoT Device. Data can have any meaning, such as control commands, readings from the sensors or any other type of digital data. All the supported functions needed to perform the read and store operations are described in Table \[tab:smart-contract-functions\], including a description of each of them, input/output arguments, permission restriction, and which module should call them.
### **BIoT Gateway** {#gateway}
One of the main modules of our proposed architectures is the novel BIoT Gateway. It acts as an interface between all the modules of the architecture, allowing three possible communications: *i)* the interface with the User/Client, usually through an API REST (web/mobile application); *ii)* the interface with the Blockchain (smart contract); and *iii)* the interface with the IoT Device module, for example, a MQTT broker or any other protocol such as HTTPS, as long as it provides a secure communication, and the authentication of both ends. It is important to remark that the inclusion of the BIoT Gateway breaks the intrinsic distributed nature of the Blockchain. Nevertheless, this approach remains valid and secure in most scenarios. In fact, if the same entity manages both the BIoT Gateway and the IoT Device, they form a logical unit from the point of view of trust, so the security obtained would be equivalent to place the Client directly on the Device.
### **IoT Device** {#IoT_device}
The last module of the proposed architecture is the IoT Device. It is important to remark that no especial hardware or software capabilities of the IoT Device are needed when considering our proposed method, making it feasible for any low-cost device. This is one of the main advantages of our proposed methods.
Configurations
--------------
### **Client-Blockchain-Gateway (CBG)** {#CBG}
The User/Client first communicates with the Blockchain module, which broadcasts an event that is captured by the BIoT Gateway, and then moved forward to the IoT Device. Later on, the command/data is sent back to the BIoT Gateway from the IoT Device, storing the command/data into the Blockchain. Finally, it is sent back to the User/Client.
This is the most secure configuration as the User/Client can always prove that one message has been sent, even if the BIoT Gateway refuses to process it (non-repudiation). In return, a considerable delay can be introduced under some circumstances. Therefore, this configuration is more suitable for scenarios without strong latency requirements such as in the certification of information.
### **Client-Gateway-Blockchain (CGB)** {#CGB}
In this case, the communication between the User/Client and the IoT Device occurs first through the BIoT Gateway. Once the command/data is received by the BIoT Gateway, it is moved forward to the IoT Device, and optionally, to the Blockchain to assure its integrity as soon as possible. Later on, the command/data is first sent back to the BIoT Gateway and then to the User/Client and optionally, again to the Blockchain, to assure its integrity as soon as possible.
This approach reduces the overhead due to the optional use of the Blockchain, but in return, security can be affected. Therefore, this configuration should only be considered in scenarios where real-time communications are needed, such as a smart-home scenario as the security of the messages exchanged is in theory not critical.
Experimental Setup {#sec:experimentalSetup}
==================
IoT Scenarios {#IoTScenarios}
-------------
Our proposed method has been implemented and validated in a real environment. Two different IoT scenarios are recreated, considering both unidirectional and bidirectional configurations:
- **Refrigerated container**: the IoT Device periodically sends its temperature and other metadata (24bytes) to the BIoT Gateway, once per minute. A unidirectional communication is considered in this scenario.
- **Smart light**: the IoT Device can receive and send simple commands (20 per day) to turn on/off the light (24 bytes), simulated with a LED matrix. A bidirectional communication is considered in this scenario.
Implementation Details {#ImplementationDetails}
----------------------
For the implementation of each scenario, the following details are considered in our experiments:
- **User/Client**: a personal computer is used to receive and send commands to the IoT Device.
- **Blockchain**: a smart contract[^2] has been developed in Solidity language, and deployed in Ethereum Ropsten testnet[^3]. This platform is functionally identical to the main platform, but allows development and testing of applications without economic cost.
- **BIoT Gateway**: it has been implemented using a Raspberry Pi 4 [@RaspberryPi4], running the official Ethereum client (Geth), and connected to the Ropsten testnet in light mode.
- **IoT Device**: we consider the Wemos D1 mini [@wemos], a popular low-cost microcontroller based on the ESP8266 platform, connected to a temperature sensor and a LED matrix. All the elements are operated with a battery.
In our laboratory setup, both the BIoT Gateway and the IoT Device are connected through the MQTT protocol, although, as stated before, any other secure protocol may be used. Of course, this protocol must allow the mutual authentication of both elements, because they form a single “logical unit” in terms of trust. In our case, both BIoT Gateway and IoT Device are issued with a x509 certificate by a common CA. The fingerprint of the certificate of the BIoT Gateway is hard-coded (certificate pinning), to avoid man-in-the-middle (MITM) attacks.
Blockchain Storage Schemes {#BlockchainStorage}
--------------------------
One of the main potential limitations for the integration of both IoT and Blockchain technologies is the economic cost of running an IoT system (totally or partially) in Blockchain. It is therefore crucial to properly estimate and minimize the cost that, to a large extent, is due to the storage of data. In our experimental framework we analyze the three different storage schemes proposed in [@delgado2019biometric], which can be ordered in terms of complexity (from lower to higher), and economic cost (from higher lo lower) as follows:
- **Full on-chain storage**: all data is stored, as-is, in the Blockchain.
- **Data hashing**: the Blockchain only stores a hash of the data that guarantees its immutability. The data itself is stored off-chain in another system: distributed (e.g., IPFS [@Benet2014]), cloud or even existing traditional databases.
- **Merkle trees**: data is also stored off-chain, but it is preprocessed by constructing a Merkle tree structure, which reduces storage costs and increases the bandwidth.
These alternatives are discussed in terms of economic cost and execution time in the next section.
Experimental Results {#experimentalResults}
====================
Our proposed method has been evaluated according to the following measures:
- **Economic Cost**: associated to the use of Blockchain, due to the data storage and smart contract execution. Both CBG and CGB configurations are analyzed.
- **Performance**: associated to the execution time of the smart contract. Transmission times are not included as they are negligible with respect to the smart contract.
Table \[tab:results\] shows the economic costs and performance of each Blockchain storage scheme (i.e., full on-chain, data hashing, and Merkle trees) and architecture configuration (i.e., CBG and CGB). We also include in the last two rows of the table the economic cost and performance results obtained in our two IoT scenarios studied (i.e., refrigerated container and smart light). The economic cost is described in terms of units of gas and US dollars at the time of writing (January, 2020). Performance is described in terms of seconds.
### **Economic Cost**
We first analyze how the different Blockchain storage schemes affect the feasibility of our proposed approach in terms of economic cost. In general, the results depicted in Table \[tab:results\] remark that Merkle trees seem to be the only viable Blockchain storage scheme. The remaining storage schemes would quickly become prohibitive for the volume of data typically exchanged in a real environment. We now analyze in detail each Blockchain alternative.
The full on-chain or direct storage scheme is specially expensive (e.g., \$11.52 per day for the refrigerated container scenario) as all data is stored, as-is, in the Blockchain. The reason of this high economic cost is due to the pricing storage in Blockchain, which is intentionally discouraged to minimize its uncontrolled growth. For example, protecting the security of one million messages with this approach would cost between \$8,756 and \$22,565 for messages of 8 and 128 bytes, respectively. Depending on the final scenario and the value of the protected information, this could be reasonable but, in general, these figures are not affordable for general purpose applications.
The next storage scheme considered is data hashing. This approach slightly improves the economic cost figures compared with the full on-chain storage scheme, but only when the size of the messages is bigger than 32 bytes (for smaller messages the direct storage is still cheaper). Despite the improvement, this storage scheme is still prohibitive in terms of economic cost (e.g., \$17.28 per day for the refrigerated container scenario).
The last storage scheme studied is Merkle trees. In this case, all messages and data received during a period of time are grouped under a single tree. As a result, only the root of the tree must be secured in the Blockchain. Therefore, an arbitrarily large volume of data can be secured at the cost of only 256 bits, having a fixed cost of \$0.0122 per day and IoT Device (in our experimental setup), being this storage scheme the only one viable in real environments. The exact duration of this period (window) of time in which data is grouped under a single tree must be determined taking into account the volume of messages processed, their importance, and the cost that can be assumed. Depending on these parameters, the duration of the window can range from a few minutes to several hours or days. Finally, despite the economic cost advantages of this storage scheme, it is important to remark that if the BIoT Gateway, for whatever reason, is lost or corrupted before the root of the tree can be secured in the Blockchain, then the security of the previous messages is lost. In addition, this scheme slightly complicates the verification of data integrity, since, apart from the message itself, it is necessary to save a cryptographic proof that allows its reconstruction.
Finally, and although the economic cost of this last approach is very low for a single IoT Device (\$0.0122), it could be too high in a realistic IoT environment composed of potentially millions of devices. To mitigate this aspect, a Merkle tree meta-structure could be generated by aggregating the corresponding trees to the individual BIoT Gateways. This way, it would be possible to authenticate and process an arbitrarily large volume of data and messages at a very low fixed cost.
### **Performance**
We now analyze how the architecture configuration selected (i.e., CBG and CGB) affects the feasibility of our proposed approach in terms of execution time (last column of Table \[tab:results\]).
In general, the experiments show that this hybrid approach based on our proposed BIoT Gateway is also viable. As can be seen, the execution time is slightly higher than 10 seconds for the *sendMessageToDevice()* operation, which is used in both CBG and CGB configurations.
This time delay could be acceptable or not depending on the final IoT scenario. For the refrigerated container scenario considered in our experimental setup (unidirectional communication), where the container periodically sends its temperature to the BIoT Gateway, it seems feasible a time delay between 10 and 15 seconds as no hard time constrains are needed in this specific scenario. Therefore, the CBG configuration should be chosen to increase the security.
For the smart light scenario considered (bidirectional communication), in which the IoT Device can receive and send simple commands to turn on/off the light, time delay seems much more sensitive due to usability reasons. Therefore, in this specific scenario, the CGB configuration should be considered as it does not add virtually delays. However, the messages exchanged are not secured in the Blockchain until the end of the window period configured for the system.
Finally, the message retrieval operation, i.e., *receiveMessagesFromDevice()*, is free of charge, as it is a read-only operation and does not include/modify anything of the Blockchain. Furthermore, this operation can be considered immediate in terms of execution time, since the request is processed by the local Ethereum node, and does not reach the network.
Conclusions
===========
In this study we have explored new architectures for incorporating Blockchain into current IoT scenarios. In particular, we have proposed new methods based on a novel module named BIoT Gateway that allows both unidirectional and bidirectional communications with IoT devices on real scenarios, allowing to exchange any kind of data.
Our proposed methods have been fully implemented and validated on two different real-life IoT scenarios: *i)* a refrigerated container with a unidirectional communication, and *ii)* a smart light with a bidirectional communication. Also, three different Blockchain storage schemes are evaluated in order to minimize the economic cost of data storage.
The results achieved prove that straightforward schemes such as the direct storage of the IoT templates on-chain, or direct data hashing, are not feasible for practical IoT scenarios. Nevertheless, when the Merkle tree scheme is included as an intermediate data structure, the economic cost is significantly reduced and also fixed regardless of the volume of data to store. Regarding the performance, times between 10-20 seconds are obtained for store operations whereas for read operations, they are usually free of cost and very fast to run as they are processed locally. These figures prove the viability of our proposed approach on current IoT scenarios, overcoming some limitations of most IoT devices such as the limited processing capabilities and storage space.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work has been supported by projects: PRIMA (H2020-MSCA-ITN-2019-860315), TRESPASS-ETN (H2020-MSCA-ITN-2019-860813), BIBECA (RTI2018-101248-B-I00 MINECO/FEDER), and COPCIS (TIN2017-84844-C2-1-R MINECO/FEDER). Ruben Tolosana is supported by Consejería de Educación, Juventud y Deporte de la Comunidad de Madrid y Fondo Social Europeo.
[^1]: 0x89f04bFE1c8dbbdbA7c2A7b7815A4A3b229989f8
[^2]: It is a basic contract that should be considered only for research purposes.
[^3]: 0x89f04bFE1c8dbbdbA7c2A7b7815A4A3b229989f8
|
[**Quasi-Fuchsian Surfaces In Hyperbolic Link Complements**]{}
Joseph D. Masters and Xingru Zhang
[**Abstract.**]{} We show that every hyperbolic link complement contains closed quasi-Fuchsian surfaces. As a consequence, we obtain the result that on a hyperbolic link complement, if we remove from each cusp of the manifold a certain finite set of slopes, then all remaining Dehn fillings on the link complement yield manifolds with closed immersed incompressible surfaces.
Introduction
============
By a [*link complement*]{} we mean, in this paper, the complement of a link in a closed connected orientable $3$-manifold. A link complement is said to be hyperbolic if it admits a complete hyperbolic metric of finite volume. By a *surface* we mean, in this paper, the complement of a finite (possibly empty) set of points in the interior of a compact, orientable $2$-manifold (which may not be connected). By a *surface in a $3$-manifold* $W$, we mean a continuous, proper map $f:S \ra W$ from a surface $S$ into $W$. A surface $f:S\ra W$ in a 3-manifold $W$ is said to be *connected* if and only if $S$ is connected. A surface $f:S\ra W$ in a 3-manifold $W$ is said to be *incompressible* if each component $S_j$ of $S$ is not a $2$-sphere and the induced homomorphism $f^*: \pi_1(S_j, s)\ra \pi_1(W, f(s))$ is injective for any choice of base point $s$ in $S_j$. A surface $f:S\ra W$ in a 3-manifold $W$ is said to be [*essential*]{} if it is incompressible and for each component $S_j$ of $S$, the map $f: S_j\ra W$ cannot be properly homotoped into a boundary component or an end component of $W$.
Connected essential surfaces in hyperbolic link complements can be divided into three mutually exclusive geometric types: quasi-Fuchsian surfaces, geometrically infinite surfaces, and essential surfaces with accidental parabolics. Geometrically these three types of surfaces can be characterized by their limit sets as follows: a connected essential surface $f:S\ra M$ in a hyperbolic link complement $M$ is Quasi-Fuchsian if and only if the limit set of the subgroup $f^*(\pi_1(S))\subset \pi_1(M)$ is a Jordon circle in the boundary $2$-sphere of the hyperbolic $3$-space $\mathbb H^3$; is geometrically infinite if and only if the limit set of $f^*(\pi_1(S))$ is the whole $2$-sphere; and is having accidental parabolics otherwise. Topologically these three types of surfaces can be characterized as follows: a connected essential surface $f:S\ra M$ in a hyperbolic link complement $M$ is geometrically infinite if and only if it can be lifted (up to homotopy) to a fiber in some finite cover of $M$; is having accidental parabolics if and only $S$ contains a closed curve which cannot be freely homotoped in $S$ into a cusp of $S$ but can be freely homotoped in $M$ into a cusp of $M$, and is quasi-Fuchsian otherwise.
In [@MZ] it was shown that every hyperbolic knot complement contains closed quasi-Fuchsian surfaces. In this paper we extend this result to hyperbolic link complements.
\[main\] Every hyperbolic link complement contains closed quasi-Fuchsian surfaces.
This yields directly the following consequence.
\[cor\] For every given hyperbolic link complement $M$, if we remove certain finitely many slopes from each cusp of $M$, then all remaining Dehn fillings produce manifolds which contain closed incompressible surfaces.
We note that Corollary \[cor\] would also be a consequence of Khan and Markovic’s recent claim that every closed hyperbolic 3-manifold contains a surface subgroup.
This paper is an extension of [@MZ] where the existence of closed quasi-Fuchsian surfaces in any hyperbolic knot complement was proved. The proof of Theorem \[main\] follows essentially the approach given in [@MZ]. To avoid repetition, we shall assume the reader is familiar with the machinery laid out in [@MZ]. In particular we shall use most of the terms and properties about hyperbolic $3$-manifolds and about groups established in [@MZ], without recalling them in detail, and shall omit details of constructions and proof of assertions whenever they are natural generalization of counterparts of [@MZ].
To help the reader to get a general idea about which parts of our early arguments are needed to be adjusted nontrivially, we first very briefly recall how a closed quasi-Fuchsian surface was constructed in a hyperbolic knot complement. We started with a pair of connected bounded embedded quasi-Fuchsian surfaces in a given hyperbolic knot exterior $M^-$ (which is truncation of a hyperbolic knot complement $M$) with distinct boundary slopes in $\p M^-$. We then considered two hyperbolic convex $I$-bundles resulting from the two corresponding quasi-Fuchsian surface groups. By a careful “convex gluing” of two suitable finite covers of some truncated versions of the two $I$-bundles, and then “capping off convexly” by a solid cusp, we constructed a convex hyperbolic $3$-manifold $Y$ with a local isometry $f$ into the given hyperbolic knot complement $M$. The manifold $Y$ had non-empty boundary each component of which provided a closed quasi-Fuchsian surface in $M$ under the map $f$. To find the required finite covers of the truncated $I$-bundles and at the same time to lift certain immersions to embeddings, we needed a stronger version of subgroup separability property for surface groups with boundary, which was proved using Stallings’ folding graph techniques.
Now to extend the construction to work for hyperbolic link complements, we first need to prove, for any given hyperbolic link exterior $M^-$, the existence of two properly embedded bounded quasi-Fuchsian surfaces $S_i^-$, $i=1,2$, in $M^-$, each of which is not necessarily connected, with the crucial property that for each of $i=1,2$ and each component $T_j$ of $\p M^-$, $S_i^-\cap T_j$ is a non-empty set of simple closed essential curves, and furthermore the slope of the curves $S_1^-\cap T_j$ is different from that of $S_2^-\cap T_j$ for each $T_j$. The proof of this result, given in Section \[bs\], is based on work of Culler-Shalen [@CS] and Cooper-Long [@CL] and Thurston [@T], making use of the $SL_2(\c)$ character variety of the link exterior $M^-$ and some special properties of essential surfaces in hyperbolic $3$-manifolds with accidental parabolics. With the given two surfaces $S_i^-$, $i=1,2$, we may construct two corresponding convex $I$-bundles (in the current situation each $I$-bundle may not be connected). Following the approach in [@MZ] we still want to choose a suitable cover for each component of each of the truncated $I$-bundles, and “convexly glue” them all together in certain way and “convexly cap off” with $m$ (which is the number of components of $\p M^-$) solid cusps, to form a convex hyperbolic $3$-manifold $Y$, with a local isometry $f$ into $M$, such that the boundary of $Y$ is a non-empty set, each component of which is mapped by $f$ to a closed quasi-Fuchsian surface in $M$.
As before, we want to choose the cover so that the boundary components of $S_i^-$ unwrap as much as possible. And if the two surfaces $S_i^-$ are connected, our previous arguments go through with very little change. However, if the surfaces $S_i^-$ are disconnected, complications arise. In order to piece the different covers together, we need to know that they all have the same degree. And this turns out to require a non-trivial strengthening of our previous separability result; see Theorem \[each large n\]. The proof of this property uses a careful refinement of the folding graph arguments used in [@MZ].
Cusped qausi-Fuchsian surfaces in hyperbolic link complements {#bs}
=============================================================
From now on let $M$ be a given hyperbolic link complement of $m\geq 2$ cusps. For each of $i=1,...,m$, let $C_i$ be a fixed $i$-th cusp of $M$ which is geometric, embedded and small enough so that $C_1,...,C_m$ are mutually disjoint. The complement of the interior of $C_1\cup...\cup C_m$ in $M$, which we denote by $M^-$, is a compact, connected and orientable $3$-manifold whose boundary is a set of $m$ tori. We call $M^-$ a *truncation* of $M$. Let $T_k=\p C_k$, $k=1,...,m$. Then $\p
M^-=T_1\cup...\cup T_m$.
\[slopes\] There are two embedded essential quasi-Fuchsian surfaces $S_1$ and $S_2$ in $M$ (each $S_i$ may not be connected) such that for each of $i=1,2$ and each of $k=1,...,m$, $S_i\cap T_k$ is a nonempty set of parallel simple closed essential curves in $T_k$ of slope $\lambda_{i,k}$ and $\lambda_{1,k}\ne \lambda_{2,k}$.
It is equivalent to show that the truncation $M^-$ of $M$ contains two properly embedded bounded essential surfaces $S_1^-$ and $S_2^-$ such that: (i) For each of $i=1,2$, each component of $S_i^-$ is not a fiber or semi-fiber of $M^-$. (ii) For each of $i=1,2$, any closed curve in $S_i^-$ that can be freely homotoped in $M^-$ into $\p M^-$ can also be freely homotoped in $S_i^-$ into $\p S_i^-$. (iii) For each of $i=1,2$ and each of $k=1,...,m$, $S_i^-$ has non-empty boundary on $T_k$ of boundary slope $\lambda_{i,k}$ and $\lambda_{1,k}\ne \lambda_{2,k}$.
Let $\{\g_k\subset T_k; k=1,...,m\}$ be any given set of $n$ slopes. By [@CS Theorem 3], there is a properly embedded essential surface $S_1^-$ (maybe disconnected) in $M^-$ with the following properties (in fact the surface $S_1^-$ is obtained through a nontrivial group action on a simplicial tree associated to an ideal point of a curve in a component of the $SL(2,\c)$-character variety of $M^-$ which contains the character of a discrete faithful representation of $\pi_1(M^-)$):
\(1) No component of $S_1^-$ is a fiber or semi-fiber of $M^-$.
\(2) For each of $k=1,...,m$, $S_1^-$ has non-empty boundary on $T_k$ of boundary slope $\lambda_{1,k}$ which is different from $\g_k$.
\(3) If an element of $\pi_1(M^-)$ is freely homotopic to a curve in $M^-\setminus S_1^-$, then it is contained in a vertex stabilizer of the action on the tree.
\(4) If an element of $\pi_1(M^-)$ is freely homotopic to $\g_k$, then it is not contained in any vertex stabilizer of the action on the tree and thus must intersect $S_1^-$.
\(5) If an element of $\pi_1(M^-)$ is freely homotopic to a curve in $S_1^-$, then it is contained in an edge stabilizer of the tree. It follows that
\(6) If an element of $\pi_1(M^-)$ is freely homotopic to a simple closed essential curve in $T_k$ whose slope is different from $\lambda_{1,k}$, then it is not contained in any vertex stabilizer of the action on the tree.
Let $S^-_{1,j}$, $j=1,...,n_1$, be the components of $S^-_1$. If some $S^-_{1,j}$ has a closed curve which cannot be freely homotoped in $S_1^-$ into $\p S_1^-$ but can be freely homotoped in $M^-$ into $\p M$, then arguing as in [@CL Lemma 2.1], we see that there is an embedded annulus $A$ in $M^-\setminus S^-_{1,j}$ such that one boundary component, denoted $a_1$, of $A$ lies in $S_{1,j}^-$ and is not boundary parallel in $S^-_{1,j}$, and the other boundary component, denoted $a_2$, of $A$ is contained in some boundary component $T_k$ of $M^-$. By Properties (5) and (6) listed above, we have
\(7) $a_2\subset T_k$ must have the slope $\lambda_{1,k}$.
Now consider in $A$ the intersection set $A\cap (S^-_1-S^-_{1,j})$ of $A$ with other components of $S_1^-$. By Property (7), we may assume that $\p A\cap
(S^-_1-S^-_{1,j})=\emptyset$. Thus by proper isotopy of $(S_1^--S^-_{1,j}, \p
(S^-_1-S^-_{1,j}))\subset (M^-,\p M^-)$ and surgery (if necessary) we may assume that each component of $A\cap (S^-_1-S^-_{1,j})$ is a circle which is isotopic in $A$ to the center circle of $A$ and if the component is contained in $S^-_{1,j'}$, then it is not boundary parallel in $S^-_{1,j'}$. So the component of $A\cap (S^-_1-S^-_{1,j})$, denoted $a_1'$, which is closest to $a_2$ in $A$, cuts out from $A$ an sub-annulus $A'$ which is properly embedded in $M^-\setminus S^-_1$ such that $a_1'$ lies in $S^-_{1,j'}$, for some $j'$, and is not boundary parallel in $S^-_{1,j'}$. So we may perform the annulus compression on $S^-_{1,j'}$ along $A'$ to get an essential surface which still satisfies the properties (1)-(6) above (because the new resulting surface can be considered as a subsurface of the old surface $S_1^-$ and because of property (7)) but has larger Euler characteristic. Thus such annulus compression must terminate in a finite number of times. So eventually we end up with a surface, which we still denote by $S_1^-$, satisfying the condition
\(8) Any closed curve in $S_1^-$ that can be freely homotoped in $M^-$ into $\p M$ can be freely homotoped in $S_1^-$ into $\p S_1^-$.
Now letting $\g_k=\lambda_{1,k}, k=1,...,m$, and repeating the above arguments, we may get another properly embedded essential surface $S_2^-$ such that
(1’) Each component of $S_2^-$ is not a fiber or semi-fiber of $M^-$.
(2’) For each of $k=1,...,m$, $S_2^-$ has non-empty boundary on $T_k$ of boundary slope $\lambda_{2,k}$ which is different from $\lambda_{1,k}$.
(8’) Any closed curve in $S_2^-$ that can be freely homotoped in $M^-$ into $\p M$ can be freely homotoped in $S_2^-$ into $\p S_2^-$.
So $S_1^-$ and $S_2^-$ satisfy conditions (i), (ii) and (iii) listed above. The lemma is thus proved.
Let $S_i,i=1,2$ be the two surfaces provided by Lemma \[slopes\]. By taking disjoint parallel copies of some components of $S_i$ (if necessary), we may and shall assume
\[at least two\] [For each $i=1,2$ and $k=1,...,m$, $S_i\cap T_k$ has a positive, even number of components]{}.
\[dik\] [Let $S_{i,j}$, $j=1,...,n_i$, be components of $S_i$, $i=1,2$. Let $i_*$ be the number such that $\{i,i_*\}=\{1,2\}$ for $i=1,2$. Let $S_i^-=S_i\cap M^-$ and let $\p_k
S_{i,j}^-$ be the boundary components of $S^-_{i,j}$ on $T_k$ (which may be empty for some $j$’s) and let $\p_k S_i^-=\cup_j \p_k S_{i,j}^-$. Now for each $i=1,2, k=1,...,m$, let $d_{i,k}$ be the geometric intersection number in $T_k$ between a component of $\p_k S_i^-$ and the whole set $\p_k S_{i_*}$. Obviously $d_{i, k}$ is independent of the choice of the component of $\p_k S_i^-$. By Condition \[at least two\], $d_{i,k}\geq 2$ is even for each $i,k$. Now set $$d_i=lcm\{d_{i,k}; \;k=1,...,m\},$$ the (positive) least common multiple. Then $d_i\geq 2$ is even for each $i=1,2$.]{}
Let $\mathbb H^3$ be the hyperbolic $3$-space in the upper half space model, let $S_{\infty}^2$ be the $2$-sphere at $\infty$ of $\mathbb H^3$ and let $\overline{\mathbb H}^3 =\mathbb H^3\cup S_{\infty}^2$.
By Mostow-Prasad rigidity, the fundamental group of $M$ (for any fixed choice of base point) can be uniquely identified as a discrete torsion free subgroup $\G$ of $Isom^+(\mathbb H^3)$ up to conjugation in $Isom(\mathbb H^3)$ so that $M=\mathbb H^3/\G$. We shall fix one such identification. Let $p:\mathbb
H^3\ra M$ be the corresponding covering map.
For the given surface $S_{i,j}$ in $M$ (for each $i, j$), we identify its fundamental group with a quasi-Fuchsian subgroup $\G_{i,j}$ of $\G$ as follows. As $S_{i, j}$ is embedded in $M$ we may consider it as a submanifold of $M$. Fix a component $\tilde S_{i,j}$ of $p^{-1}(S_{i,j})$ (topologically $\tilde S_{i,j}$ is an open disk in $\mathbb H^3$), there is a subgroup $\G_{i,j}$ in the stabilizer of $\tilde S_{i,j}$ in $\G$ such that $S_{i,j}=\tilde S_{i,j}/\G_{i,j}$.
Note that the limit set $\L_{i,j}$ of $\G_{i,j}$ is a Jordan circle in the $2$-sphere $S^2_{\infty}$ at the $\infty$ of $\mathbb H^3$. Let $H_{i,j}$ be the convex hull of $\L_{i,j}$ in $\mathbb H^3$.
Let ${\cal B}_k=p^{-1}(C_k)$, $k=1,...,m$, and ${\cal B}=p^{-1}(C)$. Then by our assumption on $C$, ${\cal B}$ is a set of mutually disjoint horoballs in $\mathbb H^3$. Let $B$ be a component of ${\cal B}$ and let $\p B$ be the frontier of $B$ in $\mathbb H^3$. Then $\p B$ with the induced metric is isometric to a Euclidean plane. We shall simply call $\p B$ a Euclidean plane. A strip between two parallel Euclidean lines in $\p B$ will be called a Euclidean strip in $\p B$. Note that every Euclidean line in $\p B$ bounds a totally geodesic half plane in $B$ (which is perpendicular to $\p B$). By a [*$3$-dimensional strip region*]{} in $B$ we mean a region in $B$ between two totally geodesic half planes in $B$ bounded by two parallel disjoint Euclidean lines in $\p B$.
\[strip\] If the cusp set $C=C_1\cup ...\cup C_m$ of $M$ is small enough, then for each component $B$ of ${\cal B}$ whose point at $\infty$ is a parabolic fixed point of $\G_{i,j}$, $H_{i,j}\cap B$ is a $3$-dimensional strip region in $B$.
The proof is similar to that of [@MZ Lemma 5.2].
From now on we assume that $C$ has been chosen so that Lemma \[strip\] holds for all $i=1,2,
j=1,...,n_i$. For a fixed small $\e>0$, let $X_{i,j}$ be the $\e$-collared neighborhood of $H_{i,j}$ in $\mathbb H^3$. Then it follows from Lemma \[strip\] that for each component $B$ of ${\cal B}$ whose point at $\infty$ is a parabolic fixed point of $\G_{i,j}$,, $X_{i,j}\cap B$ is a $3$-dimensional strip region in $B$, for all $i=1,2, j=1,..., n_i$, by geometrically shrinking $C$ further if necessary.
Note that $X_{i,j}$ is a metrically complete and strictly convex hyperbolic $3$-submanifold of $\mathbb{H}^3$ with $C^1$ boundary, invariant under the action of $\G_{i,j}$. Let $${\cal
B}_{i,j}=\{X_{i,j}\cap B; \mbox{$B$ a component of $ {\cal B}$ based at a
parabolic fixed point of $\G_{i,j}$}\}.$$ We call ${\cal B}_{i,j}$ the *horoball region* of $X_{i,j}$. Let $X_{i,j}^-=X_{i,j}\setminus {\cal B}_{i,j}$, and call $X_{i,j}^-\cap \p {\cal B}_{i,j}$ the *parabolic boundary* of $X_{i,j}^-$, denoted by $\p_p X_{i,j}^-$. Note that $ X_{i,j}^-$ is locally convex everywhere except on its parabolic boundary.
Each of $X_{i,j}$, ${\cal B}_{i,j}$, $X_{i,j}^-$ and $\p_p X_{i,j}^-$ is invariant under the action of $\G_{i,j}$. Let $Y_{i,j}=X_{i,j}/\G_{i,j}$, which is a metrically complete and strictly convex hyperbolic $3$-manifold with boundary. Topologically $Y_{i,j}=S_{i,j}\times I$, where $I = [-1, 1]$. There is a local isometry $f_{i,j}$ of $Y_{i,j}$ into $M$, which is induced from the covering map $\mathbb H^3/\G_{i,j}\;\lra \;M$ by restriction on $Y_{i,j}$, since $Y_{i,j}=X_{i,j}/\G_{i,j}$ is a submanifold of $\mathbb H^3/\G_{i,j}$. Also $p|_{X_{i,j}}=f_{i,j}\circ p_{i,j}$, where $p_{i,j}$ is the universal covering map $X_{i,j}\ra Y_{i,j}=X_{i,j}/\G_{i,j}$. Let $Y_{i,j}^-=X_{i,j}^-/\G_{i,j}$, let ${\cal C}_{i,j}={\cal B}_{i,j}/\G_{i,j}$, and let $\p_p Y_{i,j}^-=\p_p X_{i,j}^-/\G_{i,j}$. We call ${\cal C}_{i,j}$ the cusp part of $Y_{i,j}$, and call $\p_pY_{i,j}^-$ the parabolic boundary of $Y_{i,j}^-$, which is the frontier of $Y_{i,j}^-$ in $Y_{i,j}$ and is also the frontier of ${\cal C}_{i,j}$ in $Y_{i,j}$. Each component of $\p_p Y_{i,j}^-$ is a Euclidean annulus. The manifold $Y_{i,j}^-$ is locally convex everywhere except on its parabolic boundary. Topologically $Y_{i,j}^-=S_{i,j}^-\times I$.
As in [@MZ Section 5], we fix a product $I$-bundle structure for $Y_{i,j}=S_{i,j}\times I$ such that each component of ${\cal C}_{i,j}$ has the induced $I$-bundle structure which is the product of a totally geodesic cusp annulus and the $I$-fiber (i.e. we assume that $(S_{i,j}\times \{0\})\cap {\cal C}_{i,j}$ is a set of totally geodesic cusp annuli). We let every free cover of $Y_{i,j}$ have the induced $I$-bundle structure. In particular $X_{i,j}$ has the induced $I$-bundle structure from that of $Y_{i,j}$, and this structure is preserved by the action of $\G_{i,j}$; i.e. every element of $\G_{i,j}$ sends an $I$-fiber of $X_{i,j}$ to an $I$-fiber of $X_{i,j}$. Similar to [@MZ Corollary 5.6], we have
\[length\] For each of $i=1,2, j=1,...,n_i$, there is an upper bound for the lengths of the $I$-fibers of $X_{i,j}$.
The restriction of the map $f_{i,j}$ on the center surface $S_{i,j}\times \{0\}$ of $Y_{i,j}=S_{i,j}\times I$ may not be an embedding in general but by Lemma \[strip\] we may and shall assume that the map is an embedding when restricted on $(S_{i,j}\times \{0\})\cap {\cal
C}_{i,j}$. We now replace our original embedded surface $S_{i,j}$ by the center surface $f_{i,j}:S_{i,j}\times\{0\}$ and we simply denote $S_{i,j}\times \{0\}$ by $S_{i,j}$.
The restriction map $f_{i,j}:(Y_{i,j}^-, \p_p Y_{i,j}^-)\ra (M^-,\p M^-)$ is a proper map of pairs and $f_{i,j}:
(S_{i,j}^-,\p S_{i,j}^-)\ra (M^-, \p M^-)$ is a proper map which is an embedding on $\p S_{i,j}^-$ (This property will remain valid if we shrink the cusp $C$ of $M$ geometrically). In fact $f_{i,j}(\p S_{i,j}^-)$ are embedded Euclidean circles in $\p M^-$. Hence boundary slopes of the new quasi-Fuchsian surfaces $f_{i,j}:
(S_{i,j}^-,\p S_{i,j}^-)\ra (M^-, \p M^-)$ are defined and are the same as those of the original embedded surfaces $S_{i,j}^-$.
\[still defined\] [As $f_{i, j}: \p S_{i, j}^-\ra \p M^-$ is an embedding, we sometimes simply consider $\p S_{i, j}^-$ as subset of $\p M^-$, for each $i=1,2,
j=1,..., n_i$. By choosing a slightly different center surface for $Y_{i, j}$ (if necessary), we may assume that the components of $\{\p S_{i, j}^-, j=1,..., n_i\}$ are mutually disjoint in $\p M^-$, for each fixed $i=1,2$. So the numbers $d_{i, k}$, $d_i$ defined in Notation \[dik\] remain well defined for the current surface $f_{i, j}:(S_{i, j}^-, \p S_{i, j}^-)\ra (M, \p M)$ and are the same numbers as given there, for all $i, j$. Also $\p_k S_{i, j}^-$ remain defined as before for all $i, j$. ]{}
Let $\tilde S_{i, j}$ and $\tilde S_{i,j}^-$ be the corresponding center surfaces of $X_{i,j}$ and $X_{i, j}^-$ respectively.
Note also that if a component of $\p S_{i,j}^-$ intersects a component of $\p S_{i_*,j'}^-$ in some component $T_k$ of $\p M^-$, then they intersect geometrically in $T_k$, and their intersection points in $T_k$ are one-to-one corresponding to the geodesic rays of $f_{i,j}(S_{i,j})\cap f_{i_*,j'}(S_{i_*,j'})\cap C_k$.
We fix an orientation for $S_{i,j}$, and let $S_{i,j}^-$ and $\p S_{i,j}^-$ have the induced orientation.
Construction of intersection pieces $K_{i,j}$
=============================================
Suppose that $\p S_{i,j}^-$ intersects $\p S_{i_*,j'}^-$ for some $j,j'$. We construct the “intersection pieces” $K_{i, j,j'}$ and $K_{i_*,j',j}$ between $Y_{i,j}$ and $Y_{i_*,j'}$ in a similar fashion as in [@MZ Section 6] such that (1) $K_{i, j,j'}$ and $K_{i_*,j',j}$ are isometric. (2) Each component of $K_{i, j,j'}$ or of $K_{i_*,j',j}$ is a metrically complete convex hyperbolic $3$-manifold. (3) There are local isometries $g_{i,j,j'}: K_{i,j,j'}\ra Y_{i,j}$ and $g_{i_*,j',j}:K_{i_*,j',j}\ra Y_{i_*,j'}$. (4) $K^-_{i,j,j'}$ and $K^-_{i_*,j',j}$ (which are the truncated versions of $K_{i, j,j'}$ and $K_{i_*,j',j}$ respectively) are compact. (5) Each component of the parabolic boundary $\p_p K_{i,j,j'}^-$ of $K_{i,j,j'}^-$ is a Euclidean parallelogram, the number of cusp ends of $K_{i,j,j'}$ is precisely the number of intersection points between $f_{i,j}(\p S_{i,j}^-)$ and $f_{i_*,j'}(\p S_{i_*,j'}^-)$. Similar properties hold for $K_{i_*,j',j}$. (6) The restriction of $g_{i,j,j'}$ to $K_{i,j,j'}\setminus K_{i,j,j'}^-$ is an embedding and so is the restriction of $g_{i_*,j',j}$ to $K_{i_*,j',j}\setminus K_{i_*,j',j}^-$. (7) $f_{i,j}(g_{i,j,j'}(K_{i,j,j'}\setminus K_{i,j,j'}^-))$ contains $f_{i,j}(S_{i,j})\cap f_{i_*,j'}(S_{i_*,j'})
\cap C$ (the latter is a set of geodesic rays) and so does $f_{i_*,j'}(g_{i_*,j',j}(K_{i_*,j',j}\setminus K_{i_*,j',j}^-))$.
Let $K_{i,j}$ be the disjoint union of these $K_{i,j,j'}$ over such $j'$. Then the number of components of $\p_p K_{i,j}^-$ is precisely the number of intersection points between $\p S_{i,j}^-$ and $\p
S_{i_*}^-$. In fact there is a canonical one-to-one correspondence between components of $\p_p K_{i,j}^-$ and the intersection points between $\p S_{i,j}^-$ and $\p S_{i_*}^-$.
Let $K_{i}$ be the disjoint union of these $K_{i,j}$. Then the number of cusp ends of $K_{i}$ is precisely the number of intersection points between $\p S_{i}^-$ and $\p S_{i_*}$ and there is an isometry between $K_1$ and $K_2$.
Construction of $J_{i,j}$, $J_{i,j}^-$ , $\hat J_{i,j}$ and $C_n(J_{i,j}^-)$
============================================================================
As in [@MZ Section 6], we fix a number $R>0$ bigger than the number $R(\e)$ provided in [@MZ Proposition 4.5] and also bigger than the upper bound provided by Lemma \[length\] for the lengths of $I$-fibers of $X_{i,j}$ (for each of $i=1,2$, $j=1,...,n_i$). As in [@MZ Section 6], we define and construct the [*abstract $R$-collared neighborhood of $K_{i,j}$ with respect to $X_{i,j}$*]{} which is denoted by $AN_{(R,X_{i,j})}(K_{i,j})$. Also define the truncated version $(AN_{(R,X_{i,j})}(K_{i,j}))^-$, the parabolic boundary $\p_p (AN_{(R,X_{i,j})}(K_{i,j}))^-$ and the cups region $AN_{(R,X_{i,j})}(K_{i,j})\setminus(AN_{(R,X_{i,j})}(K_{i,j}))^-$ accordingly.
Now as in [@MZ Section 7], we construct a connected metrically complete, convex, hyperbolic $3$-manifold $J_{i,j}$ with a local isometry $g_{i,j}:J_{i,j}\ra Y_{i,j}$ such that $J_{i,j}$ contains $AN_{(R,X_{i,j})}(K_{i,j})$ as a hyperbolic submanifold, and $J_{i,j}\setminus AN_{(R,X_{i,j})}(K_{i,j})$ is a compact $3$-manifold $W_{i,j}$ (which may not be connected). Also $W_{i,j}$ is disjoint from $AN_{(R,X_{i,j})}(K_{i,j})\setminus
(AN_{(R,X_{i,j})}(K_{i,j}))^-$, the parabolic boundary $\p_p J_{i,j}^-$ of $J_{i,j}^-$ is equal to the parabolic boundary of $(AN_{(R,X_{i,j})}(K_{i,j}))^-$, and $g_{i,j}|: (J_{i,j}^-, \p_p J_{i,j}^-)\ra
(Y_{i,j}^-, \p_pY_{i,j}^-)$ is a proper map of pairs.
Each component of $\p_p J_{i,j}^-$ is a Euclidean parallelogram and thus can be capped off by a convex $3$-ball. Let $\hat J_{i,j}$ be the resulting manifold after capping off all components of $\p_p J_{i,j}^-$. Then $\hat J_{i,j}$ is a connected, compact, convex $3$-manifold with a local isometry (which we still denote by $g_{i,j}$) into $Y_{i,j}$.
The number of components of $\p_p J_{i,j}^-$ is equal to the number of components of $\p_p K_{i,j}^-$, and the former is an abstract $R$-collared neighborhood of the latter with respect to $\p_p X_{i,j}^-$.
[The components of $\p_p J_{i,j}^-$ are canonically one-to-one correspond to the intersection points of $\p S_{i, j}^-$ with $\p S_{i_*}^-$.]{}
Now as in [@MZ Section 8], we construct, for each sufficiently large integer $n$, a connected, compact, convex, hyperbolic $3$-manifold $C_n(J_{i,j}^-)$ with a local isometry (still denoted as $g_{i,j}$) into $Y_{i,j}$ such that $C_n(J_{i,j}^-)$ contains $J_{i,j}^-$ as a hyperbolic submanifold. The manifold $C_n(J_{i,j}^-)$ is obtained by gluing together $J_{i,j}^-$ with $n_{i,j}$ “multi-$1$-handles” $H_{i,j,a}(n), a=1,...,n_{i,j}$, along the attaching region $\p_p J_{i,j}^-$, where $n_{i,j}$ is the number of components of $\p S_{i,j}^-$. But there is a subtle difference from the construction of [@MZ Section 8] in choosing “the wrapping numbers” of the handles $H_{i,j,a}(n)$.
\[diff lengths\] [If $\b$ is a component of $\p S_{i,j}^-$ which lies in the component $T_k$ of $\p M$, then the multi-$1$-handle associated to it, say the $a$-th one $H_{i,j,a}(n)$, will have “wrapping number” $\frac{n d_i}{d_{i,k}}$ (instead of $n$ given in [@MZ Section 8]), where $d_{i,k}$ and $d_i$ were defined in Notation \[dik\].]{}
Finding the right covers {#lifting}
========================
Recall the definitions of $n_i$ and $d_i$ given in Notation \[dik\]. The main task of this section is to prove the following
\[each large n\]Given $S_{i,j}^-$, there is a positive even integer $N_{i,j}$ such that for each even integer $N_*\geq N_{i,j}$, we have (1) $S_{i,j}^-$ has an $$m_i=N_* d_i+1$$ fold cover $\breve S_{i,j}^-$ with $|\p \breve S_{i,j}^-|=|\p S_{i,j}^-|$ (i.e. each component of $\p
\breve S_{i,j}^-$ is an $m_i$-fold cyclic cover of a component of $\p S_{i,j}^-$). So equivalently each $Y_{i,j}^-$ has an $$m_i=N_* d_i+1$$ fold cover $\breve Y_{i,j}^-$ with $|\p_p \breve Y_{i,j}^-|=|\p_p Y_{i,j}^-|$ (i.e. each component of $\p_p \breve Y_{i,j}^-$ is an $m_i$-fold cyclic cover of a component of $\p_p Y_{i,j}^-$). (2) The map $g_{i,j}:J_{i,j}^-\ra Y_{i,j}^-$ lifts to an embedding $\breve g_{i,j}:J_{i,j}^-\ra \breve
Y_{i,j}^-$ and if $\tilde A$ is a component of $\p_p \breve Y_{i,j}^- $, then components of $\breve
g_{i,j}(\p_p J_{i,j}^-)\cap \breve A$ are evenly spaced along $\breve A$. More precisely if $\breve
\b$ is the component of $\p \breve S_{i,j}^-$ corresponding to $\breve A$, covering a component $\b$ of $\p S_{i,j}^-$ in $T_k$, then the topological center points of $\tilde
g_{i,j}(\p_p J_{i,j}^-)\cap \breve A$ divide $\breve \b$ into arc components each with wrapping number $N_*\frac{d_i}{d_{i,k}}$.
Of course in Theorem \[each large n\], the cover $\tilde S_{i,j}^-$ and the number $m_i$ depend on $N_*$. For simplicity, we suppressed this dependence in notation for $\tilde S_{i,j}^-$ and $m_i$. Similar suppressed notations will occur also in other places later in the paper when there is no danger of causing confusion, and we shall not remark on this all the time.
For the definition of the wrapping number see Definition \[wrapping\].
\[indep of j\]There is a positive even integer $N_0$ such that for each even integer $N_*\geq N_0$ and for each $i=1,2, j=1,...,n_i$, we have (1) $S_{i,j}^-$ has an $$m_i=N_* d_i+1$$ fold cover $\breve S_{i,j}^-$ with $|\p \breve S_{i,j}^-|=|\p S_{i,j}^-|$. So equivalently each $Y_{i,j}^-$ has an $$m_i=N_* d_i+1$$ fold cover $\breve Y_{i,j}^-$ with $|\p_p \breve Y_{i,j}^-|=|\p_p Y_{i,j}^-|$. (2) The map $g_{i,j}:J_{i,j}^-\ra Y_{i,j}^-$ lifts to an embedding $\breve g_{i,j}:J_{i,j}^-\ra \breve
Y_{i,j}^-$ and if $\tilde A$ is a component of $\p_p \breve Y_{i,j}^- $, then components of $\breve
g_{i,j}(\p_p J_{i,j}^-)\cap \breve A$ are evenly spaced along $\breve A$. More precisely if $\breve
\b$ is the component of $\p \breve S_{i,j}^-$ corresponding to $\breve A$, covering a component $\b$ of $\p S_{i,j}^-$ in $T_k$, then the topological center points of $\tilde
g_{i,j}(\p_p J_{i,j}^-)\cap \breve A$ divide $\breve \b$ into arc components each with wrapping number $N_*\frac{d_i}{d_{i,k}}$.
Apply Theorem \[each large n\] and let $N_0=max\{N_{i,j}; i=1,2,j=1,...,n_i\}$.
Corollary \[indep of j\] is to say that the number $N_*$ and thus the number $m_i$ are independent of the second index $j$ in $S_{i,j}^-$.
For notational simplicity, we shall only consider the following two cases in proving Theorem \[each large n\]:
[**Case 1**]{}. A given surface $S_{i,j}^-$ has $b_1$ boundary components $\{\b_{1,p}, p=1,...,
b_1\}$ on $T_1$ and $b_2$ boundary components $\{\b_{2,p}, p=1,...,b_2\}$ on $T_2$, and is disjoint from $T_3,...,T_m$. So $n_{i,j}=b_1+b_2$ which is the number of components of $\p S_{i,j}^-$.
[**Case 2**]{}. A given surface $S_{i,j}^-$ has only one boundary component $\{\b\}$ on $T_1$ and is disjoint from $T_2,...,T_m$. So $n_{i,j}=1$, which is the number of components of $\p
S_{i,j}^-$.
The reader will see that the proof of Theorem \[each large n\] for a general surface $S_{i,j}^-$ will be very similar to either case 1 or 2, depending on whether $S_{i,j}^-$ has multiple boundary components, or just a single one.
[**Proof of Theorem \[each large n\] in Case 1**]{}.
Again to avoid too complicated notations on indices, in the following we shall suppress the indices $i,j$ for some items depending on them, when there is no danger of causing confusion.
Recall that $\p S_{i,j}^-$ have the induced orientation from the orientation of $S_{i,j}^-$. Let $\b_{k,p}, p=1,...,b_k$ be the oriented boundary components of $\p S_{i,j}$ in $T_k$ for each $k=1,2$. Recall the number $d_{i,k}$ given in Notation \[dik\]. We list the set of intersection points of $\p S_{i,j}^-$ with $\p S_{i_*}^-$ as $t_{k,p,q}$, $k=1,2$, $p=1,...,b_k$ and $q=1,...,d_{i,k}$, so that $t_{k,p,q}, q=1,...,d_{i,k}$, appear consecutively along $\b_{k,p}$ following its orientation. We choose $t_{1,1,1}$ as the base point for $\pi_1(S_{i,j}^-)=\pi_1(Y_{i,j}^-)=\pi_1(S_{i,j})=\pi_1(Y_{i,j})$.
\[n even\] [[**From now on in this paper we assume that $n$ is a positive even number**]{}]{}
Recall that there is a local isometry $g_{i,j}:C_n(J_{i,j}^-)\ra Y_{i,j}$ which is a one-to-one map when restricted to the set of center points of $\p_p J_{i,j}^-$. We list these center points as $b_{k,p,q}$ so that $t_{k,p,q}=g_{i,j}(b_{k,p,q})$ for all $k,p,q$. We choose $b_{1,1,1}$ as the base point for each of $J_{i,j}$, $J_{i,j}^-$, $\hat J_{i,j}$ and $C_n(J_{i,j}^-)$.
Similar to the definition given in [@MZ p2144], we have
\[wrapping\][Suppose that $\breve p:\breve{\b}_{k,p}\ra
\b_{k,p}$ is a covering map, and let $\breve \b_{k,p}$ have the orientation induced from that of $\b_{k,p}$. Let $\a\subset \tilde \b_{k,p}$ be an embedded, connected, compact arc with the orientation induced from that of $\breve \b_{k,p}$, whose initial point is in $\breve p^{-1}(t_{k,p,q})$ and whose terminal point is in $\breve p^{-1}(t_{k,p,q+1})$ (here $q+1$ is defined mod $d_{i,k}$). We say that $\a$ has [*wrapping number*]{} $n$ if there are exactly $n$ distinct points of $\breve p_i^{-1}(t_{k,p,q})$ which are contained in the interior of $\a$.]{}
\[generators\] [Let $g$ be the genus of $S_{i,j}^-$. As in [@MZ Section 10], the group $\pi_1(S_{i,j}^-, t_{1,1,1})$ has a set of generators $$L=\{a_1,b_1,a_2, b_2, ...,a_g,b_g,
x_{1},x_2, ...,x_{n_{i,j}-1}\}$$ such that the elements $$x_{1},x_{2},...,x_{n_{i,j}-1},x_{n_{i,j}}=
[a_{1},b_{1}][a_{2},b_{2}]\cdots [a_{g},b_{g}]x_{1}x_{2}\cdots x_{n_{i,j}-1}$$ have representative loops, based at the point $t_{1,1,1}$, freely homotopic to the $n_{i,j}=b_1+b_2$ components $\b_{1,1},\b_{1,2},...,\b_{1,b_1},\b_{2,1},\b_{2,2},...,\b_{2,b_2}$ of $\p
S_{i,j}^-$ respectively. ]{}
As in [@MZ Section 10], we fix a generating set $$w_1,...,w_\ell$$ for $\pi_1(J_{i,j}^-,
b_{1,1,1})$ and choose a generating set $$w_1,...,w_\ell, z_{k,p,q}(n), \;k=1,2, p=1,..,b_k, q=1,...,d_{i,k}-1$$ for $\pi_1(C_n(J_{i,j}^-),b_{1,1,1})$ such that $$\pi_1(C_n(J_{i,j}^-),b_{1,1,1})=\pi_1(J_{i,j}^-, b_{1,1,1})*<z_{k,p,q}(n),
\;k=1,2, p=1,..,b_k, q=1,...,d_{i,k}-1>$$ where $*$ denotes the free product, and $<z_{k,p,q}(n),
\;k=1,2, p=1,..,b_k, q=1,...,d_{i,k}-1>$ is the free group freely generated by the $z_{k,p,q}(n)$’s.
Here are some necessary details of how $z_{k,p,q}(n)$ is defined, following [@MZ Section 10] but with different and simplified notations for indices. Let $\alpha_{k,p,q}\subset J_{i,j}^-$ be a fixed, oriented path from $b_{1,1,1}$ to $b_{k,p,q}$, for each of $k=1,2, p=1,...,b_k, q=1,...,d_{i,k}$ ($\a_{1,1,1}$ is the constant path). Recall the construction of $C_n(J_{i,j}^-)$ and Adjustment \[diff lengths\]. For $k=1,2$, $p=1,...,b_k$, let $H_{k,p}(n)$ denote the multi-1-handle of $C_n(J_{i,j}^-)$ corresponding to the component $\b_{k,p}$ of $\p S_{i,j}^-$. For $k=1,2$, $p=1,..., b_k$, $1 \leq q \leq d_{i,k}-1$, let $\delta_{k,p,q}(n)$ be the oriented geodesic arc in the multi-one-handle $H_{k,p}(n)\subset C_n(J_{i,j}^-)$ from the point $b_{k,p,q}$ to $b_{k,p,q+1}$. Then $$z_{k,p,q}(n)=
\alpha_{k,p,q}\cdot\delta_{k,p,q}(n)\cdot\overline{\alpha_{k,p,q+1}},$$ where the symbol “$\cdot$” denotes path concatenation (sometimes omitted), and $\overline{\alpha_{k,p,q+1}}$ denotes the reverse of $\alpha_{k,p,q+1}$. Also we always write path (in particular loop) concatenation from left to right.
As in [@MZ Section 10], if $\a$ is an oriented arc in $C_n(J_{i,j}^-)$, we use $\a^*$ to denote the oriented arc $g_{i,j}\circ\a$ in $Y_{i,j}$, and if $\g$ is an element in $\pi_1(C_n(J_{i,j}^-,
b_{1,1,1})$, we use $\g^*$ to denote the element $g_{i,j}^*(\g)$ where $g_{i,j}^*$ is the induced homomorphism $g_{i,j}^*: \pi_1( C_N(J_{i,j}^-), b_{1,1,1})\ra \pi_1(Y_{i,j}, t_{1,1,1})$.
The oriented path $\alpha_{k,p,q}^*$ in $Y_{i,j}^-$ runs from $t_{1,1,1}$ to $t_{k,p,q}$. For $k=1,2$, $p=1,...,b_k$, $1 \leq q \leq d_{i,k}-1$, let $\eta_{k,p,q}$ be the oriented subarc in $\beta_{k,p}$ from $t_{k,p,q}$ to $t_{k,p,q+1}$ following the orientation of $\b_{k,p}$, and let $\s_{k,p,q}\subset Y_{i,j}^-$ be the loop $\alpha_{k,p,q}^*\cdot\eta_{k,p,q}\cdot\overline{\alpha_{k,p,q+1}}^*$. Let $\s_{k,p,0}$ be the constant path based at $t_{1,1,1}$. Let $x_{b}^{\prime}$ be the loop $\alpha_{1,b,1}^*\cdot\beta_{1,b}\cdot\overline{\alpha_{1,b,1}}^*$ if $b=1,...,b_1$ and be the loop $\alpha_{2,b-b_1,1}^*\cdot\beta_{2,b-b_1}\cdot\overline{\alpha_{2,b-b_1,1}}^*$ if $b=b_1+1,..., b_1+b_2=n_{i,j}$, where $\b_{k,p}$ is considered as an oriented loop starting and ending at the point $t_{k,p,1}$. Similar to [@MZ Lemma 10.1], we have
\[diff powers\] \[lk\] Considered as an element in $\pi_1(Y_{i,j}, t_{1,1,1})=\pi_1(S_{i,j}, t_{1,1,1})
=\pi_1(S_{i,j}^-, t_{1,1,1})$, $$z_{k,p,q}(n)^* =
(\overline{\s_{k,p,q-1}}\cdots\overline{\s_{k,p,0}})
(x_{b_{k-1}+p}')^{n\frac{d_i}{d_{i,k}}}
(\s_{k,p,0}\cdots\s_{k,p,q}),$$ for each of $k=1,2, p=1,...,b_k, q=1,...,d_{i,k}-1$, where $b_0$ is defined to be $0$.
[The only real meaningful difference of this lemma from [@MZ Lemma 10.1] is that the power of $x_{b_{k-1}+p}'$ in the expression of $z_{k,p,q}(n)^*$ above depends on the indices $i$ and $k$ besides $n$, which is due to the Adjustment \[diff lengths\]. ]{}
Recall that $\hat J_{i,j}$ is a connected, compact, convex, hyperbolic 3-manifold obtained from $J_{i,j}^-$ by capping off each component of $\p_p
J_{i,j}^-$ with a compact, convex $3$-ball, and that $\pi_1(J_{i,j},
b_{1,1,1})=\pi_1(J_{i,j}^-,b_{1,1,1})=\pi_1(\hat J_{i,j},b_{1,1,1})$. Also, $\hat J_{i,j}$ is a submanifold of $C_n(J_{i,j}^-)$, so by [@MZ Lemma 4.2], $\pi_1(\hat J_{i,j},b_{1,1,1})$ can be considered as a subgroup of $\pi_1(C_n(J_{i,j}^-),b_{1,1,1})$. As $C_n(J_{i,j}^-)$ is a connected, compact, convex, hyperbolic 3-manifold, the induced homomorphism $g_{i,j}^*:\pi_1(C_n(J_{i,j}^-), b_{1,1,1})\ra \pi_1(Y_{i,j},
t_{1,1,1})
=\pi_1(S_{i,j}^-, t_{1,1,1})$ is injective by again [@MZ Lemma 4.2]. So $g_{i,j}^*(\pi_1(C_n(J_{i,j}^-), b_{1,1,1}))= g_{i,j}^*(\pi_1(J_{i,j}^-, b_{1,1,1}))
*<z_{k,p,q}(n),
\;k=1,2, p=1,..,b_k, q=1,...,d_{i,k}-1>$ is a subgroup of $\pi_1(Y_{i,j}, t_{1,1,1})=\pi_1(S_{i,j},
t_{1,1,1}) =\pi_1(S_{i,j}^-, t_{1,1,1})$.
By [@MZ Proposition 4.7] there is a set of elements $y_1,..., y_r$ in $$\pi_1(Y_{i,j}, t_{1,1,1}) -g_{i,j}^*(\pi_1(\hat J_{i,j},b_{1,1,1}))$$ such that, if $G$ is a finite index subgroup of $\pi_1(Y_{i,j}, t_{1,1,1})$ which separates $g_{i,j}^*(\pi_1(\hat J_{i,j},b_{1,1,1}))$ from $y_1,...,y_r$, then the local isometry $g_{i,j}:\hat J_{i,j}\ra Y_{i,j}$ lifts to an embedding $\breve g_{i,j}$ in the finite cover $\breve Y_{i,j}$ of $Y_{i,j}$ corresponding to $G$.
To prove Theorem \[each large n\] in Case 1, we just need to prove the following
\[large n\] There is a positive even integer $N_{i,j}$ such that for each even integer $N_*\geq N_{i,j}$, there is a finite index subgroup $G$ of $\pi_1(Y_{i,j}, t_{1,1,1})=\pi_1(S_{i,j}, t_{1,1,1}) =\pi_1(S_{i,j}^-, t_{1,1,1})$ such that (i) $G$ has index $m_i=N_*d_i+1$; (ii) $G$ contains the elements $w_{1}^*,...,w_{l}^*$, and thus contains the subgroup $g_{i,j}^*(\pi_1(\hat
J_{i,j},b_{1,1,1}))=g_{i,j}^*(\pi_1(\hat J_{i,j}^-,b_{1,1,1}))$; (iii) $G$ contains the elements $z_{k,p,q}(N_*)^*$, $k=1,2$, $p=1,...,b_k$, $q=1,...,d_{i,k}-1$; (iv) $G$ does not contain any of $x_b^d$, $b=1,...,n_{i,j}$, and $d=1,...,m_i-1$; (v) $G$ does not contain any of $y_1,...,y_r$.
Theorem \[each large n\] in Case 1 follows from Theorem \[large n\].
The proof is similar to that of [@MZ Proposition 11.1].
Let $\breve Y_{i,j}$ be the finite cover of $Y_{i,j}$ corresponding the subgroup $G$ provided by Theorem \[large n\], and let $\breve S_{i,j}$ be the corresponding center surface of $\breve Y_{i,j}$ covering $S_{i,j}$.
As noted earlier, Conditions (ii) and (v) of Theorem \[large n\] imply that the map $g_{i,j}:\hat J_{i,j}\ra Y_{i,j}$ lifts to an embedding $\breve g_{i,j}:\hat J_{i,j}\ra \breve Y_{i,j}$. Conditions (i) and (iv) of Theorem \[large n\] imply that $|\p \breve S_{i,j}^-|=|\p S_{i,j}|$. So part (1) of Theorem \[each large n\] holds in Case 1.
We may now let $\breve \b_{k,p}$ be the component of $\p \breve S_{i,j}^-$ covering $\b_{k,p}$ for each of $k=1,2, p=1,..., b_k$.
Conditions (ii) and (iii) of Theorem \[large n\] imply that the group $g_{i,j}^*(\pi_1(C_{N^*}(J_{i,j}^-), b_{1,1,1}))$ is contained in $G$. Therefore the map $g_{i,j}:(C_{N_*}(J_{i,j}^-),b_{1,1,1})\ra (Y_{i,j}, t_{1,1,1})$ lifts to a map $$\breve g_{i,j}:(C_{N_*}(J_{i,j}^-), b_{1,1,1})\ra
(\breve Y_{i,j}, \breve g_{i,j}(b_{1,1,1})).$$ Recall the notations established earlier. Consider the multi-1-handle $H_{k,p}(N_*) \subset C_{N_*}(J_{i,j}^-)$ containing the points $b_{k,p,q}, q=1,...,d_{i,k}$, and the geodesic arcs $\d_{k,p, q}(N_*)\subset H_{k,p}(N_*)$, $q=1,...,d_{i,k}-1$. By our construction the immersed arc $g_{i,j}:\d_{k,p,q}(N_*)\ra S_{i,j}$ is homotopic, with end points fixed, to the path in $\b_{k,p}$ which starts at the point $t_{k,p,q}$, wraps $N_*\frac{d_i}{d_{i,k}}$ times around $\b_{k,p}$ and then continues to the point $t_{k,p,q+1}$, following the orientation of $\b_{k,p}$. This latter (immersed) path lifts to an embedded arc in $\breve \b_{k,p}$ connecting $\breve g_{i, j}(b_{k,p,q})$ and $\breve g_{i, j}(b_{k,p,q+1})$, because $\breve \b_{k,p}$ is an $N_*d_i+1$-fold cyclic cover of $\b_{k,p}$. This shows that part (2) of Theorem \[each large n\] holds in Case 1.
Theorem \[large n\] is proved using the technique of folded graphs. We shall follow as closely as possible the approach used in [@MZ] and we assume the terminologies used there concerning $L$-directed graphs. Recall that $L$ is the generating set chosen in Notation \[generators\] for the free group $\pi_1(Y_{i,j},
t_{1,1,1})=\pi_1(S_{i,j}, t_{1,1,1}) =\pi_1(S_{i,j}^-, t_{1,1,1})$. From now on any group element in $\pi_1(Y_{i,j},
t_{1,1,1})=\pi_1(S_{i,j}, t_{1,1,1}) =\pi_1(S_{i,j}^-, t_{1,1,1})$ will be considered as a word in $L\cup L^{-1}$.
First we translate Theorem \[large n\] into the following theorem, in terms of folded graphs.
\[large\] There is a positive even integer $N_{i,j}$ such that for each even integer $N_*\geq N_{i,j}$ there is a finite, connected, $L$-labeled, directed graph ${\cal G}(N_*)$ (with a fixed base vertex $v_0$) with the following properties: (0) ${\cal G}(N_*)$ is $L$-regular;(1) The number of vertices of ${\cal G}(N_*)$ is $m_{i}=N_*d_i+1$; (2) Each of the words $w_1^*,..., w_\ell^*$ is representable by a loop, based at $v_0$, in ${\cal G}(N_*)$; (3) ${\cal G}(N_*)$ contains a closed loop, based at $v_0$, representing the word $z_{k,p,q}(N_*)^*$, for each $k=1,2, p=1,...,b_k, q=1,...,d_{i,k}-1$; (4) ${\cal G}(N_*)$ contains no loop representing the word $x_b^d$ for any $b=1,...,n_{i,j}$ and $d=1,...,m_{i}-1$; (5) each of the words $y_1,...,y_r$ is representable by a non-closed path, based at $v_0$, in ${\cal G}(N_*)$.
[The procedure for constructing the graphs described in Theorem \[large\] follows mostly that given in [@MZ Section 11]. In the current case we need to deal with two major complications. One is due to the fact that the number $d_{i, k}$ of intersection points in a boundary component $\b_{k,p}$ of $\p S_{i,j}^-$ depends on $k$; the other is due to the requirement of showing that such a graph ${\cal G}(N_*)$ exists for each even integer $N_*\geq N_{i,j}$. Actually our adjustment has begun as early as in Adjustment \[diff lengths\]. ]{}
[@MZ Remark 9.7] was one of the main group theoretical results obtained in [@MZ] and it will also play a fundamental role in our current case. We quote this result below as Theorem \[regular\] in the current notations.
\[regular\][([@MZ Remark 9.7])]{} If ${\cal G}_\#$ is a finite, connected, $L$-labeled, directed, folded graph with base vertex $v_0$, with corresponding subgroup $G_\# = L({\cal G}_\#,v_0) \subset \pi_1(S_{i,j}^-,t_{1,1,1})$, such that\
$\bullet\;$ ${\cal G}_\#$ does not contain any loop representing the word $x_b^d$ for any $b=1,...,n_{i,j}$, $d\in \z-\{0\}$, and\
$\bullet\;$ $y_1,...,y_r$ are some fixed, non-closed paths based at $v_0$ in ${\cal G}_\#$,\
then there is a finite, connected, $L$-regular graph ${\cal G}_*$ such that\
$\bullet\;$ ${\cal G}_*$ contains ${\cal G}_\#$ as an embedded subgraph, and thus in particular $y_1,...,y_r$ remain non-closed paths based at $v_0$ in ${\cal G}_*$, and\
$\bullet\;$ ${\cal G}_*$ contains no loops representing the word $x_b^d$, for each of $b=1,...,n_{i,j}$, $d=1,...,m_*-1$, where $m_*$ is the number of vertices of ${\cal G}_*$.
We now begin our constructional proof of Theorem \[large\]. Let ${\cal G}_{1}$ be the connected, finite, $L$-labeled, directed graph which results from taking a disjoint union of embedded loops– representing the reduced versions of the words $w_{1}^*, ..., w_{\ell}^*$ respectively– and non-closed embedded paths– representing the reduced versions of the words $y_{1}, ..., y_{r}$ respectively– and then identifying their base points to a common vertex $v_{0}$. Then $L({\cal G}_{1}, v_{0})$ represents the subgroup $g_{i,j}^*(\pi_1( J_{i,j}^-, b_{1,1,1}))$ of $\pi_1(S_{i,j}^-, t_{1,1,1})$. Since the folding operation does not change the group that the graph represents, $L({\cal G}_{1}^f, v_{0})=g_{i,j}^*(\pi_1(J_{i,j}^-,b_{1,1,1}))$ (where ${\cal G}_1^f$ denotes the folded graph of ${\cal G}_1$).
Recall from Lemma \[diff powers\] that $$z_{k,p,q}(n)^* =
(\overline{\s_{k,p,q-1}}\cdots\overline{\s_{k,p,0}})(x_{b_{k-1}+p}')^{n\frac{d_i}{d_{i,k}}}
(\s_{k,p,0}\cdots\s_{k,p,q}),$$ $k=1,2, p=1,...,b_k, q=1,...,d_{i,k}-1$. Note that $x_{b}'$ is conjugate to $x_b$ in $\pi_1(S_{i,j}^-, t_{1,1,1})$, for $b=1,..., n_{i,j}$. Let $\t_{b}$ be an element of $\pi_1(S_{i,j}^-, t_{1,1,1})$ such that $x_{b}^{\prime} = \t_{b} x_{b}\t_{b}^{-1}$, $b=1,..., n_{i,j}$. Let ${\cal G}_{2}$ be the connected graph which results from taking the disjoint union of ${\cal G}_{1}^f$ and non-closed embedded paths representing the reduced version of the words $\overline{\s_{k,p,q-1}}\cdots\overline{\s_{k,p,0}}\t_{b_{k-1}+p}$, $k=1,2, p=1,...,b_k, q=1,...,d_{i,k}-1$, respectively, and then identifying their base vertices into a single base vertex which we still denote by $v_{0}$. Then obviously we have $L({\cal G}_{2}^f, v_{0}) =
L({\cal G}_{2}, v_{0})
= L({\cal G}_{1}^f, v_{0})=g_{i,j}^*(\pi_1(J_{i,j}^-,b_{1,1,1}))$.
Let $v_{k,p,q}$ be the terminal vertex of the path $\overline{\s_{k,p,q-1}}\cdots\overline{\s_{k,p,0}}\t_{b_{k-1}+p}$ in ${\cal G}_{2}^f$, for each $k=1,2, p=1,...,b_k, q=1,...,d_{i,k}$. For each of $(1, p, q)$, $p=1,...,b_1, q=1,...,d_{i,1}$, and $(2, p,q)$, $p=1,..., b_2-1$, $q=1,...,d_{i,2}$, let $Q_{k,p,q}$ be the maximal $x_{b_{k-1}+p}$-path in $\widehat{{\cal G}_{2}^f}$ (a maximal $x_{b}$-path was defined in [@MZ Section 9]) which contains the vertex $v_{k,p,q}$. For each of $(2, b_2, q)$, $q=1,..., d_{i,2}$, let $Q_{2,b_2,q}$ be the maximal $x_{n_{i,j}}$-path in $\widehat{{\cal G}_{2}^f}$ determined by (1) if there is a directed edge of $\widehat{{\cal G}_{2}^f}$ with $v_{2,b_2,q}$ as its initial vertex and with the first letter of the word $x_{n_{i,j}}$ as its label, then $Q_{2, b_2,q}$ contains that edge; (2) if the edge described in (1) does not exists, then $v_{2,b_2,q}$ is the terminal vertex of $Q_{2,b_2,q}$ and the first letter of the word $x_{n_{i,j}}$ is the terminal missing label of $Q_{2,b_2,q}$. Note that each $Q_{k, p,q}$ is uniquely determined. Also no $Q_{k, p,q}$ can be an $x_{b}$-loop, since the group $L({\cal G}_{2}^f, v_{0})=g_{i,j}^*(\pi_1(J_{i,j}^-,b_{1,1,1}))$ does not contain non-trivial peripheral elements of $\pi_1(S_{i,j}^-, t_{1,1,1})$ by [@MZ Lemma 4.2]. Let $v_{k,p,q}^-$ and $v_{k,p,q}^+$ be the initial and terminal vertices of $Q_{k,p,q}$ respectively. Note that if $p\ne b_2$ and $Q_{k,p,q}$ is not a constant path, then $v_{k,p,q}^-$ and $v_{k,p,q}^+$ must be distinct vertices; however $v_{2,b_2, q}^-$ and $v_{2,b_2,q}^+$ may possibly be the same vertex, even if $Q_{2, b_2, q}$ is a non-constant path.
Let $Q_{k,p,q}^-$ be the embedded subpath of $Q_{k,p,q}$ with $v_{k,p,q}^-$ as the initial vertex and with $v_{k,p,q}$ as the terminal vertex, and let $Q_{k,p,q}^+$ be the embedded subpath of $Q_{k,p,q}$ with $v_{k,p,q}$ as the initial vertex and with $v_{k,p,q}^+$ as the terminal vertex.
Note that the number $max\{Length(Q_{k,p,q}): k=1,2, p=1,...,b_k, q=1,...,d_{i,k}\}$ is independent of $n$, and thus is bounded. So we may assume that $$n>40|L|+2 max\{Length(Q_{k,p,q}): k=1,2, p=1,...,b_k, q=1,...,d_{i,k}\}.$$ We shall also assume that $n$ has been chosen large enough so that $C_n(J_{i,j}^-)$ is convex.
Now for each $k=1,2, p=1,...,b_k$, $q=1,...,d_{i,k}-1$, we make a new non-closed embedded path $\Theta_{k,p,q}(n)$ representing the word $x_{b_{k-1}+p}^{n\frac{d_i}{d_{i,k}}}$, and we add it to the graph ${\cal
G}_{2}^f$, by identifying the initial vertex of $\Theta_{k,p,q}(n)$ with $v_{k,p,q}$ and the terminal vertex with $v_{k,p,q+1}$.
[For each $k=1,2, p=1,...,b_k$, $q=d_{i,k}$, we make a new non-closed embedded path $\Theta_{k,p,q}(n)$ representing the word $x_{b_{k-1}+p}^{n\frac{d_i}{d_{i,k}}}$, and we add it to the graph ${\cal G}_{2}^f$, by identifying the initial vertex of $\Theta_{k,p,q}(n)$ with $v_{k,p,q}$. ]{}
\[total\][For each fixed $k=1,2, p=1,...,b_k$, the paths $\{\Theta_{k,p,q}(n), q=1,...,d_{i,k}\}$, are connected together and form a connected path representing the word $x_{b_{k-1}+p}^{n d_i}$.]{}
In the resulting graph there are some obvious places one can perform the folding operation: for each $k=1,2, p=1,...,b_k, q=1,...,d_{i,k}-1$, the path $Q_{k,p,q}^+$ can be completely folded into the added new path $\Theta_{k,p,q}(n)$, and likewise the path $Q_{k,p,q+1}^-$ can be completely folded into $\Theta_{k,p,q}(n)$. Let ${\cal G}_{3}(n)$ be the resulting graph after performing these specific folding operations for each $k=1,2, p=1,...,b_k, q=1,...,d_{i,k}$.
From the explicit construction, it is clear that ${\cal G}_{3}(n)$ has the following properties: (1) ${\cal G}_{3}(n)$ is a connected, finite, $L$-labeled, directed graph; (2) ${\cal G}_{3}(n)$ contains loops, based at $v_{0}$, representing the word $z_{k,p,q}(n)^*$ for each $k=1,2, p=1,...,b_k, q =1,...,d_{i,k}-1$; (3) ${\cal G}_{3}(n)$ contains ${\cal G}_{2}^f$ as an embedded subgraph.
It follows from Property (3) that the paths in ${\cal G}_{2}^f$ representing the words $y_{1}, ..., y_{r}$ remain each non-closed in ${\cal G}_{3}(n)$, and it follows from Properties (2) and (3) and the construction that $L({\cal G}_{3}(n),v_{0})
=g_{i,j}^*(\pi_1(C_n(J_{i,j}^-), b_{1,1,1}))$. So $\widehat{{\cal G}_{3}(n)}$ cannot have $x_{b}$-loops for any $b=1,...,n_{i,j}$ (again by [@MZ Lemma 4.2]).
Now we consider the remaining folding operations on ${\cal G}_{3}(n)$ that need to be done, in order to get the folded graph ${\cal G}_{3}(n)^f$.
For each $k=1,2, p=1,...,b_k, q=1,...,d_{i,k}-1$, let $$\Theta_{k,p,q}(n)' =\Theta_{k,p,q}(n)\setminus
(Q_{k,p,q}^+\cup Q_{k,p,q+1}^-),$$ and for each $k=1,2, p=1,...,b_k, q=d_{i,k}$, let $$\Theta_{k,p,q}(n)' =\Theta_{k,p,q}(n)\setminus Q_{k,p,q}^+.$$ Then by our construction each $\Theta_{k,p,q}(n)'$ is an embedded $x_{b_{k-1}+p}$-path with $v_{k,p,q}^+$ as its initial vertex and with $v_{k,p,q+1}^-$ (when $q\ne
d_{i,k}$) as the terminal vertex. Also all these paths $\Theta_{k,p,q}(n)'$,$k=1,2, p=1,...,b_k,
q=1,...,d_{i,k}$, are mutually disjoint in their interior, and their disjoint union is equal to ${\cal
G}_{3}(n)\setminus {\cal G}_{2}^f$. Since $\widehat{{\cal G}_{3}(n)}$ has no $x_{b}$-loops, we see immediately that when $(k,p)\ne (2, b_2)$, all the vertices $v_{k,p,q}^{\pm}$, $q=1,...,d_{i,k}$, are mutually distinct.
It follows that the only remaining folds are at the vertices $v_{2, b_2,q}^{\pm}$. At such a vertex there is at most one edge from $\Theta_{2,b_2,q}(n)'$ which may be folded with one $x_{b_{k-1}+p}$-edge of $\Theta_{k,p,q_*}(n)'$ at its initial or terminal vertex, for some $(k,p)\ne (2,
b_2)$ and some $1\leq q_*\leq d_{i,k}$. Thus ${\cal G}_{3}(n)^f$ is obtained from ${\cal G}_{3}(n)$ by performing at most $2d_{i,2}$ folds (which occur at some of the vertices $v_{2, b_2,q}^{\pm}$, $q=1,...,d_{i,2}$), and every non-closed, reduced path in ${\cal G}_{3}(n)$ which is based at $v_{0}$ will remain non-closed in ${\cal G}_{3}(n)^f$. In particular, the paths representing the words $y_{1}, ... y_{r}$ are each non-closed in ${\cal G}_{3}(n)^f$.
Let $f_3:{\cal G}_{3}(n)\ra {\cal G}_{3}(n)^f$ be the natural map and we fix a number $$s>
2d_{i,2}+Diameter({\cal G}_{2}^f).$$ Then the map $f_3: {\cal G}_{3}(n) \ra {\cal G}_{3}(n)^f$ is an embedding on ${\cal G}_{3}(n) - N_{s}(v_{0})$, where $N_s(v_0)$ denotes the $s$-neighborhood of $v_0$ in ${\cal G}_3(n)$ considering a graph as a metric space, by making each edge isometric to the interval $[0,1]$. Obviously the number $s$ is independent of $n$.
\[large power\] [ We may assume further that $n$ is large enough so that the components of ${\cal G}_{3}(n)^f\setminus f(N_{v_{0}}(s))$ can be denoted by $\Phi_{k,p,q}(n)$, $k=1,2, p=1,...,b_k, q=1,...,d_{i,k}$, such that $\Phi_{k,p,q}(n)$ is an embedded subpath in $\Theta_{k,p,q}(n)'$ containing a power of $x_{b_{k-1}+p}$ which is larger than $\frac{n d_i }{d_{i,k}}-\frac{n}{4}$. This is clearly possible from the construction.]{}
The next step is to modify the graph ${\cal G}_{3}(n)^f$, by inserting copies of a certain graph $\Omega$, pictured in Figure \[fg1\], and then performing folding operations, to obtain a graph (the graph ${\cal
G}_{4}(n)$ given below) which contains loops, based at the base vertex $v_{0}$, representing the words $$w_{1}^*, ...,
w_{\ell}^*, z_{k,p,q}(n+1)^*, k=1,2, p=1,...,b_k,q=1,..., d_{i,k}-1,$$ respectively, and which contains non-closed paths, based at $v_{0}$, representing the words $$y_{1}, ..., y_{r}$$ respectively. In Figure \[fg1\], single edge loops at a vertex have label one each from $L^*=\{a_{1}, b_{1},..., a_{g}, b_{g}\}$. The edges in part (a) and (b) connecting two adjacent vertices are $x_{b}$-edges, $b=1,2,...,n_{i,j}-1$, (precisely $n_{i,j}-1$ edges). In part (a) of the figure, an $x_{b}$-edge points from the left vertex to the right vertex iff $b$ is odd, and in part (b) of the figure, an $x_{b}$-edge points from left to right iff $b$ is $1$ or an even number.
=4.5in
The method for constructing ${\cal G}_{4}(n)$ breaks into two subcases, i.e. (a) when $n_{i,j}$ is even, (b) when $n_{i,j}$ is odd.
[**Subcase (a):**]{} $n_{i,j}$ is even.
Recall that for each $k=1,2$, $p=1,...,b_k$, $\cup_{q=1}^{d_{i,k}}\Theta_{k,p,q}(n)$ is a connected path in ${\cal{G}}_{3}(n)^f$ representing the word $$x_{b_{k-1}+p}^{nd_i}$$ and thus we can divide the path equally into $d_i$ subpaths $$\Psi_{k,p,a}(n);\;\;\;a=1,...,d_i$$ each representing the word $$x_{b_{k-1}+p}^{n}.$$ Now we pick a vertex $u_{k,p,a}$ in $\Psi_{k,p,a}(n)$ for each $k=1,2, p=1,...., b_k, a=1,...,d_i$ as follows – if $(k,p)\ne (2, b_2)$, then $u_{k,p,a}$ is the middle vertex of $\Psi_{k,p,a}(n)$ (recall that $n$ is even); – if $(k,p)=(2,b_2)$, then $u_{2, b_2,a}$ is a vertex around the middle vertex of $\Psi_{k,p,a}(n)$ which is the initial vertex of a $x_1$-edge.
By Note \[large power\], for each $k=1,2$ and $p=1,...,b_k$, the set of $d_i$ points $$\{u_{k,p,a}; a=1,...,d_i\}$$ is contained in the set of $d_{i,k}$ paths $$\{\Phi_{k,p,q}(n); q=1,..., d_{i,k}\}.$$
Now we cut ${\cal{G}}_{3}(n)^f$ at the vertices $u_{k,p,a}$, $k=1,2, p=1,..., b_k, a=1,..., d_i$, that is, we form a cut graph ${\cal{G}}_{3}(n)^f_c={\cal{G}}_{3}(n)^f
\setminus \{u_{k,p,a},
k=1,2, p=1,..., b_k, a=1,..., d_i\}$, whose vertex set is obtained from the vertex set of ${\cal{G}}_{3}(n)^f$ by replacing each $u_{k,p,a}$ with a pair of vertices $u_{k,p,a}^{\pm}$ (where $u_{k,p,a}^{+}$ is the terminal vertex and $u_{k,p,a}^{-}$ is the initial vertex).
Now we take $d_i$ copies of the graph $\Omega$ shown in Figure \[fg1\] (a), which we denote by $\Omega_a$, $a=1,...,d_i$. For each fixed $a=1,...,d_i$, we identify the vertex set $$\{u_{k,p,a}^{\pm}, k=1,2, p=1,...,b_k\}$$ of ${\cal{G}}_{3}(n)^f_c$ with the vertices of $\Omega_a$ as follows:\
– if $(k,p)\ne (2, b_2)$, identify $u_{k,p,a}^+$ with the left vertex of $\O_a$ if $b_{k-1}+p$ is odd and to the right vertex if $b_{k-1}+p$ is even, and identify $u_{k,p,a}^-$ with the right vertex of $\O_a$ if $b_{k-1}+p$ is odd and to the left vertex if $b_{k-1}+p$ is even,\
– identify $u_{2,b_2,a}^+$ with the left vertex of $\O_a$ and identify $u_{2,b_2,a}^-$ with the right vertex of $\O_a$.
=4.7in
The resulting graph is not folded, but becomes folded graph after the following obvious folding operation around each inserted $\O_a$:\
– fold the subpath $x_{n_{i,j}-1}a_{1}b_{1}a_{1}^{-1}b_{1}^{-1}\cdots a_{g}
b_{g}a_{g}^{-1}b_{g}^{-1}$ whose terminal vertex is the vertex $u_{2,b_2,a}^+$ with the loops of $\O_a$ at the left vertex of $\O_a$ and then with the $x_{n_{i,j}-1}$-edge of ${\cal G}_{4}(n)$ whose terminal vertex is the left vertex of $\O_a$, and\
– fold the two $x_1$-edges whose initial vertices are the right vertex of $\O_a$.\
The resulting folded graph, denoted ${\cal G}_{4}(n)$, around the inserted $\O_a$ is shown in Figure \[fg2\]. By our construction we see that ${\cal G}_{4}(n)$ is a folded, $L$-labeled, directed graph, with no $x_{b}$-loops, with each of the words $w_{1}^*,...,w_{\ell}^*$ still representable by a loop based at $v_{0}$, and with each of the words $y_{1},..., y_{r}$ still representable by a non-closed path based at $v_{0}$. Also we see that the graph ${\cal G}_{4}(n)$ contains loops based $v_{0}$ representing the words $$z_{k,p,q}(n+1)^*, \;\;
k=1,2, p=1,...,b_k, q=1,...,d_{i,k}-1.$$
The graph ${\cal G}_{4}(n)$ is not $L$-regular yet since it does not contain any $x_{b}$-loops. So it must contain a missing label. Let $x\in L$ be a missing label at a vertex $v$ of ${\cal G}_{4}(n)$. Let $\a$ be a finite directed graph consisting of a single path of edges all labeled with $x$, as shown in Figure \[fg3\]. We identify the left end vertex of $\a$ to the vertex $v$ of ${\cal G}_{4}(n)$. The resulting graph ${\cal G}_{5}(n)$ is obviously still folded, contains ${\cal G}_{4}(n)$ as an embedded subgraph, and contains no $x_{b}$-loops for any $b=1,...,n_{i,j}$. By choosing a long enough path $\alpha$, we may assume that the number of vertices of ${\cal G}_{5}(n)$ is bigger than $(n+1) d_i+1$.
Now by Theorem \[regular\], we can obtain an $L$-regular graph ${\cal G}_{6}(n)$ such that (1) ${\cal G}_{5}(n)$ is an embedded subgraph of ${\cal G}_{6}(n)$; thus in particular in ${\cal G}_{6}(n)$ each of the words $w_{1}^*,...,w_{\ell}^*$, $z_{k.p.q}(n+1)^*$, $k=1,2, p=1,...,b_k$, $q=1,...,d_{i,k}-1$ is representable by a loop based at $v_{0}$, and each of the words $y_{1},..., y_{r}$ is representable by a non-closed path based at $v_{0}$; (2) ${\cal G}_{6}(n)$ contains no loops representing the word $x_{b}^d$ for any $b=1,...,n_{i,j}$, $d=1,...,m_*-1$, where $m_*$ is the number of vertices of ${\cal G}_{6}(n)$ (note that $m^*$ depends on $i$ and $j$).
Note that $m_*$ is some integer larger than $(n+1)d_i+1$. Let $N_{i,j}=m_*-(d_i-1)(n+1)-1$. Then $N_{i,j}>(n+1)$.
During the transformation from ${\cal G}_{4}(n)$ to ${\cal G}_{6}(n)$, the subgraph of ${\cal G}_{4}(n)$ consisting of the edges which intersect the subgraph $\Omega_a$ (for each fixed $a=1, ..., d_i$) remained unchanged since ${\cal G}_{4}(n)$ was locally $L$-regular already at the two vertices of $\O_a$. Now we replace $\O_a$, for each of $$a=1, ..., d_i-1,$$ by a graph $\O_{a}(N_{i,j}-n+1)$ which is similar to $\Omega_a$ but with $N_{i,j}-n+1 \geq 3$ vertices (Figure \[fg4\] illustrates such a graph with four vertices). Then the resulting graph, which we denote by ${\cal G}(N_{i,j})$, has the following properties. (1) ${\cal G}(N_{i,j})$ is $L$-regular; (2) each of the words $y_{1},...,y_{r}$ is still representable by a non-closed path based at $v_{0}$ in ${\cal G}(N_{i,j})$, (3) each of the words $w_{1}^*,..., w_{\ell}^*$ is still representable by a loop based at $v_{0}$ in ${\cal G}(N_{i,j})$, (4) ${\cal G}(N_{i,j})$ contains no loops representing the word $x_{b}^d$ for each $b=1,...,n_{i,j}$ and each $d=1,...,m_\#-1$, where $m_\#$ is the number of vertices of ${\cal G}(N_{i,j})$, (5) ${\cal G}(N_{i,j})$ contains a closed loop based at $v_{0}$ representing the word $z_{k,p,q}(N_{i,j})^*$, for each $k=1,2, p=1,...,b_k, q=1,...,d_i-1$, and (6) $m_\#$, the number of vertices of ${\cal G}(N_{i,j})$, is equal to $ N_{i,j} d_i+1$. Properties (1)-(5) are obvious by the construction, while property (6) follows by a simple calculation. Indeed $$\begin{aligned}
m_\# &=& m_* + (N_{i,j}-n+1-2)(d_i-1)\\
&=&[N_{i,j} + (d_i-1)(n+1)+1] + (N_{i,j} -(n+1))(d_i-1)\\
&=& N_{i,j} d_i+1.\end{aligned}$$
Now for each integer $N_*\geq N_{i,j}$ we construct a finite, connected, $L$-labeled, directed graph ${\cal G}(N_*)$ (with a fixed base vertex $v_0$) with the properties (0)-(5) listed in Theorem \[large\]. In the graph ${\cal G}(N_{i,j})$ above, for each $a=1,...,d_i-1$, replace the subgraph $\O_a(N_{i,j}-n+1)$ by the graph $\O_a(N_*-n+1)$, and replace subgraph $\O_{d_i}$ by the graph $\O_{d_i}(N_*-N_{i,j}+2)$. The resulting graph is ${\cal G}(N_*)$.
[**Subcase (b)**]{} $n_{i,j}>1$ is odd.
We modify the graph ${\cal G}_{3}(n)^f$ as follows. Besides the vertices $u_{k,p,a}$ we have chosen before, we choose, for each $a=1,...,d_i$, a vertex $u_{2,b_2,a}^{\prime}$ in the directed subpath $\Psi_{2,b_2,a}$ such that\
– $u_{2,b_2,a}^{\prime}$ is the initial vertex of an edge with label $x_2$,\
– $u_{2,b_2,a}'$ appears after the vertex $u_{2,b_2,a}$ in the directed subpath $\Psi_{2,b_2,a}$,\
– there are precisely five edges with label $x_1$ between $u_{2,b_2,a}$ and $u_{2, b_2, a}^{\prime}$ in the directed subpath $\Psi_{2,b_2,a}$ (this is possible as $n$ is large).
Again as $n$ is large, the set of $d_i$ vertices $\{u_{2,b_2,a}'; a=1,...,d_i\}$ is contained in the set of $d_{i,k}$ paths $\{\Phi_{2,b_2,q}(n); q=1,..., d_{i,k}\}$ (cf. Note \[large power\]).
Now cut ${\cal{G}}_{3}(n)^f$ at the vertices $\{u_{k,p,a}, k=1,2, p=1,...,b_k, a=1,...,d_i\}$, and $\{u_{2,b_2,s}', a=1,...,d_i\}$, and for each $a=1,...,d_i$, insert the graph $\O_a$, which is a copy of the graph $\Omega$ shown in Figure \[fg1\] (b). That is, we\
(1) Form a cut graph ${\cal{G}}_{3}(n)^f_c={\cal{G}}_{3}(n)^f\setminus
\{u_{k,p,a},u_{2,b_2,a}', k=1,2, p=1,...,b_k, a=1,...,d_i\}$, defined as in Subcase (a), with obvious modifications, i.e. we have similarly defined pairs of vertices $u_{k,p,a}^{\pm}$, $u_{2,b_2,a}^{'\pm}$ for ${\cal{G}}_{3}(n)^f_c$ such that if each such $\pm$ pair of vertices are identified, then the resulting graph is the original ${\cal{G}}_{3}(n)^f$.\
(2) For each fixed $a=1,...,d_i$, we identify the vertex set $\{u_{k,p,a}^{\pm}, u_{2,b_2,a}^{'\pm}, k=1,2, p=1,...,b_k\}$ of ${\cal{G}}_{3}(n)^f_c$ with the left-most and right-most vertices of $\Omega_a$ as follows:\
– if $(k,p)\ne (2,b_2)$, and $(k, p)=(1,1)$ or $b_{k-1}+p$ is even, then identify $u_{k,p,a}^+$ with the left-most vertex of $\O_a$ and $u_{k,p,a}^-$ with the right-most vertex.\
– if $(k,p)\ne (2,b_2)$, $(k,p)\ne (1,1)$ and $b_{k-1}+p$ is odd, then identify $u_{k,p,a}^+$ with the right-most vertex of $\O_a$ and $u_{k,p,a}^-$ with the left-most vertex,\
– identify $u_{2,b_2, a}^+$ with the left-most vertex of $\O_a$ and identify $u_{2,b_2,a}^-$ with the right-most vertex of $\O_a$,\
– identify $u_{2,b_2, a}^{'+}$ with the left-most vertex of $\O_a$ and identify $u_{2,b_2,a}^{'-}$ with the right-most vertex of $\O_a$.
The resulting graph is not folded, but becomes folded graph after the following folding operations are performed around each inserted $\O_a$:\
– fold the path $x_{n_{i,j}-1}a_{1}b_{1}a_{1}^{-1}b_{1}^{-1}\cdots a_{g}
b_{g}a_{g}^{-1}b_{g}^{-1}$ whose terminal vertex is the vertex $u_{2,b_2,a}^+$ with the loops of $\O_a$ at the left-most vertex of $\O_a$ and then with the $x_{n_{i,j}-1}$-edge of ${\cal G}_{4}(n)$ whose terminal vertex is the left-most vertex of $\O_a$,\
– fold the two $x_{1}$-edges whose initial vertices are the right-most vertex of $\O_a$,\
– fold the two $x_{1}$-edges whose terminal vertices are the left-most vertex of $\O_a$,\
– fold the two $x_{2}$-edges whose initial vertices are the right-most vertex of $\O_a$.\
The resulting folded graph ${\cal G}_{4}(n)^f$ around the inserted $\O_a$ is shown in Figure \[fg5\]. By our construction we see that ${\cal G}_{4}(n)^f$ is a folded, $L$-labeled, directed graph, with no $x_{b}$-loops, with each of the words $w_{1}^*,...,w_{\ell}^*$ still representable by a loop based at $v_{0}$, and with each of the words $y_{1},..., y_{r}$ still representable by a non-closed path based at $v_{0}$. Also we see that the graph ${\cal G}_{4}(n)$ contains loops based $v_{0}$ representing the words $z_{k,p,q}(n+2)^*$, for all $k=1,2, p=1,...,b_k, q=1,...,d_{i,k}-1$.
=5.5in
We then define ${\cal G}_{5}(n)$ and ${\cal G}_{6}(n)$ in a similar manner as in Subcase (a); here we may assume that ${\cal G}_{5}(n)$ has at least $(d_i-1)(n+2)-1$ vertices. Let $m_*$ be the number of vertices of ${\cal G}_{6}(n)$, and let $N_{i,j}= m_*-(d_i-1)(n+2)-1$. To form ${\cal G}(N_{i,j})$, we replace each subgraph $\Omega_a$, $a=1,...,d_i-1$ in ${\cal G}_{6}(n)$ with a graph $\O_a(1+N_{i,j}- n)$ similar to Figure \[fg1\](b) but with $1+N_{i,j}- n$ vertices. In the current case, we need $1+N_{i,j}-n$ to be an odd integer in order for the construction to work. This is made possible by the following
$N_{i,j} - n$ is even, i.e. $N_{i,j}$ is even (since we have chosen $n$ to be even (see Note \[n even\])).
The proof of this lemma is similar to that of [@MZ Lemma 11.3], noticing in the current case the number $d_i$ is even for each $i=1,2$ by Notation \[dik\].
The rest of the argument proceeds by obvious analogy with the Subcase (a). That is, the graph ${\cal G}(N_{i,j})$ is a graph with the properties listed as (1)-(6) in Subcase (a). Indeed, Properties (1)-(5) are immediate. To verify Property (6), we let $m_\#$ be the number of vertices of ${\cal G}(N_{i,j})$, and then we have: $$\begin{aligned}
m_\#&=& m_*+ (1+N_{i,j}-n-3)(d_i-1)\\
&=& N_{i,j} + (d_i-1)(n+2) +1+ (N_{i,j}-n-2)(d_i-1)\\
&=& N_{i,j}d_i+1.\end{aligned}$$
Now for each even integer $N_*\geq N_{i,j}$, the graph ${\cal G}(N_*)$ required by Theorem \[large\] is obtained from the graph ${\cal G}(N_{i,j})$ by replacing each subgraph $\O_a(N_{i,j}-n+1)$, $a=1,...,d_i-1$, by the graph $\O_a(N_*-n+1)$, and replacing the subgraph $\O_{d_i}$ by the graph $\O_{d_i}(N_*-N_{i,j}+3)$.
[**Proof of Theorem \[each large n\] in Case 2**]{}.
The proof is similar to that of Case 1 and much simpler notationally, and we shall be very brief. In this case $n_{i,j}=1$, i.e. the surface $S_{i,j}$ has only one boundary component, which we denote by $\b$ and may assume lying in $T_1$. $\b$ has $d_{i,1}$ intersection points with $\p S_{i_*}$, which we denote by $t_q$, $q=1,...,d_{i,1}$. Similarly as in Case 1, we define the points $b_q, q=1,...,d_{i,1}$ in $\p_p J_{i,j}$. The group $\pi_1(S_{i,j}^-,t_1)$ has a set of generators $$L=\{a_1, b_1,...,a_g, b_g\}$$ ($g$ must be larger than $0$) such that $$x_1=a_1 b_1 a_1^{-1}b_1^{-1}\cdots a_g b_g a_g^{-1}b_g^{-1}$$ is an embedded loop which is homotopic to $\b$. As in Case 1, we similarly define the elements $w_1,...,w_{\ell}$, the element $y_1,...,y_r$, and the elements $z_q(n)$, $q=1,...,d_{i,1}-1$, and we reduce the proof of Theorem \[each large n\] in Case 2 to the proof of the following theorem which is an analogue of Theorem \[large\].
\[large2\] There is a positive even integer $N_{i,j}$ such that for each even integer $N_*\geq N_{i,j}$ there is a finite, connected, $L$-labeled, directed graph ${\cal
G}(N_*)$ (with a fixed base vertex $v_0$) with the following properties: (0) ${\cal G}(N_*)$ is $L$-regular;(1) The number of vertices of ${\cal G}(N_*)$ is $m_{i}=N_*d_i+1$; (2) Each of the words $w_1^*,..., w_\ell^*$ is representable by a loop, based at $v_0$, in ${\cal G}(N_*)$; (3) ${\cal G}(N_*)$ contains a closed loop, based at $v_0$, representing the word $z_{q}(N_*)^*$, for each $q=1,...,d_{i,1}-1$; (4) ${\cal G}(N_*)$ contains no loop representing the word $x_1^d$ for any $d=1,...,m_{i}-1$; (5) each of the words $y_1,...,y_r$ is representable by a non-closed path, based at $v_0$, in ${\cal G}(N_*)$.
To prove this theorem, we construct, similar as in Case 1, the analogue graph ${\cal G}_3(n)^f$ and its subgraphs $\Phi_q(n), q=1,...,d_{i,1}$, $\Psi_a(n), a=1,...,d_i$, with similar properties. We modify the graph ${\cal G}_{3}(n)^f$ as follows. For each of $a=1,...,d_i$, we pick a pair vertices $\{u_a,
u_{a}'\}$ in the path $\Psi_a(n)$ as follows:\
–$u_a$ is closed to the middle vertex of $\Psi_a(n)$;\
–$u_a$ is the terminal vertex of an edge with label $a_{1}$; and\
–$u_{a}^{\prime}$ is the terminal vertex of an edge with label $b_{1}$ which appears after the vertex $u_{a}$;\
–there are precisely five edges with label $b_{1}$ between $u_{a}$ and $u_{a}^{\prime}$ in the path $\Psi_{a}(n)$.
We may assume that the set $$\{u_a, u_a'; a=1,...,d_i\}$$ is contained in the set $$\{\Phi_q(n); q=1,...,d_{i,1}\}.$$
Now cut the graph ${\cal{G}}_{3}(n)^f$ at all the pairs of vertices $\{u_{a}, u_{a}'\}$, $a=1,...,d_i$, and for each $a$, insert the graph $\O_a$– which is a copy of the graph $\Omega$ shown in Figure \[fg6\] – as follows. Form a cut graph ${\cal{G}}_{3}(n)^f_c={\cal{G}}_{a}(n)^f\setminus
\{u_{a}, u_{a}'; a=1,...,d_i\}$, and let $u_{a}^{\pm}$, $u_{a}^{'\pm}$ be the corresponding vertices for ${\cal{G}}_{3}(n)^f_c$. For each fixed $a=1,...,d_i$, we identify the vertex $u_{a}^+$ with the left-most vertex of $\O_a$, identify $u_{a}^-$ with the right-most vertex of $\O_a$, identify $u_{a}^{'+}$ with the right-most vertex of $\O_a$ and identify $u_{a}^{'-}$ with the left-most vertex of $\O_a$.
The resulting graph is not folded, but becomes folded graph after a single folding operation around each inserted $\O_a$: fold the two $a_{1}$-edges whose terminal vertices are the right-most vertex of $\O_a$. The resulting folded graph ${\cal G}_{4}(n)^f$ around the inserted $\O_a$ is shown in Figure \[fg7\]. By our construction we see that ${\cal G}_{4}(n)^f$ is a folded $L$-labeled directed graph, with no $x_1$-loops, with each of the words $w_{1}^*,...,w_{\ell}^*$ still representable by a loop based at $v_{0}$, and with each of the words $y_{1},..., y_{r}$ still representable by a non-closed path based at $v_{0}$. Also we see that the graph ${\cal G}_{4}(n)$ contains loops based at $v_{0}$ representing the words $z_{q}(n+4)^*$, for all $q=1,...,d_{i,1}-1$.
=5.5in
As in Case 1, we get ${\cal G}_{5}(n)$ and ${\cal G}_{6}(n)$. In the current case, $N_{i,j}= m_*-
(d_i-1)(n+4)-1$, which is larger than $n+4$, where $m_*$ is the number of vertices of ${\cal G}_6(n)$. To form ${\cal G}(N_{i,j})$, we replace the left half (with three vertices) of $\Omega_a$, for each $a=1,...,d_i-1$, with a graph $\O_a(N_{i,j}- n-1)$ which is similar to Figure \[fg6\] but with $N_{i,j}- n-1$ vertices. In the current case, we also need $N_{i,j}$ to be an even integer in order for the construction to work. This is true, and can be proved as in Subcase (b) of Case 1. It is easy to see that ${\cal G}(N_{i,j})$ has all the Properties (0)-(5) listed in Theorem \[large2\] (when $N_*=N_{i,j}$). For instance to check Property (1), we have: $$\begin{aligned}
m_i &=& m_*+(N_{i,j}-n-1-3)(d_i-1)\\
&=& N_{i,j}+(d_i-1)(n+4)+1+ (N_{i,j}-n-4)(d_i-1)\\
&=& N_{i,j}d_i+1\end{aligned}$$ To show that Theorem \[large2\] holds for any even integer $N_*\geq N_{i,j}$, we simply let ${\cal
G}(N_*)$ be the graph obtained from the graph ${\cal G}(N_{i,j})$ by replacing each subgraph $\O_a(N_{i,j}-n-1)$, $a=1,...,d_i-1$, by the graph $\O_a(N_*-n-1)$, and replacing the subgraph $\O_{d_i}$ by the graph $\O_{d_i}(N_*-N_{i,j}+5)$.
The final assembly {#hsmf}
==================
Fix an even integer $N_*$ satisfying Corollary \[indep of j\] (later on we may need $N_*$ to have been chosen large enough). Then as given in Corollary \[indep of j\], for each $i=1,2$ and $j=1,.., n_i$, the manifold $Y_{i,j}^-=S_{i,j}^-\times I$ has an $m_i=N_* d_i+1$ fold cover $\breve Y_{i,j}^-=\breve S_{i,j}^-\times I$ such that $|\p_p \breve Y_{i,j}^-|= |\p_p Y_{i,j}^-|$, i.e. each component of $\p_p \breve Y_{i,j}^-$ is an $m_i$ fold cyclic cover of a component of $\p_p Y_{i,j}^-$. Moreover the map $g_{i,j}: (J_{i,j}^-, \p_p J_{i,j}^-)
\ra (Y_{i,j}^-, \p_p Y_{i,j}^-)$ lifts to an embedding $\breve g_{i,j}: (J_{i,j}^-, \p_p J_{i,j}^-)
\ra (\breve Y_{i,j}^-, \p_p \breve Y_{i,j}^-)$ such that if $\breve A$ is a component of $\p_p\breve Y_{i,j}^-$ then the components of $\breve g_{i,j}(\p_p J_{i,j}^-)\cap \breve A$ are evenly distributed along $\breve A$. More precisely if $S_{i,j}^-$, for instance, is the surface given in the proof of Theorem \[each large n\] in Case 1, then with the notations given there, we may index the boundary components of $\breve S_{i,j}^-$ as $\breve\b_{k,p}$, $k=1,2,p=1,...,b_k$, so that each $\breve\b_{k,p}$ is an $m_i$ fold cyclic cover of $\b_{k,p}$ and the points $\{\breve g_{i,j}(b_{k,p,q}); q=1,...., d_{i,k}\}$ divide $\breve \b_{k,p}$ into $d_{i,k}$ segments each having wrapping number $N_* d_i/d_{i,k}$. Also note that the $d_{i,k}$ points $\{\breve g_{i,j}(b_{k,p,q}); q=1,...., d_{i,k}\}$ are mapped to the $d_{i,k}$ points $\{t_{k,p,q}; q=1,...., d_{i,k}\}$ respectively under the covering map $\breve\b_{k,p}\ra \b_{k,p}$. As $N_*$ can be assumed to be arbitrarily large, we may assume that the wrapping number $N_* d_i/d_{i,k}$ be as large as needed for each $i=1,2$ and $k=1,...,m$.
Also recall that $(K_{i,j}^-, \p_p K_{i,j}^-)$ is properly embedded in the pair $(J_{i,j}^-, \p_p J_{i,j}^-)$, with a relative $R$-collared neighborhood. It follows that the pair $(\breve g_{i,j}(K_{i,j}^-), \breve
g_{i,j}(\p_p K_{i,j}^-))$ has a relative $R$-collared neighborhood in $(\breve Y_{i,j}^-, \p_p\breve
Y_{i,j}^-)$. Also $K_1^-=\cup_{j=1}^{n_1} K_{1,j}$ and $K_2^-=\cup_{j=1}^{n_2} K_{2,j}$ are isometric under the isometry $h:K_1\ra K_2$. Now let $\breve Y^-$ be the union of $\breve Y_1^-=\cup_{j=1}^{n_1}
Y_{1,j}^-$ and $\breve Y_2^-=\cup_{j=1}^{n_2} Y_{2,j}^-$ with $\cup_{j=1}^{n_1} (\breve g_{1,j}(K_{1,j}^-), \breve
g_{1,j}(\p_p K_{1,j}^-))$ and $\cup_{j=1}^{n_2} (\breve g_{2,j}(K_{2,j}^-), \breve g_{2,j}(\p_p
K_{2,j}^-))$ identified by the corresponding isometry and let $(U^-, \p_p U^-)$ be the identification space of $\cup_{j=1}^{n_1} (\breve g_{1,j}(K_{1,j}^-), \breve g_{1,j}(\p_p K_{1,j}^-))$ and $\cup_{j=1}^{n_2} (\breve g_{2,j}(K_{2,j}^-), \breve g_{2,j}(\p_p K_{2,j}^-))$ in $\breve Y^-$. Then $\breve Y^-$ is a connected metric space, with a path metric induced from the metrics on $\breve Y_1^-$ and $\breve Y_2^-$. There is an induced local isometry $f:\breve{Y}^- \ra M$ which extends the local isometry $\breve{Y}_{i,j}^- \ra Y_{i,j}^-\ra M$ for each $i,j$.
Define the parabolic boundary, $\p_p\breve Y^-$, of $\breve
Y^-$ to be the union of $\p_p\breve Y_1^-$ and $\p_p\breve Y_2^-$, with $\cup_{j=1}^{n_1} \breve
g_{1,j}(\p_p K_{1,j}^-)$ and $\cup_{j=1}^{n_2}\breve g_{2,j}(\p_p K_{2,j}^-)$ identified by the isometry. Then $(U^-, \p_p U^-)$ has a relative $R$-collared neighborhood in $(\breve Y^-, \p_p\breve
Y^-)$.
Now for each $k=1,...,m$, let $\breve C_K$ be the cover of the $k$-th cusp $C_k$ of $M$ corresponding to the subgroup of $\pi_1(C_k)$ generated by the $m_1$-th power of a component of $\p_k S_1^-$ and the $m_2$-th power of a component of $\p_k S_2^-$. Then each oriented component, say $\b$, of $\p_k S_i^-$ has its inverse image in $\p\breve C_k$, denoted $\breve \b$, a connected oriented circle. So we may and shall identify $\breve\b$ with the oriented component of $\p \breve S_i^-$ which covers $\b$. This way we embed naturally all components of $\p\breve S_i^-$ into $\p \breve C=\cup _{k=1}^m \p \breve C_k$, for each $i=1,2$. We denote by $\p_k \breve S_i^-$ those components of $\p \breve S_i^-$ which are embedded in $\p \breve C_k$. Then we have $|\p_k \breve S_i^-|=|\p_k S_i|$, and the components of $\p_k
\breve S_i^-$ are far apart from each other in $\p \breve C_k$, for each $i=1,2$ and $k=1,...,m$. So we may and shall embed the corresponding components of $\p_p\breve Y_i^-$ into $\p \breve C$ along $\p
\breve S_i^-$, for each $i=1,2$. After such identification, we get a connected hyperbolic $3$-manifold $$\breve Y^-\cup(\cup_{k=1}^m \breve C_k)$$ with $m$ rank two cusps and with a local isometry into $M$.
As in [@MZ Section 13] we construct the thickening $\bar U^-$ of $U^-$ so that $\p_p \bar U$ is embedded in $\p \breve C$ (Note that each component of $\bar U^-$ is a handlebody, with a similar proof as that of [@MZ Lemma 13.2]) and let $$Y^-=\breve Y_1^-\cup \bar U^-\cup \breve Y_2^-.$$ Then $Y^-$ is a connected, compact, hyperbolic $3$-manifold, locally convex everywhere except on its parabolic boundary $\p_p Y^-=\p_p \breve Y_1^-\cup \p_p \bar U^-\cup \p_p\breve Y_2^-$. The complement of $\p_p(Y^-)$ in $\p \breve C$ is a set of “round-cornered parallelograms” with very long sides in $\p \breve
C$. As in [@MZ Section 13], we scoop out from $\breve C=\cup_{k=1}^m C_k$ the convex half balls based on these round-cornered Euclidean parallelograms and denote the resulting cusps by $\cup_{k=1}^m
\breve C_k^0$. Then $$Y=Y^-\cup (\cup_{k=1}^m\breve C_k^0)$$ is a connected, metrically complete, convex hyperbolic $3$-manifold, with a local isometry $f$ into $M$. Thus the local isometry $f$ induces an injection of $\pi_1(Y)$ into $\pi_1(M)$ by [@MZ Lemma 4.2].
Each boundary component of $Y$ provides a Quasi-Fuchsian surface in $M$. To prove this claim, it suffices to show, with a similar reason as that given in [@MZ Section 13], that every Dehn filling of $Y$ along its cusps gives a $3$-manifold with incompressible boundary.
Let $Y(\a_1,...,\a_m)$ be any Dehn filling of $Y$ with slopes $\a_1,...,\a_m$. Then $Y(\a_1,...,\a_m)$ is an $HS$-manifold (see [@MZ Section 12] for its definition). The handlebody part $H$ of $Y(\a_1,...,\a_m)$ is $\bar U^-\cup (\cup_{k=1}^m \breve
C_k^0(\a_k))$ (which may have several components in the current case but each has genus at least two) and the $S\times
I$ part of $Y(\a_1,...,\a_m)$ is $Y(\a_1,...,\a_m)\setminus H=Y^-\setminus \bar U^-$. This $HS$ decomposition of $Y(\a_1,...,\a_m)$ satisfies the conditions of [@MZ Lemma 12.1] and thus $Y(\a_1,...,\a_m)$ has incompressible boundary by that lemma. The proof of this last claim is similar to that of [@MZ Lemma 13.6], for which we only need to note the following: (1) With a similar proof as that of [@MZ Lemma 13.5] we have that each component of $Y^-\setminus \bar U^-$ is not simply connected. (2) $|\p_k \breve S^-_i|=|\p_k S_i^-|\geq 2$ for each $i=1,2, k=1,...,m$, by Condition \[at least two\].
The proof of Theorem \[main\] is now finished.
[2]{}
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|
---
abstract: |
Let $D$ be a division algebra such that $D\t D^o$ is a Noetherian algebra, then any division subalgebra of $D$ is a [*finitely generated*]{} division algebra. Let $\D $ be a finite set of commuting derivations or automorphisms of the division algebra $D$, then the group $\Ev (\D )$ of common eigenvalues (i.e. [*weights*]{}) is a [*finitely generated abelian*]{} group. Typical examples of $D$ are the quotient division algebra ${\rm Frac} (\CD
(X))$ of the ring of differential operators $\CD (X)$ on a smooth irreducible affine variety $X$ over a field $K$ of characteristic zero, and the quotient division algebra ${\rm Frac} (U (\Gg ))$ of the universal enveloping algebra $U(\Gg )$ of a finite dimensional Lie algebra $\Gg $. It is proved that the algebra of differential operators $\CD (X)$ is isomorphic to its opposite algebra $\CD (X)^o$.
[*Mathematics subject classification 2000:*]{} 16S15, 16W25, 16S32, 16P40, 16K40.
author:
- 'V. Bavula'
title: Finite generation of division subalgebras and of the group of eigenvalues for commuting derivations or automorphisms of division algebras
---
Introduction
============
Throughout this paper, $K$ is a field, $\t=\t_K$. Noetherian means left and right Noetherian. For a $K$-algebra $A$, $A^o$ denotes the [*opposite*]{} algebra to $A$ (recall that $A^o=A$ as abelian groups but the multiplication in $A^o$ is given by the rule: $a*b=ba$), and $A^e:=A\t A^o$ is called the [*enveloping* ]{} algebra of $A$. The expressions ${}_AM$, $M_A$, and ${}_AM_A$ means that $M$ is respectively a left, right $A$-module, and an $A$-bimodule. [*Finitely generated division algebra*]{} means a division algebra which is generated (as a division algebra) by a finite set of elements (i.e. $x_1, \ldots , x_n$ is a set of generators for a division $K$-algebra $D$ if $D$ is the only division $K$-subalgebra of $D$ that contains $x_1, \ldots , x_n$).
For division algebras [*finite dimensional*]{} over $K$ there is a well-developed theory where (commutative) subfields play a fundamental role. By contrast, if a division algebra is infinite dimensional little is known about its division subalgebras.
[*Question. Suppose that $D$ is a finitely generated division $K$-algebra, is any division $K$-subalgebra of $D$ finitely generated?*]{}
Certainly this is the case when $D$ is a field. We will see that the answer is [*affirmative*]{} for many popular division algebras. For a similar question about [*subfields*]{} ($=$ commutative division $K$-subalgebras), Resco, Small and Wadsworth give an affirmative answer in [@RSW79]: [*Let $D$ be a division algebra over a field $K$ such that $ D\t D^o$ is Noetherian, then every (commutative) subfield of $D$ containing $K$ is finitely generated.*]{} One of the crucial steps in their proof is the following result of Vamos [@Va78]: [*Let $L$ be a field extension of $K$. Then $L\t L$ is Noetherian iff $L$ is a finitely generated over $K$.*]{} M. Smith [@MSmith73] showed that there is a division algebra $D$ with centre $K$, containing two maximal subfields whose transcendence degrees are any two prescribed cardinal numbers.
Let $\D =\{ \d_1, \ldots , \d_t\}$ be a set of [*commuting*]{} $K$-derivations of a division $K$-algebra $D$. The set $\Ev (\D
):=\{ \l =(\l_1, \ldots , \l_t)\in K^t\, | \, \d_i (u)=\l_i u,
i=1, \ldots , t$ for some $0\neq u\in D\}$ of common eigenvalues is an [*additive*]{} subgroup of $K^t$, and the $\D$-[*eigen-algebra*]{} $D(\D ):=\bigoplus_{\l \in \Ev (\D )}D_\l $ is a $\Ev (\D )$-graded algebra where $D_\l :=\{ u\in D\, | \,
\d_i(u)=\l_i u, i=1, \ldots , t\}$, $D_\l D_\mu \subseteq D_{\l
+\mu} $ for all $\l , \mu \in \Ev (\D )$, and $0\neq u\in D_\l $ implies $u^{-1}\in D_{-\l }$.
Let $\D =\{ \d_1, \ldots , \d_t\}$ be a set of [*commuting*]{} $K$-automorphisms of a division $K$-algebra $D$, and let $K^*:=K\backslash \{ 0\}$ be the multiplicative group of the field $K$. The set $\Ev (\D ):=\{ \l =(\l_1, \ldots , \l_t)\in K^{*t}\,
| \, \d_i (u)=\l_i u, i=1, \ldots , t$ for some $0\neq u\in D\}$ of common eigenvalues is an [*multiplicative*]{} subgroup of $K^{*t}$, and the $\D$-[*eigen-algebra*]{} $D(\D ):=\bigoplus_{\l
\in \Ev (\D )}D_\l $ is a $\Ev (\D )$-graded algebra where $D_\l
:=\{ u\in D\, | \, \d_i(u)=\l_i u, i=1, \ldots , t\}$, $D_\l D_\mu
\subseteq D_{\l \mu} $ for all $\l , \mu \in \Ev (\D )$, and $0\neq u\in D_\l $ implies $u^{-1}\in D_{\l^{-1} }$.
The first statement of the next result is an extension of the mentioned above result of Resco-Small-Wadsworth to division subalgebras (with a short [*different*]{} proof given in Section \[PRTHM\]).
\[DDaccd\]Let $D$ be a division $K$-algebra such that $D\t D$ is a Noetherian $D$-bimodule, and let $\D =\{ \d_1, \ldots , \d_t\}$ be either a set of commuting $K$-derivations or commuting $K$-automorphisms of the division $K$-algebra $D$. Then
1. $D$ satisfies the ascending chain condition on division $K$-subalgebras, or equivalently, every division $K$-subalgebra of $D$ is a finitely generated division $K$-algebra.
2. The group of eigenvalues $\Ev (\D )$ is a finitely generated abelian group, and so $\Ev (\D )=\CT \oplus \mathbb{Z}^r$ where $r$ is the rank of the group $\Ev (\D )$ and $\CT $ is a finite abelian group.
3. The eigen-algebra $D(\D )$ is a Noetherian domain which isomorphic to an iterated skew Laurent extension. In more detail, $D_\CT :=\oplus_{\l \in \CT } D_\l $ is a division algebra of right and left dimension $|\CT |$ over the division algebra $D_0$, $D(\D )$ is isomorphic to the iterated skew Laurent extension $D_\CT [x_1, x_1^{-1}; \s_1]\cdots [x_r,
x_r^{-1}; \s_r]$ with coefficients from the division algebra $D_\CT$.
4. For each subgroup $F$ of $\,\Ev (\D )$, $\CF (F):=\oplus_{\l
\in F}D_\l $ is a Noetherian domain the quotient division algebra ${\rm Frac}(\CF (F))$ of which is $\D $-invariant and $\Ev ({\rm
Frac}(\CF (F)))=F$, any $\D$-eigenvector $v\in {\rm Frac}(\CF
(F))_\l$ has the form $u^{-1}w$ for some $0\neq u\in D_\mu$, $w\in D_{\l +\mu }$, and $\l , \mu \in F$.
[*Remark 1.*]{} $D\t D$ is a [*Noetherian*]{} $D$-bimodule iff the algebra $D\t D^o$ is [*Noetherian*]{} iff the algebra $D\t D^o$ is [*left*]{} Noetherian iff the algebra $D\t D^o$ is [*right*]{} Noetherian as it follows from
$$\label{D4D}
{}_DD\t D_D\simeq {}_DD\t ({}_{D^o}D^o)\simeq {}_{D\t D^o}D\t D^o,
\; {}_DD\t D_D\simeq D^o_{D^o}\t D_D\simeq D_D\t D^o_{D^o}\simeq
(D\t D^o)_{D\t D^o}.$$
[*Remark 2.*]{} ‘Finite generation’ is built in in the structure of the eigen-algebra $D(\D )$ in the sense that it is a finitely generated algebra over a finitely generated division algebra.
In Section \[APLDX\], it is proved that many division algebras that appear naturally in applications satisfy the conditions of Theorem \[DDaccd\] (Proposition \[constg\], Lemma \[itfgd\]), [*eg*]{} ${\rm Frac}(\CD (X))$ (Corollary \[divDXfg\]) and ${\rm Frac} (U(\Gg ))$ (Corollary \[DUGN\]).
Proof of Theorem \[DDaccd\] {#PRTHM}
===========================
Recall that any torsion free finitely generated abelian group is a free abelian group of finite rank, and vice versa. Any finitely generated abelian group $G$ is isomorphic to $\CT \oplus
\mathbb{Z}^r$ where $r:=\dim_\mathbb{Q}(\mathbb{Q}\t_\mathbb{Z}G)$ is the [*rank*]{} of the group $G$, $\CT $ is the [*torsion*]{} subgroup of $G$, that is the subgroup of $G$ that contains all the elements of finite order, it is a finite group.
[**Proof of Theorem \[DDaccd\]**]{}. $1$. Suppose that inside $D$ one can pick a strictly ascending chain of division $K$-subalgebras $\G_1\subset \G_2\subset \cdots \subset
\G_n\subset \cdots $, we seek a contradiction; this gives a strictly ascending chain of $D$-sub-bimodules, $K_1\subset
K_2\subset \cdots \subset K_n\subset \cdots $, where $K_n=\ker
(\phi_n)$ where $\phi_n: D\t D\ra D\t_{\G_{n}}D$, $x\t y\ra
x\t_{\G_{n}} y$, (use the fact that $D$ is a free left and right $\G_n$-module and tensor product commutes with direct sum), a contradiction. Hence $D$ satisfies the [*acc*]{} on division $K$-subalgebras.
$2$, $3$, and $4$. The proof of two cases are very similar, so we will treat them simultaneously by making some adjustments to our notation. So, let $\D =\{ \d_1, \ldots , \d_t\}$ be either a set of commuting $K$-derivations or a set of commuting $K$-automorphisms of the division algebra $D$. In the first case, $\Ev (\D )$ is an [*additive*]{} subgroup of $K^t$, in the second case, $\Ev (\D )$ is a [*multiplicative*]{} subgroup of $K^{*t}$. In the second case, we still will write the group operation [*additively*]{}, i.e. $\l +\mu$ means $\l \mu$, $-\l $ means $\l^{-1}$, $0$ means $1$. Let $D_0$ be the set of $\D $-[*constants*]{}: $D_0:=\cap_{i=1}^t\ker_D(\d_i)$, in the case of derivations; and $D_0:=\{ d\in D\, | \, \d_i(d)=d, i=1, \ldots , t\}$, in the case of automorphisms. In both cases, $D_0$ is a division subalgebra of $D$.
Given a division algebra $\G $, a group $G$, a group homomorphism $\v :G\ra \Aut_K(\G )$, and a ‘2-cocycle’ $G\times G\ra \G^*:=\G
\backslash \{ 0\}$, $(g,h)\mapsto (g,h)$. A [*generalized crossed product*]{} is an algebra $\G *G=\oplus_{g\in G}\G g$ which is a free left $\G $-module with multiplication given by the rule $$ag\cdot bh=a \v (g)(b)(g,h)gh, \;\; a,b\in \G, \; g,h\in G.$$ It follows from $ag=g \v (g)^{-1}(a)$ that $\G g=g\G\simeq
\G_\G$, and so $\G *G=\oplus_{g\in G}g\G$ is a free right $\G
$-module. A ‘2-cocycle’ means that the multiplication of the generalized crossed product is associative. When $G=\mathbb{Z}$ and $(i,j)=1$ for all $i,j\in
\mathbb{Z}$, we have, so-called, a [*skew Laurent extension*]{} with coefficients from $\G $ denoted $\G [x, x^{-1}; \s]$ where $x$ is the group generator $1$ for $\mathbb{Z}$ and $\s =\v
(1)\in \Aut_K(\G )$. So, the skew Laurent extension generated by $\G $ and $x$, $x^{-1}$ subject to the defining relations $x^{\pm
1}a=\s^{\pm 1}(a)x^{\pm 1}$ for all $a\in \G$. An [*iterated skew Laurent extension* ]{} $A_n:= \G [x_1, x_1;\s_1]\cdots [x_n,
x_n^{-1}; \s_n]$ is defined inductively as $A_n=A_{n-1}[x_n,
x_n^{-1}; \s_n]$. Since the division algebra $\G $ is a Noetherian algebra then the iterated skew Laurent extension $A_n$ is a Noetherian algebra (1.17, [@GW]).
For each $\l \in \Ev (\D )$, fix $0\neq u_\l \in D_\l $. Then it is easy to see that the $\D$-[*eigen-algebra*]{} is a free (left and right) $D_0$-module: $$\label{D=dcpz}
\CD =D(\D ):= \bigoplus_{\l \in \Ev (\D )}D_0u_\l =\bigoplus_{\l
\in \Ev (\D )}u_\l D_0, \;\; u_\l u_\mu =(\l , \mu )u_{\l +\mu },
\; (\l , \mu ):= u_\l u_\mu u_{\l +\mu}^{-1}\in D_0,$$ for $a,b\in D_0$: $au_\l bu_\mu=a(u_\l bu^{-1}_\l )u_\l u_\mu =
a(u_\l au^{-1}_\l )(u_\l u_\mu u^{-1}_{\l +\mu })u_{\l +\mu }$ (In general, this is not a generalized crossed product but if $\Ev (\D
)\simeq \mathbb{Z}^r$ one can choose generators in such a way that it is). Given any finitely generated subgroup $F=\CT \oplus (\oplus_{i=1}^s\mathbb{Z}v_i)$ of $\Ev (\D )$ where $\CT $ is the torsion part of $F$. The algebra $D_\CT :=\oplus_{\l \in \CT}D_0u_\l =\oplus_{\l \in
\CT} u_\l D_0$ has left and right dimension $|\CT |<\infty$ over the division algebra $D_0$ where $|\CT |$ is the order of the group $\CT $. The map $$l: D_\CT \ra {\rm
End}_{D_\CT}(D_{\CT}),\; a\mapsto (l_a:x\mapsto ax),$$ is an algebra isomorphism where ${\rm End}_{D_\CT}(D_{\CT})$ is the endomorphism algebra of the [*right*]{} $D_\CT $-module $D_\CT $. For each nonzero element $a\in D_\CT $, $l_a$ is a monomorphism, hence it is an isomorphism since (the right dimension over $D_0$) ${\rm r.dim}_{D_0}(D_\CT )={\rm r.dim}_{D_0}(aD_\CT )=| \CT |<\infty$. Therefore, ${\rm End}_{D_\CT}(D_{\CT})$ is a division algebra, hence so is its isomorphic copy $D_\CT $. Let $F'=\oplus_{i=1}^s\mathbb{Z}v_i$, and so $F=\CT \oplus F'$. The subalgebra $\CF' =\CF (F'):= \oplus_{\l \in F'}D_0u_\l$ of $D (\D )$ is isomorphic to the iterated skew Laurent extension $\CL :=
D_0[x_1, x_1^{-1}; \s_1]\cdots [x_s, x_s^{-1}; \s_s]$ where $\s_i
(d)=u_{v_i}du_{v_i}^{-1}$ $(d\in D_0)$ and $\s_i (x_j)=\l_{ij}x_j$, $j<i$, where $\l_{ij}:=u_{v_i}u_{v_j}u_{v_i}^{-1}u_{v_j}^{-1}\in
D_0$ (via the $K$-algebra epimorphism $\CL \ra \CF' $, $d\mapsto d$, $x_i\mapsto u_{v_i}$, where $d\in D_0$). This follows easily from a definition of an iterated skew Laurent extension and the facts that $D_0$ is a division algebra, $u_{v_1}^{n_1}\cdots
u_{v_s}^{n_s}\in D_{n_1v_1+\cdots +n_sv_s}$, and $F'=\oplus_{i=1}^s\mathbb{Z}v_i$. Then, by a similar reasoning (since $D_\CT$ is a division algebra), the algebra $\CF$ is isomorphic to the iterated skew Laurent extension $D_\CT [x_1, x_1^{-1}; \s_1]\cdots [x_s, x_s^{-1}; \s_s]$ where $\s_i
(d)=u_{v_i}du_{v_i}^{-1}$ $(d\in D_\CT)$ and $\s_i (x_j)=\l_{ij}x_j$, $j<i$, $\l_{ij}$ are as above.
Since $D_\CT$ is a Noetherian algebra so is the algebra $\CF$. So, $\CF
$ is a Noetherian domain, let ${\rm Frac}(\CF )$ be its quotient division algebra, so any element of ${\rm Frac}(\CF )$ is a fraction $a^{-1}b$ for some $0\neq a, b\in \CF $. Note that the elements $a $ and $b$ are finite sums $\sum a_\l $ and $\sum b_\l
$ of eigenvectors $a_\l , b_\l \in D_\l $, $\l \in F$. If $0\neq c=a^{-1}b\in
D_\mu$ for some $\mu \in \Ev (\D )$, then $ac=b\neq 0$ implies that $a_\l c=b_\nu$ for some $a_\l\neq 0$ and $b_\nu\neq 0$ such that $\l +\mu =\nu$, and so $c=a_\l^{-1}b_\nu$ and $\mu =\nu -\l $. This proves that any $\D $-eigenvector of ${\rm Frac}(\CF )$ is a fraction of the eigenvectors of $\CF $ and that $$\label{EvDFrF}
\Ev (\D , {\rm Frac}(\CF (F) ))=F.$$ It follows immediately from this fact and statement 1 that $\Ev
(\D )$ is finitely generated: otherwise one can find in $\Ev (\D
)$ a strictly ascending chain of subgroups: $F_1\subset F_2\subset
\cdots $, which gives, by (\[EvDFrF\]), the strictly ascending chain of division subalgebras: ${\rm Frac}(\CF (F_1) )\subset {\rm
Frac}(\CF (F_2) )\cdots $, a contradiction. This finishes the proof of statement 2 and 4. Then statement 3 has, in fact, been proved above. This finishes the proof of Theorem \[DDaccd\]. $\Box $
For an abelian monoid $E$, the set $\tor (E)$ of all the elements $e\in E$ such that $ne=0$ is a group, so-called, the [*torsion subgroup*]{} of $E$.
\[1DDaccd\]Let a $K$-algebra $A$ be a Noetherian domain with $D:={\rm
Frac}(A)$ such that $D\t D$ is a Noetherian $D$-bimodule, $\D =\{
\d_1, \ldots , \d_t\}$ be either a set of commuting $K$-derivations or commuting $K$-automorphisms of the algebra $A$. Then the abelian monoid of eigenvalues $\Ev (\D , A)$ for $\D $ in $A$ is a submonoid of a finitely generated abelian group, and so the rank of $\Ev (\D , A)$ is finite, and the torsion subgroup $\tor (\Ev (\D, A))$ is a finite group.
[*Proof*]{}. Note that each derivation (resp. an automorphism) $\d_i$ of $A$ can be uniquely extended to a derivation (resp. an automorphism) of the division algebra $D$ by the rule $\d_i
(s^{-1}a)=s^{-1}a-s^{-1}\d_i (s)s^{-1}a$ (resp. $\d_i(s^{-1}a)=\d_i(s)^{-1}\d_i(a)$). So, the zero derivation $[\d_i, \d_j]=0$ of $A$ has zero extension to $D$, and by uniqueness it must be zero on $D$. Similarly, the identity automorphism $[\d_i, \d_j]=\d_i\d_j\d_i^{-1}\d_j^{-1}$ of $A$ has the obvious extension to $D$, and by uniqueness it must be the identity map on $D$. So, $\D$ is a set of commuting derivations (resp. automorphisms) of $D$. Clearly, $\Ev (\D , A)\subseteq \Ev
(\D, D)$, and the result follows from Theorem \[DDaccd\].(2). $\Box $
\[2DDaccd\]Let a $K$-algebra $A$ be a commutative affine domain with $D:={\rm Frac}(A)$, $\D =\{ \d_1, \ldots , \d_t\}$ be either a set of commuting $K$-derivations or commuting $K$-automorphisms of the algebra $A$. Then the abelian monoid of eigenvalues $\Ev (\D
, A)$ for $\D $ in $A$ is a submonoid of a finitely generated abelian group, and so the rank of $\Ev (\D , A)$ is finite, and the torsion subgroup $\tor (\Ev (\D, A))$ is a finite group.
In general, the eigen-algebra $D(\D )$ is not a finitely generated algebra even in the case of a commutative affine domain $A$ since, in general, the $\D$-constants $D_0$ is not a finitely generated algebra (Hilbert 14’th problem, etc).
Applications {#APLDX}
============
Let $\G $ be a $K$-algebra, $\s$ be a $K$-automorphism of $\G $, and $\d \in
\Der_K(\G )$ be a $\s $-[*derivation*]{} of $\G$: $\d (ab)=\d (a)
b +\s (a) \d (b)$ for $a,b\in \G$. The [*Ore*]{} extension $A=\G
[x; \s , \d ]$ is a $K$-algebra generated freely by $\G $ and an element $x$ satisfying the defining relations: $xa=\s (a)x+\d (a)$ for $a\in \G$. Let $\G^o$ be the opposite algebra with multiplication given by the rule $a*b=ba$. Then $\s \in
\Aut_K(\G^o)$ as $\s (a*b)=\s (ba)=\s (b)\s (a)=\s (a)*\s (b)$, and so $\s^{-1}\in \Aut_K(\G^o)$, and finally $\d \s^{-1}\in
\Der_K(\G^o)$ is a $\s^{-1}$-derivation of the algebra $\G^o$: $$\begin{aligned}
\d\s^{-1}(a*b)&=&\d\s^{-1}(ba)=\d (\s^{-1}(b)\s^{-1}(a))=
\d\s^{-1}(b)\s^{-1}(a)+b\d\s^{-1}(a)\\
&=&\d\s^{-1}(a)*b+\s^{-1}(a)*\d\s^{-1}(b).\end{aligned}$$ $$\label{Oreop=Ore}
A^o=\G^o[x;\s^{-1}, -\d\s^{-1}].$$ [*Proof*]{}. The $K$-algebra $A$ is generated by $\G $ and $x$ that satisfy the defining relations: $x\s^{-1}(a)=ax+\d\s^{-1}(a)$, $a\in \G $, since $\s $ an [*automorphism*]{} of $\G $. Hence the $K$-algebra $A^o$ is generated by the $\G^o$ and $x$ that satisfy the defining relations: $x*a=\s^{-1}(a)*x-\d\s^{-1}(a)$, $a\in \G^o$, and we are done. $\Box$
The [*iterated Ore*]{} $A=\G [x_1; \s_1 , \d_1 ]\cdots [x_n; \s_n
, \d_n ]$ is defined inductively as $$(\G [x_1; \s_1 , \d_1
]\cdots [x_{n-1}; \s_{n-1} , \d_{n-1} ]) [x_n; \s_n , \d_n ].$$ By (\[Oreop=Ore\]) and induction on $n$, $$\label{itOreop}
(\G [x_1; \s_1 , \d_1 ]\cdots [x_n; \s_n , \d_n ])^o\simeq \G^o
[x_1; \s_1^{-1} , -\d_1\s_1^{-1} ]\cdots [x_n; \s_n^{-1} ,
-\d_n\s_n^{-1} ].$$ The tensor product of two iterated Ore extensions $A=\G [x_1; \s_1
, \d_1 ]\cdots [x_n; \s_n , \d_n ]$ and $B=\D [y_1; \tau_1 ,
\der_1 ]\cdots [y_m; \tau_m , \der_m ]$ is again an iterated Ore extension $$A\t B=\G \t \D [x_1; \s_1
, \d_1 ]\cdots [x_n; \s_n , \d_n ][y_1; \tau_1 , \der_1 ]\cdots
[y_m; \tau_m , \der_m ]$$ where $\s_i$, $\d_i$ and $\tau_j$, $\der_j$ act [*trivially*]{} on the elements where they have not been defined. Recall that if $\G $ is a domain (resp. a Noetherian algebra) then so is the iterated Ore extension $A$. If $\G^e=\G\t\G^o$ is a Noetherian algebra then so is the algebra $\G
$.
\[itfgd\]Suppose that $\G^e=\G\t\G^o$ is a Noetherian domain (then so is $\G $ and an iterated Ore extension $A=\G [x_1; \s_1 , \d_1
]\cdots [x_n; \s_n , \d_n ]$). Let $D={\rm Frac} (A)$. Then $D^e=D\t D^o$ is a Noetherian domain, and so the results of Theorem \[DDaccd\] hold.
[*Proof*]{}. $A^e=A\t A^o$ is an iterated Ore extension with coefficients from the Noetherian domain $\G^e$, hence $A^e$ is a Noetherian domain (1.12, [@GW]), and so is its localization $D^e$. $\Box $
\[divDXfg\]Let $X$ be a smooth irreducible affine variety over a field $K$ of characteristic zero, $\CD (X)$ be the ring of differential operators on $X$, and $D(X)={\rm
Frac}(\CD (X))$ be its quotient division algebra. Then $D(X)\t
D(X)^o$ is a Noetherian domain, and so the results of Theorem \[DDaccd\] hold.
[*Proof*]{}. The coordinate algebra $\OO=\OO (X)$ of the variety $X$ is a finitely generated domain with the field of fractions, say $\G ={\rm Frac}(\OO )$. Let $S=\OO \backslash \{ 0\}$. Then, by 15.2.6, [@MR], $\S1 \CD (X)\simeq \G [t_1;
\frac{\der}{\der x_1}]\cdots [t_n; \frac{\der}{\der x_n}]$ is an iterated Ore extension (with trivial automorphisms: $\s_i={\rm
id}_\G $) where $n=\dim (X)$ (the dimension of $X$), $\G $ contains a rational function field $Q_n:=K(x_1, \ldots, x_n)$ where the $\frac{\der}{\der x_i}$ are partial derivations (extended uniquely from $Q_n $ to $\G$). Note that $\G \t \G^o=\G \t \G$ is a Noetherian domain as the localization of the domain $\OO(X\times
X)\simeq \OO (X)\t \OO (X)$, the variety $X\times X$ is a smooth irreducible affine variety. By Lemma \[itfgd\], $D(X)^e$ is a Noetherian domain and every division $K$-subalgebra of $D(X)$ is a finitely generated division $K$-algebra. This proves the first two statements. Then statement 3 follows from Theorem \[DDaccd\].(2). $\Box $
\[AAoFFo\]Let $A$ be a $K$-algebra and $A\ra A^o$, $a\mapsto a^o$, be the canonical anti-isomorphism $((\l a +\mu b)^o=\l a^o+\mu b^o$ and $(ab)^o=b^o*a^o$ for all $\l , \mu \in K$ and $a,b\in A$). Then
1. $(s^{-1})^o=(s^o)^{-1}$ for each unit $s\in A$.
2. If $S$ is a left (resp. right) Ore subset of $A$ then $S^o$ is a right (resp. left) Ore subset of $A^o$ and $(\S1 A)^o\simeq A^o(S^o)^{-1}$, $s^{-1}a\mapsto a^o* (s^o)^{-1}$ (resp. $( A\S1 )^o\simeq (S^o)^{-1}A^o$, $as^{-1}\mapsto (s^o)^{-1}*a^o$) is the $K$-algebra isomorphism.
3. If $A$ is a Noetherian domain then ${\rm
Frac}(A^o)\simeq {\rm Frac}(A)^o$, $s^{-1}a\mapsto a^o*
(s^o)^{-1}$, is the isomorphism of division $K$-algebras.
4. If $A$ is a Noetherian domain such that $A\simeq A^o$ then ${\rm Frac}(A^o)\simeq {\rm Frac}(A)^o$.
[*Proof*]{}. $1$. $ss^{-1}=s^{-1}s=1$ implies $s^o*(s^{-1})^o=(s^{-1})^o*s^o=1$, and so $(s^{-1})^o=(s^o)^{-1}$.
$2$. Straightforward.
$3$. It is a particular case of statement 2.
$4$. By the universal property of localization, $A\simeq A^o$ implies ${\rm Frac}(A)\simeq {\rm Frac}(A^o)$, and by statement 3, ${\rm Frac}(A^o)\simeq {\rm Frac}(A)^o$. $\Box $
\[DUGN\]Let $\Gg$ be a finite dimensional Lie algebra over a field $K$, $U=U(\Gg )$ be its universal enveloping algebra, $D(\Gg )={\rm
Frac}(U)$ be its quotient division algebra. Then $D(\Gg )^e\simeq
D(\Gg )\t D( \Gg )$ is a Noetherian domain, and so the results of Theorem \[DDaccd\] hold.
[*Proof*]{}. $U\simeq U^o$, $g\mapsto -g$, $g\in \Gg $. Hence, ${\rm Frac} (U)\simeq {\rm Frac} (U^o)\simeq{\rm Frac} (U)^o$ (Lemma \[AAoFFo\]) and $U^e=U\t U^o\simeq U\t U\simeq U(\Gg
\oplus \Gg )$, and so $D(\Gg )^e\simeq D(\Gg ) \t D(\Gg )$ is a Noetherian domain as a localization of $U(\Gg \oplus \Gg )$, the rest follows from Theorem \[DDaccd\]. $\Box $
We have a great similarity in the proofs of the last two statements, one can repeat this pattern for other ‘constructions’ of algebras and their division algebras. To formalize the proofs in many similar situations let us introduce a concept of a [*good construction*]{} of algebras. We say that we have a [*construction*]{} of $K$-algebras, say $\CA $, if, for a given $K$-algebra $\G $, one attaches a set (class) of $K$-algebras $\CA
(\G )$. Examples in mind are Ore extensions $\CA (\G )=\{ \G
[x;\s, \d ]\}$, iterated Ore extensions, iterated skew Laurent polynomial algebras, etc. We say that the construction $\CA $ is [*good*]{} if the following three properties hold:
$(G1)$ if $\G $ is a Noetherian domain then so is each algebra from the set $\CA
(\G )$,
$(G2)$ $\CA (\G )^o\subseteq \CA (\G^o )$, and
$(G3)$ $\CA (\G )\t \CA (\G ')\subseteq \CA (\G \t \G')$,
where $\CA (\G )^o:=\{ A^o\, | \, A\in \CA (\G )\}$ and similarly $\CA (\G )\t \CA (\G '):=\{ A\t A'\, | \, A\in \CA (\G ), A'\in \CA ( \G')
\}$.
For the definitions and properties of the algebras from the examples below the reader is refereed to [@GW] and [@MR].
[*Examples of good constructions*]{}. $(1)$ Iterated Ore extensions.
$(2)$ Iterated skew Laurent extensions: $\G [x_1, x_1^{-1}; \s_1]
\cdots [x_n, x_n^{-1}; \s_n]$ where $\s_i$ are $K$-automorphisms ($(G1)$ - use the leading term and 1.17, [@GW]; $(G2)$ - Exercise 1P, p. 17, [@GW]; $(G3)$ - obvious).
\[constg\]Suppose that the enveloping algebra $\G^e=\G \t \G^o$ of an algebra $\G $ is a Noetherian domain, $\CA $ is a good construction, then each algebra $A\in \CA (\G )$ is a Noetherian domain and so is $D^e=D\t D^o$ where $D={\rm Frac}(A)$. Hence, Theorem \[DDaccd\] is true for $D$.
[*Proof*]{}. $\G^e$ is a Noetherian domain, then so is $\G $, and then each algebra $A\in \CA (\G )$ is a Noetherian domain since the construction $\CA $ is good. $A^e=A\t A^o\in \CA (\G) \t \CA
(\G )^o\subseteq \CA (\G) \t \CA (\G^o )\subseteq \CA (\G \t
\G^o)$, and so $A^e$ is a Noetherian domain, as $\CA$ is good. Hence, so is its localization $D^e$. The rest is obvious. $\Box $
So, the algebras that satisfy conditions of Theorem \[DDaccd\] are fairly common.
\[DNGN\]Let $D$ be a division algebra over a field $K$ such that $D\t D$ is a Noetherian $D$-bimodule. Let $\G $ be a division $K$-subalgebra of $D$. Then
1. $\G \t \G$ is a Noetherian $\G $-bimodule, and
2. $\Kdim ({}_\G(\G \t \G )_\G )\leq \Kdim ({}_D(D\t D)_D)$.
[*Remark*]{}. $\Kdim ({}_\G M_\G)$ means the Krull dimension of a $\G $-bimodule $M$.
[*Proof*]{}. Suppose that $\G \t \G$ is not a Noetherian $\G
$-bimodule, we seek a contradiction. Then one can find a strictly ascending chain of $\G$-sub-bimodules: $I_1\subset I_2\subset
\cdots $. Note that $D\t D\simeq D \t_\G (\G \t \G )\t_\G D$, an isomorphism of $D$-bimodules. Since $D_\G$ is free, $D\t_\G
I_1\subset D\t_\G I_2\subset \cdots $ is a strictly ascending chain of $(D, \G )$-bimodules. Similarly, since ${}_\G D$ is free, $D\t_\G I_1\t_\G D\subset D\t_\G I_2\t_\G D\subset \cdots $ is a strictly ascending chain of $D$-bimodules, a contradiction. We have proved that any strictly ascending chain of $\G $-bimodules $\{ I_i\}$ gives (by tensoring as above) the strictly ascending chain of $D$-bimodules $\{ D\t_\G I_i\t_\G D\}$, hence $\Kdim ({}_\G(\G \t \G )_\G )\leq \Kdim ({}_D(D\t D)_D)$. $\Box $
Given a $K$-algebra $A$, a $K$-linear map $\s :A\ra A$ is called an [*anti-isomorphism*]{} iff $\s (ab)=\s (b) \s (a)$ for all $a,b\in A$. Clearly, $\s $ is an anti-isomorphism iff $\s : A\ra
A^o$, $a\mapsto \s (a)^o$, is a $K$-algebra isomorphism.
\[AFFDual\]Let $A_i$, $i\in I$, be subalgebras of a $K$-algebra $A$, $B:=\cap_{i\in I}A_i$, $\s : A\ra A$ be an anti-isomorphism such that $\s (A_i)=A_i$ for all $i$. Then $\s $ induces the anti-isomorphism of the algebra $B$.
[*Proof*]{}. Clearly, $\s^{-1}$ is an anti-isomorphism of the algebra $A$ such that $\s^{-1}(A_i)=A_i$ for all $i\in I$. Then $\s (B)\subseteq (B)$ and $\s^{-1}(B)\subseteq B$ for all $i\in
I$, hence $\s (B)=B$, and we are done. $\Box $
\[KDXDX\]Let $X$ be a smooth irreducible affine variety over a field $K$ of characteristic zero, $\CD (X)$ be the ring of differential operators on $X$, and $D(X)={\rm
Frac}(\CD (X))$ be its quotient division algebra. Then
1. $\CD (X)\simeq \CD(X)^o$, $d\mapsto d$, $\der \mapsto -\der
$, where $d\in \OO (X)$ and $\der \in \Der_K(\OO (X))$.
2. $D(X)\simeq D(X)^o$.
3. if $X$ is a smooth affine variety (not necessarily irreducible) over $K$ then still $\CD (X)\simeq \CD
(X)^o$.
[*Remark*]{}. $1$. The algebra $\CD (X)$ is generated by the coordinate algebra $\OO (X)$ and the $\OO (X)$-module $\Der_K(\OO
(X))$ of $K$-derivations of the algebra $\OO (X)$ (5.6, [@MR]).
$2$. So, the ring of differential operators on a smooth irreducible algebraic variety is [*symmetric*]{} object indeed. If $A$ is [*not*]{} smooth then, in general, the algebra $\CD
(A)$ need not be a finitely generated algebra nor a left or right Noetherian algebra, [@BGGDiffcone72], the algebra $\CD (A)$ can be finitely generated and right Noetherian yet not left Noetherian, [@SmStafDifopcurve].
[*Proof*]{}. $1$. We keep the notation of the proof of Corollary \[divDXfg\]. In particular, the algebra $\CD (X)$ is a subalgebra of its localization $A:= \S1 \CD (X)\simeq \G [t_1;
\frac{\der}{\der x_1}]\cdots [t_n; \frac{\der}{\der x_n}]$.
By 2.13 and 2.6, [@MR], there is a finite set of elements of the coordinate algebra $\OO (X)$, say $c_1, \ldots ,
c_s$, such that the natural inclusion $\CD (X)\ra \prod_{i=1}^s
\CD (X)_{c_i}$ is a [*faithfully flat*]{} extension where $A_i:=\CD (X)_{c_i}$ is the localization of $\CD (X)$ at the powers of the element $c_i$ such that $\CD (X)_{c_i}=\OO (X)_{c_i}[t_1;
\frac{\der}{\der x_1}]\cdots [t_n; \frac{\der}{\der x_n}]$. Note that $A_i\subseteq A$. Hence $\CD (X)=\cap_{i=1}^sA_i$ (let $B:=\cap_{i=1}^sA_i$ then $\CD (X)\subseteq B\subseteq A_i$ for each $i$, then the localization of the chain of inclusions above at $c_i$ gives $A_i:=\CD (X)_{c_i}\subseteq B_{c_i}\subseteq A_i$, and so, by the faithful flatness, we must have $\CD (X)=B$). The map $\s :A\ra A$ given by $d\ra d$ $(d\in \G )$, $t_i\mapsto -t_i$, gives an anti-isomorphism of $A$ such that $\s (A_i)=A_i$ for all $i$. By Lemma \[AFFDual\], $\s $ gives the anti-isomorphism of the algebra $\CD (X)$.
$2$. By Lemma \[AAoFFo\], $D(X)^o={\rm Frac}(\CD (X))^o\simeq
{\rm Frac}(\CD (X)^o)\simeq {\rm Frac}(\CD (X))=D(X)$.
$3$. Then $X\simeq \prod_{i=1}^sX_i$ is a direct product of smooth irreducible affine varieties $X_i$ over $K$. Then $$\CD (X)=\CD (\prod_{i=1}^sX_i)\simeq \prod_{i=1}^s\CD (X_i)\simeq \prod_{i=1}^s\CD
(X_i)^o \simeq (\prod_{i=1}^s\CD (X_i))^o\simeq \CD (X)^o. \;\;\;
\Box$$
It is well-known fact that if $C$ is a commutative $K$-subalgebra of the ring of differential operators $\CD (X)$ (see Theorem \[KDXDX\]) then the Gelfand-Kirillov dimension $\GK (C)\leq
n:=\dim (X)$ (i.e. the transcendence degree ${\rm tr.deg}_K ({\rm Frac}
(C))\leq n$). It follows from Corollary 3.12, [@BavCr], that $\Ev (\D )\simeq \mathbb{Z}^r$ and $r\leq n$, where $\D =\{ \d_1,
\ldots , \d_t\}$ is a set of commuting [*locally finite*]{} $K$-derivations of the algebra $\CD (X)$. A $K$-derivation $\d$ of an algebra $A$ is called [*locally finite*]{} if, for each $a\in A$, $\dim_K(\sum_{i\geq 0}K\d^i(a))<\infty$. In a view of Corollary 3.12, [@BavCr] and Theorem \[KDXDX\], the author propose the following conjecture.
[*Conjecture. If $\D =\{ \d_1,
\ldots , \d_t\}$ is a set of commuting $K$-derivations or $K$-automorphisms of the division algebra $D(X)={\rm Frac}(\CD
(X))$ then the rank of the abelian group $\Ev (\D )\leq \dim
(X)$.*]{}
[*Question 1. Given a division $K$-algebra $D$ over a field $K$ of characteristic zero such the algebra $D\t D^o$ is Noetherian and a set $\D =\{ \d_1,
\ldots , \d_t\}$ of commuting $K$-derivations or $K$-automorphisms of the division algebra $D$. Is it true that the rank of the abelian group $\Ev (\D )\leq
\Kdim (D\t D^o)$?*]{}
[*Remark*]{}. If, in the question above, $D$ is a (finitely generated) [*field*]{} then the result is obviously true (as it follows from (\[D=dcpz\]) and $\GK (D(\D ))\leq \GK (D)$ that $$\GK (D(\D ))={\trdeg}_K(D_0)+{\rm rank}(\Ev (\D ))\leq \trdeg_K(D)=\Kdim(D\t
D)$$ where $D_0$ is the subfield of $\D $-constants in $D$).
[*Question 2. For a singular irreducible affine variety find a necessary and sufficient condition that $\CD (X)\simeq \CD
(X)^o$.*]{}
[*Conjecture 2. Let $X$ be a smooth irreducible affine curve over an algebraically closed field $K$ of characteristic zero, and $\d \in \Der_K(\CD (X))$. Then the eigen-algebra $D(\d )$ is a finitely generated Noetherian algebra*]{}.
[99]{}
V. V. Bavula, Gelfand-Kirillov dimension of commutative subalgebras of simple infinite dimensional algebras and their quotient division algebras, (arXiv:math.RA/0401263).
I. N. Bernstein, I. M. Gelfand and S. I. Gelfand, Differential operators on a cubic cone. [*Uspehi Mat. Nauk*]{} [**27**]{} (1972), no. 1, 185–190.
K. R. Goodearl and R. B. Warfield, Jr. [*An introduction to noncommutative Noetherian rings*]{}. Cambridge University Press, Cambridge, 1989.
J. C. McConnell and J. C. Robson, [*Noncommutative Noetherian rings*]{}. With the cooperation of L. W. Small. Revised edition. Graduate Studies in Mathematics, 30. American Mathematical Society, Providence, RI, 2001.
R. Resco, L. W. Small, and A. R. Wadsworth, Tensor products of division rings and finite generation of subfields. [*Proc. Amer. Math. Soc.*]{} [**77**]{} (1979), no. 1, 7–10.
M. Smith, Centralizers in rings of quotients of group rings. [ *J. Algebra*]{}, [**25**]{} (1973), 158–164.
P. Vamos, On the minimal prime ideal of a tensor product of two fields. [*Math. Proc. Cambridge Philos. Soc.*]{} [**84**]{} (1978), no. 1, 25–35.
S. P. Smith and J. T. Stafford, Differential operators on an affine curve. [*Proc. London Math. Soc.*]{} [**56**]{} (1988), no. 2, 229–259.
Department of Pure Mathematics
University of Sheffield
Hicks Building
Sheffield S3 7RH
UK
email: v.bavula@sheffield.ac.uk
|
= 6.6truein = 8.7truein = 0.9 in \#1\#2[3.6pt]{} = -.7truein = -.7truein plus 0.2pt minus 0.1pt = 42by .5cm .5cm plus 1pt
SU-ITP-93-37\
UG-8/93\
hep-th/9401025
[**EXACT DUALITY\
.8 cm IN STRING EFFECTIVE ACTION** ]{}\
1.7 cm [**Eric Bergshoeff [^1] and Ingeborg Entrop**]{} [^2] 0.05cm Institute for Theoretical Physics, University of Groningen\
Nijenborgh 4, 9747 AG Groningen, The Netherlands .5truecm [**Renata Kallosh**]{} [^3] 0.05cm Physics Department, Stanford University, Stanford CA 94305\
0.7 cm
1.5 cm
**ABSTRACT**
> We formulate sigma-model duality transformations in terms of spin connection. This allows to investigate the symmetry of the string action including higher order $\alpha'$ corrections. An important feature of the new duality transformations is a simple homogeneous transformation rule of the spin connection (with torsion) and specifically adjusted transformation of the Yang-Mills field.
>
> We have found that under certain conditions this duality is a symmetry of the full effective string action in the target space, free of $\alpha'$ corrections. We demonstrate how the exact duality generates new fundamental string solutions from supersymmetric string waves.
Introduction
============
In recent years an active field of research has been the study of modified Einstein-Maxwell equations. The modifications that have been considered include additional scalar or antisymmetric tensor fields (called dilatons and axions) or modifications in which the electromagnetic field is replaced by non-Abelian Yang-Mills fields. These modified Einstein-Maxwell theories admit new solutions whose consistency crucially depends on the presence of the new fields and/or on the non-Abelian nature of the Yang-Mills fields. For examples of such new solutions see the review articles [@Ca1; @Ho2; @Se1] and references therein.
One motivation for studying the above-mentioned modifications to Einstein-Maxwell theory is that they arise in string theory. The zero slope limit $\alpha'\to 0$ of string theory corresponds to a modified Einstein-Maxwell theory of the type discussed above. The complete effective action also includes contributions which are of higher order in the Riemann tensor and the Yang-Mills field strength. Since string theory claims to give a consistent description of quantum gravity, solutions of the string effective action will contribute to our understanding of quantum gravity.
Not many exact solutions to the string equations of motion are known. One of the reasons for this is that knowledge about the explicit form of the higher order $\alpha'$ corrections to the string effective action have become available only fairly recently [@Be2]. The higher order terms in $\alpha'$ in the effective action are for instance crucial in the construction of the five-brane soliton [@St1].
In general, it appears difficult to find exact solutions to the string equations of motion. Fortunately, if one considers spacetimes with a symmetry, there exist transformations which generate new solutions from old ones. Examples of solution generating transformations specific to the string effective action are the target space duality transformations [@BUS; @Ve1] and the symmetry transformations of [@Se2].
So far, the target space duality transformations have only been derived in a sigma model formulation of string theory [@BUS]. The explicit form of the duality transformation is only known to lowest order in $\alpha'$. In general one expects that the duality transformations will be modified with an infinite number of terms of increasingly higher order in $\alpha'$ but no information is available about these higher order corrections. Recently, Klimčík and Tseytlin found an exact duality between some pp-waves and flat space with non-vanishing antisymmetric tensor and dilaton fields [@KT]. A discussion, from the sigma model point of view, of other examples of situations where the duality transformations are exact can be found in [@KG], [@GE].
A discussion of less explicit examples of situations where the leading-order duality transformations do not acquire $\alpha'$ corrections for a special choice of field redefinitions can also be found in [@KT].
The purpose of this paper is to find duality transformations which form a symmetry of the theory in the zero slope limit, and remain a symmetry of the theory even with account taken of $\alpha'$ corrections. However, $\alpha'$ corrections include terms involving spin connection. This forced us to formulate duality transformations not in terms of metric, as is usually done, but in terms of spin connection with torsion. The resulting transformations have a rather simple structure, which allowed us to investigate higher order $\alpha'$ corrections.
We will also investigate special field configurations for which these duality transformations do [*not*]{} receive higher order corrections, i.e. are [*exact*]{} transformations to all orders in $\alpha'$. To be specific, our starting point will be the supersymmetric string wave (SSW) solutions to ten-dimensional string theory [@Be1]. These solutions are characterized by an arbitrary vector field $A_\mu$. It has been shown in [@Ho4] that, for the special case of plane fronted waves where only $A_u\ne 0$, these wave solutions are dual to the field outside a straight fundamental string [@Da1]. Furthermore, it is known that the plane fronted waves are an [*exact*]{} solution of the string equations of motion [@Gu1; @Am1; @Ho1]. We will show that the fundamental string (FS) solution is also an exact solution. As a consequence, we find that the duality transformation, applied to a plane fronted wave, is an exact transformation.
The above result can be generalized in the following way. We will derive a theorem stating that, given a field configuration that solves the zero slope limit of the string equations, the same field configuration can be promoted to an exact solution provided that all $\alpha'$ corrections to the space-time supersymmetry transformation rules vanish. The latter condition is equivalent to the requirement that for our solution (i) the (Lorentz + Yang-Mills) Chern-Simons forms vanish and (ii) the so-called $T$-tensors (see eqs. (\[eq:t1\])–(\[eq:t3\])) vanish. In order to fulfill these two conditions, it will be necessary in certain cases to make a nontrivial Ansatz for the Yang-Mills gauge fields[^4].
A corollary of our theorem is that, given two solutions to the zero slope limit of the string equations, which are connected to each other by a duality transformations, then the duality transformation is exact to all orders in $\alpha'$ provided both solutions satisfy the two conditions in the theorem given above. The main application of our theorem in this paper will be to show that the SSW solutions, after duality, lead to a [*generalized*]{} FS solution, which is again an exact solution. Therefore, the duality transformation connecting the two solutions is exact to all orders in $\alpha'$.
Our approach is different from the one developed in [@Se2] in the treatment of the duality transformation of vector fields. We do not include vector fields in the zero slope limit of the effective action but treat the non-abelian vector fields at the level of $\alpha'$ corrections. These corrections cancel against gravitational $\alpha'$ corrections for the special configurations which we are considering.
The organization of this paper is as follows. In section 2 we will review the sigma model derivation of the target space duality transformations in the zero slope limit. As a new result we will also present the duality transformation of the Yang-Mills gauge fields since we will need them in the following. In section 3 we will explicitly show how the target space duality invariance works in the zero slope limit. In the next section this duality transformation will be applied to construct the generalized FS solution. In section 5 we will give the derivation of our theorem. In section 6 we will show that the FS solution is an exact solution and hence that the duality rotation connecting the SSW and generalized FS solution is exact. Details about our notations and conventions can be found in the Appendix A. Finally, in Appendices B and C we prove that the generalized FS solution and the solution of [@KT] are space-time supersymmetric.
Sigma model duality
===================
We consider the sigma model action of the form $$S = {1 \over 2\pi} \int d^2 z \;\left [(g_{\mu\nu} +
B_{\mu\nu})\;\partial
x^{\mu} \bar \partial x^{\nu} + i\; \psi_I(\bar \partial
\psi^I +
V_{\mu}{}^{IK} \bar \partial x^{\mu} \psi_K)\right ] \ .$$ This action is a truncation of a supersymmetric sigma model [@HW] related to the heterotic string. We assume that the background fields $g_{\mu\nu},\; B_{\mu\nu}$ and $V_{\mu}{}^{IK}$ are independent of some coordinate $x$ and may depend on all remaining bosonic coordinates $x^\alpha$, where $\{x^\mu\} = \{x, x^\alpha \}$. This sigma model allows a straightforward generalization of the discrete target space duality transformations rules [@BUS] in presence of the vector fields in the background. We proceed in the standard way [@RV] by presenting the first order action $$\begin{aligned}
S_1& =& {1 \over 2\pi} \int d^2 z \Bigl[ g_{xx} A\bar A +
(g_{x\alpha}
+B_{x\alpha}) A\; \bar \partial x^{\alpha} + \Bigl((g_{\alpha x}
+B_{\alpha x }) \partial x^{\alpha} + V_x \Bigr)\bar A\nonumber\\
\nonumber\\
&+& (g_{\alpha\beta} + B_{\alpha\beta})\;\partial
x^{\alpha} \bar \partial x^{\beta} +
i\psi_I \bar \partial \psi^I +
V_{\alpha}\; \bar \partial
x^{\alpha}
+ \tilde \theta ( \partial \bar A - \bar \partial A) \Bigr] \ .\end{aligned}$$ The following simplifying notation have been used $$\begin{aligned}
V_x &\equiv &i \psi_I V_{x}{}^{IJ} \psi_J\ , \nonumber\\
\nonumber\\
V_\alpha &\equiv &i \psi_I V_{\alpha}{}^{IJ} \psi_J\ .\end{aligned}$$ By integrating out the Lagrange multiplier field $\tilde \theta$ on a topologically trivial world-sheet, one recovers the original action since the solution to the equation $( \partial \bar A - \bar \partial A)=0$ is $A= \partial x, \bar A= \bar \partial x$. It is important to stress that this dualization procedure requires the non-vanishing value of $g_{xx}$. If one integrates out the gauge fields $A, \bar A$ one gets the dual model $$\tilde S = {1 \over 2\pi} \int d^2 z \;\left [(\tilde g_{\mu\nu} +
\tilde B_{\mu\nu})\;\partial \tilde x^{\mu} \bar \partial \tilde
x^{\nu} + i\;
\psi_I(\bar \partial \psi^I +
\tilde V_{\mu}{}^{IK} \bar \partial \tilde x^{\mu} \psi_K)\right ]
\ .$$ The new coordinates are $\{\tilde x^\mu\} = \{\tilde \theta ,
x^\alpha \}$ and the dual metric and antisymmetric tensor field are $$\begin{aligned}
\tilde g_{xx} & =& 1/g_{xx}\ , \qquad \tilde g_{x\alpha} =
B_{x\alpha}/
g_{xx}\ , \nonumber\\
\tilde g_{\alpha\beta} & =& g_{\alpha\beta} -
(g_{x\alpha}g_{x\beta} -
B_{x\alpha}B_{x\beta})/g_{xx}\ , \nonumber\\
\tilde B_{x\alpha} & =& g_{x\alpha}/g_{xx}\ , \qquad
\tilde B_{\alpha\beta} = B_{\alpha\beta} +2
g_{x[\alpha}
B_{\beta]x}/g_{xx}\ .
\label{bus}\end{aligned}$$ The dual vector field is $$\label{eq:dilaton}
\nonumber\\
\tilde V_{x}{}^{JK} = - V_{x}{}^{JK} /g_{xx} \ , \qquad
\tilde V_{\alpha}{}^{JK} = V_{\alpha}{}^{JK} -
(g_{x\alpha} + B_{x\alpha} ) V_{x}{}^{JK} /g_{xx}\ .
\label{vec}$$ Taking into account the one-loop jacobian from integrating out $A,
\bar
A$-fields one finds, as usual, the dilaton transformation rules [@BUS][^5] $$\tilde \phi = \phi - {1\over 2} \log g_{xx} \ .
\label{eq:dil}$$ Thus by using the truncated version of a supersymmetric sigma model it is easy to supplement the well known target space duality transformations of the metric and of the antisymmetric tensor field and of the dilaton by the accompanying transformations of the vector fields.
Target-Space Duality in the Zero Slope Limit
============================================
Our starting point is the bosonic part of the action of $N=1,\, d=10$ supergravity [@Ch1][^6]: $$\label{eq:action}
S(g_{\mu\nu}, B_{\mu\nu}, \phi)
={1\over 2}\int d^{10}x\ \sqrt {-g}e^{-2\phi}\biggl (
-R+4(\partial\phi)^2-
{3\over 4}H^2\biggr )\ ,$$ where $g_{\mu\nu}$ is the metric, $\phi$ the dilaton and $H$ the field strength of the axion $B_{\mu\nu}$. We now consider the special class of field configurations which have an isometry generated by a Killing vector $k^\mu$. It is convenient to use a special coordinate system where $k^\mu$ is constant and is only nonzero in one direction, let us say the $x$ direction. Furthermore we assume that $k^\mu$ is a non-null Killing vector, i.e. $k^2\ne 0$. The isometry property then amounts to the following condition on the field configuration $\{g_{\mu\nu}, B_{\mu\nu}, \phi\}$: $$\label{eq:cond}
k^\mu\partial_\mu \{g_{\mu\nu}, B_{\mu\nu}, \phi\}
= \partial_x \{g_{\mu\nu}, B_{\mu\nu}, \phi\} = 0\ .$$ We want to show that the action (\[eq:action\]) is invariant under target space duality transformations modulo terms which contain the derivative of one of the fields with respect to $x$. In other words we want to show that $$S(g_{\mu\nu}, B_{\mu\nu}, \phi)=
S({\tilde g}_{\mu\nu}, {\tilde B}_{\mu\nu}, {\tilde \phi}) +
\int d^{10}x\ A(g,B,\phi) \partial_x B(g,B,\phi)\ ,$$ where $A$ and $B$ are some expressions in terms of $g_{\mu\nu}, B_{\mu\nu}$ and $\phi$. If this is the case for [*any*]{} field configuration that satisfies (\[eq:cond\]) then the target-space duality transformation serves as a truly solution generating transformation: the dual of [*any*]{} solution of the field equations that is independent of the coordinate $x$ will automatically be another, inequivalent, solution of the same field equations.
In order to show that the action (\[eq:action\]) is invariant under the duality transformations given in the previous section it is convenient to use the zehnbein instead of the metric. The zehnbein transforms under the following duality transformations: $$\begin{aligned}
{\tilde e} _x^a &=& {1\over g_{xx}}e_x^a\ ,\nonumber\\
{\tilde e}_{\alpha}^a &=& e_\alpha^a - {1\over g_{xx}}
\bigl (g_{x\alpha} - B_{x\alpha}\bigr )e_x^a\ .
\label{zehnbein}\end{aligned}$$
From the point of view of reproducing the duality transformation of the metric, given in equation (\[bus\]), we could equally well use a different duality transformation for the zehnbeins. We could have used the transformation $${\tilde e} _x^a = - {1\over g_{xx}}e_x^a\ ,\qquad
{\tilde e}_{\alpha}^a = e_\alpha^a - {1\over g_{xx}}
\bigl (g_{x\alpha} + B_{x\alpha}\bigr )e_x^a\ .$$ This transformation coincides exactly with the duality transformation of the vector field in equation (\[vec\]). We have found, however, that to provide the absence of $\alpha'$ corrections we have to use the duality transformation of the zehnbein given in equation (\[zehnbein\]) if the one for the vector field is given in equation (\[vec\]). Another possibility is to change both of them to the opposite one. This will be a necessary condition for the duality symmetry to preserve the embedding condition of the spin connection into a subgroup of the gauge group.
Under a duality transformation the spin-connection and the axion field strength transform as: $$\begin{aligned}
{\tilde \omega}_c{}^{ab} &=& \omega_c{}^{ab} - {1\over k^2}k_ck^d
\Omega_{d-}{}^{ab} -{2\over k^2}k^dk^{[a}\Omega_{d-,c}{}^{b]}\ ,
\nonumber\\
{\tilde H}_{abc} &=& H_{abc} -{2\over k^2}k^dk_{[a}
\Omega_{d-,bc]}\ ,\end{aligned}$$ where the torsionful spin connections $\Omega_{\mu \pm }{}^{ab}$ are defined by[^7] $$\label{eq:torsion}
\Omega_{\mu\pm}{} ^{ab} = \omega_\mu{}^{ab}(e) \mp {3\over 2}
H_\mu{}^{ab}\ .$$ For later convenience we note that the dual of the torsionful spin connections are given by $$\begin{aligned}
\label{eq:dualtorsion}
{\tilde \Omega}_{c-}{}^{ab} &=& \Omega_{c-}{}^{ab} - {2\over
k^2}k_ck^d
\Omega_{d-}{}^{ab}\ ,\nonumber\\
{\tilde \Omega}_{c+}{}^{ab} &=& \Omega_{c+}{}^{ab} - {4\over k^2}k^d
k^{[a}
\Omega_{d-,c}{}^{b]}\ .
\label{spin}\end{aligned}$$ Using the identity $$\label{eq:Palatini}
- \int d^{10}x\ \sqrt {-g}e^{-2\phi}
R \, =\, \int d^{10}x\ \sqrt {-g}e^{-2\phi}\biggl (
\omega_a{}^{ac}\omega_b{}^{bc} + \omega_a{}^{bc}\omega_b{}^{ca}
+ 4 (\partial_a\phi)\omega_b{}^{ba}
\biggr ) \ ,$$ we can now rewrite the action (\[eq:action\]) in the following convenient form: $$\label{eq:action2}
S={1\over 2}\int d^{10}x\ ee^{-2\phi}\biggl (
4(\partial^a\phi + {1\over 2}\omega_b{}^{ba})^2 +
\omega_a{}^{bc}\omega_b{}^{ca} - {3\over 4} H_{abc}H^{abc}
\biggr )\ ,$$ where $e= {\rm det}\ e_\mu^a$. Using the additional identity $$k^bk^c\omega_{b,ca} = -{1\over 2}\partial_a k^2\ ,$$ and transformation rule $$\tilde e = {1\over k^2} e\ ,$$ it is straightforward to show that the action (\[eq:action2\]) is invariant under target-space duality transformations. We note that the action consists of three parts which are separately duality invariant: the combinations $ee^{-2\phi}, \ (\partial^a\phi +
{1\over 2}\omega_b{}^{ba})$ and $(\omega_a{}^{bc}\omega_b{}^{ca} -
{3\over 4} H_{abc}H^{abc})$ are all three duality invariant.
Thus we have shown that the target space action in the zero slope limit is invariant under the sigma model duality transformations, given in equations (\[bus\]) and (\[eq:dil\]). The vector fields are not present in the effective action in the zero slope limit.
New Solutions in the Zero Slope Limit
=====================================
The purpose of this section is to show how the duality transformations can be used to generate new solutions in the zero slope limit. Higher-order $\alpha'$ corrections will be considered in the next two sections. Our starting point is the Supersymmetric String Wave (SSW) solution of [@Be1]. In [@Be1] it was shown that the SSW solves the string equations of motion to all orders in $\alpha'$. Here we will only consider the zero slope limit. In particular, we will set the vector gauge fields equal to zero. The solution thereby reduces to the one given in [@Ru1], [@Ho3]. A crucial feature of the SSW solution is the existence of a null Killing vector $l^\mu$ with $l^2=0$. This Killing vector generates an isometry in the $v$ direction where we use light-cone coordinates $x^\mu = (u,v,x^i)\ (i=1,\dots ,8)$. Since this Killing vector is null we cannot use it for a duality transformation. We therefore make the additional assumption that the fields occurring in the SSW solution are also independent of the $u$ coordinate. Since the Ansatz for the dilaton in the SSW solution only depends on $u$, it must be a constant and for simplicity we will take this constant to be zero. The remaining nonzero fields $g_{\mu\nu}$ and $B_{\mu\nu}$ are both described in terms of one vector function of the transverse coordinates $x^i$: $$A_\mu(x^i) = \{A_u(x^i), A_v=0, A_i(x^i)\}\ .$$ They are given by the Brinkmann metric [@Br1] and the following 2-form[^8]: $$\begin{aligned}
\label{eq:SSW}
ds^2 &=& 2dudv + 2A_\mu dx^\mu du - dx^idx^i\ ,\nonumber \\
\label{eq:SSW2}
B &=& 2 A_\mu dx^\mu \wedge du\ .\end{aligned}$$ The equations that $A_u(x^i)$ and $A_i(x^i)$ have to satisfy are: $$\label{eq:Lapl}
\triangle A_u = 0\ , \hskip 1.5truecm \triangle\partial^{[i}A^{j]} =
0\ ,$$ where the Laplacian is taken over the transverse directions only. Since $g_{\mu\nu}$ and $B_{\mu\nu}$ are independent of $u$ and $v$ they are independent of $x$ and $t$. For our duality transformation we will use the isometry in the space-like $x$ direction. The non-null Killing vector generating this isometry will be denoted by $k^\mu$. Note that we have now two isometries given by $$l^\mu\partial_\mu \{g_{\mu\nu}, B_{\mu\nu}\} = \partial_v
\{g_{\mu\nu}, B_{\mu\nu}\} = 0 \ ,\hskip 1truecm
k^\mu\partial_\mu \{g_{\mu\nu}, B_{\mu\nu}\} = \partial_x
\{g_{\mu\nu}, B_{\mu\nu}\} = 0\ .$$ A straightforward application of the sigma-model duality transformations given in (\[bus\]), (\[eq:dil\]) on the SSW solution given in eq. (\[eq:SSW\]) leads to the following new solution of the zero slope limit equations of motion: $$\begin{aligned}
\label{eq:new}
ds^2 &=& - (A_u-1)^{-1}\bigl \{ 2dudv + 2 A_i dudx^i
\bigr \} - dx^idx^i\ ,\nonumber\\
B &=& (A_u-1)^{-1} \bigl
\{ 2A_u du \wedge dv + 2 A_idu \wedge dx^i \bigr \}\ ,\\
\phi &=& -{1\over 2} {\rm log} (A_u -1)\ .\nonumber\end{aligned}$$ Note that we can make the following particular choice of the vector function $A_\mu$[^9]: $$A_u = -{\tilde c M\over r^6}\ , \hskip 1.5truecm A_i = 0\ ,$$ where $r^2 = x^ix_i$ and $\tilde c$ a constant. The solution given in (\[eq:new\]) reduces then to the solution of [@Da1] corresponding to the field outside a fundamental string with total mass $M$. We will hence refer to the solution (\[eq:new\]) as the generalized FS solution[^10].
It was shown in [@Be1] that the SSW solution is supersymmetric under 8 of the 16 ten-dimensional supersymmetries. In [@Da1], it has been shown that the dual FS solution, for the special case that $A_i=0$, is again supersymmetric. In Appendix B we will show that also the generalized FS solution, with $A_i\ne 0$, is supersymmetric.
$\alpha'$ corrections
======================
In this section we will consider the $\alpha'$ corrections to the zero slope limit. In particular, we will derive a theorem stating that any solution to the zero slope limit string equations can be promoted to an exact solution to all orders in $\alpha'$ provided that (i) the $T$-tensors defined by eqs. (\[eq:t1\]) - (\[eq:t3\]) vanish and (ii) a particular combination of the (Lorentz + Yang-Mills) Chern-Simons terms vanish. We note that the discussion in this section is independent of any particular solution to the field equations.
There exists one remarkable property of the duality transformations considered above, which is of crucial importance for the understanding of $\alpha'$ corrections. This property is best see in terms of the duality transformations of the torsionful spin connections $\Omega_{c-}{}^{ab}$, defined above in eq. (\[eq:torsion\]). The useful form of this transformation is $$\label{eq:tro}
{\tilde \Omega}_{c-}{}^{ab} = \Pi _c{}^d \; \Omega_{d-}{}^{ab} \ ,$$ where we have introduced the projector $$\Pi _c{}^d \equiv \delta _c{}^d - {2\over k^2}k_ck^d \ .$$ The square of this projector is a unit operator: $$\Pi _c{}^d \Pi _d{}^e = \delta _c{}^e \ .$$ This property of the projector nicely confirms the fact that we are performing a discrete operation and that two such operations bring us back to the original configuration since $${\tilde {\tilde \Omega}}_{c-}{}^{ab} = \Pi _c{}^d \; {\tilde
\Omega}_{d-}{}^{ab} =
\Pi _c{}^d \; \Pi _d{}^e \; \Omega_{e-}{}^{ab} =
\Omega_{c-}{}^{ab} \ .$$
Now let us consider the $\alpha^{\prime}$ corrections to the string effective action. It is well known that one has to add to $S^{(0)}$ the Lorentz and Yang-Mills Chern-Simons terms which play a crucial role in the Green-Schwarz anomaly cancellation mechanism[^11]. These terms break supersymmetry. To restore supersymmetry order by order in $\alpha^{\prime}$, one has to add to $S^{(0)}$ an infinite series of higher order terms in $\alpha^{\prime}$. By the procedure of adding terms to restore supersymmetry, the effective action was obtained in [@Be1] up to order $O(\alpha^{\prime 4})$ terms: $$\begin{aligned}
\label{eric}
S & = & {1\over 2}\int d^{10}x\ \sqrt {-g}e^{-2\phi}\biggl (
-R+4(\partial\phi)^2- {3\over 4}H^2+ \nonumber \\ &&+{1\over 2}T +
2\,
\alpha^{\prime} T^{\mu\nu} T_{\mu\nu}
+6 \, \alpha^{\prime} T^{\mu\nu\lambda\rho}
T_{\mu\nu\lambda\rho}+ O(\alpha^{\prime 4})\biggr )\ ,\end{aligned}$$ where the antisymmetric tensor $T_{\mu\nu\lambda\rho}$, the symmetric tensor $T_{\mu\nu}$, and the scalar $T$ are given by $$\begin{aligned}
T_{\mu\nu\lambda\rho} &=&2 \alpha^{\prime} \biggl ( \,
R_{[\mu\nu}{}^{ab}\bigl
(\Omega_-)
\, R_{\lambda\rho]}{}^{ab}\bigl (\Omega_-)
+
\frac{1}{30}\, {\rm tr} F_{[\mu\nu}
F_{\lambda\rho]} \biggr )\label{eq:t1} \
,\\
T_{\mu\nu} &= &2 \alpha^{\prime} \, \biggl (
R_{\mu\lambda}{}^{ab}\bigl
(\Omega_-)
R^{\lambda } {}_{ \nu} {}^{ ab}(\Omega_-)
+ \frac{1}{30} \, {\rm tr} F_{\mu \lambda}
F^{\lambda}{}_{\nu} \biggr )\label{eq:t2} \ ,\\
T & = & T_\mu{}^\mu \ \ .
\label{eq:t3}\end{aligned}$$ In the above expression there are explicit [*and*]{} implicit $\alpha^{\prime}$ corrections. The explicit corrections always appear via $T$-tensors, and they are essentially the $\alpha^{\prime}$ factors in front of eqs. (\[eq:t1\])-(\[eq:t3\]). The implicit $\alpha^{\prime}$ corrections always appear via the torsion $H$ which is defined by the following iterative procedure: At the lowest order $H$ is just $H^{(0)}_{[\mu\nu\rho]}=
\partial_{[\mu}B_{\nu\rho]}$. With $H^{(0)}$ we calculate the lowest order $\Omega_{\pm}=\Omega_{\pm}^{(0)}$, as given in eq. (\[eq:torsion\]). At first order in $\alpha^{\prime}$, $H=H^{(1)}$ is $H^{(0)}$ corrected by the Yang-Mills Chern-Simons term and the Lorentz Chern-Simons term corresponding to the zero-order $\Omega_{-}=\Omega_{-}^{(0)}$: $$H^{(1)}_{\mu\nu\rho} = \partial_{[\mu}B_{\nu\rho]}
+\alpha^{\prime}( \omega_{\mu\nu\rho}^{Y.M.} +
\omega_{\mu\nu\rho}^{L}) \ ,$$ where $$\omega_{\mu\nu\rho}^{Y.M.}= -\frac{1}{5}
{\rm tr}
\biggl \{ V_{[\mu}\partial_\nu V_{\rho]}-
{1\over 3}V_{[\mu}V_\nu V_{\rho]}\biggr \} \ ,$$ and $$\label{eq:LCS}
\omega_{\mu\nu\rho}^{L} = -6 \, \biggl
\{ \Omega^{(0)}_{[\mu-}{}^{ab}\partial_\nu
\Omega^{(0)}_{\rho]-}{}^{ab}-
{1\over 3}\Omega^{(0)}_{[\mu-}{}^{ab}\Omega^{(0)}_{\nu-}{}^{ac}
\Omega^{(0)}_{\rho-]}{}^{cb}\biggr \} \ .$$
With $H^{(1)}$ one would get $\Omega^{(1)}$ using again its definition eq. (\[eq:torsion\]) and $H^{(2)}$ would be given by the above expression but with $\Omega^{(0)}$ replaced by $\Omega^{(1)}$. Iterating this procedure one gets the all-order expression $H$ for the torsion which involves the promised infinite series of corrections.
In short, to be able to understand the properties of $\alpha^{\prime}$ corrections to specific configurations we have to calculate the value of the Lorentz and Yang-Mills Chern-Simons term and the values of all $T$-tensors, presented above.
To study the corrections to the equations of motion we will use the corresponding analysis, performed in [@Be1]. We first have to vary the action (\[eric\]) over all the fields present in the theory, and only then substitute the solutions in the corrected equations. We will study the linear corrections separately. The equations of motion corrected up to first order come from the action $$S^{(1)}={1\over 2}\int d^{10}x\ \sqrt
{-g}e^{-2\phi}\biggl (-R+4(\partial\phi)^{2}
-{3\over 4}H^{2} +\frac{1}{2}T \biggr )\ .$$ All terms of order $\alpha^{\prime 2}$ and higher are neglected at this stage. The corrections linear in $\alpha^{\prime}$ to the lowest order equations of motion are derived from the variation $\delta (S^{(1)}-S^{(0)})\equiv \delta
\Delta S$. It is convenient to perform this variation with respect to the explicit dependence of the action on $g_{\mu\nu}$, $V_{\mu}$, $\phi$ and $B_{\mu\nu}$, and then with respect to the implicit dependence on these fields through the torsionful spin connection $\Omega_{-}$, that is $$\begin{aligned}
\delta \Delta S & = &
\frac{\delta\Delta S}{\delta g_{\mu\nu}} \delta g_{\mu\nu}
+\frac{\delta\Delta S}{\delta B_{\mu\nu}}\delta B_{\mu\nu}
+\frac{\delta\Delta S}{\delta\phi}\delta\phi+ \nonumber \\
&&+\frac{\delta\Delta S}{\delta V_{\mu}}\delta V_{\mu}
+\frac{\delta\Delta S}{\delta \Omega_{\mu -}{}^{ab}}
\delta\Omega_{\mu -}{}^{ab}\ ,\label{DS}\end{aligned}$$ where $$\delta\Omega_{\mu -}{}^{ab}=
\frac{\delta\Omega_{\mu -}{}^{ab}}{\delta g_{\mu\nu}} \delta
g_{\mu\nu}
+\frac{\delta\Omega_{\mu -}{}^{ab}}{\delta B_{\mu\nu}}\delta
B_{\mu\nu}
+\frac{\delta\Omega_{\mu -}{}^{ab}}{\delta V_{\mu}}\delta V_{\mu}\ .$$ The explicit variations are $$\begin{aligned}
\frac{\delta\Delta S}{\delta g_{\mu\nu}} & = &
-\frac{1}{4}\sqrt{-g}e^{-2\phi}(T^{\mu\nu}-g^{\mu\nu}T)\ ,
\\
\frac{\delta\Delta S}{\delta \phi} & = &
-\frac{1}{2}\sqrt{-g}e^{-2\phi}T\ , \\
\frac{\delta\Delta S}{\delta B_{\mu\nu}}
& = & \frac{3}{4}\,
\partial_{\lambda}[\sqrt{-g}e^{-2\phi}
(H^{(1)}{}^{\lambda\mu\nu}
-H^{(0)}{}^{\lambda\mu\nu})] \ , \\
\frac{\delta\Delta S}{\delta V_{\mu}} & =
&\frac{1}{15}\alpha^{\prime}\biggl\{
\partial_\lambda
\bigl ( \sqrt{-g}e^{-2\phi}
F^{\lambda\mu}\bigr
)
+\sqrt{-g}e^{-2\phi}[V_\lambda
,F^{\lambda \mu}]+
\nonumber \\
&&-\frac{3}{2} \sqrt{-g}e^{-2\phi}
H^{(0)\mu}{}_{\lambda\rho}
F^{\lambda\rho}
+\frac{3}{2}V_{\rho}
\partial_{\lambda}(\sqrt{-g}e^{-2\phi}
H^{(0)}{}^{\lambda\mu\rho})\biggr
\}\ .\end{aligned}$$ If some solution of the zero slope limit equations has the property that the $T$-tensors vanish on this solution, the first two equations above do not get linear $\alpha'$ corrections. The third equation is not corrected, provided the combination of Lorentz and Yang-Mills Chern-Simons term is vanishing for the configuration. The last equation is satisfied if the classical equation of motion for the vector field[^12] as well as the equation of motion for the $B_{\mu\nu}$-field in the zero slope limit are satisfied. We next consider the implicit variations represented by the last term in eq. (\[DS\]). In [@Be2] a rather non-trivial property of the string effective action was analysed. This property is that all the implicit variations are proportional to the lowest order equations of motion of the different fields[^13]. We therefore conclude that, to linear order in $\alpha'$, any solution of the lowest order equations of motion are also solutions of the equations of motion corrected to order $\alpha^{\prime}$ provided that (i) the T-tensor $T_{\mu\nu}$ and (ii) the (Lorentz + Yang-Mills) Chern-Simons terms vanish.
Now let us consider the higher order $\alpha'$ corrections. The general structure of the bosonic part of the effective action, which can be obtained by the procedure outlined before based upon the restoration of supersymmetry, has been conjectured in ref. [@Be1]. New terms in the action are quadratic or of higher degree in the $T$-tensors. Therefore their contribution to the string equations of motion automatically vanishes for the configurations with vanishing $T$-tensors. This concludes the proof of the theorem stated at the beginning of this section.
Finally, we would like to note that the vanishing of a combination of Lorentz and Yang-Mills Chern-Simons term in $H_{\mu \nu \lambda}$ and the vanishing of all $T$-tensors is sufficient for the absence of corrections to the supersymmetry transformation laws and the Killing spinors [@Be1], [@Be2]. As a corollary of this we conclude that the condition for a field configuration that solves the zero slope limit equations of motion to be exact, i.e. to be free of $\alpha^{\prime}$-corrections, coincides with the property of the configuration to have vanishing $\alpha^{\prime}$-corrections to classical supersymmetry transformations.
Exact duality between pp-waves and strings
==========================================
The complete discussion of the supersymmetric string waves and $\alpha'$ corrections has been performed in [@Be1]. Our final conclusion was that the on-shell action, the fields that solve the lowest order equations of motion and the Killing spinors for the SSW solutions do not receive any higher order string corrections.
In this section we would like to use the pp-wave solution, i.e. the SSW solution with $A_i=0$, as the starting point for investigation of the corrections to duality symmetry in the target space. In the next section we will consider the general case with $A_i \ne 0$.
Gravitational plane fronted waves [@Br1] have a null Killing vector $\nabla_{\mu} l_{\nu} = 0\ , \quad l^{\nu}l_{\nu}= 0$ and very special dependence of this null vector. This simplifies the analysis of higher order corrections to field equations.
Consider the class of 10-dimensional pp-waves with metrics of the form $$\label{d}
ds^2 = 2 du dv + K (u, x^i ) du^2 - dx^i d x^i \ ,$$ where $i = 1,2, ..., 8$, the Riemann curvature is [@Ho1; @Ho3] $$R_{\mu\nu\rho\sigma} = - 2 l_{[\mu}( \partial_{\nu]}
\partial_{[\rho} K )
l_{\sigma]}\ .$$ The curvature is orthogonal to $l_{\mu}$ in all its indices. This fact is of crucial importance in establishing that all higher order in $\alpha^{\prime}$ terms in the equations of motion are zero due to the vanishing of all the possible contractions of curvature tensors. The dilaton, the antisymmetric tensor field and the vector field are absent in this solution. The only non-trivial function in the metric has to satisfy the equation $$\triangle K = 0\ ,$$ where $\triangle=\partial_{i}\partial_{i}$ is the flat space Laplacian. The spin connection of these pp-waves is given by the following expression: $$\omega_\mu{}^{ab} = - l_\mu \; l^{[a}\partial^{b]} K \ .$$ Note also that the indices $ab$ related to the fact that the spin connection is a Lorentz-Lie-algebra valued object$$\omega \equiv dx^\mu \omega_\mu {} ^{ab} M_{ab}=
- dx^\mu l_\mu \;l^{[a}\partial^{b]} K \; M_{ab}$$ have at least one null vector. In this case the spin connection coincides with the torsionful spin connection since the antisymmetric field strength tensor $H_{abc}$ vanishes: $$\Omega_{c-}{}^{ab}= - l_c \; l^{[a}\partial^{b]} K\ .$$ This spin connection is orthogonal to $l_a$ in all indices and what is very important, has at least one $l$-vector in $ab$-type indices.
In the discussion below we also have to take into account that $$\partial_\mu l^a =0\ .$$ The structures which we need, Lorentz Chern-Simons term given in eq. (\[eq:LCS\]) and all T-tensors, given in eqs. (\[eq:t1\])-(\[eq:t3\]), always have all $ab$-type indices contracted. This means that for the pp-waves all these structures vanish. There is no way to contract those null vectors, if they are contracted with another $l$ we get zero and if they are contracted with $\partial$ we have zero again, since this a Killing condition, the solution is independent of $v$. Thus for pp-waves the $\alpha^{\prime }$ corrections to the action, to the equation of motion and to supersymmetric transformations of all fermion fields vanish.
As explained in [@Ho4] the dual version of pp-waves (without a dilaton, antisymmetric field and vector field) is a string solution of [@Da1]. As explained above, the pp-waves have no $\alpha'$ corrections since the Lorentz Chern-Simons term and all T-tensors vanish for pp-waves. Now that we have a simple duality transformation rule for the torsionful spin connection we can investigate the corresponding features of the solution dual to the pp-waves.
As stated in the beginning of the previous section, the duality transformation ${\tilde \Omega}_{c-}{}^{ab} = \Pi _c{}^d \; \Omega_{d-}{}^{ab}$ does not affect the structure of the $ab$-type indices in this expression. Therefore the proof that the Lorentz Chern-Simons term and all T-tensors vanish for pp-waves is immediately extended to the dual version, i.e. to the string solution of [@Da1]. The reason for that is that all those terms always have at least one null Killing vector in $a$ or $b$ direction, which has to be contracted with another $a$- or $b$-type index. Although the contraction with new Killing vector $k^\mu$ is possible in principle and would give a non-vanishing contribution since $k^2\ne 0$ and $k\cdot l \ne 0$, we see from the structure of Chern-Simons term and T-tensors that after duality transformation no such contraction actually occurs.
Having established the vanishing of $\alpha'$ corrections to Lorentz Chern-Simons term and all T-tensors for the FS string solution of [@Da1] we may apply the general analysis performed in the previous section. We find that there are no corrections to the FS solution of the zero slope limit equations. We therefore conclude that both the FS solution of [@Da1] and the pp-waves are exact solutions of the string effective action.
A simplifying feature of the pp-waves is the fact that they cannot have corrections independently of the specific form of the $\alpha'$ corrections. For the dual version, the FS solution, we have under complete control only the $\alpha'$ corrections described above which are related to anomalies. [*A priori*]{} they could receive an infinite set of corrections, in principle, unless the iterative procedure requires them to vanish on each step as we have seen above.
The conclusion of this section is that the duality transformation, which relates the pp-waves to the FS solution is an exact duality transformation, i.e. the $\alpha'$ corrections to the duality transformation vanish for this case.
Exact duality between supersymmetric string waves and generalized FS solutions
==============================================================================
In this section we extend the discussion of the previous section to the generalized FS solution given in eq. (\[eq:new\]) for the case that $A_i \ne 0$. It has been shown in [@Be1] that even for this case the corresponding SSW solution is an exact solution to all orders in $\alpha'$. This is a nontrivial result since with $A_i\ne 0$ both the Lorentz Chern-Simons term as well as the $T$ tensors do [*not*]{} vanish. To make both vanish one must make a nontrivial Ansatz for the vector gauge fields $V_\mu{}^{IJ}$ and embed the torsionful spin connection $\Omega_-$ in an $SO(8)$ subgroup of the gauge group, thereby identifying the spin connection with the gauge field: $$\label{eq:embedding}
{1\over \sqrt{30}}V_\mu{}^{IJ}
= l_\mu V^{IJ}\hskip .5truecm \equiv \hskip .5truecm
\Omega_{\mu-}{}^{ab} = l_\mu {\cal A}^{ab}
\ \ \ \ (a,b,I,J=1,\dots ,8)\ .$$ Here the Yang-Mills indices refer to the adjoint representation of $SO(8)$. One then makes use of the fact that with a non-zero gauge field the Lorentz Chern-Simons term always occurs together with a Yang-Mills Chern-Simons term. The same applies to the $T$ tensors where the $R^2$ terms are always accompanied by similar $F^2$ terms. The above identification then leads to a cancellation between the spin connection and gauge field terms such that even with $A_i\ne 0$ the (Lorentz + Yang-Mills ) Chern-Simons terms and the (generalized) $T$ tensors do vanish. Of course this cancellation only involves terms with the function $A_i$. The terms involving $A_u$ in the Lorentz sector already cancel by themselves as has been shown in the previous section.
To investigate what happens with the (Lorentz + Yang-Mills) Chern-Simons terms and the $T$ tensors for the generalized FS solution we apply a duality transformation on the torsionful spin connection corresponding to the SSW solution. This leads to the following expression of $\Omega_-$ for the generalized FS solution: $$\tilde {\Omega}_{c-}{}^{ab} = \{l_c -
{2 k\cdot l\over k^2}k_c\}{\cal A}^{ab} = \Pi_c{}^d l_d {\cal
A}^{ab}\
{}.$$ Note that the structure of the $ab$ indices remains unchanged under a duality rotation.
In order to show that the generalized FS solution is again a solution of the field equations to all orders in $\alpha'$ it is essential that the above-mentioned cancellation between the spin connection and gauge field terms in the (Lorentz + Yang-Mills) Chern-Simons terms and the $T$ tensors is not spoiled by the duality transformation. We therefore require that the embedding is duality invariant. In order to obtain a duality-invariant embedding we want that $\Omega_-$ transforms in the same way as the Yang-Mills gauge fields. Luckily enough it turns out that this is indeed the case. Using the duality transformation of the vector fields given in (\[eq:dilaton\]) one can show that the duality transformations of the spin connection and the gauge fields have precisely the same form:[^14]: $$\tilde {\Omega}_{c -}{}^{ab} = \Pi_c{}^d l_d {\cal A}^{ab}\hskip
1truecm
{\underline {{\rm and}}} \hskip 1truecm
{1\over \sqrt{30}}\tilde {V}_c{}^{IJ} = \Pi_c{}^d l_d {\cal A}^{IJ}\
{}.$$ This means that the embedding condition (\[eq:embedding\]) is indeed duality invariant and the cancellations which took place between the spin connection and gauge field terms in the SSW solution again take place, [*after*]{} the duality rotation, for the generalized FS solution. Hence, for the generalized FS solution we can again derive that the (Lorentz + Yang-Mills) Chern-Simons terms and the $T$-tensors are zero. At this point we can use the results of section 5 where we have shown that the vanishing of the (Lorentz + Yang-Mills) Chern-Simons form and the $T$ tensors is enough to ensure that the zero-slope limit solution extends to a solution to all orders in $\alpha'$.
The conclusion is therefore that the generalized FS solution given by $$\begin{aligned}
\label{eq:generalised}
ds^2 &=& - (A_u-1)^{-1}\bigl \{ 2dudv + 2 A_i dudx^i
\bigr \} - dx^idx^i\ ,\nonumber\\
B &=& (A_u-1)^{-1} \bigl
\{ 2A_u du \wedge dv + 2 A_idu \wedge dx^i \bigr \}\ ,\\
\phi &=& -{1\over 2} {\rm log} (A_u -1)\ ,\nonumber\\
V_\mu^{IJ} &=& - (A_u-1)^{-1} l_\mu {\cal A}^{IJ}\hskip 1truecm
(I,J=1,\dots ,8)\ ,\nonumber
\end{aligned}$$ with ${\cal A}_{\mu\nu}=\partial_\mu A_\nu -
\partial_\nu A_\mu$ solves the string equations of motion to all orders in $\alpha'$. The equations that $A_u(x^i)$ and $A_i(x^i)$ have to satisfy are: $$\triangle A_u = 0\ , \hskip 1.5truecm \triangle\partial^{[i}A^{j]} =
0\ ,$$ where the Laplacian is taken over the transverse directions. Furthermore, the duality transformation connecting the SSW solution and the generalized FS solution is exact to all orders in $\alpha'$.
Conclusion
==========
In this paper we presented a set of duality transformations for the zehnbein, spin connection (with torsion) and vector fields. They are given in equations (\[bus\]), (\[vec\]), (\[zehnbein\]), (\[spin\]). The most elegant duality transformation of the spin connection with torsion is given in equation (\[eq:tro\]). This specific transformation plays a crucial role in the analysis of $\alpha'$ corrections.
We have found the conditions under which a target space duality symmetry is exact, i.e. does not acquire $\alpha'$ corrections. Two configurations can be qualified as being exactly dual to each other if:
i\) there exists a non-null Killing vector in the original as well as in the dual configuration, which allows one to identify the corresponding sigma-model duality transformation. This transformation defines a symmetry of the zero slope limit of the string effective action in the target space.
ii\) the condition for the action and equations of motion not to acquire $\alpha'$ corrections is provided by the vanishing of the combination of the Lorentz and Yang-Mills Chern-Simons term as well as by the vanishing of the $T$-tensors for the original as well as for the dual configuration.
iii\) exact duality in all explicit examples known to us brings one supersymmetric configuration to another supersymmetric configuration. The zero slope supersymmetric transformation rules are not affected by the $\alpha'$ corrections for the original as well as the final configurations related by exact duality.
Exact duality defined above serves as an exact solution generating transformation. As an example of configurations related by exact duality we have analysed the pp-waves and fundamental string solutions [@Da1]. Both solutions are free of $\alpha'$ corrections [^15].
As a further application of our general results we have also checked that the examples of exact duality investigated by Klimč' ik and Tseytlin [@KT] also have some unbroken space-time supersymmetry. Their original configuration is supersymmetric, since it is equivalent to one of the Güven [@Gu1] solutions. The dual configuration turns out to be also supersymmetric, as we have shown in Appendix C.
A more general example of exact duality is given by the supersymmetric string waves [@Be1] and the generalized fundamental strings given in (\[eq:generalised\]). The last solution to the best of our knowledge is new. The proof that it is free of anomaly related $\alpha'$ corrections is provided by the corresponding properties of its dual partner, supersymmetric string waves, and by special properties of duality symmetry to preserve the condition of the embedding of the spin connection in a subgroup of a gauge group. Our generalized fundamental strings are different from the charged heterotic string solution obtained by Sen [@Se2] by twisting the uncharged string solution of [@Da1]. For instance, in Sen’s solution the charge-dependent terms in the metric occur in the $du^2 + dv^2$ sector, whereas in our case the $A_i$-dependent terms occur in the $dudx^i$ sector. Another difference is that in our solution the antisymmetric tensor contains $A_i$-dependent terms, whereas in Sen’s solution the antisymmetric tensor has no charge-dependent terms. Sen’s solution is known to be supersymmetric in the zero slope limit [@Se2; @Wa1]. However, no information is available about the $\alpha'$ corrections to this solution.
The advantage of our method of using sigma-model duality is in the fact that the structure of $\alpha'$ corrections is under control for the dual solution if it was under control for the original solution. In this way the nice properties of the plane waves are carried over to the string-type solutions via sigma model duality, acting as a symmetry of the target space action.
Acknowledgements {#acknowledgements .unnumbered}
================
We are grateful to G. Horowitz for a useful discussion. The work of E.B.and R.K. was partially supported by a NATO Collaborative Research Grant. The work of E.B. has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences (KNAW). E.B. would like to thank the Physics Department of Stanford University for its hospitality. The work of R.K. was supported in part by NSF grant PHY-8612280.
Notation and conventions
========================
We use a metric with mostly minus signature. Our conventions for the Riemann tensor and the spin connection are given in an appendix of [@Be1]. We often use a light-cone basis for the ten-dimensional coordinates $x^\mu$: $$x^\mu = (u,v,x^i),\ i=1,\dots ,8, \ \ \ u={1\over \sqrt
2}(t+x),\ v
={1\over \sqrt 2}(t-x)\ ,$$ where $t\equiv x^0$ and $x\equiv x^9$. In this paper, all the indices are raised and lowered with the full ten-dimensional metric $g_{\mu\nu}$. In the case in which the metric corresponds to the SSW solution given in eq. (\[eq:SSW\]), the relation between upper and lower indices is $$\begin{aligned}
\xi^{u} & = & \xi_{v}\ \ , \\
\xi^{v} & = & \xi_{u}-(2A_{u}+\sum_{i=1}^{8}A_{i}^{2})\xi_{v}+
\sum_{i=1}^{8}A_{i}\xi_{i}\ \ ,\\
\xi^{i} & = &A_{i}\xi_{v}-\xi_{i}\ \ .\end{aligned}$$ The constant Killing vectors $k^\mu$ and $l^\mu$ are given in the light-cone basis by: $$\begin{aligned}
k^\mu &=& {1\over \sqrt 2} (1,-1,0,\dots,0) \ ,\nonumber\\
l^\mu &=& (0,1,\dots,0)\ .\end{aligned}$$ The expressions for these Killing vectors with down indices depend on the metric we are using. For the SSW metric (\[eq:SSW\]) they are given by: $$\begin{aligned}
k_\mu &=& {1\over 2}\sqrt 2 (2A_u-1, 1, A_i)\ ,\nonumber\\
l_\mu &=& (1,0,\dots , 0)\ .\end{aligned}$$ The inner products between $k$ and $l$ for the SSW metric are given by $$l^2 = 0\ ,\hskip 1truecm k^2=A_u-1\ ,\hskip 1truecm k \cdot l =
{1\over 2}
\sqrt 2\ .$$
Proof of Supersymmetry of the Generalized FS Solution
=====================================================
In this appendix we will show that the generalized FS solution is supersymmetric. Since the supersymmetry transformation rules involve fermions it is necessary to reformulate the SSW Ansatz for the metric in terms of zehnbein fields: $$e_\mu{}^a = \delta_\mu{}^a + A_\mu l^a\ .$$ The unbroken supersymmetries of the SSW solution are given by $$\label{eq:g1}
l^\mu\gamma_\mu\epsilon_0 = 0 \hskip .5truecm {\rm or}
\hskip .5truecm (\gamma^0+\gamma^9)\epsilon_0 = 0\ ,$$ where $\epsilon_0$ is a constant ten-dimensional spinor. It is instructive to see how the sigma model duality transformation leads to unbroken supersymmetries for the generalized FS solution as well.
In order to investigate the existence of unbroken supersymmetries for our purely bosonic solutions we only need to consider the bosonic terms in the supersymmetry transformation rules of the fermions. They are given by $$\begin{aligned}
\delta \psi_\mu &=& \bigl (\partial_\mu - {1\over 4}
\Omega_{\mu +}{}^{ab}\gamma_{ab}\bigr)
\epsilon \ ,
\label{eq:susy1}\\
\delta\lambda &=& \bigl (\gamma^\mu\partial_\mu\phi + {1\over 4}
H_{\mu\nu\rho}
\gamma^{\mu\nu\rho}\bigr)\epsilon \ ,
\label{eq:susy2}\\
\delta\chi &=& - {1\over 4} F_{\mu\nu}\gamma^{\mu\nu}\epsilon\ .\end{aligned}$$ We first consider the $\lambda$ transformation rule. Requiring that $\delta\lambda = 0$ leads to the equation $$\label{eq:dl=0}
\gamma^i(\partial_i\phi)\epsilon + {3\over
2}H_{iuv}\gamma^{iuv}\epsilon
+ {3\over 4}H_{iju}\gamma^{iju}\epsilon
= 0\ ,$$ where it is understood that in this equation the generalized FS solution is substituted. To investigate this equation we need the form of the (inverse) zehnbein fields of the generalized FS solution. They are given by: $$\begin{aligned}
e_a^u &=& \delta^v_a \ ,\nonumber\\
e^v_a &=& (1-A_u)\delta^u_a - A_i\delta^i_a\ .\\
e^i_a &=& \delta^i_a\nonumber\ .\end{aligned}$$ It is now not too difficult to show that the first two and the last terms in (\[eq:dl=0\]) vanish separately provided that the supersymmetry parameter $\epsilon$ satisfies[^16] $$(\gamma^0 - \gamma^9)\epsilon = 0\ .
\label{eq:condsusy}$$ We next consider the gravitino transformation rule. Instead of substituting the generalized FS solution into the equation $\delta\psi_\mu=0$ it is easier, and equivalent, to use the SSW solution and to require that after a duality transformation $\delta\psi_\mu=0$. This leads to the equation $$\delta\psi_\mu = \bigl (\partial_\mu - {1\over 4}
\tilde {\Omega}_{\mu+}{}^{ab}\gamma_{ab}\bigr )\epsilon = 0\ .$$ Applying the duality rotation of the torsionful spin connection given in (\[eq:dualtorsion\]) we find $$\partial_\mu\epsilon -{1\over
4}\Omega_{\mu+}{}^{ab}\gamma_{ab}\epsilon
+ {1\over k^2}k^\lambda k^a\Omega_{\lambda -,\mu}{}^b\gamma_{ab}
\epsilon = 0
\ ,$$ where it is now understood that the SSW solution is substituted. We next substitute the expression for the torsionful spin connections corresponding to the SSW solution [@Be1]: $$\Omega_{\mu+}{}^{ab} = -2 l^{[a}{\cal A}^{b]}{}_\mu\ ,\hskip 1truecm
\Omega_{\mu-}{}^{ab} = {\cal A}^{ab}l_\mu\ ,$$ where ${\cal A}_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$. We thus find that the following equation must be satisfied: $$\partial_\mu\epsilon + {1\over 2} l_\rho {\cal A}_{\sigma\mu}
\gamma^{\rho\sigma}\epsilon + {k \cdot l\over k^2} k_\rho
{\cal A}_{\mu\sigma}\gamma^{\rho\sigma}\epsilon = 0\ .$$ The $\mu=v$ component of this equation is automatically satisfied. Using the condition (\[eq:condsusy\]) we find that the $\mu = u$ component is satisfied as well. Finally, for $\mu = i$ we find that the equation is satisfied provided that $$\epsilon = (A_u-1)^{-1/2}\epsilon_0
\label{eq:condsusy2}$$ where $\epsilon_0$ is constant.
Finally, by using the condition (\[eq:condsusy\]) in the form $\gamma^u\epsilon=0$ and the fact that the gauge field corresponding to the generalized FS solution only has a nonvanishing $u$-component, it is not too difficult to show that also $\delta\chi = 0$.
We thus conclude that the generalized FS solution has 8 unbroken supersymmetries given by eqs. (\[eq:condsusy\]) and (\[eq:condsusy2\]). We should stress that it is not obvious that duality transformations [*always*]{} preserve the supersymmetry of a given solution. Note that the unbroken supersymmetries before and after the duality transformation differ from each other.
Proof of the space-time supersymmetry of the K-T Solution
=========================================================
The solution of [@KT] contains a flat metric describing a four-dimensional spacetime with coordinates $x^\mu=\{u,v,x^1,x^2\}$, a 2-form field $$B= - 2 f(u) dx^1\wedge dx^2\ ,$$ and a $u$-dependent dilaton $\phi(u)$. The only nonzero component of $H$ is given by $H_{u12}=-1/3 \partial_u f$.
The supersymmetry transformation rules for the dilatino $$\delta\lambda = \bigl (\gamma^\mu\partial_\mu\phi + {1\over 4}
H_{\mu\nu\rho}
\gamma^{\mu\nu\rho}\bigr)\epsilon = 0$$ have a non-trivial solution under the condition that $$\gamma^u \epsilon = 0\ .$$ The supersymmetry transformation rules for the gravitino have a $v$- component $$\delta \psi_v = \bigl (\partial_v - {1\over 4}
\Omega_{v +}{}^{ab}\gamma_{ab}\bigr)
\epsilon = 0\ .$$ This equation is satisfied under the condition that $\epsilon$ is $v$-independent, since $\Omega_{v +}{}^{ab}=0$. The $i$ component of this equation is solved if $\epsilon$ is $x^i$-independent and $\gamma^u \epsilon = 0$: $$\delta \psi_i = \bigl (\partial_i - {1\over 4}
\Omega_{i +}{}^{ab}\gamma_{ab}\bigr)
\epsilon = 0 \ .$$ The $u$-component of this equation $$\delta \psi_u = \bigl (\partial_u + {3\over 4}
H_{u}{}^{12}\gamma_{12}\bigr)
\epsilon = 0 \ ,$$ is satisfied if the spinor of unbroken supersymmetry $\epsilon$ has a specific dependence on $u$-coordinate of the form $$\epsilon (u) = e^{{1\over 4}f(u) \gamma^1 \gamma^2 } \epsilon _0\ .$$
After a duality transformation in the $x^1$-direction, the dual solution is given by a zero axion field and the following non-flat metric: $$ds^2 = 2dudv + 2f(u)dx^1dx^2 - dx^1dx^1 - (1+f^2)dx^2dx^2\ .$$ The dilaton field remains unchanged: $\tilde\phi = \phi$. The supersymmetry of the dual solution is proven as follows. The equation $\delta\lambda=0$ again leads to the condition that $\gamma^u\epsilon=0$. To investigate the supersymmetry transformation rule of the gravitino it is convenient to first consider the dual of $\Omega_+$: $${\tilde \Omega}_{c+}{}^{ab} = -{3\over 2} (H_c{}^{ab} - 4 k^{[a}
H_{1c}{}^{b]})\ .$$ Here $k^\mu$ is the vector $k^\mu = (0,0,1,0)$. Using this expression it follows that the equation $\delta \psi_v =
0$ is satisfied under the condition that $\epsilon$ is $v$-independent. Furthermore, using $\gamma^u\epsilon = 0$, it follows that $\delta\psi_i=0$ if $\epsilon$ is $x^i$-independent. Finally, the equation $\delta\psi_u=0$ is satisfied if $$\delta \psi_u = \bigl (\partial_u - {3\over 4}
H_{u}{}^{12}\gamma_{12}
\bigr)
\epsilon = 0 \ ,$$ or $$\epsilon (u) = e^{-{1\over 4} f(u) \gamma^1 \gamma^2 } \epsilon _0\
{}.$$
Finally, one can show that both before and after the duality transformation all the $T$-tensors and Lorentz Chern-Simons terms vanish. We therefore conclude that both the Klimčík-Tseytlin solution as well as its dual are supersymmetric to all orders in $\alpha'$.
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[^1]: E-mail: bergshoe@th.rug.nl
[^2]: E-mail: entrop@th.rug.nl
[^3]: E-mail: kallosh@physics.stanford.edu
[^4]: Note that the Yang-Mills vector fields do not occur in the zero slope limit of the string effective action.
[^5]: Strictly speaking, we should take in (\[eq:dil\]) the absolute value $|g_{xx}|$ instead of $g_{xx}$.
[^6]: We use the same conventions as in [@Be1], except that we have redefined the axion field with $B_{\mu\nu}
\rightarrow -{3\over 2} B_{\mu\nu}$ in order to agree with the duality transformations given in the previous section. Further details about our notation and conventions can be found in the appendix of [@Be1].
[^7]: Note that with this definition the $\Omega_\pm$ of this paper is identical to the $\Omega_\pm$ of [@Be1] [*after*]{} the redefinition $B_{\mu\nu} \rightarrow - {3\over 2}B_{\mu\nu}$.
[^8]: We use the following form notation: $ds^2 = g_{\mu\nu}dx^\mu dx^\nu$ and $B = B_{\mu\nu}dx^\mu\wedge
dx^\nu$.
[^9]: In order to solve the equations (\[eq:Lapl\]), it is understood that a source term at $r=0$, representing a fundamental string, has been added to these equations.
[^10]: Note that the solution (\[eq:new\]) does not yet include the vector fields. The complete generalized FS solution, including the vector fields, is given in (\[eq:generalised\]).
[^11]: Not much is known about the properties of $\alpha^{\prime}$ corrections, which are not related to anomalies. For some recent results about these additional corrections, see [@deRoo] and references therein. We will not consider these corrections in this paper.
[^12]: One can include the vector field action in the classical Lagrangian of supergravity interacting with the Yang-Mills multiplet. We treated the vector field here as coming at the level of $\alpha'$ corrections, which is natural in the framework of string effective action.
[^13]: For more details, see [@Be2] and the Appendix of [@Be1].
[^14]: In the equations given below it is understood that the SSW metric is used. Note that the dual gauge fields are given with flat indices. To convert them into curved indices one should use the metric corresponding to the FS solution.
[^15]: As explained above, in the string case we have control only on anomaly-related $\alpha'$ corrections.
[^16]: Note that both (\[eq:g1\]) and (\[eq:condsusy\]) can be written as $\gamma^u\epsilon =0$, using curved indices.
|
---
abstract: 'Proponents of the Everett interpretation of Quantum Theory have made efforts to show that to an observer in a branch, everything happens as if the projection postulate were true without postulating it. In this paper, we will indicate that it is only possible to deduce this rule if one introduces another postulate that is logically equivalent to introducing the projection postulate as an extra assumption. We do this by examining the consequences of changing the projection postulate into an alternative one, while keeping the unitary part of quantum theory, and indicate that this is a consistent (although strange) physical theory.'
author:
- |
Patrick Van Esch\
*Institut Laue Langevin, Grenoble*\
France
title: 'On the Everett programme and the Born rule.'
---
\[firstpage\]
Introduction
============
An important part in the programme of ’the Everett interpretation’ (first proposed in Everett 1957) or the relative state interpretation of quantum theory is to avoid postulating the Projection Postulate (PP) which von Neuman (1955) entitled ’process (1)’, so that we can assume that only unitary evolution is necessary, and that this induces, for an observer, a measurement history *as if* the Projection Postulate were true (Dewitt & Graham, 1973). The relative state view on quantum theory needs to address two issues in order to produce the effects of the Born rule: (i) it needs to show *in what basis* an effective Born rule will emerge and (ii) it needs to show that the correct probabilities will emerge for an observer. A good recent overview of the history of the subject can for example be found in the work of Rubin (2003).
Much progress has been made on (i), mainly through the decoherence programme, thoroughly described in a book by Joos et al. (2005). Its relevance for the Everett programme is for instance described in work by Zurek (1998). However, (ii) seems to be much more problematic.
As an example of recent work on (ii), Deutsch (1999) has proven an interesting theorem, which states, under additional ’reasonable’ assumptions, that the only way a rational decider can assign probabilities to outcomes of future quantum measurements, is through the Born rule. Several papers then argued on the validity of this proof (Hanson 2003 ; Wallace 2002, 2003 ; Gill 2003 ; Greaves 2004). Finkelstein (2000) claims that Gleason’s theorem solves the issue ; but this theorem also contains an additional assumption.
We will argue in this paper that the extra assumptions, in all these cases, are logically equivalent to introducing the PP. This doesn’t affect the value of Deutsch’s and other’s work, which allow us to reformulate the PP in other terms, and thus to understand better what exactly are its essential ingredients. But it means that there is no hope of deriving the PP directly from the rest of the machinery of quantum theory, and hence puts that part of the Everett programme to an end.
The situation is in certain ways reminiscent of attempts, during more than 2 millennia, of deducing Euclid’s Fifth Postulate (Trudeau, 1987 gives a marvelous account on that history) from the other postulates of Euclidean geometry, until it was resolved by Gauss and Bolyai and independently by Lobachebsky, by showing that there was a consistent way of building Non-Euclidean geometry by explicitly introducing an alternative Fifth Postulate. We will try to apply the same strategy: we will postulate an Alternative Projection Postulate (APP) and see that this gives rise to a consistent theory on the same level as the standard theory — even if the theory is experimentally of course completely wrong. The very logical existence of this theory then indicates the independence of the PP from the unitary part of quantum theory.
Apart from proving the logical independence of the PP from the unitary part of quantum theory, constructing such a logical alternative has another practical advantage: it allows one more easily to find out where “proofs” of the PP make hidden (or explicit) extra assumptions. We will then examine where exactly it is in disagreement with Deutsch’s ’reasonable assumptions’, or with Gleason’s theorem.
The Alternative Projection Postulate.
=====================================
Let us propose a quantum theory à la von Neumann (1955), except for the projection postulate (which he calls process (1)), which we replace by the Alternative Projection Postulate (APP). It has to be said that the APP can seem slightly more limited in scope than the original PP, in that only measurements with a finite number of different outcomes are handled. However, this is not a physical shortcoming, because any true measurement can result only in a finite amount of information, and hence in a finite number of discrete outcomes. We propose the APP:
> Let $\{\hat{X_k}\}$ be a set[^1] of commuting self-adjoint operators with a finite, common discrete spectrum (the different possible outcomes of the measurement). This finite spectrum is given by a finite series of sets of eigenvalues $\{x^k\}_i$. The full set of $\{\hat{X_k}\}$ defines the measurement to be performed. To each different set of eigenvalues $\{x^k\}_i$ of $\{\hat{X_k}\}$ corresponds a projector $P_i$ on the space of common eigenvectors belonging to $\{x^k\}_i$. There are by hypothesis only a finite number of such projectors, which form a complete, orthogonal set [^2]. Let $N$ be the (finite) number of projectors $P_i$. Let $|\psi\rangle$ be the state of the system in Hilbert space before the measurement. Let $n_{\psi}$ be the number of projectors for which $P_i|\psi\rangle$ is different from 0. We obviously have: $1 \le n_{\psi} \le N$. If the system is in state $|\psi\rangle$, each of the set of values $\{x^k\}_i$ corresponding to such a projector has a probability $1/n_{\psi}$ to be realized. The other sets of values have probability 0 to be realized. If the outcome of the measurement equals $\{x^k\}_u$, then the state after measurement equals $P_u |\psi\rangle$, properly normalized.
One notices the difference with the original PP as found in most standard texts on quantum mechanics, such as Cohen-Tannoudji (1997): the probability equals $1/n_{\psi}$ instead of $\langle\psi|P_i|\psi\rangle$[^3].
We should point out that the APP is in fact the most natural probability rule that goes with the Everett interpretation: on each “branching” of an observer due to a measurement, all of its alternative ’worlds’ receive an equal probability.
Consistency of Quantum Theory based upon the APP.
=================================================
The alternative quantum theory (which is normal quantum theory, with the PP replaced by the APP, for short AQT) will turn out to be a physical theory which is completely different from standard quantum theory (SQT) and also experimentally totally wrong. However, we will try to show that it is a consistent theory on the same level as SQT. It is a priori very difficult to prove that a physical theory is consistent. However, the bulk of the mathematical machinery of SQT and AQT is the same (the unitary evolution). The intervention of the APP on the mathematical machinery is the same as the PP (indeed, it is a projection of the state vector on an eigenspace, followed by a normalization of the projection, in both cases). So on the purely mathematical side, both theories are identical concerning the evolution of the state vector.
The subtler aspects are related the physical interpretation. Indeed, the PP is the only link to experimental quantities, and this is replaced by the APP. We have to ensure that through the APP, we arrive at an operational definition of the mathematical entities which is consistent. We will show that it is in fact exactly the same as in SQT. Furthermore, we have to prove that different mathematical descriptions describing the same physical situation give identical results. This means invariance under unitary transformations, and invariance under different ways of formulating the same measurement process.
Respect of unitary transformations.
-----------------------------------
The representation of the state space, and all of the unitary evolution machinery, can undergo a unitary transformation without changing their interpretation. We have to ensure that our AQT gives identical results when such an isomorphism is applied. So we need to show:
> Any unitary transformation of the Hilbert space of states, such that $|\psi\rangle$ is mapped upon $U|\psi\rangle$ and every observable $O$ is mapped upon $UOU^{\dagger}$, leaves the results and effects of measurements, such as they are introduced by the APP, invariant.
The proof is straightforward. First of all, the projectors $P_i$ are transformed into $UP_iU^{\dagger}$, so that the projections of $U|\psi\rangle$ are transformed into $U P_i |\psi\rangle$. This projection is zero if and only if $P_i|\psi\rangle = 0$, so the number of non-zero projections $n$ is conserved, as well as the eigenvalues $\{x^k\}_i$ which belong to such projections. The probabilities of the measurement results are hence the same before and after the transformation. Also any further evolution, after the measurement, is equivalent to the evolution before transformation, given that the state after the measurement (with result $\{x^k\}_u$) is now $U P_u |\psi\rangle$, properly normalized, which is nothing else but the transformation, under $U$, of the state we would have obtained under the same circumstances.
Measurement results predicted with certainty in AQT and SQT are the same.
-------------------------------------------------------------------------
In SQT, the interpretation of the mathematical entities (state vector, observable etc...) is completely fixed by the experimental results predicted with certainty. We will show that this interpretation is exactly the same in AQT.
> If $|\psi\rangle$ is an eigenvector of the different $\hat{X}^k$ with respective eigenvalues $\{x^k\}_i$, then the measurement will give with certainty the result $\{x^k\}_i$ for this observable, and the state after measurement will still be $|\psi\rangle$.
The proof is trivial and based upon the fact that $P_i
|\psi\rangle = |\psi\rangle$ and $P_j |\psi\rangle = 0$ for $i
\neq j$.
We also have:
> If $|\psi\rangle$ doesn’t have any component with eigenvalues $\{x^k\}_i$, then the measurement will give a result which is different from $\{x^k\}_i$, with certainty.
This is a direct consequence of the APP.
Equivalence of physically identical measurements.
-------------------------------------------------
Two measurements are physically identical if exactly the same information is extracted by either of both measurements. We will show that two mathematically different sets of observables, $\{X_k\}$ and $\{Y_k\}$, which correspond to physically identical measurements, result in operationally identical results.
If $\{X_k\}$ and $\{Y_k\}$ extract the same information, this means that to each distinct set of eigenvalues $\{x^k\}_i$ corresponds exactly one set of distinct eigenvalues $\{y^k\}_i$ and vice versa ; and that for each case where the result of measurement of $\{X_k\}$ gives with certainty $\{x^k\}_i$, then the result of the measurement if we measure $\{Y_k\}$ should give with certainty $\{y^k\}_i$. But this means that $P_i^X = P_i^Y$. As only the projectors play a role in the APP, the two measurements with equivalent sets of observables yield exactly the same results under the APP.
Strange properties of AQT.
==========================
We will discuss some strange properties of AQT, which immediately disqualify it as a possible candidate of a physical theory of our world. However, we want to emphasize that such a world is a logical possibility including the unitary part of quantum theory, even if it is a very strange one to our standards. By examining some very strange and ’unreasonable’ properties, we also show how easy it is to eliminate AQT by introducing ’reasonable assumptions’.
If we take the topology induced by the Hilbert in product, then an arbitrary small change of $|\psi\rangle$ (by adding a small component of an eigenspace that was orthogonal to $|\psi\rangle$) can induce a discrete change in probabilities of outcomes. But this, as such, is not an internal contradiction of the theory. Note also that in general, the state of a system is never strictly orthogonal to an eigenspace of a set of observables (except immediately after measurement), so one can usually assume that $N
= n$, except immediately after a measurement.
This also means that if there is a small time lapse between two identical measurements, that the two results are uncorrelated except in the case where all the $\{X^k\}$ commute with the full Hamiltonian of the system. Although this result seems very strange indeed, it does not necessarily indicate an internal inconsistency, but just means that measurements incompatible with the full $H$ are a waste of time, because the information is immediately lost. As we usually don’t know the full $H$, this means that most measurements are a waste of time. AQT describes a very random world indeed, in which, most of the time, the outcomes of measurements are independent of the state the system is in!
Another strange property of AQT is the following. Imagine that we consider two different, commuting observables, measuring the same quantity. We have a course-grained one, $X$, and a fine-grained one, $Y$. Let us assume that $Y$ has 5 distinct eigenvalues, namely 1,2,3,4, and 5. Let us assume that $X$ has two eigenvalues, 10 and 20, and that $X$ takes on value 10 when $Y$ takes on value 1, and that $X$ takes on the value 20 when $Y$ takes on the values 2,3,4, and 5. For a general state $|\psi\rangle$ which isn’t ’particular’ with respect to $X$ or $Y$ (meaning, has non-zero components for all of their eigenstates), a measurement consisting purely of $X$ will result in a probability 0.5 for value 10, and 0.5 for value 20. A measurement consisting purely of $Y$ will give a probability of 0.2 for each of (1,2,3,4 and 5), and hence, if we calculate X from it, a probability of 0.2 for finding 10 and a probability 0.8 for finding 20. So, depending on whether we measure also $Y$ or not, the result $x = 10$ has a probability of 0.5 or 0.2. At first sight, this is a strange result ; however it is not an inconsistency, and just extends the “strangeness” already present in SQT. In SQT, incompatible measurements influence each other’s outcome probabilities ; in AQT, even compatible, but different, measurements influence each other’s otucome probabilities. Indeed, the measurement consisting purely of ${X}$ is not physically equivalent with the measurement consisting of ${X,Y}$, because a different amount of information is extracted. On the other hand, the measurement ${X,Y}$ and the measurement ${Y}$ are identical, because the measurement of $Y$ is also a measurement of $X$ ; this is an illustration of the equivalence of measurements. It is a property of AQT that changing the resolution of a measurement can change the probabilities of the outcome of the crude measurement, which is not the case under SQT. Note that this property of SQT is the ’non-contextuality’ needed in the application of Gleason’s theorem. Indeed, in AQT, the fact that the probability of measuring 10 for $X$ depends on whether we have measured $Y$ or not (and which corresponds physically to two different measurement situations), means that AQT is not non-contextual. So an axiom of non-contextuality can be seen as an axiom, equivalent to the Born rule.
We now see where some of Deutsch’s ’reasonable’ assumptions (made more explicit in Wallace (2003)) explicitly rule out AQT. One instance is requiring identical probabilities under Payoff Equivalence, when the function $f(x_i)$ is not invertible (meaning, for $x_i \neq x_j$ we can have $f(x_i) = f(x_j)$. Indeed, in AQT, the measurement $f(X)$ and $X$ are not considered equivalent because $f(X)$ extracts less information from the system than $X$. Using payoff equivalence with $f$ non-bijective is a crucial point in the proof of Deutsch’s theorem, as made clear in steps (35) and (36) in Wallace (2003).
We want again to emphasize that these strange results are a logical possibility of a theory evolving according to “unitary quantum theory”. They are simply different from those given by SQT, in the same way that geometrical results in hyperbolic geometry are different from the geometrical results by Euclidean geometry, and are “strange” as compared to everyday “geometrical measurements”.
Discussion
==========
In this paper we tried to show that quantum theory, with the projection postulate replaced by an alternative one, gives rise to a consistent physical theory, at least at the same level as standard quantum theory. This theory has very strange consequences and can certainly not describe our world, but its consistency (in relation to standard theory) proves that it is not possible to deduce the projection postulate from the ’unitary’ part of quantum theory, in the same way it is not possible to deduce Euclid’s Fifth Axiom from the four other ones. All attempts to do so by introducing extra assumptions just indicate that those extra assumptions are logically equivalent to the projection postulate. This, by itself, is not necessarily a meaningless exercise.
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\[lastpage\]
[^1]: We need a set, only because we want to be able to label the outcomes with several different real numbers. As long as there are only a finite number of different outcomes, one single operator could in principle be sufficient.
[^2]: This means that $\sum_i P_i = 1$ and that $P_i P_j = 0$ if $i \neq
j$.
[^3]: We could even go further and postulate the probability to be equal to $\alpha/n_{\psi} +
(1-\alpha)\langle\psi|P_i|\psi\rangle$ with $\alpha$ a real number between 0 and 1, which defines a generalized APP for each value of $\alpha$.
|
---
abstract: 'Nearly all aspects of earthquake rupture are controlled by the friction along the fault that progressively increases with tectonic forcing, but in general cannot be directly measured. We show that fault friction can be determined at any time, from the continuous seismic signal. In a classic laboratory experiment of repeating earthquakes, we find that the seismic signal follows a specific pattern with respect to fault friction, allowing us to determine the fault’s position within its failure cycle. Using machine learning, we show that instantaneous statistical characteristics of the seismic signal are a fingerprint of the fault zone shear stress and frictional state. Further analysis of this fingerprint leads to a simple equation of state quantitatively relating the seismic signal power and the friction on the fault. These results show that fault zone frictional characteristics and the state of stress in the surroundings of the fault can be inferred from seismic waves, at least in the laboratory.'
title: Estimating Fault Friction from Seismic Signals in the Laboratory
---
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Machine learning models can discern the frictional state of a laboratory fault from the statistical characteristics of the seismic signal
The use of machine learning uncovers a simple relation between fault frictional state and statistical characteristics of the seismic signal
The discovery of this equation of state also uncovers the hysterectic behavior of the laboratory fault
This equation of state between seismic signal power and friction generalizes to different stress conditions with the appropriate scaling
Plain language summary {#plain-language-summary .unnumbered}
======================
In a laboratory setting that closely mimics Earth faulting, we show that the most important physical properties of a fault can be accurately estimated using machine learning to analyze the sound that the fault broadcasts. The artificial intelligence identifies telltale sounds that are characteristic of the physical state of the fault, and how close it is to failing. A fundamental relation between the sound emitted by the fault and its physical state is thus revealed.
Introduction
============
Most tectonic earthquakes take place when juxtaposed crustal blocks that are locked or slowly slipping overcome the static fault friction and abruptly slide past one another. A rupture initiates and propagates along the fault plane, eventually coming to a stop as the dynamic fault friction puts a brake on continued slip. It is the frictional state that controls how the fault ruptures, its nucleation and how big the earthquake will ultimately become. The fault frictional state also controls when the next event may take place under a given tectonic (or anthropogenic) forcing ([@scholz2002mechanics; @Marone1998]).
Inferring the frictional state on faults, and where a fault is within its seismic cycle, is extremely challenging. Seismic wave recordings at the time of an earthquake can inform us about characteristics such as rupture velocity and can be used to calculate fundamental parameters such as earthquake magnitude ([@aki2002quantitative]), the evolution of elasticity following an earthquake ([@2007GeoRL..34.2305B; @brenguier2008postseismic; @curtis2006seismic; @nakata2011near]) and slip distribution for instance ([@manighetti2005evidence]). However, seismic waves have not been used to directly examine the frictional state throughout the entire seismic cycle, nor its distribution along the fault. In fact, no geophysical data set has enabled the direct and continuous quantification of the fault frictional state.
Frictional characteristics are determined primarily from theory, simulations and laboratory experiments ([@Rabinowicz1956; @Fineberg2004; @Dorostkar2017; @scholz1968microfracturing; @scholz2002mechanics; @JGRB:JGRB51267; @mclaskey2011micromechanics; @MORGAN1997209; @Madariaga2016; @Kaproth2013]). Large scale stress simulations based on plate movements can provide estimates of stress and frictional state on a fault, but within significant error bounds ([@zoback1991tectonic; @townend2013]). Computer models, including state-of-the-art simulations can be powerful but currently fall short in regards to predicting actual fault behavior. Nonetheless, simulations of the complex behavior of faulting are improving rapidly ([@Richards-Dinger983]) and laboratory experiments provide tremendous insight into frictional processes ([@scholz1968microfracturing; @scholz2002mechanics; @JGRB:JGRB51267; @mclaskey2011micromechanics; @brantut2008high]). Laboratory shearing experiments, involving an apparatus identical to that which produced the data that we analyze here, have been instrumental in the development of rate and state friction laws ([@scholz1998; @Marone1998]).
In laboratory shear experiments that use fault blocks separated by fault gouge, many slip behaviors that resemble those observed in Earth can be induced, including stick-slip and slow-slip ([@Kaproth2013; @Scuderi2016; @mclaskey2011micromechanics; @GRL:GRL27642]). In particular, the fundamental Gutenberg-Richter relation for laboratory events ([@johnson2013]) is very similar to small-scale earth observations such as in mines ([@GRL:GRL25912]), tectonic regions ([@GJI:GJI5595]), and to the whole Earth ([@GJI:GJI135]), showing that event amplitudes in the laboratory scale in the same way as in Earth.
Our goal is to determine if the continuous seismic signal from the laboratory fault contains information about its frictional state. Recently, a seismic signal previously thought to be noise has been identified in the laboratory ([@Rouet2017]). This new signal has strong predictive ability regarding upcoming failures over the entire seismic cycle, suggesting that the seismic signal is imprinted with information about the fault frictional state.
Machine learning finds the frictional state of the laboratory fault from the seismic signal it emits
====================================================================================================
The experimental apparatus, a biaxial shear device, is a double direct shear device with an adjustable normal load (Fig. \[fig:stress\_prediction\]A). A piston mimicking tectonic forcing drives a central block relative to two fixed side blocks. The two side blocks are separated from the central block by two layers of granular material, the fault gouge. The gouge layer thicknesses, shear stress, normal load and shear displacement are all recorded. The fault frictional state $\mu$, is given by the shear stress divided by the normal stress ($\mu=\sigma_{\rm S}/\sigma_{\rm N}$). In the following, we use shear stress and friction or frictional state interchangeably, as they are proportional at constant normal load. In the first experiment we analyze, the normal load is fixed at 2.5 MPa. In the second experiment, that we analyze in the next sections, the normal load is constant at load levels of 4, 5, 6, and 7 MPa. The fault gouge is comprised of class IV glass beads with diameters 105-149 microns. The seismic signal (also known as acoustic emission) coming from the fault is recorded by piezoceramics embedded in the side blocks (see Methods for more details). The apparatus has been broadly discussed in the literature ([@johnson2013; @Kaproth2013; @Scuderi2016; @Marone1998]).
In order to study the fundamental friction physics of the fault system, we analyze the continuous seismic signal recorded during the experiment using a machine learning (ML) approach that is explicit and can thus be used to obtain physical information about the shear system. Our primary goal is to infer at all times the current frictional state of the fault, using information from short moving time windows of the seismic data (Fig. \[fig:stress\_prediction\]E, solid blue window). In each time window, we compute a set of potentially relevant statistical features that describe the distribution of the seismic signal. The ML model uses the features calculated in a time window to estimate the average shear stress (or friction) during that time window. The time windows we consider are 1.33 s in duration. The laboratory seismic cycle varies from 7 s to 17 s, with an average of $\approx$12s (Fig. \[fig:stress\_prediction\]D), and thus the time windows are snapshots of the instantaneous state of the fault system.
We used a ML algorithm known as gradient boosted trees (XGBoost implementation) ([@Chen2016; @Friedman2000]), which is a decision tree ensemble method ([@Breiman1999]). The hyper-parameters of the gradient boosted trees model are determined using the EGO method ([@Jones1998; @Rouet2016; @Rouet2017a]), maximizing the performance in 5-fold cross-validation on the training set (see Methods for details). The training set, used to build the model, corresponds to the first $60\%$ of the experimental data, shown as the green shaded region in Fig. \[fig:stress\_prediction\]C and \[fig:stress\_prediction\]D. The testing set, used to evaluate the model’s performance, corresponds to the remaining 40$\%$ of the data, shown as the blue shaded region in Fig. \[fig:stress\_prediction\]C and \[fig:stress\_prediction\]D. Each decision tree estimates the frictional state using a sequence of decisions based on the statistical features derived from the time windows (see Methods). We train the gradient boosted trees model by providing the algorithm with both the time series of the measured friction and features of the measured seismic signal. We then test the resulting ML model on a portion of data not used in training (shown on Fig. \[fig:stress\_prediction\], (E) and (F)). It is important to note that during the testing procedure, the ML model has access only to the features of the seismic data. In order to quantify the quality of the model’s estimates of the frictional state compared to the experimental values, we use the coefficient of determination (R$^2$) as our evaluation metric.
Figure \[fig:stress\_prediction\]F shows that the ML model can accurately determine the instantaneous shear stress, *i.e.* the frictional state, directly from instantaneous features of the seismic data. The statistical characteristics of any arbitrary segment of seismic data are a fingerprint of the associated fault frictional state. Despite the fact that the stress cycles are aperiodic, the ML model can determine the instantaneous frictional state of the fault from the seismic signal it emits, at all times. Importantly, the connection between instantaneous (local in time) seismic features and instantaneous frictional state works throughout the entire seismic cycle.
In our experiments, the seismic signals come from grain fracture, rotation and displacement, or brittle failure of adhesive grain contact junctions within the laboratory fault gouge. Ongoing Discrete Element simulations ([@Dorostkar2017; @Ferdowsi2015]) and Finite Element plus Discrete Element simulations are being applied to study the role of granular processes during shearing.
Using machine learning, we showed that we are able to precisely infer the friction of a laboratory fault from statistical characteristics of the continuous seismic signal it emits. In the next section we will show that by probing the most important statistical feature identified in the seismic signal, we can extract a simpler model that does not have the same level of accuracy, but that is easier to interpret, can be generalized across experimental conditions, and from which we can uncover an equation of state linking fault friction and properties of the seismic signal.
The laboratory fault exhibits a simple equation of state linking friction to seismic power, and exhibits a hysteretic behavior
==============================================================================================================================
The frictional state determined by the ML model from the seismic data is highly accurate (R$^2 > 0.9$). A key characteristic of the ML decision tree models, and what makes them so valuable for the analysis of scientific data, is their simplicity and the fact that they are constructed explicitly from the features of the data they are provided with. This allows for a straight-forward ranking of the features based on their importance for the ML model (see Methods). Used in this way, the decision tree procedure enables us to determine which characteristic of the seismic signal is the most important to estimate the fault friction. Following this approach, we find that the key feature of the seismic signal is its variance. By definition, the variance of an elastic wave signal is proportional to the average energy per unit of time, thus it is proportional to the average power in the elastic wave signal during a time window. Therefore, it is straightforward to rebuild the frictional state ML model based solely on this single feature of the seismic signal. We show such a model, determined solely by the power in the seismic signal from the fault, in Figure 2. Note that the estimated friction values $\mu$ remain accurate (R$^2 > 0.8$), which demonstrates a strong link between the power in the seismic signal from the fault and its frictional state. Other features of the seismic signal are important, but less so than the variance (*i.e.* seismic power).
Fig. 2A shows the shear stress as a function of seismic power. The ML model built in training (where the ML model uses both the shear stress and seismic signals) is shown as a bold blue line. The testing data are shown for the nine stress cycles (thin dashed lines) shown in Figure \[fig:stress\_prediction\]E and F. Fig. 2B shows several of the stick-slip cycles as a function of time, with colors corresponding to the data of Figure 2A. The time window analysis (e.g., see Figure 1E) used to construct the ML model of the frictional state during the training phase is established point-wise in time, over 1.33s intervals that are displaced by increments of 0.133s (90 percent overlap). Therefore, the model inputs contain no information about the timing of the failure events seen in Figure 2B. The ML algorithm is able to estimate the frictional state, and therefore the position within the seismic cycle, based solely on the continuous seismic signal radiated by the fault. Surprisingly, even though experimental shear stress trajectories differ in time, they are identical in seismic power-shear stress space. The training data can be scrambled in time and the frictional state model we find is unchanged. If we take the seismic signal to be generated by a spectrum of abrupt grain rearrangements driven by the shear stress, the relationship between shear stress or frictional state and the power in the seismic signal can be regarded as an equation of state.
Our results shown in Fig. 2A demonstrate a robust, predictive relationship between fault zone friction (or shear stress) and the power of the seismic signal coming from shear deformation within the fault gouge. This relation between seismic signal power and friction can be estimated by training the ML model on both seismic and shear stress data sets. This model is the bold blue line shown in Fig. 2A. In comparison, the thin dashed color lines in Fig. 2A and 2B come from the testing data that the ML model has never seen. The friction for any and all laboratory earthquake cycles can be calculated from this relation. Moreover, we find that this predictive relation holds for a broad range of conditions, including when the laboratory earthquake cycles are periodic, aperiodic and during the transient failure episodes as friction evolves (see Fig. S1 in the Methods section). The results show that in the case of the laboratory fault, failure does not occur randomly, but on the contrary follows a very specific pattern given by an equation of state that links the friction on the fault to the power of the seismic signal it emits.
Interestingly, the laboratory seismic cycles show a complex behavior, with segments of quasi-steady stress prior to failure (Figure 2B). During the critical stress state preceding failure the shear stress occasionally decreases, reflecting a small gouge failure, and then recovers (Figures 1D and 1F). This is manifested in a hysteresis loop in the friction vs. seismic power space (Figure 2C). The inset of Fig. 2 shows two stress cycles in seismic power-shear stress space, one with no inner loop (corresponding to no small stress drop during the cycle), and the other exhibiting an inner loop (corresponding to a small stress drop during the stress cycle). We draw a parallel between the hysterectic behavior that we find here and quasi-static experiments on rock (where ‘discrete memory’, also termed ‘end point memory’ may occur when small stress cycles take place during a larger stress cycle ([@JGRB:JGRB3711])).
The equation of state linking friction to seismic power generalizes across load levels
======================================================================================
The bi-axial apparatus enables us to study the laboratory seismic cycle for different normal loads (Fig. \[fig:normal\_loads\]B). In this section, we analyze a second experiment from the same apparatus, during which the normal load is progressively stepped up and then down. Equations of state similar to that in Fig.\[fig:normal\_loads\]A can be constructed for each normal load. The thick colored lines correspond to the equation of state linking friction or shear stress to seismic power estimated by the ML model for each load level (determined on the training set). The light colored crosses show the experimental trajectories the laboratory fault has gone through in seismic power-shear stress space (in the testing set, not used to build the ML model). In Fig. \[fig:normal\_loads\]C we show the estimated equation of state for each load level where seismic power is now plotted against frictional state instead of shear stress. The different relations partially collapse onto one another. As a final step we scale the seismic power by the cube of the normal stress (Fig. \[fig:normal\_loads\]D). We find this scaling empirically from the data. A single *universal* equation of state results.
The scaling of the equation of state linking friction to seismic power can be understood as arising from the properties of the fault gouge. The seismic signal is due to elastic waves broadcasts from the interior of the system that come from abrupt particle rearrangements. These rearrangements occur as the configurations of the granular material evolve to support larger and larger shear stress imposed by the drive. The granular material, modeled as a Hertzian material ([@Johnson1987]), involves particle-particle bonds that have energy, $e_{\rm B}$, that scales with the normal load as $e_{\rm B}\propto \sigma_{\rm N}^{5/3}$ (see Methods more details). We can assume that the elastic wave broadcasts that accompany rearrangements carry energy that scales as the bond energy $e_{\rm B}$. If we also assume that the set of particle configurations that unfold in a slip cycle are statistically the same for all values of the normal stress, then at a point in the slip cycle the elastic broadcasts will differ primarily due to the event rate $r$, as the slip cycle unfolds. Thus, the seismic power $P$ scales as: $P \propto e_{\rm B}r \propto \sigma_{\rm N}^{8/3} \approx \sigma_{\rm N}^{3}$. To derive the third term, we use the observation that $r \propto \sigma_{\rm N}$: in the bi-ax experiment, the inter-event time is inversely proportional to the normal stress.
The frictional state law derived by machine learning at one load level can therefore be transferred to any arbitrary load level by normalizing the seismic power by the cube of the normal stress. This simple relation can give accurate estimations of the stress (or friction) on the fault for any stress cycle, at any load level. Moreover, once the machine learning analysis has established the direct relationship between seismic power and friction on the fault, we can use a simpler exponential fit to visualize this relationship. Such a simple fit is shown in Fig. \[fig:fit\]E: $$\mu=\mu_0-b \exp \left(-a\frac{P}{\sigma^3_{\rm N}}\right)$$ \[eq:fit\] with $\mu_0$ the asymptotic friction (reached at the end of the stress cycles and during stable sliding), $P$ the seismic power during a time window, $\sigma_{\rm N}$ the normal load, and $a=0.25$ and $b=0.1$ the parameters of the fit.
The laboratory fault assembly is opaque and therefore we cannot see inside to examine the behavior of the fault gouge. However, a simple interpretation of the seismic power we measure as coming from elastic energy stored in the granular material reproduces the scaling of the seismic power-friction law that we find.
We have demonstrated that certain statistics of the seismic signal over short windows of time provide a fingerprint of the shear stress and frictional state of the fault. It is well established that failure in granular materials ([@Michlmayr2013]) is frequently accompanied by impulsive acoustic/seismic precursors. Precursors are also routinely observed soon before failure of a spectrum of industrial ([@Huang1998]) and Earth materials ([@Schubnel2013; @jaeger2007poroelasticity]). Precursors are observed in laboratory faults ([@johnson2013; @Goebel2013]) as well as models of faults ([@Daub2011; @Latour2011]), and are widely but not systematically observed preceding earthquakes ([@bouchon2013; @bouchon2016; @McGuire2015; @Mignan2014; @Wyss1997; @Geller1997]). The fingerprint that we find in the seismic signal emitted by the fault extends the observation of precursory seismic activity that often takes place soon before failure: we show that characteristics of the seismic signal can tell us about the frictional state of the laboratory fault not only right before failure, but at any time during the slip cycle.
Conclusion
==========
Our results show that the laboratory fault does not fail randomly but in a highly predictable manner. The observations also demonstrate that key properties of the laboratory earthquake cycle can be inferred from the continuous seismic signal emitted by the fault. In particular, the instantaneous frictional state, the critical stress state and therefore where the fault is within the earthquake cycle can be determined using exclusively an equation of state that links the power of the continuous seismic signal to the friction on the fault. This tells us that at least in the laboratory, earthquake catalog approaches for analyzing fault physical characteristics are discarding critical information. Similar approaches using the continuous signal from seismic waves may yield new insight into faults in Earth.
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Acknowledgements {#acknowledgements .unnumbered}
----------------
We acknowledge funding from Institutional Support (LDRD) at Los Alamos National Laboratory, as well as funding from the US DOE Office of Fossil Energy. We thank Andrew Delorey, Kipton Barros, James Theiler, Marian Anghel, Jamal Mohd-Yusof and Nick Lubbers for helpful comments and/or discussions. All the data used are freely available on the data repository hosted by Chris Marone at the Pennsylvania State University.
|
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abstract: '[*Evidence and Evolution: the Logic behind the Science*]{} was published in [-@sober:2008] by Elliott Sober. It examines the philosophical foundations of the statistical arguments used to evaluate hypotheses in evolutionary biology, based on simple examples and likelihood ratios. The difficulty with reading the book from a statistician’s perspective is the reluctance of the author to engage into model building and even less into parameter estimation. The first chapter nonetheless constitutes a splendid coverage of the most common statistical approaches to testing and model comparison, even though the advocation of the Akaike information criterion against Bayesian alternatives is rather forceful. The book also covers an examination of the “intelligent design" arguments against the Darwinian evolution theory, predictably if unnecessarily resorting to Popperian arguments to correctly argue that the creationist perspective fails to predict anything. The following chapters cover the more relevant issues of assessing selection versus drift and of testing for the presence of a common ancestor. While remaining a philosophy treatise, [*Evidence and Evolution*]{} is written in a way that is accessible to laymen, if rather unusual from a statistician viewpoint, and the insight about testing issues gained from [*Evidence and Evolution*]{} makes it a worthwhile read.'
author:
- 'Robert, C.P.'
title: |
[*[**Evidence and Evolution**]{}*]{}:\
A review
---
[ [Christian P. Robert]{}\
Université Paris-Dauphine, CEREMADE, and CREST, Paris ]{}
[**Keywords:**]{} Foundations, frequentist statistics, Bayesian statistics, likelihood, evolution, Darwin, cladistic parsimony, random drift, selection, hypothesis testing, model comparison, model, data.
Introduction
============
Acknowledgements {#acknowledgements .unnumbered}
================
The author’s research is partly supported by the Agence Nationale de la Recherche (ANR, 212, rue de Bercy 75012 Paris) through the 2007–2010 grant ANR-07-BLAN-0237 “SPBayes".
Evide.bbl
|
---
abstract: 'Many pathogens spread primarily via direct contact between infected and susceptible hosts. Thus, the patterns of contacts or *contact network* of a population fundamentally shapes the course of epidemics. While there is a robust and growing theory for the dynamics of single epidemics in networks, we know little about the impacts of network structure on long term epidemic or endemic transmission. For seasonal diseases like influenza, pathogens repeatedly return to populations with complex and changing patterns of susceptibility and immunity acquired through prior infection. Here, we develop two mathematical approaches for modeling consecutive seasonal outbreaks of a partially-immunizing infection in a population with contact heterogeneity. Using methods from percolation theory we consider both *leaky immunity*, where all previously infected individuals gain partial immunity, and *perfect immunity*, where a fraction of previously infected individuals are fully immune. By restructuring the epidemiologically active portion of their host population, such diseases limit the potential of future outbreaks. We speculate that these dynamics can result in evolutionary pressure to increase infectiousness.'
author:
- 'Shweta Bansal[^1] [^2] [^3], Lauren Ancel Meyers[^4] [^5]'
bibliography:
- 'ms1\_with\_figs.bib'
title: The Impact of Past Epidemics on Future Disease Dynamics
---
Introduction
============
Immunity acquired via infection gives an individual protection from subsequent infection by the same or similar pathogen for some period of time. For diseases such as measles, varicella (chickenpox), mumps and rubella, complete immunity lasts a lifetime; therefore an individual who has been infected by one of these pathogens, once recovered, cannot be reinfected, nor transmit the infection again. For other diseases, immunity wanes with time, leaving previously infected individuals only partially protected against reinfection (called *partial immunity*). This degradation of immunity may be caused by antigenic variation in the circulating pathogen or loss of antibodies over time. The transition from complete to partial immunity can happen over different timescales: over a few weeks as with norovirus and rotavirus [@rotavirus], over months or a few years as with influenza [@flu_book], or over many years as with pertussis [@pertussis]. Here, we present new methods for modeling the epidemiological consequences of partial immunity.
Partial immunity may impact the host in multiple ways, and have far-reaching implications for the transmission of a disease through a population. Specifically, it can decrease one or both of two fundamental epidemiological quantities: *infectivity*, the probability that an infected individual will infect a susceptible individual with whom he or she has contact; and *susceptibility*, the probability that a susceptible individual will be infected if exposed to disease via contact with an infected individual. In mathematical models, the probability of transmission (*transmissibility*) during a contact between an infected and susceptible individual is often represented as a product of the infectivity of the infected node and the susceptibility of the susceptible node. Partial immunity can limit transmissibility either by lowering the probability of reinfection or reducing the degree to which an infected individual sheds the pathogen. Both, for example, occur in the case of influenza [@flu_immunity; @flu_immunity_2].
Mathematical modeling of infectious disease dynamics has been dominated by the Susceptible-Infected-Recovered (SIR) compartmental model [@kermack] which considers infectious disease transmission in a closed population of individuals who enjoy complete immunity following infection. The SIR model has been extended to Susceptible-Infected-Recovered-Susceptible (SIRS) dynamics to model the full loss of complete immunity after a temporary period of protection [@hoppen; @waltman], and has been applied successfully in several situations (e.g. [@grassly]). Models of partially immunizing pathogens are less common, and have primarily been developed for particular pathogens, such as influenza [@recker_flu_immunity; @levin_dushoff_plotkin; @nuno_flu_models]. They consider the impacts of antigenic variation and the resulting complex patterns of cross-immunity on epidemic dynamics, but are limited by the assumptions of homogeneous-mixing.
Contact network epidemiology is a tractable and powerful mathematical approach that goes beyond homogeneous-mixing and explicitly captures the diverse patterns of interactions that underlie disease transmission [@barbour; @watts; @pastor_complex; @meyers_sars; @shirley; @bansal_interface]. In this framework, the host population is represented by a network of individuals (each represented by a node) and the disease-causing contacts (represented by edges) between them (Figure \[fig:network\](a)). The number of contacts (edges) of a node is called its *degree*, and the distribution of degrees throughout the network fundamentally influences where and when a disease will spread [@meyers_sars; @mejn; @bansal_interface]. The traditional SIR model has been mapped to a bond percolation process on a contact network, in which individuals independently progress through S, I, and R stages if and when disease reaches their location in the network [@mejn]. The bond percolation threshold corresponds to the epidemic threshold, above which an epidemic outbreak is possible (i.e. one that infects a non-zero fraction of the population, in the limit of large populations); and the size of the percolating cluster (or giant component) above this transition corresponds to the size of the epidemic. The standard bond percolation model for disease spread through a network, however, assumes a completely naive population without immunity from prior epidemics [@mejn].
In this paper, we extend the bond percolation framework to consider the impact of infection-acquired immunity on epidemiological dynamics. We model both perfect (Section 2.1) and leaky (Section 2.2) partial immunity, and show that the two models are identical in the cases of no immunity or complete immunity, but make very different predictions for partial immunity. The evolution of infectiousness, virulence and a pathogen’s antigenic characteristics are in part driven by the epidemiological environment. Although significant attention has been paid to the interaction between contact network structure and pathogen evolution and competition [@boots_sasaki; @read_net_evolution; @van_baalen; @buckee_straindiversity; @nunes_pathdiversity], we do not yet understand the inter-seasonal interactions via modification to the immunological structure of the host contact network. Feedback from an evolving organism to its own ecological and evolutionary environment is generally known as niche construction [@niche_feldman; @boni_niche]. Here, we use our models to explore a particular instance of niche construction: the impacts of prior epidemics on the future dynamics of the pathogen.
Methods: Incorporating Infection-Acquired Immunity into a Network Model
=======================================================================
We present two mathematical approaches to modeling partial immunity. First, we model perfect partial immunity by completely removing a fraction of the individuals (their nodes and edges) who are infected during an epidemic (Figure \[fig:network\](b)) from the network.Using the bond percolation model, we then derive epidemiological quantities for a subsequent outbreak in the immunized population. Second, we model leaky partial immunity using a new two-type percolation model. The underlying contact network topology remains intact, but nodes are classified either as partially immune or susceptible (Figure \[fig:network\](c)). In both models, we assume that both infectivity and susceptibility are reduced due to immunity, but the leaky partial immunity model can be easily adapted to model other effects of immunity.
Below, we use both models to consider dynamics in three network types: (a) Poisson, with degree distribution $p_k = e^{-\lambda}\lambda^k/k!$; (b) exponential, with degree distribution $p_k = (1-e^{\kappa})e^{-\kappa (k-1)}$; and (c) scale-free, with degree distribution $p_k = k^{-\gamma}/\zeta(\gamma)$, each with a mean degree of 10. All model predictions are verified using stochastic simulations which assume a simple percolation process with parameters to match the model.
Perfect Partial Immunity
------------------------
Perfect partial immunity, sometimes known as “all-or-nothing” partial immunity or polarized immunity, implies that for a partial immunity level $\left(1-\alpha\right)$, a fraction $\left(1-\alpha\right)$ of the infected population are fully immune to reinfection (and thus transmitting to others) and the remaining proportion $\alpha$ are fully vulnerable to reinfection (and transmission to others thereafter.) In terms of a contact network, this means that a fraction of the previously infected nodes are now completely removed (along with its edges) from the contact network and are no longer a part of the transmission process. The residual network, introduced in [@ferrari; @bansal_residual] models this phenomenon. Previously, we characterized the residual network as the network made up of uninfected individuals and the edges connecting them, as we assumed that all infected individuals had gained full immunity to infection and thus could be fully removed (along with their edges) from the transmission chain of future epidemics. Now, we extend the description of the residual network to include not only uninfected nodes, but also nodes that were previously infected but have already lost immunity. We apply bond percolation methods to this extended residual network to model the spread of a subsequent outbreak in a population that has already suffered an initial outbreak.
The simple Susceptible-Infectious-Recovered (SIR) bond percolation model allows us to derive fundamental epidemiological quantities based on the average transmissibility $T$ of the pathogen (that is, the average probability that an infected node will transmit to a susceptible contact sometime during its infectious period) and the degree distribution of the host contact network, denoted $\{p_{k}\}$ where $p_{k}$ is the fraction of nodes with degree $k$ [@mejn]. This assumes that the probabilities of transmission from infected nodes to susceptible nodes are *iid* random variables. We can then calculate the epidemic threshold for a given network ($T_{c})$, above which a large scale epidemic is possible; this is closely related to the traditional epidemiological quantity, $R_{0}$. We can also find the probability and expected size of an epidemic above that threshold as well as the probability that an individual at the end of a randomly chosen edge (contact) does not become infected during an epidemic ($u$) [@mejn]. We will apply this method to calculate epidemic quantities for two consecutive seasons, and use subscripts $1$ and $2$ to denote initial and subsequent outbreak, respectively. Specifically, $T_{1}$ and $T_{2}$ denote the average transmissibilities of the pathogen in each season, respectively, and allow for evolution of infectiousness from one season to the next; $p_{1}(k)$ and $p_{2}(k)$ denote the fraction of nodes with $k$ susceptible contacts prior to the first and second seasons, respectively; and $u_{1}$ and $u_{2}$ denote the fraction of contacts that remain uninfected following the each outbreak.
The probability that an individual of degree $k$ will remain uninfected after the first epidemic can be calculated as $\left(1-T_{1}+T_{1}u_{1}\right)^{k}$ [@meyers_sars]. We denote this probability $\eta_{1}(k)$. We next derive the degree distribution of the epidemiologically active portion of the network following the initial outbreak. This includes both nodes that were not infected and nodes that were infected and subsequently lost immunity, as well as all edges connecting them. The fraction of *active* nodes with $k$ *active* edges just prior to the second outbreak is given by
$$p_{2}\left(k\right)=\frac{p_{2}^{uninfected}\left(k\right)+\alpha p_{2}^{infected}\left(k\right)}{\sum\limits _{j}p_{1}(j)\eta_{1}(j)+\alpha\sum\limits _{j}p_{1}(j)\left(1-\eta_{1}(j)\right)}\label{eq:1}$$
where $p_{2}^{uninfected}\left(k\right)$ and $p_{2}^{infected}\left(k\right)$ are the fractions of susceptible nodes with $k$ susceptible neighbors among previously uninfected and infected nodes, respectively and $\alpha$ is the proportion of infected individuals who have lost immunity prior to the second outbreak. The denominator of Equation \[eq:1\] gives the proportion of the network that is susceptible prior to the second outbreak, where the first term considers previously uninfected nodes, and the second term gives the proportion $\alpha$ of previously infected nodes.
The probability that a node in the residual network has $k$ remaining edges (i.e. edges that connect them to other susceptible nodes), given that it had $\kappa$ edges in the initial network is the following:
$$p_{2}(k\vert k_{init}=\kappa)=\left(\begin{array}{c}
\kappa\\
k\end{array}\right)\left(u_{1}+(1-u_{1})\alpha\right)^{\kappa}\left((1-u_{1})(1-\alpha)\right)^{\kappa-k}$$ For every node in the residual network, remaining edges include (a) those that lead to nodes that were uninfected in the previous epidemic (which occurs with probability $u_{1}$[@bansal_residual]) and (b) those that lead to nodes that were infected but have lost immunity (which occurs with probability $\left(1-u_{1}\right)\alpha$). Then the degree distribution prior to season two can thus be rewritten as,
$$p_{2}\left(k\right)=\frac{\sum\limits _{\kappa\ge k}p_{1}(\kappa)\eta_{1}(\kappa)p_{2}\left(k|k_{init}=\kappa\right)+\alpha\sum\limits _{\kappa\ge k}p_{1}(\kappa)\left(1-\eta_{1}(\kappa)\right)p_{2}\left(k|k_{init}=\kappa\right)}{\sum\limits _{j}p_{1}(j)\eta_{1}(j)+\alpha\sum\limits _{j}p_{1}(j)\left(1-\eta_{1}(j)\right)}$$ We provide the full derivation of this equation in the Supplementary Information.
The residual degree distribution $\{p_{2}(k)\}$ reflects the epidemiologically active portion of the population following the initial epidemic. Although the residual network differs from the original contact network in degree distribution, component structure and other topological characteristics, it is still reasonable to model it as a semi-random graph (as shown in [@bansal_residual]) and thus apply bond percolation methods [@mejn]. Additionally, we show in Supplementary Information that both immunity models also perform well on non-random realistic or empirical networks. We next derive epidemiological quantities that predict the fate of a subsequent outbreak through the residual network.
The probability generating function (PGF) for the second season degree distribution in terms of the PGF for the initial degree distribution, $\Gamma_{1}\left(x\right)$ is given by $$\Gamma_{2}(x)=\frac{\Gamma_{1}\left(r\left(x\left(1-s\right)+s\right)\right)+\alpha\Gamma_{1}\left(\left(1-r\right)\left(x\left(1-s\right)+s\right)\right)}{\Gamma_{1}\left(r\right)+\alpha\left(1-\Gamma_{1}\left(r\right)\right)}.$$ where $r=\left(1-T_{1}+T_{1}u_{1}\right)$ is the probability that disease was not transmitted along a uniform random edge in the first epidemic; and $s=\left(1-u_{1}\right)\left(1-\alpha\right)$ is the probability that a node at the end of a uniform random edge was infected gained full immunity.
This allows us to derive the epidemic threshold for the subsequent outbreak, that is, the critical value of transmissibility above which a second epidemic is possible, given that some previously infected individuals have perfect immunity. It is a function of the original network topology (via the PGF $\Gamma_{1}\left(x\right)$) and the loss of immunity, $\alpha$, and is given by $$\begin{array}{c}
\left(T{}_{2_{c}}\right)_{perfect}=\frac{\Gamma_{2}\prime\left(1\right)}{\Gamma_{2}\prime\prime\left(1\right)}=\frac{\Gamma_{1}^{\prime}\left(r\right)r\left(1-s\right)+\alpha\Gamma_{1}^{\prime}\left(1-r\right)\left(1-r\right)\left(1-s\right)}{\Gamma_{1}^{\prime\prime}\left(r\right)r^{2}\left(1-s\right)^{2}+\alpha\Gamma_{1}^{\prime\prime}\left(1-r\right)\left(1-r\right)^{2}\left(1-s\right)^{2}}\end{array}$$ where $\Gamma_{1}^{\prime}\left(r\right),\Gamma_{1}^{\prime}\left(1-r\right)$ are the average degrees among previously uninfected nodes and infected nodes, respectively. If the second strain is above this epidemic threshold, then the following equation gives the expected fraction of the residual population infected during the resulting epidemic
$$S_{2}=1-\Gamma_{2}\left(u_{2}\right)$$ where $u_{2}$ is the probability that a random edge in the residual network leads to a node which was uninfected in the second outbreak. (See Supplementary Information.) Thus the overall fraction expected to become infected during a second epidemic, assuming perfect partial immunity at a level $\left(1-\alpha\right)$ is given by $$\left(S_{2}\right)_{perfect}=S_{2}\left(\sum_{k}p_{1}(k)\eta_{1}(k)+\alpha\left(1-\sum_{k}p_{1}(k)\eta_{1}(k)\right)\right)$$ where $\sum p_{k}\eta_{k}$ represents the size of the population which was uninfected in the previous outbreak and $\alpha\left(1-\sum p_{k}\eta_{k}\right)$ is the proportion of the population that was infected in the previous outbreak but has lost immunity.
Leaky Partial Immunity
----------------------
To model leaky partial immunity, we reduce the probabilities of reinfection and transmission for nodes infected in the first epidemic. Rather than deleting nodes and attached edges entirely (as above), we introduce a two-type percolation approach in which the parameters of disease transmission depend on the epidemiological history of both nodes involved in any contact.
### Two-type Percolation
The standard bond percolation model of [@mejn] assumes that, all nodes of a given degree $k$ are homogeneous with respect to disease susceptibility and all edges are homogeneous (probabilities of transmission along edges are i.i.d. random variables with mean $T$). We extend the basic model to allow for two types of nodes, we call them $A$ and $B$; and four types of edges, $AA,\: AB,\: BA,\: BB$, connecting all combinations of nodes. (A similar model was recently introduced in [@babak_multitype].) We use $p_{ij}$ to denote the joint probability that a uniform random type $A$ node has $i$ edges leading to other type $A$ nodes and $j$ edges leading to type $B$ nodes (where $i$ the $A$-degree of the node and $j$ the $B$-degree of the node). Similarly, $q_{ij}$ denotes the joint probability of a type $B$ node having an $A$-degree of $i$ and a $B$-degree of $j$. The multivariate probability generating functions (PGFs) for these probability distributions are given by $$f_{A}(x,y)=\sum p_{ij}x^{i}y^{j}$$ $$f_{B}(x,y)=\sum q_{ij}x^{i}y^{j}$$
While $f_{A}$ and $f_{B}$ describe the distribution of degrees of randomly chosen $A$ and $B$ nodes, the degree of a node reached by following a randomly chosen edge is measured by the its excess degree [@mejn]. The PGFs for the $A$-excess degree and the $B$-excess degree of $A$ and $B$ nodes are given by
$$\begin{aligned}
f_{AA}(x,y)=\frac{\sum ip_{ij}x^{i-1}y^{j}}{\sum iq_{ij}} & & f_{BA}(x,y)=\frac{\sum jp_{ij}x^{i}y^{j-1}}{\sum jq_{ij}}\\
f_{AB}(x,y)=\frac{\sum iq_{ij}x^{i-1}y^{j}}{\sum iq_{ij}} & & f_{BB}(x,y)=\frac{\sum jq_{ij}x^{i}y^{j-1}}{\sum jq_{ij}}\end{aligned}$$
as illustrated in Figure \[fig:diag\].
Having formalized the structure of the contact network in PGFs, we can now derive the distributions for the number of infected edges, which are edges over which disease has been successfully transmitted. We assume that for each edge type ($XY$), transmission probabilities are i.i.d. random variables with averages denoted $T_{XY}$, and that these values can vary among the four edge types. Then the PGFs for the number of occupied edges emanating from a node of type $A$ and $B$ are, respectivel $$f_{A}(x,y;T_{AA},T_{AB})=f_{A}((1+(x-1)T_{AA}),(1+(y-1)T_{AB})$$ $$f_{B}(x,y;T_{BA},T_{BB})=f_{B}((1+(x-1)T_{BA}),(1+(y-1)T_{BB})$$
Each of these generating functions was derived following the arguments outlined in [@mejn] for the simple bond percolation SIR model. We can similarly derive the PGFs for the number of infected excess edges emanating from a node of type $A$ ($B$), at which we arrived by following a uniform random edge from a node of type $A$ ($B$): $$f_{AA}(x,y;T_{AA},T_{AB})=f_{AA}((1+(x-1)T_{AA}),(1+(y-1)T_{AB}))$$ $$f_{BA}(x,y;T_{AA},T_{AB})=f_{BA}((1+(x-1)T_{AA}),(1+(y-1)T_{AB}))$$ $$f_{AB}(x,y;T_{BA},T_{BB})=f_{AB}((1+(x-1)T_{BA}),(1+(y-1)T_{BB}))$$ $$f_{BB}(x,y;T_{BA},T_{BB})=f_{BB}((1+(x-1)T_{BA}),(1+(y-1)T_{BB}))$$
The PGFs for outbreak sizes starting from a node of type $A$ or $B$, respectively, are then given by $$F_{A}(x,y;T_{AA},T_{AB})=xf_{A}(F_{AA}(x,y;\left\{ T\right\} ),F_{AB}(x,y;\left\{ T\right\} );T_{AA},T_{AB})$$ $$F_{B}(x,y;T_{BA},T_{BB})=yf_{B}(F_{BA}(x,y;\left\{ T\right\} ),F_{BB}(x,y;\left\{ T\right\} );T_{BA},T_{BB})$$
where, $F_{AA}$ and $F_{BA}$ are the PGFs for the outbreak size distribution starting from an (infected) node of type $A$ which has been reached by following an edge from another (infected) node of type $A$ or $B$, respectively. Similarly, $F_{AB}$ and $F_{BB}$ are the PGFs for the outbreak size distribution starting from an (infected) node of type $B$ which has been reached by following an edge from another (infected) node of type $A$ or $B$, respectively. These PGFs are as follows
$$F_{AA}(x,y;\left\{ T\right\} )=xf_{AA}(F_{AA}(x,y;\left\{ T\right\} ),F_{AB}(x,y;\left\{ T\right\} );T_{AA},T_{AB})$$ $$F_{BA}(x,y;\left\{ T\right\} )=xf_{BA}(F_{BA}(x,y;\left\{ T\right\} ),F_{BB}(x,y;\left\{ T\right\} );T_{AA},T_{AB})$$ $$F_{AB}(x,y;\left\{ T\right\} )=yf_{AB}(F_{BA}(x,y;\left\{ T\right\} ),F_{BB}(x,y;\left\{ T\right\} );T_{BA},T_{BB})$$ $$F_{BB}(x,y;\left\{ T\right\} )=yf_{BB}(F_{BA}(x,y;\left\{ T\right\} ),F_{BB}(x,y;\left\{ T\right\} );T_{BA},T_{BB})$$
Again following the method of [@mejn], we can derive the expected size of a small outbreak and the epidemic threshold (given in the Supplementary Information). The expected numbers of $A$ and $B$ nodes infected in a small outbreak are found by taking partial derivatives of the PGF for the outbreak size distribution: $$\left\langle s\right\rangle _{A}=\frac{\partial F_{A}}{\partial x}|_{x=1,y=1}+\frac{\partial F_{B}}{\partial x}|_{x=1,y=1}$$
$$\left\langle s\right\rangle _{B}=\frac{\partial F_{A}}{\partial y}|_{x=1,y=1}+\frac{\partial F_{B}}{\partial y}|_{x=1,y=1}$$
Finally, we can find the size of a large-scale epidemic among $A$ nodes and among $B$ nodes as: $$S_{A}\left(T_{AA},T_{AB}\right)=1-F_{A}(1,1;T_{AA},T_{AB})=1-\sum p_{ij}(1+(a-1)T_{AA})^{i}(1+(c-1)T_{AB})^{j}\label{eq:4}$$ $$S_{B}\left(T_{BA},T_{BB}\right)=1-F_{B}B(1,1;T_{BA},T_{BB})=1-\sum q_{ij}(1+(b-1)T_{BA})^{i}(1+(d-1)T_{BB})^{j}\label{eq:5}$$
where, $a=F_{AA}(1,1;\left\{ T\right\} ),b=F_{BA}(1,1;\left\{ T\right\} ),c=F_{AB}(1,1;\left\{ T\right\} ),d=F_{BB}(1,1;\left\{ T\right\} )$. The probability of a large-scale epidemic can be derived similarly. The numerical values for the size and probability of an outbreak will be equal if $T_{AB}=T_{BA}$. Further details are provided in the Supplementary Information.
This two-type percolation model provides a general framework for modeling pathogens with variable transmissibility and host populations with immunological heterogeneity.
### Modeling Leaky Immunity with Two-Type Percolation
We now apply the two-type percolation method to model leaky partial immunity. In this model, type $A$ nodes represent individuals who were not infected in the initial epidemic and thus have no prior immunity, and type $B$ nodes represent those who were infected and maintain partial immunity (at a level $1-\alpha$). (Note that $\alpha$ gives the fraction of immunity lost in both models.) Here, we assume that prior immunity causes equal sized reductions in both infectivity and susceptibility ($\alpha$); but the approach can be extended easily to include more complex models of immunity. Specifically, during the subsequent epidemic, type A individuals (previously uninfected) have a susceptibility of one and an infectivity of $T_{2},$while type B individuals (previously infected) have a susceptibility of $\alpha$ and an infectivity of $T_{2}\alpha$. Correspondingly, $T_{AA}=T_{2}$, $T_{AB}=T_{2}\alpha$, $T_{BA}=T_{2}\alpha$, and $T_{BB}=T_{2}\alpha^{2}$ .
The joint degree distributions for type A and type B nodes depend on the course of the initial epidemic, and are given by
$$p_{ij}=p_{1}(i+j)\eta_{1}(i+j)\left(\begin{array}{c}
i+j\\
i\end{array}\right)u_{1}^{i}\left(1-u_{1}\right)^{j}$$
$$q_{ij}=p_{1}(i+j)\left(1-\eta_{1}(i+j)\right)\left(\begin{array}{c}
i+j\\
i\end{array}\right)u_{1}^{i}\left(1-u_{1}\right)^{j}$$
respectively, where $i$ is the A-degree and $j$ is the B-degree. Further explanation can be found in Supplementary Information.
Using the quantities derived above, we can model epidemics that leave varying levels of individual-level partial immunity. Using equations \[eq:4\] and \[eq:5\], for example, we can solve for the size of the epidemic in a second epidemic with (individual-level) leaky partial immunity, $\left(1-\alpha\right)$:
$$\left(S_{2}\right)_{leaky}=\left(\sum_{k}p_{1}(k)\eta_{1}(k)\right)S_{A}\left(T_{2},T_{2}\alpha\right)+\left(1-\sum_{k}p_{1}(k)\eta_{1}(k)\right)S_{B}\left(T_{2}\alpha,T_{2}\alpha^{2}\right).$$
Results
=======
Impact of One Epidemic on the Next
----------------------------------
We have introduced two distinct mathematical approaches for modeling the epidemiological consequences of naturally-acquired immunity. The residual network model probabilistically removes nodes and edges corresponding to the fraction ($\alpha$) of infected nodes expected to lose immunity entirely. The two-type percolation model tracks the epidemiological history of all individuals and reduces the infectivity and susceptibility of all previously infected nodes by a fraction ($\alpha$). By adjusting $\alpha$, both models can explore the entire range of immunity from none to complete. At $\alpha=1$ , these model the absence or complete loss of immunity and thus would apply when the second season strain is entirely antigenically distinct from the prior strain. At $\alpha=0$, these model full or no loss of prior immunity and might apply when a secondary epidemic is caused by the same or very similar pathogen as caused the first epidemic. Values of $\alpha$ between 0 and 1 represent partial immunity to the second pathogen, with the level of protection increasing as $\alpha$ approaches 0.
In Figure \[fig:compare\_imm\], we compare the predicted sizes of a second epidemic for both the perfect and leaky models against simulations for a Poisson, exponential and scale-free random network (of the same mean degree) and under the conditions of no prior immunity ($\alpha=1$), partial immunity ($\alpha=0.5$), and full immunity ($\alpha=0$) for values of transmissibility between $0$ and $0.5$ . T Assuming no immunity (Figure \[fig:compare\_imm\](a)), the two models simplify to the standard bond percolation model on the original network, and thus make identical predictions. Assuming full immunity (Figure \[fig:compare\_imm\](c)), the perfect immunity model removes all previously infected nodes (and the corresponding edges) before the second outbreak; and the leaky immunity model sets transmissibility along all edges leading from and to previously infected nodes to zero, thus de-activating those nodes entirely. Consequently, the models also converge at this extreme. The two models are, however, fundamentally different for any level of intermediate partial immunity between $\left({0<\alpha<1}\right)$ as they assume different models of immunological protection (Figure \[fig:compare\_imm\](b)). At $\alpha=0.5$, leaky immunity confers greater herd immunity than perfect immunity at low values of transmissibility, while the reverse is true for more infectious pathogens. The makeup of the previously infected population is identical in both models and biased towards high degree individuals. When the pathogen is only mildly contagious, leaky immunity goes a long way towards protecting all previously infected hosts whereas perfect immunity protects only a fraction of these hosts; when it is more highly contagious, however, leaky immunity is insufficient to protect hosts with large numbers of contacts whereas perfect immunity is not diminished. We find further that network heterogeneity acts consistently across different levels of immunity. The Poisson network has the most homogeneous degree distribution followed by the exponential network and finally the scale-free network with considerable heterogeneity. Holding mean degree constant, variance in degree increases the vulnerability of the population (allowing epidemics to occur at lower rates of transmissibility), yet generally reduces the ultimate size of epidemics when they occur. At high levels of immunity, the susceptible network at the start of the second season becomes more sparse and homogeneous. Thus the impact of network variance on the second epidemic diminishes as immunity increases, that is, as $\alpha\rightarrow0$. (We elaborate further on these results in the Supplementary Information.)
We explore intermediate levels of immunity further in Figure \[res-fig7\], and again find reasonable agreement between our analytic predictions and simulations. As expected, increasing levels of immunity (from left to right) decrease the epidemic potential of a second outbreak.At these intermediate values of transmissibility ($T_{1}=0.15$ and $T_{2}=0.3$), leaky immunity tends to confer lower herd immunity than perfect immunity, except at extremely high levels of immunity. The level of immunity at which the predicted epidemic sizes for two immunity models cross represents the point at which leaky partial immunity for all prior cases effectively protects more individuals than the complete removal of a fraction of those cases. This transition point occurs at a higher level of immunity in the exponential network than the Poisson network, and never occurs in the scale-free network, perhaps because the immunized individuals in the more heterogeneous networks tend to have anomalously high numbers of contacts thus limiting the efficacy of partial protection.
Pathogen Re-invasion and Immune Escape
--------------------------------------
When a pathogen enters a population that has experienced a prior outbreak, its success will depend on the extent and pattern of naturally-acquired immunity in the host population. The new pathogen may not be able to invade unless it is significantly different from the original strain. If it is antigenically distinct from the prior strain, then prior immunity may be irrelevant; and if it is more transmissible than the original strain, then it may have the potential to reach previously unexposed individuals.
Figure \[fig:reinvasion\] indicates the minimum transmissibility required for the new strain to cause an epidemic (that is, its critical transmissibility $T_{2_{c}}$), as a function of the transmissibility of the original strain ($T_{1}$) and the level of leaky immunity ($\alpha$). The leakier the immunity (high $\alpha$) and the lower the infectiousness of the original strain (low $T_{1}$), the more vulnerable the population to a second epidemic (light coloration in Figure \[fig:reinvasion\]). Generally the homogeneous Poisson network is less vulnerable to re-invasion than the heterogeneous scale-free network. The blue curves in figure \[fig:reinvasion\] show combinations of $T_{1}$ and $\alpha$ where the epidemic threshold for the new strain equals the transmissibility of the original strain ($T_{2_{c}}=T_{1}$) and have two complementary interpretations. First, if we assume that the new strain is exactly as transmissible as the original strain ($T=T_{2}=T_{1}$), then the curves indicate the critical level of cross-immunity ($\alpha_{c}(T)$) below which the strain can never invade and above which the strain can invade with some probability that increases with $\alpha$. This threshold indicates the extent of antigenic evolution (or intrinsic decay in immune response) required for a second epidemic to occur. The more heterogeneous the contact patterns (scale-free versus Poisson network), the lower the amount of immune escape required for a pathogen of the same transmissibility to re-invade. Second, if we assume a fixed level of immune decay ($\alpha$), then the curves indicate the critical initial transmissibility ($T_{1}$) above which the new strain can only invade if it is more contagious than the original strain ($T_{2}>T_{1}$). Below this point, the network topology and preexisting immunity create a selective environment that excludes the original strain and favors more transmissible variants. The perfect immunity model yields similar results (Supplementary Information.)
Some epidemiologists have speculated that there are ’trade-offs’ between virulence and infectiousness, implying that more infectious pathogens will necessarily be more virulent [@anderson_82; @ewald; @bull; @frank]. If true, figure \[fig:reinvasion\] suggests that naturally-acquired immunity, by opening niches for more infectious variants, may indirectly lead to the evolution of greater virulence. This is consistent with a previous study showing that that host populations with high levels of immunity maintain more virulent pathogens than na�ve host populations [@gandon].
Discussion & Conclusion
=======================
In this work, we have considered the impact of pathogen spread on future outbreaks of the same or similar pathogen and on pathogen invasion and evolution. We have compared two standard models for immunity, perfect and leaky, and found that the extent of herd immunity varies with the pathogen transmissibility and the degree and nature of immunity. Leaky immunity appears to confer greater herd immunity at moderate levels of pathogen infectiousness for all levels of partial immunity, whereas perfect immunity is more effective at higher transmissibilities.
This analysis also has implications for public health intervention strategies. Contact-reducing interventions (e.g., patient quarantine and social distancing) and vaccination often result in complete removal of a fraction of individuals from the network (akin to perfect immunity), whereas transmission-reducing interventions (e.g., face-masks and other hygienic precautions) typically reduce transmissibility along edges leading to and from a fraction of individuals (akin to leaky immunity) [@pourbohloul]. These results thus suggest that contact reductions will be more effective than a comparable degree of transmission reductions at higher levels of pathogen infectiousness.
The evolution of new antigenic characteristics in a pathogen that escape prior immunity and the evolution of higher transmissibility both depend on genetic variation. Thus, the more infections there are in the first season, the greater the opportunity for evolutionary change [@boni]. This poses a trade-off for the pathogen: a large initial epidemic may generate variation that fuels evolution yet wipes out the susceptible pool for the subsequent season; while a small initial outbreak leaves a large fraction of the network susceptible to future transmission yet may fail to generate sufficient antigenic or other variation for future adaptation. We have shown that the trade-off between generating immunity via infections and escaping immunity via antigenic drift will depend not only on the size of the susceptible population, but also on its connectivity. Although we have focused primarily on the role of antigenic drift, these models also apply to loss of immunity through decay in immunological memory, as occurs following pertussis and measles infections [@mossong_measles; @pertussis].
Much epidemiological work, particularly the analysis of intervention strategies, ignores the immunological history of the host population. Thus our effort to incorporate host immune history into a flexible individual-based network model will potentially advance our understanding of the epidemiological and evolutionary dynamics of partially-immunizing infections such as influenza, pertussis, or rotavirus. However, these provide just an initial step in this direction, as the models consider only two consecutive seasons and do not yet allow for replenishment or depletion of susceptibles due to births and deaths.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported by the RAPIDD program of the Science & Technology Directorate, Department of Homeland Security, and the Fogarty International Center, National Institutes of Health, and grants from the James F. McDonnell Foundation and National Science Foundation (DEB-0749097) to L.A.M.
![Epidemiological contact networks. (a) Prior to an initial epidemic, all individuals are fully susceptible to disease (gray nodes). Then some individuals become infected during the epidemic (red nodes). (b) Perfect partial immunity (at 50%) means that half of the previously infected individuals are fully protected against reinfection, while the other half are fully susceptible again. (c) Leaky partial immunity (at 50%) means that all nodes remain in the network, but the edges leading to and/or from previously infected individuals are half as likely to transmit disease (illustrated here with the lighter edges.)[]{data-label="fig:network"}](newfig.pdf){width="11cm"}
![The probability generating functions give the numbers of A and B contacts for each type of vertex (top). The four excess degree distributions give the numbers of each type of contact for a vertex chosen by following a uniform random edge (bottom).[]{data-label="fig:diag"}](partial_immunity_model_diag.pdf){width="12cm"}
![Expected size of a second epidemic as infectiousness increases. We compare the predictions of our mathematical models for perfect (dashed line) and leaky (solid line) immunity to corresponding numerical simulations (crosses and circles indicate perfect and leaky immunity, respectively). Calculations are for three types of networks: Poisson (red), exponential (blue), and power law (green) with mean degree 10, for three levels of immunity: (a) no immunity ($\alpha=1$), (b) partial immunity ($\alpha=0.5$), and (c) full immunity ($\alpha=0$), and for a range of second strain transmissibility values ($T_{2}$) along each x-axis (assuming $T_{1}=0.15$ in all cases). []{data-label="fig:compare_imm"}](methods_compare_T.pdf){width="14cm"}
![Expected size of a second epidemic as immunity increases. We compare predictions of the perfect immunity model (gray dashed lines), leaky immunity model (black solid lines) and simulations for each model (gray cross and black circle markers, respectively). Calculations and simulations are for networks with (a) Poisson, (b) exponential, and (c) scale-free degree distributions with mean degree 10, at transmissibilities $T_{1}=0.15$ and $T_{2}=0.3$.[]{data-label="res-fig7"}](methods_compare_alpha.pdf){width="14cm"}
\[h\] ![Epidemic threshold ($T_{2_{c}})$ in the second season. The colors indicate the level of transmissibility required for the second strain to invade the population (cause an epidemic), assuming leaky partial immunity for (a) a Poisson-distributed network and (b) a scale-free network, each with mean degree of 10. The x-axis gives the first season transmissibility ($T_{1}$) and y-axis gives the loss of immunity ($\alpha$). The blue line denotes $T_{2_{c}}=T_{1}$,: above the line $T_{2_{c}}<T_{1},$ and invasion by the original pathogen is possible.\[fig:reinvasion\]](ms1_threshold.pdf "fig:"){width="14cm"}
[^1]: Center for Infectious Disease Dynamics, The Pennsylvania State University, 208 Mueller Lab, University Park PA 16802
[^2]: Fogarty International Center, National Institutes of Health, Bethesda, MD 20892, USA
[^3]: Corresponding Author: shweta@sbansal.com
[^4]: Section of Integrative Biology and Institute for Cellular and Molecular Biology, University of Texas at Austin, 1 University Station, C0930, Austin, TX 78712, USA
[^5]: Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA
|
---
title: 'Symmetry Energy in the Equation of State of Asymmetric Nuclear Matter\'
---
Introduction
============
The equation of state of isospin asymmetric nuclear matter is a fundamental quantity that determines the properties of systems as small and light as an atomic nucleus, and as large and heavy as a neutron star. The key unknown in the EOS of asymmetric nuclear matter is the symmetry energy. Recently the possibility of extracting information on the symmetry energy and the isospin (neutron-to-proton ratio) of the fragments in a multifragmentation reaction has gained tremendeous importance [@SHE05; @FEV05]. Such information is of importance for understanding key problems in astrophysics[@DAN02], and various aspects of nuclear physics such as the structure of exotic nuclei (the binding energy and rms radii) [@BRO00; @HORO01] and the dynamics of heavy ion collisions [@BAL01].
Traditionally, the symmetry energy of nuclei has been extracted by fitting the binding energy of the ground state with various versions of the liquid drop mass formula. The properties of nuclear matter are then determined by theoretically extrapolating the nuclear models designed to study the structure of real nuclei. However, real nuclei are cold, nearly symmetric ($N \approx Z$) and found at equilibrium density. It is not known how the symmetry energy behaves at temperatures, isospin (neutron-to-proton ratio) and densities away from the normal nuclear matter. Theoretical many-body calculations [@DIE03] and those from the empirical liquid drop mass formula [@MYE66] predict symmetry energy near normal nuclear density ($\approx$ 0.17 fm$^{-3}$) and temperature ($T$ $\approx$ 0 MeV), to be around 28 - 32 MeV.
In a multifragmentation reaction, an excited nucleus expands to a sub-nuclear density and disintegrates into various light and heavy fragments. The fragments are highly excited and neutron-rich ; their yields depend on the available free energy, which in turn depends on the strength of the symmetry energy and the extent to which the fragments expand. By studying the isotopic yield distribution of these fragments, one can extract important information about the symmetry energy and the properties of the fragments at densities, excitation energies and isospin away from those of ground state nuclei. Here we present some of the results obtained from various measurements carried out at the Cyclotron Institute of Texas A$\&$M University (TAMU). We present these results in the framework of both, the statistical and the dynamical multifragmentation models.
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Isoscaling and Symmetry Energy
==============================
In a multifragmentation reaction, the ratio of isotope yields in two different reactions, 1 and 2, $R_{21}(N,Z) = Y_{2}(N,Z)/Y_{1}(N,Z)$, has been shown to obey an exponential dependence on the neutron number ($N$) and the proton number ($Z$) of the isotopes, an observation known as isoscaling [@BOT02; @TSAN01; @TSANG01]. The dependence is characterized by a simple relation,
$$R_{21}(N,Z) = Y_{2}(N,Z)/Y_{1}(N,Z) = C exp({\alpha N + \beta Z})$$
where, $Y_{2}$ and $Y_{1}$ are the yields from the neutron-rich and neutron-deficient systems, respectively. $C$ is an overall normalization factor, and $\alpha$ and $\beta$ are the parameters characterizing the isoscaling behavior.
Theoretically, isoscaling has been predicted by both, statistical [@TSAN01] as well as dynamical [@ONO03] multifragmentation models. In these models, the difference in the chemical potential of systems with different $N/Z$ is directly related to the scaling parameter $\alpha$. The scaling parameter $\alpha$ is proportional to the symmetry energy through the relation,
$$\alpha = \frac{4 C_{sym}}{T} \bigg (\frac{Z_{1}^{2}}{A_{1}^{2}} - \frac{Z_{2}^{2}}{A_{2}^{2}} \bigg )$$
In the above equation, $Z_{1}$, $A_{1}$ and $Z_{2}$, $A_{2}$ are the charge and the mass numbers of the fragmenting systems, $T$ is the temperature of the system and $C_{sym}$, the symmetry energy. The parameter $\alpha$, has been shown to be independent of the complex nature of the secondary de-excitation of the primary fragments, and is thus a robust observable for studying the symmetry energy [@TSAN01]. While it is well established that many versions of statistical models show very little or no difference between the $\alpha$ values for the primary and the secondary fragments, the same may not be true for the $\alpha$ values obtained from dynamical models. The origin of this discrepancy between the two approaches is currently being debated and not fully understood. It will be shown from the present study that the difference between the primary and the secondary $\alpha$’s in statistical model is very small. We will presume the difference to be insignificant in the dynamical model framework.
Fig. 1 shows the experimentally determined isotopic yield ratio as a function of neutron number $N$, for some of the reactions studied using beams from the K500 Cyclotron at TAMU. The figure on the left shows the ratios for the ($^{40}$Ar + $^{58}$Fe)/($^{40}$Ca + $^{58}$Ni) and ($^{40}$Ar + $^{58}$Ni)/($^{40}$Ca + $^{58}$Ni) pairs of reactions. The one on the right is for the deep-inelastic reactions of $^{64}$Ni + Ni, Sn, Th, Pb at 20 AMeV. One observes that the ratios for various elements in a given reaction pair lie along a straight line in the logarithmic plot and align with the neighboring elements in accordance with the relation given in equation 1. This feature is observed for all the beam energies and the pairs of reactions studied. The alignment of the data points varies with beam energies as well as the pairs of reaction. The ratio for each elements ($Z$) were simultaneously fitted using an exponential relation (shown by the solid lines) to obtain the slope parameter $\alpha$. In the following sections, we use these experimentally determined $\alpha$’s to study the symmetry energy using the statistical and dynamical model interpretation of the multifragmentation reaction.
Symmetry Energy from a Statistical Model Approach
=================================================
The Statistical Multifragmentation Model (SMM) [@BON95; @BOT95] is the most widely used model for describing multfragmentation reactions. It is based on the assumption of statistical equilibrium at a low density freeze-out stage. All breakup channels composed of nucleons and excited fragments are taken into account and considered as partitions. During each partition the conservation of mass, charge, energy and angular momentum is taken into account, and the partitions are sampled uniformly in the phase space according to their statistical weights using Monte Carlo sampling. The Coulomb interaction between the fragments is treated in the Wigner-Seitz approximation. Light fragments with mass number $A$ $\leq$ 4 are considered as elementary particles with only translational degrees of freedom (“nuclear gas"). Fragments with $A$ $>$ 4 are treated as heated nuclear liquid drops, and their individual free energies $F_{A,Z}$ are parametrized as a sum of the volume, surface, Coulomb and symmetry energy,
$$F_{A,Z} = F^{V}_{A,Z} + F^{S}_{A,Z} + E^{C}_{A,Z} + E^{sym}_{A,Z}$$
where $F^{V}_{A,Z} = (-W_{o} - T^{2}/\epsilon_{o})A,$ with parameter $\epsilon_{o}$ related to the level density and $W_{o}$ = 16 MeV being the binding energy of infinite nuclear matter. $F^{S}_{A,Z} = B_{o}A^{2/3}[(T^{2}_{c} -
T^{2})/(T^{2}_{c} + T^{2})]^{5/4}$, with $B_{o}$ = 18 MeV being the surface co-efficient and $T_{c}$ = 18 MeV being the critical temperature of infinite nuclear matter. $E^{C}_{A,Z} = c Z^{2}/A^{1/3}$, where $c = (3/5)(e^{2}/r_{o})[1 - (\rho/\rho_{o})^{1/3}]$ is the Coulomb parameter obtained in the Wigner-Seitz approximation with charge unit $e$, and $r_{o}$ = 1.17 fm. $E^{sym}_{A,Z} = C_{sym}(A - 2Z)^{2}/A$, where $C_{sym}$ = 25 MeV is the symmetry energy co-efficient. These parameters are adopted from the Bethe-Weizsacker mass formula and correspond to the assumption of isolated fragments with normal density in the freeze-out configuration. The value of the symmetry energy co-efficient $C_{sym}$ is taken from the fit to the binding energies of isolated cold nuclei in their ground states. In a multifragmentation process the primary fragments are not only excited but also expanded.
Figure 2 shows a comparison between the SMM calculated and the experimentally observed values of $\alpha$. The left side of the figure corresponds to the ($^{40}$Ar + $^{58}$Ni)/($^{40}$Ca + $^{58}$Ni) and the ($^{40}$Ar + $^{58}$Fe)/($^{40}$Ca + $^{58}$Ni) pairs of reactions. The one on the right corresponds to ($^{58}$Fe + $^{58}$Fe)/($^{58}$Ni + $^{58}$Ni) and ($^{58}$Fe + $^{58}$Ni)/($^{58}$Ni + $^{58}$Ni) pairs of reactions. The dotted lines in the $\alpha$ versus excitation energy plot corresponds to $\alpha$ calculated from the primary fragment distribution and the solid lines to those calculated from the secondary fragment distribution. The symbols correspond to the experimentally determined $\alpha$’s. The figure on the left clearly shows that the experimentally determined $\alpha$’s are significantly lower than the calculated values using the standard value of the symmetry energy, $C_{sym}$ = 25 MeV. To explain the observed dependence of the isoscaling parameter $\alpha$ on excitation energy, the $C_{sym}$ of the hot primary fragment in the SMM calculation was varied in the range 25 - 15 MeV. As shown in the center and the bottom panel of the left figure, the isoscaling parameter decreases slowly with decreasing symmetry energy. The experimentally determined $\alpha$ can be reproduced for both pairs of systems at all excitation energies using a symmetry energy value of $C_{sym}$ = 15 MeV. This value of the symmetry energy is significantly lower than the value of $C_{sym}$ = 25 MeV often used for the isolated cold nuclei in their ground states. On the right side of the figure, we show the comparisons for the ($^{58}$Fe + $^{58}$Fe)/($^{58}$Ni + $^{58}$Ni) and ($^{58}$Fe + $^{58}$Ni)/($^{58}$Ni + $^{58}$Ni) pairs of reactions. Once again a lower value of symmetry energy $C_{sym}$ is required to explain the experimental data. Furthermore, one also observes a small dependence of the symmetry energy with increasing excitation energy.
{width="52.00000%" height="0.34\textheight"} {width="52.00000%"}
{width="52.00000%"}
A similar behavior of the $C_{sym}$ is also observed from the deep-inelastic reactions studies of $^{86}$Kr + $^{124,112}$Sn, $^{64,58}$Ni (25 AMeV), $^{64}$Ni + Ni, Sn, Th, Pb (25 AMeV) and $^{136}$Xe + Ni, Sn, Th, Au (20 AMeV) [@SOU05]. Fig. 3 shows the values of the $C_{sym}$ obtained from these reactions. The values were obtained using equation 2 with two different assumptions for temperature determination ; the Fermi gas temperature (closed symbols) and the expanding mononucleus temperatures(open symbols). The symmetry values for both set of temperatures show decreasing trend with increasing excitation energy.
The above comparison of the experimentally observed isoscaling parameter with the statistical multifragmentation model therefore shows that a significantly lower value of the symmetry energy is required to explain the isotopic composition of the fragments produced in a fragmentation reaction. This indicates that the properties of nuclei at high excitation energy, isospin and reduced density are very sensitive to the symmetry energy. Similar hot and neutron-rich nuclei are routinely produced in the interior of a collapsing star and subsequent supernova explosion, where a slight decrease in the symmetry energy can significantly alter the elemental abundance and the synthesis of heavy elements [@BOT04]. The present observations can provide important inputs for the understanding of the nuclear composition of supernova matter.
Symmetry Energy from a Dynamical Model Approach
===============================================
In the following, we analyze the above results in the dynamical model framework using the Anti-symmetrized Molecular Dynamic (AMD) calculation [@ONO03]. AMD is a microscopic model that simulates the time evolution of a nuclear collision, where the colliding system is represented in terms of a fully antisymmetrized product of Gaussian wave packets. During the evolution, the wave packet centroids move according to the deterministic equation of motion. The followed state of the simulation branches stochastically and successively into a huge number of reaction channels. The interactions are parameterized in terms of an effective force acting between nucleons and the nucleon-nucleon collision cross-sections. The beauty of the dynamical models is that it allows one to understand the functional form of the density dependence of the symmetry energy at a very fundamental level i.e., from the basic nucleon-nucleon interactions. Theoretical studies [@DIE03] based on microscopic many-body calculations and phenomenological approaches predict various forms of the density dependence of the symmetry energy. In general, two different forms have been identified. One, where the symmetry energy increases monotonically with increasing density (“ stiff " dependence) and the other, where the symmetry energy increases initially up to normal nuclear density and then decreases at higher densities (“ soft " dependence).
Determining the exact form of the density dependence of the symmetry energy is important for studying the structure of neutron-rich nuclei [@BRO00; @HORO01], and studies relevant to astrophysical problems, such as the structure of neutron stars and the dynamics of supernova collapse [@LAT01]. For example, a “ stiff " density dependence of the symmetry energy is predicted to lead to a large neutron skin thickness compared to a “ soft " dependence [@OYA98]. Similarly, a “ stiff " dependence of the symmetry energy can result in rapid cooling of a neutron star, and a larger neutron star radius, compared to a soft density dependence [@LAT94].
Recently, a linear relation between the isoscaling parameter $\alpha$, and the difference in the isospin asymmetry ($Z/A)^{2}$ of the fragments, with appreciably different slopes, was predicted for two different forms of the density dependence of the symmetry energy ; a “ stiff " dependence (obtained from Gogny-AS interaction) and a “ soft " dependence (obtained from Gogny interaction).
Fig. 4 (left) shows a comparison between the experimentally observed $\alpha$ and those from the AMD model calculations plotted as a function of the difference in the fragment asymmetry for the beam energy of 35 MeV/nucleon. The solid and the dotted lines are the AMD predictions using the “ soft " (Gogny) and the “ stiff " (Gogny-AS) density dependence of the symmetry energy, respectively. The solid and the hollow symbols (squares, stars, triangles and circles) are the results of the present study for the two different values of the fragment asymmetry, assuming Gogny and Gogny-AS interactions, respectively. Also shown in the figure are the scaling parameters (asterisks, crosses, diamond and inverted triangle) taken from various other works in the literature. It is observed that the experimentally determined $\alpha$ parameter increase linearly with increasing difference in the asymmetry of the two systems as predicted by the AMD calculation. Also, the data points are in closer agreement with those predicted by the Gogny-AS interaction (dotted line) than those from the usual Gogny force (solid line). The slightly lower values of the symbols from the present measurements with respect to the Gogny-AS values (dotted line) could be due to the small secondary de-excitation effect of the fragments not accounted for in this comparison. It has been shown [@TSANG01] that the experimentally determined $\alpha$ values can be lower by about 10 - 15 $\%$ for the systems and the energy studied here. Accounting for this effect results in a slight increase in the $\alpha$ values bringing them even closer to the dotted line. The observed agreement of the experimental data with the Gogny-AS type of interaction therefore appears to suggest a stiffer density dependence of the symmetry energy. However, as mentioned in section 2, the effect of secondary de-excitation in the dynamical model calculations is currently under study [@ONO] and the predicted sensitivity may be significantly diminished by the secondary decay.
Recently, Chen [*[et al.]{}*]{} [@CHE05] also showed, using the isospin dependent Boltzmann-Uehling-Uhlenbeck (IBUU04) transport model calculation, that a stiff density dependence of the symmetry energy parameterized as E$_{sym}$ $\approx$ 31.6 ($\rho/\rho_{\circ})^{1.05}$ explains well the isospin diffusion data [@TSA04] from NSCL-MSU (National Superconducting Cyclotron Laboratory at Michigan State University). Their calculation was also based on a momentum-dependent Gogny effective interaction. However, the present measurements on isoscaling gives a slightly softer density dependence of the symmetry energy at higher densities than those obtained by Chen [*[et al.]{}*]{}
The difference in stifness is clear from figure 4 (right), which shows the parameterization of various theoretical predictions of the density dependence of the nuclear symmetry energy in isospin asymmetric nuclear matter. The dot-dashed, dotted and the dashed curve corresponds to those from the momentum dependent Gogny interactions used by Chen [*[et al.]{}*]{} to explain the isospin diffusion data. These are given as, E$_{sym}$ $\approx$ 31.6 ($\rho/\rho_{\circ})^{\gamma}$, where, $\gamma$ = 1.6, 1.05 and 0.69, respectively. The solid curves and the solid points correspond to those from the Gogny and Gogny-AS interactions used to compare with the present isoscaling data. As shown by Chen [*[et al.]{}*]{}, the dependence parameterized by E$_{sym}$ $\approx$ 31.6 ($\rho/\rho_{\circ})^{1.05}$ (dotted curve) explains the NSCL-MSU data on isospin diffusion quite well. On the other hand, the isoscaling data from the present work can be explained well by the Gogny-AS interaction (solid points).
{width="52.00000%"} {width="52.00000%"}
Both measurements yield similar results at low densities with significant difference at higher densities. It is interesting to note that by parameterizing the density dependence of the symmetry energy that explains the present isoscaling data, one gets, E$_{sym}$ $\approx$ 31.6 ($\rho/\rho_{\circ})^{\gamma}$, where $\gamma$ = 0.69. This form of the density dependence of the symmetry energy is consistent with the parameterization adopted by Heiselberg and Hjorth-Jensen in their studies on neutron stars [@HEI00]. By fitting earlier predictions of the variational calculations by Akmal [*[et al.]{}*]{} [@AKM98], where the many-body and special relativistic corrections are progressively incorporated, Heiselberg and Hjorth-Jensen obtained a value of E$_{sym}$($\rho_{\circ}$) = 32 MeV and $\gamma$ = 0.6, similar to those obtained from the present measurements. The present form of the density dependence is also consistent with the findings of Khoa [*[et al.]{}*]{} [@KHO05], where a comparison of the experimental cross-sections in a charge-exchange reaction with the Hartree-Fock calculation using the CDM3Y6 interaction [@KHO97], reproduces well the empirical half-density point of the symmetry energy obtained from the present work (see fig. 2 of Ref. [@KHO05]).
The observed difference in the form of the density dependence of the symmetry energy between the present measurement and those obtained by Chen [*[et al.]{}*]{} is not surprising. Both measurements probe the low density part of the symmetry energy and are thus less sensitive to the high density region. But the important point to be noted is that both measurements clearly favor a stiff density dependence of the symmetry energy at higher densities, ruling out the very “ stiff ” (dot-dashed curve) and very “ soft ” (solid curve) predictions. These results can thus be used to constrain the form of the density dependence of the symmetry energy at supranormal densities relevant for the neutron star studies.
In view of the findings from the present measurements and those of Chen [*[et al.]{}*]{}, we believe that the best estimate of the density dependence of the symmetry energy that can be presently extracted from heavy ion reaction studies is, E$_{sym}$ $\approx$ 31.6 ($\rho/\rho_{\circ})^{\gamma}$, where $\gamma$ = 0.6 - 1.05. Measurements at higher densities should be able to constrain the density dependence of the symmetry energy further.
Conclusions
===========
In conclusion, a number of studies have been carried out at TAMU to study the symmetry energy in the equation of state of isospin asymmetric nuclear matter. The results were analyzed within the framework of statistical and dynamical model calculations. It is observed that the properties of nuclei at excitation energy, isospin and density away from the normal ground state nuclei are significantly different and sensitive to the symmetry energy. The symmetry energy required to explain the isoscaling parameter of the fragments produced in multifragmentation reactions are significantly lower, and as small as 15 MeV. The dynamical model calculation of the isoscaling parameter shows that a stiffer form of the density dependence of the symmetry energy is preferred over a soft dependence. A dependence of the form E$_{sym}$ $\approx$ 31.6 ($\rho/\rho_{\circ})^{0.69}$ appears to agree better with the present data. Recently it has been shown that this form of the density dependence of symmetry energy provides an accurate description of several collective modes having different neutron-to-proton ratios. Among the predictions from this dependence are a symmetric nuclear-matter incompressibility of $K$ = 230 MeV and a neutron skin thickness in $^{208}$Pb of 0.21 fm. Further, this dependence leads to a neutron star mass of $M_{max}$ = 1.72 $M_{\odot}$ and a radius of $R$ = 12.66 km for a “canonical" $M$ = 1.4 $M_{\odot}$ neutron star. These results have significant implications for nuclear astrophysics and future experiments probing the properties of nuclei using beams of neutron-rich nuclei.
Acknowledgment(s) {#acknowledgments .unnumbered}
=================
This work was supported in parts by the Robert A. Welch Foundation (grant No. A-1266) and the Department of Energy (grant No. DE-FG03-93ER40773). We also thank A. Botvina for fruitful discussions on statistical multifragmentation model.
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|
---
abstract: 'In this paper, we compute the asymptotic average of the decimals of some real numbers. With the help of this computation, we prove that if a real number cannot be represented as a finite decimal and the asymptotic average of its decimals is zero, then it is irrational. We also show that the asymptotic average of the decimals of simply normal numbers is 9/2.'
address: |
Department of Engineering Science\
Golpayegan University of Technology \
Golpayegan\
Iran
author:
- Peyman Nasehpour
title: A Computational Criterion for the Irrationality of Some Real Numbers
---
Introduction
============
One of the interesting topics in the theory of real numbers is to decide the irrationality of a real number based on the properties of the sequence of its decimal expansions [@Bundschuh1984; @CsapodiHegyvari2018; @Hegyvari1993; @Mahler1981; @Martinez2001; @Mercer1994; @Sander1995]. We mention two of such beautiful results:
- Bundschuh in [@Bundschuh1984], as a generalization of Mahler’s theorem in [@Mahler1981], proves that if $g,h\geq 2$ are two fixed integers, then the positive real number $0.(g^0) (g^1) (g^2) \ldots (g^n) \ldots$ is irrational, where $(g^n)$ is to mean the number $g^n$ written in base $h$.
- Hegyvári in [@Hegyvari1993] proves that if $(a_n)_{n\in \mathbb N}$ is a strictly increasing sequence of positive integers for which $\sum_{n=1}^{\infty} 1/a_n = \infty$, then the decimal fraction $\alpha = 0.(a_1) (a_2) \ldots (a_n) \ldots$ is irrational, where $(a_n)$ is to mean the positive integer number in base 10.
Hardy and Wright (cf. [@HardyWright1975 Theorem 137]) show that the real number $$r = 0.r_1r_2r_3 \ldots r_n \ldots = 0.011010100010 \ldots,$$ is irrational, where $r_n = 1$ if $n$ is prime and $r_n = 0$ otherwise. Their beautiful proof is based on this fact that no non-constant polynomial in $\mathbb Z[X]$ is a prime-representing function [@HardyWright1975 Theorem 21]. We recall that a function $f(x)$ is said to be a prime-representing function if $f(x)$ is a prime number for all positive integral values of $x$ [@Mills1947]. In Section \[sec:criterion\], by The Chebychev’s Estimate Theorem [@FineRosenberger2007 Theorem 4.2.1] and calculating the asymptotic average of the decimals of some real numbers, we give an alternative proof for Hardy and Wright’s theorem (see Theorem \[HardyWright\]). Note that this is a corollary of the main theorem of this paper which says that if a real number $r$ cannot be represented as a finite decimal and the asymptotic average of its decimals is zero, then $r$ is irrational (see Theorem \[AverageDecimalsThm2\]).
In Corollary \[AverageDecimalsCor\], we show that if $(a_n)_{n\in \mathbb N}$ is a strictly increasing sequence of positive integers such that $\displaystyle \lim_{n\to+\infty}\frac{n}{a_n} =0$, then $\displaystyle r=\sum_{n=1}^{+\infty} \frac{b_n}{10^{a_n}}$ is irrational, where each $1 \leq b_n \leq 9$ is a positive integer.
Section \[sec:normal\] is devoted to the asymptotic average of the decimals of simply normal numbers. Let us recall that a real number $r$ is a simply normal number to base $b$ if for the decimals $(r_n)_{n\in \mathbb N}$ of the fractional part $(0.r_1r_2r_3 \dots r_n \dots)_b $ of the real number $r$, we have the following property:
$$\lim_{n\to+\infty} \frac{\card \{j : 1\leq j \leq n, r_j=d\}}{n} = \frac{1}{b},$$ where $d \in \{0,1,2,\dots,b-1\}$ [@Bugeaud2012 Definition 4.1]. In Theorem \[SimplyNormalCor\], we prove that if $r$ is a simply normal number to base $b$, then the asymptotic average of the decimals of $r$ is equal to $\displaystyle \frac{b-1}{2}$. We need to recall that a fractional part of a real number $r$ is the non-negative real number $\operatorname{frac}(r) :=|r| - \lfloor |r| \rfloor$, where $|r|$ is the absolute value of $r$ and $\lfloor r \rfloor$ is the integer part of $r$.
In Proposition \[ChampernowneLiouville\], we show that the asymptotic average of the decimals of Champernowne number is 9/2. Note that Champernowne number is the number $C_{10} =0.1234567891011121314151617181920\dots,$ whose sequence of decimals is the increasing sequence of all positive integers. Since we are not aware of this point if the asymptotic average of the decimals of all irrational numbers exists and if it exists we do not know of a systematic method to calculate it, we propose a couple of questions at the end of the paper (check Questions \[AsymptoticAveragesQ\]).
A Criterion for the Irrationality of Some Real Numbers {#sec:criterion}
======================================================
First we recall some facts related to real numbers in order to fix some definitions and terminologies. Let us recall that every real number to base $b$ can be expressed by a decimal expansion, and this expansion can be performed in only one way [@StewartTall2015 p. 38]. To be more precise, a real number is regular with respect to some base number $b$ when it can be expanded in the corresponding number system with a finite number of negative powers of $b$ [@Ore1948 p. 316]. A regular number is also called a real number with finite decimal [@Havil2003 p. 25]. It is easy to see that a real number $r$ with respect to some base number $b$ is regular if and only if there are coprime integer numbers $p$ and $q$ such that $\displaystyle r = \frac{p}{q}$ and $q$ contains no other prime factors than those that divide $b$ [@Ore1948 p. 316]. In this paper, if the fractional part of a regular number to base $b$ is $$\alpha = (0.r_1 r_2 \dots r_n)_b,$$ we only consider the representation with an infinite series of $(b-1)$: $$\alpha = (0.r_1 r_2 \dots (r_n - 1) \overline{(b-1)} \dots)_b .$$ Therefore, if we agree always to pick the non-terminating expansion in the case of regular numbers, then fractional part of each real number to base $b$ corresponds uniquely to an infinite decimal $(0.r_1r_2r_3\dots r_n\dots)_b$. Finally, we assert that if $(r_n)_{n\in \mathbb N}$ is a sequence in real numbers, the sequence of the averages is defined as follows: $$a_n = \frac{r_1 + r_2 + \dots + r_n}{n}.$$
\[AverageDecimalsDef\]
Let the fractional part of a real number $r$ to base $b$ be $$(0.r_1 r_2 \dots r_n \dots)_b.$$ Then, we define the asymptotic average of the decimals of the real number $r$ by $$\operatorname{Av}_b(r) = \lim_{n\to+\infty} \frac{r_1 + r_2 + \dots + r_n}{n},$$ if it exists. Usually, we denote $\operatorname{Av}_{10}(r)$ by $\operatorname{Av}(r)$ if there is no fear of any ambiguity.
In the following, we calculate the asymptotic average of the decimals of some real numbers:
\[AverageDecimalsThm1\]
Let the decimals $(r_n)_{n\in \mathbb N}$ of the fractional part $$(0.r_1r_2r_3 \dots r_n \dots)_b$$ of a real number $r$ satisfy the following:
$$\lim_{n\to+\infty} \frac{\card \{j : 1\leq j \leq n, r_j=d\}}{n} = \omega_d,$$ where $d \in \{0,1,2,\dots,(b-1)\}$, $0 \leq \omega_d \leq 1$, and $\displaystyle \sum^{b-1}_{d=0} \omega_d = 1$. Then, $$\operatorname{Av}_b(r) = \sum^{b-1}_{d=1} (d \cdot \omega_d) .$$
Let $A(d,n) = \{j : 1\leq j \leq n, r_j=d\}$. By assumption, $$\lim_{n\to+\infty} \frac{\card A(d,n)}{n} = \omega_d.$$ This means that for any $\epsilon > 0$, there is a natural number $N_d$ such that if $n > N_d$, then $$\left| \frac{\card A(d,n)}{n} - \omega_d\right| < \epsilon.$$ Now, if we define $N = \max\{N_d\}^{b-1}_{d=0}$, for each $n > N$, we have the following:$$\label{inequality} \displaystyle \omega_d - \epsilon < \frac{\card A(d,n)}{n} < \omega_d + \epsilon. $$
Since $$\frac{r_1 + r_2 + \dots + r_n}{n} =\displaystyle \frac{\sum^{b-1}_{d=0} \sum_{i\in A(d,n)} r_i}{n} = \frac{\sum^{b-1}_{d=1} d\card A(d,n)}{n},$$ by using the inequality (\[inequality\]), we have the following: $$ \left| \frac{r_1 + r_2 + \dots + r_n}{n} - \sum^{b-1}_{d=1} d \cdot \omega_d \right| < \frac{b(b-1)\epsilon}{2}.$$ Hence, $$\operatorname{Av}_b(r) = \sum^{b-1}_{d=1} d\cdot \omega_d$$ and the proof is complete.
Let us recall that any rational number is expressible as a finite decimal (if it is regular) or an infinite periodic decimal. Conversely, any decimal expansion which is either finite or infinite periodic is equal to some rational number [@Niven1961 p. 32]. Since in this paper, we only consider the infinite decimal representation of a finite decimal number, we have the following:
\[AverageDecimalsLem\]
Let $r$ be a rational number. Then $\operatorname{Av}_b(r)$ exists and is a positive rational number. Moreover, if the fractional part of the rational number $r$ to base $b$ is $$(0.r_1r_2 \dots r_n \overline{p_1 p_2 \dots p_m})_b,$$ then $$\operatorname{Av}_b(r) = \frac{p_1 + p_2 + \dots + p_m}{m}.$$ In particular, if $r$ is regular to base $b$, then $$\operatorname{Av}_b(r) = b-1.$$
Now we prove the main theorem of our paper:
\[AverageDecimalsThm2\]
Let $r$ be a real number such that $\operatorname{Av}_b(r) =0$. Then $r$ is irrational.
From Definition \[AverageDecimalsDef\], it is clear that $\operatorname{Av}_b(r)$ is non-negative. Also by Corollary \[AverageDecimalsLem\], $\operatorname{Av}_b(r)$ is positive if $r$ is rational. Therefore, if $\operatorname{Av}_b(r) = 0$, then $r$ is irrational and the proof is complete.
\[AverageDecimalsCor\]
Let $(a_n)_{n\in \mathbb N}$ be a strictly increasing sequence of positive integers such that $$\lim_{n\to+\infty}\frac{n}{a_n} =0.$$ Define $\displaystyle r=\sum_{n=1}^{+\infty} \frac{b_n}{10^{a_n}},$ where $1 \leq b_n \leq 9$ is a positive integer, for each $n\in \mathbb N$. Then, $r$ is irrational.
$\displaystyle \operatorname{Av}(r) \leq 9\cdot \lim_{n\to+\infty}\frac{n}{a_n} =0$.
By Corollary \[AverageDecimalsCor\], the Liouville’s constant $$\displaystyle \ell=\sum_{n=1}^{+\infty} \frac{1}{10^{n!}},$$ [@Stark1987 Theorem 6.6] is irrational.
Let us recall that $\pi(n) = \card\{p \in \mathbb P: p\leq n\}$ is the prime counting function, where $\mathbb P$ is the set of all prime numbers. The Chebychev’s Estimate Theorem states that there exist positive constants $A_1$ and $A_2$ such that $$A_1 \cdot \frac{n}{\ln n} < \pi(n) < A_2 \cdot \frac{n}{\ln n},$$ for all $n \geq 2$ [@FineRosenberger2007 Theorem 4.2.1]. One of the nice corollaries of this theorem says that $\lim_{n\to+\infty} \displaystyle \frac{\pi(n)}{n} = 0$ [@FineRosenberger2007 Corollary 4.2.3]. We use this to give an alternative proof for the following result brought in the book by Hardy and Wright [@HardyWright1975]:
\[HardyWright\] [@HardyWright1975 Theorem 137] The real number $$r = 0.r_1r_2r_3 \dots r_n \dots = 0.011010100010\dots,$$ where $r_n = 1$ if $n$ is prime and $r_n = 0$ otherwise, is irrational.
It is clear that $\displaystyle \frac{r_1 + r_2 + r_3 + \dots + r_n}{n} = \frac{\pi(n)}{n}.$ Since $\displaystyle \operatorname{Av}(r) = \lim_{n\to+\infty} \frac{\pi(n)}{n} = 0$ [@FineRosenberger2007 Corollary 4.2.3], $r$ is irrational (Corollary \[AverageDecimalsCor\]). This finishes the proof.
The Asymptotic Average of the Decimals of Simply Normal Numbers {#sec:normal}
===============================================================
Let us recall that a real number $r$ is a simply normal number to base $b$ if for the decimals $(r_n)_{n\in \mathbb N}$ of the fractional part $(0.r_1r_2r_3 \dots r_n \dots)_b $ of the real number $r$, we have the following property:
$$\lim_{n\to+\infty} \frac{\card \{j : 1\leq j \leq n, r_j=d\}}{n} = \frac{1}{b},$$ where $d \in \{0,1,2,\dots,b-1\}$ [@Bugeaud2012 Definition 4.1].
\[SimplyNormalCor\]
Let $r$ be a simply normal number to base $b$. Then $\displaystyle \operatorname{Av}_b(r) = \frac{b-1}{2}$.
By Theorem \[AverageDecimalsThm1\], $\displaystyle \operatorname{Av}_b(r) = \sum^{b-1}_{d=1} (d \cdot \frac{1}{b}) = \frac{b-1}{2}$.
Let us recall that a real number $r$ is algebraic if it is the root of a polynomial $f(X) \in \mathbb Z[X]$, otherwise it is transcendental [@ErdosDudley1983; @Niven1956]. The set of transcendental numbers is uncountable [@Cantor1874]. For a masterful exposition of some central results on irrational and transcendental numbers, refer to [@Niven1956].
\[ChampernowneLiouville\]
The following statements hold:
1. There is a transcendental number $r$ such that $\operatorname{Av}(r) \neq 0$.
2. There is a transcendental number $r$ such that $\operatorname{Av}(r) = 0$.
$(1)$: The Champernowne number, $$C_{10} =0.1234567891011121314151617181920\dots,$$ whose sequence of decimals is the increasing sequence of all positive integers, is a simply normal number (cf. [@Champernowne1933] and [@Bugeaud2012 Theorem 4.2]). So, by Theorem \[SimplyNormalCor\], $\displaystyle \operatorname{Av}(r) \neq 0$. On the other hand, $C_{10}$ is transcendental [@Mahler1937].
$(2)$: The Liouville’s constant $\displaystyle \ell=\sum_{n=1}^{+\infty} \frac{1}{10^{n!}}$ is transcendental [@Stark1987 Theorem 6.6] while by Corollary \[AverageDecimalsCor\], we have $\operatorname{Av}(\ell) = 0$.
With the help of Corollary \[AverageDecimalsLem\], it is easy to see that if $s\in (0,9] \cap \mathbb Q$, then there is a rational number $r$ such that $\operatorname{Av}(r) = s$. Based on this, the following questions arise:
\[AsymptoticAveragesQ\]
1. Is there any irrational (transcendental) number $r$ such that $\operatorname{Av}(r)$ does not exist?
2. Is there any irrational (transcendental) number $r$ such that $\operatorname{Av}(r)=s$, for an arbitrary $s$ in $(0,9] \cap (\mathbb R - \mathbb Q)$,?
3. Does the asymptotic avergage of the decimals of the irrational number $\sqrt{2}$ exist and if it exists, what is that? The same question arises for other celebrated irrational numbers such as $e$, $\pi$, $\gamma$, and $\log^3_2$, where $$\gamma = \lim_{n\to+\infty} \big( -\ln n + \sum^n_{k=1} \frac{1}{k}\big)$$ is the Euler-Mascheroni constant?
Acknowledgments {#acknowledgments .unnumbered}
---------------
This work is supported by the Golpayegan University of Technology. Our special thanks go to the Department of Engineering Science at the Golpayegan University of Technology for providing all the necessary facilities available to us for successfully conducting this research.
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|
---
abstract: 'The exclusive processes are considered, where a point-like source of heavy quark-antiquark pairs $Q \bar Q$, e.g. their electromagnetic current, produces a pair consisting of a heavy quarkoniumlike exotic meson (tetraquark) or baryon (pentaquark) and a light meson or an antibaryon. For a sufficiently large mass of the heavy quark $m_Q$ there is a range of the energy $E$ above the $Q \bar Q$ threshold, where $E \ll m_Q$ and still the energy is large compared to the strong interaction scale, $E \gg \Lambda_{QCD}$. It is shown that in this energy range, where the heavy quarks are nonrelelativistic, a specific ‘intermediate asymptotic’ behavior sets in determined by the number $n$ of the pairs of constituent quarks, with the rate scaling as $E^{1-n}$.'
---
[**William I. Fine Theoretical Physics Institute\
University of Minnesota\
**]{}
FTPI-MINN-16/22\
UMN-TH-3534/16\
July 2016\
[**Constituent counting rule for exclusive production of heavy quarkoniumlike exotic resonance and a light hadron.\
**]{} William I. Fine Theoretical Physics Institute, University of Minnesota,\
Minneapolis, MN 55455, USA\
School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, USA\
and\
Institute of Theoretical and Experimental Physics, Moscow, 117218, Russia\
Studies of the asymptotic high energy behavior of amplitudes of exclusive hadronic processes go back to the early days of the development of QCD [@mmt; @bf]. In particular, it has then been derived that in the ultrarelativistic regime, i.e. at the energy that is larger than any hadron masses and the scale of the strong interaction, the power of the energy in the scaling law for the fall-off the amplitudes for such processes is determined by the minimal number of constitutuent quarks in the hadrons involved in such processes. This understanding proved to be of a great practical value in numerous studies, e.g. in constructing models of hadronic form factors and in the studies of analytic properties of the amplitudes. Lately there has been a revival of interest to application of the same ideas to processes with the recently found manifestly exotic hadrons containing a heavy quark-antiquark pair in addition to light constituents, such as the isovector mesonic resonances $Z^\pm_b(10610)$ and $Z^\pm_b(10650)$ [@bellez] in the bottomonium sector, the charmoniumlike charged states $\psi^\pm(4430)$ [@belle443; @lhcb443], $Z^\pm_c(3900)$ [@besz1], $Z^\pm_c(4020)$ [@besz2], and the hidden-charm pentaquarks $P_c$ [@lhcbp]. In particular it has been argued on the basis of the constituent counting rules [@bll; @bl; @cks] that studies of the kinematic behavior of processes with exotic hadrons can resolve between theoretical models of their internal dynamics. These arguments however were critically analyzed in a recent paper [@gmw].
The most basic exclusive process of a practical interest involving a heavy exotic resonance is the hard production of a pair consisting of the exotic hadron and an ordinary light meson or baryon. Particularly, the production of such pairs in $e^+e^-$ annihilation, $e^+e^- \to Z_Q \, \pi$ or $e^+e^- \to P_Q \bar p$ with $P_Q$ standing for a heavy pentaquark and $\bar p$ is the antiproton, is potentially observable in experiments at electron-positron colliders, and in fact has been observed with the mesonic resonances $Z_c$ and $Z_b$. Furthermore, the constituent counting rule in its original form [@mmt; @bf] was applied [@kv; @mv16] to description of the analytic properties of the production amplitudes.
Clearly, the scaling behavior, based on neglecting masses of all hadrons in an exclusive process, becomes applicable only at very high energies if that process involves a heavy quark-antiquark pair. In particular, at asymptotically high energies a heavy hidden-flavor quark pair cannot be counted as ‘constituent’, as is pointed out in Ref. [@gmw], since in the leading order in the energy scale its production by gluons carries no suppression in comparison with light quark-antiquark pairs. It is clear however that, although formally correct, this conclusion appears to be only of an academic as opposed to practical interest. Indeed, the production amplitude falls off with the energy and becomes extremely small in the asymptotic region where the ultrarelativistic behavior for heavy quarks sets in. At a ‘moderate’ excitation energy $E$ above the $Q \bar Q$ threshold, $\sqrt{s}=2m_Q+E$, where the amplitudes are possibly measurable in practice and where the creation and annihilation of heavy quark pairs is not essential, the behavior of the amplitudes is determined by relation between $E$ and a hadronic momentum scale $\mu$ that determines the dynamics inside the exotic states and inside ordinary light hadrons [^1]. In light hadrons the scale $\mu$ is of order $\Lambda_{QCD}$, while in the hadrons containing a heavy $Q \bar Q$ pair this scale depends on the QCD parameters and the mass $m_Q$. In particular it becomes proportional to $\alpha_s \, m_Q$ in the limit of asymptotically heavy quark. In exotic hadrons with hidden flavors the characteristic momenta can be a mixture of low scales, that can go to very low values in loosely bound molecular states. Any detailed discussion of the internal structure of exotic heavy resonances is beyond the scope of this paper, and the notation $\mu$ is used here for a combination of those low momentum scales. It is important for the present treatment that $\mu$ is considered to be much smaller than $m_Q$, which approximation appears to be reasonably applicable for the bottomonium sector. In the limit $E \gg \mu$ the behavior of the amplitudes becomes, to an extent, tractable by the standard in QCD methods of separation of the short- and long-distance dynamics (see e.g. a discussion of factorization in a similar context in Ref. [@gmw]).
The treatment is further simplified for sufficiently heavy quarks $Q$ if simultaneously with the condition of $E$ being large as compared to $\mu$, one can also require that the excitation energy is much smaller than the heavy quark mass, $E \ll m_Q$. Clearly, the range of energy where both these restrictions apply is only marginal for the charmed quarks whose mass $m_c$ is not sufficiently larger than $\mu$, but may well be of relevance for the production of bottomonium-like exotic resonances. The condition $E \ll m_Q$ allows one to treat the heavy quarks as nonrelativistic. In what follows it will be shown that under these assumptions the rate $\Gamma$ of production by a local source $(\bar Q \Gamma Q)$ of an exclusive state $X + h$ with $h$ being a light hadron and $X$ – an exotic resonance containing the $Q \bar Q$ heavy pair as well as light (anti)quarks scales as E\^[1-n]{} , \[gres\] where $n$ is the number of constituent light quark-antiquark pairs in the final state $X+h$. In particular, for the production cross section in $e^+e^-$ annihilation this implies the relations , [(e\^+e\^- P\_Q |p) (e\^+e\^- \^+ \^-)]{} . \[res\]
The basic ingredients that lead to the scaling formula (\[gres\]) can be illustrated starting with the simplest case $n=0$ and then increasing the number of constituent fermions. The case $n=0$ can be considered as corresponding to the process $e^+ e^- \to (Q \bar Q) + \gamma$ with the pair $(Q \bar Q)$ forming a bound (non-exotic) quarkonium state. The graph for this process is shown in Fig. 1. The propagation of the heavy quark pair is shown by a single thick box, rather than by individual lines for the quark and antiquark, reflecting the fact that for a nonrelativistic pair only the relative distance $\vec r$ between them (as a function of time) is essential. Also, due to the condition $E \ll m_Q$ the whole excess energy $E$ is carried away by the emitted photon, and any recoil of the quarkonium as whole can be neglected. Furthermore, the electromagnetic vertex for the creation of the quark pair reduces in the nonrellativistic limit to a local $\delta$-function operator ${\cal O} \to C \delta^{(3)}(\vec r)$ with the normalization constant $C$ being inessential for the present discussion of the scaling behavior. Finally, the filled circle in Fig. 1 describes the interaction of the quark pair with the electromagnetic field. For a nonrelativistic pair this interaction can be described by the Pauli Hamiltonian H\_[EM]{}= -[2 Q m\_Q]{} (p A) - [Q 2 m\_Q]{} (\_Q - \_[|Q]{})\_i B\_i , \[hem\] where $Q$ is the electric charge of the quark, $\vec A$ and $\vec B$ are the vector potential and the magnetic field strength for the emitted photon, $\vec \sigma_Q$ ($\vec \sigma_{\bar Q}$) are the spin operators for the quark (antiquark), and $\vec p$ stands for the momentum in the center-of-mass system. It can be noted that any spatial variation of the field of the emitted photon, set by the distance scale $\sim 1/E$, can be neglected, since the Green’s function for the propagation of the pair between the local creation by the virtual photon and the emission vertex constrains the contributing distances to a much shorter scale $\sim 1/\sqrt{m_Q E}$.
The ratio of the amplitudes generated by the first term in the Hamiltonian (\[hem\]), the electric dipole $E1$, and the second term, the magnetic dipole $M1$, is of order $p/E \sim \mu/E$. Thus at $E \gg \mu$ the dominant contribution arises from the $M1$ interaction [^2]. It is important for arriving at this conclusion that it is the momentum $p \sim \mu$ that determines the emission amplitude rather than the momentum of the photon $q \approx E$, due to the gauge condition $(\vec q \cdot \vec A) = 0$. Retaining only the $M1$ term in the interaction, one readily finds that the amplitude for the process in Fig. 1 is constant in the energy: A\_ = (Q |Q) | [O]{} | 0 E\^0 , \[m1g\] since the Green’s function between the vertices in Fig. 1 is of order $1/E$. The rate for the considered process is then evaluated as |A\_|\^2 2 (E- q\_0) [d\^3 q (2 )\^3 2 q\_0]{} E , \[gg\] which estimate agrees with Eq.(\[gres\]) at $n=0$.
A somewhat more complex, but still simplified example, corresponding to $n=1$ is the rather artificial process shown in Fig. 2. In this process the vector particle emitted by the heavy quark pair is virtual and produces a pair of light fermions, of which one (the fermion for definiteness) forms an ‘exotic’ bound state $X_f$ with the $Q \bar Q$ pair and the other (antifermion) is emitted as a free particle. Since this example, discussed here purely as an illustration, is not realistic in either QED or QCD the notation ‘vector’ (i.e. neither a photon nor gluon) and ‘fermion’ (i.e. neither a lepton nor quark) is used. Noting that the fermion in the bound state has momentum of order $\mu$, while the antifermion carries the energy $E$, one can conclude that for the vector propagator $q^2 \sim E \mu$. Taking into account the spinor normalization factor $\sqrt{E}$ for the fast antifermion, it can be readily seen that, as far as the scaling with $E$ is concerned, the amplitude for the process in Fig. 2 contains an extra factor proportional to $1/\sqrt{E}$ in comparison with that for a real photon emission in Fig. 1. Thus the rate for the (unrealistic) process $e^+e^- \to X_f \, \bar f$ scales as $1/E$ also in agreement with Eq.(\[gres\]).
The simplest process involving production of an actual exotic quarkoniumlike resonance and a light meson is $e^+ e^- \to Z_Q \pi$. This is the process that, for concreteness, is discussed here, since the treatment is trivially generalized to any similar production of a heavy exotic four-quark resonance in association with a light meson. The graphs with hard production of light quark pairs in this process are shown in Fig. 3. The relevant terms in the interaction of a nonrelativistic heavy quark pair with gluons are described by the Hamiltonian H\_[QCD]{}= -[t\^a\_Q - t\^a\_[|Q]{} m\_Q]{} (p A\^a) - [t\^a\_Q - t\^a\_[|Q]{} 4 m\_Q]{} (\_Q - \_[|Q]{})\_i B\^a\_i + T\^a A\^a\_0 , \[hqcd\] where $A^a$ and $B^a$ are the potential and the magnetic strength of the gluon field, $t^a_Q$ ($t^a_{\bar Q}$) are the color generators for the heavy quark (antiquark) and $T^a = t^a_Q + t^a_{\bar Q}$ is the total color generator for the $Q \bar Q$ system.
The last term in the Hamiltonian (\[hqcd\]) is the monopole term. Unlike the first two terms, its contribution is not suppressed by the heavy quark mass and it would be dominant for a color octet pair. However, the source (the electromagnetic current) produces a color singlet $Q \bar Q$ pair, hence the first emission of a hard gluon is possible only due to the first two terms. These terms contain the operator $t^a_Q - t^a_{\bar Q}$ converting the pair to color octet state, so that in the subsequent emissions from the heavy system only the monopole term can be retained in the leading order in $m_Q$. (In particular, the chromomagnetic term proportional to $T^a$, not shown in Eq.(\[hqcd\]), is totally negligible because of its suppression by $1/m_Q$.) Finally, the graph in Fig. 3$b$ arises from the quadratic in $A^a$ term in the chromomagnetic field $B^a$.
It can be further noticed that the large components, proportional to $E$, of the momenta of the fast light quarks as well as of the virtual gluons are collinear and proportional to the momentum of the emitted pion. For this reason the virtuality of each of the gluon propagators is $q^2 \sim E \mu$. Another consequence of the collinearity of the large components is that, as in the previously discussed simplified cases, due to the gauge condition the contribution of the chromoelectric $E1$ term from Eq.(\[hqcd\]) does not contain a large momentum proportional to $E$ and is thus suppressed relative to that of the $M1$ chromomagnetic dipole.
One can readily find that the contribution of the graphs of Figs. 3$a$ and 3$b$ to the amplitude is of the same order in the energy $E$, and including the spinor normalization factors, proportional to $\sqrt{E}$ for each fast (anti)quark, the energy dependence of this contribution can be evaluated as A(Z\_Q ) \~[1 E]{} , \[azpi\] where it is also taken into account that the three gluon vertex in Fig. 3$a$ is proportional to $E$. On the other hand, the contribution from graph in Fig. 3$c$ is only of the order $1/E^2$ and is thus subdominant. This is because the extra propagator of the heavy pair introduces the factor $1/E$ with no energy dependence of the monopole vertex, while an extra hard gluon propagator and the three gluon vertex (in the graph of Fig. 3$a$) result in the factor $E/(E \mu) = 1/\mu$. It can also be readily verified that the graphs where the additional light quark pair is emitted by a gluon attached to another light quark line are suppressed relative to (\[azpi\]) by a factor $1/\sqrt{E}$, and for this reason are not considered in the present discussion.
The $E$ dependence of the rate generated by the amplitude (\[azpi\]) can be estimated as (e\^+e\^- Z\_Q ) |A(Z\_Q )|\^2 (E-\_1 - \_2) [d\^3 k\_1 d\^3 k\_2 \_1 \_2]{} , \[n2\] where $k_1$ and $k_2$ ($\omega_1$ and $\omega_2$) are the momenta (energies) of the fast quark and antiquark. The large longitudinal components of the momenta cancel against the energies in the denominator, while the integration over the relative transverse momentum is constrained at $\mu^2$ by the condition that the light quark and antiquark make a pion. The only large factor remaining in the integration arises from the integration over the total momentum of the pion, and, together with the energy conservation $\delta$ function gives a factor of $E$, i.e. the same as in the previously considered cases $n=0$, and $n=1$. The final estimate of the energy dependence in Eq.(\[n2\]) is obviously the one given by the general formula (\[gres\]).
The generalization of the derivation of Eq.(\[gres\]) to the case of arbitrary $n$ is quite straightforward. Indeed, as argued for the case of $n=2$, the dominant $E$ dependence arises from a single hard $M1$ interaction on the line of the heavy pair, while graphs with any additional vertices on this line produce only a subdominant contribution. Thus emission of additional constituent light quark pairs proceeds through the branching of the gluons in the graphs of Figs. 3$a$ and 3$b$. Each such branching gives in the amplitude an extra factor proportional to $1/\sqrt{E}$. On the other hand, the phase space integration does not introduce new energy dependence once the condition that $n$ produced fast (anti)quarks are constituents in a fast hadron. Thus one concludes that each extra pair of produced constituent light quarks brings the factor $1/E$ in the rate, and thus arrives at the general formula (\[gres\]).
Before concluding, two points related to the derived here scaling rule and the mechanism leading to the derivation merit a brief discussion. One point is regarding the spin state of the heavy quark pair corresponding to the dominant at large $E$ production mechanism in $e^+e^-$ annihilation. Namely, the electromagnetic current produces the $Q \bar Q$ pair in the spin triplet state. The spin operator $(\vec \sigma_Q - \vec \sigma_{\bar Q})$ flips the total spin into the siglet state. Thus the dominance of the $M1$ interaction in the considered energy region implies that in the exclusive production of the heavy exotic resonances in a pair with a light hadron there should be mostly the states with a spin singlet heavy quark pair. It is not clear at present whether this behavior can be studied in experiments. Indeed, the only so far known bottomoniumlike exotic resonances $Z_b(10610)$ and $Z_b(1065)$ are mixed states with regards to the total spin of the $b \bar b$ pair [@bgmmv], and can thus be produced through the spin singlet component. It would however be possible to study the predicted behavior if some or all of the expected [@mvwb] isovector $G$-negative bottomoniumlike resonances $W_{bJ}$ are found and become accessible to observation in $e^+e^-$ annihilation through $e^+e^- \to W_{bJ} \rho$. Two of these resonances with $J=0$: $W_{b0}$ and $W'_{b0}$, also contain a spin singlet heavy quark component and thus their exclusive production at energy well above the threshold, should have a higher yield than for the resonances $W_{b1}$ and $W_{b2}$ containing only pure spin triplet $b \bar b$ quark pair.
Another point that merits mentioning is that the rather slow $1/E$ fall off of the cross section for $e^+e^- \to Z_b \, \pi$ generally implies that there should be some production of this exclusive final state in the continuum at energies above the region of the $\Upsilon(nS)$ resonances. At present it does not appear possible to reliably estimate the rate beyond the simple remark that it contains an extra suppression by the inverse of the mass $m_b$ inherent in the $M1$ interaction in Eq.(\[hqcd\]). Namely, the relations (\[res\]) with proper dimensional parameters restored should read as \~[\^3 m\_Q\^2 E]{} , [(e\^+e\^- P\_Q |p) (e\^+e\^- \^+ \^-)]{} \~[\^4 m\_Q\^2 E\^2]{} . \[resd\] As of yet the production of the final states $Z_b \, \pi$ has been observed [@belle1508] only in the $\Upsilon(5S)$ and $\Upsilon(6S)$ resonances. It would thus be quite interesting if a nonresonant production of the $Z_b \, \pi$ pairs could be studied experimentally at energies above the $\Upsilon(6S)$ resonance.
This work is supported in part by U.S. Department of Energy Grant No. DE-SC0011842.
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[^1]: The effective ‘quenching’ of the heavy quark pairs in the intermediate range of $E$ can be readily effected, for the purpose of theoretical discussion, by considering the quark and antiquark in the pair as being of different flavor.
[^2]: This is opposite to the relation for transitions between states of a nonrelativistic bound system, where $E \sim \mu^2/m \ll m$. It can be also noted that in the discussed here process the $E1$ term describes the production of $P$-wave quarkonium, while the dominant $M1$ term corresponds to the production of $S$-wave states.
|
---
address:
- 'ARTEMIS (OCA/CNRS) & IRAP (OMP/CNRS)'
- Observatory of Rome
author:
- 'B. Gendre'
- 'G. Stratta'
- on behalf of the FIGARO collaboration
title: 'Models and possible progenitors of gamma-ray bursts at the test field of the observations'
---
Introduction
============
With the discovery that Gamma-Ray Bursts (GRBs) are cosmological events [@ghe12], a common picture has emerged to explain these events. There are two classes of GRBs: short and long events, separated by the canonical duration value of $T_{90} = 2$ s [@kou93]. Short events are thought to originate from the merging of a binary system of compact objects [@eic89], even if some magnetar formation models can also explain the observations [@uso92]. Long events are associated with the death of a certain kind of Wolf-Rayet stars in a cataclysmic collapse of the core, the collapsar model [@woo86]. Several pieces of evidence have confirmed the collapsar model: observation of stellar winds around the progenitors [@gen04; @gen06], association of type Ib/c supernovae [@hjo03],...
Independently on the progenitor nature, a fireball, collimated within a jet and with a Lorentz factor of the order of several hundreds, is produced [@ree92; @mes97; @pan98]. The fireball is responsible of the observed radiation through three different classes of shocks: internal shocks (the origin of the prompt phase), external shocks (responsible of the afterglow), and the reverse shock (that cause a rebrightening at low energy) as explained in details in the review of [@pir05]. A good example can be seen in Fig. \[fig1\] where the light curve of GRB 110205A is shown. This burst is the archetypal event that fit the fireball model, displaying all the components well separated in time. More details on this burst can be found in Gendre [*et al.*]{} 2012 [@gen11].
An example of “non-fitting” burst: GRB 090102
=============================================
One of the best example of burst that cannot be fitted by the fireball model is GRB 090102. This work has been presented in Gendre [*et al.*]{} 2010 [@gen10], and can be summarized as follow. In X-rays, it presents a very smooth light curve with no hint of a temporal break. In the optical, the light curve presents a steep-flat behavior, with a break time at $\sim$ 1 ks after the burst. When taken alone, each of the observation band results can be explained by the standard model. In X-ray, this is a typical afterglow expanding in the interstellar medium. In optical, the data could be interpreted as either due to a termination shock, locating the end of the free-wind bubble at the position of the optical break; or as a normal fireball expanding in an ISM, with a reverse shock present at an early time (before the break time). However, once combined together, the flux levels are not compatible between the optical and X-ray band. The cannonball model [@dad09] can partly reproduce the data. It appears clear, however, that in order to explain the broad-band emission, some fine-tuning of this model is mandatory, likewise for the fireball model. Another very good example of “non-fitting” burst is GRB 061126 that also feature an unusual afterglow, and again a non standard model has been proposed for explaining this event [@per08].
Peculiar progenitor: the case of GRB 111209A
============================================
We now turn our focus to GRB 111209A, a very peculiar event. It was discovered by the Swift satellite, producing two triggers of the Burst Alert Telescope, and also followed by Konus-Wind. As shown by a re-analysis on ground, the burst started about 5400 seconds before $T_0$, and lasted in gamma-ray about 15 000 seconds. In X-ray, the start of the steep decline is supposed to be the true end of the prompt phase [@zha06]: taking this stop time we reached a total duration for this event of more than 25 000 seconds [@gen12]. Browsing all available archives and catalog of GRBs, it was impossible to find another burst with such a large duration. Because this burst occurred at a redshift of *z* = 0.677 [@vre11], its intrinsic duration is larger than 10 000 seconds, making it the first ultra-long burst studied. Its light curve is displayed in Fig. \[fig2\].
The origin of this event is not clear. As one can clearly see in Fig. \[fig3\], it is very different from normal long GRBs. It presents a thermal component at the start of the XRT observation. Two super long GRBs (GRB 060218, z = 0.033, and GRB 100316D, z = 0.059) were associated with a supernova shock breakout [@cam06; @sta11], and presented a strong thermal component. However the thermal component for GRB 111209A disappears very soon, and most of the prompt phase is free of thermal emission. Moreover, there is no clear evidence of any SN emission. We can therefore discard this hypothesis, and test unusual progenitors. The main difficulty in explaining the nature of the progenitor of GRB 111209A is its duration. In [@gen12] we discuss how a magnetar model cannot reproduce the energetics and spectral characteristics. The most probable scenario is a single supergiant star with low metallicity. The hypothesized progenitors of long duration GRBs are Wolf-Rayet stars (stars with the outer layers expelled during stellar evolution). When these layers are still present, as in low metallicity super-giant stars with weak stellar winds, the stellar envelope may fall-back and accretion can fuel the central engine for a much longer time. In this scenario, blue super-giant stars can produce GRBs with prompt emission lasting about $10^4$ seconds [@woo12].
The afterglow analysis uses the observations of XMM-Newton and Swift in the X-ray, TAROT and GROND in the optical, and ACTA data in radio. From the optical-to-gamma-ray prompt spectral energy distribution we find evidence of dust extinction of the order of $A_V\sim0.3-1.5$ mag in the rest frame of the GRB, depending on the assumed spectral continuum, that however is not confirmed during the afterglow emission. We find that our results point against a low metallicity environment possibly challenging the low-metallicity progenitor solution. Despite the unusual progenitor nature, the standard fireball model can fit the afterglow data.
Conclusion
==========
We have presented several cases that does not follow the standard paradigms of GRBs. These cases show that not all GRBs can be explained using the standard fireball, and that not all long GRBs are due to the same kind of progenitor. We have also shown that an unusual progenitor does not imply an unusual afterglow. The exact explanation of GRBs is still not clearly set, and further works are still needed.
Acknowledgments {#acknowledgments .unnumbered}
===============
The FIGARO collaboration is funded by the Programme National des Hautes Energies in France.
References {#references .unnumbered}
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---
abstract: 'In this paper we study the super-critical $2$D dissipative quasi-geostrophic equation. We obtain some regularization effects allowing us to prove global well-posedness result for small initial data lying in critical Besov spaces constructed over Lebesgue spaces $L^p,$'
address:
- |
IRMAR, Université de Rennes 1\
Campus de Beaulieu\
35 042 Rennes cedex\
France
- |
IRMAR, Université de Rennes 1\
Campus de Beaulieu\
35 042 Rennes cedex\
France
author:
- Taoufik Hmidi
- Sahbi Keraani
title: 'On the global solutions of the super-critical $2$D quasi-geostrophic equation in Besov spaces'
---
Introduction
============
This paper deals with the Cauchy problem for the two-dimensional dissipative quasi-geostrophic equation $${(\textnormal{QG}_{\alpha})}\left\lbrace
\begin{array}{l}\partial_{t}\theta+v\cdot\nabla\theta+|\hbox{D}|^{{\alpha}} \theta=0\\
\theta_{\vert t =0}=\theta^0,
\end{array}
\right.$$ where the scalar function $\theta$ represents the potential temperature and $\alpha \in[0,2].$ The velocity $v=(v^1,v^2)$ is determined by $\theta$ through a stream function $\psi$, namely $$v=(-\partial_{2}\psi,\partial_{1}\psi), \quad\textnormal{with}\quad|\hbox{D}|\psi=\theta.$$ Here, the differential operator $|\hbox{D}|=\sqrt{-\Delta}$ is defined in a standard fashion through its Fourier transform: $\mathcal{F}(|D|u)=|\xi|\mathcal{F}u.$ The above relations can be rewritten as $$v=(-\partial_{2}|\hbox{D}|^{-1}\theta,\partial_{1}|\hbox{D}|^{-1}\theta)=(-{R}_{2}\theta,{R}_{1}\theta),$$ where ${R}_{i} (i=1,2)$ are Riesz transforms.
First we notice that solutions for ${(\textnormal{QG}_{\alpha})}$ equation are scaling invariant in the following sense: if $\theta$ is a solution and $\lambda>0$ then $\theta_{\lambda}(t,x)=\lambda^{\alpha-1}\theta(\lambda^\alpha t,\lambda x)$ is also a solution of ${(\textnormal{QG}_{\alpha})}$ equation. From the definition of the homogeneous Besov spaces, described in next section, one can show that the norm of $\theta_{\lambda}$ in the space $\dot B_{p,r}^{1+\frac{2}{p}-\alpha},$ with $p,\,r\in[1,\infty],$ is quasi-invariant. That is, there exists a pure constant $C>0$ such that for $$C^{-1}\|\theta_{\lambda}(t)\|_{\dot B_{p,r}^{1+\frac{2}{p}-\alpha}}\leq\|\theta(\lambda^\alpha t)\|_{\dot B_{p,r}^{1+\frac{2}{p}-\alpha}}\leq C\|\theta_{\lambda}(t)\|_{\dot B_{p,r}^{1+\frac{2}{p}-\alpha}}.$$
Besides its intrinsic mathematical importance the ${(\textnormal{QG}_{\alpha})}$ equation serves as a $2$D models arising in geophysical fluid dynamics, for more details about the subject and the references therein. Recently the ${(\textnormal{QG}_{\alpha})}$ equation has been intensively investigated and much attention is carried to the problem of global existence. For the sub-critical case $(\alpha>1)$ the theory seems to be in a satisfactory state. Indeed, global existence and uniqueness for arbitrary initial data are established in various function spaces However in the critical case, that is $\alpha=1,$ Constantin showed the global existence in Sobolev space $H^1$ under smallness assumption of the $L^\infty$-norm of the initial temperature $\theta^0$ but the uniqueness is proved for initial data in $H^2.$ Many other relevant results can be found in [@C-C; @Ju; @Ju1]. The super-critical case $\alpha<1$ seems harder to deal with and work on this subject has just started to appear. In [@Cha] the global existence and uniqueness are established for data in critical Besov space $B_{2,1}^{2-\alpha}$ with a small $\dot{B}_{2,1}^{2-\alpha}$ norm. This result was improved by N. Ju [@Ning1] for small initial data in $H^s$ with $s\geq2-\alpha.$ We would like to point out that all these spaces are constructed over Lebesgue space $L^2$ and the same problem for general Besov space $B_{p,r}^s$ is not yet well explored and few results are obtained in this subject. In [@Wu1], Wu proved the global existence and uniqueness for small initial data in $C^r\cap L^q$ with $r>1$ and $q\in]1,\infty[,$ which is not a scaling space. We can also mention the paper [@Wu2] in which global well-posedness is established for small initial data in $B_{2,\infty}^s\cap B_{p,\infty}^s,$ with $s>2-\alpha$ and $p=2^N.$
The main goal of the present paper is to study existence and uniqueness problems in the super-critical case when initial data belong to inhomogeneous critical Besov with $p\in[1,\infty].$
Our first main result reads as follows.
\[Thm1\] Let $\alpha\in[0,1[,$ $p\in[1,\infty]$ and $s\geq s_{c}^p,$ with $ s_{c}^p=1+\frac{2}{p}-\alpha$ and define $${\mathcal{X}}_{p}^s=\left\lbrace
\begin{array}{l}B_{p,1}^s,\quad\hbox{if}\quad p<\infty,\\
B_{\infty,1}^s\cap \dot{B}_{\infty,1}^0,\quad\hbox{otherwise}. \
\end{array}
\right.$$ Then for $\theta^0\in {\mathcal{X}}_{p}^s$ there exists $T>0$ such that the ${(\textnormal{QG}_{\alpha})}$ equation has a unique solution $\theta $ belonging to ${\it C}([0,T];{\mathcal{X}}_{p}^s)\cap L^1([0,T];\dot{B}_{p,1}^{s+\alpha}).$
In addition, there exists an absolute constant $\eta>0$ such that if $$\|\theta^0\|_{\dot{B}_{\infty,1}^{1-\alpha}}\leq\eta,$$ then one can take $T=+\infty$.
We observe that in our global existence result we make only a smallness assumption of the data in Besov space $\dot B_{\infty,1}^{1-\alpha}$ which contains the increasing Besov chain spaces $\{\dot{B}_{p,1}^{s_{c}^p}\}_{p\in[1,\infty]}.$
In the case of $s>s_{c}^p$ we have the following lower bound for the local time existence. There exists a nonnegative constant $C$ such that $$T\geq {C}{\|\theta^0\|_{\dot{B}_{\infty,1}^{s-\frac{2}{p}}}^{\frac{-\alpha}{s-s_{c}^p}}}.$$ However in the critical case $s=s^p_{c}$ the local time existence is bounded from below by $$\sup\Big\{t\geq 0,\,\sum_{q\in{\mathbb{Z}}}(1-e^{-ct2^{q\alpha}})^{\frac{1}{2}}2^{q(1-\alpha)}\|\Delta_{q}\theta^0\|_{L^\infty}\leq \eta \Big\},$$ where $\eta$ is an absolute nonnegative constant.
The proof relies essentially on some new estimates for transport-diffusion equation
$$(\textnormal{TD}_{\alpha})\left\lbrace
\begin{array}{l}
\partial_t \theta+v\cdot\nabla \theta+|\textnormal{D}|^{\alpha} \theta=f\\
{\theta}_{| t=0}=\theta^{0},\\
\end{array}
\right.$$ where the unknown is the scalar function $\theta.$ Our second main result reads as follows
\[Thm3\]
Let $s\in]-1,1[,\,\alpha\in[0,1[,\,(p,r)\in[1,+\infty]^2,$ $f\in L^1_{\textnormal{loc}}({\mathbb{R}}_{+}; \dot{B}_{p,1}^s)$ and $v$ be a divergence free vector field belonging to $L^1_{\textnormal{loc}}({\mathbb{R}}_{+};\textnormal{Lip}({\mathbb{R}}^d)).$ We consider a smooth solution $\theta$ of the transport-diffusion equation $(\textnormal{TD}_{\alpha})$ , then there exists a constant $C$ depending only on $s$ and $\alpha$ such that $$\|\theta\|_{\widetilde{L^r_{t}}\dot{B}_{p,1}^{s+\frac{\alpha}{r}}}\leq C e^{C\int_{0}^t\|\nabla v(\tau)\|_{L^\infty}d\tau}\Big( \|\theta^0\|_{\dot{B}_{p,1}^s}+\|f\|_{L^1_{t}\dot{B}_{p,1}^s}\Big).$$ Besides if $v=\nabla^\bot|\textnormal{D}|^{-1}\theta$ then the above estimate is valid for all $s>-1.$
We use for the proof a new approach based on Lagrangian coordinates combined with paradifferential calculus and a new commutator estimate. This idea has been recently used by the first author to treat the two-dimenional Navier-Stokes vortex patches [@Hmidi].
The estimates of Theorem $\ref{Thm3}$ hold true for Besov spaces $\dot B_{p,m}^s,$ with $m\in[1,\infty].$ The proof can be done strictly in the same line as the case $m=1.$ It should be also mentioned that we can derive similar results for inhomogeneous Besov spaces.
[**Notation:**]{} Throughout the paper, $C$ stands for a constant which may be different in each occurrence. We shall sometimes use the notation $A\lesssim B$ instead of $A\leq CB$ and $A\approx B$ means that $A\lesssim B$ and $B\lesssim A.$
The rest of this paper is structured as follows. In next section we recall some basic results on Littlewood-Paley theory and we give some useful lemmas. Section $3$ is devoted to the proof of a new commutator estimate while sections $4$ and $5$ are dealing successively with the proofs of Theorem \[Thm3\] and \[Thm1\]. We give in the end of this paper an appendix.
Preliminaries
=============
In this preparatory section, we provide the definition of some function spaces based on the so-called Littlewood-Paley decomposition and we review some important lemmas that will be used constantly in the following pages.
We start with the dyadic decomposition. Let $\varphi\in C^\infty_{0}({\mathbb{R}}^d)$ be supported in the ring $\mathcal{C}:=\{ \xi\in{\mathbb{R}}^d,\frac{3}{4}\leq|\xi|\leq\frac{8}{3}\}$ and such that $$\sum_{q\in{\mathbb{Z}}}\varphi(2^{-q}\xi)=1 \quad\hbox{for}\quad \xi\neq 0.$$ We define also the function $\;\chi(\xi)=1-\sum_{q\in{\mathbb{N}}}\varphi(2^{-q}\xi).$ Now for $u\in{\mathcal S}'$ we set $$\Delta_{-1}u=\chi(\hbox{D})u;\, \forall
q\in{\mathbb{N}},\;\Delta_qu=\varphi(2^{-q}\hbox{D})u\quad\hbox{ and }\;\forall\,\,q\in{\mathbb{Z}},\,\,\dot{\Delta}_{q}u=\varphi(2^{-q}\textnormal{D})u.$$ The following low-frequency cut-off will be also used: $$S_q u=\sum_{-1\leq j\leq q-1}\Delta_{j}u\quad\hbox{and}\quad \dot{S}_{q}u=\sum_{j\leq q-1}\dot\Delta_{j}u.$$ We caution that we shall sometimes use the notation $\Delta_{q}$ instead of $\dot\Delta_{q}$ and this will be tacitly understood from the context.\
Let us now recall the definition of Besov spaces through dyadic decomposition.
Let $(p,m)\in[1,+\infty]^2$ and $s\in{\mathbb{R}},$ then the inhomogeneous is the set of tempered distribution $u$ such that $$\|u\|_{B_{p,m}^s}:=\Big( 2^{qs}
\|\Delta_q u\|_{L^{p}}\Big)_{\ell ^{m}}<\infty.$$ To define the homogeneous Besov spaces we first denote by $\mathcal{S}'/\mathcal{P}$ the space of tempered distributions modulo polynomials. Thus we define the space $\dot B_{p,r}^s$ as the set of distribution $u\in\mathcal{S}'/\mathcal{P}$ such that $$\|u\|_{\dot{B}_{p,m}^s}:=\Big( 2^{qs}
\|\dot{\Delta}_q u\|_{L^{p}}\Big)_{\ell ^{m}}<\infty.$$ We point out that if $s>0$ then we have $B_{p,m}^s=\dot B_{p,m}^s\cap L^{p}$ and $$\|u\|_{B_{p,m}^s}\approx\|u\|_{\dot B_{p,m}^s}+\|u\|_{L^{p}}.$$ Another characterization of homogeneous Besov spaces that will be needed later is given as follows (see [@Triebel]). For $s\in]0,1[, p,m\in[1,\infty]$ $$\label{equivalence}
C^{-1}\|u\|_{\dot{B}_{p,m}^s}\leq
\Big(\int_{{\mathbb{R}}^d}\frac{\|u(\cdot-x)-u(\cdot)\|^{m}_{L^p}}{|x|^{sm}}\frac{dx}{|x|^d}\Big)^{\frac{1}{m}}\leq C\|u\|_{\dot{B}_{p,m}^s},$$ with the usual modification if $m=\infty$.\
In our next study we require two kinds of coupled space-time Besov spaces. The first one is defined in the following manner: for $T>0$ we denote by $L^r_{T}\dot B_{p,m}^s$ the set of all tempered distribution $u$ satisfying $$\|u\|_{L^r_{T}\dot B_{p,r}^s}:= \Big\|\Big( 2^{qs}
\|\dot\Delta_q u\|_{L^{p}}\Big)_{\ell ^{m}}\Big\|_{L^r_{T}}<\infty.$$ The second mixed space is $\widetilde L^r_{T}{\dot B_{p,m}^s}$ which is the set of tempered distribution $u$ satisfying $$\|u\|_{ \widetilde L^r_{T}{\dot B_{p,m}^s}}:= \Big( 2^{qs}
\|\dot\Delta_q u\|_{L^r_{T}L^{p}}\Big)_{\ell ^{m}}<\infty .$$ We can define by the same way the spaces $L^r_{T} B_{p,m}^s$ and $\widetilde L^r_{T}{ B_{p,m}^s}.$
The following embeddings are a direct consequence of Minkowski’s inequality.
Let $s\in{\mathbb{R}},$ $r\geq1$ and $\big(p,m\big)\in[1,\infty]^2,$ then we have $$\begin{aligned}
\label{lemm4}
L^r_{T}\dot B_{p,m}^s&\hookrightarrow&\widetilde L^r_{T}{\dot B_{p,m}^s},\,\textnormal{if}\quad m\geq r\quad\hbox{and}\\
\nonumber\widetilde L^r_{T}{\dot B_{p,m}^s}&\hookrightarrow& L^r_{T}\dot B_{p,m}^s,\, \textnormal{if}\quad
r\geq m.\end{aligned}$$ Another classical result that will be frequently used here is the so-called Bernstein inequalities (see [@Ch1] and the references therein): there exists $C$ such that for every function $u$ and for every $q\in{\mathbb{Z}},$ we have $$\begin{aligned}
\sup_{|\alpha|=k}\|\partial ^{\alpha}S_{q}u\|_{L^b}&\leq& C^k\, 2^{q(k+d(\frac{1}{a}-\frac{1}{b}))}\|S_{q}u\|_{L^a},\quad\hbox{for}\quad b\geq a,\\
C^{-k}2^{qk}\|\dot\Delta_{q}u\|_{L^a}&\leq&\sup_{|\alpha|=k}\|\partial ^{\alpha}\dot\Delta_{q}u\|_{L^a}\leq C^k2^{qk}\|\dot\Delta_{q}u\|_{L^a}.\end{aligned}$$ It is worth pointing out that the above inequalities hold true if we replace the derivative $\partial^\alpha$ by fractional derivative $|\textnormal{D}|^\alpha.$ According to Bernstein inequalities one can show the following embeddings $$\dot{B}_{p,m}^s\hookrightarrow\dot{B}_{p_{1},m_{1}}^{s-d(\frac{1}{p}-\frac{1}{p_{1}})},\quad\hbox{for}\quad p\leq p_{1}\quad\hbox{and}\quad m\leq m_{1}.$$
Now let us we recall the following commutator lemma (see [@Ch1; @rd23] and the references therein).
\[lemm12\] Let $p,r\in[1,\infty], 1=\frac{1}{r}+\frac{1}{\bar r}, \rho_{1}<1, \rho_{2}<1$ and $v$ be a divergence free vector field of ${\mathbb{R}}^d.$ Assume in addition that $$\rho_{1}+\rho_{2}+d\min\{1,{2}/{p}\}>0\quad\hbox{and}\quad\rho_{1}+{d}/{p}>0.$$ Then we have $$\sum_{q\in{\mathbb{Z}}}2^{q(\frac{d}{p}+\rho_{1}+\rho_{2}-1)}\big\|[\dot \Delta_q, v\cdot \nabla ]u\big\|_{L^1_{t}L^{p}}\lesssim \|
v\|_{\widetilde L^r_{t}\dot B_{p,1}^{\frac{d}{p}+\rho_{1}}}\|u \|_{\widetilde L^{\bar r}_{t}\dot B_{p,1}^{\frac{d}{p}+\rho_{2}}}.$$ Moreover we have for $s\in]-1,1[$ $$\sum_{q\in{\mathbb{Z}}}2^{qs}\big\|[\dot \Delta_q, v\cdot \nabla ]u\big\|_{L^{p}}\lesssim \|\nabla
v\|_{L^{\infty}}\|u\|_{\dot B_{p,1}^s}.$$ In addition this estimate holds true for all $s>-1$ if $v=\nabla^\perp|\textnormal{D}|^{-1}u.$
The following result describes the action of the semi-group operator $e^{t|\textnormal{D}|^\alpha}$ on distributions whose Fourier transform is supported in a ring.
\[l:5\] Let $\mathcal{C}$ be a ring and $\alpha\in{\mathbb{R}}_{+}.$ There exists a positive constant $C$ such that for any $p\in[1;+\infty],$ for any couple $(t,\lambda)$ of positive real numbers, we have $$\textnormal{supp} \mathcal{ F}u\subset\lambda\mathcal{C}\Rightarrow
\|e^{t|\textnormal{D}|^\alpha}u\|_{L^p}\leq Ce^{-C^{-1}t\lambda^{\alpha}}\|u\|_{L^p}.$$
We will imitate the same idea of [@Ch2]. Let $\phi\in \mathcal{D}({\mathbb{R}}^d\backslash\{ 0\}),$ radially and whose value is identically $1$ near the ring $\mathcal{C}.$ Then we have $$\begin{aligned}
e^{t|\textnormal{D}|^\alpha }u=\phi(\lambda^{-1}|\textnormal{D}|) u=h_{\lambda}*u,\end{aligned}$$ where $$h_{\lambda}(t,x)=\frac{1}{(2\pi)^d}\int_{{\mathbb{R}}^d}\phi(\lambda^{-1}\xi)e^{-t\vert\xi\vert^\alpha}e^{i<x,\xi>}d\xi.$$ We set $$\bar h_{\lambda}(t,x):=\lambda^{-d}h(t,\lambda^{-1}x)=\frac{1}{(2\pi)^d}
\int_{{\mathbb{R}}^d}\phi(\xi)e^{-t\lambda^\alpha\vert\xi\vert^\alpha}e^{i<x,\xi>}d\xi.$$ Now to prove the proposition it suffices to show that $\|\bar h_{\lambda}(t)\|_{L^1}\leq Ce^{-C^{-1}t\lambda^{\alpha}}.$ For this purpose we write with the aid of an integration by parts $$(1+\vert x\vert^2)^d{\bar h_{\lambda}}(x)=\frac{1}{(2\pi)^d}
\int_{{\mathbb{R}}^d}(\textnormal{Id}-\Delta_{\xi})^d\big(\phi(\xi)e^{-t\lambda^\alpha\vert\xi\vert^\alpha}\big)e^{i<x,\xi>}d\xi.$$ From Leibnitz’s formula, we have $$(\textnormal{Id}-\Delta_{\xi})^d\big(\phi(\xi)e^{-t\lambda^\alpha \xi^\alpha}\big)=\sum_{\vert
\gamma\vert\leq 2d\\
\atop \beta\leq\gamma}C_{\gamma,\beta}\partial^{\gamma-\beta}\phi(\xi)
\partial^\beta e^{-t\lambda^\alpha\xi^\alpha}.$$ As $\phi$ is supported in a ring that does not contain some neighbourhood of zero then we get for $\xi\in \hbox{supp }\phi$ $$\begin{aligned}
\arrowvert\partial^\beta e^{-t\lambda^\alpha\vert\xi\vert^\alpha}
\arrowvert&\leq& C_{\beta}(1+t\lambda^\alpha)^{\vert\beta\vert}
e^{-t\lambda^\alpha\vert\xi\vert^\alpha}, \;\forall\, \xi\in \hbox{ supp }\phi\\
&\leq& C_{\beta}e^{-C^{-1}t\lambda^\alpha}.\end{aligned}$$ Thus we find that $$\big{|}(\textnormal{Id}-\Delta_{\xi})^d\big(\phi(\xi)e^{-t\lambda^\alpha \xi^\alpha}\big)\big{|}\leq Ce^{-C^{-1}t\lambda^\alpha}\sum_{\vert
\gamma\vert\leq 2d\\
\atop \beta\leq\gamma}C_{\gamma,\beta}|\partial^{\gamma-\beta}\phi(\xi)|.$$ Since the term of the right-hand side belongs to $L^1({\mathbb{R}}^d),$ then we deduce that $$(1+\vert x\vert^2)^d\arrowvert{\bar h_{\lambda}}(x)\arrowvert\leq Ce^{-C^{-1}t\lambda^\alpha}.$$ This completes the proof of the proposition.
Commutator estimate
===================
The main result of this section is the following estimate that will play a crucial role for the proof of Theorem \[Thm3\].
\[pr:1\] Let $f\in \dot{B}_{p,1}^\alpha$ with $\alpha\in [0,1[$ and $p\in[1,+\infty],$ and let $\psi$ be a Lipshitz measure-preserving homeomorphism Then there exists $C:=c(\alpha)$ such that $$\begin{aligned}
\big{\|}|\textnormal{D}|^\alpha(f\circ\psi)-(|\textnormal{D}|^\alpha f)\circ\psi\big{\|}_{L^p}&\leq&
C\max\Big(\vert 1-\|\nabla\psi^{-1}\|_{L^\infty}^{d+\alpha}\vert ; \vert 1-
{\| \nabla\psi \|_{{L^\infty}}^{-d-\alpha}}\vert\Big) \\
&&
\|\nabla \psi \|_{L^\infty}^\alpha\| f \|_{\dot{B}_{p,1}^\alpha}.\end{aligned}$$
First we rule out the obvious case $\alpha=0$ and let us recall the following formula detailed in [@C-C] which tells us that for all $\alpha\in]0,2[$ $$|\textnormal{D}|^\alpha f(x)=C_{\alpha}\,\textnormal{P. V.}\int\frac{f(x)-f(y)}{\vert x-y \vert^{d+\alpha}}dy.$$ Now we claim from (\[equivalence\]) that if $g\in\dot{B}_{p,1}^\alpha,$ with $\alpha\in]0,1[,$ then the above identity holds as an $L^p$ equality$$\label{E1}
|\textnormal{D}|^\alpha f(x)=C_{\alpha}\int_{{\mathbb{R}}^d}\frac{f(x)-f(y)}{\vert x-y \vert^{d+\alpha}}dy,\quad a.e.w.$$ and moreover, $$\label{E2}
\||\textnormal{D}|^\alpha f\|_{L^p}\lesssim \|f\|_{\dot{B}_{p,1}^\alpha}.$$ Indeed, the $L^p$ norm of the integral function satisfies in view of Minkowski inequalities $$\big{\|}\int_{{\mathbb{R}}^d}\frac{f(\cdot)-f(y)}{\vert \cdot-y \vert^{d+\alpha}}dy\big{\|}_{L^p}\leq
\int_{{\mathbb{R}}^d}\frac{\| f(\cdot)-f(\cdot-y)\|_{L^p}}{\vert y \vert^{d+\alpha}}dy\approx \|f\|_{\dot{B}_{p,1}^\alpha}.$$ Thus we find that the left integral term is finite almost every where.\
Inasmuch as the flow preserves Lebesgue measure then the formula (\[E1\]) yields $$(|\textnormal{D}|^\alpha f)\circ\psi(x)=C_{\alpha}\int_{{\mathbb{R}}^d}\frac{f(\psi(x))-f(y)}{\vert \psi(x)-y \vert^{d+\alpha}}dy=
C_{\alpha}\int_{{\mathbb{R}}^d}
\frac{ f(\psi(x))-f(\psi(y))}{\vert \psi(x)-\psi(y) \vert^{d+\alpha}}dy
.$$ Applying again (\[E1\]) with $f\circ\psi$, we obtain $$|\textnormal{D}|^\alpha (f\circ\psi)(x)=C_{\alpha}\int_{{\mathbb{R}}^d}\frac{f(\psi(x))-f(\psi(y))}{\vert x-y \vert^{d+\alpha}}dy.$$
Thus we get $$\begin{aligned}
|\textnormal{D}|^\alpha(f\circ\psi)(x)-(|\textnormal{D}|^\alpha f)\circ\psi(x)
&=& C_{\alpha}\int_{{\mathbb{R}}^d}\frac{f(\psi(x))-f(\psi(y))}{\vert x-y \vert^{d+\alpha}}\times\\
&&\Big( 1-\frac{\vert x-y \vert^{d+\alpha}}{\vert \psi(x)-\psi(y) \vert^{d+\alpha}} \Big) dy.\end{aligned}$$ Taking the $L^p$ norm and using (\[equivalence\]) we obtain $$\label{hamouda1}
\big{\|}|\textnormal{D}|^\alpha(f\circ\psi)-(|\textnormal{D}|^\alpha f)\circ\psi\big{\|}_{L^p}\lesssim
\| f \circ\psi\|_{\dot{B}_{p,1}^\alpha}
\sup_{x,y}\big{\vert} 1-\frac{\vert x-y \vert^{d+\alpha}}{\vert \psi(x)-\psi(y) \vert^{d+\alpha}} \big{\vert}.$$ According to [@Marcel] one has the following composition result $$\|f\circ\psi\|_{\dot{B}_{p,1}^\alpha}\leq c_{\alpha}\|\nabla\psi \|_{L^\infty}^\alpha \|f\|_{\dot{B}_{p,1}^\alpha}, \quad\textnormal{for}\quad\alpha\in]0,1[.$$ Therefore (\[hamouda1\]) becomes $$\begin{aligned}
\big{\|}|\textnormal{D}|^\alpha(f\circ\psi)-(|\textnormal{D}|^\alpha f)\circ\psi\big{\|}_{L^p} & \leq &C\|\nabla \psi \|_{L^\infty}^\alpha
\| f \|_{\dot{B}_{p,1}^\alpha} \times\\
&&\sup_{x,y}\big{\vert} 1-\frac{\vert x-y \vert^{d+\alpha}}{\vert \psi(x)-\psi(y) \vert^{d+\alpha}}\big{\vert}.
\end{aligned}$$ It is plain from mean value Theorem that $$\frac{1}{\|\nabla \psi\|_{L^\infty}^{d+\alpha}}\leq\frac{|x-y|^{d+\alpha}}{|\psi(x)-\psi(y)|^{d+\alpha}}\leq \|\nabla\psi^{-1}\|_{L^\infty}^{d+\alpha},$$
which gives easily the inequality $$\sup_{x,y}\big{\vert} 1-\frac{\vert x-y \vert^{d+\alpha}}{\vert \psi(x)-\psi(y) \vert^{d+\alpha}} \big{\vert}
\leq \max\Big(\vert 1-\|\nabla \psi^{-1} \|_{{L^\infty}}^{d+\alpha}\vert;
\vert 1-{\|\nabla \psi \|_{{L^\infty}}^{-d-\alpha}}\vert \Big).$$ This concludes the proof.
Proof of Theorem \[Thm3\]
=========================
We shall divide our analysis into two cases: $r=+\infty$ and $r$ is finite. The first case is more easy and simply based upon a maximum principle and a commutator estimate. Before we move on let us mention that in what follows we will work with the homogeneous Littlewood-Paley operators but we take the same notation of the inhomogeneous operators.
Set $\theta_{q}:=\Delta_{q}\theta,$ then localizing the equation through the operator $\Delta_{q}$ gives $$\label{eq:1}
\partial_{t}\theta_{q}+v\cdot\nabla\theta_{q}+|\hbox{D}|^{\alpha}\theta_{q}=-[\Delta_{q},v\cdot\nabla]\theta+f_{q}:=\mathcal{R}_{q}.$$ According to Proposition \[maximum\] we have $$\label{eq:2}
\| \theta_{q}(t)\|_{L^p}\leq\|\theta_{q}^0\|_{L^p}+\int_{0}^t\|\mathcal{R}_{q}(\tau)\|_{L^p}d\tau.$$ Multiplying both sides by $2^{qs}$ and summing over $q$ $$\|\theta\|_{\widetilde L^\infty_{t}\dot{B}_{p,1}^s}\leq \|\theta^0\|_{\dot{B}_{p,1}^s}+\|f\|_{L^1_{t}\dot{B}_{p,1}^s}+\int_{0}^t\sum_{q}2^{qs} \|\mathcal{R}_{q}(\tau) \|_{L^p}d\tau.$$ This yields in view of Lemma \[lemm12\] $$\|\theta\|_{\widetilde L^\infty_{t}\dot{B}_{p,1}^s}\leq \|\theta^0\|_{\dot{B}_{p,1}^s}+\|f\|_{L^1_{t}\dot{B}_{p,1}^s}+C\int_{0}^t\|\nabla v(\tau)\|_{L^\infty} \|\theta\|_{\widetilde L^\infty_{\tau}\dot{B}_{p,1}^s}d\tau.$$ To achieve the proof in the case of $r=\infty,$ it suffices to use Gronwall’s inequality.\
We shall now turn to the proof of the finite case $r<\infty$ which is more technical. Let $\psi$ denote the flow of the velocity $v$ and set $$\bar\theta_{q}(t,x)=\theta_{q}(t,\psi(t,x))\quad\hbox{and}\quad \bar {\mathcal R}_{q}(t,x)=\mathcal{R}_{q}(t,\psi(t,x)).$$
Since the flow preserves Lebesgue measure then we obtain $$\label{eq:5}
\|\bar{\mathcal{R}}_{q}\|_{L^p}\leq \|[\Delta_{q},v\cdot\nabla]\theta\|_{L^p}+\|f_{q}\|_{L^p}.$$ It is not hard to check that the function $\bar\theta_{q}$ satisfies $$\label{T1}
\partial_{t}\bar\theta_{q}+|\hbox{D}|^\alpha\bar\theta_{q}=
|\hbox{D}|^\alpha(\theta_{q}\circ\psi)-(|\hbox{D}|^\alpha\theta_{q})\circ\psi+\bar {\mathcal R}_{q}:=\bar{\mathcal R}_{q}^1.$$ From Proposition \[pr:1\] we find that for $q\in{\mathbb{Z}}$ $$\begin{aligned}
\label{T:5}
\nonumber\| |\hbox{D}|^\alpha(\theta_{q}\circ\psi)-(|\hbox{D}|^\alpha\theta_{q})\circ\psi\|_{L^p}
&\leq& Ce^{CV(t)} \Big(e^{CV(t)}-1\Big)\|\theta_{q}(t)\|_{\dot B_{p,1}^\alpha}\\
&\leq&Ce^{CV(t)} \Big(e^{CV(t)}-1\Big) 2^{q\alpha}\|\theta_{q}\|_{L^p},
\end{aligned}$$ where $V(t):=\|\nabla v\|_{L^1_{t}L^\infty}.$ Notice that we have used here the classical estimates $$e^{-CV(t)}\leq\|\nabla\psi^{\mp 1}(t)\|_{L^\infty}\leq
e^{CV(t)}.$$ Putting together (\[eq:5\]) and (\[T:5\]) yield $$\|\bar{\mathcal R}_{q}^1(t)\|_{L^p}\leq \| f_{q}(t)\|_{L^p}+\|[\Delta_{q},v\cdot\nabla]\theta\|_{L^p}
+Ce^{CV(t)
} (e^{CV(t)}-1)2^{q\alpha}\|\theta_{q}(t)\|_{L^p}.$$ Applying the operator $\Delta_{j},$ for $j\in{\mathbb{Z}},$ to the equation (\[T1\]) and using Proposition \[l:5\] $$\begin{aligned}
\|\Delta_{j}\bar\theta_{q}(t)\|_{L^p}&\leq& Ce^{-ct2^{j\alpha}}\|\Delta_{j}\theta_{q}^0\|_{L^p}+C\int_{0}^te^{-c (t-\tau)2^{j\alpha}}\| f_{q}(\tau)\|_{L^p}d\tau \\
\nonumber&+& Ce^{CV(t)}(e^{CV(t)}-1)2^{q\alpha}
\int_{0}^te^{-c (t-\tau)2^{j\alpha}}\|\theta_{q}(\tau)\|_{L^p}d\tau\\
\nonumber &+&C\int_{0}^te^{-c (t-\tau)2^{j\alpha}}\|[\Delta_{q},v\cdot\nabla]\theta(\tau)\|_{L^p}d\tau.
\end{aligned}$$ Integrating this estimate with respect to the time and using Young’s inequality $$\begin{aligned}
\label{eres}
\nonumber \|\Delta_{j}\bar\theta_{q}\|_{L^r_{t}L^p}&\leq& C
2^{-j\alpha/r}\big((1-e^{-crt2^{j\alpha}})^{\frac{1}{r}}\|\Delta_{j}\theta_{q}^0\|_{L^p}+\|f_{q}\|_{L^1_{t}L^p}\big)\\
\nonumber&+&
Ce^{CV(t)}(e^{CV(t)}-1)2^{(q-j)\alpha}\|\theta_{q}\|_{L^r_{t}L^p}\\
&+&C2^{-j\alpha/r}\int_{0}^t \|[\Delta_{q},v\cdot\nabla]\theta(\tau)\|_{L^p}d\tau.\end{aligned}$$ Since the flow $\psi$ preserves Lebesgue measure then one writes $$\begin{aligned}
2^{q(s+\alpha/r)}\|\theta_{q}\|_{L^r_{t}L^p}&=&2^{q(s+\alpha/r)}
\|\bar\theta_{q}\|_{L^r_{t}L^p}\\
&\leq &2^{q(s+\alpha/r)}\Big(
\sum_{\vert j-q \vert >N}\|\Delta_{j}\bar\theta_{q}\|_{L^r_{t}L^p}+
\sum_{\vert j-q \vert\leq N}\|\Delta_{j}\bar\theta_{q}\|_{L^r_{t}L^p}\Big)\\
&:=&\textnormal{I}+\textnormal{II}.\end{aligned}$$ To estimate the term $\textnormal I$ we make appeal to Lemma \[l400\] $$\begin{aligned}
\|\Delta_{j}\bar\theta_{q}\|_{L^r_{t}L^p}&\leq& C2^{-\vert q-j\vert}
e^{\int_{0}^t
\|\nabla v(\tau)\|_{L^\infty}d\tau}\|\theta_{q}\|_{L^r_{t}L^p}\\
&\leq&C2^{-\vert q-j\vert}
e^{V(t)}\|\theta_{q}\|_{L^r_{t}L^p}.\end{aligned}$$ Therefore we get $$\label{e:11}
\textnormal{I}\leq C2^{-N}e^{V(t)}2^{q(s+\alpha/r)}\|\theta_{q}\|_{L^r_{t}L^p}.$$ In order to bound the second term $\textnormal{II}$ we use (\[eres\]) $$\begin{aligned}
\label{Meth1}
\nonumber \textnormal{II}&\leq& C(1-e^{-crt2^{q\alpha}})^{\frac{1}{r}}2^{qs}\|\theta_{q}^0\|_{L^p}+C2^{N\frac{\alpha}{r}}2^{qs}\|f_{q}\|_{L^1_{t}L^p}\\
\nonumber&+&
C2^{N\alpha}e^{CV(t)}(e^{CV(t)}-1)2^{q(s+\alpha/r)}\|\theta_{q}\|_{L^r_{t}L^p}\\
&+&C2^{N\alpha/r}2^{qs}\int_{0}^t \|[\Delta_{q},v\cdot\nabla]\theta(\tau)\|_{L^p}d\tau.\end{aligned}$$ Denote $Z_{q}^r(t):=2^{q(s+\alpha/r)}\|\theta_{q}\|_{L^r_{t}L^p},$ then we obtain in view of (\[e:11\]) and (\[Meth1\]) $$\begin{aligned}
Z_{q}^r(t)&\leq &C(1-e^{-crt2^{q\alpha}})^{\frac{1}{r}}2^{qs}\|\theta_{q}^0\|_{L^p}+C2^{N\frac{\alpha}{r}}2^{qs}\|f_{q}\|_{L^1_{t}L^p}\\
&+&C\big(2^{N\alpha}e^{CV(t)}
(e^{CV(t)}-1)+2^{-N}e^{CV(t)}\big)Z_{q}^r(t)\\
&+&C2^{N\alpha/r}2^{qs}\int_{0}^t \|[\Delta_{q},v\cdot\nabla]\theta(\tau)\|_{L^p}d\tau.\end{aligned}$$ We can easily show that there exists two pure constants $N$ and $C_{0}$ such that $$V(t)\leq C_{0}\Rightarrow C2^{-N}e^{CV(t)}+C2^{N\alpha}e^{CV(t)}(e^{CV(t)}-1)\leq \frac{1}{2}\cdot$$ Thus we obtain under this condition $$\begin{aligned}
\label{mahma00}
\nonumber Z_{q}^r(t)&\leq& C(1-e^{-crt2^{q\alpha}})^{\frac{1}{r}}2^{qs}\|\theta_{q}^0\|_{L^p}+C2^{qs}\|f_{q}\|_{L^1_{t}L^p}\\
&+&C2^{qs}\int_{0}^t \|[\Delta_{q},v\cdot\nabla]\theta(\tau)\|_{L^p}d\tau.\end{aligned}$$ Summing over $q$ and using Lemma \[lemm12\] lead for $V(t)\leq C_{0},$ $$\begin{aligned}
\|\theta\|_{\widetilde L^r_{t}\dot{B}_{p,1}^{s+\frac{\alpha}{r}}}&\leq& C \|\theta^0\|_{\dot{B}_{p,1}^s}+C\|f\|_{L^1_{t}\dot B_{p,1}^s}+
C\int_{0}^t\| \nabla v(\tau)\|_{L^\infty}
\|\theta(\tau)\|_{\dot{B}_{p,1}^s}d\tau\\
&\leq&C \|\theta^0\|_{\dot{B}_{p,1}^s}+C\|f\|_{L^1_{t}\dot B_{p,1}^s}+CV(t)\| \theta\|_{L^\infty_{t}\dot{B}_{p,1}^s}.\end{aligned}$$ Thus we get in view of the estimate of the case $r=\infty$ $$\label{EDF}
\|\theta\|_{\widetilde L^r_{t}\dot{B}_{p,1}^{s+\frac{\alpha}{r}}}\leq C \|\theta^0\|_{\dot{B}_{p,1}^s}+C\|f\|_{L^1_{t}\dot B_{p,1}^s}.$$ This gives the result for a short time.
For an arbitrary positive time $T$ we make a partition $(T_{i})_{i=0}^M$ of the interval $[0,T],$ such Then proceeding for (\[EDF\]), we obtain $$\|\theta\|_{\widetilde L^r_{[T_{i},T_{i+1}]}\dot{B}_{p,1}^{s+\frac{\alpha}{r}}}\leq
C \|\theta(T_{i})\|_{\dot{B}_{p,1}^s}+C\int_{T_{i}}^{T_{i+1}}\|f(\tau)\|_{\dot{B}_{p,1}^s}d\tau.$$ Applying the triangle inequality gives $$\|\theta\|_{\widetilde L^r_{T}\dot{B}_{p,1}^{s+\frac{\alpha}{r}}}\leq C\sum_{i=0}^{M-1}\|\theta(T_{i})\|_{\dot{B}_{p,1}^s}+C\int_{0}^T\|f(\tau)\|_{\dot B_{p,1}^s}d\tau.$$ On the other hand the estimate proven in the case $r=\infty$ allows us to write $$\|\theta\|_{\widetilde L^r_{T}\dot{B}_{p,1}^{s+\frac{\alpha}{r}}}\leq
CM\big(\|\theta^0\|_{\dot{B}_{p,1}^s}+\|f\|_{L^1_{T}\dot B_{p,1}^s}\big)e^{CV(T)}+\|f\|_{L^1_{T}\dot B_{p,1}^s}.$$ Thus the following observation $C_{0}M\approx 1+V(t)$ completes the proof of the theorem.
Proof of Theorem [\[Thm1\]]{}
=============================
For the sake of a concise presentation, we shall just provide the [*a priori*]{} estimates supporting the claims of the theorem. To achieve the proof one must combine in a standard way these estimates with a standard approximation procedure such as the following iterative scheme $$\left\lbrace
\begin{array}{l}\partial_{t}\theta_{n+1}+v^n\cdot\nabla\theta_{n+1}+|\hbox{D}|^{{\alpha}} \theta_{n+1}=0,\\
v_{n}=(-R_{2} \theta_{n},R_{1}\theta_{n}),\\
\theta_{n+1}(0,x)=S_{n}\theta^0(x),\\
(\theta_{0},v_{0})=(0,0).
\end{array}
\right.$$
Global existence
-----------------
It is plain from Theorem \[Thm3\] that to derive global [*a priori*]{} estimates it is sufficient to bound globally in time the quantity $
V(t):=\|\nabla v\|_{L^1_{t}L^\infty}. $ First, the embedding $\dot{B}_{\infty,1}^0\hookrightarrow L^\infty$ combined with the fact that Riesz transform maps continuously homogeneous Besov space into itself $$\label{important}
\|\nabla v\|_{L^1_{t}L^\infty}\leq \|\nabla v\|_{L^1_{t}\dot B_{\infty,1}^0}\leq C\|\theta\|_{L^1_{t}\dot B_{\infty,1}^1}.$$ Combined with Theorem \[Thm3\] this yields $$V(t)\leq C\|\theta^0\|_{\dot{B}_{\infty,1}^{1-\alpha}} e^{CV(t)}.$$ Since the function $V$ depends continuously in time and $V(0)=0$ then we can deduce that for small initial data $V$ does not blow up, and there exists $C_{1},\eta>0$ such that $$\label{TA1}
\|\theta^0\|_{\dot B_{\infty,1}^{1-\alpha}}<\eta\Rightarrow\| \nabla v\|_{L^1({\mathbb{R}}_{+};L^\infty)}\leq C_{1}\|\theta^0\|_{\dot{B}_{\infty,1}^{1-\alpha}},\forall t\in{\mathbb{R}}_{+}.$$ Let us now show how to derive the [*a priori*]{} estimates. Take $s\geq s_{c}^p:=1+\frac{2}{p}-\alpha.$ Then combining Theorem \[Thm3\] with (\[TA1\]) we get $$\begin{aligned}
\|\theta\|_{\widetilde L^\infty_{{\mathbb{R}}_{+}}\dot{B}_{p,1}^{s}}+ \|\theta\|_{ L^1_{{\mathbb{R}}_{+}}\dot{B}_{p,1}^{s+\alpha}} &\leq& C\|\theta^0\|_{\dot{B}_{p,1}^s}e^{C\|\theta^0\|_{\dot{B}_{\infty,1}^{1-{\alpha}}} }\\
&\leq&
C\|\theta^0\|_{\dot{B}_{p,1}^s}.
\end{aligned}$$ On the other hand we have from Proposition \[maximum\] $$\forall t\in{\mathbb{R}}_{+},\, \|\theta(t)\|_{L^p}\leq\|\theta^0\|_{L^p}.$$ Therefore we get an estimate of $\theta$ in the inhomogeneous Besov space as follows $$\|\theta\|_{\widetilde L^\infty_{{\mathbb{R}}_{+}}{B}_{p,1}^{s}}\leq C\|\theta^0\|_{{B}_{p,1}^s}.$$ Using again Theorem \[Thm3\] yields $$\|\theta\|_{\widetilde L^\infty_{t}\dot{B}_{\infty,1}^0}\leq C\|\theta^0\|_{\dot{B}_{\infty,1}^0}e^{CV(t)}\leq
C\|\theta^0\|_{\dot{B}_{\infty,1}^0},$$ Thus we obtain for $p\in[1,\infty]$ $$\label{Jui1}
\|\theta\|_{\widetilde L^\infty_{{\mathbb{R}}_{+}}{\mathcal{X}}_{p}^s}\leq C\|\theta^0\|_{{\mathcal{X}}_{p}^s}.$$ For the velocity we have the following result.
For $p\in]1,\infty]$ there exists $C_{p}$ such that $$\|v\|_{\widetilde L^\infty_{{\mathbb{R}}_+}B_{p,1}^s}\leq C_{p}\|\theta^0\|_{\mathcal{X}_{p}^s}.$$ However, for $p=1$ we have $$\|v\|_{\widetilde L^\infty_{{\mathbb{R}}_{+}}\dot B_{1,1}^s}+\|v\|_{L^\infty_{{\mathbb{R}}_{+}}L^{p_{1}}}\leq C_{p_{1}}\|\theta^0\|_{B_{1,1}^s},\,\forall p_{1}>1.$$
Let $p\in]1,\infty[.$ Then we can write in view of (\[Jui1\]) $$\begin{aligned}
\|v\|_{\widetilde L^\infty_{{\mathbb{R}}_{+}}B_{p,1}^s}&\leq &\|v\|_{\widetilde L^\infty_{{\mathbb{R}}_{+}}\dot{B}_{p,1}^s}+\|\Delta_{-1}v
\|_{L^\infty_{{\mathbb{R}}_{+}}L^p}\\
&\leq& C\|\theta\|_{\widetilde L^\infty_{{\mathbb{R}}_{+}}\dot{B}_{p,1}^s}+C\|v\|_{L^\infty_{{\mathbb{R}}_{+}}L^p}\\
&\leq&C\|\theta^0\|_{B_{p,1}^s}+C\|v\|_{L^\infty_{{\mathbb{R}}_{+}}L^p}.\end{aligned}$$ Combining the boundedness of Riesz transform with the maximum principle $$\|v\|_{L^\infty_{{\mathbb{R}}_{+}}L^p}\leq C_{p}\|\theta^0\|_{L^p}.$$ Thus we obtain $$\|v\|_{\widetilde L^\infty_{{\mathbb{R}}_{+}}B_{p,1}^s}\leq C_{p}\|\theta^0\|_{B_{p,1}^s}.$$ To treat the case $p=\infty$ we write according to the embedding $\dot{B}_{\infty,1}^0\hookrightarrow L^\infty$ and the continuity of Riesz transform $$\|\Delta_{-1}v(t)\|_{L^\infty}\leq C\|\theta(t)\|_{\dot{B}_{\infty,1}^0}.$$ Combining this estimate with (\[Jui1\]) yields $$\|v\|_{\widetilde L^\infty_{{\mathbb{R}}_{+}}B_{\infty,1}^s}\leq C\|\theta^0\|_{B_{\infty,1}^s\cap \dot{B}_{\infty,1}^0}.$$ Hence we get for all $p\in]1,\infty]$ $$\label{acigne}
\|v\|_{\widetilde L^\infty_{{\mathbb{R}}_{+}}B_{p,1}^s}\leq C_{p}\|\theta^0\|_{{\mathcal{X}}_{p}^s}.$$ Let us now move to the case $p=1.$ Since $B_{1,1}^s\hookrightarrow L^{p_{1}}$ for all $p_{1}\geq1$ then we get in view of Bernstein’s inequality and the maximum principle $$\|\Delta_{-1}v\|_{L^{p_{1}}}\leq C_{p_{1}}\|\theta^0\|_{L^{p_{1}}}\leq C_{p_{1}}\|\theta^0\|_{B_{1,1}^s}.$$ We eventually find that $v\in\widetilde L^\infty_{{\mathbb{R}}_{+}}\dot B_{1,1}^s\cap L^\infty_{{\mathbb{R}}_{+}} L^{p_{1}}.$
Let us now briefly sketch the proof of the continuity in time, that is $\theta\in C({\mathbb{R}}_{+};{\mathcal{X}}_{p}^s).$ We should only treat the finite case of $p$ and similarly one can show the case $p=\infty$. From the definition of Besov spaces we have $$\|\theta(t)-\theta(t')\|_{B_{p,1}^s}\leq \sum_{q< N}2^{qs}\|\theta_{q}(t)-\theta_{q}(t')\|_{L^p}+2\sum_{q\geq N}2^{qs}\|\theta_{q}\|_{L^\infty_{{\mathbb{R}}_{+}}L^p}$$ Let $\epsilon>0$ then we get from (\[Jui1\]) the existence of a number $N$ such that $$\sum_{q\geq N}2^{qs}\|\theta_{q}\|_{L^\infty_{{\mathbb{R}}_{+}}L^p}\leq\frac{\epsilon}{4}.$$ Thanks to Taylor’s formula $$\begin{aligned}
\sum_{q< N}2^{qs}\|\theta_{q}(t)-\theta_{q}(t')\|_{L^p}&\leq &|t-t'|\sum_{q<N}2^{qs}\|\partial_{t}\theta_{q}\|_{L^\infty_{{\mathbb{R}}_{+}}L^p}\\
&\leq& C|t-t'|2^N\|\partial_{t}\theta\|_{L^\infty_{{\mathbb{R}}_{+}}B_{p,1}^{s-1}}.\end{aligned}$$ To estimate the last term we write $$\partial_{t}\theta=-|\textnormal{D}|^\alpha\theta-v\cdot\nabla\theta.$$ In one hand we have $|\textnormal{D}|^\alpha\theta\in B_{p,1}^{s-\alpha}\hookrightarrow B_{p,1}^{s-1}$. On the other hand since the space $B_{p,1}^s$ is an algebra ($s>\frac{2}{p}$) and $v$ is zero divergence then $$\|v\cdot\nabla\theta\|_{B_{p,1}^{s-1}}\leq C\|v\,\theta\|_{B_{p,1}^{s}}\leq C\|v\|_{B_{p,1}^{s}}\|\theta\|_{B_{p,1}^{s}}.$$ Thus we get $\partial_{t}\theta\in L^\infty_{{\mathbb{R}}_{+}}B_{p,1}^{s-1}$ and this allows us to finish the proof of the continuity.
Local existence
----------------
The local time existence depends on the control of the quantity $V(t):=\|\nabla v\|_{L^1_{t}L^\infty}.$ In our analysis we distinguish two cases:
$\bullet$ [*First case:* ]{} $s>s_{c}^p=1+\frac{2}{p}-\alpha.$
We observe first that there exists $r>1$ such that $1+\frac{2}{p}-\frac{\alpha}{r}\leq s.$ From (\[important\]) and according to the said Hölder’s inequality we have $$\begin{aligned}
V(t)&\leq& C\|\theta\|_{L^1_{t}\dot{B}_{\infty,1}^1}\\
&\leq&Ct^{\frac{1}{\bar{r}}} \|\theta\|_{L^r_{t}\dot{B}_{\infty,1}^1}.
\end{aligned}$$ Using Theorem (\[Thm3\]) we obtain $$\begin{aligned}
V(t)\leq Ct^{\frac{1}{\bar{r}}}
\|\theta^0\|_{\dot{B}_{\infty,1}^{1-\frac{\alpha}{r}}}e^{CV(t)}.
\end{aligned}$$ Thus we conclude that there exists $C_{0},\,\eta>0$ such that $$\label{Tr23}
t^{\frac{1}{\bar{r}}}
\|\theta^0\|_{\dot{B}_{\infty,1}^{1-\frac{\alpha}{r}}}\leq\eta\Rightarrow V(t)\leq C_{0},$$ and this gives from Theorem \[Thm3\] $$\label{Tr24}
\|\theta\|_{L^\infty_{t
}B_{p,1}^s}+\|\theta\|_{L^1_{t}\dot B_{p,1}^{1+\frac{2}{p}}}\leq C\|\theta^0\|_{B_{p,1}^s}.$$ We point out that one can deduce from (\[Tr23\]) that the time existence is bounded below $$T\gtrsim \|\theta^0\|_{\dot{B}_{\infty,1}^{1-\frac{\alpha}{ r}}}^{-\bar{r}}.$$
$\bullet$ [*Second case:* ]{}$ s=s_{c}^p=1+\frac{2}{p}-\alpha.$
By applying (\[mahma00\]) to the ${(\textnormal{QG}_{\alpha})}$ equation with $r=1,\,p=\infty$ and $s=1-\alpha$ we have under the condition $V(t)\leq C_{0}$ $$\|\theta\|_{L^1_{t}\dot{B}_{\infty,1}^1}\leq C\sum_{q\in{\mathbb{Z}}}(1-e^{-ct2^{q\alpha}})2^{q(1-\alpha)}\|\theta_{q}^0\|_{L^\infty}+C\sum_{q\in{\mathbb{Z}}}2^{q(1-\alpha)}\|[\Delta_{q},v\cdot\nabla]\theta\|_{L^1_{t}L^\infty}.$$ The second term of the right-hand side can be estimated from Lemma \[lemm12\] as follows $$\begin{aligned}
\label{All-Port}
\nonumber\sum_{q\in{\mathbb{Z}}}2^{q(1-\alpha)}\|[\Delta_{q},v\cdot\nabla]\theta\|_{L^1_{t}L^\infty}&\leq& C\|v\|_{\widetilde L^2_{t}\dot B_{\infty,1}^{1-\frac{\alpha}{2}}}\|\theta\|_{\widetilde L^2_{t}\dot B_{\infty,1}^{1-\frac{\alpha}{2}}}\\
&\leq&
C\|\theta\|_{\widetilde L^2_{t}\dot B_{\infty,1}^{1-\frac{\alpha}{2}}}^2.\end{aligned}$$ Notice that we have used in the above inequality the fact that Riesz transform maps continuously homogeneous Besov space into itself. Hence we get $$\label{RT}
\|\theta\|_{L^1_{t}\dot{B}_{\infty,1}^1}\leq C\sum_{q\in{\mathbb{Z}}}(1-e^{-ct2^{q\alpha}})2^{q(1-\alpha)}\|\theta_{q}^0\|_{L^\infty}+C\|\theta\|_{\widetilde L^2_{t}\dot B_{\infty,1}^{1-\frac{\alpha}{2}}}^2.$$ Using again (\[mahma00\]) with $r=2,\,p=\infty$ and $s=1-{\alpha},$ we obtain $$\|\theta\|_{\widetilde L^2_{t}\dot B_{\infty,1}^{1-\frac{\alpha}{2}}}\leq
C\sum_{q\in{\mathbb{Z}}}(1-e^{-ct2^{q\alpha}})^{\frac{1}{2}}2^{q(1-\alpha)}\|\theta_{q}^0\|_{L^\infty}+C\sum_{q\in{\mathbb{Z}}}2^{q(1-\alpha)}\|[\Delta_{q},v\cdot\nabla]\theta\|_{L^1_{t}L^\infty}.$$ Thus (\[All-Port\]) yields $$\|\theta\|_{\widetilde L^2_{t}\dot B_{\infty,1}^{1-\frac{\alpha}{2}}}\leq
C\sum_{q\in{\mathbb{Z}}}(1-e^{-ct2^{q\alpha}})^{\frac{1}{2}}2^{q(1-\alpha)}\|\theta_{q}^0\|_{L^\infty}+C \|\theta\|_{\widetilde L^2_{t}\dot B_{\infty,1}^{1-\frac{\alpha}{2}}}^2.$$ By Lebesgue theorem we have $$\lim_{t\to 0^+}\sum_{q\in{\mathbb{Z}}}(1-e^{-ct2^{q\alpha}})^{\frac{1}{2}}2^{q(1-\alpha)}\|\theta_{q}^0\|_{L^\infty}=0.$$ Let $\eta$ be a sufficiently small constant and define $$T_{0}:=\sup\Big\{t> 0,\sum_{q\in{\mathbb{Z}}}(1-e^{-ct2^{q\alpha}})^{\frac{1}{2}}2^{q(1-\alpha)}\|\theta_{q}^0\|_{L^\infty}\leq \eta\Big\}.$$ Then we have under the assumptions $t\leq T_{0}$ and $V(t)\leq C_{0}$ $$\|\theta\|_{\widetilde L^2_{t}\dot B_{\infty,1}^{1-\frac{\alpha}{2}}}\leq 2C\sum_{q\in{\mathbb{Z}}}(1-e^{-ct2^{q\alpha}})^{\frac{1}{2}}2^{q(1-\alpha)}\|\theta_{q}^0\|_{L^\infty}.$$ Inserting this estimate into (\[RT\]) gives $$\begin{aligned}
V(t)\leq C\|\theta\|_{L^1_{t}\dot{B}_{\infty,1}^1}&\leq&
C\sum_{q\in{\mathbb{Z}}}(1-e^{-ct2^{q\alpha}})^{\frac{1}{2}}2^{q(1-\alpha)}\|\theta_{q}^0\|_{L^\infty}\\
&+&C\Big(\sum_{q\in{\mathbb{Z}}}(1-e^{-ct2^{q\alpha}})^{\frac{1}{2}}2^{q(1-\alpha)}\|\theta_{q}^0\|_{L^\infty}\Big)^2.
\end{aligned}$$ For sufficiently small $\eta$ we obtain $V(t)<C_{0}$ and this allows us to prove that the time $T_{0}$ is actually a local time existence. Thus we obtain from Theorem \[Thm3\] $$\|\theta\|_{\widetilde L^\infty_{T}B_{p,1}^{s_{c}^p}}+\|\theta\|_{L^1_{T}\dot{B}_{p,1}^{1+\frac{2}{p}}}\leq C\|\theta^0\|_{B_{p,1}^{s_{c}^p}}.$$
Uniqueness
-----------
We shall give the proof of the uniqueness result which can be formulated as follows. There exists at most one solution for the system ${(\textnormal{QG}_{\alpha})}$ in the functions space $X_{T}:=L^\infty_{T}\dot{B}_{\infty,1}^0\cap L^1_{T}\dot{B}_{\infty,1}^1.$ We stress out that the space $L^\infty_{T}X^s_{p}\cap L^1_{T}\dot B_{p,1}^{s+\alpha},$ with $p\in[1,\infty],$ is continuously embedded in $X_{T}.$\
Let $\theta^{i}, i=1,2$ (and $v^{i}$ the corresponding velocity) be two solutions of the ${(\textnormal{QG}_{\alpha})}$ equation with the same initial data and belonging to the space $X_{T}.$ We set $\theta=\theta^{1}-\theta^{2}$ and $v=v^1-v^2,$ then it is plain that $$\partial_{t}\theta+v^1\cdot\nabla\theta+|\textnormal{D}|^\alpha \theta=-v\cdot\nabla\theta^{2},\,\,\theta_{|t=0}=0.$$ Applying Theorem \[Thm3\] to this equation gives $$\label{Coupe1}
\|\theta(t)\|_{\dot{B}_{\infty,1}^0}\leq Ce^{C\|\nabla v^1\|_{L^1_{t}L^\infty}} \int_{0}^t\|v\cdot\nabla\theta^2(\tau)\|_{\dot{B}_{\infty,1}^0}d\tau.$$ We will now make use of the following law product and its proof will be given later. $$\label{Ac1}
\|v\cdot\nabla\theta^2\|_{\dot{B}_{\infty,1}^0}\leq C \|v\|_{\dot{B}_{\infty,1}^0}\|\theta^2\|_{\dot{B}_{\infty,1}^1}.$$ Since Riesz transform maps continuously $\dot{B}_{\infty,1}^0$ into itself, then we get $$\|v\cdot\nabla\theta^2\|_{\dot{B}_{\infty,1}^0}\leq C \|\theta\|_{\dot{B}_{\infty,1}^0}\|\theta^2\|_{\dot{B}_{\infty,1}^1}.$$ Inserting this estimate into (\[Coupe1\]) and using Gronwall’s inequality give the wanted result.\
Let us now turn to the proof of (\[Ac1\]) which is based on Bony’s decomposition $$v\cdot\nabla\theta^2=T_{v}\nabla\theta^2+T_{\nabla\theta^2}v+R(v,\nabla\theta^2), \quad\hbox{with}$$ $$T_{v}\nabla\theta^2=\sum_{q\in{\mathbb{Z}}}\dot S_{q-1}v\cdot\nabla\dot\Delta_{q}\theta^2\quad\hbox{and}\quad
R(v,\nabla\theta^2)=\sum_{q\in{\mathbb{Z}}\atop
i\in\{\mp1,0\} }\dot\Delta_{q}v\cdot\dot\Delta_{q+i}\nabla\theta^2.$$ Using the quasi-orthogonality of the paraproduct terms one obtains $$\begin{aligned}
\|T_{v}\nabla\theta^2\|_{\dot B_{\infty,1}^0}&\leq& C\sum_{q\in{\mathbb{Z}}}\|\dot S_{q-1}v\|_{L^\infty}\|\dot\Delta_{q}\nabla\theta^2\|_{L^\infty}\\
&\leq&C\|v\|_{\dot B_{\infty,1}^0}\|\theta^2\|_{\dot B_{\infty,1}^1}.
\end{aligned}$$ By the same way we get $$\begin{aligned}
\|T_{\nabla\theta^2}v\|_{\dot B_{\infty,1}^0}&\leq& C\sum_{q\in{\mathbb{Z}}}\|\dot S_{q-1}\nabla\theta^2\|_{L^\infty}\|\dot\Delta_{q}v\|_{L^\infty}\\
&\leq& C\|\nabla\theta^2\|_{L^\infty}\|v\|_{\dot B_{\infty,1}^0}\\
&\leq&C\|\theta^2\|_{\dot B_{\infty,1}^1}\|v\|_{\dot B_{\infty,1}^0}.
\end{aligned}$$ For the remainder term we write in view of the incompressibility of the velocity and the convolution inequality $$\begin{aligned}
\|R(v,{\nabla\theta^2})\|_{\dot B_{\infty,1}^0}&=&\sum_{j\in{\mathbb{Z}}} \|\dot\Delta_{j}R(v,\nabla\theta^2)\|_{L^\infty} \leq C\sum_{q\geq j-3
\atop i\in\{\mp1,0\}}2^j\|\dot \Delta_{q}v\|_{L^\infty}\|\dot\Delta_{q+i}\theta^2\|_{L^\infty}\\
&\leq&C\sum_{q\geq j-3
\atop i\in\{\mp 1,0 \}}2^{j-q}\|\dot \Delta_{q}v\|_{L^\infty}2^q\|\dot\Delta_{q+i}\theta^2\|_{L^\infty}\\
&\leq& C\|\nabla\theta^2\|_{L^\infty}\|v\|_{\dot B_{\infty,1}^0}\\
&\leq&C\|\theta^2\|_{\dot B_{\infty,1}^1}\|v\|_{\dot B_{\infty,1}^0}.
\end{aligned}$$ This completes the proof of (\[Ac1\]).
Appendix
========
The following result is due to Vishik [@v1] and was used in a crucial way for the proof of Theorem [\[Thm3\]]{}. For the convenience of the reader we will give a short proof based on the duality method.
\[l400\] Let $f$ be a function in Schwartz class and $\psi$ a diffeomorphism preserving Lebesgue measure, then we have for all $p\in[1,+\infty]$ and for all [ $j,q\in{\mathbb{Z}},$]{} $$\|\dot\Delta_j(\dot\Delta_q f\circ\psi)\|_{L^p}\leq
C2^{-\vert j-q\vert}\|\nabla\psi ^{\epsilon(j,q)}\|_{L^{\infty}}\|\dot\Delta_q
f\|_{L^p},$$ with $$\epsilon(j,q)=\hbox{sign}(j-q).$$
We shall begin with the proof of Lemma \[l400\]
We distinguish two cases: $j\geq q$ and $j<q.$ For the first one we simply use Bernstein’s inequality $$\|\dot\Delta_j(\dot\Delta_q f\circ\psi)\|_{L^p}\lesssim 2^{-j}\|\nabla\dot\Delta_j(\dot\Delta_q f\circ\psi)\|_{L^p}.$$ It suffices now to combine Leibnitz formula again with Bernstein’s inequality $$\begin{aligned}
\|\nabla\dot\Delta_j(\dot\Delta_q f\circ\psi)\|_{L^p}&\lesssim& \|\nabla\dot\Delta_q f\|_{L^p}\|\nabla\psi\|_{L^\infty}\\
&\lesssim& 2^q\|\dot\Delta_q f\|_{L^p}\|\nabla\psi\|_{L^\infty}.\end{aligned}$$ This yields to the desired inequality. Let us now move to the second case and use the following duality result $$\label{duality1}
\|\dot\Delta_j(\dot\Delta_q f\circ\psi)\|_{L^p}=\sup_{\|g\|_{L^{\bar p}}\leq1}\big{|}\langle\dot\Delta_j(\dot\Delta_q f\circ\psi),g\rangle\big{|},\,\hbox{with}\quad \frac{1}{p}+\frac{1}{\bar p}=1.$$ Let $\bar\varphi\in C^\infty_{0}({\mathbb{R}}^d)$ be supported in a ring and taking value $1$ on the ring $\mathcal{C}$ (see the definition of the dyadic decomposition). We set $\bar{\dot\Delta}_{q}f:=\bar\varphi(2^{-q}\textnormal{D})f.$ Then we can see easily that $\dot\Delta_{q}f=\bar{\dot\Delta}_{q}\dot\Delta_{q}f.$ Combining this fact with Parseval’s identity and the preserving measure by the flow $$\big{|}\langle\dot\Delta_j(\dot\Delta_q f\circ\psi),g\rangle\big{|}=\big{|}\langle\dot\Delta_q f,\bar{\dot\Delta}_{q}\big((\dot\Delta_jg)\circ\psi^{-1}\big)\rangle\big{|}.$$ Therefore we obtain $$\big{|}\langle\dot\Delta_j(\dot\Delta_q f\circ\psi),g\rangle\big{|}\leq \|\dot\Delta_{q}f\|_{L^p}
\|\bar{\dot\Delta}_{q}\big((\dot\Delta_jg)\circ\psi^{-1}\big)\|_{L^{\bar p }}.$$ This implies in view of the first case $$\begin{aligned}
\big{|}\langle\dot\Delta_j(\dot\Delta_q f\circ\psi),g\rangle\big{|}&\lesssim& \|\dot\Delta_{q}f\|_{L^p}2^{j-q}\|\nabla\psi^{-1}\|_{L^\infty}\|\dot\Delta_{j}g\|_{L^{\bar p}}\\
&\lesssim&
\|\dot\Delta_{q}f\|_{L^p}2^{j-q}\|\nabla\psi^{-1}\|_{L^\infty}\|g\|_{L^{\bar p}}.\end{aligned}$$ Thus we get in view of (\[duality1\]) the wanted result.
Next we give a maximum principle estimate for the equation $(TD_{\alpha})$ extending a recent result due to [@C-C] for the partial case $f=0$. The proof uses the same idea and will be briefly described.
\[maximum\] Let $v$ be a smooth divergence free vector field and $f$ be a smooth function. We assume that $\theta$ is a smooth solution of the equation $$\partial_{t}\theta+v\cdot\nabla \theta+\kappa |\textnormal{D}|^\alpha \theta=f, \quad\textnormal{with}\quad\kappa\geq 0\quad\textnormal{and}\quad
\alpha\in[0,2].$$ Then for $p\in[1,+\infty]$ we have $$\|\theta(t)\|_{L^p}\leq\|\theta(0)\|_{L^p}+\int_{0}^t\|f(\tau)\|_{L^p}d\tau.$$
Let $p\geq 2$, then multiplying the equation by $|\theta|^{p-2}\theta$ and integrating by parts lead to $$\frac{1}{p} \frac{d}{dt}\|\theta(t)\|_{L^p}^p+\kappa\int|\theta|^{p-2}\theta\,|\textnormal{D}|^\alpha\theta dx=\int f |\theta|^{p-2}\theta dx.$$ On the other hand it is shown in [@C-C] that $$\int|\theta|^{p-2}\theta\,|\textnormal{D}|^\alpha\theta dx\geq 0.$$ Now using Hölder’s inequality for the right-hand side $$\int f |\theta|^{p-2}\theta dx\leq \|f\|_{L^p}\|\theta\|_{L^p}^{p-1}.$$ Thus we obtain $$\frac{d}{dt}\|\theta(t)\|_{L^p}\leq \|f(t)\|_{L^p}.$$ We can deduce the result by integrating in time. The case $p\in[1,2[$ can be obtained through the duality method.
When this paper was finished we had been informed that similar results were obtained by Chen et [*al*]{} [@Chen]. In fact they obtained global well-posedness result for small initial data in $\dot B_{p,q}^{s_{c}^p},$ with $p\in[2,\infty[$ and $q\in[1,\infty[.$ For the particular case $q=\infty$ our result is more precise. Indeed, first, we can extend their result to $p\in[1,\infty]$ and second our smallness condition is given in the space $\dot B_{\infty,1}^{1-\alpha}$ which contains Besov spaces $\{\dot{B}_{p,1}^{s_{c}^p}\}_{p\in[1,\infty]}.$
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|
---
author:
- 'L. Decin[^1]'
- 'I. Cherchneff'
- 'S. Hony'
- 'S. Dehaes[^2]'
- 'C. De Breuck'
- 'K. M. Menten'
date: 'Received September 15, 2007; accepted October 15, 2007'
title: 'Detection of ‘parent’ molecules from the inner wind of AGB stars as tracers of non-equilibrium chemistry'
---
[Asymptotic Giant Branch (AGB) stars are typified by strong dust-driven, molecular outflows. For long, it was believed that the molecular setup of the circumstellar envelope of AGB stars is primarily determined by the atmospheric C/O ratio. However, recent observations of molecules such as HCN, SiO, and SO reveal gas-phase abundances higher than predicted by thermodynamic equilibrium (TE) models. UV-photon initiated dissociation in the outer envelope or non-equilibrium formation by the effect of shocks in the inner envelope may be the origin of the anomolous abundances.]{} [We aim at detecting *(i)* a group of ‘parent’ molecules (CO, SiO, HCN, CS), predicted by the non-equilibrium study of Cherchneff (2006) to form with almost constant abundances independent of the C/O ratio and the stellar evolutionary stage on the Asymptotic Giant Branch (AGB), and *(ii)* few molecules, such as SiS and SO, which are sensitive to the O- or C-rich nature of the star.]{} [Several low and high excitation rotational transitions of key molecules are observed at mm and sub-mm wavelengths with JCMT and APEX in four AGB stars: the oxygen-rich Mira , the S star , and the two carbon stars and . A critical density analysis is performed to determine the formation region of the high-excitation molecular lines.]{} [We detect the four ‘parent’ molecules in all four objects, implying that, indeed, these chemical species form whatever the stage of evolution on the AGB. High-excitation lines of SiS are also detected in three stars with APEX, whereas SO is only detected in the oxygen-rich star . ]{} [ This is the first multi-molecular observational proof that periodically shocked layers above the photosphere of AGB stars show some chemical homogeneity, whatever the photospheric C/O ratio and stage of evolution of the star. ]{}
Introduction {#Introduction}
============
Circumstellar envelopes of Asymptotic Giant Branch stars (AGBs) have long been known to be efficient sites of molecule formation. While the outer layers of such envelopes experience penetration of interstellar UV photons and cosmic rays resulting in a fast ion-molecule chemistry, the deepest layers are dominated by a non-equilibrium chemistry due to the passage of shocks generated by stellar pulsation. Dust forms in those inner gas layers, still bound to the star, and grains couple to the gas to accelerate it, thereby generating stellar wind and mass loss phenomena. The described processes greatly modify the abundances established by the equilibrium chemistry in the dense, hot photosphere .
For a long time, the gas chemical composition was believed to be dominated entirely by the C/O ratio of the photosphere. A C/O ratio greater than one implied that all the oxygen was tied in CO, leading to an oxygen-free chemistry, whereas a C/O ratio less than one meant that no carbon bearing molecules apart from CO could ever form in an oxygen-rich (O-rich) environment. This picture, based essentially on thermal equilibrium considerations applied to the gas, has been first disproved by the detection of SiO at millimeter (mm) wavelength in carbon-rich (C-rich) AGBs . As for O-rich AGBs, CO$_2$ infrared (IR) transition lines were detected in various objects with the Short-Wavelength Spectrometer (SWS) onboard the Infrared Space Observatory (ISO) .
Theoretical modeling describing the chemistry in the inner wind of the extreme carbon star showed that the formation of SiO was due primarily to hydroxyl OH reaction with atomic silicon close to the photosphere as a result of shock activity and therefore non-equilibrium chemistry . Later on, showed that CO$_2$ formation in the O-rich Mira results from the reaction of OH radicals with CO in the shocked regions, implying again that non-equilibirum chemistry was paramount to the formation of C-bearing species in O-rich Miras. It was then recently proposed that the inner wind of AGBs shows a striking homogeneity in chemical composition, despite their photospheric C/O ratio and stage of stellar evolution . In particular, showed that when taking shock chemistry into account, molecules such as SiO, HCN and CS are present in comparable amount in the inner layers of M, S, and C AGBs, whereas specific molecules (e.g. SO and HS for O-rich Miras and C$_2$H$_2$ for carbon stars) are typical for O-rich or C-rich chemistries.
In this letter, we present observations carried out with the JCMT and the APEX telescope of four AGBs: one O-rich, , one S star ($\equiv$ C/O $\approx$ 1), , and two carbon stars, and . We focus on the detection of (sub)mm transitions of CO, SiO, HCN, CS, SiS and SO in order to confirm or disprove the above hypothesis and to check for homogeneity in AGB winds.
Observations and line profiles {#Observations}
==============================
The observations were performed in October 2006 with the 15m JCMT[^3] for , , and , and in the period from September till October 2006 with the APEX[^4] 12m telescope for , and . Due to technical problems with the RxB3 JCMT receiver, only low frequency lines within the RxA3 receiver (211 – 276GHz) were obtained. For the APEX observations, both the APEX-2A receiver (279–381GHz) and FLASH receiver (460–495GHz and 780–887GHz) were used. The observations were carried out using a position-switching mode. The JCMT data reduction was performed with the SPLAT devoted routines of STARLINK, the APEX-data with CLASS. A polynomial was fitted to an emision free region of the spectral baseline and subtracted. The velocity resolution for the JCMT-data equals to 0.0305MHz, for the APEX to 0.1221MHz. For , , and the data were rebinned to a resolution of 1km/s, for to 0.75km/s in order to have at least 40 independent resolution elements per line profile. The antenna temperature, $T_A^*$, was converted to the main-beam temperature ($T_{mb} =
T_A^*/\eta_{mb}$), using a main-beam efficiency $\eta_{mb}$ of 0.69 for the JCMT RxA3 receiver, of 0.73 for the APEX-2A receiver, and of 0.60 and 0.43 for the 460–495GHz and 780–887GHz FLASH channels respectively .
The observed molecular emission lines of four AGB stars in our sample are displayed in the Figs. \[fig\_WXPsc\_all\] – \[fig\_VCyg\_all\].
{width=".95\textwidth"}
{width="\textwidth" height=".45\textheight"}
{width="\textwidth" height=".45\textheight"}
In all stars molecular emission lines of CO, SiO, HCN, and CS are detected. This confirms the prediction of homogeneity by as these species being ‘parent’ molecules which form in the inner layers of the CSE. This can be understood in terms of the chemistry of these four molecules being determined by shock propagation and not by the photospheric C/O ratio and the stellar evolutionary stage.
SiS is also detected in all stars and we were able to detect with APEX the high-excitation SiS(19-18) line in the O-rich , the C-rich and the S-rich . This is in good agreement with recent SiS OSO, JCMT, and APEX observations of @Schoier2007arXiv0707.0944S in a large sample of M and C stars, including and . This implies that SiS also forms close to the star, whatever the stage of stellar evolution.
Both the SO($6_5$ – $5_4$) and the high-excitation SO($10_{11}-10_{10}$) line were detected in O-rich . SO was neither found in the S-type , nor in the two carbon stars and . SO appears to be typical for O-rich AGBs only, supporting the non-detection of SO in C-stars by .
Both optically thin and optically thick lines occur (e.g., $^{13}$CO(2-1) versus the $^{12}$CO(2-1) line in , see e.g.Fig. \[fig\_WAql\_all\]). The line parameters, i.e., the main-beam brightness temperature at the line centre ($T_{mb}$), the line centre velocity ($v_*$), and half the full line width ($v_e$), are obtained by fitting the ‘soft parabola’ line profile function to the data [@Olofsson1993ApJS...87..267O] $$T(v) = T_{\mathrm{mb}} \left[ 1 - \left( \frac{v-v_*}{v_{\mathrm e}} \right) ^2
\right] ^{\beta/2}\,,$$ where $\beta$ describes the shape of the line. The velocity-integrated intensities are obtained by integrating the emission between $v_* \pm
v_e$ and are listed in Table \[Table\_overview\]. A value of $\beta\,=\,2$ represents a parabolic line shape, expected for optically thick expanding spherical envelopes in case the source is much smaller than the radio telescope beam size; $\beta < 0$ results in a profile with horns at the extreme velocity, expected for optically thin lines when the source is resolved [@Morris1975ApJ...197..603M]. In our sample, several lines have a $\beta$-value distinctly larger than 2, indicating that the line formation occurs partially in the inner region, where the stellar wind has not yet reached its full terminal velocity. This is notably the case for the CS, SiO, and HCN lines in our sample, corroborating their formation close to the star. The estimated terminal velocities for , , , and are 19.2km/s, 18.8km/s, 20.1km/s, and 12.4km/s, respectively. A maximum of 5% difference is found in the derived terminal velocity values for each target, being within the velocity binsize, indicating that all of the detected lines at least partly survive the dust formation and wind acceleration processes. An exception is SiS(19-18) in (16km/s), but particularly the SO($10_{11}$-$10_{10}$), with an estimated terminal velocity of only 10km/s, traces a much smaller geometrical region.
---------------- ------------------ ----------------- ------------------- ----------------- ------------------ ----------------- ------------------------------------- ----------------- ------------------------------------- ------------------------- ------------------ --
*transition* *$^{12}$CO(2-1)* $^{12}$CO(3-2) $^{12}$CO(4-3) $^{12}$CO(7-6) *$^{13}$CO(2-1)* $^{13}$CO(3-2) *$^{12}$CS(5-4)* $^{12}$CS(6-5) $^{12}$CS(7-6) $^{12}$CS(10-9) $^{12}$CS(17-16)
*frequency* *230.538* 345.795 461.040 806.651 *220.398* 330.587 *244.935* 293.912 342.882 489.759 832.061
$E_{\rm{low}}$ *3.84* 11.54 23.07 80.74 *3.67* 11.03 *16.34* 24.51 34.32 75.53 222.15
WX Psc *58.0* 64.3 40.2 22.4 *9.9* 11.3 *$<$2.4* \[0.8\] 1.3 $<$8.1 $<$42.6
W Aql *80.6* 118.4 95.9 $<$178 *4.6* 8.1 $-$ 2.4 4.6 $<$54.1 $<$203
II Lup $-$ 145.0 130.0 99.7 $-$ 32.5 $-$ 20.4 35.2 $-$ $-$
V Cyg *45.6* $-$ $-$ $-$ *4.2* $-$ *7.4* $-$ $-$ $-$ $-$
*transition* $^{13}$CS(7-6) $^{13}$CS(10-9) *H$^{12}$CN(3-2)* H$^{12}$CN(4-3) H$^{12}$CN(9-8) H$^{13}$CN(4-3) *SiO(5-4)* *SiO(6-5)* SiO(7-6) SiO(11-10)
*frequency* 323.684 462.334 *265.886* 354.505 797.433 345.340 *217.105* *260.518* 303.926 477.504
$E_{\rm{low}}$ 32.39 69.41 *8.87* 17.74 106.42 17.28 *14.48* *21.73* 30.42 79.65
WX Psc $<$67.4 $<$12.7 *9.9* 8.9 $<$117 \[2.6\] *9.2* *11.1* 10.4 $<$15.0
W Aql $-$ $<$13.0 *21.7* 12.2 \[34.1\] 3.1 *7.1* *8.9* 5.2 13.8
II Lup $-$ $-$ $-$ 60.7 $-$ 64.2 $-$ $-$ 19.6 $-$
V Cyg $-$ $-$ *25.7* $-$ $-$ $-$ *3.8* *5.1* $-$ $-$
*transition* SiO(19-18) SiS(12-11) *SiS(14-13)* SiS(16-15) SiS(19-18) SO($1_1$-$1_0$) *SO($\mathit{6_5}$-$\mathit{5_4}$)* SO($7_8$-$6_7$) *SO($\mathit{8_9}$-$\mathit{8_8}$)* SO($10_{11}$-$10_{10}$)
*frequency* 824.235 217.817 *254.102* 290.380 344.778 286.34 *219.949* 340.714 *254.573* 336.597
$E_{\rm{low}}$ 247.57 39.96 *55.10* 72.66 103.53 1.00 *16.98* 45.10 *60.80* 88.08
WX Psc $<$92.2 *3.2* *$<$1.2* 5.4 4.8 $<$0.9 *1.6* $<$1.4 $<$2.0 \[0.9\]
W Aql \[22.3\] $-$ $-$ 1.2 \[3.4\] $-$ *$<$1.2* $<$3.4 $-$ $-$
II Lup $-$ $-$ $-$ $-$ 7.8 $-$ $-$ $<$4.2 $-$ $-$
V Cyg $-$ *0.9* *$<$14.8* $-$ $-$ $-$ *$<$1.3* $-$ $-$ $-$
---------------- ------------------ ----------------- ------------------- ----------------- ------------------ ----------------- ------------------------------------- ----------------- ------------------------------------- ------------------------- ------------------ --
Excitation analysis {#Analysis}
===================
In case many transitions of an individual molecule can be observed, it is possible to assess whether collisional or radiative excitation mechanisms can produce the observed line intensities. While it is well known that CO is formed in both M, C and S-type stars and survives dust condensation, it is of interest to study the excitation requirements for the three other ‘parent’ molecules SiO, HCN and CS predicted by to be abundant in the inner winds of M, S and C AGBs.
For all lines, except for those from the CO molecule, the emission distribution is expected to be much smaller than the FWHM beam size of the used telescope. To calculate column densities we need to correct for the different beam filling factor, $f = \theta_S^2/(\theta_S^2 +
\theta_B^2)$, with $\theta_B$ and $\theta_S$ being the FWHM of the beam and the source respectively. To do this, we define a beam-averaged brightness temperature, $T_b$, scaled by correcting our main-beam brightness temperatures for a fictitious $10''$ FWHM source, i.e, $T_b = 1/f \times T_{mb}$.
Little interferometric data exist for the molecules that we have observed in *any* circumstellar envelope. We note, however, that $4''$–$6''$ resolution observations of the HCN $J =1-0$ line in the Mira variables TX Cam and IK Tau [@Marvel2005AJ....130..261M] barely resolve the emission distibutions in these objects. Since, first, those objects are closer to the Sun than our target stars and, second, our higher exciation lines most likely arise from more compact regions than the $1 - 0$ line, the column densities derived are strict lower limits.
Assuming that the lines are optically thin and that the excitation temperature, $T_{ex}$, between upper and lower level is such that $T_{ex}$$\gg$$T_{bg}$ (with $T_{bg}$ the temperature of any background source, e.g. 2.7K), the integration of the standard radiative transfer equation shows that [@Goldsmith1999ApJ...517..209G] $$N_u = \frac{1.67 \times 10^{14}}{\nu \ \mu^2}
\left(\int{T_{b}\,dv}\right)\,,
\label{EqNup}$$ with $N_u$ the column density of the upper transition state in cm$^{-2}$, $\mu$ the transition dipole moment in Debye, $\nu$ the transition frequency in GHz, $T_{mb}$ the main-beam temperature in Kelvin and $v$ the velocity in km/s. For rotational transitions, the transition moment for diatomic and polyatomic molecule is given by $$\mu^2(J+1,J) = \frac{J+1}{2J+3}\, \mu_e^2\,,$$ where $\mu_e$ is the permanent dipole moment of the molecule. Using the velocity–integrated intensities of Table \[Table\_overview\], we calculate the resulting upper state column densities (see Table \[Table\_results\]).
Assuming purely collisional excitation, and ignoring radiation trapping, one can derive from solving the statistical equilibrium equation that [@Tielens2005] $$\frac{N_l}{N_u} = \frac{g_l}{g_u}
\left(\frac{n_{\rm{crit}}}{n_{\rm{H_2}}}+1\right)\,
\exp(h\nu/kT_{\rm{kin}})\,,
\label{Eqcol}$$ where $N_u$, $N_l$, $g_u$, $g_l$ are the upper and lower state column densities and degeneracies, $T_{\rm{kin}}$ the kinetic temperature, $n_{\rm{crit}}$ the critical density, and $n_{\rm{H_2}}$ the hydrogen density. For a multi-level system, the critical density is given by [@Tielens2005] $$n_{\rm{crit}} = \frac{\sum_{l<u} A_{ul}}{\sum_{l \ne u}
\gamma_{ul}}\,,$$ where $A_{ul}$ represents the Einstein coefficient, and $\gamma_{ul}$ the collisional rate coefficients. Critical densities were calculated at differing temperatures using the data available in the [LAMDA]{}-database (see Table \[Table\_results\]). Applying Eq. (\[Eqcol\]) together with the column densities from Table \[Table\_results\], the minimum density requirements for $n_{\rm{H_2}}$ to achieve the observed number-density ratios are tabulated in Table \[Table\_NH2\]. Note that the listed values are at $T_{\rm{kin}}\,=\,300$K, and that for lower temperatures the critical density increases.
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
SiO (5-4) SiO (6-5) SiO (7-6) SiO (8-7) HCN(3-2) HCN(4-3) CS(5-4) CS(6-5) CS(7-6)
------------------- ------------------- ----------------------- ----------------------- ----------------------- ----------------------- ------------------------ ----------------------- ------------------------ ----------------------- ------------------------
$\int T_{mb}\,dv$ 9.2 11.1 10.4 9.9 8.9 2.4 \[0.8\]$^{\dagger}$ 1.3
\[0pt\][WX Psc]{} $N_u$ $7.96 \times 10^{12}$ $7.91 \times 10^{12}$ $5.15 \times 10^{12}$ $7.96 \times 10^{12}$ $4.22 \times $ 2.53 \times 10^{12}$ $1.08 \times 10^{12}$ $1.41
10^{12}$ \times 10^{12}$
$\int T_{mb}\,dv$ 7.1 8.9 5.2 21.7 12.2 *8.5*$^a$ 2.3 4.6
\[0pt\][W Aql]{} $N_u$ $6.19 \times 10^{12}$ $6.36 \times 10^{12}$ $2.57 \times 10^{12}$ $1.74 \times 10^{13}$ $5.81 \times 10^{12}$ $8.95 \times $3.03 \times 10^{12}$ $5.00 \times 10^{12}$
10^{12}$
$\int T_{mb}\,dv$ 19.62 *20.4*$^b$ *139.76*$^c$ 60.69 *32.34*$^c$ 20.37 35.24
\[0pt\][II Lup]{} $N_u$ $6.96 \times 10^{12}$ $8.74 \times 10^{12}$ $1.24 \times 10^{14}$ $2.88 \times 10^{13}$ $6.85 \times 10^{13}$ $2.61 \times 10^{13}$ $3.83 \times
10^{13}$
$\int T_{mb}\,dv$ 3.8 5.1 *8.4*$^d$ 25.7 *31.9*$^d$ 7.4
\[0pt\][V Cyg]{} $N_u$ $3.33 \times 10^{12}$ $3.51 $5.28 \times 10^{12}$ $2.06 \times 10^{13}$ $1.51 \times 10^{13}$ $1.41 \times 10^{13}$
\times 10^{12}$
at 40K $n_{\rm{crit}}$ $3.06 \times 10^{6}$ $5.26 \times 10^{6}$ $8.41 $1.27 \times 10^{7}$ $5.83 \times 10^{6}$ $1.32 $1.85 \times 10^{6}$ $3.22 \times 10^{6}$ $5.09
\times 10^{6}$ \times 10^{7}$ \times 10^{6}$
at 100K $n_{\rm{crit}}$ $2.00 \times 10^{6}$ $3.44 \times 10^{6}$ $5.57 $8.18 \times 10^{6}$ $4.22 \times 10^{6}$ $9.63 $1.35 \times 10^{6}$ $2.37 \times 10^{6}$ $3.76
\times 10^{6}$ \times 10^{6}$ \times 10^{6}$
at 300K $n_{\rm{crit}}$ $1.22 \times 10^{6}$ $2.10 \times 10^{6}$ $3.44 $5.12 \times 10^{6}$ $2.35 \times 10^{6}$ $5.65 $8.35 \times 10^{5}$ $1.48 \times 10^{6}$ $2.37
\times 10^{6}$ \times 10^{6}$ \times 10^{6}$
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
$^a$ ; $^b$ @Schoier2006ApJ...649..965S; $^c$ ; $^d$ @Bieging2000ApJ...543..897B $^{\dagger}$ line detected however with small S/N so that the integrated intensity is quite uncertain
The gas temperature, density and velocity structure was calculated in a self-consistent way using the GASTRoNOoM-code to determine where in the envelope the gas density falls below the density requirements given in Table \[Table\_NH2\] (see Fig. \[Figstructure\]). The assumed stellar parameters are listed in Table \[stellar\_parameters\], and the derived (maximum) radius of the emitting region is listed in Table \[Table\_NH2\]. If collisional excitation is assumed, it appears from Table \[Table\_NH2\] that the ‘parent’ molecules SiO, HCN and CS are excited in the inner ($\la 20$R$_*$) and intermediate ($\la 70$R$_*$ ) regions of the circumstellar envelope and trace regions after the dust condensation zone where they have been injected from deeper layers.
------------------------------------------------------------------------------------------------
SiO HCN CS
-------- ----------------------- -------------------- -------------------- ---------------------
$n_{\rm{H_2}}$ $ 5.0 \times 10^6$ $ 4.4 \times $9.0 \times 10^5$
10^6$
WX Psc R \[Eq. (\[Eqcol\])\] $ 17$R$_*$ $ 19$R$_*$ $ 33$R$_*$
R \[Eq. (\[Eqrad\])\] $ 4.5$R$_*$ $ 6$R$_*$ $ 7.5$R$_*$
$n_{\rm{H_2}}$ $ 2.1 \times 10^6$ $ 2.1 \times $ 6.34 \times 10^5$
10^6$
W Aql R \[Eq. (\[Eqcol\])\] $ 40$R$_*$ $ 40$R$_*$ $ 70$R$_*$
R \[Eq. (\[Eqrad\])\] $ 9$R$_*$ $ 10$R$_*$ $ 11$R$_*$
$n_{\rm{H_2}}$ $ 2.7 \times 10^7$ $ 1.3 \times $ 7.6 \times 10^5$
10^6$
II Lup R \[Eq. (\[Eqcol\])\] $ 13$R$_*$ $ 53$R$_*$ $ 70$R$_*$
R \[Eq. (\[Eqrad\])\] $ 2.5$R$_*$ $ 13$R$_*$ $
10$R$_*$
$n_{\rm{H_2}}$ $ 2.8 \times 10^7$ $ 8.7 \times 10^6$
V Cyg R \[Eq. (\[Eqcol\])\] $ 5$R$_*$ $ 9$R$_*$
R \[Eq. (\[Eqrad\])\] $ 4$R$_*$
------------------------------------------------------------------------------------------------
: For each target, the first row lists the minimum number density for $n_{\rm{H_2}}$ in cm$^{-3}$ as derived using Eq. (\[Eqcol\]) at $T_{\rm{kin}}\,=\,300$K and the second row gives the corresponding maximum radius for the emission regions obtained using the GASTRoNOoM-code. The third row gives the maximum radius for the emitting region calculated using Eq. (\[Eqrad\]).[]{data-label="Table_NH2"}
WX Psc W Aql II Lup V Cyg
--------------------------------------------- ------------------ --------- ---------- ----------
T$_*$ \[K\] 2000$^a$ 2800 2400$^f$ 1900$^f$
R$_*$ \[$10^{13}$cm\] 5.5 2.4 3.8 5.1
L$_*$ \[$10^3$L$_{\odot}$\] 10$^a$ 6.8$^b$ 8.8$^f$ 6.3$^f$
[\[CO\]]{}/\[H$_2$\] \[$10^{-4}$\] 3$^c$ 6$^c$ 8$^c$ 8$^c$
distance \[pc\] 833$^a$ 230$^d$ 500$^f$ 310$^f$
R$_{\rm{inner}}$ \[R$_*$\] 5$^a$ 8$^e$ 4$^f$ 2$^f$
$v_{\infty}$ \[km/s\] 18 17.5 21 10.5
$\dot{M}$ \[$10^{-6}$M$_{\odot}$yr$^{-1}$\] 6$^{a, \dagger}$ 2.5$^b$ 9$^f$ 1.2$^g$
: Stellar parameters used as input for the GASTRoNOoM-code. The terminal velocity, $v_{\infty}$, is derived from the CO lines; the stellar radius from the stellar luminosity and temperature. The envelope density falls off as $\sim r^{-2}$. Literature references are given in the footnote.[]{data-label="stellar_parameters"}
\
$^a$ @Decin2007; $^b$ ; $^c$ @Knapp1985ApJ...292..640K; $^d$ @Bieging2000ApJ...543..897B; $^e$ @Danchi1994AJ....107.1469D; $^f$ @Schoier2006ApJ...649..965S; $^g$ $^{\dagger}$ refers to the inner region (A) in @Decin2007
It is also of interest to consider the case where collisional excitation is ignored, and the molecules are excited by infrared radiation from the star. One can derive that [@Tielens2005] $$\frac{N_l}{N_u} = \frac{g_l}{g_u} \frac{\exp(h \nu /k T_*) -1}{W} +
1\,,
\label{Eqrad}$$ where $W$ is the geometrical dilution, being $(R_*/2R)^2$ when R$\gg$R$_*$. As seen from Table \[Table\_NH2\], radiative excitation constrains the emitting zone for SiO being $\la$9R$_*$, for HCN $\la$ 13R$_*$, and for CS $\la$11R$_*$. In general, these regions are smaller than derived in case of collisional excitation, indicating that the radiation field of the star does not have enough energy to sustain the excitation. An analogous expression as for Eq. (\[Eqrad\]) can be derived for a radiation field characterized by a dust temperature $T_d$ at the condensation radius $R_{\rm{inner}}$, with the dilution factor then being $(R_{\rm{inner}}/2R)^2$. Using the dust condensation radii listed in Table \[stellar\_parameters\] and a dust temperature of 800K the derived emitting regions are a factor 3.2 larger for , a factor 4.3 for , a factor 4 for , and a factor 2 for , resulting in similar radii as in case of collisional excitation.
Although the numbers in Table \[Table\_NH2\] can only be used as rough guidelines, they suggest in both cases a sequence in the excitation pattern, SiO being the species emitting the closest to the star, followed by HCN and CS.
Conclusions {#Conclusions}
===========
From the above analysis, one can draw the following conclusions:
1. The observations reported in this letter confirm the status of ‘parent’ molecules for CO, SiO, HCN, and CS in AGB stars, whose observed molecular lines form close to the star in support of the theoretical predictions of . The excitation analysis suggests that SiO is emitted closest to the star, followed by HCN and CS.
2. High-excitation lines of SiS are detected in all stars, implying that SiS too forms close to the star, whatever the stage of stellar evolution. However, a thorough line analysis is necessary to prove or disprove that SiS is more abundant in C stars than in O-rich Miras, as predicted by .
3. SO appears to be typical of O-rich AGBs only. With only two excitation lines detected, no information can be drawn on the locus of SO formation. However, the SO($10_{11}$-$10_{10}$) line has an estimated velocity of only 10km/s in when the wind terminal velocity for this object is 19.2km/s. This fact suggests that the line excitation occurs close to star. In any case, the inner chemistry of O-rich AGB envelopes appears to be as rich, if not richer, than that of C-rich stars.
A detailed line analysis of the present data coupled to further observational campaigns with JCMT and APEX are planned to corroborate these results.
LD and SD acknowledge financial support from the Fund for Scientific Research - Flanders (Belgium), IC acknowledges support from the Swiss National Funds for Science through a Marie-Heim-V[ö]{}gtlin Fellowship, and SH acknowledges financial support from the Interuniversity Attraction Pole of the Belgian Federal Science Policy P5/36. We thank Remo Tilanus (JCMT) for his support during the observations and reduction of the data.
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[^1]: *Postdoctoral Fellow of the Fund for Scientific Research, Flanders*
[^2]: *Scientific Researcher of the Fund for Scientific Research, Flanders*
[^3]: The James Clerk Maxwell Telescope (JCMT) is operated by The Joint Astronomy Centre on behalf of the Science and Technology Facilities Council of the United Kingdom, the Netherlands Organisation for Scientific Research, and the National Research Council of Canada. Program ID is m06bn03 (34h).
[^4]: APEX, the Atacama Pathfinder Experiment, is a collaboration between the Max-Planck-Institut fur Radioastronomie, the European Southern Observatory, and the Onsala Space Observatory. Program ID is ESO.078.D-0534 (44h).
|
---
abstract: 'Machine learning (ML) algorithms and machine learning based software systems implicitly or explicitly involve complex flow of information between various entities such as training data, feature space, validation set and results. Understanding the statistical distribution of such information and how they flow from one entity to another influence the operation and correctness of such systems, especially in large-scale applications that perform classification or prediction in real time. In this paper, we propose a visual approach to understand and analyze flow of information during model training and serving phases. We build the visualizations using a technique called Sankey Diagram - conventionally used to understand data flow among sets - to address various use cases of in a machine learning system. We demonstrate how the proposed technique, tweaked and twisted to suit a classification problem, can play a critical role in better understanding of the training data, the features, and the classifier performance. We also discuss how this technique enables diagnostic analysis of model predictions and comparative analysis of predictions from multiple classifiers. The proposed concept is illustrated with the example of categorization of millions of products in the e-commerce domain - a multi-class hierarchical classification problem.'
author:
- 'Abon Chaudhuri, Walmart Labs, Sunnyvale, CA, USA'
bibliography:
- 'template.bib'
title: A Visual Technique to Analyze Flow of Information in a Machine Learning System
---
Introduction {#sec:intro}
============
Machine learning (ML) algorithms and machine learning based software systems implicitly or explicitly involve complex flow of information between various entities such as training data, feature space, validation set and results. Each of them contains valuable information. The quality, quantity, and distribution of such information across different containers, and their flow from one to another influence the operation and the correctness of machine learning based systems. Certain algorithms such as probabilistic graphical models or deep neural networks explicitly rely on the flow of information. Employing statistical and visual methods to understand the distributions and the flow of information is critical to the success of large-scale data science applications.
Let us consider a real application - product categorization - a large-scale classification task commonly encountered in the e-commerce domain (for classifying commodities into thousands of categories) for example. The problem is to assign every product a category from a multi-level hierarchy of categories such as “home$\rightarrow$kitchen$\rightarrow$appliances$\rightarrow$microwave". To develop an accurate model (a classifier) for such a task, it is crucial to answer a number of key questions at every step - starting from data collection to feature engineering, training, and finally, evaluation. A few examples are: is every class well represented in the training data? Are there redundant features or collinearity among features? Does the evaluation strategy cover examples from all the classes? In addition to statistical analysis, the use of visual analytics to answer these questions effectively is becoming increasingly popular.
Going one step deeper, we observe that the flow of information across various entities can often be formulated as joint or conditional probability distributions. A few examples are: distribution of class labels in the training data, conditional distribution feature values given a label, comparison between distribution of classes in test and training data. Statistical measures such as mean and variance have well-known limitations in understanding distributions. On the other hand, visualization based techniques allow a human expert to analyze information at different levels of granularity. To give a simple example, a histogram can be used to examine different sub-ranges of a probability distribution. In this paper, we present how Sankey diagrams [^1] can represent probability distributions at various levels of detail. While this is not a new technique, we reinvent it as a visual encoding for joint and conditional distributions. Use of this particular technique along with supporting visualization techniques can lead to effective visualization-enhanced machine learning systems.
We also discuss the use of this technique in model comparison and diagnostics. For a given task, multiple models with different features and parameters are usually trained at the same time or over a period of time. When it comes to select the best one for large-scale use, the proposed technique allows a human expert to study relevant questions such as if they were trained on near identical data, if their performance varied significantly across certain categories - as opposed to relying only on the overall accuracy number.
Related Work {#sec_rel_work}
============
#### Visualization in Machine Learning:
Recently, both machine learning and visualization research communities have started to adopt techniques from each other, leading to publications and open source software systems for visually exploring different stages of a machine learning pipeline.
Alsallakh et al. [@visual1] propose to use multiple box plots to visualize the feature distribution of training samples and the ability of a feature to separate data into different classes. FeatureInsight [@featureinsight] is a system that combines human and machine intelligence to examine classification errors to identify and build potentially useful features. Infuse [@infuse] is another visual analytic system that allows the human analyst to visually compare and rank different features based on their usefulness in model building.
Another area of active research is understanding of machine learning and deep learning algorithms. Visual methods have been proposed to construct and understand classifiers using traditional machine learning algorithms such as Bayesian modeling [@bayes_vis], Decision Tree [@dctree_vis] and Support Vector Machine [@svm_vis]. Of late, visualization has turned out to be a candidate approach to understand deep learning models which are inherently harder to explain. A growing body of work [@nodelink_cnn; @hidden; @activis] explore different ways to understand the evolution of feature maps, activations, and gradients in deep neural nets. t-SNE [@tsne2008] a powerful technique for visual exploration of high-dimensional embeddings of data often produced by deep learning algorithms.
A number of recent works explore visual techniques to interpret the results of a machine learning algorithm. Augmented confusion matrices and confusion wheels [@visual1] can effectively highlight instances or classes that a classifier is more likely to classify incorrectly. Modeltracker [@modeltracker] presents an enhanced visual error analysis tool that complements and enhances the traditional measures for model performance.
Chen et al. [@vis_survey] conducts a design study of the usefulness of a number of visual techniques used in machine learning and presents a system called VizML that allows diagnostic analysis of machine learning models.
Besides academic research articles, resources such as tutorials and blogs [@mlvisblog] summarizing the visualization techniques commonly used in machine learning are widely available.
#### Visualization of Data Distributions:
Understanding the data distribution often holds the key to statistical modeling. Visualization is often more descriptive compared to summary statistics such as central and higher order moments. Histograms and box plots continue to be the most popular techniques to visualize discretized probability distributions. Violin plots have been used in certain cases. They are the popular choices when building more complex visualization software involving distribution data. Potter et al. [@distvis2] presents a comprehensive summary of the techniques used to visualize probability density functions and cumulative distribution functions.
Proposed Visualization Methodology {#sec_main_technique}
==================================
A machine learning algorithm tries to learn the parameters of a mathematical model from known examples of a dataset, and uses the same model to predict information about unknown examples of that dataset. Hence, maximizing the use of the information present in the known examples is critical for successful modeling. In this paper, we propose to consider the input data, the intermediate transformations of the data, and the output (the predictions) as sets containing information.
The training data contains a set of instances $X$ and a set of labels $Y$. Usually, all or some of the instances have labels. In a standard supervised classification task, many instances map to a label.
The training instances are often converted to features using some mathematical function. We denote the features obtained by applying a function $f$ on $X$ by $F=f(x)$. Usually, a large number of features is computed from the data. In the presence of $m$ features, $F_k=f_k(x)$ denotes the $k^{th}$ feature dimension where $k$ varies from 1 to $m$.
After training an algorithm, it is usually evaluated on a held-out set or evaluation set of instances. We denote that set as $R$. The true labels of these instances (denoted by $Y_g$) are already known, the predictions or the actual labels produced by the algorithm (denoted by $Y_p$) is compared against these ground truth information.
Table \[tbl:common\_sets\] lists the above mentioned entities or information containers. Technically speaking, we present them as multi-sets or bags so that the duplicates (example: the training labels repeat many times in a dataset). Also, given that the training and evaluation is inherently an iterative process, we add the notion of time or instance using a superscripted suffix $t$. Depending on the context, this suffix may denote a particular date or simply the iteration number.
Set Notation Size
---------------------------- ---------- ---------
Training Set $X^t$ $N^t$
Training Set Labels $Y^t$ $N^t$
$k^{th}$ Feature $F_k^t$ $N^t$
Evaluation Set $R^t$ $N_E^t$
Evaluation Set Labels $Y_g^t$ $N_R^t$
Evaluation Set Predictions $Y_p^t$ $N_E^t$
: Common Notations to Represent Information Containers in a Machine Learning System[]{data-label="tbl:common_sets"}
In a machine learning system, the relationships between such sets or multi-sets are often captured by conditional or joint probability distributions. We propose to visually capture these relationships using a technique called Sankey diagrams [^2] that was traditionally used to visualize flow from one set to another. We re-purpose this technique to meaningfully visualize flow of information some of which can be formulated as joint and conditional probability distributions.
Let us consider an example with two multi-sets, each containing 100 elements - $S = \{a: 90, b:10\}$ and $T = \{x:40, y:30, z:40\}$. In other words, $S$ is a bag of 90 instances of $a$ and 10 instances of $b$ and so on. Suppose, we are interested in the conditional distribution $P(T|S)$. More specifically, we would like to compare $P(T=y|S=a)$ (value=0.2) with $P(T=y|S=b)$ (value=1.0). The probability values hardly explain anything about the data. On the contrary, the Sankey diagram in Figure \[fig:sankey\_basic\] (follow the two purple flows) reveals much more about the data than the the probabilities. For example, it clearly shows that despite the higher value of $P(T=y|S=b)$, it includes only a narrow subset of elements since $P(S=b)$ is quite low. Rest of paper will present examples of such insights using Sankey diagrams.
![Sankey diagram as a visual expression of a conditional probability distribution. In this example, the purple lines highlight $P(T=y|S=a)$ (value=0.2) and $P(T=y|S=b)$ (value=1.0)[]{data-label="fig:sankey_basic"}](images/sankey_basic.pdf){width="0.75\linewidth"}
Application Scenario {#sec_application}
--------------------
The product catalog of an e-commerce company typically contains millions of products that need to be placed into categories structured as multi-level hierarchies (such as electronics$\rightarrow$appliances$\rightarrow$kitchen appliances$\rightarrow$microwave). An oversimplified example is shown in Figure \[fig\_taxonomy\]. Categorization ensures that the product is visible on the correct segment of the website and searchable by the users.
![A simplified illustration of a multi-level product category hierarchy.[]{data-label="fig_taxonomy"}](images/taxonomy.pdf){width="0.98\linewidth"}
From a machine learning point of view, this is a classic multi-class multi-label hierarchical classification problem. Typically, a training data of a few million products is used to train four to five level hierarchy of categories containing a few thousand leaf level categories. Product name, description, image and a number of other properties are leveraged to compute features for the model [@wmt_classification]. The trained model is evaluated using a controlled and sampled set of at least a few thousands products.
This application can benefit heavily from visual exploration of data because the scale and complexity of the data is enormous, the category hierarchy is deep and ever-changing. Finally, visual exploration leads to data cleaning or filtering actions, causing incremental yet quick improvements. The following sections uses this as the primary case study.
Visual Analysis of Training Data {#subsec_training_data}
--------------------------------
The training data consists of items and labels. In the context of product categorization, each instance of the training data contains product information such as title and other attributes, and category name which serves as the class label. Understanding the distribution of these labels in terms of quantity and quality is a key to building an accurate classifier.
### Distribution of Relative Label Quantity {#subsec_trdata_dist}
The training dataset should have adequate examples from each class for the algorithm to learn. Given a hierarchical training dataset, the quantity of data should be adequate at each level of the hierarchy. Also, the distribution of labeled data among the children (classes) of each class should be nearly uniform. Let us assume a class $C_l$ has three children $C^1_{l+1}$, $C^2_{l+1}$, and $C^3_{l+1}$. Even if $C_l$ has enough training instances, if most of it comes from one of the child classes, the model may become biased towards that class and tend to mis-classify the other two classes to that one.
To give an example from our application, it is not enough to have sufficient training examples for a category called “Camera”. If “Camera” has two finer sub-categories “DSLR” and “Point-n-shoot”, the total number of labels should be near equally distributed between these two. Otherwise, the classifier will tend to classify most cameras to the dominant sub-category.
The absolute quantity of labels at each node can be visualized in many ways. To capture the relative distribution among the child classes, we propose to use Sankey diagrams. One possible implementation is to determine the width of the Sankey diagram’s $i^{th}$ right side node based on $\frac{N_i}{\sum{N_k}}$ where $k$ iterates over all the child nodes in consideration. Figure \[fig\_tr\_balanced\] presents a near uniform label distribution among the child classes under “Baby”. On the contrary, Figure \[fig\_tr\_unbalanced\] shows an example where certain classes under “Camping” do not have enough training examples.
We extend this idea to visually inspect an entire path of the hierarchy using multi-level flow diagrams (Figure \[fig\_sankey\_trflow\]). This is reveal imbalance at inner levels. For example, even though “DSLR” and “Point-n-shoot” both have sufficient training data, “Cameras” itself may have low relative quantity compared to other electronics categories.
[\[fig\_tr\_balanced\]![Use of Sankey diagram to visualize the relative distribution of training labels among finer sub-categories[]{data-label="fig_tr_label_dist"}](images/trdata_sankey_balanced.pdf "fig:"){width="0.45\linewidth"}]{}
[\[fig\_tr\_unbalanced\]![Use of Sankey diagram to visualize the relative distribution of training labels among finer sub-categories[]{data-label="fig_tr_label_dist"}](images/trdata_sankey_unbalanced.pdf "fig:"){width="0.45\linewidth"}]{}
![Use of multi-level Sankey diagram to show the distribution of labels from top to level of the class hierarchy. “Tripod” and “Flash memory” is sparsely populated categories compared to “Camcorders” and “Lenses”. The prefix of the label indicates the level at which the category is located in the hierarchy. For example, “Cameras” is at the intermediate level where “Lenses” is at the leaf level.[]{data-label="fig_sankey_trflow"}](images/trdata_relative_flow.pdf){width="\linewidth"}
[\[fig:labeldist\_1\]![Use of Sankey diagram to visualize the distribution of labelS collected from different sources and the distribution of positive and negative labels.[]{data-label="fig_tr_label_dist"}](images/source_dist_balanced.pdf "fig:"){width="0.45\linewidth"}]{}
[\[fig:labeldist\_2\]![Use of Sankey diagram to visualize the distribution of labelS collected from different sources and the distribution of positive and negative labels.[]{data-label="fig_tr_label_dist"}](images/source_dist_unbalanced.pdf "fig:"){width="0.45\linewidth"}]{}
### Distribution of Label Quality {#subsec_trdata_dist}
Training instances in real applications are not curated. They are often noisy and incorrect, demanding quality assessment and control before they are used to train an algorithm. They often come from variety of sources marked with accuracy levels, making the inspection of quality relatively easier. For example, crowd-sourced labels are usually less trusted than labels annotated by trained experts. Also, labels may become outdated over time. In case of our e-commerce application, this happens because products are continuously incorporated into and taken off the catalog.
It is common to have a training dataset with both positive and negative examples. For our product categorization example, the dataset often has labels that indicate that a product does not fall into a category. For example, “this item is not a DSLR camera”. These type of labels are usually generated by curating the predictions of an existing model by human experts. If the training algorithm cannot leverage negative labels, then distribution biased towards positive labels is welcome. Otherwise, a balanced distribution is favorable.
We inspect the quality of training labels at least from two aspects: decision type (positive or negative) and source. For each category, we use Sankey diagram to inspect the distribution of labels from various sources and for each source, various decisions. The goal here is two-fold:
- to identify the classes for which the distribution is well-balanced: High accuracy is expected off these classes. If not, further investigation is recommended post evaluation. An example is shown in Figure \[fig:labeldist\_1\].
- to identify the classes for which the distribution has high proportion of less reliable source(s). These classes require immediate attention. An example is shown in Figure \[fig:labeldist\_2\].
Visual Analysis of Features {#subsec_feature}
---------------------------
In a typical modeling problem, hundreds, if not thousands, of features are extracted from the training dataset. To prevent over-fitting of the model and to make the best use of computation power and storage, it is important to train only on features that are important. However, the notion of importance is not well-defined. Some statistical measures such as Welch’s statistic can be used to see if a feature has the ability to separate data into classes. We propose that visual methods provide certain types of insight that cannot be derived from statistical methods.
### Feature Importance Estimation {#sec_feature_imp}
**Numeric Features:** In general, a feature can help in classifying data if its presence is distinctive in different classes. For example, if the range of a numeric feature over class $A$ do not overlap with the range of the same feature values for class $B$, this feature can help differentiate the two classes. Hence, visualizing the distributions of a numeric feature for different categories can be immensely helpful. Our product categorization example relies heavily on product titles. Length of product title is an important numeric feature. We propose to examine the distribution of the length of product titles for different categories using violin plots (Figure \[fig:trdata\_violin\]).
![Comparative visualization of the distributions of lengths of product titles from a few semantically close classes. For example, *Notebook batteries* and *Notebook cases* have almost identical distributions. The title length is not a good feature between these two classes.[]{data-label="fig:trdata_violin"}](images/trdata_violin_1.pdf){width="\linewidth"}
**Categorical Features:** Categorical features assume a number of discrete values. Let us begin with the common scenario of involving the feature and the label spaces in a two-class classification problem. Let us denote the class labels by $y_1$ and $y_2$. $f_1$ and $f_2$ are two possible values of a categorical feature $F$. How the $f_i$s are related to $c_j$s for all $i,j$ indicates if $F$ will be an important contributor to a classification model for this problem. For example, If $n(f_1 \rightarrow y_1) \sim n(f_1 \rightarrow y_2)$ and $n(f_2 \rightarrow y_1) \sim n(f_2 \rightarrow y_2)$, then we can infer that $F$ does not contain a strong signal to distinguish between the two classes. Figure \[fig:sankey\_nr\] shows how Sankey diagram can be used to capture this. An example contrary to this is shown in Figure \[fig:sankey\_rel\] where the feature is potentially useful for classification.
[\[fig:sankey\_nr\]![Use of Sankey diagram to explore feature-label relationship[]{data-label="fig:example_1"}](images/sankey_example_1.pdf "fig:"){width="0.45\linewidth"}]{}
[\[fig:sankey\_rel\]![Use of Sankey diagram to explore feature-label relationship[]{data-label="fig:example_1"}](images/sankey_example_2.pdf "fig:"){width="0.45\linewidth"}]{}
We propose to use Sankey diagrams to study the distribution of the labels for one or more values of a categorical feature. In this particular section, we deviate from our running example of product categorization dataset and use a publicly available dataset related to the publicly available movie genre classification problem [^3]. We study two potential features: “Director" and “Color" (takes two values: “Color" or “B&W") are two features. Sankey diagrams in Figures \[fig:director1\] and \[fig:director2\] clearly highlight that the distribution of genre vary considerably from director to director, which makes it a useful feature for classifying genre. On the other feature, Figures \[fig:color\] and \[fig:bw\] show that the distribution of genre remains largely unchanged regardless of the value of “Color”.
[\[fig:director1\]![Use of Sankey diagram to demonstrate relative feature importance for a movie genre classification problem using IMDB dataset. **Top Row:** The distribution of genres (labels) vary significantly for different values of “Director” - a categorical feature. **Bottom Row:** On the contrary, the label distributions are very similar for color and B&W - two values of “Color”.[]{data-label="fig:use_case_1"}](images/feature_to_label_3.pdf "fig:"){width="0.4\linewidth"}]{}
[ \[fig:director2\]![Use of Sankey diagram to demonstrate relative feature importance for a movie genre classification problem using IMDB dataset. **Top Row:** The distribution of genres (labels) vary significantly for different values of “Director” - a categorical feature. **Bottom Row:** On the contrary, the label distributions are very similar for color and B&W - two values of “Color”.[]{data-label="fig:use_case_1"}](images/feature_to_label_4.pdf "fig:"){width="0.4\linewidth"}]{}
[ \[fig:color\]![Use of Sankey diagram to demonstrate relative feature importance for a movie genre classification problem using IMDB dataset. **Top Row:** The distribution of genres (labels) vary significantly for different values of “Director” - a categorical feature. **Bottom Row:** On the contrary, the label distributions are very similar for color and B&W - two values of “Color”.[]{data-label="fig:use_case_1"}](images/feature_to_label_1.pdf "fig:"){width="0.45\linewidth"}]{}
[\[fig:bw\]![Use of Sankey diagram to demonstrate relative feature importance for a movie genre classification problem using IMDB dataset. **Top Row:** The distribution of genres (labels) vary significantly for different values of “Director” - a categorical feature. **Bottom Row:** On the contrary, the label distributions are very similar for color and B&W - two values of “Color”.[]{data-label="fig:use_case_1"}](images/feature_to_label_2.pdf "fig:"){width="0.45\linewidth"}]{}
**Text Features:** Many classification problems, including our product categorization use case, is largely driven by text data such as product name and description. Text data is usually converted into appropriate numeric or categorical features using suitable methods such as tf-idf or word2vec. After that, the relevance distributions of prominent keywords across categories can be visually investigated.
Visual Analysis of Results {#subsec_results}
--------------------------
### Overview of Model Evaluation {#sec_model_eval}
![**Visual model diagnostics:** Visual comparison between the predicted outcomes of the classifier and the actual true labels.[]{data-label="fig:model_diagnosis"}](images/error_analysis.pdf){width="\linewidth"}
Once trained, a model is evaluated on an evaluation set that consists of a relatively small yet representative set of items with known labels. The known labels that serve as ground truth in the evaluation are collected using human experts or crowd-sourcing. The model prediction for each item in the evaluation set is compared against the ground truth. The cost of each mistake is accumulated based on a distribution of importance of the items in the evaluation set. Details of the evaluation process is beyond the scope of this paper and can be found here [^4].
Typically, a model evaluation produces accuracy (or similar metrics) and sample size per category (Table \[tbl:accu\_report\]). A list of mis-classified products per category is also generated, allowing fine-grained analysis (Table \[tbl:err\_report\]).
Category Sample Size Accuracy
------------------------ ------------- ----------
POP EASY LISTENING 5 0.82
POP LOUNGE 8 0.92
POP ADULT CONTEMPORARY 3 0.96
... ... ...
: A Segment of the Accuracy Report of a Model[]{data-label="tbl:accu_report"}
ID True Category Predicted Category
----- --------------- --------------------
A1 Workwear Sleepwear
B23 Workwear Men’s Jumpsuit
C98 Movie TV Show
... ... ...
: A Segment of the Misclassification Report of a Model[]{data-label="tbl:err_report"}
### Visual Error Analysis {#sec_err_report}
The class-level accuracy is relatively straightforward to visualize. A node-link visualization color-coded with class accuracies is used to represent the performance of the classifier on the different classes.
In this paper, we emphasize on visualizing the misclassification report because it has the potential to guide the human analyst to the categories where the model consistently performs poorly. Repeated mis-classifications of similar items often indicates either a problem with the category definition or with the data.
{width="0.65\linewidth"} \[fig:accu\_trend\]
[\[fig:multi\_model\_1\]{width="0.4\linewidth"}]{}
[\[fig:multi\_model\_2\]{width="0.41\linewidth"}]{}
We propose to draw flow diagrams between the bag of predicted labels and the bag of true labels for the mis-classified items for each category. Figure \[fig:model\_diagnosis\] shows various interesting observations about the model performance. In may be noted that majority the items in a category called “Wall Decor” are consistently misclassified to another called “Art and Wall Decor” (region A1). The diagram also reveals that the misclassification is happening in both directions - a sizable portion of “Art and Wall Decor” items are being classified into “Wall Decor” (region A2). The diagnosis for the problem is that the category definitions are too close. Another example is notable where items in various clothing categories are mis-classified into “KnitTops”, indicating the “KnitTops” category is too broad (region B). Also, the classifier does not have enough information to perform finer categorization. The visualization reveals more obvious mistakes such as “MensCasualDress” being mis-categorized in “LadiesCasuals” indicates that the model is being trained on wrongly labeled data (region C).
{width="0.8\linewidth"}
Comparative Analysis of Multiple Models {#subsec_results_multi_models}
---------------------------------------
Machine learning models are often improved incrementally. It is a common practice to train multiple models simultaneously, or over a period of time for the same problem by varying one or more of the following: training data, features, hyper-parameters, and algorithm. In the context of e-commerce, product data come with two strong textual sources of information: product title and product description. Product image is another rich source that can be leveraged. Hence, it is worth experimenting if product categorization improves if both title and description are used as opposed to only title. Transitioning to a deep learning approach is another step that requires extensive comparison with the current model.
When comparing two or more models, the overall accuracy number does not provide enough information for comparison, especially when the class structure is large. Models need to be analyzed and compared at class level and at instance level. Visual methods are ideal for such detailed analysis.
Our case study is based on three models $M_0$, $M_1$, and $M_2$ where $M_0$ and $M_1$ differ in the feature set, $M_2$ uses a deep neural network as opposed to the first two that are logistic regression based.
The first step is to compare class-level accuracies. We choose to use line charts for each class to highlight the trend of accuracy change (Figure \[fig:accu\_trend\]). Accuracy trends of multiple classes are observed together by presenting these line charts together as small multiple plots. The line charts are color coded based on the type of the trend to immediately draw attention to classes that require human attention. An example in Figure \[fig:accu\_trend\] is “HomeHardware” that is shown in red because of its strictly decreasing accuracy.
The accuracy numbers at class level can be misleading. For example, the accuracy of a class can stay more or less same while the items misclassified by the model for that class can change significantly. Hence, with a new model in hand, it is important to study the major changes in predicted labels across various classes. A model that brings forward drastic changes to large groups of items often needs more scrutiny before deployment. However, given that the evaluation set often contains hundreds of items per category, it is impossible for a human analyst to go over each individual mis-prediction or change of prediction. We propose to use a visualization that aims to guide the analyst to the potentially interesting subsets of the evaluation set.
Figure \[fig:multi\_model\_1\] and Figure \[fig:multi\_model\_2\] shows that Sankey diagrams can be concatenated to capture if the predictions of a large bag of items in the evaluation set change as we switch from one model to another. Items from models $M_1$, $M_2$, and $M_3$ are plotted from left to right. If $M_2$ is able to assign finer sub-categories for to certain items that were in a broader category in $M_0$, the diagram is able to highlight that as a splitting flow. The split of “KnitTops” in Figure \[fig:multi\_model\_1\] into three categories is one such example. On the other hand, Figure \[fig:multi\_model\_2\] shows that a large group of items classified as “ArtAndWallDecor” by $M_1$ joined a broader category in $M_2$. Such changes often demand further investigation of individual instances by a human expert.
Application in a Machine Learning System {#sec_application}
========================================
This section outlines how the proposed techniques are currently in use in accordance with machine learning algorithms, and how it can be extended further.
Implementation Details {#sec:impl}
----------------------
At present, the visualization modules are implemented in Javascript using Google Charts API[^5] (for Sankey diagrams), d3.js[^6] (for circular node-link layout), and Plotly[^7] (for small multiple plots). The data scientists often use Python with Jupyter for rapid prototyping. The JS-based visualization modules run based on data produced by the Python experiments and present the visualizations on stand-alone web-based interfaces.
To enable the data scientists to iterate between model building and visualizing even faster, a Python version of Plotly, along with a widget for Sankey diagrams [^8], is used to integrate the visualizations directly within the Jupyter workflow.
Outline of Interactive System {#sec:vis_components}
-----------------------------
The system currently has three modules: one for the exploring the training data, one for exploring the classifier performance, and one for comparing multiple classifiers.
The interface for training data exploration consists of three views - a node-link style network visualization of the training data with radial layout, a Sankey diagram visualizer for a group of nodes, and another Sankey diagram visualizer for the label distribution of a single node. Figure \[fig\_visp\_1\] shows an example flow of exploration using these views. In this case, the user first explores that there are relatively large number of labels for “Camcorders” as opposed to some other categories that have very few such as “Batteries” and “Drones”. However, further exploration with the bottom right view reveals that most of the labels for “Camcorders Traditional” came from rules and none came from trained experts. This is a clear signal for requesting more labels for this category.
This design suits our application where the product catalog is a large hierarchy, and it should generalize well to most classification problems that are inherently tied to a hierarchical class structure. The node-link visualization serves as the main overview driving the system. Class-specific properties such as sample size and classification accuracy can be encoded as node color, size etc so that the user can make informed selection of nodes.
The interface for exploring the model results contains two views: a node-link visualization for showing the class-wise accuracies, and a Sankey diagram highlighting the relationship between predictions and ground truth labels.
The interface for exploring multiple models contains two views: a collection of trend charts arranged in a grid layout where each chart reflects the accuracy trend of a class. An accompanying Sankey diagram presents the major prediction changes.
Conclusion and Future Work {#sec_conc}
==========================
In this paper, we present a set of techniques to enhance machine learning driven classification systems with visualization and interactive data analysis. The visualizations are designed around a central theme of understanding the flow of information across different entities such as training set, features, and results. In particular, we explore the potential of using flow diagrams, namely Sankey diagram, to capture the flow of information in a machine learning system. Other types of flow diagrams such as chord diagram can be applied as well depending on the application. The examples and datasets in the paper are derived from a real application: large-scale product categorization in e-commerce. The proposed technique can benefit similar hierarchical classification systems in other domains as well. Also, the proposed technique is not tied to any particular machine learning algorithm.
We are in the process of improving our system in many ways. We plan to integrate the web-based visual interfaces into a central system. At present, most of the data selection operations are driven from the Python scripts running at the back end. We plan to add more interactive capability to the front end. We also plan to allow the user to trigger a data filtering task or a model training task directly from the interface. We have received positive feedback about the system from various corners. However, we plan to conduct more formal experiments and user studies to quantify the benefit of such a system. We also plan to extend this system for understanding other machine learning driven systems, especially the deep learning based systems.
[^1]: https://developers.google.com/chart/interactive/docs/gallery/sankey
[^2]: http://www.sankey-diagrams.com
[^3]: https://www.kaggle.com/orgesleka/imdbmovies
[^4]: https://www.youtube.com/watch?v=wkky7k4scbQ
[^5]: https://developers.google.com/chart/
[^6]: https://d3js.org/
[^7]: https://plot.ly/
[^8]: https://github.com/ricklupton/ipysankeywidget
|
---
abstract: 'We propose a modification of the classical Black-Derman-Toy (BDT) interest rate tree model, which includes the possibility of a jump with small probability at each step to a practically zero interest rate. The corresponding BDT algorithms are consequently modified to calibrate the tree containing the zero interest rate scenarios. This modification is motivated by the recent 2008–2009 crisis in the United States and it quantifies the risk of a future crises in bond prices and derivatives. The proposed model is useful to price derivatives. This exercise also provides a tool to calibrate the probability of this event. A comparison of option prices and implied volatilities on US Treasury bonds computed with both the proposed and the classical tree model is provided, in six different scenarios along the different periods comprising the years 2002–2017.'
author:
- 'Grzegorz Krzy[ż]{}anowski[^1], Ernesto Mordecki[^2], Andrés Sosa[^3]'
title: 'A zero interest rate Black-Derman-Toy model'
---
=1
Introduction
============
The Federal Funds Rate (i.e., the interest rate at which depository institutions lend reserve balances to other depository institutions overnight on an uncollateralized basis) is an important benchmark in financial markets. This interest rate affects monetary and financial conditions which influence certain aspects of the general economy in the United States, such as employment, growth, inflation and term structure interest rates.
Following the 2007–2008 financial crisis in United States, the Federal Reserve reduced the Fed Funds Rate by 425 basis points to practically zero (targeting interest rates in the interval $0$-$0.25\%$) in one year. This decision was preserved for nine years and was called the Zero Interest Rate Policy (ZIRP policy). Figure \[figure:fed\] shows the evolution of the Federal Funds Rate between the years 2002 and 2017.
Motivated by this phenomena, and inspired by Lewis’s (2016) ZIRP models in continuous time, and by using the default models of Duffie and Singleton (1999), we propose a modification of the classical Black-Derman-Toy (BDT) model on interest rates (Black, Derman & Toy, 1990).
![Federal Fund Rate (2002-2017).[]{data-label="figure:fed"}](fed.png){width="13.5cm" height="9.0cm"}
Different approaches to model the ZIRP
--------------------------------------
Recently, several approaches to model the ZIRP have appeared. Lewis (2016) makes two proposals, which he summarizes as: (i) slowly-reflecting boundaries, also known as sticky boundaries; and (ii) jump-returns from a boundary. The first model consists in the utilization of a resource used in diffusions considered as Markovian processes, consisting of the introduction of sticky points. The sticky point retains the process for a longer time than the other points. To produce this phenomena, in the continuous time model, an atomic point is introduced in the speed measure of the diffusion (Borodin & Salminen, 2002). The second model consists of the introduction of a delayed start of the process. This delay time is modeled by an exponential random variable. The process stays at the $x=0$ level until this exponential time. It then jumps to an independent state, from which it continues its dynamics as a diffusion. The bond prices for these models are given in (Lewis, 2016).
An alternative approach was proposed by Tian and Zhang (2018). These authors depart from the classical CIR process (Cox, Ingresoll & Ross, 1985), and add one skew point at a certain relatively small level of the interest rate. The skew phenomena in diffusion models represents a permeable barrier. When the process reaches the skew point, the probability of upwards and downwards movements is modified according to a certain probability. In this way, if the probability of downwards continuation is higher that 1/2, as the CIR process never reaches zero, then the proposed process remains below the skew point for a longer time than the CIR process. The skew diffusions can be constructed by departing from the excursion theory for diffusions, and in many other ways (Lejay, 2006). The discrete analogue of this model is a binary random walk with symmetric probabilities at all states with the exception of one—the skew point. At this point there is a higher probability of going downwards. This produces a process that stays longer below the critical threshold than the original. It also can be seen that the weak limit of this process, properly normalized, goes to a skew diffusion (Lejay, 2006). In the paper (Tian & Zhang, 2018), based on stochastic calculus arguments, the authors give bond prices for this model.
Another approach to model the ZIRP phenomena was introduced by Eberlein et al.(2018). This proposal is in the context of Lévy modeling of Libor rates, and the modification allows negative interest rates. This model is especially suited for calibration in the presence of extremely low rates, it is presented in the framework of the semimartingale theory, and includes derivatives pricing, particularly caplets. As an application, European caplets market prices are used to calibrate the proposed model, with the help of Normal inverse Gaussian Lévy processes.
Martin (2018) made an alternative proposal, which considers that the financial crisis changes the modeling perspective of the term structure. The main reason is that there are differences between interest rates that were previously linked. Therefore, the proposal is to use several interest rate curves in the same model, which reflect the different types of risk observed in the fixed income markets. The paradigm of the valuation that the authors use is based on intensity models. The dynamics of the term structure is given by exponential affine factor models. The hazard rate incorporates the risk observed in the interbank sector that affects the corresponding interest rate. The author states that the approach is important for long-term assets, such as swaps and swaptions.
Our proposal
------------
In view of the need of adequate models to the ZIRP, we propose to depart from the Black-Derman-Toy (BDT) binary tree model, incorporating into its dynamics the possibility of a downwards jump with a small probability at each time step to a practically zero interest rate value. Additionally, we assume that once the process reaches the zero interest rate zone, it remains there with high probability. This proposal mimics the intensity approach in default bond models proposed by Duffie and Singleton (1999), by jumping to near zero according to a geometric random variable with a small rate. In addition, the sticky phenomena described by Lewis (2016), as the interest rate process, once this jump is realized, stays with high probability in this close to zero zone. In practical terms, the initial BDT binary tree model is modified to a mixed binary-ternary tree model to find consistent interest rates with the market term structure. The new model is called the ZBDT (for Zero interest rate Black-Derman-Toy).
The rest of this paper is structured as follows. In Section \[section:bdt\], we introduce the main ideas of the classical BDT model with an emphasis on calibration, with the aim of introducing the ZBDT model in Section \[section:zbdt\], together with its respective calibration equations. Section \[section:empirical\] has an empirical content. It contains a detailed account of critical financial events during the period of study (2002–2017) in the United States, which provides information about interest rates with their respective volatilities, and we choose six representative different scenarios to compare the results given by the BDT and the ZBDT models. In Section \[section:conclusions\], we conclude with a brief discussion of the results and comments on some possible future work.
The Black-Derman-Toy model {#section:bdt}
==========================
The Black-Derman-Toy model (Black et al. 1990) is one of the most popular and celebrated models in fixed income interest rate theory. It consists in a binary tree with equiprobable transitions, which makes it simple and flexible to use. More precisely, the model departs from the current interest rate curve, from where the yields for different maturities are extracted, and it uses a series of consecutive historical interest rate curves during a certain time interval to compute this yield volatilities. The model assumes that the volatility only depends on time and not on the value of the interest rate. A calibration procedure is implemented to obtain the interest rates acting during the respective time intervals defined in the model.
The model assumes that the future interest rates evolve randomly in a binomial tree with two scenarios at each node, labeled, respectively, by $u$ (for “up") and $d$ (for “down"), with the particularity that an $u$ followed by a $d$ take us to the same value as a $d$ followed by an $u$. In this way, after $n$ periods, we have $n+1$ possible states for our stochastic process modeling the interest rate. With the aim of simplifying the presentation, we consider that one period is equivalent to one year. The corresponding modification to shorter periods is straightforward. In Figure \[figure:bdtp\], we present the tree corresponding to the prices of a zero-coupon bond with expiration in $n=3$ years, where we denote by $P_{ij}$ the zero coupon bond price corresponding to the period $i$ and state $j$ for the same values of $i$ and $j$. Additionally, $P_{nj}=100$ for $j=1,\dots,n+1$.
= \[circle, fill=gray!0,draw\] (11) at (0,0) [$P_{0,1}$]{}; (21) at (3,0) [$P_{1,1}$]{}; (22) at (3,2) [$P_{1,2}$]{}; (31) at (6,0) [$P_{2,1}$]{}; (32) at (6,2) [$P_{2,2}$]{}; (33) at (6,4) [$P_{2,3}$]{}; (41) at (9,0) [$100$]{}; (42) at (9,2) [$100$]{}; (43) at (9,4) [$100$]{}; (44) at (9,6) [$100$]{}; = \[circle, fill=gray!40\] (1121) at (1.5,0) [$\frac12$]{}; (1122) at (1.5,1) [$\frac12$]{}; (2131) at (4.5,0) [$\frac12$]{}; (2132) at (4.5,1) [$\frac12$]{}; (2232) at (4.5,2) [$\frac12$]{}; (2233) at (4.5,3) [$\frac12$]{}; (3141) at (7.5,0) [$\frac12$]{}; (3142) at (7.5,1) [$\frac12$]{}; (3242) at (7.5,2) [$\frac12$]{}; (3243) at (7.5,3) [$\frac12$]{}; (3343) at (7.5,4) [$\frac12$]{}; (3344) at (7.5,5) [$\frac12$]{}; /in [11/1121,11/1122]{} () – (); /in [1121/21,1122/22]{} () – (); /in [21/2131,21/2132,22/2232,22/2233]{} () – (); /in [2131/31,2132/32,2232/32,2233/33]{} () – (); /in [31/3141,31/3142,32/3242,32/3243,33/3343,33/3344]{} () – (); /in [3141/41,3142/42,3242/42,3243/43,3343/43,3344/44]{} () – ();
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The evolution of this bond is associated to a tree with the interest rates that apply to each time period, as shown in Figure \[figure:bdtr\]. In the BDT model the probability of each $u$ or $d$ scenario at each node is $1/2$, the evolutions are independent, and the values of the interest rates are obtained through calibration.
Calibration of the BDT model
----------------------------
In a model with $n$ time periods, we calibrate a tree of order $n$ departing from the following data: the yields on zero coupon bonds $y(k),\ k=1,\dots,n$, corresponding to the respective periods $[0,k]$ (the first $k$ periods), and the yield volatilities for the same bonds $\beta(k),\ k=2,\dots,n$, under the same convention.
The interest rates of the tree, are $\{r_{i,j}\colon i=0,\dots,n-1; j=1,\dots,i+1\}$, and correspond to each time period in the up and down scenarios, giving $n(n+1)/2$ unknowns to be calibrated.
The first step uses only $y(1)$ and concludes that $r_{0,1}=y(1)$: $$\begin{aligned}
P_{1,1}&=P_{1,2}=100,\\
P_{0,1}&=\frac{100}{1+y(1)}=\frac12\,\frac{1}{1+r_{0,1}}\left(P_{1,1}+P_{1,2}\right).\end{aligned}$$ When $n>1$, we introduce the yields $y_u$ (up) and $y_d$ (down) one year from now, corresponding to prices $P_u$ and $P_d$. The relevant relations that this quantities satisfy are $$P_u=\frac1{(1+y_u)^{n-1}},\quad P_d=\frac1{(1+y_d)^{n-1}}.$$
### Variance equation at a node {#variance-equation-at-a-node .unnumbered}
Consider a tree with $n$ steps. We introduce a random variable $Y$ that takes two values: $$Y=
\begin{cases}
y_u,&\text{ with probability $1/2$},\\
y_d, &\text{ with probability $1/2$}
\end{cases}$$ Then, $\log Y$ has a variance $
{{\operatorname{\mathbf{var}}}}\log Y=\beta^2(n),\text{ if and only if } y_u=y_de^{2\beta(n)},
$ equivalent to $$\label{eq:var}
\beta(n)=\frac12\log{y_u\over y_d},$$ as follows from the following computation: $$\begin{gathered}
{{\operatorname{\mathbf{var}}}}\log Y=\frac{1}{2}\log^2y_u+\frac{1}{2}\log^2y_d-\left(\frac{1}{2}(\log y_u+\log y_d)\right)^2\\
=\left(\frac{1}{2}(\log y_u -\log y_d)\right)^2=\left(\frac{1}{2}\log\frac{y_u}{y_d}\right)^2
=\beta(n)^2.\end{gathered}$$ The BDT model assumes that the variance of the log-interest rate with fixed time is constant for all nodes. The respective interest rates at each node at time $n-1$ are represented by an auxiliar random variable $R_{n-2,j}$. $$R_{n-2,j}=
\begin{cases}
r_{n-1,j+1},&\text{ with probability $1/2$},\\
r_{n-1,j}, &\text{ with probability $1/2$},\\
\end{cases}$$ for $j=1,\dots,n-1$. The variance of this random variable is assumed to be constant for all nodes at the same time period, and satisfies $$\label{eq:sigma}
\sigma(n)=\frac12\log{r_{n-1,j+1}\over r_{n-1,j}},\quad j=1,\dots,n-1.$$ In the second step of the calibration, the new data are $y(2)$ and $\beta(2)$. The unknowns are $r_{1,1},r_{1,2}$ and $\sigma(2)$. In this case $\sigma(2)=\beta(2)$, because the local variation of the interest rate for one year coincides with the global variation. Accordingly, $y_u=r_{1,2}$ and $y_d=r_{1,1}$. The bond prices then satisfy $$\begin{aligned}
P_{0,1}&=\frac{100}{{(1+y(2))}^2}=\frac12\frac{1}{(1+r_{0,1})}\left(P_{u}+P_{d}\right),\\
P_{2,j}&=100, j=1,2,3,\\
P_{u}&=\frac{100}{{(1+y_u)}},\quad P_{d}=\frac{100}{{(1+y_d)}},\\
\beta(2)&=\frac12\log{y_u\over y_d}.\\\end{aligned}$$ For general $n$ the new data are $y(n)$ and $\beta(n)$. The unknowns are $r_{n-1,j}$ for $j=1\dots,n$ and $\sigma(n)$. The value $\sigma(n)^2$ is the variance of the interest rate at each node (see ). The bond prices then satisfy
$$\begin{aligned}
\frac{100}{{(1+y(n))}^{n}}&=\frac12\frac{1}{(1+r_{0,1})}\left(P_{u}+P_{d}\right),\\
P_{n,j}&=100, \ j=1,\dots,(n+1),\\
P_{i,j}&=\frac12\frac{1}{(1+r_{i,j})}\left({P_{i+1,j}+P_{i+1,j+1}}\right),\ i=0,\dots,(n-1), \ j=1,\dots,i+1,\\
P_{u}&=\frac{100}{{(1+y_u)}^{n-1}},\quad P_{d}=\frac{100}{{(1+y_d)}^{n-1}},\\
\beta(n)&=\frac12\log{y_u\over y_d},\\
\sigma(n)&=\frac12\log{r_{n-1,j}\over r_{n-1,j-1}}, \ \ j=2,\dots,n.\end{aligned}$$
The ZBDT model {#section:zbdt}
==============
Our modification of the classical BDT interest rate tree model adds to the dynamics the possibility of a downwards jump with a small probability at each time step to a practically zero interest rate, where, after its arrival, the process remains with high probability. More precisely, in the new model, the nodes labeled $(i,j)$ with $j\geq 2$ have the same characteristics as in the BDT model (up and down probabilities $1/2$, and jump to values to be calibrated). In addition, the nodes of the form $(1,j)$ add a third possible downwards jump with a small probability $p$ and the other two possible jumps have probability $\hat{p}=(1-p)/2$. If this downwards jump is realized, then the process enters the so called ZIRP zone, meaning that interest rate becomes a small value $x_0$ (close to the target of the policy). When the process is in the ZIRP zone, it remains there with a high probability $(1-q)$ and exits with probability $q$. Finally, to calibrate the tree, following the same convention as in the classical BDT model, we further impose that the variance at each node for the same time period remains the same (to be determined by calibration, denoted below by $\sigma(n)$ for the period $n$). To the previous $r_{i,j}$ and $P_{i,j}$ corresponding to the BDT model, the ZBDT model adds the (unknown) bond prices $P_{i,0}$ for $i=1,\dots,n-1$ and $P_{0,n}=100$. The corresponding interest rates $r_{i,0}$ for $1,\dots,i+1$ are fixed to $x_0$. In Figure \[figure:zbdtp\], we present the tree corresponding to the prices of a zero-coupon bond with expiration in $n=3$ years.
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Calibration of the ZBDT model
-----------------------------
For the calibration, we use the same data as in the BDT model. The strategy is modified to cope with the new unknowns, but follows the same ideas. The first step uses only $y(1)$ and we conclude that $r_{01}=y(1)$. The equations are $$P_{1,0}=P_{1,1}=P_{1,2}=100,$$ $$P_{0,1}=\frac{100}{1+y(1)}=\frac{1}{1+r_{0,1}}\left(\frac{1-p}2\left(P_{1,0}+P_{1,1}\right)+pP_{1,2}\right).$$
### Variance equation at a node {#variance-equation-at-a-node-1 .unnumbered}
In the present situation, the random variable $y$ takes three values: $$Y=
\begin{cases}
y_u,&\text{ with probability}\ \hat{p},\\
y_d, &\text{ with probability}\ \hat{p},\\
y_0, &\text{ with probability}\ p.\\
\end{cases}$$ Then, $\log Y$ has the same variance as the random variable $$\log {Y\over y_0}=
\begin{cases}
\log {y_u \over y_0},&\text{ with probability}\ \hat{p},\\
\log {y_d \over y_0}, &\text{ with probability}\ \hat{p},\\
0, &\text{ with probability}\ p.\\
\end{cases}$$ The mean of $\log(Y/y_0)$ is $$\frac{1-p}2\left(\log{y_u\over y_0}+\log{y_d\over y_0}\right),$$ then $${{\operatorname{\mathbf{var}}}}\log {Y\over y_0} =\frac{1-p}2\left(\left(\log{y_u\over y_0}\right)^2+\left(\log{y_d\over y_0}\right)^2\right)-
\left(
\frac{1-p}2\left(\log{y_u\over y_0}+\log{y_d\over y_0}\right)
\right)^2.$$ Introducing the notation $$\label{eq:ells}
\ell_u=\log\frac{y_{u}}{y_{0}},\quad
\ell_d=\log\frac{y_{d}}{y_{0}}.$$ we obtain $$\label{eq:tres}
{{\operatorname{\mathbf{var}}}}\log y=
\frac{1-p^2}4\left(\ell_u^2+\ell_d^2\right)-\frac{(1-p)^2}2\ell_u\ell_d.$$ Considering now the interest rates, if the node $n-1,j$ has two edges (i.e. $j=2,\dots,n$), then the variance at the node satisfies equation , (the same as in the BDT case). If the node has three edges (when $j=1$), then the variance satisfies equation .
In the second step of calibration, the new data are $y(2)$ and $\beta(2)$. The unknowns are $r_{1,1},r_{1,2}$ and $\sigma(2)$. In this case $r_{1,1}=y_d$, $r_{1,2}=y_u$, $y_0=x_0$ and $\sigma(2)=\beta(2)$ (because in this case the local variation of the interest rate for one year coincides with the global variation). Accordingly, $y_d=r_{1,1}$ and $y_u=r_{1,2}$.
$$\begin{aligned}
P_{0,1}&=\frac{1}{{(1+y(2))}^2}=\frac{1}{1+r_{0,1}}\left(\frac{1-p}2\left(P_{u}+P_{d}\right)+pP_{0}\right),\\
P_{2,j}&=100, \quad j=1,2,3,\\
P_{u}&=\frac{100}{{(1+y_u)}},\quad
P_{d}=\frac{100}{{(1+y_d)}},\quad
P_{0}=\frac{100}{{(1+y_0)}},\\
\beta(2)^2&=\frac{1-p^2}4\left(\ell_u^2+\ell_d^2\right)-\frac{(1-p)^2}2\ell_u\ell_d.\\\end{aligned}$$
with $\ell_u$ and $\ell_d$ given in equation .
For general $n$, the new data are $y(n)$ and $\beta(n)$. The unknowns are $r_{n-1,j}$ for $j=1,\dots,n$, and $\sigma(n)$. The calibration equations are: [$$\begin{aligned}
P_{0,1}&=\frac{1}{{(1+y(n))}^n}=\frac{1}{1+r_{0,1}}\left(\frac{1-p}{2}P_{u}+\frac{1-p}{2}P_{d}+pP_{0}\right),\\
P_{n,j}&=100, \ j=0,\dots,(n+1),\\
P_{i,j}&=\frac12\frac{1}{1+r_{i,j}}\left(P_{i+1,j+1}+P_{i+1,j}\right),\ i=1,\dots,(n-1), \ j=2,\dots,i+1,\\
P_{i,1}&=\frac{1}{1+r_{i,j}}\left(\frac{1-p}{2}P_{i+1,2}+\frac{1-p}{2}P_{i+1,1}+pP_{i+1,0}\right),\ i=1,\dots,(n-1),\\
P_{i,0}&=\frac{1}{1+x_{0}}\left(qP_{i+1,1}+(1-q)P_{i+1,0}\right),\ i=1,\dots,(n-1),\\
P_{u}&=\frac{100}{{(1+y_u)}^{n-1}},
\quad
P_{d}=\frac{100}{{(1+y_d)}^{n-1}},
\quad
P_{0}=\frac{100}{{(1+y_0)}^{n-1}},\\
\ell_u&=\log\frac{y_{u}}{y_{0}},\quad
\ell_d=\log\frac{y_{d}}{y_{0}},\\
\beta(n)^2&=\frac{1-p^2}4\left(\ell_u^2+\ell_d^2\right)-\frac{(1-p)^2}2\ell_u\ell_d,\\
\sigma(n)&=\frac{1}{2}\log{\frac{r_{n-1,j+1}}{r_{n-1,j}}},\ j=2,\dots,n,\\
\ell_{1}&=\log\frac{r_{n-1,1}}{x_{0}},\qquad\ell_{2}=\log\frac{r_{n-1,2}}{x_{0}},\\
\sigma(n)^2&=\frac{1-p^2}4\left(\ell_1^2+\ell_2^2\right)-\frac{(1-p)^2}2\ell_1\ell_2.\\\end{aligned}$$]{}
Empirical analysis of different scenarios with US treasury bonds data {#section:empirical}
=====================================================================
The main motivation of our work is to analyze the new features observed in bond prices as a consequence of the ZIRP implemented by the US Government in 2008. In the Timeline \[timeline\], we give an account of the main events related with the US economy during the period of the study.
------------------------------------------------------------------------
[\
]{} [\
]{} [\
]{} [\
]{} [ US financial crisis. The Federal Reserve decreased the interest rate to 0-0.25%]{} [\
]{} [\
]{} [\
]{} [ Economic growth. The Federal Reserve increased its interest rate twice by 0.25%]{}
------------------------------------------------------------------------
Interest rates yields and volatilities 2002-2017
------------------------------------------------
We present the yields and its volatilities used to calibrate the ZBDT model (including interest rates and bond prices) with the aim of computing bond option prices. The daily interest rates correspond to the period from August 6, 2002 to April 28, 2017, and were obtained from the Federal Reserve Board of the United States. The data are denoted by $y(t,k)$, where $t$ denotes the day and $k$ the corresponding six maturities used in this study ($k=1/2,1,2,3,4,5$ in years). In the previous sections, $t$ was omitted because the analysis was performed for a fixed time. To compute the volatility $\beta(t,k)$ corresponding to these values, we use the formulas $$\aligned
\bar{\ell}(t,k)&=\frac1{252}\sum_{i=0}^{251}\log{y(t-i,k)\over y(t-i-1,k)},\\
\beta^2(t,k)&=\frac1{252}\sum_{i=0}^{251}\left(\log{y(t-i,k)\over y(t-i-1,k)}-\bar{\ell}(t,k)\right)^2,
\endaligned$$ where the factor 252 corresponds to the number of business day of one year; that is, the window chosen to compute the volatilities. The obtained data is presented in Figure \[figure:vol\].
![Yield rates and yield volatilities for different maturities (2002-2017).[]{data-label="figure:vol"}](yield.png "fig:"){width="8.4cm" height="6.0cm"} ![Yield rates and yield volatilities for different maturities (2002-2017).[]{data-label="figure:vol"}](vola.png "fig:"){width="8.4cm" height="6.0cm"}
Six observed typical scenarios
------------------------------
We choose six different days, corresponding to six different periods, expecting to analyze the impact of the downwards jump in the interest rate included in the ZBDT model. To select each of these days, the interest rates depicted in the Figure \[figure:vol\] and the Timeline \[timeline\] were taken into account.
In each of the six chosen scenarios, we calibrate the BDT and ZBDT models, presenting the corresponding interest rates and bond prices. These numerical results can be seen in Tables \[table:rp2\], \[table:rp4\], \[table:rp6\], \[table:rp8\] \[table:rp10\] and \[table:rp12\] in the Appendix.
With this information, we compute vanilla call option prices along strikes of bond prices ranging from 80 to 100, obtaining the respective implied volatility. To compute the implied volatility at time $t$ of an option written on a zero coupon bond that expires at time $T$, with strike $K$ and maturity $S$ ($t<S<T$), we use Black’s formula (see (Black, 1976)), which states $$C=P(t,T)\Phi(d_1)-KP(t,S)\Phi(d_2),$$ where $$d_{1,2}={\log\left({P(t,T)\over KP(t,S)}\right)\over\sigma\sqrt{S-t}}\pm{\sigma\sqrt{S-t}\over 2}.$$ For more details see (McDonald, 2006). In our empirical exercise, we consider a zero coupon bond with expiration in five years ($T=5$) and European call options written at $t=0$ with exercise time two years ($S=2$). The results are presented in Tables \[table:rp3\], \[table:rp5\], \[table:rp7\], \[table:rp9\], \[table:rp11\] and \[table:rp13\] in the Appendix. A primary conclusion is that, in contrast to the BDT model, the ZBDT allows us to price options with strikes close to the face value of the bond, which corresponds to low interest rate periods. This gives more accurate option prices in pre-crisis periods.
Conclusions {#section:conclusions}
===========
In the present work, we propose a novel and practical approach to model the possibility of a drop in the interest rates structure of sovereign bonds. This modification is motivated by the recent 2008–2009 crisis in United States.
Our approach is inspired by Lewis’s (2016) ZIRP models in continuous time, and also in Duffie and Singleton’s (1999) default framework of bond pricing models. Our proposal consists in adding a new branch at each period to the classical Black-Derman-Toy tree model that takes into account the small probability of this drop event to happen. We name this the ZBDT model, the “Z” standing for (close to) zero interest rate. To the best of our knowledge, our model is the first discrete space–time model proposed for the ZIRP, and it shares the motivation of including this phenomena as previously considered in continuous time models through sticky diffusions (such as in (Lewis, 2016)) and skew diffusions (Tian & Zhang, 2018).
This paper includes a development of the corresponding modified calibration scheme (that, naturally, happens to be more complex than the classical BDT calibration, and uses the same information) to obtain the interest rate tree and corresponding bond prices. With this information, we valuate European option prices provided by both models. The comparison between the two models is carried out though the implied volatility analysis provided by the Black option pricing formula. Our proposal opens the possibility of correcting option prices in different scenarios, especially under the risk of future zero interest rates. The analysis of implied volatility curves provided by the US bond market is a tool that can reveal in which situation this drop probability is not negligible. Our main conclusion is that the ZIRP models allows u to price options with high strikes. All of the observed implied volatilities are higher in the ZBDT model than in the BDT model. This gives more accurate option prices in pre-crisis periods.
Further research includes the consideration of American bond options market prices (and possible other usual derivatives in the bond markets) to calibrate the parameters of the proposed model: the probability of drop and the probability of staying in the ZIRP zone, and the more complex task of proposing a continuous time model analog.
[99]{}
Black, F. (1976), *The Pricing of Commodity Contracts.* Journal of Financial Economics , Vol 3, pp 167-179.
Black, F.; Derman, E. & Toy, W. (1990), *A One-Factor Model of Interest Rates and Its Application to Treasury Bond Options* Financial Analysts Journal, Vol 46, No 1, pp 33-39.
Borodin, A. N.; Salminen, P. (2002), *Handbook of Brownian motion-facts and formulae.* Second edition. Probability and its Applications. Birkhäuser Verlag, Basel, 2002.
Brigo, D., Mercurio, F. (2006), *Interest Rate Models Theory and Practice with Smile, Inflation and Credit.* Second Edition Springer Verlag.
Cox, J., Ingresoll J. & Ross S. (1985), *A Theory of the Term Structure of Interest Rate.* Econometrica 53, pp. 385-407.
Duffie, D., Singleton, K. (1999), *Modeling Term Structure of Defaultable Bonds.* Review of Financial Studies, Vol 12, pp 687-720.
Eberlein, E., Gerhart, G. & Grbac, Z. (2018), *Multiple Curve Lévy Forward Price Model Allowing for Negative Interest Rates.* Quantitative Finance, Vol 18, Issue 4.
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Lejay, A. (2006), *On the Constructions of the Skew Brownian Motion.* Probability Surveys, Vol 3, pp 413-466.
Lewis, A. (2016), *Option Valuation under Stochastic Volatility II.* Finance Press, Newport Beach, California, USA.
Martin, M. (2018), *An Overview of Post-crisis Term Structure Models.* New Methods in Fixed Income Modeling, Springer, pp 85-97.
McDonald, R. (2006), *Derivatives Markets.* Third edition, Pearson Series in Finance, Boston: Addison-Wesley, 2006
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Appendix
========
### Scenario I (May 23, 2003): Expanding economy, normal term structure. {#scenario-i-may-23-2003-expanding-economy-normal-term-structure. .unnumbered}
### Scenario II (August 07, 2006): Flat term structure curves. {#scenario-ii-august-07-2006-flat-term-structure-curves. .unnumbered}
### Scenario III (November 14, 2007): Start of financial crisis. {#scenario-iii-november-14-2007-start-of-financial-crisis. .unnumbered}
### Scenario IV (August 08, 2008): US crisis. {#scenario-iv-august-08-2008-us-crisis. .unnumbered}
### Scenario V (August 03, 2010): European crisis. {#scenario-v-august-03-2010-european-crisis. .unnumbered}
### Scenario VI (May 20, 2015): End of US-crisis. {#scenario-vi-may-20-2015-end-of-us-crisis. .unnumbered}
[^1]: Faculty of Pure and Applied Mathematics, Wroc[ł]{}aw University of Science and Technology. email: grzegorz.krzyzanowski@pwr.edu.pl
[^2]: Centro de Matemática, Facultad de Ciencias, Universidad de la República. email: mordecki@cmat.edu.uy
[^3]: Centro de Matemática, Facultad de Ciencias, Universidad de la República. email: asosa@cmat.edu.uy
|
---
abstract: 'We consider a sequence of polynomials appearing in expressions for the derivatives of the Lambert W function. The coefficients of each polynomial are shown to form a positive sequence that is log-concave and unimodal. This property implies that the positive real branch of the Lambert W function is a Bernstein function.'
address: |
Department of Applied Mathematics\
The University of Western Ontario\
London, Ontario, Canada N6A 5B7
author:
- 'G. A. Kalugin and D. J. Jeffrey'
title: Unimodal sequences show Lambert W is Bernstein
---
Introduction {#sect1}
============
The Lambert $W$ function was defined and studied in [@Corless1996Lam]. It is a multivalued function having branches $W_k(z)$, each of which obeys $W_k\exp(W_k) = z$. The principal branch $W_0$ maps the set of positive reals to itself, and is the only branch considered here. Therefore we omit the subscript $0$ for brevity. The $n$th derivative of $W$ is given implicitly by $$\label{eq:nthderiv}
\frac{d^n W(x)}{d x^n} = \frac{ \exp(-n W(x)) p_n(W(x)) } { (1 + W(x))^{2n-1} }
\mbox{\qquad for\ } n\ge 1\ ,$$ where the polynomials $p_n(w)$ satisfy $p_1(w)=1$, and the recurrence relation $$\label{eq:polyrecurr}
p_{n+1}(w) = -(nw + 3n - 1)p_n(w) + (1+w)p_n'(w)\mbox{ \qquad for\ }n\ge 1\ .$$ In [@Corless1997seqseries], the first 5 polynomials were printed explicitly: $$\begin{aligned}
p_1(w)&=1\ ,\ p_2(w)=-2-w\ ,\ p_3(w) = 9+8w+2w^2\ ,\\
p_4(w)&=-64-79w-36w^2-6w^3\ ,\\
p_5(w)&=625 + 974w+ 622w^2+ 192w^3+ 24w^4\ .\end{aligned}$$ These initial cases suggest the conjecture that each polynomial $(-1)^{n-1}p_n(w)$ has all positive coefficients, and if this is true, then $dW(x)/dx$ is a completely monotonic function [@SokalComm2008]. We here prove the conjecture and prove in addition that the coefficients are unimodal and log-concave.
Formulae for the coefficients {#sect2}
=============================
In view of the conjecture, we write $$\label{eq:Polydef}
p_n(w) = (-1)^{n-1}\sum_{k=0}^{n-1}\beta_{n,k}w^k\ .$$ We now give several theorems regarding the coefficients.
\[th:Formulae\] The coefficients $\beta_{n,k}$ defined in obey the recurrence relations $$\begin{aligned}
&\beta_{n,0}=n^{n-1}\ ,\quad \beta_{n,1}=3n^n-(n+1)^n-n^{n-1} \ , \label{eq:begin} \\
&\beta_{n,n-1}=(n-1)!\ ,\quad \beta_{n,n-2}=(2n-2)(n-1)! \ , \label{eq:end} \\
&\beta_{n+1,k}=(3n-k-1) \beta_{n,k} +n\beta_{n,k-1}-(k+1)\beta_{n,k+1}\ , \quad 2\le k \le n-3 \ . \label{eq:recurrence}\end{aligned}$$
By substituting (\[eq:Polydef\]) into (\[eq:polyrecurr\]) and equating coefficients.
An explicit expression for the coefficients $\beta_{n,k}$ is $$\label{eq:ExplicitOne}
\beta_{n,k}= \sum_{m=0}^k\frac{1}{m!} \binom{2n-1}{k-m} \sum_{q=0}^m \binom{m}{q} (-1)^q (q+n)^{m+n-1} \ .$$
We rewrite (\[eq:nthderiv\]) in the form $$p_n(W(x)) = (1+W(x))^{2n-1} e^{nW(x)} \frac{d^n W(x)}{dx^n}\ .$$ From the Taylor series of $W(x)$ around $x=0$, given in [@Corless1996Lam], we obtain $$\frac{d^n W(x)}{dx^n} = \sum_{m=n}^\infty \frac{(-m)^{m-1}}{(m-n)!} x^{m-n}\ .$$ Substituting this into the expression of $p_n$, using $x=We^W$ and changing the index of summation, we obtain the equation $$\label{eq:polynom}
p_n(w) = (1+w)^{2n-1}\sum_{s=0}^\infty(-1)^{n+s-1}(n+s)^{n+s-1}\frac{w^s}{s!}e^{(n+s)w}\ .$$ We expand the right side around $w=0$ and equate coefficients of $w$.
\[r1\] The polynomials $p_n(w)$ can be expressed in terms of the diagonal Poisson transform $\mathbf{D}_n[f_s;z]$ defined in [@PobleteViolaMunro1997], namely, by $$\label{eq:DPT}
p_n(w)=(-1)^{n-1}(1+w)^{2(n-1)}\mathbf{D}_n[(n+s)^{n-1};-w]\ .$$
\[Th:specnum\] The coefficients can equivalently be expressed either in terms of shifted $r$-Stirling numbers of the second kind ${ \genfrac{\{}{\}}{0pt}{}{n+r}{m+r} }_r$ defined in [@Broder1984], $$\label{eq:r-Stir}
\beta_{n,k} = \sum_{m=0}^k(-1)^m \binom{2n-1}{k-m}
{ \genfrac{\{}{\}}{0pt}{}{2n -1 + m}{ n + m} }_n\quad \ ,$$ or in terms of Bernoulli polynomials of higher order $B_n^{(z)}(\lambda)$ defined in [@Norlund1924], $$\label{eq:Bernoulli}
\beta_{n,k}= \sum_{m=0}^k(-1)^m \binom{2n-1}{k-m}
\binom{m+n-1}{n-1}B_{n-1}^{(-m)}(n) \ ,$$ or in terms of the forward difference operator $\Delta$ [@GrahamKnuthPatashnik p.188], $$\beta_{n,k}=\sum_{m=0}^k
\binom{2n-1}{k-m}
\frac{(-1)^m}{m!}\Delta^m n^{m+n-1} \ .$$
We convert using identities found in [@Broder1984] and [@LopezTemme2010] respectively. $${ \genfrac{\{}{\}}{0pt}{}{n+r}{m + r} }_r= \frac{1}{m!}\sum_{q=0}^m (-1)^{m-q}\binom{m}{q}(q+r)^n$$ and $$B_n^{(-m)}(r)= \frac{n!}{(m+n)!}\sum_{q=0}^m (-1)^{m-q}\binom{m}{q}(q+r)^{m+n}\ .$$
Properties of the coefficients {#sect3}
==============================
We now give theorems regarding the properties of the $\beta_{n,k}$. We recall the following definitions [@Stanley1989]. A sequence $c_0, c_1 \ldots c_n$ of real numbers is said to be *unimodal* if for some $0\le j\le n$ we have $c_0\le c_1\le \ldots \le c_j \ge c_{j+1} \ge \ldots \ge c_n$, and it is said to be *logarithmically concave* (or log-concave for short) if $c_{k-1}c_{k+1} \le c_k^2$ for all $1\le k\le n-1$. We prove that for each fixed $n$, the $\beta_{n,k}$ are unimodal and log-concave with respect to $k$. Since a log-concave sequence of positive terms is unimodal [@Wilf2005], it is convenient to start with the log-concavity property.
\[Th:LC\] For fixed $n\geq3$ the sequence $\left\{k!\beta_{n,k}\right\}_{k=0}^{n-1}$ is log-concave.
Using we can write $$k!\beta_{n,k}=(2n-1)!\sum_{m=0}^k \binom{k}{m}x_m y_{k-m} \ , \notag$$ where $$\label{eq:x}
x_m= \sum_{j=0}^j \binom{m}{j}a_j \ , \quad a_j=(-1)^j (n+j)^{m+n-1} \ ,$$ and $y_m=1/(2n-1-m)!$ . Since the binomial convolution preserves the log-concavity property [@Walkup1976; @WangYeh2007LC], it is sufficient to show that the sequences $\left\{x_m\right\}$ and $\left\{y_m\right\}$ are log-concave. We have $$\begin{split}
a_{j-1}a_{j+1}=(-1)^{j-1}(n+j-1)^{m+n-1}(-1)^{j+1}(n+j+1)^{m+n-1} \\
=(-1)^{2j}\left((n+j)^2-1\right)^{m+n-1}<(-1)^{2j}(n+j)^{2(m+n-1)}=a_j^2 \ . \notag
\end{split}$$ Thus the sequence $\left\{a_j\right\}$ is log-concave and so is $\left\{x_m\right\}$ due to and the afore-mentioned property of the binomial convolution. The sequence $\left\{y_m\right\}$ is log-concave because $$\begin{split}
y_{m-1}y_{m+1}&=\frac{1}{(2n-1-m+1)!}\frac{1}{(2n-1-m-1)!} \\
&=\frac{2n-1-m}{2n-1-m+1}\frac{1}{(2n-1-m)!}\frac{1}{(2n-1-m)!}<y_m^2 \ .
\end{split}$$
Now we prove that the coefficients $\beta_{n,k}$ are positive. The following two lemmas are useful.
\[Lemma:Gen\] If a positive sequence $\left\{k!c_k\right\}_{k\geq0}$ is log-concave, then
1. $\left\{(k+1)c_{k+1}/c_k\right\}$ is non-increasing;
2. $\left\{c_k\right\}$ is log-concave;
3. the terms $c_k$ satisfy
$$\label{ineq:genlemma}
c_k c_m \geq \binom{k+m}{k} c_0 c_{k+m} \quad (0 \leq m \leq k+1) \ .$$
The statements (i) and (ii) are obvious. To prove (iii) we apply a method used in [@AsaiKuboKuo2000]. Specifically, by (i) we have for $0\leq p\leq k$ $$\frac{c_{p+1}}{c_p}\geq\frac{k+p+1}{p+1}\frac{c_{k+p+1}}{c_k} \ . \notag$$ Apply the last inequality for $p=0,1,2,...,m$ with $m \leq k+1$, and form the products of all left-hand and right-hand sides. As a result, after the cancellation we obtain $$\frac{c_m}{c_0}\geq\frac{k+1}{1}\frac{k+2}{2} \ldots \frac{k+m}{m}\frac{c_{k+m}}{c_k} \ ,$$ which is equivalent to .
If the coefficients $\beta_{n,k}$ are positive, then for fixed $n\geq3$ they satisfy $$\label{ineq:lemma}
\frac{(k+1)\beta_{n,k+1}}{\beta_{n,k}}<n-1 \ .$$
By Theorem \[Th:LC\] and under the assumption of lemma, for fixed $n\geq3$ the sequence $\left\{k!\beta_{n,k}\right\}_{k=0}^{n-1}$ meet the conditions of Lemma \[Lemma:Gen\]. Applying the inequality with $m=1$ to this sequence gives $(k+1)\beta_{n,k+1}/\beta_{n,k}\leq \beta_{n,1}/\beta_{n,0}$. Then the lemma follows as due to $$\frac{\beta_{n,1}}{\beta_{n,0}}=\frac{3n^n-(n+1)^n-n^{n-1}}{n^{n-1}}=3n-n\left(1+\frac{1}{n}\right)^n-1<3n-2n-1=n-1 \ . \notag$$
\[Th:positive\] The coefficients $\beta_{n,k}$ are positive.
We prove the statement by induction on $n$. It is true for $n\le 5$ (see §1). Assume that for some fixed $n$ all the members of the sequence $\left\{\beta_{n,k}\right\}_{k=0}^{n-1}$ are positive. Since $\beta_{n+1,0}=(n+1)^n>0$ and $\beta_{n+1,n}=n!>0$ by and , we only need to consider $k=1,2,...,n-1$.
Substituting inequalities $\beta_{n,k+1}<(n-1)\beta_{n,k}/(k+1)$ and $\beta_{n,k-1}>k\beta_{n,k}/(n-1)$, which follow from , in the recurrence immediately gives the result $$\beta_{n+1,k}>(3n-k-1)\beta_{n,k}+n\frac{k}{n-1}\beta_{n,k}-(k+1)\frac{n-1}{k+1}\beta_{n,k}
=\left(2n+\frac{k}{n-1}\right)\beta_{n,k}>0 \ .$$ Thus the proof by induction is complete.
The sequence $\left\{\beta_{n,k}\right\}_{k=0}^{n-1}$ is unimodal for $n\geq3$.
By Theorem \[Th:positive\] the sequence $\left\{\beta_{n,k}\right\}_{k=0}^{n-1}$ is positive, therefore by Theorem \[Th:LC\] and Lemma \[Lemma:Gen\](ii) it is log-concave and, hence, unimodal.
Relation to Carlitz’s numbers {#sect4}
=============================
There is a relation between the coefficients $\beta_{n,k}$ and numbers $B(\kappa,j,\lambda)$ introduced by Carlitz in [@Carlitz1980WII]. Comparing the formulae and with the corresponding [@Carlitz1980WII eq.(6.3)] and [@Carlitz1980WII eq.(2.9)], taking into account that he uses the notation $R(n,m,r)={ \genfrac{\{}{\}}{0pt}{}{n+r}{m+r} }_r$, we find $$\label{eq:termsB}
\beta_{n,k}=(-1)^kB(n-1,n-1-k,n) \ .$$ It follows that for $n\geq3$, the sequence $\left\{B(n-1,k,n)\right\}_{k=0}^{n-1}$ is log-concave together with $\left\{\beta_{n,k}\right\}_{k=0}^{n-1}$.
Using the property [@Carlitz1980WII eq.(2.7)] that $\sum_{j=0}^\kappa B(\kappa,j,\lambda)=(2\kappa-1)!!$, we can compute $p_n(w)$ at the singular point where $W=-1$ (cf. ). Thus, substituting $w=-1$ in gives $p_n(-1)=(-1)^{n-1}(2n-3)!!$. Thus $w=-1$ is not a zero of $p_n(w)$.
We also note that the numbers $B(\kappa,j,\lambda)$ are polynomials of $\lambda$ and satisfy a three-term recurrence [@Carlitz1980WII eq.(2.4)] $$\label{eq:recB}
B(\kappa,j,\lambda)=(\kappa+j-\lambda)B(\kappa-1,j,\lambda)+(\kappa-j+\lambda)B(\kappa-1,j-1,\lambda)$$ with $B(\kappa,0,\lambda)=(1-\lambda)^{\bar{k}}$,$\quad B(0,j,\lambda)=\delta_{j,o}$. This gives one more way to compute the coefficients $\beta_{n,k}$, specifically, for given $n$ and $k$ we find a polynomial using and then set $\lambda=n$ to use .
Concluding remarks {#sect5}
==================
It has been established that the coefficients of the polynomials $(-1)^{n-1}p_n(w)$ are positive, unimodal and log-concave. These properties imply an important property of $W$. In particular, it follows from formula and Theorem \[Th:positive\] that $(-1)^{n-1}(dW/dx)^{(n-1)}>0$ for $n\geq1$. Since $W(x)$ is positive for all positive $x$ [@Corless1996Lam], this means that the derivative $W'$ is completely monotonic and $W$ itself is a Bernstein function [@Berg2008].
Some additional identities can be obtained from the results above. For example, computing $\beta_{n,n-1}$ by and comparing with gives $$\sum_{m=0}^{n-1}(-1)^m \binom{2n-1}{n-m-1}
{ \genfrac{\{}{\}}{0pt}{}{2n -1 + m}{ n + m} }_n
=(n-1)! \ .$$ A relation between ${ \genfrac{\{}{\}}{0pt}{}{2n -1 + m}{ n + m} }_n$ and $B_{n-1}^{(-m)}(n)$ can be obtained from and , but this is a special case of [@Carlitz1980WII eq.(7.5)]. It is finally interesting to note that and can be inverted. Indeed, in these formulae for fixed $n$, the sequence $(-1)^k\beta_{n,k}$ is a convolution of two sequences and the corresponding relation between their generating functions is $G(w)=(1-w)^{2n-1}F(w)$. Since $F(w)=G(w)(1-w)^{-(2n-1)}=G(w)\sum_{k\geq0}\binom{2n-2+k}{2n-2}w^k$, the inverse of, for example, is $$\label{eq:inverse}
{ \genfrac{\{}{\}}{0pt}{}{2n -1 + m}{ n + m} }_n = \sum_{k=0}^{n-1}(-1)^k\beta_{n,k}\binom{2n-2+m-k}{2n-2} \ .$$
We thank Prof. Alan Sokal for sending us his conjecture and for his interest and encouragement. The work was supported by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada.
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abstract: '*N*-continuous orthogonal frequency division multiplexing (NC-OFDM) is a promising technique to obtain significant sidelobe suppression for baseband OFDM signals, in future 5G wireless communications. However, the precoder of NC-OFDM usually causes severe interference and high complexity. To reduce the interference and complexity, this paper proposes an improved time-domain *N*-continuous OFDM (TD-NC-OFDM) by shortening the smooth signal, which is linearly combined by rectangularly pulsed OFDM basis signals truncated by a smooth window. Furthermore, we obtain an asymptotic spectrum analysis of the TD-NC-OFDM signals by a closed-form expression, calculate its low complexity in OFDM transceiver, and derive a closed-form expression of the received signal-to-interference-plus-noise ratio (SINR). Simulation results show that the proposed low-interference TD-NC-OFDM can achieve similar suppression performance but introduce negligible bit error rate (BER) degradation and much lower computational complexity, compared to conventional NC-OFDM.'
author:
- 'Peng Wei, Lilin Dan, Yue Xiao, Wei Xiang, and Shaoqian Li[^1]'
title: 'Improved *N*-continuous OFDM for 5G Wireless Communications'
---
Orthogonal frequency division multiplexing (OFDM), sidelobe suppression, *N*-continuous OFDM (NC-OFDM).
Introduction
============
Orthogonal frequency division multiplexing (OFDM) [@Ref1] has been one of the most popular multicarrier transmission techniques in future 5G wireless communications [@Ref18; @Ref19] due to its high-speed data transmission and inherent robustness against the inter-symbol interference (ISI). However, in rectangularly pulsed OFDM systems, the signal possesses a discontinuous pulse edge and thus exhibits large spectral sidelobes. Thus, power leakage due to sidelobes, which is also known as out-of-band power emission, causes severe interference to adjacent channels [@Ref2], especially in cognitive radio (CR) and carrier aggregation (CA) combined 5G systems [@Ref20; @Ref21].
For improving conventional OFDM in out-of-band emission, various methods have been proposed for sidelobe suppression \[7–18\]. The windowing technique in [@Ref3] extends the guard interval in the price of a reduction in spectral efficiency. Cancellation carriers [@Ref4; @Ref5] consume extra power and incur a signal-to-noise ratio (SNR) loss with high complexity. In the precoding methods \[10–12\], complicate decoding algorithms are required to eliminate the interference caused by the precoders.
NC-OFDM techniques \[13–19\] smooth the amplitudes and phases of the OFDM signal by making the OFDM signal and its first *N* derivatives continuous (so-called *N*-continuous). Conventional NC-OFDM [@Ref9] obtains the *N*-continuous signal at the expense of high interference. Aiming at optimizing the frequency domain precoder in [@Ref9], Beek et al. proposed the memoryless scheme in [@Ref10] and the improved scheme [@Ref11] nulling the spectrum at several chosen frequencies. On the other hand, to enable low-complexity signal recovery in NC-OFDM, several approaches have been proposed \[16–18\]. However, similar to the precoding techniques, the existing NC-OFDM techniques need robust signal recovery algorithms for reception. Among them, some methods degrade system performance, such as peak-to-average-power ratio (PAPR) growth in [@Ref10] and high complexity of transmitter in [@Ref12; @Ref13].
In this paper, to reduce the interference of the NC-OFDM signals and obtain a low-complexity transceiver, we propose a low-interference time-domain *N*-continuous OFDM (TD-NC-OFDM) based on the conventional TD-NC-OFDM [@Ref15]. A smooth signal is superposed in the front part of each OFDM symbol to achieve *N*-continuous OFDM signal, including smoothing both edges of the transmitted signal. The smooth signal is linearly combined by the basis vectors in a basis set, which is composed of the rectangularly pulsed OFDM basis signals truncated by a smooth window function for an example. Furthermore, we give rise to analyses of spectrum, complexity and signal-to-interference-plus-noise ratio (SINR) in the low-interference TD-NC-OFDM. Among them, an asymptotic expression of PSD of the TD-NC-OFDM signal is first obtained, where the sidelobes asymptotically decay with $f^{-N-2}$, when the first *N* derivatives of the OFDM signal are all continuous. Then, we compare the complexity among NC-OFDM, TD-NC-OFDM and its low-interference scheme, to show the complexity reduction of the proposed low-interference scheme. The closed-form expression of the received SINR of the low-interference scheme is also calculated to show the slight SNR loss. Simulation results show that the low-interference scheme can achieve as notable sidelobe suppression as NC-OFDM method [@Ref9] with excellent bit error rate (BER) performance and low complexity.
The remainder of the paper is organized as follows. In Section 2, the OFDM signaling is briefly introduced, the traditional NC-OFDM is reviewed. Section 3 proposes the low-interference TD-NC-OFDM model, gives the linear combination design of the smooth signal with a new basis set, and describes the transmitter. In Section 4, the effect of the low-interference TD-NC-OFDM on sidelobe decaying and the received SINR is analyzed as well as the computational complexity of the transceiver. Finally, Section 5 draws concluding remarks.
*Notation*: Boldfaced lowercase and uppercase letters represent column vectors and matrices, respectively. $\{\mathbf{A}\}_{m,n}$ indicates the element in the *m*th row and *n*th column of matrix **A**. The $M\times M$ identity matrix and $M\times N$ zero matrix are denoted by $\mathbf{I}_M$ and $\mathbf{0}_{M\times N}$, respectively. $|\cdot|$ represents the absolute value. The trace and expectation of a matrix are represented by $\mathrm{Tr}\{\cdot\}$ and $E\{\cdot\}$, respectively. $\mathbf{A}^T$, $\mathbf{A}^{\ast}$, $\mathbf{A}^H$ and $\mathbf{A}^{-1}$ denote the transposition, conjugate, conjugate transposition, and inverse of matrix **A**, respectively.
System aspects and *N*-continuous OFDM
======================================
OFDM signaling
--------------
In a baseband OFDM system, the input bit stream of the *i*th OFDM symbol is first modulated onto an uncorrelated complex-valued data vector $\mathbf{x}_i={[ x_{i,k_0}, x_{i,k_1},\ldots,x_{i,k_{K-1}}]}^T$ drawn from a constellation, such as phase-shift keying (PSK) or quadrature amplitude modulation (QAM). The complex-valued data vector is mapped onto *K* subcarriers with the index set $\mathcal{K}=\left\{k_0,k_1,\ldots,k_{K-1}\right\}$. An OFDM signal is formed by summing all the *K*-modulated orthogonal subcarriers with equal frequency spacing $\Delta f=1/T_s$, where $T_s$ is the OFDM symbol duration. The *i*th OFDM time-domain symbol, assuming a normalized rectangular time-domain window $R(t)$ [@Ref6], can be expressed as $$y_i(t)=\sum\limits^{K-1}_{r=0}{x_{i,k_r}e^{j2\pi k_r\Delta ft}}, -T_{cp}\leq t <T_s
\label{Eqn:1}$$ where $T_{cp}$ is the cyclic prefix (CP) duration. Then, in the time range of $(-\infty,+\infty)$, the transmitted OFDM signal $s(t)$ can be written as $$s(t)=\sum\limits^{+\infty}_{i=-\infty}{y_i\left(t-iT\right)}.
\label{Eqn:2}$$ where $T=T_s+T_{cp}$.
After the OFDM signal is oversampled by a time-domain sampling interval $T_{samp}=T_s/M$, the discrete-time OFDM signal is expressed as $$y_i(m)=\frac{1}{M}\sum\limits^{K-1}_{r=0}{x_{i,k_r}e^{j2\pi \frac{k_r}{M}m}},
\label{Eqn:3}$$ where $m\in \mathcal{M}=\left\{-M_{cp},\ldots,0,\ldots,M-1\right\}$, and $M_{cp}$ is the length of CP samples.
*N*-continuous OFDM
-------------------
To improve the continuity of the time-domain OFDM signal, the conventional NC-OFDM [@Ref9] introduces a frequency-domain precoder to making the OFDM signal and its first *N* derivatives continuous. NC-OFDM follows that $$\bar{y}_i(t)=\sum\limits^{K-1}_{r=0}{\bar{x}_{i,k_r}e^{j2\pi k_r\Delta ft}}, -T_{cp}\leq t <T_s
\label{Eqn:4}$$ $$\left. \bar{y}^{(n)}_i(t)\right|_{t=-T_{cp}}=\left. \bar{y}^{(n)}_{i-1}(t)\right|_{t=T_s},
\label{Eqn:5}$$ where $\bar{x}_{i,k_r}$ is the precoded symbol on the *r*th subcarrier, and $\bar{y}^{(n)}_{i}(t)$ is the *n*th-order derivative of $\bar{y}_{i}(t)$ with $n\in\mathcal{U}_N=\{0,1,\ldots,N\}$.
Based on and , the precoding process can be summarized as $$\begin{cases}
\bar{\mathbf{x}}_i=\mathbf{x}_0, & i=0 \\
\bar{\mathbf{x}}_i=(\mathbf{I}_K-\mathbf{P})\mathbf{x}_i+\mathbf{P}\mathbf{\Phi}^H\bar{\mathbf{x}}_{i-1}, & i>0
\end{cases}
\label{Eqn:6}$$ where $\mathbf{I}_K$ is the identity matrix, $\mathbf{P}=\mathbf{\Phi}^H\mathbf{A}^H(\mathbf{A}\mathbf{A}^H)^{-1}\mathbf{A}\mathbf{\Phi}$, $\left\{\mathbf{A}\right\}_{n+1,r+1}=k^n_r$, $\mathbf{\Phi}=diag(e^{j\varphi k_0},e^{j\varphi k_1},\ldots,e^{j\varphi k_{K-1}})$, and $\varphi=-2\pi\beta$ with $\beta=T_{cp}/T_s$. Figure \[Fig:1\] depicts the spectrally precoded NC-OFDM transmitter. The *i*th frequency-domain data vector $\mathbf{x}_i$ is first precoded. The precoded data vector $\bar{\mathbf{x}}_i=\left[\bar{x}_{i,k_0},\bar{x}_{i,k_1},\ldots,\bar{x}_{i,k_{K-1}}\right]^T$ then undergoes the inverse fast Fourier transform (IFFT), and finally the CP is added to generate the transmission signal.
![Block diagram of the *N*-continuous OFDM transmitter.[]{data-label="Fig:1"}](NC_OFDM_transmitter "fig:"){width="3in"} .
Proposed Low-Interference Scheme of TD-NC-OFDM
==============================================
Conventional TD-NC-OFDM [@Ref15] and Conventional NC-OFDM [@Ref9] cause high interference, which is needed to be reduced by complicate signal recovery algorithms \[13, 16, 18\]. As show in Figure \[Fig:2\] (a), the interference term, defined as the smooth signal $w_i(m)$, is located in the whole time-domain in the conventional TD-NC-OFDM as well as NC-OFDM . To eliminate the interference and simplify the receiver, we truncate $w_i(m)$ with a window function. As illustrated in in Figure \[Fig:2\] (b), the truncated term $\tilde{w}_i(m)$ only locates in the front section of each OFDM symbol.
![Conventional and proposed ways of of adding the smooth signal in the time domain: (a) Addition in the whole OFDM symbol; (b) Addition in the front of each OFDM symbol.[]{data-label="Fig:2"}](Distribution_smooth_signals "fig:"){width="3.5in"} .
To make the OFDM signal *N*-continuous, $\tilde{w}_i(m)$ should satisfy $$\bar{y}_i(m)=y_i(m)+\tilde{w}_i(m),
\label{Eqn:7}$$ $$\left.\tilde{w}^{(n)}_i\!(m)\right|_{m=-M_{cp}}\!\!=\left.y^{(n)}_{i-1}\!(m)\right|_{m=M}
\!-\left.y^{(n)}_{i}\!(m)\right|_{m=-M_{cp}}\!.
\label{Eqn:8}$$
$\tilde{w}_i(m)$ is the linear combination of the first *N*+1 basis vectors in a basis set $\mathcal{Q}$, written as $$\tilde{w}_i(m)=\left\{\begin{matrix}
\sum\limits^{N}_{n=0}{{b}_{i,n}\tilde{f}_n(m)}, & m\in \mathcal{L} \\
0, & m\in \mathcal{M}-\mathcal{L}
\label{Eqn:9}
\end{matrix}\right.,$$ where $\mathcal{L}=\left\{-M_{cp},-M_{cp}+1,\ldots,-M_{cp}+L-1\right\}$ indicates the location of $\tilde{w}_i(m)$ with the length of *L*, and the basis set $\mathcal{Q}$ is given by $$ {\mathcal{Q}}=\left\{{\mathbf{q}}_{\tilde{n}}\left|{\mathbf{q}}_{\tilde{n}}=\left[\tilde{f}_{\tilde{n}}(-M_{cp}),\tilde{f}_{\tilde{n}}(-M_{cp}+1),\ldots, \right. \tilde{f}_{\tilde{n}}(-M_{cp}+L-1)\right]^T, \tilde{n}\in \mathcal{U}_{2N}\right\},
\label{Eqn:10}$$ where $\mathcal{U}_{2N}=\{0,1,\ldots,2N\}$. In , the design of the basis signal $\tilde{f}_{\tilde{n}}(m)$ and the calculation of the coefficients $b_{i,n}\in\mathbf{b}_i=[b_{i,0},b_{i,1},\ldots,b_{i,N}]^T$ will be specified as follows.
To guarantee the smoothness of $\tilde{w}_i(m)$ and to obtain the *N*-continuous signal, an example of designing $\tilde{f}_{\tilde{n}}(m)$ is given by $$\tilde{f}_{\tilde{n}}(m)=f^{(\tilde{n})}(m)g_h(m),
\label{Eqn:11}$$ for $m\in\mathcal{L}$, and $ \tilde{f}_{\tilde{n}}(m)=0$ for $m\in\mathcal{M}-\mathcal{L}$. In $\mathcal{L}$, $$f^{(\tilde{n})}(m)= 1/M\left(j2\pi/M\right)^{\tilde{n}}\!\!\sum\limits_{k_r\in\mathcal{K}}\!\!\!{k^{\tilde{n}}_re^{j\varphi k_r}e^{j2\pi\frac{k_r}{M}m}},
\label{Eqn:12}$$ and $g_h(m)$ is the right half part of a baseband-equivalent window function $g(m)$. Thus, the discontinuity at the adjacent point between two consecutive OFDM symbols can be eliminated by $\tilde{w}_i(m)$ from only a single side of the adjacent point. Under the constraint of $g_h(m)$, the interference caused by $\tilde{w}_i(m)$ can be limited to the front section of the OFDM symbol. In order to satisfy , $g(m)$ should be considered as a smooth and zero-edged window function, such as a triangular, Hanning, or Blackman window function.
On the other hand, the linear combination coefficients $b_{i,n}$ in can be calculated as $${\mathbf{b}}_i=\mathbf{P}^{-1}_{\tilde{f}}\begin{bmatrix}
y_{i-1}(M)-y_i(-M_{cp})\\
{\mathbf{P}}_1\mathbf{x}_{i-1}-\mathbf{P}_2\mathbf{x}_i
\end{bmatrix},
\label{Eqn:13}$$ where $\mathbf{P}_{\tilde{f}}$ is a $(N+1)\times(N+1)$ symmetric matrix related to ${\mathcal{Q}}$ with element $\left\{\mathbf{P}_{\tilde{f}}\right\}_{n+1,v+1}$ is $\tilde{f}^{(n+v)}(-M_{cp})$, and element $\left\{{\mathbf{P}}_1\right\}_{n,r+1}=1/M\left(j2\pi k_r/M\right)^n$ and element $\left\{\mathbf{P}_2\right\}_{n,r+1}=1/M\left(j2\pi k_r/M\right)^ne^{j\varphi k_r}$ for $n\neq 0$.
Figure \[Fig:3\] shows the block diagram of the proposed low-interference TD-NC-OFDM. According to $\mathcal{K}$ and $\mathcal{Q}$, the matrices $\mathbf{Q}_{\tilde{f}}=[\mathbf{q}_0, \mathbf{q}_1, \ldots, \mathbf{q}_N ]$, $\mathbf{P}^{-1}_{\tilde{f}}$, $\mathbf{P}_{1}$, and $\mathbf{P}_{2}$, can be calculated and stored in advance. Then, the data is first mapped and transformed to the time domain by the IFFT. Furthermore, the oversampled OFDM signal is appended by a CP. Finally, under the initialization of $\mathbf{y}_{-1}=\mathbf{0}$, the smooth signal is added onto the OFDM signal constructed by $M_s$ OFDM symbols $\mathbf{y}_i$ to generate the following transmit signal $$\bar{\mathbf{y}}_i=\left\{\begin{matrix}
\mathbf{y}_i+\mathbf{Q}_{\tilde{f}}{\mathbf{b}}_i, & 0\leq i\leq M_s \\
\mathbf{Q}_{\tilde{f}}{\mathbf{b}}_i, & i=M_s+1
\end{matrix}\right..
\label{Eqn:14}$$
![Block diagram of the OFDM transmitter with the proposed low-interference TD-NC-OFDM.[]{data-label="Fig:3"}](Figure1.eps "fig:"){width="3.6in"} .
Analysis of Spectrum and Complexity
===================================
Spectral Analysis
-----------------
According to the definition of PSD [@Ref8] and the relationship between spectral roll-off and continuity [@Ref16], the PSD of the OFDM signal processed by the low-interference scheme is achieved as follows.
All the derivatives of the OFDM signal $s(t)$ are known to exist except for around the two edges and on the points between adjacent OFDM symbols. Meanwhile, except for these non-differentiable points, the smooth signal also possesses derivatives of all orders, according to the existence of all the derivatives of the basis function $\tilde{f}_n(m)$. Then, we assume that at non-differentiable points, the first *N*-1 derivatives of the smoothed OFDM signal $\bar{s}(t)$ are continuous, and the *N*th-order derivative $\bar{s}^{(N)}(t)$ has finite amplitude discontinuity. We also suppose that all the derivatives of $\bar{s}(t)$ approach zeroes , which corresponds to . Firstly, based on [@Ref16], we can obtain $$\mathcal{F}\left\{\bar{s}(t)\right\}=\frac{1}{(j2\pi f)^{N-1}}\int\limits^{+\infty}_{-\infty}{\bar{s}^{(N-1)}(t)e^{-j2\pi ft}dt}.
\label{Eqn:15}$$
Furthermore, since $\bar{s}^{(N)}(t)$ has finite amplitude discontinuities, by setting $u=\bar{s}^{(N-1)}(t)$ and $dv=e^{-j2\pi ft}dt$ in the above expression, we arrive at $$\begin{aligned}
\mathcal{F}\left\{\bar{s}(t)\right\}
&=\frac{1}{(j2\pi f)^{N-1}}\left(\!\left.\frac{\bar{s}^{(N-1)}(t)e^{-j2\pi ft}}{-j2\pi f}\right|^{t=+\infty}_{t=-\infty}\!-\!\int\limits^{+\infty}_{-\infty}{\frac{s^{(N)}(t)e^{-j2\pi ft}}{-j2\pi f}dt}\!\right) \nonumber \\
&=\frac{1}{(j2\pi f)^{N}}\int\limits^{+\infty}_{-\infty}{\bar{s}^{(N)}(t)e^{-j2\pi ft}dt}.
\label{Eqn:16}
\end{aligned}$$
Because $\bar{s}^{(N)}(t)$ has finite amplitude discontinuities at the adjacent points, from , $\bar{s}^{(N)}(t)$ can be written as $$\bar{s}^{(N)}(t)=\sum\limits^{+\infty}_{i=-\infty}{\bar{y}^{(N)}_i\left(t-iT\right)}.
\label{Eqn:17}$$
It is inferred in that the *N*th derivative $\bar{y}^{(N)}_i(t)$ can be assumed being windowed by the rectangular function $R(t)$. Therefore, based on the definition of PSD, , and , the PSD of $\bar{s}(t)$ can be expressed as $$\begin{aligned}
{\Psi}(f) \!=\!\! \lim\limits_{U\rightarrow \infty}{\frac{1}{2UT}E\left\{\left|\mathcal{F}\left\{
\sum\limits^{U-1}_{i=-U}{\!\frac{\bar{y}^{(N)}_{i}(t-iT)}{\left(j2\pi f\right)^N}}\right\}\right|^2\!\right\}} \!=\!\! \lim\limits_{U\rightarrow \infty}{\!\frac{1}{2UT}E\left\{\left|\sum\limits^{U-1}_{i=-U}{\frac{\mathcal{F}
\left\{\bar{y}^{(N)}_{i}(t)\right\}}{\left(j2\pi f\right)^N}e^{-j2\pi fiT}}\right|^2\!\right\}}.
\label{Eqn:18}\end{aligned}$$
Eq. indicates that the spectrum of the TD-NC-OFDM signal is related to the expectation of $\bar{y}^{(N)}_i(t)$ and $f^{-N}$. In this paper, the conventional Blackman window function is used as an example, given as $g(t)=0.42-0.5\cos{(2\pi \rho t)}+0.08\cos{(4\pi \rho t)}$ where $\rho=1/\left((2L-2)T_{samp}\right)$. By substituting , , , and into , the PSD of the smoothed OFDM signal in low-interference scheme is expressed by $$\begin{aligned}
{\Psi}(f)
&=\lim\limits_{i\rightarrow \infty}\frac{1}{2UT}E\Bigg\{\bigg|\sum\limits^{U-1}_{i=-U}e^{-j2\pi fiT}\left(j2\pi f\right)^{-N}
\sum\limits_{k_r\in \mathcal{K}}{\left(\frac{j2\pi k_r}{T_s}\right)^Nx_{i,k_r}\mathrm{sinc}\left(f_r(1+\beta)\right)e^{j\pi f_r(1-\beta)}} \nonumber \\
& \quad +\frac{1}{T}\sum\limits^{N}_{n=0}{b}_{i,n}\sum\limits^{N}_{\bar{n}=0}\left(\begin{matrix}
N \\ \bar{n} \end{matrix}\right)\sum\limits_{k_r\in \mathcal{K}}\left(\frac{j2\pi k_r}{T_s}\right)^{N-\bar{n}+n}
\int\limits^{-T_{cp}+T_p}_{-T_{cp}}{g^{(\bar{n})}_{h}(t)e^{j2\pi f_r t/T_s}dt}\bigg|^2\Bigg\} \nonumber \\
&=\lim\limits_{i\rightarrow \infty}\frac{1}{2UT}E\Bigg\{\bigg|\sum\limits^{U-1}_{i=-U}e^{-j2\pi fiT}\left(fT_s\right)^{-N}
\times \sum\limits_{k_r\in \mathcal{K}}{k^N_rx_{i,k_r}\mathrm{sinc}\left(f_r(1+\beta)\right)e^{j\pi f_r(1-\beta)}} \nonumber \\
& \quad +\!\frac{1}{T}\!\sum\limits^{N}_{n=0}{b}_{i,n}\!\sum\limits^{N}_{\bar{n}=0}\!\!\left(\begin{matrix}
N \\ \bar{n} \end{matrix}\right)\!\!\left(\!\dfrac{j2\pi}{T_s}\!\right)^{n-\bar{n}} \!\!\! \sum\limits_{k_r\in \mathcal{K}}\!{k^{N-\bar{n}+n}_rG_{\bar{n}}(f)}\bigg|^2\!\Bigg\}
\label{Eqn:19}
\end{aligned}$$ where $G_{\bar{n}}(f)$ is given by $$\begin{aligned}
\!\!\!&\!\!\! G_{\bar{n}}(f)=e^{j\tilde{f}_r}\Bigg(0.42^{(\bar{n})}T_p\mathrm{sinc}\left(\!\mu f_r\right)-\dfrac{0.5(2\pi\rho)^{\bar{n}}\cos{\left(\pi\mu f_r\right)}}{1-\left(\rho T_s/f_r\right)^2} \left(\dfrac{\cos{\left(\pi\bar{n}/2\right)}}{j\pi f_r/T_s}-\pi\rho\dfrac{\sin{\left(\pi\bar{n}/2\right)}}{\left(\pi f_r/T_s\right)^2}\right) \\
\!\!\!&\!\!\! + \dfrac{0.08(4\pi\rho)^{\bar{n}}\sin{\left(\pi\mu f_r\right)}}{1-\left(2\rho T_s/f_r\right)^2}
\!\left(\!\!\dfrac{\cos{\left(\pi\bar{n}/2\right)}}{\pi f_r/T_s}\!-j2\pi\rho\dfrac{\sin{\left(\pi\bar{n}/2\right)}}{\left(\pi f_r/T_s\right)^2}\!\right)\!\!\!\Bigg),\end{aligned}$$ where $\text{sinc}(x)\triangleq \sin(\pi x)/(\pi x)$, and $\tilde{f}_r=\pi f_r(T_p-2T_{cp})/T_s$ with $f_r=k_r-T_sf$, $T_p=(L-1)T_{samp}$, and $\mu=T_p/T_s$.
Eq. shows that the power spectral roll-off of the smoothed signal, whose first *N*-1 derivatives are continuous, decays with $f^{-2N-2}$. Moreover, $G_{\bar{n}}(f)$ reveals that the sidelobe is affected by the length of $g(t)$, so that a rapid sidelobe decaying can be achieved by increasing the length of $g(t)$. Figure \[Fig:4\] compares the theoretical and simulation results of the low-interference scheme with a window length of 144. It is shown that the simulation results match well with the theoretical analyses.
![PSD comparison between the analytical and simulation results for the TD-NC-OFDM signal in the low-interference TD-NC-OFDM with *L*=144.[]{data-label="Fig:4"}](Figure2.eps "fig:"){width="4in"} .
Complexity Comparison
---------------------
Firstly, we consider the complexity of the transmitter. In NC-OFDM, its frequency-domain precoder requires $2K^2$ complex multiplications and $2K^2$ complex additions as indicated in . In TD-NC-OFDM [@Ref15], $2NK+(N+1)(2N+1)$ complex multiplications and $2NK+N(2N+1)$ complex additions are required. However, for the generation and overlapping of the smooth signal, $M(N+1)$ complex multiplications and $M(N+1)$ complex additions are needed. By shortening the length of the smooth signal, the low-interference scheme just requires $L(N+1)$ complex multiplications and $L(N+1)$ complex additions. At the same time, the complexity of calculating its linear combination coefficients in is $2NK+(N+1)^2$ complex multiplications and $2NK+N^2+1$ complex additions.
Secondly, the complexity of the receiver is shown in Table \[Tab1\]. For NC-OFDM, an iterative signal recovery algorithm [@Ref9] is used to eliminate the interference. Due to the equivalence between NC-OFDM and TD-NC-OFDM, the signal recovery algorithm is also desired in TD-NC-OFDM with identical complexity. In the low-interference scheme, the receiver is the same as in original OFDM without extra signal recover processing.
Assume that a complex addition is equivalent to two real additions; a complex multiplication to four real multiplications plus two real additions; and a real-complex multiplication to two real multiplications. The complexity comparison among NC-OFDM, TD-NC-OFDM and the low-interference scheme is shown in Table \[Tab1\], where $L_R$ denotes the number of iterations in the signal recovery algorithm [@Ref9].
[13.65cm]{}[|m[3.5cm]{}|m[1.5cm]{}|m[3.45cm]{}|m[3.45cm]{}|]{} & **[Real multiplication]{} & **[Real addition]{}\
**NC-OFDM & **[Transmitter]{} & $O(8K^2)$ & $O(8K^2)$\
& **[Receiver]{} & $O(16(N+1)K L_R)$ & $O(16(N+1)K L_R)$\
**TD-NC-OFDM & **[Transmitter]{} & $O(8NK+4(N+1)M)$ & $O(8NK+4(N+1)M)$\
& **[Receiver]{} & $O(16(N+1)K L_R)$ & $O(16(N+1)K L_R)$\
**Low-interference scheme & **[Transmitter]{} & $O(8NK+4(N+1)L)$ & $O(8NK+4(N+1)L)$\
& **[Receiver]{} & $0$ & $0$\
& $O(2M\log_2 M)$ & $O(2M\log_2 M)$\
**********************
The low-interference scheme considerably suppresses the sidelobes shown in Figure \[Fig:4\]. Simultaneously, since the length of the smooth signal *L* is often as short as the CP length or shorter in real systems, the low-interference scheme is of lower transmitter complexity than NC-OFDM and the conventional TD-NC-OFDM. Moreover, its complexity is comparable to *M*-point IFFT. On the other hand, the low-interference scheme just requires the receiver of original OFDM, and avoiding extra processing in NC-OFDM and TD-NC-OFDM receivers.
SINR Analysis
-------------
One disadvantage of NC-OFDM is that the transmit signal is easily interfered by the smooth signal. Thus, a measure is needed to evaluate the interference in terms of the SNR loss. In this section, we investigate the SINR of NC-OFDM, and demonstrate the effectiveness of the proposed low-interference scheme in reducing the SNR loss, based on the analysis of the average power of the smooth signal.
In a multipath channel with time-domain coefficients $h_{\tilde{l}}$ in the $\tilde{l}$th path, the *i*th received time-domain OFDM symbol $r_i(t)$ is given by $$r_i(t)=\sum\limits^{\tilde{L}}_{\tilde{l}=1}{h_{\tilde{l}}\bar{y}_i(t-\tau_{\tilde{l}})+n_i(t)}
\label{Eqn:20}$$ where $\tau_{\tilde{l}}$ is the time delay in the $\tilde{l}$th path, and $n_i(t)$ is the AWGN noise with mean zero and variance $\sigma^2_n$.
Because $\mathbf{x}_i$ is uncorrelated, i.e., $E\left\{\mathbf{x}_i\mathbf{x}^H_i\right\}=\mathbf{I}_K$ and $E\left\{\mathbf{x}_{i-1}\mathbf{x}^H_i\right\}=\mathbf{0}_{K\times K}$, we can obtain $E\left\{\mathbf{x}^H_i\mathbf{x}_i\right\}=
\mathrm{Tr}\left\{E\left\{\mathbf{x}_i\mathbf{x}^H_i\right\}\right\}=K$. Thus, the average power of the OFDM symbol vector $\mathbf{y}_i$ is $$E\left\{\mathbf{y}^H_i\mathbf{y}_i\right\}
=\frac{1}{M^2}E\left\{\mathbf{x}^H_i\mathbf{F}_f\mathbf{F}^H_f\mathbf{x}_i\right\}
=E\left\{\mathbf{x}^H_i\mathbf{x}_i\right\}
={K}/{M}.
\label{Eqn:21}$$
To mitigate the performance degradation in conventional TD-NC-OFDM, the proposed low-interference scheme is analyzed by exploring the distribution of $\tilde{w}_i(l)$ in the multipath fading channel. As illustrated in Figure \[Fig:5\], different channel paths with varying time delays lead to varying $\tilde{w}_i(l)$. With the increased time delay, the delayed tail of $\tilde{w}_i(l)$ is prolonged and the interference increases. The interferences $\tilde{w}_i(l)$ are composed of the delayed tails in all the paths, whose powers are much smaller than the conventional TD-NC-OFDM and NC-OFDM.
![A time-domain illustration of the effect of the smooth signal on the multipath channel without power attenuation and Gaussian noise.[]{data-label="Fig:5"}](Figure5 "fig:"){width="4in"} .
Thus, the average power of $\tilde{w}_i(l)$ in the $\tilde{l}$th path is calculated by $$\begin{aligned}
E\left\{\left(h_{\tilde{l}}\tilde{\mathbf{w}}_i\right)^H h_{\tilde{l}}\tilde{\mathbf{w}}_i\right\}
&= \mathrm{Tr}\left\{E\left\{h_{\tilde{l}}\tilde{\mathbf{w}}^H_i
\tilde{\mathbf{w}}_ih^H_{\tilde{l}}\right\}\right\}
= E\left\{\left|h_{\tilde{l}}\right|^2\right\}
\mathrm{Tr}\left\{E\left\{\tilde{\mathbf{w}}^H_i\tilde{\mathbf{w}}_i\right\}\right\} \nonumber\\
&= \frac{2}{M^2}E\left\{\left|h_{\tilde{l}}\right|^2\right\}
\mathrm{Tr}\left\{\mathbf{P}^{-1}_{\tilde{f}}\mathbf{B}_2\mathbf{B}^H_2
\left(\mathbf{P}^{-1}_{\tilde{f}}\right)^H
\mathbf{Q}^H_{\tilde{f}_{\tilde{l}}}\mathbf{Q}_{\tilde{f}_{\tilde{l}}}\right\}
\label{Eqn:22}\end{aligned}$$ where $\mathbf{g}_{\tilde{l}}$ is the delayed tail of $\mathbf{g}$ in the $\tilde{l}$th path and $\mathbf{Q}_{\tilde{f}_{\tilde{l}}}=\mathbf{g}_{\tilde{l}}\mathbf{Q}_f$.
The derivation of $\mathrm{Tr}\left\{E\left\{\tilde{\mathbf{w}}^H_i\tilde{\mathbf{w}}_i\right\}\right\}$ is shown in Appendix A.
Therefore, in all the paths, the average power of these delayed tails is expressed by $$\sum\limits^{\tilde{L}}_{\tilde{l}=1}
{E\left\{\left(h_{\tilde{l}}\tilde{\mathbf{w}}_i\right)^Hh_{\tilde{l}}\tilde{\mathbf{w}}_i\right\}}
= \frac{2}{M^2}\sum\limits^{\tilde{L}}_{\tilde{l}=1}{E\left\{\left|h_{\tilde{l}}\right|^2\right\}
\mathrm{Tr}\left\{\mathbf{P}^{-1}_{\tilde{f}}\mathbf{B}_2\mathbf{B}^H_2
\left(\mathbf{P}^{-1}_{\tilde{f}}\right)^H
\mathbf{Q}^H_{\tilde{f}_{\tilde{l}}}\mathbf{Q}_{\tilde{f}_{\tilde{l}}}\right\}}.
\label{Eqn:23}$$
Finally, from -, the received SINR ${\gamma}_{SINR}$ is obtained by $$\begin{aligned}
{\gamma}_{SINR}
&=\dfrac{\sum\limits^{\tilde{L}}_{\tilde{l}=1}
{E\left\{\left|h_{\tilde{l}}\right|^2\right\}E\left\{\mathbf{y}^H_i\mathbf{y}_i\right\}}}
{\sigma^2_n+\sum\limits^{\tilde{L}}_{\tilde{l}=1}
{E\left\{\left(h_{\tilde{l}}\tilde{\mathbf{w}}_i\right)^Hh_{\tilde{l}}\tilde{\mathbf{w}}_i\right\}}} \nonumber \\
& =\dfrac{K/M}{\dfrac{\sigma^2_n}
{\sum\limits^{\tilde{L}}_{\tilde{l}=1}{E\left\{\left|h_{\tilde{l}}\right|^2\right\}}}
\!+\!\dfrac{2\sum\limits^{\tilde{L}}_{\tilde{l}=1}{E\left\{\left|h_{\tilde{l}}\right|^2\right\}
\mathrm{Tr}\!\!\left\{\mathbf{P}^{-1}_{\tilde{f}}\mathbf{B}_2\mathbf{B}^H_2
\left(\mathbf{P}^{-1}_{\tilde{f}}\right)^H
\!\!\mathbf{Q}^H_{\tilde{f}_{\tilde{l}}}
\mathbf{Q}_{\tilde{f}_{\tilde{l}}}\!\!\right\}}}
{M^2\sum\limits^{\tilde{L}}_{\tilde{l}=1}{E\left\{\left|h_{\tilde{l}}\right|^2\right\}}}}.
\label{Eqn:24}
\end{aligned}$$
![SINR analysis and simulations of the TD-NC-OFDM signals in the EVA fading channel, where the signal is modulated by 16-QAM.[]{data-label="Fig:6"}](Figure6.eps "fig:"){width="4in"} .
Figure \[Fig:6\] compares theoretical analysis in and the simulation results in terms of the SINR. The simulated Rayleigh channel employs the Extended Vehicular A (EVA) channel model [@Ref17], whose excess tap delay is \[0, 30, 150, 310, 370, 710, 1090, 1730, 2510\] ns with relative power \[0, -1.5, -1.4, -3.6, -0.6, -9.1, -7, -12, -16.9\] dB. It is shown that the theoretical analysis aligns well with the simulations. It also reveals that the SNR loss is negligible for the low-interference scheme. Moreover, the low-interference scheme has much better SINR than NC-OFDM and TD-NC-OFDM. In general, even if the length of $\tilde{w}_i(l)$ is increased, there is no extra need for signal recovery, which reduces the heavy computation load in original NC-OFDM.
Numerical Results
=================
This section presents simulation results to evaluate the PSD, complexity, and BER performance of NC-OFDM, TD-NC-OFDM and proposed low-interference schemes. Simulations are performed in a baseband-equivalent OFDM system with 256 subcarriers mapped onto the subcarrier index set $\left\{-128,-127,\ldots,127\right\}$. 16-QAM digital modulation is employed with a symbol period $T_s=1/15$ms, time-domain oversampling interval $T_{samp}=T_s/2048$ and CP duration $T_{cp}=144T_{samp}$. The PSD is evaluated by Welch’s averaged periodogram method with a 2048-sample Hanning window and 512-sample overlap after observing 105 symbols. To investigate the BER performance, the signal is transmitted through the Extended Vehicular A (EVA) channel model.
Figure \[Fig:7\] compares the PSD of NC-OFDM transmit signals with different *N* and different *L*. As *N* increases, the sidelobe suppression performance is further improved in the three methods. Moreover, the low-interference scheme can obtain as good sidelobe suppression performance as the conventional TD-NC-OFDM and NC-OFDM. Figure \[Fig:7\] also shows that with the increase of *L*, a steeper spectral roll-off can be obtained in the low-interference scheme. With a relatively small *L*, the sidelobe suppression of the low-interference scheme can approach that of TD-NC-OFDM, such as *N*=2 and 3 with *L*=144.
![PSDs of the transmit signals of NC-OFDM, TD-NC-OFDM and its low-interference scheme with varying *N* and varying *L*.[]{data-label="Fig:7"}](Figure7.eps "fig:"){width="4in"} .
Figure \[Fig:8\] presents the BER performances of NC-OFDM, TD-NC-OFDM and its low-interference scheme with varying *N* and varying *L* in the EVA channel. It is shown that the BER performance of the received signal is significantly degraded as *N* increases in NC-OFDM and TD-NC-OFDM. By contrast, the low-interference scheme causes slight BER performance degradation. Compared to the BER performance of NC-OFDM and TD-NC-OFDM with the high-complexity signal recovery [@Ref9], the increased length of the smooth signal just results in slight performance degradation for the low SNR loss in the low-interference scheme as mentioned in Section 4.3. Meanwhile, the low-interference scheme exhibits promising sidelobe suppression performance as shown in Figure \[Fig:7\].
![BERs of NC-OFDM, TD-NC-OFDM and its low-interference scheme with varying *N* and varying *L* in the EVA channel.[]{data-label="Fig:8"}](Figure8.eps "fig:"){width="4in"} .
Conclusion
==========
In this paper, a low-interference TD-NC-OFDM was proposed to reduce the interference and implementation complexity as opposed to the original TD-NC-OFDM and NC-OFDM. By adding the smooth signal, the *N*-continuous signal was obtained by the low-interference scheme. The smooth signal was designed by the linear combination of a basis set, which is generated by rectangularly pulsed OFDM basis signals truncated by a smooth window. Furthermore, using the continuity criterion, a closed-form spectrum expression was derived in the low-interference TD-NC-OFDM, which had more rapid decaying than [@Ref16]. Then the complexity of the low-interference scheme was measured. The received SINR is also measured by deriving the closed-form expression. Simulation results showed that the low-interference scheme was capable of effectively suppressing sidelobes as well as NC-OFDM and TD-NC-OFDM but with much better BER performance and much lower complexity. In this sense, the low-interference TD-NC-OFDM is a promising alternative to conventional NC-OFDM in future cognitive radio and carrier aggregation combined 5G systems.
Acknowledgment {#acknowledgment .unnumbered}
==============
This work was supported by the open research fund of National Mobile Communications Research Laboratory, Southeast University (No. 2013D05).
Appendix: Derivation of $\mathrm{Tr}\left\{E\left\{\tilde{\mathbf{w}}^H_i\tilde{\mathbf{w}}_i\right\}\right\}$ {#appendix-derivation-of-mathrmtrleftelefttildemathbfwh_itildemathbfw_irightright .unnumbered}
==============================================================================================================
For ease of exposition, Eq. is rewritten as $$\mathbf{b}_i
=\mathbf{P}^{-1}_{\tilde{f}}\left({\mathbf{P}}_1\mathbf{x}_{i-1}-\mathbf{P}_2\mathbf{x}_i\right)
\label{Eqn:25}$$ where ${\mathbf{P}}_1$ and $\mathbf{P}_2$ include a row of $n=0$.
According to the construction of ${\mathbf{P}}_1$ and $\mathbf{P}_{2}$, $\mathrm{Tr}\left\{E\left\{\tilde{\mathbf{w}}^H_i\tilde{\mathbf{w}}_i\right\}\right\}$ can be expressed as $$\begin{aligned}
\mathrm{Tr}\left\{E\left\{\tilde{\mathbf{w}}^H_i\tilde{\mathbf{w}}_i\right\}\right\} &=& \mathrm{Tr}\left\{E\left\{\mathbf{Q}_{\tilde{f}}\mathbf{b}_i
\mathbf{b}^H_i\mathbf{Q}^H_{\tilde{f}}\right\}\right\} \nonumber \\
&=& \mathrm{Tr}\left\{E\left\{\mathbf{Q}_{\tilde{f}}\mathbf{P}^{-1}_{\tilde{f}}{\mathbf{P}}_{1}
\mathbf{x}_{i-1}\mathbf{x}^H_{i-1}{\mathbf{P}}^H_{1}
\left(\mathbf{P}^{-1}_{\tilde{f}}\right)^H\mathbf{Q}^H_{\tilde{f}}\right\}\right\} \nonumber\\
& & + \mathrm{Tr}\left\{E\left\{\mathbf{Q}_{\tilde{f}}\mathbf{P}^{-1}_{\tilde{f}}\mathbf{P}_{2}
\mathbf{x}_{i}\mathbf{x}^H_{i}\mathbf{P}^H_{2}
\left(\mathbf{P}^{-1}_{\tilde{f}}\right)^H\mathbf{Q}^H_{\tilde{f}}\right\}\right\} \nonumber \\
&=& \mathrm{Tr}\left\{\mathbf{P}^{-1}_{\tilde{f}}{\mathbf{P}}_{1}{\mathbf{P}}^H_{1}
\left(\mathbf{P}^{-1}_{\tilde{f}}\right)^H
\mathbf{Q}^H_{\tilde{f}}\mathbf{Q}_{\tilde{f}}\right\} + \mathrm{Tr}\left\{\mathbf{P}^{-1}_{\tilde{f}}\mathbf{P}_{2}\mathbf{P}^H_{2}
\left(\mathbf{P}^{-1}_{\tilde{f}}\right)^H
\mathbf{Q}^H_{\tilde{f}}\mathbf{Q}_{\tilde{f}}\right\}.
\label{Eqn:26}\end{aligned}$$
Then, we rewrite ${\mathbf{P}}_1$ and $\mathbf{P}_2$ as $\mathbf{P}_1={1}/{M}\mathbf{B}_2$ and $\mathbf{P}_2={1}/{M}\mathbf{B}_2\mathbf{\Phi}$ with $\left\{\mathbf{B}_2\right\}_{n+1,r+1}=\left(j 2\pi k_r/M\right)^n$. We obtain $$\mathbf{B}_1=\mathbf{\Phi}^H\mathbf{B}^T_2,
\label{Eqn:27}$$ with $\left\{\mathbf{B}_1\right\}_{r+1,n+1}=\left(j2\pi k_r/M\right)^ne^{-j\varphi k_r}$.
According to , we arrive at $$\begin{aligned}
{\mathrm{Tr}\left\{E\left\{\tilde{\mathbf{w}}^H_i\tilde{\mathbf{w}}_i\right\}\right\}} &=& \frac{1}{M^2}\mathrm{Tr}\left\{\mathbf{P}^{-1}_{\tilde{f}}\mathbf{B}_2\mathbf{B}^H_2
\left(\mathbf{P}^{-1}_{\tilde{f}}\right)^H
\mathbf{Q}^H_{\tilde{f}}\mathbf{Q}_{\tilde{f}}\right\} + \frac{1}{M^2}\mathrm{Tr}\left\{\mathbf{P}^{-1}_{\tilde{f}}\mathbf{B}_2\mathbf{\Phi}\mathbf{\Phi}^H
\mathbf{B}^H_2\left(\mathbf{P}^{-1}_{\tilde{f}}\right)^H
\mathbf{Q}^H_{\tilde{f}}\mathbf{Q}_{\tilde{f}}\right\} \nonumber \\
&=& \frac{2}{M^2}\mathrm{Tr}\left\{\mathbf{P}^{-1}_{\tilde{f}}\mathbf{B}_2\mathbf{B}^H_2
\left(\mathbf{P}^{-1}_{\tilde{f}}\right)^H
\mathbf{Q}^H_{\tilde{f}}\mathbf{Q}_{\tilde{f}}\right\}.
\label{Eqn:28}\end{aligned}$$
[99]{}
Hwang T, Yang C, Wu G, et al. OFDM and its wireless applications: A survey. IEEE Trans Veh Technol, 2009, 58: 1673-1694 Wang C X, Haider F, Gao X, et al. Cellular architecture and key technologies for 5G wireless communication networks. IEEE Commun Mag, 2014, 52: 122-130 Wang Y, Li J, Huang L, et al. 5G mobile: spectrum broadening to higher-frequency bands to support high data rates. IEEE Trans Veh Technol, 2014, 9: 39-46 Hong X, Wang J, Wang C X, et al. Cognitive radio in 5G: a perspective on energy-spectral efficiency trade-off. IEEE Commun Mag, 2014, 52: 46-53 Yuan G, Zhang X, Wang W, et al. Carrier aggregation for LTE-advanced mobile communication systems. IEEE Commun Mag, 2010, 48: 88-93 Bogucka H, Wyglinski A M, Pagadarai S, et al. Spectrally agile multicarrier waveforms for opportunistic wireless access. IEEE Commun Mag, 2011, 49: 108-115 Weiss T, Hillenbrand J, Krohn A, et al. Mutual interference in OFDM-based spectrum pooling systems. In: The 59th IEEE Vehicular Technology Conference Spring (VTC 2004-Spring), Milan, 2004. 1873-1877 Brandes S, Cosovic I, Schnell M. Reduction of out-of-band radiation in OFDM systems by insertion of cancellation carriers. IEEE Commun Lett, 2006, 10: 420-422 Qu D, Wang Z, Jiang T. Extended active interference cancellation for sidelobe suppression in cognitive radio OFDM systems with cyclic prefix. IEEE Trans Veh Technol, 2010, 59: 1689-1695 Ma M, Huang X, Jiao B, et al. Optimal orthogonal precoding for power leakage suppression in DFT-based systems. IEEE Trans Commun, 2011, 59: 844-853 Zhang J, Huang X, Cantoni A, et al. Sidelobe suppression with orthogonal projection for multicarrier systems. IEEE Trans Commun, 2012, 60: 589-599 Chung C D. Spectrally precoded OFDM. IEEE Trans Commun, 2006, 54: 2173-2185 Beek de van J, Berggren F. *N*-continuous OFDM. IEEE Commun Lett, 2009, 13: 1-3 Beek de van J, Berggren F. EVM-constrained OFDM precoding for reduction of out-of-band emission. In: The 70th Vehicular Technology Conference Fall (VTC 2009-Fall), Anchorage, 2009. 1-5 Beek de van J. Sculpting the multicarrier spectrum: a novel projection precoder. IEEE Commun Lett. 2009, 13: 881-883 Ohta M, Iwase A, Yamashita K. Improvement of the error characteristics of an *N*-continuous OFDM system with low data channels by SLM. In: The 2011 IEEE International Conference on Communications (ICC 2011), Kyoto, 2011. 1-5 Ohta M, Okuno M, Yamashita K. Receiver iteration reduction of an *N*-continuous OFDM system with cancellation tones. In: The 2011 IEEE Global Telecommunications Conference (GLOBECOM 2011), Kathmandu, 2011. 1-5 Zheng Y, Zhong J, Zhao M, et al. A precoding scheme for *N*-continuous OFDM. IEEE Commun Lett, 2012, 16: 1937-1940 Wei P, Dan L, Xiao Y, et al. A Low-Complexity Time-Domain Signal Processing Algorithm for *N*-continuous OFDM. In: The 2013 IEEE International Conference on Communications. (ICC 2013), Budapest, 2013. 5754-5758 Bracewell R, The Fourier Transform and its applications, 2nd ed. New York: McGraw-Hill, 1978. 143-146 User Equipment (UE) radio transmission and reception (Release 12), 3GPP TS 36.101, v12.3.0, 2014. \[Online\]. Available: http: //www.3gpp.org/
[^1]: The authors are with the school of National Key Laboratory of Science and Technology on Communications, University of Electronic Science and Technology of China, Chengdu, China (e-mail: wpwwwhttp@163.com; {lilindan, xiaoyue}@uestc.edu.cn). Wei Xiang is with the school of Mechanical and Electrical Engineering Faculty of Health, Engineering and Sciences, University of Southern Queensland, Austrialia (e-mail: wei.xiang@usq.edu.au).
|
---
abstract: 'In this paper, we explore the problem of identifying substitute relationship between food pairs from real-world food consumption data as the first step towards the healthier food recommendation. Our method is inspired by the distributional hypothesis in linguistics. Specifically, we assume that foods that are consumed in similar contexts are more likely to be similar dietarily. For example, a turkey sandwich can be considered a suitable substitute for a chicken sandwich if both tend to be consumed with french fries and salad. To evaluate our method, we constructed a real-world food consumption dataset from MyFitnessPal’s public food diary entries and obtained ground-truth human judgements of food substitutes from a crowdsourcing service. The experiment results suggest the effectiveness of the method in identifying suitable substitutes.'
author:
- |
Palakorn Achananuparp\
\
\
Ingmar Weber\
\
\
bibliography:
- 'main.bib'
title: Extracting Food Substitutes From Food Diary via Distributional Similarity
---
<ccs2012> <concept> <concept\_id>10002951.10003317.10003347.10003350</concept\_id> <concept\_desc>Information systems Recommender systems</concept\_desc> <concept\_significance>500</concept\_significance> </concept> </ccs2012>
Introduction {#sec:introduction}
============
Food Substitute Extraction {#sec:method}
==========================
Data Collection & Processing {#sec:data}
============================
Experiments {#sec:experiments}
===========
Results & Discussion {#sec:results}
====================
Conclusion {#sec:conclusion}
==========
Acknowledgements {#sec:ack}
================
This work is supported by the National Research Foundation under its International Research Centre @ Singapore Funding Initiative and administered by the IDM Programme Office.
|
---
author:
- |
\
GSI Helmholtzzentrum für Schwerionenforschung GmbH,\
Planckstraße 1, 64291 Darmstadt, Germany\
E-mail:
- |
Yonggoo Heo\
Suranaree University of Technology, Nakhon Ratchasima, 30000, Thailand\
E-mail:
- |
Matthias F.M. Lutz\
GSI Helmholtzzentrum für Schwerionenforschung GmbH,\
Planckstraße 1, 64291 Darmstadt, Germany\
Technische Universität Darmstadt, D-64289 Darmstadt, Germany\
E-mail:
bibliography:
- 'thesis.bib'
title: 'On chiral extrapolations of coupled-channel reaction dynamics for charmed mesons'
---
Introduction
============
Open-charm meson systems are believed to be of crucial importance in understanding the low-energy behavior of QCD [@Yan:1992gz; @Lutz:2015ejy]. A heavy charm quark is surrounded by a light quark, either a $u, d$ or $s$ quark. The interplay of the heavy-quark spin symmetry for the charm quark and the chiral SU(3) symmetry for the $u, d$ or $s$ quarks make such systems unique and effective field theory approaches particularly predictive. The leading order chiral Lagrangian already produces significant short-range and attractive forces that may dynamically generate open-charm resonances. Within the coupled-channel approach [@Lutz:2001yb], the scalar meson $D_{s0}^*(2317)$ is well described by the leading order chiral interaction [@Kolomeitsev:2003ac; @Hofmann:2003je; @Lutz:2007sk; @Guo:2006fu; @Guo:2015dha; @Du:2017ttu]. At this leading order further predictions are made on exotic scalar resonances in a flavour sextet [@Kolomeitsev:2003ac; @Hofmann:2003je; @Lutz:2007sk]. The existence of such states depends on the precise form of chiral forces generated by higher order chiral counter terms [@Hofmann:2003je; @Lutz:2007sk; @Liu:2012zya; @Lutz:2015ejy]. The aim of this work is to report on the impact of lattice data from [@Mohler:2011ke; @Na:2012iu; @Kalinowski:2015bwa; @Moir:2016srx] on such counter terms and their implications for coupled-channel open-charm systems.
We notice that the counter terms have significant impact not only on the open-charm coupled channel systems but also on the quark-mass dependence of the $D$-meson masses. The lattice data sets on charmed meson masses from [@Mohler:2011ke; @Na:2012iu; @Kalinowski:2015bwa] are considered and play a crucial role in our derivation of a set of the low-energy parameters. Such data are supplemented by first lattice data from HSC on the s-wave $\pi D$ scattering process at a pion mass of about 400 MeV [@Moir:2016srx]. Based on our fit scenarios, we present the pole positions found for the scalar open-charm mesons in the exotic flavour sextet channels as expected for the physical choice of the quark masses. A striking quark-mass dependence of such states is illustrated by presenting the form of the s-wave $\pi D$ phase shift as extrapolated down from the considered HSC ensemble at a pion mass of about 400 MeV to the physical point.
The chiral Lagrangian for open-charm mesons {#sec:2}
===========================================
We recall the SU(3) chiral Lagrangian formulated in the presence of anti-triplets of $D$ mesons with $J^P =0^-$ and $J^P =1^-$. In the relativistic version the Lagrangian was developed in [@Kolomeitsev:2003ac; @Hofmann:2003je; @Lutz:2007sk]. The chiral Lagrangian used in [@Guo:2018kno] considers the anti-triplets fields, $D$ and $D_{\mu \nu}$, of charmed mesons with $J^P= 0^-$ and $J = 1^-$ quantum numbers respectively. The terms relevant here read $$\begin{aligned}
&& \mathcal{L}_{}=({\hat \partial}_\mu D)({\hat \partial}^\mu \bar D)- M^2\, D \, \bar D
+ 2\,g_P\,\big\{D_{\mu \nu}\,U^\mu\,({\hat \partial}^\nu \bar D)
- ({\hat \partial}^\nu D )\,U^\mu\,\bar D_{\mu \nu} \big\}
\nonumber\\
&& \quad - \,\big( 4\,c_0-2\,c_1\big)\, D \,\bar{D} \,{{\rm tr \,}} \chi_+ -2\,c_1\,D \,\chi _+\,\bar{D}
+ \, 4\,\big(2\,c_2+c_3\big)\,D\bar{D}\,{{\rm tr \,}} \big(U_{\mu }\,U^{\mu \dagger }\big)- 4\,c_3\, D \,U_{\mu }\,U^{\mu \dagger }\,\bar{D}
\nonumber\\
&& \quad +\,\big( 4\,c_4+2\,c_5\big)\, ({{\hat \partial}_\mu } D) ({{\hat \partial}_\nu }\bar{D}) \,{{\rm tr \,}} \big[ U^{\mu }, \,U^{\nu \dagger }\big]_+ /M^2
-2\,c_5\,({{\hat \partial}_\mu } D) \big[ U^{\mu }, \,U^{\nu \dagger }\big]_+({{\hat \partial}_\nu }\bar{D}) /M^2
\nonumber\\
&& \quad + \, 4\,g_1\,{D}\,[\chi_-,\,{U}_\nu]_- {\hat \partial}^\nu \,\bar{D}/M
- 4\,g_2\,{D}\,\big([{U}_\mu,\,[{\hat \partial}_\nu,\,{U}^\mu]_-]_- + [{U}_\mu,\,[{\hat \partial}^\mu,\,{U}_\nu]_-]_-\big)\,{\hat \partial}^\nu\bar{D}/M
\nonumber\\
&& \quad - \,4\,g_3\,{D}\,[{U}_\mu,\,[{\hat \partial}_\nu,\,{U}_\rho]_-]_-\,[{\hat \partial}^\mu,\,[{\hat \partial}^\nu,\,{\hat \partial}^\rho]_+]_+\bar{D}/M^3
+ {\rm h.c.}\,,
\label{def-kin}\end{aligned}$$ where $$\begin{aligned}
&& U_\mu = {\textstyle \frac{1}{2}}\,e^{-i\,\frac{\Phi}{2\,f}} \left(
\partial_\mu \,e^{i\,\frac{\Phi}{f}} \right) e^{-i\,\frac{\Phi}{2\,f}} \,, \qquad \qquad
\Gamma_\mu ={\textstyle \frac{1}{2}}\,e^{-i\,\frac{\Phi}{2\,f}} \,\partial_\mu \,e^{+i\,\frac{\Phi}{2\,f}}
+{\textstyle \frac{1}{2}}\, e^{+i\,\frac{\Phi}{2\,f}} \,\partial_\mu \,e^{-i\,\frac{\Phi}{2\,f}}\,,
\nonumber\\
&& \chi_\pm = {\textstyle \frac{1}{2}} \left(
e^{+i\,\frac{\Phi}{2\,f}} \,\chi_0 \,e^{+i\,\frac{\Phi}{2\,f}}
\pm e^{-i\,\frac{\Phi}{2\,f}} \,\chi_0 \,e^{-i\,\frac{\Phi}{2\,f}}
\right) \,, \qquad \chi_0 =2\,B_0\, {\rm diag} (m_u,m_d,m_s) \,,
\nonumber\\
&& {\hat \partial}_\mu \bar D = \partial_\mu \, \bar D + \Gamma_\mu\,\bar D \,, \qquad \qquad \qquad \quad \;\;
{\hat \partial}_\mu D = \partial_\mu \,D - D\,\Gamma_\mu \,.
\label{def-chi}\end{aligned}$$ The quark masses enter via $\chi_\pm$ and the octet of the Goldstone bosons is encoded into the $3\times3$ matrix $\Phi$. The covariant derivative ${\hat \partial}_\mu$ in the kinetic term of the $D$ mesons generates the leading order two-body chiral interaction, recognized as the Weinberg-Tomozawa interaction. Its interaction strength is determined by the parameter $f$, the chiral limit value of the pion-decay constant. The parameter $M$ measures the mass of the $D$ mesons in the chiral limit, provided that a suitable renormalization scheme is applied [@Lutz:2018cqo; @Guo:2018kno]. We consider the isospin limit with $m_u = m_d = m$. The hadronic decay width of the charged $D^*$-meson implies $|g_P| = 0.57 \pm 0.07 $ [@Lutz:2007sk]. Further symmetry breaking counter terms involving two $\chi_+$ fields are not shown in (\[def-kin\]) but are systematically considered in [@Guo:2018kno]. In the latter work it was illustrated in great detail that a chiral decomposition of the charmed meson masses is well converging if it is organized in terms of on-shell meson masses, rather than bare masses. This implies that the derivation of the charmed mesons masses on a given lattice ensemble requires the solution of a coupled and non-linear set of four equations.
The role of subleading operators of chiral order $Q^3$ as introduced first in [@Du:2017ttu] was explored. Such terms, proportional to $g_i$ in (\[def-kin\]), do not affect the charmed-meson masses but do impact the scattering phase shifts.
-0.2cm
Fit to QCD lattice data
=======================
We determine the LECs of the chiral Lagrangian from lattice QCD simulations of the $D$-meson masses. HPQCD provides a data set for the pseudoscalar $D$-meson masses [@Na:2012iu], based on the MILC AsqTad ensembles [@Bazavov:2009bb]. Those ensembles are also used in [@Liu:2012zya], however employing domain-wall quarks as set up by LHPC. This work provides the pseudoscalar charmed meson masses but also some s-wave scattering lengths, that characterize the low-energy interaction of the charmed mesons with the Goldstone bosons. Based on the PACS-CS ensembles, the masses for charmed mesons with both $J^P=0^-$ and $1^-$ are calculated in [@Mohler:2011ke; @Lang:2014yfa]. An even richer data set derived on the ETMC ensembles is published in [@Kalinowski:2015bwa]. So far charmed meson masses were computed on one HSC ensemble only [@Moir:2016srx]. However, besides the masses, this work also considers the $\pi D$ scattering process on that ensemble.
Fit 1 Fit 2 Fit 3 Fit 4
------------ --------- --------- --------- ---------
$ M\;\;$ 1.8762 1.9382 1.9089 1.8846
$ c_0$ 0.2270 0.3457 0.2957 0.3002
$ c_1$ 0.6703 0.9076 0.8765 0.8880
$ c_2$ -0.6031 -2.2299 -1.6630 -1.3452
$ c_3$ 1.2062 4.5768 3.3260 3.0206
$ c_4$ 0.3644 2.0012 1.2436 0.9122
$ c_5$ -0.7287 -4.1445 -2.4873 -2.1393
$ g_1\;\;$ 0 0 0.4276 0.4407
$ g_2\;\;$ 0 0 1.0318 0.8788
$ g_3\;\;$ 0 0 0.2772 0.2003
: The low-energy constants (LEC) from four fit scenarios as explained in [@Guo:2018kno]. Each parameter set reproduces the isospin average of the empirical D and $D^*$ meson masses from the PDG. The value $f = 92.4$ MeV was used in [@Guo:2018kno]. []{data-label="tab:1"}
We perform chiral extrapolations based on the finite-box framework originally set up for chiral extrapolations of the baryon masses [@Lutz:2014oxa]. Discretization effects are not implemented so far. Only the data sets with the pion and kaon masses smaller than about 600 MeV are considered. The empirical $D$-meson masses from the PDG are used as constraints in our analysis defining our non-standard lattice scale settings. The low-energy constants (LEC) are obtained by a global fit to the QCD lattice data set. Most ensembles suffer from a sizeable uncertainty in the choice of their charm quark mass, not always hitting its physical value. Therefore our fits consider in most cases mass splittings of the charmed mesons only. An *ad hoc* systematic error for the charmed-meson masses is imposed. By requiring that the $\chi^2$ per data point turns close to one, we arrive at our estimate of 5-10 MeV for the latter.
For a given ensemble the quark masses, $m= (m_u+ m_d)/2$ and $m_s$, are determined from their published pion and kaon masses. This involves the low-energy parameters of Gasser and Leutwyler, for which we derived particular values in [@Guo:2018kno]. The physical quark-mass ratio $m_s/m$ is compatible with the latest result of ETMC [@Carrasco:2014cwa] with $m_s/m = 26.66(32)$ always. In Fig.\[fig:ratio\], the quark-mass ratio $m_s/m$ on various lattice ensembles are shown. All our four fit scenarios in [@Guo:2018kno] lead to a good agreement with the lattice results, even though such ratios did not enter any of our chisquare functions. This is an important result, since it justifies our choices for the quark masses on the HSC ensemble. Note that such ratios are not available from HSC directly.
In Tab.\[tab:1\] we recall our values for the low-energy parameters $M $, $c_i$ and $g_i$ from [@Guo:2018kno]. All four scenarios recover the open-charm meson masses on the various ensembles as discussed above but also the s-wave scattering lengths from [@Liu:2012zya]. In addition, Fit 2-4 are adjusted to the scattering phase shifts of HSC [@Moir:2016srx]. In Fit 3 and 4, the subleading counterterms (\[def-kin\]) are activated. We note that Fit 1 and 3 imposes the relations $c_2 = -c_3/2$ and $c_4 = -c_5/2$ which hold in the large $N_c$ limit of QCD.
Phase shifts and poles in the complex plane
===========================================
In this section we discuss the coupled-channel dynamics of $J^P=0^+$ charmed meson. We apply the on-shell reduction scheme as developed in[@Lutz:2001yb] to derive the coupled-channel unitarized scattering amplitude. This approach rests on a matching scale $\mu_M$, the natural value of which is given in [@Kolomeitsev:2003ac]. Given the set of low-energy parameters in Tab. \[tab:1\] phase shifts and inelasticity parameters are determined in all flavous sectors characterized by isospin ($I$) and strangeness ($S$) quantum numbers.
$(I,S)= (1,1)$ $(I,S)= (1/2,0)$ $(I,S)= (0,-1)$
------- ---------------------------------------- -------------------------------------- --------------------------------------
WT $2.488_{-19}^{+22}-0.083_{-5}^{+14}i$ $2.390_{-17}^{+20}-0.038_{-1}^{+0}i$ $2.335_{+15}^{-43}$
Fit 1 $2.542_{-16}^{+15}-0.114_{-9}^{+19}i$ $2.471_{-7}^{+8}-0.046_{-3}^{+7}i$ $2.360_{-0}^{-1}-0.143_{-14}^{+17}i$
Fit 2 $2.450_{-9}^{+8}-0.297_{-8}^{+10}i$ $2.460_{-11}^{+17}-0.152_{+2}^{-5}i$ $2.287_{+2}^{-4}-0.124_{-12}^{+14}i$
Fit 3 $2.389_{-9}^{+6}-0.336_{-6}^{+11}i$ $2.463_{-27}^{+37}-0.106_{+6}^{-8}i$ $2.230_{+3}^{-4}-0.121_{-11}^{+13}i$
Fit 4 $2.382_{-10}^{+10}-0.322_{-10}^{+12}i$ $2.439_{-32}^{+42}-0.092_{+3}^{-7}i$ $2.229_{+3}^{-4}-0.083_{-11}^{+13}i$
: Pole masses of the $0^+$ meson resonances in the flavour sextet channels, in units of GeV. The $(1,1), (1/2,0), (0,-1)$ poles are located on the $(-,+), (-,-,+), (-)$ sheets respectively according to the notation used in [@Guo:2018gyd]. The asymmetric errors are estimated by varying the matching scale $\mu_M$ around its natural value by 0.1 GeV. With ’WT’ we refer to the leading order scenario that relies on the parameter $f = 92.4$ MeV only.[]{data-label="tab:2"}
-0.2cm
In this Proceeding we focus on the $\pi D$ phase shift with $(I, S) =(1/2, 0)$. The phase shift and inelasticities generated by the leading order Weinberg-Tomozawa interaction are shown in Fig.\[fig:phases\_a\]. The rapid rise of the phase shift through 90$^\circ$ and 180$^\circ$ reflects the presence of a broad and a narrow resonance state in this channel. The narrow one seen around the $\eta D$ threshold is a member of the flavour sextet. The uncertainty band is implied by a variation of $\pm 0.1$ GeV in the matching scale $\mu_M$ around its natural value [@Kolomeitsev:2003ac]. In comparison, the $\pi D$ phase shift is shown in Fig.\[fig:phases\_b\] from our preferred Fit 4. The result at physical quark masses is shown in a black sold line. We clearly see a signal of a resonance in between the $\eta D$ and $\bar K D_s$ thresholds. Most striking are our predictions for the quark-mass dependence of the $\pi D$ phase shift. We present the phase shifts at different unphysical quark masses in dashed and dotted lines in Fig.\[fig:phases\_b\].
In all the sextet channels, poles are found in the lower complex plane, following the analytic continuation method illustrated in[@Guo:2018gyd]. The pole masses are listed in Tab.\[tab:2\]. The pole at $(I,S)=(1/2,0)$ sector is lying well between the $\eta D$ and $\bar K D_s$. The width depends on fitting scenarios, but is always significantly smaller than the antitriplet partner in the same channel. The latter exhibits a pole at $(2.082_{+2}^{-9} -0.187_{-28}^{+44}i)$GeV (Fit 4) where the asymmetric error is implied by a $\pm 0.1$GeV deviation of $\mu_M$ from their natural values [@Kolomeitsev:2003ac].
-0.2cm
Summary
=======
We studied the chiral extrapolation of charmed meson masses based on the three-flavour chiral Lagrangian. About 80 lattice data points from 5 different lattice groups are analyzed. Such data pose a significant constraint on the low-energy constants of the chiral Lagrangian formulated for charmed meson fields. The implication of higher order counter terms in the coupled-channel dynamics of the open-charm sector of QCD is explored. A striking quark-mass dependence of phase-shifts and inelasticity parameters is derived. At the physical point we predict a clear signal for the flavour sextet in the $\pi D$ phase shift with a pole lying in the complex plane between the $\eta D$ and $\bar K D_s$ thresholds.
|
---
abstract: 'We report on our search for very-long-term variability (weeks to years) in X-ray binaries (XRBs) in the giant elliptical galaxy M87. We have used archival [*Chandra*]{} imaging observations to characterise the long-term variability of 8 of the brightest members of the XRB population in M87. The peak brightness of some of the sources exceeded the ultra luminous X-ray source (ULX) threshold luminosity of $\sim 10^{39}$ erg s$^{-1}$, and one source could exhibit dips or eclipses. We show that for one source, if it has similar modulation amplitude as in SS433, then period recoverability analysis on the current data would detect periodic modulations, but only for a narrow range of periods less than 120 days. We conclude that a dedicated monitoring campaign, with appropriately defined sampling, is essential if we are to investigate properly the nature of the long-term modulations such as those seen in Galactic sources.'
author:
- |
D. L. Foster,$^{1,2}$[^1] P. A. Charles,$^{3}$ D. A. Swartz,$^{4}$ R. Misra,$^{5}$ and K. G. Stassun,$^{2,6}$\
\
$^{1}$South African Astronomical Observatory, PO Box 9, Observatory, 7935, South Africa\
$^{2}$Vanderbilt University, Department of Physics & Astronomy, 1807 Station B, Nashville, TN, 37235, USA\
$^{3}$University of Southampton, School of Physics & Astronomy, Southampton, Hampshire, SO17 1BJ, UK\
$^{4}$Universities Space Research Association, NASA Marshall Space Flight Center, ZP12, Huntsville, AL, 35805, USA\
$^{5}$Inter University Centre for Astronomy and Astrophysics, Pune University Campus, Pune, India\
$^{6}$Fisk University, Department of Physics, 1000 17th Avenue North, Nashville, TN, 37208, USA
date: 'Accepted 2013 March 27. Received 2013 March 26; in original form 2013 February 16'
title: 'Monitoring the Very-Long-Term Variability of X-ray Sources in the Giant Elliptical Galaxy M87'
---
\[firstpage\]
accretion, accretion discs – black hole physics – X-rays: binaries
Introduction
============
[*Chandra’s*]{} sub-arcsecond resolution routinely images tens to hundreds of X-ray sources in individual observations of galaxies out to Virgo-cluster distances down to detection limits of $\sim 10^{37}$ erg s$^{-1}$. In analogy with the Milky Way, these are mostly X-ray binaries [@Fabbiano:2006a; @Fabbiano:2006b]. Detection limits are of order 10 counts, corresponding to count rates of several $10^{-4}$ c s$^{-1}$ for typical exposure times of 10 to 100 ks so that meaningful lightcurves can only be obtained for the most luminous sources and only over time intervals of order 1 day at most. In rare cases, these lightcurves display periodic dips or other structures that can be used to constrain properties of the XRBs [e.g. @Trudolyubov:2002].
Multi-epoch X-ray observations of nearby galaxies allows for the study of the long-term variability of their X-ray source populations. We know from previous temporal and spectral analysis on the most luminous XRBs in nearby galaxies that they exhibit behaviours similar to what is seen in the Galactic XRBs [@Fabbiano:2006a and references therein]. For example, [@Kong:2002] find that among the 204 sources detected in the nearby spiral galaxy M31, 50 per cent are variable on the time-scale of several weeks and 13 are transients. We expect that ultraluminous X-ray sources [ULXs, defined as extranuclear point-like sources of X-rays with $L_X \gtrsim 10^{39}$ erg s$^{-1}$; e.g. @Fabbiano:2006b] exhibit similar very-long-term modulations to those seen in the Galactic XRB population.
Owing to the lack of regular, multi-epoch observations of extragalactic XRBs, not much is known about their variability on time-scales of tens to hundreds of days [@Fridriksson:2008]. Shorter time-scales can also be challenging for a variety of reasons: (a) count rates are always too low to see very short fluctuations like pulse periods; (b) a few short, presumably orbital, modulations have been detected in rare cases in which the modulations occur 2–3 or more times within a single observation; however, these required exposure times in the 10–100 ks range, yet the typical [*Chandra*]{} exposure times reported here are less than this; and (c) transient-like behaviour–in which variations in luminosity are by a factor of 100 or greater–can almost always only be seen in multi-epoch exposures [typically separated by of order 100 days; e.g. @Barnard:2012a]. Notable exceptions are some of the super-soft sources that vary rapidly within a given observation as well as between observations [@Swartz:2002; @Fabbiano:2003; @Mukai:2003; @Fabbiano:2006a; @Orio:2006; @Carpano:2007]. Thus, there are only a handful of examples of category (b) and (c). However, [@Kong:2011] demonstrated that regular pointed X-ray observations can be successfully deployed as monitoring instruments for extragalactic XRBs. This is particularly true of [*Chandra X-ray Observatory*]{} because of its superior resolution.
Longer term, non-orbital periodicities have been observed in both LMXBs and HMXBs, are termed [*superorbital*]{}, and range from tens to hundreds of days [@Wen:2006; @Charles:2010]. The study of these modulations are a probe of the physical processes occurring within both the donor star and the accretion disc, and also a probe of the interaction of the intense X-ray luminosity with the accretion flow. These superorbital modulations display different properties depending on the type of interacting binary in which they are found [@Charles:2010]. Highly relevant here are systems which display long-term superorbital modulations due to the coupled precession of their accretion discs and relativistic jets, such as the 35-d modulation of [@Petterson:1977] and the 162-d modulation of SS433 [@Margon:1984]; this mechanism may be responsible for the 115-d modulation seen in the ULX [@Foster:2010]. Only with a properly designed long-term monitoring program can we distinguish between the competing physical explanations of these superorbital periods.
To date, there have been extensive investigations of low-mass X-ray binaries (LMXBs) in M87 [@Jordan:2004b; @Jordan:2004]. Some studies contain observations separated by $\sim$ years, but without regularly repeated visits (typical of monitoring campaigns) extended over a baseline of many years. These have primarily focused on the pointing-to-pointing variability of the spectra and fluxes of LMXBs, and their association with globular cluster systems within M87. Other important work on these XRBs includes the nature of the apparently years-long transient outbursts discovered in these sources [@Irwin:2006]. Better samplings have been obtained for LMXBs in globular clusters near the central region of M31 [@Barnard:2012b].
Monitoring of XRBs in the Milky Way has led to numerous major discoveries. For example, the discovery within our own Galaxy of relativistic jets and their superluminal motion in an accreting compact binary [@Mirabel:1994] provided a link to the physics of distant quasars that are thought to behave in a similar (yet much more powerful) manner. The discovery of warping and precessing of the accretion discs was revealed through monitoring the long-term periodic variations in systems like SS433, , and [@Margon:1980; @Cowley:1991; @Gerend:1976]. And, the discovery of sub-millisecond quasi-periodic oscillations (QPOs) in [@Klis:1996] demonstrated a probe of the dynamics of the inner accretion flow in a XRB. As was shown in [@Margon:1980], only with regular, sustained monitoring is it possible to measure longer-term (quasi-) periodicities that range from weeks to hundreds of days, with greater precision for the longer variations such as that achieved for SS433 coming only after years (up to decades) of follow-up observations [@Eikenberry:2001].
[*Chandra*]{} has now studied M87 many times over $\sim$ 10 years; this has the potential for showing long-term modulations or transient behaviour in several XRBs in M87 [NGC 4486, $D = 17$ Mpc; @Tully:2008]. We have selected a handful of the brightest sources in this galaxy and searched archival data to study their very long-term variability.
Methods
=======
Observations and data reduction
-------------------------------
The [CIAO]{} data reduction threads were used to process the data, generating level-2 events lists for each observation. We used version 4.4.1 of [CIAO]{} and version 4.5.1 of [CALDB]{} to reprocess the data. All the data presented here were recorded with [*Chandra’s*]{} Advanced CCD Imaging Spectrometer (ACIS) S3 chip.
The data span the period 2000 July 29 to 2010 May 14. With a handful of exceptions, the 85 independent observations included in this analysis were typically 5 ks in duration. Chandra detects photons at a rate of about $10^{-4}$ c s$^{-1}$ for a source of about $10^{-15}$ erg cm $^{-2}$ s$^{-1}$ or only about 15 counts from a borderline ULX ($\sim 10^{39}$ erg s$^{-1}$) in these 5 ks observations. Also, many of the observations were performed in subarray mode (mostly due to the brightness of the core and jet of M87) and with varying roll angles. This led to somewhat sporadic coverage for a few sources.
To test for the presence of periods of high background or strong flares, we excluded the regions of the chip containing the core and jet of M87, as well as regions containing point sources detected by the [CIAO]{} tool `wavdetect`, then examined the lightcurve of the remaining events of the observation (binning of 200s) using the tool `lc_sigma_clip`, rejecting any events that are beyond 5$\sigma$ from the mean count rates. No significant flares or periods of high background are detected. This means that the total exposure accumulated during the good time intervals (GTIs) as recomputed by `lc_sigma_clip` are nearly identical to the value of the keyword `EXPOSURE` in each observation’s FITS file, and this time was used to compute the mean count rate.
Source selection
----------------
Eight sources were visually selected for analysis based on their apparent relative brightness in the X-ray images, their higher sampling given the roll angles of the observations, and their being distinct from either the nucleus of the galaxy or the jet (including knots or other clumps of diffuse emission). All of these sources appear in the catalogue of sources analysed in [@Jordan:2004]. The X-ray sources are listed in Table \[summary\], and are hereafter individually referred to by the labels indicated therein (X1, X2, et cetera).
Extraction of source count rates
--------------------------------
Source regions were defined as 1.5-arcsec circles centred on the identified point sources. The background regions were defined for the sources as annuli centred on the sources with inner and outer radii of 1.5 and 2.5 arcsec, respectively, to account for the diffuse emission that surrounds some of the sources while minimizing the effects of overlapping source and background regions. A spatial extraction of the total number of counts in the source and background regions, in the 0.5–8.0 keV range, was performed using the [CIAO]{} tool `dmextract`, with the error in the counts estimated using Gehrels’ method [@Gehrels:1986]. The larger of the asymmetric Poisson errors is used for all calculations, so as to be conservative. Each point on the lightcurves represents the mean count rate (in c s$^{-1}$) of an individual observation. Data points on the lightcurves that are consistent with zero counts per second to within twice their 1$\sigma$ Poisson errors are denoted by arrow markers on all lightcurves, and we treat these data as upper limits. An example is shown in Figure \[lightcurves\], which is the long-term lightcurve for the source X7.
The largest distance of any of these sources from the focal point (i.e. the narrowest point-spread function, or PSF) of the ACIS S3 chip is $\approx 3$ arcmin. We estimate the 90 per cent encircled energy radius of a 1.5-keV event to be 1.0 arcsec on-axis and 2.0 arcsec at 3 arcmin off-axis, and that for a 6.4-keV event to be 2.0 arcsec on-axis and 2.6 arcsec at 3 arcmin off-axis. We checked the most off-axis sources for any differences in period detection arising from the factor of 2 increase in the radii of the extraction apertures and background annuli, finding none. In the event that a particular source was not on the active subarray, there was no data point recorded for such a source. We confirmed each off-subarray instance visually with after inspecting all extractions returning exactly zero counts for both source and background regions.
Analysis & Results {#analysis}
==================
Peak X-ray Flux and X-ray Luminosity {#flux}
------------------------------------
Table \[summary\] shows the observations having the largest X-ray flux and luminosity for each source. $F_{\rmn{max}}$ (in units of $10^{-14}$ erg cm$^{-2}$ s$^{-1}$) and $L_{\rmn{max}}$ (in units of $10^{39}$ erg s$^{-1}$) are the maximum values of the net model-independent (i.e., directly summed) X-ray flux and luminosity, respectively, over the entire baseline of observations. The sum of the flux within each source and background region was computed using the [CIAO]{} tools `eff2evt` and `dmstat`. The source regions are defined exactly as in the lightcurve extraction step and background fluxes were estimated for a region of the same radius near the source using . The events were filtered on the 0.5–8 keV band. The fluxes are estimates based on the most likely energy of each arriving photon, and do not take into account a spectral model and the redistribution matrix. The luminosities are scaled to the assumed distance to M87 of 17 Mpc and uncorrected for absorption. All of our analysis below uses only the relative fluxes to search for variability and periodicity. The OBSID of the data sets corresponding to the peak values $F_{\rmn{max}}$ and $L_{\rmn{max}}$ for each point source is given as a reference.
Statistical analysis
--------------------
Having invoked the Gehrels error estimation method in the counts extraction step, the Poisson errors in the case of low counts are properly taken into account. To be conservative, the larger of the asymmetric Poisson errors is used.
Each lightcurve was fitted with a horizontal line representing the median count rate for the individual sources to determine how well they agree with being constant. The results are shown in Table \[summary\]. In the goodness-of-fit analysis for $\chi^2$, the null hypothesis is rejected for each source except for X3, indicating that the sources are very unlikely to be non-variable.
![Long-term lightcurve for X7. Events were extracted in the 0.5–8.0 keV band. Each point on the lightcurves represents the mean count rate (in c s$^{-1}$) of an individual observation. The level of zero counts per second is represented by the dashed line. Arrow markers represent data that are consistent with zero counts per second, and the arrow length is the measurement error associated with the particular data point. In our analysis, we treat these points as upper limits. Appendix A contains the information for all the sources.[]{data-label="lightcurves"}](plot_cwzero_net_counts_N96.eps){width="90mm"}
Period search {#periodsearch}
-------------
Lomb-Scargle periodogram (LSP) analysis [@Lomb:1976; @Scargle:1982; @Press:1989] was applied to all extracted lightcurves and the resulting power spectra were searched for periods ranging from 30 days to 5 years. Figure \[lsp\] is the LSP of the data in Figure \[lightcurves\]. In order to identify spurious period detections due to sampling, each power spectrum was compared to its corresponding window function power spectrum. Figure \[windowN96\] is the window function of the data in Figure \[lightcurves\] corresponding to the power spectrum in Figure \[lsp\].
To determine the white noise levels, Monte Carlo simulations of 10,000 randomly generated data sets for each source were made using the time values, and the mean and standard deviation of the count rate values. LSPs were computed for the random data sets, and the resulting maxima in the power spectra are used to construct a cumulative probability distribution function, which describes the false-alarm probability (FAP) associated with each value of the power. Hence, a 3$\sigma$ confidence level is the power associated with FAP of 0.27 per cent. For our lightcurves, the power typically associated with FAP of 0.27 per cent is $\approx$ 6–10. For the source X6, the number of data points is very small, and the computed 3$\sigma$ confidence level is suspiciously low and perhaps not the “true” value.
![Lomb-Scargle periodogram for the data plotted in Figure \[lightcurves\]. The frequencies correspond to periods ranging from 30 d to 5 yr. The highest peak corresponds to a period of $\sim$ 180 d. The dashed line represents the 3$\sigma$ Monte Carlo confidence level. Appendix B contains the information for all the sources.[]{data-label="lsp"}](LS_n96.eps){width="90mm"}
![Window function for the data plotted in Figure \[lightcurves\]. The frequencies correspond to periods ranging from 30 d to 5 yr. Appendix C contains the information for all the sources.[]{data-label="windowN96"}](n96_window.eps){width="90mm"}
Period recoverability analysis
------------------------------
Following an identical period search algorithm in section \[periodsearch\], we used our observed lightcurves to perform Monte Carlo simulations to calculate the recoverability of an extended range of periods. The recoverability chart for the data plotted in Figure \[lightcurves\] is shown in Figure \[recover\]. We conducted period recoverability analysis for amplitudes ranging from 25–200 per cent for the source X7. Generally, the recoverability charts show the expected result that the sampling of the available data is adequate for recovery of longer-term periods for higher amplitude modulations and for higher count rates. Interestingly, the charts also show that for amplitudes of $\sim$ 20 per cent, which is similar to that found in SS433, long-term periods would have low recoverability with the current sampling, and hence we would not have expected to see them had they been present. However, a more extreme amplitude variation of $\gtrsim$ 50 per cent could have enabled recovery of a wide range of periods in the existing data for the source X7 had they been present.
Discussion
==========
None of the sources show strong evidence for periodic variations, although the source X7 does show marginally significant periodic variability. The remaining sources may show variability of a more general nature (see Table \[summary\]). Henceforth, we will use the case of X7 (see Figure \[lightcurves\]) as an exemplar in our discussion, although the analysis discussed in Section \[analysis\] was applied to all sources.
{width="130mm"}
{width="84mm"} {width="84mm"}
{width="84mm"} {width="84mm"}
----- ------------------ ------------------------------------- ----------------------------- ------- ------------------------ -------------- -------------
No. Name $F_{\rmn{max}}$ $L_{\rmn{max}}$ \[D=17Mpc\] OBSID Median Rate $\chi^2/dof$ $p$
(CXOU) (10$^{-14}$ erg cm$^{-2}$ s$^{-1}$) (10$^{39}$ erg s$^{-1}$) ($10^{-3}$ c s$^{-1}$)
X1 J123047.7+122334 12.56 4.34 8576 3.42 178/79 $< 10^{-4}$
X2 J123049.2+122334 6.81 2.36 8513 3.69 193/79 $< 10^{-4}$
X3 J123049.6+122333 1.73 0.60 8516 1.18 79/79 0.54
X4 J123047.1+122415 8.98 3.10 8513 3.51 1199/32 $< 10^{-4}$
X5 J123053.2+122356 2.68 0.93 11513 1.01 115/37 $< 10^{-4}$
X6 J123044.7+122434 3.15 1.09 2707 0.56 297/7 $< 10^{-4}$
X7 J123050.8+122502 6.54 2.26 4918 4.09 577/59 $< 10^{-4}$
X8 J123049.1+122604 4.40 1.52 2707 3.64 557/13 $< 10^{-4}$
----- ------------------ ------------------------------------- ----------------------------- ------- ------------------------ -------------- -------------
Maximum flux and luminosity (at the distance to M87) for each source, calculated as described in section \[flux\] and uncorrected for absorption. The observation identification numbers (OBSID) corresponding to the peak values $F_{\rmn{max}}$ (in 10$^{-14}$ erg cm$^{-2}$ s$^{-1}$) and $L_{\rmn{max}}$ (in 10$^{39}$ erg s$^{-1}$) for each source are shown for reference. Each source’s data are fitted to a constant (the median count rate), and the resulting $\chi^2$ statistic is tabulated, along with the degrees of freedom $(dof)$. $p$ is the null hypothesis probability.
The need for a long-term campaign
---------------------------------
The nature of ULXs in nearby galaxies continues as a subject of intense speculation and debate. Once considered to be the long-sought intermediate-mass BHs, current models invoke slightly heavier than stellar-mass BHs (20–50M$_\odot$) to account for their observed spectra and luminosities [@Gladstone:2009]. Their controversial status has been maintained with contemporary mass estimates for the ULX ranging from 20 to 5000M$_\odot$ [@Strohmayer:2009]. Its 115-d modulation is either orbital, or superorbital and related to a precessing jet, as in the hyperaccreting SS433. If the latter is the case then ULXs might be expected to display superorbital variations in the 50–200 day range, and these should be visible in X-ray lightcurves. Such variations require multi-year long term systematic studies and could only be carried out by [*Chandra*]{} due to its superior spatial resolution and its more flexible scheduling capabilities compared with other X-ray telescopes. Rather than having random sampling of the XRBs, if we design a sampling plan such as is proposed here (section \[propsim\]) then we are able to recover these long-term periods.
Simulation of a long-term variability {#propsim}
-------------------------------------
To demonstrate the potential of a long-term X-ray campaign, we have used the observed X-ray modulations of NGC 5408 X-1 and SS 433 (described above as indicative of the periodicities we wish to search for) as input to simulate the outcomes of such a campaign. The range that we wish to be sensitive to (40–200 days) infers a data sampling rate of 12 observations per year (i.e. one every 3–4 weeks), which is then repeated over the subsequent two years. We found that the observations within each year’s observing window can be pseudo-random, in that they should not be closer together than 14 days.
To test the viability of such a program, we have simulated the expected lightcurves for ULXs having a mean count rate of 0.03 counts per second. A time series is generated with an aperiodic, white noise component with root mean square variability $rms_{N}$ to which we add a coherent sinusoidal signal with amplitude $A$ and period $P$. The average of the series is taken to be the actual detected count rate of the ULX being simulated. The time series is binned over the typical observation duration of $\Delta T = 10^4$ s and a Poisson realisation is undertaken to obtain a simulated light-curve of counts per time bin. The simulated light-curve is then sampled on the 36 observations spread over the 3 years, and these data points are input to periodogram analysis, in order to see whether the input periodicity $P$ can be recovered.
We have constructed simulations with periodic modulations of 50, 100 and 150 days, with amplitudes of 20 per cent (typical of , , and SS433), and for a variety of samplings. All recover the input periodicity, but with varying degrees of significance. Best results are obtained with 12 observations per year (spacings of 2–4 weeks), for all the periodicities, and a typical example, including both the lightcurve and the power spectrum, is shown in Figure \[ulxsim\]. Compare this with the power spectrum of the 15-year [*RXTE*]{}/ASM 1-day averages data [@Wen:2006] for SS433 in Figure \[ss433resample\], which shows that if we have the random sampling of the M87 data (right panel), the well-established 162-d period [e.g. @Eikenberry:2001] is not recovered.
All previous attempts to study long-term variability with [*Chandra*]{} and/or [*XMM-Newton*]{} have been based on relatively few archival observations randomly distributed in time. They serve only to indicate why such a study needs to be undertaken in a well-defined, systematic manner over at least a 3-year baseline, as simulated here.
Conclusions & Future Work
=========================
We identified 8 XRBs in M87 to search for very-long-term periods ($P \sim$ weeks to years). Using Lomb-Scargle periodogram analysis, we found no statistically significant periods. We also found no visible evidence of transient outburst events. However, using period recoverability analysis we found that for amplitude modulations of $\sim$ 20 per cent (similar to what is observed for SS433, NGC5408, and M82), period recovery with the current data is very low for a wide range of periods. Only if the amplitude of the modulation is $\gtrsim$ 50 per cent could a wide range of periods be found.
We continue to monitor the [*Chandra*]{} archive for sources observed over many years. Perhaps with better sampling of these sources with facilities such as [*Chandra*]{}, we can begin to understand their longer-term variations and its consequences for the dynamics of XRBs.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank the referee for providing thoughtful suggestions which have improved this article. This research has made use of , developed by Smithsonian Astrophysical Observatory. The [STARLINK]{} package `PERIOD` was used to make all periodograms and window functions. Automation of the analysis process was achieved with scripts written in [PYTHON]{}, and `matplotlib` was used to generate the figures included with this article.
We thank Dr Alan Levine for providing the [FORTRAN]{} program used to obtain the white noise estimates for the significance tests. Furthermore, we thank Dr Mark Finger, Dr Allyn Tennant, and Dr Dan Harris for many helpful comments and suggestions.
DLF acknowledges support from NASA in the form of a Harriett G. Jenkins Fellowship; the Vanderbilt-Cape Town Partnership Program; the Astrophysics, Cosmology & Gravity Centre (ACGC) at the University of Cape Town; and the International Academic Programmes Office (IAPO) at the University of Cape Town.\
(ACIS), [*RXTE*]{} (ASM).
Lightcurves
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Window Functions
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Period Detection Sensitivity
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[^1]: E-mail: deatrick@saao.ac.za
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abstract: 'Because the opacity of clouds in substellar mass object (SMO) atmospheres depends on the composition and distribution of particle sizes within the cloud, a credible cloud model is essential for accurately modeling SMO spectra and colors. We present a one–dimensional model of cloud particle formation and subsequent growth based on a consideration of basic cloud microphysics. We apply this microphysical cloud model to a set of synthetic brown dwarf atmospheres spanning a broad range of surface gravities and effective temperatures (= $1.78\times 10^3$ – $3\times 10^5$ cm s$^{-2}$ and = 600 – 1600 K) to obtain plausible particle sizes for several abundant species (Fe, Mg$_2$SiO$_4$, and Ca$_2$Al$_2$SiO$_7$). At the base of the clouds, where the particles are largest, the particle sizes thus computed range from $\sim$$5\,\micron$ to over $300\,\micron$ in radius over the full range of atmospheric conditions considered. We show that average particle sizes decrease significantly with increasing brown dwarf surface gravity. We also find that brown dwarfs with higher effective temperatures have characteristically larger cloud particles than those with lower effective temperatures. We therefore conclude that it is unrealistic when modeling SMO spectra to apply a single particle size distribution to the entire class of objects. \[abstract\]'
author:
- 'Curtis S. Cooper, David Sudarsky, John A. Milsom, Jonathan I. Lunine, Adam Burrows'
title: Modeling the Formation of Clouds in Brown Dwarf Atmospheres
---
Introduction {#Introduction}
============
Substellar mass objects (SMOs) are fundamentally more complex than Sun-like stars because of the formation of molecules in their cool outer layers. These molecular species cause the spectra to deviate strongly from blackbody values in a multitude of spectral bands [@Leggett:1999; @Kirkpatrick:1999; @Burrows:1997 and references therein]. Because of the complex chemistry occurring in their atmospheres, therefore, a comprehensive theory of SMOs requires detailed knowledge of the opacities of all the abundant molecular species. Obtaining complete opacity data has been a major challenge for the field of SMO spectral synthesis [@Burrows:2001].
The chemistry of brown dwarf and giant planet atmospheres is further complicated by the condensation of gaseous molecules into liquid or solid cloud particles, a process which occurs naturally at low temperatures. Because clouds significantly affect spectral features, a satisfactory theory for the structure of substellar atmospheres must address cloud particle formation and subsequent growth. Detailed knowledge of the distribution of particle sizes near the photosphere is a basic requirement for properly modeling the optical effects of clouds [@Lunine:1989; @Ackerman:2001].
Clouds influence brown dwarf and giant planet spectra in several important ways. First, cloud formation can deplete the atmosphere of refractory elements that become sequestered in condensed form and then rain out from the upper atmosphere. This effect is hypothesized to be important for the interpretation of the L to T dwarf spectral transition [@Burrows:1999; @Burrows:2001]. Second, clouds near the photosphere will have the general effect of smoothing out prominent spectral features [@Jones:1997]. Third, the optical albedo of irradiated objects will be increased substantially by the presence of clouds [@Sudarsky:2000]. Fourth, the presence of an optically thick cloud layer causes a back-warming effect that results in heating of the atmosphere.
Clouds may be manifest in brown dwarf spectra in a more subtle way. @Bailer-Jones:2001 reported photometric I-band variability of up to $7\%$ in L dwarf spectra. The effect is a possible signature of variations due to the patchiness of clouds. Unfortunately, cloud patchiness in SMOs is not well understood because even though much is known about the dynamic meteorology of the Earth, these gaseous bodies are dynamically very different. For example, general circulation models of the Earth, although they are of great utility for studying Mars, completely fail to reproduce the observed dynamics in Jupiter’s atmosphere. A detailed, three–dimensional cloud model paralleling the state of the art in Earth climate models would require detailed meteorological measurements for proper parameter calibration that are simply not available for any other planet.
The present cloud model at its core employs the basic microphysical timescale arguments of @Rossow:1978 to determine the most probable particle size in a cloud at each vertical pressure level. The model is one–dimensional only. We make no attempt to treat the intricacies of cloud patchiness or the effects of winds and horizontal advection. The present cloud model is not intended to calculate the detailed meteorology of planetary atmospheres but to offer a simple prescription for estimating the opacity of clouds in SMO atmospheres. A detailed but computationally cumbersome SMO climate model, if it could be developed, would be difficult to incorporate into spectral synthesis models. The philosophy behind a simple, one–dimensional approach to cloud modeling is to develop a prescription based on well–established physical principles that can be used to guide and inform spectral synthesis calculations. The present model aims to characterize SMO cloud particle sizes and densities in a global and time averaged sense. We therefore assume the particle number density of the cloud to be uniform across the spherical surface of an object at a given pressure and temperature, and that there has been sufficient time in these systems to establish chemical equilibrium throughout the atmosphere.
Our timescale arguments compute an average effectiveness for each of the various competing physical processes. Therefore, the output particle sizes clearly depend directly on our assumptions of the values of the four unknown parameters in the microphysical timescales (see Section \[subsection:Cloud\_Code\]). The calculation represents a first–order estimate of SMO cloud particle sizes, and we therefore present our results for SMO particle sizes with the realization that the procedure employed could potentially overestimate or underestimate the true particle sizes by some unknown factor, which we expect will be of order unity. Nevertheless, since we have been consistent in the assumed values of the unknown physical parameters throughout the calculation, we expect that the errors in the computed particle sizes will be uniform among the SMOs studied. We are therefore confident, despite the inaccuracies of the approach, that the sizes computed are correct to within an order of magnitude and that the *trends *we demonstrate, in which typical cloud particle sizes vary systematically with brown dwarf effective temperature and gravity, do represent physically meaningful results.**
In this paper, we reiterate the general conclusion arrived at in @Lunine:1989 and @Ackerman:2001: it is not satisfactory when modeling brown dwarf spectra to assume *a priori *a single particle size distribution because the sizes of cloud particles vary strongly with effective temperature, surface gravity, and height in the atmosphere. We extend the previous efforts to incorporate clouds into spectral models by calculating particle sizes based on a one-dimensional model of cloud particle growth for several abundant species over a range of realistic atmospheric conditions. We compute particle radii spanning a broad range from about $5\,\micron$ to over $300\,\micron$. Therefore, in the context of this discussion, we hereafter refer to particles less than $10\,\micron$ in radius as small; we refer to particles in the range from $10-100\,\micron$ in radius as medium-sized; and we consider large particles to be those having radii greater than $100\,\micron$.**
In Section \[section:Cloud\_Model\], we present an improved model of cloud formation and droplet growth to determine the composition, abundance, and distribution of cloud particles in brown dwarf atmospheres. In Section \[section:Model\_Results\], we obtain modal cloud particle sizes for several representative cloud–forming species of high abundance for atmospheric models spanning a broad range of effective temperatures and surface gravities. In Section \[section:Cloud\_Model\_Comparisons\], we compare our particle sizes with those of two other recent papers that have also addressed clouds [@Ackerman:2001; @Helling:2001]. In Section \[section:Discussion\], we apply our cloud model to brown dwarf spectra by computing the opacity of clouds resulting from the wavelength–dependent effects of Mie scattering and absorption of radiation. We also demonstrate the importance of clouds as an opacity source by comparing the spectrum of a cloudy brown dwarf atmosphere with that of a cloud–free atmosphere.
The Cloud Model {#section:Cloud_Model}
===============
Condensation Level {#subsection:Condensation_Level}
------------------
The present cloud model follows much of the formalism of @Lewis:1969, @Rossow:1978, @Stevenson:1988, and @Lunine:1989. To aid the reader, we list in Table \[table:Model\_Symbols\] all the symbols used in our cloud model as described in this section of the text.
[ll]{} $\rm g_{surf},\, T_{eff}$ & SMO surface gravity and effective temperature.\
$\rm P_{cond},\,P_{sat}$ & Partial pressure of the condensing vapor and its saturation vapor pressure.\
$\rm \mathcal{S}_{max},\,S$ & Assumed maximum supersaturation and the saturation ratio.\
$\rm \tau_{cond},\,\tau_{nuc}$ & Timescales for heterogeneous and homogeneous nucleation.\
$\rm \tau_{coal},\,\tau_{coag}$ & Timescales for coalescence and coagulation.\
$\rm \tau_{fall},\,\tau_{conv}$ & Timescales for gravitational fallout and convective upwelling.\
$\rm \tau_{rad}$ & Timescale for particles to cool by radiation.\
T, P, $\rm \rho,\, \eta$ & Temperature, pressure, mass density, and dynamic viscosity of the atmosphere.\
$\rm R, k_b, N_A$ & Universal Gas Constant, Boltzmann’s Constant, and Avogadro’s Number.\
$\rm \mu, \,\mu_p$ & Mean molecular weight of atmosphere and condensate molecular weight.\
Kn, Re & Knudsen and Reynolds numbers.\
$\rm \lambda,\,c_{sound}$ & Mean free path of atmospheric gas particles and atmospheric sound speed.\
v & Relative velocity between particle and fluid in Re expression.\
$\rm v_{fall}(r),\,g$ & Terminal velocity of particles of radius r in local acceleration of gravity, g.\
$\rm v_{conv}$ & Upward velocity of convecting gas parcels.\
H & Atmospheric scale height; i.e., pressure e-folding distance.\
$\rm \epsilon_{surf}$, L & Surface tension of condensed molecules and the latent heat of vaporization.\
$\rm r_c$, Z & Critical radius of particles and the Zeldovich factor of classical nucleation theory.\
N & Number density of cloud particles.\
$\rm \epsilon_{coal},\,\epsilon_{coag}$ & Coalescence and coagulation efficiencies.\
$\rm q_c$ & Moles of condensate per mole of atmosphere.\
$\rm q_v$ & Moles of condensable vapor per mole of atmosphere.\
$\rm q_t$ & Total moles of condensable material per mole of atmosphere.\
$\rm q_{below}$ & Mole fraction of condensable vapor below the cloud base.\
$\rm P_{c,1}$ & Condensation curves as shown in Figure \[figure:Condensation\_Level\].\
$\rm n(r),\, r_0$ & Particle size distribution about modal particle size, $\rm r_0$.\
As in @Lewis:1969, we compare the partial pressure of a given condensable species to its saturation vapor pressure. Only at levels where the partial pressure exceeds the saturation vapor pressure can condensation begin. However, the required level of supersaturation for efficient condensation depends on the physics of the nucleation process itself. We discuss the two relevant nucleation processes—homogeneous and heterogeneous nucleation—in more detail in Section \[subsection:Microphysical\_Timescales\]. In a clean atmosphere, in which the availability of condensation nuclei is low (the case of homogeneous nucleation), condensation will not begin until the vapor becomes highly supersaturated. Thus, the partial pressure will greatly exceed—often by a factor of two or more—the saturation vapor pressure [@Rogers:1989]. This situation is hypothesized to occur in methane clouds in Titan’s atmosphere [@Tokano:2001]. The condensation level appears where $$\label{equation:Condensation_Level}
\textrm {P}_{\rm cond} >
\textrm {P}_{\rm sat} \, (1+\mathcal{S}_{\rm max}).$$ In Equation \[equation:Condensation\_Level\], $\textrm {P}_{\rm cond}$ is the partial pressure of the condensing vapor, and $\textrm {P}_{\rm sat}$ is its saturation vapor pressure. The parameter is the assumed maximum supersaturation. For example, if $\rm \mathcal{S}_{max} = 0.01$, then the partial pressure of the condensing vapor must be $\rm 1\%$ larger than the saturation vapor pressure before condensation can begin. To aid future discussions, we also define the *saturation ratio *as $\rm S =
P_{cond}/P_{sat}$.**
In addition to the original @Lewis:1969 cloud model, we also treat the case of heterogeneous (or chemical) condensation, in which chemical constituents different in composition from the cloud react chemically to produce the condensate. For example, we allow Mg, Si, and O to produce forsterite, Mg$_2$SiO$_4$. In such cases of heterogeneous condensation, the concept of a saturation vapor pressure is replaced by the heterogeneous condensation pressure. We apply the chemical equilibrium model of @Burrows:1999 to a solar–composition gas [@Anders:1989] to determine which chemical species are thermodynamically favored. The calculation determines the equilibrium compositions by minimizing the Gibbs free energy of the system.
The maximum supersaturation, $\rm \mathcal{S}_{max}$ in Equation \[equation:Condensation\_Level\], appears in the calculation as an essentially free parameter, though we have some guidance as to its likely value from measurements of the supersaturation of water clouds on Earth. We assume in these calculations that the abundance of condensation nuclei is sufficient for heterogeneous nucleation to begin at only $1\%$ maximum supersaturation, though we will explore the effect of this parameter on the computed particle sizes subsequently (see Section \[subsection:Varying\_Free\_Parameters\]). We have not attempted herein to calculate the precise composition or abundance of these condensation nuclei. But even for iron, which is the most refractory of the homogeneously condensing species, we argue based on chemical equilibrium considerations that there exist abundant refractory condensates that can potentially serve as nucleation sites for heterogeneous nucleation, including corundum ($\rm Al_2O_3$), hibonite ($\rm CaAl_{12}O_{19}$), the calcium titanates (e.g., $\rm CaTiO_3$ and $\rm Ca_4Ti_3O_{10}$), grossite ($\rm CaAl_4O_7$) and other calcium-aluminum oxides, and spinel ($\rm MgAl_2O_4$).
As in @Rossow:1978, we ignore variations in $\rm \mathcal{S}_{max}$ within the cloud. Although in reality the supersaturation should decrease with altitude, we ignore this effect in the present calculations. Furthermore, it is possible to have a small amount of condensate present between $\rm S = 1$ and $\rm S = 1 + \mathcal{S}_{max}$, even though condensate cannot possibly form there (since the initial condensation of vapor into embryonic particles in general requires at least a modest level of supersaturation). However, particles can settle into the region between $\rm S = 1$ and $\rm S = 1 + \mathcal{S}_{max}$ once they have formed and yet still be thermodynamically stable. Since it depends on a balance between the processes of gravitational settling and convective upwelling, the amount of cloud material present in the barely supersaturated region is difficult to calculate with certainty. The clouds we treat in this article have a very low $\rm \mathcal{S}_{max}$, and therefore the errors introduced by neglecting these effects are small. In a cloud having very few condensation nuclei, which as we have stated would require a large $\rm \mathcal{S}_{max}$, considerations such as the variation of the supersaturation with altitude and the exact location of the cloud base become more crucial for accurate cloud modeling. The present cloud model is not configured to account for these effects realistically, though they are relatively unimportant for the low supersaturations generally assumed throughout this paper.
Figure \[figure:Condensation\_Level\] shows the condensation curve for two condensates, forsterite (Mg$_2$SiO$_4$) and gehlenite (Ca$_2$Al$_2$SiO$_7$), as well as the “effective condensation curve” of iron derived from the iron saturation vapor pressure curve, and several model brown dwarf atmospheres. As the caption to Figure \[figure:Condensation\_Level\] explains, the curve labeled “Fe” is not the true saturation vapor pressure curve of iron. It is an “effective condensation curve” derived by taking the saturation vapor pressure of iron and dividing by the number mixing ratio of iron vapor just below the cloud base (parameter $\rm q_{below}$ of Table \[table:Condensate\_Properties\]), which is the maximum allowed number mixing ratio of gaseous iron in a solar composition mixture when hydrogen is all in molecular form ($\rm H_2$). We discuss this further in Section \[subsection:Cloud\_Code\].
The intersection points between the four long–dashed curves of brown dwarf temperature–pressure profiles and the condensation curves of forsterite and gehlenite represent levels in the atmosphere at which the relevant species may appear via one or more chemical reactions.
The more familiar case of homogeneous condensation, or direct condensation from a vapor of the same composition, does occur for water and iron. Thus, for the condensation of iron and water, we follow @Lewis:1969 and employ the well–known saturation vapor pressure relations of iron and water [@Barshay:1976; @Lunine:1989]. In Figure \[figure:Condensation\_Level\], the iron curve is thus shown as a thick, short–dashed line as a reminder that the iron cloud condenses directly from iron vapor. As we discuss in more detail in Section \[subsection:Coupling\_Clouds\_With\_SMO\_Atmosphere\_Models\], in which we explore the coupling between clouds and the radiative transfer problem, the atmospheres used in this stage are computed on the basis of an independent, dust–free stellar atmosphere code.
Microphysical Timescales {#subsection:Microphysical_Timescales}
------------------------
Although the chemistry applying to brown dwarf atmospheres is radically different from Earth’s atmospheric chemistry, we nevertheless treat the various competing effects in the cloud formation process as microphysical timescales [@Rossow:1978]. In our model, five microphysical processes compete with gravitational fallout to increase the modal particle size: 1) convective uplifting of condensable vapor, 2) heterogeneous nucleation, 3) homogeneous nucleation, 4) coagulation, and 5) coalescence. Each process is characterized by a single-valued timescale which expresses its relative importance.
The expressions for four of the microphysical timescales are computed in @Rossow:1978: $\tau_{\rm cond}$ (heterogeneous condensation), $\tau_{\rm coal}$ (coalescence), $\tau_{\rm coag}$ (coagulation), and $\tau_{\rm fall}$ (gravitational fallout). These expressions depend on the values of both the Knudsen and Reynolds numbers, which characterize how particles of different sizes interact physically with the surrounding fluid and with each other. It is therefore necessary to calculate these values during the particle growth phase of the calculation (see Section \[subsection:Cloud\_Code\]).
We also require an equation of state relating the gas density to its temperature and pressure. In this model, we employ the ideal gas equation of state to relate the gas mass density $\rm \rho$ to the atmospheric temperature and pressure (T, P): $\rm \rho = \mu P/RT$, where R is the universal gas constant, $\rm 8.314\times
10^7\,ergs\,mol^{-1}\,K^{-1}$. The equation of state depends on the mean molecular weight of the atmospheric gas mixture. For these atmospheres, we use $\rm \mu = 2.36\;g\,mol^{-1}$, the mean molecular weight of a solar composition gas in which hydrogen is present in the molecular state, $\rm H_2$. We have used the chemical equilibrium model of @Burrows:1999 to verify that—as for Jupiter—molecular hydrogen is the dominant phase of hydrogen in the upper atmospheres of SMOs. The ideal gas equation of state reproduces to within 1% the value of $\rm \rho$ obtained using the more sophisticated thermodynamic equation of state described in @Burrows:1989. The dimensionless Knudsen number differentiates between the two regimes—classical and gas kinetic—of physical interaction between cloud particles and the surrounding medium: $$\rm Kn = \lambda/r,
\label{equation:Knudsen_Number}$$ where $\rm r$ is the cloud particle’s radius and $\lambda$ is the mean free path of atmospheric gas particles. The transition between the two regimes occurs where the Knudsen number is near unity (i.e., where $\rm \lambda \sim r$). The high Knudsen number regime, in which the mean free path of gas molecules is much larger than the size of a typical cloud particle, will normally not occur in SMO clouds. However, there are two situations in which large Knudsen numbers could arise: 1) when particles are smaller than $\sim$$1\,\micron$ in radius, and 2) high up in the atmosphere of a planet or brown dwarf at pressures much less a bar. We include the Knudsen number in the cloud model to account for these possibilities.
The dimensionless Reynolds number characterizes the behavior of a spherical particle moving through a fluid. The equation for the Reynolds number is $$\rm Re = \frac{2\rho vr}{\eta},
\label{equation:Reynolds_Number}$$ where $\rm \rho$ is the mass density of gas in the atmosphere, $\rm r$ is the radius of the particle, v is the velocity of the particle relative to the fluid, and $\eta$ is the dynamic viscosity of the fluid. The flow will be turbulent for $\rm Re\gg10$ but laminar for low Reynolds numbers [@Landau:1959]. For the purposes of these calculations, we assume laminar flow unless $\rm Re > 70$ (see Section \[subsection:Cloud\_Code\]). This choice for the laminar to turbulent cutoff is convenient because above $\rm Re = 70$, the drag coefficient describing the variation in terminal velocity of a falling particle becomes roughly a constant at $\rm C_D\approx 0.2$ [@Rossow:1978].
To calculate the Reynold’s number and the timescale for gravitational fallout, we introduce an approximate value for $\eta$, the dynamic viscosity. The viscosity of a hydrogen and helium gas mixture at low densities is roughly constant over a broad range of temperatures. Because the viscosity increases with the square root of temperature at low pressures, it varies by less than a factor of two over the full range of SMO effective temperatures discussed herein (see Section \[subsection:Cloud\_Code\]). For simplicity, our model therefore assumes a constant value for the dynamic viscosity of $\rm \eta=2\cdot10^{-4}\,poise$, which is consistent with the measured values of $\eta$ for hydrogen and helium at low densities and $\rm T\sim 1000\,K$ [@CRC:1991 CRC Handbook].
The timescale represents gravitational fallout of particles, which is the fundamental process limiting particle growth. As the particles become large, they fall out. We take the gravitational fallout timescale to be the time required for particles to fall through a pressure scale height of the atmosphere at terminal velocity.
The terminal velocity of falling particles depends on their size, shape, density, rigidity, and the nature of the fluid flow around them (e.g., Stokes flow, free molecular flow, or turbulent flow). We assume rigid spherical particles throughout this paper. While it is true that liquid particles of a given size will fall more slowly through the atmosphere than solid particles because of the disruption of their shape, the effect will not significantly alter the fall speeds for particles smaller than about a millimeter [@Rogers:1989]. The particles we obtain using this model—see Section \[section:Model\_Results\]—never grow large enough for the difference in terminal velocity between solid and liquid particles to be noticeable. Thus, taking the particles to be rigid spheres is in this case a reasonable simplification.
For the common case of low Knudsen and low Reynolds number flow, the terminal velocity will be given by the Stokes solution for a falling rigid sphere [@Rogers:1989]. The Cunningham factor, $\rm 1 + \case{4}{3}$Kn, accounts for the variation in the terminal velocity with Knudsen number. There is also a variation in terminal velocity with Reynold’s number. These combined effects yield three different forms for the terminal velocity as a function of the radius r of the particle: $$\rm v_{fall}(r) = r^2\cdot\left( \frac{2\rho_{cond}g}{9\eta} \right) \;\;\;\;\;Re < 70,\;Kn < 1,
\label{equation:Stokes_Terminal_Velocity}$$ $$\rm v_{fall}(r) = r^{1/2}\cdot\sqrt{\frac{40\rho_{cond} g}{3\rho} }
\;\;\;\;\;Re > 70,\;Kn < 1,\;and
\label{equation:Turbulent_Terminal_Velocity}$$ $$\rm v_{fall}(r) = r\cdot\sqrt{\frac{\pi \mu}{2N_A k_b T}}\,
\left(\frac{8 \rho_{cond} g}{27 \rho}\right) \;\;\;\;\;Kn > 1.
\label{equation:Gas_Kinetic_Terminal_Velocity}$$
In Equations \[equation:Stokes\_Terminal\_Velocity\], \[equation:Turbulent\_Terminal\_Velocity\], and \[equation:Gas\_Kinetic\_Terminal\_Velocity\], $\rm N_A$ is Avogadro’s number ($\rm 6.022\times 10^{23}\,mol^{-1}$), $\rm k_b$ is Boltzmann’s constant ($\rm 1.38\times 10^{-16}\,erg\,K^{-1}$), and g is the local gravitational acceleration. The quantity $\rm \rho_{cond}$ represents the mass density of the particle itself, for which we use the bulk density of a solid or liquid phase having the composition of the condensate. The parameters used for each condensate we have treated—iron, forsterite, gehlenite, and water—are listed in Table \[table:Condensate\_Properties\]. The mass density of the gas $\rm \rho$ depends on temperature and pressure through the equation of state. The terminal velocity of particles depends also on the local acceleration of gravity, $\rm g$, which varies considerably among the SMO population (see Section \[section:Model\_Results\]). In the Stokes solution, Equation \[equation:Stokes\_Terminal\_Velocity\], the terminal velocity is proportional to the square of the radius of the falling sphere. In the turbulent regime, Equation \[equation:Turbulent\_Terminal\_Velocity\], it is proportional to the square root of the radius. In the gas kinetic (or large Knudsen number) regime, Equation \[equation:Gas\_Kinetic\_Terminal\_Velocity\], the terminal velocity increases linearly with particle radius. The gravitational fallout timescale is inversely proportional to the terminal velocity: $\rm \tau_{fall} =
H/v_{fall}$, where $\rm H = \case{\rm RT}{\rm \mu g}$ is the atmospheric scale height. We describe in Section \[subsection:Cloud\_Code\] our procedure for differentiating between the Reynolds number regimes, which is necessary to calculate and the other microphysical timescales.
The timescale represents the convective upwelling of particles. Although convective upwelling is not itself a particle growth process, convective updrafts hold small cloud particles aloft in the atmosphere so that they accumulate more material via coagulation, coalescence, etc. We define the timescale characterizing convective updrafts as = $\rm H/\rm v_{\rm conv}$. The convective timescale is thus the time required for gas parcels to rise through a pressure scale height. The upward convective velocity, $\rm v_{conv}$, is estimated via a mixing length prescription in the atmosphere code.
The nucleation timescale, $\rm \tau_{nuc}$, characterizes particle growth via homogeneous nucleation, in which highly supersaturated vapor condenses spontaneously due to rapid collisions between the vapor molecules. For the nucleation timescale, we employ the classical homogeneous nucleation theory, in which droplets form in free space as a result of chance collisions between the molecules of the supersaturated vapor. This process is rarely observed to occur in nature because its timescale will be longer in almost all cases than , which characterizes the condensation growth of a population of particles via heterogeneous nucleation. Condensation growth means the conversion of vapor to particles, whether they be in the liquid or solid phase. For example, the phase diagram of iron indicates that iron will condense into liquid particles above 1800 K. However, we ignore the distinction between liquid and solid particles because the particles are small enough that the flattening effect during free–fall is negligible. In heterogeneous nucleation, particles nucleate onto seed particles, called condensation nuclei, which are presumed to populate the region of condensation. As discussed, when heterogeneous nucleation is efficient, condensation may begin even when the level of supersaturation is quite low ($<1\%$). Neither nucleation process represented by and apply to the species that appear due to chemical reactions (e.g., forsterite and gehlenite) rather than from a supersaturated vapor. The nucleation timescale expressions apply only to the species (e.g., iron and water) for which the condensate forms directly from a vapor of the same composition.
For the condensation timescale, , we consider nucleation onto seed aerosols, also known as condensation nuclei. The assumption of a low maximum supersaturation of the condensable vapor—$\rm \mathcal{S}_{max} = 1\%$ throughout (see Section \[subsection:Condensation\_Level\])—is equivalent to the assumption that condensation nuclei are abundant in the upper atmosphere; i.e., these are “dirty” atmospheres. We have not attempted in the present model to calculate the compositions or exact abundances of these condensation nuclei. Our assumed value for is based rather on the recognition that the complex chemistry occurring in SMO atmospheres will likely generate a plethora of molecules suitable as seeds for the onset of heterogeneous nucleation. The expressions for the condensation timescale, which depend on the Knudsen number regime, are adapted from @Rossow:1978: $$\rm \tau_{cond} = \rho r^2\left(\frac{\rho_{cond}RT}{4\eta
\mu P_{sat}\mathcal{S}_{max}}\right),\;\;\;\;\; Kn < 1,
\label{equation:Condensation_Timescale_Classical_Gas}$$ $$\rm \tau_{cond} = r\left(\frac{\rho_{cond}RT}{3P_{sat}\mathcal{S}_{max}}\right)
\left(\frac{\pi}{2\mu N_A k_b T}\right)^{1/2},\;\;\;\;\; Kn > 1.
\label{equation:Condensation_Timescale_Gas_Kinetic}$$ In Equations \[equation:Condensation\_Timescale\_Classical\_Gas\] and \[equation:Condensation\_Timescale\_Gas\_Kinetic\], $\rm P_{sat}$ is the saturation vapor pressure defined in Equation \[equation:Condensation\_Level\], and $\rm \rho_{cond}$ is the mass density of the condensate. The other variables are the same as in Equations \[equation:Stokes\_Terminal\_Velocity\], \[equation:Turbulent\_Terminal\_Velocity\], and \[equation:Gas\_Kinetic\_Terminal\_Velocity\]. The mass densities of the four condensates treated in this paper are given in Table 1.
The condensation timescale given in Equations \[equation:Condensation\_Timescale\_Classical\_Gas\] and \[equation:Condensation\_Timescale\_Gas\_Kinetic\] assumes that heterogeneous nucleation dominates over homogeneous nucleation; i.e. $\rm \mathcal{S}_{max}\ll 1$. We relax this assumption by incorporating the process of homogeneous nucleation explicitly ($\rm \tau_{nuc}$). It should be noted, however, that the inclusion of $\rm \tau_{nuc}$ will only be important if the maximum supersaturation parameter is assumed to be large ($\rm \mathcal{S}_{max}\gg 1$). We present homogeneous nucleation as a feature of the cloud model so that particle growth can be treated in the extreme case of very high supersaturations. As we have indicated, however, this situation is not likely in brown dwarf atmospheres. In most cases, homogeneous nucleation will be orders of magnitude slower than condensation growth by heterogeneous nucleation (i.e., $\rm \tau^{-1}_{nuc} \ll
\tau^{-1}_{cond}$).
For the timescale of homogeneous nucleation, we have adapted expressions from @Stevenson:1988 for large Knudsen numbers: $$\rm \tau_{\rm nuc}= \frac{\rho_{cond}rL^2\mu}{c_{sound}RTP\left
[exp\left (2\epsilon_{surf}\mu/RT\rho_{cond}r\right) - 1\right ]}
\;\; (\textrm {Kn} > 1).
\label{equation:tau_nuc1}$$ Here, $\rm r$ is the particle radius, $\rm L$ is the latent heat of vaporization, $\rm \epsilon_{surf}$ is the surface tension of the condensed liquid molecules, $\rm c_{sound} = \sqrt{RT/\mu}$ is the sound speed in the atmosphere, and P and T are the ambient temperature and pressure. We list the values of these parameters for iron and water in Table \[table:Condensate\_Properties\]. This expression is derived from Equations 3-5 of @Stevenson:1988 by taking $\rm \tau_{nuc} = r\left[\frac{dr}{dt}\right]^{-1}$ and then substituting in from the ideal gas equation of state and the Clausius-Clapeyron equation to eliminate the particle number density, saturation vapor pressure, and the vapor pressure temperature gradient. We take the efficiency of heat exchange in Equation 3 of @Stevenson:1988 to be $\rm 100\%$ and the average relative velocity of cloud particles colliding with the local hydrogen to be $\rm c_{sound}$.
The $\rm \tau_{nuc}$ expression for large Knudsen numbers represents the timescale for the growth of an embryonic cloud particle that must release the latent heat of condensation through collisions with the surrounding hydrogen [@Stevenson:1988]. It is possible, however, for particles in high-temperature environments to grow by releasing the latent heat of condensation simply through radiatively cooling. We therefore include in the model an estimate of the radiative cooling timescale, which is compared with Equation \[equation:tau\_nuc1\] in the calculation. If $\rm \tau^{-1}_{rad} > \tau^{-1}_{nuc}$, and the mean free path of particles is greater than the particle size, we use $\rm \tau_{rad}$ in place of $\rm \tau_{nuc}$. The equation we use to approximate $\rm \tau_{rad}$ is adapted from @Woitke:1999: $$\begin{aligned}
\rm \tau_{rad} \approx \left(2\times10^{-2}\;sec\right)\cdot\left(\frac{\rho_{cond}}{1\;g\,cm^{-3}}\right)\times \nonumber\\
\left(\frac{\mu_p}{1\;g\,mol^{-1}}\right)^{-1}\left(\frac{T}{1000\;K}\right)^{-4}.
\label{equation:tau_rad}\end{aligned}$$ In Equation \[equation:tau\_rad\], $\rm \mu_p$ is the molecular weight of the condensate (as distinct from $\rm \mu\approx 2.36\;g\,mol^{-1}$, the mean molecular weight of the atmospheric gas mixture). The values of $\rm \mu_p$ for the four condensates treated herein are given in Table \[table:Condensate\_Properties\].
It should be noted that in a very high temperature, low density environment, in which the release of latent heat is extremely fast, the limiting timescale for homogeneous nucleation will be neither $\rm \tau_{nuc}$ nor $\rm \tau_{rad}$ but the timescale for the diffusion of vapor molecules to the surface of the grain. We do not calculate this timescale in the present model. For the purposes of SMO atmospheres, however, this regime will never be realized in practice because the condensation temperatures of even the most refractory compounds rarely exceed 2000 K. Therefore, although the limit of $\rm \tau_{rad} \to 0$ in this formulation appears to be potentially problematic, the temperature constraints ensure that $\rm \tau_{rad}$ cannot be arbitrarily small. Hence, for use in the calculation of SMO clouds, it is reasonble to assume that the shorter of the two timescales, $\rm \tau_{nuc}$ and $\rm \tau_{rad}$, determines the particle’s homogeneous nucleation rate.
The constant preceding Equation \[equation:tau\_rad\] is strictly correct only for highly opaque species having high extinction efficiencies. This is appropriate for the condensates we focus on here, although the timescale for radiative cooling will be longer by a factor of ten or a hundred than the timescale shown in Equation \[equation:tau\_rad\] if the particles are mostly transparent. In cases in which the $\rm \tau_{rad}$ timescale dominates the particle sizes, a more accurate approximation than we have made in Equation \[equation:tau\_rad\] should be calculated by integrating the extinction efficiency over wavelength; e.g., see Equation 17 of @Woitke:1999.
The more common case of small Knudsen numbers, which corresponds to high atmospheric gas density, is a commonly used meteorological expression; see @Rogers:1989. The low Knudsen number expression for is written in terms of the critical radius at which the equilibrium vapor pressure over the surface of a particle equals the ambient vapor pressure. The equilibrium vapor pressure over the surface of a liquid particle, because of its finite curvature, is in general higher than the equilibrium vapor pressure over a flat surface of the liquid, which is the saturation vapor pressure normally measured in the laboratory. Therefore, the critical radius of the liquid particle will only be attained when the saturation ratio exceeds unity; i.e., when the actual ambient pressure exceeds the saturation vapor pressure. The critical radius therefore depends on the supersaturation [@Rogers:1989]: $$\rm r_c = \frac{2 \mu_p\epsilon_{surf}}{\rho_{cond} R T \,
\textrm{ln} (1 + \mathcal{S}_{\rm max})}.
\label{equation:r_crit}$$
Unlike the gas kinetic regime, in which the removal of latent heat limits the rate at which particles can grow by nucleation, in the classical gas regime, the nucleation rate of particles is determined by the rate at which supercritical droplets are formed; i.e., droplets larger than the critical radius above which particles grow spontaneously. The expression for $\rm \tau_{nuc}$ in terms of the critical droplet radius $\rm r_c$, adapted from @Rogers:1989, reads $$\rm \tau_{\rm nuc} = \frac{1}{P}\,\sqrt{ \frac{\mu_pk_bT}{8\pi N_A r_c^4} }
{\textrm{Z}}^{-1}\;exp\left(\frac{4\pi
r_c^2\epsilon_{surf}}{3k_bT}\right)\;\;(\textrm {Kn} < 1).
\label{equation:tau_nuc2}$$ In Equation \[equation:tau\_nuc2\], Z is the dimensionless Zeldovich or non–equilibrium factor, which depends on temperature and the physical properties of the condensate. The equation for Z, given by @Jacobson:1999, is $$\rm Z = \frac{\mu_p}{2\pi r_c^2 \rho_{cond} N_A} \, \sqrt{\frac{\epsilon_{surf}}{k_b T}}.
\label{equation:Zeldovich_Factor}$$ The Zeldovich factor accounts for the differences between equilibrium and non–equilibrium cluster concentrations in the classical homogeneous nucleation theory used to derive Equations \[equation:r\_crit\] and \[equation:tau\_nuc2\].
Coagulation and coalescence represent particle growth due to collisions. The coagulation timescale, $\rm \tau_{coag}$, refers to the formation of larger particles by the collision of smaller particles. It therefore depends primarily on the thermal temperature, viscosity, and the number density of cloud particles. The coalescence timescale, $\rm \tau_{coal}$, characterizes the growth of particles by coalescence, in which large particles having high downward velocities in the fluid overtake and merge with small, slowly falling particles. Thus, coalescence refers to the collisional process caused by the different fall speeds of different sized particles, whereas coagulation is collisional growth resulting from Brownian motion. Both processes proceed at a rate proportional to the particle number density, N, since the total amount of material available for condensation is conserved. The number density of cloud particles is given in terms of the gas density as $$\rm N = q_c\left(\frac{\rho}{\mu}\right)\left(\frac{3\mu_p}{4\pi\rho_{cond}r^3}\right).
\label{equation:Particle_Number_Density}$$ In Equation \[equation:Particle\_Number\_Density\], r is the cloud particle radius, $\rm \rho$ is the mass density of the surrounding gas (as given by the equation of state), $\rm \rho_{cond}$ is the mass density of the condensate, and $\rm q_c$ is the condensate mixing ratio by number, which depends on the specific rainout prescription applied. The rainout scheme we employ to compute the variation of $\rm q_c$ with height above the cloud base is explained in Section \[subsection:Cloud\_Code\].
Given the particle number density, N, the timescale for coagulation in terms of the radius of the condensed particle, $\rm r$, is given by $$\rm \tau_{coag} \equiv \frac{N}{dN/dt}=
\frac{3\eta}{4k_bT\epsilon_{coag}}\,\frac{1}{N},\;\;\;\;\;Kn<1
\label{equation:Coagulation_Timescale_Classical_Gas}$$ $$\rm \tau_{coag} = \frac{1}{4\epsilon_{coag}}\sqrt{\frac{\rho_{cond}}{3rk_bT}}\,\frac{1}{N},\;\;\;\;\;Kn>1.
\label{equation:Coagulation_Timescale_Gas_Kinetic}$$ Here, $\rm \epsilon_{coag}$ is the coagulation efficiency, which is a free parameter of the model. We discuss this further in Section \[subsection:Free\_Parameters\].
Our final microphysical timescale, $\rm \tau_{coal}$, characterizes the process of coalescence, in which larger particles overtake and coalesce with smaller particles. For coalescence, the Reynolds number again plays an important role because—unlike coagulation—coalescence pertains specifically to the merging of particles of greatly different radii whose interaction depends on the nature of the surrounding flow fields. The coalescence expressions are adapted from @Rossow:1978, although we have introduced an extra free parameter into the equations, the efficiency for coalescence, $\rm \epsilon_{coal}$: $$\rm \tau_{coal} =
\frac{1}{r^4\epsilon_{coal}}\left(\frac{9\eta}{\pi\rho_{cond}g}\right)\,\frac{1}{N},\;\;\;\;\;Re<70, Kn<1
\label{equation:Coalescence_Timescale_Laminar}$$ $$\rm \tau_{coal} = \frac{1}{\epsilon_{coal}}\sqrt{\frac{3\rho}{10\pi^2\rho_{cond} r^5 g}}\,\frac{1}{N},
\;\;\;\;\;Re>70, Kn<1
\label{equation:Coalescence_Timescale_Turbulent}$$ $$\rm \tau_{coal} =
\frac{1}{r^3\epsilon_{coal}}\left(\frac{27\rho}{4\pi\rho_{cond}g}\right)
\sqrt{\frac{2RT}{\pi \mu}}\,\frac{1}{N},\;\;\;\;\;Kn>1.
\label{equation:Coalescence_Timescale_Gas_Kinetic}$$
As we discuss in Section \[subsection:Free\_Parameters\], coalescence is important only for liquid particles. This is the only case in our model in which the distinction between liquid and solid particle formation is important. Thus, for iron cloud formation above 1800 K, coalescence will potentially be important, and it can be important for water clouds, but we ignore it for the silicate clouds.
Our Cloud Code {#subsection:Cloud_Code}
--------------
The code proceeds in two stages: (1) determination of the cloud base globally in the atmosphere, and then (2) particle growth at each atmospheric temperature–pressure level. The algorithm is run for each of the atmospheric temperature–pressure profiles, which differ in effective temperature and gravity, and for each condensable species separately. In our model, we ignore *core–mantle *grains composed of molecules of two or more of the major species. These types of condensates could potentially be important in regions in which several major constituents in the equilibrium vapor mixture form at similar temperatures and pressures. In the present cloud model, we do not treat the growth of grains composed of multiple chemical phases.**
We list in Table \[table:Condensate\_Properties\] the input parameters for the four condensable species treated herein.
[lcccc]{} Molecular weight, $\rm \mu_p$ \[$\rm g\,mol^{-1}$\] & 55.8 & 140.7 & 274.2 & 18.0\
Mass Density, $\rm \rho_{cond}$ \[$\rm g\,cm^{-3}$\] & 7.9 & 3.2 & 3.0 & 1.0\
Mole Fraction, $\rm q_{below}$ & $5.4\times 10^{-5}$ & $3.2\times 10^{-5}$ & $1.8\times
10^{-6}$ & $1.4\times 10^{-3}$\
Surface Tension, $\epsilon_{surf}$ \[$\rm ergs\,cm^{-2}$\] & 200 & & & 75\
Latent Heat of Vaporization, L \[$\rm ergs\,g^{-1}$\] & $6.34\times 10^{10}$ & & & $4.87\times 10^{12}$
In stage (1), two cases must be treated separately, as explained above. In the case of homogeneous condensation, we calculate the intersection between the partial pressure of the condensable vapor and its saturation vapor pressure curve and identify the condensation level using Equation \[equation:Condensation\_Level\]. In the case of heterogeneous condensation, in which the condensing chemical is not present in vapor form below the cloud base, we determine the cloud base by using the chemical equilibrium model of @Burrows:1999 (see Figure \[figure:Condensation\_Level\]).
In stage (2), we calculate the modal size of particles, which are assumed to be spherical, at each atmospheric temperature–pressure level. The model outputs the modal particle radius. This radius is determined by growing embryonic particles of radius $\sim$$10\,\textrm{\AA}$ slowly until the various competing particle growth timescales are balanced by the timescale of gravitational fallout. For the most probable particle radius, the sedimentation timescale equals the shortest of the growth timescales: $$\tau_{\rm fall} = \textrm{Min}\,\{\tau_{\rm nuc},\,\tau_{\rm cond},\,
\tau_{\rm coag},\,\tau_{\rm coal},\,\tau_{\rm conv}\}.
\label{equation:Min_Timescale}$$ Note that for homogeneously condensing species in the gas kinetic regime ($\rm Kn >
1$), it may be that radiative cooling, rather than the release of latent heat by collisions with atmospheric hydrogen, is the dominant process limiting the rate of particle growth. Therefore, if the Knudsen number is greater than one and $\rm \tau^{-1}_{rad} > \tau^{-1}_{nuc}$, we replace $\rm \tau_{\rm nuc}$ in Equation \[equation:Min\_Timescale\] with the timescale for particles to release the latent heat of condensation by radiation, $\rm \tau_{rad}$. Additionally, as we have stated in Section \[subsection:Microphysical\_Timescales\], nucleation does not apply to the case of chemical clouds [@Rossow:1978], and coalescence is inefficient between solid particles. The condensed particles can only accumulate more material by coagulating under the influence of convective updrafts. Hence, for forsterite and gehlenite, the fall timescale must balance the minimum of the two timescales relevant for heterogeneously condensing clouds: $\rm
\tau_{coag}$ and $\rm \tau_{conv}$.
The timescales in Equation \[equation:Min\_Timescale\] depend on the Knudsen and Reynolds numbers (see Section \[subsection:Microphysical\_Timescales\]). We therefore need a prescription to determine which physical regime applies as the particles are grown. Because the Reynolds number depends on velocity, and the velocity depends on the Reynolds number, calculating the two quantities independently for a given particle size and atmospheric parameters is a circular problem. There are no difficulties if the Knudsen number is larger than one, since the gas kinetic terminal velocity in that case does not depend on the Reynolds number; the velocity to use in this case is given by Equation \[equation:Gas\_Kinetic\_Terminal\_Velocity\]. To differentiate between Reynolds number regimes in the small Knudsen number case (i.e., Equations \[equation:Stokes\_Terminal\_Velocity\] and \[equation:Turbulent\_Terminal\_Velocity\]), we use the following procedure based on the fact that each of the two expressions for $\rm v_{fall}$ can be calculated independently of the Reynolds number.
At each point in the particle growth phase, we use *both *Equations \[equation:Stokes\_Terminal\_Velocity\] and \[equation:Turbulent\_Terminal\_Velocity\], along with the known atmospheric parameters and the current particle size. In solving for the terminal velocity, we substitute the SMO surface gravity, $\rm g_{surf}$, in for g, the local gravitation acceleration. We then calculate two temporary values for the Reynolds numbers, one for each of the velocities calculated. If these two independently computed Reynolds numbers are both smaller than our Reynolds number cutoff point of 70 between laminar and turbulent flow [@Rossow:1978], we take the flow around the particles to be laminar and adopt the Stokes terminal velocity, Equation \[equation:Stokes\_Terminal\_Velocity\], as the actual terminal velocity. Likewise, if they are both larger than the cutoff, we take the flow around the particles to be turbulent and adopt the turbulent terminal velocity, Equation \[equation:Turbulent\_Terminal\_Velocity\], as the actual terminal velocity.**
The intermediate case, in which one temporary Reynolds number is larger than the cutoff but the other is smaller than the cutoff, is more difficult to resolve. We settled on the choice of using the lesser of the two terminal velocities as the actual terminal velocity. Although this could have been handled in a variety of ways, the inherent ambiguities could not have been eliminated without detailed knowledge of the drag coefficient, which unfortunately depends in general on the Reynolds number. Ours was the most conservative choice for a first–order estimate of the terminal velocity. With this procedure in place to calculate the terminal velocity of particles, the effective Reynolds number—which is used to calculate all the other timescales—is computed directly from its definition in Equation \[equation:Reynolds\_Number\], with $\rm v_{fall}$ substituted in for the relative velocity, v.
Equation \[equation:Min\_Timescale\] shows the role of convection in this model. Convection directly opposes gravitational sedimentation. In the presence of sufficiently vigorous convection, therefore, the particles may continue to accumulate material until their terminal velocities become equal to the convective updraft velocity.
The simple cloud model delineated above is a first–order estimate of SMO particle sizes. The true particle sizes could very well deviate from the values computed by a factor of order unity, though we believe that this factor will be the same for all objects studied because the systematic errors depend on our assumptions of the unknown parameters of the model, assumptions which we have applied consistently throughout the computations.
Our treatment of rainout, in which condensates settle gravitationally and thus become depleted from the upper atmosphere, differs between homogeneously and heterogeneously condensing species. Depletion of the condensate with height must differ between these two different types of clouds because homogeneous condensation involves a pressure equilibrium between the vapor and the condensed phase (See Section \[subsection:Microphysical\_Timescales\]), whereas heterogeneous condensation produces the condensate directly through exothermic chemical reactions.
For the purposes of this discussion, we adopt the notation of @Ackerman:2001 for the relevant mixing ratios: $\rm q_v =$ moles of vapor per mole of atmosphere, $\rm q_c=$ moles of condensate per mole of atmosphere. The total mixing ratio is $\rm q_t=q_v + q_c$. We also define $\rm q_{below}$ as the mixing ratio by number (or mole fraction) of condensable vapor just below the cloud base, which equals $\rm q_t$ at that level, since $\rm q_c = 0$ by definition below the cloud base (i.e., at levels in the atmosphere where Equation \[equation:Condensation\_Level\] is not satisfied). The values of $\rm q_{below}$ for each condensable treated herein are given in Table \[table:Condensate\_Properties\].
For both homogeneously and heterogeneously condensing species, we assume aggressive rainout such that the total material available to condense into particles drops off with pressure much more steeply than the gas pressure. That is, at the condensation level, the mixing ratio by number (or equivalently, the mole fraction) of the condensing species is taken to be equal to the maximum partial pressure attained by that species in a gas of solar composition (i.e., parameter $\rm q_{below}$ in Table \[table:Condensate\_Properties\]). For heterogeneously condensing species, $\rm q_{below}$ is constrained by the maximum partial pressure of the least abundant of the constituent molecules combining to form the condensate. For example, in the case of forsterite, the magnesium abundance in the gas controls the value of $\rm q_{below}$. We have assumed a solar abundance of magnesium in the atmosphere below the forsterite condensation level. High above the condensation level, however, the mixing ratio of cloud–forming material is significantly reduced from the equilibrium ratio, and the cloud becomes more and more tenuous with decreasing pressure.
For homogeneously condensing species, our rainout prescription is based on the assumption that all of the supersaturated vapor above the base of the cloud will condense. As we have discussed in Section \[subsection:Condensation\_Level\], the onset of particle formation requires a supersaturated environment, which thus elevates the cloud base above the level where precise saturation of condensable vapor is attained. Once present, however, particles remain thermodynamically stable unless the saturation ratio drops to below one, which will not happen without a temperature inversion in the atmosphere’s thermal profile. For already existent particles or cloud particle embryos, the vapor in equilibrium with the particles remains exactly at saturation. All excess material goes into the condensate. Hence, throughout the cloud, we take $\rm q_v = P_{sat}/P$, where $\rm P_{sat}$ is the saturation vapor pressure and P is the gas pressure. Notice that this ratio decreases with altitude because $\rm P_{sat}$ decreases with pressure much more steeply than the gas pressure, P. It therefore follows that the partial pressure of the condensable vapor required to maintain a saturated environment will be extremely small high above the condensation level. Though the mole fraction of condensing vapor, $\rm q_v$, is constrained by the requirement that the condensed phase remain thermodynamically stable during particle growth, we still need the total number mixing ratio (i.e., mole fraction) of condensable material, $\rm q_t$, in order to determine $\rm q_c$, the condensate number mixing ratio.
The character of the rainout itself derives from the assumption that the total material available for condensation above the cloud base follows the saturation vapor pressure and *not *the gas pressure. We use $\rm q_t = (1 + \mathcal{S}_{max})\,P_{sat}/P$, which implies that the total condensable material is constrained to drop off with altitude proportionally to the saturation vapor pressure. This is an assumption of the cloud model. It is not based on a rigorous derivation of the total mixing ratio with altitude. The justification for the rainout scheme is that once condensation begins in the atmosphere, with subsequent growth and gravitational settling of particles, the condensable material can no longer remain well–mixed in the atmosphere, as we assume it had been below the cloud base. Therefore, the total number of moles of condensable material per mole of gas will decrease rapidly from its initial value with increasing height above the condensation level. By contrast with our rainout scheme, a no–rainout model would be one in which the fraction of condensed material follows the gas pressure; i.e, $\rm q_t$ equals a constant throughout the cloud. In this case, the ratio of condensate to condensable vapor, $\rm q_c/q_v$, will increase with height above the base of the cloud.**
With the above considerations, it is straightforward to calculate $\rm q_c$ as simply the difference between the total number mixing ratio of condensable material and the number mixing ratio of condensable vapor: $\rm q_c = q_t - P_{sat}/P$. We therefore obtain the following equation for $\rm q_c$: $$\rm q_c = \mathcal{S}_{\rm max}\cdot\frac{P_{sat}}{P}
= \mathcal{S}_{\rm max}\cdot q_{below}\cdot \frac{P_{c,\,1}}{P}.
\label{equation:Rainout_Homogeneous}$$ In Equation \[equation:Rainout\_Homogeneous\], $\rm P_{c,\,1}$ denotes the saturation vapor pressure curve—given by $\rm P_{sat}$, as in Equation \[equation:Condensation\_Level\]—adjusted to resemble a condensation curve by applying the mixing ratio in a solar composition gas; i.e., $\rm P_{sat} = q_{below}\cdot P_{c,\,1}$. For example, the dotted curve labeled “Fe” in Figure \[figure:Condensation\_Level\] shows the $\rm P_{c,\,1}$ for iron, *not *the iron saturation vapor pressure curve. The subscript notation for $\rm P_{c,\,1}$ simply refers to the “effective” iron condensation curve as derived from the saturation vapor pressure curve. The rainout is expressed in terms of $\rm P_{c,\,1}$ for comparison with the analogous prescription for rainout in a heterogeneous cloud (Equation \[equation:Rainout\_Heterogeneous\]). The quantity $\rm P$ in Equation \[equation:Rainout\_Homogeneous\] is the total gas pressure in the atmosphere. As is clear from comparing the thick dotted Fe line in Figure \[figure:Condensation\_Level\] with one of the gas pressure profiles, $\rm P$ decreases with height above the cloud base much less steeply than $\rm P_{c,\,1}$. The assumption of a constant supersaturation throughout the cloud, which we have made in fixing the value of $\rm \mathcal{S}_{max}$, leads to a cloud with a very low ratio of condensed material to total condensable material unless $\rm \mathcal{S}_{max} \gg 0.01$.**
For heterogeneously condensing clouds, we refer to the condensation curves (denoted $\rm P_{c,\,1}$ in Equations \[equation:Rainout\_Homogeneous\] and \[equation:Rainout\_Heterogeneous\]) of forsterite and gehlenite as shown in Figure \[figure:Condensation\_Level\]. We make the assumption for a heterogeneously condensing cloud that the total mixing ratio, $\rm q_{t}$, follows the condensation curve above the cloud base. This assumption is analogous to the assumption used to derive Equation \[equation:Rainout\_Homogeneous\] for homogeneously condensing clouds that the total mixing ratio follows the saturation vapor pressure curve. However, in a heterogeneous (or chemical) cloud, once the condensate appears in chemical equilibrium, we argue that the solid phase will then be strongly favored, and hence the product chemical will be fully condensed. Thus, for heterogeneous condensates, $\rm q_v=0$ and $\rm q_c=q_t$. The rainout formula for heterogeneous species is therefore quite similar to Equation \[equation:Rainout\_Homogeneous\], without the reduction of $\rm q_c$ relative to $\rm q_t$ by the supersaturation factor, which is not relevant for chemical clouds: $$\rm q_c = q_{below}\cdot \frac{P_{c,\,1}}{P}.
\label{equation:Rainout_Heterogeneous}$$ In Equation \[equation:Rainout\_Heterogeneous\], the quantity $\rm P_{c,\,1}$ represents the chemical equilibrium condensation curve of the condensing species, as shown for forsterite and gehlenite in Figure \[figure:Condensation\_Level\]. It is apparent from Figure \[figure:Condensation\_Level\] that this rainout prescription sequesters most of the condensate within a scale height of the cloud base, since the condensation pressure drops off so steeply relative to the gas pressure. For example, consider the forsterite cloud shown in Figure \[figure:Condensation\_Level\] for the brown dwarf model at $\rm T_{eff} = 900\,K$ and $\rm g_{surf} = 3\times 10^5\;cm\,s^{-2}$. At the forsterite cloud base, which is at about $\rm T \approx 2000\,K$ and $\rm P \approx 125\,bars$, the mixing ratio of forsterite is just determined by the abundance of magnesium in the mixture, or $\rm q_c = q_{below} = 3.2\times 10^{-5}$. Higher up in the atmosphere (in altitude), at $\rm T \approx 1700\,K$ and $\rm P \approx 70$ bars, Equation \[equation:Rainout\_Heterogeneous\] shows that $\rm q_c$, which follows the condensation curve, is reduced by a factor $\rm \sim$$100$ from its value at the cloud base. However, the gas pressure is down by only about $55\%$, or $\sim$$\case{1}{2}$ a scale height.
Equations \[equation:Rainout\_Homogeneous\] and \[equation:Rainout\_Heterogeneous\] show that, in the present model, virtually all of the atoms in the solar–composition gas that are initially available for condensation become sequestered in the condensate within a scale height of the atmosphere. For example, because the limiting atomic species for the formation of forsterite is magnesium, and forsterite is favored chemically below about 2000 K, magnesium will have rained out near the forsterite cloud level, and no magnesium will be available for further condensation into other chemicals high above the forsterite cloud level. Rainout thus depletes the atmosphere of the most refractory elements at progressively lower temperatures and pressures.
Each timescale characterizes the average (or characteristic) effectiveness of each of the competing microphysical processes in the system. Our computed particle sizes thus represent the most probable particle size in the distribution of particle sizes appearing at each pressure level. To the order of the accuracy of the particle sizes themselves, therefore, the resulting particle sizes in each pressure level will equal the mode of the particle size distribution.
We make no attempt in the present model to calculate the particle size distribution for each cloud layer. Rather, for the purposes of calculating the opacity of clouds and using these opacities in SMO spectral models, we assume a functional form for the particle size distribution that is consistent with measurements of particle size distributions attained in Earth’s water clouds [@Deirmendjian:1964]. We use the form of the @Deirmendjian:1964 exponentially decaying power law employed by @Sudarsky:2000: $$\rm n(r) \propto \left(\frac{r}{r_0}\right)^6 exp\left[-6\left(\frac{r}{r_0}\right)\right],
\label{equation:Size_Distribution}$$ where n(r)dr is the number of particles per cubic centimeter having radii $\rm r \to r+dr$. We employ the cloud model described above to determine the mode of the distribution, $\rm r_0$.
Free Parameters {#subsection:Free_Parameters}
---------------
In addition to $\rm \mathcal{S}_{max}$, the assumed maximum supersaturation, the free parameters of the cloud model include the elemental abundances and the sticking coefficients for coagulation and coalescence. The vapor mixing ratios have been calculated assuming solar [@Anders:1989] abundances of the elements in the atmosphere.
Coagulation and coalescence are not expected to be $100\%$ efficient in all cases because some molecules stick together when they collide more easily than others. Therefore, we have divided the timescales for coagulation and coalescence, given in @Rossow:1978 as and $\tau_{\rm coal}$, by efficiency parameters $\epsilon_{\rm coag}$ and $\epsilon_{\rm coal}$.
Coalescence between solid particles is extremely inefficient [@Rossow:1978]. Coalescence will therefore not be important for forsterite and gehlenite because the solid phase is strongly favored once the temperature decreases sufficiently for the chemical to appear (see Figure \[figure:Condensation\_Level\]). Therefore, for the heterogeneously condensing materials, we have ignored coalescence ($\epsilon_{\rm coal} = 0$). For the species condensing into liquid particles (iron and water), we have taken $\epsilon_{\rm coag}$ = $10^{-1}$ and $\epsilon_{\rm coal}$ = $10^{-3}$ [@Lunine:1989]. We have also used $\epsilon_{\rm coag}$ = $10^{-1}$ for the coagulation efficiency of heterogeneously condensing species. Subsequently, we will argue that this choice is relatively unimportant because coagulation and coalescence generally operate more slowly, for the atmospheres we are considering here, than heterogeneous nucleation and convection. We discuss in Section \[subsection:Varying\_Free\_Parameters\] how varying these parameters affects the general features of the cloud.
Model Results {#section:Model_Results}
=============
Modal Particle Sizes {#subsection:Modal_Particle_Sizes}
--------------------
We compute particle sizes for each of four species on a set of atmospheric models spanning the ranges = 600 – 1600 K and = $1.78\times 10^3$ – $3\times 10^5\;\textrm{cm s}^{-2}$. We chose the most abundant condensable species in the solar–composition mixture: iron, forsterite, and water. Iron and forsterite are the most abundant high–temperature condensates [@Lunine:1989]. Although we have included one of the calcium–aluminum silicates, gehlenite, and these refractory species do condense into clouds, their abundance in a solar–composition gas is lower than that of iron or forsterite by a factor of about ten [@Lunine:1989]. Thus, the contribution of the calcium and aluminum silicates to the total opacity is expected to be relatively minor. Nevertheless, the condensation of the calcium and aluminum silicates is important for sequestering calcium and aluminum at depth beneath the photospheres of cooler SMOs. Moreover, the most refractory calcium and aluminum silicates can potentially serve as nucleation sites for the condensation of iron vapor via heterogeneous nucleation at low supersaturations.
Trends With Brown Dwarf Gravity and Effective Temperature {#subsection:Trends}
---------------------------------------------------------
Figures \[figure:Iron\_Sizes\], \[figure:Forsterite\_Sizes\], and \[figure:Gehlenite\_Sizes\] show the particle sizes computed by our model for a range of brown dwarf atmospheric profiles. Each circled point represents, for the atmospheric profile to which it applies, the grain size computed by our model at the initial cloud level (i.e., the cloud base). As we discuss further in Section \[subsection:Cloud\_Decks\_Rad\_vs\_Conv\], these are in general the largest modal particle sizes for the cloud, since particle sizes decrease with height above the cloud base. The model atmosphere corresponding to each circled point is indicated by its position on the diagram: gravity increases from left to right, whereas effective temperature increases from bottom to top in the field. These graphs say nothing about the distribution of material above the cloud base. They apply only to the initial condensation level where the mass density of material is greatest. Figure \[figure:Iron\_Sizes\] shows the results for iron grains, which condense directly from iron vapor. Figure \[figure:Forsterite\_Sizes\] shows the analogous results for forsterite grains, which appear as a result of chemistry (i.e., heterogeneous condensation). Figure \[figure:Gehlenite\_Sizes\] shows the same for gehlenite, which also condenses heterogeneously.
Figures \[figure:Iron\_Sizes\], \[figure:Forsterite\_Sizes\], and \[figure:Gehlenite\_Sizes\] show that cloud grains in hot brown dwarfs, for a fixed surface gravity, are systematically larger than in cold brown dwarfs. Similarly, cloud particles in objects with high surface gravities, for a fixed effective temperature (e.g., older brown dwarfs), are systematically smaller than in objects having low surface gravities.
The trend with surface gravity is simple to explain: particles settle more quickly in high–gravity environments. This appears to be the overriding effect governing particle size. Although both the atmospheric pressure and temperature at the cloud base are higher in high–gravity objects, and the convection therefore more rapid, gravity dominates and the resulting particles are smaller. For the lowest gravity objects, the forsterite particles grow quite large. They therefore contribute only negligibly to the total optical depth [@Sudarsky:2000].
The trend with effective temperature results from the difference in convective updraft velocity between colder and hotter objects. Objects with higher effective temperatures must transport a higher flux of thermal energy to the surface via interior turbulent convection. Therefore, the updrafts holding cloud particles aloft will be more rapid in the higher $\textrm {T}_{\rm eff}$ objects, resulting in larger particle sizes.
Cloud Decks in Convective vs. Radiative Regions {#subsection:Cloud_Decks_Rad_vs_Conv}
-----------------------------------------------
Brown dwarfs are almost fully convective, with only a thin radiative atmosphere [@Basri:2000; @Burrows:2001]. Convective uplifting sustains the particles against gravitational fallout. The effectiveness of this process depends on the velocity scale of the convection. Hence, the gradient in the convective velocity plays a role in determining particle size. Deep in the atmosphere, the brown dwarf is fully convective, with a relatively high updraft velocity (approaching $\rm 5\times 10^3\;cm/s$). At higher altitudes, the updraft speed is small, but *not *zero. Therefore, clouds at depth are convective; clouds at altitude are quiescent. By quiescent, we mean clouds formed in convectively stable layers where the true lapse rate is shallower than the adiabatic lapse rate.**
The updraft speeds observed in stably stratified layers in Earth’s atmosphere typically range from $\rm 1-10\;cm/s$. Clouds analogous to Earth’s stratus clouds are possible in quiescent layers of planetary atmospheres, even though the updraft speeds are smaller by a factor of a hundred or more than the typical speeds of convective updrafts [@Rogers:1989]. In the case of these stratiform clouds, small particle sizes (generally less than $\rm 10\,\mu m$) lead to relatively slow sedimentation rates. Large–scale uplifting of a stable layer can thus replenish the cloud material lost by evaporating particles on a timescale of hours or days, which is rapid enough to sustain the cloud. Slow, large–scale updrafts, which we assume are occurring in these atmospheres, are too slow to affect the particles directly. Therefore, we do not include them as a growth process in the cloud model, as we have done for convective updrafts, but their existence is an important feature of the formation of quiescent clouds by maintaining the supersaturated environment necessary for particle growth.
Particles in quiescent clouds are generally smaller than particles in convective clouds. Figure \[figure:Cloud\_Decks\_Rad\_vs\_Conv\] compares the particle sizes of condensed iron for a brown dwarf model having = 1500 K and = $5.62\times10^4\;\textrm{cm s}^{-2}$. The modal particle sizes in the cloud are shown for iron under two contrasting assumptions: (1) the cloud is convective (i.e., $\rm \tau_{conv} = H/v_{conv}$), and (2) the cloud is radiative (i.e., $\tau_{\rm conv} = \infty$). Our calculations suggest that for this atmosphere, a convective iron cloud is more realistic. We thus predict the solid curve for the actual particle size distribution in the cloud deck. Figure \[figure:Cloud\_Decks\_Rad\_vs\_Conv\] compares the particle sizes obtained by activating and deactivating the convective uplifting mechanism.
Figure \[figure:Cloud\_Decks\_Rad\_vs\_Conv\] shows, for this particular brown dwarf atmosphere, that cloud particles growing in a region of strong convective upwelling are largest near the cloud base but then diminish rapidly with height. This decrease in size is due to the proximity of the iron cloud base, for this particular model, to the radiative–convective boundary of the atmosphere, where the gradient in the upwelling velocity is steep. For convective clouds deep within the convective zone, where the upwelling velocity is nearly constant, the particle sizes will be more constant throughout the cloud deck. However, deep clouds will not strongly influence the object’s emergent spectrum.
If the cloud is radiative, the particles will not grow as large as they would under the influence of convection. Furthermore, their sizes will not decrease by more than a factor of two from base to cloud top. This result appears quite general for the radiative clouds we have investigated.
The structure of the cloud decks (e.g., as shown in Figure \[figure:Cloud\_Decks\_Rad\_vs\_Conv\]) of brown dwarfs is crucial for computing the optical depth of the cloud. For clouds in which the overall particle sizes decrease significantly with altitude above the cloud base, it may be that the bulk of the optical depth in the near–infrared will be contributed not by the lowest cloud layers but by intermediate layers having significant abundances of smaller particles.
Varying the Free Parameters {#subsection:Varying_Free_Parameters}
---------------------------
Among the model free parameters discussed above, the level of maximum supersaturation is the most problematic. The value of is particularly difficult to calculate independently, and because the condensation timescale depends on it, has a significant effect on the resulting particle size. This effect will be manifest in quiescent clouds but will be less important in convective clouds. This is because in rapidly convecting regions, convective upwelling can directly sustain particle growth.
Figure \[figure:Particle\_Sizes\_vs\_Smax\] shows, for one brown dwarf model, the effect of varying on the modal particle size of a water cloud. For low levels of supersaturation, the particles grow to a radius of $\sim$50 $\mu
\textrm {m}$. In this case, abundant condensation nuclei raise the rate of heterogeneous nucleation high above the rate of homogeneous nucleation, and condensation can begin as soon as the atmosphere becomes slightly supersaturated. For high levels of supersaturation ($>$$10\%$), they grow much larger, to $\sim$$150\,
\textrm {\micron}$. In this clean atmosphere case, heterogeneous nucleation does not occur, and the partial pressure of water in the atmosphere must greatly exceed the saturation vapor pressure in order for condensation to begin. We have assumed an intermediate value of $=10^{-2}$ for iron. This value is our best estimate for the of vaporous iron based on a knowledge of the maximum supersaturations attained in terrestrial water clouds [@Rossow:1978].
The other free parameters of the cloud model are less difficult. The mixing ratios depend on the metallicity, which we have taken to be solar. The chemical equilibrium code requires the metallicity to be specified. Once the assumption of elemental composition is made, however, we can calculate the partial pressures of all the gases in the mixture. We defer a discussion of the effect of brown dwarf metallicity on cloud structure to future work.
The efficiency parameters for coagulation and coalescence, $\epsilon_{\rm coag}$ and $\epsilon_{\rm coal}$, are potentially important. However, our model results show that the timescales for these processes, whatever their efficiency, are consistently longer than the timescale of growth by heterogeneous nucleation at 1$\%$ supersaturation. The model suggests that coalescence is much faster than coagulation for the two species—iron and water—in which coalescence operates efficiently. The coalescence timescale, for the largest particles we have grown ($\sim$500 $\mu \textrm {m}$), can approach the timescale of heterogeneous nucleation. However, for the clouds in most of the atmospheres we have treated, coalescence is still slower than heterogeneous nucleation by a factor of ten or more. Indeed, unless the efficiency of coalescence is greater than one, requiring charged aerosols [@Rossow:1978], coalescence will never dominate particle growth.
Comparison With Other Cloud Models {#section:Cloud_Model_Comparisons}
==================================
Ackerman & Marley (2001) {#subsection:Ackerman_Marley_model}
------------------------
@Ackerman:2001 also employ a one–dimensional cloud model in which cloud layers are treated as horizontally uniform; i.e., in a globally averaged sense. The @Ackerman:2001 cloud model arrives at particle sizes by balancing the upward transport of condensable material due to convective updrafts with the downward sedimentation of condensate. In the case in which the convective timescale, $\rm \tau_{conv}$, dominates over the other growth timescales in the present cloud model, which will occur in vigorously convecting regions of the atmosphere, the results of the present cloud model are expected to produce very similar results to the @Ackerman:2001 treatment. This is because our formulation then reduces to a similar comparison between the characteristic velocities of turbulent convective updrafts and the terminal velocity of particles.
In their treatment of rainout, @Ackerman:2001 introduce an additional parameter, $f_{\rm rain}$, which characterizes the vertical distribution of condensate in the cloud. The $f_{\rm rain}$ parameter, as @Ackerman:2001 state, is difficult to calculate from basic principles. They leave it as an adjustable parameter in their model. It depends on the mass–weighted sedimentation velocity of the cloud droplets and the convective velocity scale. The difficulty in calculating $f_{\rm rain}$ lies in the complexity of modeling fully the nature of the convection within the cloud. This problem has yet to be successfully attacked for brown dwarf atmospheres.
If it can be computed for these convective clouds, however, knowledge of $f_{\rm rain}$ can potentially provide a more realistic measure of the height and distribution of material in the cloud. For quiescent clouds, $f_{\rm rain}$ does not play a role. @Ackerman:2001 have assumed $f_{\rm rain}$ = 2–3 in their calculations. The cloud decks they derive are somewhat more compact than those produced in the present cloud model, although both prescriptions lead to rapid depletion of cloud material within a fraction of an atmospheric scale height above the condensation level.
The particle sizes we obtain compare favorably with the sizes predicted by the @Ackerman:2001 model. They predict modal particle radii in the intermediate size range of 40-80 $\mu \textrm {m}$ for both iron and silicate grains, in good agreement with the particle size ranges shown in Figures \[figure:Iron\_Sizes\], \[figure:Forsterite\_Sizes\], and \[figure:Gehlenite\_Sizes\].
Helling *et al. *(2001)** {#subsection:Helling_model}
-------------------------
@Helling:2001 study the onset of cloud particle growth via acoustic waves. They show that small ($1-10\,\micron$ in radius) sized particles can nucleate rapidly, normally within a few seconds. These values are consistent with our particle sizes in radiative regions under the assumption of very low supersaturations ($\mathcal{S}_{\rm max} \ll 10^{-2}$). Our larger particles form from sustained particle growth by convective uplifting. The grains generated in [@Helling:2001] are not grown to the maximum size allowed gravitationally, which we assume can occur.
The predictions of the two models in terms of particle radii will be in qualitative agreement in the absence of the $\rm \tau_{conv}$ timescale employed in our model. Nevertheless, we emphasize caution in comparing directly the results of these two models. They employ vastly different physical approaches and are therefore not expected to agree in many cases, even qualitatively. The model of @Helling:2001 attempts a detailed simulation of particle growth in brown dwarf atmospheres, including the complicated effects of hydrodynamics. The goal of the present cloud model, rather, is not to directly simulate the dynamics of particle growth but to develop a computationally economical cloud model that can be incorporated easily into spectral synthesis models.
Discussion {#section:Discussion}
==========
Cloud Opacity {#subsection:Cloud_Opacity}
-------------
The effects of clouds on the emergent spectra of substellar objects depend strongly on grain sizes. We derive the wavelength–dependent absorption and scattering opacities of grains with a full Mie theory approach, utilizing the formalism of @vandeHulst:1957 and @Deirmendjian:1969.
Figure \[figure:Cloud\_Opacity\] shows the results of such a calculation for forsterite grains in a brown dwarf atmosphere (T$_{\rm eff}$ = 1500 K, g$_{\rm surf}$ = $10^5$ cm s$^{-2}$) based on our cloud model results and optical constants from @Scott:1996. Also shown are the results for a larger grain size distribution peaked at 50 $\rm \mu m$, as well as for a size distribution peaked at 0.1 $\rm \mu m$, which is representative of an interstellar particle size [@Mathis:1977] often assumed to be appropriate for substellar objects [@Allard:2001; @Barman:2001]. For comparison, we have plotted the atomic and molecular gaseous opacities [@Burrows:2001 and references therein] within the cloud region. The substantial absorption and scattering differences between the size distributions, and relative to the gaseous absorption, convey the importance of proper cloud modeling.
Effect on Spectra {#subsection:Spectral_Effects}
-----------------
Figure \[figure:Model\_Spectra\] depicts the effects of our modeled forsterite cloud on the emergent spectrum of a brown dwarf (= 1500 K, = $10^5$ cm s$^{-2}$). This model atmosphere was obtained using the self–consistent stellar atmosphere code, TLUSTY [@Hubeny:1988; @Hubeny:1995]. The base of the forsterite cloud resides at approximately 4 bars and 1800 K, which is the highest temperature in this atmosphere at which forsterite grains can form.
We made two simplifying approximations in producing the cloudy model spectrum shown in Figure \[figure:Model\_Spectra\]. First, a cutoff for the cloud deck was desirable to facilitate the calulation of optical depth. We chose this cutoff to be one scale height above the cloud base, which is the atmospheric pressure e-folding distance. Since the remaining cloud–forming material is highly depleted of condensate a scale height above the condensation level (see Section \[subsection:Cloud\_Code\], Equation \[equation:Rainout\_Heterogeneous\]), the cloud in this region is optically thin. This simplification is therefore not a concern, and the spectral model shown in Figure \[figure:Model\_Spectra\] incorporates the full opacity of the cloud.
Second, owing to difficulties in computing the Mie theory scattering and absorption opacities iteratively, we chose to simplify the radiative transfer problem by using a uniform modal particle radius of $\rm 11\,\mu m$ throughout the cloudy region. That is, at every atmospheric pressure level for which we computed the opacity of particles, we employed the @Deirmendjian:1964 particle size distribution, given by Equation \[equation:Size\_Distribution\], with value of $\rm r_0$ equal to $\rm 11\,\mu m$. The scattering and absorption opacities of a cloud of $\rm 11\,\mu m$ forsterite particles are shown in Figure \[figure:Cloud\_Opacity\]. This value represents the typical particle size in the cloud deck, accounting for the decrease in density of cloud material with height above the cloud base (i.e., a number density weighted average of the particle sizes predicted by the cloud model).
A cloud–free model of the same effective temperature and gravity is plotted for comparison. Within the $B$ ($\sim$ 0.45 $\mu$m) and $Z$ ($\sim$ 1 $\mu$m) bands, the emergent flux is lowered by as much as one dex. However, the strong absorption by the wings of the sodium and potassium resonance lines is mitigated somewhat due to the clouds. Also of interest are the differences in the $J$ ($\sim 1.25 \, \mu$m), $H$ ($\rm \sim 1.6 \, \mu
m$), and $K$ ($\rm \sim 2.2 \, \mu m$) infrared bands. The presence of clouds reduces the emergent flux in the otherwise relatively clear $J$ and $H$ bands, allowing more flux to escape between these bands and at longer wavelengths. In fact, the $J-K$ colors differ by over 1.5 magnitudes between the cloudy and cloud–free models.
The cloudy spectrum presented in Figure \[figure:Model\_Spectra\] is intended to demonstrate the potential importance of clouds as an opacity source in SMO atmospheres. We defer to future work the problem of more realistically incorporating the opacity of clouds into spectral synthesis models.
Coupling Clouds With SMO Atmosphere Models {#subsection:Coupling_Clouds_With_SMO_Atmosphere_Models}
------------------------------------------
Developing fully self–consistent atmosphere models with clouds is potentially problematic because clouds perturb the radiative balance of the upper atmosphere. Scattering and absorption of radiation by the cloud causes the temperature–pressure structure of a cloudy atmosphere to deviate from the structure of a cloud–free atmosphere having the same effective temperature and surface gravity. The extent of the effect is strongly dependent on the location of the cloud in the atmosphere and the vertical variation of cloud particle sizes and number densities. Calculating an atmospheric profile that is in radiative equilibrium including both the gaseous opacity and the opacity of the cloud is straightforward. The potential inconsistency arises from the fact that the nature of the cloud itself, including the cloud base location and the particle sizes, depends in general on the temperature–pressure profile of the atmosphere, as we show in Section \[section:Cloud\_Model\]. This is a problem faced by all research groups attempting to include clouds completely self–consistently into SMO spectral synthesis models.
We explored this problem quantitatively by comparing the temperature–pressure profile of two atmosphere models, a “cloudy” model atmosphere and a “cloud–free” model atmosphere, at the same effective temperature and surface gravity. The cloud–free model used for this test was obtained using TLUSTY, our self–consistent model atmosphere code [@Hubeny:1988; @Hubeny:1995], by including only the opacity from gaseous atomic and molecular absorption. We then applied the cloud model described in Section \[section:Cloud\_Model\] to this cloud–free atmosphere to obtain the distribution of particle sizes and densities of the forsterite cloud predicted to form near the object’s visible surface. For simplicity, we took a sensible average of the particle sizes output by the cloud model and calculated, for a cloud of uniform modal particle size, the absorption and scattering of radiation by the particles using the Mie theory approach outlined in Section \[subsection:Cloud\_Opacity\]. We then incorporated this opacity back into the atmosphere code to recompute the temperature–pressure structure, thus obtaining the cloudy model atmosphere. We found the cloudy atmosphere to be hotter by several hundred degrees than the cloud–free atmosphere at the same pressure, a significant change. This is the back-warming effect of the cloud alluded to previously.
We then applied the cloud model to the new cloudy atmosphere to see whether the forsterite grain sizes varied significantly as a result of the change in the temperature–pressure profile. We found that the particles did change in size: their radius increased by about a factor of three. That is, the forsterite cloud looks quite different for the perturbed atmosphere model than it does for the original cloud–free model. This change is a result of the fact that the forsterite cloud straddles the convective–radiative boundary of the atmosphere. Thus, in the original model, the forsterite cloud appears in a convectively stable region of the brown dwarf, but in the perturbed model, the forsterite cloud forms in the convective zone of the atmosphere. Thus, the particles in the perturbed model came out larger than the particles in the original model (Figure \[figure:Cloud\_Decks\_Rad\_vs\_Conv\] shows how particles in radiative regions are systematically smaller than particles forming in convective regions).
We performed a similar test for iron clouds. Unlike for the forsterite cloud, we found only a small change in the particle sizes—a decrease of about $15\%$ in radius—resulting from the perturbation in the atmospheric temperature–pressure structure. The iron cloud forms deeply enough in the convective region of the brown dwarf studied to not be strongly affected by the heating due to the cloud. The degree of inconsistency between cloud–free and cloudy profiles therefore depends on the details of the calculation itself. In many cases, the discrepancies will not be a major concern, but they can be particularly large if the condensation curve happens to intersect the atmospheric profile near the convective–radiative boundary.
We emphasize that both the “cloud–free” and “cloudy” model atmospheres described above satisfy the radiative equilibrium boundary condition, and in that sense, the profiles are self–consistent. The remaining inconsistency arises from not accounting properly for possible adjustments to the opacity of the cloud when the particle sizes change after the profile is perturbed. A future challenge will be developing the machinery to generate model atmospheres that incorporate clouds fully self–consistently.
Conclusions {#section:Conclusions}
===========
General Results {#subsection:General_Results}
---------------
We have addressed the condensation and subsequent growth of cloud particles in the atmospheres of brown dwarfs. We present optimal particle sizes for three abundant species—iron, forsterite, and gehlenite—for a broad range of brown dwarf effective temperatures and surface gravities.
High-gravity brown dwarfs exhibit clouds with typical particle sizes in the $5-20\,\micron$ range. The particles grow much larger, however, in low-gravity objects, often greater than $100\,\micron$ in radius. We discovered a similar trend with effective temperature: hot brown dwarfs have characteristically larger particle sizes than cool brown dwarfs because of the increased energy flux that must be transported by convection. The distribution of cloud particle sizes depends strongly on the atmospheric parameters, and it is therefore unrealistic in spectral models to assume a single particle size distribution for the entire class of SMOs.
We demonstrate that particle size crucially affects the optical depth of the cloud. Unlike clouds having a particle size distribution centered at $0.1\,\micron$, these cloud decks do not dominate the opacity. Rather, they smooth the emergent spectrum and partially redistribute the radiative energy (see Figure \[figure:Model\_Spectra\]).
Improved Cloud Models: Morphology and Patchiness {#subsection:Improving_Cloud_Models}
------------------------------------------------
A complete theory of brown dwarf and giant planet atmospheres will require detailed modeling of clouds. Although obtaining plausible particle sizes is a good start, we have been unable to say anything yet about the morphology or patchiness of clouds. For example, on Jupiter, clouds appear in banded structures that vary latitudinally, while Neptune and Uranus show less latitudinal banding in their surface cloud patterns. The structure and patchiness of clouds on brown dwarfs is not known, though the putative variability reported by @Bailer-Jones:2001 suggests that cloud patches are not necessarily static.
A simple approach to understanding surface cloud distributions is the moist entraining plume model. Such a model has been put forth by @Lunine:1989 based on the work of @Stoker:1986 for Jupiter’s equatorial plumes. The model generates plumes from atmospheric material whose buoyancy is increased by the release of latent heat from condensation.
More elaborate calculations would involve three–dimensional modeling of the general atmospheric circulation. Such an approach would be quite difficult to implement, especially without accurate meteorological data for SMO atmospheres. General circulation models would also be too intensive computationally to directly incorporate into SMO spectral models.
Future Work in Spectral Synthesis {#subsection:Future_Work}
---------------------------------
The results of the current work will be useful in developing more elaborate spectral models of substellar atmospheres. We plan to follow up the present work with an exploration of the spectral effects of a variety of cloud compositions and distributions. Although Figure \[figure:Model\_Spectra\] shows the general effects of introducing a forsterite cloud on the spectrum of a brown dwarf, we have not yet varied the particle size distribution from the base to the top of the cloud. A large drop in the particle sizes with height above the cloud base will produce greater optical depth than a cloud deck having uniformly large particles. It is also likely that clouds of more than one species will form near the photosphere simultaneously. A major future challenge of this field will be to model the time dependent dust formation processes of all the major condensable species together, including the possible formation of *core–mantle *grains, which are composed of multiple chemical phases, and then to incorporate them fully self–consistently into substellar model atmospheres.**
We thank Ivan Hubeny, Chris Sharp, Jonathan Fortney, Jason Barnes, Adam Showman, Bill Hubbard, Jim Liebert, and the referee for insightful discussions and advice. This work was supported in part by NASA grants NAG5-10450, NAG5-10760 and NAG5-10629.
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|
---
abstract: 'Parity-violating asymmetries in polarized electron scattering have been interpreted as the asymmetries between opposite helicities of incoming fermion based on the approximation of the spin polarization operator. Here exact calculations of cross sections for parity-violating asymmetries in SLAC E158 and SLD have been performed using spin projection operators. And the parity-violating factor incorporating with spin polarization and momentum has been identified and shown that its sign depends on the spin polarization of incoming particle and the relative velocity of incoming and target particles. Therefore, I suggest a new concept of relative spin polarization to interpret the parity-violating asymmetry as contributed by the antisymmetric nature of the weak interactions depending on whether the spin direction of the incoming electron is inward or outward relative to the target electron.'
author:
- 'J.C. Yoon'
title: 'Relative Spin Polarization of Parity-Violating Asymmetries'
---
Introduction
============
The parity-violating asymmetries of SLAC E158 and SLD experiments have been interpreted as the asymmetries between opposite helicities for the Standard Model [@Glashow; @Weinberg; @Salam; @Prescott:1978tm; @SLACE158; @SLD2003] using the chiral projection, approximated from the spin projection operator.
Helicity $h\pm$ is defined by the momentum of a fermion and its spin orientation; if the spin orientation is in the same direction as its momentum, it is called right-handed helicity($h+$). And helicity can be observed to be opposite depending on the reference frame and is especially indefinite in the rest frame where momentum is zero. For example, a right-handed helicity massive fermion $u_{h+}(+p_{z}, +s_{z})$ in one reference frame can be observed at the same time as $u_{h-}(-p_{z}, +s_{z})$ with left-handed helicity by observers in other reference frames. Chirality (L,R) is defined to indicate either of the two-component objects in a massive fermion field, and no physical measurement is available for the chirality of massive fermions and only when fermions are massless chirality becomes helicity.
After a review of spin polarization, we investigate the validity of the chiral projection as an approximation of the spin polarization operator in the practical calculation of matrix elements [@PeskinSchroeder; @BjorkenDrell; @JCYoon].
Spin Polarization and Chiral Projection
=======================================
Spin polarization is defined by the direction of the spin relative to a given coordinate system, whereas helicity is relative to the momentum direction of the particle. The spin polarization vector $s_{\mu}$ can be derived from the spin projection operator $P(\mathbf{s})$ for an electron with $\mathbf{s}$ being the spin direction at rest: $$\begin{aligned}
P(\mathbf{s}) &\equiv& c u \overline{u} \nonumber \\
&=& {1 \over 4}\left(1 + {p\!\!\slash \over m}\right)\left(1 -
\gamma^{5} {{s}\!\!\!\slash}\,\right)\end{aligned}$$ where the spin polarization 4-vector $s_{\mu}$ is $$\begin{aligned}
s_{\mu} = \left( {\mathbf{s} \cdot \mathbf{p} \over m}, \mathbf{s}
+ {\mathbf{p}(\mathbf{s} \cdot \mathbf{p}) \over m (E + m)}
\right),\end{aligned}$$ satisfying that $$\begin{aligned}
s^{2} = -1 \qquad \mathrm{and} \qquad s^{\mu}p_{\mu} = 0,\end{aligned}$$ where it is normalized to one particle per unit volume in the rest frame by the normalization factor $c$ [@OkunLQ; @OkunWI; @Tolhoek]. For example, the spin projection operator for an electron with the spin direction $+s_{z}$ and momentum $+p_{z}$ is: $$\begin{aligned}
P(+s_{z}) &\equiv& c u_{h+}(+p_{z},+s_{z})
\overline{u}_{h+}(+p_{z},+s_{z}) \nonumber \\
&=& {1 \over 2m }\left( \begin{array}{cccc} m & 0 & E-p_{z} & 0 \\
0 & 0 & 0 & 0 \\ E+p_{z} & 0 & m & 0 \\ 0 & 0 & 0 & 0
\end{array} \right)
\nonumber \\
&=& {1 \over 4}\left(1 + {p\!\!\slash \over m}\right)\left(1 -
\gamma^{5} {{s}\!\!\!\slash}\,\right)\end{aligned}$$ and the spin polarization vector is: $$\begin{aligned}
s_{\mu} &=& \left( {+p_{z} \over m}, 0, 0, 1 + {(+p_{z}) p_{z}
\over m (E + m) }\right) \nonumber \\
&=& \left( {+p_{z} \over m}, 0, 0, {E \over m}\right)\end{aligned}$$ In general, the spin projection operators of a fermion with $\pm
s_{z}$ can be given by: $$\begin{aligned}
P(+s_{z}) &\equiv& c u_{h\pm}(\pm p_{z},+s_{z})
\overline{u}_{h\pm}(\pm p_{z},+s_{z}) \nonumber \\
&=& {1 \over 2m }\left( \begin{array}{cccc} m & 0 & E \mp p_{z} & 0 \\
0 & 0 & 0 & 0 \\ E \pm p_{z} & 0 & m & 0 \\ 0 & 0 & 0 & 0
\end{array} \right)
\nonumber \\
P(-s_{z}) &\equiv& c u_{h\mp}(\pm p_{z},-s_{z})
\overline{u}_{h\mp}(\pm p_{z},+s_{z}) \nonumber \\
&=& {1 \over 2m }\left( \begin{array}{cccc} 0 & 0 & 0 & 0 \\
0 & m & 0 & E \pm p_{z} \\ 0 & 0 & 0 & 0 \\ 0 & E \mp p_{z} & 0 &
m
\end{array} \right)\end{aligned}$$ Note that the location of the non-zero matrix elements is determined by the spin direction from the product of the two-component spinors $\xi \xi^{\dagger}$.
The approximation of the spin projection operator $P(\mathbf{s})$ should be performed after the full evaluation of the matrix element, taking into account the normalization factor. For example, the trace of the matrix element calculation $\mathrm{tr}[{P(+s_{z})}] = 1$ vanishes for the massless approximation ($m \rightarrow 0$) ignoring the normalization factor $1 / 2m$: $$\begin{aligned}
\mathrm{tr}[{P(+s_{z})}] \rightarrow {1 \over 2m }\left(
\begin{array}{cccc} 0 & 0 & E-p_{z} & 0 \\
0 & 0 & 0 & 0 \\ E+p_{z} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0
\end{array} \right) \rightarrow 0.\end{aligned}$$ The common practice of spin polarization approximation is inaccurate compared with the accurate evaluation of the full spin projection operator. For the relativistic limit of $m/E
\rightarrow 0$, the spin projection operators for longitudinally polarized electrons with $\mathbf{s}$ parallel to $\mathbf{p}$ can be reduced as: $$\begin{aligned}
s^{\mu} &=& {1 \over m \beta}p^{\mu} - {\sqrt{1 - \beta^{2}} \over
\beta}g^{\mu 0} \nonumber \\
&\rightarrow& {p^{\mu} \over m} \qquad \qquad \textrm{for} \;\;
\beta \rightarrow 1,\end{aligned}$$ whereas the explicit expression of the spin polarization ${P(+s_{z})}$ is now given by: $$\begin{aligned}
{P(+s_{z})} &\rightarrow& {1 \over 4}\left(1 + {p\!\!\slash \over
m}\right)\left(1 - \gamma^{5} {{{p}\!\!\!\slash} \over
m}\,\right)\nonumber \\
&=& {1 \over 4m }\left(
\begin{array}{cccc} m & 0 & 2(E-p_{z}) & 0 \\
0 & m & 0 & 2(E+p_{z}) \\ 0 & 0 & m & 0 \\ 0 & 0 & 0 & m
\end{array} \right)\end{aligned}$$ and although the approximation holds true for some specific calculations such as $\mathrm{tr}[{P(+s_{z})}] = 1$ it is inconsistent with the accurate evaluation of the matrix element in general. The omission of the normalization factor is also misleading because it neglects the electron mass term in the relativistic limit [@LampeReya]: $$\begin{aligned}
{1 \over 2}\left({p\!\!\slash } + m \right)\left(1 - \gamma^{5}
{{s}\!\!\!\slash}\,\right) \rightarrow {1 \over
2}\left({p\!\!\slash } \right)\left(1 - \gamma^{5}
{{s}\!\!\!\slash}\,\right).\end{aligned}$$ whereas the contribution of the $m$ term to the matrix element should be the same regardless of the energy $E$ once the proper normalization is considered, since the mass factor in $ms^{\mu}$ is normalized to be one by $m^{-1}$. The simplification of the matrix element calculation in the relativistic limit can be given by: $$\begin{aligned}
\left( {{1 \pm \gamma^{5} {s \!\!\slash}_{h+}} \over 2
}\right)\left( {{p\!\!\slash + m} \over 2m} \right) \rightarrow
\left( {{1 \pm \gamma^{5}} \over 2 }\right)\left( {{p\!\!\slash +
m} \over 2m} \right), \label{EqExactApprox}\end{aligned}$$ whereas the explicit expression of the spin polarization ${P(+s_{z})}$ is now given by: $$\begin{aligned}
\left( {{1 + \gamma^{5}} \over 2 }\right)\left( {{p\!\!\slash + m}
\over 2m} \right) = {1 \over 2m } \left(
\begin{array}{cccc} m & 0 & E-p_{z} & 0 \\
0 & m & 0 & E+p_{z} \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0
\end{array} \right) \nonumber\end{aligned}$$ This is inconsistent with the accurate evaluation of the general matrix element calculation, since the spin direction represented by a nonzero matrix element is inconsistent for massive electrons. The matrix element for a right-handed chirality relativistic electron to scatter to a left-handed chirality electron with interaction $\gamma^{\mu}$ is proportional to: $$\begin{aligned}
\lefteqn{\overline{u}_{L}(p')\gamma^{\mu}u_{R}(p)}\nonumber \\
&& {} = \overline{u}(p')\left({{1 + \gamma_{5}} \over 2}
\right)\gamma_{\mu}\left( { {1 + \gamma_{5}} \over 2}
\right)u(p)
\rightarrow 0\end{aligned}$$ whereas from the exact representation of massive fermions in terms of helicity, not of chirality, the corresponding matrix element is in general nonzero: $$\begin{aligned}
\lefteqn{\mathrm{tr}\left[\overline{u}_{h-}(p')\gamma^{\mu}u_{h+}(p)
\overline{u}_{h+}(p)\gamma^{\nu}u_{h-}(p') \right]} \nonumber \\
&&{} = \mathrm{tr}\left[\overline{u}_{h-}(p')\gamma^{\mu}u(p)
\overline{u}(p)\left({1 - \gamma^{5} {{s}\!\!\!\slash} \over 2}
\right)\gamma^{\nu}u(p')_{h-} \right]. \nonumber\end{aligned}$$ Note that $(1 \pm \gamma^{5} {{s}\!\!\!\slash})/2$ is supposed to act on $({p\!\!\slash + m})/2m$, not on $u$ alone, and the chiral projection $(1 \pm \gamma^{5})$ on $u$ is inconsistent with being a proper approximation of $(1 \pm \gamma^{5} {{s}\!\!\!\slash})/2$ for a massive fermion.
Since the chiral projection as an approximation of spin projection operators could differ from the exact calculation of spin projection operators, the matrix element should be carefully evaluated comparing with the full representation of the spin projection operators.
The Parity-Violating Asymmetry in SLAC E158 and SLD
===================================================
The parity-violating asymmetry for polarized electrons scattering on an unpolarized target is given by: $$\begin{aligned}
A_{PV} &\equiv& {{d\sigma_{h+}(+p_{z},+s_{z}) -
d\sigma_{h-}(+p_{z},-s_{z})} \over {d\sigma_{h+}(+p_{z},+s_{z}) +
d\sigma_{h-}(+p_{z},-s_{z})}}\end{aligned}$$ where $d\sigma_{\lambda}$ denotes the differential cross section for an incoming electron of helicity $\lambda$ on an unpolarized target. For the parity-violating asymmetry in SLAC E158 and SLD [@SLACE158; @SLD2003], left- and right-handed helicity massive fermions are considered as distinguishable particles with the weak interaction structure of $\gamma^{\mu}(v - a\gamma^{5})$, as suggested in the Standard Model [@Swartz; @Derman; @Clarke], and the asymmetry is accounted for as parity violation between opposite helicities. For the electron scattering of $$\begin{aligned}
e^{-}(p_{1},\lambda_{1}) + e^{-}(p_{2},\lambda_{2}) \rightarrow
e^{-}(p_{3},\lambda_{3})+ e^{-}(p_{4},\lambda_{4}),\end{aligned}$$ where $(p_{i},\lambda_{i})$ denote the four momenta and helicities of incoming and outgoing electrons, respectively, the interaction Lagrangian for the electromagnetic and weak interactions with the Z boson is given by: $$\begin{aligned}
\mathcal{L}_{int} &\sim&
-g_{\gamma}\overline{e}\gamma_{\mu}eA^{\mu}
\nonumber \\
&+& g_{z}\bigg[c_{L}\overline{e}\gamma_{\mu}(1 - \gamma_{5})e +
c_{R}\overline{e}\gamma_{\mu}(1 + \gamma_{5})e \bigg]Z^{\mu}
\nonumber \\
&=& -g_{\gamma}\overline{e}\gamma_{\mu}eA^{\mu} +
g_{Z}\overline{e}\gamma_{\mu}(v - a\gamma_{5})eZ^{\mu},
\label{WeakLagEq}\end{aligned}$$ where $g_{\gamma} = e$, $g_{z} = {e [2 \cos \theta_{W} \sin
\theta_{W}]^{-1}}$ and $v \equiv c_{L} + c_{R} = {1 / 2} - 2
\sin^{2} \theta_{W}$, $a \equiv c_{L} - c_{R} = {1 / 2}$. Note that $(1 \pm \gamma^{5})$ in Eq. (\[WeakLagEq\]) are not chiral projections but chiral interaction structures, and electrons $e$ are later to be classified as left- and right-handed helicity electrons $e_{\lambda}$ using the spin polarization projection operator $(1 \mp \gamma^{5}{{s_{1}}\!\!\!\!\!\slash}\,)$ [@BjorkenDrell; @OkunLQ; @OkunWI; @Tolhoek; @Derman]. The total tree-level amplitude for $ee$ scattering via $\gamma$ and $Z$ exchange in the center-of-mass (CM) frame is then given by: $$\begin{aligned}
\lefteqn{\mathcal{M} = \mathcal{M}_{\gamma}^{d} +
\mathcal{M}_{\gamma}^{c}+ \mathcal{M}_{Z}^{d}+\mathcal{M}_{Z}^{c}
} \nonumber \\
&&{}= {- g^{2}_{\gamma} \over ys}
\overline{u}(p_{3})\gamma_{\mu}u(p_{1}) \cdot
\overline{u}(p_{4})\gamma^{\mu} u(p_{2}) \nonumber
\\
&&{} +{ g^{2}_{\gamma} \over (1 -
y)s}\overline{u}(p_{4})\gamma_{\mu}u(p_{1})\cdot
\overline{u}(p_{3})\gamma^{\mu}u(p_{2}) \nonumber \\
&&{}- { g^{2}_{z} \over m^{2}_{z}}
\overline{u}(p_{3})\gamma_{\mu}(v-a\gamma_{5}) u(p_{1}) \cdot
\overline{u}(p_{4})\gamma^{\mu}(v-a\gamma_{5}) u(p_{2}) \nonumber
\\
&&{} + { g^{2}_{z} \over m^{2}_{z}}
\overline{u}(p_{4})\gamma_{\mu}(v-a\gamma_{5}) u(p_{1}) \cdot
\overline{u}(p_{3})\gamma^{\mu}(v-a\gamma_{5}) u(p_{2}) \nonumber\end{aligned}$$ where $y \equiv - {(p_{1}-p_{3})^{2}/ s} = \sin^{2}
(\theta_{cm}/2)$ with the CM scattering angle $\theta_{cm}$ and the CM energy $\sqrt{s}=[(p_{1}+p_{2})^{2}]^{1/2}$.
For the explicit spin polarization calculation with the operator $(1 \mp \gamma^{5}{{s_{1}}\!\!\!\!\!\slash}\,)$ for inward and outward relative spin polarization corresponding to $(1 \pm
\gamma^{5})$, the nonzero parity-violation asymmetry arises from such terms as: $[\mathcal{M}_{\gamma}^{d}\mathcal{M}_{z}^{d*}]'_{\lambda_{1}}
+[\mathcal{M}_{\gamma}^{d}\mathcal{M}_{z}^{c*}]'_{\lambda_{1}}
+[\mathcal{M}_{\gamma}^{c}\mathcal{M}_{z}^{d*}]'_{\lambda_{1}}
+[\mathcal{M}_{\gamma}^{c}\mathcal{M}_{z}^{c*}]'_{\lambda_{1}}$, $$\begin{aligned}
\lefteqn{[\mathcal{M}_{\gamma}^{d}\mathcal{M}_{z}^{d*}]'_{\lambda_{1}}
%{2 g_{\gamma}^{2} g_{z}^{2} \over ysm_{z}^{2}}
}\hspace{3.3in} \nonumber \\
\lefteqn{={ g_{\gamma}^{2} g_{z}^{2} \over ysm_{z}^{2}}
\mathrm{tr}[\overline{u}(p_{3})\gamma^{\mu}(1 \mp
\gamma^{5}{{s_{1}}\!\!\!\!\!\slash}\,)u(p_{1})
\overline{u}(p_{1})\gamma^{\nu}(v - a
\gamma^{5})u(p_{3})]}\hspace{3.1in}
\nonumber \\
\lefteqn{ \mathrm{tr}[\overline{u}(p_{4})\gamma_{\mu}u(p_{2})
\overline{u}(p_{2})
\gamma_{\nu}(v - a\gamma^{5})u(p_{4})]}\hspace{2.7in} \nonumber \\
\lefteqn{= [\mathcal{M}_{\gamma}^{d}\mathcal{M}_{z}^{d*}] \mp 32
{g_{\gamma}^{4} \over y}\beta (-E_{1}p_{2z}+E_{2}p_{1z}) v a
}\hspace{3.1in}\end{aligned}$$ $$\begin{aligned}
\lefteqn{[\mathcal{M}_{\gamma}^{d}\mathcal{M}_{z}^{c*}]'_{\lambda_{1}}
}\hspace{3.3in} \nonumber \\
\lefteqn{={ g_{\gamma}^{2} g_{z}^{2} \over ysm_{z}^{2}}
\mathrm{tr}[\overline{u}(p_{3})\gamma^{\mu}(1 \mp
\gamma^{5}{{s_{1}}\!\!\!\!\!\slash}\,)u(p_{1})
\overline{u}(p_{1})\gamma^{\nu}(v - a
\gamma^{5})u(p_{4})}\hspace{3.1in}
\nonumber \\
\lefteqn{ \overline{u}(p_{4})\gamma_{\mu}u(p_{2})
\overline{u}(p_{2})
\gamma_{\nu}(v - a\gamma^{5})u(p_{3})]}\hspace{2.45in} \nonumber \\
\lefteqn{= [\mathcal{M}_{\gamma}^{d}\mathcal{M}_{z}^{c*}] \mp 32
{g_{\gamma}^{4} \over y}\beta (-E_{1}p_{2z}+E_{2}p_{1z}) v a
}\hspace{3.1in}\end{aligned}$$ where the incoming and target electrons are given by $p_{1} =
(E_{1}, p_{1z})$ and $p_{2} = (E_{2}, p_{2z})$. The parity-violating asymmetry $A_{PV}$ is given by: $$\begin{aligned}
A_{PV} &=& - 16 \beta (-E_{1}p_{2z}+E_{2}p_{1z}) v a {y(1-y) \over
1 + y^{4} + (1-y)^{4}} \nonumber\end{aligned}$$ where $(-E_{1}p_{2z}+E_{2}p_{1z}) = E_{1} E_{2} (\vec{v}_{1} -
\vec{v}_{2}) = \sqrt{(p_{1} \cdot p_{2}) - m_{1}^{2}m_{2}^{2}}$ is Lorentz invariant holding for any arbitrary frame when $\vec{v}_{1}$ is parallel to $\vec{v}_{2}$ and the massless approximation of $A_{PV}$ in the CM frame $p_{1} \simeq (E, E)$ and $p_{2} \simeq (E, -E)$ is consistent with the approximated parity-violating asymmetry of the SLAC E158 denoted as $A^{approx}_{PV}$ [@Derman]. In SLD, the polarized differential cross section of the $e^{-}_{h\pm}e^{+} \rightarrow Z^{0} \rightarrow f \overline{f}$ process with longitudinally polarized electrons and unpolarized positrons is $$\begin{aligned}
\lefteqn{
\bigg[{d \sigma^{Z}_{f} \over d \Omega}\bigg]_{h\pm}
= {N_{c} \alpha^{2} \over 8 s \sin^{4} \theta_{W} \cos^{4} \theta_{W} [(s-m^{2}_{Z})^{2} + s^{2}\Gamma^{2}_{Z}/m^{2}_{Z}]} }\hspace{3.3in}
\nonumber \\
\lefteqn{\times [2(v_{e}^2+a_{e}^2)(v_{f}^2+a_{f}^2)(E_{1}E_{2}E_{4}E_{3}
+p_{1z}p_{2z}p_{4z}p_{3z})}\hspace{3.in}
\nonumber \\
\lefteqn{\mp2v_{e}a_{e}(v_{f}^2+a_{f}^2)(2E_{1}p_{2z}p_{4z}p_{3z}
+2p_{1z}E_{2}E_{4}E_{3}}\hspace{2.9in} \nonumber \\
-(E_{1}E_{2}+p_{1z}p_{2z})(p_{4z}E_{3}+E_{4}p_{3z}))
\nonumber \\
\lefteqn{+4v_{e}a_{e}v_{f}a_{f}(p_{1z}E_{2}-E_{1}p_{2z})
(E_{4}p_{3z}-p_{4z}E_{3})}\hspace{2.9in} \nonumber \\
\lefteqn{\mp2(v_{e}^2+a_{e}^2)v_{f}a_{f}(E_{1}E_{2}-p_{1z}p_{2z})
(E_{4}p_{3z}-p_{4z}E_{3})}\hspace{2.9in}
\nonumber \\
\lefteqn{-(v_{e}^2+a_{e}^2)(v_{f}^2+a_{f}^2)(E_{1}p_{2z}
+p_{1z}E_{2}) }\hspace{2.9in}
\nonumber \\ \lefteqn{\times (E_{4}p_{3z}+p_{4z}E_{3})
]}\hspace{1.5in}\end{aligned}$$ Note that its massless approximation in the CM frame $p_{1} \simeq (E, E)$ and $p_{2} \simeq (E, -E)$ is consistent with the approximated parity-violating calculation of the SLD [@SLD2003] and the total cross sections are Lorentz invariant.
Relative Spin Polarization for two-particle systems
===================================================
Let us investigate the exact cross section calculation under parity regarding to spin polarization and momentum of incoming and target particles. In the exact calculations, the factor that determines the signs of parity-violating terms depending on spin polarization and momentum is given by $\pm (E_{1}p_{2z} - E_{2} p_{1z}) = \mp E{1}E_{2}(\vec{v}_{1} - \vec{v}_{2})$. The signs of parity-violating terms become reversed when either the spin polarization ($\pm s_{z}$) of the incoming particle or the relative velocity between two particles $\vec{v}_{r} = (\vec{v}_{1} - \vec{v}_{2})$ is reversed. Note that the signs of parity-violating terms can remain constant even when the helicity of incoming electron becomes opposite as its momentum is reversed ($\vec{v}_{1} \rightarrow -\vec{v}_{1}$) under the Lorentz transformations such as in Fig. \[fig:LT2Particle\], since the factor $\pm (E_{1}p_{2z} - E_{2} p_{1z}$) is Lorentz invariant along the $z$ direction.
For a two-particle system, the spin polarization can be characterized by its direction relative to the velocity difference between the incoming and target particles $\vec{v}_r$, indicating whether the spin direction of the incoming particle points outward from or toward the target particle. Under Lorentz transformations, the differential cross sections for a right-handed helicity incoming electron can be observed as a left-handed helicity electron, but the relative spin polarization remains inward as in Fig. \[fig:LT2Particle\]. Thus, the exact evaluation of parity-violating asymmetry $A_{PV}$ remains asymmetric in general: $$\begin{aligned}
A_{PV} &\equiv& {{d\sigma_{h+}(+p_{z},+s_{z},in) -
d\sigma_{h-}(+p_{z},-s_{z},out)} \over
{d\sigma_{h+}(+p_{z},+s_{z},in) +
d\sigma_{h-}(+p_{z},-s_{z},out)}} \nonumber \\
&=& {{d\sigma_{h-}(-p_{z},+s_{z},in) -
d\sigma_{h-}(+p_{z},-s_{z},out)} \over
{d\sigma_{h-}(-p_{z},+s_{z},in) +
d\sigma_{h-}(+p_{z},-s_{z},out)}} \nonumber\end{aligned}$$ whereas the approximated calculation of parity-violating asymmetry, interpreted as the asymmetry between opposite helicities, vanishes $A^{approx}_{PV}=0$.
Therefore, the parity-violating asymmetry measured in SLAC E158 and SLD should be interpreted in terms of relative spin polarization not of helicity, since the asymmetry between opposite helicities vanishes under Lorentz transformations whereas the exact calculation does not.
Conclusion
==========
Since the approximate spin polarization $(1 \pm \gamma^{5})$ may significantly differ from the exact spin polarization in evaluating the matrix element, here exact calculations of cross sections for parity-violating asymmetries in SLAC E158 and SLD have been performed using the full expression of spin projection $(1 \mp \gamma^{5}{{s}\!\!\!\slash}\,)$. And the parity-violating factor incorporating with spin polarization and momentum $\pm (E_{1}p_{2z} - E_{2} p_{1z}) = \mp E{1}E_{2}(\vec{v}_{1} - \vec{v}_{2})$ has been identified and shown that its sign depends on the spin polarization of incoming particle and the relative velocity of incoming and target particles, not on the helicity of incoming particle. Therefore, I suggest a new concept of relative spin polarization to interpret the parity-violating asymmetry as contributed by the antisymmetric nature of the weak interactions depending on whether the spin direction of the incoming electron is inward or outward relative to the target electron.
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|
---
abstract: 'We study the adjunction property of the Jacquet-Emerton functor in certain neighborhoods of critical points in the eigencurve. As an application, we construct two-variable $p$-adic $L$-functions around critical points via Emerton’s representation theoretic approach.'
author:
- |
Yiwen Ding\
yiwen.ding@bicmr.pku.edu.cn
title: 'A note on critical $p$-adic $L$-functions'
---
Introduction
============
Let $N\in {\mathbb Z}_{>1}$, $p\nmid N$, $k\in {\mathbb Z}_{\geq 0}$, and $f$ be a classical newform of level $\Gamma_1(N)$ of weight $k+2$ over $E$ (which is a finite extension of ${\mathbb Q}_p$ sufficiently large). Let $a_p$ (resp. $b_p$) be the eigenvalue of the Hecke operator $T_p$ (resp. $S_p$) on $f$, and $\alpha$ be a root of the Hecke polynomial $X^2-a_p X+pb_p$. To $f$ and $\alpha$, one can associate an eigenform $f_{\alpha}$ of level $\Gamma_1(N)\cap \Gamma_0(p)$ satisfying that $f_{\alpha}$ has the same prime-to-$p$ Hecke eigenvalues as $f$, and that $U_p(f_{\alpha})=\alpha f_{\alpha}$. The eigenform $f_{\alpha}$ is called a *refinement* (or a *$p$-stabilization*) of $f$. We have $\operatorname{\mathrm val}_p(a_p)\geq 0$ and $\operatorname{\mathrm val}_p(b_p)=k$ (where $\operatorname{\mathrm val}_p$ the additive $p$-adic valuation on $\overline{{\mathbb Q}_p}$ normalized with $\operatorname{\mathrm val}_p(p)=1$). Since $\alpha^2-a_p \alpha+pb_p=0$, we easily deduce $\operatorname{\mathrm val}_p(\alpha)\leq k+1$. The refinement $f_{\alpha}$ is called
- *of non-critical slope* if $\operatorname{\mathrm val}_p(\alpha)<k+1$,
- *of critical slope* if $\operatorname{\mathrm val}_p(\alpha)=k+1$,
- *critical* if $\rho_{f,p}$ is split (which implies $\operatorname{\mathrm val}_p(\alpha)=k+1$),
where $\rho_{f,p}$ is the $2$-dimensional $\operatorname{\mathrm Gal}_{{\mathbb Q}_p}$-representation associated to $f$.
To the form $f_{\alpha}$, we can associate a $p$-adic $L$-function $L(f_{\alpha}, -)$, which is a distribution on ${\mathbb Z}_p^{\times}$ (for example see [@MTT] [@PSc] [@Bel] [@LLZ]... see also Proposition \[intpl\] of this note). When $f_{\alpha}$ is non-critical, $L(f_{\alpha},-)$ interpolates the critical values of the classical $L$-function attached to $f_{\alpha}$ (e.g. see (\[equ: clp-int\])). If $f_{\alpha}$ is moreover of non-critical slope, we know $L(f_{\alpha}, -)$ can be determined (up to non-zero scalars) by the interpolation property. When $f_{\alpha}$ is of critical slope, the interpolation property is however not enough to determine $L(f_{\alpha}, -)$. And in this case, the problem of proving that the $p$-adic $L$-functions constructed by different methods coincide is not completely settled yet (to the author’s knowledge). We remark that when $f_{\alpha}$ is critical, then $L(f_{\alpha}, \phi x^j)=0$ for all smooth characters $\phi$ on ${\mathbb Z}_p^{\times}$ and $j\in \{0,\cdots, k\}$ where $x^j$ denotes the algebraic character $a\mapsto a^j$ on ${\mathbb Z}_p^{\times}$ (e.g. see Proposition \[cri0\]). In all cases, $L(f_{\alpha}, -)$ fits into the so-called two-variable $p$-adic L-functions $L(-,-)$ where the first variable runs around points on the eigencurve (we recall that such points correspond to overconvergent eigenforms $g$), and $L(g,-)$ is, up to non-zero scalars, equal to the $p$-adic $L$-function of $g$ when $g$ is classical.
In [@Em05], Emerton provided a representation theoretic approach of the construction of $p$-adic $L$-functions, using results in the $p$-adic Langlands program. Via this approach, in [@Em1 § 4.5], Emerton also constructed two-variable $p$-adic $L$-functions in certain neighborhoods of non-critical points in the eigencurve (we call a point critical if its associated eigenform is critical)). A key ingredient in this construction is an adjunction property of the Jacquet-Emerton functor around non-critical points. In this note, we study the adjunction property around critical points in the eigencurve. We show that by adding “poles", the adjunction can extend to critical points (cf. Theorem \[thm: clp-rps\] and Theorem \[adj003\]). Using this result, together with the smoothness of the eigencurve at critical points (due to Bellaïche [@Bel]), we construct two-variable $p$-adic $L$-functions (see (\[padicL\])) in some neighborhoods of critical points via Emerton’s approach. Evaluating the two-variable $p$-adic $L$-functions at the critical points then allows us to associate $p$-adic $L$-functions to the corresponding critical eigenforms.
The note is organised as follows. In § \[sec: clp-1.1\], §\[sec: clp-1.2\], we recall some basic facts on the completed $H^1$ of modular curves, and recall Emerton’s construction of the eigencurves. Nothing is new in these twos sections. In § \[sec: clp-1.3\], we show an adjunction property of the Jacquet-Emerton functor in neighborhoods of a critical point on the eigencurve. In § \[sec: 21\], we use this adjunction property to construct two-variable $p$-adic $L$-functions around critical points. Finally, we study some properties of our $p$-adic $L$-functions in § \[sec: 22\].
Notations
---------
In this note, $E$ will be a finite extension of ${\mathbb Q}_p$ with ${\mathcal O}_E$ its ring of integers, $\varpi_E\in {\mathcal O}_E$ a uniformiser and $k_E:={\mathcal O}_E/\varpi_E$ its residue field. We let $B$ denote the Borel subgroup of $\operatorname{\mathrm GL}_2$ of upper triangular matrices, $T\subseteq B$ the subgroup of diagonal matrices, and $N\subseteq B$ the unipotent radical. We use ${\mathfrak t}$ to denote the Lie algebra of $T({\mathbb Q}_p)$ over $E$, ${\mathfrak g}$ the Lie algebra of $\operatorname{\mathrm GL}_2({\mathbb Q}_p)$ over $E$. For a continuous character $\chi: T({\mathbb Q}_p) {\rightarrow}E^{\times}$, we denote by $d\chi: {\mathfrak t}{\rightarrow}E$ the induced $E$-linear map with $$d\chi (X)=\lim_{t{\rightarrow}0} \frac{\chi(\exp{tX})-1}{t} |_{t=0},$$ for $X\in {\mathfrak t}$. Similarly, for a continuous character $\chi: {\mathbb Q}_p^{\times} {\rightarrow}E^{\times}$, we denote by $\operatorname{\mathrm wt}(\chi): {\mathbb Q}_p {\rightarrow}E$ the induced $E$-linear map, called the weight of $\chi$. For $a\in E^{\times}$, we denote by $\operatorname{\mathrm unr}(a): {\mathbb Q}_p^{\times} {\rightarrow}E^{\times}$ the unramified character (i.e. $\operatorname{\mathrm unr}(a)|_{{\mathbb Z}_p^{\times}}=1$) with $\operatorname{\mathrm unr}(a)(p)=a$.
For an $E$-vector space $V$ that is equipped with an $E$-linear action of $A$ (with $A$ a set of operators), and for a system of eigenvalues $\chi$ of $A$, we denote by $V[A=\chi]$ the $\chi$-eigenspace for $A$, and $V\{A=\chi\}$ the generalized $\chi$-eigenspace for $A$.
Acknowledgement {#acknowledgement .unnumbered}
---------------
I want to thank Matthew Emerton for suggesting the problem of extending the adjunction formula to critical points on the eigencurve, that led to the note. I thank Daniel Barrera Salazar, John Bergdall, Xin Wan, Shanwen Wang for helpful discussions or remarks. I also thank the anonymous referee for the reading and helpful suggestions. This work was supported by EPSRC grant EP/L025485/1 and by Grant No. 7101500268 from Peking University.
Eigencurves
===========
Completed cohomology of modular curves {#sec: clp-1.1}
--------------------------------------
Let ${\mathbb A}^{\infty}$ denote the finite adeles of ${\mathbb Q}$. For a compact open subgroup $K$ of $\operatorname{\mathrm GL}_2({\mathbb A}^{\infty})$, we denote by $Y_K$ the affine modular curve over ${\mathbb Q}$ such that the ${\mathbb C}$-points of $Y_K$ are given by $$Y_K({\mathbb C})=\operatorname{\mathrm GL}_2({\mathbb Q})\backslash \operatorname{\mathrm GL}_2({\mathbb A})/\big({\mathbb R}_+^{\times}\operatorname{\mathrm SO}_2({\mathbb R})K\big).$$ Let $K^p=\prod_{\ell\neq p} K_{\ell}$ be a compact open subgroup of $\operatorname{\mathrm GL}_2({\mathbb A}^{\infty,p})$, and $$\Sigma(K^p):=\{p\}\cup \{\ell\neq p, \ \text{$K_{\ell}$ is not maximal}\}.$$Let ${\mathcal H}(K^p)$ denote the Hecke ${\mathcal O}_E$-algebra of $K^p$ double cosets in $\operatorname{\mathrm GL}_2({\mathbb A}^{\infty,p})$ (which is non-commutative in general). Following Emerton (cf. [@Em1 (2.1.1)]), we put $$\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,{\mathcal O}_E):=\varprojlim_{n} \varinjlim_{K_p} H^1_{\operatorname{\text{\'et}},c}\big(Y_{K_pK^p,\overline{{\mathbb Q}_p}}, {\mathcal O}_E/\varpi_E^n\big)$$ which is a complete ${\mathcal O}_E$-module equipped with a continuous action of $\operatorname{\mathrm GL}_2({\mathbb Q}_p)\times {\mathcal H}(K^p)\times \operatorname{\mathrm Gal}_{{\mathbb Q}}\times \pi_0$, where $\pi_0$ is the $2$-element group generated by the archimedean Hecke operator $$T_{\infty}:=({\mathbb R}^{\times}_+ \operatorname{\mathrm SO}_2({\mathbb R}))\begin{pmatrix} 1 & 0 \\ 0 & -1\end{pmatrix}\operatorname{\mathrm SO}_2({\mathbb R}).$$ Let ${\mathcal H}^p$ be a commutative ${\mathcal O}_E$-subalgebra of ${\mathcal H}(K^p)$ containing ${\mathcal H}(K^p)^{\operatorname{\mathrm sph}}:=\otimes'_{\ell\notin \Sigma(K^p)} {\mathcal O}_E[T_{\ell},S_{\ell}^{\pm}]$ with $T_{\ell}:=K^p\begin{pmatrix} \ell & 0 \\ 0 & 1 \end{pmatrix}K^p$, $S_{\ell}:=K^p\begin{pmatrix}\ell & 0 \\ 0 & \ell \end{pmatrix}K^p$. One has the Eichler-Shimura relations on $\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,{\mathcal O}_E)$ (e.g. see [@BE Prop. 3.2.3] and the proof): $$\operatorname{\mathrm Frob}_{\ell}^{-2}-T_{\ell} \operatorname{\mathrm Frob}_{\ell}^{-1}+\ell S_{\ell}=0$$ for $\ell\notin \Sigma(K^p)$, where $\operatorname{\mathrm Frob}_{\ell}$ denotes an arithmetic Frobenius at $\ell$. Put $$\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E):=\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,{\mathcal O}_E)\otimes_{{\mathcal O}_E} E,$$ which is a unitary admissible $E$-Banach space representation of $\operatorname{\mathrm GL}_2({\mathbb Q}_p)$ equipped with an action of ${\mathcal H}^p \times \operatorname{\mathrm Gal}_{{\mathbb Q}}\times \pi_0$, commuting with $\operatorname{\mathrm GL}_2({\mathbb Q}_p)$.
\(1) For an ${\mathcal O}_E$-module $M$ equipped with a $\pi_0$-action, denote by $M^{\pm}$ the $\pm 1$-eigenspace for $T_{\infty}$, one has thus $M=M^{+}\oplus M^-$.
\(2) Let $\overline{\rho}$ be a $2$-dimensional continuous representation of $\operatorname{\mathrm Gal}_{{\mathbb Q}}$ over $k_E$, unramified outside $\Sigma(K^p)$, we denote by ${\mathfrak{m}}(\overline{\rho})$ the maximal ideal of ${\mathcal H}(K^p)^{\operatorname{\mathrm sph}}$ attached to $\overline{\rho}$ satisfying that the quotient map ${\mathcal H}(K^p)^{\operatorname{\mathrm sph}}\twoheadrightarrow {\mathcal H}(K^p)^{\operatorname{\mathrm sph}}/{\mathfrak{m}}(\overline{\rho})\cong k_E$ sends $T_{\ell}$ to $\operatorname{\mathrm Tr}(\overline{\rho}(\operatorname{\mathrm Frob}_{\ell}^{-1}))$ and $S_{\ell}$ to $\ell^{-1}\det(\overline{\rho}(\operatorname{\mathrm Frob}_{\ell}^{-1}))$ for $\ell\notin \Sigma(K^p)$.
\(3) For a finite $\varpi_E$-torsion ${\mathcal O}_E$-module $M$ that is equipped with an ${\mathcal H}(K^p)^{\operatorname{\mathrm sph}}$-action, denote by $M_{\overline{\rho}}$ the localisation of $M$ at ${\mathfrak{m}}(\overline{\rho})$. We put $$\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,{\mathcal O}_E)_{\overline{\rho}}:=\varprojlim_{n} \varinjlim_{K_p} H^1_{\operatorname{\text{\'et}},c}\big(Y_{K_pK^p,\overline{{\mathbb Q}_p}}, {\mathcal O}_E/\varpi_E^n\big)_{\overline{\rho}},$$ and $\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}:=\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,{\mathcal O}_E)_{\overline{\rho}} \otimes_{{\mathcal O}_E} E$.
In the sequel, we fix a $2$-dimensional continuous representation $\overline{\rho}$ of $\operatorname{\mathrm Gal}_{{\mathbb Q}}$ over $k_E$, which is absolutely irreducible, unramified outside $\Sigma(K^p)$ and satisfies $\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}\neq 0$. We summarize some properties of $\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}$:
\[thm: clp-cco\]
\(1) $\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}$ is a direct summand of $\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)$ stable under $\operatorname{\mathrm GL}_2({\mathbb Q}_p)\times \operatorname{\mathrm Gal}_{{\mathbb Q}}\times {\mathcal H}^p\times \pi_0$. For any compact open pro-$p$-subgroup $K_p$ of $\operatorname{\mathrm GL}_2({\mathbb Z}_p)$, there exists $r\in {\mathbb Z}_{>0}$ such that we have an isomorphism of $K_p$-representations: $$\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}|_{K_p}\cong {\mathcal C}(K_p,E)^{\oplus r},$$ where ${\mathcal C}(K_p,E)$ denotes the space of continuous functions on $K_p$ with values in $E$, equipped with the right regular $K_p$-action.
\(2) We have: $$\label{localg}
\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm lalg}}\xlongrightarrow{\sim} \oplus_{k\in {\mathbb Z}_{\geq 2}, w\in {\mathbb Z}} H^1_{\operatorname{\text{\'et}},c}(K^p,{\mathcal F}_{\operatorname{\mathrm alg}(k,w)})_{\overline{\rho}} \otimes_E \operatorname{\mathrm alg}(k,w)^{\vee}.$$ where $\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm lalg}}$ denotes the locally algebraic vectors for $\operatorname{\mathrm GL}_2({\mathbb Q}_p)$ inside $\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}$, $\operatorname{\mathrm alg}(k,w)$ denotes the algebraic representation $\operatorname{\mathrm Sym}^{k-2} E^2 \otimes_E \operatorname{\mathrm det}^w$ of $\operatorname{\mathrm GL}_2({\mathbb Q}_p)$, ${\mathcal F}_{\operatorname{\mathrm alg}(k,w)}$ denotes the local system associated to $\operatorname{\mathrm alg}(k,w)$ on $Y_{K_pK^p}$ for compact open subgroups $K_pK^p$ of $\operatorname{\mathrm GL}_2({\mathbb A}^{\infty})$, and where $$H^1_{\operatorname{\text{\'et}},c}(K^p,{\mathcal F}_{\operatorname{\mathrm alg}(k,w)})_{\overline{\rho}}:=\varinjlim_{K_p} H^1_{\operatorname{\text{\'et}},c}(Y_{K^pK_p,\overline{{\mathbb Q}}},{\mathcal F}_{\operatorname{\mathrm alg}(k,w)})_{\overline{\rho}}$$ denotes the (classical) étale cohomology with compact support of modular curves (localized at ${\mathfrak{m}}(\overline{\rho})$) with tame level $K^p$ and coefficients in ${\mathcal F}_{\operatorname{\mathrm alg}(k,w)}$ (with $K_p$ running through compact open subgroups of $\operatorname{\mathrm GL}_2({\mathbb Z}_p)$).
For (1), see the discussion in [@Em4 § 5.3] and [@Em4 Cor. 5.3.19]. For (2), see [@Br11b Thm. 4.1] (see also [@Em1 Cor. 2.2.18, (4.3.4)]).
Eigencurves (eigensurfaces) {#sec: clp-1.2}
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We first briefly recall Emerton’s construction of the eigencurves (eigensurfaces), and we refer to [@Em1 § 2.3] for details. We let $\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an}}$ be the locally analytic subrepresentation of $\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}$. By [@Em11 Thm. 0.5], applying the Jacquet-Emerton functor to $\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an}}$, one gets an essentially admissible locally analytic representation $J_B\big(\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an}}\big)$ of $T({\mathbb Q}_p)$. Here $J_B\big(\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an}}\big)$ being essentially admissible means that there exists a coherent sheaf ${\mathcal M}$ over $\widehat{T}$ such that (cf. [@Em04 §6.4], [@Em1 Prop. 2.3.2]) $$\label{jacEss}{\mathcal M}(\widehat{T})\cong J_B\big(\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an}}\big)^{\vee}_b,$$ where “$-^{\vee}_b$" denotes the continuous dual equipped with the strong topology, and where $\widehat{T}$ denotes the rigid space parametrizing locally analytic characters of $T({\mathbb Q}_p)$ (e.g. see [@Em04 Prop. 6.4.5]). Moreover, $J_B\big(\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an}}\big)$ inherits from $\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an}}$ a continuous action of ${\mathcal H}^p\times \pi_0$ commuting with $T({\mathbb Q}_p)$. Hence ${\mathcal M}$ is equipped with a natural ${\mathcal O}(\widehat{T})$-linear action of ${\mathcal H}^p\times \pi_0$ such that the isomorphism in (\[jacEss\]) is ${\mathcal H}^p\times \pi_0$-equivariant.
From $\{{\mathcal M},\widehat{T},{\mathcal H}^p\}$, one can construct as in [@Em1 § 2.3] a rigid analytic space ${\mathcal S}$ over $E$ equipped with a natural finite morphism $\kappa_1: {\mathcal S}{\rightarrow}\widehat{T}$ such that for any admissible affinoid open $U=\operatorname{\mathrm Spm}A\subset \widehat{T}$, we have $\kappa_1^{-1}({\mathcal U})\cong \operatorname{\mathrm Spm}B$ where $B$ is the finite $A$-subalgebra of $\operatorname{\mathrm End}_A({\mathcal M}({\mathcal U}))$ generated by ${\mathcal H}^p$ (noting that ${\mathcal M}({\mathcal U})$ is equipped with an $A$-linear ${\mathcal H}^p$-action). The rigid space ${\mathcal S}$ is referred to as the eigensurface of tame level $K^p$. An $E$-point $z$ of ${\mathcal S}$ can be parametrized as $(\chi_z,\lambda_z)$ where $\chi_z$ is a locally analytic character of $T({\mathbb Q}_p)$ over $E$, and $\lambda_z: {\mathcal H}^p {\rightarrow}E$ is a system of eigenvalues of ${\mathcal H}^p$. Moreover, such a point $(\chi_z,\lambda_z)$ lies in ${\mathcal S}$ if and only if the corresponding eigenspace $$J_B\big(\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an}}\big)[T({\mathbb Q}_p)=\chi_z,{\mathcal H}^p=\lambda_z]\neq 0.$$ The ${\mathcal O}(\widehat{T})$-module ${\mathcal M}$ has a natural ${\mathcal O}({\mathcal S})$-action, which makes ${\mathcal M}$ to be a coherent ${\mathcal O}({\mathcal S})$-module. For any $z=(\chi_z,\lambda_z)\in {\mathcal S}$ with $k_z$ the residue field, we have a natural isomorphism of finite dimensional $k_z$-vector spaces (cf. [@Em1 Prop. 2.3.3 (iii)]) $$\label{fib0}
(z^*{\mathcal M})^{\vee} \cong J_B\big(\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an}}\big)[T({\mathbb Q}_p)=\chi_z,{\mathcal H}^p=\lambda_z].$$ Since the action of the center ${\mathbb Q}_p^{\times}$ of $\operatorname{\mathrm GL}_2({\mathbb Q}_p)$ on $\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}$ is unitary, we easily see $$\label{center}
\operatorname{\mathrm val}_p(\chi_z(p))=0$$ if $(\chi_z, \lambda_z)\in {\mathcal S}$. The following definition is standard.
\[class00\] (1) A point $z=(\chi_z,\lambda_z)\in {\mathcal S}$ is called classical if $$J_B\big(\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm lalg}}\big)[T({\mathbb Q}_p)=\chi_z,{\mathcal H}^p=\lambda_z]\neq 0.$$ A point $z$ is called very classical if $z$ is classical and the natural injection $$J_B\big(\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm lalg}}\big)[T({\mathbb Q}_p)=\chi_z,{\mathcal H}^p=\lambda_z]
{\lhook\joinrel\longrightarrow}J_B\big(\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an}}\big)[T({\mathbb Q}_p)=\chi_z,{\mathcal H}^p=\lambda_z]$$ is bijective.
\(2) Let $z=(\chi_z, \lambda_z)$ be a point in ${\mathcal S}$ with $\chi_z=(\psi_{z,1}x^{k_1})\otimes (\psi_{z,2} x^{k_2})$ where $\psi_{z,i}$ are smooth characters of ${\mathbb Q}_p^{\times}$, $k_1$, $k_2\in {\mathbb Z}$ and $k_2\geq k_2$ (we call such character locally algebraic of dominant weight). We call $z$ of non-critical slope if $\operatorname{\mathrm val}_p(p\psi_{z,1}(p))<1-k_2$.
We have by [@Em1 Prop. 2.3.6]:
\[prop: clp-cla\] Let $z=(\chi_z,\lambda_z)\in {\mathcal S}$ with $\chi_z$ locally algebraic of dominant weight. If $z$ is of non-critical slope, then $z$ is very classical.
Using Theorem \[thm: clp-cco\] (1) and [@Em11 Prop. 4.2.36], one can actually reformulate the construction of ${\mathcal S}$ using the spectral theory of compact operators (e.g. see [@BHS1 Lem. 3.10]). Let $\widehat{T}_0$ be the rigid space over $E$ parametrizing locally analytic characters of $T({\mathbb Z}_p)$, and we denote by $\kappa$ the composition $$\kappa: {\mathcal S}\xlongrightarrow{\kappa_1} \widehat{T} {\longrightarrow}\widehat{T}_0.$$ Let $\varpi_1:=\begin{pmatrix}
p & 0 \\ 0 & 1 \end{pmatrix}$, $\varpi_2:=\begin{pmatrix}
p & 0 \\ 0 & p
\end{pmatrix} \in T({\mathbb Q}_p)$. By the same argument as in the proof of [@BHS1 Prop. 3.11], we have:
\[prop: eigna\] There exists an admissible covering $\{{\mathcal U}_i\}_{i\in I}$ of ${\mathcal S}$ by affinoids ${\mathcal U}_i$ such that for all $i$ there exits an open affinoid $W_i$ of $\widehat{T}_0$ such that
- the morphism $\kappa$ induces a finite surjective morphism from each irreducible component of ${\mathcal U}_i$ onto $W_i$,
- ${\mathcal M}({\mathcal U}_i)$ is a finite projective ${\mathcal O}(W_i)$-module equipped with ${\mathcal O}(W_i)$-linear operators $\varpi_1$, $\varpi_2$ and $T\in {\mathcal H}^p$,
- ${\mathcal O}({\mathcal U}_i)$ is isomorphic to a ${\mathcal O}(W_i)$-subalgebra of $\operatorname{\mathrm End}_{{\mathcal O}(W_i)}({\mathcal M}({\mathcal U}_i))$ generated by $\varpi_1$, $\varpi_2$ and the operators in ${\mathcal H}^p$.
We assume that the ${\mathcal H}^p$-action on $\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm lalg}}$ is semi-simple (e.g. this holds when ${\mathcal H}^p={\mathcal H}(K^p)^{\operatorname{\mathrm sph}}$). We summarize some (well-known) properties of ${\mathcal M}$ and ${\mathcal S}$ in the following theorem.
\[thm: clp-eac\] (1) The coherent sheaf ${\mathcal M}$ is Cohen-Macauly over ${\mathcal S}$.
\(2) The rigid space ${\mathcal S}$ is equidimensional of dimension $2$, and the points of non-critical slope are Zariski-dense in ${\mathcal S}$, and accumulate at points $(\chi,\lambda)$ with $\chi$ locally algebraic.
\(3) The rigid space ${\mathcal S}$ is reduced.
\(1) follows from Proposition \[prop: eigna\] and the argument in the proof of [@BHS2 Lem. 3.8]. Since $\widehat{T}_0$ is equidimensional of dimension $2$, by [@Che Prop. 6.4.2] and Proposition \[prop: eigna\], ${\mathcal S}$ is also equidimensional of dimension $2$. The density and the accumulation property of the points of non-critical slope follow from standard arguments as in [@Che § 6.4.5]. Finally, since the action of ${\mathcal H}^p$ on $\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm lalg}}$ is semi-simple, (3) follows by the same argument as in [@Che05 Prop. 6.4].
\[dens00\] By Theorem \[thm: clp-eac\] (2) and Proposition \[prop: clp-cla\], the very classical points are Zariski-dense in ${\mathcal S}$.
Let $z=(\chi_z=\chi_{z,1}\otimes \chi_{z,2},\lambda_z)\in {\mathcal S}$. Recall that one can associate (by the theory of pseudo-characters) to $z$ a semi-simple continuous representation $\rho_z: \operatorname{\mathrm Gal}_{{\mathbb Q}}{\rightarrow}\operatorname{\mathrm GL}_2(k_z)$ (where $k_z$ denotes the residue field of $z$, which is a finite extension of $E$) satisfying that
1. the mod $p$ reduction of $\rho_z$ is isomorphic to $\overline{\rho}$,
2. the restriction $\rho_{z,\ell}:=\rho_z|_{\operatorname{\mathrm Gal}_{{\mathbb Q}_{\ell}}}$ is unramified for all $\ell \notin \Sigma(K^p)$, and $$\rho_z(\operatorname{\mathrm Frob}_{\ell}^{-2})-\lambda_z(T_{\ell})\rho_z(\operatorname{\mathrm Frob}_{\ell}^{-1})+\ell \lambda_z(S_{\ell})=0.$$
Note that the first property together with our assumption on $\overline{\rho}$ actually imply that $\rho_z$ is absolutely irreducible. Note also that $\rho_z$ is determined by the second property by Chebotarev’s density theorem. Let $z=(\chi_z, \lambda_z)\in {\mathcal S}$ be such that $\operatorname{\mathrm wt}(z):=\operatorname{\mathrm wt}(\chi_{z,1})-\operatorname{\mathrm wt}(\chi_{z,2})\in {\mathbb Z}_{\geq 0}$, we put $$\chi_z^c:=\chi_z(x^{-\operatorname{\mathrm wt}(z)-1} \otimes x^{\operatorname{\mathrm wt}(z)+1}).$$ If $z^c:=(\chi_z^c, \lambda_z)\in {\mathcal S}$, we call $z^c$ a *companion point* of $z$. Suppose $z$ is classical, then we call $z$ *critical* if $z$ admits a companion point. By [@Em1 Prop. 4.5.5] (and the proof), we have (noting that the bad points in *loc. cit.* are exactly the points admitting companion points in our terminology, see [@Em1 Def. 4.5.4])
\[sloNC\] If $z$ is of non-critical slope, then $z$ is not critical.
\[ThmBE\]The point $z$ is critical if and only if the Galois representation $\rho_{z,p}:=\rho_z|_{\operatorname{\mathrm Gal}_{{\mathbb Q}_p}}$ splits.
We recall the relation between Emerton’s eigensurface ${\mathcal S}$ and Coleman-Mazur eigencurve. Let ${\mathcal W}$ denote the rigid space parameterizing locally analytic characters of ${\mathbb Z}_p^{\times}$, the decomposition $$\label{decomp0}
T({\mathbb Q}_p)\cong \begin{pmatrix}
{\mathbb Z}_p^{\times} & 0 \\ 0 & 1
\end{pmatrix}\times \begin{pmatrix}
1 & 0 \\ 0 & {\mathbb Z}_p^{\times}
\end{pmatrix}\times \begin{pmatrix}
p & 0 \\ 0 & 1
\end{pmatrix}^{{\mathbb Z}} \times \begin{pmatrix}
1 & 0 \\ 0 & p
\end{pmatrix}^{{\mathbb Z}}$$ induces $\widehat{T} \cong {\mathcal W}_+\times {\mathcal W}_-\times {\mathbb G}_m \times {\mathbb G}_m$ with ${\mathcal W}_{\pm}\cong {\mathcal W}$. The trivial character on ${\mathcal W}_+$ induces an injection $${\mathcal W}_-\times {\mathbb G}_m \times {\mathbb G}_m{\lhook\joinrel\longrightarrow}\widehat{T}.$$ Denote by ${\mathcal C}$ the pull-back of ${\mathcal S}$ over ${\mathcal W}_-\times {\mathbb G}_m \times {\mathbb G}_m$, by $\kappa$ the induced map ${\mathcal C}{\rightarrow}{\mathcal W}_-\cong {\mathcal W}$. We still use ${\mathcal M}$ to denote the pull-back of the ${\mathcal O}({\mathcal S})$-module ${\mathcal M}$ over ${\mathcal C}$. We have $$\label{curve0}
{\mathcal M}({\mathcal C})\cong \big(J_B(\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an}})^{T_1}\big)^{\vee}_b$$ where $T_1= \begin{pmatrix}
{\mathbb Z}_p^{\times} & 0 \\ 0 & 1
\end{pmatrix}$. By [@Em1 Prop. 4.4.6], we have ${\mathcal S}\cong {\mathcal C}\times {\mathcal W}$. Moreover, one can show that Proposition \[prop: eigna\], Theorem \[thm: clp-eac\] and Remark \[dens00\] also hold for $({\mathcal C}, {\mathcal M})$ (except that ${\mathcal C}$ is equidimensional of dimension $1$). By [@Em1 Prop. 4.4.2], we have an explicit relation between the classical points of ${\mathcal C}$ and those of Coleman-Mazur eigencurve (constructed from the finite slope overconvergent modular forms). Using the reducedness of ${\mathcal C}$, the density of (very) classical points and the same argument as in [@BCh Prop. 7.2.8], one can deduce that ${\mathcal C}$ is isomorphic to the corresponding Coleman-Mazur eigencurve. Finally we have by [@Em1 Prop. 4.5.5]:
\[prop: clp-neigh\] Let $z=(\chi_{z,1}\otimes \chi_{z,2},\lambda_z)\in {\mathcal C}$, then there exists an admissible open $U\subset {\mathcal S}$ containing $z$ such that any point in $U\setminus \{z\}$ does not admit companion points in ${\mathcal S}$.
Adjunctions {#sec: clp-1.3}
===========
In this section, we study some adjunction properties of the Jacquet-Emerton functor. Let $z=(\chi_z=\chi_{z,1}\otimes \chi_{z,2},\lambda_z)$ be an $E$-point of ${\mathcal C}$. By Proposition \[prop: eigna\] (with ${\mathcal S}$ replaced by ${\mathcal C}$ as discussed below Theorem \[ThmBE\]) and Proposition \[prop: clp-neigh\], there exists an affinoid neighborhood ${\mathcal U}_0$ of $z$ in ${\mathcal C}$ such that
- $\kappa: {\mathcal U}_0 {\rightarrow}\kappa({\mathcal U}_0)$ is a finite morphism of affinoids;
- ${\mathcal M}({\mathcal U}_0)$ is finitely generated locally free over ${\mathcal O}(\kappa({\mathcal U}_0))$;
- any $z'\in {\mathcal U}_0(\overline{E})\setminus \{z\}$ does not admit companion points;
- $\kappa^{-1}(\kappa(z))^{\operatorname{\mathrm red}}=\{z\}$.
Let ${\mathcal U}:=\kappa^{-1}({\mathcal V})$ with ${\mathcal V}$ an admissible strictly quasi-Stein neighborhood of $\kappa(z)$ in $\kappa({\mathcal U}_0)$ (cf. [@Em04 Def. 2.1.17 (iv)]. Thus ${\mathcal U}$ also satisfies the above listed properties. Since ${\mathcal M}({\mathcal U})$ is a finitely generated locally free ${\mathcal O}({\mathcal V})$-module and ${\mathcal V}$ is strictly quasi-Stein, we have that ${\mathcal M}({\mathcal U})^{\vee}_b$ is an *allowable* locally analytic representation of $T({\mathbb Q}_p)$ in the sense of [@Em2 Def. 0.11] (e.g. using similar arguments as in [@Ding Ex. 6.3.15 (ii)]). Note also ${\mathcal M}({\mathcal U})^{\vee}_b$ is naturally equipped with a continuous action of ${\mathcal H}^p$. The restriction maps ${\mathcal M}({\mathcal S}) {\rightarrow}{\mathcal M}({\mathcal C}){\rightarrow}{\mathcal M}({\mathcal U})$ induce ${\mathcal H}^p$-invariant morphisms of locally analytic $T({\mathbb Q}_p)$-representations: $$\label{cplf: inja}{\mathcal M}({\mathcal U})^{\vee}_b {\longrightarrow}{\mathcal M}({\mathcal C})^{\vee}_b{\longrightarrow}{\mathcal M}({\mathcal S})^{\vee}_b\cong J_B\big(\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an}}\big).$$ When $z$ does not have companion points (hence all points in ${\mathcal U}$ do not admit companion points by assumption), one can prove as in [@Em1 Lem. 4.5.12] that the composition in (\[cplf: inja\]) is *balanced* in the sense of [@Em2 Def. 0.8] and induces an ${\mathcal H}^p$-invariant morphism of locally analytic $\operatorname{\mathrm GL}_2({\mathbb Q}_p)$-representations $$\label{equ: clp-seB}
\big(\operatorname{\mathrm Ind}_{\overline{B}({\mathbb Q}_p)}^{\operatorname{\mathrm GL}_2({\mathbb Q}_p)} {\mathcal M}({\mathcal U})^{\vee}_b\otimes_E \delta_B^{-1}\big)^{\operatorname{\mathrm an}} {\longrightarrow}\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an}},$$ where $\delta_B:=\operatorname{\mathrm unr}(p^{-1})\otimes \operatorname{\mathrm unr}(p)$ is the modulus character of $B({\mathbb Q}_p)$ (which we also view as a character of $\overline{B}({\mathbb Q}_p)$ via $\overline{B}({\mathbb Q}_p)\twoheadrightarrow T({\mathbb Q}_p)$). The morphism (\[equ: clp-seB\]) plays a crucial role in Emerton’s construction of two-variable $p$-adic $L$-functions (cf. [@Em1 Thm. 4.5.7]). However, if $z$ admits a companion point (e.g. if $z$ is critical), then (\[cplf: inja\]) does not (directly) induce a such morphism (e.g. see Proposition \[prop: clp-ntr\] below). The main result in this section is to establish a similar adjunction result in the locus of such $z$.
Suppose henceforward $k:=\operatorname{\mathrm wt}(\chi_{z,1})-\operatorname{\mathrm wt}(\chi_{z,2})\in {\mathbb Z}_{\geq 0}$ (note $\operatorname{\mathrm wt}(\chi_{z,1})=0$ by (\[curve0\])), and let $h=\begin{pmatrix}
1 & 0 \\ 0 & -1
\end{pmatrix}\in {\mathfrak t}$. We have a natural injection ${\mathfrak t}\hookrightarrow {\mathcal O}(\widehat{T}_0)$, and we view hence $h$ as an element in ${\mathcal O}(\widehat{T}_0)$. By (\[curve0\]), the ${\mathfrak t}$-action on ${\mathcal M}({\mathcal U})^{\vee}_b$ factors through the projection ${\mathfrak t}=\begin{pmatrix} {\mathbb Q}_p & 0 \\ 0 & {\mathbb Q}_p\end{pmatrix} \twoheadrightarrow {\mathbb Q}_p$ onto the bottom right factor. Shrinking ${\mathcal V}$ (and hence ${\mathcal U}$), we assume $\kappa(z)$ is the only point in ${\mathcal V}$ such that the associated character of ${\mathbb Z}_p^{\times}$ is of weight $-k$ (noting $\kappa(z)=\chi_{z,2}$). Consider ${\mathcal M}({\mathcal U})^{\vee}_b[h=k]$ which is a finite dimensional $E$-vector space, and is a $T({\mathbb Q}_p)\times {\mathcal H}^p$-stable subspace of ${\mathcal M}({\mathcal U})^{\vee}_b$. By assumption, we have in fact: $$\label{equ: clp-fiber}
{\mathcal M}({\mathcal U})^{\vee}_b[h=k]=J_B\big(\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an}}\big)[{\mathfrak t}=d\chi_z]\{T({\mathbb Q}_p)=\chi_z, {\mathcal H}^p=\lambda_z\}.$$ We call a vector $v\in J_B\big(\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an}}\big)$ *classical* if $v$ lies in $J_B\big(\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm lalg}}\big)$ (compare with Definition \[class00\]), and we call a vector $v\in {\mathcal M}({\mathcal U})^{\vee}_b$ classical if it is sent to a classical vector in $J_B\big(\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an}}\big)$ via the map (\[cplf: inja\]).
\[lem: clp-ber\] Let $\gamma \in \operatorname{\mathrm End}_{T({\mathbb Q}_p)}\big({\mathcal M}({\mathcal U})^{\vee}_b\big)$, the composition $$\label{equ: clp-wUe}
{\mathcal M}({\mathcal U})^{\vee}_b \xlongrightarrow{\gamma} {\mathcal M}({\mathcal U})^{\vee}_b \xlongrightarrow{\text{(\ref{cplf: inja})}} J_B\big(\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an}}\big)$$ is balanced if and only if any vector in $\gamma({\mathcal M}({\mathcal U})^{\vee}_b[h=k])$ is classical.
By definition and the same argument as in the proof of [@Em1 Lem. 4.5.12], (\[equ: clp-wUe\]) is balanced if and only if for any $k'\in {\mathbb Z}_{\geq 0}$, the image of the composition $$\label{equ: clp-oge}
{\mathcal M}({\mathcal U})^{\vee}_b[h=k'] \xlongrightarrow{\text{(\ref{equ: clp-wUe})}} J_B\big(\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an}}\big) \longrightarrow (\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an}})^{N_0} {\lhook\joinrel\longrightarrow}\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an}}$$ is annihilated by the operator $X_-^{k'+1}$ where $X_-:=\begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}\in \text{U}({\mathfrak g})$ naturally acts on the locally analytic representation $\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an}}$, $N_0=N({\mathbb Z}_p)=\begin{pmatrix}
1 & {\mathbb Z}_p \\ 0 & 1
\end{pmatrix}$, and where the second map in (\[equ: clp-oge\]) is the “canonical lifting" of [@Em2 (3.4.8)] (with respect to $N_0$), equivariant under the action of $$T({\mathbb Q}_p)^+:=\bigg\{\begin{pmatrix}
a & 0 \\ 0 & d
\end{pmatrix}\in T({\mathbb Q}_p)\ |\ a/d \in {\mathbb Z}_p\setminus \{0\}\bigg\}.$$
One can show as in the proof of [@Em1 Lem. 4.5.12] that if $k'\neq k$, then any vector $v$ in the image of the following composition (which is obtained in the same way as (\[equ: clp-oge\]) replacing (\[equ: clp-wUe\]) by (\[cplf: inja\])) $$\label{equ: hk'}
{\mathcal M}({\mathcal U})^{\vee}_b[h=k'] \xlongrightarrow{\text{(\ref{cplf: inja})}} J_B\big(\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an}}\big) \longrightarrow \widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an}}$$ is annihilated by $X_-^{k'+1}$. Indeed, using the fact that ${\mathcal M}({\mathcal U})^{\vee}_b[h=k']$ is finite dimensional (and $T({\mathbb Q}_p) \times {\mathcal H}^p$-stable), we can assume without loss of generality that $v$ is a $(\chi',\lambda')$-eigenvector for $T({\mathbb Q}_p)^+\times {\mathcal H}^p$ with $(\chi',\lambda')\in {\mathcal U}$. In this case, by [@Ding Lem. 7.3.15] (which is an easy variation of [@Em11 Prop. 4.4.4]), $X_-^{k'+1} \cdot v \in \big(\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an}}\big)^{N_0}$ is a generalized $((\chi')^c, \lambda')$-eigenvector for $T({\mathbb Q}_p)^+\times {\mathcal H}^p$. If $X_-^{k'+1} \cdot v \neq 0$, we deduce $\big(\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an}}\big)^{N_0}\{T({\mathbb Q}_p)^+=(\chi')^c, {\mathcal H}^p=\lambda'\}\neq 0$, and hence (see [@Em11 Prop. 3.4.9] for the first equality): $$J_B\big(\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an}}\big)[T({\mathbb Q}_p)=(\chi')^c, {\mathcal H}^p=\lambda']
=\big(\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an}}\big)^{N_0}[T({\mathbb Q}_p)^+=(\chi')^c, {\mathcal H}^p=\lambda']\neq 0.$$ But this implies $((\chi')^c, \lambda')\in {\mathcal S}$, contradicting the fact $(\chi',\lambda')$ does not admit any companion point. Since (\[equ: clp-oge\]) factors through (\[equ: hk’\]), we deduce the image of (\[equ: clp-oge\]) for $k'\neq k$ is also annihilated by $X_-^{k'+1}$.
Now it suffices to show that for $k'=k$ and a vector $v$ in the image of (\[equ: clp-oge\]), $X_-^{k+1} \cdot v=0$ if and only $v$ is classical. Using the fact $v$ is fixed by $N_0$, ${\mathfrak t}v=d\chi_z v$ and the highest weight theory, it is not difficult to see that that followings are equivalent
- $v$ is classical,
- $\text{U}({\mathfrak g}) v\cong (\operatorname{\mathrm Sym}^{k} E^2)^{\vee}$,
- $X_-^{k+1} \cdot v=0$.
The lemma then follows.
Recall that in [@Em2 (2.8)], Emerton introduced a closed $\operatorname{\mathrm GL}_2({\mathbb Q}_p)$-subrepresentation $$\label{equ: cpl-IPG}I_{\overline{B}({\mathbb Q}_p)}^{\operatorname{\mathrm GL}_2({\mathbb Q}_p)}\big({\mathcal M}({\mathcal U})^{\vee}_b\otimes_E \delta_B^{-1}\big){\lhook\joinrel\longrightarrow}\big(\operatorname{\mathrm Ind}_{\overline{B}({\mathbb Q}_p)}^{\operatorname{\mathrm GL}_2({\mathbb Q}_p)} {\mathcal M}({\mathcal U})^{\vee}_b\otimes_E \delta_B^{-1}\big)^{\operatorname{\mathrm an}},$$ which, by [@Em2 Lem. 2.8.3], is the closed $\operatorname{\mathrm GL}_2({\mathbb Q}_p)$-subrepresentation of $$\big(\operatorname{\mathrm Ind}_{\overline{B}({\mathbb Q}_p)}^{\operatorname{\mathrm GL}_2({\mathbb Q}_p)} {\mathcal M}({\mathcal U})^{\vee}_b\otimes_E \delta_B^{-1}\big)^{\operatorname{\mathrm an}}$$ generated by ${\mathcal M}({\mathcal U})^{\vee}_b$ via the following composition (see [@Em2 Lem. 0.3] for the first map, and [@Em11 (3.4.8)] for the second map) $${\mathcal M}({\mathcal U})^{\vee}_b {\lhook\joinrel\longrightarrow}J_B\big(\big(\operatorname{\mathrm Ind}_{\overline{B}({\mathbb Q}_p)}^{\operatorname{\mathrm GL}_2({\mathbb Q}_p)} {\mathcal M}({\mathcal U})^{\vee}_b\otimes_E \delta_B^{-1}\big)^{\operatorname{\mathrm an}}\big) \longrightarrow \big(\operatorname{\mathrm Ind}_{\overline{B}({\mathbb Q}_p)}^{\operatorname{\mathrm GL}_2({\mathbb Q}_p)} {\mathcal M}({\mathcal U})^{\vee}_b\otimes_E \delta_B^{-1}\big)^{\operatorname{\mathrm an}}.$$ However, by the assumption that ${\mathcal M}({\mathcal U})$ is locally free over ${\mathcal O}(\kappa({\mathcal U}))$, one can prove as in [@Em1 Lem. 4.5.12 (ii)] that (\[equ: cpl-IPG\]) is an isomorphism. The following theorem thus follows from Emerton’s adjunction formula [@Em2 Thm. 0.13] combined with Lemma \[lem: clp-ber\].
\[thm: clp-rps\] Keep the notation of Lemma \[lem: clp-ber\], and suppose $\gamma$ commutes with ${\mathcal H}^p$ (so that(\[equ: clp-wUe\]) is ${\mathcal H}^p$-equivariant), then the followings are equivalent:
\(i) there exists an ${\mathcal H}^p$-equivariant morphism of locally analytic $\operatorname{\mathrm GL}_2({\mathbb Q}_p)$-representations $$\big(\operatorname{\mathrm Ind}_{\overline{B}({\mathbb Q}_p)}^{\operatorname{\mathrm GL}_2({\mathbb Q}_p)} {\mathcal M}({\mathcal U})^{\vee}_b\otimes_E \delta_B^{-1}\big)^{\operatorname{\mathrm an}}{\longrightarrow}\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)^{\operatorname{\mathrm an}}_{\overline{\rho}}$$ such that the induced morphism $$\label{Adj002}
{\mathcal M}({\mathcal U})^{\vee}_b {\lhook\joinrel\longrightarrow}J_B\big(\big(\operatorname{\mathrm Ind}_{\overline{B}({\mathbb Q}_p)}^{\operatorname{\mathrm GL}_2({\mathbb Q}_p)} {\mathcal M}({\mathcal U})^{\vee}_b\otimes_E \delta_B^{-1}\big)^{\operatorname{\mathrm an}}\big){\longrightarrow}J_B\big(\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)^{\operatorname{\mathrm an}}_{\overline{\rho}}\big)$$ is equal to (\[equ: clp-wUe\]);
\(ii) any vector in $\gamma({\mathcal M}({\mathcal U})^{\vee}_b[h=k])$ is classical.
Let ${\mathcal M}({\mathcal U})^{\vee}_b[h=k]^{\operatorname{\mathrm cl}} \subseteq {\mathcal M}({\mathcal U})^{\vee}_b[h=k]$ be the subspace of classical vectors. Recall we have ${\mathcal M}({\mathcal U})^{\vee}_b[h=k]\cong ({\mathcal M}({\mathcal U})/(h-k))^{\vee}$ on which the ${\mathcal O}({\mathcal U})$-action is determined by the action of $T({\mathbb Q}_p)\times {\mathcal H}^p$. By definition (see also (\[equ: clp-fiber\])), one easily verifies that ${\mathcal M}({\mathcal U})^{\vee}_b[h=k]^{\operatorname{\mathrm cl}} $ is stable under the action of $T({\mathbb Q}_p)$ and ${\mathcal H}^p$. Hence ${\mathcal M}({\mathcal U})^{\vee}_b[h=k]^{\operatorname{\mathrm cl}}$ is stable under the ${\mathcal O}({\mathcal U})$-action. Put $$\label{Iz}
{\mathcal I}_z:=\{\gamma \in {\mathcal O}({\mathcal U})\ |\ \gamma({\mathcal M}({\mathcal U})^{\vee}_b[h=k]) \subseteq {\mathcal M}({\mathcal U})^{\vee}_b[h=k]^{\operatorname{\mathrm cl}}\}.$$ We see ${\mathcal I}_z$ is either an ideal of ${\mathcal O}({\mathcal U})$ of ${\mathcal I}_z={\mathcal O}({\mathcal U})$. It is also clear that $h-k\in {\mathcal I}_z$.
\[prop: clp-ntr\]Suppose $\chi_z$ is a product of an algebraic character and an unramified character, then ${\mathcal I}_z={\mathcal O}({\mathcal U})$ if and only if $z$ does not admit companion points.
If $z$ does not have companion points, by the same argument as in the proof of Lemma \[lem: clp-ber\], we see any vector in ${\mathcal M}({\mathcal U})^{\vee}_b[h=k]$ is classical, hence ${\mathcal I}_z={\mathcal O}({\mathcal U})$.
Suppose now ${\mathcal I}_z={\mathcal O}({\mathcal U})$. By Theorem \[thm: clp-rps\], any vector in ${\mathcal M}({\mathcal U})^{\vee}_b[h=k]$ is classical. By (\[equ: clp-fiber\]), any vector in $J_B\big(\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an}}\big)[h=k]\{T({\mathbb Q}_p)=\chi_z, {\mathcal H}^p=\lambda_z\}$ is classical (in particular, the point $z$ is classical). Assume first $\chi_z=\chi_{z,1} \otimes \chi_{z,2}$ satisfies $\chi_{z,1} \chi_{z,2}^{-1} \neq \operatorname{\mathrm unr}(p^{-1}) x^k$. In this case, by similar (and easier) argument as in the proof of [@Ding3 Thm. 7] (using Chenevier’s method [@Che11 § 4.4]), we can deduce that ${\mathcal C}$ is étale over ${\mathcal W}$ at the point $z$. However, by [@Bergd0 Thm. 1.1], if $z$ is critical, then ${\mathcal C}$ will be ramified over ${\mathcal W}$ at $z$, that leads to a contradiction. Now assume $\chi_{z,1} \chi_{z,2}^{-1} = \operatorname{\mathrm unr}(p^{-1}) x^k$. Together with (\[center\]), we easily deduce $\operatorname{\mathrm val}_p(\chi_{z,1}(p))=(k-1)/2$. Since $z\in {\mathcal C}$, the character $\chi_{z,1}$ is actually smooth. By Definition \[class00\] (2), the point $z$ is of non-critical slope, and hence non-critical by Proposition \[sloNC\]. This concludes the proof.
Under the $\pi_0$-action, the coherent sheaf ${\mathcal M}$ naturally decomposes into ${\mathcal M}^+\oplus {\mathcal M}^-$. We deduce that ${\mathcal M}^{\pm}({\mathcal U})$ are both locally free over ${\mathcal O}(\kappa({\mathcal U}))$. Moreover by considering their fibers at classical points, we can see both of them are non-zero. The precedent results also hold for $\{{\mathcal M}^{\pm}({\mathcal U}), \widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an},\pm}\}$. In particular, we can define ideals ${\mathcal I}_z^{\pm}$ in a similar way, and we can prove that the same statement in Proposition \[prop: clp-ntr\] holds with ${\mathcal I}_z$ replaced by ${\mathcal I}_z^{\pm}$.
Two-variable $p$-adic $L$-functions
===================================
We use the results in § \[sec: clp-1.3\] to construct two-variable $p$-adic $L$-functions in a neighborhood of critical points in ${\mathcal C}$.
Constructions {#sec: 21}
-------------
Let $N>1$, $p\nmid N$, and let $$K^p=\bigg\{\begin{pmatrix} a & b \\ c & d \end{pmatrix}\in \operatorname{\mathrm GL}_2(\hat{{\mathbb Z}}^p) \ \bigg| \ c\equiv 0 \pmod{N},\ d\equiv 1 \pmod N\bigg\}.$$ In this section, we let ${\mathcal H}^p$ denote the ${\mathcal O}_E$-algebra generated by $T_{\ell}$, $S_{\ell}$ for $\ell \nmid N$, $\ell \neq p$, and the diamond operators $\langle a \rangle$ for $a\in ({\mathbb Z}/N{\mathbb Z})^{*}$. Let $k\in {\mathbb Z}_{\geq 0}$, $f$ be a newform of weight $k+2$, level $\Gamma_1(N)$ over $E$, i.e. $f$ is an eigenform for ${\mathcal H}^p$ and $T_p$, $S_p$, such that there does not exist an eigenform (for ${\mathcal H}^p$) of weight $k+2$, of level $\Gamma_1(N')$ with $N'$ is a proper divisor of $N$, having the same eigenvalue as $f$ for the operators in ${\mathcal H}^p$. Let $\rho_f$ be the associated $\operatorname{\mathrm Gal}_{{\mathbb Q}}$-representation over $E$ (enlarge $E$ if necessary), and we assume that the mod $\varpi_E$ reduction of $\rho_f$ is isomorphic to $\overline{\rho}$. Let ${\mathcal C}$, ${\mathcal M}$ be as in § \[sec: clp-1.2\]. Let $\epsilon: (Z/NZ)^{\times} {\rightarrow}E^{\times}$ be the character with $\langle a\rangle f=\epsilon(a) f$. Let $a_p, b_p\in E$ (enlarge $E$ if necessary) be the eigenvalues of $T_p$, $S_p$ of $f$ respectively. Let $\alpha$ be a root of $X^2-a_p X+pb_p=0$, and put $$f_{\alpha}(z)=f(z)-\frac{pb_p}{\alpha} f(pz),\\$$ which is a modular form of weight $k+2$, of level $\Gamma_1(N)\cap \Gamma_0(p)$, and is an eigenform of the same eigenvalues of $f$ for ${\mathcal H}^p$, an eigenform for $U_p$ of eigenvalues $\alpha$. The form $f_{\alpha}$ is referred to as a *refinement* (or a *$p$-stabilization*) of $f$. By [@Em1 Prop. 4.4.2], $f_{\alpha}$ corresponds to a classical point $z_{\alpha}=(\chi_{z_{\alpha}},\lambda_f)\in {\mathcal C}\subset {\mathcal S}$ where $\lambda_f$ denotes the system of eigenvalues of $f$ for operators in ${\mathcal H}^p$, and where $$\chi_{z_{\alpha}}=\operatorname{\mathrm unr}(\alpha/p)\otimes x^{-k} \operatorname{\mathrm unr}(pb_p/\alpha).\\$$ In this section, we construct two-variable $p$-adic $L$-functions in a neighborhood of $z_{\alpha}$ (especially in the case where $z_{\alpha}$ is critical), via Emerton’s method [@Em05][@Em1].
\[locfree\] ${\mathcal M}^{\pm}$ are locally free of rank $1$ in a neighborhood of $z_{\alpha}$ in ${\mathcal C}$.
Suppose first $\alpha\neq pb_p/\alpha$ (i.e. $X^2-a_p X+pb_p=0$ has two distinct roots). By [@Bel § 2.3], the eigencurve ${\mathcal C}$ is smooth at the classical point $z_{\alpha}$. Since ${\mathcal M}$ is Cohen-Macauly (see Theorem \[thm: clp-eac\] (1) and the discussion below (\[curve0\])), by [@EGAiv1 Cor. 17.3.5 (i)], ${\mathcal M}$ is locally free around $z_{\alpha}$. Hence ${\mathcal M}^{\pm}$ are also locally free around $z_{\alpha}$. For any classical point $z'=(\chi_{z'},\lambda_{z'})$ of non-critical slope in ${\mathcal C}$, we have $$\begin{gathered}
\label{multi00}
\dim_{k_{z'}} (z')^* {\mathcal M}^{\pm}= \dim_{k_{z'}} J_B\big(\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an},\pm}\big)[T({\mathbb Q}_p)=\chi_{z'},{\mathcal H}^p=\lambda_{z'}]\\
=\dim_{k_{z'}} J_B\big(\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm lalg},\pm}\big)[T({\mathbb Q}_p)=\chi_{z'},{\mathcal H}^p=\lambda_{z'}]=1.\end{gathered}$$ where the first equality follows from (\[fib0\]), the second follows from Proposition \[prop: clp-cla\], and the last equality follows from the multiplicity one result (see [@Em1 Prop. 4.4.18]). Since such points accumulate at $z_{\alpha}$ (see Theorem \[thm: clp-eac\] (2) and the discussion below (\[curve0\])), we deduce that ${\mathcal M}^{\pm}$ are locally free of rank $1$ in a neighborhood of $z_{\alpha}$ in ${\mathcal C}$. If $\alpha=pb_p/\alpha$, as in the proof of Proposition \[prop: clp-ntr\], we know $z_{\alpha}$ is of non-critical slope. The proposition in this case then follows from [@Em1 Prop. 4.4.20] (and the proof).
Let ${\mathcal U}$ be a neighborhood of $z_{\alpha}$ in ${\mathcal C}$ as in § \[sec: clp-1.3\]. If $\alpha\neq pb_p/\alpha$, we shrink ${\mathcal U}$ such that the maximal ideal ${\mathfrak{m}}_{z_{\alpha}}$ associated to $z_{\alpha}$ is generated by one element $r_{z_{\alpha}}\in {\mathcal O}({\mathcal U})$ (using the fact ${\mathcal U}$ is one-dimensional and smooth at the point $z_{\alpha}$). We let $e$ be the ramification degree of ${\mathcal C}$ over ${\mathcal W}$ at $z_{\alpha}$ in this case. If $\alpha=pb_p/\alpha$, we put $r_{z_{\alpha}}=1$ and $e=1$.
\[adj003\] The composition (that is ${\mathcal H}^p \times T({\mathbb Q}_p)$-equivariant) $$\label{equ: clp-cEp}
{\mathcal M}^{\pm}({\mathcal U})^{\vee}_b \xlongrightarrow{r_{z_{\alpha}}^{e-1}} {\mathcal M}^{\pm}({\mathcal U})^{\vee}_b {\longrightarrow}J_B\big(\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an}}\big)$$ induces an ${\mathcal H}^p$-equivariant morphism of locally analytic representations of $\operatorname{\mathrm GL}_2({\mathbb Q}_p)$ $$\label{equ: clp-ppA}
\iota^{\pm}: \big(\operatorname{\mathrm Ind}_{\overline{B}({\mathbb Q}_p)}^{\operatorname{\mathrm GL}_2({\mathbb Q}_p)} {\mathcal M}^{\pm}({\mathcal U})^{\vee}_b\otimes_E \delta_B^{-1}\big)^{\operatorname{\mathrm an}} {\longrightarrow}\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an},\pm}.$$
If $\alpha=pb_p/\alpha$, then $z_{\alpha}$ is of non-critical slope (see the proof of Proposition \[prop: clp-ntr\]), and hence non-critical by Proposition \[sloNC\]. By Proposition \[prop: clp-ntr\] (and the discussion that follows), we see ${\mathcal I}_z^{\pm}={\mathcal O}({\mathcal U})$. The theorem then follows from Theorem \[thm: clp-rps\].
We suppose $\alpha\neq pb_p/\alpha$. We prove first $$\label{chaofcl}
{\mathcal M}^{\pm}({\mathcal U})^{\vee}_b[r_{z_{\alpha}}]={\mathcal M}^{\pm}({\mathcal U})_b^{\vee}[h=k]^{\operatorname{\mathrm cl}}.$$ We have $$\begin{gathered}
\label{ss00}{\mathcal M}^{\pm}({\mathcal U})_b^{\vee}[h=k]^{\operatorname{\mathrm cl}}=J_B\big(\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm lalg},\pm}\big)[{\mathfrak t}=d\chi_z]\{T({\mathbb Q}_p)=\chi_z,{\mathcal H}^p=\lambda_z\} \\=J_B\big(\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm lalg},\pm}\big)[T({\mathbb Q}_p)=\chi_z,{\mathcal H}^p=\lambda_z]\end{gathered}$$where the first equality follows from (\[equ: clp-fiber\]), and the second from the assumption $\alpha\neq pb_p/\alpha$ and the semi-simplicity of the ${\mathcal H}^p$-action on $\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm lalg}}$ (using the classical fact that the ${\mathcal H}^p$-action on the right hand side of (\[localg\]) is semi-simple). By (the proof of) [@Em1 Prop. 4.4.18], the $E$-vector space in (\[ss00\]) is one dimensional. It is clear that we have an injection $$\begin{gathered}
\label{inja}J_B\big(\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm lalg},\pm}\big)[T({\mathbb Q}_p)=\chi_z,{\mathcal H}^p=\lambda_z] \\ {\lhook\joinrel\longrightarrow}J_B\big(\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an},\pm}\big)[T({\mathbb Q}_p)=\chi_z,{\mathcal H}^p=\lambda_z]\cong (z_{\alpha}^* {\mathcal M}^{\pm})^{\vee}={\mathcal M}^{\pm}({\mathcal U})^{\vee}_b[r_{z_{\alpha}}].\end{gathered}$$ By Proposition \[locfree\], $\dim_E {\mathcal M}^{\pm}({\mathcal U})^{\vee}_b[r_{z_{\alpha}}]=1$. We see (\[inja\]) is actually bijective. Combined with (\[ss00\]), (\[chaofcl\]) follows.
We have $(h-k){\mathcal O}({\mathcal U})=r_{z_{\alpha}}^e {\mathcal O}({\mathcal U})$. By Proposition \[locfree\], we have ${\mathcal M}^{\pm}({\mathcal U})_b^{\vee}[h=k]\cong ({\mathcal O}({\mathcal U})/(h-k))^{\vee}$ and by (\[chaofcl\]) ${\mathcal M}^{\pm}({\mathcal U})_b^{\vee}[h=k]^{\operatorname{\mathrm cl}}\cong ({\mathcal O}({\mathcal U})/r_{z_{\alpha}})^{\vee}$. We deduce (see (\[Iz\]) and the discussion at the end of § \[sec: clp-1.3\]) $$\label{Iz2}{\mathcal I}_z={\mathcal I}_z^{\pm}=(\frac{h-k}{r_{z_\alpha}}) {\mathcal O}({\mathcal U})=r_{z_{\alpha}}^{e-1} {\mathcal O}({\mathcal U}).$$ The theorem follows then from Theorem \[thm: clp-rps\].
We have ${\mathcal M}^{\pm}({\mathcal U})^{\vee}_b[r_{z_{\alpha}}]\cong \chi_{z_{\alpha}}$ as $T({\mathbb Q}_p)$-representation, thus (\[equ: clp-ppA\]) induces a morphism $$\label{equ: clp-adj}
\big(\operatorname{\mathrm Ind}_{\overline{B}({\mathbb Q}_p)}^{\operatorname{\mathrm GL}_2({\mathbb Q}_p)} \chi_{z_{\alpha}} \delta_B^{-1}\big)^{\operatorname{\mathrm an}} {\longrightarrow}\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an},\pm}[{\mathcal H}^p=\lambda_f].$$ The locally analytic representation $ (\operatorname{\mathrm Ind}_{\overline{B}({\mathbb Q}_p)}^{\operatorname{\mathrm GL}_2({\mathbb Q}_p)}\chi_{z_{\alpha}}\delta_B^{-1})^{\operatorname{\mathrm an}}$ sits in a non-splitting exact sequence (e.g. see [@Br Thm. 4.1]) $$\begin{gathered}
\label{equ: clp-exact}
0 {\longrightarrow}(\operatorname{\mathrm Ind}_{\overline{B}({\mathbb Q}_p)}^{\operatorname{\mathrm GL}_2({\mathbb Q}_p)} \chi_{z_{\alpha}}^{\operatorname{\mathrm sm}}\delta_B^{-1})^{\infty}\otimes_E (\operatorname{\mathrm Sym}^k E^2)^{\vee} \\ {\longrightarrow}(\operatorname{\mathrm Ind}_{\overline{B}({\mathbb Q}_p)}^{\operatorname{\mathrm GL}_2({\mathbb Q}_p)}\chi_{z_{\alpha}}\delta_B^{-1})^{\operatorname{\mathrm an}} {\longrightarrow}(\operatorname{\mathrm Ind}_{\overline{B}({\mathbb Q}_p)}^{\operatorname{\mathrm GL}_2({\mathbb Q}_p)} \chi_{z_{\alpha}}^c\delta_B^{-1})^{\operatorname{\mathrm an}}{\rightarrow}0,\end{gathered}$$ where $\chi_{z_{\alpha}}^{\operatorname{\mathrm sm}}:=\operatorname{\mathrm unr}(\alpha/p)\otimes \operatorname{\mathrm unr}(pb_p/\alpha)$ is the “smooth" part of $\chi_{z_{\alpha}}$, $(\operatorname{\mathrm Ind}-)^{\infty}$ denotes the smooth induction, and recall $\chi_{z_{\alpha}}^c=\chi_{z_{\alpha}} (x^{1-k} \otimes x^{k-1})$.
\[prop: clp-rest\] The map (\[equ: clp-adj\]) is non-zero, and its restriction on the locally algebraic vectors is zero if and only if $z_{\alpha}$ is critical.
We first prove the second part of the proposition. Put $$I(\chi_{z_{\alpha}} \delta_B^{-1}):=(\operatorname{\mathrm Ind}_{\overline{B}({\mathbb Q}_p)}^{\operatorname{\mathrm GL}_2({\mathbb Q}_p)} \chi_{z_{\alpha}}^{\operatorname{\mathrm sm}}\delta_B^{-1})^{\infty}\otimes_E (\operatorname{\mathrm Sym}^k E^2)^{\vee}$$ which is in fact the locally algebraic subrepresentation of $\big(\operatorname{\mathrm Ind}_{\overline{B}({\mathbb Q}_p)}^{\operatorname{\mathrm GL}_2({\mathbb Q}_p)} \chi_{z_{\alpha}} \delta_B^{-1}\big)^{\operatorname{\mathrm an}}$ (cf. (\[equ: clp-exact\])). Applying the (left exact) Jacquet-Emerton functor to the composition $$I(\chi_{z_{\alpha}} \delta_B^{-1}){\lhook\joinrel\longrightarrow}\big(\operatorname{\mathrm Ind}_{\overline{B}({\mathbb Q}_p)}^{\operatorname{\mathrm GL}_2({\mathbb Q}_p)} \chi_{z_{\alpha}} \delta_B^{-1}\big)^{\operatorname{\mathrm an}} \xlongrightarrow{\text{(\ref{equ: clp-adj})}} \widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an},\pm}[{\mathcal H}^p=\lambda_f],$$ one gets an injection of locally analytic representations of $T({\mathbb Q}_p)$ (see [@Em2 Lem. 0.3]): $$\label{equ: clp-ggb}
\chi_{z_{\alpha}}{\lhook\joinrel\longrightarrow}J_B\big(\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an},\pm}\big)[{\mathcal H}^p=\lambda_f].$$ We have actually a commutative diagram (see (\[Adj002\]) for the bottom maps) $$\begin{CD}
\chi_{z_{\alpha}}@>>> J_B\big(\operatorname{\mathrm Ind}_{\overline{B}({\mathbb Q}_p)}^{\operatorname{\mathrm GL}_2({\mathbb Q}_p)} \chi_{z_{\alpha}} \delta_B^{-1}\big)^{\operatorname{\mathrm an}} @>>> J_B\big(\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an},\pm}\big)[{\mathcal H}^p=\lambda_f] \\@VVV @VVV @VVV\\
{\mathcal M}({\mathcal U})^{\vee}_b @>>> J_B\big(\big(\operatorname{\mathrm Ind}_{\overline{B}({\mathbb Q}_p)}^{\operatorname{\mathrm GL}_2({\mathbb Q}_p)} {\mathcal M}({\mathcal U})^{\vee}_b\otimes_E \delta_B^{-1}\big)^{\operatorname{\mathrm an}}\big) @>>> J_B\big(\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)^{\operatorname{\mathrm an}}_{\overline{\rho}}\big)
\end{CD}$$ Using Theorem \[thm: clp-rps\], we see (\[equ: clp-ggb\]) is equal to the composition $$\chi_{z_{\alpha}}\cong {\mathcal M}^{\pm}({\mathcal U})^{\vee}_b[r_{z_{\alpha}}] {\lhook\joinrel\longrightarrow}{\mathcal M}^{\pm}({\mathcal U})^{\vee}_b \xlongrightarrow{\text{(\ref{equ: clp-cEp})}} J_B\big(\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an},\pm}\big).$$ It is straightforward to see the above map is non-zero if and only if $e=1$, which is equivalent to that $z_{\alpha}$ is non-critical by Proposition \[prop: clp-ntr\]. The second part follows.
Now we prove (\[equ: clp-adj\]) is non-zero. The non-critical case is an easy consequence of the second part proved above. Now assume $z_{\alpha}$ is critical (in particular $\alpha\neq pb_p/\alpha$). If (\[equ: clp-adj\]) is zero, we see the morphism $$(\operatorname{\mathrm Ind}_{\overline{B}({\mathbb Q}_p)}^{\operatorname{\mathrm GL}_2({\mathbb Q}_p)} {\mathcal M}^{\pm}({\mathcal U})^{\vee}_b \otimes_E \delta_B^{-1})^{\operatorname{\mathrm an}} {\longrightarrow}\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an},\pm}$$ factors as (noting the kernel of the first map of the following composition is $$(\operatorname{\mathrm Ind}_{\overline{B}({\mathbb Q}_p)}^{\operatorname{\mathrm GL}_2({\mathbb Q}_p)} {\mathcal M}^{\pm}({\mathcal U})^{\vee}_b[r_{z_{\alpha}}] \otimes_E \delta_B^{-1})^{\operatorname{\mathrm an}} \cong \big(\operatorname{\mathrm Ind}_{\overline{B}({\mathbb Q}_p)}^{\operatorname{\mathrm GL}_2({\mathbb Q}_p)} \chi_{z_{\alpha}} \delta_B^{-1}\big)^{\operatorname{\mathrm an}}\Big):$$ $$(\operatorname{\mathrm Ind}_{\overline{B}({\mathbb Q}_p)}^{\operatorname{\mathrm GL}_2({\mathbb Q}_p)} {\mathcal M}^{\pm}({\mathcal U})^{\vee}_b \otimes_E \delta_B^{-1})^{\operatorname{\mathrm an}} \xlongrightarrow{r_{z_{\alpha}}} (\operatorname{\mathrm Ind}_{\overline{B}({\mathbb Q}_p)}^{\operatorname{\mathrm GL}_2({\mathbb Q}_p)} {\mathcal M}^{\pm}({\mathcal U})^{\vee}_b \otimes_E \delta_B^{-1})^{\operatorname{\mathrm an}} {\longrightarrow}\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an},\pm},$$ where the second morphism is also $\operatorname{\mathrm GL}_2({\mathbb Q}_p)\times {\mathcal H}^p$-equivariant. Applying the Jacquet-Emerton functor, we deduce (\[equ: clp-cEp\]) can factor as $$\label{comp11}
{\mathcal M}^{\pm}({\mathcal U})^{\vee}_b \xlongrightarrow{r_{z_{\alpha}}} {\mathcal M}^{\pm}({\mathcal U})^{\vee}_b \xlongrightarrow{\iota'} J_B\big(\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an},\pm}\big),$$ where $\iota'$ is $T({\mathbb Q}_p)\times {\mathcal H}^p$-equivariant. By [@Em2 Thm. 0.13], the map $\iota'$ is balanced. Using the same argument as in the proof of Lemma \[lem: clp-ber\], we see that any vector in $\iota'({\mathcal M}^{\pm}({\mathcal U})^{\vee}_b[h=k])$ is classical. The composition $$\label{pol0}r_{z_{\alpha}} \circ \iota': {\mathcal M}^{\pm}({\mathcal U})^{\vee}_b[h=k] {\longrightarrow}{\mathcal M}^{\pm}({\mathcal U})^{\vee}_b[h=k]$$is equal to $r_{z_{\alpha}}^{e-1}$ (since the two maps (\[comp11\]) and (\[equ: clp-cEp\]) are equal, and their restriction on ${\mathcal M}^{\pm}({\mathcal U})^{\vee}_b[h=k]$ is injective using (\[equ: clp-fiber\])). However, $\iota'({\mathcal M}^{\pm}({\mathcal U})^{\vee}_b[h=k])={\mathcal M}^{\pm}({\mathcal U})^{\vee}_b[h=k]^{\operatorname{\mathrm cl}}={\mathcal M}^{\pm}({\mathcal U})^{\vee}_b[r_{z_{\alpha}}]$ and hence the morphism (\[pol0\]) is zero, which contradicts $r_{z_{\alpha}}^{e-1}\neq 0$ on ${\mathcal M}^{\pm}({\mathcal U})^{\vee}_b[h=k]\cong ({\mathcal O}({\mathcal U})/r_{z_\alpha}^e)^{\vee}$. The proposition follows.
\[rem: clp-classical\] If $z_{\alpha}$ is not critical then (\[equ: clp-adj\]) is injective, and its restriction on the locally algebraic subrepresentations is given by the local-global compatibility in classical local Langlands correspondence. If $z_{\alpha}$ is critical then (\[equ: clp-adj\]) factors as $$\label{compan} \big(\operatorname{\mathrm Ind}_{\overline{B}({\mathbb Q}_p)}^{\operatorname{\mathrm GL}_2({\mathbb Q}_p)} \chi_{z_{\alpha}} \delta_B^{-1}\big)^{\operatorname{\mathrm an}} {\relbar\joinrel\twoheadrightarrow}(\operatorname{\mathrm Ind}_{\overline{B}({\mathbb Q}_p)}^{\operatorname{\mathrm GL}_2({\mathbb Q}_p)} \chi_{z_{\alpha}}^c\delta_B^{-1})^{\operatorname{\mathrm an}}{\lhook\joinrel\longrightarrow}\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an},\pm}[{\mathcal H}^p=\lambda_f].$$ The existence of the second injection was proved in [@BE], which is an important fact on local-global compatibility in $p$-adic local Langlands program in critical case.
Let $D^0$ be the ${\mathcal O}_E$-module of degree zero divisors in ${\mathbb P}^1({\mathbb Q})$. Recall that by [@Em05 Prop. 4.2], one has the following pairing which interpolates the classical modular symbols $$\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,{\mathcal O}_E)_{\overline{\rho}}\times D^0 {\longrightarrow}{\mathcal O}_E.$$Evaluating at $\{\infty\}-\{0\}$ gives a continuous $E$-linear map (of norm less than $1$) $$\label{equ: clp-cpa}
\{\infty\}-\{0\}: \widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}} {\longrightarrow}E.$$ We fix an isomorphism of ${\mathcal O}({\mathcal U})$-modules ${\mathcal M}^{\pm}({\mathcal U})\cong {\mathcal O}({\mathcal U})$. The following composition $$\begin{gathered}
\label{padicL}
{\mathcal C}^{\operatorname{\mathrm la}}({\mathbb Z}_p^{\times},E)\widehat{\otimes}_E {\mathcal O}({\mathcal U})^{\vee}_b \cong {\mathcal C}^{\operatorname{\mathrm la}}({\mathbb Z}_p^{\times},{\mathcal O}({\mathcal U})^{\vee}_b) {\lhook\joinrel\longrightarrow}\big(\operatorname{\mathrm Ind}_{\overline{B}({\mathbb Q}_p)}^{\operatorname{\mathrm GL}_2({\mathbb Q}_p)} {\mathcal O}({\mathcal U})^{\vee}_b\otimes_E \delta_B^{-1}\big)^{\operatorname{\mathrm an}}
\\ \xlongrightarrow{\sim} \big(\operatorname{\mathrm Ind}_{\overline{B}({\mathbb Q}_p)}^{\operatorname{\mathrm GL}_2({\mathbb Q}_p)} {\mathcal M}^{\pm}({\mathcal U})^{\vee}_b\otimes_E \delta_B^{-1}\big)^{\operatorname{\mathrm an}}
\xlongrightarrow{\iota^{\pm}}\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\pm, \operatorname{\mathrm an}} \xlongrightarrow{\{\infty\}-\{0\}} E,\end{gathered}$$ thus gives a global section $L^{\pm}$ of ${\mathcal U}\times{\mathcal W}$, where the second map sends $F$ to $I(F)$ with $I(F)$ supported in $\overline{B}({\mathbb Q}_p)N({\mathbb Z}_p^{\times})$ with $N({\mathbb Z}_p^{\times}):=\begin{pmatrix}1 & {\mathbb Z}_p^{\times} \\ 0 & 1 \end{pmatrix}$, and $I(F)\bigg(\begin{pmatrix} 1 & x \\ 0 & 1 \end{pmatrix}\bigg)=F(x)$ for $x\in {\mathbb Z}_p^{\times}$. The function $L^{\pm}$ was constructed in [@Em1 Thm. 4.5.7] when $z_{\alpha}$ is of non-critical slope. We also remark that $L^{\pm}$ depends on the (fixed) isomorphism ${\mathcal M}^{\pm}({\mathcal U})\cong {\mathcal O}({\mathcal U})$, and hence is naturally defined up to units in ${\mathcal O}({\mathcal U})$.
Denote by $L^{\pm}(z_{\alpha},-): {\mathcal C}^{\operatorname{\mathrm la}}({\mathbb Z}_p^{\times}, E) {\rightarrow}E$ the evaluation of $L^{\pm}$ at $z_{\alpha}$ which is a distribution of ${\mathbb Z}_p^{\times}$. Up to non-zero scalars, $L^{\pm}(z_{\alpha},-)$ equals the distribution given by the following composition $$\label{equ: clp-mea}
{\mathcal C}^{\operatorname{\mathrm la}}({\mathbb Z}_p^{\times}, E){\lhook\joinrel\longrightarrow}\big(\operatorname{\mathrm Ind}_{\overline{B}({\mathbb Q}_p)}^{\operatorname{\mathrm GL}_2({\mathbb Q}_p)} \chi_{z_{\alpha}} \delta_B^{-1}\big)^{\operatorname{\mathrm an}} \xlongrightarrow{(\ref{equ: clp-adj})} \widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an},\pm} \xlongrightarrow{(\ref{equ: clp-cpa})} E,$$ where the first map sends $F$ to $I(F)$ with $I(F)$ supported in $\overline{B}({\mathbb Q}_p)N({\mathbb Z}_p^{\times})$, and $I(F)\bigg(\begin{pmatrix} 1 & x \\ 0 & 1 \end{pmatrix}\bigg)=F(x)$ for $x\in {\mathbb Z}_p^{\times}$. The following proposition is due to Emerton.
\[intpl\] (1) The distribution $L^{\pm}(z_{\alpha},-)$ is $\alpha$-tempered (see [@Em05 Def. 3.12] for the definition of $\alpha$-tempered distributions).
\(2) Suppose $z_{\alpha}$ is not critical, then we have up to non-zero scalars (independent of $\phi x^j$), $$\label{equ: clp-int}
L^{\pm}(z_{\alpha},\phi x^j)=e_p(\alpha,\phi x^j)\frac{m^{j+1}}{(-2\pi i)^j}\frac{j!}{\tau(\phi^{-1})}\frac{L_{\infty}(f\phi^{-1},j+1)}{\Omega_f^{\pm}},$$ where $\phi: {\mathbb Z}_p^{\times}{\rightarrow}{\mathbb Q}_p^{\times}$ is a smooth character of conductor $p^v$ satisfying $\phi(-1)=\pm 1$, $j\in\{0, \cdots, k\}$, $$e_p(\alpha,\phi x^j):=\frac{1}{\alpha^v}\big(1-\frac{\phi^{-1}(p)\epsilon(p) p^{k-j}}{\beta}\big)\big(1-\frac{\phi(p) p^j}{\beta}\big),$$ $\tau(\phi^{-1})$ is the Gauss sum of $\phi$, $L_{\infty}$ is the archimedean $L$-function and $\Omega_f^{\pm}$ are the two archimedean periods of $f$.
\(1) By [@Em05 Lem. 3.22], the composition $${\mathcal C}^{\operatorname{\mathrm la}}({\mathbb Z}_p^{\times}, E){\lhook\joinrel\longrightarrow}\big(\operatorname{\mathrm Ind}_{\overline{B}({\mathbb Q}_p)}^{\operatorname{\mathrm GL}_2({\mathbb Q}_p)} \chi_{z_{\alpha}} \delta_B^{-1}\big)^{\operatorname{\mathrm an}} \xlongrightarrow{(\ref{equ: clp-adj})} \widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\pm}$$ is $\alpha$-tempered (in the sense of [@Em05 Def. 3.12]). Since (\[equ: clp-cpa\]) sends $\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,{\mathcal O}_E)_{\overline{\rho}}$ to ${\mathcal O}_E$, we deduce by [@Em05 Lem. 3.20] that $L^{\pm}(z_{\alpha},-)$ is $\alpha$-tempered.
\(2) The interpolation result for the distribution (\[equ: clp-mea\]) in non-critical slope case (i.e. $\operatorname{\mathrm val}_p(\alpha)<k+1$) was proved in [@Em05 Prop. 4.9], and the critical slope (i.e. $\operatorname{\mathrm val}_p(\alpha)=k+1$) but non-critical case follows by the same argument.
By results of Amice-Vélu and Vishik (e.g. see [@Em05 Lem. 3.14]), when ${z_{\alpha}}$ is not of critical slope, $L^{\pm}(z_{\alpha},-)$ is determined (up to non-zero scalars) by the interpolation property.
\[cri0\] If $z_{\alpha}$ is critical, then $L^{\pm}(z_{\alpha},\phi x^j)=0$ for all $\phi x^j$ given as in (\[equ: clp-int\]).
We denote by ${\mathcal C}^{\operatorname{\mathrm lp},\leq k}(N({\mathbb Z}_p^{\times}), E)$ the closed subspace of ${\mathcal C}^{\operatorname{\mathrm la}}(N({\mathbb Z}_p^{\times}), E)$ consisting of functions that are locally polynomial of degree $\leq k$. We have a natural commutative diagram (e.g. see [@Em05 (3.15) (3.16)]): $$\begin{CD}
{\mathcal C}^{\operatorname{\mathrm lp},\leq k}(N({\mathbb Z}_p^{\times}), E) @>>> (\operatorname{\mathrm Ind}_{\overline{B}({\mathbb Q}_p)}^{\operatorname{\mathrm GL}_2({\mathbb Q}_p)} \chi_{z_{\alpha}}^{\operatorname{\mathrm sm}}\delta_B^{-1})^{\infty}\otimes_E (\operatorname{\mathrm Sym}^k E^2)^{\vee} \\
@VVV @VVV \\
{\mathcal C}^{\operatorname{\mathrm la}}(N({\mathbb Z}_p^{\times}), E) @>>> \big(\operatorname{\mathrm Ind}_{\overline{B}({\mathbb Q}_p)}^{\operatorname{\mathrm GL}_2({\mathbb Q}_p)} \chi_{z_{\alpha}} \delta_B^{-1}\big)^{\operatorname{\mathrm an}}
\end{CD}$$ where the horizontal maps are injective and are given as in the first map in (\[equ: clp-mea\]). The commutative diagram, together with Proposition \[prop: clp-rest\], imply that $L^{\pm}(z_{\alpha}, \psi)=0$ for all $\psi \in {\mathcal C}^{\operatorname{\mathrm lp},\leq k}(N({\mathbb Z}_p^{\times}), E) $. It is easy to see that all the $\phi x^j$ (given as in (\[equ: clp-int\])) lie in ${\mathcal C}^{\operatorname{\mathrm lp},\leq k}(N({\mathbb Z}_p^{\times}), E) $. The proposition follows.
\(1) It is not clear to the author whether $L^{\pm}(z_{\alpha}, -)$ is zero or not.
\(2) We can also directly construct the critical $p$-adic $L$-functions $L^{\pm}(z_{\alpha},-)$ without using the two-variable $p$-adic $L$-functions $L^{\pm}(-,-)$. In fact, we have: $$\begin{gathered}
\label{mult1}
\dim_E \operatorname{\mathrm Hom}_{\operatorname{\mathrm GL}_2({\mathbb Q}_p)}\big((\operatorname{\mathrm Ind}_{\overline{B}({\mathbb Q}_p)}^{\operatorname{\mathrm GL}_2({\mathbb Q}_p)} \chi_{z_{\alpha}}^c\delta_B^{-1})^{\operatorname{\mathrm an}}, \widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an},\pm}[{\mathcal H}^p=\lambda_f]\big) \\
=\dim_E \operatorname{\mathrm Hom}_{\operatorname{\mathrm GL}_2({\mathbb Q}_p)}\big( (\operatorname{\mathrm Ind}_{\overline{B}({\mathbb Q}_p)}^{\operatorname{\mathrm GL}_2({\mathbb Q}_p)} \chi_{z_{\alpha}}^{\operatorname{\mathrm sm}}\delta_B^{-1})^{\infty}\otimes_E (\operatorname{\mathrm Sym}^k E^2)^{\vee}, \widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an},\pm}[{\mathcal H}^p=\lambda_f]\big)=1
\end{gathered}$$ where the first equation is a consequence of the local-global compatibility of $p$-adic Langlands correspondence [@Em4 Thm. 1.2.1], and the second equation follows from the multiplicity one result. Any non-zero element $j$ in the $1$-dimensional $E$-vector space on the left hand side of (\[mult1\]) induces (where the first map is the same as the first map in (\[equ: clp-mea\])) $$\label{padicL2}
{\mathcal C}^{\operatorname{\mathrm la}}({\mathbb Z}_p^{\times}, E){\lhook\joinrel\longrightarrow}\big(\operatorname{\mathrm Ind}_{\overline{B}({\mathbb Q}_p)}^{\operatorname{\mathrm GL}_2({\mathbb Q}_p)} \chi_{z_{\alpha}} \delta_B^{-1}\big)^{\operatorname{\mathrm an}}
{\relbar\joinrel\twoheadrightarrow}(\operatorname{\mathrm Ind}_{\overline{B}({\mathbb Q}_p)}^{\operatorname{\mathrm GL}_2({\mathbb Q}_p)} \chi_{z_{\alpha}}^c\delta_B^{-1})^{\operatorname{\mathrm an}} \xlongrightarrow{j} \widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an},\pm} \xlongrightarrow{(\ref{equ: clp-cpa})} E.$$ By (\[mult1\]), $j$ is equal to the second morphism in (\[compan\]) up to non-zero scalars. We deduce that (\[padicL2\]) is equal to $L^{\pm}(z_{\alpha},-)$ up to non-zero scalars. We also remark that this construction does not rely on the smoothness of the eigencurve.
Properties {#sec: 22}
----------
Keep the notation in § \[sec: 21\], and assume $z_{\alpha}$ is critical. In this section, we study some properties of $L^{\pm}(z_{\alpha},-)$ and the two-variable $p$-adic $L$-functions $L^{\pm}(-,-)$.
### Relations with Bellaïche’s critical $p$-adic $L$-functions
Recall in [@Bel], Bellaiche constructed $2$-variable $\pm$-$p$-adic $L$-functions ${\mathscr L}^{\pm}$ in a neighborhood of $z_{\alpha}$. In this section, we compare ${\mathscr L}^{\pm}$ with our $L^{\pm}$ constructed in § \[sec: 21\].
Let ${\mathcal V}\subseteq {\mathcal U}$ be an affinoid open neighborhood of $z_{\alpha}$ in ${\mathcal C}$ such that both $L^{\pm}$ and ${\mathscr L}^{\pm}$ are defined, that the points of non-critical slope are Zariski-dense in ${\mathcal V}$, and that ${\mathcal O}({\mathcal V})$ is a PID.
The isomorphism ${\mathbb Z}_p^{\times} \cong ({\mathbb Z}/q {\mathbb Z})^{\times}\times (1+q {\mathbb Z}_p)$ (with $q=p$ if $p\neq 2$, and $q=2p$ if $p=2$) induces an isomorphism of rigid spaces ${\mathcal W}\cong \sqcup_{i\in ({\mathbb Z}/q {\mathbb Z})^{\times}} {\mathcal W}_i$ where all the ${\mathcal W}_i$ are isomorphic to the rigid space over $E$ parametrizing continuous characters of $1+q{\mathbb Z}_p\cong {\mathbb Z}_p$. Note that the latter rigid space is isomorphic to the open unit disc: a point $z$ of ${\mathbb C}_p$ with $|z|<1$ corresponds to the character $(1+z)^a$ for $a\in {\mathbb Z}_p$. For $\lambda\in {\mathcal O}({\mathcal W})$ (resp. $\Lambda\in {\mathcal O}({\mathcal V}\times {\mathcal W})$, we denote by $\lambda^i\in {\mathcal O}({\mathcal W}_i)$ (resp. $\Lambda^i\in {\mathcal O}({\mathcal V}\times {\mathcal W}_i)$) its restriction on ${\mathcal W}_i$ (resp. on ${\mathcal O}({\mathcal V}\times {\mathcal W}_i)$).
Let $i\in ({\mathbb Z}/q {\mathbb Z})^{\times}$. Assume ${\mathscr L}^{\pm}(z_{\alpha},-)^i\neq 0$ (resp. $L^{\pm}(z_{\alpha}, -)^i\neq 0$), then there exist an admissible affinoid ${\mathcal V}'\subset {\mathcal V}$ containing $z_{\alpha}$ and $a_{\pm}\in {\mathcal O}({\mathcal V}')$ such that $L^{\pm}(-,-)^i=a_{\pm} {\mathscr L}^{\pm}(-,-)^i$ (resp. ${\mathscr L}^{\pm}(-,-)^i=a_{\pm} L^{\pm}(-,-)^i$).
We only prove the case where ${\mathscr L}^{\pm}(z_{\alpha},-)^i\neq 0$, the other case being symmetric. Let $w_1, w_2\in {\mathcal W}_i$, put: $$d_{w_1,w_2}^{\pm}(-):=L^{\pm}(-,w_1){\mathscr L}^{\pm}(-,w_2)-L^{\pm}(-,w_2){\mathscr L}^{\pm}(-,w_1)\in {\mathcal O}({\mathcal V}).$$ We claim $d_{w_1,w_2}^{\pm}=0$ (thus independent of $w_1,w_2$). Indeed for any point $z\in {\mathcal V}$ of non-critical slope, we know the distributions $L^{\pm}(z,-)$, ${\mathscr L}^{\pm}(z,-)$ equal up to non-zero scalars (by the interpolation property), thus $d_{w_1,w_2}^{\pm}(z)=0$. Since such points are Zariski-dense in ${\mathcal V}$, the claim follows.
For $w\in {\mathcal W}_i$ such that ${\mathscr L}^{\pm}(z_{\alpha}, w)={\mathscr L}^{\pm}(z_{\alpha}, w)^i\neq 0$ (by assumption, such $w$ exists), we put $a_{\pm}':=\frac{L^{\pm}(-,w)}{{\mathscr L}^{\pm}(-,w)}\in \operatorname{\mathrm Frac}({\mathcal O}({\mathcal V}))$. By the above claim, we see $a_{\pm}'$ is independent of the choice of $w$. We put $a_{\pm}:=\frac{L^{\pm}(-,-)^i}{{\mathscr L}^{\pm}(-,-)^i}\in \operatorname{\mathrm Frac}({\mathcal O}({\mathcal V}\times {\mathcal W}_i))$. We claim $a_{\pm}'=a_{\pm}$ (in other words $a_{\pm}\in \operatorname{\mathrm Frac}({\mathcal O}({\mathcal V}))$). We view $a_{\pm}'$ as an element in $\operatorname{\mathrm Frac}({\mathcal O}({\mathcal V}\times {\mathcal W}_i))$ by the natural inclusion. To prove $a_{\pm}'=a_{\pm}$, it is sufficient to prove $$\mathfrak{d}:={\mathscr L}^{\pm}(-,w)L^{\pm}(-,-)^i-L^{\pm}(-,w){\mathscr L}^{\pm}(-,-)^i=0$$ where $w$ satisfies ${\mathscr L}^{\pm}(z_{\alpha},w)\neq 0$, ${\mathscr L}^{\pm}(-,w)$ and $L^{\pm}(-,w)$ are viewed as elements in ${\mathcal O}({\mathcal V}\times{\mathcal W}_i)$ via the natural injection ${\mathcal O}({\mathcal V})\hookrightarrow {\mathcal O}({\mathcal V}\times {\mathcal W}_i)$ (and so $\mathfrak{d}\in {\mathcal O}({\mathcal V}\times{\mathcal W}_i)$). Let $Z$ be the set of classical points of non-critical slope in ${\mathcal V}$. As discussed above, we know $$\mathfrak{d}(z,w')={\mathscr L}^{\pm}(z,w)L^{\pm}(z,w')^i-L^{\pm}(z,w) {\mathscr L}^{\pm}(z,w')^i=0$$for all $z\in Z$, and $w'\in {\mathcal W}_i$. Since $Z\times {\mathcal W}_i$ is Zariski-dense in ${\mathcal V}\times{\mathcal W}_i$, we deduce $\mathfrak{d}=0$. Thus $a_{\pm}=a_{\pm}'\in \operatorname{\mathrm Frac}({\mathcal O}({\mathcal V}))$. The proposition follows by shrinking ${\mathcal V}$ (and using the assumption ${\mathscr L}^{\pm}(z_{\alpha},-)^i\neq 0$).
Recall that the restriction ${\mathscr L}^{\pm}(z_{\alpha},-)$ is equal, up to non-zero scalars, to the $\pm$-$p$-adic $L$-function for $f_{\alpha}$ constructed in [@Bel], we can thus deduce from the proposition the following corollary.
Let $i\in ({\mathbb Z}/q{\mathbb Z})^{\times}$. Assume ${\mathscr L}^{\pm}(z_{\alpha},-)^i\neq 0$ (resp. $L^{\pm}(z_{\alpha},-)^i\neq 0$), then there exists $a^{\pm}\in E$ (enlarge $E$ if necessary) such that $L^{\pm}(z_{\alpha},-)^i=a^{\pm} {\mathscr L}^{\pm}(z_{\alpha}, -)^i$ (resp. ${\mathscr L}^{\pm}(z_{\alpha},-)^i=a^{\pm} L^{\pm}(z_{\alpha}, -)^i$).
### Secondary critical $p$-adic $L$-functions
For $i=1,\cdots, e-1$, we put (shrinking ${\mathcal U}$ if necessary) $L^{\pm}_{i}\in {\mathcal O}({\mathcal U}\times {\mathcal W})$ such that (noting that $r_{z_{\alpha}}$ is a uniformiser of ${\mathcal O}({\mathcal U})$ at $z_{\alpha}$) $$L^{\pm}_{i}(t, \sigma):=\frac{\partial^i L^{\pm}}{\partial r_{z_{\alpha}}^{i}}(t,\sigma), \forall \ (t,\sigma)\in {\mathcal U}\times {\mathcal W}.$$ The statement in the following proposition was proved in [@Bel § 4.4] for Bellaïche’s two-variable $p$-adic $L$-functions. We show that it also holds for our $L^{\pm}(-,-)$.
\(1) For $i=1,\cdots, e-2$, $L^{\pm}_i(z_{\alpha}, \phi x^j)=0$ for all $\phi x^j$ given as in (\[equ: clp-int\]).
\(2) With the notation of (\[equ: clp-int\]), we have up to non-zero scalars (independent of $\phi x^j$): $$\label{int2}
L^{\pm}_{e-1}(z_{\alpha},\phi x^j)=e_p(\alpha,\phi x^j)\frac{m^{j+1}}{(-2\pi i)^j}\frac{j!}{\tau(\phi^{-1})}\frac{L_{\infty}(f\phi^{-1},j+1)}{\Omega_f^{\pm}}.$$
Step (a). We unwind a bit Emerton’s adjunction formula ([@Em2 Thm. 0.13]). By [@Em11 Thm. 3.5.6], the composition in (\[equ: clp-cEp\]) first induces $B({\mathbb Q}_p)$-equivariant morphisms $$\label{adj000}
{\mathcal C}^{\operatorname{\mathrm sm}}_c(N({\mathbb Q}_p), {\mathcal M}^{\pm}({\mathcal U})^{\vee}_b \otimes_E \delta_B^{-1}) \xlongrightarrow{r_{z_{\alpha}}^{e-1}} {\mathcal C}_c^{\operatorname{\mathrm sm}}(N({\mathbb Q}_p), {\mathcal M}^{\pm}({\mathcal U})^{\vee}_b\otimes_E \delta_B^{-1})\xlongrightarrow{\iota_0} \widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an}, \pm},$$ where ${\mathcal C}^{\operatorname{\mathrm sm}}_c(N({\mathbb Q}_p),{\mathcal M}^{\pm}({\mathcal U})^{\vee}_b \otimes_E \delta_B^{-1})$ denotes the space of locally constant, compactly supported functions on $N({\mathbb Q}_p)$ with values in ${\mathcal M}^{\pm}({\mathcal U})^{\vee}_b \otimes_E \delta_B^{-1}$ (which is equipped with a natural $B({\mathbb Q}_p)$-action as in [@Em11 § 3.5]). The morphisms in (\[adj000\]) further induce $({\mathfrak g}, B({\mathbb Q}_p))$-equivariant morphisms (cf. [@Em07 (5.11)]) $$\begin{gathered}
\label{adj001}
\text{U}({\mathfrak g}) \otimes_{\text{U}({\mathfrak b})} {\mathcal C}^{\operatorname{\mathrm sm}}_c(N({\mathbb Q}_p), {\mathcal M}^{\pm}({\mathcal U})^{\vee}_b \otimes_E \delta_B^{-1})\xlongrightarrow{r_{z_{\alpha}}^{e-1}}\text{U}({\mathfrak g}) \otimes_{\text{U}({\mathfrak b})} {\mathcal C}_c^{\operatorname{\mathrm sm}}(N({\mathbb Q}_p), {\mathcal M}^{\pm}({\mathcal U})^{\vee}_b\otimes_E \delta_B^{-1})\\ \xlongrightarrow{\iota_1^{\pm}} \widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an}, \pm}.\end{gathered}$$ On the other hand, we have a natural $({\mathfrak g}, B({\mathbb Q}_p))$-equivariant morphism (cf. [@Em2 (2.8.7)]) $$\begin{gathered}
\label{Verma}
\text{U}({\mathfrak g}) \otimes_{\text{U}({\mathfrak b})} {\mathcal C}^{\operatorname{\mathrm sm}}_c(N({\mathbb Q}_p), {\mathcal M}^{\pm}({\mathcal U})^{\vee}_b \otimes_E \delta_B^{-1}) {\longrightarrow}{\mathcal C}^{\operatorname{\mathrm lp}}_c(N({\mathbb Q}_p), {\mathcal M}^{\pm}({\mathcal U})^{\vee}_b \otimes_E \delta_B^{-1}) \\
\cong {\mathcal C}^{\operatorname{\mathrm sm}}_c(N({\mathbb Q}_p, {\mathcal M}^{\pm}({\mathcal U})^{\vee}_b \otimes_E \delta_B^{-1}) \otimes_E E[z],\end{gathered}$$ where ${\mathcal C}^{\operatorname{\mathrm lp}}_c(N({\mathbb Q}_p), {\mathcal M}^{\pm}({\mathcal U})^{\vee}_b \otimes_E \delta_B^{-1})$ denotes the space of locally polynomial functions on $N({\mathbb Q}_p)$ with values in ${\mathcal M}^{\pm}({\mathcal U})^{\vee}_b \otimes_E \delta_B^{-1}$ and we refer to [@Em2 Def. 2.5.21] for the precise definition and for the second isomorphism in (\[Verma\]). As in [@Em1 (4.5.16)], the morphism in (\[Verma\]) is given by $$X_-^{l} \otimes f \mapsto ((\prod_{i=0}^{l-1} (h-i)) \cdot f) z^l$$ for $l\in {\mathbb Z}_{\geq 1}$, and $f\in {\mathcal C}^{\operatorname{\mathrm sm}}_c(N({\mathbb Q}_p), {\mathcal M}^{\pm}({\mathcal U})^{\vee}_b \otimes_E \delta_B^{-1})$ (and $X_-^0 \otimes f \mapsto f$). As in the last paragraph of the proof of [@Em1 Lem. 4.5.12], the first morphism in (\[Verma\]) is surjective. Since the composition in (\[equ: clp-cEp\]) is balanced, the composition in (\[adj001\]) factors through (\[Verma\]) (noting that in contrary $\iota_1^{\pm}$ does not factor through (\[Verma\])). In summary, we have a $({\mathfrak g}, B({\mathbb Q}_p))$-equivariant commutative diagram $$\label{comt00}
\begin{CD}
\text{U}({\mathfrak g}) \otimes_{\text{U}({\mathfrak b})} {\mathcal C}^{\operatorname{\mathrm sm}}_c(N({\mathbb Q}_p), {\mathcal M}^{\pm}({\mathcal U})^{\vee}_b \otimes_E \delta_B^{-1})@> r_{z_{\alpha}}^{e-1}>> \text{U}({\mathfrak g}) \otimes_{\text{U}({\mathfrak b})} {\mathcal C}^{\operatorname{\mathrm sm}}_c(N({\mathbb Q}_p), {\mathcal M}^{\pm}({\mathcal U})^{\vee}_b\otimes_E \delta_B^{-1}) \\
@VVV @V\iota_1^{\pm} VV \\
{\mathcal C}^{\operatorname{\mathrm lp}}_c(N({\mathbb Q}_p), {\mathcal M}^{\pm}({\mathcal U})^{\vee}_b \otimes_E \delta_B^{-1}) @>>> \widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an}, \pm}
\end{CD}.$$ We remark that the morphism (\[equ: clp-ppA\]) is actually induced by the bottom map in (\[comt00\]) (e.g. see [@Em2 Cor. 4.3.3]). For $N({\mathbb Z}_p^{\times})=\begin{pmatrix} 1 & {\mathbb Z}_p^{\times} \\ 0 & 1\end{pmatrix}$, we have a natural injection ${\mathcal C}^*(N({\mathbb Z}_p^{\times}), -) \hookrightarrow {\mathcal C}^*_c(N({\mathbb Q}_p),-)$, sending a function $F$ to the function whose value at $x\in N({\mathbb Z}_p^{\times})$ is $F(x)$ and $0$ outside $N({\mathbb Z}_p^{\times})$. We then easily deduce from (\[comt00\]) a commutative diagram (which is $\text{U}({\mathfrak g})$-equivariant) $$\label{diagK}
\begin{CD}
\text{U}({\mathfrak g}) \otimes_{\text{U}({\mathfrak b})} {\mathcal C}^{\operatorname{\mathrm sm}}(N({\mathbb Z}_p^{\times}), {\mathcal M}^{\pm}({\mathcal U})^{\vee}_b \otimes_E \delta_B^{-1})@> r_{z_{\alpha}}^{e-1}>> \text{U}({\mathfrak g}) \otimes_{\text{U}({\mathfrak b})} {\mathcal C}^{\operatorname{\mathrm sm}}(N({\mathbb Z}_p^{\times}), {\mathcal M}^{\pm}({\mathcal U})^{\vee}_b\otimes_E \delta_B^{-1}) \\
@VVV @V\iota_1^{\pm} VV \\
{\mathcal C}^{\operatorname{\mathrm lp}}(N({\mathbb Z}_p^{\times}), {\mathcal M}^{\pm}({\mathcal U})^{\vee}_b \otimes_E \delta_B^{-1}) @>>> \widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an}, \pm}
\end{CD}.$$ The bottom map of (\[diagK\]) is in fact equal to the composition $$\begin{gathered}
{\mathcal C}^{\operatorname{\mathrm lp}}(N({\mathbb Z}_p^{\times}), {\mathcal M}^{\pm}({\mathcal U})^{\vee}_b \otimes_E \delta_B^{-1}) {\lhook\joinrel\longrightarrow}{\mathcal C}^{\operatorname{\mathrm la}}(N({\mathbb Z}_p^{\times}), {\mathcal M}^{\pm}({\mathcal U})^{\vee}_b \otimes_E \delta_B^{-1})\\ {\longrightarrow}\big(\operatorname{\mathrm Ind}_{\overline{B}({\mathbb Q}_p)}^{\operatorname{\mathrm GL}_2({\mathbb Q}_p)} {\mathcal M}^{\pm}({\mathcal U})^{\vee}_b\otimes_E \delta_B^{-1}\big)^{\operatorname{\mathrm an}} \xlongrightarrow{\iota^{\pm}} \widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an}, \pm},\end{gathered}$$ where the middle map is given in the same way as in the discussion below (\[padicL\]).\
Step (b). Let ${\mathcal V}$ be a compact open subset of ${\mathbb Z}_p^{\times}$, and $1_{{\mathcal V}}\in {\mathcal C}^{\infty}({\mathbb Z}_p^{\times},E)$ be the function with $1_{{\mathcal V}}(x)=\begin{cases}
1 & x\in {\mathcal V}\\ 0 & \text{otherwise}
\end{cases}$. For $j\in {\mathbb Z}_{\geq 0}$ and $i=1,\cdots, e-1$, we have $$\label{deri0}
L^{\pm}_i(-,1_{{\mathcal V}} z^j)=\frac{d L^{\pm}(-,1_{{\mathcal V}} z^j)}{d^i r_{z_{\alpha}}}.$$Recall that we have fixed an isomorphism ${\mathcal O}({\mathcal U})\cong {\mathcal M}^{\pm}({\mathcal U})$, which induces an isomorphism ${\mathcal O}({\mathcal U})^{\vee}_b \cong {\mathcal M}^{\pm}({\mathcal U})^{\vee}_b\otimes_E \delta_B^{-1}$, that we fix in the sequel. By definition, $L^{\pm}(-,1_{{\mathcal V}} z^j)\in {\mathcal O}({\mathcal U})$ is characterized by the following composition $$\label{Lfunval}
{\mathcal O}({\mathcal U})^{\vee}_b \cong {\mathcal M}^{\pm}({\mathcal U})^{\vee}_b \otimes_E \delta_B^{-1} \xlongrightarrow{h_{{\mathcal V}, j}} {\mathcal C}^{\operatorname{\mathrm lp}}(N({\mathbb Z}_p^{\times}), {\mathcal M}^{\pm}({\mathcal U})^{\vee}_b \otimes_E \delta_B^{-1}) \\ {\longrightarrow}\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an}, \pm} \xlongrightarrow{\{\infty\}-\{0\}} E,$$ where the map $h_{{\mathcal V},j}$ sends $m \in {\mathcal M}^{\pm}({\mathcal U})^{\vee}_b \otimes_E \delta_B^{-1}$ to the element: $$m\otimes 1_{{\mathcal V}} \otimes z^j \in {\mathcal C}^{\operatorname{\mathrm sm}}(N({\mathbb Z}_p^{\times}), E) \otimes_E {\mathcal M}^{\pm}({\mathcal U})^{\vee}_b \otimes_E \delta_B^{-1}) \otimes_E E[z]
\cong {\mathcal C}^{\operatorname{\mathrm lp}}(N({\mathbb Z}_p^{\times}), {\mathcal M}^{\pm}({\mathcal U})^{\vee}_b \otimes_E \delta_B^{-1}).$$ Denote by $F_{{\mathcal V}, j}\in {\mathcal O}({\mathcal U})$ the element given by the following composition $$\begin{gathered}
\label{Fvj}
{\mathcal O}({\mathcal U})^{\vee}_b \cong {\mathcal M}^{\pm}({\mathcal U})^{\vee}_b \otimes_E \delta_B^{-1} \xlongrightarrow{g_{{\mathcal V},j}} \text{U}({\mathfrak g}) \otimes_{\text{U}({\mathfrak b})} {\mathcal C}^{\operatorname{\mathrm sm}}(N({\mathbb Z}_p^{\times}), {\mathcal M}^{\pm}({\mathcal U})^{\vee}_b \otimes_E \delta_B^{-1})\\ \xlongrightarrow{\iota_1^{\pm}\circ r_{z_{\alpha}}^{e-1}} \widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an}, \pm}\xlongrightarrow{\{\infty\}-\{0\}} E,\end{gathered}$$ where $g_{{\mathcal V},j}$ sends $m$ to $X_-^{j} \otimes 1_{{\mathcal V}} \otimes m$. By the description of (\[Verma\]), we see $$\label{fomul1}F_{{\mathcal V},j}=\Delta_j L^{\pm}(-,1_{{\mathcal V}} z^j)$$with $\Delta_j=\prod_{l=0}^{j-1}(h-l)$ for $j\in {\mathbb Z}_{>0}$, and $\Delta_0=1$. Similarly, the composition $$\begin{gathered}
\label{Gvj}
{\mathcal O}({\mathcal U})^{\vee}_b \cong {\mathcal M}^{\pm}({\mathcal U})^{\vee}_b \otimes_E \delta_B^{-1} \xlongrightarrow{g_{{\mathcal V},j}} \text{U}({\mathfrak g}) \otimes_{\text{U}({\mathfrak b})} {\mathcal C}^{\operatorname{\mathrm sm}}(N({\mathbb Z}_p^{\times}), {\mathcal M}^{\pm}({\mathcal U})^{\vee}_b \otimes_E \delta_B^{-1})\\ \xlongrightarrow{\iota_1^{\pm}} \widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an}, \pm} \xlongrightarrow{\{\infty\}-\{0\}} E\end{gathered}$$ gives an element $G_{{\mathcal V},j}\in {\mathcal O}({\mathcal U})$ as well. Comparing (\[Fvj\]) with (\[Gvj\]), we see $$\label{fomul2}F_{{\mathcal V},j}=G_{{\mathcal V},j} r_{z_{\alpha}}^{e-1}.$$ Suppose $j\leq k$, then $\Delta_j$ and $r_{z_{\alpha}}$ are coprime (using $(r_{z_{\alpha}}^{e})=(h-k)$). By (\[fomul1\]) and (\[fomul2\]), we deduce (shrinking ${\mathcal U}$ if necessary) that there exists $H_{{\mathcal V},j}\in {\mathcal O}({\mathcal U})$ such that $L^{\pm}(-,1_{{\mathcal V}} z^j)=r_{z_{\alpha}}^{e-1} H_{{\mathcal V},j}$ and $G_{{\mathcal V},j}=\Delta_j H_{{\mathcal V},j}$. Together with (\[deri0\]), the part (1) follows and we have $L_{e-1}^{\pm}(z_{\alpha},1_{{\mathcal V}}z^j)=(e-1)!H_{{\mathcal V},j}(z_{\alpha})$.\
Step (c). We prove the part (2). We fix an isomorphism of $E$-vector space $\chi_{z_{\alpha}}\cong E$. Denote by $i_{z_{\alpha}}: {\mathcal O}({\mathcal U})\twoheadrightarrow {\mathcal O}({\mathcal U})/r_{z_{\alpha}}=E$ the natural projection, which induces $i_{z_{\alpha}}^*: \chi_{z_{\alpha}} \hookrightarrow {\mathcal O}({\mathcal U})^{\vee}_b$. We remark that for $s\in {\mathcal O}({\mathcal U})$ the evaluation $s(z_{\alpha})$ is given by $i_{z_{\alpha}}(s)$. Since $i_{z_{\alpha}}^*$ has image in ${\mathcal M}^{\pm}({\mathcal U})^{\vee}_b[h=k]^{\operatorname{\mathrm cl}}$ (by (\[chaofcl\])), as in [@Em07 Ex. 5.22], the composition $$\chi_{z_{\alpha}} \hookrightarrow {\mathcal O}({\mathcal U})^{\vee}_b\cong {\mathcal M}^{\pm}({\mathcal U})^{\vee}_b {\longrightarrow}J_B\big(\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an}, \pm}\big)$$ is balanced and induces an injection $$\iota_{z_{\alpha}}^{\pm}: (\operatorname{\mathrm Ind}_{\overline{B}({\mathbb Q}_p)}^{\operatorname{\mathrm GL}_2({\mathbb Q}_p)} \chi_{z_{\alpha}}^{\operatorname{\mathrm sm}}\delta_B^{-1})^{\infty}\otimes_E (\operatorname{\mathrm Sym}^k E^2)^{\vee} {\lhook\joinrel\longrightarrow}\widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an},\pm}.$$ We have hence a commutative diagram $$\label{comt01}
\begin{CD}
\text{U}({\mathfrak g}) \otimes_{\text{U}({\mathfrak b})} {\mathcal C}^{\operatorname{\mathrm sm}}(N({\mathbb Z}_p^{\times}), \chi_{z_{\alpha}} \otimes_E \delta_B^{-1}) @> i_{z_{\alpha}}^*>> \text{U}({\mathfrak g}) \otimes_{\text{U}({\mathfrak b})} {\mathcal C}^{\operatorname{\mathrm sm}}_c(N({\mathbb Z}_p^{\times}), {\mathcal O}({\mathcal U})^{\vee}_b\otimes_E \delta_B^{-1})\\
@VVV @V \iota_1^{\pm} VV \\
(\operatorname{\mathrm Ind}_{\overline{B}({\mathbb Q}_p)}^{\operatorname{\mathrm GL}_2({\mathbb Q}_p)} \chi_{z_{\alpha}}^{\operatorname{\mathrm sm}})^{\infty}\otimes_E (\operatorname{\mathrm Sym}^k E^2)^{\vee} @> \iota_{z_{\alpha}}^{\pm} >> \widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an},\pm}
\end{CD}$$ where the top morphism is induced by $i_{z_{\alpha}}^*$, and the left hand side map factors as $$\begin{gathered}
\text{U}({\mathfrak g}) \otimes_{\text{U}({\mathfrak b})} {\mathcal C}^{\operatorname{\mathrm sm}}(N({\mathbb Z}_p^{\times}), \chi_{z_{\alpha}} \otimes_E \delta_B^{-1}) {\longrightarrow}{\mathcal C}^{\operatorname{\mathrm lp}, \leq k}(N({\mathbb Z}_p^{\times}), \chi_{z_{\alpha}} \otimes_E \delta_B^{-1})\\
{\lhook\joinrel\longrightarrow}(\operatorname{\mathrm Ind}_{\overline{B}({\mathbb Q}_p)}^{\operatorname{\mathrm GL}_2({\mathbb Q}_p)} \chi_{z_{\alpha}}^{\operatorname{\mathrm sm}}\delta_B^{-1})^{\infty}\otimes_E (\operatorname{\mathrm Sym}^k E^2)^{\vee} \Big({\lhook\joinrel\longrightarrow}(\operatorname{\mathrm Ind}_{\overline{B}({\mathbb Q}_p)}^{\operatorname{\mathrm GL}_2({\mathbb Q}_p)} \chi_{z_{\alpha}}\delta_B^{-1})^{\operatorname{\mathrm an}}\Big),\end{gathered}$$ with the first map given in the same way as in (\[Verma\]), and the second map given in the same way as in the discussion below (\[padicL\]). By (\[comt01\]), $G_{{\mathcal V},j}(z_{\alpha})$ is equal to the evaluation at $1\in E$ of the following composition $$E\cong \chi_{z_{\alpha}} \otimes_E \delta_B^{-1} \xrightarrow{g_{{\mathcal V},j}} \text{U}({\mathfrak g}) \otimes_{\text{U}({\mathfrak b})} {\mathcal C}^{\operatorname{\mathrm sm}}(N({\mathbb Z}_p^{\times}), \chi_{z_{\alpha}} \otimes_E \delta_B^{-1}) \\
\xlongrightarrow{\iota_1^{\pm}\circ i_{z_{\alpha}}^*} \widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an},\pm} \xlongrightarrow{\{\infty\}-\{0\}} E.$$ Consider the following composition (recall $j\leq k$, and the map $h_{{\mathcal V},j}$ is given in the same way as in (\[Lfunval\])) $$\begin{gathered}
\mu_{{\mathcal V},j}: E\cong \chi_{z_{\alpha}} \otimes_E \delta_B^{-1} \xlongrightarrow{h_{{\mathcal V}, j}} {\mathcal C}^{\operatorname{\mathrm lp}, \leq k}(N({\mathbb Z}_p^{\times}), \chi_{z_{\alpha}} \otimes_E \delta_B^{-1})\\ {\lhook\joinrel\longrightarrow}(\operatorname{\mathrm Ind}_{\overline{B}({\mathbb Q}_p)}^{\operatorname{\mathrm GL}_2({\mathbb Q}_p)} \chi_{z_{\alpha}}^{\operatorname{\mathrm sm}}\delta_B^{-1})^{\infty}\otimes_E (\operatorname{\mathrm Sym}^k E^2)^{\vee} \xlongrightarrow{\iota_{z_{\alpha}}^{\pm}} \widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an},\pm} \xlongrightarrow{\{\infty\}-\{0\}} E.\end{gathered}$$ By the commutative diagram (\[comt01\]) (and using the description of (\[Verma\])), we see $$G_{{\mathcal V},j}(z_{\alpha})=\Delta_j(z_{\alpha}) \mu_{{\mathcal V},j}(1)$$ and hence (by step (b)) $L_{e-1}^{\pm}(z_{\alpha}, 1_{{\mathcal V}}z^j)=(e-1)! \mu_{{\mathcal V},j}(1)$. In summary, the values of $L_{e-1}^{\pm}(z_{\alpha},-)$ at functions in ${\mathcal C}^{\operatorname{\mathrm lp}, \leq k} ({\mathbb Z}_p^{\times},E)$ (in particular, at $\phi x^j$ as in (\[equ: clp-int\])) are characterized by the composition $$\begin{gathered}
{\mathcal C}^{\operatorname{\mathrm lp}, \leq k}(N({\mathbb Z}_p^{\times}), \chi_{z_{\alpha}} \otimes_E \delta_B^{-1}) {\lhook\joinrel\longrightarrow}(\operatorname{\mathrm Ind}_{\overline{B}({\mathbb Q}_p)}^{\operatorname{\mathrm GL}_2({\mathbb Q}_p)} \chi_{z_{\alpha}}^{\operatorname{\mathrm sm}}\delta_B^{-1})^{\infty}\otimes_E (\operatorname{\mathrm Sym}^k E^2)^{\vee} \\ \xlongrightarrow{\iota_{z_{\alpha}}^{\pm}} \widetilde{H}^1_{\operatorname{\text{\'et}},c}(K^p,E)_{\overline{\rho}}^{\operatorname{\mathrm an},\pm} \xlongrightarrow{\{\infty\}-\{0\}} E.\end{gathered}$$ Now we are in the same situation as in [@Em05 (4.14)(4.15)]. The proof of [@Em05 Prop. 4.9] carries over verbatim to this setting, and the part (2) follows.
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abstract: 'In this paper we present a new construction of analytic analogues of quantum groups over non-Archimedean fields and construct braided monoidal categories of their representations. We do this by constructing analytic Nichols algebras and use Majid’s double-bosonisation construction to glue them together. We then go on to study the rigidity of these analytic quantum groups as algebra deformations of completed enveloping algebras through bounded cohomology. This provides the first steps towards a $p$-adic Drinfel’d-Kohno Theorem, which should relate this work to Furusho’s $p$-adic Drinfel’d associators. Finally, we adapt these constructions to working over Archimedean fields.'
author:
- Craig Smith
title: On analytic analogues of quantum groups
---
\[section\] \[theorem\][Corollary]{} \[theorem\][Example]{} \[theorem\][Lemma]{} \[theorem\][Observation]{} \[theorem\][Assumptions]{} \[theorem\][Proposition]{} \[theorem\][Definition]{}
\[theorem\][Remark]{}
Introduction
============
In 2007, Soibelman gave a rough introduction to $p$-adic analogues of quantum groups in [@QpSaQpG] as examples of non-commutative spaces over non-Archimedean fields. Inspired by this, Lyubinin explicitly constructs a $p$-adic quantum hyperenveloping algebra in [@pQHAfSL2] in the case of $\mathfrak{sl}_{2}$. His construction involves using Skew-Tate algebras to construct completions of the positive and negative parts of the quantum enveloping algebra. The disadvantage of this construction is that it requires some work to generalise this to arbitrary Kac-Moody Lie algebras. In this paper we present an alternative construction of analytic analogues of quantum groups over non-Archimedean fields that works for any Kac-Moody Lie algebra and construct braided monoidal categories of their representations. With this we hope to exhibit interesting new analytic representations of braid groups. We then go on to use bounded cohomology to study the rigidity of these analytic quantum groups as algebra deformations of completed enveloping algebras. We hope that this will provide the first steps towards a $p$-adic Drinfel’d-Kohno Theorem, relating this work to Furusho’s $p$-adic Drinfel’d associators in [@pMPatpKZE].\
In [@ItQG], Lusztig constructs the positive and negative parts of quantum enveloping algebras as quotients of tensor algebras by the radical of a duality pairing. This is an example of more a general construction, a *Nichols algebra*, discussed in detail in [@PHA]. Section 1 of this paper is devoted to presenting the definitions and results required to define analytic analogues of Nichols algebras. All of this is done in the categories of IndBanach spaces over both Archimedean and non-Archimedean fields.\
Majid’s construction in [@DBoBG] brings together dually paired braided Hopf algebras $B$ and $C$ with compatible respective right and left actions of a Hopf algebra $H$ to form a new Hopf algebra $U(B,H,C)$, the *double-bosonisation*. The motivation behind this construction is that one can recover the quantum enveloping algebra $U_{q}(\mathfrak{g})$ from $U(B,H,C)$ if $B=U_{q}^{+}(\mathfrak{g})$ and $C=U_{q}^{-}(\mathfrak{g})$ are the respective positive and negative parts of a quantum enveloping algebra and $H=U_{q}^{0}(\mathfrak{g})$ is the Cartan part. Section 2 of this paper recalls and rephrases Majid’s double-bosonisation construction in the context of IndBanach spaces, which will allow us to construct analytic analogues quantum enveloping algebras from analytic Nichols algebras in the subsequent sections.\
In Section 3 we restrict ourselves to working over non-Archimedean fields. We begin Subsection 3.1 by proving the existence of the analytic Nichols algebras defined in Section 1 through two different constructions. The first, given in the proof of Proposition \[Radius1NicholsAlgebrasExists\], exhibits the quotient of a completed tensor algebra by a certain universal Hopf ideal as an analytic Nichols algebra. The second, given in Proposition \[BilinearFormGivesNicholsAlgebra\], constructs an analytic Nichols algebra as the quotient of a completed tensor algebra by the radical of a duality pairing. In particular this second construction gives the duality pairing between Nichols algebras that allows us to use Majid’s double-bosonisation construction. We show in Proposition \[NicholsAlgebrasEquivalentDefinition\] that these two constructions are equivalent. In Subsection 3.2 we apply these constructions to obtain completions of the positive and negative parts of quantum enveloping algebras and use Majid’s double-bosonisation to glue them together into completions of quantum enveloping algebras. We call the resulting IndBanach Hopf algebras *analytic quantum groups*.\
Unfortunately, we see in Subsection 3.3 that the R-matrix of $U_{q}(\mathfrak{g})$ still does not converge in any of our analytic quantum groups. Nonetheless, in Subsection 3.4 we use an alternate description of our analytic quantum groups as quotients of a Drinfel’d doubles to obtain a braided monoidal category of representations analogous to the BGG category $\mathcal{O}$. We then present an example in the case of $\mathfrak{g}=\mathfrak{sl}_{2}$ of such a braided representation with no highest weight vectors. The further study of these braided representations should produce interesting new examples of braid group representation on Banach spaces and may give a new context in which some special analytic functions, such as $p$-adic multiple polylogarithms, naturally arise.\
The classical rigidity results of Chevalley, Eilenberg and Cartan from the 1940s assert that there are no non-trivial formal deformations (as an algebra) of the universal enveloping algebra of a semisimple Lie algebra $\mathfrak{g}$. The proof relies on the vanishing of certain Lie algebra cohomology groups. In Theorems \[Rigidity1\] and \[Rigidity2\] of Subsection 3.5 we prove an analogous result that, provided an unproven bounded cohomology vanishing result holds, any algebra deformation of a completed enveloping algebra is isomorphic to the trival one. In particular this implies Corollary \[RigidityApplied\] that asserts that, modulo a bounded cohomology vanishing result, our analytic quantum groups are isomorphic to the trivial algebra deformation of a completed enveloping algebra. Furthermore, both of these isomorphisms are unique up to conjugation. In Subsection 3.6 we highlight the benefits of working over formal powerseries $k \llbracket \hslash \rrbracket$ as opposed to convergent powerseries. In particular this allows us to prove Theorems \[Rigidity1b\] and \[Rigidity2b\], rigidity results analogous to Theorems \[Rigidity1\] and \[Rigidity2\] that require weaker assumptions on bounded cohomology.\
In [@QG], Kassel uses algebraic analogues of the rigidity theorems of Subsections 3.5 and 3.6 to present a proof of the Drinfel’d-Khono Theorem over $\mathbb{C}$. This theorem states that the category of representations of the quantum enveloping algebra is equivalent, as a braided monoidal category, to the category of $U(\mathfrak{g})$-modules with associativity constraint given by the Drinfel’d associator and braiding given by the associated R-matrix. As a result of this, the associated braid group representations are equivalent. This can be interpreted as a statement about the monodromy of the Knizhnik-Zamolodchikov (KZ) equations that govern the Drinfel’d associator. In [@pMPatpKZE], Furusho uses $p$-adic multiple polylogarithms to construct solutions to the $p$-adic KZ equations and a $p$-adic Drinfel’d associator. In the future the author hopes to expand upon the work in Subsections 3.5 and 3.6 to prove a $p$-adic analogue of the Drinfel’d-Khono theorem and to investigate links to Furusho’s work.\
Finally, in Section 4 we adapt these constructions to working over Archimedean fields. We begin by proving the existence of some analytic Nichols algebras and then use Majid’s double-bosonisation to form Archimedean analytic quantum groups. We finish by constructing a braided monoidal category of representations as in Subsection 3.4. Again, we hope that the further study of these representations will produce interesting new braid group representations in which we might see some special analytic functions arising, such as the quantum dilogarithms that appear in [@PRftQDaQM].
Funding {#funding .unnumbered}
-------
This work was supported by the Engineering and Physical Sciences Research Council \[EP/M50659X/1\].
Acknowledgements {#acknowledgements .unnumbered}
----------------
[^1]
Braided IndBanach Hopf algebras, analytic gradings and Nichols algebras
=======================================================================
Preliminary definitions
-----------------------
Let $k$ be a complete valued field throughout.
\[BanachCategory\] Let $\text{Ban}_{k}$ denote the category of $k$-Banach spaces, each equipped with a specific norm, and bounded linear transformations between them, and let $\text{Ban}_{k}^{\leq 1}$ denote the wide subcategory whose morphisms are bounded linear transformations of norm at most 1, the *contracting category of Banach spaces*. By *wide* we mean that $\text{Ban}_{k}^{\leq 1}$ contains all objects of $\text{Ban}_{k}$. If our field is non-Archimedean then Banach spaces may be defined in two ways, depending on whether we require norms to satisfy the usual triangle inequality or the strong triangle inequality. We shall therefore distinguish between the following two cases:
- $k$ is non-Archimedean and we require all norms in $\text{Ban}_{k}$ to satisfy the strong triangle inequality; and
- $k$ is not necessarily non-Archimedean and we only require norms in $\text{Ban}_{k}$ to satisfy the weak triangle inequality.
\[Contracting(Co)Products\] Let $(V_{i})_{i \in I}$ be a family of Banach spaces. Let us define the *contracting product* of this family as the Banach space $$\prod\nolimits_{i \in I}^{\leq 1}V_{i}=\{(v_{i})_{i \in I} \in \times_{i \in I} V_{i} \mid \text{Sup}_{i \in I} \|v_{i}\| \leq \infty\}$$ with norm $\|(v_{i})\|=\text{Sup}_{i \in I} \|v_{i}\|$ in both the [**(NA)**]{} and [**(A)**]{} cases, and the *contracting coproduct* as the Banach space $$\coprod\nolimits_{i \in I}^{\leq 1}V_{i} = \{(v_{i})_{i \in I} \in \times_{i \in I} V_{i} \mid \sum_{i \in I} \|v_{i}\| \leq \infty\}$$ with norm $\|(v_{i})\|=\sum_{i \in I} \|v_{i}\|$ in the case of [**(A)**]{} and $$\coprod\nolimits_{i \in I}^{\leq 1}V_{i} = \{(v_{i})_{i \in I} \in \times_{i \in I} V_{i} \mid \text{lim}_{i \in I} \|v_{i}\| =0\}$$ with norm $\|(v_{i})\|=\text{Sup}_{i \in I} \|v_{i}\|$ in the case of [**(NA)**]{}.
The category $\text{Ban}_{k}^{\leq 1}$ has small limits and colimits.
Indeed, it has kernels and cokernels inhereted from $\text{Ban}_{k}$, and it is easy to verify that Definition \[Contracting(Co)Products\] describes products and coproducts in this category.
For a set $I$, let $\text{Ban}_{k}^{I,\text{bd}}$ be the category whose objects are collections $(V_{i})_{i \in I}$ of Banach spaces $V_{i}$ indexed by $i \in I$ and whose morphisms are uniformly bounded, $$\text{Hom}((V_{i})_{i \in I},(V'_{i})_{i \in I}):=\prod\nolimits_{i \in I}^{\leq 1}\underline{\text{Hom}}(V_{i},V'_{i}).$$
\[ContractingUniversalProperty\] $\prod_{i \in I}^{\leq 1}$ and $\coprod_{i \in I}^{\leq 1}$ define functors from $\text{Ban}_{k}^{I,\text{bd}}$ to $\text{Ban}_{k}$. Furthermore, contracting products are right adjoints to the diagonal functors $$\Delta^{I}:\text{Ban}_{k} \rightarrow \text{Ban}_{k}^{I,\text{bd}}, \quad V \mapsto (V)_{i \in I},$$ and likewise contracting coproducts are left adjoints to $\Delta^{I}$.
This follows from Lemma 2.5 of [@TRTfIBS].
Let us denote by $\text{IndBan}_{k}$ the Ind completion of $\text{Ban}_{k}$. That is, objects of $\text{IndBan}_{k}$ are filtered diagrams of Banach spaces, $X:I \rightarrow \text{Ban}_{k}$, and morphisms are given by $$\text{Hom}(X,Y)=\text{colim}_{j \in J} \text{lim}_{i \in I} \text{Hom}(X(i),Y(j)).$$ We think of these objects as formal colimits, and use the notation $\text{"colim"}_{i \in I} X(i)$ for the diagram $X$. For a Banach space $V$ we denote by $\text{"}V\text{"}$ the object in $\text{IndBan}_{k}$ represented by the constant diagram at $V$, and often just as $V$ when there is no ambiguity.
The category $\text{IndBan}_{k}$ is a complete and cocomplete quasi-abelian category, and can be given a closed monoidal structure extending that of $\text{Ban}_{k}$ by defining $$(\text{"colim"}_{i \in I} X_{i})\hat{\otimes}(\text{"colim"}_{j \in J} Y_{j}):= \text{"colim"}_{\substack{i \in I \\ j \in J}} X_{i} \hat{\otimes} Y_{j},$$ $$\underline{\text{Hom}}(\text{"colim"}_{i \in I} X_{i},\text{"colim"}_{j \in J} Y_{j}):= \text{colim}_{j \in J} \text{lim}_{i \in I} \underline{Hom}(X_{i},Y_{j}).$$
Since $\text{Ban}_{k}$ has cokernels and finite direct sums, $\text{IndBan}_{k}$ is cocomplete. An explicit construction of limits in $\text{IndBan}_{k}$ can be found in Section 1.4.1 of [@LaACH]. Proposition 2.1.17 of [@QACaS] asserts that $\text{IndBan}_{k}$ is quasi-ableian.
For an account of Ind completions see [@CaS], and more on $\text{IndBan}_{k}$ can be found in [@TRTfDM], [@SDiBAG], [@NAAGaRAG] and [@LaACH] and numerous other excellent sources.
We extend the definition of contracting (co)products to $\text{IndBan}_{k}$ as follows. The contracting product and coproduct functors $$\prod\nolimits^{\leq 1}_{I},\coprod\nolimits^{\leq 1}_{I}:\text{Ban}_{k}^{I,\text{bd}} \rightarrow \text{Ban}_{k}$$ induce functors from the Ind completion of $\text{Ban}_{k}^{I,\text{bd}}$, $$\text{IndBan}_{k}^{I,\text{bd}}:=\text{Ind}(\text{Ban}_{k}^{I,\text{bd}}),$$ to $\text{IndBan}_{k}$, which we will continue to denote as $\prod^{\leq 1}_{I}$ and $\coprod^{\leq 1}_{I}$ respectively. There is a faithful diagonal embedding functor $\Delta^{I}:\text{IndBan}_{k} \rightarrow \text{IndBan}_{k}^{I,\text{bd}}$ induced by $\Delta^{I}:\text{Ban}_{k} \rightarrow \text{Ban}_{k}^{I,\text{bd}}$.
With the above definitions, there are adjunctions $$\text{Hom}(\coprod\nolimits_{I} ^{\leq 1}X_{I},Y) \cong \text{Hom}(X_{I},\Delta^{I}Y)$$ and $$\text{Hom}(Y,\prod\nolimits_{I} ^{\leq 1}X_{I}) \cong \text{Hom}(\Delta^{I}Y,X_{I})$$ for $X_{I} \in \text{IndBan}_{k}^{I,\text{bd}}$, $Y \in \text{IndBan}_{k}$.
This follows from the adjunction given in Lemma \[ContractingUniversalProperty\] by taking filtered colimits.
Braided IndBanach Hopf algebras
-------------------------------
\[Braiding\] Let $V$ be an IndBanach space. We say that a morphism $c:V \hat{\otimes} V \rightarrow V \hat{\otimes} V$ is a *pre-braiding* on $V$ if it satisfied the *hexagon axiom*, *i.e.* the diagram
\(A) [$V \hat{\otimes} V \hat{\otimes} V$]{}; (B) \[right=1.3cm of A\] [$V \hat{\otimes} V \hat{\otimes} V$]{}; (C’) \[below=1cm of A\] ; (C) \[left=0.5cm of C’\] [$V \hat{\otimes} V \hat{\otimes} V$]{}; (D’) \[below=1cm of B\] ; (D) \[right=0.5cm of D’\] [$V \hat{\otimes} V \hat{\otimes} V$]{}; (E) \[below=1cm of C’\] [$V \hat{\otimes} V \hat{\otimes} V$]{}; (F) \[below=1cm of D’\] [$V \hat{\otimes} V \hat{\otimes} V$]{}; (A) to node [$c \otimes \text{Id}_{V}$]{} (B); (B) to node [$\text{Id}_{V} \otimes c$]{} (D); (D) to node [$c \otimes \text{Id}_{V}$]{} (F); (A) to node \[swap\][$\text{Id}_{V} \otimes c$]{} (C); (C) to node \[swap\][$c \otimes \text{Id}_{V}$]{} (E); (E) to node \[swap\][$\text{Id}_{V} \otimes c$]{} (F);
commutes. We say that the pair $(V,c)$ is a *pre-braided IndBanach space*. If $c$ is an isomorphism then $c$ is a *braiding* and $(V,c)$ is a *braided IndBanach space*.
Let $V$ be a Banach space. We define the *Banach tensor algebra*, $T(V)$, to be the contracting coproduct $$T(V):=\coprod\nolimits_{n \in \mathbb{Z}_{\geq 0}}^{\leq 1}V^{\hat{\otimes} n}$$ where $V^{\hat{\otimes}0}:=k$. For $r>0$ we will use the notation $T_{r}(V)$ for the Banach space $T(V_{r})$, where $V_{r}$ is the Banach space $V$ with its norm rescaled by $r$. For $r \leq r'$ there is a natural map $T_{r'}(V) \rightarrow T_{r}(V)$, and for all $\rho \geq 0$ we will denote by $T_{\rho}(V)^{\dagger}$ the colimit $\text{"colim"}_{r>\rho}T_{r}(V)$ of this system. We will call this the *dagger tensor algebra* or *overconvergent tensor algebra* of radius $\rho$.
Given a pre-braiding $c$ of a Banach space $V$, there is an induced map $c_{n,m}:V^{\hat{\otimes} n} \hat{\otimes} V^{\hat{\otimes} m} \rightarrow V^{\hat{\otimes} m} \hat{\otimes} V^{\hat{\otimes} n}$ with $\|c_{n,m}\| \leq \|c\|^{mn}$ for each $n,m \geq 0$ satisfying the commutative diagram below:
\(A) [$V^{\hat{\otimes} l} \hat{\otimes} V^{\hat{\otimes} m} \hat{\otimes} V^{\hat{\otimes} n}$]{}; (B) \[right=1.2cm of A\] [$V^{\hat{\otimes} m} \hat{\otimes} V^{\hat{\otimes} l} \hat{\otimes} V^{\hat{\otimes} n}$]{}; (C’) \[below=1cm of A\] ; (C) \[left=0.2cm of C’\] [$V^{\hat{\otimes} l} \hat{\otimes} V^{\hat{\otimes} n} \hat{\otimes} V^{\hat{\otimes} m}$]{}; (D’) \[below=1cm of B\] ; (D) \[right=0.2cm of D’\] [$V^{\hat{\otimes} m} \hat{\otimes} V^{\hat{\otimes} n} \hat{\otimes} V^{\hat{\otimes} l}$]{}; (E) \[below=1cm of C’\] [$V^{\hat{\otimes} n} \hat{\otimes} V^{\hat{\otimes} l} \hat{\otimes} V^{\hat{\otimes} m}$]{}; (F) \[below=1cm of D’\] [$V^{\hat{\otimes} n} \hat{\otimes} V^{\hat{\otimes} m} \hat{\otimes} V^{\hat{\otimes} l}$]{}; (A) to node [$c_{l,m} \otimes \text{Id}_{V}$]{} (B); (B) to node [$\text{Id}_{V} \otimes c_{l,n}$]{} (D); (D) to node [$c_{m,n} \otimes \text{Id}_{V}$]{} (F); (A) to node \[swap\][$\text{Id}_{V} \otimes c_{m,n}$]{} (C); (C) to node \[swap\][$c_{l,n} \otimes \text{Id}_{V}$]{} (E); (E) to node \[swap\][$\text{Id}_{V} \otimes c_{l,m}$]{} (F);
Hence if $\|c\| \leq 1$ then there is an induced pre-braiding $\tilde{c}$ on $T(V)$ with $\|\tilde{c}\| = \|c\|$. Furthermore, $\tilde{c}$ is a braiding if and only if $c$ is an isometry.
Applying successively $\text{Id}_{V}^{\otimes n-i} \otimes c \otimes \text{Id}_{V}^{\otimes i-1}$ for $i=1, \ldots, n$ we obtain a map $c_{n}:V^{\hat{\otimes} n} \hat{\otimes} V \rightarrow V \hat{\otimes} V^{\hat{\otimes} n}$ with $\|c_{n}\| \leq \|c\|^{n}$. Then successive applications of $\text{Id}_{V}^{\otimes i-1} \otimes c_{n} \otimes \text{Id}_{V}^{\otimes m-i}$ for $i=1, \ldots, m$ gives a map $c_{n,m}:V^{\hat{\otimes} n} \hat{\otimes} V^{\hat{\otimes} m} \rightarrow V^{\hat{\otimes} m} \hat{\otimes} V^{\hat{\otimes} n}$ with $\|c_{n,m}\| \leq \|c_{n}\|^{m}\leq \|c\|^{nm}$. The commutativity of the given diagram follows from repeated applications of the hexagon axiom from Definition \[Braiding\].
\[BraidedBialgebra\] A *pre-braided IndBanach bialgebra* is an IndBanach space $A$ with both the structure of an algebra, $(A, \mu, \eta)$, and a coalgebra, $(A, \Delta, \varepsilon)$, and equipped with a pre-braiding $c$ on $A$ such that $$c \circ (\eta \otimes \text{Id}) = \text{Id} \otimes \eta, \quad c \circ (\text{Id} \otimes \eta) = \eta \otimes \text{Id},$$ $$(\text{Id} \otimes \mu)(c \otimes \text{Id})(\text{Id} \otimes c)=c \circ (\mu \otimes \text{Id}), \quad (\mu \otimes \text{Id})(\text{Id} \otimes c)(c \otimes \text{Id})=c \circ (\text{Id} \otimes \mu),$$ $$(\varepsilon \otimes \text{Id}) \circ c = \text{Id} \otimes \varepsilon, \quad (\text{Id} \otimes \varepsilon) \circ c = \varepsilon \otimes \text{Id},$$ $$(c \otimes \text{Id})(\text{Id} \otimes c)(\Delta \otimes \text{Id})=(\text{Id} \otimes \Delta) \circ c, \quad (\text{Id} \otimes c)(c \otimes \text{Id})(\text{Id} \otimes \Delta)=(\Delta \otimes \text{Id}) \circ c,$$ the diagram
\(A) [$A \hat{\otimes} A$]{}; (B) \[below=2.045cm of A\] [$A$]{}; (C) \[right=2cm of A\] [$A \hat{\otimes} A \hat{\otimes} A \hat{\otimes} A$]{}; (D) \[below=2cm of C\] [$A \hat{\otimes} A$]{}; (E) \[below=0.75cm of C\] [$A \hat{\otimes} A \hat{\otimes} A \hat{\otimes} A$]{}; (A) to node \[swap\][$\mu$]{} (B); (C) to node [$\text{Id} \otimes c \otimes \text{Id}$]{} (E); (A) to node [$\Delta \otimes \Delta$]{} (C); (E) to node [$\mu \otimes \mu$]{} (D); (B) to node [$\Delta$]{} (D);
commutes and $$\Delta \circ \eta = \eta \otimes \eta, \quad \varepsilon \circ \eta = \text{Id}_{k}, \quad \varepsilon \circ \mu = \varepsilon \circ \varepsilon.$$ If $c$ is an isomorphism then $A$ is a *braided IndBanach* *bialgebra*.
For any IndBanach algebra $A$ with a (pre-) braiding $c$, we may define morphisms $$A \hat{\otimes} A \hat{\otimes} A \hat{\otimes} A \overset{\text{Id} \otimes c \otimes \text{Id}}{\xrightarrow{\hspace*{1cm}}} A \hat{\otimes} A \hat{\otimes} A \hat{\otimes} A \overset{\mu \otimes \mu}{\longrightarrow} A \hat{\otimes} A,$$ $$A \hat{\otimes} A \overset{\Delta \otimes \Delta}{\longrightarrow} A \hat{\otimes} A \hat{\otimes} A \hat{\otimes} A \overset{\text{Id} \otimes c \otimes \text{Id}}{\xrightarrow{\hspace*{1cm}}} A \hat{\otimes} A \hat{\otimes} A \hat{\otimes} A.$$ The first four relations in Definition \[BraidedBialgebra\] are equivalent to the first of these maps making $A \hat{\otimes} A$ an associative algebra with unit $\eta \otimes \eta$. The next four relations are equivalent to the second of these maps making $A \hat{\otimes} A$ an associative coalgebra with counit $\varepsilon \otimes \varepsilon$. Then the diagram and final three relations are equivalent to $\Delta$ and $\varepsilon$ being algebra homomorphisms, or equivalently $\mu$ and $\eta$ being coalgebra homomorphisms.
\[TensorBialgebra\] Assume [**(NA)**]{}. For $V$ a Banach space, $T_{r}(V)$ is a Banach algebra with multiplication $\mu$ induced by the maps $V_{r}^{\hat{\otimes}n} \hat{\otimes} V_{r}^{\hat{\otimes}m} \overset{\sim}{\longrightarrow} V_{r}^{\hat{\otimes}n+m}$. Given a pre-braiding $c$ on $V$ with $\|c\| \leq 1$, $T_{r}(V)$ is a pre-braided Banach bialgebra with the comultiplication $\Delta$ uniquely determined by $\Delta(v)=1 \otimes v + v \otimes 1$ for all $v \in V_{r}=V_{r}^{\hat{\otimes} 1} \subset T_{r}(V)$ and the pre-braiding $\tilde{c}$ induced by $c$. Furthermore, $\Delta$ has norm at most 1.
The fact that $T_{r}(V)$ forms a Banach algebra is clear from construction, with unit $k \overset{\sim}{\longrightarrow} V^{\hat{\otimes} 0}$. We have a map $\Delta_{1}:V_{r}^{\hat{\otimes} 1} \rightarrow T_{r}(V) \hat{\otimes} T_{r}(V)$, given by $v \mapsto 1 \otimes v + v \otimes 1$, with $\|\Delta_{1}\| \leq 1$. Suppose we have a map $\Delta_{n}:V_{r}^{\hat{\otimes} n} \rightarrow T_{r}(V) \hat{\otimes} T_{r}(V)$ with $\|\Delta_{n}\| \leq \|c\|^{n-1}$. Then we define $\Delta_{n+1}$ as the composition $$\begin{array}{rcl}
V_{r}^{\hat{\otimes}n+1}= V_{r}^{\hat{\otimes}n} \hat{\otimes} V_{r} &\overset{\Delta_{n} \otimes \Delta_{1}}{\xrightarrow{\hspace*{1cm}}}& T_{r}(V) \hat{\otimes} T_{r}(V) \hat{\otimes} T_{r}(V) \hat{\otimes} T_{r}(V)\\
&\overset{\text{Id} \otimes \tilde{c} \otimes \text{Id}}{\xrightarrow{\hspace*{1cm}}}& T_{r}(V) \hat{\otimes} T_{r}(V) \hat{\otimes} T_{r}(V) \hat{\otimes} T_{r}(V)\\
&\overset{\mu \otimes \mu}{\xrightarrow{\hspace*{1cm}}}& T_{r}(V) \hat{\otimes} T_{r}(V).
\end{array}$$ Then since the multiplication on $T_{r}(V)$ is of norm 1, we have $\|\Delta_{n+1}\| \leq \|\Delta_{n}\| \cdot \|\Delta_{1}\| \cdot \|\tilde{c}\| \leq \|c\|^{n}$.
\[ArchimedeanTensorBialgebra\] Assume [**(A)**]{}. For $V$ a Banach space, $T_{r}(V)$ is a Banach algebra with multiplication $\mu$ induced by the maps $V_{r}^{\hat{\otimes}n} \hat{\otimes} V_{r}^{\hat{\otimes}m} \overset{\sim}{\longrightarrow} V_{r}^{\hat{\otimes}n+m}$. Given a pre-braiding $c$ on $V$ with $\|c\| \leq 1$, $T_{0}(V)^{\dagger}$ is a pre-braided IndBanach bialgebra with comultiplication $\Delta$ whose restriction to $V$ is $$\text{"colim"}_{r>0}V_{r} \cong V \rightarrow T(V) \hat{\otimes} T(V) \rightarrow T_{0}(V)^{\dagger} \hat{\otimes} T_{0}(V)^{\dagger}, \quad v \mapsto 1 \otimes v + v \otimes 1,$$ for the pre-braiding $\tilde{c}$ induced by $c$.
The given map $V \rightarrow T(V) \hat{\otimes} T(V)$ induces maps $V_{r} \rightarrow T_{\frac{r}{2}}(V) \hat{\otimes} T_{\frac{r}{2}}(V)$ of norm at most $1$. By the same construction as in the proof of the previous proposition, we obtain maps $T_{r}(V) \rightarrow T_{\frac{r}{2}}(V) \hat{\otimes} T_{\frac{r}{2}}(V)$ which induce the desired comultiplication on $T_{0}(V)^{\dagger}$.
Note that $V \mapsto T(V)$ is only functorial on the contracting category $\text{Ban}_{k}^{\leq 1}$. So, in the [**(NA)**]{} case, the diagonal embedding $V \rightarrow V \oplus V$ induces the map $T(V) \rightarrow T(V \oplus V) \cong T(V) \hat{\otimes} T(V)$. However, in the [**(A)**]{} case, the diagonal embedding is not contracting, thus $T(V)$ does not form a coalgebra.
Let $A$ be a (pre-) braided IndBanach bialgebra. We say that $A$ is a (pre-) braided IndBanach Hopf algebra if the identity on $A$ is convolution invertible. That is, $\text{Id}_{A}$ is invertible with respect to the convolution product $\ast$ on $\text{Hom}(A,A)$, $$f \ast g:A \overset{\Delta}{\longrightarrow} A \hat{\otimes} A \overset{f \otimes g}{\longrightarrow} A \hat{\otimes} A \overset{\mu}{\longrightarrow} A$$ for $f,g \in \text{Hom}(A,A)$, whose unit is $\eta \circ \varepsilon$. We will call the convolution inverse of $\text{Id}_{A}$ the antipode, and often denote it by $S$ or $S_{A}$.
\[GradedPiecesOfTensorAlgebraCoideals\] Assume [**(NA)**]{}. For a Banach space $V$ with a pre-braiding $c$ on $V$ with $\|c\| \leq 1$, we have that, in $T(V)$, $\Delta(V^{\hat{\otimes} n}) \subset \sum_{i=0}^{n} V^{\hat{\otimes} i} \hat{\otimes} V^{\hat{\otimes} n-i}$, and so in particular $$\Delta(\coprod\nolimits_{i \leq n}^{\leq 1}V^{\hat{\otimes} i}) \subset (\coprod\nolimits_{i \leq n-1}^{\leq 1}V^{\hat{\otimes} i}) \hat{\otimes} T(V) + T(V) \hat{\otimes} (\coprod\nolimits_{i \leq n-1}^{\leq 1}V^{\hat{\otimes} i})$$ for all $n \geq 0$.
This follows by induction using the proof of Proposition \[TensorBialgebra\].
\[TensorHopfAlgebra\] Assume [**(NA)**]{}. Given a Banach space $V$ with a pre-braiding $c$ on $V$ with $\|c\| \leq 1$, $T(V)$ is a pre-braided Banach Hopf algebra.
We proceed as in Takeuchi’s proof of Lemma 5.2.10 presented in [@HAatAoR] to find a convolution inverse to $\text{Id}_{T(V)}$. Let $\gamma = \eta \circ \varepsilon - \text{Id}_{T(V)}$. Then $\gamma|_{V^{\hat{\otimes} 0}} =0$. Now suppose that $\gamma^{\ast n}|_{\coprod_{i \leq n-1}^{\leq 1}V^{\hat{\otimes} i}} =0$ for some $n \geq 1$ and let $x \in \coprod_{i \leq n}^{\leq 1}V^{\hat{\otimes} i}$. Here we use the notation $\gamma^{\ast n}$ for the $n$-fold product of $\gamma$ under the convolution product. Since $$\Delta(x) \in (\coprod\nolimits_{i \leq n-1}^{\leq 1}V^{\hat{\otimes} i}) \hat{\otimes} T(V) + T(V) \hat{\otimes} (\coprod\nolimits_{i \leq n-1}^{\leq 1}V^{\hat{\otimes} i}),$$ by Lemma \[GradedPiecesOfTensorAlgebraCoideals\], and since $\gamma^{\ast (n+1)}=\gamma^{\ast n} \ast \gamma = \gamma \ast \gamma^{\ast n}$ we see that $\gamma^{\ast n+1}(x)=0$. Hence, inductively, $\gamma^{\ast n+1}|_{\coprod_{i \leq n}^{\leq 1}V^{\hat{\otimes} i}} =0$ for all $n \geq 0$. It follows that $\sum_{n=0}^{\infty} \gamma^{\ast n}$ is well defined on the direct sum of vector spaces $\bigoplus_{n \in \mathbb{Z}_{\geq 0}}V^{\hat{\otimes} n}$. Furthermore, since $\|c\| \leq 1$ and so $\|\Delta\| \leq 1$ we see that $\| \gamma \| \leq 1$, $\| \gamma^{\ast n} \| \leq 1$ and so $\|\sum_{n=0}^{N} \gamma^{\ast n}\| \leq 1$ for all $N \geq 0$. It then follows that $\sum_{n=0}^{\infty} \gamma^{\ast n}$ converges to a well defined function on $T(V)$, which is convolution inverse to $\text{Id}_{T(V)}$ since $$\begin{array}{rcl}
\text{Id}_{T(V)}\ast \sum_{n=0}^{\infty} \gamma^{\ast n} &=& (\eta \circ \varepsilon) \ast \sum_{n=0}^{\infty} \gamma^{\ast n} - \gamma \ast \sum_{n=0}^{\infty} \gamma^{\ast n}\\
&=& \sum_{n=0}^{\infty} \gamma^{\ast n} - \sum_{n=1}^{\infty} \gamma^{\ast n}\\
&=& \eta \circ \varepsilon.
\end{array}$$
Assume [**(NA)**]{}. Given a (pre-) braided Banach space $(V,c)$ with $\|c\| \leq 1$ and $r>0$ we denote by $T_{r}^{c}(V)$ the (pre-) braided Banach Hopf algebra described in Proposition \[TensorBialgebra\] and Proposition \[TensorHopfAlgebra\], or just $T_{r}(V)$ if the (pre-) braiding is implicit.
\[TensorHopfAlgebraA\] Assume [**(A)**]{}. Given a Banach space $V$ with a braiding $c$ on $V$ with $\|c\| \leq 1$, $T_{0}(V)^{\dagger}$ is a pre-braided Banach Hopf algebra.
We take a slightly different approach to the proof of Proposition \[TensorHopfAlgebra\]. The linear maps $S_{r}:V_{r} \rightarrow T_{r}(V)$ defined by $v \mapsto -v$ determine a unique algebra homomorphisms $S:T_{r}(V) \rightarrow T_{r}(V)^{\text{op}}$, where $T_{r}(V)^{\text{op}}$ is the opposite algebra whose multiplication is $\mu \circ \tilde{c}$. The compositions $$T_{r}(V) \overset{\Delta}{\longrightarrow} T_{\frac{r}{2}}(V) \hat{\otimes} T_{\frac{r}{2}}(V) \overset{S_{\frac{r}{2}} \otimes \text{Id}}{\longrightarrow} T_{\frac{r}{2}}(V) \hat{\otimes} T_{\frac{r}{2}}(V) \rightarrow T_{\frac{r}{2}}(V)$$ agree with $\eta \circ \varepsilon$ when restricted to $V$. It is then easy to check that they agree on the subalgebra generated by $V$, which is dense in $T_{r}(V)$. So they agree. Likewise this is true for $\text{Id} \otimes S_{\frac{r}{2}}$ in place of $S_{\frac{r}{2}} \otimes \text{Id}$. Taking colimits we obtain the antipode $S:T_{0}(V)^{\dagger} \rightarrow T_{0}(V)^{\dagger}$.
Assume [**(A)**]{}. Given a (pre-) braided Banach space $(V,c)$ with $\|c\| \leq 1$ we denote by $T_{0}^{c}(V)^{\dagger}$ the (pre-) braided Banach Hopf algebra described in Proposition \[ArchimedeanTensorBialgebra\] and Proposition \[TensorHopfAlgebraA\], or just $T_{0}(V)^{\dagger}$ if the (pre-) braiding is implicit.
The main distinction between the cases [**(NA)**]{} and [**(A)**]{} is that the tensor Hopf algebra can be defined on any radius in the non-Archimedean setting but in the Archimedean setting can only be defined at radius 0. This is not entirely unexpected. Analogously we see that, for a non-Archimedean field $k$, the balls of each radius in $k$ form additive subgroups, however the same cannot be said for Archimedean fields.
Analytic gradings
-----------------
Let $C$ be an IndBanach bialgebra. We will say that an IndBanach space $V$ is *graded by* $C$ if it is a $C$-comodule. An IndBanach space $V$ with a (pre-) braiding $c$ is a *graded (pre-) braided IndBanach space* (graded by $C$) if $c$ is a morphism of $C\hat{\otimes} C$-comodules. A *graded (pre-) braided IndBanach bialgebra* is a (pre-) braided IndBanach bialgebra $A$ such that $A$ is a $C$-comodule, $\eta$, $\mu$, $\varepsilon$ and $\Delta$ are $C$-comodule homomorphisms, and $c$ is a $C \hat{\otimes} C$-comodule homomorphism. If, in addition, the identity on $A$ has a convolution inverse $S$ that is a $C$-comodule homomorphism then $A$ is a *graded (pre-) braided IndBanach Hopf algebra*.
The results in Sections 4.2 and 4.3 of [@TRTfIBS] justify our definition of grading above.
\[FunctionsOnOpenDisk\] Let $k\{\underline{t}\}=k\{t_{1},\ldots,t_{N}\}$ be the bialgebra $\coprod_{\underline{n} \in \mathbb{N}^{N}}^{\leq 1} k \cdot \underline{t}^{\underline{n}}$ where the comultiplication maps $\underline{t}^{\underline{n}} \mapsto \underline{t}^{\underline{n}} \otimes \underline{t}^{\underline{n}}$ and the counit is $\underline{t}^{\underline{n}} \mapsto 1$, and the multiplication maps $\underline{t}^{\underline{m}} \otimes \underline{t}^{\underline{n}} \mapsto \underline{t}^{\underline{m}+ \underline{n}}$ with unit $\underline{t}^{\underline{0}}$.
IndBanach spaces graded by $k\{\underline{t}\}$ are of the form $\coprod_{\mathbb{N}^{N}}^{\leq 1}M$ for $M$ in $\text{IndBan}_{k}^{\mathbb{N}^{N},\text{bd}}$. The space of morphisms that respect the grading is $$\underline{\text{Hom}}_{k\{\underline{t}\}}(\coprod\nolimits_{\mathbb{N}^{N}}^{\leq 1}M,\coprod\nolimits_{\mathbb{N}^{N}}^{\leq 1}M')= \text{colim}_{j \in J} \text{lim}_{i \in I} \prod\nolimits_{\underline{n} \in \mathbb{N}^{N}}^{\leq 1}\underline{\text{Hom}}(M_{\underline{n}}(i),M'_{\underline{n}}(j))$$ where $M=\text{"colim"}_{i \in I}(M_{\underline{n}}(i))_{\underline{n} \in \mathbb{N}^{N}}$ and $M'=\text{"colim"}_{j \in J}(M'_{\underline{n}}(j))_{\underline{n} \in \mathbb{N}^{N}}$ in $\text{IndBan}_{k}^{\mathbb{N}^{N},\text{bd}}$.
This is Proposition 4.3 of [@TRTfIBS].
Let $k\{\underline{t}\}^{\dagger}:=\text{"colim"}_{\underline{r} > 1} k\{\underline{t}/ \underline{r}\}$, where the colimit is taken over all polyradii $\underline{r}=(r_{1},..,r_{N})$ with $1 < r_{i}$ and $$k\{\underline{t}/ \underline{r}\}=\coprod\nolimits_{\underline{n} \in \mathbb{N}^{N}}^{\leq 1} k_{\underline{r}^{\underline{n}}} \cdot t^{\underline{n}}=\left\lbrace \ \sum a_{\underline{n}} \underline{t}^{\underline{n}} \ \middle| \ \|a_{\underline{n}}\| \underline{r}^{\underline{n}} \rightarrow 0 \ \right\rbrace.$$ The algebra structure given as in Definition \[FunctionsOnOpenDisk\] on each $k\{\underline{t}/ \underline{r}\}$ makes $k\{\underline{t}\}^{\dagger}$ an IndBanach algebra. Furthermore, the maps $k\{\underline{t}/ \underline{r}^{2}\} \rightarrow k\{\underline{t}/\underline{r}\} \hat{\otimes} k\{\underline{t}/ \underline{r}\}$, $\underline{t}^{\underline{n}} \mapsto \underline{t}^{\underline{n}} \otimes \underline{t}^{\underline{n}}$, and $k\{\underline{t}/\underline{r}\} \rightarrow k$, $\underline{t}^{\underline{n}} \mapsto 1$, induce an IndBanach bialgebra structure on $k\{\underline{t}\}^{\dagger}$. Likewise we define the IndBanach bialgebra $k\{\underline{t}/0\}^{\dagger}:=\text{"colim"}_{\underline{r} > 0} k\{\underline{t}/\underline{r}\}$.
IndBanach spaces graded by $k\{\underline{t}\}^{\dagger}$ are of the form $$M=\text{"colim"}_{\substack{\underline{r} > 1 \\ i \in I}}\coprod\nolimits_{\underline{n} \in \mathbb{N}^{N}}^{\leq 1} M(\underline{n},i)_{\underline{r}^{\underline{n}}}$$ for $\text{"colim"}_{i \in I} (M(\underline{n},i))_{\underline{n} \in \mathbb{N}^{N}}$ in $\text{IndBan}_{k}^{\mathbb{N}^{N},\text{bd}}$. The space of morphisms that respect the grading is $$\underline{\text{Hom}}_{k\{\underline{t}\}^{\dagger}}(M,M')= \text{colim}_{j \in J} \text{lim}_{i \in I}\text{lim}_{\underline{r}<1}\prod\nolimits_{\underline{n} \in \mathbb{N}^{n}}^{\leq 1} \text{Hom}(M(\underline{n},i),M'(\underline{n},j))_{\underline{r}^{\underline{n}}},$$ for $$M=\text{"colim"}_{\underline{r} > 1, i \in I}\coprod\nolimits_{\underline{n} \in \mathbb{N}^{N}}^{\leq 1} M(\underline{n},i)_{\underline{r}^{n}}$$ and $$M'=\text{"colim"}_{\underline{r} > 1, j \in J}\coprod\nolimits_{\underline{n} \in \mathbb{N}^{N}}^{\leq 1} M'(\underline{n},j)_{\underline{r}^{n}}.$$ Similarly, IndBanach spaces graded by $k\{\underline{t}/0\}^{\dagger}$ are of the form $$M=\text{"colim"}_{\substack{\underline{r} > 0 \\ i \in I}}\coprod\nolimits_{\underline{n} \in \mathbb{N}^{N}}^{\leq 1} M(\underline{n},i)_{\underline{r}^{\underline{n}}}$$ for $\text{"colim"}_{i \in I} (M(\underline{n},i))_{\underline{n} \in \mathbb{N}^{N}}$ in $\text{IndBan}_{k}^{\mathbb{N}^{N},\text{bd}}$. The space of morphisms that respect the grading is $$\underline{\text{Hom}}_{k\{\underline{t}/0\}^{\dagger}}(M,M')= \text{colim}_{j \in J} \text{lim}_{i \in I}\text{lim}_{\underline{r}>0}\prod\nolimits_{\underline{n} \in \mathbb{N}^{n}}^{\leq 1} \text{Hom}(M(\underline{n},i),M'(\underline{n},j))_{\underline{r}^{\underline{n}}},$$ for $$M=\text{"colim"}_{\underline{r} > 0, i \in I}\coprod\nolimits_{\underline{n} \in \mathbb{N}^{N}}^{\leq 1} M(\underline{n},i)_{\underline{r}^{n}}$$ and $$M'=\text{"colim"}_{\underline{r} > 0, j \in J}\coprod\nolimits_{\underline{n} \in \mathbb{N}^{N}}^{\leq 1} M'(\underline{n},j)_{\underline{r}^{n}}.$$
This is Proposition 4.8 and Proposition 4.10 of [@TRTfIBS].
We say that an IndBanach space is *analytically $\mathbb{N}^{N}$-graded*, or just *analytically graded* if $N$ is implicit, if it is graded over $k\{\underline{t}\}$. Likewise, we say that an IndBanach space is *dagger-1 $\mathbb{N}^{N}$-graded*, or just *dagger-1 graded*, (respectively *dagger-0 $\mathbb{N}^{N}$-graded*, or *dagger-0 graded*) if it is graded over $k\{\underline{t}\}^{\dagger}$ (respectively over $k\{\underline{t}/0\}^{\dagger}$). We will occasionally just use the term *dagger graded* when there is no ambiguity.
Assuming [**(NA)**]{}, for each $0 < r$ and each pre-braided Banach space $(V,c)$ with $\|c\| \leq 1$ the Banach tensor algebra $T_{r}^{c}(V)$ is naturally an analytically graded pre-braided Hopf algebra. For any $0 < \rho$, $T_{\rho}^{c}(V)^{\dagger}:=\text{"colim"}_{r > \rho} T_{r}^{c}(V)$ is naturally a dagger-1 graded pre-braided Hopf algebra, and $T_{0}^{c}(V)^{\dagger}:=\text{"colim"}_{r > 0} T_{r}^{c}(V)$ is a dagger-0 graded pre-braided Hopf algebra. Likewise, assuming [**(A)**]{}, for each pre-braided Banach space $(V,c)$ with $\|c\| \leq 1$ the dagger tensor algebra $T_{0}^{c}(V)^{\dagger}$ is naturally a dagger-0 graded pre-braided Hopf algebra.
We define $T_{r}(V) \rightarrow k\{\frac{t}{s}\} \hat{\otimes} T_{r'}(V)$ by $x \mapsto t^{n} \otimes x$ for $x \in V^{\hat{\otimes} n}$ whenever $r \geq r's$. For $r=r'$, $s=1$, this gives $T_{r}(V)$ an analytic grading for which it is an analytically graded braided Hopf algebra. For fixed $\rho >0$ and each $r>\rho$ there exists $s>1$ such that $r>s\rho$, so we may define a map $T_{\rho}(V)^{\dagger} \rightarrow k\{t\}^{\dagger} \hat{\otimes} T_{\rho}(V)^{\dagger}$. Likewise we can define a map $T_{0}(V)^{\dagger} \rightarrow k\{t/0\}^{\dagger} \hat{\otimes} T_{0}(V)^{\dagger}$. These give $T_{\rho}(V)^{\dagger}$ and $T_{0}(V)^{\dagger}$ dagger-1 and dagger-0 gradings respectively. By construction all of the structure maps for the pre-braided Hopf algebras $T_{r}^{c}(V)$, $T_{\rho}^{c}(V)^{\dagger}$ and $T_{0}^{c}(V)^{\dagger}$ are graded.
Let $C$ be an IndBanach coalgebra. A *generalised element* of $C$ is a morphism $\lambda:k \rightarrow C$. We say that a generalised element is *grouplike* if it is a coalgebra homomorphism. Let $G(C)$ denote the set of grouplike generalised elements of $C$.
Let $C$ be an IndBanach coalgebra. Then $G(C)$ forms a monoid under the composition law where $\lambda \ast \lambda'$ is the composition $k \cong k \hat{\otimes} k \overset{\lambda \otimes \lambda'}{\longrightarrow} C \hat{\otimes} C \rightarrow C$. If $C$ is an IndBanach Hopf algebra then $G(C)$ is a group.
Note that this is just the restriction of the convolution product on $\text{Hom}(k,C)$. As with Hopf algebras over vector spaces, it is clear that the inverse to $\lambda$ is the composition $k \overset{\lambda}{\longrightarrow} C \overset{S}{\longrightarrow} C$.
Note that $G(k\{\underline{t}\}) \cong G(k\{\underline{t}\}^{\dagger}) \cong G(k\{\underline{t}/0\}^{\dagger}) \cong \mathbb{N}^{N}$ as monoids.
Let $C$ be an IndBanach coalgebra and $V$ be an IndBanach space graded over $C$. Given $\lambda\in G(C)$ we define $V(\lambda)=\text{eq}(V \rightrightarrows C \hat{\otimes} V)$ as the equaliser of the given coaction of $C$ on $V$ and the map $$V \cong k \hat{\otimes} V \overset{\lambda \otimes \text{Id}_{V}}{\longrightarrow} C \hat{\otimes} V.$$ We think of this as the degree $\lambda$ graded piece. If $\eta_{C}$ is the unit of $C$ then we use the notation $V(0):=V(\eta_{C})$.
\[TensorOfBraidedPieces\] Let $C$ be an IndBanach coalgebra and $(V,c)$ a $C$-graded, pre-braided IndBanach space and let $\lambda, \lambda' \in G(C)$. Assume further that $V$ and $V(\lambda')$ are flat as IndBanach spaces. Then $(V \hat{\otimes} V)(\lambda \otimes \lambda') = V(\lambda) \hat{\otimes} V(\lambda')$ where $\lambda \otimes \lambda'$ is the morphism $k \cong k \hat{\otimes} k \xrightarrow{\lambda \otimes \lambda'} C \otimes C$.
The map $V(\lambda) \hat{\otimes} V(\lambda') \rightarrow V \hat{\otimes} V$ induces a map $$V(\lambda) \hat{\otimes} V(\lambda') \rightarrow (V \hat{\otimes} V)(\lambda \otimes \lambda').$$ Suppose we are given a morphism $f: W \rightarrow V \hat{\otimes} V$ such that the compositions $$W \xrightarrow{f} V \hat{\otimes} V \cong k \hat{\otimes} k \hat{\otimes} V \hat{\otimes} V \xrightarrow{\lambda \otimes \lambda' \otimes \text{Id} \otimes \text{Id}} C \hat{\otimes} C \hat{\otimes} V \hat{\otimes} V$$ and $$W \xrightarrow{f} V \hat{\otimes} V \xrightarrow{\Delta_{V} \otimes \Delta_{V}} C \hat{\otimes} V \hat{\otimes} C \hat{\otimes} V \xrightarrow{\text{Id} \otimes \tau \otimes \text{Id}} C \hat{\otimes} C \hat{\otimes} V \hat{\otimes} V$$ agree. Postcomposing both with $\varepsilon \otimes \text{Id}\otimes \text{Id}\otimes \text{Id}$ we see that the compositions $$W \xrightarrow{f} V \hat{\otimes} V \cong V \hat{\otimes} k \hat{\otimes} V \xrightarrow{\text{Id} \otimes \lambda' \otimes \text{Id}} V \hat{\otimes} C \hat{\otimes} V$$ and $$W \xrightarrow{f} V \hat{\otimes} V \xrightarrow{\text{Id} \otimes \Delta_{V}} V \hat{\otimes} C \hat{\otimes} V$$ agree, so we obtain a map from $W$ to the equaliser of $\text{Id} \otimes \Delta_{V}$ and $\text{Id} \otimes \lambda' \otimes \text{Id}$. Since $V$ is flat, this equaliser is $V \hat{\otimes} V(\lambda')$. Thus we have a unique map $f':W \rightarrow V \hat{\otimes}V(\lambda')$ such that the compositions $$W \xrightarrow{f'} V \hat{\otimes} V(\lambda') \cong k \hat{\otimes} V \hat{\otimes} V(\lambda') \xrightarrow{\lambda \otimes \text{Id} \otimes \text{Id}} C \hat{\otimes} V \hat{\otimes} V(\lambda')$$ and $$W \xrightarrow{f'} V \hat{\otimes} V(\lambda') \xrightarrow{\Delta_{V} \otimes \text{Id}} C \hat{\otimes} V \hat{\otimes} V(\lambda')$$ agree. Again, since $V(\lambda')$ is assumed to be flat, the equaliser of $\lambda \otimes \text{Id} \otimes \text{Id}$ and $\Delta_{V} \otimes \text{Id}$ is $V(\lambda) \hat{\otimes} V(\lambda')$ and we obtain a unique map $f'':W \rightarrow V(\lambda) \hat{\otimes} V(\lambda')$. This exhibits $V(\lambda) \hat{\otimes} V(\lambda')$ as the equaliser $(V \hat{\otimes} V)(\lambda \otimes \lambda')$ as required.
\[RestrictedBraiding\] Let $C$ be an IndBanach coalgebra and $(V,c)$ a $C$-graded, (pre-) braided IndBanach space and let $\lambda\in G(C)$. Assume further that $V$ and $V(\lambda)$ are flat as an IndBanach space. Then $c$ restricts to a (pre-) braiding of $C(\lambda)$.
The braiding $c$ restricts to a morphism $(V \hat{\otimes} V)(\lambda \otimes \lambda) \rightarrow (V \hat{\otimes} V){(\lambda \otimes \lambda)}$. By Lemma \[TensorOfBraidedPieces\], this gives a braiding $V(\lambda) \hat{\otimes} V(\lambda) \rightarrow V(\lambda) \hat{\otimes} V(\lambda)$.
Analytic and Dagger Nichols algebras
------------------------------------
Theorem 4.3 of [@PHA] shows that the positive and negative parts of quantum enveloping algebras arise as Nichols algebras in the category of vector spaces. In this section we introduce analytic and dagger analogues of Nichols algebras that will allow us to construct the positive and negative parts of analytic quantum groups in Sections \[NAQGSection\] and \[AQGSection\].
Given an IndBanach Hopf algebra $H$, we denote by $P(H)$ the equaliser of the two maps $H \rightarrow H \hat{\otimes} H$, given by the comultiplication $\Delta$ on $H$ and the sum of the maps $$H \cong k \hat{\otimes} H \xrightarrow{\eta \otimes \text{Id}_{H}} H \hat{\otimes} H \text{ and } H \cong H \hat{\otimes} k \xrightarrow{\text{Id}_{H} \otimes \eta} H \hat{\otimes} H,$$ the *primitive subspace* of $H$.
\[GeneratedBy\] Given an IndBanach algebra $A$ and an IndBanach space $V$ equipped with a strict monomorphism $V \hookrightarrow A$, we say that $A$ *is generated by* $V$ if, for any diagram
\(A) [$V$]{}; (B) \[right=1cm of A\] [$A$]{}; (C) \[below=0.5cm of B\] [$A'$]{}; (A) to node (B); (A) to node (C); (C) to node \[swap\][$f$]{} (B);
where $A'$ is an IndBanach algebra and $f$ is a morphism of algebras, $f$ is an epimorphism.
Given an IndBanach algebra $A$ and an IndBanach space $V$ equipped with a strict monomorphism $V \hookrightarrow A$, $A$ is generated by $V$ if and only if the induced map $\coprod_{n \geq 0}V^{\hat{\otimes}n} \rightarrow A$ is an epimorphism.
If $V$ generates $A$ then the induced algebra map $\coprod_{n \geq 0}V^{\hat{\otimes}n} \rightarrow A$ is epic by assumption. Conversely, suppose that we have an epimorphism $\coprod_{n \geq 0}V^{\hat{\otimes}n} \rightarrow A$ and a diagram
\(A) [$V$]{}; (B) \[right=1cm of A\] [$A$]{}; (C) \[below=0.5cm of B\] [$A'$]{}; (A) to node (B); (A) to node (C); (C) to node \[swap\][$f$]{} (B);
as in Definition \[GeneratedBy\]. Then for each $n \geq 0$ we obtain a commutative diagram
\(A) [$V^{\hat{\otimes} n}$]{}; (B) \[right=1cm of A\] [$A^{\hat{\otimes} n}$]{}; (C) \[below=0.5cm of B\] [$(A')^{\hat{\otimes} n}$]{}; (D) \[right=0.7cm of B\] [$A$]{}; (E) \[below=0.7cm of D\] [$A'$]{}; (A) to node (B); (A) to node (C); (C) to node \[swap\][$f^{\hat{\otimes} n}$]{} (B); (E) to node \[swap\][$f$]{} (D); (B) to node (D); (C) to node (E);
which induces a diagram
\(A) [$\coprod_{n \geq 0}V^{\hat{\otimes}n}$]{}; (B) \[right=1cm of A\] [$A$]{}; (C) \[below=0.5cm of B\] [$A'$]{}; (A) to node (B); (A) to node (C); (C) to node \[swap\][$f$]{} (B);
which ensures that $f$ is an epimorphism since the map $\coprod_{n \geq 0}V^{\hat{\otimes}n} \rightarrow A$ is epic.
Let $V$ be a flat IndBanach space with pre-braiding $c$. Fix an IndBanach bialgebra $C$ and a grouplike generalised element $\lambda:k \rightarrow C$. Then a flat braided graded IndBanach Hopf algebra $R$, graded over $C$, is called an *IndBanach Nichols algebra* of $V$ if $R(0) \cong k$, $P(R) = R(\lambda) \cong V$ as a braided IndBanach space, and $R$ is generated by $R(\lambda)$. If $C=k\{t\}$, $\lambda:1 \mapsto t$, we say that $R$ is an *analytic Nichols algebra*, and likewise if $C=k\{t\}^{\dagger}$ (respectively $C=k\{t/0\}^{\dagger}$), $\lambda:1 \mapsto t$, we say that $R$ is a *dagger-1 Nichols algebra* (respectively a *dagger-0 Nichols algebra*).
We require flatness in the above definition so that the braiding on $R$ automatically restricts to $R(\lambda)$ by Lemma \[RestrictedBraiding\]. In the [**(NA)**]{} case flatness is automatic by Lemma 3.49 of [@SDiBAG].
We will prove existence of these Nichols algebras in Sections \[NANicholsAlegbraSection\] and \[ANicholsAlegbraSection\], following a discussion of how to form quantum groups using Majid’s double-bosonisation construction.
Double-bosonisation {#Analytic Bosonisation}
===================
In [@DBoBG], Majid introduces a construction, *double-bosonisation*, which he uses to reconstruct Lusztig’s form of the quantum enveloping algebra $U_{q}(\mathfrak{g})$. We present here an adaptation of this construction to the setting of IndBanach spaces. For more on these ideas, see [@AaHAiBC], a brief review of which is given in [@DBoBG].
Let $H$ and $A$ be IndBanach Hopf algebras. A duality pairing is a bilinear form $\langle -,- \rangle:H \hat{\otimes} A \rightarrow k$ such that the following diagrams commute:
\(A) [$H \hat{\otimes} H \hat{\otimes} A$]{}; (A’) \[below=0.75cm of A\] [$H \hat{\otimes} H \hat{\otimes} A$]{}; (B) \[below=2cm of A\] [$H \hat{\otimes} A$]{}; (C) \[right=3cm of A\] [$H \hat{\otimes} H \hat{\otimes} A \hat{\otimes} A$]{}; (D) \[below=2.05cm of C\] [$k$]{}; (E) \[below=0.75cm of C\] [$H \hat{\otimes} A$]{}; (A’) to node \[swap\][$\mu_{H} \otimes \text{Id}$]{} (B); (A) to node \[swap\][$\tau \otimes \text{Id}$]{} (A’); (C) to node \[swap\][$\text{Id} \otimes \langle - , - \rangle \otimes \text{Id}$]{} (E); (A) to node [$\text{Id} \otimes \text{Id} \otimes \Delta_{A}$]{} (C); (E) to node \[swap\][$\langle - , - \rangle$]{} (D); (B) to node [$\langle - , - \rangle$]{} (D);
\(A) [$H$]{}; (B) \[right=1cm of A\] [$H \hat{\otimes} A$]{}; (C) \[below=2cm of B\] [$k$]{}; (A) to node [$\eta_{A}$]{} (B); (A) to node \[swap\][$\varepsilon_{H}$]{} (C); (B) to node [$\langle -,- \rangle$]{} (C);
\(A) [$H \hat{\otimes} A \hat{\otimes} A$]{}; (A’) \[below=0.75cm of A\] [$H \hat{\otimes} A \hat{\otimes} A$]{}; (B) \[below=2cm of A\] [$H \hat{\otimes} A$]{}; (C) \[right=3cm of A\] [$H \hat{\otimes} H \hat{\otimes} A \hat{\otimes} A$]{}; (D) \[below=2.05cm of C\] [$k$]{}; (E) \[below=0.75cm of C\] [$H \hat{\otimes} A$]{}; (A’) to node \[swap\][$\text{Id} \otimes \mu_{A}$]{} (B); (A) to node \[swap\][$\text{Id} \otimes \tau$]{} (A’); (C) to node \[swap\][$\text{Id} \otimes \langle - , - \rangle \otimes \text{Id}$]{} (E); (A) to node [$\Delta_{H} \otimes \text{Id} \otimes \text{Id}$]{} (C); (E) to node \[swap\][$\langle - , - \rangle$]{} (D); (B) to node [$\langle - , - \rangle$]{} (D);
\(A) [$A$]{}; (B) \[right=1cm of A\] [$H \hat{\otimes} A$]{}; (C) \[below=2cm of B\] [$k$]{}; (A) to node [$\eta_{H}$]{} (B); (A) to node \[swap\][$\varepsilon_{A}$]{} (C); (B) to node [$\langle -,- \rangle$]{} (C);
\(A) [$H \hat{\otimes} A$]{}; (B) \[right=2cm of A\] [$H \hat{\otimes} A$]{}; (C) \[below=0.6cm of A\] [$H \hat{\otimes} A$]{}; (D) \[below=0.65cm of B\] [$k$]{}; (A) to node [$\text{Id} \otimes S_{A}$]{} (B); (B) to node [$\langle -,- \rangle$]{} (D); (A) to node \[swap\][$S_{H} \otimes \text{Id}$]{} (C); (C) to node \[swap\][$\langle -,- \rangle$]{} (D);
If $H$ and $A$ have respective right and left actions $\zeta_{H}$ and $\zeta_{A}$ of an algebra $\mathcal{A}$ then we say that $\langle-,- \rangle$ is $\mathcal{A}$-equivariant if the diagram
\(A) [$H \hat{\otimes} \mathcal{A} \hat{\otimes} A$]{}; (B) \[right=2cm of A\] [$H \hat{\otimes} A$]{}; (C) \[below=1cm of A\] [$H \hat{\otimes} A$]{}; (D) \[below=1cm of B\] [$k$]{}; (A) to node [$\zeta_{H}\otimes \text{Id}$]{} (B); (B) to node [$\langle -,- \rangle$]{} (D); (A) to node \[swap\][$\text{Id} \otimes \zeta_{A}$]{} (C); (C) to node \[swap\][$\langle -,- \rangle$]{} (D);
commutes. Likewise, if $H$ and $A$ have respective left and right coactions $\zeta_{H}$ and $\zeta_{A}$ of a coalgebra $C$ then we say that $\langle-,- \rangle$ is $C$-equivariant if the diagram
\(A) [$H \hat{\otimes} A$]{}; (B) \[right=2cm of A\] [$C \hat{\otimes} H \hat{\otimes} A$]{}; (C) \[below=1cm of A\] [$H \hat{\otimes} A \hat{\otimes} C$]{}; (D) \[below=1cm of B\] [$C$]{}; (A) to node [$\zeta_{H}\otimes \text{Id}$]{} (B); (B) to node [$\text{Id} \otimes \langle -,- \rangle$]{} (D); (A) to node \[swap\][$\text{Id} \otimes \zeta_{A}$]{} (C); (C) to node \[swap\][$\langle -,- \rangle \otimes \text{Id}$]{} (D);
commutes.
Let $H$ and $A$ be IndBanach Hopf algebras with a duality pairing $\langle -,- \rangle:H \hat{\otimes} A \rightarrow k$. Suppose we have a pair of convolution invertible maps $\mathscr{R}$ and $\overline{\mathscr{R}}$ in $\text{Hom}(A,H)$ that are both algebra homomorphisms and anti-coalgebra homomorphisms, such that the following diagrams commute:
\(A) [$A \hat{\otimes} A$]{}; (B) \[below=1.25cm of A\] [$H \hat{\otimes} A$]{}; (C) \[right=3cm of A\] [$A \hat{\otimes} H$]{}; (D) \[below=1.3cm of C\] [$k$]{}; (E) \[below=0.35cm of C\] [$H \hat{\otimes} A$]{}; (A) to node \[swap\][$\overline{\mathscr{R}} \otimes \text{Id}$]{} (B); (C) to node \[swap\][$\tau$]{} (E); (A) to node [$\text{Id} \otimes \mathscr{R}^{-1}$]{} (C); (E) to node (D); (B) to node [$\langle - , - \rangle$]{} (D);
\
\(A) [$H \hat{\otimes} A$]{}; (B) \[below=3.85cm of A\] [$H \hat{\otimes} H \hat{\otimes} A$]{}; (C) \[right=5cm of A\] [$H \hat{\otimes} H \hat{\otimes} A \hat{\otimes} A \hat{\otimes} A$]{}; (D) \[below=3.9cm of C\] [$H$]{}; (E) \[below=0.5cm of C\] [$H \hat{\otimes} A \hat{\otimes} H \hat{\otimes} A \hat{\otimes} A$]{}; (F) \[below=0.5cm of E\] [$A \hat{\otimes} H \hat{\otimes} A \hat{\otimes} H \hat{\otimes} A$]{}; (G) \[below=0.5cm of F\] [$H \hat{\otimes} H \hat{\otimes} H$]{}; (A) to node \[swap\][$\Delta_{H} \otimes \text{Id}$]{} (B); (C) to node \[swap\][$\text{Id} \otimes \tau \otimes \text{Id}\otimes \text{Id}$]{} (E); (A) to node [$\Delta_{H} \otimes ((\text{Id} \otimes \Delta_{A})\circ\Delta_{A})$]{} (C); (E) to node \[swap\][$\tau \otimes \tau \otimes \text{Id}$]{} (F); (F) to node \[swap\][$\mathscr{R} \otimes \langle -,- \rangle \otimes \text{Id} \otimes \mathscr{R}^{-1}$]{} (G); (G) to node \[swap\][$\mu_{H} \circ(\mu_{H} \otimes \text{Id})$]{} (D); (B) to node \[swap\][$\text{Id} \otimes \langle -,- \rangle$]{} (D);
\
\(A) [$H \hat{\otimes} A$]{}; (B) \[below=2.95cm of A\] [$H \hat{\otimes} H \hat{\otimes} A$]{}; (C) \[right=5cm of A\] [$H \hat{\otimes} H \hat{\otimes} A \hat{\otimes} A \hat{\otimes} A$]{}; (D) \[below=3cm of C\] [$H$]{}; (E) \[below=1.15cm of A\] [$H \hat{\otimes} H \hat{\otimes} A$]{}; (F) \[below=0.5cm of C\] [$A \hat{\otimes} H \hat{\otimes} H \hat{\otimes} A \hat{\otimes} A$]{}; (G) \[below=0.6cm of F\] [$H \hat{\otimes} H \hat{\otimes} H$]{}; (A) to node \[swap\][$\Delta_{H} \otimes \text{Id}$]{} (E); (E) to node \[swap\][$\tau \otimes \text{Id}$]{} (B); (A) to node [$\Delta_{H} \otimes ((\text{Id} \otimes \Delta_{A})\circ\Delta_{A})$]{} (C); (C) to node \[swap\][$((\tau \otimes \text{Id}) \circ (\text{Id} \otimes \tau)) \otimes \text{Id} \otimes \text{Id}$]{} (F); (F) to node \[swap\][$\overline{\mathscr{R}} \otimes \text{Id} \otimes \langle -,- \rangle \otimes \overline{\mathscr{R}}^{-1}$]{} (G); (G) to node \[swap\][$\mu_{H} \circ(\mu_{H} \otimes \text{Id})$]{} (D); (B) to node \[swap\][$\text{Id} \otimes \langle -,- \rangle$]{} (D);
In this case we call $H$ and $A$ a *weakly quasi-triangular dual pair*.
For an IndBanach Hopf algebra $H$, let us denote by $H\text{-Mod}$ and $\text{Mod-}H$ the categories of left and right $H$-modules respectively. Given a dual pair of IndBanach Hopf algebras, $H$ and $A$, let us denote by $A\text{-Comod}$ and $\text{Comod-}A$ the categories of left and right $A$-comodules respectively. There are faithful functors $A\text{-Comod} \rightarrow \text{Mod-}H$ and $\text{Comod-}A \rightarrow H\text{-Mod}$ induced by the duality pairing.
\[WeaklyQTPairComodulesBraided\] For a weakly quasi-triangular dual pair of IndBanach Hopf algebras, $H$ and $A$, $A\text{-Comod}$ and $\text{Comod-}A$ are braided monoidal, with braidings given by the compositions $$M \hat{\otimes} M' \longrightarrow A \hat{\otimes} M \hat{\otimes} A \hat{\otimes} M' \xrightarrow{\mathscr{R} \otimes \text{Id} \otimes \text{Id} \otimes \text{Id}} H \hat{\otimes} M \hat{\otimes} A \hat{\otimes} M' \longrightarrow M \hat{\otimes} M' \xrightarrow{\tau} M' \hat{\otimes} M,$$ where the third map is $(\langle -,- \rangle \otimes \text{Id}) \circ (\text{Id} \otimes \tau \otimes \text{Id})$, and $$N \hat{\otimes} N' \xrightarrow{\tau} N' \otimes N \longrightarrow N' \hat{\otimes} A \hat{\otimes} N \hat{\otimes} A \xrightarrow{\text{Id} \otimes \mathscr{R} \otimes \text{Id} \otimes \text{Id}} N' \hat{\otimes} H \hat{\otimes} N \hat{\otimes} A \longrightarrow N' \hat{\otimes} N,$$ where the last map is $(\text{Id} \otimes \langle -,- \rangle) \circ (\text{Id} \otimes \tau \otimes \text{Id})$, for left $A$-comodules $M$ and $M'$ and right $A$-comodules $N$ and $N'$.
This follows from Theorem 1.16 of [@AaHAiBC], using the remark from the preliminary section of [@DBoBG] that $A$ is dual quasi-triangular under the composition $$A \hat{\otimes} A \xrightarrow{\mathscr{R} \otimes \text{Id}} H \hat{\otimes} A \xrightarrow{\langle-,-\rangle} k.$$
For a weakly quasi-triangular dual pair of IndBanach Hopf algebra, $H$ and $A$, and an algebra $B$ in $\text{Comod-}A$ there is an algebra structure on $B \hat{\otimes} H$ with multiplication defined by $$\begin{array}{rcl}
B \hat{\otimes} H \hat{\otimes} B \hat{\otimes} H & \overset{\text{Id} \otimes \Delta_{H} \otimes \text{Id} \otimes \text{Id}}{\xrightarrow{\hspace*{2cm}}} & B \hat{\otimes} H \hat{\otimes} H \hat{\otimes} B \hat{\otimes} H \\
& \overset{\text{Id} \otimes \text{Id} \otimes \tau \otimes \text{Id}}{\xrightarrow{\hspace*{2cm}}} & B \hat{\otimes} H \hat{\otimes} B \hat{\otimes} H \hat{\otimes} H \\
& \overset{\text{Id} \otimes \zeta_{B} \otimes \text{Id} \otimes \text{Id}}{\xrightarrow{\hspace*{2cm}}} & B \hat{\otimes} B \hat{\otimes} H \hat{\otimes} H \\
& \overset{\mu_{M} \otimes \mu_{H}}{\xrightarrow{\hspace*{2cm}}} & B \hat{\otimes} H.
\end{array}$$ Here, $\zeta_{B}$ is the left action of $H$ on $B$. Furthermore, if $B$ is a braided IndBanach Hopf algebra then we can give $B \hat{\otimes} H$ a braided IndBanach Hopf algebra structure with comultiplication defined by $$\begin{array}{rcccl}
B \hat{\otimes} H & \overset{\Delta_{B} \otimes \Delta_{H}}{\xrightarrow{\hspace*{1.3cm}}} & B \hat{\otimes} B \hat{\otimes} H \hat{\otimes} H
& \overset{\text{Id} \otimes \Psi_{B,H} \otimes \text{Id}}{\xrightarrow{\hspace*{2cm}}} & B \hat{\otimes} H \hat{\otimes} B \hat{\otimes} H
\end{array}$$ where $\Psi$ is the braiding on $\text{Comod-}A$. Likewise, for $C$ in $\text{Mod-H}$ there is a (Hopf) algebra structure on $H \hat{\otimes} C$ defined analogously.
This follows from Theorem 2.1 of [@CPbBGaB].
We denote by $B \rtimes H$ the IndBanach (Hopf) algebra on $B \hat{\otimes} H$ as described above, and likewise we denote by $H \ltimes C$ the (Hopf) algebra on $H \hat{\otimes} C$. These are the *bosonisations* of $B$ and $H$ or $H$ and $C$ respectively.
\[OverlineHopfAlgebras\] Let $H$ and $A$ be a weakly quasi-triangular dual pair of IndBanach Hopf algebras, with maps $\mathscr{R}_{H,A}$, $\overline{\mathscr{R}}_{H,A}$ giving the weakly quasi-triangular structure. Let $C$ be a braided IndBanach Hopf algebra in $\text{Comod-}A$ with invertible antipode $S_{C}$. Then there is a weakly quasi-triangular dual pair $\overline{H}$ and $\overline{A}$ with the same Hopf algebra structures as $H$ and $A$ but with weakly quasi-triangular structure given by $\mathscr{R}_{\overline{H},\overline{A}}=\overline{\mathscr{R}}^{-1}_{{H},{A}}$ and $\overline{\mathscr{R}}_{\overline{H},\overline{A}}=\mathscr{R}^{-1}_{{H},{A}}$. Furthermore, there is a braided IndBanach Hopf algebra $\overline{C}$ in $\text{Comod-}\overline{A}$ with the same algebra structure as $C$ but with the opposite comultiplication $\Delta_{\overline{C}}=\Psi^{-1}_{C,C} \circ \Delta_{C}$, where $\Psi$ is the braiding on $\text{Comod-}A$, and antipode $S_{\overline{C}}=S_{C}^{-1}$.
This follows from Lemma 3.1 in [@DBoBG], which was proven in [@AaHAiBC], using Remark 3.9 of [@DBoBG].
\[DoubleBosonisationVariation\] Let $H$ and $A$ be a weakly quasi-triangular dual pair of IndBanach Hopf algebras, let $B$ be a braided IndBanach Hopf algebra in $A\text{-Comod}$ and let $C$ be a braided IndBanach Hopf algebra in $\text{Comod-}A$ with respective induced right and left actions $\zeta_{B}$ and $\zeta_{C}$ of $H$. Suppose further that we have a $H$-equivariant duality pairing between $B$ and $C$, $\langle - , - \rangle: B \hat{\otimes} C \rightarrow k$. Then there is an IndBanach Hopf algebra structure on $C \hat{\otimes} H \hat{\otimes} B$ such that the maps $\overline{C} \rtimes \overline{H} \rightarrow C \hat{\otimes} H \hat{\otimes} B$ and $H \ltimes B \rightarrow C \hat{\otimes} H \hat{\otimes} B$ are morphisms of Hopf algebras, and the multiplication restricts to the composition $$\begin{array}{rcl}
B \hat{\otimes} \overline{C} &\xrightarrow{\hspace*{2.5cm}}& B \hat{\otimes} B \hat{\otimes} B \hat{\otimes} \overline{C} \hat{\otimes} \overline{C} \hat{\otimes} \overline{C}\\
&\overset{\tau_{(1 \, \, 2 \, \, 4)(3 \, \, 6 \, \, 5)}}{\xrightarrow{\hspace*{2.5cm}}}& \overline{C} \hat{\otimes} B \hat{\otimes} \overline{C} \hat{\otimes} B \hat{\otimes} \overline{C} \hat{\otimes} B\\
&\overset{\text{Id} \otimes \text{Id} \otimes \text{Id} \otimes \text{Id} \otimes S_{\overline{C}} \otimes \text{Id}}{\xrightarrow{\hspace*{2.5cm}}}& \overline{C} \hat{\otimes} B \hat{\otimes} \overline{C} \hat{\otimes} B \hat{\otimes} \overline{C} \hat{\otimes} B\\
&\overset{\text{Id} \otimes R_{B,\overline{C}} \otimes R^{-1}_{B,\overline{C}} \otimes \text{Id}}{\xrightarrow{\hspace*{2.5cm}}}& \overline{C} \hat{\otimes} B \hat{\otimes} \overline{C} \hat{\otimes} B \hat{\otimes} \overline{C} \hat{\otimes} B\\
&\overset{\langle -,- \rangle \otimes \text{Id} \otimes \text{Id} \otimes \langle -,- \rangle}{\xrightarrow{\hspace*{2.5cm}}}& \overline{C} \hat{\otimes} B \longrightarrow C \hat{\otimes} H \hat{\otimes} B\\
\end{array}$$ between $B \hookrightarrow C \hat{\otimes} H \hat{\otimes} B$ and $C \hookrightarrow C \hat{\otimes} H \hat{\otimes} B$. Here, the first map is induced by the coproducts $\Delta_{B}$ and $\Delta_{\overline{C}}$, $\tau_{(1 \, \, 2 \, \, 4)(3 \, \, 6 \, \, 5)}$ is a reordering given by the permutation $(1 \, \, 2 \, \, 4)(3 \, \, 6 \, \, 5) \in S_{6}$, and $R_{B,\overline{C}}$ and $R_{B,\overline{C}}^{-1}$ are the respective compositions $$B \hat{\otimes} \overline{C} \rightarrow A \hat{\otimes} B \hat{\otimes} \overline{C} \hat{\otimes} A \overset{\mathscr{R} \otimes \text{Id} \otimes \text{Id} \otimes \overline{\mathscr{R}}}{\xrightarrow{\hspace*{2.5cm}}} H \hat{\otimes} B \hat{\otimes} \overline{C} \hat{\otimes} H \rightarrow B \hat{\otimes} \overline{C}$$ and $$B \hat{\otimes} \overline{C} \rightarrow A \hat{\otimes} B \hat{\otimes} \overline{C} \hat{\otimes} A \overset{\mathscr{R}^{-1} \otimes \text{Id} \otimes \text{Id} \otimes \overline{\mathscr{R}}^{-1}}{\xrightarrow{\hspace*{2.5cm}}} H \hat{\otimes} B \hat{\otimes} \overline{C} \hat{\otimes} H \rightarrow B \hat{\otimes} \overline{C}$$ where the first maps are the coactions of $A$ and the last maps are the induced actions of $H$.
This follows from Theorem 3.2, along with Remark 3.9, of [@DBoBG].
We will denote by $U(C,H,B)$ the Banach Hopf algebra $C \hat{\otimes} H \hat{\otimes} B$ as described in Proposition \[DoubleBosonisationVariation\], the *double bosonisation* of $B$, $H$ and $C$ over $A$.
Let $H$ be an IndBanach Hopf algebra. An *R-matrix* for $H$ is a convolution invertible generalised element of $H \hat{\otimes} H$, $\mathscr{R}:k \rightarrow H \hat{\otimes} H$, such that $(\Delta \otimes \text{Id}) \circ \mathscr{R} = \mathscr{R}_{13} \ast \mathscr{R}_{23}$, $(\text{Id} \otimes \Delta) \circ \mathscr{R} = \mathscr{R}_{13} \ast \mathscr{R}_{12}$, and $\tau \circ \Delta = \mathscr{R}(\Delta)\mathscr{R}^{-1}$. Here we use the notation $\mathscr{R}_{12}$ for the composition $$k \overset{\mathscr{R}}{\longrightarrow} H \hat{\otimes} H \cong H \hat{\otimes} H \hat{\otimes} k \overset{\text{Id} \otimes \text{Id} \otimes \eta_{H}}{\xrightarrow{\hspace*{1.2cm}}} H \hat{\otimes} H \hat{\otimes} H$$ and likewise for $\mathscr{R}_{13}$ and $\mathscr{R}_{23}$, and $\mathscr{R}(\Delta)\mathscr{R}^{-1}$ for the composition $$H \cong k \hat{\otimes} H \hat{\otimes} k \overset{\mathscr{R} \otimes \Delta_{H} \otimes \mathscr{R}^{-1}}{\xrightarrow{\hspace*{1.4cm}}} (H \hat{\otimes} H) \hat{\otimes} (H \hat{\otimes} H) \hat{\otimes} (H \hat{\otimes} H) \rightarrow H \hat{\otimes} H$$ where the last map is the multiplication on $H \hat{\otimes} H$. A *quasi-triangular IndBanach Hopf algebra* is an IndBanach Hopf algebra $H$ equipped with an R-matrix $\mathscr{R}$.
Let $H$ and $A$ be Hopf algebras with a dual pairing $\langle-,-\rangle:H \hat{\otimes} A \rightarrow k$. Suppose further that $H$ is quasi-triangular, with R-matrix $\mathscr{R}':k \rightarrow H \hat{\otimes} H$. Then the maps $$\mathscr{R}:A \cong k \hat{\otimes} A \xrightarrow{\mathscr{R}' \otimes \text{Id}} H \hat{\otimes} H \hat{\otimes} A \xrightarrow{\text{Id} \otimes \tau} H \hat{\otimes} A \hat{\otimes} H \xrightarrow{\langle -,- \rangle \otimes \text{Id}} k \hat{\otimes} H \cong H$$ and $$\overline{\mathscr{R}}:A \cong k \hat{\otimes} A \xrightarrow{(\mathscr{R}')^{-1} \otimes \text{Id}} H \hat{\otimes} H \hat{\otimes} A \xrightarrow{\text{Id} \otimes \langle -,- \rangle} H \hat{\otimes} k \cong H$$ induce a weak quasi-triangular structure on the dual pair $H$ and $A$.
This follows from the remarks in the preliminary section of [@DBoBG].
\[QuasiTriangularBraiding\] For a quasi-triangular IndBanach Hopf algebra $H$, $H\text{-Mod}$ and $\text{Mod-}H$ are braided monoidal. The braiding on $M$ and $M'$ in $H\text{-Mod}$ is given by the composition $$M \hat{\otimes}M' \cong k \hat{\otimes} M \hat{\otimes}M' \overset{\mathscr{R} \otimes \text{Id} \otimes \text{Id}}{\xrightarrow{\hspace*{1.2cm}}} H \hat{\otimes} H\hat{\otimes} M \hat{\otimes}M' \overset{\tau \circ (\zeta_{M} \otimes \zeta_{M'})\circ(\text{Id} \otimes \tau \otimes \text{Id})}{\xrightarrow{\hspace*{3.2cm}}}M' \hat{\otimes}M,$$ whilst the braiding on $N$ and $N'$ in $\text{Mod-}H$ is given by the composition $$N \hat{\otimes}N' \overset{\tau}{\rightarrow} N' \hat{\otimes}N \overset{ \text{Id} \otimes \text{Id} \otimes \mathscr{R}}{\xrightarrow{\hspace*{1.4cm}}} N' \hat{\otimes}N\hat{\otimes}H \hat{\otimes} H \overset{(\zeta_{N'} \otimes \zeta_{N})\circ(\text{Id} \otimes \tau \otimes \text{Id})}{\xrightarrow{\hspace*{2.8cm}}}N' \hat{\otimes}N.$$
This is Theorem 1.10 of [@AaHAiBC].
Non-Archimedean Analytic quantum groups {#NAQGSection}
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We will assume [**(NA)**]{} throughout this section. The [**(A)**]{} case will be covered in the subsequent section.
Constructing Non-Archimedean Nichols algebras {#NANicholsAlegbraSection}
---------------------------------------------
Let $V$ be an analytically $\mathbb{N}$-graded Banach space, $V(n):=V(t^{n})$, $V=\coprod_{n \in \mathbb{N}}^{\leq 1} V(n)$. For $W \subset V$ a subspace, not necessarily closed, let $W(n)=V(n) \cap W$. We say that $W$ is homogeneous if $$\bigoplus\nolimits_{n \in \mathbb{N}}W(n) \subset W \subset \coprod\nolimits_{n \in \mathbb{N}}^{\leq 1} W(n).$$ Note that $W=\coprod_{n \in \mathbb{N}}^{\leq 1} W(n)$ if and only if it is closed and homogeneous.
For $C$ a Banach coalgebra, we say that a closed subspace $I\subset C$ is a coideal if $\Delta(C) \subset \overline{I \hat{\otimes} C + C \hat{\otimes} I}$. Then $C/I$ is again a Banach coalgebra.
\[CoidealsOfTensorAlgebras\] Let $V$ be a Banach space with pre-braiding $c$ of norm at most 1. Suppose we have closed homogeneous coideals $I \subset J \subset T(V)$. If the induced map $T(V)/I \rightarrow T(V)/J$ is injective on $P(T(V)/I)$ then $I=J$.
We proceed similarly to Lemma 5.3.3 of [@HAatAoR]. Let us denote by $R=T(V)/I$, $R'=T(V)/J$, by $R(n)$, $R'(n)$ their respective $n$th graded pieces, and by $R(\leq n)=\bigoplus_{i \leq n}R(i)$ and $R'(\leq n)=\bigoplus_{i \leq n}R'(i)$. Since $I \subset J$ we have a strict graded epimorphism $f:R \rightarrow R'$ which restricts to an isometry $R(1) \rightarrow R'(1)$. Suppose that we know that $f$ restricts also to an isometry $R(\leq n) \rightarrow R'(\leq n)$, and let $x \in R(\leq n+1)$. We know that, by Lemma \[GradedPiecesOfTensorAlgebraCoideals\], $$\begin{array}{rcl}
\Delta(R(n+1)) &\subset& \sum_{i=0}^{n+1} R(i) \hat{\otimes} R(n-i)\\
&\subset& R(n+1) \hat{\otimes} k + k \hat{\otimes} R(n+1) + R(\leq n) \hat{\otimes} R(\leq n),
\end{array}$$ so $\Delta(x)=y \otimes 1 + 1 \otimes y' + z$ for some $y,y' \in R(n+1)$, $z \in R(\leq n) \hat{\otimes} R(\leq n)$. But then $x-y = (\text{Id} \otimes \varepsilon)\Delta(x)-y= \varepsilon(y)\cdot 1 + (\text{Id} \otimes \varepsilon)(z) \in R(\leq n)$ and likewise $x-y' \in R(\leq n)$. So $\Delta(x)=x \otimes 1 + 1 \otimes x + z'$ for some $z' \in {R(\leq n)} \hat{\otimes} {R(\leq n)}$. If $f(x)=0$ then $(f \otimes f)(z')=0$, but, by assumption and by Lemma 3.49 of [@SDiBAG], $f \otimes f$ is injective on $R(\leq n) \hat{\otimes} R(\leq n)$. So $z'=0$ and $x$ is primitive, hence $x=0$. Thus $f$ is an isometry ${R(\leq n)} \rightarrow {R'(\leq n)}$ since the norms on $R(\leq n)$ and $R'(\leq n)$ are the quotient norms from $\coprod_{i \leq n}^{\leq 1} V^{\hat{\otimes} i}$. Taking contracting colimits over $n$ we see that $f$ is isometric. Hence $I=J$.
\[Radius1NicholsAlgebrasExists\] Let $V$ be a Banach space with pre-braiding $c$ of norm at most $1$. Then an analytic Nichols algebra of $V$ exists.
Let $\mathcal{I}(V)$ be the set of all homogeneous ideals of $T(V)$ contained in $\coprod_{n \geq 2}^{\leq 1} V^{\hat{\otimes}n}$ that are also coideals. Let $\mathcal{I}'(V)$ be the subset of ideals $I$ in $\mathcal{I}(V)$ for which $\overline{I} \hat{\otimes} T(V) + T(V) \hat{\otimes} \overline{I}$ is preserved by $c$. Let $I(V)$ and $I'(V)$ be the sums of all ideals in $\mathcal{I}(V)$ and $\mathcal{I}'(V)$ respectively, and let $\overline{I}(V)$ and $\overline{I}'(V)$ be their respective closures. Clearly then $\overline{I}(V)$ is a homogeneous ideal contained in $\coprod_{n \geq 2}^{\leq 1} V^{\hat{\otimes}n}$. Also, $$\Delta(\overline{I}(V)) \subset \overline{\Delta(I(V))} \subset \overline{T(V) \hat{\otimes} \bar{I}(V) + \bar{I}(V) \hat{\otimes} T(V)},$$ so $\overline{I}(V)$ is also a coideal in $T(V)$, and hence is also in $\mathcal{I}(V)$. So $\overline{I}(V)=I(V)$ is closed. Likewise $I'(V)$ is closed. We must check that $$P(T(V)/I(V))=(T(V)/I(V))(1).$$ This follows as in Lemma 5.3.3 of [@PHA] since the closed ideal in $T(V)$ generated by $I(V)$ and $$\left\lbrace x \in \coprod\nolimits_{n \geq 2}^{\leq 1} V^{\hat{\otimes} n} \middle| \Delta(x) \in x \otimes 1 + 1 \otimes x + I(V) \otimes T(V) + T(V) \otimes I(V)\right\rbrace$$ must be in $\mathcal{I}(V)$. Likewise $P(T(V)/I'(V))=(T(V)/I'(V))(1)$. But $I'(V) \subset I(V)$, so by Lemma \[CoidealsOfTensorAlgebras\] we have $I'(V) = I(V)$. Hence $c$ descends to a braiding on $T(V)/I(V)$. We then have that $T(V)/I(V)$ is an analytically graded braided IndBanach Hopf algebra with $(T(V)/I(V))(0) \cong k$ and generated by $(T(V)/I(V))(1) \cong V$.
For a Banach space $V$ with pre-braiding $c$ of norm at most $1$ we will denote by $\mathfrak{B}^{c}(V)$, or $\mathfrak{B}(V)$ when the braiding is implicit, the Banach Nichols algebra defined in the proof of Proposition \[Radius1NicholsAlgebrasExists\]. For $0< r$, let us denote by $\mathfrak{B}_{r}(V)$, or $\mathfrak{B}_{r}^{c}(V)$, the analytically graded pre-braided Banach Hopf algebra $\mathfrak{B}^{c}(V_{r})$. For $0\leq \rho$, let us denote by $\mathfrak{B}_{\rho}(V)^{\dagger}$, or $\mathfrak{B}_{\rho}^{c}(V)^{\dagger}$, the dagger graded pre-braided Banach Hopf algebra $\text{"colim"}_{r > \rho}\mathfrak{B}_{r}^{c}(V)$.
\[DaggerNicholsAlgebrasInFiniteDimensions\] Let $0< r$, $0\leq \rho$ and let $V$ be a finite dimensional Banach space with pre-braiding $c$ of norm at most $1$. Then $\mathfrak{B}_{r}^{c}(V)$ is an analytic Nichols algebra of $V$ and $\mathfrak{B}_{\rho}^{c}(V)^{\dagger}$ is a dagger Nichols algebra of $V$.
The fact that $\mathfrak{B}_{r}^{c}(V)$ is an analytic Nichols algebra follows from Proposition \[Radius1NicholsAlgebrasExists\]. If $R=\mathfrak{B}_{\rho}(V)^{\dagger}$ then it follows from Proposition \[Radius1NicholsAlgebrasExists\] that $R(0)=\text{"colim"}_{r>\rho}k_{r} \cong k$ and $P(R)=R(1)=\text{"colim"}_{r>\rho}V_{r} \cong V$. It remains to check that $R(0)$ generates $R$. This follows since, for each $r>\rho$, the composition $\coprod_{n \geq 0}V^{\hat{\otimes}n} \rightarrow T_{r}(V) \rightarrow \mathfrak{B}_{r}^{c}(V)$ is an epimorphism.
\[NicholsAlgebraUniversalProperty\] Let $V$ be a Banach space with pre-braiding $c$ of norm at most $1$. Let $R$ be an analytically graded pre-braided Banach Hopf algebra with contracting multiplication (*i.e* of norm at most $1$) such that $R(0)\cong k$, $R(1) \cong V$ as pre-braided Banach spaces, and $R$ is generated as an algebra by $R(1)$. Then there is an epimorphism of analytically graded braided Hopf algebras $\mathfrak{B}_{r}^{c}(V) \rightarrow R$ extending $V \overset{\sim}{\longrightarrow} R(1)$ where $r$ is the norm of this isomorphism.
Since $V \overset{\sim}{\longrightarrow} R(0)$ is of norm $r$, the map $V_{r} \overset{\sim}{\longrightarrow} R(0)$ is of norm 1. Since the multiplication is contracting, we obtain contracting maps $V_{r}^{\hat{\otimes} n} \rightarrow R^{\hat{\otimes} n} \rightarrow R$ which induce an algebra homomorphism $T_{r}(V) \rightarrow R$ through which $V \cong R(0) \rightarrow R$ factors. Since $R(0)$ generates $R$, this morphism is epic. Furthermore, by construction of the analytically graded pre-braided bialgebra structure on $T_{r}(V)$ and since $$\Delta(R(1)) \subset R(0)\hat{\otimes} R(1) + R(1) \hat{\otimes} R(0)$$ implies that $R(1)\subset P(R)$, the map $T_{r}(V) \rightarrow R$ is a morphism of bialgebras. Hence the kernel of this map is a closed homogeneous ideal and coideal, so is contained in $I(V_{r})$. Hence we obtain an epimorphism $\mathfrak{B}_{r}^{c}(V)=T(V_{r})/I(V_{r}) \rightarrow R$.
Note that the assignment $(V,c) \mapsto \mathfrak{B}_{r}^{c}(V)$ is functorial if we restrict to morphisms $(V,c) \rightarrow (V',c')$ given by maps $V \rightarrow V'$ of norm at most $1$ that respect the braiding. Hence different norms on finite dimensional Banach spaces and different values of $r$ may give non-isomorphic analytic Nichols algebras. This differs from the algebraic case where Nichols algebras always exist uniquely.
\[BilinearForm\] A *bilinear form* on a braided IndBanach space $(V,c)$ is a morphism $\langle -,- \rangle : V \hat{\otimes} V \rightarrow k$ such that the diagram
\(A) [$V \hat{\otimes} V$]{}; (B) \[right=0.5cm of A\] [$V^{\ast} \hat{\otimes} V^{\ast}$]{}; (C) \[below=0.7cm of A\] [$V \hat{\otimes} V$]{}; (D) \[below=0.7cm of B\] [$V^{\ast} \hat{\otimes} V^{\ast}$]{}; (E) \[right=0.5cm of B\] [$(V \hat{\otimes} V)^{\ast}$]{}; (F) \[below=0.62cm of E\] [$(V \hat{\otimes} V)^{\ast}$]{}; (A) to node (B); (E) to node [$c^{\ast}$]{} (F); (A) to node \[swap\][$c$]{} (C); (C) to node \[swap\] (D); (B) to node (E); (D) to node (F);
commutes for both of the induced maps $V \rightarrow V^{\ast}$. A bilinear form is *symmetric* if $\langle - , - \rangle \circ \tau = \langle - , - \rangle$, and is *non-degenerate* if the induced maps $V \rightarrow V^{\ast}$ are both injective.
\[ExtendBilinearForm\] Let $V$ be a Banach space with pre-braiding $c$ of norm at most 1, and suppose we have a non-degenerate symmetric bilinear form $\langle - , - \rangle: V \hat{\otimes} V \rightarrow k$ of norm $C>0$. Then for each $r,s >0$ with $C \leq rs$ there is a unique extension of this bilinear form to a dual pairing of Hopf algebras $T_{r}^{c}(V) \hat{\otimes} T_{s}^{c}(V) \rightarrow k$. Furthermore, the restriction to $V^{\hat{\otimes} n} \hat{\otimes} V^{\hat{\otimes} m}$ is symmetric if $n=m$ and is zero if $n \neq m$.
This is proved as in [@ItQG]. We construct this extension as follows. Our bilinear form on $V$ induces a continuous map $V_{r} \rightarrow V_{s}^{\ast}$ of norm $\frac{C}{rs}$, whilst the natural projection $T_{s}^{c}(V) \rightarrow V_{s}$ induces a map $V_{s}^{\ast} \rightarrow T_{s}^{c}(V)^{\ast}$ of norm $1$, and composition gives us a map $V_{r} \rightarrow T_{s}^{c}(V)^{\ast}$ of norm at most $\frac{C}{rs} \leq 1$. The coalgebra structure on $T_{s}^{c}(V)$ induces an algebra structure on $T_{s}^{c}(V)^{\ast}$ whose multiplication is contracting, and we get a unique continuous algebra homomorphism $T_{r}^{c}(V) \rightarrow T_{s}^{c}(V)^{\ast}$ extending $V_{r} \rightarrow T_{s}^{c}(V)^{\ast}$. This gives us our desired bilinear form on $T_{r}^{c}(V) \hat{\otimes} T_{s}^{c}(V)$. For this to be a dual pairing we must also check that the diagrams
\(A) [$T_{r}^{c}(V)$]{}; (B) \[right=2cm of A\] [$T_{s}^{c}(V)^{\ast}$]{}; (C) \[below=1cm of A\] [$T_{r}^{c}(V) \hat{\otimes} T_{r}^{c}(V)$]{}; (D) \[below=1cm of B\] [$(T_{s}^{c}(V) \hat{\otimes} T_{s}^{c}(V))^{\ast}$]{}; (A) to node (B); (B) to node [$\mu^{\ast}$]{} (D); (A) to node \[swap\][$\Delta$]{} (C); (C) to node \[swap\] (D);
\(A) [$T_{r}^{c}(V)$]{}; (B) \[right=0.7cm of A\] [$T_{s}^{c}(V)^{\ast}$]{}; (C) \[below=1cm of A\] [$k$]{}; (D) \[below=1cm of B\] [$k^{\ast}$]{}; (A) to node (B); (B) to node [$\eta^{\ast}$]{} (D); (A) to node \[swap\][$\varepsilon$]{} (C); (C) to node [$\sim$]{} (D);
commute. By assumption all of these morphisms are algebra homomorphisms, and so it is enough to check these diagrams commute on $V$, which is trivial. Since the algebra homomorphism $T_{r}^{c}(V) \rightarrow T_{s}^{c}(V)^{\ast}$ is graded we have $\langle V^{\hat{\otimes} n}, V^{\hat{\otimes} m} \rangle=\{0\}$ for $n \neq m$. Since this is a duality pairing, symmetry on $V^{\hat{\otimes} n} \hat{\otimes} V^{\hat{\otimes} n}$ can be reduced to the case where $n=1$ where it is true by assumption.
\[BilinearFormGivesNicholsAlgebra\] Let $V$ be a Banach space with pre-braiding $c$ of norm at most 1, and suppose we have a non-degenerate symmetric bilinear form $\langle - , - \rangle: V \hat{\otimes} V \rightarrow k$ of norm $C$. Then for each $0< r$, let $I_{r}$ be the radical in $T_{r}^{c}(V)$ of the extension of this bilinear form to $T_{r}^{c}(V)\hat{\otimes} T_{s}^{c}(V)$ for some $s>0$ such that $C\leq rs$. Then $I_{r}$ is a closed ideal and coideal of $T_{r}^{c}(V)$, independent of the choice of $s$, and $P(T_{r}^{c}(V)/I_{r})=V$. Hence $T_{r}^{c}(V)/I_{r}$ is an analytic Nichols algebra of $V$.
The fact that $I_{r}$ is a closed homogeneous ideal and coideal of $T_{r}^{c}(V)$ follows from Lemma \[ExtendBilinearForm\]. It is independent of the choice of $s$ since the vector subspace $\bigoplus_{n \geq 0} V^{\otimes n}$ is dense in $T_{s}^{c}(V)$ for all $s$, hence $$I_{r}=\left\lbrace x \in T_{r}^{c}(V) \middle| \langle x,y \rangle = 0 \text{ for all } y \in \bigoplus\nolimits_{n \geq 0} V^{\otimes n}\right\rbrace.$$ Since the bilinear form on $V$ is non-degenerate, $I_{r} \subset \coprod_{n \geq 2}^{\leq 1} V_{r}^{\hat{\otimes}n}$. Clearly the quotient $T_{r}(V)/I_{r}$ is generated by $V$, and so it remains to check that the subspace of primitive elements is just $V$. Given $x \in T_{r}^{c}(V)$ homogeneous of degree $n \geq 2$ (*i.e.* in $V_{r}^{\hat{\otimes}n}$) such that its image in $T_{r}^{c}(V)/I_{r}$ is primitive, we must have that $\langle x,yy' \rangle = \langle 1,y \rangle \langle x,y' \rangle + \langle x,y \rangle \langle 1,y' \rangle =0$ for all $y,y'$ homogeneous of degree at least $1$. It then follows that $x$ must be in the radical $I_{r}$ since $\langle x,z \rangle=0$ for any $z$ homogeneous of degree at most $1$. By the assumption in Definition \[BilinearForm\] the diagram
\(A) [$T_{r}(V) \hat{\otimes} T_{r}(V)$]{}; (B) \[right=0.5cm of A\] [$(T_{s}(V) \hat{\otimes} T_{s}(V))^{\ast}$]{}; (C) \[below=0.5cm of A\] [$T_{r}(V) \hat{\otimes} T_{r}(V)$]{}; (D) \[below=0.5cm of B\] [$(T_{s}(V) \hat{\otimes} T_{s}(V))^{\ast}$]{}; (A) to node (B); (B) to node [$\tilde{c}^{\ast}$]{} (D); (A) to node \[swap\][$\tilde{c}$]{} (C); (C) to node \[swap\] (D);
commutes, and so $I_{r} \hat{\otimes} T_{r}(V) + T_{r}(V) \hat{\otimes} I_{r}$ is preserved by $\tilde{c}$ and the braiding on $T_{r}^{c}(V)$ descends to a braiding of $T_{r}^{c}(V)/I_{r}$. Hence this is an analytic Nichols algebra of $V$.
\[NicholsAlgebrasEquivalentDefinition\] Let $V$ be a Banach space with pre-braiding $c$ of norm at most $1$, and suppose we have a non-degenerate symmetric bilinear form $\langle - , - \rangle: V \hat{\otimes} V \rightarrow k$. Retaining the notation of Proposition \[BilinearFormGivesNicholsAlgebra\], the induced map $\mathfrak{B}_{r}^{c}(V) \rightarrow T_{r}^{c}(V)/I_{r}$ is an isomorphism for each $0< r$. In particular, $I_{r}$ is independent of the choice of bilinear form.
This follows from Lemma \[CoidealsOfTensorAlgebras\], noting that $I_{r} \subset I(V_{r})$.
Let $V$ be a Banach space with pre-braiding $c$ of norm at most $1$, and suppose we have a non-degenerate symmetric bilinear form $\langle - , - \rangle: V \hat{\otimes} V \rightarrow k$. Then for each $0 \leq \rho$, $\mathfrak{B}_{\rho}^{c}(V)^{\dagger}$ is a dagger Nichols algebra of $V$.
This follows as in the proof of Proposition \[DaggerNicholsAlgebrasInFiniteDimensions\].
\[WeakClassicalNicholsAlgebrasDense\] Let $V$ be a Banach space with a pre-braiding $c$ of norm at most 1 that restricts to an algebraic braiding $V \otimes V \rightarrow V \otimes V$ of vector spaces, and let $\langle - , - \rangle: V \hat{\otimes} V \rightarrow k$ be a non-degenerate symmetric bilinear form. Then the algebraic Nichols algebra of $V$, as defined in Definition 2.1 of [@PHA], is dense in the Banach space $\mathfrak{B}_{r}^{c}(V)$ for each $r> 0$.
The algebraic Nichols algebra can be constructed as the quotient of the tensor algebra of vector spaces, $\bigoplus_{n \geq 0} V^{\otimes n}$, by the radical of the restriction of the induced bilinear form on $T_{r}(V) \hat{\otimes} T_{s}(V)$ for some sufficiently large $s>0$. The result then follows since $\bigoplus_{n \geq 0} V^{\otimes n}$ is dense in $T_{r}(V)$ for all $r>0$.
\[ClassicalNicholsAlgebrasDense\] Let $V$ be a Banach space with pre-braiding $c$ of norm at most $1$, and suppose that $R$ is a Banach analytic Nichols algebra of $V$. Then the algebraic Nichols algebra of $V$, as defined in Definition 2.1 of [@PHA], is dense in $R$. Furthermore, if $V$ is finite dimensional, $R(n)$ is isomorphic as a vector space to the $n$th graded piece of the algebraic Nichols algebra.
In the category of vector spaces, there is a unique braided $\mathbb{N}$-graded Hopf algebra structure on $\bigoplus_{n \geq 0} V^{\otimes n}$ for which $V$ is primitive and the braiding restricts to $c$. The inclusion $V \hookrightarrow R$ induces a morphism of braided graded Hopf algebras $\bigoplus_{n \geq 0} V^{\otimes n} \rightarrow R$. The image of this morphism, which we denote by $R_{0}$, is a braided graded Hopf algebra, generated by $V$, that is dense in $R$. Then $V \subset P(R_{0}) \subset P(R) = V$ and so $P(R_{0})=V$. Likewise $R_{0}(0)=k$ and $R_{0}(1)=V$. Thus $R_{0}$ is an algebraic Nichols algebra. Furthermore, each $R_{0}(n)$ must be dense in $R(n)$. If these are finite dimensional then they must be equal.
Constructing non-Archimedean analytic quantum groups {#Constructingnon-Archimedeananalyticquantumgroups}
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The main motivation behind Majid’s work in [@DBoBG], which we summarised in Section \[Analytic Bosonisation\], is the reconstruction of Lusztig’s from of the quantum enveloping algebra. We will use the same technique to construct analytic analogues of these quantum enveloping algebras.\
Throughout the following, we will fix an element $q \in k \setminus \{0\}$ of norm 1 that is not a root of unity. We fix the following root datum of a Lie algebra.
\[KacMoodyRootDatum\] Let $\mathfrak{g}$ be the Lie algebra defined by the data of
- a free $\mathbb{Z}$-module $\Phi$, the *weight lattice*, a submodule $\Psi \subset \Phi$, the *root lattice*, and a free basis $\{\alpha_{i} \mid i \in I\}$ of $\Psi$, the *simple roots*, indexed over some set $I$;
- a symmetric bilinear form $( \cdot , \cdot ) : \Phi \times \Phi \rightarrow \mathbb{Q}$ such that $( \alpha_{i}, \alpha_{i}) \in 2 \mathbb{N}$ and $( \alpha_{i}, \alpha_{j}) \leq 0$ for $i, j \in I, i \neq j$; and
- *simple coroots* $\lambda_{i} \in \Phi^{\ast}=\text{Hom}_{\mathbb{Z}}(\Phi, \mathbb{Z})$ such that $\lambda_{i}(\alpha) = \frac{2(\alpha_{i}, \alpha)}{(\alpha_{i}, \alpha_{i})}$ for $i \in I, \alpha \in \Phi$.
Then $\mathfrak{g}$ is generated over $\mathbb{Q}$ by elements $e_{i}, f_{i}, h_{i}$ for $i \in I$ subject to the relations $$\begin{array}{cccc}
[h_{i},h_{j}] = 0, & [e_{i},f_{i}] = \delta_{ij} h_{i}, &
[h_{i},e_{j}] = \lambda_{i}(\alpha_{j})e_{j}, & [h_{i},f_{j}] = - \lambda_{i}(\alpha_{j})f_{j} \end{array},$$ and for $i \neq j$, $$\begin{array}{cc}
(\text{ad}e_{i})^{1-\lambda_{i}(\alpha_{j})}e_{j} = 0, & (\text{ad}f_{i})^{1-\lambda_{i}(\alpha_{j})}f_{j} = 0, \end{array}$$ where ad is the *adjoint map* $(\text{ad}x)(y) = [x,y]$.
\[CartanPart\] Let $H=\coprod_{\lambda \in \Phi^{\ast}}^{\leq 1} k \cdot K_{\lambda}$ be the Banach group Hopf algebra of $\Phi^{\ast}$ with $$K_{\lambda} \cdot K_{\lambda'}=K_{\lambda + \lambda'}, \quad \Delta_{H}(K_{\lambda})=K_{\lambda} \otimes K_{\lambda}\quad\text{and}\quad S(K_{\lambda})=K_{-\lambda}.$$ We use the notation $$t_{i}:=K_{\frac{(\alpha_{i},\alpha_{i})}{2}\lambda_{i}}$$ which we borrow from [@OCB]. Let $H'$ be the closed sub-Hopf algebra generated by $\{t_{i} \mid i \in I\}$, $H'=\coprod_{\underline{n} \in \mathbb{Z}^{I}}^{\leq 1} k \cdot \underline{t}^{\underline{n}}$.
\[HAPairingWeakQuasiTriangular\] There is a duality pairing $H \hat{\otimes} H' \rightarrow k$ defined by $$K_{\lambda} \otimes \underline{t}^{\underline{n}} \mapsto q^{\lambda(\sum n_{i} \alpha_{i})}$$ and simultaneous algebra homomorphisms and coalgebra anti-homomorphisms $$\mathscr{R}:H' \rightarrow H, t_{i} \mapsto t_{i}, \, \, \, \overline{\mathscr{R}}:H' \rightarrow H, t_{i} \mapsto t_{i}^{-1},$$ for $i \in I$, making $H$ and $H'$ a weakly quasi-triangular dual pair.
This is Lemma 4.1 of [@DBoBG].
We will say that a Banach $H$-modules $M$ is a *Banach weight space* of weight $\alpha \in \Phi$ if $K_{\lambda} \cdot m = q^{\lambda(\alpha)}m$ for all $m \in M$. We say that an Banach $H$-module $M$ *decomposes into Banach weight space* if it is a contracting coproduct of Banach weight spaces $M_{\alpha}$ of weights $\alpha \in \Phi$, $M \cong \coprod_{\alpha \in \Phi}^{\leq 1} M_{\alpha}$. The weights of $M$ will be those $\alpha \in \Phi$ such that $M_{\alpha} \neq 0$. We will say that an IndBanach $H$-module $M$ *decomposes locally into Banach weight spaces* if it can be written as a colimit of Banach $H$-modules that decompose into Banach weight spaces. Given a subset $X \subset \Phi$ we denote by $H\text{-Mod}_{X}$ the full subcategory of $H\text{-Mod}$ consisting of modules that decompose locally into Banach weight spaces with weights in $X$.
Lemma \[HAPairingWeakQuasiTriangular\] above, along with Proposition \[WeaklyQTPairComodulesBraided\] and Proposition 4.4 of [@TRTfIBS], says precisely that $H\text{-Mod}_{\Psi}$ is braided monoidal. In the next section we shall extend this braiding, under certain conditions, to $H\text{-Mod}_{\Phi}$.
\[AnalyticQuantumGroupPositivePart\] Let $V=\coprod_{i \in I}^{\leq 1} k \cdot v_{i}$ with basis $\{v_{i} \mid i \in I \}$. This is a left $H'$-comodule with coaction $v_{i} \mapsto t_{i} \otimes v_{i}$. The weakly quasi-triangular structure of $H$ and $H'$ then induces a braiding $c$ on $V$ given by $$c(v_{i} \otimes v_{j})=q^{(\alpha_{i},\alpha_{j})}v_{j} \otimes v_{i}$$ of norm $\|c\| = 1$. Let $\langle -,- \rangle$ be the non-degenerate bilinear form on $V$ defined by $$\langle v_{i},v_{j} \rangle = \delta_{i,j}\frac{1}{(q_{i}-q_{i}^{-1})} \quad \text{for} \quad q_{i}=q^{\frac{(\alpha_{i},\alpha_{i})}{2}}.$$ Given $0 < r$ or $0 \leq \rho$ we denote by $\mathbf{f}_{r}^{\text{an}}$ and $\mathbf{f}_{\rho}^{\dagger}$ the Nichols algebras $\mathfrak{B}_{r}^{c}(V)$ and $\mathfrak{B}_{\rho}^{c}(V)^{\dagger}$ respectively. We will also use the notation $U_{q}^{+}(\mathfrak{g})_{r}^{\text{an}}=U_{q}^{-}(\mathfrak{g})_{r}^{\text{an}}=\mathbf{f}_{r}^{\text{an}}$ and $U_{q}^{+}(\mathfrak{g})_{\rho}^{\dagger}=U_{q}^{-}(\mathfrak{g})_{\rho}^{\dagger}=\mathbf{f}_{\rho}^{\dagger}$. Note that these are braided IndBanach Hopf algebras in $H'\text{-Comod}=\text{Comod-}H'$.
\[PositivePartDense\] For each $0<r$, the positive part of the quantum group, as defined in [@ItQG] and [@DBoBG], is dense in the Banach space $\mathbf{f}_{r}^{\text{an}}$.
This follows from Proposition \[ClassicalNicholsAlgebrasDense\] along with Theorem 4.2 of [@PHA], which is a restatement of the constructions in Chapter 1 of [@ItQG].
Suppose $0< r,s$ such that $1 \leq |q_{i}-q_{i}^{-1}|rs$ for all $i \in I$. Then there is a duality pairing $\mathbf{f}_{s}^{\text{an}} \hat{\otimes} \mathbf{f}_{r}^{\text{an}} \rightarrow k$ as Banach Hopf algebras in $H\text{-Mod}=\text{Mod-}H$ extending $\langle-,-\rangle$ in Definition \[AnalyticQuantumGroupPositivePart\]. Likewise, for $0 \leq \rho, \sigma$ with $1 \leq |q_{i}-q_{i}^{-1}|\rho \sigma$ for all $i \in I$ there is a duality pairing $\mathbf{f}_{\sigma}^{\dagger} \hat{\otimes} \mathbf{f}_{\rho}^{\dagger} \rightarrow k$.
This follows from Lemma \[ExtendBilinearForm\].
\[AnalyticQuantumGroups\] \[DaggerQuantumGroups\] For $0< r,s$ with $1 \leq |q_{i}-q_{i}^{-1}|rs$ for all $i \in I$ we denote by $U_{q}(\mathfrak{g})_{r,s}^{\text{an}}$ the *analytic quantum group* $U(\mathbf{f}_{r}^{\text{an}},H,\mathbf{f}_{s}^{\text{an}})$. We will denote by $U_{q}^{\leq 0}(\mathfrak{g})_{r}^{\text{an}}$ and $U_{q}^{\geq 0}(\mathfrak{g})_{r}^{\text{an}}$ the respective sub-Hopf algebras $\mathbf{f}_{r}^{\text{an}} \rtimes H$ and $\overline{H} \ltimes \overline{\mathbf{f}_{s}^{\text{an}}}$ of $U_{q}(\mathfrak{g})_{r,s}^{\text{an}}$. Let us denote by $F_{i}$ the element $v_{i} \otimes 1 \in \mathbf{f}_{r}^{\text{an}} \rtimes H$, and by $E_{i}$ the element $1 \otimes v_{i}\in \overline{H} \ltimes \overline{\mathbf{f}_{s}^{\text{an}}}$ for $i \in I$. For $0 \leq \rho, \sigma$ with $1 \leq |q_{i}-q_{i}^{-1}|\rho \sigma$ for all $i \in I$ we denote by $U_{q}(\mathfrak{g})_{\rho,\sigma}^{\dagger}$ the *dagger quantum group* $U(\mathbf{f}_{\rho}^{\dagger},H,\mathbf{f}_{\sigma}^{\dagger})$. We will denote by $U_{q}^{\leq 0}(\mathfrak{g})_{\rho}^{\dagger}$ and $U_{q}^{\geq 0}(\mathfrak{g})_{\sigma}^{\dagger}$ the respective sub-Hopf algebras $\mathbf{f}_{\rho}^{\dagger} \rtimes H$ and $\overline{H} \ltimes \overline{\mathbf{f}_{\sigma}^{\dagger}}$ of $U_{q}(\mathfrak{g})_{\rho,\sigma}^{\dagger}$.
\[NAQGGraded\] $U_{q}(\mathfrak{g})_{r,s}^{\text{an}}$ and $U_{q}(\mathfrak{g})_{\rho,\sigma}^{\dagger}$ are analytically graded by $\mathbb{Z}I \cong \Psi$ (*i.e* are graded by the Banach group Hopf algebra of $\Psi$).
Note that $\mathbf{f}_{r}^{\text{an}}$ and $\mathbf{f}_{s}^{\text{an}}$ are $H'$-comodules, both left and right since $H'$ is cocommutative. If we give $H$ the trivial $H'$-coaction, $$H \cong k \hat{\otimes} H \overset{\eta_{H'} \otimes \text{Id}_{H}}{\longrightarrow} H' \hat{\otimes} H,$$ then all of the morphisms involved in defining $U(\mathbf{f}_{r}^{\text{an}},H,\mathbf{f}_{s}^{\text{an}})$ are $H'$-comodule homomorphisms. The result then follows since $H'$ is isomorphic to the Banach group Hopf algebra of $\Phi$.
\[QuantumGroupDenseInNAAnalyticVersion\] For each $0< r,s$ with $1 \leq |q_{i}-q_{i}^{-1}|rs$ for all $i \in I$, the quantum enveloping algebra $U_{q}(\mathfrak{g})$ is dense in the Banach space $U_{q}(\mathfrak{g})_{r,s}^{\text{an}}$, and the Hopf structure on $U_{q}(\mathfrak{g})_{r,s}^{\text{an}}$ restricts to the usual Hopf structure on $U_{q}(\mathfrak{g})$.
This follows from Proposition 4.3 of [@DBoBG], Lemma \[PositivePartDense\] and the triangular decomposition of $U_{q}(\mathfrak{g})$ given by the Poincaré-Birkhoff-Witt Theorem.
\[EpimorphismFromAnalyticQuantumGroup\] Suppose $\mathcal{U}$ is a Banach Hopf algebra in which $U_{q}(\mathfrak{g})$ is a dense sub-Hopf algebra, with $\mathcal{U}$ analytically $\mathbb{Z}I$ graded (*i.e* are graded by the Banach group Hopf algebra of $\Psi$) extending the grading on $U_{q}(\mathfrak{g})$. Then there is an epimorphism of Banach Hopf algebras $$U_{q}(\mathfrak{g})_{r,s}^{\text{an}} \rightarrow \mathcal{U}$$ for some $r,s>0$ with $|q_{i}-q_{i}^{-1}|rs\geq 1$ for all $i \in I$.
Let $\mathcal{U}^{-}$, $\mathcal{U}^{0}$, and $\mathcal{U}^{+}$ be the respective closured of $U_{q}^{-}(\mathfrak{g})$, $U_{q}^{0}(\mathfrak{g})$, and $U_{q}^{+}(\mathfrak{g})$ in $\mathcal{U}$. The $\mathbb{Z}I$ grading on $U_{q}^{+}(\mathfrak{g})$ is concentrated in degrees in the submonoid $\mathbb{N}I$, and hence so is the analytic grading on $\mathcal{U}^{+}$. The homomorphism of monoids from this submonoid to $\mathbb{N}$, $\sum_{i \in I}n_{i} \cdot i \mapsto \sum_{i \in I}n_{i}$ gives $\mathcal{U}^{+}$ an analytic $\mathbb{N}$ grading compatible with that of the algebraic Nichols algebra $U_{q}^{+}(\mathfrak{g})$. Since $\mathcal{U}^{+}$ is a completion of the algebraic Nichols algebra $U_{q}^{+}(\mathfrak{g})$ it must be an analytic Nichols algebra of $V$ as in Definition \[AnalyticQuantumGroupPositivePart\], and hence by Proposition \[EpimorphismFromAnalyticQuantumGroup\] we have an epimorphism $\mathbf{f}_{s} \rightarrow \mathcal{U}^{+}$ for some $s>0$. Likewise we obtain an epimorphism $\mathbf{f}_{r} \rightarrow \mathcal{U}^{-}$ for some $r>0$, and without loss of generality we may take $r$ and $s$ such that $rs|q_{i}-q_{i}^{-1}| \geq 1$ for all $i \in I$. Finally, each $K_{\lambda} \in \mathcal{U}_{0}$ is grouplike, so since comultiplication on $\mathcal{U}_{0}$ is bounded by some constant $C>0$ we have $\|K_{\lambda}\|^{2}=\|K_{\lambda} \otimes K_{\lambda}\| \leq C\|K_{\lambda}\|$, so $\|K_{\lambda}\|\leq C$. Thus we can define a bounded epimorphism $H \rightarrow \mathcal{U}^{0}$ by mapping each $K_{\lambda}$ to the associated element of $U_{q}^{0}(\mathfrak{g})$ inside $\mathcal{U}^{0}$. This gives us an epimorphism $$U_{q}(\mathfrak{g})^{\text{an}}_{r,s} = \mathbf{f}^{\text{an}}_{r} \hat{\otimes} H \hat{\otimes} \mathbf{f}^{\text{an}}_{s} \twoheadrightarrow \mathcal{U}^{-} \hat{\otimes} \mathcal{U}^{0} \hat{\otimes} \mathcal{U}^{+} \overset{\mu}{\twoheadrightarrow} \mathcal{U}$$ that restricts to a Hopf algebra homomorphism on the dense subspace $U_{q}(\mathfrak{g})$.
Quasi-triangularity and the quasi-R-matrix
------------------------------------------
Suppose throughout this section that the symmetrised Cartan matrix associated to our root datum in Definition \[KacMoodyRootDatum\], $A=((\alpha_{i},\alpha_{j}))_{i,j \in I}$, is invertible (over $\mathbb{Q}$). Suppose further that $q=\text{exp}(\hslash)$ for some $\hslash \in k$ of sufficiently small norm such that ${\frac{1}{|n!|}(|\hslash| \cdot \text{max}_{i,j}|A^{-1}_{i,j}|)^{n}}$ converges to $0$, where $A_{i,j}^{-1}$ are the entries of the inverse of the Cartan matrix.
If $k$ is an extension of $\mathbb{Q}_{p}$, the requirement that ${\frac{1}{|n!|}(|\hslash| \cdot \text{max}_{i,j}|A^{-1}_{i,j}|)^{n}}$ converges to $0$ is equivalent to $|\hslash| \cdot \text{max}_{i,j}|A^{-1}_{i,j}|<p^{\frac{1}{1-p}}$. If $k$ is such that $|n|=1$ for all integers $n \in \mathbb{Z}$ then this is just the assumption that $|\hslash|<1$.
Let us denote by $\mathscr{H}$ the Banach space ${\mathscr{H}:=\coprod_{\alpha \in \mathbb{Z}I}^{\leq 1} k \cdot H_{\alpha}}$. This has an algebra structure induced by the group structure of $\mathbb{Z}I$, and we make $\mathscr{H}$ a Hopf algebra by defining $$\Delta_{\mathscr{H}}(H_{i})=1 \otimes H_{i} + H_{i} \otimes 1 \quad \text{for} \quad i \in I.$$ We will denote by $e:H' \rightarrow \mathscr{H}$ the morphism of Hopf algebras determined by $$t_{i} \mapsto \text{exp}\left(\frac{(\alpha_{i},\alpha_{i})}{2}\hslash H_{i}\right)=\sum_{n \geq 0} \frac{(\alpha_{i},\alpha_{i})^{n}\hslash^{n}}{2^{n} \cdot n!}H_{ni}.$$
\[CartanPartQuasiTriangular\] Under our assumptions, $\mathscr{H}$ is quasi-triangular with R-matrix $$\mathscr{R}_{\mathscr{H}}=\text{exp}(\hslash \sum_{i,j}A^{-1}_{i,j}H_{i} \otimes H_{j}).$$
Since $\mathscr{H}$ is both commutative and cocommutative, it is trivial to check that $\mathscr{R}_{\mathscr{H}}$ is an R-matrix. It remains only to check that this converges, which follows from our assumptions since $$\|\hslash \sum_{i,j}A^{-1}_{i,j}H_{i} \otimes H_{j}\| \leq |\hslash| \cdot \text{max}_{i,j}|A^{-1}_{i,j}|.$$
\[HModuleBraiding\] There is a braiding on $H\text{-Mod}_{\Phi}$ extending that of Lemma \[HAPairingWeakQuasiTriangular\].
Suppose $M_{\alpha}$ and $N_{\alpha'}$ are Banach $H$-modules, and hence $H'$-modules, of weights $\alpha$ and $\alpha'$ in $\Phi$ respectively. Then $M_{\alpha}$ is a $\mathscr{H}$-module where $H_{i}$ acts by the scalar $(\alpha_{i},\alpha)$, and likewise so is $N_{\alpha'}$. The action of $\mathscr{R}_{\mathscr{H}}$ induces the braiding $$M_{\alpha} \hat{\otimes} N_{\alpha'} \rightarrow N_{\alpha'} \hat{\otimes} M_{\alpha}, \quad m \otimes n \mapsto q^{\sum_{i,j} A_{i,j}^{-1}(\alpha_{i},\alpha)(\alpha_{j},\alpha')}n \otimes m.$$ This braiding commutes not only with the action of $H'$ but also with that of $H$. If $\alpha=\sum n_{i} \alpha_{i}$ and $\alpha'=\sum n'_{i} \alpha_{i}$ in $\Psi$ then $$\begin{array}{rcl}
\sum_{i,j} A_{i,j}^{-1}(\alpha_{i},\alpha)(\alpha_{j},\alpha') &=& \sum_{i,j,i',j'} A_{i,j}^{-1}n_{i'}n'_{j'}(\alpha_{i},\alpha_{i'})(\alpha_{j},\alpha_{j'})\\
&=& \sum_{i,j,i',j'} A_{i,j}^{-1}n_{i'}n'_{j'}(\alpha_{i},\alpha_{i'})A_{j,j'}\\
&=& \sum_{i,i'} n_{i'}n'_{i}(\alpha_{i},\alpha_{i'}) = (\alpha,\alpha'),
\end{array}$$ so the braiding restricts to the one given by Lemma \[HAPairingWeakQuasiTriangular\] on $H\text{-Mod}_{\Psi}$.
\[RMatrixDoesn’tConverge\] Proposition 3.6 of [@DBoBG] gives criterion for when a double-bosonisation is quasi-triangular. The Banach version of this would be the following.
Let $H$ be a quasi-triangular Banach Hopf algebra, let $B$ be a braided Banach Hopf algebra $B$ in $H\text{-Mod}$ and let $C$ be a braided Banach Hopf algebra in $\text{Mod-}H$, equipped with a duality pairing between $B$ and $C$, $\langle - , - \rangle: B \hat{\otimes} C \rightarrow k$. Suppose there exists linearly independent subsets $\{b_{n} \mid n \in \mathbb{N}\} \subset B$ and $\{c_{n} \mid n \in \mathbb{N}\} \subset C$ that span dense subspaces and satisfy $$\langle b_{n},c_{m} \rangle = \delta_{n,m} \quad \text{and} \quad \|b_{n}\|\cdot \|c_{n}\| \rightarrow 0.$$ Then $U(C,H,B)$ is quasi-triangular with R-matrix $$\mathscr{R}_{U(C,H,B)}:=\text{exp}_{C,B} \cdot \mathscr{R}_{H}=\sum_{x \in X} (c_{x} \otimes \mathscr{R}_{2}^{(1)}\mathscr{R}_{1}^{(1)} \otimes 1) \otimes (1 \otimes \mathscr{R}_{1}^{(2)} \otimes S_{B}(b_{x}) \cdot \mathscr{R}_{2}^{(2)})$$ where $\text{exp}_{C,B}=\sum_{x \in X} c_{x} \otimes S_{B}(b_{x})$, and $\mathscr{R}_{1}=\sum \mathscr{R}_{1}^{(1)} \otimes \mathscr{R}_{1}^{(2)}$ and $\mathscr{R}_{2}=\sum \mathscr{R}_{2}^{(1)}\otimes \mathscr{R}_{2}^{(2)}$ are copies of the R-matrix $\mathscr{R}_{H}$ on $H$.
Unfortunately, as in [@DBoBG], the conditions required for this theorem do not hold if $B$ and $C$ are infinite dimensional, hence it does not apply to $U(\mathbf{f}_{r}^{\text{an}}, \mathscr{H}, \mathbf{f}_{s}^{\text{an}})$. Indeed, if such sets were to exist for infinite dimensional $B$ and $C$ then $\sum b_{n} \otimes c_{n}$ would converge in $B \otimes C$ but its image under $\langle -,- \rangle$ would not.\
If we allow ourselves to work over formal powerseries in $\hslash$ then we may define an R-matrix for some analytic quantum groups, which we will see in Section \[WorkingOverk\[\[h\]\]\].
Quantum groups as Drinfel’d doubles
-----------------------------------
\[Drinfel’dDoubleConstruction\] Suppose we have IndBanach Hopf algebras $B$ and $C$ with a duality pairing $\langle -,- \rangle : B \hat{\otimes} C^{\text{op}} \rightarrow k$. Then there is a Hopf algebra structure on $C \hat{\otimes} B$ such that $B \rightarrow C \hat{\otimes} B$ and $C \rightarrow C \hat{\otimes} B$ are morphisms of Hopf algebras and the multiplication restricts to the composition $$\begin{array}{rcl}
B \hat{\otimes} C &\longrightarrow& B \hat{\otimes} B \hat{\otimes} B \hat{\otimes} C \hat{\otimes} C \hat{\otimes} C\\
&\longrightarrow& B \hat{\otimes} C \hat{\otimes} C \hat{\otimes} B \hat{\otimes} B \hat{\otimes} C\\
&\longrightarrow& C \hat{\otimes} B
\end{array}$$ where the first morphism is given by the respective comultiplications, the second is a reordering given by the permutation $(2 \quad 4)(3 \quad 5) \in S_{6}$ and the last is given by $\langle -,- \rangle \otimes \text{Id} \otimes \text{Id} \otimes \langle -,- \rangle$.
This construction is outlined in Section 8.2.1 of [@QGaTR] and Section IX.5 of [@QG] for (finite dimensional) vector spaces, but is easily generalised to the category of IndBanach spaces.
We will denote by $D(B,C)$ the Hopf algebra on $C \hat{\otimes} B$ described in the previous lemma, the *relative Drinfel’d double* of $B$ and $C$.
\[DualityPairingforQuantumDouble\] There is a duality pairing $(\overline{H} \ltimes \overline{\mathbf{f}^{\text{an}}_{s}}) \hat{\otimes} (\mathbf{f}^{\text{an}}_{r} \rtimes H')^{\text{op}} \rightarrow k$ such that $$\begin{array}{rclrcl}
K_{\lambda} \otimes t_{j} &\mapsto& q^{-\lambda(\alpha_{j})},&
K_{\lambda} \otimes F_{j} &\mapsto& 0,\\
E_{i} \otimes t_{j} &\mapsto& 0,&
E_{i} \otimes F_{j} &\mapsto& -\delta_{i,j} \frac{1}{q_{i}-q_{i}^{-1}}.
\end{array}$$
This follows as in Section 6.3.1 of [@QGaTR]. We first define bounded algebra homomorphisms $\phi_{i}$ and $\psi_{i}$ from $H \ltimes \overline{\mathbf{f}^{\text{an}}_{s}}$ to $k$ such that $$\phi_{i}(K_{\lambda} \otimes v_{i})=\frac{1}{q_{i}^{-1}-q_{i}}, \quad \phi_{i}(K_{\lambda} \otimes x)=0$$ and $$\psi_{i}(K_{\lambda} \otimes y)=q^{-\lambda(\alpha_{i})} \varepsilon(y)$$ for all $i \in I$, $x \in \mathbf{f}_{s}^{\text{an}}(\alpha)$, $\alpha_{i} \neq \alpha \in \Psi$, and $y \in \mathbf{f}_{s}^{\text{an}}$. Here $\varepsilon$ is the counit on $\mathbf{f}_{s}^{\text{an}}$. We define contracting morphisms $V_{r} \rightarrow (\overline{H} \ltimes \overline{\mathbf{f}^{\text{an}}_{s}})^{\ast}$ and $\coprod_{i\in I}^{\leq 1}t_{i} \rightarrow (\overline{H} \ltimes \overline{\mathbf{f}^{\text{an}}_{s}})^{\ast}$ defined respectively by $v_{i} \mapsto \phi_{i}$ and $t_{i} \mapsto \psi_{i}$. These induce algebra homomorphisms $T_{r}(V) \rightarrow (\overline{H} \ltimes \overline{\mathbf{f}^{\text{an}}_{s}})^{\ast}$, $H' \rightarrow (\overline{H} \ltimes \overline{\mathbf{f}^{\text{an}}_{s}})^{\ast}$. It follows from the proof of Proposition 34 in *loc. cit.* that the map $T_{r}(V) \rightarrow (\overline{H} \ltimes \overline{\mathbf{f}^{\text{an}}_{s}})^{\ast}$ factors as a composition $T_{r}(V) \rightarrow \mathbf{f}_{r}^{\text{an}} \rightarrow (H \ltimes \overline{\mathbf{f}^{\text{an}}_{s}})^{\ast}$. From this we obtain a morphism $\mathbf{f}^{\text{an}}_{r} \rtimes H' \rightarrow (\overline{H} \ltimes \overline{\mathbf{f}^{\text{an}}_{s}})^{\ast}$ and hence a bilinear form as desired. The fact that this is a duality pairing is checked on a dense subspace in [@QGaTR], and extends by continuity.
It is shown in Proposition 34 of [@QGaTR] that this pairing can be expressed as the composition $$\begin{array}{rcccl}
H \hat{\otimes} \mathbf{f}^{\text{an}}_{s} \hat{\otimes} \mathbf{f}^{\text{an}}_{r} \hat{\otimes} H' &\overset{\text{Id} \otimes \text{Id} \otimes S \otimes\text{Id}}{\xrightarrow{\hspace*{1.5cm}}}& H \hat{\otimes} \mathbf{f}^{\text{an}}_{s} \hat{\otimes} \mathbf{f}^{\text{an}}_{r} \hat{\otimes} H'
&\overset{\text{Id} \otimes \langle -,- \rangle \otimes\text{Id}}{\xrightarrow{\hspace*{1.5cm}}}& H \hat{\otimes} k \hat{\otimes} H' \\
&\overset{\text{Id} \otimes S}{\xrightarrow{\hspace*{1.5cm}}}& H \hat{\otimes} H'
&\overset{\langle -,- \rangle}{\xrightarrow{\hspace*{1.5cm}}}& k.
\end{array}$$
\[CrossedBimodules\] For an IndBanach Hopf algebra $C$, we will denote by $_{C}\text{Cross}^{C}$ the category of IndBanach spaces $V$ equipped with both a left action and right coaction of $C$, $\mu_{V}: C \hat{\otimes} V \rightarrow V$ and $\Delta_{V}: V \rightarrow V \hat{\otimes} C$, such that the following diagram commutes:
\(A) [$C \hat{\otimes} V$]{}; (B) \[right=3cm of A\] [$C \hat{\otimes} C \hat{\otimes} V$]{}; (C) \[below=0.5cm of B\] [$C \hat{\otimes} V$]{}; (D) \[below=0.5cm of A\] [$C \hat{\otimes} C \hat{\otimes} V \hat{\otimes} C$]{}; (E) \[below=0.5cm of C\] [$V \hat{\otimes} C$]{}; (F) \[below=0.5cm of D\] [$C \hat{\otimes} V \hat{\otimes} C \hat{\otimes} C$]{}; (G) \[below=0.5cm of F\] [$V \hat{\otimes} C$]{}; (H) \[below=0.5cm of E\] [$V \hat{\otimes} C \hat{\otimes} C.$]{}; (A) to node [$\Delta_{C} \otimes \text{Id}$]{} (B); (B) to node [$\text{Id} \otimes \mu_{V}$]{} (C); (C) to node [$\tau$]{} (E); (E) to node [$\Delta_{V} \otimes \text{Id}$]{} (H); (H) to node [$\text{Id} \otimes \mu_{C}$]{} (G); (A) to node [$\Delta \otimes \Delta_{V}$]{} (D); (D) to node [$\text{Id} \otimes \tau \otimes \text{Id}$]{} (F); (F) to node [$\mu_{V} \otimes \mu_{C}$]{} (G);
We will refer to these as *crossed bimodules*, although they are sometimes also referred to as *Yetter-Drinfel’d modules*.
\[CrossedBimodulesBraided\] The category $_{C}\text{Cross}^{C}$ is pre-braided monoidal with braiding given by $$M \hat{\otimes} N \overset{\tau}{\rightarrow} N \hat{\otimes} M \overset{\Delta_{N} \otimes \text{Id}}{\longrightarrow} N \hat{\otimes} C \hat{\otimes} M \overset{\text{Id} \otimes \mu_{M}}{\longrightarrow} N \hat{\otimes} M$$ for $M$ and $N$ in $_{C}\text{Cross}^{C}$. If $C$ has an invertible antipode then this is a braiding.
This is Theorem 5.2 and Theorem 7.2 of [@QGaRoMC].
\[ModulesofQuantumDouble\] There is a faithful functor $_{C}\text{Cross}^{C} \rightarrow D(B,C)\text{-Mod}$.
This follows from the proof of Proposition 6 in Section 13.1 of [@QGaTR]. An object $V$ of $_{C}\text{Cross}^{C}$ already has an action of $C$. It gains an action of $B$ using the duality pairing, giving a map $$\mu_{V}':B \hat{\otimes} V \overset{\text{Id} \otimes \Delta_{V}}{\xrightarrow{\hspace*{0.8cm}}} B \hat{\otimes} V \hat{\otimes} C \overset{(\text{Id} \hat{\otimes} \langle -,- \rangle) \circ (\tau \otimes \text{Id})}{\xrightarrow{\hspace*{2.5cm}}} V.$$ Then $B$ and $C$ generate $D(B,C)$, and the fact that $\mu_{V}$ and $\mu'_{V}$ together give a well defined action of $D(B,C)=C \hat{\otimes} B$ on $V$ (given by $\mu_{V} \circ (\text{Id} \otimes \mu'_{V})$) follows from the commutativity of the diagram in Definition \[CrossedBimodules\] as in the proof of Proposition 6 of *loc. cit.*
\[NAAnalyticQuantumGroupAsDrinfel’dDouble\] There is a strict epimorphism of braided analytically graded Banach Hopf algebras $$D(\overline{H} \ltimes \overline{\mathbf{f}^{\text{an}}_{s}},\mathbf{f}^{\text{an}}_{r} \rtimes H') \rightarrow U_{q}(\mathfrak{g})^{\text{an}}_{r,s}$$ whose kernel is the closed two sided ideal generated by $$\{1 \otimes t_{i} \otimes 1 \otimes 1 - 1 \otimes 1 \otimes t_{i} \otimes 1 \mid i \in I\} \subset \mathbf{f}^{\text{an}}_{r} \hat{\otimes} H \hat{\otimes} H' \hat{\otimes} \mathbf{f}^{\text{an}}_{s}.$$
This strict epimorphism can be written as $$\mathbf{f}^{\text{an}}_{r} \hat{\otimes} H' \hat{\otimes} H \hat{\otimes} \mathbf{f}^{\text{an}}_{s} \xrightarrow{\text{Id} \otimes \mu_{H} \otimes \text{Id}} \mathbf{f}^{\text{an}}_{r} \hat{\otimes} H \hat{\otimes} \mathbf{f}^{\text{an}}_{s}.$$ Corollary 15 of Section 8.2.4 of [@QGaTR], along with Proposition \[QuantumGroupDenseInNAAnalyticVersion\], show that this restricts to a morphism of braided graded Hopf algebras between the dense subspaces $$U_{q}^{\leq 0}(\mathfrak{g}) \otimes U_{q}^{\geq 0}(\mathfrak{g}) \rightarrow U_{q}(\mathfrak{g})$$ whose kernel is generated by $t_{i} \otimes 1 - 1 \otimes t_{i}$. The result then follows by continuity.
Let us denote by $\mathcal{C}$ the full subcategory of $_{(\mathbf{f}^{\text{an}}_{r} \rtimes H')}\text{Cross}^{(\mathbf{f}^{\text{an}}_{r} \rtimes H')}$ consisting of IndBanach spaces $V$ equipped with both a left action and right coaction of $\mathbf{f}^{\text{an}}_{r} \rtimes H'$, $\mu_{V}: (\mathbf{f}^{\text{an}}_{r} \rtimes H') \hat{\otimes} V \rightarrow V$ and $\Delta_{V}: V \rightarrow V \hat{\otimes} (\mathbf{f}^{\text{an}}_{r} \rtimes H')$, such that the following additional diagram commutes:
\(A) [$H' \hat{\otimes} V$]{}; (B) \[below=1cm of A\] [$(\mathbf{f}^{\text{an}}_{r} \rtimes H') \hat{\otimes} V$]{}; (C) \[right=1cm of A\] [$(\overline{H} \ltimes \overline{\mathbf{f}^{\text{an}}_{s}}) \hat{\otimes} V$]{}; (D) \[below=1.05cm of C\] [$V$]{}; (A) to node (B); (C) to node [$\mu_{V}'$]{} (D); (A) to node (C); (B) to node [$\mu_{V}$]{} (D);
where $\mu_{V}'$ is the action described in the proof of Lemma \[ModulesofQuantumDouble\].
\[FullyFaithfulFunctorCtoUqanMod\] There is a fully faithful functor $\mathcal{C} \rightarrow U_{q}(\mathfrak{g})^{\text{an}}_{r,s}\text{-Mod}$.
By Lemma \[ModulesofQuantumDouble\] and Proposition \[NAAnalyticQuantumGroupAsDrinfel’dDouble\] there is a faithful functor $\mathcal{C} \rightarrow U_{q}(\mathfrak{g})^{\text{an}}_{r,s}\text{-Mod}$. Let $f:M \rightarrow N$ be a morphism in $U_{q}(\mathfrak{g})^{\text{an}}_{r,s}\text{-Mod}$ between two objects in the image of $\mathcal{C}$. Then the action of $\overline{H} \ltimes \overline{\mathbf{f}_{s}^{\text{an}}}$ on both $M$ and $N$ is induced by a coaction of $\mathbf{f}_{r}^{\text{an}} \rtimes H'$ via their dual pairing. Furthermore, $f$ commutes with the action of $U_{q}(\mathfrak{g})^{\text{an}}_{r,s}$, and hence with the actions of $\mathbf{f}_{r}^{\text{an}} \rtimes H'$ and $\overline{H} \ltimes \overline{\mathbf{f}_{s}^{\text{an}}}$. The morphism $f$ must therefore preserve the locally Banach weight space decompositions of $M$ and $N$, so it preserves their respective coactions of $H'$. Since the pairing of $\mathbf{f}_{r}^{\text{an}}$ and $\mathbf{f}_{s}^{\text{an}}$ is non-degenerate, $f$ must also preserve the coaction of $\mathbf{f}_{r}^{\text{an}}$ that induces the action of $\mathbf{f}_{s}^{\text{an}}$. Hence $f$ preserves the coaction of $\mathbf{f}_{r}^{\text{an}} \rtimes H'$, so comes from a morphism in $\mathcal{C}$. So the functor is fully faithful.
\[IntegrableRepresentations\] We will denote by $\mathcal{O}_{\Psi}$ the essential image of $\mathcal{C}$ in $U_{q}(\mathfrak{g})^{\text{an}}_{r,s}\text{-Mod}$. This is the full subcategory of $U_{q}(\mathfrak{g})^{\text{an}}_{r,s}$ modules whose action of $H$ gives a representation in $\text{Comod-}H' = H\text{-Mod}_{\Psi}$ and whose action of $U_{q}^{-}(\mathfrak{g})^{\text{an}}_{s}$ comes from a coaction of $\mathbf{f}_{r}^{\text{an}}$ via their dual pairing. Let us denote by $\mathcal{O}$ the full subcategory of $U_{q}(\mathfrak{g})^{\text{an}}_{r,s}\text{-Mod}$ consisting of modules whose action of $U_{q}^{-}(\mathfrak{g})^{\text{an}}_{s}$ comes from a coaction of $\mathbf{f}_{r}^{\text{an}}$ and whose action of $H$ gives a representation in $H\text{-Mod}_{\Phi}$. Note that $\mathcal{O}_{\Psi} \subset \mathcal{O}$.
Note that the conditions for a $U_{q}(\mathfrak{g})^{\text{an}}_{r,s}$ module to be in $\mathcal{O}$ resemble the conditions for a $U_{q}(\mathfrak{g})$ module to be integrable, with the requirement that the action of $U_{q}^{+}(\mathfrak{g})^{\text{an}}_{s}$ comes from a coaction of $\mathbf{f}_{r}^{\text{an}}$ taking the place of a locally finite dimensional action of $U_{q}^{+}(\mathfrak{g})$.
The category $\mathcal{O}_{\Psi}$ is braided monoidal.
This is a consequence of Lemma \[CrossedBimodulesBraided\] and Lemma \[FullyFaithfulFunctorCtoUqanMod\].
A short computation shows that the braiding on $\mathcal{O}_{\Psi}$ can be expressed as performing the braiding on $H'$-comodules given by Lemma \[HAPairingWeakQuasiTriangular\] followed by the action of $\text{exp}_{\mathbf{f}_{r}^{\text{an}},\mathbf{f}_{r}^{\text{an}}}$ as described at the end of Section \[RMatrixDoesn’tConverge\], which is well defined despite $\text{exp}_{\mathbf{f}_{r}^{\text{an}},\mathbf{f}_{r}^{\text{an}}}$ not converging. This is expected given the description of the R-matrix at the end of Section \[RMatrixDoesn’tConverge\].
Suppose that the symmetrised Cartan matrix associated to our root datum, $A=((\alpha_{i},\alpha_{j}))_{i,j \in I}$, has an inverse over $\mathbb{Q}$ with entries $A_{i,j}^{-1}$. Suppose further that $q=\text{exp}(\hslash)$ for some $\hslash \in k$ of sufficiently small norm such that ${\frac{1}{|n!|}(|\hslash| \cdot \text{max}_{i,j}|A^{-1}_{i,j}|)^{n}}$ converges to $0$. Then there is a braiding on the category $\mathcal{O}$ extending that of $\mathcal{O}_{\Psi}$.
Given $M$ and $N$ in $\mathcal{O}$, the braiding is given by the composition $$M \hat{\otimes} N \xrightarrow{\tau} N \hat{\otimes} M \longrightarrow N \hat{\otimes} \mathbf{f}_{r}^{\text{an}} \hat{\otimes} M \longrightarrow N \hat{\otimes} M \xrightarrow{\mathscr{R}_{\mathscr{H}}} N \hat{\otimes} M$$ where the second morphism is the coaction of $\mathbf{f}_{r}^{\text{an}}$ on $N$, the third is its action on $M$, and the last in the action of the R-matrix as in the proof of Proposition \[HModuleBraiding\]. The computations in the proof of Proposition 3.6 of [@DBoBG], alongside the previous remark, ensure that this is a braiding. This is an extension of the braiding in Lemma \[CrossedBimodulesBraided\].
Let $\mathfrak{g}=\mathfrak{sl}_{2}$, and fix $r,s>0$ with $|q_{i}-q_{i}^{-1}|rs\geq 1$ for all $i \in I$. Let $\lambda \in \Phi \cong \mathbb{Z}$, and let $W_{\lambda}=k\{\frac{x}{r}\}=\coprod_{n \geq 0}^{\leq 1} k_{r^{n}} \cdot x^{n}$ with the action of $U_{q}(\mathfrak{sl}_{2})_{r,s}^{\text{an}}$ given by $$K \cdot x^{n} = q^{\lambda -2n}x^{n}, \quad F \cdot x^{n}= x^{n+1}, \quad E \cdot x^{n} = [n][\lambda - (n-1)] x^{n-1}$$ where $x^{-1}:=0$. Alternatively, taking $y^{n}=\frac{x^{n}}{[n]!}$, $W=\{\sum a_{n}y^{n} \mid \frac{|a_{n}|r^{n}}{[n]!} \rightarrow 0\}$ with action $$K \cdot y^{n} = q^{\lambda -2n}y^{n}, \quad F \cdot y^{n}= [n+1]y^{n+1}, \quad E \cdot y^{n} = [\lambda - (n-1)] y^{n-1}$$ where $y^{-1}:=0$. Note that $W_{\lambda}$ is isomorphic to the quotient of $U_{q}(\mathfrak{sl}_{2})_{r,s}^{\text{an}}$ by the closed left ideal generated by $E$ and $K-q^{\lambda}$, and so we call it an *analytic Verma module* of weight $\lambda$. This gives a representation in $\mathcal{O}$ where the coaction of $\mathbf{f}_{r}^{an}$ is given by $$\begin{array}{rrl}
x^{n} &\longmapsto& \sum (-1)^{k}\frac{(q-q^{-1})^{k}}{[k]!} F^{k} \otimes E^{k}x^{n}\\
&& =\sum_{k=0}^{n} (-1)^{k}(q-q^{-1})^{k}\frac{[n]![\lambda -n +k]!}{[k]![n-k]![\lambda-n]!} F^{k} \otimes x^{n-k},\\
y^{n} &\longmapsto& \sum_{k=0}^{n} (-1)^{k}(q-q^{-1})^{k}\frac{[\lambda -n +k]!}{[k]![\lambda-n]!} F^{k} \otimes y^{n-k}.
\end{array}$$ Given $\lambda, \lambda' \in \Phi$ the braiding $W_{\lambda} \hat{\otimes} W_{\lambda'} \rightarrow W_{\lambda'} \hat{\otimes} W_{\lambda}$ is given by $$\begin{array}{rrl}
x^{n} \otimes x^{m} &\longmapsto& \sum_{k=0}^{n} (-1)^{k}(q-q^{-1})^{k}\frac{[n]![\lambda -n +k]!}{[k]![n-k]![\lambda-n]!} q^{2(m+k)(n-k)} \ x^{m+k} \otimes x^{n-k},\\
y^{n} \otimes y^{m} &\longmapsto& \sum_{k=0}^{n} (-1)^{k}(q-q^{-1})^{k}\frac{[m+k]![\lambda -n +k]!}{[k]![m]![\lambda-n]!} q^{2(m+k)(n-k)} \ y^{m+k} \otimes y^{n-k}.
\end{array}$$
Note that objects in $\mathcal{O}$ are not necessarily generated by highest weight vectors. We give an example of a representation with no such highest weights. Let $\mathfrak{g}=\mathfrak{sl}_{2}$, fix $r,s>0$ with $|q_{i}-q_{i}^{-1}|rs\geq 1$ for all $i \in I$, and suppose that $q=\text{exp}(\hslash)$ for $|\hslash| \ll 1$. Then $$q-q^{-1}=\left(\sum_{n=0}^{\infty}\frac{(\hslash)^{n}}{n!}\right)-\left(\sum_{n=0}^{\infty}\frac{(-\hslash)^{n}}{n!}\right)=2\hslash \left(\sum_{k=0}^{\infty}\frac{(\hslash)^{2k}}{k!}\right)$$ so $|q-q^{-1}|=|2\hslash|$, and $$\begin{array}{rcl}
[n]-n&=&q^{n-1}+q^{n-3}+...+q^{-n+1}-n\\
&=& \hslash\sum_{k=1}^{\infty} \frac{1}{k!}[(n-1)^{k}+(n-3)^{k}+...+(-n+1)^{k}]\hslash^{k-1}
\end{array}$$ has norm strictly smaller than 1 for $|\hslash|$ sufficiently small, so $|[n]|=|n|$. Fix some $\lambda \in \mathbb{Z}_{\geq 0}$. Let us define $$M_{\lambda}:= \coprod\nolimits^{\leq 1}_{i,j \geq 0} k_{r^{j-i}} \cdot x_{i,j}=\left\lbrace\sum \alpha_{i,j}x_{i,j} \middle| |\alpha_{i,j}|r^{j-i} \rightarrow 0\right\rbrace.$$ Then $M_{\lambda}$ becomes a $U_{q}(\mathfrak{sl}_{2})^{\text{an}}_{r,s}$ module with $$\begin{array}{rclr}
K \cdot x_{i,j} &=& q^{\lambda+2i-2j}x_{i,j},\\
E \cdot x_{i,j} &=& x_{i+1,j},\\
F \cdot x_{0,j} &=& x_{0,j+1},\\
F \cdot x_{i,j} &=& x_{i,j+1}-[i][\lambda+i-1-2j]x_{i-1,j} & \text{for } i>0,
\end{array}$$ so that $x_{i,j} = E^{i}F^{j}x_{0,0}$ and $x_{0,0}$ is of weight $\lambda$. Note that this action is bounded since $$\|K \cdot x_{i,j}\| = \|x_{i,j}\|,$$ $$\|E \cdot x_{i,j}\| = r^{j-i-1} = \frac{1}{rs} \|E\| \cdot \|x_{i,j}\| \leq \|E\| \cdot \|x_{i,j}\|,$$ $$\|F \cdot x_{i,j}\| \leq \text{max}\{1, |[i][\lambda+i-1-2j]|\}r^{j+1-i} \leq r^{j+1-i} = \|F\|\cdot \|x_{i,j}\|.$$ The map $M_{\lambda} \rightarrow \mathbf{f}^{\text{an}}_{r} \hat{\otimes} M_{\lambda}$ given by $$x_{i,j} \mapsto \sum_{k \geq 0} (-1)^{k} \frac{(q-q^{-1})^{k}}{[k]!} F^{k} \otimes x_{i+k,j}$$ is well defined and bounded since $|\frac{(q-q^{-1})^{k}}{[k]!}|\frac{r^{k}r^{j-i-k}}{r^{j-i}}=\frac{|2\hslash|^{k}}{|k!|} \rightarrow 0$. This shows that $M_{\lambda}$ is indeed in $\mathcal{O}$, since $$\sum_{k \geq 0} (-1)^{k} \frac{(q-q^{-1})^{k}}{[k]!} \langle E^{n},F^{k} \rangle \otimes x_{i+k,j}=x_{i+n,j}=E^{n}\cdot x_{i,j}.$$
Rigidity results
----------------
Classical rigidity results of Chevalley, Eilenberg and Cartan from the 1940s assert that there are no non-trivial formal deformations (as an algebra) of the universal enveloping algebra of a semisimple Lie algebra $\mathfrak{g}$. The proof relies on the vanishing of the second Lie algebra cohomology group. In this section we prove an analogous result that relies on a bounded cohomology vanishing result that has yet to be proven. We proceed as in Chapter XVIII of [@QG].\
We fix a set of root datum as in Definition \[KacMoodyRootDatum\].
Let $\mathscr{H}_{0}:=\coprod^{\leq 1}_{\alpha \in \mathbb{Z}I}k H_{\alpha}$ be the Banach Hopf algebra generated by $H_{i}$ for $i \in I$, where $$\Delta_{\mathscr{H}_{0}}(H_{i})=1 \otimes H_{i} + H_{i} \otimes 1.$$ Let $V_{0}:=\coprod^{\leq 1}_{i \in I}k \cdot v_{i}$ and let $T_{r}(V_{0})=\coprod_{n \geq 0}^{\leq 1} (V_{0}^{\hat{\otimes} n})_{r^{n}}$. For $r>0$ we will denote by both $U^{-}(\mathfrak{g})_{r}^{\text{an}}$ and $U^{+}(\mathfrak{g})_{r}^{\text{an}}$ the quotient of $T_{r}(V_{0})$ by the closed ideal generated by the Serre relations $$\sum_{k=0}^{1-(\alpha_{i},\alpha_{j})}(-1)^{k} {1-(\alpha_{i},\alpha_{j}) \choose k} v_{i}^{1-(\alpha_{i},\alpha_{j})-k}v_{j}v_{i}^{k}=0$$ for $i \neq j$.
Let $r,s>0$ such that $1 \leq rs$. Then there is a Banach algebra structure on $$U(\mathfrak{g})_{r,s}^{\text{an}}:=U^{-}(\mathfrak{g})_{r}^{\text{an}} \hat{\otimes} \mathscr{H}_{0} \hat{\otimes} U^{+}(\mathfrak{g})_{s}^{\text{an}}$$ such that $U^{-}(\mathfrak{g})_{r}^{\text{an}}$, $\mathscr{H}_{0}$ and $U^{+}(\mathfrak{g})_{s}^{\text{an}}$ are all subalgebras, and $$[H_{i},E_{j}]=(\alpha_{i},\alpha_{j})E_{j}, \quad [H_{i},F_{j}]=-(\alpha_{i},\alpha_{j})F_{j}, \quad [E_{i},F_{j}]=\delta_{i,j} H_{i},$$ where $F_{i}=v_{i} \otimes 1 \otimes 1$ and $E_{i}=1 \otimes 1 \otimes v_{i}$. This becomes a Banah Hopf algebra with $\mathscr{H}_{0}$ as a sub-Hopf algebra and $$\begin{array}{rclrcl}
\Delta(E_{i})&=&E_{i} \otimes 1 + 1 \otimes E_{i},& S(E_{i})&=&-E_{i},\\
\Delta(F_{i}) &=& F_{i} \otimes 1 + 1 \otimes F_{i},& S(F_{i})&=&-F_{i}.
\end{array}$$
By construction and the triangular decomposition given by the Poincaré-Birkhoff-Witt Theorem the enveloping algebra $U(\mathfrak{g})$ sits as a dense subspace of $U(\mathfrak{g})_{r,s}^{\text{an}}$ on which this Hopf algebra structure is well defined. It is therefore enough to check that it extends continuously. Since $$\begin{array}{rcccccl}
\|H_{i} \cdot F_{j}\| &=& \|F_{j}H_{i} - (\alpha_{i},\alpha_{j})F_{j}\| &\leq& r &=& \|H_{i}\| \|F_{j}\|\\
\|E_{i} \cdot H_{j}\| &=& \|H_{j}E_{i} + (\alpha_{i},\alpha_{j})E_{j}\| &\leq& s &=& \|E_{i}\| \|H_{j}\|\\
\|E_{i} \cdot F_{j}\| &=& \|F_{j}E_{i} - \delta_{i,j}H_{i}\| &\leq& rs &=& \|E_{i}\| \|F_{j}\|
\end{array}$$ and since $$\|\Delta(x)\|=\|1 \otimes x + x \otimes 1\| \leq \|x\|$$ for each generator $x \in \{F_{i},H_{i},E_{i}\mid i \in I\}$ we may define bounded linear transformations $$\left( T_{r}(V_{0}) \hat{\otimes} \mathscr{H}_{0} \hat{\otimes} T_{s}(V_{0}) \right) \hat{\otimes} \left( T_{r}(V_{0}) \hat{\otimes} \mathscr{H}_{0} \hat{\otimes} T_{s}(V_{0}) \right) \rightarrow U(\mathfrak{g})_{r,s}^{\text{an}}$$ $$T_{r}(V_{0}) \hat{\otimes} \mathscr{H}_{0} \hat{\otimes} T_{s}(V_{0}) \rightarrow U(\mathfrak{g})_{r,s}^{\text{an}} \hat{\otimes} U(\mathfrak{g})_{r,s}^{\text{an}}$$ which descend to the described multiplication and comultiplication maps.
We will denote by $\mathfrak{g}_{r,s}$ the closed Lie subalgebra of $U(\mathfrak{g})_{r,s}^{\text{an}}$ generated by $\{F_{i},H_{i},E_{i} \mid i \in I\}$. This is a Banach Lie algebra with $\| [x,y]\| \leq \|x\| \cdot \|y\|$ for $x,y \in \mathfrak{g}_{r,s}$.
Suppose we have a Banach algebra $A$ and a morphism of Banach Lie algebras $\mathfrak{g}_{r,s} \rightarrow A$ of norm at most 1. Then this extends to a unique contracting morphism of Banach algebras $U(\mathfrak{g})_{r,s}^{\text{an}} \rightarrow A$. In particular, given a Banach $\mathfrak{g}_{r,s}$ module $M$ whose action satisfies $\| x \cdot m\| \leq \|x\| \cdot \|m\|$ for $x \in \mathfrak{g}$, $m \in M$, this extends to a unique action of $U(\mathfrak{g})_{r,s}^{\text{an}}$ on $M$ such that the morphism $U(\mathfrak{g})_{r,s}^{\text{an}} \hat{\otimes} M \rightarrow M$ is contracting.
Taking the images of $F_{i}$, $H_{i}$ and $E_{i}$ in $A$ gives us a contracting map $$(T_{r}(V_{0}) \hat{\otimes} \mathscr{H}_{0} \hat{\otimes} T_{s}(V_{0})) \rightarrow A.$$ The images of $F_{i}$ and $E_{i}$ must satisfy the Serre relations in $A$, and hence this descends to a map $U(\mathfrak{g})_{r,s} \rightarrow A$. This restricts to a well defined algebra homomorphism on the dense subspace $U(\mathfrak{g})$, hence the result follows by continuity. Applying this to a morphism $\mathfrak{g}_{r,s} \rightarrow \text{Hom}(M,M)$ gives the rest of this result.
Let $M$ be a left Banach $\mathfrak{g}_{r,s}$ module whose action satisfies $\| x \cdot m\| \leq \|x\| \cdot \|m\|$ for $x \in \mathfrak{g}_{r,s}$, $m \in M$. Then we define the complex $$C_{b}^{n}(\mathfrak{g}_{r,s},M):=\text{Hom}(\Lambda^{n}\mathfrak{g}_{r,s},M),$$ the space of bounded antisymmetric n-linear maps from $\mathfrak{g}_{r,s}$ to $M$, and let $\delta_{n}:C_{b}^{n}(\mathfrak{g}_{r,s},M) \rightarrow C_{b}^{n+1}(\mathfrak{g}_{r,s},M)$ be the map $$\begin{array}{rcl}
\delta_{n}(f)(x_{1},.., x_{n+1})&=&\sum_{1 \leq i \leq j \leq n+1} (-1)^{i+j}f([x_{i},x_{j}],x_{1},..,\hat{x_{i}},..,\hat{x_{j}},..,x_{n+1})\\
&& \quad + \sum_{1 \leq i \leq n+1} (-1)^{i+1}x_{i}f(x_{1},..,\hat{x_{i}},..,x_{n+1}).
\end{array}$$ Note that $\|\delta_{n}\| \leq 1$ and $\delta_{n+1} \circ \delta_{n}=0$ for all $n \geq 0$. We define $H_{b}^{n}(\mathfrak{g}_{r,s},M)$ to be the seminormed space $\text{Ker}(\delta_{n})/\text{Im}(\delta_{n-1})$, the *$n$th bounded Lie algebra cohomology of $\mathfrak{g}_{r,s}$ with coefficients in $M$*. We will say that $C_{b}^{\bullet}(\mathfrak{g}_{r,s},M)$ is *strictly exact* at $C_{b}^{n}(\mathfrak{g}_{r,s},M)$ if $H_{b}^{n}(\mathfrak{g}_{r,s},M)=0$ and the induced map $$C_{b}^{n-1}(\mathfrak{g}_{r,s},M)/\text{Ker}(\delta_{n-1}) \rightarrow \text{Im}(\delta_{n-1})=\text{Ker}(\delta_{n})$$ is an isomorphism.
Let $M$ be as above. Then we say that a strict epimorphism $p:\tilde{\mathfrak{g}} \rightarrow \mathfrak{g}_{r,s}$ of Banach Lie algebras is an extension with kernel $M$ if $\text{Ker}(p) \cong M$ as $\mathfrak{g}_{r,s}$ modules, where $\text{Ker}(p) \subset \tilde{\mathfrak{g}}$ has the adjoint action. This extension is split if there exists a morphism of Banach Lie algebras $s:\mathfrak{g}_{r,s} \rightarrow \tilde{\mathfrak{g}}$ with $p \circ s = \text{Id}$.
\[SplittingLemma\] If $H^{2}_{b}(\mathfrak{g}_{r,s},M)=0$ then any extension $p:\tilde{\mathfrak{g}} \rightarrow \mathfrak{g}_{r,s}$ with kernel $M$ and $\|p\| \leq 1$ which is already split in $\text{Ban}_{k}$ is split as an extension of Lie algebras. Moreover, if $C_{b}^{\bullet}(\mathfrak{g}_{r,s},M)$ is strictly exact at $C_{b}^{2}(\mathfrak{g}_{r,s},M)$, the splitting $s$ has norm $\|s\| \leq C$ where $C$ is the norm of the isomorphism $\text{Ker}(\delta_{2})=\text{Im}(\delta_{1}) \overset{\sim}{\rightarrow} C_{b}^{1}(\mathfrak{g}_{r,s},M)/\text{Ker}(\delta_{1})$.
This proceeds as in Proposition XVIII.1.2. in [@QG]. Fix a splitting $\tilde{\mathfrak{g}}\cong \mathfrak{g}_{r,s} \oplus M$ as Banach spaces, and let $$f(x,y)=p([(x,0),(y,0)]) \text{ for } x,y \in \mathfrak{g}_{r,s},$$ so that $f \in C_{b}^{2}(\mathfrak{g}_{r,s},M)$ and $\|f\| \leq 1$. Then it is easily checked that $f \in \text{Ker}(\delta_{2})$ and hence $f=\delta_{1}(\alpha)$ for some $\alpha \in C_{b}^{1}(\mathfrak{g}_{r,s},M)$. If $C_{b}^{\bullet}(\mathfrak{g}_{r,s},M)$ is strictly exact at $C_{b}^{2}(\mathfrak{g}_{r,s},M)$ then $\|\alpha\| \leq C\|f\| \leq C$. Then, as in *loc. cit.*, $s(x)=(x,-\alpha(x))$ gives a splitting, and $\|s\| \leq \text{max}\{C,1\}$. Note that, as the inverse $C_{b}^{1}(\mathfrak{g}_{r,s},M)/\text{Ker}(\delta_{1}) \overset{\sim}{\rightarrow} \text{Ker}(\delta_{2})$ is contracting, we automatically have $C \geq 1$.
Let $M$ be a Banach $\mathfrak{g}_{r,s}$ bimodule. Then we denote by $\overline{M}$ the $\mathfrak{g}_{r,s}$ module whose underlying Banach space is $M$ with action $x \cdot m := xm-mx$.
\[VanishingofHb2Consequence\] Let $M$ be a Banach $\mathfrak{g}_{r,s}$ bimodule with $\|x m\| \leq \|x\| \|m \|$ and $\|m x\| \leq \|x\| \|m \|$ for all $x \in \mathfrak{g}_{r,s}$, $m \in M$. Let $f:U(\mathfrak{g})_{r,s}^{\text{an}} \hat{\otimes} U(\mathfrak{g})_{r,s}^{\text{an}} \rightarrow M$ be a bounded linear map such that $$f(1,x)=f(x,1)=0, \quad xf(y,z)-f(xy,z)+f(x,yz)-f(x,y)z=0,$$ for all $x,y,z \in U(\mathfrak{g})_{r,s}^{\text{an}}$, and $\|f\| \leq 1$. Suppose that $H_{b}^{2}(\mathfrak{g}_{r,s},\overline{M})=0$. Then there is a bounded bilinear map $\alpha:U(\mathfrak{g})_{r,s}^{\text{an}} \rightarrow M$ such that $$\alpha(1)=0 \text{ and } f(x,y)=x\alpha(y)-\alpha(xy)+\alpha(x)y \text{ for all } x,y \in U(\mathfrak{g})_{r,s}^{\text{an}}.$$ Furthermore, if we assume that $C_{b}^{\bullet}(\mathfrak{g}_{r,s},\overline{M})$ is strictly exact at $C_{b}^{2}(\mathfrak{g}_{r,s},\overline{M})$, and the isomorphism $\text{Ker}(\delta_{2})=\text{Im}(\delta_{2}) \overset{\sim}{\rightarrow} C_{b}^{1}(\mathfrak{g}_{r,s},\overline{M})/\text{Ker}(\delta_{1})$ is contracting, then we can take $\alpha$ such that $\|\alpha\| \leq 1$ .
We proceed as in Proposition XVIII.1.3. in [@QG]. We may define a contracting multiplication on $U(\mathfrak{g})_{r,s}^{\text{an}} \oplus M$ by $$(x,m) \cdot (y,n) = (xy, xn+my+f(x,y)) \text{ for } x,y \in U(\mathfrak{g})_{r,s}^{\text{an}}, m,n \in M.$$ Under the commutator bracket, $\tilde{\mathfrak{g}}=\mathfrak{g}_{r,s} \oplus M \subset U(\mathfrak{g})_{r,s}^{\text{an}} \oplus M$ is a Banach Lie algebra, and an extension of $\mathfrak{g}_{r,s}$ with kernel $\overline{M}$. Thus by Lemma \[SplittingLemma\], there exists $s:\mathfrak{g}_{r,s} \rightarrow \tilde{\mathfrak{g}}$ splitting the projection $p:\tilde{\mathfrak{g}} \rightarrow \mathfrak{g}_{r,s}$, with $\|s\| \leq 1$ under the stronger assumption. Then the map $\mathfrak{g}_{r,s} \overset{s}{\rightarrow} \tilde{\mathfrak{g}} \rightarrow U(\mathfrak{g})_{r,s}^{\text{an}} \oplus M$ induces a unique algebra homomorphism $s':U(\mathfrak{g})_{r,s}^{\text{an}} \rightarrow U(\mathfrak{g})_{r,s}^{\text{an}} \oplus M$ which splits the first projection. This map must be of the form $s'(x)=(x,-\alpha(x))$ for some $\alpha:U(\mathfrak{g})_{r,s}^{\text{an}} \rightarrow M$. But then, for all $x,y \in U(\mathfrak{g})_{r,s}^{\text{an}}$, $$(xy,-\alpha(xy))=s'(xy)=s'(x)s'(y)=(xy,-x\alpha(y)-\alpha(x)y+f(x,y))$$ which completes the proof.
\[Rigidity1\] Let $\mathfrak{g}_{r,s}$ and $\mathfrak{g}'_{r',s'}$ be Banach Lie algebras, each coming from some root datum. Let $1>\varepsilon>0$. Suppose we have two morphisms of $k\{\frac{\hslash}{\epsilon}\}$ algebras $\alpha, \alpha':U(\mathfrak{g})_{r,s}^{\text{an}}\{\frac{\hslash}{\varepsilon}\} \rightarrow U(\mathfrak{g}')_{r',s'}^{\text{an}}\{\frac{\hslash}{\varepsilon}\}$ such that $\alpha \equiv \alpha'$ modulo $\hslash$ and $\|\alpha\| \leq 1$, $\|\alpha'\| \leq 1$. Suppose that $C_{b}^{\bullet}(\mathfrak{g}_{r,s},U(\mathfrak{g}')_{r',s'}^{\text{an}})$ is strictly exact at $C_{b}^{1}(\mathfrak{g}_{r,s},U(\mathfrak{g}')_{r',s'}^{\text{an}})$ and the isomorphism $$\text{Ker}(\delta_{1})=\text{Im}(\delta_{0}) \overset{\sim}{\rightarrow} U(\mathfrak{g}')_{r',s'}^{\text{an}}/\text{Ker}(\delta_{0})$$ is contracting. Then, for any $\varepsilon>\varepsilon'>0$, there exists an invertible $F \in U(\mathfrak{g}')_{r',s'}^{\text{an}}\{\frac{\hslash}{\varepsilon'}\}$ such that $\alpha'(x)=F\alpha(x)F^{-1}$ for all $x \in U(\mathfrak{g})_{r,s}^{\text{an}}\{\frac{\hslash}{\varepsilon'}\}$ where we now view $\alpha$ and $\alpha'$ as maps from $U(\mathfrak{g})_{r,s}^{\text{an}}\{\tfrac{\hslash}{\varepsilon'}\} = U(\mathfrak{g})_{r,s}^{\text{an}}\{\tfrac{\hslash}{\varepsilon}\} \hat{\otimes}_{k\{\frac{\hslash}{\varepsilon}\}}k\{\tfrac{\hslash}{\varepsilon'}\}$ to $U(\mathfrak{g}')_{r',s'}^{\text{an}}\{\tfrac{\hslash}{\varepsilon'}\} = U(\mathfrak{g}')_{r',s'}^{\text{an}}\{\tfrac{\hslash}{\varepsilon}\} \hat{\otimes}_{k\{\frac{\hslash}{\varepsilon}\}}k\{\tfrac{\hslash}{\varepsilon'}\}$.
We proceed as in Theorem XVIII.2.1 of [@QG]. Fix $1>\varepsilon>\varepsilon'>0$ and a sequence $(\varepsilon_{n})_{n \geq 0}$ with $$\varepsilon>\varepsilon_{1}>\varepsilon_{2}>...>\varepsilon_{n}>\varepsilon_{n+1}>...>\varepsilon'.$$ Since $\alpha$ is $k\{\frac{\hslash}{\varepsilon}\}$-linear, it is uniquely determined by its restriction to $U(\mathfrak{g})_{r,s}^{\text{an}}$. We may write $\alpha$ in the form $$\alpha(x)=\sum \alpha_{i}(x)\hslash^{i} \text{ for } x \in U(\mathfrak{g})_{r,s}^{\text{an}}$$ for $\alpha_{i}:U(\mathfrak{g})_{r,s}^{\text{an}} \rightarrow U(\mathfrak{g}')_{r',s'}^{\text{an}}$, $\|\alpha_{i}\|\varepsilon^{i} \leq 1$. Now, suppose we have $u_{0},u_{1},...,u_{n} \in U(\mathfrak{g}')_{r',s'}^{\text{an}}$ such that $U_{i}\alpha \equiv \alpha_{0} U_{i}$ modulo $\hslash^{i+1}$, where $$U_{i}=(1+u_{i}\hslash^{i})(1+u_{i-1}\hslash^{i-1})...(1+u_{0}),$$ and $\|u_{i}\|\varepsilon_{i}^{i}< 1$. Note that $(1+u_{i}\hslash^{i})$ is then invertible in $U(\mathfrak{g})_{r,s}^{\text{an}}\{\tfrac{\hslash}{\varepsilon_{n}}\}$ for each $i=0,..,n$, with inverse $\sum_{j\geq 0} (-u_{i}\hslash^{i})^{j}$, and hence so is $U_{i}$, in which case $$\alpha^{(i)}(x):=U_{i}\alpha(x) U_{i}^{-1} \equiv \alpha_{0}(x) \text{ modulo } \hslash^{i+1}$$ for all $x \in U(\mathfrak{g})_{r,s}^{\text{an}}$. Note also that $$\|1+u_{i}\hslash^{i}\| = \|(1+u_{i}\hslash^{i})^{-1}\| = \|U_{i}\| = \|U_{i}^{-1}\|=1,$$ and so $\|\alpha^{(n)}\| \leq \|\alpha\|$ as maps $U(\mathfrak{g})_{r,s}^{\text{an}}\{\frac{\hslash}{\varepsilon_{n}}\} \rightarrow U(\mathfrak{g}')_{r',s'}^{\text{an}}\{\frac{\hslash}{\varepsilon_{n}}\}$. Again, $\alpha^{(n)}$ is $k\{\frac{\hslash}{\varepsilon_{n}}\}$-linear, and we may write the restricted map $\alpha^{(n)}:U(\mathfrak{g})_{r,s}^{\text{an}} \rightarrow U(\mathfrak{g}')_{r',s'}^{\text{an}}\{\tfrac{\hslash}{\varepsilon_{n}}\}$ as $$\alpha^{(n)}(x)=\sum \alpha^{(n)}_{i}(x)\hslash^{i} \text{ for } x \in U(\mathfrak{g})_{r,s}^{\text{an}}$$ for $\alpha_{i}^{(n)}:U(\mathfrak{g})_{r,s}^{\text{an}} \rightarrow U(\mathfrak{g}')_{r',s'}^{\text{an}}$, $\|\alpha_{i}^{(n)}\|\varepsilon_{n}^{i} \leq \|\alpha^{(n)}\|\leq\|\alpha\|\leq 1$. By assumption, $\alpha_{0}^{(n)}=\alpha_{0}$ and $\alpha_{i}^{(n)}=0$ for $i=1,...,n$. Looking at the $\hslash^{n+1}$ coefficient of $\alpha^{(n)}(xy)=\alpha^{(n)}(x)\alpha^{(n)}(y)$ we see that $$\begin{array}{rcl}
\alpha^{(n)}_{n+1}(xy)&=&\alpha^{(n)}_{0}(x)\alpha^{(n)}_{n+1}(y)+\alpha^{(n)}_{n+1}(x)\alpha^{(n)}_{0}(y)\\
&=&\alpha_{0}(x)\alpha^{(n)}_{n+1}(y)+\alpha^{(n)}_{n+1}(x)\alpha_{0}(y)
\end{array}$$ for all $x,y \in U(\mathfrak{g})_{r,s}^{\text{an}}$. Thus $$\alpha^{(n)}_{n+1}([x,y])=[\alpha_{0}(x),\alpha^{(n)}_{n+1}(y)]-[\alpha_{0}(y),\alpha^{(n)}_{n+1}(x)]$$ for all $x,y \in \mathfrak{g}_{r,s}$. Given that $x \in \mathfrak{g}_{r,s}$ acts on $U(\mathfrak{g}')_{r',s'}^{\text{an}}$ via $[\alpha_{0}(x),-]$, this is precisely the fact that $\alpha_{n+1}^{(n)}$ restricted to $\mathfrak{g}_{r,s}$ is in $\text{Ker}(\delta_{1})$. Hence there is a $u_{n+1} \in U(\mathfrak{g}')_{r',s'}^{\text{an}}$ such that $\alpha^{(n)}_{n+1}(x)=[\alpha_{0}(x),u_{n+1}]$ for all $x \in \mathfrak{g}_{r,s}$ and $\|u_{n+1}\| \leq \|\alpha_{n+1}^{(n)}\|$, so that $\|u_{n+1}\|\varepsilon_{n}^{n+1} \leq 1$. Then, in $U(\mathfrak{g}')_{r',s'}^{\text{an}}\{\tfrac{\hslash}{\varepsilon_{n+1}}\}$, $\|u_{n+1}\|\varepsilon_{n+1}^{n+1} < 1$ and $(1+u_{n+1}\hslash^{n+1})$ is invertible with $$\begin{array}{rcll}
\alpha^{(n+1)}(x)&:=& (1+u_{n+1}\hslash^{n+1})\alpha(x)(1+u_{n+1}\hslash^{n+1})^{-1}&\\
&\equiv& \alpha_{0}(x) + (u_{n+1}\alpha_{0}(x)-\alpha_{0}(x)u_{n+1}+\alpha_{n+1}^{(n)}(x))\hslash^{n+1} &\text{ mod } \hslash^{n+2}\\
&\equiv&\alpha_{0}(x) &\text{ mod } \hslash^{n+2}.
\end{array}$$ Taking $u_{0}:=0$ as our base case, we obtain inductively sequences $(u_{n})_{n \geq 0}$ and $(U_{n})_{n \geq 0}$ in $U(\mathfrak{g}')_{r',s'}^{\text{an}}\{\tfrac{\hslash}{\varepsilon'}\}$. The sequence $(U_{n})_{n \geq 0}$ converges to $U:=1+\sum_{n=1}^{\infty} v_{n}\hslash^{n}$ where $v_{n}=\sum u_{i_{1}}u_{i_{2}}...u_{i_{k}}$ whose sum is taken over all finite sequences $i_{1}>i_{2}>...>i_{k}$ with $i_{1}+i_{2}+...+i_{k}=n$. Since $\|v_{n}\|(\varepsilon')^{n}<1$ for all $n \geq 0$, $U$ is invertible in $U(\mathfrak{g}')_{r',s'}^{\text{an}}\{\tfrac{\hslash}{\varepsilon'}\}$ with inverse $U^{-1}=\sum_{i=0}^{\infty}(-\sum_{n=1}^{\infty} v_{n}\hslash^{n})^{i}$. It follows from the fact that $$U_{n}\alpha(x) U_{n}^{-1} \equiv \alpha_{0}(x) \text{ modulo } \hslash^{n+1}$$ for each $n\geq 0$ that $U\alpha(x)U^{-1}=\alpha_{0}(x)$ for each $x \in U(\mathfrak{g})_{r,s}^{\text{an}}\{\tfrac{\hslash}{\varepsilon'}\}$. Similarly, there exist mutual inverses $U'$, $U'^{-1}$ in $U(\mathfrak{g}')_{r',s'}^{\text{an}}\{\tfrac{\hslash}{\varepsilon'}\}$ such that $U'\alpha'(x)U'^{-1}=\alpha'_{0}(x)=\alpha_{0}(x)$ for each $x \in U(\mathfrak{g})_{r,s}^{\text{an}}\{\tfrac{\hslash}{\varepsilon'}\}$. Thus taking $F=U'^{-1}U$ gives our result.
\[Rigidity2\] Let $\varepsilon >0$. Suppose $A$ is a Banach $k\{\frac{\hslash}{\varepsilon}\}$ algebra with contracting multiplication such that there is a bounded $k\{\frac{\hslash}{\varepsilon}\}$-linear isomorphism of Banach spaces $A \cong U(\mathfrak{g})_{r,s}^{\text{an}}\{\frac{\hslash}{\varepsilon}\}$ that preserves the unit, and that $A/\hslash A \cong U(\mathfrak{g})_{r,s}^{\text{an}}$ is an isomorphism of Banach algebras. Suppose that $C_{b}^{\bullet}(\mathfrak{g}_{r,s},U(\mathfrak{g})_{r,s}^{\text{an}})$ is strictly exact at $C_{b}^{2}(\mathfrak{g}_{r,s},U(\mathfrak{g})_{r,s}^{\text{an}})$ and the isomorphism $$\text{Ker}(\delta_{2})=\text{Im}(\delta_{1}) \overset{\sim}{\rightarrow} C_{b}^{1}(\mathfrak{g}_{r,s},U(\mathfrak{g})_{r,s}^{\text{an}})/\text{Ker}(\delta_{1})$$ is contracting. Then for any $\varepsilon > \varepsilon' > 0$, there is an isomorphism of $k\{\frac{\hslash}{\varepsilon'}\}$ algebras $$\alpha:A_{\varepsilon'}:=A \hat{\otimes}_{k\{\frac{\hslash}{\varepsilon}\}}k\{\tfrac{\hslash}{\varepsilon'}\} \overset{\sim}{\rightarrow} U(\mathfrak{g})_{r,s}^{\text{an}}\{\tfrac{\hslash}{\varepsilon'}\}$$ inducing the given isomorphism $A_{\varepsilon'}/\hslash A_{\varepsilon'} \cong A/\hslash A \cong U(\mathfrak{g})_{r,s}^{\text{an}}$.
We proceed as in Theorem XVIII.2.2 of [@QG]. As before, fix $1>\varepsilon>\varepsilon'>0$ and a sequence $(\varepsilon_{n})_{n \geq 0}$ with $$\varepsilon>\varepsilon_{1}>\varepsilon_{2}>...>\varepsilon_{n}>\varepsilon_{n+1}>...>\varepsilon'.$$ The given isomorphism $A \cong U(\mathfrak{g})_{r,s}^{\text{an}}\{\frac{\hslash}{\varepsilon}\}$ induces a multiplication $$\mu:(U(\mathfrak{g})_{r,s}^{\text{an}} \hat{\otimes} U(\mathfrak{g})_{r,s}^{\text{an}})\{\tfrac{\hslash}{\varepsilon}\} \cong U(\mathfrak{g})_{r,s}^{\text{an}}\{\tfrac{\hslash}{\varepsilon}\} \hat{\otimes} U(\mathfrak{g})_{r,s}^{\text{an}}\{\tfrac{\hslash}{\varepsilon}\} \rightarrow U(\mathfrak{g})_{r,s}^{\text{an}}\{\tfrac{\hslash}{\varepsilon}\}$$ which is $k\{\frac{\hslash}{\varepsilon}\}$-linear and hence is determined by its restriction to $U(\mathfrak{g})_{r,s}^{\text{an}} \hat{\otimes} U(\mathfrak{g})_{r,s}^{\text{an}}$. We may write this restriction as $$\mu = \sum \mu_{i} \hslash^{i}, \quad \mu_{i}:U(\mathfrak{g})_{r,s}^{\text{an}} \hat{\otimes} U(\mathfrak{g})_{r,s}^{\text{an}} \rightarrow U(\mathfrak{g})_{r,s}^{\text{an}}, \quad \|\mu_{i}\|\varepsilon^{i} \leq 1.$$ Suppose we have maps $\alpha_{0},\alpha_{1},...,\alpha_{n}:U(\mathfrak{g})_{r,s}^{\text{an}} \rightarrow U(\mathfrak{g})_{r,s}^{\text{an}}$ such that $\|\alpha_{i}\|\varepsilon_{i}^{i}< 1$ and $V_{i}(\mu(x,y)) \equiv \mu_{0}(V_{i}(x),V_{i}(y))$ modulo $\hslash^{i+1}$ for all $x,y \in U(\mathfrak{g})_{r,s}^{\text{an}}$, where $V_{i}$ is the endomorphism of $U(\mathfrak{g})_{r,s}^{\text{an}}\{\frac{\hslash}{\varepsilon}\}$ with $$V_{i}|_{U(\mathfrak{g})_{r,s}^{\text{an}}}=(\text{Id}+\hslash^{i}\alpha_{i})(\text{Id}+\hslash^{i-1}\alpha_{i-1})...(\text{Id}+\alpha_{0}).$$ Note that, since $\|\alpha_{i}\|\varepsilon_{n}^{i} < \|\alpha_{i}\|\varepsilon_{i}^{i}\leq 1$, the endomorphism of $U(\mathfrak{g})_{r,s}^{\text{an}}\{\frac{\hslash}{\varepsilon_{n}}\}$ whose restriction to $U(\mathfrak{g})_{r,s}^{\text{an}}$ is $(\text{Id}+\hslash^{i}\alpha_{i})$ is then invertible for each $i=0,..,n$, with inverse $\sum_{j\geq 0} (-\hslash^{i}\alpha_{i})^{j}$, and hence so is $V_{i}$ as an endomorphism of $U(\mathfrak{g})_{r,s}^{\text{an}}\{\frac{\hslash}{\varepsilon_{n}}\}$, in which case $$\mu^{(i)}(x,y):=V_{i}(\mu(V_{i}^{-1}(x),V_{i}^{-1}(y))) \equiv \mu_{0}(x) \text{ modulo } \hslash^{i+1}$$ for all $x,y \in U(\mathfrak{g})_{r,s}^{\text{an}}$. Note also that $$\|1+\hslash^{i}\alpha_{i}\| = \|(1+\hslash^{i}\alpha_{i})^{-1}\| = \|V_{i}\| = \|V_{i}^{-1}\|=1,$$ and so $\|\mu^{(n)}\| \leq \|\mu\|$ as multiplication maps on $U(\mathfrak{g})_{r,s}^{\text{an}}\{\frac{\hslash}{\varepsilon_{n}}\}$. Again, $\mu^{(n)}$ is $k\{\frac{\hslash}{\varepsilon_{n}}\}$-linear, and we may write $$\mu^{(n)}(x,y)=\sum \mu^{(n)}_{i}(x,y)\hslash^{i} \text{ for } x,y \in U(\mathfrak{g})_{r,s}^{\text{an}}$$ where $\mu_{i}^{(n)}:U(\mathfrak{g})_{r,s}^{\text{an}}\hat{\otimes}U(\mathfrak{g})_{r,s}^{\text{an}} \rightarrow U(\mathfrak{g})_{r,s}^{\text{an}}$, $\|\mu_{i}^{(n)}\|\varepsilon_{n}^{i} \leq \|\mu^{(n)}\|\leq\|\mu\|\leq 1$. By assumption, $\mu_{0}^{(n)}=\mu_{0}$ and $\mu_{i}^{(n)}=0$ for $i=1,...,n$. Looking at the $\hslash^{n+1}$ coefficient of $\mu^{(n)}(\mu^{(n)}(x,y),z)=\mu^{(n)}(x,\mu^{(n)}(y,z))$ we see that $$\mu^{(n)}_{n+1}(xy,z)+\mu_{n+1}^{(n)}(x,y)z=\mu^{(n)}_{n+1}(x,yz)+x\mu^{(n)}_{n+1}(y,z)$$ for all $x,y,z \in U(\mathfrak{g})_{r,s}^{\text{an}}$. Here, we are using the simplified notation $xy:=\mu_{0}(x,y)$. So, by Lemma \[VanishingofHb2Consequence\], there exists $\alpha_{n+1}:U(\mathfrak{g})_{r,s}^{\text{an}} \rightarrow U(\mathfrak{g})_{r,s}^{\text{an}}$ such that $\|\alpha_{n+1}\| \leq \|\mu^{(n)}_{n+1}\|$, $\alpha_{n+1}(1)=0$ and $$\mu_{n+1}^{(n)}(x,y)=x\alpha_{n+1}(y)-\alpha_{n+1}(xy)+\alpha_{n+1}(x)y.$$ Setting $V_{n+1}$ as the endomorphism of $U(\mathfrak{g})_{r,s}^{\text{an}}\{\frac{\hslash}{\varepsilon_{n+1}}\}$ whose restriction to $U(\mathfrak{g})_{r,s}^{\text{an}}$ is $(\text{Id}+\hslash^{n+1}\alpha_{n+1})V_{n}$, we have that $V_{n+1}$ is invertible since $\|\alpha_{n+1}\|\varepsilon_{n+1}^{n+1} < 1$. Let $$\mu^{(n+1)}(x,y):=V_{n+1}(\mu(V_{n+1}^{-1}(x),V_{n+1}^{-1}(y))).$$ Then, modulo $\hslash^{n+2}$, $$\begin{array}{rcl}
\mu^{(n+1)}(x,y) &\equiv& (\text{Id}+\alpha_{n+1}\hslash^{n+1}) \circ (\mu_{0}+\mu^{(n)}_{n+1}\hslash^{n+1})\\
&& \qquad (x-\alpha_{n+1}(x)\hslash^{n+1},y-\alpha_{n+1}(y)\hslash^{n+1})\\
&\equiv& xy + (\alpha_{n+1}(xy) + \mu^{(n)}_{n+1}(x,y)-\alpha_{n+1}(x)y-x\alpha_{n+1}(y))\hslash^{n+1}\\
&\equiv& xy.
\end{array}$$ Taking $\alpha_{0}=0$ as a base case, we inductively obtain sequences $(\alpha_{n})_{n \geq 0}$ and $(V_{n})_{n \geq 0}$. The sequence $(V_{n})_{n \geq 0}$ converges on $U(\mathfrak{g})_{r,s}^{\text{an}}\{\frac{\hslash}{\varepsilon'}\}$ to $V:=\text{Id}+\sum_{n=1}^{\infty} \hslash^{n}\beta_{n}$ where $\beta_{n}=\sum \alpha_{i_{1}}\alpha_{i_{2}}...\alpha_{i_{k}}$ whose sum is taken over all finite sequences $i_{1}>i_{2}>...>i_{k}$ with $i_{1}+i_{2}+...+i_{k}=n$. Since $\|\alpha_{n}\|(\varepsilon')^{n}<1$, $V$ is invertible on $U(\mathfrak{g})_{r,s}^{\text{an}}\{\tfrac{\hslash}{\varepsilon'}\}$ with inverse $$V^{-1}=\sum_{i=0}^{\infty}(-\sum_{n=1}^{\infty} \hslash^{n}\beta_{n})^{i}.$$ It follows from the fact that $V_{n}\mu(V_{n}^{-1}(x),V_{n}^{-1}(y)) \equiv \mu_{0}(x,y) \text{ modulo } \hslash^{n+1}$ for each $n\geq 0$ that $V\mu(V^{-1}(x),V^{-1}(y)) = \mu_{0}(x,y)$ for each $x,y \in U(\mathfrak{g})_{r,s}^{\text{an}}\{\tfrac{\hslash}{\varepsilon'}\}$. It then follows that $$A_{\varepsilon'} \cong U(\mathfrak{g})_{r,s}^{\text{an}}\{\tfrac{\hslash}{\varepsilon'}\} \overset{V}{\longrightarrow} U(\mathfrak{g})_{r,s}^{\text{an}}\{\tfrac{\hslash}{\varepsilon'}\}$$ is our desired isomorphism of algebras.
The following are slight variations of the above theorems.
\[Rigidity1a\] Let $\mathfrak{g}_{r,s}$ and $\mathfrak{g}'_{r',s'}$ be Banach Lie algebras, each coming from some root datum. Let $\varepsilon \geq 0$. Suppose we have two morphisms of $k\{\frac{\hslash}{\varepsilon}\}^{\dagger}$ algebras $\alpha, \alpha':U(\mathfrak{g})_{r,s}^{\text{an}}\{\frac{\hslash}{\varepsilon}\}^{\dagger} \rightrightarrows U(\mathfrak{g}')_{r',s'}^{\text{an}}\{\frac{\hslash}{\varepsilon}\}^{\dagger}$ such that $\alpha \equiv \alpha'$ modulo $\hslash$. Suppose that $C_{b}^{\bullet}(\mathfrak{g}_{r,s},U(\mathfrak{g}')_{r',s'}^{\text{an}})$ is strictly exact at $C_{b}^{1}(\mathfrak{g}_{r,s},U(\mathfrak{g}')_{r',s'}^{\text{an}})$ and the isomorphism $$\text{Ker}(\delta_{1})=\text{Im}(\delta_{0}) \overset{\sim}{\rightarrow} U(\mathfrak{g}')_{r',s'}^{\text{an}}/\text{Ker}(\delta_{0})$$ is contracting. Then there exists a convolution invertible generalised element $F:k \rightarrow U(\mathfrak{g}')_{r',s'}^{\text{an}}\{\frac{\hslash}{\varepsilon}\}^{\dagger}$ such that $\alpha'=F \ast \alpha \ast F^{-1}$.
The morphisms $\alpha$ and $\alpha'$ are $k\{\frac{\hslash}{\varepsilon}\}^{\dagger}$-linear, so are determined by their restrictions $U(\mathfrak{g})_{r,s}^{\text{an}} \rightrightarrows U(\mathfrak{g}')_{r',s'}^{\text{an}}\{\frac{\hslash}{\varepsilon}\}^{\dagger}$. Since $U(\mathfrak{g})_{r,s}^{\text{an}}$ is Banach, there is $\varepsilon'>\varepsilon$ such that the restrictions of $\alpha$ and $\alpha'$ are determined by morphisms of Banach algebras $U(\mathfrak{g})_{r,s}^{\text{an}} \rightrightarrows U(\mathfrak{g}')_{r',s'}^{\text{an}}\{\frac{\hslash}{\varepsilon'}\}$. By the proof of Theorem \[Rigidity1\] there is $\varepsilon' > \varepsilon'' > \varepsilon$ and an invertible element $F \in U(\mathfrak{g}')_{r',s'}^{\text{an}}\{\frac{\hslash}{\varepsilon''}\}$ such that $\alpha'(x)=F\alpha(x)F^{-1} \in U(\mathfrak{g}')_{r',s'}^{\text{an}}\{\frac{\hslash}{\varepsilon''}\}$ for all $x \in U(\mathfrak{g})_{r,s}^{\text{an}}$. It then follows that $\alpha = F \ast \alpha' \ast F^{-1}$ as maps $U(\mathfrak{g})_{r,s}^{\text{an}}\{\frac{\hslash}{\varepsilon}\}^{\dagger} \rightrightarrows U(\mathfrak{g}')_{r',s'}^{\text{an}}\{\frac{\hslash}{\varepsilon}\}^{\dagger}$, where we denote by $F$ the generalised element $k \xrightarrow{1 \mapsto F} U(\mathfrak{g}')_{r',s'}^{\text{an}}\{\frac{\hslash}{\varepsilon''}\} \rightarrow U(\mathfrak{g}')_{r',s'}^{\text{an}}\{\frac{\hslash}{\varepsilon}\}^{\dagger}$.
\[Rigidity2a\] Let $\varepsilon \geq 0$. Suppose we have a Banach $k\{\frac{\hslash}{\varepsilon'}\}$-algebra $A_{\varepsilon'}$, for some $\varepsilon'>\varepsilon$, equipped with a $k\{\frac{\hslash}{\varepsilon'}\}$-linear isomorphism $A_{\varepsilon'} \cong U(\mathfrak{g})_{r,s}^{\text{an}}\{\frac{\hslash}{\varepsilon'}\}$ that preserves the unit such that $A_{\varepsilon'}/\hslash A_{\varepsilon'} \cong U(\mathfrak{g})_{r,s}^{\text{an}}$ is an isomorphism of Banach algebras. Suppose further that $C_{b}^{\bullet}(\mathfrak{g}_{r,s},U(\mathfrak{g})_{r,s}^{\text{an}})$ is strictly exact at $C_{b}^{2}(\mathfrak{g}_{r,s},U(\mathfrak{g})_{r,s}^{\text{an}})$ and the isomorphism $$\text{Ker}(\delta_{2})=\text{Im}(\delta_{1}) \overset{\sim}{\rightarrow} C_{b}^{1}(\mathfrak{g}_{r,s},U(\mathfrak{g})_{r,s}^{\text{an}})/\text{Ker}(\delta_{1})$$ is contracting. Then there is an isomorphism of $k\{\frac{\hslash}{\varepsilon}\}^{\dagger}$ algebras $$\alpha:A \overset{\sim}{\rightarrow} U(\mathfrak{g})_{r,s}^{\text{an}}\{\tfrac{\hslash}{\varepsilon}\}^{\dagger},$$ where $A=k\{\frac{\hslash}{\varepsilon}\}^{\dagger} \hat{\otimes}_{k\{\frac{\hslash}{\varepsilon'}\}} A_{\varepsilon'}$, inducing the given isomorphism $A/\hslash A \cong A_{\varepsilon'}/\hslash A_{\varepsilon'} \cong U(\mathfrak{g})_{r,s}^{\text{an}}$.
By Theorem \[Rigidity2\] there is $\varepsilon' > \varepsilon'' > \varepsilon$ and an isomorphism of $k\{\frac{\hslash}{\varepsilon''}\}$-algebras $$\alpha_{\varepsilon''}:A_{\varepsilon''} \overset{\sim}{\rightarrow} U(\mathfrak{g})_{r,s}^{\text{an}}\{\tfrac{\hslash}{\varepsilon''}\},$$ where $A_{\varepsilon''}=k\{\frac{\hslash}{\varepsilon''}\} \hat{\otimes}_{k\{\frac{\hslash}{\varepsilon'}\}} A_{\varepsilon'}$, inducing the given isomorphism $A_{\varepsilon''}/\hslash A_{\varepsilon''} \cong A_{\varepsilon'}/\hslash A_{\varepsilon'} \cong U(\mathfrak{g})_{r,s}^{\text{an}}$. Then $\alpha$ is the composition $$A \cong k\{ \tfrac{\hslash}{\varepsilon}\}^{\dagger} \hat{\otimes}_{k\{\frac{\hslash}{\varepsilon''}\}} A_{\varepsilon''} \xrightarrow{\text{Id} \otimes \alpha_{\varepsilon''}} k\{\tfrac{\hslash}{\varepsilon}\}^{\dagger} \hat{\otimes}_{k\{\frac{\hslash}{\varepsilon''}\}} U(\mathfrak{g})_{r,s}^{\text{an}}\{\tfrac{\hslash}{\varepsilon''}\} \cong U(\mathfrak{g})_{r,s}^{\text{an}}\{\tfrac{\hslash}{\varepsilon}\}^{\dagger}.$$
If we let $k=\mathbb{C}$ with the trivial valuation then $k\{\frac{\hslash}{\varepsilon}\}=k\{\frac{\hslash}{\varepsilon}\}^{\dagger}=\mathbb{C}\llbracket\hslash\rrbracket$ and we recover the classical rigidity results as stated in Section XVIII.2 of [@QG].
We may construct analytic quantum groups over $k\{\frac{\hslash}{\varepsilon}\}$ and $k\{\frac{\hslash}{\varepsilon}\}^{\dagger}$ for a formal parameter $\hslash$, where $q=e^{\hslash}$, to which the above deformation theory applies. For the remainder of this section, assume that $\varepsilon>0$ is sufficiently small such that $\text{exp}(\hslash)$ converges in $k\{\frac{\hslash}{\varepsilon}\}$.
Let $\mathscr{H}_{\frac{\hslash}{\varepsilon}}:=\coprod^{\leq 1}_{\alpha \in \mathbb{Z}I}k\{\frac{\hslash}{\varepsilon}\} \cdot H_{\alpha}$ be the Banach Hopf algebra over $k\{\frac{\hslash}{\varepsilon}\}$ generated by $H_{i}$ for $i \in I$, where $$\Delta_{\mathscr{H}}(H_{i})=1 \otimes H_{i} + H_{i} \otimes 1.$$ Let $V_{\frac{\hslash}{\varepsilon}}:=\coprod^{\leq 1}_{i \in I}k\{\frac{\hslash}{\varepsilon}\} \cdot v_{i}$ and define $$c:V_{\frac{\hslash}{\varepsilon}} \hat{\otimes}_{k\{\frac{\hslash}{\varepsilon}\}} V_{\frac{\hslash}{\varepsilon}} \rightarrow V_{\frac{\hslash}{\varepsilon}} \hat{\otimes}_{k\{\frac{\hslash}{\varepsilon}\}} V_{\frac{\hslash}{\varepsilon}}, \quad v_{i} \otimes v_{j} \mapsto q^{\lambda_{i}(\alpha_{j})} v_{j} \otimes v_{i},$$ where $q=e^{\hslash}$. Let $T_{r}(V_{\frac{\hslash}{\varepsilon}})$ be the resulting braided analytically graded Hopf algebra on $\coprod_{n \geq 0}^{\leq 1} (V_{\frac{\hslash}{\varepsilon}}^{\hat{\otimes} n})_{r^{n}}$ defined using the tensor product $\hat{\otimes}_{k\{\frac{\hslash}{\varepsilon}\}}$ over $k\{\tfrac{\hslash}{\varepsilon}\}$. We define a bilinear form $\langle-,- \rangle: V_{\frac{\hslash}{\varepsilon}} \hat{\otimes}_{k\{\frac{\hslash}{\varepsilon}\}} V_{\frac{\hslash}{\varepsilon}} \rightarrow k\{\frac{\hslash}{\varepsilon}\}$ by $$\langle v_{i},v_{j} \rangle = \delta_{i,j} \frac{\hslash}{(q_{i}-q_{i}^{-1})}$$ where $q_{i}=q^{\frac{(\alpha_{i},\alpha_{i})}{2}}$. By Lemma \[ExtendBilinearForm\] this extends to a bilinear form on $T_{r}(V_{\frac{\hslash}{\varepsilon}})$. Let $\mathfrak{B}_{r}(V_{\frac{\hslash}{\varepsilon}})$ be the quotient of $T_{r}(V_{\frac{\hslash}{\varepsilon}})$ by the radical of this bilinear form, which again is a braided analytically graded Banach Hopf algebra.
Note that $$q_{i}-q_{i}^{-1}=(\alpha_{i},\alpha_{i})\hslash \left(\sum_{k=0}^{\infty}\frac{(\frac{(\alpha_{i},\alpha_{i})}{2}\hslash)^{2k}}{k!}\right)$$ is not invertible in $k\{\frac{\hslash}{\varepsilon}\}$, but $\frac{1}{\hslash}(q_{i}-q_{i}^{-1})$ is. Thus we have had to rescale the inner product from Definition \[AnalyticQuantumGroupPositivePart\] in order to define it over $k\{\frac{\hslash}{\varepsilon}\}$.
Let $r,s>0$ such that $1 \leq |q_{i}-q_{i}^{-1}|rs$. Then there is a Banach algebra structure on $$U_{\frac{\hslash}{\varepsilon}}(\mathfrak{g})_{r,s}^{\text{an}}:=\mathfrak{B}_{r}(V_{\frac{\hslash}{\varepsilon}}) \hat{\otimes}_{k\{\frac{\hslash}{\varepsilon}\}} \mathscr{H}_{\frac{\hslash}{\varepsilon}} \hat{\otimes}_{k\{\frac{\hslash}{\varepsilon}\}} \mathfrak{B}_{s}(V_{\frac{\hslash}{\varepsilon}})$$ such that $\mathfrak{B}_{r}(V_{\frac{\hslash}{\varepsilon}})$, $\mathscr{H}_{\frac{\hslash}{\varepsilon}}$ and $\mathfrak{B}_{s}(V_{\frac{\hslash}{\varepsilon}})$ are all subalgebras, and $$[H_{i},E_{j}]=(\alpha_{i},\alpha_{j})E_{j}, \quad [H_{i},F_{j}]=-(\alpha_{i},\alpha_{j})F_{j}, \quad [E_{i},F_{j}]=\delta_{i,j} \frac{t_{i}-t_{i}^{-1}}{q_{i}-q_{i}^{-1}},$$ where $F_{i}=v_{i} \otimes 1 \otimes 1$, $E_{i}=1 \otimes 1 \otimes v_{i}$ and $t_{i}=\text{exp}(\frac{(\alpha_{i},\alpha_{i})}{2} \hslash H_{i})$. This becomes a Banah Hopf algebra with $\mathscr{H}_{\frac{\hslash}{\varepsilon}}$ as a sub-Hopf algebra and $$\begin{array}{rclrcl}
\Delta(E_{i})&=&E_{i} \otimes t_{i} + 1 \otimes E_{i},& S(E_{i})&=&-E_{i}t_{i}^{-1},\\
\Delta(F_{i}) &=& F_{i} \otimes 1 + t_{i}^{-1} \otimes F_{i},& S(F_{i})&=&-t_{i}F_{i}.
\end{array}$$
The fact that this is a Hopf algebra is checked on a dense subspace in Proposition 7 of Section 6.1.3 of [@QGaTR]. We must check that it extends continuously to $U_{\frac{\hslash}{\varepsilon}}(\mathfrak{g})_{r,s}^{\text{an}}$. By construction $$U_{\frac{\hslash}{\varepsilon}}^{\leq 1}(\mathfrak{g})_{r,s}^{\text{an}}:=\mathfrak{B}_{r}(V_{\frac{\hslash}{\varepsilon}}) \rtimes \mathscr{H}_{\frac{\hslash}{\varepsilon}} \quad \text{and} \quad U_{\frac{\hslash}{\varepsilon}}^{\geq 1}(\mathfrak{g})_{r,s}^{\text{an}}:=\mathscr{H}_{\frac{\hslash}{\varepsilon}} \ltimes \mathfrak{B}_{s}(V_{\frac{\hslash}{\varepsilon}})$$ sit as sub-Hopf algebras in $U_{\frac{\hslash}{\varepsilon}}(\mathfrak{g})_{r,s}^{\text{an}}$, it is enough to check that the restriction of the multiplication map $$\mathfrak{B}_{s}(V_{\frac{\hslash}{\varepsilon}}) \hat{\otimes} \mathfrak{B}_{r}(V_{\frac{\hslash}{\varepsilon}}) \rightarrow U_{\frac{\hslash}{\varepsilon}}(\mathfrak{g})_{r,s}^{\text{an}}, \quad E_{i} \otimes F_{j} \mapsto F_{i}E_{j} + \delta_{i,j} \frac{t_{i}-t_{i}^{-1}}{q_{i}-q_{i}^{-1}},$$ is continuous. This follows from the assumption that $$\left\|\frac{t_{i}-t_{i}^{-1}}{q_{i}-q_{i}^{-1}}\right\|=|q_{i}-q_{i}^{-1}|^{-1} \leq rs.$$
\[QuantumGroupModuloh\] Let $r,s>0$ such that $1 \leq |q_{i}-q_{i}^{-1}|rs$. Then there is an isomorphism of $k\{\frac{\hslash}{\varepsilon}\}$-modules $U_{\frac{\hslash}{\varepsilon}}(\mathfrak{g})_{r,s}^{\text{an}} \cong U(\mathfrak{g})_{r,s}^{\text{an}}\{\frac{\hslash}{\varepsilon}\}$ that descends to an isomorphism of Banach Hopf algebras $U_{\frac{\hslash}{\varepsilon}}(\mathfrak{g})_{r,s}^{\text{an}} / \hslash U_{\frac{\hslash}{\varepsilon}}(\mathfrak{g})_{r,s}^{\text{an}} \cong U(\mathfrak{g})_{r,s}^{\text{an}}$.
By Theorem 33.1.3 of [@ItQG] and Proposition \[WeakClassicalNicholsAlgebrasDense\], $\mathfrak{B}_{r}(V_{\frac{\hslash}{\varepsilon}})$ is the quotient of $T_{r}(V_{\frac{\hslash}{\varepsilon}})$ by the closed homogeneous ideal generated by the quantum Serre relations $$\sum_{k=0}^{1-(\alpha_{i},\alpha_{j})}(-1)^{k} \frac{[1-(\alpha_{i},\alpha_{j})]_{q}}{[k]_{q}[1-(\alpha_{i},\alpha_{j})-k]_{q}} v_{i}^{1-(\alpha_{i},\alpha_{j})-k}v_{j}v_{i}^{k}=0$$ for $i \neq j$. Since $V_{\frac{\hslash}{\varepsilon}} \rightarrow V_{0}\{\frac{\hslash}{\varepsilon}\}$, $v_{i} \mapsto v_{i}$, is an isomorphism there is an isomorphism of $k\{\frac{\hslash}{\varepsilon}\}$-modules $T_{r}(V_{\frac{\hslash}{\varepsilon}}) \cong T_{r}(V_{0})\{\frac{\hslash}{\varepsilon}\}$. By the Poincaré-Birkhoff-Witt Theorem this descends to isomorphisms between the graded pieces $\mathfrak{B}_{r}(V_{\frac{\hslash}{\varepsilon}})(n) \cong U^{-}(\mathfrak{g})_{r}^{\text{an}}(n)\{\frac{\hslash}{\varepsilon}\}$, hence $\mathfrak{B}_{r}(V_{\frac{\hslash}{\varepsilon}}) \cong U^{-}(\mathfrak{g})_{r}^{\text{an}}\{\frac{\hslash}{\varepsilon}\}$. Likewise $\mathscr{H}_{\frac{\hslash}{\varepsilon}} \cong \mathscr{H}_{0}\{\frac{\hslash}{\varepsilon}\}$. So, as Banach spaces, $$\mathfrak{B}_{r}(V_{\frac{\hslash}{\varepsilon}})/\hslash \mathfrak{B}_{r}(V_{\frac{\hslash}{\varepsilon}}) \cong U^{-}(\mathfrak{g})_{r}^{\text{an}} \quad \text{and} \quad \mathscr{H}_{\frac{\hslash}{\varepsilon}}/\hslash \mathscr{H}_{\frac{\hslash}{\varepsilon}} \cong \mathscr{H}_{0},$$ and so $U_{\frac{\hslash}{\varepsilon}}(\mathfrak{g})_{r,s}^{\text{an}} / \hslash U_{\frac{\hslash}{\varepsilon}}(\mathfrak{g})_{r,s}^{\text{an}} \cong U(\mathfrak{g})_{r,s}^{\text{an}}$. By Remark 4 of Section 6.1.3 this restricts to a Hopf algebra isomorphism on a dense subspace, hence is a Hopf algebra isomorphism by continuity.
Let $r,s>0$ such that $1 \leq |q_{i}-q_{i}^{-1}|rs$ and let $\varepsilon >0$. Then for any $\varepsilon'>\varepsilon$ we define $$U_{\frac{\hslash}{\varepsilon}}^{\dagger}(\mathfrak{g})_{r,s}^{\text{an}}:=U_{\frac{\hslash}{\varepsilon'}}(\mathfrak{g})_{r,s}^{\text{an}} \hat{\otimes}_{k\{\frac{\hslash}{\varepsilon'}\}} k\{\hslash/\varepsilon\}^{\dagger}.$$
\[QuantumGroupModuloDaggerh\] Let $r,s>0$ such that $1 \leq |q_{i}-q_{i}^{-1}|rs$. Then there is an isomorphism of $k\{\frac{\hslash}{\varepsilon}\}^{\dagger}$-modules $U_{\frac{\hslash}{\varepsilon}}^{\dagger}(\mathfrak{g})_{r,s}^{\text{an}} \cong U(\mathfrak{g})_{r,s}^{\text{an}}\{\frac{\hslash}{\varepsilon}\}^{\dagger}$ that descends to an isomorphism of Banach Hopf algebras $U^{\dagger}_{\frac{\hslash}{\varepsilon}}(\mathfrak{g})_{r,s}^{\text{an}} / \hslash U^{\dagger}_{\frac{\hslash}{\varepsilon}}(\mathfrak{g})_{r,s}^{\text{an}} \cong U(\mathfrak{g})_{r,s}^{\text{an}}$.
This follows from Theorem \[QuantumGroupModuloh\].
\[RigidityApplied\] Suppose that the complex $C_{b}^{\bullet}(\mathfrak{g}_{r,s},U(\mathfrak{g})_{r,s}^{\text{an}})$ is strictly exact at both $C_{b}^{1}(\mathfrak{g}_{r,s},U(\mathfrak{g})_{r,s}^{\text{an}})$ and $C_{b}^{2}(\mathfrak{g}_{r,s},U(\mathfrak{g})_{r,s}^{\text{an}})$, and that the isomorphisms $$\text{Ker}(\delta_{1})=\text{Im}(\delta_{0}) \overset{\sim}{\longrightarrow} U(\mathfrak{g})_{r,s}^{\text{an}}/\text{Ker}(\delta_{0})$$ and $$\text{Ker}(\delta_{2})=\text{Im}(\delta_{1}) \overset{\sim}{\longrightarrow} C_{b}^{1}(\mathfrak{g}_{r,s},U(\mathfrak{g})_{r,s}^{\text{an}})/\text{Ker}(\delta_{1})$$ are contracting. Then there is an isomorphism of algebras $$U_{\frac{\hslash}{\varepsilon}}^{\dagger}(\mathfrak{g})_{r,s}^{\text{an}} \overset{\sim}{\longrightarrow} U(\mathfrak{g})_{r,s}^{\text{an}}\{\tfrac{\hslash}{\varepsilon}\}^{\dagger}$$ that induces the isomorphism $U_{\frac{\hslash}{\varepsilon}}^{\dagger}(\mathfrak{g})_{r,s}^{\text{an}} / \hslash U_{\frac{\hslash}{\varepsilon}}^{\dagger}(\mathfrak{g})_{r,s}^{\text{an}} \cong U(\mathfrak{g})_{r,s}^{\text{an}}$ of Corollary \[QuantumGroupModuloDaggerh\]. Furthermore, this isomorphism is unique up to conjugation by a convolution invertible generalised element of $U(\mathfrak{g})_{r,s}^{\text{an}}\{\tfrac{\hslash}{\varepsilon}\}^{\dagger}$.
This follows from Corollary \[Rigidity1a\], Corollary \[Rigidity2a\] and Corollary \[QuantumGroupModuloDaggerh\].
If $\mathfrak{g}$ is finite dimensional then, as vector spaces, bounded Lie algebra cohomology $H_{b}^{n}(\mathfrak{g}_{r,s},M)$ agrees with the unbounded Lie algebra cohomology $H^{n}(\mathfrak{g},M)$ as defined in Chapter XVIII of [@QG]. Furthermore, if we also assume that $k$ is algebraically closed and $\mathfrak{g}$ is semisimple then $H^{n}(\mathfrak{g},U(\mathfrak{g}))=0$ by Corollary XVIII.3.3 of [@QG] for $n=1,2$. Unfortunately, it is unclear whether the same argument can be extended to show that $H_{b}^{n}(\mathfrak{g}_{r,s},U(\mathfrak{g})^{\text{an}}_{r,s})$ vanishes when $n=1,2$, or whether we have the strict exactness required for Corollary \[RigidityApplied\] to hold. This is a goal of future work by the author. At the end of the following section we show that if we allow ourselves to work over formal powerseries in $\hslash$ then we may weaken these assumptions on bounded cohomology to remove the requirement of strict exactness.\
In [@QG], Kassel uses an algebraic analogue of the rigidity theorems of this Section, alongside another based on results of Drinfel’d from 1989 regarding quasi-Hopf algebra structures on universal enveloping algebra, to present a proof of the Drinfel’d-Khono theorem. This theorem states that the category of representations of the quantum enveloping algebra is equivalent, as a braided monoidal category, to the category of $U(\mathfrak{g})$-modules with associativity constraint given by the Drinfel’d associator and braiding given by the associated R-matrix. As a result of this, the associated braid group representations are equivalent. This can be interpreted as a statement about the monodromy of the Knizhnik-Zamolodchikov (KZ) equations that govern the Drinfel’d associator. In [@pMPatpKZE], Furusho uses $p$-adic multiple polylogarithms to construct solutions to the $p$-adic KZ equations and a $p$-adic Drinfel’d associator. In the future the author hopes to prove a $p$-adic analogue of the Drinfel’d-Khono theorem and to investigate this link to Furusho’s work.
Analytic quantum groups over $k\llbracket \hslash \rrbracket$ {#WorkingOverk[[h]]}
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Let $k \llbracket \hslash \rrbracket$ denote the IndBanach algebra of powerseries in $\hslash$, $$k \llbracket \hslash \rrbracket := \lim_{n \geq 0} k[\hslash]/(\hslash^{n}) = \prod_{n \geq 0} k \cdot \hslash^{n}.$$ For an IndBanach space $V$ let $V\llbracket \hslash \rrbracket := \prod\nolimits _{n\geq 0} V\cdot \hslash^{n}$, which forms an IndBanach algebra if $V$ is an algebra.
\[CountableProductsCommuteWithTensor\] Let $(V(n))_{n \in \mathbb{N}}$ be a countable collection of Banach spaces, and $W$ be a Banach space. Then the natural map $$\left(\prod_{n \geq 0}V(n)\right) \hat{\otimes} W \rightarrow \prod_{n \geq 0} \left(V(n) \hat{\otimes} W\right)$$ is an isomorphism.
Fix a summable sequence of positive real numbers $a_{n} \in \mathbb{R}_{>0}$. By the explicit description of products in Section 1.4.1 in [@LaACH], we have that $$\prod\nolimits_{n \geq 0}V(n) \cong \text{colim}_{r_{n}>0} \prod\nolimits^{\leq 1}_{n \geq 0} V(n)_{r_{n}}.$$ The maps $$\coprod\nolimits^{\leq 1}_{n \geq 0} V(n)_{r_{n}} \rightarrow \prod\nolimits^{\leq 1}_{n \geq 0} V(n)_{r_{n}}, \quad (v_{n})_{n \geq 0} \mapsto (v_{n})_{n \geq 0},$$ and $$\prod\nolimits^{\leq 1}_{n \geq 0} V(n)_{r_{n}} \rightarrow \coprod\nolimits^{\leq 1}_{n \geq 0} V(n)_{a_{n}r_{n}}, \quad (v_{n})_{n \geq 0} \mapsto (v_{n})_{n \geq 0},$$ induce an isomorphism $$\text{colim}_{r_{n}>0}\prod\nolimits^{\leq 1}_{n \geq 0} V(n)_{r_{n}} \overset{\sim}{\longrightarrow} \text{colim}_{r_{n}>0}\coprod\nolimits^{\leq 1}_{n \geq 0} V(n)_{r_{n}}.$$ Hence $$\begin{array}{rcl}
\left(\prod_{n \geq 0}V(n)\right) \hat{\otimes} W &\cong& \left( \text{colim}_{r_{n}>0}\prod\nolimits^{\leq 1}_{n \geq 0} V(n)_{r_{n}} \right) \hat{\otimes} W\\
&\cong& \left( \text{colim}_{r_{n}>0}\coprod\nolimits^{\leq 1}_{n \geq 0} V(n)_{r_{n}} \right) \hat{\otimes} W\\
&\cong& \text{colim}_{r_{n}>0}\coprod\nolimits^{\leq 1}_{n \geq 0} (V(n) \hat{\otimes} W)_{r_{n}}\\
&\cong& \text{colim}_{r_{n}>0}\prod\nolimits^{\leq 1}_{n \geq 0} (V(n) \hat{\otimes} W)_{r_{n}}\\
&\cong& \prod_{n \geq 0} (V(n) \hat{\otimes} W).
\end{array}$$
\[V\[\[h\]\]=Votimesk\[\[h\]\]\] For a Banach space $V$, $V\llbracket \hslash \rrbracket \cong V \hat{\otimes}_{k} k \llbracket \hslash \rrbracket$. Hence $$V\llbracket \hslash \rrbracket \hat{\otimes}_{k\llbracket \hslash \rrbracket} W\llbracket \hslash \rrbracket \cong (V \hat{\otimes} W)\llbracket \hslash \rrbracket$$ for all Banach spaces $V$ and $W$.
This follows from Lemma \[CountableProductsCommuteWithTensor\].
Let $r,s>0$ such that $1 \leq |q_{i}-q_{i}^{-1}|rs$. Then for any sufficiently small $\varepsilon>0$ we define $$U_{\llbracket \hslash \rrbracket}(\mathfrak{g})_{r,s}^{\text{an}}:=U_{\frac{\hslash}{\varepsilon}}(\mathfrak{g})_{r,s}^{\text{an}} \hat{\otimes}_{k\{\frac{\hslash}{\varepsilon}\}} k\llbracket \hslash \rrbracket$$ as an IndBanach Hopf algebra over $k \llbracket \hslash \rrbracket$.
### Quasi-triangularity
The following is essentially a restatement of a well known result of Drinfel’d.
Suppose that $\mathfrak{g}$ is a simple Lie algebra. Then the IndBanach Hopf algebra $U_{\llbracket \hslash \rrbracket}(\mathfrak{g})_{r,s}^{\text{an}}$ is quasi-triangular.
Fix $\varepsilon>0$. By the proof of Lemma \[CountableProductsCommuteWithTensor\] we may write $U_{\llbracket \hslash \rrbracket}(\mathfrak{g})_{r,s}^{\text{an}}$ as the colimit $$U_{\llbracket \hslash \rrbracket}(\mathfrak{g})_{r,s}^{\text{an}} = \text{"colim"}_{(\varepsilon_{n})} \ U_{\frac{\hslash}{\varepsilon}}(\mathfrak{g})_{r,s}^{\text{an}} \hat{\otimes}_{k\{\frac{\hslash}{\varepsilon}\}} \left( \coprod\nolimits_{n \geq 0}^{\leq 1} k_{\varepsilon_{n}} \cdot \hslash^{n} \right)$$ where the colimit is taken over all sequences of positive real numbers $(\varepsilon_{n})_{n \geq 0}$ such that $\varepsilon^{n} \varepsilon_{n'} \geq \varepsilon_{n+n'}$ for all $n,n' \geq 0$. Note that the requirement on $(\varepsilon_{n})_{n \geq 0}$ ensures that $\coprod\nolimits_{n \geq 0}^{\leq 1} k_{\varepsilon_{n}} \cdot \hslash^{n}$ is naturally a $k\{\frac{\hslash}{\varepsilon}\}$-module. Theorem 17 of Section 8.3.2 of [@QGaTR] exhibits a formula for an R-matrix in the $\hslash$-adic qunatum enveloping algebra over $\mathbb{C}$. As in *loc. cit.* this formula naturally converges in $U_{\frac{\hslash}{\varepsilon}}(\mathfrak{g})_{r,s}^{\text{an}} \hat{\otimes}_{k\{\frac{\hslash}{\varepsilon}\}} \left( \coprod\nolimits_{n \geq 0}^{\leq 1} k_{\varepsilon_{n}} \cdot \hslash^{n} \right)$ for some sufficiently small sequence $(\varepsilon_{n})_{n \geq 0}$, which gives a generalised element of $U_{\llbracket \hslash \rrbracket}(\mathfrak{g})_{r,s}^{\text{an}}$ that makes $U_{\llbracket \hslash \rrbracket}(\mathfrak{g})_{r,s}^{\text{an}}$ quasi-triangular.
### Rigidity
\[Rigidity1b\] Let $\mathfrak{g}$ and $\mathfrak{g}'$ be Banach Lie algebras. Suppose we have two morphisms of $k\llbracket \hslash \rrbracket$ algebras $\alpha, \alpha':U(\mathfrak{g})_{r,s}^{\text{an}}\llbracket \hslash \rrbracket \rightarrow U(\mathfrak{g}')_{r',s'}^{\text{an}}\llbracket \hslash \rrbracket$ such that $\alpha \equiv \alpha'$ modulo $\hslash$. Suppose that $H_{b}^{1}(\mathfrak{g},U(\mathfrak{g}')_{r',s'}^{\text{an}})=0$. Then there exists a convolution invertible generalised element $F: k \rightarrow U(\mathfrak{g}')_{r',s'}^{\text{an}}\llbracket \hslash \rrbracket$ such that $\alpha'=F \ast \alpha \ast F^{-1}$.
This follows as in the proof of Theorem \[Rigidity1\]. By Corollary \[V\[\[h\]\]=Votimesk\[\[h\]\]\], $\alpha$ is uniquely determined by its restriction to $U(\mathfrak{g})_{r,s}^{\text{an}}$, which may be written as a formal sum $\sum \hslash^{i}\alpha_{i}$ for $\alpha_{i}:U(\mathfrak{g})_{r,s}^{\text{an}} \rightarrow U(\mathfrak{g}')_{r',s'}^{\text{an}}$ bounded. We suppose we have $u_{0},u_{1},...,u_{n} \in U(\mathfrak{g}')_{r',s'}^{\text{an}}$ such that $U_{i}\alpha \equiv \alpha_{0} U_{i}$ modulo $\hslash^{i+1}$ as before, where $U_{i}=(1+u_{i}\hslash^{i})(1+u_{i-1}\hslash^{i-1})...(1+u_{0})$ is now a generalised element of $U(\mathfrak{g}')_{r',s'}^{\text{an}}\llbracket \hslash \rrbracket$. Since we are working with formal powerseries, both $(1+u_{i}\hslash^{i})$ and $U_{i}$ are automatically convolution invertible. Again, if $\alpha^{(n)}:=U_{n} \ast \alpha \ast U_{n}^{-1}$ whose restriction to $U(\mathfrak{g})_{r,s}^{\text{an}}$ is given by the formal sum $\sum \hslash^{i}\alpha^{(n)}_{i}$, then $\alpha_{n+1}^{(n)}$ restricts to a bounded 1-cocycle on $\mathfrak{g}$, hence is a 1-coboundary. We therefore obtain $u_{n+1} \in U(\mathfrak{g}')_{r',s'}^{\text{an}}$ such that $\alpha^{(n)}_{n}(x)=[\alpha_{0}(x),u_{n+1}]$ for all $x \in \mathfrak{g}_{r,s}$. Then $(1+u_{n+1}\hslash^{n+1})$ is an invertible generalised element of $U(\mathfrak{g}')_{r',s'}^{\text{an}}\llbracket \hslash \rrbracket$ and as in the proof of Theorem \[Rigidity1\] we have $$(1+u_{n+1}\hslash^{n+1})\alpha(x)(1+u_{n+1}\hslash^{n+1})^{-1}\equiv\alpha_{0}(x) \text{ mod } \hslash^{n+2}.$$ Taking $u_{0}:=0$ as our base case, as before, inductively gives a sequence of convolution invertible generalised elements $(U_{n})_{n \geq 0}$ that converge in each $U(\mathfrak{g})_{r,s}^{\text{an}} \cdot \hslash^{n}$ to give $U:k \rightarrow U(\mathfrak{g}')_{r',s'}^{\text{an}}\llbracket \hslash \rrbracket$ such that $U \ast \alpha \ast U^{-1}=\alpha_{0}$. Similarly there is a convolution invertible generalised element $U'$ of $U(\mathfrak{g}')_{r',s'}^{\text{an}}\llbracket \hslash \rrbracket$ such that $U' \ast \alpha' \ast U'^{-1}=\alpha'_{0}=\alpha_{0}$. Taking $F=U'^{-1} \ast U$ gives our result.
\[Rigidity2b\] Suppose $A$ is an IndBanach $k\llbracket \hslash \rrbracket$ algebra equipped with a $k\llbracket \hslash \rrbracket$-linear isomorphism $A \cong U(\mathfrak{g})_{r,s}^{\text{an}}\llbracket \hslash \rrbracket$ that preserves the unit, and that the induced isomorphism $A/\hslash A \cong U(\mathfrak{g})_{r,s}^{\text{an}}$ is an isomorphism of algebras. Suppose that $H_{b}^{2}(\mathfrak{g},U(\mathfrak{g})_{r,s}^{\text{an}})=0$. Then there is an isomorphism of $k\llbracket \hslash \rrbracket$-algebras $A \cong U(\mathfrak{g})_{r,s}^{\text{an}}\llbracket \hslash \rrbracket$ inducing the given algebra isomorphism modulo $\hslash$.
This proof follows as for Theorem \[Rigidity2\]. The alternate multiplication $\mu$ on $U(\mathfrak{g})_{r,s}^{\text{an}}$ induced by $A$ is again determined by its restriction to $U(\mathfrak{g})_{r,s}^{\text{an}} \hat{\otimes} U(\mathfrak{g})_{r,s}^{\text{an}}$, which may written as a formal sum $\mu = \sum \hslash^{i}\mu_{i}$ for $$\mu_{i}:U(\mathfrak{g})_{r,s}^{\text{an}} \hat{\otimes} U(\mathfrak{g})_{r,s}^{\text{an}} \rightarrow U(\mathfrak{g})_{r,s}^{\text{an}}.$$ We again assume we have maps $\alpha_{0},\alpha_{1},...,\alpha_{n}:U(\mathfrak{g})_{r,s}^{\text{an}} \rightarrow U(\mathfrak{g})_{r,s}^{\text{an}}$ such that $V_{i}(\mu(x,y)) \equiv \mu_{0}(V_{i}(x),V_{i}(y))$ modulo $\hslash^{i+1}$ for all $x,y \in U(\mathfrak{g})_{r,s}^{\text{an}}$, where $$V_{i}=(\text{Id}+\hslash^{i}\alpha_{i})(\text{Id}+\hslash^{i-1}\alpha_{i-1})...(\text{Id}+\alpha_{0}):U(\mathfrak{g})_{r,s}^{\text{an}}\llbracket \hslash \rrbracket \rightarrow U(\mathfrak{g})_{r,s}^{\text{an}}\llbracket \hslash \rrbracket.$$ Again, since we are working with formal powerseries, each $V_{i}$ is automatically invertible and give new multiplication maps $\mu^{(i)}:=V_{i} \circ \mu \circ (V_{i}^{-1} \otimes V_{i}^{-1})$ on $U(\mathfrak{g})_{r,s}^{\text{an}}\llbracket \hslash \rrbracket$. We write the restriction of $\mu^{(i)}$ to $U(\mathfrak{g})_{r,s}^{\text{an}}$ as a formal sum $\sum \hslash^{j} \mu^{(i)}_{j}$. As in the proof of Theorem \[Rigidity2\], $\mu_{n+1}^{(n)}$ satisfies the conditions of Lemma \[VanishingofHb2Consequence\], so gives $\alpha_{n+1}:U(\mathfrak{g})_{r,s}^{\text{an}} \rightarrow U(\mathfrak{g})_{r,s}^{\text{an}}$ such that $$(\text{Id}+\alpha_{n+1}\hslash^{n+1}) \circ \mu^{(n)} \circ \left( (\text{Id}+\alpha_{n+1}\hslash^{n+1})^{-1} \otimes (\text{Id}+\alpha_{n+1}\hslash^{n+1})^{-1} \right)$$ is equivalent to $\mu_{0}$ modulo $\hslash^{n+1}$. Taking $\alpha_{0}=0$ as a base case, we inductively obtain a sequence of morphisms $(V_{n})_{n \geq 0}$. Their restrictions to $U(\mathfrak{g})_{r,s}^{\text{an}}$ converges in each $U(\mathfrak{g})_{r,s}^{\text{an}} \cdot \hslash^{n}$ to give a morphism $V: U(\mathfrak{g})_{r,s}^{\text{an}} \llbracket \hslash \rrbracket \rightarrow U(\mathfrak{g})_{r,s}^{\text{an}} \llbracket \hslash \rrbracket$. It then follows that $$A \cong U(\mathfrak{g})_{r,s}^{\text{an}} \llbracket \hslash \rrbracket \xrightarrow{V} U(\mathfrak{g})_{r,s}^{\text{an}}\llbracket \hslash \rrbracket$$ is our desired isomorphism of algebras.
\[QuantumGroupModuloPowerseriesh\] Let $r,s>0$ such that $1 \leq |q_{i}-q_{i}^{-1}|rs$. Then there is an isomorphism of $k\llbracket \hslash \rrbracket$-modules $U_{\llbracket \hslash \rrbracket}(\mathfrak{g})_{r,s}^{\text{an}} \cong U(\mathfrak{g})_{r,s}^{\text{an}}\llbracket \hslash \rrbracket$ that descends to an isomorphism of Banach Hopf algebras $U_{\llbracket \hslash \rrbracket}(\mathfrak{g})_{r,s}^{\text{an}} / \hslash U_{\llbracket \hslash \rrbracket}(\mathfrak{g})_{r,s}^{\text{an}} \cong U(\mathfrak{g})_{r,s}^{\text{an}}$.
This follows from Theorem \[QuantumGroupModuloh\].
Suppose that $$H_{b}^{1}(\mathfrak{g},U(\mathfrak{g})_{r,s}^{\text{an}})=0=H_{b}^{2}(\mathfrak{g},U(\mathfrak{g})_{r,s}^{\text{an}}).$$ Then there is an isomorphism of algebras $$U_{\llbracket \hslash \rrbracket}(\mathfrak{g})_{r,s}^{\text{an}} \overset{\sim}{\longrightarrow} U(\mathfrak{g})_{r,s}^{\text{an}} \llbracket \hslash \rrbracket$$ that induces the isomorphism $U_{\llbracket \hslash \rrbracket}(\mathfrak{g})_{r,s}^{\text{an}} / \hslash U_{\llbracket \hslash \rrbracket}(\mathfrak{g})_{r,s}^{\text{an}} \cong U(\mathfrak{g})_{r,s}^{\text{an}}$. Furthermore, this isomorphism is unique up to conjugation by a convolution invertible generalised element of $U(\mathfrak{g})_{r,s}^{\text{an}}\llbracket \hslash \rrbracket$.
This follows from Theorem \[Rigidity1b\], Theorem \[Rigidity2b\] and Corollary \[QuantumGroupModuloPowerseriesh\].
Archimedean analytic quantum groups {#AQGSection}
===================================
Throughout this section we shall assume [**(A)**]{}.
Constructing Archimedean Analytic Nichols algebras {#ANicholsAlegbraSection}
--------------------------------------------------
\[ACoidealsOfTensorAlgebras\] Let $V$ be a Banach space with pre-braiding $c$ of norm at most 1. Suppose we have closed homogeneous subspaces $I_{r} \subset J_{r} \subset T_{r}(V)$ for each $r>0$ such that, whenever $r \geq r'$, $T_{r}(V) \rightarrow T_{r'}(V)$ maps $I_{r}$ and $J_{r}$ to $I_{r'}$ and $J_{r'}$ respectively. Suppose further that $\Delta(I_{r}) \subset \overline{I_{\frac{r}{2}} \hat{\otimes} T_{\frac{r}{2}}(V) + T_{\frac{r}{2}}(V) \hat{\otimes} I_{\frac{r}{2}}}$ and $\Delta(J_{r}) \subset \overline{J_{\frac{r}{2}} \hat{\otimes} T_{\frac{r}{2}}(V) + T_{\frac{r}{2}}(V) \hat{\otimes} J_{\frac{r}{2}}}$. If the induced map $$P(\text{colim}_{r>0} T_{r}(V)/I_{r}) \rightarrow \text{colim}_{r>0} T_{r}(V)/I_{r} \rightarrow \text{colim}_{r>0} T_{r}(V)/J_{r}$$ is injective then $I_{r}=J_{r}$ for all $r>0$.
This is similar to Lemma \[CoidealsOfTensorAlgebras\]. We will denote by $R_{r}=T_{r}(V)/I_{r}$, $R_{r}'=T_{r}(V)/J_{r}$, by $R_{r}(n)$, $R'_{r}(n)$ their respective $n$th graded pieces, and by $R_{r}(\leq n)=\bigoplus_{i \leq n}R_{r}(i)$ and $R'_{r}(\leq n)=\bigoplus_{i \leq n}R'_{r}(i)$. Since $I_{r} \subset J_{r}$ we have strict analytically graded epimorphisms $f_{r}:R_{r} \rightarrow R'_{r}$ which restricts to isometries $R_{r}(1) \rightarrow R'_{r}(1)$. Suppose that $f_{r}$ restricts also to an isometry $R_{r}(\leq n) \rightarrow R'_{r}(\leq n)$ for all $r>0$. Let $r>0$ and $x \in {R_{r}(\leq n+1)}$. By the proof of Lemma \[GradedPiecesOfTensorAlgebraCoideals\], we see that $$\begin{array}{rcl}
\Delta(R_{r}(n+1)) &\subset& \sum_{i=0}^{n+1} R_{\frac{r}{2}}(i) \hat{\otimes} R_{\frac{r}{2}}(n-i)\\
&\subset& R_{\frac{r}{2}}(n+1) \hat{\otimes} k + k \hat{\otimes} R_{\frac{r}{2}}(n+1) + R_{\frac{r}{2}}(\leq n) \hat{\otimes} R_{\frac{r}{2}}(\leq n),
\end{array}$$ so $\Delta(x)=y \otimes 1 + 1 \otimes y' + z$ for some $y,y' \in R_{\frac{r}{2}}(n+1)$, $z \in R_{\frac{r}{2}}(\leq n) \hat{\otimes} R(\leq n)$. But then $x-y = (\text{Id} \otimes \varepsilon)\Delta(x)-y= \varepsilon(y)\cdot 1 + (\text{Id} \otimes \varepsilon)(z) \in R_{\frac{r}{2}}(\leq n)$ and likewise $x-y' \in R_{\frac{r}{2}}(\leq n)$. So $\Delta(x)=x \otimes 1 + 1 \otimes x + z'$ for some $z' \in {R_{\frac{r}{2}}(\leq n)} \hat{\otimes} {R_{\frac{r}{2}}(\leq n)}$. If $f_{r}(x)=0$ then $(f_{\frac{r}{2}} \otimes f_{\frac{r}{2}})(z')=0$, but, since each $R_{\frac{r}{2}}(\leq n)$ is finite dimensional and hence flat, $f_{\frac{r}{2}} \otimes f_{\frac{r}{2}}$ is injective on $R_{\frac{r}{2}}(\leq n) \hat{\otimes} R_{\frac{r}{2}}(\leq n)$. So $z'=0$ and $x$ is primitive, hence $x=0$. Thus $f_{r}$ is an isometry ${R_{r}(\leq n)} \rightarrow {R'_{r}(\leq n)}$ since the norms on $R_{r}(\leq n)$ and $R'_{r}(\leq n)$ are the quotient norms from $\coprod_{i \leq n}^{\leq 1} V_{r}^{\hat{\otimes} i}$. Taking contracting colimits over $n$ we see that $f_{r}$ is isometric. Hence $I_{r}=J_{r}$.
\[ANicholsAlgebrasAreFlat\] Let $W=\coprod_{n \geq 0}^{\leq 1} W(n)$ be a Banach space where each $W(n)$ is finite dimensional. Then $W$ is flat.
Let $f:A \rightarrow B$ be a morphism of Banach spaces. Then $$\begin{array}{rcl}
\text{Ker}\left(W \hat{\otimes} A \xrightarrow{\text{Id} \otimes f} W \hat{\otimes} B \right) &\cong& \text{Ker}\left(\coprod^{\leq 1} (W(n) \hat{\otimes} A) \rightarrow \coprod^{\leq 1} (W(n) \hat{\otimes} B)\right)\\
&\cong& \coprod^{\leq 1}\text{Ker}\left( W(n) \hat{\otimes} A \xrightarrow{\text{Id} \otimes f} W(n) \hat{\otimes} B\right)\\
&\cong& \coprod^{\leq 1} \left( W(n) \hat{\otimes} \text{Ker}\left( A \xrightarrow{f} B\right) \right)\\
&\cong& W \hat{\otimes} \text{Ker}\left( A \xrightarrow{f} B\right),
\end{array}$$ where the second isomorphism is an easy computation and the third is because each $W(n)$ is finite dimensional. Hence $W$ is flat.
\[Radius1NicholsAlgebrasExistsA\] Let $V$ be a finite dimensional Banach space with pre-braiding $c$ of norm at most $1$. Then a dagger-0 Nichols algebra of $V$ exists.
As a variation on the proof of Proposition \[Radius1NicholsAlgebrasExists\], let $\mathcal{I}(V)$ be the set consisting of collections $(I_{r})_{r \in \mathbb{R}_{>0}}$ where each $I_{r}$ is a homogeneous ideals of $T_{r}(V)$ contained in $\coprod_{n \geq 2}^{\leq 1} V_{r}^{\hat{\otimes}n}$ such that the maps $T_{r}(V) \rightarrow T_{r'}(V)$ for $r>r'$ send $I_{r}$ to a subspace of $I_{r'}$ and the maps $\Delta:T_{r}(V) \rightarrow T_{\frac{r}{2}}(V) \hat{\otimes} T_{\frac{r}{2}}(V)$ giving the comultiplication send $I_{r}$ to a subspace of $I_{\frac{r}{2}} \hat{\otimes} T_{\frac{r}{2}}(V) + T_{\frac{r}{2}}(V) \hat{\otimes} I_{\frac{r}{2}}$. Let $\mathcal{I}'(V)$ be the subset of $\mathcal{I}(V)$ consisting of $(I_{r})_{r \in \mathbb{R}_{>0}}$ for which $\overline{I_{r}} \hat{\otimes} T(V) + T(V) \hat{\otimes} \overline{I_{r}}$ is preserved by $\tilde{c}$. Let $I_{r}(V)$ be the sum of all ideals $I_{r}$ for $(I_{r})_{r \in \mathbb{R}_{>0}} \in \mathcal{I}(V)$, which is closed by a similar argument to the proof of Proposition \[Radius1NicholsAlgebrasExists\]. Likewise let $I'_{r}(V)$ be the sum of all ideals $I'_{r}$ for $(I'_{r})_{r \in \mathbb{R}_{>0}} \in \mathcal{I}'(V)$, which is again closed. We will denote by $I(V)$ and $I'(V)$ the collection $(I_{r}(V))_{r \in \mathbb{R}_{>0}}$ and $(I'_{r}(V))_{r \in \mathbb{R}_{>0}}$, and let $T_{0}(V)^{\dagger}/I(V):=\text{"colim"}_{r>0}T_{r}(V)/I_{r}(V)$ and $T_{0}(V)^{\dagger}/I'(V):=\text{"colim"}_{r>0}T_{r}(V)/I'_{r}(V)$. We must check that $$P(T_{0}(V)^{\dagger}/I(V))=(T_{0}(V)^{\dagger}/I(V))(1).$$ As before, the closed ideals generated by $I_{r}(V)$ and $$\left\lbrace x \in \coprod\nolimits_{n \geq 2}^{\leq 1} \middle| \substack{\Delta(x) \in x \otimes 1 + 1 \otimes x + I_{r'}(V) \otimes T_{r'}(V) + T_{r'}(V) \otimes I_{r'}(V) \\ \text{for some sufficiently small } \tfrac{r}{2} \geq r'>0} \right\rbrace$$ form an element of $\mathcal{I}(V)$. So $P(T_{0}(V)^{\dagger}/I(V))=(T_{0}(V)^{\dagger}/I(V))(1)$, and likewise $P(T_{0}(V)^{\dagger}/I'(V))=(T_{0}(V)^{\dagger}/I'(V))(1)$. So, by Lemma \[ACoidealsOfTensorAlgebras\], we have $I_{r}(V)=I'_{r}(V)$ for all $r>0$, and so the braiding $\tilde{c}$ descends to braidings of $T_{r}(V)/I_{r}(V)$. Hence $T_{0}(V)^{\dagger}/I(V):=\text{"colim"}_{r>0}T_{r}(V)/I_{r}(V)$ forms a $k\{t\}^{\dagger}$-graded braided IndBanach Hopf algebra with $(T_{0}(V)^{\dagger}/I(V))(0) \cong k$ and generated by $(T_{0}(V)^{\dagger}/I(V))(1) \cong V$. Finally, by Lemma \[ANicholsAlgebrasAreFlat\], $T_{r}(V)/I_{r}(V)$ is flat for each $r>0$. Hence the filtered colimit $\text{"colim"}_{r>0}T_{r}(V)/I_{r}(V)$ is flat.
Let $\mathfrak{B}_{r}(V):=T_{r}(V)/I_{r}(V)$. We shall use the notation $\mathfrak{B}_{0}(V)^{\dagger}$ for the dagger Nichols algebra $\text{colim}_{r>0} \mathfrak{B}_{r}(V)$ defined in the proof of Proposition \[Radius1NicholsAlgebrasExistsA\].
Note that, in the proof of Proposition \[Radius1NicholsAlgebrasExistsA\], each $I_{r}(V)$ is a closed ideal of $T_{r}(V)$, and hence each $\mathfrak{B}_{r}(V)$ is a Banach algebra. So $\mathfrak{B}_{0}(V)^{\dagger}$ is locally Banach as an algebra.
\[ExtendBilinearFormA\] Let $V$ be a Banach space with pre-braiding $c$ of norm at most 1, and suppose we have a symmetric non-degenerate bilinear form $\langle - , - \rangle: V \hat{\otimes} V \rightarrow k$. Then this extends to a dual pairing $(\coprod_{n \geq 0} V^{\hat{\otimes} n}) \hat{\otimes} T_{0}^{c}(V)^{\dagger} \rightarrow k$ such that the composition $$V^{\hat{\otimes} n} \hat{\otimes} V^{\hat{\otimes} m} \longrightarrow \left( \coprod_{n \geq 0} V^{\hat{\otimes} n} \right) \hat{\otimes} T_{0}^{c}(V)^{\dagger} \longrightarrow k$$ is symmetric when $n=m$ and is $0$ if $n \neq m$.
Again, we proceed as in the proof of Proposition 1.2.3 of [@ItQG], as we did for Lemma \[ExtendBilinearForm\]. The given bilinear form induces a morphism $V \rightarrow V^{\ast}$ whilst the projections $T_{r}(V) \twoheadrightarrow V$ induce a morphism $V^{\ast} \rightarrow (T_{0}(V)^{\dagger})^{\ast}$. The coalgebra structure on $T_{0}(V)^{\dagger}$ gives $(T_{0}(V)^{\dagger})^{\ast}$ an algebra structure, and so the composition $V \rightarrow V^{\ast} \rightarrow (T_{0}(V)^{\dagger})^{\ast}$ induces a unique algebra homomorphism $\coprod_{n \geq 0} V^{\hat{\otimes} n} \rightarrow (T_{0}(V)^{\dagger})^{\ast}$ that gives our bilinear form. The fact that this is is a dual pairing that is symmetric on $V^{\hat{\otimes} n} \hat{\otimes} V^{\hat{\otimes} n}$ with $V^{\hat{\otimes} n}$ perpendicular to $V^{\hat{\otimes} m}$ for $m \neq n$ follows as in the proof of Lemma \[ExtendBilinearForm\].
Suppose we have a Banach space $V$ with pre-braiding $c$ of norm at most 1 and a non-degenerate symmetric bilinear form $\langle - , - \rangle: V \hat{\otimes} V \rightarrow k$. Let $r,s>0$ with $\|\langle - , - \rangle\| \leq rs$. The $n$-fold comultiplication $T_{r}(V) \rightarrow T_{r/2^{n}}(V)^{\hat{\otimes} n}$ given by Proposition \[ArchimedeanTensorBialgebra\] induces an $n$-fold multiplication $(T_{r/2^{n}}(V)^{\ast})^{\hat{\otimes} n} \rightarrow T_{r}(V)^{\ast}$. Also, for each $n >0$ the bilinear form $\langle - , - \rangle$ and the projection $T_{r/2^{n}}(V) \twoheadrightarrow V_{r/2^{n}}$ induce contracting homomorphisms $V_{2^{n}s} \rightarrow V_{r/2^{n}}^{\ast}$ and $V_{r/2^{n}}^{\ast} \twoheadrightarrow T_{r/2^{n}}(V)^{\ast}$. Then the compositions $$V_{2^{n}s}^{\hat{\otimes} n} \rightarrow (V_{r/2^{n}}^{\ast})^{\hat{\otimes} n} \rightarrow (T_{r/2^{n}}(V)^{\ast})^{\hat{\otimes} n} \rightarrow T_{r}(V)^{\ast}$$ induce a bilinear pairing $\mathscr{T}_{s}(V) \hat{\otimes} T_{r}(V) \rightarrow k$, where $$\mathscr{T}_{s}(V):=\coprod\nolimits^{\leq 1}_{n \geq 0} (V_{2^{n}s})^{\hat{\otimes} n} = \left\lbrace \sum x_{n} \middle| x_{n} \in V^{\hat{\otimes} n} \text{ and } \sum_{n}\|x\|2^{n^{2}}s^{n} < \infty \right\rbrace,$$ and hence a bilinear pairing $\mathscr{T}_{\infty}(V) \hat{\otimes} T_{0}(V)^{\dagger} \rightarrow k$, where $\mathscr{T}_{\infty}(V):= \text{lim}_{s>0} \mathscr{T}_{s}(V)$. Furthermore, the restriction of these bilinear forms to $\bigoplus_{n \geq 0} V^{\otimes n}$ is a dual pairing by Proposition 1.2.3 of [@ItQG]. Unfortunately, the algebra structure of $\bigoplus_{n \geq 0} V^{\otimes n}$ neither extends to $\mathscr{T}_{s}(V)$ nor $\mathscr{T}_{\infty}(V)$. It does, however, extend to $$\text{lim}_{s_{n}>0} \coprod\nolimits^{\leq 1}_{n \geq 0} (V_{s_{n}})^{\hat{\otimes} n} = \text{lim}_{s_{n}>0} \left\lbrace \sum x_{n} \middle| x_{n} \in V^{\hat{\otimes} n} \text{ and } \sum_{n}\|x\|s_{n}^{n} < \infty \right\rbrace$$ which we may pair with $T_{0}(V)^{\dagger}$ via the composition $$\left( \text{lim}_{s_{n}>0} \coprod\nolimits^{\leq 1}_{n \geq 0} (V_{s_{n}})^{\hat{\otimes} n}\right) \hat{\otimes} T_{0}(V)^{\dagger} \rightarrow \mathscr{T}_{\infty}(V) \hat{\otimes} T_{0}(V)^{\dagger} \rightarrow k.$$ The following lemma shows that $\text{lim}_{s_{n}>0} \coprod\nolimits^{\leq 1}_{n \geq 0} (V_{s_{n}})^{\hat{\otimes} n}$ is just $\coprod_{n \geq 0} V^{\hat{\otimes} n}$ as in the previous lemma.
The natural morphism $$\coprod_{n \geq 0} V^{\hat{\otimes} n} \rightarrow \text{lim}_{s_{n}>0} \coprod\nolimits^{\leq 1}_{n \geq 0} (V_{s_{n}})^{\hat{\otimes} n}$$ is an isomorphism.
As in the proof of Lemma \[CountableProductsCommuteWithTensor\], for a sequence of positive real numbers $a_{n}>0$ such that $(a_{n}^{n})_{n \geq 0}$ is summable, the maps $$\coprod\nolimits^{\leq 1}_{n \geq 0} (V_{s_{n}})^{\hat{\otimes} n} \rightarrow \prod\nolimits^{\leq 1}_{n \geq 0} (V_{s_{n}})^{\hat{\otimes} n}, \quad (x_{n})_{n \geq 0} \mapsto (x_{n})_{n \geq 0},$$ and $$\prod\nolimits^{\leq 1}_{n \geq 0} (V_{s_{n}})^{\hat{\otimes} n} \rightarrow \coprod\nolimits^{\leq 1}_{n \geq 0} (V_{a_{n}s_{n}})^{\hat{\otimes} n}, \quad (x_{n})_{n \geq 0} \mapsto (x_{n})_{n \geq 0},$$ induce an isomorphism $$\text{lim}_{s_{n}>0}\coprod\nolimits^{\leq 1}_{n \geq 0} (V_{s_{n}})^{\hat{\otimes} n} \cong \text{lim}_{s_{n}>0}\prod\nolimits^{\leq 1}_{n \geq 0} (V_{s_{n}})^{\hat{\otimes} n}.$$ For a Banach space $W$, $$\begin{array}{rcl}
\text{Hom}(W,\text{lim}_{s_{n}>0}\coprod\nolimits^{\leq 1}_{n \geq 0} (V_{s_{n}})^{\hat{\otimes} n}) &\cong& \text{Hom}(W,\text{lim}_{s_{n}>0}\prod\nolimits^{\leq 1}_{n \geq 0} (V_{s_{n}})^{\hat{\otimes} n})\\
&\cong& \text{lim}_{s_{n}>0}\prod\nolimits^{\leq 1}_{n \geq 0} \text{Hom}(W,(V_{s_{n}})^{\hat{\otimes} n})\\
&=& \left\lbrace (f_{n})_{n \geq 0} \middle| \substack{f_{n}:W \rightarrow V^{\hat{\otimes} n} \text{ and } (\|f_{n}\|s_{n})_{n \geq 0}\\ \text{ is bounded for all } s_{n}>0} \right\rbrace\\
&=& \left\lbrace (f_{n})_{n \geq 0} \middle| \substack{f_{n}:W \rightarrow V^{\hat{\otimes} n} \text{ and}\\ f_{n} = 0 \text{ for } n\gg 0} \right\rbrace
\end{array}$$ where the last equality is because if $s_{n}:=\frac{n}{\|f_{n}\|}$ when $f_{n} \neq 0$ and $s_{n}:=1$ if $f_{n}=0$ then $(\|f_{n}\|s_{n})_{n \geq 0}$ is only bounded if $f_{n}=0$ for $n \gg 0$. Hence the map of sets $$\text{Hom}(W,\coprod_{n \geq 0}V^{\hat{\otimes} n}) \cong \bigoplus_{n \geq 0}\text{Hom}(W,V^{\hat{\otimes} n}) \rightarrow \text{Hom}(W,\text{lim}_{s_{n}>0}\coprod\nolimits^{\leq 1}_{n \geq 0} (V_{s_{n}})^{\hat{\otimes} n})$$ is bijective. Since this holds for all Banach spaces $W$ we must have $$\coprod_{n \geq 0} V^{\hat{\otimes} n} \cong \text{lim}_{s_{n}>0} \coprod\nolimits^{\leq 1}_{n \geq 0} (V_{s_{n}})^{\hat{\otimes} n}.$$
\[BilinearFormGivesNicholsAlgebraA\] Let $V$ be a Banach space with pre-braiding $c$ of norm at most 1, and suppose we have a non-degenerate bilinear form $\langle - , - \rangle: V \hat{\otimes} V \rightarrow k$. Then for each $0< r$, let $I_{r}$ be the radical in $T_{r}^{c}(V)$ of $$\left( \coprod_{n \geq 0} V^{\hat{\otimes} n} \right) \hat{\otimes} T_{r}^{c}(V) \longrightarrow \left( \coprod_{n \geq 0} V^{\hat{\otimes} n} \right) \hat{\otimes} T_{0}^{c}(V)^{\dagger} \longrightarrow k.$$ Then the $I_{r}$ are closed homogeneous ideals in $T_{r}^{c}(V)$ such that $T_{r}(V) \rightarrow T_{r'}(V)$ maps $I_{r}$ to $I_{r'}$ and $\Delta:T_{r}(V) \rightarrow T_{\frac{r}{2}}(V) \hat{\otimes} T_{\frac{r}{2}}(V)$ maps $I_{r}$ to $I_{\frac{r}{2}} \hat{\otimes} T_{\frac{r}{2}}(V) + T_{\frac{r}{2}}(V) \hat{\otimes} I_{\frac{r}{2}}$. Furthermore, $P(\text{colim}_{r>0} T_{r}^{c}(V)/I_{r}) = V$, hence $\text{colim}_{r>0} T_{r}^{c}(V)/I_{r}$ is a dagger Nichols algebra of $V$.
This is analogous to the proof of Proposition \[BilinearFormGivesNicholsAlgebra\]. The fact that $I_{r}$ are closed homogeneous ideals compatible under restrictions to smaller radii with $\Delta(I_{r}) \subset I_{\frac{r}{2}} \hat{\otimes} T_{\frac{r}{2}}(V) + T_{\frac{r}{2}}(V) \hat{\otimes} I_{\frac{r}{2}}$ follows from Lemma \[ExtendBilinearFormA\]. As the bilinear form on $V$ is non-degenerate, $I_{r} \subset \coprod_{n \geq 2}^{\leq 1} V_{r}^{\hat{\otimes}n}$, and the quotient $T_{r}(V)/I_{r}$ is generated by $V$. It remains to check that the subspace of primitive elements is just $V$, which follows from the same argument as for Proposition \[BilinearFormGivesNicholsAlgebra\]. Again, $\tilde{c}$ preserves $I_{r} \hat{\otimes} T_{r}(V) + T_{r}(V) \hat{\otimes} I_{r}$ for all $r>0$, so the braidings on $T_{r}^{c}(V)$ descend to braidings of $T_{r}^{c}(V)/I_{r}$ and $\text{colim}_{r>0} T_{r}^{c}(V)/I_{r}$ is a dagger Nichols algebra of $V$.
\[NicholsAlgebraUniversalPropertyA\] Let $V$ be a finite dimensional Banach space with pre-braiding $c$ of norm at most $1$. Let $R=\text{colim}_{r>0} \coprod^{\leq 1}R(n)_{r^{n}}$ be a dagger graded pre-braided IndBanach Hopf algebra. Suppose further that the algebra structure on $R$ is determined by algebra structures on $\coprod^{\leq 1}R(n)_{r^{n}}$ for each $r>0$ which are generated by $R(1)_{r}$. Then, if $R(0)\cong k$, $P(R)=R(1) \cong V$ as pre-braided Banach spaces, there is an epimorphism of dagger graded braided Hopf algebras $\mathfrak{B}_{0}^{c}(V) \rightarrow R$ extending $V \overset{\sim}{\longrightarrow} R(1)$.
Without loss of generality, we may assume that the isomorphism $V \xrightarrow{\sim} R(1)$ is of norm at most 1. Hence we obtain morphisms of Banach algebras $T_{r}(V) \rightarrow \coprod^{\leq 1}_{n \geq 0} R(n)_{r^{n}}$ that are epic since $R(1)_{r}$ generate $\coprod^{\leq 1}_{n \geq 0} R(n)_{r^{n}}$. Since $P(R)=R(1)$, this induces a morphism of Hopf algebras $T_{0}(V)^{\dagger} \rightarrow R$. Hence the kernels of the maps $T_{r}(V) \rightarrow \coprod^{\leq 1}_{n \geq 0} R(n)_{r^{n}}$ form an element of $\mathcal{I}(V)$, in the notation of the proof of Proposition \[Radius1NicholsAlgebrasExistsA\]. Thus we obtain a well-defined epimorphism $\mathfrak{B}_{0}^{c}(V) \rightarrow R$.
\[NicholsAlgebrasEquivalentDefinitionA\] Let $V$ be a finite dimensional Banach space with pre-braiding $c$ of norm at most $1$, and suppose we have a non-degenerate symmetric bilinear form $\langle - , - \rangle: V \hat{\otimes} V \rightarrow k$. Retaining the notation of Proposition \[BilinearFormGivesNicholsAlgebraA\], there is an isomorphism $\mathfrak{B}_{0}^{c}(V)^{\dagger} \rightarrow \text{colim}_{r>0}T_{r}^{c}(V)/I_{r}$.
By Proposition \[NicholsAlgebraUniversalPropertyA\] there is an epimorphism $\mathfrak{B}_{0}^{c}(V) \rightarrow \text{colim}_{r>0}T_{r}^{c}(V)/I_{r}$. By Lemma \[ACoidealsOfTensorAlgebras\] the maps $\mathfrak{B}_{r}^{c}(V) \rightarrow T_{r}^{c}(V)/I_{r}$ are all isomorphisms. Hence $\mathfrak{B}_{0}^{c}(V) \rightarrow \text{colim}_{r>0}T_{r}^{c}(V)/I_{r}$ is an isomorphism.
Let $V$ be a finite dimensional Banach space with pre-braiding $c$ of norm at most $1$, and suppose we have a non-degenerate symmetric bilinear form $\langle - , - \rangle: V \hat{\otimes} V \rightarrow k$. For each $n \geq 0$ let $I(n)$ be the radical in $V^{\hat{\otimes} n}$ of the composition $$V^{\hat{\otimes} n} \hat{\otimes} T_{0}^{c}(V)^{\dagger} \longrightarrow \left( \coprod_{n \geq 0} V^{\hat{\otimes} n} \right) \hat{\otimes} T_{0}^{c}(V)^{\dagger} \longrightarrow k$$ and let $\mathfrak{B}_{\text{alg}}^{c}(V)$ be the braided graded Hopf algebra $\coprod_{n \geq 0} V^{\hat{\otimes} n}/I(n)$. This is the algebraic Nichols algebra of $V$, as defined in [@PHA], by Proposition 2.10 of *loc. cit.* By Lemma \[ExtendBilinearFormA\] there is a dual pairing $$\mathfrak{B}_{\text{alg}}^{c}(V) \hat{\otimes} \mathfrak{B}_{0}^{c}(V)^{\dagger} \rightarrow k,$$ which we also denote by $\langle - , - \rangle$.
Constructing Archimedean analytic quantum groups {#ConstructingArchimedeanAnalyticQuantumGroups}
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Again, we will use $q$ to denote an element of $k \setminus \{0\}$ of norm 1 and we fix root datum as in Definition \[KacMoodyRootDatum\] for a Lie algebra $\mathfrak{g}$.
\[CartanPartA\] Let $H=\coprod_{\lambda \in \Phi^{\ast}}^{\leq 1} k \cdot K_{\lambda}$ denote the Banach group Hopf algebra of $\Phi^{\ast}$ with $$K_{\lambda} \cdot K_{\lambda'}=K_{\lambda + \lambda'}, \quad \Delta_{H}(K_{\lambda})=K_{\lambda} \otimes K_{\lambda}\quad\text{and}\quad S(K_{\lambda})=K_{-\lambda}.$$ We continue to use the notation $$t_{i}:=K_{\frac{(\alpha_{i},\alpha_{i})}{2}\lambda_{i}}$$ and let $H'$ be the closed sub-Hopf algebra generated by $\{t_{i} \mid i \in I\}$. As before, there is a duality pairing $H \hat{\otimes} H' \rightarrow k$, which we continue to denote by $\langle-,- \rangle$, defined by $$K_{\lambda} \otimes \underline{t}^{\underline{n}} \mapsto q^{\lambda(\sum n_{i} \alpha_{i})}$$ and simultaneous algebra homomorphisms and coalgebra anti-homomorphisms $$\mathscr{R}:H' \rightarrow H, \ t_{i} \mapsto t_{i}, \quad \overline{\mathscr{R}}:H' \rightarrow H, \ t_{i} \mapsto t_{i}^{-1},$$ for $i \in I$, making $H$ and $H'$ a weakly quasi-triangular dual pair.
\[HAPairingWeakQuasiTriangularA\] Proposition 4.4 of [@TRTfIBS] says that the category of IndBanach $H$-modules that decompose locally into Banach weight spaces with weights in the root lattice $\Psi \subset \Phi$, which we will continue to denote by $H\text{-Mod}_{\Psi}$ as before, is braided.
\[QuantumBraidingA\] \[AnalyticQuantumGroupPositivePart\] Let $V=\coprod_{i \in I}^{\leq 1} k \cdot v_{i}$ with basis $\{v_{i} \mid i \in I \}$ have the $H'$-coaction $v_{i} \mapsto t_{i} \otimes v_{i}$. Then $V$ is a $H$-module with braiding $c(x_{i} \otimes x_{j})=q^{(\alpha_{i},\alpha_{j})}x_{j} \otimes x_{i}$. Let $\langle -,- \rangle$ be the non-degenerate bilinear form on $V$ where $$\langle v_{i},v_{j} \rangle = \delta_{i,j}\frac{1}{(q_{i}-q_{i}^{-1})} \quad \text{for} \quad q_{i}=q^{\frac{(\alpha_{i},\alpha_{i})}{2}}.$$ Given $0 < r$ we denote by $\mathbf{f}_{r}^{\text{an}}$ the algebras $\mathfrak{B}_{r}^{c}(V)$. We then use the notation $\mathbf{f}_{0}^{\dagger}$ for the dagger Nichols algebra $\mathfrak{B}_{0}^{c}(V)^{\dagger}$ and $\mathbf{f}$ for the algebraic Nichols algebra $\mathfrak{B}_{\text{alg}}^{c}(V)$.
\[PositivePartDenseA\] For each $0<r$, the positive part of the quantum group is dense in the Banach space $\mathbf{f}_{r}^{\text{an}}$.
The proof of this is the same as for Lemma \[PositivePartDense\].
Suppose $0< r$. Then there is a $H'$-equivariant dual pairing $\langle-,-\rangle:\mathbf{f}_{0}^{\dagger} \hat{\otimes} \mathbf{f} \rightarrow k$ extending $\langle-,-\rangle$ in Definition \[AnalyticQuantumGroupPositivePart\] such that $\langle \mathbf{f}_{0}^{\dagger}(n) , \mathbf{f}(m) \rangle = \{0\}$ for $n \neq m$.
This follows from Lemma \[ExtendBilinearFormA\].
\[AnalyticQuantumGroupsA\] We denote by $U_{q}(\mathfrak{g})_{0}^{\dagger}$ the double bosonisation $U(\mathbf{f}_{0}^{\dagger},H,\mathbf{f})$. Let us denote by $F_{i}$ the generalised element in $\mathbf{f}_{0}^{\dagger} \rtimes H$ represented by ${v_{i} \otimes 1} \in \mathbf{f}_{r}^{\text{an}} \hat{\otimes} H$, and by $E_{i}$ the generalised element in $\overline{H} \ltimes \overline{\mathbf{f}}$ represented by $1 \otimes v_{i}\in H \hat{\otimes} V$ for $i \in I$. We retain this notation when viewing $\mathbf{f}_{0}^{\dagger} \rtimes H$ and $\overline{H} \ltimes \overline{\mathbf{f}}$ as sub-Hopf algebras of $U_{q}(\mathfrak{g})_{0}^{\dagger}$.
$U_{q}(\mathfrak{g})_{0}^{\dagger}$ is analytically graded by $\mathbb{Z}I \cong \Psi$.
As in the proof of Proposition \[NAQGGraded\] we have that $\mathbf{f}_{r}^{\text{an}}$, $\mathbf{f}_{0}^{\dagger}$ and $\mathbf{f}$ are $H'$-comodules, and if we give $H$ the trivial $H'$-coaction $H \cong k \hat{\otimes} H \xrightarrow{\eta_{H'} \otimes \text{Id}_{H}} H' \hat{\otimes} H$ then all of the morphisms involved in defining $U(\mathbf{f}_{0}^{\dagger},H,\mathbf{f})$ are $H'$-comodule homomorphisms.
Quantum groups as Drinfel’d doubles and braided monoidal representations
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There is a duality pairing $$\langle -,- \rangle:(\overline{H} \ltimes \overline{\mathbf{f}}) \hat{\otimes} (\mathbf{f}^{\dagger}_{0} \rtimes H')^{\text{op}} \rightarrow k$$ given by the composition $$\begin{array}{rcccl}
H \hat{\otimes} \mathbf{f}^{\text{an}}_{\frac{s}{4}} \hat{\otimes} \mathbf{f}^{\text{an}}_{\frac{r}{4}} \hat{\otimes} H' &\overset{\text{Id} \otimes \text{Id} \otimes S \otimes\text{Id}}{\xrightarrow{\hspace*{1.5cm}}}& H \hat{\otimes} \mathbf{f}^{\text{an}}_{\frac{s}{4}} \hat{\otimes} \mathbf{f}^{\text{an}}_{\frac{r}{4}} \hat{\otimes} H'
&\overset{\text{Id} \otimes \langle -,- \rangle \otimes\text{Id}}{\xrightarrow{\hspace*{1.5cm}}}& H \hat{\otimes} k \hat{\otimes} H' \\
&\overset{\text{Id} \otimes S}{\xrightarrow{\hspace*{1.5cm}}}& H \hat{\otimes} H'
&\overset{\langle -,- \rangle}{\xrightarrow{\hspace*{1.5cm}}}& k.
\end{array}$$
As with Lemma \[DualityPairingforQuantumDouble\], this follows from Proposition 34 of Section 6.3.1 of [@QGaTR].
We will denote by $D(\overline{H} \ltimes \overline{\mathbf{f}},\mathbf{f}^{\dagger}_{0} \rtimes H')$ the relative Drinfel’d double of $\overline{H} \ltimes \overline{\mathbf{f}}$ and $\mathbf{f}^{\dagger}_{0} \rtimes H'$ as constructed in Lemma \[Drinfel’dDoubleConstruction\].
Recall the definition of crossed bimodules from Definition \[CrossedBimodules\].
\[ModulesofQuantumDoubleA\] There is a fully faithful functor $$_{\mathbf{f}^{\dagger}_{0} \rtimes H'}\text{Cross}^{\mathbf{f}^{\dagger}_{0} \rtimes H'} \rightarrow D(\overline{H} \ltimes \overline{\mathbf{f}},\mathbf{f}^{\dagger}_{0} \rtimes H')\text{-Mod}.$$
This functor is constructed in Lemma \[ModulesofQuantumDouble\]. Given a morphism ${M \xrightarrow{f} N}$ in $D(\overline{H} \ltimes \overline{\mathbf{f}},\mathbf{f}^{\dagger}_{0} \rtimes H')\text{-Mod}$ between objects in the image of $_{\mathbf{f}^{\dagger}_{0} \rtimes H'}\text{Cross}^{\mathbf{f}^{\dagger}_{0} \rtimes H'}$ then $f$ commutes with both the action of $H'$ and $\mathbf{f}$. Hence $f$ must preserve the locally Banach weight space decomposition, hence commutes with the coaction of $H'$. Also, since the bilinear pairing between the corresponding graded pieces of $\mathbf{f}$ and $\mathbf{f}^{\dagger}_{0}$ is nondegenrate, $f$ must also commute with the coaction of $\mathbf{f}^{\dagger}_{0}$. Hence $f$ is a morphism in $_{\mathbf{f}^{\dagger}_{0} \rtimes H'}\text{Cross}^{\mathbf{f}^{\dagger}_{0} \rtimes H'}$.
\[AnalyticQuantumGroupAsDrinfel’dDoubleA\] There is a strict epimorphism $$D(\overline{H} \ltimes \overline{\mathbf{f}},\mathbf{f}^{\dagger}_{0} \rtimes H') \rightarrow U_{q}(\mathfrak{g})^{\dagger}_{0}$$ whose kernel is $$\mathbf{f}^{\dagger}_{0} \hat{\otimes} \overline{\langle t_{i} \otimes 1 - 1 \otimes t_{i} \mid i \in I \rangle} \hat{\otimes} \mathbf{f} \hookrightarrow \mathbf{f}^{\dagger}_{0} \hat{\otimes} H' \hat{\otimes} H \hat{\otimes} \mathbf{f}.$$
This follows as in the proof of Proposition \[NAAnalyticQuantumGroupAsDrinfel’dDouble\]. Again, this morphism can be written as $$\mathbf{f}^{\dagger}_{0} \hat{\otimes} H' \hat{\otimes} H \hat{\otimes} \mathbf{f} \xrightarrow{\text{Id} \otimes \mu_{H} \otimes \text{Id}} \mathbf{f}^{\dagger}_{0} \hat{\otimes} H \hat{\otimes} \mathbf{f}.$$ By Proposition \[ANicholsAlgebrasAreFlat\], $\mathbf{f}^{\dagger}_{0}$ is flat. Also, since $\mathbf{f}$ is a colimit of finite dimensional spaces it is also flat. The result then follows from the fact that $\overline{\langle t_{i} \otimes 1 - 1 \otimes t_{i} \mid i \in I \rangle}$ is the kernel of $\mu_{H}: H' \hat{\otimes}H \rightarrow H$.
\[SubcatOfCrossedBimodulesA\] Let us denote by $\mathcal{C}$ the full subcategory of $_{(\mathbf{f}^{\dagger}_{0} \rtimes H')}\text{Cross}^{(\mathbf{f}^{\dagger}_{0} \rtimes H')}$ consisting of IndBanach spaces $V$ equipped with both a left action and right coaction of $\mathbf{f}^{\dagger}_{0} \rtimes H'$, $\mu_{V}: (\mathbf{f}^{\dagger}_{0} \rtimes H') \hat{\otimes} V \rightarrow V$ and $\Delta_{V}: V \rightarrow V \hat{\otimes} (\mathbf{f}^{\dagger}_{0} \rtimes H')$, such that the diagram
\(A) [$H' \hat{\otimes} V$]{}; (B) \[below=1cm of A\] [$(\mathbf{f}^{\dagger}_{0} \rtimes H') \hat{\otimes} V$]{}; (C) \[right=1cm of A\] [$(\overline{H} \ltimes \overline{\mathbf{f}}) \hat{\otimes} V$]{}; (D) \[below=1.05cm of C\] [$V$]{}; (A) to node (B); (C) to node [$\mu_{V}'$]{} (D); (A) to node (C); (B) to node [$\mu_{V}$]{} (D);
commutes, where $\mu_{V}'$ is the action of $\overline{H} \ltimes \overline{\mathbf{f}}$ on $V$ induced by $\Delta_{V}$.
There is a fully faithful functor $\mathcal{C} \rightarrow U_{q}(\mathfrak{g})_{0}^{\dagger}\text{-Mod}$.
This follows from Lemma \[ModulesofQuantumDouble\] and Proposition \[AnalyticQuantumGroupAsDrinfel’dDoubleA\].
\[IntegrableRepresentations\] Let us denote by $\mathcal{O}_{\Psi}$ the essential image of $\mathcal{C}$ in $U_{q}(\mathfrak{g})_{0}^{\dagger}\text{-Mod}$. By the previous Lemma this is precisely the full subcategory of $U_{q}(\mathfrak{g})_{0}^{\dagger}\text{-Mod}$ consisting of modules, $M$, such that the action of $H$ gives a module in $H\text{-Mod}_{\Psi}$ and the action of $\mathbf{f}$ is induced by a coaction of $\mathbf{f}_{0}^{\dagger}$ via their pairing.
The category $\mathcal{O}_{\Psi}$ is braided.
This follows from Lemma \[CrossedBimodulesBraided\].
In time, the author hopes to study the representations in $\mathcal{O}_{\Psi}$ further and compute examples of the braiding. The hope is that this will produce interesting new braid group representations in which we might see some special analytic functions arising. For example, in [@PRftQDaQM], Goncharov exhibits an automorphism of a Schwarz space using the quantum dilogarithm that satisfies a pentagon relation. This Schwarz space is equipped with an action of the algebra of regular functions on a quantised cluster variety, and this automorphism of the Schwarz space intertwines an automorphism of this algebra of regular functions. The action of these regular functions gives a natural action of the positive part of $U_{q}(\mathfrak{sl}_{2})$. It would be interesting to see whether we can use this to obtain a representation in $\mathcal{O}_{\Psi}$ for $\mathfrak{g}=\mathfrak{sl}_{2}$ whose braiding is related to the quantum dilogarithm and this work of Goncharov.
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[^1]: The author would like to thank Kobi Kremnitzer for his expert supervision and continued support throughout this research, without which writing this paper would not have been possible. He would also like to thank Dan Ciubotaru and Shahn Majid for their advice and encouragement.
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---
abstract: 'The cosmological constant, also known as dark energy, was believed to be caused by vacuum fluctuations, but naive calculations give results in stark disagreement with fact. In the Casimir effect, vacuum fluctuations cause forces in dielectric media, which is very well described by Lifshitz theory. Recently, using the analogy between geometries and media, a cosmological constant of the correct order of magnitude was calculated with Lifshitz theory \[U. Leonhardt, Ann. Phys. (New York) [**411**]{}, 167973 (2019)\]. This paper discusses the empirical evidence and the ideas behind the Lifshitz theory of the cosmological constant without requiring prior knowledge of cosmology and quantum field theory.'
author:
- |
Ulf Leonhardt\
Department of Physics of Complex Systems,\
Weizmann Institute of Science,\
Rehovot 7610001, Israel
title: The case for a Casimir cosmology
---
The problem
===========
Einstein [@Einstein] introduced the cosmological constant $\Lambda$ in 1917 for having a static universe as solution of his equations of the gravitational field [@LL2]. What was the problem? The gravity of matter is attractive, so the matter distribution the universe is made of may first expand and then collapse — like a stone thrown on Earth first rises and then falls, or it may expand forever, like a spacecraft on a voyage into space. In order to have an equilibrium, a repulsive force was required that acts on cosmological scales, but is otherwise too small to play a significant role. This repulsive background was provided for by Einstein’s cosmological term [@Einstein]. Hubble’s measurements [@Hubble] of the Doppler shift in the light coming from galaxies revealed a different picture however: the universe has been expanding at a positive rate. There was no need for an equilibrium solution. In the 1990s measurements on the light coming from certain supernovae [@Supernovae1; @Supernovae2] became precise enough to calculate the second derivative of the expansion, which turned out to be positive, too (Fig. \[expansion\]). This implies that there is indeed a repulsive component to the gravitational force (more on this in Sec. 2.1). Recent measurements on the Cosmic Microwave Background (CMB) [@CMBPlanck] have established that the repulsive component takes the form of Einstein’s cosmological term with constant $\Lambda$. The latest measurements on bright stars in galaxies [@Cepheids] seem to indicate, however, that $\Lambda$ is not constant, but has varied from the time the CMB was created to the present era. In any case, there is clear empirical evidence for the existence of the cosmological term and there are good quantitative measurements of $\Lambda$.
The state of the theory could hardly be more different. The standard prediction of $\Lambda$ from quantum field theory deviates from the observed value by 120 orders of magnitude [@Weinberg]. This state of affairs is reflected in the contemporary scientific term for $\Lambda$: dark energy. The term conjures up a picture of some dark force filling the universe, while in reality it just says that current theoretical physics is unable to shed light on what “dark energy” really is. While there are many attempts [@Weinberg; @DarkEnergy] to derive the cosmological term, none have been fully convincing in the following sense. Physical theories do not just give quantitative explanations of measured facts, but they connect the facts across several different areas of empirical investigation. The larger the range of explanation the more valueable is the theory and the more likely it is that the theory captures a portion of the truth. The attempts of deriving the cosmological constant from theory seem to fall into two categories: either are they off by many orders of magnitude, if they originate from other known physics, or they are highly specialized and therefore highly speculative. Yet a physically well-motivated explanation [@Zeldovich] has been put forward right from the beginning.
![ \[expansion\]](fig1.pdf){width="32pc"}
Zel’dovich [@Zeldovich] suggested in 1968 that Einstein’s cosmological term [@Einstein] comes from the physics of the quantum vacuum [@Milonni]. The quantum vacuum is the ground state of the physical fields — the electromagnetic field and the fields of the weak and strong interaction. Fields are thought of being spanned by modes of harmonic oscillators, and the ground state of a harmonic oscillator is known to carry a zero–pointy energy and to appear with a fluctuating amplitude. An antenna put in vacuum would pick up noise that could be attributed to the fluctuating amplitudes of the modes it samples. The physics of the quantum vacuum has been well–tested and explains a huge set of phenomena, from the stickiness of materials on the nano scale to the limit trees can grow [@Koch]. Recent precision measurements of vacuum forces [@Rodriguez] including measurements of repulsive forces [@Levitation; @CasimirEquilibrium] have tested the theory with an accuracy only limited by the knowledge of the material data involved. Zel’dovich’s proposal [@Zeldovich] thus connects the cosmological constant to a range of completely different phenomena, which would give great strength to his theory. Unfortunately, while Zel’dovich derived the correct structure of the cosmological term, the predicted quantitative value for $\Lambda$ is in stark disagreement with the facts.
The facts are that the universe is indeed dominated by vacuum fluctuations, simply because it is rather empty, and that the spectrum of these fluctuations is almost unlimited, because of the equivalence principle of general relativity [@LL2]. The emptiness of space is quantified in the number for the average mass density of space on cosmological scales at the present time: $$\rho\approx 10^{-27} \mathrm{g}/\mathrm{cm}^{3} \,.
\label{density}$$ The equivalence principle [@LL2] states that gravity acts equally on all bodies and on all scales. In particular, it implies that gravity acts equally on all fields on all wavelengths, down to the Planck scale where — presumably — classical general relativity [@LL2] does no longer hold. The equivalence principle has been well–tested, although not to the Planck scale, but it is a principle that connects a wide range of gravitational phenomena and therefore seems to capture a significant portion of the truth. The potential limitation of the principle is characterized by two numbers worth remembering, the Planck length $\ell_\mathrm{p}$ and the Planck mass $m_\mathrm{p}$ with the expressions $$\ell_\mathrm{p} = \sqrt{\frac{\hbar G}{c^3}} \approx 1.6 \times 10^{-33} \mathrm{cm} \,,\quad
m_\mathrm{p} = \sqrt{\frac{\hbar c}{G}} \approx 2.1 \times 10^{-5} \mathrm{g} \,.
\label{planck}$$ These are the length and the mass units one can construct by taking combinations of the fundamental constants of nature, Planck’s $\hbar$, Newton’s gravitational constant $G$ and the speed of light $c$. Their precise physical meanings are not known yet, but one may get some insight from the thermodynamics of causal horizons [@Jacobson] (generalizing black–hole thermodynamics [@Bekenstein]). Here the entropy increment is given by the area element divided by $4\ell_\mathrm{p}^2$, which suggests that $2\ell_\mathrm{p}$ plays the role of a fundamental length scale in gravitational physics. The Planck mass follows from the classical physics of black holes [@LL2] as the mass of a black hole of Schwarzschild radius $2\ell_\mathrm{p}$. It would be larger by a factor of $\sqrt[4]{5120 \pi} \approx 10$ than the minimal mass required for a quantum–mechanically stable black hole [@LeoBook] (that is not immediately evaporated [@Hawking]). Vacuum modes with wavelengths comparable to the Planck length, if they were to exist, would thus immediately generate black holes and could no longer be taken as passive objects obeying the equivalence principle, but as active agents dissolving the structure of space–time.
Let me give a simple argument, based on dimensional analysis, why the numbers (\[density\]) and (\[planck\]) appear to be in conflict with each other. The energy density $\varepsilon_\mathrm{vac}$ of the quantum vacuum must be proportional to $\hbar$, because it is made by the zero–point energy of all the field modes combined that goes with $\hbar$. In order to get an energy one should multiply $\hbar$ by $c$ and divide by the forth power of a length. This length would correspond to the minimal wavelength of the field modes, which one may take as the Planck length. This gives $$\varepsilon_\mathrm{vac} \sim \frac{\hbar c}{\ell_\mathrm{p}^4} \,.
\label{prediction}$$ Converted into a mass density $\rho_\mathrm{vac} = \varepsilon_\mathrm{vac}\,c^{-2}$ one gets $$\rho_\mathrm{vac} \sim \frac{m_\mathrm{p}}{\ell_\mathrm{p}^3}$$ and hence, using the numerical values (\[density\]) and (\[planck\]) for the actual mass density $\rho$ and the Planck length and mass: $$\log_{10} (\rho_\mathrm{vac}/\rho) \approx 120 \,.$$ The elementary prediction (\[prediction\]) of quantum field theory disagrees with astronomical observations on a truly astronomical scale. Resolving this conflict between theory and reality, while keeping Zel’dovich’s connection between the cosmological constant and the quantum vacuum, requires a re–examination of the empirical evidence and the supporting theory, as follows.
The evidence
============
Cosmology
---------
Astronomical observations have shown [@Survey] that the universe is homogeneous and isotropic on cosmological scales (larger than $100 \mathrm{Mpc}$). This implies [@LL2] that infinitesimal spatial distances can only change with a universal factor $n$ that is uniform in space but may depend on time $t$. In the spirit of optical analogues of gravity [@Gordon; @Plebanski; @Schleich; @LeoPhil] we may regard the factor $n$ as a uniform refractive index that varies in time. The matter and energy in the universe at large acts, on average, as a fluid [@LL6]. This fluid must be at rest relative to the expanding universe, for otherwise the direction of its movement would violate the condition of isotropy. The fluid is therefore solely characterized by its energy density $\varepsilon$ and pressure $p$ where the energy density is related to the mass density by $\varepsilon=\rho c^2$. These postulates, supported by empirical evidence [@Survey], constitute the cosmological principle [@LL2].
One may formulate the cosmological principle in terms of a space–time metric [@LL2] and deduce from Einstein’s equations the Friedmann equations of cosmic evolution [@LL2]. It is however possible — and instructive — to deduce the laws of cosmology almost exclusively from Newtonian physics [@Milne]. Take one point in space and imagine a sphere around this point filled with the cosmic fluid of uniform mass density $\rho$. Consider the Newtonian gravitational potential inside the sphere. Gauss’ theorem implies that it is the potential of a harmonic oscillator with spring constant given by Newton’s constant $G$ times the mass density $\rho$ multiplied by the volume $4\pi/3$ of the unit sphere. Relativity [@LL2] makes only one correction: in addition to the mass density the pressure also causes gravity. To be precise, relativity adds the term $3p/c^2$ to $\rho$ in the effective mass density of gravity in cosmology. One gets from the equation of motion of the harmonic oscillator: $$\frac{\ddot{n}}{n} = - \frac{4\pi G}{3} \left(\rho+\frac{3p}{c^2}\right) ,
\label{newton}$$ which is known as Newton’s equation of the universe [@Telephone]. In the case when the pressure is much smaller than the rest energy density the gravity of matter thus establishes a restoring force with negative acceleration $\ddot{n}$. The observation [@Supernovae1; @Supernovae2] of $\ddot{n}>0$ in the present era reveals however that there is a substantial negative pressure with $p<-\rho c^2/3$. The analysis of CMB fluctuations [@CMBPlanck] identifies a constant contribution of $\varepsilon_\Lambda$ to the total energy density $\varepsilon$ with $$p_\Lambda = -\varepsilon_\Lambda \,.
\label{lambda}$$ This constant negative pressure combined with the constant positive energy density $\varepsilon_\Lambda$ acts like a uniform source of repulsive gravity — as if the matter of the universe were embedded in a uniform background of opposite gravitational charge [@Kolomeisky]. With increasing expansion the mass density of matter decreases such that this uniform background becomes increasingly important. At the present time $\varepsilon_\Lambda/c^2$ amounts to about $70\%$ of the total mass density (\[density\]). This constant background is called dark energy. More prosaically — and more accurately — it corresponds to the cosmological constant introduced by Einstein [@Einstein].
One more ingredient is needed for completing the laws of cosmic evolution, and that comes from thermodynamics [@LL5]. The cosmic fluid is assumed to be isentropic — entropy is conserved, such that a change in the total energy $E$ of a volume $V$ is given by $\mathrm{d}E=-p\,\mathrm{d}V$. As the volume $V$ varies with $n^3$ and $E=\varepsilon V$ one gets the second Friedmann equation [@LL2]: $$\dot{\varepsilon} = -3(\varepsilon + p) H
\label{f2}$$ where $H$ denotes the Hubble constant $$H=\frac{\dot{n}}{n} \,.
\label{hubble}$$ Note that $H$ is not actually constant in general, except when the universe is exponentially expanding (in the case of de Sitter space [@deSitter]).
Having established the equations of motion for the expanding universe, Eqs. (\[newton\]) and (\[f2\]), we integrate them. Consider the quantity $K$ defined by the relation $$H^2 + \frac{K}{n^2} = \frac{8\pi G}{3c^2}\, \varepsilon
\label{f1}$$ with $\varepsilon=\rho c^2$. Differentiating Eq. (\[f1\]) and making use of the Newton equation (\[newton\]) for $\dot{H}+H^2=\ddot{n}/n$ as well as the thermodynamic relation (\[f2\]) and the undifferentiated Eq. (\[f1\]) reveals that $\dot{K}=0$. In other words, $K$ is a constant. It turns out in general relativity [@LL2] that $K$ describes the spatial curvature; $K$ is positive for positively curved space and negative for negative spatial curvature, and in the marginal case of flat space $$K = 0 \,.
\label{flat}$$ Newtonian cosmology does account for the curvature of space in principle [@MM]. The analysis of CMB data [@CurveMeas] has shown that in practice space is approximately flat on cosmological scales such that Eq. (\[flat\]) holds to a good approximation. Equation (\[f1\]) is known as the first Friedmann equation [@LL2], which concludes all the cosmology we will need in this article.
Vacuum
------
Having collected the evidence from cosmology, consider now the case for forces of the quantum vacuum [@Forces]. Without exception, the empirical evidence for vacuum forces comes from Atomic, Molecular and Optical Physics (AMO) and here from quantum fluctuations of the electromagnetic field. They appear as the van der Waals and Casimir forces [@Rodriguez]. The Casimir force is typically (Fig. \[typical\]) a force between electrically neutral bodies of refractive indices $n_i$ immersed in a uniform background with index $n_0$. Strictly speaking [@LL8], dielectrics are characterized by the refractive index $n$ and the impedance $Z$; here we assume $Z=1$ for simplicity. At sufficiently low temperatures (for thermal wavelengths larger than the characteristic distances) the Casimir force originates from fluctuations of the quantum vacuum [@LL9]. These are in general not fluctuations of the electromagnetic field in empty space, but inside the media the bodies and the background are made of. The term “vacuum” is used to state that they are quantum fluctuations of the field in the ground state, given the arrangement of dielectrics. These fluctuations carry energy and exert stress that does mechanical work; the divergence $\nabla\cdot\sigma$ of the stress tensor $\sigma$ gives the force density. For an arrangement of dielectric bodies of uniform refractive indices in a uniform background (Fig. \[typical\]) the force density is entirely concentrated at the surfaces of the bodies, causing the Casimir force [@Casimir].
![ \[typical\]](fig2.pdf){width="20pc"}
There are various theories for the Casimir effect of the quantum vacuum. Casimir’s original theory [@Casimir] considers the total sum $E$ of all the ground state energies $\hbar\omega_m/2$ of all electromagnetic modes involved. This theory presumes the existence of stationary modes oscillating with the circular frequencies $\omega_m$. In realistic media the concept of modes become questionable, because real materials are dissipative such that stationary modes no longer exist, strictly speaking. The theory explaining Casimir forces in realistic dielectrics, Lifshitz theory [@LL9; @Lifshitz; @LDP; @Scheel] takes a different starting point: the fluctuation–dissipation theorem [@Scheel]. The theorem allows to express the energy density and stress tensor [@Pita; @Burger] in terms of the electromagnetic Green functions. Lifshitz theory [@LL9; @Lifshitz; @LDP; @Scheel] has become the theoretical workhorse in AMO Casimir physics [@Rodriguez]. There it agrees with precision experiments [@Levitation; @CasimirEquilibrium] with an accuracy in the percent level that appears to be only limited by the spectral range and accuracy of the material data needed for calculating the force.
The theory of Casimir forces, whether it is Lifshitz theory [@LL9; @Lifshitz; @LDP; @Scheel] or any other formulation [@Milonni; @BKMM] involves one crucial step: renormalization. Without it the energy density and the stress of the quantum fluctuations are infinite. To see this in its most elementary form, consider a one–dimensional toy model, a waveguide of length $a$ with reflecting end cups. The spectrum this one–dimensional cavity supports is given by the discrete circular frequencies $\omega_m = (\pi c/a) m$ with positive integers $m$. One thus gets for the density of the total zero–point energy $$\varepsilon_{\mathrm{\small 1D}} = \frac{E}{a} = \frac{1}{a}\sum_{m=1}^\infty \frac{\hbar\omega_m}{2} = \frac{\hbar c}{a^2} \frac{\pi}{4}\sum_{m=1}^\infty m = \infty \,.
\label{toy}$$ Renormalization extracts the part of the energy and stress that does mechanical work. In the case of the one–dimensional toy model of Eq. (\[toy\]) one obtains [@BKMM] the renormalized vacuum energy density $$\varepsilon_{\mathrm{vac 1D}} = -\frac{\pi}{48} \,\frac{\hbar c}{a^2}$$ as if the sum of all positive integers were effectively $-1/12$. In its most physically convincing and numerically efficient form [@Reid] the renormalization is done by comparing the energy of the arrangement of dielectric bodies at finite distances with the energy they had if they were infinitely far apart. It is important to keep the immersion at the same index $n_0$, for otherwise the difference in energies would diverge. This simple renormalization procedure is consistent with Lifshitz theory [@Reid] and hence has been well–tested by precision experiments.
In cosmology, the expansion factor $n$ plays the role of the refractive index. Here $n$ is uniform in space but varies in time, whereas in the arrangement of dielectric bodies $n$ varies between the bodies and the background — it varies in space and not in time. The energies and stresses of vacuum fluctuations should be just the same though, as they are both given by the energy–momentum tensor of the electromagnetic field (in the presence of the gravitational field or in media). What differs is only the way energies and stresses act. In AMO physics, the divergence of the stress produces directly the force density. In cosmology, the energy density and pressure contribute to the cosmic expansion due to their gravity, as described in the Friedmann equations (\[f2\]) and (\[f1\]). Yet in general relativity the entire energy and stress is supposed to gravitate, and not just the renormalized part, which produces a figure that disagrees with fact by 120 orders of magnitude, as shown in Sec. I. In view of this obvious conflict with reality, why should we not take the empirical fact of the astronomically small mass density, Eq. (\[density\]), as evidence that the bare vacuum energy, for whatever reason, does not gravitate?
The theory
==========
The bare vacuum energy and stress has not appeared in any experimental test of Casimir forces, nor does it appear in cosmology. Should we still take it as real? Or should we rather regard it as an artefact of the theory? Assume that renormalization is real, that the bare vacuum and stress always needs to be removed, even in gravity. The universe would still create a Casimir energy $\varepsilon_\mathrm{vac}$ and pressure $p_\mathrm{vac}$, because it is evolving. Instead of changing in space (on cosmological scales) the expansion factor $n$ changes in time, as if the spatial refractive index profile (Fig. \[typical\]) would be turned into a space–time diagram (Fig. \[rot\]). This time–dependent refractive–index profile produces a renormalized $\varepsilon_\mathrm{vac}$ and pressure $p_\mathrm{vac}$ that will be significantly smaller than the notorious 120 orders of magnitudes of the bare theory [@Weinberg].
![ \[rot\]](fig3.jpg){width="20pc"}
Yet there is one important subtlety to consider: the cosmic expansion $n(t)$ is continuous, it does not happen in jumps from one constant $n$ to the next, whereas in the typical setting for the Casimir effect (Fig. \[typical\]) the refractive index $n$ is piecewise homogenous. The Casimir force in inhomogeneous media is no longer concentrated at the surfaces of dielectric bodies — because there are no distinct bodies one could distinctly identify in gradually varying media. The force density will be a continuous function. One might think [@Simpson] one could simply discretise a continuous variation of $n$ and then calculate the force at each discrete transition from one $n$ to the next. This discretisation procedure would create force densities localized at the interfaces that, in the continuum limit, approach a smooth force distribution. It has been one of the surprises in theoretical Casimir physics [@SimpsonSurprise] that this simple discretisation of the refractive index does not work [@Simpson]. It then took some time and effort to find a workable solution [@Grin1] for the case of planar media where $n$ varies in one direction in space.
The problem is that the physically intuitive renormalization procedure of the standard Casimir effect (Fig. \[typical\]) is not applicable anymore. In a continuously varying medium it is simply impossible to move the bodies of the medium to infinity, for working out their bare Casimir energy, because the medium consists of [*one*]{} indivisible body. A local renormalization procedure is needed. For finding this procedure, insight and intuition comes from Schwinger’s source theory [@SchwingerSource]. Schwinger assumed here [@SchwingerSource] that the quantum fluctuations of fields originate from quantum fluctuations of their sources. In the case of dielectric media, these sources are the electric and magnetic polarizations of the dielectrics. Their response to the electromagnetic field forms the refractive–index profile. According to the dissipation–fluctuation theorem [@Scheel] the polarizations fluctuate, even at zero temperature when they are purely quantum. The fluctuating charges and currents of the medium create electromagnetic fields that propagate them in time across space. The fields are the messengers, not the sources of the quantum fluctuations. This picture agrees with the modern quantization procedure [@KSW; @PhilbinQ1; @PhilbinQ2; @Buhmann; @Horsley; @HorsleyPhilbin] of the electromagnetic field in dispersive and dissipative media. It also agrees with the starting point of Lifshitz theory [@Lifshitz]: Rytov’s fluctuating sources [@Rytov]. The picture suggests a local renormalization procedure as follows.
Each point of the medium is mentally split into two points, one is the emitter and the other the receiver of electromagnetic fields. Driven by quantum fluctuations, the emitter sends out an electromagnetic wave. The wave propagates, is partially reflected in the medium and is then picked up by the receiving end of the cell. Of course, all the other cells of the medium are also emitting electromagnetic waves the receiver is responding to. But averaged over the fluctuations, each cell can only interact with the field emitted by itself, because the different cells of the medium are not correlated, their emissions are independent from each other and random. Only the wave emitted by one and the same cell has a well–defined phase relation with itself and so, on average, only this field will have a physical effect onto the cell. The emitted field consists of two part, an outgoing wave and a scattered wave that is reflected at the inhomogeneities of the medium. The receiver does only interact with the scattered wave. Yet the theoretical description of the field does also contain the outgoing wave. In order to describe the idea that the receiver interacts only with the scattered radiation, one should subtract in the energy and stress the contribution coming from the outgoing wave. This subtraction procedure of the unphysical interaction of the point with itself corresponds to the renormalization procedure in Schwinger’s picture [@Grin1]. It is implemented in Lifshitz theory [@LL9; @Lifshitz; @LDP; @Scheel] by subtracting the outgoing Green function from the total Green function. This is how renormalization was conceived in Lifshitz’, Pitaevskii’s and Dzyaloshinskii’s original theory [@LDP], whereas the intuitive idea of comparing the Casimir energy between dielectric bodies at finite and infinite distances was Casimir’s [@Casimir]. Casimir’s method has made it into Lifshitz theory in its modern numerical implementation [@Reid], whereas the original renormalization procedure in Lifshitz theory has been the tool for analytical calculations.
Now, the outgoing wave emitted at a given point in the medium depends on the dielectric environment of that point: it depends on the local refractive index $n$. Without taking the local $n$ into account, the difference between the total and the bare stress would not converge to a finite value. Lifshitz theory with the original renormalization procedure [@LDP] agrees with experiments [@Levitation; @CasimirEquilibrium] where the dielectric backgrounds differ from vacuum ($n_0\neq 1$). This gives experimental evidence supporting the concept of local renormalization and ruling out any idea that renormalization amounts to subtracting the large, global energy of the bare quantum vacuum. Renormalization is local, but how local is it? The outgoing wave needs at least one cycle of oscillation for establishing itself as a wave — the oscillating magnetic field made by the source current induces an electric field that, in turn, drives the magnetic field, which induces an electric field and so on. Therefore it seems plausible that the outgoing wave depends not only on the local value of $n$, but also on the first two derivatives of $n$. For piece–wise homogeneous media (Fig. \[typical\]) the dependence of the renormalization on derivatives of $n$ does not matter, as those derivatives are zero. For inhomogeneous media, the appearance of the extra derivatives explains why the naive discretisation of the medium fails to give a converging Casimir force [@Simpson]. For planar media, the renormalization to second order was proven [@Grin1] to converge. The theory [@Grin1] has not been tested in experiments yet, but it predicts effects [@Grin2] that seem measurable with current experimental techniques, and hence are testable.
The theory [@Grin1] of the Casimir stress inside inhomogeneous planar materials makes one more prediction that, when widely extrapolated to cosmological scales, explains why the Casimir effect might play a role in cosmology: the convergence of the renormalization relies on dispersion. Ordinary dielectric materials are dispersive in the sense that the refractive index $n$ depends on frequency. The Casimir effect is a broadband electromagnetic phenomenon [@Rodriguez] depending on the entire frequency window of the material. For large frequencies all materials become completely transparent, $n\rightarrow 1$. Without this feature, the renormalized stress would contain a logarithmically diverging contribution [@Grin1]. On the other hand, the “material” of general relativity — the geometry of space and time — acts on all frequencies equally, as a consequence of the equivalence principle [@LL2]. Therefore even the renormalized $\varepsilon_\mathrm{vac}$ would still diverge, although significantly weaker than the unrenormalized one. The wavelength range contributing to the forces of the quantum vacuum would go to the Planck length (\[planck\]) where, presumably, the equivalence principle ceases to hold. So $\varepsilon_\mathrm{vac}$ would not grow with the inverse forth power of the Planck length as in Eq. (\[prediction\]) that produces the wrong 120 orders of magnitudes, but significantly weaker. The logarithmic divergence [@Grin1] is not sufficient though, for the following reason.
The divergence of the vacuum energy density and stress with the cutoff scale occurs only in inhomogeneous media. The prefactor of the divergence is therefore not a universal constant, but vanishes for constant $n$. The prefactor must depend on derivatives of $n$. In cosmology, $n$ varies on the time scale of the inverse of the Hubble constant $H$. For being able to influence the cosmic evolution described by the Friedmann equation (\[f1\]), the energy density $\varepsilon_\mathrm{vac}$ should go like $H^2$. Being a quantum energy $\varepsilon_\mathrm{vac}$ must be proportional to $\hbar$; having the physical dimensions of an energy density suggests that $\varepsilon_\mathrm{vac}$ should go like $(\hbar/c) H^2/\ell^2$. Indeed, identifying $\ell$ with the Planck length (\[planck\]) satisfies the Friedmann equation (\[f1\]) for $K=0$. This simple dimensional analysis [@Tractatus] indicates that $\varepsilon_\mathrm{vac}$ must diverge with an inverse square length for having a case for the Casimir effect in cosmology.
In planar media [@Grin1] the refractive–index profile depends only on one direction of space (Fig. \[typical\]), in spatially–flat cosmology it depends only on time (Fig. \[rot\]). This apparently simple rotation of the spatial profile into a space–time diagram involves two important subtleties that distinguishes the cosmological case from the planar case: horizons and causality. Horizons are a consequence of Hubble’s law [@LL2]: the space around a point appears to expand with a velocity that grows with the Hubble constant (\[hubble\]) times the distance. Hubble’s law is a simple kinematic feature of spatially uniform expansion where distances $r$ grow as $n(t)r$. Differentiation shows that a distant point moves away with velocity $H$ times the distance $nr$. This apparent velocity may become arbitrarily large. At the horizon the expansion speed is equal to the speed of light such that the interior of the horizon is causally disconnected from the rest of the universe. No wave from outside of the horizon can reach the point. The cosmological horizon is completely analogous [@Tractatus] to the event horizon of the black hole [@Brout]. In particular, the horizon is predicted [@Tractatus; @GibbonsHawking] to emit the analogue of Hawking radiation [@Hawking] with temperature $$k_\mathrm{B} T = \frac{\hbar H}{2\pi}
\label{gh}$$ where $k_\mathrm{B}$ denotes Boltzmann’s constant. Although the temperature (\[gh\]) lies significantly below the $2.7\mathrm{K}$ of the CMB ($2\times 10^{-29}\mathrm{K}$ for the current inverse Hubble constant of about $10^{10}$ years) it turns out [@Tractatus] to play a dominant role in the regularization of the vacuum energy.
Another extremely subtle but crucial feature of the cosmological case is causality. Recall that in the local renormalization method each point is mentally split into two: the emitter and the receiver. While emitter and receiver can change places in space, they cannot alter their order in time: emission must precede reception. This subtle constraint from causality, combined with the radiation of the cosmological horizon, creates [@Tractatus] the cosmologically relevant divergence of the vacuum energy in the second–order renormalization procedure established for planar media [@Grin1]. The Casimir energy may become cosmologically relevant. But does it explain dark energy?
The anomaly
===========
The characteristic feature, Eq. (\[lambda\]), of the cosmological constant is the negative pressure $p_\Lambda$ equal in magnitude to the positive energy density $\varepsilon_\Lambda$. Is this $\varepsilon_\Lambda$ the renormalized energy density $\varepsilon_\mathrm{vac}$ of the quantum vacuum? No, because the corresponding pressure $p_\mathrm{vac}$ must be $\varepsilon_\mathrm{vac}/3$. This is a consequence of the fact that the energy of an electromagnetic wave is equal to its momentum times the speed of light. The energy of a volume element filled with incoherent radiation must therefore be equal to the momentum transfer, [*i.e.*]{} the pressure, in the three directions of space, which gives $\varepsilon_\mathrm{vac}=3p_\mathrm{vac}$.
So where does $\varepsilon_\Lambda$ and $p_\Lambda$ come from? Consider the conservation of energy and momentum as expressed in the second Friedmann equation (\[f2\]). With $p_\mathrm{vac}=\varepsilon_\mathrm{vac}/3$ one gets the differential equation $\dot{\varepsilon}_\mathrm{vac}=-4\varepsilon_\mathrm{vac}$ with the solution $\varepsilon_\mathrm{vac}\propto n^{-4}$. This would imply that $\varepsilon_\mathrm{vac}$ is only a function of the expansion factor $n$. Yet if $\varepsilon_\mathrm{vac}$ is a Casimir energy it must also depend on derivatives of $n$. The vacuum energy $\varepsilon_\mathrm{vac}$ violates energy–momentum conservation. Wald [@Wald] understood the root of the problem: the lack of reciprocity in the renormalization procedure. Recall the point–splitting picture: the emitter sends out fluctuating electromagnetic waves, the emitter receives the reflected waves. The outgoing waves depend on the refractive–index profile around the emitter, which differs from the profile around the receiver when $n$ varies. Emitter and receiver and not reciprocal. Wald realized that this lack of reciprocity violates the conservation of energy and momentum [@Wald]. Intuitively one may see this as a recoil imbalance [@Tractatus]: the recoil of the wave on the emitter is not the same as the recoil on the receiver.
Consider the energy density of the recoil imbalance — just add a term $\varepsilon_\mathrm{recoil}$ to $\varepsilon_\mathrm{vac}$. The term should describe exactly the missing energy in the energy–momentum balance, [*i.e.*]{} on the left–hand side of the Friedmann equation (\[f2\]). This is only possible if the right–hand side is not changed by the associated pressure. The recoil pressure must therefore be the exact opposite of the recoil energy, precisely as for $\varepsilon_\Lambda$ and $p_\Lambda$ in Eq. (\[lambda\]). One may thus identify $\varepsilon_\Lambda$ with $\varepsilon_\mathrm{recoil}$ and arrives at a physical picture for the energy of the cosmological constant — dark energy. The energy and pressure of $\Lambda$ is created in the recoil imbalance of vacuum fluctuations emitted and received in the medium of space–time [@Tractatus]. The technical term for this imbalance is called trace anomaly [@Wald].
Not only gives this idea a physical picture for the cosmological constant $\Lambda$, it also establishes a method for calculating $\varepsilon_\Lambda$ (and $p_\Lambda=-\varepsilon_\Lambda$). Consider for completeness the rest of matter and radiation described by some energy density $\varepsilon_\mathrm{m}$ and pressure $p_\mathrm{m}$ in addition to the energy and pressure of the quantum vacuum. The total energy density and pressure is then given by $$\varepsilon = \varepsilon_\mathrm{m} + \varepsilon_\mathrm{vac} + \varepsilon_\Lambda \,,\quad
p = p_\mathrm{m} + \frac{1}{3}\, \varepsilon_\mathrm{vac} - \varepsilon_\Lambda \,.$$ Assume for simplicity and in agreement with observation [@CurveMeas] that the universe is spatially flat. Differentiating the first Friedmann equation (\[f1\]) and using the second Friedmann equation (\[f2\]) establishes an evolution equation of the universe: $$\dot{H} = -\frac{8\pi G}{c^2} \left(\varepsilon_\mathrm{m} + p_\mathrm{m} + \frac{4}{3}\,\varepsilon_\mathrm{vac}\right)
\label{evolution}$$ similar to the Newton equation (\[newton\]). Given $\varepsilon_\mathrm{m}$ and $p_\mathrm{m}$ as functions of $n$ and $\varepsilon_\mathrm{vac}$ as functions of $n$ and its derivatives, Eq. (\[evolution\]) defines an equation of motion for the universe on cosmological scales. This equation of motion is independent of the cosmological term $\varepsilon_\Lambda$. Given a solution of the cosmic dynamics, the energy density $\varepsilon_\Lambda$ follows from the Friedmann equation (\[f1\]) with $\varepsilon = \varepsilon_\mathrm{m} + \varepsilon_\mathrm{vac} + \varepsilon_\Lambda$.
It remains to calculate $\varepsilon_\mathrm{vac}$ for spatially uniform refractive index profiles $n$ changing in time $t$. Assuming exactly the same renormalization as for refractive index profiles changing in one direction of space [@Grin1] but taking into account the temperature (\[gh\]) of cosmic horizons and causality in the point–splitting method produces after renormalization the vacuum energy density [@Tractatus] $$\varepsilon_\mathrm{vac} = - \frac{\hbar}{12\pi^2c}\,\frac{\Delta}{\ell^2}
\label{result}$$ where $\Delta$ depends on the time derivatives of the Hubble constant $H$ and has units of $H^2$. Assuming in the renormalization procedure that the horizon temperature should be taken with respect to conformal time [@Tractatus], gives [@Tractatus] $$\Delta = \partial_t^3\frac{1}{H} + H \partial_t^2\frac{1}{H} \,.
\label{resultdelta}$$ The energy density (\[result\]) goes with the inverse square of the cutoff length $\ell$. Setting this length to the order of the Planck length (\[planck\]) gives an $\varepsilon_\mathrm{vac}$ that contributes at the right level to the cosmic evolution described by Eq. (\[evolution\]). This vacuum energy is neither too large nor too small, neither is it off by the notorious 120 orders of magnitude nor is it as insignificant as one might expect for the Casimir force. Note that Eqs. (\[result\]) and (\[resultdelta\]) hold not only in a spatially flat cosmology, but also in homogeneous and isotropic spaces with curvature [@Tractatus]. The calculation [@Tractatus] was done for electromagnetic fields. Assuming the same principal behaviour for the other fields of the Standard Model would amount to multiplying Eq. (\[result\]) with the number of the independent field components divided by two (the polarizations of the electromagnetic field). As the precise cutoff length is not precisely known, one can express both the multitude of fields and the cutoff in an effective constant prefactor $\eta$ and write Eq. (\[result\]) as $$\varepsilon_\mathrm{vac} = - \frac{\hbar\eta}{12\pi^2c}\,\frac{\Delta}{\ell_\mathrm{p}^2} \,.
\label{eta}$$ The following picture emerges from the theory [@Tractatus]. The cosmic evolution, described by the scaling factor $n$ and its derivatives, generates the energy density (\[eta\]) of the quantum vacuum. The vacuum energy acts back on the cosmic dynamics as described in the evolution equation (\[evolution\]). Consistent with this evolution is the effective cosmological constant with energy density $\varepsilon_\Lambda$ given by the Friedmann equation (\[f1\]) with $\varepsilon = \varepsilon_\mathrm{m} + \varepsilon_\mathrm{vac} + \varepsilon_\Lambda$. The cosmological constant corresponds to a trace anomaly [@Wald] one may interpret as a recoil imbalance of vacuum fluctuation in the “material” of space–time [@Tractatus]. The vacuum energy of Eqs. (\[resultdelta\]) and (\[eta\]) vanishes if the Hubble constant (\[hubble\]) is indeed constant, [*i.e.*]{} for exponential expansion. In this case, the cosmological constant $\Lambda$ is constant as well. Otherwise it will vary, which might explain the observed variation of $\Lambda$ [@Cepheids]. This picture shares some features with the theory of quintessence [@Caldwell] but it does not require any new fields, just new concepts in fields as old as quantum electromagnetism.
Conclusions
===========
There is no empirical evidence for the bare vacuum energy of fields. Neither does the bare vacuum energy do mechanical work [@Forces], nor does it gravitate. Therefore it seems wise to take for the vacuum energy in the universe a renormalized energy density. Assuming the same renormalization procedure as for the Lifshitz theory in planar media [@Grin1] while taking into account cosmological horizons and causality, gives a vacuum energy density of the right order of magnitude [@Tractatus]. The resulting theory makes concrete predictions about the cosmic evolution, Eq. (\[evolution\]), apart from one parameter in Eq. (\[eta\]) that cannot be determined yet. The result seems encouraging, but of course it remains to be seen whether the theory reproduces the astronomical facts in detail, and not only their order of magnitude. The theory was developed [@Tractatus] for the electromagnetic field as carrier of vacuum fluctuations; it therefore remains to be checked whether it can be extended to other fields.
Analogues of gravity [@Gordon; @Plebanski; @Schleich; @LeoPhil] have played a decisive role in developing the theory [@Tractatus] and analogues may be important for testing crucial components of the theory in experiments. The theory extends the renormalization procedure of the Casimir stress in inhomogeneous planar media [@Grin1] to spatially uniform, time–dependent materials. Direct measurements of the Casimir force inside planar media are difficult, the easiest is perhaps a test of the extreme behaviour predicted in Ref. [@Grin2]. But one could emulate Casimir forces with other fluctuations, as long as they have a similar spectrum than the vacuum fluctuations, replacing the $\hbar$ in the Casimir force by an effective noise parameter of significantly larger magnitude, which would enhance the effect. Measuring Casimir forces in time–dependent media is not simple either. Here also analogues may play an important role for crucial tests.
While the physics of the Casimir effect of separate bodies is well–understood, the Casimir force inside materials has remained a fairly underdeveloped subject. There is enormous scope for research in both theory and experiment. If the Lifshitz theory of the cosmological constant [@Tractatus] does indeed agree with the astronomical facts in detail, either directly or after minor modifications, it would not only shed light on a rather dark subject in cosmology, but also motivate a better understanding of the forces acting on the nanoscale in the everyday world.
[**Funding.**]{} European Research Council, Israel Science Foundation, Murray B. Koffler Professorial Chair.\
[**Acknowledgements.**]{} This paper would be impossible without my students (in alphabetical order) Yael Avni, Nimrod Nir, Itay Griniasty, Sahar Sahebdivan, and William Simpson. I also thank Mikhail Isachenkov, Ephraim Shahmoon, and Anna and Yana Zilberg for discussions, support and inspiration.
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---
abstract: 'We demonstrate an implementation scheme for constructing quantum gates using unitary evolutions of the one-dimensional spin-$J$ ferromagnetic XXZ chain. We present numerical results based on simulations of the chain using the time-dependent DMRG method and techniques from optimal control theory. Using only a few control parameters, we find that it is possible to implement one- and two-qubit gates on a system of spin-3/2 XXZ chains, such as Not, Hadamard, Pi-8, Phase, and C-Not, with fidelity levels exceeding $99\%$.'
author:
- Tom Michoel
- Jaideep Mulherkar
- Bruno Nachtergaele
title: 'Implementing Quantum Gates using the Ferromagnetic Spin-${J}$ XXZ Chain with Kink Boundary Conditions'
---
Introduction
============
For quantum computers to become a reality we need to find or build physical systems that faithfully implement the quantum gates used in the algorithms of quantum computation. The basic requirement is that the experimenter has access to two states of a quantum system that can be effectively decoupled from environmental noise for a sufficiently long time, and that transitions between these two states can be controlled to simulate a number of elementary quantum gates (unitary transformations). Systems that have been investigated intensively are atomic levels in ion traps [@CZ1995; @MKIW1995], superconducting device physics using Josephson rings [@MOL1999], nuclear spins [@CVZ1998](using NMR in suitable molecules) and quantum dots [@LD1998]. In this paper we demonstrate the implementation of quantum gates using one-dimensional spin-$J$ systems. The results are obtained using a computer simulation of these systems.
The Hamiltonian of the XXZ model with kink boundary conditions is given by $$\begin{aligned}
\label{eqn:Hamiltonian}
H_L^{\rm k}(\Delta^{-1})= &\sum_{\alpha=-L+1}^{L-1} \Big[({J}^2-S_\alpha^3 S_{\alpha+1}^3) - \Delta^{-1}(S_\alpha^1 S_{\alpha+1}^1\\ \nonumber
&+ S_\alpha^2 S_{\alpha+1}^2)\Big] + {J}\sqrt{1-\Delta^{-2}}(S_{-L+1}^3 - S_L^3)\nonumber\end{aligned}$$ where $S_\alpha^1$, $S_\alpha^2$ and $S_\alpha^3$ are the spin-${J}$ matrices acting on the site $\alpha$. Apart from the magnitude of the spins, $J$, the main parameter of the model is the anisotropy $\Delta>1$ and the limit $\Delta\rightarrow\infty$ is referred to as the *Ising limit*. In the case of $J=1/2$ kink boundary conditions were first introduced in [@PS1990]. They lead to ground states with a domain wall between down spins on the left portion of the chain and up spins on the right. The third component of the magnetization, $M$, is conserved, and there is exactly one ground state for each value of $M$. Different values of $M$ correspond to different positions of the domain walls, which in one dimension are sometimes referred to as kinks. In [@KNS2001], Koma, Nachtergaele, and Starr showed that there is a spectral gap above each of the ground states in this model for all values of ${J}$. Recently [@MNSS2008] it was shown that for spin values $J\ge \frac{3}{2}$ and for sufficiently large value of the anisotropy $\Delta$ the low lying spectrum of (\[eqn:Hamiltonian\]) for each value of $M$ has isolated eigenvalues that persist in the thermodynamic limit.
The presence of isolated eigenvalues is ideal from the point of view of quantum computation. The idea is to use the subspace, denoted by $\mathcal{D}$, of the ground state and the first excited state of the Hamiltonian to encode a qubit. We let the system evolve under its own unitary time evolution generated by the Hamiltonian (\[eqn:Hamiltonian\]) with the addition of a few local control fields. We have two requirements to fulfill: the time evolution should leave the qubit space $\mathcal{D}$ approximately invariant, and the (approximately) unitary matrix describing the dynamics restricted to $\mathcal{D}$ and stopped at a suitable time should coincide with the desired quantum gate.
The control inputs needed to drive the system such that high fidelity gates are obtained are determined using techniques from optimal control theory. The simulation of the time evolution of the chain that is large enough to resemble the properties in the thermodynamic limit is carried out using the Density Matrix Renormalization Group (DMRG) algorithm. Figure \[fig:proftrans\] shows the transition of the magnetic profiles in the z-direction from the ground to the first excited state using the Not gate constructed from a spin-$\frac{3}{2}$ XXZ spin chain of length 50 sites. We also demonstrate the construction of Pi-8, Hadamard, and Phase gates that form a set of universal single qubit gates.
In order to have a viable quantum computing scheme one needs to implement at least one 2-qubit gate. Here we have implemented the C-Not gate which, in combination with the 1-qubit gates, is known to be universal [@AB1995].
Our scheme capitalizes on the kink nature of the excitations of the XXZ Hamiltonian, which are rather sharply localized. We imagine a setup with two parallel chains with the location of the kink lined up in their ground states. The subspace for the 2-qubit state space is then $\mathcal{D}_1\otimes\mathcal{D}_2$, where $\mathcal{D}_1$ represents the space of isolated eigenvalues of the first chain and $\mathcal{D}_2$ for the second chain. A set of three controls localized near the kinks is used to generate the single qubit gates acting on $\mathcal{D}_1$ and $\mathcal{D}_2$ and a C-Not gate on $\mathcal{D}_1\otimes\mathcal{D}_2$. This scheme produces a universal set of gates necessary for two-qubit computation. It is clear how to generalize this scheme to implement n-qubit computation. Since a universal set of single qubit gates and nearest neighbor C-Not gates are universal for n-qubit computation, this can be achieved by using $n$ parallel chains and controls that are localized and act on neighboring chains only.
In the next section we describe the model and review some of the past results. Then, in section \[sec:Control\], the optimal control problem to construct the quantum gates is described. Section \[sec:DMRG\] is devoted to the DMRG algorithm and the specific adaptations to the XXZ spin chain. Finally, in section \[sec:Results\] we present our results based on numerical simulations of the XXZ Hamiltonian using the DMRG algorithm.
The Model {#sec:Model}
=========
In this section we describe in detail the spin-$J$ ferromagnetic XXZ model with kink boundary conditions on the one-dimensional lattice $\mathbb{Z}$. The local Hilbert space for a single site $\alpha$ is $\mathcal{H}_\alpha = \mathbb{C}^{2{J}+1}$ with ${J}\in \frac{1}{2} \mathbb{N} = \{0,\frac{1}{2},1,\frac{3}{2},2,\dots\}$. We consider the Hilbert space for a finite chain on the sites $[-L+1,L] = \{-L+1,-L+2,\dots,+L\}$. This is $\mathcal{H}_{[-L+1,L]}=\bigotimes_{\alpha=-L+1}^L\mathcal{H}_\alpha$. The Hamiltonian of the spin-${J}$ XXZ model is given by equation (\[eqn:Hamiltonian\]). Note that, by a telescoping sum, we can absorb the boundary fields into the local interactions: $$\begin{aligned}
\label{XXZ Hamiltonian}
H_L^{\rm k}(\Delta^{-1}) &=& \sum_{\alpha=-L+1}^{L-1} h^{\rm k}_{\alpha,\alpha+1}(\Delta^{-1})\\
h^{\rm k}_{\alpha,\alpha+1}(\Delta^{-1}) &=& {J}^2-S_\alpha^3 S_{\alpha+1}^3 - \Delta^{-1}(S_\alpha^1 S_{\alpha+1}^1 + S_\alpha^2 S_{\alpha+1}^2)\\
&+& {J}\sqrt{1-\Delta^{-2}}\, (S_{{\alpha}}^3 - S_{{\alpha}+1}^3)\end{aligned}$$ The main parameter of the model is the anisotropy $\Delta>1$ and we get the Ising limit as $\Delta\rightarrow\infty$. It is mathematically more convenient to work with the parameter $\Delta^{-1}$, which we then assume is in the interval $[0,1]$. As we said, $\Delta^{-1}=0$ is the Ising limit, and $\Delta^{-1}=1$ is the isotropic XXX Heisenberg model. The Hamiltonian commutes with the total magnetization $$S^3_{\rm tot}\, =\, \sum_{\alpha =-L}^L S^3_\alpha\, .$$ As indicated in the introduction, for each $M \in \{-2{J}L,-2{J}L +1 ,\dots,2{J}L\}$, the corresponding sector is defined to be the eigenspace of $S^3_{\rm tot}$ with eigenvalue $M$; clearly, these are invariant subspaces for all the Hamiltonians introduced above. These subspaces are called “sectors”.
It was shown [@PS1990; @ASW1995; @GW1995; @KN1998] that the kink boundary conditions lead to a family of ground states. It was also shown in [@ASW1995; @GW1995; @Mat1996; @KN1998] that for each sector there is a unique ground state of $H_L^{\rm k}(\Delta^{-1})$ with eigenvalue 0. Moreover, this ground state, $\psi_M$, is given by the following expression: $$\psi_M = \sum
\bigotimes_{\alpha \in [-L+1,L]}\binom{2{J}}{{J}-m_\alpha}^{1/2} q^{\alpha({J}-m_\alpha)}{\vert{m_\alpha}\rangle}_\alpha\, ,$$ where the sum is over all configurations for which $\sum_\alpha m_\alpha = M$ and the relationship between $\Delta>1$ and $q\in(0,1)$ is given by $
\Delta=(q+q^{-1})/2$. A straightforward calculation shows a sharp transition in the magnetization from fully polarized down at the left to fully polarized up at the right. For this reason they are called kink ground states. In [@KNS2001], Koma, Nachtergaele, and Starr showed that there is a spectral gap above each of the ground states in this model for all values of ${J}$. Recently [@MNSS2008] we were able to prove the following theorem.
For spin values $J \ge 3/2$, there exists a finite $\Delta_0$ so that for all $\Delta > \Delta_0$, the first few excitations of $H_L^{\rm k}(\Delta^{-1})$ when restricted to any sector of magnetization, are isolated eigenvalues that persist in the thermodynamic limit.
In this paper it was also proved that in certain values of spin and sector, for example ${J}= \frac{3}{2}$ and $M=0$ both the ground and excited states are non-degenerate (simple eigenvalues). This is the qubit space we work with and our quantum gates will be unitaries on this space.
Quantum gates using quantum control {#sec:Control}
===================================
The problem of constructing quantum gates can be formulated as a problem in quantum control theory [@MK2005]. The goal is to steer the system using a small number of control parameters such that the unitary operator describing the quantum dynamics after a finite time $T$, has maximal overlap with a desired target unitary (the gate). >From a control perspective these problems reduce to control of bilinear systems evolving on finite dimensional Lie groups. This is an optimal control problem on a two-level system which has been studied widely with exact results known in some cases. For example, time optimal implementation of single and two qubit quantum gates was studied [@KBG2001] when the Lie algebra $\rm{g}$ of $su(2)$ ($su(4)$) can be decomposed as a Cartan pair $\rm{g} = \rm {k} \oplus {p}$ with $ \rm{k}$ is the Lie subalgebra generated by a the drift Hamiltonian and $\rm{p}$ is the Lie sub algebra generated by the control Hamiltonian’s. Finding the time optimal trajectories is reduced to finding geodesics on the coset space $G/K$ ($G$ and $K$ being the Lie Groups corresponding to $\rm{g}$ and [k]{}). The problem of driving the evolution operator while minimizing an energy-type quadratic cost was studied in [@DD2001]. In this case the optimal solutions can be expressed as Elliptic functions. The time optimal problem of population transfer problem of a two-level quantum system and bounded controls was studied in [@BM2005] and again explicit expressions for the optimal trajectories. In this paper we follow a numerical gradient based approach to optimal control [@DD2008; @KR2005].
Single qubit gates {#single qubit gates}
------------------
We consider the problem of time evolution of the one-dimensional XXZ chain under external controls. The equation of motion for the unitary evolution of the XXZ chain isolated from the environment is given by Schrödinger’s equation $$\label{eqn:control1}
\dot{U}(t) = -i\Big(H_L^{\rm k}(\Delta^{-1}) + v(t)H^{\rm ext}\Big)U,\quad U(0) = {{1\hskip -3pt \rm{I}}}$$ In control terminology $H_L^{\rm k}(\Delta^{-1})$ is the free or drift Hamiltonian and $H^{\rm ext}$ is the control Hamiltonian corresponding to the control field $v(t)$. We require that $\mathcal{D}$ is an invariant subspace of $H^{\rm ext}$, so that the time evolution of the system \[eqn:control1\] given by the unitary $U(t)$ starting from an initial state in $\mathcal{D}$ will be constrained to $\mathcal{D}$ at all future times. The induced evolution on $\mathcal{D}$ at any specified final time $T$ will be the quantum gate on the qubit space $\mathcal{D}$ and is given by the 2x2 matrix
$$(U_{xxz})_{ij}:= {\langle{\psi_i}|{U(T)|\psi_j}\rangle}\qquad i=0,1$$
The control Hamiltonian we choose is the two site operator $H^{\rm ext}=S_0^3S_1^3$. In practice for $S_0^3S_1^3$ there is a very small error probability for states to move out of $\mathcal{D}$ and the matrix $U_{xxz}$ is not exactly unitary. The matrix elements ${\langle{\psi_0 | H^{\rm ext}}|{\psi_k}\rangle}$ and ${\langle{\psi_1 | H^{\rm ext}}|{\psi_k}\rangle}$ $k \ne 0,1$ are proportional to the transition probabilities to move from states $\psi_0$ and $\psi_1$ to other eigenstates of $H_L^{\rm k}(\Delta^{-1})$. We calculate the error probability to move out of the subspace $\mathcal{D}$ by the following estimates of these matrix elements $$\label{errprob}
\text{ErrProb} = \|H^{\rm ext}\psi_i\|^2 - |{\langle{\psi_0}|{H^{\rm ext}\psi_i}\rangle}|^2 - |{\langle{\psi_1}|{H^{\rm ext}\psi_i}\rangle}|^2$$ for $i=0,1$. Figure \[fig:err\_prob\] shows that the probabilities of transitioning out of the subspace $\mathcal{D}$ are extremely small for $\Delta^{-1} \le 0.3$.
Implementing two-qubit gates {#two qubit gates}
----------------------------
The idea for implementing two-qubit gates is to use two copies of the XXZ chain. The Hilbert space for two-qubit quantum computation is $\mathcal{D}_1\otimes\mathcal{D}_2 \cong \mathbb {C}^4$, where $\mathcal{D}_1$ and $\mathcal{D}_2$ are the subspaces spanned by the ground state and first excited state of the first chain and second chain respectively. The Hamiltonian of an uncoupled two chain system is given by $$H_{L}^{\rm k}(\Delta^{-1})^{(1,2)} := H_L^{\rm k}(\Delta^{-1})^{(1)} + H_L^{\rm k}(\Delta^{-1})^{(2)}$$ Here the notation $H_L^{\rm k}(\Delta^{-1})^{(1)}$ is to be interpreted as $\underbrace{H_L^{\rm k}(\Delta^{-1})}_{chain 1}\otimes (\underbrace{{{1\hskip -3pt \rm{I}}}\otimes\cdots\otimes{{1\hskip -3pt \rm{I}}}}_{chain 2})$ and $H_L^{\rm k}(\Delta^{-1})^{(2)}$ is to be interpreted as $(\underbrace{{{1\hskip -3pt \rm{I}}}\otimes\cdots\otimes{{1\hskip -3pt \rm{I}}}}_{chain 1})\otimes \underbrace{H_L^{\rm k}(\Delta^{-1})}_{chain 2}$. The two-qubit space is spanned by the four vectors $\psi_{mn}:=\psi_m\otimes\psi_n$ for $m,n=0,1$ which are eigenvectors of the above Hamiltonian. If we consider the control system $$\begin{aligned}
\label{eqn:control2}
\qquad\dot{U} =& -i\Big(H_{L}^{\rm k}(\Delta^{-1})^{(1,2)} + v_1(t) (S_0^3.S_1^3)^{(1)}\\ \nonumber
&+ v_2(t) (S_0^3.S_1^3)^{(2)}\Big)\nonumber\end{aligned}$$ with $U(0) = {{1\hskip -3pt \rm{I}}}$, then by selectively turning on $v_1(t)$ and $v_2(t)$ for certain time periods, the above system is equivalent to the control system (\[eqn:control1\]) on chains 1 and 2 respectively during those time intervals. This can be used to generate single qubit gates on $\mathcal{D}_1$ and $\mathcal{D}_2$. Moreover by simultaneously using $v_1(t)$ and $v_2(t)$ the local gates i.e. gates of the kind $X_1\otimes Y_2$ can be generated on $\mathcal{D}_1\otimes\mathcal{D}_2$. To implement a two-qubit quantum computing scheme we need to also implement perfectly entangling gates i.e. a gate that can take a product state to a maximally entangled state. It is known that single qubit gates and any perfectly entangling gate are universal for two-qubit quantum computing [@ZVSW2003]. Clearly such a gate cannot be implemented by the control scheme (\[eqn:control2\]) alone. In this paper we choose to implement the C-Not gate, which is an example of a perfectly entangling gate. For this purpose we make use of an additional control namely $(S_0^3 S_1^3)^{(1)}\otimes (S_0^3 S_1^3)^{(2)}$.
We demonstrate the C-Not gate to high precision by using following control system $$\begin{aligned}
\label{eqn:control3}
\dot{U} = &-i\Big(H_{L}^{\rm k}(\Delta^{-1})^{(1,2)} + v_1(t) (S_0^3S_1^3)^{(1)} \\\nonumber
+& v_2(t) (S_0^3S_1^3)^{(2)} + v_3(t)(S_0^3S_1^3)^{(1)}\otimes (S_0^3S_1^3)^{(2)}\Big)\nonumber\end{aligned}$$ with $U(0) = {{1\hskip -3pt \rm{I}}}$ by selectively turning on and off some or all of the control fields $v_1(t)$, $v_2(t)$ and $v_3(t)$ for specified time periods. Figure \[fig:xxz\_schematic\] shows a diagrammatic representation of the two-qubit scheme. The C-Not gate is then given by the $4\times 4$ matrix with elements $$({\rm C-Not}^{xxz})_{mn;rs}:= {\langle{\psi_{mn}}|{U(T)|\psi_{rs}}\rangle}\, i=0,1$$
Optimal control {#sec:optimal control}
---------------
We first solve the control problems (\[eqn:control1\]) and (\[eqn:control3\]) for the projected system on $\mathcal{D}$ for the single chain and $\mathcal{D}_1\otimes\mathcal{D}_2$ for two chain system. $$\label{eqn:control4}
\dot{U}(t) = -i\Big(H + \sum_k v_k(t)B_k\Big)U,\qquad U(0) = {{1\hskip -3pt \rm{I}}}$$ For the projected system on $\mathcal{D}$ the $H$ and $B_k$’s are given by the $2\times 2$ matrices $$\begin{aligned}
\label{eqn:matrix2}
H_{ij} &=& {\langle{\psi_i}|{H_L^{\rm k}(\Delta^{-1})|\psi_j}\rangle}\\ \nonumber
(B_1)_{ij} &=& {\langle{\psi_i}|{H_L^{\rm k}(\Delta^{-1})|\psi_j}\rangle}\, i,j=0,1 \nonumber\end{aligned}$$ whereas the the projected system on $\mathcal{D}_1\otimes\mathcal{D}_2$ the control problem involves $4\times 4$ matrices $$\begin{aligned}
\label{eqn:matrix4}
H_{mn;rs}&=& {\langle{\psi_{mn}}|{H_{L}^{\rm k}(\Delta^{-1})^{(1,2)}|\psi_{rs}}\rangle}\\ \nonumber
(B_1)_{mn;rs}&=& {\langle{\psi_{mn}}|{(S_0^3.S_1^3)^{(1)}|\psi_{rs}}\rangle}\\ \nonumber
(B_2)_{mn;rs}&=& {\langle{\psi_{mn}}|{(S_0^3.S_1^3)^{(2)}|\psi_{rs}}\rangle}\\ \nonumber
(B_3)_{mn;rs}&=& {\langle{\psi_{mn}}|{(S_0^3.S_1^3)^{(1)}\otimes (S_0^3.S_1^3)^{(2)}|\psi_{rs}}\rangle} \nonumber\end{aligned}$$ where $m,n,r,s =0,1$. The overlap between a desired unitary gate $U_f$ and the solution of (\[eqn:control3\]) at time $T$, $U(T)$, is measured as the difference in the norm square $\|U_f-U(T)\|^2$, and the norm is defined in terms of the standard inner product ${\langle{V}|{W}\rangle} := Tr(V^{\dag}W)$. The norm can be written as $$\|U_f-U(T)\|^2 = \|U_f\|^2 -2Re{\langle{U_f}|{U(T)}\rangle} + \|U(T)\|^2$$ and hence minimizing this norm is equivalent to maximizing $$\label{eqn:cost}
\Phi:= Re{\langle{U_f}|{U(T)}\rangle} = Tr(U_f^{\dagger}U(T))$$ We define the gate fidelity as $$\label{eqn:fidelity}
\mathcal{F}_{\text{Gate}}:= \frac{|Tr(U_f^{\dag}{U(T))}|}{Tr({{1\hskip -3pt \rm{I}}})}$$ To select the optimal control fields $v_i(t)$ we use the numerical gradient ascent approach described in many books on control theory. This approach was applied to the quantum setting in [@KR2005]. We start with the necessary conditions for optimality called the Pontryagin maximum principle which is a generalization of the Euler-Lagrange equations from calculus of variations. In the problems with costs of type (\[eqn:cost\]) and no a priori bound on controls, Pontryagin’s maximum principle takes the following form
(Pontryagin maximum principle ) If $v_i(t)$’s are optimal controls of the system (\[eqn:control3\]) and $U(t)$ the corresponding trajectory solution, then there exists a nonzero operator valued Lagrange multiplier $\lambda$ which is the solution of the adjoint equations $$\begin{aligned}
\dot{\lambda}(t) &=& -iH(t)\lambda(t) \qquad \text{with terminal condition} \\
\lambda^{'}(T) &=&-\frac{\partial{\Phi(T)}}{\partial{U(T)}} = -U_f\end{aligned}$$ and a scalar valued Hamiltonian function $h(U(t),v_i(t)):= Re\ Tr(-i \lambda^{'}(t)H(t)U(t))$ such that, for every $\tau\in (0,T]$ we have $$\label{eqn:gradient}
\frac{\partial {h(U)}}{\partial{v_i}}= Im\ Tr(\lambda^{'}(t) B_iU(t)) = 0$$
The algorithm to find the optimal controls is as follows
1. A suitable gate time $T$ is chosen and discretized in $N$ equal steps of duration $\Delta t = \frac{T}{N}$. The initial control $v_i^{(0)}(t_k)$ for all the discretized time intervals is based on a guess or at random.
2. For these piecewise constant controls, from $U(0) = {{1\hskip -3pt \rm{I}}}$ and $\lambda(T)= -U_f$, compute the forward and backward propagation respectively as follows $$\begin{aligned}
\label{eqn:fequation}
U^{(r)}(t_k) &=& F^{(r)}(t_k)F^{(r)}(t_{k-1})\ldots F^{(r)}(t_1)\\
\label{eqn:bequation}
\lambda^{(r)}(t_k)&=& F^{(r)}(t_k)F^{(r)}(t_{k+1})\ldots F^{(r)}(t_N)\lambda(T) \end{aligned}$$ for all $t_1,\ldots,t_N$ and where $r$ is an iteration number of the algorithm initially set to 0 and $$\label{eqn:evolution1}
F^{(r)}(t_k) = exp\Big\{-i\Delta t\Big(H + \sum_i v_i^{(r)}(t_k)B_i\Big)\Big\}$$
3. Substitute the equations (\[eqn:fequation\]) and (\[eqn:bequation\]) into equation (\[eqn:gradient\]) to evaluate the gradient, and then update the controls as $$v_i^{(r+1)}(t_k) = v_i^{(r)}(t_k) + \tau \frac{\partial h(U(t_k),v_i(t_k))}{\partial v_i}$$ where $\tau$ is a small step size.
4. if $\mathcal{F}_{\text{Gate}}< \gamma$ ($\gamma$ being the level of accuracy) then done, otherwise goto step (2) for the next iteration with the updated controls.
Having solved the control problem on the projected systems to get the optimal controls $v_1(t)$, $v_2(t)$ and $v_3(t)$ we would like to apply them to a large system and see their effects on the projected system. However simulating even a moderately sized spin chain is hard because of the exponentially growing dimension of the Hilbert space. In the next section we describe an algorithm by which we are able to simulate the XXZ chain of 50 sites.
DMRG simulations for quantum gates {#sec:DMRG}
==================================
To see the effect of the evolution of the XXZ chain with external magnetic controls we numerically simulate the XXZ chain using the DMRG algorithm. The dynamics of the interfaces of the XXZ chain using DMRG was studied recently in [@MNS2007]. The standard DMRG algorithm is a numerical algorithm originally developed by Steven White [@SW1993] that has worked successfully in providing very accurate results for ground state energies and correlation functions in strongly correlated systems. Modifications to this method [@GV2004; @WF2004] allow to address the physics of time-dependent and out of equilibrium systems. The crux of the DMRG algorithm is a decimation procedure that chooses the physically most relevant states to describe the target states. It is now known that DMRG works well because the ground states of non-critical quantum chains like the XXZ chain are only slightly entangled, i.e. they obey an area law of entanglement that says that the entanglement between a distinguished block of the chain and the rest of the chain is bounded by the boundary area of the block. In fact the DMRG procedure is a variational ansatz over states known as Matrix product states (MPS) [@FNW1992]. The standard DMRG procedure and its connection with MPS and entanglement is described in detail in [@US2005]. For a single XXZ chain our target states are the ground state $\psi_0$ and first excited state $\psi_1$ restricted to a sector of magnetization. We use the standard DMRG procedure with the adaptation that we grow the chain while restricting the blocks to the sector of zero magnetization using the symmetry of the Hamiltonian (see [@MNS2007]).
For the two-qubit gates we convert the two chain system to a one dimensional spin chain by a spin ladder construction. $$\begin{array}{ccccccccccc}
{\mathcal{H}}_{-L+1}^{(1)} & \otimes & {\mathcal{H}}_{-L+2}^{(1)} & \otimes
& \dots & \otimes & {\mathcal{H}}_L^{(1)} & = & {\mathcal{H}}_{[-L+1,L]}^{(1)}\\
\otimes & & \otimes & & & & \otimes && \otimes \\
{\mathcal{H}}_{-L+1}^{(2)} & \otimes & {\mathcal{H}}_{-L+2}^{(2)} & \otimes
& \dots & \otimes & {\mathcal{H}}_L^{(2)} & = & {\mathcal{H}}_{[-L+1,L]}^{(2)}
\end{array}$$ The single site Hilbert space for the DMRG is the rung composed of ${\mathcal{H}}_{\alpha}^{(1)}\otimes{\mathcal{H}}_{\alpha}^{(2)}$ for $\alpha\in[-L+1..L]$. On this site we define the local operators $$S^{i(1)}_{\alpha}= S^i_{\alpha}\otimes{{1\hskip -3pt \rm{I}}}_{\alpha}, \qquad S^{i(2)}_{\alpha}= {{1\hskip -3pt \rm{I}}}_{\alpha}\otimes S^i_{\alpha} \qquad \text{for i= 1,2,3}$$ We can then write the Hamiltonian of this single chain using the above construction $$\begin{aligned}
H_{L}^{\rm k}(\Delta^{-1})^{(1,2)} &:= \sum_{\alpha =-L+1}^{L-1} h^{(1)}_{\alpha,\alpha+1}(\Delta^{-1}) + h^{(2)}_{\alpha,\alpha+1}(\Delta^{-1}) \\
h^{(k)}_{\alpha,\alpha+1}(\Delta^{-1})&= {J}^2-S_\alpha^{3(k)} S_{\alpha+1}^{3(k)} - \Delta^{-1}\big(S_\alpha^{1(k)} S_{\alpha+1}^{1(k)}\\
&+ S_\alpha^{2(k)} S_{\alpha+1}^{2(k)}\big) + {J}\sqrt{1-\Delta^{-2}} (S_{{\alpha}}^{3(k)} - S_{{\alpha}+1}^{3(k)})\end{aligned}$$ for $k=1,2$. We carry out the DMRG procedure as described in the algorithm with the Hamiltonian $H_{L}^{\rm k}(\Delta^{-1})^{(1,2)}$ but we ensure that we keep both the chains in the magnetization sector 0 by simultaneously diagonalizing $H_{L}^{\rm k}(\Delta^{-1})^{(1,2)}$ with the total magnetization operators $$S_{tot}^{(k)} = \sum_{\alpha=-L+1}^L S^{3(k)}_{\alpha} \qquad \text{for k=1,2}$$ The target states $\psi_0\otimes\psi_0$, $\psi_0\otimes\psi_1$, $\psi_1\otimes\psi_0$, $\psi_1\otimes\psi_1$ are the simultaneous eigenvectors of the these operators and form the computational basis ${\vert{00}\rangle}$, ${\vert{01}\rangle}$,${\vert{10}\rangle}$ and ${\vert{11}\rangle}$ for two-qubit quantum computation.
To compute the time evolution of the chain under the controlled evolution by the external fields we use the time-dependent DMRG procedure. The idea is that a two site operator can be applied to a DMRG state most effectively by expressing the state in the basis where the left block has length $x-1$ so the two middle sites that are untruncated are the the sites where the operator is acting. We can write the time evolution in the Trotter decomposition $$\begin{aligned}
e^{-iH\delta} \cong e^{-\frac{i}{2}h_{-L+1,-L+2}}e^{-\frac{i}{2}h_{-L+2,-L+3}}\cdots \\ \cdots e^{-\frac{i}{2}h_{L-2,L-1}}e^{-\frac{i}{2}h_{L-1,L}} +O(\delta^3)\end{aligned}$$ To apply $e^{-iH\delta}$ to the ground and excited states in the basis with the center sites all the way to the left we apply $e^{-\frac{i}{2}h_{-L+1,-L+2}}$. After shifting one site to the right we apply $e^{-\frac{i}{2}h_{-L+2,-L+3}}$ etc. Since all our controls are two site controls at the center, only the interaction $h_{0,1}$ is time-dependent. In the adaptive time-dependent methods the Hilbert space is continuously modified as time progresses by carrying out reduced basis transformations on the evolved state. In our case since the gates are obtained in a relatively short period of time our Hilbert space remains unchanged resembling the static DMRG methods.
Results {#sec:Results}
=======
In this section we present numerical results of the construction of quantum gates using the spin-3/2 XXZ spin chain. Our results are for the universal set of single qubit gates consisting of the Not (X), Hadamard (H), Pi-8 (T) and Phase (S) gates and the two-qubit C-Not gate. All results are obtained using the DMRG algorithm and the optimal control methods described in the previous sections.
Not(X) Hadamard(H) Pi-8(T) Phase(S)
--------- ------------- --------- ----------
0.1874 -0.2182 -0.1152 -0.0797
-0.0533 -0.1176 -0.2544 -0.1889
-0.2447 -0.0631 -0.3310 -0.2579
-0.3587 -0.0670 -0.3613 -0.2945
-0.3764 -0.1296 -0.3632 -0.3085
-0.2901 -0.2396 -0.3524 -0.3091
-0.1075 -0.3766 -0.3410 -0.3031
0.1376 -0.5154 -0.3358 -0.2943
0.3712 -0.6286 -0.3383 -0.2836
0.4908 -0.6917 -0.3443 -0.2691
0.4355 -0.6899 -0.3441 -0.2466
0.2359 -0.6241 -0.3222 -0.2103
-0.0246 -0.5099 -0.2590 -0.1538
-0.2681 -0.3723 -0.1328 -0.0715
-0.4399 -0.2404 0.0709 0.0386
-0.5065 -0.1424 0.3368 0.1704
-0.4553 -0.1001 0.5877 0.3053
-0.2959 -0.1225 0.7029 0.4128
-0.0605 -0.2033 0.6294 0.4642
0.1909 -0.3236 0.4374 0.4516
: Numerical simulation of the construction of the Not, Pi-8, Hadamard and Phase gates. Results are obtained using DMRG and time-dependent DMRG for a spin-$\frac{3}{2}$ chain of $L=50$ sites and at $\Delta^{-1}=0.3$ in the sector corresponding to $M=0$. The table shows the values of the control field $v_1(t)$ with gate time $T=10$ discretized with $\Delta t = 0.5$.[]{data-label="table:qbitcontrol"}
$v_1(t)$ $v_2(t)$ $v_3(t)$
---------- ---------- ----------
-0.4040 0.3953 0.0540
2.4494 0.2588 0.0501
3.7163 0.1886 -0.1314
3.0455 0.1766 -0.2677
1.6565 0.2185 -0.0872
0.5583 0.3085 0.1455
-0.1036 0.4206 0.2346
-0.5117 0.5196 0.1648
-0.8215 0.5716 -0.0282
-1.0083 0.5578 -0.3599
-0.8630 0.4926 -0.9102
-0.3774 0.3612 -1.4955
-0.0810 0.0978 -1.4905
-0.0940 -0.1214 -0.9196
-0.0881 -0.1138 -0.3217
-0.0408 0.0794 0.0922
-0.1805 0.3399 0.3374
-0.8094 0.5868 0.4146
-2.2183 0.8522 0.2021
-4.2425 1.2694 -0.4760
: The C-Not gate controls using two spin-$\frac{3}{2}$ XXZ-chains of length $L=50$ at $\Delta^{-1} =0.25$ and $M=0$ for both the chains. The gate time $T=3.5$ is discretized into $N=20$ time steps. The table shows the values for the three control fields $v_1$, $v_2$ and $v_3$ that are constant during any one of the time intervals.[]{data-label="table:cnotcontrol"}
$$\begin{array}{cc}
\epsfig{file=Not_Controls.eps,height=3.1cm,width=4cm}&\epsfig{file=Hadamard_Controls.eps,height=3.1cm,width=4cm}\\\\
\epsfig{file=Phase_Controls.eps,height=3cm,width=4cm}&\epsfig{file=Pi-8_Controls.eps,height=3cm,width=4cm}
\end{array}$$
The steps carried out to obtain the single qubit gates are as follows
1. We use ground state DMRG of the XXZ chain to obtain the lowest eigenvectors $\psi_0$ and $\psi_1$ of $H_L^{\rm k}(\Delta^{-1})$ in the sector corresponding to $M =0$.
2. We obtain the projected $2\times 2$ control system of equation (\[eqn:control4\]) with matrices $H$ and $B_1$ with matrix elements given by (\[eqn:matrix2\]). For a target gate $U_f$ and a suitable final time $T$ we find the optimal control $v_1(t)$ on this $2\times 2$ system using the technique described in section \[sec:optimal control\].
3. Finally we apply the time-dependent DMRG procedure of section \[sec:DMRG\] to the chain of (\[eqn:control1\]) for a specified time $T$ starting from $\psi_0$ and $\psi_1$ and using the $v_1(t)$ found in step 2 to get the time evolved states $\psi_0(T) = U(T)\psi_0$ and $\psi_1(T) = U(T)\psi_1$. We compute the induced evolution on the subspace $\mathcal{D}$ to obtain the gate $U_{xxz}$ given by the matrix elements ${\langle{\psi_i}|{\psi_j(T)}\rangle}$ for $i,j=0,1$ and compare the overlap with $U_f$ using equation (\[eqn:fidelity\]).
Our desired single qubit target gates are given by the unitaries $$\begin{array}{ll}
X=
\begin{pmatrix}
0&i\\
i&0
\end{pmatrix}&
H=
\frac{1}{\sqrt{2}}\begin{pmatrix}
i&i\\
i&-i
\end{pmatrix}\\\\
T=
\begin{pmatrix}
e^{-i\pi/4}&0\\
0&e^{-i\pi/4}
\end{pmatrix}&
S=
\begin{pmatrix}
e^{-i\pi/8}&0\\
0&e^{-i\pi/8}
\end{pmatrix}
\end{array}$$ The gates obtained using the XXZ chain and their fidelities are as follows. $$\begin{aligned}
X_{xxz}&=&
\begin{pmatrix}
0.0016 - 0.0011i&0.0033 + 0.9997i\\
-0.0017 + 0.9997i&0.0017 + 0.0011i
\end{pmatrix}\\
\mathcal{F}_{X} &=& 0.9997,\\\\
H_{xxz}&=&
\begin{pmatrix}
-0.0027 + 0.7081i& 0.0011 + 0.7053i\\
-0.0016 + 0.7052i&-0.0022 - 0.7085i
\end{pmatrix}\\
\mathcal{F}_{H} &=& 0.9995,\\\\
T_{xxz}&=&
\begin{pmatrix}
0.9221 - 0.3859i&-0.0037 + 0.0038i\\
0.0037 + 0.0038i&0.9216 + 0.3871i
\end{pmatrix}\\
\mathcal{F}_{T} &=& 0.9995,\\\\
S_{xxz}&=&
\begin{pmatrix}
0.7043 - 0.7095i & -0.0046 + 0.0015i\\
0.0045 + 0.0016i&0.7017 + 0.7121i
\end{pmatrix}\\
\mathcal{F}_{S} &=& 0.9997\end{aligned}$$ The optimal controls $v_1(t)$ used to get the gate results are shown in Table \[table:qbitcontrol\] and Figure \[fig:controls\]. For the C-Not gate the procedure described earlier is only slightly modified. We do the ground state DMRG of a one dimensional chain built from the spin ladder described in section \[sec:DMRG\] to get four eigenvectors $\psi_{mn}$ for $m,n = 0,1$. The optimal control procedure is applied to the $4\times 4$ control system (\[eqn:control4\]) with $H$, $B_1$, $B_2$, $B_3$ given by equations (\[eqn:matrix4\]) to find the controls $v_1(t)$, $v_2(t)$ and $v_3(t)$. The time-dependent DMRG procedure is applied to the chain of equation (\[eqn:control3\]) for time $T$ with the controls $v_1(t)$, $v_2(t)$ and $v_3(t)$ to get the time evolved states $\psi_{mn}(T) = U(T)\psi_{mn}$. The induced evolution on the subspace $ \mathcal{D}_1\otimes\mathcal{D}_2$ gives the ${\rm C-Not}_{xxz}$ gate with matrix elements ${\langle{\psi_{mn}}|{\psi_{rs}(T)}\rangle}$. Table \[table:cnotcontrol\] shows the optimal controls $v_1(t)$, $v_2(t)$ and $v_3(t)$ used to obtain the C-not gate. The gate obtained using the $XXZ$ chain and gate fidelity is as follows
$$\begin{aligned}
\rm {C-Not}&=&
\begin{pmatrix}
1&0&0&0\\
0&1&0&0\\
0&0&0&1\\
0&0&1&0
\end{pmatrix}\\\\
\rm {C-Not}_{xxz} &=&
\begin{pmatrix}
0.9959+0.0001i&-0.0015+0.0006i&0.0003-0.0003i&-0.0010-0.0001i\\\\
-0.0014-0.0010i&0.9939-0.0003i&0.0015+0.0000i&-0.0005+0.0005i\\\\
0.0013-0.0001i&0.0004+0.0003i&0.0004-0.0003i &0.9945-0.0008i\\\\
-0.0003-0.0003i&-0.0013+0.0002i&0.9954-0.0004i&0.0003+0.0003i
\end{pmatrix}\\\\
\mathcal{F}_{\rm{C-Not}} &=& 0.9949\\\end{aligned}$$
Based upon work supported in part by the National Science Foundation under Grants DMS-0605342 and DMS-0757581. J.M. also received support from NSF VIGRE grant DMS-0636297.
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|
---
abstract: 'An axisymmetric magnetic field is applied to a spherical, turbulent flow of liquid sodium. An induced magnetic dipole moment is measured which cannot be generated by the interaction of the axisymmetric mean flow with the applied field, indicating the presence of a turbulent electromotive force. It is shown that the induced dipole moment should vanish for any axisymmetric laminar flow. Also observed is the production of toroidal magnetic field from applied poloidal magnetic field (the $\omega$–effect). Its potential role in the production of the induced dipole is discussed.'
author:
- 'E.J.Spence'
- 'M.D.Nornberg'
- 'C.M.Jacobson'
- 'R.D.Kendrick'
- 'C.B.Forest'
title: 'Observation of a Turbulence–Induced Large Scale Magnetic Field'
---
Many stars and planets generate their own nearly–axisymmetric magnetic fields. Understanding the mechanism by which these fields are generated is a problem of fundamental importance to astrophysics. These dynamos are sometimes modeled using two components: a process which generates toroidal magnetic field from poloidal field and a feedback mechanism which reinforces the poloidal field [@Parker_1955]. The first process is easily modeled in an axisymmetric system: toroidal differential rotation of a highly–conducting fluid sweeps the pre-existing poloidal field in the toroidal direction creating toroidal field. This phenomenon, known as the $\omega$–effect, is efficient at producing magnetic field and has been observed experimentally [@Lehnert_1957; @Odier_et_al_1998; @Bourgoin_et_al_2002]. The second ingredient to the model is more subtle, as toroidal currents must be generated to reinforce the original axisymmetric poloidal field. Cowling’s theorem [@Cowling_1933] excludes the possibility of an axisymmetric flow generating such currents so some symmetry–breaking mechanism is required.
The usual mechanism invoked [@Moffatt] is a turbulent electromotive force (EMF), $\cal{E}=\left<\tilde{\mathbf{v}}\times\tilde{\mathbf{b}}\right>$, whereby small scale fluctuations in the velocity and magnetic fields break the symmetry and interact coherently to generate the large scale magnetic field. This EMF is sometimes expanded [@Krause_and_Raedler] in terms of transport coefficients about the mean magnetic field: ; $\alpha$ is characterized by helicity in the turbulence, $\beta$ by enhanced diffusion and $\bm{\gamma}$ by a gradient in the intensity of the turbulence. $\alpha$ is of particular interest as it results in current flowing parallel to a magnetic field, and when coupled with the $\omega$–effect can generate the toroidal currents needed to reinforce the poloidal field.
![Schematic of the Madison Dynamo Experiment showing a cut-away view of the sphere, impellers, external field coils, surface and internal Hall probes.[]{data-label="fig:schematic"}](fig1.eps){width="0.8\columnwidth"}
Experimental evidence for mean–field EMFs (such as the $\alpha$–effect) in turbulent flows has been scarce. Three experiments, relying on a laminar $\alpha$–effect, have generated an EMF [@Steenbeck_et_al_1968] and dynamo action [@Gailitis_et_al_2000; @Stieglitz_and_Muller_2001], but heavily-constrained flow geometries were used to produce the needed helicity; the role of turbulence was ambiguous. Experiments with unconstrained flows have provided evidence for turbulent EMFs, though not the turbulent $\alpha$–effect. Reighard and Brown [@Reighard_and_Brown_2001] have attributed a measured reduction in the conductivity of a turbulent flow of sodium to the $\beta$–effect. Pétrélis et al. have observed [@Petrelis_et_al_2003] distortion of a magnetic field similar to an $\alpha$–effect (currents generated in the direction of an applied magnetic field) and postulate that turbulence may be responsible for disagreement between a laminar model and observations. Not all liquid–metal experiments have had such results: Frick et al.have reported [@Frick_et_al_2004] that the mean flow accounts for all magnetic fields in their torus–shaped gallium experiment, and Peffley, Cawthorne and Lathrop [@Peffley_and_Cawthorne_and_Lathrop_2000] have observed no such effects. It should also be noted that an $\alpha$–effect has been observed in the core of magnetically–confined plasmas [@Ji_et_al_1996; @Redd_et_al_2002].
In this Letter we report measurements of the magnetic field induced by applying an axisymmetric magnetic field to a turbulent, axisymmetric flow of liquid sodium. An induced dipole moment is measured which cannot be generated by the mean flow, indicating the presence of a turbulent EMF.
![[*Upper half:*]{} color contours of induced toroidal magnetic field, $B_\phi(s,z)$, measured by sets of internal Hall probes, for $Rm_{tip}=100$. Induced poloidal flux surfaces, $\Psi(s,z)$, are in black. The positions of the internal Hall probes are indicated with dots. The cylindrical axis of symmetry is horizontal. [*Lower half:*]{} velocity field measured in a water model of the Madison Dynamo Experiment, for an impeller rotation rate of 16.7Hz. Contours of toroidal flow, $v_\phi(s,z)$, are in color and poloidal stream function, $\Phi(s,z)$, are in black. The arrows indicate the direction of the poloidal flow, and the rectangles indicate the positions and size of the impellers which drive the flow.[]{data-label="fig:flows"}](fig2.eps){width="0.9\columnwidth"}
The study is conducted in the Madison Dynamo Experiment, a 1m diameter stainless steel sphere containing liquid sodium. As shown in Fig. \[fig:schematic\], two drive shafts enter the sphere through each pole and drive 30.5cm diameter impellers which generate an axisymmetric mean flow. The shafts are coupled to two 75kW motors which are independently controlled by variable–frequency drives. The radial component of the magnetic field is measured by an array of 74 temperature–compensated Hall probes mounted to the sphere’s surface, allowing resolution of spherical harmonic components of the external magnetic field up to polar order of $\ell=7$ and azimuthal order of $m=5$. Magnetic fields within the sphere are measured by seven linear arrays of Hall probes inserted into the sodium within stainless steel sheaths. These probes are oriented to measure either the axial or toroidal component of the field. Finally, two external electromagnets, in a Helmoltz configuration coaxial with the impellers, apply a nearly uniform magnetic field throughout the sphere. The applied field is between 0 and 60G, and dominated by spherical harmonic content of $\ell=1,m=0$; the largest measured $m\ne0$ component of the applied field is less than 2% of the axisymmetric part.
The study is conducted in the kinematic regime—the magnetic field is not strong enough to affect the flow. The strength of the Lorentz force relative to the inertial forces acting on the fluid is characterized by the interaction parameter (also called the Stuart number), $N=\sigma a B_0^2 / \rho v_0$, where $a$ is the radius of the sphere, $\sigma$ and $\rho$ are the conductivity and density of the fluid, respectively, and $B_0$ and $v_0$ are characteristic magnetic and velocity field magnitudes. $N\sim10^{-2}$ for a total magnetic field of 100G and $v_0=16.0$m/s, so the magnetic field is not expected to alter the flow. This is confirmed by the linear dependence of the induced magnetic field with respect to the applied field. To affect the flow we would expect $N\approx 0.1$, or $B_0\approx 180$G, a field magnitude not yet achieved. We note that the fluctuations, which are characterized by slower velocities, may be in a regime that is affected by the magnetic field. The axisymmetric part of the velocity field generated by the impellors can be expressed in cylindrical coordinates $(s,\phi,z)$ as $$\mathbf{v}=\nabla\Phi\times\nabla\phi+v_\phi(s,z){\hat{\bm{\phi}}},
\label{eq:Veq}$$ where $\Phi(s,z)$ is the poloidal stream function. The flow consists of two large cells, one in the northern and one in the southern hemisphere. An example of this flow, based on measurements made in a water model of the sodium apparatus [@Forest_et_al_2002], can be seen in the lower half of Fig. \[fig:flows\]. The poloidal cells flow inward at the equator and outward at the poles. The two toroidal cells flow in opposing directions. The flow is similar to the $t2s2$ flow proposed by Dudley and James [@Dudley_and_James_1989]; a flow which is calculated to magnetically self-excite at sufficiently high magnetic Reynolds number, $Rm=\mu_0\sigma a v_0$, where $\mu_0$ is the vacuum magnetic permeability ($Rm_{tip}=\mu_0\sigma a v_{tip}$, where $v_{tip}$ is the impeller tip speed). This study is conducted below the critical $Rm$ for self-excitation, as demonstrated by the lack of observed growing magnetic fields. The Reynolds number of the fluid is $Re\sim 10^7$; turbulent fluctuations of the measured flow can be as large as 20% of the mean, depending on location.
Once the sphere is full of sodium the motors are started and a constant magnetic field is applied to the sphere. Hall probes sample the magnetic field at 1kHz for 5 minutes; the applied field is then subtracted from these data to determine the induced field. Measurements of the induced field are presented in the upper half of Fig. \[fig:flows\]. The field is represented by a toroidal component, $B_\phi(s,z)$, and poloidal flux function, $\Psi(s,z)$, such that $$\mathbf{B} = \nabla \Psi \times \nabla \phi+B_\phi(s,z){\hat{\bm{\phi}}}.
\label{eq:Beq}$$ The toroidal magnetic field, undetectable by probes outside the sphere and orthogonal to the applied poloidal field, is measured within the sphere by internal Hall probes, confirming the presence of the $\omega$–effect. The peak amplitude of the toroidal magnetic field scales linearly with $Rm$, and can be larger than the magnitude of the applied field.
The external induced poloidal magnetic field is decomposed into its spherical harmonic components to reveal its spatial structure. Since the Hall probes on the sphere’s surface lie outside regions containing currents the magnetic field can be expressed as the gradient of a scalar magnetic potential, $\mathbf{B}=-\nabla\Phi_m$, which solves Laplace’s equation. In spherical coordinates the solution to the potential, for the region excluding the origin, is well known: $\Phi_m(r,\theta,\phi)=\sum_{\ell,m}D_{\ell,m}r^{-(\ell+1)}Y_\ell^m(\theta,\phi)$, where $Y_\ell^m(\theta,\phi)$ is the spherical harmonic. The coefficients in the expansion, $D_{\ell,m}$, which fit the mean induced field are calculated using singular value decomposition. The induced poloidal magnetic field is predominantly axisymmetric; the largest components are given in Tab. \[tab:spectral\_content\]. The dominant components with $\ell$ equal to 3 and 5 are expected due to the structure of the applied field and mean flow; the large measured dipole component is not expected, as it cannot be generated by the axisymmetric mean flow, as will be shown below.
[@e@@dd@d]{} & & &\
1,0([dipole]{}) & 1.6 & 11.4[G]{} & 1.8[G]{}\
2,0 & 0.2 & 3.1 & 3.5\
3,0 & 0.5 & 13.6 & 2.4\
4,0 & 0.1 & 7.1 & 3.5\
5,0 & 0.4 & 18.3 & 3.5\
1,1 & 0.0 & 0.8 & 7.8\
2,1 & 0.0 & 0.6 & 3.3\
![Induced dipole moment versus time, for $Rm_{tip}=100$ and an applied magnetic field of 60G. 1Gm$^3$ corresponds to 13.2G at the sphere’s pole.[]{data-label="fig:S1signal"}](fig3.eps){width="0.9\columnwidth"}
The induced dipole moment fluctuates dramatically in time around a well–defined mean, as seen in Fig. \[fig:S1signal\]. Measurements indicate that the induced dipole depends on $Rm$ (Fig. \[fig:Dipole\_scaling\]a) and upon the magnitude of the externally–applied field (Fig. \[fig:Dipole\_scaling\]b). The dipole moment’s dependence on $Rm$ eliminates the possibility of the measurement being a systematic error in the analysis. The EMF depends linearly on the applied field, indicating that it is a kinematic effect and not due to the back reaction.
![ (a) Mean induced dipole moment versus $Rm_{tip}$, for an applied field of 60G with a quadratic fit valid at low $Rm$. (b) Mean induced dipole moment versus applied magnetic field, for $Rm_{tip}=100$. A linear fit is plotted for comparison. Error bars are RMS fluctuation levels about the mean; the uncertainties in the mean values are very small (less than 0.01Gm$^3$) due to long averaging times.[]{data-label="fig:Dipole_scaling"}](fig4a-fit.eps "fig:"){width="0.9\columnwidth"}\
![ (a) Mean induced dipole moment versus $Rm_{tip}$, for an applied field of 60G with a quadratic fit valid at low $Rm$. (b) Mean induced dipole moment versus applied magnetic field, for $Rm_{tip}=100$. A linear fit is plotted for comparison. Error bars are RMS fluctuation levels about the mean; the uncertainties in the mean values are very small (less than 0.01Gm$^3$) due to long averaging times.[]{data-label="fig:Dipole_scaling"}](fig4b.eps "fig:"){width="0.9\columnwidth"}
While Cowling’s theorem demonstrates that self-excitation is not possible in axisymmetric systems, it is not obvious that a dipole moment cannot be induced by an axisymmetric velocity field exposed to an axisymmetric magnetic field. The proof of this is as follows. Consider a bounded, steady–state, axisymmetric system described by Eq. \[eq:Beq\]. For axisymmetric fields, the only non-trivial component of the dipole moment, $\boldsymbol{\mu}\equiv\int\mathbf{x}\times\mathbf{J}\,d^3x$, is oriented along the symmetry axis and results from currents flowing in the toroidal direction, $$\mu_z = \int s J_\phi\, d^3x.
\label{eq:muz}$$ These currents can only be generated by the $\mathbf{v}\times\mathbf{B}$ force due to the mean fields, so using Ohm’s law gives $$\begin{aligned}
\nonumber
s J_\phi & = & s\sigma\left[\mathbf{v}\times\left( \nabla \Psi \times
\nabla \phi \right)\right]\cdot{\hat{\bm{\phi}}}
\\ \nonumber & = &
\sigma\left[v_\phi\nabla\Psi-\left(\mathbf{v}\cdot\nabla\Psi\right){\hat{\bm{\phi}}}\right]\cdot{\hat{\bm{\phi}}}
\\ & = &
-\sigma\nabla\cdot\left(\mathbf{v}\Psi\right),\label{eqn:ohm}\end{aligned}$$ where use has been made of and the fluid has been assumed incompressible, . Inserting Eq. \[eqn:ohm\] into Eq. \[eq:muz\] and making use of Gauss’ theorem and $\mathbf{v}\cdot{\hat{\bm{n}}}=0$, where ${\hat{\bm{n}}}$ is the unit vector normal to the vessel’s surface, one finds that $\mu_z=0$. It is interesting to note that it is only the dipole moment that vanishes; moments which include different powers of $s$ in Eq. \[eq:muz\] are nonzero in general. This conclusion is also independent of geometry; any simply–connected axisymmetric system gives the same result.
It is possible that an induced dipole could be generated if mean non-axisymmetric magnetic and velocity field modes interacted. The stainless steel tubes which contain the internal Hall probes could potentially break the symmetry and create a mean non-axisymmetric flow. However, if this were the case one would expect higher–order non-axisymmetric induced field components, which are not observed (see Tab. \[tab:spectral\_content\]). The mean induced dipole moment is present both with and without the tubes.
Since it cannot be generated by the mean flow, the dipole moment must be the result of turbulence breaking the symmetry of the system, likely a turbulent EMF of some form. Any of the terms in the mean–field expansion of the EMF have the potential to yield the observed mean dipole moment. A toroidal $\alpha$–effect could produce large scale toroidal currents by interacting with the observed $\omega$–effect. The small scale helicity needed for the $\alpha$–effect might come from either a turbulent cascade or be produced directly by the impellers. The $\beta$–effect leads to turbulent modifications of the fluid conductivity [@Krause_and_Raedler]. A nonuniform $\beta$–effect could cause uneven distributions of currents to generate the dipole moment. A third possibility is the $\gamma$–effect [@Krause_and_Raedler], which expels magnetic field from regions of high–intensity turbulence, resulting in diamagnetism. The intensity of the turbulence varies with position, so the $\beta$–effect and the $\gamma$–effect are both candidates to explain the field.
Expanding the EMF in terms of the mean magnetic field may not be appropriate, since the largest fluctuations in the magnetic field do not satisfy the scale–separation and homogeneity requirements usually imposed in the expansion of the mean–field EMF. The largest turbulent magnetic fluctuations are $m=1$. Their Gaussian probability distribution is centered at zero, consistent with a passively–advected magnetic field in a turbulent cascade of velocity fluctuations. These $m=1$ fluctuations in $\mathbf{B}$ could, in principle, interact with $m=1$ fluctuations in the flow and average to give a net toroidal current.
In summary, a mean dipole moment is induced in the experiment which cannot be produced by the mean flow. The induced currents are of the correct form to create a poloidal magnetic field, as required in the $\alpha\omega$–dynamo model [@Parker_1955]. This is the first observation of this effect in a laboratory experiment. Explicit characterization of the EMF is impossible without more detailed knowledge of the form of the turbulence and direct measurement of the fluctuating components of $\mathbf{v}$ and $\mathbf{B}$. Future work will be directed towards identifying the characteristics of the fluctuations responsible for producing the dipole field. We also note that no saturation of the mechanism has yet been definitively observed, as might be expected from numerical simulations and theory [@Gruzinov_and_Diamond_1994; @Cattaneo_and_Hughes_1996]. Future experiments with larger magnetic fields may provide insight into the saturation mechanism.
We express our gratitude to A. Bayliss for helpful dialogue and C. Parada for assistance with data acquisition. CBF would like to thank S. Prager and P. Terry for their continued support and useful discussions. This work is funded by the US Department of Energy, the National Science Foundation, and David and Lucille Packard Foundation.
[17]{}
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